Free spectral range

The free spectral range (FSR), also known as the mode spacing, is the frequency separation between consecutive resonant modes in an optical cavity or the interval between adjacent interference maxima in a Fabry-Pérot interferometer, representing the range over which the device can uniquely resolve spectral features without ambiguity.^[1]^[2]^[3] In a Fabry-Pérot cavity consisting of two parallel mirrors separated by a distance L, the FSR in frequency \Delta \nu is given by \Delta \nu = \frac{c}{2nL}, where c is the speed of light in vacuum and n is the refractive index of the medium between the mirrors (often n=1 for air).^[3]^[4] For confocal configurations, where the mirror separation equals the radius of curvature, the effective FSR adjusts to \Delta \nu = \frac{c}{4L} due to the doubled optical path in the round trip.^[2]^[5] In wavelength terms, the FSR is \Delta \lambda = \frac{\lambda^2}{2nL}, highlighting its inverse dependence on cavity length.^[4] The FSR plays a critical role in determining the performance of optical devices, as it sets the scale for spectral resolution in applications like laser mode selection, high-precision spectroscopy (e.g., measuring Zeeman shifts), and optical filtering in telecommunications.^[4]^[6] It is also integral to the finesse F of a resonator, defined as F = \frac{\Delta \nu}{\delta \nu}, where \delta \nu is the full width at half maximum of a resonance peak, quantifying the cavity's sharpness and ability to distinguish closely spaced frequencies.^[3] In modern contexts, such as microresonators and photonic integrated circuits, tuning the FSR enables compact devices for wavelength-division multiplexing and sensing.^[7]^[8]

Fundamentals

Definition

The free spectral range (FSR) is defined as the frequency or wavelength interval between consecutive maxima or minima in the transmission or reflection spectrum of periodic optical structures, such as resonators or diffractive elements.^[9]^[10] This periodic spacing arises from the round-trip phase accumulation in resonant cavities or the periodic nature of diffraction, resulting in constructive interference conditions at regular intervals.^[9] Physically, the FSR represents the inverse of the round-trip group delay in cavities, determining the separation of supported modes.^[9] The FSR is typically expressed in the frequency domain as Δν (in Hz) for resonators, reflecting the mode spacing, or in the wavelength domain as Δλ (in nm) for applications like grating spectrometers, where it denotes the range before order overlap.^[9]^[10] For small spectral ranges, the two are related approximately by Δλ ≈ (λ² / c) Δν, where λ is the central wavelength and c is the speed of light in vacuum. The term "free spectral range" originated in the context of interferometry during the early 20th century, following the development of the Fabry-Pérot interferometer in 1899, with initial applications in high-resolution spectroscopy to measure spectral line separations.^[11] The FSR holds fundamental significance in optics, as it sets the operational bandwidth over which devices like spectrometers and optical filters can function without mode overlap or ambiguity, while also imposing limits on spectral resolution that scale inversely with cavity length or grating parameters.^[9] In resonant cavities, for instance, a larger FSR enables broader tuning ranges in lasers but requires higher finesse to maintain resolution.^[9]

Mathematical Formulation

The free spectral range (FSR) of an optical cavity arises from the condition for constructive interference after a round trip, where the round-trip phase shift is an integer multiple of $2\pi. For a cavity with round-trip geometric length L, the resonance condition is m \lambda = n L, with m an integer mode number, \lambda the wavelength, and n the phase refractive index of the medium.^[9]^[3] To derive the frequency spacing, express the resonance in terms of optical frequency \nu = c / \lambda, yielding \nu_m = m c / (n L), assuming n is constant for adjacent modes. For dispersive media where n = n(\nu), differentiate the phase condition \delta = 2\pi \nu n(\nu) L / c = 2\pi m:

\frac{d\delta}{d\nu} = 2\pi \frac{L}{c} \left( n + \nu \frac{dn}{d\nu} \right) = 2\pi \frac{dm}{d\nu}^{-1}.

The mode spacing \Delta \nu_\text{FSR} for \Delta m = 1 is thus

\Delta \nu_\text{FSR} = \frac{c}{n_g L},

where n_g = n - \lambda \, dn/d\lambda = n + \nu \, dn/d\nu is the group refractive index, accounting for dispersion.^[9]^[12]^[1] The corresponding wavelength FSR follows from the relation \Delta \lambda / \lambda \approx - \Delta \nu / \nu under the approximation \Delta \nu / \nu \ll 1, giving

\Delta \lambda_\text{FSR} = \frac{\lambda^2}{n_g L}.

