3.4 Cuts and coupling

Rotate the cutting plane of an LN boule and the surface wave changes speed and strength. At a cut rotated about 128° from the Y axis, the ordinary surface wave is close to its strongest coupling (5.4%) and the competing fast wave has no isolated solution derived; 128° rotated Y wafers are the LN SAW product of crystal suppliers (Sumitomo Metal Mining P). Near 41 and 64° the fast, leaky wave dominates instead, with two to three times the coupling.

Slide through 128°: the Rayleigh coupling peaks (5.42% at 129°) where the leaky branch has faded out; near 60 to 70° the reverse happens, the Rayleigh coupling falls to about zero while the leaky wave carries 9 to 11%. The industrial cuts sit where one wave is strong and the other absent.

Cut notation (IEEE 176 Sec. 3.6 P): a singly rotated plate (YXl) θ starts as a Y-cut plate with length along X and is rotated by θ about its length (X). “128° Y-X” = (YXl) 128°, wave propagating along X. Plate normal Y′ = Y cos θ + Z sin θ. The bulk thresholds along X do not depend on θ: slow shear 4028.96 m/s, fast shear 4750.09 m/s, quasi-longitudinal 6544.61 m/s derived.

Kushibiki et al. 1999 constants P, partial-wave solver, 0.5° sweep derived. Indicative: one of the two surfaces has no isolated leaky root (threshold-pinned), so k² uses a window-edge velocity.

The Rayleigh coupling peaks at 128 to 129°, where the leaky wave's coupling has fallen to 1.3% at 120° and no leaky root is resolved at 128 to 130°; the Rayleigh coupling falls to about zero near 60 to 70°, where the leaky wave carries 9 to 11%. One cut gives a clean Rayleigh filter, the other a clean, strongly coupled leaky-wave filter. Shibayama et al. 1976 reported an optimum rotated Y cut near 128° for SAW filters S.

A k² quoted for a cut is meaningful only with its constant set named: the older Warner set gives about 17% and 11% for 41 and 64°, the Kushibiki et al. 1999 and Ogi et al. 2002 sets 15.6 to 15.7% and 10.3 to 10.4% (switch the constant set in the figure). Cuts in use: LN 128° rotated Y wafers (350 µm at 4 in, 500 µm at 6 in) and LT 36 to 50° rotated Y wafers, mainly for smartphone SAW filters and duplexers (Sumitomo Metal Mining P). Thin-film LN uses X-cut and Z-cut films (and LN on sapphire) for phononic circuits; the in-plane propagation angle matters in addition to the cut (Sarabalis et al. 2020, abstract reported).

Leaky branch. The leaky column needs the complex-velocity solver. Points marked “no isolated root” are where the leaky mode merges with the slow-shear threshold (4029 m/s along X) and the 4×4 boundary determinant has no isolated minimum. Near 36 to 41° the free-surface leaky root lies 0.7 to 0.8 m/s below the fast-shear threshold (4750.1 m/s for the Kushibiki et al. constants); after a minimum ends at the 1 m/s exclusion window, a search inside a 0.05 m/s window resolves it as an isolated complex root (41°: 4749.31 − 0.017i m/s, loss 2.0×10⁻⁴ dB per wavelength; metallized 4376.20 − 1.28i m/s, 0.016 dB per wavelength) derived. The same near-threshold position occurs for the Ogi et al. constants (4721.3 vs 4722.1) and the Warner et al. constants (4784.7 vs 4785.5). The vendor 4750 m/s for 41° equals this velocity to the stated digits V.

Spurious minima. The determinant has false minima at bulk thresholds; a 1 m/s window around each bulk velocity along x₁ is excluded, a minimum that ends at a window edge is searched again inside a 0.05 m/s window, and only if no isolated zero exists there is it reported as threshold-pinned, never as a root.

Axes. Convention here: rotation matrix rows = new axes in crystal coordinates; cut = (YXl) θ; global x₁ = propagation, x₃ = outward plate normal.

Vendor values. Fomos-Materials lists 3980 m/s (128° Y-X), 4690 (64°) and 4750 (41°), within 3 m/s of the Kushibiki et al. 1999 results; its coupling percentages are for an undefined quantity and are not compared V.