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

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1/e 2 its ma.'{imum value. This distribution can be converted to the angular frequency
distribution
l2.26)
through the vacuum dispersion relationship w = kc = 2rrc/ A. The angular frequency
bandwidth varied as the pulse duration changed. Typically, ~w ranged from 0.021
to 0.053 fs- 1 over the spectral range of the Ti:Ah03 laser.
Also note. the wavevectors are expressed explicitly as a function of the angular
frequency, k(wd, in order to exhibit the dispersive nature of the medium. vVe can
simplify equations (2.23) and (2.24) and facilitate the calculation of the nonlinear
polarization induced by an ultrafast fundamental source by expanding the wavevector
k(wd to first-order as a power series in (WI - w o ), i.e.
( 'J ·r) -.-1
In Appendix A. the expansion up to first-order is rigorously justified for the systems
of interest. The second-order expansion term accounts for the broadening of the
fundamental pulse. vVe calculate that for our quartz sample thicknesses, the pulse is
broadened less than 0.3%. For the GaN samples, the broadening is even less. Hence.
a first-order expansion is valid. The symbols in equation (2.27) have their usual
meanings; n(wo) is the index of refraction at frequency Wo and c is the speed of light.
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