Long-term Hybrid Stabilization of the Carrier-Envelope Phase
Jack Hirschman, Randy Lemons, Evan Chansky, Günter Steinmeyer, Sergio Carbajo
LLong-term Hybrid Stabilization of the Carrier-Envelope Phase
Jack Hirschman,
1, 2, ∗ Randy Lemons,
2, 3
Evan Chansky,
4, 2
G¨unter Steinmeyer, and Sergio Carbajo
2, 3 Department of Applied Physics, Stanford University, 348 Via Pueblo, Stanford, CA 94305, USA SLAC National Accelerator Laboratory and Stanford University,2575 Sand Hill Rd, Menlo Park, CA 94025, USA Colorado School of Mines, 1500 Illinois St, Golden, CO 80401, USA Lehigh University, 27 Memorial Drive W, Bethelehem, PA 18015 Max Born Institute for Nonlinear Optics and Short Pulse Spectroscopy, Max-Born-Straße 2a, 12489 Berlin, Germany (Dated: June 5, 2020)Controlling the carrier envelope phase (CEP) in mode-locked and associated laser systems overpractically long timescales is crucial for real-world applications in ultrafast optics and precisionmetrology. We present a hybrid solution that combines a feed-forward (FF) technique to stabilizethe phase offset in fast timescales and a feedback (FB) technique that addresses slowly varyingsources of interference. We experimentally realize the hybrid stabilization system in an Er:Yb:glassmode-locked laser and demonstrate 75 hours of stabilization with integrated phase noise of 14 mrad,corresponding to around 11 as of timing jitter. Additionally, we examine the impact of environmentalfactors, such as humidity and pressure, on the long-term stability and performance of the system.
I. INTRODUCTION
With lasers operating in the femtosecond and attosec-ond regimes becoming increasingly prevalent, preciselycontrolling the offset of the underlying electric field withrespect to the envelope of the pulse, known as the carrierenvelope offset phase (CEP), will continue to rise in im-portance. Myriads of applications including optical fre-quency metrology [1, 2], high harmonic generation [3, 4],coherent synthesis of distributed fiber-laser arrays [5, 6],and laser-based [7–9] and on-chip accelerators [10, 11]rely on stabilizing and controlling the CEP. Achievingthis stabilization and control over practically long dura-tions is just as important if it is to be used in real-worldapplications.There are two common CEP locking schemes in mode-locked lasers, feedback (FB) and feed-forward (FF). FBtechniques measure fluctuations in the carrier-envelopeoffset (CEO) frequency and make adjustments to thelaser cavity or to the pump power in order to alterdifferences in the phase and group velocities [12–14].Specifically, changes to the cavity directly affect thepath length and changes to pump power alter the phaseand group velocities via nonlinear intracavity processes.These FB methods require electronics, usually in theform of proportional-integral-derivative (PID) controllersor phase-locked loops, designed to maintain the lockedCEO frequency.FF techniques instead allow for phase changes inthe cavity to develop naturally and then provide ameans for modulating the pulse phase dowstream [15–17]. Here, CEP can be stabilized and controlled usingan acousto-optic frequency shifter (AOFS). Furthermore,FF techniques often require relatively simple electronicsto achieve stabilization and do not tie short-term phase ∗ [email protected] stabilization to longer-term system performance. How-ever, phase modulators like an AOFS have limited oper-ation bandwidths (BW), typically on the order of a fewhundred kHz, which can compromise long-term stabiliza-tion if the modulating frequency drifts away from the BWrange.Prior work has explored retaining stability using FFtechniques over runs as long as a day. One of the longestruns came from L¨ucking, et al., who report less than 30mrad of phase noise with a CEP-locked run of more than24 hours [15] using a Ti:sapphire oscillator. Zhang, etal., using a Kerr-lens mode-locked Yb-CYA 1 µ m solid-state laser, report a run stabilized over two hours witha residual integrated phase noise of 79.3 mrad (1 Hzto 1 MHz) [18] using purely FF techniques. For solid-state lasers in the 1 µ m range, slightly longer runs canbe achieved using FF