Abstract

Scattering hinders the passage of light through random media and consequently limits the usefulness of optical techniques for sensing and imaging. Thus, methods for increasing the transmission of light through such random media are of interest. Against this backdrop, recent theoretical and experimental advances have suggested the existence of a few highly transmitting eigen-wavefronts with transmission coefficients close to 1 in strongly backscattering random media. Here, we numerically analyze this phenomenon in 2D with fully spectrally accurate simulators and provide rigorous numerical evidence confirming the existence of these highly transmitting eigen-wavefronts in random media with periodic boundary conditions that are composed of hundreds of thousands of nonabsorbing scatterers. Motivated by bio-imaging applications in which it is not possible to measure the transmitted fields, we develop physically realizable algorithms for increasing the transmission through such random media using backscatter analysis. We show via numerical simulations that the algorithms converge rapidly, yielding a near-optimum wavefront in just a few iterations. We also develop an algorithm that combines the knowledge of these highly transmitting eigen-wavefronts obtained from backscatter analysis with intensity measurements at a point to produce a near-optimal focus with significantly fewer measurements than a method that does not utilize this information.

© 2013 Optical Society of America

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    [CrossRef]
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2012 (2)

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

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]

2011 (5)

2010 (5)

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]

2004 (2)

R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. 67, 717–754 (2004).
[CrossRef]

M. F. Yanik and S. Fan, “Time reversal of light with linear optics and modulators,” Phys. Rev. Lett. 93, 173903 (2004).
[CrossRef]

2001 (1)

2000 (1)

R. Carminati, J. J. Saenz, J.-J. Greffet, and M. Nieto-Vesperinas, “Reciprocity, unitarity, and time-reversal symmetry of the s matrix of fields containing evanescent components,” Phys. Rev. A 62, 012712 (2000).
[CrossRef]

1999 (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]

1994 (1)

C. Prada and M. Fink, “Eigenmodes of the time reversal operator: a solution to selective focusing in multiple-target media,” Wave Motion 20, 151–163 (1994).
[CrossRef]

1993 (1)

M. Fink, “Time-reversal mirrors,” J. Phys. D 26, 1333–1350 (1993).
[CrossRef]

1991 (2)

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).
[CrossRef]

C. Barnes and J. B. Pendry, “Multiple scattering of waves in random media: a transfer matrix approach,” Proc. R. Soc. London Ser. A Math. Phys. Eng. Sci. 435, 185–196 (1991).
[CrossRef]

1990 (1)

J. B. Pendry, A. MacKinnon, and A. B. Pretre, “Maximal fluctuations—a new phenomenon in disordered systems,” Phys. 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,” JETP Lett. 36, 458 (1982).

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,” Proc. R. Soc. London Ser. A Math. Phys. Eng. Sci. 435, 185–196 (1991).
[CrossRef]

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 preprint 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]

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]

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]

R. Carminati, J. J. Saenz, J.-J. Greffet, and M. Nieto-Vesperinas, “Reciprocity, unitarity, and time-reversal symmetry of the s matrix of fields containing evanescent components,” Phys. Rev. A 62, 012712 (2000).
[CrossRef]

Cassereau, D.

M. Fink, C. Prada, F. Wu, and D. Cassereau, “Self focusing in inhomogeneous media with time reversal acoustic mirrors,” in Proceedings of 1989 IEEE Ultrasonics Symposium (IEEE, 1989), pp. 681–686.

Choi, W.

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

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

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.-Han. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photonics 6, 581–585 (2012).

Cui, M.

De Nicola, S.

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,” JETP Lett. 36, 458 (1982).

Fan, S.

M. F. Yanik and S. Fan, “Time reversal of light with linear optics and modulators,” Phys. Rev. Lett. 93, 173903 (2004).
[CrossRef]

Ferraro, P.

Finizio, A.

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]

C. Prada and M. Fink, “Eigenmodes of the time reversal operator: a solution to selective focusing in multiple-target media,” Wave Motion 20, 151–163 (1994).
[CrossRef]

M. Fink, “Time-reversal mirrors,” J. Phys. D 26, 1333–1350 (1993).
[CrossRef]

M. Fink, C. Prada, F. Wu, and D. Cassereau, “Self focusing in inhomogeneous media with time reversal acoustic mirrors,” in Proceedings of 1989 IEEE Ultrasonics Symposium (IEEE, 1989), pp. 681–686.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes 3rd Edition: The Art of Scientific Computing (Cambridge University, 2007).

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.

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]

Greffet, J.-J.

