single-longitudinal-mode laser as a discrete dynamical system

218 J. Opt. Soc. Am. B/Vol. 2, No. 1/January 1985

Single-longitudinal-mode laser as a discrete dynamicalsystem

F. Hollinger and Chr. Jung

Department of Physics, University of Kaiserslautern, Postfach D-6750 Kaiserslautern, Federal Republic ofGermany

Received May 11, 1984; accepted October 1, 1984A solid-state laser running on a single longitudinal mode can show chaotic time behavior if several transverse modesare excited. The temporal development of the laser is approximated by a Kirchhoff integral together with a rateequation for the inversion. The qualitative structure of the solution of this discrete dynamical system is studiedfor various values of the pump rate by numerical as well as by analytical methods. With increasing pump rate wefind constant, quasi-periodic, and irregular time behavior for the output power.

1. INTRODUCTION

It has been found experimentally that instabilities occur inalmost any laser systems if the output power is increasedabove some critical value. Above this value, it is not possibleto obtain a total output power that is constant in time. Inmany cases, the spatial distribution is not stable, either. Atthe beginning of the investigations of these instabilities, it wasnot clear whether the instabilities were caused by mechanicalor thermal instabilities of the experimental setup or whetherthey were inherent properties of the system. Subsequently,it became clear that instabilities are properties of lasers. Onereason for the occurrence of irregular behavior is the nonlinearinteraction of several longitudinal modes with the longitudinalgain structure of the active medium. Such coupled longitu-dinal-mode behavior has been investigated theoreticallyl- 3

and experimentally.4-7 Irregular behavior can also be causedby the interaction of several transverse modes. These systemshave been considered theoretically, 8 and some experimentalobservations of periodic and chaotic interactions betweenmodes have been reported. 9 -'2 For some recent reports oninstabilities in single-mode systems, see Refs. 13-16.

Our purpose in this paper is to develop a simple mathe-matical model for a single longitudinal, multitransversal-modelaser, which is able to reproduce the transition from stable toirregular behavior with increasing output power. In Section2, we set up the model. Section 3 contains the numerical re-sults. In Section 4, we study the qualitative stability prop-erties of our model by analytical methods. Section 5 presentsconclusions and final remarks.

2. EQUATIONS OF MOTION

We consider only rotationally symmetric resonators and ro-.tationally symmetric field configurations. Let S be a two-dimensional section through the resonator, which is transverseto the direction of propagation of the field. Then theKirchlloff-Fresnel itegral mapg the transverse electric dis-tribution E on S onto the field distribution on S after oneround trip:

E(x, n + 1) = P(x, y, n)E(y, n)d2y,

P(x, y, n) = exp [lN(y, n)]Po(x, y),

(1)

(2)

wher x and y are both two-dimensional coordinates for therunning point on S, n is the discrete time measured in unitsof the round-trip time, o is the cross section for inducedemission, is the length of the active medium, and N(y, n) isthe space- and time-dependent inversion projected onto thesurface S. Po is the propagator for the empty resonator.

We have given Eqs. (1) and (2) in a form that applies di-rectly to a ring laser. For a linear resonator, the field on onemirror would be mapped onto the other mirror by a Kirch-hoff-Fresnel integral of the form of Eq. (1), and, in a secondstep, it would be mapped back onto the first mirror. Thistwo-step map can again be written in the form of an integraltransformation, and it gives the mapping of the field for acomplete round trip in a linear resonator. In any case, weassume that an equation of form (1) gives the development ofthe field for a complete round trip.

