Abstract

An interferometric method to measure the slope of phase objects is presented. The analysis was performed by implementing a polarizing phase-shifting cyclic shear interferometer coupled to a 4-f Fourier imaging system with crossed high-frequency Ronchi gratings. This system can obtain nine interference patterns with adjustable phase shifts and variable lateral shear. In order to extract the slope of a phase object, it is only analyzed using four patterns obtained in a single shot, and applying the classical method of phase extraction.

© 2010 Optical Society of America

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References

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  1. V. Ronchi, “Forty years of history of a grating interferometer,” Appl. Opt. 3, 437–451 (1964).
    [Crossref]
  2. A. Cornejo Rodriguez, “Ronchi test,” in Optical Shop Testing, D.Malacara, ed. (Wiley, 1992), Chap. 9.
  3. M. C. Hutley, Diffraction Gratings (Academic, 1982).
  4. D.Malacara, ed., Optical Shop Testing (Wiley, 1978).
  5. C. Meneses-Fabián, G. Rodríguez-Zurita, J. F. Vázquez Castillo, C. Robledo Sánchez, and V. Arrizon, “Common-path phase-shifting interferometer with binary grating,” Opt. Commun. 264, 13–17 (2006).
    [Crossref]
  6. G. Rodríguez-Zurita, N. I. Toto-Arellano, C. Meneses-Fabian, and J. C. Vázquez-Castillo, “Adjustable lateral-shear single-shot phase-shifting interferometry for moving phase distributions,” Meas. Sci. Technol. 20, 115902(2009).
    [Crossref]
  7. M. V. R. K. Murty, “Rotational shearing interferometry,” Appl. Opt. 5, 615–619 (1966).
    [Crossref] [PubMed]
  8. D. Malacara, A. Cornejo, and M. V. R. K. Murty, “A shearing interferometer for convergent or divergent beams,” Boletin del Instituto de Tonantzintla 1, 223–226 (1975).
  9. W. H. Steel, “A radial shear interferometer for testing microscope objectives,” J. Sci. Instrum. 42, 102–104(1965).
    [Crossref]
  10. G. Rodriguez-Zurita, C. Meneses-Fabian, N. I. Toto-Arellano, J. F. Vázquez-Castillo, and C. Robledo-Sánchez, “One-shot phase-shifting phase-grating interferometry with modulation of polarization: case of four interferograms,” Opt. Express 16, 7806–7817 (2008).
    [Crossref] [PubMed]
  11. T. Nomura, S. Murata, E. Nitanai, and T. Numata, “Phase-shifting digital holography with a phase difference between orthogonal polarizations,” Appl. Opt. 45, 4873–4877(2006).
    [Crossref] [PubMed]
  12. F. Bai, Z. Liu, and X. Bao, “Two-shot point-diffraction interferometer with an unknown phase shift,” J. Opt. 12, 045702(2010).
    [Crossref]
  13. A. Cordero-Dávila, E. Luna-Aguilar, S. Vázquez-Montiel, S. Zárate-Vázquez, and M. E. Percino-Zacarías, “Ronchi test with a square grid,” Appl. Opt. 37, 672–675 (1998).
    [Crossref]
  14. 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] [PubMed]
  15. Y. Y. Hung and A. J. Durelli, “Simultaneous measurement of three displacement derivatives using a multiple image-shearing interferometric camera,” J. Strain Anal. 14, 81–88(1979).
    [Crossref]
  16. P.K.Rastogi, ed., Digital Speckle Pattern Interferometry and Related Techniques (Wiley, 2001).
  17. P. K. Rastogi, “Measurement of in-plane strains using electronic speckle and electronic speckle-shearing pattern interferometry,” J. Mod. Opt. 43, 1577–1581 (1996).
  18. T. W. Ng, “Digital speckle pattern interferometer for combined measurements of out-of-plane displacement and slope,” Opt. Commun. 116, 31–35 (1995).
    [Crossref]
  19. B. Bhaduri, N. Krishna Mohan, and M. P. Kothiyal, “A dual-function ESPI system for the measurement of out-of-plane displacement and slope,” Opt. Lasers Eng. 44, 637–644(2006).
    [Crossref]
  20. A. Martínez, J. A. Rayas, and R. Cordero, “Measurement of in plane strain with shearography and electronic speckle pattern interferometry,” in Proceedings of OPTIMESS2009 (2009), p. 217, ISBN 978-90-423-0366-9.
  21. P. Hariharan and D. Sen, “Cyclic shearing interferometer,” J. Sci. Instrum. 37, 374–376 (1960).
    [Crossref]
  22. J. Wyant, “White light extended source shearing interferometer,” Appl. Opt. 13, 200–202 (1974).
    [Crossref] [PubMed]
  23. M. P. Kothiyal and C. Delisle, “Shearing interferometer for phase shifting interferometry with polarization phase shifter,” Appl. Opt. 24, 4439–4447 (1985).
    [Crossref] [PubMed]
  24. D. K. Sharma, R. S. Sirohi, and M. P. Kothiyal, “Simultaneous measurement of slope and curvature with a three-aperture speckle shearing interferometer,” Appl. Opt. 23, 1542–1546(1984).
    [Crossref] [PubMed]
  25. B. Barrientos-García, A. J. Moore, C. Pérez-López, L. Wang, and T. Tschudi, “Transient deformation measurement with electronic speckle pattern interferometry by use of a holographic optical element for spatial phase stepping,” Appl. Opt. 38, 5944–5947 (1999).
    [Crossref]
  26. D. Malacara, M. Servin, and Z. Malacara, “Phase detection algorithms,” in Interferogram Analysis for Optical Testing(Dekker, New York, 1998), Chap. 6.
  27. V. Ronchi, “On the phase grating interferometer,” Appl. Opt. 4, 1041–1042 (1965).
    [Crossref]

