ADVANCED LASER STABILISATION

Pound–Drever–Hall Locking

A complete derivation — from the Fabry–Pérot cavity reflection coefficient through phase modulation, the phasor picture, and all the way to the electronic servo chain. Every step interactive. Based on E. D. Black, Am. J. Phys. 69, 79 (2001).

<1 kHzachievable linewidth
PDHPound–Drever–Hall
Im[F]error signal origin
∞ BWnot cavity-limited
Jump to: FP Cavity Complex Plane Why Intensity Fails Phase Modulation Error Signal Phasor Picture PDH Explorer Electronics Practical Offset Lock Noise Deep Questions
§0 — The Problem

Why Lock a Laser to a Cavity?

Free-running diode lasers are notoriously unstable — they drift, they jitter, and they cannot achieve the spectral purity demanded by precision AMO experiments.

Free-Running Laser Noise

A free-running external-cavity diode laser (ECDL) typically drifts by 1–10 MHz on minute timescales due to fluctuations in injection current, chip temperature, and external-cavity mechanical length. The instantaneous linewidth (coherence time ~μs) is 100 kHz–1 MHz.

For comparison: the Cs D₂ natural linewidth is Γ/2π = 5.2 MHz, the Li D₁ linewidth is 5.9 MHz, and the Cs electric-quadrupole transition at 685 nm is only Γ/2π ≈ 117 Hz — six orders of magnitude narrower.

Unstabilised operation means you cannot reliably sit on a transition, cannot do coherent spectroscopy, and cannot reproduce results day-to-day.

The PDH Solution

The Pound–Drever–Hall technique (Drever et al. 1983; Black 2001) stabilises the laser to a reference Fabry–Pérot cavity whose length is mechanically and thermally isolated. The cavity acts as a flywheel, providing a stable short-term frequency reference.

A key insight: the cavity's reflected field — not its transmission — carries a dispersive (antisymmetric) error signal. Using RF phase modulation and heterodyne detection, the PDH scheme extracts a signal whose slope is set by the cavity finesse: higher finesse = steeper slope = tighter lock.

Achieved linewidths: 1 kHz for commercial ECDL + moderate cavity; sub-Hz for research-grade setups. Our 685 nm laser achieves ~1 kHz locked to a ULE cavity with finesse 𝒻 ≈ 1.5 × 10⁴.

§1 — The Reference

The Fabry–Pérot Cavity

Before deriving the PDH error signal, we need to understand exactly how a Fabry–Pérot cavity reflects and transmits light as a function of frequency.

Cavity Geometry and Round-Trip Phase

A linear cavity of length \(L\) with two mirrors (amplitude reflectivity \(r\), transmissivity \(t = \sqrt{1-r^2}\) for a lossless mirror) supports resonant modes separated by the free spectral range:

$$\nu_{\rm FSR} = \frac{c}{2nL}$$

Light accumulates a round-trip phase \(\phi = 2\pi\nu/\nu_{\rm FSR}\). Resonance occurs when this phase is a multiple of \(2\pi\), constructing a standing wave inside the cavity. The finesse \(\mathcal{F}\) measures the sharpness of these resonances:

$$\mathcal{F} = \frac{\pi\sqrt{R}}{1-R} = \frac{\pi r}{1-r^2} \approx \frac{\pi}{1-R} \quad (R \to 1)$$

where \(R = r^2\) is the power reflectivity. The cavity linewidth (FWHM) is:

$$\delta\nu_{\rm cav} = \frac{\nu_{\rm FSR}}{\mathcal{F}}$$

Reflection Coefficient

The total reflected field is the coherent sum of the promptly reflected beam (bounces off the first mirror without entering) and the leakage field (cavity field that leaks back through the input mirror). For a lossless symmetric cavity, the complex reflection coefficient is:

$$F(\nu) = \frac{r\left(e^{i\phi} - 1\right)}{1 - r^2 e^{i\phi}}, \qquad \phi = \frac{2\pi\nu}{\nu_{\rm FSR}}$$

The reflected power \(|F(\nu)|^2\) is the familiar Airy function — it reaches zero at each resonance (complete destructive interference) and approaches \(r^2 \approx 1\) far between resonances. The transmitted power is \(T(\nu) = 1 - |F(\nu)|^2\) (Airy transmission function) peaked at resonance.

Fig 1 — Airy Function: Cavity Reflection & Transmission
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Physical Interpretation of the Airy Dip

At resonance, the promptly reflected beam and the leakage beam are equal in amplitude and exactly 180° out of phase — they cancel completely. Off resonance, the phase mismatch is incomplete, and some light reflects. The high-finesse cavity makes this cancellation exquisitely sensitive to frequency, giving a very narrow dip. The ring-down time \(\tau_c = \mathcal{F}/(π\,\nu_{\rm FSR})\) is the 1/e decay time of the cavity field — it sets how long the cavity "remembers" the incident frequency.

$$\tau_c = \frac{\mathcal{F}}{\pi\,\nu_{\rm FSR}} = \frac{\mathcal{F}\cdot 2L}{\pi c}$$
Key number: For our 685 nm ULE cavity: L = 77.5 mm, 𝒻 = 1.5 × 10⁴, ν_FSR ≈ 1.93 GHz, δν_cav ≈ 130 kHz, τ_c ≈ 2.5 μs.
§2 — Geometry of Reflection

F(ν) in the Complex Plane

One of the most elegant properties of the Fabry–Pérot reflection coefficient is that as the frequency varies, F(ν) traces a perfect circle in the complex plane.

