Abstract

Light propagating through a scattering medium generates a random field, which is also known as a speckle. The scattering process hinders the direct retrieval of the information encoded in the light based on the randomly fluctuating field. In this study, we propose and experimentally demonstrate a method for the imaging of polarimetric-phase objects hidden behind a scattering medium based on two-point intensity correlation and phase-shifting techniques. One advantage of proposed method is that it does not require mechanical rotation of polarization elements. The method exploits the relationship between the two-point intensity correlation of the spatially fluctuating random field in the observation plane and the structure of the polarized source in the scattering plane. The polarimetric phase of the source structure is determined by replacing the interference intensity in traditional phase shift formula with the Fourier transform of the cross-covariance of the intensity. The imaging of the polarimetric-phase object is demonstrated by comparing three different phase-shifting techniques. We also evaluated the performance of the proposed technique on an unstable platform as well as using dynamic diffusers, which is implemented by replacing the diffuser with a new one during each phase-shifting step. The results were compared with that obtained with a fixed diffuser on a vibration-isolation platform during the phase-shifting process. A good match is found among the three cases, thus confirming that the proposed intensity-correlation-based technique is a useful one and should be applicable with dynamic diffusers as well as in unstable environments.

© 2020 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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    [Crossref]
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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]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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2020 (1)

2019 (2)

2018 (4)

2016 (1)

2015 (4)

C. Luo, L. Zhong, P. Sun, H. Wang, J. Tian, and X. Lu, “Two-step demodulation algorithm based on the orthogonality of diamond diagonal vectors,” Appl. Phys. B 119(2), 387–391 (2015).
[Crossref]

A. Dogariu and R. Carminati, “Electromagnetic field correlations in three-dimensional speckles,” Phys. Rep. 559, 1–29 (2015).
[Crossref]

R. V. Vinu and R. K. Singh, “Synthesis of statistical properties of a randomly fluctuating polarized field,” Appl. Opt. 54(21), 6491–6497 (2015).
[Crossref]

R. V. Vinu and R. K. Singh, “Experimental determination of generalized Stokes parameters,” Opt. Lett. 40(7), 1227–1230 (2015).
[Crossref]

2014 (1)

R. K. Singh, R. V. Vinu, and M. A. Sharma, “Recovery of complex valued objects from two-point intensity correlation measurement,” Appl. Phys. Lett. 104(11), 111108 (2014).
[Crossref]

2013 (1)

2011 (2)

2010 (1)

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum Entanglement Resolves a Basic Issue in Polarization Optics,” Phys. Rev. Lett. 104(2), 023901 (2010).
[Crossref]

2009 (1)

2005 (1)

2004 (1)

J. Millerd, N. Brock, J. Hayes, M. North-Morris, B. Kimbrough, and J. Wyant, “Pixelated phase-mask dynamic interferometers,” Proc. SPIE 5531, 304–314 (2004).
[Crossref]

2003 (2)

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003).
[Crossref]

J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11(10), 1137–1143 (2003).
[Crossref]

1998 (1)

1997 (1)

1956 (1)

R. H. Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177(4497), 27–29 (1956).
[Crossref]

1954 (1)

U. Fano, “A Stokes-parameter technique for the treatment of polarization in quantum mechanics,” Phys. Rev. 93(1), 121–123 (1954).
[Crossref]

Borghi, R.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum Entanglement Resolves a Basic Issue in Polarization Optics,” Phys. Rev. Lett. 104(2), 023901 (2010).
[Crossref]

Brock, N.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, B. Kimbrough, and J. Wyant, “Pixelated phase-mask dynamic interferometers,” Proc. SPIE 5531, 304–314 (2004).
[Crossref]

Brown, R. H.

R. H. Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177(4497), 27–29 (1956).
[Crossref]

Brundabanam, M. M.

Carminati, R.

A. Dogariu and R. Carminati, “Electromagnetic field correlations in three-dimensional speckles,” Phys. Rep. 559, 1–29 (2015).
[Crossref]

Chen, L.

Chen, Z.

DiMarzio, C. A.

Dogariu, A.

A. Dogariu and R. Carminati, “Electromagnetic field correlations in three-dimensional speckles,” Phys. Rep. 559, 1–29 (2015).
[Crossref]

Fano, U.

U. Fano, “A Stokes-parameter technique for the treatment of polarization in quantum mechanics,” Phys. Rev. 93(1), 121–123 (1954).
[Crossref]

Friberg, A. T.