2010 (1)

F. Bai, Z. Liu, and X. Bao, “Two-shot point-diffraction interferometer with an unknown phase shift,” J. Opt. 12, 045702(2010).
[Crossref]

2009 (1)

G. Rodríguez-Zurita, N. I. Toto-Arellano, C. Meneses-Fabian, and J. C. Vázquez-Castillo, “Adjustable lateral-shear single-shot phase-shifting interferometry for moving phase distributions,” Meas. Sci. Technol. 20, 115902(2009).
[Crossref]

2008 (1)

2006 (3)

T. Nomura, S. Murata, E. Nitanai, and T. Numata, “Phase-shifting digital holography with a phase difference between orthogonal polarizations,” Appl. Opt. 45, 4873–4877(2006).
[Crossref] [PubMed]

C. Meneses-Fabián, G. Rodríguez-Zurita, J. F. Vázquez Castillo, C. Robledo Sánchez, and V. Arrizon, “Common-path phase-shifting interferometer with binary grating,” Opt. Commun. 264, 13–17 (2006).
[Crossref]

B. Bhaduri, N. Krishna Mohan, and M. P. Kothiyal, “A dual-function ESPI system for the measurement of out-of-plane displacement and slope,” Opt. Lasers Eng. 44, 637–644(2006).
[Crossref]

2005 (1)

1999 (1)

1998 (1)

1996 (1)

P. K. Rastogi, “Measurement of in-plane strains using electronic speckle and electronic speckle-shearing pattern interferometry,” J. Mod. Opt. 43, 1577–1581 (1996).

1995 (1)

T. W. Ng, “Digital speckle pattern interferometer for combined measurements of out-of-plane displacement and slope,” Opt. Commun. 116, 31–35 (1995).
[Crossref]

1985 (1)

1984 (1)

1979 (1)

Y. Y. Hung and A. J. Durelli, “Simultaneous measurement of three displacement derivatives using a multiple image-shearing interferometric camera,” J. Strain Anal. 14, 81–88(1979).
[Crossref]

1975 (1)

D. Malacara, A. Cornejo, and M. V. R. K. Murty, “A shearing interferometer for convergent or divergent beams,” Boletin del Instituto de Tonantzintla 1, 223–226 (1975).

