Abstract

Recent theoretical and experimental advances have shed light on the existence of so-called “perfectly transmitting” wavefronts with transmission coefficients close to 1 in strongly backscattering random media. These perfectly transmitting eigen-wavefronts can be synthesized by spatial amplitude and phase modulation. Here, we consider the problem of transmission enhancement using phase-only-modulated wavefronts. Motivated by biomedical applications, in which it is not possible to measure the transmitted fields, we develop physically realizable iterative and non-iterative algorithms for increasing the transmission through such random media using backscatter analysis. We theoretically show that, despite the phase-only modulation constraint, the non-iterative algorithms will achieve at least about 25π%78.5% transmission with very high probability, assuming that there is at least one perfectly transmitting eigen-wavefront and that the singular vectors of the transmission matrix obey the maximum entropy principle such that they are isotropically random. We numerically analyze the limits of phase-only-modulated transmission in 2D with fully spectrally accurate simulators and provide rigorous numerical evidence confirming our theoretical prediction in random media, with periodic boundary conditions, that is composed of hundreds of thousands of non-absorbing scatterers. We show via numerical simulations that the iterative algorithms we have developed converge rapidly, yielding highly transmitting wavefronts while using relatively few measurements of the backscatter field. Specifically, the best performing iterative algorithm yields 70% transmission using just 15–20 measurements in the regime, where the non-iterative algorithms yield 78.5% transmission, but require measuring the entire modal reflection matrix. Our theoretical analysis and rigorous numerical results validate our prediction that phase-only modulation with a given number of spatial modes will yield higher transmission than amplitude and phase modulation with half as many modes.

© 2014 Optical Society of America

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2014 (1)

2013 (1)

2012 (2)

C. Stockbridge, Y. Lu, J. Moore, S. Hoffman, R. Paxman, K. Toussaint, and T. Bifano, “Focusing through dynamic scattering media,” Opt. Express 20, 15086–15092 (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, 583–587 (2012).
[CrossRef]

2011 (6)

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]

E. G. van Putten, A. Lagendijk, and A. P. Mosk, “Optimal concentration of light in turbid materials,” J. Opt. Am. Soc. B 28, 1200–1203 (2011).

M. Cui, “A high speed wavefront determination method based on spatial frequency modulations for focusing light through random scattering media,” Opt. Express 19, 2989–2995 (2011).
[CrossRef]

M. Cui, “Parallel wavefront optimization method for focusing light through random scattering media,” Opt. Lett. 36, 870–872 (2011).
[CrossRef]

F. Kong, R. H. Silverman, L. Liu, P. V. Chitnis, K. K. Lee, and Y.-C. Chen, “Photoacoustic-guided convergence of light through optically diffusive media,” Opt. Lett. 36, 2053–2055 (2011).
[CrossRef]

W. Choi, A. P. Mosk, Q.-H. Park, and W. Choi, “Transmission eigenchannels in a disordered medium,” Phys. Rev. B 83, 134207 (2011).
[CrossRef]

2010 (4)

T. W. Kohlgraf-Owens and A. Dogariu, “Transmission matrices of random media: means for spectral polarimetric measurements,” Opt. Lett. 35, 2236–2238 (2010).
[CrossRef]

X. Wang, W. W. Roberts, P. L. Carson, D. P. Wood, and B. J. Fowlkes, “Photoacoustic tomography: a potential new tool for prostate cancer,” Biomed. Opt. Express 1, 1117–1126 (2010).
[CrossRef]

Z.-Q. Luo, W.-K. Ma, A. M.-C. So, Y. Ye, and S. Zhang, “Semi-definite relaxation of quadratic optimization problems,” IEEE Signal Process. Mag. 27(3), 20–34 (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]

2008 (3)

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]

E. G. van Putten, I. M. Vellekoop, and A. P. Mosk, “Spatial amplitude and phase modulation using commercial twisted nematic LCDs,” Appl. Opt. 47, 2076–2081 (2008).
[CrossRef]

2007 (1)

A. M.-C. So, J. Zhang, and Y. Ye, “On approximating complex quadratic optimization problems via semi-definite programming relaxations,” Math. Program. 110, 93–110 (2007).
[CrossRef]

2006 (1)

S. Zhang and Y. Huang, “Complex quadratic optimization and semi-definite programming,” SIAM J. Optimiz. 16, 871–890 (2006).

