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Bistable operation of multimode lasers containing saturable absorbers
Volume 33, number 3
OPTICS COMMUNICATIONS
June 1980
BISTABLE OPERATION OF MULTIMODE LASERS CONTAINING SATURABLE ABSORBERS Roland MIJLLER
Zentralinstitut ffir Optik und Spektroskopie, Akademie der Wissenschaften der DDR, DDR -1199 Berlin Received 22 October 1979 Revised manuscript received 3 March 1980 Continuous-wave operation of many longtudinal modes in a unidirectional ring cavity containing a laser medium and a saturable absorber is investigated. The phases of the modes are assumed to be randomly distributed. The system is described by a set of rate equations including noise terms from both media. The steady-state total intensity (as an average value) and the linewidth of the laser output are calculated for the region where bistable operation and hysteresis can occur. Moreover, the hysteresis cycles of individual modes are studied. 1. Introduction
2. The basic equations
It is well-known that a laser system like that investigated here can have two stable intensity levels corresponding to the unsaturated and strongly saturated state, respectively, o f the absorber. Associated with it a hysteresis cycle o f the intensity versus pump power occurs. After the discovery of hysteresis effects in lasers with a saturable absorber [1,2] a lot o f papers dealing theoretically with this phenomenon has been published. According to the first experiments some of them are concerned with the gas laser (see, for example [3,4] ). In the following we will assume a homogneously broadened laser line in b o t h media. Therefore, we especially refer to [5,6] where this case has been extensively studied b y means o f semiclassical and fully quantum-mechanical treatment. The authors have presupposed a single-mode regime. However, at least below threshold, and also in a small region above it, multimode operation should be taken into account, in particular, when the noise level is rather high as for semiconductor and dye lasers. In this paper we will discuss some aspects of the problem in the frame o f a rate-equation approximation. This means that phase relations between different modes are not considered. Thus the model approximately describes a case where the phases are randomly distributed and the calculated steady-state intensities correspond to average values of a real system.
The following set of equations for the number o f p h o t o n s M q in the qth mode and the inversion densities na, n b of the amplifying and absorbing medium, respectively, will b e investigated:
326
dMq Mq dt
-
+
r p t- T(i--+rq2 ) ( O a n a / a - Obnblb)
1
rO+rq 2)
(Oanal a +½ [nb0 - nb] Oblb) ,
(la)
o~
dn a _ nao - n a dt
Tla
na ~ Mq daTTlalsa q=_~o l + r q 2 '
(lb)
oo
dnb_ nbO-n b dt
Tlb
,finb
~
Mq
(lc)
daTTlblsa q = _ ~ l+rq2"
The index "a" indicates parameters o f the active medium, " b " those o f the absorber. The used symbols have the following meaning: Tp is the p h o t o n mean lifetime in the cavity, T its round-trip time, l the length o f the cell containing the amplifying and absorbing medium, and d is the cross-sectional area o f the light beam. r = (26co/P) 2 contains the ratio o f mode-spacing 6co (circular frequency) to the width F o f the lorentzian atomic lines which are assumed to be equal for b o t h media. Further, o is the atomic cross-section of stimulated emission or absorption at line centre, T 1
Volume 33, number 3
the longitudinal relaxation time, and I s = ( a T 1 )-1 is the photon flux that saturates the medium. Finally, we put/~ = lsada/lsbd b . The last term at the righthand side of eq. (la) is due to spontaneous emission from both media, whose atoms are considered as fourlevel (amplifier) oI two-level systems (absorber), respectively. Because rib0 denotes the density of absorbing atoms if no field is in the cavity the expression 1 (rib0 - n b ) is the density of those in the upper level. First of all we calculate the steady-state solutions ~tq, ha, and h b after putting their derivatives to zero. Then the following relation can be found easily: .~q = s/(1 +rq 2 + o - u).
(2)
Besides, h a = ha0/(1 + R) and ~ib = nb0/(1 +/~R). Here we have used the abbreviations oo
R
_
1 ~ isadaT q = - ~
/~q l+rq 2'
U = "Ya0[(1 + R ) n ] - i
(3a, b) °='fbo[(l+lJR)l¢]-l,
s = u +31 vlSR.
