Computer simulation study of image speckle patterns with relation to object surface profile

H. Fujii; J. Uozumi; T. Asakura

doi:10.1364/JOSA.66.001222

Journal of the Optical Society of America
Vol. 66,
Issue 11,
pp. 1222-1236
(1976)
•https://doi.org/10.1364/JOSA.66.001222

Computer simulation study of image speckle patterns with relation to object surface profile

H. Fujii, J. Uozumi, and T. Asakura

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H. Fujii, J. Uozumi, and T. Asakura, "Computer simulation study of image speckle patterns with relation to object surface profile," J. Opt. Soc. Am. 66, 1222-1236 (1976)

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Abstract

The intensity distribution and contrast of image speckle patterns for various objects having different surface profiles are investigated on a computer as a function of the surface roughness properties of the objects and the point spread of an imaging system. The present computer simulation study for the objects having a random surface gives a theoretical background for the experimental results obtained by Fujii and Asakura. The same study for the objects having a periodic surface of two different profiles indicates that the contrast variation takes a complicated form as a function of the point spread of the imaging system, while the intensity distribution of images simply follows a periodic variation. It becomes clear from the simulation study that the maximum value of the image contrast variation is dependent mainly on the rms surface roughness of the objects but not on their surface profile.

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a	I(x′)	x′	Equation
$0 ≦ a < \frac{δ}{2}$	$\frac{4}{c^{2}} {sin}^{2} (c a)$	$a - \frac{δ}{2} ≦ x^{'} < \frac{δ}{2} - a$	(9.1)
	$\frac{4}{c^{2}} {1 - 2 cos (c a) cos [c (x^{'} - \frac{δ}{2})] + {cos}^{2} [c (x^{'} - \frac{δ}{2})]}$	$\frac{δ}{2} - a ≦ x^{'} < \frac{δ}{2} + a$	(9.1)
$\frac{δ}{2} ≦ a < δ$	$\frac{4}{c^{2}} {{sin}^{2} [c (a - δ)] + 4 {sin}^{2} (\frac{c δ}{2}) + 4 sin [c (a - δ)] sin (\frac{c δ}{2}) cos (c x^{'})}$	$\frac{δ}{2} - a ≦ x^{'} < a - \frac{δ}{2}$	(9.2)
	$\frac{4}{c^{2}} {1 - 2 cos (c a) cos [c (x^{'} - \frac{δ}{2})] + {cos}^{2} [c (x^{'} - \frac{δ}{2})]}$	$a - \frac{δ}{2} ≦ x^{'} < \frac{3 δ}{2} - a$	(9.2)
$δ ≦ a < \frac{3 δ}{2}$	$\frac{4}{c^{2}} [{sin}^{2} [c (a - δ)] + 4 {sin}^{2} (\frac{c δ}{2}) + 4 sin [c (a - δ)] sin (\frac{c δ}{2}) cos (c x^{'})]$	$a - \frac{3 δ}{3} ≦ x^{'} < \frac{3 δ}{2} - a$	(9.3)
	$\frac{4}{c^{2}} (1 + 8 {sin}^{2} (\frac{c δ}{2}) + 2 cos [c (x^{'} - \frac{δ}{2})] {2 sin (\frac{c δ}{2}) sin [c (a - \frac{3 δ}{2})] - cos [c (a - δ)]} + {cos}^{2} [c (x^{'} - \frac{δ}{2})])$	$\frac{3 δ}{2} - a ≦ x^{'} < a - \frac{δ}{2}$	(9.3)
$\frac{3 δ}{2} ≦ a < 2 δ$	$\frac{4}{c^{2}} [{sin}^{2} [c (a - 2 δ)] + 16 {sin}^{2} (\frac{c δ}{2}) + 8 sin [c (a - 2 δ)] sin (\frac{c δ}{2}) cos (c x^{'})]$	$\frac{3 δ}{2} - a ≦ x < a - \frac{3 δ}{2}$	(9.4)
	$\frac{4}{c^{2}} (1 + 8 {sin}^{2} (\frac{c δ}{2}) + 2 cos [c (x^{'} - \frac{δ}{2})] {2 sin (\frac{c δ}{2}) sin [c (a - \frac{3 δ}{2})] - cos [c (a - δ)]} + {cos}^{2} [c (x^{'} - \frac{δ}{2})])$	$a - \frac{3 δ}{2} ≦ x^{'} < \frac{5 δ}{2} - a$	(9.4)

