Instabilities in a Simple Model of a Laser with a Fast Saturable Absorber

JOURNAL OF MODERN OPTICS, 1988, VOL . 35, NO. 8, 1 281-1296

Instabilities in a simple model of a laser with a fast saturableabsorber

L . A. ZENTENOf , K . E. ALDOUS and G. H . C. NEW

Laser Optics Group, Department of Physics, Imperial College,London SW7 2BZ, England

(Received 11 January 1988 ; revision received 5 February 1988)

Abstract . We study the effect of adding a fast saturable absorber to the single-mode Maxwell-Bloch equations which are otherwise isomorphic to the Lorenzequations . The modified system, which includes a nonlinear loss term in thefield equation, exhibits both the Lorenz instability and the Q-switching insta-bility within a single framework . We show that adding the saturable absorberlowers the threshold for the Lorenz instability, making it more accessible toexperimental observation .

1 . IntroductionThere has been intense interest recently among experimentalists in making laser

systems to test the results of semi-classical single-mode laser theory as expressed bythe Maxwell-Bloch equations . The equations are isomorphic to the well studiedLorenz model [1-5] and predict unstable and chaotic motion under certainconditions .

Attempts have been unsuccessful so far because of the severe experimentalconstraints imposed on a potentially feasible laser system by the assumptionsunderlying the dynamical model, namely a single-mode homogeneously broadenedline, relaxation times of comparable magnitude for all three field variables and theplane-wave approximation. However, even if these conditions could be met inpractice, it might still be difficult to find a high gain laser than would operate under`bad cavity' conditions, where the rate of extraction of photons K exceeds the rate ofatomic emission processes [1] . Furthermore, the laser would have to be operated farabove threshold before instabilities could be observed .

The dynamics of lasers with saturable absorbers have been studied extensivelyover many years . Earlier papers were mostly concerned with passive mode-lockingthrough a multimode instability; Garside and Lim [6] analysed the case where boththe amplifying and absorbing media were treated in the full semi-classical approxi-mation, while New and Rea [7] studied the simpler case where the rate equationapproximation was made. However, the latter paper also identified the condition forsingle-mode instability and demonstrated limit cycle behaviour (repetitiveQ-switching) when these conditions were fulfilled . This instability occurs for verylow pumping, in the case where the absorber saturates more easily than the ampli-fier. The leading and trailing edges of each pulse are formed respectively by theQ-switching effect of the absorber and the saturation of the gain .

Tachikawa et al . [8-9] have developed a comprehensive theory of lasers withsaturable absorbers, that includes (albeit in the rate-equation approximation) the

t Currently with Polaroid Corporation, Cambridge, Massachusetts 02139, U .S .A .

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L. A . Zenteno et al .

effects of three vibrational levels in the gain medium and gives results that are inexcellent agreement with experimental work on passive Q-switching .

In the present paper (see also [10]) on the other hand, the approach is different .We consider the dynamics of the Lorenz equations as modified by the presence ofthe simplest possible saturation term, representing an absorber in the rate-equationapproximation and the fast (or 'inertialess') limit . As might be expected, this modelincludes both the fundamental (Lorenz-type) instability and the Q-switchinginstability within a single framework . We investigate whether the same effects thatcause periodic Q-switching can produce aperiodic pulsing (chaos) for laser para-meters that can be realised in practice . We find that the fast absorber promoteschaotic behaviour of the Lorenz type at pumping levels that are lower, andtherefore more accessible experimentally, than those required by the unmodifiedLorenz equations .

2 . ModelThe single-mode Maxwell-Bloch equations for the case of perfect tuning can be

modified to represent a laser with a saturable absorber (LSA) as

where barred quantities refer to the absorber . In equations (1)-(5), the saturationparameter a, determines whether the absorber saturates more easily (a > 1) or lesseasily (a<1) than the amplifier ; E, P, and D are respectively the slowly-varyingelectric field, polarizations and population differences normalized to the stationarystate (E and P are assumed to be real and D is real in any case) ; A is a measure of thepumping relative to the first (or lasing) threshold (A=0); A=D„D./D, is the amplifierunsaturated inversion normalized with respect to the first threshold in a laserwithout absorber; and y l and y 11 are respectively the transverse and longitudinalrelaxation rates . In previous work on the semi-classical laser theory, 2 has normallybeen regarded as the pumping parameter . However, the pumping power requiredto reach the first threshold increases with the saturable absorber concentration, andA, which is directly related to the power supplied to a real laser, is therefore a moreappropriate control parameter than A in the present context .

