Text from the interactive explanation

An Introduction to Pound-Drever-Hall laser frequency stabilization

Eric D. Black

LIGO Project · 1999 · Eric D. Black

A laser’s frequency — how fast its light waves oscillate, which our eyes read as colour — never sits perfectly still. Gravitational-wave hunters needed it to. This primer from the LIGO Project explains the trick they rely on: Pound–Drever–Hall locking . Below, a simulated laser drifts in and out of tune with a pair of mirrors. Try to keep the mirrors glowing.

Pale curve: light bouncing back from the mirrors · Teal curve: the steering signal this paper explains · Red trail: where the laser’s frequency has been, newest at the top · Teal bar on the right edge: when the lock was on

Top row, left to right: laser → modulator → beam splitter → two-mirror cavity. Reflected light drops to the detector; the dashed line carries the signal through the mixer and servo back to the laser.

Simulation recomputed from Eqs 4.1–4.3 of Black (1999) with modulation at 6 cavity linewidths; bench after Fig. 3. Laser drift and servo are illustrative.

8 × 10−21 m/√Hz

In the paper’s worked example — a 20 cm mirror cavity and a half-watt laser — this method could, at its quantum limit, sense changes in mirror spacing of about 8 × 10−21 metres per √Hz: more than a hundred thousand times smaller than a proton. (Calculated limit, Section VI.)

01 · Fig. 1

A ruler made of mirrors

Start with the ruler. Face two very good mirrors toward each other and you have a Fabry–Perot cavity. Light that gets in bounces back and forth many times, meeting copies of itself on every round trip. Only when twice the spacing between the mirrors holds a whole number of wavelengths do the copies line up crest to crest. Then light builds up inside and streams out through the far mirror. Otherwise the copies cancel, and the cavity turns the light away.

So a cavity passes only a comb of narrow “windows” in frequency, evenly spaced by the free spectral range, c/2L — 750 MHz for mirrors 20 cm apart. Better mirrors mean more round trips and narrower windows: a sharper ruler. The finesse measures that sharpness: window spacing divided by window width.

Your task Sweep the frequency slowly. The transmission curve is drawn only where you have been — find two windows.

What you just saw

Find two windows and the explanation appears here.

Light gets through only when ν/FSR is a whole number — when the wave fits the round trip exactly. Each window is one free spectral range from the next, and its width is that spacing divided by the finesse: raise the mirror reflectivity and the windows sharpen. A laser parked in one window is pinned to the cavity’s length. Everything that follows is about keeping it there.

02 · Section III

Perch on a slope

Before Pound–Drever–Hall, a common trick was to park the laser partway up one side of a window — say where half the light gets through. Drift up in frequency and more light comes through; drift down and less does. An electronic servo watches the transmitted light and nudges the laser’s tuning to hold it at half.

Black names the flaw: the servo cannot tell a change in the laser’s frequency from a change in the laser’s own brightness. If the laser simply dims, the servo thinks it has slid along the slope and “corrects” the frequency — moving it away from where it should be.

Your task With the slope lock on, add brightness flicker. Does the trail stay on the dashed line?

What you just saw

Add flicker and the explanation appears here.

With a steady laser the slope lock holds well. Add flicker and the trail drifts off the dashed line although the laser’s frequency noise has not changed: the servo is holding the brightness constant, not the frequency. You could stabilise the laser’s power separately — that was tried in the late 1970s with limited success — or look instead at a signal that is zero on resonance however bright the laser is.

03 · Figs 4–5

Aim for the dark

So watch the light that bounces back instead. What comes back from the cavity is really two beams added together: the prompt reflection, which bounces straight off the front mirror and never enters, and the leakage, a little of the light stored inside that seeps back out through the front mirror.

Draw each light wave as an arrow: its length is the wave’s strength, its angle is the wave’s timing — its phase. Adding waves means placing arrows tip to tail. On resonance the two arrows are equally long and point in opposite directions, so they cancel and the reflection goes dark. Zero is zero at any brightness, so flicker can no longer fool us. But…

Your task Detune the laser and watch the leakage arrow swing. Where does the sum vanish?

Predict The detector reads 40% reflected. Which way did the laser slip?

What you just saw

Move the detuning slider and the explanation appears here.

The reflected light falls to zero on resonance and rises the same way on both sides — brightness is symmetric. Its phase is not: the sum arrow points up above resonance and down below, and the phase jumps by 180° as you cross. That phase is exactly the “which way” information a lock needs — if only we could measure it.

