298 Ultrashort Sources I: Fundamentals
5.2.3. Elements of a Numerical Treatment
Because of its central importance in the description of the pulse evolution
let us first elaborate on the successive treatment of processes in the time and
frequency domains. If a single cavity element represents several processes it is
often convenient to divide it into thin slices. The term thin means that the change
in the complex pulse envelope caused by one slice is small. In this case the order
of processes considered in one slice is unimportant. The choice about which
processes are treated in which domain (time, frequency, or spatial frequency) is
made based on numerical or analytical convenience and feasibility. Gain and the
Kerr effect are typically dealt with in the time domain ((x, y, z, t ) space) while
free-space propagation and dispersion are usually treated in the frequency domain
((k
x
, k
y
, z , t)or(x, y, z, ) space). For example, if gain and dispersion occur in
one element, for each slice one has to solve a differential equation in the time
domain to deal with the gain and subsequently treats the effect of the dispersion
and diffraction in the frequency domain.
Figure 5.10 illustrates the procedure for the sequence of an element (or slice)
with gain, free-space propagation, phase modulation, and dispersion. The illus-
tration starts with an electric field
˜
E
1
(x, y, z, t) =
˜
E
1
(x, y, z, t)e
iω
t
entering the
gain medium. This could be noise if we want to simulate the pulse evolution.
This first step where all processes but the gain are neglected can formally be
written using a transfer operator T
g
.
˜
E
2
(x, y, t) = T
g
(t)
˜
E
1
(x, y, t). (5.24)
In practice one solves the differential equations derived in Chapters 3 and 4 for a
medium with population inversion. The next step involves propagation over a
distance L
P
. This diffraction problem is best described in the frequency domain.
FFT algorithms are applied to obtain
˜
E
3
(, x, y). In Fresnel approximation the
Noise
12 3 4 5
L
P
L
M
L
D
Gain Free-space
propagation
Phase
modulator
Dispersion
N
Figure 5.10 Illustration of some of the main elements and processes in a circulation model that
describes pulse evolution in a laser. The numbers refer to the subscript of the electric field before
and after a certain element or process.
Circulating Pulse Model 299
propagation step through free space is described by Eq. (1.203):
˜
E
3
(x, y, ) =
i
2πcL
P
e
−iL
P
/c
dx
dy
˜
E
2
(x
, y
, )
× exp
−
i
2L
P
c
(x
− x)
2
+ (y
− y)
2
1/2
. (5.25)
Inverse FFT then produces the output in the time domain
˜
E
3
(x, y, t). Except for
pulses of a few optical cycles or shorter the approximation ≈ ω
can be
made in the terms preceding the integral and in the exponent of the integrand.
As explained in Chapter 1 this is equivalent to separating the space and time
effects on propagation.
The next element introduces a phase modulation. Let us assume that through
some effect the (nondispersive) refractive index of the material is modulated
in time and/or space, n = n(x, y , t). Its effect on the pulse is advantageously
described in the time picture
˜
E
4
(x, y, t) =
˜
E
3
(x, y, t) exp
−i
ω
c
n(x, y, t)L
M
. (5.26)
Note that the pulse envelope |E(t)| does not change while the pulse spectrum and
spatial frequency spectrum are modified because of the action of such a phase
modulator.
As detailed in Chapter 1, cf. Eq. (1.166), a dispersive element is characterized
by its (linear) transfer function, which for a dispersive path of length L
D
is simply
the propagator exp[−ik()L
D
] with k = n()/c. Thus
˜
E
5
(x, y, ) =
˜
E
4
(x, y, ) exp
−i
c
n()L
D
. (5.27)
The necessary input field is obtained after FFT of the output of the phase
modulator.
This procedure is continued until all resonator elements are taken into account.
The final output pulse
˜
E
N
(x, y, t) is then coupled back into the first element (gain in
our case) to start the next round-trip.
As pointed out previously, the treatment of a single cavity element may
require a procedure as just described. This, for example, is true for the gain
crystal in a Kerr lens mode-locked laser. This element is responsible for gain,
dispersion, self-lensing and diffraction (beam propagation). The procedure is
exemplified in Figure. 5.11. The crystal is divided into slices of thickness z
and the various effects are dealt with one at a time in each slice. At the begin-
ning of each slice the pulse properties are defined by the complex amplitude
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