The Circle Theorem

As the laser frequency \(\nu\) is tuned continuously, the complex number \(F(\nu)\) traces a circle in the complex plane. For our symmetric lossless cavity, this circle passes through the origin (\(F = 0\) at resonance) and through \(-r^2/(1+r^2) \cdot 2 \approx -1\) at half-FSR (completely anti-resonant).

The magnitude \(|F(\nu)|\) gives the reflected amplitude, and the phase of \(F\) is antisymmetric about resonance — this is what allows us to determine which side of resonance the laser is on.

$$F(\nu) \in \left\{\,z \in \mathbb{C} \;:\; \left|z - z_0\right| = \rho\,\right\} \quad \text{(a circle in } \mathbb{C}\text{)}$$
Fig 2a — F(ν) Tracing a Circle (Complex Plane)
Fig 2b — Phase of F(ν): The Antisymmetric Discriminator

Why Phase is the Key Quantity

Look at Fig 2b: the phase of \(F(\nu)\) is antisymmetric about the resonance — it is positive above resonance, negative below, and zero exactly on resonance. This antisymmetry is exactly what we need for a frequency discriminator. The reflected power \(|F|^2\) (Fig 1) is symmetric and cannot distinguish left from right. The phase breaks this symmetry.

The PDH technique is essentially a method to measure the phase of \(F(\nu)\) using RF heterodyne detection. Near resonance, the imaginary part of \(F(\nu)\) linearises:

$$\mathrm{Im}[F(\nu)] \approx \frac{r}{1-r^2} \cdot \frac{2\pi\Delta\nu}{\nu_{\rm FSR}} = \frac{\mathcal{F}}{\pi} \cdot \frac{\Delta\nu}{\nu_{\rm FSR}} \quad (\Delta\nu \ll \delta\nu_{\rm cav})$$
Note: Near resonance, Re[F] ≈ 0 (it's quadratic in Δν) while Im[F] is linear. The PDH error signal measures Im[F] — so it's automatically linear near lock.
§3 — The Symmetry Problem

Why Reflected Intensity Alone Fails

A natural first idea: lock the laser to the minimum of reflected power. This doesn't work — and understanding why leads directly to the PDH solution.

The Symmetry Argument

The Airy dip in reflected power (Fig 1) is symmetric about resonance: \(|F(\nu_{\rm res} + \delta)| = |F(\nu_{\rm res} - \delta)|\). Sitting exactly on the minimum, a small frequency perturbation \(\delta\nu\) changes the reflected power by \(\propto \delta\nu^2\) — the first derivative is zero at resonance.

This means: (a) the error signal has zero slope at the lock point — terrible for servo gain; (b) you cannot tell whether the laser has drifted above or below resonance from the intensity alone — no sign information.

What you need is a signal that is odd in frequency detuning — positive on one side, negative on the other, with a steep slope through zero at resonance. This is the derivative of the reflected power, or equivalently, Im[F(ν)].

The Conceptual Fix: Frequency Dithering

The simplest solution: modulate the laser frequency sinusoidally at a low frequency Ω, and look at how the reflected power responds.

  • Above resonance: reflected power increases with ν → response is in-phase with the dither.
  • Below resonance: reflected power decreases with ν → response is 180° out-of-phase.
  • On resonance: dP/dν = 0 → no response at Ω (but response at 2Ω).

Demodulating the reflected signal at Ω recovers a signal proportional to \(dP_{\rm ref}/d\nu\) — an antisymmetric error signal. This is the "slow modulation" (adiabatic) limit of PDH.

Limitation of slow modulation: If the dither frequency Ω is comparable to the cavity linewidth δν_cav, the cavity cannot respond fast enough — the standing wave doesn't have time to build up/decay. The PDH insight is to use fast modulation (Ω ≫ δν_cav) and RF detection, which bypasses this limitation entirely.
§4 — Creating the Probe

Phase Modulation and Sidebands

Instead of slowly dithering the laser frequency, PDH uses an EOM to imprint RF phase modulation — creating sidebands that act as a fixed phase reference.

The Electro-Optic Modulator (EOM)

An EOM (or Pockels cell) applies a sinusoidally varying voltage \(V = V_0\sin\Omega t\) across an electro-optic crystal, modulating the crystal's refractive index and hence the optical path length. The transmitted field acquires a phase modulation:

$$E_{\rm in}(t) = E_0\, e^{i(\omega t + \beta\sin\Omega t)}$$

where \(\beta = \pi V_0/V_\pi\) is the modulation depth (typically 0.1–1 rad) and \(V_\pi\) is the half-wave voltage of the EOM. Expanding using the Jacobi–Anger identity:

$$e^{i\beta\sin\Omega t} = \sum_{n=-\infty}^{\infty} J_n(\beta)\, e^{in\Omega t}$$

