- Email: [email protected]
Fluctuations and transitions at instabilities in a ring laser with optical back-scattering
Physica 111C (1981) 111-126 North-Holland Publishing Company
111
F L U C T U A T I O N S A N D TRANSITIONS AT INSTABILITIES IN A RING LASER WITH OPTICAL BACK-SCATTERING D. K U H L K E Department of Physics, University of Jena, 6900 Jena, (}DR
R. H O R A K Laboratory of Optics, Palacky University, 77146 Olomouc, CSSR Received 24 November 1980 For the stationary states of a two-mode ring laser the two-time correlations and the irreversible circulation of the intensity and phase-difference fluctuations are derived and the influence of the spectral characteristics of the gain line, of the strength and the phase of back-reflections on them is considered. Depending on the phase, the coupling of the counter-rotating waves due to back-reflections can diminish the magnitude of the negative cross-correlation of the intensity fluctuations and even cause the cross-correlationto becomepositive. This couplingcan also lead to a non-vanishingcorrelation between the intensity and phase-difference fluctuations. The behaviour of the irreversible circulation in the vicinity of the marginal points corresponding to the first or second kind phase transitions shows that both soft mode and hard mode instabilities can occur. For non-vanishing back-reflection even the laser threshold can be associated with a hard mode instability.
1. Introduction The treatment of the fluctuations in a ring laser from various aspects has been the subject of a great deal of attention in recent years [1-4]. Back-scattering of the counter-running waves at optical elements of the ring resonator essentially changes the amplitude and frequency characteristics of the single-frequency ring laser. Depending on the strength and the phase difference of the back-reflected radiation as well as on the spectral characteristics of the gain medium (homogeneously or inhomogeneously broadened gain line; see [5] and references cited therein), various stable stationary states as well as limit cycles can occur. The transitions between these laser states correspond to the phase transitions of first and second kind, respectively [5]. In this paper a general fluctuation theory is applied to this example. Statements are made on the following points. (1) The two-time correlation functions and the variance of the fluctuations of the intensities and the phase difference of the counter-running waves are calculated, and their dependence on the coupling due to the active medium, on the strength and the phase of the back-scattered radiation is discussed. The coupling of the counter-running waves due to the active medium is determined by the spectral characteristics of the gain line (inhomogeneously broadened line as in the case of a gas laser and homogeneously broadened gain line with spatial saturation grating induced by the standing wave pattern of counter-running waves [6]). (2) The two-time correlation functions, the variance and the irreversible circulation of fluctuation are studied in the vicinity of the critical points. Besides the features of phase transitions such as critical 0378-4363/81/0000-0000/$02.50 O 1981 North-Holland
112
D. Kiihlke and R. Horak / Fluctuations in a ring laser with back-scattering
fluctuations and critical slowing down [7, 8], questions of general interest are connected with the nature of the instabilities. In addition to the conventional soft mode instability familiar at equilibrium there is a hard mode instability which is met only in a system of thermodynamically coupled degrees of freedom far from equilibrium [9, 10]. In the latter the detailed balance does not hold and the irreversible circulation which is proposed by Tomita et al. [9, 10], to be used as a measure of deviation from detailed balance, shows a critical behaviour, too. It is shown in this paper that both soft and hard mode instabilities are met in a ring laser with optical back-scattering. In section 2 a general formalism is given to calculate the two-time correlation matrix of the fluctuations of a non-linear system with several degrees of freedom, provided the fluctuations can be described by linearized Langevin equations. The general formulas for the two-time correlation matrix, the variance and the irreversible circulation allow us, according to the form of the eigenvalues of the drift matrix, to state the behaviour of the system in marginal situations. Model and basic equations of a ring laser with back-reflections, the stationary solutions of the mean values of the amplitudes and the phase difference for equal pump rates of the counter-running waves and their region of stability are recapitulated from [5] in the first part of section 3. In the second part the two-time correlation matrix and the irreversible circulation for the various laser states are calculated. The influence of the coupling of the counter-rotating waves due to the active medium and due to back-reflections on these quantities are discussed in detail especially in the vicinity of marginal points.
2. Two-time correlation functions, variance and irreversible circulation of fluctuation
yj(t) (i = 1. . . . . N) are supposed to be a set of fluctuating physical quantities obeying the Langevin equations y,,(t) = g,({yj}, t) + F~(t).
(2.1)
F~(t) are fluctuating forces with the property (F~(t)~(t')) = D,~(t-
t'),
(F~(t)) = 0,
(2.2)
where D0 is the diffusion matrix Dij = d i s ~ ,
(2.3)
with 8o being the identity matrix and di the noise strength. Here we use the sum convention, i.e. in place of Ig~.l ajb~ we write a~b~. Equal indices for which this convention shall not be valid will be underlined. Under the assumption that the fluctuations around the secular motion of the system are small compared to the mean values of the physical quantities, i.e. 18,(01 = ly,(t) - (y~(t))l "~ [(yi(t))],
(2.4)
eq. (2.1) can be linearized with respect to the fluctuation [11]. From eq. (2.1) we obtain the equations of
D. Kiihlke and R. Horak / Fluctuations in a ring laser with back-scattering
113
motion of the mean values X~(t) = (y~(t)): = g,({X/}, t),
i = 1. . . . . N ,
(2.5)
and the evolution equations of the fluctuations 8~(t)= y ~ ( t ) - X ~ ( t ) : 6,(t) = K~j({Xk}, t)8/(t) + F~(t),
i = 1. . . . . N ,
(2.6)
where
=0g, is the drift matrix. By means of (2.4) the two-time correlation functions of the second (amplitude correlations) and fourth (intensity correlations) order, respectively, can be approximated by G~])(t, t') = ((y,(t) - (y,(t)))(yi(t') - (yj(t')))) - (8,(t)Sj(t')) -- o'~j(t, t'),
(2.7)
G[~)(t, t') = ((y,(O2 - (y,(t))2)(yi (t')2_ (yj(t,))2)) __ 4X i (t)X~(t')(Si(t)S/(t')).
(2.8)
They are essentially determined by the two-time correlation matrix of the fluctuations o'~j(t', t ) = (Si(t')Sj(t)) which is considered in the following. Using the formal integral of (2.6) with the initial condition 8~(0)= 0,
8,(0= (exp(f' X(s)ds)),jf' du (exp(- i 0
0
Y((v)dv))/kFk(U ) ,
(2.9)
0
where Yt' denotes the matrix with the elements K~, the correlation matrix of the fluctuations can be written as I+'/'
i
t
0
t u
t
×o..
, 10, u
where relation (2.2) have been used. The tilde " denotes the transposed matrix. When the stationary solutions of eqs. (2.5) are considered (.~ -- 0), the drift matrix becomes time-independent and eq. (2.10) can be further simplified. If the eigenvalues and the eigenvectors of the drift matrix are known, ~ can be transformed in the diagonal form At8 u = (U-1)aKtmUmj.
(2.11)
114
D. Kiihlke and R. Horak / Fluctuations in a ring laser with back-scattering
A~ are the eigenvalues and the columns of the transformation matrix q/, the eigenvectors of ~. Taking into account that
(U-')l. (e~).,.U.k = e*l~k, eq. (2.10) can be written as trij(t + r, t) = Uik eAk'B~k(t)(O)~j,
(2.12)
where
B..(t) = (U-'),,D~k(O-'),~
eC*-+~o) ' - 1 A m + h.o
(2.13)
This representation of the two-time correlation matrix of fluctuations allows us to draw some conclusions. (1) The stationary solutions of (2.5) are stable if Re Aj < 0 for all i = 1. . . . . N. In this case the correlation matrix (2.12) tends to its stationary value o'~i(r) for t--* %
cr,j(r) = U,, e * " B k . ( t - * = ) ( 0 ) . ~ ,
(2.14)
where B m , ( t - - , oo) = - - ( U - ' ) m t D l k ( O - x ) k ~ ( A ~
+ A~)-'.
(2.15)
For r = 0 we obtain the second moment or the variance o-~(0) - ~r~j from (2.14), trij
=
U~kBk,(t ~ oo)(O),j .
(2.16)
(2) A zero real part of at least one eigenvalue leads to the points of marginal stability. Passing through such critical points corresponds to the first or second kind phase transitions [8]. In the vicinity of the critical points typical phenomena occur which are obvious from (2.12)-(2.16). For Re Ai ~ 0, both the correlation time and the relaxation time of the system, which are proportional to (Re A;)-1, tend to infinity corresponding to critical slowing down. On the other hand, Re Ai--* 0 leads to a divergency of the matrix elements of the variance (see eq. (2.16)) corresponding to critical fluctuations. If a real eigenvalue is vanishing in the vicinity of the critical point one eigensolution (fluctuating mode) corresponding to Ai becomes unstable and dominates the whole system at the marginal situation (soft mode instability). From eq. (2.13), for & ~ 0 we obtain lim B~(t) = (U-~)itDa((J-l)kit,
Ai..~O
(2.17)
which makes it evident that at the marginal point the elements of the two-time correlation matrix are growing linearly in time. If the eigenvalues are complex in the vicinity of the critical point, there always exist a couple of conjugate eigenvalues, since X is real, denoted by Ai and A, = A*, where Re ai = 0
D. Kiihlke and R. Horak / Fluctuations in a ring laser with back-scattering
115
determines the marginal point. In this case the term A, + At = 2 Re Ai in eqs. (2.13) and (2.15) leads to divergencies of the correlation time, the life time of the fluctuations and the elements of the variance; that means that in the vicinity of the marginal point two fluctuating modes corresponding to At and A,, respectively, become unstable. The transition has essentially a two-dimensional character (hard mode instability). Due to the complex exponents in formulas (2.13) and (2.14), the elements of the two-time correlation matrix oscillate with a frequency proportional to Im At for varying t and ~-. (3) The antisymmetric matrix a,j = ~(o',,( g ) o -
K,~r,j)
(2.18)
has been shown to be a measure of the deviation from the detailed balance [9, 10]. It characterized a directed flux in a thermodynamically coupled system of more than one degree of freedom and has, therefore, been termed as the irreversible circulation of fluctuation [9]. Inserting (2.16) into (2.18) one obtains OtO"= ½[Ujn((U)mjAm)
- -
(2.19)
(UinAn)(U)mj]Bnr# (t --, oo).
