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Disappearance of laser instabilities in a Gaussian cavity mode
Volume 46, number 1
OPTICS COMMUNICATIONS
1 June 1983
DISAPPEARANCE OF LASER INSTABILITIES IN A GAUSSIAN CAVITY MODE L.A. LUGIATO and M. MILANI
Istituto di l;~sicadeU'Uniw'rsit~, 20133 Milano, Italy Received 28 February 1983
We consider a homogeneously-broadened ring laser with spherical mkrors. We show that if ones assumes a gaussian transverse profile for the electric field, all the instabilities predicted by the plane wave theory vanish. The analysis is performed in the mean field limit, assuming that a suitably defined Fresnel number is much larger than unity and that perfect tuning between atoms and cavity exist. In 1966 Haken [1] and Risken et al. [2] showed that under proper conditions the stationary state of a homogeneously broadened, single-mode ring laser can become unstable. This instability requires a "bad cavity" situation, i.e. the cavity damping constant k must be larger than the sum of the transverse and longitudinal atomic decay rates 7 and 7 . Later Haken [3] proved that such an instability coincides with the well known Lorenz instability [4], which leads to chaotic behaviour. On the other hand, Risken and Nummedal [5] and Graham and Haken [6] analysed the homogeneously broadened ring laser taking into account all the longitudinal cavity modes. They showed that under appropriate conditions in the good cavity case, k <7~ + 711, some cavity modes, different from the one resonant with the atomic system, can become unstable. In this case, one has the formation of a pulse that travels in the cavity. In all of refs. [ 1 - 3 , 5 , 6 ] , the electric field is treated in the plane wave approximation. However, in a real laser the electric field has a radial profile which is typically gaussian, hence one naturally asks what is the effect of this transverse structure on the laser instabilities in homogeneously broadened lasers. In this paper, we assume that the electric field has the radial profile of a TEM00 mode of the cavity, and show that this transverse variation has unfortunately the effect of completely washing out both the good and the bad cavity instabilities. We consider the ring cavity with spherical mirrors shown in fig. 1. Mirrors 3 and 4 have 10(Y~breflectivity, whereas mirrors 1 and 2 have 0 030-4018/83/0000-0000/$ 03.00 © 1983 North-Holland
transmissivity coefficient T. We call £ the total length of the cavity and L the length of the atomic sample. We assume that the transverse dimension of this sample is much larger than the beam waist w 0, and that the Fresnel number nw2/~q3L, where X0 is the wavelength of the radiation, is much larger than unity. Hence the radius of the beam is uniform along the active region. Precisely, we assume that the beam has the radial shape exp(-r2/w2), where r is the radial variable, independent of the longitudinal variable z along the sample. Our starting point is the "one transverse mode" model derived and discussed in ref. [7], to which we refer for all details. The equations of this model are
Of/~z + C- 1 3f[3t
dr
:ao
0
4r 2 2 2 ex P( - r [ w O) P r( , z, t) , w0
(1.1)
3P/3 t =71 (Dr exp ( - r 2/w~) - P } ,
(1.2)
3D/3t =-')'If {Pf exp (-r2/w 2) + D - 1 } ,
(1.3)
where.f(z, t)exp(-r2/w 2) is the amplitude of the elec tric field, P(r, z, t) is the normalized atomic polarization and D(r, z, t) is the normalized population difference between the lower and the upper level of the laser transition, a is the unsaturated absorption coefficient per unit length, o is the population inversion per atom created by the pump. Note that eqs. (1) incorpo57
Volume
46, number
1
OPTICS
COMMUNICATIONS
I June
I
A
K(X, v) -
F
k 2
O
..f
-..>
t_
7 , ( I - v) + X
I +vCy, + X)(%I + X) +7±7,1v
(5.3)
where
2
*2
k = cT/£,
i
a n = 2rrcn/£
(n = O, -+ 1 .... ).
((~)
The index n corresponds to the longitudinal frequencies of the cavity, in particular n = 0 corresponds to the cavity frequency that is exactly resonant with the atomic system. In the plane wave theory, the eigenvalt,e equation has tile form (5.1) but with
~'kQ_ __
H(),,x) = x2K(X,x 2) . Fig. 1. Unidirectional ring cavity with spherical mirrors. The arrows indicate the output of the laser. rate all tile longitudinal modes of the cavity; the onemode assumption concerns only tile transverse modes. The boundary condition for eq. (1.1)is (fig. 1) f ( - L / 2 , t) = (1 . T ) f ( L / 2 , t - (£ - L ) / c ) .
