Abstract

An optimized method of accurately extracting the arbitrary unknown and unequal phase steps in phase-shifting interferometry is proposed. The approximate phase steps are first calculated based on the statistical nature of the diffraction field, and the mutated adaptive quantum-behaved particle swarm optimization is used to further extract the arbitrary unknown and unequal phase steps. We improve the mutated adaptive quantum-behaved particle swarm optimization by adding mutation operator with Gaussian probability distribution, thereby increasing the population diversity. The developed method here is fast, highly accurate, and can effectively overcomes the “sawtooth-solution phenomenon” often encountered in traditional direct search approach. We demonstrate the speed and quality of the solution by measuring the transmissive object with the four-step phase-shifting interference method, as is verified by both the computer simulations and optical experiments.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
OSA Recommended Articles
Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects

L. Z. Cai, Q. Liu, and X. L. Yang
Opt. Lett. 29(2) 183-185 (2004)

Fast blind extraction of arbitrary unknown phase shifts by an iterative tangent approach in generalized phase-shifting interferometry

X. F. Xu, L. Z. Cai, X. F. Meng, G. Y. Dong, and X. X. Shen
Opt. Lett. 31(13) 1966-1968 (2006)

References

  • View by:
  • |
  • |
  • |

  1. W. M. Ash, L. Krzewina, and M. K. Kim, “Quantitative imaging of cellular adhesion by total internal reflection holographic microscopy,” Appl. Opt. 48(34), H144–152 (2009).
    [Crossref]
  2. V. Srivastava, T. Anna, and D. S. Mehta, “Full-field Hilbert phase microscopy using nearly common-path low coherence off-axis interferometry for quantitative imaging of biological cells,” J. Opt. 14(12), 125707 (2012).
    [Crossref]
  3. R. J. Edwards, S. W. Zhou, S. J. Hwang, K. Y. Mckeown, B. Wang, R. Bhaduri, P. J. Ganti, A. G. Yunker, J. A. Yodh, L. L. Rogers, G. Goddard, and Popescu, “Diffraction phase microscopy: monitoring nanoscale dynamics in materials science [invited],” Appl. Opt. 53(27), G33–43 (2014).
    [Crossref]
  4. N. A. Ochoa and J. M. Huntley, “Convenient method for calibrating nonlinear phase modulators for use in phase-shifting interferometry,” Opt. Eng. 37(9), 2501–2505 (1998).
    [Crossref]
  5. Y. J. Shen and J. M. Huntley, “Simple method to calibrate phase modulators for use in dynamic phase-shifting interferometry,” Opt. Eng. 43(12), 2998–3002 (2004).
    [Crossref]
  6. X. Chen, M. Gramaglia, and J. A. Yeazell, “Phase-shifting interferometry with uncalibrated phase shifts,” Appl. Opt. 39(4), 585–591 (2000).
    [Crossref]
  7. H. Kadono, Y. Bitoh, and S. Toyoka, “Statistical interferometry based on a fully developed speckle field: an experimental demonstration with noise analysis,” J. Opt. Soc. Am. A 18(6), 1267–1274 (2001).
    [Crossref]
  8. Z. T. Cheng and D. Liu, “Fast and accurate wavefront reconstruction in two-frame phase-shifting interferometry with unknown phase step,” Opt. Lett. 43(13), 3033–3036 (2018).
    [Crossref]
  9. Y. Zhang, X. B. Tian, and R. G. Liang, “Accurate and fast two-step phase shifting algorithm based on principle component analysis and Lissajous ellipse fitting with random phase shift and no pre-filtering,” Opt. Express 27(14), 20047–20063 (2019).
    [Crossref]
  10. Y. Zhang, X. B. Tian, and R. G. Liang, “Random two-step phase shifting interferometry based on Lissajous ellipse fitting and least squares technologies,” Opt. Express 26(12), 15059–15072 (2018).
    [Crossref]
  11. L. Z. Cai, Q. Liu, X. L. Yang, and Y. R. Wang, “Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. 28(19), 1808–1810 (2003).
    [Crossref]
  12. L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett. 29(2), 183–185 (2004).
    [Crossref]
