February 15, 1999 / Vol. 24, No. 4 / OPTICS LETTERS 229

Self-mode-locked pulsed monomode laser

Marc Brunel, Olivier Emile, Mehdi Alouini, Albert Le Floch, and Fabien Bretenaker

Laboratoire d’Electronique Quantique—Physique des Lasers, Unite Mixte de Recherche du Centre National de la RechercheScientifique 6627, Universite de Rennes I, Campus de Beaulieu, F-35042 Rennes Cedex, France

Received October 5, 1998

We predict the existence of a new pulsed-laser operation regime, when the phases and polarizations of the twocoupled cold-cavity eigenstates of a monomode solid-state laser are taken into account in the derivation of theMaxwell–Bloch equations. This monomode pulsed regime is experimentally observed, without any normalmode locking or Q switching occurring inside the cavity. We obtain close agreement between experimentsand theory, even in the simple case of a Nd:YAG microchip laser, for which sech2 pulses at nearly megahertzrepetition rates are readily observed. 1999 Optical Society of America

OCIS codes: 060.5530, 140.4050, 140.3580, 140.3570.

Because of the large variety of their applications,pulsed-laser operation regimes have long been thesubject of constant investigation. Gain switching, Qswitching, and mode locking have proved to be satisfac-tory methods of generating light pulses.1,2 However,apart from Kerr-lens mode locking, which appears inhighly multimode lasers only,3 pulsed operation is ob-tained at the expense of either modulating the cav-ity parameters4 or introducing extra components, e.g.,saturable absorbers or active Q switches,5 inside thecavity. Monomode lasers in their simplest form—cwpump, active medium, optical resonator—have, to ourbest knowledge, never been shown to yield a stable self-pulsed output. However, solid-state lasers do exhibitresonances in their output intensity spectra because ofthe existence of relaxation oscillations at low frequen-cies, e.g., in the 10-kHz–1-MHz range.6 – 8 Besides,weak coupling of two such lasers is known to leadto nonlinear self-pulsations.9,10 Moreover, the oscilla-tion of modes with tunable frequencies in a solid-statelaser is permitted by the low value of the couplingconstant between two orthogonally linearly polarizedeigenstates.11 One can hence wonder whether twolaser eigenmodes beating in that frequency range couldnonlinearly interact with these resonances frequenciesof the system and then induce interesting temporalbehavior in the output intensity. Our aim in this Let-ter is consequently to explore theoretically and experi-mentally the new pulsed behavior obtainable from asolid-state laser when relaxation oscillations and dual-eigenstate operation are taken into account.

First we consider the longitudinally pumped lasershown schematically in Fig. 1(a). The active mediumis a 1.1-mm-long Nd:YAG crystal. One of its ends,mirror M1, is highly transmitting sT . 95%d at 809 nmand highly ref lecting sR . 99.5%d at 1064 nm. Theresonator is L ­ 385 mm long and is terminated with500-mm radius-of-curvature mirror M2 with trans-mission T ­ 1% at 1064 nm. The pump laser isa fiber-coupled laser diode emitting at 809 nm. A1.5-mm-diameter aperture placed against mirror M2ensures TEM00 operation. A continuously adjustablelinear phase retardance Dw between the x and y polar-ization directions is created by two intracavity quarter-wave plates whose neutral axes are not aligned. Withthis phase anisotropy alone, the laser eigenstates are

0146-9592/99/040229-03$15.00/0

linearly polarized along the x and y axes, with angularfrequencies vx and vy , respectively. Their differenceis vy 2 vx ­ cDwyL, where c is the velocity of light,as shown by the straight lines in Fig. 1(b). A 2-mm-thick uncoated silica etalon forces the two eigenstatesto oscillate in a single longitudinal mode. Owing tothe Fresnel ref lections on both faces of this etalon,whose normal axis is tilted with respect to cavity axisz, a small loss anisotropy Dt' ,, 1 exists. When theetalon’s low-loss axis is oriented at 45± with respectto the x and y axes, a locking region DvL ­ cDt'yLappears [see Fig. 1(b)]. Indeed, for jvy 2 vxj , DvLthe two eigenstates become elliptical, lock to the sameeigenfrequency, and experience different losses. Onthe other hand, for jvy 2 vxj . DvL the two cold-cavityeigenstates are also elliptical, but their angular fre-quencies v1 and v2 are no longer degenerate, lead-ing to

v1 2 v2 ø svy 2 vxd f1 2 DvL2ysvy 2 vxd2g1/2. (1)

