Incorporation of achromatic compensator into a dual rotating compensator multichannel ellipsometer

Transcription

1 The University of Toledo The University of Toledo Digital Repository Theses and Dissertations 212 Incorporation of achromatic compensator into a dual rotating compensator multichannel ellipsometer Balaji Ramanujam The University of Toledo Follow this and additional works at: Recommended Citation Ramanujam, Balaji, "Incorporation of achromatic compensator into a dual rotating compensator multichannel ellipsometer" (212). Theses and Dissertations This Thesis is brought to you for free and open access by The University of Toledo Digital Repository. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of The University of Toledo Digital Repository. For more information, please see the repository's About page.

2 A Thesis entitled Incorporation of Achromatic Compensator into a Dual Rotating Compensator Multichannel Ellipsometer by Balaji Ramanujam Submitted to the Graduate Faculty as partial fulfillment of the requirements for the Master of Science Degree in Electrical Engineering Dr. Robert W. Collins, Committee Chair Dr. Daniel G. Georgeiv, Committee Member Dr. Rashmi Jha, Committee Member Dr. Patricia R. Komuniecki, Dean College of Graduate Studies The University of Toledo August 212

3 Copyright 212, Balaji Ramanujam This document is copyrighted material. Under copyright law, no parts of this document may be reproduced without the expressed permission of the author.

4 An Abstract of Incorporation of Achromatic Compensator into a Dual Rotating Compensator Multichannel Ellipsometer by Balaji Ramanujam Submitted to the Graduate Faculty as partial fulfillment of the requirements for the Master of Science Degree in Electrical Engineering The University of Toledo August 212 A dual rotating compensator multichannel ellipsometer with an achromatic compensator has been developed in this Thesis research. The sequence of polarization modifying optical elements for this ellipsometer is denoted PC 1r ( SC 2r ( )A where P, S, and A represent polarizer, sample, and analyzer. C 1r ( ) and C 2r ( ) represent the achromatic compensator and a biplate compensator, respectively, that rotate at frequencies of /2= 1 Hz and /2= 6 Hz, synchronized for a ratio ( : of 5:3. The light source consists of a xenon lamp for broadband operation over a wavelength range from 28 nm to 73 nm, and the detection system consists of a grating spectrograph and a 124 pixel silicon photodiode array detector. This ellipsometer provides spectra (28-73 nm) in the complete 4x4 Mueller matrix of a reflecting sample placed between the two rotating compensators with a minimum acquisition time of.25 s. In this Thesis, the procedures for building such an ellipsometer are explained in detail starting from the initial steps that involve calibrating the photodiode array pixel numbers versus wavelength, positioning filters within the detection system for suppression of second order reflections from the grating, performing pixel dependent iii

5 image persistence corrections of the detector output, and aligning the internal biplate compensator. The performance of the single rotating compensator system using the biplate as the single compensator is tested prior to development of the dual rotating compensator system. The testing involves measurement of the Stokes vector parameters (Q,, p) in straight through without a sample after performing biplate retardance, biplate phase angle, and analyzer offset angle calibrations, the latter relative to the polarizer angle. In straight through, (Q,, p), the polarization tilt angle, the polarization ellipticity angle, and the degree of polarization, respectively, are independent of wavelength and given by (Q=, =, p=1). The quality of the single rotation compensator instrument is evaluated based on the closeness of the spectra in the experimental Stokes vector values to these predicted values. In this research a novel procedure for mounting and aligning the achromatic compensator, which is based on the three-reflection King-type rhomb, has been proposed and successfully implemented, enabling continuous rotation of the rhomb in the dual rotating compensator ellipsometer configuration. Instrumentation calibration procedures are demonstrated for the dual rotating compensator system, including achromatic compensator calibration, biplate compensator calibration, and phase and azimuthal angle calibrations for the compensators and analyzer, respectively. The instrument performance for the measurement of sample Mueller matrices has been evaluated in the straight through configuration, which results in an identity sample Mueller matrix. The quality of the dual rotation compensator instrument is evaluated based on the closeness of the experimental Mueller matrix element spectra to the predicted identity matrix element iv

6 values. Optical modeling of a Krasilov-type rhomb compensator has been completed for possible manufacture and application as a replacement for the King-type rhomb. v

7 For my grandmother S. Rajalakshmi

8 Acknowledgements I would like to express my sincere gratitude to Prof. Robert Collins for his guidance and support in the development of this thesis. His expertise in ellipsometry has made my work an invaluable experience. I would like to thank Jian Li for helping me to understand the basics of ellipsometry and optics. I would like to sincerely thank Bob Lingohr who is very patient in helping me by making the mounts and aluminum parts whenever necessary. I would like to thank Carl Salupo, Terry Kahle, Shan Ambalanath, and Rick Irving who helped me with necessary inputs based on their expertise. I would like to thank all my colleagues and group members for all their valuable support. vii

9 Table of Contents Abstract Acknowledgement Contents List of Tables List of Figures iii vii viii xiii xiv 1 Introduction Background and motivation Thesis structure Basic Concepts and Instrumentation Basic concepts in ellipsometry Polarized light The Jones vector and the Jones matrix formalism The Stokes vector and Mueller matrix formalism Interaction of light with a sample Instrumentation Light source Collimating optics Polarizer module...16 viii

10 Polarizer Rotator block Stepper motor system Compensator modules Biplate compensator Achromatic compensator Low cogging, hollow shaft, continuously rotating motors Shutter system Objective Spectrograph Detector system Detector Detector controller Detector interface Data acquisition timing Modeling the Performances of Dual Biplate and Krasilov Compensators Retardance optimization of the dual biplate compensator Theory of the dual biplate compensator Simulation methods for the dual biplate compensator Results and discussion for the dual biplate compensator First simulation approach: constrained spectral average of the effective retardance Second simulation approach: unconstrained spectral average ix

11 of the effective retardance Optical modeling of Krasilov type compensator Theory of Krasilov compensator Optical modeling of the first reflection Optical modeling of the second reflection Optical modeling of the third reflection Determining effective retardance and effective dichroism of the Krasilov type compensator Initial Steps in the Development of the Multichannel Mueller Matrix Ellipsometer System electrical component operation Sample alignment Collimator alignment Wavelength calibration Alignment of polarizer Order sorting filters Positioning of order sorting filters Selection of order sorting filters Procedure for inserting order sorting filters Pulse divider circuit design for single rotating compensator ellipsometer Rotating polarizer configuration Data collection principles Correction of systematic errors...89 x

12 Dark current and ambient corrections Image persistence correction Single Rotating Compensator Multichannel Ellipsometer Introduction Instrument design and theory Internal alignment of biplate compensator Calibration of single rotating compensator ellipsometer Retardance calibration Offset angle calibration Compensator phase angle calibration Data reduction and results Dual Rotating Compensator Multichannel Ellipsometer Introduction Instrument design and theory Data collection Pulse dividing circuit for dual rotating compensator ellipsometer Data reduction Mounting and alignment of achromatic King rhomb compensator Achieving condition (1) Achieving condition (2) Achieving condition (3) Calibration of dual rotating compensator ellipsometer xi

13 6.4.1 Compensator calibration Offset and phase angle self-calibration Determination of Mueller matrix elements Conclusions and future work Conclusions Future work References 171 A User drawing of spectrograph CP B Optical properties of materials 179 C Program to view live irradiance at the detector 24 D Basic data acquisition program 218 E Inversion matrix to determine Fourier coefficients for dual rotating compensator ellipsometer 244 F Program to determine roots of quartic equation 256 xii

14 List of Tables 6.1 Average and standard deviation for each Mueller matrix element over the entire spectral range of 68 pixel groups ( ev) xiii

15 List of Figures 2-1 Schematic representation of reflection of polarized light from a sample surface Schematic representation of a Rochon polarizer Schematic representation of a biplate compensator Schematic representation of an achromatic King rhomb Optical system of a dual biplate compensator Quarter wave wavelength of the second biplate 2 selected based on first simulation approach and max min plotted with respect to quarter wave wavelength of the first biplate a Individual retardance spectra of two biplates selected for the dual biplate compensator based on first simulation approach b Spectra in the effective Q, C, and e for the dual biplate compensator based on first simulation approach Dispersion of the effective retardance e for different 2 1 values. In the legend, the max min values are provided for each of the 2 1 values Quarter wave wavelength of the second biplate 2 selected based on second simulation approach, minimum value of max min and e plotted with respect to quarter wave wavelength of the first biplate xiv

16 3-6a Individual retardance spectra of two biplates selected for the dual biplate compensator based on second simulation approach b Spectra in the effective Q, C, and e for the dual bi-plate compensator based on second simulation approach Dispersion in the effective retardance e for different 2 1 values. In the legend, the max min values are provided for each of the 2 1 values Krasilov compensator design Beam path through the first prism of the Krasilov type compensator Dispersion of the dichroism angle c1 and the retardance c1 induced by the first reflection, which is a total internal reflection from a fused silica/ ambient interface Beam path through the first and second prisms Dispersion of the dichroism angle c2 and the retardance c2 induced by the second reflection, which is from the fused-silica/aluminum interface Dispersions in the effective dichroism angle c and in the effective retardance c produced by three reflections of the Krasilov type compensator a Degrees of freedom for orienting an alignment laser b Degrees of freedom for aligning a sample c Photograph of the sample stage with a thin glass sample mounted on it for alignment purposes a The sample is arbitrarily mounted and tilt adjustments are performed to align the sample surface parallel to the goniometer axis (C).[4-3] b The sample surface is positioned parallel to the goniometer axis (C).[4-3]...72 xv

17 4-3 Horizontal adjustments are performed to ensure that the sample surface coincides with the axis of rotation of the goniometer (C) [4-3] Schematic of an aligned collimator Plot showing the linear fit relationship between pixel number and wavelength Transmission spectra of order sorting filters measured using the J.A. Woollam Company V-VASE ellipsometer Intensity spectrum of the xenon lamp after insertion of order sorting filters IPCF obtained experimentally and by calculation from the residual function, plotted as a function of pixel group number Photon energy dependence of the residual function obtained after application of the IPCF correction to acquired data Schematic of the single rotating compensator multichannel ellipsometer, indicating the reference axes and azimuthal angles that describe the three optical elements Schematic of misaligned biplate compensator The dichroic angle C for a zero order MgF 2 biplate compensator measured using a J.A. Woollam Company M2 ellipsometer (a) Measured and best-fit retardance spectra for the internally aligned biplate compensator, and (b) difference between the two fits Analyzer angular offset A S plotted versus photon energy; a constant result should be obtained in the absence of instrument errors (a) Rotating compensator phase angle C S obtained in calibration and the best fit linear relationship, (b) difference between the two fits xvi

18 5-7 Spectra in the azimuth and ellipticity angles and the degree of polarization, (Q,, p), respectively, that characterize the Stokes vector of the light beam Ellipsometry angles (, ) plotted vs. photon energy for a straight through measurement Schematic of the dual rotating compensator ellipsometer Wavelength as a function of pixel number indicating the grouping mode used for the dual rotating compensator ellipsometer Schematic showing the King-type rhomb mounted to an aluminum plate a Photograph of the King-type rhomb and mounting stages required for its alignment relative to the motor b Photograph of the achromatic compensator motor and the mounting stages required for its alignment (a) Retardance C1 of achromatic compensator; (b) Retardance C2 of the biplate compensator its polynomial fit; (c) Difference between the fits of C The difference between the best fit polynomials used to fit the retardance of the biplate compensator, as measured in the PC 1r C 2r A and PC r A configuration Measured dichroic angles for the achromatic rhomb compensator ( C1 ) and the biplate compensator ( C2 ) Schematic of Mueller matrix calculations and self-calibration equations from the Fourier coefficients for a dual rotating compensator ellipsometer Analyzer offset angle A S plotted as a function of photon energy in the dual rotating compensator configuration xvii

19 6-1 Rotating compensator phase angles (C S1, C S2 ) and their differences (C S1,C S2 ) plotted from k = 12 to k = 46 for detector pixels grouped by Rotating compensator phase angles (C S1, C S2 ) and their differences (C S1,C S2 ) plotted from k = 47 to k = 59 for detector pixels grouped by Mueller matrix elements deduced experimentally in the straight through dual rotating compensator configuration Off-diagonal Mueller matrix elements on an expanded scale Diagonal Mueller matrix elements on an expanded scale Random depolarization D p assumed to occur at the position of the sample, as evaluated through the consistency check on the Mueller matrix A-1 User drawing of spectrograph Jobin Yvon CP C-1 Snapshot of user interface for live irradiance scan...24 D-1 Snapshot of basic data acquisition user interface F-1 Snapshot of user interface to determine roots of quartic equation xviii

20 Chapter 1 Introduction 1.1 Background and motivation Thin film growth processes, the properties of thin films, their surfaces and interfaces, as well as the associated process-property relationships have become increasingly important research topics in science and engineering. Even the most widely studied thin film materials applied in many technologies, examples being thermallygrown silicon oxide and amorphous, microcrystalline, and polycrystalline silicon, still provide challenges in terms of understanding the process-property relationships at a fundamental level. Advanced nanostructured thin films are being relied upon increasingly to meet the needs of various technologies; these include multilayered and gradient thin films, nanoscale sculptured films, and self-assembled organic/inorganic nanocomposites. As the functionality of thin films increases, a growing demand must be met for contactless, non-invasive probes often optical in nature, to determine the thickness and other basic properties of thin films either in individual layers or in multilayer stacks [1-1]. 1

21 Ellipsometry is a non-invasive, non-destructive optical technique that can be applied in a reflection geometry to determine key characteristics of thin films starting with thickness [1-2]. The critical advantage of ellipsometry compared to reflectometry is that, in the ellipsometry measurement, the change in the polarization state of the reflected light is measured -- yielding two parameters, whereas in the reflectometry measurement, the change in irradiance is measured -- yielding one parameter. Ellipsometric measurements provide the ratio of the complex amplitude reflection coefficients of the sample defined by r p /r s tan exp(i), where is the ratio of the amplitudes and is the difference between the phases of the coefficients r p and r s. Here p and s denote parallel and perpendicular to the plane of incidence. As a ratio and phase difference, the two angles () are independent of the irradiance in the incident light beam. These parameters provide information on the multilayer film structure and complex dielectric functions of the layer components, the latter in turn leading to other important properties of these components such as band gap. The first ellipsometric measurements were performed by Prof. Paul Drude ( ) [1-3], even though the term ellipsometry was coined much later in 1944 in an article by Alexandre Rothen [1-4]. Drude derived the basic equations used in ellipsometry, and performed experimental studies of both absorbing and transparent solids. Ellipsometry requires intensive data collection and processing; hence the widespread use of laboratory computers enabled development of automated ellipsometers for different purposes. Another breakthrough which enabled fast spectroscopic ellipsometry measurements is the invention of photodiode array detectors, with such detectors the measurement of an entire ellipsometric spectrum can be completed in much 2

22 less time than would be possible with single element detectors. Now commercial multichannel ellipsometers exploiting charge coupled device and photodiode arrays as detectors are available that span a wide spectral range from the near infrared to near ultraviolet region. Among the different types automated ellipsometers developed in the time frame, two major types still in use are the (i) the rotating element ellipsometers [1-5] and the (ii) the phase modulation ellipsometers [1-6]. In rotating element ellipsometers, as their name implies, optical components such as polarizers or optical retarders (or compensators) are rotated to introduce polarization modulation to the incident light beam. In phase modulation ellipsometers, piezobirefringent modulators are used to induce polarization variations in the incident light beam. The subject of focus in this Thesis research is rotating element ellipsometry. The capability of rotating element ellipsometers in analyzing complex sample heterogeneities has continuously improved with the continuous advance of the ellipsometry instrumentation. The rotating polarizer ellipsometer is designed in the P r SA configuration, where P represents the polarizer, subscript r indicates that the polarizer is continuously rotating, S represents the reflecting sample, and A represents the analyzer (another polarizer in front of detector). This instrument measures cos and so cannot detect the sign of the phase difference in the reflection of polarized light from a sample. This in turn implies that such an instrument configuration cannot detect the handedness of rotation of the polarization state, meaning that it cannot distinguish left and right elliptical polarization having the same ellipse shape [1-7]. This and other disadvantages of the rotating polarizer ellipsometer were overcome by the single rotating compensator 3

23 ellipsometer in the PSC r A configuration, where C r represents the rotating compensator. This instrument can measure the full Stokes vector of the light beam reflected from the sample, but it cannot detect the full Mueller matrix of the sample [1-8]. Presently, this rotating compensator ellipsometer is the most widely used ellipsometer for high speed multichannel spectroscopy. By introduction of another compensator in front of the sample, the dual rotating compensator ellipsometer was developed in the PC 1r SC 2r A configuration, where C 1r and C 2r denote the two rotating compensators. In this mode, the polarization state of light incident on the detector is highly modulated, and from the amplitude of the different frequency components -- up to 12 in number plus dc, the full 4x4 Mueller matrix of the sample can be determined [1-9]. The (1,1)-normalized Mueller matrix with its 15 elements can over-determine the 6 quantities that define the non-diagonal complex Jones matrix, the latter providing a means for the analysis of non-depolarizing, specularly-reflecting surfaces exhibiting optical anisotropy. One specific application of relevance for in situ and real time studies involves the vacuum deposition of cubic single crystalline materials by epitaxial methods. In this application, one can perform simultaneous measurements of the bulk optical properties and the surface-induced anisotropic optical properties [1-1] using the dual rotating compensator multichannel ellipsometer. In fact, the real and imaginary parts of the surface-induced dielectric function anisotropy s can be determined simultaneously with the real and imaginary parts of the bulk isoropic dielectric function b. As a result, one can analyze monolayer level surface phenomena, while at the same time characterizing the underlying bulk isotropic material. Thus, multichannel Mueller matrix 4

24 spectroscopy using the dual rotating compensator configuration can serve as a unique and powerful optical probe in materials science and engineering. Because the signature of surface-induced dielectric function anisotropy is very weak, as observed from Mueller matrix elements and cross-polarization reflection ratios that reach maximum values on the order of 1 3, such a measurement demands very high precision and accuracy instrumentation [1-11]. The compensators used to this point in dual rotating compensator ellipsometric spectroscopy are MgF 2 biplates, whose retardances exhibit spectral dependences of the of the form C = 2 nd/. Thus, these compensators are highly chromatic with retardance that is inversely proportional to the wavelength, where n describes the birefringence and d describes the thickness or thickness difference. By simulation, it has been proposed [1-11] that the use of achromatic compensators, which can exhibit a retardance of ~ 9 o over the entire spectral range of the dual rotating compensator system, minimizes the effect of random and systematic errors on cross polarization reflection ratios. At the start of this Thesis research, achromatic compensators have not been used in the dual rotating compensator ellipsometer because of their difficulty in alignment and the challenge of mounting and rotating these elements without strain within the fused silica optical material [1-12]. The motivation of this Thesis research is the development of a dual rotating compensator ellipsometer system by incorporation of a rotating achromatic compensator on the polarization generation side of the ellipsometer and checking its performance in straight through (i.e. without a sample in place). A standard rotating biplate compensator is used on the polarization detection side. A step by step development procedure is followed to evaluate the performance of the system in each stage, by first developing a 5

25 rotating polarizer ellipsometer as described in Chapter 4, then the single rotating compensator system with biplate compensator as described in Chapter 5, and finally, a dual rotating compensator system with achromatic compensator as described in Chapter 6. The achromatic compensator is mounted on the polarization generation side before the sample because mounting and dismounting samples can generate greater challenges in the alignment of achromatic compensator if it is placed on polarization detection side. The achromatic compensator used in this thesis research is a three reflection King-type rhomb [1-13]. 1.2 Thesis structure In this Thesis, Chapter 1, Section 1.1 introduces the concept of ellipsometry and introduces different ellipsometer configurations. This Chapter also provides motivation for the development of the dual rotating compensator ellipsometer with an achromatic compensator, A chapter-by-chapter list of the thesis structure is provided, as well. Chapter 2 introduces the basic concepts and principles of ellipsometry in Section 2.1 and the instrumentation used in building the ellipsometer in Section 2.2. Chapter 3 addresses the modeling of chromatic effects in compensators, the dual biplate compensator in Section 3.1, and a Krasilov-type compensator in Section 3.2. In Chapter 4, the intial setup required for development of the ellipsometer is explained in detail, with the rotating polarizer ellipsometer being explained in Section 4.8. In Chapter 5, construction of a single rotating compensator system with a biplate compensator is explained in detail, including basic instrument theory, internal alignment 6

26 of the biplate compensator, calibration of optical components, and data reduction to extract Stokes vector components and ellipsometric parameters (, ). In Chapter 6, construction of the dual rotating compensator system is explained in detail, including basic instrument theory, mounting and alignment of the achromatic King-type rhomb, pulse dividing circuitry for the dual rotating compensator system, data collection, calibration of optical components, and data reduction to extract Mueller matrix elements. Chapter 7 summarizes the performance of the system, and outlines future work that must be done to improve the performance of the system. 7

27 Chapter 2 Basic Concepts and Instrumentation 2.1 Basic concepts in ellipsometry Polarized light Polarization can be defined in terms of the time evolution of one of the field vectors that describes the light wave, observed at a fixed point in space. For light waves, there are four field vectors that can be used to describe polarization. They are E (electric field), D (displacement field), B (magnetic induction), and H (magnetic field). E and D describe the electric forces on the medium in which the wave propagates, whereas B and H describe the magnetic forces on the medium. The latter two are of interest for light wave interaction with ferromagnetic materials, for example. In the materials analysis of interest in this thesis either E or D is selected. In studies of wave propagation in isotropic media in which case D and E are parallel, the electric field E is further selected to describe the state of polarization. The time evolution of each of the other field vectors is determined from Maxwell s Equations and the constitutive relations. 8

28 2.1.2 The Jones vector and the Jones matrix formalism The representation of the electromagnetic wave in an arbitrary polarization state propagating along the direction z is given by: E(z, t) E ˆ ˆ x cos( t kz x)x Eycos( t kz y)y (2.1) where ˆx, ŷ represent two orthogonal unit vectors along the x and y axes, respectively. E x and E y are the amplitudes of the electric fields vibrating along the x and y axes, respectively. is the angular frequency of the light wave; k is the magnitude of the wave vector; x and y are the absolute phase angles of the electric field components vibrating along the x and y axes, respectively. Equation (2.1) can be expressed in column vector form as: E x(z, t) Ex cos( t kz x) E(z,t) E (z, t) E cos( t kz ) y y y (2.2) Then, phasor notation can be used to simplify Eq. (2.2) further: E(z,t) e ikz Ee x Ee y i x i y (2.3) It can be seen that the time dependent part of the wave exp(it) is removed in the phasor notation. The measurable electric field can be restored by multiplying each field component phasor by exp(it) and taking the real part. It can be shown [2-1] that, at any fixed point (x,y,z) in space, the end point of the electric field vector described by Eq. (2.1) to Eq. (2.3) sweeps out an ellipse in the most general situation. This ellipse is located within the x-y plane with a tilt angle Q (9 < Q 9 ), which is measured relative to the x axis, and an ellipticity e (1 e 1), which 9

29 is defined as the ratio of the semi-minor axis to the semi-major axis of the ellipse. The right or left handedness of the polarization state is determined by the rotation direction of the electric field vector, either clockwise or counterclockwise, respectively, when viewing the beam toward the direction opposite to that of wave propagation. With this convention, the ellipticity e is assigned a positive value for right handed polarization, and a negative value for left handed polarization. In summary, Q and e (including its sign) uniquely define the polarization ellipse and handedness. Returning to Eq. (2.3), the prefactor e ikz describes the absolute phase of the electric field at point z, which is of no interest in conventional ellipsometry in the optical wavelength range. It can be proven [2-1] that the complex vector in Eq. (2.3) i x Ee x E x E i Ee y E y y, which is called the Jones vector, also uniquely defines the polarization ellipse and handedness. In fact, relationships can be derived between the shape parameters Q and e and the phasor parameters E y /E x and y x, which are the Jones vector parameters of relevance in conventional ellipsometry. Only in three parameter ellipsometry, which involves measurement of the reflectance as well, are the absolute field amplitudes E y and E x of interest (as opposed to their ratio). The function of a linear component of an optical system in modifying the polarization state of light can be fully described by a 2x2 complex Jones matrix. This matrix left-multiplies the Jones vector of the input light to generate the Jones vector of the output light: E o JE i or o i E o x J E 11 J 12 x E o i E y J E 21 J 22 y (2.4) 1

30 The Jones vectors of some special polarization states, the Jones matrices of some frequently used polarization modifying optical system components, as well as the Jones matrix for a coordinate rotation operation have been tabulated in [2-1] The Stokes vector and Mueller matrix formalism The Jones vector and matrix formalism has the advantage of being concise but the disadvantage of being able to describe only completely polarized light, or light of a single definite polarization state. Thus, the Jones vector and matrix formalism is not sufficiently powerful to describe (i) unpolarized light, which includes a random superposition of polarization states, (ii) partially polarized light, which includes one component having a single definite polarization state and another component characterized by a random mixture of polarization states, or (iii) depolarizing optical systems that act to randomize a component of the polarization state of light. To deal with these challenges, the Stokes vector formalism and the Mueller matrix formalism have been introduced. The Stokes vector is a 4x1 column vector defined as: S I S1 Ix Iy S S 2 I/4 I /4 S3 IR IL (2.5) where I is the total irradiance of light including all polarization components, I x and I y are the irradiances of the components of light that are linearly polarized along the x and y directions, respectively. I /4 and I /4 are the irradiances of the components that are linearly polarized along /4 and /4 (radians) relative to the x axis, respectively; and I R and I L are the irradiances of the components that are right and left circularly polarized, respectively. The degree of polarization p can be defined as 11

31 S S S p 1. (2.6) S If a beam is completely polarized, then S S S S (2.7) and p = 1. Thus, 1 p can be used to define the fraction of fully unpolarized light. In the Stokes vector formalism, an optical component that modifies the polarization state of light is the 4x4 Mueller matrix. The Stokes vector of the input beam ( S i ) leftmultiplied by the Mueller matrix of an optical component gives the Stokes vector of o output beam ( S ), according to S o M M M M M M M M M M M M M M M M i S (2.8) The Mueller matrices of frequently used optical components, the Mueller matrix for a coordinate rotation operation, and the transformations that convert between the Jones and Mueller matrix formalisms have been described in [2-1] Interaction of light with a sample In the most common geometry for ellipsometry, a plane wave of light is specularly reflected at an oblique angle from a sample surface (i.e. the angle of incidence is equal to the angle of reflection), although the transmission geometry is also used in some circumstances. The schematic of oblique light reflection from a sample surface is shown in Fig

32 E p Reflecting surface s E ~ i ~ J plane of incidence E ~ o s p E Figure 2-1: Reflection of polarized light from a sample surface. The p and s and directions of both the incident and reflected beams are identified. The incident and reflected beams are described by ~ the sample surface is described by a Jones matrix J. i E ~ and o E ~, respectively, and As shown in the figure, the plane of incidence (plane formed by the incident and reflected light waves) is perpendicular to the plane of the sample surface. For an isotropic sample the linear p and s Jones vectors are the eigenvectors, meaning that they are unchanged in direction relative to the propagation vector upon reflection. The vector p is the direction lying within the plane of incidence, and the vector s is the direction perpendicular to the plane of incidence and parallel to the sample surface. The modification of the polarization imposed by the most general non-depolarizing sample upon reflection from its surface can be described by the Jones matrix J where: o i i o E J p pp J ps E r p pp rps E p E o i i E J r s sp J ss E s sp rss E s. (2.9) 13

33 Here r jk (j, k = p, s) are the j-to-k complex amplitude reflection coefficients defined in terms of the incident and reflected beam electric fields by the following equations: r (E /E ), (2.1a) o i pp p p E i s r (E /E ), (2.1b) o i ps p s E i p r (E /E ), (2.1c) o i sp s p E i s r (E /E ), (2.1d) o i ss s s E i p where o E p and incidence respectively and, i E p are output and input electric fields in a direction parallel to plane of o E s and i E s are output and input electric field in a direction perpendicular to plane of incidence respectively. As indicated earlier in this chapter, the absolute phase and amplitude of an electromagnetic wave are of no interest in conventional ellipsometry. Thus, the sample s Jones matrix can be rewritten without loss of information as: pp ps J sp 1, (2.11) where r ij ij tan r ss iij ije is the ratio of complex reflection coefficients and generally describes the quantities of interest. When the reflecting sample is isotropic, the off-diagonal ratios vanish. Then the parameters pp and pp that describe the relative (p-to-s) amplitude ratio 14

34 (reflected-to-incident) and the phase difference (p-to-s) shift (upon reflection) are called the ellipsometric angles. 2.2 Instrumentation The instrumentation for the dual rotating compensator multichannel ellipsometer consists of optical components in the polarization generation arm, as well as in the polarization detection arm, along with a laboratory computer to process the data. The sample to be measured is placed between the two arms. The components in the polarization generation arm include a light source, collimator, shutter, polarizer module, and achromatic compensator module. The polarization detection arm consists of a chromatic biplate compensator, the analyzer module, focusing objective, spectrograph, and detector system. Each of these components is described briefly below Light source The light source used in the ellipsometer is a high pressure super-quiet 75 W Xe arc lamp manufactured by Hamamatsu. The lamp model number is L Since it is a high pressure lamp, proper shielding must be provided during its operation. Hence, this lamp is mounted within a lamp housing, model number E2419, and driven by a power supply, model number C2177, both manufactured by Hamamatsu. The lamp bulb is made of fused silica, which transmits the full spectrum emitted from the arc light source, as compared to ozone free silica which cuts the shortest wavelength UV region from the emission. The ultraviolet to near-infrared radiation spectrum available from this lamp is continuous in the range from 2 nm to 85 nm. 15

35 When the light source is started, the lamp is subjected to a very high voltage on the order of 2 kv to initiate the arc and then the voltage is dropped back to its operating value. Hence, all the other components of ellipsometer should be turned off when starting the lamp to avoid any electrical damage. To minimize any fluctuation in the irradiance during ellipsometry measurements, it is advisable to take these measurements at least 5 minutes after starting the lamp Collimating optics Optical elements are used to collimate the light from the lamp in the focusing direction. The elements include a lens and an achromatic doublet with a pinhole in between. The collimator is manufactured by Rudolph Research. Alignment of the collimator plays a crucial role in the precise operation of ellipsometer. Its alignment is explained in Chapter Polarizer module There are two polarizer modules used in this ellipsometer. Each consists of a polarizer, rotator block, and stepper motor system. The polarizer placed before the detector is called an analyzer Polarizer Rochon polarizers made from MgF 2 crystals are used in the ellipsometer. MgF 2 is a birefringent material, meaning that a light beam is split into ordinary and extraordinary rays in general when passing from an ambient medium into the crystal. MgF 2 also exhibits a wide spectral range without any optical activity. A Rochon polarizer consists of two MgF 2 crystals, each cut at an angle S, having mutually perpendicular optic axes. These two cut crystals are optically contacted to one 16

