- Email: [email protected]
Optical bistability via a phase fluctuation effect of the control field
4 November 19%
PHYSICS LETTERS A
ELSEVIER
Physics Letters A 222 (1996) 237-240
Optical bistability via a phase fluctuation effect of the control field Shaug-qing Gong a,b,Si-de Du a*b,Zhi-zhan Xu a7b,Shao-hua Pan arc a China Cenfer of Aduanced Science and Technology (World Laboratory), P.O. Box 8730, Beijing 100080, China b Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China ’ ’ Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China ’ Received 2 July 1996; accepted for publication 13 August 1996 Communicated by B. Fricke
The steady-state behavior in a unidirectional cavity has been analyzed by taking into account the effect of phase fluctuation in the control field based on a A-type atomic model. It is shown that even if both fiefds are in resonance with the corresponding atomic transitions, optical bistability can still be realized due to a phase fluctuation in the control field. Furthermore, the bistable hysteresis loop becomes wider as the linewidth increases. PACS: 42.65.P~; 42.50.Md; 42.50.H~
1. Introduction Quaff many new
coherence and interference have led to optical effects, such as lasing without
inversion [l-3], enhancement of the refractive index [4-61, electromagnetically induced transparency [7-91 and spontaneous emission reduction and cancellation [lo]. Very recently, Harshawardhan and Agarwal [l l] investigated the role of atomic coherence and interference in optical bistability. They found that these effects result in a considerable lowering of the threshold intensity. It is well known [12] that phase fluctuation of the driving field can significantIy influence the optical
’ Mailing address.
properties of driven atomic systems. Fleischhauer et al. 1131have shown that, in a double-A noninversion laser, the pump-meld phase ~f~sion leads to a decay of the coherent trapping state and leads to a fluctuating oscillation frequency in the coherence which thereby reduces the laser gain. We found that in a simple three-level atomic system, when the driving field is not very strong, a change from an inversion laser to a noninversion laser action can occur as the driving-field linewidth increases [14]. We also found that the linewidth of the driving field precludes the medium becoming transparent [ 151. In this paper, we investigate the steady-state behavior in a unidirection~ cavity (Fig. 1) by taking into account phase ~uc~a~on of the control field based on a A-type model. For simplicity, we assume that mirrors 3 and 4 have 100% reflectivity, and call
03759601/96/$12.00 Copyright 0 1996 Elsevier Science B.V. All rights reserved. PII SO375-9601(96)00637-S
S. Gong et al./Physics
238
Letters A 222 (19961237-240
Under the rotating wave approximation, the interaction ~~iltoni~ between the atom and the two fields can be written as H= -[Ee-i~‘~1)(31+0e-i”“11)(21]
+c.c., (I)
Fig. 1. Unidir~tion~ ring cavity. E1 and ET are the incident and transmitted fields, respectively. a represents the control field.
R and T (with R + T = 1) the reflection and trans-
mission coefficient of mirrors 1 and 2. We describe the dynamics of the coupled system (atoms plus radiation fields) by the well-known Maxwell-Bloch equations. We find that, owing to the phase fluctuation of the control field, optical bistability can still be realized even if both the fields are in resonance with the corresponding atomic transitions. Furthermore, the bistable hysteresis loop becomes wider as the linewidth of the control field increases.
2. Equations of dynamics Consider a closed A-type medium with excited state 11) and lower states 12) and 13), as illustrated in Fig. 2. The excited state i I> decays to j 3) and 12) with decay rates 2 R, and 2R,, respectively. A coherent field with Rabi frequency 2E drives the transition between states 13) and I l}, and a control field with Rabi frequency 2ti is applied to the transition 1l} t, 12). The control field does not circulate in the unidirectional cavity, and thus its dynamical evolution can be ignored.