Because outside the locking region the two eigen-states have the same losses and because coupling con-stant C between linearly polarized eigenstates in such

Fig. 1. (a) Experimental arrangement. Gain, Nd:YAGcrystal; Dw, x y phase anisotropy; Dt', slightly tiltedetalon. (b) Evolution of the angular eigenfrequenciesv1 and v2 of the cold-cavity eigenstates versus phaseanisotropy Dw. v0 ­ svx 1 vy dy2 is the average laserangular frequency. Notice the locking region DvL ­cDt'yL.

1999 Optical Society of America

230 OPTICS LETTERS / Vol. 24, No. 4 / February 15, 1999

a laser is weak11 sC ­ 0.16d, in this case we expect thetwo eigenstates to oscillate simultaneously. This os-cillation is experimentally verified in Fig. 2(a), whichreproduces the intensities along the x and y directionsand their beat note at v1 2 v2 ø 2.6 MHz. We ver-ify that, because jvy 2 vxj .. DvL, the eigenstates arealmost perfectly x and y linearly polarized, apart fromthe small modulations of their output powers at thebeat frequency. The spectrum of beat note I45 is dis-played in Fig. 2(b). The undamped relaxation oscil-lation at frequency vry2p ­ 65 kHz can be seen. Asexpected from Eq. (1), the adjustment of Dw permitsthe precise and continuous control of the eigenfre-quency difference v1 2 v2.

To fully describe theoretically the temporal behaviorof a vectorial solid-state laser we developed the follow-ing set of six coupled field–atom equations,12 follow-ing Ref. 13, which treated the dynamics of fiber lasers.The phase-sensitive interactions that were previouslyapplied to longitudinal or transverse modes9,14,15 aretaken into account here in the dynamics of both polar-ization eigenstates:ÙIx, y ­ 2Gx, yIx, y 1 DvL

qIxIy cos C 1 2kax, y sD0 6 D1d

3sIx, y 1 ex, y d 1 2ksax, y cos C 6 gax, y sin CdD2

qIxIy ,

(2a)

ÙC ­ vx 2 vy 2 2DvL

≥qIxyIy 1

qIy yIx

¥sin C

1kffaxsD0 1 D1d 2 fay sD0 2 D1dg

2 k

hsax sin C 2 fax cos Cd

qIyyIx

2say sin C 1 fay cos Cdq

IxyIy

iD2 ,

(2b)

ÙD0 ­ gksP0 2 D0d 2 z faxsD0 1 D1dIx 1 ay sD0 2 D1dIy g

2 z fsax 1 aydcos C 1 sfax 2 fayd sin CgD2

qIxIy ,

(2c)

ÙD1 ­ 2gkD1 2 z faxsD1 1 D0y2dIx 1 ay sD1 2 D0y2dIyg ,

(2d)

ÙD2 ­ 2gkD2 2 z saxD2Ix 1 ayD2Iyd 2 z fsax 1 ay dcos C

1sfax 2 fayd sin CgD0y2q

IxIy . (2e)

Ix and Iy are the intensities of the x- and y-polarizedcomponents of the field, C is their relative dephas-ing, Gx and Gy are the cavity decay rates, gk is thepopulation-inversion decay rate, k and z are the field–atom coupling coefficients, ex and ey hold for sponta-neous emission, and P0 is the pumping rate. If v isthe center frequency of the transition and Ts is the life-time of optical coherences (which we eliminate adia-batically13), then aj ­ 1yf1 1 Ts