(2)
Eqs. (1,2) incorporate the assumption that the atomic frequency coincides with a longitudinal cavity frequency. In tile following, we shall assume the "mean field limit" [8] aL ,~ 1 ,
T'~ I ,
C - e.L/2T arbitrary .
(3)
In this limit, tile stationary solution of (1) becomes uniform in space. As shown in [7], by settingx = f ( L / 2 ) we obtain the steady state equation 2C=x2/ln{l +x2).
(4)
Eq. (4) must be compared with the well known relation x 2 = 2C -. 1 of the plane wave theory. Note that the laser threshold occurs for C = 1/2 in both cases. Let us now analyse the stability of the steady state. To this aim, we introduce the deviations from the stationary values 8f(z, t) =-f(z, t) - fst (z), etc. and linearize eqs. (1). As usually, we introduce the ansatz 6 f ( z , t ) = exp(Xt) 6 f ( z ) . etc., which under the assumptions (3) and using (2) leads to the following equation for the eigenvalues X X =-.i~ n-- k{1 - ( 2 C / x 2 ) T x H ( X , x 2 ) } ,
(5.1)
x2
tI(X, x 2 ) =
f 0
58
dvK(X,v),
1983
(5.2)
(5.4)
The stationary state is stable provided that all the solutions of eq. (5.1) have a negative real part. Let us consider first the good cavity case k "~ 7.1 • ")'ll, in which the eigenvalues are obtained by solving (5.1) perturbatively Xn =
i% - k { 1 - ( 2 C / x 2 ) y l t l ( - i % , x 2 ) }
.
(6)
Hence the instabili O, condition is 1 < ( 2 C / x 2 ) y t Re t t ( - i e n , X 2 ) ,
(7)
at least for one n = 0. + 1 ..... In tile plane wave case, i.e. using (5.4), (5.3) and the relation x 2 = 2C - I. one finds [5] that fo, 2 C > 5 + 3('/,/%.) + 2[4 + 6(yl,/7± ) + 2(,71i/y±)2] 1/2 condition (7) is satisfied for two ranges of values of the variable % , symmetrically placed with respect to e n = 0 (fig. 2). On the other hand, in the gaussian case using (5.2) and (5.3) we find
171 "7±ReH(--io%,x 2) = l n ( l + x 2 ) - [a,~ +(/3+ 1) 2 ] 1 X [h~n)(x 2) - h{n)(o) + h(n)(x2)- - h~n)(o)],(8.1) where ~n = % / Y - ,
13= 7!,/Yi.
h{n )(x2)= I ( S 2 + 132 + 13 + 2 )
(8.2) (8.3)
(1 +X2) 2 X In . . . . . . . . . . . . . . . . . . . . . . . . _ - 2 +(/3+1 )2 ] ' t32(1 +X2)2 2&23( 1 + x 2 ) + a -n 2 [~n -2+213(13+I) / 3 ( l + x 2) -~ h~n)(x2) = ~n tan-1 - c~,~ ~n ~n(~ + l) (8.4) By taking into account (4), we have numerically verified that condition (7) is never satisfied (see e.g. figs.
Volume 46, number 1
OPTICS COMMUNICATIONS
1 June 1983
2,3), which means that t h e s t e a d y s t a t e is a l w a y s stable. On the o t h e r hand, let us n o w consider the bad cavity case k >'TL + ")'N and, like in refs. [ 1 - 3 ] , let us restrict ourselves to the cavity m o d e resonant with the atoms (n = 0). This a m o u n t s to consider eq. (5.1) with a n = 0. Let us concentrate ourselves on the b o u n d a r y o f the stability region, that is defined by the condition X = ip, with p real. By substituting this ansatz in (5.1) with a n = 0 and equating real and imaginary part, we obtain two equations
-Re~,~
. m
oi ....
--
~,~..
1 = ( 2 C / x 2 ) Re HOp, x 2 ) ,
(9.1)
k -1 v = ( 2 C / x 2 ) 7 ± lm H(iv, x 2 ) .
