An 80 MHz TeO₂ Bragg cell on a tilt stage: focus a HeNe into the crystal, find the Bragg angle, watch the orders on a screen, switch the beam with TTL, shift its frequency, and identify an unknown modulator from its spot spacing
In short: An acousto-optic modulator is a clear crystal with a tiny loudspeaker glued to one side. Driving it at 80 MHz fills the crystal with sound waves that act like a moving diffraction grating. A laser beam passing through is split into a few beams at slightly different angles, and the deflected beam comes out with its frequency shifted by exactly the sound frequency. Switching the sound on and off switches the deflected beam in tens of nanoseconds, which is why these devices are used to pulse, steer and frequency-shift lasers.
In 3D: drag the AOM's tilt knob (turn) to rock the crystal through the Bragg angle; drag the RF driver to change the RF power; click the iris to pick an order; click the photodiode to flip it into the beam; click the recollimating lens to take it out. Diffraction angles in the 3D view are drawn ×5 (they are only milliradians); the screen panel below shows them to scale.
Shine a HeNe through an unknown modulator at its RF frequency and measure the spacing between the 0 and +1 spots on a card.
1. Find the Bragg angle. Start at normal incidence with the RF on: the ±1 orders are weak and equal. Rock the crystal (tilt knob or slider): at about +6 mrad the +1 spot takes most of the light and the −1 spot vanishes. That is the Bragg condition, sin θ = λf/(2v).
2. Turn up the RF. Efficiency rises as sin² of the square root of the power and peaks at the "π power" (about 0.5 W here); beyond it light flows back into the zero order.
3. Focus harder for speed. A smaller waist switches faster (the sound has less beam to cross: rise time ≈ 0.64·d/v), but the focused beam's angular spread starts to exceed the cell's acceptance and the peak efficiency drops.
4. Thin cell. Shorten L until Q drops below 1: many orders appear with Bessel-function strengths (Raman–Nath diffraction), and tilting hardly matters.
5. Frequency shift. The +1 order is shifted up by the RF frequency, the −1 down. Overlap 0 and +1 on a fast photodiode and you see an 80 MHz beat (too fast for a Red Pitaya input). A double pass shifts by 2f and keeps the output pointing fixed when f changes.
6. Identify an unknown AOM. Measure the 0 to +1 spacing at a known distance: at 80 MHz and 633 nm, 1 m away, about 12 mm means TeO₂, 14 mm PbMoO₄, 8.5 mm fused silica.
A first bench for acousto-optic modulators (AOMs): the Bragg cells that pulse, shift, stabilise and steer laser beams in almost every atomic-physics and interferometry lab. It walks through what a student actually does with a new AOM: focus a beam into it, rock it to the Bragg angle, turn up the RF until the first order takes most of the light, pick that order off with an iris, and watch it switch on a photodiode. It also helps identify an unknown modulator, such as the lab's four 80 MHz H-211 units, from a simple spot-spacing measurement.
1. Bragg angle and order spacing. Sound of frequency f and velocity v makes a moving index grating of period Λ = v/f. Light of vacuum wavelength λ diffracts strongly when it meets the grating at the Bragg angle sin θ_B = λ/(2nΛ) inside the crystal, which is θ_B = λf/(2v) outside. Adjacent orders leave λf/v apart. Handled: braggExt() and sepExt(). At 633 nm and 80 MHz in TeO₂: θ_B = 6.03 mrad, spacing 12.05 mrad (12 mm at 1 m, 1.81 mm after the f = 150 mm lens).
2. Bragg or Raman–Nath: the Klein–Cook parameter. Q = 2πλLf²/(nv²). For Q ≳ 10 only one first order can be phase-matched (Bragg regime); for Q ≲ 1 the cell acts as a thin phase grating that throws light into many orders (Raman–Nath regime). The default cell has Q = 10.2.
3. Diffraction efficiency. The sound gives a peak phase modulation Δφ = (2π/λ)√(M₂LP/(2H)). In the Bragg limit at the Bragg angle η₁ = sin²(Δφ/2) = sin²[(π/λ)√(M₂LP/(2H))], reaching 100 % at the π power P_π = λ²H/(2M₂L) (0.44 W for the default TeO₂ cell at 633 nm, 1.25 W at 1064 nm). In the Raman–Nath limit the order powers are J_m²(Δφ).
