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Increasing the electromagnetic attenuation below a quasi-matched surface with use of passive hyperbolic metamaterials
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Photonics and Nanostructures – Fundamentals and Applications xxx (2013) xxx–xxx www.elsevier.com/locate/photonics
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On increasing the electromagnetic attenuation below a quasi-matched surface with use of passive hyperbolic metamaterials
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Constantinos A. Valagiannopoulos *, Igor S. Nefedov
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Department of Radio Science and Engineering, School of Electrical Engineering, Aalto University, P.O. Box 13000, FI-00076 Aalto, Finland
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Received 24 April 2013; received in revised form 24 May 2013; accepted 28 May 2013
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Abstract Electromagnetic attenuation effect within a short distance can be very useful in numerous devices and occasions. It is exploited for experimental (anechoic chambers in laboratories), military (anti-radar coatings of aircrafts and ships) and computational (realization of absorbing boundary conditions in software simulations) reasons. In this work, we compare the attenuation inflicted by a hyperbolic metamaterial with that occurred into an ordinary lossy dielectric. In order for the comparison to be fair, we use the same magnitude of permittivity and the same loss tangent in both cases; similarly, the reflection coefficient is kept low in all the regarded examples. The results indicate that the hyperbolic metamaterial vastly outperforms the commonly used dielectric and one can use these media in order to construct very thin and efficient attenuators or absorbers by considering moderate thermal losses. # 2013 Elsevier B.V. All rights reserved. Keywords: Absorber; Attenuation; Hyperbolic Metamaterial; Lossy material
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1. Introduction 25 26 27 28 29 30 31 32 33 34 35 36
Electromagnetic attenuation is the gradual loss in intensity of a wave propagating into the host medium. In most cases, it is an unwanted effect since it constitutes a power leakage for the electromagnetic flow into a device. However, there are certain applications that attenuation is desirable, since electromagnetic absorption and insulation is necessary for the operation of a component. For example, passive mode locking of solid-state lasers by saturable absorbers based on carbon nanotubes fabricated by spin-coating a polymer combined with dielectric laser-mirrors, are studied
Q2 * Corresponding author. Tel.: þ358 504205858.
E-mail address: [email protected] (C.A. Valagiannopoulos).
and demonstrated in [1]. Total frequency and polarization selective absorber for mid-IR, made of metallic carbon nanotubes, was considered in [2]. In addition, very high absorption regardless of the field polarization and the angle of incidence has been reported [3], with a concept that utilizes the device as plasmonic sensor for refractive index sensing, which maintains its high performance even in non-laboratory environments. Furthermore, absorbers constituted by a matched layered structure has been employed [4] to attenuate a Gaussian beam. Finally, exceptional absorbing performances have been reported in [5], where the mechanism of Brewster angle is imitated by exploiting anisotropy, and in [6], where plasmonic metamaterials make the creation of topological darkness, feasible. Hyperbolic metamaterials are called the uniaxial anisotropic media whose real part of its transverse permittivity is opposite to that of the longitudinal
1569-4410/$ – see front matter # 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.photonics.2013.05.002
Please cite this article in press as: C.A. Valagiannopoulos, I.S. Nefedov, On increasing the electromagnetic attenuation below a quasi-matched surface with use of passive hyperbolic metamaterials, Photon Nanostruct: Fundam Appl (2013), http://dx.doi.org/ 10.1016/j.photonics.2013.05.002
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permittivity. Such a feature yields to a hyperbolically shaped dispersion surface instead of an ellipsoidally shaped which corresponds to the common dielectrics. The hyperbola has a big difference compared to the ellipse: it is extended to infinity. This means that in a hyperbolic medium can be supported very highly oscillated waves which are applicable in numerous practical cases. A review of the modern trends for structures and devices utilizing hyperbolic substrates such as fluorescence effect, nano-imaging and subsurface sensing, has been presented in [7]. In addition, the unusual properties of the hyperbolic metamaterials are exploited in order to convert the evanescent modes into propagating ones, by passing through a prismatic wedge [8]. Finally, planar multilayer hyperbolic metamaterials are investigated in [9], where the effective-medium approach is thoroughly discussed. In this work, we combine the two aforementioned topics (attenuation, hyperbolic metamaterials) by considering a very simple structure of a half space region illuminated by an oblique incident wave. This region is filled either by a common lossy medium or with the aforementioned hyperbolic metamaterial. The two boundary value problems are solved and the angle of incidence wave which leads to a (quasi-)matched wave along the discontinuity plane (zero or negligible reflections), have been determined. In this sense, the attenuation, that the transmitted wave is subjected to, is computed in the two cases and the results are compared each other. We find that, under certain conditions, the attenuation in the hyperbolic case can be several thousands more than in a typical lossy case. Furthermore, the effective thickness required in order to vanish the incident radiation, tends to be infinitesimal in terms of the free-space wavelength. Our work differs from [10], where the appearance of a surface plasmon mainly
at the surfaces of a slab filled with hyperbolic metamaterials, is observed. In [11], a similar issue is examined and certain resolutions about the analogous of Brewster angle in hyperbolic metamaterials have been extracted. However, in the work at hand we sketch diagrams for the supported waves into the absorbing region, under the assumption of minimum reflections. This feature provides the reader with an illustration of field attenuation, when the device is operating optimally each time by choosing properly the excitation. Furthermore, our motivation was to compare the absorbing efficiency of the hyperbolic metamaterial with the corresponding ordinary lossy substance, unlike [11], where the analysis is exclusively focused on the mechanism of the plasmonic material itself. The absorption into indefinite media has been also studied also in [12], where analysis mainly focuses on controlling the attenuation and not to its maximization as in this paper. Finally, super absorption of hyperbolic metamaterials has been reported in [13] but without the condition for matched surfaces as in the present study. 2. Problem statement Consider the two-dimensional (2D) physical configurations of Fig. 1, where the common Cartesian coordinate system (x, y, z) is also defined. In both structures the upper half space z < 0 is vacuum (Region 0), while the bottom half space z > 0 is filled with homogeneous material (Region 1). In both cases the structures are excited by an obliquely incident TMpolarized plane wave possessing a single y magnetic component given below: H0;inc ¼ yexp½ jk0 ð xsinu þ zcosuÞ:
E0,inc
E0,inc
H0,inc
Region 0
(1) x’
H0,inc
θ
O
(ε0, μ0)
x
Region 0
y Region 1
(a)
(ε0, μ0)
O
x
y=y’ (εr,losε0, μ0)
z
θ
Region 1
([εr,hyp]ε0, μ0)
ξ
z’
z
(b)
Fig. 1. The physical configurations of the two examined structures. An obliquely incident plane wave traveling into vacuum, meets a homogeneous half space filled with: (a) lossy isotropic material and (b) passive hyperbolic medium with a tilt angle.