In the case of a real eigenvalue At ~ 0 at the marginal situation, i.e. the soft mode instability, the irreversible circulation remains finite (the summands of (2.19) where m = n = i are zero because the square bracket vanishes and the other summands remain finite) and plays no important role compared to the diverging variance. In the special case of ~ = ~-1 which implies 0//= 0//-1, the irreversible circulation vanishes identically. In the case of complex eigenvalues occurring only for the non-symmetric ~, i.e. the hard mode instability, the matrix elements of the irreversible circulation diverge, too. Let A1, Az = AT be a couple of complex conjugate eigenvalues with Re AI~ 0. Then the summand of eq. (2.19) which contains hi, A2 takes the form
-½(U/l(~r)2.~2- U/1/~I(O)2j)(/~ 2 + i~l)-l((u-1)llDik(O-1)k2)
=½Uil(O)2j(U-1)l~Dtk(O-1)k2ImAJRehl--)o~,
for
Re AI-*0.
That means that, unlike the soft mode instability, the irreversible circulation diverges and the detailed balance does not hold. It is worth noting that these divergencies are a non-physical outcome of the linear approximation, but they serve as an indication of unusually large fluctuations in the vicinity of the marginal points.
3. Fluctuations in a ring laser with optical back-scattering
3.1. Basic equations and stationary solutions To take into account the influence of optical back-scattering, the Langevin equations for the single-frequency ring laser [2] are extended by terms describing the reflection from one direction of propagation into the other (see [5] and references therein). Under the assumption that the fluctuations are caused by the spontaneous emission the influence of back-reflections on the fluctuating forces can be
D. Kiihlke and R. Horak / Fluctuations in a ring laser with back-scattering
116
neglected: ~
- [ a l - a(l
d +
1
12)1 i -
=
(3.1)
El(t),
(3.2) /3t (i = 1, 2) are the complex amplitudes of the counter-running waves, • describes their coupling due to the active medium and, depending on the spectral characteristic of the active medium, can take the following values: 0 < ~ ~< 1 for the Doppler-broadened gain line, 1 ~< e ~<2 for the homogeneously broadened gain line with spatial saturation grating, and in the intermediate case, 0 ~< ~ ~< ~ where ~ > 1 [2]. a~ = g 0 - K~- r~ is the pump parameter, go the linear gain of the active medium, rj are the cavity losses, d is the saturation parameter, ~ = r~ exp(-i0~) with r~ being the coupling coefficient due to back-reflections and 0~ the phase of the back-reflected radiation. The fluctuating forces have the property
(F~(t')F'~(t)) = 4 q S : ( t - t'),
(E(t)Fj(t')} = 0 ,
(Fdt)) = 0 ,
(3.3)
where q is the noise strength. Inserting/3j(t) = Ej(t) exp(-icj(t)) into (3.1) and (3.2), with Ei(t) and ~0~(t) being the real amplitudes and phase of the counter-running waves, respectively, and taking the real and imaginary part we obtain, with the notations $ = 02 - 0~ - ~', ~0 = ¢2 - ,~ + 01 + ½or, f~l - [al - d ( E ~
(3.4)
+ ~E~)]E1 - E2r sin ~o = F l ( t ) ,
/~2 - [az - d(E~ + ~E~)]E2 + E l f sin(¢ + 0) = F2(t),
(3.5)
?0 + r[(EJE2) cos(¢ + 0) - (E2/E1)cos ¢] = F3(t),
(3.6)
where 16L2= Re(F1.2 e i~.2) ,
1 a63 = E1 Im(Fl e i~') -
Ira(F2 e i ~2).
H e r e we have specified rl = r2 = r. If the fluctuations are small enough, eqs. (3.4)-(3.6) can be linearized around the mean values (Ei.2(t)) and (¢ (t))
El.2(t) = (E1.2(t)) + 81.2(0,
~ (t) = (~ (t)) + 83(0.
(3.7)
8,(0
(i = 1, 2, 3) are small deviations caused by the noise source. Inserting (3.7) into (3.4)-(3.6), retaining only the terms linear in 8t and adopting the notations Xl.2(t)= (El.z(t)), X3(t)= (¢(t)) we get: X~ = g~({X~}) = [al - d ( X ] + eX~)]X1 + rX2 sin X3,
(3.8)
Jr2 = g2({X~}) = [a2 - d ( X ~ + eX~)]X2 - rXl sin(X3 + 0 ) ,
(3.9)
X3 = g3({X/}) =
~, = K,flj + £ ,
-r[(Xl/X2) cos(X3 +
0 ) - (X2/Xl) cos X31 ,
(3.10)
(3.11)
117
D. Kiihlke and R. Horak / Fluctuations in a ring laser with back-scattering
where ~r (Ko)= (oggoxj) =
=
at - d(3X$ + eXl) -2deXiX2 - r sin(X3 + O)
r[(x~/x~)cos x3 + (l/X2)cos(X3+ q')l
-2edXiX2 + r sin X3 a2 -- d(3X] + eX~) r[(1/Xl) cos Xa + (Xt/X]) cos(X3 + 0)]
X2r cos X3 -X,r cos(X3+ O)
\
r[-(XzIXl) sin X3 + (XdX2) sin(X3 + qt)]
)
(3.12)
is the drift or regression matrix of the system. Assuming that exp(-i~j) can be absorbed into the fluctuating forces as a new phase factor, the correlations of ~ ( t ) in this approximation become (F/(t')~(t)) = DoS(t- t'),
(3.13)
(F/(t)) = 0 ,
with the diffusion matrix
= (Do) = 2q
(i0 0) 1
0
0
(1/X~+l/X2,)
(3.14)
.
The stationary solutions and their stability as well as the possible limit cycles of eqs. (3.8)-(3.10) are discussed in detail for 0 = 2nor and ~ = (2n + 1)¢r in [5]. Let us recapitulate the steady-states X0i and their region of stability for equal pump rates al = a2 = a. In the following the abbreviations 77 Xo2/Xol and ~"= dX~l will be used. (A) 0 = (2n + 1)~': In this case eqs. (3.8)-(3.10) have the following stationary solutions (77 ~>0): =
'11 = 1 ,
'i'~2'3
For
=
~'1 = a + r l+e '
X o 3 = ~" 2 '
a(•2 r 1)+4(a -
(3.15)
f- 1 ,
a +/'v~2,3 ~2,3 ~ ~ ,
,/7X03 = "~-.
(3.16)
0 ~ • < 1, solution (3.15) is stable if r>0,
a>0
and
a<0,
r > -a,
(3.17)
whereas solution (3.16) is unstable. The laser threshold is given by a = - r . For 1 < • ~ 2 solution (3.15) is stable in the region a>0,
r > r c = ~1a ( • - l ) ;
a<0,
r>-a,
(3.18)
and (3.16) in the region a > 0, (see fig. 1).
1
r < rc = ~a(e - 1),
(3.19)
118
D. Kiihlke and R. Horak / Fluctuations in a ring laser with back-scattering
"7
3
2
1
i
qz
o~
q6 r_.. a
Fig. 1. Steady states of the ring laser for ¢ = (2n + 1)~r and ~ = 2. T h e solid and dashed lines show the stable and unstable range, respectively.
(B) ¢ = 2nw: The stationary solutions of (3.8)-(3.10) are (7 ~ 0): 71 =
1,
~1 = al(~ +
1),
sin X03 = 0,
(3.20)
~. = a - (l/7/)r sin X03
(3.21)
1,/2 + ~
1
X03 = -~-,
+ X03 =
[a(1-,)÷ .l(aX_fl. h=_ 2(, +
72,3=-~[
-~/2,
74,5 :
2r - Y \ 2r / 1 _ ~/(~)2 l+~[a(12;')+
1)l J
2 - 2 ( ¢ + 1)] ,
1/72,3 ,
Contrary to (3.15), the laser threshold is given by a = 0. For 0 ~ • < 1, solution (3.20) is stable in the
range r > 0,
(3.22)
whereas solution (3.21) is unstable. For 1 < • ~ 2, solution (3.20) is unstable and the stability of solutions (3.21) is determined by ~" (72,"2"),
('04,--2)
(compare figs. 2 and 3).
stable for
r < Pc = ~a/ 8( (~~-' 1)
(3.23)
D. Kiihlke and R. Horak / Fluctuations in a ring laser with back-scattering
119
3
,I g-2
To
r0-÷ r
3 I
/ 2
I I
\ /
//
\,..
t"
....
.........i
f.o
q.l,
E(I )*o-0 \
/
a)
&
. . . .
r
1
A
i
0,5
t
t,5
f
Q
Fig. 2. Steady states for 0 = 2mr and • = 2 as well as • < 1 (the unstable stationary solutions for • < 1 are omitted).
Fig. 3. Steady states for q J = 2 m r depending on the parameter e, which describes the coupling of the counterrunning waves due to the active medium, for a constant ratio of the back-scattering coefficient to the p u m p rate (r/a = 0.05).
In the following section the two-time correlation matrix and the irreversible circulation will be calculated for these steady states.
