Ultrashort Laser Pulse Phenomena, 2nd Edition

Introduction 291

favor pulsed over continuous radiation. The problem with the frequency domain

picture is the difficulty to treat the various processes in the laser including the

coupling between the large number of modes to predict the pulse parameters

M, E

m

, v

m

, and φ

0

. Recall that the mode frequencies v

m

are not given by the dis-

persion of the cold cavity, cf. Eq. (5.7), but establish themselves in the process

of mode-locking. Therefore the simple picture presented in the beginning of this

chapter can only serve as a qualitative description of the mode-locking process.

We have seen in the previous section that the frequency domain picture,

in which all longitudinal modes within the gain bandwidth oscillate in phase,

is an oversimplification. The ratio of the laser cavity length to the pulse dura-

tion would be a measure of the number of modes oscillating in phase. Typically,

for a meter long laser producing a train of 100 fs pulses, there would be over

100,000 longitudinal modes contributing to the pulse bandwidth. Because of the

dispersion of the intracavity elements, the longitudinal modes are not equidis-

tant over that range. Moreover, because a real laser resonator is not infinitely

rigid, one cannot even talk of a fixed set of modes. Therefore, the most common

approach to model a mode-locked laser is to analyze, in the time domain, the

shaping mechanisms of one (sometimes more) pulse(s) traveling back and forth

in a linear cavity, or circulating in a ring cavity. This is the description that will

generally be followed in this book, and in particular in this chapter, Sections 5.2

and 5.3. In this picture the function of the cavity is not to establish a comb of

modes but rather to force the circulating field to interact periodically with the

cavity elements. Most analyses follow one of two main routes—(a) the evolution

of the pulse from noise (spontaneous emission) and (b) the characterization of a

steady state where the circulating pulse reproduces itself after an integer num-

ber (ideally one) of round trips. In either case, the result is the complex pulse

envelope

˜

E(t) rather than the frequency domain parameters M, v

m

, and E

m

.

In the case of passive mode-locking, some intensity dependent loss or

dispersion mechanism is used to favor operation of pulsed over continuous

radiation. Another type of mode-locking mechanism is active: A coupling is

introduced between cavity modes, “locking” them in phase. Between these two

classes are “hybrid” and “doubly mode-locked” lasers in which both mecha-

nisms of mode-locking are used. In parallel to this categorization in “active” and

“passive” lasers, one can also classify the lasers as being modulated inside (the

most common approach) or outside (usually in a coupled cavity) the resonator.

5.1.5. Basic Elements and Operation of a fs Laser

There are a few basic elements essential to a fs laser:

• a broadband (v

g

 1 THz) gain medium,

• a laser cavity,