Please cite this article in press as: C.A. Valagiannopoulos, I.S. Nefedov, On increasing the electromagnetic attenuation below a quasi-matched surface with use of passive hyperbolic metamaterials, Photon Nanostruct: Fundam Appl (2013), http://dx.doi.org/ 10.1016/j.photonics.2013.05.002
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The wave has a unitary magnitude and is propagating along a direction forming angle u 2 p2 ; p2 with the vertical axis. A harmonic time dependence exp(+ j2pft) is adopted and suppressed throughout the analysis. The pffiffiffiffiffiffiffiffiffiffi symbol k0 ¼ 2p l0 ¼ 2p f e0 m0 is reserved for the wavenumber into free space (e0, m0 are the constituent parameters of vacuum). In the case of Fig. 1(a), this material is isotropic and magnetically inert with a complex relative permittivity: er,los = a jd with a >> d > 0, where the subscript los corresponds to the lossy nature (I[er,los] = d) of the Region 1. We additionally assume that the real part of the permittivity is greater than unity, namely a > 1 (ordinary material presumption). In the case of Fig. 1(b), the material of Region 1 is anisotropic. In particular, the medium is uniaxially anisotropic with reference to the longitudinal direction z0 (the plane x0 y0 plays the role of the transversal surface); thus, the relative permittivity tensor expressed in the auxiliary Cartesian coordinate system (x0 , y0, z0 ) is given by: 2 3 a 0 0 5: ½e0r;hyp ¼ 4 0 a 0 (2) 0 0 a jd Such an anisotropic substance is a passive hyperbolic metamaterial; that is why the subscript hyp is employed in the aforementioned relation. Attempts to describe the materialization procedure and the design of similar artificial media, have been made in [14–16]. Moreover, technology of fabrication of zigzag silicon nanowire structures was developed by Sivakov et al. and described in [17]. We add the same losses (d) and use the same magnitude a for the real dielectric constants as in the isotropic medium; in this way, the two cases become comparable each other. As indicated in Fig. 1(b), the primed (auxiliary) coordinate system (x0 , y0, z0 ) is produced by rotating the unprimed (main) coordinate system (x, y, z) by angle j with respect to y = y0 axis. Accordingly, the permittivity tensor in the latter Cartesian coordinate system takes the form: 2
165 164 166 167
3 0 sinj cosj ½er;hyp ¼ 4 0 1 0 5 ½e0r;hyp sinj 0 cosj 2 3 cosj 0 sinj 5: 40 1 0 (3) sinj 0 cosj The angle j 2 p2 ; p2 corresponds to the tilt angle of the cluster of scatterers constituting the hyperbolic
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metamaterial. For example silicon (Si) nanorods [18] have been found to possess this opposite-sign property for their permittivities at the infrared frequency range. By titling them properly, one can obtain characteristics similar to that of the medium used to fill the Region 1 of Fig. 1(b). The scope of this problem is to evaluate the electromagnetic field transmitted into Region 1 and compare its attenuation due to the passivity of the materials in each structure (of Fig. 1(a) and (b)) under the assumption that the reflected field magnitude (back into Region 0) is kept very small. In this way, we can evaluate the comparative capacity of the hyperbolic metamaterial in absorbing the electromagnetic illumination below a matched surface against the case of using a simple isotropic lossy substance. The feature of rapid attenuation is very important since determines the required effective thickness of a substrate when modeling, designing and constructing numerous absorbing or fading devices. 3. Isotropic lossy material Let us focus on the structure depicted in Fig. 1(a) (all the quantities would have a subscript los). After imposing the necessary boundary conditions, the reflecting ray into Region 0 has an amplitude equal to the following reflection coefficient:
Rlos
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a jd sin2 u pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ¼ ða jdÞcosu þ a jd sin2 u ða jdÞcosu
(4)
If one takes into account that d ! 0, the angle u which corresponds to zero reflection (Rlos = 0, Brewster’s angle) is readily given by: rffiffiffiffiffiffiffiffiffiffiffi a ulos ¼ arcsin (5) ; 1þa where arcsinðwÞ 2 p2 ; p2 , which is compatible with our assumption for the incidence angle: u 2 p2 ; p2 . The assumption that d = 0 is mandatory in order to obtain a real optimal incidence angle u. The transmitted magnetic field into the lossy Region 1 possesses the expression: H1;los ðx; zÞ ¼ yT los exp½ jðk0 xsinu þ kz;los zÞ;
(6)
where Tlos = 1 + Rlos is the transmission coefficient and the longitudinal wavenumber kz,los is given by:
Please cite this article in press as: C.A. Valagiannopoulos, I.S. Nefedov, On increasing the electromagnetic attenuation below a quasi-matched surface with use of passive hyperbolic metamaterials, Photon Nanostruct: Fundam Appl (2013), http://dx.doi.org/ 10.1016/j.photonics.2013.05.002
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kz;los
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ k0 a jd sin2 u:
(7)
It should be stressed that I[kz,los] < 0 in order not to violate the energy preservation principle. The transverse wave impedance at the discontinuity plane z = 0 (for a wave traveling along the positive z semi-axis with electric field E1,los) has been found equal to: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a jd sin2 u x E1;los ðx; 0Þ ¼ z0 ; (8) Z los ¼ y H1;los ðx; 0Þ a jd qffiffiffiffi where z0 ¼ me 0 ¼ 120pV is the free-space wave im0 pedance. The wave impedance for u = ulos is computed as: Zlos ! z0 cos ulos , d ! 0, which is equal to the transverse impedance at z = 0 of the incident wave: Zinc = z0 cos u. In other words, a perfect matching (or a quasi-perfect matching, since d > 0) is achieved along the discontinuity plane z = 0; accordingly, zero (or very small since d > 0) back reflections into Region 0 are occurred.
with arccosðwÞ 2 ½0; p. It should be stressed that the angles u hyp should be real, namely:
0<
acosð2jÞ þ 1 <1) a2 þ 1 1 1 1 1 arccos < j < arccos : 2 a 2 a
The latter double inequality is compatible with our convention that j 2 p2 ; p2 because we supposed that a > 1. Note that in this case, both angles of (10) belong within the assumed interval u 2 p2 ; p2 , because 0 < w < 1 ) 0 < arccosðwÞ < p2 . But which of the two angles (10) is the correct since both yield to zero reflection? The answer to the question posed above is given by the sign of the imaginary part of the supported longitudinal wavenumber. In particular, this quantity is defined by:
4. Anisotropic hyperbolic metamaterial Let us examine the configuration shown in Fig. 1(b) (all the quantities would have a subscript hyp). After solving the vectorial Helmholtz equation and imposing the necessary boundary conditions we determine the two supported waves by the hyperbolic metamaterial. The reflection coefficient for these two waves, is evaluated as follows: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2aða þ jdÞcosu ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2a þ jdÞcosð2jÞ þ 2sin2 u þ jd pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (9) : Rhyp ¼ þ jdÞ cosu 2aða qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2a þ jdÞcosð2jÞ þ 2sin2 u þ jd
(11)
kz;hyp
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2sin2 u 2aða þ jdÞ þð2a þ jdÞcosð2jÞ þ jd ð2a þ jdÞsinusinð2jÞ : ¼ k0 ð2a þ jdÞcosð2jÞ þ jd (12)
We have found that only one of the two angles u hyp makes the selected wave not to violate the energy preservation principle (I[kz,hyp] < 0). In other words, the optimal angle in the hyperbolic case is given by: ( uhyp ¼
uþ hyp u hyp
; I½kz;hyp ju¼uþ < 0 hyp ; I½kz;hyp ju¼u < 0
(13)
hyp
239 240 241 242 243 244 245 246
In our attempt to find that incidence angle u which nullifies the reflection coefficient for d ! 0, we realize that this is possible only for one of the two modes (the one corresponding to the upper sign), due to our convention that u 2 p2 ; p2 . Therefore, we reject the second mode (the one corresponding to the lower sign) and for the first solution, we set the numerator of the Rhyp for d ! 0 equal to zero. The related condition is written as follows: acosð2jÞ þ 1 ) u ¼ u hyp a2 þ 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! acosð2jÞ þ 1 ¼ arccos ; a2 þ 1
Waves at this angle have been also examined in [19,20]. Anisotropic materials have been recently utilized to mitigate the reflections from a surface by exploiting the extreme differences between the permittivities along each axis [21,22]. On the other hand, extreme properties (perfect absorption) are acquired without extreme parameters [5,6,23,24]. The magnetic component of the transmitted field into Region 1 of Fig. 1(b) is derived as follows:
cos2 u ¼ 248 247
H1;hyp ðx; zÞ ¼ yT hyp exp½ jðk0 xsinu þ kz;hyp zÞ; (10)
(14)
where Thyp = 1 + Rhyp. The transverse wave impedance at the discontinuity plane z = 0 (for the transmitted wave
Please cite this article in press as: C.A. Valagiannopoulos, I.S. Nefedov, On increasing the electromagnetic attenuation below a quasi-matched surface with use of passive hyperbolic metamaterials, Photon Nanostruct: Fundam Appl (2013), http://dx.doi.org/ 10.1016/j.photonics.2013.05.002
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traveling along the positive z semi-axis, with electric field E1,hyp), is given by: x E1;hyp ðx; 0Þ y H1;hyp ðx; 0Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2a þ jdÞcosð2jÞ þ 2sin2 u þ jd pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ z0 : 2aða þ jdÞ
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perfect matching in both cases). Another crucial quantity determining the functionality of each structure is the longitudinal wavenumber when the reflection is not significant, namely:
Z hyp ¼
288 287 289 290 291 292 293
(15)
Note again that for u = uhyp, the surface is matched with the free space since Zhyp ! z0 cos uhyp = Zinc, d ! 0. All the electromagnetic power of the incident field (or the largest portion of it, since d > 0) would be transmitted in Region 1, instead of being reflected into vacuum area. 5. Numerical results
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5.1. Input and output parameters 295 296
