  Spectra of the degenerate two-photon absorption coefficient $\beta(\omega)$
and the anisotropy parameters $\sigma(\omega)$ and $\eta(\omega)$ 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 model. In
indirect-gap GaP and Si, these components tend to increase with $\hbar \omega$
over our tuning range. Two-photon absorption spectral shapes differ
significantly from predictions of semi-empirical models, motivating
first-principles calculations of $\chi^{(3)}(\omega)$. 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.
