304 Ultrashort Sources I: Fundamentals
is recovered, which takes a time of the order of the gain material lifetime (typi-
cally microseconds). The output of such a laser consists in bursts of Q-switched
mode-locked pulse trains.
This first subsection is dedicated to straightforward linear cavities. The case
of ring cavities and some linear cavities with two pulses per cavity round-trip is
more complex because it involves mutual coupling between counter propagating
pulses in an absorber or nonlinear loss element.
5.3.1. Rate Equations for the Evolution of the
Pulse Energy
Nonlinear Element
The hypothetical laser to be considered here consists of a gain and a loss
medium whose parameters vary with the intensity and the energy of the evolving
pulse, depending on the time constant of the nonlinearity. Examples are saturable
gain and loss as described in detail in Chapter 3. A nonlinear element will be
said to provide negative feedback if it enhances the net cavity losses with increas-
ing energy or intensity. The reverse (cavity losses decreasing with intensity or
energy) occurs for a nonlinear element that provides positive feedback. “Saturable
absorption” is an example of positive feedback: the loss decreases with increas-
ing intensity. Positive feedback is needed for the establishment of a pulse train.
It is generally desirable to have a positive feedback dominating the nonlineari-
ties of the cavity at higher intensities. Examples of negative feedback are two
photon absorption and intracavity SHG. It will be shown in Section 5.4 that Kerr
lensing contains both types of feedback. Another important example of positive
and negative feedback is found with semiconductor absorbers, as discussed in
Section 6.5.
We will consider in this section a combination of positive and negative passive
feedback nonlinearities. The nonlinear losses can be expressed through their
dependence on the pulse energy density W . We assume that at a certain energy,
a negative feedback takes over, i.e., the loss start increasing with energy. The sim-
plest form of nonlinear loss that will show a transition from positive to negative
feedback is:
L(W) = L
L
+ a(W − W
0
)
2
, (5.35)
where W
0
defines the energy at which the nonlinear losses switch over from
saturable losses (positive feedback) to induced losses (negative feedback). As we
will see when discussing specific examples of cavities, Eq. (5.35) is a second-
order fit for the actual energy dependence of the losses, hence L
L
is not simply
Evolution of the Pulse Energy 305
a sum of the linear losses but may also contain a contribution from the nonlinear
elements.
The saturable gain is another factor that determines the dynamics of the pulse
evolution in the cavity. We will show in an example of saturable absorption and
intracavity two photon absorption (cf. Section 5.3.2) how the parameters L
L
,
a, and W
0
are related to those material parameters. In the case of mode-locking
dominated by self-lensing, we will show in Section 5.4.3 the connection between
the phenomenological parameters L
L
, a, and W
0
and properties such as the
magnitude of the nonlinearity, the transverse dimension of the beam, the length,
and position of the nonlinear element.
Rate Equations
In the present derivation of the evolution of the pulse energy we will use a rate
equation approximation for the gain medium. Referring for instance to Eq. (4.18)
for a two-level system, we can write for the population difference N :
dN
dt
=−
I(t)N
I
s
T
1
−
N −N
0
T
1
−
N +N
0
2
R (5.36)
where I(t) is the laser intensity, R is a constant pumping rate,
6
and W
s
= I
s
T
1
is
the saturation energy density. N
0
is the equilibrium population difference in the
absence of the pump and laser field. For most gain media, the energy relaxation
time T
1
is longer than the pulse duration. The preceding equation is equivalent
to the rate equation often used to model a gain medium:
dN
dt
=−
IN
I
s
T
p
−
N −N
0
T
p
+ R
(5.37)
which has a constant pump rate R
=−RN
0
, and where the energy relaxation
time T
1
has been replaced by a shorter characteristic constant T
p
given by:
1
T
p
=
1
T
1
+
R
2
. (5.38)
A modified saturation intensity was introduced as I
s
= (T
1
/T
p
)I
s
. Without laser
field (I = 0) the population difference, according to Eq. (5.37), approaches an
6
The pumping term is proportional to the population of the ground state which is N
1
= (N +
N
0
)/2. R is an effective (assumed to be constant) pump rate that also contains the properties of a
third energy level involved in the pumping process.
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