Optical phase-locked loop (OPLL)

An optical phase-locked loop (OPLL) makes one laser, the slave, follow the phase of another, the master, at a fixed frequency offset. The two beams are combined on a fast photodiode, which produces a heterodyne beat note at their difference frequency. A phase detector compares that beat with a stable RF reference at the desired offset, and the resulting error signal tunes the slave laser, usually through its injection current for fast corrections and its temperature or a piezo for slow ones. Once locked, the slave's optical frequency equals the master's plus the reference frequency, with the master's phase noise copied onto the slave inside the loop bandwidth. Offsets from a few MHz up to tens of GHz are common, limited by the photodiode and the RF electronics.

Injection locking also makes a slave follow a master, but optically and at zero offset unless the master is shifted first; Pound-Drever-Hall locking and dither locking lock a laser to a cavity or atomic reference. The OPLL is the standard tool for laser-to-laser locking with an arbitrary, tunable offset.

Loop components

A typical OPLL contains:

• a fiber or free-space combiner with matched polarizations, so the beat amplitude is maximal;
• a photodiode and RF amplifier with bandwidth above the offset frequency;
• a phase detector: a double-balanced mixer for a pure phase lock, or a digital phase-frequency detector, which also pulls the loop in from large frequency errors;
• a loop filter (proportional-integral, often with a second integrator) driving the slave laser's current and a slower actuator.

Frequency dividers ahead of a digital phase detector let a low-frequency reference lock a beat at several GHz. The lock is verified on an electrical spectrum analyzer: a locked beat collapses to a narrow coherent carrier, flanked by "servo bumps" at roughly the loop bandwidth, where the residual noise is pushed.

Residual phase error and loop bandwidth

The beat of two lasers with Lorentzian linewidths has a linewidth equal to their sum, $\Delta\nu$. Its phase noise falls as $1/f^2$, so a loop with a first-order response and bandwidth $f_c$ leaves a residual phase variance

$$\sigma_\varphi^2 = \frac{\Delta\nu}{2 f_c}.$$

Two semiconductor lasers of 100 kHz each give $\Delta\nu$ = 200 kHz. A loop bandwidth of 1 MHz leaves $\sigma_\varphi \approx$ 0.32 rad; holding the error to 0.1 rad needs

$$f_c = \frac{2 \times 10^5\ \text{Hz}}{2 \times 0.01} = 10\ \text{MHz}.$$

The limit on $f_c$ is the loop delay $\tau$, the total time from the optical combiner through photodiode, electronics, cables and the laser's tuning response. A first-order loop with delay becomes unstable at $f_c = 1/(4\tau)$, where its phase margin reaches zero, and a usable loop keeps $f_c\tau$ near 0.1 or below. A 10 MHz bandwidth therefore needs $\tau$ of a few nanoseconds, under about 1 m of total signal path at 2 × 10⁸ m/s, well below the 25 ns stability limit. This is why OPLLs for diode lasers are built with short cables, fast electronics and sometimes on a single chip, and why narrow-linewidth sources such as external-cavity or fiber lasers, with kHz linewidths, are much easier to lock.

Applications

• Offset locking for atomic physics. Two lasers separated by an atomic hyperfine splitting, 6.834 GHz for rubidium-87, drive Raman transitions in atom interferometers and clocks; the OPLL makes their difference phase as stable as the RF synthesizer.
• Locking to a comb. Phase-locking a continuous-wave laser to one line of a frequency comb transfers the comb's stability and absolute frequency to the laser, or locks the comb to an optical reference.
• Coherent receivers. Homodyne coherent detection needs the local oscillator phase-locked to the incoming carrier, often with a Costas-type loop that is insensitive to the data modulation. Most modern links use a free-running local oscillator and digital phase recovery instead.
• Microwave photonics and swept lasers. Two locked lasers beating on a fast photodiode generate a low-noise microwave or millimeter-wave carrier at the offset frequency, and a loop referenced to a delay interferometer linearizes the chirp of an FMCW lidar source.

Pitfalls

The beat at $+f$ and $-f$ looks identical on the photodiode, so the slave can lock on either side of the master; the sign of the feedback picks one, and a wrong sign gives a lock on the mirror frequency or no lock at all. A diode laser's current tuning changes sign between the thermal regime at low modulation frequencies and the carrier regime above a crossover typically between 100 kHz and a few MHz, adding phase lag right where many loops cross over. Cycle slips, jumps of $2\pi$ in the phase error during a brief disturbance, are invisible to a slow frequency counter but corrupt coherent measurements. A drifting polarization reduces the beat amplitude and with it the loop gain.

Common questions

What is the difference between an OPLL and a frequency offset lock?

A frequency lock (often built with a delay-line discriminator) holds the average beat frequency and lets the phase wander, so each laser keeps its own linewidth. An OPLL holds the phase difference itself, so within its bandwidth the slave inherits the master's coherence.

How fast must the loop be?

Fast enough that $\Delta\nu/(2f_c)$, the residual phase variance, is small: for 0.1 rad, the loop bandwidth must be about 50 times the summed linewidth.

References: L. G. Kazovsky, "Balanced phase-locked loops for optical homodyne receivers: performance analysis, design considerations, and laser linewidth requirements," J. Lightwave Technol. 4, 182 (1986); F. M. Gardner, Phaselock Techniques, 3rd ed. (Wiley, 2005); E. D. Black, "An introduction to Pound-Drever-Hall laser frequency stabilization," Am. J. Phys. 69, 79 (2001).