15June 1997
OPTICS
COMMUNICATIONS
ELSlSVIER
Optics Communications I39 (1997) 125- 13I
Full length article
Localized synchronization of two coupled solid state lasers
Rachel Kuske a31,Thomas Emeux b
’ Department of Mathematics, Stanford University, Stanford, CA 94305-2125, USA
b UniuersitL:Libre de Bruxelles, Optique Nonline’aire ThPorique, Campus Plaine, C.P. 231, 1050 Bruxelles, Belgium
Received 18 November 1996; accepted 22 January 1997
Abstract
Two coupled lasers exhibiting oscillatory intensities are known to synchronize in phase or out-of-phase and with equal
intensities. But a different form of synchronization - called localization - has been discussed recently in the literature of
coupled oscillators. Localization means that the two lasers may exhibit different intensities. We show that this phenomenon
is possible in a system of two coupled solid state lasers differing only by their detunings. We determine the bifurcation
diagram of the localized states and obtain analytical conditions for stable localization.
1. Introduction
Arrays of coupled lasers have been proposed as devices
for applications that require high optical power from a
laser source such as high speed optical recording, high
speed printing, and space communications
[I]. Maximum
power is achieved provided that all lasers are perfectly
synchronized in phase but experiments and linear stability
theories have shown that laser arrays have a natural tendency for antiphasing. This is particularly dramatic for
arrays of coupled semiconductor
lasers because of the
strong phase/amplitude
coupling [2]. Recent theoretical
activities have proposed external synchronization
mechanisms such as injecting an external electrical field into all
lasers [3] or by coupling lasers in parallel [4]. These
methods assume very weak coupling between lasers so that
the intensities of each laser are nearly steady. In Ref. [5],
the response of two laterally coupled Nd:YAG lasers
differing only by their detunings [6] is analyzed in detail. It
is demonstrated theoretically and experimentally
that an
amplitude-phase instability may appear for weak coupling.
This is possible because the natural damping of the laser
oscillations is relatively slow compared to the typical time
scale of the oscillations. This particular property on the
’ Present address: Tufts University, Department of Mathematics, Bromfield Pearson Bldg. Medford. MA 02 155, USA.
solid state laser allows different forms of amplitude/phase
synchronization
between lasers. This contrasts to weakly
coupled limit-cycle oscillators modeling chemical or biological systems which only exhibit phase instabilities [ 121.
The long time synchronization of two coupled identical
oscillators may occur in phase or out-of-phase and with
equal amplitudes. But a different form of synchronization
- called localization - has been investigated recently [8,9].
Localization means that one or more oscillators in the
population exhibit large amplitude oscillations while the
amplitude of the oscillations of the remaining oscillators is
small. For a system of two coupled oscillators, a localized
state thus means that the first and the second oscillators
exhibit large and small amplitude oscillations, respectively.
Stable localized states may coexist with equal intensities
states in the bifurcation diagram. But they may dominate if
a parameter controlling the amplitude of the oscillations is
different for each laser [lo]. The main purpose of this
paper is to find general conditions for stable localization in
a system of coupled lasers. Although, localization is a
phenomena that was first described for a large population
of oscillators, we concentrate on a system of two lasers for
which detailed quantitative observations are possible. In
Refs. [5] and [6], a simple model of two coupled solid state
lasers has been investigated and solutions characterized by
identical intensities have been analyzed. In this paper, we
are specifically interested in finding solutions exhibiting
different intensities. To this end, we analyze the two
W30-4018/97/$17.00 Copyright 0 1997 Elsevier Science B.V. All rights reserved.
PI1 SOO30-4018(97)00062-X
126
R. Kuske. T. Erneux/Oprics
Communications
coupled solid state laser equations using asymptotic techniques and derive bifurcation equations for localized solutions. These equations are then investigated and analytical
results are compared with the numerical bifurcation solutions. We find that branches of localized states are limited
in the bifurcation diagram but that this limitation gradually
disappears as the difference between individual pumps is
changed. Because our bifurcation analysis applies to a
large class of lasers exhibiting similar relaxation properties
(solid state, CO, and semiconductor lasers), our results are
relevant for other coupled lasers systems.
The plan of the paper is as follows. We introduce the
model in Section 2 and describe the bifurcation equations
for the localized states. In Section 3, we investigate these
equations in detail. All mathematical details are concentrated in the Appendix but are not needed for the comprehension of the main results.
