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Scaling properties of switching pulses
Volume
55, number
SCALING
4
15 September
OPTICS COMMUNICATIONS
PROPERTIES
OF SWITCHING
1985
PULSES
Paul MANDEL Uniuersitk LIbre de Bruxelles, Received
18 March
Campus Plaine, CP 231, Brussels 1050, Belgium
1985
We investigate analytically the switching process of a bistable system which is described near each limit point by a single ordinary nonlinear differential equation. For a holding beam near the limit point and a rectangular switching pulse, we prove that the requirement of switching imposes a constraint only on the pulse area. This results holds for bistability in general and is applied to purely dispersive optical bistability.
1. Introduction Because optical bistability arises from physical nonlinearities, it yields nonlinear equations [ 11. They are usually not soluble exactly except in steady state. Therefore the dynamical (i.e., time-dependent) properties have to be studied numerically. Quite often, however, the experimental conditions in which observations are made suggest the use of asymptotic methods. The two best known methods are the good and bad cavity limits which allow for the adiabatic elimination of all but one dependent variable [ 11. An alternative scheme [2] was proposed in absorptive optical bistability with fully developed hysteresis when all decay rates are of the same order of magnitude. In this case adiabatic elimination schemes are still possible near each limit point though the remaining variables is not the same near the upswitching and downswitching points. These asymptotic methods lead to equations of the form Zt = Z% Z, P) 3
(1)
where z is the remaining physical variable which rules the time evolution of the system, Z is the input intensity and p is the parameter which characterises bistability. Typically bistability will occur for p larger than some critical value pc. In absorptive optical bistability, p is the b&ability parameter C and pc = 4. In purely dispersive optical bistability p is proportional to the detuning and pc = 3112. 0 0304018/85/$03.30 0 Elsevier Science Publishers B.V (North-Holland Physics Publishing Division) .
To study the switching process we must integrate (1) which is often not possible except in the simplest case of absorptive optical bistability [3,4]. We therefore propose the following approximate method. When p > pc there exists a unique scaling x=G~(z,P),
p=G2Kp),
(2)
which will map any of the two limit points onto the (1,l) point in the plane (x, p). For p Spc we may approximate the resulting curve by a single parabola. For the upswitching process it is the limit point of the lower branch in the bistable curve which is centered at (1,l) in the scaled plane and the upper branch goes to infinity. The equation for the parabolic approximation is x,=x2-2xQl. For the and the
(3)
the downswitching process it is the limit point of upper branch which is at (1,l) in the scaled plane the lower branch becomes zero. The equation for parabolic approximation is
x,=-x2+2/U-l.
(4)
2. Upswitching pulse In this section we analyse eq. (3). Its stationary lutions are (see the upper part of fig. 1):
so-
293
15 September 1985
OPTICS COMMUNICATIONS
Volume 55, number 4
Here t*(O,iI)is the jump duration when the system is initially at x = /J = 1. Hence there is a critical time T(E, a) = t*(.c, Q) - t*(O,Cl)which is the time necessary to bring the system from its initial state up to the point of instability where it can only switch up. A switching pulse of amplitude. p1 - p. must last longer than T(e, L-L).Conversely a nonswitching perturbation of amplitude /.Q - /.I~ must last less than T(E, LZ). The corresponding critical pulse area is A, = (/$ -&))T(E,
a)=
e[1+(2e2/3G)+O(&)]. (11)
Fig. 1. Steady state solutions and pulse evolution in scaled plane for upswitching process.
x,=1+(1-~)1~?
(5)
A linear stability analysis shows that 0
/.l=/+)
t>O:
/.l=/_Q> 1.
x(0)=x_(&)),
This describes a situation in sity maintains the device on in the absence of additional pulse has an amplitude rl(l of (3) with (6) is x(t)=(l
The relevant property ofA, is that, to dominant order in E, it is equal to e: In other terms the only constraint on the minimum switching pulse is its area and not its duration or amplitude separately. Such a property of a pulse area scaling had already been noticed by Hopf and Meystre in a numerical study of gaussian switching pulses for absorptive optical bistability in the good cavity limit [5].
