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In the last decade, the scattering medium has been gradually attacking attention from researchers. Among the proposed approaches, the transmission matrix (TM) is considered as an effect way to describe the scattering properties which relate to input optical and output optical fields. However, the acquired transmission matrix and its eigenvalues strongly depend on the experimental conditions, such as the numbers of input channels (limited numerical aperture and illumination area, or the pixel number of the spatial light modulator) and output channels. In other words, the actual transmission matrix of the scattering medium is the acquired transmission matrix with infinite numbers of the input and output channels. We propose an approach to obtaining the actual matrix by evaluating its eigenvalues. First, the matrix is expressed by the singular value decomposition to obtain its inverse matrix. Then first level optimization is to dispose of some extreme singular values to remove the ill-conditioned problem of the matrix, and then, as a second level optimization, the genetic algorithm is to remove the eigenvalues which have the negative contributions to the intensity of the selected focal point. Our experiments show that the gray value of the intensity and the signal-to-noise ratio (SNR) of the focal point after employing the phase pattern are 129 and 7.54, respectively. After the first level optimization, the gray value of the intensity and the SNR rise to 172 and 9.73, respectively. Then, they reach to 192 and 10.29, respectively, after adopting the genetic algorithm. After the second level optimizations, the intensity at the focal point increases 48.8% compared with the case with just the optimized phase pattern from the acquired TM, and the SNR increases by nearly 36.5%. The reason behind the increase of the intensity after the optimizations, we believe, is that the transmission matrix of the scattering medium reaches its actual matrix in certain conditions. The proposed approach opens the way to obtaining the actual transmission matrix by mathematic optimizations without increasing the experimental levels.
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Keywords:
- phase modulation /
- transmission matrix /
- scattering media /
- singular value
[1] Vellekoop I M, Mosk A P 2007 Opt. Lett. 32 2309
[2] Vellekoop I M, Mosk A P 2008 Opt. Commun. 281 3071
[3] Conkey D B, Brown A N, Caravaca-Aguirre A M, Piestun R 2012 Opt. Express 20 4840
[4] Booth M J, Neil M A, Juskaitis R, Wilson T 2002 Proc. NAS USA 99 5788
[5] Vellekoop I M 2008 Ph. D. Dissertation (Enschede:Univeristy of Twente) (in Netherlands)
[6] Fang L, Zhang X, Zuo H, Pang L 2018 Opt. Commun. 407 301
[7] Fang L, Zhang C, Zuo H, Zhu J, Pang L 2017 Chin. Opt. Lett. 15 102901
[8] Vellekoop I M, Aegerter C M 2010 Opt. Lett. 35 1245
[9] Vellekoop I M, Aegerter C M 2010 Proc. SPIE 7554 755430
[10] Vellekoop I M, Cui M, Yang C 2012 Appl. Phys. Lett. 101 2309
[11] Vellekoop I M, Lagendijk A, Mosk A P 2010 Nat. Photon. 4 320
[12] Vellekoop I M, Putten E G V, Lagendijk A, Mosk A P 2008 Opt. Express 16 67
[13] Popoff S, Lerosey G, Fink M, Boccara A C, Gigan S 2010 Nat. Commun. 1 81
[14] Popoff S M, Lerosey G, Carminati R, Fink M, Boccara A C, Gigan S 2010 Phys. Rev. Lett. 104 100601
[15] Popoff S M, Lerosey G, Fink M, Boccara A C, Gigan S 2011 New J. Phys. 3 1
[16] Chaigne T, Katz O, Boccara A C, Fink M, Bossy E, Gigan S 2013 Nat. Photon. 8 58
