Abstract

Wavefront shaping allows for ultimate control of light propagation in multiple-scattering media by adaptive manipulation of incident waves. We shine two separate wavefront-shaped beams on a layer of dry white paint to create two enhanced output spots of equal intensity. We experimentally confirm by interference measurements that the output spots are almost correlated like the two outputs of an ideal balanced beam splitter. The observed deviations from the phase behavior of an ideal beam splitter are analyzed with a transmission matrix model. Our experiments demonstrate that wavefront shaping in multiple-scattering media can be used to approximate the functionality of linear optical devices with multiple inputs and outputs.

© 2014 Optical Society of America

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  1. E. Hecht, Optics (Addison Wesley, 4, 2002).
  2. B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley-Interscience, 2, 2007).
  3. M. A. Nielsen, I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 1, 2000).
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    [CrossRef] [PubMed]
  5. P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, G. J. Millburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
    [CrossRef]
  6. J. L. O’Brien, A. Furusawa, J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009).
    [CrossRef]
  7. I. Freund, “Looking through walls and around corners,” Physica A 168, 49–65 (1990).
    [CrossRef]
  8. I. M. Vellekoop, A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Optics Lett. 32, 2309–2311 (2007).
    [CrossRef]
  9. A. P. Mosk, A. Lagendijk, G. Lerosey, M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics 6, 283–292 (2012).
    [CrossRef]
  10. I. M. Vellekoop, A. Lagendijk, A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322 (2010).
    [CrossRef]
  11. E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, A. P. Mosk, “Scattering lens resolves sub-100 nm structures with visible light,” Phys. Rev. Lett. 106, 193905 (2011).
    [CrossRef] [PubMed]
  12. J. Aulbach, B. Gjonaj, P. M. Johnson, A. P. Mosk, A. Lagendijk, “Control of light transmission through opaque scattering media in space and time,” Phys. Rev. Lett. 106, 103901 (2011).
    [CrossRef] [PubMed]
  13. O. Katz, E. Small, Y. Bromberg, Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics 5, 372–377 (2011).
    [CrossRef]
  14. D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, B. Chatel, “Spatio-temporal focussing of an ultrafast pulse through a multiply scattering medium,” Nat. Comm. 2, 447 (2011).
    [CrossRef]
  15. Y. Guan, O. Katz, E. Small, J. Zhou, Y. Silberberg, “Polarization control of multiply scattered light through random media by wavefront shaping,” Opt. Lett. 37, 4663–4665 (2012).
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  16. S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, 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]
  17. R. A. Campos, B. E. A. Saleh, M. C. Teich, “Quantum-mechanic lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A 40, 1371–1384 (1989).
    [CrossRef] [PubMed]
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  21. L. Knöll, S. Scheel, E. Schmidt, D. -G. Welsch, A.V. Chizhov, “Quantum-state transformation by dispersive and absorbing four-port devices,” Phys. Rev. A 59, 4716–4726 (1999).
    [CrossRef]
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    [CrossRef]
  23. I. M. Vellekoop, A. P. Mosk, “Phase control algorithms for focusing light through turbid media,” Opt. Commun. 281, 3071–3080 (2008).
    [CrossRef]
  24. We use Matlab 2013 for creating random unitary matrices S with dimension Dim(S) using the following syntax: Random_Matrix=rand(N,N)+1i*rand(N,N); %generates uniformly distributed random complex numbers [S, r]=qr(Random Matrix); % S is the output of an orthogonal-triangular decomposition.
  25. N. P. Puente, E. I. Chaikina, S. Herath, A. Yamilov, “Fabrication, characterization and theoretical analysis of controlled disorder in the core of the optical fibers,” Appl. Optics 50, 802–810 (2011).
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  26. S. R. Huisman, N. Jain, S. A. Babichev, F. Vewinger, A. N. Zhang, S. H. Youn, A. I. Lvovsky, “Instant single-photon Fock state tomography,” Opt. Lett. 34, 2739–2741 (2009).
    [CrossRef] [PubMed]
  27. T. J. Huisman, S. R. Huisman, A. P. Mosk, P. W. H. Pinkse, “Controlling single-photon Fock-state propagation through opaque scattering materials,” Appl. Phys. B. (2013).
    [CrossRef]
  28. P. Lodahl, A. P. Mosk, A. Lagendijk, “Spatial quantum correlations in multiple scattered light,” Phys. Rev. Lett. 95, 173901 (2005).
    [CrossRef] [PubMed]
  29. S. Smolka, A. Huck, U. L. Andersen, A. Lagendijk, P. Lodahl, “Observation of spatial quantum correlations induced by multiple scattering of nonclassical light,” Phys. Rev. Lett. 102, 193901 (2009).
    [CrossRef] [PubMed]
  30. Y. Bromberg, Y. Lahini, R. Morandotti, Y. Silberberg, “Quantum and classical correlations in waveguide lattices,” Phys. Rev. Lett. 102, 253904 (2009).
    [CrossRef] [PubMed]
  31. A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. -Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, J. L. O’Brien, “Quantum walks of correlated photons,” Science 329, 1500–1503 (2010).
    [CrossRef] [PubMed]
  32. Y. Lahini, Y. Bromberg, D. N. Christodoulides, Y. Silberberg, “Quantum correlations in Anderson localization of indistinguishable particles,” Phys. Rev. Lett. 105, 163905 (2010).
    [CrossRef]
  33. J. R. Ott, N. A. Mortensen, P. Lodahl, “Quantum interference and entanglement induced by multiple scattering of light,” Phys. Rev. Lett. 105, 090501 (2010).
    [CrossRef] [PubMed]
  34. W. H. Peeters, J. J. D. Moerman, M. P. van Exter, “Observation of two-photon speckle patterns,” Phys. Rev. Lett. 104, 173601 (2010).
    [CrossRef] [PubMed]
  35. D. Bonneau, M. Lobino, P. Jiang, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, M. G. Thompson, J. L. O’Brien, “Fast path and polarization manipulation of telecom wavelength single photons in lithium niobate waveguide devices,” Phys. Rev. Lett. 108, 053601 (2012).
    [CrossRef] [PubMed]
  36. S. R. Huisman, Light Control with Ordered and Disordered Nanophotonic Media (PhD. thesis, University of Twente, 2013).

