6324
OPTICS LETTERS / Vol. 39, No. 21 / November 1, 2014
Focusing and energy deposition inside random media Xiaojun Cheng1,2,* and Azriel Z. Genack1,2 1
Department of Physics, Queens College, The City University of New York, Queens, New York, New York 11367, USA 2
The Graduate Center, The City University of New York, New York, New York 10016, USA *Corresponding author: [email protected] Received September 8, 2014; revised October 2, 2014; accepted October 3, 2014; posted October 6, 2014 (Doc. ID 222649); published October 29, 2014
The degree of control over waves transmitted through random media is determined by characteristics of the singular values of the transmission matrix. This Letter explores focusing and energy deposition in the interior of disordered samples and shows that these are determined by the singular values of the matrix relating the field channels inside a medium to the incident channels. Through calculations and simulations, we discovered that the variation with depth of the maximal energy density and the contrast in optimal focusing are determined by the participation number Mz of the energy density eigenvalues, while its inverse gives the variance of the energy density at z in a single configuration. © 2014 Optical Society of America OCIS codes: (030.6600) Statistical optics; (290.4210) Multiple scattering; (290.7050) Turbid media. http://dx.doi.org/10.1364/OL.39.006324
The depth of imaging in disordered samples is often limited by the scattering mean free path in which the incident phase is scrambled to create a random speckle pattern of intensity [1,2]. Because the wave is temporally coherent in static samples so that the random speckle pattern created is stable, the scrambling of the wave may be overcome by manipulating the incident wavefront to control specific characteristics of the transmitted field [3–13]. Vellekoop and Mosk demonstrated that light could be focused at a selected point in the transmitted field by maximizing the intensity at the point by sequentially adjusting the phase of the wave reflected by elements of the spatial light modulator (SLM). When the intensity at the point is maximized, the field from all pixels of the SLM are brought into phase so they interfere constructively to yield the maximal possible intensity at the focal point [3,4]. The incident waveform also may be adjusted to significantly enhance or suppress transmission. It is possible to obtain the limits of control over the transmitted wave and the full statistics of transmission by understanding the transmission matrix, whose elements are the coupling coefficients relating the fields in incident and outgoing channels [11,14–21]. Analogously, it might be possible to relate the field inside the sample to the incident waveform. Though such measurements are generally impossible in threedimensional (3D) samples, it may be possible to focus at selected points inside the sample when the response to external excitation can be sensed via an opening into or a probe within the sample or via fluorescence or second harmonic generation at a sensitized point [10,22–24]. Considering the relationship of the fields produced inside the sample would enable the characterization and control over the field in the interior of random samples. In this Letter, we describe focusing inside scattering systems. We introduce the field matrix e relating the field at a depth z inside the system to the incident field. The energy density is greatly enhanced at z when the incident wavefront is coupled to the highest energy density eigenchannel inside random media. The contrast in optimal focusing is determined by the energy density eigenchannel participation number Mz of the energy density matrix ee† and the number of channels N. The integrated energy 0146-9592/14/216324-04$15.00/0
density over the cross section at z for a single incident channel relative to the average over all incident channels is shown to be M −1 z. The statistics of energy density mimic those of transmitted flux even though the eigenvalues of ee† do not have an absolute maximum as do the eigenvalues of tt† , which cannot exceed unity. The energy density in a given energy density eigenchannel may greatly exceed the incident energy density since wave trajectories can pass many times through a given cross section of the sample. We therefore reviewed the properties of the transmission matrix before discussing the field matrix. The transmittance T, which is the sum of the flux transmission coefficients between all of the N incident and outgoing propagation channels, may be expressed P as the sum of all the transmission eigenvalues T N n τn [25]. Here, τn is the square of the nth singular value λn of t obtained from the singular value decomposition of t, t UΛV † , where U and V are unitary matrices and Λ is a diagonal matrix with singular values λn . The transmittance is related to the dimensionless conductance, which is the average conductance P in units of the quantum conductance, via g hTi h n τn i [26,27]. For diffusive samples, g ξ∕L Nl∕L, where ξ Nl is the localization length. Transmission in diffusive samples is governed by open