Steady-state solutions of equations (1)-(5) are the `off' state

where the only permitted solutions are real non-negative roots and we haveintroduced the parameter C=1-A according to a popular convention . Note that

dE/dt+KE=KAP/(1 +A)+KAP/(1 +a2), (1)

dP/dt+ylP=y1DE, (2)dD/dt+yIID=y11(1 +2)-y1IAPE, (3)dP/dt+y1P=y1DE, (4)

dD/dt+311D=311(1 +a2)-T 11 a)1PE, (5)

2=0 (6)and

1-A/(1 +A)-A/(1 +al)=0 . (7)Equation (7) has the solutions

Af={[-(1-A)-C/a]±J{ [(1-A)+C/a]2-4(C-A)/a}}/2, (8)

Instabilities in a simple laser model

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these solutions are independent of any adiabatic elimination scheme that may beapplied to equations (1)-(5), since the steady state will not depend on relaxationrates .

'Examination of equations (6)-(8) shows that there are two different regimes of

laser operation (see figure 1) :

a=a

A+ A=C-

A2 A(b)

Figure 1 . Schematic representation of the stationary states of an LSA for(a) 1 < a < QC -1) (for 0 < a < 1 a similar graph is obtained but the curvature of theupper branch is negative), (b) a > C/(C-1) . Solid (dashed) lines denote stable(unstable) states . First and second thresholds are denoted by A, and A2 respectively .

For a < QC -1), the LSA goes continuously from the off-state to c .w. operation atA=C. Lugiato et al. [11] showed that at this first (lasing) threshold, the off-statebecomes non-stable and the A + state becomes stable .

For a > C/(C-1), the LSA jumps abruptly from the off-state to the on-state atA = C . A range of values of A exists for which three equilibrium states coexist, butA_ is always non-stable [11] . In this paper, the emphasis is on the stability of theupper A + branch .

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The laser equations can be simplified by adiabatic elimination . In general, thismay lead to a loss of information and the disappearance of some instabilities withinthe model . In particular, if the rate-equation approximation is used (y1, Y1 > K, Y it >Y II ), the cavity is necessarily `good' (K <y1 + y II) and the Lorenz-type of instabilitywill not occur . Instead, in this paper, we make the rate- equation approximation forthe absorber only, and in addition assume that the absorber response is fast, so that

K,: YII zYl<YII ^-Y1 •

In this limit, the steady-state solutions of equations (4) and (5) apply, and equations(1)-(5) then reduce to

which are the modified Lorenz equations and the basic equations that we consider .In the limit of no absorber, equations (14)-(16) reduce exactly to the Lorenz model .

A linear stability analysis about the stationary c .w . solution (E=P=D= 1) leadsto a characteristic equation of the form

(9)

which defines the boundary between stable and unstable stationary (c .w .) states . Inthe limit of no saturable absorber, A--+0 (C-->1) and the equations revert to theLorenz model . In this case, equation (21) can be written

r>rm;,,=Q(6+b+3)/[Q-(b+1)]

(22)This is the familiar second-threshold formula for a single-mode laser (see forexample Risken and Nummedal [12]) from which it follows that rm ; n cannot be lessthan 9, a value that is realized when a = 3 and in the limit b << 1 .

We note that the Lorenz model has been extensively studied over many years,

dE/dt+KE=KAP/(1 +A)+KAE/(1 +a2E2), (10)

dP/dt+y1P=y1DE, (11)

dD/dt+y1 D=y 11 (1+))-y 112PE . (12)

By making the transformation first suggested by Haken [1] namely :

U=K/Y1, b=YII/Y1, r=1+), t-*y1t,E=x/.J(b1), P=yl,l(bA), D=1+2-z, (13)

equations (10)-(12) becomedx/dt=oAy/r+aAx/(1 +ax 2/b)-ax, (14)

dy/dt=rx-zx-y, (15)

dz/dt=xy-bz, (16)

