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Laser with a saturable absorber-dispersive effects
Physica 111C (1981) 353-364 North-Holland Publishing Company
LASER WITH A SATURABLE A B S O R B E R - D I S P E R S I V E EFFECTS T e m b a S. D L O D L O * Department of Physics, University of Helsinki, Finland Received 23 September 1980 Revised 13 April 1981 We study the characteristics of the gain function of a laser with an intracavity saturable absorber. Included within the laser cavity is a Fabry-Perot etalon. A semiclassical description of a single mode homogeneously broadened laser system is used. The laser frequency is tuned to the amplifier transition frequency while the absorber detuning is nonzero. In contrast with the usual models, the loss is a simple function of the cavity detuning which in turn depends on the absorber detuning. The etalon characteristic is approximated by an inverted Lorentzian whose relative width proves to be an important parameter. From an analysis of the gain function we conclude that for a suitable combination of parameters the bistability can be purely dispersive. Comparisons are made with the dispersive case of the ordinary optical bistability. We also show that such an arrangement can behave as a filter, suppressing fields within a certain frequency range.
1. Introduction Since S z 6 k e et al. [1] first p r o p o s e d t h e use of a n o n l i n e a r a b s o r b i n g F a b r y - P e r o t (FP) interf e r o m e t e r for o p t i c a l b i s t a b l e o p e r a t i o n s , a lot of w o r k has b e e n d o n e on t h e t h e o r e t i c a l u n d e r s t a n d i n g of t h e t r a n s m i s s i o n c h a r a c t e r i s t i c s of a F P cavity c o n t a i n i n g a t w o - l e v e l a t o m i c system [2]. Such a s y s t e m was e x p e r i m e n t a l l y shown t o e x h i b i t b i s t a b l e b e h a v i o u r w h e n i r r a d i a t e d with laser light [3]. This i n t e r e s t has b e e n p r o m p t e d by t h e r e a l i z a t i o n t h a t t h e s y s t e m has a p o t e n t i a l for d e v i c e a p p l i c a t i o n s [4], such as m e m o r i e s , o p t i c a l a m p l i f i e r s a n d switches. This o p t i c a l bistability has b e e n f o u n d t o b e d u e to b o t h abs o r p t i v e a n d d i s p e r s i v e effects [5]. T h e first semiclassical analysis of b o t h t y p e s of o p t i c a l b i s t a b i l i t y was given b y M c C a l l [6]. T h e t h e o r y has b e e n d e v e l o p e d within t h e m e a n field app r o x i m a t i o n of ref. [2], t h e validity of which has b e e n t h e o r e t i c a l l y d e m o n s t r a t e d [7]. O n t h e o t h e r h a n d , f o r a laser with an int r a c a v i t y s a t u r a b l e a b s o r b e r , which also e x h i b i t s b i s t a b i l i t y [8, 9], t h e analysis has, in g e n e r a l , n o t * Present address: Physics Department, University of Zimbabwe, P.O. Box MP 167, Salisbury, Zimbabwe.
b e e n c o n c e r n e d with d i s p e r s i v e bistability. A s a result t h e laser f r e q u e n c y is always a s s u m e d to be n e a r l y e q u a l to t h e cavity f r e q u e n c y . It is d u e to this f e a t u r e t h a t t h e cavity losses a r e always a s s u m e d c o n s t a n t in laser t h e o r y . In this p a p e r we i n t e n d to c a l c u l a t e t h e effective gain of t h e laser with an i n t r a c a v i t y s a t u r a b l e a b s o r b e r as a f u n c t i o n of t h e field s t r e n g t h f o r different v a l u e s of t h e d e t u n i n g , for a m o d e l of t h e laser s y s t e m in which a F P e t a l o n is i n c l u d e d within t h e laser cavity. T h i s a r r a n g e m e n t i n t r o d u c e s t h e possibility of losses which a r e v a r i a b l e within t h e oscillation b a n d w i d t h of t h e laser. T h e inclusion of t h e F P e t a l o n within t h e l a s e r cavity i n t r o d u c e s a n u m b e r of n e w p a r a m e t e r s in t h e calculations, such as t h e s e p a r a t i o n a n d t h e reflectivity of t h e e n d faces of t h e F P e t a l o n . F o r simplicity, w e will l o o k at a s y s t e m in which t h e a m p l i f i e r d e t u n i n g is z e r o while t h a t of t h e abs o r b e r is not. In section 2 we p r e s e n t a g e n e r a l f o r m u l a t i o n which i n c l u d e s a t h e o r e t i c a l m o d e l a n d t h e d e r i v a t i o n of t h e gain function for t h e h o m o geneously broadened amplifier and absorber transitions. F o r t h e l a t e r discussions w e also inc l u d e t h e e x p r e s s i o n for t h e gain f u n c t i o n in t h e
0378-4363/81/0000-0000/$02.75 O 1981 N o r t h - H o l l a n d
354
T.S. Dlodlo / Laser with a saturable absorber - dispersive effects
case w h e n the a b s o r b e r is i n h o m o g e n e o u s l y b r o a d e n e d . In section 3 we discuss the stability of t h e s y s t e m with r e s p e c t to v a r i o u s p a r a m e t e r s . W e also include an i n v e s t i g a t i o n into t h e effect of a m p l i f i e r d e t u n i n g . Section 4 gives t h e results a n d section 5 c o n t a i n s a discussion of the results a n d s o m e conclusions.
S
/ 0°0
.
(~--RI
Fig. 2. Losses versus cavity detuning (to- O). The function represented by this figure is: L(to - / ~ ) = L0{l + (k - 1)(to -
2. General formulation
~ ) : / [ ~ 2 + (to x.1)21/. 2.1.
Theoretical model
O u r t h e o r e t i c a l m o d e l consists of a p a i r of t w o - l e v e l m e d i a s p a t i a l l y s e p a r a t e d b e t w e e n two m i r r o r s . A F a b r y - P e r o t e t a l o n is i n s e r t e d within t h e laser cavity as s h o w n in fig. l. T h e t w o m e d i a a r e p u m p e d in such a w a y that o n e is an amplifier while t h e o t h e r o n e is an a b s o r b e r . W e f o r m u l a t e o u r a p p r o a c h in t e r m s of L a m b ' s s e m i c l a s s i c a l t h e o r y of t h e laser [10]. W e a s s u m e that t h e g e n e r a t e d single m o d e laser field is t u n e d to r e s o n a n c e with t h e t r a n s i t i o n f r e q u e n c y of t h e a m p l i f i e r while t h a t of t h e a b s o r b e r is not. T h e two m e d i a c o u l d consist of t h e s a m e o r different m a t e r i a l s . F o r simplicity w e c o n s i d e r an a r r a n g e m e n t in which b o t h t r a n s i t i o n s a r e homogeneously broadened and we neglect the i n h o m o g e n e o u s b r o a d e n i n g d u e to t h e a t o m i c p o s i t i o n s in t h e s t a n d i n g w a v e ; t h e t r e a t m e n t w o u l d , in fact, b e e x a c t in a o n e - d i r e c t i o n a l ring cavity. T h e t r e a t m e n t can b e e x t e n d e d in a s t r a i g h t f o r w a r d w a y to i n c l u d e t h e s e a d d i t i o n a l f e a t u r e s . T h e m e d i a cells a n d t h e r e s o n a t o r all c o n t r i b u t e to t h e losses. W e l u m p t h e s e losses as if t h e y c a m e f r o m a single s o u r c e a n d a s s u m e t h a t t h e r e s u l t i n g loss f a c t o r d e p e n d s on t h e
M,
\
A
ttttt
/
pump(amplifier)
z\
"'
pump(absorber)
Fig. 1. Laser with saturable absorber with MI and M2 ordinary laser mirrors. The FP etalon is used to introduce dispersive effects, d is the thickness of the etaion, A is the amplifier and B the absorber.
cavity d e t u n i n g which in turn, d e p e n d s on t h e a b s o r b e r d e t u n i n g a n d t h e field intensity. T h e d e p e n d e n c e of t h e loss f a c t o r on t h e cavity d e t u n i n g and, h e n c e , on t h e field intensity, which is a b s e n t in t h e usual laser r e s o n a t o r m o d e l is a result of t h e F P e t a l o n i n c l u d e d in t h e laser cavity. F o r such an a r r a n g e m e n t even small d e t u n i n g s a w a y f r o m t h e e i g e n f r e q u e n c y of t h e e m p t y cavity m a y b e significant. L a s e r t h e o r y , in c o n t r a d i s t i n c t i o n , usually a s s u m e s that the cavity Q f a c t o r can be e v a l u a t e d at the e i g e n f r e q u e n c y . O u r c h o i c e for this d e p e n d e n c e is g u i d e d by t h e k n o w l e d g e of t h e t r a n s m i s s i o n c h a r a c t e r i s t i c s of a F P e t a l o n [11] a n d w e c h o o s e t h e loss function in such a w a y that t h e loss is a m i n i m u m at z e r o cavity d e t u n i n g a n d a large c o n s t a n t at large v a l u e s of the d e t u n i n g . A suita b l e a p p r o x i m a t i o n for the F P c h a r a c t e r i s t i c s n e a r t h e t r a n s m i s s i o n r e s o n a n c e is an i n v e r t e d L o r e n t z i a n as shown in fig. 2. T h e F P transmission is a s s u m e d to b e c e n t e r e d at the laser AnnD]ifier
Absorber
'°'I,,o: Fig. 3. Energy level diagrams for the amplifier and absorber media, a, b, c and d indicate the excited levels and [0) is the ground state, yi(A/) is the decay (pump) rate of the level [i) where i = a, b, c, d.