This expression highlights the inverse scaling with cavity size and group index.^[9]^[3] Intuitively, the FSR equals the inverse of the round-trip group delay T_R = n_g L / c, so \Delta \nu_\text{FSR} = 1 / T_R. This periodicity reflects the cavity's response repeating every time the input frequency advances by one full cycle relative to the round-trip time.^[9]^[12] In dispersive media, n_g typically exceeds the phase index n under normal dispersion (dn/d\lambda > 0), leading to a smaller FSR than predicted by n alone; this effect is pronounced in broadband applications where mode spacing varies across the spectrum.^[12]^[9] For units, \Delta \nu_\text{FSR} is in Hz with c = 3 \times 10^8 m/s, n_g dimensionless, and L in meters; \Delta \lambda_\text{FSR} is in meters (often converted to nm). For a linear cavity with mirror separation d = 0.1 mm (so L = 2d = 0.2 mm) in GaAs (n_g \approx 3.6) at \lambda = 808 nm, \Delta \nu_\text{FSR} \approx 417 GHz and \Delta \lambda_\text{FSR} \approx 0.9 nm. For a longer air-filled cavity (n_g \approx 1, L = 3 m), \Delta \nu_\text{FSR} = 100 MHz and \Delta \lambda_\text{FSR} \approx 0.80 pm at 1550 nm.^[3]^[9]

Resonant Cavities

Fabry–Pérot Interferometer

The Fabry–Pérot interferometer consists of two parallel highly reflective mirrors separated by a fixed distance l, with the space between them filled by a medium of refractive index n, forming a linear resonant cavity that supports standing waves through multiple reflections.^[13] Light incident on the cavity at an angle \theta undergoes interference, producing sharp transmission or reflection peaks at resonant frequencies where the round-trip phase shift is an integer multiple of $2\pi. This geometry enables high-resolution spectral analysis, with the cavity length l typically ranging from millimeters to centimeters depending on the desired resolution.^[13] The free spectral range (FSR) in frequency for the Fabry–Pérot interferometer is given by

\Delta \nu_{\text{FSR}} \approx \frac{c}{2 n l \cos \theta},

where c is the speed of light in vacuum and \theta is the angle of incidence inside the cavity.^[14] This formula adapts the general expression for the FSR of optical cavities, \Delta \nu_{\text{FSR}} = c / L, by defining the effective round-trip optical path length as L = 2 n l \cos \theta, which accounts for the oblique propagation reducing the effective path compared to normal incidence (\theta = 0).^[14] Equivalently, the FSR in wavelength, centered at a reference wavelength \lambda_0, is

\Delta \lambda_{\text{FSR}} \approx \frac{\lambda_0^2}{2 n l \cos \theta}.

This wavelength form arises from the relation \Delta \lambda \approx (\lambda^2 / c) \Delta \nu, emphasizing how smaller cavities or higher indices yield larger FSR values, limiting the unambiguous spectral range to one FSR without order ambiguity.^[9] The finesse F quantifies the sharpness of the resonances and is defined as the ratio of the FSR to the full width at half maximum (FWHM) of a resonance peak, F = \Delta \nu_{\text{FSR}} / \Delta \nu_{\text{FWHM}}.^[15] For mirrors with power reflectivity R > 0.5, the finesse is approximated by

F \approx \frac{\pi \sqrt{R}}{1 - R},

which increases with higher reflectivity, enabling resolutions exceeding $10^5 for R \approx 0.999, though limited by mirror imperfections and alignment.^[15] This relation highlights how finesse sets the interferometer's ability to resolve closely spaced spectral lines within the FSR, with the product F \times m determining the overall resolving power, where m is the mode number. Practical implementations require careful control of cavity parameters, as the FSR is highly sensitive to changes in length l or refractive index n, which can be tuned mechanically or via piezo actuators for scanning.^[16] Temperature fluctuations induce thermal expansion and index variations, shifting the FSR; for instance, fused silica etalons show a sensitivity of approximately 3600 MHz/°C, necessitating stabilization to within 0.001°C for sub-MHz precision in applications like Doppler wind lidars.^[16] Length stability is equally critical, with vibrations or misalignment altering \theta and broadening peaks, often mitigated by mounting on low-expansion materials like Invar or Zerodur.^[17] Unlike optical ring resonators, which use continuous looped waveguides without discrete mirrors, the Fabry–Pérot's linear design introduces explicit angular dependence but simplifies fabrication for bulk optics.^[13]

