# Tolerance on material inhomogenity and surface irregularity

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1 Opti 521 Wenrui Cai Tolerance on material inhomogenity and surface irregularity Abstract In this tutorial, a case study on tolerance for a focusing doublet is performed by using ZEMAX. First, how to perform a general tolerance analysis is briefly introduced and compared with results from manual calculation. Then methods to tolerance the material inhomogenity and surface irregularity in ZEMAX are discussed. And method to understand the results of the tolerance analysis is introduced. Introduction to ZEMAX tolerance Tolerance analysis is an important step in optical system design, since all the optical elements cannot be made perfectly. Tolerancing provides information about the sensitivity of an optical system to typical fabrication and mounting errors. Tolerancing can also help determine which design to make if you have a selection of lens designs to choose from, as well as determine the manufacturing tolerances you need to maintain to achieve a particular level of performance. Criterions In ZEMAX, tolerances may be evaluated by several different criterions: RMS spot radius for low quality optics RMS wavefront error MTF Merit function for superb quality optics for photographic or moderate quality optics for custom requirements Analysis options Tolerances may be computed and analyzed three ways in ZEMAX: Used to determine the change in performance for a given set of Sensitivity tolerances individually. Used to reduce individual tolerances to meet a maximum Inverse Sensitivity allowable change in performance. Used to determine the change in performance when all tolerances Monte Carlo are combined together randomly. The statistical distribution may be Normal (Gaussian), Uniform, or Parabolic. Tolerancing procedure The procedure of tolerancing usually consists of the following steps. 1) Define an appropriate set of tolerances for the lens. The default tolerance is usually a good place to start, or you can use the different tolerance levels given in appendix table a1. 2) Modify the default tolerances or add new ones to suit the system requirements. 3) Add compensators and set allowable ranges for the compensators. The default compensator is

2 the back focal distance, which controls the position of the image plane. Other compensators, such as image surface tilt and decenter, may be defined. 4) Select appropriate criteria, such as RMS spot radius, wavefront error, MTF or boresight error. More complex criteria may be defined using a user defined merit function. 5) Select the desired mode, either sensitivity or inverse sensitivity. For inverse sensitivity, choose criteria limits or increments, and whether to use averages or computer each field individually. 6) Perform an analysis of the tolerances. 7) Review the data generated by the tolerance analysis, and consider the budgeting of tolerances. If required, modify the tolerances and repeat the analysis. Case study From here an example of focusing doublet in Opti 521 HW4 part3 will be used to perform a tolerance analysis, especially tolerancing on inhomogenity and surface irregularity. Fig 1 Focusing Doublet layout and surface summary Lens errors include: radii of curvature, lens thickness, Wedge, surface irregularity, index error, and inhomogenity, decenter, tilt, and lens spacing. Use built-in tolerance operand There are many fabrication and mounting errors to consider when tolerancing an optical system. ZEMAX s tolerancing capabilities can model a number of different tolerances, including tolerance on radius, thickness, tilts and decenters of surfaces or elements, surface irregularity, and much more. Each of these is supported via their own tolerance operand in ZEMAX. Tolerance information in Zemax is inserted into the Tolerance Data Editor (TDE) (Editors -> tolerance data, Shift + F2). Each line of the TDE spreadsheet is one operand that represents one degree of freedom. In the case of the default tolerances (TDE window -> Tools -> default tolerances), it lists the alignment variables for each element, as well as for each surface.

3 Fig 2 default tolerances Since this system is rotationally symmetric, tilts and decenters in the x and y directions turn out to be the same. It is proper to delete all of the y direction entries. Then According to different tolerance levels, specifically refine minimum and maximum values for every surfaces or elements surface in TDE. Or you can type in different operand from the TDE. Each tolerance operand has a four letter mnemonic. Next step is to go to button TOL or to the Tools drop down window and choose Tolerancing and then Tolerancing Fig 3 tolerancing

