Acousto-Optic Bragg Cell
Acousto-optic Bragg cell design: Bragg and deflection angles, scan range, Klein–Cook Q, first-order diffraction efficiency from the figure of merit M₂ and acoustic power, access time and resolvable spots, for TeO₂, LiNbO₃, PbMoO₄ and fused silica.
TeO₂ slow-shear cells use anisotropic Bragg diffraction (light near the optic axis, polarization changes); their Bragg angles differ from these isotropic values.
n = 2.26, v = 617 m/s, M₂ = 793 ×10⁻¹⁵ s³/kg.
Bragg angle outside θ_B
2.352°
sin θ_B = λf/(2v)
Deflection 2θ_B
4.704°
angle between the zero and first order
Bragg angle inside the cell
1.04°
asin(λf/(2nv)); refraction restores the outside angle
Scan range for Δf = 40 MHz
41.04 mrad
Δθ = λΔf/v, outside the cell
Klein–Cook parameter Q
295.9
Bragg regime (Q ≫ 1)
First-order efficiency η
97.03 %
at 0.02 W acoustic power
Power for 100 %
25.26 mW
λ²H/(2M₂L)
Access time τ = D/v
3.241 µs
time for the sound to cross the beam
Resolvable spots N = τΔf
129.7
for the swept bandwidth
Efficiency against acoustic power
Presets
| Material | Mode | n | v (m/s) | M₂ (×10⁻¹⁵ s³/kg) | Source |
|---|---|---|---|---|---|
| TeO₂ | slow shear along [110], light along [001] | 2.26 | 617 | 793 | US 3,661,441 Table 1 |
| TeO₂ | longitudinal along [001], light ⊥ [001] | 2.26 | 4260 | 34.5 | US 3,661,441 Table 1 |
| LiNbO₃ | longitudinal along [11̄20], light along [0001] | 2.29 | 6570 | 7 | US 3,661,441 Table 2 |
| PbMoO₄ | longitudinal along [001], light ⊥ [001] | 2.39 | 3630 | 36.3 | Isomet AN4820 (vendor) |
| Fused silica | longitudinal | 1.46 | 5960 | 1.51 | Isomet AN4820 (vendor) |
M₂ = n⁶p²/(ρv³) for the light direction and polarization given (TeO₂ and LiNbO₃ measured at 632.8 nm); using it at other wavelengths ignores the dispersion of n and p. TeO₂ and LiNbO₃: Uchida, Ohmachi, Niizeki & Inamura, US Patent 3,661,441 (1972); PbMoO₄ and fused silica: Isomet application note AN4820 (vendor table; the patent lists PbMoO₄ as 3750 m/s and 37). n is the ordinary index at 632.8 nm (Uchida 1971; Malitson; Zelmon 1997).
Model
sin θ_B = λ₀f/(2v) outside, λ₀f/(2nv) inside, Δθ = λ₀Δf/v
Refraction at the entrance face restores the outside value of the Bragg angle. The diffracted beam leaves at 2θ_B.
Q = 2πλ₀Lf²/(nv²), η = sin²[(π/(√2 λ₀))√(M₂LP/H)], P₁₀₀ = λ₀²H/(2M₂L)
The Klein–Cook parameter (Klein & Cook 1967) separates the Bragg regime (Q ≫ 1) from Raman–Nath diffraction (Q ≪ 1). The efficiency and the power for 100 % follow Yariv & Yeh, ch. 9.
τ = D/v, N = τΔf
Acoustic attenuation is ignored. The Bragg acoustic bandwidth is not modelled: an isotropic cell passes about Δf ≈ 1.8 n v²/(λ₀f₀L) (Chang, Handbook of Optics vol. II ch. 12), here ≈ 3.06 MHz, which is why wide-band TeO₂ deflectors use anisotropic (tangential) phase matching. The resolvable-spot count assumes the full sweep diffracts.
Model and references
Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers
Isotropic Bragg diffraction of plane waves with acoustic attenuation ignored. M₂ and the sound velocity hold for the listed acoustic mode and light polarization at 632.8 nm; the PbMoO₄ and fused-silica values are from a vendor table. TeO₂ slow-shear deflectors use anisotropic diffraction, so their Bragg angles differ from the isotropic ones. The acoustic bandwidth of Bragg matching is not modelled, and below Q = 1 (Raman–Nath) the sin² efficiency is flagged as invalid.
- Uchida N., Ohmachi Y., Niizeki N., Inamura T. (1972). Arrangement for acousto-optical light deflection. US Patent 3,661,441, Tables 1 and 2 (TeO₂, LiNbO₃). patents.google.com
- Isomet Corp. AO Scanning, application note AN4820 (PbMoO₄, fused silica; vendor table). isomet.com
- Uchida N. (1971). Optical properties of single-crystal paratellurite (TeO₂). Phys. Rev. B 4, 3736–3745. doi:10.1103/PhysRevB.4.3736
- Klein W. R., Cook B. D. (1967). Unified approach to ultrasonic light diffraction. IEEE Trans. Sonics Ultrason. 14, 123–134. doi:10.1109/T-SU.1967.29423
- Yariv A., Yeh P. (1984). Optical Waves in Crystals. Wiley. Ch. 9, acousto-optics.