2012 (3)

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

Y. Guan, O. Katz, E. Small, J. Zhou, Y. Silberberg, “Polarization control of multiply scattered light through random media by wavefront shaping,” Opt. Lett. 37, 4663–4665 (2012).
[CrossRef] [PubMed]

D. Bonneau, M. Lobino, P. Jiang, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, M. G. Thompson, J. L. O’Brien, “Fast path and polarization manipulation of telecom wavelength single photons in lithium niobate waveguide devices,” Phys. Rev. Lett. 108, 053601 (2012).
[CrossRef] [PubMed]

2011 (5)

N. P. Puente, E. I. Chaikina, S. Herath, A. Yamilov, “Fabrication, characterization and theoretical analysis of controlled disorder in the core of the optical fibers,” Appl. Optics 50, 802–810 (2011).
[CrossRef]

E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, A. P. Mosk, “Scattering lens resolves sub-100 nm structures with visible light,” Phys. Rev. Lett. 106, 193905 (2011).
[CrossRef] [PubMed]

J. Aulbach, B. Gjonaj, P. M. Johnson, A. P. Mosk, A. Lagendijk, “Control of light transmission through opaque scattering media in space and time,” Phys. Rev. Lett. 106, 103901 (2011).
[CrossRef] [PubMed]

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

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, B. Chatel, “Spatio-temporal focussing of an ultrafast pulse through a multiply scattering medium,” Nat. Comm. 2, 447 (2011).
[CrossRef]

2010 (6)

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, 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]

I. M. Vellekoop, A. Lagendijk, A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322 (2010).
[CrossRef]

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. -Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, J. L. O’Brien, “Quantum walks of correlated photons,” Science 329, 1500–1503 (2010).
[CrossRef] [PubMed]

Y. Lahini, Y. Bromberg, D. N. Christodoulides, Y. Silberberg, “Quantum correlations in Anderson localization of indistinguishable particles,” Phys. Rev. Lett. 105, 163905 (2010).
[CrossRef]

J. R. Ott, N. A. Mortensen, P. Lodahl, “Quantum interference and entanglement induced by multiple scattering of light,” Phys. Rev. Lett. 105, 090501 (2010).
[CrossRef] [PubMed]

W. H. Peeters, J. J. D. Moerman, M. P. van Exter, “Observation of two-photon speckle patterns,” Phys. Rev. Lett. 104, 173601 (2010).
[CrossRef] [PubMed]

2009 (4)

S. R. Huisman, N. Jain, S. A. Babichev, F. Vewinger, A. N. Zhang, S. H. Youn, A. I. Lvovsky, “Instant single-photon Fock state tomography,” Opt. Lett. 34, 2739–2741 (2009).
[CrossRef] [PubMed]

S. Smolka, A. Huck, U. L. Andersen, A. Lagendijk, P. Lodahl, “Observation of spatial quantum correlations induced by multiple scattering of nonclassical light,” Phys. Rev. Lett. 102, 193901 (2009).
[CrossRef] [PubMed]

Y. Bromberg, Y. Lahini, R. Morandotti, Y. Silberberg, “Quantum and classical correlations in waveguide lattices,” Phys. Rev. Lett. 102, 253904 (2009).
[CrossRef] [PubMed]

J. L. O’Brien, A. Furusawa, J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009).
[CrossRef]

2008 (1)

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

2007 (2)

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, G. J. Millburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[CrossRef]

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

2005 (1)

P. Lodahl, A. P. Mosk, A. Lagendijk, “Spatial quantum correlations in multiple scattered light,” Phys. Rev. Lett. 95, 173901 (2005).
[CrossRef] [PubMed]

2001 (1)

E. Knill, R. Laflamme, G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46 (2001).
[CrossRef] [PubMed]

2000 (1)

J. Jeffers, “Interference and the lossless lossy beam splitter,” J. Mod. Optic. 47, 1819–1824 (2000).
[CrossRef]

1999 (1)

L. Knöll, S. Scheel, E. Schmidt, D. -G. Welsch, A.V. Chizhov, “Quantum-state transformation by dispersive and absorbing four-port devices,” Phys. Rev. A 59, 4716–4726 (1999).
[CrossRef]

1998 (1)

S. M. Barnett, J. Jeffers, A. Gatti, R. Loudon, “Quantum optics of lossy beam splitters,” Phys. Rev. A 57, 2134–2145 (1998).
[CrossRef]

1995 (1)

T. Kiss, U. Herzog, U. Leonhardt, “Compensation of losses in photodetection and in quantum-state measurements,” Phys. Rev. A 52, 2433–2435 (1995).
[CrossRef] [PubMed]

1990 (1)

I. Freund, “Looking through walls and around corners,” Physica A 168, 49–65 (1990).
[CrossRef]

1989 (1)

R. A. Campos, B. E. A. Saleh, M. C. Teich, “Quantum-mechanic lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A 40, 1371–1384 (1989).
[CrossRef] [PubMed]

Akbulut, D.