2. Two coupled solid state lasers
The dimensionless equations for a system of two spatially coupled, single transverse and longitudinal mode
solid state lasers are given by [6]
Ei=(Fj-1
+~co~)E~-KE,,
F;=r[Aj-(l+IEi12)~],
(1)
(2)
where j,k = 1,2 and k f j. The variables Ej(t) and q(t)
are the complex electrical field and gain for the jth laser,
respectively. y = r,/r5 is defined as the ratio of the cavity
round trip time 7, and the fluorescence time of the laser
medium rr. K > 0 measures the coupling between the
lasers due entirely through spatial overlap of the fields [6].
Aj is the dimensionless pump which is assumed larger than
its threshold value (i.e., A, > 1). wi denotes the dimensionless detuning of the jth laser (wj = Oj/r,>. Note that
y is typically an 0(10d3) small quantity for solid state
lasers. Its low value explains why amplitude/phase
instabilities are observed as soon as K = O(y) [5,7]. In Ref. [5],
the bifurcation diagram of the time-periodic
states was
studied assuming equal pumps (A, = AZ).
In this paper, we concentrate on the phenomenon of
localization for arbitrary values of A, and A,. However, a
stable localized state already exists for A, = A, as shown
numerically in Fig. 1. The figure represents the intensities
li = 1Ej12 as functions of the scaled time s = nt, where
0= (2(A, - 1)~) “2 is defined as the laser relaxation
oscillations frequency. As we shall demonstrate, the amplitude of the oscillations at laser 1 and laser 2 are typically
O(1) and O(h),
respectively.
The laser problem depends on several parameters and
we wish to understand their specific effects on the localized states. It will be convenient for our asymptotic analysis to rewrite Eqs. (1) and (2) in dimensionless form. In
terms of s = Ot, Eq. (1) and Eq. (2) become Eqs. (28)-(32)
I39 (1997) 125-131
411
3
2
1
12
0
0
S
Fig. 1. Localized oscillatory intensities. The solution has been
obtained by integrating Eqs. (28)-(32)
and then by evaluating
I, = ISj( 1+ yj) = (A j - 1Xl + ?;I. Note that the oscillations are
out of phase. The values of the parameters are A, = A, = 2,
p = 0.03,~ = 0.01.6 = 0.9. The initial conditions were x,(O) =
- 1.6, y,(O)=O.23,
s,(O)= -0.06, y,(O)= -0.11, $(0)=4.586
and the solution is represented after an interval of time equal to
400.
in the Appendix. These equations
less parameters given by
e = y0-‘,
exhibit four dimension-
S= AR-‘,
The parameter E = O( y I/‘) -=z 1 measures the natural weak
damping of the laser oscillations. The parameter 6 is the
difference between the two detunings normalized by the
laser relaxation oscillations frequency. The parameter p is
a scaled coupling coefficient which we assume O(y ‘1’)
small. The parameter q is the ratio of steady state intensities and equals 1 if A I = A,. Note that we may determine
the steady state solutions of Eqs. (1) and (2) analytically
and we may reduce the number of equations if we assume
equal intensity solutions and q = 1 [S]. But we are interested in determining if stable periodic solutions characterized by different intensities are possible. To this end, our
analysis of the laser equations will be based on the asymptotic limit p = O(E) + 0, keeping 6 and q fixed. Specifically, we seek a solution characterized by O(1) intensity
oscillations for laser 1 and small intensity oscillations for
laser 2. All mathematical
details are described in the
Appendix. We investigate two possibilities corresponding
to the case S f & and the case S = &.
If 6 + \/(;;, we find the solution
(u)
62 &I,
4 = 42p
+
=Z,,[l
+ Y(s+
PY21W)lr
O,J)],
(4)
(9
where Z, = Aj - 1 (j = 1 or 2) is the steady state intensity
R. Kuske, T. Emeu.~/Optics
Communicarivrls
139 11997) 125-131
127
and S = 6s. In (4). the oscillations are O(1) large and the
2n-periodic function Y(S.6) satisfies Eq. (38). In Eq. (5).
the oscillations are O( p 1 small and the function _v,,(S,S)
is passively related to Y(S,S) (see Eq. (41)). The solution
(4) depends on the phase 0, which satisfies the bifurcation
equation (43) or equivalently
As 1 - S increases, the amplitude of the oscillations increases and can be obtained numerically from Eq. (38).