\l+--+d~-u]
- coth(Tr/x/7)},
coth ~" (4)
(5)
where I~ denotes the atomic linewidth. Secondly, we write down the formula for the total photon flux 2
I t : ( u - v ) R + (Fs/4dalsa) coth(Tr/x/r).
(6)
When the noise terms in eq. ( l a ) a r e neglected then we have s = 0 and eq. (4) is reduced to ( l + v - u ) R =0 i.e. 31q = 0 for q 4:0 and I t = R (single-mode regime).
3. Calculation of the total intensity Now we will solve eq. (4) approximately. The factor f, at its right-hand side is in the range o f 10 - 4 10 -3 for a GaAs laser [8]. Concerning dye lasers an estimation of f gives 1 0 - 7 - 1 0 - 6 i f d a ~ 3 X 10 -5 cm 2. Presuming r ' ~ 1 (many modes within the range of F) two extreme cases can be distinguished. The first is characterized by the inequality x - ( l + v - u ) / r >> 1 that leads to c o t h x ~ 1. Using eq. (5) one findsx -----(AH/266o)2 >> 1 i.e. many modes also within the range of A H. From eq. (4) then follows (l+v-u)R
= -~ •
1 +
"
This particular form of eq. (7) has been chosen to facilitate a comparison with the equation corresponding to single-mode regime (cf. the end of ch. 2). The second case, corresponding to high intensities, is connected with x ,~ 1, so that eq. (4) delivers now
O + v - u ) R - f s ( x / 7 - ~(~+v-u)) ~ ( u - v)
where the relations 6w = 27r/T and f = P/4Isad a have been used. The s y m b o l " c o t h " means the hyperbolic cotangent. When we put o = O, i.e. eliminating the absorber, eq. (4) then agrees with a corresponding expression of [8]. Before deriving solutions of eq. (4) in the next chapter we give two other relations needed in the following. At first we calculate the linewidth AiLof the field (spectral width), defined by 3~qH = ~-M 0 with 2qn 6w = AH, assuming at least several modes to be within the range o f A H. From eq. (2) one gets • A H = (1 + u - u)l/2I ",
art = (lsadaT)-I ~,qMq using eq. (4):
(3c, d)
Moreover, 7a = °a~Tala and 7b = Ob rib lb define the coefficients of small-signal gain and loss of the amplifier and absorber, respectively. The cavity loss is denoted by K = T I T v. After introducing ~tq from eq. (2) the infinite sum of eq. (3a) can be calculated [7]. The result is an implicit expression for R : u-V'
June 1980
OPTICS COMMUNICATIONS
(8)
We will construct approximate solutions of eqs. (7), (8) in using those for the single-mode regime. Eq. (7) with s = 0 immediately yields the following three solutions f o r R (cf. also [5] ), marked by a tilde, /~0 = 0
(9a)
and o
9b,
Here we have used the abbreviations go = 7aO--Tb0--K,
a =go +(1--1//5)Tb0--r//~" (10a, b)
The relation go = Odetermines the threshold of the centralmode (q = 0). R _ has no physical meaning and/~0 is not stable i f g 0 > 0. On the other hand there is a cer327
Volume 33, number 3
OPTICS COMMUNICATIONS
tain range go ~< 0 where all the three functions/~i are positive if/6 ~> 1 +~/3'b0. Then/~0 is always stable,/~+ can be stable but R _ is certainly not [5]. Thereby a range of parameters exist where bistability can occur. Now we introduce the ansatz Ri = R i + Pi with ]Oil /~i into the left-hand sides of eqs. (7), (8). At their right-hand sides, which are called e i in short, we substitute R i for R. Then we find for the right-hand side of eq. (7) ei
=
(fsi)2/R i
(11 a)
and for eq. (8) ei
=
fsix/'r/Tr,
(1 lb)
where the quantity s i is defined by eqs. ( 3 b - d ) after substituting R i f o r R there. Moreover, we confine ourselves to linear terms o f Pi. Then from eqs. (7) and (8) f o r g O < 0 follows a simple but somewhat rough approximation: (1 +/~_+)(1 +lSR±)e~ R± ~/~+ +-
2 ((c~/2K)2 + go/K/~} 1/2/5/~+ "
(12a)
It should be noted that eq. (12a) is not valid in the vicinity of the points where R+ coincides w i t h / ~ because the square root tends to zero there. Finally, an expression f o r R 0 corresponding to/~0 = 0 has to be found. To this end we pass over to the complete eq. (7) without the restriction described by eq. (1 la). AssumingR 0 ~ 1 and/?R 0 ~ 1 one gets Roof
7a0 (l_(_K/go)l/2) ")'b0 --')'a0
'
(lZb)
where go < 0. Fig. 1 shows the dependence of R on the small-signal gain coefficient 7a0" The curve has been calculated from eq. (4). The used parameters are f = 5 × 10 - 4 , "Yb0 = 2, ~ = 0.5 ,i6 = 5, and r = 1010 (i.e. a ring-cavity length of 3 m with I" ~ 1014 s -1 for GaAs). The dashed line marks the threshold g 0 = 0. The lower branch of the curve (from left below to A) corresponds to R 0 the upper one (from B to right above) to R+. The medium part (from A to B) belongs to R . At the crossing points of the lower and upper branch with the medium one the system becomes unstable and passes from R 0 to R+ at A and vice versa at B. This hysteresis cycle is indicated by arrows in fig. 1. Because R 0 > / ~ and R+ > / ~ + (cf. eq. (12a))but R _