T.S. Dlodlo / Laser with a saturable absorber - dispersive effects
cavity eigenfrequency. The additional parameters introduced by this choice of the loss function are the width of the Lorentzian which is the bandwidth of the FP e t a l o n - i n our a p p r o x i m a t i o n - a n d the etalon parameter k which is a measure of the losses (kLo) at large cavity detuning. The polarization which is the driving force is the sum of the polarizations of the two media
355
and is inserted into Maxwell's equations to derive the field amplitude and the laser frequency. The energy level diagram is shown in fig. 3 where the various decay (pump) rates are shown with arrows down (up), and tOab(tOed)is the transition frequency between the levels la) (Ic)) and Ib) (Id)). The laser frequency is given by tO while /2 is the eigenfrequency of the empty cavity.
2.2. T h e g a i n ]:unction
The semiclassical laser theory [10] gives us the following self-consistent equations for the electric field amplitude and the laser frequency: (1)
= - (goEI2Q + ~Pd2e),
(2)
(tO - O ) = - (&Pr/2EE ,
where t3 = tO, Q is the quality factor and Pi(Pr) the imaginary (real) part of the sum of the average polarizations induced in the two media. The derivation of Pi and Pr will not be repeated here since it is found in most of the literature on lasers. We only give the results relevant to our laser model: DOabAA
Pi = --hrabO~ab '1 + X2 + Z2b
[ DOabZabAA
e r : --hl abOiab~l"~-'~-~-"~b
~ P' " D O ,~A s "~
1-~X~-Z'Z2d) ~ '
(3)
~'flD°cz¢,rA.]
(4)
l + flXz + Z~d] ~ ,
w h e r e AA and AB are constants proportional to the lengths of the amplifier and the absorber medium,
respectively, Zab = (toab- to)/%b, Zcd = (tOed- tO)/Ycd, 7/' = F¢d/F~b, ~ = Ct~/a~b is a saturation parameter,
y,yj
Imijl~
1"° = Yi + Yj'
°t° = h2Yi%'
O o (Ai ~], =
\ ~ / - Yi!
Vii = 21('Yi+ Yj),
m e is the element of the electric dipole moment, X 2 = a a b E 2 and i, j = a, b, c, d. Now we substitute (3) and (4) into (1) and (2) respectively and obtain the general expressions for the rate of change of the field amplitude and the laser frequency, which are
1{ ic=-~
L0A L(to-gl) 1
[
LoB}
1+xe+z2b÷1+/3X2+Z~
AZab
Bz~
( t o - ° ) = ~ L 0 ~ I + ~ ¥ z~b l+~x2+z~) ,
X,
(5) (6)
where L(to - O ) = &/Q, B = rl[3 if the absorber strength, L0 gives the cavity losses at to - / 2 = 0 and
T.S. Dlodlo / Laser with a saturable absorber- dispersive effects
356
A = h&Fab~,bD°~AA/eoLo and "r/= ~'h&~abFabD°acAB/EoLo are p r o p o r t i o n a l to the amplifier and a b s o r b e r p u m p i n g respectively. L(w - f2) is the function shown in fig. 2, i.e. (k - 1)(w - 12)2 /
f
L(oJ - I2) = Lo i1 + ~4F~-~_~-~-- ~-~ j .
(7)
If we substitute (6) into (7) and put the resulting expression for L into (5) we obtain 1
f
A
2=~L0/-l+x
B
2+z~b
l + f l x 2+z2~
1
(k - 1)e2{AZab(l + fiX 2 + zZd) -- BZco(1 + x 2 + Za2b)}2 - (1 + x 2 +
} r-+
x,
(Sa)
where the width p a r a m e t e r ~ = Lo/F measures the influence of the FP width F in (7) and k is the etalon p a r a m e t e r in fig. 2 which measures the losses when the cavity detuning ~o - ~ 2 ~- F. W e n o w consider the special case of a r e s o n a n t amplifier for which Zah = 0 and replace z,,,~ by z so that (8a) b e c o m e s
1 { A 2=~Lo l+x2
B l+flx2+z 2
1
(k --I)E2B2z 2 I {(l+~x~_z--~B2z2}jx.
(88)
T h e first three terms on the right-hand side of (8) represent the situation when the loss L is i n d e p e n d e n t of the d e t u n i n g o)-£2, i.e. k = 1. T h e amplification factor, which we shall call the gain function, is defined by
G(x, z, e) =
{ A
B
1+ x2
(k~l)E2B~z2
1 q- j~X2 -}- -72
1
I
{(1 q- j~X2 -}- Z2)2 -{- E2B2z2}J"
(9)
In the case of an i n h o m o g e n e o u s l y b r o a d e n e d a b s o r b e r transition the macroscopic polarization P(r, t) induced in the m e d i u m gets its contribution f r o m all a t o m s regardless of the.Jr velocities or positions at the time they are excited. T h u s in deriving P(r, t) f r o m the density matrix elements for a single a t o m it is necessary to integrate over the velocity distribution and excitation position. W e neglect collisions and recoil and calculate the polarization of the a b s o r b e r m e d i u m , following L a m b ' s t h e o r y for the gas laser [10], in the rate e q u a t i o n approximation. T h e integration over position yields a constant. T h e expressions similar to (3) and (4) are
Pi
D°abAA
7q'flD°~FiAs } E ,
(10)
DObz'~A n,BDOFrAB} E , P~ = - hF.ba.b t i ¥ ~ -~ab
(11)
~ h~b~ab
t l + x 2+ Z2b
where F~ =
If
W(s){L+ + L_}
do
{1 + lzOxZ(L+ + L_)} o,
W(s) = ~ exp(--//,2s2), vTr
F, =
fo® W(s){L+(z + s) + L_(z - s)} ds {1 + 12OxZ(L++ L_)} '
S = KV/Tca,
tx = ycduK,
L+ = 1/{l + (z + s)2},
T.S. Diodlo / Laser with a saturableabsorber- dispersiveeffects
357
u is the width of the Maxwell velocity distribution function, K is the wave number and V the velocity. In the usual model of the laser with an intracavity saturable absorber where the detuning is neglected one can express Fr and Fi in terms of the plasma dispersion function as is done in [8]. In our case we carry out a numerical integration, and we want to consider all values of the detuning z. For a ring laser, F, and Fi are easily evaluated. On substituting (10) and (11) into (1) and (2) and setting Zab = 0 we obtain an expression for the gain function when the absorber transition is inhomogeneously broadened: G ( x , z, ~) =
A
---4-~x2 - BF, - 1 -
(k - 1)~2B2F~/
(12)
1 + ~2B2F~ J"
An extension to the situation where both transitions are inhomogeneously broadened can easily be made. Since we mainly consider an arrangement in which both amplifier and absorber transitions are homogeneously broadened, our further analysis revolves around the cavity term of (9), i.e. the term which contains the FP cavity parameters ~, k and the detuning z.
3. A n a l y s i s of s t e a d y state s o l u t i o n s
A discussion of (9) with k = 1 and zero absorber detuning can be found in the literature [8, 9] and some of the earliest work on lasers containing intracavity saturable absorbers is reported in [12]. H e r e we will examine the effect of the cavity term with the etalon parameter k ~ 1 and the absorber detuning z # 0 for various values of the width parameter E. We will begin by examining the gain function for constant amplifier and absorber pumping. In steady state ~ = 0 ; hence we obtain the gain equation (G(x, z, E) = 0) whose solutions we would like to analyse. This gain equation has been analysed in detail [9] when z = 0 and k = 1. When the system is bistable the gain equation has two real solutions one of which is the stable solution. For nonzero detuning both the stable and unstable solutions are seen to depend on the absorber detuning parameter. There are three steady state solutions for the electric field: the trivial x = 0 and the two solutions, obtained from the gain equation. Furthermore, by defining a "free energy" like function
U(x) = -X2Lofo G(x, z, ~)x dx,
(13)
we are able to see which of the two solutions is
the stable one. The stable solution is at a minimum of U(x) while the unstable solution is at a maximum of U(x). From (9) and (13) we find that 1 L 0 { x 2+ , / 1 + / 3 x 2+z2\ U(x)=~ 'Tm~ iTz~ ) - A In(1 + x 2)
+(k - 1 ) , ~ z {tan-' ( l + flx2 ~Bz + z2)
- tan-' \ ~--~Bz-z] J "
(14)
The use of U(x) for conventional laser theory was first proposed in [13] where a continuum mode laser was treated and in [14] where the single mode laser was treated. Scott et al. [15] showed that U(x) could be used in the stability analysis of the laser with a saturable absorber. Calculations of the fir.st passage time for a laser with a saturable absorber have been done by Bulsara and Schieve [16] using the pseudopotential U(x) and analysis of G(x) using U(x) has been done in the description of bistability in the dye laser [17]. The values of the parameters z, ~ for which G(0, z, E) < 0 and Gmax(x, z, ~) > 0 yield two solutions. When G(0, z, e) < 0 and Gmax(X,z, e) < 0 then there are no solutions. In all cases when
358
T.S. D l o d l o I L a s e r with a saturable absorber - dispersive effects
G(0, z , ~ ) > 0 there is only one solution: the stable solution. The border between the one stable solution situation and the bistable situation is given by the simulataneous conditions G(0, z, ~) = 0 and Gm~(X, z, e) > 0 and that between the bistable and the no solution situation by the simultaneous conditions Gmax(X, z, e ) = 0 and G(0, z, E ) < 0. In fig. 4, curve 1 satisfies the f o r m e r conditions while curve 2 satisfies the latter. The simultaneous conditions satisfied by curve 2 imply that the stable and unstable solutions should coincide or equivalently that gain and loss are equal at a point x = x0 where their derivatives with respect to x are also equal, i.e. OG(x, z, e)[I = 0, OX x=x,
G(Xo, z, e) = O,
(15)
Xo = Xs = Xu •
From the above conditions it is possible to derive expressions or inequalities which define a region in the e, z plane, where bistable behaviour is possible. W e find the relationship between e and z by eliminating x0 in (15). This procedure is a mathematically c u m b e r s o m e exercise. W e will therefore find the relationship between e and z numerically below.