Optical Ring Resonators

Optical ring resonators are compact photonic devices consisting of a closed-loop waveguide, such as a microring with radius R and circumference L = 2\pi R, evanescently coupled to straight input and output waveguides for light injection and extraction. Unlike linear cavities, the ring forms a traveling-wave resonator where light circulates unidirectionally, enabling integration into planar photonic circuits. Resonance in an optical ring resonator occurs when the accumulated phase shift over one round trip equals an integer multiple of $2\pi, expressed as $2\pi m = \beta L, where m is the mode number, \beta = 2\pi n_\mathrm{eff} / \lambda is the propagation constant, n_\mathrm{eff} is the effective refractive index, \lambda is the wavelength, and L is the round-trip length.^[9] The free spectral range (FSR), representing the frequency spacing between consecutive resonant modes m and m+1, is derived by considering the change in frequency \Delta \nu that produces a $2\pi phase shift, accounting for dispersion via the group index n_g = n_\mathrm{eff} - \lambda \, dn_\mathrm{eff}/d\lambda. This yields the frequency-domain FSR as

\Delta \nu_\mathrm{FSR} = \frac{c}{n_g L},

where c is the speed of light, reflecting the inverse of the round-trip group delay \tau_g = n_g L / c.^[9] Equivalently, in the wavelength domain, the FSR is

\Delta \lambda_\mathrm{FSR} = \frac{\lambda^2}{n_g L}.

For a typical microring with R \approx 5 \, \mu\mathrm{m} at \lambda = 1550 \, \mathrm{nm} and n_g \approx 4 in silicon, this results in \Delta \lambda_\mathrm{FSR} \approx 20 \, \mathrm{nm} or \Delta \nu_\mathrm{FSR} \approx 2.5 \, \mathrm{THz}. The compact dimensions of optical ring resonators enable exceptionally large FSR values, spanning from GHz to THz scales, which is advantageous for applications requiring broad operational bandwidths, such as dense wavelength-division multiplexing (DWDM) in integrated photonics. In add-drop filter configurations, critical coupling—achieved by balancing the intrinsic loss and external coupling rates—allows near-complete transfer of resonant wavelengths from the through port to the drop port, with extinction ratios exceeding 20 dB. This contrasts with bulk linear cavities by offering footprint reductions by orders of magnitude while maintaining high performance in wavelength-selective routing. Bending losses in the curved waveguide of ring resonators, arising from radiation leakage in high-index-contrast materials like silicon (n \approx 3.5), can perturb the effective and group indices by altering mode confinement, thereby influencing the FSR and limiting the quality factor Q = \nu_0 / \Delta \nu, where \Delta \nu is the resonance linewidth. In silicon rings with radii below 5 \mu\mathrm{m}, these losses contribute propagation attenuations of 0.1–1 dB/cm, reducing Q to $10^4–$10^5, though optimized designs using partial cladding or larger bends mitigate this to achieve Q > 10^6. Polymer-based rings, with lower index contrast (\Delta n \approx 0.1), require larger radii (>50 \mu\mathrm{m}) to suppress bending losses, resulting in smaller FSR (e.g., <1 nm) but potentially higher Q due to reduced scattering. In modern photonic integrated circuits, optical ring resonators serve as key components in WDM filters, where thermo-optic tuning adjusts the resonance position across the FSR to align with ITU grids or compensate fabrication variations.^[18] Integrated microheaters exploit the high thermo-optic coefficient of silicon (dn/dT \approx 1.8 \times 10^{-4} \, \mathrm{K^{-1}}), enabling wavelength shifts of ~0.01 nm/mW with total powers as low as 4 mW/nm for full FSR traversal in micromachined designs.^[18] Such tunability supports dynamic reconfiguration in optical communication systems, with demonstrated channel spacings of 200 GHz.^[18]