(9.2)
In eqs. (9), x 2 is a function o f C defined by x 2 = 2C 1 in the plane wave case and by (4) in the gaussian case. By eliminating v between (9.1) and (9.2), one obtains the equation that d e f i n e s the b o u n d a r y in the space of the parameters C, k, ~'i, '711. F u r t h e r m o r e , the solution o f eqs. (9) gives the values o f the frequency on the boundary. This corresponds to the pulsation frequency at the onset o f instability. In the plane wave case, using (5.4) and (5.3) one finds the b o u n d a r y equation [ 1 - 3 ] 2C = 1 + [(q¢± + -
Fig. 2. Plot of - R e hn (see eq. (6)) as a function of fin for = ~ll/-'t.l_= 1 and (a) C = 20 (x 2 = 39), plane wave case; (b) C = 20 (x2 = 220), gaussian case; (c) C-'- 10.57 (x 2 = 39), gaussian case. The instability condition (7) is satisfied when - R e hn < O.
]
a .
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.
c
g
lb
'
2'0
'--
~,, "
Fig. 3. Same as fig. 2 but C = 40 (x 2 = 500) and (a)/3 = 2; (b)/3 = 1 ; (c)/3 = 0.1 ; (d)/3 = 0.01 ; (e)/3 = 0.001 ; (f)/3 ~ 0, in which case 2 - R e hn tends to an/(a n2 + 1). All curves axe for the gaussian case. 59
Volume 46, number 1
OPTICS COMMUNICATIONS
7i' + k)(71 + k)l~li (k - ~/j "/,i)]- On the ottler hand ill in the gaussian case we can immediately decide that no stability boundary exists. In fact, we observe that (9.1) coincides with (7) after replacing 'cin' by ' - v ' and ' < ' by '='. Now in the gaussian case (2C'),±/x 2) × Re H(ilJ, x 2) is always strictly smaller than unity and therefore eq. (9.1) can never be satisfied. Hence we cannot find any stability boundary, which since the stability domain is nonvoid means that again the
steady state is always stable. 11 is now clear that, using the same procedure which leads to eqs. (9.1,2), we can prove that in the gaussian case one does not find any instability in the homogeneous broadened ring laser,for any value o f the ratios k/7±, k/Tli and even including all cavity modes (i.e., leaving a n arbitrary). In fact, if we look for the boundary of the stability domain in the space of the parameters C, k,"/l,Tii, % we set X = iv in eq. (5.1). By separating real and imaginary part, we derive two equations, one of which coincides with (9.1), while the other is identical to the equation obtained from (9.2) by replacing i, by v + c~n in the lefthand side. Since as we said eq. (9.1) can never be satisfied, no instability boundary exist in any case. In conclusion, we have shown that the gaussian transverse shape of the electric field destroys all instabilities in the homogeneously broadened ring laser. On the other hand, we know that if the laser is inhomogeneously broadened the instability of refs. [1--3] not only remains, but also becomes more accessible because it occurs the nearer to laser threshold, the larger is the ratio between the inhomogeneous and the homogeneous linewidth [ 9 - 12]. Curiously enough, the fact that the bad cavity instabilities in inhomogeneously broadened lasers are but the continuation of the Haken-Risken-Schmid-Weidlich instability has been recognized only recently. In the case of inhomogeneously broadened laser, the instability has been experimentally observed [13,14], and therefore the gaussian profile of the electric field does not destroy it. In a future paper, we plan to discuss the theory of the bad cavity instabili-
60
1 Junc 1983
ties in inhomogeneously broadened lasers assuming a gaussian transverse structure of the electric field. We observe that the analysis of this paper does not exclude that other cavity modes, different from the TEMo0 mode, can become unstable thereby recovering some instability even in the case of homogeneous broadening. More in general, one might conceive suitable procedures to obtain a beam profile different from gaussian, and in this case the instability might be restored. Finally, we have considered only the case of perfect tuning between the cavity mode and the atoms, and cannot exclude that in case of detuning some instability can survive even in presence of a gaussian profile. We thank N. Abraham, H. Haken, L.M. Narducci and M. Sargent I11 for stimulating discussions. Paper supported by CNR contribution CT 82.00031.02.
