Abstract

We study the energy density of shaped waves inside a quasi-1D disordered waveguide. We find that the spatial energy density of optimally shaped waves, when expanded in the complete set of eigenfunctions of the diffusion equation, is well described by considering only a few of the lowest eigenfunctions. Taking into account only the fundamental eigenfunction, the total internal energy inside the sample is underestimated by only 2%. The spatial distribution of the shaped energy density is very similar to the fundamental eigenfunction, up to a cosine distance of about 0.01. We obtain the energy density of transmission eigenchannels inside the sample by numerical simulation of the scattering matrix. Computing the transmission-averaged energy density over all transmission channels yields the ensemble averaged energy density of shaped waves. From the averaged energy density, we reconstruct its spatial distribution using the eigenfunctions of the diffusion equation. The results of our study have exciting applications in controlled biomedical imaging, efficient light harvesting in solar cells, enhanced energy conversion in solid-state lighting, and low threshold random lasers.

© 2016 Optical Society of America

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References

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2016 (2)

M. L. Meretska, A. Lagendijk, H. Thyrrestrup, A. P. Mosk, W. L. IJzerman, and W. L. Vos, “How to distinguish elastically scattered light from stokes shifted light for solid-state lighting?” J. Appl. Phys. 119, 093102 (2016).
[Crossref]

O. S. Ojambati, H. Yılmaz, A. Lagendijk, A. P. Mosk, and W. L. Vos, “Coupling of energy into the fundamental diffusion mode of a complex nanophotonic medium,” New J. Phys. 18, 043032 (2016).
[Crossref]

2015 (4)

2014 (5)

S. M. Popoff, A. Goetschy, S. F. Liew, A. D. Stone, and H. Cao, “Coherent control of total transmission of light through disordered media,” Phys. Rev. Lett. 112, 133903 (2014).
[Crossref] [PubMed]

S. A. Goorden, M. Horstmann, A. P. Mosk, B. Škoríc, and P. W. H. Pinkse, “Quantum-secure authentication of a physical unclonable key,” Optica 1, 421–424 (2014).
[Crossref]

V. Y. F. Leung, A. Lagendijk, T. W. Tukker, A. P. Mosk, W. L. IJzerman, and W. L. Vos, “Interplay between multiple scattering, emission, and absorption of light in the phosphor of a white light-emitting diode,” Opt. Express 22, 8190–8204, (2014).
[Crossref] [PubMed]

B. Gérardin, J. Laurent, A. Derode, C. Prada, and A. Aubry, “Full transmission and reflection of waves propagating through a maze of disorder,” Phys. Rev. Lett. 113, 173901 (2014).
[Crossref] [PubMed]

F. T. Si, D. Y. Kim, R. Santbergen, H. Tan, R. A. C. M. M. van Swaaij, A. H. M. Smets, O. Isabella, and M. Zeman, “Quadruple-junction thin-film silicon-based solar cells with high open-circuit voltage,” Appl. Phys. Lett. 105, 063902 (2014).
[Crossref]

2013 (2)

T. Ogi, A. B. D. Nandiyanto, K. Okino, F. Iskandar, W.-N. Wang, E. Tanabe, and K. Okuyama, “Towards better phosphor design: Effect of SiO2 nanoparticles on photoluminescence enhancement of YAG:Ce,” ECS J. Solid State Sci. Technol. 2(5), R91–R95 (2013).
[Crossref]

R. Horstmeyer, B. Judkewitz, I. M. Vellekoop, S. Assawaworrarit, and C. Yang, “Physical key-protected one-time pad,” Sci. Rep. 3, 3543 (2013).
[Crossref] [PubMed]

2012 (11)

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,” Nature Photon. 6, 581 (2012).
[Crossref]

Y. M. Wang, B. Judkewitz, C. A. DiMarzio, and C. Yang, “Deep-tissue focal fluorescence imaging with digitally time-reversed ultrasound-encoded light,” Nat. Commun. 3, 928 (2012).
[Crossref] [PubMed]

