Glossary entry

Longitudinal modes

The resonant frequencies of a laser cavity, at which a whole number of wavelengths fits into one round trip. They form an evenly spaced comb, c/(2n_gL) apart in a linear cavity, and a laser oscillates on those where its gain exceeds the loss.

Lasers & gain Optics fundamentalsUpdated October 2026

Longitudinal modes are the resonant frequencies of a laser cavity: the frequencies at which a whole number of wavelengths fits into one round trip, so that the light returns to any point in phase with itself. In a linear cavity of length $L$ they are evenly spaced by

Δ ν = \frac{c}{2 n_{g} L},

where $n_{g}$ is the group index of the material inside. The spacing, equal to the cavity's free spectral range, is 500 MHz for a 30 cm helium-neon laser, 150 MHz for a 1 m solid-state laser, and about 139 GHz, or 1.1 nm at 1550 nm, for a 300 µm semiconductor laser chip. A laser oscillates on the modes that fall where its gain exceeds the cavity loss, so whether it runs on one frequency or several depends on how the mode spacing compares with the width of the gain spectrum, and on how the modes compete for that gain. Transverse modes, by contrast, describe the pattern of the beam across its cross-section.

Resonance condition

Light making one round trip of a linear cavity of length $L$ , filled with a medium of refractive index $n$ , accumulates a phase $4 π n L ν / c$ . Resonance requires this to be a whole number $q$ of $2 π$ , which gives

ν_{q} = q \frac{c}{2 n L}, q = 1, 2, 3, \dots

The phase on reflection at the mirrors and the Gouy phase of the beam are ignored here; both shift the whole comb without changing its spacing. The mode number $q$ is the number of half-wavelengths that fit in the cavity, and it is large: about 950,000 for a 30 cm helium-neon laser at 632.8 nm, and about 1,240 for a 300 µm InGaAsP chip at 1550 nm ( $n \approx 3.2$ ). Only the modes near the gain peak matter, so lasers are described by the spacing and position of their modes rather than by $q$ .

In a ring cavity the light passes each point once per round trip, and the modes are spaced by $c / (n_{g} L)$ , where $L$ is the circumference. Every Fabry-Perot resonator, whether a laser cavity, an etalon or a scanning interferometer, has the same comb of resonances.

Mode spacing

The spacing between adjacent modes is the inverse of the round-trip time, $Δ ν = 1 / T_{rt}$ with $T_{rt} = 2 n_{g} L / c$ , and in wavelength it is $Δ λ = λ^{2} / (2 n_{g} L)$ . The group index enters rather than the phase index because the spacing depends on how fast the round-trip phase changes with frequency. In an InGaAsP laser chip at 1550 nm, $n_{g} \approx 3.6$ against $n \approx 3.2$ , so using the phase index would overstate the spacing by 12%.

Cavity	Length	Mode spacing
InGaAsP Fabry-Perot laser chip, 1550 nm ( $n_{g} = 3.6$ )	300 µm	139 GHz (1.11 nm)
InGaAsP Fabry-Perot laser chip, 1550 nm ( $n_{g} = 3.6$ )	1 mm	41.6 GHz (0.33 nm)
External-cavity diode laser, 1550 nm	5 cm optical length	3.0 GHz (24 pm)
Helium-neon laser, 632.8 nm	30 cm	500 MHz (0.67 pm)
Solid-state laser, 1064 nm	1 m	150 MHz (0.57 pm)
Fiber ring laser ( $n_{g} = 1.468$ ), 1550 nm	100 m circumference	2.04 MHz
Silicon microring ( $n_{g} = 4.2$ ), 1550 nm	50 µm circumference	1.43 THz (11.4 nm)

The longitudinal mode spacing entry tabulates more cavities, and the Fabry-Perot cavity calculator gives the spacing, mode number and mode width for any length and group index. The relation also runs in reverse: the mode spacing measured on a laser chip of known length is the usual way to find the group index of its waveguide.

How many modes a laser runs on

The cavity supports an unlimited comb of modes, but only those within the part of the gain medium's spectrum where the gain exceeds the loss can oscillate. That width divided by the mode spacing is an upper limit on the number of modes; how many actually oscillate depends on how the modes share the gain.

In an inhomogeneously broadened medium each mode draws on its own group of atoms. The gain of a helium-neon laser at 632.8 nm is Doppler broadened to about 1.5 GHz by the thermal motion of the neon atoms, and a mode interacts only with the atoms whose velocity Doppler-shifts them into resonance with it. Each mode therefore burns its own spectral holes in the gain curve, and every mode above threshold oscillates. A 30 cm tube, with modes 500 MHz apart, typically runs on two or three modes, and shorter tubes on fewer.

In a homogeneously broadened medium every atom or carrier contributes gain across the whole line, so in steady state the mode with the highest net gain should clamp the gain and suppress the others. Standing-wave cavities undermine this through spatial hole burning: each mode's standing wave saturates the gain at its antinodes and leaves it unused at its nodes, where a mode of slightly different wavelength finds gain. A standing-wave Nd:YAG laser therefore usually oscillates on several longitudinal modes unless it contains a mode-selecting element, while a unidirectional ring laser, whose traveling wave saturates the gain evenly along its length, runs on a single mode far more readily.

Semiconductor lasers are a partial exception. Carriers diffuse over 1 to 2 µm, several times the 0.24 µm period of the standing wave, which washes most of this fine-scale hole burning out. A Fabry-Perot laser chip still runs on several modes because its gain spectrum, some tens of nanometers wide, is nearly flat on the scale of the mode spacing, so neighboring modes have almost the same net gain, and spontaneous emission, which feeds every mode, keeps them populated. A 300 µm chip at 1550 nm, with modes 1.1 nm apart, shows a spectrum of several modes spread over a few nanometers. Under modulation the power moves randomly between those modes; this mode partition noise, combined with fiber dispersion, confines multimode lasers to short links.

