Free spectral range (FSR) is the spacing in optical frequency or wavelength between two successive reflected or transmitted optical intensity maxima or minima of an interferometer or diffractive optical element.[1]
The FSR is not always represented by
\Delta\nu
\Deltaλ
The free spectral range (FSR) of a cavity in general is given by [2]
\left|\DeltaλFSR\right|=
2\pi | \left|\left( | |
L |
\partial\beta | |
\partialλ |
\right)-1\right|
\left|\Delta\nuFSR\right|=
2\pi | \left|\left( | |
L |
\partial\beta | |
\partial\nu |
\right)-1\right|
\Delta\betaL=2\pi
\Delta\beta
\beta=k0n(λ)=
2\pi | |
λ |
n(λ)
k0
λ
n
L
L
Given that
\left|\left( | \partial\beta |
\partialλ |
\right)\right|=
2\pi | |
λ2 |
\left[n(λ)-λ
\partialn | |
\partialλ |
\right]=
2\pi | |
λ2 |
ng
\DeltaλFSR=
λ2 | |
ngL |
,
ng
\Delta\nuFSR=
c | |
ngL |
,
c
If the dispersion of the material is negligible, i.e.
\partialn | |
\partialλ |
≈ 0
\DeltaλFSR ≈
λ2 | |
n(λ)L |
,
\Delta\nuFSR ≈
c | |
n(λ)L |
.
A simple intuitive interpretation of the FSR is that it is the inverse of the roundtrip time
TR
TR=
ngL | |
c |
=
1 | |
\Delta\nuFSR |
.
\DeltaλFSR=
λ2 | |
ngL |
,
λ
L=2l
L
l
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
λ
(λ+\Deltaλ)
\Deltaλ=
λ | |
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
\Deltaλ=
| |||||||
2nl\cos\theta+λ0 |
≈
| |||||||
2nl\cos\theta |
,
where λ0 is the central wavelength of the nearest transmission peak, n is the index of refraction of the cavity medium,
\theta
l
\Deltaf ≈
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:
l{F}=
\Deltaλ | |
\deltaλ |
=
\pi | |
2\arcsin(1/\sqrtF) |
,
where
F=
4R | |
{(1-R)2 |
This is commonly approximated (for R > 0.5) by
l{F} ≈
\pi\sqrt{F | |