channels with eigenvalues τn > 1∕e first introduced by Dorokhov to describe the scaling of conductance [28,29]. The contrast between the peak and background intensity in optimal focusing P is essentially P the channel participation number M n τn 2 ∕ n τ2n [30]. However, the transmission matrix only deals with the transmission coefficient and it therefore cannot describe the field inside a random sample, which is of interest for focusing and energy depositing inside the medium. The field matrix ez with elements eba z relates the field in channel b at a depth z inside a random sample to the fields in an incident channel a as E b z eba zE a . In practice, a and b represent the locations of sources and detectors, respectively, instead of extended transverse propagation channels. The singular value decomposition ez UzΛzVz† yields the singular values λn z. The square of the λn z are the © 2014 Optical Society of America
November 1, 2014 / Vol. 39, No. 21 / OPTICS LETTERS
energy density eigenvalues ϵn z. The energy density uz at depth z when all incident channels are excited each with unit energy density, P is the sum of the energy density eigenvalues uz N n1 ϵn z. Note that at z L, where L is the length of the sample, eL is essentially the field transmission matrix t. As z → L, the energy density eigenvalues ϵn z correspond to the transmission eigenvalues τn and ux approaches the transmittance T. We found the field inside the random sample from simulations of wave propagation in disordered media using the recursive Green’s functions method [31]. Simulations were carried out in an ensemble of 300 configurations with g 16 and in 2000 configurations with g 0.3 and g 0.195. The dimensionless conductance g depends upon N, L and the transport mean free path l. The diffusive samples we studied were composed of 800 × 500 units with the dielectric function at each site drawn randomly from a rectangular distribution from 0.8 to 1.2. The localized samples were composed of 750 × 100 units and 2250 × 100 units with the values of the dielectric function drawn randomly from the rectangular distribution between 0.5 and 2.5. Waves with wavelength of 2π in units of the lattice spacing were launched from one side and the fields were calculated at all locations. For the diffusive system, L∕ξ 0.0625, while L∕ξ 3.3; 10 for the two localized samples. Figure 1 shows uz for the samples supporting diffusive and localized waves. For diffusive waves, the energy density falls off linearly; for localized waves, there is a deviation from linearity. This result can be explained in terms of a position-dependent diffusion coefficient Dz, which reflects the increasing renormalization of transport for points more remote from the sample boundaries [32–34]. For localized waves, the results are in good agreement with the integrated intensity obtained with the position dependent diffusion coefficient Dz∕D0 e−zL−z∕Lξ . Figure 2 shows the maximum energy at depth z, which is ϵ1 z, and the average energy per channel. The energy is greatly enhanced at z for the highest energy density eigenchannel at z compared to the average energy at z. It is possible to obtain the condition for optimal focusing at p a target point β with wavefront P the incident 2 eβa z∕ I β z, where I β z N a jeβa zj . This is similar to the results for focusing obtained at the output [4]. Since E β z eβa zE a for a given a, jE β zj ≤ jeβa zE a j. The maximum of jEβ zj can only be reached
Fig. 1. Ensemble average of the energy density inside the sample for incident radiation with unit energy density.
6325
Fig. 2. Ensemble average of the highest energy density eigenvalue at z, hϵ1 zi, and the average energy per channel huz∕Ni for diffusive samples with L∕ξ 0.0625.
p when E a ∝ eβa z. We choose E a eβa z ∕ I β z so that the incident flux is unity and the fields are brought into phase at the focal point. Figure 3 shows the profiles of the focused beam at two depths into the sample in a diffusive system with g 16 for two values of z. The contrast in focusing is greater at z L∕4 than at z L∕2 because the participation number of energy density eigenchannels is greater close to the input surface. The contrast at the output is determined eigenPby the P channels participation number M n τn 2 ∕ n τ2n [30]. We also found that focusing inside a random medium is determined by the energy density eigenchannels participation number at depth z:
Mz
X n
2 X 2 ϵn z ∕ ϵn z ;
(1)
n
which is demonstrated below. Figure 4 shows the comparisons of the contrast μz and the energy density eigenchannel participation number Mz for diffusive and localized samples. μz is the ratio of the average over all configurations of the intensity at the focal points at depth z and the average background intensity within the cross section at depth z,
Fig. 3. Intensity distribution over the cross sections at z L∕4 and z L∕2 for a locally two-dimensional system with L∕ξ 0.0625 and the focal point at the center of the cross section x 0. Here, x is the coordinate in the transverse direction ranges from −A∕2; A∕2 while A is the width of the system. I b is the average background intensity at x ≠ 0.