4 Adjust the parameters in the Tolerancing window (shown in Figure 3). 1 Choose mode: sensitivity. 2 Check Force Ray Aiming On (which makes it more accurate, but slower). 3 Choose the Criteria: (RMS Spot Radius, RMS Wavefront, Merit Function, Boresight Error, MTF and more). We need to select RMS Wavefront. 4 Choose the Compensator: (Paraxial focus, Optimize All, None). We want the paraxial focus to be the compensator, which is already the default. Show Compensators (for example to see how much focus changes for example). At last, a results window will open, showing many results (shown in appendix). Depending on the results, one may wish to loosen or tighten the tolerances. Each tolerance operand listed with the change in criterion for its maximum and minimum values. These are then ordered in a list called Worst Offenders, which lists the operands from most to least effect on the criterion. A statistical analysis is then performed on the data, estimating the change in criterion using a Root Sum Square calculation. Now we can compare the result with the manual calculation in HW4 ZEMAX Operand Table 1 Sensitivities (hand calculation) Sensitivities (ZEMAX tolerance) Compare (%) Lens1 Decenter (mm) TEDX/TEDY Tilt (deg) TETX/TETY R1 (mm) TRAD R2 (mm) TRAD Thickness (mm) TTHI Index TIND Wedge 1 (deg) TSTX/TSTY Wedge 2 (deg) TSTX/TSTY Irregularity 1 (waves) TIRR Irregularity 2 ( waves) TIRR Inhomogenity no Lens2 Decenter (mm) TEDX/TEDY Tilt (deg) TETX/TETY R1 (mm) TRAD R2 (mm) TRAD Thickness (mm) TTHI Index TIND Wedge 1 (deg) TSTX/TSTY Wedge 2 (deg) TSTX/TSTY Irregularity 1 ( waves) TIRR Irregularity 2 ( waves) TIRR Inhomogenity no We can see in the table above, sensitivities of most lens error are within one order of magnitude using two different methods. However the operand for surface irregularity (TIRR) obviously fails to analysis the sensitivity correctly. And there is no as-built operand to calculate the tolerance of material inhomogenity.

5 Tolerance on surface irregularity using different methods Modeling irregularity is somewhat more problematic than other types of tolerances. This is primarily because irregularity by nature is random, and not deterministic such as a change in radius. Therefore, some assumptions about the nature of the irregularity need to be made in order to perform the analysis. Use rule of thumb Change in RMS wavefront error (WFE) due to P-V surface irregularity on one surface is W 0.25 S ( n 1) cos RMS Where:α is the ratio of beam foot print to the surface diameter. Ф is the beam incidence angle. Lens diameter is 25mm and stop diameter is 20mm, so assume α = 1 and normal incidence here, given 1 wave P-V surface irregularity (0.1582umrms), W S ( n 1) ( ) ( wave) rms pv P V Use operand TIRR (S+A Irreg) TIRR is used to analyze irregularity of a Standard surface. Surface irregularity can be defined as a sum of spherical and astigmatism (which is usually what is done when test-plate interferograms are 'eyeballed'). The assumption ZEMAX makes when using TIRR is that the irregularity is half spherical aberration, and half astigmatism. This is less restrictive model than assuming 100% astigmatism, because astigmatism cannot be compensated by focus, and is therefore a more serious defect in the lens. The min and max values are the irregularity in units of fringes measured at the maximum radial aperture of the surface where the maximum radial aperture is defined by the semi-diameter of the surface. More detail information can be found in ZEMAX User s Guide p502. Use operand TEXI TEXI is used to analyze random irregular deviations of small amplitude on a surface that is either a Standard, Even Aspheric, or Zernike Fringe Sag surface. TEXI uses the Zernike Fringe Sag surface (see ZEMAX User s Guide p319) to model the irregularity rather than using the third order aberration formulas used by TIRR. Table 2 Extra data definitions for Zernike fringe sag surfaces When using TEXI, the min and max tolerance values are interpreted to be the approximate magnitude of the zero to peak error of the surface in double-pass fringes at the test wavelength.

6 The zero to peak is only a very rough measure of the irregularity. Whether the zero to peak and peak to valley are the same depends upon the particular Zernike term used. The "Number of terms" is used to specify the maximum Zernike polynomial term to be used in calculating the surface sag. This number is provided to speed the ray tracing calculation; terms beyond this number are ignored. Generally speaking, if lower order terms are used, the irregularity will be of low frequency, with fewer "bumps" across the surface. If higher order terms are used, there will be higher frequency irregularity, with more "bumps" across the surface. (see table 3) For example, let s use surface 2 of the system to illustrate the difference of two methods. In the Tolerance Data Editor, insert a SAVE tolerance control operand under the operand you want to deal with. Here is TEXI of Surf.#2. If we want to see the RMS wave front error due to 1 wave P-V Surface irregularity, we need to enter 2 fringes. (it is wrong to enter 1 fringe to represent 1 wave p-v in table 1, but the result will still off two orders of magnitude) And we can use different Zernike polynomial terms to represent different kinds of surface irregularity. (see fig 4). The SAVE command allows you to save the previous tolerance to a ZEMAX Lens File with the specified File #. A file will be saved for both the maximum and minimum tolerance. The file names will be TSAV_MIN_xxxx.ZMX and TSAV_MAX_xxxx.ZMX for the min and max tolerance analysis, respectively, where xxxx is the integer number specified in the Int1 column. In this case, the integer number is 1, so the maximum tolerance file will be TSAV_Max_0001.ZMX. Fig 4 Save a tolerance situation Much like the SAVE tolerance control operand (which is useful for evaluating one tolerance at a time), you may also save each individual Monte Carlo file generated during the tolerance analysis. This option exists in the Tolerancing dialog.(shown in fig 5) Fig 5 Saving Monte Carlo Tolerance Files