E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, A. P. Mosk, “Scattering lens resolves sub-100 nm structures with visible light,” Phys. Rev. Lett. 106, 193905 (2011).
[CrossRef] [PubMed]

Andersen, U. L.

S. Smolka, A. Huck, U. L. Andersen, A. Lagendijk, P. Lodahl, “Observation of spatial quantum correlations induced by multiple scattering of nonclassical light,” Phys. Rev. Lett. 102, 193901 (2009).
[CrossRef] [PubMed]

Aulbach, J.

J. Aulbach, B. Gjonaj, P. M. Johnson, A. P. Mosk, A. Lagendijk, “Control of light transmission through opaque scattering media in space and time,” Phys. Rev. Lett. 106, 103901 (2011).
[CrossRef] [PubMed]

Austin, D. R.

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, B. Chatel, “Spatio-temporal focussing of an ultrafast pulse through a multiply scattering medium,” Nat. Comm. 2, 447 (2011).
[CrossRef]

Babichev, S. A.

Barnett, S. M.

S. M. Barnett, J. Jeffers, A. Gatti, R. Loudon, “Quantum optics of lossy beam splitters,” Phys. Rev. A 57, 2134–2145 (1998).
[CrossRef]

Bertolotti, J.

E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, A. P. Mosk, “Scattering lens resolves sub-100 nm structures with visible light,” Phys. Rev. Lett. 106, 193905 (2011).
[CrossRef] [PubMed]

Boccara, A. C.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, 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]

Bondareff, P.

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, B. Chatel, “Spatio-temporal focussing of an ultrafast pulse through a multiply scattering medium,” Nat. Comm. 2, 447 (2011).
[CrossRef]

Bonneau, D.

D. Bonneau, M. Lobino, P. Jiang, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, M. G. Thompson, J. L. O’Brien, “Fast path and polarization manipulation of telecom wavelength single photons in lithium niobate waveguide devices,” Phys. Rev. Lett. 108, 053601 (2012).
[CrossRef] [PubMed]

Bromberg, Y.

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

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. -Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, J. L. O’Brien, “Quantum walks of correlated photons,” Science 329, 1500–1503 (2010).
[CrossRef] [PubMed]

Y. Lahini, Y. Bromberg, D. N. Christodoulides, Y. Silberberg, “Quantum correlations in Anderson localization of indistinguishable particles,” Phys. Rev. Lett. 105, 163905 (2010).
[CrossRef]

Y. Bromberg, Y. Lahini, R. Morandotti, Y. Silberberg, “Quantum and classical correlations in waveguide lattices,” Phys. Rev. Lett. 102, 253904 (2009).
[CrossRef] [PubMed]

Campos, R. A.

R. A. Campos, B. E. A. Saleh, M. C. Teich, “Quantum-mechanic lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A 40, 1371–1384 (1989).
[CrossRef] [PubMed]

Carminati, R.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, 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]

Chaikina, E. I.

N. P. Puente, E. I. Chaikina, S. Herath, A. Yamilov, “Fabrication, characterization and theoretical analysis of controlled disorder in the core of the optical fibers,” Appl. Optics 50, 802–810 (2011).
[CrossRef]

Chatel, B.

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, B. Chatel, “Spatio-temporal focussing of an ultrafast pulse through a multiply scattering medium,” Nat. Comm. 2, 447 (2011).
[CrossRef]

Chizhov, A.V.

L. Knöll, S. Scheel, E. Schmidt, D. -G. Welsch, A.V. Chizhov, “Quantum-state transformation by dispersive and absorbing four-port devices,” Phys. Rev. A 59, 4716–4726 (1999).
[CrossRef]

Christodoulides, D. N.

Y. Lahini, Y. Bromberg, D. N. Christodoulides, Y. Silberberg, “Quantum correlations in Anderson localization of indistinguishable particles,” Phys. Rev. Lett. 105, 163905 (2010).
[CrossRef]

Chuang, I. L.

M. A. Nielsen, I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 1, 2000).

Dorenbos, S. N.

D. Bonneau, M. Lobino, P. Jiang, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, M. G. Thompson, J. L. O’Brien, “Fast path and polarization manipulation of telecom wavelength single photons in lithium niobate waveguide devices,” Phys. Rev. Lett. 108, 053601 (2012).
[CrossRef] [PubMed]

Dowling, J. P.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, G. J. Millburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[CrossRef]

Fink, M.

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

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, 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]

Freund, I.

I. Freund, “Looking through walls and around corners,” Physica A 168, 49–65 (1990).
[CrossRef]

Furusawa, A.

J. L. O’Brien, A. Furusawa, J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009).
[CrossRef]

Gatti, A.

S. M. Barnett, J. Jeffers, A. Gatti, R. Loudon, “Quantum optics of lossy beam splitters,” Phys. Rev. A 57, 2134–2145 (1998).
[CrossRef]

Gigan, S.

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, B. Chatel, “Spatio-temporal focussing of an ultrafast pulse through a multiply scattering medium,” Nat. Comm. 2, 447 (2011).
[CrossRef]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, 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]

Gjonaj, B.

J. Aulbach, B. Gjonaj, P. M. Johnson, A. P. Mosk, A. Lagendijk, “Control of light transmission through opaque scattering media in space and time,” Phys. Rev. Lett. 106, 103901 (2011).
[CrossRef] [PubMed]

Guan, Y.