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hI β i
hI b ib≠β
N X
ϵn z∕N;
(5)
n
PN 2 N 1 ϵ z∕N 1 X PbN n ϵ z∕N: − N − 1 b ϵn z∕N N − 1 b n
(6)
In the limit of N ≫ 1, we take N − 1 → N, μz
Fig. 4. Comparison of the contrast in focusing and the eigenchannel participation number along the sample length for (a) diffusive waves with g 16, N 166, and (b) localized waves with g 0.3, N 32, respectively.
μz hIziβ ∕hIzib≠β :
(2)
The results are in good agreement after a depth at which the waves are randomized and spread throughout the cross section. We were able to express μz in terms of Mz. We have shown the key steps below. More details of the calculations can be found from those carried out at the output [11,30], with all the quantities now inside random media. The intensity at the focal P 2 2 point is I β z N ∕I β x I β z. For any a jeβa zj β and a, eβa z can be expressed as eβa z PN n λn zunβ zυna z from the singular value decomposition. The focused intensity can be expressed as
I β z
N X
ϵn zjunβ zj2 ;
1 : 1∕Mz − 1∕N
(7)
For N ≫ Mz, this expression reduces to μz Mz. In contrast to transmission eigenchannels that cannot exceed unity, the energy density eigenvalues can be much larger than unity. Figure 4 shows that Mz falls rapidly with increasing depth into the sample. We found that for large n, the energy density eigenvalues ϵn z fall quickly with increasing z so that fewer incident channels contribute to the energy density as the output boundary of the sample is approached. For localized waves, Mz approaches a constant close to unity near the sample output since in localized samples transmission is then dominated by the highest eigenchannel. The statistics of intensity inside the random medium also is determined by Mz. For N ≫ 1, the variance of the total energy relative to its average Nua z∕uz at depth z, equals Mz−1 within one configuration, P which is similar at P the output [11,30]. Here ua z b jeba zj2 and uz a ua z. Figure 5 shows the comparison for a random sample of length 4500 and width 1500 with all other parameters the same as the diffusive samples in Fig. 4. The P total energy of an incident channel is 2 ua z N n ϵzjυna zj . varNua z∕uz hNua z∕ 2 uz i − 1, since hNua z∕uzi 1. We obtain hNua z∕uz2 i
N X
ϵn z∕uz2 hN 2 jυna zj4 i
n
N X
ϵn zϵn0 zhN 2 jυna zj2 jυn0 a zj2 i∕u2 z:
(8)
n≠n0
(3)
n
PN where we have used a υna zυn0 a z δnn0 since the energy density eigenchannels are orthonormal. The intensity at b ≠ β is N 2 X I b z ϵn zunb zunβ ∕I β z:
(4)
n
Since hjunβ j2 i 1∕N this gives
Fig. 5. Mz and varNua z∕uz for one configuration with N 500.