  13. X. F. Xu, L. Z. Cai, X. F. Meng, G. Y. Dong, and X. X. Shen, “Fast blind extraction of arbitrary unknown phase shifts by an iterative tangent approach in generalized phase-shifting interferometry,” Opt. Lett. 31(13), 1966–1968 (2006).
    [Crossref]
  14. X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: Direct wave-front reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90(12), 121124 (2007).
    [Crossref]
  15. X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, W. J. Sun, H. Zhang, X. C. Cheng, G. Y. Dong, and X. X. Shen, “Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments,” Opt. Lett. 33(8), 776–778 (2008).
    [Crossref]
  16. X. F. Xu, L. Z. Cai, Y. R. Wang, and D. L. Li, “Accurate phase shift extraction for generalized phase-shifting interferometry,” Chin. Phys. Lett. 27(2), 024215 (2010).
    [Crossref]
  17. X. F. Xu, L. Z. Cai, Y. R. Wang, and R. S. Yan, “Direct phase shift extraction and wavefront reconstruction in two-step generalized phase-shifting interferometry,” J. Opt. 12(1), 015301 (2010).
    [Crossref]
  18. X. F. Xu, L. Z. Cai, F. Gao, Y. L. Jia, and H. Zhang, “Detection and correction of wavefront errors caused by slight reference tilt in two-step phase-shifting digital holography,” Appl. Opt. 54(32), 9591 (2015).
    [Crossref]
  19. J. Sun, B. Feng, and W. B. Xu, “Particle swarm optimization with particles having quantum behavior,” Cong. Evolution. Comp.1, 325–331 (2004).
  20. J. Sun, W. B. Xu, and B. Feng, “Adaptive parameter control for quantum-behaved particle swarm optimization on individual level,” IEEE Int. Conf. Systems, Man and Cybernetics4, 3049–3054 (2005).
  21. Q. Niu, Z. Zhou, H. Y. Zhang, and J. Deng, “An Improved Quantum-Behaved Particle Swarm Optimization Method for Economic Dispatch Problems with Multiple Fuel Options and Valve-Points Effects,” Energies 5(9), 3655–3673 (2012).
    [Crossref]
  22. H. Mirvaziri and Z. S. Mobarakeh, “Improvement of EEG-based motor imagery classification using ring topology-based particle swarm optimization,” Biomed. Signal Proces. 32, 69–75 (2017).
    [Crossref]
  23. X. L. Yu, Y. Yao, J. J. Xiao, and J. J. Tian, “Optimal design of short fiber Bragg gratings with triangular spectrum,” Opt. Commun. 285(5), 631–637 (2012).
    [Crossref]
  24. J. He and H. Guo, “A Modified Particle Swarm Optimization Algorithm,” Int. J. Comput. Int. Sys. 11(10), 151–155 (2013).
  25. C. Wang, T. T. Yu, J. L. Curiel-Sosa, N. G. Xie, and T. Q. Bui, “Adaptive chaotic particle swarm algorithm for isogeometric multi-objective size optimization of FG plates,” Struct Multidisc Optim 60(2), 757–778 (2019).
    [Crossref]
  26. S. Khan, S. Yang, and O. U. Rehman, “A dynamic particle swarm optimization method applied to global optimizations of engineering inverse problem,” COMPEL 37(1), 98–117 (2018).
    [Crossref]
  27. Z. H. Zhan, J. Zhang, and Y. Li, “Adaptive particle swarm optimization,” IEEE Trans. Syst., Man, Cybern. B 39(6), 1362–1381 (2009).
    [Crossref]
  28. J. Kennedy and R. C. Eberhart, “Particle swarm optimization,” IEEE Int. Conf. on Neural Networks, 1942–1948 (1995).

2019 (2)

C. Wang, T. T. Yu, J. L. Curiel-Sosa, N. G. Xie, and T. Q. Bui, “Adaptive chaotic particle swarm algorithm for isogeometric multi-objective size optimization of FG plates,” Struct Multidisc Optim 60(2), 757–778 (2019).
[Crossref]

Y. Zhang, X. B. Tian, and R. G. Liang, “Accurate and fast two-step phase shifting algorithm based on principle component analysis and Lissajous ellipse fitting with random phase shift and no pre-filtering,” Opt. Express 27(14), 20047–20063 (2019).
[Crossref]

2018 (3)

2017 (1)

H. Mirvaziri and Z. S. Mobarakeh, “Improvement of EEG-based motor imagery classification using ring topology-based particle swarm optimization,” Biomed. Signal Proces. 32, 69–75 (2017).
[Crossref]

2015 (1)

2014 (1)

2013 (1)

J. He and H. Guo, “A Modified Particle Swarm Optimization Algorithm,” Int. J. Comput. Int. Sys. 11(10), 151–155 (2013).

2012 (3)