Hadfield, R. H.

D. Bonneau, M. Lobino, P. Jiang, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, M. G. Thompson, J. L. O’Brien, “Fast path and polarization manipulation of telecom wavelength single photons in lithium niobate waveguide devices,” Phys. Rev. Lett. 108, 053601 (2012).
[CrossRef] [PubMed]

Hecht, E.

E. Hecht, Optics (Addison Wesley, 4, 2002).

Herath, S.

N. P. Puente, E. I. Chaikina, S. Herath, A. Yamilov, “Fabrication, characterization and theoretical analysis of controlled disorder in the core of the optical fibers,” Appl. Optics 50, 802–810 (2011).
[CrossRef]

Herzog, U.

T. Kiss, U. Herzog, U. Leonhardt, “Compensation of losses in photodetection and in quantum-state measurements,” Phys. Rev. A 52, 2433–2435 (1995).
[CrossRef] [PubMed]

Huck, A.

S. Smolka, A. Huck, U. L. Andersen, A. Lagendijk, P. Lodahl, “Observation of spatial quantum correlations induced by multiple scattering of nonclassical light,” Phys. Rev. Lett. 102, 193901 (2009).
[CrossRef] [PubMed]

Huisman, S. R.

S. R. Huisman, N. Jain, S. A. Babichev, F. Vewinger, A. N. Zhang, S. H. Youn, A. I. Lvovsky, “Instant single-photon Fock state tomography,” Opt. Lett. 34, 2739–2741 (2009).
[CrossRef] [PubMed]

T. J. Huisman, S. R. Huisman, A. P. Mosk, P. W. H. Pinkse, “Controlling single-photon Fock-state propagation through opaque scattering materials,” Appl. Phys. B. (2013).
[CrossRef]

S. R. Huisman, Light Control with Ordered and Disordered Nanophotonic Media (PhD. thesis, University of Twente, 2013).

Huisman, T. J.

T. J. Huisman, S. R. Huisman, A. P. Mosk, P. W. H. Pinkse, “Controlling single-photon Fock-state propagation through opaque scattering materials,” Appl. Phys. B. (2013).
[CrossRef]

Ismail, N.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. -Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, J. L. O’Brien, “Quantum walks of correlated photons,” Science 329, 1500–1503 (2010).
[CrossRef] [PubMed]

Jain, N.

Jeffers, J.

J. Jeffers, “Interference and the lossless lossy beam splitter,” J. Mod. Optic. 47, 1819–1824 (2000).
[CrossRef]

S. M. Barnett, J. Jeffers, A. Gatti, R. Loudon, “Quantum optics of lossy beam splitters,” Phys. Rev. A 57, 2134–2145 (1998).
[CrossRef]

Jiang, P.

D. Bonneau, M. Lobino, P. Jiang, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, M. G. Thompson, J. L. O’Brien, “Fast path and polarization manipulation of telecom wavelength single photons in lithium niobate waveguide devices,” Phys. Rev. Lett. 108, 053601 (2012).
[CrossRef] [PubMed]

Johnson, P. M.

J. Aulbach, B. Gjonaj, P. M. Johnson, A. P. Mosk, A. Lagendijk, “Control of light transmission through opaque scattering media in space and time,” Phys. Rev. Lett. 106, 103901 (2011).
[CrossRef] [PubMed]

Katz, O.

Y. Guan, O. Katz, E. Small, J. Zhou, Y. Silberberg, “Polarization control of multiply scattered light through random media by wavefront shaping,” Opt. Lett. 37, 4663–4665 (2012).
[CrossRef] [PubMed]

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

Kiss, T.

T. Kiss, U. Herzog, U. Leonhardt, “Compensation of losses in photodetection and in quantum-state measurements,” Phys. Rev. A 52, 2433–2435 (1995).
[CrossRef] [PubMed]

Knill, E.

E. Knill, R. Laflamme, G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46 (2001).
[CrossRef] [PubMed]

Knöll, L.

L. Knöll, S. Scheel, E. Schmidt, D. -G. Welsch, A.V. Chizhov, “Quantum-state transformation by dispersive and absorbing four-port devices,” Phys. Rev. A 59, 4716–4726 (1999).
[CrossRef]

Kok, P.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, G. J. Millburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[CrossRef]

Laflamme, R.

E. Knill, R. Laflamme, G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46 (2001).
[CrossRef] [PubMed]

Lagendijk, A.

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

E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, A. P. Mosk, “Scattering lens resolves sub-100 nm structures with visible light,” Phys. Rev. Lett. 106, 193905 (2011).
[CrossRef] [PubMed]

J. Aulbach, B. Gjonaj, P. M. Johnson, A. P. Mosk, A. Lagendijk, “Control of light transmission through opaque scattering media in space and time,” Phys. Rev. Lett. 106, 103901 (2011).
[CrossRef] [PubMed]

I. M. Vellekoop, A. Lagendijk, A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322 (2010).
[CrossRef]

S. Smolka, A. Huck, U. L. Andersen, A. Lagendijk, P. Lodahl, “Observation of spatial quantum correlations induced by multiple scattering of nonclassical light,” Phys. Rev. Lett. 102, 193901 (2009).
[CrossRef] [PubMed]

P. Lodahl, A. P. Mosk, A. Lagendijk, “Spatial quantum correlations in multiple scattered light,” Phys. Rev. Lett. 95, 173901 (2005).
[CrossRef] [PubMed]

Lahini, Y.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. -Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, J. L. O’Brien, “Quantum walks of correlated photons,” Science 329, 1500–1503 (2010).
[CrossRef] [PubMed]

Y. Lahini, Y. Bromberg, D. N. Christodoulides, Y. Silberberg, “Quantum correlations in Anderson localization of indistinguishable particles,” Phys. Rev. Lett. 105, 163905 (2010).
[CrossRef]

Y. Bromberg, Y. Lahini, R. Morandotti, Y. Silberberg, “Quantum and classical correlations in waveguide lattices,” Phys. Rev. Lett. 102, 253904 (2009).
[CrossRef] [PubMed]

Leonhardt, U.

