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

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for materials with no free charge. This implies
V'. E(w) = _ 47rV' . P(2)(W).
€(w)
(2.7)
2.2 Properties of X~~k(w = 2wo)
A full expression for the second-order susceptibility element that describes secondharmonic
generation is
+ (1Ir,ln)(nlr.lm)(mlrk II)
[(W-Wmn )+hmn][(Wo+Wnl )-hnl J
+ (1Ir,ln)(nlr;lm)(mlrkll)
[(W+Wnm )-hnm ][(wo -Wml )+hml J
+ (1Ir,ln)(nlrklm)(mlr.II») (? 8)
[(w+wmd-t"fmd[(wo+wnd+hnd . _.
Equation (2.8) can be derived from first principles using quantum statistical mechanics
with the proper choice of the interaction Hamiltonian [34]. In equation (2.8)
It), 1m), andln) are quantum mechanical states of the system; pf?) is the unperturbed
density of the state It), i.e. p(O) is the equilibrium population distribution: -eri is
the dipole moment operator along direction i; liw nm is the energy difference between
states In) and 1m); and 'Ynm is a dephasing term. The different terms in equation
(2.8) correspond to different time ordering of the photons. This is discussed further
in section 5.-1.1.
There are a few salient features of equation (2.8) that bear discussion.
First.
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