Ultrashort Laser Pulse Phenomena, 2nd Edition

Evolution of the Pulse Energy 303

be taken into account by using the operator defined in Eq. (3.79) for the media

instead of Eq. (5.29) derived from the rate equations.

The operator describing the pulse change at each round-trip is obtained by

multiplying the transfer functions of all elements of the cavity, neglecting prod-

ucts of small quantities. To evaluate the steady state (5.28),

˜

E(t + h ) can be

conveniently written as (1 + h

d

dt

+ …)

˜

E(t), leading to an integro-differential

steady-state equation for the complex pulse envelope. The type of laser to be

modeled determines the actual elements (operators) that need to be included.

A parametric approach is generally taken to solve the steady-state equation.

An analytical expression is chosen for the pulse amplitude and phase, depending

on a number of parameters. This ansatz is substituted in the steady-state equation,

leading to a set of algebraic equations for the unknown pulse parameters. Several

types of mode-locked fs lasers have been modeled by this approach [21–24].

Changes in the beam profile because of the self-lensing effect have been incor-

porated [25,26]. The transverse dimension is included through a modification of

the pulse matrices introduced in Chapter 2 to include the action of the various

active resonator elements [26].

5.3. EVOLUTION OF THE PULSE ENERGY

Before proceeding with a discussion of the various processes of pulse

formation and compression, we will consider only the evolution of pulse energy

in the presence of saturable gain and nonlinear losses. Based on the continu-

ous model we will derive rate equations that describe the evolution of the pulse

energy on time scales of the cavity round-trip time and longer. The rate equations

will be written in terms of derivatives with respect to time. To relate this to the

spatial derivatives used in the continuous model we apply

d

dz

≈

1

ν

g

d

dt

=

τ

RT

2L

d

dt

, (5.34)

where L is the cavity length. Through most of this section we will neglect the

transverse variation of the beam intensity (flat top beam) and diffraction effects.

If we assume a beam cross section area A we may refer either to the total pulse

energy W or the energy density W = W/A.

We will concentrate first on parameters that may lead to a continuous

mode-locked pulse train, as opposed to Q-switched mode-locking. Most broad-

band solid-state laser media being used for short pulse operation have a long gain

lifetime. As a result, there is a tendency for the intracavity pulse to grow until

the gain has been depleted. The laser operation thereafter ceases until the gain