T. Kiss, U. Herzog, U. Leonhardt, “Compensation of losses in photodetection and in quantum-state measurements,” Phys. Rev. A 52, 2433–2435 (1995).
[CrossRef] [PubMed]

U. Leonhardt, Measuring the Quantum State of Light (Cambridge University, 1, 1997).

Lerosey, G.

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

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, 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]

Lobino, M.

D. Bonneau, M. Lobino, P. Jiang, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, M. G. Thompson, J. L. O’Brien, “Fast path and polarization manipulation of telecom wavelength single photons in lithium niobate waveguide devices,” Phys. Rev. Lett. 108, 053601 (2012).
[CrossRef] [PubMed]

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. -Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, J. L. O’Brien, “Quantum walks of correlated photons,” Science 329, 1500–1503 (2010).
[CrossRef] [PubMed]

Lodahl, P.

J. R. Ott, N. A. Mortensen, P. Lodahl, “Quantum interference and entanglement induced by multiple scattering of light,” Phys. Rev. Lett. 105, 090501 (2010).
[CrossRef] [PubMed]

S. Smolka, A. Huck, U. L. Andersen, A. Lagendijk, P. Lodahl, “Observation of spatial quantum correlations induced by multiple scattering of nonclassical light,” Phys. Rev. Lett. 102, 193901 (2009).
[CrossRef] [PubMed]

P. Lodahl, A. P. Mosk, A. Lagendijk, “Spatial quantum correlations in multiple scattered light,” Phys. Rev. Lett. 95, 173901 (2005).
[CrossRef] [PubMed]

Loudon, R.

S. M. Barnett, J. Jeffers, A. Gatti, R. Loudon, “Quantum optics of lossy beam splitters,” Phys. Rev. A 57, 2134–2145 (1998).
[CrossRef]

Lvovsky, A. I.

Matsuda, N.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. -Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, J. L. O’Brien, “Quantum walks of correlated photons,” Science 329, 1500–1503 (2010).
[CrossRef] [PubMed]

Matthews, J. C. F.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. -Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, J. L. O’Brien, “Quantum walks of correlated photons,” Science 329, 1500–1503 (2010).
[CrossRef] [PubMed]

McCabe, D. J.

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, B. Chatel, “Spatio-temporal focussing of an ultrafast pulse through a multiply scattering medium,” Nat. Comm. 2, 447 (2011).
[CrossRef]

Milburn, G. J.

E. Knill, R. Laflamme, G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46 (2001).
[CrossRef] [PubMed]

Millburn, G. J.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, G. J. Millburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[CrossRef]

Moerman, J. J. D.

W. H. Peeters, J. J. D. Moerman, M. P. van Exter, “Observation of two-photon speckle patterns,” Phys. Rev. Lett. 104, 173601 (2010).
[CrossRef] [PubMed]

Morandotti, R.

Y. Bromberg, Y. Lahini, R. Morandotti, Y. Silberberg, “Quantum and classical correlations in waveguide lattices,” Phys. Rev. Lett. 102, 253904 (2009).
[CrossRef] [PubMed]

Mortensen, N. A.

J. R. Ott, N. A. Mortensen, P. Lodahl, “Quantum interference and entanglement induced by multiple scattering of light,” Phys. Rev. Lett. 105, 090501 (2010).
[CrossRef] [PubMed]

Mosk, A. P.

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

E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, A. P. Mosk, “Scattering lens resolves sub-100 nm structures with visible light,” Phys. Rev. Lett. 106, 193905 (2011).
[CrossRef] [PubMed]

J. Aulbach, B. Gjonaj, P. M. Johnson, A. P. Mosk, A. Lagendijk, “Control of light transmission through opaque scattering media in space and time,” Phys. Rev. Lett. 106, 103901 (2011).
[CrossRef] [PubMed]

I. M. Vellekoop, A. Lagendijk, A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322 (2010).
[CrossRef]

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

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

P. Lodahl, A. P. Mosk, A. Lagendijk, “Spatial quantum correlations in multiple scattered light,” Phys. Rev. Lett. 95, 173901 (2005).
[CrossRef] [PubMed]

T. J. Huisman, S. R. Huisman, A. P. Mosk, P. W. H. Pinkse, “Controlling single-photon Fock-state propagation through opaque scattering materials,” Appl. Phys. B. (2013).
[CrossRef]

Munro, W. J.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, G. J. Millburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[CrossRef]

Natarajan, C. M.

D. Bonneau, M. Lobino, P. Jiang, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, M. G. Thompson, J. L. O’Brien, “Fast path and polarization manipulation of telecom wavelength single photons in lithium niobate waveguide devices,” Phys. Rev. Lett. 108, 053601 (2012).
[CrossRef] [PubMed]

Nemoto, K.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, G. J. Millburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[CrossRef]

Nielsen, M. A.

M. A. Nielsen, I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 1, 2000).

O’Brien, J. L.