3. Downswitching pulse We now analyse eq. (4). Its stationary (see upper part of fig. 2):
which the holding intenthe stable lower branch perturbation and the p. (fig. 1). The solution
-ia)(a+iQ)-(l+iSZ)(a-iQ)exp(-2iSlt) cw+is1 -(cw-iQ)exp(-2Xl2t)
x, = /J + (/.$ - 1)1/Z .
solutions are
(12)
A linear stability analysis shows that 1
(7)’
where (Y= (1 - po) U2 and a = (/+ - 1)li2. This solution diverges in a finite time t* given by tan2at*
= 2(1 - /.L~)~/~(P, - 1)1/2/~1
- p. - 2). (8)
Since the parabolic approximation is the best near x = j,t = 1, we introduce a smallness parameter e through &)=l-e2,
(9)
with p1 - 1 = O(1). Then t*(e, i-22)= lr/2s2 + (E/&)[
1 - (&/3@)+
O(P)]
. (10)
294
Fig. 2. Steady state solutions and pulse evolution in scaled plane for dowswitching process.
Volume 55, number 4
t=o:
/J=/.Q>l,
OPTICS COMMUNICATIONS
x(O)=x+(l.co), (13)
t>o:
15 September 1985
strength. The other variables have their usual meaning. We first make the following change of variables:
@=&
This corresponds to a situation where the system is initially on the stable upper branch. At t = 0 the holding intensity is suddenly decreased by an amplitude ~1~ - j.tl (fig. 2). It is fairly simple to solve (4) with (13) and to determine the dowswitching time t* for which x(t*) = 0. The result is tan 2m*
c(t(u2-@)t,(~~
= -2G?
i14)
where 52 = (1 - pf)‘j2 and u = x+&o) - /..Q. Here again we concentrate our analysis on a holding intensity which lies near the limit point. Consequently we define a smallness parameter e through /Jo = 1+ e2 )
(15) From(14)and(lS)we
derive
+ 2)/4~1(1 -PI)+
o(e3).
The critical pulse area which separates switching and nonswitching pulses is A, = OctJ- PI) [t*(e, PI) - t*(0, P1 )I
= w)[ 1 + k/a
(3~~ + 2)/p1 + 0 (e2)1 .
l/2
&’
(y=-d K’
-6 e’K+d’
which transforms (17) into
nt = -yn
tT(1
+(u)21x12.
The steady state equation n[l + (n -fJ)2J
(18)
is given by
= b12,
(19)
which displays optical bistability for 0 2 > 3. When this last condition is fulfnled the two limit points are located at n, = 3 [2e + (e2 - 3)lj2] .
t*(e, S2)= n/2Q - (1/2SZ)tan-1(S2/~1) + e/2(1 -~t)+e~(3~~
E(t) >
x,=--K(lto)[l+i@-n)]xt~~,
-s22)
(P; - J22)(&-Q2)-4uf.@
with 1 -$=0(l).
n(t) = AAn(t).
(16)
Again it is found that for E < 1 the dominant contribution to A, is e/2 which expresses a constraint only on the pulse area.
In the limit e2 % 3, the coordinates points are n+ = 013,
IYI: = 4(e/3)3
n_=e,
blZ_ = 8 .
of the limit
,
(20)
To apply the method developed in the previous sections we concentrate on the bad cavity limit in which case the electric field can be adiabatically eliminated: n 7 =-n+ly12/[1+(n-O)2],
7=7t.
(21)
Using (20) and (2 1) we see that the scaling (2) near the upswitching point is 4. Application
n = (e/3)x
As an example we apply the above method to purely dispersive optical bistability which is governed by the equation [6] :
In scaled variables, A, = E. Transforming this result back into the physical variables of eq. (17) leads to
Et=-K[E-Ei/(l An, = -yAn
-R)‘J2] t yn21E12 .
