carrier-envelope phase stabilization of single and...

Carrier-Envelope Phase Stabilization

of Single and Multiple Femtosecond Lasers

David J. Jones, Steve T. Cundiff, Tara M. Fortier, John L. Hall, and Jun Ye

JILA, University of Colorado and National Institute of Standards and Technology,Boulder, CO 80309-0440, [email protected]

Abstract. The basic concepts, technical implementation, and known limitationsof actively stabilizing the carrier-envelope phase of a few-cycle pulse train arediscussed. The route toward determining the “absolute” carrier-envelope phase,thereby enabling electronic waveform synthesis at optical frequencies, is reviewed.Lastly, techniques and applications of stabilizing the relative carrier-envelope phasebetween two (or more) femtosecond lasers are also covered.

1 Introduction

The advent of few-cycle laser pulse generation has heightened interest inmeasuring and controlling the phase between the optical carrier wave andthe pulse intensity envelope. There are a number of physical processes thatare dependent on the electric field, rather than just the intensity envelope ofa pulse [1]. Accordingly, such processes, including coherent (quantum) con-trol of atomic and molecular systems [2], optimization of high-harmonic (softX-ray) generation [3], and investigation of atomic systems on femtosecondand attosecond timescales, will benefit from control over the carrier-envelopephase. Combined with well-established methods of conventional amplitudeand chirp pulseshaping [4], control over the carrier-envelope phase will en-able us to synthesize electronic waveforms at optical frequencies. This typeof waveform synthesis could be used with the above mentioned physical in-vestigations, but it can also be employed in more of an application-orientedmanner. Phase-coherent operations, such as analog signal processing, thathave historically been operating at microwave frequencies can now be per-formed at optical frequencies. Some of these possibilities are discussed in moredetail in other chapters of this book as well as in other books on high-fieldphysics [5].

1.1 Definition of the Carrier-Envelope Phasefor a Few-Cycle Pulse

Figure 1 displays a few-cycle pulse with the carrier-envelope phase (CEP)defined as φCE. Mathematically, the electric field of a pulse can be expressed

F. X. Kartner (Ed.): Few-Cycle Laser Pulse Generation and Its Applications,Topics Appl. Phys. 95, 315–340 (2004)c© Springer-Verlag Berlin Heidelberg 2004

316 David J. Jones et al.

φCE

Fig. 1. Definition of carrier-envelopephase. Ultrashort pulse where theperiod of the carrier frequency ap-proaches the pulse width. The pulseintensity envelope is shown as a dot-ted line, and the solid line is the oscil-lating electric field. φCE is the phasebetween the peak of the pulse inten-sity envelope and the peak of the car-rier wave

as

E(t) = A(t) cos(ωct + φCE) , (1)

where A(t) is the pulse envelope, ωc is the carrier frequency, and φCE is thecarrier-envelope phase. The reliable periodicity of a train of optical pulsesgenerated by a mode-locked laser allows identification of a phase referencedto the pulse envelope. Relative to this frame, the phase (φCE) of the oscillatingelectric field can vary, depending on conditions both within and outside thelaser cavity. For a clear understanding of the dynamics of φCE, the CEP canbe broken into two components,

φCE = φo + ∆φCE , (2)

where φo is the “static” offset CEP and ∆φCE represents the pulse-to-pulsechange in CEP due to conditions inside the cavity of the laser oscillator.1 Asthe pulse propagates through any medium outside the laser cavity (exceptvacuum), a difference between the phase and group velocities (caused bydispersion) will cause φo to vary; so in reality, φo is not truly static. Ina similar vein, the physical origin of ∆φCE results from dispersion of theoptical elements inside a laser cavity. In the case of ∆φCE, the pulse is sampledonce per round-trip when it hits the output coupler, and it is only the phasechange modulo 2π that matters. Specifically,Please check the

mod spacing!

∆φCE =(

1vg

− 1vp

)lcωc mod [2π] , (3)

1 Often the term absolute phase has been used to label φo, which can be misleadingas there is nothing absolute about the peak of the pulse envelope that is used asthe reference. This terminology has probably arisen to help distinguish betweenφo and ∆φCE and to emphasize the fact that φo is not relative to a second opticalreference beam

Carrier-Envelope Phase Stabilization 317

where vg(vp) is the group (phase) velocity and lc is the length of the lasercavity.

As discussed in detail in Sect. 2, it is now possible to detect and con-trol ∆φCE. However, as (2) indicates, to define φCE completely, φo must alsobe measured (and stabilized). Although not yet completely realized, stepstowards this latter requirement are covered in Sect. 3. Recent work will bedescribed in locking together the carrier-envelope phases of two independentfs lasers in Sect. 4. Finally, an outlook on future developments and applica-tions is presented in Sect. 5.

2 Pulse-to-Pulse Carrier-Envelope Phase

The techniques used to stabilize the pulse-to-pulse evolution of the CEP arebest understood in the frequency domain. Figure 2a displays three pulsesthat are part of an infinite train that has a constant ∆φCE. The frequency-domain representation of this pulse train, given in Fig. 2b, is a frequencycomb with tooth spacing equal to the pulse repetition rate (fr). The entirecomb is offset from exact harmonics of fr by an offset frequency (fo). Froma careful derivation [6], the relation between fo and ∆φCE can be expressedas

∆φCE = 2πfo

fr. (4)

Thus, the task of stabilizing ∆φCE is reduced to stabilization of fo.

2.1 Detection of the Offset Frequency

As each comb element is shifted by the (same) offset frequency by opti-cally heterodyning different harmonics of the frequency comb together, itis not possible to extract the value of the offset frequency. Instead, scalingof the comb spectrum must be implemented before the heterodyne compar-ison. A straightforward method is to frequency double the red end of thecomb spectrum and compare it with the existing spectrum at the blue endwhere these two beams spectrally overlap. Thus, the simplest heterodyneprocedure requires an octave of optical bandwidth and is known as ν-to-2ν self-referencing. The initial demonstrations of this technique [7, 8] usedthe continuum generated from microstructure (photonic crystal) fiber [9] andnormal optical fiber, respectively. More recently, an octave of bandwidth hasbeen generated directly from a mode-locked laser [10,11,12]; the latter resulthas enough energy in the spectral extremes to provide a suitable signal-to-noise ratio for a tight phase lock of fo.

