Section 13:
Ultrafast Dispersion:
When It Matters
While the e ect of dispersion is minimal for many types of laser systems,
it is especially problematic in ultrafast laser applications. Ultrafast
lasers are characterized by short pulse durations on the order of
picoseconds, femtoseconds, or attoseconds. Due to the Heisenberg
uncertainty principle, transform-limited ultrafast pulses reaching the
lower limit of their pulse duration have a wide wavelength bandwidth
(Figure 13.1). As these wide bandwidth pulses transmit through optical
media, chromatic dispersion lengthens the pulse duration, which is detrimental
in ultrafast applications.
Section 13.1:
Overview of Chromatic Dispersion
The way in which a laser pulse travels through an optical medium is
described by group velocity (vg ) - the variation of the phase velocity of
light in a medium relative to its wavenumber (k):
ω is the light’s angular frequency, c is the speed of light in a vacuum, and
n is the refractive index of the medium. The wavenumber k is 2/λ - this
concept is sometimes referred to as the spatial frequency of the wave.
The di erence between phase velocity and group velocity is illustrated
in Figure 13.2.
When light of multiple wavelengths travels through a material, it is common
for the longer wavelength (low frequencies) to travel slightly faster
than the shorter wavelengths due to a frequency (or wavelength) dependence
of the group velocity,1 This causes a spectral variation of the
phase of the wavefront in the same way that light travelling through a
prism is broken into its component colors from spectral dispersion of the
material. As the group velocity is given as the rst derivative of phase
velocity with respect to frequency, the group velocity dispersion (GVD)
is the derivative of the inverse group velocity with respect to frequency:
The inverse group velocity is known as rst-order dispersion, and GVD
is known as second-order dispersion. Just as the group velocity is similar
to spectral dispersion, in that both correspond to the rst derivative
of refractive index with wavelength or frequency, the GVD is used similarly
to the partial dispersion as they are both second derivatives with
respect to wavelength or frequency. Designing optics for low-GVD is
similar to designing for good chromatic performance, except the focus is
placed on group velocity and GVD rather than the related Abbe number
and partial dispersion.
GVD is independent of the length of the given optical medium. Group
delay dispersion (GDD) considers the length of the medium and can be
found by multiplying the GVD by the length.
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1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Pulse Bandwidth for Different Pulse Durations
1020 1030 1040 1050 1060 1070 1080 1090 1100 1110
Relative Intensity
Wavelength (nm)
50 fs
500 fs
10 ps
Figure 13.1: As the pulse duration of an ultrafast laser decreases, the
wavelength bandwidth increases
Group velocity: motion of envelope
Phase velocity: motion of individual point
Figure 13.2: The group velocity de nes the motion of the envelope,
or wave packet, highlighted in blue, while the phase velocity de nes the
higher frequency motion of each individual point of the wave itself, highlighted
in red
13.1
13.3
0
-50
-100
-150
Fused Silica GVD vs Wavelength
1 1.2 1.4 1.6 1.8
GVD (fs2/mm)
(m)
Figure 13.3: GVD vs wavelength for fused silica with a zero-dispersion
wavelength around 1,3 m
13.2