Schawlow–Townes linewidth

The Schawlow–Townes linewidth is the narrowest linewidth a laser can have once every technical noise source has been removed. Spontaneous emission into the lasing mode keeps adding photons of random phase to the field, so the phase of the laser light wanders, and its spectrum becomes a Lorentzian line of full width at half maximum

$$\mathrm{\Delta }{\nu }_{\text{ST}}=\frac{\pi h\nu (\mathrm{\Delta }{\nu }_c{)}^2{n}_{\text{sp}}}{P_{\text{out}}},$$

where $h\nu$ is the photon energy, $\mathrm{\Delta }{\nu }_c$ the linewidth of the passive (cold) cavity, set by its mirror and internal losses without the gain or absorption of the active medium, $n_{\text{sp}}$ the population inversion factor and $P_{\text{out}}$ the output power. The linewidth falls as the power rises and grows as the square of the cavity's loss rate. For a 300 µm long diode laser at 1550 nm emitting 10 mW it is about 0.22 MHz once the internal loss is included, as described below, and in a semiconductor laser the linewidth enhancement factor $\alpha$ multiplies it by $1+{\alpha }^2$, to 3.7 MHz for $\alpha =4$. Arthur Schawlow and Charles Townes derived the limit in 1958, in the paper that proposed extending the maser to optical wavelengths, two years before the first laser operated.

The formula and its symbols

• $h\nu$ is the photon energy, $1.28\times {10}^{-19}$ J (0.80 eV) at 1550 nm.
• $\mathrm{\Delta }{\nu }_c=1\mathrm{/}(2\pi {\tau }_p)$ is the cold-cavity linewidth, written through the photon lifetime ${\tau }_p$ of the passive resonator. For a laser chip, whose resonances are too broad to be Lorentzian, this photon-lifetime form is the one that belongs in the formula.
• $n_{\text{sp}}=N_2\mathrm{/}(N_2-N_1)$ is the population inversion factor. It equals 1 for complete inversion, as in a four-level laser whose lower level stays empty, and exceeds 1 in a semiconductor, where it is $1\mathrm{/}\{1-\exp [(h\nu -\mathrm{\Delta }{E}_F)\mathrm{/}{k}_{B}T]\}$ with $\mathrm{\Delta }{E}_F$ the separation of the quasi-Fermi levels: 1.58 when $\mathrm{\Delta }{E}_F$ exceeds the photon energy by $k_{B}T$.
• $P_{\text{out}}$ is the total power leaving the cavity through its mirrors. The formula as written assumes that all the cavity loss is output coupling. With an internal loss ${\alpha }_i$ beside the mirror loss ${\alpha }_m$, the right side is multiplied by ${\alpha }_m\mathrm{/}({\alpha }_i+{\alpha }_m)$, because the same output power then means more photons inside the cavity.

C. H. Henry's form of the result makes the mechanism plainer. The linewidth is the rate $R_{\text{sp}}$ of spontaneous emission into the lasing mode divided by $4\pi$ times the number of photons $N_p$ in the cavity:

$$\mathrm{\Delta }{\nu }_{\text{ST}}=\frac{R_{\text{sp}}}{4\pi N_p},R_{\text{sp}}=\frac{n_{\text{sp}}}{{\tau }_p}\mathrm{.}$$

Every spontaneous photon moves the phase of the field by a random amount, and the more photons the field already holds, the smaller that step is. Doubling the output power doubles $N_p$ and halves the linewidth.

Worked examples

The first two rows use the cleaved 1550 nm InGaAsP chip of the Fabry-Perot resonator and photon lifetime entries: facet reflectance 0.31, internal loss 10 cm⁻¹, group index 3.6, with $n_{\text{sp}}=1.6$ and 10 mW leaving through the two facets. The third is a solid-state laser at 1064 nm with a cavity of 10 cm optical length, a 98% output coupler, 1% of other loss per round trip, 1 W of output and $n_{\text{sp}}=1$.

For the 300 µm chip the mirror loss is 39 cm⁻¹, so 80% of the loss is output coupling. The 3.7 MHz result lies within the 1 to 10 MHz typical of telecom DFB lasers. Making the same chip four times longer lowers the mirror loss to 9.8 cm⁻¹ and the linewidth by a factor of ten, which is why narrow-linewidth DFB lasers have long cavities. The solid-state laser's limit is ${10}^{10}$ times smaller than the short chip's, mostly because its cavity linewidth is 9,000 times narrower and enters squared; for such a laser, whose gain at the center of a symmetric atomic line has almost no amplitude-phase coupling, $\alpha$ is close to zero, and the measured linewidth is set entirely by technical noise.