Selecting a single longitudinal mode

A single-longitudinal-mode, or single-frequency, laser adds enough wavelength selectivity that only one mode reaches threshold:

DFB lasers and DBR lasers replace one or both mirrors with a Bragg grating that reflects only a narrow band; DFB lasers are the standard telecom transmitter.
External-cavity lasers feed back one wavelength from a diffraction grating in the Littrow or Littman-Metcalf arrangement, and tune by turning the grating or a mirror.
An intracavity etalon adds a transmission peak narrow enough to favor one cavity mode, the usual method in solid-state and ion lasers.
A very short cavity pushes the neighboring modes outside the gain spectrum: a VCSEL, with a cavity about one wavelength long, has a single longitudinal mode within its gain.
A unidirectional ring cavity avoids spatial hole burning altogether.
Injection seeding, used in pulsed lasers, injects a weak single-frequency beam so that the seeded mode builds up first and takes the gain (see injection locking).

The purity of the result is quoted as the side-mode suppression ratio, the power in the lasing mode over that in the strongest other mode; telecom DFB lasers are typically specified above 35 dB. A single-mode laser whose selectivity is overcome, by a drifting gain peak or by optical feedback, jumps to a neighboring mode in a mode hop.

Tuning and mode hops

Each mode frequency follows the optical length of the cavity: a change in length of half a wavelength moves every mode by one spacing. A helium-neon tube sweeps its modes through one free spectral range for every 316 nm it expands, many times over as it warms up, which is why the power of an unstabilized tube fluctuates during warm-up and why frequency-stabilized helium-neon lasers servo the cavity length.

In semiconductor lasers, heating moves the gain peak by 0.3 to 0.5 nm/K but the cavity modes by less than 0.1 nm/K. As the temperature rises the lasing wavelength follows its cavity mode slowly until the neighboring mode has more gain, then hops by one mode spacing; the wavelength temperature coefficient entry gives the numbers.

Observing longitudinal modes

Optical spectrum analyzer. A grating optical spectrum analyzer resolves the 1 nm modes of a laser chip easily. At a resolution of 0.02 nm it separates the modes only of cavities with less than about 6 cm of air path, which excludes most gas and solid-state lasers.
Scanning Fabry-Perot interferometer. An interferometer whose free spectral range exceeds the width of the laser's spectrum displays each mode as a separate peak as its length is scanned, and resolves the 500 MHz modes of a helium-neon laser.
RF beat note. Two modes falling on a fast photodiode beat at their difference frequency, so a multimode laser shows a line at the mode spacing, and at its harmonics, on an RF spectrum analyzer: 500 MHz for a 30 cm helium-neon laser. The line disappears when the laser runs on one mode, which makes it a quick test. For laser chips the beat lies between tens of gigahertz and above 100 GHz, beyond most photodiodes.

Mode locking and frequency combs

When many longitudinal modes oscillate with fixed relative phases, their sum is a train of short pulses, one per round trip, at a repetition rate equal to the mode spacing: a 1.5 m linear cavity gives 100 MHz, a pulse every 10 ns. The pulse duration is roughly the inverse of the total bandwidth of the locked modes. Mode locking is the technique that fixes the phases, and the spectrum of a mode-locked laser is a frequency comb: a titanium-sapphire laser with a 30 nm bandwidth at 800 nm and an 80 MHz repetition rate oscillates on about 175,000 modes at once.

Longitudinal and transverse modes

Each transverse mode of a cavity has its own comb of longitudinal modes. The combs of the higher-order transverse modes are offset from the fundamental's by the extra Gouy phase those modes accumulate between the mirrors, a fraction of the mode spacing set by the mirror curvatures, so a laser that also runs on higher-order transverse modes shows additional spectral lines and beat notes. The two are independent: most helium-neon lasers produce a clean Gaussian TEM₀₀ beam that contains two or three frequencies.

Common questions

What is the difference between longitudinal and transverse modes?

Longitudinal modes are the frequencies a cavity resonates at, set by its length. Transverse modes are the field patterns across the beam, set by the mirror curvatures, apertures or waveguide that confine it. A laser can run on one transverse mode and several longitudinal modes, or on several of each.

How many longitudinal modes does a laser have?

The cavity has an unlimited comb of them; the number that oscillate is at most the width of the gain above threshold divided by the mode spacing, and fewer when the modes compete for the same gain. A 30 cm helium-neon laser runs on two or three, a Fabry-Perot laser chip on several, a DFB laser on one, and a mode-locked laser on tens of thousands or more.

What is a single-longitudinal-mode laser?

A laser that oscillates on one cavity mode, with the others suppressed by a wavelength-selective element such as a Bragg grating, an etalon or an external diffraction grating. Its output has a single frequency, a narrow linewidth and a long coherence length.

How do you calculate the longitudinal mode spacing?

Divide the speed of light by twice the optical length of the cavity, measured with the group index: $Δ ν = c / (2 n_{g} L)$ , or $Δ λ = λ^{2} / (2 n_{g} L)$ in wavelength. For a 1 m air-spaced cavity this gives 150 MHz; for a 500 µm InGaAsP chip with $n_{g} = 3.6$ it gives 83 GHz, or 0.67 nm at 1550 nm.

References: A. E. Siegman, Lasers (University Science Books, 1986). O. Svelto, Principles of Lasers, 5th ed. (Springer, 2010). B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019). L. A. Coldren, S. W. Corzine and M. L. Mašanović, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012).