36 another. In this polarizer, the ordinary wave experiences the same index of refraction in the two orthogonally oriented crystals and is undeviated, whereas the extraordinary wave in the first prism experiences an index of refraction change and undergoes an angular deviation after exiting, given approximately by the expression [2-2] n e n o n a sin tans = +, (2.12) n a sin 2n e where n e and n o are the extraordinary and ordinary indices of refraction of MgF 2 and n a is the index of refraction of the ambient. Typical polarizers with a cut angle of S = 1 will experience an angular deviation within the range from 4 to 5, and decreasing with increasing wavelength from.2 to 2 m. A schematic of the Rochon polarizer is shown in Figure 2.2. f e t S Figure 2-2: Schematic representation of a Rochon polarizer showing its semi-field angle ( f ), cut angle (S), and the deviation angle of the first prism's extraordinary beam (). 17

37 The quality of a polarizer can be described by its extinction ratio, given by P = T Pe /T Pt, which is the ratio of the transmittance for fields vibrating along the extinction axis (e) to that for fields vibrating along the transmission axis (t). A higher ratio indicates a poorer polarizer. The extinction ratio is generally poorer for Rochon polarizers, ~ 5 x 1 4, in comparison with Glan-type crystal polarizers made from calcite. The semi-field angle for the Rochon polarizer is defined as the cone half-angle of converging light on the polarizer at which the deviated beam of one incident ray of the cone overlaps with the undeviated beam of the opposing ray, and is given in radians approximately by [2-3]: n e n o f = cot S. (2.13) 2n a For a MgF 2 Rochon polarizer with a cut angle of S = 1, f ranges from 2.2 for.2 m to 1.8 for 2 m. In practice, however, owing to the non-zero beam size and the need for instrument compactness, the operational field angle is generally lower. As a result, beam collimation must be carefully controlled. The Mueller matrix of an ideal polarizer is given by [2-1] MP R( P) R(P) (2.14) where P is the azimuthal angle of the transmission axis of the polarizer with respect to a reference plane that contains the optic axis of the polarizer, and R is a Mueller matrix describing rotation [2-1]. 18

38 Rotator block The rotator block was manufactured by Rudolph Research. This block has the advantage of being able to rotate the polarizer with an accuracy and repeatability of.1 degrees Stepper motor system The stepper motor system consists of a stepper motor and a stepper motor controller. The system is manufactured by Superior Electric Danaher Motion. The motor is a synchronous NEMA 23 motor, model number M61-LS8. NEMA is the acronym for "National Electric Manufacturers Association" which specifies the distance between the center of the shaft and edge of the frame of the motor used for mounting. The Superior Electric model number of the controller is SS2D6i. This stepper motor controller can be driven by computer by using ASCII commands. The stepper motor controllers of both polarizer and analyzer are connected in a daisy chain RS485 configuration. The outputs of this controller are also used to control the shutter operation Compensator modules There are two compensator modules present in the ellipsometer developed in this thesis research. They include an achromatic compensator module on the polarization generation side and a biplate compensator module on the polarization detection side. Each compensator module consists of its respective compensator with a similar motors for continuously rotating each of them Biplate compensator The general linear retarder, also described as a compensator, incorporates orthogonal fast (F) and slow (S) axes and is defined operationally by two quantities C', 19

39 the azimuthal orientation of the fast axis with respect to a reference plane, and the retardance C = F S. Thus, the quantity C is the shift upon transmission (transmitted incident) in the phase differences for orthogonal linear polarization states with fields vibrating along the fast and slow axes. This quantity, relevant for the transmission of light through an anisotropic medium, is analogous to the quantity for the oblique reflection of light from an isotropic medium. In fact, the quantity is the shift upon reflection (reflected incident) in the phase differences for orthogonal linear polarization states with fields vibrating along the p and s axes. The biplate compensator used in the ellipsometer is a zero order, retarder, which consists of two MgF 2 plates of thickness d 1 and d 2 separated by a distance greater than the coherence length of the light wave. MgF 2 is a positive uniaxial material meaning that and n e > n o, where n e and n o are the extraordinary and ordinary refractive indices, respectively. The fast axis (F) is orthogonal to the optic axis direction c, as shown in Fig. 2-3: F C S C F C Reference plane (d 1 >) d 2 d 1 Figure 2-3. Schematic representation of the biplate compensator 2

40 The two plates of the compensator should be internally aligned in such a manner that the fast axis of the two plates are exactly perpendicular. Then under these ideal conditions, the biplate exhibits no apparent dichroism; ( BC 45) [2-5]. The dichroic angle of a transmitting compensator is defined analogously to the ellipsometric angle of a reflecting sample as the relative (fast-to-slow) field amplitude ratio (transmitted-toincident). A small thickness difference between the plates of the compensator, d 1 d 2, gives a zero order retarder with retardance given by [2-1] BC 2 n n d d e o 1 2 (2.15) where is the wavelength of light. The fast axis of the thicker plate defines the fast axis of the biplate compensator. In Fig. 2-3, C is the angle made by the fast axis of the thicker biplate relative to the reference plane. Precise internal alignment of the biplates is necessary for high precision operation of the ellipsometer. In particular, when misorientations between the fast and slow axes of the two plates occur, rapid oscillations in the retardance are observed [2-4], [2-5]. These must be eliminated by carefully rotating one plate with respect to the other, and locking it into the proper position. The biplate is a chromatic compensator, i.e. the retardance spectrum ( BC ) of the compensator varies with the wavelength of light. This 1/ dependence leads to a variation of retardance far from the desired C = /2 quarter wave value at both ends of the spectrum. At long wavelengths, BC decreases slowly toward zero -- thus losing its polarization modulation capability as a rotaing element. At short wavelengths, however, BC approaches and crosses the half wave value of BC =, also resulting in a loss of 21

41 polarization analysis capability. The Mueller matrix of an ideal biplate compensator is given by [2-1] 1 1 MBC R( C) R(C) cosbc sin BC sinbc cosbc. (2.16) Achromatic compensator The problem of chromaticity exhibited by the biplate compensator can be overcome by using an achromatic compensator based on the rhomb design. The King-type rhomb described in [2-6] is used in the ellipsometer developed in this thesis research. There are three reflections in the King-type rhomb -- one intermediate reflection between two internal reflections from fused-silica/ambient interfaces. The intermediate reflection occurs internally from a silica/(275 Å MgF 2 )/ambient optical structure. A schematic of the King-type rhomb is shown in Fig F o S 275 Å MgF 2 Figure 2-4. Schematic representation of the achromatic King rhomb For total internal reflection at a single interface between a medium of index of refraction n s and ambient air, the p-direction parallel to the plane of incidence represents 22

42 the fast axis F for incident linearly polarized waves. The F and S phase shifts are given by [2-1]: n 2 s (sin 2 s sin 2 c ) 1/2 F = 2 tan 1,.. (2.17a) cos s (sin 2 s sin 2 c ) 1/2 S = 2 tan 1, (2.17b) cos s where s is the angle of incidence at the interface and c (< s ) is the critical angle for total internal reflection. For the King-type rhomb design, the angle of incidence at the two silica/ambient interfaces is 72.25, leading to a two-reflection F-S phase difference shift of 5.5 at =.4 m [2-6]. The angle of incidence at the silica/mgf 2 interface is s = This angle of incidence is below the critical angle for the interface, however, the refracted beam with f = is above the critical angle for the MgF 2 /ambient interface. The phase shift for total internal reflection at the second interface in the two interface problem is given by: (r af ) sin( ) 2 = 2 tan 1 + ( ), (2.18) 1 + (r af ) cos( ) where the subscript '2' indicates a two interface problem and the associated '' = (F, S) identifies the linear polarization mode. In addition, are the phase shifts for total internal reflection at the film/ambient interface as given in Eqs. (2-17); (r af ) are the amplitude reflection coefficients for the first silica/film interface; and is the phase shift 23

43 upon transmission through the film. Thus, = 4d f n f cos f /where d f, n f, and f are the film thickness and index, and the film/ambient angle of incidence, respectively. Applying this equation leads to a phase difference shift C = 2F 2S of at.4 m for the intermediate silica/mgf 2 /ambient reflection. The total F-S phase difference shift at.4 m is upon transmission through this device as designed with a first reflection angle of incidence of In the ideal device, when all internal reflections are total and no absorption in the rhomb coating or surfaces occurs, then the AC value should be 45. Because measured values of C deviate from 45 as shown as shown in Chapter 6, then non-idealities must exist. One possible problem arises from non-idealities due to ultrathin absorbing layers at the reflecting interfaces. Another potential non-ideality in rhombs results from strain in the fused silica induced upon mounting that alters the polarization state as the beam travels between internal reflections. The Mueller matrix of the device is given by [2-1] 1 cos 2AC cos 2 1 AC MAC R( C) R(C) sin 2AC cos AC sin 2AC sin AC sin 2 sin sin 2 cos AC AC AC AC (2.19) C is the angle of the fast axis of the rhomb measured with respect to the reference plane, which is the plane of incidence in a reflection measurement Low cogging, hollow shaft, continuously rotating motors The motor system consists of two low cogging, hollow shaft, continuous rotating motors, a motor controller operating the motors in a master-slave mode, and a power supply. All of these components are manufactured by Airex Motor Corporation. The achromatic compensator serves as the master and rotates at 1 Hz, whereas the biplate 24

44 compensator serves as the slave and rotates at 3/5 of the master frequency, or 6 Hz. The frequencies of these motors are fixed by the motor controllers, which are connected to the power supply. In stand-alone operation, i.e. when operating not in master-slave mode, both the motors rotate at 1 Hz. The head of each motor consists of an encoder module, which is used to track the position of the motor shaft while in continuous rotation. The encoder module produces 124 encoder pulses per mechanical cycle of the motor. One-half mechanical cycle is generally described as an optical cycle since the range -18, although mechanically inequivalent to the range 18-36, is optically equivalent. These pulses are transferred to the motor controller where they are used as the feedback signal for control of the motor stability. The motor controller also incorporates BNC sockets at the back of the unit to transmit these pulses externally for other applications, such as for triggering of data collection Shutter system The shutter system used in the ellipsometer consists of a shutter and shutter driver, both manufactured by Vincent Associates with model designation Uniblitz and model numbers VS25S2TO and VCM-D1, respectively. The shutter is placed after the collimator in the polarization generation arm. This shutter has an aperture of 25 mm, a 3 ms transfer time upon opening, i.e. time to open after the pulse is applied and a corresponding 5 ms transfer time upon closing [2-7]. This shutter is encased for protection from dust and mechanical damage. 25

45 2.2.6 Objective The objective used in the ellipsometer is manufactured by OFR, and the model designation is LMU-5X-. It is an UV achromatic objective with a 1 mm aperture. This objective is used to focus the light exiting from the analyzer onto the entrance slit of the spectrograph. The objective is inserted into an optical mount which has multiple degrees of freedom for ease of alignment Spectrograph The spectrograph used in the ellipsometer is manufactured by Jobin-Yvon, and the model number is CP The spectrograph consists of an internal mirror and grating, the latter to diffract the light and thus disperse it into its component wavelengths over the length of the detector array. The CP spectrograph incorporates a grating with 285 grooves per mm; the grating reference number is The wavelength range of the grating is 19 8 nm, and it is blazed at 25 nm by ion etching. The length of the spectrum at the focal plane is 25.2 mm, and the average dispersion is 24.2 nm/mm, yielding the total spectral range of 61 nm. The desired output of the diffraction grating is the first order spectrum, which is overlapped by the undesirable second order spectrum. Order sorting optical filters must be introduced in order to remove the second order. This procedure is explained in detail in Chapter 4. Slits are present at the entrance of the spectrograph which are able to control its spectral resolution. Decreasing the slit width increases the resolution, but in a trade-off, decreases the irradiance entering the spectrograph. Slits of different widths (.1,.1,.25,.5 and 1 mm) are available, and each slit is 8mm in height. With a 1:1 magnification of this instrument, the spectral resolution with the minimum and maximum 26

46 slit widths of.1 and 1 mm are ~.24 and 24 nm, respectively. Given that a single pixel of the detector spans 5 m (see Sec ), the spectral resolution of the detector, and that of spectrograph with the smallest slit width are well matched; however, if the detector is grouped by 8 for example (see Sec ), then a slit width closer to.1 mm should be used to achieve this condition. A user drawing of the spectrograph obtained from the manufacturer is shown in the Appendix A Detector system The detector system consists of a photodiode array (PDA) detector, a detector interface, and a detector controller, all manufactured by EG&G Instruments, Princeton Applied Research (PAR; model numbers 1412, 1461, and 1463, respectively) Detector The detector consists of a 124 pixel Si photodiode array system with a fused silica faceplate. The detector incorporates a Peltier cooler for normal (ambient temperature of +2 o C) operation down to 5 o C when the detector is backfilled with dry N 2 and 2 o C, when evacuated to 5 m Hg by using a vacuum pump. Even lower temperature can be achieved with supplemental ice water cooling. The operable wavelength range for the EG&G PAR model 1412 detector is < 18 nm to 11 nm (i.e. 1.1 ev to > 6.8 ev). The pixel size is 25 m x 2.5 mm on 25 m centers and the array length is 2.75 cm (1.24 ) [2-8]. Elements can be grouped in multiples of two to increase the effective width of the detector elements, at the expense of spectral resolution. The individual silicon photodiodes of the linear array used in the model 1412 consist of p-type bars diffused into an n-type silicon substrate such that both n and p-type areas are photosensitive [2-9]. Incident light generates charges which are collected and stored on 27

47 the p-type bars during exposure. These charges are sequentially switched to a video output for scanning during read-out Detector controller The EG&G PAR model 1463 detector controller consists of (i) an analog card, (ii) a 14 bit analog-to-digital converter (A/D) board, and (iii) a scan control card. The analog card converts the output of the model 1412 detector into a composite video signal so that it can be digitized by the 14 bit A/D board. A socket exists on the analog card to obtain a video output of the detected signal from the detector. The analog current measured by the detector is converted to a digital signal by the 14 bit A/D board. So the maximum operable light irradiance is that yielding 2 14 counts from the detector (i.e in decimal). The converted signal is stored in the memory of the detector controller. It is advisable to operate the detector at the maximum possible irradiance level without saturation for the highest sensitivity. The primary purpose of the scan control card is to provide the timing and control signals for the analog card. The analog card uses these signals as it controls sequential detector scanning and analog signal processing. Once the scanning is complete, the analog card waits for another trigger (external or internal as the case may be) to start another scan. The scan control card also synchronizes the A/D board. The SYNC input signal enables the synchronization of scanning of the detector s photodiode array to the timing circuitry of the external device Detector interface The EG&G PAR model 1461 detector interface is an buffered controller. It incorporates a microprocessor controlled data acquisition system that can collect, store, 28

48 and preprocess data before sending it to the host computer by GPIB interface. A National Instruments model NI GPIB interface card is installed in the computer in order to obtain data from the model 1461 detector interface. For communicating with the host computer, the GPIB address of the detector interface is set by DIP switches to 1 which has the decimal equivalent of 8 [2-1]. There are three basic methods of scanning available when the model 1461 detector interface is used with the model 1463 detector controller. They are (i) "normal" scanning, (ii) fast access scanning, and (iii) grouped scanning. In normal scanning mode, all the active photodiodes are scanned at the same speed for spectral data collection. The time required to scan a single pixel in this mode is 16 s. If the user does not need to acquire information from a group of contiguous pixels then that group of pixels is "fast scanned", i.e. no spectral data are acquired from them. The time required for scanning a single pixel in this mode is.5 s. Grouped scanning gives the user the ability to increase the model 1461 s sensitivity to low irradiance incident light, while at the same time proportionately decreasing the system spectral resolution. In this mode, the user defined groups of contiguous channels are scanned, and data collected from these photodiodes are combined into a single unit. The sensitivity of the group is increased by a factor equal to the number of units in that group and the resolution is reduced by the same factor. The time required for scanning a group of two pixels is 32 s. For each additional pixel in the group, the scanning time increases by.5 s. The group size must be an even number. For example, the time required for scanning a group of 8 pixels is [32 + 6(.5)] s = 35 s. 29

49 Data acquisition timing The basic unit of time for the model 1461 detector interface is the exposure time, which is the total time between the successive read-outs of a given pixel. Each pixel continuously integrates the photon flux falling on it throughout its exposure time. The total exposure time is divided into three sections [2-1]. Scan time is the interval during which pixels are read/reset and the data are transferred to the computer. The scan time duration can be varied by changing the hardware configuration and changing the set parameters in the software. Overhead time immediately follows each scan time and is provided to allow the host computer to make decisions regarding the next scan. Overhead time is fixed at 25 s. The variable integration time follows the overhead time. During this time the array is not being scanned and the system is basically at rest. This time contributes to the exposure time and is generally set to zero for the instrument developed in this Thesis. In all modes of data acquisition described above, a 17 s pre-read interval exists before starting each scan, hence [2-1] Scan time = [Pre-read time (17 s)] + [(# pixels) (pixel read time)] + [2 pixel read time] When the detector pixels are uniformly grouped by 8, for example, 128 grouped pixels are formed from 124 individual pixels. In this example, the scan time is given by Scan time = [17 s] + [(128) (35 s)] + [32 s] = ms When using external synchronization, the actual exposure time equals the period of the externally derived SYNC pulses and must be greater than the scan time. 3

50 Leakage current occurs in the detector elements due to thermally generated charge carriers and is also integrated during exposure time. These data should be subtracted from the original scanned data obtained during data acquisition. Modes of scanning, data acquisition, and data storage can be controlled through ASCII commands given to the detector controller by the computer. Commands and the data acquisition program for the ellipsometer are written in Labwindows/CVI. 31

51 Chapter 3 Modeling the Performances of Dual Biplate and Krasilov Compensators 3.1 Retardance optimization of the dual biplate compensator The retardance produced by a biplate compensator is strongly dependent on wavelength, varying as 1/. Hence, this compensator design is not optimum for use in rotating compensator ellipsometry over a wide spectral range. In order to develop a less chromatic compensator, a dual biplate design was proposed by Johs et al. [3-1]. The design consists of two biplates with their fast axes oriented such that the retardance produced by one can be counterbalanced by that produced by the other so as to get a relatively flatter retardance spectrum. In the article by Johs et al. [3-1], the authors presented only experimental results without any theoretical explanation. In this section, a theoretical analysis of the dual biplate compensator is presented, enabling design of such a device with a relatively flatter retardance spectrum Theory of the dual biplate compensator A dual biplate compensator is constructed from two biplates as shown in Fig This device can be expressed mathematically by using Jones matrices [3-2]. According to a theorem in Jones matrix algebra, any combination of non-depolarizing optical 32

52 components such as this can be described as a single optical component which itself acts as a rotator, that rotates the polarization state by an angle Q with respect to a reference axis, and then as a retarder with dichroic parameter ( c ) and effective retardance ( e ), with its fast axis oriented along a reference axis [3-3]. Applying this theorem to a pair of biplates, one can define a dual biplate compensator consisting of a common reference plane with the fast axis of the compensators oriented at angles 1 and 2 with respect to the reference plane, as shown in Fig The frame of reference of the dual biplate may be different than those of the component elements, hence the need for angle C. 2 1 \ Figure 3-1. An optical system consisting of two biplate compensators with retardance values of 1 and 2. 1, 2, and C are the angles made by the fast axes of the two individual compensators and the effective rotator-compensator, respectively, measured with respect to the reference plane, shown as horizontal. The unknown quantities of the new element are described on the left side of the following equation: z -C Q C 2 1 cot e i e e i i c e (3.1),

53 and the quantities associated with the known components from which the new device is constructed are described on the right side. In Equation (3.1), C is the angle between the fast axis frame of reference of the new rotating-retarding element and the reference plane, 1 and 2 are the retardance contributions from the individual biplates, and 1 and 2 are the orientations of the fast axes of the two biplates measured with respect to the reference plane. The quantity z is a complex number defined by z = re i. Equating the real and imaginary parts of the both sides of Eq. (3.1) gives eight equations. However, it turns out that two equations are identical to another two of the eight, and so there are only six independent equations. It is necessary to solve the six equations for the six unknowns (r,, C, c, Q, and e ) in terms of the known quantities ( 1, 2, 1, and 2 ). In order to solve Eq. (3.1), the first simplification is to multiply both sides of the equation by the rotation matrix C on the left and by C simplifies the left side of the equation, yielding: on the right. This 1 i 1 1 re Q C 2. (3.2) C i e i i cot c e e e Assuming = C 1 and = C 2 as wavelength dependent angles and = 2 1 as a wavelength independent angle, then re i cos Q sin Q 1 1. (3.3) i e i e 2 1 sin Qcot cos Qcot i i ce ce e e The real and imaginary parts of Equation (3.3) are equated to determine the equations to be solved. The linear combinations of these equations eventually provide the 34

54 six independent equations. From those six independent equations the quantities r,, C, c, Q, and e can be detemined and the results for C are given as follows 2sin 1tan C2 tan 2 2 sin 2 1 tan 1 2 sin 1 2. (3.4) Equation (3.4) shows that C requires only the design parameters of the two retarders ( 1, 2, 1, and 2 ). The results for e and Q are given as follows sin cos sin sin cos sin ½cos cos sin 2 1 e 2tan (3.5) where + = 2C ( ) ; = 2 1, ½ ½ cos cos tan Q tan. (3.6) 2 cos tan 2 1 cos The other three parameters in Equation (3.3) are given by: r = 1, (3.7a) = ½ e (3.7b) c (3.7c) In the calculations, the parameters 1 and 2 were chosen versus wavelength such that they describe quarter wave retardance values at particular wavelengths, which are determined through the general relation for the retardance () of a single biplate compensator, given by 2 n o n e d. 35

55 Here n o and n e are the ordinary and extraordinary indices of refraction of the component material and d is the thickness difference between the two plates Simulation methods for the dual biplate compensator The retardance of the biplate compensator depends on the thickness difference for the component plates, applying the expressions: Q = 4 n o n e d () = (/2)( Q /) (3.8a) (3.8b) So in this analysis, d is used to define Q for each biplate in addition to the relative angular orientation of two biplates ( 2 1 ), as the three principal variables in the optimization of the flattening of the retardance spectrum. To assess optimized performance, one can use the spectral average of the effective retardance e of the dual biplate compensator and the difference between the maximum and minimum in the effective retardance ( max min ). Since MgF 2 is widely used for ultraviolet optics applications, it is assumed that the biplates are manufactured from MgF 2, which fixes n o n e in the simulation. The optical properties n o and n e of MgF 2 are obtained from the literature [3-4] for use in determining the birefringence n o n e this analysis. These are provided in Appendix B. Two approaches are followed to determine the optimum thickness differences. The first approach is to assign the first biplate thickness difference independently such that the biplate operates as a quarter wave plate at a relatively short wavelength, whereas the second biplate thickness is determined by simulation such that an ~ 9 o average effective retardance is obtained over the entire spectral range according to Eq. (3.5). The same 36

56 procedure is repeated for various choices of the first thickness difference so as to obtain the flattest possible spectral dependence of the effective retardance from the dual biplate system, i.e. a minimum in max min. In the second approach, both thickness differences are determined independently such that the effective retardance is as flat as possible over the spectral range of interest. In this second approach, the second biplate thickness difference is chosen as that yielding a minimum in max min. More importantly, in this approach, the average retardance over the spectral range of interest is unconstrained and so is allowed to be different than 9. After each approach, the relative angular orientation of the biplates ( 2 1 ) is varied around 45 o so as to determine the optimum angular difference for a flat retardance dispersion. The orientation angle of 45 o is ultimately selected based on results obtained in the second approach. Johs et al. [3-1] used this orientation for their design, as well. A series of simulations is performed for each of the approaches, and the results of these simulations are presented in next section Results and discussion for the dual biplate compensator First simulation approach: constrained spectral average of the effective retardance The quarter wave wavelength of the first biplate 1 is assigned based on d from Eq. (3.8a), starting from 15 nm, and the associated quarter wave wavelength of the second biplate 2 is determined such that the average effective retardance of the dual biplate system is ~9 o. Then the difference between the maximum and minimum in the effective retardance, max min, is determined as shown in Fig This simulation approach is 37

57 performed over the wavelength range of 15 nm to 2 nm, and initially 2 1 is set at 45. Figure 3-2. (a) Quarter wave wavelength of the second biplate 2 selected such that the average effective retardance is ~ 9 for the different assigned values of along the abscissa; (b) corresponding max min for the range 15-2 nm as a function of the quarter wave wavelength of the first biplate is set at 45 for these simulations. 38

58 It has been found that, only for a very narrow range of 1 from nm to 26 nm, does there exists a corresponding 2 for which the average effective retardance is ~ 9 o. From Fig. 3-2, it can be observed that max min varies with 1 in the same way as 2. A higher value of max min, however indicates that the ( 1, 2 ) pair is a better choice for flattening the effective retardance spectrum of the dual biplate retarder. In Fig. 3-3a, the retardance spectra of the individual biplates 1 and 2 are shown for the ( 1, 2 ) pair for which max min is minimum. The respective quarter wave points for the individual biplates are mentioned in the legend. Figure 3-3a: Individual retardance spectra of the two biplates selected for the dual biplate compensator. The quarter wave points of the biplates are specified. 39

59 From the individual biplates with their respective retardance values selected as described above, the effective retardance e of the dual biplate system is simulated based on Eq. (3.5). Also shown in Fig. 3-3b are the azimuthal angle C and the optical rotation Q generated by dual biplate system as determined from Eq. (3.4) and Eq. (3.6), respectively. The reference plane of the device from which C is measured is established by the fast axis of the second biplate. This implies that 2 =, and 1 is taken as 45 in these simulations. From Fig. 3-3b it can be seen that effective retardance e is 9 o at both quarter wave points of the individual retarders. That is, at 1 there is no contribution to the effective retardance from the second biplate, and at 2 there is no contribution to the effective retardance from first biplate. 4

60 Figure 3-3b: (a) Spectra in the effective a) optical rotation Q ; b) azimuthal angle C, and c) retardance e for the hypothetical optical element that describes the dual bi-plate compensator system shown in Fig The results described above assume 2 1 =45 o. As the last step in this first approach, the angular difference 2 1 is varied and the corresponding results are shown in Fig From the figure it can be seen that the flatness of retardance spectrum is continuously increasing as 2 1 is increasing. This suggests that the above procedure that has led to Figure 3.3a and Figure 3.3b should be repeated at different angular differences 2 1. Specifically, this procedure involves selecting a range of quarter wave wavelength for the first retarder such that the quarter wave wavelength for the second retarder can be chosen to ensure an average effective retardance of 9. Then for these 41

61 quarter wave wavelength pairs, the effective retardance difference max min can be minimized. This procedure would then be incorporated into an iteration loop as a function of 2 1 in order to minimize max min globally. Figure 3-4: Dispersion of the effective retardance e for different 2 1 values. In the legend, the max min values are provided for each of the 2 1 values Second simulation approach: unconstrained spectral average of the effective retardance The quarter wave wavelength of first biplate 1 is varied randomly, and the quarter wave wavelength of the second biplate 2 is determined such that effective retardance e is as flat as possible with wavelength, meaning that max min is a minimum. The spectral range over which the retardance flattening is calculated extends from 15 nm to 2 nm. 42

62 Thus, in this approach the average value of e is not constrained to be ~ 9 o. This simulation approach is performed over the same wavelength range as the calculation of the retardance flattening, 15 nm to 2 nm.. Figure 3-5 depicts the set of pairs of 1 and 2 for which max min is found to be less than 1 o. Figure 3-5: (a) Quarter wave wavelength of the second biplate 2 selected such that max min is minimized for the given values plotted on the abscissa, applying no constraint on the spectral average of e ; (b) the minimum value of max min as a function of the quarter wave wavelength 1 ; (c) Spectral average of e for the ( pair defined by the abscissa and by the ordinate in (a). 43

63 From Fig. 3-5, it can be seen that the range of 1 which yields a corresponding 2 with a minimum in max min less than 1 is nm, where 15 nm is the lower limit of the values attempted. A total of 356 pairs of 1 and 2 are available for which max min < 1 o, using a grid spacing of ~.15 nm for 1 from 15 nm to 23 nm. Also, the 2 value yielding minimum max min is found to increase monotonically with 1. On the other hand max min decreases up to 1 ~ 22 nm and then increases again. As before, the retardance spectra are shown in Fig 3-6a for the individual biplates, 1 and 2, that yield the overall minimum value of max min for the hypothetical optical element. As given in the legend of Fig 3-6a, the best quarter wave pair ( 1, 2 ) for which max min is minimum is ( nm, nm). Also it should be noted from Fig. 3.5(c) that for this pair of biplates, the average effective retardance over over the spectral range of 15-2 nm is ~8, which is not significantly different from the constraint of 9 used in the first simulation approach. 44

64 Figure3-6a: Individual retardances of two biplates selected for the dual biplate compensator system with the quarter wave points specified in the legend. These are the values that flatten the effective retardance spectrum of the hypothetical optical element, meaning minimization of max min. 45

65 Figure 3-6b: Spectra in the effective a) optical rotation Q; b) azimuthal angle C, and c) retardance e generated by the dual bi-plate compensator system as obtained by minimizing max min without constraint according to the second simulation approach. The effective retardance is shown in Fig. 3-6b as generated by a dual biplate system described by the individual quarter wave values 1, 2. It can be seen here again that at 1, e is 9 o and there is no contribution to the retardance from the second biplate. Similarly at 2, e is 9 o and there is no contribution to the retardance from the first biplate. As before, C, measured with respect to the second biplate fast axis orientation, as well as Q, both depicted in Fig. 3-6b, are the effective azimuthal angle and rotation angle, respectively, generated by the dual biplate system. 46