, WI2= 4(e/3)$.
(22)
-SE-dE+iaEAn, (17)
In these equations, An is the field-induced refraction index, E is the complex slowly varying electric field envelope, d measures the linear absorption, 6 is the filled cavity eigenfrequency minus the.external field frequency and a measures the field-matter interaction
Likewise the scaling near the dowswitching n=ex,
point is
l.~1~=0~,
and the critical pulse area becomes in physical variables
295
Volume 55, number 4
OPTJCS COMMUNICATIONS
(23) To sum up, we see that it is possible to set up a very simple procedure which allows for the evaluation of the switching time and therefore for the minimum area of a switching pulse. The method is limited since it requires that one deals with a single differential equation and widely separated upper and lower branches. The parabolic approximation will hold in a domain around x = p = 1 which increases with p - pc > 0. The interest of the procedure is to show that the pulse area scaling law is not related to the detailed physical mechanism underlying a particular type of bistability but is a generic property of S-shaped curves.
Acknowledgement The author is Senior Research Associate with the
296
15 September 1985
FNRS (Belgium). This research has been carried out in the framework of an operation launched by the Commission of the European Community under the experimental phase of the European Community Stimulation Action (1983-1985).
References [ l] L.A. Lugiato, in: Progress in optics, ed. E. Wolf (NorthHolland, Amsterdam, 1984) pp. 71-216. [ 21 T. Erneux and P. Mandel, Phys. Rev. 28A (1983) 896. [ 31 V. Benza and L.A. Lugiato, Lett. Nuovo Cimento 26 (1979) 405. [4] P. Mandel and T. Erneux, Optics Comm. 42 (1982) 362. [5] F.A. Hopf and P. Meystre, Optics Comm. 29 (1979) 235. [6] L.A. Lugiato and R.J. Horowitz, J. Opt. Sot. Am. B, to be published.
55, number
SCALING
4
15 September
OPTICS COMMUNICATIONS
PROPERTIES
OF SWITCHING
1985
PULSES
Paul MANDEL Uniuersitk LIbre de Bruxelles, Received
18 March
Campus Plaine, CP 231, Brussels 1050, Belgium
1985
We investigate analytically the switching process of a bistable system which is described near each limit point by a single ordinary nonlinear differential equation. For a holding beam near the limit point and a rectangular switching pulse, we prove that the requirement of switching imposes a constraint only on the pulse area. This results holds for bistability in general and is applied to purely dispersive optical bistability.
1. Introduction Because optical bistability arises from physical nonlinearities, it yields nonlinear equations [ 11. They are usually not soluble exactly except in steady state. Therefore the dynamical (i.e., time-dependent) properties have to be studied numerically. Quite often, however, the experimental conditions in which observations are made suggest the use of asymptotic methods. The two best known methods are the good and bad cavity limits which allow for the adiabatic elimination of all but one dependent variable [ 11. An alternative scheme [2] was proposed in absorptive optical bistability with fully developed hysteresis when all decay rates are of the same order of magnitude. In this case adiabatic elimination schemes are still possible near each limit point though the remaining variables is not the same near the upswitching and downswitching points. These asymptotic methods lead to equations of the form Zt = Z% Z, P) 3
(1)
where z is the remaining physical variable which rules the time evolution of the system, Z is the input intensity and p is the parameter which characterises bistability. Typically bistability will occur for p larger than some critical value pc. In absorptive optical bistability, p is the b&ability parameter C and pc = 4. In purely dispersive optical bistability p is proportional to the detuning and pc = 3112. 0 0304018/85/$03.30 0 Elsevier Science Publishers B.V (North-Holland Physics Publishing Division) .