R. Carminati, J. J. Saenz, J.-J. Greffet, and M. Nieto-Vesperinas, “Reciprocity, unitarity, and time-reversal symmetry of the s matrix of fields containing evanescent components,” Phys. Rev. A 62, 012712 (2000).
[CrossRef]

Grilli, S.

Haus, H. A.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984), Vol. 1.

Hoffman, S.

Horn, R. A.

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

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Wiley, 1999), Vol. 12.

Jin, C.

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 2012 IEEE Statistical Signal Processing Workshop (SSP) (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]

Kim, J.

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

Kim, M.

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

Kohlgraf-Owens, T. W.

Kong, J. A.

J. A. Kong, Electromagnetic Wave Theory (Wiley, 1986).

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.

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

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

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]

Lu, Y.

MacKinnon, A.

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

McDowell, E. J.

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]

Meucci, R.

Michielssen, E.

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 2012 IEEE Statistical Signal Processing Workshop (SSP) (IEEE, 2012), pp. 97–100.

Moore, J.

Mosk, A. P.

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

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

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

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]

Nadakuditi, R. R.

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 2012 IEEE Statistical Signal Processing Workshop (SSP) (IEEE, 2012), pp. 97–100.

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]

Nieto-Vesperinas, M.

R. Carminati, J. J. Saenz, J.-J. Greffet, and M. Nieto-Vesperinas, “Reciprocity, unitarity, and time-reversal symmetry of the s matrix of fields containing evanescent components,” Phys. Rev. A 62, 012712 (2000).
[CrossRef]

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, 1991).

Park, Q.-H.

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

Park, Q.-Han.

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

Paxman, R.

Pendry, J. B.

C. Barnes and J. B. Pendry, “Multiple scattering of waves in random media: a transfer matrix approach,” Proc. R. Soc. London Ser. A Math. Phys. Eng. Sci. 435, 185–196 (1991).
[CrossRef]

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

Pereyra, P.

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

Pierattini, G.

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]

Potton, R. J.

R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. 67, 717–754 (2004).
[CrossRef]

Prada, C.

C. Prada and M. Fink, “Eigenmodes of the time reversal operator: a solution to selective focusing in multiple-target media,” Wave Motion 20, 151–163 (1994).
[CrossRef]

M. Fink, C. Prada, F. Wu, and D. Cassereau, “Self focusing in inhomogeneous media with time reversal acoustic mirrors,” in Proceedings of 1989 IEEE Ultrasonics Symposium (IEEE, 1989), pp. 681–686.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes 3rd Edition: The Art of Scientific Computing (Cambridge University, 2007).

Pretre, A. B.

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

Rand, S.

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 2012 IEEE Statistical Signal Processing Workshop (SSP) (IEEE, 2012), pp. 97–100.

Robinson, P. 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]

Saenz, J. J.

R. Carminati, J. J. Saenz, J.-J. Greffet, and M. Nieto-Vesperinas, “Reciprocity, unitarity, and time-reversal symmetry of the s matrix of fields containing evanescent components,” Phys. Rev. A 62, 012712 (2000).
[CrossRef]

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W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes 3rd Edition: The Art of Scientific Computing (Cambridge University, 2007).

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[CrossRef]

Opt. Commun. (1)

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Figures (13)

Fig. 1.
Fig. 1.

Geometry of the scattering system considered.

Fig. 2.
Fig. 2.

Theoretical distribution in Eq. (7) for L / l = 3 .

Fig. 3.
Fig. 3.

Relationship between wavefronts in a medium that exhibits reciprocity. Reciprocity tells us that S 11 H · a ̲ is obtained by time reversing the wave before and after sending a ̲ into the medium, and we call this sequence of operations double phase conjugation.

Fig. 4.
Fig. 4.

Empirical transmission coefficients distribution from a scattering system with D = 197 λ , L = 1.2 × 10 4 λ , r = 0.11 λ , N c = 14,000 (dielectric), n d = 1.3 , M = 395 , l ¯ = 6.7 λ , where l ¯ is the mean of the minimum-inter-scatterer-distances.

Fig. 5.
Fig. 5.

Wavefield plot of the incident-plus-backscatter wave corresponding to (a) normally incident wavefront and (b) optimal wavefront, which were sent to a scattering system with D = 14 λ , L = 5.4 λ , r = 0.11 λ , N c = 50 PEC , M = 27 , l ¯ = 0.8 λ . The normally incident wavefront has τ normal = 0.483 , while the optimal wavefront yields τ opt = 0.9997 .

Fig. 6.
Fig. 6.

Modal coefficients of the optimal wavefront corresponding to Fig. 5(b) are shown.

Fig. 7.
Fig. 7.