November 1, 2014 / Vol. 39, No. 21 / OPTICS LETTERS
For N ≫ 1, hjυna zj2 i 1∕N, hjυna zj4 i 2∕N 2 , so that PN hNua z∕uz2 i
n
P N ϵ z 2 ϵ2n z n n P : N ϵ z 2 n n
(9)
This gives varNua z∕uz−1 Mz:
(10)
Note that when N is not much larger than unity, the relation hjυna zj4 i 2∕N 2 is not satisfied. We therefore carried out the simulations shown in Fig. 5 in a sample with N 500. In conclusion, we have described the characteristics of optimal focusing and maximal energy deposition inside disordered systems in terms of the energy density eigenvalues at a depth z. Our future work will consider the probability density of the energy density eigenvalues inside random media, which approaches the bimodal distribution [17] toward the output of the sample for diffusive waves. We thank Zhou Shi for stimulating discussions and the simulation program used to obtain the field inside random media. This work is supported by the National Science Foundation (No. DMR-1207446). References 1. I. Freund, Physica A 168, 49 (1990). 2. A. K. Dunn and D. A. Boas, J. Biomed. Opt. 15, 011109 (2010). 3. I. M. Vellekoop and A. P. Mosk, Opt. Lett. 32, 2309 (2007). 4. I. M. Vellekoop and A. P. Mosk, Phys. Rev. Lett. 101, 120601 (2008). 5. S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, Nat. Commun. 1, 81 (2010). 6. I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, Nat. Photonics 4, 320 (2010). 7. O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, Nat. Photonics 5, 372 (2011).
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8. J. Aulbach, B. Gjonaj, P. M. Johnson, A. P. Mosk, and A. Lagendijk, Phys. Rev. Lett. 106, 103901 (2011). 9. Y. Sivan and J. B. Pendry, Phys. Rev. Lett. 106, 193902 (2011). 10. A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, Nat. Photonics 6, 283 (2012). 11. M. Davy, Z. Shi, J. Wang, and A. Z. Genack, Opt. Express 21, 10367 (2013). 12. Z. Shi, M. Davy, J. Wang, and A. Z. Genack, Opt. Lett. 38, 2714 (2013). 13. S. M. Popoff, A. Goetschy, S. F. Liew, A. D. Stone, and H. Cao, Phys. Rev. Lett. 112, 133903 (2014). 14. K. A. Muttalib, J. L. Pichard, and A. D. Stone, Phys. Rev. Lett. 59, 2475 (1987). 15. P. A. Mello, P. Pereyra, and N. Kumar, Ann. Phys. 181, 290 (1988). 16. E. Kogan and M. Kaveh, Phys. Rev. B 52, R3813 (1995). 17. C. W. J. Beenakker, Rev. Mod. Phys. 69, 731 (1997). 18. Z. Shi and A. Z. Genack, Phys. Rev. Lett. 108, 043901 (2012). 19. M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q. H. Park, and W. Choi, Nat. Photonics 6, 581 (2012). 20. S. Tripathi, R. Paxman, T. Bifano, and K. C. Toussaint, Opt. Express 20, 16067 (2012). 21. H. Yu, T. R. Hillman, W. Choi, J. O. Lee, M. S. Feld, R. R. Dasari, and Y. Park, Phys. Rev. Lett. 111, 153902 (2013). 22. K. M. Yoo, Q. Xing, and R. R. Alfano, Opt. Lett. 16, 1019 (1991). 23. V. Ntziachristos, Annu. Rev. Biomed. Eng. 8, 1 (2006). 24. C. L. Hsieh, Y. Pu, R. Grange, and D. Psaltis, Opt. Express 18, 12283 (2010). 25. Y. Imry and R. Landauer, Rev. Mod. Phys. 71, S306 (1999). 26. R. Landauer, Philos. Mag. 21(172), 863 (1970). 27. D. S. Fisher and P. A. Lee, Phys. Rev. B 23, 6851 (1981). 28. O. N. Dorokhov, Solid State Commun. 51, 381 (1984). 29. Y. Imry, Europhys. Lett. 1, 249 (1986). 30. M. Davy, Z. Shi, and A. Z. Genack, Phys. Rev. B 85, 035105 (2012). 31. H. Baranger, D. P. DiVincenzo, R. A. Jalabert, and A. D. Stone, Phys. Rev. B 44, 10637 (1991). 32. C. Tian, Phys. Rev. B 77, 064205 (2008). 33. C. S. Tian, S. K. Cheung, and Z. Q. Zhang, Phys. Rev. Lett. 105, 263905 (2010). 34. A. G. Yamilov, R. Sarma, B. Redding, B. Payne, H. Noh, and H. Cao, Phys. Rev. Lett. 112, 023904 (2014).
