15 October 1998

Ž .Optics Communications 155 1998 292–296

Class-B two-photon Fabry–Perot laser´Vıctor Espinosa a, Fernando Silva b, German J. de Valcarcel b, Eugenio Roldan b´ ´ ´ ´

a Departament de Fısica Aplicada, Escola UniÕersitaria de Gandıa, UniÕersitat Politecnica de Valencia, 46730 Gandıa, Spain´ ´ ` ` ´b `Departament d’Optica, UniÕersitat de Valencia, Dr. Moliner 50, 46100 Burjassot, Spain`

Received 29 May 1998; revised 24 June 1998; accepted 30 June 1998

Abstract

We study the stationary operation and stability properties of a class-B two-photon Fabry–Perot laser. We show that,´differently from the one-photon laser, the intensity emitted by the two-photon laser is larger in a Fabry–Perot than in a ring´cavity. The lasing solution loses stability through a subcritical Hopf bifurcation, as it occurs in the unidirectional ring laser.The stability domain in the parameter space is larger in the Fabry–Perot than in the ring cavity configuration. q 1998´Elsevier Science B.V. All rights reserved.

Ž .Two-photon lasers TPLs and usual one-photon lasersare a priori very different systems since the former arebased in an intrinsic nonlinear process, the two-photonstimulated emission, which depends on the field intensity.It is thus not surprising that the emission and stabilityproperties of TPLs be very different from those of one-photon lasers. The most salient distinctive features of TPLs

Ž . Žare: i the laser-off solution is always stable thus imply-. Ž .ing the necessity of triggering for laser action and ii the

Žlaser-on solution is stable for pump values aboÕe and not. Ž .below the laser second or instability threshold. More-

over, self-pulsing emission is still possible in autonomousŽclass-B TPLs lasers for which the polarization decay rate

largely exceeds the population and photon decay rates and.on which no external modulation is exerted , a behaviour

w xthat is in contrast with most laser models 1,2 .In the past, the stability and dynamical properties of

w xTPLs have been the subject of several studies 3–10 . Theinterest of such studies is not only mathematical since theycould find direct experimental test after achieving two-photon amplification and lasing in strongly driven two-level

w xatoms 11–14 . Among the factors influencing the laserstability that have been considered in detail are AC-Stark

w x w xshifts 4,8,9 , cavity mistuning 6,8,9 , the presence ofŽintermediate largely detuned atomic levels i.e., the use of

. w xmicroscopic instead of effective Hamiltonians 8,9 , andw xthe use of an injected signal 10,15 .

As far as we know the influence of the type of cavityon the emission and stability properties of TPLs has notbeen previously considered. The only exception known by

w xus is the work by Wang and Haken 16 in which, assum-ing small intensity values, the steady state of a Fabry–Perot´

w xTPL was analyzed. One of the results of Ref. 16 is thatthe intensity emitted by TPLs is smaller in Fabry–Perot´than in ring-cavities, as it also occurs in one-photon lasersw x17,18 . As we show below, we obtain the opposite result,i.e., that a TPL can be more efficient in the Fabry–Perot´than in the ring cavity configuration. We also extend ourstudy to the stability of the lasing solution and show that itloses stability through a subcritical Hopf bifurcation, as itoccurs in unidirectional ring lasers. The stability domain inthe parameter space is larger in the Fabry–Perot than in´the ring cavity configuration. These last results are similar

w xto those corresponding to the one-photon laser 17,18 . Inthe present paper we will limit our analysis to class-BTPLs because they are more tractable and in TPLs this

w xlimit retains the essential behaviour of class-C lasers 10 .We leave the extension of our work to the class-C laser toa future publication.

The model equations for a TPL inside a Fabry–Perot´cavity can be easily obtained from the ring-cavity modelw x8,9 by referring the Bloch equations to a point z insidethe laser cavity and introducing the total medium polariza-

Ž .tion integrated along the cavity axis in the field equation.

