  Pump-probe supercontinuum (SC) spectral interferometry in reflection is used
to measure time-dependent variations in the complex dielectric function ($n$,
$k$) over the visible wavelength range in bulk silicon. Changes in $n$ ($\Delta
n$) due to a near infrared pump pulse are derived from transient fluctuations
in the reflected SC probe amplitude, as is measured in conventional pump-probe
experiments. Additionally, modifications to $k$ ($\Delta k$) are derived from
phase shifts in the reflected SC. The ultrafast response is attributed to
two-photon absorption ($\Delta k$) and the optical Kerr effect ($\Delta n$),
enabling measurement of effective two-photon ($\beta$) and Kerr ($n_2$)
coefficients. The onset of strong two-photon absorption as the two-photon
energy is tuned through the $E_1$ = 3.4 eV direct band edge of silicon is
observed for the first time to our knowledge. These constitute the first
measurements of the complex nonlinear response of a bulk medium from
reflectivity alone. The technique allows straightforward spectroscopic
measurements of the $\chi^{(3)}$ nonlinear response at the surface of absorbing
materials.
