Which pulse maximizes resonant nonlinear conversion?

Abstract: At fixed pulse energy, which drive waveform extracts the most $n$th-order nonlinear conversion from a resonator ($n=2$ for second harmonic)? A short pulse couples poorly to a narrow resonance, a long one dilutes its energy, and no linear rule fixes the compromise. We solve the problem exactly for a single mode of amplitude decay rate $κ$. Eliminating the drive turns fixed incident energy into a constraint on the stored field alone, and the optimization becomes a sharp Gagliardo--Nirenberg inequality whose extremal is the ground-state soliton of the nonlinear Schrödinger equation. The optimal stored field is $\mathrm{sech}<sup>{1/(n-1)}[(n-1)κt]$, sustained by an asymmetric input that rises as $e<sup>{κt}$ and falls as $e<sup>{-(2n-1)κt}$; the largest converted energy follows in closed form. A rising exponential, the time-reversal recipe, retains at most $79.0\%$ of the bound at $n=2$ and $2/e$ at large $n$; a two-rate pulse retains above $97\%$. Critical coupling generalizes to $n$-fold overcoupling, with optimal input coupling $n$ times the intrinsic loss rate. The bound applies from microrings to superconducting circuits and caps the per-pulse brightness of broadband photon-pair sources.