The iterated Kirchhoff integral defines a discretized timeevolution of the electric field on S. For each step in thisprocess, we must insert into P the inversion N taken at thetime of this individual step. We assume that the active me-dium is concentrated in a thin slab lying directly beside S. Tocomplete the equations of motion for the total system, we mustadd an equation for the time development of the inversion.We take a rate equation for a four-level system

N(y, n + 1) = (1 - A)N(y, n) + W[N(y) - N(y, n)]

- A N(y, n)IE(y, n)12,ES

(3)

where A is the probability for spontaneous decay during onetime step. W is the pump rate and gives the relative fractionof the unexcited atoms, which become excited during one timestep. No is the density of the active atoms, and E is thesaturation parameter of the active medium. The discretiza-tion of time is a good approximation as long as the round-triptime is small compared with the lifetime of the excited atoms

0740-3224/85/010218-08$02.00 © 1985 Optical Society of America


Vol. 2, No. 1/January 1985/J. Opt. Soc. Am. B 219

and as long as the amplification of the field in one round tripis not much bigger than 1. W is a free parameter, which de-termines the total output power. All other parameters arefixed either by the geometry of the system or by natural con-stants. For varying values of W, Eqs. (1) and (3) togetherform a one-parameter family of discrete dynamical systems.It is an open system with friction, and it is driven from out-side.

Starting from some arbitrary values for E(x) and N(x), wefirst expect to observe some transient motion (relaxation os-cillations); afterward, the system will settle down into somesteady state of long-time behavior. The central question is:What is the qualitative structure of the long-time behaviorfor the various values of W? Are there critical values of Wat which this qualitative structure changes, i.e., are therevalues of W at which bifurcations occur?

In Section 3, we show by numerical computations that bi-furcations do indeed show up. In Section 4, we give a math-ematical classification of the sequence of bifurcations that leadto instabilities in our model.

PUMP RATE (a)

W 5 96 periodicWth emission

a: [

L

_' 0• ~ (b)

W periodic a. a l l a~IMWMEMIWWth= emission D _ WWW1m,

0 054f

O ~ (C)w stable 0 _ __

Wth emission 00O 1.00 ' 1.02

g1=0.5

Fig. 2. Magnified plots of the three

TIME msecl

small squares in Fig. 1.

3. NUMERICAL RESULTS

The numerical calculations are done for the example of alinear resonator with a length of 1 m. For the parameters, weuse the values corresponding to a Nd:glass rod: wavelengthX = 1.06 X 10-4 cm, spontaneous decay time 1/A = 156 Aisec,diffraction index q = 1.8, o = 4.7 X 10-20 cm 2. g's are pa-rameters of the resonator: g = 0.5, 0.4, as indicated in theresults shown below; g2 = 1.0. The length of the active me-dium 1 = 7.62 cm; mirror reflectivities R1 = 1.0, R2 = 0.5; lossesof the resonator without pinholes (caused by scattered lightat the surfaces of the rod) V = 0.83; pinhole radii r1 = 0.3 cm,r2 = 0.085 cm.

For the numerical calculations we add to Eq. (1) a termsimulating spontaneous emission. The amplitude of this termis given by the spontaneous decay term in Eq. (3). The phaseis given by a random generator. The initial values of E andN are identically zero. Because only rotationally symmetricfield configurations are considered, polar coordinates on S are

E-

zLU

z

RADIAL INTENSITY DISTRIBUTION

Fig. 3. Time dependence of the spatial energy distribution in thebeam in the steady state (transients are already damped away).Examples for W/Wth = 2, in which only the TEMOO mode is excited.g = 0.5.

q6 1110I (bt (C)ski vl L

l31 t1 o E0.5 1.0

TIME (rmsec]

Fig. 1. Numerical-model calculations for the total output power asa function of time. Shown are examples for three values of the pumprate W measured in units of the first threshold Wth and g, = 0.5.Magnifications of the three small squares are given in Fig. 2.

used, and the angle integration in Eq. (1) can be done ana-lytically in closed form and gives the Bessel function Jo for P0.For radial integration, the 16-point Gauss procedure is used.For several values of W and two values of g, we have done200,000 iterations.