Consider a single comb component on the red side of the spectrum. Theelectrical field of this (the nth) comb line will have a phase,

φn = 2π(fn + fo)t + φon = 2π(nfr + fo)t + φon , (5)


2∆φCE

1/ frep = τ

t

E(t)

(a) Time domain∆φCE

I(f)

f

fo

νn = n frep + fo

frep

(b) Frequency domain

∆φCE = 2π fo/ frep

“Extra” phase accumulated inone cavity round trip:

Fig. 2. Time-frequency correspondence and relationship between ∆φCE and fo.(a) In the time domain, the relative phase between the carrier and the envelopeevolves from pulse to pulse by the amount ∆φCE due to an inequality of intracav-ity group and phase velocities. (b) In the frequency domain, the elements of thefrequency comb of a mode-locked pulse train are spaced by fr. The entire combis offset from integral multiples of fr by an offset frequency fo. Without activestabilization, fo is a dynamic quantity, which is sensitive to perturbations of thelaser. Hence ∆φCE changes in a nondeterministic manner from pulse to pulse in anunstabilized laser

where φon is the optical phase constant of the nth comb line. Similarly, anoctave away (2 times the frequency) on the blue side of the spectrum, the2nth comb line will have the phase,

φ2n = 2π(f2n + fo)t + φo2n = 2π(2nfr + fo)t + φo2n . (6)

If the electric field of the nth line is doubled (with any standard second-harmonic nonlinear crystal) and the optical heterodyne beat between thedoubled signal and the original field at 2n is detected on a photodiode, thesignal will have an interference term with the phase,

φdetect = 2πfot + 2φon − φo2n . (7)

From the photodiode signal, simple filtering in the radio-frequency domainyields fo. The meaning and use of the phase term (2φon − φo2n) in (7) isdiscussed in Sect. 3, and a complete derivation of (5–7) can be found else-where [6].


The actual experimental setup implementing the ν-to-2ν interferometeris shown in Fig. 3. In reality, a group of comb lines contributes phase coher-ently to the photodiode signal, so even with at most 10 nW per comb line (atypical amount for a 100 MHz spaced comb after broadening in microstruc-ture fiber) there is enough signal-to-noise (S/N) to lock fo properly. In ourexperience, the S/N should be > 25 dB to 30 dB (in a 100 kHz bandwidth)to achieve a tight phase lock. Normally, a collection of comb lines spanningapproximately 10 nm centered at ν2n is used for the heterodyne beat. Thereare two other important characteristics regarding the ν-to-2ν interferometershown in Fig. 3. First, to observe the heterodyne beat, the path lengths ofeach arm must be matched so that the field at 2n and the (doubled) fieldat 2 × n overlap in time. Second, by including an acousto-optical modulator(AOM) in one arm of the interferometer, the comb lines at 2n are shiftedin frequency by fAOM. Thus, the observed beats (7) are shifted by fAOM aswell. This condition allows us to avoid processing fo around the troublesomedc or fr frequency range when fo is locked to zero. With the AOM, fo iseasily locked to zero by mixing the photodiode signal with fAOM to gener-ate an error signal, although one does have to be careful to avoid electronicpickup noise at fAOM. Locking fo to zero forces every pulse to have the sameCEP and allows one to let fr float while still generating pulses with a stablecarrier-envelope phase. Recently, an alternate implementation of an ν-to-2νinterferometer has also been used [12] which has less differential noise.

There are other methods capable of measuring fo, such as detecting thebeat from 2ν and 3ν [11]. This latter method requires only one-half of anoctave. However, a cascaded nonlinear process is required and producingthe offset frequency with suitable signal-to-noise is rather difficult. Recently,using a high repetition rate mode-locked laser (1 GHz), an S/N of 25 dB (in a100 kHz bandwidth) was observed on fo derived from a 2ν-to-3ν scheme [13].A more thorough examination of these and other (similar) techniques toobtain fo is presented in [14]. Heterodyne beating of two harmonics froma CW laser stabilized against the comb can also be used to detect fo [15].

2.2 Stabilization of the Offset Frequency

To complete the stabilization loop, an accessible adjustment (or “knob”) onthe laser is required that can be used to adjust fo. In this case, such a knobmust be capable of changing the difference between the intracavity group andphase velocities. One such technique is to swivel the end mirror in the arm ofthe laser cavity that contains the prism sequence [16], as shown in Fig. 4. Sincethe spectrum is spatially dispersed on this mirror, a small tilt produces a lin-ear phase delay with frequency, which is equivalent to a group delay [17]. Analternate method of controlling fo is via modulation of the pump power [8,18].Empirically, this, it was found, causes a change in fo [19], although at thepresent time the exact physical mechanism is not clearly understood. Sev-eral possibilities may cause the sensitivity of fo to change with respect to


1064nm

532nm

LBO

AOM

Polarizer

532 nminterference

filter

fAOM

Cos [ 2π(nfrep+ (fo+fAOM))t +2φon-φo2n]

frep

fo+fAOM frep-(fo+fAOM)

0

Fig. 3. ν-to-2ν interferometer used to measure fo. The incoming octave-spanningcomb is spectrally separated using a dichroic mirror. Typically, we used wavelengthscentered at 1064 nm for ν and 532 nm for 2ν. In reality, the S/N of fo is opti-mized by experimentally adjusting ν and 2ν. The infrared portion (ν) is frequencydoubled with a lithium triborate (LBO) crystal and then polarization multiplexedwith the existing 2ν signal. The combined beam passes through an interferencefilter (centered at 2ν) to reject any nonspectrally overlapped comb components. Anacousto-optic modulator (AOM) is placed in the visible arm to enable measuringthe heterodyne beat unambiguously. An example of an observed rf spectrum is alsoshown

the laser pump power, including a nonlinear phase shift in the Ti:sapphirecrystal [8, 19], spectral shifts combined with wavelength-dependent groupvelocity dispersion, [8] or a differential change in the phase and group ve-locities [20, 21]. The pump power is usually modulated with an AOM orelectro-optic modulator (EOM) which typically has greater bandwidths thana mirror mounted on a piezoelectric transducer (PZT). However, the modula-tion depth required to lock fo successfully can induce a significant amount ofamplitude noise on the laser output; the consequences of this added noise arediscussed in Sect. 3.3. By carefully designing the mechanical structure of thePZT, a 3 dB bandwidth of approximately 50 kHz can be realized, which canbe as effective as an AOM for stabilization. Indeed, when we have directlycompared both PZT- and AOM-based stabilization schemes, little differencein locking performance was observed. These latter results are discussed inmore detail in Sect. 2.4.