3.2. Two-time correlations, variance and irreversible circulation of fluctuations (A) ~, = (2n + 1)~': First solution (3.15) is considered. The drift matrix is obtained by inserting (3.15) into (3.12):
- ~
1
(2a + r(e +3))
- ~ ( 21 • a + r ( • - l ) ) 1+•
1 (2ca + r ( a - 1)) - ~ 1 (2a + r(• + 3)) l+a l+a
(Kq)= 0
0
\
0 0 -2r
)
(3.24)
The drift matrix (3.24) has the eigenvalues Az= - 2 r ,
A 2 = - 2 ( a + r),
Xa=
1 +2 • ( a ( 1 - e)+2r).
(3.25)
From the eigenvectors of (3.24) we get the transformation matrix
=
-1 o
, x/2
(3.26)
D. Kiihlke and R. Horak / Fluctuations in a ring laser with back-scattering
120
for which the relation ~ / = o~ = ~d-1 holds. Putting (3.24)-(3.26) and (3.14) with (3.15) into (2.12) yields the two-time correlation matrix with the following elements:
r)[h,(t, ~)+(a+
o'11(t + % t ) = (Tm(t + r, t ) = 4 ( 2 +
r ) ( e + 1)h2(t, ~')] 2(r - re)
tr21(t + z' t)= (T12(t+ z' t)= 4(2+ r) [hl(t' r ) - ( a + r)(e - re)
(T33(t+ r, t) = qd(e + 1) . 2r(a + r) [1 -
1)
(3.27)
h2(t, z)] ,
(3.28)
e - " ] e -2"
(3.29)
0"13(t + ~', t) = (T23(t + r, t) = (T3,(t + r, t) = (T32(t + r, t) = O,
(3.30)
where
hi(t, "r) =
[1 - e x p ( - 4 ( a + r)t)] e x p ( - 2 ( a + r)~-),
h2(t,r)=[1-exp(-S(r-r¢)t)]exp(-4(r-r~)~'~ 1+~
1+~
]'
and rc = a(~ - 1)/2. Within the ranges of stability 0 ~< ~ < 1, r > 0 and 1 < ~ ~< 2, r > re, we obtain the variance from (3.27)-(3.29) by t--* oo and r = 0
((T0) : 8(a +
rq)(r- re)
I
2a + r(e + 3)
- ( 2 a ~ + r(E - 1))
- ( 2 a ~ + r(e - 1)) 0
2a + r(~ + 3) 0
o
)
0
(3.31)
4d(r - ro)(e + 1)/r
T h e irreversible circulation vanishes identically. A b o v e all it is obvious from (3.31) that for e > 1 the cross m o m e n t s o-n, (TEl are negative, i.e. the fluctuations of the counter-running waves are anticorrelated (anti-bunching). T h e strength of the anti-correlation increases with increasing ~. Within the range 0 ~< • < 1, the back-scattering counter-acts the anti-correlation of the intensity fluctuations: if r < 2a~/(1 - ~), O"12 and O"21 are negative but the a m o u n t decreases with increasing r; if r = 2 a e / ( 1 - ~), o"12 and o'21 vanish, i.e. the intensity fluctuations of the counter-running waves are uncorrelated; if r > 2 a ~ / ( 1 - e), o'12 and (T2t are positive and the fluctuations are correlated. T h e fluctuations of the intensities and the phase difference are uncorrelated. At the critical point r = 0, 0 ~< e < 1, only the variance of the phase-difference fluctuations diverges. T h a t reflects the fact that the solution rt = 1 does not change for r ~ 0 but the phase locking is abolished. This b e c o m e s evident if we take the limit r ~ 0 in the formulas for the two-time correlations (3.27)-(3.29) (Tn(t + ~', t) = o'22(t + ~', t) = ~
xexp-
( (I-
(T33(t + % t) = {2qd(~ +
e+l
1)/a}t.
[/(1 - e x p ( - 4 a t ) ) e x p ( - 2 a ' r ) + , ~ + 1 \( 1 _ e x p (\ _ 2 (e1)-a+t ' ~1 ,
]]
I.
]J'
(3.32) (3.33)
D. Kiihlke and R. Horak / Fluctuations in a ring laser with back-scattering
121
Eq. (3.33) describes the well-known phase-diffusion which does not influence the stability of the steady state of the amplitudes (t ~ o0 in (3.32) leads to the stationary autocorrelation functions of the intensity fluctuations). In the range 1 < ~ ~< 2, solution (3.15) is stable for r > re. At the critical point r = r e all those elements of the variance diverge which are associated with the fluctuations of the amplitudes. The phase difference does not change on passing over to the new solution (3.16) and the variance of their fluctuations remains finite. The behaviour at the marginal point r = - a , a < 0 is analogous. T h e drift matrix for solution (3.16) which is symmetric, too, has the following eigenvalues A1.2=-(a+rc)±
}1/2,
4a
(a+r~)2---~(r2-r 2)
A3 = - 2 r o .
(3.34)
Because of the lengthy formulas only the variance as well as the two-time correlation matrix at the critical point are given. T h e variance is
12ar~AA(r- 2-r r2(a2)- re) r(r~ + a) (~,;) q r2~- r 2
r(rc r ~ -+r a) 2 2r2A + r2(a - r¢) A(r~- r 2) 0 -
0
0 0
2ar2d
)
'
(3.35)
where A = rc ___~/r~--~--~ (+ sign belongs to */2 and - to ~73). T h e irreversible circulation vanishes identically. T h e intensity fluctuations are anti-correlated and show critical behaviour at r = re, whereas the variance of the phase-difference fluctuations remains finite. T h e two-time correlations of the intensities at the critical point r = rc are of the same form as in the case of solution (3.15) , / = 1 for r = re; only the magnitude of the phase-difference fluctuations is different tr11(t (12) + r, t) = trz2(t (21) + r, t) = 4(a q+ re) {[1 -
e -4(a+rc)`]e-2(a+'°)"+ 4(a (-)
o'33(t + r, t) = qda (1 - e -4'°') e -2'o" 2re
+
re)t}, (3.36)
Resuming, we can state that in the case of qt = (2n + 1)~-, the critical points corresponding to second kind phase transitions are characterized by a soft m o d e instability. All eigenvalues of the drift matrix are real within the range of stability. At the critical points only one of the eigenvalues becomes zero and this m o d e dominates the whole system. The irreversible circulation as a measure of deviation from the detailed balance vanishes throughout. (B) ~ = 2nrr: First solution (3.20) is considered. The drift matrix belonging to this solution is
Yl=
2a 1+~
2a, -1+~
2a~
2a
-2"
/
a
r~d(l~E)
2"X/dOf
I----a-- [
°
/
(3.37)
D. Kiihlke and R. Horak / Fluctuations in a ring laser with back-scattering
122
which has the eigenvalues At,2 =
- ¢~ a -----~ / D , - 11 +-"--e
A3 = - 2 a
(3.38)
where
If the eigenvectors of (3.37) are known we can calculate the transformation matrix ~ which is of the form
/rX/a/d(1+') -r~/a/d(l+') i ) (Uo) = ~-rX/a/d(1 + e) r~/a/d(1 + e) ~--t~ 2
(3.39)
I~ 1
From (3.39) 0~, q/-1, q~-i are determined and inserted together with (3.38), (3.39) and (3.14) with (3.20) into (2.12). After some algebra we come to the two-time correlation matrix which has the following elements: crn(t + z, t) = o'z2(t + ~', t) = (+) (12)
e) (/~l(t, r) + ~2(t, r))
(2t)
+ A (--) h3(t, z) -
4r2(1 + e ) ' ^ ( t , "r)+ ~2(t, r))] D(1 - e) tg (
0"13(t + ~', t) = -cr23(/t + ~', t) =
q - 4rD
(3.40)
4 d ( l + e) ~'a(1 - e) [AJ*l(/, ~') + At/~2(t, r)]
[---i--+-~-e
4r2(1 a ' ~ ' - -+E e) [A,~,(t, "r)+ A2~2(t, r)]} ,
o'3,(t + r, t) = -o'32(t + % t) =
•
q - 4rD
4r2( 1 + e) r x ~ , -
o'33(t + % t) =
a(1-e)
4 ~
(3.41)
I'a(1 - ") [A2/~,(t, r) + A,/~2(t,r)]
[ ~
r ) + Al~2(t, ~')]}
(3.42)
t,,ml~,,
qd(14r2D-e) [A 2/~l(t ' ~.) + A~/~2(t, r) -
/4r2(1 \ " h ' ~ - -+- ~~)~2 ' ] (~l(t, ~') + ~2(t, 7))]
where / ~ i ( t , ~ ' ) = [ 1 - e z~'t]e ~,*,
~,(t,~')=
[1 - e x p
-
l+e
t
.
(3.43)
D. Kiihlke and R. Horak / Fluctuations in a ring laser with back-scattering
123
For r > 0, a < 1, where solution (3.20) is stable we get from (3.40)-(3.43) the variance of fluctuation by t ~ o o and ~- = 0,
q
/
3+e 1-~ 1 + 3e 1-e
1+~3 1-~ 3+
2 V a d ( 1 + ~)
- 2 V a d ( 1 + ~)
2 ~ / a d ( l + ¢)
)])
_ 2 ~ / a d ( i + E)
1-E
r
(3.44)
r(1 + ~) 2 4ad(lr2 - ' ) [1 + 2 ( a ( - ~ - ~)
r
Substituting (3.44) and (3.37) in (2.18) leads to the irreversible circulation
(aiD= a ( 1 - ~ )
0
0
1
1
-1
0
.