In this section, W is measured in units of Wth, which is thefirst laser threshold. First, we take the values W = 2 X Wthand gl = 0.5. Figure 1(c) displays the total power outside asa function of the iteration step, i.e., as a function of time.After regular spiking, the intensity converges rapidly to aconstant value. Also, in the magnification of Fig. 1(c), shownin Fig. 2(c), we do not see any fluctuations. Figure 3 shows thespatial distribution of the energy. This plot demonstratesthat only the TEMoo mode is excited. For g1 = 0.4, there

PUMP RATE (a)

Wth

w 3 Wth 30

CL

I-a.I

0

2.

0.1

O.:

0



PUMP RATE

w periodicWth = 596 emission

w 3 periodicWth emission

, = 0.5

(a)

100001000

- 10010

a2 1 . _

I-

zwI-Z

100

10

(b)

0 10 20 30 40 50 60 70FREQUENCY [MHz

Fig. 4. Fourier spectra of the time series shown in Fig. 2 forgl = 0.5.The absolute values of the Fourier coefficients are plotted in a loga-rithmic scale.

I-

zg,0.5

PUP RATE: w 3Wth

PERIODIC EMISSION

/ ~~~~~TEMQ0 + EM10 MODE


Fig. 5. Time dependence of the spatial energy distribution in thebeam in the steady state (transients are already damped away).Example for W/Wth = 3, in which the TEMoo and the TEM10 modesare excited. Each tenth value of the time-discretized numericalcalculation is plotted. g = 0.5.

would be no difference so far; therefore we do not repeat thesefigures for the case of g1 = 0.4.

Next we set W = 3 X Wth and g = 0.5. Figure 1(b) givesthe total power output as function of time. First, we see thesuperposition of two relaxation oscillations. As soon as theserelaxation oscillations are damped away, the value of thepower is confined within a small strip. Figure 2(b) showsmagnification of the fluctuations of the power within this strip.

I-

znwI-z

9, = 0.5

Wh 5.96


Fig. 6. Time dependence of the spatial energy distribution in thebeam in the steady state (transients are already damped away).Example for W/Wth = 5.96, in which at least three modes are excited.Each tenth value of the time-discretized numerical calculation isplotted. g = 0.5.

U)~~~~~~~~~~4

I

PUMP RATE: :3.0Wth

PERIODIC EMISSION

RADIAL INTENSITY DISTRIBUTIONFig. 7. Time dependence of the spatial energy distribution in thebeam in the steady state (transients are already damped away).Example for W/Wth = 3, in which the TEMoo mode and the TEM10mode are excited. Each tenth value of the time-discretized numericalcalculation is plotted. Resonance parameter g = 0.4.



PUMP RATE: w =5.96

CHOTIC EMISSION


Fig. 8. Time dependence of the spatial energy distribution in thebeam in the case of a strange attractor. Example for W/Wth = 5.96and g, = 0.4. Each tenth value of the time-discretized numericalcalculation is plotted. At least three modes are excited.

16PUMP RATE

W0

0

I-3a.

n

4

'I.2.8

C

(a)

k kW A L

shows the time development of the spatial energy distribution.We see a time-dependent interference between the TEMOOand the TEM1 0 modes. Sometimes the ring of the TEM10mode can be seen clearly. Figure 5 (and also Figs. 6-8) hasbeen created by plotting the value of the power density as afunction of the distance from the resonator axis for each tenthiteration.

Next we show the results for W = 3 X Wth and g1 = 0.4.The relaxation oscillations for small times look nearly thesame as for g1 = 0.5 in Fig. 1(b). Therefore we do not showanother plot for g1 = 0.4. Figure 9(b) gives the time depen-dence of the total power in the steady state, and Fig. 10(b)gives the Fourier transformation. Figure 7 gives the timedevelopment of the spatial structure. Again we see a periodicsuperposition of the TEMoo and the TEM1 0 modes, and cor-respondingly the intensity fluctuates with one discrete fre-quency.

