Free spectral range

The free spectral range (FSR) of a cavity in general is given by ^[2]

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

or, equivalently,

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

These expressions can be derived from the resonance condition ${\displaystyle \Delta \beta L=2\pi }$ by expanding ${\displaystyle \Delta \beta }$ in Taylor series. Here, ${\displaystyle \beta =k_{0}n(\lambda )={\frac {2\pi }{\lambda }}n(\lambda )}$ is the wavevector of the light inside the cavity, ${\displaystyle k_{0}}$ and ${\displaystyle \lambda }$ are the wavevector and wavelength in vacuum, ${\displaystyle n}$ is the refractive index of the cavity and ${\displaystyle L}$ is the round trip length of the cavity (notice that for a standing-wave cavity, ${\displaystyle L}$ is equal to twice the physical length of the cavity).

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

${\displaystyle \Delta \lambda _{\text{FSR}}={\frac {\lambda ^{2}}{n_{\text{g}}L}},}$

being ${\displaystyle n_{\text{g}}}$ is the group index of the media within the cavity. or, equivalently,

${\displaystyle \Delta \nu _{\text{FSR}}={\frac {c}{n_{\text{g}}L}},}$

where ${\displaystyle c}$ is the speed of light in vacuum.

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

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

and

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

A simple intuitive interpretation of the FSR is that it is the inverse of the roundtrip time ${\displaystyle T_{R}}$:

${\displaystyle T_{R}={\frac {n_{\text{g}}L}{c}}={\frac {1}{\Delta \nu _{\text{FSR}}}}.}$

In wavelength, the FSR is given by

${\displaystyle \Delta \lambda _{\text{FSR}}={\frac {\lambda ^{2}}{n_{\text{g}}L}},}$

where ${\displaystyle \lambda }$ is the vacuum wavelength of light. For a linear cavity, such as the Fabry-Pérot interferometer^[3] discussed below, ${\displaystyle L=2l}$, where ${\displaystyle L}$ is the distance travelled by light in one roundtrip around the closed cavity, and ${\displaystyle l}$ 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 ${\displaystyle \lambda }$ and m-th order of ${\displaystyle (\lambda +\Delta \lambda )}$ lie at the same angle, then

${\displaystyle \Delta \lambda ={\frac {\lambda }{m}}.}$

In a Fabry–Pérot interferometer^[3] or etalon, the wavelength separation between adjacent transmission peaks is called the free spectral range of the etalon and is given by

${\displaystyle \Delta \lambda ={\frac {\lambda _{0}^{2}}{2nl\cos \theta +\lambda _{0}}}\approx {\frac {\lambda _{0}^{2}}{2nl\cos \theta }},}$

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

${\displaystyle \Delta f\approx {\frac {c}{2nl\cos \theta }}.}$

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

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

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

This is commonly approximated (for R > 0.5) by

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