04 · Figs 2 & 6

Wiggle and listen

Here is a way to read the slope of that dip: wiggle the laser’s frequency a little, back and forth, and watch the reflected light. Above resonance the reflection rises and falls in step with the wiggle. Below resonance it does the opposite. Right on resonance it barely changes — and what change there is comes at twice the wiggle rate, because a nudge either way brightens it.

A mixer multiplies the wiggle by the detector’s signal, and a low-pass filter averages the product. In step gives a positive average; opposite gives a negative one; on resonance it averages to zero. Sweep the laser across the dip and these averages trace out its slope — an error signal that tells how far, and which way.

Your task Move the operating point across the dip — or let it sweep — and watch the error signal build, point by point.

What you just saw

Build a few points of the error signal and the explanation appears here.

The averaged product is the slope of the dip: zero on resonance, positive above, negative below, and its size says how far off you are. That antisymmetric curve is the error signal of Fig. 6. This picture works when the wiggle is slow enough for the light stored in the cavity to keep up at every instant.

05 · Figs 7–10

Faster than the mirrors can follow

Real setups wiggle far faster — so fast the stored light cannot keep up. Surprisingly, the method still works there, and usually works better: less noise, a faster servo.

In practice the wiggle is a rapid jiggle of the laser’s phase from a crystal modulator (a Pockels cell). That splits the light into three: the original carrier and two faint sidebands, one a frequency Ω above it and one Ω below. With a fast wiggle the sidebands land far outside the cavity’s window and simply bounce off the front mirror, while the carrier sits on resonance.

Now picture each as an arrow in a frame that turns along with the carrier. The carrier’s reflection stands still: a short arrow pointing up above resonance, down below. The sidebands spin in opposite directions, so together they bob up and down. The detector sees the length of the total arrow, which now flickers at the wiggle rate — and the timing of that flicker flips across resonance. The mixer reads it.

Your task Sweep the detuning across resonance and watch the red arrow flip. Then knock the phase shifter off and bring it back by eye until the curve looks clean again — and try a slow wiggle.

What you just saw

Sweep the detuning and the explanation appears here.

The reflected carrier beats against the sidebands. Whether it points up or down decides the timing of the flicker at Ω, and mixing with the oscillator turns that timing into a signed error. With a slow wiggle the cosine part of Eq. 4.3 survives and the curve looks like Fig. 6. With a fast wiggle the sine part survives: the curve is steep through resonance, with small opposite-sign features where a sideband itself crosses resonance (Fig. 7). Between those features the sign always points home.

06 · Sections V–VI

Close the loop

Now feed the error signal back to the laser’s tuning. Near resonance the signal is a straight line, ε = D·δf, where δf is how far the laser is off and D = −8√(PcPs)/δν is the frequency discriminant (Eq. 5.2): Pc and Ps are the carrier and sideband powers and δν the cavity’s linewidth. A straight line is what standard control theory needs.

And because we lock where the reflected carrier vanishes, many imperfections drop out to first order: the laser’s power, the photodiode’s response, the modulation depth, the mixer’s phase and the modulation frequency (Section VI.A). What the lock still feels: fluctuations in sideband power at Ω, and shot noise.

Your task Turn on brightness flicker, then switch between the two locks. Which one keeps the trail on the line?

What you just saw

Compare the two locks with flicker on and the explanation appears here.

With flicker, the slope lock wanders, because brightness changes move its lock point. The Pound–Drever–Hall lock stays put: flicker only changes how hard its error signal pushes, never where it crosses zero. Try the gain, too: more gain squeezes the laser’s own drift harder in both locks, but no amount of gain removes the slope lock’s flicker error — the servo is faithfully following a misleading signal.

Faster than the cavity

One more advantage (Section V.C). The light stored in the cavity is an average of the laser’s recent past — a steady reference. The prompt reflection is the laser right now. The error signal compares the two, so a sudden jump shows up at once, long before the stored light — and the light coming through the cavity — has caught up. The lock is not limited by how slowly the cavity responds.

Your task Jump the laser’s frequency and compare when each trace starts to move.

07 · Sections VI–VII

How still can it get?

With enough gain, the servo squashes almost any noise in the laser. What it cannot squash is noise in the measurement itself — it would faithfully copy that onto the laser. The deepest such noise is shot noise: light arrives as separate photons, so even a perfect detector sees a little graininess.

Black works out the floor this sets (Section VI.B). Surprisingly, it depends on the power in the carrier, not in the sidebands. And it agrees with Mizuno’s general “sensitivity theorem” for interferometers (Section VI.C).

Sf = √(hc³) / (8 𝓕 L √(λ Pc))

SL = √(hc) √λ / (8 𝓕 √Pc)

𝓕 finesse · L cavity length · λ wavelength · Pc carrier power · h Planck’s constant · c speed of light

How much light to put in the sidebands?

