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Two-dimensional wave-front reconstruction from lateral shearing interferograms

Peiying Liang, Jianping Ding, Zhou Jin, Cheng-Shan Guo, and Hui-tian Wang

Open Access Open Access

Abstract

An algorithm is proposed to reconstruct two-dimensional wave-front from phase differences measured by lateral shearing interferometer. Two one-dimensional phase profiles of object wave-front are computed using Fourier transform from phase differences, and then the two-dimensional wave-front distribution is retrieved by use of least-square fitting. The algorithm allows large shear amount and works fast based on fast Fourier transform. Investigations into reconstruction accuracy and reliability are carried out by numerical experiments, in which effects of different shear amounts and noises on reconstruction accuracy are evaluated. Optical measurement is made in a lateral shearing interferometer based on double-grating.

©2006 Optical Society of America

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Figures (5)
Fig. 1.
Fig. 1. Computer simulation of wave-front reconstruction from two phase differences in orthogonal directions: (a) is the original phase function, (b) and (c) are the x- and y-directional phase difference distributions, respectively, and (d) is the reconstructed phase distribution.

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Fig. 2.
Fig. 2. Phase difference imposed by uniformly random noise with a level of 30%, where noise level is defined as the ratio between the average of noise absolute value and the one of the phase difference. (a) and (b) are the x- and y-directional phase difference distributions, respectively. (c) is the reconstructed phase distribution.

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Fig. 3.
Fig. 3. Deviation of the reconstructed to the original phase vs. noise level, where noise level is defined as the ratio between the average of noise absolute value and the one of the phase difference

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Fig. 4.
Fig. 4. Double-grating lateral shearing interferometer: Rotating ground glass is used to lower spatial coherence of the illumination light for reducing speckle; Two Ronchi gratings produce both shear and phase shift.

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Fig. 5.
Fig. 5. Result of the optic surface testing with the double-grating lateral shearing interferometer. (a) The x-directional lateral-shearing interferogram. (b) The y-directional lateral-shearing interferogram. (c) The reconstructed phase distribution.

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Tables (1)

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Table 1. Deviation of the reconstructed phase to the original phase vs. the shear amount s. (λ=632.8nm)

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Equations (17)

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D x m n = φ m n − φ m − s n … m , n = 0,1,2 , . N − 1 .
D y m n = φ m n − φ ( m , n − s ) . m , n = 0,12 , … N − 1 .
FT x { φ m n } = FT x { D x m n } 1 − exp ( − i 2 π v x s N ) .
FT y { φ m n } = FT y { D y m n } 1 − exp ( − i 2 π v y s N ) .
f x m n = FT x − 1 { FT x { D x m n } 1 − exp ( − i 2 π v x s N ) } .
f y m n = FT y − 1 { FT y { D y m n } 1 − exp ( − i 2 π v y s N ) } .
f x m n + c n = φ m n + N x m n .
f x m n + d m = φ m n + N y m n .
c n = − 1 N ∑ n = 0 N − 1 [ f x N − 1 n − f y N − 1 n ] − 1 N ∑ m = 0 N − 1 [ f x m n − f n m n ] + 1 N 2 ∑ m = 0 N − 1 ∑ n = 0 N − 1 [ f x m n − f n m n ] . n = 0,1,2 , … , N − 1 .
d m = − 1 N ∑ n = 0 N − 1 [ f x m n − f y m n ] − 1 N ∑ m = 0 N − 1 [ f x N − 1 n − f y N − 1 n ] . m = 0,1,2 , … N − 1 .
d N − 1 = 0 .
φ m n = [ f x m n + c n ] + [ f y m n + d m ] 2 .
∑ p = 0 ( N s ) − 1 D x ( m + ps , n ) = 0 . m = 0,1,2 , … s − 1 ; n = 0,1,2 , … , N − 1 .
∑ q = 0 ( N s ) − 1 D y ( m , n + qs ) = 0 . n = 0,1,2 , … s − 1 ; m = 0,1,2 , … , N − 1 .
D x m n = − ∑ p = 1 N s − 1 D x ( m + ps , n ) . m = 0,1,2 , … , s − 1 ; n = 0,1,2 , … , N − 1
D y m n = − ∑ q = 1 N s − 1 D y ( m , n + qs ) . n = 0,1,2 , … , s − 1 ; m = 0,1,2 , … , N − 1
φ x y = 2 π × [ 0.1665 × ( x 2 + y 2 − 2 ) ( x 2 + y 2 + 1 ) − 0.8325 × ( x 2 − y 2 ) − 0.6660 × ( x 2 + y 2 ) ( x 2 − y 2 ) ] .
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