Schawlow–Townes linewidth

The Schawlow–Townes linewidth is the fundamental lower bound on how spectrally pure a laser oscillator can be. Each spontaneous emission event into the lasing mode adds a photon of random phase; the accumulated phase diffusion makes the emission line a Lorentzian of nonzero width, even with every technical noise source silenced.

The standard form:

$\Delta\nu_{ST} = \frac{\pi h \nu \, (\Delta\nu_c)^2 \, n_{sp}}{P_{out}}$

where $\Delta\nu_c$ is the cold-cavity linewidth (inverse photon lifetime over $2\pi$), $n_{sp} \geq 1$ the population-inversion (spontaneous emission) factor, and $P_{out}$ the output power. Two scalings deserve memorization: linewidth falls as $1/P$ (more coherent photons dilute each phase kick) and rises as the square of cavity loss rate, which is why long or low-loss cavities are the route to narrow lines.

Semiconductor lasers break the naive formula in a characteristic way. In a semiconductor gain medium, gain and refractive index are coupled: a carrier-density fluctuation that perturbs amplitude also perturbs phase. Henry's linewidth enhancement factor $\alpha$ captures this, multiplying the linewidth:

$\Delta\nu = \Delta\nu_{ST}\,(1+\alpha^2)$

With $\alpha$ of 2–5, the penalty is a factor of 5–25. That is the central reason a solid-state laser can sit at hertz-level linewidth while a bare DFB of similar output sits at hundreds of kilohertz to megahertz.

Representative magnitudes: DFB diode, ~0.1–1 MHz; external-cavity or SiN-resonator-based hybrid lasers, kHz to sub-kHz (the external cavity attacks the $(\Delta\nu_c)^2$ term); bulk solid-state and fiber lasers, Hz-class intrinsic. Measured linewidths often sit above the intrinsic value because technical noise (current source, temperature, vibration) adds a Gaussian envelope. Instruments report the combination.