To study the switching process we must integrate (1) which is often not possible except in the simplest case of absorptive optical bistability [3,4]. We therefore propose the following approximate method. When p > pc there exists a unique scaling x=G~(z,P),
p=G2Kp),
(2)
which will map any of the two limit points onto the (1,l) point in the plane (x, p). For p Spc we may approximate the resulting curve by a single parabola. For the upswitching process it is the limit point of the lower branch in the bistable curve which is centered at (1,l) in the scaled plane and the upper branch goes to infinity. The equation for the parabolic approximation is x,=x2-2xQl. For the and the
(3)
the downswitching process it is the limit point of upper branch which is at (1,l) in the scaled plane the lower branch becomes zero. The equation for parabolic approximation is
x,=-x2+2/U-l.
(4)
2. Upswitching pulse In this section we analyse eq. (3). Its stationary lutions are (see the upper part of fig. 1):
so-
293
15 September 1985
OPTICS COMMUNICATIONS
Volume 55, number 4
Here t*(O,iI)is the jump duration when the system is initially at x = /J = 1. Hence there is a critical time T(E, a) = t*(.c, Q) - t*(O,Cl)which is the time necessary to bring the system from its initial state up to the point of instability where it can only switch up. A switching pulse of amplitude. p1 - p. must last longer than T(e, L-L).Conversely a nonswitching perturbation of amplitude /.Q - /.I~ must last less than T(E, LZ). The corresponding critical pulse area is A, = (/$ -&))T(E,
a)=
e[1+(2e2/3G)+O(&)]. (11)
Fig. 1. Steady state solutions and pulse evolution in scaled plane for upswitching process.
x,=1+(1-~)1~?
(5)
A linear stability analysis shows that 0
/.l=/+)
t>O:
/.l=/_Q> 1.
x(0)=x_(&)),
This describes a situation in sity maintains the device on in the absence of additional pulse has an amplitude rl(l of (3) with (6) is x(t)=(l
The relevant property ofA, is that, to dominant order in E, it is equal to e: In other terms the only constraint on the minimum switching pulse is its area and not its duration or amplitude separately. Such a property of a pulse area scaling had already been noticed by Hopf and Meystre in a numerical study of gaussian switching pulses for absorptive optical bistability in the good cavity limit [5].
3. Downswitching pulse We now analyse eq. (4). Its stationary (see upper part of fig. 2):
which the holding intenthe stable lower branch perturbation and the p. (fig. 1). The solution
-ia)(a+iQ)-(l+iSZ)(a-iQ)exp(-2iSlt) cw+is1 -(cw-iQ)exp(-2Xl2t)
x, = /J + (/.$ - 1)1/Z .
solutions are
(12)
A linear stability analysis shows that 1
(7)’
where (Y= (1 - po) U2 and a = (/+ - 1)li2. This solution diverges in a finite time t* given by tan2at*
= 2(1 - /.L~)~/~(P, - 1)1/2/~1
- p. - 2). (8)
Since the parabolic approximation is the best near x = j,t = 1, we introduce a smallness parameter e through &)=l-e2,
(9)
with p1 - 1 = O(1). Then t*(e, i-22)= lr/2s2 + (E/&)[
1 - (&/3@)+
O(P)]
. (10)
294
Fig. 2. Steady state solutions and pulse evolution in scaled plane for dowswitching process.
Volume 55, number 4
t=o:
/J=/.Q>l,
OPTICS COMMUNICATIONS
x(O)=x+(l.co), (13)
t>o:
15 September 1985
strength. The other variables have their usual meaning. We first make the following change of variables:
@=&
This corresponds to a situation where the system is initially on the stable upper branch. At t = 0 the holding intensity is suddenly decreased by an amplitude ~1~ - j.tl (fig. 2). It is fairly simple to solve (4) with (13) and to determine the dowswitching time t* for which x(t*) = 0. The result is tan 2m*
c(t(u2-@)t,(~~
= -2G?
i14)
where 52 = (1 - pf)‘j2 and u = x+&o) - /..Q. Here again we concentrate our analysis on a holding intensity which lies near the limit point. Consequently we define a smallness parameter e through /Jo = 1+ e2 )
(15) From(14)and(lS)we
derive
+ 2)/4~1(1 -PI)+
o(e3).
