On maximum left/right reflectance asymmetry exhibited by a gyrotropic dielectric slab

Gyrotropic dielectric materials, being Lorentz non-reciprocal, exhibit scientifically and technologically interesting reflection asymmetries. On numerically characterizing left/right asymmetries in linear reflectances exhibited by a gyrotropic dielectric slab, we found these asymmetries to be highly sensitive to: (i) the constitutive parameters of the gyrotropic dielectric material, (ii) the thickness of the slab, (iii) the direction of incidence, and (iv) the refractive indexes of the isotropic dielectric materials above and below the slab. In particular, left/right reflectance asymmetries increase as (i) dissipation in the gyrotropic dielectric material decreases and (ii) the anti-symmetric component of the relative permittivity dyadic of that material increases. Generally, the cross-polarized left/right reflectance asymmetry is an order of magnitude smaller than the co-polarized left/right reflectance asymmetries.

In this communication, we numerically characterize left/right asymmetries in linear reflectances for a slab of a gyrotropic dielectric material, and show how these asymmetries can be maximized-depending upon the choice of the constitutive parameters of the gyrotropic dielectric material, as well as the thickness of the slab, the direction of incidence, and the refractive indexes of the isotropic dielectric materials above and below the slab.In the following, an w -i t exp( )dependence on time t is implicit, with = - i 1 and ω as the angular frequency.The permittivity and permeability of free space are ò 0 and μ 0 , respectively; the intrinsic impedance of free space is h m =  ; 0 0 0 the freespace wavelength is λ 0 ; and the free-space wavenumber is w m p l = =  k 2 0 0 0 0 .A Cartesian coordinate system (x, y, z) is adopted with u x ˆ, u y ˆ, and u z ˆas unit vectors.Vectors are underlined and dyadics [3] are double underlined.
with the relative-permittivity scalars . This material is nondissipative for ν = 0 and dissipative for ν > 0. In the limit ò 3 → 0, the material ceases to be gyrotropic (and Lorentz non-reciprocal) and becomes a uniaxial dielectric material [1,3] which is Lorentz reciprocal.
The half-space z < 0 is filled by an isotropic dielectric material with refractive index n 1 and the half-space z > d is filled by an isotropic dielectric material with refractive index n 2 .It is assumed that n 1 > 0 and n 2 > 0.
In figure 1 a schematic representation of the reflection-transmission problem is displayed.An incident plane wave propagates in the half-space z < 0 towards the plane z = 0. Its direction of propagation is oriented at an angle θ inc ä [0, π/2), (i.e., 0 θ inc < π/2), relative to the z axis and at an angle ψ ä [0, 2π), (i.e., 0 ψ < 2π), relative to the x axis in the xy plane.Thus, the plane of incidence is defined by the unit vectors y y + u u cos sin x y ˆˆand u z ˆ.Since the materials below and above the slab are isotropic dielectric materials, the most natural decomposition of the incident plane wave is in terms of its components parallel and perpendicular to the plane of incidence.Accordingly, the usual convention is adopted in which the field phasors associated with the incident plane wave are expressed in terms of s-and p-polarized components as [20,21] y y q h y y q wherein the real-valued wavenumber The amplitudes of the s-and the p-polarized components of the incident plane wave are a s and a p , respectively, while the unit vectors y y y y q q The field phasors for the corresponding reflected plane wave are given as [20,21] y y q h y y q with the unit vector The field phasors for the corresponding transmitted plane wave need not concern us here, as our focus is on reflection.
The transfer-matrix method (TMM) is well suited to reflection-transmission problems involving one or more slabs of anisotropic (or bianisotropic) material.No coordinate transformations are required to accommodate complex materials.Indeed, the TMM was first developed for a uniaxial dielectric-magnetic slab [22] and subsequently extended for multilayered slabs [20].
By invoking the standard boundary conditions at the interfaces z = 0 and z = d The elements of the 2 × 2 matrix herein are the linear reflection coefficients, with co-polarized coefficients having identical subscripts and cross-polarized coefficients having nonidentical subscripts.Full details are available elsewhere [20].
The squared magnitudes of the linear reflection coefficients yield the corresponding linear reflectances.For example, is the linear reflectance corresponding to the linear reflection coefficient r sp , which is associated with a ppolarized incident wave and an s-polarized reflected wave.