OPTICS LETTERS / Vol. 39, No. 21 / November 1, 2014
Focusing and energy deposition inside random media Xiaojun Cheng1,2,* and Azriel Z. Genack1,2 1
Department of Physics, Queens College, The City University of New York, Queens, New York, New York 11367, USA 2
The Graduate Center, The City University of New York, New York, New York 10016, USA *Corresponding author: [email protected] Received September 8, 2014; revised October 2, 2014; accepted October 3, 2014; posted October 6, 2014 (Doc. ID 222649); published October 29, 2014
The degree of control over waves transmitted through random media is determined by characteristics of the singular values of the transmission matrix. This Letter explores focusing and energy deposition in the interior of disordered samples and shows that these are determined by the singular values of the matrix relating the field channels inside a medium to the incident channels. Through calculations and simulations, we discovered that the variation with depth of the maximal energy density and the contrast in optimal focusing are determined by the participation number Mz of the energy density eigenvalues, while its inverse gives the variance of the energy density at z in a single configuration. © 2014 Optical Society of America OCIS codes: (030.6600) Statistical optics; (290.4210) Multiple scattering; (290.7050) Turbid media. http://dx.doi.org/10.1364/OL.39.006324
The depth of imaging in disordered samples is often limited by the scattering mean free path in which the incident phase is scrambled to create a random speckle pattern of intensity [1,2]. Because the wave is temporally coherent in static samples so that the random speckle pattern created is stable, the scrambling of the wave may be overcome by manipulating the incident wavefront to control specific characteristics of the transmitted field [3–13]. Vellekoop and Mosk demonstrated that light could be focused at a selected point in the transmitted field by maximizing the intensity at the point by sequentially adjusting the phase of the wave reflected by elements of the spatial light modulator (SLM). When the intensity at the point is maximized, the field from all pixels of the SLM are brought into phase so they interfere constructively to yield the maximal possible intensity at the focal point [3,4]. The incident waveform also may be adjusted to significantly enhance or suppress transmission. It is possible to obtain the limits of control over the transmitted wave and the full statistics of transmission by understanding the transmission matrix, whose elements are the coupling coefficients relating the fields in incident and outgoing channels [11,14–21]. Analogously, it might be possible to relate the field inside the sample to the incident waveform. Though such measurements are generally impossible in threedimensional (3D) samples, it may be possible to focus at selected points inside the sample when the response to external excitation can be sensed via an opening into or a probe within the sample or via fluorescence or second harmonic generation at a sensitized point [10,22–24]. Considering the relationship of the fields produced inside the sample would enable the characterization and control over the field in the interior of random samples. In this Letter, we describe focusing inside scattering systems. We introduce the field matrix e relating the field at a depth z inside the system to the incident field. The energy density is greatly enhanced at z when the incident wavefront is coupled to the highest energy density eigenchannel inside random media. The contrast in optimal focusing is determined by the energy density eigenchannel participation number Mz of the energy density matrix ee† and the number of channels N. The integrated energy 0146-9592/14/216324-04$15.00/0
density over the cross section at z for a single incident channel relative to the average over all incident channels is shown to be M −1 z. The statistics of energy density mimic those of transmitted flux even though the eigenvalues of ee† do not have an absolute maximum as do the eigenvalues of tt† , which cannot exceed unity. The energy density in a given energy density eigenchannel may greatly exceed the incident energy density since wave trajectories can pass many times through a given cross section of the sample. We therefore reviewed the properties of the transmission matrix before discussing the field matrix. The transmittance T, which is the sum of the flux transmission coefficients between all of the N incident and outgoing propagation channels, may be expressed P as the sum of all the transmission eigenvalues T N n τn [25]. Here, τn is the square of the nth singular value λn of t obtained from the singular value decomposition of t, t UΛV † , where U and V are unitary matrices and Λ is a diagonal matrix with singular values λn . The transmittance is related to the dimensionless conductance, which is the average conductance P in units of the quantum conductance, via g hTi h n τn i [26,27]. For diffusive samples, g ξ∕L Nl∕L, where ξ Nl is the localization length. Transmission in diffusive samples is governed by open