Abstract

We propose and experimentally verify a method to program the effective transmission matrix of general multiport linear optical circuits in random multiple-scattering materials by phase modulation of incident wavefronts. We demonstrate the power of our method by programming linear optical circuits in white paint layers with 2 inputs and 2 outputs, and 2 inputs and 3 outputs. Using interferometric techniques we verify our ability to program any desired phase relation between the outputs. The method works in a deterministic manner and can be directly applied to existing wavefront-shaping setups without the need of measuring a transmission matrix or to rely on sensitive interference measurements.

© 2015 Optical Society of America

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References

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  5. S. W. Leonard, H. M. van Driel, J. Schilling, and R. B. Wehrspohn, “Ultrafast band-edge tuning of a two-dimensional silicon photonic crystal via free-carrier injection,” Phys. Rev. B. 66, 161102 (2002).
    [Crossref]
  6. W. Zhang, K. Aljasem, H. Zappe, and A. Seifert, “Completely integrated, thermo-pneumatically tunable microlens,” Opt. Express 19, 2347–2362 (2011).
    [Crossref] [PubMed]
  7. P. J. Shadbolt, M. R. Verde, A. Peruzzo, A. Politi, A. Laing, M. Lobino, J. C. F. Matthews, M. G. Thompson, and J. L. O’Brien, “Generating, manipulating and measuring entanglement and mixture with a reconfigurable photonic circuit,” Nat. Photonics 6, 45–49 (2012).
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    [Crossref] [PubMed]
  9. E. Yüce, G. Ctistis, J. Claudon, E. Dupuy, K. J. Boller, J. M. Gérard, and W. L. Vos, “Competition between electronic Kerr and free-carrier effects in an ultimate-fast optically switched semiconductor microcavity,” J. Opt. Soc. Am. B 29, 2630–2642 (2012).
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    [Crossref]
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    [Crossref]
  13. E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, and A. P. Mosk, “Scattering lens resolves sub-100 nm structures with visible light,” Phys. Rev. Lett. 106, 193905 (2011).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  15. O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics 5, 372–377 (2011).
    [Crossref]
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    [Crossref] [PubMed]
  17. S. R. Huisman, T. J. Huisman, S. A. Goorden, A. P. Mosk, and P. W. H. Pinkse, “Programming balanced optical beam splitters in white paint,” Opt. Express 22, 8320–8332 (2014).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
  20. I. M. Vellekoop and A. P. Mosk, “Phase control algorithms for focusing light through turbid media,” Opt. Commun. 281, 3071–3080 (2008).
    [Crossref]
  21. H. Yilmaz, W. L. Vos, and A. P. Mosk, “Optimal control of light propagation through multiple-scattering media in the presence of noise,” Biomed. Opt. Express 4, 1759–1768 (2013).
    [Crossref] [PubMed]
  22. P. Sheng, Introduction to wave scattering, localization, and mesoscopic phenomena, 1st Edition (Academic, 1995).
  23. E. Akkermans and G. Montambaux, Mesoscopic physics of electrons and photons, 1st Edition (Cambridge University, 2007).
    [Crossref]
  24. I. M. Vellekoop and A. P. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett. 101, 120601 (2008).
    [Crossref] [PubMed]
  25. R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantum-mechanic lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A 40, 1371–1384 (1989).
    [Crossref] [PubMed]
  26. T. J. Huisman, S. R. Huisman, A. P. Mosk, and P. W. H. Pinkse, “Controlling single-photon Fock-state propagation through opaque scattering materials,” Appl. Phys. B 116, 603–607 (2014).
    [Crossref]
  27. S. A. Goorden, M. Horstmann, A. P. Mosk, B. Skoric, and P. W. H. Pinkse, “Quantum-secure authentication of a physical unclonable key,” Optica 1, 421–424 (2014).
    [Crossref]
  28. S. A. Goorden, J. Bertolotti, and A. P. Mosk, “Superpixel-based spatial amplitude and phase modulation using a digital micromirror device,” Opt. Express 22, 17999–18009 (2014).
    [Crossref] [PubMed]

