PHYSICAL REVIEW A VOLUME 38, NUMBER 11

Frequency loc»ag of xylo transverse optical modes of a laser

DECEMBER 1, 1988

Chr. TammPhysikalisch Tec-hnische Bundesanstalt, D-3300 Braunschweig Berlin, Federal &epuhlic of germany

The interaction of nearly degenerate TEMp& and TEMlp modes of a laser is investigated.When the difference between the oscillation frequencies of the two modes is decreased, a transi-tion occurs from a nearly sinusoidal "mode-beating" intensity modulation via periodic and irregu-lar pulsing to frequency-locked steady-state emission in a TEMpl hybrid mode. The formation ofthis hybrid mode by frequency locking is a simple example of the recently predicted emergence ofstable transverse patterns in nonlinear optical resonators, brought about by frequency locking ofnear-degenerate, competing transverse modes.

Recently, instabilities forming or changing stable trans-verse radiation patterns were predicted for passive' andactive23 nonlinear optical resonators. We report on asimple experiment in which a stable spatial pattern isformed by "cooperative frequency locking" 2 of the nearlydegenerate TEMO| and TEMio modes of a helium-neonlaser.

Although there have been early observations of fre-quency locking of transverse modes in multiple-transverse- and multiple-longitudinal-mode lasers, thisstate of operation seems to have been regarded as ratheratypical up to now. Instead, particularly for the case thattransverse laser modes are only weakly coupled by an in-homogeneously broadened gain medium, it is more com-mon to find that the laser-output intensity contains oscil-lating "mode-beat" components whose frequencies can berelated in a simple manner to frequency differences be-tween pairs of transverse modes. 6 A very different situa-tion arises if the frequencies of different oscillating trans-verse modes lie within the homogeneous linewidth of thegain medium, and if the modes have sufficient spatialoverlap to be strongly coupled by competition for theavailable inversion: Recent experiments with lasers oscil-lating in three or more transverse modes have shown thatunder these conditions, mode interaction can lead toquasiperiodic, intermittent, and chaotic emission. 7

The experiment reported here demonstrates how thenonlinear interaction of two nearly degenerate transversemodes TEMO| and TEMio in a single-longitudinal-modeHe-Ne laser changes the spectral composition and shapeof the mode-beat signal if the oscillation frequencydifference of the two modes is tuned towards zero. It isobserved that under these conditions, the time dependenceof the laser-output intensity changes qualitatively fromnearly sinusoidal mode beating via regular and irregularpulsing of the mode intensities to a state where the opticalphases of the two modes are locked to a constantdifference of ~ tt/2 to yield a stationary TEMO| hybridmode" pattern. It may be noted that the commontheoretical models on mode interaction in inhomogeneous-ly broadened gas lasers'2'3 do not permit a suitabledescription of this case since here, the transverse field dis-tribution of different modes is not equal but ortho onal.However, both chaotic two-transverse-mode lasing' and,

particularly interesting, the formation of stable stationaryspatial patterns that arise from the coexistence offrequency-locked transverse modes have been treated re-cently. 2'3

In order to relate the theoretical model of Refs. 2 and 3more closely to our experimental situation, the coupled-mode equations given in Ref. 2 were reformulated to de-scribe the modal geometry discussed here. The resultingequations describe a homogeneously broadened traveling-wave laser with circular-symmetric pumping that oscil-lates in the two first-order transverse modes. The equa-tions were solved numerically to study the temporal evolu-tion of the amplitudes and optical phases of the modes fordifferent parameter sets. In particular, the two cases of a"bad-cavity" laser (yi y& x'/3, where yi is the popula-tion decay rate, y~ is the polarization decay rate, and x' isthe cavity field decay rate) and of a laser with slow inver-sion relaxation (yt y~/loo-n'/l0) were studied for con-ditions of symmetrical tuning and weak pumping (pumpparameter two times larger than threshold value). Themain results of the calculations may be summarized asfollows.