For small modulation depth \(\beta \ll 1\), only the zeroth and first-order Bessel functions matter:

$$E_{\rm in} \approx E_0\left[\underbrace{J_0(\beta)\,e^{i\omega t}}_{\rm carrier} + \underbrace{J_1(\beta)\,e^{i(\omega+\Omega)t}}_{\rm upper\;sideband} - \underbrace{J_1(\beta)\,e^{i(\omega-\Omega)t}}_{\rm lower\;sideband}\right]$$

We now have three distinct frequency components incident on the cavity: the carrier at \(\omega\), and two sidebands at \(\omega \pm \Omega\). Their powers are \(P_c = J_0^2(\beta)P_0\) and \(P_s = J_1^2(\beta)P_0\) each. For \(\beta \ll 1\): \(J_0(\beta) \approx 1 - \beta^2/4\), \(J_1(\beta) \approx \beta/2\), so the sideband power fraction is approximately \(P_s/P_0 \approx \beta^2/4\).

Fig 3 — Sideband Spectrum: Carrier + Sidebands

The Critical Frequency Placement

The PDH scheme works best when the sidebands are placed outside the cavity resonance: \(\Omega \gg \delta\nu_{\rm cav}\). In this regime:

  • The carrier at \(\omega\) is near resonance and interacts strongly with the cavity — its reflection is frequency-dependent.
  • The sidebands at \(\omega \pm \Omega\) are far from any resonance — they are totally reflected with reflection coefficient \(F \approx -1\) (the minus sign comes from the prompt reflection phase).

The sidebands thus act as a fixed phase reference that does not depend on the carrier frequency. The beat between the reflected carrier and the reflected sidebands carries the phase information we need.

Design rule: Choose Ω such that Ω/δν_cav ≫ 1 (typically Ω ≈ 5–50 × δν_cav). For our 685 nm cavity (δν_cav ≈ 130 kHz), Ω = 27 MHz gives Ω/δν_cav ≈ 200 — well in the fast-modulation regime. For a lower-finesse cavity (δν_cav ~ 10 MHz), you need Ω > 100 MHz.
§5 — The Core Calculation

Deriving the PDH Error Signal

With the three-component incident field and the cavity reflection coefficient in hand, we can compute the exact reflected power and extract the error signal.

Reflected Field

Each spectral component is reflected with its own reflection coefficient \(F(\omega)\), \(F(\omega+\Omega)\), \(F(\omega-\Omega)\). The total reflected field is:

$$E_{\rm ref} = E_0\!\left[F(\omega)\,J_0(\beta)\,e^{i\omega t} + F(\omega{+}\Omega)\,J_1(\beta)\,e^{i(\omega+\Omega)t} - F(\omega{-}\Omega)\,J_1(\beta)\,e^{i(\omega-\Omega)t}\right]$$

Reflected Power and Beat Terms

The photodetector measures the power \(P_{\rm ref} = |E_{\rm ref}|^2\). Expanding and collecting terms by oscillation frequency:

$$P_{\rm ref}(t) = \underbrace{P_c|F(\omega)|^2 + P_s|F(\omega+\Omega)|^2 + P_s|F(\omega-\Omega)|^2}_{\rm DC} + \underbrace{2\sqrt{P_c P_s}\,\mathrm{Re}\!\left[\left(F(\omega)F^*(\omega{+}\Omega) - F^*(\omega)F(\omega{-}\Omega)\right)e^{i\Omega t}\right]}_{\rm beat\;at\;\Omega} + \underbrace{\cdots}_{2\Omega\;\rm terms}$$

The term oscillating at \(\Omega\) is the one we want. Writing \(\mathcal{Z} \equiv F(\omega)F^*(\omega{+}\Omega) - F^*(\omega)F(\omega{-}\Omega)\):

$$P_\Omega(t) = 2\sqrt{P_c P_s}\left[\mathrm{Re}(\mathcal{Z})\cos\Omega t - \mathrm{Im}(\mathcal{Z})\sin\Omega t\right]$$

Demodulation: Mixing with the Local Oscillator

We multiply the photodetector output by a reference \(\cos(\Omega t + \varphi)\) from the same local oscillator that drives the EOM, then low-pass filter. Using \(\langle\cos\Omega t\cdot\cos(\Omega t+\varphi)\rangle_{\rm LPF} = \tfrac{1}{2}\cos\varphi\) and \(\langle\sin\Omega t\cdot\cos(\Omega t+\varphi)\rangle_{\rm LPF} = -\tfrac{1}{2}\sin\varphi\):

$$\boxed{\varepsilon(\omega,\varphi) = \sqrt{P_c P_s}\left[\mathrm{Re}(\mathcal{Z})\cos\varphi + \mathrm{Im}(\mathcal{Z})\sin\varphi\right]}$$

Fast-Modulation Limit: \(\Omega \gg \delta\nu_{\rm cav}\)

When the sidebands are far outside the cavity resonance, \(F(\omega\pm\Omega) \approx -1\) (fully reflected with a \(\pi\) phase shift for a high-finesse lossless cavity). Then:

$$\mathcal{Z} \approx F(\omega)(-1) - F^*(\omega)(-1) = -F(\omega) + F^*(\omega) = -2i\,\mathrm{Im}[F(\omega)]$$

