Abstract

A novel, single-shot, low-cost, multidirectional lateral shear interferometer for extended range wave front phase gradient sensing has been developed. It exploits the Fresnel diffraction field, which is formed by the five lowest diffraction orders of a simple binary amplitude checker grating. The Fresnel intensity pattern encodes information on four directional partial derivatives of the wave front under test. It has been theoretically, numerically, and experimentally shown that for larger gradient phase objects or shear amounts only the diagonal derivative information is easily accessible. The horizontal and vertical direction gradient maps are strongly amplitude modulated. Therefore, their demodulation becomes a challenging task. The same feature has been found in widely used quadriwave interferometer, which was developed at ONERA, France. The results of analytical and numerical studies and experimental works, including fringe pattern processing and phase demodulation, are presented.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref]
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2016 (1)

2015 (2)

2014 (2)

K. Patorski, M. Trusiak, and K. Pokorski, “Single-shot two-channel Talbot interferometry using checker grating and Hilbert-Huang fringe pattern processing,” Proc. SPIE 9132, 91320Z (2014).
[Crossref]

K. Pokorski and K. Patorski, “Wavelet transform - capabilities expanded,” Proc. SPIE 9203, 92030N (2014).
[Crossref]

2012 (1)

2011 (1)

R. Legarda-Saenz and A. Espinosa-Romero, “Wavefront reconstruction using multiple directional derivatives and Fourier transform,” Opt. Eng. 50(4), 040501 (2011).
[Crossref]

2009 (1)

2007 (1)

B. Toulon, J. Primot, N. Guérineau, R. Haidar, S. Velghe, and R. Mercier, “Step-selective measurement by grating-based lateral shearing interferometry for segmented telescopes,” Opt. Commun. 279(2), 240–243 (2007).
[Crossref]

2005 (2)

S. Velghe, J. Primot, N. Guerineau, R. Haidar, M. Cohen, and B. Wattellier, “Accurate and highly resolving quadri-wave lateral shearing interferometer, from visible to IR,” Proc. SPIE 5776, 134–143 (2005).
[Crossref]

S. Velghe, J. Primot, N. Guérineau, M. Cohen, and B. Wattellier, “Wave-front reconstruction from multidirectional phase derivatives generated by multilateral shearing interferometers,” Opt. Lett. 30(3), 245–247 (2005).
[Crossref] [PubMed]

2000 (2)

1999 (1)

J. A. Quiroga, D. Crespo, and E. Bernabeu, “Fourier transform method for automatic processing of moiré deflectograms,” Opt. Eng. 38(6), 974–982 (1999).
[Crossref]

1997 (1)

1995 (1)

1988 (1)

1987 (1)

1978 (1)

Bai, J.

Bernabeu, E.

J. A. Quiroga, D. Crespo, and E. Bernabeu, “Fourier transform method for automatic processing of moiré deflectograms,” Opt. Eng. 38(6), 974–982 (1999).
[Crossref]

Bhattacharya, S.

Bon, P.

Cohen, M.

S. Velghe, J. Primot, N. Guérineau, M. Cohen, and B. Wattellier, “Wave-front reconstruction from multidirectional phase derivatives generated by multilateral shearing interferometers,” Opt. Lett. 30(3), 245–247 (2005).
[Crossref] [PubMed]

S. Velghe, J. Primot, N. Guerineau, R. Haidar, M. Cohen, and B. Wattellier, “Accurate and highly resolving quadri-wave lateral shearing interferometer, from visible to IR,” Proc. SPIE 5776, 134–143 (2005).
[Crossref]

Crespo, D.

J. A. Quiroga, D. Crespo, and E. Bernabeu, “Fourier transform method for automatic processing of moiré deflectograms,” Opt. Eng. 38(6), 974–982 (1999).
[Crossref]

Czarnek, R.

Espinosa-Romero, A.

R. Legarda-Saenz and A. Espinosa-Romero, “Wavefront reconstruction using multiple directional derivatives and Fourier transform,” Opt. Eng. 50(4), 040501 (2011).
[Crossref]

Guerineau, N.

S. Velghe, J. Primot, N. Guerineau, R. Haidar, M. Cohen, and B. Wattellier, “Accurate and highly resolving quadri-wave lateral shearing interferometer, from visible to IR,” Proc. SPIE 5776, 134–143 (2005).
[Crossref]

Guérineau, N.

Guo, Y.