Gautam, S. K.

S. Mishra, S. K. Gautam, D. N. Naik, Z. Chen, J. Pu, and R. K. Singh, “Tailoring and analysis of vectorial coherence,” J. Opt. 20(12), 125605 (2018).
[Crossref]

Gezhi, Z.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 2000).

Gori, F.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum Entanglement Resolves a Basic Issue in Polarization Optics,” Phys. Rev. Lett. 104(2), 023901 (2010).
[Crossref]

F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23(4), 241–243 (1998).
[Crossref]

Goshtasby, A. A.

A. A. Goshtasby, Image Registration: Principle, Tools and Methods (Springer, 2012).

Guo, M.

Hassinen, T.

Hayes, J.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, B. Kimbrough, and J. Wyant, “Pixelated phase-mask dynamic interferometers,” Proc. SPIE 5531, 304–314 (2004).
[Crossref]

Hu, X.

Itou, H.

Kimbrough, B.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, B. Kimbrough, and J. Wyant, “Pixelated phase-mask dynamic interferometers,” Proc. SPIE 5531, 304–314 (2004).
[Crossref]

Korotkova, O.

Lu, X.

C. Luo, L. Zhong, P. Sun, H. Wang, J. Tian, and X. Lu, “Two-step demodulation algorithm based on the orthogonality of diamond diagonal vectors,” Appl. Phys. B 119(2), 387–391 (2015).
[Crossref]

Luo, C.

C. Luo, L. Zhong, P. Sun, H. Wang, J. Tian, and X. Lu, “Two-step demodulation algorithm based on the orthogonality of diamond diagonal vectors,” Appl. Phys. B 119(2), 387–391 (2015).
[Crossref]

Luo, M.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Millerd, J.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, B. Kimbrough, and J. Wyant, “Pixelated phase-mask dynamic interferometers,” Proc. SPIE 5531, 304–314 (2004).
[Crossref]

Mishra, S.

S. Mishra, S. K. Gautam, D. N. Naik, Z. Chen, J. Pu, and R. K. Singh, “Tailoring and analysis of vectorial coherence,” J. Opt. 20(12), 125605 (2018).
[Crossref]

Miyamoto, Y.

Mukunda, N.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum Entanglement Resolves a Basic Issue in Polarization Optics,” Phys. Rev. Lett. 104(2), 023901 (2010).
[Crossref]

Naik, D. N.

North-Morris, M.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, B. Kimbrough, and J. Wyant, “Pixelated phase-mask dynamic interferometers,” Proc. SPIE 5531, 304–314 (2004).
[Crossref]

Pu, J.

Santarsiero, M.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum Entanglement Resolves a Basic Issue in Polarization Optics,” Phys. Rev. Lett. 104(2), 023901 (2010).
[Crossref]

Sasaki, O.

Setälä, T.

Sharma, M. A.

R. K. Singh, R. V. Vinu, and M. A. Sharma, “Recovery of complex valued objects from two-point intensity correlation measurement,” Appl. Phys. Lett. 104(11), 111108 (2014).
[Crossref]

Simon, B. N.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum Entanglement Resolves a Basic Issue in Polarization Optics,” Phys. Rev. Lett. 104(2), 023901 (2010).
[Crossref]

Simon, R.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum Entanglement Resolves a Basic Issue in Polarization Optics,” Phys. Rev. Lett. 104(2), 023901 (2010).
[Crossref]

Simon, S.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum Entanglement Resolves a Basic Issue in Polarization Optics,” Phys. Rev. Lett. 104(2), 023901 (2010).
[Crossref]

Singh, D.

Singh, R. K.

L. Chen, R. K. Singh, Z. Chen, and J. Pu, “Phase shifting digital holography with the Hanbury Brown–Twiss approach,” Opt. Lett. 45(1), 212–215 (2020).
[Crossref]

D. Singh and R. K. Singh, “Lensless Stokes holography with the Hanbury Brown-Twiss approach,” Opt. Express 26(8), 10801–10812 (2018).
[Crossref]

S. Mishra, S. K. Gautam, D. N. Naik, Z. Chen, J. Pu, and R. K. Singh, “Tailoring and analysis of vectorial coherence,” J. Opt. 20(12), 125605 (2018).
[Crossref]

