Abstract

Transmission matrices (TMs) have become a powerful and widely used tool to describe and control wave propagation in complex media. In certain scenarios the TM is partially uncontrollable, complicating its identification and use. In standard optical wavefront shaping experiments, uncontrollable reflections or imperfect illumination may be the cause; in reverberating cavities, uncontrollable reflections off the walls have that effect. Here we employ phase retrieval techniques to identify such a partially uncontrollable TM solely based on random intensity-only reference measurements. We demonstrate the feasibility of our method by focusing both on a single target as well as on multiple targets in a microwave cavity, using a phase-binary Spatial-Microwave-Modulator.

© 2016 Optical Society of America

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References

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2015 (6)

P. Schniter and S. Rangan, “Compressive phase retrieval via generalized approximate message passing,” IEEE Trans. Signal Process. 63, 1043–1055 (2015).
[Crossref]

M. Dupré, P. del Hougne, M. Fink, F. Lemoult, and G. Lerosey, “Wave-field shaping in cavities: Waves trapped in a box with controllable boundaries,” Phys. Rev. Lett. 115, 017701 (2015).
[Crossref] [PubMed]

I. Waldspurger, A. d’Aspremont, and S. Mallat, “Phase recovery, maxcut and complex semidefinite programming,” Math. Prog. 149, 47–81 (2015).
[Crossref]

P. Netrapalli, P. Jain, and S. Sanghavi, “Phase retrieval using alternating minimization,” IEEE Trans. Signal Process. 63, 4814–4826 (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, 11898–11911 (2015).
[Crossref] [PubMed]

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, 12648–12668 (2015).
[Crossref] [PubMed]

2014 (5)

C. Jin, R. R. Nadakuditi, E. Michielssen, and S. C. Rand, “Backscatter analysis based algorithms for increasing transmission through highly scattering random media using phase-only-modulated wavefronts,” J. Opt. Soc. Am. A 31, 1788–1800 (2014).
[Crossref]

N. Kaina, M. Dupré, M. Fink, and G. Lerosey, “Hybridized resonances to design tunable binary phase metasurface unit cells,” Opt. Express 22, 18881–18888 (2014).
[Crossref] [PubMed]

B. Alexeev, A. Bandeira, M. Fickus, and D. Mixon, “Phase retrieval with polarization,” SIAM J. Imaging Sci. 7, 35–66 (2014).
[Crossref]

N. Kaina, M. Dupré, G. Lerosey, and M. Fink, “Shaping complex microwave fields in reverberating media with binary tunable metasurfaces,” Sci. Rep. 4, 6693 (2014).
[Crossref] [PubMed]

A. Liutkus, D. Martina, S. Popoff, G. Chardon, O. Katz, G. Lerosey, S. Gigan, L. Daudet, and I. Carron, “Imaging with nature: Compressive imaging using a multiply scattering medium,” Sci. Rep. 4, 5552 (2014).
[Crossref] [PubMed]

2013 (7)

J.-H. Park, C. Park, H. Yu, J. Park, S. Han, J. Shin, S. H. Ko, K. T. Nam, Y.-H. Cho, and Y. Park, “Subwavelength light focusing using random nanoparticles,” Nat. Photonics 7, 454–458 (2013).
[Crossref]

A. Goetschy and A. D. Stone, “Filtering random matrices: The effect of incomplete channel control in multiple scattering,” Phys. Rev. Lett. 111, 063901 (2013).
[Crossref] [PubMed]

H. Yu, T. R. Hillman, W. Choi, J. O. Lee, M. S. Feld, R. R. Dasari, and Y. Park, “Measuring large optical transmission matrices of disordered media,” Phys. Rev. Lett. 111, 153902 (2013).
[Crossref] [PubMed]

M. Nixon, O. Katz, E. Small, Y. Bromberg, A. A. Friesem, Y. Silberberg, and N. Davidson, “Real-time wavefront shaping through scattering media by all-optical feedback,” Nat. Photonics 7, 919–924 (2013).
[Crossref]

E. Candes, T. Strohmer, and V. Voroninski, “Phaselift: Exact and stable signal recovery from magnitude measurements via convex programming,” Commun. Pure Appl. Math. 66, 1241–1274 (2013).
[Crossref]

