Ultrashort Laser Pulse Phenomena, 2nd Edition

Introduction 283

allows the measurement to extend far into the wings of the mode-locked spectrum,

greater than 50 dB down from the center peak. The repetition rate of the different

wavepackets does not vary as a function of frequency or wavelength over a total

span of 250 nm. It is the nonlinear phase shift because of the mode-locking

mechanism (in this case the Kerr modulation) that compensates for the GVD.

Such a result is expected, because the different wavepackets formed with any

group of modes should all travel at the same group delay, or they will not produce

a pulse that “stays together” after several round-trips. It is also consistent with the

Fourier transform of an infinite train of equally spaced pulses, which produces a

comb of equally spaced spectral components.

A recent experiment with a stabilized laser has confirmed that “teeth” of the

frequency comb, which is the Fourier transform of the pulse train, are equally

spaced throughout the pulse bandwidth to 3.0 parts in 10

17

[4].

5.1.3. The “Perfect” Mode-Locked Laser

The perfect mode-locked laser produces a continuous train of identical pulses

at a constant repetition rate. Such a perfect mode-locked laser has to be stabilized

to minimize, for example, length fluctuations because of thermal expansion and

vibrations.

A mode-locked fs laser requires a broadband gain medium, which will

typically sustain over 100,000 longitudinal modes. The train of pulses results

from the leakage (outcoupling) of a single pulse traveling back and forth in

a cavity of constant length. The round-trip time of the cavity is thus a con-

stant, implying a perfectly regular comb of pulses in the time domain. The

frequency spectrum of such a pulse train is a perfect frequency comb, with

equally spaced teeth, at variance with the unequal comb of longitudinal modes

of a nonmode-locked cw laser.

The historical and standard textbook definition of mode-locking presented in

the previous section originates from the description of the laser in the frequency

domain, where the emission is considered to be made up of the sum of the

radiation of each of these (longitudinal) modes. This description can still be

applied to the ideal mode-locked laser considered in this section, if a fictitious

perfect comb with equal tooth spacing is substituted to the real longitudinal

modes of the cavity. This frequency description of mode-locking is equivalent

to having, in the time domain, a continuous single frequency carrier, sampled

at equal time intervals τ

RT

by an envelope function, as shown in the top part of

Figure 5.4.

Our ideal mode-locked laser emits a train of equally spaced pulses with a

period τ

RT

, which corresponds to a comb of modes in the spectral domain whose

spacing is constant,  = 1/τ

RT

. Consequently the mode frequency can be

284 Ultrashort Sources I: Fundamentals

Fre

q

uenc

y

ν

N

Figure 5.4 Top: a pure carrier at a frequency v

N

is modulated periodically by envelopes, at regular

time intervals τ

RT

. Bottom: the corresponding frequency picture. A comb of δ functions in frequency,

is extended to near zero frequency. The frequency f

0

of the first mode is the carrier to envelope offset.

expressed as

v

m

= f

0

+ m = f

0

+

m

τ

RT

, (5.10)

where m is the mode index that now starts at m = 0. Note that f

0

<is nonzero

in general. This is different from the cold cavity referred to in the introduction

of this chapter, where the mode frequencies are solely determined by the optical

pathlength of the cavity Ln(v). In cases where the index can be approximated

by a constant over the gain bandwidth, the group velocity is equal to the phase

velocity, and the mode frequencies are integer multiples of .

While the pulse envelope peaks again exactly after one round-trip time τ

RT

the phase of a mode with index m changes by

2πv

m

τ

RT

= 2πf

0

τ

RT

+ 2πmτ

RT

= 2πf

0

τ

RT

+ 2πm. (5.11)

Apart from multiples of 2π each mode acquires an additional phase with respect

to the pulse envelope

φ

CE

= 2πf

0

τ

RT

. (5.12)

This is illustrated in Fig. 5.4. Because the phase shift φ

CE

is independent of the

mode index it leads to a slippage of the phase of the carrier frequency with

respect to the pulse envelope. The frequency f

0

responsible for this slippage

is called carrier to envelope offset (CEO). One can also interpret the relative