Abstract

Performing linear operations using optical devices is a crucial building block in many fields ranging from telecommunications to optical analog computation and machine learning. For many of these applications, key requirements are robustness to fabrication inaccuracies, reconfigurability, and scalability. We propose a way to perform linear operations using complex optical media such as multimode fibers or scattering media as a computational platform driven by wavefront shaping. Given a large random transmission matrix representing light propagation in such a medium, we can extract any desired smaller linear operator by finding suitable input and output projectors. We demonstrate this concept by finding input wavefronts using a spatial light modulator that cause the complex medium to act as a desired complex-valued linear operator on the optical field. We experimentally build several 16×16 operators and discuss the fundamental limits of the scalability of our approach. It offers the prospect of reconfigurable, robust, and easy-to-fabricate linear optical analog computation units.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2018 (2)

P. del Hougne and G. Lerosey, “Leveraging chaos for wave-based analog computation: Demonstration with indoor wireless communication signals,” Phys. Rev. X 8, 041037 (2018).
[Crossref]

P. L. Mennea, W. R. Clements, D. H. Smith, J. C. Gates, B. J. Metcalf, R. H. S. Bannerman, R. Burgwal, J. J. Renema, W. S. Kolthammer, I. A. Walmsley, and P. G. R. Smith, “Modular linear optical circuits,” Optica 5, 1087–1090 (2018).
[Crossref]

2017 (6)

W. Xiong, P. Ambichl, Y. Bromberg, B. Redding, S. Rotter, and H. Cao, “Principal modes in multimode fibers: exploring the crossover from weak to strong mode coupling,” Opt. Express 25, 2709–2724 (2017).
[Crossref]

R. Fickler, M. Ginoya, and R. W. Boyd, “Custom-tailored spatial mode sorting by controlled random scattering,” Phys. Rev. B 95, 161108 (2017).
[Crossref]

A. Y. Piggott, J. Petykiewicz, L. Su, and J. Vučković, “Fabrication-constrained nanophotonic inverse design,” Sci. Rep. 7, 1786 (2017).
[Crossref]

A. Annoni, E. Guglielmi, M. Carminati, G. Ferrari, M. Sampietro, D. A. B. Miller, A. Melloni, and F. Morichetti, “Unscrambling light—automatically undoing strong mixing between modes,” Light: Sci. Appl. 6, e17110 (2017).
[Crossref]

Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljačić, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11, 441–446 (2017).
[Crossref]

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

2016 (7)

S. Diamond and S. Boyd, “Cvxpy: a python-embedded modeling language for convex optimization,” J. Mach. Learn. Res. 17, 1–5(2016).

H. Defienne, M. Barbieri, I. A. Walmsley, B. J. Smith, and S. Gigan, “Two-photon quantum walk in a multimode fiber,” Sci. Adv 2, e1501054(2016).
[Crossref]

T. A. W. Wolterink, R. Uppu, G. Ctistis, W. L. Vos, K.-J. Boller, and P. W. H. Pinkse, “Programmable two-photon quantum interference in 103 channels in opaque scattering media,” Phys. Rev. A 93, 053817 (2016).
[Crossref]

M. Mounaix, D. Andreoli, H. Defienne, G. Volpe, O. Katz, S. Grésillon, and S. Gigan, “Spatiotemporal coherent control of light through a multiple scattering medium with the multispectral transmission matrix,” Phys. Rev. Lett. 116, 253901 (2016).
[Crossref]

P. del Hougne, F. Lemoult, M. Fink, and G. Lerosey, “Spatiotemporal wave front shaping in a microwave cavity,” Phys. Rev. Lett. 117, 134302 (2016).
[Crossref]

R. Bruck, K. Vynck, P. Lalanne, B. Mills, D. J. Thomson, G. Z. Mashanovich, G. T. Reed, and O. L. Muskens, “All-optical spatial light modulator for reconfigurable silicon photonic circuits,” Optica 3, 396–402 (2016).
[Crossref]

