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Low threshold instabilities in unidirectional ring lasers
Volume 64, number
OPTICS
2
LOW THRtiSHOLD
INSTABILITIES
15 October
COMMUNICATIONS
IN UNIDIRECTIONAL
1987
RING LASERS
L.A. LUGIATO lstituto di Fisica. Politecnico
di Torino, 10129 Torino, Italy
F. PRATI Dipartimento
di Fisica, Universitb di Milano, 20133 Milano. Ita1.v
D.K. BANDY, L.M. NARDUCCI, PhJ?rics Department, Received
P. RU and J.R. TREDICCE
Drexel University, Philadelphra,
PA I91 04, USA
I8 May I987
We analyze the Maxwell-Block equations for a unidirectional ring laser after inclusion of transverse effects. We prove that, in the uniform field limit, the steady state of the laser has a spatial structure corresponding to a single longitudinal and radial mode. Under the additional assumption that the cavity linewidth is much smaller than the polarization relaxation time, we calculate the eigenvalues of the linearized equations and show that dynamical instabilities can emerge even in the close neighborhood of the ordinary laser threshold. This result is in much closer agreement with the available experimental facts and in striking contrast with the predictions of the traditional plane wave models.
The common modern framework for the description of laser dynamics rests on the plane-wave Maxwell-Bloch model which has been analyzed in detail especially in the case of the unidirectional ring laser geometry. This formulation provides a solid base for understanding the operation of a laser when the parameters favor the emission of a stable output. When instabilities become a dominant dynamical feature, the Maxwell-Bloch equations yield, instead, a consistent pattern of quantitative disagreement with the experimental facts 111. In this paper we produce evidence that transverse degrees of freedom play a far more important role in laser dynamics than previously anticipated. We have known for some time that transverse effects have a strong influence on the stationary and dynamic features of passive driven systems [ 21. McLaughlin, Moloney and Newell [ 31, for example, have shown that plane-wave and transverse models produce very different routes to chaos in bistable systems. In addition, the plane-wave numerical simulations of singlemode optical bistability [4] displays complex dynamical behaviors which are not reflected in the 0 030-4018/87/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
results of the careful experiments carried out by Kimble and collaborators [ 51. The experimental pulsations are instead in much better agreement with numerical simulations that include a gaussian field profile under single-mode operation [ 61. Because it is clear that the plane-wave model removes, by definition, important physical aspects, we wish to examine a more realistic setting that includes the effects of diffraction, of the wavefront curvature introduced by the spherical mirrors, and of the transverse variations of gain introduced by the pump [7]. We limit our considerations to the paraxial approximation and, for the sake of simplicity, we assume axial symmetry and exact resonance for the unidirectional ring laser. These approximations are not entirely realistic, but already lead to significant and novel theoretical deviations from the predictions of the plane-wave models. The resonator of interest for this work is shown schematically in fig. 1. The spherical mirrors, both having a radius of curvature RO,are separated by a distance L;the ring, whose total length is /i is completed by two plane mirrors adjusted in such a way B.V.
167
Volume
64. number
OPTICS
2
r
,,I.-
Fig. I. Schematic
rcprcsen~a~~on
spherical
hale
mirrors
power rcflectlvity length
based on an active \phcrlcal
radll
R. and arc separated
of the resonator
Z
of the ring resonator.
identical
of curvature by a dlstancr
is ,l. The calculations
medium
that
The two
R,,. Ihe same 1.. The full
in this paper al-c
tills the distance
between
F(p.11.7) =exp( -iSLV)
that the angle of incidence of the light beams on all reflecting surfaces is small in order to minimize aberrations. The Maxwell-Bloch equations for a homogencously broadened two-level system have the form (see. for example, ref. [ 81) (la)
dD/ils=
[FD+
In general, the completeness relation of this basis set provides only a way to transform the field differential equation into an infinite set of coupled equations for the modal amplitudes. This is not much of an improvement from the point of view of the calculations. However. with an appropriate generalization of the well known uniform field limit of the plane-wave theory. the empty cavity eigenfunctions are also exact modes of the filled resonator. In this limit one can carry out most of the calculations in analytic terms. .As our first step we let
the
mirrors.
ilP/iir= -
Ii ocloher I’)‘(.