Haidar, R.

B. Toulon, J. Primot, N. Guérineau, R. Haidar, S. Velghe, and R. Mercier, “Step-selective measurement by grating-based lateral shearing interferometry for segmented telescopes,” Opt. Commun. 279(2), 240–243 (2007).
[Crossref]

S. Velghe, J. Primot, N. Guerineau, R. Haidar, M. Cohen, and B. Wattellier, “Accurate and highly resolving quadri-wave lateral shearing interferometer, from visible to IR,” Proc. SPIE 5776, 134–143 (2005).
[Crossref]

Harvey, J. E.

Kozak, S.

Legarda-Saenz, R.

R. Legarda-Saenz and A. Espinosa-Romero, “Wavefront reconstruction using multiple directional derivatives and Fourier transform,” Opt. Eng. 50(4), 040501 (2011).
[Crossref]

Legarda-Sáenz, R.

Ling, T.

Liu, D.

Maucort, G.

Mercier, R.

B. Toulon, J. Primot, N. Guérineau, R. Haidar, S. Velghe, and R. Mercier, “Step-selective measurement by grating-based lateral shearing interferometry for segmented telescopes,” Opt. Commun. 279(2), 240–243 (2007).
[Crossref]

Monneret, S.

Patorski, K.

Pokorski, K.

K. Patorski, M. Trusiak, and K. Pokorski, “Diffraction grating three-beam interferometry without self-imaging regime contrast modulations,” Opt. Lett. 40(6), 1089–1092 (2015).
[Crossref] [PubMed]

K. Patorski, M. Trusiak, and K. Pokorski, “Single-shot two-channel Talbot interferometry using checker grating and Hilbert-Huang fringe pattern processing,” Proc. SPIE 9132, 91320Z (2014).
[Crossref]

K. Pokorski and K. Patorski, “Wavelet transform - capabilities expanded,” Proc. SPIE 9203, 92030N (2014).
[Crossref]

Post, D.

Primot, J.

B. Toulon, J. Primot, N. Guérineau, R. Haidar, S. Velghe, and R. Mercier, “Step-selective measurement by grating-based lateral shearing interferometry for segmented telescopes,” Opt. Commun. 279(2), 240–243 (2007).
[Crossref]

S. Velghe, J. Primot, N. Guerineau, R. Haidar, M. Cohen, and B. Wattellier, “Accurate and highly resolving quadri-wave lateral shearing interferometer, from visible to IR,” Proc. SPIE 5776, 134–143 (2005).
[Crossref]

S. Velghe, J. Primot, N. Guérineau, M. Cohen, and B. Wattellier, “Wave-front reconstruction from multidirectional phase derivatives generated by multilateral shearing interferometers,” Opt. Lett. 30(3), 245–247 (2005).
[Crossref] [PubMed]

J. Primot and N. Guérineau, “Extended Hartmann test based on the pseudoguiding property of a Hartmann mask completed by a phase chessboard,” Appl. Opt. 39(31), 5715–5720 (2000).
[Crossref] [PubMed]

J. Primot and L. Sogno, “Achromatic three-wave (or more) lateral shearing interferomeetr,” J. Opt. Soc. Am. A 12(12), 2679–2685 (1995).
[Crossref]

Quiroga, J. A.

J. A. Quiroga, D. Crespo, and E. Bernabeu, “Fourier transform method for automatic processing of moiré deflectograms,” Opt. Eng. 38(6), 974–982 (1999).
[Crossref]

Rivera, M.

Rodríguez-Vera, R.

Shack, R. V.

Shen, Y.

Sirohi, R. S.

Sluzewski, L.

Sogno, L.

Toulon, B.

B. Toulon, J. Primot, N. Guérineau, R. Haidar, S. Velghe, and R. Mercier, “Step-selective measurement by grating-based lateral shearing interferometry for segmented telescopes,” Opt. Commun. 279(2), 240–243 (2007).
[Crossref]

Trujillo-Schiaffino, G.

Trusiak, M.

Velghe, S.

