Analysis of lateral shearing interferograms by use of Zernike polynomials

G. Harbers; P. J. Kunst; G. W. R. Leibbrandt

doi:10.1364/AO.35.006162

Applied Optics
Vol. 35,
Issue 31,
pp. 6162-6172
(1996)
•https://doi.org/10.1364/AO.35.006162

Analysis of lateral shearing interferograms by use of Zernike polynomials

G. Harbers, P. J. Kunst, and G. W. R. Leibbrandt

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G. Harbers, P. J. Kunst, and G. W. R. Leibbrandt, "Analysis of lateral shearing interferograms by use of Zernike polynomials," Appl. Opt. 35, 6162-6172 (1996)

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Abstract

A modal phase-reconstruction method for wave-front analysis in lateral shearing interferometry is presented. Pseudo-Zernike polynomial functions describe the differential wave fronts and are related to a Zernike polynomial description of the original wave front. We show that this reconstruction is robust for shear ratios in the range 0.15–0.50. The error propagation properties of this differential Zernike polynomial matrix-inversion method are discussed on the basis of both analysis and simulation. It is concluded that the method allows wave-front analysis with an absolute inaccuracy of 2 mλ rms for diffraction-limited wave fronts and with 1% relative inaccuracy for more strongly aberrated wave fronts.

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n	m	Z^nm (r, θ)	X^nm (x, y)
0	0	1	1
1	1	r cos θ	x
1	−1	r sin θ	y
2	2	r ² cos 2θ	x ² − y ²
2	0	−1 + 2r ²	−1 + 2x ² + 2y ²
2	−2	r ² sin 2θ	2xy
3	3	r ³ cos 3θ	x ³ − 3xy ²
3	1	(−2r + 3r ³)cos θ	−2x + 3x ³ + 3xy ²
3	−1	(−2r + 3r ³)sin θ	−2y + 3x ² y + 3y ³
3	−3	r ³ sin 3θ	3x ² y − y ²
4	2	(−3r ² + 4r ⁴)cos 2θ	−3x ² + 4x ⁴ + 3y ² − 4y ⁴
4	0	1 − 6r ² + 6r ⁴	1 − 6x ² + 6x ⁴ − 6y ² + 12x ² y ² + 6y ⁴
4	−2	(−3r ² + 4r ⁴)sin 2θ	2(−3xy + 4x ³ y + 4xy ³)
5	1	(3r − 12r ³ +10r ⁵)cos θ	3x − 12x ³ + 10x ⁵ − 12xy ² + 20x ² y ² + 10xy ⁴
5	−1	(3r − 12r ³ + 10r ⁵)sin θ	3y − 12x ² y + 10x ⁴ y −12y ³ + 20x ² y ³ + 10y ⁵
6	0	−1 + 12r ² − 30r ⁴ + 20r ⁶	−1 + 12x ² − 30x ⁴ + 20x ⁶ + 12y ² − 60x ² y ² + 60x ⁴ y ² − 30y ⁴ + 60x ² y ⁴ + 20y ⁶

n	m	ΔX_x ^nm (x, y) x-Shear Polynomials	ΔX_y ^nm (x, y) y-Shear Polynomials
0	0	0	0
1	1	2s	0
1	−1	0	2s
2	2	4sx	− 4sy
2	0	8sx	8sy
2	−2	4sy	4sx
3	3	2s ³ + 6sx ² − 6sy ²	−12 sxy
3	1	−4s + 6s ³ + 18sx ² + 6sy ²	12 sxy
3	−1	12sxy	−4s + 6s ³ + 6sx ² + 18sy ²
3	−3	12sxy	−2s ³ + 6sx ² − 6sy ²
4	2	−12sx + 32s ³ x + 32sx ³	12sy − 32s ³ y − 32sy ³
4	0	−24sx + 48s ³ x + 48sx ³ + 48sxy ²	−24sy + 48s ³ y + 48sx ² y + 48sy ³
4	−2	−12sy + 16s ³ y + 48sx ²y + 16sy ³	−12sx + 16s ³ x + 16sx ³ + 48sxy ²