(3.45)
Analogously to the cases discussed in (A) the intensity fluctuations of the counter-running waves are anti-correlated, o't2 = o'2~< 0 , however, the back-scattering does not influence the magnitude of this anti-correlation. Contrary to the solutions discussed in (A) the cross-correlation of the intensity and the phase-difference fluctuations do not vanish. This is quite clear. The phase difference of the reflected radiation causes that the energy from one sense of propagation is pumped into the other. But the phase difference of the waves counter-rotating in the ring resonator takes such a value that d u e to interferences between the reflected and original waves the energy of the counter-running waves remains equal (even if the pump rates of the two directions are different [1]). Therefore the threshold (see eq. (3.20)), the intensities and the intensity fluctuations do not explicitly depend on the back-scattering coefficient but the cross-correlations of the intensity and phase-difference fluctuations and the variance of the phase-difference fluctuations depend on r. The laser threshold a = 0 (see (3.20)), r = 0 and ~ = 1 (the laser frequency is tuned to the centre of the inhomogeneously broadened gain line) are the critical points. At r = 0 we have a soft mode instability: the circulation vanishes and one mode dominates the system. A different behaviour shows the system in the vicinity of the critical point ~ = 1 corresponding to a first kind phase transition. For r >2~a(1 - e)/(1 + e),
(3.46)
a couple of the eigenvalues (3.38) are complex: AI.2 = a(1 - ~)/(1 + ~ ) ± i g2,
(3.47)
with
12
.]4r2_ {a(1 - ~)~2 ¥
/
In this region the elements of the two-time correlation matrix oscillate with the frequency 12. The
D. Kiihlke and R. Horak I Fluctuations in a ring laser with back-scattering
124
stationary elements o'11(~-)= 0-~0") and (12)
O'33('1") of the two-time correlation matrix are given for example.
(21)
Taking the limit t -* oo for (a - 2r)/(a + 2r) < e < 1, we obtain from (3.40) and (3.43) with (3.47) 0-11(~.) = (12)
0.22(,)= (21)
+ (_)~_~.
•
q d ( 1 - e) exp(_ ( 1 - e)ar~
:
÷
? + ~ a ) cos £2~-(-)exp(-2a~')
1+,
,{[1
.
_[r(1 +,)\2"1
]
a(1-,)
/
(1 + , ) O
(3.48)
,
sin O~'}
"
(3,49)
At the marginal point e = 1 the diagonal elements of the variance (3.44) as well as the cross-moments of the amplitudes diverge. The corresponding elements of the two-time correlation matrix (3.40) and (3.43) increase with time t: 1 (1 0-11(t + r, t ) = 0-z2(t + z, t ) = ( +- ) q [ t cos(2r'r) ( +- ) ~--da 02)
e_,at) e -~" ] ,
(3.50)
(21)
0-33(t + r, r) = qd(1 + E) t cos(2r'r). a
(3.51)
Contrary to the cases considered up to now the irreversible circulation (3.45) does not vanish but diverges, too. This marginal situation is associated with a hard mode instability: A couple of eigenvalues become complex, both the irreversible circulation and the variance of fluctuation diverge and the elements of the two-time correlation matrix oscillate with a frequency determined by the imaginary part of that couple of complex conjugate eigenvalues. Beyond this marginal point within the range 1 < e < ¢c with
(compare eq. (3.23)), there is no stable stationary state (see fig. 3) and the system tends to a limit cycle [51. An analogous behaviour shows the system in the vicinity of the laser threshold a = 0, ¢ < 1 corresponding to a second kind phase transition. For a < 2r(~ + 1)/(1 - ~),
(3.52)
the couple of eigenvalues AL2 (3.38) are complex, too, and, contrary to the usual lasers, the threshold of the ring laser with optical back-scattering of a phase difference ~ = 2n~" is characterized by a hard mode instability. The state beyond this hard mode instability is given by X1 -~ X2 -- 0 (a ~<0), i.e. there is no limit cycle. For these two cases the detailed balance does not hold. The irreversible circulation as a measure of the deviation of the detailed balance diverges at ~ = 1 and a = 0. This becomes evident, too, from the two-time correlation matrix (3.40)-(3.43) which is not symmetric even in the stationary case 0-13(t + ~', t) = (Sl(t + r)83(t)) ~ 0-31(t + r, t) = (83(t + r)81(t)) .
(3.53)
D. Kiihlke and R. Horak / Fluctuations in a ring laser with back-scattering
125
The condition of detailed balance is equivalent to requiring a symmetric stationary two-time correlation matrix (see, e.g. [12]). Since the formulas of the two-time correlation matrix according to solution (3.21) are very extensive this case is qualitatively discussed for the marginal situation Po= a ( e - 1)/~/8-(~ + 1), ~ > 1. The eigenvalues of the drift matrix (3.12) belonging to solution (3.21) at the critical point r = rc are A1 =
2(~a+ 1) (~2 + 2~ + 5) ,
A2 = 0 ,
A3 = -½a(~ - 1).
(3.54)
As the eigenvalues are real at this critical point, corresponding to a first kind phase transition, we have the case of a soft mode instability. Only one of the eigenvalues is zero and this unstable mode dominates the system. According to the general statements of section 2, the irreversible circulation remains finite and, therefore, is unimportant. However, it is worth noting that beyond this soft mode instability there exists no stable stationary state (see fig. 2) and the ring laser tends to a limit cycle [5]. This contradicts the expectation that the newly emerging state is expected to be a mode which is asymptotically stable [10].
4. Concluding remark For a nonlinear system of several degrees of freedom, the fluctuations around the secular motion of the system have been calculated in linear approximation. In the stationary case the expressions for the two-time correlation functions, the variance and the irreversible circulation of fluctuation allow us to draw conclusions on the behaviour of the system at marginal situations. Besides phenomena closely resembling phase transitions, such as critical fluctuations and critical slowing down, we are also interested in the types of instabilities. Essentially there are two typical cases, i.e. the soft mode instability familiar in thermodynamic equilibrium and the hard mode instability met only in dissipative systems. The latter is associated with periodic variation in time of the two-time correlations and the critical behaviour of the irreversible circulation as a measure of deviation from detailed balance. On this basis the single-frequency ring laser with optical back-scattering has been treated as a system of three degrees of freedom. The dependence of the two-time correlation matrix, the variance and the irreversible circulation on the coupling of the counter-running waves, due to the active medium which is described by the coupling parameter ~, and on the strength r and the phase ~p of the back-reflected radiation has been discussed and their behaviour at marginal situations considered. The coupling by the active medium is determined by its spectral characteristic (inhomogeneously broadened gain line, 0~<~ ~ 1; homogeneous broadening with spatial saturation grating, 1 ~<~ ~<2; and the intermediate case, 0 ~ • ~< ~: where ~: > 1). In all cases the elements of the two-time correlation matrix consist of several exponentials. The back-scattering influences the cross-correlation of the fluctuations in different ways. In the case of a phase difference of the back-reflected radiation of ¢, = (2n + 1)~" and 0 ~< ~ < 1, it counteracts the anti-correlation (anti-bunching) of the intensity fluctuations of the counter-rotating waves and, if the back-reflection coefficient r exceeds a certain value, the cross-correlation becomes positive. In the case of 0 = (2n + 1)zr and 1 < ~ ~< 2, the magnitude of the negative crosscorrelation depends strongly on r only in the vicinity of the marginal point r-- rc and shows a critical behaviour there. For 0 = 2n~r and 0 ~< ~ < 1, the anti-correlated intensity fluctuations do not depend on r.
126
D. Kiihlke and R. Horak / Fluctuations in a ring laser with back-scattering
For ~b = (2n + 1)¢r, all transitions between different stable states correspond to the second kind phase transitions. The irreversible circulation vanishes throughout and the instabilities occurring there are characterized by a soft mode instability. The fluctuations of the intensities and the phase difference are uncorrelated. On the contrary, the corresponding cross-correlation functions do not vanish in the case of 0 = 2nor. For an inhomogeneously broadened gain line, i.e. E < 1 and a non-vanishing back-scattering coefficient r, we have marginal situations at the laser threshold (vanishing pump-parameter) corresponding to a second kind phase transition, and at ~ = 1 (laser frequency tuned to the line centre) we have a situation corresponding to a first kind phase transition. Both instabilities are characterized by a hard mode instability, the irreversible circulation diverges at the critical points and the two-time correlation functions vary periodically in time. But only the state above E = 1, which can occur in the intermediate case 0 ~< 6 < 6, tends to a limit cycle. The critical point r = 0 associated with a second kind phase transition corresponds to a soft mode instability. In the case of a homogeneously broadened gain line 1 < • ~< 2 the marginal point r = rc (re: critical back-scattering coefficient), corresponding to a first kind phase transition, is associated with a soft mode instability. But despite that, the state beyond this instability tends to a limit cycle.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
M.M. Tehrani and L. Mandel, Phys. Rev. A17 (1978) 677, 694. D. Kiihlke and R. Horak, Opt. Quant. Elect. 11 (1979) 485. S. Singh and L. Mandel, Phys. Rev. A20 (1979) 2459. E.B. Rockower and N.B. Abraham, Optica Acta 26 (1979) 1289. D. Kiihlke and G. Jetschke, Physica 106C (1981) 287. J.B. Hambenne and M. Sargent, Phys. Rev. A13 (1976) 784. H. Haken, Rev. Mod. Phys. 47 (1975) 67. A. Nitzan, P. Ortoleva, J. Deutch and J. Ross, J. Chem. Phys. 61 (1974) 1056. K. Tomita and H. Tomita, Prog. Theor. Phys. 51 (1974) 1731. K. Tomita, T. Ohta and H. Tomita, Prog. Theor. Phys. 52 (1974) 1744. R.L. Stratonovich, Topics in the Theory of Random Noise, Vol. II (New York, 1967). S.R. de Groot, Thermodynamics of Irreversible Processes (North-Holland, Amsterdam, 1951).