An alternate method of generating CEP stable pulses, recently reportedby Baltuska and co-workers [22], is discussed in the Chapter by Baltuska et al.in this book. Using the phase relations of optical parametric amplification,Dear author, I’ve

changed the ref-erence to the firstauthor name inthe other chapter.Please confirm.


LOCKING

ELECTRONICS

MS

MS- microstructure fiberDBM Double Balanced Mixer

MS

KLM Ti:sapphire laser

Pump

Out of Loop Anaysis

ν to 2ν interferometer (lock)

ν to 2ν interferometer (measure)

AOM

In Loop Anaysis

DBM

DBM

AOM

Fig. 4. Experimental setup showing stabilization of ∆φCE. As the pulse spectrumis spatially resolved after the second prism, feedback via a small tilt on the endmirror produces a linear phase delay with frequency (a group delay) which changes∆φCE. A second ν-to-2ν interferometer provides an out-of-loop measurement offo which is critical for measuring the true coherence time of φCE (see Sect. 2.4)

they demonstrated an elegant technique capable of passively producing pulseswith zero pulse-to-pulse CEP change (∆φCE = 0). As this is an amplifiedsystem with kilohertz repetition rates, determination of φo is difficult in thefrequency domain (as discussed in Sect. 3). Rather, a time-domain, high-field process [3, 23] or a measurement using spectral interferometry [24] isnecessary.

2.3 Time-Domain Measurement of Phase-Stable Pulses

The relation between the offset frequency fo and the pulse-to-pulse evolu-tion of the CEP given in (4) was confirmed by examining the second-ordercross-correlation between pulse i and i + 2. By locking fo to various rationalfractions of fr, the subsequent change in ∆φCE, it was found, accurately fol-lows the prediction of (4) [7], as shown in Fig. 5. However, such a time-domainmeasurement with a single time delay does not yield a timescale (beyond twosuccessive pulses or 20 ns) over which the CEP remains coherent. Methodsand results for determining the coherence time of the CEP are addressed inSect. 2.4.


14

12

10

8

6

4

2

∆φ

1412108642

4πfo/ frep

Experiment Linear Fit (slope =1.06)

CE

Fig. 5. Experimental measurement of ∆φCE between the ith and ith + 2 versusvarious rational fractions of 4πfo/fr. According to (2), the slope should = 1 (anextra factor of 2 comes from correlating every second pulse). A linear fit of the datayields a slope of 1.06

2.4 Coherence of the Carrier-Envelope Phase

Although control of a pulse train’s ∆φCE is a critical first step, the utilityof CEP-stable pulses in many actual nonlinear experiments will be signifi-cantly enhanced by long-term phase coherence of the CEP. As seen from (4),slight excursions from fo = 0 (or from a rational fraction with respect to fr)cause an accumulated phase error in ∆φCE. Thus, phase noise of ∆φCE ismanifested as frequency noise of fo and leads to broadening of its linewidth.The rms fluctuations in the carrier-envelope phase, ∆φCErms, can be foundby integrating the frequency noise power spectral density (PSD), Sfo(ν), offo [25],

∆φCErms |τobs= 2

√∫ 1/2πτobs

−∞

Sfo(ν)dν

ν2= 2

√∫ 1/2πτobs

−∞Sφ(ν)dν , (8)

where τobs is the observation time, Sφ(ν) is the phase noise spectral densityof fo, and ν is an offset frequency relative to the optical carrier. PhysicallySfo(ν) is the frequency-domain representation of the fluctuations in fo at fre-quencies about its carrier. Thus, Sfo(ν) gives the spectrum of the frequency-noise sidebands present on the offset frequency’s linewidth, which may beconverted to the phase noise spectrum using the relation Sφ(ν) = Sfo(ν)/ν2.Following a convention from the frequency metrology field, the coherencetime is defined as

τcoh ≡ τobs |∆φCErms≈1 rad . (9)

The experimental setup used to measure Sφ(ν) is shown in Fig. 4. Twoν-to-2ν interferometers are used simultaneously, each with its own piece of


10-10

10

-8

10-6

10-4

10-2

100

102

104

106

108

rad

2 /Hz

10-3

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100

101

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103

104

105

Frequency

0.8

0.6

0.4

0.2

0.0

Accum

ulated phase noise (rad)10

-510

-410

-310

-210

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010

110

2

Observation Time (s)

Sφ 2 unlocked

Sφ 2out of loop

Sφ 2 in loop

∆φCErms in loop

∆φCErms out of loop

.

Fig. 6. Phase power spectral density (Sφ) vs. offset frequency (ν) from the car-rier (fo), left axis. Accumulated phase noise as a function of observation time isobtained via integration of Sφ(ν), right axis. The in-loop (black) and out-of-loop(gray) spectra (0.488 mHz to 102 kHz) were compiled from five different spectra ofdecreasing span and increasing resolution to obtain greater resolution close to thecarrier (displayed here as zero frequency). The stabilization process adds noise pastroughly 5 kHz, and roll-off in the out-of-loop spectrum at approximately 30 kHz isconsistent with the stabilization servo bandwidth. An unlocked spectrum (spanning31.25 mHz to 102 kHz) is included to indicate the effectiveness of the stabilizationloop. The first data points for the two spectra are artifacts as they include the dcoffset given by the carrier

microstructure fiber. One interferometer is employed to lock ∆φCE; the sec-ond interferometer is used for an authentic evaluation of Sφ(ν) independentof the feedback loop (out-of-loop). An in-loop measurement of Sφ(ν) (fromthe locking interferometer) quantifies only the stabilization capability of thefeedback loop. Furthermore, the loop (in particular, the locking interferome-ter itself) could possibly write noise onto ∆φCE that can be diagnosed onlywith an out-of-loop measurement. Both the in-loop and out-of-loop Sfo(ν)can be measured with a dynamic signal analyzer (such as the Stanford Re-search Systems SR785 or the Agilent 35670A) [26]. fig 6 left and right

borders are croppedFigure 6 displays Sφ(ν) for the unlocked, in-loop locked, and out-of-looplocked cases (left axis). The difference between the lock and unlocked casesshows an excellent improvement in the frequency deviations of fo. As ex-pected, the in-loop measurement has a lower phase noise spectral densitycompared to the out-of-loop case. We attribute most of this difference to dif-


ferential noise between the in-loop and out-of-loop ν-to-2ν interferometers.Any interferometer noise in the in-loop setup will be written on the laser bythe feedback loop, creating phase noise in the carrier-envelope phase.