Why semiconductor lasers exceed it

In a semiconductor the carrier density sets both the gain and the refractive index. A spontaneous emission event changes the intensity of the field as well as its phase, and as the laser returns to its steady state the carrier density, and with it the refractive index and the optical frequency, changes too, adding a further phase shift $\alpha$ times the first. Henry showed in 1982 that this multiplies the linewidth by $1+{\alpha }^2$:

$$\mathrm{\Delta }\nu =\mathrm{\Delta }{\nu }_{\text{ST}}(1+{\alpha }^2)\mathrm{.}$$

For $\alpha$ between 2 and 5, typical of InGaAsP quantum-well lasers, the factor is 5 to 26. Lasers whose modes are not orthogonal, such as gain-guided stripes and unstable resonators, exceed the linewidth by a further Petermann factor. Even so, the larger difference between diode lasers and other lasers is the cavity: a photon lifetime of picoseconds rather than nanoseconds, as the table shows.

Narrowing the line

The formula names the available levers: a longer or lower-loss cavity, which lowers $\mathrm{\Delta }{\nu }_c$, and more output power. An external-cavity laser adds a passive section, such as a bulk grating, a fiber Bragg grating or ring resonators on a chip, so that the light spends most of its round trip outside the gain medium; for the same loss per round trip, its cold-cavity linewidth falls roughly in inverse proportion to the cavity's total optical length, and the Schawlow–Townes linewidth with the square of that length. Hybrid lasers with long, low-loss silicon nitride feedback circuits have reached an intrinsic linewidth of 40 Hz (Fan et al., 2020), and self-injection locking to ultra-high-Q silicon nitride resonators has brought diode lasers to hertz-level fundamental linewidths (Jin et al., 2021). Raising the power helps less than the formula suggests: the measured linewidths of diode lasers fall as $1\mathrm{/}P$ at moderate power but level off at high power, where other noise sources take over.

Intrinsic and measured linewidth

The Schawlow–Townes linewidth, with the $1+{\alpha }^2$ factor where it applies, is the intrinsic or Lorentzian linewidth: the width that white frequency noise alone produces. A white frequency-noise level $S_0$, one-sided in Hz²/Hz, gives a Lorentzian of width $\mathrm{\Delta }\nu =\pi S_0$, so the 3.7 MHz laser above has $S_0=1.2\times {10}^6$ Hz²/Hz. A measured linewidth is usually wider. Current-source noise, temperature drift, vibration and 1/f frequency noise make the frequency wander slowly, and that wander broadens the line into a roughly Gaussian shape whose width grows with the observation time. Datasheets for narrow-linewidth lasers therefore often quote the intrinsic linewidth, read from the white floor of a frequency-noise spectrum, separately from an integrated linewidth over a stated time, which the β-separation line gives a rule for estimating. How to measure laser linewidth compares the delayed self-heterodyne, heterodyne and frequency-noise methods.

Common questions

What is the Schawlow-Townes limit?

The narrowest linewidth a free-running laser can reach, set by spontaneous emission into the lasing mode. Its value follows from the photon energy, the cavity linewidth and the output power: 0.22 MHz for the short diode laser in the table and 20 µHz for the solid-state laser.

Why does laser linewidth decrease with output power?

More output power means more photons in the cavity, so each spontaneous emission event shifts the phase of the total field by less, the phase diffuses more slowly, and the line narrows as $1\mathrm{/}P$.

Can a laser be narrower than the Schawlow-Townes linewidth?

A free-running laser in the usual regime, where the cavity linewidth is much narrower than the gain bandwidth, cannot. A laser locked to an external frequency reference by a fast servo can, because the servo corrects its frequency noise within the servo bandwidth: lasers locked to ultrastable optical cavities reach linewidths below 1 Hz.

What is the difference between the Schawlow-Townes linewidth and the intrinsic linewidth?

For a laser without amplitude-phase coupling they are the same. For a semiconductor laser the intrinsic linewidth is the Schawlow–Townes value times $1+{\alpha }^2$, and times the Petermann factor where that applies. Both are Lorentzian widths from white frequency noise, and both are narrower than a linewidth measured over a long observation time.

References: A. L. Schawlow and C. H. Townes, "Infrared and optical masers," Physical Review 112, 1940 (1958). C. H. Henry, "Theory of the linewidth of semiconductor lasers," IEEE Journal of Quantum Electronics 18, 259 (1982). L. A. Coldren, S. W. Corzine, and M. L. Mašanović, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012). G. Di Domenico, S. Schilt, and P. Thomann, "Simple approach to the relation between laser frequency noise and laser line shape," Applied Optics 49, 4801 (2010). Y. Fan et al., "Hybrid integrated InP-Si3N4 diode laser with a 40-Hz intrinsic linewidth," Optics Express 28, 21713 (2020). W. Jin et al., "Hertz-linewidth semiconductor lasers using CMOS-ready ultra-high-Q microresonators," Nature Photonics 15, 346 (2021).