X3 3 +C2f2 +C1f3+C0=0, (17)where

CO =2ab).{1-A(1-a)l[1 +a(r-1)]2}, (18)C1=b(v+r)+{aA/[1+a(r-1)]}{1-(b+1)[1-a(r-1)]/[1+a(r-1)]} (19)C2 =(a+b+ 1)-AA[1-a(r-1)]/[1 +a(r-1)] 2 . (20)

The c.w. state is non-stable if (using Hurwitz criteria, or putting fi=M)O=C1 C2 -C0 <0, (21)


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and has several characteristic features that make it possible to identify an instabilityas being of the Lorenz type . These include a sudden transition to chaos at a Hopfbifurcation (the second threshold), a homoclinic boundary at which unstable orbitsare produced, and a heteroclinic bifurcation just below the second threshold atwhich the chaotic attractor is formed .

3 . Results and DiscussionWe have used quation (21) to plot stability boundaries as a function of A and C

for fixed values of K, Y,, Y II (and hence a and b), and we have solved equations(14)-(16) numerically using a standard Runge-Kutta method in some interestingregions of these phase diagrams . We consider in particular the effect of addingabsorber (increasing C) on the laser properties and we show that the primary effectis to lower the value of the pumping parameter r necessary for the onset of chaos .

The case where a < 1 (for which the amplifier saturates more easily than theabsorber) is considered first, and results for a=0. 3 are presented in figures 2 and 3in which the solid line marks the boundary below which the c .w . state is stable. Alsoshown is the first threshold (A=C), as well as the second threshold for a=0, thelimit in which the absorber simply acts as an additional linear loss . In this lattercase, the c .w . state goes unstable when

r>aC(aC+b+3)/[aC-(b+1)]

(23)

which is just equation (22) with a replaced by an effective loss aC. It is clear fromfigures 2 and 3 that the pump level required to reach the second threshold in a laserwith a saturable absorber can be lowered considerably with the appropriate choiceof absorber concentration and that there is an optimum amount of absorber C forwhich the second threshold is a minimum .

a

0

1

2

3ABSORBER CONCENTRATION (C-1)

Figure 2. Stability diagram for an LSA with a=0 .3, a=2 and b=0 . 5, --- Firstthreshold ;

second threshold (a=0-3) --- homoclinic ; . . . . heteroclinic ;second threshold (a=0) ; x numerical runs .

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a

20-

10


0

1

2

3

4ABSORBER CONCENTRATION (C-1)

Figure 3 . Stability diagram for an LSA with a=0 .3, v=1 .2 and b=0 . 5 .

Zaaa

8

1

0

1

2

3 IX4

Figure 4 . Numerical solution of equations (14)-(16) for parameter values as in figure 2 withA=7.68, C=1 . 7 .

Figure 2 corresponds to a bad cavity (r > b+ 1) . As expected, it shows thatinstability of the c.w . state can occur for any value of C, but that higher values of thepump parameter A are needed in the limit (C-1)->0, in which the system revertsto the Lorenz model. Also plotted (as a dashed line) in figure 2, is the locus of points


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at which a homoclinic bifurcation occurs in this laser model . Above this boundary,the two fixed points in phase space (x, y, z) representing steady-state solutions arestill stable, but trajectories starting close to the origin in one half-space spiral to thefixed point in the half-space with the opposite sign of x . Many trajectories (althoughnot those in the special category just mentioned) cross repeatedly from one half-space to the other before settling down to one or other of the fixed points, a processknown as preturbulence (or unstable chaos) .

Numerical solutions of equations (14)-(16) at points labelled 4 and 5 in figure 2are presented in figures 4 and 5 respectively . As expected, just above the secondthreshold (where the Hopf bifurcation occurs), the motion is chaotic . As notedalready, this sudden transition to chaos as r is increased across the second thresholdis characteristic of the Lorenz model [5], where the chaotic attractor is formed in aheteroclinic bifurcation just below the second threshold boundary. Below thispoint, trajectories starting out along the unstable eigenvector of the origin alwaysspiral immediately to one of the fixed points without any preturbulence, and justabove the heteroclinic bifurcation, this trajectory lies on a chaotic attractor andnever approaches the fixed points even though these are stable . This provides aconvenient indication of the birth of the chaotic attractor. In our modified model,we found similar behaviour and we have plotted several points at which heteroclinicbifurcations occur in figure 2 .

