Highly Efficient Broadband Light Absorber Based on Nonuniform Hyperbolic Metamaterial Film

We develop a concept of highly efficient broadband light absorber based on nonuniform hyperbolic metamaterial. We suggest a gradual bending of the anisotropy axis inside the metamaterial that results in spatial shift of the area of resonant absorption depending on the incidence angle. In this resonant region the wavevector of light is parallel to the generatrix of resonant cone and the radiation losses are maximal because of extremely high value of refraction index. Changing the radiation frequency also shifts the spatial position of the resonant region so that high level of absorption may be achieved in wide frequency range. Using the model of nanowire medium (silver wires in silica host) we predict that 200 nm film of this hyperbolic metamaterial allows reaching almost total absorption of radiation throughout the visible band.


Introduction
Exotic and interesting electromagnetic properties of the socalled hyperbolic metamaterials (HMs) cause their extensive study [1][2][3][4][5][6][7].HM is an anisotropic resonance medium with hyperboloid-shaped isofrequency surface.In optics HM can be realized, for example, as a metal-dielectric planar nanostructure or a lattice of metallic nanowires embedded in a dielectric matrix.
The ability to support high-index volume electromagnetic modes is one of the main advantages of HM, and it results from the subwavelength light localization, which is also a characteristic feature of surface plasmon polaritons.The similarity comes from the fact that the volume electromagnetic mode in HM is actually the superposition of interacting plasmon modes localized at the nearby metal-dielectric interfaces.The collective character of the modes results in a hybridized broadband response [8,9].
Hyperbolic character of the metamaterial (MM) can be achieved only for rather high volume fraction of metal that causes high losses.The radiation absorption is especially strong in the case of high refractive index, when a wave beam propagates with its phase velocity directed along the resonance cone generatrix.Unavoidable ohmic loss is a great drawback for applications in high-resolution imaging and lithography; however this effect can be used to engineer absorbing coatings.
The functionality of the absorber (let it be an absorbing film) is defined by its absorptivity, its spatial extent (preferably, smallest), and its broadband and omnidirectional operation.The absorptivity, in turn, depends on the thickness of the film, on the radiation wavelength, and on the incidence angle.In the case of anisotropic material of the film one can also consider the orientation of the anisotropy axis.Some of the requirements are contradictory.Indeed, thin films often demonstrate resonant behaviour that leads to strong angular and wavelength dependence of the absorption coefficient; severely absorbing material with large imaginary part of the permittivity (able to absorb the radiation in small volume) strongly reflects the incident wave.These contradictions make it difficult to design an absorber.
The engineering problem is twofold: first, it is necessary to avoid strong reflection when launching light from free space, and second, it is desirable to transform the incident light into small-scale radiation which is strongly damped and can be almost totally absorbed in a quite thin MM layer.
Usual way to improve the coupling between free space radiation and the medium is a modification of the airmedium interface, for example, using a diffraction grating.If the grating period is less than the wavelength in free space, then the scattered small-scale radiation can be effectively absorbed in the metamaterial.An interesting approach for increasing the absorption in hyperbolic media was proposed in [27] where the authors experimentally observed strongly reduced reflection from roughened metamaterial surface.
Another approach for increasing the absorption is to turn the optical axis of HM in such a way that the direction of the resonance cone generatrix would coincide with the normal to the HM interface [5][6][7].In this case normally incident light directly transforms into small-scale radiation; however, rather high reflection takes place (see Figure 4) which has a detrimental effect on the absorptivity of the HM layer.In both cases, the spatial spectrum of radiation is changed mainly at the interfaces.On the other hand, the absorption can be increased due to the medium inhomogeneity.

Statement of the Problem
In this paper, we study the absorbing properties of the HM layer when the material structure changes inside the slab.We use the effective medium approximation and consider the propagation of the extraordinary wave in the wire medium: dielectric host with metal wires.In this case the components of effective permittivity tensor ε along the wires  ‖ and perpendicular to them  ⊥ can be defined as [28] where   is the permittivity of dielectric host,   is the permittivity of metal, and   is the volume fraction of metal.Below we limit ourselves to the TM polarization of the light and use for computations the parameters of silica (  = 2.1590) and silver with the volume fraction   = 0.25.The dispersion dependence of silver   () is taken from [29] and depicted in Figure 1(a): notice different scales for real and imaginary parts of permittivity.
Figure 1 shows also the dependence of  ‖ ,  ⊥ (see Figure 1(b)), and the normalized critical angle,  * / (see Figure 1(c)), on the radiation wavelength; the latter can be found from the condition tan( * ) = Re(√− ‖ / ⊥ ). * is the angle of the resonance cone opening: the hyperbolic medium is transparent only for the waves with the wave vector k inside the resonance cone, i.e., when the angle between the optical axis s of the MM and k is less than  * , ∠(k, s) ≤  * .The refraction index for normally incident light tends to infinity when the longitudinal (along the wires) optical axis of the metamaterial makes an angle  * with the normal to the interface (the resonance cone generatrix is along the normal).Dashed line marks the critical wavelength ( * ≈ 437 nm) at which the topological transition takes place: the effective medium transforms from common elliptical to hyperbolic state.

