AM and high-harmonic FM laser mode locking

Ryan P. Scott, Corey V. Bennett, and Brian H. Kolner

We demonstrate a new technique of active mode locking that combines amplitude-modulated ~AM! modelocking at the cavity fundamental repetition rate with frequency-modulated ~FM! mode locking at a highharmonic. This method combines the advantages of pulse shortening by high-harmonic mode lockingwhile preserving the higher peak powers available at the fundamental repetition rate. We demonstratethis technique using a Nd:YAG laser that is simultaneously AM mode locked at 80 MHz and FM modelocked at the 22nd harmonic ~1.76 GHz!. Pulses as short as 16 ps with a peak power of 6.25 kW weremeasured. © 1997 Optical Society of America

Key words: Active mode locking, mode-locked laser, simultaneous mode locking, amplitude-modulated mode locking, frequency-modulated mode locking, harmonic mode locking.

1. Introduction

Modern approaches to ultrashort laser-pulse genera-tion have been dramatically successful by relying onnonlinear effects such as Kerr lens mode locking1 inTi:sapphire, soliton pulse shaping in erbium-dopedfiber lasers,2,3 and saturation effects in semiconduc-tors.4 Early approaches to laser mode locking suchas intracavity amplitude modulation or frequencymodulation suffered from a requirement of high drivesignals andyor frequencies to exploit the full gainbandwidth available in the host laser medium. Bytoday’s standards, active mode locking is not compet-itive with passive mode locking, yet there are still afew applications in which active mode locking, andimprovements to the method, may still find useful-ness. These would include, for example, low-powerlaser systems and gain media that do not possess auseful Kerr effect.

In this paper we describe a new approach to activemode locking that overcomes some of the traditionalbarriers to shorter pulse formation by combiningmultiple ~in this case, two! mode lockers within thecavity. It provides a method for reducing pulsewidth while simultaneously increasing peak power.

Consider the generic actively mode-locked lasershown in Fig. 1. We can drive the arbitrary mode

The authors are with the Department of Applied Science, Uni-versity of California, Davis, 228 Walker Hall, Davis, California95616.

Received 20 December 1996; revised manuscript received 25April 1997.

0003-6935y97y245908-05$10.00y0© 1997 Optical Society of America

5908 APPLIED OPTICS y Vol. 36, No. 24 y 20 August 1997

locker with one of the following two methods: 1! acw signal or 2! short pulses at the laser’s fundamentalrepetition rate. Method 1 will generate short pulsesif we use a high frequency, but this leads to a highrepetition rate and lower peak powers. Method 2would be ideal for generating short pulses, but it isextremely difficult to implement because of the re-quired modulator bandwidth.

Many years ago Siegman suggested placing multi-ple mode lockers in the laser cavity,5,6 each driven atsuccessively higher harmonics of the fundamentalrepetition rate. With superposition, these modelockers represent terms in a Fourier series that, inthe limit of large numbers, approximate a delta func-tion. This leads to a very short gain window andhence short pulses. Our technique accomplishessome of the advantages of this approach using twomode lockers. We run one amplitude-modulated~AM! mode locker at the fundamental repetition rateand a frequency-modulated ~FM! mode locker at the22nd harmonic.7 As an historical note, Kinsel useda single FM mode locker driven simultaneously at thefundamental and the second harmonic of the cavityrepetition rate.8 This is equivalent to two separatemodulators, but his goal was to improve the stabilityof the FM mode-locking process rather than obtain-ing shorter pulses. Nonetheless, it shares a commonprinciple with our approach. That is, the superpo-sition of harmonics in a Fourier series can enhance aneffect at the fundamental while suppressing un-wanted effects at higher harmonics.

2. Active AM and FM Mode Lockers

To understand how these mode lockers work togetherin the same cavity, consider how a single-mode locker

influences laser operation. From standard theory,6the pulse width is proportional to two parameters,the modulation index and the modulation frequency.This can be written as

tp } S 1Dm

D1y4S1fmD1y2

, (1)

where Dm is the modulation index and fm is the mod-ulation frequency. The mode locker drive power Pmvaries the modulation index as

Dm } HPm,Pm

1y2,acousto-optic AM mode lockerselectro-optic FM mode lockers. (2)

For an AM mode locker, the dependence is linear, andfor a FM mode locker it depends on the square root ofthe drive power. Therefore the pulse width dependsweakly on the mode locker drive power ~Pm

21y4 forAM or Pm

21y8 for FM!, but more strongly on the drivefrequency ~ fm

21y2!. If we want to shorten the pulsewidth, we win fastest by increasing the modulationfrequency.