References [1] II. 11aken, Z. Physik 190 (1966) 327. [2] H. Risken, C. Schmid and W. Weidlich, Z. Physik 194 (1966) 337. [3] H. Haken, Phys. Left. 53A (1975) 77. [4] E.N. Lorenz, J. Atmos. Sci. 20 (1963) 130. [5] H. Risken and K. Nummedal, J. Appl. Plays. 39 (1968) 4662. [6] R. Graham and H. Haken, Z. Physik 213 (1968) 420. [7] L.A. Lugiato and M. Milani, Zeits. f. Physik B, in press and references quoted therein. [8] R. Bonifacio and L.A. Lugialo, Lett. Nuovo Cimento 21 (1978) 505. [9] M.L. Minden and L.\V. Casperson. IH'~E J. Quantmn Electron. QE-18 (1982) 1952. [ 10] S.T. Hendow and M. Sargent III, Optics Comm. 40 (1982) 385. [ 11 ] P. Mandel, Optics Comm., to appear. [121 L.A. Lugiato, L.M. Narducci, D. Bandy and N. Abraham, submitted for publication. [131 L.W. Casperson, IEEE J. Quantum Electron, QE-14 (1978) 756. [14] J. Bentley and N.B. Abraham, Optics Comm. 41 (1982) 52.
OPTICS COMMUNICATIONS
1 June 1983
DISAPPEARANCE OF LASER INSTABILITIES IN A GAUSSIAN CAVITY MODE L.A. LUGIATO and M. MILANI
Istituto di l;~sicadeU'Uniw'rsit~, 20133 Milano, Italy Received 28 February 1983
We consider a homogeneously-broadened ring laser with spherical mkrors. We show that if ones assumes a gaussian transverse profile for the electric field, all the instabilities predicted by the plane wave theory vanish. The analysis is performed in the mean field limit, assuming that a suitably defined Fresnel number is much larger than unity and that perfect tuning between atoms and cavity exist. In 1966 Haken [1] and Risken et al. [2] showed that under proper conditions the stationary state of a homogeneously broadened, single-mode ring laser can become unstable. This instability requires a "bad cavity" situation, i.e. the cavity damping constant k must be larger than the sum of the transverse and longitudinal atomic decay rates 7 and 7 . Later Haken [3] proved that such an instability coincides with the well known Lorenz instability [4], which leads to chaotic behaviour. On the other hand, Risken and Nummedal [5] and Graham and Haken [6] analysed the homogeneously broadened ring laser taking into account all the longitudinal cavity modes. They showed that under appropriate conditions in the good cavity case, k <7~ + 711, some cavity modes, different from the one resonant with the atomic system, can become unstable. In this case, one has the formation of a pulse that travels in the cavity. In all of refs. [ 1 - 3 , 5 , 6 ] , the electric field is treated in the plane wave approximation. However, in a real laser the electric field has a radial profile which is typically gaussian, hence one naturally asks what is the effect of this transverse structure on the laser instabilities in homogeneously broadened lasers. In this paper, we assume that the electric field has the radial profile of a TEM00 mode of the cavity, and show that this transverse variation has unfortunately the effect of completely washing out both the good and the bad cavity instabilities. We consider the ring cavity with spherical mirrors shown in fig. 1. Mirrors 3 and 4 have 10(Y~breflectivity, whereas mirrors 1 and 2 have 0 030-4018/83/0000-0000/$ 03.00 © 1983 North-Holland
transmissivity coefficient T. We call £ the total length of the cavity and L the length of the atomic sample. We assume that the transverse dimension of this sample is much larger than the beam waist w 0, and that the Fresnel number nw2/~q3L, where X0 is the wavelength of the radiation, is much larger than unity. Hence the radius of the beam is uniform along the active region. Precisely, we assume that the beam has the radial shape exp(-r2/w2), where r is the radial variable, independent of the longitudinal variable z along the sample. Our starting point is the "one transverse mode" model derived and discussed in ref. [7], to which we refer for all details. The equations of this model are
Of/~z + C- 1 3f[3t
dr
:ao
0
4r 2 2 2 ex P( - r [ w O) P r( , z, t) , w0
(1.1)
3P/3 t =71 (Dr exp ( - r 2/w~) - P } ,
(1.2)
3D/3t =-')'If {Pf exp (-r2/w 2) + D - 1 } ,
(1.3)
where.f(z, t)exp(-r2/w 2) is the amplitude of the elec tric field, P(r, z, t) is the normalized atomic polarization and D(r, z, t) is the normalized population difference between the lower and the upper level of the laser transition, a is the unsaturated absorption coefficient per unit length, o is the population inversion per atom created by the pump. Note that eqs. (1) incorpo57
Volume
46, number
1
OPTICS
COMMUNICATIONS
I June
I
A
K(X, v) -
F
k 2
O
..f
-..>
t_
7 , ( I - v) + X
I +vCy, + X)(%I + X) +7±7,1v
(5.3)
where
2
*2
k = cT/£,
i
a n = 2rrcn/£
(n = O, -+ 1 .... ).
