Optical filter

An optical filter transmits a chosen part of the spectrum and suppresses the rest. Four spectral shapes cover most uses: longpass (transmits above a cut-on wavelength), shortpass (transmits below a cut-off), bandpass (transmits one band) and notch or band-stop (blocks one band). Neutral-density filters are the fifth category, attenuating evenly across a range instead of selecting wavelengths. Typical figures for commercial interference filters are a bandwidth of 1–3 nm for laser-line cleanup filters and 20–50 nm for fluorescence bandpass filters, peak transmittance above 90% for hard-coated designs, and out-of-band blocking of optical density 4–6, meaning $10^{-4}$ to $10^{-6}$ of the light is transmitted.

Absorptive and interference filters

Absorptive filters are glasses or polymers doped with dyes or semiconductor particles. Their edges are gradual, typically tens of nanometers wide, but their transmission hardly changes with angle of incidence, they reflect little and they are inexpensive. Absorbed power becomes heat, and some colored glasses fluoresce under short-wavelength excitation, which matters in fluorescence and Raman work.

Interference filters are stacks of dielectric thin films, often tens to over a hundred layers, whose partial reflections add or cancel by interference. They can have edges a few nanometers wide, flat passbands and very deep blocking, and they reject light mainly by reflection. A bandpass filter of this kind is a thin-film Fabry-Perot cavity, or several coupled cavities, related to the etalon. Hard-coated filters made by ion-beam or magnetron sputtering have dense layers that barely absorb water and hold their wavelength over time; older soft-coated filters shift and degrade with humidity. A dichroic mirror is an edge filter designed for 45° use, and a notch filter is the band-stop case.

Specifications

A filter datasheet normally gives the center or edge wavelength, the full width at half maximum (FWHM) of a passband, the edge steepness (often the distance from 50% transmission to the blocking level), the average or peak transmission, the blocking OD and the range over which it holds, the design angle of incidence, and the clear aperture. Transmission curves are measured with a spectrophotometer in collimated light. Blocking beyond OD 4 is hard to verify directly, because stray light and the instrument's dynamic range limit the measurement; the procedure is described in measuring optical density beyond OD 4.

Angle dependence

Tilting an interference filter moves its spectrum to shorter wavelengths. For angle of incidence $\theta$ in air,

$$\lambda(\theta) = \lambda_0\sqrt{1-\frac{\sin^2\theta}{n_\text{eff}^2}},$$

where $n_\text{eff}$ is an effective index of the layer stack, given on some datasheets and typically between about 1.5 and 2.2. For a 532 nm bandpass filter with $n_\text{eff} = 2.0$, tilting by 10° shifts the center to 530.0 nm, a change of 2.0 nm; at 15° the shift is 4.5 nm. A 1 nm filter is therefore tuned by tilt, and a filter placed in a converging beam sees a spread of angles that broadens the passband, lowers its peak and moves it to shorter wavelengths. Filters are best placed in collimated space; where that is impossible the cone half-angle should be kept to a few degrees. At oblique incidence the s and p passbands also separate.

Where filters matter

In fluorescence microscopy, an excitation filter, a dichroic and an emission filter together must suppress the excitation light by many orders of magnitude, since fluorescence is weak by comparison. In Raman spectroscopy a longpass or notch filter removes the laser line while passing Stokes-shifted light within a few hundred wavenumbers of it. Laser-line filters remove spontaneous emission and plasma lines from diode and gas lasers. In imaging and machine vision, bandpass filters matched to an LED or laser reject ambient light.

Pitfalls

Two reflective interference filters facing each other form a weak cavity; tilting one by a few degrees avoids this and keeps the reflection from returning to a laser. Stacked filters add their ODs only in principle: scattered light, pinholes and leakage paths around the edges of the mount usually set the practical limit. The back reflection from a filter can also produce ghost images. Filters have a preferred direction, usually marked on the mount, because the reflective coating should face the source, so that less light reaches, heats and excites fluorescence in any absorbing glass behind it.

Common questions

What is the difference between a longpass and a shortpass filter?

A longpass filter transmits wavelengths longer than its cut-on edge and blocks shorter ones; a shortpass filter does the opposite. The names refer to wavelength: a longpass filter passes lower frequencies.

Why does a bandpass filter pass a different wavelength than specified?

The most common causes are tilt, a converging or diverging beam, and use at a different temperature than the design. Each of these shifts or broadens the passband, as in the tilt example above.

Can two filters be stacked to get more blocking?

Yes, in principle the ODs add, so two OD 3 filters give OD 6. In practice the result is limited by stray light, reflections between the filters and light leaking around them, so stacked filters should be tilted relative to each other and baffled.

References: H. A. Macleod, Thin-Film Optical Filters, 4th ed. (CRC Press, 2010); E. Hecht, Optics, 5th ed. (Pearson, 2017), Ch. 9; J. R. Lakowicz, Principles of Fluorescence Spectroscopy, 3rd ed. (Springer, 2006), Ch. 2.