Q. Niu, Z. Zhou, H. Y. Zhang, and J. Deng, “An Improved Quantum-Behaved Particle Swarm Optimization Method for Economic Dispatch Problems with Multiple Fuel Options and Valve-Points Effects,” Energies 5(9), 3655–3673 (2012).
[Crossref]

X. L. Yu, Y. Yao, J. J. Xiao, and J. J. Tian, “Optimal design of short fiber Bragg gratings with triangular spectrum,” Opt. Commun. 285(5), 631–637 (2012).
[Crossref]

V. Srivastava, T. Anna, and D. S. Mehta, “Full-field Hilbert phase microscopy using nearly common-path low coherence off-axis interferometry for quantitative imaging of biological cells,” J. Opt. 14(12), 125707 (2012).
[Crossref]

2010 (2)

X. F. Xu, L. Z. Cai, Y. R. Wang, and D. L. Li, “Accurate phase shift extraction for generalized phase-shifting interferometry,” Chin. Phys. Lett. 27(2), 024215 (2010).
[Crossref]

X. F. Xu, L. Z. Cai, Y. R. Wang, and R. S. Yan, “Direct phase shift extraction and wavefront reconstruction in two-step generalized phase-shifting interferometry,” J. Opt. 12(1), 015301 (2010).
[Crossref]

2009 (2)

Z. H. Zhan, J. Zhang, and Y. Li, “Adaptive particle swarm optimization,” IEEE Trans. Syst., Man, Cybern. B 39(6), 1362–1381 (2009).
[Crossref]

W. M. Ash, L. Krzewina, and M. K. Kim, “Quantitative imaging of cellular adhesion by total internal reflection holographic microscopy,” Appl. Opt. 48(34), H144–152 (2009).
[Crossref]

2008 (1)

2007 (1)

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: Direct wave-front reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90(12), 121124 (2007).
[Crossref]

2006 (1)

2004 (2)

L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett. 29(2), 183–185 (2004).
[Crossref]

Y. J. Shen and J. M. Huntley, “Simple method to calibrate phase modulators for use in dynamic phase-shifting interferometry,” Opt. Eng. 43(12), 2998–3002 (2004).
[Crossref]

2003 (1)

2001 (1)

2000 (1)

1998 (1)

N. A. Ochoa and J. M. Huntley, “Convenient method for calibrating nonlinear phase modulators for use in phase-shifting interferometry,” Opt. Eng. 37(9), 2501–2505 (1998).
[Crossref]

Anna, T.

V. Srivastava, T. Anna, and D. S. Mehta, “Full-field Hilbert phase microscopy using nearly common-path low coherence off-axis interferometry for quantitative imaging of biological cells,” J. Opt. 14(12), 125707 (2012).
[Crossref]

Ash, W. M.

Bhaduri, R.

Bitoh, Y.

Bui, T. Q.

C. Wang, T. T. Yu, J. L. Curiel-Sosa, N. G. Xie, and T. Q. Bui, “Adaptive chaotic particle swarm algorithm for isogeometric multi-objective size optimization of FG plates,” Struct Multidisc Optim 60(2), 757–778 (2019).
[Crossref]

Cai, L. Z.

X. F. Xu, L. Z. Cai, F. Gao, Y. L. Jia, and H. Zhang, “Detection and correction of wavefront errors caused by slight reference tilt in two-step phase-shifting digital holography,” Appl. Opt. 54(32), 9591 (2015).
[Crossref]

X. F. Xu, L. Z. Cai, Y. R. Wang, and D. L. Li, “Accurate phase shift extraction for generalized phase-shifting interferometry,” Chin. Phys. Lett. 27(2), 024215 (2010).
[Crossref]

X. F. Xu, L. Z. Cai, Y. R. Wang, and R. S. Yan, “Direct phase shift extraction and wavefront reconstruction in two-step generalized phase-shifting interferometry,” J. Opt. 12(1), 015301 (2010).
[Crossref]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, W. J. Sun, H. Zhang, X. C. Cheng, G. Y. Dong, and X. X. Shen, “Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments,” Opt. Lett. 33(8), 776–778 (2008).
[Crossref]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: Direct wave-front reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90(12), 121124 (2007).
[Crossref]

X. F. Xu, L. Z. Cai, X. F. Meng, G. Y. Dong, and X. X. Shen, “Fast blind extraction of arbitrary unknown phase shifts by an iterative tangent approach in generalized phase-shifting interferometry,” Opt. Lett. 31(13), 1966–1968 (2006).
[Crossref]

L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett. 29(2), 183–185 (2004).
[Crossref]

L. Z. Cai, Q. Liu, X. L. Yang, and Y. R. Wang, “Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. 28(19), 1808–1810 (2003).
[Crossref]

Chen, X.

