Abstract

Characterizing the transmission matrix (TM) of a multimode fiber (MMF) benefits many fiber-based applications and allows in-depth studies on the physical properties. For example, by modulating the incident field, the knowledge of the TM allows one to synthesize any optical field at the distill end of the MMF. However, the extraction of optical fields usually requires holographic measurements with interferometry, which complicates the system design and introduces additional noise. In this work, we developed an efficient method to retrieve the TM of the MMF in a referenceless optical system. With pure intensity measurements, this method uses the extended Kalman filter (EKF) to recursively search for the optimum solution. To facilitate the computational process, a modified speckle-correlation scatter matrix (MSSM) is constructed as a low-fidelity initial estimation. This method, termed EKF-MSSM, only requires 4N intensity measurements to precisely solve for N unknown complex variables in the TM. Experimentally, we successfully retrieved the TM of the MMF with high precision, which allows optical focusing with the enhancement (>70%) close to the theoretical value. We anticipate that this method will serve as a useful tool for studying physical properties of the MMFs and potentially open new possibilities in a variety of applications in fiber optics.

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

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References

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2019 (5)

2018 (8)

M. N’Gom, T. B. Norris, E. Michielssen, and R. R. Nadakuditi, “Mode control in a multimode fiber through acquiring its transmission matrix from a reference-less optical system,” Opt. Lett. 43(3), 419–422 (2018).
[Crossref]

A. Turpin, I. Vishniakou, and J. d Seelig, “Light scattering control in transmission and reflection with neural networks,” Opt. Express 26(23), 30911–30929 (2018).
[Crossref]

N. Borhani, E. Kakkava, C. Moser, and D. Psaltis, “Learning to see through multimode fibers,” Optica 5(8), 960–966 (2018).
[Crossref]

L. Deng, J. D. Yan, D. S. Elson, and L. Su, “Characterization of an imaging multimode optical fiber using a digital micro-mirror device based single-beam system,” Opt. Express 26(14), 18436–18447 (2018).
[Crossref]

T. Zhao, L. Deng, W. Wang, D. S. Elson, and L. Su, “Bayes’ theorem-based binary algorithm for fast reference-less calibration of a multimode fiber,” Opt. Express 26(16), 20368–20378 (2018).
[Crossref]

A. Hemphill, Y. Shen, J. Hwang, and L. Wang, “High-speed alignment optimization of digital optical phase conjugation systems based on autocovariance analysis in conjunction with orthonormal rectangular polynomials,” J. Biomed. Opt. 24(03), 1 (2018).
[Crossref]

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[Crossref]

C. Ma, J. Di, Y. Zhang, P. Li, F. Xiao, K. Liu, X. Bai, and J. Zhao, “Reconstruction of structured laser beams through a multimode fiber based on digital optical phase conjugation,” Opt. Lett. 43(14), 3333–3336 (2018).
[Crossref]

2017 (7)

2016 (4)

K. Lee and Y. Park, “Exploiting the speckle-correlation scattering matrix for a compact reference-free holographic image sensor,” Nat. Commun. 7(1), 13359 (2016).
[Crossref]

W. Xiong, P. Ambichl, Y. Bromberg, B. Redding, S. Rotter, and H. Cao, “Spatiotemporal Control of Light Transmission through a Multimode Fiber with Strong Mode Coupling,” Phys. Rev. Lett. 117(5), 053901 (2016).
[Crossref]

M. Azimipour, F. Atry, and R. Pashaie, “Calibration of digital optical phase conjugation setups based on orthonormal rectangular polynomials,” Appl. Opt. 55(11), 2873–2880 (2016).
[Crossref]

A. Yamilov, S. Petrenko, R. Sarma, and H. Cao, “Shape dependence of transmission, reflection, and absorption eigenvalue densities in disordered waveguides with dissipation,” Phys. Rev. B 93(10), 100201 (2016).
[Crossref]

2015 (8)

M. Kim, W. Choi, Y. Choi, C. Yoon, and W. Choi, “Transmission matrix of a scattering medium and its applications in biophotonics,” Opt. Express 23(10), 12648–12668 (2015).
[Crossref]

