Spatial Kramers–Kronig relations and the reflection of waves

Abstract

When a planar dielectric medium has a permittivity profile that is an analytic function in the upper or lower half of the complex position plane x = x′ + ix″ then the real and imaginary parts of its permittivity are related by the spatial Kramers-Kronig relations. We find that such a medium will not reflect radiation incident from one side, whatever the angle of incidence. Using the spatial Kramers-Kronig relations, one can derive a real part of a permittivity profile from some given imaginary part (or vice versa) such that the reflection is guaranteed to be zero. This result is valid for both scalar and vector wave theories and may have relevance for designing materials that efficiently absorb radiation or for the creation of a new type of anti-reflection surface.A wave propagating through an inhomogeneous medium is usually partially reflected, which is often undesirable from a practical point of view. Although the reflection from a sharp interface can be suppressed by applying an anti-reflection coating 1 , less is understood about what is required in order for a generic inhomogeneous medium not to reflect any radiation. There are some well-known examples of non-reflecting material profiles, such as the hyperbolic secant profile described by Landau and Lifshitz 2 , which has been discussed by Lekner 3 (see ref. 4 for an experimental realization). More recently, the design technique of transformation optics 5,6 has made significant strides forward, giving us a strategy for finding inhomogeneous, anisotropic materials (transformation media) that reflect no radiation, whatever the incident field 5,7,8 . In the same vein, perfectly matched layers 9 are a family of anisotropic lossy media that are often used in computer simulations to mimic an infinitely extended system and are closely connected to transformation media, absorbing a wave without producing any reflection [10][11][12] . The last few years have seen increasing interest in the property of parity-time (PT) symmetry in optics [13][14][15] , partly because media with this property can suppress reflection 16,17 . For complex permittivities this requires regions of gain (Im[ϵ(x)] < 0) as well as loss (Im[ϵ(x)] > 0) and has been connected to the use of complex coordinates in transformation optics 18 . Realizing any of these non-reflecting materials is challenging, but suitably structured metamaterials 19 are promising, because a wide range of material parameters can be achieved through the use of specially designed sub-wavelength elements. In particular, recent work on 'dispersion engineering' 20 has seen simultaneous control of the real and imaginary parts of the permittivity and permeability, which is necessary to realize PT-symmetric media, as well as the materials proposed in this work.Here, we find a new and very general relation between the real and imaginary parts of a (locally isotropic) planar permittivity profile ϵ(x) that guarantees zero reflection. The result makes use of the properties of ϵ(x) at complex values of the spatial coordinate x = x′ + ...

Perfect absorption and no reflection in disordered photonic crystals

“…In this case, the correlated modulations in N j (z) and δ ds (z) result in a P T -antisymmetric susceptibility with its real (imaginary) part being an odd (even) function of the position z [2,43]. This implies, in particular, both that the system is a pseudo-Hermitian one [46] and that it satisfies the spatial Kramers-Kronig relations [3]. In the case of φ d = ±π/4 or p = 0, however, χ p j (z) will no longer be pseudo-Hermitian.…”

“…Among them, unidirectional reflectionless (URL) media [1][2][3], where reflection from one side is significantly suppressed, and coherent perfect absorption (CPA) media [4][5][6][7][8][9][10], where the two-sided incident waves are completely quenched, are extensively studied wave phenomena, especially in optics [6,[9][10][11][12][13][14][15][16][17]. On-chip implementation of URL media, e.g., is expected to underpin a new generation of photonic devices [18], while CPA media clearly provides an additional flexibility to tune absorption when compared to perfect one-port absorbers.…”

Front Matter

“…It is interesting that some periodic materials with damping can have trivial dispersion relations, with a dispersion diagram equivalent to that of a homogeneous damped material [27,28]: this happens when the moduli are analytic functions, not of the frequency, but of the complex variable x 1 + ix 2 , where i = √ −1 and for a 2D material x 1 and x 2 are the Cartesian spatial coordinates. Closely related materials were discovered by Horsley, Artoni, and La Rocca, who realized they would not reflect radiation incident from one side, whatever the angle of incidence [29].…”

Unidirectional reflectionless light propagation at exceptional points

“…Moreover, Horsley et al completely suppressed scattering in one direction in a planar inhomogeneous dielectric structure, in which the spatial distributions of the real and imaginary parts of the dielectric permittivity are related by Kramers-Kronig relations [100]. Specifically, they considered a monochromatic electromagnetic wave propagating in the x-y plane within a medium with a positive background contribution ε b plus a spatially varying part α in the dielectric permittivity ε: ε(x) = ε b + α(x) ( Figure 7A).…”