C’(_)MMIJNIC‘.L\l-IONS
(1 +iis,,
) P],
-;’ [ -f(F*P+FP*)+h~(p,;rl)l.
(lb) (Ic)
=exp(
-
(3aj
I;yl~.~/.s)
30:)s) C_4,,(p,t1) /,,(,~.s) /a
f'(/Vf,T)=exp(
-iriQr)
&L,/.T)
(3b)
.
where the modal functions A,,(p.q) for the resonator of fig. 1 are discussed in ref. [ 91. and r?Q is the unknown offset, measured in units of 1’ between the carrier frequency of the stationary laser field and the reference frequency. Because the modal functions are orthonormal solutions of the stationary field eq. (la) in the absence of the medium (cd=O). the Maxwell-Bloch equations expressed in terms of the modal amplitudes become
where r/=-_/L. r-y,/.
p= (71/L&,) ’ 2r. 2s=clLy
.
(2)
The symbol A0 denotes the wavelength of a cavity mode taken as a reference and 1’ is the atomic lincwidth. The quantities F, P and D represent the normalized slowly varying envelope of the electric field. the atomic polarization envelope, and the population difference, respectively. The parameter CYis the gain coefficient per unit length; 6,, is the difference between the atomic transition frequency and the reference frequency, measured in units of y : y is the damping rate of the population difference, also in units of y , , and ~(p,?) is the unsaturated population inversion. In our calculation we adopt the following strategy: we first calculate the modal functions and eigenfrequencies of the empty resonator [ 91 and then expand the cavity field in a series of radial modes.
iiP/ilr
=
df>/dT=
--
[ i;D+ ( 1 + iJ)Js] .
-;I[
- i(F*P+I.‘P*)
(3b) +I_)-x(p.q)]
where d = Cs.,(. - 652. The boundary modal amplitudes ./;,( rl.7) are /;,( -
112,s) =
R exp( -id,,)
x,f;,[I/l,T-;',
(/I-L)/C]
conditions
(4c) for the
exp[ idRy I (A - I j/c] (4d)
where R is the power reflection coefficient of the curved mirrors (the flat mirrors are assumed to be perfect reflectors for simplicity). and
Volume 64, number
d,,=(o,--w,,())
2
OPTICS
Ale.
COMMUNICATIONS
T-0,
2C=oLIT=arbitrary,
(T is the transmittivity S,=O(l)
of the curved mirrors)
>
(6a) and (6b)
we have the following rigorous result (the proof of this statement will be given in a forthcoming extended publication [ 131) &=0(l),
.&o=O(T).
(7)
I his implies that the steady state field has the structure of a TEMoo mode, as already assumed in ref. [ 21 on the basis of qualitative considerations. The modal intensity If0 I* of the TEMoo mode in steady state is given by the state equation II2
1=2c
cc
IA0 I *x(P,rl) s drl s dPPl+d*+lAo&]*’ _ I/2 0
while the detuning parameter mode-pulling formula A=AC6
l+R’
K I?-=E=z’
(8)
A obeys the familiar
CT
where K is the cavity linewidth. The specific choice of the pump profile and the corresponding equilibrium population difference x(p,q) depend on the details of the excitation system. A reasonable choice is given by x(p,V) =exp( -p2/2PZ),
=o,
1987
(5)
In eq. (5), o,, is the transverse contribution to the cavity eigenfrequencies which for the chosen resonator geometry are calculated in ref. [ 91. The reference frequency has been selected to correspond to a TEMoo mode (p=O). In steady state and in the uniform field limit, defined by the constraints aL+O,
15 October
lrll < l/2 ,
(10)
otherwise,
corresponding to a medium that becomes transparent at the edges of the discharge region (a case in point is provided by a CO, laser). The threshold gain for laser action is given by
xtan’
(
l/2 %(l Sy2)“2
>’
(11)
where v =pJ VA’* is a measure of the pump waist relative to the width ?,$* of the field mode; an expression for ‘lo as a function of Ro, L and A is given in ref. [9]. The steady state intensity, solution of eq. (8) grows monotonically for (2C) > (2C),,,,. We have looked for possible bistability effects of the type reported in ref. [ lo] near threshold for laser action, but found no evidence of this type of behavior in our model. A detailed study of the steady state behavior does not reveal unusual effects, or significant qualitative differences from the predictions of the plane-wave theory (except of course in matters of details such as the dependence of the output intensity on the resonator parameters and on the profile of the pump). The situation is quite different, instead, with regard to the linear stability