1999 (2)

S. T. Smith, “Optimum phase-only adaptive nulling,” IEEE Trans. Signal Process. 47, 1835–1843 (1999).

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, and C. M. De Sterke, “Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders,” Phys. Rev. E 60, 7614–7617 (1999).
[CrossRef]

1998 (2)

T. J. Dougherty, C. J. Gomer, B. W. Henderson, G. Jori, M. Kessel, D. Korbelik, J. Moan, and Q. Peng, “Photodynamic therapy,” J. Natl. Cancer Inst. 90, 889–905 (1998).

Y. Nesterov, “Semi-definite relaxation and nonconvex quadratic optimization,” Optim. Method Softw. 9, 141–160 (1998).

1996 (1)

L. Vandenberghe and S. Boyd, “Semi-definite programming,” SIAM Rev. 38, 49–95 (1996).
[CrossRef]

1995 (1)

M. Goemans and D. P. Williamson, “Improved approximation algorithms for maximum cut and satisfiability problems using semi-definite programming,” J. ACM 42, 1115–1145 (1995).
[CrossRef]

1991 (2)

C. Barnes and J. B. Pendry, “Multiple scattering of waves in random media: a transfer matrix approach,” P. Roy. Soc. London A Mat. 435, 185–196 (1991).

S. Singh and R. Singh, “On the use of Shank’s transform to accelerate the summation of slowly converging series,” IEEE Trans. Microwave Theor. Tech. 39, 608–610 (1991).

1990 (1)

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

1988 (1)

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

1982 (1)

O. N. Dorokhov, “Transmission coefficient and the localization length of an electron in N bound disordered chains,” J. Exp. Theor. Phys. Lett. 36, 318–322 (1982).

1962 (1)

M. M. Siddiqui, “Some problems connected with Rayleigh distributions,” J. Res. Nat. Bur. Stand. 660, 167–174 (1962).

Asatryan, A. A.

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, and C. M. De Sterke, “Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders,” Phys. Rev. E 60, 7614–7617 (1999).
[CrossRef]

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]

Barnes, C.

C. Barnes and J. B. Pendry, “Multiple scattering of waves in random media: a transfer matrix approach,” P. Roy. Soc. London A Mat. 435, 185–196 (1991).

Bau, D.

L. N. Trefethen and D. Bau, Numerical Linear Algebra (Society for Industrial Mathematics, 1997).

Beenakker, C. W. J.

C. W. J. Beenakker, “Applications of random matrix theory to condensed matter and optical physics,” arXiv:0904.1432 (2009).

Bifano, T.

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]

Bossy, E.

Botten, L. C.

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, and C. M. De Sterke, “Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders,” Phys. Rev. E 60, 7614–7617 (1999).
[CrossRef]

Boyd, S.

L. Vandenberghe and S. Boyd, “Semi-definite programming,” SIAM Rev. 38, 49–95 (1996).
[CrossRef]

M. Grant and S. Boyd, “Graph implementations for nonsmooth convex programs,” in Recent Advances in Learning and Control: Lecture Notes in Control and Information Sciences, V. Blondel, S. Boyd, and H. Kimura, eds. (Springer, 2008), pp. 95–110.

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]

Carson, P. L.

Chaigne, T.

Chen, Y.-C.

Chitnis, P. V.

Choi, W.

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, 583–587 (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, 583–587 (2012).
[CrossRef]

W. Choi, A. P. Mosk, Q.-H. Park, and W. Choi, “Transmission eigenchannels in a disordered medium,” Phys. Rev. B 83, 134207 (2011).
[CrossRef]

W. Choi, A. P. Mosk, Q.-H. Park, and W. Choi, “Transmission eigenchannels in a disordered medium,” Phys. Rev. B 83, 134207 (2011).
[CrossRef]

Choi, Y.

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, 583–587 (2012).
[CrossRef]

Cui, M.

De Sterke, C. M.

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, and C. M. De Sterke, “Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders,” Phys. Rev. E 60, 7614–7617 (1999).
[CrossRef]

Dogariu, A.

Dorokhov, O. N.

O. N. Dorokhov, “Transmission coefficient and the localization length of an electron in N bound disordered chains,” J. Exp. Theor. Phys. Lett. 36, 318–322 (1982).

Dougherty, T. J.

T. J. Dougherty, C. J. Gomer, B. W. Henderson, G. Jori, M. Kessel, D. Korbelik, J. Moan, and Q. Peng, “Photodynamic therapy,” J. Natl. Cancer Inst. 90, 889–905 (1998).

Evans, M.

C. Forbes, M. Evans, N. Hastings, and B. Peacock, Statistical Distributions (Wiley, 2011).

Fink, 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]

Forbes, C.

