Localized monochromatic and pulsed waves in hyperbolic media

Consider a uniaxially anisotropic material with diagonal electric permittivity and magnetic permeability tensor elements ε_xx, ε_yy = ε_xx, ε_zz and μ_xx, μ_yy = μ_xx, μ_zz , respectively. If all four elements are positive, the dispersion relation is described by an ellipsoid. Within a certain frequency band, however, it may turn out that not all diagonal elements are positive and the dispersion relation is a hyperboloid. Under these conditions, the material is referred to as a hyperbolic medium. For simplicity, the discussion will be confined to a nonmagnetic material. Source-free, transverse magnetic electromagnetic fields in the frequency domain are expressed Gin terms of an appropriately defined Hertz vector potential Π (r, ω) = Π_e (r, ω) z governed by the equation (∇² + (ε_zz/ε_xx) (∂²/∂z²) + (ε_zz/ε₀)k²)Π_e(r,w) 0; k≡ ω/C. For ε_xx <;0 and ε_zz > 0, the expression above is a de Broglie-like equation, with the coordinate z being timelike. On the other hand, for ε_xx >0 and ε_zz <;0, one has a Klein-Gordon-like equation, again with a timelike z coordinate. For both cases, large classes of spatially localized solutions Π_e (r,ω) are available. A parabolic approximation of the de Broglie-like equation along the y direction yields an equation analogous to that arising in the study of bidispersion. Using hyperbolic rotations, a broad class of skewed, nonspreading, “accelerating” Airy solutions can be obtained. Suppose the permittivity tensor elements are constant within a narrow fre- uency regime. Then, approximately, one has (∇_t² - (ε_zz/ε_xx) (∂²/∂z²) - (ε_zz/ε₀) (1/c²) (∂²/∂t²))Π_e (r,t) = 0 in the time domain for ε_xx <;0 and ε_zz > 0. A large class of spatiotemporally localized luminal, subluminal and superluminal pulsed solutions to this equation can be derived. These solutions differ substantially from the analogous ones in isotropic free space.