methods by manually adjustingcertain system parameters. For instance, Lemons, etal., using an Er:Yb:glass laser oscillator, could achieveeight hours of stabilization using manual adjustments tothe pump power during run time but introducing largeamounts of low-frequency phase noise as a result [19].Replacing the manual adjustments with adequate FB en-ables longer stabilization durations. Recently, utilizing amix of FF and FB techniques, Musheghyan, et al. pre-sented 75 hours of locked CEP in a Ti:Sapphire amplifierwith integrated phase noise of 150 mrad [20].As one possible solution to achieve ultra-low noise CEPstabilization over practically long periods of time, in thisletter, we elaborate on prior work to examine a FF tech-nique combined with a slow-drift compensation FB sys-tem on a Er:Yb:glass mode-locked laser which does notdetrimentally affect the short-term phase noise perfor-mance. After providing further motivation for combin-ing FF and FB, we will present an overview of the CEPstabilization system and will describe in detail the slow-drift compensation FB system design and performance,which is susceptible to environmental variations, such astemperature, humidity, and pressure changes. a r X i v : . [ phy s i c s . op ti c s ] J un Figure 1: The system contains seven functional blocks: Mode-locked laser, IL f -to-2 f , IL RF conditioning, OOL f -to-2 f , OOL RF conditioning, FF system, and FB system. Signals in red are optical , signals in black are RFpaths, and signals in light blue are electrical. II. MOTIVATION FOR A HYBRIDFEED-FORWARD AND FEEDBACK APPROACH
CEP stabilization in Ti:Sapphire lasers and certainfiber lasers has become increasingly common with somecommercial systems available today. On the other hand,stabilization of all-solid-state lasers in the 1 to 2 µ mwavelength range has proven more difficult to accom-plish. An important factor to consider is the relativelylong upper-state lifetime of Yb ranging around 1 to2 ms for the F / to F / transition [21] or that ofEr:doped host materials, typically around 8 to 10 ms for I / [22]. But considering that Er:fiber lasers rely onthe same ionic transition in Er as the Er:Yb:glass laserin this study motivates examining the interplay betweenthese time constants and their role in gain dynamics ofmode-locked lasers in careful detail. For this exercise, weemploy a textbook model [23–26], which describes thegain dynamics as an effective resonant RLC circuit. Inthis model, the population inversion in the gain mediumis equivalent to the energy stored in the capacitor C, theintracavity pulse energy translates into magnetic energystored in an inductor L, and output coupling (OC) along with other intracavity losses are expressed as a resistanceR OC . Additionally, for this class of mode-locked lasers,the model requires a second resistance for saturated ab-sorption (SA) represented as R SA , which has a negativeeffective value and is in parallel with R OC .Most mode-locked fiber lasers rely on an instantaneousreactive nonlinearity, e.g., a nonlinear optical loop mir-ror [27] whereas all-solid-state lasers are typically mode-locked by soliton mode-locking [28] with a semiconduc-tor saturable absorber. While this method is widespreadand very successfully commercialized, it strongly relies onsuitable suppression of q-switching instabilities [29]. Tothis end, the relative spot size in the gain medium and onthe absorber have to be carefully adapted to avoid pos-itive feedback within the effective RLC circuit. For sus-tained mode-locking, the nonlinear feedback cannot bearbitrarily reduced, and, as a result, the gain dynamicsof mode-locked all-solid-state lasers are either criticallydamped or even slightly underdamped.Therefore, it is worth considering solutions for CEPstabilizing these lasers other than feedback alone, suchas employing a feed-forward scheme. In the FF scheme,an acousto-optic frequency shifter is used to correct theCEP outside the laser cavity [16]. Specifically, the AOFSmodulates the phase of the pulses once they are cou-pled out of the mode-locked laser. Using the first orderdiffracted beam from the AOFS, the