Birefringent (Lyot) filter

A birefringent filter transmits light at wavelengths where a plate of birefringent crystal, placed between polarizers, returns the polarization to its original state. The plate's retardance, $\Gamma = 2\pi\,\Delta n\,d/\lambda$, varies with wavelength, so the transmission through the plate and an analyzing polarizer is a periodic function of optical frequency, with a peak whenever $\Gamma$ is a multiple of $2\pi$. A crystalline quartz plate, $\Delta n = 0.009$, of thickness 0.533 mm gives 6.0 waves of retardance at 800 nm and transmission peaks at 686, 800 and 960 nm. B. Lyot introduced stacks of such plates in the 1930s for imaging the solar corona in a narrow spectral line; the same principle, in a simpler form, is the standard coarse tuning element of continuous-wave Ti:sapphire, dye and VECSEL lasers.

Single plate

With the plate's axes at 45° to the polarizers, the transmission is

$$T = \cos^2\!\left(\frac{\pi\,\Delta n\,d}{\lambda}\right),$$

and the spacing between adjacent peaks, the free spectral range, is approximately

$$\Delta\lambda_\text{FSR} \approx \frac{\lambda^2}{\Delta n\,d}.$$

For the 0.533 mm plate this is 133 nm at 800 nm; the exact spacings, 114 nm below and 160 nm above, differ because the formula is evaluated at a single wavelength. The full width at half maximum of each peak is about half the free spectral range, 67 nm for this plate. A single plate therefore selects a broad band, which is sufficient to pick one region of a wide gain spectrum but not to fix one longitudinal mode.

Lyot stacks

A Lyot filter places several plates in series, each between parallel polarizers, with thicknesses $d$, $2d$, $4d$ and so on. Every plate transmits at the peaks of the thinnest plate, so those peaks survive, while the thickest plate narrows each of them. The total transmission is the product of the individual $\cos^2$ factors. Computed for quartz plates starting at 0.533 mm, ignoring the dispersion of $\Delta n$:

The free spectral range is set by the thinnest plate and the width by the thickest, so each added plate halves the passband. A Šolc filter achieves a similar response with identical plates at alternating azimuth angles and only two polarizers.

Intracavity birefringent filters

In a laser the separate polarizers are replaced by the plate's own surfaces. The plates are cut with the optic axis in the plane of the plate and set at Brewster's angle, so the p-polarized lasing mode passes without reflection loss, while any polarization converted by the plate is partly reflected at each surface. The loss per pass away from the peak is modest, but the light makes many round trips, and the effective selectivity is much sharper than a single-pass calculation suggests. Plate sets of one to three plates with integer thickness ratios are common. The filter is tuned by rotating it about its surface normal: this changes the angle between the optic axis and the beam, the effective birefringence and so the retardance, which sweeps the transmission peak across the gain band of the laser.

A birefringent filter is often combined with one or two etalons for single-frequency operation: the filter selects a band of a fraction of a nanometer to a few nanometers, and the etalons pick one longitudinal mode within it.

Pitfalls

• The transmission of a Brewster-angle filter has side peaks where neighboring orders come close to a full wave; on a gain medium as broad as Ti:sapphire, the laser can jump to a neighboring order if the plates are poorly matched.
• The birefringence of quartz depends on temperature and wavelength; a filter calibrated at one temperature shifts at another.
• Each plate must be oriented accurately: a small error in the axis angle leaves residual polarization rotation at the peak, which costs gain and can cause polarization instability.
• An imperfect plate surface or a wedge between plates walks the beam, so plate stacks are optically contacted or held in precision mounts.

Common questions

What is a Lyot filter used for?

Narrowband imaging in astronomy, especially of the Sun in the hydrogen-alpha line, and coarse wavelength selection inside tunable lasers. Liquid-crystal versions with electrically tuned retarders are used in multispectral cameras.

How does a birefringent filter tune a laser?

Rotating the plate about its normal changes the effective birefringence seen by the beam, so the wavelength at which the plate is a whole number of waves, and the transmission peak, moves across the gain band.

What sets the bandwidth of a Lyot filter?

The thickest plate: the passband is about half its free spectral range. The thinnest plate sets the spacing between transmitted bands.

References: A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984); A. L. Bloom, "Modes of a laser resonator containing tilted birefringent plates," Journal of the Optical Society of America 64, 447 (1974); B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019); A. E. Siegman, Lasers (University Science Books, 1986).