Free spectral range

The free spectral range (FSR) of a cavity in general is given by {{Cite book|title=Integrated Ring Resonators|last=Rabus|first=Dominik.G.|date = 26 April 2007|isbn=978-3-540-68788-7|oclc=123893382}}

:$\backslash left|\backslash Delta\backslash lambda\_\backslash text\{FSR\}\backslash right|\; =\; \backslash frac\{2\backslash pi\}\{L\}\backslash left|\backslash left(\backslash frac\{\backslash partial\; \backslash beta\}\{\backslash partial\; \backslash lambda\}\backslash right)^\{-1\}\; \backslash right|$

or, equivalently,

:$\backslash left|\backslash Delta\backslash nu\_\backslash text\{FSR\}\backslash right|\; =\; \backslash frac\{2\backslash pi\}\{L\}\backslash left|\backslash left(\backslash frac\{\backslash partial\; \backslash beta\}\{\backslash partial\; \backslash nu\}\backslash right)^\{-1\}\backslash right|$

These expressions can be derived from the resonance condition $\backslash Delta\; \backslash beta\; L\; =\; 2\backslash pi$ by expanding $\backslash Delta\; \backslash beta$ in Taylor series. Here, $\backslash beta\; =\; k\_0\; n(\backslash lambda)\; =\; \backslash frac\{2\backslash pi\}\{\backslash lambda\}n(\backslash lambda)$ is the wavevector of the light inside the cavity, $k\_0$ and $\backslash lambda$ are the wavevector and wavelength in vacuum, $n$ is the refractive index of the cavity and $L$ is the round trip length of the cavity (notice that for a standing-wave cavity, $L$ is equal to twice the physical length of the cavity).

Given that $\backslash left|\backslash left(\backslash frac\{\backslash partial\; \backslash beta\}\{\backslash partial\; \backslash lambda\}\backslash right)\; \backslash right|\; =\; \backslash frac\{2\backslash pi\}\{\backslash lambda^2\}\backslash left[n(\backslash lambda)-\backslash lambda\; \backslash frac\{\backslash partial\; n\}\{\backslash partial\; \backslash lambda\}\backslash right]\; =\; \backslash frac\{2\backslash pi\}\{\backslash lambda^2\}n\_g$, the FSR (in wavelength) is given by

:$\backslash Delta\backslash lambda\_\backslash text\{FSR\}\; =\; \backslash frac\{\backslash lambda^2\}\{n\_\backslash text\{g\}\; L\},$

being $n\_\backslash text\{g\}$ is the group index of the media within the cavity.

or, equivalently,

:$\backslash Delta\backslash nu\_\backslash text\{FSR\}\; =\; \backslash frac\{c\}\{n\_\backslash text\{g\}\; L\},$

where $c$ is the speed of light in vacuum.

If the dispersion of the material is negligible, i.e. $\backslash frac\{\backslash partial\; n\}\{\backslash partial\; \backslash lambda\; \}\backslash approx\; 0$, then the two expressions above reduce to

:$\backslash Delta\backslash lambda\_\backslash text\{FSR\}\; \backslash approx\; \backslash frac\{\backslash lambda^2\}\{n(\backslash lambda)\; L\},$

and

:$\backslash Delta\backslash nu\_\backslash text\{FSR\}\; \backslash approx\; \backslash frac\{c\}\{n(\backslash lambda)\; L\}.$

A simple intuitive interpretation of the FSR is that it is the inverse of the roundtrip time $T\_R$:

:$T\_R\; =\; \backslash frac\{n\_\backslash text\{g\}\; L\}\{c\}\; =\; \backslash frac\{1\}\{\backslash Delta\backslash nu\_\backslash text\{FSR\}\}.$

In wavelength, the FSR is given by

:$\backslash Delta\backslash lambda\_\backslash text\{FSR\}\; =\; \backslash frac\{\backslash lambda^2\}\{n\_\backslash text\{g\}\; L\},$

where $\backslash lambda$ is the vacuum wavelength of light. For a linear cavity, such as the Fabry-Pérot interferometer{{cite journal | last1 = Ismail | first1 = N. | last2 = Kores | first2 = C. C. | last3 = Geskus | first3 = D. | last4 = Pollnau | first4 = M. | year = 2016 | title = Fabry-Pérot resonator: spectral line shapes, generic and related Airy distributions, linewidths, finesses, and performance at low or frequency-dependent reflectivity | journal = Optics Express | volume = 24 | issue = 15| pages = 16366–16389 | doi = 10.1364/OE.24.016366 | pmid = 27464090 | bibcode = 2016OExpr..2416366I | url = http://kth.diva-portal.org/smash/get/diva2:948682/FULLTEXT01 | doi-access = free }}

discussed below, $L\; =\; 2\; l$, where $L$ is the distance travelled by light in one roundtrip around the closed cavity, and $l$ is the length of the cavity.

The free spectral range of a diffraction grating is the largest wavelength range for a given order that does not overlap the same range in an adjacent order. If the (m + 1)-th order of $\backslash lambda$ and m-th order of $(\backslash lambda\; +\; \backslash Delta\; \backslash lambda)$ lie at the same angle, then

:$\backslash Delta\; \backslash lambda\; =\; \backslash frac\{\backslash lambda\}\{m\}.$

In a Fabry–Pérot interferometer or etalon, the wavelength separation between adjacent transmission peaks is called the free spectral range of the etalon and is given by

:$\backslash Delta\backslash lambda\; =\; \backslash frac\{\backslash lambda\_0^2\}\{2nl\; \backslash cos\backslash theta\; +\; \backslash lambda\_0\}\; \backslash approx\; \backslash frac\{\backslash lambda\_0^2\}\{2nl\; \backslash cos\backslash theta\},$

where λ₀ is the central wavelength of the nearest transmission peak, n is the index of refraction of the cavity medium, $\backslash theta$ is the angle of incidence, and $l$ is the thickness of the cavity. More often FSR is quoted in frequency, rather than wavelength units:

:$\backslash Delta\; f\; \backslash approx\; \backslash frac\{c\}\{2nl\; \backslash cos\backslash theta\}.$

Image:Etalon-2.png

The FSR is related to the full-width half-maximum δλ of any one transmission band by a quantity known as the finesse:

:$\backslash mathcal\{F\}\; =\; \backslash frac\{\backslash Delta\backslash lambda\}\{\backslash delta\backslash lambda\}\; =\; \backslash frac\{\backslash pi\}\{2\; \backslash arcsin(1/\backslash sqrt\; F)\},$

where $F\; =\; \backslash frac\{4R\}\{\{(1\; -\; R)^2\}\}$ is the coefficient of finesse, and R is the reflectivity of the mirrors.

This is commonly approximated (for R > 0.5) by

:$\backslash mathcal\{F\}\; \backslash approx\; \backslash frac\{\backslash pi\; \backslash sqrt\{F\}\}\{2\}\; =\; \backslash frac\{\backslash pi\; R^\{1/2\}\}\{(1\; -\; R)\}.$

Category:Physical optics