n	m	j	k	X_x,jk ^nm (s)
1	1	0	0	2s
2	2	1	1	4(s − s ²)
2	0	1	1	8(s − s ²)
2	−2	1	−1	$4 \sqrt{1 - s s}$
3	3	0	0	−3/2s ² + 7s ³
3	3	2	2	3(2s − 3s ² + s ³)
3	3	2	0	3/2(−s ² + s ³)
3	1	0	0	(4s − 21s ² + 21s ³)/2
3	1	2	2	3(2s − 5s ² + 3s ³)
3	1	2	0	3/2(4s − 7s ² + 3s ³)
3	−1	2	−2	$6 (\sqrt{1 - s s} - {\sqrt{1 - s s}}^{2})$
3	−3	2	−2	$6 (\sqrt{1 - s s} - {\sqrt{1 - s s}}^{2})$
4	2	1	1	4(s − 9s ² + 20s ³ − 12s ⁴)
4	2	3	3	8(s − 3s ² + 3s ³ − s ⁴)
4	2	3	1	8(s − 3s ² + 3s ³ − s ⁴)
4	0	1	1	8(s − 8s ² + 16s ³ − 9s ⁴)
4	0	3	3	12(−s ² + 2s ³ − s ⁴)
4	0	3	1	4(4s − 11s ² + 10s ³ − 3s ⁴)
4	−2	1	−1	$4 (\sqrt{1 - s s} - 6 {\sqrt{1 - s s}}^{2} + 6 {\sqrt{1 - s s}}^{3})$
4	−2	3	−1	$4 (2 \sqrt{1 - s s} - 3 {\sqrt{1 - s s}}^{2} + {\sqrt{1 - s s}}^{3})$
4	−2	3	−3	$4 (2 \sqrt{1 - s s} - 5 {\sqrt{1 - s s}}^{2} + 3 {\sqrt{1 - s s}}^{3})$
6	0	1	1	8s − 148s ² + 815s ³ − 1885s ⁴ + 1925s ⁵ − 715s ⁶
6	0	3	3	4(−9s ² + 77s ³ − 192s ⁴ + 189s ⁵ − 65s ⁶)
6	0	3	1	4(4s − 53s ² + 205s ³ − 332s ⁴ + 241s ⁵ − 65s ⁶)
6	0	5	1	3(8s − 36s ² + 65s ³ − 59s ⁴ + 27s ⁵ − 5s ⁶)

n	m	j	k	X_x,jk ^nm (s)
1	−1	0	0	2s
2	2	1	−1	4(−s + s ²)
2	0	1	−1	8(s − s ²)
2	−2	1	1	$4 \sqrt{1 - s s}$
3	3	2	−2	$6 [- (\sqrt{1 - s s}) + {\sqrt{1 - s s}}^{2}]$
3	1	2	−2	$6 (\sqrt{1 - s s} - {\sqrt{1 - s s}}^{2})$
3	−1	0	0	(4s − 21s ² + 21s ³)/2
3	−1	2	2	3(−2s + 5s ² − 3s ³)
3	−1	2	0	3/2(4s − 7s ² + 3s ³)
3	−3	0	0	(3s ² − 7s ³)/2
3	−3	2	2	3(2s − 3s ² + s ³)
3	−3	2	0	3/2(s ² − s ³)
4	2	1	−1	4(−s + 9s ² − 20s ³ + 12s ⁴)
4	2	3	−1	8(−s + 3s ² − 3s ³ + s ⁴)
4	2	3	−3	8(s − 3s ² + 3s ³ − s ⁴)
4	0	1	−1	8(s − 8s ² + 16s ³ − 9s ⁴)
4	0	3	−1	4(4s − 11s ² + 10s ³ − 3s ⁴)
4	0	3	−3	12(s ² − 2s ³ + s ⁴)
4	−2	1	1	$4 (\sqrt{1 - s s} - 6 {\sqrt{1 - s s}}^{2} + 6 {\sqrt{1 - s s}}^{3})$
4	−2	3	3	$4 (- 2 \sqrt{1 - s s} + 5 {\sqrt{1 - s s}}^{2} - 3 {\sqrt{1 - s s}}^{3})$
4	−2	3	1	$4 (2 \sqrt{1 - s s} - 3 {\sqrt{1 - s s}}^{2} + 1 {\sqrt{1 - s s}}^{3})$
6	0	1	−1	8s − 148s ² + 815s ³ − 1885s ⁴ + 1925s ⁵ − 715s ⁶
6	0	3	−1	4(4s − 53s ² + 205s ³ − 332s ⁴ + 241s ⁵ − 65s ⁶)
6	0	3	−3	4(9s ² − 77s ³ + 192s ⁴ − 189s ⁵ + 65s ⁶)
6	0	5	−1	3(8s − 36s ² + 65s ³ − 59s ⁴ + 27s ⁵ − 5s ⁶)

Front	a ²²	a ³¹	a ³³	a ⁴⁰	a ⁴²	a ⁴⁴	a ⁶⁰
Front				mλ (rms)
1	79	36	13	191	15	3	31
2	424	18	6	27	8	4	10

Abstract

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Applied Optics