a	$\bar{I (x^{'})}, \bar{I {(x^{'})}^{2}}$	Equation
$0 ≦ a < \frac{δ}{2}$	$\bar{I (x^{'})} = \frac{4}{δ c^{2}} [(2 a + \frac{δ}{2}) - \frac{3}{2 c} sin (2 a c) - \frac{1}{2} (δ - 2 a) cos (2 a c)]$	(10.1)
	$\bar{I {(x^{'})}^{2}} = \frac{16}{δ c^{4}} [6 a + \frac{3 δ}{8} - \frac{29}{6 c} sin (2 a c) + (3 a - \frac{δ}{2}) cos (2 a c) + \frac{11}{48 c} sin (4 a c) + \frac{1}{8} (δ - 2 a) cos (4 a c)]$	(10.2)
$\frac{δ}{2} ≦ a < δ$	$\bar{I (x^{'})} = \frac{4}{δ c^{2}} (\frac{1}{2} (δ - 2 a) cos [2 c (a - δ)] + 2 (δ - 2 a) cos (c δ) + \frac{3}{2 c} sin [2 c (a - δ)] + 2 a + \frac{δ}{2})$	(10.3)
	$\begin{array}{l} \bar{I {(x^{'})}^{2}} = \frac{16}{δ c^{4}} (16 (2 a - δ) {sin}^{4} (\frac{c δ}{2}) + 16 (2 a - δ) {sin}^{2} (\frac{c δ}{2}) {sin}^{2} [c (a - δ)] + (2 a - δ) {sin}^{4} [c (a - δ) + \frac{16}{c} [4 {sin}^{3} (\frac{c δ}{2}) sin [c (a - δ)] \\ + sin (\frac{c δ}{2}) {sin}^{3} [c (a - δ)]] sin [c (a - \frac{δ}{2})] + \frac{8}{c} {sin}^{2} (\frac{c δ}{2}) {sin}^{2} [c (a - δ)] sin [c (2 a - δ)] + \frac{1}{16 c} sin [4 c (δ - a)] \\ - \frac{2}{3 c} cos (c a) sin [3 c (δ - a)] - \frac{14}{c} cos (c a) sin [c (δ - a)] + \frac{1}{c} (\frac{5}{2} + cos (2 c a)) sin [2 c (δ - a)] + (δ - a) [\frac{27}{4} + 2 cos (2 c a)]) \end{array}$	(10.4)
$δ ≦ a < \frac{3 δ}{2}$	$\bar{I (x^{'})} = \frac{4}{δ c^{2}} [4 (2 a - δ) {sin}^{2} (\frac{c δ}{2}) + (3 δ - 2 a) {sin}^{2} [c (a - δ)] - \frac{3}{2 c} sin [2 c (a - δ)] + 3 (a - δ)]$	(10.5)
	$\begin{array}{l} \bar{I {(x^{'})}^{2}} = \frac{16}{δ c^{4}} (16 (6 a - 5 δ) {sin}^{4} (\frac{c δ}{2}) + (3 δ - 2 a) {sin}^{4} [c (a - δ)] + (16 (3 δ - 2 a) + \frac{8}{c} sin [c (3 δ - 2 a)]) {sin}^{2} (\frac{c δ}{2}) {sin}^{2} [c (a - δ)] \\ + \frac{4}{c} sin [c (\frac{3 δ}{2} - a)] sin (\frac{c δ}{2}) sin [c (a - δ)] [4 {sin}^{2} [c (a - δ)] - 16 {sin}^{2} (\frac{c δ}{2}) - 7] + {48 (a - δ) - \frac{24}{c} sin [2 c (a - δ)] \\ + 8 {sin}^{2} [c (a - \frac{3 δ}{2})] [\frac{1}{c} sin [2 c (a - δ)] + 2 (a - δ)]} {sin}^{2} (\frac{c δ}{2}) + [\frac{4}{3 c} sin (3 c (a - δ)] - 8 cos [c (a - δ)] \\ \times (\frac{1}{c} sin [2 c (a - δ)] + 2 (a - δ))] sin [c (a - \frac{3 δ}{2})] sin (\frac{c δ}{2}) + \frac{1}{16 c} sin [4 c (a - δ)] - \frac{2}{3 c} sin [3 c (a - δ)] cos [c (a - δ)] \\ + \frac{1}{c} (2 {cos}^{2} [c (a - δ)] - \frac{11}{2}) sin [2 c (a - δ)] + 4 (a - δ)] {cos}^{2} [c (a - δ)] + \frac{19}{4} (a - δ)) \end{array}$	(10.6)
$\frac{3 δ}{2} ≦ a < 2 δ$	$\begin{array}{l} \bar{I (x^{'})} = \frac{4}{δ c^{2}} {16 (a - δ) {sin}^{2} (\frac{c δ}{2}) + (2 a - 3 δ) {sin}^{2} [c (2 δ - a)] - \frac{8}{c} sin [c (a - \frac{3 δ}{2})] sin (\frac{c δ}{2}) sin [c (2 δ - a)] + \frac{1}{2 c} sin [2 c (2 δ - a)] \\ - \frac{4}{c} cos [c (a - δ)] sin [c (2 δ - a)] + 3 (2 δ - a)} \end{array}$	(10.7)
	$\begin{array}{l} \bar{I {(x^{'})}^{2}} = \frac{16}{δ c^{4}} [128 (3 a - 4 δ) {sin}^{4} (\frac{c δ}{2}) + 16 {sin}^{2} (\frac{c δ}{2}) (2 (2 δ - a) + {1 + {sin}^{2} [c (a - \frac{3 δ}{2})]} (\frac{1}{2 c} sin [2 c (2 δ - a)] + (2 δ - a))) \\ + 4 [\frac{1}{3 c} sin [3 c (2 δ - a)] - 4 cos [c (a - δ)] (\frac{1}{2 c} sin [2 c (2 δ - a)] + (2 δ - a))] sin [c (a - \frac{3 δ}{2})] sin (\frac{c δ}{2}) \\ + 32 (2 (2 a - 3 δ + \frac{1}{c} sin [c (2 a - 3 δ)]) {sin}^{2} (\frac{c δ}{2}) {sin}^{2} [c (a - 2 δ)] + {sin [c (a - \frac{3 δ}{2})] [\frac{384}{2} {sin}^{2} (\frac{c δ}{2}) \\ + \frac{32}{c} {sin}^{2} [c (a - 2 δ)] - \frac{28}{c}] + \frac{64}{c} sin (\frac{c δ}{2}) cos [c (a - δ)]} sin (\frac{c δ}{2}) sin [c (a - 2 δ)] + (2 a - 3 δ) {sin}^{4} [c (a - 2 δ)] \\ + \frac{1}{16 c} sin [4 c (2 δ - a)] - \frac{2}{3 c} cos [c (a - δ)] sin [3 c (2 δ - a)] + \frac{1}{2 c} {3 + 4 {cos}^{2} [c (a - δ)]} sin [2 c (2 δ - a)]] \\ - \frac{14}{c} cos [c (a - δ)] sin [c (2 δ - a)] + 4 (2 δ - a) {cos}^{2} [c (a - δ)] + \frac{19}{4} (2 δ - a) \end{array}$	(10.8)

