Epsilon

Linear anisotropic media are considered from a viewpoint of not necessarily crystal-aligned external device coordinates. The script calculates optical axes and effective refractive index eigenvalues associated with a general optical permittivity, and provides controls for arbitrary 3-D rotations of the tensor. Facilities for the rudimentary illustration of the wave propagation through the oriented crystalline medium, and of the associated power flow, are available.

Permittivity

The permittivity tensor is specified with respect to a Cartesian crystal coordinate system (x, y, z) that is fixed to the medium, but not necessarily aligned with any of the axes (if applicable) of the medium. Following the reasoning of Ref. [1], we split the permittivity ϵ into its refractive part ϵ_r and an absorptive part ϵ_a. These tensors satisfy the relations

for imaginary unit i, where ^† indicates the adjoint. By construction, both ϵ_r and ϵ_a are Hermitian. Depending on the state of the checkboxes Transparent medium (refractive part only / refractive and absorptive part of the permittivity), Diagonal permittivity (diagonal entries only / diagonal and off-diagonal elements), and Real off-diagonal elements (real / complex off-diagonal entries), the input mask accepts separately values for the real diagonal and real or complex off-diagonal elements of ϵ_r and ϵ_a. Note that the association of optical losses or gain with negative or positive imaginary parts of effective indices, of effective permittivity values, and with the signs of related entries in the permittivity tensor depends on the sign convention adopted for the time dependence in the frequency domain description (see the remarks in the next paragraph).

The Fill selection offers pre-defined data for a few media (that have been relevant for the work of the author at some time). Values are provided for

• Vacuum: air / vacuum, meant as a reset to a transparent, isotropic medium, the default,
• Eps_11: a transparent, isotropic medium with given scalar permittivity; fills the entry present in the field ϵ₁₁ (real part) into the fields ϵ₂₂ and ϵ₃₃, all other components are set to zero,
• Glass: values for fused silica (transparent, isotropic) at about 600nm vacuum wavelength,
• Uniaxial: an artificial transparent uniaxial medium with strong birefringence, directional refractive indices n_o = 2.5, n_e = 2.0; included for purposes of demonstration,
• Biaxial: an artificial transparent biaxial medium with strong anisotropy, directional refractive indices n₁ = 1.5, n₂ = 1.75, n₃ = 2.0; included for purposes of demonstration,
• Nonreciprocal: a transparent nonreciprocal medium with artificially large offdiagonal elements, set α to ±90° for an illustration of the nonreciprocal Faraday rotation; included for purposes of demonstration,
• LiNbO3: transparent birefringent crystalline bulk lithium niobate, at a vacuum wavelength of 1.55μm, adapted from Ref. [2] (temperature 20°C); ordinary refractive index n_o = 2.2112, extraordinary refractive index n_e = 2.1379,
• MO garnet: a magnetooptic iron garnet crystal, with certain substitutions, at a vacuum wavelength of 1.3μm, adapted from Ref. [3] (losses omitted); refractive index n = 2.317; specific Faraday rotation Θ_F = -1360°/cm, leading to offdiagonal permittivity elements ±iξ with ξ = 0.0023.

Note that the input fields also accept simple mathematical expressions (see the Teval-script): given e.g. a refractive index of 1.234, you might wish to enter the term 1.234^2 for the respective permittivity element.

Optical axes and effective indices

We refer to a frequency domain description with dependence ∼exp(iωt) on time t, for fixed positive angular frequency ω = kc = 2πc/λ, vacuum wavelength λ, vacuum wavenumber k, and speed of light in vacuum c. One considers plane waves with an electric field of the form

that propagate with effective index n along direction κ, with |κ| = 1, through the medium; r denotes the position argument. The Maxwell equations then require that the polarization vector E₀ and direction vector κ satisfy the relations

The solver characterizes the potentially crystalline optical medium in terms of pairs of effective index / effective permittivity values n_j, ϵ_j, with n_j² = ϵ_j, and optical axes vectors a_j, j = 1, 2, 3. Assuming a mostly transparent medium, the optical axes are determined through the refractive part of the permittivity only, as orthogonal solutions of the Hermitian 3×3 eigenvalue problem ϵ_r a_j = ϵ_j a_j. Subsequently, the absorptive part ϵ_a of the permittivity enters in the form of perturbational corrections to the effective permittivity eigenvalues.

The optical axis vectors, determined in the aforementioned way, thus indicate the polarization E_0,j = E₀a_j of potential plane waves with amplitudes E₀ that can propagate through the medium in directions κ that are perpendicular to the axes, κ·a_j = 0. Note that, even after normalization and phase adjustment, the axes vectors can be essentially complex, i.e. the vectors do not necessarily describe some physical spatial direction that could be visualized directly. However, at some fixed position, the physical electric field associated with a plane wave of the former type with polarization vector E_0,j = E₀a_j oscillates in time as

Hence, the solver illustrates the "axes" by plotting a trace of E(t) over one period of time. The "amplitude" E₀, set to a length of √|ϵ_j|, is meant to hint at the associated effective index.