66 As in the first approach, all results reported so far for this second approach assume 2 1 = 45. The difference max min in effective retardance e also has been determined for different fixed values of 2 1. In the second optimization approach, minimum results for max min are found for 2 1 = 45 o. These results are shown in Fig Figure 3-7: Dispersion in the effective retardance e for different 2 1 values. In the legend, the max min values are provided for each of the 2 1 values. By comparing the above results it can be concluded that the effective retardance of dual biplate system is much flatter in the second approach, and this approach constitutes only a 1 sacrifice of the constraint on the average retardance. In this method, in which case max min is minimized without a constraint on the average value of e, the angular 47

67 difference 2 1 at which the minimum occurs appears to be 45 o. This is in contrast to the behavior observed using the first optimization approach. Also, it appears that Johs et al. adopted an optimization method similar to this second approach because the spectral dependence of the retardance in their report [3-1] is quite similar to the results presented in the second approach. 3.2 Optical modeling of Krasilov type compensator In 1967 Krasilov proposed a three reflection type compensator which he called an "achromatic device (AD)" [3-5]. This device produces a retardance that is relatively weakly dependent on the wavelength of light. It consists of two prisms and an aluminum mirror, in which all the components are separated from one another without a common means of support. Because this type of compensator is very difficult to construct and continuous rotation is not easily accomplished, a new type of compensator based on the Krasilov design has been proposed by Prof. Robert Collins, Univ. of Toledo. The advantage of this new design is that it is relatively easier to construct and may be readily usable as a continuously rotating element. The proposed design is shown in Fig. 3.8 and discussed in the following paragraphs. 48

68 3.2.1 Theory of Krasilov compensator Opaque aluminum coating prism Light beam path prisms Figure 3-8: Krasilov compensator design. The Krasilov type compensator is constructed using three prisms made from vitreous silica. Two prisms are of the standard equilateral type (6-6-6) and the third is an isoceles (3-12-3) prism. The side opposite to the 12 o angle of the isoceles prism has an opaque aluminum coating on its surface. All three prisms are optically contacted as shown in Fig The output beam is optically in line with the input beam, which is a necessary condition for continuous rotation in rotatingcompensator spectroscopic ellipsometry applications. Nearly quarter wave retardance is produced after the light has been reflected with the designed angles of incidence at the three internal interfaces, i.e. two fused-silica/air interfaces and a single fusedsilica/aluminum interface, as shown in Fig The reflection processes are assumed to be specular, i.e. the incident plane wave is transformed to a reflecting plane wave such 49

69 that the angle of incidence is equal to the angle of reflection. This design has been optically modeled to determine the wavelength dependence of the relative amplitude ratio and phase shift difference for two orthogonal linearly polarized p and s waves. The optical properties of fused silica and aluminum from the literature [3-4] are used for modeling purposes and these values are provided in Appendix B In the reflection process, a change in the polarization state of the light beam occurs abruptly at the interface between two optically dissimilar media. Each reflection can be modeled by a Jones matrix for the isotropic reflecting interface [3-2]. The Jones matrix of the device shown in Fig. 3-8 is given by ~ T ~ rp3 ~ r ~ r s3 p2 ~ r ~ r s2 p1 ~ r s1 (3.9) ~ r ~ ~ p3 rp2 rp1 ~ r ~ ~ ~ ~ ~ s 3rs2 rs1 rs3 rs2 rs1.. (3.1) Here, r pj, r sj where j= 1,2,3 are the Fresnel complex amplitude reflection coefficients for the first, second, and third reflections. These Fresnel reflection coefficients can be expressed in terms of a dichroism angle cj, which is the inverse tangent of the modulus ratio of r pj and r sj, and in terms of a phase shift difference (cj ) which is the difference in the phase shifts associated with the Fresnel reflection coefficients. The dichroism angles and phase shift differences are key components for optical modeling, and are given explicitly by 5

70 cj tan 1 r pj r sj, (3.11a) cj = pj sj. (3.11b) Optical modeling of the first reflection 6 o Light beam 6 o 9 o i i 9 o 3 o 6 o Figure 3-9. Beam path through the first prism of the Krasilov type compensator. The first reflection occurs at the fused-silica/air interface. From Fig. 3-9, it can be noted that i + 3 o = 9 o ; hence the angle of incidence for the first reflection is given by i = 6 o. Using Snell s law, the critical angle ( c ) of reflection at the fused-silica/air interface is calculated to be o for a wavelength of nm at which the refractive index of fused silica is given by n a =1.4584; the refractive index of air is given by n s =1, independent of wavelength. Since the angle of incidence for the first reflection in the device i = 6 o is greater than critical angle c =43.29 o, there exists total internal reflection at the interface. In fact, as long as the index of refraction of the incident medium is greater than 1.155, which is the case for fused silica throughout the wavelength range of interest, total internal reflection will occur. Thus, the first interface behaves as desired throughout the full spectral range of interest. 51

71 The change in the polarization state of the light upon this first reflection is described by the Fresnel reflection coefficients. The ratio of these coefficients can be expressed in terms of the dichroism angle c1 and phase shift difference c1 using Equations (3.11). This ratio of Fresnel reflection coefficients ( ) is given as [3-2] ~ ~ rp ~ r s n n a a sin sin 2 2 i i cos cos i i N ~ N ~ 2 s 2 s n n 2 a 2 a sin sin 2 2 i i. (3.12) The dielectric functions of the reflecting medium ~ s and the incident medium or ambient a are given by ~ 2 s = N ~ = n 2 s = 1 for air and a = n 2 a for a non-absorbing ambient s medium such as fused silica. Using these expressions in Eq. (3.12) yields ~ n n a a tan i tan i sin i sin i ~ ~ s s a a sin sin 2 2 i i. (3.13) To determine the sign in this equation, and ultimately the sign of c1, Eq. (3.13) is converted to the complex form using the definition, A p exp(i p ) = ~ s a sin 2 i. This results in the equation ~ n n a a tan i tan i sin i sin i A A p p 1/ 2 1/ 2 p cos ia 2 p cos ia 2 p p 1/ 2 1/ 2 p sin 2 p sin 2 (3.14) 2 where A sin p sr a i 2 2 si and p 1 si tan. In these 2 sr a sin i expressions, ~ s = 2 2 sr i si, sr = n s k s and si = 2n s k s, where sr and si are the real and imaginary parts of the dielectric function of the reflecting medium, respectively. In 52

72 addition, n s and k s, are the associated refractive index and extinction coefficient, respectively. Because the electric field must decrease with increasing distance into the reflecting medium after the beam has crossed the interface, it can be shown that the upper sign in Eq. (3.14) is appropriate as long as s 2 sr 2 a sin i. For the first reflection, 2 s = 1, sr = 1, si =, and a = , the latter for a wavelength of nm. It can seen for this first reflection that, s 2 sr 2 a sin i ; hence the lower sign should be used in Eq. (3.12) for determining c1. As a result, Equation (3.14) can be written as 1/ 2 p 1/ 2 p n a tan i sin i A p cos ia p sin ~ 2 2. (3.15) 1/ 2 p 1/ 2 p n a tan i sin i A p cos ia p sin 2 2 For = nm, n a = , n s = 1, and i = 6 o. Evaluating the individual terms in Equation (3.15) yields sr = n s 2 k s 2 = 1, si = 2n s k s =, a = n a 2 = ; A p sr a sin i si = ; A 1/2 p = , (3.16) p 1 si tan = 2 sr a sin i tan 1 =. (3.17) A quadrant correction in Eq. (3.17) is performed since the denominator is negative, yielding p = + 18 o = 18 o, and p /2 = 18/2 = 9 o, n a sin i tan i = ; 53

73 1/2 p p p A sin ; 1/2 A cos, 2 2 p ~ [ (n a sin i tan i + r p A cos 2 ) 2 + 1/2 p p 1/2 p 2 (Ap sin ) ] 1/2 = , (3.18) 2 ~ [ (n a sin i tan i r s A cos 2 ) 2 + 1/2 p p 1/2 p 2 (Ap sin ) ] 1/2 = , (3.19) 2 tan c1 = ~ rp ~ r s = 1; c1 = 45 o, p = tan -1 p 1/2 [ (n a sin i tan i + Ap cos ) / 1/2 p (Ap sin ) ] (3.2) 2 2 = tan -1 [ /(2.1876) ] = o, s = tan -1 [ (n a sin i tan i 1/2 p Ap cos ) / 1/2 p p 2 (A sin ) ] (3.21) 2 = tan -1 [ /(2.1876) ] = o, p s = ( ) = c1 = o. The results obtained above are repeated for the entire photon energy range of.74 ev 6.51 ev (or spectral range of nm), and the dispersions of dichroism angle c1 and retardance c1 induced by the first reflection are shown in Fig This first reflection is also modeled in J.A. Woollam Company software WVASE32 by constructing a model using fused silica on top of void (air) substrate and considering fused silica as the ambient. The angles c1 and c1 obtained by this model are also shown in Fig

74 Figure 3-1. Dispersion of the dichroism angle c1 and the retardance c1 induced by the first reflection, which is a total internal reflection from a fused silica/ambient interface. 55

75 Optical modeling of the second reflection 3 o 6 o 6 o 6 o o 9 o Light beam 9 o 6 o 9 o 12 o 6 o 6 o 3 o 6 o 3 o Figure Beam path through the first and second prisms. The second reflection occurs at the fused silica/aluminum interface. From Fig. 3.11, it can be established that 2 is 3 o at this interface. Similar to the calculation for the first reflection, the change in polarization state of light induced by this reflection is characterized in terms of ( c2, c2 ). Since aluminum is a metal with large extinction coefficient, absorption at the interface will occur that depends on the polarization mode (p or s), which leads to a dichroism angle for the reflection that differs from 45. For a wavelength of nm, the real refractive index n a of the ambient fused silica is , and the complex refractive index of the reflecting aluminum is given by N s = n s ik s, where n s =1.1389, k s = , ~ 2 = N ~, and a = n 2 a. As described s s previously when discussing the first reflection, the upper sign in Eq. (3.12) is selected when 2 s. This selection in turn determines the sign of c2. sr 2 a sin i 56

76 2 2 Here, = so that s sr si s 2 = , and = sr 2 a sin i Thus, s 2 and hence upper sign must be used, which leads to sr 2 a sin i 1/ 2 p 1/ 2 p n a tan i sin i A p cos ia p sin ~ 2 2. (3.22) 1/ 2 p 1/ 2 p n a tan i sin i A p cos ia p sin 2 2 For = nm, n a = , N s = i (7.1263), and 2 = 3 o. 2 Individual terms in Eq. (3.22) can then be determined, using sr = n s k 2 s = , si = 2n s k s = , and a = n 2 a = These substitutions yield A p sr a sin i si = ; A 1/2 p = , (3.23) p 1 si tan = 2 sr a sin i tan 1 = o. (3.24) [ ) (2.1269)(.25 ] Since the denominator in Eq. (3.24) is negative, a quadrant correction is required so that p = o + 18 o = o, p /2 = /2 = o, n a sin i tan i =.4213, 1/2 p Ap sin ; 1/2 p p 2 A cos , ~ [ (n a sin i tan i r p A cos 2 ) 2 + 1/2 p p 1/2 p 2 (Ap sin ) ] 1/2 (3.25) 2 = , 57

77 ~ [ (n a sin i tan i + r s A cos 2 ) 2 + 1/2 p p 1/2 p 2 (Ap sin ) ] 1/2 = , (3.26) 2 tan c2 = ~ rp ~ r s =.9827 ; c2 = o, p = tan -1 ( (n a sin i tan i 1/2 p Ap cos ) / 1/2 p p 2 (A sin ) ) = , (3.27) 2 s = tan -1 ( (n a sin i tan i + 1/2 p Ap cos ) / 1/2 p p 2 (A sin ) ) = (3.28) 2 p s = c2 = o. The results obtained above for the single wavelength have been extended over the entire photon energy range of.74 ev 6.51 ev (or wavelength range of nm). Thus the dispersion in the dichroism angle c2 and the retardance c2 induced by the second reflection are shown in Fig This second reflection is modeled using J.A. Woollam Company software WVASE32 by constructing an optical model of fused silica on top of an aluminum substrate and considering fused silica as the ambient. The angles c2 and c2 obtained by this model are also shown in Fig

78 Figure 3-12: Dispersion of the dichroism angle c2 and the retardance c2 induced by the second reflection, which is from the fused-silica/aluminum interface Optical modeling of the third reflection The third reflection is identical to the first reflection, occurring at the fusedsilica/air interface with the same angle of incidence as the first reflection ( 3 = 6 o ). As a result, the output beam is in line with the input beam entering the compensator. For a wavelength of nm, c3 = 45, c3 = , as in the first reflection. 59

79 3.2.2 Determining effective retardance and effective dichroism of the Krasilov type compensator Since the first reflection is identical to the third reflection, Eq. (3.1) can be rewritten as T 2 r p2 r p1 2 rs2 r s1 r s2 r s1 (3.29) 1 1 = Applying Equations (3.11a) and (3.11b), Eq. (3.29) can be rewritten as 2 rp2 r p1 i 2 c2 ic1 2 rs2 r s1 r 2 c2 s2 r s1 = c1 rs2 r s i( c 2 2* c1 ) = ~ ~ 2 (tan c1 tan c2 )e r s2 r s1 o i( ) 2 = tan e tan e (3.3) = o o i( * ) 2 r s2 r s1.9827e e rs2 r s1 (3.31) 1 From the above Eq. (3.31), for a wavelength of nm, c = o and c = o. Also fast axis is the p-direction and slow axis is the s-direction. The results obtained above for the single wavelength can be extended over the entire photon energy range of ev (or wavelength range of nm). Shown in Fig are the dispersions in the effective dichroism angle c and in the effective retardance c induced by all three reflections within the Krasilov type compensator. From Fig. 3-13, it can noted that the device exhibits a relatively weak dependence on the wavelength of light. In addition, the quarter wave wavelength occurs 6

80 near the center of the photon energy range at 3.47 ev, and the effective dichroism angle differs from 45 by no more than 1. Figure 3-13: Dispersions in the effective dichroism angle c and in the effective retardance c produced by three reflections of the Krasilov type compensator. Internal alignment of this device in fabrication is critical for proper operation, and external alignment by the user is a critical task as well. For internal alignment the device must be fabricated such that the planes of incidence for all three reflections are parallel. Only in such a circumstance can the Jones matrix of the device be described in terms of Eq. (3.9). Another critical feature is left-right symmetry about the plane depicted as the broken line in Fig This helps ensure that the exiting beam is collinear with the incident beam. In addition, the six interface transmissions must all occur at normal 61

81 incidence to avoid modifying the polarization of the transmitting beam. In external alignment, the face of the compensator should be aligned exactly perpendicular to input beam in order to avoid wobbling of the output beam upon rotation. Small errors in misalignment will also change the effective dichroism angle c and retardance c of the compensator. 62

82 Chapter 4 Initial Steps in the Development of the Multichannel Mueller Matrix Ellipsometer 4.1 System electrical component operation Before initiating optical system alignment, it is necessary to test all the electrical systems of the ellipsometer for proper operation. These systems are essential components for data collection. Programs for controlling the shutter, stepper motor, and detection system are written in Labwindows/CVI; CVI is the acronym of "C for Virtual Instrumentation". Labwindows/CVI is an ANSI C environment developed by National Instruments for virtual instrumentation. The controllers are connected to a laboratory computer and are in turn controlled by the Virtual Instruments Software Architecture (VISA) developed by National Instruments for Labwindows/CVI. Programs in the LabWindows/CVI environment consist of a source code and a User Interface (UI) which points to the source code. When any button is clicked on the UI, the source code -- which is connected to the UI -- begins execution [4-1]. This environment provides a user with the robustness of executing any part of the program any number of times without halting execution of the 63

83 entire program. By using the VISA architecture, multiple instruments can be controlled simultaneously from a single source code file. Stepper motor controllers are connected in an RS485 daisy chain configuration to the computer through a serial port. The controlling program is written in Labwindows/CVI in such a manner that either the polarizer or analyzer rotator module (Rudolph Research Corp.) can be selected at any given time. These components can then be rotated under program control with an angular resolution of.1 degrees in either clockwise or counter-clockwise directions. The stepper motor controllers have outputs which can be used to control other instruments, as well. In fact, the shutter controller is given commands through the outputs from the stepper motor controller. The commands serve to initiate the open and close operations of the shutter. Pin 2 of the shutter controller input is given the actuation signal and pin 3 is grounded. Programs are written in Labwindows/CVI for controlling the stepper motors and shutter. Multiple programs are written to instruct the detector controller and detector to perform its required operations. There are 14 types data acquisition modes available based on different combinations of data collection, data processing, and storage. In fact, mode 8 and mode 9 have been used predominantly in the development of this ellipsometer [4-2]. In mode 8, data from each scan are stored at consecutive memory locations. For example, when the detector must be scanned 8 times per optical cycle for waveform analysis, the detector continuously acquires the data, and data output from each sequential scan is stored in consecutive memory locations. Hence in this mode, 64

84 memory usage is not efficient when data acquisition involves averaging more than one optical cycle. In mode 9, data scanned from one optical cycle can be added to the data from the preceding optical cycle in a cumulative manner, i.e. the first scan of the first optical cycle can be added to the first scan of the second optical cycle and this process can be repeated depending on the number of optical cycles identified in the program. These cumulative additions of data from each scan of the optical cycle are stored in separate memory locations. For example, if 8 scans are performed per optical cycle, and data collection over 2 optical cycles is desired, then the data are cumulatively added and stored in only 8 memory locations, rather than 16 as would occur in mode 8. A program based on mode 8 was written to view the live irradiance that strikes the detector. In this program, the detector operates in normal mode without any external synchronization pulses. The detector is scanned sequentially, and the acquired data are updated on the screen continuously, in the form of irradiance counts versus pixel number. Collection and display of the data continue until the user interface stop button is clicked. Since only one scan is used for each disply, the refresh rate is high such that even a small variation in the light signal can be easily noted. This program plays a major role in the alignment of optical components, wavelength calibration, etc. This program is provided in Appendix C. Programs have been written to check the status of the detector controller each time the detector is turned on, to check the detector temperature, and to determine if the detector cooler is locked. Additional programs have been written to perform all the 65

85 required operations of the detector when performing single and dual rotating compensator measurements. 4.2 Sample alignment The polarization generation and detection arms of the ellipsometer are mounted onto a goniometer base which defines a fixed vertical central axis of the ellipsometer. The goniometer base allows the polarization detection arm to be rotated relative to the fixed polarization generation arm. This rotation occurs about the vertical central axis such that the horizontal axis of the polarization detection arm sweeps out a plane normal to the vertical central axis, which is the plane of incidence in the ellipsometry measurement [4-4]. A sample stage is set at the vertical axis enabling a sample to be rotated through 36 about the vertical axis -- as well as aligned at that location. For a well constructed goniometer system, the two horizontal axes of the polarization generation and detection arms must intersect at a point. Furthermore, this intersection point must lie on the goniometer vertical central axis. This condition must be maintained even as the polarization detection arm is rotated. Thus, the plane generated by the horizontal central axis of the polarization detection arm, when it is rotated about the vertical central axis, must contain the central axis of the polarization generation arm. If the above conditions as described are not met, then ellipsometer alignment independent of the goniometer setting, i.e. the angle of incidence setting, is not possible. In a well constructed ellipsometer set in the straight-through configuration, a laser beam can be aligned so that it progresses along the horizontal axis of polarization generation arm, intersects the vertical central rotation axis of the goniometer and 66

86 continues along the horizontal axis of polarization detection arm as described in the previous paragraph [4-3]. In the reflection configuration, the same beam progression occurs, however, in this case a sample, aligned so that its surface contains the vertical central rotation axis, redirects the beam along the axis of the polarization detection arm. Sample alignment refers to the process by which the sample surface is located at the preestablished intersection point of the axes of the polarization generation and detection arms such that the vertical rotation axis of the goniometer lies in the plane of the sample surface [4-4]. In addition to sample alignment, it is also important to align all optical components, so that the light beam impinges on each of the components at normal incidence; otherwise systematic errors will result. Two components are required for performing sample alignment. First, an alignment laser is mounted onto a stage on the polarization generation arm. The mounting hardware includes a) two linear translation stages one for horizontal movement of the beam within the plane of incidence normal to the axis of the polarization generation arm and another for vertical movement normal to the plane of incidence, b) a single tilt stage that serves to tilt the laser beam about an axis of rotation that is in the plane of incidence and normal to the axis of the polarization generation arm, and, c) a rotation stage with rotation in the plane of incidence about an axis normal to the plane of incidence. These degrees of freedom are shown in Fig. 4-1a. The laser along with its alignment stage are placed on the source end of the polarization generation arm. Second, a thin glass sample is used that can reflect light from both its front and back surfaces. This sample is mounted on a stage consisting of two linear translations, both in the plane of incidence. Only one translation is needed for sample alignment, for movement along 67

87 the axis of the fixed polarization generation arm in straight through. The sample stage also includes a tilt stage with an axis of rotation within the plane of incidence perpendicular to x y z Figure 4-1a: Degrees of freedom for orienting an alignment laser including two translations (x, y) to locate a point on a plane, and a rotation and a tilt to identify a direction in three dimensions. z z Figure 4-1b: Degrees of freedom for aligning a sample including a translation z to locate a plane, and a rotation and a tilt to identify the normal to the plane in three dimensions. 68

88 the polarization generation arm to align the sample surface parallel to goniometer axis. The sample stage finally enables a full 36 degree rotation about an axis normal to the plane of incidence. These degrees of freedom are shown in Fig. 4-1b, and a photograph of the sample stage is shown in Figure 4-1c. Thin glass plate Tilt stage Horizontal linear stage Horizontal linear stage Figure 4-1c: Photograph of the sample stage with a thin glass sample mounted on it for alignment purposes. Alignment of light source and sample can be performed in 4 iterative steps [4-3]. 1) Laser alignment of a series of apertures defines the central axes of the polarization generation and detection arms of the straight-through ellipsometer. 2) Alignment of sample surface first ensures that it is parallel to the axis of rotation 3) Translation of the sample surface second ensures that it exactly coincide with the axis of rotation of the goniometer. 69

89 4) Adjustment of the light source used in the ellipsometer is performed as the last step. Step 1: All optical components except the apertures and alignment laser are removed from arms and goniometer of the ellipsometer. A minimum of four apertures is required for proper alignment, two on the polarization generation arm and two on the polarization detection arm. Two reference apertures are provided by the goniometer manufacturer, Rudolph Instruments Inc. These apertures are used to define a line in the plane of incidence which should pass through the vertical goniometer axis. Initially the reference apertures are placed on both the polarization generation arm and the polarization detection arm and the alignment laser is aligned. The laser beam is parallel to the axes of the two arms when the beam passes through the entrance and exit reference apertures without being eclipsed. Custom made apertures are aligned to further define the laser beam as an alignment reference. Step 2: This next step is to ensure that the glass sample surface is parallel to the vertical goniometer axis of the ellipsometer (C) as shown in Figure 4-2. This step relies on the capability of attaching the sample to a stage such that the mounting post of the stage can be inserted into the goniometer fixture. Then in this configuration, the stage can be rotated a full 36 o about the goniometer axis. The glass sample must be mounted in such a way that both plane parallel sides are accessible and so can reflect light from the laser. Since the glass sample is arbitrarily mounted initially, its surface is not necessarily parallel to the goniometer axis. To ensure that this surface is parallel, light from one face 7

90 of the sample in position (a) in Figure 4-2a is reflected onto a distant screen, identified as position B 1 in the figure. By rotating the sample by 18 o, light from other side of the sample when in position (b) in Figure 4-2a is reflected to position B 2 as shown in the figure. Iterative tilt adjustments on the sample holder are performed until the light beams reflected from both sides of sample surface, B 1 and B 2 are parallel to one another as shown in Figure 4-2b. This can be checked by measuring the spacing between the two beams, which should be constant as a function of distance from the goniometer axis. In fact, each reflection consists of two slightly displaced beams due to the two glass surfaces and the non-normal angle of incidence. In this step, the two brightest beams can be selected. B 1 (b) C (a) B 2 Figure 4-2a: The sample is arbitrarily mounted and tilt adjustments are performed to align the sample surface parallel to the goniometer axis (C) [4-3] 71

91 C (b) (a) B 1 B 2 Figure 4-2b: The sample surface is positioned parallel to the goniometer axis (C) [4-3] Step 3: Once sample surface is aligned parallel to the goniometer axis of ellipsometer (C) the next step is to ensure that the axis lies within the sample surface. In this step, a translation only is performed until the light beam reflected form one face of the sample (B 1 ) and the light beam reflected from the same face of the sample after rotation by 18 o (B 2 ) exactly coincide with one another, such that the distance z from the central goniometer axis to the selected sample surface vanishes [4-3]. A schematic of this step is shown in Figure 4-3. If this cannot be done, one can conclude that the beam does not intersect at the goniometer axis, which indicates that the alignment apertures from the manufacturer are not properly fabricated. One must then return to Step 1, translate the laser beam and all apertures on the polarization generation side and repeat Steps 2 and 3 until it becomes possible to rotate the sample and ensure that the beams before and after the 18 rotation can be superimposed. This condition can be checked by ensuring that the light beam precisely grazes the surface of the sample. 72

92 Precision obtained in this third step depends on the sensitivity of the alignment, the distance to the screen, and the size of the substrate, the latter in the case of the grazing condition. C (b) (a) z z B 1 B 2 Figure 4-3: Horizontal adjustments are performed to ensure that the sample surface coincides with the axis of rotation of the goniometer (C) [4-3] These three steps complete the sample alignment. From this procedure it can be observed that sample alignment is independent of alignment of the light source as long as the collimated beam from that source intersects the goniometer axis. Thus the glass sample aligned by this method can be used for aligning other components confidently. Step 4: A final step is required for aligning the light source, for example the laser in this case, so that its beam is perpendicular to the aligned sample. In this step, the two apertures on the polarization generation arm must be removed. The light from the laser is reflected by the glass sample back onto itself. If the laser is not perpendicular to the sample, the reflected light beam forms a spot that can be visualized on a screen at the output end of the laser head. The screen incorporates an aperture that allows the incident 73

93 beam to reach the sample. In this step, the sample is rotated so as to be normal to the beam and ensure that the light beam returns through the aperture on the screen. If this condition cannot achieved, then it means that the reference apertures form a line that, although intersects the vertical goniometer axis, is not precisely perpendicular to it. In this case the laser is tilted to bring about this condition. Because the tilt may translate the beam as well, however, Steps 1-4 must be iterated until the two conditions on the incident beam are met, namely that it passes through the goniometer axis and can be made normal to the surface of the properly aligned sample. Once these two conditions are met, then the two apertures on the polarization generation arm can be returned to their proper positions, and the apertures on the polarization detection arm can be placed in their proper positions with respect to the beam in the straight through configuration. 4.3 Collimator alignment As explained in Chapter 2, the source collimator consists of a lens and an achromatic doublet with pinhole between them. The laser, properly aligned with respect to sample, is also used for collimator alignment. Precise collimator alignment is a critical requirement because it determines the degree to which the light beam from the broadband source, in this case a xenon lamp, is perpendicular to the faces of all the optical components used in the ellipsometer. The lens and achromatic doublet used in the source collimator each have the property of a convex lens with a defined focal length. The focal length of the first lens is denoted f L and that of the achromatic doublet is denoted f A. The focal lengths f L and f A at the laser wavelength are measured before alignment. The distance from the lens to the 74

94 pinhole is adjusted according to the focal length of the lens f L such that the beam from the alignment laser converges at the pinhole. The appropriate equation is given by (4.1) f x x L i o where x o is the distance from a general source to the center of lens and x i is the distance from the center of lens to the pinhole as shown in Figure 4-4. A laser beam is collimated meaning that x o ; thus, x i is set at f L when the laser is used Because a collimated output beam is sought, the achromatic doublet is placed similarly such that the distance from the pinhole to the achromatic doublet is equal to the focal length of the doublet f A. This latter condition is set irrespective of the nature of the source, e.g. alignment laser, multi-wavelength calibration source, or the broadband xenon source used for ellipsometry A test of the alignment of the collimator is an undeviated laser beam with maximum output when the device is placed on the polarization generation arm. The pinhole is inserted first without the two lenses. The pinhole is adjusted to allow the maximum laser irradiance to pass though it. Next, the achromatic doublet is added, ensuring that the recollimated laser beam continues to travel though all the apertures. Finally, the first lens is added, making sure that the laser beam focuses exactly onto the pinhole. The positioning of the first lens along the beam path, i.e. the value of x o in this last step, will depend on the source as well as the collection optics used for the source. In general, for the broadband xenon source, the first lens of the collimator does not cover a sufficient solid angle; as a result additional collection optics are needed. 75

95 Lens Pinhole Achromatic doublet Source x o x i Focal length of achromatic doublet (f A ) Collimated beam Figure 4-4: Schematic of an aligned collimator. 4.4 Wavelength calibration Wavelength calibration is defined as the process through which wavelength values are assigned to each numbered pixel of the detector. The laser used for collimator alignment should be replaced by a multi-wavelength source, such as a low pressure Hg lamp designed specifically for performing wavelength calibrations. The spectrographdetector module is mounted onto an alignment stage, providing multiple degrees of freedom similar to the laser alignment stage described previously. An entrance slit of minimum width.1 mm is inserted at the front of the spectrograph when performing wavelength calibration. Alignment of the spectrograph itself involves ensuring that the collimated light source strikes the planar entrance slit at its vertical and horizontal center. Also tilt and rotation degrees of freedom ensure the spectrograph slit and body are normal to the 76

96 incident beam. The detector also must be aligned relative to the spectrograph, a procedure that can be performed using the Hg calibration lamp as a source with the spectrograph-detector module placed on the end of the polarization detection arm. The position of the detector is adjusted relative to the spectrograph such that the plane of the photodiode array is perpendicular to the plane of the incidence and the linear direction of the array lies in the plane of incidence. These conditions are defined when the maximum irradiance is registered on the illuminated pixels. When the detection system achromatic objective is added for focusing the collimated ellipsometry beam onto the slits of the spectrograph, it is inserted and aligned ensuring that the collimated beam is not deviated, and the focused beam remains at the geometric center of the slits. If the f-number of the achromatic objective lens used for beam focusing is chosen to match the f-number of the spectrograph, then a diverging broad-band light beam internal to the spectrograph from the slit exactly fills the internal optics, including the mirror, grating, and active area of the detector. This is also a test of the spectrograph slit alignment normal to the incident beam. If the incident cone of light focused on the slit exceeds the f-number matching condition, then the beam internal to the spectrograph will overfill the internal optics, leading to detrimental stray light within the spectrograph. A low pressure mercury lamp, which provides six distinct peaks in its irradiance spectrum versus wavelength, is used for performing the wavelength calibration. The six calibrated wavelength lines are located at nm, nm, nm, nm, nm, and nm [4-5]. After alignment of the spectrograph-detector module and the achromatic objective, light from the mercury lamp is focused onto the slit 77