0030-4018r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved.Ž .PII: S0030-4018 98 00376-9

( )V. Espinosa et al.rOptics Communications 155 1998 292–296 293

Within the uniform field approximation, that consists inneglecting any slow variation – as compared to the laserwavelength – of the model variables along the propagation

w xdirection 1,2 , and the rotating wave approximation, andŽ .assuming an effective Hamiltonian interaction see below ,

these equations read

L 2q z ,t sin kz d zŽ . Ž .Hd 0)e t syk eq ie , 1Ž . Ž .

Ld t 2sin kz d zŽ .H0

d 2q z ,t sy g qiD qq iD esinkz , 2Ž . Ž . Ž .Ž .Hd t

d 21 )D z ,t sg D yD q i q e sinkz yc.c. ,Ž . Ž .Ž .I 0 2d t3Ž .

if Ž t .Ž . 'where e t s I t e is proportional to the field am-Ž .Ž Ž . Ž .plitude I t and f t are proportional to the field inten-

. Ž . Ž .sity and phase, respectively and D z,t and q z,t sŽ . 2 ifQ z,t e are proportional to the population inversion

and complex two-photon coherence at position z, respec-Žtively the variables I,Q and D have the same scalings as

w x .those used in Refs. 8,9 . The parameters k ,g and gI Hare the field, population and two-photon atomic coherencedecay rates, D is the incoherent pump rate and D is the0

Žtwo-photon cavity detuning. Finally Lsmlr2 with m a.positive integer number is the length of the laser-cavity

with l the field wavelength and ks2prl its wavenum-ber kLsmp .Ž .

Although homogeneous broadening has been assumedŽ . Ž .in writing Eqs. 1 – 3 , due to the well known Doppler

w xcompensation in two-photon processes 19 , we can beconfident that the model gives also a good description ofan inhomogeneously broadened Fabry–Perot TPL.´

Before going on some comments on the degree ofŽ . Ž .applicability of the model are in order. Eqs. 1 – 3 have

been derived by assuming an effective interaction Hamilto-nian, i.e., by assuming a pure two-photon interaction be-tween the two-level atom and the electromagnetic field.This approach neglects residual effects of any largelydetuned one-photon transitions between the lasing levels

w xand other atomic levels 20 . A more precise approachconsists in assuming an exact or microscopic interactionHamiltonian that describes the interaction of the electro-magnetic field with a three-level cascade atomic schemew x8,9 . When the intermediate atomic level is far fromone-photon resonance the one-photon coherences can beadiabatically eliminated and the resulting TPL equations

Ž . Ž .are similar to Eqs. 1 – 3 but include three additionaldetuning terms describing frequency shifts. One of theseshifts is of static nature and is related to the amount of

Žequilibrium population in the intermediate level it merely.adds a constant shift to D . The other two shifts are of

dynamic nature and are proportional to the field intensity

Ž .AC-Stark shift and to the two-photon population inver-Ž w x .sion see Refs. 8,9 for a detailed discussion .

w xIt was shown in Ref. 9 that if the two one-photontransitions involved in the cascade atomic scheme havecouplings similar to the electromagnetic field the effect ofthe dynamic shifts are rather small and hence can beignored in a first approximation. Thus in that case theeffective Hamiltonian constitutes a good approximation.Here we are assuming that this holds also in the Fabry–Perot configuration.´

Ž . Ž .Now we consider the class-B limit of Eqs. 1 – 3 , i.e.,we assume that g 4k ,g . In this case the two-photonH Icoherence q can be adiabatically eliminated. Equating to

Ž .zero Eq. 2 , and substituting the obtained expression forŽ .q z,t into the remaining equations, it is easy to obtain the

simplified model,d g H ² :I t s2k I I D y1 , 4Ž . Ž .2 2ž /d t g qDH

d g H 2 4D z ,t sg D yD y DI sin kz , 5Ž . Ž .Ž .I 0 2 2d t g qDH

with

L 4D z ,t sin kz d zŽ . Ž .H0² :D ' , 6Ž .

