Ultrashort Laser Pulse Phenomena, 2nd Edition

Nonlinear Refractive Index Effects 409

In amplifier chains operating up to high saturation parameters, saturable

absorbers will generally be used to isolate stages of high gain, reducing the

effect of ASE. In addition the saturable absorber can counteract the broadening

effect of the amplifier because of:

• pulse shaping (steepening of the leading edge) by saturable absorption

and thus elimination of subsequent broadening by the saturable gain

(cf. Fig. 7.4); and

• pulse compression because of the combination of downchirping (if the pulse

has a longer wavelength than that of the peak of the absorption band) by the

absorber and propagation in a gain medium of normal (linear) dispersion.

7.3.2. Self-Focusing

The situation of pulse amplification is more complex if we consider self-

focusing effects which were introduced in Chapter 3. In this section, to obtain

some order of magnitude estimations, we will use the relations derived for cw

Gaussian beams. If the instantaneous peak power of the amplified pulse exceeds

the critical power for self-focusing defined in Eq. (3.171)

P

cr

=

(1. 22λ



)

2

π

32 n

0

¯n

2

, (7.19)

particular attention has to be given to the beam profile. Even weak ripples in the

transverse beam profile may get strongly amplified, and lead to breaking up of the

beam in filaments. The critical transverse dimension of these beam fluctuations,

w

cr

, below which a beam of intensity I (and power P > P

cr

) becomes unstable

against transverse intensity irregularities can be estimated from:

P

cr

=

πw

2

cr

2

I. (7.20)

If a smooth transverse beam profile is used, the amplifier may be operated above

the critical power, provided the optical path through the amplifier L does not

exceed the self-focusing length L

SF

:

L

SF

(t) =

ρ

0

√

P/P

cr

− 1

, (7.21)

where P = P(t) refers to the instantaneous power on axis of the Gaussian beam,

and the beam waist is at the sample input (cf. Eq. [3.189]).