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Let us clarify the ranges of the input parameters in the first place. The real permittivity a is chosen greater than unity (ordinary material presumption) but of moderate magnitude, namely: 1 < a < 10. The losses d are kept quite small in order to fulfill our hypothesis a >> d; in particular, 0 < d < 0.3. The tilt angle j is selected with reference to (11). The operational frequency f is not a crucial parameter in our consideration since the defined configurations of Fig 1 are comprised by two half spaces of infinite dimensions; therefore, no finite thickness is regarded. In other words, an increase in the operational wavelength would simply make the field quantities to spatially oscillate less rapidly into the two infinite regions, without modifying at all the reflection and the transmission phenomenon. That is why we do not consider frequency-dependent permittivities er,los and er,hyp; in this study we prefer to adopt a macroscopic approach treating the electromagnetic wave interaction instead of focusing in the microscopic material properties. As far as the output quantities are concerned, we would certainly represent the variations of the reflection coefficients in each case evaluated at u = ulos for the lossy problem and at u = uhyp for the hyperbolic problem. The used notations are shown below:
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r los ¼ Rlos ju¼ulos ;
322
The aforementioned reflection parameters (16) should be very small since we examine each case under the assumption of a quasi-perfect matching with the free space (the prefix ‘‘quasi-’’ is used due to the presence of losses which rules out the possibility of
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r hyp ¼ Rhyp ju¼uhyp :
(16)
klos ¼
kz;los ju¼ulos ; k0
khyp ¼
kz;hyp ju¼uhyp k0
:
(17)
The imaginary parts of these wavenumbers are a measure of how rapidly the transmitted wave is decaying into I½k Region 1. Therefore, the ratio I½khyp shows how more los effective is the absorption of the hyperbolic metamaterial compared to the case of an ordinary lossy substance. It is important to be clarified that the representation of the quantities rlos, rhyp, klos, khyp with respect to the material parameters (real permittivity or losses) implies that for each point the incidence angle of the plane wave changes to that value which assures a minimum level of reflections. In this way, we show the behavior of the optimal operation of the device when the problem configuration changes and can estimate its maximum performance for variable media. On the contrary, in [11] the perfect absorption is discussed for a specific realizable case (glass doped with silver) without examining the situation when the dielectric properties of the employed media (host and wire) are perturbed. 5.2. Graphs and comments In Fig. 2(a), we show the variation of the optimal incidence angles ulos, uhyp with respect to the tilt angle j for various real permittivities a. The thick lines (being invariant to the angle j) correspond to the lossy case, while the thin curves (defined at different j-interval as indicated by (11)) connecting isolated points represent the hyperbolic case. One can clearly observe from the graph that higher dielectric constants dictate more oblique incidence (larger u) in order to achieve matching. The curves of uhyp are even with respect to j and reaches a sole minimum at j = 0 which is smaller than ulos. Such a characteristic makes the hyperbolic medium preferable since it can absorb incoming waves with more extended incidence angle range. Note also that ulos gets closer to the minimum of the corresponding curve for increasing a. In Fig. 2(b), the optimal incidence angles are represented as functions of the real permittivity a for various tilt angles j. The increasing trend of the ulos and uhyp with respect to a is again recorded. The spread between the curves is maximized for small dielectric constants, where the Brewster’s angle is quite high.
Please cite this article in press as: C.A. Valagiannopoulos, I.S. Nefedov, On increasing the electromagnetic attenuation below a quasi-matched surface with use of passive hyperbolic metamaterials, Photon Nanostruct: Fundam Appl (2013), http://dx.doi.org/ 10.1016/j.photonics.2013.05.002
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optimal incidence angle θ (deg)
optimal incidence angle θ (deg)
90
80
70
60 θ
los
for α=2
θhyp for α=2
50
θlos for α=5 θhyp for α=5
40
θlos for α=8 θ
hyp
30 −60
−40
−20
0
20
60 50 40 θlos
30
tilt angle ξ (deg)
376 377 378 379 380 381 382 383 384 385 386
o o
1
2
3
4
5
6
7
8
9
10
real relative permittivity α
(b)
In Fig. 3(a), we depict the magnitude of the reflection coefficients |rlos|, |rhyp| in the quasi-matched regime as function of the relative dielectric constant a for several tilt angles j. It is evident that the reflection in all cases is negligible (below 1%) and thus we can focus in the attenuation factors in the following. It should be stressed that for most a, the ordinary lossy medium scatters more than the hyperbolic metamaterial regardless of the angle j. What seems rather peculiar is the decreasing trend of the curves with increasing a, namely when the material magnifies its contrast with the background vacuum medium. This happens mainly because the losses are kept constant (d = 0.1) and a
larger a makes the material less ‘‘relatively lossy’’ (decreases the loss tangent d/a) and thus more compatible with the lossless assumption, onto which the derivation of (10) and (16) is based. In Fig. 3(b), the reflection coefficients are represented with respect to the losses d for various tilt angles j. Obviously, the magnitudes |rlos|, |rhyp| are (almost linearly) increasing with d, since the optimal incidence angles have been computed for d ! 0. Again, all the configurations correspond to very low reflections and the results are better in the hyperbolic case than in the lossy one. In Fig. 4(a), we represent the variation of ( I [khyp]) expressed in dB, in a contour plot with
−3
7
x 10
0.012
|rlos| 6
reflection coefficient magnitude
375
for ξ=20
Fig. 2. The optimal incidence angles u (for the lossy and the hyperbolic case) that correspond to minimum reflection coefficients (a) as functions of the tilt angle j, for several real relative permittivities a, (b) as functions of the real relative permittivity a, for several tilt angles j. Plot parameter: d = 0.001.
reflection coefficient magnitude
374
θ
θhyp for ξ=40
(a)
373
for ξ=0o
hyp
0
60
θ
hyp
20 10
for α=8
40
70
o
|rhyp| for ξ=0 |r
| for ξ=20o
|r
| for ξ=40o
hyp
5
hyp
4
3
2
|rlos|
0.01
o
|rhyp| for ξ=0 |r
o
| for ξ=20
hyp
0.008
o
|rhyp| for ξ=40
0.006
0.004
0.002
1
0
0
1
2
3
4
5
6
7
real relative permittivity α (a)
8
9
10
0
0.05
0.1
0.15
0.2
0.25
0.3
dielectric losses δ (b)
Fig. 3. The magnitude of the (minimized) reflection coefficients |r| as functions of: (a) the real permittivity a (d = 0.1) and (b) the losses d (a = 5), in the lossy and in the hyperbolic case for various tilt angles j.
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Fig. 4. The imaginary part of the longitudinal waveguide expressed in dB 20 log( I [khyp]), in contour plot with respect to the dielectric losses d and the tilt angle j for (a) a ffi 1 and (b) a = 10. 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430
respect to the losses d and the tilt angle j for a ! 1. Mind that the quantity khyp defined by (17), is a relative one since the wavenumber is divided by k0. In other words, a specific ( I [khyp]) corresponds to a decay of the transmitted field by a factor of exp(2p I [khyp]) along a free-space wavelength l0. To put it alternatively, if we aim at a field whose magnitude is attenuated Q times into the volume of Region 1, we should choose a minimum thickness Wmin of the absorbing medium given by: Wlmin ¼ 2p IlnQ in terms of l0. Therefore, the 0 j ½khyp j knowledge of I[khyp] (or I[klos]), automatically indicates the required distance Wmin that the wave ought to travel into the absorbing material to get attenuated by a factor Q. In this way, the considered configuration, despite the fact that it is not realizable in physical reality due to the semi-infinite half space, can certainly give consistent information for the finite structure. It is remarkable that the level of losses d does not affect significantly the value of ( I [khyp]). The quantity is minimized at j ffi p4 (where I [khyp] ! 0) and maximized at j ffi p4 (where I [khyp] ffi 2). Namely, the attenuation of the input signal into Region 1 is moderate. The situation is different in Fig. 4(b), where a = 10. Once again, maximal attenuation is achieved in the vicinity of j ¼ p4 , but with enormous values of ( I [khyp]) > 300. In other words, after propagating within an infinitesimal fraction of l0 into Region 1, the the free-space wavelength 1000 transmitted field has lost about 99% of its power. In Fig. 5, we represent the same quantities with respect to the same variables (d, j) as in Fig. 4, but by
‘‘zooming’’ at the region of maximal absorption (j ffi p4 ). One can point out that even when a ! 1, the attenuation can be substantial in case of small losses. In fact, what really plays a role is the ratio ad (loss tangent) and not the losses d themselves. That is why the maximal attenuation area in Fig. 4(b) (where a = 10 and d a smaller) has been horizontally ‘‘stretched’’ compared to Fig. 4(a) (where ad larger). In Fig. 6, we compare the two materials (lossy and hyperbolic) in terms of how rapidly they attenuate the electromagnetic field under the assumption that only negligible reflections are occurred. In particular, we restrict our consideration at tilt angles close to j ffi p4 , which assures maximal absorption by the anisotropic I½k
medium and we represent the ratio I½khyp (in dB) on the los 2D map of (d, j). In Fig. 6(a), we take a ! 1 and we can note that the improvement for small losses is huge: I½khyp I½klos
> 106 . The results are even more impressive in Fig. 6(b) (a = 10), where the maximal ratio region is much more extended as it gets ‘‘stretched’’ exactly as in Fig. 5(b). But across the entire window of the considered parameters (d, j), the improvement in the I½k
attenuation factor is quite high: I½khyp > 30. That feature los demonstrates the superiority of the hyperbolic medium over the common lossy materials when it comes to the attenuation of the electromagnetic radiation within its volume. It is clear that the absorption is much more spatially rapid when using hyperbolic metamaterials and thus the required effective thickness of Region 1 could be much smaller.