I. N. Papadopoulos, S. Farahi, C. Moser, and D. Psaltis, “High-resolution, lensless endoscope based on digital scanning through a multimode optical fiber,” Biomed. Opt. Express 4, 260–270 (2013).
[Crossref] [PubMed]

J. Jang, J. Lim, H. Yu, H. Choi, J. Ha, J.-H. Park, W.-Y. Oh, W. Jang, S. Lee, and Y. Park, “Complex wavefront shaping for optimal depth-selective focusing in optical coherence tomography,” Opt. Express 21, 2890–2902 (2013).
[Crossref] [PubMed]

2012 (4)

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, 581–585 (2012).
[Crossref]

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-free and wide-field endoscopic imaging by using a single multimode optical fiber,” Phys. Rev. Lett. 109, 203901 (2012).
[Crossref] [PubMed]

S. Bianchi and R. Di Leonardo, “A multi-mode fiber probe for holographic micromanipulation and microscopy,” Lab Chip 12, 635–639 (2012).
[Crossref]

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics 6, 283–292 (2012).
[Crossref]

2011 (2)

Y. Choi, T. D. Yang, C. Fang-Yen, P. Kang, K. J. Lee, R. R. Dasari, M. S. Feld, and W. Choi, “Overcoming the diffraction limit using multiple light scattering in a highly disordered medium,” Phys. Rev. Lett. 107, 023902 (2011).
[Crossref] [PubMed]

T. Ćižmár and K. Dholakia, “Shaping the light transmission through a multimode optical fibre: complex transformation analysis and applications in biophotonics,” Opt. Express 19, 18871–18884 (2011).
[Crossref] [PubMed]

2010 (3)

S. Popoff, G. Lerosey, R. Carminati, M. Fink, A. 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, 100601 (2010).
[Crossref] [PubMed]

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 81 (2010).
[Crossref] [PubMed]

I. Vellekoop, A. Lagendijk, and A. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322 (2010).

2008 (2)

I. M. Vellekoop and A. P. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett. 101, 120601 (2008).
[Crossref] [PubMed]

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

2007 (4)

S. Marchesini, “Invited article: A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. 78, 011301 (2007).
[Crossref]

O. Bunk, A. Diaz, F. Pfeiffer, C. David, B. Schmitt, D. K. Satapathy, and J. F. Veen, “Diffractive imaging for periodic samples: Retrieving one-dimensional concentration profiles across microfluidic channels,” Acta Crystallograph. Sect. A: Found. Crystallogr. 63, 306–314 (2007).
[Crossref]

I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32, 2309–2311 (2007).
[Crossref] [PubMed]

G. Lerosey, J. De Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-field time reversal,” Science 315, 1120–1122 (2007).
[Crossref] [PubMed]

2006 (1)

J. Corbett, “The pauli problem, state reconstruction and quantum-real numbers,” Rep. Math. Phys. 57, 53–68 (2006).
[Crossref]

2005 (1)

S. Hemmady, X. Zheng, T. M. Antonsen, E. Ott, and S. M. Anlage, “Universal statistics of the scattering coefficient of chaotic microwave cavities,” Phys. Rev. E 71, 056215 (2005).
[Crossref]

2004 (1)

G. Lerosey, J. De Rosny, A. Tourin, A. Derode, G. Montaldo, and M. Fink, “Time reversal of electromagnetic waves,” Phys. Rev. Lett. 92, 193904 (2004).
[Crossref] [PubMed]

2003 (3)

H. Cao, “Lasing in disordered media,” Prog. Optics 45, 317 (2003).
[Crossref]

J. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419–1421 (2003).
[Crossref] [PubMed]

S. Skipetrov, “Information transfer through disordered media by diffuse waves,” Phys. Rev. E 67, 036621 (2003).
[Crossref]

2002 (1)

1997 (1)

M. Fink, “Time reversed acoustics,” Phys. Today 50, 34–40 (1997).
[Crossref]

1993 (1)

1978 (1)

1972 (1)

R. Gerchberg, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237 (1972).

Alexeev, B.

B. Alexeev, A. Bandeira, M. Fickus, and D. Mixon, “Phase retrieval with polarization,” SIAM J. Imaging Sci. 7, 35–66 (2014).
[Crossref]

Anlage, S. M.