(a) For both parameter combinations, there exist criti-cal values for the difference

~b

~of the empty-cavity mode

eigenfrequencies below which both modes oscillate withequal amplitude and frequency, maintaining a constantphase difference close to + tr/2 (stable steady-stateTEMoi operation). For

~ b~ above the critical value, themodes are not phase locked and show temporal variationsof frequency and amplitude due to their mutual interac-tion.

(b) For the bad-cavity parameters, decreasing ( b ( tothe critical value results in a continuous transition fromsynchronous oscillation of the mode amplitudes at 2vs(vsis the mode beat frequency) to synchronous regular puls-

ing; during each pulse, the phase difference between themodes changes by ~ tr.

(c) For the "slow-inversion" parameters, approachingthe frequency-locked state by decreasing ) b ( leads to aperiod doubling of the mode-amplitude modulation at2vtt. In this bifurcation, the synchronicity of the mode-amplitude variations is lost. A further small decrease of 8'

results in irregular, unsynchronized pulsing of the modeamplitudes. The period-doubled and chaotic solutions

5960 1988 The American Physical Society

FREQUENCY LOCKING OF TWO TRANSVERSE OPTICAL. . . 5961

coexist for a certain interval of~b

~with stable steady-

state solutions that correspond to TEMot operation.Small deviations from symmetry in the coupled-mode

model considered, e.g., by asymmetric tuning or inhomo-geneous pumping, did not qualitatively change the resultsof the calculations.

The experimental setup consisted of a 22-cm long com-mercial He-Ne laser tube terminated by Brewster win-dows (Melles-Griot type 05-LHB-290). The linear,near-confocal laser resonator had a length of 25 cm andused mirrors with a radius of curvature of 60 cm and areflectivity of 99.2% at X 633 nm. The astigmatism ofthe —1.5-mm thick Brewster windows was calculated'5 tolead to a difference in resonance frequency for the TEMp&and TEMto modes of ( b ( 0.6 MHz for an orientationof the nodal planes of the modes parallel and perpendicu-lar, respectively, to the plane of incidence of the Brewsterwindows. The mode eigenfrequencies can, however, be-come degenerate (8 0) if the nodal planes of the modesare rotated by 45' from this position. Qne output beam ofthe laser was used to determine visually the time-averagedtransverse-mode pattern of the laser. A section of the oth-er output beam illuminated a fast photodetector to allowfor generation of transverse-mode-beating signaL

With a discharge current about four times larger thanthe threshold current, the laser operated in a mixture ofTEMoo, TEMp~, and TEM~o modes that were linearly po-larized along a common axis defined by the Brewster win-dows. For the experiments described below a resonatorlength was chosen at which only one longitudinal modewas oscillating (oscillation in two longitudinal modes didnot seem to significantly affect the observed behavior).

The transverse-mode structure of the laser and the fre-quency separation ) b ( was controlled by a dot of absorb-ing material (approximate size, 50 pm) on the surface ofan intracavity Brewster plate which could be positionedprecisely with respect to the resonator axis by means of anx-y translation stage. Suitable positioning of the absorb-ing dot suppressed oscillation in the TEMoo mode. Smalltranslations of some tens of micrometers also reproduciblypermitted us to define the losses and, together with theeffect of the slightly noncentral aperture of the laser tubebore, the orientation of the modal planes of the first-ordertransverse modes. In this way h could be varied throughzero while maintaining the time-averaged intensities ofboth modes approximately equal.

Figure 1 shows the output-intensity pattern of the laserfor oscillation in a single first-order transverse mode [Fig.1(a)] and for approximately equally strong TEMot andTEMto modes [Fig. 1(b)]. For the recording of the datashown below, the dot absorber was first positioned to ob-tain the "doughnut" pattern of Fig. 1(b) which usually re-sulted in a mode-beat frequency of vn 1-2 MHz. Thedot was then moved in small steps of a few pm along anapproximately straight line in a direction that led to a de-crease of vn but left the visual shape of the intensity pat-tern unchanged.

