🔐 Build 04 · Laser Locking Guide

Laser Stabilization Guide

Free-running diode lasers drift by MHz on minute timescales, comparable to atomic linewidths. Three complementary techniques solve this: SAS for absolute references, beat-note for precise offsets, and PDH for sub-kHz linewidths, plus the vacuum, thermal, and vibration engineering that makes the reference cavity itself stable enough to lock to.

Laser frequency stability is foundational to every AMO experiment, a coherence time of 100 μs requires laser linewidth below ~10 kHz, and Rydberg transitions need sub-MHz stability. Saturated absorption spectroscopy (SAS) provides Doppler-free atomic references with ~1 MHz width; Pound-Drever-Hall (PDH) locking to an optical cavity achieves sub-Hz linewidths by using phase-modulated sidebands to generate a dispersive error signal. This guide explains the physics of both techniques and gives you tools for exploring SAS spectra, PDH error signals, and offset locking, the three pillars of practical laser stabilization. A fourth tab, Ultrastable Cavities, goes one level deeper: PDH only works as well as the cavity you lock to, and that cavity's finesse, vacuum, thermal, and vibration engineering deserve their own treatment.

offset: —

P Gain 0.18

How hard the servo pushes back per MHz of error

D Damping 0.72

Resists velocity — prevents overshoot and ringing

✓ Well-tuned — fast snap, no overshoot

Try: high P + low D to see ringing · low P + high D for overdamped crawl · hit Perturb while locked

Saturated-Absorption Spectroscopy (SAS) Locking

SAS provides an absolute optical frequency reference tied directly to an atomic transition, no external frequency standard needed. The technique exploits the narrow Lamb dip hidden beneath the broad Doppler-broadened absorption profile of a room-temperature vapour cell.

How it works

Step 1, Counter-propagating beams.
A strong pump beam and weak probe beam travel in opposite directions through the vapour cell. An atom moving at velocity v sees the pump Doppler-shifted to ν₀(1 − v/c) and the probe to ν₀(1 + v/c).

Step 2, Velocity selectivity.
Only atoms with v ≈ 0 (zero-velocity class) are simultaneously resonant with both beams at the unshifted line centre ν₀. The pump saturates these atoms, depleting the ground-state population and creating a "Bennett hole" in the velocity distribution.

Step 3, Lamb dip.
The saturated zero-velocity atoms absorb less probe power at ν₀. A narrow Lamb dip appears in probe transmission, with FWHM ≈ Γ√(1 + I/Iₛₐₜ), set by the natural linewidth, not by the 300–500 MHz Doppler envelope.

Step 4, FM error signal.
The laser frequency (or probe EOM) is weakly frequency-modulated. Lock-in demodulation at the modulation frequency yields a dispersive first derivative of the Lamb dip, a zero-crossing exactly at ν₀ to use as the servo error signal.

$$V_{\rm err} \propto \frac{dS(\nu)}{d\nu} \quad \text{[first-harmonic demodulation]}$$ $$\text{Lamb dip FWHM} \approx \frac{\Gamma}{2\pi}\sqrt{1 + I/I_{\rm sat}} \quad \text{[power-broadened]}$$ $$\text{Doppler FWHM} = \frac{\nu_0}{c}\sqrt{\frac{8k_{\rm B}T\ln 2}{m}}$$ $$I_{\rm sat} = \frac{\pi h c \Gamma}{3\lambda^3} \quad \text{[two-level, natural linewidth]}$$

Crossover resonances

Why they appear

In multi-level atoms (real alkalis), multiple excited states ν_a, ν_b share a common ground state. An atom moving at velocity v* = c(ν_b−ν_a)/(ν_a+ν_b) sees the pump resonant with ν_a and the counter-propagating probe resonant with ν_b (or vice versa). This produces an extra Lamb dip at the midpoint between the two transitions.

$$\nu_{\rm cross} = \frac{\nu_a + \nu_b}{2}$$ For pump resonant with $\nu_a$: $v^* = c(\nu_{\rm laser} - \nu_a)/\nu_a$
Probe seen by same atom at $v^*$: $\nu_{\rm probe} = \nu_{\rm laser} + \nu_a v^*/c = \nu_b$ ✓

Key properties

Crossover features are often stronger and narrower than true Lamb dips because two velocity classes contribute, one class has pump resonant with ν_a and probe with ν_b; a second class has the roles reversed. Both pathways deplete the same ground state, doubling the effective saturation.

In Cs D₂, the crossover at (F=4→F'=3 + F=4→F'=5)/2 is the most commonly used lock point: it is sharp, high-contrast, and well-separated from neighbours.