111
F L U C T U A T I O N S A N D TRANSITIONS AT INSTABILITIES IN A RING LASER WITH OPTICAL BACK-SCATTERING D. K U H L K E Department of Physics, University of Jena, 6900 Jena, (}DR
R. H O R A K Laboratory of Optics, Palacky University, 77146 Olomouc, CSSR Received 24 November 1980 For the stationary states of a two-mode ring laser the two-time correlations and the irreversible circulation of the intensity and phase-difference fluctuations are derived and the influence of the spectral characteristics of the gain line, of the strength and the phase of back-reflections on them is considered. Depending on the phase, the coupling of the counter-rotating waves due to back-reflections can diminish the magnitude of the negative cross-correlation of the intensity fluctuations and even cause the cross-correlationto becomepositive. This couplingcan also lead to a non-vanishingcorrelation between the intensity and phase-difference fluctuations. The behaviour of the irreversible circulation in the vicinity of the marginal points corresponding to the first or second kind phase transitions shows that both soft mode and hard mode instabilities can occur. For non-vanishing back-reflection even the laser threshold can be associated with a hard mode instability.
1. Introduction The treatment of the fluctuations in a ring laser from various aspects has been the subject of a great deal of attention in recent years [1-4]. Back-scattering of the counter-running waves at optical elements of the ring resonator essentially changes the amplitude and frequency characteristics of the single-frequency ring laser. Depending on the strength and the phase difference of the back-reflected radiation as well as on the spectral characteristics of the gain medium (homogeneously or inhomogeneously broadened gain line; see [5] and references cited therein), various stable stationary states as well as limit cycles can occur. The transitions between these laser states correspond to the phase transitions of first and second kind, respectively [5]. In this paper a general fluctuation theory is applied to this example. Statements are made on the following points. (1) The two-time correlation functions and the variance of the fluctuations of the intensities and the phase difference of the counter-running waves are calculated, and their dependence on the coupling due to the active medium, on the strength and the phase of the back-scattered radiation is discussed. The coupling of the counter-running waves due to the active medium is determined by the spectral characteristics of the gain line (inhomogeneously broadened line as in the case of a gas laser and homogeneously broadened gain line with spatial saturation grating induced by the standing wave pattern of counter-running waves [6]). (2) The two-time correlation functions, the variance and the irreversible circulation of fluctuation are studied in the vicinity of the critical points. Besides the features of phase transitions such as critical 0378-4363/81/0000-0000/$02.50 O 1981 North-Holland
112
D. Kiihlke and R. Horak / Fluctuations in a ring laser with back-scattering
fluctuations and critical slowing down [7, 8], questions of general interest are connected with the nature of the instabilities. In addition to the conventional soft mode instability familiar at equilibrium there is a hard mode instability which is met only in a system of thermodynamically coupled degrees of freedom far from equilibrium [9, 10]. In the latter the detailed balance does not hold and the irreversible circulation which is proposed by Tomita et al. [9, 10], to be used as a measure of deviation from detailed balance, shows a critical behaviour, too. It is shown in this paper that both soft and hard mode instabilities are met in a ring laser with optical back-scattering. In section 2 a general formalism is given to calculate the two-time correlation matrix of the fluctuations of a non-linear system with several degrees of freedom, provided the fluctuations can be described by linearized Langevin equations. The general formulas for the two-time correlation matrix, the variance and the irreversible circulation allow us, according to the form of the eigenvalues of the drift matrix, to state the behaviour of the system in marginal situations. Model and basic equations of a ring laser with back-reflections, the stationary solutions of the mean values of the amplitudes and the phase difference for equal pump rates of the counter-running waves and their region of stability are recapitulated from [5] in the first part of section 3. In the second part the two-time correlation matrix and the irreversible circulation for the various laser states are calculated. The influence of the coupling of the counter-rotating waves due to the active medium and due to back-reflections on these quantities are discussed in detail especially in the vicinity of marginal points.
2. Two-time correlation functions, variance and irreversible circulation of fluctuation
yj(t) (i = 1. . . . . N) are supposed to be a set of fluctuating physical quantities obeying the Langevin equations y,,(t) = g,({yj}, t) + F~(t).
(2.1)
F~(t) are fluctuating forces with the property (F~(t)~(t')) = D,~(t-
t'),
(F~(t)) = 0,
(2.2)
where D0 is the diffusion matrix Dij = d i s ~ ,
(2.3)
with 8o being the identity matrix and di the noise strength. Here we use the sum convention, i.e. in place of Ig~.l ajb~ we write a~b~. Equal indices for which this convention shall not be valid will be underlined. Under the assumption that the fluctuations around the secular motion of the system are small compared to the mean values of the physical quantities, i.e. 18,(01 = ly,(t) - (y~(t))l "~ [(yi(t))],
(2.4)
eq. (2.1) can be linearized with respect to the fluctuation [11]. From eq. (2.1) we obtain the equations of
D. Kiihlke and R. Horak / Fluctuations in a ring laser with back-scattering
113
motion of the mean values X~(t) = (y~(t)): = g,({X/}, t),
i = 1. . . . . N ,
(2.5)
and the evolution equations of the fluctuations 8~(t)= y ~ ( t ) - X ~ ( t ) : 6,(t) = K~j({Xk}, t)8/(t) + F~(t),
i = 1. . . . . N ,
(2.6)
where
=0g, is the drift matrix. By means of (2.4) the two-time correlation functions of the second (amplitude correlations) and fourth (intensity correlations) order, respectively, can be approximated by G~])(t, t') = ((y,(t) - (y,(t)))(yi(t') - (yj(t')))) - (8,(t)Sj(t')) -- o'~j(t, t'),
(2.7)
G[~)(t, t') = ((y,(O2 - (y,(t))2)(yi (t')2_ (yj(t,))2)) __ 4X i (t)X~(t')(Si(t)S/(t')).
(2.8)
They are essentially determined by the two-time correlation matrix of the fluctuations o'~j(t', t ) = (Si(t')Sj(t)) which is considered in the following. Using the formal integral of (2.6) with the initial condition 8~(0)= 0,
8,(0= (exp(f' X(s)ds)),jf' du (exp(- i 0
0
Y((v)dv))/kFk(U ) ,
(2.9)
0
where Yt' denotes the matrix with the elements K~, the correlation matrix of the fluctuations can be written as I+'/'
i
t
0
t u
t
×o..
, 10, u
where relation (2.2) have been used. The tilde " denotes the transposed matrix. When the stationary solutions of eqs. (2.5) are considered (.~ -- 0), the drift matrix becomes time-independent and eq. (2.10) can be further simplified. If the eigenvalues and the eigenvectors of the drift matrix are known, ~ can be transformed in the diagonal form At8 u = (U-1)aKtmUmj.
(2.11)
114
D. Kiihlke and R. Horak / Fluctuations in a ring laser with back-scattering
A~ are the eigenvalues and the columns of the transformation matrix q/, the eigenvectors of ~. Taking into account that
(U-')l. (e~).,.U.k = e*l~k, eq. (2.10) can be written as trij(t + r, t) = Uik eAk'B~k(t)(O)~j,
(2.12)
where
B..(t) = (U-'),,D~k(O-'),~
eC*-+~o) ' - 1 A m + h.o
(2.13)
This representation of the two-time correlation matrix of fluctuations allows us to draw some conclusions. (1) The stationary solutions of (2.5) are stable if Re Aj < 0 for all i = 1. . . . . N. In this case the correlation matrix (2.12) tends to its stationary value o'~i(r) for t--* %
cr,j(r) = U,, e * " B k . ( t - * = ) ( 0 ) . ~ ,
(2.14)
where B m , ( t - - , oo) = - - ( U - ' ) m t D l k ( O - x ) k ~ ( A ~
+ A~)-'.
(2.15)
For r = 0 we obtain the second moment or the variance o-~(0) - ~r~j from (2.14), trij
=
U~kBk,(t ~ oo)(O),j .
(2.16)
(2) A zero real part of at least one eigenvalue leads to the points of marginal stability. Passing through such critical points corresponds to the first or second kind phase transitions [8]. In the vicinity of the critical points typical phenomena occur which are obvious from (2.12)-(2.16). For Re Ai ~ 0, both the correlation time and the relaxation time of the system, which are proportional to (Re A;)-1, tend to infinity corresponding to critical slowing down. On the other hand, Re Ai--* 0 leads to a divergency of the matrix elements of the variance (see eq. (2.16)) corresponding to critical fluctuations. If a real eigenvalue is vanishing in the vicinity of the critical point one eigensolution (fluctuating mode) corresponding to Ai becomes unstable and dominates the whole system at the marginal situation (soft mode instability). From eq. (2.13), for & ~ 0 we obtain lim B~(t) = (U-~)itDa((J-l)kit,
Ai..~O
(2.17)
which makes it evident that at the marginal point the elements of the two-time correlation matrix are growing linearly in time. If the eigenvalues are complex in the vicinity of the critical point, there always exist a couple of conjugate eigenvalues, since X is real, denoted by Ai and A, = A*, where Re ai = 0
D. Kiihlke and R. Horak / Fluctuations in a ring laser with back-scattering
115
determines the marginal point. In this case the term A, + At = 2 Re Ai in eqs. (2.13) and (2.15) leads to divergencies of the correlation time, the life time of the fluctuations and the elements of the variance; that means that in the vicinity of the marginal point two fluctuating modes corresponding to At and A,, respectively, become unstable. The transition has essentially a two-dimensional character (hard mode instability). Due to the complex exponents in formulas (2.13) and (2.14), the elements of the two-time correlation matrix oscillate with a frequency proportional to Im At for varying t and ~-. (3) The antisymmetric matrix a,j = ~(o',,( g ) o -
K,~r,j)
(2.18)
has been shown to be a measure of the deviation from the detailed balance [9, 10]. It characterized a directed flux in a thermodynamically coupled system of more than one degree of freedom and has, therefore, been termed as the irreversible circulation of fluctuation [9]. Inserting (2.16) into (2.18) one obtains OtO"= ½[Ujn((U)mjAm)
- -
(2.19)
(UinAn)(U)mj]Bnr# (t --, oo).