Cheng, X. C.

Cheng, Z. T.

Curiel-Sosa, J. L.

C. Wang, T. T. Yu, J. L. Curiel-Sosa, N. G. Xie, and T. Q. Bui, “Adaptive chaotic particle swarm algorithm for isogeometric multi-objective size optimization of FG plates,” Struct Multidisc Optim 60(2), 757–778 (2019).
[Crossref]

Deng, J.

Q. Niu, Z. Zhou, H. Y. Zhang, and J. Deng, “An Improved Quantum-Behaved Particle Swarm Optimization Method for Economic Dispatch Problems with Multiple Fuel Options and Valve-Points Effects,” Energies 5(9), 3655–3673 (2012).
[Crossref]

Dong, G. Y.

Eberhart, R. C.

J. Kennedy and R. C. Eberhart, “Particle swarm optimization,” IEEE Int. Conf. on Neural Networks, 1942–1948 (1995).

Edwards, R. J.

Feng, B.

J. Sun, W. B. Xu, and B. Feng, “Adaptive parameter control for quantum-behaved particle swarm optimization on individual level,” IEEE Int. Conf. Systems, Man and Cybernetics4, 3049–3054 (2005).

J. Sun, B. Feng, and W. B. Xu, “Particle swarm optimization with particles having quantum behavior,” Cong. Evolution. Comp.1, 325–331 (2004).

Ganti, P. J.

Gao, F.

Goddard, G.

Gramaglia, M.

Guo, H.

J. He and H. Guo, “A Modified Particle Swarm Optimization Algorithm,” Int. J. Comput. Int. Sys. 11(10), 151–155 (2013).

He, J.

J. He and H. Guo, “A Modified Particle Swarm Optimization Algorithm,” Int. J. Comput. Int. Sys. 11(10), 151–155 (2013).

Huntley, J. M.

Y. J. Shen and J. M. Huntley, “Simple method to calibrate phase modulators for use in dynamic phase-shifting interferometry,” Opt. Eng. 43(12), 2998–3002 (2004).
[Crossref]

N. A. Ochoa and J. M. Huntley, “Convenient method for calibrating nonlinear phase modulators for use in phase-shifting interferometry,” Opt. Eng. 37(9), 2501–2505 (1998).
[Crossref]

Hwang, S. J.

Jia, Y. L.

Kadono, H.

Kennedy, J.

J. Kennedy and R. C. Eberhart, “Particle swarm optimization,” IEEE Int. Conf. on Neural Networks, 1942–1948 (1995).

Khan, S.

S. Khan, S. Yang, and O. U. Rehman, “A dynamic particle swarm optimization method applied to global optimizations of engineering inverse problem,” COMPEL 37(1), 98–117 (2018).
[Crossref]

Kim, M. K.

Krzewina, L.

Li, D. L.

X. F. Xu, L. Z. Cai, Y. R. Wang, and D. L. Li, “Accurate phase shift extraction for generalized phase-shifting interferometry,” Chin. Phys. Lett. 27(2), 024215 (2010).
[Crossref]

Li, Y.

Z. H. Zhan, J. Zhang, and Y. Li, “Adaptive particle swarm optimization,” IEEE Trans. Syst., Man, Cybern. B 39(6), 1362–1381 (2009).
[Crossref]

Liang, R. G.

Liu, D.

Liu, Q.

Mckeown, K. Y.

Mehta, D. S.

V. Srivastava, T. Anna, and D. S. Mehta, “Full-field Hilbert phase microscopy using nearly common-path low coherence off-axis interferometry for quantitative imaging of biological cells,” J. Opt. 14(12), 125707 (2012).
[Crossref]

Meng, X. F.

Mirvaziri, H.

H. Mirvaziri and Z. S. Mobarakeh, “Improvement of EEG-based motor imagery classification using ring topology-based particle swarm optimization,” Biomed. Signal Proces. 32, 69–75 (2017).
[Crossref]

Mobarakeh, Z. S.

H. Mirvaziri and Z. S. Mobarakeh, “Improvement of EEG-based motor imagery classification using ring topology-based particle swarm optimization,” Biomed. Signal Proces. 32, 69–75 (2017).
[Crossref]

Niu, Q.