J. Yoon, K. Lee, J. Park, and Y. Park, “Measuring optical transmission matrices by wavefront shaping,” Opt. Express 23(8), 10158–10167 (2015).
[Crossref]

P. Lai, L. Wang, J. W. Tay, and L. V. Wang, “Photoacoustically guided wavefront shaping for enhanced optical focusing in scattering media,” Nat. Photonics 9(2), 126–132 (2015).
[Crossref]

D. Wang, E. H. Zhou, J. Brake, H. Ruan, M. Jang, and C. Yang, “Focusing through dynamic tissue with millisecond digital optical phase conjugation,” Optica 2(8), 728–735 (2015).
[Crossref]

H. Ruan, M. Jang, and C. Yang, “Optical focusing inside scattering media with time-reversed ultrasound microbubble encoded light,” Nat. Commun. 6(1), 8968 (2015).
[Crossref]

B. Judkewitz, R. Horstmeyer, I. M. Vellekoop, I. N. Papadopoulos, and C. Yang, “Translation correlations in anisotropically scattering media,” Nat. Phys. 11(8), 684–689 (2015).
[Crossref]

E. E. Morales-Delgado, S. Farahi, I. N. Papadopoulos, D. Psaltis, and C. Moser, “Delivery of focused short pulses through a multimode fiber,” Opt. Express 23(7), 9109–9120 (2015).
[Crossref]

A. Drémeau, A. Liutkus, D. Martina, O. Katz, C. Schülke, F. Krzakala, S. Gigan, and L. Daudet, “Reference-less measurement of the transmission matrix of a highly scattering material using a DMD and phase retrieval techniques,” Opt. Express 23(9), 11898–11911 (2015).
[Crossref]

2014 (3)

2013 (1)

T. R. Hillman, T. Yamauchi, W. Choi, R. R. Dasari, M. S. Feld, Y. Park, and Z. Yaqoob, “Digital optical phase conjugation for delivering two-dimensional images through turbid media,” Sci. Rep. 3(1), 1909 (2013).
[Crossref]

2012 (5)

Y. M. Wang, B. Judkewitz, C. A. DiMarzio, and C. Yang, “Deep-tissue focal fluorescence imaging with digitally time-reversed ultrasound-encoded light,” Nat. Commun. 3(1), 928 (2012).
[Crossref]

K. Si, R. Fiolka, and M. Cui, “Fluorescence imaging beyond the ballistic regime by ultrasound-pulse-guided digital phase conjugation,” Nat. Photonics 6(10), 657–661 (2012).
[Crossref]

D. B. Conkey, A. N. Brown, A. M. Caravaca-Aguirre, and R. Piestun, “Genetic algorithm optimization for focusing through turbid media in noisy environments,” Opt. Express 20(5), 4840–4849 (2012).
[Crossref]

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q.-H. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photonics 6(9), 581–585 (2012).
[Crossref]

D. B. Conkey, A. M. Caravaca-Aguirre, and R. Piestun, “High-speed scattering medium characterization with application to focusing light through turbid media,” Opt. Express 20(2), 1733–1740 (2012).
[Crossref]

2011 (2)

O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics 5(6), 372–377 (2011).
[Crossref]

Y. Chong and A. D. Stone, “Hidden black: Coherent enhancement of absorption in strongly scattering media,” Phys. Rev. Lett. 107(16), 163901 (2011).
[Crossref]

2010 (2)

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the Transmission Matrix in Optics: An Approach to the Study and Control of Light Propagation in Disordered Media,” Phys. Rev. Lett. 104(10), 100601 (2010).
[Crossref]

M. Cui and C. Yang, “Implementation of a digital optical phase conjugation system and its application to study the robustness of turbidity suppression by phase conjugation,” Opt. Express 18(4), 3444–3455 (2010).
[Crossref]

2008 (1)

I. M. Vellekoop and A. P. Mosk, “Phase control algorithms for focusing light through turbid media,” Opt. Commun. 281(11), 3071–3080 (2008).
[Crossref]

2007 (2)

Ambichl, P.