channels with eigenvalues τn > 1∕e first introduced by Dorokhov to describe the scaling of conductance [28,29]. The contrast between the peak and background intensity in optimal focusing P is essentially P the channel participation number M n τn 2 ∕ n τ2n [30]. However, the transmission matrix only deals with the transmission coefficient and it therefore cannot describe the field inside a random sample, which is of interest for focusing and energy depositing inside the medium. The field matrix ez with elements eba z relates the field in channel b at a depth z inside a random sample to the fields in an incident channel a as E b z eba zE a . In practice, a and b represent the locations of sources and detectors, respectively, instead of extended transverse propagation channels. The singular value decomposition ez UzΛzVz† yields the singular values λn z. The square of the λn z are the © 2014 Optical Society of America
November 1, 2014 / Vol. 39, No. 21 / OPTICS LETTERS
energy density eigenvalues ϵn z. The energy density uz at depth z when all incident channels are excited each with unit energy density, P is the sum of the energy density eigenvalues uz N n1 ϵn z. Note that at z L, where L is the length of the sample, eL is essentially the field transmission matrix t. As z → L, the energy density eigenvalues ϵn z correspond to the transmission eigenvalues τn and ux approaches the transmittance T. We found the field inside the random sample from simulations of wave propagation in disordered media using the recursive Green’s functions method [31]. Simulations were carried out in an ensemble of 300 configurations with g 16 and in 2000 configurations with g 0.3 and g 0.195. The dimensionless conductance g depends upon N, L and the transport mean free path l. The diffusive samples we studied were composed of 800 × 500 units with the dielectric function at each site drawn randomly from a rectangular distribution from 0.8 to 1.2. The localized samples were composed of 750 × 100 units and 2250 × 100 units with the values of the dielectric function drawn randomly from the rectangular distribution between 0.5 and 2.5. Waves with wavelength of 2π in units of the lattice spacing were launched from one side and the fields were calculated at all locations. For the diffusive system, L∕ξ 0.0625, while L∕ξ 3.3; 10 for the two localized samples. Figure 1 shows uz for the samples supporting diffusive and localized waves. For diffusive waves, the energy density falls off linearly; for localized waves, there is a deviation from linearity. This result can be explained in terms of a position-dependent diffusion coefficient Dz, which reflects the increasing renormalization of transport for points more remote from the sample boundaries [32–34]. For localized waves, the results are in good agreement with the integrated intensity obtained with the position dependent diffusion coefficient Dz∕D0 e−zL−z∕Lξ . Figure 2 shows the maximum energy at depth z, which is ϵ1 z, and the average energy per channel. The energy is greatly enhanced at z for the highest energy density eigenchannel at z compared to the average energy at z. It is possible to obtain the condition for optimal focusing at p a target point β with wavefront P the incident 2 eβa z∕ I β z, where I β z N a jeβa zj . This is similar to the results for focusing obtained at the output [4]. Since E β z eβa zE a for a given a, jE β zj ≤ jeβa zE a j. The maximum of jEβ zj can only be reached
Fig. 1. Ensemble average of the energy density inside the sample for incident radiation with unit energy density.
6325
Fig. 2. Ensemble average of the highest energy density eigenvalue at z, hϵ1 zi, and the average energy per channel huz∕Ni for diffusive samples with L∕ξ 0.0625.
p when E a ∝ eβa z. We choose E a eβa z ∕ I β z so that the incident flux is unity and the fields are brought into phase at the focal point. Figure 3 shows the profiles of the focused beam at two depths into the sample in a diffusive system with g 16 for two values of z. The contrast in focusing is greater at z L∕4 than at z L∕2 because the participation number of energy density eigenchannels is greater close to the input surface. The contrast at the output is determined eigenPby the P channels participation number M n τn 2 ∕ n τ2n [30]. We also found that focusing inside a random medium is determined by the energy density eigenchannels participation number at depth z:
Mz
X n
2 X 2 ϵn z ∕ ϵn z ;
(1)
n
which is demonstrated below. Figure 4 shows the comparisons of the contrast μz and the energy density eigenchannel participation number Mz for diffusive and localized samples. μz is the ratio of the average over all configurations of the intensity at the focal points at depth z and the average background intensity within the cross section at depth z,
Fig. 3. Intensity distribution over the cross sections at z L∕4 and z L∕2 for a locally two-dimensional system with L∕ξ 0.0625 and the focal point at the center of the cross section x 0. Here, x is the coordinate in the transverse direction ranges from −A∕2; A∕2 while A is the width of the system. I b is the average background intensity at x ≠ 0.