B. Toulon, J. Primot, N. Guérineau, R. Haidar, S. Velghe, and R. Mercier, “Step-selective measurement by grating-based lateral shearing interferometry for segmented telescopes,” Opt. Commun. 279(2), 240–243 (2007).
[Crossref]

S. Velghe, J. Primot, N. Guerineau, R. Haidar, M. Cohen, and B. Wattellier, “Accurate and highly resolving quadri-wave lateral shearing interferometer, from visible to IR,” Proc. SPIE 5776, 134–143 (2005).
[Crossref]

S. Velghe, J. Primot, N. Guérineau, M. Cohen, and B. Wattellier, “Wave-front reconstruction from multidirectional phase derivatives generated by multilateral shearing interferometers,” Opt. Lett. 30(3), 245–247 (2005).
[Crossref] [PubMed]

Wattellier, B.

Yang, Y.

Yue, X.

Appl. Opt. (5)

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

Opt. Commun. (1)

B. Toulon, J. Primot, N. Guérineau, R. Haidar, S. Velghe, and R. Mercier, “Step-selective measurement by grating-based lateral shearing interferometry for segmented telescopes,” Opt. Commun. 279(2), 240–243 (2007).
[Crossref]

Opt. Eng. (2)

J. A. Quiroga, D. Crespo, and E. Bernabeu, “Fourier transform method for automatic processing of moiré deflectograms,” Opt. Eng. 38(6), 974–982 (1999).
[Crossref]

R. Legarda-Saenz and A. Espinosa-Romero, “Wavefront reconstruction using multiple directional derivatives and Fourier transform,” Opt. Eng. 50(4), 040501 (2011).
[Crossref]

Opt. Express (1)

Opt. Lett. (5)

Proc. SPIE (3)

K. Patorski, M. Trusiak, and K. Pokorski, “Single-shot two-channel Talbot interferometry using checker grating and Hilbert-Huang fringe pattern processing,” Proc. SPIE 9132, 91320Z (2014).
[Crossref]

K. Pokorski and K. Patorski, “Wavelet transform - capabilities expanded,” Proc. SPIE 9203, 92030N (2014).
[Crossref]

S. Velghe, J. Primot, N. Guerineau, R. Haidar, M. Cohen, and B. Wattellier, “Accurate and highly resolving quadri-wave lateral shearing interferometer, from visible to IR,” Proc. SPIE 5776, 134–143 (2005).
[Crossref]

Other (4)

K. Patorski and M. Kujawińska, Handbook of the Moiré Fringe Technique (Elsevier).

D. Malacara, ed., Optical Shop Testing, 3rd ed. (Wiley).

D. L. Fried, “Atmospheric turbulence optical effects: understanding the adaptive-optics implications”, in Adaptive Optics for Astronomy, C243 of NATO Advanced Study Institute Series (Kluwer Academic).

K. Patorski, in Progress in Optics, E. Wolf, ed. (North-Holland), vol. 27, pp. 1–108 (1989).

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Figures (13)