W. Xiong, P. Ambichl, Y. Bromberg, B. Redding, S. Rotter, and H. Cao, “Spatiotemporal Control of Light Transmission through a Multimode Fiber with Strong Mode Coupling,” Phys. Rev. Lett. 117(5), 053901 (2016).
[Crossref]

Atry, F.

Azimipour, M.

Bai, X.

Bishop, G.

G. Welch and G. Bishop, “An introduction to the Kalman filter,” (1995).

Blochet, B.

Boccara, A. C.

T. Chaigne, O. Katz, A. C. Boccara, M. Fink, E. Bossy, and S. Gigan, “Controlling light in scattering media non-invasively using the photoacoustic transmission matrix,” Nat. Photonics 8(1), 58–64 (2014).
[Crossref]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the Transmission Matrix in Optics: An Approach to the Study and Control of Light Propagation in Disordered Media,” Phys. Rev. Lett. 104(10), 100601 (2010).
[Crossref]

Borhani, N.

Bossy, E.

T. Chaigne, O. Katz, A. C. Boccara, M. Fink, E. Bossy, and S. Gigan, “Controlling light in scattering media non-invasively using the photoacoustic transmission matrix,” Nat. Photonics 8(1), 58–64 (2014).
[Crossref]

Bourdieu, L.

Brake, J.

Bromberg, Y.

W. Xiong, P. Ambichl, Y. Bromberg, B. Redding, S. Rotter, and H. Cao, “Spatiotemporal Control of Light Transmission through a Multimode Fiber with Strong Mode Coupling,” Phys. Rev. Lett. 117(5), 053901 (2016).
[Crossref]

O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics 5(6), 372–377 (2011).
[Crossref]

Brown, A. N.

Cao, H.

R. Sarma, A. Yamilov, and H. Cao, “Enhancing light transmission through a disordered waveguide with inhomogeneous scattering and loss,” Appl. Phys. Lett. 110(2), 021103 (2017).
[Crossref]

W. Xiong, P. Ambichl, Y. Bromberg, B. Redding, S. Rotter, and H. Cao, “Spatiotemporal Control of Light Transmission through a Multimode Fiber with Strong Mode Coupling,” Phys. Rev. Lett. 117(5), 053901 (2016).
[Crossref]

A. Yamilov, S. Petrenko, R. Sarma, and H. Cao, “Shape dependence of transmission, reflection, and absorption eigenvalue densities in disordered waveguides with dissipation,” Phys. Rev. B 93(10), 100201 (2016).
[Crossref]

Caravaca-Aguirre, A. M.

Carminati, R.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the Transmission Matrix in Optics: An Approach to the Study and Control of Light Propagation in Disordered Media,” Phys. Rev. Lett. 104(10), 100601 (2010).
[Crossref]

Chaigne, T.

T. Chaigne, O. Katz, A. C. Boccara, M. Fink, E. Bossy, and S. Gigan, “Controlling light in scattering media non-invasively using the photoacoustic transmission matrix,” Nat. Photonics 8(1), 58–64 (2014).
[Crossref]

Chen, S.-Y.

Chen, W.-H.

Chen, Y.-H.

Choi, W.

M. Kim, W. Choi, Y. Choi, C. Yoon, and W. Choi, “Transmission matrix of a scattering medium and its applications in biophotonics,” Opt. Express 23(10), 12648–12668 (2015).
[Crossref]

M. Kim, W. Choi, Y. Choi, C. Yoon, and W. Choi, “Transmission matrix of a scattering medium and its applications in biophotonics,” Opt. Express 23(10), 12648–12668 (2015).
[Crossref]

T. R. Hillman, T. Yamauchi, W. Choi, R. R. Dasari, M. S. Feld, Y. Park, and Z. Yaqoob, “Digital optical phase conjugation for delivering two-dimensional images through turbid media,” Sci. Rep. 3(1), 1909 (2013).
[Crossref]

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q.-H. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photonics 6(9), 581–585 (2012).
[Crossref]

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q.-H. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photonics 6(9), 581–585 (2012).
[Crossref]

Choi, Y.