C. Forbes, M. Evans, N. Hastings, and B. Peacock, Statistical Distributions (Wiley, 2011).

Fowlkes, B. J.

Gateau, J.

Genack, A. Z.

Z. Shi, J. Wang, and A. Z. Genack, “Measuring transmission eigenchannels of wave propagation through random media,” in Frontiers in Optics (Optical Society of America, 2010).

Gigan, S.

T. Chaigne, J. Gateau, O. Katz, E. Bossy, and S. Gigan, “Light focusing and two-dimensional imaging through scattering media using the photoacoustic transmission matrix with an ultrasound array,” Opt. Lett. 39, 2664–2667 (2014).
[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]

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]

Goemans, M.

M. Goemans and D. P. Williamson, “Improved approximation algorithms for maximum cut and satisfiability problems using semi-definite programming,” J. ACM 42, 1115–1145 (1995).
[CrossRef]

Golub, G. H.

G. H. Golub and C. F. Van Loan, Matrix Computations, 4th ed. (JHU, 2012).

Gomer, C. J.

T. J. Dougherty, C. J. Gomer, B. W. Henderson, G. Jori, M. Kessel, D. Korbelik, J. Moan, and Q. Peng, “Photodynamic therapy,” J. Natl. Cancer Inst. 90, 889–905 (1998).

Grant, M.

M. Grant and S. Boyd, “Graph implementations for nonsmooth convex programs,” in Recent Advances in Learning and Control: Lecture Notes in Control and Information Sciences, V. Blondel, S. Boyd, and H. Kimura, eds. (Springer, 2008), pp. 95–110.

Hastings, N.

C. Forbes, M. Evans, N. Hastings, and B. Peacock, Statistical Distributions (Wiley, 2011).

Henderson, B. W.

T. J. Dougherty, C. J. Gomer, B. W. Henderson, G. Jori, M. Kessel, D. Korbelik, J. Moan, and Q. Peng, “Photodynamic therapy,” J. Natl. Cancer Inst. 90, 889–905 (1998).

Hoffman, S.

Horn, R. A.

R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University, 1990).

Huang, Y.

S. Zhang and Y. Huang, “Complex quadratic optimization and semi-definite programming,” SIAM J. Optimiz. 16, 871–890 (2006).

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media, IEEE/OUP Series on Electromagnetic Wave Theory (IEEE, 1997).

Jin, C.

C. Jin, R. R. Nadakuditi, E. Michielssen, and S. Rand, “Iterative, backscatter-analysis algorithms for increasing transmission and focusing light through highly scattering random media,” J. Opt. Soc. Am. A 30, 1592–1602 (2013).
[CrossRef]

C. Jin, R. R. Nadakuditi, E. Michielssen, and S. Rand, “An iterative, backscatter-analysis-based algorithm for increasing transmission through a highly backscattering random medium,” in Statistical Signal Processing Workshop (SSP), 2012 (IEEE, 2012), pp. 97–100.

Johnson, C. R.

R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University, 1990).

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]

Jori, G.

T. J. Dougherty, C. J. Gomer, B. W. Henderson, G. Jori, M. Kessel, D. Korbelik, J. Moan, and Q. Peng, “Photodynamic therapy,” J. Natl. Cancer Inst. 90, 889–905 (1998).

Katz, O.

Kessel, M.

T. J. Dougherty, C. J. Gomer, B. W. Henderson, G. Jori, M. Kessel, D. Korbelik, J. Moan, and Q. Peng, “Photodynamic therapy,” J. Natl. Cancer Inst. 90, 889–905 (1998).

Kim, J.

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, 583–587 (2012).
[CrossRef]

Kim, M.

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, 583–587 (2012).
[CrossRef]

Kohlgraf-Owens, T. W.

Kong, F.

Korbelik, D.

T. J. Dougherty, C. J. Gomer, B. W. Henderson, G. Jori, M. Kessel, D. Korbelik, J. Moan, and Q. Peng, “Photodynamic therapy,” J. Natl. Cancer Inst. 90, 889–905 (1998).

Kumar, N.

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

Lagendijk, A.

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]

E. G. van Putten, A. Lagendijk, and A. P. Mosk, “Optimal concentration of light in turbid materials,” J. Opt. Am. Soc. B 28, 1200–1203 (2011).

Ledoux, M.

M. Ledoux, The Concentration of Measure Phenomenon (American Mathematical Society, 2005), Vol. 89.

Lee, K. K.

Lerosey, G.

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]

Liu, L.

Lu, Y.

Luo, Z.-Q.

