Even-k Suppression

Even‑k Suppression in Parametrically Driven Spherical Cavities This paper establishes four analytical results for parametric instabilities in spherical optical cavities with sinusoidally oscillating boundaries R(t) = R_0(1 + ε cos Ωt), ε ≪ 1. 1. Even‑k suppression: The inter‑mode coupling mechanism (Mechanism 2) contributes identically zero to all even‑k parametric resonances Ω = 2ω_0/k at leading order in ε. The complete O(ε^k) instability for even k is carried entirely by Mechanism 1 (Mathieu frequency modulation). 2. Exact growth rate at k=2: γ_2 = (√5/6) ε^2 ω_0 ≈ 0.37268 ε^2 ω_0, arising from competition between two off‑resonant Floquet mediators. Standard single‑mediator approximations underestimate the correct value by a factor of 2√5/3 ≈ 1.491. 3. Drive contamination threshold: Of all harmonic contaminations, only a sin(2Ωt) component in the boundary motion seeds the k=2 instability. The exact contamination growth rate is γ_contam = (1/2) δ ω_0, and the dominance threshold is δ < (√5/3) ε^2, requiring –111 dBc at 2Ω for ε = 2×10^{-3}. 4. All‑orders suppression: The ε^{k-1} suppression of even‑k relative to odd‑k resonances extends to all orders in ε and to arbitrarily many coupled modes. The governing principle is the temporal ℤ_2 symmetry t → t + π/Ω, under which every interaction vertex changes sign; k is the corresponding symmetry quantum number. The proofs rest on a closed‑form evaluation of the energy‑transfer overlap integral I_k = ∫_0^{2π/Ω} sin(Ωt) cos(kΩt/2) dt, which vanishes identically for all even k. All results are classical; a quantum generalisation (photon statistics, squeezing spectra, role of the degeneracy factor 2l+1) is deferred to future work. This work provides a rigorous extended framework for Even‑K suppression in Ref. 1 and has direct implications for cavity optomechanics, the dynamical Casimir effect, and gravitational‑wave detectors such as LIGO.