So \(\mathrm{Re}(\mathcal{Z}) \approx 0\) and \(\mathrm{Im}(\mathcal{Z}) \approx -2\,\mathrm{Im}[F(\omega)]\). With optimal demodulation phase \(\varphi = \pi/2\) (sine demodulation):

$$\varepsilon(\omega) \approx -2\sqrt{P_c P_s}\cdot\mathrm{Im}(\mathcal{Z})\cdot\sin\frac{\pi}{2} = 4\sqrt{P_c P_s}\,\mathrm{Im}[F(\omega)]$$

Linear Region Near Resonance

Near resonance, \(\mathrm{Im}[F(\omega)] \approx \frac{\mathcal{F}}{\pi}\frac{\Delta\nu}{\nu_{\rm FSR}}\), giving a linear error signal:

$$\varepsilon \approx \frac{4\mathcal{F}}{\pi\nu_{\rm FSR}}\sqrt{P_c P_s}\cdot\Delta\nu = \underbrace{\frac{8\sqrt{P_c P_s}}{\delta\nu_{\rm cav}}}_{D \;=\;\rm discriminant}\cdot\Delta\nu$$
The discriminant \(D\) has units of V/Hz. It grows with finesse (smaller \(\delta\nu_{\rm cav}\)) and with optical power. For \(P_c = 1\,\text{mW}\), \(P_s = 0.1\,\text{mW}\), \(\delta\nu_{\rm cav} = 130\,\text{kHz}\): \(D = 8\sqrt{10^{-3}\times 10^{-4}}/130\times10^3 \approx 1.7\,\mu\text{V/Hz}\).

Slow-Modulation Limit: \(\Omega \ll \delta\nu_{\rm cav}\)

When \(\Omega\) is much smaller than the cavity linewidth, the cavity can follow adiabatically. In this limit:

$$\mathcal{Z} \approx \frac{d|F|^2}{d\nu}\cdot\Omega \quad\Rightarrow\quad \varepsilon \approx 2\sqrt{P_c P_s}\,\frac{d|F(\omega)|^2}{d\nu}\cdot\Omega$$
The slow-modulation error signal is proportional to the derivative of reflected power — exactly the conceptual model from §3. The fast-modulation PDH signal goes beyond this, accessing frequencies faster than the cavity storage time.
§6 — Rotating Frame Intuition

The Phasor Picture

Black's paper offers a beautiful geometric picture in the rotating reference frame that makes the physics immediately clear without any algebra.

The Rotating Reference Frame

Boost to a frame rotating at the carrier frequency \(\omega\). In this frame, the incident carrier appears as a fixed vector along the real axis. The sidebands at \(\omega \pm \Omega\) become vectors rotating at \(\pm\Omega\):

  • Upper sideband \((\omega+\Omega)\): rotates counter-clockwise (CCW) at \(\Omega\).
  • Lower sideband \((\omega-\Omega)\): rotates clockwise (CW) at \(\Omega\).

When both sidebands are totally reflected (\(F = -1\)), they each acquire a \(\pi\) phase flip. Their vector sum oscillates along the imaginary axis:

$$E_{\rm sidebands}(t) = -J_1 e^{i\Omega t} + J_1 e^{-i\Omega t} = -2i J_1 \sin\Omega t \quad \text{(purely imaginary oscillation)}$$

Now add the reflected carrier. Near resonance with the laser slightly above resonance, \(F(\omega) \approx i\,\mathrm{Im}[F]\) (purely imaginary, positive). This small imaginary vector adds to the sideband oscillation during half its cycle and subtracts in the other half — creating an asymmetry in the intensity that oscillates at \(\Omega\).

Below resonance, Im\([F] < 0\) and the asymmetry flips sign — giving the opposite phase at \(\Omega\). This is the origin of the antisymmetric error signal.

Fig 5 — Phasor Animation: Rotating Frame at Carrier Frequency
Photodetector signal (reflected power) vs time. The Ω-frequency component is the PDH beat signal.
Upper sideband (+Ω)
Lower sideband (−Ω)
Sideband sum
Reflected carrier
Total reflected field
§7 — Everything Together

PDH Explorer: Full Interactive Tool

Adjust all parameters simultaneously and see how the Airy dip, reflected signal, and PDH error signal change.

Fig 6 — PDH Master Explorer
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§8 — Implementation

The Electronic Chain

From laser to lock: every component in the PDH signal chain has a specific role.

LASER ECDL Isolator 30–40 dB EOM phase mod. at Ω Mode Match 2-lens telescope TEM₀₀ → cavity ULE CAVITY L = 77.5 mm 𝒻 = 1.5 × 10⁴ δν ≈ 130 kHz in vacuum + vibrational iso. QWP + PBS deflect reflected beam Fast PD ~GHz bandwidth Mixer × RF demodulate LPF error signal ε Local Oscillator RF synth at Ω (e.g. 27 MHz) + phase shifter φ PID Servo fast + slow fast (current) slow (PZT) transmitted beam (not used for PDH) reflected beam