Q. Niu, Z. Zhou, H. Y. Zhang, and J. Deng, “An Improved Quantum-Behaved Particle Swarm Optimization Method for Economic Dispatch Problems with Multiple Fuel Options and Valve-Points Effects,” Energies 5(9), 3655–3673 (2012).
[Crossref]

Ochoa, N. A.

N. A. Ochoa and J. M. Huntley, “Convenient method for calibrating nonlinear phase modulators for use in phase-shifting interferometry,” Opt. Eng. 37(9), 2501–2505 (1998).
[Crossref]

Popescu,

Rehman, O. U.

S. Khan, S. Yang, and O. U. Rehman, “A dynamic particle swarm optimization method applied to global optimizations of engineering inverse problem,” COMPEL 37(1), 98–117 (2018).
[Crossref]

Rogers, L. L.

Shen, X. X.

Shen, Y. J.

Y. J. Shen and J. M. Huntley, “Simple method to calibrate phase modulators for use in dynamic phase-shifting interferometry,” Opt. Eng. 43(12), 2998–3002 (2004).
[Crossref]

Srivastava, V.

V. Srivastava, T. Anna, and D. S. Mehta, “Full-field Hilbert phase microscopy using nearly common-path low coherence off-axis interferometry for quantitative imaging of biological cells,” J. Opt. 14(12), 125707 (2012).
[Crossref]

Sun, J.

J. Sun, W. B. Xu, and B. Feng, “Adaptive parameter control for quantum-behaved particle swarm optimization on individual level,” IEEE Int. Conf. Systems, Man and Cybernetics4, 3049–3054 (2005).

J. Sun, B. Feng, and W. B. Xu, “Particle swarm optimization with particles having quantum behavior,” Cong. Evolution. Comp.1, 325–331 (2004).

Sun, W. J.

Tian, J. J.

X. L. Yu, Y. Yao, J. J. Xiao, and J. J. Tian, “Optimal design of short fiber Bragg gratings with triangular spectrum,” Opt. Commun. 285(5), 631–637 (2012).
[Crossref]

Tian, X. B.

Toyoka, S.

Wang, B.

Wang, C.

C. Wang, T. T. Yu, J. L. Curiel-Sosa, N. G. Xie, and T. Q. Bui, “Adaptive chaotic particle swarm algorithm for isogeometric multi-objective size optimization of FG plates,” Struct Multidisc Optim 60(2), 757–778 (2019).
[Crossref]

Wang, Y. R.

X. F. Xu, L. Z. Cai, Y. R. Wang, and R. S. Yan, “Direct phase shift extraction and wavefront reconstruction in two-step generalized phase-shifting interferometry,” J. Opt. 12(1), 015301 (2010).
[Crossref]

X. F. Xu, L. Z. Cai, Y. R. Wang, and D. L. Li, “Accurate phase shift extraction for generalized phase-shifting interferometry,” Chin. Phys. Lett. 27(2), 024215 (2010).
[Crossref]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, W. J. Sun, H. Zhang, X. C. Cheng, G. Y. Dong, and X. X. Shen, “Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments,” Opt. Lett. 33(8), 776–778 (2008).
[Crossref]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: Direct wave-front reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90(12), 121124 (2007).
[Crossref]

L. Z. Cai, Q. Liu, X. L. Yang, and Y. R. Wang, “Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. 28(19), 1808–1810 (2003).
[Crossref]

Xiao, J. J.

X. L. Yu, Y. Yao, J. J. Xiao, and J. J. Tian, “Optimal design of short fiber Bragg gratings with triangular spectrum,” Opt. Commun. 285(5), 631–637 (2012).
[Crossref]

Xie, N. G.

C. Wang, T. T. Yu, J. L. Curiel-Sosa, N. G. Xie, and T. Q. Bui, “Adaptive chaotic particle swarm algorithm for isogeometric multi-objective size optimization of FG plates,” Struct Multidisc Optim 60(2), 757–778 (2019).
[Crossref]

Xu, W. B.

J. Sun, W. B. Xu, and B. Feng, “Adaptive parameter control for quantum-behaved particle swarm optimization on individual level,” IEEE Int. Conf. Systems, Man and Cybernetics4, 3049–3054 (2005).

J. Sun, B. Feng, and W. B. Xu, “Particle swarm optimization with particles having quantum behavior,” Cong. Evolution. Comp.1, 325–331 (2004).

Xu, X. F.