S. Hemmady, X. Zheng, T. M. Antonsen, E. Ott, and S. M. Anlage, “Universal statistics of the scattering coefficient of chaotic microwave cavities,” Phys. Rev. E 71, 056215 (2005).
[Crossref]

Antonsen, T. M.

S. Hemmady, X. Zheng, T. M. Antonsen, E. Ott, and S. M. Anlage, “Universal statistics of the scattering coefficient of chaotic microwave cavities,” Phys. Rev. E 71, 056215 (2005).
[Crossref]

Bandeira, A.

B. Alexeev, A. Bandeira, M. Fickus, and D. Mixon, “Phase retrieval with polarization,” SIAM J. Imaging Sci. 7, 35–66 (2014).
[Crossref]

Bauschke, H. H.

Bianchi, S.

S. Bianchi and R. Di Leonardo, “A multi-mode fiber probe for holographic micromanipulation and microscopy,” Lab Chip 12, 635–639 (2012).
[Crossref]

Boccara, A.

S. Popoff, G. Lerosey, R. Carminati, M. Fink, A. 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, 100601 (2010).
[Crossref] [PubMed]

Boccara, A. C.

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 81 (2010).
[Crossref] [PubMed]

Bromberg, Y.

M. Nixon, O. Katz, E. Small, Y. Bromberg, A. A. Friesem, Y. Silberberg, and N. Davidson, “Real-time wavefront shaping through scattering media by all-optical feedback,” Nat. Photonics 7, 919–924 (2013).
[Crossref]

Bunk, O.

O. Bunk, A. Diaz, F. Pfeiffer, C. David, B. Schmitt, D. K. Satapathy, and J. F. Veen, “Diffractive imaging for periodic samples: Retrieving one-dimensional concentration profiles across microfluidic channels,” Acta Crystallograph. Sect. A: Found. Crystallogr. 63, 306–314 (2007).
[Crossref]

Candes, E.

E. Candes, T. Strohmer, and V. Voroninski, “Phaselift: Exact and stable signal recovery from magnitude measurements via convex programming,” Commun. Pure Appl. Math. 66, 1241–1274 (2013).
[Crossref]

Cao, H.

H. Cao, “Lasing in disordered media,” Prog. Optics 45, 317 (2003).
[Crossref]

Carminati, R.

S. Popoff, G. Lerosey, R. Carminati, M. Fink, A. 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, 100601 (2010).
[Crossref] [PubMed]

Carron, I.

A. Liutkus, D. Martina, S. Popoff, G. Chardon, O. Katz, G. Lerosey, S. Gigan, L. Daudet, and I. Carron, “Imaging with nature: Compressive imaging using a multiply scattering medium,” Sci. Rep. 4, 5552 (2014).
[Crossref] [PubMed]

Chardon, G.

A. Liutkus, D. Martina, S. Popoff, G. Chardon, O. Katz, G. Lerosey, S. Gigan, L. Daudet, and I. Carron, “Imaging with nature: Compressive imaging using a multiply scattering medium,” Sci. Rep. 4, 5552 (2014).
[Crossref] [PubMed]

Cho, Y.-H.

J.-H. Park, C. Park, H. Yu, J. Park, S. Han, J. Shin, S. H. Ko, K. T. Nam, Y.-H. Cho, and Y. Park, “Subwavelength light focusing using random nanoparticles,” Nat. Photonics 7, 454–458 (2013).
[Crossref]

Choi, 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, 12648–12668 (2015).
[Crossref] [PubMed]

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, 12648–12668 (2015).
[Crossref] [PubMed]

H. Yu, T. R. Hillman, W. Choi, J. O. Lee, M. S. Feld, R. R. Dasari, and Y. Park, “Measuring large optical transmission matrices of disordered media,” Phys. Rev. Lett. 111, 153902 (2013).
[Crossref] [PubMed]

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-free and wide-field endoscopic imaging by using a single multimode optical fiber,” Phys. Rev. Lett. 109, 203901 (2012).
[Crossref] [PubMed]

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, 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, 581–585 (2012).
[Crossref]

Y. Choi, T. D. Yang, C. Fang-Yen, P. Kang, K. J. Lee, R. R. Dasari, M. S. Feld, and W. Choi, “Overcoming the diffraction limit using multiple light scattering in a highly disordered medium,” Phys. Rev. Lett. 107, 023902 (2011).
[Crossref] [PubMed]

