Spatially-varying permittivity tensors in buff-em

buff-em supports objects with arbitrary user-specified spatially-varying frequency-dependent permittivity tensors. These tensors are described by simple text files conventionally given the file extension .SVTensor.

The .SVTensor file is thought of as describing a 3 $\times$ 3 matrix-valued function $\mathbf{Q}$ of frequency and space:

$\mathbf{Q}(\omega, \mathbf x)=\left(\begin{array}{ccc} Q_{xx}(\omega, \mathbf x) & Q_{xy}(\omega, \mathbf x) & Q_{xz}(\omega, \mathbf x) \\ Q_{yx}(\omega, \mathbf x) & Q_{yy}(\omega, \mathbf x) & Q_{yz}(\omega, \mathbf x) \\ Q_{zx}(\omega, \mathbf x) & Q_{zy}(\omega, \mathbf x) & Q_{zz}(\omega, \mathbf x) \end{array}\right)$

Table of Contents

1. Syntax of the .SVTensor file
- Functions and variables
  - Referring to scuff-em material designations
2. Examples of .SVTensor files

1. Syntax of the `.SVTensor` file

The .SVTensor file contains lines of the form

Q = function of space and frequency

to define a spatially-varying but isotropic permittivity (proportional to the $3 \times 3$ identity matrix), or

Qxx = function of space and frequency
Qxy = function of space and frequency
...
Qzz = function of space and frequency

to define the individual cartesian components of the permittivity tensor.

Note: Because .SVTensor files are most commonly used to define permittivities $\boldsymbol{\epsilon}$ , you can alternatively use the syntax

Eps = function of space and frequency

or

EpsXX = function of space and frequency
EpsXY = function of space and frequency
...
EpsZZ = function of space and frequency

If you leave any off-diagonal components unspecified, they will be assumed to be zero. If you do specify an off-diagonal component function, you only need to specify either the above-diagonal or the below-diagonal component, e.g. Qxy but not also Qyx; the code will automatically set $Q_{yx}=Q_{xy}$ . If you do specify two separate functions for the two off-diagonal components of $\mathbf{Q}$ , the code will symmetrize by setting both components equal to their average.

For diagonal components, if you specify Qxx but omit specifications for Qyy and Qzz then the code will set $Q_{yy}=Q_{zz}=Q_{xx}.$

Functions and variables

The user-defined function of space and frequency in the definition of permittivity components is a character string that may refer to any of the following variables:

w (the angular frequency in units of 3e14 rad/sec)
x,y,z (cartesian coordinates of points in space)
r,Theta,Phi (spherical coordinates of points in space)

Referring to scuff-em material designations

In many cases your permittivity functions will want to refer to scuff-em material designations describing frequency-dependent homogeneous isotropic materials. You can do this by including the string MP_MATNAME in the function definition, where MATNAME is the name of the scuff-em material. For example, here is a description of a material tensor that varies continuously from 100% gold to 100% silicon dioxide as the z coordinate runs from 0 to 1:

 Eps = MP_SIO2 * z + MP_GOLD*(1-z)

Note that there is no need to refer to the frequency w here; the GOLD and SIO2 permittivities will automatically by evaluated at the correct frequency.

The definitions for the scuff-em materials (SIO2 and GOLD in this case) may appear in MATERIAL...ENDMATERIAL sections within the .SVTensor file; alternatively, they may be defined in the global database file ${HOME}/.matprop.dat or the local data file matprop.dat in the current working directory.

2. Examples of `.SVTensor` files

1. An isotropic but spatially-varying permittivity

Here's an isotropic material whose dielectric constant varies linearly as a function of the z-coordinate, from a value of $\epsilon=5+\frac{4i}{\omega}$ at $z=-5$ to a value of $\epsilon=10+\frac{8i}{\omega}$ at $z=+5$ :

 Eps = (5 + 4*i/w)*(1 + (z+5)/10 )

2. A non-isotropic permittivity

This example describes the constant permittivity tensor $\boldsymbol{\epsilon}= \left(\begin{array}{ccc} 2+3i & 0.01 & 0 \\ 0.01 & 2+3i & 0.00 \\ 0 & 0 & 4.5i \\ \end{array}\right)$ :

EpsXX=2+3i
EpsXY=0.1
EpsZZ=4+5i

3. A gold-coated SiO2 sphere

This example is used to describe a silicon dioxide sphere of radius 2 um coated with a layer of gold. Note that step refers to the Heaviside step function.

 Eps = step(2-r)*MP_SIO2 + step(r-2)*MP_GOLD

4. A gold/SiO2 Janus particle

In this case the material is gold at points above the xy plane and SiO2 at points below the xy plane.

 Eps = step(z)*MP_GOLD + step(-z)*MP_SIO2