((~)
The index n corresponds to the longitudinal frequencies of the cavity, in particular n = 0 corresponds to the cavity frequency that is exactly resonant with the atomic system. In the plane wave theory, the eigenvalt,e equation has tile form (5.1) but with
~'kQ_ __
H(),,x) = x2K(X,x 2) . Fig. 1. Unidirectional ring cavity with spherical mirrors. The arrows indicate the output of the laser. rate all tile longitudinal modes of the cavity; the onemode assumption concerns only tile transverse modes. The boundary condition for eq. (1.1)is (fig. 1) f ( - L / 2 , t) = (1 . T ) f ( L / 2 , t - (£ - L ) / c ) .
(2)
Eqs. (1,2) incorporate the assumption that the atomic frequency coincides with a longitudinal cavity frequency. In tile following, we shall assume the "mean field limit" [8] aL ,~ 1 ,
T'~ I ,
C - e.L/2T arbitrary .
(3)
In this limit, tile stationary solution of (1) becomes uniform in space. As shown in [7], by settingx = f ( L / 2 ) we obtain the steady state equation 2C=x2/ln{l +x2).
(4)
Eq. (4) must be compared with the well known relation x 2 = 2C -. 1 of the plane wave theory. Note that the laser threshold occurs for C = 1/2 in both cases. Let us now analyse the stability of the steady state. To this aim, we introduce the deviations from the stationary values 8f(z, t) =-f(z, t) - fst (z), etc. and linearize eqs. (1). As usually, we introduce the ansatz 6 f ( z , t ) = exp(Xt) 6 f ( z ) . etc., which under the assumptions (3) and using (2) leads to the following equation for the eigenvalues X X =-.i~ n-- k{1 - ( 2 C / x 2 ) T x H ( X , x 2 ) } ,
(5.1)
x2
tI(X, x 2 ) =
f 0
58
dvK(X,v),
1983
(5.2)
(5.4)
The stationary state is stable provided that all the solutions of eq. (5.1) have a negative real part. Let us consider first the good cavity case k "~ 7.1 • ")'ll, in which the eigenvalues are obtained by solving (5.1) perturbatively Xn =
i% - k { 1 - ( 2 C / x 2 ) y l t l ( - i % , x 2 ) }
.
(6)
Hence the instabili O, condition is 1 < ( 2 C / x 2 ) y t Re t t ( - i e n , X 2 ) ,
(7)
at least for one n = 0. + 1 ..... In tile plane wave case, i.e. using (5.4), (5.3) and the relation x 2 = 2C - I. one finds [5] that fo, 2 C > 5 + 3('/,/%.) + 2[4 + 6(yl,/7± ) + 2(,71i/y±)2] 1/2 condition (7) is satisfied for two ranges of values of the variable % , symmetrically placed with respect to e n = 0 (fig. 2). On the other hand, in the gaussian case using (5.2) and (5.3) we find
171 "7±ReH(--io%,x 2) = l n ( l + x 2 ) - [a,~ +(/3+ 1) 2 ] 1 X [h~n)(x 2) - h{n)(o) + h(n)(x2)- - h~n)(o)],(8.1) where ~n = % / Y - ,
13= 7!,/Yi.
h{n )(x2)= I ( S 2 + 132 + 13 + 2 )
(8.2) (8.3)
(1 +X2) 2 X In . . . . . . . . . . . . . . . . . . . . . . . . _ - 2 +(/3+1 )2 ] ' t32(1 +X2)2 2&23( 1 + x 2 ) + a -n 2 [~n -2+213(13+I) / 3 ( l + x 2) -~ h~n)(x2) = ~n tan-1 - c~,~ ~n ~n(~ + l) (8.4) By taking into account (4), we have numerically verified that condition (7) is never satisfied (see e.g. figs.
Volume 46, number 1
OPTICS COMMUNICATIONS
1 June 1983
2,3), which means that t h e s t e a d y s t a t e is a l w a y s stable. On the o t h e r hand, let us n o w consider the bad cavity case k >'TL + ")'N and, like in refs. [ 1 - 3 ] , let us restrict ourselves to the cavity m o d e resonant with the atoms (n = 0). This a m o u n t s to consider eq. (5.1) with a n = 0. Let us concentrate ourselves on the b o u n d a r y o f the stability region, that is defined by the condition X = ip, with p real. By substituting this ansatz in (5.1) with a n = 0 and equating real and imaginary part, we obtain two equations
-Re~,~
. m
oi ....
