Longitudinal mode spacing

The longitudinal mode spacing of an optical cavity is the frequency separation between adjacent longitudinal modes — the discrete frequencies satisfying the round-trip phase condition. It is numerically equal to the free spectral range (FSR) of the cavity and is set by the cavity round-trip time.

Definition. For a linear cavity of length $L$ filled with a medium of group index $n_g$, the longitudinal mode spacing is:

$$\Delta\nu_\text{LMS} \;=\; \frac{c}{2 n_g L} \;=\; \frac{1}{T_\text{rt}},$$

where $T_\text{rt} = 2 n_g L / c$ is the cavity round-trip time. For a ring cavity of total round-trip length $L$:

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

(No factor of 2 because light traverses the ring only once per round trip, not back-and-forth.)

Why group index matters. The mode-spacing formula uses the group index $n_g$, not the phase index $n$. This distinction is essential in dispersive media:

$$n_g \;=\; n + \omega \frac{dn}{d\omega} \;=\; n - \lambda \frac{dn}{d\lambda}.$$

For semiconductor laser materials at 1550 nm, $n_g \approx 3.6$ – 4.0 while $n \approx 3.2$ – 3.5. Using $n$ instead of $n_g$ in the mode-spacing formula gives errors of 15 – 25%.

The physical reason: the mode spacing is the inverse of the round-trip group delay, which is what an optical pulse experiences when bouncing through the cavity.

Typical mode spacings.

Cavity	Length	Medium	$n_g$	$\Delta\nu_\text{LMS}$
FP diode laser	300 μm	InGaAsP	3.8	132 GHz
FP diode laser	500 μm	InGaAsP	3.8	79 GHz
FP diode laser	1000 μm	InGaAsP	3.8	39 GHz
VCSEL	1 μm	GaAs+DBR	$\sim 4$	37 THz
External-cavity diode laser	5 cm	air + chip	$\sim 1.05$ effective	2.9 GHz
HeNe (typical)	30 cm	air	1.00	500 MHz
Ti:sapphire	1.8 m (folded)	air	1.00	83 MHz
Bench-top Nd:YAG	30 cm	mostly air	1.0	500 MHz
Fiber laser ring	10 m	silica	1.47	20 MHz
Fiber laser ring	100 m	silica	1.47	2 MHz
Microring (Si photonics)	100 μm circumference	Si	4.2	700 GHz
Microsphere WGM resonator	1 mm diameter	silica	1.45	65 GHz

Relation to gain bandwidth. A critical design relationship is between mode spacing and gain bandwidth:

Regime	$\Delta\nu_\text{LMS}$ vs gain BW	Behavior
$\Delta\nu_\text{LMS} \gg$ gain BW	Single longitudinal mode	VCSELs; only one mode falls within gain
$\Delta\nu_\text{LMS} \sim$ gain BW	Few modes (2 – 10)	Typical Fabry-Perot diode laser
$\Delta\nu_\text{LMS} \ll$ gain BW	Many modes (100s)	HeNe at long cavity, fiber lasers

This explains why VCSELs are naturally single-longitudinal-mode (1 μm cavity, FSR > 30 THz, much larger than the 5 THz gain bandwidth), while edge-emitting Fabry-Perot diodes are typically multimode (300 μm cavity, FSR = 100 GHz < 5 THz gain bandwidth).

Mode index. The longitudinal mode number $m$ is the integer of optical wavelengths fitting in a round trip:

$$m \;=\; \frac{2 n L}{\lambda} \;=\; \frac{\nu T_\text{rt}}{1}.$$

For a 300 μm InGaAsP laser at 1550 nm: $m \approx 1100$. Modes are densely numbered; the physically meaningful quantity is the mode position relative to the gain peak, not the absolute mode index.

Wavelength spacing. Converting to wavelength:

$$\Delta\lambda_\text{LMS} \;\approx\; \frac{\lambda^2}{2 n_g L}.$$

For a 500 μm InGaAsP laser at 1550 nm: $\Delta\lambda_\text{LMS} \approx 0.63$ nm.

Cavity	$\Delta\lambda_\text{LMS}$ at 1550 nm
300 μm InGaAsP	1.05 nm
500 μm InGaAsP	0.63 nm
1000 μm InGaAsP	0.32 nm
5 cm external cavity	0.024 nm
30 cm air cavity	0.004 nm
1 m air cavity	0.0012 nm

For OSA wavelength resolution: typical commercial OSAs resolve 0.02 nm, sufficient to resolve modes of cavities 1 cm or longer.

Side-mode suppression. In a multimode laser, the longitudinal modes near the gain peak compete for inversion. Side-mode suppression ratio (SMSR) measures the ratio of the dominant mode to the largest side mode. Standard SMSR values:

Laser type	Typical SMSR
Multi-mode Fabry-Perot	0 – 5 dB
Mode-selected FP with intracavity filter	20 – 30 dB
DFB laser	35 – 50 dB
DBR laser	30 – 45 dB
External-cavity diode laser	40 – 60 dB
VCSEL	30 – 50 dB

Mode hops. Discrete jumps between longitudinal modes occur when the gain peak shifts relative to the mode positions (typically through temperature or current changes). The hop is by one mode spacing — $\sim 1$ nm for typical FP diodes, $\sim 0.6$ nm for DFB lasers.

Mode beat note. Adjacent longitudinal modes simultaneously incident on a photodetector beat at $\Delta\nu_\text{LMS}$. For diode lasers this is $\sim 100$ GHz — usually beyond detector bandwidth. For external cavity lasers (a few GHz), the beat is observable on RF spectrum analyzers and provides a diagnostic of mode purity.

Cavity FSR engineering. Choosing cavity length to set FSR is a primary design knob:

• Communications lasers: 200 – 500 μm cavity for moderate FSR (60 – 150 GHz)
• Single-frequency lasers: short cavity (FSR > gain BW) or DFB/DBR mode selection
• Mode-locked lasers: cavity length sets the repetition rate; e.g., 100 MHz rate requires 1.5 m cavity
• Frequency-comb sources: FSR = comb tooth spacing

Mode spacing in monolithic vs external cavities. Monolithic semiconductor cavities have FSR set by the chip length (typically 100 GHz). External cavities lengthen the effective cavity, reducing FSR — useful for narrowing linewidth (Schawlow-Townes formula gives $\Delta\nu \propto 1/L^2$) at the cost of mode density.

References: Saleh & Teich, Fundamentals of Photonics (3rd ed., 2019), Ch. 11 (laser resonators); Coldren, Corzine & Mašanović, Diode Lasers and PICs (2nd ed., 2012), Ch. 3 for the group-index treatment; Siegman, Lasers (1986), Ch. 11.