June 1980
' /og R 1
o
B~.____
-2
A', l
-3 ~Z----~~ Z5
20
-
2' ~'a°K~
Fig. 1. Sum of the weighted photon numbers of all modes, R (defined by eq. (3a)), versus small-signalgain coefficient, ~'ao" The arrow indicate the hysteresis cycle. R in the upper branch is almost equal to the normalized total photon flux or intensity, I t .
modes oscillate compared to a single-mode regime. For the latter A coincides with the crossing point of the 3'a0"axis and the dashed line. The numerical calculation shows, however, that B is only slightly delayed compared to A. Therefore, the hysteresis curve is shortened by nearly the delay of A. In our example the latter is of about 10 percent related to the distance between A and B.
4. Linewidth of the field Now we have to deal with eq. (5). Fig. 2 shows AH/P for the lower (left scale) and the upper (right scale) branch of R in fig. 1. The linewidths in the two branches differ by a factor of about 103 at the transition points. Therefore, in a long cavity like that in our
~A,
..aA, k3
-, 8
2
7~ U
-2 7 -1
I
.,
O.5
/.0
¢5
20
2.5
~o_
-
Fig. 2. Linewidth of the field over atomic linewidth, AH/F, versus small-signal gain coefficient for the lower (left scale) and upper (right scale) branch in fig. 1.
Volume 33, number 3
OPTICS COMMUNICATIONS
example are still about 100 modes within A H at the upper branch between A and B. The value in fig. 2, which have been calculated in using eq. (4), and those which can be found from approximate relations in connection with eq. (7) are in good agreement. For the upper branch this relation takes the form (cf. eqs. (7) and (! la)): (13)
A H ~ (fs+/R+)I"
as long as (AIt/P) 2 N r. For the lower branch we get A H ~ {1 +(Tb0 --TaO)/K} 1/2 F, where AH/F > 1 when 7a0 ~" 0 because of stronger damping of modes near the line centre compared to those lying more distant to it.
5. Intensities of individual modes Because s and A H depend o n R the value of/~q (cf. eq. (2)) abruptly changes when R passes from one branch to the other. Thus, we have also to distinguish three parts/~qi corresponding to R i. As long as many modes are within A n eq. (2) yieldsthe following expression for the photon flux I 0 = Mo/Tdalsa at the upper branch of the central mode:
I 0 ~ (vrr/Irfs+)R2+,
(14)
where the relation f = n(x/~-d a Tlsa)-1 has been used. The other extreme case, characterized by eq. (8) (where e+ of eq. (1 ~ ) is substituted for its right-hand side), leads to I 0 ~ R+, i.e. practically to a single-mode regime. Fig. 3 shows the upper branch of the photon flux, normalized to Isa, for q = 0 and q = 1000. For compar-
752
10 5
/
------_
2.5" lOS],ooo
/
June 1980
ison the upper branch of the total photon flux of all modes, It, is also drawn. In accordance with the narrowing of the linewidth (fig. 2) I 0 increases much stronger than I t with growing pumping while the curve corresponding to I1000 decreases slowly, excepted the outer left part, and tends to a constant. Whereas the shape of the curve Iq f o r q ~ U/26w (= 105 in our example), in principle, looks like that of R in fig. 1 this is not so for greater q. Fig. 4 shows 1105 where ~ corresponds to the lower branch of R, B-C to the upper branch and AB to the medium one of R. The physical reason of this difference is that modes with a large q are essentially fed by spontaneous emission. The latter becomes smaller, however, when stimulated emission increases, i.e. when R goes from the lower to the upper branch. Therefore, the photon flux in fig. 4 decreases at point A contrary to R in fig. 1. On the other hand the photon flux at B is higher than in the lower branch in accordance with R. Obviously this is due to strong saturation o f the absorber which in common with the amplifier delivers a greater portion o f spontaneously emitted photons, within a certain range of relatively low pumping, when R is in the upper branch than the amplifier alone when R is in the lower one and the absorber is hardly inverted. The intensity of this mode is extremely small indeed but there is a large number of, say, 104 neighbouring modes with almost the same properties. Thus the resultant effect due to all of these modes should yet be measurable. Moreover, the curve of fluorescence radiation from a cell containing a mixture of amplifying and absorbing atoms, measured perpendicular to the laser axis, should show the same behaviour.
