Abstract

The optical transmission matrix (TM) characterizes the transmission properties of a sample. We show a novel experimental procedure for measuring the TM of light waves in a slab geometry based on sampling the light field on a hexagonal lattice at the Rayleigh criterion. Our method enables the efficient measurement of a large fraction of the complete TM without oversampling while minimizing sampling crosstalk and the associated distortion of the statistics of the matrix elements. The procedure and analysis described here is demonstrated on a clear sample, which serves as an important reference for other systems and geometries, such as dense scattering media.

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

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2019 (5)

D. A. B. Miller, “Waves, modes, communications, and optics: a tutorial,” Adv. Opt. Photonics 11(3), 679–825 (2019).
[Crossref]

P. Fang, C. Tian, L. Zhao, Y. P. Bliokh, V. Freilikher, and F. Nori, “Universality of eigenchannel structures in dimensional crossover,” Phys. Rev. B 99(9), 094202 (2019).
[Crossref]

H. Yilmaz, C. W. Hsu, A. Goetschy, S. Bittner, S. Rotter, A. Yamilov, and H. Cao, “Angular memory effect of transmission eigenchannels,” Phys. Rev. Lett. 123(20), 203901 (2019).
[Crossref]

H. Yilmaz, C. W. Hsu, A. Yamilov, and H. Cao, “Transverse localization of transmission eigenchannels,” Nat. Photonics 13(5), 352–358 (2019).
[Crossref]

A. Boniface, I. Gusachenko, K. Dholakia, and S. Gigan, “Rapid broadband characterization of scattering medium using hyperspectral imaging,” Optica 6(3), 274–279 (2019).
[Crossref]

2018 (2)

2017 (2)

C. W. Hsu, S. F. Liew, A. Goetschy, H. Cao, and A. D. Stone, “Correlation-enhanced control of wave focusing in disordered media,” Nat. Phys. 13(5), 497–502 (2017).
[Crossref]

S. Rotter and S. Gigan, “Light fields in complex media: Mesoscopic scattering meets wave control,” Rev. Mod. Phys. 89(1), 015005 (2017).
[Crossref]

2016 (3)

2015 (5)

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

M. Plöschner, T. Tyc, and T. Čižmár, “Seeing through chaos in multimode fibres,” Nat. Photonics 9(8), 529–535 (2015).
[Crossref]

M. Kim, W. Choi, C. Yoon, G. H. Kim, S. hyun Kim, G.-R. Yi, Q.-H. Park, and W. Choi, “Exploring anti-reflection modes in disordered media,” Opt. Express 23(10), 12740 (2015).
[Crossref]

M. Davy, Z. Shi, J. Park, C. Tian, and A. Z. Genack, “Universal structure of transmission eigenchannels inside opaque media,” Nat. Commun. 6(1), 6893 (2015).
[Crossref]

X. Hao, L. Martin-Rouault, and M. Cui, “A self-adaptive method for creating high efficiency communication channels through random scattering media,” Sci. Rep. 4(1), 5874 (2015).
[Crossref]

2014 (3)

S. Liew, S. Popoff, A. Mosk, W. Vos, and H. Cao, “Transmission channels for light in absorbing random media: from diffusive to ballistic-like transport,” Phys. Rev. B 89(22), 224202 (2014).
[Crossref]

B. Gérardin, J. Laurent, A. Derode, C. Prada, and A. Aubry, “Full transmission and reflection of waves propagating through a maze of disorder,” Phys. Rev. Lett. 113(17), 173901 (2014).
[Crossref]

M. Plöschner, B. Straka, K. Dholakia, and T. Čižmár, “GPU accelerated toolbox for real-time beam-shaping in multimode fibres,” Opt. Express 22(3), 2933–2947 (2014).
[Crossref]

2013 (4)

R. N. Mahalati, R. Y. Gu, and J. M. Kahn, “Resolution limits for imaging through multi-mode fiber,” Opt. Express 21(2), 1656–1668 (2013).
[Crossref]

A. Goetschy and A. D. Stone, “Filtering random matrices: The effect of incomplete channel control in multiple scattering,” Phys. Rev. Lett. 111(6), 063901 (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(15), 153902 (2013).
[Crossref]

M. Davy, Z. Shi, J. Wang, and A. Z. Genack, “Transmission statistics and focusing in single disordered samples,” Opt. Express 21(8), 10367–10375 (2013).
[Crossref]

2012 (4)

Z. Shi and A. Z. Genack, “Transmission eigenvalues and the bare conductance in the crossover to Anderson localization,” Phys. Rev. Lett. 108(4), 043901 (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(5), 283–292 (2012).
[Crossref]

S. Tripathi, R. Paxman, T. Bifano, and K. C. Toussaint, “Vector transmission matrix for the polarization behavior of light propagation in highly scattering media,” Opt. Express 20(14), 16067–16076 (2012).
[Crossref]

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

2011 (1)

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(2), 023902 (2011).
[Crossref]

2010 (3)

