Prism
A block of glass with flat faces: a right-angle prism, an equilateral prism, a window or a Dove prism. Rays refract at each face by Snell's law in real glass, reflect by Fresnel or total internal reflection, and disperse by the glass's index against wavelength.
The model
Shapes
In the prism’s own frame, with its size \(a\):
| Shape | Faces |
|---|---|
| Right-angle prism | an isosceles right triangle, legs \(a\), the right angle at the lower left |
| Equilateral prism | an equilateral triangle of side \(a\), its apex pointing along the prism’s angle |
| Window | a plate of thickness \(t\) and height \(a\) |
| Dove prism | a trapezoid of height \(a\) and length \(4.224\,a\) (as catalogue N-BK7 ones are), its base below and its ends cut at \(45°\) |
The prism turns about its centre.
A Dove prism sends light along it in at one end, turns it back up by total internal reflection off its base and out of the other end, so the image leaves upside down across the base. Rolled about the beam by \(\rho\) (its Roll), it turns the image by \(2\rho\): a ray that went in at \(\mathbf v\) from the axis leaves at
\[\mathbf v' = 2(\mathbf v\cdot\hat m)\,\hat m - \mathbf v, \qquad \hat m = (-\sin\rho,\ \cos\rho),\]across and up, a mirror image about a line rolled \(\rho\) from straight up. Its s and p are taken from each face’s own plane of incidence, so it turns the polarisation too, as a real one does.
At each face
Each face is a flat surface between the glass and the air. Rays refract by Snell’s law, with the glass’s index at their own wavelength from its Sellmeier coefficients:
\[n_1 \sin\theta_1 = n_2 \sin\theta_2.\]Each face reflects by its Fresnel coefficients or its coating, and the reflections become branches of their own when weak reflections are followed. Beyond the critical angle, \(\sin\theta_c = 1/n\), the light is totally internally reflected. That is how a right-angle prism folds or retro-reflects a beam.
Dispersion
The angle a prism deviates light by depends on the wavelength, through \(n(\lambda)\). At minimum deviation an equilateral prism of apex angle \(A = 60°\) deviates light by \(D\), where
\[n = \frac{\sin\!\left(\frac{A + D}{2}\right)}{\sin\!\left(\frac{A}{2}\right)}.\]JEKray does not use this formula: it traces each wavelength through the real faces, so the formula comes out of the trace. For a short pulse, the optical path through the glass, at 33 wavelengths across its spectrum, gives its dispersion. Two prisms make a prism compressor.
Settings
| Field | Default | Meaning |
|---|---|---|
shape | "right" | one of the prism shapes below. |
size | 20 | mm. |
thickness | 5 | mm (for a window). |
material | {"glass":"N-BK7","abs":0} | e.g. {"glass": "N-BK7"}. |
coat | {"mode":"bare","R":4} | coating on its faces. |
From the tool's own tables, file format version 1: the names are those in a .jekray file.
Assumptions
- The faces are perfectly flat, the corners sharp, and the glass homogeneous.
- Light leaves only through the faces. It is not scattered at the corners.
Related
- Lens, for curved surfaces in glass.
- Free-space optics: short pulses, for prism compressors.