Dither locking (top-of-fringe lock)

Dither locking, also called top-of-fringe or peak locking, keeps a laser frequency at the center of a resonance (a cavity transmission peak, an absorption line, a fringe of an interferometer) or keeps a tunable resonator centered on a fixed laser. A small sinusoidal modulation, the dither, is applied to the laser frequency or to the resonance position, the transmitted or reflected power is detected, and a lock-in amplifier or digital equivalent demodulates it at the dither frequency. The output is approximately the derivative of the resonance lineshape, which serves as the error signal for a feedback loop. Typical dither amplitudes are 5–20% of the resonance linewidth, at frequencies from about a kilohertz for thermally tuned ring resonators to hundreds of kilohertz for current-modulated diode lasers.

The derivative error signal

If the detuning from the resonance center is $\delta$ and the dither adds $a\sin\omega t$, the detected power is $P(\delta + a\sin\omega t)$. For a dither small compared with the linewidth, a Taylor expansion gives a component at $\omega$ with amplitude

$$e(\delta) \approx a\,\frac{dP}{d\delta},$$

which is zero at the peak and changes sign across it. For a Lorentzian of full width $\Gamma$ and peak power $P_0$, $P = P_0/[1 + (2\delta/\Gamma)^2]$, and the slope of the error signal at the center is

$$\left.\frac{de}{d\delta}\right|_0 = -\frac{8\,a\,P_0}{\Gamma^2}.$$

The error signal is close to linear only out to the points of steepest slope on the lineshape, at $\delta = \pm\Gamma/(2\sqrt{3}) = \pm 0.29\,\Gamma$; beyond them it keeps the correct sign but shrinks, so the loop still pulls in from farther out with falling gain. Because the zero crossing is at the peak whatever the power, the lock point does not move with laser intensity, unlike a side-of-fringe lock, which holds the power at half maximum and converts every intensity fluctuation into a frequency error. The loop gain does scale with power.

Dither amplitude trade-off

A larger dither gives a steeper error signal and a better signal-to-noise ratio; it also modulates the very quantity being stabilized. Numerically, for a Lorentzian:

The slope is greatest at $a = \Gamma/(2\sqrt{2}) \approx 0.35\,\Gamma$, but there the average transmission has dropped by 18%. At $a = 0.1\,\Gamma$ the slope is half its maximum and the cost is a 1.9% average loss, with a residual power modulation at $2\omega$ of about 1.9% of the peak. In a ring resonator with a quality factor of 10⁵ at 1550 nm, whose linewidth is 15.5 pm (1.93 GHz), that dither is 1.55 pm. This residual modulation appears on every signal passing through the locked element, which is why dither locks are avoided where the output must be clean.

Bandwidth

The demodulated signal must be low-pass filtered to remove the components at $\omega$ and $2\omega$, so the loop bandwidth is limited to a fraction of the dither frequency. For a thermally tuned ring dithered at a few kilohertz, this gives loop bandwidths of tens of hertz to about a kilohertz, adequate for drift and too slow for acoustic noise or laser frequency noise, as discussed in thermal tuning and locking of ring resonators. The dither must also be slow compared with the resonance's own response, which matters for cavities with kilohertz linewidths. Pound-Drever-Hall locking avoids both limits by modulating above the linewidth, at the cost of an electro-optic modulator and faster electronics.

Offsets and residual amplitude modulation

Anything that produces a signal at $\omega$ other than the slope of the resonance shifts the lock point. Dithering a diode laser through its current modulates its power as well as its frequency; a relative power modulation $m$ in phase with the frequency dither adds $m P$ to the error signal, and the lock settles where the two cancel, near

$$\delta_\text{off} \approx \frac{m\,\Gamma^2}{8a}.$$

With $m$ = 1% and $a = 0.1\,\Gamma$, the offset is 0.013 $\Gamma$, or 0.2 pm on the ring above. A sloping background, such as an absorption line sitting on a tilted baseline in tunable diode laser absorption spectroscopy, has the same effect. Demodulating at $3\omega$ (third-harmonic locking) makes the error signal proportional to the third derivative of the lineshape, which still crosses zero at the peak and is insensitive to linear and quadratic backgrounds, at the cost of a weaker signal.

Common questions

What is the difference between dither locking and Pound-Drever-Hall locking?

Dither locking modulates slowly, within the resonance, and reads the slope of the transmitted power. PDH modulates far outside the linewidth and reads the phase of the reflected field, which allows a faster lock with sidebands that fall outside the resonance.

Why lock at 1f rather than 2f?

The first-harmonic signal is the first derivative and crosses zero at the peak, which is what a servo needs. The second-harmonic signal, proportional to the second derivative, peaks at line center and is used for spectroscopy and for finding a line.

References: R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley and H. Ward, "Laser phase and frequency stabilization using an optical resonator," Applied Physics B 31, 97 (1983); E. D. Black, "An introduction to Pound-Drever-Hall laser frequency stabilization," American Journal of Physics 69, 79 (2001); W. Demtröder, Laser Spectroscopy 1: Basic Principles, 5th ed. (Springer, 2014).