Inter-harmonic ratio structure and saturation of Bernstein modes in graphene

Bernstein modes (BM) in graphene are finite-wavevector magnetoplasmons excited by contact near fields, whereas ordinary cyclotron resonance (CR) probes $q\approx0$. We derive the BM peak absorption in the quasiclassical ballistic regime and show that it factorizes into a launch spectrum, Bernstein-mode splitting, turning-point enhancement, and residual dielectric-response factor. At fixed excitation frequency, BM overtones ($n\ge2$) are sampled, to leading order, at the same momentum $q\simeq\omega/v_F$. Smooth launch and screening factors therefore cancel in inter-harmonic peak ratios, yielding $I_n/I_m\simeq m/n$, modified by linewidth corrections and one residual response ratio for each harmonic pair. In smooth-launcher synthetic tests, noisy full-$q$ spectra recover the residual ratio within errors: moderate launcher/dielectric misspecification within this benchmark family shifts it by only $\sim\!1$--$2\%$, whereas linewidth assumptions shift it by $\sim\!10$--$30\%$. The same factorization connects low-power amplitudes to nonlinear saturation. If BM harmonics share the same cooling region and bolometric readout, the low-power slope times onset intensity is harmonic independent, while BM and CR power sweeps obey distinct normalized saturation curves with linewidth scalings $\Gamma^{-1/2}$ and $\Gamma^{-1}$.