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Pitchfork bifurcation, crisis, and flip-flop operation in a two-element optical bistable device
Volume 152, number 1,2
PHYSICS LETTERS A
7 January 1991
Pitchfork bifurcation, crisis, and flip-flop operation in a two-element optical bistable device Ching-Sheu Wang a Jyh-Long Chern b Jow-Tsong Shy Yih-Shun Gou a and John K. Mclver b b
Institute ofOptical-Electronics, National Chiao-Tung University, Hsinchu, Taiwan 30050, ROC Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM 87131, USA Department ofPhysics, National TsingHua University, Hsinchu, Taiwan 30043, ROC
Received 24 April 1990; revised manuscript received 29 August 1990; accepted for publication 8 November 1990 Communicated by A.P. Fordy
Flip-flop operation and the pitchfork bifurcation in a coupled element optical bistable device are studied. In the long delay limit, the dynamics is described by a two-dimensional mapping. Within a pitchfork bifurcation structure, a flip-flop operation is found. However, this flip-flop operation cannot be realized when a boundary crisis appears due to the increase of the input intensity.
All-optical flip-flop operation is a key function for ultrafast optical signal processing. In normal optical bistable devices [1], bistability is realized through an “S-shaped” hysteresis. However, the switching time from up to down will include the switchingtime of the bias laser because there is no “negative” optical pulse to switch down. Therefore, utilizing such an S-shaped hysteresis limits possible application on ultrafast optical signal processing. Recently Otsuka proposed an all-optical flip-flop operation utilizing the pitchfork bifurcation structure in a two-element optical bistable device and showed that several ultrafast all-optical binary logic functions can be realized [2,3]. Because of its potential applications, it is worthwhile to further investigate the connection between the bifurcation structure and the flip-flop operation in all-optical systems. The analysis by Otsuka is based on a short transit time, or short delay limit. Because there is a delay in a real device, it must be considered. In general, a delay will increase the dimension of the system, and may also destabilize the possible final steady states, i.e. it may destroy the pitchfork bifurcation structure which has been used to realize the desired all-optical flip-flop operation. In this paper, we study the flip-flop operation of Otsuka but with a long delay approximation. The dy0375-960l/91/$ 03.50 © 1991
—
namics is then described by a two-dimensional mapping. In the mapping form it is not obvious that within a pitchfork bifurcation structure a flip-flop operation can be realized for the output. Instead such a structure may exhibit a periodic oscillation similar to that which occurs in the logistic mapping [4,5] and the one-dimensional Ikeda mapping [61. The dynamics that occurs in the short delay limit, which is described by a set of differential equations [21, and the long delay limit are expected to be dramatically different. Unlike the one-dimensional mapping case, it will be shown that the flip-flop operation can be created when the system is derived within the regime where a pitchfork bifurcation structure exists. As the input intensity increases, the desired flip-flop operation may disappear even though the system is still within the pitchfork bifurcation structure. The termination ofthe flip-flop operation is related to the onset of a boundary crisis which is generated by the collision between a chaotic attractor and the coexisting unstable orbit [7]. Such a feature is dramatically different from that exhibited in the short delay limit [21. Following the approach of Otsuka and Ikeda [81, the dynamical equations of a ring cavity containing
Elsevier Science Publishers B.V. (North-Holland)
21
Volume 152, number 1,2
PHYSICS LETTERS A
coupled nonlinear elements as shown in fig. 1 can be described as follows [2,3]:
two
El(l)=A!
+BE2(t—T)
exp{i[Ø2(t)+Ø0]}
,
E2(t)=A2+BE~(t—T)exp{i[01(t)+Øo]}, rØ~(t)=—Ø(t)+ E1(t—T)H, T02(t)=
—02(t)+
1E2(l—T)
(1)
2
02(1) =
7 January 1991
I E2 ( t T) —
(2)
2
After rescaling the time by the delay T and considering a small coupling (or a large dissipation), i.e. B<< 1, and A~,B=O(l),eqs. (2) can be further reduced to a two-dimensional mapping as follows: 2cos[Ø Ø,(n+l)=11+2B(I12)” 2(n)+Ø0] 2cos[Ø Ø2(n+ 1 )=12+2B(1211 )“ 1(n)+Ø0] (3) 2 and ~ (m=l, 2). Symbolically can write eqs. (3) as X(n+l)=F(X(n)). Here we Om= IEmI The stationary solutions X” of eqs. (3) obey r=F(r), and period-n stationary solutions obey ~ (~)=F(F(...F(r)...)). Linear stability analysis shows that the period-n stationary solutions will be stable when they satisfy detiJi <1 with J,~=ôF(n)(X~”)/öX~.To describe a real optical device with such a coupled mapping requires justifi.