where u and w’ are the frequencies of the coherent field and the control field, respectively. Let #(t) represent the phase fluctuation of the control field, i.e., 0 = 0, exp[i+( t)] , (2) where 0, is assumed to be real. The phase is characterized by the following random equation of motion [ 161, 4(t) = P(T), (3) with zero average, i.e., { I.L(~))= 0. IIere p(_tct> is a S-correlated Langevin-noise term, whose diffusion coefficient gives the linewidth 2R, of the control field, i.e., (EL(~)~LI~))=~RL~(~-~‘). (4) In this case, the Maxwell-Bloch equations should be averaged over the randomly fluctuating phase. That is, the density-matrix elements pii, pij and the coherent field E must be replaced by their stochastic averaged values ( pii), ( Pij) and (E), respectively. With the usual phenomenological inclusion of the damping processes and by using the methods in Refs. [ 14,15,17], the corresponding Maxwell-Bloch equations can be derived as follows, ( PZ2) = 2R,( p,t) + i%((
pi2) - (~%t))~
(5)
(if33)=2R1(~il)+i((Ef)(~13)-(E)(~31))r (6) (Pr2)=
-(R,+R,+&,+~A,)(P~~) + ifl,((
pz2) - ( pi,)) +i(EX
pa2), (7)
< P13) = --(Et
-t-RS+iAl)(~r3)
+ i(E)((
pX3) - ( pi,)) f iai2,( i&.
12)
(8) t&j=
-[R,+i(A,-Az)l(p,,)-i(E)(pzi) + ia,< p,J,
Fig. 2. A-type three-level atomic system. w (CO’) is the frequency of a coherent field (control field); ZR, and 2R, are the decay rates.
a(E)
-+C-=:
at
a(E)
az
2rriwd,,(P(
(9) w)).
(10)
S. Gong et al./ Physics Lerters A 222 119961237-240
are thefreHere A, = u,~ - o and A2=o,z-~’ quency detnnings. (P(w)> =Nd,,( P,~), where N is the density of atoms and d3, is a matrix element of the atomic dipole moment. From Eqs. (7) and (91, we can see that the phase quotation in the control field modify the off-diagonal rates R,, = R, + R, and R,, = 0 to R,, = R, + R, + R, and R,, = R,, respectively, In other words, because of the phase fluctuation of the control field, the off-diagonal decay rates now have additional diffusion terms along with the usual decay rates. For a monochromatic control field (RL = 01, the situation reduces to the case given by Ref. ill], which corresponds to Eq. (7) of Ref. ill]. The coherent field Ei enters into the cavity from the left and drives the atoms. For a perfectly tuned cavity, the relations between E,, and the transmitted field ET, and the fields EiO) and E(L) are [l&19]
E(L) = ET/e,
(11)
E(O) = J?;E; f RE( L).
(12)
where L is the length of the atomic sample, and the second contribution on the right-hand side of Eq. (12) describes a feedback mechanisms due to the mirror, which is essential to give rise to bistability, namely, there will be no bistability if R = 0. 3. Steady-state behavior We consider here the situation A, = A, = 0, that is, the coherent field and the control field are assumed to be in resonance with the corresponding atomic transitions. Setting all the time derivatives in Eqs. (5)-(10) equal to zero for the steady state, we can obtain the field amplitude (E) as follows, a(E) = -2rNmd$( E) Im( P,~), ( 13) 32 where Im( p,s) can be obtained from the steady-state solution of Eqs. 6)-(9). It depends on the atomic decay rates, the Rabi frequencies of the coherent field and the control field as well as the linewidth. In the mean-field limit [ 181, using the boundary conditions (11) and (121, we obtain the mean-field state equation Y=X+2CXR,
Im( P,~(X)),
where Y = E,/ fi,
X = ET/ fi
(14) and C is the usual
239
Cooper-action parameter [ 11,181. The nonlinear term of Eq. (14) is essential to give rise to bistability. For a monochromatic control field (R, = 01, with the help of electromagnetically induced interference, both Im( pi3) and Re( p,s) vanish at zero detunings 1151.In this case E, = ET, thus we can say that when A, = A, = 0, bistability is impossible for R, = 0, in agreement with the result of Ref. ill]. While for R, f 0, the finite linewidth of the control field dephases the atomic coherence and leads to the destmction of trapping [12,13]. Due to the decoherence effect, the transparency of the medium for zero detunings is destroyed, i.e., there is absorption of the medium [15]. In this case, the nonlinear term of Rq. (14) has a nonzero value, and optical bistability may be realized for certain conditions. This behavior of the system can be clearly shown by the following examples. In numerical calculations, we will choose the parameters to be dimensionless (i&,/R, and R, = 1, etc.). Choosing the atomic decay rate R, = 1.0, the Rabi frequency L?, = 1.O and the cooperation parameter C = 400.0, we obtain the plots of transmitted light versus incident light for various values of linewidth R,, as shown in Fig. 3. In this figure, curve 1 corresponds to R, = 0, curve 2 to R, = 0.1, curve 3 to R, = 0.5 and curve 4 to R, = 1.0. It is easy to see from Fig. 3 that for a monochromatic driving field (RL = 0), the relation between the **
1
J
0
IO
20
30
40
50
60
70
80
incident Field Fig. 3. Plots of transmitted linewidths R,. The values 1.0 and C = 400.0. Curve R, = 0.1, curve 3 to R, =
light versus incident tight for various of the parameters are R, = 1.O, 0, = 1 corresponds to R, = 0.0, curve 2 to 0.5 and curve 4 to R, = 1.O.