1974 (1)

1966 (1)

1965 (2)

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

V. Ronchi, “On the phase grating interferometer,” Appl. Opt. 4, 1041–1042 (1965).
[Crossref]

1964 (1)

1960 (1)

P. Hariharan and D. Sen, “Cyclic shearing interferometer,” J. Sci. Instrum. 37, 374–376 (1960).
[Crossref]

Arrizon, V.

C. Meneses-Fabián, G. Rodríguez-Zurita, J. F. Vázquez Castillo, C. Robledo Sánchez, and V. Arrizon, “Common-path phase-shifting interferometer with binary grating,” Opt. Commun. 264, 13–17 (2006).
[Crossref]

Bai, F.

F. Bai, Z. Liu, and X. Bao, “Two-shot point-diffraction interferometer with an unknown phase shift,” J. Opt. 12, 045702(2010).
[Crossref]

Bao, X.

F. Bai, Z. Liu, and X. Bao, “Two-shot point-diffraction interferometer with an unknown phase shift,” J. Opt. 12, 045702(2010).
[Crossref]

Barrientos-García, B.

Bhaduri, B.

B. Bhaduri, N. Krishna Mohan, and M. P. Kothiyal, “A dual-function ESPI system for the measurement of out-of-plane displacement and slope,” Opt. Lasers Eng. 44, 637–644(2006).
[Crossref]

Brock, N.

Cordero, R.

A. Martínez, J. A. Rayas, and R. Cordero, “Measurement of in plane strain with shearography and electronic speckle pattern interferometry,” in Proceedings of OPTIMESS2009 (2009), p. 217, ISBN 978-90-423-0366-9.

Cordero-Dávila, A.

Cornejo, A.

D. Malacara, A. Cornejo, and M. V. R. K. Murty, “A shearing interferometer for convergent or divergent beams,” Boletin del Instituto de Tonantzintla 1, 223–226 (1975).

Delisle, C.

Durelli, A. J.

Y. Y. Hung and A. J. Durelli, “Simultaneous measurement of three displacement derivatives using a multiple image-shearing interferometric camera,” J. Strain Anal. 14, 81–88(1979).
[Crossref]

Hariharan, P.

P. Hariharan and D. Sen, “Cyclic shearing interferometer,” J. Sci. Instrum. 37, 374–376 (1960).
[Crossref]

Hayes, J.

Hung, Y. Y.

Y. Y. Hung and A. J. Durelli, “Simultaneous measurement of three displacement derivatives using a multiple image-shearing interferometric camera,” J. Strain Anal. 14, 81–88(1979).
[Crossref]

Hutley, M. C.

M. C. Hutley, Diffraction Gratings (Academic, 1982).

Kothiyal, M. P.

Liu, Z.

F. Bai, Z. Liu, and X. Bao, “Two-shot point-diffraction interferometer with an unknown phase shift,” J. Opt. 12, 045702(2010).
[Crossref]

Luna-Aguilar, E.

Malacara, D.

D. Malacara, A. Cornejo, and M. V. R. K. Murty, “A shearing interferometer for convergent or divergent beams,” Boletin del Instituto de Tonantzintla 1, 223–226 (1975).

D. Malacara, M. Servin, and Z. Malacara, “Phase detection algorithms,” in Interferogram Analysis for Optical Testing(Dekker, New York, 1998), Chap. 6.

Malacara, Z.

D. Malacara, M. Servin, and Z. Malacara, “Phase detection algorithms,” in Interferogram Analysis for Optical Testing(Dekker, New York, 1998), Chap. 6.

Martínez, A.

A. Martínez, J. A. Rayas, and R. Cordero, “Measurement of in plane strain with shearography and electronic speckle pattern interferometry,” in Proceedings of OPTIMESS2009 (2009), p. 217, ISBN 978-90-423-0366-9.

