Longitudinal modes

Longitudinal modes are the set of optical resonances supported by a Fabry-Perot cavity along its propagation axis. They arise from the requirement that the round-trip phase accumulated by light bouncing between the two mirrors equal an integer multiple of $2\pi$ — the resonance condition that determines which optical frequencies the cavity supports.

Resonance condition. For a Fabry-Perot cavity of length $L$ with refractive index $n$, the round-trip phase is $\phi_\text{rt} = 2 k L = 2 n \omega L / c$. The condition $\phi_\text{rt} = 2\pi m$ for integer $m$ gives:

$$\omega_m \;=\; \frac{m \pi c}{n L}, \quad \text{or} \quad \nu_m \;=\; \frac{m c}{2 n L},$$

where $m = 1, 2, 3, \ldots$ The longitudinal modes form a uniformly-spaced "comb" in frequency.

Free spectral range. The frequency spacing between adjacent longitudinal modes is the free spectral range (FSR):

$$\Delta\nu_\text{FSR} \;=\; \nu_{m+1} - \nu_m \;=\; \frac{c}{2 n L} \;=\; \frac{1}{T_\text{rt}},$$

where $T_\text{rt} = 2nL/c$ is the cavity round-trip time. The FSR is inversely proportional to cavity length:

Cavity type	$L$	FSR at 1550 nm
Fabry-Perot laser diode	300 μm	138 GHz
Edge-emitter DFB laser	500 μm	83 GHz
HeNe laser tube	30 cm	500 MHz
External-cavity diode laser	5 cm	3 GHz
Solid-state laser	1 m	150 MHz
Fiber laser ring	100 m	1.5 MHz
Microring resonator	50 μm	800 GHz
Whispering-gallery resonator	1 mm	50 GHz

Group-index correction. The expression above uses the phase index $n$. For dispersive media, the FSR is set by the group index $n_g = n + \omega(dn/d\omega)$:

$$\Delta\nu_\text{FSR} \;=\; \frac{c}{2 n_g L}.$$

This distinction is essential for semiconductor lasers, where $n_g$ is typically 3.6 – 4.0 while $n$ is 3.2 – 3.5. Using the wrong index gives FSR estimates off by 15 – 25%.

Why multiple modes are populated. In a homogeneously-broadened gain medium (e.g., a typical semiconductor laser), only the longitudinal mode closest to the gain peak should lase — all others should be suppressed by gain competition. In practice, several modes near the gain peak are simultaneously populated due to:

• Spatial hole burning: the standing-wave pattern of each mode burns its own gain profile, leaving gain available for adjacent modes
• Carrier diffusion: averages out spatial hole burning only partially
• Frequency-modulated gain dynamics: pulsed or modulated lasers transiently populate side modes
• Mode partition noise: stochastic energy transfer between modes during operation

In an unfiltered Fabry-Perot laser, 10 – 100 longitudinal modes can be simultaneously above threshold, distributed across the gain bandwidth.

Single-mode operation requires mode selection. A "single longitudinal mode" laser uses a wavelength-selective element to favor one specific mode:

• DFB laser: a Bragg grating along the cavity provides selective feedback only at the Bragg wavelength
• DBR laser: one or both mirrors are wavelength-selective Bragg reflectors
• Intracavity etalon: a thin etalon in the cavity provides narrowband transmission
• External cavity Littrow/Littman: a diffraction grating selects one wavelength
• VCSEL: extremely short cavity has FSR larger than gain bandwidth, allowing only one longitudinal mode

Side-mode suppression ratio (SMSR) quantifies how strongly the dominant mode dominates: typical single-mode lasers achieve SMSR $> 30$ dB; high-quality DFB lasers achieve $> 50$ dB.

Mode beat note in detection. When two longitudinal modes are simultaneously incident on a photodetector, they beat at the difference frequency (= FSR for adjacent modes). This produces RF noise at the FSR frequency. For Fabry-Perot lasers, this beat note (typically 50 – 200 GHz, well above electronic bandwidth) is usually invisible. For external-cavity lasers with longer cavities (FSR in the GHz range), the beat note may interfere with the signal of interest.

Why FSR equals 1/round-trip time. The longitudinal modes can be viewed alternatively as a discrete decomposition of the cavity's frequency response. The cavity's impulse response is a sequence of decaying pulses spaced by $T_\text{rt}$. The Fourier transform of this impulse train is a comb of frequencies spaced by $1/T_\text{rt}$ — exactly the FSR. This connection makes the mode comb intuitive: the cavity remembers a roundtrip time, and modes are the frequencies that constructively interfere with themselves after each roundtrip.

Frequency combs. A mode-locked laser actively populates many longitudinal modes with a fixed phase relationship. The resulting "frequency comb" with $> 10^5$ teeth uniformly spaced by FSR is the basis of modern optical clocks and absolute-frequency metrology.

References: Saleh & Teich, Fundamentals of Photonics (3rd ed., 2019), Ch. 11 (laser resonators); Siegman, Lasers (University Science Books, 1986), Ch. 11 for the rigorous cavity analysis; Yariv & Yeh, Photonics (6th ed., 2007), Ch. 7.