M. Kim, W. Choi, Y. Choi, C. Yoon, and W. Choi, “Transmission matrix of a scattering medium and its applications in biophotonics,” Opt. Express 23(10), 12648–12668 (2015).
[Crossref]

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q.-H. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photonics 6(9), 581–585 (2012).
[Crossref]

Chong, Y.

Y. Chong and A. D. Stone, “Hidden black: Coherent enhancement of absorption in strongly scattering media,” Phys. Rev. Lett. 107(16), 163901 (2011).
[Crossref]

Conkey, D. B.

Cui, M.

K. Si, R. Fiolka, and M. Cui, “Fluorescence imaging beyond the ballistic regime by ultrasound-pulse-guided digital phase conjugation,” Nat. Photonics 6(10), 657–661 (2012).
[Crossref]

M. Cui and C. Yang, “Implementation of a digital optical phase conjugation system and its application to study the robustness of turbidity suppression by phase conjugation,” Opt. Express 18(4), 3444–3455 (2010).
[Crossref]

d Seelig, J.

Dasari, R. R.

T. R. Hillman, T. Yamauchi, W. Choi, R. R. Dasari, M. S. Feld, Y. Park, and Z. Yaqoob, “Digital optical phase conjugation for delivering two-dimensional images through turbid media,” Sci. Rep. 3(1), 1909 (2013).
[Crossref]

Daudet, L.

Deng, L.

Di, J.

DiMarzio, C. A.

Y. M. Wang, B. Judkewitz, C. A. DiMarzio, and C. Yang, “Deep-tissue focal fluorescence imaging with digitally time-reversed ultrasound-encoded light,” Nat. Commun. 3(1), 928 (2012).
[Crossref]

Ding, Y.

Drémeau, A.

Elson, D. S.

Estakhri, N. M.

M. N’Gom, M.-B. Lien, N. M. Estakhri, T. B. Norris, E. Michielssen, and R. R. Nadakuditi, “Controlling light transmission through highly scattering media using semi-definite programming as a phase retrieval computation method,” Sci. Rep. 7(1), 1–9 (2017).
[Crossref]

Fan, P.

Farahi, S.

Feld, M. S.

T. R. Hillman, T. Yamauchi, W. Choi, R. R. Dasari, M. S. Feld, Y. Park, and Z. Yaqoob, “Digital optical phase conjugation for delivering two-dimensional images through turbid media,” Sci. Rep. 3(1), 1909 (2013).
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K. Lee and Y. Park, “Exploiting the speckle-correlation scattering matrix for a compact reference-free holographic image sensor,” Nat. Commun. 7(1), 13359 (2016).
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I. M. Vellekoop and A. P. Mosk, “Phase control algorithms for focusing light through turbid media,” Opt. Commun. 281(11), 3071–3080 (2008).
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Z. Yu, J. Huangfu, F. Zhao, M. Xia, X. Wu, X. Niu, D. Li, P. Lai, and D. Wang, “Time-reversed magnetically controlled perturbation (TRMCP) optical focusing inside scattering media,” Sci. Rep. 8(1), 2927 (2018).
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A. Hemphill, Y. Shen, J. Hwang, and L. Wang, “High-speed alignment optimization of digital optical phase conjugation systems based on autocovariance analysis in conjunction with orthonormal rectangular polynomials,” J. Biomed. Opt. 24(03), 1 (2018).
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P. Lai, L. Wang, J. W. Tay, and L. V. Wang, “Photoacoustically guided wavefront shaping for enhanced optical focusing in scattering media,” Nat. Photonics 9(2), 126–132 (2015).
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Wang, W.

