Lyot filter

The Lyot filter , named after its inventor, the French astronomer Bernard Ferdinand Lyot , is an optical filter that uses birefringence to generate a narrow pass band of the transmitted wavelengths . The field of application of the Lyot filter is astronomy , laser physics , in order to realize tunable lasers, as well as optical data transmission .

construction

A Lyot filter consists of a birefringent crystal, usually quartz , followed by a polarizing filter .

To increase the free spectral range , several Lyot filters are connected in series. The thickness of the crystal plates is halved for each subsequent filter.

Physical principle

Principle sketch of a Lyot filter. See text for an explanation

Due to the birefringent properties of the plates, the ordinary and extraordinary components of a light beam are subject to different refractive indices and therefore have different phase velocities . For different wavelengths, this leads to different phase differences between ordinary and extraordinary rays after passing through the crystal. Looking at linearly polarized light that hits the filter, the light is generally elliptically polarized through the plate . The light behind the filter is only linearly polarized in the same way ( a natural number) if the phase difference between the two partial beams corresponds . This is only the case at certain wavelengths. ${\displaystyle \delta }$ ${\displaystyle \delta =m\cdot 2\pi }$ ${\displaystyle m}$

The time- dependent field strength of the field propagating in the x direction with angular frequency and magnitude of the wave vector is . This is broken down into the components parallel to the optical axis (extraordinary ray) and perpendicular to the optical axis (ordinary ray) ${\displaystyle t}$ ${\displaystyle \omega }$ ${\displaystyle k=\left|{\vec {k}}\right|}$ ${\displaystyle {\vec {E}}(x,t)={\vec {E}}_{0}\cos(\omega t-kx)}$

${\displaystyle {\vec {E_{0}}}=E_{0}\cos(\alpha ){\vec {e}}_{z}+E_{0}\sin(\alpha ){\vec {e}}_{y}=E_{0z}{\vec {e}}_{z}+E_{0y}{\vec {e}}_{y}}$

where the unit vector in the x-direction is parallel to the direction of propagation, parallel to the optical axis and is the angle that the polarization plane of the light and the optical axis enclose (see figure). If the birefringent crystal is placed in the beam path so that it begins at and ends at , the field strength behind the crystal is through ${\displaystyle {\vec {e}}_{x}}$ ${\displaystyle {\vec {e}}_{z}}$ ${\displaystyle \alpha }$ ${\displaystyle x=0}$ ${\displaystyle x=L}$

${\displaystyle {\vec {E}}(t,L)=E_{0z}\cos(\omega t-kn_{a}L){\vec {e}}_{z}+E_{0y}\cos(\omega t-kn_{o}L){\vec {e}}_{y}}$

described. Here is the refractive index of the ordinary ray and the refractive index of the extraordinary ray. ${\displaystyle n_{o}}$ ${\displaystyle n_{a}}$

By comparing with the field strength before hitting the crystal, the phase difference of the two partial beams follows:

${\displaystyle {\begin{aligned}\delta &=k&\left(n_{o}-n_{a}\right)L\\&={\frac {2\pi }{\lambda }}&\left(n_{o}-n_{a}\right)L\end{aligned}}}$

After passing through the crystal, the light is only in the same state of polarization as at the point of incidence if the phase difference is an integral multiple of : ${\displaystyle 2\pi }$

${\displaystyle {\begin{aligned}\delta &=m\cdot 2\pi \\{\frac {2\pi }{\lambda }}(n_{o}-n_{a})L&=m\cdot 2\pi \\\lambda &={\frac {L(n_{o}-n_{a})}{m}}\end{aligned}}}$

The following polarization filter attenuates all parts of the light whose wavelength does not meet the above condition. The Lyot filter is therefore a wavelength-dependent optical filter.

A quantitative statement can also be made about the transmitted portion. Let us now be the angle between the optical axis of the birefringent crystal and that of the subsequent polarization filter, under which the linearly polarized light emerging from the crystal is optimally transmitted (maximum transmission). The polarization filter rotated around any desired then only lets through the component . This corresponds to an intensity ${\displaystyle \varphi =0}$ ${\displaystyle {\vec {E}}}$ ${\displaystyle \varphi }$ ${\displaystyle |{\vec {E}}|\cdot \cos(\varphi )}$

${\displaystyle I(\varphi )=\left|{\vec {E}}(\varphi )\right|^{2}=|{\vec {E}}|^{2}\cos ^{2}(\varphi ).}$

The intensity transmission coefficient, which is defined as the ratio of the incident intensity to the output intensity of the filter, is then ${\displaystyle I_{0}}$ ${\displaystyle I}$ ${\displaystyle T={\frac {I}{I_{0}}}}$

${\displaystyle T(\lambda )=\cos ^{2}\left({\frac {\pi L(n_{o}-n_{a})}{\lambda }}\right)}$

or depending on the light frequency ${\displaystyle \nu ={\frac {\omega }{2\pi }}}$

${\displaystyle T(\nu )=\cos ^{2}\left({\frac {\pi L(n_{o}-n_{a})\nu }{c}}\right).}$

The free spectral range of the filter results from the distance between two maxima ${\displaystyle \Delta \nu }$

${\displaystyle \Delta \nu ={\frac {c}{L(n_{o}-n_{a})}}.}$

Cascading

Transmission of Lyot filters connected in series. The thickness of the birefringent crystal is halved with each subsequent filter

The total transmission of filters connected in series is obtained by multiplying the individual transmissions : ${\displaystyle M}$ ${\displaystyle T_{m}}$

${\displaystyle T_{tot}(\lambda )=\prod \limits _{m=1}^{M}T_{m}(\lambda )}$

In the picture opposite, four Lyot filters were connected in series. The thickness of the plates (birefringent crystal) was halved for each additional filter.

Tunability

The transmitted wavelengths of the Lyot filter are through the thickness of the crystal, and or set, the refractive indices of the ordinary and extraordinary beams of birefringent material. If these parameters are changed, the pass band of the filter changes. ${\displaystyle L}$ ${\displaystyle n_{o}}$ ${\displaystyle n_{a}}$

The easiest way to detune the Lyot filter is to turn the crystal around the z-axis, which leads to a change in . For example, if it is a cube-shaped crystal, it is minimal if the light hits a side surface perpendicularly. If the crystal is rotated around the z-axis, the light has to travel a greater distance in the crystal, which leads to a change in the phase difference of the two partial beams and thus to a change in the filter's pass band. ${\displaystyle L}$ ${\displaystyle L}$

By rotating the crystal by the angle around the x-axis, the transmission maximum of the Lyot filter changes, since it is independent but dependent on ( refractive index ellipsoid ). ${\displaystyle \vartheta }$ ${\displaystyle \lambda _{m}={\frac {L(n_{o}-n_{a})}{m}}}$ ${\displaystyle n_{o}}$ ${\displaystyle n_{a}}$ ${\displaystyle \vartheta }$

The use of electrically changeable birefringence elements (e.g. liquid crystals ) results in an “electrically tunable Lyot filter”. By varying the field strength of an external electric field, the refractive index of special crystals such as KDP ( potassium dihydrogen phosphate ) changes due to the electro-optical effect . This in turn results in a detunable Lyot filter, the tunable range being small.

literature

• Wolfgang Demtröder : Laser Spectroscopy. Basics and Techniques. 4th, expanded and revised edition. Springer, Berlin a. a. 2000, ISBN 3-540-64219-6 .