Fig. 1
Fig. 1 Schematic representation of the principle of multidirectional 5-beam lateral shear interferometry using an amplitude checker grating G1. φ(x,y) is the phase distribution of the impinging aberrated quasi-plane wave front under test. OP indicates one of the observation planes (coinciding with a CCD/CMOS matrix).
Fig. 2
Fig. 2 Simulated intensity patterns generated by interfering four side orders of the checker grating [see first 5 rows of Eq. (3)) in the case of (a) the beam under test with spherical aberration and (b) without it].
Fig. 3
Fig. 3 Simulated intensity pattern generated by interference of four side orders of the checker grating taking into consideration only the first two rows of Eq. (3); the beam under test with spherical aberration.
Fig. 4
Fig. 4 Simulated intensity distribution of two-beam interference providing one of the two second harmonic terms of intensity distribution given by Eq. (3). Here the pattern along the x + y diagonal direction, see the fourth row of Eq. (3), is presented.
Fig. 5
Fig. 5 Simulated intensity patterns generated by interference of five lowest diffraction orders of the checker grating and described Eq. (3) in the case of the presence of spherical aberration, Fig. 5(a), and its absence, Fig. 5(b), respectively.
Fig. 6
Fig. 6 Simulated intensity distribution according to the one before last term of Eq. (3). It displays the first harmonic of intensity distribution along the diagonal direction x + y. The pattern is effectively generated by three-beam interference which leads to characteristic contrast modulations due to the Talbot effect (self-imaging phenomenon).
Fig. 7
Fig. 7 Schematic geometry of our experimental amplitude checker grating based 5-beam lateral shear interferometer. Five lowest diffraction orders of the checker grating G1 are passed through the circular opening filter in the spatial frequency (light source image) plane. OL1 and OL2 – objectives with focal lengths f1 and f2, respectively, OP – output plane.
Fig. 8
Fig. 8 The situation in the central part of the spatial frequency plane of the amplitude checker grating; the position of the square shaped filter is indicated be red dashed line.
Fig. 9
Fig. 9 Multidirectional interference pattern influenced by the aberration of lens OL1 tested in the optical system shown in Fig. 7 for the axial separation distance between the checker grating G1 and the spatial frequency plane equal to Md2/λ ≈63 mm (M = 4). Opening mask in the frequency plane passes 5 lowest diffraction orders of the checker grating to form a fringe pattern.
Fig. 10
Fig. 10 Modulus of the Fourier spectrum of the multidirectional fringe pattern shown in Fig. 9. The comatic shape of diffraction spots is characteristic to the wave front spherical aberration tested by lateral shear interferometry [20,23]. See text for detailed explanation of rectilinear bands modulating comatic shape diffraction spots. Two diagonal diffraction spots are encircled with dashed lines to symbolically indicate filtering masks used in the fringe pattern Fourier transform processing described below.
Fig. 11
Fig. 11 Demodulated diagonal derivatives of the tested aberrated wave front. (a) and (b) correspond to 45 and 135 degrees directions, respectively, phase maps presented in radians; (c) and (d) cosine functions of demodulated phase maps – 45 and 135 degrees directions, respectively.
Fig. 12
Fig. 12 (a) amplitude modulation map of the second harmonic along 45 deg direction of the intensity distribution of the aberrated 5-beam fringe pattern, Fig. 9; (b) the x direction harmonic phase distribution, and (c) cosine function of this phase map presenting phase 2π jumps along dark amplitude modulation bands; (d) amplitude modulation map of the x harmonic. Additional high spatial frequency fringes seen in all the amplitude modulation images result from parasitic interferences present in the optical experimental system.
Fig. 13
Fig. 13 Amplitude modulation maps of (a) the y direction harmonic and (b) first 45 deg direction harmonic of the intensity distribution of the aberrated 5-beam fringe pattern, Fig. 9. Additional high spatial frequency fringes seen in all the amplitude modulation images result from parasitic interferences present in the optical experimental system.

Equations (4)

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t( x,y )= m n [ 1+ ( 1 ) m+n ] a m a n exp[ i 2π d ( mx+ny ) ],
E( x,y,z )=2 a 0 2 exp[ iφ( x,y ) ] + 2 a 1 2 exp{ i( 2π d [ ( xΔ )+( yΔ ) ]2π λz d 2 +φ( xΔ,yΔ ) ) } + 2 a 1 2 exp{ i( 2π d [ ( xΔ )( y+Δ ) ]2π λz d 2 +φ( xΔ,y+Δ ) ) } + 2 a 1 2 exp{ i( 2π d [ ( x+Δ )+( yΔ ) ]2π λz d 2 +φ( x+Δ,yΔ ) ) } + 2 a 1 2 exp{ i( 2π d [ ( x+Δ )( y+Δ ) ]2π λz d 2 +φ( x+Δ,y+Δ ) ) },
I( x,y,z )=16 a 1 4 + 8 a 1 4 cos[ 2π d 2x2Δ φ( x,yΔ ) x ]+8 a 1 4 cos[ 2π d 2x2Δ φ( x,y+Δ ) x ]  + 8 a 1 4 cos[ 2π d 2y2Δ φ( xΔ,y ) y ]+8 a 1 4 cos[ 2π d 2y2Δ φ( x+Δ,y ) y ] + 8 a 1 4 cos[ 2π d ( 2x+2y )2Δ φ( x,y ) ( x+y ) ] + 8 a 1 4 cos[ 2π d ( 2x2y )2Δ φ( x,y ) ( xy ) ]  + 4 a 0 4 + 16 a 0 2 a 1 2 cos[ 2π d ( x+y )Δ φ( x,y ) ( x+y ) ]cos[ 6π λz d 2 +( Δ 2 2 ) 2 φ( x,y ) ( x+y ) 2 ] + 16 a 0 2 a 1 2 cos[ 2π d ( xy )Δ φ( x,y ) ( xy ) ]cos[ 6π λz d 2 +( Δ 2 2 ) 2 φ( x,y ) ( xy ) 2 ].
φ( x,y )φ( xΔ,y )+Δ φ( x,y ) x Δ 2 2 2 φ( x,y ) x 2

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