Also shown in Fig. 6 is ∆φCErms (right axis) for in-loop and out-of-loopcases, calculated as prescribed by (8) and displayed as a function of τobs.At an observation time of 320 s, the out-of-loop ∆φCErms has accumulatedonly 0.8 rad of phase corresponding to a coherence time of at least 320 s.At present, this is a lower limit on the coherence time that is limited bymeasurement details. It should be specifically noted that the coherence timequoted above is not a coherence time of the optical carrier wave with succes-sive pulses; rather it is the coherence time of the carrier-envelope phase. Therepetition rate of the laser was not stabilized, so the carrier frequency is freeto shift, when fr changes, to maintain coherence of the CEP. However, if fr

is stabilized, then optical coherence can be realized. Considering that there isa rather large multiplication factor of 106 from radio to optical frequencies,simply locking fr to an rf synthesizer will not be adequate. The phase noiseof the synthesizer will be written on the pulse envelope and effectively multi-plied and written onto the optical carrier as well (because the CEP remainslocked). This will severely limit any chance of realizing optical coherence.Rather, locking fr to an optical transition or optical cavity will most likelybe necessary to achieve this goal.

The contribution of fluctuations in the ν-to-2ν interferometers to the∆φCErms was investigated by threading a single-frequency HeNe laser throughone interferometer. From the resulting transmission signal, a measurement ofthe phase spectral density can quantify the interferometers’ effect on theCEP noise. And in a manner analogous to (8), the phase noise added to the∆φCErms by the ν-to-2ν interferometer can be calculated. Figure 7 displaysthese results. If fluctuations in the in-loop and out-of-loop interferometersare assumed to be uncorrelated, then they add roughly

√2× 0.12 = 0.17 rad

of phase noise to ∆φCErms at 0.01 Hz, an amount consistent with the resultsin Fig. 6.

The results shown in Fig. 6 were obtained by controlling (swiveling) thelaser cavity mirror following the prism sequence that was mounted on a spe-cially designed, high-speed PZT that had a resonance frequency of 50 kHz.The design consisted of two 1/8 in PZT discs that were mounted with oppo-site polarity to a large copper mass (to reduce recoil and damp mechanicalresonances of the mirror mount). On the front side of the PZTs, a rectangu-lar mirror (with as little mass as possible) was mounted with a stiff wax. Weobserved slightly worse performance using an AOM to modulate the pumppower. Between the two techniques (AOM vs. PZT), we prefer the PZT-basedmethod as the AOM tends to place amplitude noise on the output of the laser.However, if a prismless laser is stabilized, then, at the present time, an AOMis the only option to control fo with a reasonably large bandwidth. However,when a laser (with intracavity prisms) that is locked with an AOM is included


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rad

2 /Hz

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100

frequency (Hz)

0.12

0.10

0.08

0.06

0.04

0.02

0.00

accumulated

phasenoise

(rad)

Fig. 7. Phase spectral density and accumulated phase noise resulting from an unsta-bilized ν-to-2ν interferometer. This data is measured by examining the fluctuationsof an independent laser (a single-frequency HeNe) through the interferometer

in a system with microstructure fiber, there can be severe consequences forthe coherence of φCE as discussed in Sect. 3.3.

To obtain a tight lock on fo and realize long-term coherence of φCE, itis necessary to pay close attention to the mechanical construction of thelaser cavity. Anything that can passively reduce the environmental pertur-bations to the laser relieves the burden placed on servoloops and leads tomore successful stabilization. Cavity designs with dispersion-compensatingmirrors (DCMs) used in place of prisms are reportedly more stable [27]. Thisis understandable as DCM-based cavities are free of beam-pointing fluctu-ations through the prism sequence (which lead to CEP fluctuations). How-ever, DCM-based cavities are by no means a requirement to realize long-termcoherence of φCE. The long-term coherence results reported above were ob-tained by carefully constructing a laser using intra-cavity prisms. Some of themore obvious measures include using the lowest practical beam height (weuse 2.5 in); employing high quality, solid mounts with stiff springs; enclosingthe cavity in a sealed PlexiglassTM(> 0.5 in thick) box; and using a single-frequency, diode-pumped, solid-state, 532 nm laser as the pump source. Build-ing the cavity on a solid cast aluminum breadboard (as opposed to a hon-eycomb or rolled aluminum breadboard) helps enormously. Plate modes ofthe cast breadboard can be reduced by attaching lead plates with suitableimpedance matching. Low-frequency vibrations (to 10 Hz) are attenuated byany standard optical table with an air flotation system. Frequencies up to200 Hz to 300 Hz can be attenuated by placing the breadboard on rubberstoppers or other suitable low-frequency springs. Frequencies above 500 Hzare typically airborne; thus, it is helpful to place the Plexiglass box around(rather than on top of) the breadboard. Elimination (or attenuation) of any


noise sources in the audio-frequency range (such as chillers or air compres-sors) is also a crucial step.

2.5 Application to Optical-Frequency Metrology

In addition to providing control over the CEP, the successful stabilizationof fo has led to a revolution in optical-frequency metrology by using thefrequency combs from femtosecond (fs) lasers as optical rulers. Building onproposals and preliminary experimental demonstrations with mode-lockedpicosecond pulses first made in the 1970s [28, 29, 30], in 1999, workers fromHansch’s group at MPQ used a frequency comb from a femtosecond laser tospan 20 THz in a frequency chain [31]. Following their lead, at JILA, a fem-tosecond comb-spanning 104 THz was demonstrated [32]. Our JILA workculminated in an octave-spanning frequency comb [33]. This latter develop-ment enabled the demonstration of an optical-frequency synthesizer [7] thatprovided the first self-referenced, phase-coherent, direct link between opti-cal and radio frequencies. This work was independently confirmed a shorttime later by the MPQ group [34]. A complete discussion of stabilized fsfrequency combs and their application to optical frequency metrology can befound in [35] and in the Chapter by Udem et al. in this book.