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Fig. 5. The imaginary part of the longitudinal waveguide expressed in dB 20 log( I [khyp]), in contour plot with respect to the dielectric losses d and the tilt angle j for (a) a ffi 1 and (b) a = 10. Zoom at the maximal-value region on the map (d, j).
I½k Fig. 6. The ratio of the imaginary parts of the supported wavenumbers, expressed in dB, 20log I½khyp , in contour plot with respect to the dielectric los losses d and the tilt angle j for (a) a ffi 1 and (b) a = 10. Zoom at the maximal-value region on the map (d, j). 460
6. Conclusion 461 462 463 464 465 466 467 468 469 470 471
A comparison between the attenuation inflicted by a common lossy medium and a hyperbolic metamaterial to an incident plane wave has been made. The incidence angle is selected in each case in order to substantially mitigate the reflections and almost all the power is channeled into the half space filled with the corresponding material. It has been found that, under certain conditions, the effective thickness that attenuates 90% of the power could be as small as the one thousandth of the free-space wavelength. In addition, we reached the
conclusion that the hyperbolic metamaterial can be 106 times more effective than the corresponding isotropic medium in absorbing the incoming power. References [1] T.R. Schibli, K. Minoshima, H. Kataura, E. Itoga, N. Minami, S. Kazaoui, K. Miyashita, M. Tokumoto, Y. Sakakibara, Ultrashort pulse-generation by saturable absorber mirrors based on polymer-embedded carbon nanotubes, Optics Express 13 (2005) 8025–8031. [2] S.M. Hashemi, I.S. Nefedov, Wideband perfect absorption in arrays of tilted carbon nanotubes, Physical Review B 86 (2012) 195411.
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C.A. Valagiannopoulos, I.S. Nefedov / Photonics and Nanostructures – Fundamentals and Applications xxx (2013) xxx–xxx 483 484 [3] N. Liu, M. Mesch, T. Weiss, M. Hentschel, H. Giessen, Infrared 485 perfect absorber and its application as plasmonic sensor, Nano 486 Letters 10 (2010) 2342–2348. 487 Q3 [4] C.A. Valagiannopoulos, Electromagnetic absorption of Gaussian 488 beams by a grounded layered structure, Radioengineering 489 (2013), in press. 490 [5] V.G. Kravets, F. Schedin, A.N. Grigorenko, Plasmonic black491 body: almost complete absorption of light in nanostructured 492 metallic coatings, Physical Review B 78 (2008) 205405. 493 [6] V.G. Kravets, F. Schedin, R. Jalil, L. Britnell, R.V. Gorbachev, D. 494 Ansell, B. Thackray, K.S. Novoselov, A.K. Geim, A.V. Kaba495 shin, A.N. Grigorenko, Singular phase nano-optics in plasmonic 496 metamaterials for label-free single-molecule detection, Nature 497 Materials 12 (2013) 304–309. 498 [7] Y. Guo, W. Newman, C.L. Cortes, Z. Jacob, Applications of 499 hyperbolic metamaterial substrates, Advances in Optoelectron500 ics 2012 (2012) 452502. 501 [8] C.A. Valagiannopoulos, C.R. Simovski, An iterative semi-ana502 lytical technique solving the boundary value problem of trans503 mission through an anisotropic wedge, Radio Science 47 (2012) 504 RS6004. 505 [9] O. Kidwai, S.V. Zhukovsky, J.E. Sipe, Effective-medium 506 approach to planar multilayer hyperbolic metamaterials: 507 strengths and limitations, Physical Review A 85 (2012) 508 053842. 509 [10] D.G. Baranova, A.P. Vinogradova, C.R. Simovski, I.S. Nefedov, 510 S.A. Tretyakov, On the electrodynamics of an absorbing uniaxial 511 nonpositive determined (indefinite) medium, Journal of Experi512 mental and Theoretical Physics 114 (2012) 568–574. 513 [11] D.G. Baranov, A.P. Vinogradov, C.R. Simovski, Perfect absorp514 tion at Zenneck wave to plane wave transition, Metamaterials 6 515 (2012) 70–75. 516 [12] T.U. Tumkur, L. Gu, J.K. Kitur, E.E. Narimanov, M.A. Noginov, 517 Control of absorption with hyperbolic metamaterials, Applied 518 Physics Letters 100 (2012) 161103. 519 [13] C. Guclu, S. Campione, F. Capolino, Hyperbolic metamaterial as 520 super absorber for scattered fields generated at its surface, 521 Physical Review B 86 (2012) 205130.
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[14] X. Ni, G.V. Naik, A.V. Kildishev, Y. Barnakov, A. Boltasseva, V.M. Shalaev, Effect of metallic and hyperbolic metamaterial surfaces on electric and magnetic dipole emission transitions, Applied Physics B 103 (2011) 553–558. [15] T.U. Tumkur, J.K. Kitur, B. Chu, L. Gu, V.A. Podolskiy, E.E. Narimanov, M.A. Noginov, Control of reflectance and transmittance in scattering and curvilinear hyperbolic metamaterials, Applied Physics Letters 101 (2012) 091105. [16] M. Albooyeh, C.R. Simovski, Huge local field enhancement in perfect plasmonic absorbers, Optics Express 20 (2012) 21888– 21895. [17] V.A. Sivakov, G. Bronstrup, B. Pecz, A. Berger, G.Z. Radnoczi, M. Krause, S.H. Christiansen, Realization of vertical and zigzag single crystalline silicon nanowire architectures, Journal of Physical Chemistry C 114 (2010) 3798–3803. [18] G. Masetti, M. Severi, S. Solmi, Modeling of carrier mobility against carrier concentration in arsenic-, phosphorous-, and boron-doped silicon, IEEE Transactions on Electron Devices 30 (1983) 764–769. [19] C. Argyropoulos, K.Q. Le, N. Mattiucci, G. D’Aguanno, A. Alu, Broadband absorbers and selective emitters based on plasmonic Brewster metasurfaces, Physical Review B 87 (2013) 205112. [20] M.A. Noginov, Y.A. Barnakov, G. Zhu, T. Tumkur, H. Li, E.E. Narimanov, Bulk photonic metamaterial with hyperbolic dispersion, Applied Physics Letters 94 (2009) 151105. [21] P.B. Catrysse, S. Fan, Transverse electromagnetic modes in aperture waveguides containing a metamaterial with extreme anisotropy, Physical Review Letters 106 (2011) 223902. [22] P.B. Catrysse, S. Fan, Routing of deep-subwavelength optical beams and images without reflection and diffraction using infinitely anisotropic metamaterials, Advanced Materials 25 (2013) 194–198. [23] J. Kim, V.P. Drachev, Z. Jacob, G.V. Naik, A. Boltasseva, E.E. Narimanov, V.M. Shalaev, Improving the radiative decay rate for dye molecules with hyperbolic metamaterials, Applied Physics B 20 (2012) 161992. [24] C.M. Watts, X. Liu, W.J. Padilla, Metamaterial electromagnetic wave absorbers, Advanced Materials 24 (2012) OP98–OP120.