K. Si, R. Fiolka, and M. Cui, “Fluorescence imaging beyond the ballistic regime by ultrasound-pulse-guided digital phase conjugation,” Nature Photon. 6, 657–661 (2012).
[Crossref]

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491, 232–234, (2012).
[Crossref] [PubMed]

J.-H. Park, C. Park, H. Yu, Y.-H. Cho, and Y. Park, “Dynamic active wave plate using random nanoparticles,” Opt. Express 20, 17010–17016, (2012).
[Crossref]

Y. Guan, O. Katz, E. Small, J. Zhou, and Y. Silberberg, “Polarization control of multiply scattered light through random media by wavefront shaping,” Opt. Lett. 37, 4663–4665 (2012).
[Crossref] [PubMed]

J.-H. Park, C. Park, H. Yu, Y.-H. Cho, and Y. Park, “Active spectral filtering through turbid media,” Opt. Lett. 37, 3261–3263 (2012).
[Crossref] [PubMed]

E. Small, O. Katz, Y. Guan, and Y. Silberberg, “Spectral control of broadband light through random media by wavefront shaping,” Opt. Lett. 37, 3429–3431 (2012).
[Crossref]

M. Davy, Z. Shi, and A. Z. Genack, “Focusing through random media: Eigenchannel participation number and intensity correlation,” Phys. Rev. B 85, 035105 (2012).
[Crossref]

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nature Photon. 6, 283–292 (2012).
[Crossref]

A. Polman and H. Atwater, “Photonic design principles for ultrahigh-efficiency photovoltaics,” Nature Mater. 11, 174 (2012).
[Crossref]

2011 (3)

O. Yizhar, L. E. Fenno, T. J. Davidson, M. Mogri, and K. Deisseroth, “Optogenetics in neural systems,” Neuron 71, 9–34 (2011).
[Crossref] [PubMed]

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

S. M. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Controlling light through optical disordered media: transmission matrix approach,” New J. Phys. 13, 123021 (2011).
[Crossref]

2010 (2)

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] [PubMed]

T. Čižmar, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nature Photon. 4, 388–394 (2010).
[Crossref]

2008 (2)

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nature Photon. 2, 110–115 (2008).
[Crossref]

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

2007 (4)

I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32, 2309–2311 (2007).
[Crossref] [PubMed]

G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-field time reversal,” Science 315, 1120–1122 (2007).
[Crossref] [PubMed]

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” J. Display Technol. 3, 160–175 (2007).
[Crossref]

J. M. Phillips, M. E. Coltrin, M. H. Crawford, A. J. Fischer, M. R. Krames, R. Mueller-Mach, G. O. Mueller, Y. Ohno, L. E. S. Rohwer, J. A. Simmons, and J. Y. Tsao, “Research challenges to ultra-efficient inorganic solid-state lighting,” Laser Photon. Rev. 1(4), 307–333 (2007).
[Crossref]

2004 (1)

G. Lerosey, J. de Rosny, A. Tourin, A. Derode, G. Montaldo, and M. Fink, “Time reversal of electromagnetic waves,” Phys. Rev. Lett. 92, 193904 (2004).
[Crossref] [PubMed]

1999 (1)

M. C. W. van Rossum and Th. M. Niewenhuizen, “Multiple scattering of classical waves,” Rev. Mod. Phys. 71, 313–371 (1999).
[Crossref]

1997 (1)

C. W. J. Beenakker, “Random-matrix theory of quantum transport,” Rev. Mod. Phys. 69, 731–808 (1997).
[Crossref]

1996 (1)

D. S. Wiersma and A. Lagendijk, “Light diffusion with gain and random lasers,” Phys. Rev. E 54, 4256–4265 (1996).
[Crossref]

1995 (1)

