Introduction 285
shift of envelope and carrier as the result of the difference of phase and group
velocity. An average group velocity can be defined as ¯ν
g
= 2L/τ
RT
. The time
a phase front of a mode of index N needs to complete one round-trip (2L)is
N/v
N
, which suggest to define an average phase velocity ¯ν
p
= 2Lv
n
/N. The
delay between the pulse envelope and an arbitrary point on the phase front can
now be written as
τ
CE
= 2L
1
¯ν
g
−
1
¯ν
p
= (τ
RT
− N/v
N
), (5.13)
which yields for the phase
φ
CE
= 2πv
N
τ
CE
= 2π(τ
RT
v
N
− N). (5.14)
It is only when f
0
= 0 that the repetition rate is an integer number of optical
cycles of an oscillating mode, cf. Eqs. (5.14) and (5.12).
The ability to measure (or control) f
0
implies that one is able to establish a link
between the optical frequencies of the mode comb (v
m
) and the radio frequency
(1/τ
RT
). Let us assume for instance that one optical mode at v
N
of the laser is
linked to an optical frequency standard and that f
0
= 0 so that there are N optical
cycles 1/v
N
within the pulse period τ
RT
. Under these conditions, the repetition
rate can be considered to be a radio frequency standard with a relative linewidth,
v/v, N times narrower than that of the optical reference.
The existence of a perfectly regular frequency comb has revolutionized the
field of metrology. Such a comb can be used as a ruler to measure the spacing
between any pairs of optical frequencies v
1
and v
2
. The technique is similar to
a standard measurement of length with a ruler. One measures the beat note v
1
between the source at v
1
and the closest tooth—assigned the index m
1
—of the
frequency comb, as well as the beat note v
2
between the source at v
2
and the
neighboring tooth m
2
of the frequency comb. The frequency difference between
the two sources is v
2
− v
1
= v
2
− v
1
+ (m
2
− m
1
)/τ
RT
.
We will discuss the frequency rulers and the mode-locked laser as time
standard in Chapter 13. Details on stabilization techniques as well as frequency
standards can be found in Ye and Cundiff [5].
5.1.4. The “Common” Mode-Locked Laser
The expression mode-locking suggests equidistant longitudinal modes of the
laser cavity emitting in phase. As mentioned in the previous section, this fre-
quency description of mode-locking is equivalent to having, in the time domain,
a continuous single frequency carrier, sampled at equal time intervals by an
286 Ultrashort Sources I: Fundamentals
envelope function. Unless sophisticated stabilization techniques as described in
Section 13.4 are used, an ordinary mode-locked laser does not at all fit the above
description. We shall use the term common mode-locked laser when the cavity
length is not stabilized across the spectrum. In such a common situation, each
cavity mirror is subject to vibrational motions. A typical mechanical resonance
is around 100 Hz, with a motion amplitude L of up to 1 µm. Because of that
motion, the position of the longitudinal modes of the cavity is not fixed in time.
As the cavity length L drifts, so does the mode frequency v
m
and the repetition
rate 1/τ
RT
. From Eq. (5.10), we can express the change in mode frequency v
m
because of a change in cavity length L:
v
m
=
df
0
dL
+ m
d
dL
L =
df
0
dL
−
m
τ
2
RT
dτ
RT
dL
L. (5.15)
It depends on the specifics of the mode-locked laser how the CEO f
0
and the
roundtrip time (group velocity) vary individually with L.
Pulse Train Coherence
Because of this change of the carrier frequency, the repetition rate and the
carrier to envelope offset, one can no longer talk of an output pulse train made of
identical pulses. The difference between the properties of the radiation from an
ultrastable “frequency comb” as opposed to the common mode-locked laser can
be established in a coherence measurement. Coherence can be measured with a
Mach–Zehnder interferometer, as sketched in Figure 5.5. In the case of a sin-
gle pulse, the interference contrast approaches zero for optical delays x of the
interferometer exceeding the coherence length of the pulse. The interferogram
will resemble that shown in Fig. 2.3. In the case of a pulse train from a perfect
mode-locked laser, as the delay of the interferometer is being scanned, an iden-
tical fringe pattern reappears at delays equal to an integer multiple q of the pulse
spacing τ
RT
. In a common mode-locked laser, the visibility of these reoccurring
fringes will decay with increasing q. To explain this loss in fringe contrast let
us assume that at each delay x we measure a signal from N pulse pairs. The
signal at the detector
S(q, x) = η
2
N
i=1
8
E
2
(t)
6
cos(ω
t + φ
i
) + cos(ω
t + φ
i+q
+ kx)
7
2
9
. (5.16)
Here φ
i
is the relative phase of the carrier with respect to the peak of the pulse
envelope and denotes time integration over the pulse envelope and carrier
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