- Email: [email protected]
Mode instabilities in a homogeneously broadened ring laser
Volume 42, number 4
OPTICS COMMUNICATIONS
15 July 1982
MODE INSTABILITIES IN A HOMOGENEOUSLY BROADENED RING LASER Paul MANDEL * UniversitJ Libre de Bruxelles, Service de Chimie-Physique II, CP. 231, 1050 Brussels, Belgium Govind P. AGRAWAL Physics Department, City College of the City University of New York, N Y 10031, USA and Quantel, B.P. 23, 91941 Les Ulis Orsay, France Received 6 April 1982
A nonperturbative theory of a homogeneously broadened ring laser is presented after incorporating spatial holeburning effects arising due to mutual interference of two counterpropaga0ng cavity modes, It is shown that under suitable conditions the continuous-wave unidirectional lasing mode becomes unstable due to Hopf bifurcations. The time-periodic state associated with these bifurcations corresponds to self-induced pulsations either in one direction or in both directions..
1. Introduction Ring lasers are, in general, capable of sustaining two counterpropagating travelling-wave modes, a feature exploited for their use in laser gyroscopes [ 1 - 3 ] . These modes, however, are coupled by the gain medium and the operating characteristics of a ring laser are highly sensitive to various factors such as the linebroadening mechanism. When the gain medium is homogeneously broadened, mode coupling is so strong that one of the modes is altogether suppressed [4--6]. It is well known that spatial hole.burning, arising due to mutual interference of two counterpropagating waves, plays an important role [7-9] in determining the extent of mode coupling. In recent years mode instabilities and associated phase transitions in a homogeneously broadened ring laser have been studied theoretically and experimentally [10--15]. In most of the previous work the analysis is simplified using third-order perturbation theory for the induced polarisation. Moreover, stability criteria for the steady-state mode intensities are often formulated after adiabatically eliminating the gain medium atomic variables. This is a severe limitation and it is known that a more general stability analysis is required to investigate some self-pulsing instabilities [16-21 ]. In this paper a theoretical analysis of a homogeneously broadened ring laser is presented by relaxing both of the above-mentioned limitations. The theory predicts a new Hopf bifurcation which has previously gone unnoticed.
2. General formulation We consider a ring cavity containing the gain medium which is modeled as a homogeneously broadened and inverted two-level system with the atomic transition frequency Wa" The laser is assumed to oscillate at the single frequency w. The operating conditions are taken to be such that the ring cavity behaves in a symmetric manner with * Maitre de Reeherches FNRS.
~~.: 269
Volume 42, number 4
OPTICSCOMMUNICATIONS
15 July 1982
respect to its counter-propagating modes at ¢o. Although the dispersive effects may play a significant role under certain conditions [19], in the following we assume exact resonance so that co = coa = coe where coc is the cavity resonance frequency. The laser characteristics are obtained by solving the Maxwell-Bloch equations extended to incorporate the counter-propagating nature of the cavity field. Making the usual plane wave and slowly varying envelope approximations and following the approach and notation of ref. [19], the semi-classical equations are: Z i(a t
+ X) ~/(t) = N~. f gT(x) a(x, t) dx
(la)
o i(a t
+ 2ti) a(x, t) = -D(x, t)1..~1,2 gj(x) [3l(t) ,
l o t + "rll)D(x, t) = itrt u + 2 ~
j: 1,2
[gj(x) a*(x, t) [3j(t) - c.c.l ,
(lb)
(lc)
where ~ ( t ) is the slowly varying field envelope, a(x, t) is the induced polarization and D(x, t) = (N+-N )/(N+ +N The mode index isj = 1, 2 for the forward and the backward wave, respectively. The quantities N+(x, t),N_(x, t ) , K, 711 and 7x correspond to the upper and lower atomic numbers, the cavity decay rate, the longitudinal and transverse relaxation rates. Further, the coupling constant is
gl(x) = --i(21rco)l/20a/h) exp(+-ikx)
(2)
where kc = co,/z is the dipole matrix element and x is measured along the cavity perimeter. The upper and lower sign refer t o j =1 andj = 2, respectively. Eqs. (1) fully incorporate the effects of spatial hole-burning arising due to interference between the counterpropagating waves. It is convenient to define
~j(t) = Ei(t ) + iFj(t) ,
g;(x) ~(x, t) = iVj(x, t) - Uj(x, t) ,
(3)
and obtain the following set of nine coupled equations:
N L
(a t + K) Ej (t) = I, f 1~ (x, t) dx , 0 N L Ot + x) Fj(t) = -[. f o
(4)
(a t + ~j.) U1(x, t) = g~-D(x, t) [Fj(t) + ,%_,dt) cos :Zkx + (-)JE3w(t) sin 2 ~ ] ,
(6)
(at + Vj_) V)(x, t) = g2D(x, t)[ej(t) + e3w(t) cos 2 ~ + (-)iF3 q(t) sin 2 ~ ] ,
(7)
(5)
o
(a t + ~,ll)D(x, t) = o~ll - 4 ~
1:1,2
[Ej(t) Vj(x, t) ÷ Fj(t) Uj(x, t)] ,
(8)
where g2 = [gl 12 = ig212 = 21rco/a2flt 2 is x-independent.