Component-by-Component

  • Faraday isolator (30–40 dB): blocks retro-reflected light from reaching the laser diode. Without it, back-reflections destabilise the laser's mode and make locking impossible.
  • EOM (electro-optic modulator): a Pockels cell driven at Ω (typically 10–100 MHz) imprints phase modulation. The RF drive voltage \(V_0\) sets \(\beta = \pi V_0/V_\pi\). Keep β small (0.2–0.5 rad) to maximise carrier power.
  • Mode-matching telescope: two lenses that transform the free-space Gaussian beam to match the cavity's TEM₀₀ mode (same waist size and position). Poor mode matching reduces coupling, signal size, and PDH slope.
  • QWP + PBS reflector: picks off the reflected beam without interfering with the input. QWP rotates the polarisation by 90° after two passes (reflection), allowing the PBS to separate the reflected beam cleanly. Acts as a "Faraday-free" optical isolator — see §9.
  • Fast photodetector: bandwidth must exceed Ω. For Ω = 27 MHz, need BW ≥ 50 MHz; for Ω = 200 MHz, need BW ~ 500 MHz. InGaAs or Si PIN detectors typical.
  • Mixer (RF multiplier): multiplies PD signal × LO signal. Output contains sum and difference frequencies — LPF keeps only the DC / low-frequency difference (the error signal).
  • Phase shifter: adjusts relative phase between PD signal and LO before mixing. Critical for extracting the Im[F] dispersive component; wrong phase gives absorptive (Re[F]) signal with worse properties.
  • LPF (low-pass filter): cuts the 2Ω component, passing only the audio-frequency error signal to the servo.
  • PID servo: two feedback paths — fast (current modulation, bandwidth 100 kHz–1 MHz) and slow (PZT, bandwidth 100 Hz–10 kHz). See §9 for servo design.

Why \(\varphi = 90°\) for PDH?

The carrier-sideband beat produces both cosine and sine terms at Ω (Re[ℤ] and Im[ℤ] respectively). In the fast-modulation limit:

  • \(\varphi = 90°\) (sine demodulation): extracts Im[ℤ] ∝ Im[F] — the dispersive signal, antisymmetric, linear at resonance. ✓ PDH-optimal.
  • \(\varphi = 0°\) (cosine demodulation): extracts Re[ℤ] ≈ 0 near resonance — the absorptive signal, nearly zero at the lock point. ✗

In practice, path-length delays between the EOM drive and the photodetector shift the effective phase. You scan the laser across a resonance, adjust the phase shifter until you see a clean antisymmetric (dispersive) error signal, and you're done.

Common pitfall — wrong demodulation phase: If φ is off by 45°, you get a mixture of absorptive and dispersive features. At 90° offset from optimal you see the absorptive signal (zero slope at resonance — the lock will be very noisy).

Try it in Fig 6: set φ = 0° and watch the error signal shape change.
§9 — In the Lab

Practical Implementation Guide

Everything that matters when you actually set up a PDH lock.

Spatial Mode Matching to the Cavity

The cavity only accepts its own TEM₀₀ mode. The fraction of power coupled into the cavity mode is the mode-matching efficiency:

$$\eta_{\rm MM} = \left(\frac{2 w_1 w_2}{w_1^2 + w_2^2}\right)^2$$

where \(w_1\) is the incident beam waist and \(w_2\) is the cavity waist at the input mirror. This formula applies when the waist positions are coincident; any longitudinal offset reduces \(\eta\) further.

Practical procedure:

  • Calculate the cavity waist: for a plano-plano linear cavity, \(w_{\rm cav} = \sqrt{\lambda L/\pi} \cdot (g_1 g_2 (1-g_1 g_2))^{1/4}\) where \(g_i = 1 - L/R_i\), but for a plano-concave cavity the waist sits at the flat mirror.
  • Use an ABCD matrix to propagate your beam from the fiber/collimator output through the telescope lenses to the cavity input. Adjust lens positions to minimise beam waist mismatch.
  • A quick check: scan the laser across the cavity and look at the transmitted power. If you see only one strong resonance and very weak higher-order modes (HG₁₀, etc.), mode matching is good.
Target: Mode-matching efficiency η > 80%. Below 50%, the PDH signal is weak and lock acquisition is unreliable.

The QWP as a Faraday-Free Reflector

Why not just use a normal beamsplitter to pick off the reflected beam? Because a 50:50 BS wastes half the input power. Instead:

  • Start with PBS (linearly polarised input, say H-polarisation).
  • After PBS, place a QWP with fast axis at 45° to H. The beam is converted to circular polarisation going into the cavity.
  • The cavity reflects with the same handedness. The reflected beam passes through the QWP again, rotating the polarisation by another 45° → now V-polarised.
  • V-polarised light is reflected by the PBS into the detector path. The input H-beam and the reflected V-beam are now completely separated.

This works because two passes through a QWP at 45° = one HWP (90° rotation). It's efficient (no power wasted) but not as robust as a true Faraday isolator (it's sensitive to the QWP orientation).

