3.6 Acousto-optics

A sound wave in a crystal is a moving grating of compressed and stretched regions, and the refractive index follows the strain (photoelastic effect). Light crossing it is diffracted and shifted in frequency by exactly the sound frequency. In LN the same crystal can launch the sound piezoelectrically, so a modulator, frequency shifter or deflector needs no glued-on transducer (acousto-optic effect).

Sweep the RF frequency: the diffracted beam swings like a scanner, the basis of an acousto-optic deflector, and the light frequency of the +1 order rises by exactly the RF frequency.

Photoelastic (strain-optic) relation (Weis & Gaylord 1985 Eq. 18 P). Nelson and Lax showed the independent variable is the displacement gradient, not only strain: rotations of volume elements contribute in birefringent crystals, so p_ijkl is not symmetric in kl for shear waves; in LN p₄₄ = p₅₅ has symmetric and antisymmetric parts, not resolved by the available data (Weis & Gaylord 1985 p. 200 P-comp).

3m form, eight independent coefficients p₁₁, p₁₂, p₁₃, p₁₄, p₃₁, p₃₃, p₄₁, p₄₄, with p₂₂ = p₁₁, p₂₃ = p₁₃, p₂₄ = −p₁₄, p₃₂ = p₃₁, p₄₂ = −p₄₁, p₅₅ = p₄₄, p₅₆ = p₄₁, p₆₅ = p₁₄, p₆₆ = (p₁₁ − p₁₂)/2 (Weis & Gaylord 1985 Eq. 19 P). p_IJ is not a symmetric matrix (p₁₄ ≠ p₄₁).

Avakyants and Andrushchak at 633 nm; the wavelengths of the Coquin and Krupych sets are not recorded in the sources held. No set at 1550 nm is known; TFLN devices at 1550 nm reuse 633 nm values. Uncertainties of Andrushchak 2009: ±0.007 to ±0.029 (via Sarabalis et al. 2020 Table I).

Indirect (piezo-electro-optic) contribution. Strain makes a field through e, the field changes the index through r. It depends on the acoustic propagation direction and matters for p₁₃ and p₃₃: p₁₃(primary) = p₁₃(measured) − 0.043, p₃₃(primary) = p₃₃(measured) − 0.154 (Weis & Gaylord 1985 p. 201 P-comp; propagation direction not stated). Applied to Set B this gives p₃₃(primary) of about −0.08 derived, so which p₃₃ a model uses matters. The offsets are shown, not applied.

Figures of merit

LN, longitudinal wave along X with extraordinary light at 35° to y: n = 2.20, ρ = 4.7×10³ kg/m³, v = 6.57×10³ m/s (Dixon 1967 and Chang 1976 via Weis & Gaylord 1985 p. 201 P-comp).

The four definitions with the stated n, ρ, v reproduce M₁ and M₄; the printed M₃ exponent and unit are inconsistent with them, so the printed M₃ is not used derived.

Bragg example. λ₀ = 1550 nm, f = 1 GHz, X-longitudinal v = 6544.6 m/s: Λ = 6.54 µm and θ_B = 6.8° in air derived.

Integrated acousto-optics on LN. Collinear TE0-TE1 conversion by an SH0 wave at 440 MHz in a 1.25 µm wide suspended X-cut waveguide, up to 18% conversion (Sarabalis et al. 2021 reported); acousto-optic modulators in X-cut LN on sapphire (Sarabalis et al. 2020 reported); AO modulator V_πL of 0.03 V cm in TFLN with chalcogenide, unsuspended (Wan et al. 2022 reported); a bulk-wafer longitudinal piezoelectric resonant photoelastic modulator (Atalar et al. 2022 S).

1. Pair signs only within one frame. The p₁₄ and p₄₁ signs flip with c₁₄ (and e₂₂, d₂₂) under a 180° rotation about Z. Never combine p₁₄, p₄₁ from one paper with c₁₄, e₂₂ from another without checking the frame.

2. What is known. Weis & Gaylord 1985 list Avakyants' p₁₄, p₄₁ < 0 next to Smith and Welsh's c₁₄ > 0 in a paper that states the IRE frame, without saying it checked Avakyants' own frame; Krupych 2014 pairs c₁₄ > 0 with p₁₄, p₄₁ > 0 (via Mys et al. 2015 S); Mytsyk et al. 2021 use S₁₄ > 0 (c₁₄ < 0) S, a frame that is non-standard for LN if d₂₂ > 0 is imposed (inference). The common TFLN pairing (Kushibiki et al. c₁₄ > 0 with Set B) matches the Weis & Gaylord compilation but is not confirmed at the primary level.

3. What a mismatch does. For a film with in-plane unit vector b (light polarization and dominant strain both along b), the longitudinal coefficient is p′_bbbb = p_ijklb_ib_jb_kb_l. On an X-cut film with b = (0, cos φ, sin φ), φ from +Y toward +Z derived:

On X-cut the flip mirrors the optimum orientation from about 45° to about 135°; the magnitude is unchanged. On Y-cut (b = (cos φ, 0, sin φ)) the flip changes nothing (Andrushchak: maximum +0.230 at 50.7 and 129.3°; Set B maximum +0.238 at 48.3 and 131.7°). On Z-cut p′_bbbb = p₁₁ for every in-plane direction. Which orientation is right on X-cut depends on the unresolved pairing in item 2.

The Andrushchak values at 45 and 135°, 0.304 (X-cut) and 0.223 (Y-cut), equal the “p₂₂′” values in the author's public 2018 note on photoelastic tensor rotation (outside5sigma.com, 2018 S), where the X-cut angle was relabelled from 45 to 135°. A p₁₄/p₄₁ sign or frame inconsistency produces such a relabelling; a change of the θ = 0 reference, also recorded in that note, is an alternative explanation (inference).

Toggle the film cut: on X-cut the two wafers disagree about the bright direction; on Y-cut they agree.

4. Optical wavelength. The Avakyants and Andrushchak sets are at 633 nm (Weis & Gaylord 1985 P-comp; Sarabalis et al. 2020 Table I S); no 1550 nm set is known.

5. Rotation (Nelson-Lax) term. For shear waves in a birefringent crystal it adds Δε terms proportional to (ε_ii − ε_jj) times the rotation; an estimate for one LN nanobeam geometry put it below 4% of the photoelastic coupling (author's public 2018 note, outside5sigma.com S; unreviewed). Optomechanical coupling calculations on TFLN include moving-boundary and photoelastic terms; the rotation term is usually omitted.