Optical propagation through anisotropic metamaterials: Application to metallo-dielectric stacks

Optics Communications 425 (2018) 71–79

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Optical propagation through anisotropic metamaterials: Application to metallo-dielectric stacks Rudra Gnawali a , Partha P. Banerjee a,b, *, Joseph W. Haus b , Victor Reshetnyak c , Dean R. Evans d a b c d

Department of Electrical and Computer Engineering, University of Dayton, Dayton, OH 45469, USA Department of Electro-Optics and Photonics, University of Dayton, Dayton, OH 45469, USA Physics Faculty, Taras Shevchenko National University of Kyiv , Volodymyrska street 64, Kyiv, 01601, Ukraine Air Force Research Laboratory, Materials and Manufacturing Directorate, Wright-Patterson Air Force Base, OH 45433, USA

ARTICLE

INFO

Keywords: Anisotropic metamaterial Hyperbolic metamaterial Berreman matrix method Transfer matrix method Effective medium

ABSTRACT We perform numerical simulations to compare the Berreman matrix method using effective medium results for an anisotropic material with exact calculations of a multi-layer metallo-dielectric stack using the transfer matrix method and finite element techniques. Results are given for a wide band of wavelengths and incident angles. For fixed sample thickness the number of layers is increased to study convergence of the optical characteristics (transmittance and reflectance). It is shown that the Berreman matrix method with effective medium results for an anisotropic material provides a fast and reliable estimate of the optical characteristics of the composite material. The Berreman technique readily leads to the transfer function matrix for propagation in anisotropic materials.

1. Introduction Metamaterials are widely known in the field of optics because of their unique electromagnetic (EM) properties [1–3]. Metamaterials are artificially engineered structures designed to interact with EM radiation to achieve exotic material properties such as negative permittivity, negative permeability, negative refractive index, etc. leading to applications such as perfect imaging, optical filters, and coatings for special applications [2]. Such materials can be constructed, for instance, in the form a multilayered metallo-dielectric (MD) structure comprising alternating layers of metal and dielectric which can be modeled as a bulk anisotropic medium using effective medium theory [1]. These anisotropic metamaterials are believed to display interesting properties, including negative refraction and super-resolution in the near and/or far-field [1]. The propagation of EM waves in a medium is determined by its electric permittivity, 𝜖 and magnetic permeability, 𝜇 [3,4]. In an anisotropic metamaterial 𝜖 and 𝜇 are tensor quantities [5,6]. The anisotropy in these materials can be expressed as a diagonal matrix of 𝜖 and 𝜇 with their principal components having different values [6,7]. Hyperbolic metamaterials (HMMs) are a form of anisotropic material where the dielectric tensor elements have opposite signs [7–9]. There are two types of HMMs which can be distinguished by the signs of the principal elements of the diagonal permittivity matrix. Important dispersion characteristics of the hyperboloid are determined by whether the medium dielectric tensor principal components satisfy 𝜖𝑧𝑧 < 0; 𝜖𝑥𝑥 , 𝜖𝑦𝑦 > 0 or *

𝜖𝑥𝑥 , 𝜖𝑦𝑦 < 0; 𝜖𝑧𝑧 > 0 [10]. Negative refraction can be achieved through the hyperbolic dispersion of these materials [11,12]. Anisotropic metamaterials are fabricated as a stack comprising alternating layers of metal and dielectric films, often referred to as metallodielectrics [2,6]. As shown by Argyropoulos et al. [1] for a hypothetical case, and using effective medium theory, the MD stack can be represented as a homogeneous anisotropic bulk material where the permittivities along the principal diagonal can have opposite signs owing to the negative (real part of the) permittivity of the metal. Conceptually, the reason for the anisotropy along the nominal direction of propagation z can be attributed to the (periodic) changes in the permittivities along this direction. The physical process that allows a layered metamaterial to mimic an anisotropic material is that surface plasmons are supported at an interface where the permittivity changes sign. When the metal permittivity is negative, the sign change occurs at every interface; the wave is transmitted via coupled surface plasmons [13]. MD stacks have potential applications, such as super-resolution with sub-wavelength focusing, negative refraction, harmonic generation, photonic bandgap structures and filters, sensing etc. [1,2,14–17]. Typically, EM propagation through MD stacks can be analyzed using the transfer matrix method (TMM) which is formulated for plane wave incidence, applied to the multilayered structure, or by using numerical methods, such as a finite element method, e.g. COMSOL. However, using effective medium theory and considering the MD stack to be an inhomogeneous bulk material, it is possible to use simpler and faster methods, such as the Berreman matrix method (BMM) to approximately

Corresponding author at: Department of Electro-Optics and Photonics, University of Dayton, Dayton, OH 45469, USA. E-mail address: [email protected] (P.P. Banerjee).

https://doi.org/10.1016/j.optcom.2018.04.069 Received 13 March 2018; Received in revised form 25 April 2018; Accepted 27 April 2018 0030-4018/© 2018 Elsevier B.V. All rights reserved.