A. Derode, P. Roux, and M. Fink, “Robust acoustic time reversal with high-order multiple scattering,” Phys. Rev. Lett. 75, 4206–4209 (1995).
[Crossref] [PubMed]

1994 (1)

N. M. Lawandy, R. M. Balachandran, A. S. L. Gomes, and E. Sauvain, “Laser action in strongly scattering media,” Nature 368, 436 (1994).
[Crossref]

1992 (2)

J. B. Pendry and A. MacKinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. 69, 2772–2775 (1992).
[Crossref] [PubMed]

D. R. Dowling and D. R. Jackson, “Narrow ÂŘband performance of phase ÂŘconjugate arrays in dynamic random media,” J. Acoust. Soc. Am. 91, 3257 (1992).
[Crossref]

1990 (1)

I. Freund, “Looking through walls and around corners,” Physica A 168(1), 49–65 (1990).
[Crossref]

1989 (1)

A. Lagendijk, R. Vreeker, and P. de Vries, “Influence of internal reflection on diffusive transport in strongly scattering media,” Phys. Lett. A 136, 81–88 (1989).
[Crossref]

1988 (1)

D. Y. K. Ko and J. C. Inkson, “Matrix method for tunneling in heterostructures: Resonant tunneling in multilayer systems,” Phys. Rev. B 38, 9945–9951 (1988).
[Crossref]

1977 (1)

1966 (1)

Akkermans, E.

E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University, 2007).
[Crossref]

Assawaworrarit, S.

R. Horstmeyer, B. Judkewitz, I. M. Vellekoop, S. Assawaworrarit, and C. Yang, “Physical key-protected one-time pad,” Sci. Rep. 3, 3543 (2013).
[Crossref] [PubMed]

Atwater, H.

A. Polman and H. Atwater, “Photonic design principles for ultrahigh-efficiency photovoltaics,” Nature Mater. 11, 174 (2012).
[Crossref]

Aubry, A.

B. Gérardin, J. Laurent, A. Derode, C. Prada, and A. Aubry, “Full transmission and reflection of waves propagating through a maze of disorder,” Phys. Rev. Lett. 113, 173901 (2014).
[Crossref] [PubMed]

Balachandran, R. M.

N. M. Lawandy, R. M. Balachandran, A. S. L. Gomes, and E. Sauvain, “Laser action in strongly scattering media,” Nature 368, 436 (1994).
[Crossref]

Beenakker, C. W. J.

C. W. J. Beenakker, “Random-matrix theory of quantum transport,” Rev. Mod. Phys. 69, 731–808 (1997).
[Crossref]

Bence, S. J.

K. F. Riley, M. P. Hobson, and S. J. Bence, Mathematical Methods for Physics and Engineering (Cambridge University, 2006).
[Crossref]

Bertolotti, J.

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491, 232–234, (2012).
[Crossref] [PubMed]

Blum, C.

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491, 232–234, (2012).
[Crossref] [PubMed]

Boccara, A. C.

S. M. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Controlling light through optical disordered media: transmission matrix approach,” New J. Phys. 13, 123021 (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] [PubMed]

Cao, H.

S. F. Liew and H. Cao, “Modification of light transmission channels by inhomogeneous absorption in random media,” Opt. Express 23, 11043–11053 (2015).
[Crossref] [PubMed]

S. M. Popoff, A. Goetschy, S. F. Liew, A. D. Stone, and H. Cao, “Coherent control of total transmission of light through disordered media,” Phys. Rev. Lett. 112, 133903 (2014).
[Crossref] [PubMed]

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] [PubMed]

Cho, Y.-H.

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,” Nature Photon. 6, 581 (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,” Nature Photon. 6, 581 (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.