a	I(x′)	x′	Equation
$0 ≦ a < \frac{δ}{2}$	$\frac{4}{c^{2}} {sin}^{2} (c a)$	$a - \frac{δ}{2} ≦ x^{'} < \frac{δ}{2} - a$	(9.1)
	$\frac{4}{c^{2}} {1 - 2 cos (c a) cos [c (x^{'} - \frac{δ}{2})] + {cos}^{2} [c (x^{'} - \frac{δ}{2})]}$	$\frac{δ}{2} - a ≦ x^{'} < \frac{δ}{2} + a$	(9.1)
$\frac{δ}{2} ≦ a < δ$	$\frac{4}{c^{2}} {{sin}^{2} [c (a - δ)] + 4 {sin}^{2} (\frac{c δ}{2}) + 4 sin [c (a - δ)] sin (\frac{c δ}{2}) cos (c x^{'})}$	$\frac{δ}{2} - a ≦ x^{'} < a - \frac{δ}{2}$	(9.2)
	$\frac{4}{c^{2}} {1 - 2 cos (c a) cos [c (x^{'} - \frac{δ}{2})] + {cos}^{2} [c (x^{'} - \frac{δ}{2})]}$	$a - \frac{δ}{2} ≦ x^{'} < \frac{3 δ}{2} - a$	(9.2)
$δ ≦ a < \frac{3 δ}{2}$	$\frac{4}{c^{2}} [{sin}^{2} [c (a - δ)] + 4 {sin}^{2} (\frac{c δ}{2}) + 4 sin [c (a - δ)] sin (\frac{c δ}{2}) cos (c x^{'})]$	$a - \frac{3 δ}{3} ≦ x^{'} < \frac{3 δ}{2} - a$	(9.3)
	$\frac{4}{c^{2}} (1 + 8 {sin}^{2} (\frac{c δ}{2}) + 2 cos [c (x^{'} - \frac{δ}{2})] {2 sin (\frac{c δ}{2}) sin [c (a - \frac{3 δ}{2})] - cos [c (a - δ)]} + {cos}^{2} [c (x^{'} - \frac{δ}{2})])$	$\frac{3 δ}{2} - a ≦ x^{'} < a - \frac{δ}{2}$	(9.3)
$\frac{3 δ}{2} ≦ a < 2 δ$	$\frac{4}{c^{2}} [{sin}^{2} [c (a - 2 δ)] + 16 {sin}^{2} (\frac{c δ}{2}) + 8 sin [c (a - 2 δ)] sin (\frac{c δ}{2}) cos (c x^{'})]$	$\frac{3 δ}{2} - a ≦ x < a - \frac{3 δ}{2}$	(9.4)
	$\frac{4}{c^{2}} (1 + 8 {sin}^{2} (\frac{c δ}{2}) + 2 cos [c (x^{'} - \frac{δ}{2})] {2 sin (\frac{c δ}{2}) sin [c (a - \frac{3 δ}{2})] - cos [c (a - δ)]} + {cos}^{2} [c (x^{'} - \frac{δ}{2})])$	$a - \frac{3 δ}{2} ≦ x^{'} < \frac{5 δ}{2} - a$	(9.4)