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Available online at www.sciencedirect.com
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Photonics and Nanostructures – Fundamentals and Applications xxx (2013) xxx–xxx www.elsevier.com/locate/photonics
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On increasing the electromagnetic attenuation below a quasi-matched surface with use of passive hyperbolic metamaterials
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4 5 6 7
Constantinos A. Valagiannopoulos *, Igor S. Nefedov
Q1
Department of Radio Science and Engineering, School of Electrical Engineering, Aalto University, P.O. Box 13000, FI-00076 Aalto, Finland
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Received 24 April 2013; received in revised form 24 May 2013; accepted 28 May 2013
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Abstract Electromagnetic attenuation effect within a short distance can be very useful in numerous devices and occasions. It is exploited for experimental (anechoic chambers in laboratories), military (anti-radar coatings of aircrafts and ships) and computational (realization of absorbing boundary conditions in software simulations) reasons. In this work, we compare the attenuation inflicted by a hyperbolic metamaterial with that occurred into an ordinary lossy dielectric. In order for the comparison to be fair, we use the same magnitude of permittivity and the same loss tangent in both cases; similarly, the reflection coefficient is kept low in all the regarded examples. The results indicate that the hyperbolic metamaterial vastly outperforms the commonly used dielectric and one can use these media in order to construct very thin and efficient attenuators or absorbers by considering moderate thermal losses. # 2013 Elsevier B.V. All rights reserved. Keywords: Absorber; Attenuation; Hyperbolic Metamaterial; Lossy material
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1. Introduction 25 26 27 28 29 30 31 32 33 34 35 36
Electromagnetic attenuation is the gradual loss in intensity of a wave propagating into the host medium. In most cases, it is an unwanted effect since it constitutes a power leakage for the electromagnetic flow into a device. However, there are certain applications that attenuation is desirable, since electromagnetic absorption and insulation is necessary for the operation of a component. For example, passive mode locking of solid-state lasers by saturable absorbers based on carbon nanotubes fabricated by spin-coating a polymer combined with dielectric laser-mirrors, are studied
Q2 * Corresponding author. Tel.: þ358 504205858.
E-mail address: [email protected] (C.A. Valagiannopoulos).
and demonstrated in [1]. Total frequency and polarization selective absorber for mid-IR, made of metallic carbon nanotubes, was considered in [2]. In addition, very high absorption regardless of the field polarization and the angle of incidence has been reported [3], with a concept that utilizes the device as plasmonic sensor for refractive index sensing, which maintains its high performance even in non-laboratory environments. Furthermore, absorbers constituted by a matched layered structure has been employed [4] to attenuate a Gaussian beam. Finally, exceptional absorbing performances have been reported in [5], where the mechanism of Brewster angle is imitated by exploiting anisotropy, and in [6], where plasmonic metamaterials make the creation of topological darkness, feasible. Hyperbolic metamaterials are called the uniaxial anisotropic media whose real part of its transverse permittivity is opposite to that of the longitudinal
1569-4410/$ – see front matter # 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.photonics.2013.05.002
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permittivity. Such a feature yields to a hyperbolically shaped dispersion surface instead of an ellipsoidally shaped which corresponds to the common dielectrics. The hyperbola has a big difference compared to the ellipse: it is extended to infinity. This means that in a hyperbolic medium can be supported very highly oscillated waves which are applicable in numerous practical cases. A review of the modern trends for structures and devices utilizing hyperbolic substrates such as fluorescence effect, nano-imaging and subsurface sensing, has been presented in [7]. In addition, the unusual properties of the hyperbolic metamaterials are exploited in order to convert the evanescent modes into propagating ones, by passing through a prismatic wedge [8]. Finally, planar multilayer hyperbolic metamaterials are investigated in [9], where the effective-medium approach is thoroughly discussed. In this work, we combine the two aforementioned topics (attenuation, hyperbolic metamaterials) by considering a very simple structure of a half space region illuminated by an oblique incident wave. This region is filled either by a common lossy medium or with the aforementioned hyperbolic metamaterial. The two boundary value problems are solved and the angle of incidence wave which leads to a (quasi-)matched wave along the discontinuity plane (zero or negligible reflections), have been determined. In this sense, the attenuation, that the transmitted wave is subjected to, is computed in the two cases and the results are compared each other. We find that, under certain conditions, the attenuation in the hyperbolic case can be several thousands more than in a typical lossy case. Furthermore, the effective thickness required in order to vanish the incident radiation, tends to be infinitesimal in terms of the free-space wavelength. Our work differs from [10], where the appearance of a surface plasmon mainly
at the surfaces of a slab filled with hyperbolic metamaterials, is observed. In [11], a similar issue is examined and certain resolutions about the analogous of Brewster angle in hyperbolic metamaterials have been extracted. However, in the work at hand we sketch diagrams for the supported waves into the absorbing region, under the assumption of minimum reflections. This feature provides the reader with an illustration of field attenuation, when the device is operating optimally each time by choosing properly the excitation. Furthermore, our motivation was to compare the absorbing efficiency of the hyperbolic metamaterial with the corresponding ordinary lossy substance, unlike [11], where the analysis is exclusively focused on the mechanism of the plasmonic material itself. The absorption into indefinite media has been also studied also in [12], where analysis mainly focuses on controlling the attenuation and not to its maximization as in this paper. Finally, super absorption of hyperbolic metamaterials has been reported in [13] but without the condition for matched surfaces as in the present study. 2. Problem statement Consider the two-dimensional (2D) physical configurations of Fig. 1, where the common Cartesian coordinate system (x, y, z) is also defined. In both structures the upper half space z < 0 is vacuum (Region 0), while the bottom half space z > 0 is filled with homogeneous material (Region 1). In both cases the structures are excited by an obliquely incident TMpolarized plane wave possessing a single y magnetic component given below: H0;inc ¼ yexp½ jk0 ð xsinu þ zcosuÞ:
E0,inc
E0,inc
H0,inc
Region 0
(1) x’
H0,inc
θ
O
(ε0, μ0)
x
Region 0
y Region 1
(a)
(ε0, μ0)
O
x
y=y’ (εr,losε0, μ0)
z
θ
Region 1
([εr,hyp]ε0, μ0)
ξ
z’
z
(b)
Fig. 1. The physical configurations of the two examined structures. An obliquely incident plane wave traveling into vacuum, meets a homogeneous half space filled with: (a) lossy isotropic material and (b) passive hyperbolic medium with a tilt angle.