••.
t - z . _ . --...
//,.---.,,--%, ~
o.~" ~ _ / ./
-o..
Y/"
-"-,, o ! i " - . ~ ~!o "~'-'~2
/
/ Fig. 4. The gain function G ( x ) as a function of the field strength with the absorber detuning z and the width parameter e such that curve 1 gives the border between I - t h e monostable (single solution) s i t u a t i o n - and I I - the bistable s i t u a t i o n - and curve 2 represents the border between II and I I I - t h e no solution situation. I, II and III are given by the dotted curves.
An analysis of the loss function in (7) is necessary for a better understanding of the behaviour of G ( x ) . Let us start with the situation when z # 0 and to - / 2 ~> F. In this case the cavity loss is kLo, which is taken to exceed the gain so that oscillations are impossible. In this connection the p a r a m e t e r B in (6) is important because it determines the magnitude of the m a x i m u m possible value of the cavity detuning (to - 1 2 ) . If we look at the case with x > 0 then (6) shows that the field intensity pulls the laser frequency towards the cavity eigenfrequency (to ~ / 2 ) and the loss decreases towards L0. It is then possible that the gain exceeds the loss and oscillations are stable. The change of the refractive index, due to the presence of laser oscillations, makes the laser wavelength fit the geometry of the etalon and reduces the loss further below the gain. This can happen also when the gain is a monotonic function of x, i.e.
A-B>
A - -
1 + x2
B 1 +/~x 2
and no absorptive bistability is possible. As we are interested in analysing the dispersive bistability we will choose our p a r a m e t e r s such that there is no bistability when z -- 0. Since laser oscillations occur near zero cavity detuning, we expect the gain function to behave in such a way that at large z there is no bistability because the cavity detuning ( t o - / 2 ) ~ 0 as z ~ 0 and z~ as can be seen in (6) if we put Zab = 0 and Zcd=Z. When z # 0 , e # 0 and k # l the cavity term L ( t o - / 2 ) appears in the expression for the gain function and when x is varied we find a typical bistable situation emerging which is due to purely dispersive effects. At x = 0 the absorber detuning makes the laser want to oscillate well off the cavity resonance; the loss is large, but an increase in intensity makes the detuning approach zero for large intensities and a stable steady state emerges. W e now look at the effect of amplifier detuning on the gain function. In eq. (8a) we see that
T.S. Dlodlo / Laser with a saturable absorber - dispersive effects
there are two terms which involve zab,- the amplifier detuning. For all values of Zab the gain t e r m is reduced but can still be positive for very small Z,b. T h e cavity t e r m must be sensitive to zab because, depending on the values of A, B and z, it could approach zero or be very large. In the latter case there would be no oscillations while in the former, oscillations would only take place for very small Z~b and no bistability would be observed.
4. Results
W e present here the numerical results of the effective gain (the gain function) as a function of the dimensionless electric field strength x, with z, and B as parameters. In all the numerical calculations the p a r a m e t e r proportional to the amplifier pumping rate has been chosen to be A = 2.0 and the etalon p a r a m e t e r k = 5. For this choice, fig. 5 shows us for which values of the absorber strength B, there is no absorptive bistability. W e have chosen B = 0.5 and B = 1.0 for comparison. B = 1.0 is at the border between a
a :o.5 as.
A=20 z--()43 B:"I13
one stable solution situation and a bistable situation when only the absorptive bistability (z = 0) case is considered for the chosen amplifier pumping p a r a m e t e r A = 2.0. For every A there is an approximate B giving the b o r d e r between the one stable solution situation and a bistable situation. In [8, 9] regions of bistability are identified in the A, B plane. The curves in fig. 6 give the results for the width p a r a m e t e r ~ = 3.0. In fig. 6a the absorber strength B = 0.5 while B = 1.0 in fig. 6b. It is shown in this figure that when the absorber detuning is increased from zero the system moves m o r e into the bistable region where there
4:20
0.5
~= 3.0
~/ 1
B
.o
~'~
z =1.5
a
Q5
/ G(X)f
359
/
~
A = 2.0
\
B=I.0
"
£=3.0
•
0.0
I
"
o~
/--2.0 Fig. 5. The gain function G(x) as a function of the field strength x at absorber strength parameters B = ,//3 = 0.5, 1.0 and 1.5. Absorptive bistability does not take place when B < 1 for the chosen value of the amplifier pump parameter A = 2.0. In [8a, 9a] regions of bistability are identified in the A, B planes.
,0
x
b
Fig. 6. The gain function O ( x ) with A = 2.0 and ~ = 3.0 for various values of the absorber detuning z. (a) B = 0.5 and 0a) B = 1.0. As z is increased from z = 0.0 the system moves, via the bistable region, into the no solution region and back into the single solution region at still higher z values. The U(x) function corresponding to (a) is found in fig. 7.
364~
T.S. Dlodlo / Laser with a saturable absorber - dispersive effects
are two solutions. A s the detuning is further increased a situation is first r e a c h e d w h e r e there are no solutions at all. At still higher a b s o r b e r detunings z, leaves the no solution region to enter the o n e solution region. T h e n the a b s o r b e r is t o o far d e t u n e d to influence the laser operation. T h e influence of the a b s o r b e r strength B, and h e n c e of the m a x i m u m of the cavity d e t u n i n g ( t o - O ) can be seen by c o m p a r i n g the results of fig. 6a with those of fig. 6b w h e r e a higher cavity detuning results in a higher loss. Fig. 7 shows the "free e n e r g y " function curves c o r r e s p o n d i n g to fig. 6a. For a large absorber detuning z the gain function has no maximum. Since is s y m m e t r i c about x = 0, we see f r o m fig. 7 that the curves for z = 0.5, 1.0 and 1.5 have two m i n i m a (bistable situation) and the curve z = 3.0 has o n e m i n i m u m (monostable situation). For the curve z = 0.5 the system is
G(x,z, e)
more
likely
to be in x = xs since U(0, 0.5, e). In the curve z = 1.5 the laser is unlikely to oscillate since the minim u m of at x = 0 is lower than the o t h e r m i n i m u m at x # 0. T h e curve z = 3.0 has o n e m i n i m u m and represents the situation with o n e stable solution. F o r some values of the a b s o r b e r detuning z between 1.5 and 3.0, U(0, z, e) is the only m i n i m u m and in that case the laser will definitely not oscillate. Fig. 8 shows a situation where the detuning is
U(xs,0.5, e) <
U(x)
U(x) A=2.0
U(x)
"~.0 0.5 G(x)
(no etaton)
a=o5
z= 15
X
E=2
A=2.0 B ~0.5 £ =3.0
x U
)
0.C~
.
0
x?
-0.5
oo[J~ ,
A-2.0 B:I.0 z=1.5
~.Jl
x~z=3O x~
z=3O
G(x)
b
z=0.5
I~°z°~ Fig. 7. T h e " f r e e e n e r g y " function U ( x ) c o r r e s p o n d i n g to t h e gain f u n c t i o n G ( x ) s h o w n in fig. 6a. U ( x ) is s y m m e t r i c a b o u t x = 0. T w o m i n i m a i n d i c a t e bistability w h i l e o n e m i n i m u m at x = xs i n d i c a t e s a s i n g l e s o l u t i o n s i t u a t i o n a n d a s i n g l e m i n i m u m at x = 0 i n d i c a t e s t h a t t h e l a s e r will n o t o s c i l l a t e since its o n l y s t a b l e s o l u t i o n is the trivial x = 0 solution.
Fig. 8. T h e gain function G ( x ) with A = 2.0 a n d z = 1.5 for v a r i o u s v a l u e s of t h e w i d t h p a r a m e t e r e. (a) B = 0.5 a n d (b) B = 1.0. A s ¢ is i n c r e a s e d the s y s t e m p r o g r e s s i v e l y m o v e s f r o m the s i n g l e s o l u t i o n s i t u a t i o n i n t o the b i s t a b l e r e g i o n a n d t h r o u g h i n t o t h e n o s o l u t i o n region. T h e U ( x ) f u n c t i o n c o r r e s p o n d i n g to (a) is f o u n d in fig. 9.