e 8~_/'IA 70~"J~oS
/
/
70~2°
-
~c
4
7.5 20 2.5 Fig. 3. Normalized photon flux of all modes, It, and that of the modes with q = 0 (central mode) and q = 103 at the upper branches of their hysteresis curves.
~5
2.0
25
Fig. 4. Normalized photon flux of a m o d e (q = 105) far away
from line centre. The arrows indicate the hysteresis cycle whose shape differs remarkably from that of R in fig. 1. 329
Volume 33, number 3
OPTICS COMMUNICATIONS
6. Final remark From a semiclassical description in [9] follows that a homogeneously broadened laser (without absorber) in a unidirectional ring cavity exhibits pulse evolution, i.e. the amplitudes of some modes are growing, at least far above threshold. Rate equations cannot describe this effect, which leads to multimode operation also for very high intensity despite of mode competition, because interference terms between modes are neglected. On the other hand, from [10] (which includes the absorber but neither noise nor hysteresis) one can conclude that in a certain region above threshold cw operation can be stable especially if the longitudinal relaxation time of the passive medium is longer than that of the active one. One may surely assume that also in the presence of noise cw operation can be stable, which means there is a stable stationary intensity with fluctuations superimposed, at the upper branch o f the hysteresis cycle if Tlb ~ Tla and the intensity is not too high. However, instability could also appear when the pump power (-~ 7a0) becomes small as on the outer left part o f the upper branch near B in fig. 1. Then a negative deviation from the mean value of the amplifier population inversion possibly cannot be equalized quickly enough (which again depends on Tla ) so that cw operation breaks down. On the assumption that instability does not occur we have derived relations for the stationary total intensity and spectral width o f all
330
June 1980
modes at the lower and upper branch. A numerical example relating to GaAs as an active medium has been given. It shows that the linewidths o f b o t h branches differ b y a factor of about 103. Moreover, the intensities for some individual modes have been calculated. The influence o f spatial hole-burning on bistable behaviour will be discussed in another paper. I want to thank Prof. W. Brunner and Prof. H. Paul for useful discussions.
References [1 ] V.P. Chebotayev, J.M. Beterov and V.N. Lisitsyn, IEEE J. Quant. El. QE-4 (1968) 788. [2] P.H. Lee, P.B. Schoefer and W.B. Barker, Appl. Phys. Lett. 13 (1968) 373. [3] A.P. Kasantsev, S.G. Rautian and G.J. Surdutovich, Zh. Eksp. Teor. Fiz. 54 (1968) 1409. [4] R. Salomaa and S. Stenholm, Phys. Rev. A8 (1973) 2695. [5 ] L.A. Lugiato, P. Mandel, S.T. Dembinski and A. Kossakowski, Phys. Rev. A18 (1978) 238. .[6] S.T. Dembinski, A. Kossakowski, L.A. Lugiato and P. Mandel, Phys. Rev. A18 (1978) 1145. [7] I.M. Ryshik and I.S. Gradstein, Tafeln VEB Verlag d. Wiss. 1963. [8] L.W. Casperson, J. Appl. Phys. 46 (1975) 5194. [9] H. Risken and K. Nummedal, J. Appl. Phys. 39 (1968) 4662. [10] H. Knapp, H. Risken and H.D. Vollmer, Appl. Phys. 15 (1978) 265.