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Optics Communications 425 (2018) 71–79

find the EM fields inside the structure and determine the transmission and reflection coefficients. Such an approach may be advantageous during the design of such MD stacks, where, say, the tuning behavior as a function of the wavelength or angle of incidence can be preliminarily assessed quickly using BMM. In this paper, illustrative examples of EM analysis of such structures using BMM are given and compared with results from TMM and COMSOL. Although it has been shown that the effective permittivity of such structures can be determined to within the Wiener bounds [18], it is instructive to determine how accurately effective medium theory accurately determines the transmittance and reflectance from MD stacks structures through direct simulations. It is concluded that BMM with effective medium results for an anisotropic material provides a fast and reliable estimate of the optical characteristics of the composite material. In the process, the concept of the angular plane wave matrix for propagation in anisotropic materials using BMM is introduced, and its application to beam propagation (including 𝑧polarized beams) in such materials is discussed. The organization of the paper is as follows. In Section 2, EM propagation using BMM in a bulk medium modeled as an effective medium is summarized, with emphasis on TM polarization and hyperbolic metamaterials. The concept of the transfer function matrix for propagation in such anisotropic metamaterials is introduced. In Section 3, TMM is summarized, along with a scheme to compare TMM with BMM for MD stacks. In Section 4, numerical results of BMM and TMM along with COMSOL are presented for transmittance and reflectance of MD stacks, especially those with dimensions which give rise to hyperbolic dispersion when modeled as an effective medium. It is shown that BMM provides a simple and fast way to get a general estimate of the spectral properties of such stacks for possible applications such as tunable filters, which can aid in the design of such structures for a given application. At the same time, through numerical simulations, it is shown why and how TMM results converge to that obtained from BMM using effective medium theory, as is expected from theoretical limits. Section 5 concludes the paper.

of layers approaches infinity) and the metallo-dielectric patterning has a spatial scale which is much smaller than radiation wavelength, then one can treat the system as a bulk anisotropic medium with an effective dielectric permittivity tensor [1,13,21] 𝜖 0 0⎤ ] ⎡ 𝑥𝑥 𝜖𝑒𝑓 𝑓 = ⎢ 0 𝜖𝑦𝑦 0 ⎥, ⎢ ⎥ 0 𝜖𝑧𝑧 ⎦ ⎣ 0 ( )( ) 𝜖1 𝜖2 𝑑1 + 𝑑2 𝜖𝑧𝑧 = 𝜖0 ( ) , 𝑑1 𝜖2 + 𝑑2 𝜖1 [

𝑘2𝑥

𝜔0

−𝜇𝑦𝑦 +

𝜇𝑦𝑧 𝜇𝑧𝑦 𝜇zz

+

𝑘𝑥 2

( 𝑘𝑥

𝜖𝑦𝑧 𝜖zz

−

𝜇zy 𝜇zz

𝜖𝑥𝑥

𝑘20

=

𝜖0

(2)

,

where, 𝑘0 and 𝜃𝑖 are the amplitude of the incident magnetic field, the free space wavenumber, and the angle of incidence, respectively. The reflected and transmitted magnetic fields, 𝑯𝒓 and 𝑯𝒕 , respectively, can be represented in a similar way. The corresponding incident, reflected and transmitted electric fields, 𝑬𝒊 , 𝑬𝒓 , and 𝑬𝒕 can be similarly written as

(

𝑬𝒊 = (𝒂̂ 𝒙 cos 𝜃𝑖 − 𝒂̂ 𝒛 sin 𝜃𝑖 )𝐸𝑎+ 𝑒−𝑗𝑘0 (𝑥𝑠𝑖𝑛𝜃𝑖 +𝑧𝑐𝑜𝑠𝜃𝑖 ) .

(4)

𝑬𝒓 =

(𝒂̂ 𝒙 cos 𝜃𝑖 + 𝒂̂ 𝒛 sin 𝜃𝑖 )𝐸𝑎− 𝑒−𝑗𝑘0 (𝑥𝑠𝑖𝑛𝜃𝑟 −𝑧𝑐𝑜𝑠𝜃𝑟 ) ,

(5)

𝑬𝑡 =

(𝒂̂ 𝒙 cos 𝜃𝑡 − 𝒂̂ 𝒛 sin 𝜃𝑡 )𝐸𝑐+ 𝑒−𝑗𝑘0 (𝑥𝑠𝑖𝑛𝜃𝑡 +(𝑧−𝐿)𝑐𝑜𝑠𝜃𝑡 ) .

(6)

𝜖𝑧𝑦 𝜖zz

𝜇𝑦𝑧

−

𝜔0 −𝜖𝑥𝑦 +

)

𝜇zz 𝜖𝑥𝑧 𝜖𝑧𝑦

)

𝜖zz

𝜇 𝑘𝑥 𝑥𝑧 𝜇zz (

) 𝜔0

(3)

𝐻𝑎+ ,

(

𝜖𝑥𝑧 𝜖zz ( ) 𝜇𝑥z 𝜇zy 𝜔0 𝜇𝑥𝑦 − 𝜇33

𝑘2𝑧

𝑯𝒊 = 𝒂̂ 𝒚 𝐻𝑎+ 𝑒−𝑗𝑘0 (𝑥𝑠𝑖𝑛𝜃𝑖 +𝑧𝑐𝑜𝑠𝜃𝑖 ) ,

𝑘𝑥

𝑘𝑥

(1)

where 𝑘𝑥 and 𝑘𝑧 are the transverse and longitudinal components of the wave vector. If the condition 𝜖𝑧𝑧 < 0 and 𝜖𝑥𝑥 > 0, the dispersion relation is hyperbolic [1]. It is remarked that assuming real values for 𝜖1 and 𝜖2 , 𝜖 𝑑 hyperbolic dispersion can be obtained when the condition 𝜖1 ≥ − 𝑑1 is 2 2 satisfied [1,22,23]. We consider a plane wave obliquely incident from an isotropic ambient medium (assumed to be free space) onto the anisotropic medium, finally exiting into free space once again. It is remarked that this technique can be readily extended to the case of arbitrary incident and transmitted media. The plane of incidence is the x–z plane, and we assume there is no variation in y direction and wave is propagating in the x–z direction. The bounds of the effective medium are 𝑧 = [0, 𝐿]. The x- variation of all fields in all regions (a,b,c) is in the form exp(−𝑗𝑘𝑥 𝑥) where 𝑘𝑥 = 𝑘0 𝑠𝑖𝑛𝜃𝑖 , where 𝑘0 is the propagation constant in free space, and 𝜃𝑖 is the angle of incidence from free space onto the medium, as shown in Fig. 1(b). This is because the momentum of the waves along the x-direction is unchanged since there is no interface normal to the xdirection. The incident magnetic field for TM polarization can be written as [22,23]