The phase 0, is determined from Eq. (6) until a limit point
S = S,, < I is reached. The limit point corresponds to
0, = rr and is the solution of the following equation
(assuming G > 0)
\1;7cos(H,)G(S)
- fiG(
+EA,K(G)
=O.
(6)
In Eq. (6), G( S ) and K(S) > 0 denote functions of 6
which are defined by (44). The coefficient E is related to
the damping coefficient E as
E= E/T’.
(7)
S,,)
+ RA, R( S,,)
(14)
= 0.
Thus, the oscillations of laser 1 are possible for the
interval S,, < 6 < 1, or equivalently, in terms of the deviation D defined by (1 I), the interval D,, < D < DC, where
D,!and DC are defined by
D,, z @-Z/3y-‘/“(
S,, - fi)
Thus. the bifurcation problem for solution (4) and (5) is
reduced to the solutions of the phase equation (6). It is
laser 1 that controls the oscillations of the coupled lasers
system. The oscillations of laser 2 are simply entrained by
the oscillations of laser I.
However. the solution (4) and (5) is mathematically
valid only if 6 f 6, = \l;7/k (k = 1,2,...). The points 6,
are ~oirrts of reronunce.
We investigate the principal
resonance 6 = S, = fi.
The intensity of laser 1 is still
given by (4) (with 6 = 6, ) but the intensity of laser 2 takes
a different form. We now find
There are no restrictions on the oscillations of laser 2 since
they are passively related to the oscillations of laser I if
6 # 6,. If S = S,, the oscillations of laser 2 are described
by Eq. (10). From the real and imaginary parts, we obtain
two conditions given by
(h)
sin( &) = 0.
S= A:/,
/,=/,?[I
=[,,[I
+ Y(S+
e,,s,>].
+2/3”7R,sin(S+H,)].
(8)
(9)
By contrast to (5). the oscillations of laser 2 are O( p ‘/j)
and nearly harmonic in time. The amplitude R, and the
phase 0z satisfy the bifurcation equation (55) for cr2 =
R,exp(0?) or equivalently
2Da?+
+;a,-
iF(t),)
= 0.
(IO)
and D,,=/L~‘/~~-‘/~(~
2DR,
- &).
+ f R; - +,)cos(&)
(‘5)
= 0,
(16)
(17)
We analyze these conditions in terms of D assuming
F( 0, ) > 0. From (16) and (17), we find two branches of
solutions given by
(i)
H?=Oand
D(RZ)=
-iRi+-
F(O,)
2qR,
(ii)
H,=aand
D(R?)=
-iRi--
’
F(B,)
2qRz
In (IO). the function F( 0, ) is defined by (56). The coefficient D = D( 6) is proportional to the deviation S - 6,
and is given by
D s p-V4-‘/?(S
- S,),
(II)
Thus, we determine H, from Eq. (6) and then obtain R2
and tl? from the real and imaginary parts of Eq. (IO). The
bifurcation problem for S near 6, is richer than the
general case 6 + S, because it depends on both laser 1 and
laser 2.
3. Bifurcation diagram of the localized states
We wish to find the amplitude of the oscillations for
each laser as a function of the control parameter 6. From
(46). we note that the oscillations of laser I exist only if
Sr
1.
(12)
As I - 6 approaches zero, the oscillations are of the form
X= -2R,sin(S)
and Y= 2R,cos(S) where
R, = [6(1 - 6)]““.
(‘3)
Assuming F(B,) > 0, the branch of solutions
(19) exhibits a limit point at D = DL2 where
1
DLy2
3F(0,)
___
2q
i
(18)
(19)
given
by
7’3
(20)
1
These branches of solutions are represented in Fig. 2 for
the case q = I (same pumps). Specifically, Fig. 2 shows
the bifurcation diagram of the localized states in terms of
the amplitudes R, and R, of the oscillations of laser 1 and
laser 2, respectively (these amplitudes are defined as R, 5
max(x,)/2,
R, = max(.x2)/(2P’/3)
where x, and xa are
obtained nume&lly
from Eqs. (28)-(32). We have found
that (13) is a good approximation of R, while the implicit
expression (19) with F(B,) = l/2 is a good approximations of R,. Both amplitudes are shown in terms of the
deviation D defined by (11). The localized states obtained
numerically from Eqs. (28)-(32) are shown by dots in the
bifurcation diagram. Note that these states are competing
with solutions exhibiting equal intensities. They are not
shown in the bifurcation diagram of Fig. 2. Furthermore,
we did not look for localized states that belong to the
R. Kuske. T. Emeu.x/Optics
128
Communications
2
second branch of solutions (specifically, the branch of
solutions given by (18) and corresponding to the upper line
in Fig. 2b).
