Carrier-Envelope Phase Control
of Femtosecond Mode-Locked
Lasers and Direct Optical
Frequency Synthesis
David J. Jones,
1
* Scott A. Diddams,
1
* Jinendra K. Ranka,
2
Andrew Stentz,
2
Robert S. Windeler,
2
John L. Hall,
1
* Steven T. Cundiff
1
*†
We stabilized the carrier-envelope phase of the pulses emitted by a femto-
second mode-locked laser by using the powerful tools of frequency-domain
laser stabilization. We confirmed control of the pulse-to-pulse carrier-envelope
phase using temporal cross correlation. This phase stabilization locks the ab-
solute frequencies emitted by the laser, which we used to perform absolute
optical frequency measurements that were directly referenced to a stable
microwave clock.
Progress in femtosecond pulse generation has
made it possible to generate optical pulses that
are only a few cycles in duration (1– 4). This
has resulted in rapidly growing interest in con-
trolling the phase of the underlying carrier wave
with respect to the envelope (1, 5–7). The
absolute carrier phase is normally not important
in optics; however, for such ultrashort pulses, it
can have physical consequences (6, 8). Concur-
rently, mode-locked lasers, which generate a
train of ultrashort pulses, have become an im-
portant tool in precision optical frequency mea-
surement (9 –14). There is a close connection
between these two apparently disparate topics.
We exploited this connection to develop a fre-
quency domain technique that stabilizes the
carrier phase with respect to the pulse envelope.
Using the same technique, we performed abso-
lute optical frequency measurements using a
single mode-locked laser with the only input
being a stable microwave clock.
Mode-locked lasers generate a repetitive
train of ultrashort optical pulses by fixing the
relative phases of all of the lasing longitudi-
nal modes (15). Current mode-locking tech-
niques are effective over such a large band-
width that the resulting pulses can have a
duration of 6 fs or shorter, i.e., approximately
two optical cycles (2– 4). With such ultra-
short pulses, the relative phase between peak
of the pulse envelope and the underlying
electric-field carrier wave becomes relevant.
In general, this phase is not constant from
pulse to pulse because the group and phase
velocities differ inside the laser cavity (Fig.
1A). To date, techniques of phase control of
femtosecond pulses have employed time do-
main methods (5). However, these techniques
have not used active feedback, and rapid
dephasing occurs because of pulse energy
fluctuations and other perturbations inside the
cavity. Active control of the relative carrier-
envelope phase prepares a stable pulse-to-
pulse phase relation, as presented below, and
will dramatically impact extreme nonlinear
optics.
Although it may be natural to think about
the carrier-envelope phase in the time do-
main, it is also apparent in a high-resolution
measurement of the frequency spectrum. The
output spectrum of a mode-locked laser con-
sists of a comb of optical frequencies sepa-
rated by the repetition rate. However, the
comb frequencies are not necessarily integer
multiples of the repetition rate; they may also
have an offset (Fig. 1B). This offset is due to
the difference between the group and phase
velocities. Control of the carrier-envelope
phase is equivalent to control of the absolute
optical frequencies of the comb, and vice
versa. This means that the same control of the
carrier-envelope phase will also result in a
revolutionary technique for optical frequency
metrology that directly connects the micro-
wave cesium frequency standard to the opti-
cal frequency domain with a single laser (14).
We used a self-referencing technique to
control the absolute frequencies of the optical
comb generated by a mode-locked laser.
Through the relation between time and fre-
quency described below, this method also
1
JILA, University of Colorado and National Institute of
Standards and Technology, Boulder, CO 80309– 0440,
USA.
2
Bell Laboratories, Lucent Technologies, Murray
Hill, NJ 07733, USA.
*These authors contributed equally to this work.
†To whom correspondence should be addressed. E-
mail: cundiffs@jila.colorado.edu
Fig. 1. Time-frequency correspondence and relation between ⌬ and ␦.(A) In the time domain, the
relative phase between the carrier (solid) and the envelope (dotted) evolves from pulse to pulse by
the amount ⌬. Generally, the absolute phase is given by ⫽⌬(t/) ⫹
0
, where
0
is an
unknown overall constant phase. (B) In the frequency domain, the elements of the frequency comb
of a mode-locked pulse train are spaced by f
rep
. The entire comb (solid) is offset from integer
multiples (dotted) of f
rep
by an offset frequency ␦⫽⌬f
rep
/2. Without active stabilization, ␦ is
a dynamic quantity, which is sensitive to perturbation of the laser. Hence, ⌬ changes in a
nondeterministic manner from pulse to pulse in an unstabilized laser.