a	$\bar{I (x^{'})}, \bar{I {(x^{'})}^{2}}$	Equation
$0 ≦ a < \frac{δ}{2}$	$\bar{I (x^{'})} = \frac{4}{δ c^{2}} [(2 a + \frac{δ}{2}) - \frac{3}{2 c} sin (2 a c) - \frac{1}{2} (δ - 2 a) cos (2 a c)]$	(10.1)
	$\bar{I {(x^{'})}^{2}} = \frac{16}{δ c^{4}} [6 a + \frac{3 δ}{8} - \frac{29}{6 c} sin (2 a c) + (3 a - \frac{δ}{2}) cos (2 a c) + \frac{11}{48 c} sin (4 a c) + \frac{1}{8} (δ - 2 a) cos (4 a c)]$	(10.2)
$\frac{δ}{2} ≦ a < δ$	$\bar{I (x^{'})} = \frac{4}{δ c^{2}} (\frac{1}{2} (δ - 2 a) cos [2 c (a - δ)] + 2 (δ - 2 a) cos (c δ) + \frac{3}{2 c} sin [2 c (a - δ)] + 2 a + \frac{δ}{2})$	(10.3)
	$\begin{array}{l} \bar{I {(x^{'})}^{2}} = \frac{16}{δ c^{4}} (16 (2 a - δ) {sin}^{4} (\frac{c δ}{2}) + 16 (2 a - δ) {sin}^{2} (\frac{c δ}{2}) {sin}^{2} [c (a - δ)] + (2 a - δ) {sin}^{4} [c (a - δ) + \frac{16}{c} [4 {sin}^{3} (\frac{c δ}{2}) sin [c (a - δ)] \\ + sin (\frac{c δ}{2}) {sin}^{3} [c (a - δ)]] sin [c (a - \frac{δ}{2})] + \frac{8}{c} {sin}^{2} (\frac{c δ}{2}) {sin}^{2} [c (a - δ)] sin [c (2 a - δ)] + \frac{1}{16 c} sin [4 c (δ - a)] \\ - \frac{2}{3 c} cos (c a) sin [3 c (δ - a)] - \frac{14}{c} cos (c a) sin [c (δ - a)] + \frac{1}{c} (\frac{5}{2} + cos (2 c a)) sin [2 c (δ - a)] + (δ - a) [\frac{27}{4} + 2 cos (2 c a)]) \end{array}$	(10.4)
$δ ≦ a < \frac{3 δ}{2}$	$\bar{I (x^{'})} = \frac{4}{δ c^{2}} [4 (2 a - δ) {sin}^{2} (\frac{c δ}{2}) + (3 δ - 2 a) {sin}^{2} [c (a - δ)] - \frac{3}{2 c} sin [2 c (a - δ)] + 3 (a - δ)]$	(10.5)
	$\begin{array}{l} \bar{I {(x^{'})}^{2}} = \frac{16}{δ c^{4}} (16 (6 a - 5 δ) {sin}^{4} (\frac{c δ}{2}) + (3 δ - 2 a) {sin}^{4} [c (a - δ)] + (16 (3 δ - 2 a) + \frac{8}{c} sin [c (3 δ - 2 a)]) {sin}^{2} (\frac{c δ}{2}) {sin}^{2} [c (a - δ)] \\ + \frac{4}{c} sin [c (\frac{3 δ}{2} - a)] sin (\frac{c δ}{2}) sin [c (a - δ)] [4 {sin}^{2} [c (a - δ)] - 16 {sin}^{2} (\frac{c δ}{2}) - 7] + {48 (a - δ) - \frac{24}{c} sin [2 c (a - δ)] \\ + 8 {sin}^{2} [c (a - \frac{3 δ}{2})] [\frac{1}{c} sin [2 c (a - δ)] + 