In Fig. 3, we examine
the effect of q by slightly
changing the values of the pump parameters. Its main
effect is noted by comparing Fig. 3a and Fig. 2a. We
observe a shift of the point D, defined in (15): DC = 0 if
q = 1 but is O(1) as soon as the deviation 1q - l( is
O( p ‘13). The localized states obtained numerically from
Fqs. (28)-(32) are shown by dots. We note that the branch
of numerical solutions shown in Fig. 3a does not terminate
2
R2
I39 (1997) 125-131
RI
(4
Fig. 3. Bifurcation diagram of the localized states for two lasers
differing by their pumps. Same values of the fixed parameters as
in Fig. 2 except A, = 2.15 and A, = 2. The dots corresponds to
stable localized states obtained by integrating the laser equations
(28)-(32) from 6 = 0.96 to 6 = 1.1. The critical points D,,and
0, are shown in the figure and have been determined from (15).
We find D,, = -0.02 and 0,. = 0.76.
point D = DC: a different analysis assuming both R,
and /3 ‘j3R2 small is needed to explain the D > DC part
of the bifurcation diagram.
at the
Fig. 2. Bifurcation diagram of the localized states for two identical
lasers. We represent the bifurcation diagram of the localized states
in terms of R, = max(x,)/2
and R, = max(x2)/(2/3’/3).
The
values of the fixed parameters are A, = A, = 2, /3 = 0.03, E =
0.01. The dots correspond to stable localized states obtained by
integrating numerically the laser equations (28)-(32) from 6 = 0.9
to 6 = 0.96. The lines are approximations described in Section 2.
The approximation for R,(D) is given by (13) with 6 = fi (1 +
p ‘/ ‘D) and the approximation for R,(D) is given by the implicit
expressions (18) or (19). If 6 < 0.9 or if 6 > 0.96, we have found
numerically that the laser system approaches a state characterized
by equal intensities. The critical points DL,,DC and DL2 are
shown in the figure and have been determined from (15) and (20)
with F(B,)=1/2.
We find D,,=-0.95,D,=O
and DL,=
- 0.42.
4. Discussion
We have determined localized states in a system of two
coupled solid state lasers. Specifically, we derived bifurcation equations for time-periodic solutions exhibiting O(1)
amplitude oscillations for laser 1 and small amplitude
oscillations for laser 2.
For laser 1, the amplitude of the oscillations is large
and is determined by a simple periodicity condition. It
does not depend on the small parameters E and p proportional to the damping and coupling coefficients, respectively. However, these parameters influence the phase of
the oscillations. For laser 2, the small amplitude oscillations depend passively on the oscillations of laser 1 unless
129
R. Kuske, T. Erneux/Optics Communications 139 (1997) 125-131
the detuning difference is close to resonance points. Then,
the amplitude of-the oscillations of laser 2 strongly depends on the coupling coefficient as we demonstrated for
the principal resonance.
Our bifurcation analysis showed that stable localized
states may compete with states exhibiting equal intensities
(if q = 1) or nearly equal intensities (if q = 1). If the
lasers have identical pumps (q = 1), the branch of stable
localized states is limited in the bifurcation diagram by the
condition
Eqs. (23)-(25), we remove the y term in the resulting
equations by introducing the deviations xj and Yi given by
I, =
R
and F, = 1 + 2~j,
Isj( 1 + y,)
where lsi E Aj - 1 denotes the steady state intensity.
laser equations (23)-(25) then become
x; = -y,
- ex,(l+Z,,(l
+Y,,),
Y; = (1 + Y,)X, - P&b
D,, CD CD,:
The
(28)
+ Y,)(l
+ Yz) COS((cr)~
(21)
(see Fig. 2). This explains why the localized states are
difficult to find numerically if both lasers are identical.
However, if the pumps are slightly changed (q f l), the
domain of stable localized states can be larger because
(29)
x;=
-qyr-
Ex2(1+1s,(l
+Y*))7
+Yz)xz-PLli’(l
6
Y;=(l
(22)
D>DL,
(27)
+y,)(1
(30)
+Y*)
cos(+),
(31)
then becomes the only restriction (see Fig. 3).