)

𝜔20 𝜖zz

+

𝜖𝑧𝑧

As stated earlier, a metamaterial with hyperbolic dispersion can be built, for instance, as a multilayer structure consists of alternating layers of dielectric and metal (see Fig. 1(a)), and modeled as an anisotropic bulk medium BM (see Fig. 1(b)), based on the effective medium theory. Effective medium theory is a consequence of the homogenization technique [19,20]. This technique is based on the averaging of the EM field in the unit cell of metamaterial and can be applied to multilayered periodic systems. In addition, mean-field homogenization theories can also explain the effective parameters from the distribution of fundamental metamaterial enclosures, such as in Lorentz, Clausius–Mossotti, and Maxwell–Garnett approximations [18,19]. Rapidly-varying spatial scales and spatial periodicity are the two basic ingredients for homogenization approach of a metamaterial [20]. If the multilayered MD stack system, such as that shown in Fig. 1, is indeed periodic (i.e., the number

(

𝜖1 𝑑1 + 𝜖2 𝑑2 , 𝑑1 + 𝑑2

where 𝑑1 and 𝑑2 are the thickness of the dielectric and metal layers, respectively, 𝜖0 is the permittivity of free space and 𝜖1 = 𝑛21 , 𝜖2 = 𝑛22 are relative permittivities of the dielectric and the metal, respectively. For future reference, it is useful to note that as long as the ratio 𝑑1 ∕𝑑2 is maintained a constant, the values of 𝜖𝑥𝑥 and 𝜖𝑧𝑧 remain unchanged. The dispersion relation for this anisotropic metamaterial is

2. EM analysis using BMM for effective medium

⎡ 𝜖 ⎢ 𝑘𝑥 𝑧𝑥 𝜖zz ⎢ ) ⎢ ( ⎢𝜔 −𝜖 + 𝜖𝑥𝑧 𝜖z𝑥 𝑥𝑥 ⎢ 0 𝜖zz ⎢ 𝑀𝐵 = 𝑗 ⎢ 0 ⎢ ⎢ ⎢ ) ⎢ ( ⎢ 𝜔 −𝜖 + 𝜖𝑦𝑧 𝜖z𝑥 𝑦𝑥 ⎢ 0 𝜖zz ⎣

𝜖𝑥𝑥 = 𝜖𝑦𝑦 = 𝜖0

−𝜖𝑦𝑦 +

Box I. 72

𝜖yz 𝜖𝑧𝑦 𝜖zz

+

𝑘𝑥 2 𝜔20 𝜇zz

)

( ) 𝜇𝑦𝑧 𝜇z𝑥 ⎤ ⎥ 𝜔0 𝜇𝑦𝑥 − 𝜇zz ⎥ ⎥ ⎥ 0 ⎥ ( )⎥ 𝜇𝑥𝑧 𝜇z𝑥 ⎥ 𝜔0 −𝜇𝑥𝑥 + ⎥ 𝜇𝑧𝑧 ⎥ ⎥ ⎥ 𝜇z𝑥 ⎥ −𝑘𝑥 ⎥ 𝜇zz ⎦

(7)

R. Gnawali et al.

Optics Communications 425 (2018) 71–79

Fig. 1. (a) Geometry of multilayered dielectric (red) - metal (blue) metamaterial structure; (b) same but modeled as a bulk medium (BM) using effective medium theory. Figure (b) also shows oblique plane wave incidence onto the structure. The angle of incidence is 𝜃𝑖 , the angle of reflection is 𝜃𝑟 = 𝜃𝑖 , and the angle of transmission is 𝜃𝑡 = 𝜃𝑖 . The incident and transmitted media are assumed to be free space. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Alternatively, from Eqs. (8) and (9), one can easily derive a set of two coupled equations involving 𝐸𝑥 and 𝐻𝑦 . Upon solving this set of coupled equations, the EM fields inside the bulk medium can be expressed as [21,22]

In general, the Berreman 4 × 4 matrix (see Eq. (7) in Box I) can be used to calculate the EM fields inside the bulk medium (BM) [24]. Assuming magnetic isotropy and 𝜇 = 𝜇0 for simplicity, and also assuming that the crystal axes are oriented so that the permittivity tensors are diagonal with 𝜖𝑥𝑥 = 𝜖𝑦𝑦 ≠ 𝜖𝑧𝑧 (as is in our case to be discussed below), the Berreman matrix for the anisotropic metamaterial structure can be written as [22,23] ⎡ 0 ⎢ ⎢ ⎢−𝜔0 𝜖𝑥𝑥 𝑀𝐵 = 𝑗 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎣

−𝜔0 𝜇0 +

𝑘𝑥 2 𝜔0 𝜖𝑧𝑧

0

0

0

0

0

0

−𝜔0 𝜖𝑥𝑥 +

⎤ ⎥ ⎥ 0 ⎥ ⎥. −𝜔0 𝜇0 ⎥ ⎥ 0 ⎥⎦ 0

𝑘𝑥 2 𝜔0 𝜇0

(9)

In the general case, the solution of Eq. (9), symbolically expressed as ⎡ 𝐸𝑥 (𝑧) ⎤ ⎡ 𝐸𝑥 (0) ⎤ ⎢ ⎥ ⎢ ⎥ 𝐻 (𝑧) ⎢ 𝑦 ⎥ ⎢ 𝐻𝑦 (0) ⎥ ⎢ ⎥ = [exp(𝑀𝐵 𝑧)] ⎢ ⎥, ⎢ 𝐸𝑦 (𝑧) ⎥ ⎢ 𝐸𝑦 (0) ⎥ ⎢ ⎥ ⎢ ⎥ ⎣−𝐻𝑥 (𝑧)⎦ ⎣−𝐻𝑥 (0)⎦