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The wave has a unitary magnitude and is propagating along a direction forming angle u 2 p2 ; p2 with the vertical axis. A harmonic time dependence exp(+ j2pft) is adopted and suppressed throughout the analysis. The pffiffiffiffiffiffiffiffiffiffi symbol k0 ¼ 2p l0 ¼ 2p f e0 m0 is reserved for the wavenumber into free space (e0, m0 are the constituent parameters of vacuum). In the case of Fig. 1(a), this material is isotropic and magnetically inert with a complex relative permittivity: er,los = a jd with a >> d > 0, where the subscript los corresponds to the lossy nature (I[er,los] = d) of the Region 1. We additionally assume that the real part of the permittivity is greater than unity, namely a > 1 (ordinary material presumption). In the case of Fig. 1(b), the material of Region 1 is anisotropic. In particular, the medium is uniaxially anisotropic with reference to the longitudinal direction z0 (the plane x0 y0 plays the role of the transversal surface); thus, the relative permittivity tensor expressed in the auxiliary Cartesian coordinate system (x0 , y0, z0 ) is given by: 2 3 a 0 0 5: ½e0r;hyp ¼ 4 0 a 0 (2) 0 0 a jd Such an anisotropic substance is a passive hyperbolic metamaterial; that is why the subscript hyp is employed in the aforementioned relation. Attempts to describe the materialization procedure and the design of similar artificial media, have been made in [14–16]. Moreover, technology of fabrication of zigzag silicon nanowire structures was developed by Sivakov et al. and described in [17]. We add the same losses (d) and use the same magnitude a for the real dielectric constants as in the isotropic medium; in this way, the two cases become comparable each other. As indicated in Fig. 1(b), the primed (auxiliary) coordinate system (x0 , y0, z0 ) is produced by rotating the unprimed (main) coordinate system (x, y, z) by angle j with respect to y = y0 axis. Accordingly, the permittivity tensor in the latter Cartesian coordinate system takes the form: 2
165 164 166 167
3 0 sinj cosj ½er;hyp ¼ 4 0 1 0 5 ½e0r;hyp sinj 0 cosj 2 3 cosj 0 sinj 5: 40 1 0 (3) sinj 0 cosj The angle j 2 p2 ; p2 corresponds to the tilt angle of the cluster of scatterers constituting the hyperbolic
3
metamaterial. For example silicon (Si) nanorods [18] have been found to possess this opposite-sign property for their permittivities at the infrared frequency range. By titling them properly, one can obtain characteristics similar to that of the medium used to fill the Region 1 of Fig. 1(b). The scope of this problem is to evaluate the electromagnetic field transmitted into Region 1 and compare its attenuation due to the passivity of the materials in each structure (of Fig. 1(a) and (b)) under the assumption that the reflected field magnitude (back into Region 0) is kept very small. In this way, we can evaluate the comparative capacity of the hyperbolic metamaterial in absorbing the electromagnetic illumination below a matched surface against the case of using a simple isotropic lossy substance. The feature of rapid attenuation is very important since determines the required effective thickness of a substrate when modeling, designing and constructing numerous absorbing or fading devices. 3. Isotropic lossy material Let us focus on the structure depicted in Fig. 1(a) (all the quantities would have a subscript los). After imposing the necessary boundary conditions, the reflecting ray into Region 0 has an amplitude equal to the following reflection coefficient:
Rlos
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a jd sin2 u pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ¼ ða jdÞcosu þ a jd sin2 u ða jdÞcosu
(4)
If one takes into account that d ! 0, the angle u which corresponds to zero reflection (Rlos = 0, Brewster’s angle) is readily given by: rffiffiffiffiffiffiffiffiffiffiffi a ulos ¼ arcsin (5) ; 1þa where arcsinðwÞ 2 p2 ; p2 , which is compatible with our assumption for the incidence angle: u 2 p2 ; p2 . The assumption that d = 0 is mandatory in order to obtain a real optimal incidence angle u. The transmitted magnetic field into the lossy Region 1 possesses the expression: H1;los ðx; zÞ ¼ yT los exp½ jðk0 xsinu þ kz;los zÞ;
(6)
where Tlos = 1 + Rlos is the transmission coefficient and the longitudinal wavenumber kz,los is given by:
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229 230 231 232 233 234 235 236
238 237
kz;los
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ k0 a jd sin2 u:
(7)
It should be stressed that I[kz,los] < 0 in order not to violate the energy preservation principle. The transverse wave impedance at the discontinuity plane z = 0 (for a wave traveling along the positive z semi-axis with electric field E1,los) has been found equal to: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a jd sin2 u x E1;los ðx; 0Þ ¼ z0 ; (8) Z los ¼ y H1;los ðx; 0Þ a jd qffiffiffiffi where z0 ¼ me 0 ¼ 120pV is the free-space wave im0 pedance. The wave impedance for u = ulos is computed as: Zlos ! z0 cos ulos , d ! 0, which is equal to the transverse impedance at z = 0 of the incident wave: Zinc = z0 cos u. In other words, a perfect matching (or a quasi-perfect matching, since d > 0) is achieved along the discontinuity plane z = 0; accordingly, zero (or very small since d > 0) back reflections into Region 0 are occurred.
with arccosðwÞ 2 ½0; p. It should be stressed that the angles u hyp should be real, namely:
0<
acosð2jÞ þ 1 <1) a2 þ 1 1 1 1 1 arccos < j < arccos : 2 a 2 a
The latter double inequality is compatible with our convention that j 2 p2 ; p2 because we supposed that a > 1. Note that in this case, both angles of (10) belong within the assumed interval u 2 p2 ; p2 , because 0 < w < 1 ) 0 < arccosðwÞ < p2 . But which of the two angles (10) is the correct since both yield to zero reflection? The answer to the question posed above is given by the sign of the imaginary part of the supported longitudinal wavenumber. In particular, this quantity is defined by:
4. Anisotropic hyperbolic metamaterial Let us examine the configuration shown in Fig. 1(b) (all the quantities would have a subscript hyp). After solving the vectorial Helmholtz equation and imposing the necessary boundary conditions we determine the two supported waves by the hyperbolic metamaterial. The reflection coefficient for these two waves, is evaluated as follows: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2aða þ jdÞcosu ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2a þ jdÞcosð2jÞ þ 2sin2 u þ jd pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (9) : Rhyp ¼ þ jdÞ cosu 2aða qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2a þ jdÞcosð2jÞ þ 2sin2 u þ jd
(11)
kz;hyp
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2sin2 u 2aða þ jdÞ þð2a þ jdÞcosð2jÞ þ jd ð2a þ jdÞsinusinð2jÞ : ¼ k0 ð2a þ jdÞcosð2jÞ þ jd (12)
We have found that only one of the two angles u hyp makes the selected wave not to violate the energy preservation principle (I[kz,hyp] < 0). In other words, the optimal angle in the hyperbolic case is given by: ( uhyp ¼
uþ hyp u hyp
; I½kz;hyp ju¼uþ < 0 hyp ; I½kz;hyp ju¼u < 0
(13)
hyp
239 240 241 242 243 244 245 246
In our attempt to find that incidence angle u which nullifies the reflection coefficient for d ! 0, we realize that this is possible only for one of the two modes (the one corresponding to the upper sign), due to our convention that u 2 p2 ; p2 . Therefore, we reject the second mode (the one corresponding to the lower sign) and for the first solution, we set the numerator of the Rhyp for d ! 0 equal to zero. The related condition is written as follows: acosð2jÞ þ 1 ) u ¼ u hyp a2 þ 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! acosð2jÞ þ 1 ¼ arccos ; a2 þ 1
Waves at this angle have been also examined in [19,20]. Anisotropic materials have been recently utilized to mitigate the reflections from a surface by exploiting the extreme differences between the permittivities along each axis [21,22]. On the other hand, extreme properties (perfect absorption) are acquired without extreme parameters [5,6,23,24]. The magnetic component of the transmitted field into Region 1 of Fig. 1(b) is derived as follows:
cos2 u ¼ 248 247
H1;hyp ðx; zÞ ¼ yT hyp exp½ jðk0 xsinu þ kz;hyp zÞ; (10)
(14)
where Thyp = 1 + Rhyp. The transverse wave impedance at the discontinuity plane z = 0 (for the transmitted wave
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traveling along the positive z semi-axis, with electric field E1,hyp), is given by: x E1;hyp ðx; 0Þ y H1;hyp ðx; 0Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2a þ jdÞcosð2jÞ þ 2sin2 u þ jd pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ z0 : 2aða þ jdÞ
5
perfect matching in both cases). Another crucial quantity determining the functionality of each structure is the longitudinal wavenumber when the reflection is not significant, namely:
Z hyp ¼
288 287 289 290 291 292 293
(15)
Note again that for u = uhyp, the surface is matched with the free space since Zhyp ! z0 cos uhyp = Zinc, d ! 0. All the electromagnetic power of the incident field (or the largest portion of it, since d > 0) would be transmitted in Region 1, instead of being reflected into vacuum area. 5. Numerical results
294
5.1. Input and output parameters 295 296
319
Let us clarify the ranges of the input parameters in the first place. The real permittivity a is chosen greater than unity (ordinary material presumption) but of moderate magnitude, namely: 1 < a < 10. The losses d are kept quite small in order to fulfill our hypothesis a >> d; in particular, 0 < d < 0.3. The tilt angle j is selected with reference to (11). The operational frequency f is not a crucial parameter in our consideration since the defined configurations of Fig 1 are comprised by two half spaces of infinite dimensions; therefore, no finite thickness is regarded. In other words, an increase in the operational wavelength would simply make the field quantities to spatially oscillate less rapidly into the two infinite regions, without modifying at all the reflection and the transmission phenomenon. That is why we do not consider frequency-dependent permittivities er,los and er,hyp; in this study we prefer to adopt a macroscopic approach treating the electromagnetic wave interaction instead of focusing in the microscopic material properties. As far as the output quantities are concerned, we would certainly represent the variations of the reflection coefficients in each case evaluated at u = ulos for the lossy problem and at u = uhyp for the hyperbolic problem. The used notations are shown below:
321 320
r los ¼ Rlos ju¼ulos ;
322
The aforementioned reflection parameters (16) should be very small since we examine each case under the assumption of a quasi-perfect matching with the free space (the prefix ‘‘quasi-’’ is used due to the presence of losses which rules out the possibility of
297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318
323 324 325 326
r hyp ¼ Rhyp ju¼uhyp :
(16)
klos ¼
kz;los ju¼ulos ; k0
khyp ¼
kz;hyp ju¼uhyp k0
:
(17)
The imaginary parts of these wavenumbers are a measure of how rapidly the transmitted wave is decaying into I½k Region 1. Therefore, the ratio I½khyp shows how more los effective is the absorption of the hyperbolic metamaterial compared to the case of an ordinary lossy substance. It is important to be clarified that the representation of the quantities rlos, rhyp, klos, khyp with respect to the material parameters (real permittivity or losses) implies that for each point the incidence angle of the plane wave changes to that value which assures a minimum level of reflections. In this way, we show the behavior of the optimal operation of the device when the problem configuration changes and can estimate its maximum performance for variable media. On the contrary, in [11] the perfect absorption is discussed for a specific realizable case (glass doped with silver) without examining the situation when the dielectric properties of the employed media (host and wire) are perturbed. 5.2. Graphs and comments In Fig. 2(a), we show the variation of the optimal incidence angles ulos, uhyp with respect to the tilt angle j for various real permittivities a. The thick lines (being invariant to the angle j) correspond to the lossy case, while the thin curves (defined at different j-interval as indicated by (11)) connecting isolated points represent the hyperbolic case. One can clearly observe from the graph that higher dielectric constants dictate more oblique incidence (larger u) in order to achieve matching. The curves of uhyp are even with respect to j and reaches a sole minimum at j = 0 which is smaller than ulos. Such a characteristic makes the hyperbolic medium preferable since it can absorb incoming waves with more extended incidence angle range. Note also that ulos gets closer to the minimum of the corresponding curve for increasing a. In Fig. 2(b), the optimal incidence angles are represented as functions of the real permittivity a for various tilt angles j. The increasing trend of the ulos and uhyp with respect to a is again recorded. The spread between the curves is maximized for small dielectric constants, where the Brewster’s angle is quite high.