3. Steady-state mode intensifies The calculation of the steady-state solution of eqs. ( 4 ) - ( 8 ) is simplified if we note that in the absence of any external reference the phase of the electric fields is arbitrary and one can choose F/= 0 without any loss of gener270
).
Volume 42, ]lumber 4
OPTICS COMMUNICATIONS
15 J u l y 1 9 8 2
ality. From eqs. (6)-(8) we then obtain
D(x) = o/[1 + S(E 2 + E22 + 2EIE 2 cos 2kx)] ,
(9)
= ¢a2D(x)/9',)(E/+ E3_ / cos 2gx), where S = 4g2h'llg'± is the saturation parameter. Spatial hole-burning is evident in the expression for D and has its origin in the formation of a population grating induced by the counter-propagating waves. We substitute (9) in (4) and perform the integration. The reduced mode intensities//= SE? verify 2/i +A [1 +13_ / - / / ) [ D ] = 0 ,
(10)
where D = 1 + 2(• 1 +•2) + (I1 - •2) 2 and the pump parameter A =Ng2o/rVl is def'med such thatA = 1 corresponds to the laser threshold [19]. The solutions of (10) can be grouped in three distinct classes: (i) the trivial solution 11 = 12 = 0 , (ii) the symmetric solution I 1 =12 = ] [4/1 - 1 - (8/1 + 1)1/2], (iii) the asymmetric unidirectional solutions//=A - 1,13_ / = 0 f o r / = 1 or 2. A linear stability analysis is performed to examine the stability of each solution under small perturbations. In general, the complete set ( 4 ) - (8) of nine dynamical variables should be considered and the results will be presented in the next section. Here we examine the stability in the good cavity limit, assuming r "<9'1, 9'11 SO that the atomic variables can be adiabatically eliminated. Under these conditions the stability analysis can be based on
a:/=rI/+ U
"
_
I/)/O].
½ A[1 - (1 +I3_ j st
i
.
.
.
.
~
.
.
Ass mmg li(t) - I ) +lj exp(Xt) and lmear]zang ln1~ we obtain a quadratic polynomial in ?~. In order for 1 .st to be stable it is sufficient that Re X < 0 for both roots. 1?he following picture emerges: the trivial solution is si/able below thresholdA ~< 1. Above thresholdA > 1, the trivial and symmetric solutions are unstable while the unidirectional solutions are stable. However since the ring laser is assumed to have no prefered direction, it can oscillate in either of the modes determined by internal fluctuations. As a matter of fact, mode hopping, where in a ring dye laser spontaneously switches its lasing direction, is a wellknown phenomenon [15]. It should be remarked that mode hopping, often an undesirable device feature, can be eliminated by breaking the forward-backward symmetry. Among other means, it can be achieved by differential cavity toss mechanisms, external injection or the use of an additional mirror [22]. For an asymmetric ring laser, an analytic treatment is, unfortunately, not feasible. We considered numerically the case of an additional mirror and the results can be summerized as follows. Below threshold, 11 = I 2 = 0 as before. Above threshold the solution 11 > t 2 ~ O, arising from the perturbation of the symmetric branch, remains unstable so that bidirectional lasing is ruled out. The only remaining unidirectional solution 11 = A - 1 and 12 = 0 is stable and thus the laser operates without mode hopping.
4. General linear stability analysis We now examine the stability of the three classes of steady state solutions without imposing any restriction on the cavity and atomic decay rates, i.e. by considering the full set of eqs. (4) to (8). Assuming X(t) = X st + X'exp(Xt) where X is any of the nine dynamical variables in (4)-(8) with its stationary value X st andX' is a small perturbation, the decay constant Satisfies a ninth degree polynomal. A solution will be stable iff Re ), < 0 for all nine roots. For the trivial solution I 1 =12 = 0, there is a single nondegenerate root X = - 7 u andtwo fourfold degenerate roots given by
CA + ~i)(~ + K) =AK~I.
