A simplified transfer function approach to beam propagation in anisotropic metamaterials

Abstract

The Berreman analysis for plane wave propagation through an anisotropic metamaterial is modified to derive the transfer function matrix for propagation in such a structure. The eigenvalues of the Berreman matrix, which determine the transfer function, depend on the anisotropy. Using the transfer function, propagation of TM and TE polarized beams in hyperbolic metamaterials is analyzed theoretically, and numerically based on the developed theory. For TM polarization, self-focusing of Gaussian beams in a hyperbolic metamaterial can be analytically explained using the q-parameter approach. Furthermore, the transmission coefficient for plane wave propagation through an anisotropic metamaterial slab is extended to a transmission transfer function, which is then used to analyze the propagation of beams through hyperbolic metamaterials and the ensuing spatial shifts. For TM polarized beams, negative refraction is verified. The main objective of this paper is to provide simple analytical tools for understanding self-focusing and negative refraction in anisotropic hyperbolic metamaterials as an alternative to, and check the validity of, rigorous numerical computations. Propagation of the longitudinally (z-) polarized optical field profile is derived. The transfer function approach should be useful for the analysis of arbitrary beam profiles through composite metamaterials.

Introduction

There has been enormous interest in the development of periodic or almost periodic electromagnetic (EM) and optical structures and heterogeneous structures that can exhibit interesting properties, such as prescribed band gaps and tuning [1] sub-wavelength imaging [2], efficient harmonic generation [2], invisibility cloaking [2], [3], etc. The artificial structures that can achieve these properties are referred to as metamaterials. These metamaterials interact with EM radiation to achieve exotic material properties, such as negative permittivity, negative permeability, and negative refractive index, leading to effects such as negative refraction, etc., many of which has been experimentally demonstrated [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15]. While most of the theory is based on plane waves, the behavior of beams is more important from a practical standpoint. As is well-known, the beam propagation method (BPM) is employed to study the properties of various guided wave optics structures, such as optical waveguides with bends and tapers, gratings and spectrum analyzers, etc. [9], [10]. The propagation of beams in metamaterials can be modeled using the fast Fourier transform (FFT-BPM), or finite element and finite difference methods [9], [10], [11], [12], [13], [14], [15].

While analyzing beam propagation in anisotropic media, theanisotropy can be expressed as matrices for the electric permittivity, $ϵ$ and magnetic permeability, $\mu$ [2], [13]. For any FFT based BPM, the transfer function of propagation is required to relate the angular plane wave spectrum at one plane to the spectrum at another plane during propagation [9], [10], [11], [12], [13], [14], [15]. In anisotropic media, the transfer function matrix is required.

Hyperbolic metamaterials (HMMs) are a form of anisotropic material where the dielectric tensor elements have opposite signs [16], [17]. There are two types of HMMs which can be distinguished by the signs of the principal elements of the diagonal permittivity matrix: ${ϵ}_{zz}<0;{ϵ}_{xx},{ϵ}_{yy}>0$ or ${ϵ}_{xx},{ϵ}_{yy}<0;{ϵ}_{zz}>0$ [15]. The dispersion occurs in these metamaterials when the principal components of diagonalized matrices change the sign [4], [15]. It has been argued that negative refraction can be achieved through the hyperbolic dispersion of these materials [16], [17], [18], [19], [20]. HMMs have many potential applications in EM and optics. HMMs are used in several optical applications, such as waveguiding, imaging, sensing, quantum, and thermal engineering [16], [17]. In this work, beam propagation through HMMs is analyzed.

Incidentally, anisotropic metamaterials can be built as a multilayer structure comprising alternating layers of dielectric and metal, and modeled as an anisotropic bulk medium (BM) with an effective dielectric permittivity tensor based on effective medium theory if the thickness of the metal and dielectric are much smaller than the wavelength [21], [22], [23], [24], [25]. Reflection and transmission from slabs of anisotropic metamaterials comprised of metallo-dielectric structures have been analyzed as a function of the angle of incidence, number of layers, and layer thicknesses, and it has been shown that effective medium theory can give results that are sufficiently close to, and computationally much faster than, rigorous computations using the transfer matrix method and finite element methods such as COMSOL [25], [26], [27], [28], [29], [30].