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serves to stabilize the carrier phase with re-
spect to the envelope. This phase stabilization
was verified with temporal cross correlation.
The utility of this method for absolute optical
frequency metrology was demonstrated by
measuring the frequency of a stable single-
frequency laser directly from a microwave
clock.
Time versus frequency. The connection
between the pulse-to-pulse carrier-envelope
phase shift and the absolute frequency spec-
trum can be understood by considering how
the spectrum is built up by a temporal train of
pulses (7, 9, 14 ). It is easily shown with
Fourier transforms that a shift in time corre-
sponds to a phase shift that is linear with
frequency; that is, the phase at angular fre-
quency is t for a time shift of t. Within a
narrow spectral bandwidth, successive pulses
interfere, but a signal will be observed only at
frequencies where they add constructively,
i.e., have a phase shift of 2n. For a pulse
train with time between pulses, these fre-
quencies are f
n
⫽ n/⫽nf
rep
, where n is an
integer and f
rep
⫽ 1/ is the repetition fre-
quency of the pulse train. Thus, we recover
the fact that the frequency spectrum consists
of a comb of frequencies spaced by the rep-
etition rate of the pulse train. If we include
the pulse-to-pulse phase shift ⌬, then the
phase difference between successive pulses at
angular frequency is now ⫺⌬. Again,
for constructive interference, this phase dif-
ference is set equal to 2n, which shows that
now the frequencies are f
n
⫽ nf
rep
⫹␦, where
2␦⫽⌬f
rep
. Hence, a pulse-to-pulse phase
shift between the carrier and envelope corre-
sponds to an offset of the frequency comb
from simple integer multiples of the repeti-
tion rate.
In a mode-locked laser, the carrier slips
through the envelope as the pulse circulates
in the cavity. Because an output pulse is only
produced once per cavity round trip, what
matters is the accumulated carrier phase, with
respect to the envelope, per round trip. If this
is an integer multiple of 2, then there is no
pulse-to-pulse phase shift for the emitted
pulses, and the frequencies are all integer
multiples of the repetition rate, without an
offset. If the accumulated carrier phase is an
integer multiple of 2 plus a rational fraction
of 2, then the phase will periodically shift,
and the frequency comb will be offset by the
repetition rate times the rational fraction. For
example, if the pulse-to-pulse phase shift is
2/8, then every eighth pulse will have the
same phase, and the frequency offset will be
f
rep
/8. In the most general case, the phase and
frequency offsets are arbitrary. Furthermore,
in a free-running laser (even one with its
repetition rate locked), the phase-group ve-
locity difference drifts with time.
Self-referencing technique. Choosing to
control the carrier-envelope phase by locking
the frequency domain offset ␦ enabled us to use
the powerful techniques developed for stabili-
zation of single-frequency lasers. It is possible
to lock the position of the frequency comb to a
known optical frequency, such as that of a
single-frequency laser (12). However, employ-
ing this approach to determine and lock ␦ is
problematic because the optical frequencies are
⬎10
6
times the repetition rate. In addition, it
introduces the complication of a highly stabi-
lized single-frequency laser, not to mention the
uncertainty in the single-frequency continuous
wave (CW) laser itself.
A more elegant approach is to use a self-
referencing technique, which is based on
comparing the frequency of comb lines on the
low-frequency side of the optical spectrum to
those on the high-frequency side that have
approximately twice the frequency. Let the
frequency of comb line n, which is on the red
side of the spectrum, have a frequency f
n
⫽
nf
rep
⫹␦. The comb line corresponding to 2n,
which will be the blue side of the spectrum,
will have frequency f
2n
⫽ 2nf
rep
⫹␦.We
obtain ␦ by frequency doubling f
n
and then
taking the difference 2f
n
– f
2n
⫽␦. To imple-
ment the technique in a simple way, we need
an optical spectrum that spans a factor of 2 in
frequency, known as an optical octave. This
is obtained by spectrally broadening the laser
pulse in air-silica microstructure fiber (16).
Full experimental detail is given below.
Optical frequency metrology tech-
niques. Optical frequencies are preferred for
the precision measurements that test funda-
mental physical theories (17, 18). In part, this
preference is associated with the very narrow
fractional linewidths displayed by optical res-
onances, allowing precise measurements to
be made in relatively short times. For similar
reasons, optical resonances will probably be
used in future atomic clocks. However, any
absolute frequency measurement must be de-
rived from the 9.193-GHz cesium hyperfine
transition, which defines the second as one of
the SI base units. The frequencies involved
and the large ratio between the cesium tran-
sition frequency and optical frequencies (a
factor of ⬃5 ⫻ 10
4
times the transition fre-
quency) represent serious obstacles.