frequency comb ofthe laser is altered and the CEO frequency becomes ef-fectively locked [19, 30]. This has proven to be an elegantsolution on short time scales, resulting in residual jittersas low as a few milliradians [19].On longer time scales, however, the FF scheme ex-hibits significant phase drift, the origin of which mani-fests in analyzing the signal path from the laser to theAOFS, and ultimately resulting in a limited operationbandwidth for the AOFS. Slow drifts can alter f CEO tothe extent that the RF drive frequency deviates so farfrom the center frequency of the AOFS that the Braggangle is significantly altered [17], thus negatively impact-ing the diffracted efficiency. This phenomenon can beunderstood by analyzing timing within a FF setup, asdepicted in Figure 1. Travel times from the laser to ra-dio frequency (RF) output of the f -to-2 f interferometerare generally kept in the few nanosecond range. Com-paratively large group delays occur in the AOFS, whichrelies on the generation of an ultrasonic wave with a typ-ical travel velocity on the order of 10 − of the vacuumspeed of light. For example, bringing the input beam tothe AOFS as close as a few hundred microns to the piezo-electric transducer on the AOFS still generates a signaldelay greater than 1 µ s. Furthermore, a similar delayis generated by resonant circuitry in the AOFS used toconvert the output current of a microwave amplifier intohigh voltage amplitude necessary for driving the trans-ducer for a high diffraction efficiency of the AOFS.The lag in the signal path to the AOFS does not posea problem when used at constant frequency as result-ing phase shifts can be compensated. However, with theCEP freely drifting inside the laser, an error in CEP cor-rection can occur at the AOFS. With an assumed 2 µ s asan optimistic assumption, a drift in frequency of 100 kHztranslates to a drift in phase of over 1 radian. To mit-igate this effect, feedback stabilization can be combinedwith feed-forward stabilization [31] (as depicted in Fig-ure 1 and more thoroughly described in Section III). FFstabilization ensures high CEP stability down to millisec-ond time scales, and then at longer durations, a simplefrequency stabilization takes over to keep the drive fre-quency of the AOFS near its center frequency. III. SYSTEM OVERVIEW
The system achieves CEP stabilization via FF short-term stabilization combined with a slow-drift FB forstabilization over many hours. The detailed schematicdiagram is shown in Figure 1. There are seven mainfunctional blocks: the mode-locked laser (an Origami-15Er:Yb:glass laser), the in-loop (IL) f -to-2 f interferom-eter, the IL radio frequency (RF) conditioning, the FFAOFS system, the AOFS-based feed-forward system, the PID-based FB system, the out-of-loop (OOL) f -to-2 f in-terferometer, and the OOL RF diagnostics. Within thesystem there is one main signal path with two auxiliarypaths (FF and FB). The main signal path is predomi-nately optical. It is generated in the laser cavity, feedsinto the AOFS, goes through the OOL f -to-2 f interfer-ometer, and is then converted into RF for diagnostics.The feed-forward auxiliary path is a mix of optical andRF. It starts with an optical output from the laser cav-ity, goes through the IL f -to-2 f interferometer where it isconverted into an RF signal, and then enters the IL RFconditioning section where it feeds into the AOFS andcan affect the signal path. The feedback auxiliary pathis a mix of optical, RF, and electrical signals. It at firstfollows the FF auxiliary path but, instead of going intothe AOFS, branches-off and feeds into a PID controllerwhich modulates pump power to the cavity.The IL and OOL segments operate in nearly identi-cal fashion, utilizing the f -to-2 f heterodyning technique.For the path through the IL System, the laser spectrumcan be quantified as f n = nf REP + f CEO , (1)where f n is the comb spacing of the mode-locked laserand f REP is the repetition rate of the mode-locked laser.This optical comb from the original spectrum is amplifiedand compressed in order to reach intensities necessary fornon-linear broadening of the frequency