As stated above, FFT-BPM is a powerful tool for the computation of EM wave propagation in anisotropic media [26]. The common analysis technique is to decompose the beam into an infinite number of plane waves traveling in different directions, or plane wave spectra [31], [32], [33], [34], [35], using the Fourier transform. Propagation of the plane wave spectra is achieved by multiplying the initial angular spectrum with the transfer function for propagation, which can be done analytically or computationally [33]. The transfer function matrix for propagation in an anisotropic medium can be derived by starting from the Berreman matrix [36], [37].

Also, in related earlier work, it has been shown by Banerjee et al. [38] that propagation of radially polarized Gaussian beams can be readily analyzed rather simply by knowing the q-transformation of the underlying scalar Gaussian beam. The exact profiles of the longitudinal and transverse components of initially radially polarized lowest-order Laguerre–Gaussian beams have been derived using the transfer function approach which leads to the laws of q-transformation [39].

The objective of this work is to extend existing plane wave theory to beam propagation in anisotropic metamaterials using the notion of transfer functions in order to help understand effects such as self-focusing, negative refraction, etc. of beams in hyperbolic metamaterials from an analytical and physical standpoint. The transfer function matrix for beam propagation can be readily found from the Berreman matrix. This is summarized in Section 2, leading to the derivation of simplified transfer functions, which are subsequently used to analyze propagation of one transverse-dimensional (1TD) TE and TM polarized beams in semi-infinite HMMs. The paraxial version of the transfer function is used to analytically derive the propagation of Gaussian beams using the q-parameter approach. In Section 3, the transmission through a slab of a HMM is analyzed by extending the plane wave approach to an angular spectrum approach, by simply relating the angle of propagation to the corresponding spatial frequency to derive the transmission transfer function. The transfer function approach can be readily extended to determine the propagation of the longitudinal or z-component of the optical field. Section 4 shows illustrative examples. For instance, it is shown that in such semi-infinite HMMs, 1TD TM polarized Gaussian beams can undergo linear self-focusing, while 1TD TE polarized beams spread due to diffraction, and the results from paraxial theory match well with rigorous non-paraxial calculations. Simulations using the transfer functions include examples of realistic HMMs with complex permittivities derived from exact (complex) values of the permittivities of the metal and the dielectric; and the results are compared with those obtained from setting the imaginary part of the effective permittivity to zero. Also, it is shown that a slab of a HMM can cause negative refraction only of a 1TD TM polarized beam, evidenced from the spatial shift of the beam upon exit from the material.

Liu et al. [11] have a full vectorial description of the focusing of EM beams in media with hyperboloid and ellipsoid dispersion surfaces, and show that a paraxial beam undergoes “focusing in both x and y axes for the slab with an ellipsoid dispersion relation”, and undergoes “focusing or dispersion in different axes for the slab exhibiting a dispersion surface of a single-sheeted hyperboloid”. A nonparaxial correction to the paraxial solution is then derived. In our case, we start from the unidirectional nonparaxial solution and make the paraxial approximation which leads to the q-transformation of a Gaussian beam in HMMs. This formulation enables us to analytically predict the location of the self-focusing distance, which is also verified by numerical simulations. Comparison of non-paraxial and paraxial simulations based on the respective transfer functions can clearly show when the paraxial approximation is valid. Kivijärvi et al. [12] have a detailed theory of vectorial beam propagation in anisotropic metamaterials showing the intensity profiles of a focused beam through metamaterials composed of, for instance, a period array of silver disk and dimer metamolecules. Although based on angular spectrum decomposition, the rigorous computations are performed using COMSOL; furthermore, arbitrary radii of curvatures and oblique incidence are not examined. While complete in treatment, the physics of self-focusing and spatial shift due to propagation in HMMs for different polarizations are not clearly apparent. Zhou et al. have examined extraordinary reflection and refraction in natural hyperbolic materials for wave propagation [40]. As discussed above, our work extends explicitly to beam propagation, and the anticipated beam shift is verified using the spatial frequency dependence of the transmission function, which can be regarded as the transmission transfer function for the hyperbolic system. Incidentally, beam shift in HMMs has also been shown by Argyropoulos et al. [2], but using finite element based simulations from the outset. 2-D Gaussian beams have also been studied in the past for uniaxial crystal and uniaxial inhomogeneous media for transmission properties [41], [42]. Reflection and transmission properties have been derived using the Berreman matrix [42], [43] for a homogeneous slab, but these studies have not discussed Gaussian beam propagation in a hyperbolic medium. We feel the uniqueness of our work lies in explaining the details about beam propagation and negative refraction using BMM along with the effective medium theory along with a transfer function approach based on angular spectral decomposition of beams.