Optical black hole: Broadband omnidirectional light absorber Evgenii E. Narimanov and Alexander V. Kildishev Citation: Applied Physics Letters 95, 041106 (2009); doi: 10.1063/1.3184594 View online: http://dx.doi.org/10.1063/1.3184594 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/95/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Development of metamaterials with desired broadband optical properties Appl. Phys. Lett. 101, 071907 (2012); 10.1063/1.4746400 Optical scanning system for light-absorption measurement of deep biological tissue Rev. Sci. Instrum. 82, 093101 (2011); 10.1063/1.3632133 A cylindrical optical black hole using graded index photonic crystals J. Appl. Phys. 109, 103104 (2011); 10.1063/1.3590336 Polarization conversion and “focusing” of light propagating through a small chiral hole in a metallic screen Appl. Phys. Lett. 86, 201105 (2005); 10.1063/1.1925759 The shadow of the black hole at the galactic center AIP Conf. Proc. 522, 317 (2000); 10.1063/1.1291730
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APPLIED PHYSICS LETTERS 95, 041106 共2009兲
Optical black hole: Broadband omnidirectional light absorber Evgenii E. Narimanova兲 and Alexander V. Kildisheva兲 Department of Electrical Engineering, Purdue University, West Lafayette, Indiana 47903, USA
共Received 19 May 2009; accepted 1 July 2009; published online 27 July 2009兲 We develop an approach to broad-band omnidirectional light absorption, based on light propagation in a metamaterial structure forming an effective “black hole.” The proposed system does not rely on magnetic response, is nonresonant, and can be fabricated from existing materials. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3184594兴 While the theoretical concept of “black body” radiation proved remarkably useful for modern science and engineering—from its role in the creation of quantum mechanics to its applications to actual light sources, few of the actual materials come close to 100% absorption for all angles in broad bandwidth. Even though many applications would greatly benefit from such a perfect absorber—from cross-talk reduction in optoelectronic devices to thermal light emitting sources to solar light harvesting—it remains an elusive dream. Although it is possible to reach nearly total absorption by existing methods, this is generally limited to a specific range of incidence angles,1–7 or to a narrow bandwidth.8,9 In the present work, we develop an alternative approach to highly efficient light absorption, based on metamaterial structures. The proposed system relies on the local control of the electromagnetic response of the structure, with the resulting “effective potential” that determines the dynamics of the light wave, forming an effective “black hole.” Note that, as opposed to the earlier proposal for the optical analog of the gravitational black hole,10 our device does not involve any mechanical motion of its media. Furthermore, our approach does not rely on any resonance, and leads to omnidirectional high absorption in broad bandwidth. To ensure for equal performance at any direction, we consider spherically symmetric shell—see Fig. 1共a兲—with the radial variation in the dielectric permittivity ⑀共r兲 matching those of the outer medium and of the internal core at the corresponding interfaces
冦
⑀c + i␥ , r ⬍ Rc , ⑀ = ⑀共r兲, Rc ⬍ r ⬍ R, ⑀0 , r ⬎ R,
冧
共1兲
where the core radius Rc = R冑⑀0 / ⑀c. The inner core 共r ⬍ Rc兲 represents the “payload” of the device—e.g., a photovoltaic system for solar power applications. Similarly, for applications that deal with the waveguide propagation, we introduce the cylindrical equivalent of Eq. 共1兲—see Fig. 1共b兲. The radial variation in the dielectric permittivity of the 共composite兲 shell can be achieved with, e.g., changing the relative volume fractions of the component materials. While the exact functional form of the dielectric permittivity strongly depends on the topological structure of the composite 共e.g., a layered system versus fractal mixture兲, it is gena兲
Electronic addresses:
[email protected] and
[email protected].
erally a monotonic function and thus the desired variation in ⑀共r兲 can always be realized with the proper radial dependence of the component densities. Note that matching the dielectric permittivity of the system to a low-index environment 共e.g., air兲 would require metallic components 共with ⑀m ⬍ 0兲 in the composite forming the spherical shell, with the concomitant losses leading to a nonzero imaginary part of ⑀共r兲. On the other hand, if the dielectric permittivity of the outer medium ⑀0 ⬎ 1, an all-dielectric design is possible. To incorporate the 共application-dependent兲 inner structure of the core, the dimensions of the proposed device would significantly exceed the light wavelength
Rc ⱖ ,R Ⰷ .