2sv 2 vj d2g and faj ­Tssv 2 vj daj for j ­ x, y. D0, D1, and D2 are thefirst three angular Fourier components of populationinversion Dsu, td, which depends on the orientationangle u of the dipole moment of the emitting atom with

respect to x. They read as D0std ­ kDsu, tdl, D1std ­kcos 2uDsu, tdl, and D2std ­ ksin 2uDsu, tdl, where k lholds for angular averaging for 0 # u # 2p. As inRef. 13 we neglect higher-order angular Fourier com-ponents of D. Moreover, owing to the position andthe thinness of the active medium with respect to thebeat length of the two eigenstates, we neglect spatialhole-burning effects. Finally, we define the relativeexcitation rate h ­ 4kP0ysGxyax 1 Gyyay d. With theexperimental parameters of Fig. 2(a), we numericallyintegrate Eqs. (2), leading to Figs. 2(c) and 2(d), ingood agreement with the experiment. In particular,the modulations of the intensities Ix and Iy are wellreproduced, and the residual peak of the relaxation os-cillations can be seen in Fig. 2(d).

Let us now turn to the opposite situation, wherejvy 2 vxj , DvL, i.e., when the frequencies of the twoeigenstates of the cold cavity are locked. Then, if thelaser behavior followed the predictions of the cold cavitysummarized in Fig. 1(b), we would expect only one ofthe eigenstates to oscillate continuously, with its polar-ization elliptical and roughly aligned with the low-lossaxis of the loss anisotropy Dt'. However, Eqs. (2) ex-emplify the predominant role of the active medium inthe laser behavior as a result of the long decay timeof the population-inversion components. The inf lu-ence of the relaxation-oscillation resonance frequencyis then illustrated by the predictions of Figs. 3(a)and 3(b) obtained for svy 2 vxdy2p ­ 500 kHz andDvLy2p ­ 1.24 MHz. Figure 3(a) shows that the fre-quency locking of the two eigenstates sk ÙCl ­ 0d leads toa self-pulsing operation of the laser, the pulse’s dura-tion being of the order of 2 ms and its polarization be-ing elliptical. To check this prediction, we further tiltthe intracavity etalon to increase the locking thresholdDvL. For jvy 2 vxj , DvL, we then obtain the ex-perimental results of Fig. 3(c) and 3(d), which confirmthe predictions of a pulsed regime. In particular, thisregime is clearly not due to spiking effects related to

Fig. 2. (a) Experimental time evolution of the laser outputpower observed behind a polarizer oriented along the x axissIxd, at 45± of the x and y axes sI45d, and along the y axis sIy dwhen jvy 2 vxj . DvL. (b) Experimental power spectrumcorresponding to (a). (c), (d) Corresponding theoreticalresults obtained with h ­ 2.0, Ts ­ 80 ps, 1ygk ­ 230 ms,svy 2 vxdy2p ­ 2.6 MHz, DvLy2p ­ 100 kHz, v0y2p ­1 MHz, and Gyy0.971 ­ Gx ­ 3.8 3 107 s21.

February 15, 1999 / Vol. 24, No. 4 / OPTICS LETTERS 231

Fig. 3. (a) Theoretical evolutions of I45 (solid curve) andof phase C (dashed curve) versus time, obtained with thesame parameters as in Fig. 2, except that svy 2 vxdy2p ­500 kHz and DvLy2p ­ 1.24 MHz. (b) Correspondingtheoretical spectrum. (c), (d) Corresponding experimentalresults. Notice the theoretical as well as experimentalincrease of the laser peak power with respect to the resultsof Fig. 2.

Fig. 4. Experimental time evolution of the microchip laseroutput power. Repetition rate, 800 kHz; pulse width,140 ns FWHM.

relaxation oscillations but is a stable pulsed regime.We verify experimentally that the pulses are quasi-circularly polarized and that their duration is 2 ms, asexpected. We further verify experimentally and theo-retically that the handedness of the pulse polarizationfollows the sign of vy 2 vx. It is worth noting that,because the phase difference C between the x- and they-polarized components of the laser field is locked, thisoperation regime can really be called monomode, unlikethe usual pulsed phase locking in multimode lasers.1 – 3

