Reflectance of a layered system with a gaussian distribution of the refractive index and inserted metamaterial
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1 Reflectance of a layered system with a gaussian distribution of the refractive index and inserted metamaterial Xóchitl Saldaña Saldaña Instituto de Física de la Benemérita Universidad Autónoma de Puebla David Armando Contreras Solorio Agustín Enciso uñoz Jesús adrigal elchor Unidad Académica de Física Universidad Autónoma de acatecas E mail:dacs10@yahoo.com.mx Resumen Calculamos la reflectancia de una estructura dieléctrica con 41 capas dispuestas alternadamente con índices de refracción positivos y negativos. El índice de refracción de las capas impares es positivo y sigue una distribución de valores gaussiana, mientras que las capas pares intercaladas tienen un índice de refracción negativo constante n = -1. Para varios ángulos de incidencia, realizamos un estudio de la reflectancia variando la longitud de onda, para ambas polarizaciones TE y T. Comparamos este estudio con otro realizado cuando las capas pares tienen índice de refracción positivo constante n = 1. La reflectancia para ondas TE en ambos casos es excelente para longitudes de onda largas, siendo mucho meor en longitudes de onda cortas para capas intercaladas con n = -1. Para ondas T no es buena la reflectancia en ambos casos. Palabras clave: ateriales izquierdos, Reflectancia, ulticapas. Abstract We calculate the electromagnetic reflectance for a dielectric structure made by stacking alternately 41 layers of positive and negative refractive indices. The refractive index of the odd layers is positive and follows a Gaussian distribution, while the inserted 1
2 even layers have a negative refractive index n = -1. For several incident angles, we carry out a study of the reflectance varying the wavelength, for both polarizations TE and T. We compare this study with another one when the even layers have positive refractive index n = 1. The reflectance for TE waves in both cases is excellent for long wavelengths, but for inserted layers with negative refraction the reflectance for short wavelengths is much better. For T waves the reflectance is not good for both cases. Keywords: Left-handed materials, Reflectance, Layered systems. Introduction V. Veselago predicted the possibility to have materials with negative refractive index [1], for which both the magnetic permeability and the electric permittivity were simultaneously negative. If ever found these materials, they would exhibit extraordinary electromagnetic properties. These materials, also known as left-handed materials or metamaterials, were only a theoretical curiosity for 30 years. At the end of the decade of the 90, materials with negative refraction were fabricated [2-5], confirming the predictions of Veselago. Initially, these materials had negative refraction for frequencies in the range of microwaves. However, there are recent reports of materials with negative refraction for optical frequencies [6]. On the other hand, there are proposals of electronic filters using quantum superlattices where the height, or the width, of the barriers potentials follows a Gaussian distribution [7,8]. The transmittance for these superlattices is practically perfect for an interval of electron energies. Out of that interval, the transmittance is zero, i. e. the reflectance is perfect. The interval can be adusted modifying the parameters of the superlattice or of the Gaussian distribution. Therefore, for other type of waves it is reasonable that a layered system with a Gaussian distribution of some property could also have excellent transmittance or reflectance. In this work, we calculate the electromagnetic reflectance for a structure of 41 dielectric layers made by stacking alternately layers of positive and negative refractive 2
3 indices. The index for the odd positive refraction layers is modulated by a Gaussian function, while the even inserted layers have a constant negative refractive index n = -1. For a fixed width of the structure, and for several incident angles of the electromagnetic radiation, we made a study of the dependence of the reflectance with the wavelength, for both polarizations TE and T. In order to compare, we also made the study inserting even layers with constant positive refractive index n = 1. The calculations were made following the procedure described in the book of P. Yeh [9]. Theoretical model and calculation method To calculate the transmittance, we use the theory of Ref. [9]. We consider a structure of N plane dielectric multilayers. The layers are perpendicular to the x axis. Each layer has a width d, electric permitivity ε, and magnetic permeability μ. Each medium is homogeneous and isotropic. For each layer the refractive index n is given by n ± ε μ = (1) ε 0 μ 0 Where ε 0 and μ 0 are, respectively, the permittivity and the permeability of the vacuum. In Eq. 1 the plus sign is for normal materials with positive refraction which have ε and μ simultaneously positive. The minus sign is taken for metamaterials, which is congruent with the boundary conditions for the electromagnetic fields when both ε and μ are simultaneously negative. We consider an electromagnetic plane wave propagating in the xz plane, coming from the left in a medium with refraction index n 0. The plane wave is incident on the structure of N plane multilayers. In the right side of the structure, the wave propagates in a medium S with refraction index n S. We can write the wave function for each layer in the following form i( k r ωt) i( k' r ωt) Ae Be Ψ = + (3) 3