Transmitted power versus the number of iterations is shown for steepest descent algorithm with μ = 0.5037 and for conjugate gradient in the setting with D = 197 λ , L = 3.4 × 10 5 λ , r = 0.11 λ , N c = 430 , 000 dielectric cylinders with n d = 1.3 , M = 395 , l ¯ = 6.69 λ . The conjugate gradient algorithm converged to the optimal transmitted power slightly faster than the steepest descent algorithm. However, since the steepest descent algorithm requires a line search for setting the optimal step size μ , it requires more measurements than the conjugate gradient method, which does not require any parameters to be set.

Fig. 8.
Fig. 8.

Transmitted power at the tenth iteration as a function of the step size μ used in Algorithm 1 for the same setting as in Fig. 7.

Fig. 9.
Fig. 9.

Gain ( τ opt / τ normal ) versus the number of control modes for the same setting as in Fig. 7. Here we compute the realized gain for algorithms that control only part of the total number of modes, but capture (1) all modes in the backscatter field, (2) only as many modes in the transmitted field as the number of control modes, and (3) only as many modes in the backscatter field as the number of control modes. For the last algorithm, we transmit the eigen-wavefront of the (portion of the) S 11 matrix that yields the highest transmission.

Fig. 10.
Fig. 10.

Intensity plot around the target at ( D / 2 , L + 5.4 λ ) for the scattering system defined in Fig. 7. The optimal focusing wavefront forms a sharp focus of 1 λ around the target. The unoptimized wavefront solution corresponds to an incident wavefront that would have produced a focus at the target if there were no intervening scattering medium.

Fig. 11.
Fig. 11.

Here, we depict the magnitude of the coefficients of the optimal focusing wavefront, corresponding to the situation in Fig. 10, in terms of two choices of basis vectors. In (a) we decompose the optimal focusing wavefront with respect to the basis vectors corresponding to plane waves; in (b) we decompose the optimal focusing wavefront with respect to the basis vectors associated with the eigen-wavefronts of the S 11 matrix. A particular important observation is that the eigen-wavefront decomposition yields a sparse representation of the optimal focusing wavefront.

Fig. 12.
Fig. 12.

Intensity at target as a function of the number of basis vectors for the new algorithm [which uses the basis vectors estimated using Eq. (21) and the algorithm described in Table 3] for different number of control modes versus the standard coordinate descent method, which uses the plane wave associated basis vectors (see Section 6) for the same setting as in Fig. 10. The sparsity of the optimal wavefront’s modal coefficient vector when expressed using the basis of the eigen-wavefronts [shown in Fig. 11(b)] leads to the rapid convergence observed. The optimal wavefront was constructed as described in Section 3.B using time reversal. The number of basis vectors needed to attain 95% of the optimal focus intensity for a given number of control modes is indicated with a vertical line highlighting the fast convergence of the algorithm and the ability to obtain a near-optimal focus using significantly fewer measurements than the coordinate descent approach.

Fig. 13.
Fig. 13.

Gain ( τ opt / τ normal ) versus thickness L / λ in a setting with D = 197 λ , r = 0.11 λ , N c = 430 , 000 Absorbing Dielectric, n d = 1.3 j κ , M = 395 , l ¯ = 6.69 λ , for different values of κ . The solid line represents the maximum possible gain, and the dashed line represents the gain obtained using the backscatter-minimizing algorithm discussed in Section 5.

Tables (4)

Tables Icon

Table 1 Algorithm 1: Steepest Descent Algorithm for Finding a _ opt

Tables Icon

Table 1. Steepest Descent Algorithm for Transmission Maximizationa

Tables Icon

Table 2. Conjugate Gradient Algorithm for Transmission Maximizationa

Tables Icon

Table 3. Lanzcos Algorithm and Its Physical Counterpart That Computes a Tridiagonal Matrix H Whose Eigenvalues and Eigenvectors Are Closely Related to the Eigenvalues and Eigenvectors of S 11 H · S 11 a

Equations (28)