97 of spectrograph, diffracted by a grating into a spectrum of component wavelengths, and directed onto the active area of the detector. The pixel numbers at which the six spectral lines reach their maxima are noted and are associated with the corresponding wavelengths. To obtain the wavelength positions of all other pixels in the detector, a linear fit of wavelength versus pixel number is performed as shown in Fig The fitting equation provides the wavelength positions of all other pixels. The fitting equation obtained in the calibration is given by k = * k nm (4.2) where k is the pixel number. Thus, k ranges from 1 to 124 and k is the wavelength position of the k th pixel. 78

98 Figure 4-5: Plot showing the linear fit of the relationship between pixel number and wavelength. Six mercury lamp wavelengths are the data points that were fit, indicated by the markers that lie on the best fit line. In another possible approach, which also works if the Hg source is sufficiently bright, the focusing object is not added in front of the spectrograph until after the wavelength calibration. When the wavelength calibration is completed, the objective is attached to an optical mount and placed in front of spectrograph slit. The collimated light from the polarization generating arm is converged by the objective to a point at the center of the slit as described previously. Alignment of the objective is further verified by ensuring that no change occurs in the pixel positions of the Hg source spectral lines at the detector. 79

99 4.5 Alignment of polarizer Alignment of a polarizer can be performed in two steps. First, the two parallel faces of the polarizer should be made exactly perpendicular to axis of rotation of the polarizer. After this first condition is met, then the face of the polarizer should be made perpendicular to the optical axis of the instrument. The first step can be achieved by rotating the polarizer using its rotator while reflecting the beam from the entrance face of the polarizer. The reflected beam is viewed on a screen, and the polarizer is adjusted in its mount with the goal being to eliminate the rotation of the laser beam on the screen [4-4]. The quality of the polarizer can be assessed by performing this check on the exit face of the polarizer, as well; alignment of the two faces must be possible simultaneously. The second polarizer alignment step can be achieved by making tilt adjustments in the position of the rotator until the light reflected from the face of polarizer is reflected back through the aperture used to define the incident beam. A screen can also be placed at the laser head which itself has a narrow aperture allowing only the beam to pass through. If the polarizer is slightly misaligned with respect to the alignment laser, it may be noticeable in reflection through the apertures and on the screen. This latter method enhances the sensitivity of the second step. 4.6 Order sorting filters When using a grating spectrograph for dispersing the broadband light into its component wavelengths, multiple orders in the diffracted spectrum are produced. The second order overlaps the first, i.e. light of wavelength in order m=1 is diffracted in the same direction as light of wavelength /2 in order m=2 [4-6]. The aim of inserting the 8

100 order sorting filter, which is placed directly in front of the detector, is to remove the overlapping second and possibly higher order spectra from the first order spectrum. First, one must decide what type of filter is to be selected, based on the spectral range of the instrument. One must also decide on a location within the spectrograph where the filter should be attached, for example, on top of the faceplate or directly on top of the detector element. The latter placement requires removing the faceplate. In fact, removing the faceplate breaks the vacuum seal in the detector, which may cause condensation on top of the detector element. Thus, in this ellipsometer, the filter is attached on top of the faceplate of the detector. Before inserting the order sorting filter, its optimum location must be established. This determines the pixels that the filter must cover, and the pixels that can be left uncovered Positioning of order sorting filters The wavelength range of the spectrograph used in this research is 19 nm to 8 nm. Since a xenon lamp is used as the broadband source, this reduces the actual wavelength range from ~ 23 nm to 8 nm. The free spectral range F of a diffraction grating is defined as the range of wavelengths for a given spectral order over which superposition of light from adjacent orders does not occur [4-6]. Hence the free spectral range of the grating in first order is from 23 nm to 46 nm. Thus, at the first order wavelengths of 46 nm and longer, the second order light of 23 nm and longer begins to overlap the first order spectrum. In fact, the second order wavelengths from 23 to 4 nm are superimposed on the first order wavelengths from 46 to 8 nm, which leads to corruption of the data. The third order spectrum superimposes on the first two orders over the range from 69 to 8 nm; however, typically the grating efficiency in third 81

101 order will be low. In any case, the filter selection made to eliminate the second order will automatically eliminate third order, which spans the wavelengths from 23 to 267 nm. The goal then is to insert the order sorting filter to remove the second order spectrum from 23 nm to 4 nm so that it no longer overlaps the first order from 46 nm to 8 nm. This is accomplished by means of two filters Selection of order sorting filters Using the V-VASE ellipsometer supplied by J.A. Woollam Company, transmission measurements were performed on thin film plastic Roscolux filters of varying wavelength transmission ranges, available from the manufacturer, Rosco Laboratories, Inc. Based on those transmission measurements, Filter # Clear and Filter #312 Canary were selected. The transmission spectra of the selected filters are shown in Figure 4-6. From Figure 4-6 it can be observed that Filter # absorbs all wavelengths shorter than 28 nm and transmits wavelengths longer than 35 nm, with a gradual onset in transmission between the two wavelengths. Similarly Filter #312 absorbs all wavelengths shorter than 46 nm and transmits wavelengths longer than 56 nm, with a gradual onset in transmission between the two wavelengths. When Filter # is placed over the pixels that register first order wavelengths of 46 nm and longer, the second order spectrum arising from light of wavelengths below 28 nm are cut off, no longer overlapping the first order within the 46-8 nm range. Because very little light is present from the xenon source below 23 nm, the spectral range that Filter # cuts is nm. The second order range of these wavelengths is nm, and this defines the selection of 46 nm as the edge of Filter #. One 82

102 must remain concerned, however, with the light over the range 28-4 nm which appears in second order over the range 56-8 nm. To address this remaining problem, Filter #312 is placed over the pixels that register first order wavelengths of 56 nm and longer, overlapping the clear filter #. Filter #312 removes the second order spectrum arising from light of wavelengths below 46 nm, so that they no longer overlap the first order. Because of the first filter eliminating light below 28 nm, the spectral range that Filter #312 cuts is nm. The second order range of these wavelengths is nm, and this defines the selection of 56 nm as the edge of Filter #312. The third order spectra that these two filters cut are nm for Filter # and nm for Filter #312. Thus, there is no concern that third order diffraction, even though it is very weak, will affect the data. Finally, it should be noted that by placing the filters in the specified locations as described above, the requirement presented for positioning the filters in Section is justified. 83

103 Figure 4-6: Transmission spectra of order sorting filters measured using the J.A. Woollam Company V-VASE ellipsometer Procedure for inserting order sorting filters The procedure for inserting Filter #312 Canary is explained here; a similar procedure can be followed for insertion of Filter # Clear. 1) The number of the pixel registering 56 nm is noted from the wavelength calibration; this pixel is number ) Although the spectrograph and detector are separated, their operation is maintained. 3) The low pressure mercury lamp is placed directly in front of the spectrograph. 84

104 4) The position of the detector is moved such that one of the three mercury lamp peaks at a wavelength less than 46 nm falls on the 526 th pixel. In this case, it is the nm peak of the mercury lamp that falls on the desired pixel. 5) The filter is attached to a micrometer and slowly inserted from the region of higher wavelength moving towards that of lower wavelength on top of faceplate. 6) Because the reference mercury peak of nm is detected at the 526 th pixel, and because the filter cuts out all wavelengths less than 46 nm, the peak is blocked when the filter just reaches the pixel. Then the filter is attached at that position to the rim of the faceplate by adhesive tape and the remaining parts of the filter are cut away. 7) After this step, the spectrograph and detector are reconnected, aligned, and wavelength calibration is performed again according to the method explained above. The irradiance spectrum of the xenon lamp after insertion of the order sorting filters is shown in Figure 4-7. Inserting the order sorting filter on top of detector faceplate has its disadvantages. First, multiple reflections between the faceplate and filter can occur which may generate stray light. Second, obtaining precise alignment so that the filter edge is parallel to the lines that define the pixel width is a challenge. The goal is to ensure that the edge of the filter will disrupt the signal of only one pixel, at the maximum; however, multiple pixels can in fact be affected. 85

105 Figure 4-7: Irradiance spectrum of the xenon lamp after insertion of order sorting filters. 4.7 Pulse divider circuit design for single rotating compensator ellipsometer In the single rotating compensator mode, the biplate compensator is attached to the slave motor which rotates at 6 Hz. In this mode, the detector is scanned 8 times per optical cycle. The pulse divider circuit for this mode is constructed such that it divides the 124 encoder pulses per mechanical cycle by 64 to produce the output synchronization pulses which are transmitted to the input of the detector during data acquisition. In other words, for every 64 pulses from the encoder of the slave motor, a single output pulse is generated. Hence 8 synchronization pulses are produced per optical cycle and 16 pulses are produced per mechanical cycle. 86

106 In the single rotating compensator mode, the time period for a single mechanical cycle of the motor is 1/(6 s -1 ) =.167 s. The time period for a single optical cycle is (.167 s)/2 = ms. Since there are 8 pulses produced per optical cycle, then the time between successive synchronization pulses is (83.33 ms)/8 = 1.42 ms. When 124 individual pixels are uniformly grouped by 8 for detector scanning, the result is 128 grouped pixels. As described in Chapter 2, the time required for scanning a single group of 8 pixel is 35 s. Hence, the time required for scanning 128 grouped pixels is 128 x 35 s = 4.48 ms. It can be observed that, out of the 1.42 ms fixed exposure time, the detector scanning time takes up 4.48 ms. Hence, scanning can be performed easily in the time available; thus, there is no chance that synchronization pulses will be missed. 4.8 Rotating polarizer configuration Before developing the single rotating compensator configuration using the biplate compensator, it is necessary to correct for systematic errors present in the system. Since the rotating polarizer configuration is simpler than the rotating compensator configuration, it is much easier to determine and correct for systematic errors in this mode. In order to correct for systematic errors, the polarizer is rotated at the same speed as the compensator when the latter is set up in the single rotating compensator mode [4-7]. Thus, the polarizer is attached to the slave motor and aligned using the procedure described previously. The detector can be scanned 8 times per optical cycle of the rotating polarizer, hence the same pulse divider circuit is used for the rotating polarizer configuration as was used for the single rotating compensator configuration. This circuit produces the synchronization pulses for the detector. The overall program, written in 87

107 Labwindows/CVI, controls grouping and scanning of pixels in the same configuration as in the single rotating compensator mode. Either mode 8 or mode 9 can be selected for data acquisition based number of optical cycles required Data collection principles For a polarizer rotating at a frequency of the irradiance at any pixel k exhibits the waveform I' (t) I (1 ' cos2t ' sin2 t). (4.3) ' k k k k Here k, k are the normalized 2 Fourier coefficients of the irradiance waveform, uncorrected for the absolute phase of polarizer rotation. The integer index k is the pixel number with the value of k ranging from 1 to 128. The Fourier coefficients in Equation (4.3) can be obtained according to [4-4] k = (S 1k + S 2k S 3k S 4k S 5k S 6k + S 7k + S 8k )/ (2I k), (4.4) k = (S 1k + S 2k + S 3k + S 4k S 5k S 6k S 7k S 8k )/ (2I k), (4.5) where I k = (S 1k + S 2 k + S 3k + S 4k + S 5k + S 6k + S 7k + S 8k )/. (4.6) The photodiode array is an integrating detector, and S jk, j= 1,..., 8, represent eight consecutive readout scans performed over a single optical cycle, each with an identical exposure time of /8 [4-7] In the straight through configuration, i.e. in transmission mode without any sample in place, the residual function of the error free system, given by [4-8] R 1 ( ' ' ), (4.7) 2 2 1/2 k k k 88

108 should vanish for all pixels, k. Deviations of R k from zero are attributed to various errors present in the ellipsometer Correction of systematic errors Systematic errors inherent in the ellipsometry system are of several types and separate procedures must be followed to correct each systematic error Dark current and ambient corrections The acquired data must be corrected for the ambient light present in the laboratory as well as for the dark current. The dark current arises from thermally generated charge carriers and is observed even when there is no light incident on the detector. Ambient light and dark current corrections are performed simultaneously by closing the shutter and collecting the data in the same grouping mode and with the same of number of optical cycles as is used for data collection with the shutter open. The data collected with the shutter closed must be subtracted from the data collected with the shutter open in order to correct for ambient light and dark current Image persistence correction The detector used in the system is not ideal; image persistence is another problem that occurs in addition to dark current. Image persistence present in the detector can be observed using a constant irradiance source focused on the entrance slit of the spectrograph. In this configuration, a high speed shutter with a full on off time less than the detector exposure time is used to block the slit. It is found that some counts persist from the previous readout to affect the next readout, since the shutter is known to be closed over the full exposure time of the second readout. Thus, these counts can be identified as image persistence. This image persistence signal is cleared as a result of the 89

109 second readout and with the shutter closed no readouts after the second one register measurable counts. In other words, for some pixels, two successive exposures are identified such that the first occurs when the shutter is open during most of the exposure time and thus scanning reads a high count level, and then the second occurs after the shutter has been completely closed. The scanning associated with this second exposure leads to residual counts which arise from the incomplete readout of counts accumulated during the previous scan [4-7]. The image persistence correction factor (IPCF) is determined from a collection of two such successive readouts, as described next. A program for determining the IPCF has been written in Labwindows/CVI, such that data from individual optical cycles are collected without averaging, and during a given exposure the shutter is closed. Even when the shutter is closed over the full exposure time, some fraction of the counts measured in the previous scan remain unread and are left over for collection in the succeeding scans. In order to obtain the IPCF, the data which are left over (obtained when the shutter is fully closed during the full exposure) are divided by the data collected in the previous scan during which the shutter is open for at least a fraction of the exposure time [4-8]. The resulting IPCF is found to depend on the pixel number, indicated by IPCF k. Then data correction can be performed for all 128 pixels according to S jk, c = S jr + IPCF k *S jk,r IPCF k *S (j-1)k,r, (4.8) where S jk,c are the data corrected for image persistence of pixel k in scan j, S jk,r are the data uncorrected for image persistence of pixel k in scan j, and (IPCF) k is the IPCF for the k th pixel. A plot of IPCF as a function of pixel number is shown in Figure

110 The IPCF can also be determined independently based on a calculation assuming that the entire error in the measured residual function is due to image persistence. The corresponding IPCF for each pixel is determined by numerical inversion of the equation that results when setting the residual function to zero. In other words, the correct IPCF value is that which leads to a zero residual function [4-8]. The IPCF obtained in this calculation is plotted along with the experimental IPCF in Figure 4-8. From the plot, one can observe a close match between the IPCF obtained by the experimental method and that by the residual function calculation. Hence it can be concluded that the dominant systematic error that distorts the residual function is image persistence. It should be noted that systematic errors exist that do not appear in the residual function. For example, spectrograph stray light has the same polarization characteristics as the light that follows the specified optical path, and thus does not affect the residual function. Stray light does distort the ellipsometric spectra of samples with a spectrally dependent polarization change upon reflection. 91

111 Figure 4-8: Image persistence correction factor (IPCF) obtained experimentally and by calculation from the residual function, plotted as a function of pixel group number. The residual function spectrum R k, determined using Eq. (4-7) after incorporating the IPCF in the scanned data, is shown in Figure 4-9. The results in Figure 4-9 are comparable to those published elsewhere [4-7]. 92

112 Figure 4-9: Photon energy dependence of the residual function obtained after application of the IPCF correction to acquired data. 93

113 Chapter 5 Single Rotating Compensator Multichannel Ellipsometer 5.1 Introduction Two designs are possible for the single rotating compensator multichannel ellipsometer. One design places the compensator before the sample and the second places it after the sample. In this research, however, the plan has involved the incorporation of an achromatic rotating compensator before the sample due to the reproducible alignment possible in that location. As a result, the biplate rotating compensator must be mounted after the sample holder in the single rotating compensator configuration. Thus, the single rotating compensator multichannel ellipsometer using the biplate retarder has been constructed in the [P (fixed polarizer)]-[s (sample)]-[c r (rotating compensator)]-[a (fixed analyzer)] configuration, or the PSC r A configuration, for short. The schematic of this single rotating compensator ellipsometer configuration is shown in Figure 5-1. This ellipsometer has been constructed as an intermediate step in the development of the dual rotating compensator multichannel ellipsometer. High performance of the ellipsometer in this configuration, if possible, will ensure that the dual 94

114 rotating compensator ellipsometer can be constructed with greater confidence using an advanced achromatic compensator in front of the sample. The rotating-compensator ellipsometer has been developed in order to overcome the inability of the rotating polarizer/analyzer ellipsometer configurations to provide accurate characterization of samples that generate reflected polarized light with low ellipticity angle. Not only does it provide high precision and accuracy spectra in the ellipsometric angle of the sample over the full range from 18 o through o to 18 o, but it also provides the degree of polarization of the reflected wave. Thus, the instrument can be used to characterize samples exhibiting non-uniformities that are unrecognized in advance and lead to depolarization of the reflected wave. The rotating compensator ellipsometer provides all four components of the unnormalized Stokes vector, which can be defined in general as: S1 1 S p cos2q cos2 S 2 I S 3 psin2qcos2 S4 psin2, (5.1) where I is the irradiance; Q is the tilt angle, defined over the range 9 o < Q 9 o ; and is ellipticity angle, given by 1 b o o tan ( ). In the definition of, b and a a are the semi-minor and semi-major axes of the ellipse, respectively. Finally in Eq. (5.1), p is the degree of polarization defined in Eq. (2.6). In fact, p spans the allowable range of p 1, where p = represents a completely unpolarized state and p=1 represents a completely polarized one. Both represent ideal cases; in the experiment, however, p always lies between and 1, describing the partially polarized state. With the rotating 95

115 compensator ellipsometer, all four Stokes parameters describing the polarization ellipse of the reflected wave can be obtained without any problem. In addition, the sign of can be obtained, which determines the sign of, a measurement capability not possible with the rotating polarizer and rotating analyzer instruments. Thin film/ Substrate P= P P S p t s s F C' (t) = t CS p s t A= A AS Xe light source Collimating optics Fixed polarizer Rotating biplate compensator Fixed Analyzer p Spectrograph with photodiode array Figure 5-1: Schematic of the single rotating compensator multichannel ellipsometer, indicating the reference axes and azimuthal angles that describe the three optical elements. 5.2 Instrument design and theory The single rotating compensator ellipsometer system in the PC r A configuration (straight through mode without a sample) consists of the following elements: (i) a 75 W xenon lamp; (ii) a collimator; (iii) a MgF 2 fixed Rochon polarizer with stepper motor controlled rotator; (iv) a MgF 2 biplate compensator under continuous rotation by the slave motor at 6 Hz; (v) a MgF 2 fixed Rochon analyzer with stepper motor controlled 96

116 rotator; (vi) a grating spectrograph coupled to a 124 pixel silicon photodiode array detector with controller. A pulse dividing circuit has been constructed as described in Chapter 4 to convert motor encoder pulses to the frequency needed to trigger detector read-out. The data acquisition program written in Labwindows/CVI is shown in Appendix D. The output irradiance at each pixel group of the array has the following time dependence [5-4]. I(t) = I o (1+ 2 cos2 C t + 2 sin2 C t+ 4 cos4 C t+ 4 sin4 C t), (5.2) where I o is the time averaged irradiance at the pixel group and C =/T C is the angular frequency of compensator mechanical rotation. T C is the optical period of compensator rotation which is one-half of the mechanical period. In Eq. (5.2), [( m, m ); m=2,4] are the normalized four Fourier coefficient spectra measured in the experiment. These spectra can be related to theoretical coefficients by m = m cos (mc S ) + m sin (mc S ) ; m= 2,4, (5.3a) m = m sin (mc S ) + m cos (mc S ) ; m= 2,4, (5.3b) which are used in expressions that describe the polarization state of light reflected from the sample. In these expressions C S is the azimuthal angle of the compensator fast axis with respect to plane of incidence at the onset of data collection, which defines time t =. The relationships of these theoretical coefficients to the Stokes parameters (Q,, p) are given by following equations [5-5] ' 2iA [ ipsinc sin2e ] 2 i2, (5.4a) 97

117 psin [ i 4 4 / 2]cos 2e 2 2i(A' Q) C, (5.4b) 1 [p cos ( / 2)cos 2cos 2(A Q)], (5.4c) 2 ' C where C is the retardance of biplate compensator and A= AA S is the azimuthal angle of the transmission axis of the analyzer with respect to plane of polarization. To determine the experimental spectra in the Fourier coefficients ( m, m ), the photodiode array is read 8 times per optical cycle. Thus, the readout for a given pixel group is the integral of the irradiance I(t) over one eighth of its optical cycle [5-5] as follows: S j jt /8 C I(t)dt, (5.5a) (j1)t C /8 jt 2 C /8 ' ' ' Sj I 1 [ (j 1)T 2ncos2nct2nsin2nct] ; j= (5.5b) C /8 n1 Equations (5.5b) can be inverted to yield the unknown coefficients as follows [5-1]: 2 = (S 1 + S 2 S 3 S 4 S 5 S 6 + S 7 + S 8 )/ 2I, 2 = (S 1 + S 2 + S 3 + S 4 S 5 S 6 S 7 S 8 )/ 2I, 4 = (S 1 S 2 S 3 + S 4 + S 5 S 6 S 7 + S 8 )/ 2I, 4 = (S 1 + S 2 S 3 S 4 + S 5 + S 6 S 7 S 8 )/ 2I, I = (S 1 + S 2 + S 3 + S 4 + S 5 + S 6 + S 7 + S 8 )/, (5.6a) (5.6b) (5.6c) (5.6d) (5.6e) The uncorrected Fourier coefficients determined from Eq. (5.6) are in turn converted to Fourier coefficients corrected for the compensator phase by using Eq. (5.3). 98

118 5.3 Internal alignment of biplate compensator Before using the MgF 2 biplate compensator in the ellipsometer, it is necessary to perform an internal alignment of the two plates. Such a procedure is necessary if it was not performed carefully in advance by the manufacturer. Internal alignment of the biplates can be defined as the process by which the individual plates of the biplate are oriented with respect to one another such that their fast axes are perpendicular. This is a precise alignment procedure; hence, it must be performed in a high accuracy, high precision ellipsometer. A multichannel ellipsometer in the PC r SA configuration (J.A. Woollam Company M-2 ) mounted on an ex-situ base was used in performing the internal alignment of the biplate compensator. A schematic of the alignment error exhibited by a biplate compensator is shown in Figure 5-2. From Figure 5-2, C is the misalignment angle, which is assumed to be shared equally between the two plates. S 1 F 1 C/2 C/2 S 2 Plate 1 Plate 2 F 2 Figure 5-2: Schematic of misaligned biplate compensator. 99

119 The objective of this alignment procedure is to ensure that the misalignment angle C vanishes. A first order calculation of the effect of a small misalignment error C (in radians) has been performed. The Jones matrix of misaligned compensator is given by [5-2] C 1 C C1 C 2 e 2 2 e 2 o o T R R(9 ) R( 9 ) R R R i 2 i 1, (5.7) where R(C/2) are the rotation matrices for the angles C/2 and R(9) are the corresponding matrices for rotations by (9). In Eq. (5.7), 1 and 2 are the retardances of first and second plates, respectively. The right side of Equation (5.7) can be multiplied to obtain a Jones matrix of the form T e i 2 1 C2 1 2 C 1 C1 C 1 2 C C C, (5.8) where i( 12) i C e 1 C1 e and C2 i e 2 Equation (5.8) reduces to the proper Jones matrix in diagonal form when C=. Before performing the internal alignment using the J.A. Woollam Company M-2 ellipsometer, this instrument is calibrated using coarse and fine calibration methods in succession [5-3]. Both arms of the ellipsometer are set in the straight through configuration, its sample stage is removed, and the azimuthal angle of the polarizer is set at 45 o. The biplate compensator to be aligned is placed at the position of the sample with 1

120 its individual plates separated and attached to two individual rotators using custom made aluminum mounts. In the first step, the entire compensator assembly with coarse internal alignment of the plates is rotated incrementally and an anisotropic analysis of each data set is performed. In this procedure, the angle of the assembly is found at which the off-diagonal terms are minimized. At this angle, either the fast or slow axis of the biplate lies within the reference plane of the instrument. In the second step, fine internal alignment of the plates is performed. In this procedure, one of the plates is adjusted with respect to the other in.1 o increments, and this process is repeated until the off-diagonal elements of the Jones matrix of the biplate compensator sample vanish. Iteration of the first and second steps may be needed. The principle of this procedure is as follows. First, for a compensator with perfect internal alignment (C = ), the off-diagonal elements of the Jones matrix are ½sin2C'(1 C ), where C' is the angle of the biplate fast axis relative to the reference plane. In the first step, these terms are eliminated. In fact, any Jones matrix element terms in C neno d 2i e are slowly varying with wavelength due to the small factor d in the phase. In contrast, the terms C1 and C2 appearing only in the off-diagonal Jones matrix elements in the first order expansion are very rapidly varying with wavelength, and these terms remain after the first alignment step. Second order terms in C introduce C1 and C2 into the diagonal Jones matrix elements, giving rise to weak, but rapid variations in the apparent retardance c and the dichroic angle C. These oscillations appear most clearly in C owing to its wavelength independence in the absence of errors [5-4]. Figure 11

121 5-3 clearly shows those high frequency oscillations before the internal alignment. The oscillations are eliminated after a successful alignment, and the small off-diagonal terms in the Jones matrix disappear under these conditions, as well. As a summary of the compensator internal alignment data, Figure 5.3 shows a comparison between C for a misaligned compensator and for the same compensator, but after internal alignment. Also shown for comparison are the results of straight through measurements without the compensator in place. Figure 5-3: The dichroic angle C for a zero order MgF 2 biplate compensator measured using a J.A. Woollam Company M2 ellipsometer. The results in Fig. 5-3 demonstrate that this MgF 2 compensator is nearly ideal, exhibiting no significant dichroic behavior. In fact the residual variations in C in Fig. 5-3 from 45 appear to be generated by the measurement ellipsometer itself, not by the aligned 12

122 compensator. After the alignment procedure is completed, the relative orientations of two plates are carefully fixed using external screws. 5.4 Calibration of single rotating compensator ellipsometer Calibration of the single rotating compensator ellipsometer developed in this Thesis research is the most important step before data reduction. The sequence of calibration procedures for the single rotating compensator ellipsometer involves (i) measurement of retardance spectrum ( c ) of the biplate compensator, and (ii) determination of the polarizer offset angle (P S ), analyzer offset angle (A S ), and phase angle of the compensator fast axis (C S ), the latter defined relative to the reference plane at the beginning of data acquisition, i.e. at the onset of the S 1 integration for the given pixel Retardance calibration Measurement of the retardance of the aligned biplate compensator in the single rotating compensator multichannel ellipsometer (PC r A) was performed in the straight through configuration after a dc irradiance function calibration that determines P'A'. Initially, the transmission axis of the analyzer is fixed, and an irradiance function calibration is performed with P=PP S in the vicinity of A=AA S and with the rotating compensator removed from the instrument [5-6]. The angles P and A are the nominal angular settings; P S and A S are corrections to these settings to give the true angles P' and A' of the polarizer and analyzer transmission axes, measured with respect to a common reference plane. This reference plane is formed from the two reference axes and also contains the collinear optic axes of the elements. In the absence of a sample, it is reasonable to select the analyzer transmission axis A' as defining the reference plane. The 13

123 irradiance function calibration involves measuring the dc irradiance level at the detector I (P) for several closely spaced polarizer settings P such that the transmission axes of the polarizer and analyzer are nearly parallel. According to Malus s law, in the absence of instrument errors, I(P') = I cos 2 (P'A') = ½I [1+cos2(P'A')] I [1 (P'A') 2 ], the latter approximation valid when P'A' (with P' and A' in radians). Thus, I (P) exhibits a maximum when PA=, which enables identification of the angle P'A', as is needed for the retardance calibration. Based on the dc irradiance function calibration, the polarizer transmission axis is kept almost parallel to analyzer transmission axis with a known offset angle of P'A' [5-6]. Finally, the compensator is added to the slave motor, which rotates continuously at 6 Hz, and aligned with respect to the ellipsometer while maintaining the straight through configuration. Data acquisition is performed and the Fourier coefficients are calculated based on the above Eqs. (5.6). In the straight through configuration the 2 C ac Fourier coefficients ( 2, 2 ) vanish, whereas the 4 C ac Fourier coefficients ( 4, 4 ) yield the compensator retardance in accordance with the expression ' 1 B 1 4 C 2cos ' ' ' 1B4 cos2(p A) 1/2, (5.9) where B 4 ( ) 1/2. The retardance C from this expression is plotted as a function of photon energy in Fig

124 Figure 5-4: (a) Measured and best-fit retardance spectra for the internally aligned biplate compensator; (b) difference between the experimental and best fit spectra, showing fluctuations of.15, considering the photon energy range of ev. The retardance spectrum was fitted to the polynomial equation given by [5-6] 5 k 1 C(E) (36 /1239.8)d E a k E, (5.1) k1 where a k ; k = 1,...,5, are the polynomial coefficients that serve as fitting parameters; E is the photon energy; and d is the thickness difference between the two MgF 2 plates. The units of C, d, and E in Eq. (5.1) are degrees, nanometers, and electron volts, respectively. 15