Please cite this article in press as: C.A. Valagiannopoulos, I.S. Nefedov, On increasing the electromagnetic attenuation below a quasi-matched surface with use of passive hyperbolic metamaterials, Photon Nanostruct: Fundam Appl (2013), http://dx.doi.org/ 10.1016/j.photonics.2013.05.002
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optimal incidence angle θ (deg)
optimal incidence angle θ (deg)
90
80
70
60 θ
los
for α=2
θhyp for α=2
50
θlos for α=5 θhyp for α=5
40
θlos for α=8 θ
hyp
30 −60
−40
−20
0
20
60 50 40 θlos
30
tilt angle ξ (deg)
376 377 378 379 380 381 382 383 384 385 386
o o
1
2
3
4
5
6
7
8
9
10
real relative permittivity α
(b)
In Fig. 3(a), we depict the magnitude of the reflection coefficients |rlos|, |rhyp| in the quasi-matched regime as function of the relative dielectric constant a for several tilt angles j. It is evident that the reflection in all cases is negligible (below 1%) and thus we can focus in the attenuation factors in the following. It should be stressed that for most a, the ordinary lossy medium scatters more than the hyperbolic metamaterial regardless of the angle j. What seems rather peculiar is the decreasing trend of the curves with increasing a, namely when the material magnifies its contrast with the background vacuum medium. This happens mainly because the losses are kept constant (d = 0.1) and a
larger a makes the material less ‘‘relatively lossy’’ (decreases the loss tangent d/a) and thus more compatible with the lossless assumption, onto which the derivation of (10) and (16) is based. In Fig. 3(b), the reflection coefficients are represented with respect to the losses d for various tilt angles j. Obviously, the magnitudes |rlos|, |rhyp| are (almost linearly) increasing with d, since the optimal incidence angles have been computed for d ! 0. Again, all the configurations correspond to very low reflections and the results are better in the hyperbolic case than in the lossy one. In Fig. 4(a), we represent the variation of ( I [khyp]) expressed in dB, in a contour plot with
−3
7
x 10
0.012
|rlos| 6
reflection coefficient magnitude
375
for ξ=20
Fig. 2. The optimal incidence angles u (for the lossy and the hyperbolic case) that correspond to minimum reflection coefficients (a) as functions of the tilt angle j, for several real relative permittivities a, (b) as functions of the real relative permittivity a, for several tilt angles j. Plot parameter: d = 0.001.
reflection coefficient magnitude
374
θ
θhyp for ξ=40
(a)
373
for ξ=0o
hyp
0
60
θ
hyp
20 10
for α=8
40
70
o
|rhyp| for ξ=0 |r
| for ξ=20o
|r
| for ξ=40o
hyp
5
hyp
4
3
2
|rlos|
0.01
o
|rhyp| for ξ=0 |r
o
| for ξ=20
hyp
0.008
o
|rhyp| for ξ=40
0.006
0.004
0.002
1
0
0
1
2
3
4
5
6
7
real relative permittivity α (a)
8
9
10
0
0.05
0.1
0.15
0.2
0.25
0.3
dielectric losses δ (b)
Fig. 3. The magnitude of the (minimized) reflection coefficients |r| as functions of: (a) the real permittivity a (d = 0.1) and (b) the losses d (a = 5), in the lossy and in the hyperbolic case for various tilt angles j.
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Fig. 4. The imaginary part of the longitudinal waveguide expressed in dB 20 log( I [khyp]), in contour plot with respect to the dielectric losses d and the tilt angle j for (a) a ffi 1 and (b) a = 10. 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430
respect to the losses d and the tilt angle j for a ! 1. Mind that the quantity khyp defined by (17), is a relative one since the wavenumber is divided by k0. In other words, a specific ( I [khyp]) corresponds to a decay of the transmitted field by a factor of exp(2p I [khyp]) along a free-space wavelength l0. To put it alternatively, if we aim at a field whose magnitude is attenuated Q times into the volume of Region 1, we should choose a minimum thickness Wmin of the absorbing medium given by: Wlmin ¼ 2p IlnQ in terms of l0. Therefore, the 0 j ½khyp j knowledge of I[khyp] (or I[klos]), automatically indicates the required distance Wmin that the wave ought to travel into the absorbing material to get attenuated by a factor Q. In this way, the considered configuration, despite the fact that it is not realizable in physical reality due to the semi-infinite half space, can certainly give consistent information for the finite structure. It is remarkable that the level of losses d does not affect significantly the value of ( I [khyp]). The quantity is minimized at j ffi p4 (where I [khyp] ! 0) and maximized at j ffi p4 (where I [khyp] ffi 2). Namely, the attenuation of the input signal into Region 1 is moderate. The situation is different in Fig. 4(b), where a = 10. Once again, maximal attenuation is achieved in the vicinity of j ¼ p4 , but with enormous values of ( I [khyp]) > 300. In other words, after propagating within an infinitesimal fraction of l0 into Region 1, the the free-space wavelength 1000 transmitted field has lost about 99% of its power. In Fig. 5, we represent the same quantities with respect to the same variables (d, j) as in Fig. 4, but by
‘‘zooming’’ at the region of maximal absorption (j ffi p4 ). One can point out that even when a ! 1, the attenuation can be substantial in case of small losses. In fact, what really plays a role is the ratio ad (loss tangent) and not the losses d themselves. That is why the maximal attenuation area in Fig. 4(b) (where a = 10 and d a smaller) has been horizontally ‘‘stretched’’ compared to Fig. 4(a) (where ad larger). In Fig. 6, we compare the two materials (lossy and hyperbolic) in terms of how rapidly they attenuate the electromagnetic field under the assumption that only negligible reflections are occurred. In particular, we restrict our consideration at tilt angles close to j ffi p4 , which assures maximal absorption by the anisotropic I½k
medium and we represent the ratio I½khyp (in dB) on the los 2D map of (d, j). In Fig. 6(a), we take a ! 1 and we can note that the improvement for small losses is huge: I½khyp I½klos
> 106 . The results are even more impressive in Fig. 6(b) (a = 10), where the maximal ratio region is much more extended as it gets ‘‘stretched’’ exactly as in Fig. 5(b). But across the entire window of the considered parameters (d, j), the improvement in the I½k
attenuation factor is quite high: I½khyp > 30. That feature los demonstrates the superiority of the hyperbolic medium over the common lossy materials when it comes to the attenuation of the electromagnetic radiation within its volume. It is clear that the absorption is much more spatially rapid when using hyperbolic metamaterials and thus the required effective thickness of Region 1 could be much smaller.
