# Free Spectral Range

Author: the photonics expert Dr. Rüdiger Paschotta

Definition: frequency spacing of the axial modes of an optical resonator

Alternative term: axial mode spacing

Units: Hz

Formula symbol: <$\Delta \nu$>

The free spectral range of an optical resonator (cavity) is the spacing of its axial (Gaussian-shaped) resonator modes in terms of optical frequency. It is also called *axial mode spacing*. For an empty standing-wave resonator of length <$L$>, it can be calculated as

For a ring resonator, the factor of 2 is removed, since there is no double pass.

For a standing-wave resonator filled with a dispersive medium, the free spectral range is determined by the group index, rather than by the ordinary refractive index:

$$\Delta \nu = \frac{c}{{2{n_{\rm{g}}}L}}$$This follows from the fact that the round-trip phase shift <$2 k L = 2 L n \omega /c$> must change by <$2\pi$> from one resonance to the next one, and only through the group index one considers the full frequency dependence of the round-trip phase shift (including the frequency dependence of the refractive index). Due to chromatic dispersion, the group index can substantially deviate from the refractive index, and it generally is frequency-dependent.

For a waveguide resonator, one would have to calculate the group index using the frequency-dependent effective refractive index.

More generally, e.g. for an optical resonator containing different transparent media, the free spectral range is the inverse of the round-trip time (more precisely, the round-trip group delay) of a light pulse.

The free spectral range of a Fabry–Pérot interferometer (or a Lyot filter) often limits the optical frequency range in which it can be used as a spectrometer. A large free spectral range can thus be desirable. It can be obtained simply by making the resonator shorter – which, however, also leads to a larger bandwidth of the resonances, thus to poorer spectral resolution, as long as the same mirrors are used. For better resolution, one then needs to increase the finesse by minimizing the round-trip power losses of circulating light.

For a wavelength-tunable single-frequency laser, the free spectral range often (but not always) limits the achievable mode-hop-free tuning range.

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2023-01-18

What should be the FSR and finesse of a Fabry Perot scanning interferometer to scan a 1 nm wide laser line?

The author's answer:

The FSR should then be several nanometers, and the finesse high enough to obtain the required spectral resolution.