(10)

−𝑗

− 𝑗𝛽TM 𝑧 + −𝑗𝛽TM 𝑧 −𝑗𝑘𝑥 𝑥 𝐻𝑦 = (𝐻𝑏𝑦 𝑒 + 𝐻𝑏𝑦 𝑒 )𝑒 ,

(14)

𝐸𝑏𝑧 = −

𝛽TM sin 𝛽𝑇 𝑀 𝑧⎞ ⎟ 𝜖𝑥𝑥 𝜔0 ⎟, ⎟ cos 𝛽𝑇 𝑀 𝑧 ⎠

𝑘𝑥 𝐻 ∕𝜖 , 𝜔0 𝑏𝑦 𝑧𝑧

𝐻𝑏𝑧 = 0.

(15)

Propagation of 𝑧-polarized beams in isotropic media has been analyzed using transfer function approach by Banerjee et al. [28]; this can be readily extended to propagation in anisotropic media using the transfer function matrix and the relationship between the 𝑧-polarized components of the 𝐸 and 𝐻 fields and the transverse components — this will also be pursued in the future. Using EM boundary conditions, e.g., continuity of the tangential components of the electric and magnetic fields, and the normal components of the electric displacement and magnetic flux density (assuming no sources) at the boundaries, all EM field amplitudes inside the material can be calculated. After extensive but straightforward algebra, the transmission coefficient (𝑡𝑇 𝑀 ) and the reflection coefficient (𝑟𝑇 𝑀 ) for TM incidence can be written as [22,23]

determines the 𝑧 dependence of the EM fields inside the anisotropic material. The matrix operation exp(𝑀𝐵 𝑧) can be achieved, for instance, by first determining the eigenvalues of the Berreman matrix and then the eigenvectors, or by other techniques for matrix exponentiation [25,26]. The matrix exp(𝑀𝐵 𝑧) can be regarded as the spatial transfer function matrix for propagation 𝑯(𝑘𝑥 ; 𝑧) in the anisotropic material relating the angular plane wave spectra of the initial and final EM field components during propagation. For instance, for the simple case of TM polarization, 𝑯(𝑘𝑥 ; 𝑧) becomes a 2 × 2 matrix given by ⎛ cos 𝛽𝑇 𝑀 𝑧 ( ) ⎜ 𝑯 𝑘𝑥 ; 𝑧 = ⎜ 𝜖 𝜔 ⎜−𝑗 𝑥𝑥 0 sin 𝛽𝑇 𝑀 𝑧 ⎝ 𝛽TM

(13)

+ − are the forward and backward propagating xwhere 𝐸𝑏𝑥 and 𝐸𝑏𝑥 + − are the component of the electric fields respectively, and 𝐻𝑏𝑦 and 𝐻𝑏𝑦 forward and backward propagating magnetic fields, respectively, inside the bulk anisotropic metamaterial. The TE case can be similarly solved using the set of coupled equations relating 𝐸𝑦 and −𝐻𝑥 . In an isotropic material and for unidirectional propagation, it can be shown, using (11) and (13), that 𝑒−𝑗𝛽TM 𝑧 ∝ 𝑒𝑥𝑝𝑗𝑘2𝑥 𝑧, which is the scalar transfer function for propagation and effectively models diffraction of a beam [27]. For 𝜖𝑧𝑧 < 0 (with 𝜖𝑥𝑥 > 0), as in a hyperbolic metamaterial, the expression for 𝛽𝑇 𝑀 suggests possible focusing instead of spreading due to diffraction since the second term under the square root sign is positive; this is outside the scope of this paper and will be pursued in the future. For TE polarization, the corresponding expression for 𝛽𝑇 𝐸 does not have any dependence on 𝜖𝑧𝑧 (see below). The longitudinal (𝑧) components of E and H in the Berreman approach are expressed as functions of the transverse components using Maxwell’s equations. In our TM case, these simplify to

(8)

The differential equation for the EM fields can be expressed as ⎡ 𝐸𝑥 ⎤ ⎡ 𝐸𝑥 ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ 𝐻𝑦 ⎥ 𝜕 ⎢ 𝐻𝑦 ⎥ = 𝑀 ⎥. 𝐵⎢ 𝜕𝑧 ⎢⎢ 𝐸𝑦 ⎥⎥ ⎢ 𝐸𝑦 ⎥ ⎢ ⎥ ⎢ ⎥ ⎣−𝐻𝑥 ⎦ ⎣−𝐻𝑥 ⎦

− 𝑗𝛽TM 𝑧 + −𝑗𝛽TM 𝑧 −𝑗𝑘𝑥 𝑥 𝐸𝑥 = (𝐸𝑏𝑥 𝑒 + 𝐸𝑏𝑥 𝑒 )𝑒

(11)

where 𝛽TM represents the 𝑧 component of the wave vector in the metamaterial for TM polarization and can be written as [21,22] √ 𝑘2 𝜖𝑥𝑥 𝛽𝑇 𝑀 = 𝜖𝑥𝑥 𝜔20 𝜇0 − 𝑥 . (12) 𝜖𝑧𝑧

𝑡𝑇 𝑀 =

𝐸𝑐+ 𝐸𝑎+

4𝜂0 𝛽𝑇 𝑀 𝜔0 𝜖𝑥𝑥 cos 𝜃𝑖 𝑒−𝑗𝛽𝑇 𝑀 𝐿 = ( , (16) )2 ( )2 𝛽𝑇 𝑀 + 𝜔0 𝜖𝑥𝑥 𝜂0 cos 𝜃𝑖 − 𝛽𝑇 𝑀 − 𝜔0 𝜖𝑥𝑥 𝜂0 cos 𝜃𝑖 𝑒−2𝑗𝛽𝑇 𝑀 𝐿 73

R. Gnawali et al.

Optics Communications 425 (2018) 71–79

Fig. 2. Multilayer metamaterial structure, with black dotted rectangle representing the portion of the metamaterial of length 𝐿′ modeled using (a) BMM and (b) TMM. 𝑀𝐵′ is the modified BMM matrix, being compared with the TMM matrix 𝑀𝑇 𝑚 .