X. F. Xu, L. Z. Cai, F. Gao, Y. L. Jia, and H. Zhang, “Detection and correction of wavefront errors caused by slight reference tilt in two-step phase-shifting digital holography,” Appl. Opt. 54(32), 9591 (2015).
[Crossref]

X. F. Xu, L. Z. Cai, Y. R. Wang, and D. L. Li, “Accurate phase shift extraction for generalized phase-shifting interferometry,” Chin. Phys. Lett. 27(2), 024215 (2010).
[Crossref]

X. F. Xu, L. Z. Cai, Y. R. Wang, and R. S. Yan, “Direct phase shift extraction and wavefront reconstruction in two-step generalized phase-shifting interferometry,” J. Opt. 12(1), 015301 (2010).
[Crossref]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, W. J. Sun, H. Zhang, X. C. Cheng, G. Y. Dong, and X. X. Shen, “Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments,” Opt. Lett. 33(8), 776–778 (2008).
[Crossref]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: Direct wave-front reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90(12), 121124 (2007).
[Crossref]

X. F. Xu, L. Z. Cai, X. F. Meng, G. Y. Dong, and X. X. Shen, “Fast blind extraction of arbitrary unknown phase shifts by an iterative tangent approach in generalized phase-shifting interferometry,” Opt. Lett. 31(13), 1966–1968 (2006).
[Crossref]

Yan, R. S.

X. F. Xu, L. Z. Cai, Y. R. Wang, and R. S. Yan, “Direct phase shift extraction and wavefront reconstruction in two-step generalized phase-shifting interferometry,” J. Opt. 12(1), 015301 (2010).
[Crossref]

Yang, S.

S. Khan, S. Yang, and O. U. Rehman, “A dynamic particle swarm optimization method applied to global optimizations of engineering inverse problem,” COMPEL 37(1), 98–117 (2018).
[Crossref]

Yang, X. L.

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: Direct wave-front reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90(12), 121124 (2007).
[Crossref]

L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett. 29(2), 183–185 (2004).
[Crossref]

L. Z. Cai, Q. Liu, X. L. Yang, and Y. R. Wang, “Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. 28(19), 1808–1810 (2003).
[Crossref]

Yao, Y.

X. L. Yu, Y. Yao, J. J. Xiao, and J. J. Tian, “Optimal design of short fiber Bragg gratings with triangular spectrum,” Opt. Commun. 285(5), 631–637 (2012).
[Crossref]

Yeazell, J. A.

Yodh, J. A.

Yu, T. T.

C. Wang, T. T. Yu, J. L. Curiel-Sosa, N. G. Xie, and T. Q. Bui, “Adaptive chaotic particle swarm algorithm for isogeometric multi-objective size optimization of FG plates,” Struct Multidisc Optim 60(2), 757–778 (2019).
[Crossref]

Yu, X. L.

X. L. Yu, Y. Yao, J. J. Xiao, and J. J. Tian, “Optimal design of short fiber Bragg gratings with triangular spectrum,” Opt. Commun. 285(5), 631–637 (2012).
[Crossref]

Yunker, A. G.

Zhan, Z. H.

Z. H. Zhan, J. Zhang, and Y. Li, “Adaptive particle swarm optimization,” IEEE Trans. Syst., Man, Cybern. B 39(6), 1362–1381 (2009).
[Crossref]

Zhang, H.

Zhang, H. Y.

Q. Niu, Z. Zhou, H. Y. Zhang, and J. Deng, “An Improved Quantum-Behaved Particle Swarm Optimization Method for Economic Dispatch Problems with Multiple Fuel Options and Valve-Points Effects,” Energies 5(9), 3655–3673 (2012).
[Crossref]

Zhang, J.

Z. H. Zhan, J. Zhang, and Y. Li, “Adaptive particle swarm optimization,” IEEE Trans. Syst., Man, Cybern. B 39(6), 1362–1381 (2009).
[Crossref]

Zhang, Y.

Zhou, S. W.

Zhou, Z.

Q. Niu, Z. Zhou, H. Y. Zhang, and J. Deng, “An Improved Quantum-Behaved Particle Swarm Optimization Method for Economic Dispatch Problems with Multiple Fuel Options and Valve-Points Effects,” Energies 5(9), 3655–3673 (2012).
[Crossref]

Appl. Opt. (4)

Appl. Phys. Lett. (1)

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: Direct wave-front reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90(12), 121124 (2007).
[Crossref]

Biomed. Signal Proces. (1)

H. Mirvaziri and Z. S. Mobarakeh, “Improvement of EEG-based motor imagery classification using ring topology-based particle swarm optimization,” Biomed. Signal Proces. 32, 69–75 (2017).
[Crossref]

Chin. Phys. Lett. (1)