Meneses-Fabian, C.

G. Rodríguez-Zurita, N. I. Toto-Arellano, C. Meneses-Fabian, and J. C. Vázquez-Castillo, “Adjustable lateral-shear single-shot phase-shifting interferometry for moving phase distributions,” Meas. Sci. Technol. 20, 115902(2009).
[Crossref]

G. Rodriguez-Zurita, C. Meneses-Fabian, N. I. Toto-Arellano, J. F. Vázquez-Castillo, and C. Robledo-Sánchez, “One-shot phase-shifting phase-grating interferometry with modulation of polarization: case of four interferograms,” Opt. Express 16, 7806–7817 (2008).
[Crossref] [PubMed]

Meneses-Fabián, C.

C. Meneses-Fabián, G. Rodríguez-Zurita, J. F. Vázquez Castillo, C. Robledo Sánchez, and V. Arrizon, “Common-path phase-shifting interferometer with binary grating,” Opt. Commun. 264, 13–17 (2006).
[Crossref]

Millerd, J.

Mohan, N. Krishna

B. Bhaduri, N. Krishna Mohan, and M. P. Kothiyal, “A dual-function ESPI system for the measurement of out-of-plane displacement and slope,” Opt. Lasers Eng. 44, 637–644(2006).
[Crossref]

Moore, A. J.

Murata, S.

Murty, M. V. R. K.

D. Malacara, A. Cornejo, and M. V. R. K. Murty, “A shearing interferometer for convergent or divergent beams,” Boletin del Instituto de Tonantzintla 1, 223–226 (1975).

M. V. R. K. Murty, “Rotational shearing interferometry,” Appl. Opt. 5, 615–619 (1966).
[Crossref] [PubMed]

Ng, T. W.

T. W. Ng, “Digital speckle pattern interferometer for combined measurements of out-of-plane displacement and slope,” Opt. Commun. 116, 31–35 (1995).
[Crossref]

Nitanai, E.

Nomura, T.

North-Morris, M.

Novak, M.

Numata, T.

Percino-Zacarías, M. E.

Pérez-López, C.

Rastogi, P. K.

P. K. Rastogi, “Measurement of in-plane strains using electronic speckle and electronic speckle-shearing pattern interferometry,” J. Mod. Opt. 43, 1577–1581 (1996).

Rayas, J. A.

A. Martínez, J. A. Rayas, and R. Cordero, “Measurement of in plane strain with shearography and electronic speckle pattern interferometry,” in Proceedings of OPTIMESS2009 (2009), p. 217, ISBN 978-90-423-0366-9.

Robledo-Sánchez, C.

Rodriguez, A. Cornejo

A. Cornejo Rodriguez, “Ronchi test,” in Optical Shop Testing, D.Malacara, ed. (Wiley, 1992), Chap. 9.

Rodriguez-Zurita, G.

Rodríguez-Zurita, G.

G. Rodríguez-Zurita, N. I. Toto-Arellano, C. Meneses-Fabian, and J. C. Vázquez-Castillo, “Adjustable lateral-shear single-shot phase-shifting interferometry for moving phase distributions,” Meas. Sci. Technol. 20, 115902(2009).
[Crossref]

C. Meneses-Fabián, G. Rodríguez-Zurita, J. F. Vázquez Castillo, C. Robledo Sánchez, and V. Arrizon, “Common-path phase-shifting interferometer with binary grating,” Opt. Commun. 264, 13–17 (2006).
[Crossref]

Ronchi, V.

Sánchez, C. Robledo

C. Meneses-Fabián, G. Rodríguez-Zurita, J. F. Vázquez Castillo, C. Robledo Sánchez, and V. Arrizon, “Common-path phase-shifting interferometer with binary grating,” Opt. Commun. 264, 13–17 (2006).
[Crossref]

Sen, D.

P. Hariharan and D. Sen, “Cyclic shearing interferometer,” J. Sci. Instrum. 37, 374–376 (1960).
[Crossref]

Servin, M.