3. Comparison of BMM for effective medium with TMM TMM is a popular technique to analyze the propagation of EM waves through a multilayer MD stack [2,29]. In TMM, EM propagation through such a multilayer structure can therefore be written as a product of matrices and optical propagation through the stack can be readily treated using this overall matrix approach. The reflectance and transmittance can also be calculated from the matrix elements [29–32]. In general, the forward and backward traveling components of electric fields, 𝑬𝑙+ and 𝑬𝑙− in layer 𝑙 are related to similar electric field components in another layer 𝑚 > 𝑙 by means of a transfer matrix 𝑀𝑇 as [27–30] ( +) ( +) [ ] 𝐸𝑙 𝐸𝑚 𝑀11 𝑀12 = 𝑀 , 𝑀 = . (21) 𝑇 𝑇 𝐸𝑚− 𝑀21 𝑀22 𝐸𝑙− Now from the overall transfer matrix 𝑀𝑇 𝐿 relating the input (region a) to the output (region c) for a MD stack of total thickness 𝐿, the reflection (𝑟) and transmission (𝑡) coefficients can be determined as Fig. 3. Plots of n and k vs. wavelength for metal (silver, Ag) and dielectric (titanium dioxide, TiO2 ).

𝑟=

𝑀𝑇 𝐿21 , 𝑀𝑇 𝐿11

(22)

1 . 𝑀𝑇 𝐿11

(23)

and 𝑡= 𝑟𝑇 𝑀 =

𝐸𝑎−

The transfer matrix 𝑀𝑇 is composed of dynamical matrices 𝐷𝑙,𝑙+1 which take into account propagation across the interface between 𝑙 and 𝑙 + 1 and given by, for instance,

𝐸𝑎+ ( )( )( ) 𝛽𝑇 𝑀 − 𝜔0 𝜖𝑥𝑥 𝜂0 cos 𝜃𝑖 𝛽𝑇 𝑀 + 𝜔0 𝜖𝑥𝑥 𝜂0 cos 𝜃𝑖 1 − 𝑒−2𝑗𝛽𝑇 𝑀 𝐿 = ( , )2 ( )2 𝛽𝑇 𝑀 + 𝜔0 𝜖𝑥𝑥 𝜂0 cos 𝜃𝑖 − 𝛽𝑇 𝑀 − 𝜔0 𝜖𝑥𝑥 𝜂0 cos 𝜃𝑖 𝑒−2𝑗𝛽𝑇 𝑀 𝐿 (17)

where 𝜂0 =

√𝜇

0

𝜖0

𝐷𝑙,𝑙+1

is the free space impedance.

For completeness, the transmission coefficient (𝑡𝑇 𝐸 ) and the reflection coefficient (𝑟𝑇 𝐸 ) for TE incidence can be written as 4𝜂0 𝛽𝑇 𝐸 𝜔0 𝜇 cos 𝜃𝑖 𝑒−𝑗𝛽𝑇 𝐸 𝐿 𝑡𝑇 𝐸 = + = ( , )2 ( )2 𝐸𝑎 𝜂0 𝛽𝑇 𝑀 + 𝜔0 𝜇 cos 𝜃𝑖 − 𝜔0 𝜇 cos 𝜃𝑖 − 𝜂0 𝛽𝑇 𝑀 𝑒−2𝑗𝛽𝑇 𝐸 𝐿 (18) ( )( )( ) −2𝑗𝛽 𝐿 − 𝑇𝐸 𝜔0 𝜇 cos 𝜃𝑖 − 𝜂0 𝛽𝑇 𝐸 𝜔0 𝜇 cos 𝜃𝑖 + 𝜂0 𝛽𝑇 𝐸 1 − 𝑒 𝐸 𝑟𝑇 𝐸 = 𝑎+ = ( )2 ( )2 −2𝑗𝛽 𝐿 , 𝐸𝑎 𝑇𝐸 𝜔0 𝜇 cos 𝜃𝑖 + 𝜂0 𝛽𝑇 𝐸 − 𝜔0 𝜇 cos 𝜃𝑖 − 𝜂0 𝛽𝑇 𝐸 𝑒 (19)

𝑛2𝑙 𝑘(𝑙+1)𝑧 𝑛2𝑙+1 𝑘𝑙𝑧 𝑛2𝑙 𝑘(𝑙+1)𝑧

𝑛2𝑙+1 𝑘𝑙𝑧 ⎞ ⎟ 𝑛2𝑙 𝑘(𝑙+1)𝑧 ⎟ ⎟ 𝑛2𝑙+1 𝑘𝑙𝑧 ⎟ 1+ ⎟ 𝑛2𝑙 𝑘(𝑙+1)𝑧 ⎠ 1−

(24)

In Eq. (25), 𝑑𝑙 is either 𝑑1 or 𝑑2 , depending on whether propagation is [( ]1 )2 2 through the dielectric or the metal, and 𝑘𝑙𝑧 = 𝑛𝑙 𝜔𝑐 − 𝑘2𝑙𝑥 , where 𝑐 is the velocity of the EM wave in vacuum, and 𝑘𝑙𝑥 = 𝑘𝑥 = 𝑘0 sin 𝜃𝑖 . For instance,

where 𝛽TE represents the 𝑧 component of the wave vector in the metamaterial for TE polarization and can be written as 𝛽𝑇 𝐸

𝑛2𝑙+1 𝑘𝑙𝑧

for TM polarization, and propagation matrices 𝑃𝑙 through layer 𝑙 given by ( 𝑗𝑘 𝑑 ) 𝑒 𝑙𝑧 𝑙 0 𝑃𝑙 = (25) −𝑗𝑘𝑙𝑧 𝑑𝑙 . 0 𝑒

𝐸𝑐+

√ = 𝜖𝑥𝑥 𝜔20 𝜇 − 𝑘2𝑥 .

