Lyot filter

This section describes how the Lyot filter's wavelength dependent transmission of light arises from birefrigence.

The Lyot filter relies on light's polarization property, a vector (Jones vector) perpendicular to light's path that has a fixed direction (linear polarization) or a time-varying rotating direction (circular or elliptical polarization). For a typical single-plate Lyot filter, light passes through three consecutive optical elements that modify the light's polarization: the first horizontal polarizer, a waveplate (retarder) and a second horizontal polarizer.^{[10][11]: 95–96} Linearly polarized light travels fastest when aligned with the waveplate's fast F direction, and slowest when aligned with the waveplate's orthogonal slow S direction.^{[12]: 125 : 127} The speed difference depends on the difference between the waveplate's ordinary refractive index and extraordinary refractive index.^{[11]: 95–96} This example assumes that the horizontal makes a 45-degree angle with the waveplate's F and S directions.^{[11]: 95–96}

The first horizontal polarizer transforms the incoming light's polarization to horizontally polarized light by passing only the incoming light's horizontal polarization component. The waveplate may modify the incoming horizontally polarized light to a different polarization based on the light's wavelength. The second horizontal polarizer passes only the horizontal polarization component of the light exiting the waveplate. For example, at one wavelength, if the light exiting the waveplate is horizontally polarized, then the light passes through the second horizontal polarizer fully, exiting the optical filter with no attenuation. At a different wavelength, if the light exiting the waveplate is vertically polarized, then no light passes through the second horizontal polarizer, and no light exits the optical filter. At most wavelengths, some wavelength-dependent attenuation will occur.

This single-plate optical filter transmits light intensity ${\displaystyle I_{T}}$  from an input of horizontally polarized light intensity ${\displaystyle I_{X}}$  with wavelength ${\displaystyle \lambda }$ , waveplate thickness ${\displaystyle d}$ , waveplate ordinary refractive index ${\displaystyle n_{o}}$  and waveplate extraordinary refractive index ${\displaystyle n_{e}}$ :^{[11]: 95–96 [10]}

${\displaystyle I_{T}=I_{X}\cdot \operatorname {cos} ^{2}\left({\frac {\pi (n_{o}-n_{e})d}{\lambda }}\right)}$ 

Multiplate filters are a series of consecutive single-plate filters, with each waveplate half the thickness of the preceding plate.^[10] Using this design, a graph describing the transmitted light intensity at each wavelength will show sharper major peaks (narrower bandwidth) of transmitted light and a greater wavelength interval between the major peaks of transmitted light (free spectral range).^{[11]: 95–96 [1]: 108} As an example, extending the single-plate equation to a 3-plate optical filter with maximum waveplate thickness ${\displaystyle d}$ , this multiplate optical filter transmits light intensity ${\displaystyle I_{T}}$  from an input of horizontally polarized light intensity ${\textstyle I_{X}}$ :^[10]

${\displaystyle I_{T}=I_{X}\cdot \operatorname {cos} ^{2}\left({\frac {2\pi (n_{o}-n_{e})d}{2\lambda }}\right)\cdot \operatorname {cos} ^{2}\left({\frac {2\pi (n_{o}-n_{e})d}{2^{2}\lambda }}\right)\cdot \operatorname {cos} ^{2}\left({\frac {2\pi (n_{o}-n_{e})d}{2^{3}\lambda }}\right)}$ 

The waveplates are commonly quartz or calcite.^[1]: 109 Rotating the waveplate may shift the wavelength of the transmission peaks.^[4]: 2.5.2 Splitting the crystals in half and adding a 1⁄2 waveplate in the middle increases the filter's field of view.^[10] The separation and narrowness of the transmission peaks depends on the number, thicknesses, and orientation of the plates.^[12]: 125 Due to the temperature dependent birefringent properties of quartz and calcite, the Lyot filter requires a thermostat to minimize temperature fluctuations.^[1]: 109

An electrically tunable Lyot filter contains tuneable electro-optic or liquid crystal birefringent elements. ^[13]: 30 The tunable electro-optic Lyot filter uses lead magnesium niobate-lead titanate (PMN-PT) opto-ceramic to tune the filter.^[6]: 5.8.3 Liquid crystal tunable filters allow analog tuning of the transmitted wavelength by carefully adjusting the voltage over the liquid crystal cells. Liquid crystal Lyot filter spectral bandpass may range from 30 nm to 0.05 nm.^[3]: 177 The two categories of Lyot liquid crystal filters are polarizing interference retardance filters and electro-optical photonic crystals.^[5]: 167 Often these filters are based on the original Lyot design, but many other designs exist in order to achieve other properties such as narrow or broad band transmission, or polarization selectivity.^[14]

The Lyot filter and Fabry-Perót filter are the most common tunable electro-optic filters.^[6]: 5.8.3 In comparison to the Fabry-Perót filter, the tunable Lyot filter has broader and more stable adjustable range, but the Lyot filter transmits less light.^[6]: 5.8.3 Poor transmittance occurs due to the large number of highly absorbing polarizers and imperfect waveplate action. ^[3]: 177 Lyot filters may contain up to 12 individual filters, making the Lyot filter expensive, limiting its use in compact instruments.^[5]: 166 In contrast to Lyot filters, the Solc filter relies on only two polarizers, leading to less light reduction.^[10]

In solar astronomy, viewing the sun's chromosphere, the sun's second atmospheric layer, requires narrowband optical filters (spectroheliograph), such as a Lyot filter, using wavelengths for viewing solar flares, prominences, filaments, and plages arising from calcium and hydrogen.^{[8]: 327 [9]: 14}

Single- and multi-plate Lyot filters are often used inside the optical cavity of lasers to allow tuning of the laser.^[7]: 202 In this case, Brewster losses from the plate and other intracavity elements are usually sufficient to produce the polarizing effect, and no additional polarizers are required. Lyot filters are used also in broadband Ti—sapphire lasers, and dye laser oscillators for wavelength selection.^[11]: 96

Although their mechanisms are different, modelocking lasers and Lyot-filter lasers both produce a comb of multiple wavelengths which can be placed on the ITU channel grid for dense wave division multiplexing (DWDM) or used to give each suburban home its own return-signal laser wavelength in a passive optical network (PON) used to provide FTTH (Fiber To The Home).^[6]: 5.8.3

Another application has been use of Lyot filters is for Raman chemical imaging.^[15]: 205 Other applications have been in microspectrometer and hyperspectral imaging devices and biomedical photonics.^[5]: 163

• Lyot stop
• Lyot depolarizer
• Liquid crystal tunable filter

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