Abstract

In this research, we present an interferometric system to analyze transparent samples using interferograms generated by a phase-shifting radial shear grating interferometer for two cases: the first obtaining $n$ simultaneous phase-shifting interferograms using a coherent light source and the second one using sequential phase steps with a white light source. For the first case, the simultaneous interferograms are generated using two optical systems: the first one generates the polarized pattern while the second one consists of a ${4}f$ system creating replicas of the output interferograms. By using a 2D sinusoidal phase grating, we have the advantage of obtaining up to nine replicated interferograms, all of them with comparable intensities and having amplitudes modulated by the 2D sinusoidal phase grating diffraction orders as zero-order Bessel’s functions. To obtain the optical phase map, several phase shifts are generated by placing a polarizing filter covering each replicated interferogram. We highlight the advantage of using $n$ simultaneous interferograms by comparing resulting optical phases processed by a conventional four-step algorithm against those obtained by an implemented ${n}={N}+{1}$ method, reducing errors with noisy interferograms. Results for ${n}={7}$ and ${n}={9}$ cases are presented. In addition, we have tested the setup with white light interference techniques by employing the polarizer radial shearing interferometer; for this case, the optical phase is calculated with the four-step and the three-step algorithms. Results of testing the developed system to examine static and dynamic phase objects are also included.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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Corrections

30 March 2020: A correction was made to Eq. (2).


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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]

2020 (1)

V. H. Flores, A. Reyes-Figueroa, C. Carrillo-Delgado, and M. Rivera, “Computation of the phase step between two-step fringe patterns based on Gram--Schmidt algorithm,” Optics & Laser Technology 126, 106105 (2020).
[Crossref]

2019 (3)

Z. Wang, S. Wang, P. Yang, and B. Xu, “Three-wave radial shearing interferometer,” Opt. Lett. 44, 1996–1999 (2019).
[Crossref]

J. B. Mitchell, G. W. Roberts, and P. C. T. Rees, “Full-field, high-frequency, heterodyne interferometry for dynamic metrology based on phase detection using a modified time-of-flight camera,” Proc. SPIE 11056, 110560U (2019).
[Crossref]

M. T. Nguyen, Y. Ghim, and H. Rhee, “Single-shot deflectometry for dynamic 3D surface profile measurement by modified spatial-carrier frequency phase-shifting method,” Sci. Rep. 9, 3157 (2019).
[Crossref]

2018 (1)

2017 (1)

N. I. Toto-Arellano, “4D measurements of biological and synthetic structures using a dynamic interferometer,” J. Mod. Opt. 64, S20–S29 (2017).
[Crossref]

2016 (3)

T. D. Yang, H.-J. Kim, K. J. Lee, B.-M. Kim, and Y. Choi, “Single-shot and phase-shifting digital holographic microscopy using a 2-D grating,” Opt. Express 24, 9480–9488 (2016).
[Crossref]

P. K. Upputuri, M. Pramanik, K. M. Nandigana, and M. P. Kothiyal, “Multi-colour microscopic interferometry for optical metrology and imaging applications,” Opt. Lasers Eng. 84, 10–25 (2016).
[Crossref]

M. Rivera, O. Dalmau, A. Gonzalez, and F. Hernandez-Lopez, “Two-step fringe pattern analysis with a Gabor filter bank,” Opt. Lasers. Eng. 85, 29–37 (2016).
[Crossref]

2014 (1)

M. Trusiak, M. Wielgus, and K. Patorski, “Advanced processing of optical fringe patterns by automated selective reconstruction and enhanced fast empirical mode decomposition,” Opt. Lasers Eng. 52, 230–240 (2014).
[Crossref]

2013 (1)

S. Sarkar and K. Bhattacharya, “Polarization phase shifting cyclic interferometer for surface profilometry of nonbirefringent phase samples,” J. Mod. Opt. 60, 185–189 (2013).
[Crossref]

2012 (3)

U. P. Kumar, W. Haifeng, N. K. Mohan, and M. P. Kothiyal, “White light interferometry for surface profiling with a colour CCD,” Opt. Lasers Eng. 50, 1084–1088 (2012).
[Crossref]

Y. Kim, J. Jeong, J. Jang, M. W. Kim, and Y. Park, “Polarization holographic microscopy for extracting spatio-temporally resolved Jones matrix,” Opt. Express 20, 9948–9955 (2012).
[Crossref]

D. I. Serrano-García, N. I. Toto-Arellano, A. Martínez-García, and G. Rodríguez Zurita, “Radial slope measurement of dynamic transparent samples,” J. Opt. 14, 045706 (2012).
[Crossref]

2009 (1)

2008 (2)

2007 (2)

D. Liu, Y. Yang, Y. Shen, J. Weng, and Y. Zhuo, “System optimization of radial shearing interferometer for aspheric testing,” Proc. SPIE 6834, 68340U (2007).
[Crossref]

D. Liu, Y. Yang, L. Wang, and Y. Zhuo, “Real-time diagnosis of transient pulse laser with high repetition by radial shearing interferometer,” Appl. Opt. 46, 8305–8314 (2007).
[Crossref]

2006 (1)

2005 (2)

M. N. Morris, J. Millerd, N. Brock, J. Hayes, and B. Saif, “Dynamic phase-shifting electronic speckle pattern interferometer,” Proc. SPIE 5869, 58691B (2005).
[Crossref]

M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, “Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer,” Appl. Opt. 44, 6861–6868 (2005).
[Crossref]

