## Abstract

We study complex eigenmodes of layered metal-dielectric metamaterials. Varying losses from weak to realistic, we analyze band structure of the metamaterial and clarify effect of lossess on its intrinsic electromagnetic properties. The structure operates in a regime with infinite numbers of eigenmodes, whereas we analyze dominant ones.

© 2013 Optical Society of America

## 1. Introduction

Multilayered metamaterials, which can be defined as one-dimensional structures of two periodically alternating optically thin layers with sufficient optical contrast, are known to have unusual electromagnetic properties being promising for many applications especially in the optical frequency range. Optical multilayered metal-dielectric nanostructures (MMDN) comprising the periodic arrays of alternating dielectric and metal layers with thicknesses of several tens of nanometers, are known to have striking electromagnetic properties. A number of applications exploiting unique properties of MMDNs were proposed, for references see Ref. [1].

Recently, multilayered metamaterials have become widely known as a simple realization of the so-called hyperbolic media [2] with resulting ultra-high values of the Purcell factor [3, 4]. It is also known that MMDNs demonstrate a strong nonlocal response [1, 5]. However, there is a question of how these phenomena behave in real structures where realistic losses should be taken into account, hence we need to study complex modes of the multilayered metamaterial and clarify the effect of losses.

In order to describe the structure with two periodically alternating layers having permittivities *ε*_{1}, *ε*_{2} and thicknesses *d*_{1}, *d*_{2} (see Fig. 1, left side), we employ a local effective medium model along with the transfer matrix method [6]. The former describes the structure as an uniaxial anisotropic crystal with the effective parameters,

*ε*

_{||}and

*ε*

_{⊥}are of opposite sign, i.e.

*ε*

_{||}

*ε*

_{⊥}< 0, a material is called indefinite [7] or hyperbolic [8, 9]. Since hyperbolic metamaterials allow high-

*k*modes. they may have very high photonic density of states [10, 11].

However, it should be noticed that the local effective medium model is not accurate, and it is used in the present paper only to show nonlocal behavior of the multilayered metamaterial. The nonlocal effective medium model (see Ref. [12, 13]) is capable of providing analytic results coinciding with ones of transfer matrix method which is the most correct electromagnetic description of multilayers.

Analysis of eigenmodes of the lossless periodic layered nanostructures has been accomplished in Refs. [1, 14, 15]. In the present paper complex eigenmodes of lossless and lossy multilayered metamaterials are under study.

## 2. Complex band structure

For MMDN with two layers forming its period, using effective medium model and transfer matrix method we obtain two dispersion relations *ω*(**k**), where **k** = (*k _{x}*,

*k*, 0) is the wave vector. TM polarization is assumed to take advantage of plasmonic behavior. The dispersion relation obtained by the transfer matrix method, assuming that ${k}_{x}^{\left(i\right)}$ is the wave number in the

_{y}*i*-th layer, is given by:

In the paper graphic representation of the dispersion equations is chosen in the form of dispersion diagrams for waves travelling along *y*-direction.

The analyzed structure consists of metal and dielectric layers and infinite in all directions. Period of the structure is *D* = *d*_{1} + *d*_{2} = 62.5nm, where *d*_{1} and *d*_{2} are the thicknesses of the layers. Dielectric permittivity of the dielectric layers is constant and equals to 4.6, while metal layers are described by dielectric function of the Drude form,
${\epsilon}_{2}\left(\omega \right)=1-{\omega}_{p}^{2}/\left[\omega \left(\omega +i\mathrm{\Gamma}\right)\right]$, with the plasma wavelength *λ _{p}* = 2

*πc/ω*= 4

_{p}*D*and the damping coefficient Γ. Layers thickness are

*d*

_{1}= 25nm and

*d*

_{2}= 37.5nm and vice versa.

Dispersion relation given by Eq. (2) allows to study complex modes of the structure immediately if the complex value of *k _{y}* is considered. To find the roots of Eq. (2) in the complex plane we employ the bisection method.

In Figs. 1, 2 dispersion diagrams of lossless (Γ → 0) and lossy structures are presented. We show positive real part of the wave vector with corresponding imaginary part. Due to the periodicity and unboundedness of the structure its complex eigenvalue spectrum contains an infinite number of modes. However, only modes with low imaginary parts of its wave number are of interest as they can travel significant distances. We restrict imaginary part of *k _{y}* by the value of 8

*π/D*in our calculations.

In the absence of losses the structure supports the set of waves having wavevectors ±*k* and ±*k*^{*}. Thus, there are forward and backward waves along with their complex conjugations, and imaginary parts of the modes are symmetric in Figs. 1(a), 2(a). It follows from the fact that if dielectric permittivities of layers are real then the right part of Eq. (2) is a holomorphic function whose restriction to the real numbers is real-valued.

The three most important modes are marked in Figs. 1, 2 with different colors while high-order complex modes are in gray color. Effective medium model results are presented in the figures as well (thin curves). At the surface plasmon polariton (SPP) resonance real part of the propagating mode goes to infinity, while imaginary part degenerates at this point which can be seen from the example of mode II in the case of *d*_{1} < *d*_{2} and from the mode I in the case of *d*_{1} > *d*_{2}. One can also observe an effect of decay switching in high-order modes: after a mode undergoes the SPP resonance its imaginary part jumps from one branch to another. In the case of *d*_{1} < *d*_{2} presented in Fig. 1(a) there is a very special mode III that is fully imaginary in the presented frequency range, and its real part is dispersionless and equals to zero. This feature is very unique since that mode is not sensitive to the resonance. Its presence is highly related to effective medium model as the latter has the resonance at another frequency and at higher frequencies converges to the entirely imaginary mode III while at low frequencies it converges to the mode II.