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Y. M. Wang, B. Judkewitz, C. A. DiMarzio, and C. Yang, “Deep-tissue focal fluorescence imaging with digitally time-reversed ultrasound-encoded light,” Nat. Commun. 3(1), 928 (2012).
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Z. Yu, J. Huangfu, F. Zhao, M. Xia, X. Wu, X. Niu, D. Li, P. Lai, and D. Wang, “Time-reversed magnetically controlled perturbation (TRMCP) optical focusing inside scattering media,” Sci. Rep. 8(1), 2927 (2018).
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W. Xiong, P. Ambichl, Y. Bromberg, B. Redding, S. Rotter, and H. Cao, “Spatiotemporal Control of Light Transmission through a Multimode Fiber with Strong Mode Coupling,” Phys. Rev. Lett. 117(5), 053901 (2016).
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R. Sarma, A. Yamilov, and H. Cao, “Enhancing light transmission through a disordered waveguide with inhomogeneous scattering and loss,” Appl. Phys. Lett. 110(2), 021103 (2017).
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A. Yamilov, S. Petrenko, R. Sarma, and H. Cao, “Shape dependence of transmission, reflection, and absorption eigenvalue densities in disordered waveguides with dissipation,” Phys. Rev. B 93(10), 100201 (2016).
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Yang, C.

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H. Ruan, M. Jang, and C. Yang, “Optical focusing inside scattering media with time-reversed ultrasound microbubble encoded light,” Nat. Commun. 6(1), 8968 (2015).
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M. Jang, H. Ruan, H. Zhou, B. Judkewitz, and C. Yang, “Method for auto-alignment of digital optical phase conjugation systems based on digital propagation,” Opt. Express 22(12), 14054–14071 (2014).
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E. H. Zhou, H. Ruan, C. Yang, and B. Judkewitz, “Focusing on moving targets through scattering samples,” Optica 1(4), 227–232 (2014).
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Y. M. Wang, B. Judkewitz, C. A. DiMarzio, and C. Yang, “Deep-tissue focal fluorescence imaging with digitally time-reversed ultrasound-encoded light,” Nat. Commun. 3(1), 928 (2012).
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Yang, T.-H.

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M. Kim, W. Choi, Y. Choi, C. Yoon, and W. Choi, “Transmission matrix of a scattering medium and its applications in biophotonics,” Opt. Express 23(10), 12648–12668 (2015).
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Yu, Y.-W.

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Z. Yu, J. Huangfu, F. Zhao, M. Xia, X. Wu, X. Niu, D. Li, P. Lai, and D. Wang, “Time-reversed magnetically controlled perturbation (TRMCP) optical focusing inside scattering media,” Sci. Rep. 8(1), 2927 (2018).
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Zhang, Y.

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Z. Yu, J. Huangfu, F. Zhao, M. Xia, X. Wu, X. Niu, D. Li, P. Lai, and D. Wang, “Time-reversed magnetically controlled perturbation (TRMCP) optical focusing inside scattering media,” Sci. Rep. 8(1), 2927 (2018).
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Appl. Opt. (1)

Appl. Phys. Lett. (1)

R. Sarma, A. Yamilov, and H. Cao, “Enhancing light transmission through a disordered waveguide with inhomogeneous scattering and loss,” Appl. Phys. Lett. 110(2), 021103 (2017).
[Crossref]

J. Biomed. Opt. (1)

A. Hemphill, Y. Shen, J. Hwang, and L. Wang, “High-speed alignment optimization of digital optical phase conjugation systems based on autocovariance analysis in conjunction with orthonormal rectangular polynomials,” J. Biomed. Opt. 24(03), 1 (2018).
[Crossref]

Nat. Commun. (3)

K. Lee and Y. Park, “Exploiting the speckle-correlation scattering matrix for a compact reference-free holographic image sensor,” Nat. Commun. 7(1), 13359 (2016).
[Crossref]

Y. M. Wang, B. Judkewitz, C. A. DiMarzio, and C. Yang, “Deep-tissue focal fluorescence imaging with digitally time-reversed ultrasound-encoded light,” Nat. Commun. 3(1), 928 (2012).
[Crossref]

H. Ruan, M. Jang, and C. Yang, “Optical focusing inside scattering media with time-reversed ultrasound microbubble encoded light,” Nat. Commun. 6(1), 8968 (2015).
[Crossref]

Nat. Photonics (5)

K. Si, R. Fiolka, and M. Cui, “Fluorescence imaging beyond the ballistic regime by ultrasound-pulse-guided digital phase conjugation,” Nat. Photonics 6(10), 657–661 (2012).
[Crossref]