spectrum to en-sure f n is reached (seen in Figure 1 as SCG, super con-tinuum generation). The pulse then undergoes frequencydoubling to obtain 2 f n (seen in Figure 1 as SHG, secondharmonic generation). Filters select these two signals atthe same frequency, and the heterodyning occurs at thephotodiode (PD) where the signal is output as an RFsignal. This part of the system is described in detail inRef. [19].The RF output signal from IL now contains the f n and 2 f n mixing products and is fed into the IL RF con-ditioning segment before feeding into the AOFS and theFB loop. The RF conditioning is required because theoperating range of the AOFS is 80 ± . f CEO may not be in this range. The first set of components inthe IL RF conditioning use low pass filters to retain thedifference mixing term:2 f n − f n = 2 nf REP + 2 f CEO − (2 nf REP + f CEO ) = f CEO . (2)The f CEO signal is amplified and obtains a 40 dB signal-to-noise ratio (SNR) before being mixed with a local os-cillator (LO) signal to ensure the output signal is in thecorrect band for the AOFS.The LO signal comes from a RF frequency comb gen-erated from an ultra-low phase noise, 10 MHz rubidiumcrystal oscillator (Stanford Research Systems PRS10)signal fed into a divider that creates a comb spacing of1.465 MHz starting from 1.4 MHz. The LO signal usedis only one line of this comb. It is chosen for mixing with f CEO via a band-pass (BP) filter such that f AOFS = f CEO + f LO = 80 MHz . (3)The mixed f AOFS signal then follows two branches: oneinto the AOFS for FF, the other into the FB system.Using the − st diffraction order of the AOFS yields f OOL = f n − f AOFS = ( nf REP + f CEO ) − ( f CEO + f LO )= nf REP − f LO , (4)where negative and positive frequencies are physicallyequivalent. This output is sent into the OOL segmentwhere the signal goes through a similar process as inthe IL segment. The output from the OOL PD has themixed f REP and f LO products as well as the constituents.The RF output of the OOL PD then undergoes low passfiltering in the OOL RF Diagnostics block, yielding ameasurement of f LO .The repetition rate of the mode-locked laser, f REP , is204 MHz, and f LO was chosen in order to keep f AOFS at80 MHz. Since f CEO and f LO are related via f AOFS (seeequation 3), then the stability of f CEO can be determinedfrom the stability of f LO and the stability of f AOFS , andin turn one can monitor drifts in f OOL and f AOFS as ameans for characterizing how stable the system is (seeequation 4).The other branch out of the IL RF conditioning goesinto the FB block. Section IV discusses this system indetail.
IV. SLOW DRIFT FB STABILIZATION
The FB system works by generating an error signalfrom the f AOFS signal and feeding this error signal to aPID controller, which in turn affects the pump power tothe cavity. Figure 2 shows how this FB is accomplished.The f AOFS signal is produced by the IL RF condi-tioning block, as discussed in Section III. It is also thesum of f CEO and the constant f LO and is set for opti-mal operation at 80 MHz. Therefore, the deviation in f AOFS away from 80 MHz can be used as the error sig-nal. To generate the error signal from these deviations, f AOFS is mixed with a local oscillator ( f SET ) of 79.80MHz, whose filtered mixing products would be f ERROR : f ERROR = f AOFS − f SET . The operation is limited by thefrequency-to-voltage converter chip with input frequencyrange of 0 to 500 kHz, where f AOFS between 79.80 MHzand 80.30 MHz yields f ERROR = 0 kHz and f ERROR =500 kHz, respectively.After amplification, f ERROR is converted to a voltagevia a frequency-to-voltage converter which feeds into aStanford Research Systems SIM 960 Analog PID con-troller. An internal set point on the PID controller is setto a voltage corresponding to f ERROR of 200 kHz, which maps to f AOFS of 80 MHz. The PID controller then al-ters the pump power, accordingly, which in turn affectsthe center wavelength of the beam.
A. Performance
The proceeding sections will present short- and long-term performance characterization of the system as de-scribed above with an emphasis on long-term stabiliza-tion performance.
