# How does an etalon work

## Characteristic Parameters of an Etalon

The performance of etalon is characterized by several main parameters: including visibility (V), free spectral range （FSR), full width half maximum (FWHM), and central wavelength.

Suppose that the $a_{1}$ , $a_{2}$ , $a_{3}$ are the electrical field intensity of input, oscillating, output light. The relationship is as follows

$a_{2}=ta_{1}+r^{2}a_{2}e^{-i\phi }$ $a_{3}=ta_{2}$ Where \phi means that phase delay in the cavity and the gap between two reflective surfaces is d.

$\phi =2kd=2{\frac {2\pi }{\lambda }}d$ From the upper two equations, we have

$a_{2}=a_{1}{\frac {t}{1-r^{2}e^{-i\phi }}}$ $a_{3}=a_{1}{\frac {t^{2}}{1-r^{2}e^{i\phi }}}$ Then, we can calculate the transmission ratio:

$T_{R}=|{\frac {a_{3}}{a_{1}}}|=|{\frac {t^{2}}{1-r^{2}e^{i\phi }}}|^{2}={\frac {t^{4}}{(1-r^{2}e^{i\phi })^{2}+(r^{2}sin\phi )^{2}}}={\frac {t^{4}}{1-r^{2}-2rcos\phi }}$ Here, we have $cos\phi =1-2sin^{2}{\frac {\phi }{2}}$ So, $T_{R}$ can be simplified as follows

$T_{R}={\frac {t^{4}}{(1-r^{2})^{2}+4r^{2}sin^{2}{\frac {\phi }{2}}}}={\frac {1}{1+{\frac {4r^{2}}{t^{4}}}sin^{2}{\frac {\phi }{2}}}}$ Here, we define a new parameter: $F={\frac {4r^{2}}{t^{4}}}$ , which is called coefficient finesse.

So, $T_{R}={\frac {1}{1+Fsin^{2}{\frac {\phi }{2}}}}$ Since we already have the expression of transmission (T), we can also derive some important parameters:

1.Visibility

The interferometric visibility quantifies the contrast of interference in an optical system. The ratio of the amplitude of oscillations to the sum of the powers of the individual waves is defined as the visibility.

Assume $I_{max}$ , $I_{min}$ are the maximum intensity of the oscillations and the minimum intensity of the oscillations, $V$ is the visibility of the interference pattern.

$V={\frac {I_{max}-I_{min}}{I_{max}+I_{min}}}$ Suppose the intensity of incident light of etalon is $I$ ,the minimum transmission is $T_{min}$ , the maximum transmission is $T_{max}$ , we can rewrite the visibility

$V=I{\frac {T_{max}^{2}-T_{min}^{2}}{T_{max}^{2}+T_{min}^{2}}}$ $T_{min}={\frac {1}{1+0}}$ , $T_{max}={\frac {1}{1+F}}$ $V={\frac {(1+F)^{2}-1}{(1+F)^{2}+1}}$ This indicate the visibility of interference pattern is associated with coefficient finesse. When $V_{max}=1$ , $F$ is approximately equal to infinite, get the best interference pattern; when $V_{min}=0$ , $F=0$ , can’t observe the interference pattern.

2.Free spectral range(FSR)

The free spectral range(FSR) of a cavity, in general, is given by

$|\Delta \lambda _{FSR}|={\frac {2\pi }{L}}|{\frac {1}{\frac {\partial \beta }{\partial \lambda }}}|$ Where $\beta$ is the wavevector of the light inside the cavity, $\beta =\kappa _{0}n(\lambda )={\frac {2\pi }{\lambda }}n(\lambda )$ . $\kappa _{0}$ and $\lambda$ are the wavevector and wavelength in vacuum, $n$ is the refractive index of the cavity, $L$ is the length of the cavity(for a standing-wave cavity, $L$ is equal to twice the physical length of the cavity)

$|{\frac {\partial \beta }{\partial \lambda }}|={\frac {2\pi }{\lambda ^{2}}}[n(\lambda )-\lambda {\frac {\partial n}{\partial \lambda }}]={\frac {2\pi }{\lambda ^{2}}}n_{g}$ The FSR is $\Delta \lambda _{FSR}={\frac {\lambda ^{2}}{n_{g}L}}$ , $n_{g}$ is the group index of the media within the cavity.

In etalon, the FSR is $\Delta \lambda _{FSR}={\frac {\lambda _{0}^{2}}{2nl\cos \theta }}$ Where $\lambda _{0}$ is the central wavelength of the nearest transmission peak, $n$ is the index of refraction of the cavity, $l$ is the thickness of the cavity, $\theta$ is the angle of incidence.

3. Full width at half maximum

The full width at half maximum (FWHM) is a parameter commonly used to describe the width of a "bump" on a curve or function. It is given by the distance between points on the curve at which the function reaches half its maximum value.

4.Central wavelength

Central Wavelength, used in defining bandpass filters, describes the midpoint of spectral bandwidth over which the filter transmits.

5.The relationship between FSR and FWHM

The FSR is related to the full-width half-maximum $\delta \lambda$ of any one transmission band by a quantity known as the finesse

${\mathcal {F}}={\frac {\Delta \lambda }{\delta \lambda }}={\frac {\pi }{2\arcsin {\frac {1}{\sqrt {F}}}}}$ In a word, If we want to observe more clear interference pattern, we should make $F$ large, or make $\Delta \lambda (FSR)$ large and $\delta \lambda (FWHM)$ small.