Although frequency-domain locking techniques for both optical-frequencymetrology and carrier-envelope phase stabilization are closely related, it is in-teresting to note key differences in their requirements. In metrology, the widthof the optical comb lines presents one of the primary limits on measurementprecision. The dominant contribution to the optical linewidth is fluctuationsof fr (typically 100 MHz to 1000 MHz) which are multiplied by a large inte-ger, of order 106, to reach optical frequencies. Fluctuations in fo, however,are typically negligible from the standpoint of optical metrology. Conversely,for ultrafast applications where CEP plays a role, small fluctuations in fo

quickly lead to phase errors in ∆φCE [as indicated in (4)], whereas stabiliza-tion of fr is of secondary concern (as long as fo is locked to zero or a rationalfraction of fr).

3 Absolute Carrier-Envelope Phase

Consider a train of pulses with high carrier-envelope phase coherence, asdescribed in the previous section. The logical next step is to measure andcontrol φo. A careful derivation [6] shows that, in principle, the phase of thesignal detected from an ν-to-2ν interferometer is φo when ∆φCE = 0. How-ever, an interferometer, such as that shown in Fig. 3, that has distinct armsfor ν and 2ν light introduces arbitrary phase shifts that make the detectedphase no longer equal to φo. In the following, we will present three alterna-tives that preserve the phase relation between the detected signal and φo.At this point, none of them has been demonstrated as a viable means formeasuring φo.


3.1 Chirp Compensation

The two-arm ν-to-2ν interferometer shown in Fig. 3 is used to achieve tem-poral overlap between the ν and 2ν spectral components. Since the requiredbroad spectrum is often obtained by nonlinear broadening in microstructurefiber, there is often substantial third-order chirp on the pulse. This meansthat there can be significant temporal separation between the ν and 2ν com-ponents. If an octave-spanning laser is used [10, 12], the cubic chirp can becompensated for by standard techniques [12]. If close to a flat phase can beobtained, an interferometer is not needed at all, but just passing the beamthrough a second harmonic crystal will produce ν-to-2ν beats [11].

Although chirp compensation removes the need for an explicit interfer-ometer, phase shifts in the second-harmonic (SHG) crystal can still preventthe phase of the ν-to-2ν signal from being an accurate measure of the pulsecarrier-envelope phase because dispersion in the SHG crystal causes the CEPto evolve in a known manner, but with a large degree of uncertainty. Al-though second-harmonic generation requires phase matching, which in thiscase is obtained via birefringence to cancel dispersion, this is true only withina very limited range of wavelengths. Since the pulse inherently has a broadbandwidth, sum-frequency generation is truly the relevant process when theν-to-2ν signal is generated. Calculations show that even a small discrepancybetween the phase matching and detection wavelengths can cause large errorsin the phase. Of course, a very thin second-harmonic crystal can overcomethis uncertainty, although the resulting signal may be unusably weak.

3.2 Quantum Interference

An alternative approach to optical interference [14, 36] from the detection offo is to use quantum interference. If a final state can be reached by both one-and two-photon transitions, quantum mechanical interference between thepathways will produce a phase-dependent population. However, if these arediscrete states, say, in an atom, parity results in selection rules that preventboth one- and two-photon transitions from a given initial state to any finalstate. However, if the final state does not have parity as a good quantumnumber, this is a possible scenario. Continuum states are an example of suchstates. Thus, this scheme can be implemented if the final state is in theionization continuum of an atom [37] or is in band states in a solid.

Such quantum interference has been demonstrated in semiconductors atthe University of Toronto [38, 39]. In this work, two laser pulses with a fac-tor of 2 difference in frequency and adjustable relative phase were generated.They were then combined and illuminated a low-temperature-grown GaAssample. The resulting interference between one- and two-photon absorptionproduces a detectable current with a relative phase-dependent direction. Ifa single pulse with an octave-spanning spectrum is used, the current gener-ated will depend on the carrier-envelope phase. Simulations show that this


should generate a weak, but detectable signal. Since the absorption occursin a very thin semiconductor layer, approximately 1 µm, unmeasured phaseslippage of φo due to propagation effects is minimized. Recently, using thistechnique, we detected the CEP using a GaAs sample [40]. An alternatequantum interference approach in the multiphoton regime uses photoemis-sion from a gold cathode [41] and has also been recently demonstrated.

3.3 Effects of External Broadening

At the present time, two Ti:sapphire (Ti:s) oscillators have been demon-strated which generate enough energy in the spectral wings to provide suffi-cient S/N for a good ν-to-2ν lock [12, 42]. However, it is useful to considerthe possible deleterious effects on the stabilization of ∆φCE and φo arisingfrom external broadening.

One area of obvious concern is amplitude-to-phase conversion in the mi-crostructure fiber. When the feedback loop for stabilizing the CEP is closed,phase noise generated in the fiber will be written back onto the laser by theaction of the loop as it tries to correct for this extracavity phase error. As theorigin of this noise is outside the cavity, this represents a noise term addedto the laser. In Fig. 8, the spectral density of relative power fluctuations[Sp(ν)] is shown on the left axis for the unlocked laser, PZT-locked laser, andAOM-locked laser. It is clear that though there is little difference between un-locked and PZT-locked lasers (indicating that the PZT lock does not increasethe amplitude noise of the laser), there is a significant increase in amplitudenoise with an AOM-locked laser. Using a setup similar to Fig. 4 with two,parallel ν-to-2ν interferometers, the amplitude-to-phase coefficient of the mi-crostructure fiber was measured [43] at2 6.0 rad/mW. With this coefficient,Should I put foot-

note citation in theReferences?

the accumulated fiber phase noise on the CEP can be easily calculated [usingan expression similar to (8)] as a function of frequency. The result of thisintegration is shown in Fig. 8 for both the PZT-lock and AOM-lock. Thoughthe amplitude fluctuations in the PZT-locked case contribute a negligibleamount of phase noise, it is entirely a different story for the AOM-lockedlaser. After integration from +∞ to 100 Hz, the accumulated phase noise hasalready reached 2π. Thus after an observation time of only (2π100 Hz−1) =1.6 ms, coherence of the CEP is lost when an AOM is used to stabilize fo ina laser with intracavity prisms in combination with microstructure fiber. Incontrast, it should be noted that with DCM cavities, the amplitude noise ofthe laser decreases when fo is locked via feedback to an AOM located in thepump beam [8]. This difference in behavior between prism and DCM fs lasers

2 As a side note, this value of an amplitude-to-phase coefficient is significantlylarger than that of standard optical fiber. Consequently, the microstructure fibermay be an ideal medium to generate amplitude-squeezed light. For example,see Marco Fiorentino, Jay E. Sharping, Prem Kumar, Alberto Porzio Robert S.Windeler, Opt. Letter. 27, 649–651 (2002)


10-6

10-5

10-4

(∆P

/Po)

/Hz0.

