Group-velocity dispersion

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 ${\displaystyle \sigma }$ 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,

${\displaystyle E(t)=Ae^{-{\frac {t^{2}}{4\sigma ^{2}}}}e^{-i\omega _{0}t},}$

or equivalently, as a function of frequency,

${\displaystyle E(\omega )=Be^{-{\frac {(w-w_{0})^{2}}{4(1/2\sigma )^{2}}}}}$

(the parameters A and B are normalization constants). Passing through the medium results in a frequency-dependent phase accumulation ${\displaystyle \Delta \phi (\omega )=k(\omega )d}$, such that the post-medium pulse can be described by

${\displaystyle E(\omega )=Be^{-{\frac {(w-w_{0})^{2}}{4(1/2\sigma )^{2}}}}e^{ik(\omega )d}.}$

In general, the refractive index ${\displaystyle n(\omega )}$, and therefore the wave vector ${\displaystyle k(\omega )=n(\omega )\omega /c}$, can be an arbitrary function of ${\displaystyle \omega }$, 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 ${\displaystyle n}$, then good approximations of the impact of the refractive index can be obtained by replacing ${\displaystyle k(\omega )}$ with its Taylor expansion centered about ${\displaystyle \omega _{0}}$:

${\displaystyle {\frac {n(\omega )\omega }{c}}=\underbrace {\frac {n(\omega _{0})\omega _{0}}{c}} _{k(\omega _{0})}+\underbrace {\left[{\frac {n(\omega _{0})+n'(\omega _{0})\omega _{0}}{c}}\right]} _{k'(\omega _{0})}(\omega -\omega _{0})+{\frac {1}{2}}\underbrace {\left[{\frac {2n'(\omega _{0})+n''(\omega _{0})\omega _{0}}{c}}\right]} _{\text{GVD}}(\omega -\omega _{0})^{2}+\dots }$

Truncating this expression and inserting it into the post-medium frequency-domain expression results in a post-medium time-domain expression

${\displaystyle E_{\text{post}}(t)=A_{\text{post}}\exp \left[-{\frac {\left(t-k'(\omega _{0})d\right)^{2}}{4\left(\sigma ^{2}-i\,{\text{GVD}}\,d/2\right)}}\right]e^{i[k(\omega _{0})d-\omega _{0}t]}.}$

On balance, the pulse is lengthened to an intensity standard deviation value of

${\displaystyle \sigma _{\text{post}}={\sqrt {\sigma ^{2}+\left[d\,{\textrm {GVD}}(\omega _{0}){\frac {1}{2\sigma }}\right]^{2}}},}$

thus validating the initial expression. Note that a transform-limited pulse has ${\displaystyle \sigma _{\omega }\sigma _{t}=1/2}$, which makes it appropriate to identify 1/(2σ_t) as the bandwidth.

An alternate derivation of the relationship between pulse chirp and GVD, which more immediately illustrates the reason why GVD can be defined by the derivative of inverse group velocity, can be outlined as follows. Consider two transform-limited pulses of carrier frequencies ${\displaystyle \omega _{1}}$ and ${\displaystyle \omega _{2}}$, which are initially overlapping in time. After passing through the medium, these two pulses will exhibit a time delay between their respective pulse-envelope centers, given by

${\displaystyle \Delta T=d\left({\frac {1}{v_{g}(\omega _{2})}}-{\frac {1}{v_{g}(\omega _{1})}}\right).}$

The expression can be approximated as a Taylor expansion, giving

${\displaystyle \Delta T=d\left({\frac {1}{v_{g}(\omega _{1})}}+{\frac {\partial }{\partial \omega }}\left({\frac {1}{v_{g}(\omega ')}}\right)_{\omega '=\omega _{1}}(\omega _{2}-\omega _{1})-{\frac {1}{v_{g}(\omega _{1})}}\right),}$

or

${\displaystyle \Delta T=d\times {\textrm {GVD}}(\omega _{1})\times (\omega _{2}-\omega _{1}).}$

From here it is possible to imagine scaling this expression up two pulses to infinitely many. The frequency difference ${\displaystyle \omega _{2}-\omega _{1}}$ must be replaced by the bandwidth, and the time delay ${\displaystyle \Delta T}$ evolves into the induced chirp.