1. Performance in the Short Term
Without the FB loop, the integrated phase noise den-sity (IPND) for f OOL is 3.13 mrad (1 Hz to 3 MHz),corresponding to an rms jitter of 2.57 as. With the FBsystem in place, the IPND increases to about 13.72 mrad,corresponding to an rms jitter of 11.29 as (see second halfof Table I). While this increase is significant, the addi-tion of the FB system is not detrimental to the short-termphase noise. Figure 3 (on the right) shows the PND andIPND of the system with and without the FB loop inplace along with the PND and IPND for the local oscil-lator.Notwithstanding such low noise performance, the sys-tem without the FB loop is unable to stabilize the CEPbeyond around thirty minutes. Figure 3 (on the left)compares the drifts in f AOFS over a thirty-five hour runfor an exclusively feed-forward system (labeled in figureas No PID Feedback) to our system with both feed-forward and feedback (labeled in figure as PID Feed-back). Without FB, f AOFS drifts by hundreds of kHz,whereas the FB constrains it to less than tens of kHz,which is comfortably within the bandwidth of the AOFS.From these results it is clear that the addition of theFB system to the FF system does not impose a significantdetriment to the phase noise in exchange for constrain-ing frequency drifts in f AOFS that otherwise would beunwieldy over practically long durations.
2. Performance in the Long Term
Using the FB system described in section IV, over 75hours of stabilization was recorded. Figure 4 shows theresults from f AOFS , f OOL , f CEO , and f LO , all with theraw data and with a moving average of 1000. The movingaverage reveals slower trends in the data.Both f LO and f OOL describe drifts in the local oscil-lator frequency. The drifts are on the order of hundredsof mHz to several Hz, respectively. Since f AOFS is thecombination of f CEO and f LO and since f LO has verysmall drifts, the drifts in f AOFS are dominated by driftsin f CEO . These drifts are contained to less than tens ofkHz with an rms jitter of 0.27 kHz (see Table I for rmsfrequency jitter for all signals). Furthermore, for this run,Figure 2: The FB system consists predominantly of RF conditioning, which generates an error frequency from theinput f AOFS signal, and a FB control system, which converts this error frequency to a voltage and produces avoltage to drive the pump power in order to lock f AOFS at a particular frequency.Figure 3: The plot on the left is the variation in f AOFS with FB and without FB over a 35 hour run. The plot onthe right is the phase noise and integrated phase noise of LO, OOL with FB, and OOL without FB. These plotsshow how FB system affects performance of experiment in regards to added phase noise in short-term andstabilization of f AOFS in long-term. On the left, drifts in f AOFS are shown for two signals: one with exclusivelyfeed-forward (in green labeled No PID Feedback) and the other with both feed-forward and feedback (in dark bluelabeled PID Feedback). On the right, three signals are shown: local oscillator frequency (grey), OOL frequency withFB (blue), and OOL frequency without FB (orange). For each of these signals both PND and IPND are shown,solid and dashed lines, respectively. f CEO was directly collected from within the IL RD con-ditioning block, and its drifts closely match f AOFS driftbehavior.Figure 5 shows the f AOFS signal in the frequency do-main, where low frequency noise is dominant. Aside from1/f noise, environmental factors such as temperature, hu-midity, and pressure can cause fluctuations in the half-hour to few hour range. In particular, there are largedrifts in the absolute value of f AOFS for frequencies be-tween 1 mHz to 0.1 mHz, corresponding to time periodsof a half-day to a day that match with where the largestamplitudes in the drift of the environmental variables are.Figure 6 shows the variation of these variables aroundtheir means, collected over 75 hours uninterrupted.Within the first few hours of the first day, temperaturedrops by nearly 2 ◦ C. Humidity has a change of around12 % RH over the second 24-hour period, and pressurehas a peak-to-peak change of nearly 6 mbar. In thesemeasurements, the laboratory was not temperature- orhumidity-stabilized to high precision optical laboratorystandards. Additionally, weather conditions, includingrelatively heavy rain within the first twenty-four hours,affected humidity and pressure.Cross-covariance analysis was performed on each envi-ronmental variable with each frequency signal. The datawere normalized before performing the correlation anal-ysis and the cross-covariance values were normalized tovalues between ±
1. The first half of Table I shows thetime lag values for the maximum correlation values fromthese analyses. These time lag values correspond to wheneffects from the environment are correlated to changes inthe data. For instance, the maximum correlation be-tween humidity and f AOFS occurs roughly after a 3.85hour delay between when the change in temperature po-tentially manifests as a change in the f AOFS data. f AOFS and f CEO have similar correlation values with tempera-ture and pressure. The time lag of these correlations areon the order of minutes. f LO with temperature and f OOL with pressure have the strongest correlations relative tothe whole set.We believe the step in the f LO around the 30 hourmark is likely caused by the drop in humidity around thesame time. Additionally, we suspect that the oscillatorybehavior in f OOL could be coupling in from RF noise orsignals in the FB system imprinted on the optical combdriving the AOFS.