L 2sin kz d zŽ .H0

together with the pulling formulad kD

² :Dv t ' f t s I D . 7Ž . Ž . Ž .2 2d t g qDH

Ž .Note that the instantaneous frequency shift Dv t is not atrue dynamical variable since it is given by the instanta-neous values of I and D.

Finally, the introduction of the adimensional quantities2k g gI HX

tsg t , zskz , gs , D s D ,I 0 02 2(g g qDI H

g gI HXD z ,t s D z ,t ,Ž . Ž .2 2(g qDH

g HXI t s I t 8Ž . Ž . Ž .2 2(g g qDŽ .I H

leads to the class-B TPL model equations,d

² :I t sg I I D y1 , 9Ž . Ž . Ž .dt

d2 4D z ,t sD yD 1q I sin z , 10Ž . Ž .Ž .0dt

where primes have been removed, withmp

4dz D z ,t sin zŽ .H0² :D ' , 11Ž .

mp2dz sin zH

0

where the equality kLsmp has been used.

( )V. Espinosa et al.rOptics Communications 155 1998 292–296294

Ž . Ž .Eqs. 9 and 10 have two sets of stationary solutions,the laser-off solution

I s0, D z sD , 12Ž . Ž .off off 0

and the lasing solution

D0D z s , 13Ž . Ž .on 2 41q I sin kzŽ .on

A2(D s I , As 1q I , 14Ž .0 on on'2 Ay 2 1qAŽ .

Ž .see the Appendix for details of the integration . Thew Ž .xfrequency shift Dv see Eq. 7 corresponding to this

solution is

DDvsk , 15Ž .

g H

w xin agreement with Ref. 16 in the limit k<g .HIt is interesting to compare the stationary intensity

corresponding to a TPL with a Fabry–Perot cavity, Eq.´Ž .14 , with that corresponding to a ring cavity TPL which is

w xgiven by 3,5,8

1ring 2(I s D 1q 1y4rD . 16Ž .on 0 02 ž /In Fig. 1, both I and I ring are displayed and two factson on

Ž .can be clearly appreciated: i the pump threshold for theexistence of the lasing solution is smaller for the case of a

Ž w x . Ž .ring-cavity in agreement with Ref. 16 and ii althoughfor pump values close to threshold the ring-cavity solutionis larger, from a given value of the pump on, the intensityemitted by the Fabry–Perot TPL is larger than that of the´

Fig. 1. Lasing intensity as a function of the pump parameter D0

for the Fabry–Perot and ring cavity TPLs. The dashed line´indicates the unstable branch of the solution.

ring-cavity. This result is in contradiction with the resultw xobtained by Wang and Haken 16 . The reason lies in the

w xfact that in Ref. 16 an approximate solution was derivedby assuming small values of the laser intensity, the validityof their result being limited to the neighbourhood of thethreshold. Thus it seems that the spatial average introducedby the Fabry–Perot cavity acts in a different way for´one-photon and two-photon transitions, what constitutes anunexpected result.

Next we analyze the stability of the solutions. ByŽ . mt Ž .making I t s I q d I e and D z ,t s D qon on

Ž . mt Ž . Ž .dD z e and linearizing Eqs. 9 and 10 we obtain thefollowing characteristic equation for the evolution of theLyapunov exponents m,

1ymrgŽ .2 D0

2 I 3 sin8zpony dzs0,H 2 4 2 4p 1q I sin z 1qmq I sin z0 Ž . Ž .on on

17Ž .

Ž .which, after integration see the Appendix for details , canbe rewritten in the compact form

2 ' '(0s mqA 2 Ay 1qA 1ymrgŽ .Ž .

' 'q2 2 Amy2 1qA

5r4 2('(y2 1qm A 1qm q mqA , 18Ž . Ž .

