Pound-Drever-Hall locking (PDH)

Locking a laser to a cavity needs an error signal that says not only how far the laser is from resonance but on which side. Transmitted or reflected power alone cannot: the resonance is symmetric, so a drop in transmission looks the same whichever way the laser moved. The Pound-Drever-Hall technique supplies the sign. The laser is phase-modulated at a radio frequency $\Omega$, usually with an electro-optic modulator, which puts sidebands at $\pm\Omega$ on either side of the carrier; $\Omega$ is chosen well outside the cavity linewidth, so when the carrier is near resonance the sidebands are reflected almost completely. The reflected beam falls on a fast photodiode, and the beat between the sidebands and the reflected carrier is demodulated at $\Omega$ with a mixer.

Near resonance the cavity imprints a phase on the reflected carrier that is proportional to the detuning, and the sidebands serve as a phase reference that the cavity barely touches. The demodulated signal is therefore dispersive: zero at line center, positive on one side, negative on the other. For a detuning $\delta f$ small compared with the cavity linewidth $\delta\nu$ (full width at half maximum), it is

with $P_c$ and $P_s$ the power in the carrier and in each first-order sideband (Black 2001). The slope scales inversely with the linewidth, which is why the method pairs naturally with high-finesse cavities. For a phase-modulation depth $\beta$, $\sqrt{P_cP_s}$ is proportional to $J_0(\beta)J_1(\beta)$, which is largest at $\beta = 1.08$ rad; there 53% of the power stays in the carrier and 21.7% goes into each first-order sideband. A 10 cm Fabry-Perot cavity has a free spectral range of 1.5 GHz; with a finesse of 10 000 its linewidth is 150 kHz, and modulation at a few tens of megahertz keeps the sidebands far outside it.

Three properties explain the method's dominance in precision work. The error signal is measured in reflection and vanishes on resonance, so to first order it is immune to fluctuations of the laser's intensity. The loop bandwidth is not limited by the cavity's storage time: a fast phase excursion appears promptly in the reflected light, because the promptly reflected field interferes with the field leaking out of the cavity, which cannot follow. And the capture range extends out to the sidebands, far beyond the linewidth. The laser frequency is corrected through whichever actuators it offers, typically injection current or a piezo for slow, large corrections and an acousto-optic or electro-optic element for fast ones.

The dominant systematic error is residual amplitude modulation: a modulator that also modulates intensity, through polarization misalignment or etalons in the crystal, produces a signal at $\Omega$ that the mixer cannot tell from the cavity's, which shifts the lock point and drifts with temperature. Wedged, temperature-controlled crystals, careful polarization alignment, and active cancellation reduce it. PDH is the servo behind every ultrastable optical cavity and gravitational-wave interferometer, and it applies equally to on-chip resonators, where it complements the slower dither locking described in Thermal tuning and locking of ring resonators; the resulting linewidth is measured as in How to measure laser linewidth.

References: R. W. P. Drever et al., Appl. Phys. B 31, 97 (1983); E. D. Black, Am. J. Phys. 69, 79 (2001).