Z.-Q. Luo, W.-K. Ma, A. M.-C. So, Y. Ye, and S. Zhang, “Semi-definite relaxation of quadratic optimization problems,” IEEE Signal Process. Mag. 27(3), 20–34 (2010).
[CrossRef]

Ma, W.-K.

Z.-Q. Luo, W.-K. Ma, A. M.-C. So, Y. Ye, and S. Zhang, “Semi-definite relaxation of quadratic optimization problems,” IEEE Signal Process. Mag. 27(3), 20–34 (2010).
[CrossRef]

MacKinnon, A.

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

McPhedran, R. C.

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, and C. M. De Sterke, “Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders,” Phys. Rev. E 60, 7614–7617 (1999).
[CrossRef]

Mello, P. A.

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

Michielssen, E.

C. Jin, R. R. Nadakuditi, E. Michielssen, and S. Rand, “Iterative, backscatter-analysis algorithms for increasing transmission and focusing light through highly scattering random media,” J. Opt. Soc. Am. A 30, 1592–1602 (2013).
[CrossRef]

C. Jin, R. R. Nadakuditi, E. Michielssen, and S. Rand, “An iterative, backscatter-analysis-based algorithm for increasing transmission through a highly backscattering random medium,” in Statistical Signal Processing Workshop (SSP), 2012 (IEEE, 2012), pp. 97–100.

Moan, J.

T. J. Dougherty, C. J. Gomer, B. W. Henderson, G. Jori, M. Kessel, D. Korbelik, J. Moan, and Q. Peng, “Photodynamic therapy,” J. Natl. Cancer Inst. 90, 889–905 (1998).

Moore, J.

Mosk, A. P.

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]

E. G. van Putten, A. Lagendijk, and A. P. Mosk, “Optimal concentration of light in turbid materials,” J. Opt. Am. Soc. B 28, 1200–1203 (2011).

W. Choi, A. P. Mosk, Q.-H. Park, and W. Choi, “Transmission eigenchannels in a disordered medium,” Phys. Rev. B 83, 134207 (2011).
[CrossRef]

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

E. G. van Putten, I. M. Vellekoop, and A. P. Mosk, “Spatial amplitude and phase modulation using commercial twisted nematic LCDs,” Appl. Opt. 47, 2076–2081 (2008).
[CrossRef]

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

Nadakuditi, R. R.

C. Jin, R. R. Nadakuditi, E. Michielssen, and S. Rand, “Iterative, backscatter-analysis algorithms for increasing transmission and focusing light through highly scattering random media,” J. Opt. Soc. Am. A 30, 1592–1602 (2013).
[CrossRef]

C. Jin, R. R. Nadakuditi, E. Michielssen, and S. Rand, “An iterative, backscatter-analysis-based algorithm for increasing transmission through a highly backscattering random medium,” in Statistical Signal Processing Workshop (SSP), 2012 (IEEE, 2012), pp. 97–100.

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Y. Nesterov, H. Wolkowicz, and Y. Ye, “Semi-definite programming relaxations of nonconvex quadratic optimization,” in Handbook of Semidefinite Programming (Springer, 2000), pp. 361–419.

Nicorovici, N. A.

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, and C. M. De Sterke, “Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders,” Phys. Rev. E 60, 7614–7617 (1999).
[CrossRef]

Park, Q.-H.

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, 583–587 (2012).
[CrossRef]

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C. Barnes and J. B. Pendry, “Multiple scattering of waves in random media: a transfer matrix approach,” P. Roy. Soc. London A Mat. 435, 185–196 (1991).

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

Peng, Q.

T. J. Dougherty, C. J. Gomer, B. W. Henderson, G. Jori, M. Kessel, D. Korbelik, J. Moan, and Q. Peng, “Photodynamic therapy,” J. Natl. Cancer Inst. 90, 889–905 (1998).

Pereyra, P.

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

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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).
[CrossRef]

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

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C. Jin, R. R. Nadakuditi, E. Michielssen, and S. Rand, “Iterative, backscatter-analysis algorithms for increasing transmission and focusing light through highly scattering random media,” J. Opt. Soc. Am. A 30, 1592–1602 (2013).
[CrossRef]

C. Jin, R. R. Nadakuditi, E. Michielssen, and S. Rand, “An iterative, backscatter-analysis-based algorithm for increasing transmission through a highly backscattering random medium,” in Statistical Signal Processing Workshop (SSP), 2012 (IEEE, 2012), pp. 97–100.