Modulation Transfer Spectroscopy (MTS), superior alternative

How MTS differs from FM-SAS

In standard FM-SAS, the laser frequency is modulated, the derivative of the entire absorption profile is detected, including the broad Doppler background. Residual amplitude modulation (RAM) and etalon fringes superimpose on the useful signal.

In MTS, the pump beam is phase-modulated at Ω via an EOM. Through four-wave mixing in the resonant medium, the pump modulation transfers to the probe beam only at atomic resonances, the Doppler background does not participate. Demodulating the probe at Ω gives a dispersive signal on a near-zero baseline.

Four-wave mixing origin

The modulated pump (fields at ω±Ω) drives a coherent population grating at Ω. A counter-propagating probe scatters off this grating, acquiring sidebands at ω±Ω. Beat detection at Ω produces the dispersive error signal without a lock-in amplifier.

Advantages: no Doppler pedestal, immune to laser FM-to-AM conversion (RAM), 5–20× better SNR than FM-SAS in practice.

Limitation: requires a pump EOM and collinear geometry. Not all transitions support efficient four-wave mixing (weak ones don't).

Method	Baseline	Typical SNR	Setup complexity
Direct absorption	Doppler envelope	Low (~1:100)	Simplest, beamsplitter + PD
FM-SAS (current/freq. mod.)	Doppler slope + Lamb dip derivative	Medium (~1:20)	Modulate laser current, demodulate PD
MTS (pump EOM)	Near-zero (flat)	High (~1:3)	Pump EOM + demodulate probe at Ω

Species-specific considerations

Species	Wavelength	Γ/2π	Iₛₐₜ	Cell temp.	Key notes
Cs	852 nm (D₂)	5.23 MHz	1.1 mW/cm²	25–40 °C	Rich HFS; (3+5)/2 crossover is best lock; 9192 MHz GS splitting (primary standard)
⁸⁷Rb	780 nm (D₂)	6.07 MHz	1.67 mW/cm²	25–50 °C	Most used BEC species; 6.835 GHz GS split; easy SAS at room temperature
Na	589 nm (D₂)	9.80 MHz	6.3 mW/cm²	100–130 °C	High Iₛₐₜ → needs more pump power; yellow → dye laser or SHG; 1772 MHz GS split
⁶Li	671 nm (D₁+D₂)	5.87 MHz	2.54 mW/cm²	300–400 °C	Ground hyperfine splitting is 228 MHz; D₁/D₂ fine-structure splitting is ~10 GHz; excited-state HFS is only a few MHz (barely resolved); heated oven cell essential; MTS strongly preferred; congested spectrum
⁴⁰K	767 nm (D₂)	6.04 MHz	1.75 mW/cm²	60–100 °C	Natural abundance 0.012% → use enriched cell or long path length; 1286 MHz GS split

⚠ Why SAS fails for weak (E2, M1) transitions: forbidden or weak transitions usually give poor room-temperature SAS signals because the oscillator strength, vapor pressure, optical pumping dynamics, and transit-time/collision broadening work against a clean Lamb dip. Do not estimate their visibility from D-line saturation-intensity intuition alone. In practice, PDH locking to a high-finesse cavity or direct spectroscopy in a cold-atom/lattice system is the cleaner route.

Calculator, Doppler, Lamb dip, saturation & pump power

Parameters

Species

Vapour cell temperature (K): 300

Saturation I/Iₛₐₜ: 1.0

FM mod. freq. (MHz): 5.0

Pump beam diameter (mm): 1.0

SAS Transmission (Doppler + Lamb dip)

FM Error Signal dS/dν

Vapour cell temperature: 25–60 °C for Cs/Rb; 300–400 °C for Li (oven cell). Higher T → stronger absorption but broader Doppler, more collisional broadening, and risk of alkali deposition on windows.
Pump power: needs I ≳ Iₛₐₜ to produce a visible Lamb dip. Excess power broadens the dip (FWHM ∝ √(1 + I/Iₛₐₜ)). Optimum: 3–10× Iₛₐₜ.
Pump:probe ratio: probe should be ~10–20% of pump to avoid probe saturation. Route pump through a retro-reflector or PBS to generate counter-propagating beams.
Polarisation: σ⁺ pump + σ⁻ probe (or vice versa) works well for MTS; linear polarisations work for FM-SAS. Tilt beams 5–10 mrad to avoid pump back-scatter into probe path.
Modulation frequency: 1–50 MHz typical. Higher pushes 1/f noise below shot-noise floor. For FM-SAS: Ω should be ≤ Lamb dip FWHM for maximum slope. MTS can use Ω up to ~Γ/2 for an optimum.
Crossover resonances: often strongest features (2× signal contribution); Cs F=4→(3+5)/2 and Rb F=2→(2+3)/2 are common lock points.
Li-6 special case: the ground hyperfine splitting is 228 MHz, while the D₁/D₂ fine-structure splitting is ~10 GHz and the excited-state hyperfine splittings are only a few MHz. Use a heated cell at ~350°C. MTS is often preferred. If D1 and D2 lasers are both needed, lock one laser to the cell and offset/beat-note lock the other at the required fine-structure plus hyperfine offset.
SAS fails for E2/M1 transitions where I_sat is ≫ W/cm², use PDH cavity locking.