240
S. Gong et al. / Physics Letters A 222 (19961237-240
transmitted light and the incident light is linear, and hence there is no optical bistability. However, for a given linewidth (e.g., R, = 0.5 in Fig. 31, Em{ P,~) # 0 (absorption of the medium exists), the relation between the transmitted light and the incident light is nonlinear, and optical bistability occurs. In the resonant coupled system of atoms and radiation fields, the physical origin of bistability is the linewidth of the control field. Thus we can say that optical bistability can be realized in the unidirectional cavity system due to a phase fluctuation effect of the control field. We can also see from this figure that the bistable hysteresis loop becomes wider as the linewidth increases.
4. Summary In this paper, we investigate steady-state behavior in a unidirectional cavity by taking into account the effects of phase ~uc~ation of control field based on the A model. We find that optical bistability can still be realized even if both the fields are in resonance with the corresponding atomic transition. Furthermore, the bistable hysteresis loop becomes wider as the linewidth increases.
This work was supported by the Natural Science Foundation of China.
References [l] SE. Harris, Phys. Rev. Lett. 62 (1989) 1033. 121 M.O. Scully, S.Y. Zhu and A. Gavrielides, Phys. Rev. Lett. 62 (1989) 2813. [3] 0. Kccharovskaya and Ya.L Khanin, Pis’ma Zh. Eksp. Tear. Fiz. 48 (1988) 581 [JEPT Lett. 48 0988) 6301. [4] M.O. Scully, Phys. Rev. Lett. 67 (1991) 1855. [51 M.O. Scully and S.Y. Zhu, Opt. Commun. 87 (1992) 134. f61 U. Rathe, M. Fleischhauer, S.Y. Ztm, T.W. Hansch and M.O. Scully, Phys. Rev. A 47 (1993) 4994. [7] A. Imamoght, I.E. Field and S.E. Harris, Phys. Rev. Lett. 66 (1991) 1154. IS] SE. Harris, J.E. Field and A. Kasapi, Phys. Rev. A 46 (1992) R29. [9] J.C. Petch, C.H. Keitel, P.L. Knight and J.P. Marcngous, Phys. Rev. A 53 (1996) 543. [lo] S.Y. Zhu and M.O. Scully, Phys. Rev. Lett. 76 (1996) 388. Ill] W. Harshawardhan and G.S. Agarwal, Phys. Rev. A 53 (1996) 1812. 1121 B.J. Dalton and P.L. Knight, Opt. Commun. 42 (1982) 411; J. Phys. B 15 (1982) 3997. [13] M. Fleischhauer, D. Lukin, D.E. Nikonov and M.O. Scully, Opt. Commun. 110 (1994) 351. Zhaug and [I41 Shang-qiug Gong, Zhi-zhan Xu, Zheng-quan Shao-hua Pan, Phys. Rev. A 52 (1995) 4787. [15] Shang-qing Gong and Zhi-zhan Xu, Opt. Commun. 115 (1995) 65; Shang-qing Gong, Zttan-zhan Xu and Shao-hua Pan, J. Mod. Optics 42 (1995) 1397. [I61 W.H. Louisell, Quantum statistical properties of radiation (Wiley, New York, 1973). [17] M. Fleischhauer and M.O. Scully, Phys. Rev. A 49 (1994) 1973. [IS] R. Bonifacio and L.A. Lugiato, Len. Nuovo Cimento 21 (1978) 505. 1191 Shang-qing Gong, Shaehua Pan and Guo-zhen Yang, Phys. Rev. A 45 (1992) 6655.
