William Angerer - Department of Physics and Astronomy - University ...

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by a plane wave for both the surface and bulk, the polarization cannot radiate a
second-harmonic wave in the vacuum. Note that SHG is forbidden from Si (100) in
this geometry for the electric quadrupole and magnetic dipole contributions to the
nonlinear polarization.
The discrepancy between theory and experiment is resolved by incorporating a
more sophisticated description of the fundamental beam in the focal plane. The
electric field of a Gaussian beam in the focal plane (z = 0) has the form
~ (X2 + y2)
E(x, y, z = 0) = xEexp - w~ , (7.1)
where Wo is the spot size or beam waist of the focused beam. Because silicon has
inversion symmetry, its nonlinear polarization is related to the fundamental fields as
( - .J) 1.-
Equation (7.2) is a nonlocal response of the nonlinear polarization to the fundamental
fields. The nonlocal response is mediated by a fourth rank tensor, Cjkl, which includes
electric quadrupole and magnetic dipole terms. A plane wave at normal incidence
induces only a nonlinear polarization that has a component perpendicular to the
silicon surface, i.e. the silicon surface is defined by the xy plane with z normal to the
plane, and the only nonlinear polarization is directed along z. From equation (7.2),
the second-order nonlinear polarization induced in Si (100) by an x polarized plane
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