2014 (4)

2013 (2)

2012 (5)

P. J. Shadbolt, M. R. Verde, A. Peruzzo, A. Politi, A. Laing, M. Lobino, J. C. F. Matthews, M. G. Thompson, and J. L. O’Brien, “Generating, manipulating and measuring entanglement and mixture with a reconfigurable photonic circuit,” Nat. Photonics 6, 45–49 (2012).
[Crossref]

D. Bonneau, M. Lobino, P. Jiang, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, M. G. Thompson, and 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]

E. Yüce, G. Ctistis, J. Claudon, E. Dupuy, K. J. Boller, J. M. Gérard, and W. L. Vos, “Competition between electronic Kerr and free-carrier effects in an ultimate-fast optically switched semiconductor microcavity,” J. Opt. Soc. Am. B 29, 2630–2642 (2012).
[Crossref]

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

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 (4)

E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, and 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, and 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, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics 5, 372–377 (2011).
[Crossref]

W. Zhang, K. Aljasem, H. Zappe, and A. Seifert, “Completely integrated, thermo-pneumatically tunable microlens,” Opt. Express 19, 2347–2362 (2011).
[Crossref] [PubMed]

2010 (1)

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, 100601 (2010).
[Crossref] [PubMed]

2008 (2)

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

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

2007 (1)

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

2002 (1)

S. W. Leonard, H. M. van Driel, J. Schilling, and R. B. Wehrspohn, “Ultrafast band-edge tuning of a two-dimensional silicon photonic crystal via free-carrier injection,” Phys. Rev. B. 66, 161102 (2002).
[Crossref]

1995 (1)

L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13, 615–627 (1995).
[Crossref]

1994 (1)

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
[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, and 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, and A. P. Mosk, “Scattering lens resolves sub-100 nm structures with visible light,” Phys. Rev. Lett. 106, 193905 (2011).
[Crossref] [PubMed]

Akkermans, E.

E. Akkermans and G. Montambaux, Mesoscopic physics of electrons and photons, 1st Edition (Cambridge University, 2007).
[Crossref]

Aljasem, K.

Aulbach, J.

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

Bernstein, H. J.

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
[Crossref] [PubMed]

Bertani, P.

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
[Crossref] [PubMed]

Bertolotti, J.

S. A. Goorden, J. Bertolotti, and A. P. Mosk, “Superpixel-based spatial amplitude and phase modulation using a digital micromirror device,” Opt. Express 22, 17999–18009 (2014).
[Crossref] [PubMed]

E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, and 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, 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]

Boller, K. J.

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

Campos, R. A.

R. A. Campos, B. E. A. Saleh, and 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, 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]

Claudon, J.

Ctistis, G.

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, and 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]

Dupuy, E.

Fink, M.

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]

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, 100601 (2010).
[Crossref] [PubMed]

Freund, I.

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

Gérard, J. M.

Gigan, S.

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, 100601 (2010).
[Crossref] [PubMed]

Gjonaj, B.

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

Goorden, S. A.

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, and 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, 4th Edition (Addison Wesley, 2002).

Horstmann, M.

Huisman, S. R.

S. R. Huisman, T. J. Huisman, S. A. Goorden, A. P. Mosk, and P. W. H. Pinkse, “Programming balanced optical beam splitters in white paint,” Opt. Express 22, 8320–8332 (2014).
[Crossref] [PubMed]

T. J. Huisman, S. R. Huisman, A. P. Mosk, and P. W. H. Pinkse, “Controlling single-photon Fock-state propagation through opaque scattering materials,” Appl. Phys. B 116, 603–607 (2014).
[Crossref]

Huisman, T. J.

