Abstract

Here, we consider that a novel regime of propagation occurs for Transverse Magnetic (TM) waves propagating, in the long wavelength regime, through rapidly modulated stratified media with a large dielectric modulation depth (Kapitza media). The considered electromagnetic situation is conceptually equivalent to that of the mechanical inverted pendulum whose pivot point is subjected to high-frequency vertical oscillations (Kapitza pendulum) [1], since these oscillations produce a rapidly varying contribution to the lagragian function with a large modulation depth. We consider a monochromatic TM electromagnetic field of complex amplitudes E = E_x(x,z)ê_x +E_z(x,z)ê_z, H= H_y(x,z)ê_y (assuming the time dependence e^−iωt) in a dielectric medium modulated along the z-axis whose dielectric permittivity admits the Fourier series expansion $\epsilon ={\epsilon }_m+{\displaystyle {\sum }_{n\ne 0}\left(a_n+\frac{b_n}{\eta }\right)}{e}^{in\frac{K}{\eta }z}$, where ε_m and (a_n + b_n/η) are the Fourier coefficients whereas 2πη/K is the spatial period. Here η is a dimensionless parameter we have introduced to explore the asymptotic electromagnetic behavior pertaining the limit η → 0 where both the grating amplitude and its spatial frequency are very large (assuming that K ≈ k₀ = ω/c). Exploiting the multiscale technique (where we represent each electromagnetic field component as a Taylor expansion up to first order in η and introducing two different coordinates, namely z and Z = z/η), we obtain that the leading term of the TM electromagnetic field is slowly varying and rigorously Transverse Electro-Magnetic (TEM), i.e. $E\simeq {\overline{E}}_x^{(0)}(x,z){\widehat{e}}_x$, $H\simeq {\overline{H}}_y^{(0)}(x,z){\widehat{e}}_y$. In addition, its components satisfy the equations ${\partial }_z{\overline{E}}_x^{(0)}=i\omega {\mu }_0{\overline{H}}_y^{(0)}$, ${\partial }_z{\overline{H}}_y^{(0)}=-i\omega {\epsilon }_0{\epsilon }_{eff}{\overline{E}}_x^{(0)}$, ${\overline{E}}_z^{(0)}=0$, where ${\epsilon }_{eff}={\epsilon }_m+\frac{k_0^2}{K^2}{\displaystyle {\sum }_{n\ne 0}\frac{b^{(-n)}{b}^{(n)}}{n^2}}$ and the superscript (0) labels the order η⁰ of each term whereas the overline labels the averaged contributions. It is worth noting that the field rigorouslyundergoes diffractionless propagation and experiences the uniform effective permittivity ε_eff. In order to check the predictions of the proposed Kapitza effective medium theory (EMT), we have considered reflection and transmission of TM plane waves from a slab filled by a Kapitza stratified medium, as sketched in Fig.1(a). The dielectric modulation is along the z-axis with period ηλwhere λ= 2π/k₀ is the vacuum radiation wavelength and ηis the above introduced small parameter, whereas the slab thickness L is a multiple of the period ηλ. The unit cell comprises N homogeneous layers of thicknesses ηλ /N and the dielectric permittivity of the j −th layer (j = 1,...,N) is ${\epsilon }_j={\epsilon }_m+\left(\frac{1}{\eta }+i\delta \epsilon \right)\cos \left[\frac{2\pi }{N}(j-1)\right]$ where ε_m is the mean value of the dielectric permittivity and δε is responsible for the (not large) modulation of medium absorption (see Figs.1 (b) and (c)). The check of the Kapitsa EMT has been performed by choosing λ = 100µm, ε_m = 0.05+0.05i, δε = 0.025, N = 10 and η = 1/60. In Fig.1(d) we have plotted the exact profiles (solid line) of the transmissivity T (evaluated by means of the transfer-matrix method) as function of the transverse wave vector k_x = k₀ sinθfor various slab thicknesses L. In Fig.1(d) we have reported various profiles (dashed lines) of the resulting transmissivity pertaining to the Kapitza EMT. In addition, the standard EMT would describe the slab as an anisotropic medium with dielectric permittivities ε_x = 〈ε〉 and ε_z = 〈ε⁻¹〉⁻¹ and, in Fig.1(d), we have plotted various profiles (dashed dot lines) of the the corresponding transmissivity. Note that the agreement between the exact predictions and those based on the Kapitza EMT is remarkable whereas the standard EMT completely fails.

© 2013 IEEE