7 With this capability, we can clearly review what ZEMAX has done to ensure any given tolerance is performed the way we expect. Most importantly, we can thoroughly investigate any tolerance which we find to produce curious results. Table 3 different kinds of surface irregularity Zernike polynomial terms Wavefront Map RMS WFE (wave) Term #2-#8 Irregularity of low spatial frequencies Term #10-#18 Irregularity of medium spatial frequencies Term #37-#30 Irregularity of high spatial frequencies Now we can see that. The optical performance of a surface depends not only on the RMS amplitude of the irregularity but also on the frequency of those peaks and valleys, because it is the

8 slope of the surface that bends rays. As we polish a surface from l/5 wave to l/10 wave to l/20 wave to l/50 wave, the spatial frequency of the irregularity increases. Surfaces polished to say l/5 are often quite "slow" in terms of the spatial frequency of the irregularity, whereas super-polished surfaces often have a very high spatial freqeuncy of irregularity. To illustrate this, we use the example from ZEMAX s knowledge base online. (see fig 6) The surface #2 type is Periodic with a periodic structure in Y direction only. The 3D layout shows the difference in the ray trace results when the frequency of the periodic structure is increased while keeping the amplitude constant. Fig 6 Zernike polynomials tend to diverge quite rapidly beyond the normalization radius, and so care should be taken that rays do not strike the surface beyond this radius. Although the ray tracing algorithm may work, the data may be inaccurate. The extrapolate flag may be set to zero to ignore the Zernike terms for rays that land outside the normalization radius. Compare TIRR with TEXI Now compare the result of the sensitivity calculated in ZEMAX using different operand. Use Zernike polynomial terms #10-#18 for example. Table 4 comparison between TIRR & TEXI Surface ZEMAX Sensitivities Sensitivities Irregularity Operand (hand calculation) (ZEMAX tolerance) Compare (%) Surf 2 TIRR TEXI Surf 3 TIRR TEXI Surf 4 TIRR TEXI Surf 5 TIRR TEXI

9 The TIRR irregularity operand models the lowest frequency form of irregularity, with just a quadratic and quartic deviation across the surface. TEXI can model much more irregular surfaces, and with 30 or more terms used, about 5-15 "bumps" will typically be seen over the surface. So we can model the surface irregularity that more close to the RMS wavefront error the rule of thumb predicted. Irregularity generated by TIRR Irregularity generated by TEXI Fig 7 Tolerance on inhomogenity Given a single number or value representing the inhomogeneity of a material, it is impossible to exactly predict the index profile of the glass. Therefore, the most accurate and superior approach to modeling the inhomogeneity of a material can be performed via the statistical results of Monte Carlo Tolerance Analysis using tolerances on surface irregularity. For example, the TEZI tolerance operand in ZEMAX is used to analyze random irregular deviations of small amplitude on a surface. Within the Monte Carlo Analysis, the specified surface is converted to a Zernike Standard Sag surface, and each polynomial term is assigned a coefficient randomly chosen between zero and one. The resulting coefficients are normalized to yield the exact specified RMS tolerance. Assume we have a perturbation of refractive index as +/-1e-4 Lens1: Wrms 0.25 n t / e 5 5mm / m ( wave) Lens2: Wrms 0.25 n t / e 5 4mm / m ( wave) According to the article How to tolerance for material inhomogeneity, we first calculate: At lens 1, OPL1 t n 5mm 2 1e 5 1E 4mm At lens 2, OPL2 t n 4mm 2 1e 5 8E 5mm. Then we use TEZI tolerance operand in the tolerance data editor. The number of Zernike terms used for the analysis may be between 0 and 231. Generally