D. Bonneau, M. Lobino, P. Jiang, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, M. G. Thompson, J. L. O’Brien, “Fast path and polarization manipulation of telecom wavelength single photons in lithium niobate waveguide devices,” Phys. Rev. Lett. 108, 053601 (2012).
[CrossRef] [PubMed]

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. -Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, J. L. O’Brien, “Quantum walks of correlated photons,” Science 329, 1500–1503 (2010).
[CrossRef] [PubMed]

J. L. O’Brien, A. Furusawa, J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009).
[CrossRef]

Ott, J. R.

J. R. Ott, N. A. Mortensen, P. Lodahl, “Quantum interference and entanglement induced by multiple scattering of light,” Phys. Rev. Lett. 105, 090501 (2010).
[CrossRef] [PubMed]

Peeters, W. H.

W. H. Peeters, J. J. D. Moerman, M. P. van Exter, “Observation of two-photon speckle patterns,” Phys. Rev. Lett. 104, 173601 (2010).
[CrossRef] [PubMed]

Peruzzo, A.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. -Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, J. L. O’Brien, “Quantum walks of correlated photons,” Science 329, 1500–1503 (2010).
[CrossRef] [PubMed]

Pinkse, P. W. H.

T. J. Huisman, S. R. Huisman, A. P. Mosk, P. W. H. Pinkse, “Controlling single-photon Fock-state propagation through opaque scattering materials,” Appl. Phys. B. (2013).
[CrossRef]

Politi, A.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. -Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, J. L. O’Brien, “Quantum walks of correlated photons,” Science 329, 1500–1503 (2010).
[CrossRef] [PubMed]

Popoff, S. M.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, 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]

Poulios, K.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. -Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, J. L. O’Brien, “Quantum walks of correlated photons,” Science 329, 1500–1503 (2010).
[CrossRef] [PubMed]

Puente, N. P.

N. P. Puente, E. I. Chaikina, S. Herath, A. Yamilov, “Fabrication, characterization and theoretical analysis of controlled disorder in the core of the optical fibers,” Appl. Optics 50, 802–810 (2011).
[CrossRef]

Ralph, T. C.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, G. J. Millburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[CrossRef]

Saleh, B. E. A.

R. A. Campos, B. E. A. Saleh, M. C. Teich, “Quantum-mechanic lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A 40, 1371–1384 (1989).
[CrossRef] [PubMed]

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley-Interscience, 2, 2007).

Scheel, S.

L. Knöll, S. Scheel, E. Schmidt, D. -G. Welsch, A.V. Chizhov, “Quantum-state transformation by dispersive and absorbing four-port devices,” Phys. Rev. A 59, 4716–4726 (1999).
[CrossRef]

Schmidt, E.

L. Knöll, S. Scheel, E. Schmidt, D. -G. Welsch, A.V. Chizhov, “Quantum-state transformation by dispersive and absorbing four-port devices,” Phys. Rev. A 59, 4716–4726 (1999).
[CrossRef]

Silberberg, Y.

Y. Guan, O. Katz, E. Small, J. Zhou, Y. Silberberg, “Polarization control of multiply scattered light through random media by wavefront shaping,” Opt. Lett. 37, 4663–4665 (2012).
[CrossRef] [PubMed]

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

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. -Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, J. L. O’Brien, “Quantum walks of correlated photons,” Science 329, 1500–1503 (2010).
[CrossRef] [PubMed]

Y. Lahini, Y. Bromberg, D. N. Christodoulides, Y. Silberberg, “Quantum correlations in Anderson localization of indistinguishable particles,” Phys. Rev. Lett. 105, 163905 (2010).
[CrossRef]

Y. Bromberg, Y. Lahini, R. Morandotti, Y. Silberberg, “Quantum and classical correlations in waveguide lattices,” Phys. Rev. Lett. 102, 253904 (2009).
[CrossRef] [PubMed]

Small, E.

Y. Guan, O. Katz, E. Small, J. Zhou, Y. Silberberg, “Polarization control of multiply scattered light through random media by wavefront shaping,” Opt. Lett. 37, 4663–4665 (2012).
[CrossRef] [PubMed]

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

Smolka, S.

S. Smolka, A. Huck, U. L. Andersen, A. Lagendijk, P. Lodahl, “Observation of spatial quantum correlations induced by multiple scattering of nonclassical light,” Phys. Rev. Lett. 102, 193901 (2009).
[CrossRef] [PubMed]

Tajalli, A.

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, B. Chatel, “Spatio-temporal focussing of an ultrafast pulse through a multiply scattering medium,” Nat. Comm. 2, 447 (2011).
[CrossRef]

Tanner, M. G.

D. Bonneau, M. Lobino, P. Jiang, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, M. G. Thompson, J. L. O’Brien, “Fast path and polarization manipulation of telecom wavelength single photons in lithium niobate waveguide devices,” Phys. Rev. Lett. 108, 053601 (2012).
[CrossRef] [PubMed]

Teich, M. C.

R. A. Campos, B. E. A. Saleh, M. C. Teich, “Quantum-mechanic lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A 40, 1371–1384 (1989).
[CrossRef] [PubMed]

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley-Interscience, 2, 2007).

Thompson, M. G.

D. Bonneau, M. Lobino, P. Jiang, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, M. G. Thompson, J. L. O’Brien, “Fast path and polarization manipulation of telecom wavelength single photons in lithium niobate waveguide devices,” Phys. Rev. Lett. 108, 053601 (2012).
[CrossRef] [PubMed]

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. -Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, J. L. O’Brien, “Quantum walks of correlated photons,” Science 329, 1500–1503 (2010).
[CrossRef] [PubMed]

van Exter, M. P.