2003 (3)

J. C. Wyant, “Dynamic interferometry,” Opt. Photon. News 14(4), 36–41 (2003).
[Crossref]

W. W. Kowalik, B. E. Garncarz, and H. T. Kasprzak, “Corneal topography measurement by means of radial shearing interference: part II—measurements errors,” Optik 114, 199–206 (2003).
[Crossref]

J. A. Quiroga and M. Servin, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224, 221–227 (2003).
[Crossref]

2002 (1)

W. W. Kowalik, B. E. Garncarz, and H. T. Kasprzak, “Corneal topography measurement by means of radial shearing interference: part I—theoretical considerations,” Optik 113, 39–45 (2002).
[Crossref]

2001 (1)

T. Shirai, T. H. Barnes, and T. G. Haskell, “Real-time restoration of a blurred image with a liquid-crystal adaptive optics system based on all-optical feedback interferometry,” Opt. Commun. 188, 275–282 (2001).
[Crossref]

2000 (3)

1988 (1)

R. F. Horton, “Design of a white light radial shear interferometer for segmented mirror control,” Opt. Eng. 27, 1063–1066 (1988).
[Crossref]

1985 (1)

1984 (1)

P. Hariharan, B. F. Oreb, and Z. Wanzhi, “Measurement of aspheric surfaces using a microcomputer controlled digital radial-shear interferometer,” J. Mod. Opt. 31, 989–999 (1984).
[Crossref]

1974 (4)

1973 (1)

1972 (1)

1965 (1)

W. H. Steel, “A radial shear interferometer for testing microscope objectives,” J. Sci. Instrum. 42, 102–104 (1965).
[Crossref]

1964 (1)

1962 (1)

D. S. Brown, “Radial shear interferometry,” J. Sci. Intrum. 39, 71–72 (1962).
[Crossref]

1961 (1)

P. Hariharan and D. Sen, “Radial shearing interferometer,” J. Sci. Instrum. 38, 428–432 (1961).
[Crossref]

Baker, K. L.

Barnes, T. H.

T. Shirai, T. H. Barnes, and T. G. Haskell, “Real-time restoration of a blurred image with a liquid-crystal adaptive optics system based on all-optical feedback interferometry,” Opt. Commun. 188, 275–282 (2001).
[Crossref]

T. Shirai, T. H. Barnes, and T. G. Haskell, “Adaptive wave-front correction by means of all-optical feedback interferometry,” Opt. Lett. 25, 773–775 (2000).
[Crossref]

Bhattacharya, K.

S. Sarkar and K. Bhattacharya, “Polarization phase shifting cyclic interferometer for surface profilometry of nonbirefringent phase samples,” J. Mod. Opt. 60, 185–189 (2013).
[Crossref]

Brangaccio, D. J.

Brock, N.

M. N. Morris, J. Millerd, N. Brock, J. Hayes, and B. Saif, “Dynamic phase-shifting electronic speckle pattern interferometer,” Proc. SPIE 5869, 58691B (2005).
[Crossref]

M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, “Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer,” Appl. Opt. 44, 6861–6868 (2005).
[Crossref]

Brown, D. S.

D. S. Brown, “Radial shear interferometry,” J. Sci. Intrum. 39, 71–72 (1962).
[Crossref]

D. S. Brown, Interferometry, Symposium 11 at National Physical Laboratory (Her Majesty’s Stationery Office, 1960), p. 253.

Bruning, J. H.

Carrillo-Delgado, C.

V. H. Flores, A. Reyes-Figueroa, C. Carrillo-Delgado, and M. Rivera, “Computation of the phase step between two-step fringe patterns based on Gram--Schmidt algorithm,” Optics & Laser Technology 126, 106105 (2020).
[Crossref]

Choi, Y.

Collier, J. L.

Creath, K.

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1998), Vol. 26, pp. 349–393.

Dalmau, O.

M. Rivera, O. Dalmau, A. Gonzalez, and F. Hernandez-Lopez, “Two-step fringe pattern analysis with a Gabor filter bank,” Opt. Lasers. Eng. 85, 29–37 (2016).
[Crossref]

Danson, C. N.

Delisle, C.

Edwards, C. B.

Estrada, J. C.

Flores, V. H.

V. H. Flores, A. Reyes-Figueroa, C. Carrillo-Delgado, and M. Rivera, “Computation of the phase step between two-step fringe patterns based on Gram--Schmidt algorithm,” Optics & Laser Technology 126, 106105 (2020).
[Crossref]

Fouéré, J. C.

Gallagher, J. E.

García, A. M.

Garncarz, B. E.

W. W. Kowalik, B. E. Garncarz, and H. T. Kasprzak, “Corneal topography measurement by means of radial shearing interference: part II—measurements errors,” Optik 114, 199–206 (2003).
[Crossref]

W. W. Kowalik, B. E. Garncarz, and H. T. Kasprzak, “Corneal topography measurement by means of radial shearing interference: part I—theoretical considerations,” Optik 113, 39–45 (2002).
[Crossref]

Geary, J. M.

J. M. Geary, “Wavefront sensors,” in Adaptive Optics Engineering Handbook, R. K. Tyson, ed. (2000), pp. 123–150.

Ghiglia, C.

C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998), Chap. 4.

Ghim, Y.

M. T. Nguyen, Y. Ghim, and H. Rhee, “Single-shot deflectometry for dynamic 3D surface profile measurement by modified spatial-carrier frequency phase-shifting method,” Sci. Rep. 9, 3157 (2019).
[Crossref]

Gonzalez, A.

