Ultrashort Laser Pulse Phenomena, 2nd Edition

314 Ultrashort Sources I: Fundamentals

For a thin saturable absorber the transmitted pulse energy, according to

Eq. (5.52), is:

W

out

= W

in



1 − α

0

d

1



1 −

β

2

W

in

W

sa

+

θ

6



W

in

W

sa



2



. (5.63)

We have introduced the coefficients β and θ to describe the colliding pulse

effect on the saturation. If the absorber is geometrically thin compared to the

pulse length and the pulses interact incoherently, for example because of crossed

polarizations, W

in

should be replaced by 2W

in

and θ = β

2

= 4.

In the case of coherent overlap the two counter-propagating pulses produce

an intensity grating in the absorber.

I(z) = I

0

[

1 + cos(2kz)

]

. (5.64)

While at the nodes of the intensity modulation there is no saturation, the saturation

at the maxima corresponds to an energy density of 4W

in

. An analysis of the

grating and its effect on the propagating field gives β = 3 and θ = 5in

Eq. (5.63) [28].

5.4. PULSE SHAPING IN INTRACAVITY ELEMENTS

In any description of a laser that follows the round-trip model, either

numerically or analytically, each element is taken in sequence, represented either

by a matrix or a scalar function. Essential intracavity elements are analyzed in this

section. In the sections that follow and in Appendix E, we will derive expressions

for the most essential combinations of intracavity elements. The term element

refers here more to a function than a physical element, because each compo-

nent of a laser will have generally a plurality of properties which are most easily

treated separately. For instance, the Ti:sapphire crystal in a laser may serve simul-

taneously as a gain medium, dispersive element, nonlinear nonresonant element,

astigmatism compensator. In each subsection characterizing an element, we will

give expressions for its function at various levels of approximation, either in the

time domain or in the frequency domain, as appropriate.

The various elements are organized in resonant, nonresonant passive elements

and active elements. Under resonant elements we include saturable absorbers and

gain, because they are generally associated with a near-resonant transition.

The organization of this section is as follows:

1. Saturable absorbers and gain

2. Nonlinear nonresonant elements