of the steady state. The physical setting appropriate to the following calculations corresponds to a laser that operates in steady state in a TEMoo mode. We ask whether sidebands characterized by a modal index p # 0 and the same axial label as the steady state field can become unstable. The stability of the gaussian modes (p=O) corresponding to different axial indices was already considered by Lugiato and Milani [ 1 l] with the surprising result that these sidebands are always stable (in resonance, i.e. for S,,=O) in striking contrast with the plane-wave predictions of Risken and Nummedal [ 121. Our calculation reflects the approach of the Risken-Nummedal theory with the following main differences: (i) We look for unstable radial sidebands corresponding to the same axial index. (ii) The radial sidebands lie on one side of the resonant mode in contrast with the axial sidebands which are distributed symmetrically around the resonant frequency. (iii) Each sideband is characterized by a different modal profile. We study the linear stability of the Maxwell-Bloch equations (a) in the uniform field limit defined by 169
Volume 64. number
OPTICS
7
(‘OMMlINIC.~TI(
I5 (kYoher198’
INS
conditions (6), (b) in the so called good cavity limit K < j’ I . and (c) with a longitudinal mode spacing of the same order of magnitude as the atomic linewidth. The details of this analysis will be reported in ref. [ 131: here we describe only the results. the eigenvalues, in units of ;’ . have For KSY! the form j.=).t”l
+pfj
(I) L
.
(I’)
where A,,I ,,(“) is the pure imaginary ^(OI _ I.,, I ,, - -i(cl/iy,
quantity
.
) n‘,,=-iprSw,
(13)
where 6w, is the frequency separation between the first radial mode (p= 1) and the fundamental cavity resonance (p=O) in units of ;I 1 Hence, the instability condition for the sideband p# 0 is given by Reij,“>O.
p=1,2 . . ..
(14)
A simple perturbation calculation yields the required result
in powers of ~1;’
Re i”,‘,’’ = - 1 iI,?
I
-2CRe
dppI,$ I;’
I ’ 7’, (p.q)
(15)
0
where
T, (PJL~.;,“’ 1=
x(m) l+d’+/F,,l
F1g. 1. Dependence of the second laser threshold on the ticqucncy spacing between the radtal modes (measured in units cl! ,I )forCa);‘-2. (b) :’ I. and (c) ;I 0 I. The horltontal dashed llnc IS the lirst laser threshold for the chosen parameter\. Thr \aluc oft,/ 1n this simulation IS 7.5. the dctunlng CT,, 15 equal to Iwo. the radius ofcurvature of the mirrors in units of I IS 5. and /./.l=O.Z: these parameters yield ?~,,=4.5X.
small values of ;’ and reasonably small values of rirc~, the second laser threshold can be only a few tens of percent above the ordinary laser threshold. A key parameter that controls the stability properties of the system is the ratio v =pol$,” between the pump waist and the width of the fundamental gaussian mode. Fig. 3 shows the dependence of the ratio 2(““/2C”’ on I// for the first radial sideband. and demonstrates that instabilities are enhanced b, broadening the pump profile relative to the beam waist. This could lead to potentially important conclusions with respect to the design of a laser resonator if stable operation is the main objective. In closing this brief survey of our studies. WCmention that. unlike the plane-wave Maxwell-Bloch model. our system maintains low threshold instabilities in the rate equation limit (dP/dr= 0 ) and even
(16) When the inequality (14) is satisfied for one or more radial sidebands, the stationary TEMoo mode of the laser becomes unstable. One of the main results of our analysis is displayed in fig. 2. Here the dashed horizontal line denotes the threshold gain 2C”’ required by the laser to begin operation in a TEM,,,, mode. The solid lines represent the gain 2C’ ‘I required to produce an instability (second threshold) in the first radial sideband for different values of the parameter y plotted as a function of the frequency separation 60,. Our theory predicts that for 170
20
x:
t
(2)
. tb,
-mm-_
tf-8---b+----Q__e
i)
“
‘:
I
()
L0
A__/
__~~-_
_
4
_~~_
..~_
j
y/
H
t-g. 3. Dependence of the radio ZC“"il("" on the parameter fy for (a ) the ,I= I. and (b) for the p= 2 radial mode. The selected parameters arc ;a= 0. I. d ,( 7 0. h’tu,=0.46. and r~,,=4.58.
