Pulse Shaping in Intracavity Elements 323
This reflects the fact that the shortest focal length is produced at the inten-
sity maximum leading to the smallest beam size at the position of the aperture.
Obviously, such a lens aperture sequence is just another example of a fast element
whose overall transmission is controlled by the pulse intensity.
Laser pulse induced lensing does take place in nearly all ultrashort pulse
mode-locked lasers. In femtosecond pulse lasers, there is always a pulse shaping
mechanism in the cavity, involving a balance of dispersion and temporal SPM.
The latter effect implies necessarily a spatial modulation of the wavefront, hence
self-lensing. As a result of self-lensing, the size of the cavity modes is modi-
fied, leading to an increase or reduction of losses because (a) there is a change
in transmission through an aperture (hard aperture) or (b) there is a change in
spatial overlap between the cavity mode and the pump beam in the gain medium
(soft aperture).
5.4.4. Summary of Compression Mechanisms
Figure 5.16 summarizes the compression mechanisms and the associated pulse
shaping that were discussed in the previous sections.
5.4.5. Dispersion
The effect of dispersion is most simply treated in the frequency domain.
Using the notations of Chapter 2, a dispersive element is characterized by a
frequency dependent phase factor ψ(). In the particular case where the disper-
sion is because of propagation through a thickness d of a homogeneous medium
of index n(), the dispersive phase factor is simply ψ() = k()d. The most
rigorous procedure to model dispersion is to take the temporal Fourier transform
of the pulse,
˜
E
in
(), and multiply by the dispersion factor, to find the field
˜
E
out
()
after the dispersive element:
˜
E
out
() =
˜
E
in
()e
−iψ()
. (5.97)
An inverse Fourier transform will lead to the field
˜
E
out
(t) in the time domain.
When analytical expressions are sought to model the evolution of a pulse
in a cavity and the dispersion per round-trip is small, one can use an
approximate analytical solution in the time domain. We approximate
ψ() ≈ ψ
ω
( − ω
)
2
/2. This is the lowest order of a Taylor expansion
that produces a change in pulse shape. Expanding the exponential function
exp[−i ψ
ω
( − ω
)
2
/2] up to first order and Fourier transform to the time
324 Ultrashort Sources I: Fundamentals
tt
s
W
sg
I(t)
t
t
t
ttt
“Fast”
nonlinear
material
“Slow”
nonlinear
material
t
t
s
ϕ
ϕ
t
s
W
sa
I(t)
I(t)
I(t)
(a) (b)
(c) (d)
(e) (f)
Figure 5.16 Various compression mechanisms. (a) Gain saturation: The original pulse and its pulse
width are indicated by the solid line. The leading edge of the pulse is amplified, until the accumulated
energy equals the saturation energy density W
sg
at time t
s
, as indicated by the dashed area. In the
case of the figure, the pulse tail is not amplified, resulting in a shorter amplified pulse (dashed line).
(b) Saturable absorption: The leading edge of the pulse is attenuated, until the saturation energy
density W
sa
is reached at t
s
. (c) Frequency modulation (dotted line) because of saturation peaks at
the time t
s
when the saturation energy density W
sa
is reached. For t > t
s
, the pulse experiences
a downchirp, if the carrier frequency of the pulse is smaller than the resonance frequency of the
absorber. (d) Frequency modulation (dotted line) produced by the Kerr effect. The central part of
the pulse (intensity profile indicated by the solid line) experiences an upchirp. (e) Self-focusing by a
fast nonlinearity combined with an aperture, leads to a compression by attenuating both leading and
trailing edges. (f) In the case of self-focusing by a slow nonlinearity combined with an aperture, only
the trailing edge is trimmed.
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