  The demand for nonlinear effects within a silicon platform to support
photonic circuits requiring phase-only modulation, frequency doubling, and/or
difference frequency generation, is becoming increasingly clear. However, the
symmetry of the silicon crystal inhibits second order optical nonlinear
susceptibility, $\chi^{(2)}$. Here, we show that the crystalline symmetry is
broken when a DC field is present, inducing a $\chi^{(2)}$ in a silicon
waveguide that is proportional to the large $\chi^{(3)}$ of silicon. First,
Mach-Zehnder interferometers using the DC Kerr effect optical phase shifters in
silicon ridge waveguides with p-i-n junctions are demonstrated with a
$V_{\pi}L$ of $2.4Vcm$ in telecom bands $({\lambda}_{\omega}=1.58{\mu}m)$
without requiring to dope the silicon core. Second, the pump and second
harmonic modes in silicon ridge waveguides are quasi-phase matched when the
magnitude, spatial distribution of the DC field and $\chi^{(2)}$ are controlled
with p-i-n junctions. Using these waveguides, second harmonic generation at
multiple pump wavelengths are observed with a maximum efficiency of
$P_{2{\omega}}/P_{\omega}^2$=12%/W at ${\lambda}_{\omega}=2.29{\mu}m$ in a 1mm
long waveguide. This corresponds to a field-induced $\chi^{(2)}=41pm/V$,
comparable to non-centrosymmetric media (LiNbO3, GaAs, GaN). The field-induced
nonlinear silicon photonics will lead to a new class of CMOS compatible
integrated devices spanning from near to mid infrared spectrum.