Volume 64, number
2
OPTICS
COMMUNICATIONS
when both the atomic variables are eliminated adiabatically (dP/dt = dDldt = 0 ). Thus the inclusion of transverse effects leads to significant qualitative and quantitative changes in the behavior of a homogeneously broadened ring laser by bringing the instability threshold much closer to the usual laser threshold than predicted by the plane-wave theory and by introducing a sensitive dependence of the dynamic behavior of the laser on the cavity and pump parameters. This work was partially supported by the EEC twinning project on Instabilities in Nonlinear Optical Systems, by a NATO travel grant and by a contract with the U.S. Army Research Office (Durham, N.C.). One of us (L.M.N.) is grateful to the Institute of Scientific Interchange in Torino, Italy for their support and hospitality during some phases of this research. It is also a pleasure to acknowledge useful discussions with Professor N.B. Abraham.
References [ I ] For a survey of recent results see, for example, Optical instabilities, eds. R.W. Boyd, M.G. Raymer L.M. Narducci (Cambridge University Press, 1986);
and
15 October
1987
R.G. Harrison and D.J. Biswas, Progr. Quantum Electron. 10 (1985) 147. [2] R.J. Ballagh, J. Cooper. M.W. Hamilton, W.J. Sandle and D.M. Warrington, Optics Comm. 37 (1981) 143; P.D. Drummond, IEEE J. Quantum Electron. QE-17 (198 1) 301; W.J. Firth and E.M. Wright, Optics Comm. 40 (1982) 223; J.V. Moloney and H.M. Gibbs, Phys. Rev. Lett. 48 (1982) 1607; L.A. Lugiato and M. Milani, 2. Phys. B50 (1983) 17 1. [3] D.W. McLaughlin, J.V. Moloney and A.C. Newell, Phys. Rev. Lett. 54 (1985) 681. [4] L.A. Lugiato, L.M. Narducci, D.K. Bandy and C.A. Pennise, Optics Comm. 43 (1982) 281. [ 5 ] A.T. Rosenberger, L.A. Orozco and H.J. Kimble, Phys. Rev. A28 (1983) 2569; L.A. Orozco, A.T. Rosenberger and H.J. Kimble, Phys. Rev. Lett. 53 (1984) 2547. [ 61 L.A. Lugiato and M.L. Asquini, unpublished. [ 71 A preliminary account of these results can be found in L.A. Lugiato, L.M. Narducci, D.K. Bandy, J.R. Tredicce and P. Ru, in: Lasers and synergetics, eds. A. Wunderlin and R. Graham (Springer Verlag, Berlin, to be published). [ 81 L.M. Narducci, J.R. Tredicce, L.A. Lugiato, N.B. Abraham and D.K. Bandy, Phys. Rev. A33 (1986) 1842. [9] P. Ru, L.M. Narducci, J.R. Tredicce, D.K. Bandy and L.A. Lugiato, Optics Comm. 63 (1987) 310 [lo] W.J. Witteman and G.J. Ernst, IEEE J. Quantum Electron. QE-11 (1975) 198. [ 111L.A. Lugiato and M. Milani, Optics Comm. 46 (1983) 57. [ 121 H. Risken and K. Nummedal, J. Appl. Phys. 39 (1968) 4662. [ 131 L.A. Lugiato, F. Prati, L.M. Narducci, P. Ru, J.R. Tredicce and D.K. Bandy, in preparation.