4 Ψ represents the electric or magnetic field. The first and the second terms on the righthand side of Eq. 3 represent propagation to the right and to the left, respectively, i.e. the forward and the backward waves. The index = 0 represents the medium at the left side of the structure with refractive index n 0. In the medium S with = N + 1 we consider only propagation to the right, consequently B S = 0. k y k are the forward and backward wavevectors for medium, t is the time, ω the angular frequency, and i the imaginary unit. If the wave is incident at an angle θ 0 with the normal to the structure, the Law of Snell gives n sinθ = n sin θ =... = n sin θ =... n sinθ. (4) S S The wave propagation from medium 0 to medium S through the multilayer structure is described by A B = S A. (5) 0 Where the (2 2) transfer matrix is given by = D 1 0 N l= 1 D P D l l 1 l D S. (6) The matrix D is called the dynamical or transmission matrix, and arises from the continuity conditions on the electric and magnetic fields at the interface between media 1 and. The matrix P is the kinematical or propagation matrix inside the layer. If it is supposed that the media are lossless, the matrices are given by D TE 1 1 =, D T = 1 1, (7) P ik d e 0 =. ik d 0 e 4
5 The two different polarizations TE and T are indicated. is the wave impedance, given by. μ = (8) ε The transmittance T is given by the ratio of the Poynting power flow of the transmitted wave to that of the incident wave, and is given in terms of the transfer matrix by T = 0 S S (9) The reflectance R is given by R = 1 T (10) The structure we consider has 41 layers and is situated in the vacuum. The refractive index for the odd layers is positive and modulated by the Gaussian function n ( x) = ( n σ + n (11) 2 2 max nmin )exp( ( x x0 ) / ) min Results and discussion We use in Eq. (11) n max = 3.5 and n min = The total width of the structure is D tot = 10 arbitrary units of length (meters, nm, Å, etc.). We use the value σ = D tot / 4. The thickness for all the odd and even layers is the same. The reflectance is calculated using Eqs. (9) and (10) when the inserted even layers have positive refractive index n = 1 or negative refraction index n = -1, in order to compare the reflectances for both cases. In Fig. 1 is shown the profile of refractive indices for the structure, when the inserted layers have positive refraction. 5
6 Figure 1. Diagram of the Gaussian distribution of refractive indices for the structure with even inserted layers of n = 1. Fig. 2 shows the results for the calculations of the dependence of the reflectance with wavelength for incident angles 0 O, 45º y 85º, when the even inserted layers have positive refraction. We observe that the reflectance por TE waves is practically perfect for the three angles, for extensive intervals of long wavelengths. However, the reflectance is poor for short wavelengths, around the value D tot. (a) (b) 6
7 (c) Figure 2. Reflectance vs. wavelength for inserted layers with n = 1. (a) incident angle of 0 O, (b) incident angle of 45 O, (c) incident angle of 85º. In Fig. 3 we show the results for the dependence of the reflectance with the wavelength, when the even layers have negative refraction, for the same angles than in Fig. 2. In this case, we notice that the reflectance for TE waves also is practically perfect for long wavelengths, but is good for short wavelengths as well. For inserted layers with positive or negative refraction, the reflectance is not good for T waves.. (a) (b) 7
8 (c ) Figure 3. Reflectance vs. wavelength for inserted layers with n = -1. (a) incident angle of 0 O, (b) incident angle of 45 O, (c) incident angle of 85º. Conclusion We have made a theoretical study of the reflectance for a structure of 41 dielectric layers where the odd layers have a positive refractive index modulated by a Gaussian distribution, while the even inserted layers can have positive refraction index n = 1 or negative refraction index n = -1. For several incident angles, the reflectance for TE waves is excellent for both cases of even layers with negative or positive refraction for long wavelengths. However, the reflectance for short wavelengths is much better for inserted layers with negative refraction than that produced by inserted layers with positive refraction. The reflectance for T waves is not good for both cases. Therefore, for inserted layers with negative refraction, the structure acts as an omnidirectional mirror for TE waves. Acknowledgments This work has been supported partially by Proect 37/EXC/G/06VIEP-BUAP and by Secretaría General de la Universidad Autónoma de acatecas. 8
9 References 1. V. G. Veselago, The electrodynamics of substances with simultaneously negative values of ε and μ, Sov. Phys. Usp. 10 (1968) J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, agnetism from conductors and enhanced nonlinear phenomena, IEEE Trans. icrowave Theory Tech. 47 (1999) D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser, and S. Schultz, Composite medium with simultaneously negative permeability and permittivity, Phys. Rev. Lett. 84 (2000) R.A. Shelby, D.R.Smith, S.Schultz, Experimental verification of a negative index of refraction, Science 292 (2001) R. Ruppin, Surface polaritons of a left-handed medium, Phys. Lett. A 277 (2000) C.. Soukoulis, S. Linden, and. Wegener, Negative refractive index at optical wavelengths, Science 315 (2007) H. H. Tung and C. P. Lee, An energy band-pass filter using superlattice structures, IEEE J. Quantum Electron. 32 (1996) I. Gómez, F. Domínguez-Adame, E. Diez, and V. Bellani, Electron transport across a Gaussian superlattice, J. Appl. Phys. 85(7) (1999) P. Yeh, Optical Waves in Layered edia (Wiley, New York, 1988). 9
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