Numerical studies of linear reflectance asymmetries
For a given polar angle of incidence θ inc , we wish to compare the reflectance for a plane wave with (i) an azimuthal angle of incidence ψ and (ii) an azimuthal angle of incidence ψ + π.Accordingly, let us make explicit the dependency of the reflectances on the angles of incidence θ inc and ψ and introduce the asymmetries q y q y p q y q y p q y q y p q y q y p With ψ ä [0, π), the appellation LEFT/RIGHT REFLECTANCE ASYMMETRY is appropriate for d R ss , etc.In the following discussion on reflectance asymmetries there is no need to present graphs of d R ps since it was observed that º sp ps .The free-space wavelength is fixed at λ 0 = 560 nm, without loss of generality.Also, the refractive indexes of the half-spaces are fixed at n 1 = n 2 = 1 for figures 2-5, but this constraint is relaxed for figure 6.

Plots of d
R ss , d R sp , and d R pp versus θ inc ä (0, π/2) and ψ ä (0, π) are provided in figure 2 for the case in which the gyrotropic dielectric material is specified by the constitutive parameters ò 1 = 3, ò 2 = 1.5, ò 3 = 0.4, and ν = 0.01, with the slab thickness d = 0.9λ 0 .At large values of θ inc the left/right reflectance asymmetries are acutely sensitive to ψ, whereas the opposite is the case at small values of θ inc .The greatest values of left/right reflectance asymmetries arise when ψ takes values from outside the neighborhoods of 0, π/2, and π.The plots for d R ss and d R pp are both qualitatively and quantitatively similar; the plots for d R sp are qualitatively similar to those for d  and d R pp are all highly sensitive to changes in slab thickness at high values of θ inc but much less sensitive at low values of θ inc , when ψ = 45°.Furthermore, at θ inc = 75°, each of d R ss , d R sp , and d R pp varies dramatically as both d and ψ are varied.Generally, the cross-polarized left/right reflectance asymmetry d R sp is an order of magnitude smaller than the co-polarized left/right reflectance asymmetries d R ss and d R pp .We turn now to the influence of the constitutive parameters of the gyrotropic dielectric material on the left/ right reflectance asymmetries.In figure 4, d R ss , d R sp , and d R pp are plotted against n Î 0, 0.3 ( )and Î  0, 0.5 )for the gyrotropic dielectric material specified by ò 1 = 3 and ò 2 = 1.5.The slab thickness is d = 0.9λ 0 and the angles of incidence are fixed at θ inc = 75°and ψ = 45°.At fixed values of ν, each of the left/right reflectance asymmetries increases monotonically as ò 3 increases; and at fixed values of ò 3 , each of the left/right reflectance asymmetries increases monotonically as ν decreases.Thus, the greatest values of d R ss , d R sp , and d R pp are achieved at the maximum value of ò 3 and minimum value of ν.Notably, all three left/right reflectance asymmetries vanish in the limit ò 3 → 0, regardless of the value of ν, in accord with known results for uniaxial dielectric materials [23].)in figure 6.The gyrotropic dielectric material is specified by ò 1 = 3, ò 2 = 1.5, ò 3 = 0.4, and ν = 0.01.The slab thickness and the angles of incidence are the same as for figures 4 and 5.All three left/right reflectance asymmetries are acutely sensitive to changes in n 1 .This sensitivity is most obvious for 0.5 < n 1 < 2; for larger values of n 1, the left/right reflectance asymmetries decay monotonically as n 1 increases.All three left/right reflectance asymmetries are much less sensitive to changes in n 2 .

Discussion
The Lorentz non-reciprocal nature of gyrotropic dielectric materials gives rise to scientifically and technologically interesting left/right reflection asymmetries exhibited by a gyrotropic dielectric slab.The choice of the constitutive parameters of the gyrotropic dielectric material, as well as the thickness of the slab, the direction of incidence, and the refractive indexes of the materials above and below the slab, all were found to exert a strong influence on these asymmetries.Most notably, left/right reflectance asymmetries increase as (i) dissipation in the gyrotropic dielectric material decreases and (ii) the anti-symmetric component of the relative permittivity dyadic of the gyrotropic dielectric material increases.These results may be useful in the design of optical reflectors with maximal left/right reflectance asymmetries.
In particular, through a parametric study we have demonstrated how left/right reflection asymmetries exhibited by a gyrotropic dielectric slab can be maximized as the constitutive parameters of the materials

Figure 1 .
Figure 1.Schematic representation of the reflection-transmission problem.

[ 3 ,
21], the TMM delivers the reflection amplitudes r s and r p as linear combinations of the incidence amplitudes a s and a p per
R ss and d R pp but the greatest values of d R sp are almost 10 times smaller than the greatest values of d R ss and d R pp .The dependency of the left/right reflectance asymmetries on the slab thickness can be gathered from figure 3 wherein d R ss , d R sp , and d R pp are plotted against l Î d 0θ inc = 75°.The constitutive parameters of the gyrotropic dielectric material have the same values as those used for figure 2. The left/right reflectance asymmetries d R ss , d R sp ,