T.S. Dlodlo / Laser with a saturable absorber - dispersive effects
kept constant while the FP width parameter • is varied. The system goes from the o n e solution region through the bistable region to the no solution o n e as ~ is increased starting from • = 0.0. This represents a very wide FP resonance when it does not affect the operation. For very large ~ the cavity term in (9) b e c o m e s a constant and whether the effective gain is positive or negative will depend on the value of the etalon parameter k of fig. 2. Fig. 8a shows the results when B = 0.5 while fig. 8b shows the results when B = 1.0 for comparison. Fig. 9 shows U(x) curves, corresponding to fig. 8a. The curves e = 2.0 and E = 4.0 have one minimum x = x, and x = 0, respectively. For a range of values of the width parameter ~ between 2.0 and 4.0,
361
U(x,z,~)
has two minima one of which is U(0, z, ~) and the other is either lower or higher than U(0, z, ~). When the other minimum is higher, as in the curve E = 3.0, the more probable solution is the trivial one while the solution is x = x, when the other minimum is lower than
U(0, z, ~). T h e behaviour of the gain function in figs. 6 and 8 implies that one can identify regions in the E, z plane where bistability does or does not occur. Fig. 10a and 10b show such regions in the ~, z plane where there is: (I) one stable solution,
~,= 0.5 10
~t
r ~'.4.0 0.6
U(x) 0.2
I
Cl
0.0
z~ 0.2 B=I.0
1C
5 1E
:.2.0.
1
Fig. 9. T h e " f r e e e n e r g y " f u n c t i o n U(x) c o r r e s p o n d i n g t o the g a i n f u n c t i o n G(x) in fig. 8a. U(x) is s y m m e t r i c a b o u t x = 0. T w o m i n i m a i n d i c a t e b i s t a b l e b e h a v i o u r w h i l e o n e m i n i m u m a t x = x, i n d i c a t e s a single s o l u t i o n s i t u a t i o n a n d a s i n g l e m i n i m u m at x = 0 indicates that the laser will n o t o s c i l l a t e s i n c e its o n l y s t a b l e s o l u t i o n is the trivial x = 0 solution.
2
3
4 z~
5
Fig. 10. T h e ~, z p l a n e s h o w i n g the three regions: I - monostability (one solution situation), II-bistability, and I I I - n o
solutions. A = 2.0. (a) B = 0.5 and (b) B = 1.0. In (a) we see that for certain values of • the system can go through all three regions I, II and III as the absorber detuning z is varied.
362
T.S. Dlodlo / Laser with a saturable absorber - dispersive effects
.~, = 2 . 0
Fig. 11. T h e g a i n f u n c t i o n G ( x ) w i t h A = 2.0, B = 0.5, z = 1.5 a n d E = 3.0 w i t h zab as a p a r a m e t e r . G ( x ) is v e r y s e n s i t i v e t o a Zab v a r i a t i o n . S w i t c h i n g t h r e s h o l d s a r e d r a s t i c a l l y r e d u c e d . T h e r e is n o b i s t a b i l i t y . F o r z a b ~ 0.5 t h e r e is n o lasing.
(II) bistable behaviour and (III) no solution possible. Fig. 10a shows the results when B = 0.5, fig. 10b shows the same when B = 1.0 for comparison. From eq. (8) it is seen that, if 2"ab~ 0, the sum of the first three terms on the right-hand side must decrease in value but the cavity term then depends on both A and B as well as on Zab, zca = z and e. Fig. 11 shows the influence of Zab on the gain behaviour.
5. Discussion and conclusions In the previous sections we have discussed the gain function of a laser with an intracavity saturable absorber when a F a b r y - P e r o t etalon is inserted into the laser cavity. We have looked at the situation where absorptive bistability is excluded. We have introduced a loss which varies with the cavity detuning, which in turn varies with the absorber detuning. This has resulted in the appearance of additional significant parameters into the usual gain equation of the conventional laser with a saturable
absorber. The behaviour of both the gain function and the pseudo free-energy function resulting from it, is that of systems characterised by bistability and hysteresis effects. The results show that a purely dispersive bistability is indeed possible if appropriate parameter values are chosen. Figs. 6 and 8 also show that bistability is only possible within a certain parameter range. Beyond a certain absorber detuning z the system is monostable and beyond a certain width parameter e there are no solutions, i.e. G(x, z, ~ ) < 0 for all x and z. We observe that the switching thresholds are lower in the dispersive (z ¢ 0) case than in the absorptive (z = 0) case as can be seen from fig. 6, with fig. 5 as reference. This conclusion has been reached earlier in the theory of dispersive optical bistability of a two level system [3a, 5a]. In figs. 6 and 8 when B = 1.0 the loss is larger than when B = 0.5 since the maximum cavity detuning is larger for larger B. In fig. 10 we see that the existence of the three regions I, II and III introduces the possibility of suppressing oscillations in a certain range of frequencies. For instance, if we take E - - 3 . 0 in fig. 10a we see that we can traverse the three regions in the direction of the z axis. In this situation the laser will definitely not oscillate when 1.5 < z < 2.5. It may oscillate in the range 1.5 > z > 0.25 where it is in the bistable region. This will depend on which of the stable states has the lower "free energy" function U(x); the x = 0 stable solution or the x = xs stable solution. In fig. 10b if z = 0 the stable x = 0 solution coincides with the unstable state x = xu as can be seen by referring to fig. 5 (the B = 1.0 curve). We see that provided ~ ' ~ 0 . 5 the system is monostable for all z > 0. When ~ ~ 0.5 all three regions are represented for various E, z values. We note that a change of pump parameter A and of the etalon parameter k does not change the pattern of behaviour but merely shifts the solutions xu and Xs in figs. 6 and 8 and the regions I, II and I I I in fig. 10. From fig. 11 we conclude that the gain function is very sensitive to the
T.S. Dlodio / Laser with a saturable absorber-dispersive effects
amplifier detuning Z~b and that no bistability can be attained. The switching threshold is drastically lowered as we increase Z~b. For Zab > 0.5 there is no lasing at all. This is due to the decrease of the gain and an increase in the cavity term which represents the variable part of the loss. The replacement of the homogeneously broadened absorber transition by an inhomogeneously broadened one leads to similar conclusions as reached above. This is illustrated by the response of the gain function G(x) given by (12) and shown in fig. 12 where all the parameters are the same as those in fig. 6a. The dotted curves in fig. 12 give the response in the case when the absorber transition is homogeneously broadened (fig. 6a). A closer look at fig. 12 leads to the conclusion that for the inhomogeneously broadened absorber transition, bistability sets in at a slightly higher absorber detuning z value than in the case of homogeneous broadening. A consideration of the effect of fluctuations can be carried out by adding a random noise force to (8a) as in [17] where technical noise was considered. The behaviour of the pseudo-potential U(x) in figs. 7 and 9 suggests that here too quantities such as the relaxation times from
363
metastable states to stable states can be computed when the system is in the bistable situation (region II in fig. 10). The function U(x) of figs. 7 and 9 is comparable to that derived in ordinary optical bistability [18]. The parameters A = 2.0 and B--< 1.0 can be realized in actual experimental situations. An evaluation of the width parameter E can be made by adapting the formulae in [11] to our approximate inverted Lorentzian. Its width F (fig. 2) depends on the thickness d of the FP etalon (fig. 1) and the reflectivity of its end faces. Its value is found by using formulae from [11] and is estimated to be 0 : : F ~ 1 0 ~ ° s - ~ . The value of L0 is estimated to be L0 = 106 s -1. 1/Lo is in fact the lifetime of photons in an empty laser resonator. If we choose a FP etalon 20 mm thick and with reflectivity 0.95 we find that experimentally realizable values are 0 ~ F ~ 1 0 . k is not an important parameter. It is reasonable to assume that it depends on the geometry and reflectivity of the FP etalon. In conclusion we hope we have been able to show that our model brings the laser with an intracavity saturable absorber into the same class of optically bistable systems as a FP etalon containing a nonlinear medium.
Acknowledgements ~'
~z=O.O
I would like to thank Prof. Stig Stenholm for his helpful comments and suggestions during the preparation of this work. •
"\.\
References
/ Fig. 12. The gain function G(x) in the case when the absorber transition of fig. 6a is inhomogeneously broadened. Bistability sets in at a higher absorber detuning z value than in the case of homogeneous broadening. The dotted curves give the behaviour when the absorber transition is homogeneously broadened (fig. 6a).
[1] A. Sz6ke, V. Daneu, J. Goidher and N.A. Kurnit, Appl. Phys. Lett. 15 (1969) 376. [2] R. Bonifacio and L.A. Lugiato, Opt. Comm. 19 (1976) 172; Phys. Rev. 18 (1978) 1129; T. Bischefberger and Y.R. Shen, Phys. Rev. A19 (1979) 1169. [3] H.M. Gibbs, S.L. McCall and T.N.C. Venkatesan, Phys. Rev. Lett. 36 (1976) 1135; P.W. Smith, E.H. Turner and P.J. Maloney, QE-14 (1978) 207.
364
T.S. D l o d l o / Laser with a saturable absorber - dispersive effects
[4] F.A. Hopf and P. Meystre, Opt. Comm. 27 (1978) 147; D.A.B. Miller, S.D. Smith and A. Johnson, Appl. Phys. Lett. 35 (1979) 658. [5] G.P. Agrawal and H.J. Carmichaet, Phys. Rev. A19 (1979) 2074; S.S. Hassan, P.D. Drummond and D.F. Walls, Opt. Comm. 27 (1978) 480. [6] S.L. McCall, Phys. Rev. A9 (1974) 1515. [7] R. Roy and M.S. Zubairy, Phys. Rev. A21 (1980) 274; P. Meystre, Opt. Comm. 26 (1978) 277. [8] (a) R. Salomaa and S. Stenholm, Phys. Rev. A8 (1973) 2695; (b) R. Salomaa and S. Stenholm, Phys. Rev. A8 2711 (1973). [9] (a) L.A. Lugiato, P. Mandel, S.T. Dembinski and A. Kossakowski, Phys. Rev. A18 (1978) 238; (b) L.A. Lugiato, P. Mandel, S.T. Dembinski and A. Kossakowski, Phys. Rev. A t 8 (1978) 1145.
