Crystal
A birefringent crystal, where s and p see different indices and go their own ways. It can be a beam displacer, a Wollaston prism or a Glan–Taylor polariser, cut from calcite, YVO₄ or α-BBO.
The model
Two indices
A uniaxial crystal has an ordinary index \(n_o\) and an extraordinary index \(n_e\), each from its own dispersion formula:
- calcite: Ghosh, 1999;
- YVO₄: Shi et al., 2001;
- α-BBO: Eimerl et al., 1987.
Light polarised across the optic axis sees \(n_o\). Light polarised in the plane of the optic axis sees an index between \(n_o\) and \(n_e\), set by the angle \(\theta\) between its wave normal and the axis:
\[\frac{1}{n^2(\theta)} = \frac{\cos^2\theta}{n_o^2} + \frac{\sin^2\theta}{n_e^2}.\]A ray entering the crystal splits into its s and p parts, and each is traced with its own index from there on. The beam tree splits the same way.
Beam displacer
The optic axis lies in the bench plane, tilted by \(\theta\) from the faces. So p is the extraordinary ray: its energy walks off sideways from its wave normal by
\[\tan\rho = \frac{(n_o^2 - n_e^2)\sin\theta\cos\theta}{n_e^2\cos^2\theta + n_o^2\sin^2\theta},\]while s goes straight. The two leave parallel, side by side, a distance \(L\tan\rho\) apart for a crystal of length \(L\).
Wollaston prism
Two wedges with their optic axes crossed: across the beam in the plane in the first, and out of the plane in the second. At the join, s and p swap between the ordinary and extraordinary index, so they refract opposite ways and leave diverging. The angle between them grows with the wedge angle and with \(\lvert n_e - n_o\rvert\).
Glan–Taylor polariser
Two prisms with an air gap cut between them, at an angle between the two critical angles \(\arcsin(1/n_o)\) and \(\arcsin(1/n_e)\). The polarisation with the higher index is totally reflected out of the side, and the other passes. So it is a polariser with a very high extinction ratio, and its rejected light leaves as a beam of its own.
Tipped and rolled
A crystal can be tipped or rolled out of the plane, and its optic axis \(\hat c\) turns with it. The light entering is then split along the crystal’s own two polarisations, not s and p:
- the ordinary one along \(\hat k \times \hat c\), across the axis and the wave normal \(\hat k\);
- the extraordinary one at right angles to it and to \(\hat k\).
The extraordinary index is taken at the wave normal’s own angle to the axis, \(n(\theta)\) above, and its energy walks off by \(\rho\) in the plane of \(\hat k\) and \(\hat c\). Rolled \(90°\), a beam displacer walks its beam up instead of across, and a Wollaston prism splits up and down. Rolled \(45°\), a Wollaston prism splits p light into two equal beams on the diagonal. The faces split the light by their Fresnel coefficients in their own planes of incidence, as an isotropic face would.
Settings
| Field | Default | Meaning |
|---|---|---|
shape | "displacer" | one of the crystal shapes below. |
crystal | "calcite" | one of the crystals below. |
size | 10 | mm aperture. |
length | 10 | mm length along the beam. |
axisAngle | 45 | degrees: the optic axis from the normal, for a displacer. |
wedge | 30 | degrees: the wedge angle, for Wollaston and Glan–Taylor. |
From the tool's own tables, file format version 1: the names are those in a .jekray file.
Assumptions
- Unturned, the optic axis lies in the bench plane, or straight out of it. Tipping or rolling the crystal turns it out of the plane; crystals cut with other axes are not modelled.
- The Glan–Taylor’s gap is cut for 633 nm. Far from it, both polarisations may pass or both reflect.
- The faces are uncoated: each reflects by its Fresnel coefficients for its own index.
Related
- Beam splitter, a polarising cube with a set leakage.
- Polariser, an ideal thin polariser.