M. Rivera, O. Dalmau, A. Gonzalez, and F. Hernandez-Lopez, “Two-step fringe pattern analysis with a Gabor filter bank,” Opt. Lasers. Eng. 85, 29–37 (2016).
[Crossref]

Haifeng, W.

U. P. Kumar, W. Haifeng, N. K. Mohan, and M. P. Kothiyal, “White light interferometry for surface profiling with a colour CCD,” Opt. Lasers Eng. 50, 1084–1088 (2012).
[Crossref]

Hariharan, P.

P. Hariharan, B. F. Oreb, and Z. Wanzhi, “Measurement of aspheric surfaces using a microcomputer controlled digital radial-shear interferometer,” J. Mod. Opt. 31, 989–999 (1984).
[Crossref]

P. Hariharan and D. Sen, “Radial shearing interferometer,” J. Sci. Instrum. 38, 428–432 (1961).
[Crossref]

Haskell, T. G.

T. Shirai, T. H. Barnes, and T. G. Haskell, “Real-time restoration of a blurred image with a liquid-crystal adaptive optics system based on all-optical feedback interferometry,” Opt. Commun. 188, 275–282 (2001).
[Crossref]

T. Shirai, T. H. Barnes, and T. G. Haskell, “Adaptive wave-front correction by means of all-optical feedback interferometry,” Opt. Lett. 25, 773–775 (2000).
[Crossref]

Hawkes, S. J.

Hayes, J.

M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, “Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer,” Appl. Opt. 44, 6861–6868 (2005).
[Crossref]

M. N. Morris, J. Millerd, N. Brock, J. Hayes, and B. Saif, “Dynamic phase-shifting electronic speckle pattern interferometer,” Proc. SPIE 5869, 58691B (2005).
[Crossref]

Helen, S. S.

S. S. Helen, M. P. Kothiyal, and R. S. Sirohi, “White-light interferometry with polarization phase-shifter at the input of the interferometer,” J. Mod. Opt. 47, 1137–1145 (2000).
[Crossref]

Hernandez-Gomez, C.

Hernandez-Lopez, F.

M. Rivera, O. Dalmau, A. Gonzalez, and F. Hernandez-Lopez, “Two-step fringe pattern analysis with a Gabor filter bank,” Opt. Lasers. Eng. 85, 29–37 (2016).
[Crossref]

Herriott, D. R.

Horton, R. F.

R. F. Horton, “Design of a white light radial shear interferometer for segmented mirror control,” Opt. Eng. 27, 1063–1066 (1988).
[Crossref]

Hutchin, R. A.

R. A. Hutchin, “Combined shearing interferometer and Hartmann wavefront sensor,” U.S. patent4518854 (21May1985).

Jang, J.

Jeong, J.

Kasprzak, H. T.

W. W. Kowalik, B. E. Garncarz, and H. T. Kasprzak, “Corneal topography measurement by means of radial shearing interference: part II—measurements errors,” Optik 114, 199–206 (2003).
[Crossref]

W. W. Kowalik, B. E. Garncarz, and H. T. Kasprzak, “Corneal topography measurement by means of radial shearing interference: part I—theoretical considerations,” Optik 113, 39–45 (2002).
[Crossref]

Kim, B.-M.

Kim, H.-J.

Kim, M. W.

Kim, Y.

Kothiyal, M. P.

P. K. Upputuri, M. Pramanik, K. M. Nandigana, and M. P. Kothiyal, “Multi-colour microscopic interferometry for optical metrology and imaging applications,” Opt. Lasers Eng. 84, 10–25 (2016).
[Crossref]

U. P. Kumar, W. Haifeng, N. K. Mohan, and M. P. Kothiyal, “White light interferometry for surface profiling with a colour CCD,” Opt. Lasers Eng. 50, 1084–1088 (2012).
[Crossref]

S. S. Helen, M. P. Kothiyal, and R. S. Sirohi, “White-light interferometry with polarization phase-shifter at the input of the interferometer,” J. Mod. Opt. 47, 1137–1145 (2000).
[Crossref]

M. P. Kothiyal and C. Delisle, “Shearing interferometer for phase shifting interferometry with polarization phase shifter,” Appl. Opt. 24, 4439–4442 (1985).
[Crossref]

Kowalik, W. W.

W. W. Kowalik, B. E. Garncarz, and H. T. Kasprzak, “Corneal topography measurement by means of radial shearing interference: part II—measurements errors,” Optik 114, 199–206 (2003).
[Crossref]

W. W. Kowalik, B. E. Garncarz, and H. T. Kasprzak, “Corneal topography measurement by means of radial shearing interference: part I—theoretical considerations,” Optik 113, 39–45 (2002).
[Crossref]

Kumar, U. P.

U. P. Kumar, W. Haifeng, N. K. Mohan, and M. P. Kothiyal, “White light interferometry for surface profiling with a colour CCD,” Opt. Lasers Eng. 50, 1084–1088 (2012).
[Crossref]

Lee, K. J.

Liu, D.

D. Liu, Y. Yang, L. Wang, and Y. Zhuo, “Real-time diagnosis of transient pulse laser with high repetition by radial shearing interferometer,” Appl. Opt. 46, 8305–8314 (2007).
[Crossref]

D. Liu, Y. Yang, Y. Shen, J. Weng, and Y. Zhuo, “System optimization of radial shearing interferometer for aspheric testing,” Proc. SPIE 6834, 68340U (2007).
[Crossref]

Malacara, D.