[10] M. Sargent III, M.O. Scully and W.E. Lamb, Jr., Laser Physics (Addison-Wesley, Reading, Mass. 1974). [11] A. Yariv, Introduction to Optical Electronics (Holt, Rinehart and Wilson, New York, 1971). [12] A.P. Kazantsev and G.I. Surdutovich, Progress in Quantum Electronics, Vol. 3, Part 3, J.H. Sanders and S. Stenholm, eds., see also references therein. [13] R. Graham, and H. Haken, Z. Phys. 237 (1970) 31. [14] V. De Giorgio and M.O. Scully, Phys. Rev. A2 (1970) 1170.
[15] J.F. Scott, M. Sargent III and C.D. Cantell, Opt. Comm. 15 (1975) 13. [16] A.R. Bulsara and W.C. Schieve, Opt. Comm. 26 (1978) 384. [17] T.S. Dlodlo, Physica 101C (1980) 125. [18] R. Bonifacio and P. Meystre, Opt. Comm. 29 (1979) 131.
LASER WITH A SATURABLE A B S O R B E R - D I S P E R S I V E EFFECTS T e m b a S. D L O D L O * Department of Physics, University of Helsinki, Finland Received 23 September 1980 Revised 13 April 1981 We study the characteristics of the gain function of a laser with an intracavity saturable absorber. Included within the laser cavity is a Fabry-Perot etalon. A semiclassical description of a single mode homogeneously broadened laser system is used. The laser frequency is tuned to the amplifier transition frequency while the absorber detuning is nonzero. In contrast with the usual models, the loss is a simple function of the cavity detuning which in turn depends on the absorber detuning. The etalon characteristic is approximated by an inverted Lorentzian whose relative width proves to be an important parameter. From an analysis of the gain function we conclude that for a suitable combination of parameters the bistability can be purely dispersive. Comparisons are made with the dispersive case of the ordinary optical bistability. We also show that such an arrangement can behave as a filter, suppressing fields within a certain frequency range.
1. Introduction Since S z 6 k e et al. [1] first p r o p o s e d t h e use of a n o n l i n e a r a b s o r b i n g F a b r y - P e r o t (FP) interf e r o m e t e r for o p t i c a l b i s t a b l e o p e r a t i o n s , a lot of w o r k has b e e n d o n e on t h e t h e o r e t i c a l u n d e r s t a n d i n g of t h e t r a n s m i s s i o n c h a r a c t e r i s t i c s of a F P cavity c o n t a i n i n g a t w o - l e v e l a t o m i c system [2]. Such a s y s t e m was e x p e r i m e n t a l l y shown t o e x h i b i t b i s t a b l e b e h a v i o u r w h e n i r r a d i a t e d with laser light [3]. This i n t e r e s t has b e e n p r o m p t e d by t h e r e a l i z a t i o n t h a t t h e s y s t e m has a p o t e n t i a l for d e v i c e a p p l i c a t i o n s [4], such as m e m o r i e s , o p t i c a l a m p l i f i e r s a n d switches. This o p t i c a l bistability has b e e n f o u n d t o b e d u e to b o t h abs o r p t i v e a n d d i s p e r s i v e effects [5]. T h e first semiclassical analysis of b o t h t y p e s of o p t i c a l b i s t a b i l i t y was given b y M c C a l l [6]. T h e t h e o r y has b e e n d e v e l o p e d within t h e m e a n field app r o x i m a t i o n of ref. [2], t h e validity of which has b e e n t h e o r e t i c a l l y d e m o n s t r a t e d [7]. O n t h e o t h e r h a n d , f o r a laser with an int r a c a v i t y s a t u r a b l e a b s o r b e r , which also e x h i b i t s b i s t a b i l i t y [8, 9], t h e analysis has, in g e n e r a l , n o t * Present address: Physics Department, University of Zimbabwe, P.O. Box MP 167, Salisbury, Zimbabwe.
b e e n c o n c e r n e d with d i s p e r s i v e bistability. A s a result t h e laser f r e q u e n c y is always a s s u m e d to be n e a r l y e q u a l to t h e cavity f r e q u e n c y . It is d u e to this f e a t u r e t h a t t h e cavity losses a r e always a s s u m e d c o n s t a n t in laser t h e o r y . In this p a p e r we i n t e n d to c a l c u l a t e t h e effective gain of t h e laser with an i n t r a c a v i t y s a t u r a b l e a b s o r b e r as a f u n c t i o n of t h e field s t r e n g t h f o r different v a l u e s of t h e d e t u n i n g , for a m o d e l of t h e laser s y s t e m in which a F P e t a l o n is i n c l u d e d within t h e laser cavity. T h i s a r r a n g e m e n t i n t r o d u c e s t h e possibility of losses which a r e v a r i a b l e within t h e oscillation b a n d w i d t h of t h e laser. T h e inclusion of t h e F P e t a l o n within t h e l a s e r cavity i n t r o d u c e s a n u m b e r of n e w p a r a m e t e r s in t h e calculations, such as t h e s e p a r a t i o n a n d t h e reflectivity of t h e e n d faces of t h e F P e t a l o n . F o r simplicity, w e will l o o k at a s y s t e m in which t h e a m p l i f i e r d e t u n i n g is z e r o while t h a t of t h e abs o r b e r is not. In section 2 we p r e s e n t a g e n e r a l f o r m u l a t i o n which i n c l u d e s a t h e o r e t i c a l m o d e l a n d t h e d e r i v a t i o n of t h e gain function for t h e h o m o geneously broadened amplifier and absorber transitions. F o r t h e l a t e r discussions w e also inc l u d e t h e e x p r e s s i o n for t h e gain f u n c t i o n in t h e
0378-4363/81/0000-0000/$02.75 O 1981 N o r t h - H o l l a n d
354
T.S. Dlodlo / Laser with a saturable absorber - dispersive effects
case w h e n the a b s o r b e r is i n h o m o g e n e o u s l y b r o a d e n e d . In section 3 we discuss the stability of t h e s y s t e m with r e s p e c t to v a r i o u s p a r a m e t e r s . W e also include an i n v e s t i g a t i o n into t h e effect of a m p l i f i e r d e t u n i n g . Section 4 gives t h e results a n d section 5 c o n t a i n s a discussion of the results a n d s o m e conclusions.
S
/ 0°0
.
(~--RI
Fig. 2. Losses versus cavity detuning (to- O). The function represented by this figure is: L(to - / ~ ) = L0{l + (k - 1)(to -
2. General formulation
~ ) : / [ ~ 2 + (to x.1)21/. 2.1.
Theoretical model
O u r t h e o r e t i c a l m o d e l consists of a p a i r of t w o - l e v e l m e d i a s p a t i a l l y s e p a r a t e d b e t w e e n two m i r r o r s . A F a b r y - P e r o t e t a l o n is i n s e r t e d within t h e laser cavity as s h o w n in fig. l. T h e t w o m e d i a a r e p u m p e d in such a w a y that o n e is an amplifier while t h e o t h e r o n e is an a b s o r b e r . W e f o r m u l a t e o u r a p p r o a c h in t e r m s of L a m b ' s s e m i c l a s s i c a l t h e o r y of t h e laser [10]. W e a s s u m e that t h e g e n e r a t e d single m o d e laser field is t u n e d to r e s o n a n c e with t h e t r a n s i t i o n f r e q u e n c y of t h e a m p l i f i e r while t h a t of t h e a b s o r b e r is not. T h e two m e d i a c o u l d consist of t h e s a m e o r different m a t e r i a l s . F o r simplicity w e c o n s i d e r an a r r a n g e m e n t in which b o t h t r a n s i t i o n s a r e homogeneously broadened and we neglect the i n h o m o g e n e o u s b r o a d e n i n g d u e to t h e a t o m i c p o s i t i o n s in t h e s t a n d i n g w a v e ; t h e t r e a t m e n t w o u l d , in fact, b e e x a c t in a o n e - d i r e c t i o n a l ring cavity. T h e t r e a t m e n t can b e e x t e n d e d in a s t r a i g h t f o r w a r d w a y to i n c l u d e t h e s e a d d i t i o n a l f e a t u r e s . T h e m e d i a cells a n d t h e r e s o n a t o r all c o n t r i b u t e to t h e losses. W e l u m p t h e s e losses as if t h e y c a m e f r o m a single s o u r c e a n d a s s u m e t h a t t h e r e s u l t i n g loss f a c t o r d e p e n d s on t h e
M,
\
A
ttttt
/
pump(amplifier)
z\
"'
pump(absorber)
Fig. 1. Laser with saturable absorber with MI and M2 ordinary laser mirrors. The FP etalon is used to introduce dispersive effects, d is the thickness of the etaion, A is the amplifier and B the absorber.