The radio-frequency (rf) intensity spectra obtained fora decreasing fundamental mode-beat frequency vn areshown in Fig. 2. It can be seen that the decrease of vgfirst leads to an increase in the amplitude of harmonics

(a)

FIG. 1. Photographs of the transverse intensity distributionof the helium-neon laser for the cases (a) of single first-ordertransverse-mode oscillation and (b) oscillation in the two or-thogonal Srst-order transverse modes. The stripes visible at thebottom of the patterns result from interference caused by spuri-ous re8ection off the back side of the outcoupling mirror.

[Fig. 2 (b)] and, then, to a period-doubling bifurcation[Fig. 2(c)]; after at least one more observable period dou-bling (not shown for brevity), the spectrum becomesbroad and quite featureless [Fig. 2(d)]. Finally, there is asudden transition to stationary (frequency-locked) two-mode operation [Fig. 2(e)]. It was not possible to deter-mine the value of )8( below which frequency lockingoccurs; it can only be noted that the frequency-lockedstate could be reached reproducibly and that it remainedstable for virtually infinite periods (several hours) in spiteof drifts and noise in the experimental setup.

Figure 3 shows the time dependence of the detector sig-nal under conditions similar to those of Fig. 2. In agree-ment with what can be inferred from the spectra shown inFig. 2, it turns out that for decreasing vn, the mode-beat

5962 CHR. TAMM

Ca)

1.0

Cai+,

0

C

Cl

~ o~0

Cb3

IIÃ4

?LLI

I

Z

N

O I0 05FREQUENCV CNHzl

FIG. 2. rf intensity spectra of the laser emission when thelaser simultaneously oscillates in the two first-order transversemodes. Spectra (a) - (e) are recorded when the mode-beatingfrequency is successively reduced towards frequency-locked, sta-tionary operation (e) (see text). Please note the different fre-quency scales as indicated.

signal evolves from a nearly sinusoidal shape [Fig. 3(a)]into a pulse train [Fig. 3(b) and 3(c)] which becomes ir-regular with respect to the pulse heights [Fig. 3(d)l in theregion where the corresponding spectrum is broad.Different pulse shapes of the detector signal could be ob-served for the case that the detector was scanned acrossthe transverse-beam profile of the laser and for the casethat the integral laser power was focused on the activearea of the photodetector. In all cases, however, the basicspectral and pulse-shape features remained unchanged.

An interpretation of the experimental findings maystart from the observation (see Fig. 2) that the harmoniccontent of the mode-beat signal strongly increases if itsfundamental frequency vp is reduced: Under the condi-tion that vg becomes comparable to or smaller than thehomogeneous linewidth of the gain medium, the two oscil-lating modes share a large part of the inhomogeneouslybroadened population inversion. On the one hand, this sit-uation is known to lead to an increase of the effectivemode frequency separation ("mode pushing"' ); in fact,the observed maximum mode-beat frequency (vs —1-2

TINE Kg 83FIG. 3. Time dependence of the laser emission when the laser

simultaneously oscillates in the two first-order transverse modes,showing a transition from mode-beating-type intensity modula-tion (a) via regular self-pulsing (b), (c) to "chaotic" pulsing (d)when the mode-beating frequency is successively reduced.Please note the different time scales as indicated.

MHz) is significantly larger than the calculated empty-cavity frequency spacing of

~ 8( 0.6 MHz. On the otherhand, interference of the modal fields in the nonlineargain medium also implies that gain and refractive index ofthe active medium become time dependent. More con-cretely, the theoretical model mentioned above predictsthat the intensities and the optical phases of the modes aremod'ulated with nvg (n even), while odd harmonics of vgare not excited as a result of the orthogonality of themodal field distributions. Difference-frequency mixing ofthe various spectral components at the photodetector leadsto the observed harmonics nva (n 2, 3,4. . .) of theintermode-beat signal. It is reasonable to suppose that theformation of harmonics is most effective for those har-monics where nva is small as compared to the relaxationrates of the gain medium and the linewidth of the resona-tor.