PHYSICS LETTERS A
ELSEVIER
Physics Letters A 222 (1996) 237-240
Optical bistability via a phase fluctuation effect of the control field Shaug-qing Gong a,b,Si-de Du a*b,Zhi-zhan Xu a7b,Shao-hua Pan arc a China Cenfer of Aduanced Science and Technology (World Laboratory), P.O. Box 8730, Beijing 100080, China b Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China ’ ’ Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China ’ Received 2 July 1996; accepted for publication 13 August 1996 Communicated by B. Fricke
The steady-state behavior in a unidirectional cavity has been analyzed by taking into account the effect of phase fluctuation in the control field based on a A-type atomic model. It is shown that even if both fiefds are in resonance with the corresponding atomic transitions, optical bistability can still be realized due to a phase fluctuation in the control field. Furthermore, the bistable hysteresis loop becomes wider as the linewidth increases. PACS: 42.65.P~; 42.50.Md; 42.50.H~
1. Introduction Quaff many new
coherence and interference have led to optical effects, such as lasing without
inversion [l-3], enhancement of the refractive index [4-61, electromagnetically induced transparency [7-91 and spontaneous emission reduction and cancellation [lo]. Very recently, Harshawardhan and Agarwal [l l] investigated the role of atomic coherence and interference in optical bistability. They found that these effects result in a considerable lowering of the threshold intensity. It is well known [12] that phase fluctuation of the driving field can significantIy influence the optical
’ Mailing address.
properties of driven atomic systems. Fleischhauer et al. 1131have shown that, in a double-A noninversion laser, the pump-meld phase ~f~sion leads to a decay of the coherent trapping state and leads to a fluctuating oscillation frequency in the coherence which thereby reduces the laser gain. We found that in a simple three-level atomic system, when the driving field is not very strong, a change from an inversion laser to a noninversion laser action can occur as the driving-field linewidth increases [14]. We also found that the linewidth of the driving field precludes the medium becoming transparent [ 151. In this paper, we investigate the steady-state behavior in a unidirection~ cavity (Fig. 1) by taking into account phase ~uc~a~on of the control field based on a A-type model. For simplicity, we assume that mirrors 3 and 4 have 100% reflectivity, and call
03759601/96/$12.00 Copyright 0 1996 Elsevier Science B.V. All rights reserved. PII SO375-9601(96)00637-S
S. Gong et al./Physics
238
Letters A 222 (19961237-240
Under the rotating wave approximation, the interaction ~~iltoni~ between the atom and the two fields can be written as H= -[Ee-i~‘~1)(31+0e-i”“11)(21]
+c.c., (I)
Fig. 1. Unidir~tion~ ring cavity. E1 and ET are the incident and transmitted fields, respectively. a represents the control field.
R and T (with R + T = 1) the reflection and trans-
mission coefficient of mirrors 1 and 2. We describe the dynamics of the coupled system (atoms plus radiation fields) by the well-known Maxwell-Bloch equations. We find that, owing to the phase fluctuation of the control field, optical bistability can still be realized even if both the fields are in resonance with the corresponding atomic transitions. Furthermore, the bistable hysteresis loop becomes wider as the linewidth of the control field increases.
2. Equations of dynamics Consider a closed A-type medium with excited state 11) and lower states 12) and 13), as illustrated in Fig. 2. The excited state i I> decays to j 3) and 12) with decay rates 2 R, and 2R,, respectively. A coherent field with Rabi frequency 2E drives the transition between states 13) and I l}, and a control field with Rabi frequency 2ti is applied to the transition 1l} t, 12). The control field does not circulate in the unidirectional cavity, and thus its dynamical evolution can be ignored.