Generally for a laser with a single-mode locker, thepeak optical power Ppk will depend on the modulationfrequency,

Ppk <Pavg

tpfm}

PavgDm1y4

fm1y2 , (3)

where Pavg is the average optical power. For con-stant modulation index and average power, the peakpower decreases as the pulse is shortened because ofthe increase in repetition frequency. However, thisproblem can be alleviated if we include two or moremode lockers within the same cavity which forces therepetition rate to be that of the lowest-frequencymode locker.

There are several ways of combining multiple modelockers. In the case of two mode lockers, either onebeing an amplitude or a frequency modulator, thereare four possible combinations. AM mode lockersare generally acousto-optic and are limited to modu-lation frequencies less than 1 GHz, making them

Fig. 1. Generic actively mode-locked laser. The mode locker isdriven by the signal v~t! with the goal of producing short pulses.v~t! can be 1! a high-frequency cw signal or 2! a waveform consist-ing of short pulses at the fundamental laser repetition rate.

difficult to use at high harmonic frequencies. How-ever, they have stable mode-locking properties, and,because they modulate cavity loss, AM mode lockerscan be used to fix the maximum repetition frequency.Alternatively, electro-optic FM mode lockers areadapted easily to microwave frequencies,9 but they donot have the stability of their AM counterparts.8Also, because they do not affect the gain or loss of thecavity, FM mode lockers are not well suited for pre-serving the repetition rate in a multiply mode-lockedcavity. For these reasons we chose to use an AMmode locker at the fundamental and a FM modelocker at a high harmonic.

To see the effect of using the two mode lockers, letus first look at the time domain representation of AMmode locking. Figure 2 shows the gain curve gT~t!and the periodic loss curve aT~t! for a generic AMmode-locked laser. gT~t! is the integrated round-tripamplitude gain, and it is nearly constant for mediawith long upperstate lifetimes. aT~t! is the totalround-trip amplitude loss and it oscillates at the am-plitude modulation frequency. During mode lock-ing, pulses only exist in the temporal region wherethe loss is less than the gain ~region a!. We call thisthe region of allowed oscillation and it has a width Dt.

Next we add a FM mode locker to the cavity andconsider its effects within the region a ~Fig. 3!. Inconventional FM mode locking, pulses form at a cuspof the FM signal.6 With the combination of FM andAM mode lockers we have the additional constraintthat pulses will form only at FM cusps within theregion of allowed oscillation established by the AMmode locker. This is the key to keeping the laserrunning at the fundamental repetition rate. By run-ning the FM mode locker at a higher harmonic, weobtain the pulse shortening advantages of harmonicmode locking while the AM mode locker preserves aperiodic gain window at the fundamental repetitionrate. This works, of course, provided that only asingle FM cusp resides within region a. If the FM

Fig. 2. Integrated round-trip amplitude gain gT~t! and the inte-grated round-trip amplitude loss aT~t! as a function of time for anAM mode-locked laser. When the cavity loss is greater than thegain ~region b!, the cavity will not oscillate. But when the lossdrops below the gain ~region a! the laser can oscillate. This is theregion of allowed oscillation and has a width Dt.

20 August 1997 y Vol. 36, No. 24 y APPLIED OPTICS 5909

signal frequency were, say, two or three times greaterthan that shown in Fig. 3, two or more cusps couldoccupy the oscillation region simultaneously ~Fig. 4!.A pulse could then form under each cusp creatingmultiple pulses within the oscillation window.8,10

We, in fact, observed double pulsing and unstableoperation when the laser was run in this mode.

When designing a multiply mode-locked laser wewant to choose our operating parameters such thatwe will avoid unstable operation and maximize thelaser’s performance. A basic approach to determin-ing the optimum parameters of the laser follows.

As we have discussed, the pulse will reside withina region where the gain exceeds the loss, gT $ aT~t!.We can also write this as the product of three trans-mission terms:

tAMt0tg . 1. (4)

These terms are defined as

tAM 5 exp@2DAM~1 2 cos vAMt!#,

t0 5 exp~2a0!,

tg 5 exp~gT!, (5)

where tAM is the transmission through the AM modelocker, DAM is the peak modulation index ~single passfor a traveling-wave cavity and double pass for stand-ing wave!, vAM is the angular modulation frequency,t0 is the fixed round-trip cavity loss, and tg is theintegrated round-trip amplitude gain. We also con-clude that aT~t! 5 a0 1 DAM~1 2 cos vAMt!.