Reflection from Homogeneous HM Slab
In the following, we consider the reflection of incident radiation from the MM slab with the thickness   placed on the perfectly reflecting substrate.The geometry is shown in Figure 2.
Hyperbolic medium is highly anisotropic, and the wave propagation and, in particular, the reflection coefficient depend on the mutual arrangement of the incident light wave (vector k) and the metamaterial's normal n and symmetry axis s.In this paper, we consider reflection from a uniform slab with fixed orientation of the optical axis, ∠(n, s) ≡  = const, and reflection from the MM slab in which  varies linearly with the coordinate .
Figure 3 shows the dependence of the reflection coefficient on the angle  in the case of the uniform slab.The results are given for three values of the film thickness,  (1,2,3)  = 10, 100, and 1000 nm, and also for the reflection from the HM half-space,  (4)   = ∞.One can see that for the angles  ≈  * the curves 2, 3, and 4 nearly coincide.This means that rather thin MM films behave as (infinitely) thick ones when the relation  =  * is satisfied.This statement is verified for most of the visible domain, as one can see in Figures 4(a Figure 4 shows the spectral dependence of absorption of normally incident light.Solid lines in the panel (a) show the wavelength dependence of the absorption coefficient  = 1 − || 2 ( is the reflection coefficient, i.e., the ratio of the complex amplitudes of reflected and incident waves) in the case of metamaterial layer of finite thickness   = 200 nm, while the dotted lines correspond to the absorption coefficient  ∞ = 1 − | ∞ | 2 of the MM half-space.The orientation angle is  = 0 (curve 1),  = 0.1 (curve 2), and  =  * (curve 3; note that  * also depends on , see Figure 1(c)).Panel (b) demonstrates the wavelength dependence of the ratio / ∞ for these three cases.One can see that a thin layer with  =  * absorbs as an infinite one (the curve 3 actually coincides with the corresponding dotted line in panel (a) and the ratio / ∞ = 1 in the whole visible band), but considerable part of radiation is reflected.The absorption of the finite layer with fixed orientation angle (curves 1, 2) strongly depends on the wavelength and degrades for longer waves.