W. H. Peeters, J. J. D. Moerman, M. P. van Exter, “Observation of two-photon speckle patterns,” Phys. Rev. Lett. 104, 173601 (2010).
[CrossRef] [PubMed]

van Putten, E. G.

E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, A. P. Mosk, “Scattering lens resolves sub-100 nm structures with visible light,” Phys. Rev. Lett. 106, 193905 (2011).
[CrossRef] [PubMed]

Vellekoop, I. M.

I. M. Vellekoop, A. Lagendijk, A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322 (2010).
[CrossRef]

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

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

Vewinger, F.

Vos, W. L.

E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, A. P. Mosk, “Scattering lens resolves sub-100 nm structures with visible light,” Phys. Rev. Lett. 106, 193905 (2011).
[CrossRef] [PubMed]

Vuckovic, J.

J. L. O’Brien, A. Furusawa, J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009).
[CrossRef]

Walmsley, I. A.

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, B. Chatel, “Spatio-temporal focussing of an ultrafast pulse through a multiply scattering medium,” Nat. Comm. 2, 447 (2011).
[CrossRef]

Welsch, D. -G.

L. Knöll, S. Scheel, E. Schmidt, D. -G. Welsch, A.V. Chizhov, “Quantum-state transformation by dispersive and absorbing four-port devices,” Phys. Rev. A 59, 4716–4726 (1999).
[CrossRef]

Wörhoff, K.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. -Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, J. L. O’Brien, “Quantum walks of correlated photons,” Science 329, 1500–1503 (2010).
[CrossRef] [PubMed]

Yamilov, A.

N. P. Puente, E. I. Chaikina, S. Herath, A. Yamilov, “Fabrication, characterization and theoretical analysis of controlled disorder in the core of the optical fibers,” Appl. Optics 50, 802–810 (2011).
[CrossRef]

Youn, S. H.

Zhang, A. N.

Zhou, J.

Zhou, X. -Q.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. -Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, J. L. O’Brien, “Quantum walks of correlated photons,” Science 329, 1500–1503 (2010).
[CrossRef] [PubMed]

Zwiller, V.

D. Bonneau, M. Lobino, P. Jiang, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, M. G. Thompson, J. L. O’Brien, “Fast path and polarization manipulation of telecom wavelength single photons in lithium niobate waveguide devices,” Phys. Rev. Lett. 108, 053601 (2012).
[CrossRef] [PubMed]

Appl. Optics (1)

N. P. Puente, E. I. Chaikina, S. Herath, A. Yamilov, “Fabrication, characterization and theoretical analysis of controlled disorder in the core of the optical fibers,” Appl. Optics 50, 802–810 (2011).
[CrossRef]

J. Mod. Optic. (1)

J. Jeffers, “Interference and the lossless lossy beam splitter,” J. Mod. Optic. 47, 1819–1824 (2000).
[CrossRef]

Nat. Comm. (1)

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, B. Chatel, “Spatio-temporal focussing of an ultrafast pulse through a multiply scattering medium,” Nat. Comm. 2, 447 (2011).
[CrossRef]

Nat. Photonics (4)

J. L. O’Brien, A. Furusawa, J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009).
[CrossRef]

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

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

I. M. Vellekoop, A. Lagendijk, A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322 (2010).
[CrossRef]

Nature (1)

E. Knill, R. Laflamme, G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46 (2001).
[CrossRef] [PubMed]

Opt. Commun. (1)

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

Opt. Lett. (2)

Optics Lett. (1)

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

Phys. Rev. A (4)

R. A. Campos, B. E. A. Saleh, M. C. Teich, “Quantum-mechanic lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A 40, 1371–1384 (1989).
[CrossRef] [PubMed]

T. Kiss, U. Herzog, U. Leonhardt, “Compensation of losses in photodetection and in quantum-state measurements,” Phys. Rev. A 52, 2433–2435 (1995).
[CrossRef] [PubMed]

S. M. Barnett, J. Jeffers, A. Gatti, R. Loudon, “Quantum optics of lossy beam splitters,” Phys. Rev. A 57, 2134–2145 (1998).
[CrossRef]

L. Knöll, S. Scheel, E. Schmidt, D. -G. Welsch, A.V. Chizhov, “Quantum-state transformation by dispersive and absorbing four-port devices,” Phys. Rev. A 59, 4716–4726 (1999).
[CrossRef]

Phys. Rev. Lett. (10)

P. Lodahl, A. P. Mosk, A. Lagendijk, “Spatial quantum correlations in multiple scattered light,” Phys. Rev. Lett. 95, 173901 (2005).
[CrossRef] [PubMed]

S. Smolka, A. Huck, U. L. Andersen, A. Lagendijk, P. Lodahl, “Observation of spatial quantum correlations induced by multiple scattering of nonclassical light,” Phys. Rev. Lett. 102, 193901 (2009).
[CrossRef] [PubMed]

Y. Bromberg, Y. Lahini, R. Morandotti, Y. Silberberg, “Quantum and classical correlations in waveguide lattices,” Phys. Rev. Lett. 102, 253904 (2009).
[CrossRef] [PubMed]

Y. Lahini, Y. Bromberg, D. N. Christodoulides, Y. Silberberg, “Quantum correlations in Anderson localization of indistinguishable particles,” Phys. Rev. Lett. 105, 163905 (2010).
[CrossRef]