P. Lai, L. Wang, J. W. Tay, and L. V. Wang, “Photoacoustically guided wavefront shaping for enhanced optical focusing in scattering media,” Nat. Photonics 9(2), 126–132 (2015).
[Crossref]

O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics 5(6), 372–377 (2011).
[Crossref]

T. Chaigne, O. Katz, A. C. Boccara, M. Fink, E. Bossy, and S. Gigan, “Controlling light in scattering media non-invasively using the photoacoustic transmission matrix,” Nat. Photonics 8(1), 58–64 (2014).
[Crossref]

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q.-H. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photonics 6(9), 581–585 (2012).
[Crossref]

Nat. Phys. (1)

B. Judkewitz, R. Horstmeyer, I. M. Vellekoop, I. N. Papadopoulos, and C. Yang, “Translation correlations in anisotropically scattering media,” Nat. Phys. 11(8), 684–689 (2015).
[Crossref]

Opt. Commun. (1)

I. M. Vellekoop and A. P. Mosk, “Phase control algorithms for focusing light through turbid media,” Opt. Commun. 281(11), 3071–3080 (2008).
[Crossref]

Opt. Express (18)

J. Xu, H. Ruan, Y. Liu, H. Zhou, and C. Yang, “Focusing light through scattering media by transmission matrix inversion,” Opt. Express 25(22), 27234–27246 (2017).
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Y. Huang, Y. Shen, C. Min, and G. Veronis, “Switching of the direction of reflectionless light propagation at exceptional points in non-PT-symmetric structures using phase-change materials,” Opt. Express 25(22), 27283 (2017).
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M. I. Mishchenko, L. Liu, and J. W. Hovenier, “Effects of absorption on multiple scattering by random particulate media: exact results,” Opt. Express 15(20), 13182–13187 (2007).
[Crossref]

M. Cui and C. Yang, “Implementation of a digital optical phase conjugation system and its application to study the robustness of turbidity suppression by phase conjugation,” Opt. Express 18(4), 3444–3455 (2010).
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D. B. Conkey, A. M. Caravaca-Aguirre, and R. Piestun, “High-speed scattering medium characterization with application to focusing light through turbid media,” Opt. Express 20(2), 1733–1740 (2012).
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D. B. Conkey, A. N. Brown, A. M. Caravaca-Aguirre, and R. Piestun, “Genetic algorithm optimization for focusing through turbid media in noisy environments,” Opt. Express 20(5), 4840–4849 (2012).
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M. Jang, H. Ruan, H. Zhou, B. Judkewitz, and C. Yang, “Method for auto-alignment of digital optical phase conjugation systems based on digital propagation,” Opt. Express 22(12), 14054–14071 (2014).
[Crossref]

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Figures (7)

