Ultrashort Laser Pulse Phenomena, 2nd Edition

5

Ultrashort Sources I: Fundamentals

5.1. INTRODUCTION

The standard source of ultrashort pulses is a mode-locked laser. Fundamental

properties of the radiation emitted by such a source, both in time and frequency

domains, are presented in this first section. Section 5.2 exposes the main theoret-

ical models to predict the shape of the pulses generated in such a laser. General

considerations about the evolution of the pulse energy are given in Section 5.3.

Section 5.4 is dedicated to the analysis of the main components of the laser, out-

lining the mechanism of pulse shaping of each element (or groups of elements).

Of course, the laser resonator itself has its role in the mode-locked operation.

The remainder of this chapter, Section 5.5 is therefore dealing with the properties

of the laser cavity.

5.1.1. Superposition of Cavity Modes

Central to the generation of ultrashort pulses is the laser cavity with its

longitudinal and transverse modes. A review of the mode spectrum of a laser

cavity is contained in Section 5.5.1. Mode-locked operation requires a well-

defined mode structure. As will be shown, mode-locking refers to establishing

a phase relationship between longitudinal modes. A transverse mode structure

will generally contribute to amplitude noise (at frequencies corresponding to the

differences between mode frequencies). Most fs lasers operate in a single TEM

00

transverse mode. A typical laser cavity can support a large number of longitudi-

nal modes. In the absence of transverse mode structure, we can consider that the

277

278 Ultrashort Sources I: Fundamentals

laser can operate on any of the longitudinal modes of index m, whose frequency

v

m

satisfies the condition

v

m

=

mc

2

:

i

n

i

(v

m

)L

i

≡

mc

2n(v

m

)L

(5.1)

where m is a positive integer and n

i

(v

m

)L

i

is the optical pathlength at the

frequency v

m

of the cavity element i of length L

i

. The total pathlength OL =

:

i

n

i

(v

m

)L

i

is the sum of the optical pathlengths of all cavity elements. We will

formally write OL = n(v)L, where L is the geometrical cavity length and n is an

effective average refractive index. We will first consider the ideal textbook case

where the mode spacing  = v

m+1

− v

m

= c/(2nL) is constant, which implies

that n is nondispersive for frequencies within the laser gain bandwidth.

The electric field of a laser that oscillates on M adjacent longitudinal modes

of frequency ω

m

= 2πv

m

= ω



+ 2πm with equal field amplitude E

0

can be

written as

˜

E

+

(t) =

1

2

˜

E(t)e

iω



t

=

1

2

E

0

e

iω



t

(M−1)/2



m=(1−M)/2

e

i(2mπt+φ

m

)

, (5.2)

where we now count m from (1 − M)/2to(M − 1)/2. Here φ

m

is the phase of

mode m, which is random for a free-running laser. The mode spectrum is centered

about a cavity mode of frequency ω



= 2πp, where p is a large positive integer.

The laser field, except for a phase factor, is a repeating pattern with a periodicity

of 1/, because, for any integer q,

˜

E

+



t +

1



q



=

˜

E

+

(t)e

iω



q/

(5.3)

as can be verified using Eq. (5.2). This periodicity is the cavity round-trip time

τ

RT

= 1/. Figure 5.1 compares the laser output for random and constant

phase φ

m

.

For any particular distribution of phases, the time-dependent laser power can

be written as:

P(t) ∝ E

2

0

[M + f (t)]. (5.4)

Here |f (t)|< M carries the information on the time dependence of the periodic

laser output. This follows from the fact that the length of the sum vector of

M unit vectors of random phase is equal to

√

M (random walk). Note that each

member of the sum in Eq. (5.2) represents such unit vector. For random phases