where A.,w E
2
1,{(l—R)icn2[l—exp(_~,I)/ii]}~ (m= 1,2) is the normalized amplitude of the incident bias laser light from the mth channel and B (RR 2x 0)’~’ exp( —p1/2) indicates the coupling between the elements. Here E,~is the electric field of the incident bias laser light from the mth channel, ‘i is the absorption rate, n 2 is the nonlinear refractive-index coefficient, R and R0 are the reflectivities of the mirrors M~,M3 and M2, M4 respectively, K is the wave number and I is the medium length. Meanwhile, the variable 0,,, (m = 1, 2) denotes the phase shift across the mth medium, 0,,, is the time derivative of 0~~~ Oo is the linear phase across each element, t is the medium response time, and T=L/c (2L is the total length of the ring cavity, c is the velocity of light) is the transit time through each element. The differential terms in eqs. (1) can be omitted in the long delay limit, i.e. t/T—*0. In this case, the dynamical equations obey
cation as it does for the one-element case [9,10]. Nevertheless, a recent study suggests that such a mapping approach can provide information about the dynamics in a more realistic situation [11]. First we will consider the system at the same bias intensities, i.e. i~ = ‘2 = Fig. 2 follows from eqs. (3), for B=0.2, Oo= —it and with the condition that the last value of iteration for a control parameter is the ~.
10
E~(t)=A~ +BE2(t—T) exp{i[02(t)+0o]}
2
_
E2(t)=A2+BE1(l—T) Oi(t)= 1E1(t—T)1
5
/ I
Ei.
I
Ia
M
1J~______ 1
/11 I
exp{i[Ø~(t)+Øo]}
~
lb
.2
0
2
3.220
I 4.793
6I
8I
10
Bias Intensity 1:11 12 M 4
M3
Fig. 1. The conceptual geometry of a two-element optical bistable device.
22
Fig. 2. Final outputs 0,,, versus bias intensity I (=Ii =12) with B=0.2 and Oo= —it and under a slow input-sweeping process. The same figure can be obtained using equal initial conditions, e.g. 0~(0)‘~02(0) 0, for each!.
Volume 152, number 1,2
PHYSICS LETTERS A
initial value of the next parameter and the number of time steps of the iteration is large enough to obtam the final states, i.e. the system is under a slowly sweeping input process. Here, 0~and 02 are equal at any bias intensities j=ji =12. In fact, this situation is exactly equivalent to the one-dimensional Ikeda mapping and the transition scenario is following a period-doubling cascade. However, as the input intensities ‘=11=12 biased within the period-doubling regime, the stationary solution of the system shows a pitchfork bifurcation under proper initial condition for 0 and 02, i.e. the symmetric solution (0l=02) bifurcates to an asymmetrical solution. In the short delay limit [2,3] when the system is biased within the first pitchfork bifurcation regime, the possibility that each element can be set to the upper or lower state of the bifurcated branch is dependent on the negligible unbalance in I or Oo for two elements. For the mapping approach, the output of the system depends on the initial values Ø~(0)and 02(0). There are two modes for the outputs of the system, one is the asymmetrical solution that each element stays at the upper or lower state separately and the other is a “hopping” solution in which the outputs of both channels are the same and hop between upper and lower states together. These two modes are distinguished by the initial conditions of 0~(0) and 02(0). When the initial condition is 0l(0)=02(0)=0, the system output is always in the hopping mode. Such a phenomenon is a general feature of a higher dimensional mapping [12]. In order to investigate the ‘2flip-flop operation, 2) versus (Ii) when Il (Ia)we i5 plot Om (i.e. Em! fixed at a value within the first bifurcated regime of fig. 2. A typical case for I~(‘2) =4.5 is shown in fig. 3, where the points U and L correspond to the upper and lower states at 1= 4.5. If the system is in the “hopping” mode, i.e. Ø~and 02 are the same and Osciliate, we can set one element to the upper state by applying a trigger pulse with proper intensity and duration to the other channel. This is shown in fig. 4. In fig. 5 it is shown that switchingbetween the upper and lower states (i.e. flip-flop operation) can be realized by applying a trigger optical pulse to the element which is in the lower state. This can be understood by looking at fig. 3. Notice that in fig. 3, in the region 5<12 (1 ) <6 the hysteresis of the output is
7 January 1991 10
(a)
-
unstable stable
~--
u / N
~
5
/ //
~
- -
~ I
I
0
2
I
i2u
4
I
)6
8
10
10
(b)
U
~~-._--~ A
5
/3
- - -
I
-~
,‘‘
----~‘.--“
~1 I 00
‘ I
10
Fig. 3. Final outputs versus ‘2 (Ii) where I (12)=4.5; (a) 02 (0) versus 12 (Ii); (b) Oi (02) versus 12 (I~); B= 0.2 and 00= — It.