where u and w’ are the frequencies of the coherent field and the control field, respectively. Let #(t) represent the phase fluctuation of the control field, i.e., 0 = 0, exp[i+( t)] , (2) where 0, is assumed to be real. The phase is characterized by the following random equation of motion [ 161, 4(t) = P(T), (3) with zero average, i.e., { I.L(~))= 0. IIere p(_tct> is a S-correlated Langevin-noise term, whose diffusion coefficient gives the linewidth 2R, of the control field, i.e., (EL(~)~LI~))=~RL~(~-~‘). (4) In this case, the Maxwell-Bloch equations should be averaged over the randomly fluctuating phase. That is, the density-matrix elements pii, pij and the coherent field E must be replaced by their stochastic averaged values ( pii), ( Pij) and (E), respectively. With the usual phenomenological inclusion of the damping processes and by using the methods in Refs. [ 14,15,17], the corresponding Maxwell-Bloch equations can be derived as follows, ( PZ2) = 2R,( p,t) + i%((
pi2) - (~%t))~
(5)
(if33)=2R1(~il)+i((Ef)(~13)-(E)(~31))r (6) (Pr2)=
-(R,+R,+&,+~A,)(P~~) + ifl,((
pz2) - ( pi,)) +i(EX
pa2), (7)
< P13) = --(Et
-t-RS+iAl)(~r3)
+ i(E)((
pX3) - ( pi,)) f iai2,( i&.
12)
(8) t&j=
-[R,+i(A,-Az)l(p,,)-i(E)(pzi) + ia,< p,J,
Fig. 2. A-type three-level atomic system. w (CO’) is the frequency of a coherent field (control field); ZR, and 2R, are the decay rates.
a(E)
-+C-=:
at
a(E)
az
2rriwd,,(P(
(9) w)).
(10)
S. Gong et al./ Physics Lerters A 222 119961237-240
are thefreHere A, = u,~ - o and A2=o,z-~’ quency detnnings. (P(w)> =Nd,,( P,~), where N is the density of atoms and d3, is a matrix element of the atomic dipole moment. From Eqs. (7) and (91, we can see that the phase quotation in the control field modify the off-diagonal rates R,, = R, + R, and R,, = 0 to R,, = R, + R, + R, and R,, = R,, respectively, In other words, because of the phase fluctuation of the control field, the off-diagonal decay rates now have additional diffusion terms along with the usual decay rates. For a monochromatic control field (RL = 01, the situation reduces to the case given by Ref. ill], which corresponds to Eq. (7) of Ref. ill]. The coherent field Ei enters into the cavity from the left and drives the atoms. For a perfectly tuned cavity, the relations between E,, and the transmitted field ET, and the fields EiO) and E(L) are [l&19]
E(L) = ET/e,
(11)
E(O) = J?;E; f RE( L).
(12)
where L is the length of the atomic sample, and the second contribution on the right-hand side of Eq. (12) describes a feedback mechanisms due to the mirror, which is essential to give rise to bistability, namely, there will be no bistability if R = 0. 3. Steady-state behavior We consider here the situation A, = A, = 0, that is, the coherent field and the control field are assumed to be in resonance with the corresponding atomic transitions. Setting all the time derivatives in Eqs. (5)-(10) equal to zero for the steady state, we can obtain the field amplitude (E) as follows, a(E) = -2rNmd$( E) Im( P,~), ( 13) 32 where Im( p,s) can be obtained from the steady-state solution of Eqs. 6)-(9). It depends on the atomic decay rates, the Rabi frequencies of the coherent field and the control field as well as the linewidth. In the mean-field limit [ 181, using the boundary conditions (11) and (121, we obtain the mean-field state equation Y=X+2CXR,
Im( P,~(X)),
where Y = E,/ fi,
X = ET/ fi
(14) and C is the usual
239
Cooper-action parameter [ 11,181. The nonlinear term of Eq. (14) is essential to give rise to bistability. For a monochromatic control field (R, = 01, with the help of electromagnetically induced interference, both Im( pi3) and Re( p,s) vanish at zero detunings 1151.In this case E, = ET, thus we can say that when A, = A, = 0, bistability is impossible for R, = 0, in agreement with the result of Ref. ill]. While for R, f 0, the finite linewidth of the control field dephases the atomic coherence and leads to the destmction of trapping [12,13]. Due to the decoherence effect, the transparency of the medium for zero detunings is destroyed, i.e., there is absorption of the medium [15]. In this case, the nonlinear term of Rq. (14) has a nonzero value, and optical bistability may be realized for certain conditions. This behavior of the system can be clearly shown by the following examples. In numerical calculations, we will choose the parameters to be dimensionless (i&,/R, and R, = 1, etc.). Choosing the atomic decay rate R, = 1.0, the Rabi frequency L?, = 1.O and the cooperation parameter C = 400.0, we obtain the plots of transmitted light versus incident light for various values of linewidth R,, as shown in Fig. 3. In this figure, curve 1 corresponds to R, = 0, curve 2 to R, = 0.1, curve 3 to R, = 0.5 and curve 4 to R, = 1.0. It is easy to see from Fig. 3 that for a monochromatic driving field (RL = 0), the relation between the **
1
J
0
IO
20
30
40
50
60
70
80
incident Field Fig. 3. Plots of transmitted linewidths R,. The values 1.0 and C = 400.0. Curve R, = 0.1, curve 3 to R, =
light versus incident tight for various of the parameters are R, = 1.O, 0, = 1 corresponds to R, = 0.0, curve 2 to 0.5 and curve 4 to R, = 1.O.
