1074 J. Opt. Soc. Am. B/Vol. 5, No. 5/May 1988

Space inhomogeneity and detuning effects in a laserwith a saturable absorber: a first-order approximation

Priscila Garcia-Fernandez and Manuel G. Velarde

Departamento de Fisica Fundamental, Universidad Nacional de Educaci6n a Distancia, Apartado 60.141,Madrid-28.071, Spain

Received October 12, 1987; accepted January 27, 1988To a first approximation the effects of detuning and/or space inhomogeneity on the stability domain of a model for alaser with a saturable absorber are presented. It appears that the space dependence increases the domain of theemissionless state, thus delaying the laser action.

INTRODUCTION

The basic features of a laser with a saturable absorber oper-ating in a single-mode regime can be obtained from a modelrecently discussed in the literature.'- 9 If, following the ap-proach developed in Ref. 10, we assume that a coherent,stationary, plane-wave field EL is longitudinally injectedinto the cavity and that both the input and the internalcavity field are linearly polarized in the y direction, the laserproblem, taking into account the space dependence, can beexpressed as

aE 22t K(E + EL) + NV+NV +a 2 (la)

dv= zlv + IgI2DE - iOy, (lb)

-t = v+ IgI2DE - ij,

dD -y11D - 4E + -yla, (Id)

dD _- 1 -1D - 4E + (le)where N is the number of two-level excitable (active) atoms;E is the electric field complex amplitude, with EL taken to bereal and positive; v is the polarization of the atoms; D is theperturbed atomic inversion (population inversion); and g isthe field-matter (active or passive) coupling constant. K isthe damping constant of the field in the cavity, o- is theunsaturated inversion, and oyl and -yI are the longitudinaland transverse relaxation constants, respectively. A barover a quantity indicates the corresponding variable for thepassive atoms. We call z the longitudinal coordinate and x,y the transverse coordinates. B is the distance between thetwo mirrors considered orthogonal to the x axis, and a = 1/27rTF, where T is the transmission coefficient and F = b2/XLis the Fresnel number (X is the avelength and L is thedistance between the two mirrors orthogonal to the z axis).

The model considered in Eq. (la) can be derived from theMaxwell-Bloch equations for a two-level system by intro-duction of the mean field limit T << 1. Because T << 1 with aof order unity, we assume that F >> 1 (generalized mean field

limit).9 Finally, in Eqs. (lb) and (c) and denote atomicdetuning parameters,10 and the frequency of the input fieldis assumed to coincide with the frequency of the cavitymode.

For simplicity, we take 2E/ax2 = G2E. This drastic ap-proximation simplifies the problem greatly, and although itdoes not permit rigorous conclusions about the space depen-dence problem, it does permit a first-order estimate of itsinfluence." It introduces a characteristic length scale,11G, whose influence on the dynamics of the laser we caneasily explore. Space dependence below this cutoff length isnot considered. Note also that with this approximation theproblem [Eqs. (1)] becomes that of a single-mode model.12

For universality in the analysis, we change units in orderto write dimensionless equations. Following Refs. 2-6, weset

E = -(ll-yy)12 a/lgJ2, EL =-(,Y17,)1/2aL/1g12,

v = -0a1g1('Yi/4Y 1)1/2p,

v -Ig(-y 11/4y) 1/2p, D = (- d), D = (1 -d),

= aYt, = ei/'eL, r, = 1 /-y, r2 = 7Y1/711,

p = K/y , A = NgJ 2 /Ky 1, C = 1 - Nfg 2G/K -.

A and C are the control parameters that account for thepumping rates of active and passive atoms, respectively.Then the system (1) becomes

dtt = p[-a + aL + Ap + r1(1 - C)p] + a,

dt' ldp P-+a(l-d)-i P,

=dt= _rlp + (1 - d)a-i-p,

dd = w[-d + 1/2(ap* + a*p)],

dd == wf-r 2a + /2 (ap* + a*1)].