The success of our analysis is based on the fact that our
laser problem is equivalent to a problem of two lasers
controlled by the same periodic modulation. As a result,
branches of periodic states do not emerge as Hopf bifurcation branches as they do in other coupled lasers systems
+fi
r1
E = yfi-‘.
6mAfl-‘,
El.
(’ +“)
____
(1 fY,)
sin($)
(32)
’
where
5. Appendix
/3=2Kfi-‘,
A,-]
andq=---
5. I. Bifurcation equations
In this appendix, we derive the bifurcations equations
for the localized states. We reformulate the laser equations
in terms of amplitude and phase variables. Substituting
E, = fi exp(i+,) into Eqs. (1) and (2) gives
1i=2(F,-
l)fi-2K\iil,l,cos(~),
Prime now means differentiation with respect to s. If p is
small and if 6 = O(l), we find $= $(O) + SS from Eq.
(32). Inserting I,!J= 6s into (29) and (31), we obtain equations for two weakly modulated oscillators which we
investigate.
(23)
5.2. Leading approximation
F,‘=y[Aj
$‘=A+K
(33)
A,-1’
(25)
We determine a periodic solution Eqs. (28)-(32) assuming that (x,,y,)
is O(1) and that (x2,y2) is small.
Specifically, we seek a 2~periodic
solution of the form
where A = w2 - w, is the difference between the detunings and 4 = I#+ - 4, is the difference between the phases.
In order to determine time-periodic solutions of Eqs. (23)(25), it will be useful to introduce a new basic time s
defined by
(XZPY?) = P(
s = nt,
$=S++$,(S)+
(26)
where fi = (2( A, - 1)~) I” is known as the laser relaxation oscillation frequency. This new time is suggested by
the linearized problem for the steady state solution (Z,,Fj)
= ( Aj - 1,l) which admits slowly decaying and 2n/fi
periodic solutions if y is small. After inserting (26) into
(X,?Y,)
= (X,O(~)~Y,&))
+ P(G%Y,,(S))
~,,(~)~?J,,(~))
+
+ ‘.‘,
(34)
..‘>
(35)
.. ..
(36)
where S = 8s. We consider
scale E as
the case S = O(1) fixed and
e=PE.
We introduce (34)-(37)
(37)
into Eqs. (28)-(32)
and equate to
130
R. Kuske. T. Erneux/Optics
Communications
zero the coefficients of each power of p. The leading
order equations for (xn,,yn,) and (x2,,Y2,) are given by
sx;,
= -y,,,
6.x;, = -qy2,,
6Y’,o = (1 -fY,O)X,O~
6Y’,, =x2,
- +,;m
(38)
COS(S)>
v4
(39)
where prime means differentiation with respect to S. Eq.
(38) forms a conservative system of equations which admits an one-parameter family of periodic solutions. Its first
integral is given by C=xfo +.v,~ - ln(1 + ylo) where C
is the constant of integration (0 5 C < m). If S < 1, the
2rr-periodicity
condition selects a specific value of C
which determines the amplitude of the oscillations but not
its phase. We thus write the solution as
(~,o,Y,o)=(X(~-t~,,~),y(~+~,,~)),
(40)
where (X,Y) is defined as the 2n=periodic solution of Eqs.
(38) and 0, is an arbitrary phase. Without loss of generalities, we may define S such that ylo = Y(S,S) is an even
function of S (this property will be useful when we
evaluate integrals of Y(S, S )>. We next consider Eq. (39)
and determine a 2n-periodic
solution of Eq. (39) with
y,,, = Y(S + 0,,6). We find
6
x2, = xf, ( pk eikS + c.c.),
yz, = --x;,,
(41)
9
where pk is defined by
____Pk= k2@-q
fi
2n1 IZVJ
o
1 + Y(S+
@,,S)
x cos(S) eeikSdS.