Choosing the Cavity Finesse

Higher finesse gives a steeper PDH discriminant (larger D) — but there are trade-offs:

Finesse 𝒻 δν_cav (𝒻_FSR=1.93 GHz) D slope ∝ 1/δν Pros Cons
100–500 4–20 MHz ×1 Easy mode-matching; large capture range Weak signal; limited linewidth reduction
1,000–5,000 400 kHz–2 MHz ×10 Good for locking near D-lines; compatible with SAS pre-lock Requires vacuum isolation
10,000–50,000 40–200 kHz ×100 Sub-kHz linewidth; our 685 nm laser Small capture range; sensitive to alignment; higher mirror cost
>100,000 <20 kHz ×1000+ Sub-Hz linewidth; frequency comb / gravitational wave detectors Alignment extremely critical; cavity thermal drift dominates; superpolished mirrors

Servo Design: Fast and Slow Paths

The PDH error signal goes to a servo (PID) that feeds back to the laser. Two feedback paths are almost always used:

  • Fast path → injection current: The laser frequency shifts by ~3–5 MHz/mA. Current modulation has a bandwidth of several MHz (limited by the diode's response time). This path suppresses high-frequency noise.
  • Slow path → external-cavity PZT: The grating angle tunes the frequency more coarsely (~1 GHz/V) but with much larger dynamic range. PZT actuators are limited to ~kHz bandwidth but can compensate slow thermal drift, keeping the fast path within its range.

Setting the crossover frequency between fast and slow paths requires careful Bode plot analysis. A common approach: set the slow integrator's corner frequency at ~10–100 Hz, fast path flat gain up to ~100 kHz with a roll-off above the cavity linewidth.

Lock Acquisition

The PDH error signal is only well-defined near a cavity resonance (within ~±3δν_cav). Acquiring the lock requires the laser to first be brought close enough to resonance:

  • Use a wavemeter to coarse-tune the laser to within a few cavity FSRs of the target resonance.
  • Ramp the slow PZT slowly while monitoring the transmitted power. When you see the Airy peak, the cavity is in transmission.
  • Enable the servo with the fast path only, and check that the error signal zeroes with the expected dispersive shape.
  • Once locked, engage the slow path to remove long-term drift from the fast path.
§10 — Tuning the Lock Point

Offset Locking to the Cavity

A PDH lock fixes the laser to a cavity mode — but what if you need the laser at a specific frequency offset from that mode?

Method 1: AOM in the Feedback Path

Place an AOM between the laser and the cavity. The laser frequency is \(\nu_{\rm laser}\), but the light entering the cavity is at \(\nu_{\rm laser} + \nu_{\rm AOM}\) (double-pass AOM: \(\nu_{\rm laser} + 2\nu_{\rm AOM}\)).

The PDH lock zeros the cavity input frequency to a resonance, so \(\nu_{\rm laser} + 2\nu_{\rm AOM} = \nu_{\rm cav}\), giving \(\nu_{\rm laser} = \nu_{\rm cav} - 2\nu_{\rm AOM}\). Tuning \(\nu_{\rm AOM}\) (typically 50–250 MHz) shifts the laser frequency continuously by up to ~500 MHz, while the PDH lock remains active.

Advantage: Smooth, electronically controlled tuning without breaking the lock.

Limitation: Tuning range limited by AOM bandwidth (~50–100 MHz from centre).

Method 2: EOM Sideband Lock

Add a second EOM (or use the first at high modulation depth) to create a sideband at frequency \(\nu_{\rm laser} + \Omega_{\rm offset}\). Lock this sideband to the cavity resonance instead of the carrier.

The carrier then sits at \(\nu_{\rm cav} - \Omega_{\rm offset}\), which is tunable by adjusting \(\Omega_{\rm offset}\) over a large range (potentially many GHz if the EOM supports it).

Advantage: Large tuning range (GHz scale).

Limitation: Only a fraction of the power is in the sideband being locked, so SNR is reduced.

In our lab (685 nm): The PDH lock fixes the laser to the ULE cavity mode. A double-pass AOM in the beam path then shifts the frequency to the target \(6S_{1/2} \to 5D_{5/2}\) transition frequency for each experimental sequence. This cleanly separates the frequency stabilisation function (PDH) from the tuning function (AOM).
§11 — Fundamental Limits

Noise and Performance Limits

How good can a PDH lock actually be? Several noise sources set the floor.

Shot Noise

The fundamental quantum limit. Photon shot noise on the photodetector current produces a frequency noise floor:

$$S_\nu^{1/2} = \frac{\delta\nu_{\rm cav}}{2\mathcal{F}}\sqrt{\frac{h\nu}{P_c}}$$

Achievable shot-noise limited linewidth: \(\lesssim 1\,\text{mHz}/\sqrt{\text{Hz}}\) for mW-class power and finesse \(\sim 10^4\).

RAM (Residual Amplitude Modulation)

An EOM that is not purely a phase modulator also amplitude-modulates the beam (due to etalon effects, birefringence, etc.). This RAM appears as a DC offset in the error signal, shifting the lock point away from the cavity resonance.

Suppression: temperature-control the EOM crystal; use an amplitude-stabilisation loop on the RAM signal.

Cavity Thermal Noise

Brownian motion of the mirror coatings and spacer causes length fluctuations even in a perfect vacuum. For a ULE spacer at 300 K:

$$S_L^{1/2} \sim 10^{-16}\,\text{m}/\sqrt{\text{Hz}}$$

This sets an ultimate limit of ~0.1–1 Hz/√Hz for sub-Hz laser stabilisation. Cryogenic cooling or crystalline coating mirrors push this lower.