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Fig. 5. The imaginary part of the longitudinal waveguide expressed in dB 20 log( I [khyp]), in contour plot with respect to the dielectric losses d and the tilt angle j for (a) a ffi 1 and (b) a = 10. Zoom at the maximal-value region on the map (d, j).
I½k Fig. 6. The ratio of the imaginary parts of the supported wavenumbers, expressed in dB, 20log I½khyp , in contour plot with respect to the dielectric los losses d and the tilt angle j for (a) a ffi 1 and (b) a = 10. Zoom at the maximal-value region on the map (d, j). 460
6. Conclusion 461 462 463 464 465 466 467 468 469 470 471
A comparison between the attenuation inflicted by a common lossy medium and a hyperbolic metamaterial to an incident plane wave has been made. The incidence angle is selected in each case in order to substantially mitigate the reflections and almost all the power is channeled into the half space filled with the corresponding material. It has been found that, under certain conditions, the effective thickness that attenuates 90% of the power could be as small as the one thousandth of the free-space wavelength. In addition, we reached the
conclusion that the hyperbolic metamaterial can be 106 times more effective than the corresponding isotropic medium in absorbing the incoming power. References [1] T.R. Schibli, K. Minoshima, H. Kataura, E. Itoga, N. Minami, S. Kazaoui, K. Miyashita, M. Tokumoto, Y. Sakakibara, Ultrashort pulse-generation by saturable absorber mirrors based on polymer-embedded carbon nanotubes, Optics Express 13 (2005) 8025–8031. [2] S.M. Hashemi, I.S. Nefedov, Wideband perfect absorption in arrays of tilted carbon nanotubes, Physical Review B 86 (2012) 195411.
Please cite this article in press as: C.A. Valagiannopoulos, I.S. Nefedov, On increasing the electromagnetic attenuation below a quasi-matched surface with use of passive hyperbolic metamaterials, Photon Nanostruct: Fundam Appl (2013), http://dx.doi.org/ 10.1016/j.photonics.2013.05.002
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PNFA 393 1–9
C.A. Valagiannopoulos, I.S. Nefedov / Photonics and Nanostructures – Fundamentals and Applications xxx (2013) xxx–xxx 483 484 [3] N. Liu, M. Mesch, T. Weiss, M. Hentschel, H. Giessen, Infrared 485 perfect absorber and its application as plasmonic sensor, Nano 486 Letters 10 (2010) 2342–2348. 487 Q3 [4] C.A. Valagiannopoulos, Electromagnetic absorption of Gaussian 488 beams by a grounded layered structure, Radioengineering 489 (2013), in press. 490 [5] V.G. Kravets, F. Schedin, A.N. Grigorenko, Plasmonic black491 body: almost complete absorption of light in nanostructured 492 metallic coatings, Physical Review B 78 (2008) 205405. 493 [6] V.G. Kravets, F. Schedin, R. Jalil, L. Britnell, R.V. Gorbachev, D. 494 Ansell, B. Thackray, K.S. Novoselov, A.K. Geim, A.V. Kaba495 shin, A.N. Grigorenko, Singular phase nano-optics in plasmonic 496 metamaterials for label-free single-molecule detection, Nature 497 Materials 12 (2013) 304–309. 498 [7] Y. Guo, W. Newman, C.L. Cortes, Z. Jacob, Applications of 499 hyperbolic metamaterial substrates, Advances in Optoelectron500 ics 2012 (2012) 452502. 501 [8] C.A. Valagiannopoulos, C.R. Simovski, An iterative semi-ana502 lytical technique solving the boundary value problem of trans503 mission through an anisotropic wedge, Radio Science 47 (2012) 504 RS6004. 505 [9] O. Kidwai, S.V. Zhukovsky, J.E. Sipe, Effective-medium 506 approach to planar multilayer hyperbolic metamaterials: 507 strengths and limitations, Physical Review A 85 (2012) 508 053842. 509 [10] D.G. Baranova, A.P. Vinogradova, C.R. Simovski, I.S. Nefedov, 510 S.A. Tretyakov, On the electrodynamics of an absorbing uniaxial 511 nonpositive determined (indefinite) medium, Journal of Experi512 mental and Theoretical Physics 114 (2012) 568–574. 513 [11] D.G. Baranov, A.P. Vinogradov, C.R. Simovski, Perfect absorp514 tion at Zenneck wave to plane wave transition, Metamaterials 6 515 (2012) 70–75. 516 [12] T.U. Tumkur, L. Gu, J.K. Kitur, E.E. Narimanov, M.A. Noginov, 517 Control of absorption with hyperbolic metamaterials, Applied 518 Physics Letters 100 (2012) 161103. 519 [13] C. Guclu, S. Campione, F. Capolino, Hyperbolic metamaterial as 520 super absorber for scattered fields generated at its surface, 521 Physical Review B 86 (2012) 205130.
9
[14] X. Ni, G.V. Naik, A.V. Kildishev, Y. Barnakov, A. Boltasseva, V.M. Shalaev, Effect of metallic and hyperbolic metamaterial surfaces on electric and magnetic dipole emission transitions, Applied Physics B 103 (2011) 553–558. [15] T.U. Tumkur, J.K. Kitur, B. Chu, L. Gu, V.A. Podolskiy, E.E. Narimanov, M.A. Noginov, Control of reflectance and transmittance in scattering and curvilinear hyperbolic metamaterials, Applied Physics Letters 101 (2012) 091105. [16] M. Albooyeh, C.R. Simovski, Huge local field enhancement in perfect plasmonic absorbers, Optics Express 20 (2012) 21888– 21895. [17] V.A. Sivakov, G. Bronstrup, B. Pecz, A. Berger, G.Z. Radnoczi, M. Krause, S.H. Christiansen, Realization of vertical and zigzag single crystalline silicon nanowire architectures, Journal of Physical Chemistry C 114 (2010) 3798–3803. [18] G. Masetti, M. Severi, S. Solmi, Modeling of carrier mobility against carrier concentration in arsenic-, phosphorous-, and boron-doped silicon, IEEE Transactions on Electron Devices 30 (1983) 764–769. [19] C. Argyropoulos, K.Q. Le, N. Mattiucci, G. D’Aguanno, A. Alu, Broadband absorbers and selective emitters based on plasmonic Brewster metasurfaces, Physical Review B 87 (2013) 205112. [20] M.A. Noginov, Y.A. Barnakov, G. Zhu, T. Tumkur, H. Li, E.E. Narimanov, Bulk photonic metamaterial with hyperbolic dispersion, Applied Physics Letters 94 (2009) 151105. [21] P.B. Catrysse, S. Fan, Transverse electromagnetic modes in aperture waveguides containing a metamaterial with extreme anisotropy, Physical Review Letters 106 (2011) 223902. [22] P.B. Catrysse, S. Fan, Routing of deep-subwavelength optical beams and images without reflection and diffraction using infinitely anisotropic metamaterials, Advanced Materials 25 (2013) 194–198. [23] J. Kim, V.P. Drachev, Z. Jacob, G.V. Naik, A. Boltasseva, E.E. Narimanov, V.M. Shalaev, Improving the radiative decay rate for dye molecules with hyperbolic metamaterials, Applied Physics B 20 (2012) 161992. [24] C.M. Watts, X. Liu, W.J. Padilla, Metamaterial electromagnetic wave absorbers, Advanced Materials 24 (2012) OP98–OP120.
Please cite this article in press as: C.A. Valagiannopoulos, I.S. Nefedov, On increasing the electromagnetic attenuation below a quasi-matched surface with use of passive hyperbolic metamaterials, Photon Nanostruct: Fundam Appl (2013), http://dx.doi.org/ 10.1016/j.photonics.2013.05.002