271
Volume 42, number 4
OPTICS COMMUNICATIONS
15 July 1982
It is easy to see that one of the roots becomes positive forA > 1 indicating an instability of the trivial solution above threshold. For the symmetric solution 11 = I 2 > 0, the ninth degree polynomial factors out into two qua&aries and a quintic. The quadratics are (X + 7±)0~ + ~) = K7±,
(11)
CA+ 7±)(X + g) = ~71(1 + 4/1).
(12)
The roots of (I I) are X = 0 and ~, = - ~ - 7±. The presence of a zero root indicates the existence of an invariant in the set (4)-(8). Its origin ~an be traced back to the fact that the phase difference, and not the individual phases of the counter-propagating waves, is a relevant variable and implies that eqs. (4)-(8) can be reformulated in terms of eight variables only. One of the roots of (12) is always positive for any I 1 > 0 indicating that the symmetric steady solution is always unstable forA > 1. We may remark that this result is a direct consequence of spatial hole-burning enforced by homogeneous broadening. Indeed for a Doppler-broadened gain medium, spatial holes are washed aut by atomic motion [7,8], mode competition is reduced and under certain conditions both modes can osdllate simultaneously [1,3] whereas the asymmetric solution becomes unstable. We now discuss the stability of one of the two unidirectional solutions, e.g. I 1 =A - 1,12 = 0. It is governed by a ninth degree polynomial which factors into a quadratic, a cubic and a quartic. The quadratic is identical to (11) and requires no further considerations. The cubic and the quartie are (X+ r)(X + 7± + b) = ~-/.,
(13)
(?, + 71)CA + ~)(X + 7 , + b) = -~rT± (2k + 27± + b ) ,
(14)
b = 7H(A - 1)(X + 27±)/(X + 2711 - ATll) .
(15)
where
It is useless to solve these equations in dosed form for X. We therefore first show that in the vicinity of the threshold,A = 1 + e (0 < e ,< 1), the unidirectional solution is stable. To the leading order in e, the roots of (13) are -711, - r - 71 and - e r 7 j / ( r + 7±). Three roots of the qtmrtic (14) are roots of (13), the fourth being X = -7±. Both unidirectional solutions are therefore stable above but near threshold. Well above threshold the unidirectional solution will be destabilized if the real part of a root of (13)-(14) vanishes. It is important to note that (13) and (14) govern the stability of the nonzero- and the zero-intensity mode, respectively. Let us first consider the cubic. It is readily verified that no real root can vanish for A > 1. This leaves open the possibility of a Hopf bifurcation where the real part of a pair of complex conjugate roots vanishes. To check the occurrence of such a bifurcation we solve (13) with the ansatz (X-/z)(X 2 + ~212)= 0 and find /a = --r -- 7± -- 7 u ,
~12 = 7U(r + ~'IJT) ,
(16)
A" = KCK+ 371 + 711)/[7±(r --711- 7±)] •
(17)
where
A Hopf bifurcation therefore takes place at the critical value A =.4. The expression (17) for/T is identical to the one obtained previously [16,17] while investigath'ag self-pulsing in a unidirectional ring laser. This is not surprising because when fluctuations of the zero.intensity mode are ignored, the bidirectional case reduces to the unidirectional case. Let us now consider the stability of the zero-intensity mode. Again, no real root of the quartic (14) can vanish for A > 1. We therefore solve (14) with the ansatz (X - ;11) Q, - ta2) Ca2 + ~22) = 0. A straightforward analysis shows that a Hopf bifurcation is possible at A = A given by
272
Volume 42, number 4
OPTICS COMMUNICATIONS
= 1 + KI2__~ ,,72 ['~7117±+ (r + 7.t)(7.t + 711)-- D'2] ,
15 July 1982
(lS)
where 122 = 71171(~K.4 -- ~r + A T t ) / ( r + 711 + 27±).
(19)
Note that (18)__provides an implicit relation to obtain,4 as a function of the decay rates r , 71[ and 7i. It is easy to show that A andA exist only in the bad cavity limit so that r > 7i + 711. This is why these instabilities are not present when the atomic variables are adiabatically eliminated.