(2a)

(2b)

(2c)

(2d)

(2e)

Taking into account that field amplitudes are complex, we

0740-3224/88/051074-03$02.00 © 1988 Optical Society of America

P. Garcia-Fernandez and M. G. Velarde

Vol. 5, No. 5/May 1988/J. Opt. Soc. Am. B 1075

obtain for the real and imaginary parts of the variables thefollowing set of ordinary differential equations:

a5 b2 G 2

dar = p[-ar + aL + APr + r(I-C)] + p aidt' l

dai a b2 G 2

d i p[-ai + Api + r(1 - C)pi] + ar,dt' 'YJ

dPr + a6(I-d)- Pi,dt'

dt -Pi+ai(1- d) - Pr,

d~r - - 6 -

dt' I- r + r- Pi,

Pi, = -rj)i + (1d)ai - iPr,

ddd= = w(-d + arpr + aipi),

dd +dt'=-co(-r2d + arpr + aipi),

(3a)

(3b)

(3c)

(3d)

(3e)

(3f)

(dp/dt') = a(l - d) - p(r + i'),

(dd/dt') = [-d + 1/2(ap* + a*p)],

(dd/dt') = w[-r 2d + 1/2(ap* + a*p)),

(6c)

(6d)

(6e)

where it is apparent that the role of the parameter s is incavity detuning. The stationary solutions verify that

-ibcoa' =p[(-l + is)a' + Ap5 + rj(I - Qps],

-ibcopS= a'(1 - dS) - pS(1 + if'),

j-ps = a'(1 -d) -p(r, + %'),o =-d' + 1/2(aSps* + as* ps),

0 = .dS + 1/2(aSpS* + aS*pS)

and 05 = 0 - bw. Equations (7b) and (7c) become

0 = a( - d) - pS( + is),

0 = a(1 - d) - p5 (r + iOs),

(7a)

(7b)

(7c)

(7d)

(7e)

(7b')

(7c')

(3g) and we obtain the following relations:

P = 1 - s aspS=1 + 6s' + I' as(3h)

where the subscripts r and i have the obvious meaning of realand imaginary parts of the variables.

ds la S12 1 + Os2 + lasl2

2 2 + s2 + I 2r1

(8a)

_s= as. i

P r 2 + s2 + r1la 2 a

(8b)RESULTS

The steady solution of Eqs. (2) is a nonlinear emission that

satisfies

(-1 +is)a +A a +rj(1 -) 1 1 + O2 + a 12 r1

2 + 6,2 + r1lal2

a-= -aL, (4)

where s = asbG2/pyt, 0' = /)y, and 6' = /'y. Here thequantity s plays the role of a cavity-detuning parameter [seeEqs. (2)]. Obviously, when aL = 0 and we disregard space

dependence and detuning, Eq. (4) reduces to X4 + X2(1 - A +rC) + ri(C - A) = 0, where x2

= ar2 + ai 2(x2

5d 0), inagreement with Refs. 2-4.

Following the arguments developed by Lugiato,12'13 thesteady solutions at aL = 0 must be calculated by introducinginto Eqs. (2)

a = a' exp(-ibwt), (5a)

p = ps exp(-ibcot), (5b)

p = p 5 exp(-iecot), (5c)

d = ds, d = ds, (5d)

with (as, ps, ps, ds, ds) constant. No confusion is expected,with superscripts here denoting steady state. 6o is a fre-quency offset (frequency pulling) that must be determinedsimultaneously with the values of lasl2. On setting aL = 0,Eqs. (2) become

(da/dt') = p[(-1 + is)a + Ap + rj(1 - C)p], (6a)

(dp/dt') = a(1 - d) - p(1 + i6'), (6b)

From Eqs. (8) and (7a), we obtain the result

F ~~A(l - i65) rj(1 - C)(r1 - OD)1-ibwa' = pas (-1 + is) + 2 + IasI2 r1

2 + _ s2 + rIa2 ][ ~ ~~1+O2 + aI rI. , +rls(9)

The nonzero stationary solution is obtained by equatingthe real and the imaginary parts of Eq. (9):

1 = A + r12 (1-C)

1 + s2 + Iasl2 r 2 + s2 + rlas 12

-( /°) = s- AOs |5 2( 1 - '1 + Os + as 12 -r 1

2 + Ps2 + r1IasI2

(1Oa)

(lOb)

where the frequency pulling inside the parameters s, ssatisfies Eq. (lOb).

The variation of the stationary solutions [Eq. (9)] with thespace parameter s and the atomic detunings 6', 0' are given inFigs. 1 and 2, for a and b, 6' = 6' = 1.5; a, s = 0.5 and b, s = 2.0;andforcandd,'=' = 1.9;c,s=0.5andd,s=2.0. InbothFigures rj = 0.4 and p = 0.1. In Fig. 1 lines a-d (ANL) are thelines of transition from the emissionless state to the steadylasing states. For C > ANL, only the solution as = 0 exists.On the other hand, it appears that ANL decreases as s and 6',

0' increase. Thus the region of values of the control parame-ters A and C where nonvanishing, positive values of the fieldintensity exist becomes smaller as s and 6', 0' increase theirvalues.

The variation of the stationary intensity, |as2, versus thepumping parameter A, for C = 16, is given in Fig. 2. We cansee that the intensity decreases as the space parameter andthe atomic detunings increase.