A. Ribeiro, A. Ruocco, L. Vanacker, and W. Bogaerts, “Demonstration of a 4 × 4-port universal linear circuit,” Optica 3, 1348–1357 (2016).
[Crossref]

2015 (7)

S. R. Huisman, T. J. Huisman, T. A. W. Wolterink, A. P. Mosk, and P. W. H. Pinkse, “Programmable multiport optical circuits in opaque scattering materials,” Opt. Express 23, 3102–3116 (2015).
[Crossref]

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

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

A. Liutkus, D. Martina, S. Popoff, G. Chardon, O. Katz, G. Lerosey, S. Gigan, L. Daudet, and I. Carron, “Imaging with nature: compressive imaging using a multiply scattering medium,” Sci. Rep. 4, 5552 (2015).
[Crossref]

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

B. Shen, P. Wang, R. Polson, and R. Menon, “An integrated-nanophotonics polarization beamsplitter with 2.4 × 2.4 μm 2 footprint,” Nat. Photonics 9, 378–382 (2015).
[Crossref]

J. Carolan, C. Harrold, C. Sparrow, E. Martin-Lopez, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref]

2014 (4)

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alu, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343, 160–163 (2014).
[Crossref]

S. M. Popoff, A. Goetschy, S. F. Liew, A. D. Stone, and H. Cao, “Coherent control of total transmission of light through disordered media,” Phys. Rev. Lett. 112, 133903 (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]

G. Labroille, B. Denolle, P. Jian, P. Genevaux, N. Treps, and J.-F. Morizur, “Efficient and mode selective spatial mode multiplexer based on multi-plane light conversion,” Opt. Express 22, 15599–15607 (2014).
[Crossref]

2013 (4)

B. Redding, S. M. Popoff, and H. Cao, “All-fiber spectrometer based on speckle pattern reconstruction,” Opt. Express 21, 6584–6600 (2013).
[Crossref]

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

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

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

2012 (1)

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)

J. S. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photon. Rev. 5, 308–321 (2011).
[Crossref]

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]

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]

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

2010 (4)

J.-F. Morizur, L. Nicholls, P. Jian, S. Armstrong, N. Treps, B. Hage, M. Hsu, W. Bowen, J. Janousek, and H.-A. Bachor, “Programmable unitary spatial mode manipulation,” J. Opt. Soc. Am. A 27, 2524–2531 (2010).
[Crossref]

S. M. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 1–5 (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, 100601 (2010).
[Crossref]

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

2007 (2)

2003 (1)

2002 (1)

V. Doya, O. Legrand, F. Mortessagne, and C. Miniatura, “Speckle statistics in a chaotic multimode fiber,” Phys. Rev. E 65, 056223(2002).
[Crossref]

1999 (1)

1997 (1)

C. Draeger and M. Fink, “One-channel time reversal of elastic waves in a chaotic 2D-silicon cavity,” Phys. Rev. Lett. 79, 407–410 (1997).
[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]

1967 (1)

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

Alu, A.

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alu, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343, 160–163 (2014).
[Crossref]

Ambichl, P.

Andreoli, D.

M. Mounaix, D. Andreoli, H. Defienne, G. Volpe, O. Katz, S. Grésillon, and S. Gigan, “Spatiotemporal coherent control of light through a multiple scattering medium with the multispectral transmission matrix,” Phys. Rev. Lett. 116, 253901 (2016).
[Crossref]

Annoni, A.

A. Annoni, E. Guglielmi, M. Carminati, G. Ferrari, M. Sampietro, D. A. B. Miller, A. Melloni, and F. Morichetti, “Unscrambling light—automatically undoing strong mixing between modes,” Light: Sci. Appl. 6, e17110 (2017).
[Crossref]

Armstrong, S.

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]

Bachor, H.-A.

Baehr-Jones, T.

Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljačić, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11, 441–446 (2017).
[Crossref]

Bannerman, R. H. S.

Baraniuk, R. G.