5. Discussion The previous analysis shows that under suitable operating conditions the continuous wave unidirectional mode of a ring laser becomes unstable at two critical pump parameters.4 and A" leading to Hopf bifurcations. The Hopf theorem [23] asserts that a time periodic (self-pulsing) solution does exist beyond each bifurcation point. However, it remains to be shown that the resulting periodic solution is stable beyond the bifurcation point. This can be achieved using a well-known technique [23] which was recently applied to intracavi~ second harmonic generation [21]. Our analysis shows that beyond but near the Hopf bifurcation point A = A < , 4 the unidirectional cw mode gives way to a stable unidirectional self.pulsing mode with the modulation frequency ~1 given by (16). On the other hand, beyond but near the Hopf bifurcation point A =,4 < A" the instability of the zero mode gives rise to a stable bidirectional self-pulsing mode with the modulation fr__equeney~22 given by (19). Which one of ~esebifurcations occurs first depends on the relative magnitude of A and A. Our estimates show that generally A < A and unidirectional pulsing is likely to occur first. However, since the bifurcation atA =A" does not affect the stability of the zero mode, we conjecture that unidirectional pulsing will eventually be destabilized to give way to our predicted bidirectional pulsing. Note that in both cases, in a symmetric ring laser mode hopping will make the direction of pulsation to switch randomly. It should be mentioned that the stability of the timeperiodic solutions has been examined only in the vicinity of the bifurcation points. As the pump parameter A is further increased it is probable that further instabilities will arise. In particular the occurence of chaos is n o t ruled out [24--27]. An investigation of this phenomenon will require numerical solutions of our complete set of eqs. (4)-(8). The present work can be generalized in several directions. The dispersive effects are known to give rise to new instabilities even for a unidirectional single-mode laser [19]. The influence of spatial hole-burning on the dispersive effects is also worth investigating. In the particular set-up used to build gyro lasers [1,2,3] an additional effect is likely to appear. In such devices, the gas is excited by an electric discharge through two electrodes inserted in the cavity. Therefore in part of the cavity the gas population is inverted leading to amplification, but the remaining part of the cavity contains the unexcited gas which will act as a saturable absorber. It is known that a laser with a saturable absorber displays optical bistability [19] even if both amplifying and absorbing atoms are of the same kind [28].
Acknowledgement It is a pleasure to acknowledge E. Wollast for her interest and support. The work at City College of New York was partially supported by the US Army Research Office. The work in Brussels was partially supported by the Association Euratom-Etat beige.
273
Volume 42, number 4
OPTICS COMMUNICATIONS
References [1] F. Aronowitz, Phys. Rev. 139 (1965) A635. [2] V.E. Privalov and S.A. Fridrikbov, Sov. Phys. Uspekhi 12 (1969) 153. [3] L.N. Menegozzi and W.E. Lamb Jr., Phys. Rev. A8 (1973) 2103. [4] J.A. White, Phys. Rev. 137 (1965) A1651. [5] B.L. Zhelnov, A.P. Kazantsev and V.S. Smirnov, Soy. Phys. Solid State 7 (1966) 2276. [6] W.W.Rigrod and T.J. Bridges, IEEE J. QE 1 (1965) 298. [7] J.B. Hambenne and M. Sargent III, IEEE J. QE 11 (1975) 90; Phys. Rev. A13 (1976) 797. [8] M. Sargent III, AppL Phys. 9 (1976) 127. [9] D. Kiihlke and W. Dietel, Opt. Quant. Electron. 9 (1977) 305. [10] M.M. Tehrani and L. Mandel, Phys. Rev. A17 (1978) 677 and 694. [11] D. Kfihlke and R. Horak, Opt. Quant. Electron. 11 (1979) 485. [12] R. Roy, R. Short, J. Durnin and L Mandel, Phys. Rev. Lett. 45 (1980) 1486. [13] D. Kfihlke and G. Jetsehke, Physica 106C (1981) 1287. [14] D. Kfthlke and R. Horak, Physiea l l l C (1981) 111. [15] P. Lett, W. Christian, S. Sing and L. Mandel, Phys. Rev. Lett. 47 (1981) 1892. [16] H. Risken and K. Nummedal~ J. Appl. Plays. 39 (1968) 4662. [17] R. Graham and H, Hakon, Z. Physik 213 (1968) 420. [18] P. Mandel, Phys. Lett. 83A (1981) 207. [19] P. Mandel, Phys. Rev. A21 (1980) 2020. [20] T. Ernaux and P. Mandel, Z. Physik B44 (1981) 353 and 365. [21 ] P. Mandel and R. Erneux, optiea Aeta 29 (1982) 7. [221 See e.g. ref. [61. [23] D.D. Joseph and D.H. Sattinger, Arehs. ration. Mech. Analysis 45 (1972) 79. [24] M.L Rabinovich, Soy. Phys. Uspekhi 21 (1978) 443. [25] A.N. Oraevski , Soy. J. Quantum Electron. 11 (1981) 71. [26] K. Ikeda, H. Daido and O. Akimoto, Phys. Rev. Lett. 45 (1980) 709. [27] R.R. Snapp, HJ. Carmiehael and W.C. Sehieve, Optics Comrn. 40 (1981) 68. [28] W. Heudorfer and G. Marowski, Appl. Phys. 17 (1978) 181.