5

4 5 6 7 8 9

101

2 3 4 5 6 7 8 9

102

2 3 4 5 6 7 8 9

103

Frequency (Hz)

6

4

2

0

accumulated

fibernoise

(rad)

Sp AOM lockSp PZT lockSp unlockedaccumulated phase error (AOM)accumulated phase error (PZT)

Fig. 8. Spectral density of normalized power fluctuations (left axis) for unlocked(solid gray), PZT-locked (dashed gray), and AOM-locked (solid black) fo. The ac-cumulated phase noise due to amplitude-to-phase conversion in a microstructurefiber is shown on the right axis. Note that using an AOM to lock fo leads to deco-herence of the CEP after only an observation time of (2π100 Hz−1) = 1.6ms, fora laser using intracavity prisms

clearly indicates variations in noise processes and stabilization dynamics offo between the two cavity designs and is a topic of other work [44].

Another matter of possible concern is the degree of coherence within thecontinuum that is generated via a microstructure fiber. Without a high de-gree of coherence throughout the broadened spectra, the true values of bothφo and ∆φCE could not be determined from the detected heterodyne beat atfo. However, the experimental time-domain results presented in Fig. 5 displaythe expected linear relationship between fo and ∆φCE within 6%. Further-more, a lack of coherence in the comb would be manifested as uncorrelatedbroadening of individual comb lines. If coherence in the broadened frequencycomb were a limiting factor, our measurement of the linewidth of fo wouldhave produced a finite linewidth that could not be corrected (narrowed) byfeedback to the laser. As the linewidth of fo is presently measurement-limitedat 0.488 mHz, coherence of the comb does not appear to be a limiting factorfor establishing stable CEP pulses or determining φo from the phase of theν-to-2ν signal.


4 Synchronizing the Carrier-Envelope Phaseof Two Independent Femtosecond Lasers

A natural extension beyond establishing the long-term CEP coherence of apulse train generated by a single mode-locked laser is to lock the relativeCEP of two or more fs lasers coherently. In the frequency domain, this isequivalent to coherently stitching together each independent frequency combinto a single comb. One application of this technology is immediately obvi-ous: the resulting comb represents a single coherent pulse stream that hasa broader bandwidth than the individual combs by themselves. Thus, it ispossible to synthesize a shorter pulse than can be generated by a single laser(with the appropriate spectral coverage). This concept is discussed in moredetail in the Chapter on few-cycle pulse generation by Kartner et al. Otherapplications include significant (and possibly enabling) flexibility to controlmolecular systems [2] and other two-photon processes coherently that arebest implemented using two independent, time-synchronized lasers such ascoherent anti-Stokes Raman spectroscopy [45].

4.1 Repetition Rate Synchronization

To establish phase coherence between two separate ultrafast lasers, it is nec-essary first to achieve a level of pulse repetition rate synchronization betweenthe two lasers such that the remaining timing jitter is less than the oscilla-tion period of the optical carrier wave, namely, 2.7 fs for Ti:sapphire laserscentered around 800 nm. This requirement is illustrated in Fig. 9a. Thoughother techniques are available for synchronization, such as using cross-phasemodulation to synchronize passively two mode-locked lasers that share thesame intracavity gain medium [46, 47], we employed a flexible all-electronicapproach for active stabilization of repetition rates to achieve an unprece-dented level of synchronization for fs lasers [48, 49]. An even tighter lock canbe realized by first using this electronic approach and then switching to anerror signal generated by an optical cross-correlation [50].

Two Kerr-lens, mode-locked Ti:sapphire lasers are located in a mechani-cally and thermally stable environment for the synchronization experiment.To synchronize the two lasers, two phase-locked loops (PLL) are employedthat work at different timing resolutions, as shown in Fig. 10. One PLLcompares and locks the fundamental repetition frequencies (100 MHz) of thelasers. An rf phase shifter between the two 100 MHz signals can be used tocontrol the (coarse) timing offset between the two pulse trains with a full dy-namic range of 10 ns. The second, high-resolution PLL compares the phaseof high-order harmonics of the two repetition frequencies, for example, the140th harmonic at 14 GHz. This second loop provides enhanced phase sta-bility of the repetition frequency when it supplements and then replaces thefirst PLL. A transition of control from the first PLL to the second PLL can


(b) Frequency domain

t

E1(t)

t

E2(t)

Carrier-envelopephase locked

Synchronized rep. rates

tr.t.= 1/frep

∆f1=2πfo1 / frep1(a) Time domain

~ 3 fs

I(f)

f

f1(n) = n frep1 - fo1frep1

0

frep1f2(n) = n frep2 - fo2

fo2 fo1fo1 - fo2

Fig. 9. (a) Time- and (b) frequency-domain representations of the required condi-tions on pulse synchronization and carrier-envelope phase lock to establish phasecoherence between two independent femtosecond lasers

cause a jump in the timing offset by at most 35.7 ps (1/2 of one 14 GHz cy-cle), whereas the adjustable range of the 14 GHz phase shifter is 167 ps. Theservo action on the slave laser is carried out by a combination of transducers,including a small mirror mounted on a fast PZT, a regular mirror mountedon a slow piezo with a large dynamic range, and an acousto-optic modulatorplaced in the pump beam to help with fast noise. The unity gain frequency ofthe servoloop is about 200 kHz and the loop employs three integrator stagesin the low-frequency region [49].