Malacara, Z.

D. Malacara, M. Servin, and Z. Malacara, Phase Detection Algorithms in interferomgram Analysis for Optical Testing (Wiley, 2005).

Martínez-García, A.

D. I. Serrano-García, N. I. Toto-Arellano, A. Martínez-García, and G. Rodríguez Zurita, “Radial slope measurement of dynamic transparent samples,” J. Opt. 14, 045706 (2012).
[Crossref]

Meneses-Fabian, C.

Millerd, J.

M. N. Morris, J. Millerd, N. Brock, J. Hayes, and B. Saif, “Dynamic phase-shifting electronic speckle pattern interferometer,” Proc. SPIE 5869, 58691B (2005).
[Crossref]

M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, “Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer,” Appl. Opt. 44, 6861–6868 (2005).
[Crossref]

Mitchell, J. B.

J. B. Mitchell, G. W. Roberts, and P. C. T. Rees, “Full-field, high-frequency, heterodyne interferometry for dynamic metrology based on phase detection using a modified time-of-flight camera,” Proc. SPIE 11056, 110560U (2019).
[Crossref]

Mohan, N. K.

U. P. Kumar, W. Haifeng, N. K. Mohan, and M. P. Kothiyal, “White light interferometry for surface profiling with a colour CCD,” Opt. Lasers Eng. 50, 1084–1088 (2012).
[Crossref]

Morris, M. N.

M. N. Morris, J. Millerd, N. Brock, J. Hayes, and B. Saif, “Dynamic phase-shifting electronic speckle pattern interferometer,” Proc. SPIE 5869, 58691B (2005).
[Crossref]

Murty, M. V. R. K.

Nandigana, K. M.

P. K. Upputuri, M. Pramanik, K. M. Nandigana, and M. P. Kothiyal, “Multi-colour microscopic interferometry for optical metrology and imaging applications,” Opt. Lasers Eng. 84, 10–25 (2016).
[Crossref]

Nguyen, M. T.

M. T. Nguyen, Y. Ghim, and H. Rhee, “Single-shot deflectometry for dynamic 3D surface profile measurement by modified spatial-carrier frequency phase-shifting method,” Sci. Rep. 9, 3157 (2019).
[Crossref]

North-Morris, M.

Novak, M.

Oreb, B. F.

P. Hariharan, B. F. Oreb, and Z. Wanzhi, “Measurement of aspheric surfaces using a microcomputer controlled digital radial-shear interferometer,” J. Mod. Opt. 31, 989–999 (1984).
[Crossref]

Otani, Y.

Park, Y.

Parra-Escamilla, G.-A.

Patorski, K.

M. Trusiak, M. Wielgus, and K. Patorski, “Advanced processing of optical fringe patterns by automated selective reconstruction and enhanced fast empirical mode decomposition,” Opt. Lasers Eng. 52, 230–240 (2014).
[Crossref]

Pepler, D. A.

Phenis, A.

R. Tansey, A. Phenis, and K. Shu, “Use of a radial shear interferometer as a self-reference interferometer in adaptive optics,” in Advanced Maui Optical Space Surveillance Technologies Conference, S. Ryan, ed. (The Maui Economic Development Board, 2006) pp. 10–14.

Pramanik, M.

P. K. Upputuri, M. Pramanik, K. M. Nandigana, and M. P. Kothiyal, “Multi-colour microscopic interferometry for optical metrology and imaging applications,” Opt. Lasers Eng. 84, 10–25 (2016).
[Crossref]

Pritt, M. D.

C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998), Chap. 4.

Quiroga, J. A.

M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express 17, 21867–21881 (2009).
[Crossref]

J. A. Quiroga and M. Servin, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224, 221–227 (2003).
[Crossref]

Rees, P. C. T.

J. B. Mitchell, G. W. Roberts, and P. C. T. Rees, “Full-field, high-frequency, heterodyne interferometry for dynamic metrology based on phase detection using a modified time-of-flight camera,” Proc. SPIE 11056, 110560U (2019).
[Crossref]

Reyes-Figueroa, A.

V. H. Flores, A. Reyes-Figueroa, C. Carrillo-Delgado, and M. Rivera, “Computation of the phase step between two-step fringe patterns based on Gram--Schmidt algorithm,” Optics & Laser Technology 126, 106105 (2020).
[Crossref]

Rhee, H.

M. T. Nguyen, Y. Ghim, and H. Rhee, “Single-shot deflectometry for dynamic 3D surface profile measurement by modified spatial-carrier frequency phase-shifting method,” Sci. Rep. 9, 3157 (2019).
[Crossref]

Rivera, M.

V. H. Flores, A. Reyes-Figueroa, C. Carrillo-Delgado, and M. Rivera, “Computation of the phase step between two-step fringe patterns based on Gram--Schmidt algorithm,” Optics & Laser Technology 126, 106105 (2020).
[Crossref]

M. Rivera, O. Dalmau, A. Gonzalez, and F. Hernandez-Lopez, “Two-step fringe pattern analysis with a Gabor filter bank,” Opt. Lasers. Eng. 85, 29–37 (2016).
[Crossref]

Roberts, G. W.

J. B. Mitchell, G. W. Roberts, and P. C. T. Rees, “Full-field, high-frequency, heterodyne interferometry for dynamic metrology based on phase detection using a modified time-of-flight camera,” Proc. SPIE 11056, 110560U (2019).
[Crossref]

Rodríguez Zurita, G.

