  Spectra of the degenerate two-photon absorption coefficient $\beta(\omega)$,
anisotropy parameter $\sigma(\omega)$, and dichroism parameter $\delta(\omega)
= \big[\sigma(\omega) + 2\eta(\omega)\big]/2$ of crystalline 110-cut GaAs, GaP,
and Si at 300 K were measured using femtosecond pump-probe modulation
spectroscopy over an excitation range in the vicinity of each material's
half-band gap $E_g/2$ (overall $0.62 < \hbar \omega < 1.91$ eV, or $2000 >
\lambda > 650$ nm). Together these three parameters completely characterize the
three independent components of the imaginary part of the degenerate
third-order nonlinear optical susceptibility tensor
$\mathrm{Im}\{\chi^{(3)}_{abcd}(\omega)\}$. In direct-gap GaAs, these
components peak at $\hbar \omega \approx 0.78 E_g$ which is close to the peak
at $\hbar \omega = 0.71 E_g$ predicted by the Jones-Reiss phenomenological
model. The dispersion is comparable to ab initio calculations. In indirect-gap
GaP and Si, these components tend to increase with $\hbar \omega$ over our
tuning range. In Si, the dispersion differs significantly from predictions of
semi-empirical models and ab initio calculations do not account for transitions
below the two-photon direct band gap, motivating further investigation.
Kleinman symmetry was observed to be broken in all three materials. We also
note anomalies observed and their possible origins, emphasizing the advantages
of a 2-beam experiment in identifying the contribution of various nonlinear
effects.
