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

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