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e i ± ( ρ ̲ ) = n = N N h n a i , n ± e j k ̲ n ± · ρ ̲ .
[ a ̲ 1 a ̲ 2 + ] = [ S 11 S 12 S 21 S 22 ] S [ a ̲ 1 + a ̲ 2 ] ,
τ ( a ̲ 1 + ) S 21 · a ̲ 1 + 2 2 a ̲ 1 + 2 2
Γ ( a ̲ 1 + ) S 11 · a ̲ 1 + 2 2 a ̲ 1 + 2 2 ,
a ̲ opt = arg max a ̲ 1 + τ ( a ̲ 1 + ) = arg max a ̲ 1 + S 21 · a ̲ 1 + 2 2 a ̲ 1 + 2 2 = arg max a ̲ 1 + 2 = 1 S 21 · a ̲ 1 + 2 2 ,
a ̲ opt = v ̲ 1 .
f ( τ ) = lim M 1 M i = 1 M δ ( τ τ ( v ̲ i ) ) = l 2 L 1 τ 1 τ , for 4 exp ( L / 2 l ) τ 1 .
a ̲ opt = arg max a ̲ 1 + 2 = 1 ( a ̲ 1 + ) H · S 21 H · S 21 · a ̲ 1 + = ( a ̲ 1 + ) H · ( I S 11 H · S 11 ) · a ̲ 1 + = arg min a ̲ 1 + 2 = 1 S 11 · a ̲ 1 + 2 2 = arg min a ̲ 1 + Γ ( a ̲ 1 + ) .
a ̲ opt = v ̲ ˜ M .
e 2 + ( ρ ̲ 0 ) = [ h N e j k ̲ N + · ρ ̲ 0 h N e j k ̲ N + · ρ ̲ 0 ] f ̲ ( ρ ̲ 0 ) H · S 21 · a ̲ 1 + .
a ̲ foc = arg max a ̲ 1 + e 2 + ( ρ ̲ 0 ) 2 2 a ̲ 1 + 2 2 = arg max a ̲ 1 + 2 = 1 f ̲ H ( ρ ̲ 0 ) · S 21 c ̲ ( ρ ̲ 0 ) H · a ̲ 1 + 2 2 ,
a ̲ foc = c ̲ ( ρ ̲ 0 ) c ̲ ( ρ ̲ 0 ) 2 = S 21 H · f ̲ ( ρ ̲ 0 ) S 21 H · f ̲ ( ρ ̲ 0 ) 2 .
a ̲ foc S 21 H · f ̲ ( ρ ̲ 0 ) = i = 1 M σ i ( u ̲ i H · f ̲ ( ρ ̲ 0 ) ) w i v ̲ i = i = 1 M σ i w i v ̲ i .
( e 1 ( ρ ̲ ) ) * = ( n = N N h n a 1 , n e j k ̲ n · ρ ̲ ) * = n = N N h n * ( a 1 , n ) * e j k ̲ n · ρ ̲ = n = N N h n ( a 1 , n ) * e j k ̲ n + · ρ ̲ .
S 11 H = F · S 11 * · F .
S 11 H · a ̲ 1 = F · S 11 * · F · a ̲ 1 = F · ( S 11 · ( F · ( a ̲ 1 ) * ) ) * .
a ̲ 1 , ( k + 1 ) + = a ̲ 1 , ( k ) + μ S 11 · a ̲ 1 + 2 2 a ̲ 1 + | a ̲ 1 + = a ̲ 1 , ( k ) + = a ̲ 1 , ( k ) + 2 μ S 11 H · S 11 · a ̲ 1 , ( k ) + ,
a ̲ ˜ 1 , ( k ) + = a ̲ 1 , ( k ) + 2 μ S 11 H · S 11 · a ̲ 1 , ( k ) + = a ̲ 1 , ( k ) + 2 μ F · S 11 * · F · S 11 · a ̲ 1 , ( k ) + .
a ̲ 1 , ( k + 1 ) + = a ̲ 1 , ( k ) + + μ ( k + 1 ) d ̲ ( k ) ,
μ ( k + 1 ) = r ̲ ( k ) 2 2 / S 11 · d ̲ ( k ) 2 2
d ̲ ( k + 1 ) = r ̲ ( k + 1 ) + β ( k + 1 ) d ̲ ( k )
β ( k + 1 ) = r ̲ ( k + 1 ) 2 2 / r ̲ ( k ) 2 2 .
r ̲ ( k + 1 ) = S 11 H · S 11 · a ̲ 1 , ( k + 1 ) + .
r ̲ ( k + 1 ) = r ̲ ( k ) μ ( k + 1 ) S 11 H · S 11 · d ̲ ( k )
r ̲ ( k + 1 ) = r ̲ ( k ) r ̲ ( k ) 2 2 S 11 · d ̲ ( k ) 2 2 S 11 H · S 11 · d ̲ ( k ) .
a ̲ 1 , ( 0 ) + Backscatter a ̲ 1 PCM a ̲ 1 + Backscatter a ̲ 1 PCM d ̲ ( 0 ) = r ̲ ( 0 ) .
a ̲ 1 + = l = 1 N B p l e j ϕ l b ̲ l ,
B = Q · U ,

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