Beat-Note (Offset) Locking

Many experiments need two lasers separated by a precise and stable frequency offset , for example, a cooling laser and repumper pair, or two Raman beams. Beat-note locking stabilises the difference frequency between a well-stabilised master laser and a slave laser, without locking each independently to an atomic reference.

Frequency discriminator vs optical phase-locked loop (OPLL)

Frequency discriminator lock

An RF chain converts the instantaneous beat frequency to a voltage. Error signal: V_err ∝ (ν_beat − ν_ref). The servo corrects frequency but does not track phase — slave and master are not phase-coherent.

Servo bandwidth: a few hundred kHz suffices.
Slave linewidth: inherits master's long-term frequency stability; short-term phase wanders freely.
Applications: cooling + repumper pairs, probe lasers, any non-interferometric use where absolute phase doesn't matter.

Optical phase-locked loop (OPLL)

The servo tracks the phase of the beat: d(φ_beat − φ_ref)/dt = 0. The slave becomes a phase-coherent offset copy of the master: their relative phase is constant and set entirely by the RF reference.

Servo bandwidth: must exceed the free-running laser linewidth (500 kHz – 5 MHz for ECDLs → bandwidth ≥ 5–20 MHz).
Slave linewidth: transfers master's linewidth at the offset.
Applications: Raman spectroscopy, atom interferometry, STIRAP, EIT, optical lattice clocks, quantum gates.

Physics of the beat note

$$E_1(t) = E_{01}\cos(2\pi\nu_1 t + \varphi_1) \qquad E_2(t) = E_{02}\cos(2\pi\nu_2 t + \varphi_2)$$ $$i_{\rm PD}(t) \propto E_{01}\cdot E_{02}\cdot\cos\!\left[2\pi(\nu_2 - \nu_1)t + (\varphi_2 - \varphi_1)\right]$$ Freq. discriminator: $V_{\rm err} \propto (\nu_2 - \nu_1) - \nu_{\rm ref}$
OPLL: $\dfrac{d}{dt}[\varphi_{\rm beat} - \varphi_{\rm ref}] = 0 \;\Rightarrow\; \nu_2 - \nu_1 \equiv \nu_{\rm ref}$ (phase-coherent)

Signal-to-noise ratio & detector requirements

Beat SNR formula

The beat photocurrent has a DC component I_dc = R(P₁+P₂) and an AC component at Δν. Shot-noise-limited SNR in bandwidth B:

SNR_amp = 2R√(P₁P₂) / √(2eI_dcB)

where R is responsivity (A/W), e is the electron charge, and the numerator is the beat-current amplitude. In RF power terms at 50 Ω, using RMS beat current:

P_RF (dBm) = 10 log₁₀[2R²P₁P₂Z₀ / 1 mW]

Rule of thumb: need ≥ 20 dB SNR (linear factor 100) for a reliable lock. With 0.5 mW on each arm and R = 0.5 A/W, you get ~+5 dBm RF power in 1 MHz bandwidth, typically sufficient.

Beat freq.	Req. PD BW	Detector type	Example
1–100 MHz	≥ 300 MHz	Si PIN	Thorlabs FDS010
100 MHz–1 GHz	≥ 1.5 GHz	Si/InGaAs PIN fast	Hamamatsu G4176
1–3 GHz	≥ 5 GHz	InGaAs PIN	Thorlabs DET08CFC
3–10 GHz	≥ 15 GHz	InGaAs balanced	Discovery DSC-R402
>10 GHz	≥ 25 GHz	UTC-PD or resonant	NTT IOD-PMF-22

Phase noise & coherence transfer

Phase noise in a frequency lock

In a frequency discriminator lock, the slave phase drifts freely relative to the master outside the servo bandwidth. The slave linewidth is roughly equal to the free-running slave linewidth reduced by the servo suppression within its bandwidth.