In the case of a real eigenvalue At ~ 0 at the marginal situation, i.e. the soft mode instability, the irreversible circulation remains finite (the summands of (2.19) where m = n = i are zero because the square bracket vanishes and the other summands remain finite) and plays no important role compared to the diverging variance. In the special case of ~ = ~-1 which implies 0//= 0//-1, the irreversible circulation vanishes identically. In the case of complex eigenvalues occurring only for the non-symmetric ~, i.e. the hard mode instability, the matrix elements of the irreversible circulation diverge, too. Let A1, Az = AT be a couple of complex conjugate eigenvalues with Re AI~ 0. Then the summand of eq. (2.19) which contains hi, A2 takes the form
-½(U/l(~r)2.~2- U/1/~I(O)2j)(/~ 2 + i~l)-l((u-1)llDik(O-1)k2)
=½Uil(O)2j(U-1)l~Dtk(O-1)k2ImAJRehl--)o~,
for
Re AI-*0.
That means that, unlike the soft mode instability, the irreversible circulation diverges and the detailed balance does not hold. It is worth noting that these divergencies are a non-physical outcome of the linear approximation, but they serve as an indication of unusually large fluctuations in the vicinity of the marginal points.
3. Fluctuations in a ring laser with optical back-scattering
3.1. Basic equations and stationary solutions To take into account the influence of optical back-scattering, the Langevin equations for the single-frequency ring laser [2] are extended by terms describing the reflection from one direction of propagation into the other (see [5] and references therein). Under the assumption that the fluctuations are caused by the spontaneous emission the influence of back-reflections on the fluctuating forces can be
D. Kiihlke and R. Horak / Fluctuations in a ring laser with back-scattering
116
neglected: ~
- [ a l - a(l
d +
1
12)1 i -
=
(3.1)
El(t),
(3.2) /3t (i = 1, 2) are the complex amplitudes of the counter-running waves, • describes their coupling due to the active medium and, depending on the spectral characteristic of the active medium, can take the following values: 0 < ~ ~< 1 for the Doppler-broadened gain line, 1 ~< e ~<2 for the homogeneously broadened gain line with spatial saturation grating, and in the intermediate case, 0 ~< ~ ~< ~ where ~ > 1 [2]. a~ = g 0 - K~- r~ is the pump parameter, go the linear gain of the active medium, rj are the cavity losses, d is the saturation parameter, ~ = r~ exp(-i0~) with r~ being the coupling coefficient due to back-reflections and 0~ the phase of the back-reflected radiation. The fluctuating forces have the property
(F~(t')F'~(t)) = 4 q S : ( t - t'),
(E(t)Fj(t')} = 0 ,
(Fdt)) = 0 ,
(3.3)
where q is the noise strength. Inserting/3j(t) = Ej(t) exp(-icj(t)) into (3.1) and (3.2), with Ei(t) and ~0~(t) being the real amplitudes and phase of the counter-running waves, respectively, and taking the real and imaginary part we obtain, with the notations $ = 02 - 0~ - ~', ~0 = ¢2 - ,~ + 01 + ½or, f~l - [al - d ( E ~
(3.4)
+ ~E~)]E1 - E2r sin ~o = F l ( t ) ,
/~2 - [az - d(E~ + ~E~)]E2 + E l f sin(¢ + 0) = F2(t),
(3.5)
?0 + r[(EJE2) cos(¢ + 0) - (E2/E1)cos ¢] = F3(t),
(3.6)
where 16L2= Re(F1.2 e i~.2) ,
1 a63 = E1 Im(Fl e i~') -
Ira(F2 e i ~2).
H e r e we have specified rl = r2 = r. If the fluctuations are small enough, eqs. (3.4)-(3.6) can be linearized around the mean values (Ei.2(t)) and (¢ (t))
El.2(t) = (E1.2(t)) + 81.2(0,
~ (t) = (~ (t)) + 83(0.
(3.7)
8,(0
(i = 1, 2, 3) are small deviations caused by the noise source. Inserting (3.7) into (3.4)-(3.6), retaining only the terms linear in 8t and adopting the notations Xl.2(t)= (El.z(t)), X3(t)= (¢(t)) we get: X~ = g~({X~}) = [al - d ( X ] + eX~)]X1 + rX2 sin X3,
(3.8)
Jr2 = g2({X~}) = [a2 - d ( X ~ + eX~)]X2 - rXl sin(X3 + 0 ) ,
(3.9)
X3 = g3({X/}) =
~, = K,flj + £ ,
-r[(Xl/X2) cos(X3 +
0 ) - (X2/Xl) cos X31 ,
(3.10)
(3.11)
117
D. Kiihlke and R. Horak / Fluctuations in a ring laser with back-scattering
where ~r (Ko)= (oggoxj) =
=
at - d(3X$ + eXl) -2deXiX2 - r sin(X3 + O)
r[(x~/x~)cos x3 + (l/X2)cos(X3+ q')l
-2edXiX2 + r sin X3 a2 -- d(3X] + eX~) r[(1/Xl) cos Xa + (Xt/X]) cos(X3 + 0)]
X2r cos X3 -X,r cos(X3+ O)
\
r[-(XzIXl) sin X3 + (XdX2) sin(X3 + qt)]
)
(3.12)
is the drift or regression matrix of the system. Assuming that exp(-i~j) can be absorbed into the fluctuating forces as a new phase factor, the correlations of ~ ( t ) in this approximation become (F/(t')~(t)) = DoS(t- t'),
(3.13)
(F/(t)) = 0 ,
with the diffusion matrix
= (Do) = 2q
(i0 0) 1
0
0
(1/X~+l/X2,)
(3.14)
.
The stationary solutions and their stability as well as the possible limit cycles of eqs. (3.8)-(3.10) are discussed in detail for 0 = 2nor and ~ = (2n + 1)¢r in [5]. Let us recapitulate the steady-states X0i and their region of stability for equal pump rates al = a2 = a. In the following the abbreviations 77 Xo2/Xol and ~"= dX~l will be used. (A) 0 = (2n + 1)~': In this case eqs. (3.8)-(3.10) have the following stationary solutions (77 ~>0): =
'11 = 1 ,
'i'~2'3
For
=
~'1 = a + r l+e '
X o 3 = ~" 2 '
a(•2 r 1)+4(a -
(3.15)
f- 1 ,
a +/'v~2,3 ~2,3 ~ ~ ,
,/7X03 = "~-.
(3.16)
0 ~ • < 1, solution (3.15) is stable if r>0,
a>0
and
a<0,
r > -a,
(3.17)
whereas solution (3.16) is unstable. The laser threshold is given by a = - r . For 1 < • ~ 2 solution (3.15) is stable in the region a>0,
r > r c = ~1a ( • - l ) ;
a<0,
r>-a,
(3.18)
and (3.16) in the region a > 0, (see fig. 1).
1
r < rc = ~a(e - 1),
(3.19)
118
D. Kiihlke and R. Horak / Fluctuations in a ring laser with back-scattering
"7
3
2
1
i
qz
o~
q6 r_.. a
Fig. 1. Steady states of the ring laser for ¢ = (2n + 1)~r and ~ = 2. T h e solid and dashed lines show the stable and unstable range, respectively.
(B) ¢ = 2nw: The stationary solutions of (3.8)-(3.10) are (7 ~ 0): 71 =
1,
~1 = al(~ +
1),
sin X03 = 0,
(3.20)
~. = a - (l/7/)r sin X03
(3.21)
1,/2 + ~
1
X03 = -~-,
+ X03 =
[a(1-,)÷ .l(aX_fl. h=_ 2(, +
72,3=-~[
-~/2,
74,5 :
2r - Y \ 2r / 1 _ ~/(~)2 l+~[a(12;')+
1)l J
2 - 2 ( ¢ + 1)] ,
1/72,3 ,
Contrary to (3.15), the laser threshold is given by a = 0. For 0 ~ • < 1, solution (3.20) is stable in the
range r > 0,
(3.22)
whereas solution (3.21) is unstable. For 1 < • ~ 2, solution (3.20) is unstable and the stability of solutions (3.21) is determined by ~" (72,"2"),
('04,--2)
(compare figs. 2 and 3).
stable for
r < Pc = ~a/ 8( (~~-' 1)
(3.23)
D. Kiihlke and R. Horak / Fluctuations in a ring laser with back-scattering
119
3
,I g-2
To
r0-÷ r
3 I
/ 2
I I
\ /
//
\,..
t"
....
.........i
f.o
q.l,
E(I )*o-0 \
/
a)
&
. . . .
r
1
A
i
0,5
t
t,5
f
Q
Fig. 2. Steady states for 0 = 2mr and • = 2 as well as • < 1 (the unstable stationary solutions for • < 1 are omitted).
Fig. 3. Steady states for q J = 2 m r depending on the parameter e, which describes the coupling of the counterrunning waves due to the active medium, for a constant ratio of the back-scattering coefficient to the p u m p rate (r/a = 0.05).
In the following section the two-time correlation matrix and the irreversible circulation will be calculated for these steady states.