171
OPTICS
2
LOW THRtiSHOLD
INSTABILITIES
15 October
COMMUNICATIONS
IN UNIDIRECTIONAL
1987
RING LASERS
L.A. LUGIATO lstituto di Fisica. Politecnico
di Torino, 10129 Torino, Italy
F. PRATI Dipartimento
di Fisica, Universitb di Milano, 20133 Milano. Ita1.v
D.K. BANDY, L.M. NARDUCCI, PhJ?rics Department, Received
P. RU and J.R. TREDICCE
Drexel University, Philadelphra,
PA I91 04, USA
I8 May I987
We analyze the Maxwell-Block equations for a unidirectional ring laser after inclusion of transverse effects. We prove that, in the uniform field limit, the steady state of the laser has a spatial structure corresponding to a single longitudinal and radial mode. Under the additional assumption that the cavity linewidth is much smaller than the polarization relaxation time, we calculate the eigenvalues of the linearized equations and show that dynamical instabilities can emerge even in the close neighborhood of the ordinary laser threshold. This result is in much closer agreement with the available experimental facts and in striking contrast with the predictions of the traditional plane wave models.
The common modern framework for the description of laser dynamics rests on the plane-wave Maxwell-Bloch model which has been analyzed in detail especially in the case of the unidirectional ring laser geometry. This formulation provides a solid base for understanding the operation of a laser when the parameters favor the emission of a stable output. When instabilities become a dominant dynamical feature, the Maxwell-Bloch equations yield, instead, a consistent pattern of quantitative disagreement with the experimental facts 111. In this paper we produce evidence that transverse degrees of freedom play a far more important role in laser dynamics than previously anticipated. We have known for some time that transverse effects have a strong influence on the stationary and dynamic features of passive driven systems [ 21. McLaughlin, Moloney and Newell [ 31, for example, have shown that plane-wave and transverse models produce very different routes to chaos in bistable systems. In addition, the plane-wave numerical simulations of singlemode optical bistability [4] displays complex dynamical behaviors which are not reflected in the 0 030-4018/87/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
results of the careful experiments carried out by Kimble and collaborators [ 51. The experimental pulsations are instead in much better agreement with numerical simulations that include a gaussian field profile under single-mode operation [ 61. Because it is clear that the plane-wave model removes, by definition, important physical aspects, we wish to examine a more realistic setting that includes the effects of diffraction, of the wavefront curvature introduced by the spherical mirrors, and of the transverse variations of gain introduced by the pump [7]. We limit our considerations to the paraxial approximation and, for the sake of simplicity, we assume axial symmetry and exact resonance for the unidirectional ring laser. These approximations are not entirely realistic, but already lead to significant and novel theoretical deviations from the predictions of the plane-wave models. The resonator of interest for this work is shown schematically in fig. 1. The spherical mirrors, both having a radius of curvature RO,are separated by a distance L;the ring, whose total length is /i is completed by two plane mirrors adjusted in such a way B.V.
167
Volume
64. number
OPTICS
2
r
,,I.-
Fig. I. Schematic
rcprcsen~a~~on
spherical
hale
mirrors
power rcflectlvity length
based on an active \phcrlcal
radll
R. and arc separated
of the resonator
Z
of the ring resonator.
identical
of curvature by a dlstancr
is ,l. The calculations
medium
that
The two
R,,. Ihe same 1.. The full
in this paper al-c
tills the distance
between
F(p.11.7) =exp( -iSLV)
that the angle of incidence of the light beams on all reflecting surfaces is small in order to minimize aberrations. The Maxwell-Bloch equations for a homogencously broadened two-level system have the form (see. for example, ref. [ 81) (la)
dD/ils=
[FD+
In general, the completeness relation of this basis set provides only a way to transform the field differential equation into an infinite set of coupled equations for the modal amplitudes. This is not much of an improvement from the point of view of the calculations. However. with an appropriate generalization of the well known uniform field limit of the plane-wave theory. the empty cavity eigenfunctions are also exact modes of the filled resonator. In this limit one can carry out most of the calculations in analytic terms. .As our first step we let
the
mirrors.
ilP/iir= -
Ii ocloher I’)‘(.