D. Malacara, M. Servin, and Z. Malacara, “Phase detection algorithms,” in Interferogram Analysis for Optical Testing(Dekker, New York, 1998), Chap. 6.

Sharma, D. K.

Sirohi, R. S.

Steel, W. H.

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

Toto-Arellano, N. I.

G. Rodríguez-Zurita, N. I. Toto-Arellano, C. Meneses-Fabian, and J. C. Vázquez-Castillo, “Adjustable lateral-shear single-shot phase-shifting interferometry for moving phase distributions,” Meas. Sci. Technol. 20, 115902(2009).
[Crossref]

G. Rodriguez-Zurita, C. Meneses-Fabian, N. I. Toto-Arellano, J. F. Vázquez-Castillo, and C. Robledo-Sánchez, “One-shot phase-shifting phase-grating interferometry with modulation of polarization: case of four interferograms,” Opt. Express 16, 7806–7817 (2008).
[Crossref] [PubMed]

Tschudi, T.

Vázquez Castillo, J. F.

C. Meneses-Fabián, G. Rodríguez-Zurita, J. F. Vázquez Castillo, C. Robledo Sánchez, and V. Arrizon, “Common-path phase-shifting interferometer with binary grating,” Opt. Commun. 264, 13–17 (2006).
[Crossref]

Vázquez-Castillo, J. C.

G. Rodríguez-Zurita, N. I. Toto-Arellano, C. Meneses-Fabian, and J. C. Vázquez-Castillo, “Adjustable lateral-shear single-shot phase-shifting interferometry for moving phase distributions,” Meas. Sci. Technol. 20, 115902(2009).
[Crossref]

Vázquez-Castillo, J. F.

Vázquez-Montiel, S.

Wang, L.

Wyant, J.

Zárate-Vázquez, S.

Appl. Opt. (10)

V. Ronchi, “Forty years of history of a grating interferometer,” Appl. Opt. 3, 437–451 (1964).
[Crossref]

M. V. R. K. Murty, “Rotational shearing interferometry,” Appl. Opt. 5, 615–619 (1966).
[Crossref] [PubMed]

T. Nomura, S. Murata, E. Nitanai, and T. Numata, “Phase-shifting digital holography with a phase difference between orthogonal polarizations,” Appl. Opt. 45, 4873–4877(2006).
[Crossref] [PubMed]

A. Cordero-Dávila, E. Luna-Aguilar, S. Vázquez-Montiel, S. Zárate-Vázquez, and M. E. Percino-Zacarías, “Ronchi test with a square grid,” Appl. Opt. 37, 672–675 (1998).
[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] [PubMed]

J. Wyant, “White light extended source shearing interferometer,” Appl. Opt. 13, 200–202 (1974).
[Crossref] [PubMed]

M. P. Kothiyal and C. Delisle, “Shearing interferometer for phase shifting interferometry with polarization phase shifter,” Appl. Opt. 24, 4439–4447 (1985).
[Crossref] [PubMed]

D. K. Sharma, R. S. Sirohi, and M. P. Kothiyal, “Simultaneous measurement of slope and curvature with a three-aperture speckle shearing interferometer,” Appl. Opt. 23, 1542–1546(1984).
[Crossref] [PubMed]

B. Barrientos-García, A. J. Moore, C. Pérez-López, L. Wang, and T. Tschudi, “Transient deformation measurement with electronic speckle pattern interferometry by use of a holographic optical element for spatial phase stepping,” Appl. Opt. 38, 5944–5947 (1999).
[Crossref]

V. Ronchi, “On the phase grating interferometer,” Appl. Opt. 4, 1041–1042 (1965).
[Crossref]

Boletin del Instituto de Tonantzintla (1)

D. Malacara, A. Cornejo, and M. V. R. K. Murty, “A shearing interferometer for convergent or divergent beams,” Boletin del Instituto de Tonantzintla 1, 223–226 (1975).