Handled: orders() integrates the coupled-mode (Raman–Nath) equations for orders −M…M through the cell:
dE_m/dz = (Δφ/2)(E_{m−1} − E_{m+1}) + i(Q/2)m(m − α)E_m, with α = θ/θ_B and z from 0 to 1 (Runge–Kutta 4). This one model covers both regimes and the angular detuning: for large Q it reproduces Kogelnik's two-wave result η = ν² sin²√(ν² + ξ²)/(ν² + ξ²); for Q → 0 it gives J_m²(Δφ).
4. Focusing, acceptance and speed. A beam focused to waist w₀ has an angular spread ≈ λ/(πnw₀) inside the crystal; the cell accepts about Λ/L. diffraction() averages the coupled-mode result over the beam's Gaussian angular spectrum, so a tight focus lowers the peak efficiency (70 % at w₀ = 50 µm vs 94 % at 1 mm for the default cell). The 10–90 % rise time for a Gaussian beam is t_r ≈ 0.64·d/v with d = 2w₀ (15 ns at w₀ = 50 µm, 7.6 ns at 25 µm), plus an acoustic delay x_T/v from the transducer to the beam (238 ns for 1 mm).
5. Frequency shift and beating. Diffraction off a moving grating Doppler-shifts each order by m·f: the +1 order is at f₀ + 80 MHz, the −1 at f₀ − 80 MHz. Overlapping 0 and +1 on a fast photodiode gives an 80 MHz beat: above the Red Pitaya's 62.5 MHz Nyquist limit and the PDA36A2's bandwidth, so a 150 MHz detector is listed.
6. Scanning. Changing f by Δf swings the deflected beam by λΔf/v. The number of resolvable spots is N = Δf·τ with τ = D/v the sound's transit time across the beam (D = 2w₀). The RF bandwidth is taken as the smaller of the Bragg acceptance 1.8nv²/(λfL) and 0.5f for the transducer. A focused modulator beam gives N ≈ 1; a 2 mm beam gives N ≈ 19 (the "deflector" preset).
7. Double pass. The +1 order is recollimated by a lens one focal length from the AOM and retro-reflected from a mirror at its focal plane (cat's eye) through a quarter-wave plate. It re-enters the AOM along its own path, diffracts again and leaves along the input beam, shifted by 2f, with efficiency ≈ η₁², and its direction no longer depends on f. The polarizing cube sends it to the photodiode.
8. Identifying a crystal. The measured spacing s between the 0 and +1 spots at distance d gives v = λfd/s. identify() compares v with TeO₂ (4.20 km/s), PbMoO₄ (3.63), crystal quartz (5.75), fused silica (5.96) and slow-shear TeO₂ (0.62). At 80 MHz and 633 nm, 1 m away, expect about 12 mm for TeO₂, 14 mm for PbMoO₄ and 8.5 mm for fused silica.
src/ao_model.js: lasers and crystals, Bragg angle, Q, Δφ and P_π, the coupled-mode solver with beam averaging, rise time, bandwidth and resolvable spots, the TTL trace and the crystal identifier. Pure functions; runs in Node.src/ao_app.js: the 3D table (AOM with crystal, transducer and moving sound fronts, platforms, driver, TTL source, lenses, iris, photodiode, screen and the double-pass arm), handles, readouts and the five canvases.src/qkit.js: the shared optics-table kit.src/ao_shell.html + src/build.py: the page. python build.py writes aom_basics_bench.html.| Parameter | Value | Source |
|---|---|---|
| TeO₂ (longitudinal) | v = 4.20 km/s, n ≈ 2.26, M₂ ≈ 34 × 10⁻¹⁵ s³/kg | standard tables (e.g. Yariv & Yeh; Chang 1976) |
| PbMoO₄ | v = 3.63 km/s, n ≈ 2.39, M₂ ≈ 36 × 10⁻¹⁵ s³/kg | standard tables |
| Fused silica | v = 5.96 km/s, n = 1.457, M₂ ≈ 1.5 × 10⁻¹⁵ s³/kg | standard tables |
| Default cell | L = 16 mm, H = 1.2 mm, 80 MHz | typical 80 MHz modulator (ASSUMED) |
| Default beam | w₀ = 50 µm in the crystal (f = 75 mm lens) | design choice |
| Rise time | t_r ≈ 0.64·d/v (Gaussian beam, 10–90 %) | AOM vendor application notes |
| RF bandwidth | min(1.8nv²/(λfL), 0.5f) | Bragg acceptance; transducer cap ASSUMED |
| Transducer-to-beam distance | 1 mm | ASSUMED |
Shapes are representative stand-ins for the listed parts; verify part numbers, prices and lead times with the vendors (Thorlabs, Gooch & Housego, Isomet, Mini-Circuits).