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S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: An approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
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M. Davy, Z. Shi, J. Park, C. Tian, and A. Z. Genack, “Universal structure of transmission eigenchannels inside opaque media,” Nat. Commun. 6, 6893 (2015).
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K. Si, R. Fiolka, and M. Cui, “Fluorescence imaging beyond the ballistic regime by ultrasound-pulse-guided digital phase conjugation,” Nature Photon. 6, 657–661 (2012).
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A. Derode, P. Roux, and M. Fink, “Robust acoustic time reversal with high-order multiple scattering,” Phys. Rev. Lett. 75, 4206–4209 (1995).
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G. Lerosey, J. de Rosny, A. Tourin, A. Derode, G. Montaldo, and M. Fink, “Time reversal of electromagnetic waves,” Phys. Rev. Lett. 92, 193904 (2004).
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I. M. Vellekoop and A. P. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett. 101, 120601 (2008).
[Crossref] [PubMed]

B. Gérardin, J. Laurent, A. Derode, C. Prada, and A. Aubry, “Full transmission and reflection of waves propagating through a maze of disorder,” Phys. Rev. Lett. 113, 173901 (2014).
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Science (1)

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The form of Eqs. 11 and 12 is different from the ones obtained in [55] because we have considered here the extrapolation lengths from both surfaces possibly to be different, while in [55] they were considered to be the same.

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Figures (9)

Fig. 1
Fig. 1 The first three eigenfunctions of the diffusion equation plotted using equation Eq. (9). The wave vectors κm are presented in table 1 and the used parameters can be found in the caption of that table.
Fig. 2
Fig. 2 Schematic of the numerical sample: a quasi-1D waveguide. In the region z = 0 to z = L, scatterers (red cirles) are randomly distributed in the waveguide. The width W and length L of the scattering part of the waveguide are such that WL. The waveguide is illuminated with an incident field E0 that is scattered in the waveguide. The waveguide is divided into slices (vertical dashed line) and in each slice there is at most one scattering particle. The scattering matrix of each of the slices is computed and concatenated to give the scattering matrix of the whole sample.
Fig. 3
Fig. 3 (a) Probability distribution of the transmission from the simulation (blue dots) and the expected Dorokhov-Mello-Pereyra-Kumar (DMPK) distribution (red curve). (b) An equally-weighted summation over the energy density of all transmission channels (blue dots) and the expectation from diffusion theory (red line) as a function of the normalized depth z/ℓ. In (a) and (b), we ensemble averaged over 8000 waveguides and the sample thickness is L = 12.1.
Fig. 4
Fig. 4 (a) Transmission-weighted averaged energy density versus normalized sample depth z/ℓ. The transmission-weighted averaged energy density is equivalent to wavefront-shaped or phase-conjugated light [7]. The blue circles are the energy density obtained from simulation, the green line is the diffusion m = 1 eigenfunction, and the red line is the summation over the first seven eigenfunctions. (b) Cosine distance COSD versus eigenfunction index m. The vertical and horizontal dashed-lines shows the position of our figure-of-merit COSD 10−4 and the eigenfunctions that fulfill the criterion m = 7 respectively. In both (a) and (b), the sample is the same as in Fig. 3.
Fig. 5
Fig. 5 The cumulative contribution of the eigenfunctions of the diffusion equation to the total internal energy density of shaped waves relative to a sum over the first 100 eigenfunctions versus the eigenfunction index m.
Fig. 6
Fig. 6 (a) Ensemble averaged energy density of an open channel in a scattering medium versus normalized depth z/ltr with transmission τ in the range 0.99 < τ < 1. The blue circles are the energy density data obtained from simulation, and the red line is the fundamental diffusion eigenfunction m = 1. The black dashed-dotted curve is a parabolic fit. (b) Ensemble averaged energy density of transmission channels in a scattering medium versus the normalized depth z/ltr for transmission τ in the range 0.49 < τ < 0.51. The blue circles are the energy density data obtained from simulation, the green line is the diffusion m = 1 eigenfunction, and the red line is a M = 7 summation of diffusion eigenfunctions. (c) The blue circles are obtained as in (a) and the transmission τ is in the range of 0.09 < τ < 0.11. The green and the red lines are summations over M = 8 and M = 16 diffusion eigenfunctions respectively. (d) The blue circles are obtained as in (a) and the transmission τ is in the range: 0 < τ < 0.02, which signifies closed channels. The green and the red lines are summations over M = 8 and M = 33 diffusion eigenfunction respectively.
Fig. 7
Fig. 7 Contribution of eigenfunctions of the diffusion equation relative to a sum of the first 100 eigenfunctions versus transmission τ for eigenfunctions m = 1,2,3, and 4 plotted with red circles, blue squares, green triangles, and black diamonds respectively.
Fig. 8
Fig. 8 (a) Cosine distance COSD versus eigenfunction index m for 4 samples with different thickness: L = 5, 12.1, 18.8 and 39.5, which are plotted as pink stars, green squares, blue triangles, and orange circles respectively. The vertical and horizontal dashed-lines shows position of our figure-of-merit COSD 10−4 and the eigenfunction, which fulfills the criterion m = 7 respectively. (b) Contribution of the fundamental diffusion mode m = 1 relative to the sum of the first 100 eigenfunctions as a function of transmission τ.
Fig. 9
Fig. 9 The values of functions f1 (blue line) and f2 (red dashed line) in Eqs. 27 and 28. as a function of reduced diffusion wave vector κmL/π. The green circles are the roots of Eq. 25.