cavity d e t u n i n g which in turn, d e p e n d s on t h e a b s o r b e r d e t u n i n g a n d t h e field intensity. T h e d e p e n d e n c e of t h e loss f a c t o r on t h e cavity d e t u n i n g and, h e n c e , on t h e field intensity, which is a b s e n t in t h e usual laser r e s o n a t o r m o d e l is a result of t h e F P e t a l o n i n c l u d e d in t h e laser cavity. F o r such an a r r a n g e m e n t even small d e t u n i n g s a w a y f r o m t h e e i g e n f r e q u e n c y of t h e e m p t y cavity m a y b e significant. L a s e r t h e o r y , in c o n t r a d i s t i n c t i o n , usually a s s u m e s that the cavity Q f a c t o r can be e v a l u a t e d at the e i g e n f r e q u e n c y . O u r c h o i c e for this d e p e n d e n c e is g u i d e d by t h e k n o w l e d g e of t h e t r a n s m i s s i o n c h a r a c t e r i s t i c s of a F P e t a l o n [11] a n d w e c h o o s e t h e loss function in such a w a y that t h e loss is a m i n i m u m at z e r o cavity d e t u n i n g a n d a large c o n s t a n t at large v a l u e s of the d e t u n i n g . A suita b l e a p p r o x i m a t i o n for the F P c h a r a c t e r i s t i c s n e a r t h e t r a n s m i s s i o n r e s o n a n c e is an i n v e r t e d L o r e n t z i a n as shown in fig. 2. T h e F P transmission is a s s u m e d to b e c e n t e r e d at the laser AnnD]ifier
Absorber
'°'I,,o: Fig. 3. Energy level diagrams for the amplifier and absorber media, a, b, c and d indicate the excited levels and [0) is the ground state, yi(A/) is the decay (pump) rate of the level [i) where i = a, b, c, d.
T.S. Dlodlo / Laser with a saturable absorber - dispersive effects
cavity eigenfrequency. The additional parameters introduced by this choice of the loss function are the width of the Lorentzian which is the bandwidth of the FP e t a l o n - i n our a p p r o x i m a t i o n - a n d the etalon parameter k which is a measure of the losses (kLo) at large cavity detuning. The polarization which is the driving force is the sum of the polarizations of the two media
355
and is inserted into Maxwell's equations to derive the field amplitude and the laser frequency. The energy level diagram is shown in fig. 3 where the various decay (pump) rates are shown with arrows down (up), and tOab(tOed)is the transition frequency between the levels la) (Ic)) and Ib) (Id)). The laser frequency is given by tO while /2 is the eigenfrequency of the empty cavity.
2.2. T h e g a i n ]:unction
The semiclassical laser theory [10] gives us the following self-consistent equations for the electric field amplitude and the laser frequency: (1)
= - (goEI2Q + ~Pd2e),
(2)
(tO - O ) = - (&Pr/2EE ,
where t3 = tO, Q is the quality factor and Pi(Pr) the imaginary (real) part of the sum of the average polarizations induced in the two media. The derivation of Pi and Pr will not be repeated here since it is found in most of the literature on lasers. We only give the results relevant to our laser model: DOabAA
Pi = --hrabO~ab '1 + X2 + Z2b
[ DOabZabAA
e r : --hl abOiab~l"~-'~-~-"~b
~ P' " D O ,~A s "~
1-~X~-Z'Z2d) ~ '
(3)
~'flD°cz¢,rA.]
(4)
l + flXz + Z~d] ~ ,
w h e r e AA and AB are constants proportional to the lengths of the amplifier and the absorber medium,
respectively, Zab = (toab- to)/%b, Zcd = (tOed- tO)/Ycd, 7/' = F¢d/F~b, ~ = Ct~/a~b is a saturation parameter,
y,yj
Imijl~
1"° = Yi + Yj'
°t° = h2Yi%'
O o (Ai ~], =
\ ~ / - Yi!
Vii = 21('Yi+ Yj),
m e is the element of the electric dipole moment, X 2 = a a b E 2 and i, j = a, b, c, d. Now we substitute (3) and (4) into (1) and (2) respectively and obtain the general expressions for the rate of change of the field amplitude and the laser frequency, which are
1{ ic=-~
L0A L(to-gl) 1
[
LoB}
1+xe+z2b÷1+/3X2+Z~
AZab
Bz~
( t o - ° ) = ~ L 0 ~ I + ~ ¥ z~b l+~x2+z~) ,
X,
(5) (6)
where L(to - O ) = &/Q, B = rl[3 if the absorber strength, L0 gives the cavity losses at to - / 2 = 0 and
T.S. Dlodlo / Laser with a saturable absorber- dispersive effects
356
A = h&Fab~,bD°~AA/eoLo and "r/= ~'h&~abFabD°acAB/EoLo are p r o p o r t i o n a l to the amplifier and a b s o r b e r p u m p i n g respectively. L(w - f2) is the function shown in fig. 2, i.e. (k - 1)(w - 12)2 /
f
L(oJ - I2) = Lo i1 + ~4F~-~_~-~-- ~-~ j .
(7)
If we substitute (6) into (7) and put the resulting expression for L into (5) we obtain 1
f
A
2=~L0/-l+x
B
2+z~b
l + f l x 2+z2~
1
(k - 1)e2{AZab(l + fiX 2 + zZd) -- BZco(1 + x 2 + Za2b)}2 - (1 + x 2 +
} r-+
x,
(Sa)
where the width p a r a m e t e r ~ = Lo/F measures the influence of the FP width F in (7) and k is the etalon p a r a m e t e r in fig. 2 which measures the losses when the cavity detuning ~o - ~ 2 ~- F. W e n o w consider the special case of a r e s o n a n t amplifier for which Zah = 0 and replace z,,,~ by z so that (8a) b e c o m e s
1 { A 2=~Lo l+x2
B l+flx2+z 2
1
(k --I)E2B2z 2 I {(l+~x~_z--~B2z2}jx.
(88)
T h e first three terms on the right-hand side of (8) represent the situation when the loss L is i n d e p e n d e n t of the d e t u n i n g o)-£2, i.e. k = 1. T h e amplification factor, which we shall call the gain function, is defined by
G(x, z, e) =
{ A
B
1+ x2
(k~l)E2B~z2
1 q- j~X2 -}- -72
1
I
{(1 q- j~X2 -}- Z2)2 -{- E2B2z2}J"
(9)
In the case of an i n h o m o g e n e o u s l y b r o a d e n e d a b s o r b e r transition the macroscopic polarization P(r, t) induced in the m e d i u m gets its contribution f r o m all a t o m s regardless of the.Jr velocities or positions at the time they are excited. T h u s in deriving P(r, t) f r o m the density matrix elements for a single a t o m it is necessary to integrate over the velocity distribution and excitation position. W e neglect collisions and recoil and calculate the polarization of the a b s o r b e r m e d i u m , following L a m b ' s t h e o r y for the gas laser [10], in the rate e q u a t i o n approximation. T h e integration over position yields a constant. T h e expressions similar to (3) and (4) are
Pi
D°abAA
7q'flD°~FiAs } E ,
(10)
DObz'~A n,BDOFrAB} E , P~ = - hF.ba.b t i ¥ ~ -~ab
(11)
~ h~b~ab
t l + x 2+ Z2b
where F~ =
If
W(s){L+ + L_}
do
{1 + lzOxZ(L+ + L_)} o,
W(s) = ~ exp(--//,2s2), vTr
F, =
fo® W(s){L+(z + s) + L_(z - s)} ds {1 + 12OxZ(L++ L_)} '
S = KV/Tca,
tx = ycduK,
L+ = 1/{l + (z + s)2},
T.S. Diodlo / Laser with a saturableabsorber- dispersiveeffects
357
u is the width of the Maxwell velocity distribution function, K is the wave number and V the velocity. In the usual model of the laser with an intracavity saturable absorber where the detuning is neglected one can express Fr and Fi in terms of the plasma dispersion function as is done in [8]. In our case we carry out a numerical integration, and we want to consider all values of the detuning z. For a ring laser, F, and Fi are easily evaluated. On substituting (10) and (11) into (1) and (2) and setting Zab = 0 we obtain an expression for the gain function when the absorber transition is inhomogeneously broadened: G ( x , z, ~) =
A
---4-~x2 - BF, - 1 -
(k - 1)~2B2F~/
(12)
1 + ~2B2F~ J"
An extension to the situation where both transitions are inhomogeneously broadened can easily be made. Since we mainly consider an arrangement in which both amplifier and absorber transitions are homogeneously broadened, our further analysis revolves around the cavity term of (9), i.e. the term which contains the FP cavity parameters ~, k and the detuning z.
