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,[1] [2]
GVD(\omega0)\equiv
\partial | |
\partial\omega |
\left(
1 | |
vg(\omega) |
\right) | |
\omega=\omega0 |
,
\omega
\omega0
vg(\omega)
vg(\omega)\equiv\partial\omega/\partialk
Equivalently, group-velocity dispersion can be defined in terms of the medium-dependent wave vector
k(\omega)
GVD(\omega0)\equiv\left(
\partial2k | |
\partial\omega2 |
\right) | |
\omega=\omega0 |
,
n(\omega)
GVD(\omega0)\equiv
2 | \left( | |
c |
\partialn | |
\partial\omega |
\right) | |
\omega=\omega0 |
+
\omega0 | |
c |
\left(
\partial2n | |
\partial\omega2 |
\right) | |
\omega=\omega0 |
.
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:
chirp=(materialthickness) x GVD(\omega0) x (bandwidth).
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
\sigma
E(t)=
| ||||
Ae |
-i\omega0t | |
e |
,
E(\omega)=
| ||||||||||||
Be |
\Delta\phi(\omega)=k(\omega)d
E(\omega)=
| ||||||||||||
Be |
ei.
In general, the refractive index
n(\omega)
k(\omega)=n(\omega)\omega/c
\omega
n
k(\omega)
\omega0
n(\omega)\omega | |
c |
=\underbrace{
n(\omega0)\omega0 | |
c |
Truncating this expression and inserting it into the post-medium frequency-domain expression results in a post-medium time-domain expression
Epost(t)=Apost\exp\left[-
\left(t-k'(\omega0)d\right)2 | |
4\left(\sigma2-iGVDd/2\right) |
\right]
i[k(\omega0)d-\omega0t] | |
e |
.
On balance, the pulse is lengthened to an intensity standard deviation value of
\sigmapost=\sqrt{\sigma2+\left[drm{GVD}(\omega0)
1 | |
2\sigma |
\right]2},
\sigma\omega\sigmat=1/2
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
\omega1
\omega2
\DeltaT=d\left(
1 | |
vg(\omega2) |
-
1 | |
vg(\omega1) |
\right).
\DeltaT=d\left(
1 | |
vg(\omega1) |
+
\partial | |
\partial\omega |
\left(
1 | |
vg(\omega') |
\right) | |
\omega'=\omega1 |
(\omega2-\omega1)-
1 | |
vg(\omega1) |
\right),
\DeltaT=d x rm{GVD}(\omega1) x (\omega2-\omega1).
\omega2-\omega1
\DeltaT
A closely related yet independent quantity is the group-delay dispersion (GDD), defined such that group-velocity dispersion is the group-delay dispersion per unit length. GDD is commonly used as a parameter in characterizing layered mirrors, where the group-velocity dispersion is not particularly well-defined, yet the chirp induced after bouncing off the mirror can be well-characterized. The units of group-delay dispersion are [time]2, often expressed in fs2.
The group-delay dispersion (GDD) of an optical element is the derivative of the group delay with respect to angular frequency, and also the second derivative of the optical phase:
D2(\omega)=-
\partialTg | |
d\omega |
=
d2\phi | |
d\omega2 |
.
Dtot
D2(\omega)=-
2\pic | |
λ2 |
Dtot.