C’(_)MMIJNIC‘.L\l-IONS
(1 +iis,,
) P],
-;’ [ -f(F*P+FP*)+h~(p,;rl)l.
(lb) (Ic)
=exp(
-
(3aj
I;yl~.~/.s)
30:)s) C_4,,(p,t1) /,,(,~.s) /a
f'(/Vf,T)=exp(
-iriQr)
&L,/.T)
(3b)
.
where the modal functions A,,(p.q) for the resonator of fig. 1 are discussed in ref. [ 91. and r?Q is the unknown offset, measured in units of 1’ between the carrier frequency of the stationary laser field and the reference frequency. Because the modal functions are orthonormal solutions of the stationary field eq. (la) in the absence of the medium (cd=O). the Maxwell-Bloch equations expressed in terms of the modal amplitudes become
where r/=-_/L. r-y,/.
p= (71/L&,) ’ 2r. 2s=clLy
.
(2)
The symbol A0 denotes the wavelength of a cavity mode taken as a reference and 1’ is the atomic lincwidth. The quantities F, P and D represent the normalized slowly varying envelope of the electric field. the atomic polarization envelope, and the population difference, respectively. The parameter CYis the gain coefficient per unit length; 6,, is the difference between the atomic transition frequency and the reference frequency, measured in units of y : y is the damping rate of the population difference, also in units of y , , and ~(p,?) is the unsaturated population inversion. In our calculation we adopt the following strategy: we first calculate the modal functions and eigenfrequencies of the empty resonator [ 91 and then expand the cavity field in a series of radial modes.
iiP/ilr
=
df>/dT=
--
[ i;D+ ( 1 + iJ)Js] .
-;I[
- i(F*P+I.‘P*)
(3b) +I_)-x(p.q)]
where d = Cs.,(. - 652. The boundary modal amplitudes ./;,( rl.7) are /;,( -
112,s) =
R exp( -id,,)
x,f;,[I/l,T-;',
(/I-L)/C]
conditions
(4c) for the
exp[ idRy I (A - I j/c] (4d)
where R is the power reflection coefficient of the curved mirrors (the flat mirrors are assumed to be perfect reflectors for simplicity). and
Volume 64, number
d,,=(o,--w,,())
2
OPTICS
Ale.
COMMUNICATIONS
T-0,
2C=oLIT=arbitrary,
(T is the transmittivity S,=O(l)
of the curved mirrors)
>
(6a) and (6b)
we have the following rigorous result (the proof of this statement will be given in a forthcoming extended publication [ 131) &=0(l),
.&o=O(T).
(7)
I his implies that the steady state field has the structure of a TEMoo mode, as already assumed in ref. [ 21 on the basis of qualitative considerations. The modal intensity If0 I* of the TEMoo mode in steady state is given by the state equation II2
1=2c
cc
IA0 I *x(P,rl) s drl s dPPl+d*+lAo&]*’ _ I/2 0
while the detuning parameter mode-pulling formula A=AC6
l+R’
K I?-=E=z’
(8)
A obeys the familiar
CT
where K is the cavity linewidth. The specific choice of the pump profile and the corresponding equilibrium population difference x(p,q) depend on the details of the excitation system. A reasonable choice is given by x(p,V) =exp( -p2/2PZ),
=o,
1987
(5)
In eq. (5), o,, is the transverse contribution to the cavity eigenfrequencies which for the chosen resonator geometry are calculated in ref. [ 91. The reference frequency has been selected to correspond to a TEMoo mode (p=O). In steady state and in the uniform field limit, defined by the constraints aL+O,
15 October
lrll < l/2 ,
(10)
otherwise,
corresponding to a medium that becomes transparent at the edges of the discharge region (a case in point is provided by a CO, laser). The threshold gain for laser action is given by
xtan’
(
l/2 %(l Sy2)“2
>’
(11)
where v =pJ VA’* is a measure of the pump waist relative to the width ?,$* of the field mode; an expression for ‘lo as a function of Ro, L and A is given in ref. [9]. The steady state intensity, solution of eq. (8) grows monotonically for (2C) > (2C),,,,. We have looked for possible bistability effects of the type reported in ref. [ lo] near threshold for laser action, but found no evidence of this type of behavior in our model. A detailed study of the steady state behavior does not reveal unusual effects, or significant qualitative differences from the predictions of the plane-wave theory (except of course in matters of details such as the dependence of the output intensity on the resonator parameters and on the profile of the pump). The situation is quite different, instead, with regard to the linear stability of the