共2兲
In this regime, semiclassical analysis11,12 not only leads to a clear physical picture of the wave dynamics, but also yields a highly accurate quantitative description. Here, we will follow the semiclassical approach, which will be complimented by the exact analysis in the last part of the paper. For spherically and cylindrically symmetric distributions of the dielectric permittivity ⑀共r兲, the effective Hamiltonian that describes the light propagation,11
H=
pr2 m2 + , 2⑀共r兲 2⑀共r兲r2
共3兲
where pr is the radial momentum and m is the total angular momentum 共for the spherical system兲 or its projection on the cylinder axis 共for the cylindrical version兲. The classical equations of motion corresponding to the Hamiltonian 共3兲 outside the core 共i.e., for r ⬎ Rc兲 are identical to those of a point particle of unitary mass in the central potential
FIG. 1. 共Color online兲 The cut-out views of the spherical 关panel 共a兲兴 and cylindrical 关panel 共b兲兴 optical “black holes.” The orange core represents the “payload” of the device 共e.g., a detector, a photovoltaic element, etc.兲.
0003-6951/2009/95共4兲/041106/3/$25.00 95, 041106-1 © 2009 American Institute of Physics Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Download to IP: 128.210.106.65 On: Fri, 17 Jun 2016 19:23:58
041106-2
Appl. Phys. Lett. 95, 041106 共2009兲
E. E. Narimanov and A. V. Kildishev
FIG. 2. 共Color online兲 Ray trajectories in systems with the dielectric permittivity defined by Eq. 共7兲 for 共a兲 n = −1, 共b兲 n = 1, 共c兲 n = 2, and 共d兲 n = 3. For clarity, the core radius is assumed to be infinitesimal.
Veff共r兲 =
冉冊
1 2 c
2
关⑀0 − ⑀共r兲兴,
共4兲
where c is the light speed and is the light frequency. For ⑀共r兲 ⬀ 1 / r the proposed system thus represents an optical analog to the Kepler problem, opening an intriguing possibility to study the nontrivial aspects of celestial dynamics via their optical analogs. A straightforward solution of the Hamiltonian equations for Eq. 共3兲 yields the ray trajectories in the polar coordinates
共r兲 = 0 +
冕
m/r
m/r1
d
冑 冉冊 C 0⑀
m − 2
共5兲
,
where the constants r1, 0 and C0 are set by the initial conditions. When the effective potential Veff ⬀ ⑀0 − ⑀ is “sufficiently attractive,” the corresponding ray trajectories experience a fall onto the core of the system. If the dielectric permittivity
⑀共r兲 ⬅ ⑀0共1 + ⌬⑀兲,
共6兲
FIG. 3. 共Color online兲 A Gaussian beam incident on the “black hole” off关panel 共a兲兴 and on-center 关panel 共b兲兴. Note the lack of any visible reflection 共which would be manifested by the interference fringes between the incident and reflected light兲. The solid black lines in panel 共a兲 show the classical ray trajectories. The “black hole” is formed by n-doped silicon-silica glass composite with the inner radius of 8.4 m and the outer radius of R = 20 m. The free-space incident light wavelength is = 1.5 m.
In terms of the practical realization of such “black holes,” higher orders represent increasingly difficult fabrication problems due to their correspondingly larger values of the local dielectric permittivity and its gradients. We therefore focus on the smallest order which still captures the incident rays: n = 2. Assuming that the fabrication method chosen for a particular application of the proposed device, allows for the local values of the dielectric permittivity in the range ⑀0 ⬍ ⑀ ⬍ ⑀c, we thus define the “fundamental” black hole via
冦冉冊
⑀0 , r ⬎ R, 2 R , Rc ⬍ r ⬍ R, ⑀共r兲 = ⑀0 r ⑀c + i␥ , r ⬍ Rc ,
can be represented by a power law, ⌬⑀ ⬃ 1 / rn, any n ⱖ 2 leads to such “fall.” We thus introduce the family of such optical “black holes” defined by
冦冉 冊
r ⬎ R,
0,
⌬⑀n共r兲 =
R r
n
, r ⬍ R.