125 The thickness difference is determined independently of the fitting parameters by assuming reference data for the birefringence of MgF 2 [5-8] and fitting the experimental C (E) to extract d. In the second iteration, d is fixed at its best fit value and C (E) is fitted using Eq. (5.1). Figure 5-4 shows the experimental retardance spectrum of the internally aligned biplate compensator (circles) and the best fit to the retardardance spectrum (lines). The resulting best fit values are given by d = nm, a 1 =.1435 nm -1 ev -1, a 2 = nm -1 ev -2, a 3 =.154 nm -1 ev -3, a 4 =.4438 nm -1 ev -4, and a 5 =.456 nm -1 ev -5. The resulting quarter-wave point obtained from fit is Q = nm (E Q = 4.27 ev). The position of the quarter wave point measured using the J.A. Woollam ellipsometer is 4.29 ev. Figure 5-4b shows the difference C between the best fit and the experimental C. The associated random errors would propagate throughout the data reduction process if the experimental C (E) results were adopted; as a result, the best fit spectrum is used in subsequent data reduction. Figure 5-4b provides estimates of the systematic errors that may result when the best fit expressions for C (E) are used in place of the experimental spectra. The C spectrum obtained in this experiment is comparable to corresponding results published elsewhere [5-6] Offset angle calibration After the retardance calibration of the ellipsometer has been completed, the next step is the offset angle calibrations of the polarizer and analyzer. This calibration step is required in order to determine the corrections (P S, A S ) to the transmission axis readings (P, A) of the polarizer and analyzer, respectively. In measurements of samples, all angular measurements are made with respect to the axis defined by the intersection of the 16

126 plane of incidence of the sample with the optical element when the plane of incidence contains the optic axis of the element (which is along k of the light wave). Since this ellipsometer is operated in straight through mode without a sample, however, the reference axis can be defined by either the polarizer or the analyzer transmission axis. For the calibration of the compensator retardance described previously, the analyzer transmission axis was selected to define the reference axis for each element. In contrast, for the P S and A S calibration, the transmission axis of the polarizer (P) is selected to define the reference axis, meaning that PP S =. As a result there is no separate procedure required for determining the polarizer offset angle P S. As described elsewhere [5-1], there are many different procedures for determining analyzer offset angle A S based on nature of sample used in calibration. Different procedures may be required for strongly and weakly absorbing samples, as evaluated by the phase shift difference of light reflected from the sample. Non-absorbing, or weakly absorbing substrate samples typically exhibit near or 18, whereas strongly absorbing substrate samples exhibit near 9. Since this ellipsometer is operated without any sample at all, which implies that =, the best procedures would be those designed for weakly absorbing samples. One possible procedure is based on the measurement of B 4 ' at the reference axis of the polarizer, P'=, and also at P'=9 [5-1]. This procedure involves independent sets of measurements near the two reference polarizer angles, so thus requires rotating the polarizer through a full 9. Instead a procedure based on the irradiance function is selected, with a sensitivity that is independent of the value of, as explained in Ref. [5-1] (p. 544). This procedure is based on the measurement of the dc Fourier coefficient, or the 17

127 average irradiance, given by Eq. (5.11). When P'P S =, which occurs by definition and when the analyzer angle A is set in the neighborhood of A S, so that A', then the dependence of the dc Fourier coefficient I on the setting A is given by ' ' p C C S S I (A) r 1 cos / 2 [2cos ( / 2)(A A ) ] ;P ;A A. (5.11) The calibration proceeds from a starting point with the analyzer setting A nearly parallel to the polarizer transmission axis serving as the reference. Then the analyzer is moved by stepper motor to collect the irradiance spectra I (A) in successive A =.2 increments over a 5 range about the starting point. Then these irradiance data sets for each pixel group are fitted to a parabola in A, and A S is the point at which the the maximum of the fitted function occurs. The offset A S obtained by this procedure is shown in Fig

128 Figure 5-5: Analyzer angular offset A S plotted versus photon energy; a constant result should be obtained in the absence of instrument errors. A positive indication for the validity of these results is the very weak photon energy dependence of the offset angle. In fact, the results of Fig. 5-5 can be described as A S = for the photon energy range from 1.7 to 4. ev. In spite of these positive results, an additional (P', A') angular zone should be tested in calibration, as well. A calibration with P' = (by definition) and with a starting point of A near 9 exhibits an extremum, as well, and would provide an additional A S spectrum that could be averaged with the first, or at least applied qualitatively to evaluate the absolute accuracy of the results in Fig

129 5.4.3 Compensator phase angle calibration In the single rotating compensator configuration, partial self-calibration is possible with a sample in place since there are five dc and ac Fourier coefficients, but there are only four measured polarization state parameters that define the Stokes vector. In straight through without a sample, however, the number of dc and ac Fourier coefficients reduces to three as the 2 C coefficients vanish, thus implying no opportunity for self-calibration. Since the compensator is rotating continuously at a constant speed of 6 Hz, the angle made by the compensator fast axis at any time with respect to the reference plane is given by C(t) = t C S, where C S is the angle at time t =, defined as the onset of data acquisition. Because the zero time point of data acquisition increases linearly for increasing pixel group number owing to the sequential readout of pixel groups of the array detector, the angle C S should then be a linear function of pixel group number. The calibration procedure used for determining C S is described in detail in Ref. [5-1] (p. 54). This procedure applies the phase function definition and its small angle expansion in P' given by 4 (P) = [arctan( 4 / 4 )]/2 2C S + (A A S ) + cot cos (P P S ) ; P P S. (5.12a) (5.12b) In Eqs. (5.12), in straight through mode with the polarizer transmission axis being defined as the reference axis, then PP S = and cot cos (P P S ) =. As a result, C S can be determined simply from C S = [ 4 (P) (A A S )]/2. (5.13) The results for C S obtained using this procedure and the fit to a linear function of pixel group number are shown in Fig. 5.6(a)., The best fit equation is given by 11

130 C S = (.75238) k; k= 1,...,128, (5.14) where k is the pixel group number. The difference between C S and its linear fit is shown in Fig. 5.6(b). Figure 5-6: (a) Rotating compensator phase angle C S obtained in calibration and plotted as a function of pixel group number N. The solid line is the best fit linear relationship from N = 19 (3.78 ev) to N = 94 (1.77 ev) ; (b) difference between the experimental C S data obtained in calibration and its linear fit. The results of Fig. 5-6 show nearly linear behavior of C S. The systematic deviations from linear behavior amount to ~.12, and the noise level, which increases with pixel number reaches ~.8 at the highest pixel numbers plotted in Fig Because C S is 111

131 angle of the compensator at t= and because this value increases from ~ 6 to ~ over the 2-1 pixel group range, the resulting negative value of the slope in Fig. 5-6 indicates a counter-clockwise rotation of the compensator when the transmitted beam travels toward the observer. In reaching this conclusion, it must also be understood that the detector is read out starting with lowest pixel group number first. The slope of C S can also be calculated theoretically based on the known read-out time of each pixel group. The compensator rotates at 6 Hz; hence the mechanical period is 1/6 s = ms and the optical period is (166.67)/2 ms = ms. Thus, in ms the compensator rotates by 18. As a result, the angle swept out by the compensator in a single pixel group readout time of 35 s (assuming 124 pixels in the diode array, grouped uniformly by 8) is.756/group which is close to the slope obtained in the experiment (.752/group). The difference between the measured and predicted C S slopes of.4/group would lead to a difference in C S values of.25 over the 128 pixel range; however, it is unclear whether the slope difference is due to a difference in the motor frequency relative to the nominal value, or to a deviation in the read out time from 35 μs. When a sample is added, the C S calibration can be performed two different ways, from the ( 2 ', 2 ') coefficients in self-calibration, as well as from the method from the ( ', 4 ') coefficients used here. The former approach, however, has not been attempted in this Thesis research. 5.5 Data reduction and results After completion of the calibration steps, data acquisition was performed to determine the polarization state and the sample parameters. The data acquisition was 112

132 performed in the straight through mode; hence, there is no sample between the polarizer and the rotating compensator. The polarization state measured by the rotatingcompensator/analyzer polarization state detector is that generated by the polarizer. This state is characterized by the Stokes vector angles Q = P' =, =. and by a degree of polarization p=1. Since no polarization state change occurs between the polarization generator and polarization detector, the normalized Mueller matrix m of the non-existent sample is the identity matrix. This implies that = 45, which generates m 12 = m 21 =, and that =, which generates m 34 = m 43 =. Data reduction equations that incorporate the calibration parameters are used to convert the measured (primed) Fourier coefficients first to theoretical (unprimed) Fourier coefficients, then to the polarization parameters (Q,, p), and finally to the sample parameters (, ). Again, since there is no physical sample present in the beam path, there is no change in (Q,, p) of the ellipse as the beam passes from the polarization generation arm to polarization detection arm. Thus, in summary, the expected results are (Q,, p) = (,, 1) and () = (45, ). To review, the Stokes vector parameters can be defined by (i) Q (the tilt of the polarization ellipse defined with respect to a reference plane), (ii) (the ellipticity angle, which defines the shape of the ellipse), and (iii) p (the degree of polarization of the light beam). The directly measured d.c. normalized cosine and sine coefficients ( 2n, 2n ); n=1,2 do not take into account the compensator azimuthal phase angle. As a result, these coefficients must be subjected to a rotation transformation that provides the phase angle adjusted coefficients ( 2n, 2n ); n=1,2. The phase angle adjusted Fourier coefficients can be obtained from Eqs. (5.3). The best fit linear function of C S obtained during calibration is used in the determination of these phase corrected Fourier coefficients. 113

133 The Stokes vector polarization parameters (Q,, p) can be deduced by inverting Eq. (5-4) to obtain [5-7] Q (1/ 2) tan ( / ) A (5.15) 1 ' 4 4 = ½ tan 1 ( B 2 /2 B 4 ) tan(/2) (5.16a) (1/ 2) tan [ cos 2(A Q) tan( / 2)] / [2 sin 2A ] (5.16b) 1 ' ' 2 4 = (1/2) tan 1 [ 2 sin2(a' +Q) tan(/2)]/[2 4 cos2a'] cos 4Q sin 4Q p cos 2cos 2(A Q)[1 (1 cos 4Q sin 4Q)cos ( / 2)] 4 4 ' (5.16c) (5.17) In Eq. (5.16a), B 2j = ( 2j 2 + 2j 2 ) 1/2 ; j=1,2, are the amplitudes of the 2j Fourier coefficients. Equations (5.16) provide three different methods for determining. Equation (5.15a) is best for determining the amplitude of, whereas Eqs. (5.16b-c) are best for determining the sign of. The parameters Q,, and p obtained from Eqs. (5.15), (5.16b), and (5.17) are shown in Fig. (5-7). These measurements were performed with the analyzer set at an angle A = 2.4, implying that A' = 1.21 and keeping in mind that the reference plane is established by P' =. As noted above, in the absence of instrument errors, the tilt angle Q of the ellipse should vanish over the entire spectrum since there is no sample to change the tilt from that generated by the polarizer. In the absence of instrument errors the ellipticity angle should also vanish and the degree of polarization should be unity for the same reason. The tilt angle Q obtained from the ratio of the 4 C Fourier coefficients shows oscillations of ~ 1 in peak-to-peak amplitude at the lowest energy with an average value spanning a range of.6 versus photon energy. These variations represent errors of 114

134 (.3-.5)/18 (.2-.3)%. The ellipticity angle, although nearly constant versus energy with no significant oscillations, shows an error of.25, or.25/9.3% The degree of polarization p also shows variations, including an average offset on the order of ~.1 from unity with variations on the order.1 or.1%. The results in Fig. 5-7 in general show poor overall instrument performance as a result of this Thesis research; the errors are at least a factor of 5 higher than the highest quality multichannel instruments. The behavior in Q suggests an optical interference effect, possibly due to multiple reflections between the biplates of the compensator. These are enhanced at lower energies possibly because of the decrease in the retardance with decreasing energy well away from the optimum value of 9 required for highest instrument performance. Possible approaches for improvement include introducing a slight external misalignment of the compensator so that multiple parallel reflections within the device are suppressed. Other possible errors of this form may occur due to reflections within the detection system including the order sorting filters and the detector faceplate; however, such detection system errors generally only appear in the polarization parameters when occurring with other errors, such as detector non-linearity. The fact that the residual function in the rotating polarizer configuration shows apparently accurate results suggests that the errors in Fig. 5-7 arise from the compensator. In fact, a possible test is to extract Q and in the rotating polarizer ellipsometer configuration to see if improved results are possible when a one-to-one comparison is made comparing rotating compensator and polarizer configurations. Improvements may also occur through the purchase of a new compensator. The quarter wave point of the existing compensator occurs at a wavelength that is too short 115

135 for optimum performance, Q = 289 nm (E Q = 4.27 ev). A better selection for the quarter wave point of the compensator of this instrument is at the center of the photon energy range, giving Q = 413 nm (E Q = 3. ev). Figure 5-7: Spectra in the azimuth and ellipticity angles and the degree of polarization, (Q,, p), respectively, that characterize the Stokes vector of the light beam. To review, the sample ellipsometric parameters (, ) are the obtained from the ratio of Fresnel reflection coefficients r p and r s according to 116 tan e i r r p s. The angles

136 (, ) are confined to the ranges of 9 o and 18 o < 18 o. The ellipsometric parameters can be deduced in general from Q and using the equation [5-5] ' cos 2P cos 2Qcos 2 cos 2. (5.18) 1 cos 2Qcos 2 cos 2P ' The angle of can be deduced over its full range using the following two equations sin ' sin 2 (cos 2cos 2P 1) ' sin 2 sin 2P, (5.19) cos ' cos 2sin 2Q(1 cos 2cos 2P ) ' sin 2 sin 2P, (5.2) which together identify the correct quadrant of. One cannot extract () using polarizer angles of P'= or 9. Thus, to apply Equations (5-18) - (5-2) in the absence of a sample, a new reference plane is defined relative to P'. If we assume that the reference plane is located at 45 relative to the transmission axis of the polarizer, then one can apply the above equations with P' = 45. This yields: cos2 = cos2q cos2 tan = sin2 csc2q. (5.21a) (5.21b) These provide the expected results of (, ) = (45, ) based on input data of (Q, ) = (45, ). Applications of these equations to the (Q, ) data yield the results in Fig Although the values are relatively close to 45, i.e. within over the photon energy range of ev, the deviations become larger at low and high energies, increasing to 45.5 and 45.3, respectively, implying up to a.6% error. The error in 117

137 seems to translate to a large error in with an average value of 1.2 with a range of.2. This represents an error of up to.4%. These errors will be reduced if the actions taken to improve the (Q, ) data are successful. Figure 5-8: Ellipsometry angles (, ) plotted vs. photon energy for a straight through measurement. In the absence of instrument errors, these values should be given by (, ) = (45, ). 118

138 Chapter 6 Dual Rotating Compensator Multichannel Ellipsometer 6.1 Introduction The single rotating compensator ellipsometer described in Chapter 5 can provide the complete Stokes vector of the beam reflected from the sample. The instrument is suitable for the study of either strongly or weakly absorbing isotropic samples. It can detect and characterize simple heterogeneities through the depolarization spectrum of the light beam. This spectrum can be modeled along with (Q, ) in data analysis. With the single rotating compensator instrument, sample anisotropy or more complicated heterogeneities require performing measurements at different angles P of the polarizer. Such a measurement strategy is not suitable, however, for real time studies of materials preparation. For real time studies of anisotropic samples or samples with complicated heterogeneities, the ultimate solution is to measure spectra in the complete Mueller matrix of the sample at high speeds, specifically with < 1 s acquisition time [6-1]. This can be performed by modulating the Stokes vector of the incident beam using a fixed polarizer followed by a rotating compensator, and then detecting the reflected beam from the sample by using another rotating compensator followed by a fixed analyzer [6-1]. 119

139 The dual rotating compensator multichannel ellipsometer in this Thesis research is constructed using a rotating achromatic rhomb compensator on the polarization generation arm and a rotating chromatic biplate compensator on the polarization detection arm. Thus, its optical configuration is denoted PC 1r SC 2r A or [P (fixed polarizer)] - [C 1r (rotating achromatic compensator)] - [S (sample)] - [C 2r (rotating biplate compensator)] - [A (fixed analyzer)]. A schematic of the dual rotating compensator ellipsometer configuration is shown in Figure 6-1. Thin film/ Substrate i M ij j Xe light source 5( B tc S1 ) F P= PP S t s p p Collimating optics s Fixed polarizer Rotating achromatic compensator ( C1, C1 ) s F 3( B tc S2 ) Rotating biplate compensator ( C2, C2 ) Fixed analyzer p s t A= AAS p Spectrograph with photodiode array Figure 6-1: Schematic of the dual rotating compensator ellipsometer. An achromatic King-type rhomb compensator, which exhibits a retardance given by C1 ~ 9 o over its entire spectral range, is used in the construction of the dual rotating compensator ellipsometer in this thesis research. The achromatic quarter-wave 12

140 compensator provides the greatest sensitivity to spectra in the cross polarization complex amplitude reflection ratios, in other words the off-diagonal elements of the Jones matrix, for a weakly anisotropic sample [6-7]. The Jones matrix of most general sample is given by T pp sp ps 1, (6.1) where pp, ps, sp are the ratio of Fresnel amplitude reflection coefficients. The six quantities defined by this matrix, including the real and imaginary parts of ij (i,j = p,s), can be used to calculate the Mueller matrix [6-6]. As an example, a sample such as the (11) plane of single crystal Si exhibits very weak surface-induced optical anisotropy, with the amplitudes of the cross polarization reflection ratios ps and sp less than 1x1 3 [6-6]. In the measurement of such samples, even small random or systematic errors can make it impossible to characterize these cross polarization reflection ratios. The use of the achromatic quarterwave compensator as the optical component of one or both rotating compensators can improve measurements of the cross polarization reflection ratios due to the potential reduction of both random noise and systematic errors [6-7]. Rotating the three reflection King rhomb [6-2] is a challenge in this application due to the complexity of its mounting and alignment [6-3]. Detailed procedures for these purposes are given in this chapter. 6.2 Instrument design and theory The dual rotating-compensator configuration in straight through mode is denoted PC 1r ( 1 )C 2r ( 2 )A, where P, C 1r ( 1 ), C 2r ( 2 ), and A represent the MgF 2 fixed Rochon 121

141 polarizer, the rotating achromatic rhomb compensator, the rotating MgF 2 biplate compensator, and the MgF 2 fixed Rochon analyzer, respectively. The mechanical rotation frequencies of the first and second compensators ( 1, 2 ) are synchronized at a ratio of 5:3 so that 1 = 5 B and 2 = 3 B, where B = /T C is the base angular frequency corresponding to 2 Hz and T C =.25 s is the fundamental optical period. The time dependent irradiance waveform predicted at each pixel of the photodiode array is given by [6-1] 16, (6.2) I(t) I M a (1 cos(2nt ) sin(2nt ) ) 11 2n 2n 2n 2n n1 (dropping the subscript "B" on B for simplicity of notation). In Eq. (6.2), a describes the dc Fourier coefficient of the waveform, 2n, 2n are the dc-normalized 2n ac Fourier coefficients, and 2n are the phases of the individual Fourier components, which can be described as functions of the phases C S1 and C S2 of the individual rotating compensators. The phases of the individual compensators are defined by the two linear equations C 1 (t) = 5(t C S1 ) and C 2 (t) = 3(t C S2 ), where C 1 (t) and C 2 (t) are the true azimuthal angles of the compensator fast axes at time t. Thus, 5C S1 and 3C S2 are the angles of the fast axes at time t =, the onset of data acquisition. Among the 33 possible Fourier coefficients in Eq. (6.2), the eight corresponding to n = 9, 12, 14, and 15 vanish even for the most general Mueller matrix. Hence, there are a total of 25 (rather than 33) useful (non-zero) Fourier coefficients [6-1]. Because the phases 2n in Eq. (6.2) must be determined in calibration, the dcnormalized Fourier coefficients 2n, 2n in Eq. (6.2) are not directly accessible in an experiment. As a result, Eq. (6.2) is considered a theoretical expression from which the 122

142 normalized Mueller matrix elements m ij = M ij /M 11 ; i=1,...,4; j=1,...,4 are to be extracted after full instrument calibration is performed. The simpler experimental expression that neglects the phase terms is given by 16 ' ' ' 2n 2n n1 I(t) I [1 ( cos2nt sin2nt)], (6.3) where I, ( 2n, 2n ) are the dc and normalized ac Fourier coefficients to be determined experimentally. As in Eq. (6.2) (' 2n, ' 2n ); n= 9, 12, 14, 15 all vanish even for most general sample. Once calibration is performed to determine the phase terms, 2n, the theoretical unprimed coefficients in Eq. (6.2) can be determined form the experimental primed coefficients of Eq. (6.3) by the following 2n rotation transformation 2n = 2n cos 2n + 2n sin 2n, 2n = 2n sin 2n + 2n cos 2n. (6.4a) (6.4b) The phase terms 2n are given explicitly through the following forms [6-4] 2n = (1C S1, 6C S2 ) for the double frequencies with n = (5, 3), respectively; 2n = (2C S1, 12C S2 ) for the quadruple frequencies with n = (1, 6), respectively; (6.4c) (6.4d) 2n = (1C S1 6C S2, 1C S1 12C S2 ) for the difference frequencies with n = (2, 1), respectively; (6.4e) 2n = (2C S1 6C S2, 2C S1 12C S2 ) for the difference frequencies with n = (7, 4), respectively; (6.4f) 2n = (1C S1 + 6C S2, 1C S1 + 12C S2 ) for the sum frequencies with n = (8, 11), respectively; 123 (6.4g)

143 2n = (2C S1 + 6C S2, 2C S1 + 12C S2 ) for the sum frequencies with n = (13, 16), respectively. (6.4h) Data collection To extract all 33 possible non-zero Fourier coefficients from the experimental irradiance function given by Eq. (6.3), a minimum of 33 irradiance equations are required. Thus, the photodiode array must be scanned a minimum of 33 times per fundamental optical period (T C ) of the two rotating compensators. Because a binary encoder is used to track the motor shaft position, the detector is scanned 64 times per optical cycle for the instrument developed in this Thesis research. This is the minimum power of 2 (i.e. 2 6 ) that enables full waveform analysis (with at least 33 scans) and that can be derived from the original 124 encoder pulses per two optical cycles through division by an integer (i.e. 8). A pulse dividing circuit generates the input pulses that trigger scans (or read-outs) of the array 64 times per optical cycle. Hence there are 64 equal exposure times per optical cycle, bounded by the succession of read-outs, each one transferred to the control computer. The integrated irradiance spectrum obtained from each scan j is given by S j j /64 ' I(t)dt (j1) /64 j= 1,...,64 (6.5a) = I I n (2j1)n (2j1)n 64 n ' 16 ' ' ' sin 2n cos 2n sin, (6.5b) n1 124

144 (dropping the subscripts "" on 2n and 2n for simplicity of notation). The sum in Eq. (6.5b) can be extended to include as many unknown Fourier coefficients as there are measurements (64). As a result, 64 equations can be generated in 64 unknowns, the latter including I ', all even pairs of Fourier coefficients up to ( 62 ', 62 ), as well as 64 ' (but not including 64 ). The matrix of coefficients described by Eqs. (6.5b) with the extended summation can be inverted to extract all 64 coefficients of the unknowns, from 64 integrated irradiances given by the S j values. The results are provided in Appendix E. A different approach to waveform analysis will be described shortly; this alternative approach is more readily adaptable to the evaluation of alignment errors Pulse dividing circuit for dual rotating compensator ellipsometer In the dual rotating compensator system, the pixels are grouped in a non-uniform grouping mode as shown in detail in Fig The linear fit relating wavelength and ungrouped pixel number k in this case is given by k = (.6126 k ) nm. A total of 68 individual pixels at high energy and 28 at low energy are not scanned. Considering the remaining 676 pixels, 368 are grouped by 8, yielding 46 groups over the higher energy range, and 38 are grouped by 14, yielding 22 groups over the low energy range. Thus, the energy region covered is from 1.8 to 2.48 ev with 22 groups and from 2.48 to 4.45 ev with 46 groups. Each 14 member group covers a wavelength range of 8.6 nm, and at 1.82 ev and ev the groups collect data over energy ranges of.23 ev and.42 ev, respectively. Similarly, each 8 member group covers a wavelength range of 4.9 nm, and at ev and ev the groups collect data over photon energy ranges of.24 ev and.77 ev, respectively. Using this grouping mode, it is 125

145 important to evaluate the timing requirements for photodiode array scanning which is described in the following paragraphs. Since the two motors rotate the compensators continuously at the frequency ratio of 5:3, the pulses of the master motor ( = 5 B ) that are multiples of 5 occur simultaneously with the pulses of the slave motor ( = 3 B ) that are multiples of 3. Without a divider, an absolute synchonization (called a Z pulse) is provided when the 5 th pulse of the master motor coincides with the 3 rd pulse of the slave motor. When developing a divideby-8 circuit, the objective then becomes matching the 4 th pulse of the master with the 24 th pulse of the slave and producing the corresponding output synchronization pulse when they match. For this instrument, the time period of a single fundamental mechanical cycle is 1/(2s -1 ) = 5 ms during which there are 124 pulses from the encorder and 128 from the divider. The fundamental optical period is (5 ms)/2 = 25 ms, during which 64 pulses are generated by the divider. Thus, the time between successive divider pulses is (25 ms)/64 = 3.96 ms. The read-out time for the above grouping scheme is calculated as follows [17 s] + [(46)(35 s)] + [(22)(38 s)] + [32 s] = ms. (6.6) The calculation method is similar to that described in Chapter 2 for the single rotating compensator system. It can be observed that, considering a 3.96 ms fixed exposure time, the detector scanning time takes up ms or 64% of the time duration. Hence, scanning can be performed easily in the time available; thus, there is no chance that synchronization pulses will be missed. This suggests that the data acquisition mode can be improved, enabling collection of a larger number of spectral positions than the current 126

146 value of 68. This in turn can result in either an improved spectral resolution or a wider spectral range. Figure 6-2: Wavelength as a function of pixel number indicating the grouping mode used for the dual rotating compensator ellipsometer Data reduction Once the dc and normalized ac Fourier coefficients, I ', ( 2n ', 2n ' ); n= 1-8, 1, 11, 13, 16, are determined, the next step is the phase correction of the coefficients. Phase correction is performed through Eqs. (6.4) as described previously. From the phase corrected, dc normalized ac Fourier coefficients of Eq. (6.2), written collectively as ( 2n, 2n ); n= 1-8, 1, 11, 13, 16, the normalized Mueller matrix elements are derived 127

147 using equations developed for compensators with arbitrary rotation frequency ratio p:q. These equations also include the possibility that both compensators exhibit dichroism [6-8]. By substituting p=5 and q=3, the Mueller matrix equations required for this Thesis research were obtained. In Ref. [6-8] where low and high frequency options are described for extracting the Mueller matrix elements, the low frequency options are selected, based on the suggestion provided in the summary of Ref. [6-7]. These Mueller matrix equations are given by: m 22 + im 23 = (a /s 1 s 2 ) exp(2ip' ) Z(8)exp(2iA' ) + Z(32exp(2iA' ), (6.7) m 32 + im 33 = (ia /s 1 s 2 ) exp(2ip' ) Z(8)exp(2iA' ) Z(32exp(2iA' ), (6.8) m 12 + im 13 = (a /s 1 t 2 )exp(2ip' )t 2 Z(2)Z(8)exp(4iA' )Z(32)exp(4iA' ), (6.9) m 21 + im 31 =(a /t 1 s 2 )exp(2ia')t 1 Z(12)[Z(8)] * exp(4ip')z(32) exp(4ip'), (6.1) m 42 + im 43 = ia exp(2ip' )]/[s 1 s 2 sin2 C2 sin C2 ] 2s 2[exp(2iA' )] Z(14) + (s 2 +1)cos2 C2 [exp(4ia' )] Z(8) + (s 2 1)cos2 C2 [exp(4ia' )] Z(32) + s 2 cos2 C2 Z(2), (6.11) m 24 + im 34 = ia exp(2ia' )]/[s 1 s 2 sin2 C1 sin C1 ] 2s 1 [exp(2ip' )] [(2) i(2)] + (s 1 +1)cos2 C1 [exp(4ip' )] [Z(8)]* + (s 1 1)cos2 C1 [exp(4ip' )] Z(32) + s 1 cos2 C1 Z(12), (6.12) 128

148 m 41 = a /[t 1 sin2 C2 sin C2 ] t 1 [(6) sin2a' (6) cos2a' ] + 2 [(14) sin(4p' 2A' ) (14) cos(4p' 2A' )] +a cot2 C2 /[s 1 s 2 sin C2 ](1s 1 )(1+s 2 )[(8)sin4(P' A' )(8)cos4(P' A')] + s 1 [(12) sin4a' (12) cos4a' ] +(1s 1 ) s 2 [(2) sin4p' (2) cos4p' ] (1s 1 )(1s 2 ) [(32)sin4(P' +A' ) (32)cos4(P' +A' )], (6.13) m 14 = a /[t 2 sin2 C1 sin C1 ] t 2 [(1) sin2p' + (1) cos2p' ] + 2 [(2) sin(2p' 4A' ) (2) cos(2p' 4A' )] +a cot2 C1 /[s 1 s 2 sin C1 ](1+s 1 )(1s 2 )[(8)sin4(P' A' )(8)cos4(P' A' )] + s 2 [(2) sin4p' + (2) cos4p' ] s 1 (1s 2 ) [(12) sin4a' (12) cos4a' ] + (1s 1 )(1s 2 ) [(32)sin4(P' +A' ) (32)cos4(P' +A' )], (6.14) m 44 = 2a /[sin2 C1 sin C1 sin2 C2 sin C2 ] [(4) cos2(p' A' ) + (4) sin2(p' A' )] ½[(cos2 C1 cos2 C2 )/a ] + [(cos2 C1 )/s 1 ][(14) cos(4p' 2A' ) + (14) sin(4p' 2A' )] + [(cos2 C2 )/s 2 ][(2) cos(2p' 4A' ) + (2) sin(2p' 4A' )] + [(cos2 C1 cos2 C2 )/s 1 s 2 ][(8) cos4(p' A' )+(8) sin4(p' A')], (6.15) a = t 1 t 2 t 1 t 2 + (8)cos4(P' A' ) + (8)sin4(P' A' ) t 1 [(12) cos4a'+ (12)sin4A' ] t 2 [(2) cos4p'+ (2) sin4p' ] + (32) cos4(p' +A' ) + (32) sin4(p' +A' ) 1. (6.16a) In Eqs. (6.7)-(6.15), m ij ; i=1,..4; j=1,..4, are the Mueller matrix elements normalized by M 11, and, P=PP S and A= AA S are the true polarizer and analyzer angles. (P, A) are the angular readings and (P S, A S ) are determined by calibration as explained in the next section [6-5]. In addition, in Eqs. ( ) and (6.16a), 129