V. CONCLUSIONS
Controlling CEP for mode-locked lasers has continuedto rise in importance with a vast array of experimentsrequiring a stabilized CEP over practically long duration.A mix of feed-forward and feedback techniques providesone such avenue for accomplishing this stabilization.The system presented in this report combines a FF por-tion, consisting of an RF signal driving an AOFS whichmodulates the beam downstream, and a slow FB por- tion, which uses the deviation of f AOFS as the error sig-nal for a PID control loop modulating the pump powerto the cavity. The FF technique, alone, selects f CEO ,keeping it stable for short periods of time until environ-mental effects and other noise alter f CEO to the pointthat the AOFS cannot recover stability. The FB section,thus, keeps f AOFS at the proper frequency by alteringthe beam’s central wavelength, allowing the FF systemto remain locked. From the OOL measurements, im-plementing the FB with the FF added about 10 mradto the integrated phase noise of the FF system alone.To investigate potential drift sources, we collected dataand studied the correlation of environmental factors (lo-cal temperature, humidity, and pressure) and changes inthe performance of the system.Using these combined methods, we were able to showstabilization for over 75 hours with 13.72 mrad integratedphase noise to be used in real-world applications underunfavorable environmental conditions. The 75-hour runwas interrupted only for reporting purposes and our dataindicates that the system would have remained lockedindefinitely.
ACKNOWLEDGMENTS
This work was supported by the U.S. Department ofEnergy and Laboratory Directed Research and Develop-ment program at SLAC National Accelerator Laboratory,under Contract No. DE- AC02-76SF00515.Figure 4: Four plots show frequency drifts of f AOFS (top left), f OOL (top right), f CEO (bottom left), and f LO (bottom right) over the 75 hour run. For each, the raw signal (in grey) and averaged signal (in red) are shown.Figure 5: Plot shows absolute value of frequency drifts of f AOFS versus frequency. Time labels corresponding tospecific frequencies are shown for 1 second, 1 hour, 1 day, and 2 days in order to contextualize noise sourcetime-scales. [1] T. Udem, R. Holzwarth, and T. W. H¨ansch, Optical fre-quency metrology, Nature , 233 (2002). [2] F. Krausz and M. Stockman, Attosecond metrology:
Figure 6: Three plots show fluctuation around the mean for temperature (top), humidity (middle), and pressure(bottom) over the 75 hour run.Table I: Left half shows time lag (in hours) for the maximum correlation values from cross-covariance analysisperformed for each measured frequency signal against each environmental variable. Right half shows signal stabilitycharacterization, specifically with comparisons of IPND and timing jitter for FB versus no FB.