3.2. Two-time correlations, variance and irreversible circulation of fluctuations (A) ~, = (2n + 1)~': First solution (3.15) is considered. The drift matrix is obtained by inserting (3.15) into (3.12):
- ~
1
(2a + r(e +3))
- ~ ( 21 • a + r ( • - l ) ) 1+•
1 (2ca + r ( a - 1)) - ~ 1 (2a + r(• + 3)) l+a l+a
(Kq)= 0
0
\
0 0 -2r
)
(3.24)
The drift matrix (3.24) has the eigenvalues Az= - 2 r ,
A 2 = - 2 ( a + r),
Xa=
1 +2 • ( a ( 1 - e)+2r).
(3.25)
From the eigenvectors of (3.24) we get the transformation matrix
=
-1 o
, x/2
(3.26)
D. Kiihlke and R. Horak / Fluctuations in a ring laser with back-scattering
120
for which the relation ~ / = o~ = ~d-1 holds. Putting (3.24)-(3.26) and (3.14) with (3.15) into (2.12) yields the two-time correlation matrix with the following elements:
r)[h,(t, ~)+(a+
o'11(t + % t ) = (Tm(t + r, t ) = 4 ( 2 +
r ) ( e + 1)h2(t, ~')] 2(r - re)
tr21(t + z' t)= (T12(t+ z' t)= 4(2+ r) [hl(t' r ) - ( a + r)(e - re)
(T33(t+ r, t) = qd(e + 1) . 2r(a + r) [1 -
1)
(3.27)
h2(t, z)] ,
(3.28)
e - " ] e -2"
(3.29)
0"13(t + ~', t) = (T23(t + r, t) = (T3,(t + r, t) = (T32(t + r, t) = O,
(3.30)
where
hi(t, "r) =
[1 - e x p ( - 4 ( a + r)t)] e x p ( - 2 ( a + r)~-),
h2(t,r)=[1-exp(-S(r-r¢)t)]exp(-4(r-r~)~'~ 1+~
1+~
]'
and rc = a(~ - 1)/2. Within the ranges of stability 0 ~< ~ < 1, r > 0 and 1 < ~ ~< 2, r > re, we obtain the variance from (3.27)-(3.29) by t--* oo and r = 0
((T0) : 8(a +
rq)(r- re)
I
2a + r(e + 3)
- ( 2 a ~ + r(E - 1))
- ( 2 a ~ + r(e - 1)) 0
2a + r(~ + 3) 0
o
)
0
(3.31)
4d(r - ro)(e + 1)/r
T h e irreversible circulation vanishes identically. A b o v e all it is obvious from (3.31) that for e > 1 the cross m o m e n t s o-n, (TEl are negative, i.e. the fluctuations of the counter-running waves are anticorrelated (anti-bunching). T h e strength of the anti-correlation increases with increasing ~. Within the range 0 ~< • < 1, the back-scattering counter-acts the anti-correlation of the intensity fluctuations: if r < 2a~/(1 - ~), O"12 and O"21 are negative but the a m o u n t decreases with increasing r; if r = 2 a e / ( 1 - ~), o"12 and o'21 vanish, i.e. the intensity fluctuations of the counter-running waves are uncorrelated; if r > 2 a ~ / ( 1 - e), o'12 and (T2t are positive and the fluctuations are correlated. T h e fluctuations of the intensities and the phase difference are uncorrelated. At the critical point r = 0, 0 ~< e < 1, only the variance of the phase-difference fluctuations diverges. T h a t reflects the fact that the solution rt = 1 does not change for r ~ 0 but the phase locking is abolished. This b e c o m e s evident if we take the limit r ~ 0 in the formulas for the two-time correlations (3.27)-(3.29) (Tn(t + ~', t) = o'22(t + ~', t) = ~
xexp-
( (I-
(T33(t + % t) = {2qd(~ +
e+l
1)/a}t.
[/(1 - e x p ( - 4 a t ) ) e x p ( - 2 a ' r ) + , ~ + 1 \( 1 _ e x p (\ _ 2 (e1)-a+t ' ~1 ,
]]
I.
]J'
(3.32) (3.33)
D. Kiihlke and R. Horak / Fluctuations in a ring laser with back-scattering
121
Eq. (3.33) describes the well-known phase-diffusion which does not influence the stability of the steady state of the amplitudes (t ~ o0 in (3.32) leads to the stationary autocorrelation functions of the intensity fluctuations). In the range 1 < ~ ~< 2, solution (3.15) is stable for r > re. At the critical point r = r e all those elements of the variance diverge which are associated with the fluctuations of the amplitudes. The phase difference does not change on passing over to the new solution (3.16) and the variance of their fluctuations remains finite. The behaviour at the marginal point r = - a , a < 0 is analogous. T h e drift matrix for solution (3.16) which is symmetric, too, has the following eigenvalues A1.2=-(a+rc)±
}1/2,
4a
(a+r~)2---~(r2-r 2)
A3 = - 2 r o .
(3.34)
Because of the lengthy formulas only the variance as well as the two-time correlation matrix at the critical point are given. T h e variance is
12ar~AA(r- 2-r r2(a2)- re) r(r~ + a) (~,;) q r2~- r 2
r(rc r ~ -+r a) 2 2r2A + r2(a - r¢) A(r~- r 2) 0 -
0
0 0
2ar2d
)
'
(3.35)
where A = rc ___~/r~--~--~ (+ sign belongs to */2 and - to ~73). T h e irreversible circulation vanishes identically. T h e intensity fluctuations are anti-correlated and show critical behaviour at r = re, whereas the variance of the phase-difference fluctuations remains finite. T h e two-time correlations of the intensities at the critical point r = rc are of the same form as in the case of solution (3.15) , / = 1 for r = re; only the magnitude of the phase-difference fluctuations is different tr11(t (12) + r, t) = trz2(t (21) + r, t) = 4(a q+ re) {[1 -
e -4(a+rc)`]e-2(a+'°)"+ 4(a (-)
o'33(t + r, t) = qda (1 - e -4'°') e -2'o" 2re
+
re)t}, (3.36)
Resuming, we can state that in the case of qt = (2n + 1)~-, the critical points corresponding to second kind phase transitions are characterized by a soft m o d e instability. All eigenvalues of the drift matrix are real within the range of stability. At the critical points only one of the eigenvalues becomes zero and this m o d e dominates the whole system. The irreversible circulation as a measure of deviation from the detailed balance vanishes throughout. (B) ~ = 2nrr: First solution (3.20) is considered. The drift matrix belonging to this solution is
Yl=
2a 1+~
2a, -1+~
2a~
2a
-2"
/
a
r~d(l~E)
2"X/dOf
I----a-- [
°
/
(3.37)
D. Kiihlke and R. Horak / Fluctuations in a ring laser with back-scattering
122
which has the eigenvalues At,2 =
- ¢~ a -----~ / D , - 11 +-"--e
A3 = - 2 a
(3.38)
where
If the eigenvectors of (3.37) are known we can calculate the transformation matrix ~ which is of the form
/rX/a/d(1+') -r~/a/d(l+') i ) (Uo) = ~-rX/a/d(1 + e) r~/a/d(1 + e) ~--t~ 2
(3.39)
I~ 1
From (3.39) 0~, q/-1, q~-i are determined and inserted together with (3.38), (3.39) and (3.14) with (3.20) into (2.12). After some algebra we come to the two-time correlation matrix which has the following elements: crn(t + z, t) = o'z2(t + ~', t) = (+) (12)
e) (/~l(t, r) + ~2(t, r))
(2t)
+ A (--) h3(t, z) -
4r2(1 + e ) ' ^ ( t , "r)+ ~2(t, r))] D(1 - e) tg (
0"13(t + ~', t) = -cr23(/t + ~', t) =
q - 4rD
(3.40)
4 d ( l + e) ~'a(1 - e) [AJ*l(/, ~') + At/~2(t, r)]
[---i--+-~-e
4r2(1 a ' ~ ' - -+E e) [A,~,(t, "r)+ A2~2(t, r)]} ,
o'3,(t + r, t) = -o'32(t + % t) =
•
q - 4rD
4r2( 1 + e) r x ~ , -
o'33(t + % t) =
a(1-e)
4 ~
(3.41)
I'a(1 - ") [A2/~,(t, r) + A,/~2(t,r)]
[ ~
r ) + Al~2(t, ~')]}
(3.42)
t,,ml~,,
qd(14r2D-e) [A 2/~l(t ' ~.) + A~/~2(t, r) -
/4r2(1 \ " h ' ~ - -+- ~~)~2 ' ] (~l(t, ~') + ~2(t, 7))]
where / ~ i ( t , ~ ' ) = [ 1 - e z~'t]e ~,*,
~,(t,~')=
[1 - e x p
-
l+e
t
.
(3.43)
D. Kiihlke and R. Horak / Fluctuations in a ring laser with back-scattering
123
For r > 0, a < 1, where solution (3.20) is stable we get from (3.40)-(3.43) the variance of fluctuation by t ~ o o and ~- = 0,
q
/
3+e 1-~ 1 + 3e 1-e
1+~3 1-~ 3+
2 V a d ( 1 + ~)
- 2 V a d ( 1 + ~)
2 ~ / a d ( l + ¢)
)])
_ 2 ~ / a d ( i + E)
1-E
r
(3.44)
r(1 + ~) 2 4ad(lr2 - ' ) [1 + 2 ( a ( - ~ - ~)
r
Substituting (3.44) and (3.37) in (2.18) leads to the irreversible circulation
(aiD= a ( 1 - ~ )
0
0
1
1
-1
0
.