J. Mod. Opt. (1)

P. K. Rastogi, “Measurement of in-plane strains using electronic speckle and electronic speckle-shearing pattern interferometry,” J. Mod. Opt. 43, 1577–1581 (1996).

J. Opt. (1)

F. Bai, Z. Liu, and X. Bao, “Two-shot point-diffraction interferometer with an unknown phase shift,” J. Opt. 12, 045702(2010).
[Crossref]

J. Sci. Instrum. (2)

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

P. Hariharan and D. Sen, “Cyclic shearing interferometer,” J. Sci. Instrum. 37, 374–376 (1960).
[Crossref]

J. Strain Anal. (1)

Y. Y. Hung and A. J. Durelli, “Simultaneous measurement of three displacement derivatives using a multiple image-shearing interferometric camera,” J. Strain Anal. 14, 81–88(1979).
[Crossref]

Meas. Sci. Technol. (1)

G. Rodríguez-Zurita, N. I. Toto-Arellano, C. Meneses-Fabian, and J. C. Vázquez-Castillo, “Adjustable lateral-shear single-shot phase-shifting interferometry for moving phase distributions,” Meas. Sci. Technol. 20, 115902(2009).
[Crossref]

Opt. Commun. (2)

C. Meneses-Fabián, G. Rodríguez-Zurita, J. F. Vázquez Castillo, C. Robledo Sánchez, and V. Arrizon, “Common-path phase-shifting interferometer with binary grating,” Opt. Commun. 264, 13–17 (2006).
[Crossref]

T. W. Ng, “Digital speckle pattern interferometer for combined measurements of out-of-plane displacement and slope,” Opt. Commun. 116, 31–35 (1995).
[Crossref]

Opt. Express (1)

Opt. Lasers Eng. (1)

B. Bhaduri, N. Krishna Mohan, and M. P. Kothiyal, “A dual-function ESPI system for the measurement of out-of-plane displacement and slope,” Opt. Lasers Eng. 44, 637–644(2006).
[Crossref]

Other (6)

A. Martínez, J. A. Rayas, and R. Cordero, “Measurement of in plane strain with shearography and electronic speckle pattern interferometry,” in Proceedings of OPTIMESS2009 (2009), p. 217, ISBN 978-90-423-0366-9.

D. Malacara, M. Servin, and Z. Malacara, “Phase detection algorithms,” in Interferogram Analysis for Optical Testing(Dekker, New York, 1998), Chap. 6.

A. Cornejo Rodriguez, “Ronchi test,” in Optical Shop Testing, D.Malacara, ed. (Wiley, 1992), Chap. 9.

M. C. Hutley, Diffraction Gratings (Academic, 1982).

D.Malacara, ed., Optical Shop Testing (Wiley, 1978).

P.K.Rastogi, ed., Digital Speckle Pattern Interferometry and Related Techniques (Wiley, 2001).

Supplementary Material (1)

» Media 1: MPG (1126 KB)     

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Figures (8)

Fig. 1
Fig. 1

Experimental setup. (a) CSI with variable shear and high-frequency bi-Ronchi grating. Q, Q , quarter-wave retarders; Δ x , linear shear; x 0 , beam separation; F 0 = λ f / d is the order of separation. (b) Arrangement of polarizing filters. P i , polarizing filters; ψ i , trans mission angle of polarization.

Fig. 2
Fig. 2

Simulated diffraction pattern generated by high- frequency bi-Ronchi grating. (a) Diffraction spectra. (b) xy components of diffraction spectra of each grating component.

Fig. 3
Fig. 3

Replicas of the interference patterns. (a) Fourier coefficients for each replica of the interference pattern. (b) Experimental diffraction orders generated by high-frequency bi-Ronchi gratings ( 100 ln / mm ). (c) Replicas of the interference patterns centered on each diffraction order.

Fig. 4
Fig. 4

Interference patterns for primary aberrations. Incident wavefront (derived): (a) with tilt, (b)–(d) primary coma combined with tilt.