--
~,~..
1 = ( 2 C / x 2 ) Re HOp, x 2 ) ,
(9.1)
k -1 v = ( 2 C / x 2 ) 7 ± lm H(iv, x 2 ) .
(9.2)
In eqs. (9), x 2 is a function o f C defined by x 2 = 2C 1 in the plane wave case and by (4) in the gaussian case. By eliminating v between (9.1) and (9.2), one obtains the equation that d e f i n e s the b o u n d a r y in the space of the parameters C, k, ~'i, '711. F u r t h e r m o r e , the solution o f eqs. (9) gives the values o f the frequency on the boundary. This corresponds to the pulsation frequency at the onset o f instability. In the plane wave case, using (5.4) and (5.3) one finds the b o u n d a r y equation [ 1 - 3 ] 2C = 1 + [(q¢± + -
Fig. 2. Plot of - R e hn (see eq. (6)) as a function of fin for = ~ll/-'t.l_= 1 and (a) C = 20 (x 2 = 39), plane wave case; (b) C = 20 (x2 = 220), gaussian case; (c) C-'- 10.57 (x 2 = 39), gaussian case. The instability condition (7) is satisfied when - R e hn < O.
]
a .
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.
c
g
lb
'
2'0
'--
~,, "
Fig. 3. Same as fig. 2 but C = 40 (x 2 = 500) and (a)/3 = 2; (b)/3 = 1 ; (c)/3 = 0.1 ; (d)/3 = 0.01 ; (e)/3 = 0.001 ; (f)/3 ~ 0, in which case 2 - R e hn tends to an/(a n2 + 1). All curves axe for the gaussian case. 59
Volume 46, number 1
OPTICS COMMUNICATIONS
7i' + k)(71 + k)l~li (k - ~/j "/,i)]- On the ottler hand ill in the gaussian case we can immediately decide that no stability boundary exists. In fact, we observe that (9.1) coincides with (7) after replacing 'cin' by ' - v ' and ' < ' by '='. Now in the gaussian case (2C'),±/x 2) × Re H(ilJ, x 2) is always strictly smaller than unity and therefore eq. (9.1) can never be satisfied. Hence we cannot find any stability boundary, which since the stability domain is nonvoid means that again the
steady state is always stable. 11 is now clear that, using the same procedure which leads to eqs. (9.1,2), we can prove that in the gaussian case one does not find any instability in the homogeneous broadened ring laser,for any value o f the ratios k/7±, k/Tli and even including all cavity modes (i.e., leaving a n arbitrary). In fact, if we look for the boundary of the stability domain in the space of the parameters C, k,"/l,Tii, % we set X = iv in eq. (5.1). By separating real and imaginary part, we derive two equations, one of which coincides with (9.1), while the other is identical to the equation obtained from (9.2) by replacing i, by v + c~n in the lefthand side. Since as we said eq. (9.1) can never be satisfied, no instability boundary exist in any case. In conclusion, we have shown that the gaussian transverse shape of the electric field destroys all instabilities in the homogeneously broadened ring laser. On the other hand, we know that if the laser is inhomogeneously broadened the instability of refs. [1--3] not only remains, but also becomes more accessible because it occurs the nearer to laser threshold, the larger is the ratio between the inhomogeneous and the homogeneous linewidth [ 9 - 12]. Curiously enough, the fact that the bad cavity instabilities in inhomogeneously broadened lasers are but the continuation of the Haken-Risken-Schmid-Weidlich instability has been recognized only recently. In the case of inhomogeneously broadened laser, the instability has been experimentally observed [13,14], and therefore the gaussian profile of the electric field does not destroy it. In a future paper, we plan to discuss the theory of the bad cavity instabili-
60
1 Junc 1983
ties in inhomogeneously broadened lasers assuming a gaussian transverse structure of the electric field. We observe that the analysis of this paper does not exclude that other cavity modes, different from the TEMo0 mode, can become unstable thereby recovering some instability even in the case of homogeneous broadening. More in general, one might conceive suitable procedures to obtain a beam profile different from gaussian, and in this case the instability might be restored. Finally, we have considered only the case of perfect tuning between the cavity mode and the atoms, and cannot exclude that in case of detuning some instability can survive even in presence of a gaussian profile. We thank N. Abraham, H. Haken, L.M. Narducci and M. Sargent I11 for stimulating discussions. Paper supported by CNR contribution CT 82.00031.02.
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