As a third example, we show W = 5.96 X Wth for the pumprate. First let us look at the case of g1 = 0.5. Figure 1(a)shows the total output power as a function of time. For smalltimes we see complicated relaxation oscillations, which die outafter 0.5 msec. For longer times the power is again confinedwithin a strip. Figure 2(a) shows a magnification of this strip.The motion seems to be a superposition of several oscillations.The corresponding spectrum is given in Fig. 4(a) and showsthat only some discrete frequencies contribute. Figure 6 givesthe temporal behavior of the spatial energy distribution. Wenote that three modes contribute. Now we keep W = 5.96 XWth and set g1 = 0.4. The time dependence of the total out-put power in the steady state is shown in Figs. 9(a) and 11.Here a significant difference from the case of gi = 0.5 appears:The strip of Fig. 2(a) is broken, and instead an irregular se-quence of fluctuations is created. Correspondingly, theFourier transformation in Fig. 10(a) contains a continuum,which is a clear indication of chaotic motion. Figure 8 givesthe time dependence of the spatial structure. Again, threemodes are excited. The case of g, = 0.5 is exceptional, insofaras the phase shift between adjacent transverse modes is ex-actly 7r/2 in the round trip. Therefore, after four round trips,the total phase shift between any two modes is an integer

PUMP RATE

W chaoticWth =596 emission(b)

1 1g1 1I 111=1111111i!1=11=11 LiNlIiiIW=I=

-.1'�

1.00 1.02TIME msecl

Fig. 9. (a) Strange attractor shown in Fig. 11 in a magnified timescale. (b) Magnified plot of result similar to Fig. l(b) for gl = 0.4.

We see a periodic motion, which is caused by the interferencebetween two oscillations. Figure 4(b) gives the Fouriertransformation of the intensity and shows that the intensityfluctuates with one frequency, which is the frequency differ-ence between the two excited transverse modes. Figure 5

100001000

_ 100, 10

1

-U)

zw 3 periodic

Wth = emission 10010.

g = 0.4

(b)

o 10 20 30 410 50 60 70

FREQUENCY [MHz]

Fig. 10. Fourier spectra of the time series shown in Fig. 9 for g1 =0.4. The absolute values of the Fourier coefficients are plotted in alogarithmic scale.

W chaoticWh emission

w periodicWth emission


I


PUMP RATE W = 5.96Wth

g,= 0.4chaotic emission

1 I, I IIi,0.5 1.0

TIME [msec]

Fig. 11. Numerical-model calculation for the total output power asa function of time. Shown is an example for the pump rate W/Wth= 5.96 and g1 = 0.4.

multiple of 27r. This fact seems to lead to a quasi-periodicbehavior of the field, even in the case in which three modes areexcited and correspondingly the spectrum shows only somediscrete frequencies.

For g = 0.4, the phase shifts are irrational multiples of 7rand do not lie close to any rational number with a small de-nominator. Therefore the case g, = 0.4 is generic. The nu-merical calculations show that, in this case, the system becomeschaotic as soon as three modes are excited. This behavior isconsistent with the Ruelle-Takens mechanism of chaos. Insummary, the computations indicate that the system is genericin the sense of Ruelle and Takens if the phase shift of themodes per round trip takes generic values. In Section 4, weshow by a qualitative analysis how the model fits into theRuelle-Takens picture.

4. QUALITATIVE STABILITY ANALYSIS OFTHE LASER MODEL

In this section, we apply the methods of qualitative dynamicsto Eqs. (1) and (3). The necessary mathematical knowledgecan be found in Refs. 17-22.

Let us assume that the amplification factor is always closeto 1 so that we can make a linear approximation of P in N,i.e.,

P(x, y, n) = Po(x, y)[l + oIN(y, n)]. (4)

Next we decompose E and P into some orthonormal systemof functions on S. It is most convenient to use the eigen-functions of P0 . For simplicity, let us assume that the reso-nator is symmetric, so that Po is also symmetric. Then thisdecomposition does not cause any difficulties. 1 - Vi are theeigenvalues of Po, and fi are the corresponding eigenfunctions.Then we find that

Po(xy) = E (1 - Vi)fi(x)fi(A

E(x, n) = F e(n)fi(x).