Transformation to device coordinates

An adjustable general rotation operator R mediates between the medium-fixed crystal coordinate system (x, y, z) and an exterior system of device coordinates (x′, y′, z′). The orthogonal matrix is defined through three angles α, β, and γ, which concern rotations of the medium around the x, y, and z or x′, y′, and z′ -axes, respectively, if applied separately:

A vector v given in the crystal coordinate system transforms to a vector v′ = R v in the device system. Likewise, an operator given by a matrix M in the crystal system is represented by a matrix M′ = R M R^⊤ in the device system. The solver applies these rules to the vectors that represent the optical axes, and to the permittivity tensor, respectively.

Optical wave propagation through the oriented medium

We consider specifically a polarized plane electromagnetic wave propagating through the oriented medium along the direction z′ in the device coordinate system. Excluding singular configurations, this concerns electric fields of the form

where the complex amplitudes A₁ and A₂ are determined through a given general initial field at z′ = 0 with transverse components E_x and E_y. For the sake of brevity, the ′-accents are here omitted for component-identifying indices.

The x- and y-components of the polarization vectors p₁, p₂ and effective index eigenvalues n_eff,1, n_eff,2 are solutions of the 2×2 eigenvalue problem

Note that this model does not concern what might happen on illuminating some facet of a crystalline medium with an exterior polarized plane wave coming in from, say, vacuum. Rather, it is assumed that the initial, purely transverse field is present in the medium at z′ = 0.

If the checkbox Propagation is enabled, the solver illustrates the wave propagation by means of a plot of a time snapshot of the polarized wave, propagating along the z′-device coordinate through the oriented medium. An option for an animation of the wave is available. Further, the solver lists the polarization eigenvectors, their amplitudes for the given initial field, and values for effective indices n_eff and effective permittivities ϵ_eff = n_eff² associated with that propagation direction, and a characteristic half-beat-length L_c = λ/(2|n_eff,1-n_eff,2|) for polarization conversion, if applicable. Note that, for a general anisotropic medium, the longitudinal component p_z = - (ϵ_zx/ϵ_zz) p_x - (ϵ_zy/ϵ_zz) p_y of the polarization vectors (disregarded for the time-snapshots and the animations) can well be nonzero.

Power flow associated with the propagating wave

Assuming (and omitting) the time dependence exp(iωt) by adopting standard frequency-domain notation, the electric and magnetic parts of the former wave are

While the magnetic field is always purely transverse, for certain types of anisotropy and crystal orientation the electric field can have nonzero longitudinal components. This electromagnetic wave is thus associated with the power flow as given by the time-averaged Poynting vector

Depending on the type of anisotropy, the crystal orientation, and on the excitation conditions (which determine the amplitudes A_j), this quantity can vary along the propagation coordinate z′. Further, S can exhibit nonzero transverse x′- and y′-components, i.e. the power flow does not need to be co-aligned with the direction z′ of the wave propagation. For enabled Propagation, after selecting the checkbox Power, the script illustrates the power flow through an array of arrows at varying z′ positions, that indicate the local direction and strength of the power flow.

By construction, along with the electromagnetic field E, H of the wave, at given z′ also S is constant along x′ and y′. Note that, in cases (mostly nonphysical) of strong anisotropy that lead to a short conversion length L_c for the polarization, also S can vary rapidly. Then the moderate sampling density of the arrows along z′ (about three per wavelength) might not be sufficient to properly resolve the z′ dependence of S.

For transparent media, the longitudinal variation of S with z′ originates from the interference of the two elementary parts (indices 1,2) that contribute to the former wave. If excitation conditions are such that only one of these (index j) becomes relevant, i.e. for (near-) zero amplitude of the other, the power flow can be characterized by the z′-constant Poynting vector

associated with the elementary wave j only.

References

[1] Peter Hertel, Continuum Physics, Graduate Texts in Physics, Springer, Berlin (2012)

[2] G.J. Edwards and M. Lawrence. A temperature-dependent dispersion equation for congruently grown lithium niobate. Optical and Quantum Electronics, 16:373–375 (1984)

[3] M. Wallenhorst, M. Niemöller, H. Dötsch, P. Hertel, R. Gerhardt, B. Gather, Enhancement of the nonreciprocal magneto‐optic effect of TM modes using iron garnet double layers with opposite Faraday rotation. Journal of Applied Physics 77(7), 2902–2905 (1995)