OPTICS COMMUNICATIONS
June 1980
BISTABLE OPERATION OF MULTIMODE LASERS CONTAINING SATURABLE ABSORBERS Roland MIJLLER
Zentralinstitut ffir Optik und Spektroskopie, Akademie der Wissenschaften der DDR, DDR -1199 Berlin Received 22 October 1979 Revised manuscript received 3 March 1980 Continuous-wave operation of many longtudinal modes in a unidirectional ring cavity containing a laser medium and a saturable absorber is investigated. The phases of the modes are assumed to be randomly distributed. The system is described by a set of rate equations including noise terms from both media. The steady-state total intensity (as an average value) and the linewidth of the laser output are calculated for the region where bistable operation and hysteresis can occur. Moreover, the hysteresis cycles of individual modes are studied. 1. Introduction
2. The basic equations
It is well-known that a laser system like that investigated here can have two stable intensity levels corresponding to the unsaturated and strongly saturated state, respectively, o f the absorber. Associated with it a hysteresis cycle o f the intensity versus pump power occurs. After the discovery of hysteresis effects in lasers with a saturable absorber [1,2] a lot o f papers dealing theoretically with this phenomenon has been published. According to the first experiments some of them are concerned with the gas laser (see, for example [3,4] ). In the following we will assume a homogneously broadened laser line in b o t h media. Therefore, we especially refer to [5,6] where this case has been extensively studied b y means o f semiclassical and fully quantum-mechanical treatment. The authors have presupposed a single-mode regime. However, at least below threshold, and also in a small region above it, multimode operation should be taken into account, in particular, when the noise level is rather high as for semiconductor and dye lasers. In this paper we will discuss some aspects of the problem in the frame o f a rate-equation approximation. This means that phase relations between different modes are not considered. Thus the model approximately describes a case where the phases are randomly distributed and the calculated steady-state intensities correspond to average values of a real system.
The following set of equations for the number o f p h o t o n s M q in the qth mode and the inversion densities na, n b of the amplifying and absorbing medium, respectively, will b e investigated:
326
dMq Mq dt
-
+
r p t- T(i--+rq2 ) ( O a n a / a - Obnblb)
1
rO+rq 2)
(Oanal a +½ [nb0 - nb] Oblb) ,
(la)
o~
dn a _ nao - n a dt
Tla
na ~ Mq daTTlalsa q=_~o l + r q 2 '
(lb)
oo
dnb_ nbO-n b dt
Tlb
,finb
~
Mq
(lc)
daTTlblsa q = _ ~ l+rq2"
The index "a" indicates parameters o f the active medium, " b " those o f the absorber. The used symbols have the following meaning: Tp is the p h o t o n mean lifetime in the cavity, T its round-trip time, l the length o f the cell containing the amplifying and absorbing medium, and d is the cross-sectional area o f the light beam. r = (26co/P) 2 contains the ratio o f mode-spacing 6co (circular frequency) to the width F o f the lorentzian atomic lines which are assumed to be equal for b o t h media. Further, o is the atomic cross-section of stimulated emission or absorption at line centre, T 1
Volume 33, number 3
the longitudinal relaxation time, and I s = ( a T 1 )-1 is the photon flux that saturates the medium. Finally, we put/~ = lsada/lsbd b . The last term at the righthand side of eq. (la) is due to spontaneous emission from both media, whose atoms are considered as fourlevel (amplifier) oI two-level systems (absorber), respectively. Because rib0 denotes the density of absorbing atoms if no field is in the cavity the expression 1 (rib0 - n b ) is the density of those in the upper level. First of all we calculate the steady-state solutions ~tq, ha, and h b after putting their derivatives to zero. Then the following relation can be found easily: .~q = s/(1 +rq 2 + o - u).
(2)
Besides, h a = ha0/(1 + R) and ~ib = nb0/(1 +/~R). Here we have used the abbreviations oo
R
_
1 ~ isadaT q = - ~
/~q l+rq 2'
U = "Ya0[(1 + R ) n ] - i
(3a, b) °='fbo[(l+lJR)l¢]-l,
s = u +31 vlSR.