274
15 July 1982
OPTICS COMMUNICATIONS
15 July 1982
MODE INSTABILITIES IN A HOMOGENEOUSLY BROADENED RING LASER Paul MANDEL * UniversitJ Libre de Bruxelles, Service de Chimie-Physique II, CP. 231, 1050 Brussels, Belgium Govind P. AGRAWAL Physics Department, City College of the City University of New York, N Y 10031, USA and Quantel, B.P. 23, 91941 Les Ulis Orsay, France Received 6 April 1982
A nonperturbative theory of a homogeneously broadened ring laser is presented after incorporating spatial holeburning effects arising due to mutual interference of two counterpropaga0ng cavity modes, It is shown that under suitable conditions the continuous-wave unidirectional lasing mode becomes unstable due to Hopf bifurcations. The time-periodic state associated with these bifurcations corresponds to self-induced pulsations either in one direction or in both directions..
1. Introduction Ring lasers are, in general, capable of sustaining two counterpropagating travelling-wave modes, a feature exploited for their use in laser gyroscopes [ 1 - 3 ] . These modes, however, are coupled by the gain medium and the operating characteristics of a ring laser are highly sensitive to various factors such as the linebroadening mechanism. When the gain medium is homogeneously broadened, mode coupling is so strong that one of the modes is altogether suppressed [4--6]. It is well known that spatial hole.burning, arising due to mutual interference of two counterpropagating waves, plays an important role [7-9] in determining the extent of mode coupling. In recent years mode instabilities and associated phase transitions in a homogeneously broadened ring laser have been studied theoretically and experimentally [10--15]. In most of the previous work the analysis is simplified using third-order perturbation theory for the induced polarisation. Moreover, stability criteria for the steady-state mode intensities are often formulated after adiabatically eliminating the gain medium atomic variables. This is a severe limitation and it is known that a more general stability analysis is required to investigate some self-pulsing instabilities [16-21 ]. In this paper a theoretical analysis of a homogeneously broadened ring laser is presented by relaxing both of the above-mentioned limitations. The theory predicts a new Hopf bifurcation which has previously gone unnoticed.
2. General formulation We consider a ring cavity containing the gain medium which is modeled as a homogeneously broadened and inverted two-level system with the atomic transition frequency Wa" The laser is assumed to oscillate at the single frequency w. The operating conditions are taken to be such that the ring cavity behaves in a symmetric manner with * Maitre de Reeherches FNRS.
~~.: 269
Volume 42, number 4
OPTICSCOMMUNICATIONS
15 July 1982
respect to its counter-propagating modes at ¢o. Although the dispersive effects may play a significant role under certain conditions [19], in the following we assume exact resonance so that co = coa = coe where coc is the cavity resonance frequency. The laser characteristics are obtained by solving the Maxwell-Bloch equations extended to incorporate the counter-propagating nature of the cavity field. Making the usual plane wave and slowly varying envelope approximations and following the approach and notation of ref. [19], the semi-classical equations are: Z i(a t
+ X) ~/(t) = N~. f gT(x) a(x, t) dx
(la)
o i(a t
+ 2ti) a(x, t) = -D(x, t)1..~1,2 gj(x) [3l(t) ,
l o t + "rll)D(x, t) = itrt u + 2 ~
j: 1,2
[gj(x) a*(x, t) [3j(t) - c.c.l ,
(lb)
(lc)
where ~ ( t ) is the slowly varying field envelope, a(x, t) is the induced polarization and D(x, t) = (N+-N )/(N+ +N The mode index isj = 1, 2 for the forward and the backward wave, respectively. The quantities N+(x, t),N_(x, t ) , K, 711 and 7x correspond to the upper and lower atomic numbers, the cavity decay rate, the longitudinal and transverse relaxation rates. Further, the coupling constant is
gl(x) = --i(21rco)l/20a/h) exp(+-ikx)
(2)
where kc = co,/z is the dipole matrix element and x is measured along the cavity perimeter. The upper and lower sign refer t o j =1 andj = 2, respectively. Eqs. (1) fully incorporate the effects of spatial hole-burning arising due to interference between the counterpropagating waves. It is convenient to define
~j(t) = Ei(t ) + iFj(t) ,
g;(x) ~(x, t) = iVj(x, t) - Uj(x, t) ,
(3)
and obtain the following set of nine coupled equations:
N L
(a t + K) Ej (t) = I, f 1~ (x, t) dx , 0 N L Ot + x) Fj(t) = -[. f o
(4)
(a t + ~j.) U1(x, t) = g~-D(x, t) [Fj(t) + ,%_,dt) cos :Zkx + (-)JE3w(t) sin 2 ~ ] ,
(6)
(at + Vj_) V)(x, t) = g2D(x, t)[ej(t) + e3w(t) cos 2 ~ + (-)iF3 q(t) sin 2 ~ ] ,
(7)
(5)
o
(a t + ~,ll)D(x, t) = o~ll - 4 ~
1:1,2
[Ej(t) Vj(x, t) ÷ Fj(t) Uj(x, t)] ,
(8)
where g2 = [gl 12 = ig212 = 21rco/a2flt 2 is x-independent.