When the condition tAMt0tg . 1 is satisfied, thetime-varying loss term DAM~1 2 cos vAMt! # gT 2 a0.The phase angle Du over which this condition is sat-isfied, is easily found to be

Du ; vAMDt 5 2 cos21S1 1a0 2 gT

DAMD (6)

Fig. 3. Expanded view of region a ~Fig. 2! with the addition offrequency modulation. The relative phase between the AM andFM signals is defined as u. An optical pulse will reside within thecusp of the FM mode locker signal as long as it is also within theregion of allowed oscillation.

5910 APPLIED OPTICS y Vol. 36, No. 24 y 20 August 1997

and corresponds to region a in Fig. 2. This phaseangle is the oscillation window size in radians withrespect to the amplitude modulation frequency. Toensure that only one FM cusp resides within thiswindow, the FM period must then satisfy TFM . 2Dt,or equivalently,

fFM ,vAM

2Du, (7)

where fFM 5 1yTFM is the frequency modulation fre-quency.

3. Experimental Setup and Results

We demonstrated simultaneous AM and harmonicFM mode locking by modifying a Coherent AntaresAM mode-locked Nd:YAG laser. As a standard AMmode-locked laser, it produces 75-ps pulses at an 80-MHz repetition rate with 24 W of average opticalpower. The FM mode locker used was a microwaveresonant LiNbO3 electro-optic phase modulator11 op-erated at 1.76 GHz with a single-pass modulationefficiency of 0.59 rady~W!1y2 ~Ref. 12!. The crystalsize was 4 mm 3 4 mm 3 40 mm and antireflectioncoated at 1.06 mm.

To provide phase locking between the AM and FMmode lockers, all RF signals were derived from acommon 10-MHz low-noise crystal oscillator. TheFM mode locker was placed in the cavity with thecenter of the crystal 14.8 cm from the output coupler,and the overall cavity length was shortened to com-pensate for the added delay introduced by the 40 mmof LiNbO3. When the FM mode locker was in placewith no drive signal applied, the average opticalpower dropped from 24 to 6 W. We believe that thiswas caused by photorefractive effects within theLiNbO3 refracting energy out of the TEM00 mode.11

Also, the laser was somewhat noisy, tending to self Qswitch occasionally. We attribute the instability tointensity-dependent waveguiding within the LiNbO3.A mode locker designed for high-intensity intracavityuse would avoid this problem.

Simultaneous AM and harmonic FM operation wasstudied under constant AM mode-locking conditions.

Fig. 4. When the frequency modulation frequency is too large,multiple optical pulses may form within each oscillation region.This results in multiple pulses and unstable laser operation.

Figure 5 shows the measured pulse width as a func-tion of RF power to the FM mode locker. Also shownis a trend line indicating the 21y8 slope dependence~on a log-log scale! predicted by theory @see Eq. ~1!and note that Dm } ~Pm!1y2#. At very low RF power~,20 mW! the AM mode locking dominates, but withonly a few tens of milliwatts ~0.1 rad! the FM modelocker begins to significantly shorten the pulse widthand dominate the mode-locking process. When weincreased the modulator power to 630 mW ~0.47 rad!the pulse width was reduced to 17 ps. The best pulsesobserved had a peak power of 6.25 kW ~16-ps pulsewidth, 8 W average optical power!. This is a 56%increase over AM mode locking alone ~;4-kW peak!.

Figure 5 shows that our data are consistent withthe 21y8 trend line for FM mode locker powers aboveapproximately 30 mW. Below this region, the pulsewidth decreases much faster with increasing modelocker power. We believe this is because two pulseshortening effects occur simultaneously. One is sim-ply an increase in peak modulation depth that is dueto the increase in mode locker power, and the other isan increase in effective modulation frequency. BothAM and FM mode locking depend on the modulationfrequency according to tp } fm

21y2. When the tran-sition is made from purely AM to FM-dominatedmode locking, the modulation frequency changesfrom 80 MHz to 1.76 GHz, which contributes to therapid reduction in pulse width in the transition re-gion. After this transition, the data follow the 21y8trend line for increasing mode locker power.

The time–bandwidth product ~DnDt! for the short-est pulses was 0.54 6 0.03 or 1.2 times the transformlimit for a Gaussian pulse. Although we were notable to verify whether or not the chirp was linear infrequency, FM mode-locking theory predicts chirpedoutput pulses.6 Also, for pure FM mode locking, thetime–bandwidth product should be =2 times thetransform limit or 0.626. Because we are simulta-neously AM and FM mode locking the laser, a time–bandwidth product that falls between the pure FMand AM limits is reasonable. However, further the-oretical and experimental investigation of this rela-tionship is warranted.