Phase noise PSD S_φ(f): characterises phase fluctuations at offset frequency f from the carrier. The servo suppresses S_φ(f) for f below the unity-gain frequency, at the cost of a "servo bump" near the bandwidth edge.

Coherence transfer in OPLL

The OPLL forces φ_slave(t) = φ_master(t) + φ_ref(t). If φ_ref comes from a low-noise DDS or RF synthesiser, the slave's coherence length equals the master's, the two beams interfere with long coherence even separated by GHz.

Raman transitions require Δφ = φ₁ − φ₂ stable over the pulse duration (μs–ms scale). With OPLL, Δφ is set by the RF synthesiser (≪ 1 mrad noise), enabling high-contrast Rabi oscillations.

AOM-based frequency offsets

When to use an AOM for offsets

For offsets <30 MHz or when the beat falls in 1/f noise, route one arm through a double-pass AOM at ν_AOM ≈ 80–110 MHz. This shifts the optical frequency by 2ν_AOM ≈ 160–220 MHz, placing the beat note well into the shot-noise-limited region.

AOM + beat combination

Route one arm through a double-pass AOM, then beat against master. The beat is |Δν_laser + 2ν_AOM|. Lock to that beat. Then:

ν_slave = ν_master + ν_RF_ref − 2ν_AOM

Tuning the AOM RF frequency scans the slave without changing the lock point — powerful for spectroscopy scans without mode hops.

Calculator, beat-note frequency, SNR & detector recommendation

Parameters

Master laser wavelength (nm)

Desired offset (MHz) [slave − master]

Optical power on PD: P₁ (mW)

Optical power on PD: P₂ (mW)

PD responsivity R (A/W)

OPLL mode (phase lock)

RF Signal Chain

Typical implementation for a beat-note lock:

Master + Slave → fibre combiner or PBS → fast PD
→ RF amplifier (+10 to +20 dB, low noise)
→ RF power splitter (monitor + lock paths)
→ mixer $\times$ DDS reference (at $\nu_{\rm ref}$)
→ low-pass filter → error signal → PID controller (Vescent D2-125 or Toptica FALC) → slave: fast current path (MHz BW) + PZT (slow)

Commercial electronics

Vescent D2-125

All-in-one laser locking module. Accepts PD input (SAS or beat), provides servo outputs for current + PZT. Built-in lock-in and frequency discriminator. Beat input: 10 MHz–4.5 GHz. Servo BW: up to 10 MHz. Widely used in AMO labs as a single-box solution.

Toptica FALC 110

Fast analog laser controller. Bandwidth: DC–10 MHz (current), DC–1 MHz (PZT). Accepts any external error signal, used when you generate your own discriminator or PDH error. High dynamic range, differential inputs reduce RF pickup. Standard in high-finesse PDH setups and OPLLs.

Vescent D2-105 / D2-135

D2-105: SAS servo only (no beat-note input). D2-135: adds fast modulation input for PDH. For beat-note locking, pair with an external frequency discriminator (Mini-Circuits FM discriminator or custom DDS + mixer board).

Applications & typical offset frequencies

Application	Species	Offset	Lock type	Notes
Cooling + repumper	⁶Li	228 MHz (D₁/D₂)	Freq. discriminator	Lock D₂ repumper to D₁ SAS-locked master
Cooling + repumper	⁸⁷Rb	6835 MHz	OPLL or chain	Large offset → 10 GHz PD or divide-by-N prescaler chain
Raman pair (2-photon)	⁸⁷Rb / Cs	6.8 / 9.2 GHz	OPLL required	Phase coherence essential for Rabi contrast
EIT / STIRAP	Rb / Cs / Li	100 MHz – 1 GHz	OPLL preferred	Phase noise limits dark-state contrast and transfer efficiency
Probe laser offset	Any	Arbitrary	Freq. discriminator	AOM-accessible range: 40–400 MHz; convenient for scanning
PDH transfer cavity	Any	FSR multiple	Freq. discriminator	Lock two lasers to same ULE cavity via beat against anchor laser