⎛ ⎜1 + 1⎜ = ⎜ 2⎜ ⎜1 − ⎝

𝑀𝑇 𝛬 = 𝐷21 𝑃1 𝐷12 𝑃2

(20)

(26)

is the transfer matrix relating the forward and backward traveling components of the 𝑬 fields over one period 𝛬 of the MD stack enclosed 74

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Fig. 4. Real and imaginary parts of (a) 𝜖𝑥𝑥 and (b) 𝜖𝑧𝑧 as a function of wavelength.

Fig. 5. Numerical results for TMM showing (a) transmittance and, (b) reflectance as a function of wavelength for a metamaterial with total thickness 375 nm composed of TiO2 and Ag stacks of different number of layers and thicknesses, keeping the total thickness constant and with 𝑑1 ∕𝑑2 = 2: (i) 𝑚 = 10; 𝑑1 = 25 nm, 𝑑2 = 12.50 nm, (ii) 𝑚 = 20; 𝑑1 = 12.50 nm, 𝑑2 = 6.25 nm, (iii) 𝑚 = 40; 𝑑1 = 6.25 nm, 𝑑2 = 3.125 nm, and (iv) 𝑚 = 80; 𝑑1 = 3.125 nm, 𝑑2 = 1.5625 nm. Results from BMM are also superposed (in blue) for comparison. TM incidence at 45 degrees is assumed. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 6. Numerical results for TMM showing (a) transmittance and, (b) reflectance as a function of wavelength for a metamaterial with total thickness 375 nm composed of TiO2 and Ag stacks of different number of layers and thicknesses, keeping the total thickness constant and with 𝑑1 ∕𝑑2 = 3: (i) 𝑚 = 10; 𝑑1 = 28.12 nm, 𝑑2 = 9.37 nm, (ii) 𝑚 = 20; 𝑑1 = 14.06 nm, 𝑑2 = 4.68 nm, (iii) 𝑚 = 40; 𝑑1 = 7.03 nm, 𝑑2 = 2.34 nm, and (iv) 𝑚 = 80; 𝑑1 = 3.51 nm, 𝑑2 = 1.17 nm. Results from BMM are also superposed (in blue) for comparison. TM incidence at 45 degrees is assumed. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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Fig. 7. Numerical results for TMM showing (a) transmittance and, (b) reflectance as a function of wavelength for a metamaterial with total thickness 375 nm composed of TiO2 and Ag stacks of different number of layers and thicknesses, keeping the total thickness constant and with 𝑑1 ∕𝑑2 = 4: (i) 𝑚 = 10; 𝑑1 = 30 nm, 𝑑2 = 7.50 nm, (ii) 𝑚 = 20; 𝑑1 = 15 nm, 𝑑2 = 3.75 nm, (iii) 𝑚 = 40; 𝑑1 = 7.50 nm, 𝑑2 = 1.875 nm, and (iv) 𝑚 = 80; 𝑑1 = 3.75 nm, 𝑑2 = 0.93 nm. Results from BMM are also superposed (in blue) for comparison. TM incidence at 45 degrees is assumed. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 8. Comparison of COMSOL with BMM and TMM. (a) Transmittance and (b) reflectance as a function of wavelength for a metamaterial with total thickness 375 nm composed of TiO2 and Ag stacks with 𝑚 = 80; 𝑑1 = 3.125 nm, 𝑑2 = 1.5625 nm. TM incidence at 45 degrees is assumed.

in the dashed rectangle in Fig. 1(a), which includes the metal–dielectric interface next to the left vertical dashed line but excludes the metal– dielectric interface next to the right vertical dashed line. For 𝑚 periods, the overall transfer matrix is [ ]𝑚 𝑀𝑇 𝑚 = 𝑀𝑇 𝛬 𝑚 = 𝐷21 𝑃1 𝐷12 𝑃2 . (27)

the left and right by the same anisotropic bulk medium. For the TM polarization case considered here, this can be derived by relating the forward and backward electric fields in the region to the left of the dotted rectangular region to the forward and backward electric fields in the region to the right of the dotted rectangular region. Upon invoking the EM boundary conditions for this case and using the procedure similar to that used to derive the reflection and transmission coefficients in Section 2, this modified 2 × 2 matrix is given by ( ) ′ 𝑒𝑗𝛽TM 𝐿 0 ′ . (28) 𝑀𝐵 = ′ 0 𝑒−𝑗𝛽TM 𝐿

A procedure for comparison of TMM with BMM (for TM polarization) is now described. Assume, now, that these 𝑚 periods are represented as a bulk medium (BM) and modeled by the Berreman matrix. Conceptually speaking, for a fixed finite thickness of the slab, 𝐿′ , one can take the limit as 𝑚 → ∞ with 𝛬 = 𝑑1 + 𝑑2 → 0 such that 𝑚𝛬 = 𝐿′ (see Fig. 2) and examine the coefficients of the TMM and BMM matrices for the desired polarization(s). However the BMM matrix 𝑀𝐵 (see Eq. (8)), and consequently exp(𝑀𝐵 𝑧) which appears in the solution of this equation, relates the transverse components of E and H to their initial values, while the TMM matrix 𝑀𝑇 relates the forward and backward traveling components of the electric field. For a fair comparison, the Berreman matrix should be recast into a form that relates the forward and backward components of the electric fields on the left and right of the dotted rectangular region of length 𝐿′ which is modeled as an anisotropic bulk medium in BMM and surrounded on