Figure 6 verifies preservation of the fundamental

Fig. 5. Optical pulse width versus FM mode locker drive power ~h5 0.59 rady~W!1y2 single pass!. Deconvolution was applied to thedata because of a 17-ps photodiode impulse response.

80-MHz repetition rate when both mode lockers wereoperating. The optical pulses ~a! as measured witha fast photodiode are shown with ~b! the 40-MHz AMand ~c! the 1.76-GHz FM mode locker drive signals.We can see that the pulses occur every half-cycle ofthe AM drive signal ~characteristic of acousto-opticmode locking! and every 22nd cycle of the FM drivesignal. When the AM mode locker was turned off,the repetition rate increased to 1.76 GHz and thepeak pulse power dropped significantly, as expected.

When both mode lockers are operating, it is possi-ble for one to scan the optical pulse across the oscil-lation window by adjusting the relative phase delay ubetween the FM and the AM mode locker drive sig-nals. Time records of the output pulses as a functionof u are shown in Fig. 7. The relative time delay axisfor the pulses is referenced to the center of the gainwindow, and zero degrees relative phase delay placesone cusp of the FM modulator approximately at thecenter of this window.

As the phase delay of the FM drive signal is in-creased, the FM cusp moves later within the AM gain

Fig. 6. Verification of fundamental pulse repetition rate. a,Mode-locked pulse train ~12.5-ns separationy80 MHz!. b, 40-MHzAM mode locker drive signal. c, 1.76-GHz FM mode locker drivesignal. The signals are displayed with arbitrary relative phases.

Fig. 7. Variation of mode-locked pulses as the FM signal is de-layed with respect to the AM drive signal. This effectively scansthe FM signal through the oscillation window shown in Fig. 3.The relative phase delay is in terms of the AM period ~i.e., u 5 vAMtwhere t is the rf time delay!.

20 August 1997 y Vol. 36, No. 24 y APPLIED OPTICS 5911

window and the optical pulse follows ~i.e., increasingtime delay!. The peak power of the pulse steadilydecreases as it nears the edge of the gain window.When the FM cusp is nearly at the limit of the gainwindow ~;250 ps!, a new pulse appears close to thecenter. Because the FM period is 568 ps, this cor-responds to a new pulse forming under a cusp ofopposite curvature. These pulses were found to beless stable, had less peak power, and also appearedwithin a smaller gain window. When the delay wasfurther increased, this unstable pulse was elimi-nated, and a new stable pulse emerged at the otherend of the gain window beneath a cusp of the originalcurvature. We speculate that the correspondencebetween the sign of the FM phase curvature and themode-locking quality is related to nonuniform axialmode pulling by the closely spaced Nd:YAG lasertransition pair at 1.064 mm.10 Figure 7 also showsthat, for certain phase delays ~e.g., u ' 7°!, two pulsescan exist simultaneously in one AM cycle. The con-dition in Eq. ~7! was evidently violated by a smallamount. However, we verified that these multiplepulses can be eliminated when the gain is lowered~lamp current!, thereby decreasing Du.

4. Conclusion

Simultaneous AM mode locking at the fundamentaland harmonic FM mode locking has been shown to beeffective at decreasing the pulse width and increasingthe peak power of the actively mode-locked Nd:YAGlaser. We have demonstrated that the pulse short-ening follows the classical theory of Kuizenga andSiegman. With as little as 1 W of microwave powerdriving the FM modulator, we were able to achieve afourfold reduction in pulse width. Although the av-erage power output of our laser dropped dramati-cally, we attribute this loss to material effects and notto the fundamental characteristics of this mode-locking technique. Consequently, with a suitableelectro-optic material we should be able to preservethe average laser power and thereby gain a largeincrease in peak power while maintaining the funda-mental repetition rate. This approach as well asother combinations of mode-locking drive frequenciesshould be applicable to enhancing the performance ofother laser systems as well.

5912 APPLIED OPTICS y Vol. 36, No. 24 y 20 August 1997

The authors thank the David and Lucile PackardFoundation, the Advanced Thermionics Research Ini-tiative program of the U.S. Air Force, and the Na-tional Science Foundation for support of thisresearch. This research was undertaken at the De-partment of Electrical Engineering, University ofCalifornia, Los Angeles.

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