WORCESTER POLYTECHNIC INSTITUTE

Transcription

1 WORCESTER POLYTECHNIC INSTITUTE MECHANICAL ENGINEERING DEPARTMENT Optical Metrology and NDT ME-593L, C 2018 Introduction: Wave Optics January 2018

2 Wave optics: coherence Temporal coherence Review interference equation: Lecture 04 Michelson interferometer to test for coherence length (temporal

3 Wave optics: coherence Temporal coherence Assume 100% flat and orthogonal mirrors Detected intensities as a function of position of movable mirror 0 /4 /2 3 /4

4 Wave optics: coherence Temporal coherence Temporal coherence describes the self-correlation of a wave. Consider the normalized autocorrelation function g( * G( U ( t U( t, (1 G(0 * U ( t U( t where is the time delay, related optical path length changes introduced by the movable mirror. Equation 1 is called the degree of temporal coherence with ( 0 g ( 1. (2 Usually, g drops from its largest values g( 1 as increases and the fluctuations become uncorrelated for sufficiently large time delay.

5 Wave optics: coherence Temporal coherence Coherence length is defined as l c c c, (3 where c is the speed of light and c is the power equivalent width g( of the function given as c 2 g( d. (4

6 Wave optics: coherence Temporal coherence Magnitude of the degree of temporal coherence g( Short coherence time Long coherence time

7 Wave optics: coherence Temporal coherence Power spectral density: S ( g( exp( j2 d

8 Wave optics: coherence Spatial coherence Young s double pinhole interferometer Incident wave k Pinholes Observation plane Predicted fringe patterns corresponding to difference pinholes distances 2a = 150 µm 2a = 100 µm 2a = 50 µm

9 Wave optics: coherence Spatial coherence Actual fringe pattern: note variation of intensity along the x-axis Spatial coherence length is: x l c c x

10 Wave optics: light scattering Speckle pattern observed on the flat surface of an object illuminated with a coherent light source with characteristic wavelength = nm and horizontally polarized. Average surface roughness is 5 m, object distance is 30 mm, and f/2.8 aperture is used. The surface is illuminated and observed normal to the plane where the surface lies.

11 Wave optics: light scattering Illumination and scattering geometry from surfaces defined as z z ( x, y and z z (x. Vectors of illumination and scattering: 1 Kˆ1 K k, K2 k Kˆ 2 K1 SI 1 z 2 y SC K 2 K 1 1 z 2 K 2 z z ( x, y (a 3 x (b z z (x x

12 Wave optics: light scattering Scattering from a smooth surface The complex amplitude of the scattered light, F o, at point p, can be predicted using the Kirchoff integral theorem F o 4 1 s 1 exp( r jkr U ds s U 1 exp( r jkr ds (5 which is derived from the Helmholtz equation. For a smooth surface, z ( x 0, extending from [-L, L]: F o F (, o 1 2 sin[ kl(sin1 sin2 ] kl(sin sin 1 sin c 2 (sin 1 sin2 L 2 (6

13 Wave optics: light scattering Scattering from a smooth surface Scattering diagram for a smooth surface characterized by the ratio / L. / L 0 Note that when scattering of light concentrates in the direction of specular reflection / L K log( Fo Fo * 0

14 Wave optics: light scattering Scattering from a surface characterized by a periodic function: 10, 1 /4, / h , 1 /4, / h K K log( * F2 F log( * F 2 F 2 0

15 Wave optics: light scattering Scattering from a surface characterized by a periodic function 10, 1 /4, / h , 1 /4, / h K K * log( F2 F log( * F2 F2 0

16 Wave optics: light scattering Scattering from a surface characterized by a random function Transition from specular to diffuse scattering reflection as a function of surface roughness:

17 Speckles Speckle pattern (x,y,z Adding contribution of scattered components at point (x,y,z (U (U

18 Speckles Speckle properties: first order statistics Sum of N complex amplitude components: N U( x, y, z 1 A k exp[ jk ( x, y, z] N k 1 (7 Real an imaginary components: N { U} 1 a N k cos( k k 1 N { U} 1 a N k sin( k k 1 (8 Amplitude and phase are statistically independent.

19 Speckles Probability distribution for the intensity is (mean = 2 2, variance = <I 2 > Speckle properties: first order statistics Probability distribution for the phase is: otherwise ; ; 2 exp, ( (, 0 I 0 I 2 1 d I P I P 2 2 I I otherwise ; ;, ( (, di I P P 0 I (9 (10

20 Speckles Speckle properties: second order statistics Observed speckle size is (without imaging system: x 2 z s (11 L Speckle field formation without imaging system y K 1 y' L O z' Dx = Dy z Observation plane

21 Speckles Speckle properties: second order statistics Observed speckle size is (with imaging system: rs 2.44 D z (12 Speckle field formation with imaging system y K 1 y' O D z Observation plane