\l+--+d~-u]
- coth(Tr/x/7)},
coth ~" (4)
(5)
where I~ denotes the atomic linewidth. Secondly, we write down the formula for the total photon flux 2
I t : ( u - v ) R + (Fs/4dalsa) coth(Tr/x/r).
(6)
When the noise terms in eq. ( l a ) a r e neglected then we have s = 0 and eq. (4) is reduced to ( l + v - u ) R =0 i.e. 31q = 0 for q 4:0 and I t = R (single-mode regime).
3. Calculation of the total intensity Now we will solve eq. (4) approximately. The factor f, at its right-hand side is in the range o f 10 - 4 10 -3 for a GaAs laser [8]. Concerning dye lasers an estimation of f gives 1 0 - 7 - 1 0 - 6 i f d a ~ 3 X 10 -5 cm 2. Presuming r ' ~ 1 (many modes within the range of F) two extreme cases can be distinguished. The first is characterized by the inequality x - ( l + v - u ) / r >> 1 that leads to c o t h x ~ 1. Using eq. (5) one findsx -----(AH/266o)2 >> 1 i.e. many modes also within the range of A H. From eq. (4) then follows (l+v-u)R
= -~ •
1 +
"
This particular form of eq. (7) has been chosen to facilitate a comparison with the equation corresponding to single-mode regime (cf. the end of ch. 2). The second case, corresponding to high intensities, is connected with x ,~ 1, so that eq. (4) delivers now
O + v - u ) R - f s ( x / 7 - ~(~+v-u)) ~ ( u - v)
where the relations 6w = 27r/T and f = P/4Isad a have been used. The s y m b o l " c o t h " means the hyperbolic cotangent. When we put o = O, i.e. eliminating the absorber, eq. (4) then agrees with a corresponding expression of [8]. Before deriving solutions of eq. (4) in the next chapter we give two other relations needed in the following. At first we calculate the linewidth AiLof the field (spectral width), defined by 3~qH = ~-M 0 with 2qn 6w = AH, assuming at least several modes to be within the range o f A H. From eq. (2) one gets • A H = (1 + u - u)l/2I ",
art = (lsadaT)-I ~,qMq using eq. (4):
(3c, d)
Moreover, 7a = °a~Tala and 7b = Ob rib lb define the coefficients of small-signal gain and loss of the amplifier and absorber, respectively. The cavity loss is denoted by K = T I T v. After introducing ~tq from eq. (2) the infinite sum of eq. (3a) can be calculated [7]. The result is an implicit expression for R : u-V'
June 1980
OPTICS COMMUNICATIONS
(8)
We will construct approximate solutions of eqs. (7), (8) in using those for the single-mode regime. Eq. (7) with s = 0 immediately yields the following three solutions f o r R (cf. also [5] ), marked by a tilde, /~0 = 0
(9a)
and o
9b,
Here we have used the abbreviations go = 7aO--Tb0--K,
a =go +(1--1//5)Tb0--r//~" (10a, b)
The relation go = Odetermines the threshold of the centralmode (q = 0). R _ has no physical meaning and/~0 is not stable i f g 0 > 0. On the other hand there is a cer327
Volume 33, number 3
OPTICS COMMUNICATIONS
tain range go ~< 0 where all the three functions/~i are positive if/6 ~> 1 +~/3'b0. Then/~0 is always stable,/~+ can be stable but R _ is certainly not [5]. Thereby a range of parameters exist where bistability can occur. Now we introduce the ansatz Ri = R i + Pi with ]Oil /~i into the left-hand sides of eqs. (7), (8). At their right-hand sides, which are called e i in short, we substitute R i for R. Then we find for the right-hand side of eq. (7) ei
=
(fsi)2/R i
(11 a)
and for eq. (8) ei
=
fsix/'r/Tr,
(1 lb)
where the quantity s i is defined by eqs. ( 3 b - d ) after substituting R i f o r R there. Moreover, we confine ourselves to linear terms o f Pi. Then from eqs. (7) and (8) f o r g O < 0 follows a simple but somewhat rough approximation: (1 +/~_+)(1 +lSR±)e~ R± ~/~+ +-
2 ((c~/2K)2 + go/K/~} 1/2/5/~+ "
(12a)
It should be noted that eq. (12a) is not valid in the vicinity of the points where R+ coincides w i t h / ~ because the square root tends to zero there. Finally, an expression f o r R 0 corresponding to/~0 = 0 has to be found. To this end we pass over to the complete eq. (7) without the restriction described by eq. (1 la). AssumingR 0 ~ 1 and/?R 0 ~ 1 one gets Roof
7a0 (l_(_K/go)l/2) ")'b0 --')'a0
'
(lZb)
where go < 0. Fig. 1 shows the dependence of R on the small-signal gain coefficient 7a0" The curve has been calculated from eq. (4). The used parameters are f = 5 × 10 - 4 , "Yb0 = 2, ~ = 0.5 ,i6 = 5, and r = 1010 (i.e. a ring-cavity length of 3 m with I" ~ 1014 s -1 for GaAs). The dashed line marks the threshold g 0 = 0. The lower branch of the curve (from left below to A) corresponds to R 0 the upper one (from B to right above) to R+. The medium part (from A to B) belongs to R . At the crossing points of the lower and upper branch with the medium one the system becomes unstable and passes from R 0 to R+ at A and vice versa at B. This hysteresis cycle is indicated by arrows in fig. 1. Because R 0 > / ~ and R+ > / ~ + (cf. eq. (12a))but R _