Master must already be well-locked (SAS or PDH). The slave inherits the master's long-term stability, no reference means the whole chain drifts.
Fibre combiner vs free-space: single-mode 50:50 fibre combiner gives automatic mode overlap; free-space PBS requires careful polarisation alignment. Fibre adds ~3 dB loss per arm but produces cleaner spectra.
RF amplifier chain: in the usual low-signal beat-note setup, amplify before the mixer so the LO/RF ports meet the mixer drive specification, often +10 to +20 dBm LO. A 20 dB low-noise amp pays off immediately in SNR for sub-mW optical powers.
OPLL bandwidth requirement: servo bandwidth must exceed the free-running laser linewidth (typically 0.5–5 MHz for ECDLs). Fast current modulation (not PZT alone) is mandatory for OPLL.
Very small offsets (<30 MHz): beat note falls within 1/f noise. Use a double-pass AOM to shift one arm by 160–220 MHz before detecting; lock to the shifted beat note.
Offsets >3 GHz: need fast InGaAs PD (≥ 10 GHz BW) or a microwave prescaler chain to divide the beat to a more manageable frequency. For >10 GHz, consider a frequency comb reference.
AOM frequency scanning: scan the AOM RF frequency to scan the slave laser continuously while keeping the beat lock engaged, ideal for high-resolution spectroscopy without mode hops.
RF synthesiser sets the offset: changing ν_ref tunes the slave without touching the master. A DDS (direct digital synthesiser) allows agile, phase-continuous frequency changes.

Pound–Drever–Hall (PDH) Cavity Locking

When no convenient atomic reference exists, or when the required short-term linewidth is narrower than SAS can provide, the laser is locked to a high-finesse Fabry–Pérot cavity. The PDH technique generates an error signal from the reflected field using RF phase modulation, achieving much higher signal slope than simple transmission locking.

$$\nu_{\rm FSR} = \frac{c}{2nL} \quad \text{[free spectral range]}$$ $$\mathcal{F} = \frac{\pi\sqrt{R}}{1-R} \approx \frac{\pi}{1-R} \quad [R \approx 1]$$ $$\delta\nu_{\rm cav} = \frac{\nu_{\rm FSR}}{\mathcal{F}} \quad \text{[cavity linewidth FWHM]}$$ $$\frac{\delta\nu}{\nu} = -\frac{\delta L}{L} = -\alpha_{\rm CTE}\,\delta T \quad \text{[fractional freq. drift]}$$

PDH error signal, how it works

Step 1, Phase modulation. An EOM imprints weak sidebands at ±Ω on the carrier field:
E_in ≈ E₀ e^{iωt} + (β/2)E₀ e^{i(ω+Ω)t} − (β/2)E₀ e^{i(ω−Ω)t} [β ≪ 1]

Step 2, Frequency placement. Choose Ω ≫ δν_cav so the sidebands sit outside the cavity resonance, they reflect unchanged. The carrier acquires a strongly frequency-dependent phase shift near resonance.

Step 3, Demodulation. Detect the reflected field on a fast photodiode. The carrier beats against the sidebands at Ω. Demodulate at Ω → bipolar error signal:

$$V_{\rm err} \propto \mathrm{Im}[r(\omega)] \quad \text{[imaginary part of reflection coefficient]}$$ Near resonance: $V_{\rm err} \approx K\cdot\delta$ where $\delta = \omega - \omega_{\rm cavity}$
Discriminator slope: $K \propto \sqrt{P_{\rm carrier}\cdot P_{\rm sideband}}\cdot\mathcal{F}/\delta\nu_{\rm cav}$
($P_{\rm sideband} \approx \beta^2 P_{\rm carrier}/4$)

Cavity Parameters Calculator

Cavity length L (mm)

Finesse 𝒻

Laser wavelength (nm)

EOM modulation Ω (MHz)

Target linewidth (Hz)

Spacer material

Cavity Reflection (Airy Dip)

PDH Error Signal Im[r(ω)]

Typical optical chain:
diode laser → prism pair (astigmatism corr.) → optical isolator (30–40 dB)
→ EOM (phase mod. at $\Omega$) → mode-matching telescope → ULE cavity

Reflected beam: PBS + QWP → fast PD → RF mixer at $\Omega$ → LPF
Error signal: fast path → laser current (high BW, small range)
slow path → PZT (low BW, large range)

ULE zero-crossing temperature: ULE has CTE ≈ 0 near a specific temperature (5–25 °C depending on blank). Operating at the zero-crossing dramatically reduces thermal drift.
Vacuum and vibration isolation: cavity must be in vacuum (P < 10⁻⁵ mbar) to avoid refractive-index fluctuations. Mount on vibration-isolated platform.
Two-stage servo: fast path (current) compensates high-frequency noise; slow path (PZT) compensates slow drift.
PDH does not provide absolute frequency: the cavity resonance drifts. Beat the locked laser against a frequency comb or SAS reference for absolute knowledge.
Finesse measurement: scan laser across a resonance and fit Airy function, or measure ring-down time τ_c = 𝒻/(π·FSR).