3. Steady-state mode intensifies The calculation of the steady-state solution of eqs. ( 4 ) - ( 8 ) is simplified if we note that in the absence of any external reference the phase of the electric fields is arbitrary and one can choose F/= 0 without any loss of gener270
).
Volume 42, ]lumber 4
OPTICS COMMUNICATIONS
15 J u l y 1 9 8 2
ality. From eqs. (6)-(8) we then obtain
D(x) = o/[1 + S(E 2 + E22 + 2EIE 2 cos 2kx)] ,
(9)
= ¢a2D(x)/9',)(E/+ E3_ / cos 2gx), where S = 4g2h'llg'± is the saturation parameter. Spatial hole-burning is evident in the expression for D and has its origin in the formation of a population grating induced by the counter-propagating waves. We substitute (9) in (4) and perform the integration. The reduced mode intensities//= SE? verify 2/i +A [1 +13_ / - / / ) [ D ] = 0 ,
(10)
where D = 1 + 2(• 1 +•2) + (I1 - •2) 2 and the pump parameter A =Ng2o/rVl is def'med such thatA = 1 corresponds to the laser threshold [19]. The solutions of (10) can be grouped in three distinct classes: (i) the trivial solution 11 = 12 = 0 , (ii) the symmetric solution I 1 =12 = ] [4/1 - 1 - (8/1 + 1)1/2], (iii) the asymmetric unidirectional solutions//=A - 1,13_ / = 0 f o r / = 1 or 2. A linear stability analysis is performed to examine the stability of each solution under small perturbations. In general, the complete set ( 4 ) - (8) of nine dynamical variables should be considered and the results will be presented in the next section. Here we examine the stability in the good cavity limit, assuming r "<9'1, 9'11 SO that the atomic variables can be adiabatically eliminated. Under these conditions the stability analysis can be based on
a:/=rI/+ U
"
_
I/)/O].
½ A[1 - (1 +I3_ j st
i
.
.
.
.
~
.
.
Ass mmg li(t) - I ) +lj exp(Xt) and lmear]zang ln1~ we obtain a quadratic polynomial in ?~. In order for 1 .st to be stable it is sufficient that Re X < 0 for both roots. 1?he following picture emerges: the trivial solution is si/able below thresholdA ~< 1. Above thresholdA > 1, the trivial and symmetric solutions are unstable while the unidirectional solutions are stable. However since the ring laser is assumed to have no prefered direction, it can oscillate in either of the modes determined by internal fluctuations. As a matter of fact, mode hopping, where in a ring dye laser spontaneously switches its lasing direction, is a wellknown phenomenon [15]. It should be remarked that mode hopping, often an undesirable device feature, can be eliminated by breaking the forward-backward symmetry. Among other means, it can be achieved by differential cavity toss mechanisms, external injection or the use of an additional mirror [22]. For an asymmetric ring laser, an analytic treatment is, unfortunately, not feasible. We considered numerically the case of an additional mirror and the results can be summerized as follows. Below threshold, 11 = I 2 = 0 as before. Above threshold the solution 11 > t 2 ~ O, arising from the perturbation of the symmetric branch, remains unstable so that bidirectional lasing is ruled out. The only remaining unidirectional solution 11 = A - 1 and 12 = 0 is stable and thus the laser operates without mode hopping.
4. General linear stability analysis We now examine the stability of the three classes of steady state solutions without imposing any restriction on the cavity and atomic decay rates, i.e. by considering the full set of eqs. (4) to (8). Assuming X(t) = X st + X'exp(Xt) where X is any of the nine dynamical variables in (4)-(8) with its stationary value X st andX' is a small perturbation, the decay constant Satisfies a ninth degree polynomal. A solution will be stable iff Re ), < 0 for all nine roots. For the trivial solution I 1 =12 = 0, there is a single nondegenerate root X = - 7 u andtwo fourfold degenerate roots given by
CA + ~i)(~ + K) =AK~I.