T. J. Huisman, S. R. Huisman, A. P. Mosk, and P. W. H. Pinkse, “Controlling single-photon Fock-state propagation through opaque scattering materials,” Appl. Phys. B 116, 603–607 (2014).
[Crossref]

S. R. Huisman, T. J. Huisman, S. A. Goorden, A. P. Mosk, and P. W. H. Pinkse, “Programming balanced optical beam splitters in white paint,” Opt. Express 22, 8320–8332 (2014).
[Crossref] [PubMed]

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

Lagendijk, A.

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]

E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, and 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, and A. Lagendijk, “Control of light transmission through opaque scattering media in space and time,” Phys. Rev. Lett. 106, 103901 (2011).
[Crossref] [PubMed]

Laing, A.

P. J. Shadbolt, M. R. Verde, A. Peruzzo, A. Politi, A. Laing, M. Lobino, J. C. F. Matthews, M. G. Thompson, and J. L. O’Brien, “Generating, manipulating and measuring entanglement and mixture with a reconfigurable photonic circuit,” Nat. Photonics 6, 45–49 (2012).
[Crossref]

Leonard, S. W.

S. W. Leonard, H. M. van Driel, J. Schilling, and R. B. Wehrspohn, “Ultrafast band-edge tuning of a two-dimensional silicon photonic crystal via free-carrier injection,” Phys. Rev. B. 66, 161102 (2002).
[Crossref]

Lerosey, G.

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]

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, 100601 (2010).
[Crossref] [PubMed]

Lobino, M.

P. J. Shadbolt, M. R. Verde, A. Peruzzo, A. Politi, A. Laing, M. Lobino, J. C. F. Matthews, M. G. Thompson, and J. L. O’Brien, “Generating, manipulating and measuring entanglement and mixture with a reconfigurable photonic circuit,” Nat. Photonics 6, 45–49 (2012).
[Crossref]

D. Bonneau, M. Lobino, P. Jiang, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, M. G. Thompson, and 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]

Matthews, J. C. F.

P. J. Shadbolt, M. R. Verde, A. Peruzzo, A. Politi, A. Laing, M. Lobino, J. C. F. Matthews, M. G. Thompson, and J. L. O’Brien, “Generating, manipulating and measuring entanglement and mixture with a reconfigurable photonic circuit,” Nat. Photonics 6, 45–49 (2012).
[Crossref]

Miller, D. A. B.

D. A. B. Miller, “Self-configuring universal linear optical component,” Phot. Research 1, 1–15 (2013).
[Crossref]

Montambaux, G.

E. Akkermans and G. Montambaux, Mesoscopic physics of electrons and photons, 1st Edition (Cambridge University, 2007).
[Crossref]

Mosk, A. P.

T. J. Huisman, S. R. Huisman, A. P. Mosk, and P. W. H. Pinkse, “Controlling single-photon Fock-state propagation through opaque scattering materials,” Appl. Phys. B 116, 603–607 (2014).
[Crossref]

S. A. Goorden, M. Horstmann, A. P. Mosk, B. Skoric, and P. W. H. Pinkse, “Quantum-secure authentication of a physical unclonable key,” Optica 1, 421–424 (2014).
[Crossref]

S. R. Huisman, T. J. Huisman, S. A. Goorden, A. P. Mosk, and P. W. H. Pinkse, “Programming balanced optical beam splitters in white paint,” Opt. Express 22, 8320–8332 (2014).
[Crossref] [PubMed]

S. A. Goorden, J. Bertolotti, and A. P. Mosk, “Superpixel-based spatial amplitude and phase modulation using a digital micromirror device,” Opt. Express 22, 17999–18009 (2014).
[Crossref] [PubMed]

H. Yilmaz, W. L. Vos, and A. P. Mosk, “Optimal control of light propagation through multiple-scattering media in the presence of noise,” Biomed. Opt. Express 4, 1759–1768 (2013).
[Crossref] [PubMed]

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]

E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, and 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, and 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 and A. P. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett. 101, 120601 (2008).
[Crossref] [PubMed]

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

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

O’Brien, J. L.