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R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, and C. M. De Sterke, “Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders,” Phys. Rev. E 60, 7614–7617 (1999).
[CrossRef]

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Z. Shi, J. Wang, and A. Z. Genack, “Measuring transmission eigenchannels of wave propagation through random media,” in Frontiers in Optics (Optical Society of America, 2010).

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S. T. Smith, “Optimum phase-only adaptive nulling,” IEEE Trans. Signal Process. 47, 1835–1843 (1999).

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Z.-Q. Luo, W.-K. Ma, A. M.-C. So, Y. Ye, and S. Zhang, “Semi-definite relaxation of quadratic optimization problems,” IEEE Signal Process. Mag. 27(3), 20–34 (2010).
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A. M.-C. So, J. Zhang, and Y. Ye, “On approximating complex quadratic optimization problems via semi-definite programming relaxations,” Math. Program. 110, 93–110 (2007).
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K. H. Toh, M. J. Todd, and R. H. Tutuncu, MATLAB, SDPT3 version 4.0.

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E. G. van Putten, A. Lagendijk, and A. P. Mosk, “Optimal concentration of light in turbid materials,” J. Opt. Am. Soc. B 28, 1200–1203 (2011).

E. G. van Putten, I. M. Vellekoop, and A. P. Mosk, “Spatial amplitude and phase modulation using commercial twisted nematic LCDs,” Appl. Opt. 47, 2076–2081 (2008).
[CrossRef]

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L. Vandenberghe and S. Boyd, “Semi-definite programming,” SIAM Rev. 38, 49–95 (1996).
[CrossRef]

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E. G. van Putten, I. M. Vellekoop, and A. P. Mosk, “Spatial amplitude and phase modulation using commercial twisted nematic LCDs,” Appl. Opt. 47, 2076–2081 (2008).
[CrossRef]

I. M. Vellekoop and A. P. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett. 101, 120601 (2008).
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I. M. Vellekoop and A. P. Mosk, “Phase control algorithms for focusing light through turbid media,” Opt. Commun. 281, 3071–3080 (2008).
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Z. Shi, J. Wang, and A. Z. Genack, “Measuring transmission eigenchannels of wave propagation through random media,” in Frontiers in Optics (Optical Society of America, 2010).

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Y. Nesterov, H. Wolkowicz, and Y. Ye, “Semi-definite programming relaxations of nonconvex quadratic optimization,” in Handbook of Semidefinite Programming (Springer, 2000), pp. 361–419.

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Z.-Q. Luo, W.-K. Ma, A. M.-C. So, Y. Ye, and S. Zhang, “Semi-definite relaxation of quadratic optimization problems,” IEEE Signal Process. Mag. 27(3), 20–34 (2010).
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A. M.-C. So, J. Zhang, and Y. Ye, “On approximating complex quadratic optimization problems via semi-definite programming relaxations,” Math. Program. 110, 93–110 (2007).
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Y. Nesterov, H. Wolkowicz, and Y. Ye, “Semi-definite programming relaxations of nonconvex quadratic optimization,” in Handbook of Semidefinite Programming (Springer, 2000), pp. 361–419.

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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, 583–587 (2012).
[CrossRef]

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A. M.-C. So, J. Zhang, and Y. Ye, “On approximating complex quadratic optimization problems via semi-definite programming relaxations,” Math. Program. 110, 93–110 (2007).
[CrossRef]

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Z.-Q. Luo, W.-K. Ma, A. M.-C. So, Y. Ye, and S. Zhang, “Semi-definite relaxation of quadratic optimization problems,” IEEE Signal Process. Mag. 27(3), 20–34 (2010).
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S. Zhang and Y. Huang, “Complex quadratic optimization and semi-definite programming,” SIAM J. Optimiz. 16, 871–890 (2006).

Ann. Phys. (1)

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

Appl. Opt. (1)

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Z.-Q. Luo, W.-K. Ma, A. M.-C. So, Y. Ye, and S. Zhang, “Semi-definite relaxation of quadratic optimization problems,” IEEE Signal Process. Mag. 27(3), 20–34 (2010).
[CrossRef]

IEEE Trans. Microwave Theor. Tech. (1)

S. Singh and R. Singh, “On the use of Shank’s transform to accelerate the summation of slowly converging series,” IEEE Trans. Microwave Theor. Tech. 39, 608–610 (1991).