However, zero losses are unachievable in practice, hence we introduce attenuation in metal to study effect of losses on electromagnetic properties of MMDN. Increasing losses smoothly we find that it changes step-like behaviour of the modes at the SPP resonance. With change of Γ modes go through a number of transformations [16]. Mechanism of modes transformation is exposed in Fig. 3. When losses tend to zero at the resonance positive imaginary branches jump to their left neighbors with growth of the frequency, while negative branches jump to their right neighbors. Then, increase of the losses level forces branches to modify their topology and after a number of metamorphoses at the resonance both positive and negative imaginary branches jump to their right neighbors with growth of the frequency.

Thus we come to realistic level of losses. Γ = 1.734·10^{13}*s*^{−1} is supposed in our calculations. We analyze complex modes in the case of realistic losses comparing them with the lossless case to show how losses affect the behaviour of the structure. In the presence of losses, as soon as *ε*_{2} becomes a complex number, the right part of Eq. (2) can be complex-valued with real *k _{y}*. Consequently, the conjugated waves with ±

*k*

^{*}are no longer supported by the structure. However, correspondence between positive and negative real and imaginary parts can be quite diverse because of the absence of symmetry in the imaginary parts of eigenmodes (see Figs. 1(b), 2(b)).

In Figs. 2(b), 1(b) we still have decay switching effect that is seen better in high-order modes. It’s expected for modes we have observed in Figs. 2(a), 1(a): losses of the mode are grown up dramatically when it enters into a band gap, while for complex modes such behaviour is non-obvious. The case of *d*_{1} > *d*_{2} is shown in Fig. 1(b). Just that case corresponds to simultaneous presence of forward and backward modes in the same frequency range (*D*/*λ*: 0.105 − 0.108) for propagation along the layers. That is, the beam splitting phenomenon revealed in [1] remains even after introduction of realistic level of losses.

Finally, it’s noteworthy that effective medium model predicts resonance behaviour for imaginary part of *k _{y}*. In reality, however, the structure functions in a regime with infinite number of modes which imaginary parts have step-like behaviour at the resonance.

## 3. Profiles of Eigenmodes Fields

In order to obtain deeper understanding of the nature of the studied complex eigenmodes of the system, we have plotted profiles of the magnetic and electric fields of these modes. Fig. 4(a–c) shows the profiles of the 3 branches for the case of the thicker metal at the frequency *D*/*λ* = 0.0875. We first note that the magnetic field of the mode I averaged over each layer is equal to zero. It means that the mode I is effectively longitudinal. Such modes are well known in plasma physics as *Langmuir* modes [17]. It has been also recently shown that such exotic modes may exist in the waveguides with metal cladding and core made of hyperbolic media [18]. We can see, however, that the longitudinal nature of the mode holds only within the effective media approximation and the local magnetic field inside each layer does not vanish. Modes II and III are conventional transverse modes.

Another interesting feature that can be noted on the profiles is the complicated field distribution of the electric field inside the metal layer for mode I. Conventionally, the modes existing in metal-dielectric structures are coupled surface plasmons at the individual metal-dielectric interfaces. Thus, there may exist no more than one extremum of the field distribution function in each layer. This condition however does not hold for the case of the complex modes. Indeed, if we consider the modes with real and imaginary part of the propagation constant, the expression for the transverse component of the wavevector in each layer reads:

*k*) is large enough, the

_{y}*k*gains large real part and the mode starts to propagate inside the structure. This leads to nonzero values of the transverse component of the Poynting vector inside the layers and to the complex shape of the field distribution inside the layers. To illustrate how the imaginary part of

_{x}*k*affects the shape of the field inside the layers we have plotted the profiles of the high order complex modes [see Fig. 4]. We can see that while we are switching to higher order modes, having the larger imaginary part of

_{y}*k*, the field profile shape becomes more and more complicated. These modes can be regarded as a specific type of coupled waveguide modes. The waveguide mode condition can be roughly estimated as

_{y}*k*=

_{i,x}d_{i}*πn*, where

*i*= 1, 2 and

*n*- is an integer. Thus, higher switching between the complex modes can be regarded as switching between different waveguide modes in the structure.

We have also plotted the profiles of the three mode branches for the thinner metal case [see Fig. 4(d–f)]. We can see that in this case we also have one longitudinal mode I and two transverse modes. In this case, however, the mode II has the complicated structure of the electric field as the one defined by the larger values of Im(*k _{y}*).

## 4. Conclusions

We have analyzed complex modes of MMDN, both in the absence and with the losses. Notable phenomenon of decay switching has been demosntrated. With introduction of realistic losses into the problem, the propagating and evanescent modes becomes mixed, and the complex band structure of MMDN is revealed, however the MMDN nonlocality is slightly affected by losses.

## Acknowledgments

This work was supported by the Ministry of Education and Science of Russia, projects 11.G34.31.0020, 14.37.21.1649, 14.B37.21.1941, the Dynasty Foundation, and the Australian Research Council.

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