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 − v/λ and the probe to ν₀ + v/λ.
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 − v/λ and the probe to ν₀ + v/λ.
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, burning a population inversion hole.
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, burning a population inversion hole.
Step 3 — Lamb dip.
Because zero-velocity atoms are partially saturated by the pump, they absorb less of the probe at ν₀ → a narrow Lamb dip appears in the probe transmission, on the broad Gaussian Doppler background.
Because zero-velocity atoms are partially saturated by the pump, they absorb less of the probe at ν₀ → a narrow Lamb dip appears in the probe transmission, on the broad Gaussian Doppler background.
Step 4 — FM error signal.
The laser frequency is weakly modulated (via current or EOM). Lock-in demodulation of the probe signal yields a dispersive derivative of the Lamb dip, with a zero-crossing at ν₀.
The laser frequency is weakly modulated (via current or EOM). Lock-in demodulation of the probe signal yields a dispersive derivative of the Lamb dip, with a zero-crossing at ν₀.
V_err ∝ dS(ν)/dν [first-harmonic demodulation]
Lamb dip FWHM ≈ Γ/2π × √(1 + I/Iₛₐₜ)
Doppler FWHM = (ν₀/c) √(8 k_B T ln2 / m)
Calculator — Doppler vs Lamb dip widths
Parameters
SAS Transmission
FM Error Signal dS/dν
- Vapour cell temperature: 20–60 °C gives good SNR for alkali D lines. Higher T → stronger signal but broader Doppler background and higher collisional broadening.
- Pump power: needs I ≳ Iₛₐₜ to produce a visible Lamb dip. Excess power broadens the dip.
- Modulation frequency: typically 1–50 MHz. Higher frequencies push technical noise below the shot-noise floor.
- Cross-over resonances: at ν = (ν₁+ν₂)/2, both zero-velocity and ±v classes contribute — extra dips, often the sharpest features in multi-level spectra.
- SAS is impractical for very weak transitions (E2, M1) where I_sat is W/cm² — the Lamb dip signal is too small.
Beat-Note (Offset) Locking
Many experiments need two lasers separated by a precise and stable frequency offset — for example, a cooling laser and a repumper, or a Raman beam pair. 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.
Physics of the beat note
E₁(t) = E₀₁ cos(2π ν₁ t + φ₁)
E₂(t) = E₀₂ cos(2π ν₂ t + φ₂)
→ i_PD(t) ∝ cos[2π(ν₂ − ν₁)t + (φ₂ − φ₁)]
V_err ∝ (ν₂ − ν₁) − ν_ref [frequency discriminator]
OPLL: d/dt [φ_beat − φ_ref] = 0 → ν₂ − ν₁ ≡ ν_ref (phase coherent)
Calculator — Beat-note frequency & detector requirements
Parameters
RF Signal Chain
Typical implementation for a beat-note lock:
Slave + Master → fast PD (1–3 GHz BW) → amplifier
→ RF power splitter
→ mixer × local oscillator (RF synthesiser at ν_ref)
→ low-pass filter → error signal
→ PID controller → slave laser current / PZT
Signal-to-Noise
Beat SNR in a bandwidth B:
SNR = P₁·P₂·R² / (2e·I_dc·B)Typically need ≳ 20 dBm RF power at the detector to lock cleanly. Responsivity R ≈ 0.5–0.8 A/W for Si/InGaAs fast photodiodes at these wavelengths.
- Master must already be well-locked (SAS or PDH). The slave inherits the master's long-term stability.
- RF synthesiser sets the offset: changing ν_ref tunes the slave frequency without touching the master.
- OPLL vs frequency lock: phase-locked loops require ~1 MHz servo bandwidth but provide phase coherence — essential for Raman spectroscopy and atom interferometry.
- Very small offsets (<10 MHz): beat note may fall within 1/f noise. Use a double-pass AOM to increase the offset before detecting.
- Offsets >3 GHz: may exceed standard fast PD bandwidth. Consider wavemeter pre-stabilisation.
- Vescent D2-125: widely used electronics that handle both SAS and beat-note locking from a single unit.
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.
Free spectral range: ν_FSR = c / (2 n L)
Finesse: 𝒻 = π √R / (1 − R) ≈ π/(1−R) [R ≈ 1]
Cavity linewidth (FWHM): δν_cav = ν_FSR / 𝒻
Fractional freq. drift: δν / ν = −δL / L = −α_CTE · δT
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]
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_err ∝ Im[r(ω)] [imaginary part of reflection coefficient]
Near resonance: V_err ≈ K · δ (δ = ω − ω_cavity)
Discriminator slope: K ∝ √(P_carrier · P_sideband) · 𝒻 / δν_cav
(P_sideband ≈ β² P_carrier / 4)
Cavity Parameters Calculator
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 Ω) → mode-matching telescope → ULE cavity
Reflected beam: PBS + QWP → fast PD → RF mixer at Ω → 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).
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_err ∝ dS(ν)/dν, Lamb dip FWHM ≈ Γ/2π × √(1 + I/Iₛₐₜ)
Beat-note: i_PD ∝ cos[2π(ν₂−ν₁)t], V_err ∝ (ν₂−ν₁) − ν_ref
PDH: V_err ∝ Im[r(ω)] ≈ K·δ,
δν_cav = c/(2LF), δν/ν = −δL/L = −α_CTE · δT
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.