J. R. Ott, N. A. Mortensen, P. Lodahl, “Quantum interference and entanglement induced by multiple scattering of light,” Phys. Rev. Lett. 105, 090501 (2010).
[CrossRef] [PubMed]

W. H. Peeters, J. J. D. Moerman, M. P. van Exter, “Observation of two-photon speckle patterns,” Phys. Rev. Lett. 104, 173601 (2010).
[CrossRef] [PubMed]

D. Bonneau, M. Lobino, P. Jiang, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, M. G. Thompson, J. L. O’Brien, “Fast path and polarization manipulation of telecom wavelength single photons in lithium niobate waveguide devices,” Phys. Rev. Lett. 108, 053601 (2012).
[CrossRef] [PubMed]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, 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]

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Other (7)

We use Matlab 2013 for creating random unitary matrices S with dimension Dim(S) using the following syntax: Random_Matrix=rand(N,N)+1i*rand(N,N); %generates uniformly distributed random complex numbers [S, r]=qr(Random Matrix); % S is the output of an orthogonal-triangular decomposition.

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[CrossRef]

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U. Leonhardt, Measuring the Quantum State of Light (Cambridge University, 1, 1997).

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Figures (8)

Fig. 1
Fig. 1

Interference at an ideal lossless 50:50 beam splitter. (a) Two incident modes (1, 2) with equal intensity interfere at a 50:50 beam splitter (BS) resulting in two output modes (1′, 2′). (b) Normalized power in the output modes as a function of relative phase shift Δθ applied to incident mode 1. The powers in the two arms oscillate as a function of Δθ with a relative phase δ = π for the ideal beam splitter. All powers are normalized to the sum of the two input powers P1 + P2.

Fig. 2
Fig. 2

The allowed phase difference δ as a function of reflectivity |r|2 for the balanced beam splitter. The white region is forbidden because of energy conservation.

Fig. 3
Fig. 3

Optimization procedure for wavefront-shaped balanced beam splitters. A single SLM is divided into two sections to phase modulate incident modes 1 and 2 that are spatially separated. (a) Only mode 1 is incident and mode 2 is blocked. Two target spots 1′ and 2′ are optimized on a CCD camera. (b) Only mode 2 is incident and mode 1 is blocked, and the same target spots 1′ and 2′ are optimized. (c) Both modes 1 and 2 are incident on the SLM, using both optimized phase patterns. A relative phase differenced Δθ between mode 1 and 2 is applied with the SLM to confirm optical interference like in Fig. 1(b).

Fig. 4
Fig. 4

Setup for wavefront-shaped balanced beam splitters. (a) Two incident modes (1, 2) are phase-modulated with a spatial light modulator (SLM). Both modes are spatially overlapped with a polarizing beam splitter cube (PBS). The modes are focused on a layer of white paint (ZnO particles) that has been spray coated on a 1 mm thick microscope slide. The transmitted light is projected on a CCD camera. Two output modes 1′ and 2′ are selected. (b) Camera image for two optimized spots when mode 1 is blocked. (c) Optimized phase pattern on the SLM.

Fig. 5
Fig. 5

Interference between two optimized spots of a wavefront-shaped beam splitter. A phase shift is applied on one of the incident modes by applying an offset to the phase pattern of the SLM. (a) Camera images for different phase offsets Δθ expressed as gray values. (b) Power in the two optimized spots (red squares and blue diamonds) as a function of gray value (top) and phase (bottom). The solid and dashed lines are sinusoidal fits. When only one incident mode is present, a constant power in both target spots is observed (white diamonds and white squares).

Fig. 6
Fig. 6

Measured reduced phase difference |δ| for different realizations of the wavefront-shaped balanced beam splitter.

Fig. 7
Fig. 7

Calculated normalized probability distribution for the phase difference δ for random scattering matrices S with different dimension Dim(S) containing balanced lossy beam splitters. All 4 elements in the scattering matrix that form the beam splitter have an equal amplitude within 1% to be accepted. The green dashed line represents the boundary of allowed and forbidden phases as in Fig. 2. The black vertical dotted line represents 〈|sa,b|2〉, which is equivalent to an intensity enhancement of η = 1.

Fig. 8
Fig. 8

Simulated phase difference |δ| for the balanced beam splitter for different intensity enhancements η and Dim(S) = 300 (bars). The probability that δπ increases with η. (a) Realizations with 7.5 ≤ η < 8.5. (b) Realizations with 9.5 ≤ η < 10.5. (c) Realizations with 11.5 ≤ η < 12.5. (d) Realizations with 13.5 ≤ η < 14.5.

Equations (7)

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S = [ e i Ψ 2 0 0 e i Ψ 2 ] [ cos Θ 2 sin Θ 2 sin Θ 2 cos Θ 2 ] [ e i Δ θ 2 0 0 e i Δ θ 2 ] .
T = 1 N [ 1 e i δ 1 1 ] .
λ 1 = 1 + e i δ 2 N , λ 2 = 1 e i δ 2 N .
τ 1 2 + τ 2 2 = 4 N .
| s i j | = c 1 | s n , j | = c 2 | s i , m | = c 3 | s n , m |
S = 1 Dim ( S ) [ η η e i α in s 1 , 3 s 1 , Dim ( S ) η e i δ e i α out η e i α in e i α out s 3 , 1 s Dim ( S ) , 1 s Dim ( S ) , Dim ( S ) ] ,
η ( 1 + e i δ ) e i α in + i = 3 i = Dim ( S ) s i , 1 s i , 2 * = 0

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