Before this work, two techniques have
been used to make absolute optical frequency
measurements. The first technique is a phase-
coherent frequency chain of oscillators that
spans from the cesium reference transition to
optical frequencies (19). The complexity and
difficulty of these chains requires substantial
investment and is typically undertaken only
at national research facilities. Furthermore,
because of their complexity, the chains may
not run on a daily basis but rather are typi-
cally used to calibrate intermediate standards
such as the HeNe/I
2
stabilized laser and CO
2
lasers stabilized to OsO
4
resonances.
A newer measurement technique involves
frequency doubling a single-frequency laser
and then measuring the difference between
the fundamental and second harmonic by
subdividing it. The subdivision may be done
by optical bisection (20, 21), comb genera-
tion (22), or, most typically, a combination of
the two (9, 10, 17, 23). The goal of the
subdivision is finally to obtain a frequency
interval small enough that it can be directly
compared to the cesium frequency. A recent
series of landmark experiments by Ha¨nsch
and co-workers demonstrated that mode-
locked lasers are the preferred implementa-
tion of optical comb generators (9 –11). We
recently performed absolute frequency mea-
surements using only the comb generated by
a mode-locked laser (13). The laser and mi-
crostructure fiber were similar to those used
here, although the technique is quite different
in that the laser comb was not stabilized in
position but rather only used to divide down
an optical frequency interval.
The self-referencing technique described
here is a dramatic step beyond these previous
techniques because it uses only a single
mode-locked laser and does not need any
stabilized single-frequency lasers. We think
that it will make precision absolute optical
frequency metrology into an easily accessible
laboratory tool.
Laser and stabilization. The heart of the
experiment is a titanium-doped sapphire
(Ti:S) laser (shown in Fig. 2) that is pumped
with a single-frequency, frequency-doubled
Nd:YVO
4
laser operating at 532 nm. The Ti:S
laser generates a 90-MHz pulse train with pulse
widths as short as 10 fs using Kerr lens mode-
locking (2). The output pulse spectrum is typi-
cally centered at 830 nm with a width of 70 nm.
For the generation of a 10-fs pulse, the normal
dispersion of the Ti:S crystal is compensated by
incorporating a pair of fused silica prisms inside
the cavity (24). After the second prism, the
optical frequencies of the pulse are spatially
resolved across the high-reflector mirror; this
property will be used to stabilize the absolute
frequency of the laser.
We previously noted that the relative car-
rier-envelope phase ⌬ in successive pulses
generated by mode-locked lasers is not con-
stant because of a difference between the
group and phase velocities inside the cavity.
As shown in Fig. 1, this is represented by the
frequency offset ␦ of the frequency comb
from f
n⫽0
⫽ 0. With the pulse repetition rate
f
rep
, the relative phase is related to the
offset frequency by 2␦⫽⌬f
rep
. Thus, by
stabilizing both f
rep
and ␦, ⌬ can be con-
trolled. Toward this end, as shown in Fig.
2, the high-reflector mirror (behind the
prism) is mounted on a piezoelectric trans-
ducer tube that allows both tilt and transla-
tion. By comparing a high harmonic of the
pulse repetition rate with the output of a
high-stability radio frequency (RF) synthe-
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sizer, a feedback loop can lock the repeti-
tion rate f
rep
by translating the mirror. Be-
cause the pulse spectrum is spatially dis-
persed across the mirror, tilting of this mir-
ror provides a linear phase change with
frequency (i.e., a group delay for the pulse),
thereby controlling both the repetition rate
and the offset frequency (11). The maxi-
mum required tilt angle is 10
⫺4
rad, sub-
stantially less than the beam divergence, so
cavity misalignment is negligible.
To stabilize the offset frequency of a sin-
gle mode-locked laser, without external in-
formation, it is useful to generate a full opti-
cal octave. The typical spectral output gener-
ated by the Ti:S laser used in these experi-
ments spans 70 nm or 30 THz, whereas the
center frequency is ⬃350 THz; that is, the
spectrum spans much less than a full octave.