D. I. Serrano-García, N. I. Toto-Arellano, A. Martínez-García, and G. Rodríguez Zurita, “Radial slope measurement of dynamic transparent samples,” J. Opt. 14, 045706 (2012).
[Crossref]

Rodriguez-Zurita, G.

Rodríguez-Zurita, G.

Rosenfeld, D. P.

Ross, I. N.

Saif, B.

M. N. Morris, J. Millerd, N. Brock, J. Hayes, and B. Saif, “Dynamic phase-shifting electronic speckle pattern interferometer,” Proc. SPIE 5869, 58691B (2005).
[Crossref]

Sarkar, S.

S. Sarkar and K. Bhattacharya, “Polarization phase shifting cyclic interferometer for surface profilometry of nonbirefringent phase samples,” J. Mod. Opt. 60, 185–189 (2013).
[Crossref]

Sen, D.

P. Hariharan and D. Sen, “Radial shearing interferometer,” J. Sci. Instrum. 38, 428–432 (1961).
[Crossref]

Serrano-Garcia, D.-I.

Serrano-García, D. I.

D. I. Serrano-García, N. I. Toto-Arellano, A. Martínez-García, and G. Rodríguez Zurita, “Radial slope measurement of dynamic transparent samples,” J. Opt. 14, 045706 (2012).
[Crossref]

Servin, M.

M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express 17, 21867–21881 (2009).
[Crossref]

J. A. Quiroga and M. Servin, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224, 221–227 (2003).
[Crossref]

D. Malacara, M. Servin, and Z. Malacara, Phase Detection Algorithms in interferomgram Analysis for Optical Testing (Wiley, 2005).

Shen, Y.

D. Liu, Y. Yang, Y. Shen, J. Weng, and Y. Zhuo, “System optimization of radial shearing interferometer for aspheric testing,” Proc. SPIE 6834, 68340U (2007).
[Crossref]

Shirai, T.

T. Shirai, T. H. Barnes, and T. G. Haskell, “Real-time restoration of a blurred image with a liquid-crystal adaptive optics system based on all-optical feedback interferometry,” Opt. Commun. 188, 275–282 (2001).
[Crossref]

T. Shirai, T. H. Barnes, and T. G. Haskell, “Adaptive wave-front correction by means of all-optical feedback interferometry,” Opt. Lett. 25, 773–775 (2000).
[Crossref]

Shu, K.

R. Tansey, A. Phenis, and K. Shu, “Use of a radial shear interferometer as a self-reference interferometer in adaptive optics,” in Advanced Maui Optical Space Surveillance Technologies Conference, S. Ryan, ed. (The Maui Economic Development Board, 2006) pp. 10–14.

Silva, D. E.

Sirohi, R. S.

S. S. Helen, M. P. Kothiyal, and R. S. Sirohi, “White-light interferometry with polarization phase-shifter at the input of the interferometer,” J. Mod. Opt. 47, 1137–1145 (2000).
[Crossref]

Smartt, R. N.

Stappaerts, E. A.

Steel, W. H.

W. H. Steel, “A radial shear interferometer for testing microscope objectives,” J. Sci. Instrum. 42, 102–104 (1965).
[Crossref]

Tansey, R.

R. Tansey, A. Phenis, and K. Shu, “Use of a radial shear interferometer as a self-reference interferometer in adaptive optics,” in Advanced Maui Optical Space Surveillance Technologies Conference, S. Ryan, ed. (The Maui Economic Development Board, 2006) pp. 10–14.

Toto-Arellano, N. I.

N. I. Toto-Arellano, “4D measurements of biological and synthetic structures using a dynamic interferometer,” J. Mod. Opt. 64, S20–S29 (2017).
[Crossref]

D. I. Serrano-García, N. I. Toto-Arellano, A. Martínez-García, and G. Rodríguez Zurita, “Radial slope measurement of dynamic transparent samples,” J. Opt. 14, 045706 (2012).
[Crossref]

N. I. Toto-Arellano, G. Rodriguez-Zurita, C. Meneses-Fabian, and J. F. Vazquez-Castillo, “Phase shifts in the Fourier spectra of phase gratings and phase grids: an application for one-shot phase-shifting interferometry,” Opt. Express 16, 19330–19341 (2008).
[Crossref]

Toto-Arellano, N.-I.

Trusiak, M.

M. Trusiak, M. Wielgus, and K. Patorski, “Advanced processing of optical fringe patterns by automated selective reconstruction and enhanced fast empirical mode decomposition,” Opt. Lasers Eng. 52, 230–240 (2014).
[Crossref]

Upputuri, P. K.

P. K. Upputuri, M. Pramanik, K. M. Nandigana, and M. P. Kothiyal, “Multi-colour microscopic interferometry for optical metrology and imaging applications,” Opt. Lasers Eng. 84, 10–25 (2016).
[Crossref]

Vazquez-Castillo, J. F.

Vázquez-Castillo, J. F.

Wang, L.

Wang, S.

Wang, Z.

Wanzhi, Z.

P. Hariharan, B. F. Oreb, and Z. Wanzhi, “Measurement of aspheric surfaces using a microcomputer controlled digital radial-shear interferometer,” J. Mod. Opt. 31, 989–999 (1984).
[Crossref]

Weng, J.

D. Liu, Y. Yang, Y. Shen, J. Weng, and Y. Zhuo, “System optimization of radial shearing interferometer for aspheric testing,” Proc. SPIE 6834, 68340U (2007).
[Crossref]

White, A. D.