149 Z(2n) = 2n + i 2n, Z*(2n) = 2n i 2n, Z(2n) = Z*(2n) (6.16b) (6.16c) (6.16d) 1 s j (1 sin 2Cjcos Cj) ; j=1,2, (6.16e) 2 1 c j (1 sin 2Cjcos Cj) ; j=1,2, (6.16f) 2 and t j = s j / c j, where j=1,2 indicate the first and second compensator, respectively. The set ( Cj, Cj ), j=1,2 represents the dichroic angles and retardances of the first and second compensator, respectively, as obtained in calibration. 6.3 Mounting and alignment of achromatic King rhomb compensator A schematic of the achromatic King-type rhomb compensator is shown in Fig As stated in Chapter 2, the King-type rhomb is an optical device with three total internal reflections; hence, precise alignment of this device with respect to the incoming beam and the axis of rotation is critical for the successful operation of the dual rotating compensator ellipsometer under development. First of all, small misalignments in the device can cause the output light to deviate from spectrograph slit as the compensator is rotated. Even after such deviations are minimized or eliminated, precise alignment in the total internal reflection process is necessary to produce the intended phase change relative to the incoming light, even as the device is rotating continuously. Thus the mechanically inequivalent half-rotations of the device must be made optically equivalent. Finally, strain along the fused silica beam path can distort the polarization. As a result, 13

150 precautionary measures must be taken to reduce strain induced birefringence of this device, which if present, can cause non-idealities in compensator performance. There are three conditions that must be satisfied in order to complete the alignment of the achromatic King-type rhomb compensator. (1) The two parallel faces of the achromatic King rhomb compensator should be exactly perpendicular to the axis of rotation of the compensator motor. This ensures that the axis of rotation is parallel to the optical (or symmetry) axis of the device which itself lies in the desired plane of incidence that is common to the three total internal reflections. (2) After the first condition is met, the two faces of the achromatic compensator must be made perpendicular to the instrument axis defined by the beam direction. This ensures that the beam is parallel to the rotation axis and also parallel to the desired plane of incidence of the three reflections. (3) If the above two conditions are met, then two types of translations are needed to ensure that the two parallel lines are colinear and lie in the common plane of incidence. The first is internal to the device and involves translating the compensator in two dimensions relative to its mount to the motor to ensure that the axis of rotation coincides with the optical axis of the device. This ensures that the mechanically inequivalent halfrotations (e.g. -18, 18-36) are optically equivalent. The second translation is external to the device and ensures that the beam coincides with the coincident rotation/optical axes of the device Achieving condition (1) The two parallel faces of the achromatic King-type rhomb compensator can be made perpendicular to the axis of rotation of the compensator motor using two of three 131

151 miniature mounts. These include (i) a polar tilt stage with its axis perpendicular to the rotation axis of the compensator and (ii) an azimuthal rotation stage with its axis perpendicular to the tilt stage axis and also perpendicular to the rotation axis of compensator. A horizontal linear translation stage is also incorporated as a miniature mount. A larger size translator for vertical positioning is built into the compensator mount to the motor, as shown later. Before attaching the King-type rhomb to the mounts, it is attached initially to an aluminum plate by means of a glass to metal adhesive as shown in Fig King rhomb MgF 2 coating Adhesive Aluminum plate Figure 6-3: Schematic showing the King-type rhomb mounted to an aluminum plate. From Fig. 6-3, it can be seen that the adhesive is used only at the bottom of the rhomb and only at its two ends, since an air space is required in the center of the device. This air space enables total internal reflection from the (fused silica)/(275 Å MgF 2 )/ambient optical structure. The aluminum plate is in turn attached to miniature optical mounts that enable alignment with three degrees of freedom, two angular and one translational. Figure 6-4(a) presents a photograph of the rhomb after its attachment to the mounting assembly and the motor. 132

152 King-type rhomb Tilt stage Rotation stage Horizontal linear stage Vertical linear stage Figure 6-4(a): Photograph of the King-type rhomb and mounting stages required for its alignment relative to the motor. In order to achieve condition (1), laser light was incident on the rear face of the rhomb (i.e. the face of the rhomb opposite to the motor), and reflected light was observed on a screen at a large distance from rhomb (~25 m). Mirrors were used for achieving this beam distance. When the rhomb was mounted as shown in Fig. 6-4(a) and the motor was rotated manually, the light reflected from the face of rhomb was also found to rotate. The tilt and rotation stages as mentioned above were adjusted to position the rhomb such that no perceptible rotation of the reflected beam spot was observed on the screen Achieving condition (2) After condition (1) is met successfully, the face of achromatic compensator can be made perpendicular to the ellipsometer beam by using the optical mounts on the motor. The degrees of freedom that may be required include (i) a multi-axis tilt stage which has both tilt and rotation capabilities, the latter about an axis parallel to goniometer axis (and both about axes perpendicular to the ellipsometer axis); (ii) a horizontal linear translation 133

153 stage; and (iii) a vertical linear translation stage. Figure 6-4(b) shows a photograph of the motor after attachment to the mounting hardware which was in turn attached to ellipsometer rail. Motor with King-type rhomb attached Horizontal linear stage Multiaxis tilt stage Vertical linear stage Figure 6-4(b): Photograph of the achromatic compensator motor and the mounting stages required for its alignment. A multi-axis tilt stage provides both rotation and tilt capabilities. In order to achieve condition (2), collimated xenon light is passed through an aperture and is incident on the front face of the rhomb facing the motor. By adjusting the tilt and rotation on the multi-axis tilt stage shown in Fig. 6-4(b), and the translators as necessary, the reflected light from the rhomb can be directed back through the incident beam aperture. 134

154 6.3.3 Achieving condition (3) After the first two conditions have been successfully achieved, the height and horizontal positions of the motor relative to the beam are adjusted to ensure that the maximum average irradiance passes through the achromatic compensator during its rotational period, corresponding to the maximum I '. This step is designed to superimpose the optic axis of the compensator onto the ellipsometer axis (or light beam). Once this alignment is achieved, the position of the achromatic compensator is adjusted with respect to the motor using the height and horizontal translations, with the goal to superimpose the rotation axis onto the combined light beam and compensator optic axes. These translations are performed using the miniature horizontal translator, and the larger vertical translator shown in Fig. 6-4(a). This generates a situation in which the two mechanically inequivalent, but optically equivalent cycles associated with the fundamental frequency give the same results for the Fourier coefficients. This can be observed on an oscilloscope, but an improved way of testing the alignment in the future is to expand the Fourier series in Eq. (6.3) in terms of both odd and even coefficients. This yields the following equation, 3 I(t) = I ' n 'cos(n B t)+ n 'sin(n B t)+ 31 'cos(31 B t)+ 32 'cos(32 B t)+ 32 'sin(32 B t) n=1 (6.17) which also has 64 unknowns, but now the odd coefficients are useful for alignment evaluation. These coefficients should be zero and minimization of their 15 amplitudes collectively, given by n (' 2n ' 2n-1 2 ) 1/2, can provide a measure of the quality of the alignment. 135

155 The two translation steps require iteration to superimpose the three critical axes, which include the axis defined by the beam, the rotation axis of the compensator, and the optic axis of the compensator. The alignment of three axes poses the most significant challenge of the rotating rhomb. For the biplate compensator, only two axes need to be aligned, the beam axis and the rotation axis, since the optical behavior of the element is translationally invariant in the two directions perpendicular to the optic axis. Finally, if the four translators are not precisely aligned perpendicular to the beam axis, the conditions (1), (2), and (3) may need to be iterated. The final result places the second internal reflection at the (fused silica)/(275 Å MgF 2 )/ambient optical structure precisely in the center of the rhomb, and the output beam from the rhomb is inline with the incident beam as intended, without wobble. 6.4 Calibration of dual rotating compensator ellipsometer: The sequence of calibration steps for the dual rotating compensator ellipsometer involves (i) determination of retardances ( Cj ) and dichroic angles ( Cj ) of the achromatic King-type rhomb compensator (j=1) and the biplate compensator (j=2); (ii) determination of the polarizer offset angle (P S ) and analyzer offset angle (A S ); and (iii) phase angle determinations for the two compensator fast axes at the beginning of data acquisition for the achromatic King-type rhomb compensator (C S1 ) and the biplate compensator (C S2 ) Compensator calibration For measurements in the straight through configuration, the normalized Mueller matrix is the identity matrix. Under these circumstances, the Fourier coefficients associated with 22, 26, and 32 frequencies all vanish [6-8]. Thus the output 136

156 waveform can be simplified, as it can be described with 9 frequencies rather than 12. Under these conditions the amplitudes of 2, 8, 12, 14, and 2 Fourier coefficients can provide ( C1, C2 ) and ( C1, C2 ) as described in [6-8]. The starting point in the calculations are the following four remarkably simple expressions for the ratios of the Fourier amplitudes t 1 (1 sin2 C1 cos C1 ) / (1 sin2 C1 cos C1 ) = Z(8) / Z(12), t 2 (1 sin2 C2 cos C2 ) / (1 sin2 C2 cos C2 ) = Z(8) / Z(2), k 1 cos2 C1 / [½(1 sin2 C1 cos C1 )] = Z(2) / Z(12), k 2 cos2 C2 / [½(1 sin2 C2 cos C2 )] = Z(14) / Z(2), (6.18a) (6.18b) (6.18c) (6.18d) where Z(2n) = ( 2n ) 2 + ( 2n ) 2 1/2. The two pairs of expressions can then be combined to yield: cos Cj = 1 t j / (1 + t j ) 2 k j 2 1/2, cos2 Cj = k j / (1 + t j ) (6.19a) (6.19b) where the index j= 1,2 indicates results for each of the two compensators. Based on Eq. (6.19a), C1 and C2 are calculated and plotted in Fig It can be observed that the retardance ( C1 ) of the King-type achromatic compensator is nearly constant with wavelength, varying between 89.7 o and 9.8 o over the spectral range from 2 ev to 5 ev. Excluding a strong noise feature near 2.8 ev, the largest statistical fluctuations in these data occur at low energies, and span a range of ~.15. In contrast, the biplate -- being a birefringent retarder -- exhibits a retardance ( C2 ) that is strongly dependent on wavelength. Polynomial fitting versus photon energy is performed for this retardance spectrum using the same method as is described in Chapter 5 with the results 137

157 plotted in Fig The resulting best fit coefficients defined by Eq. (5.1) are as follows d = nm, a 1 =.149 nm -1 ev -1, a 2 =.1852 nm -1 ev -2, a 3 =.953 nm -1 ev -3, a 4 =.2378 nm -1 ev -4, and a 5 =.225 nm -1 ev -5. The quarter-wave point is nm (E Q = ev). Because this is the same biplate as is used in the single rotating compensator ellipsometer, the thickness difference d = nm is applied from that calibration, which was shown previously in Fig The difference between the best fit and experimental data for the biplate retardance is also plotted in Fig The deviations of the measured retardance from the best fit appear oscillatory in nature, spanning a range of.25; however, the statistical deviations are small on the scale of Fig. 6-5, ~.5. Because the biplate compensator is the same element in both the single and dual rotating compensator instruments, the above coefficiencts can be compared to those presented previously in Chapter 5 (a 1 =.1435 nm -1 ev -1, a 2 = nm -1 ev -2, a 3 =.154 nm -1 ev -3, a 4 =.4438 nm -1 ev -4, and a 5 =.456 nm -1 ev -5 ). The quarter-wave point deduced in the single rotating compensator configuration is given by Q = 29.4 nm (E Q = 4.27 ev), which is ~5 nm longer in wavelength (.74 ev lower in energy) than that reported here for the dual rotating compensator ellipsometer configuration. The significant differences in coefficients result from relatively small differences in the retardances. Figure 6-6 shows the difference between the best fit polynomials for the two data sets (dual - single). The difference spectrum spans the range of 1.5 to +.15, and is attributed to small differences in alignment of the compensator, in addition to possible experimental errors (as have been observed in the 138

158 single rotating compensator configuration). The difference increases with increasing photon energy as might be expected for such error sources. Figure 6-5: (a) Measured retardance C1 of the King-type achromatic compensator; (b) measured retardance C2 of the biplate compensator (points) and its best fit polynomial (line); (c) Difference between the measured biplate retardance C2 and the polynomial fit to the retardance. 139

159 Figure 6-6: The difference between the best fit polynomials used to fit the retardance of the biplate compensator, as measured in the dual rotating compensator configuration (PC 1r C 2r A) and the single rotating compensator configuration (PC r A). By using Eq. (6.19b), the dichroic angles of the two compensators were calculated and the results plotted in Fig From the figure it can be observed that the biplate compensator does not exhibit significant dichroism. In fact, the dichroic angle of the biplate compensator spans the range of C2 = ; however, the King-type rhomb exhibits a more significant dichroic effect including a photon energy dependence, ranging from C1 = 45.9 at 2.7 ev to C1 = 41.8 at 4.6 ev. There are also larger random fluctuations in C1, increasing to.4 at high energies, and apparent systematic oscillations with amplitudes as large as 1. 14

160 For both compensators, the dichroic angle should be precisely 45 under ideal operating conditions. The dichroic behavior exhibited by King rhomb can be attributed to at least three effects including (i) alignment errors, which generate off-diagonal Jones matrix elements that cannot be eliminated even by rotation into a reference frame of fastslow axes, (ii) strain in the fused silica due to its handling and mounting, which causes polarization changes upon transmission through one or more of the four fused silica beam paths within the device, and (iii) absorbing contaminant layers on the interfaces to the ambient where total internal reflection is designed to occur. The third effect can be significant even for nanometer thick contaminant layers. Finally, it should be emphasized that these calibration spectra in ( Cj, Cj ); j=1,2 must be generated as smooth curves for subsequent use in data reduction when samples are measured. In this way, the random fluctuations are not introduced into the subsequent data analyses. For the biplate, the polynomial expression as given by Eq. (5.1) with the best fit parameters above can be used. This approach is justified based on the assumption that the birefringence follows polynomial behavior. For the achromatic compensator, two approaches are possible, either an empirical approach that uses polynomials simply for smoothening the data, or a modeling approach that incorporates the spectral dependence of the optical properties of the components, incuding optical properties and thicknesses of contaminant layers on the fused silica and MgF 2 surfaces. 141

161 Figure 6-7: Measured dichroic angles for the achromatic rhomb compensator ( C1 ) and the biplate compensator ( C2 ) Offset and phase angle self calibration In general, for the dual rotating compensator ellipsometer, there are 24 non-zero ac Fourier coefficients and 15 normalized Mueller matrix elements. As a result, 9 independent expressions can be derived for self-calibration or as consistency checks [6-8]. These 9 independent expressions are based on Mueller matrix elements of the 4 th row and 4 th column (m 4j, m j4, m 44 ), j=1,2,3. As shown by the schematic in Fig. 6-8, there are a total of six quartic self calibration equations in the two angles P = P' + 5C S1 and A = A' + 3C S2, obtained by equating two independent expressions for each of the following Mueller matrix elements: m 14, m 24, m 34, m 41, m 42, and m 43. Three coupled equations in these same two angles P and A are derived from four independent expressions for m 44, as also shown in Fig

162 Fourier Coefficients Mueller Matrix Elements Self-Calibration Equations (4p4q), (4p4q) (4p+4q), (4p+4q) m 22 m 23 m 32 m 33 (4p), (4p) m 12 m 13 m 22 m 23 m 32 m 33 (4q), (4q) m 12 m 13 m 21 m 22 m 23 m 31 m 32 m 33 (4p2q), (4p2q) (4p+2q), (4p+2q) m 12 m 13 m 21 m 22 m 23 m 32 m 33 m 34 m 42 m 43 Two Quartic Equations in tan(2a' +2qC S2 ) (2p4q), (2p4q) (2p+4q), (2p+4q) (2q), (2q) m 12 m 13 m 21 m 22 m 23 m 24 m 31 m 32 m 33 m 34 m 42 m 43 m 12 m 13 m 21 m 22 m 23 m 24 m 31 m 32 m 33 m 34 m 41 m 42 m 43 Two Quartic Equations in tan(2p' +2pC S1 ) One Quartic Equation in tan(2a' +2qC S2 ) (2p), (2p) (2p2q), (2p2q) (2p+2q), (2p+2q) m 12 m 13 m 14 m 21 m 22 m 23 m 24 m 31 m 32 m 33 m 34 m 41 m 42 m 43 m 12 m 13 m 14 m 21 m 22 m 23 m 24 m 31 m 32 m 33 m 34 m 41 m 42 m 43 m 44 One Quartic Equation in tan(2p' +2pC S1 ) Three Coupled Equations in P' +pc S1 A' +qc S2 Figure 6-8: Schematic of Mueller matrix calculations and self-calibration equations from the Fourier coefficients for a dual rotating compensator ellipsometer with a general frequency ratio of p:q. For the instrument developed in this Thesis research p:q = 5:3. 143

163 A program has been written in Labwindows/CVI to extract the 4 complex roots of the quartic equations; this program is provided in Appendix F. In this Thesis research, the calibration is performed in the straight through configuration, in which case the Mueller matrix of the sample is the identity matrix. Hence calibration equations associated with the off-diagonal elements of the Mueller matrix in the fourth row and column cannot provide useful information. Hence, the three coupled equations obtained from m 44 are used for calibration purposes. In Ref. [6-5], straight through calibration is performed using two of the three dichroic versions of the coupled equations from m 44. These equations are given by [6-8] tan2 P = ['(1 6) sin2 A + <'>(1 6) cos2 A ] + cos2 C2 ['(1) '(1 12) sin4 A + <'>(1 12) cos4 A ] X [<'>(1 6) cos2 A + '(1 6) sin2 A ] + cos2 C2 ['(1) + <'>(1 12) cos4 A + '(1 12) sin4 A ] 1, (6.2) tan2 A = ['(1 6) sin2 P + '(1 6) cos2 P ] + cos2 C1 ['(6) '(2 6) sin4 P + '(2 6) cos4 P ] X [<'>(1 6) cos2 P + <'>(1 6) sin2 P ] + cos2 C1 ['(6) + <'>(2 6) cos4 P + <'>(2 6) sin4 P ] 1, (6.21) where '( 1 2 ) '( ) '( 1 2 ), (6.22a) <'>( 1 2 ) '( ) '( 1 2 ), (6.22b) and A = A-A S + 3C S2, P = P+5C S1. In the definitions of Eqs. (6.22), ' represents either Fourier coefficient ' or Fourier coefficient ' 144

164 By considering an isotropic sample such that m 23 = m 32, the following self calibration equation has been derived [6-5] 5C 3C 1 tan tan ' ' S1 S2 ' ' (6.23) Since the present calibration was performed in the straight through configuration, as was the case for the single rotating compensator system, then m 23 = m 32 = and so the condition for application of Eq. (6.23) is satisfied. In addition, in straight through, the polarizer transmission axis defines the reference plane so that P=. Hence, Eqs. (6.2, 6.21, 6.23) can be solved for the three unknown calibration angles (A S, C S1, C S2 ). The solution was obtained with a program written in Lab/Windows CVI in order to extract the three calibration angles. The resulting plot of A S as a function of photon energy is shown in Fig For an ideal system in the absence of errors, A S should be independent of photon energy. The results in Fig. 6-9, however, reveal an average value of ~ , which spans a relatively large range. The origin of this variation is unclear; however, it suggests that other calibration equations should be developed to incorporate independent methods of determining A S. 145

165 Figure 6-9: Analyzer offset angle A S plotted as a function of photon energy in the dual rotating compensator configuration. The phase angles C S1 and C S2 are also obtained in the solution of Eqs. (6.2), (6.21), and (6.23). The results are shown in Fig. 6-1 over the pixel range for which 8x grouping was used, and in Fig over the range for which 14x grouping was used. As described previously in Section 6.2, the angles made by the two compensator fast axes at time t = are 5C S1 and 3C S2 for the achromatic and biplate compensators, respectively. Since the photodiode array is scanned sequentially, the values of C S1 and C S2 should be linear functions of pixel group number over the two pixel group ranges where the grouping is constant (either 8x or 14x). Hence a linear fit is performed for each phase angle, and the differences between the linear fits and the experimental data sets are also 146

166 shown in Figs. 6-1 and For the 8x grouping range in Fig. 6-1, the resulting linear equations are given by: C S1 = (.2536) k C S2 = (.2451) k (6.24a) (6.24b) where k= 1,.,46 are the pixel group numbers for the high photon energy grouping range. For the 14x grouping range in Fig. 6-11, the corresponding linear equations are given by: C S1 = 2.58 (.248) k C S2 = (.23) k (6.24c) (6.24d) where k = 47,...,68 are the pixel group numbers for the low photon energy grouping range. Although an individual fit must be performed over each of the ranges where different grouping occurs, the error in using a single fit seems relatively small as indicated by the similarity of the two sets of equations. 147

167 Figure 6-1: Rotating compensator phase angles (a) C S1 and (c) C S2 plotted as functions of pixel group number k from k = 12 (3.72 ev) to k = 46 (2.47 ev), over the range where the detector pixels are grouped by 8. The solid lines are bestfit linear relationships. Also shown are the spectra in (b) C S1 and (d) C S2, defined as the deviations of the measured values from the best fit linear relationships for C S1 and C S2. 148

168 Figure 6-11: Rotating compensator phase angles (a) C S1 and (c) C S2 plotted as functions of pixel group number k from k = 47 (2.45 ev) to k = 59 (2.3 ev), over the range where the detector pixels are grouped by 14. The solid lines are bestfit linear relationships. Also shown are the spectra in (b) C S1 and (d) C S2, defined as the deviations of the measured values from the best fit linear relationships for C S1 and C S2. The slopes in Figs. 6-1 and 6-11 can also be understood theoretically in the following manner. The fundamental rotation frequency of the dual rotating compensator system is 2 Hz, and the fundamental mechanical and optical time periods are.5 s and.25 s, respectively. The pixel grouping used for the data acquisition in Fig. 6-1 includes 8 pixels per group, each group requiring a 35 s total scan time. Hence, the slope for the 8x grouped pixels is given by (18)(35 s/pixel-group)/(.25 x 1 6 s) =.252/pixel-group. For the pixels grouped by 14, the corresponding 38 s scan time 149

169 leads to a slope of.274/pixel-group, a small change relative to the slope for 8x grouping. Small discrepancies of the 8x slope values measured experimentally are observed from that calculated theoretically, ranging from.2/pixel-group to.7/pixel-group. Over 46 pixel groups, this amounts to very small errors ranging from.1 to.3. An order of magnitude larger discrepancy is observed for the 14x slope values measured experimentally from that calculated theoretically, ranging from.26/pixel-group to.71/pixel-group. This generates a smaller than proportionate error due to the smaller number (22) of pixel groups, amounting to errors ranging from.6 to Determination of Mueller matrix elements After determination of the compensator phase angles C S1 and C S2, the phase terms 2n can be calculated for all non-zero Fourier coefficients based on Eqs. (6.4). These phase terms are used in turn to calculate the unprimed theoretical Fourier coefficients from the primed experimental coefficients. These Fourier coefficients along with calibration data are used for calculation of Mueller matrix elements based on Eqs. (6.7)- (6.15). The Mueller matrix elements obtained in this way are shown in Figs. 6-12, 6-13, and

170 Figure 6-12: Mueller matrix elements deduced experimentally in the straight through dual rotating compensator configuration. Since the experiment is performed in the straight through configuration, diagonal matrix elements should be equal to unity and off-diagonal elements should be equal to zero. From Figure 6-12 it can be seen that all off-diagonal elements except m 12 show only small deviations from zero. The matrix element m 12 alone shows a strong systematic variation from to ~.5 with increasing photon energy. The results for all off-diagonal 151

171 Figure 6-13: Off-diagonal Mueller matrix elements deduced experimentally in the straight through dual rotating compensator configuration; top panel: m 1j ; second panel: m 2j ; third panel m 3j ; bottom panel m 4j ; all on an expanded scale relative to Fig

172 Figure 6-14: Diagonal Mueller matrix elements deduced experimentally in the straight through dual rotating compensator configuration; top panel: m 22 ; second panel: m 33 ; third panel m 44 ; all on an expanded scale relative to Fig elements are shown on an expanded scale in Fig All elements with the exception of m 12 exhibit values <.2 over the full spectral range, and the (m 4j, m j4 ; j=1, 2, 3) elements are particularly close to zero with values <.1 over the full spectral range. The inaccuracy of m 12 is surprising since no other Mueller matrix elements show a similarly large effect, not even m 13, which is calculated from the same complex expression, Eq. (6.9). Al1 three diagonal elements are reduced from unity by ~.5, as shown on expanded scales in Figure These are clearly systematic errors and the reasons for these errors are under further investigation. Table 6-1 summarizes the 153

173 average value of each Mueller matrix element and the standard deviation, both taken spectroscopically. Table 6-1: Average and standard deviation for each Mueller matrix element over the entire spectral range of 68 pixel groups ( ev). The standard deviation is calculated using the expression S.D. = k [m ij (k) <m ij >] 2 /(N1) 1/2, where <m ij > is the average value, k identifies the pixel group number, and N=68 is the total number of spectral positions. m 11 Average =.278 S.D. = m 12 Average =.1834 S.D. =.1733 m 13 Average =.168 S.D. =.157 m 14 Average =.47 S.D. =.58 m 21 Average =.9517 S.D. =.3917 m 22 Average =.267 S.D. =.63 m 23 Average =.292 S.D. =.22 m 24 Average =.373 S.D. =.492 m 31 Average =.27 S.D. =.51 m 32 Average =.9533 S.D. =.38 m 33 Average =.7482 S.D. =.61 m 34 Average =.143 S.D. =.227 m 41 Average =.215 S.D. =.137 m 42 Average =.12 S.D. =.41 m 43 Average =.9632 S.D. =.592 m 44 The fact that all diagonal elements differ from unity by nearly equal amounts seems to suggest a random depolarization effect on the light beam at some stage in the measurement. Random depolarization D p at the position of the sample can be evaluated through the following consistency check on the Mueller matrix. D p = (1 + b) (b 2 + c) 1/2, b = (m 22 m 12 m 21 )/3, (6.25a) (6.25b) 154

174 c = [(m 22 m 12 m 21 ) 2 + (m 13 m 23 ) 2 + (m 14 m 24 ) 2 + (m 31 m 32 ) 2 + (m 41 m 42 ) 2 + (m 34 m 43 ) 2 + (m 33 + m 44 ) 2 ]/3. (6.25c) In the absence of depolarization, D p =, and for complete depolarization D p = 1. Figure 6-15 shows D p as a function of photon energy. Two interesting observations can be made based on this figure. First, the average depolarization over the spectral range above 2.25 ev is D p =.4, meaning a 4% depolarization effect, which in fact is quite large for such instruments. Second, there are well-defined oscillations with a period of ~.5 ev and an amplitude that decreases from.8 at low energies to.1 at high energies. Although the origins of these oscillations are unclear, most often they result from multiple reflections, which may occur in any well-aligned optical component. Considering the overall depolarization magnitude, this effect may occur due to transmission through any of the optical elements, starting from the polarizer. In the case of the polarizer and analyzer, the most common problem in Rochon components originates either from exceeding the semi-field angle or from scattering by prism imperfections, leading to poor extinction in both cases. Generally such effects can be detected with two polarizers alone through measurement of the residual function. Another possible origin of depolarization is stress in the achromatic rhomb. This origin is favored since the single rotating compensator configuration using the biplate compensator showed little depolarization of the beam. The best way to evaluate depolarization by the rhomb is to remove the biplate compensator from the instrument and determine if the rhomb can properly measure linearly polarized light from the polarizer in the single rotating compensator mode. This 155

175 is a reasonable approach first because this component is incorporated into the master motor, and second because the biplate is much easier to align and can be more readily moved off and on the optical rails. Finally, it may ultimately be necessary to go back to the single rotating compensator configuration with the biplate and improve the performance in that configuration first. As noted in Chapter 5, even the single biplate rotating compensator instrument configuration requires improvement, but it is unclear whether the poor performance in that case is due to the hardware (e.g. polarizer, biplate compensator, analyzer, detection system) or due to error correction or calibration (e.g. C IP, C, A S, C S ). Figure 6-15: Random depolarization D p assumed to occur at the position of the sample, as evaluated through the consistency check on the Mueller matrix given in Eqs. (6.25). 156

176 Chapter 7 Conclusions and future work 7.1 Conclusions A dual rotating compensator multichannel ellipsometer with an achromatic compensator on the polarization generation arm has been developed in this Thesis research. This ellipsometer provides spectra in the complete 4x4 Mueller matrix of a reflecting sample placed between the two rotating compensators. The instrument performance for the measurement of sample Mueller matrices can be evaluated in the straight through configuration, which results in an identity "sample" Mueller matrix. Determination of the identity "sample" Mueller matrix in turn requires assigning Mueller matrices to the two rotating compensators, including the achromatic rhomb comprensator on the polarization generation arm and a chromatic biplate compensator on the polarization detection arm. For both compensators, dichroic effects are included in their Mueller matrices and are measured automatically in a calibration routine. The measured dichroism exhibited by the achromatic compensator has been included in data reduction for determination of the sample Mueller matrix elements. 157

177 Procedures for mounting and aligning the achromatic compensator, which is based on the three-reflection King-type rhomb design, have been proposed and successfully implemented, enabling its continuous rotation in the dual rotating compensator ellipsometer configuration. The challenge of alignment results from the need to ensure the coincidence of three optical axes, including the beam center, the compensator rotation axis, and the compensator optic axis. For a biplate compensator, in contrast, only two axes need to be superimposed. An extensive set of instrument control programs for calibration and data collection, as well as programs for calibration analysis and data reduction have been written in Labwindows/CVI. The polarizer stepping motor, analyzer stepping motor, shutter, detector, and detector controller can all be controlled by personal computer. Calibration analysis and data reduction programs follow the detailed procedures outlined in Chapters 5 and 6, for the single and dual rotating compensator configurations, respectively. Although this Thesis focused on evaluating a dual rotating compensator configuration in straight through mode, programs have been written for the measurement of the most general reflecting samples, and for the use of two rotating compensators exhibiting dichroic effects. Self calibration of the dual rotating compensator instrument involves extracting the four roots of quartic equations derived using multiple expressions for the elements of the fourth row and fourth column of the Mueller matrix (m j4, m 4j ), j= 1,2,3; m 44. The optical elements used in the design of the instrument ensure a wide spectral range; these include a MgF 2 Rochon polarizer and analyzer, the fused silica King-type rhomb, and the MgF 2 biplate compensator. As a result, it is possible to add a D 2 lamp to 158

