Abstract

We present theory, design, and preliminary experimental studies for a compact wavefront sensor based on lateral shearing interferometry using a binary phase grating, image sensor, and Fourier-based processing. The integrated system places a diffractive element directly onto an image sensor to generate interference fringes within overlapping diffraction orders. The shearing ratio and the interferogram signal-to-noise ratio directly affect the reconstruction accuracy of wavefronts with differing spatial variations. Optimal shearing parameters associated with the autocorrelation of the input encourage placing a spatial light modulator as the diffractive element allowing adaptive wavefront sensing. Experimental results from a fixed-grating system are presented as well as requirements for next-generation adaptive systems.

© 2008 Optical Society of America

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2007 (1)

2006 (2)

P. Liang, J. Ding, Z. Jin, C. Guo, and H. Wang, “Two-dimensional wave-front reconstruction from lateral shearing interferograms,” Opt. Express 14, 625-634 (2006).
[CrossRef] [PubMed]

S. Zhao and P. S. Chung, “Digital speckle shearing interferometer using a liquid-crystal spatial light modulator,” Opt. Eng. 45, 105606 (2006).
[CrossRef]

2005 (1)

2003 (2)

2002 (1)

2000 (2)

S. De. Nicola and P. Ferraro, “Fourier transform method of fringe analysis for moiré interferometry,” J. Opt. A 2, 228-233(2000).
[CrossRef]

G. Paez, M. Strojnik, and G. Torales, “Vectorial shearing interferometer,” Appl. Opt. 39, 5172-5178 (2000).
[CrossRef]

1999 (3)

C. Elster and I. Weingartner, “Exact wave-front reconstruction from two lateral shearing interferograms,” J. Opt. Soc. Am. A 16, 2281-2285 (1999).
[CrossRef]

C. Elster and I. Weingartner, “Solution to the shearing problem,” Appl. Opt. 38, 5024-5031 (1999).
[CrossRef]

P. S. Fairman, B. K. Ward, B. F. Oreb, D. I. Farrant, Y. Gilliand, C. H. Freund, A. J. Leistner, J. A. Seckold, and C. J. Walsh, “300-mm-aperture phase-shifting Fizeau interferometer,” Opt. Eng. 38, 1371-1380 (1999).
[CrossRef]

1991 (1)

1988 (1)

1986 (1)

K. Freichlad and C. Koliopoulos, “Modal estimation of a wave front from difference measurements using the discrete Fourier transform,” J. Opt. Soc. Am. 3, 1852-1861 (1986).
[CrossRef]

1982 (1)

1977 (2)

1975 (1)

1973 (1)

1971 (1)

1967 (1)

1964 (1)

Bitou, Y.

Bone, D.

Brase, J. M.

Bryngdahl, O.

Chung, P. S.

S. Zhao and P. S. Chung, “Digital speckle shearing interferometer using a liquid-crystal spatial light modulator,” Opt. Eng. 45, 105606 (2006).
[CrossRef]

Cohen, M.

Ding, J.

Elster, C.

Fairman, P. S.

P. S. Fairman, B. K. Ward, B. F. Oreb, D. I. Farrant, Y. Gilliand, C. H. Freund, A. J. Leistner, J. A. Seckold, and C. J. Walsh, “300-mm-aperture phase-shifting Fizeau interferometer,” Opt. Eng. 38, 1371-1380 (1999).
[CrossRef]

Farrant, D. I.

P. S. Fairman, B. K. Ward, B. F. Oreb, D. I. Farrant, Y. Gilliand, C. H. Freund, A. J. Leistner, J. A. Seckold, and C. J. Walsh, “300-mm-aperture phase-shifting Fizeau interferometer,” Opt. Eng. 38, 1371-1380 (1999).
[CrossRef]

Ferraro, P.

S. De. Nicola and P. Ferraro, “Fourier transform method of fringe analysis for moiré interferometry,” J. Opt. A 2, 228-233(2000).
[CrossRef]

Freichlad, K.

K. Freichlad and C. Koliopoulos, “Modal estimation of a wave front from difference measurements using the discrete Fourier transform,” J. Opt. Soc. Am. 3, 1852-1861 (1986).
[CrossRef]

Freund, C. H.

P. S. Fairman, B. K. Ward, B. F. Oreb, D. I. Farrant, Y. Gilliand, C. H. Freund, A. J. Leistner, J. A. Seckold, and C. J. Walsh, “300-mm-aperture phase-shifting Fizeau interferometer,” Opt. Eng. 38, 1371-1380 (1999).
[CrossRef]

Fried, D.

Gavel, D. T.

Gilliand, Y.