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OPTICS LETTERS / Vol. 39, No. 21 / November 1, 2014
hI β i
hI b ib≠β
N X
ϵn z∕N;
(5)
n
PN 2 N 1 ϵ z∕N 1 X PbN n ϵ z∕N: − N − 1 b ϵn z∕N N − 1 b n
(6)
In the limit of N ≫ 1, we take N − 1 → N, μz
Fig. 4. Comparison of the contrast in focusing and the eigenchannel participation number along the sample length for (a) diffusive waves with g 16, N 166, and (b) localized waves with g 0.3, N 32, respectively.
μz hIziβ ∕hIzib≠β :
(2)
The results are in good agreement after a depth at which the waves are randomized and spread throughout the cross section. We were able to express μz in terms of Mz. We have shown the key steps below. More details of the calculations can be found from those carried out at the output [11,30], with all the quantities now inside random media. The intensity at the focal P 2 2 point is I β z N ∕I β x I β z. For any a jeβa zj β and a, eβa z can be expressed as eβa z PN n λn zunβ zυna z from the singular value decomposition. The focused intensity can be expressed as
I β z
N X
ϵn zjunβ zj2 ;
1 : 1∕Mz − 1∕N
(7)
For N ≫ Mz, this expression reduces to μz Mz. In contrast to transmission eigenchannels that cannot exceed unity, the energy density eigenvalues can be much larger than unity. Figure 4 shows that Mz falls rapidly with increasing depth into the sample. We found that for large n, the energy density eigenvalues ϵn z fall quickly with increasing z so that fewer incident channels contribute to the energy density as the output boundary of the sample is approached. For localized waves, Mz approaches a constant close to unity near the sample output since in localized samples transmission is then dominated by the highest eigenchannel. The statistics of intensity inside the random medium also is determined by Mz. For N ≫ 1, the variance of the total energy relative to its average Nua z∕uz at depth z, equals Mz−1 within one configuration, P which is similar at P the output [11,30]. Here ua z b jeba zj2 and uz a ua z. Figure 5 shows the comparison for a random sample of length 4500 and width 1500 with all other parameters the same as the diffusive samples in Fig. 4. The P total energy of an incident channel is 2 ua z N n ϵzjυna zj . varNua z∕uz hNua z∕ 2 uz i − 1, since hNua z∕uzi 1. We obtain hNua z∕uz2 i
N X
ϵn z∕uz2 hN 2 jυna zj4 i
n
N X
ϵn zϵn0 zhN 2 jυna zj2 jυn0 a zj2 i∕u2 z:
(8)
n≠n0
(3)
n
PN where we have used a υna zυn0 a z δnn0 since the energy density eigenchannels are orthonormal. The intensity at b ≠ β is N 2 X I b z ϵn zunb zunβ ∕I β z:
(4)
n
Since hjunβ j2 i 1∕N this gives
Fig. 5. Mz and varNua z∕uz for one configuration with N 500.