178 the standard Xe lamp in future instrument development for a photon energy range of 1.5 to 6 ev. Optical modeling of the Krasilov-type compensator has been successfully completed for possible manufacture and application as a replacement for the King-type rhomb. The Krasilov-type compensator may be less strongly influenced by contamination of the external surfaces at which total internal reflection occurs. The next step is to identify an optical manufacturer for the fabrication of the compensator. Because the Krasilov-type compensator is also a three reflection rhomb, the mounting, alignment, and calibration will involve the same procedures developed here for the Kingtype rhomb. The following brief summaries enumerate the accomplishments of this Thesis research. (1) Order sorting filters Order sorting filters were successfully incorporated within the spectrograph/ detector enclosure. The filters enable a photon energy range from 1.5 to 5 ev (25-83 nm) without the problem of overlapping orders from the grating. Without the filters, overlapping orders would appear at the photodiode array detector and generate errors. (2) Image persistence correction A pixel-dependent image persistence correction factor has been determined and its validity demonstrated in straight through with a polarizer and analyzer as the polarization state generator and detector, respectively. The image persistence correction factor has been found to reduce the maximum residual function values by a factor of 2 from.5 to.25 within the photon energy range of ~ ev (28-73 nm). The residual 2 2 1/2 function at pixel group k is given by R 1 ( ' ' ), which must be zero in the k k k 159

179 absence of instrument errors when linearly polarized light from the polarizer is incident on the analyzer. (3) Compensator biplate alignment Successful procedures have been developed to align a biplate compensator using a commercial multichannel ellipsometer. These procedures have been found to completely eliminate the high frequency oscillations in the dichroic angle that arise from internal misalignment of the two plates. Residual low frequency oscillations with an amplitude.1 appear to be due to multiple reflections between plates of the biplate. (4) Biplate compensator calibration in the single rotating compensator mode Using the ellipsometer developed in this Thesis in the single rotating compensator mode, the retardance of the biplate compensator has been calibrated in the straight through configuration. The result can be fit with a polynomial function of photon energy for use in subsequent data analysis. The difference between the measured calibration data and polynomial fit, which appears to include both random and systematic components, increases from.1 at low energies (1.7 ev) to.2 at high energies (4.5 ev). (5) Analyzer offset angle calibration in the single rotating compensator mode Using the same single rotating compensator ellipsometer configuration as in (4) the fixed analyzer angle relative to the polarizer has been calibrated. Over the photon energy range from 1.7 to 4.3 ev, the analyzer offset angle shows systematic deviations of.5-.1 from a well-defined value at low energy that can be defined with a precision of.1. 16

180 (6) Compensator phase angle calibration in the single rotating compensator mode In spite of promising retardance and analyzer angle calibrations in the single rotating compensator mode, a determination the absolute phase angle of the rotating compensator is less accurate, showing systematic deviations from expected linear behavior amounting to ~.12, and the noise level, which increases with pixel number reaches ~.8 at the lowest energies of ~ 1.7 ev. Improvements in this calibration are needed in the future as described shortly. (7) Stokes vector component measurement in the single rotating compensator mode The three Stokes vector components of the light beam generated after its passage through the polarizer have been determined in the single rotating compensator configuration. With the polarizer transmission axis defining the reference plane for the optical elements, the predicted tilt angle Q, ellipticity angle, and degree of polarization p of the state generated by the polarizer are given by (Q,, p) = (,, 1). Oscillations have been observed in the measured Q with peak-to-peak amplitude of ~ 1 at the lowest energy of 1.7 ev with an average value spanning a range of.6 versus photon energy. The ellipticity angle, although nearly constant versus energy, shows an error of.25. These variations in both Q and represent systematic errors of magnitude ~.3%. The accuracy in the measurement of the degree of polarization p is somewhat better at ~.1%. (8) Theory of the dual rotating compensator ellipsometer Equations have been developed for instrument calibration and data reduction in the dual rotating compensator configuration. The equations incorporate the possibility of the most general compensators exhibiting dichroic effects. Solutions to these equations have been implemented in Labwindows/CVI. 161

181 (9) Photodiode array exposure and scanning A divide by 8 circuit has been to employed to generate a Z trigger pulse and 64 clock pulses per fundamental optical cycle of the dual rotating compensator system that initiate and provide the timing for scanning of the photodiode array. A non-uniform grouping scheme has been developed for photodiode array scanning that generates 68 spectral positions and a photon energy (wavelength) resolution ranging from.2 ev (9 nm) at 1.7 ev (73 nm) to.8 ev (5 nm) at 4.5 ev (28 nm). With the grouping scheme, detector scanning takes up 64% of the exposure time. (1) Alignment of achromatic compensator An alignment procedure has been developed for the rotating King-type achromatic compensator. This procedure is more challenging than that used for the biplate since three axes must be brought into coincidence: the light beam axis, the compensator optic axis, and the compensator rotation axis. For the biplate, only the beam and rotation axis must be coincident; and these two axes need only be parallel to the optic axis of the biplate. (11) General compensator calibration in dual rotating compensator instrument Simultaneous calibration of the retardance and dichroic angles of the two compensators has been demontrated in the dual rotating compensator configuration in straight through mode using the amplitudes of 2, 8, 12, 14, and 2 Fourier coefficients. (12) Achromatic compensator calibration in dual rotating compensator instrument In the calibration methods of (11), the retardance of the King-type achromatic compensator is found to be weakly dependent on photon energy relative to the biplate, 162

182 varying between 89.7 o and 9.8 o over the energy range from 2 ev to 5 ev. Typical statistical fluctuations in the retardance calibration are largest at low energies, and span a range of ~.15. The King-type rhomb exhibits a dichroic effect including a photon energy dependence, ranging from 45.9 at 2.7 ev to 41.8 at 4.6 ev. Random fluctuations are also observed in the dichroic angle, increasing to.4 at high energies, and apparent systematic oscillations are observed with amplitudes as large as 1. (13) Biplate compensator calibration in dual rotating compensator instrument For the biplate compensator of the dual rotating compensator instrument, the deviations of the retardance measured in calibration from that describing the best fit polynomial appear oscillatory in nature, spanning a range of.25; however, the statistical deviations are small ~.5. A comparison of the best polynomial fits to the measured retardance of the same biplate, as obtained with the single and dual rotating compensator instruments, yields agreement within the range.15 over the photon energy range of 1.7 to 3. ev. With increasing photon energy up to 4.5 ev, the difference between the two best fits increases to 1.5. (14) Azimuthal angle calibration in dual rotating compensator instrument Calibration of the three azimuthal angles (C S1, C S2, A S ) in the dual rotating compensator configuration involves solving two quartic equations for A'+3C S2 and 5C S1 and then a system of three linear equations in these two angles along with 5C S1 3C S2. For A S, a range of values of.4 is observed over the photon energy range of 2.5 to 4.5 ev. The accuracy in the measured C S1 and C S2 values, as evaluated relative to the predicted 163

183 behavior based on linear relations versus pixel group, is much better -- within random deviations of.5. (15) Mueller matrix determination by dual rotating compensator ellipsometry In measurements using the dual rotating compensator ellipsometer in straight through mode, the largest deviations of the normalized Mueller matrix from the predicted identity matrix averaged over photon energy occur consistently for the three diagonal matrix elements, ranging from.37 to.48. The largest error in an off-diagonal element occurs uniquely for m 12, with an average value of.2, which is a factor of 2.5 higher larger than those of the other off-diagonal elements. The largest standard deviation for a Mueller matrix element is ~.12 which also ccurs for m 12 due to its photon energy dependence, which is relatively strong compared to other Mueller matrix elements. The non-ideal behavior of the diagonal Mueller matrix elements may be attributed to depolarization, possibly as a result of stress or other non-ideal characteristics of the rhomb compensator. The poorer result for the least ideal off-diagonal element m 12 is unclear at present. 7.2 Future work The Mueller matrix elements measured in this Thesis research using the dual rotating compensator ellipsometer configuration in straight through deviate from the identity matrix. These deviations from ideal operation can be attributed to the presence of systematic errors. One source of systematic error may be the stress induced birefringence in the fused silica along the beam path of the achromatic King-type rhomb. Errors due to stress should be considered when performing future Mueller matrix calculations. 164

184 The procedure for mounting and aligning the King-type rhomb used in this Thesis research may not be an ideal method. In the future mounting and alignment based on nano-positioners may be needed. If the performance of the system cannot be improved, then it is likely that the existing three-reflection rhomb has internal alignment problems due to its construction. If such is the case, the three-reflection rhomb can be replaced with a four-reflection Fresnel-type rhomb which may be less susceptible to alignment problems because of the even number of reflections. The four-reflection rhomb, however, may exacerbate the stress problems due to an additional optical path through the fused silica. If the performance of the system that incorporates an achromatic compensator on the polarization generation side can be improved, then the biplate compensator can be replaced with achromatic compensator. The encoders used in the motors should be changed so as to reduce the number scans used per optical cycle from 64. If this is possible, then the resulting increased exposure time (time between two successive encoder pulses) will enable a longer photodiode array scanning time. Then either the number of collected pixel groups can be increased from the present number of 68, increasing the spectral range, or the number of pixels per pixel group can be reduced, increasing the photon energy resolution. Another modification to waveform analysis involves determining the odd Fourier coefficients for low frequencies rather than the even Fourier coefficients for high frequencies above 32. From the odd Fourier coefficients, which should vanish, the alignment of the achromatic compensator can be better evaluated. Other errors unrelated to the achromatic rhomb compensator are instead associated with the multichannel detection system, and occur in all multichannel ellipsometers from 165

185 the rotating polarizer configuration to the dual rotating compensator configuration. First, order sorting filters which are placed on the top of the detector faceplate should be removed and placed directly on top of the photodiode array element instead. The goal of this change is to minimize systematic errors due to multiple reflections of the dispersed light between the faceplate and filter. Another important step is to correct for stray light at the detector, which can be done by reconstructing its waveform measured from pixels of the detector where no "true" light impinges, typically at the highest energies. In fact, a uv blocking filter can be placed over these pixels ensure that no "true" light strikes them -- only stray light. After acceptable system performance is achieved in straight through, then measurements of samples can be performed in reflection mode. The most challenging samples exhibit weak surface anisotropy, such as [11] Si and with such samples the performance of the ellipsometer can be tested. If ex-situ measurements in reflection mode can be made with high precision and accuracy, then in-situ real time measurements can be performed by installing a deposition chamber at the location of sample holder. As such, high speed real time spectroscopic ellipsometry measurements (RTSE) can be applied to extract complete 4x4 Mueller matrix of samples whose properties evolve with time due to film growth or surface processing.. The following detailed suggestions for future instrumentation improvements are proposed here. The general goal is to achieve accuracies in calibration and Stokes vector angles of.1 and statistical variations of Determination of the degree of polarization and Mueller matrix elements within.1 is also sought. 166

186 (1) Light source improvements With 64 detector scans per optical cycle, the exposure time is relatively short, ~3.9 ms, which implies that the light irradiance passing through the ellipsometer could be increased to more completely fill the 14-bit analog to digital converter of the detection system electronics. There are three approaches for increasing the irradiance in the collimated ellipsometry beam. First, the collimation quality could be reduced by using a larger pinhole. Second, the efficiency of the light collection optics used in conjunction with the xenon source could be increased through the use of larger sized optical elements and possibly retroreflecting optics. Third, an improved source could be incorporated, for example, a laser induced plasma source which generates a brighter point source for focusing on the pinhole of the collimator. (2) Detector scanning optimization Another related issue involves reducing the number of scans per optical cycle from 64 to 36, which involves non-integer division of the encoder pulses. Because the encoder pulses are tracked and feedback is used to control the motor stability, the same circuitry could be used to generate the 36 pulses. With 36 pulses, it is possible to extract the 25 required dc and ac Fourier coefficients, along with some of the odd Fourier coefficients for compensator alignment [see (7) below]. (3) Non-linearity corrections Although corrections have been made for detector image persistence, detector nonlinearity should also be evaluated and an additional correction procedure may be incorporated as needed. Such a correction takes the form of a factor, slightly different from unity and depending on the readout counts, that multiplies the pixel group readout 167

187 count value to obtain a corrected readout value precisely proportional to the irradiance. The non-linearity evaluation can be made without any polarization optics in place. (4) Stray light corrections Corrections for detector stray light as mentioned earlier can be developed and implemented. These corrections can be linked to the integrated irradiance over the entire array. In this case, because the array is scanned in a short time and there is a small angular range of the C S1 and C S2 values across the array, the stray light correction becomes easier to implement. (5) Biplate optimization One problem encountered in this Thesis research involves the relatively poor performance of the single rotating compensator ellipsometer using the chromatic biplate, as indicated by the errors in the measured (Q,) values of the Stokes vector generated by the polarizer. The sensitivity of this configuration for the measurement of Stokes vector will improve by using a compensator with a quarterwave photon energy closer to the center of the useful spectral range, i.e. between 3 ev and 3.5 ev. An additional method for improving the single rotating compensator ellipsometry results is to use multiple zones of the analyzer angle to extract the calibration angles C S and A S. (6) Achromatic compensator in single rotating compensator ellipsometry Considering the poor performance of the single rotating compensator using the biplate compensator, it would also be of interest to explore the performance of this configuration when the achromatic compensator is used as the single rotating element. 168

188 (7) Achromatic compensator alignment Considering the dual rotating compensator instrument, alignment of the achromatic rhomb is a considerable challenge. Such alignment could be facilitated by the measurement of the odd harmonic Fourier coefficients, i.e. the Fourier coefficients that are harmonics of the mechanical frequency (and not the optical frequency). These Fourier coefficient should vanish, and if they do not, the implication is that the waveforms from two successive optical cycles are not the same due to their mechanical inequivalence. (8) Mueller matrix of the achromatic compensator The possibility that the achromatic compensator depolarizes the light beam motivates future work in the determination of the full Mueller matrix of this compensator. If this can be done, and the depolarization effect generates a weak random component, then the compensator Mueller matrix can be modified from its ideal form as a dichroic retarder to include the depolarization effect. (9) Alternative achromatic compensators Achromatic compensators alternative to the King-type rhomb should be tested. As demonstrated in this Thesis, the Krasilov-type rhomb is an alternative three reflection rhomb that can readily replace the King-type rhomb without a significant change in mounting and alignment procedures. The next step in this direction is to have a Krasilovtype compensator fabricated specifically for this instrument. (1) Improved calibration in dual rotating compensator ellipsometry For higher precision in dual rotating compensator ellipsometry, the retardance and dichroic angles must be fit with smooth functions. Improved theory of the achromatic 169

189 compensator based on simulation of the three reflection processes is needed for such simulations. The theory may uncover problems with this element that can be incorporated into its Mueller matrix. In addition, calibration procedures must be developed that over-determine the azimuthal angles C S1, C S2, and A S, so that greater confidence in the deduced results can be achieved. This may be possible by setting A S in two different zones (i.e., A and A+9) and repeating the calibration twice. It may also be possible that by performing the measurement in reflection from a sample, then the additional available Fourier components provide greater opportunities for high accuracy calibration. 17

190 References [1-1] I. P. Herman, Optical Diagnostics for Thin Film Processing, Academic, New York, [1-2] R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, North- Holland, Amsterdam, [1-3] P. Drude, Ann. Phys. Chem. (Leipzig) 189, [1-4] Dennis Goldstein, Polarized Light, CRC, Boca Raton, 23. [1-5] B. D. Cahan and R.F. Spanier, A high speed precision automatic ellipsometer, Surf. Sci 1969, 16, [1-6] S. N. Jasperson and S. E. Schnatterly, An improved method for high reflectivity ellipsometry based on a new polarization modulation technique, Rev. Sci. Instrum. 1969, 4 (6), [1-7] J. A. Zapien, R. W. Collins, R. Messier, Multichannel ellipsometer for real time spectroscopy of thin film deposition from 1.5 to 6.5 ev Rev. Sci. Inst. 2, 71 (9), [1-8] J. Lee, P.I. Rovira, Ilsin An, and R.W. Collins, "Rotating compensator multichannel ellipsometry: applications for real time Stokes vector spectroscopy of thin film growth", Rev. Sci. Inst. 1998, 69 (4),

191 [1-9] R.W. Collins and Joohyun Koh, "Dual rotating-compensator multichannel ellipsometer: an instrument design for real time Mueller matrix spectroscopy of surfaces and films", J. Opt. Soc. Am. A 1999, 16 (8), [1-1] C.Chen, I. An, and R.W. Collins, "Multichannel Mueller matrix ellipsometry for simultaneous real-time measurement of bulk isotropic and surface anisotropic complex dielectric functions of semiconductors", Phys. Rev. Lett. 23, 9 (21), article no [1-11] J. Li, B. Ramanujam, R.W. Collins, Dual rotating compensator ellipsometry: theory and simulations, Thin Solid Films 211, 519 (9), [1-12] P. Chindaudom and K. Vedam, Determination of the optical function n() of vitreous silica by spectroscopic ellipsometry with an achromatic compensator, Appl. Opt. 1993, 32 (31), [1-13] R.J. King and M.J. Downs, "Ellipsometry applied to films and dielectric substrates", Surf. Sci. 1969, 16, [2-1] R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, North- Holland, Amsterdam, [2-2] J. M. Bennett, "Polarizers". In Bass, Michael, Ed. Handbook of Optics, Volume II (2nd ed.), McGraw-Hill, New York, [2-3] D. L. Steinmetz, W. G. Phillips, M. Wirick, and F. F. Forbes, Polarizer for the vacuum ultraviolet Appl. Opt. 1967, 6 (6), [2-4] D. E. Aspnes, Alignment of an optically active biplate compensator, Appl. Opt. 1971, 1 (11),

192 [2-5] J. Lee, P. I. Rovira, I. An, and R. W. Collins, Alignment and calibration of the MgF 2 biplate compensator for applications in rotating-compensator multichannel ellipsometry, J. Opt. Soc. Am. A 21, 18 (8), [2-6] R. J. King and M. J. Downs, Ellipsometry applied to films on dielectric substrates Surf. Sci. 1969, 16, [2-7] Instruction Manual for VS25S2TO, Vincent Associates, 211. [2-8] Models 1412, 1412F, 1412XR, Photodiode detector Instruction Manual, EG&G Princeton Applied Research, [2-9] Y. Talmi and R. W. Simpson, Self-scanned photodiode array: a multichannel spectrometric detector, Appl. Opt. 198, 19 (9), [2-1] Model 1461 Detector Interface Preliminary operating Manual, EG&G Princeton Applied Research, [3-1] B.D. Johs, J. Hale, N.J. Ianno, C. Herzinger, T.E. Tiwald, and J.A. Woollam, Recent developments in spectroscopic ellipsometry for in-situ applications, Proc. SPIE, August 21, 4449, [3-2] R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, North- Holland, Amsterdam, [3-3] Dennis Goldstein, Polarized Light, CRC, Boco Raton, 23. [3-4] E.D. Palik, Handbook of Optical Constants, Academic, Orlando, Vol. 1, 1985; Vol. 2, [3-5] J. M. Bennett, A critical evaluation of rhomb-type quaterwave retarders, Applied Optics, September 197, 1, [3-6] H. G. Tompkins and E. Irene (Eds.), Handbook of Ellipsometry, William 173

193 Andrew, Norwich NY, 25. [4-1] S. F. Khalid, LabWindows/CVI Programming for Beginners, Prentice Hall PTR, Upper Saddle River, NJ, 2. [4-2] Model 1461 Detector Interface Preliminary Operating Manual, EG&G Princeton Applied Research, [4-3] J. R. Zeidler, R. B. Kohles, and N. M. Bashara, High precision alignment procedure for an ellipsometer, Appl. Opt. 1974, 13 (5), [4-4] H. G. Tompkins and E.A. Irene, editors, Handbook of Ellipsometry, William Andrew, Norwich, NY, 25. [4-5] OMA2 Applications Software Manual, EG&G Instruments Corporation, 199. [4-6] C. Palmer and E. Loewen, Diffraction Grating Handbook, Newport Corporation, Rochester, NY, 25. [4-7] I. An, Y. Cong, N.V. Nguyen, B.S. Pudliner, R. W. Collins, Instrumentation considerations in multichannel ellipsometry for real-time spectroscopy, Thin Solid Films 1991, 26 (1-2), [4-8] J. A. Zapien, R. W. Collins, R. Messier, Multichannel ellipsometer for real time spectroscopy of thin film deposition from 1.5 to 6.5 ev, Rev. Sci. Instum. 2, 71 (9), [5-1] H. G. Tompkins and E.A. Irene, editors, Handbook of Ellipsometry, William Andrew, Norwich, NY, 25. [5-2] D.E. Aspnes, Alignment of optically active biplate compensator, Appl. Opt. 1971, 1 (11), [5-3] M2 Spectroscopic Ellipsometer Hardware Manual, J.A. Woollam Co., Inc. 174

194 Lincoln NE, 27. [5-4] J. Lee, P.I. Rovira, I. An, and R.W. Collins, "Rotating compensator multichannel ellipsometry: applications for real time Stokes vector spectroscopy of thin film growth," Rev. Sci. Instrum. 1998, 69 (4), [5-5] J. Lee and R.W. Collins, "Real-time characterization of film growth on transparent substrates by rotating compensator multichannel ellipsometry," Appl. Opt. 1998, 37 (19), [5-6] J. Lee, P.I. Rovira, I. An, and R.W. Collins, "Alignment and calibration of the MgF 2 biplate compensator for applications in rotating-compensator multichannel ellipsometry, J. Opt. Soc. Am. A 21, 18 (8), [5-7] I. An, J. A. Zapien, C. Chen, A. S. Ferlauto, A. S. Lawrence, and R. W. Collins, "Calibration and data reduction for a UV-extended rotating-compensator multichannel ellipsometer," Thin Solid Films 24, , [5-8] E.D. Palik, Editor, Handbook of Optical Constants, Academic, Orlando, Vol. 1, 1985; Vol. 2, [6-1] H. G. Tompkins and E. A. Irene, Handbook of Ellipsometry, William Andrew, Norwich NY, 25. [6-2] R. J. King and M. J. Downs, "Ellipsometry applied to films and dielectric substrates", Surface Science 1969, 16, [6-3] P. Chindaudom and K. Vedam, Determination of the optical function n() of vitreous silica by spectroscopic ellipsometry with an achromatic compensator, Applied Optics 1993, 32 (31),

195 [6-4] R. W. Collins and J. Koh, "Dual rotating-compensator multichannel ellipsometer: an instrument design for real time Mueller matrix spectroscopy of surfaces and films", Journal of the Optical Society of America A 1999, 16 (8), [6-5] J. Lee, J. Koh, and R. W. Collins, "Dual rotating-compensator multichannel ellipsometer: instrument development for high-speed Mueller matrix spectroscopy of surfaces and thin films", Review of Scientific Instruments 21, 72 (3), [6-6] C. Chen, I. An, and R. W. Collins, "Multichannel Mueller matrix ellipsometry for simultaneous real-time measurement of bulk isotropic and surface anisotropic complex dielectric functions of semiconductors", Physical Review Letters 23, 9 (21), article no [6-7] J. Li, B. Ramanujam, R. W. Collins, Dual rotating compensator ellipsometry: theory and simulations, Thin Solid Films 211, 519 (9) [6-8] R.W. Collins et al. (unpublished paper, 212). 176

196 Appendix A User drawing of spectrograph CP The user drawing of spectrograph obtained from the manufacturer Jobin Yvon is shown in the Figure A

197 Figure A-1: User drawing of spectrograph Jobin Yvon CP

198 Appendix B Optical properties of materials Optical properties of materials obtained from literature [1-2] and used in this thesis research are provided in the following sections. 179

199 18 Optical properties of MgF 2 : e N = n e ik e nm n e nm k e nm n e k e k e n e

200 181 nm n e k e k e nm n e k e nm n e

201 182 nm n e k e k e nm n e k e n e nm

202 183 nm n e k e k e nm n e k e nm n e

203 184 nm n e k k e nm n e k e nm n e

204 185 nm n e k e

205 186 Optical properties of MgF 2 : o N = n o ik o nm n o nm k o nm n o k o k o n o

206 187 nm n k k n nm n nm k

207 188 nm n k k nm n k n nm

208 189 nm n k k nm n k nm n

209 19 nm n k k nm n k nm n

210 191 nm n k

211 192 Optical properties of SiO 2 nm n nm k nm n k k n

212 193 nm n k k n nm n nm k

213 194 nm n k k nm n k n nm

214 195 nm n k k nm n k nm n

215 196 nm n k k nm n k nm n

216 197 nm n k

217 198 Optical properties of Aluminum nm n nm k nm n k n k

218 199 nm n k k n nm n nm k

219 2 nm n k k nm n k n nm

220 21 nm n k nm n k nm n k

221 22 nm n k k nm n k nm n

222 23 nm n k

223 Appendix C Program to view live irradiance at the detector User interface Figure C-1: Snapshot of user interface for live irradiance scan 24

224 Source code ************************************************************************ #include <formatio.h> #include <visa.h> #include <utility.h> #include <ansi_c.h> #include <cvirte.h> #include <userint.h> #include "prog4contview.h" #define MAX_CNT 2 FILE *fpr,*fpw,*waveval; static int panelhandle; int stat,i,j,k,m,a[125],npixels[125]; double tini,time1; float wavevalue[125],wave,k1,c; ViStatus status; ViSession defaultrm, instr; ViUInt32 retcount; //ViPBuf buffer1; ViChar buffer[max_cnt]; ViUInt16 nbyte; int stat, final, Mask; int num; char st1[4],st2[4]; int main (int argc, char *argv[]) if (InitCVIRTE (, argv, ) == ) return -1; /* out of memory */ if ((panelhandle = LoadPanel (, "prog4contview.uir", PANEL)) < ) return -1; DisplayPanel (panelhandle); RunUserInterface (); DiscardPanel (panelhandle); return ; int CVICALLBACK Switch_CB (int panel, int control, int event, void *callbackdata, int eventdata1, int eventdata2) 25

225 switch (event) case EVENT_COMMIT: GetCtrlVal(PANEL,PANEL_SWITCH,&stat); // i=1; while (stat) tini=timer(); while((timer()-tini)<.1) time1=timer()-tini; SetActivePanel(PANEL); ProcessSystemEvents(); GetCtrlVal(PANEL,PANEL_SWITCH,&stat); if (stat==) break; // Open file for Writing Data if (stat==) break; if ((fpw=fopen("continuousscan.txt","w"))==null) printf("cannot open thefile"); exit(1); //Start Data Acquisition // Start data Aquistion status = viwrite (instr, "AN;DA 8;J 1;RUN",15, &retcount); if(status<vi_success) printf("error tranferring data"); return -1; // Detrmining Status Byte status = vireadstb(instr,&nbyte); stat = *(&nbyte); // printf("2\n"); if(status<vi_success) 26

226 printf("problem with Timeout"); return -1; // printf("status byte =%d\n",stat); //Wait until status=33 while(stat!=33) status = vireadstb(instr,&nbyte); stat = *(&nbyte); //printf("2\n"); if(status<vi_success) printf("problem with Timeout"); return -1; // printf("status=%d\n",stat); //Final Status before reading status = vireadstb(instr,&nbyte); stat = *(&nbyte); //printf("2\n"); if(status<vi_success) printf("problem with Timeout"); return -1; // printf("final Status before reading =%d\n",stat); // ////Start Reading from Memory //Status status = vireadstb(instr,&nbyte); stat = *(&nbyte); //printf("2\n"); if(status<vi_success) 27

227 printf("problem with Timeout"); return -1; // printf("status=%d\n",stat); // Read the Data //From memory 1 status = viwrite (instr, "DC ",11, &retcount); if(status<vi_success) printf("error tranferring data DC"); return -1; status = viread(instr, buffer, MAX_CNT, &retcount); if(status<vi_success) printf("error reading data DLEN"); return -1; fprintf(fpw,"%s",buffer); fclose(fpw); //Wait until status=33 while(stat!=33) status = vireadstb(instr,&nbyte); stat = *(&nbyte); //printf("2\n"); if(status<vi_success) printf("problem with Timeout"); return -1; // printf("from memory Status=%d\n",stat); //Status status = vireadstb(instr,&nbyte); stat = *(&nbyte); //printf("2\n"); if(status<vi_success) 28

228 printf("problem with Timeout"); return -1; // printf("status after first scan =%d\n",stat); //printf("1st scan Copied from Memory\n"); //Open file for reading if ((fpr=fopen("continuousscan.txt","r"))==null) printf("cannot open thefile"); exit(1); //Read from file //Plot graph for (i=1;i<125;i++) fscanf(fpr,"%d",&a[i]); //fscanf(pixel,"%d",&npixels[i]); PlotXY (PANEL, PANEL_GRAPH, npixels, a, 124, VAL_INTEGER, VAL_INTEGER, VAL_THIN_LINE, VAL_EMPTY_SQUARE, VAL_SOLID, 1, VAL_RED); // PlotXY (PANEL, PANEL_GRAPH, npixels, a, 124, VAL_INTEGER, VAL_INTEGER, VAL_SCATTER, VAL_EMPTY_SQUARE, // VAL_DOT, 1, VAL_RED); //PlotXY (PANEL, PANEL_GRAPH, npixels, a, 9, VAL_INTEGER, VAL_INTEGER, VAL_THIN_LINE, VAL_EMPTY_SQUARE, // VAL_SOLID, 1, VAL_RED); // printf("%d",i); /***********************WAVELENGTH CALIBRATION PART*********************************/ //Open file to save wavelength if ((waveval=fopen("wavelengthfile.txt","w"))==null) 29

229 printf("cannot open thefile"); exit(1); //Find Highest Intensity for(k=2,j=1;k<125;k++) if(a[k]> a[j]) j=k; // printf("\n\n\n\nfirst Highest = %d\n",j); //Find Second highest Intensity for(k=2,m=1;k<125;k++) if (k==j-5) k=k+1; if (a[k]>a[m]) m=k; //Determining slope and Y-Intercept k1= (11.24/(j-m)); c= k1*j; // Displaying Two peak Intensities on a Text message Fmt(st1,"%i",j); strcat(st1," FIRST PEAK IS "); SetCtrlVal (PANEL, PANEL_HIGHPEAK, st1); Fmt(st2,"%i",m); strcat(st2," SECOND PEAK IS "); SetCtrlVal (PANEL, PANEL_LOWPEAK, st2); //Doing wavelength Calibration for (i=1;i<125;i++) 21