(5)

(6)

ei are the time-dependent decomposition coefficients of thefield on S. f (x) describes the spatial structure of the modei.

For all practical purposes, it is necessary to truncate thedecompositions and to keep only a finite number of terms.Accordingly, the inversion is also decomposed into a finite sumof, say, J terms as follows. We cut S into J pieces Sj, choosein each Sj a representative point q, denote by Nj(n) thetime-dependent number of inverted. atoms lying in Sj, andapproximate N(x, n) by

This approximation does not mean that all atoms sitting inSj are assumed to be concentrated in the one point qj. It onlyhas the effect that the coupling of all atoms in Sj to the variousfield modes is approximated by the value that holds exactlyfor the atom sitting in point q. Noj is the total number ofactive atoms lying in Sj. Using the abbreviations

and inserting Eqs. (5)-(9) into Eqs. (1) and (3), we obtain thefollowing equations of motion for the coefficients ei and Nj:

-E Bjk1Nj(n)1Ek(n)Tl(n).,k,1

For some preliminary considerations, let us look first at thesimple case in which we keep only one mode of the field andone term in Eq. (7). Remembering that is complex and Nis real, we see that the phase space is real three-dimensionalin this case. As coordinates in the phase space we use , , andN. The equations of motion in these three coordinates arethen

We shall also need the abbreviations = I G and V = 1 -- VI. In general V R, and then Eqs. (12) have only onefixed point F at

The eigenvalues of the linearization of Eqs. (12) at F are

1 = - V + GNF,

X2 = Xl,

X3 = 1 - A - W.

F is asymptotically stable if I XvI < 1 for i = 1, 2, 3:

VAIX1 = IX21 < 1 if W< - = Wth, (14)GN - V

1X31 < if W < 2 - A. (15)

The last condition is physically irrelevant for the following

f 103

0

a. 5

0

0

JN(x, n) = E Nj(n)(x -qj).

j=1(7)

Gijk = (1 - Vj)alfj(qj)fk(qj),

Bjkl = fk (qj)fl (qj)

(8)

(9)

ei(n + 1) = (1 - Vi)Ei(n) + E; GijkNEk,k,j

Nj(n + 1) = (1 -A)Nj(n) + W[Noj-Nj(n)]

(10)

(11)

E(n + 1) = (1 - V)e(n) + GN(n)e(n),

E(n + 1) = (1 - V)T(n) + UN(n)T(n),

N(n + 1) = W[No - N(n)] + (1 - A)N(n)- BN(n)E(n)e(n).

(12a)

(12b)

(12c)

EF = , eF = , NF = NWNFW+a (13)



)

Unit CircleFig. 12. Position in the complex plane of the eigenvalues Xi of thefixed point F' as a function of the pump rate W. At W = Wth, thecomplex-conjugate pair X, X2 crosses the unit circle. This causes aHopf bifurcation.

reason: W is the relative fraction of 'unexcited atoms thatbecome excited by the pump light during one time step.Therefore the discretization of time in Eq. (3) is no longerapplicable as soon as W becomes of the order of 1. So thewhole model makes sense only for small W.

The threshold value in Eq. (14) exists if No > V/G, i.e., thelaser can start to operate only if the active medium exceedsthe minimal size VIG. At W = Wth, the complex-conjugatepair of eigenvalues X1 and X2 crosses the unit circle (Fig. 12).This leads to a Hopf bifurcation in which the fixed point Fbecomes unstable and the following periodic solution of Eq.(12) is created:

Np(n) = V/G, (16a)

e(n) = r X einp, (16b)

where eiP = phase of (1 - V) = phase of G:

r2 =/NB (W-Wth). (17)

By a transition to a rotating coordinate system, we can removethe phase eiP, and then the periodic solution of the three-dimensional system becomes a fixed point of a two-dimen-sional system. This construction is equivalent to the Poincar6map of the periodic solution.1 2 The eigenvalues of thePoincare map of Eq. (16) are

1,2 = 1 i[ -2(WGNo-AV-WV)X,1 2V [4V2 J

(18)

For large W, X2 runs through -1. This period doubling isagain irrelevant for the reasons that were discussed in con-nection with Eq. (15). For small W, X11 < 1 and I X21 < 1 ifNOWG - AV - WV > 0, i.e., if W > Wth. Accordingly, pe-riodic solution (16) is asymptotically stable.