P. J. Shadbolt, M. R. Verde, A. Peruzzo, A. Politi, A. Laing, M. Lobino, J. C. F. Matthews, M. G. Thompson, and J. L. O’Brien, “Generating, manipulating and measuring entanglement and mixture with a reconfigurable photonic circuit,” Nat. Photonics 6, 45–49 (2012).
[Crossref]

D. Bonneau, M. Lobino, P. Jiang, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, M. G. Thompson, and 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]

Pennings, E. C. M.

L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13, 615–627 (1995).
[Crossref]

Peruzzo, A.

P. J. Shadbolt, M. R. Verde, A. Peruzzo, A. Politi, A. Laing, M. Lobino, J. C. F. Matthews, M. G. Thompson, and J. L. O’Brien, “Generating, manipulating and measuring entanglement and mixture with a reconfigurable photonic circuit,” Nat. Photonics 6, 45–49 (2012).
[Crossref]

Pinkse, P. W. H.

Politi, A.

P. J. Shadbolt, M. R. Verde, A. Peruzzo, A. Politi, A. Laing, M. Lobino, J. C. F. Matthews, M. G. Thompson, and J. L. O’Brien, “Generating, manipulating and measuring entanglement and mixture with a reconfigurable photonic circuit,” Nat. Photonics 6, 45–49 (2012).
[Crossref]

Popoff, S. M.

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, 100601 (2010).
[Crossref] [PubMed]

Reck, M.

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
[Crossref] [PubMed]

Saleh, B. E. A.

R. A. Campos, B. E. A. Saleh, and 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 and M. C. Teich, Fundamentals of Photonics, 2nd Edition (Wiley-Interscience, 2007).

Schilling, J.

S. W. Leonard, H. M. van Driel, J. Schilling, and R. B. Wehrspohn, “Ultrafast band-edge tuning of a two-dimensional silicon photonic crystal via free-carrier injection,” Phys. Rev. B. 66, 161102 (2002).
[Crossref]

Seifert, A.

Shadbolt, P. J.

P. J. Shadbolt, M. R. Verde, A. Peruzzo, A. Politi, A. Laing, M. Lobino, J. C. F. Matthews, M. G. Thompson, and J. L. O’Brien, “Generating, manipulating and measuring entanglement and mixture with a reconfigurable photonic circuit,” Nat. Photonics 6, 45–49 (2012).
[Crossref]

Sheng, P.

P. Sheng, Introduction to wave scattering, localization, and mesoscopic phenomena, 1st Edition (Academic, 1995).

Silberberg, Y.

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

Skoric, B.

Small, E.

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

Soldano, L. B.

L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13, 615–627 (1995).
[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, and 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, and 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 and M. C. Teich, Fundamentals of Photonics, 2nd Edition (Wiley-Interscience, 2007).

Thompson, M. G.

P. J. Shadbolt, M. R. Verde, A. Peruzzo, A. Politi, A. Laing, M. Lobino, J. C. F. Matthews, M. G. Thompson, and J. L. O’Brien, “Generating, manipulating and measuring entanglement and mixture with a reconfigurable photonic circuit,” Nat. Photonics 6, 45–49 (2012).
[Crossref]

D. Bonneau, M. Lobino, P. Jiang, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, M. G. Thompson, and 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]

van Driel, H. M.

S. W. Leonard, H. M. van Driel, J. Schilling, and R. B. Wehrspohn, “Ultrafast band-edge tuning of a two-dimensional silicon photonic crystal via free-carrier injection,” Phys. Rev. B. 66, 161102 (2002).
[Crossref]

van Putten, E. G.

E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, and 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 and A. P. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett. 101, 120601 (2008).
[Crossref] [PubMed]

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

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

Verde, M. R.