271
Volume 42, number 4
OPTICS COMMUNICATIONS
15 July 1982
It is easy to see that one of the roots becomes positive forA > 1 indicating an instability of the trivial solution above threshold. For the symmetric solution 11 = I 2 > 0, the ninth degree polynomial factors out into two qua&aries and a quintic. The quadratics are (X + 7±)0~ + ~) = K7±,
(11)
CA+ 7±)(X + g) = ~71(1 + 4/1).
(12)
The roots of (I I) are X = 0 and ~, = - ~ - 7±. The presence of a zero root indicates the existence of an invariant in the set (4)-(8). Its origin ~an be traced back to the fact that the phase difference, and not the individual phases of the counter-propagating waves, is a relevant variable and implies that eqs. (4)-(8) can be reformulated in terms of eight variables only. One of the roots of (12) is always positive for any I 1 > 0 indicating that the symmetric steady solution is always unstable forA > 1. We may remark that this result is a direct consequence of spatial hole-burning enforced by homogeneous broadening. Indeed for a Doppler-broadened gain medium, spatial holes are washed aut by atomic motion [7,8], mode competition is reduced and under certain conditions both modes can osdllate simultaneously [1,3] whereas the asymmetric solution becomes unstable. We now discuss the stability of one of the two unidirectional solutions, e.g. I 1 =A - 1,12 = 0. It is governed by a ninth degree polynomial which factors into a quadratic, a cubic and a quartic. The quadratic is identical to (11) and requires no further considerations. The cubic and the quartie are (X+ r)(X + 7± + b) = ~-/.,
(13)
(?, + 71)CA + ~)(X + 7 , + b) = -~rT± (2k + 27± + b ) ,
(14)
b = 7H(A - 1)(X + 27±)/(X + 2711 - ATll) .
(15)
where
It is useless to solve these equations in dosed form for X. We therefore first show that in the vicinity of the threshold,A = 1 + e (0 < e ,< 1), the unidirectional solution is stable. To the leading order in e, the roots of (13) are -711, - r - 71 and - e r 7 j / ( r + 7±). Three roots of the qtmrtic (14) are roots of (13), the fourth being X = -7±. Both unidirectional solutions are therefore stable above but near threshold. Well above threshold the unidirectional solution will be destabilized if the real part of a root of (13)-(14) vanishes. It is important to note that (13) and (14) govern the stability of the nonzero- and the zero-intensity mode, respectively. Let us first consider the cubic. It is readily verified that no real root can vanish for A > 1. This leaves open the possibility of a Hopf bifurcation where the real part of a pair of complex conjugate roots vanishes. To check the occurrence of such a bifurcation we solve (13) with the ansatz (X-/z)(X 2 + ~212)= 0 and find /a = --r -- 7± -- 7 u ,
~12 = 7U(r + ~'IJT) ,
(16)
A" = KCK+ 371 + 711)/[7±(r --711- 7±)] •
(17)
where
A Hopf bifurcation therefore takes place at the critical value A =.4. The expression (17) for/T is identical to the one obtained previously [16,17] while investigath'ag self-pulsing in a unidirectional ring laser. This is not surprising because when fluctuations of the zero.intensity mode are ignored, the bidirectional case reduces to the unidirectional case. Let us now consider the stability of the zero-intensity mode. Again, no real root of the quartic (14) can vanish for A > 1. We therefore solve (14) with the ansatz (X - ;11) Q, - ta2) Ca2 + ~22) = 0. A straightforward analysis shows that a Hopf bifurcation is possible at A = A given by
272
Volume 42, number 4
OPTICS COMMUNICATIONS
= 1 + KI2__~ ,,72 ['~7117±+ (r + 7.t)(7.t + 711)-- D'2] ,
15 July 1982
(lS)
where 122 = 71171(~K.4 -- ~r + A T t ) / ( r + 711 + 27±).
(19)
Note that (18)__provides an implicit relation to obtain,4 as a function of the decay rates r , 71[ and 7i. It is easy to show that A andA exist only in the bad cavity limit so that r > 7i + 711. This is why these instabilities are not present when the atomic variables are adiabatically eliminated.