To characterize the timing jitter, we focus the two pulse trains so thatthey cross in a thin β-barium borate (BBO) crystal cut for Type-I sum-fre-quency generation (SFG). The crossed-beam geometry produces an intensitySFG cross-correlation signal. A Gaussian fit to the cross-correlation (obtainedwhen the two lasers are free-running) yielded a pulsewidth of approximately160 fs full-width half-maximum. (No extracavity dispersion compensation isused, and the pulses would be 20 fs in the transform limit.) The top tracein Fig. 11 shows that the SFG signal, recorded in a 2 MHz bandwidth, hasno detectable intensity fluctuations when the two laser pulses are maximallyoverlapped (at the top of the cross-correlation peak). The two middle tracesare recorded with 2 MHz and 160 Hz bandwidths, when the timing offset be-tween the two pulse trains is adjusted to yield the half-maximum intensitylevel of the SFG signal. The slope of the cross-correlation signal near half-maximum can be used to determine the relative timing jitter between the


fs Laser 1

14 GHzSynthesizer

fs Laser 2

Laser 2repetitionrate control

100MHz

14 GHz14 GHz

50 ps

SHG

SHGBBO

SFG

SFGintensityanalysis

Laser 1repetition

ratecontrol

Phase lock: fo1 -fo2 = 0

AOM

Phaseshifter

Phaseshifter

14 GHzLoop gain

100 MHzLoop gain

(Interferometric)Cross-CorrelationAuto-CorrelationSpectral interferometry

Samplingscope

Delay

Delay

Fig. 10. Experimental schematic for pulse synchronization and carrier-envelopephase locking

two lasers from the corresponding intensity fluctuations. Timing jitter is cal-culated from the intensity noise using the slope of the correlation peak, withthe scale of the jitter indicated on the vertical axis of Fig. 11. The rms timingnoise thus determined is 1.75 fs at a 2 MHz bandwidth and 0.58 fs at a 160 Hzbandwidth. For detection bandwidths above 2 MHz, the observed jitter doesnot increase. We have recorded such stable performance over several seconds.The synchronization lock can be maintained for several hours. However, theintensity stability of the SFG signal strongly correlates with the temperaturevariations in the microwave cables located in the high-speed PLL. A carefulstudy of the servo error signal inside the feedback loop reveals that a majorlimitation on the present performance is actually due to the intrinsic noise ofthe 14 GHz phase detector, a double balanced mixer. Integration of the in-trinsic noise level in the mixer produces the lowest possible rms timing jitterlimit for the synchronization loop. From 1 Hz to 160 Hz frequency range itwas calculated as√

2.6 × 10−3 fs2/Hz × 160 Hz ≈ 0.64 fs,

which is the approximate jitter performance observed. To achieve even betterperformance, one must leave the rf domain and use either a single highlystable CW laser [51] or a stable optical cavity to control a high-order harmonicof the repetition frequency, well into the terahertz or tens and hundreds ofterahertz frequency range. Timing noise below 0.1 fs should be achievable.


Fig. 11. Timing jitter between two synchronized fs lasers. The dotted curve is thecross-correlation signal of the two lasers when the relative pulse timing is scannedacross the overlap region. Timing jitter determined from the intensity fluctuationsof the SFG intensity is shown over a period of 1 s, using two different low-passbandwidths

4.2 Coherent Phase Locking of Mode-Locked Lasers

Coherent phase locking of the CEP of two separate fs lasers requires a stepbeyond tight synchronization of the two pulse trains. One also needs to de-tect and stabilize the phase difference between the two optical carrier wavesunderlying the envelope of the pulses [52]. As illustrated in Fig. 10b, afterthe synchronization procedure discussed in the previous section matches therepetition rates (fr1 = fr2), phase locking requires maintaining the spectralcombs of the individual lasers exactly coincident in the region of spectraloverlap so that the two sets of optical-frequency combs form a continuousand phase-coherent entity. In other words, the offset frequencies of the lasersmust be set such that fo12 = fo1 − fo2 = 0. fo12 is easily detected by a co-herent heterodyne beat signal between overlapping comb components of thetwo lasers. By phase locking fo12 to a frequency of zero mean value, the twopulse trains evolve with identical CEP, i.e., ∆φCE1 − ∆φCE2 = 0.

To demonstrate this coherent comb “stitching” experimentally, two inde-pendent mode-locked Ti:sapphire lasers are operated at a 100 MHz repetitionrate, one centered at 760 nm and the other at 810 nm. The bandwidth of eachlaser corresponds to a sub-20 fs transform-limited pulse. When synchronized,the heterodyne beat between the two combs can be recovered with a S/N of60 dB in a 100 kHz bandwidth, as shown in Fig. 12. Hundreds of comb pairscontribute to the heterodyne beat signal. Figure 12 also indicates that beforethe phase lock loop is activated, fluctuations in the relative difference be-tween the offset frequencies (fo12) of the two combs easily exceed megahertzlevels on timescales as short as several tens of seconds. By stabilizing fo12 ata mean value of zero hertz, the carrier-envelope phase slip per pulse of slavelaser will accurately match the master laser.


5 MHzPhase lockactivated

R.B.100 kHz

60dB

f o1

–f o

2

-1.0

-0.5

0.0

0.5

1.0

8006004002000

(fo1

–f o

2)H

z

Time (s)

sdev = 0.15 Hz (1-s averaging time)

Fig. 12. Heterodyne beat signals can be detected between the two fs lasers afterthey are tightly synchronized. However, with phase locking, the fluctuations of thebeat signal can exceed several megahertz in a short time period. The beat frequencyunder the locked condition shows a standard deviation of 0.15 Hz at 1 s averagingtime

fo12 is locked to zero Hz using an AOM. One of the laser beams is passedthrough the AOM, thereby shifting the entire comb by the drive frequencyof the AOM. This procedure avoids the need to process the beat signal inthe troublesome frequency range around dc or fr. The detected beat signal isthen phase locked to the drive frequency of the AOM, effectively removing theAOM frequency. When unlocked, the intercomb beat frequency has a stan-dard deviation of a few megahertz with 1 s averaging time. Figure 12 showsthe recorded beat frequency signal under a phase-locked condition. With anaveraging time of 1 s, the standard deviation of the beat signal is 0.15 Hz.The established phase coherence between the two femtosecond lasers is alsorevealed via a direct time-domain analysis, as depicted in Fig. 13. For thisexample, we have employed spectral interferometric analysis of the joint spec-tra of the two pulses to produce interference fringes that correspond to phasecoherence between the two pulse trains persisting over the measurement timeperiod.

The result is displayed in Fig. 14. We note that the fringe visibility isreduced when the measurement time is increased, due to the increased phasenoise between the two lasers. A cross-correlation measurement between thetwo pulse trains also manifests the phase coherence in the display of persistentfringe patterns.