X. F. Xu, L. Z. Cai, Y. R. Wang, and D. L. Li, “Accurate phase shift extraction for generalized phase-shifting interferometry,” Chin. Phys. Lett. 27(2), 024215 (2010).
[Crossref]

COMPEL (1)

S. Khan, S. Yang, and O. U. Rehman, “A dynamic particle swarm optimization method applied to global optimizations of engineering inverse problem,” COMPEL 37(1), 98–117 (2018).
[Crossref]

Energies (1)

Q. Niu, Z. Zhou, H. Y. Zhang, and J. Deng, “An Improved Quantum-Behaved Particle Swarm Optimization Method for Economic Dispatch Problems with Multiple Fuel Options and Valve-Points Effects,” Energies 5(9), 3655–3673 (2012).
[Crossref]

IEEE Trans. Syst., Man, Cybern. B (1)

Z. H. Zhan, J. Zhang, and Y. Li, “Adaptive particle swarm optimization,” IEEE Trans. Syst., Man, Cybern. B 39(6), 1362–1381 (2009).
[Crossref]

Int. J. Comput. Int. Sys. (1)

J. He and H. Guo, “A Modified Particle Swarm Optimization Algorithm,” Int. J. Comput. Int. Sys. 11(10), 151–155 (2013).

J. Opt. (2)

X. F. Xu, L. Z. Cai, Y. R. Wang, and R. S. Yan, “Direct phase shift extraction and wavefront reconstruction in two-step generalized phase-shifting interferometry,” J. Opt. 12(1), 015301 (2010).
[Crossref]

V. Srivastava, T. Anna, and D. S. Mehta, “Full-field Hilbert phase microscopy using nearly common-path low coherence off-axis interferometry for quantitative imaging of biological cells,” J. Opt. 14(12), 125707 (2012).
[Crossref]

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

X. L. Yu, Y. Yao, J. J. Xiao, and J. J. Tian, “Optimal design of short fiber Bragg gratings with triangular spectrum,” Opt. Commun. 285(5), 631–637 (2012).
[Crossref]

Opt. Eng. (2)

N. A. Ochoa and J. M. Huntley, “Convenient method for calibrating nonlinear phase modulators for use in phase-shifting interferometry,” Opt. Eng. 37(9), 2501–2505 (1998).
[Crossref]

Y. J. Shen and J. M. Huntley, “Simple method to calibrate phase modulators for use in dynamic phase-shifting interferometry,” Opt. Eng. 43(12), 2998–3002 (2004).
[Crossref]

Opt. Express (2)

Opt. Lett. (5)

Struct Multidisc Optim (1)

C. Wang, T. T. Yu, J. L. Curiel-Sosa, N. G. Xie, and T. Q. Bui, “Adaptive chaotic particle swarm algorithm for isogeometric multi-objective size optimization of FG plates,” Struct Multidisc Optim 60(2), 757–778 (2019).
[Crossref]

Other (3)

J. Kennedy and R. C. Eberhart, “Particle swarm optimization,” IEEE Int. Conf. on Neural Networks, 1942–1948 (1995).

J. Sun, B. Feng, and W. B. Xu, “Particle swarm optimization with particles having quantum behavior,” Cong. Evolution. Comp.1, 325–331 (2004).

J. Sun, W. B. Xu, and B. Feng, “Adaptive parameter control for quantum-behaved particle swarm optimization on individual level,” IEEE Int. Conf. Systems, Man and Cybernetics4, 3049–3054 (2005).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1.
Fig. 1. Flow chart of the proposed method.
Fig. 2.
Fig. 2. The device schematic diagram in the simulation experiment.
Fig. 3.
Fig. 3. Comparison of phase shifts extraction results of our method and the direct search for a four-frame PSI. (a)–(c) The iterative process of ${\alpha _1},{\alpha _2},{\alpha _3}$, (d) the number of repetitive experiments of our method and the direct search [12].
Fig. 4.
Fig. 4. Simulation results of object phase reconstruction with the four-frame method with retrieved nonstandard phase shifts. (a) Original object phase, (b)–(e) four holograms with different phase shifts obtained on the image plane of the camera, namely ${I_1}$, ${I_2}$, ${I_3}$ and ${I_4}$, (f) retrieved object phase by our method, (g) retrieved object phase by direct search, (h) the difference between (f) and (g), (i) the difference between (a) and (f), (j) the difference between (a) and (g).
Fig. 5.
Fig. 5. Optical experiment results. (a)–(d) the four interferograms ${I_1}$, ${I_2}$, ${I_3}$ and ${I_4}$ in plane ${\textrm{P}_{\textrm{H}}}$, (e) and (f) intensity ${I_o}$ of object wave and ${I_r}$ of reference wave in plane ${\textrm{P}_{\textrm{H}}}$, (g) and (h) the reconstructed images in plane ${\textrm{P}_{\textrm{O}}}$ by the direct search and the proposed method, respectively.
Fig. 6.
Fig. 6. Comparison of reconstructed images by the direct search and the proposed method. (a) and (d) the reconstructed images in plane ${\textrm{P}_{\textrm{O}}}$ by the direct search and the proposed method, respectively, (b) and (e) enlarged views of the central part of (a) and (d), respectively, (c) and (f) the intensity distribution maps corresponding to the red lines at (b) and (e), respectively.