Ž .with A given in Eq. 14 .We have not been able of obtaining any analytical

Ž .result from Eq. 18 and thus we have made a numericalstudy of its solutions. We have found that the lower lasingbranch is always unstable, as it must be, and found that theupper lasing branch undergoes a Hopf bifurcation when-ever the intensity losses g)2.20 approximately. Thisvalue is to be compared with the case of a ring-cavity, in

w xwhich the bad-caÕity condition is g)2 8–10 . Thus thebad-cavity condition is similar for both cavity configura-tions.

In Fig. 2, the stability domains corresponding to theFabry–Perot TPL are represented in the parameter space´² : Ž .g , D full line together with those corresponding to the0

w x Ž .class-B ring laser 6,8–10 dashed line . We see how theŽunstable domain which is below the Hopf-bifurcation

.curve is slightly smaller for the Fabry–Perot configura-´tion. Notice that although for the smaller values of g theinstability threshold is larger in the Fabry–Perot configura-´

Ž .tion, Fig. 2 a , the quotient between the instability and theŽ .lasing thresholds is always smaller, Fig. 2 b .

We have checked our results by numerical integrationŽ . Ž .of Eqs. 9 and 10 and found a perfect agreement with

( )V. Espinosa et al.rOptics Communications 155 1998 292–296 295

Ž . Ž . Ž .Fig. 2. a Hopf bifurcation HB and threshold of existence onŽ .of the lasing solution for the Fabry–Perot full line and ring´

Ž . Ž .cavity dashed TPLs. b Quotient of the Hopf bifurcation andexistence of the lasing solution thresholds, r, for both cavityconfigurations.

the results of the linear stability analysis. We have alsofound that the laser turns off when the instability boundaryis crossed from above what implies that the Hopf bifurca-tion is always subcritical. Thus, class-B Fabry–Perot TPLs´do not display dynamic behaviour. This result is similar tothe case of the ring-cavity TPL where for small k and g I

w xthe bifurcation is also subcritical 9 .In conclusion, we have shown that a Fabry–Perot TPL´

is more efficient than a ring-cavity TPL. This result isunexpected because it is opposite to what happens in usualone-photon lasers. We have also shown that the stabilitydomain in the parameter space is larger for the Fabry–Perot´configuration as it occurs in one-photon lasers.

Acknowledgements

We are grateful to Ramon Vilaseca for fruitful discus-sions. This work was supported by the Spanish DGICYTproject number PB95-0778-C02-01.

Appendix A

Ž .From Eq. 9 the steady laser intensity is given by² :I D s1, i.e.,

sin4zmp mp2I D dz s dz sin z . A.1Ž .H Hon 0 2 41q I sin z0 0on

Since both integrands are periodic in z their upper limitscan be replaced by p so that the right-hand side readspr2. The expansion of the integral appearing in theleft-hand side in powers of I readson

4 `sin zp ppq1 2Ž py1. 4 pdz s y1 I dz sin z .Ž .H HÝ on2 41q I sin z0 0on ps1

A.2Ž .

The integral in the right-hand side can be evaluated andthe result reads

1G 2 pqp Ž .24 p 'dz sin zs p . A.3Ž .HG 2 pq1Ž .0

Ž . Ž .By substituting Eq. A.3 into Eq. A.2 and performingŽ .the summation in p, Eq. A.1 finally reads

A2(D s I , As 1q I . A.4Ž .0 on on'2 Ay 2 1qAŽ .

The above calculations have been made with the help ofthe program Mathematica.

The integral appearing in the characteristic equationŽ .17

sin8zp

js dz , A.5Ž .H 2 4 2 41q I sin z mq1q I sin z0 Ž . Ž .on on

expands in powers of I ason

yp` 1y mq1Ž .pq1 4Ž py1.js y1 IŽ .Ý on

mps1

=p

4Ž pq1.dz sin z , A.6Ž .H0

and the integral in the right-hand side follows from Eq.Ž .A.3 . Again with the help of the program Mathematica

Ž . Ž .Eq. A.6 can be evaluated, being Eq. 18 the final result.

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