Wielgus, M.

M. Trusiak, M. Wielgus, and K. Patorski, “Advanced processing of optical fringe patterns by automated selective reconstruction and enhanced fast empirical mode decomposition,” Opt. Lasers Eng. 52, 230–240 (2014).
[Crossref]

Winstone, T. B.

Wyant, J.

Wyant, J. C.

J. C. Wyant, “Dynamic interferometry,” Opt. Photon. News 14(4), 36–41 (2003).
[Crossref]

Xu, B.

Yang, P.

Yang, T. D.

Yang, Y.

D. Liu, Y. Yang, L. Wang, and Y. Zhuo, “Real-time diagnosis of transient pulse laser with high repetition by radial shearing interferometer,” Appl. Opt. 46, 8305–8314 (2007).
[Crossref]

D. Liu, Y. Yang, Y. Shen, J. Weng, and Y. Zhuo, “System optimization of radial shearing interferometer for aspheric testing,” Proc. SPIE 6834, 68340U (2007).
[Crossref]

Zhuo, Y.

D. Liu, Y. Yang, Y. Shen, J. Weng, and Y. Zhuo, “System optimization of radial shearing interferometer for aspheric testing,” Proc. SPIE 6834, 68340U (2007).
[Crossref]

D. Liu, Y. Yang, L. Wang, and Y. Zhuo, “Real-time diagnosis of transient pulse laser with high repetition by radial shearing interferometer,” Appl. Opt. 46, 8305–8314 (2007).
[Crossref]

Appl. Opt. (12)

M. V. R. K. Murty, “A compact radial shearing interferometer based on the law of reflection,” Appl. Opt. 3, 853–857 (1964).
[Crossref]

M. V. R. K. Murty, “Radial shearing interferometers using a laser source,” Appl. Opt. 12, 2765–2767 (1973).
[Crossref]

R. N. Smartt, “Zone plate interferometer,” Appl. Opt. 13, 1093–1099 (1974).
[Crossref]

D. Malacara, “Mathematical interpretation of radial shearing interferometers,” Appl. Opt. 13, 1781–1784 (1974).
[Crossref]

J. C. Fouéré and D. Malacara, “Holographic radial shear interferometer,” Appl. Opt. 13, 2035–2039 (1974).
[Crossref]

J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
[Crossref]

M. P. Kothiyal and C. Delisle, “Shearing interferometer for phase shifting interferometry with polarization phase shifter,” Appl. Opt. 24, 4439–4442 (1985).
[Crossref]

C. Hernandez-Gomez, J. L. Collier, S. J. Hawkes, C. N. Danson, C. B. Edwards, D. A. Pepler, I. N. Ross, and T. B. Winstone, “Wave-front control of a large-aperture laser system by use of a static phase corrector,” Appl. Opt. 39, 1954–1961 (2000).
[Crossref]

D. E. Silva, “Talbot interferometer for radial and lateral derivatives,” Appl. Opt. 11, 2613–2624 (1972).
[Crossref]

M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, “Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer,” Appl. Opt. 44, 6861–6868 (2005).
[Crossref]

D. Liu, Y. Yang, L. Wang, and Y. Zhuo, “Real-time diagnosis of transient pulse laser with high repetition by radial shearing interferometer,” Appl. Opt. 46, 8305–8314 (2007).
[Crossref]

D.-I. Serrano-Garcia, N.-I. Toto-Arellano, G.-A. Parra-Escamilla, A. M. García, G. Rodríguez-Zurita, and Y. Otani, “Multiwavelength wavefront detection based on a lateral shear interferometer and polarization phase-shifting techniques,” Appl. Opt. 57, 6860–6865 (2018).
[Crossref]

J. Mod. Opt. (4)

N. I. Toto-Arellano, “4D measurements of biological and synthetic structures using a dynamic interferometer,” J. Mod. Opt. 64, S20–S29 (2017).
[Crossref]

S. S. Helen, M. P. Kothiyal, and R. S. Sirohi, “White-light interferometry with polarization phase-shifter at the input of the interferometer,” J. Mod. Opt. 47, 1137–1145 (2000).
[Crossref]

S. Sarkar and K. Bhattacharya, “Polarization phase shifting cyclic interferometer for surface profilometry of nonbirefringent phase samples,” J. Mod. Opt. 60, 185–189 (2013).
[Crossref]

P. Hariharan, B. F. Oreb, and Z. Wanzhi, “Measurement of aspheric surfaces using a microcomputer controlled digital radial-shear interferometer,” J. Mod. Opt. 31, 989–999 (1984).
[Crossref]

J. Opt. (1)

D. I. Serrano-García, N. I. Toto-Arellano, A. Martínez-García, and G. Rodríguez Zurita, “Radial slope measurement of dynamic transparent samples,” J. Opt. 14, 045706 (2012).
[Crossref]

J. Sci. Instrum. (2)

P. Hariharan and D. Sen, “Radial shearing interferometer,” J. Sci. Instrum. 38, 428–432 (1961).
[Crossref]

W. H. Steel, “A radial shear interferometer for testing microscope objectives,” J. Sci. Instrum. 42, 102–104 (1965).
[Crossref]

J. Sci. Intrum. (1)

D. S. Brown, “Radial shear interferometry,” J. Sci. Intrum. 39, 71–72 (1962).
[Crossref]

Opt. Commun. (2)