Insensitivity of PDH to Common-Mode Noise Sources

One remarkable property of the PDH scheme (Black 2001, §V) is that the error signal is first-order insensitive to: laser power fluctuations, photodetector responsivity variations, modulation depth \(\beta\), the relative phase of the two LO signals, and the modulation frequency \(\Omega\). These would all affect a simple intensity-based lock significantly. The PDH scheme suppresses them because the error signal depends on the ratio of the beat signal to the total power — a normalised quantity — to first order.

§12 — Go Deeper

Deep Questions

Things you should be able to answer if you've really understood PDH locking.

What phase exactly is applied for demodulation — sine or cosine? And how do you choose? +

In the fast-modulation regime, the beat term in the reflected power oscillates as \(\propto \mathrm{Im}(\mathcal{Z})\sin\Omega t\) near resonance (the cos term vanishes because Re[ℤ] ≈ 0 there). To extract this sin Ωt component, you need to demodulate with sin Ωt — i.e., multiply by \(\sin(\Omega t + \varphi)\) and low-pass filter, with \(\varphi = 0\) (relative to the signal). In practice people label this a "90° phase shift" because the EOM drive is a cosine, so the LO needs a 90° shift to become a sine.

How to choose: scan the laser and adjust the phase shifter until the error signal looks like a clean antisymmetric dispersive curve (Fig 6). If the phase is 90° off, you'll see an absorptive (symmetric, going-negative-on-both-sides) signal instead.

How do you perform an offset lock to the cavity — i.e., lock the laser to a point that's not exactly on a resonance? +

There are two main methods: (1) Place an AOM in the beam path between the laser and the cavity. The PDH lock zeros the cavity-input frequency to a resonance, but the actual laser frequency is shifted by the AOM frequency. Tuning the AOM tunes the laser while keeping the lock active. (2) Use the EOM to create a sideband at the desired offset, and lock that sideband to the cavity instead of the carrier. Both methods are described in §10 above.

A common subtle point: if you put the AOM before the cavity in the input path, the reflected beam returns without the AOM shift. This breaks the PDH logic unless you either double-pass the AOM (so the reflected beam is shifted back) or put the AOM in the modulation path only.

What happens if you increase the modulation depth β beyond ~1 rad? +

For \(\beta \ll 1\): almost all power is in the carrier, with \(P_s/P_0 \approx \beta^2/4\). The error signal amplitude grows as \(\sqrt{P_c P_s} \approx \beta/2 \cdot P_0\) — linearly in β. For larger β, higher-order Bessel functions (J₂, J₃, ...) start carrying significant power, and the "sideband power" J₁²(β) actually begins to decrease after β ≈ 1.84 rad (where J₁ is maximum). Beyond β ≈ 2.4 rad, J₀ ≈ 0 — almost no power in the carrier, PDH slope collapses.

Additionally, second-order sidebands at ω ± 2Ω can land inside the cavity linewidth if 2Ω ≈ ν_FSR, which causes extra noise. In practice, β = 0.3–0.5 is optimal for PDH.

Why is a QWP needed — what exactly goes wrong without it? +

The purpose of the QWP + PBS combination is to cleanly separate the forward-going input beam from the backward-going reflected beam. Without the QWP:

  • If you use a 50:50 BS: you lose 75% of your useful power (50% on the way in, 50% on the way back to the detector).
  • If you use a PBS (input is H): the reflected beam from the cavity is also H-polarised (assuming no polarisation mixing in the cavity), and the PBS transmits it — both beams share the same path, so you can't easily pick off the reflected beam without also blocking the input.

The QWP converts H to circular on the way in. After reflection from the cavity, the circular polarisation is unchanged in handedness (for a metal-free dielectric mirror) but has now been reflected, which reverses its handedness. On the return pass through the QWP, the opposite handedness rotates to V-polarisation, which is reflected by the PBS into the detector arm. The 90° rotation from two QWP passes is the key: it converts an undetectable same-polarisation beam into an orthogonally-polarised beam that the PBS can route away.

How do you spatially mode-match to the cavity? What tools do you use? +

Theoretical approach: Use ABCD matrices (or the Gaussian beam propagation tool at RP Photonics) to propagate your beam through the optical system. Your goal: match the incoming beam's waist size and position to the cavity's fundamental mode. The cavity waist can be calculated from the mirror radii of curvature using the cavity stability criterion.

Practical approach:

  • Scan the laser across the cavity resonance and monitor the transmitted power on an oscilloscope.
  • Adjust the telescope (lens positions and beam size) until the TEM₀₀ peak is dominant and all TEM_{mn} higher-order mode peaks are minimised. These appear at frequencies shifted by \((m+n)\cdot\nu_{\rm FSR}/\mathcal{F}\cdot(\text{Gouy phase})\) — they show up as satellite peaks near the main resonance.
  • A cavity scan that shows only clean Airy peaks with no shoulders or satellite peaks indicates good mode matching (η > 90%).

The Lab Calculators page has a Gaussian beam propagation tool and cavity mode-matching calculator.