A powerful demonstration of the “coherently synthesized” aspect of acombined pulse is through a second-order autocorrelation measurement of thecombined pulse. For this measurement, the two pulse trains were maximally


CCDArray

Time-gatedSpectral interferometry

fs Laser 1

fs Laser 2

Delay 1 τ

Auto-Correlationof “synthesized” pulse

BBO

Delay 2

CEP locked

Fig. 13. Time-domain analysis of the established mutual coherence between thetwo fs lasers. Shown are spectral interferometer and second-order autocorrelator

900850800750700

Spe

ctra

lInt

erfe

rom

etry

(Lin

ear

Uni

ts)

Wavelength (nm)

Laser 1

Laser2

Two lasersphase locked

Two independentlasers

Fig. 14. Spectral interferometric data of two individual lasers, along with the caseswhen both lasers are present under locked and unlocked conditions. The interferencefringes in the spectrally overlapping region between the two lasers clearly indicatephase coherence between the two pulse trains when they are locked together

overlapped in the time domain before the autocorrelator. The autocorrela-tion curves of each individual laser are shown (Fig. 15a,b, respectively). Thespectra of the lasers are centered around 760 nm and 810 nm. An interestingautocorrelation measurement is obtained when the two lasers are not evensynchronized (Fig. 15c). Basically, we obtain an autocorrelation that is an av-erage of the two individual lasers, with a sharp spike in the data at a randomposition. The spike appears because, at that particular instant, the pulsesfrom the two lasers overlapped in time and the two electric fields came intophase and coherently added together. The timescale of this random interfer-ence is related to the offset frequency difference between the two repetitionrates and is usually less than a few nanoseconds. When the two lasers are syn-chronized but not phase locked, the resulting autocorrelation measurementindicates increased signal amplitude compared to the unsynchronized case,typically by a factor of 2.7. However, as expected, this signal displays con-siderable random phase noise within the autocorrelation interference fringes.When the two femtosecond lasers are phase locked, the autocorrelation re-veals a clean pulse that is often shorter in apparent duration and larger in


Laser 1(a)

Aut

o-C

orre

latio

n

Laser 2(b)

Un-synchronized(c)Combinedbeam

Delay Time (~ 2.6 fs per fringe)

(d) Synchronized& phase lockedCombined

beam

Delay Time (~ 2.6 fs per fringe)

Aut

o-C

orre

latio

n

Aut

o-C

orre

latio

nA

uto-

Cor

rela

tion

Fig. 15. Demonstration of coherent pulse synthesis. The second-order autocorrela-tion data show that the combined pulse has a narrower width and a higher ampli-tude compared with the two original laser pulses

amplitude (Fig. 15d). Note that we did not attempt to recompress the lightpulses outside the laser cavities to an (optimal) short duration and the pulsesare dispersively broadened to 50 fs to 70 fs. The width of the central fringepattern in an interferometric autocorrelation is more characteristic of theoverall bandwidth of the pulse than of the pulse duration and can resultin a trace that appears deceptively short. However, from Fig. 15, it is clearthat both amplitude enhancement and pulsewidth reduction are present asa result of the combined synchronization and carrier phase locking. We havetherefore demonstrated successful implementation of coherent light synthe-sis: the coherent combination of output from more than one laser so that thecombined output can be viewed as a coherent, femtosecond pulse emittedfrom a single source [52].

The capability of stabilizing the pulse repetition rate and the CEP evo-lution of two mode-locked lasers to such a high degree enables many possibleapplications. It may be particularly important in the generation of tunablefemtosecond sources in otherwise previously unreachable spectral regions.Previous work in electronic synchronization of two mode-locked Ti:sapphirelasers demonstrated timing jitter of a few hundred fs at best. Therefore,the present level of synchronization would make it possible to take full ad-vantage of this time resolution for applications such as novel pulse genera-tion and shaping [52], high-power sum- and difference-frequency mixing [53],new generations of laser/accelerator based light sources, or experiments re-quiring synchronized laser light and X-rays or electron beams from syn-chrotrons [54]. Figure 16 shows the cross-correlation measurement of thetwo stabilized mode-locked Ti:sapphire lasers using both (SFG) sum- anddifference-frequency generation (DFG). The DFG signal produced by a GaSecrystal can be tuned from 6 µm to beyond 12 µm with a high repetition rate(the same as the original lasers) and a reasonable average power (tens of mi-crowatts). Arbitrary amplitude waveform generation and rapid wavelength


Fig. 16. Simultaneous sum- and difference-frequency generation from two stabilizedfemtosecond lasers

switching in these nonlinear signals are simple to implement [55]. Anotherimportant application is in the field of nonlinear-optics-based spectroscopyand nanoscale imaging. For example, using two picosecond lasers with tightlysynchronized repetition rates (≈ 20 fs of jitter), we are able to achieve sig-nificant improvements in experimental sensitivity and spatial resolutions forcoherent anti-Stokes Raman scattering (CARS) microscopy [56].

5 Outlook

Over the last three years, the timely convergence of ultrafast lasers and high-precision spectroscopy (with accompanying highly stable CW lasers) has gen-erated a number of advances in both fields. Though the most immediate (andin itself, a truly revolutionary) effect has been realized in optical-frequencymetrology, other applications are beginning to emerge based on the unprece-dented degree of control now possible over few-cycle optical pulses. This con-trol capability is now having an impact on time-domain experiments andpromises to bring about dramatic advances in this area just as it has inoptical-frequency metrology and optical clocks. The ability to synthesize ar-bitrary electronic waveforms at optical frequencies is expected to impact bothfundamental physics and more application-specific technologies such as high-speed analog signal processing. As we have described, the ability to generatea pulse train with very high carrier-envelope phase coherence forms the tech-nological basis for waveform synthesis. The next step is to develop a methodof measuring the “absolute” or static carrier-envelope phase. Work on several


methods for doing so is well underway. Once this is achieved, the final stepis to adapt current pulseshaping technology to include the carrier-envelopephase. Several applications of this technology have been mentioned in thischapter, and others are expected to emerge as the technology matures.

Acknowledgments

The authors acknowledge support from NIST, DARPA, NASA, NSF, andONR. The authors acknowledge contributions from S. A. Diddams, S. Fore-man, L.-S. Ma, and R. Shelton. J.Y. acknowledges helpful discussions withH. C. Kapteyn.

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carrier-envelope phase stabilization of single and...

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