冧
共7兲
冦
冦
册 册
冧
共9兲
where the core radius Rc is not an independent parameter, but instead determined by
For the dielectric permittivity defined by Eq. 共7兲, for the ray trajectories within the “event horizon” 共r ⱕ R兲 we obtain
冋 冋 冋冑
冧
冧
Rc = R
冑
⑀0 . ⑀c
共10兲
In Fig. 3 we compare the predictions of the semiclassical theory of the present section to the exact numerical calculation, for the black hole corresponding to Eq. 共9兲 with outer , n ⫽ 2, dielectric permittivity ⑀0 = 2.1 共silica glass兲 and the core dir共兲 = R · electric permittivity of ⑀c + i␥ = 12+ 0.7i 共n-doped silicon with the doping density n ⬇ 2.7⫻ 1020 cm−3兲. The shell is formed mRn by the glass glass-silicon composite; the outer radius R − 1共 − 0兲 , n = 2, exp − = 20 m and the core radius Rc ⬇ 8.4 m. The free-space C0 wavelength of the incident light is = 1.5 m. Note excel共8兲 lent agreement of the exact calculation with the semiclassical theory. where R is 共yet兲 another arbitrary constant, determined from Now we analyze the proposed system via a direct soluthe initial conditions of a particular trajectory. A number of tion of Maxwell’s equations. We consider the cylindrical versuch trajectories corresponding to different orders n are sion of the device, assumingDownload that the tosystem is infinite inOn: theFri, 17 Jun shown in Fig.content 2. is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Reuse of AIP Publishing IP: 128.210.106.65 n−2 cos 共 − 0兲 2 n−2 共 R − 0兲 cos 2
2/n−2
册
2016 19:23:58
041106-3
Appl. Phys. Lett. 95, 041106 共2009兲
E. E. Narimanov and A. V. Kildishev
“axial” direction zˆ—see Fig. 1. In experiment, this would correspond to either 共i兲 the length of the cylinder d Ⰷ R Ⰷ , or 共ii兲 the cylinder inside a single-mode waveguide system 共where one shall use the effective values of the dielectric permittivities, taking into account the waveguide mode structure兲. As in this case the TE and TM polarizations decouple and can be independently treated with nearly identical steps, we will limit our analysis to the TE mode where the electric field E = zˆE. Using the polar coordinates 共r , 兲, we introduce the “wavefunction”
冋
E共r,t兲 = 0,0,
1
冑r 共r兲
册
exp共im − it兲.
共11兲
For a given reflection coefficient in the angular momentum representation, the absorption cross section13 per unit length of a 共long兲 cylinder is
a =
共k0R兲2 − m2 + 1/4 = 0, r2
冋 冉 冊 冉 冊册
a = 2R 1 − 2F where F共x兲 =
共12兲
冋冑 冋冑
共k0R兲2 − m2 log
+ B冑r sin
r R
册
共k0Rc兲2 − m2 log
册
r , Rc
共13兲
where A and B are constants defined by the boundary conditions at the “inner” 共r = Rc兲 and “outer” 共r = R兲 interfaces of the shell. In the core and outer regions where the dielectric permittivity is constant, the solutions of the wave equations reduce ⫾ ⬅ Jm ⫾ iY m兲 to the standard Bessel 共Jm兲 and Hankel 共Hm functions E共r, ;t兲 = exp共im − it兲
冦
冊
冉冑
⑀c + i␥ r , r ⬍ Rc , c ⫻ − + Hm共k0r兲 + rmHm共k0r兲, r ⬎ R, CJm
冧
共14兲
where C is a constant and rm is the reflection coefficient for the angular momentum m. Note that only the Bessel function ⫾ diverge at the Jm is present in the core region as Y m and Hm origin. For the TE polarization that we consider, the boundary conditions for the electromagnetic field reduce to the continuity of E共r , 兲 and its normal derivative. Solving the resulting system of linear equations on A, B, C, and rm, for the reflection coefficient we obtain rm = −
− ⬘共k0R兲 + mHm− 共k0R兲 Hm + ⬘共k0R兲 + mHm+ 共k0R兲 Hm
where
m = −
冑共k0R兲2 − m2 k 0R
冋冑
− arctan
再
tan
冑共k0R兲2 − m2 log R
pk0R
共k0R兲 − m
where p ⬅ 冑共⑀c + i␥兲 / ⑀0.