Equations (15)

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

I 0 = A o 2 + A r 2 + 2 A r A o cos [ φ ] I 1 = A o 2 + A r 2 + 2 A r A o cos [ φ α 1 ] I 2 = A o 2 + A r 2 + 2 A r A o cos [ φ α 1 α 2 ] I 3 = A o 2 + A r 2 + 2 A r A o cos [ φ α 1 α 2 α 3 ]
p = | I 1 I 0 | = | 4 A r A o sin ( φ α 1 / α 1 2 2 ) | sin ( α 1 / α 1 2 2 ) q = | I 2 I 1 | = | 4 A r A o sin ( φ α 1 α 2 / α 2 2 2 ) | sin ( α 2 / α 2 2 2 ) r = | I 3 I 2 | = | 4 A r A o sin ( φ α 1 α 2 α 3 / α 3 2 2 ) | sin ( α 3 / α 3 2 2 )
p = c sin ( α 1 / α 1 2 2 ) , q = c sin ( α 2 / α 2 2 2 ) , r = c sin ( α 3 / α 3 2 2 )
α 1 = 2 a r c sin ( p / p c c ) , α 2 = 2 a r c sin ( q / q c c ) , α 3 = 2 a r c sin ( r / r c c )
A o exp ( i φ ) = 1 4 A r sin [ ( α 1 + α 3 ) / ( α 1 + α 3 ) 2 2 ] × { exp [ i ( α 1 + α 2 ) / ( α 1 + α 2 ) 2 2 ] sin [ ( α 2 + α 3 ) / ( α 2 + α 3 ) 2 2 ] ( I 1 I 3 ) exp [ i ( α 1 + α 2 / α 2 2 2 + α 3 / α 3 2 2 ) ] sin [ ( α 1 + α 2 ) / ( α 1 + α 2 ) 2 2 ] ( I 0 I 2 ) }
ε = { | I 0 ( A o 2 + A r 2 + 2 A r A o cos φ ) | + | I 1 [ A o 2 + A r 2 + 2 A r A o cos ( φ α 1 ) ] | + | I 2 [ A o 2 + A r 2 + 2 A r A o cos ( φ α 1 α 2 ) ] | + | I 3 [ A o 2 + A r 2 + 2 A r A o cos ( φ α 1 α 2 α 3 ) ] | }
p e d ( m ) = p e d ( m ) ± N ( 0 , ξ 1 Δ n r a n g e )
p e d ( m ) = φ d ( m ) p e d ( m ) + [ 1 φ d ( m ) ] p g d ( m )
m b e s t ( m ) = 1 M e = 1 M p g d ( m )
x e d ( m + 1 ) = p e d ( m ) ± β ( m ) | m b e s t ( m ) x e d ( m ) | ln [ 1 / 1 u e d ( m ) u e d ( m ) ]
p e d ( m ) = p e d ( m ) ± N ( 0 , ξ i Δ n r a n g e )
p e d ( m + 1 ) = { x e d ( m + 1 ) , ( f [ x e d ( m + 1 ) ] < f [ p e d ( m ) ] ) p e d ( m ) , ( f [ x e d ( m + 1 ) ] f [ p e d ( m ) ] )
p g d ( m + 1 ) = arg min 1 e M { f [ p e d ( m ) ] }
F ( m ) = [ f e ( m ) f g b e s t ( m ) ] / [ f e ( m ) f g b e s t ( m ) ] min { a b s [ f e ( m ) ] , a b s [ f g b e s t ( m ) ] } min { a b s [ f e ( m ) ] , a b s [ f g b e s t ( m ) ] }
p e d ( m ) = p e d ( m ) ± N ( 0 , ξ 1 Δ n r a n g e )