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

32
Here, x~;!Aw
= 2wo) is a constant for quartz over a wide range of frequencies.
X~;!x(w = 2wo) for quartz varies by less than 0.4% over the bandwidth of the Ti:AI 2 0 3
laser.
Integration over W2 yields
2
( )
. -2 w - 2wo
p(2)(r w) = P. elkb(W)Yexp( )
x' 0 ~W2·
(2.33 )
with
P. = fi LlwE 2 v(2) (w = 2w )
o 2V2 "-xxx 0
(2.34)
Note the use of the symbol kb (w) in the phase pP). This symbol denotes that kb (w )
is the bound wavevector. This is clear from the functional form of kb(w), i.e.
(2.35 )
which is simply twice the wavevector of the fundamental field (equation (2.28)). This
is the same result as for the the monochromatic case: the nonlinear polarization
is coupled or bound to the fundamental wave and propagates with a wavevector
twice that of the fundamental wave.
As a final note, the subscripts 1 and 2 are
no longer necessary because we integrated over the angular frequency spectrum of
the fundamental fields. Notation used henceforth is absent of these subscripts. To
summarize this section, we have calculated the nonlinear polarization generated from
an ultrafast, or broadband, fundamental light source using a first-order expansion in
the wavevectors of the fundamental fields.
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