IEEE Trans. Signal Process. (1)

S. T. Smith, “Optimum phase-only adaptive nulling,” IEEE Trans. Signal Process. 47, 1835–1843 (1999).

J. ACM (1)

M. Goemans and D. P. Williamson, “Improved approximation algorithms for maximum cut and satisfiability problems using semi-definite programming,” J. ACM 42, 1115–1145 (1995).
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J. Opt. Am. Soc. B (1)

E. G. van Putten, A. Lagendijk, and A. P. Mosk, “Optimal concentration of light in turbid materials,” J. Opt. Am. Soc. B 28, 1200–1203 (2011).

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

J. Res. Nat. Bur. Stand. (1)

M. M. Siddiqui, “Some problems connected with Rayleigh distributions,” J. Res. Nat. Bur. Stand. 660, 167–174 (1962).

Math. Program. (1)

A. M.-C. So, J. Zhang, and Y. Ye, “On approximating complex quadratic optimization problems via semi-definite programming relaxations,” Math. Program. 110, 93–110 (2007).
[CrossRef]

Nat. Photonics (1)

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, 583–587 (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 (2)

Opt. Lett. (4)

Optim. Method Softw. (1)

Y. Nesterov, “Semi-definite relaxation and nonconvex quadratic optimization,” Optim. Method Softw. 9, 141–160 (1998).

P. Roy. Soc. London A Mat. (1)

C. Barnes and J. B. Pendry, “Multiple scattering of waves in random media: a transfer matrix approach,” P. Roy. Soc. London A Mat. 435, 185–196 (1991).

Phys. Rev. B (1)

W. Choi, A. P. Mosk, Q.-H. Park, and W. Choi, “Transmission eigenchannels in a disordered medium,” Phys. Rev. B 83, 134207 (2011).
[CrossRef]

Phys. Rev. E (1)

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, and C. M. De Sterke, “Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders,” Phys. Rev. E 60, 7614–7617 (1999).
[CrossRef]

Phys. Rev. Lett. (3)

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

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]

Physica A (1)

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

SIAM J. Optimiz. (1)

S. Zhang and Y. Huang, “Complex quadratic optimization and semi-definite programming,” SIAM J. Optimiz. 16, 871–890 (2006).

SIAM Rev. (1)

L. Vandenberghe and S. Boyd, “Semi-definite programming,” SIAM Rev. 38, 49–95 (1996).
[CrossRef]

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K. H. Toh, M. J. Todd, and R. H. Tutuncu, MATLAB, SDPT3 version 4.0.

G. H. Golub and C. F. Van Loan, Matrix Computations, 4th ed. (JHU, 2012).

Y. Nesterov, H. Wolkowicz, and Y. Ye, “Semi-definite programming relaxations of nonconvex quadratic optimization,” in Handbook of Semidefinite Programming (Springer, 2000), pp. 361–419.

C. R. Rao, Linear Statistical Inference and Its Applications, 2nd ed. Probability and Mathematical Statistics Series (Wiley, 1973).

A. Sidi, “Practical extrapolation methods,” in Cambridge Monographs on Applied and Computational Mathematics: Theory and Applications (Cambridge University, 2003), Vol. 10.

C. Forbes, M. Evans, N. Hastings, and B. Peacock, Statistical Distributions (Wiley, 2011).

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L. N. Trefethen and D. Bau, Numerical Linear Algebra (Society for Industrial Mathematics, 1997).

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Z. Shi, J. Wang, and A. Z. Genack, “Measuring transmission eigenchannels of wave propagation through random media,” in Frontiers in Optics (Optical Society of America, 2010).

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C. Jin, R. R. Nadakuditi, E. Michielssen, and S. Rand, “An iterative, backscatter-analysis-based algorithm for increasing transmission through a highly backscattering random medium,” in Statistical Signal Processing Workshop (SSP), 2012 (IEEE, 2012), pp. 97–100.

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Figures (10)

Fig. 1.
Fig. 1.

Schematic for the experimental setup considered.

Fig. 2.
Fig. 2.

Geometry of the scattering system considered.

Fig. 3.
Fig. 3.

Plot of transmitted power obtained by the SVD- and SDP-based algorithms versus L/λ in a setting where D=197λ, r=0.11λ, nd=1.3, and M=395. The system was generated so that when L=3.4×105λ, Nc=430,000, and l¯=6.69λ, where l¯ is the average distance to the nearest scatterer. The empirical average and the one-standard-deviation error bars were computed over 1700 random realizations of the scattering medium.

Fig. 4.
Fig. 4.

Plot of transmitted power obtained by the a̲opt,sdp, a̲equal, and the amplitude- and phase-modulated wavefront corresponding to the largest right singular vector of the undersampled (by four) modal transmission matrix versus L/λ for the same setup, as in Fig. 3.

Fig. 5.
Fig. 5.