2 (a - δ)]} {sin}^{2} (\frac{c δ}{2}) + [\frac{4}{3 c} sin (3 c (a - δ)] - 8 cos [c (a - δ)] \\ \times (\frac{1}{c} sin [2 c (a - δ)] + 2 (a - δ))] sin [c (a - \frac{3 δ}{2})] sin (\frac{c δ}{2}) + \frac{1}{16 c} sin [4 c (a - δ)] - \frac{2}{3 c} sin [3 c (a - δ)] cos [c (a - δ)] \\ + \frac{1}{c} (2 {cos}^{2} [c (a - δ)] - \frac{11}{2}) sin [2 c (a - δ)] + 4 (a - δ)] {cos}^{2} [c (a - δ)] + \frac{19}{4} (a - δ)) \end{array}$	(10.6)
$\frac{3 δ}{2} ≦ a < 2 δ$	$\begin{array}{l} \bar{I (x^{'})} = \frac{4}{δ c^{2}} {16 (a - δ) {sin}^{2} (\frac{c δ}{2}) + (2 a - 3 δ) {sin}^{2} [c (2 δ - a)] - \frac{8}{c} sin [c (a - \frac{3 δ}{2})] sin (\frac{c δ}{2}) sin [c (2 δ - a)] + \frac{1}{2 c} sin [2 c (2 δ - a)] \\ - \frac{4}{c} cos [c (a - δ)] sin [c (2 δ - a)] + 3 (2 δ - a)} \end{array}$	(10.7)
	$\begin{array}{l} \bar{I {(x^{'})}^{2}} = \frac{16}{δ c^{4}} [128 (3 a - 4 δ) {sin}^{4} (\frac{c δ}{2}) + 16 {sin}^{2} (\frac{c δ}{2}) (2 (2 δ - a) + {1 + {sin}^{2} [c (a - \frac{3 δ}{2})]} (\frac{1}{2 c} sin [2 c (2 δ - a)] + (2 δ - a))) \\ + 4 [\frac{1}{3 c} sin [3 c (2 δ - a)] - 4 cos [c (a - δ)] (\frac{1}{2 c} sin [2 c (2 δ - a)] + (2 δ - a))] sin [c (a - \frac{3 δ}{2})] sin (\frac{c δ}{2}) \\ + 32 (2 (2 a - 3 δ + \frac{1}{c} sin [c (2 a - 3 δ)]) {sin}^{2} (\frac{c δ}{2}) {sin}^{2} [c (a - 2 δ)] + {sin [c (a - \frac{3 δ}{2})] [\frac{384}{2} {sin}^{2} (\frac{c δ}{2}) \\ + \frac{32}{c} {sin}^{2} [c (a - 2 δ)] - \frac{28}{c}] + \frac{64}{c} sin (\frac{c δ}{2}) cos [c (a - δ)]} sin (\frac{c δ}{2}) sin [c (a - 2 δ)] + (2 a - 3 δ) {sin}^{4} [c (a - 2 δ)] \\ + \frac{1}{16 c} sin [4 c (2 δ - a)] - \frac{2}{3 c} cos [c (a - δ)] sin [3 c (2 δ - a)] + \frac{1}{2 c} {3 + 4 {cos}^{2} [c (a - δ)]} sin [2 c (2 δ - a)]] \\ - \frac{14}{c} cos [c (a - δ)] sin [c (2 δ - a)] + 4 (2 δ - a) {cos}^{2} [c (a - δ)] + \frac{19}{4} (2 δ - a) \end{array}$	(10.8)

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Journal of the Optical Society of America