steady state. The physical setting appropriate to the following calculations corresponds to a laser that operates in steady state in a TEMoo mode. We ask whether sidebands characterized by a modal index p # 0 and the same axial label as the steady state field can become unstable. The stability of the gaussian modes (p=O) corresponding to different axial indices was already considered by Lugiato and Milani [ 1 l] with the surprising result that these sidebands are always stable (in resonance, i.e. for S,,=O) in striking contrast with the plane-wave predictions of Risken and Nummedal [ 121. Our calculation reflects the approach of the Risken-Nummedal theory with the following main differences: (i) We look for unstable radial sidebands corresponding to the same axial index. (ii) The radial sidebands lie on one side of the resonant mode in contrast with the axial sidebands which are distributed symmetrically around the resonant frequency. (iii) Each sideband is characterized by a different modal profile. We study the linear stability of the Maxwell-Bloch equations (a) in the uniform field limit defined by 169
Volume 64. number
OPTICS
7
(‘OMMlINIC.~TI(
I5 (kYoher198’
INS
conditions (6), (b) in the so called good cavity limit K < j’ I . and (c) with a longitudinal mode spacing of the same order of magnitude as the atomic linewidth. The details of this analysis will be reported in ref. [ 131: here we describe only the results. the eigenvalues, in units of ;’ . have For KSY! the form j.=).t”l
+pfj
(I) L
.
(I’)
where A,,I ,,(“) is the pure imaginary ^(OI _ I.,, I ,, - -i(cl/iy,
quantity
.
) n‘,,=-iprSw,
(13)
where 6w, is the frequency separation between the first radial mode (p= 1) and the fundamental cavity resonance (p=O) in units of ;I 1 Hence, the instability condition for the sideband p# 0 is given by Reij,“>O.
p=1,2 . . ..
(14)
A simple perturbation calculation yields the required result
in powers of ~1;’
Re i”,‘,’’ = - 1 iI,?
I
-2CRe
dppI,$ I;’
I ’ 7’, (p.q)
(15)
0
where
T, (PJL~.;,“’ 1=
x(m) l+d’+/F,,l
F1g. 1. Dependence of the second laser threshold on the ticqucncy spacing between the radtal modes (measured in units cl! ,I )forCa);‘-2. (b) :’ I. and (c) ;I 0 I. The horltontal dashed llnc IS the lirst laser threshold for the chosen parameter\. Thr \aluc oft,/ 1n this simulation IS 7.5. the dctunlng CT,, 15 equal to Iwo. the radius ofcurvature of the mirrors in units of I IS 5. and /./.l=O.Z: these parameters yield ?~,,=4.5X.
small values of ;’ and reasonably small values of rirc~, the second laser threshold can be only a few tens of percent above the ordinary laser threshold. A key parameter that controls the stability properties of the system is the ratio v =pol$,” between the pump waist and the width of the fundamental gaussian mode. Fig. 3 shows the dependence of the ratio 2(““/2C”’ on I// for the first radial sideband. and demonstrates that instabilities are enhanced b, broadening the pump profile relative to the beam waist. This could lead to potentially important conclusions with respect to the design of a laser resonator if stable operation is the main objective. In closing this brief survey of our studies. WCmention that. unlike the plane-wave Maxwell-Bloch model. our system maintains low threshold instabilities in the rate equation limit (dP/dr= 0 ) and even
(16) When the inequality (14) is satisfied for one or more radial sidebands, the stationary TEMoo mode of the laser becomes unstable. One of the main results of our analysis is displayed in fig. 2. Here the dashed horizontal line denotes the threshold gain 2C”’ required by the laser to begin operation in a TEM,,,, mode. The solid lines represent the gain 2C’ ‘I required to produce an instability (second threshold) in the first radial sideband for different values of the parameter y plotted as a function of the frequency separation 60,. Our theory predicts that for 170
20
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‘:
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t-g. 3. Dependence of the radio ZC“"il("" on the parameter fy for (a ) the ,I= I. and (b) for the p= 2 radial mode. The selected parameters arc ;a= 0. I. d ,( 7 0. h’tu,=0.46. and r~,,=4.58.