10 speaking, if fewer terms are used, the irregularity will be of low frequency, with fewer bumps across the surface. The maximum number of terms should be chosen accordingly. Here we use set Max# of Zernike terms is set as 37, Min# 2. Table 5 Use Monte Carl Analysis model inhomogenity LENS 1 LENS 2 Sensitivity Difference Sensitivity Difference Rule of thumb Monte Carl (%) Rule of thumb Monte Carl (%) We can see that the difference between rule of thumb and Monte Carl analysis is within one order of magnitude. Either the rule of thumb is too general for this specific case or we can manipulate the result of MC analysis by changing the number of Zernike terms. Each Monte Carlo trial will have a slightly different representation of the inhomogeneity of your glass. Therefore, a statistical listing of the entire Monte Carlo set is essential for estimating the probable effects the inhomogeneity has on your system performance. In a number of Monte Carlo Runs, we can gather a significant amount of statistical data relating to the change in RMS Wavefront Error due to the inhomogeneity of the glass. The more Monte Carlo tolerance runs that are performed, the better the statistical average of performance degradation (change in criteria) will be.(details are shown in appendix) Conclusion In this paper, we first went over the general tolerance procedure in ZEMAX. And mainly focus on the issue of surface irregularity and inhomogenity. The Zernike polynominal is used in both cases. In the tolerance of surface irregularity, the spatial frequency of the irregularity as well as its RMS amplitude must be modeled. In the tolerance of inhomogenity, the Monte Carlo tolerance analysis is used to randomize the irregularity (inhomogeneity) and provide with accurate, statistical results of how this irregularity is affecting the performance of the whole system. Reference: 1, Zemax Users Knowledge Base , Zemax Users Guide 3, Pingzhou, Tutorial of Tolerancing Analysis Using Commercial Optical Software 4, Stacie Hvisc, Tolerancing in ZEMAX, OPTI 521 Tutorial 5, OPTI 521 class notes

11 Appendix Optical element tolerances Table a1 Optical element tolerances Parameter Base Precision High precision Lens thickness 200 μm 50 μm 10 μm Radius of curvature 20 μm 1.3 μm 0.5 μm sag Value of R 1% 0.1% 0.02% Wedge (light deviation) 6 arc min (0.1 deg) 1 arc min 15 arc sec Surface irregularity 1 wave λ /4 λ /20 Refractive index departure fromnominal ± (Standard) ± (Grade 3) Refractive index ± 1 x 10-4 ± 5 x 10-6 homogeneity (Standard) (H2) Base: Typical, no cost impact for reducing tolerances beyond this. Precision: Requires special attention, but easily achievable, may cost 25% more High precision: Requires special equipment or personnel, may cost 100% more ± (Grade 1) ± 1 x 10-6 (H4) Rules of thumb for lenses Calculate sensitivities

12 Result of tolerance analysis Analysis of Tolerances Units are Millimeters. All changes are computed using linear differences. Paraxial Focus compensation only. Criterion : RMS Wavefront Error in waves Mode : Sensitivities Sampling : 20 Nominal Criterion : Test Wavelength : Fields: Y Symmetric Angle in degrees # X-Field Y-Field Weight VDX VDY VCX VCY E E E Sensitivity Analysis: Minimum Maximum Type Value Criterion Change Value Criterion Change TRAD TRAD TRAD TRAD TTHI TTHI TIND TIND TSTX TSTX TSTX TSTX TEZI TEZI E E TEXI TEXI TEXI TEXI TEDX TEDX TETX TETX TTHI Worst offenders: Type Value Criterion Change TEZI TEZI TEXI TEXI TEXI TEXI TEXI TEXI TSTX TSTX Estimated Performance Changes based upon Root-Sum-Square method: Nominal RMS Wavefront : Estimated change : Estimated RMS Wavefront : Compensator Statistics: Change in back focus: Minimum : Maximum : Mean : Standard Deviation :

13 Tolerance Data Editors

14 Result of inhomogenity using Monte Carlo Analysis Surface 2 Monte Carlo Analysis: Number of trials: 50 Initial Statistics: Normal Distribution Trial Criterion Change Number of traceable Monte Carlo files generated: 50 Nominal Best Trial 12 Worst Trial 3 Mean Std Dev % < % < % < Surface 4: Analysis of Tolerances Monte Carlo Analysis: Number of trials: 50 Initial Statistics: Normal Distribution Trial Criterion Change Number of traceable Monte Carlo files generated: 50 Nominal Best Trial 4 Worst Trial 30 Mean Std Dev % < % < % <

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