Filter
A filter passes some wavelengths and stops others: a long-pass, short-pass or band-pass filter, or a neutral density filter that cuts every wavelength alike. It can absorb what it rejects, or reflect it, as a dichroic mirror does.
The model
A filter is a thin plane across the beam with a power transmission \(T(\lambda)\), peaking at \(T_\text{max}\).
Edges are logistic, with a 10–90 % width \(s\) (the slope) about the edge wavelength \(\lambda_e\). With \(w = s/4.394\):
\[T_\text{long}(\lambda) = \frac{T_\text{max}}{1 + e^{-(\lambda - \lambda_e)/w}}, \qquad T_\text{short}(\lambda) = \frac{T_\text{max}}{1 + e^{(\lambda - \lambda_e)/w}}.\]A band is flat-topped, an eighth-order Butterworth, with its FWHM \(\Delta\lambda\) between the half points about its centre \(\lambda_0\):
\[T_\text{band}(\lambda) = \frac{T_\text{max}}{1 + u^8}, \qquad u = \frac{2(\lambda - \lambda_0)}{\Delta\lambda}.\]A neutral density filter of optical density \(D\) transmits \(10^{-D}\) at every wavelength.
Each ray and each wavelength of a source takes the filter’s transmission at its own wavelength, in amplitude \(\sqrt{T}\). A broadband source in fibre is filtered slice by slice (see Fibre optics: broadband light). In free space a broadband beam is traced at its centre wavelength, so the filter takes that.
As a dichroic
With Reflect what it blocks on, what the filter rejects is reflected as a mirror reflects it, rather than absorbed: a dichroic mirror. The reflected share is \(1 - T(\lambda)\), and it becomes a branch of its own. That is how two wavelengths are combined or separated, for example a frequency-doubled beam from its fundamental.
Settings
| Field | Default | Meaning |
|---|---|---|
length | 25 | mm aperture. |
ftype | "longpass" | one of the filter types below. |
edge | 600 | nm: the edge of a long- or short-pass. |
slope | 10 | nm: the 10–90% width of the edge. |
centre | 550 | nm: a band-pass centre. |
fwhm | 10 | nm: a band-pass width. |
od | 1 | optical density of an "nd" filter: it transmits 10^−od. |
tmax | 95 | % peak transmission. |
reflect | false | true: what it rejects is reflected (a dichroic) rather than absorbed. |
From the tool's own tables, file format version 1: the names are those in a .jekray file.
Assumptions
- The transmission does not depend on the angle of incidence or the polarisation. A real interference filter’s edge moves to shorter wavelengths as it is tilted.
- The filter is thin: no substrate, no refraction and no added path.
Related
- Mirror, a partial reflector the same at every wavelength.
- Nonlinear crystal, whose harmonic a short-pass filter separates.