5. Discussion The previous analysis shows that under suitable operating conditions the continuous wave unidirectional mode of a ring laser becomes unstable at two critical pump parameters.4 and A" leading to Hopf bifurcations. The Hopf theorem [23] asserts that a time periodic (self-pulsing) solution does exist beyond each bifurcation point. However, it remains to be shown that the resulting periodic solution is stable beyond the bifurcation point. This can be achieved using a well-known technique [23] which was recently applied to intracavi~ second harmonic generation [21]. Our analysis shows that beyond but near the Hopf bifurcation point A = A < , 4 the unidirectional cw mode gives way to a stable unidirectional self.pulsing mode with the modulation frequency ~1 given by (16). On the other hand, beyond but near the Hopf bifurcation point A =,4 < A" the instability of the zero mode gives rise to a stable bidirectional self-pulsing mode with the modulation fr__equeney~22 given by (19). Which one of ~esebifurcations occurs first depends on the relative magnitude of A and A. Our estimates show that generally A < A and unidirectional pulsing is likely to occur first. However, since the bifurcation atA =A" does not affect the stability of the zero mode, we conjecture that unidirectional pulsing will eventually be destabilized to give way to our predicted bidirectional pulsing. Note that in both cases, in a symmetric ring laser mode hopping will make the direction of pulsation to switch randomly. It should be mentioned that the stability of the timeperiodic solutions has been examined only in the vicinity of the bifurcation points. As the pump parameter A is further increased it is probable that further instabilities will arise. In particular the occurence of chaos is n o t ruled out [24--27]. An investigation of this phenomenon will require numerical solutions of our complete set of eqs. (4)-(8). The present work can be generalized in several directions. The dispersive effects are known to give rise to new instabilities even for a unidirectional single-mode laser [19]. The influence of spatial hole-burning on the dispersive effects is also worth investigating. In the particular set-up used to build gyro lasers [1,2,3] an additional effect is likely to appear. In such devices, the gas is excited by an electric discharge through two electrodes inserted in the cavity. Therefore in part of the cavity the gas population is inverted leading to amplification, but the remaining part of the cavity contains the unexcited gas which will act as a saturable absorber. It is known that a laser with a saturable absorber displays optical bistability [19] even if both amplifying and absorbing atoms are of the same kind [28].
Acknowledgement It is a pleasure to acknowledge E. Wollast for her interest and support. The work at City College of New York was partially supported by the US Army Research Office. The work in Brussels was partially supported by the Association Euratom-Etat beige.
273
Volume 42, number 4
OPTICS COMMUNICATIONS
References [1] F. Aronowitz, Phys. Rev. 139 (1965) A635. [2] V.E. Privalov and S.A. Fridrikbov, Sov. Phys. Uspekhi 12 (1969) 153. [3] L.N. Menegozzi and W.E. Lamb Jr., Phys. Rev. A8 (1973) 2103. [4] J.A. White, Phys. Rev. 137 (1965) A1651. [5] B.L. Zhelnov, A.P. Kazantsev and V.S. Smirnov, Soy. Phys. Solid State 7 (1966) 2276. [6] W.W.Rigrod and T.J. Bridges, IEEE J. QE 1 (1965) 298. [7] J.B. Hambenne and M. Sargent III, IEEE J. QE 11 (1975) 90; Phys. Rev. A13 (1976) 797. [8] M. Sargent III, AppL Phys. 9 (1976) 127. [9] D. Kiihlke and W. Dietel, Opt. Quant. Electron. 9 (1977) 305. [10] M.M. Tehrani and L. Mandel, Phys. Rev. A17 (1978) 677 and 694. [11] D. Kfihlke and R. Horak, Opt. Quant. Electron. 11 (1979) 485. [12] R. Roy, R. Short, J. Durnin and L Mandel, Phys. Rev. Lett. 45 (1980) 1486. [13] D. Kfihlke and G. Jetsehke, Physica 106C (1981) 1287. [14] D. Kfthlke and R. Horak, Physiea l l l C (1981) 111. [15] P. Lett, W. Christian, S. Sing and L. Mandel, Phys. Rev. Lett. 47 (1981) 1892. [16] H. Risken and K. Nummedal~ J. Appl. Plays. 39 (1968) 4662. [17] R. Graham and H, Hakon, Z. Physik 213 (1968) 420. [18] P. Mandel, Phys. Lett. 83A (1981) 207. [19] P. Mandel, Phys. Rev. A21 (1980) 2020. [20] T. Ernaux and P. Mandel, Z. Physik B44 (1981) 353 and 365. [21 ] P. Mandel and R. Erneux, optiea Aeta 29 (1982) 7. [221 See e.g. ref. [61. [23] D.D. Joseph and D.H. Sattinger, Arehs. ration. Mech. Analysis 45 (1972) 79. [24] M.L Rabinovich, Soy. Phys. Uspekhi 21 (1978) 443. [25] A.N. Oraevski , Soy. J. Quantum Electron. 11 (1981) 71. [26] K. Ikeda, H. Daido and O. Akimoto, Phys. Rev. Lett. 45 (1980) 709. [27] R.R. Snapp, HJ. Carmiehael and W.C. Sehieve, Optics Comrn. 40 (1981) 68. [28] W. Heudorfer and G. Marowski, Appl. Phys. 17 (1978) 181.
274
15 July 1982