P. S. Fairman, B. K. Ward, B. F. Oreb, D. I. Farrant, Y. Gilliand, C. H. Freund, A. J. Leistner, J. A. Seckold, and C. J. Walsh, “300-mm-aperture phase-shifting Fizeau interferometer,” Opt. Eng. 38, 1371-1380 (1999).
[CrossRef]

Goodman, J.

J. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

Guerineau, N.

Guo, C.

Hariharan, P.

P. Hariharan, Optical Interferometry, 2nd ed. (Academic, 2003).

Hermerschnidt, A.

Hudgin, R.

Ina, H.

Jin, Z.

Kinnstaetter, K.

Kobayashi, S.

Koliopoulos, C.

K. Freichlad and C. Koliopoulos, “Modal estimation of a wave front from difference measurements using the discrete Fourier transform,” J. Opt. Soc. Am. 3, 1852-1861 (1986).
[CrossRef]

Krüger, S.

Leistner, A. J.

P. S. Fairman, B. K. Ward, B. F. Oreb, D. I. Farrant, Y. Gilliand, C. H. Freund, A. J. Leistner, J. A. Seckold, and C. J. Walsh, “300-mm-aperture phase-shifting Fizeau interferometer,” Opt. Eng. 38, 1371-1380 (1999).
[CrossRef]

Liang, P.

Lohmann, A.

Lohmann, A. W.

Nicola, S. De.

S. De. Nicola and P. Ferraro, “Fourier transform method of fringe analysis for moiré interferometry,” J. Opt. A 2, 228-233(2000).
[CrossRef]

Oreb, B. F.

P. S. Fairman, B. K. Ward, B. F. Oreb, D. I. Farrant, Y. Gilliand, C. H. Freund, A. J. Leistner, J. A. Seckold, and C. J. Walsh, “300-mm-aperture phase-shifting Fizeau interferometer,” Opt. Eng. 38, 1371-1380 (1999).
[CrossRef]

Paez, G.

Poyneer, L.

Primot, J.

Rimmer, M. P.

Ronchi, V.

Schwider, J.

Seckold, J. A.

P. S. Fairman, B. K. Ward, B. F. Oreb, D. I. Farrant, Y. Gilliand, C. H. Freund, A. J. Leistner, J. A. Seckold, and C. J. Walsh, “300-mm-aperture phase-shifting Fizeau interferometer,” Opt. Eng. 38, 1371-1380 (1999).
[CrossRef]

Streibl, N.

Strojnik, M.

Suzuki, T.

Takeda, M.

Torales, G.

Velghe, S.

Walsh, C. J.

P. S. Fairman, B. K. Ward, B. F. Oreb, D. I. Farrant, Y. Gilliand, C. H. Freund, A. J. Leistner, J. A. Seckold, and C. J. Walsh, “300-mm-aperture phase-shifting Fizeau interferometer,” Opt. Eng. 38, 1371-1380 (1999).
[CrossRef]

Wang, H.

Ward, B. K.

P. S. Fairman, B. K. Ward, B. F. Oreb, D. I. Farrant, Y. Gilliand, C. H. Freund, A. J. Leistner, J. A. Seckold, and C. J. Walsh, “300-mm-aperture phase-shifting Fizeau interferometer,” Opt. Eng. 38, 1371-1380 (1999).
[CrossRef]

Wattellier, B.

Weingartner, I.

Wernicke, G.

Wyant, J. C.

Yokozeki, S.

Zhao, S.

S. Zhao and P. S. Chung, “Digital speckle shearing interferometer using a liquid-crystal spatial light modulator,” Opt. Eng. 45, 105606 (2006).
[CrossRef]

Appl. Opt. (9)

J. Opt. A (1)

S. De. Nicola and P. Ferraro, “Fourier transform method of fringe analysis for moiré interferometry,” J. Opt. A 2, 228-233(2000).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

J. Primot, “Theoretical description of Shack-Hartmann wave-front sensor,” Opt. Commun. 222, 81-92 (2003).
[CrossRef]

Opt. Eng. (2)

P. S. Fairman, B. K. Ward, B. F. Oreb, D. I. Farrant, Y. Gilliand, C. H. Freund, A. J. Leistner, J. A. Seckold, and C. J. Walsh, “300-mm-aperture phase-shifting Fizeau interferometer,” Opt. Eng. 38, 1371-1380 (1999).
[CrossRef]

S. Zhao and P. S. Chung, “Digital speckle shearing interferometer using a liquid-crystal spatial light modulator,” Opt. Eng. 45, 105606 (2006).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Other (2)

P. Hariharan, Optical Interferometry, 2nd ed. (Academic, 2003).

J. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

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Figures (10)

Fig. 1
Fig. 1

Diffractive elements perform LSI by overlapping diffraction orders. (a) Information about the wavefront is obtained along the shearing direction. (b) Two-dimensional gratings simultaneously record the entire field while (c) SLMs sequentially create multiple interferograms.