For TMM, the matrix considered is that for the 𝑚 periods of the metallodielectric structure of thickness 𝐿′ as in Eq. (28). The simulation results are discussed in the next Section. 4. Simulation results The multilayer metamaterial structure considered in this work is assumed to be made from the combination of metal (silver, Ag) and dielectric (titanium dioxide, TiO2 ). The complex refractive indices (𝑛̃ = 𝑛 − 𝑗𝑘) for the dielectric and metal are taken from published 76

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Fig. 9. (a) Transmittance and (b) reflectance as a function of wavelength for different incident angles for the BM with total thickness 375 nm modeled as an effective medium, computed using BMM.

Fig. 10. Numerical results showing the convergence of the TMM matrix elements to the BMM matrix elements for a metamaterial with total thickness 375 nm. The wavelength used is 330 nm and the angle of incidence (see text for details) is 45 degrees. Magnitude of the matrix coefficients of 𝑀𝐵′ and 𝑀𝑇 𝑚 , plotted vs. number of periods.

Fig. 11. Percentage difference between BMM and TMM for the transmittance derived from Fig. 5, plotted as a function of wavelength for a metamaterial with total thickness 375 nm composed of TiO2 and Ag stacks of different number of layers and thicknesses, keeping the total thickness constant and with 𝑑1 ∕𝑑2 = 2: (i) 𝑚 = 10; 𝑑1 = 25 nm, 𝑑2 = 12.50 nm, (ii) 𝑚 = 20; 𝑑1 = 12.50 nm, 𝑑2 = 6.25 nm, (iii) 𝑚 = 40; 𝑑1 = 6.25 nm, 𝑑2 = 3.125 nm, and (iv) 𝑚 = 80; 𝑑1 = 3.125 nm, 𝑑2 = 1.5625 nm. TM incidence at 45 degrees is assumed.

literature [33,34] and are plotted in Fig. 3. In our simulations, the ratio of the thicknesses of the dielectric and the metal (𝑑1 ∕𝑑2 = 2, 𝑑1 ∕𝑑2 = 3, and 𝑑1 ∕𝑑2 = 4 ) are assumed. The variations of the real and imaginary parts of 𝜖𝑥𝑥 and 𝜖𝑧𝑧 with wavelength, needed for BMM computations, are shown in Fig. 4(a,b), respectively. Note that the imaginary part of 𝜖𝑥𝑥 is approximately zero for wavelengths larger than 350 nm. The real and imaginary parts of 𝜖𝑧𝑧 are typical of the Lorentz model, with the real and imaginary parts related through the Kramers–Kronig relations. The effective medium exhibits hyperbolic dispersion over the wavelength range approximately between 325 nm and 395 nm. Simulations are now performed based on BMM along with effective medium theory. The TM wave is assumed to be obliquely incident at ◦ 45 at the interface of the anisotropic metamaterial slab as shown in Fig. 1(b). The transmittance (𝑇 = |𝑡|2 ) and reflectance (𝑅 = |𝑟|2 ) are plotted in Figs. 5–7 as a function of wavelength. The total thickness of the bulk anisotropic metamaterial slab is taken as 𝐿 = 375 nm. Simulations are performed for different number of periods, while keeping the total slab thickness 𝐿 constant. Results for ratios of the thicknesses of the dielectric and the metal 𝑑1 ∕𝑑2 = 2, 𝑑1 ∕𝑑2 = 3, and 𝑑1 ∕𝑑2 = 4 are presented. Also, for each of the cases (different values of 𝑚) discussed in Figs. 5–7, 𝑑1 + 𝑑2 is maintained a constant. As remarked in Section 3,

the transmittance and reflectance do not depend on the number of layers since the premise of effective medium theory already assumes a bulk medium with an overall effective permittivity tensor. Also, there is only minor quantitative differences between the transmittance (and reflectance) when the ratio of 𝑑1 ∕𝑑2 is varied. The same computations as above are now repeated, but using TMM. As is clear from Figs. 5–7, the TMM results converge to those from BMM as the number of layers are increased and their thicknesses are decreased, while keeping the total thickness constant (at 375 nm). The results show the potential use of such metallo-dielectric structures as an optical filter with a maximum transmittance around 360–450 nm. It is clear that while for 𝑚 = 10 with 𝑑1 = 25 nm, 𝑑2 = 12.50 nm, which are layer thicknesses commonly found using standard deposition techniques. However, while the BMM results differ from the exact values for transmittance and reflectance, the overall structure of the optical characteristics with wavelength is similar. To achieve better convergence much smaller layer thickness are required, although these would exceed practical deposition limits. Admitting small deviations between 77