What determines the achievable linewidth? Is it always limited by the cavity? +

No — the cavity linewidth \(\delta\nu_{\rm cav}\) sets the slope of the error signal (higher finesse → steeper slope → tighter servo → narrower linewidth), but the achievable linewidth is also determined by:

  • Servo bandwidth: The servo must respond faster than the noise. A 1 MHz servo can suppress noise at up to ~100 kHz (due to finite gain margin). Beyond the servo bandwidth, noise is unsuppressed.
  • Shot noise floor: The error signal has a fundamental shot-noise floor that limits how precisely you can determine the frequency (see §11).
  • RAM (residual amplitude modulation): A DC offset in the error signal shifts the lock point by \(\Delta\nu_{\rm RAM} = \epsilon_{\rm RAM}/(D)\), where \(\epsilon_{\rm RAM}\) is the RAM amplitude and D is the discriminant slope.
  • Cavity thermal drift: Even with the laser locked, the cavity length drifts thermally, dragging the locked laser frequency with it. This is corrected by beating against an absolute reference (SAS or frequency comb) with a slow drift-correction loop.
PDH locks the laser to the cavity — but the cavity itself drifts. How do you get an absolute frequency reference? +

Correct — PDH provides a relative lock: the laser tracks the cavity. The cavity resonance itself drifts due to thermal expansion (CTE of the spacer), air pressure changes (even in vacuum, below mTorr), and mirror coating creep.

Solutions:

  • ULE glass spacer at zero-crossing temperature: ULE has a CTE that passes through zero near 5–25 °C depending on the batch. Operating at the zero-crossing dramatically reduces thermal sensitivity.
  • Beat against a frequency comb: Optical frequency combs (Ti:Sapphire or fibre) provide an absolute frequency reference. Beat the locked laser against the nearest comb tooth to measure and correct long-term drift.
  • Beat against a SAS-locked laser: If another laser in the same spectral region is SAS-locked, their beat note tells you the absolute frequency of the PDH-locked laser. A slow loop corrects residual cavity drift by feeding back to the PZT (or to an AOM in the path).

In our lab: the 685 nm PDH-locked laser is occasionally checked by scanning it across the \(6S_{1/2} \to 5D_{5/2}\) transition directly, which provides an absolute calibration to within the natural linewidth (~117 Hz). For day-to-day operation, the PDH lock alone provides sufficient stability.

The PDH signal uses the reflected beam — but what about the transmitted beam? Can you do PDH on the transmission? +

In principle, yes — but it performs worse. The transmitted power follows the Airy function and is symmetric about resonance (like the reflected power dip). To get a dispersive signal from transmission, you would still need phase-sensitive detection (FM spectroscopy), but you lose several advantages of PDH:

  • The transmitted signal falls to zero far from resonance, giving a noisy baseline.
  • For a high-finesse cavity, very little power is transmitted (T ≈ 1−R ~ 10⁻⁴ for R = 99.99%). You'd be detecting very few photons.
  • Transmission-based FM spectroscopy (frequency-modulation spectroscopy) works well for moderate-finesse cavities or for spectroscopy of atomic/molecular transitions.

The reflection-based PDH scheme is preferred because the reflected power near resonance is small but nonzero (it's the difference between prompt reflection and leakage), giving a high-slope dispersive signal against a low background — optimal for signal-to-noise.

How do you measure the cavity finesse in the lab? +

Three standard methods:

  • Cavity scan + Airy fit: Ramp the laser PZT linearly across a resonance; measure the transmitted (or reflected) power on an oscilloscope. Fit the Airy function to extract δν_cav and ν_FSR. This requires knowing the PZT tuning rate in Hz/V, calibrated from the known FSR.
  • Ring-down measurement: Quickly switch off the input beam while the cavity is resonating (using an AOM). The transmitted power decays exponentially with time constant τ_c = 𝒻/(π·ν_FSR). Measuring τ_c directly gives the finesse. For our cavity: τ_c = 15000/(π × 1.93×10⁹) ≈ 2.5 μs.
  • PDH discriminant calibration: With the laser locked, inject a known frequency modulation (e.g., scan the AOM by a known amount) and measure the resulting error signal amplitude. From ε = D × Δν, you can extract D and hence 𝒻.

Ring-down is usually most reliable and doesn't require calibration of other parameters.

What is "Pound locking" and how does it relate to PDH? Why is the technique named after three people? +

R. V. Pound (1946) invented the original technique for microwave cavities: he used RF sidebands and reflection detection to stabilise a microwave oscillator to a cavity. The concept is identical to PDH, but applied at microwave frequencies where electronics can directly detect the phase of the oscillation.

R. W. P. Drever and J. L. Hall (independently, early 1980s) extended Pound's idea to optical frequencies. The challenge is that optical frequencies (~10¹⁴ Hz) are far too high for electronics to follow directly — you can't just look at the phase of a visible light field. Their key insight: use RF phase modulation to create sidebands at frequencies accessible to electronics, then use the beating of those sidebands with the reflected carrier to recover the phase information.

The name "Pound–Drever–Hall" honours all three: Pound for the original microwave concept, Drever for applying it to lasers and high-finesse cavities (primarily in the context of gravitational wave detection), and Hall for the precision measurement physics and for demonstrating it could achieve sub-Hz linewidths.