To summarize, the behaviour observed in the modified Lorenz model is verysimilar to the Lorenz instability . The only effect of the saturable absorber appearsto be to lower the second threshold and we conclude that it is indeed possible, bythe addition of a suitable absorber, to reduce the pump level required forobservation of Lorenz-type chaotic motion in a single-mode laser . If enoughabsorber is added, the second threshold is lowered so much that the homoclinicbifurcation lies above it . The unstable orbits that are biproducts of the homoclinicbifurcation do not exist at the Hopf bifurcation, which must therefore be super-critical. The motion observed at the point labelled 6 is shown in figure 6 andconsists as expected of a stable periodic orbit .

Figure 3 shows the stability diagram of an LSA with a < 1 in the case of a `good'cavity (a < b + 1) . In this case, no instability occurs within the Lorenz model, asecond threshold only appearing in the presence of a sufficiently strong absorber forwhich C>Cm;,,>I (see figure 3) . Numerical solutions of equations (14)-(16) atpoints 7 and 8 in figure 3 are presented in the corresponding figures . Just belowpoint 7, a homoclinic bifurcation occurs and chaotic motion is observed at slightlyhigher pump levels within the unstable region (see figure 8) . The behaviour is verysimilar to that around point 6 in figure 2, and also to that predicted within theLorenz model, except that there is a small region of parameter space where a stableperiodic orbit exists before the chaotic attractor is formed .

Consider now the case a> 1, for which the absorber saturates more easily thanthe amplifier . It is known [11] that for C>a/(a-1) the laser can be bistable and thec.w . state coexists with the off-state for pump levels A>A + =X+ (a, C)/a where

X+ (a, C) =a+C-2{1- J[(a-1)(C-1)]} .

(24)

The stability diagram for this case is shown in figure 9 . The laser c .w. state is stablefor A and C within the region formed by the solid line and the A and C axes. Forsmall concentrations, the saturable absorber does not reduce the pump levelnecessary to reach the second threshold as much as it did when a < 1 . Instead, a new

1 288


5x

4-

3-

2-

-3 III,,

I~~~1 1

II I

I1

Z

I I

(a)

I

I

I

I

I

I

I

II

1

1

1I

I

C00

(b)

Figure 5. As figure 4 but with A=8 .75 . X-Z and X-t plots are shown in (a) and(b) respectively .


Figure 6 . As figure 4 but with A=5 . 81, C=2. 5 .

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Figure 7. Numerical solution of equations (14)-(16) for parameter values as in figure 3 withA=9-01, C=2-5 .

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0-1

3-

2-

-3

-4

(a)

I

(b)

qua

I pI

11

ill III 11

t00

Figure 8 . As figure 7 but with A=9 .6 . X-Z and X-t plots are shown in (a) and(b) respectively .


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0

1

2

3

4ABSORBER CONCENTRATION

(C-1)

Figure 9. Stability diagram for an LSA with A=3 .0, v=2 .0 and b=0.5 showing theQ-switching instability (see text) .

instability appears at a very low pump level, even below the first threshold . Thisinstability is shown by the deep indentation in figure 9 and corresponds to the wellknown Q-switching instability which is observed even in the full rate-equation limit[7] . The cavity switches periodically between a `bad' (small signal loss z KC) and a`better' cavity (small signal loss : 1C), between and during pulse formation . This isan intensity dependent effect .

We have solved equations (14)-(16) numerically at various points in figure 9 .Keeping the absorber concentration fixed at C=3, we start at point 10 and increasethe value of A; the results obtained at points 10-13 in the figure are displayed in thecorresponding figures . The LSA jumps abruptly from the off-state to Q-switchingbecause the c .w . state is unstable; a typical Q-switched train is shown in figure 10 .At 11, the c.w. state has regained its stability and the periodic orbit is also stable .The output of the laser will be c .w. or Q-switched depending on the initialconditions. Point 12 is above the homoclinic boundary . Below the boundary, thereare two stable asymmetric periodic orbits, and above it there is a single symmetricorbit. This homoclinic bifurcation is a different type to that observed in the Lorenzmodel where two unstable asymmetric orbits exist above the boundary which donot exist below it . We therefore expect this instability to be different to those wehave discussed so far and the route to apparent chaos at point 13 may be quitecomplicated .