Fig. 2
Fig. 2

Diffraction angles create a linear fringe pattern within the interferogram. Wavefront aberrations distort the carrier pattern, which is spectrally processed.

Fig. 3
Fig. 3

Carrier modulation allows the phase difference signal to be isolated from other illumination components. The spectral width of the sidebands contains the phase information.

Fig. 4
Fig. 4

(a) Low sampling and moiré disrupt the carrier modulation seen using (b) an integrated device consisting of a fixed phase grating and image sensor. (c) Poor detection of the horizontal carrier fringes is seen in the magnified view.

Fig. 5
Fig. 5

Integrated LSI is modified by using a 4F relay with magnification to reduce the diffraction angle from θ 1 to θ 2 and increase the carrier period.

Fig. 6
Fig. 6

(a) A section of the interferogram captured from the modified LSI is used to reconstruct the spherical profile. (b) The lens profile is compared with a phase-shifting microscope.

Fig. 7
Fig. 7

OPDs between the sheared points of overlap disrupt fringe carrier periodicity. Adjusting the shearing ratio from (a) 0.01 to (b) 0.1 and (c) 0.3 increases the difference signal highlighted in (d), (e), and (f), respectively.

Fig. 8
Fig. 8

Autocorrelation maps identify the shift needed for maximum OPD. (a) Wavefronts exhibiting slow spatial variation produce (c) large difference signals with large shears while (b) increased spatial variation requires (d) smaller shifts.

Fig. 9
Fig. 9

(a) Simulated reference wavefront sheared and with Gaussian noise added to the interferogram prior to processing. Reconstructions from (a) 10 dB , (b) 20 dB , and (c) 30 dB SNRs. A low SNR loses all profile detail, while a high SNR creates reconstructions with > λ / 100 accuracy.

Fig. 10
Fig. 10

Varying the shear ratio at different SNRs leads to reconstruction error minimums that depend on the wavefront profile. (a) Slowly varying wavefronts benefit from large shears, while (b) smaller shears perform best for increased spatial variation.

Equations (17)

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g ( x , y ) = | | f ( x , y ) | exp [ j ϕ ( x , y ) ] + | f ( x s , y ) | exp [ j ϕ ( x s , y ) ] | 2
g ( x , y ) = a ( x , y ) + b ( x , y ) cos [ ϕ ( x , y ) ϕ ( x s , y ) ]
a ( x , y ) = | f ( x , y ) | 2 + | f ( x s , y ) | 2 ,
b ( x , y ) = 2 | f ( x , y ) | | f ( x s , y ) | .
sin θ m = m λ d ,
g ( x , y ) = a ( x , y ) + b ( x , y ) cos [ ϕ ( x , y ) ϕ ( x s , y ) + 2 π f o x ] ,
f o = sin ( θ α θ β ) λ .
g ( x , y ) = a ( x , y ) + c ( x , y ) exp ( i 2 π f o x ) + c * ( x , y ) exp ( i 2 π f o x ) ,
c ( x , y ) = 1 2 b ( x , y ) exp [ i ϕ ( x , y ) i ϕ ( x s , y ) ]
G ( ω x , ω y ) = A ( ω x , ω y ) + C ( ω x f o , ω y ) + C * ( ω x + f o , ω y ) .
ϕ ( x , y ) ϕ ( x s , y ) = arctan [ Re { c r ( x , y ) } Im { c m ( x , y ) } Im { c r ( x , y ) } Re { c m ( x , y ) } Im { c r ( x , y ) } Im { c m ( x , y ) } + Re { c r ( x , y ) } Re { c m ( x , y ) } ] ,
Δ ϕ x ( m , n ) = ϕ ( m , n ) ϕ ( m s , n ) , m , n = 0 , 1 , 2 , , N 1 ,
Δ ϕ y ( m , n ) = ϕ ( m , n ) ϕ ( m , n s ) , m , n = 0 , 1 , 2 , , N 1 ,
FT { Δ ϕ x ( m , n ) } = FT { ϕ ( x , y ) } [ 1 exp ( i 2 π ω x s N ) ] ,
FT { Δ ϕ y ( m , n ) } = FT { ϕ ( x , y ) } [ 1 exp ( i 2 π ω y s N ) ] .
Δ ϕ x ( m , n ) = p = 0 ( N / s ) 1 Δ ϕ x ( m + p s , n ) , m = 0 , 1 , 2 , , s 1 , n = 0 , 1 , 2 , , N 1 ;
Δ ϕ y ( m , n ) = q = 0 ( N / s ) 1 Δ ϕ y ( m , n + q s ) , n = 0 , 1 , 2 , , s 1 , m = 0 , 1 , 2 , , N 1.

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