P. J. Shadbolt, M. R. Verde, A. Peruzzo, A. Politi, A. Laing, M. Lobino, J. C. F. Matthews, M. G. Thompson, and J. L. O’Brien, “Generating, manipulating and measuring entanglement and mixture with a reconfigurable photonic circuit,” Nat. Photonics 6, 45–49 (2012).
[Crossref]

Vos, W. L.

Wehrspohn, R. B.

S. W. Leonard, H. M. van Driel, J. Schilling, and R. B. Wehrspohn, “Ultrafast band-edge tuning of a two-dimensional silicon photonic crystal via free-carrier injection,” Phys. Rev. B. 66, 161102 (2002).
[Crossref]

Yilmaz, H.

Yüce, E.

Zappe, H.

Zeilinger, A.

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
[Crossref] [PubMed]

Zhang, W.

Zhou, J.

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, and 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. Phys. B (1)

T. J. Huisman, S. R. Huisman, A. P. Mosk, and P. W. H. Pinkse, “Controlling single-photon Fock-state propagation through opaque scattering materials,” Appl. Phys. B 116, 603–607 (2014).
[Crossref]

Biomed. Opt. Express (1)

J. Lightwave Technol. (1)

L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13, 615–627 (1995).
[Crossref]

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

Nat. Photonics (3)

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]

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

P. J. Shadbolt, M. R. Verde, A. Peruzzo, A. Politi, A. Laing, M. Lobino, J. C. F. Matthews, M. G. Thompson, and J. L. O’Brien, “Generating, manipulating and measuring entanglement and mixture with a reconfigurable photonic circuit,” Nat. Photonics 6, 45–49 (2012).
[Crossref]

Opt. Commun. (1)

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

Opt. Express (3)

Opt. Lett. (1)

Optica (1)

Optics Lett. (1)

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

Phot. Research (1)

D. A. B. Miller, “Self-configuring universal linear optical component,” Phot. Research 1, 1–15 (2013).
[Crossref]

Phys. Rev. A (1)

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

Phys. Rev. B. (1)

S. W. Leonard, H. M. van Driel, J. Schilling, and R. B. Wehrspohn, “Ultrafast band-edge tuning of a two-dimensional silicon photonic crystal via free-carrier injection,” Phys. Rev. B. 66, 161102 (2002).
[Crossref]

Phys. Rev. Lett. (6)

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
[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, and 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]

E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, and 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, and 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 and A. P. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett. 101, 120601 (2008).
[Crossref] [PubMed]

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, 100601 (2010).
[Crossref] [PubMed]

Physica A (1)

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

Other (4)

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

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

P. Sheng, Introduction to wave scattering, localization, and mesoscopic phenomena, 1st Edition (Academic, 1995).

E. Akkermans and G. Montambaux, Mesoscopic physics of electrons and photons, 1st Edition (Cambridge University, 2007).
[Crossref]

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Figures (7)

Fig. 1
Fig. 1

Wavefront-shaped programmable linear optical circuits. (a) Incident light on a multiple-scattering medium results in a speckle pattern. (b) The scattered light can be described by a scattering matrix, representing a complicated linear optical circuit. The scattering matrix is here represented as light propagating through an effective medium with the same correlations as the optical circuit, however, these optical elements are not physically located at these positions in the material. (c) By phase modulation of the incident wavefront with a spatial light modulator (SLM) it becomes possible to address correlations in the scattering matrix to create an interference pattern with a desired functionality. In this picture light travels through the material as if it would have traveled through a beam splitter. Note: reflection is omitted in this figure for clarity.

Fig. 2
Fig. 2

Schematic illustration for programming a 2 × 2 linear optical circuit. The incident input modes 1 and 2 are spatially separated on the SLM. (I–III) Optimization for output mode 1′ providing phase pattern θ1′. (IV–VI) Optimization for output mode 2′ providing phase pattern θ2′. (VII) Finally one writes phase pattern θ2×2 = arg (eiθ1′ + eiθ2′) to obtain a superposition of the fields in steps I and II. The CCD pictures are snapshots of our experiments on the 2 × 2 optical circuit.