Average transmitted power versus the number of iterations is shown for steepest descent algorithm, the phase-only steepest descent algorithm and the phase-only gradient descent algorithm for the setup described in Fig. 3. Here, L=3.4×105λ. For each of these algorithms the optimal step size μ was chosen by a line search.

Fig. 6.
Fig. 6.

Average transmitted power obtained after 50 iterations of the phase-only steepest descent (SD) and gradient descent (GD) methods as a function of L/λ for the setup described in Fig. 3. For comparison, we plot the transmitted power realized by a̲opt,sdp.

Fig. 7.
Fig. 7.

Average number of iterations to get to 95% of the respective maximum transmitted power for the phase-only steepest descent and gradient descent algorithms as a function of L/λ for the setup described in Fig. 3.

Fig. 8.
Fig. 8.

Heatmap of the average transmitted power attained by the phase-only steepest descent algorithm as a function of the number of iterations and stepsize μ for the same setup as in Fig. 5.

Fig. 9.
Fig. 9.

Heatmap of the maximum transmitted power in 50 iterations of steepest descent on the plane of stepsize and the thickness L/λ for the setup in Fig. 3.

Fig. 10.
Fig. 10.

Gain in transmitted power relative to a̲equal versus thickness L/λ for a system setup as described in Fig. 3 except with nd=1.3jκ, where κ is the extinction coefficient.

Tables (3)

Tables Icon

Algorithm 1 Steepest Descent Algorithm for Finding a̲svd

Tables Icon

Table 1. Phase-only Steepest Descent Algorithma

Tables Icon

Table 2. Phase-only Gradient Descent Algorithma

Equations (64)