The time dependence of the detector signal (Fig. 3)reflects the intensity variations of a part of the mode in-terference pattern as the phase difference between themodal fields evolves according to both their average oscil-lation frequency difference and the imposed phase modu-lation. The results of the numerical studies suggest to re-late the distinct sawtoothlike asymmetries of the pulsesthat develop for vg 0 to the effect of a phase-locking

FREQUENCY LOCKING OF TWO TRANSVERSE OPTICAL. . . 5963

"force" that tends to keep the system close to states of acertain constant phase difference between the modes [thetwo low slope regions per period in Fig. 3(c)] while otherphase differences are more unstable (high slope regions).This picture, however, is not helpful in establishing a sim-ple dynamical model for the observed period-doublingtransition to irregular, "chaotic" pulsing [Fig. 3(d)].

It may be noted that the theoretical model consideredabove differs from the experimental situation in that itneglects inhomogeneous broadening, the standing-wavestructure of the field and the supposedly complicated re-laxation dynamics of the lasing transition. Nonetheless,the main observed phenomena, particularly the self puls-ing, the locking, and the chaotic dynamics close to thelocking point are also found in the model.

It is perhaps the most interesting aspect of the experi-ment that it reveals one possible route towards the forma-tion of a stable, stationary transverse structure formed bycoexisting, frequency-locked transverse cavity modes [seeFigs. 1(b) and 2(e)1. For other modal geometries2'3 thecalculated transverse patterns seem to refiect the interplayof self-focusing and local-gain saturation in the laser

medium. In the case considered here, an intensity distri-bution of circular symmetry is formed that consists of thetwo first-order transverse modes locked to a phasedifference of ~x/2; this field configuration obviouslymakes best use of the population inversion that is avail-able for the two modes in the gain medium.

Finally, it should be mentioned that like the patternsdiscussed in Ref. 2, the case of TEMoi operation also cor-responds to a state of broken spatial symmetry since theequiphase surface of the optical field of the TEMoiconfiguration must be thought to wind around the opticalaxis in an either left- or right-handed spiral, correspond-ing to the two possible phase-locking angles of the constit-uent modes of +x/2 or —x/2. Transitions between thesetwo stable states will leave the intensity pattern un-changed; however, they should be easily detectable inter-ferometrically. 's

The author gratefully acknowledges stimulating discus-sions with L. A. Lugiato and C. Q. Weiss. This work wassupported by the Deutsche Forschungsgemeinschaft.

'L. A. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209(1987).

2L. A. Lugiato, C. Oldano, and L. M. Narducci, J. Opt. Soc.Am. B 5, 879 (1988).

L. A. Lugiato, G. -L. Oppo, M. A. Pernigo, J. R. Tredicce, L.M. Narducci, and D. K. Bandy, Opt. Commun. 68, 63(1988).

4D. H. Auston, IEEE J. Quantum Electron. QE-4, 471 (1968).sY. Watanabe, T. Fujioka, and M. Kobayashi, IEEE J. Quan-

tum Electron. QE-4, 880 (1968).sJ. P. Goldsborough, Appl. Opt. 3, 267 (1964); T. Uchida, ibid

4, 129 (1964).7R. Hauck, F. Hollinger, and H. Weber, Opt. Commun. 47, 141

(1983).

C. O. Weiss, A. Godone, and A. Olafsson, Phys. Rev. A 28, 892(1983); see also N. J. Halas, S. -N. Liu, and N. B. Abraham,Phys. Rev. A 2$, 2915 (1983).

9D. J. Biswas and R. G. Harrison, Phys. Rev. A 32, 3835(1985); D. J. Biswas, Vas Dev, and U. K. Chatterjee, Phys.Rev. A 3$, 555 (1988).

'OW. Klische and C. O. Weiss (unpublished).i i W. W. Rigrod, Appl. Phys. Lett 2, 51 (1963).'2W. R. Bennet, Jr., Phys. Rev. 126, 580 (1962).'sW. E. Lamb, Jr., Phys. Rev. 134, A1429 (1964).'4M. L. Shih and P. W. Milonni, Opt. Commun. 49, 155 (1984).'sD. C. Hanna, IEEE J. Quantum Electron. QE-5, 483 (1969).' Chr. Tamm (unpublished).