Simultaneous amplification and attenuation in isotropic chiral materials

The electromagnetic field phasors in an isotropic chiral material (ICM) are superpositions of two Beltrami fields of different handedness. Application of the Bruggeman homogenization formalism to two-component composite materials delivers ICMs wherein Beltrami fields of one handedness attenuate whereas Beltrami waves of the other handedness amplify. One component material is a dissipative ICM, the other an active dielectric material.

by the frequency-domain Tellegen constitutive relations [7] ‡ D(r) = εE(r) + iξH(r) B(r) = −iξE(r) + µH(r) . (1) The scalar constitutive parameters ε, ξ, and µ are complex-valued in general. The Bohren decomposition is employed to represent E and H as superpositions of a left-handed Beltrami field Q L and a right-handed Beltrami field Q R , with η = µ 1/2 ε −1/2 [7]. In source-free regions, the two Beltrami fields obey the relations where the wavenumbers Let us consider planewave propagation along the +z direction. Then Q L = (û x + iû y ) exp (ik L z) is a left-circularly polarized (LCP) plane wave and Q R = (û x − iû y ) exp (ik R z) is a right-circularly polarized (RCP) plane wave. Could the LCP plane wave lose energy and the RCP plane wave gain energy, or vice versa, as z increases? More generally, could Beltrami waves of one handedness attenuate while Beltrami waves of the other handedness amplify, even if these Beltrami waves are not plane waves, but, say, spherical or cylindrical waves? In other words, could Im (k L ) and Im (k R ) be of opposite signs, but Re (k L ) and Re (k R ) have the same signs? If yes, then ICM research is promising for circular polarizers of a new type.
A perusal of literature on ICMs did not turn up any example for which Im (k L ) Im (k R ) < 0 but Re (k L ) Re (k R ) > 0. Hence, we decided to investigate a particulate composite material comprising an active dielectric material and a dissipative ICM. If the component materials can be regarded as being randomly distributed as electrically small particles, then the composite material could be homogenized into an ICM itself [8].
Let the component material labeled 'a' be an active isotropic dielectric material specified by the permittivity ε a such that Re (ε a ) > 0 and Im (ε a ) < 0. Let the component material 'b' be a dissipative ICM, characterized by constitutive relations of the form given in Eqs. (1), but with the superscript 'b' attached to the constitutive parameters ε, ξ, and µ therein. We used the well-established Bruggeman formalism [9] to estimate the constitutive parameters ε Br , ξ Br , and µ Br of the homogenized composite material (HCM), per Eqs. (1) but with the superscript 'Br' attached to the constitutive parameters ε, ξ, and µ therein. Let f a ∈ [0, 1] denote the volume fraction of component material 'a', the volume fraction of component material Figure 1 shows the real and imaginary parts of ε Br , ξ Br , and µ Br as functions of f a , when ε a = (2.0 − 0.02i) ε 0 , ε b = (3 + 0.01i) ε 0 , ξ b = (0.1 + 0.001i) /c 0 , and µ b = (0.95 + 0.0002i) µ 0 , with ε 0 and µ 0 being the permittivity and permeability of free space, respectively, and c 0 = 1/ √ ε 0 µ 0 . The component material 'b' is guaranteed to be dissipative since Im ξ b 2 < Im ε b Im µ b [10]. The chosen values of ε a , ε b , ξ b , and µ b are physically plausible [6,11,12].
The real and imaginary parts of ε Br , ξ Br , and µ Br vary almost linearly in Fig. 1 as f a increases from 0 to 1, with their endpoints complying to the limits (5) ‡ Here, and henceforth, the dependencies of the constitutive parameters and field phasors on ω are not explicitly displayed.
In particular, the sign of Im ε Br changes at f a ≈ 0.27. The real and imaginary parts of the wavenumbers for the HCM-namely k Br L and k Br R per Eqs. (4) with the superscript 'Br' attached to k L , k R , ε, ξ, and µ therein-are plotted against volume fraction f a in Fig. 2 Clearly then, a continuous range of values of the volume fraction f a , specifically f a ∈ (0.22, 0.33) for the example presented in Fig. 2, can exist wherein Q L attenuates whereas Q R amplifies.
If ξ is replaced by −ξ in Eqs. (4), then k L and k R in Eqs. (4) are interchanged. Also, if the sign of Re (ξ) is reversed in Eqs. (4), then Re (k L ) and Re (k R ) are interchanged, but Im (k L ) and Im (k R ) remain unchanged. Therefore, if ξ b were to be replaced by −ξ b for the homogenization scenario represented in Figs. 1 and 2, then the handedness of the Beltrami field that is amplified/attenuated for the regime wherein Im k Br L Im k Br R < 0 will be reversed. To illustrate this point, in Fig