C. A. Metzler, M. K. Sharma, S. Nagesh, R. G. Baraniuk, O. Cossairt, and A. Veeraraghavan, “Coherent inverse scattering via transmission matrices: Efficient phase retrieval algorithms and a public dataset,” in IEEE International Conference on Computational Photography (ICCP) (IEEE, 2017).

Barbieri, M.

H. Defienne, M. Barbieri, I. A. Walmsley, B. J. Smith, and S. Gigan, “Two-photon quantum walk in a multimode fiber,” Sci. Adv 2, e1501054(2016).
[Crossref]

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]

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]

Bilenberg, B.

Boccara, A. C.

S. M. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 1–5 (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, 100601 (2010).
[Crossref]

Bogaerts, W.

Boller, K.-J.

T. A. W. Wolterink, R. Uppu, G. Ctistis, W. L. Vos, K.-J. Boller, and P. W. H. Pinkse, “Programmable two-photon quantum interference in 103 channels in opaque scattering media,” Phys. Rev. A 93, 053817 (2016).
[Crossref]

Borel, P. I.

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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).
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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).
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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).
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I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322 (2010).
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I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32, 2309–2311 (2007).
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M. Mounaix, D. Andreoli, H. Defienne, G. Volpe, O. Katz, S. Grésillon, and S. Gigan, “Spatiotemporal coherent control of light through a multiple scattering medium with the multispectral transmission matrix,” Phys. Rev. Lett. 116, 253901 (2016).
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Nam, K. T.

J.-H. Park, C. Park, H. Yu, J. Park, S. Han, J. Shin, S. H. Ko, K. T. Nam, Y.-H. Cho, and Y. Park, “Subwavelength light focusing using random nanoparticles,” Nat. Photonics 7, 454–458 (2013).
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J.-H. Park, C. Park, H. Yu, J. Park, S. Han, J. Shin, S. H. Ko, K. T. Nam, Y.-H. Cho, and Y. Park, “Subwavelength light focusing using random nanoparticles,” Nat. Photonics 7, 454–458 (2013).
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J.-H. Park, C. Park, H. Yu, J. Park, S. Han, J. Shin, S. H. Ko, K. T. Nam, Y.-H. Cho, and Y. Park, “Subwavelength light focusing using random nanoparticles,” Nat. Photonics 7, 454–458 (2013).
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J.-H. Park, C. Park, H. Yu, J. Park, S. Han, J. Shin, S. H. Ko, K. T. Nam, Y.-H. Cho, and Y. Park, “Subwavelength light focusing using random nanoparticles,” Nat. Photonics 7, 454–458 (2013).
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J.-H. Park, C. Park, H. Yu, J. Park, S. Han, J. Shin, S. H. Ko, K. T. Nam, Y.-H. Cho, and Y. Park, “Subwavelength light focusing using random nanoparticles,” Nat. Photonics 7, 454–458 (2013).
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V. A. Marcenko and L. A. Pastur, “Distribution of eigenvalues for some sets of random matrices,” Mat. Sb. 1, 457–483 (1967).
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A. Y. Piggott, J. Petykiewicz, L. Su, and J. Vučković, “Fabrication-constrained nanophotonic inverse design,” Sci. Rep. 7, 1786 (2017).
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Piggott, A. Y.

A. Y. Piggott, J. Petykiewicz, L. Su, and J. Vučković, “Fabrication-constrained nanophotonic inverse design,” Sci. Rep. 7, 1786 (2017).
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A. Liutkus, D. Martina, S. Popoff, G. Chardon, O. Katz, G. Lerosey, S. Gigan, L. Daudet, and I. Carron, “Imaging with nature: compressive imaging using a multiply scattering medium,” Sci. Rep. 4, 5552 (2015).
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S. M. Popoff, A. Goetschy, S. F. Liew, A. D. Stone, and H. Cao, “Coherent control of total transmission of light through disordered media,” Phys. Rev. Lett. 112, 133903 (2014).
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B. Redding, S. M. Popoff, and H. Cao, “All-fiber spectrometer based on speckle pattern reconstruction,” Opt. Express 21, 6584–6600 (2013).
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S. M. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 1–5 (2010).
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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).
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Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljačić, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11, 441–446 (2017).
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Smith, P. G. R.

