Tsunami - Beckman Institute Laser Resource Center

Tsunami
Group Velocity Dispersion (GVD)
Nonlinear Effects
Fourier analysis (or as consequence of the Heisenberg uncertainty principle)
imposes a restriction on the bandwidth of an ultrashort pulse. For a
pulse of duration dtp and bandwidth Av, it is always true that Av -A tp will
be greater than a constant with a value of about 1. The exact nature of the
constant depends on the exact shape of the pulse (examples are given in
appendix B). It is apparent that, the shorter the pulse, the larger the
bandwidth and, thus, the greater the difference from the lowest to highest
frequency within a pulse. Since the index of refraction of any material is
frequency dependent, each frequency in a pulse experiences a slightly
different index of refraction as it propagates. This index of refraction
difference corresponds to a velocity difference, causing a time separation
between the different frequencies of a pulse. Group velocity dispersion
(GVD) is defined as the variation in transit time as a function of
wavelength. For positive GVD, the lower frequencies (red) travel faster
than higher frequencies (blue). The effect is more pronounced for shorter
pulses (because of their larger bandwidth).
Figure A-5 shows the refractive index n versus wavelength A for a typical
transparent optical material. For any given wavelength, the refractive
index n(d) determines the phase velocity. The slope of the curve, dn(A)/d;2,
determines the group velocity, or the velocity of a short pulse with a center
wavelength of A.
The second derivative of the curve, d2n(A)/d2, determines the GVD, which
is the rate at which the group velocity changes as a function of wave-
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length, i.e., it governs the rate at which the frequency components of a
pulse change their relative time. GVD can change the temporal shape of
the pulse by broadening it or narrowing it, depending on the "chq" of the
original pulse. A pulse is said to be positively chqed, i.e., it has experienced
positive GVD, if the low frequencies lead the high (red is in front),
and negatively chqed if the opposite is true. Pulses are typically positively
chqed as they pass through normal materials at visible and near ir
wavelengths.
In addition to GVD, the output pulse width and pulse shape from the
Tsunami are governed by the interaction of the pulse with the nonlinear
index of the Ti:sapphire. The nonlinear index of refraction n;! introduces
an intensity-dependent index at high intensities:
-
where no is the linear index of refraction and I is the instantaneous pulse
intensity. This results in self phase modulation (SPM) of the pulse. As the
pulse propagates through the Ti:sapphire material, the leading edge
experiences an increasing index of refraction. This causes a delay in the
individual oscillations of the electric field and results in a "red-shifted"
" -
leading edge. Conversely, the trailing edge of the pulse is "blue-shifted."