2
共15兲
,
2
册
冎
⬘ 共pk0R兲 Jm , Jm共pk0R兲
冕
/2
2k0R␥ k 0R ␥ +F ⑀c ⑀c
,
共18兲
d cos exp共− x cos 兲
0
=
where the wavenumber k1 ⬅ 冑⑀0 / c. Equation 共12兲 allows a straightforward analytical solution
共r兲 = A冑r cos
共17兲
Substituting Eqs. 共15兲 and 共16兲 into Eq. 共17兲, in the limit k0R Ⰷ 1 we obtain
For Rc ⬍ r ⬍ R we can reduce the wave equation to
⬙ +
1 兺 兩1 − 兩rm兩2兩2 . k0 m
Rc 共16兲
冦
1−
x + O共x2兲, x Ⰶ 1, 2
冉冊
1 1 , 2 +O x x4
x Ⰷ 1.
冧
As expected, in the absence of losses Eq. 共18兲 yields zero absorption cross-section, while for k0R␥ Ⰷ 1 we find that a is close to the full geometrical cross-section per unit length of the cylinder, 2R. Thus, as predicted by the semiclassical theory, the device does indeed absorb all light incident on it from every direction. Furthermore, the effect we describe is essentially nonresonant, leading to a nearly perfect absorption for arbitrary wavelength, as long as the size of the system is substantially larger than the 共free-space兲 wavelength 0
冋 冉 冊册
a ⯝ 2R 1 −
7 ⑀c 4 k 0R ␥
2
,
R␥ ⱖ 0 .
共19兲
In conclusion, we presented an approach to broad-band omnidirectional light absorption, with nearly 100% efficiency. Such devices can find multiple applications in photovoltaics, solar energy harvesting, optoelectronics, and other areas. This work was partially supported by ARO MURI Award No. 50342-PH-MUR. One of the authors 共E.N.兲 would like to thank I. Aladinski for helpful discussions. D. Maystre and R. Petit, Opt. Commun. 17, 196 共1976兲. M. C. Hutley and D. Maystre, Opt. Commun. 19, 431 共1976兲. 3 M. Neviére, D. Maystre, R. MacPhedran, G. Derrick, and M. Hutley, in Proceedings of the ICO-11 Conference, Madrid, Spain 1978 共unpublished兲, p. 609. 4 G. H. Derrick, R. C. McPhedran, D. Maystre, and M. Nevière, Appl. Phys. 共Berlin兲 18, 39 共1979兲. 5 J.-J. Greffet, R. Carminati, K. Joulain, J.-P. Mulet, S. Mainguy, and Y. Chenet, Nature 共London兲 416, 61 共2002兲. 6 S. Collin, F. Pardo, R. Teissier, and J. L. Pelouard, Appl. Phys. Lett. 85, 194 共2004兲. 7 M.-L. Kuo, D. J. Poxson, Y. S. Kim, F. W. Mont, J. K. Kim, E. F. Schubert, and S.-Y. Lin, Opt. Lett. 33, 2527 共2008兲. 8 N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, Phys. Rev. Lett. 100, 207402 共2008兲. 9 T. V. Teperik, F. J. Garc’a de Abajo, A. G. Borisov, M. Abdelsalam, P. N. Bartlett, Y. Sugawara and J. J. Baumberg, Nat. Photonics 2, 299 共2008兲. 10 U. Leonhardt and P. Piwnicki, Phys. Rev. Lett. 84, 822 共2000兲. 11 Z. Jacob and E. E. Narimanov, Opt. Express 16, 4597 共2008兲. 12 T. Tyc and U. Leonhardt, New J. Phys. 10, 115038 共2008兲. 13 L. D. Landau and E. M. Lifshits, Quantum Mechanics, 3rd ed. 共Pergamon, New York, 1999兲. 1 2
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