Besides, it is striking to notice that the pulse-repetition rate (45 kHz) lies just below the relaxation-oscillation frequency vry2p ­ 65 kHz. Moreover, weobserve that the repetition rate increases with excita-tion rate h, just like vr , which suggests that lasers withhigher relaxation-oscillation frequencies could leadto higher repetition rates and shorter pulse widths.This is what we now experimentally verify, using asimple monomode microchip laser, i.e., a 540-mm-longNd:YAG piece with directly coated mirrors.16 Theresidual phase anisotropy is controlled by mechanicalstress. We obtain an adjustable effective crossed lossanisotropy Dt' by orienting the linear polarization ofthe pump laser at 645± of the x and y axes and byslightly tilting the chip with respect to the pump laseraxis. In this case h ­ 1.4 and vry2p ­ 930 kHz. Asin the previous experiments, these small anisotropiesare sufficient to yield a pulse train at an 800-kHz repe-tition rate with a pulse width of 140 ns FWHM (see

Fig. 4). Note that these pulses are almost as short asthose emitted by common Q-switched lasers but, on theone hand, at a much higher repetition rate and, on theother hand, with a perfect sech2 temporal profile, asone can determine by fitting both experimental and nu-merical results.

In conclusion, we have put into evidence the exis-tence of a new oscillation regime in quasi-isotropicsolid-state lasers. Indeed, we have seen that the fre-quency locking of two linearly polarized eigenstatesin a solid-state laser leads not to the cw oscillationof a single eigenstate but to the appearance of a self-pulsed monomode regime. The phase-sensitive laserequations derived to treat this problem provide predic-tions in close agreement with the experimental obser-vations. Moreover, we have verified the appearance ofthis behavior in solid-state microchip lasers. This newregime provides a new and simple means to generatehigh-repetition-rate pulses from a compact monomodelaser without any Q-switching process. The univer-sality of the phenomenon suggests, for instance, theability to generate useful eye-safe pulses from erbium-doped solid-state lasers, for which saturable absorbersare still diff icult to design.

The authors thank J. Marty and E. Molva for theirhelp and the Conseil Regional de Bretagne for itssupport.

References

1. A. E. Siegman, Lasers (University Science, Mill Valley,Calif., 1986).

2. M. Sargent III, M. O. Scully, and W. E. Lamb, Jr.,Laser Physics (Addison-Wesley, Reading, Mass., 1974),Chap. 9.

3. B. E. Bouma and J. G. Fujimoto, Opt. Lett. 21, 134(1996).

4. P. D. Drummond, J. D. Harvey, J. M. Dudley, D. B.Hirst, and S. J. Carter, Phys. Rev. Lett. 78, 836 (1997).

5. R. P. M. Green, D. Udaiyan, G. J. Crofts, D. H. Kim,and M. J. Damzen, Phys. Rev. Lett. 77, 3533 (1996).

6. D. E. McCumber, Phys. Rev. 141, 306 (1966).7. P. Mandel, K. Otsuka, J.-Y. Wang, and D. Pieroux,

Phys. Rev. Lett. 76, 2694 (1996).8. A. Hohl, A. Gavrielides, T. Erneux, and V. Kovanis,

Phys. Rev. Lett. 78, 4745 (1997).9. K. S. Thornburg, Jr., M. Moller, R. Roy, T. W. Carr,

R. D. Li, and T. Erneux, Phys. Rev. E 55, 3865 (1997).10. M. Moller, B. Forsmann, and W. Lange, in Interna-

tional Quantum Electronics Conference, Vol. 7 of 1998OSA Technical Digest Series (Optical Society of Amer-ica, Washington, D.C., 1998), p. 117.

11. M. Brunnel, M. Vallet, A. Le Floch, and F. Bretenaker,Appl. Phys. Lett. 70, 2070 (1997).

12. M. Brunel, O. Emile, M. Alouini, A. Le Floch, andF. Bretenaker, ‘‘Experimental and theoretical study oflongitudinally monomode vectorial solid-state lasers,’’Phys. Rev. A (to be published).

13. H. Zeghlache and A. Boulnois, Phys. Rev. A 52, 4229(1995).

14. P. Mandel, C. Etrich, and K. Otsuka, IEEE J. QuantumElectron. 29, 836 (1993).

15. L. A. Lugiato, K. Wang, and Y. Sun, Opt. Commun. 105,397 (1994).

16. J. J. Zayhowski, Opt. Lett. 14, 24 (1989).