Temperature Humidity Pressure FF + FB FF onlyRMS Freq IPND Timing IPN TimingTime Lag (hrs) Time Lag (hrs) Time Lag (hrs) Jitter (Hz) (mrad) Jitter (as) (mrad) Jitter (as) f AOFS f OOL f CEO f LO , 205 (2014).[3] G. Sansone, E. Benedetti, F. Calegari, C. Vozzi,L. Avaldi, R. Flammini, L. Poletto, P. Villoresi, C. Al-tucci, R. Velotta, S. Stagira, S. De Silvestri, andM. Nisoli, Isolated single-cycle attosecond pulses, Science , 443 (2006).[4] I. Sola, E. M´evel, L. Elouga, E. Constant, V. Strelkov,L. Poletto, P. Villoresi, E. Benedetti, J. Caumes, S. Sta-gira, C. Vozzi, G. Sansone, and M. Nisoli, Controllingattosecond electron dynamics by phase-stabilized polar-ization, Nature Physics , 319 (2006).[5] J. Ye, S. Cundiff, S. Foreman, T. M. Fortier, K. W.Holman, D. J. Jones, J. D. Jost, H. Kapteyn, K. A. H.Leeuwen, L. Ma, M. M. Murnane, J.-L. Peng, and R. K.Shelton, Phase-coherent synthesis of optical frequenciesand waveforms, Applied Physics B , s27 (2002).[6] R. Lemons, W. Liu, J. C. Frisch, A. Fry, J. Robinson,S. Smith, and S. Carbajo, Integrated structured light ar-chitectures (2020), arXiv:2003.14400 [physics.optics]. [7] Y. I. Salamin and S. Carbajo, A simple model for thefields of a chirped laser pulse with application to electronlaser acceleration, Frontiers in Physics , 2 (2019).[8] S. Carbajo, E. A. Nanni, L. J. Wong, G. Moriena, P. D.Keathley, G. Laurent, R. J. D. Miller, and F. X. K¨artner,Direct longitudinal laser acceleration of electrons in freespace, Phys. Rev. Accel. Beams , 021303 (2016).[9] L. J. Wong, K.-H. Hong, S. Carbajo, A. Fallahi, P. Piot,M. Soljaˇci´c, J. D. Joannopoulos, F. X. K¨artner, andI. Kaminer, Laser-induced linear-field particle accelera-tion in free space, Scientific Reports , 11159 (2017).[10] N. V. Sapra, K. Y. Yang, D. Vercruysse, K. J. Leedle,D. S. Black, R. J. England, L. Su, R. Trivedi, Y. Miao,O. Solgaard, R. L. Byer, and J. Vuˇckovi´c, On-chip inte-grated laser-driven particle accelerator, Science , 79(2020).[11] D. Cesar, J. Maxson, X. Shen, K. P. Wootton, S. Tan,R. J. England, and P. Musumeci, Enhanced energy gainin a dielectric laser accelerator using a tilted pulse frontlaser, Opt. Express , 29216 (2018). [12] D. Jones, S. Diddams, J. Ranka, A. Stentz, R. Windeler,J. Hall, and S. Cundiff, Carrier-envelope phase controlof femtosecond mode-locked lasers and direct optical fre-quency synthesis, Science , 635 (2000).[13] T. J. Yu, K.-H. Hong, H.-G. Choi, J. H. Sung, I. W.Choi, D.-K. Ko, J. Lee, J. Kim, D. E. Kim, and C. H.Nam, Precise and long-term stabilization of the carrier-envelope phase of femtosecond laser pulses using an en-hanced direct locking technique, Opt. Express , 8203(2007).[14] T. Udem, J. Reichert, R. Holzwarth, and T. W. H¨ansch,Accurate measurement of large optical frequency differ-ences with a mode-locked laser, Opt. Lett. , 881 (1999).[15] F. L¨ucking, A. Assion, A. Apolonski, F. Krausz, andG. Steinmeyer, Long-term carrier-envelope-phase-stablefew-cycle pulses by use of the feed-forward method, Opt.Lett. , 2076 (2012).