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

E. van Putten and A. P. Mosk, “The information age in optics: Measuring the transmission matrix,” Physics 3, 22 (2010).
[Crossref]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: An approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104(10), 100601 (2010).
[Crossref]

2009 (1)

A. Aubry and A. Derode, “Random matrix theory applied to acoustic backscattering and imaging in complex media,” Phys. Rev. Lett. 102(8), 084301 (2009).
[Crossref]

2008 (2)

R. Sprik, A. Tourin, J. de Rosny, and M. Fink, “Eigenvalue distributions of correlated multichannel transfer matrices in strongly scattering systems,” Phys. Rev. B 78(1), 012202 (2008).
[Crossref]

I. M. Vellekoop and A. P. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett. 101(12), 120601 (2008).
[Crossref]

2007 (1)

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4(9), 717–719 (2007).
[Crossref]

2001 (1)

J. Shen, “On the singular values of Gaussian random matrices,” Linear Algebra Appl. 326(1-3), 1–14 (2001).
[Crossref]

2000 (1)

1999 (1)

M. C. W. van Rossum and T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71(1), 313–371 (1999).
[Crossref]

1997 (1)

C. W. J. Beenakker, “Random-matrix theory of quantum transport,” Rev. Mod. Phys. 69(3), 731–808 (1997).
[Crossref]

1996 (1)

A. Lagendijk and B. A. van Tiggelen, “Resonant multiple scattering of light,” Phys. Rep. 270(3), 143–215 (1996).
[Crossref]

1992 (1)

T. Martin and R. Landauer, “Wave-packet approach to noise in multichannel mesoscopic systems,” Phys. Rev. B 45(4), 1742–1755 (1992).
[Crossref]

1991 (1)

H. Baranger, D. DiVincenzo, R. Jalabert, and A. Stone, “Classical and quantum ballistic-transport anomalies in microjunctions,” Phys. Rev. B 44(19), 10637–10675 (1991).
[Crossref]

1990 (1)

J. Pendry, A. MacKinnon, and A. Pretre, “Maximal fluctuations–a new phenomenon in disordered systems,” Phys. A 168(1), 400–407 (1990).
[Crossref]

1988 (1)

P. Mello, P. Pereyra, and N. Kumar, “Macroscopic approach to multichannel disordered conductors,” Ann. Phys. 181(2), 290–317 (1988).
[Crossref]

1986 (1)

1982 (1)

1981 (2)

D. S. Fisher and P. A. Lee, “Relation between conductivity and transmission matrix,” Phys. Rev. B 23(12), 6851–6854 (1981).
[Crossref]

E. N. Economou and C. M. Soukoulis, “Static conductance and scaling theory of localization in one dimension,” Phys. Rev. Lett. 46(9), 618–621 (1981).
[Crossref]

1967 (1)

V. A. Marčenko and L. A. Pastur, “Distribution of eigenvalues for some sets of random matrices,” Math. USSR Sb. 1(4), 457–483 (1967).
[Crossref]

1964 (1)

D. P. Petersen and D. Middleton, “Reconstruction of multidimensional stochastic fields from discrete measurements of amplitude and gradient,” Inf. Control. 7(4), 445–476 (1964).
[Crossref]

1962 (1)

1959 (1)

E. Wolf, “A scalar representation of electromagnetic fields: II,” Proc. Phys. Soc., London 74(3), 269–280 (1959).
[Crossref]

1881 (1)

E. Abbe Hon., “VII.–On the estimation of aperture in the microscope,” J. R. Microsc. Soc. 1(3), 388–423 (1881).
[Crossref]

Abbe Hon., E.

E. Abbe Hon., “VII.–On the estimation of aperture in the microscope,” J. R. Microsc. Soc. 1(3), 388–423 (1881).
[Crossref]

Akbulut, D.

D. Akbulut, T. Strudley, J. Bertolotti, E. P. A. M. Bakkers, A. Lagendijk, O. L. Muskens, W. L. Vos, and A. P. Mosk, “Optical transmission matrix as a probe of the photonic strength,” Phys. Rev. A 94(4), 043817 (2016).
[Crossref]

Amitonova, L. V.

Aubry, A.

B. Gérardin, J. Laurent, A. Derode, C. Prada, and A. Aubry, “Full transmission and reflection of waves propagating through a maze of disorder,” Phys. Rev. Lett. 113(17), 173901 (2014).
[Crossref]

A. Aubry and A. Derode, “Random matrix theory applied to acoustic backscattering and imaging in complex media,” Phys. Rev. Lett. 102(8), 084301 (2009).
[Crossref]

Badizadegan, K.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4(9), 717–719 (2007).
[Crossref]

Baranger, H.

H. Baranger, D. DiVincenzo, R. Jalabert, and A. Stone, “Classical and quantum ballistic-transport anomalies in microjunctions,” Phys. Rev. B 44(19), 10637–10675 (1991).
[Crossref]

Beenakker, C. W. J.

C. W. J. Beenakker, “Random-matrix theory of quantum transport,” Rev. Mod. Phys. 69(3), 731–808 (1997).
[Crossref]

Bertolotti, J.

