Derivation
A simple illustration of how GVD can be used to determine pulse chirp can be seen by looking at the effect of a transform-limited pulse of duration passing through a planar medium of thickness d. Before passing through the medium, the phase offsets of all frequencies are aligned in time, and the pulse can be described as a function of time,
or equivalently, as a function of frequency,
(the parameters A and B are normalization constants).
Passing through the medium results in a frequency-dependent phase accumulation , such that the post-medium pulse can be described by
In general, the refractive index , and therefore the wave vector , can be an arbitrary function of , making it difficult to analytically perform the inverse Fourier transform back into the time domain. However, if the bandwidth of the pulse is narrow relative to the curvature of , then good approximations of the impact of the refractive index can be obtained by replacing with its Taylor expansion centered about :
Truncating this expression and inserting it into the post-medium frequency-domain expression results in a post-medium time-domain expression
On balance, the pulse is lengthened to an intensity standard deviation value of
thus validating the initial expression. Note that a transform-limited pulse has , which makes it appropriate to identify 1/(2σt) as the bandwidth.