The critical pulse area which separates switching and nonswitching pulses is A, = OctJ- PI) [t*(e, PI) - t*(0, P1 )I
= w)[ 1 + k/a
(3~~ + 2)/p1 + 0 (e2)1 .
l/2
&’
(y=-d K’
-6 e’K+d’
which transforms (17) into
nt = -yn
tT(1
+(u)21x12.
The steady state equation n[l + (n -fJ)2J
(18)
is given by
= b12,
(19)
which displays optical bistability for 0 2 > 3. When this last condition is fulfnled the two limit points are located at n, = 3 [2e + (e2 - 3)lj2] .
t*(e, S2)= n/2Q - (1/2SZ)tan-1(S2/~1) + e/2(1 -~t)+e~(3~~
E(t) >
x,=--K(lto)[l+i@-n)]xt~~,
-s22)
(P; - J22)(&-Q2)-4uf.@
with 1 -$=0(l).
n(t) = AAn(t).
(16)
Again it is found that for E < 1 the dominant contribution to A, is e/2 which expresses a constraint only on the pulse area.
In the limit e2 % 3, the coordinates points are n+ = 013,
IYI: = 4(e/3)3
n_=e,
blZ_ = 8 .
of the limit
,
(20)
To apply the method developed in the previous sections we concentrate on the bad cavity limit in which case the electric field can be adiabatically eliminated: n 7 =-n+ly12/[1+(n-O)2],
7=7t.
(21)
Using (20) and (2 1) we see that the scaling (2) near the upswitching point is 4. Application
n = (e/3)x
As an example we apply the above method to purely dispersive optical bistability which is governed by the equation [6] :
In scaled variables, A, = E. Transforming this result back into the physical variables of eq. (17) leads to
Et=-K[E-Ei/(l An, = -yAn
-R)‘J2] t yn21E12 .
, WI2= 4(e/3)$.
(22)
-SE-dE+iaEAn, (17)
In these equations, An is the field-induced refraction index, E is the complex slowly varying electric field envelope, d measures the linear absorption, 6 is the filled cavity eigenfrequency minus the.external field frequency and a measures the field-matter interaction
Likewise the scaling near the dowswitching n=ex,
point is
l.~1~=0~,
and the critical pulse area becomes in physical variables
295
Volume 55, number 4
OPTJCS COMMUNICATIONS
(23) To sum up, we see that it is possible to set up a very simple procedure which allows for the evaluation of the switching time and therefore for the minimum area of a switching pulse. The method is limited since it requires that one deals with a single differential equation and widely separated upper and lower branches. The parabolic approximation will hold in a domain around x = p = 1 which increases with p - pc > 0. The interest of the procedure is to show that the pulse area scaling law is not related to the detailed physical mechanism underlying a particular type of bistability but is a generic property of S-shaped curves.
Acknowledgement The author is Senior Research Associate with the
296
15 September 1985
FNRS (Belgium). This research has been carried out in the framework of an operation launched by the Commission of the European Community under the experimental phase of the European Community Stimulation Action (1983-1985).
References [ l] L.A. Lugiato, in: Progress in optics, ed. E. Wolf (NorthHolland, Amsterdam, 1984) pp. 71-216. [ 21 T. Erneux and P. Mandel, Phys. Rev. 28A (1983) 896. [ 31 V. Benza and L.A. Lugiato, Lett. Nuovo Cimento 26 (1979) 405. [4] P. Mandel and T. Erneux, Optics Comm. 42 (1982) 362. [5] F.A. Hopf and P. Meystre, Optics Comm. 29 (1979) 235. [6] L.A. Lugiato and R.J. Horowitz, J. Opt. Sot. Am. B, to be published.