Fig. 1.
Fig. 1. A flowchart of the EKF solver.
Fig. 2.
Fig. 2. Simulation results of the EKF-MSSM. (a) The enhancement as a function of $\gamma $, when the TM is retrieved by the EKF, the MSSM, and the EKF-MSSM, respectively. N = 100 is fixed. Error bar: standard error of 500 independent runs. (b) The enhancement as a function of the degrees of freedom N. $\gamma = 4$ is fixed. Error bar: standard error of 100 independent runs. (c) The histogram of the enhancement, based on the results from 500 independent runs with $\gamma = 4$. (d) The histogram of the number of iterations, based on the results from 500 independent runs with $\gamma = 4$.
Fig. 3.
Fig. 3. Quantification of the performance when $\gamma = 4$. (a) The computational time consumed by the EKF-MSSM as a function of the degrees of freedom N. Error bar: standard error of 100 independent runs. Two distinct data points are also shown to represent the performance of the SDP and the prVBEM. (b) Normalized enhancement (with respect to the theoretical value) as a function of the signal-to-noise ratio for the SDP (red) and the EKF-MSSM (blue). N = 100. Error bar: standard error of 200 independent runs.
Fig. 4.
Fig. 4. Quantification of the performance of the algorithms when the SLM can only modulate d discrete phases values. $\gamma = 4$, N = 100 are adopted to generate these results. Error bar: standard error of 200 independent runs. (a) Normalized enhancement (with respect to the theoretical value) as a function of d. (b) The computational time as a function of d.
Fig. 5.
Fig. 5. Schematics of the experiments. (a) The experimental procedure. 4N input trials with random phase maps were sent into the MMF and their corresponding 4N output speckle patterns were measured. (b) The experimental setup. BB1, BB2: beam block; BS: beam splitter; CCD: charge-coupled device; HWP: half-wave plate; L1, L2: lens; M: mirror; MMF: multimode fiber; OBJ1, OBJ2: objective lens; PBS: polarizing beam splitter; SLM: spatial light modulator; P: polarizer.
Fig. 6.
Fig. 6. Experimental demonstration of retrieving the TM using the EKF-MSSM. (a) A typical speckle pattern observed by the CCD at the output of the MMF. (b) Distributions of 2080 intensity measurements at pixel 1. The averaged intensity is denoted by the red solid line. After retrieving the TM, the intensity of the focus is denoted by a black dot encircled in red. (c) The CCD captured focus at pixel 1, when the TM was retrieved by the EKF-MSSM. $\eta \approx 286\; ({\textrm{P}1})$. Computational time: about 23 seconds. (d) The CCD captured focus at pixel 1, when the row of TM was retrieved by the MSSM. $\eta \approx 76\; ({\textrm{P}1^{\prime}} )$, Computational time: about 10 seconds.
Fig. 7.
Fig. 7. Focusing light through the MMF at different positions with the computed TM. (a)(b) The CCD captured images of the optical foci formed at pixel 2 and pixel 3, when the TM were retrieved by the EKF-MSSM. The enhancement of these two foci reach about 76% (P2) and 89% (P3) of the theoretical value. Computational time: about 21 seconds and 25 seconds, respectively. (c)(d) The CCD captured images of the optical foci formed at pixel 2 and pixel 3, when the TM were retrieved by the MSSM. The enhancement of these two foci reach about 17% (P2’) and 13% (P3’) of the theoretical value. Computational time: both around 10 seconds.

Equations (12)

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X k = X k 1 + w k 1 .
Y ^ k = C ^ k X k + v k ,
C ^ k = [ F ( 1 ) ( X ^ k ) x 1 F ( 1 ) ( X ^ k ) x N F ( M ) ( X ^ k ) x 1 F ( M ) ( X ^ k ) x N ] , Y ^ k = [ y ( 1 ) F ( 1 ) ( X ^ k ) y ( M ) F ( M ) ( X ^ k ) ] + C ^ k X ^ k .
E ( p ) = q = 1 N t ( p , q ) e i θ q .
y ( l ) = | E ( l ) | 2 = | q = 1 N [ a ( q ) + i b ( q ) ] e i θ q ( l ) | 2 , l = 1 , , M .
X ^ k = [ a ^ k ( 1 ) a ^ k ( N ) b ^ k ( 1 ) b ^ k ( N ) ] T ,
C ^ k = [ F ( 1 ) ( X ^ k ) a ( 1 ) F ( 1 ) ( X ^ k ) a ( N ) F ( M ) ( X ^ k ) a ( 1 ) F ( M ) ( X ^ k ) a ( N ) F ( 1 ) ( X ^ k ) b ( 1 ) F ( 1 ) ( X ^ k ) b ( N ) F ( M ) ( X ^ k ) b ( 1 ) F ( M ) ( X ^ k ) b ( N ) ] ,
F ( l ) ( X ^ k ) a ( q ) = 2 Re [ E ^ k ( l ) ] cos θ q ( l ) + 2 Im [ E ^ k ( l ) ] sin θ q ( l ) ,
F ( l ) ( X ^ k ) b ( q ) = 2 Im [ E ^ k ( l ) ] cos θ q ( l ) 2 Re [ E ^ k ( l ) ] sin θ q ( l ) .
Z m , n = P m P n y P m P n y | P m | 2 | P n | 2 ,
P = [ e i θ 1 ( 1 ) e i θ N ( 1 ) e i θ 1 ( M ) e i θ N ( M ) ] .
η = I foc / I avg .