Volume 64, number
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OPTICS
COMMUNICATIONS
when both the atomic variables are eliminated adiabatically (dP/dt = dDldt = 0 ). Thus the inclusion of transverse effects leads to significant qualitative and quantitative changes in the behavior of a homogeneously broadened ring laser by bringing the instability threshold much closer to the usual laser threshold than predicted by the plane-wave theory and by introducing a sensitive dependence of the dynamic behavior of the laser on the cavity and pump parameters. This work was partially supported by the EEC twinning project on Instabilities in Nonlinear Optical Systems, by a NATO travel grant and by a contract with the U.S. Army Research Office (Durham, N.C.). One of us (L.M.N.) is grateful to the Institute of Scientific Interchange in Torino, Italy for their support and hospitality during some phases of this research. It is also a pleasure to acknowledge useful discussions with Professor N.B. Abraham.
References [ I ] For a survey of recent results see, for example, Optical instabilities, eds. R.W. Boyd, M.G. Raymer L.M. Narducci (Cambridge University Press, 1986);
and
15 October
1987
R.G. Harrison and D.J. Biswas, Progr. Quantum Electron. 10 (1985) 147. [2] R.J. Ballagh, J. Cooper. M.W. Hamilton, W.J. Sandle and D.M. Warrington, Optics Comm. 37 (1981) 143; P.D. Drummond, IEEE J. Quantum Electron. QE-17 (198 1) 301; W.J. Firth and E.M. Wright, Optics Comm. 40 (1982) 223; J.V. Moloney and H.M. Gibbs, Phys. Rev. Lett. 48 (1982) 1607; L.A. Lugiato and M. Milani, 2. Phys. B50 (1983) 17 1. [3] D.W. McLaughlin, J.V. Moloney and A.C. Newell, Phys. Rev. Lett. 54 (1985) 681. [4] L.A. Lugiato, L.M. Narducci, D.K. Bandy and C.A. Pennise, Optics Comm. 43 (1982) 281. [ 5 ] A.T. Rosenberger, L.A. Orozco and H.J. Kimble, Phys. Rev. A28 (1983) 2569; L.A. Orozco, A.T. Rosenberger and H.J. Kimble, Phys. Rev. Lett. 53 (1984) 2547. [ 61 L.A. Lugiato and M.L. Asquini, unpublished. [ 71 A preliminary account of these results can be found in L.A. Lugiato, L.M. Narducci, D.K. Bandy, J.R. Tredicce and P. Ru, in: Lasers and synergetics, eds. A. Wunderlin and R. Graham (Springer Verlag, Berlin, to be published). [ 81 L.M. Narducci, J.R. Tredicce, L.A. Lugiato, N.B. Abraham and D.K. Bandy, Phys. Rev. A33 (1986) 1842. [9] P. Ru, L.M. Narducci, J.R. Tredicce, D.K. Bandy and L.A. Lugiato, Optics Comm. 63 (1987) 310 [lo] W.J. Witteman and G.J. Ernst, IEEE J. Quantum Electron. QE-11 (1975) 198. [ 111L.A. Lugiato and M. Milani, Optics Comm. 46 (1983) 57. [ 121 H. Risken and K. Nummedal, J. Appl. Phys. 39 (1968) 4662. [ 131 L.A. Lugiato, F. Prati, L.M. Narducci, P. Ru, J.R. Tredicce and D.K. Bandy, in preparation.
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