Free Spectral RangeEdit
Free spectral range (FSR) is a foundational concept in photonics and spectroscopy, describing the spacing between successive resonant frequencies in an optical cavity or interferometer. It governs how devices that rely on standing waves or circulating light separate or resolve spectral features. In its most common form for a simple linear cavity with two mirrors, the FSR is inversely proportional to the round-trip travel time of light inside the cavity.
In a typical two-mirror cavity, the light completes a round trip of length 2nL, where L is the physical length of the cavity and n is the refractive index of the medium inside. The resonant frequencies are those for which an integral number of wavelengths fit into the round trip, leading to the spacing between adjacent resonances: - Δν ≈ c / (2 n L) Here c is the speed of light in vacuum. The corresponding spacing in wavelength near a central wavelength λ is approximately - Δλ ≈ λ^2 / (2 n L)
FSR is not limited to planar Fabry-Pérot cavities; it also applies to ring or traveling-wave resonators, where light propagates around a loop. In a ring resonator of circumference L (and with group index n_g), the fundamental spacing between resonances is - Δν ≈ c / (n_g L)
Because FSR scales inversely with the cavity length, larger devices produce smaller frequency spacings, while smaller devices produce larger spacings. The concept remains central across a range of technologies, including fiber cavities, microresonators, and integrated photonic circuits.
Physical meaning
- Spectral structure: FSR sets the fundamental spectral “beat” between neighboring standing-wave modes in a cavity. It determines how many spectral lines can appear within a given bandwidth and how easily adjacent lines can be distinguished.
- Device performance: In laser physics and cavity-enhanced spectroscopy, FSR constrains mode selection, tuning ranges, and the ability to resolve spectral features. A larger FSR can simplify mode structure and stabilization, while a smaller FSR enables finer spectroscopic detail over a narrower range.
- Relationship to finesse: The finesse F of a resonator, which characterizes how narrow its resonances are, interacts with the FSR. The ratio FSR/F roughly indicates how many resolvable lines can fit within a given spectral window. A high finesse—narrow resonances—relative to the FSR allows precise discrimination between neighboring modes.
For a practical interpretation, consider a Fabry-Pérot cavity used as a wavelength reference or a sensor. The FSR sets the spacing of spectral lines that the cavity supports, while the cavity’s linewidth (often described by its FWHM) and finesse determine how sharply those lines are defined. See also Fabry-Pérot interferometer and finesse for related concepts.
Mathematical definitions and variants
Fabry-Pérot cavities (two-mirror resonators):
- Resonant frequencies: ν_m = m c / (2 n L), with m an integer.
- Free spectral range: Δν = c / (2 n L).
- Wavelength-domain form near λ0: Δλ ≈ λ0^2 / (2 n L).
- Link to dispersion: If n or n_g varies with frequency, Δν becomes frequency-dependent; the simple constant-Δν approximation is valid only over a limited spectral range.
Ring resonators and traveling-wave cavities:
- Resonant condition: m λ = n_g L, where L is the optical path length and n_g is the group index.
- Free spectral range: Δν ≈ c / (n_g L).
- Here, dispersion and the frequency dependence of n_g cause Δν to vary with wavelength, especially in broad-band operation or high-dispersion materials.
Group index and dispersion:
- Group index n_g is defined as n_g = n - λ (dn/dλ) evaluated at the frequency of interest.
- In dispersive media, n_g ≠ n, and the FSR depends on how n and dn/dλ vary with frequency.
- See group velocity and dispersion for related ideas about how light propagation changes with frequency.
Wavelength-scale interpretation:
- In wavelength-dense devices, the spacing between resonant wavelengths is tied to Δλ via Δλ ≈ λ^2 / (2 n_g L) under modest dispersion.
Practical considerations
Design trade-offs:
- A larger cavity length L yields a smaller FSR, increasing the density of resonant lines within a given spectral window. A shorter cavity increases the FSR, helping to separate modes more visibly but potentially reducing spectral resolution in some contexts.
- The choice of material impacts n_g and dispersion, affecting FSR in ways that are important for precision sensing and laser stabilization.
Temperature and mechanical stability:
- Temperature changes alter L through thermal expansion and change the refractive index via the thermo-optic effect. Both effects shift the resonant frequencies and thus shift the FSR slightly, which matters for frequency references and high-precision spectroscopy.
- Mechanical vibrations or stresses can detune resonances and alter effective cavity length, influencing FSR stability and the observed spectrum.
Dispersion and non-idealities:
- In real devices, n_g can vary with frequency, so FSR is not perfectly constant across a broad spectrum. Designers account for this by limiting the operational bandwidth or by using dispersion-compensating structures.
- Losses, mirror reflectivity, and coupling conditions shape the observed linewidths and the effective finesse, which in turn affect how well the FSR can be exploited for spectral discrimination. See finesse and Fabry-Pérot interferometer for related details.
Applications and connections:
- In spectroscopy, FSR determines the maximum interval over which a cavity-based spectrometer can resolve spectral lines without ambiguity.
- In laser stabilization, matching a laser’s frequency to a cavity mode often relies on the known FSR to select the proper mode and to control mode hopping.
- In telecommunications, FSR is a consideration in devices like ring resonators used for wavelength division multiplexing; precise control of FSR enables predictable channel spacing. See optical communication and telecommunications for broader context.
- In sensing, cavity-enhanced methods rely on the FSR to determine how many modes can be used to interrogate a sample within a given spectral window.