[17] Dai F 1992 IEEE Trans. Microw. Theor. Tech. 40 1538
[18] de Aguiar H B, Gigan S, Brasselet S 2016 Phys. Rev. A 94 043830
[19] Gao G F, Zhao J Z, Fu Z X 2014 Adv. Mat. Res. 1027 262
[20] Guillaume G, Fortin N 2014 J. Building Perform. Simulat. 7 445
[21] Han G, Wang T 2014 The Proceedings of the Second International Conference on Communications, Signal Processing, and Systems Tianjin, China, September 1, 2013 p383
[22] Kim M, Choi W, Choi Y, Yoon C, Choi W 2015 Opt. Express 23 12648
[23] Patil M B, Okuyama Y, Ohkura Y, Toyabe T, Ihara S 1994 Solid-State Electron. 37 1359
[24] Tripathi S, Paxman R, Bifano T, Toussaint K C 2012 Opt. Express 20 16067
[25] Akbulut D, Huisman T J, Putten E G V, Vos W L, Mosk A P 2011 Opt. Express 19 4017
[26] Conkey D B, Caravaca-Aguirre A M, Piestun R 2012 Opt. Express 20 1733
[27] Tao X, Bodington D, Reinig M, Kubby J 2015 Opt. Express 23 14168
[28] Zhang X, Kner P 2014 J. Opt. 16 125704
[29] Li Z, Cao J, Zhao X, Liu W 2015 Opt. Commun. 338 11
[30] Larrat B, Pernot M, Montaldo G, Fink M 2010 IEEE Trans. Ultrason. Ferroelectr. Frequency Control 57 1734
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[1] Vellekoop I M, Mosk A P 2007 Opt. Lett. 32 2309
[2] Vellekoop I M, Mosk A P 2008 Opt. Commun. 281 3071
[3] Conkey D B, Brown A N, Caravaca-Aguirre A M, Piestun R 2012 Opt. Express 20 4840
[4] Booth M J, Neil M A, Juskaitis R, Wilson T 2002 Proc. NAS USA 99 5788
[5] Vellekoop I M 2008 Ph. D. Dissertation (Enschede:Univeristy of Twente) (in Netherlands)
[6] Fang L, Zhang X, Zuo H, Pang L 2018 Opt. Commun. 407 301
[7] Fang L, Zhang C, Zuo H, Zhu J, Pang L 2017 Chin. Opt. Lett. 15 102901
[8] Vellekoop I M, Aegerter C M 2010 Opt. Lett. 35 1245
[9] Vellekoop I M, Aegerter C M 2010 Proc. SPIE 7554 755430
[10] Vellekoop I M, Cui M, Yang C 2012 Appl. Phys. Lett. 101 2309
[11] Vellekoop I M, Lagendijk A, Mosk A P 2010 Nat. Photon. 4 320
[12] Vellekoop I M, Putten E G V, Lagendijk A, Mosk A P 2008 Opt. Express 16 67
[13] Popoff S, Lerosey G, Fink M, Boccara A C, Gigan S 2010 Nat. Commun. 1 81
[14] Popoff S M, Lerosey G, Carminati R, Fink M, Boccara A C, Gigan S 2010 Phys. Rev. Lett. 104 100601
[15] Popoff S M, Lerosey G, Fink M, Boccara A C, Gigan S 2011 New J. Phys. 3 1
[16] Chaigne T, Katz O, Boccara A C, Fink M, Bossy E, Gigan S 2013 Nat. Photon. 8 58
[17] Dai F 1992 IEEE Trans. Microw. Theor. Tech. 40 1538
[18] de Aguiar H B, Gigan S, Brasselet S 2016 Phys. Rev. A 94 043830
[19] Gao G F, Zhao J Z, Fu Z X 2014 Adv. Mat. Res. 1027 262
[20] Guillaume G, Fortin N 2014 J. Building Perform. Simulat. 7 445
[21] Han G, Wang T 2014 The Proceedings of the Second International Conference on Communications, Signal Processing, and Systems Tianjin, China, September 1, 2013 p383
[22] Kim M, Choi W, Choi Y, Yoon C, Choi W 2015 Opt. Express 23 12648
[23] Patil M B, Okuyama Y, Ohkura Y, Toyabe T, Ihara S 1994 Solid-State Electron. 37 1359
[24] Tripathi S, Paxman R, Bifano T, Toussaint K C 2012 Opt. Express 20 16067
[25] Akbulut D, Huisman T J, Putten E G V, Vos W L, Mosk A P 2011 Opt. Express 19 4017
[26] Conkey D B, Caravaca-Aguirre A M, Piestun R 2012 Opt. Express 20 1733
[27] Tao X, Bodington D, Reinig M, Kubby J 2015 Opt. Express 23 14168
[28] Zhang X, Kner P 2014 J. Opt. 16 125704
[29] Li Z, Cao J, Zhao X, Liu W 2015 Opt. Commun. 338 11
[30] Larrat B, Pernot M, Montaldo G, Fink M 2010 IEEE Trans. Ultrason. Ferroelectr. Frequency Control 57 1734
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