June 1980
' /og R 1
o
B~.____
-2
A', l
-3 ~Z----~~ Z5
20
-
2' ~'a°K~
Fig. 1. Sum of the weighted photon numbers of all modes, R (defined by eq. (3a)), versus small-signalgain coefficient, ~'ao" The arrow indicate the hysteresis cycle. R in the upper branch is almost equal to the normalized total photon flux or intensity, I t .
modes oscillate compared to a single-mode regime. For the latter A coincides with the crossing point of the 3'a0"axis and the dashed line. The numerical calculation shows, however, that B is only slightly delayed compared to A. Therefore, the hysteresis curve is shortened by nearly the delay of A. In our example the latter is of about 10 percent related to the distance between A and B.
4. Linewidth of the field Now we have to deal with eq. (5). Fig. 2 shows AH/P for the lower (left scale) and the upper (right scale) branch of R in fig. 1. The linewidths in the two branches differ by a factor of about 103 at the transition points. Therefore, in a long cavity like that in our
~A,
..aA, k3
-, 8
2
7~ U
-2 7 -1
I
.,
O.5
/.0
¢5
20
2.5
~o_
-
Fig. 2. Linewidth of the field over atomic linewidth, AH/F, versus small-signal gain coefficient for the lower (left scale) and upper (right scale) branch in fig. 1.
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OPTICS COMMUNICATIONS
example are still about 100 modes within A H at the upper branch between A and B. The value in fig. 2, which have been calculated in using eq. (4), and those which can be found from approximate relations in connection with eq. (7) are in good agreement. For the upper branch this relation takes the form (cf. eqs. (7) and (! la)): (13)
A H ~ (fs+/R+)I"
as long as (AIt/P) 2 N r. For the lower branch we get A H ~ {1 +(Tb0 --TaO)/K} 1/2 F, where AH/F > 1 when 7a0 ~" 0 because of stronger damping of modes near the line centre compared to those lying more distant to it.
5. Intensities of individual modes Because s and A H depend o n R the value of/~q (cf. eq. (2)) abruptly changes when R passes from one branch to the other. Thus, we have also to distinguish three parts/~qi corresponding to R i. As long as many modes are within A n eq. (2) yieldsthe following expression for the photon flux I 0 = Mo/Tdalsa at the upper branch of the central mode:
I 0 ~ (vrr/Irfs+)R2+,
(14)
where the relation f = n(x/~-d a Tlsa)-1 has been used. The other extreme case, characterized by eq. (8) (where e+ of eq. (1 ~ ) is substituted for its right-hand side), leads to I 0 ~ R+, i.e. practically to a single-mode regime. Fig. 3 shows the upper branch of the photon flux, normalized to Isa, for q = 0 and q = 1000. For compar-
752
10 5
/
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2.5" lOS],ooo
/
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ison the upper branch of the total photon flux of all modes, It, is also drawn. In accordance with the narrowing of the linewidth (fig. 2) I 0 increases much stronger than I t with growing pumping while the curve corresponding to I1000 decreases slowly, excepted the outer left part, and tends to a constant. Whereas the shape of the curve Iq f o r q ~ U/26w (= 105 in our example), in principle, looks like that of R in fig. 1 this is not so for greater q. Fig. 4 shows 1105 where ~ corresponds to the lower branch of R, B-C to the upper branch and AB to the medium one of R. The physical reason of this difference is that modes with a large q are essentially fed by spontaneous emission. The latter becomes smaller, however, when stimulated emission increases, i.e. when R goes from the lower to the upper branch. Therefore, the photon flux in fig. 4 decreases at point A contrary to R in fig. 1. On the other hand the photon flux at B is higher than in the lower branch in accordance with R. Obviously this is due to strong saturation o f the absorber which in common with the amplifier delivers a greater portion o f spontaneously emitted photons, within a certain range of relatively low pumping, when R is in the upper branch than the amplifier alone when R is in the lower one and the absorber is hardly inverted. The intensity of this mode is extremely small indeed but there is a large number of, say, 104 neighbouring modes with almost the same properties. Thus the resultant effect due to all of these modes should yet be measurable. Moreover, the curve of fluorescence radiation from a cell containing a mixture of amplifying and absorbing atoms, measured perpendicular to the laser axis, should show the same behaviour.