Tables (1)

Tables Icon

Table 1 Wavevectors a κm for the First 10 Eigenfunctions of the Diffusion Equation.

Equations (29)

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W ( r , t ) t = D 2 W ( r , t ) ,
W ( r , t ) = W ( x , y , t ) W ( z , t ) .
W ( q , t ) = W ( q , t = 0 ) e D q 2 t .
W ( z , t ) t = D 2 W ( z , t ) z 2
2 W ( z ) z 2 + Γ W ( z ) = 0 ,
Γ γ D .
W ( z = 0 ) z e 1 W ( z ) z | z = 0 = 0
W ( z = L ) z e 2 W ( z ) z | z = L = 0 ,
W m ( z ) = A m 1 / 2 sin ( κ m z + η m ) .
0 L W m W n d z = δ m n ,
tan ( κ m L ) = ( z e 1 + z e 2 ) κ m z e 1 z e 2 κ m 2 1 .
tan η m = z e 1 κ m ,
A m = 1 2 L 1 2 κ m cos ( κ m L + 2 η m ) sin ( κ m L ) .
W s ( z ) = m = 1 M d m W m ( z ) ,
I p z = 0 z = L d z W s ( z ) W p ( z ) ,
I p = z = 0 z = L d z m = 1 M d m W m ( z ) W p ( z ) .
z = 0 z = L d z W p ( z ) W m ( z ) = δ m p ,
I p = m = 1 M d m δ m p = d p .
W s ( z ) = m = 1 M l m W m ( z ) ,
COSD 1 i = 1 N s W re ( z i ) W s ( z i ) [ i = 1 N s W re 2 ( z i ) ] 1 2 [ i = 1 N s W s 2 ( z i ) ] 1 2 .
S = ( R L T T T R R ) ,
E c = T 1 E i n + R L 2 T 1 E i n + R R 1 R L 2 T 1 E i n + R L 2 R R 1 R L 2 T 1 E i n +
= ( 1 + R L 2 ) ( 1 R R 1 R L 2 ) 1 T 1 E i n .
W sw ( z ) = n N τ n W n ( z , τ ) ,
tan ( κ m L ) = ( z e 1 + z e 2 ) κ m z e 1 z e 2 κ m 2 1 .
f ( κ m ) = f 1 ( κ m ) f 2 ( κ m ) ,
f 1 ( κ m ) tan ( κ m L )
f 2 ( κ m ) ( z e 1 + z e 2 ) κ m z e 1 z e 2 κ m 2 1 .
f ( κ m r ) = 0 .

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