Airy formula

This article covers the formula that describes the transmission of an electromagnetic wave. For the special function see Airy function .

The Airy formula specifies the transmission of a Fabry-Pérot interferometer (FPI). For higher finesse , non-resonant light is better suppressed. The line width is approximately with the free spectral range for great finesse . ${\displaystyle {\mathcal {F}}}$ ${\displaystyle \delta }$ ${\displaystyle {\frac {\Delta \nu }{\mathcal {F}}}}$ ${\displaystyle \Delta \nu }$

The Airy formula , named after the mathematician and astronomer George Biddell Airy , gives the course of the transmitted intensity of electromagnetic radiation in a Fabry-Pérot interferometer , depending on the ratio of the wavelength or frequency of the radiation to the free spectral range of the interferometer.

The Airy formula results when the electrical fields of all partial waves circulating in the interferometer are correctly added in phase and amplitude .

Derivation

The intensity of the rays circulating in the interferometer is proportional to the transmitted intensity. During the calculation, the non-ideal reflection at the two end mirrors must be taken into account with the amplitude reflection coefficient . It is linked to the amplitude transmission coefficient via . After revolutions, i.e. reflections, the amount of the electric field is smaller by a factor . ${\displaystyle r\neq 1}$ ${\displaystyle r^{2}+t^{2}=1}$ ${\displaystyle t}$ ${\displaystyle m}$ ${\displaystyle 2m}$ ${\displaystyle r^{2m}}$

During one revolution, i. H. When a partial wave has passed through the interferometer once and back, it accumulates a phase angle (i.e. per resonator length covered ). This stage depends ${\displaystyle 2\varphi }$ ${\displaystyle 1\varphi }$ ${\displaystyle L}$

• on the ratio of the resonator length to the wavelength of the light and ${\displaystyle L}$ ${\displaystyle \lambda }$
• on the refractive index of the medium between the end mirrors. ${\displaystyle n}$

This can also be expressed as the ratio of light frequency to the free spectral range (unit frequency) of the Fabry-Pérot interferometer: ${\displaystyle \nu }$ ${\displaystyle \Delta \nu ={\frac {c}{2nL}}}$

${\displaystyle \varphi =n{\frac {2\pi L}{\lambda }}=2\pi {\frac {\nu }{\Delta \nu }}=-2\pi {\frac {\lambda }{\Delta \lambda }}}$

The electric field strength inside the resonator is ${\displaystyle E}$

${\displaystyle {\begin{aligned}E&=E_{\mathrm {i} }t\left(1+\sum _{m=1}^{m=\infty }r^{2m}\exp \left(2im\varphi \right)\right)\\&=E_{\mathrm {i} }{\frac {\sqrt {1-r^{2}}}{1-r^{2}\exp \left(2i\varphi \right)}}\end{aligned}}}$

with the field strength of the incident light. ${\displaystyle E_{i}}$

In the above calculation, the geometric series was evaluated after an index shift . The square of the absolute value of this expression gives the Airy formula with different trigonometric identities :

${\displaystyle {\begin{aligned}I=E\cdot E^{*}&={\frac {I_{\mathrm {i} }}{1-R}}\cdot {\frac {1}{1+\left({\frac {2{\sqrt {R}}}{1-R}}\right)^{2}\sin ^{2}(\varphi )}}\\&={\frac {I_{\mathrm {max} }}{1+\left({\frac {2{\mathcal {F}}}{\pi }}\right)^{2}\sin ^{2}(\varphi )}}\end{aligned}}}$

In this intensity display, the following are used:

• the reflection coefficient ${\displaystyle R=r^{2}}$
• the transmission coefficient ${\displaystyle T=t^{2}}$
• the finesse . ${\displaystyle {\mathcal {F}}={\frac {\pi {\sqrt {R}}}{1-R}}}$

See also

• Fresnel formulas