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Numerical Analysis of Wave Propagation in Hyperbolic Waveguides

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Numerical Analysis of Wave Propagation in Hyperbolic Waveguides MSc in Photonics PHOTONICSBCN Universitat Politècnica de Catalunya (UPC) Universitat Autònoma de Barcelona (UAB) Universitat de Barcelona (UB) Institut de Ciències Fotòniques (ICFO) http://www.photonicsbcn.eu Master in Photonics MASTER THESIS WORK NUMERICAL ANALYSIS OF WAVE PROPAGATION IN HYPERBOLIC WAVEGUIDES Pilar Pujol Closa Supervised by Dr. Jordi Gomis-Bresco (ICFO, UPC) and Prof. David Artigas, (ICFO, UPC) Presented on August 31st, 2017 The content of this thesis is confidential Registered at Numerical analysis of wave propagation in hyperbolic waveguides Pilar Pujol-Closa1;2 1Nonlinear Optical Phenomena Group, ICFO - Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Av. Carl Friedrich Gauss, 3, 08860 Castelldefels (Barcelona), Spain 2Universitat Politecnica de Catalunya, Jordi Girona 1-3, 08034 Barcelona, Spain E-mail: [email protected] Abstract. In the present work we study wave propagation in Hyperbolic metamaterial (HMMs) waveguides. We consider two different three layer systems, one having a HMM of type I as the middle layer, and the other having a HMM of type II. We study the existence of guided modes, determine the limiting conditions for their existence, and obtain their dispersion curves. Additionally, we also study the amplitude of the fields at certain points and for certain modes. We report cutoff-less modes, anomalous scaling laws, and a divergence of the effective refractive index. Keywords: Metamaterials, Hyperbolic Metamaterials, Berreman calculus, Wave propagation, Guided modes. 1. Introduction Metamaterials are often described as artificially engineeredz materials composed of subwavelength structures that allow control of light propagation, enabling many applications that otherwise would not be possible to achieve with conventional materials. All four quadrants of electromagnetic responses can be attained: " > 0, µ > 0; " < 0, µ > 0; " > 0, µ < 0; and " < 0, µ < 0; where " is the electric permittivity and µ is the magnetic permeability. Many of their applications have been studied: magnetic mirrors, nanoscale resonators, subwavelength imaging, or cloaking devices [2,3]. Hyperbolic metamaterials (HMMs) is among others one class of metamaterials which has the particularity that exhibits hyperbolic dispersion as Figure1 shows. The principal components of the permittivity (^") or permeability tensor (^µ) in HMM do not have the z Although some authors have described nature existing metamaterials, for example Caldwell et. al [1] in 2014. Numerical analysis of wave propagation in hyperbolic waveguides 2 same sign. Therefore, showing extreme anisotropy and allowing large wave vectors and ultrahigh effective refractive indices to be supported [4]. 0 1 0 1 "x 0 0 µx 0 0 B C B C "^ = @ 0 "y 0 A ; µ^ = @ 0 µy 0 A : (1) 0 0 "z 0 0 µz For simplicity, in the present work we have considered no magnetic response, that is µ^ to be the identity matrix. Notice that hyperbolic media satisfies then "o"e < 0. There are two types of hyperbolic metamaterials: type I has the ordinary component of the permittivity tensor positive and the extraordinary negative, and the extraordinary isofrequency surface is a hyperboloid of two sheets; on the contrary, type II has a negative permittivity ordinary component and a positive extraordinary component, and its isofrequency surface is a one sheet hyperboloid. Nevertheless, ordinary waves have spherical dispersion. Figure 1. Isofrequency surfaces of extraordinary waves in hyperbolic metamaterials. At the left type I "o > 0, "e < 0; and at the right type II "o < 0, "e > 0. Their fabrication procedures also differ from one to the other. Type I are structures usually made by embedding metallic nanowire arrays in a dielectric matrix as Figure2 shows. However, type II structures are built by placing different layers of metal-dielectric along the optical axis (OA) [5]. Figure 2. Structure of hyperbolic media of type I at the left and of type II at the right. Numerical analysis of wave propagation in hyperbolic waveguides 3 Several authors presented early studies in HMMs waveguides: Xu et al. in 2008 [6], He et. al in [7], and Yang et. in [8] in 2012. However, all these works were confined to study propagation along the principal axis, and only in 2015 Boardman et al. [9] attempted to define mode existence ranges for out-of-axis propagation. In this thesis, we study wave propagation in plane HMMs waveguides of both types implementing numerical solutions of the transcendental equations, determining the existence of guided modes when the optical axis of the hyperbolic media lays parallel to the waveguide interfaces forming an arbitrary angle with the propagation direction in plane. 