Fig. 3
Fig. 3

Setup for wavefront-shaped optical circuits. (a) Two input 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 1.5 mm thick microscope slide. The transmitted light is projected on a CCD camera. Three output modes 1′, 2′, and 3′ are selected. (b) Optimized phase pattern on the SLM. A phase offset is applied to the second incident mode. (c) Camera image for three optimized spots when both input modes are incident on the phase pattern of (b).

Fig. 4
Fig. 4

Interference measurement of the effective transmission matrix. A phase difference Δθ is applied between the two input modes. The intensity in the target optimized spots is measured as a function of this phase difference.

Fig. 5
Fig. 5

Experimental realization of a programmed 2 × 2 linear optical circuit. A transmission matrix is programmed for which output mode 2′ has a programmable phase difference α with respect to output mode 1′. (a) Example of a measured (symbols) interference characterization of the transmission matrix for α = 2 rad. Sine fits (solid) are used to determine the phases δ for which maximum intensity occurs. (b) Extracted phases δ as a function of the programmed phase α. Note that the displayed phase is wrapped, so there are no actual 2π jumps in δ1′ and δ2′. (c) Extracted phase difference between the output modes (symbols) in comparison with the expected phase difference (diagonal band) based on the programmed phase α. The observed phase differences between the output modes match the programmed phase differences excellently.

Fig. 6
Fig. 6

Computational results on the phase differences of the 2 × 2 linear optical circuit. 10,000 Realizations were simulated for systems with a scattering matrix of dimension 1000 and with 50 controlled channels per input mode. (a) Obtained distribution for phase δ1′. (b) Obtained distribution for phase δ2′. (c) Extracted phase difference between the output modes. (d) Width of the phase distributions as a function of controlled channels for scattering matrices of dimension 500 and 2500.

Fig. 7
Fig. 7

Experimental realization of a programmed 2 × 3 linear optical circuit. A transmission matrix is programmed for which output mode 2′ has a programmable phase difference α with respect to output modes 1′ and 3′. (a) Extracted phases δ (symbols) as a function of the programmed phase difference α. (b) Extracted phase difference between the output modes (symbols) in comparison with the expected phase differences (horizontal and diagonal bands) based on the programmed phase α. The observed phase differences between the output modes match the programmed phase differences excellently. The pink band indicates the uncertainty in the phase determination.

Equations (11)

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

θ 1 , 1 + θ 2 , 1
θ 1 , 1 + θ 2 , 1 + β 2 , 1 .
θ 1 = i = 2 n ( θ i , 1 + β i , 1 ) + θ 1 , 1 .
E 1 = e i ϕ 1 [ | t 1 , 1 | | t 2 , 1 | | t n , 1 | ] [ E 1 E 2 E n ] ,
E 1 = T n × 1 , 1 [ E 1 E 2 E n ] .
T n × 1 , 1 = e i ϕ 1 [ | t 1 , 1 | e i α 1 , 1 | t 2 , 1 | e 1 α 2 , 1 | t n , 1 | e 1 α n , 1 ] .
θ n × m = arg ( j = 1 m c j e i θ j ) ,
[ E 1 E 2 E m ] = f 1 [ c 1 T n × 1 , 1 c 2 T n × 1 , 2 c m T n × 1 , m ] [ E 1 E 2 E n ] ,
T m × n = [ c 1 T n × 1 , 1 c 2 T n × 1 , 2 c m T n × 1 , m ] .
T 2 × 2 = [ | T 11 | | T 12 | | T 21 | | T 22 | e i α ] .
T 2 × 3 = [ | T 11 | | T 12 | | T 21 | | T 22 | e i α | T 31 | | T 32 | ] ,

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