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S. M. Popoff, A. Goetschy, S. F. Liew, A. D. Stone, and H. Cao, “Coherent control of total transmission of light through disordered media,” Phys. Rev. Lett. 112, 133903 (2014).
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I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32, 2309–2311 (2007).
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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).
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J.-H. Park, C. Park, H. Yu, J. Park, S. Han, J. Shin, S. H. Ko, K. T. Nam, Y.-H. Cho, and Y. Park, “Subwavelength light focusing using random nanoparticles,” Nat. Photonics 7, 454–458 (2013).
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Supplementary Material (1)

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» Supplement 1       Supplementary Document

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Figures (4)

Fig. 1.
Fig. 1. Schematic representation of the setup. An expended laser beam is modulated after reflection off a DMD and injected into a complex medium (CM: ground glass diffuser or multimode fiber). One polarization of the outgoing light is recorded by a digital camera. A reference arm is used only for the final estimation of the fidelity to measure the complex optical field for the MMF experiment. FC: fiber coupler; Ci (with i [ 1 4 ] ): fiber collimator; Li (with i [ 1 4 ] ): planoconvex lens; PBS: polarization beamsplitter; and P1 and P2: polarizers.
Fig. 2.
Fig. 2. Overview of the experimental procedure for performing a DFT operation of size 4 ( n = m = 4 and G = DFT 4 ). Step 1: System calibration. Acquisition of the complex TM by measuring a set of input patterns and output intensity speckles and using a phase retrieval algorithm. Step 2: Optimal input projection calculation. We illustrate the procedure by showing how to find one subpart of the input mask. The computation is done independently for each subpart of the SLM using a convex optimization solver and using the TM and the corresponding column of the target matrix G as inputs. Step 3: Analog computation. The proposed optical processing unit is composed of a spatial light modulator (SLM) and a complex medium. The SLM and the output detection take the role of the projectors P in and P out , converting the given transmission matrix H of the complex medium into a desired linear transformation G [see Eq. (2)].
Fig. 3.
Fig. 3. Comparison of experimentally implemented operators and target operators. We use as target matrix G , a 16 × 16 Hadamard matrix [(a) and (b)], and a 16 × 16 discrete Fourier transform [(c) and (d)]. Absolute values of experimental (red dots) and theoretical (cyan diamonds) results of the linear transformations for two different random input vectors drawn from { 1 , 0 , 1 } for G = Ha 16 (a) and G = DFT 16 (c). The correlation between the two signals is shown in the insert. Comparison between experimental and target operators for G = Ha 16 , F c = 0.785 (b) and G = DFT 16 , F c = 0.818 (d). The results are obtained without averaging.
Fig. 4.
Fig. 4. Amplitude of four different random output vectors for operator G = DFT 8 obtained using a ground glass diffuser as a complex medium. Cyan diamonds correspond to the theoretical outputs and red dots to the experimental data. Inserts in the top left corners give the correlation between the experimental and the predicted transforms.

Tables (1)

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Table 1. Summary of the Efficiency Results for the MMF Experiment a

Equations (8)

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| ψ out = H | ψ in .
G = P out T HP in .
P in = [ p 1 , 1 0 0 0 p N / 4 , 1 0 0 0 0 p 1 , 2 0 0 0 p N / 4 , 2 0 0 0 0 p 1 , 3 0 0 0 p N / 4 , 3 0 0 0 0 p 1 , 4 0 0 0 p N / 4 , 4 ] , P out = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 ] ,
H k [ p 1 , k p 2 , k p N / n , k ] = G k for k [ 1 , n ] ,
[ p 1 , k p 2 , k p N / n , k ] = H k + G k ,
Y = GX = P out T HP in X .
DFT n = 1 n [ ω n j k ] j , k = 0 n 1 with ω n = e 2 π i / n ,
Ha n = 1 n [ Ha n 1 Ha n 1 Ha n 1 Ha n 1 ] with Ha 1 = [ 1 ]

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