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Concept
Group-velocity dispersion
Summary
In optics, group-velocity dispersion (GVD) is a characteristic of a dispersive medium, used most often to determine how the medium affects the duration of an optical pulse traveling through it. Formally, GVD is defined as the derivative of the inverse of group velocity of light in a material with respect to angular frequency,
where and are angular frequencies, and the group velocity is defined as . The units of group-velocity dispersion are [time]2/[distance], often expressed in fs2/mm.
Equivalently, group-velocity dispersion can be defined in terms of the medium-dependent wave vector according to
or in terms of the refractive index according to
Group-velocity dispersion is most commonly used to estimate the amount of chirp that will be imposed on a pulse of light after passing through a material of interest:
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.
Official source
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