Band-Gap-Related Multiphoton Thresholds Organize Nonlinear Operating Windows for Ultrafast Subsurface Excitation: Silicon as a Test Case

Subsurface ultrafast excitation of a transparent or\( \beta^{(K)} \) semitransparent solid requires two conditions that compete during propagation: nonlinear refraction must become strong enough to concentrate the field, while multiphoton absorption must remain weak enough that the pulse is not depleted before reaching the target. We formulate this competition in terms of characteristic Kerr and multiphoton-depletion lengths and use silicon as a quantitative test case. In a single-gap energetic approximation, the threshold of a \( K \)-photon interband channel occurs near \( \lambda_K^{(0)}=Khc/E_g \); the reduced coordinate \( x=\lambda E_g/hc \) therefore provides a material-independent way to organize where the lowest energetically allowed absorption order changes. These nominal thresholds do not determine an optimum on their own: phonon assistance, excitonic and multiband structure, and the measured wavelength dependence of \( n_2 \) and \( \beta^{(K)} \) control the actual response. The nonlinear figure of merit (NFOM) satisfies \( L_{\rm MPA}/L_{\rm Kerr}=2\pi\,\mathrm{NFOM} \). At any candidate operating intensity, simultaneous Kerr engagement and limited nonlinear depletion require the pointwise condition \( 2\pi\,\mathrm{NFOM}\gtrsim1 \); for a specified focusing length \( L_f \), the actual overlap is quantified by the interval \( I_{\rm eng}\le I\le I_{\rm dep} \). Using published silicon coefficients, representative \( L_f=100 \) and \( 250~\mu \)\( m \) values give the largest multiplicative operating-band widths near 1.95–2.15 \( \mu \)\( m \), no overlap at 2.55 \( \mu \)\( m \) within this reduced criterion, and narrower bands at 1.55 and 3.25 \( \mu \)\( m \). The calculation also shows why a low-intensity NFOM can greatly exaggerate the useful dynamic range once three-photon absorption becomes important at the operating intensity. Broadband wavelength, duration, energy, and focusing trends are calculated with a spectral UPPE implementation, while a generalized nonlinear Helmholtz equation solved by the finite-element method (NHE–FEM) provides the local scalar-nonparaxial description of strongly localized field–carrier dynamics. The combined framework connects nonlinear material coefficients, field evolution, photoexcited electron–hole carriers, and processing-relevant localization while explicitly separating the calculated electronic stage from subsequent lattice transformation.