P. Garcia-Fernandez and M. G. Velarde

1076 J. Opt. Soc. Am. B/Vol. 5, No. 5/May 1988

6

4.

2-

0

I A

4 8 120

d

AN-a

C

Fig. 1. Boundaries of the emissionless state due to space depen-dence at r = 0.4, p = 0.1. ANL is the line of transition from theemissionless state to the steady lasing state. a and b correspond to 0= ' = 1.5 and a, s = 0.5; b, s = 2. c and d correspond to 0 = ' = 1.9and c, s = 0.5; d, s = 2.

8

Ias12

abCd

6 10 12

Fig. 2. Dependence of the steady-state solutions with the spaceparameter s and the atomic detuning for r = 0.4, p = 0.1, and C = 16.Values for a, b, c, and d are as in Fig. 1.

Finally, when only detuning effects3' 4.7 are considered, thestationary solutions of Eqs. (5) verify that

A A + r 2(1 -C) (Ila)1 + ,2 + a 12 r1

2 + U~ + ra 12

0= AO'1 + 0,2 +la 1 2

r12(1 -C)j'r 1

2+ 1y2 + ra'l2

CONCLUSION AND OUTLOOK

Although our results were obtained with a drastically simpli-fied description of the space dependence in the problem,they permit us to conclude that the space dependence en-larges the stability region of the emissionless state of thesystem. In alternative statements we may say that it dimin-ishes the control-parameter region where ordinary cw lasingis possible or that, for a given set of control-parameter val-ues, it delays the bifurcation from the zero-field state tolasing. Presumably, this effect favors finite amplitude bi-furcation or oscillatory phenomena. Detuning produces thesame result. For future work we leave the analysis ofsteady-state multiplicity in Eqs. (11) as well as the stabilityof the solutions. Also needed is a more rigorous quantitativestudy of the role of the external field [Eq. (4)] with the effectof both space inhomogeneity and detuning on the alreadyreported Q switches and other oscillatory solutions in thesame problem.2 -7

ACKNOWLEDGMENTS

The authors wish to express their gratitude to L. A. Lugiatofor fruitful discussions and correspondence. This researchwas sponsored by a grant from the European EconomicCommunity.

REFERENCES AND NOTES

1. L. A. Lugiato, P. Mandel, S. T. Dembinski, and A. Kossakowski,Phys. Rev. A 18, 238 (1978).

2. J. C. Antoranz, J. Gea, and M. G. Velarde, Phys. Rev. Lett. 47,1985 (1981).

3. J. C. Antoranz, L. L. Bonilla, J. Gea, and M. G. Velarde, Phys.Rev. Lett. 49, 35 (1982).

4. M. G. Velarde, in Evolution of Order and Chaos, H. Haken, ed.(Springer-Verlag, New York, 1982), p. 132.

5. T. Erneux and P. Mandel, Z. Phys. B 44, 353 (1981).6. M. G. Velarde and J. C. Antoranz, in Optical Instabilities, R. W.

Boyd, M. G. Raymer, and L. M. Narducci, eds. (Cambridge U.Press, Cambridge, 1986), p. 287.

7. D. E. Chyba, N. B. Abraham, and A. M. Albano, Phys. Rev. A 35,2936 (1987).

8. P. Peptowski and H. Haken, Phys. Lett. A 120, 138 (1987).9. Note that the model problem discussed here is equivalent to a

two-component Lorenz model for thermoconvection in binarymixtures, as discussed in Ref. 4. See also J. C. Antoranz and M.G. Velarde, Phys. Rev. A 37 (to be published, 1988).

10. L. A. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209 (1987).11. This approach is familiar in some fluid-mechanics problems

when convective rolls affect molecular diffusion. To a firstapproximation convective phenomena, in the Fourier heatequation say, can be described by means of an effective diffusi-vity. This homogenization method requires that the tempera-ture vary on a length scale much larger than the convectivevelocity field. Its use permits valuable order-of-magnitude es-timates and qualitative results about the role of space inhomo-geneity. Improvements are obtained by expanding all fields inpowers of a smallness parameter, which is the ratio between thelength scales of the velocity field and the temperature profile.See, e.g., B. Gebhart, Heat Transfer (McGraw-Hill, New York,1971), Chap. 11.

12. L. A. Lugiato, in Progress in Optics XXI, E. Wolf, ed. (North-Holland, Amsterdam, 1984), p. 71.

13. L. A. Lugiato, University of Torino, Torino, Italy, personalcommunication.

.| + w

P. Garcia-Fernandez and M. G. Velarde