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 $2\pi$. This resonance condition 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:

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.