November 1, 2014 / Vol. 39, No. 21 / OPTICS LETTERS
For N ≫ 1, hjυna zj2 i 1∕N, hjυna zj4 i 2∕N 2 , so that PN hNua z∕uz2 i
n
P N ϵ z 2 ϵ2n z n n P : N ϵ z 2 n n
(9)
This gives varNua z∕uz−1 Mz:
(10)
Note that when N is not much larger than unity, the relation hjυna zj4 i 2∕N 2 is not satisfied. We therefore carried out the simulations shown in Fig. 5 in a sample with N 500. In conclusion, we have described the characteristics of optimal focusing and maximal energy deposition inside disordered systems in terms of the energy density eigenvalues at a depth z. Our future work will consider the probability density of the energy density eigenvalues inside random media, which approaches the bimodal distribution [17] toward the output of the sample for diffusive waves. We thank Zhou Shi for stimulating discussions and the simulation program used to obtain the field inside random media. This work is supported by the National Science Foundation (No. DMR-1207446). References 1. I. Freund, Physica A 168, 49 (1990). 2. A. K. Dunn and D. A. Boas, J. Biomed. Opt. 15, 011109 (2010). 3. I. M. Vellekoop and A. P. Mosk, Opt. Lett. 32, 2309 (2007). 4. I. M. Vellekoop and A. P. Mosk, Phys. Rev. Lett. 101, 120601 (2008). 5. S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, Nat. Commun. 1, 81 (2010). 6. I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, Nat. Photonics 4, 320 (2010). 7. O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, Nat. Photonics 5, 372 (2011).
6327
8. J. Aulbach, B. Gjonaj, P. M. Johnson, A. P. Mosk, and A. Lagendijk, Phys. Rev. Lett. 106, 103901 (2011). 9. Y. Sivan and J. B. Pendry, Phys. Rev. Lett. 106, 193902 (2011). 10. A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, Nat. Photonics 6, 283 (2012). 11. M. Davy, Z. Shi, J. Wang, and A. Z. Genack, Opt. Express 21, 10367 (2013). 12. Z. Shi, M. Davy, J. Wang, and A. Z. Genack, Opt. Lett. 38, 2714 (2013). 13. S. M. Popoff, A. Goetschy, S. F. Liew, A. D. Stone, and H. Cao, Phys. Rev. Lett. 112, 133903 (2014). 14. K. A. Muttalib, J. L. Pichard, and A. D. Stone, Phys. Rev. Lett. 59, 2475 (1987). 15. P. A. Mello, P. Pereyra, and N. Kumar, Ann. Phys. 181, 290 (1988). 16. E. Kogan and M. Kaveh, Phys. Rev. B 52, R3813 (1995). 17. C. W. J. Beenakker, Rev. Mod. Phys. 69, 731 (1997). 18. Z. Shi and A. Z. Genack, Phys. Rev. Lett. 108, 043901 (2012). 19. M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q. H. Park, and W. Choi, Nat. Photonics 6, 581 (2012). 20. S. Tripathi, R. Paxman, T. Bifano, and K. C. Toussaint, Opt. Express 20, 16067 (2012). 21. H. Yu, T. R. Hillman, W. Choi, J. O. Lee, M. S. Feld, R. R. Dasari, and Y. Park, Phys. Rev. Lett. 111, 153902 (2013). 22. K. M. Yoo, Q. Xing, and R. R. Alfano, Opt. Lett. 16, 1019 (1991). 23. V. Ntziachristos, Annu. Rev. Biomed. Eng. 8, 1 (2006). 24. C. L. Hsieh, Y. Pu, R. Grange, and D. Psaltis, Opt. Express 18, 12283 (2010). 25. Y. Imry and R. Landauer, Rev. Mod. Phys. 71, S306 (1999). 26. R. Landauer, Philos. Mag. 21(172), 863 (1970). 27. D. S. Fisher and P. A. Lee, Phys. Rev. B 23, 6851 (1981). 28. O. N. Dorokhov, Solid State Commun. 51, 381 (1984). 29. Y. Imry, Europhys. Lett. 1, 249 (1986). 30. M. Davy, Z. Shi, and A. Z. Genack, Phys. Rev. B 85, 035105 (2012). 31. H. Baranger, D. P. DiVincenzo, R. A. Jalabert, and A. D. Stone, Phys. Rev. B 44, 10637 (1991). 32. C. Tian, Phys. Rev. B 77, 064205 (2008). 33. C. S. Tian, S. K. Cheung, and Z. Q. Zhang, Phys. Rev. Lett. 105, 263905 (2010). 34. A. G. Yamilov, R. Sarma, B. Redding, B. Payne, H. Noh, and H. Cao, Phys. Rev. Lett. 112, 023904 (2014).