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, 12648–12668 (2015).
[Crossref] [PubMed]

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, 581–585 (2012).
[Crossref]

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Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-free and wide-field endoscopic imaging by using a single multimode optical fiber,” Phys. Rev. Lett. 109, 203901 (2012).
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M. Dupré, P. del Hougne, M. Fink, F. Lemoult, and G. Lerosey, “Wave-field shaping in cavities: Waves trapped in a box with controllable boundaries,” Phys. Rev. Lett. 115, 017701 (2015).
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J. Jang, J. Lim, H. Yu, H. Choi, J. Ha, J.-H. Park, W.-Y. Oh, W. Jang, S. Lee, and Y. Park, “Complex wavefront shaping for optimal depth-selective focusing in optical coherence tomography,” Opt. Express 21, 2890–2902 (2013).
[Crossref] [PubMed]

J.-H. Park, C. Park, H. Yu, J. Park, S. Han, J. Shin, S. H. Ko, K. T. Nam, Y.-H. Cho, and Y. Park, “Subwavelength light focusing using random nanoparticles,” Nat. Photonics 7, 454–458 (2013).
[Crossref]

H. Yu, T. R. Hillman, W. Choi, J. O. Lee, M. S. Feld, R. R. Dasari, and Y. Park, “Measuring large optical transmission matrices of disordered media,” Phys. Rev. Lett. 111, 153902 (2013).
[Crossref] [PubMed]

Pfeiffer, F.

O. Bunk, A. Diaz, F. Pfeiffer, C. David, B. Schmitt, D. K. Satapathy, and J. F. Veen, “Diffractive imaging for periodic samples: Retrieving one-dimensional concentration profiles across microfluidic channels,” Acta Crystallograph. Sect. A: Found. Crystallogr. 63, 306–314 (2007).
[Crossref]

Popoff, S.

A. Liutkus, D. Martina, S. Popoff, G. Chardon, O. Katz, G. Lerosey, S. Gigan, L. Daudet, and I. Carron, “Imaging with nature: Compressive imaging using a multiply scattering medium,” Sci. Rep. 4, 5552 (2014).
[Crossref] [PubMed]

S. Popoff, G. Lerosey, R. Carminati, M. Fink, A. 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, 100601 (2010).
[Crossref] [PubMed]

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 81 (2010).
[Crossref] [PubMed]

Psaltis, D.

Rajaei, B.

B. Rajaei, E. W. Tramel, S. Gigan, F. Krzakala, and L. Daudet, “Intensity-only optical compressive imaging using a multiply scattering material: A double phase retrieval system,” in “Proc. IEEE Int. Conf. on Acoustics, Speech and Signal Processing (ICASSP),” (2016).

B. Rajaei, S. Gigan, F. Krzakala, and L. Daudet, “Robust phase retrieval with the swept approximate message passing (prsamp) algorithm,” arXiv preprint arXiv: 1605.07516 (2016).

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Rangan, S.

P. Schniter and S. Rangan, “Compressive phase retrieval via generalized approximate message passing,” IEEE Trans. Signal Process. 63, 1043–1055 (2015).
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P. Netrapalli, P. Jain, and S. Sanghavi, “Phase retrieval using alternating minimization,” IEEE Trans. Signal Process. 63, 4814–4826 (2015).
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Satapathy, D. K.

O. Bunk, A. Diaz, F. Pfeiffer, C. David, B. Schmitt, D. K. Satapathy, and J. F. Veen, “Diffractive imaging for periodic samples: Retrieving one-dimensional concentration profiles across microfluidic channels,” Acta Crystallograph. Sect. A: Found. Crystallogr. 63, 306–314 (2007).
[Crossref]

Schmitt, B.

O. Bunk, A. Diaz, F. Pfeiffer, C. David, B. Schmitt, D. K. Satapathy, and J. F. Veen, “Diffractive imaging for periodic samples: Retrieving one-dimensional concentration profiles across microfluidic channels,” Acta Crystallograph. Sect. A: Found. Crystallogr. 63, 306–314 (2007).
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P. Schniter and S. Rangan, “Compressive phase retrieval via generalized approximate message passing,” IEEE Trans. Signal Process. 63, 1043–1055 (2015).
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Shin, J.

