7. Technical details¶
7.1. Specifying and using anistropic materials¶
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7.1.1. Orientation of the coordinate axes in NumBAT¶
Cartesian coordinates in NumBAT are defined so that the waveguide lies in the \(x-y\) plane with propagation along \(z\). To obtain a right-handed system, one should think of the propagation as being out of the screen (though it is rare that this matters) :
\(x\) : Increases to the right. Usually the dominant electric field component for modes labelled as TE polarisation.
\(y\) : Increases up the screen. Usually the dominant electric field component for modes labelled as TM polarisation.
\(z\) : Increases out of the screen. Propagation direction.
It is common to encounter different coordinate choices in the literature, in particular the bird’s-eye view corresponding to viewing a photonic circuit from above is frequently encountered:
\(x\): Increases to the right. Usually the dominant electric field component for modes labelled as TE polarisation.
\(y\): Increases out of the screen. Propagation direction.
\(z\): Increases up the screen. Usually the dominant electric field component for modes labelled as TM polarisation.
When dealing with elastically anisotropic materials, it is important to understand the relationship between the NumBAT coordinate system and that of any literature source you may be consulting, as it may be necessary to perform a rotation on the tensor properties of the material in question. This is discussed below.
7.1.2. Supported crystal classes¶
Depending on the degree of crystal symmetry of a material, the optical and elastic response tensors obey constraints that can substantially reduce the number of independent components.
NumBAT currently supports the following crystal classes: Isotropic, Cubic, Trigonal.
Note that this choice merely reflects what has been useful to the authors and new classes are easily added as needed. Please get in touch.
The number and location of the independent elements of each tensor — stiffness tensor \(c_{ijkl}\), photolelastic tensor \(p_{ijkl}\), viscosity tensor \(p_{ijkl}\) — depends on the particular crystal class.
As all the relevant tensors are symmetric, there can be at most 6 independent elements for rank-2 tensors (rather than 9), and 36 independent elements in rank-4 tensors (rather than 81). In particular, tensor subscripts are all symmetric in pairs as follows:
where the subscripts range over all values of \(x,y,z\).
Consequently we can use the Voigt notation in which pairs of indices are represented by a single integer in the range 1 to 6 as follows:
Then a rank two tensor like strain or stress can be represented as the column
A fourth rank tensor like the stiffness or photoelastic tensor is represented in the form
This table summarises the symmetry properties for each class and the required tensor elements that must be specified in a material .json file.
Structure |
Nature |
Examples |
\(\epsilon\) |
Stiffness elements |
Photoelastic elements |
|---|---|---|---|---|---|
Isotropic |
Every direction equivalent |
Glass |
\(\epsilon_1\) |
\(c_{11}\) |
\(p_{11}, p_{12}, p_{14}\) |
Cubic |
3 equivalent perpendicular directions |
Silicon |
\(\epsilon_1\) |
\(c_{11}, c_{12}, c_{44}\) |
\(p_{11}, p_{12}, p_{44}\) |
Trigonal |
1 threefold rotation axis |
LiNbO3 |
\(\epsilon_1, \epsilon_3\) |
\(c_{11}, c_{12}, c_{13}, c_{14}, c_{33}, c_{44}\) |
\(p_{11}, p_{12}, p_{13}, p_{14}, p_{31}, p_{33}, p_{41}, p_{44}\) |
The full form of the material tensors for each crystal class is as follows:
Isotropic
where \(c_{44} = (c_{11}-c_{12})/2\) and \(p_{44} = (p_{11}-p_{12})/2\). These quantities are related to the Lame parameters \(\mu=c_{44}\) and \(\lambda=c_{11}\) which may in turn be expressed in terms of the Young’s modulus \(E\) and Poisson ratio \(\nu\):
Cubic
The matrix expressions are identical to the isotropic case except that \(c_{44}\) and \(p_{44}\) are now independent quantities that must be specified directly.
Trigonal
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7.1.3. Required tensor components¶
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