(42)
and cc. means complex conjugate. This solution is unbounded if S is close to 6, = G/k,
i.e. near points of
resonances. At and near 6 = S,, the approximation
of
(x2, y,) in power series of p fails. We study the interesting case S = 6, in the next subsection. Because 8, is still
unknown, we examine the problem for (x, ,, y, ,) and apply
a solvability condition. A similar method has been applied
for a periodically modulated laser [ 13,141 so that we
summarize the details. The solvability condition is obtained by differentiating the energy function C(x,, y,) =
$ + y, - In(1 + y, > with respect to S and then by averaging the resulting right hand side from 0 to 27r (we note
that /:rX ’ Y d 5 = - S#X 2 d X = 0 using the first equation
in (38) and we take into account the fact that Y(.S,S) is an
even function of S). We then obtain the following equation
for 6,
&cos(~,)G(~)+EA,K(~)=O,
where G(6) and K(S) are functions of 6 defined by
(43)
139 (1997) 125-131
These integrals must be evaluated numerically. Instructive
approximations are obtained by determining a small amplitude solution of Eqs. (38) and then by evaluating the
functions in (44). The small amplitude solution is constructed using the Poincare-Lindstedt’s
method [ 11,151.
We find
_lu?“X2(t)d[.
(45)
where
R = [6(1 - S)]“’
(46)
measures the small amplitude of the periodic solution
+ 0(R2)
and Y=
(X(S),Y(S))
(i.e., X= -2Rsin(S)
2 Rcos(S) + O( R2)>.
5.3. Near resonance
We now consider the near resonance problem if 6 is
close to 6, = I&. From (41) and (42), we note the limit
X?‘PXZl -
P
1
I__
(&
q
4T
[,
o2nl/l + Y(S+
0,,6,)
1
eis + cc.
xcos(S)e-‘kSdS
(47)
as IS - S, I--) 0. Thus, the solution becomes unbounded
near 6 = 6, which means that our expansion of the solution in power series of p is no longer valid. We solve this
problem by using the method of matched asymptotic expansions [ll]. To this end, we introduce the deviation D
defined by
+p2’3D),
S=&(l
(48)
and seek a solution of the form
(%,Y,)
= (Xro(S),Y,o(S))
+ V3(
(X2TY2)
= P”‘(
X,,(S),Y,,(S))
$h = s + p “3$,(S)
+ ‘..,
(49)
+ .. ..
(50)
X?,(%Y2,(S))
+ P2’3(
X2,(S)>Y2,(S))
f
.. .
(5’)
We introduce (48)-(51) into Eqs. (28)-(32) and equate to
zero the coefficients of each power of p’/3. The leading
order equations for (xlo,ylo) and (x,,,y,,)
are
fix;0
= -y109
x;, = -hv,,,
&v;o
= (1 +Y,o)x,o,
(52)
hY’2,
=x2,,
(53)
where prime means differentiation
(53) has the solution
x2, = &
and K(6)
and K(S)=4rR2,
G(S)=2rR
( o2 eis + c.c.),
with respect to S. Eq.
yZ, = -ia,
eis + c.c.,
(54)
where cr2 is an unknown complex amplitude. The solution
of Eqs. (52) is simply given by (40) but now evaluated at
R. Ku&e. T. Emeur / Optics Communications 139 f 1997) 125-131
S = fi < 1. Because 0, and (Ye are unknown, we need to
solve the higher-order problems and apply solvability conditions. The solvability condition for the (x2s,y2s) equations leads to an equation for LY, given by
2Dcr,+
&-
&O,)=O,
Y
where F(0,)
F(0,) =
(55)
is the function defined by
+Jl;1+ y(s+
o,,fi)
111 D. Botez, D.E. Ackley. Phase-locked
F(O,)/(2Dq)and using (SO), we then
As jDj-*m,a1 ---t
have
=
p
F(O)>
“3 -
2D9
e’.’ +
C.C.
(57)
We have verified that (57) is matching (47) rewritten in
terms of D using (481. The solvability condition for the
(x,s,.~,~) equations is (43) evaluated at 6= fi.
The
integral (56) needs to be solved numerically. For small Y,
it is approximately
F(Ol)= +.
(58)
In summary, the
described by Eq.
satisfies Eq. (43).
second laser are
-&=0(l))
O(1) oscillations of the first laser
(38) for all S 5 1 and the phase
The small amplitude oscillations of
O( /?> if 16 - &I z==/3’/s (and if
an d are described
grant DMS0065, the National Science Foundation
9625843, the Fonds National de la Recherche Scientifique
(Belgium) and the InterUniversity
Attraction Pole of the
Belgian government. R.K. was supported in part by an
NSF Mathematical Sciences Research Postdoctoral Fellowship.
References
cos(S) e-‘sdS.
(56)
x2
131
are
0,
the
IS
by (54) and (55) if 16
- &I = O( p V.
Acknowledgements
This research was supported by the US Air Force
Office of Scientific Research grant AFOSR F49620-95-
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