N. K. Soni, R. V. Vinu, and R. K. Singh, “Polarization modulation for imaging behind the scattering medium,” Opt. Lett. 41(5), 906–909 (2016).
[Crossref]

R. V. Vinu and R. K. Singh, “Synthesis of statistical properties of a randomly fluctuating polarized field,” Appl. Opt. 54(21), 6491–6497 (2015).
[Crossref]

R. V. Vinu and R. K. Singh, “Experimental determination of generalized Stokes parameters,” Opt. Lett. 40(7), 1227–1230 (2015).
[Crossref]

R. K. Singh, R. V. Vinu, and M. A. Sharma, “Recovery of complex valued objects from two-point intensity correlation measurement,” Appl. Phys. Lett. 104(11), 111108 (2014).
[Crossref]

R. K. Singh, D. N. Naik, H. Itou, M. M. Brundabanam, Y. Miyamoto, and M. Takeda, “Vectorial van Cittert–Zernike theorem based on spatial averaging: experimental demonstrations,” Opt. Lett. 38(22), 4809–4812 (2013).
[Crossref]

R. K. Singh, D. N. Naik, H. Itou, Y. Miyamoto, and M. Takeda, “Vectorial coherence holography,” Opt. Express 19(12), 11558–11567 (2011).
[Crossref]

Soni, N. K.

Sun, P.

C. Luo, L. Zhong, P. Sun, H. Wang, J. Tian, and X. Lu, “Two-step demodulation algorithm based on the orthogonality of diamond diagonal vectors,” Appl. Phys. B 119(2), 387–391 (2015).
[Crossref]

Takeda, M.

Tervo, J.

Tian, J.

C. Luo, L. Zhong, P. Sun, H. Wang, J. Tian, and X. Lu, “Two-step demodulation algorithm based on the orthogonality of diamond diagonal vectors,” Appl. Phys. B 119(2), 387–391 (2015).
[Crossref]

Twiss, R. Q.

R. H. Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177(4497), 27–29 (1956).
[Crossref]

Vinu, R. V.

Wang, H.

C. Luo, L. Zhong, P. Sun, H. Wang, J. Tian, and X. Lu, “Two-step demodulation algorithm based on the orthogonality of diamond diagonal vectors,” Appl. Phys. B 119(2), 387–391 (2015).
[Crossref]

Warger, W. C.

Wolf, E.

O. Korotkova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett. 30(2), 198–200 (2005).
[Crossref]

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003).
[Crossref]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

Wyant, J.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, B. Kimbrough, and J. Wyant, “Pixelated phase-mask dynamic interferometers,” Proc. SPIE 5531, 304–314 (2004).
[Crossref]

Yamaguchi, I.

Zhang, T.

Zhao, D.

Zhong, L.

C. Luo, L. Zhong, P. Sun, H. Wang, J. Tian, and X. Lu, “Two-step demodulation algorithm based on the orthogonality of diamond diagonal vectors,” Appl. Phys. B 119(2), 387–391 (2015).
[Crossref]

Zhou, Y.

Appl. Opt. (2)

Appl. Phys. B (1)

C. Luo, L. Zhong, P. Sun, H. Wang, J. Tian, and X. Lu, “Two-step demodulation algorithm based on the orthogonality of diamond diagonal vectors,” Appl. Phys. B 119(2), 387–391 (2015).
[Crossref]

Appl. Phys. Lett. (1)

R. K. Singh, R. V. Vinu, and M. A. Sharma, “Recovery of complex valued objects from two-point intensity correlation measurement,” Appl. Phys. Lett. 104(11), 111108 (2014).
[Crossref]

J. Opt. (1)

S. Mishra, S. K. Gautam, D. N. Naik, Z. Chen, J. Pu, and R. K. Singh, “Tailoring and analysis of vectorial coherence,” J. Opt. 20(12), 125605 (2018).
[Crossref]

Nature (1)

R. H. Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177(4497), 27–29 (1956).
[Crossref]

Opt. Express (8)

Opt. Lett. (7)

Phys. Lett. A (1)

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003).
[Crossref]

Phys. Rep. (1)

A. Dogariu and R. Carminati, “Electromagnetic field correlations in three-dimensional speckles,” Phys. Rep. 559, 1–29 (2015).
[Crossref]

Phys. Rev. (1)