J.-H. Park, C. Park, H. Yu, J. Park, S. Han, J. Shin, S. H. Ko, K. T. Nam, Y.-H. Cho, and Y. Park, “Subwavelength light focusing using random nanoparticles,” Nat. Photonics 7, 454–458 (2013).
[Crossref]

Silberberg, Y.

M. Nixon, O. Katz, E. Small, Y. Bromberg, A. A. Friesem, Y. Silberberg, and N. Davidson, “Real-time wavefront shaping through scattering media by all-optical feedback,” Nat. Photonics 7, 919–924 (2013).
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S. Skipetrov, “Information transfer through disordered media by diffuse waves,” Phys. Rev. E 67, 036621 (2003).
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M. Nixon, O. Katz, E. Small, Y. Bromberg, A. A. Friesem, Y. Silberberg, and N. Davidson, “Real-time wavefront shaping through scattering media by all-optical feedback,” Nat. Photonics 7, 919–924 (2013).
[Crossref]

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A. Goetschy and A. D. Stone, “Filtering random matrices: The effect of incomplete channel control in multiple scattering,” Phys. Rev. Lett. 111, 063901 (2013).
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E. Candes, T. Strohmer, and V. Voroninski, “Phaselift: Exact and stable signal recovery from magnitude measurements via convex programming,” Commun. Pure Appl. Math. 66, 1241–1274 (2013).
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Tourin, A.

G. Lerosey, J. De Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-field time reversal,” Science 315, 1120–1122 (2007).
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G. Lerosey, J. De Rosny, A. Tourin, A. Derode, G. Montaldo, and M. Fink, “Time reversal of electromagnetic waves,” Phys. Rev. Lett. 92, 193904 (2004).
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Tramel, E. W.

B. Rajaei, E. W. Tramel, S. Gigan, F. Krzakala, and L. Daudet, “Intensity-only optical compressive imaging using a multiply scattering material: A double phase retrieval system,” in “Proc. IEEE Int. Conf. on Acoustics, Speech and Signal Processing (ICASSP),” (2016).

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J. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419–1421 (2003).
[Crossref] [PubMed]

Veen, J. F.

O. Bunk, A. Diaz, F. Pfeiffer, C. David, B. Schmitt, D. K. Satapathy, and J. F. Veen, “Diffractive imaging for periodic samples: Retrieving one-dimensional concentration profiles across microfluidic channels,” Acta Crystallograph. Sect. A: Found. Crystallogr. 63, 306–314 (2007).
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I. Vellekoop, A. Lagendijk, and A. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322 (2010).

I. Vellekoop and A. Mosk, “Phase control algorithms for focusing light through turbid media,” Opt. Commun. 281, 3071–3080 (2008).
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I. M. Vellekoop and A. P. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett. 101, 120601 (2008).
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I. Waldspurger, A. d’Aspremont, and S. Mallat, “Phase recovery, maxcut and complex semidefinite programming,” Math. Prog. 149, 47–81 (2015).
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Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-free and wide-field endoscopic imaging by using a single multimode optical fiber,” Phys. Rev. Lett. 109, 203901 (2012).
[Crossref] [PubMed]

Y. Choi, T. D. Yang, C. Fang-Yen, P. Kang, K. J. Lee, R. R. Dasari, M. S. Feld, and W. Choi, “Overcoming the diffraction limit using multiple light scattering in a highly disordered medium,” Phys. Rev. Lett. 107, 023902 (2011).
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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, 12648–12668 (2015).
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Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-free and wide-field endoscopic imaging by using a single multimode optical fiber,” Phys. Rev. Lett. 109, 203901 (2012).
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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, 581–585 (2012).
[Crossref]

Yu, H.

J. Jang, J. Lim, H. Yu, H. Choi, J. Ha, J.-H. Park, W.-Y. Oh, W. Jang, S. Lee, and Y. Park, “Complex wavefront shaping for optimal depth-selective focusing in optical coherence tomography,” Opt. Express 21, 2890–2902 (2013).
[Crossref] [PubMed]

J.-H. Park, C. Park, H. Yu, J. Park, S. Han, J. Shin, S. H. Ko, K. T. Nam, Y.-H. Cho, and Y. Park, “Subwavelength light focusing using random nanoparticles,” Nat. Photonics 7, 454–458 (2013).
[Crossref]

H. Yu, T. R. Hillman, W. Choi, J. O. Lee, M. S. Feld, R. R. Dasari, and Y. Park, “Measuring large optical transmission matrices of disordered media,” Phys. Rev. Lett. 111, 153902 (2013).
[Crossref] [PubMed]

Zhang, R.

J. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419–1421 (2003).
[Crossref] [PubMed]

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J. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419–1421 (2003).
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Acta Crystallograph. Sect. A: Found. Crystallogr. (1)

O. Bunk, A. Diaz, F. Pfeiffer, C. David, B. Schmitt, D. K. Satapathy, and J. F. Veen, “Diffractive imaging for periodic samples: Retrieving one-dimensional concentration profiles across microfluidic channels,” Acta Crystallograph. Sect. A: Found. Crystallogr. 63, 306–314 (2007).
[Crossref]

Biomed. Opt. Express (1)

Commun. Pure Appl. Math. (1)

E. Candes, T. Strohmer, and V. Voroninski, “Phaselift: Exact and stable signal recovery from magnitude measurements via convex programming,” Commun. Pure Appl. Math. 66, 1241–1274 (2013).
[Crossref]

IEEE Trans. Signal Process. (2)

P. Netrapalli, P. Jain, and S. Sanghavi, “Phase retrieval using alternating minimization,” IEEE Trans. Signal Process. 63, 4814–4826 (2015).
[Crossref]

P. Schniter and S. Rangan, “Compressive phase retrieval via generalized approximate message passing,” IEEE Trans. Signal Process. 63, 1043–1055 (2015).
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I. Waldspurger, A. d’Aspremont, and S. Mallat, “Phase recovery, maxcut and complex semidefinite programming,” Math. Prog. 149, 47–81 (2015).
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Nat. Commun. (1)

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 81 (2010).
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Nat. Photonics (5)

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, 581–585 (2012).
[Crossref]

I. Vellekoop, A. Lagendijk, and A. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322 (2010).

J.-H. Park, C. Park, H. Yu, J. Park, S. Han, J. Shin, S. H. Ko, K. T. Nam, Y.-H. Cho, and Y. Park, “Subwavelength light focusing using random nanoparticles,” Nat. Photonics 7, 454–458 (2013).
[Crossref]

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics 6, 283–292 (2012).
[Crossref]

M. Nixon, O. Katz, E. Small, Y. Bromberg, A. A. Friesem, Y. Silberberg, and N. Davidson, “Real-time wavefront shaping through scattering media by all-optical feedback,” Nat. Photonics 7, 919–924 (2013).
[Crossref]

Opt. Commun. (1)

I. Vellekoop and A. Mosk, “Phase control algorithms for focusing light through turbid media,” Opt. Commun. 281, 3071–3080 (2008).
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S. Hemmady, X. Zheng, T. M. Antonsen, E. Ott, and S. M. Anlage, “Universal statistics of the scattering coefficient of chaotic microwave cavities,” Phys. Rev. E 71, 056215 (2005).
[Crossref]

S. Skipetrov, “Information transfer through disordered media by diffuse waves,” Phys. Rev. E 67, 036621 (2003).
[Crossref]

Phys. Rev. Lett. (8)

S. Popoff, G. Lerosey, R. Carminati, M. Fink, A. 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, 100601 (2010).
[Crossref] [PubMed]

Y. Choi, T. D. Yang, C. Fang-Yen, P. Kang, K. J. Lee, R. R. Dasari, M. S. Feld, and W. Choi, “Overcoming the diffraction limit using multiple light scattering in a highly disordered medium,” Phys. Rev. Lett. 107, 023902 (2011).
[Crossref] [PubMed]

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-free and wide-field endoscopic imaging by using a single multimode optical fiber,” Phys. Rev. Lett. 109, 203901 (2012).
[Crossref] [PubMed]

G. Lerosey, J. De Rosny, A. Tourin, A. Derode, G. Montaldo, and M. Fink, “Time reversal of electromagnetic waves,” Phys. Rev. Lett. 92, 193904 (2004).
[Crossref] [PubMed]