T. Shirai, T. H. Barnes, and T. G. Haskell, “Real-time restoration of a blurred image with a liquid-crystal adaptive optics system based on all-optical feedback interferometry,” Opt. Commun. 188, 275–282 (2001).
[Crossref]

J. A. Quiroga and M. Servin, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224, 221–227 (2003).
[Crossref]

Opt. Eng. (1)

R. F. Horton, “Design of a white light radial shear interferometer for segmented mirror control,” Opt. Eng. 27, 1063–1066 (1988).
[Crossref]

Opt. Express (4)

Opt. Lasers Eng. (3)

M. Trusiak, M. Wielgus, and K. Patorski, “Advanced processing of optical fringe patterns by automated selective reconstruction and enhanced fast empirical mode decomposition,” Opt. Lasers Eng. 52, 230–240 (2014).
[Crossref]

P. K. Upputuri, M. Pramanik, K. M. Nandigana, and M. P. Kothiyal, “Multi-colour microscopic interferometry for optical metrology and imaging applications,” Opt. Lasers Eng. 84, 10–25 (2016).
[Crossref]

U. P. Kumar, W. Haifeng, N. K. Mohan, and M. P. Kothiyal, “White light interferometry for surface profiling with a colour CCD,” Opt. Lasers Eng. 50, 1084–1088 (2012).
[Crossref]

Opt. Lasers. Eng. (1)

M. Rivera, O. Dalmau, A. Gonzalez, and F. Hernandez-Lopez, “Two-step fringe pattern analysis with a Gabor filter bank,” Opt. Lasers. Eng. 85, 29–37 (2016).
[Crossref]

Opt. Lett. (4)

Opt. Photon. News (1)

J. C. Wyant, “Dynamic interferometry,” Opt. Photon. News 14(4), 36–41 (2003).
[Crossref]

Optics & Laser Technology (1)

V. H. Flores, A. Reyes-Figueroa, C. Carrillo-Delgado, and M. Rivera, “Computation of the phase step between two-step fringe patterns based on Gram--Schmidt algorithm,” Optics & Laser Technology 126, 106105 (2020).
[Crossref]

Optik (2)

W. W. Kowalik, B. E. Garncarz, and H. T. Kasprzak, “Corneal topography measurement by means of radial shearing interference: part I—theoretical considerations,” Optik 113, 39–45 (2002).
[Crossref]

W. W. Kowalik, B. E. Garncarz, and H. T. Kasprzak, “Corneal topography measurement by means of radial shearing interference: part II—measurements errors,” Optik 114, 199–206 (2003).
[Crossref]

Proc. SPIE (3)

D. Liu, Y. Yang, Y. Shen, J. Weng, and Y. Zhuo, “System optimization of radial shearing interferometer for aspheric testing,” Proc. SPIE 6834, 68340U (2007).
[Crossref]

M. N. Morris, J. Millerd, N. Brock, J. Hayes, and B. Saif, “Dynamic phase-shifting electronic speckle pattern interferometer,” Proc. SPIE 5869, 58691B (2005).
[Crossref]

J. B. Mitchell, G. W. Roberts, and P. C. T. Rees, “Full-field, high-frequency, heterodyne interferometry for dynamic metrology based on phase detection using a modified time-of-flight camera,” Proc. SPIE 11056, 110560U (2019).
[Crossref]

Sci. Rep. (1)

M. T. Nguyen, Y. Ghim, and H. Rhee, “Single-shot deflectometry for dynamic 3D surface profile measurement by modified spatial-carrier frequency phase-shifting method,” Sci. Rep. 9, 3157 (2019).
[Crossref]

Other (7)

D. S. Brown, Interferometry, Symposium 11 at National Physical Laboratory (Her Majesty’s Stationery Office, 1960), p. 253.

D. Malacara, M. Servin, and Z. Malacara, Phase Detection Algorithms in interferomgram Analysis for Optical Testing (Wiley, 2005).

C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998), Chap. 4.

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1998), Vol. 26, pp. 349–393.

R. Tansey, A. Phenis, and K. Shu, “Use of a radial shear interferometer as a self-reference interferometer in adaptive optics,” in Advanced Maui Optical Space Surveillance Technologies Conference, S. Ryan, ed. (The Maui Economic Development Board, 2006) pp. 10–14.

J. M. Geary, “Wavefront sensors,” in Adaptive Optics Engineering Handbook, R. K. Tyson, ed. (2000), pp. 123–150.

R. A. Hutchin, “Combined shearing interferometer and Hartmann wavefront sensor,” U.S. patent4518854 (21May1985).

Supplementary Material (2)

NameDescription
» Visualization 1       Dynamic phase object. Flame of a candle (Visualization-1).
» Visualization 2       Dynamic phase object. Water flow (Visualization 2).