240
S. Gong et al. / Physics Letters A 222 (19961237-240
transmitted light and the incident light is linear, and hence there is no optical bistability. However, for a given linewidth (e.g., R, = 0.5 in Fig. 31, Em{ P,~) # 0 (absorption of the medium exists), the relation between the transmitted light and the incident light is nonlinear, and optical bistability occurs. In the resonant coupled system of atoms and radiation fields, the physical origin of bistability is the linewidth of the control field. Thus we can say that optical bistability can be realized in the unidirectional cavity system due to a phase fluctuation effect of the control field. We can also see from this figure that the bistable hysteresis loop becomes wider as the linewidth increases.
4. Summary In this paper, we investigate steady-state behavior in a unidirectional cavity by taking into account the effects of phase ~uc~ation of control field based on the A model. We find that optical bistability can still be realized even if both the fields are in resonance with the corresponding atomic transition. Furthermore, the bistable hysteresis loop becomes wider as the linewidth increases.
This work was supported by the Natural Science Foundation of China.
References [l] SE. Harris, Phys. Rev. Lett. 62 (1989) 1033. 121 M.O. Scully, S.Y. Zhu and A. Gavrielides, Phys. Rev. Lett. 62 (1989) 2813. [3] 0. Kccharovskaya and Ya.L Khanin, Pis’ma Zh. Eksp. Tear. Fiz. 48 (1988) 581 [JEPT Lett. 48 0988) 6301. [4] M.O. Scully, Phys. Rev. Lett. 67 (1991) 1855. [51 M.O. Scully and S.Y. Zhu, Opt. Commun. 87 (1992) 134. f61 U. Rathe, M. Fleischhauer, S.Y. Ztm, T.W. Hansch and M.O. Scully, Phys. Rev. A 47 (1993) 4994. [7] A. Imamoght, I.E. Field and S.E. Harris, Phys. Rev. Lett. 66 (1991) 1154. IS] SE. Harris, J.E. Field and A. Kasapi, Phys. Rev. A 46 (1992) R29. [9] J.C. Petch, C.H. Keitel, P.L. Knight and J.P. Marcngous, Phys. Rev. A 53 (1996) 543. [lo] S.Y. Zhu and M.O. Scully, Phys. Rev. Lett. 76 (1996) 388. Ill] W. Harshawardhan and G.S. Agarwal, Phys. Rev. A 53 (1996) 1812. 1121 B.J. Dalton and P.L. Knight, Opt. Commun. 42 (1982) 411; J. Phys. B 15 (1982) 3997. [13] M. Fleischhauer, D. Lukin, D.E. Nikonov and M.O. Scully, Opt. Commun. 110 (1994) 351. Zhaug and [I41 Shang-qiug Gong, Zhi-zhan Xu, Zheng-quan Shao-hua Pan, Phys. Rev. A 52 (1995) 4787. [15] Shang-qing Gong and Zhi-zhan Xu, Opt. Commun. 115 (1995) 65; Shang-qing Gong, Zttan-zhan Xu and Shao-hua Pan, J. Mod. Optics 42 (1995) 1397. [I61 W.H. Louisell, Quantum statistical properties of radiation (Wiley, New York, 1973). [17] M. Fleischhauer and M.O. Scully, Phys. Rev. A 49 (1994) 1973. [IS] R. Bonifacio and L.A. Lugiato, Len. Nuovo Cimento 21 (1978) 505. 1191 Shang-qing Gong, Shaehua Pan and Guo-zhen Yang, Phys. Rev. A 45 (1992) 6655.