Nonuniform Absorber
Weak wavelength dependence of absorption is attractive for broadband operation, but it is achieved only for specific choice of the angle  =  * () (curve 3 in Figures 4(a) and 4(b)).Certainly, such an absorber with the optical axis turning in accordance with the wavelength is unrealistic.However, the absorbing properties can be improved for the case of nonuniform film.For simplicity we consider the metamaterial with constant   and suppose that the orientation angle  increases linearly (from 0 to /2) with the propagation length.These situation cannot be achieved in common wire medium since changing the orientation of the wires changes also the density of wires and thus the value of   .On the other hand, within the effective medium Advances in Condensed Matter Physics approximation it is not necessary to use continuous wires.Elongated silver particles (meta-atoms) can serve as a metallic component of the MM.Such meta-atoms suspended in a viscous liquid can be easily aligned by means of constant electric field in order to create the proper orientational pattern [30,31].
Bearing in mind this possible realization of such an inhomogeneous metamaterial, further we demonstrate considerable enhancement of the radiation absorption in the film with spatially varying orientation of the optical axes.As before, we use in analysis the effective medium approximation.
In our simulation we use transfer-matrix method.The whole MM layer is formally split into the set of uniform sublayers with the components of inverse permittivity tensor σ ≡ ε−1 locally depending on the tilt angle  as where   and   are the principal-axes components of the permittivity tensor.The magnetic field H ≡ z 0  is locally governed by the equation and in each sublayer it can be represented as a sum of up-and downpropagating (along the plus-and minus- direction) waves.The relation between the amplitudes of these waves in the neighboring sublayers is found from the continuity of tangential components of electric end magnetic fields.The thickness  of these sublayers is the step of numerical discretization, and as such, the resulting reflection and absorption coefficients do not depend on  (i.e., the calculations with /2 and the same total thickness of the film give only small corrections to  and ).
The transfer matrix is obtained as a result of successive application of this procedure.Using the boundary conditions at the first and last sublayers we find the required reflection coefficient.
The results shown below are obtained assuming that the orientational angle increases as  = (/2)/  (0 ≤  ≤   ). Figure 5(a) demonstrates the wavelength dependence of the absorption coefficient () for the nonuniform MM films with the thickness   = 200 and 100 nm.The results for thicker film are close to the case of the MM half-space,  ≈  ∞ .The absorption coefficient in the case of   = 100 nm is even more than  ∞ due to the interference of the incident wave and the wave which is reflected from the substrate and not completely damped.This interference effect results in better absorbance of the film, and the broadband character of the absorption is preserved.The dependence of reflection and absorption on the incident angle is also an important characteristic of proposed nonuniform film, and Figure 5(b) demonstrates this function for specific wavelength  = 500 nm and three values of the film thickness   = 100 nm (curve 1),   = 200 nm (curve 2), and   = 50 nm (curve 3, see comments below).When the component   of the wavevector of the incident light along the air-MM interface tends to its maximum  0 = 2/ the reflection sharply increases, but this is a common feature, e.g., for the reflection from dielectric half-space.
The improved absorption and broadband functionality of the proposed device are clearly seen in Figure 6, which shows the wavelength and incident angle dependence of the absorption coefficient calculated for the MM slab with fixed orientation of the optical axis (curves 1 and 2 correspond to  = 0 and  = 0.1) and inhomogeneous hyperbolic metamaterial with  linearly growing from 0 to /2 (curve 3).
Figure 7 demonstrates magnetic field   (Figure 7(a)) and Poynting vector   (Figure 7(b)) pattern as a function on   and .One can see that |  | strongly depends on the incident angle, and this dependence is not symmetric with respect to   = 0.The asymmetry of the energy flow is apparent from the direction of the  component of Poynting vector   which is plotted in the inset of Figure 7(b), and the energy flow is mainly antialigned with the -axis.It should be noted that, in spite of this asymmetry, the absorption coefficient (  ) is symmetric as a function of   (see Figure 5(b)), and the both positive and negative   harmonics can be effectively absorbed.
Another important feature of the Poynting-vector pattern is its strong spatial localization.It is clear from Figure 7 that the main processes (reflection and absorption) take place in the first one-third of the film, and it seems that similar results could be achieved with half the film thickness.However, if we place the perfectly reflecting substrate at  = 50 nm and keep the MM structure unchanged (such that the orientation angle varies from 0 to /4), then the absorption of the incident wave is noticeably decreased, and the absorption coefficient takes values close to  ∞ (  ) (see curve 3 in Figure 5(b)).

Conclusion
In conclusion, we have proposed the design of an absorbing film made of nonuniform hyperbolic metamaterial.In the film with smoothly changing orientation of the optical axis from parallel to normal to the interface, the resonance absorption takes place for each wavelength with only the position of the resonance shifted.This results in broadband operation of the absorber and also its weaker sensitivity to the incident angle.The analysis based on effective medium approximation shows that a 200-nanometer-thick film operates as a metamaterial absorbing half-space.The realization of the proposed idea in optics seems to be complicated.On the other hand, the development of nanotechnology has already led to the emergence of complicated metamaterial structures, such as flexible hyperbolic metamaterial the deformation of which evidently causes inhomogeneous orientation of the optical axis [32].A method of growing tilted nanowires array [33] should be also noted.

2 AdvancesFigure 1 :
Figure 1: (Color online) (a) the wavelength dependence of real (curve 1) and imaginary (curve 2) parts of the permittivity of silver,   (notice different scales for Re(  ) and Im(  )); (b) the dispersion of dielectric tensor components (real part)  ‖ (curve 1) and  ⊥ (curve 2) of effective wire medium (silver wires with volume fraction   = 0.25 in silica host); (c) the normalized critical angle,  * / as a function of the wavelength.Dashed line marks the transition from common elliptical medium to the hyperbolic one with  ‖  ⊥ < 0.

Figure 2 :Figure 3 :
Figure2: (Color online) (a) the geometry of the problem: the wire-medium slab with the thickness   is placed on the perfectly reflecting substrate.The components   ,   , and   of the effective permittivity tensor depend on the orientation of the principal axes of the metamaterial, i.e., on the angle  between the wires and the normal to the air-MM interface.In the slab of homogeneous metamaterial  is fixed; however, we also consider the case of spatially changing orientation of the medium optical axis (see Figures5-7).(b) Sketch of resonance cones orientation in inhomogeneous hyperbolic medium.The wave vector k of propagating radiation is located inside the resonance cone.

Figure 6 :Figure 7 :
Figure 6: (Color online) the absorption coefficient  of the nonuniform metamaterial film (curve 3) compared with MM layer with the fixed optical axis orientation (curves 1 and 2 correspond to  = 0 and  = 0.1): (a) as a function of the wavelength (at normal incidence) and (b) as a function of the tangential wavevector component of the incident light ( = 500 nm); the thickness of the film is   = 200 nm.Dotted lines show the absorption coefficient  ∞ of the metamaterial half-space  = 0 and  = 0.1.