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 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

ϵ = ϵ_{r} - i ϵ_{a},
ϵ_{r} = (ϵ^{†} + ϵ)/2,
ϵ_{a} = (ϵ^{†} - ϵ)/(2 i),

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).

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 **p**, with |**p**| = 1, through the medium; **r** denotes the position argument.
The Maxwell equations then require that the polarization vector **E**_{0} and direction vector **p** satisfy the relations

ϵ**E**_{0} = n^{2}(**E**_{0} - (**p**·**E**_{0})**p**),
**p**·ϵ**E**_{0} = 0.

The solver characterizes the potentially crystalline optical medium in terms of pairs of effective index / effective permittivity values n_{j}, ϵ_{j}, with n_{j}^{2} = 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} = E_{0}**a**_{j} of potential plane waves with amplitudes E_{0} that can propagate through the medium in directions **p** that are perpendicular to the axes, **p**·**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} = E_{0}**a**_{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" E_{0}, 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:

R =

cos(β) cos(γ) | cos(β) sin(γ) | -sin(β) |

sin(α) sin(β) cos(γ)-cos(α) sin(γ) | sin(α) sin(β) sin(γ)+cos(α) cos(γ) | sin(α) cos(β) |

cos(α) sin(β) cos(γ)+sin(α) sin(γ) | cos(α) sin(β) sin(γ)-sin(α) cos(γ) | cos(α) cos(β) |

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 amplitudes q_{1} and q_{2} are determined through a given general initial polarization 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
**p**_{1}, **p**_{2} and effective index eigenvalues
n_{eff,1}, n_{eff,2} are solutions of the 2×2 eigenvalue problem

P **p** = n_{eff}^{2} **p**,
with
P =

.

ϵ_{xx}-ϵ_{zx}/ϵ_{zz} | ϵ_{xy}-ϵ_{zy}/ϵ_{zz} |

ϵ_{yx}-ϵ_{zx}/ϵ_{zz} | ϵ_{yy}-ϵ_{zy}/ϵ_{zz} |

If the ckeckbox *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 and values for effective indices n_{eff} and effective permittivities ϵ_{eff} = n_{eff}^{2} 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 (not shown) can well be nonzero.

Reference

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