e 8~_/'IA 70~"J~oS
/
/
70~2°
-
~c
4
7.5 20 2.5 Fig. 3. Normalized photon flux of all modes, It, and that of the modes with q = 0 (central mode) and q = 103 at the upper branches of their hysteresis curves.
~5
2.0
25
Fig. 4. Normalized photon flux of a m o d e (q = 105) far away
from line centre. The arrows indicate the hysteresis cycle whose shape differs remarkably from that of R in fig. 1. 329
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OPTICS COMMUNICATIONS
6. Final remark From a semiclassical description in [9] follows that a homogeneously broadened laser (without absorber) in a unidirectional ring cavity exhibits pulse evolution, i.e. the amplitudes of some modes are growing, at least far above threshold. Rate equations cannot describe this effect, which leads to multimode operation also for very high intensity despite of mode competition, because interference terms between modes are neglected. On the other hand, from [10] (which includes the absorber but neither noise nor hysteresis) one can conclude that in a certain region above threshold cw operation can be stable especially if the longitudinal relaxation time of the passive medium is longer than that of the active one. One may surely assume that also in the presence of noise cw operation can be stable, which means there is a stable stationary intensity with fluctuations superimposed, at the upper branch o f the hysteresis cycle if Tlb ~ Tla and the intensity is not too high. However, instability could also appear when the pump power (-~ 7a0) becomes small as on the outer left part o f the upper branch near B in fig. 1. Then a negative deviation from the mean value of the amplifier population inversion possibly cannot be equalized quickly enough (which again depends on Tla ) so that cw operation breaks down. On the assumption that instability does not occur we have derived relations for the stationary total intensity and spectral width o f all
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modes at the lower and upper branch. A numerical example relating to GaAs as an active medium has been given. It shows that the linewidths o f b o t h branches differ b y a factor of about 103. Moreover, the intensities for some individual modes have been calculated. The influence o f spatial hole-burning on bistable behaviour will be discussed in another paper. I want to thank Prof. W. Brunner and Prof. H. Paul for useful discussions.
References [1 ] V.P. Chebotayev, J.M. Beterov and V.N. Lisitsyn, IEEE J. Quant. El. QE-4 (1968) 788. [2] P.H. Lee, P.B. Schoefer and W.B. Barker, Appl. Phys. Lett. 13 (1968) 373. [3] A.P. Kasantsev, S.G. Rautian and G.J. Surdutovich, Zh. Eksp. Teor. Fiz. 54 (1968) 1409. [4] R. Salomaa and S. Stenholm, Phys. Rev. A8 (1973) 2695. [5 ] L.A. Lugiato, P. Mandel, S.T. Dembinski and A. Kossakowski, Phys. Rev. A18 (1978) 238. .[6] S.T. Dembinski, A. Kossakowski, L.A. Lugiato and P. Mandel, Phys. Rev. A18 (1978) 1145. [7] I.M. Ryshik and I.S. Gradstein, Tafeln VEB Verlag d. Wiss. 1963. [8] L.W. Casperson, J. Appl. Phys. 46 (1975) 5194. [9] H. Risken and K. Nummedal, J. Appl. Phys. 39 (1968) 4662. [10] H. Knapp, H. Risken and H.D. Vollmer, Appl. Phys. 15 (1978) 265.