Diffractive Devices

Diffraction Gratings

Diffraction gratings feature a surface etched or ruled with periodic grooves separated by a constant period d, which causes incident light to diffract into discrete angular orders denoted by the integer m.^[19] This periodic structure disperses wavelengths according to the grating equation:

d (\sin i + \sin \theta) = m \lambda,

where i is the incidence angle, \theta is the diffraction angle, \lambda is the wavelength, and m represents the diffraction order (positive for the side opposite the incident beam, negative otherwise). The equation derives from the path length difference between adjacent grooves, ensuring constructive interference at the diffraction angles. The free spectral range (FSR) for a diffraction grating in order m is given by \Delta\lambda_\text{FSR} = \frac{\lambda}{m}, defining the maximum wavelength span in that order before overlap occurs with the adjacent order m \pm 1.^[20] This formula approximates the condition where a wavelength \lambda in order m reaches the same diffraction angle as \lambda - \Delta\lambda in order m+1; substituting into the grating equation for fixed angles yields m\lambda = (m+1)(\lambda - \Delta\lambda), solving to \Delta\lambda \approx \frac{\lambda}{m} under the small-angle assumption where \sin \theta \approx \theta.^[20] For higher orders, the FSR narrows, limiting the non-overlapping spectral bandwidth. Diffraction gratings are classified as reflective, where grooves on a mirrored substrate redirect light, or transmissive, where rulings on a transparent material allow light to pass through while diffracting. These devices are integral to spectrometers for spectral dispersion, with echelle gratings serving as a specialized reflective variant featuring coarse groove spacing (low lines per mm) and steep blaze angles to favor high-order diffraction for enhanced resolution in compact setups.^[21] A key limitation of diffraction gratings is order overlap, which compresses the effective FSR and introduces crosstalk between wavelengths from different orders, reducing spectral purity.^[10] To address this, the blaze angle of the grooves is optimized to concentrate diffracted energy into a targeted order and angular range, maximizing efficiency within the desired FSR while minimizing contributions from overlapping orders. Historically, diffraction gratings facilitated early spectroscopy by enabling the isolation of spectral lines within the FSR, with the first ruled gratings constructed by Joseph von Fraunhofer in 1821 to resolve fine details in solar spectra.^[22]

Arrayed Waveguide Gratings

Arrayed waveguide gratings (AWGs) are integrated optical devices that function as wavelength (de)multiplexers, exploiting the free spectral range (FSR) to route signals in wavelength-division multiplexing (WDM) systems. The structure consists of an input slab waveguide (star coupler) that disperses light into an array of multiple waveguides with incrementally increasing lengths, typically differing by a constant path length increment ΔL between adjacent waveguides, followed by an output slab coupler that recombines the light and directs it to specific output waveguides based on wavelength. The FSR in AWGs arises from the phase array principle, analogous to a phased antenna array, where the central wavelength λ_c satisfies the condition for constructive interference: m λ_c = n_g ΔL, with m as the diffraction order and n_g the group refractive index of the waveguides. This condition repeats every spectral interval corresponding to a change in order by 1, yielding the FSR in wavelength as Δλ_FSR ≈ λ_c² / (n_g ΔL) and in frequency as Δν_FSR = c / (n_g ΔL), where c is the speed of light in vacuum; the periodicity ensures that the output pattern repeats across orders, defining the operational bandwidth. The number of channels N_ch supported within the FSR is given by N_ch ≈ Δλ_FSR / Δλ_channel, where Δλ_channel is the channel spacing, enabling dense WDM configurations such as 40 channels at 0.8 nm spacing (100 GHz) within a typical FSR. In silicon photonics, AWGs are fabricated using complementary metal-oxide-semiconductor (CMOS)-compatible processes on silicon-on-insulator platforms, involving electron-beam or deep-ultraviolet lithography to pattern the waveguide array with sub-micron precision, followed by plasma etching and cladding deposition; typical FSR values range from 40 to 100 nm in the C-band (1530–1565 nm) for telecom applications, with performance optimized for low crosstalk (often >25 dB adjacent channel isolation) through phase error minimization and apodization techniques.^[23] Compared to bulk diffraction gratings, AWGs offer advantages in compactness (millimeter-scale footprints), enabling photonic integration, and polarization-insensitive operation via symmetric waveguide designs, making them ideal for scalable fiber-optic networks.^[24]