The steeper the error signal, the better the servo can fight technical noise. Its slope D grows with √(PcPs), so you need both carrier and sidebands — and the modulation depth β decides how the light is shared. A simple estimate (Section VII.B) puts the best split at Ps/Pc = 1/2. Doing it carefully with Bessel functions gives β = 1.08 and Ps/Pc ≈ 0.42. The maximum is broad.

What you just saw

Move β and the explanation appears here.

Too little modulation and the sidebands are too faint to beat against; too much and the carrier is drained (light also leaks into higher sidebands). The slope peaks at β ≈ 1.08, but anywhere from about 0.8 to 1.4 keeps more than 90% of it — the paper’s “fairly broad” maximum.

08 · Appendix A

Go deeper

As the laser’s frequency changes, the reflection coefficient F always traces a circle in the complex plane (Appendix A). For a front mirror with amplitude reflectivity r1 (and no loss) and an end mirror r2, the circle passes through zero — total darkness on resonance — only when r1 = r2: the cavity is impedance matched (critically coupled). If the end mirror reflects more, the cavity is overcoupled and the circle encloses the origin; if it reflects less, undercoupled, and some light always comes back. Overcoupling plays a central role in interferometric gravitational-wave detectors.

F(ω) = Eref/Einc = r(eiω/Δνfsr − 1) / (1 − r² eiω/Δνfsr)

Eq. 4.1 How much of the light, and with what timing, comes back from a symmetric lossless cavity with mirror reflectivity r. Δνfsr = c/2L.

Einc = E0[J0(β)eiωt + J1(β)ei(ω+Ω)t − J1(β)ei(ω−Ω)t]

Eq. 4.2 Phase modulation splits the beam into a carrier and two sidebands Ω away; β sets how much light goes into the sidebands.

Pref = Pc|F(ω)|² + Ps{|F(ω+Ω)|² + |F(ω−Ω)|²} + 2√(PcPs){Re[χ] cos Ωt + Im[χ] sin Ωt} + (2Ω terms), χ = F(ω)F*(ω+Ω) − F*(ω)F(ω−Ω)

Eq. 4.3 What the detector sees. The part flickering at Ω carries the carrier’s phase; the mixer picks it out.

χ ≈ −2i Im{F(ω)}

Eq. 5.1 With fast modulation the sidebands are fully reflected, so only the sine term survives and the error measures the imaginary part of F.

ε = D δf, D ≡ −8√(PcPs)/δν

Eq. 5.2 Near resonance the error is proportional to the frequency offset; D, the frequency discriminant, is its slope. δν is the cavity linewidth.

Sf = √(hc³) / (8𝓕L√(λPc)), SL = √(hc)√λ / (8𝓕√Pc)

Section VI.B, Eq. 6.1 The shot-noise floor on the laser’s frequency and on the cavity’s length, per √Hz. 𝓕 is the finesse.

09

Back to the drifting laser

So how does the laser know which way to turn? It doesn’t have to. The cavity knows where the target is; the reflected light knows how far off you are; the timing of the sidebands’ flicker knows which way; and the servo does the turning — fast, and blind to the laser’s brightness.

Cavity the ruler: it passes light only in narrow windows.

Dark reflection a target that is zero at any brightness.

Sidebands + mixer a sense of direction, from the timing of a fast flicker.

Servo the steering: it turns the knob to keep the error at zero.

Shot noise the floor: photons themselves set how still it can get.

What it makes possible

The paper notes that the technique is used extensively in atomic physics, is an essential part of interferometric gravitational-wave detection, and has been used to turn a commercial laser into a frequency standard as relatively stable as a pulsar.

Later: LIGO’s detectors made the first direct detection of gravitational waves in 2015.

1946 R. V. Pound stabilises microwave oscillators with a similar idea.

1980 R. W. P. Drever proposes the optical version for gravitational-wave detection; early development in Glasgow, detailed experiments with Jan Hall’s group at JILA.

Soon after G. C. Bjorklund independently proposes the same idea for FM spectroscopy.

1999 Eric Black writes this primer for the LIGO Project.

Source

Eric D. Black. An Introduction to Pound-Drever-Hall laser frequency stabilization. LIGO Project, California Institute of Technology, LIGO-P990042-00-D (October 4, 1999).

A version was later published in the American Journal of Physics 69, 79 (2001).

All figures on this page are live simulations recomputed from the paper’s equations, after the paper’s Figs 1–12; parameters are illustrative except where the paper’s worked example is stated.