Propagation through optical fiber is common-
ly used to broaden the spectrum of mode-
locked lasers through the nonlinear process of
self-phase modulation, based on the intrinsic
intensity dependence of the refractive index
(the Kerr effect). Optical fiber offers a small
mode size and a relatively long interaction
length, both of which enhance the width of
the generated spectrum. However, chromatic
dispersion in the optical fiber rapidly stretch-
es the pulse duration, thereby lowering the
peak power and limiting the amount of gen-
erated spectra. Although zero dispersion op-
tical fiber at 1300 and 1550 nm has existed
for years, optical fiber that supports a stable,
fundamental spatial mode and has zero dis-
persion near 800 nm has been available only
in the past year. In this work, we employed a
recently developed air-silica microstructure
fiber that has zero group velocity dispersion
at 780 nm (16). The sustained high intensity
(hundreds of GW/cm
2
) in the fiber generates
a stable, single-mode, phase-coherent contin-
uum that stretches from 510 to 1125 nm (at
⫺20 dB) as shown in Fig. 3. Through four-
wave mixing processes, the original spectral
comb in the mode-locked pulse is transferred
to the generated continuum. As described
above, the offset frequency ␦ is obtained by
taking the difference between 2f
n
and f
2n
.
Figure 2 details this process of frequency
doubling f
n
in a nonlinear crystal and com-
bining the doubled signal with f
2n
on a pho-
todetector. The resulting RF heterodyne beat
is equal to ␦. In actuality, the beat arises from
a large family of comb lines, which greatly
enhances the signal-to-noise ratio. After suit-
able processing (described below), this beat
is used to actively tilt the high-reflector mir-
ror, allowing us to stabilize ␦ to a rational
fraction of the pulse repetition rate.
The experimental implementation of the
f-2f heterodyne system is shown in Fig. 2.
The continuum output by the microstructure
fiber is spectrally separated into two arms by
a dichroic beamsplitter. The visible portion of
the continuum (500 to 900 nm, containing
f
2n
) is directed through one arm that contains
an acousto-optic modulator (AOM). The
near-infrared portion of the continuum (900
to 1100 nm, containing f
n
) traverses the other
arm of the apparatus, passing through a
4-mm-thick -barium-borate frequency-dou-
bling crystal. The crystal is angle-tuned to
efficiently double at 1040 nm. The beams
from the two arms are then mode-matched
and recombined. The combined beam is fil-
tered with a 10-nm bandwidth interference
filter centered at 520 nm and focused onto an
avalanche photo diode (APD). Approximate-
ly 5 W are incident on the APD from the
arm containing the AOM, whereas the fre-
quency-doubling arm provides ⬃1 W. The
resulting RF beats are equal to ⫾(␦⫺f
AOM
),
where f
AOM
is the drive frequency of the
AOM and is generated to be 7/8f
rep
. The RF
beats are then fed into a tracking oscillator
that phase-locks a voltage-controlled oscilla-
tor to the beat to enhance the signal-to-noise
ratio by substantially reducing the noise
bandwidth. From the tracking oscillator out-
put, we generate an error signal that is pro-
grammable to be (m/16)f
rep
, thus allowing us
to lock the relative carrier-envelope phase
from0to2 in 16 steps of /8.
Temporal cross correlation. Verification
of control of ⌬ in the time domain is ob-
tained by interferometric cross correlation be-
tween two different, not necessarily adjacent,
pulses in the pulse train (5). In fact, we
performed a time-averaged cross correlation
between pulses i and i ⫹ 2 using the corre-
lator shown in Fig. 4. A multipass cell in one
arm of the correlator is used to generate the
required 20-ns delay. For the purpose of min-
imizing dispersion, the beam splitter is a
2-m-thick polymer pellicle with a thin gold
coating. To obtain a well-formed interfero-
gram, we chose the mirror curvatures and
their separations to mode-match the output
Fig. 2. Experimental setup for locking the carrier-envelope relative phase. The femtosecond laser is
located inside the shaded box. Solid lines represent optical paths, and dashed lines show electrical
paths. The high-reflector mirror is mounted on a transducer to provide both tilt and translation.
Fig. 3. Continuum gener-
ated by air-silica micro-
structure fiber. Self-phase
modulation in the micro-
structured fiber broadens
the output of the laser so
that it spans more than
one octave. Approximate-
ly 25 mW are coupled
into the fiber to generate
the displayed continuum.
The wavelengths/frequen-
cies denoted by f
n
and f
2n
are used to lock the offset frequency as described in the text. The spectra are offset from each other
for clarity.
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from both arms. The entire correlator is in a
vacuum chamber held below 300 mTorr to
minimize the effect of the dispersion of air.
The second-order cross correlation was mea-
sured with a two-photon technique (25)by
focusing the recombined beam with a spher-
ical mirror onto a windowless GaAsP photo-
diode. The band gap of GaAsP is large
enough and the material purity is high enough
so that appreciable single-photon absorption
does not occur. This yields a pure quadratic
intensity response with a very short effective
temporal resolution.