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Fig. 12. Numerical results for TMM showing (a) transmittance and, (b) reflectance as a function of wavelength for a metamaterial with total thickness 375 nm composed of TiO2 and Ag stacks of different number of layers and thicknesses, keeping the total thickness constant and with 𝑑1 ∕𝑑2 = 2: (i) 𝑚 = 10; 𝑑1 = 25 nm, 𝑑2 = 12.50 nm, (ii) 𝑚 = 20; 𝑑1 = 12.50 nm, 𝑑2 = 6.25 nm, (iii) 𝑚 = 40; 𝑑1 = 6.25 nm, 𝑑2 = 3.125 nm, and (iv) 𝑚 = 80; 𝑑1 = 3.125 nm, 𝑑2 = 1.5625 nm. Results from BMM are also superposed (in blue) for comparison. TE incidence at 45 degrees is assumed. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

the effective medium theory and the exact results, it is computationally simpler to use BMM for predicting the behavior of MD stacks as a first approximation, as long as there are a sufficient number of layers. For additional comparison, simulations have also been performed using COMSOL Multiphysics 5.2a. The results from COMSOL are in good agreement with those from TMM and BMM, as shown in Fig. 8 for a representative case. Typical computation times are on the scale of a few seconds for BMM, a few minutes for TMM, and over an hour for COMSOL, using, for instance, a PC with Intel(R) core(TM) i5-2430M CPU @ 2.40 GHz with 4 GB memory, 64 bit operating system and Windows 7. In fact, it is clear from our simulations (viz., Figs. 5–8) that effective medium theory (and therefore BMM) works well if 𝛬 = 𝑑1 + 𝑑2 ∼ 𝜆∕40 where 𝜆 is the wavelength in vacuum. Fig. 9 shows the transmittance and reflectance as a function of wavelength for different incidence angles, computed using BMM and the effective medium theory. For each angle of incidence, it has been verified that TMM simulations converge to BMM results in the limit as the number of layers are increased and their thicknesses are decreased, while keeping the total thickness constant. It is evident that the transmittance is nonzero from around 330 nm, and is fairly independent of the angle of incidence over the passband, viz., from around 395–480 nm in this case. It is remarked that over this range of wavelengths, the imaginary part of 𝜖𝑥𝑥 is approximately zero (see Fig. 4). To further investigate the convergence of the TMM results to those of BMM, the elements of the TMM matrix 𝑀𝑇 𝑚 and the Berreman matrix 𝑀𝐵′ for TM polarization are now compared. As shown in Fig. 10, the Berreman matrix elements do not depend on the number of periods, as expected. As noted from Eq. (28), the off-diagonal elements are zero, and the real parts of the diagonal elements are equal, while the imaginary parts are equal and opposite. For TMM, the real and imaginary parts of the matrix elements tend to the Berreman values with increase in the number of periods (and correspondingly, decrease in the period thickness) while maintaining the total thickness 𝐿′ constant. For these calculations, 𝐿′ has been taken to be 375 nm for illustration purposes, although any value could have been used. The wavelength used is 330 nm. The angle of incidence upon the dashed region in Fig. 2(a) and (b) is taken as 45 degrees. As demonstrated in Fig. 10, when the number of layers is increased, the magnitude of the matrix elements in TMM uniformly converge to those of the BMM. In fact, although results of transmittance and reflectance for up to 80 periods are shown in this paper, it has been numerically verified that having even more periods does not improve the results in any significant way: the TMM, the BMM and COMSOL results are almost identical. The percentage difference

between the two convergence curves, BMM–TMM, is also presented in Fig. 11. It has been verified that, an increased number of layers shown, results in a convergence error that is less. For completeness, the variation of transmittance and reflectance as a function of wavelength for TE polarization is shown in Fig. 12. Comparison with Fig. 5 (done for TM) shows that there is only minor quantitative differences between the transmittance (and reflectance) between these two cases. It is remarked that simulations have also been performed with the data for dielectric (TiO2 ) taken from Devore et al. [33] and metal (Ag) taken from Palik [34]. Although the transmittance and reflectivity variations with wavelength are somewhat different than those presented above (as is to be expected, since the values of n and k data for TiO2 are different in Ref. [35], the results again show that the TMM and COMSOL results converge to the BMM for higher number of periods. These results are not explicitly presented in this paper for the sake of brevity, interested readers are referred to similar simulations in Gnawali et al. [22]. It is also noted that in-plane, the plasma frequency of the thin film decreases with the decreasing thickness [36]. In addition, the optical permittivity tensor is highly anisotropic, and is dependent in relation to thickness [36]. However, this was not considered here since in our paper, in order to show convergence of TMM with BMM, the same material parameters should be used. For practical applications where computation of transmittance and reflectance are required, more exact values for the permittivities consistent with layer thickness need to be used. 5. Conclusions We have shown that BMM with effective medium results for an anisotropic material provides a faithful and accurate description of the optical characteristics of the composite material. Both TM and TE polarizations have been considered. Exact calculations of multi-layer MD stacks using transfer matrix methods and finite element techniques were performed to determine convergence of optical characteristics to the effective medium results. The convergence applies across a wide band of wavelengths and incident angles. Rapid convergence is found for the transmitted light, but the reflected light is more sensitive to the layer thickness and converges more slowly. The transmittance spectrum shows only small shifts as a function of incident angle, a property of MD stacks which we attribute to the fact that they are only a few hundred nanometers thick and is an unusual property of MD stacks [37]. The reflectance spectrum, on the other 78

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hand, is much more sensitive to the angle of incidence. The reflectance and absorbance, defined by 𝐴 = 1 − 𝑇 − 𝑅, are complementary in a metallo-dielectric, so that when the reflectance is smaller the absorbance is proportionately larger. As an interesting observation we note from Fig. 9(a) and (b) for wavelengths around 330 nm that the reflectance and transmittance are both small at 𝜃𝑖 = 60◦ , so that the light is almost entirely absorbed. In general, BMM applies to a wide range of systems with effective anisotropic dielectric constants. It is an efficient technique for exploring parameters in inhomogeneous systems with either a gradient in the effective dielectric constants or multiple stacks with different material compositions and physical parameters. It is clear that BMM with effective medium results for an anisotropic material provides a fast and reliable estimate of the optical characteristics of the composite material. More complex geometries are amenable to rapid exploration of the parameter space using BMM before advanced simulations are adopted. Finally it is shown that the BMM can readily lead to the concept of the transfer function matrix for propagation in anisotropic materials through which the angular plane wave spectra of all components of the electric and magnetic fields can be computed. Work on this is currently in progress.

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