4 . ConclusionWe have presented a simple model of a laser with a saturable absorber. While

the full Maxwell-Bloch equations in the case of perfect tuning are five dimensional,in the fast absorber limit these become three-dimensional and correspond to the

a

wF_wXaa

20-

0- 10-0-X

Cl-

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Xr P r

(0

10

20

30

40

50(b)

Figure 10 . Numerical solution of equations (14)-(16) for parameter values as in figure 9with A=3-4, C=3-0 . X-Z and X-t plots are shown in (a) and (b) respectively .


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Figure 11 . As figure 10 but with A=4 . 8 .

Lorenz system with an additional nonlinear loss term . In the case where theabsorber saturates less easily than the amplifier (a< 1), we have demonstrated that,in the bad cavity limit, addition of saturable absorber lowers the second laserthreshold and the behaviour observed in the numerical simulations is very similar tothat predicted within the Lorenz model . We conclude that addition of a saturableabsorber can lower the threshold for the formation of the Lorenz attractor, makingthe experimental observation of chaotic motion potentially easier .

In the good cavity case on the other hand, addition of enough absorber promotesan instability where there is no instability in the Lorenz model . The behaviour isvery similar to the Lorenz instability but involves two stable limit cycles, formed ata Hopf bifurcation, instead of two fixed points . For higher pumping levels, chaoticmotion is observed .

When the absorber saturates more easily than the amplifier (a> 1), the stabilitydiagram is drastically different and exhibits features associated with the Q-switchinginstability . In the full rate-equation approximation, this instability leads onlyto periodic behaviour, but in the present context it evolves into chaos at highenough pumping . However, this is at levels of pumping that should be attainableexperimentally .

To summarise, we have studied the properties of a laser with a saturableabsorber using standard analytical and numerical techniques. More sophisticatedmathematical tools [5] are available to follow the evolution of periodic orbits inphase space, whether stable or not, as the parameters of the equations are varied,and we plan to use these techniques in future work to give a more detailedunderstanding of the instabilities, and the Q-switching instability route to chaos inparticular .

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z

7-

-4

-3 -2

I

I

I I

0

1

2

(a)

I

I

I

(b)

Figure 12 . As figure 10 but with A=5.77. X-Z and X-t plots are shown in (a) and(b) respectively .

I

3

1

1

1

1

1

1

1

1

1

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I

I

I

I

I

I

1h00

3-

0-1

-2

-3

-4

-5

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z

(a)

(b)

Figure 13. As figure 10 but with A=8 .73 . X-Z and X-t plots(b) respectively.

I

111 1 1 1 1111

I I

I t00

I I

are shown in (a) and

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AcknowledgmentK. E. Aldous is supported by a Postgraduate Studentship from the U .K .

Science and Engineering Research Council .

ReferencesHAKEN, H., 1975, Phys. Lett ., A, 53, 77 .GRAHAM, R., 1976, Phys. Lett., A, 58, 440.WEISS, C. 0., and KLISCHE, W ., 1984, Optics Commun ., 51, 47 .LAWANDY, N . M ., and PLANT, D . V., 1986, Optics Commun ., 59, 55 .SPARROW, C., 1982, The Lorenz Equations, Chaos and Strange Attractors, SpringerSeries in Applied Mathematics, Vol . 41 (Berlin : Springer-Verlag) .

GARSIDE, B. K., and Lim, T . K ., 1973, J . appl. Phys . 44, 2335 .NEW, G. H . C., and REA, D. H ., 1976, J . appl. Phys ., 47, 3107 .TACHIKAWA, M ., TANII, K ., KAJILA, M ., and SHIMizu, T ., 1986, Appl. Phys ., B, 39, 83 .TACHIKAWA, M ., TANII, K ., SHIMIzu, T ., 1987, J . opt. Soc Am., B, 4, 387 .ZENTENO, L. A ., 1986, Ph.D. Thesis, University of London .LUGIATO, L. A ., MANDEL, P ., DEMBINSKI, S. T., and KOSSAKWOSKI, A., 1978, Phys .Rev ., A, 18, 238 .

RISKEN, H., and NUMMEDAL, K., 1968, J . appl. Phys ., 39, 4662 .

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Instabilities in a Simple Model of a Laser with a Fast Saturable Absorber