Fig. 5
Fig. 5

Microscope slide (Configuration I of polarizer’s array). The relative π shifts corresponding to the polarizing filter angles in degrees were ψ 1 = 0 , ψ 2 = 46.577 ° , ψ 3 = 92.989 ° , ψ 4 = 136.42 ° . (a) Interference patterns obtained in a single shot. (b) Slope.

Fig. 6
Fig. 6

Microscope slide (Configuration II of polarizer’s array). The relative π shifts corresponding to the polarizing filter angles in degrees were ψ 1 = 0 , ψ 2 = 46.577 ° , ψ 3 = 92.989 ° , ψ 4 = 136.42 ° . (a) Interference patterns obtained in a single shot. (b) Slope.

Fig. 7
Fig. 7

Acetate. The relative π shifts corresponding to the polarizing filter angles in degrees were ψ 1 = 0 , ψ 2 = 46.577 ° , ψ 3 = 92.989 ° , ψ 4 = 136.42 ° . (a) Interference patterns obtained in a single shot. (b) Slope.

Fig. 8
Fig. 8

Dynamic phase object. Transparent bubble moving on a microscope slide (Media 1). Animation: 25 frames ( 1 frame / s ).

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

O ( x , y ) = e i · 2 π · ϕ ( x , y ) ,
O ( x , y ) = e i 2 π ϕ ( x , y ) O ( x , y ) = 1 + i 2 π ϕ ( x , y ) ,
x O ( x , y ) = c 0 · x ϕ ( x , y ) ,
ϕ ( x , y ) x = 2 π λ · [ sin θ u x + ( 1 + cos θ ) w x ] Δ x ,
ϕ ( x , y ) x = 4 π λ · [ w ( x , y ) x ] Δ x .
w ( x , y ) x = c 1 c 0 · x O ( x , y )
O ( x + Δ x , y ) = 1 2 ( 1 i ) e i ϕ ( x + Δ x , y ) O ( x , y ) = 1 2 ( 1 i ) e i ϕ ( x , y ) .
O = P · ψ O ( x + Δ x , y ) , O = P · ψ O ( x , y ) ,
P ψ = ( cos 2 ψ sin ψ cos ψ sin ψ cos ψ sin 2 ψ ) .
I = O + O 2 I i = 2 + 2 cos { 2 ψ i [ ϕ ( x + Δ x , y ) ϕ ( x , y ) ] } I i = 2 + 2 cos [ 2 ψ i Δ x x ϕ ( x , y ) ] .
x ϕ ( x , y ) = tan 1 ( I 1 I 3 I 2 I 4 ) ,
w ( x , y ) x = c 1 · tan 1 ( I 1 I 3 I 2 I 4 ) .
G ( μ ) = rect [ μ a w ] * n = N N δ [ μ n · d a w ] ,
G ˜ ( x ) = a w d n = sinc ( a w x d ) δ ( x n d ) = n = C n ,
G ( μ , ν ) = G ( μ ) · G ( ν ) = n = N N rect [ μ n · d μ a w μ ] · l = L L rect [ ν l · d υ a w μ ] ,
G ˜ ( x , y ) = G ˜ ( x ) · G ˜ ( y ) = a w 2 d 2 n = N N sinc ( a w d · x ) · δ ( x n d ) l = L L sinc ( a w d y ) · δ ( y l d ) = n = N N l = L L C x n C y l .
I ( x , y ) = | P ψ [ J L e i ϕ ( x + Δ x , y ) + J R e i ϕ ( x , y ) ] * G ˜ ( x , y ) | 2 = I ( x , y ) * | G ˜ ( x , y ) | 2 ,
I = 1 + cos [ 2 · ψ Δ x x ϕ ( x , y ) ] .
I ( x , y ) = n = N N l = L L C y l 2 · C x n 2 · I ( x n d x , y l d y ) = n = N N l = L L C y l 2 · C x n 2 · { 1 + cos [ 2 · ψ Δ x x ϕ ( x n d x , y n d y ) ] } .

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