D. Akbulut, T. Strudley, J. Bertolotti, E. P. A. M. Bakkers, A. Lagendijk, O. L. Muskens, W. L. Vos, and A. P. Mosk, “Optical transmission matrix as a probe of the photonic strength,” Phys. Rev. A 94(4), 043817 (2016).
[Crossref]

Bifano, T.

Bittner, S.

H. Yilmaz, C. W. Hsu, A. Goetschy, S. Bittner, S. Rotter, A. Yamilov, and H. Cao, “Angular memory effect of transmission eigenchannels,” Phys. Rev. Lett. 123(20), 203901 (2019).
[Crossref]

Blanter, Y. M.

Y. V. Nazarov and Y. M. Blanter, Quantum Transport: Introduction to Nanoscience (Cambridge University Press, 2009).

Bliokh, Y. P.

P. Fang, C. Tian, L. Zhao, Y. P. Bliokh, V. Freilikher, and F. Nori, “Universality of eigenchannel structures in dimensional crossover,” Phys. Rev. B 99(9), 094202 (2019).
[Crossref]

Boccara, A. C.

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

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: An approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104(10), 100601 (2010).
[Crossref]

Boniface, A.

Borhani, N.

Cao, H.

H. Yilmaz, C. W. Hsu, A. Yamilov, and H. Cao, “Transverse localization of transmission eigenchannels,” Nat. Photonics 13(5), 352–358 (2019).
[Crossref]

H. Yilmaz, C. W. Hsu, A. Goetschy, S. Bittner, S. Rotter, A. Yamilov, and H. Cao, “Angular memory effect of transmission eigenchannels,” Phys. Rev. Lett. 123(20), 203901 (2019).
[Crossref]

C. W. Hsu, S. F. Liew, A. Goetschy, H. Cao, and A. D. Stone, “Correlation-enhanced control of wave focusing in disordered media,” Nat. Phys. 13(5), 497–502 (2017).
[Crossref]

S. Liew, S. Popoff, A. Mosk, W. Vos, and H. Cao, “Transmission channels for light in absorbing random media: from diffusive to ballistic-like transport,” Phys. Rev. B 89(22), 224202 (2014).
[Crossref]

Carminati, R.

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Figures (6)

Fig. 1.
Fig. 1. Comparison between sampling on a square (left) and hexagonal (right) lattice. The lattice constant is $a$ and the shaded area represents the unit cell.
Fig. 2.
Fig. 2. Normalized rank of a sampling basis versus the sampling density for a square and hexagonal lattice. The horizontal axis is normalized to the critical sampling density of $\pi / \lambda ^2$, represented by the vertical line at 1. The dashed vertical line indicates the Rayleigh sampling density for the hexagonal lattice. In this numerical calculation $\mbox {NA}=1$.
Fig. 3.
Fig. 3. Setup. Light from a Helium-Neon laser is split into a signal and reference beam. The signal (in red) is spatially scanned across a sample with the scan mirror (scan mir.), and for every position on the sample the resulting transmitted scattered light is imaged in real space with two cameras C1 and C2, which measure different output polarizations. The reference beam (in blue) interferes with the signal to retrieve the phase information using off-axis holography. Abbreviations: Pol., polarizer; ND, neutral density filter; beam exp., 15x beam expander; $\lambda /2$, half-wave plate; LED1 and LED2, 633 nm light emitting diodes; O1, 0.95-NA air microscope objective; O2, 1.42-NA oil immersion microscope objective; C3, reflection camera; PBS, polarizing beamsplitter; SMF, single mode fiber.
Fig. 4.
Fig. 4. Coordinates of the generated hexagonal grid (blue dots) representing the output sampling basis (a) before and (b) after performing an affine transformation to match the grid found from the centers of mass (COM) of the focused input laser spots (red dots).
Fig. 5.
Fig. 5. Results of the TM measurement and simulations. The left and right column display the measured and simulated results respectively. (a,b) Magnitude of the TM elements for the $919 \times 919$ $T_{\mathrm {HH}}$ polarization component of a zero-thickness reference. (c,d) Histogram of the magnitudes of the diagonal elements. (e,f) Magnitude of the diagonal elements $\lvert t_{mm} \rvert$ as a function of the input position $m$.
Fig. 6.
Fig. 6. Singular value histograms of the transmission matrix. (a,b,d,e) Singular values $s$ of the measured polarization sub-matrices and (c) of the full experimental TM. (f) Simulated histogram for the full TM including 0.5% RMS noise and (g) without noise. (h) Simulated TM for square lattice sampling with the same lattice constant, without noise and (i) with noise. In the cross-polarization matrices (b,d), the red curve corresponds to the Marchenko-Pastur law of a random matrix of the same size.

Equations (5)

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

asq2=λ2π(NA)2,
ahex2=23λ2π(NA)21.15λ2π(NA)2.
ahex=0.994aRay with aRay=z1λ2π(NA)1,
T=(THHTVHTHVTVV),
SG(r)=exp[2(rR)n],

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