U. Fano, “A Stokes-parameter technique for the treatment of polarization in quantum mechanics,” Phys. Rev. 93(1), 121–123 (1954).
[Crossref]

Phys. Rev. Lett. (1)

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum Entanglement Resolves a Basic Issue in Polarization Optics,” Phys. Rev. Lett. 104(2), 023901 (2010).
[Crossref]

Proc. SPIE (1)

J. Millerd, N. Brock, J. Hayes, M. North-Morris, B. Kimbrough, and J. Wyant, “Pixelated phase-mask dynamic interferometers,” Proc. SPIE 5531, 304–314 (2004).
[Crossref]

Other (4)

J. W. Goodman, Statistical Optics (Wiley, 2000).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

A. A. Goshtasby, Image Registration: Principle, Tools and Methods (Springer, 2012).

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Figures (7)

Fig. 1.
Fig. 1. Geometry of polarized field, diffuser plane, observation plane, and propagation scheme.
Fig. 2.
Fig. 2. Experimental setup for phase imaging of target behind scattering medium. Laser: He-Ne laser; MO: Microscope objective; P: Pinhole; HWP: Half-wave plate; BS: Beam splitter; M: Mirror; SLM: Spatial light modulator; L1, L2, L3: Lenses; GG: Ground glass; CCD: Charge coupled device. Propagation of object beam (red) and reference beam (blue) from L3 to GG is shown in inset.
Fig. 3.
Fig. 3. (a)-(d) Random intensities recorded by CCD camera for smiling face for phase shifts of 0, $\pi /2$ , $\pi $ , and $3\pi /2$ , respectively. (e)-(h) are cross-covariances corresponding to intensities in (a)–(d), respectively.
Fig. 4.
Fig. 4. Fourier spectrum of cross-covariance for different test objects: (a) smiling face, (b) butterfly, (c) Chinese letter “guang,” (d) letter “A,” and (e) number “2.”
Fig. 5.
Fig. 5. Reconstructed phase structures of various test objects using eight-step phase-shifting technique: (a) smiling face, (b) butterfly, and (c) Chinese letter “guang.”
Fig. 6.
Fig. 6. Comparison of performances of different phase-shifting techniques in case of letter “A:” (a) two-step, (b) four-step, and (c) eight-step methods.
Fig. 7.
Fig. 7. Imaging performance using eight-step phase-shifting method for number “2” as test object: (a) static conditions, (b) with vibrations, and (c) dynamic conditions.

Equations (13)

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

E ( r 1 ) = ( E x ( r 1 ) e i φ ( r 1 ) E y ( r 1 ) e i φ ( r 1 ) ) ,
E i ( r 2 ) = G ( r 2 , r 1 ) E i ( r 1 ) d r 1 ,
G ( r 2 , r 1 ) e x p ( i k z ) i λ z e x p ( i k | r 2 | 2 2 r 2 r 1 + | r 1 | 2 2 z ) ,
E i ( r 2 ) = E i O ( r 2 ) + E i R ( r 2 ) ,
I ( r 2 ) = | E x ( r 2 ) | 2 + | E y ( r 2 ) | 2 .
C ( Δ r ) = Δ I ( r 2 ) Δ I ( r 2 + Δ r ) ,
C ( Δ r ) = | W x x ( Δ r ) | 2 + | W y y ( Δ r ) | 2 + | W x y ( Δ r ) | 2 + | W y x ( Δ r ) | 2 ,
W i j O ( Δ r ) = E i ( r 1 ) E j ( r 1 ) e x p ( i 2 π λ z Δ r r 1 ) d r 1 ,
W i j R ( Δ r ) = c i r c ( r 1 r g a ) exp ( i 2 π λ z Δ r r 1 ) d r 1 ,
| W i j ( Δ r ) | 2 = | W i j O ( Δ r ) | 2 + | W i j R ( Δ r ) | 2 + W i j O ( Δ r ) W i j R ( Δ r ) + W i j O ( Δ r ) W i j R ( Δ r ) .
tan ( φ / 2 ) = 1 2 C 2 C 1 ,
tan ( φ ) = C 4 C 2 C 1 C 3 ,
tan ( φ O ) = ( C 1 C 3 ) ( C 8 C 6 ) ( C 4 C 2 ) ( C 5 C 7 ) ( C 1 C 3 ) ( C 5 C 7 ) + ( C 4 C 2 ) ( C 8 C 6 ) ,

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