Why the Cavity Itself Is the Hard Part

The PDH tab shows how to derive a dispersive error signal from a Fabry–Pérot cavity, but the quality of the reference you lock to is set entirely by the mechanical, thermal, and vacuum design of the cavity spacer, not by the servo electronics. A useful organizing constraint: to hold the laser's fractional frequency stability below a target value, the cavity's fractional refractive-index and length fluctuations must satisfy

$$\frac{\delta\nu}{\nu} \;\le\; \frac{\delta n}{n} + \frac{\delta L}{L} \;\lesssim\; 10^{-9} \quad \text{[target for sub-MHz stability at optical frequency]}$$

Every design choice below, vacuum level, spacer material and temperature, and support geometry, exists to drive one of the two terms on the right toward zero.

Finesse, free spectral range, and mode structure

$$\nu_{\rm FSR} = \frac{c}{2nL} \quad \text{[free spectral range]}$$ $$\mathcal{F} = \frac{\pi\sqrt{r_1r_2}}{1-r_1r_2} = \frac{\nu_{\rm FSR}}{\delta\nu_{\rm cav}} = 2\pi\,\tau\,\nu_{\rm FSR} \quad [\tau = 1/e\text{ ring-down time}]$$

Build intuition first: $|r|=98.4\%\Rightarrow\mathcal{F}\approx100\Rightarrow\delta\nu_{\rm cav}\approx15$ MHz; push the mirrors to $|r|=99.992\%\Rightarrow\mathcal{F}\approx20{,}000\Rightarrow\delta\nu_{\rm cav}\approx75$ kHz. A real ultrastable cavity in the paper (10 cm long, 780 nm) reaches $\mathcal{F}=25{,}200$, FSR $\approx1.5$ GHz, linewidth $\approx60$ kHz, and a measured ring-down time $\tau\approx2.6$–$2.7\ \mu$s, matching the manufacturer's spec.

Geometry matters as much as reflectivity. Higher-order transverse modes (HOMs) pick up an extra Gouy phase per round trip and resonate at $$\frac{\nu_{mnp}}{\nu_{\rm FSR}} = p + \frac{m+n+1}{\pi}\arccos\!\left[\sqrt{\Big(1-\tfrac{L}{R_1}\Big)\Big(1-\tfrac{L}{R_2}\Big)}\right]$$ A confocal cavity ($L=R_1=R_2$) is a bad choice, its HOMs land exactly at FSR/2, degenerate with the neighboring TEM₀₀ resonance. The standard practical choice is planoconvex (one flat mirror, one curved), with $L/R$ tuned so HOMs sit roughly 100 MHz from the TEM₀₀ peak, far enough to avoid mode-overlap-induced lock errors.

Controlling δn/n: the vacuum requirement

Air has $n-1\approx3\times10^{-4}$ at STP, and atmospheric pressure alone drifts by roughly 10% day to day, so an unevacuated cavity sees $\delta n/n\sim10^{-5}$: four orders of magnitude too large for the $10^{-9}$ budget above. Pumping to below $10^{-3}$ mbar already fixes this; in practice, ion pumps (no moving or spinning parts, so they inject no vibration) routinely hold $\sim10^{-7}$ mbar with no bakeout, pushing $\delta n/n\sim10^{-14}$, fully negligible.

Controlling δL/L: the ULE zero-crossing

$\delta L/L=\alpha(T)\,\delta T$, so the spacer's thermal-expansion coefficient matters as much as the temperature control itself. ULE (ultra-low-expansion) glass is a SiO₂/TiO₂ mix engineered so that $\alpha(T)$ crosses zero at a specific zero-crossing temperature $T_{\rm ZC}$, chosen a few degrees above room temperature purely so a resistive heater, not a cooler, can hold the setpoint (the paper's own cavity: $T_{\rm ZC}=34.49\,^\circ$C, held with a Wavelength Electronics LFI-3751 heater controller). At $T_{\rm ZC}\pm10$ mK: $\delta L/L\sim10^{-14}$ for a 10 cm spacer. Even 0.5 °C off the zero crossing it's still only $\sim10^{-11}$, which is exactly why a simple heater-plus-PID loop suffices instead of a full cryostat.

The single most non-obvious idea in cavity design: where you hold it matters more than what you hold it with. Under vertical acceleration $a$, a cavity spacer of length $L$ supported at its bottom is compressed, $\Delta L=a\rho L^2/2E$ ($\rho$ = density, $E$ = Young's modulus), while one suspended from the top stretches by the same amount. Support it instead at the horizontal plane through its midpoint and the two effects cancel to first order, $\Delta L\approx0$. This is why real ultrastable cavities are mounted horizontally on specific nodal support points, often Viton (fluoroelastomer) balls or pads for damping and thermal isolation, rather than wherever is mechanically convenient.