[16] S. Koke, C. Grebing, A. A. Harald Frei, A. Asion, andG. Steinmeyer, Long-term carrier-envelope-phase-stablefew-cycle pulses by use of the feed-forward method, Na-ture Photonics , 462 (2010).[17] G. Steinmeyer, B. Borchers, and F. L¨ucking, Carrier-envelope phase stabilization, in [36], Chap. 6, pp. 89–110.[18] Z. Zhang, H. Han, H. Wang, X. Shao, S. Fang, andZ. Wei, Ultra-low-noise carrier-envelope phase stabiliza-tion of a kerr-lens mode-locked yb:cya laser frequencycomb with a feed-forward method, Opt. Lett. , 5489(2019).[19] R. Lemons, W. Liu, I. F. de Fuentes, S. Droste, G. Stein-meyer, C. G. Durfee, and S. Carbajo, Carrier-envelopephase stabilization of an er:yb:glass laser via a feed-forward technique, Opt. Lett. , 5610 (2019).[20] M. Musheghyan, F. L¨ucking, Z. Cheng, H. Frei, andA. Assion, 0.24 tw ultrabroadband, cep-stable multipassti:sa amplifier, Opt. Lett. , 1464 (2019).[21] F. Gan, Laser Materials
Rare-earth-doped fiber lasers andamplifiers , 2nd ed., Optical Engineering, Vol. 71 (Mar-cel Dekker, New York, 2001).[23] A. Yariv,
Quantum Electronics (Wiley, New York, NewYork, 1989).[24] M. Morishita, T. Ohmi, and J.-I. Nishizawa, Impedancecharacteristics of double-hetero structure laser diodes, Solid-State Electronics , 951 (1979).[25] J. Katz, S. Margalit, C. Harder, D. Wilt, and A. Yariv,The intrinsic electrical equivalent circuit of a laser diode,IEEE Journal of Quantum Electronics , 4 (1981).[26] R. Tucker, Large-signal circuit model for simulation ofinjection-laser modulation dynamics, IEEE ProceedingsI - Solid-State and Electron Devices , 180 (1981).[27] N. J. Doran and D. Wood, Nonlinear-optical loop mirror,Opt. Lett. , 56 (1988).[28] F. X. Kartner, I. D. Jung, and U. Keller, Soliton mode-locking with saturable absorbers, IEEE Journal of Se-lected Topics in Quantum Electronics , 540 (1996).[29] C. H¨onninger, R. Paschotta, F. Morier-Genoud,M. Moser, and U. Keller, Q-switching stability limits ofcontinuous-wave passive mode locking, J. Opt. Soc. Am.B , 46 (1999).[30] Acousto-optic Theory Application Notes , AA Opto-Electronic.[31] B. Borchers, S. Koke, A. Husakou, J. Herrmann, andG. Steinmeyer, Carrier-envelope phase stabilization withsub-10 as residual timing jitter, Opt. Lett. , 4146(2011).[32] F. Helbing, G. Steinmeyer, and U. Keller, Carrier-envelope offset phase-locking with attosecond timing jit-ter, Selected Topics in Quantum Electronics, IEEE Jour-nal of , 1030 (2003).[33] L. Xu, C. Spielmann, A. Poppe, T. Brabec, F. Krausz,and T. W. H¨ansch, Route to phase control of ultrashortlight pulses, Opt. Lett. , 2008 (1996).[34] Analog PID Controller: SIM960 , Stanford Research Sys-tems.[35]
Voltage-to-Frequency and Frequency-to-Voltage Con-verter , Analog Devices.[36] K. Yamanouchi and K. Midorikawa, eds.,
Progress in Ul-trafast Intense Laser Science , Vol. 9 (Springer, Heidel-berg, Germany, 2013).[37] W. E. Kassa,
Electrical modeling of semiconductor laserfor high data wireless communication , Ph.D. thesis, Uni-versit´e Paris-Est (2015).[38] B. Borchers, F. L¨ucking, and G. Steinmeyer, Acousticfrequency combs for carrier-envelope phase stabilization,Opt. Lett.39