3. A n a l y s i s of s t e a d y state s o l u t i o n s
A discussion of (9) with k = 1 and zero absorber detuning can be found in the literature [8, 9] and some of the earliest work on lasers containing intracavity saturable absorbers is reported in [12]. H e r e we will examine the effect of the cavity term with the etalon parameter k ~ 1 and the absorber detuning z # 0 for various values of the width parameter E. We will begin by examining the gain function for constant amplifier and absorber pumping. In steady state ~ = 0 ; hence we obtain the gain equation (G(x, z, E) = 0) whose solutions we would like to analyse. This gain equation has been analysed in detail [9] when z = 0 and k = 1. When the system is bistable the gain equation has two real solutions one of which is the stable solution. For nonzero detuning both the stable and unstable solutions are seen to depend on the absorber detuning parameter. There are three steady state solutions for the electric field: the trivial x = 0 and the two solutions, obtained from the gain equation. Furthermore, by defining a "free energy" like function
U(x) = -X2Lofo G(x, z, ~)x dx,
(13)
we are able to see which of the two solutions is
the stable one. The stable solution is at a minimum of U(x) while the unstable solution is at a maximum of U(x). From (9) and (13) we find that 1 L 0 { x 2+ , / 1 + / 3 x 2+z2\ U(x)=~ 'Tm~ iTz~ ) - A In(1 + x 2)
+(k - 1 ) , ~ z {tan-' ( l + flx2 ~Bz + z2)
- tan-' \ ~--~Bz-z] J "
(14)
The use of U(x) for conventional laser theory was first proposed in [13] where a continuum mode laser was treated and in [14] where the single mode laser was treated. Scott et al. [15] showed that U(x) could be used in the stability analysis of the laser with a saturable absorber. Calculations of the fir.st passage time for a laser with a saturable absorber have been done by Bulsara and Schieve [16] using the pseudopotential U(x) and analysis of G(x) using U(x) has been done in the description of bistability in the dye laser [17]. The values of the parameters z, ~ for which G(0, z, E) < 0 and Gmax(x, z, ~) > 0 yield two solutions. When G(0, z, e) < 0 and Gmax(X,z, e) < 0 then there are no solutions. In all cases when
358
T.S. D l o d l o I L a s e r with a saturable absorber - dispersive effects
G(0, z , ~ ) > 0 there is only one solution: the stable solution. The border between the one stable solution situation and the bistable situation is given by the simulataneous conditions G(0, z, ~) = 0 and Gm~(X, z, e) > 0 and that between the bistable and the no solution situation by the simultaneous conditions Gmax(X, z, e ) = 0 and G(0, z, E ) < 0. In fig. 4, curve 1 satisfies the f o r m e r conditions while curve 2 satisfies the latter. The simultaneous conditions satisfied by curve 2 imply that the stable and unstable solutions should coincide or equivalently that gain and loss are equal at a point x = x0 where their derivatives with respect to x are also equal, i.e. OG(x, z, e)[I = 0, OX x=x,
G(Xo, z, e) = O,
(15)
Xo = Xs = Xu •
From the above conditions it is possible to derive expressions or inequalities which define a region in the e, z plane, where bistable behaviour is possible. W e find the relationship between e and z by eliminating x0 in (15). This procedure is a mathematically c u m b e r s o m e exercise. W e will therefore find the relationship between e and z numerically below.
••.
t - z . _ . --...
//,.---.,,--%, ~
o.~" ~ _ / ./
-o..
Y/"
-"-,, o ! i " - . ~ ~!o "~'-'~2
/
/ Fig. 4. The gain function G ( x ) as a function of the field strength with the absorber detuning z and the width parameter e such that curve 1 gives the border between I - t h e monostable (single solution) s i t u a t i o n - and I I - the bistable s i t u a t i o n - and curve 2 represents the border between II and I I I - t h e no solution situation. I, II and III are given by the dotted curves.
An analysis of the loss function in (7) is necessary for a better understanding of the behaviour of G ( x ) . Let us start with the situation when z # 0 and to - / 2 ~> F. In this case the cavity loss is kLo, which is taken to exceed the gain so that oscillations are impossible. In this connection the p a r a m e t e r B in (6) is important because it determines the magnitude of the m a x i m u m possible value of the cavity detuning (to - 1 2 ) . If we look at the case with x > 0 then (6) shows that the field intensity pulls the laser frequency towards the cavity eigenfrequency (to ~ / 2 ) and the loss decreases towards L0. It is then possible that the gain exceeds the loss and oscillations are stable. The change of the refractive index, due to the presence of laser oscillations, makes the laser wavelength fit the geometry of the etalon and reduces the loss further below the gain. This can happen also when the gain is a monotonic function of x, i.e.
A-B>
A - -
1 + x2
B 1 +/~x 2
and no absorptive bistability is possible. As we are interested in analysing the dispersive bistability we will choose our p a r a m e t e r s such that there is no bistability when z -- 0. Since laser oscillations occur near zero cavity detuning, we expect the gain function to behave in such a way that at large z there is no bistability because the cavity detuning ( t o - / 2 ) ~ 0 as z ~ 0 and z~ as can be seen in (6) if we put Zab = 0 and Zcd=Z. When z # 0 , e # 0 and k # l the cavity term L ( t o - / 2 ) appears in the expression for the gain function and when x is varied we find a typical bistable situation emerging which is due to purely dispersive effects. At x = 0 the absorber detuning makes the laser want to oscillate well off the cavity resonance; the loss is large, but an increase in intensity makes the detuning approach zero for large intensities and a stable steady state emerges. W e now look at the effect of amplifier detuning on the gain function. In eq. (8a) we see that
T.S. Dlodlo / Laser with a saturable absorber - dispersive effects
there are two terms which involve zab,- the amplifier detuning. For all values of Zab the gain t e r m is reduced but can still be positive for very small Z,b. T h e cavity t e r m must be sensitive to zab because, depending on the values of A, B and z, it could approach zero or be very large. In the latter case there would be no oscillations while in the former, oscillations would only take place for very small Z~b and no bistability would be observed.
4. Results
W e present here the numerical results of the effective gain (the gain function) as a function of the dimensionless electric field strength x, with z, and B as parameters. In all the numerical calculations the p a r a m e t e r proportional to the amplifier pumping rate has been chosen to be A = 2.0 and the etalon p a r a m e t e r k = 5. For this choice, fig. 5 shows us for which values of the absorber strength B, there is no absorptive bistability. W e have chosen B = 0.5 and B = 1.0 for comparison. B = 1.0 is at the border between a
a :o.5 as.
A=20 z--()43 B:"I13
one stable solution situation and a bistable situation when only the absorptive bistability (z = 0) case is considered for the chosen amplifier pumping p a r a m e t e r A = 2.0. For every A there is an approximate B giving the b o r d e r between the one stable solution situation and a bistable situation. In [8, 9] regions of bistability are identified in the A, B plane. The curves in fig. 6 give the results for the width p a r a m e t e r ~ = 3.0. In fig. 6a the absorber strength B = 0.5 while B = 1.0 in fig. 6b. It is shown in this figure that when the absorber detuning is increased from zero the system moves m o r e into the bistable region where there
4:20
0.5
~= 3.0
~/ 1
B
.o
~'~
z =1.5
a
Q5
/ G(X)f
359
/
~
A = 2.0
\
B=I.0
"
£=3.0
•
0.0
I
"
o~
/--2.0 Fig. 5. The gain function G(x) as a function of the field strength x at absorber strength parameters B = ,//3 = 0.5, 1.0 and 1.5. Absorptive bistability does not take place when B < 1 for the chosen value of the amplifier pump parameter A = 2.0. In [8a, 9a] regions of bistability are identified in the A, B planes.
,0
x
b
Fig. 6. The gain function O ( x ) with A = 2.0 and ~ = 3.0 for various values of the absorber detuning z. (a) B = 0.5 and 0a) B = 1.0. As z is increased from z = 0.0 the system moves, via the bistable region, into the no solution region and back into the single solution region at still higher z values. The U(x) function corresponding to (a) is found in fig. 7.
364~
T.S. Dlodlo / Laser with a saturable absorber - dispersive effects
are two solutions. A s the detuning is further increased a situation is first r e a c h e d w h e r e there are no solutions at all. At still higher a b s o r b e r detunings z, leaves the no solution region to enter the o n e solution region. T h e n the a b s o r b e r is t o o far d e t u n e d to influence the laser operation. T h e influence of the a b s o r b e r strength B, and h e n c e of the m a x i m u m of the cavity d e t u n i n g ( t o - O ) can be seen by c o m p a r i n g the results of fig. 6a with those of fig. 6b w h e r e a higher cavity detuning results in a higher loss. Fig. 7 shows the "free e n e r g y " function curves c o r r e s p o n d i n g to fig. 6a. For a large absorber detuning z the gain function has no maximum. Since is s y m m e t r i c about x = 0, we see f r o m fig. 7 that the curves for z = 0.5, 1.0 and 1.5 have two m i n i m a (bistable situation) and the curve z = 3.0 has o n e m i n i m u m (monostable situation). For the curve z = 0.5 the system is
G(x,z, e)
more
likely
to be in x = xs since U(0, 0.5, e). In the curve z = 1.5 the laser is unlikely to oscillate since the minim u m of at x = 0 is lower than the o t h e r m i n i m u m at x # 0. T h e curve z = 3.0 has o n e m i n i m u m and represents the situation with o n e stable solution. F o r some values of the a b s o r b e r detuning z between 1.5 and 3.0, U(0, z, e) is the only m i n i m u m and in that case the laser will definitely not oscillate. Fig. 8 shows a situation where the detuning is
U(xs,0.5, e) <
U(x)
U(x) A=2.0
U(x)
"~.0 0.5 G(x)
(no etaton)
a=o5
z= 15
X
E=2
A=2.0 B ~0.5 £ =3.0
x U
)
0.C~
.
0
x?
-0.5
oo[J~ ,
A-2.0 B:I.0 z=1.5
~.Jl
x~z=3O x~
z=3O
G(x)
b
z=0.5
I~°z°~ Fig. 7. T h e " f r e e e n e r g y " function U ( x ) c o r r e s p o n d i n g to t h e gain f u n c t i o n G ( x ) s h o w n in fig. 6a. U ( x ) is s y m m e t r i c a b o u t x = 0. T w o m i n i m a i n d i c a t e bistability w h i l e o n e m i n i m u m at x = xs i n d i c a t e s a s i n g l e s o l u t i o n s i t u a t i o n a n d a s i n g l e m i n i m u m at x = 0 i n d i c a t e s t h a t t h e l a s e r will n o t o s c i l l a t e since its o n l y s t a b l e s o l u t i o n is the trivial x = 0 solution.