230 wave= k1*i+c; wavevalue[i]=wave; fprintf(waveval,"%f\n",wave); //Plot the Wavelength Vs Intensity PlotXY (PANEL, PANEL_GRAPH2, wavevalue, a, 124, VAL_FLOAT, VAL_INTEGER, VAL_THIN_LINE, VAL_EMPTY_SQUARE, VAL_SOLID, 1, VAL_RED); //Close the Intensity file and Wavelength File fclose(waveval); fclose(fpr); return ; break; int CVICALLBACK Stop_CB (int panel, int control, int event, void *callbackdata, int eventdata1, int eventdata2) switch (event) case EVENT_COMMIT: // fclose(fp1); // fclose(pixel); status = viclose(instr); status = viclose(defaultrm); QuitUserInterface (); break; return ; int CVICALLBACK Start_CB (int panel, int control, int event, void *callbackdata, int eventdata1, int eventdata2) switch (event) case EVENT_COMMIT: status = viopendefaultrm(&defaultrm); 211

231 // printf("1\n"); if (status < VI_SUCCESS) printf("error Initializing VISA...exiting"); return -1; status = viopen(defaultrm, "GPIB::8::INSTR", VI_NULL, VI_NULL, &instr); // printf("2\n"); if (status < VI_SUCCESS) printf("error opening GPIB"); return -1; status = visetattribute(instr, VI_ATTR_TMO_VALUE, 8); if (status < VI_SUCCESS) printf("problem with Timeout"); return -1; // printf("timeout Sucessfully executed"); return ; break; int CVICALLBACK Status_CB (int panel, int control, int event, void *callbackdata, int eventdata1, int eventdata2) switch (event) case EVENT_COMMIT: ///Detmining ID status = viwrite (instr, "ID",2, &retcount); if(status<vi_success) printf("error tranferring data"); return -1; 212

232 status = viread(instr, buffer, MAX_CNT, &retcount); // printf("\nid=%s\n", buffer); // Detrmining Status Byte status = vireadstb(instr,&nbyte); stat = *(&nbyte); // printf("2\n"); if(status<vi_success) printf("problem with Timeout"); return -1; // printf("status=%d\n",stat); // Define Delimiter status = viwrite (instr, "DD 1",5, &retcount); if(status<vi_success) printf("error tranferring data"); return -1; //Fill the Array with Pixel numbers for (i=1;i<125;i++) npixels[i]=i; return ; break; int CVICALLBACK Data_CB (int panel, int control, int event, void *callbackdata, int eventdata1, int eventdata2) switch (event) case EVENT_COMMIT: 213

233 /*// Start data Aquistion status = viwrite (instr, "AN;DA 8;J 1;RUN",15, &retcount); if(status<vi_success) printf("error tranferring data"); return -1; // Detrmining Status Byte status = vireadstb(instr,&nbyte); stat = *(&nbyte); printf("2\n"); if(status<vi_success) printf("problem with Timeout"); return -1; printf("status byte =%d\n",stat); //Wait until status=33 while(stat!=33) status = vireadstb(instr,&nbyte); stat = *(&nbyte); //printf("2\n"); if(status<vi_success) printf("problem with Timeout"); return -1; printf("status=%d\n",stat); //Final Status before reading status = vireadstb(instr,&nbyte); stat = *(&nbyte); //printf("2\n"); if(status<vi_success) 214

234 printf("problem with Timeout"); return -1; printf("final Status before reading =%d\n",stat); */ return ; break; int CVICALLBACK Read_CB (int panel, int control, int event, void *callbackdata, int eventdata1, int eventdata2) switch (event) case EVENT_COMMIT: /*//Open file if ((fp1=fopen("sampscan12.txt","w"))==null) printf("cannot open thefile"); exit(1); //Status status = vireadstb(instr,&nbyte); stat = *(&nbyte); //printf("2\n"); if(status<vi_success) printf("problem with Timeout"); return -1; printf("status=%d\n",stat); // Read the Data //From memory 1 status = viwrite (instr, "DC ",11, &retcount); if(status<vi_success) printf("error tranferring data DC"); return -1; 215

235 status = viread(instr, buffer, MAX_CNT, &retcount); if(status<vi_success) printf("error reading data DLEN"); return -1; fprintf(fp1,"%s",buffer); //Wait until status=33 while(stat!=33) status = vireadstb(instr,&nbyte); stat = *(&nbyte); //printf("2\n"); if(status<vi_success) printf("problem with Timeout"); return -1; //Status printf("from memory Status=%d\n",stat); status = vireadstb(instr,&nbyte); stat = *(&nbyte); //printf("2\n"); if(status<vi_success) printf("problem with Timeout"); return -1; printf("status after first scan =%d\n",stat); printf("1st scan Copied from Memory\n");*/ return ; break; 216

236 int CVICALLBACK Poll_CB (int panel, int control, int event, void *callbackdata, int eventdata1, int eventdata2) switch (event) case EVENT_COMMIT: // Detrmining Status Byte status = vireadstb(instr,&nbyte); stat = *(&nbyte); printf("2\n"); if(status<vi_success) printf("problem with Timeout"); return -1; printf("%d\n",stat); break; return ; ************************************************************************ 217

237 Appendix D Basic data acquisition program User Interface Figure D-1: Snapshot of basic data acquisition user interface 218

238 Source code ************************************************************************ #include <visa.h> #include <ansi_c.h> #include <cvirte.h> #include <userint.h> #include "Groupedscan with corrections.h" #define MAX_CNT 2 FILE *fp1,*fp2,*fp3,*fp4,*fp5,*fp6,*fp7,*fp8,*bg1,*bg2,*bg3,*bg4,*bg5,*bg6,*bg7,*bg8; FILE *c1,*c2,*c3,*c4,*c5,*c6,*c7,*c8,*ip,*i,*b4,*a4,*teta4; static int panelhandle; ViStatus status,statusst; ViSession defaultrm, instr,instrst; ViUInt32 retcount; ViChar buffer[max_cnt]; ViUInt16 nbyte; int stat, final, Mask; int num,x,i,j,k; int sc1[129],sc2[129],sc3[129],sc4[129],sc5[129],sc6[129],sc7[129],sc8[129]; int bc1[129],bc2[129],bc3[129],bc4[129],bc5[129],bc6[129],bc7[129],bc8[129]; int cd1[129],cd2[129],cd3[129],cd4[129],cd5[129],cd6[129],cd7[129],cd8[129]; float ipf[129],f1[129],f2[129],f3[129],f4[129],f5[129],f6[129],f7[129],f8[129],intensity[129]; float alpha4[129],beta4[129],theta4[129]; int main (int argc, char *argv[]) if (InitCVIRTE (, argv, ) == ) return -1; /* out of memory */ if ((panelhandle = LoadPanel (, "Groupedscan with corrections.uir", PANEL)) < ) return -1; DisplayPanel (panelhandle); RunUserInterface (); DiscardPanel (panelhandle); return ; 219

239 int CVICALLBACK Start_CB (int panel, int control, int event, void *callbackdata, int eventdata1, int eventdata2) switch (event) case EVENT_COMMIT: // Open a session status = viopendefaultrm(&defaultrm); // printf("1\n"); if (status < VI_SUCCESS) printf("error Initializing VISA...exiting"); return -1; // Session for Detector status = viopen(defaultrm, "GPIB::8::INSTR", VI_NULL, VI_NULL, &instr); // printf("2\n"); if (status < VI_SUCCESS) printf("error opening GPIB"); return -1; status = visetattribute(instr, VI_ATTR_TMO_VALUE, 8); if (status < VI_SUCCESS) printf("problem with Timeout"); return -1; printf("timeout Sucessfully executed"); // Session for Stepper Motor statusst= viopen (defaultrm, "ASRL1::INSTR", VI_NULL, VI_NULL, &instrst); if(statusst<vi_success) printf("error opening Instr"); return -1; statusst = visetattribute(instrst, VI_ATTR_TMO_VALUE, 6); if(statusst<vi_success) 22

240 printf("error setting timeout"); return -1; // statusst = viwrite (instrst, "RESET\n",6, VI_NULL); printf("stepper Motors Initiated"); return ; break; int CVICALLBACK Status_CB (int panel, int control, int event, void *callbackdata, int eventdata1, int eventdata2) switch (event) case EVENT_COMMIT: ///Detmining ID status = viwrite (instr, "ID",2, &retcount); if(status<vi_success) printf("error tranferring data"); return -1; status = viread(instr, buffer, MAX_CNT, &retcount); printf("\nid=%s\n", buffer); // Detrmining Status Byte status = vireadstb(instr,&nbyte); stat = *(&nbyte); // printf("2\n"); if(status<vi_success) printf("problem with Timeout"); return -1; // printf("status=%d\n",stat); // Define Delimiter status = viwrite (instr, "DD 1",5, &retcount); 221

241 if(status<vi_success) printf("error tranferring data"); return -1; // Create files to save data if ((fp1=fopen("s1.txt","w"))==null) printf("cannot open thefile"); exit(1); if ((fp2=fopen("s2.txt","w"))==null) printf("cannot open thefile"); exit(1); if ((fp3=fopen("s3.txt","w"))==null) printf("cannot open thefile"); exit(1); if ((fp4=fopen("s4.txt","w"))==null) printf("cannot open thefile"); exit(1); if ((fp5=fopen("s5.txt","w"))==null) printf("cannot open thefile"); exit(1); if ((fp6=fopen("s6.txt","w"))==null) printf("cannot open thefile"); exit(1); if ((fp7=fopen("s7.txt","w"))==null) printf("cannot open thefile"); exit(1); if ((fp8=fopen("s8.txt","w"))==null) printf("cannot open thefile"); exit(1); 222

242 // Back ground Correction files if ((bg1=fopen("bgscan1.txt","w"))==null) printf("cannot open thefile"); exit(1); if ((bg2=fopen("bgscan2.txt","w"))==null) printf("cannot open thefile"); exit(1); if ((bg3=fopen("bgscan3.txt","w"))==null) printf("cannot open thefile"); exit(1); if ((bg4=fopen("bgscan4.txt","w"))==null) printf("cannot open thefile"); exit(1); if ((bg5=fopen("bgscan5.txt","w"))==null) printf("cannot open thefile"); exit(1); if ((bg6=fopen("bgscan6.txt","w"))==null) printf("cannot open thefile"); exit(1); if ((bg7=fopen("bgscan7.txt","w"))==null) printf("cannot open thefile"); exit(1); if ((bg8=fopen("bgscan8.txt","w"))==null) printf("cannot open thefile"); exit(1); // Corrected data if ((c1=fopen("scan1.txt","w"))==null) printf("cannot open thefile"); 223

243 exit(1); if ((c2=fopen("scan2.txt","w"))==null) printf("cannot open thefile"); exit(1); if ((c3=fopen("scan3.txt","w"))==null) printf("cannot open thefile"); exit(1); if ((c4=fopen("scan4.txt","w"))==null) printf("cannot open thefile"); exit(1); if ((c5=fopen("scan5.txt","w"))==null) printf("cannot open thefile"); exit(1); if ((c6=fopen("scan6.txt","w"))==null) printf("cannot open thefile"); exit(1); if ((c7=fopen("scan7.txt","w"))==null) printf("cannot open thefile"); exit(1); if ((c8=fopen("scan8.txt","w"))==null) printf("cannot open thefile"); exit(1); // Opening IPCF File if ((ip=fopen("ipcf.txt","r"))==null) printf("cannot open thefile"); exit(1); // Opening files for calculating Fourier Coefficients if ((I=fopen("As Intensity/Intensity -1.1.txt","w"))==NULL) 224

244 printf("cannot open thefile"); exit(1); if ((a4=fopen("a4.txt","w"))==null) printf("cannot open thefile"); exit(1); if ((b4=fopen("b4.txt","w"))==null) printf("cannot open thefile"); exit(1); if ((teta4=fopen("beta4 at 85..txt","w"))==NULL) printf("cannot open thefile"); exit(1); printf("all FILES OPENED\n"); return ; break; int CVICALLBACK Closeshutter_CB (int panel, int control, int event, void *callbackdata, int eventdata1, int eventdata2) switch (event) case EVENT_COMMIT: statusst= viprintf(instrst,"<1\n"); statusst = viprintf(instrst,"out(1)=\n"); break; return ; int CVICALLBACK Openshutter_CB (int panel, int control, int event, void *callbackdata, int eventdata1, int eventdata2) switch (event) 225

245 case EVENT_COMMIT: statusst= viprintf(instrst,"<1\n"); statusst = viprintf(instrst,"out(1)=1\n"); return ; break; int CVICALLBACK Readoffset_CB (int panel, int control, int event, void *callbackdata, int eventdata1, int eventdata2) switch (event) case EVENT_COMMIT: //Status status = vireadstb(instr,&nbyte); stat = *(&nbyte); //printf("2\n"); if(status<vi_success) printf("problem with Timeout"); return -1; printf("status=%d\n",stat); // Read the Data //From memory 1 status = viwrite (instr, "DC ",1, &retcount); if(status<vi_success) printf("error tranferring data DC"); return -1; status = viread(instr, buffer, MAX_CNT, &retcount); if(status<vi_success) printf("error reading data DLEN"); return -1; fprintf(bg1,"%s",buffer); //Wait until status=33 226

246 while(stat!=33) status = vireadstb(instr,&nbyte); stat = *(&nbyte); //printf("2\n"); if(status<vi_success) printf("problem with Timeout"); return -1; printf("from memory Status=%d\n",stat); //Status status = vireadstb(instr,&nbyte); stat = *(&nbyte); //printf("2\n"); if(status<vi_success) printf("problem with Timeout"); return -1; printf("status after first scan =%d\n",stat); printf("1st scan Copied from Memory\n"); //From memory 2 status = viwrite (instr, "DC ",1, &retcount); if(status<vi_success) printf("error tranferring data DC"); return -1; status = viread(instr, buffer, MAX_CNT, &retcount); if(status<vi_success) printf("error reading data DLEN"); return -1; 227

247 fprintf(bg2,"%s",buffer); printf("2nd scan Copied from Memory\n"); //From memory 3 status = viwrite (instr, "DC ",1, &retcount); if(status<vi_success) printf("error tranferring data DC"); return -1; status = viread(instr, buffer, MAX_CNT, &retcount); if(status<vi_success) printf("error reading data DLEN"); return -1; fprintf(bg3,"%s",buffer); printf("3rd scan Copied from Memory\n"); //From memory 4 status = viwrite (instr, "DC ",1, &retcount); if(status<vi_success) printf("error tranferring data DC"); return -1; status = viread(instr, buffer, MAX_CNT, &retcount); if(status<vi_success) printf("error reading data DLEN"); return -1; fprintf(bg4,"%s",buffer); printf("4th scan Copied from Memory\n"); //From memory 5 status = viwrite (instr, "DC ",1, &retcount); if(status<vi_success) printf("error tranferring data DC"); return -1; status = viread(instr, buffer, MAX_CNT, &retcount); if(status<vi_success) 228

248 printf("error reading data DLEN"); return -1; fprintf(bg5,"%s",buffer); printf("5th scan Copied from Memory\n"); //From memory 6 status = viwrite (instr, "DC ",1, &retcount); if(status<vi_success) printf("error tranferring data DC"); return -1; status = viread(instr, buffer, MAX_CNT, &retcount); if(status<vi_success) printf("error reading data DLEN"); return -1; fprintf(bg6,"%s",buffer); printf("6th scan Copied from Memory\n"); //From memory 7 status = viwrite (instr, "DC ",1, &retcount); if(status<vi_success) printf("error tranferring data DC"); return -1; status = viread(instr, buffer, MAX_CNT, &retcount); if(status<vi_success) printf("error reading data DLEN"); return -1; fprintf(bg7,"%s",buffer); printf("7th scan Copied from Memory\n"); //From memory 8 status = viwrite (instr, "DC ",1, &retcount); if(status<vi_success) printf("error tranferring data DC"); return -1; status = viread(instr, buffer, MAX_CNT, &retcount); 229

249 if(status<vi_success) printf("error reading data DLEN"); return -1; fprintf(bg8,"%s",buffer); printf("8th scan Copied from Memory\n"); return ; break; int CVICALLBACK Readdata_CB (int panel, int control, int event, void *callbackdata, int eventdata1, int eventdata2) switch (event) case EVENT_COMMIT: //Status status = vireadstb(instr,&nbyte); stat = *(&nbyte); //printf("2\n"); if(status<vi_success) printf("problem with Timeout"); return -1; printf("status=%d\n",stat); // Read the Data //From memory 1 status = viwrite (instr, "DC ",1, &retcount); if(status<vi_success) printf("error tranferring data DC"); return -1; status = viread(instr, buffer, MAX_CNT, &retcount); if(status<vi_success) printf("error reading data DLEN"); 23

250 return -1; fprintf(fp1,"%s",buffer); //Wait until status=33 while(stat!=33) status = vireadstb(instr,&nbyte); stat = *(&nbyte); //printf("2\n"); if(status<vi_success) printf("problem with Timeout"); return -1; printf("from memory Status=%d\n",stat); //Status status = vireadstb(instr,&nbyte); stat = *(&nbyte); //printf("2\n"); if(status<vi_success) printf("problem with Timeout"); return -1; printf("status after first scan =%d\n",stat); printf("1st scan Copied from Memory\n"); //From memory 2 status = viwrite (instr, "DC ",1, &retcount); if(status<vi_success) printf("error tranferring data DC"); return -1; status = viread(instr, buffer, MAX_CNT, &retcount); 231

251 if(status<vi_success) printf("error reading data DLEN"); return -1; fprintf(fp2,"%s",buffer); printf("2nd scan Copied from Memory\n"); //From memory 3 status = viwrite (instr, "DC ",1, &retcount); if(status<vi_success) printf("error tranferring data DC"); return -1; status = viread(instr, buffer, MAX_CNT, &retcount); if(status<vi_success) printf("error reading data DLEN"); return -1; fprintf(fp3,"%s",buffer); printf("3rd scan Copied from Memory\n"); //From memory 4 status = viwrite (instr, "DC ",1, &retcount); if(status<vi_success) printf("error tranferring data DC"); return -1; status = viread(instr, buffer, MAX_CNT, &retcount); if(status<vi_success) printf("error reading data DLEN"); return -1; fprintf(fp4,"%s",buffer); printf("4th scan Copied from Memory\n"); //From memory 5 status = viwrite (instr, "DC ",1, &retcount); if(status<vi_success) printf("error tranferring data DC"); return -1; 232

252 status = viread(instr, buffer, MAX_CNT, &retcount); if(status<vi_success) printf("error reading data DLEN"); return -1; fprintf(fp5,"%s",buffer); printf("5th scan Copied from Memory\n"); //From memory 6 status = viwrite (instr, "DC ",1, &retcount); if(status<vi_success) printf("error tranferring data DC"); return -1; status = viread(instr, buffer, MAX_CNT, &retcount); if(status<vi_success) printf("error reading data DLEN"); return -1; fprintf(fp6,"%s",buffer); printf("6th scan Copied from Memory\n"); //From memory 7 status = viwrite (instr, "DC ",1, &retcount); if(status<vi_success) printf("error tranferring data DC"); return -1; status = viread(instr, buffer, MAX_CNT, &retcount); if(status<vi_success) printf("error reading data DLEN"); return -1; fprintf(fp7,"%s",buffer); printf("7th scan Copied from Memory\n"); //From memory 8 status = viwrite (instr, "DC ",1, &retcount); if(status<vi_success) printf("error tranferring data DC"); 233

253 return -1; status = viread(instr, buffer, MAX_CNT, &retcount); if(status<vi_success) printf("error reading data DLEN"); return -1; fprintf(fp8,"%s",buffer); printf("8th scan Copied from Memory\n"); return ; break; int CVICALLBACK Docorrections_CB (int panel, int control, int event, void *callbackdata, int eventdata1, int eventdata2) switch (event) case EVENT_COMMIT: fclose (fp1); fclose (fp2); fclose (fp3); fclose (fp4); fclose (fp5); fclose (fp6); fclose (fp7); fclose (fp8); fclose (bg1); fclose (bg2); fclose (bg3); fclose (bg4); fclose (bg5); fclose (bg6); fclose (bg7); fclose (bg8); // Create files to save data if ((fp1=fopen("s1.txt","r"))==null) printf("cannot open thefile"); exit(1); 234

254 if ((fp2=fopen("s2.txt","r"))==null) printf("cannot open thefile"); exit(1); if ((fp3=fopen("s3.txt","r"))==null) printf("cannot open thefile"); exit(1); if ((fp4=fopen("s4.txt","r"))==null) printf("cannot open thefile"); exit(1); if ((fp5=fopen("s5.txt","r"))==null) printf("cannot open thefile"); exit(1); if ((fp6=fopen("s6.txt","r"))==null) printf("cannot open thefile"); exit(1); if ((fp7=fopen("s7.txt","r"))==null) printf("cannot open thefile"); exit(1); if ((fp8=fopen("s8.txt","r"))==null) printf("cannot open thefile"); exit(1); // Back ground Correction files if ((bg1=fopen("bgscan1.txt","r"))==null) printf("cannot open thefile"); exit(1); if ((bg2=fopen("bgscan2.txt","r"))==null) printf("cannot open thefile"); exit(1); 235

255 if ((bg3=fopen("bgscan3.txt","r"))==null) printf("cannot open thefile"); exit(1); if ((bg4=fopen("bgscan4.txt","r"))==null) printf("cannot open thefile"); exit(1); if ((bg5=fopen("bgscan5.txt","r"))==null) printf("cannot open thefile"); exit(1); if ((bg6=fopen("bgscan6.txt","r"))==null) printf("cannot open thefile"); exit(1); if ((bg7=fopen("bgscan7.txt","r"))==null) printf("cannot open thefile"); exit(1); if ((bg8=fopen("bgscan8.txt","r"))==null) printf("cannot open thefile"); exit(1); for (i=1;i<129;i++) fscanf(fp1,"%d",&sc1[i]); fscanf(fp2,"%d",&sc2[i]); fscanf(fp3,"%d",&sc3[i]); fscanf(fp4,"%d",&sc4[i]); fscanf(fp5,"%d",&sc5[i]); fscanf(fp6,"%d",&sc6[i]); fscanf(fp7,"%d",&sc7[i]); fscanf(fp8,"%d",&sc8[i]); fscanf(bg1,"%d",&bc1[i]); fscanf(bg2,"%d",&bc2[i]); fscanf(bg3,"%d",&bc3[i]); fscanf(bg4,"%d",&bc4[i]); 236

256 fscanf(bg5,"%d",&bc5[i]); fscanf(bg6,"%d",&bc6[i]); fscanf(bg7,"%d",&bc7[i]); fscanf(bg8,"%d",&bc8[i]); fscanf(ip,"%f",&ipf[i]); for (j=1;j<129;j++) cd1[j]=sc1[j]-bc1[j]; fprintf(c1,"%d\n",cd1[j]); cd2[j]=sc2[j]-bc2[j]; fprintf(c2,"%d\n",cd2[j]); cd3[j]=sc3[j]-bc3[j]; fprintf(c3,"%d\n",cd3[j]); cd4[j]=sc4[j]-bc4[j]; fprintf(c4,"%d\n",cd4[j]); cd5[j]=sc5[j]-bc5[j]; fprintf(c5,"%d\n",cd5[j]); cd6[j]=sc6[j]-bc6[j]; fprintf(c6,"%d\n",cd6[j]); cd7[j]=sc7[j]-bc7[j]; fprintf(c7,"%d\n",cd7[j]); cd8[j]=sc8[j]-bc8[j]; fprintf(c8,"%d\n",cd8[j]); for (k=1;k<129;k++) f1[k]=cd1[k]+cd1[k]*ipf[k]-cd8[k]*ipf[k]; // fprintf(c1,"%f\n",f1[k]); f2[k]=cd2[k]+cd2[k]*ipf[k]-cd1[k]*ipf[k]; // fprintf(c2,"%f\n",f2[k]); f3[k]=cd3[k]+cd3[k]*ipf[k]-cd2[k]*ipf[k]; // fprintf(c3,"%f\n",f3[k]); f4[k]=cd4[k]+cd4[k]*ipf[k]-cd3[k]*ipf[k]; // fprintf(c4,"%f\n",f4[k]); f5[k]=cd5[k]+cd5[k]*ipf[k]-cd4[k]*ipf[k]; // fprintf(c5,"%f\n",f5[k]); f6[k]=cd6[k]+cd6[k]*ipf[k]-cd5[k]*ipf[k]; // fprintf(c6,"%f\n",f6[k]); f7[k]=cd7[k]+cd7[k]*ipf[k]-cd6[k]*ipf[k]; // fprintf(c7,"%f\n",f7[k]); f8[k]=cd8[k]+cd8[k]*ipf[k]-cd7[k]*ipf[k]; // fprintf(c8,"%f\n",f8[k]); 237

257 // Calculating b4' and a4' intensity[k]=(f1[k]+f2[k]+f3[k]+f4[k]+f5[k]+f6[k]+f7[k]+f8[k])/( *4); fprintf(i,"%f\n",intensity[k]); alpha4[k]=(f1[k]-f2[k]-f3[k]+f4[k]+f5[k]-f6[k]-f7[k]+f8[k])/(2*intensity[k]); fprintf(a4,"%f\n",alpha4[k]); beta4[k]= (f1[k]+f2[k]-f3[k]-f4[k]+f5[k]+f6[k]-f7[k]-f8[k])/(2*intensity[k]); fprintf(b4,"%f\n",beta4[k]); // theta4[k]= sqrt(alpha4[k]*alpha4[k]+beta4[k]*beta4[k]); /*theta4[k]=(((atan(beta4[k]/alpha4[k]))/2)*18)/ ; fprintf(teta4,"%f\n",theta4[k]); */ printf("corrections done\n"); return ; break; int CVICALLBACK Stop_CB (int panel, int control, int event, void *callbackdata, int eventdata1, int eventdata2) switch (event) case EVENT_COMMIT: status = viclose(instr); status = viclose(instrst); status = viclose(defaultrm); fclose(fp1); fclose(fp2); fclose(fp3); fclose(fp4); fclose(fp5); fclose(fp6); fclose(fp7); fclose(fp8); 238

258 fclose(bg1); fclose(bg2); fclose(bg3); fclose(bg4); fclose(bg5); fclose(bg6); fclose(bg7); fclose(bg8); fclose(c1); fclose(c2); fclose(c3); fclose(c4); fclose(c5); fclose(c6); fclose(c7); fclose(c8); fclose(ip); fclose(i); fclose(a4); fclose(b4); fclose(teta4); return ; QuitUserInterface (); break; int CVICALLBACK Dataacquisition_CB (int panel, int control, int event, void *callbackdata, int eventdata1, int eventdata2) switch (event) case EVENT_COMMIT: // Start data Aquistion // status = viwrite (instr, "AN;DA 8;J 8;RUN",15, &retcount); // status = viwrite (instr, "SS;GS 8;GR 128;ES;DA 8;J 8;RUN",3, &retcount); status = viwrite (instr, "SS;GS 8;GR 128;ES;DA 9;I 1;J 8;K ;RUN",38, &retcount); // FOR LESS PIXEL //status = viwrite (instr, "SS;GS 8;GR 118;FA 8;ES;DA 9;I 256;J 8;K ;RUN",46, &retcount); 239

259 if(status<vi_success) printf("error tranferring data"); return -1; // Detrmining Status Byte status = vireadstb(instr,&nbyte); stat = *(&nbyte); // printf("2\n"); if(status<vi_success) printf("problem with Timeout"); return -1; // printf("status byte =%d\n",stat); //Wait until status=33 while(stat!=33) status = vireadstb(instr,&nbyte); stat = *(&nbyte); //printf("2\n"); if(status<vi_success) printf("problem with Timeout"); return -1; printf("status=%d\n",stat); //Final Status before reading status = vireadstb(instr,&nbyte); stat = *(&nbyte); //printf("2\n"); if(status<vi_success) printf("problem with Timeout"); return -1; 24

260 printf("final Status before reading =%d\n",stat); return ; break; int CVICALLBACK Exptime_CB (int panel, int control, int event, void *callbackdata, int eventdata1, int eventdata2) switch (event) case EVENT_COMMIT: status = viwrite (instr, "ET.52633",11, &retcount); if(status<vi_success) printf("error tranferring data"); return -1; // status = viread(instr, buffer, MAX_CNT, &retcount); // printf("exposure time=%s\n", buffer); printf("exposure time is set\n"); return ; break; int CVICALLBACK Checkexp_CB (int panel, int control, int event, void *callbackdata, int eventdata1, int eventdata2) switch (event) case EVENT_COMMIT: // Check Exposure time status = viwrite (instr, "ET",2, &retcount); if(status<vi_success) 241

261 printf("error tranferring data"); return -1; status = viread(instr, buffer, MAX_CNT, &retcount); printf("\n\n\nexposure time=%s\n", buffer); return ; break; int CVICALLBACK Clrdata_CB (int panel, int control, int event, void *callbackdata, int eventdata1, int eventdata2) switch (event) case EVENT_COMMIT: status = viwrite (instr, "CLR.ALL",7, &retcount); if(status<vi_success) printf("error tranferring data DC"); return -1; // Wait until the status = 33 while(stat!=33) status = vireadstb(instr,&nbyte); stat = *(&nbyte); //printf("2\n"); if(status<vi_success) printf("problem with Timeout"); return -1; //Status printf("from memory Status=%d\n",stat); status = vireadstb(instr,&nbyte); 242

262 stat = *(&nbyte); //printf("2\n"); if(status<vi_success) printf("problem with Timeout"); return -1; printf("status Now =%d\n",stat); printf("ready for Data Acquisition\n"); break; return ; ************************************************************************ 243

263 Appendix E Inversion matrix to determine Fourier coefficients for dual rotating compensator ellipsometer I ' 2 ' 4 ' 6 ' 8 ' 1 ' 12 ' 14 ' 16 '

264 ' 2 ' 22 ' 24 ' 26 ' 28 ' 3 ' 32 ' 34 '

265

266 ' 38 ' 4 ' 42 ' 44 ' 46 ' 48 ' 5 ' 52 '

267 ' 56 ' 58 ' 6 ' 62 '

268

269

270

271

272

273

274

275 Appendix F Program to determine roots of quartic equation User Interface. The IPCF corrected data obtained from the experiment can be directly Figure F-1: Snap shot of user interface to determine roots of quartic equation 256