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Figures (14)

Fig. 1.
Fig. 1. Simultaneous phase-shifting radial shear interferometer. SFS, spatial filtering system; ${\rm L}_0$ , collimated lens; ${P}_0$ , polarizer; PBS, polarizing beam splitter; ${\rm L}_1$ , ${\rm L}_2$ , telescope lenses; ${f_1}={100}\;{\rm mm}$ , ${f_2}={80}\;{\rm mm}$ ; ${{\rm M }_i}$ , mirrors; OP, ${4}f$ system object plane; QWP, quarter-wave plate; ${\rm L}_3$ , ${\rm L}_4$ , ${4}f$ system lenses; IP, image plane; ${\rm L}_5$ , imaging lens; ${G}(\mu ,\nu )$ , 2D sinusoidal phase grating with 110 grooves/mm; ${{ P }_b}$ , base pattern; PA, polarizer array. $\lambda = {532}\;{\rm nm}$ . ${f_1}={100}\;{\rm mm}$ , ${f_2}={80}\;{\rm mm}$ , $f={200}\;{\rm mm}$ ; $s={0.8}$ . (a) Auxiliary polarizer. (b) Interferogram at the output of the PRSCPI. (c) Replicas of the base pattern. (d) Polarizer array. (e) Simultaneous interferograms.
Fig. 2.
Fig. 2. ${4}f$ Fourier imaging system. (a) Diffraction orders of the 2D sinusoidal phase grating. (b) Nine replicated interference patterns. (c) Polarizer array, case of four interferograms. (e) Polarizer array, case of seven interferograms. (g) Polarizer array, case of nine interferograms. (d) Phase shifts the case of four interferograms. (f) Phase shifts the case of seven interferograms. (h) Phase shifts the case of nine interferograms.
Fig. 3.
Fig. 3. Wavefront. (a) Four simultaneous interferograms. (b) Unwrapped phase with carrier frequency: calculated OPD. (c), (d). Unwrapped phase without carrier frequency: calculated OPD. Relative phase shift: 45 deg.
Fig. 4.
Fig. 4. Wavefront. (a) Seven simultaneous interferograms. (b) Unwrapped phase with carrier frequency: calculated OPD. (c), (d) Unwrapped phase without carrier frequency: calculated OPD. Relative phase shift: 60 deg.
Fig. 5.
Fig. 5. Wavefront. (a) Nine simultaneous interferograms. (b) Unwrapped phase with carrier frequency: calculated OPD. (c), (d). Unwrapped phase without carrier frequency: calculated OPD. Relative phase shift: 45 deg.
Fig. 6.
Fig. 6. (a) OPD profiles of the (unwrapped) estimated phases using the four-step, ( ${6} + {1}$ )-step, and ( ${8} + {1}$ )-step algorithms. (b) OPD error.
Fig. 7.
Fig. 7. Red blood cells. (a) Nine simultaneous interferograms; (b) and (c) OPD; (d) OPD of two RBCs.
Fig. 8.
Fig. 8. Dynamic phase object. Flame of a candle (Visualization 1).
Fig. 9.
Fig. 9. Dynamic phase object. Water flow (Visualization 2).
Fig. 10.
Fig. 10. Polarized radial shear interferometer. SFS, spatial filtering system; ${\rm L}_0$ , collimated lens; ${P}_0$ , polarizer; PBS, polarizing beam splitter; ${\rm L}_1$ , ${\rm L}_2$ , lenses; QWP, quarter-wave Plate; ${\rm L}_3$ , imaging lens; ${P}_1$ , polarizer; ${f_1}={100}\;{\rm mm}$ , ${f_2}={80}\;{\rm mm}$ .
Fig. 11.
Fig. 11. White light interferograms. Phase steps of (a) 0°, (b) 90°, (c) 180°, and (d) 270°.
Fig. 12.
Fig. 12. Phase information obtained by the red, green, and blue channels. (a) Interferograms for the red, green, and blue channels; (b) OPD.
Fig. 13.
Fig. 13. Intensity of the central line marked on the unwrapped phase information.
Fig. 14.
Fig. 14. Phase information obtained by the red, green, and blue channels. (a) White light interferogram. (b) Interferograms for the red, green, and blue channels. (c) Normalized interferograms for the red, green, and blue channels. (d) OPD.

Equations (12)

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P b ( x , y ) = J L O ( x , y ) + J R R ( x , y ) ,
J L = 1 2 ( 1 i ) a n d J R = 1 2 ( 1 i ) .
I ( x , y ) = | P ψ [ 1 2 ( 1 i ) O ( x , y ) + 1 2 ( 1 i ) R ( x , y ) ] | 2 ,
P ψ = ( cos 2 ψ sin ψ cos ψ sin ψ cos ψ sin 2 ψ ) ,
I ( x , y ) = a 0 + a 1 + 2 a 0 a 1 cos [ 2 ψ Δ ϕ ( x , y ) ] ,
G ( μ , υ ) = q = q = J q ( 2 π A g ) e 2 π q F 0 μ r = r = J r ( 2 π A g ) e 2 π r F 0 υ ,
G ~ 2 ( x , y ) = q = q = r = r = J q ( 2 π A g ) J r ( 2 π A g ) × δ ( x q F 0 , y r F 0 ) .
T ( x , y ) = P ψ P b ( x , y ) G ~ 2 ( x , y ) ,
I = 2 J q 2 J r 2 ( a 0 + a 1 + 2 a 0 a 1 cos [ 2 ψ Δ ϕ ( x q , y r ) ] ) ,
I 1 ( x , y ) = A 2 + B 2 + A B cos [ ϕ ( x , y ) ] , I 2 ( x , y ) = A 2 + B 2 A B sin [ ϕ ( x , y ) ] , I 3 ( x , y ) = A 2 + B 2 A B cos [ ϕ ( x , y ) ] , I 4 ( x , y ) = A 2 + B 2 + A B sin [ ϕ ( x , y ) ] ,
tan ϕ ( x , y ) = [ I 4 I 2 I 1 I 3 ] .
tan ϕ ( x , y ) = i N + 1 I i sin ( 2 π i 1 N ) i N + 1 I i cos ( 2 π i 1 N ) .

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