2. Methods Figure3a) shows the schematic of the three layer system that is being studied, where "c is the permittivity of the cladding," ¯f of the film, and" ¯s of the substrate. The cladding is air, and both the film and the substrate are uniaxial materials. The film is studied to be first a HMM of type I, and later of type II. In table1 the permittivity values of each d of the layers of the system are shown. The studied system has λ = 0:6, notice that for instance with a wavelength of λ = 632 nm the height of the waveguide is also within the order of a few hundreds of nanometers which is relatively small compared to regular waveguides. In Figure3b) the coordinate system used is detailed. In this thesis we have considered θ = 90◦, and thus the optical axis is in the plane y − z. Figure 3. a) Shows the schematic of the studied system with the cladding on top, the film as the middle layer, and the substrate at the bottom. b) Coordinate reference system that we are using, the blue line shows the optical axis direction. During this master thesis a code developed in the NonLinear Optical Phenomena group at ICFO [10] has been improved to allow simulations of systems containing hyper- bolic media. The code deals with stratified media: layers of homogeneous material that extend to infinite in two of the space directions, but have well defined interfaces in the other one (see Figure3). For this, Maxwell's equations are reduced to one dimension, but still expressed in tensorial form and solved for a general bulk uniaxial case. Bound- ary conditions are enforced by assuming the continuity of the four electromagnetic field components tangential to the interfaces. In the figure case, with x direction set normal to the interfaces, this components are Ey, Hz, Ez and Hy. It is because of that that this kind of calculus, known as Berreman Calculus [11], deals with 4 x 4 matrices. Numerical analysis of wave propagation in hyperbolic waveguides 4 Table 1. Permittivity values that determine the studied system. Type I Type II System "e "o "e "o Cladding 1 1 1 1 Film -2.7889 4 4 -2.1889 Substrate 2.25 3.3856 2.25 3.3856 The total field ~m is a column vector with elements Ey, Hz, Ez and Hy. The field matrix F^ contains the four basis vectors of the waves that can propagate in a biaxial layer for a given propagation constant, that are straight solution of the reduced Maxwell's equations. Any field ~m can be obtained by linear combination of this basis vectors. Defining ~a as a complex coefficient column vector, we find that ~m = F~a^ . A phase matrix ^ Ad needs to be defined to account for the change of phase of each of the travelling waves ^ ^ ^ ^ ^−1 along a medium. The characteristic matrix M of the layer M = F AdF transforms the ^ ^ total field between two planes within the same layer specified in Ad. Finally, A is the ^ ^−1 ^ ^ ^ system matrix and satisfies A = Fc MFs, where Fc is the field matrix of the cladding, ^ and Fs is the field matrix of the substrate. By multiplying matrices properly we can study a system with an arbitrarily large number n of layers. For instance, if we take the total field at the cover, the propagation from substrate to cover can be obtained by ^ ^ ^−1 ^ ^ ^−1 ^ ^ ^−1 ~mc = F1Ad1 F1 F2Ad2 F2 ··· FnAdn Fn . Moreover, the existence of guided modes is obtained by deriving a modal equation as in [10], and solutions are found by iteration using the Newton-Raphson method in the complex plane. Hence, leaky modes can also be studied implementing this code. 3. Results and discussions 3.1. Type I First we studied the existence of guided modes in a three layer system in which the middle layer is a type I HMM as defined in Table1. The effective refractive index of the guided modes versus the propagation direction is plotted in Figure4 as colored solid lines, determining their existence with the OA in plane. The colored background shows in a normalized scale the value that the modal equation mentioned before takes at every propagation direction for different effective refractive indices. Darker areas show where the modal equation takes minimum values, indicating the possible existence of modes. In Figure5 we plot the dispersion of the found modes. As expected for a type I HMM their isofrequency surfaces are hyperboloids of two sheets. The first found mode has circular dispersion, the other found guided modes have hyperbolic dispersion, we name them extraordinary predominant modes. Remarkably, these extraordinary predominant modes seem to be cut-off less, with an infinite number of existing modes. Numerical analysis of wave propagation in hyperbolic waveguides 5 Another interesting feature of this figure is that the guided mode that has circular dispersion, the only ordinary predominant mode, is tangent to the first hyperbolic mode, this is because at that point ne (φ) = no. Moreover, from Figure4 one can infer the existence of a critical angle φc that spatially limits the existence of hyperbolic guided modes in the plane. This φc is what determines the asymptote of the hyperboloid shown in Figure1, and satisfies tan 2 φ = "e .
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