Ultrashort Laser Pulse Phenomena, 2nd Edition

Circulating Pulse Model 293

spectrum of gains and losses). The bandwidth limit of the amplifier medium

has been reached in some (glass lasers, Nd:YAG lasers), but not all, lasers.

Other pulse width limitations arise from higher-order dispersion of optical com-

ponents and nonlinear effects (four wave mixing coupling in dye jets, two photon

absorption, Kerr effect, etc…).

The pulse evolution in a cw pumped laser leads generally to a steady state,

in which the pulse reproduces itself after an integer number of cavity round trips

(ideally one). The pulse parameters are such that gain and loss, compression and

broadening mechanisms, as well as shaping effects, balance each other.

5.2. CIRCULATING PULSE MODEL

5.2.1. General Round-Trip Model

As mentioned in the previous section, the simplest model for a practical

mode-locked laser is that of a pulse circulating in the cavity. The pulse travels

successively through the different resonator elements, each contributing to the

pulse shaping in a particular manner. The block diagram of Figure 5.7 is the basis

for the most commonly used theoretical description of such lasers. Which ele-

ments need to be considered and in which order will depend on the type of laser

to be modeled. Each block of the diagram of Fig. 5.7 can represent a real physical

element or a function rather than a physical element. For instance, the “saturable

loss” in Fig. 5.7 can represent either a saturable absorber, or the contribution of

all elements that give rise to an intensity or energy dependent transmission.

Pump

Amplifier

Linear

loss

Dispersion

Saturable

absorber

Coupled

cavity

Aperture Self lensing Filter

Figure 5.7 Schematic representation of the circulating pulse model describing a fs laser.

294 Ultrashort Sources I: Fundamentals

If we describe symbolically the action of each resonator element by an operator

function T

i

, the field after the (n + 1)-th round-trip can be written in terms of

the field before that round-trip as

˜

E

(n+1)

(t) =

(

T

N

T

N−1

...T

2

T

1

)

˜

E

(n)

(t), (5.22)

where we have numbered the resonator elements from 1 to N. If the parameters of

the laser elements are suitably chosen, the fields

˜

E

(i)

will evolve toward a steady-

state pulse, which reproduces itself (apart from a constant phase factor φ

0

) after

subsequent round trips, i.e.,

˜

E

(n+1)

=

˜

E

(n)

for n being large enough. The resulting

steady-state condition

˜

E(t)e

iφ

0

=

(

T

N

T

N−1

...T

2

T

1

)

˜

E(t), (5.23)

has been the basis for numerous analytical models.

5

These models usually

assume certain beam and pulse shapes with parameters that are determined from

Eq. (5.23). In most cases the operators have to be suitably approximated to allow

for analytical treatment. We will discuss this procedure in detail in the next sec-

tions. While the analytical or semianalytical solutions give much insight into the

physical mechanisms involved in fs lasers the complexity of the processes often

calls for numerical modeling.

The round trip model illustrated in Fig. 5.7 is well-suited for a numerical

treatment. Starting from noise (spontaneous emission) the field is traced through

each cavity round-trip. The main advantages of this approach are

• Ease of incorporating various processes and optical elements with compli-

cated transfer functions, leading to the modeling of virtually any laser.

• There is no need to make restrictive approximations for the transfer func-

tions. This allows one, for example, to follow the evolution of both the

temporal and spatial field profile.

• The evolution of the mode-locked pulse from noise can be predicted as can

the response of the laser to external disturbances.

• One is not limited to the time or frequency domain. By using Fast Fourier

Transforms (FFT), one can chose to model any phenomena in the most

appropriate frame (for instance, phase modulation in the time domain,

dispersion in the frequency domain).

5

A more general definition expresses that the pulse reproduces itself every m round trips,

i.e.,

˜

E

(n+m)

e

iφ

0

=

˜

E

(n)

.