Therefore the single-mode system shows the following be-havior: For W < Wth, the fixed point F with E = 0 is stable,and there is no lasing action in the stationary case. F attractsall other points in the phase space, and, after perturbation (forexample, spontaneous emission), the system always runs backto F. For W > Wth, the fixed point F is unstable, and a stableperiodic solution is created in which Iel is constant and the

phase of e moves around with constant speed. Therefore, forthe one-mode laser above threshold, there exists a state withconstant output power that attracts all other points in thephase space. After perturbation the system always runs backto this periodic trajectory. If we keep W > Wth fixed and startthe laser in a state with N 0 0 and I El - 0, then in the two-dimensional section of the phase space, in which the phase ofE is projected away, the system spirals toward the stable stateN = Np, El = T(W) given in Eqs. (16) and (17) (see Fig. 13).This spiral motion is a relaxation oscillation, which representsregular spiking.

Next we consider the case of several modes. As a prelimi-nary special case, let us take the case in which we have J modesof the field and J terms in Eq. (7) and in which the variousmodes are decoupled by the fact that fj(qj) = 61j, i.e., thevarious modes are amplified by disjoint parts of the activemedium. Then Gijl [Eq. (8)] and Bikl [Eq. (9)] are completelydiagonal in all three indices. Then the J-mode problem is justthe Cartesian product of J copies of the one-mode case. Toavoid degeneracies, we assume that all J thresholds Wthj aredifferent and are ordered by increasing values, i.e., Wth,1 <Wth,2 < ... < Wih,J-

The phase space is real 3J dimensional. For 0 < W < Wth,1the state in which all ej = 0 is an asymptotically stable fixedpoint. At each threshold WthJ there is a Hopf bifurcation inthe subspace Pj spanned by Nj, Ej, and e,. In this bifurcation,a stable invariant closed curve yj is created in this subspacePj. The projection of the motion of the total system onto thesubspace Pj moves periodically around this curve yj. In thetotal phase space, the Cartesian product yi X 72 X ... X Ykof the curves Y1, 1 < i k forms a k-dimensional torus. ForWth,k < W < Wth,k+1 the arbitrary point of the phase spaceis attracted by this torus, and asymptotically all trajectoriesmove around this torus quasi-periodically. What happensif the restriction is dropped that G and B be diagonal? Thenwe may consider the off-diagonal parts as perturbation of thediagonal case, and the motion of the system in the phase spaceis modified in the following way:

The numerical value of the threshold Wth,1 is changedslightly to a new value Wth,1, and in the interval (0, Wth,1) the

/±

r 1I :

Fig. 13. For fixed W, a trajectory of the system is displayed, whichstarts close to the origin. If time runs continuously, the system spiralscontinuously from close to the origin to the point (Na, r). If time isdiscretized, then the system jumps from one cross to the next one inone time step. Only a few crosses are shown. This spiral motionrepresents the regular spiking of a one-mode laser. The point (Np, r)represents the stationary state with constant output power.