A typical cross correlation is shown in Fig.
5A. To determine ⌬, we fit the fringe peaks of
the interferogram to a correlation function as-
suming a Gaussian pulse envelope. From the fit
parameters, we determined the center of the
envelope and compared it with the phase of the
underlying fringes to find ⌬. A fit of the fringe
peaks assuming a hyperbolic secant envelope
produced nearly identical results. A plot of the
experimentally determined relative phases for
various offset frequencies, along with a linear
fit of the averaged data, is given in Fig. 5B.
These results show a small offset of 0.7 ⫾ 0.35
rad from the theoretically expected relation
⌬ ⫽ 4␦/f
rep
(the extra factor of 2 results
because the cross correlator compares pulses i
and i ⫹ 2). The experimental slope is within 5%
of the theoretically predicted value and demon-
strates our control of the relative carrier-enve-
lope phase. Despite our extensive efforts to
match the arms of the correlator, we attribute
the phase offset between experiment and theory
to a phase imbalance in the correlator. The
number of mirror bounces in each arm is the
same, and mirrors with the same coatings were
used for 22 of the 23 bounces in each arm.
Nevertheless, because of availability issues,
there is a single bounce that is not matched.
Furthermore, the large number of bounces nec-
essary to generate the delay means that a very
small phase difference per bounce can accumu-
late and become significant. In addition, the
pellicle beam splitter will introduce a small
phase error because of the different reflection
interface for the two arms. Together, these ef-
fects can easily account for the observed offset.
The group-phase dispersion due to the residual
air only accounts for a phase error of ⬃/100.
We think that this correlation approach repre-
sents the best measurement strategy that can be
made short of demonstration of a physical pro-
cess that is sensitive to the phase.
We think that the uncertainty in the individ-
ual phase measurements shown in Fig. 5B aris-
es both from the cross-correlation measurement
itself and from environmental perturbations of
the laser cavity that are presently beyond the
bandwidth of our stabilizing servo loops. In-
deed, a measurement in the frequency domain
made by counting a locked offset frequency
␦⫽19 MHz with 1-s gate time revealed a
standard deviation of 143 Hz, corresponding to
a relative phase uncertainty of 10 rad. The
correlator uses a shorter effective gate time,
which decreases the averaging and hence in-
creases the standard deviation. Nevertheless,
the uncertainty in the time domain is 10
3
to 10
4
times that in the frequency domain (see below),
indicating that the correlator itself contributes to
the measurement uncertainty.
With pulses generated by mode-locked
lasers now approaching the single-cycle re-
gime (3, 4), the control of the carrier-enve-
lope relative phase that we have demonstrat-
ed is expected to dramatically impact the
field of extreme nonlinear optics. This in-
cludes above-threshold ionization and high
harmonic generation/x-ray generation with
intense femtosecond pulses. Above-threshold
ionization with circularly polarized light has
recently been proposed as a technique for
determining the absolute phase (6 ). Measure-
ments of x-ray generation efficiency also
show effects that are attributed to the evolu-
tion of the pulse-to-pulse phase (8).
Absolute optical frequency metrology. In
addition to applications in the time domain, the
stabilized mode-locked laser shown in Fig. 2
has an immediate and revolutionary impact also
in optical frequency metrology. As shown sche-
matically in Fig. 1B, when both the f
rep
(comb
spacing) and the offset frequency ␦ (comb po-
sition) are stabilized, lying underneath the
broadband continuum envelope is a frequency
comb with precisely defined intervals and
known absolute frequencies. By stabilizing f
rep
Fig. 4. Cross-correlator.
The second-order cross
correlation between pulse
i and pulse i ⫹ 2 is mea-
sured to determine the
pulse-to-pulse carrier-en-
velope phase shift. The
components inside the
shaded box are enclosed
in a vacuum chamber at
300 mTorr to minimize
the dispersive effects of
air. The number of mirror
bounces in both arms are
matched, also to mini-
mize phase errors. The
second-order cross corre-
lation is measured with
two-photon absorption in
a GaAsP photodiode.
Fig. 5. Correlation re-
sults. (A) Typical cross
correlation (solid line)
between the pulse i and
pulse i ⫹ 2, along with
a fit of the correlation
envelope (dashed line).
(B) Plot of the relative
phase versus the offset
frequency (normalized
to the pulse repetition
rate). As indicated, a
linear fit of our aver-
aged data produces the
expected slope of 4
with a 5% uncertainty.
The origin of the phase
offset is discussed in
the text.
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