I. M. Vellekoop and A. P. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett. 101, 120601 (2008).
[Crossref] [PubMed]

A. Goetschy and A. D. Stone, “Filtering random matrices: The effect of incomplete channel control in multiple scattering,” Phys. Rev. Lett. 111, 063901 (2013).
[Crossref] [PubMed]

H. Yu, T. R. Hillman, W. Choi, J. O. Lee, M. S. Feld, R. R. Dasari, and Y. Park, “Measuring large optical transmission matrices of disordered media,” Phys. Rev. Lett. 111, 153902 (2013).
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N. Kaina, M. Dupré, G. Lerosey, and M. Fink, “Shaping complex microwave fields in reverberating media with binary tunable metasurfaces,” Sci. Rep. 4, 6693 (2014).
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A. Liutkus, D. Martina, S. Popoff, G. Chardon, O. Katz, G. Lerosey, S. Gigan, L. Daudet, and I. Carron, “Imaging with nature: Compressive imaging using a multiply scattering medium,” Sci. Rep. 4, 5552 (2014).
[Crossref] [PubMed]

Science (2)

G. Lerosey, J. De Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-field time reversal,” Science 315, 1120–1122 (2007).
[Crossref] [PubMed]

J. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419–1421 (2003).
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B. Rajaei, E. W. Tramel, S. Gigan, F. Krzakala, and L. Daudet, “Intensity-only optical compressive imaging using a multiply scattering material: A double phase retrieval system,” in “Proc. IEEE Int. Conf. on Acoustics, Speech and Signal Processing (ICASSP),” (2016).

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts and Company Publishers, 2007).

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Figures (5)

Fig. 1
Fig. 1

Concept of wave focusing with a partially controllable TM: A wavefront originating from an array of sources with C controllable and U uncontrollable elements propagates linearly through a complex medium and is probed by an array of N receivers. Initially each input mode has a random phase, such that their sum at one of the receivers constitutes a random walk in the Fresnel plane. To maximize the amplitude of the sum of all modes at one output position, (binary) phase alignment of the controllable modes is carried out. Note the importance of aligning the controllable phasors such that their sum has the same phase θ as the sum of all the uncontrollable modes in (g).

Fig. 2
Fig. 2

Experimental scheme: A binary phase modulation SMM (see inset for details, adapted from [14]) partially covers the walls of a metallic cavity. Electromagnetic absorbers, isotropically distributed, reduce the cavity’s quality factor. A mode-stirrer rotation of 12° creates a statistically independent, “new” disordered cavity. This enables many realizations of disorder for averaging. We emphasize that only intensity information from the network analyzer is used.

Fig. 3
Fig. 3

Closed-loop iterative focusing on a single target: The spectra before (black) and after (orange) optimization are shown, both for a single realization (top row) and averaged over 90 realizations (bottom row). Moreover, the corresponding optimization dynamics are displayed, that is how the focusing progresses with each iteration in terms of the target intensity enhancement ηE. The dashed line indicates the iteration after which all 102 SMM pixels have been tested once.

Fig. 4
Fig. 4

Comparison of TM-based identification of optimum input configuration for focusing with the experimental closed-loop iterative scheme (orange), as a function of the number of available reference measurements K for the phase retrieval algorithm that computes the TM. Note that both TM-based approaches yielded identical results. The left hand side shows the averaged spectra corresponding to the focusing results based on K reference measurements.

Fig. 5
Fig. 5

TM-based focusing on multiple targets (K = 800). The spectra averaged over realizations of disorder and the ntargets targets after focusing on the left, and the corresponding enhancement averaged over all targets on the right. The best fit with the corresponding equation is indicated.

Equations (9)

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

Y = HX .
Y i = j H i , j X j .
η E = | Y | 2 f i n a l | Y | 2 i n i t i a l .
Y T = X T H H .
Y = H c X c + H u X u .
Y = H c X c + H u X u = H c X c + H u X u + H u X u .
X c o p t = sign [ Re ( H c 1 Y o b j ) ] = sign [ Re ( H c 1 ) ] .
D = | A | Y i | | p t a r g e t s .
η E ( n t a r g e t s ) = ( η A ( n t a r g e t s ) ) 2 = ( 1 + η A ( 1 ) 1 n t a r g e t s ) 2 ,

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