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ei±(ρ̲)=n=NNhnai,n±ejk̲n±·ρ̲.
[a̲1a̲2+]=[S11S12S21S22]S[a̲1+a̲2],
τ(a̲1+):=S21·a̲1+22a̲1+22,
γ(a̲1+):=S11·a̲1+22a̲1+22,
a̲1+=[a1,N+a1,0+a1,+N+]T,
p̲(θ̲)=1M[ejθNejθ0ejθN]T,
a̲opt=argmaxa̲1+P1Mτ(a̲1+)=argmaxa̲1+P1MS21·a̲1+22.
S11H·S11+S21H·S21=I,
a̲opt=argmina̲1+P1MS11·a̲1+2=argmina̲1+P1Mγ(a̲1+).
a̲svd=argmina̲1+2=1S11·a̲1+22,
a̲svd=v̲˜M=v̲1.
a̲opt,svd=p̲(a̲svd).
S11·a̲1+22=((a̲1+)H·S11H·S11·a̲1+)=Tr(S11H·S11·a̲1+·(a̲1+)H),
Aopt=argminACM×MTr(S11H·S11·A)subject toA=AH,A0,rank(A)=1andAii=1/Mfori=1,M,
Asdp=argminACM×MTr(S11H·S11·A)subject toA=AH,A0,andAii=1/Mfori=1,M,
a̲opt,sdp=p̲(u̲1,sdp).
a̲˜opt,sdp=p̲((i=1Mλiu̲i,sdp·u̲i,sdpH)·z),
π4τ(a̲opt)Ez[τ(a̲˜opt,sdp)]τ(a̲opt)1.
f(τ)=limM1Mi=1Mδ(ττ(v̲i))=l2L1τ1τ,for4exp(L/2l)τ1.
τ(a̲opt,svd)=τ(p̲(a̲svd)=S21·p̲(a̲svd)22,
=U·Σ·VH·p̲(a̲svd)22=Σ·VH·p̲(a̲svd)22,
τ(a̲opt,svd)=Σ·p̲˜(a̲svd)22,
=i=1Mσi2|p˜i(a̲svd)|2σ12|p˜1(a̲svd)|2.
τ(a̲opt,svd)|p˜1(a̲svd)|2.
v̲1H=[|v1,1|ejv1,1|v1,M|ejv1,M],
p˜1(a̲svd)=v̲1H·p̲(v1)=1Mi=1M|v1,i|,
|p˜1(a̲svd)|2=1Mi=1M|v1,i|2+2Mi<j|v1,i|·|v1,j|.
E[|p˜1(a̲svd)|2]=1Mi=1ME[|v1,i|2]+2Mi<jE[|v1,i|·|v1,j|].
E[|p˜1(a̲svd)|2]=E[|v1,1|2]+2M(M1)2ME[|v1,1|·|v1,2|].
E[|v1,1|2]=1M.
E[|p˜1(a̲svd)|2]=(M1)E[|v1,1|·|v1,2|]+1M.
v1,i=dgi|g1|2++|gM|2,
fri(r)=rer22forr0.
fs3(r)=23M·r2M5er22Γ(M2)forr0.
E[|v1,1|·|v1,2|]=E[|g1|·|g2||g1|2+|g2|2+(|g3|2++|gM|2)]=E[r1·r2r12+r22+s32]=000r1·r2r12+r22+s32r1er122·r2er222·23M·s32M5es322Γ(M2)dr1dr2ds3=23MΓ(M2)000r12·r22·s32M5r12+r22+s32er12+r22+s322dr1dr2ds3.
E[|v1,1|·|v1,2|]=23MΓ(M2)×0π20π20cos2(p)(rcos(t))2Msec(t)sin2(p)tan4(t)r3er22drdtdp=π4M.
E[|p˜1(a̲svd)|2]=π4+4π4M.
E[τ(a̲opt,svd)]π4+4π4M.
E[τ(a̲opt,sdp)]E[τ(a̲opt,svd)]π4+4π4M.
limME[τ(a̲opt,sdp)]limME[τ(a̲opt,svd)]π4.
i=1M|xi+δi|i=1M|xi|1·i=1M|δi|.
P(M|p˜1(a̲svd)E[p˜1(a̲svd)]|ϵ)Ce(cMϵ2),
P(|p˜1(a̲svd)E[p˜1(a̲svd)]|ϵ)Ce(cM2ϵ2).
a˜̲1,(k)+=a̲1,(k)+μS11·a̲1+22a̲1+|a̲1+=a̲1,(k)+
=a̲1,(k)+2μS11H·S11·a̲1,(k)+,
a̲1,(k+1)+=(I2μS11H·S11)·a̲1,(k)+(I2μS11H·S11)·a̲1,(k)+2.
S11H=F·S11*·F,
S11H·a̲1=F·S11*·F·a̲1=F·(S11·(F·(a̲1)*))*.
a˜̲1,(k)+=p̲(a̲1,(k)+2μa¯S11H·p̲(S11·a̲1,(k)+)),
θ̲1,(k+1)+=θ̲1,(k)+MμS11·p̲(θ̲)22θ̲|θ̲=θ̲1,(k)+,
S11·p̲(θ̲)22θ̲|θ̲=θ̲1,(k)+=2Im[diag{p̲(θ̲1,(k)+)}·S11H·S11·p̲(θ̲1,(k)+)],
S11·p̲(θ̲)22θ̲|θ̲=θ̲1,(k)+2=2Im[diag{p̲(θ̲1,(k)+)}·S11H·S11·p̲(θ̲1,(k)+)]2,2diag{p̲(θ̲1,(k)+)}2·σ˜1221M·1=2M.
θ̲1,(k+1)+=θ̲1,(k)+2MμIm[diag{p̲(θ̲1,(k)+)}·S11H·S11·p̲(θ̲1,(k)+)].
θ̲1,(k+1)+=θ̲1,(k)+2Mμa¯Im[diag{p̲(θ̲1,(k)+)}·S11H·p̲(S11·p̲(θ̲1,(k)+))],
μopt=argmaxμminμμmaxk=050τ(a̲1,(k)+,μ).
Asdp=argmaxACM×MTr(S21H·S21·A)subject toA=AH,A0,andAii=1/Mfori=1,M.
B·p̲(θ̲)22θ̲=2Im[diag{p̲(θ̲)}·BH·B·p̲(θ̲)].
B·p̲(θ̲)22=n=1M|Bnmejθm|2=n=1Mm=1M|Bnm|2+2n=1Mp>qRe(BnpBnq*ej(θpθq))=n=1Mm=1M|Bnm|2+2n=1Mp>q|Bnp||Bnq|cos(θpθq+BnpBnq),
B·p̲(θ̲)22θk=2n=1MqkIm[BnkBnq*ej(θkθq)]
=2Im[ejθkn=1MBnkqkBnq*ejθq],
B·p̲(θ̲)22θk=2Im[ejθk[B1kBMk]·B*·{Ie̲k·e̲kH}·p̲(θ̲)*],
B·p̲(θ̲)22θk=2Im[ejθk[B1kBMk]·B*·p̲(θ̲)*]2Im[[B1kBMk]·B*·e̲k]
=2Im[ejθk[B1kBMk]·B*·p̲(θ̲)*].
B·p̲(θ̲)22θ̲=2Im[diag{p̲(θ̲)}·BT·B*·p̲(θ̲)*],

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