-


N


point ej = 0, N = Noj X W/(W + A), 1 < j < J is still anasymptotically stable fixed point. (Asymptotically stablefixed points are structurally stable.) At Wth,l, a supercriticalHopf bifurcation occurs, in which a periodic solution is createdthat attracts the other orbits in phase space (an asymptoticallystable limit cycle is structurally stable). The periodic solutionneed not lie exactly in the subspace Pi in the coupled case. Itmay be deformed into other directions. The numericalcomputation indicates that the motion of the field is still suchthat the total output power is constant and only the phasechanges. If the coupling of the first mode to all parts of theactive medium is strong enough, the first mode can depopulateall inversion, so that there is no inversion left over to exciteany other mode. If the coupling is smaller, then enough in-version is left so that, at a second threshold Wth,2, a secondmode can become excited. Above Wth,2, the system movesagain quasi-periodically on T2 , which need not lie exactly inP2 (D P2 in the coupled case. Other parts of the phase spaceare attracted by this T2. Because the motion of the overallphase of the field is irrelevant to the total power, only periodicmotion (the motion of the relative phase between the twoexcited modes) enters into the total power. Accordingly, weobserve one frequency in the spectra in Figs. 4(b) and 10(b).If the first and second modes do not depopulate all inversion,there is a third threshold Wth,3, at which something new mayhappen. In the decoupled case, we had a quasi-periodic mo-tion on T3. However, in Ref. 22, it was shown that quasi-periodic motion on Tk, k _ 3, is not stable. In the genericcase, perturbations create a strange attractor. Then, in thecase of three excited coupled modes, there is still a lower di-mensional subset of the phase space, which attracts nearbyparts of the phase space, but on this attractor itself chaoticmotion occurs. The generic point in the phase space movestoward the strange attractor and shows irregular motion onit. For more than three excited modes, the same type of cha-otic motion occurs. Only in some exceptional cases, which arenot generic in the sense of Ruelle and Takens, quasi-periodicmotion on a Tk, k _ 3, can persist. The numerical compu-tations indicate that g = 0.5 is such an exceptional case.Therefore, our mathematical model can explain the type ofbehavior that has been found in the numerical computationsand that has been suggested by experimental results.

5. CONCLUSIONS

In this paper, we have given a mathematical model for a laserthat can operate on a single longitudinal mode and on severaltransverse modes. The fact that the laser runs on one longi-tudinal mode only makes it possible to discretize time and touse the Kirchhoff integral for the description of the time de-velopment of the electrical field on a two-dimensional surfaceS. The predictions of this model agree with the experimentalresults (transient relaxation oscillations). Therefore ourmodel seems to be justified. It is an interesting example fora dynamical system in which instabilities are created by theRuelle-Takens mechanism.

What can we learn from our results for the design of stablehigh-power lasers? For applications, people are interestedin a constant output power and in constant spatial distributionof the energy in the beam. Our results show that these re-quirements can be met only if the laser runs on one mode only,on a single longitudinal mode and on a single transversal

mode. In addition, the single-mode operation has the ad-vantage that the divergence angle of the beam is smallest.The upper limit for the output power at which single-modeoperation occurs is determined by the second threshold Wth,2for the pump rate. Therefore a stable high-power laser shouldbe designed in such a way that Wth,2 is as high as possible. Anecessary condition is that the active medium should be de-populated rather completely by the TEM6 mode. Thereforethe active medium must be placed inside the space in whichthe TEMoo mode is strong. For high power and for good ef-ficiency we want to use a big active medium, and accordinglywe need a resonator configuration in which the TEMoo modeoverlaps a big volume (for example a concave-convex reso-nator). In addition, we can increase the threshold for highermodes by placing appropriate apertures in the resonator.

The following questions for experimentalists remain: Isit possible to realize experimentally the case in which thephase shift per round trip between the various modes is amultiple of 7r/2? And is the total output power quasi-periodicand not chaotic in this case in the experiment too, even if threeor more transversal modes are excited? This would be a newexperimental realization of an exceptional situation, in whichthe Ruelle-Takens mechanism does not apply.

For the theoretical investigations the following questionsremain: How close must g, be to 0.5, or how close must thephase shift be to 7r/2 in order that the three-mode case bequasi-periodic? Are there other values for g, or for the phaseshifts for which the three-mode case is quasi-periodic?(Possibly there are some other angles, which are rationalfractions of r with some small denominator.) We hope to givesome answers to these questions in future work.

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single-longitudinal-mode laser as a discrete dynamical system

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