Fig. 8. T h e gain function G ( x ) with A = 2.0 a n d z = 1.5 for v a r i o u s v a l u e s of t h e w i d t h p a r a m e t e r e. (a) B = 0.5 a n d (b) B = 1.0. A s ¢ is i n c r e a s e d the s y s t e m p r o g r e s s i v e l y m o v e s f r o m the s i n g l e s o l u t i o n s i t u a t i o n i n t o the b i s t a b l e r e g i o n a n d t h r o u g h i n t o t h e n o s o l u t i o n region. T h e U ( x ) f u n c t i o n c o r r e s p o n d i n g to (a) is f o u n d in fig. 9.
T.S. Dlodlo / Laser with a saturable absorber - dispersive effects
kept constant while the FP width parameter • is varied. The system goes from the o n e solution region through the bistable region to the no solution o n e as ~ is increased starting from • = 0.0. This represents a very wide FP resonance when it does not affect the operation. For very large ~ the cavity term in (9) b e c o m e s a constant and whether the effective gain is positive or negative will depend on the value of the etalon parameter k of fig. 2. Fig. 8a shows the results when B = 0.5 while fig. 8b shows the results when B = 1.0 for comparison. Fig. 9 shows U(x) curves, corresponding to fig. 8a. The curves e = 2.0 and E = 4.0 have one minimum x = x, and x = 0, respectively. For a range of values of the width parameter ~ between 2.0 and 4.0,
361
U(x,z,~)
has two minima one of which is U(0, z, ~) and the other is either lower or higher than U(0, z, ~). When the other minimum is higher, as in the curve E = 3.0, the more probable solution is the trivial one while the solution is x = x, when the other minimum is lower than
U(0, z, ~). T h e behaviour of the gain function in figs. 6 and 8 implies that one can identify regions in the E, z plane where bistability does or does not occur. Fig. 10a and 10b show such regions in the ~, z plane where there is: (I) one stable solution,
~,= 0.5 10
~t
r ~'.4.0 0.6
U(x) 0.2
I
Cl
0.0
z~ 0.2 B=I.0
1C
5 1E
:.2.0.
1
Fig. 9. T h e " f r e e e n e r g y " f u n c t i o n U(x) c o r r e s p o n d i n g t o the g a i n f u n c t i o n G(x) in fig. 8a. U(x) is s y m m e t r i c a b o u t x = 0. T w o m i n i m a i n d i c a t e b i s t a b l e b e h a v i o u r w h i l e o n e m i n i m u m a t x = x, i n d i c a t e s a single s o l u t i o n s i t u a t i o n a n d a s i n g l e m i n i m u m at x = 0 indicates that the laser will n o t o s c i l l a t e s i n c e its o n l y s t a b l e s o l u t i o n is the trivial x = 0 solution.
2
3
4 z~
5
Fig. 10. T h e ~, z p l a n e s h o w i n g the three regions: I - monostability (one solution situation), II-bistability, and I I I - n o
solutions. A = 2.0. (a) B = 0.5 and (b) B = 1.0. In (a) we see that for certain values of • the system can go through all three regions I, II and III as the absorber detuning z is varied.
362
T.S. Dlodlo / Laser with a saturable absorber - dispersive effects
.~, = 2 . 0
Fig. 11. T h e g a i n f u n c t i o n G ( x ) w i t h A = 2.0, B = 0.5, z = 1.5 a n d E = 3.0 w i t h zab as a p a r a m e t e r . G ( x ) is v e r y s e n s i t i v e t o a Zab v a r i a t i o n . S w i t c h i n g t h r e s h o l d s a r e d r a s t i c a l l y r e d u c e d . T h e r e is n o b i s t a b i l i t y . F o r z a b ~ 0.5 t h e r e is n o lasing.
(II) bistable behaviour and (III) no solution possible. Fig. 10a shows the results when B = 0.5, fig. 10b shows the same when B = 1.0 for comparison. From eq. (8) it is seen that, if 2"ab~ 0, the sum of the first three terms on the right-hand side must decrease in value but the cavity term then depends on both A and B as well as on Zab, zca = z and e. Fig. 11 shows the influence of Zab on the gain behaviour.
5. Discussion and conclusions In the previous sections we have discussed the gain function of a laser with an intracavity saturable absorber when a F a b r y - P e r o t etalon is inserted into the laser cavity. We have looked at the situation where absorptive bistability is excluded. We have introduced a loss which varies with the cavity detuning, which in turn varies with the absorber detuning. This has resulted in the appearance of additional significant parameters into the usual gain equation of the conventional laser with a saturable
absorber. The behaviour of both the gain function and the pseudo free-energy function resulting from it, is that of systems characterised by bistability and hysteresis effects. The results show that a purely dispersive bistability is indeed possible if appropriate parameter values are chosen. Figs. 6 and 8 also show that bistability is only possible within a certain parameter range. Beyond a certain absorber detuning z the system is monostable and beyond a certain width parameter e there are no solutions, i.e. G(x, z, ~ ) < 0 for all x and z. We observe that the switching thresholds are lower in the dispersive (z ¢ 0) case than in the absorptive (z = 0) case as can be seen from fig. 6, with fig. 5 as reference. This conclusion has been reached earlier in the theory of dispersive optical bistability of a two level system [3a, 5a]. In figs. 6 and 8 when B = 1.0 the loss is larger than when B = 0.5 since the maximum cavity detuning is larger for larger B. In fig. 10 we see that the existence of the three regions I, II and III introduces the possibility of suppressing oscillations in a certain range of frequencies. For instance, if we take E - - 3 . 0 in fig. 10a we see that we can traverse the three regions in the direction of the z axis. In this situation the laser will definitely not oscillate when 1.5 < z < 2.5. It may oscillate in the range 1.5 > z > 0.25 where it is in the bistable region. This will depend on which of the stable states has the lower "free energy" function U(x); the x = 0 stable solution or the x = xs stable solution. In fig. 10b if z = 0 the stable x = 0 solution coincides with the unstable state x = xu as can be seen by referring to fig. 5 (the B = 1.0 curve). We see that provided ~ ' ~ 0 . 5 the system is monostable for all z > 0. When ~ ~ 0.5 all three regions are represented for various E, z values. We note that a change of pump parameter A and of the etalon parameter k does not change the pattern of behaviour but merely shifts the solutions xu and Xs in figs. 6 and 8 and the regions I, II and I I I in fig. 10. From fig. 11 we conclude that the gain function is very sensitive to the
T.S. Dlodio / Laser with a saturable absorber-dispersive effects
amplifier detuning Z~b and that no bistability can be attained. The switching threshold is drastically lowered as we increase Z~b. For Zab > 0.5 there is no lasing at all. This is due to the decrease of the gain and an increase in the cavity term which represents the variable part of the loss. The replacement of the homogeneously broadened absorber transition by an inhomogeneously broadened one leads to similar conclusions as reached above. This is illustrated by the response of the gain function G(x) given by (12) and shown in fig. 12 where all the parameters are the same as those in fig. 6a. The dotted curves in fig. 12 give the response in the case when the absorber transition is homogeneously broadened (fig. 6a). A closer look at fig. 12 leads to the conclusion that for the inhomogeneously broadened absorber transition, bistability sets in at a slightly higher absorber detuning z value than in the case of homogeneous broadening. A consideration of the effect of fluctuations can be carried out by adding a random noise force to (8a) as in [17] where technical noise was considered. The behaviour of the pseudo-potential U(x) in figs. 7 and 9 suggests that here too quantities such as the relaxation times from
363
metastable states to stable states can be computed when the system is in the bistable situation (region II in fig. 10). The function U(x) of figs. 7 and 9 is comparable to that derived in ordinary optical bistability [18]. The parameters A = 2.0 and B--< 1.0 can be realized in actual experimental situations. An evaluation of the width parameter E can be made by adapting the formulae in [11] to our approximate inverted Lorentzian. Its width F (fig. 2) depends on the thickness d of the FP etalon (fig. 1) and the reflectivity of its end faces. Its value is found by using formulae from [11] and is estimated to be 0 : : F ~ 1 0 ~ ° s - ~ . The value of L0 is estimated to be L0 = 106 s -1. 1/Lo is in fact the lifetime of photons in an empty laser resonator. If we choose a FP etalon 20 mm thick and with reflectivity 0.95 we find that experimentally realizable values are 0 ~ F ~ 1 0 . k is not an important parameter. It is reasonable to assume that it depends on the geometry and reflectivity of the FP etalon. In conclusion we hope we have been able to show that our model brings the laser with an intracavity saturable absorber into the same class of optically bistable systems as a FP etalon containing a nonlinear medium.
Acknowledgements ~'
~z=O.O
I would like to thank Prof. Stig Stenholm for his helpful comments and suggestions during the preparation of this work. •
"\.\
References
/ Fig. 12. The gain function G(x) in the case when the absorber transition of fig. 6a is inhomogeneously broadened. Bistability sets in at a higher absorber detuning z value than in the case of homogeneous broadening. The dotted curves give the behaviour when the absorber transition is homogeneously broadened (fig. 6a).
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T.S. D l o d l o / Laser with a saturable absorber - dispersive effects
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