(3.45)
Analogously to the cases discussed in (A) the intensity fluctuations of the counter-running waves are anti-correlated, o't2 = o'2~< 0 , however, the back-scattering does not influence the magnitude of this anti-correlation. Contrary to the solutions discussed in (A) the cross-correlation of the intensity and the phase-difference fluctuations do not vanish. This is quite clear. The phase difference of the reflected radiation causes that the energy from one sense of propagation is pumped into the other. But the phase difference of the waves counter-rotating in the ring resonator takes such a value that d u e to interferences between the reflected and original waves the energy of the counter-running waves remains equal (even if the pump rates of the two directions are different [1]). Therefore the threshold (see eq. (3.20)), the intensities and the intensity fluctuations do not explicitly depend on the back-scattering coefficient but the cross-correlations of the intensity and phase-difference fluctuations and the variance of the phase-difference fluctuations depend on r. The laser threshold a = 0 (see (3.20)), r = 0 and ~ = 1 (the laser frequency is tuned to the centre of the inhomogeneously broadened gain line) are the critical points. At r = 0 we have a soft mode instability: the circulation vanishes and one mode dominates the system. A different behaviour shows the system in the vicinity of the critical point ~ = 1 corresponding to a first kind phase transition. For r >2~a(1 - e)/(1 + e),
(3.46)
a couple of the eigenvalues (3.38) are complex: AI.2 = a(1 - ~)/(1 + ~ ) ± i g2,
(3.47)
with
12
.]4r2_ {a(1 - ~)~2 ¥
/
In this region the elements of the two-time correlation matrix oscillate with the frequency 12. The
D. Kiihlke and R. Horak I Fluctuations in a ring laser with back-scattering
124
stationary elements o'11(~-)= 0-~0") and (12)
O'33('1") of the two-time correlation matrix are given for example.
(21)
Taking the limit t -* oo for (a - 2r)/(a + 2r) < e < 1, we obtain from (3.40) and (3.43) with (3.47) 0-11(~.) = (12)
0.22(,)= (21)
+ (_)~_~.
•
q d ( 1 - e) exp(_ ( 1 - e)ar~
:
÷
? + ~ a ) cos £2~-(-)exp(-2a~')
1+,
,{[1
.
_[r(1 +,)\2"1
]
a(1-,)
/
(1 + , ) O
(3.48)
,
sin O~'}
"
(3,49)
At the marginal point e = 1 the diagonal elements of the variance (3.44) as well as the cross-moments of the amplitudes diverge. The corresponding elements of the two-time correlation matrix (3.40) and (3.43) increase with time t: 1 (1 0-11(t + r, t ) = 0-z2(t + z, t ) = ( +- ) q [ t cos(2r'r) ( +- ) ~--da 02)
e_,at) e -~" ] ,
(3.50)
(21)
0-33(t + r, r) = qd(1 + E) t cos(2r'r). a
(3.51)
Contrary to the cases considered up to now the irreversible circulation (3.45) does not vanish but diverges, too. This marginal situation is associated with a hard mode instability: A couple of eigenvalues become complex, both the irreversible circulation and the variance of fluctuation diverge and the elements of the two-time correlation matrix oscillate with a frequency determined by the imaginary part of that couple of complex conjugate eigenvalues. Beyond this marginal point within the range 1 < e < ¢c with
(compare eq. (3.23)), there is no stable stationary state (see fig. 3) and the system tends to a limit cycle [51. An analogous behaviour shows the system in the vicinity of the laser threshold a = 0, ¢ < 1 corresponding to a second kind phase transition. For a < 2r(~ + 1)/(1 - ~),
(3.52)
the couple of eigenvalues AL2 (3.38) are complex, too, and, contrary to the usual lasers, the threshold of the ring laser with optical back-scattering of a phase difference ~ = 2n~" is characterized by a hard mode instability. The state beyond this hard mode instability is given by X1 -~ X2 -- 0 (a ~<0), i.e. there is no limit cycle. For these two cases the detailed balance does not hold. The irreversible circulation as a measure of the deviation of the detailed balance diverges at ~ = 1 and a = 0. This becomes evident, too, from the two-time correlation matrix (3.40)-(3.43) which is not symmetric even in the stationary case 0-13(t + ~', t) = (Sl(t + r)83(t)) ~ 0-31(t + r, t) = (83(t + r)81(t)) .
(3.53)
D. Kiihlke and R. Horak / Fluctuations in a ring laser with back-scattering
125
The condition of detailed balance is equivalent to requiring a symmetric stationary two-time correlation matrix (see, e.g. [12]). Since the formulas of the two-time correlation matrix according to solution (3.21) are very extensive this case is qualitatively discussed for the marginal situation Po= a ( e - 1)/~/8-(~ + 1), ~ > 1. The eigenvalues of the drift matrix (3.12) belonging to solution (3.21) at the critical point r = rc are A1 =
2(~a+ 1) (~2 + 2~ + 5) ,
A2 = 0 ,
A3 = -½a(~ - 1).
(3.54)
As the eigenvalues are real at this critical point, corresponding to a first kind phase transition, we have the case of a soft mode instability. Only one of the eigenvalues is zero and this unstable mode dominates the system. According to the general statements of section 2, the irreversible circulation remains finite and, therefore, is unimportant. However, it is worth noting that beyond this soft mode instability there exists no stable stationary state (see fig. 2) and the ring laser tends to a limit cycle [5]. This contradicts the expectation that the newly emerging state is expected to be a mode which is asymptotically stable [10].
4. Concluding remark For a nonlinear system of several degrees of freedom, the fluctuations around the secular motion of the system have been calculated in linear approximation. In the stationary case the expressions for the two-time correlation functions, the variance and the irreversible circulation of fluctuation allow us to draw conclusions on the behaviour of the system at marginal situations. Besides phenomena closely resembling phase transitions, such as critical fluctuations and critical slowing down, we are also interested in the types of instabilities. Essentially there are two typical cases, i.e. the soft mode instability familiar in thermodynamic equilibrium and the hard mode instability met only in dissipative systems. The latter is associated with periodic variation in time of the two-time correlations and the critical behaviour of the irreversible circulation as a measure of deviation from detailed balance. On this basis the single-frequency ring laser with optical back-scattering has been treated as a system of three degrees of freedom. The dependence of the two-time correlation matrix, the variance and the irreversible circulation on the coupling of the counter-running waves, due to the active medium which is described by the coupling parameter ~, and on the strength r and the phase ~p of the back-reflected radiation has been discussed and their behaviour at marginal situations considered. The coupling by the active medium is determined by its spectral characteristic (inhomogeneously broadened gain line, 0~<~ ~ 1; homogeneous broadening with spatial saturation grating, 1 ~<~ ~<2; and the intermediate case, 0 ~ • ~< ~: where ~: > 1). In all cases the elements of the two-time correlation matrix consist of several exponentials. The back-scattering influences the cross-correlation of the fluctuations in different ways. In the case of a phase difference of the back-reflected radiation of ¢, = (2n + 1)~" and 0 ~< ~ < 1, it counteracts the anti-correlation (anti-bunching) of the intensity fluctuations of the counter-rotating waves and, if the back-reflection coefficient r exceeds a certain value, the cross-correlation becomes positive. In the case of 0 = (2n + 1)zr and 1 < ~ ~< 2, the magnitude of the negative crosscorrelation depends strongly on r only in the vicinity of the marginal point r-- rc and shows a critical behaviour there. For 0 = 2n~r and 0 ~< ~ < 1, the anti-correlated intensity fluctuations do not depend on r.
126
D. Kiihlke and R. Horak / Fluctuations in a ring laser with back-scattering
For ~b = (2n + 1)¢r, all transitions between different stable states correspond to the second kind phase transitions. The irreversible circulation vanishes throughout and the instabilities occurring there are characterized by a soft mode instability. The fluctuations of the intensities and the phase difference are uncorrelated. On the contrary, the corresponding cross-correlation functions do not vanish in the case of 0 = 2nor. For an inhomogeneously broadened gain line, i.e. E < 1 and a non-vanishing back-scattering coefficient r, we have marginal situations at the laser threshold (vanishing pump-parameter) corresponding to a second kind phase transition, and at ~ = 1 (laser frequency tuned to the line centre) we have a situation corresponding to a first kind phase transition. Both instabilities are characterized by a hard mode instability, the irreversible circulation diverges at the critical points and the two-time correlation functions vary periodically in time. But only the state above E = 1, which can occur in the intermediate case 0 ~< 6 < 6, tends to a limit cycle. The critical point r = 0 associated with a second kind phase transition corresponds to a soft mode instability. In the case of a homogeneously broadened gain line 1 < • ~< 2 the marginal point r = rc (re: critical back-scattering coefficient), corresponding to a first kind phase transition, is associated with a soft mode instability. But despite that, the state beyond this instability tends to a limit cycle.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
M.M. Tehrani and L. Mandel, Phys. Rev. A17 (1978) 677, 694. D. Kiihlke and R. Horak, Opt. Quant. Elect. 11 (1979) 485. S. Singh and L. Mandel, Phys. Rev. A20 (1979) 2459. E.B. Rockower and N.B. Abraham, Optica Acta 26 (1979) 1289. D. Kiihlke and G. Jetschke, Physica 106C (1981) 287. J.B. Hambenne and M. Sargent, Phys. Rev. A13 (1976) 784. H. Haken, Rev. Mod. Phys. 47 (1975) 67. A. Nitzan, P. Ortoleva, J. Deutch and J. Ross, J. Chem. Phys. 61 (1974) 1056. K. Tomita and H. Tomita, Prog. Theor. Phys. 51 (1974) 1731. K. Tomita, T. Ohta and H. Tomita, Prog. Theor. Phys. 52 (1974) 1744. R.L. Stratonovich, Topics in the Theory of Random Noise, Vol. II (New York, 1967). S.R. de Groot, Thermodynamics of Irreversible Processes (North-Holland, Amsterdam, 1951).