The fundamental floor: thermal (Brownian) noise

Vacuum, thermal control, and vibration isolation all fix drift, they don't touch the fundamental floor. The spacer, substrates, and especially the dielectric mirror coatings undergo genuine thermodynamic (Brownian) motion that scales as $\sqrt{T}$, this is noise, not drift, and no amount of the engineering above removes it. State of the art gets around it by going cryogenic: single-crystal silicon spacers operated at their own zero-crossing temperature (~16 K), which both suppresses Brownian noise directly (lower $T$) and removes the amorphous-material aging that limits room-temperature ULE cavities over the long term.

Mode matching and intracavity power

$$z_R = \sqrt{L(R_2-L)}, \qquad w_0 = \sqrt{\frac{z_R\lambda}{\pi}} \quad \text{[planoconvex cavity, flat mirror at } z=0\text{]}$$ $$\frac{I_{\rm cav}}{I_{\rm inc}} = \frac{\mathcal{F}}{\pi} \quad \text{[on-resonance build-up factor]}$$

Common mistake: for $\mathcal{F}=25{,}200$, the build-up factor is $\mathcal{F}/\pi\approx8{,}000\times$, so as little as ~100 μW incident can already build up to nearly 1 W circulating. Too much incident power into a high-finesse cavity can heat or damage the mirror coating, always check the build-up factor before raising input power. Diagnose mode matching with a transmission camera (look directly for HOM spatial patterns) and the reflected-signal dip depth (aim for ≥80% of the fully-reflected level).

A concrete PDH lock, start to finish

Example RF chain (Boyd & Lahaye's setup)

20 MHz modulation via a fiber EOM, ~350 μW incident power, function-generator LO (Tektronix AFG3022C), photodiode (Thorlabs PDA10-EC), RF amplifier (Mini-Circuits ZFL-500+), mixer (Mini-Circuits ZAD-1H+), and servo (Vescent D2-125), the same servo family used on this page's Beat-Note tab.

1. Sweep the cavity resonance and adjust the demodulation LO phase to minimize/optimize the reflected sideband beat.

2. Add a 90° phase shift from that optimum, the phase that minimizes the in-phase sideband beat is 90° away from the phase that maximizes the actual PDH error-signal slope.

3. Zero the DC offset of the error signal.

4. Slowly shrink the scan amplitude while re-centering on the target resonance (usually the fundamental TEM₀₀ peak).

5. Engage the lock and watch the transmitted-power photodiode, confirm it sits at the peak, not on a nearby HOM wing, and stays there.

6. Tune the PID gains for stability without ringing (see the P/D sliders in the lock animator above for the same intuition).

Measuring finesse independently: cavity ring-down

$$I(t) = I_0\,e^{-t/\tau}, \qquad \mathcal{F} = 2\pi\,\tau\,\nu_{\rm FSR}$$

Typical design requirements at a glance

Parameter	Requirement	Typical value (paper's cavity)
Mirror reflectivity	as high as practical	99.992% (𝒻 ≈ 25,200)
Vacuum pressure	< 10⁻³ mbar minimum	~10⁻⁷ mbar (ion pump)
Temperature stability	hold near T_ZC	±10 mK
Spacer material	near-zero CTE	ULE glass (T_ZC = 34.49 °C here)
Spacer length	typical	10 cm
Resulting linewidth	kHz-level or below	~60–75 kHz
Drift	slow, characterizable	~hundreds of kHz/day
Fundamental floor	thermal (Brownian) noise in coatings	overcome only cryogenically (~16 K, Si)

Cross-check the spec sheet: always measure finesse independently by ring-down rather than trusting the mirror coating vendor's number alone.
Mount at the nodal points: Viton balls or pads at the calculated midpoint support locations damp vibration and thermally isolate the spacer at the same time.
Budget the HOM spacing, not just the finesse: a planoconvex geometry with HOMs ~100 MHz from TEM₀₀ avoids mode-overlap lock errors that a purely high-finesse, badly-chosen-geometry cavity would suffer.
Ion pumps over turbo/scroll pumps wherever possible: no moving parts means no vibrational or acoustic noise injected into the very cavity you're trying to keep still.
This tab complements, not replaces, the PDH tab: the PDH tab covers the servo and error-signal electronics; this tab covers the mechanical, thermal, and vacuum design of what you're locking to. If you're looking for atom-cavity coupling (cavity QED) rather than laser frequency stabilization, see the Cavity QED Coupling Lab instead, a different physics problem that happens to share the word "cavity."

The Locking Hierarchy

In practice, the three techniques form a hierarchy. SAS provides the absolute anchor; beat-note locks propagate stability to other lasers at controllable offsets; PDH locks provide narrow-linewidth operation wherever no atomic reference is available.

Technique	Absolute?	Typical linewidth	Tunable offset?	Best for	Limitation
SAS lock	✓ Atomic line	100 kHz – 1 MHz	✗ Fixed to transition	Primary absolute reference (D lines)	Needs strong transition; vapour-cell lines only
Beat-note lock	Via master	Same as master	✓ RF synthesiser	Cooling/repump pairs, Raman beams	Needs pre-stabilised master; fast PD + RF chain
PDH cavity lock	✗ Cavity drifts	1 Hz – 10 kHz	Via AOM after lock	Narrow-linewidth spectroscopy, weak transitions	Thermal cavity drift; expensive; needs vacuum

Typical laboratory hierarchy

Layer 0 (Absolute): D-line laser → SAS lock → ν known to ~1 MHz Layer 1 (Derived): Laser B → beat-note lock to Layer 0 → ν₀ ± ν_RF Laser C → beat-note lock to Layer 0 → ν₀ ± ν_RF' Layer 2 (Narrow): Spectroscopy laser → PDH lock to ULE cavity → ~kHz linewidth (Long-term drift corrected by beating against a Layer 0 laser)

Key equations at a glance

SAS: $V_{\rm err} \propto dS(\nu)/d\nu$, Lamb dip FWHM $\approx (\Gamma/2\pi)\sqrt{1+I/I_{\rm sat}}$

Beat-note: $i_{\rm PD} \propto \cos[2\pi(\nu_2-\nu_1)t]$, $V_{\rm err} \propto (\nu_2-\nu_1) - \nu_{\rm ref}$

PDH: $V_{\rm err} \propto \mathrm{Im}[r(\omega)] \approx K\cdot\delta$, $$\delta\nu_{\rm cav} = \frac{c}{2L\mathcal{F}}, \qquad \frac{\delta\nu}{\nu} = -\frac{\delta L}{L} = -\alpha_{\rm CTE}\,\delta T$$

Error signal slopes compared

Technique	Slope K (typical)	Notes
SAS (direct lock-in)	~0.1–1 mV/MHz	Limited by Doppler background contrast
SAS (modulation transfer)	~1–10 mV/MHz	Better baseline; uses four-wave mixing
Beat-note (freq. discriminator)	~1–10 mV/MHz	Scales with RF power and mixer gain
PDH (high finesse)	~10–1000 mV/MHz	Scales as √(P_c·P_s) × 𝒻/δν_cav

Our lab: Cs + Li system

Lab Laser Hierarchy (Cs/Li Tweezer Experiment)

852 nm Cs D₂: SAS-locked in Cs vapour cell → primary absolute reference.
685 nm Cs E₂ (6S→5D₅/₂): PDH-locked to ULE cavity (L = 77.5 mm, 𝒻 ≈ 1.5×10⁴). FSR ≈ 1.93 GHz, δν_cav ≈ 130 kHz, laser linewidth ≈ 1 kHz. Thermal drift ≈ 2.5 kHz per 10 mK.
671 nm Li D₁: SAS-locked in heated Li vapour cell.
671 nm Li D₂: Beat-note locked to Li D₁ (Vescent D2-125), offset set by RF synthesiser.

04 Sources

Black (2001), an introduction to Pound-Drever-Hall locking Drever et al. (1983), laser phase/frequency stabilization using optical resonators Steck alkali data sheets for Cs, Rb, Li, Na, K line constants NIST Atomic Spectra Database Wieman & Hollberg (1991), RSI 62 1 — using diode lasers in atomic physics; the definitive practical guide Hänsch & Couillaud (1980), Opt. Comm. 35 441 — polarization spectroscopy for cavity locking without RF modulation Ludlow et al. (2015), Rev. Mod. Phys. 87 637 — optical atomic clocks; covers ultra-narrow laser stabilization at the frontier Salomon et al. (1988), JOSA B 5 1576 — laser stabilization at the shot-noise limit; spectroscopy signal optimization Boyd & Lahaye (2024), Am. J. Phys. 92 50 — a basic introduction to ultrastable optical cavities for laser stabilization; source for this page's Ultrastable Cavities tab (peer-reviewed) Boyd & Lahaye, arXiv:2308.08015 — free preprint of the above