connected with a portion of the period-doubling cascade. Fig. 3 also indicates that an all-optical flip-flop operation is possible when the two bias intensities are different and ‘2 (Ii) is within the hysteresis region. A stronger coupling can make the width of the hysteresis loop larger and therefore relaxing the constraints on the performance of an all-optical flip-flop operation. A larger bias intensity I~(Ii) also increases the width of the hysteresis loop for switching. In fact, when the two bias intensities are different and within the hysteresis region, the behavior of the outputs of the coupled system are similar to that at the same bias intensities. Both “asymmetrical” and 23
Volume 152, number 1,2
trigger I
PHYSICS LETTERS A
7 January 1991
struction and creation of chaotic attractors with their basins occurs, and the crisis observed here belongs to the boundary crisis [7]. The typical transition scenario of crisis for I~(‘2) =4.8 is shown in fig. 6. The chaotic boundary intersects with the unstable orbit at points a and b. When the input intensity is
~
trIgger 2
increased to a, the coexisting three different attractor basins suddenly collapse into a single attractor basin, i.e. the strange attractor of the upper (or lower) branch is destroyed. The chaotic attractor is
~ (fl)
LiJ1~
~
2(n)
5020
5000
5040
TIme Step n Fig. 4. The system outputs 0 (02) change from hopping mode into the lower (upper) state with a trigger pulse. The intensity of the trigger is 1.5 and its duration is 9 iterative cycles. Here the bias intensity is 4.5.
interrupted before the input intensity is increased to b. At point b, the strange attractor of the upper (or lower) branch suddenly appears and the system recovers to coexisting three different attractors. It is interesting to note that the hysteresis structure is interrupted by the crisis and the all optical flip-flop operation of this coupled system is not possible. The attractors and their corresponding basins for .
(1k) biased just below the crisis of sudden destruction of chaotic attractors are shown in figs. 7a and 7b respectively. Fig. 7a shows that the system ex‘2
tn gen I g
____________________________________
trigger 2
I
___________________________________ 5000
5200
5400
TIme
5600
5800
6000
Step n
Fig. 5. Flip-flop operation through a trigger pulse injection; assuming B=0.2, Oo= —it, and the bias intensity is 4.5. The intensity of the trigger is 1.5 and its duration is 9 iterative cycles.
“hopping” solutions are possible depending on the initial conditions of Ø~(0)and 02(0); however, the upper and lower states of one output are different from the other output. When the input intensity 11 (Ia) is set between the first bifurcation regime of fig. 2 and beyond 4.793, a collision between a chaotic attractor and coexisting unstable orbit can be observed. Therefore, as the bias intensity ‘2 (Ii) is increased, the crisis of sudden de24
hibits three attractors, attractorA corresponds to both channels staying in the chaotic branches, attractor L corresponds to the hopping solution and attractor P corresponds to both channels converging to a stable fixed point. Actually, the attractor of L is the lowestdimensional spatio-temporal intermittency [13]. As the bias intensity ‘2 (Ii) reaches a point such that the attractors A and L touch the basin of attractor P the boundary crisis happens and both attractors A and L are destroyed. It is worthwhile to note that when the bias intensity 1~(12) is increased beyond a certain value between 4.805 and 4.81 the boundary crisis of sudden creation of chaotic attractors will be replaced by a tangent bifurcation (i.e. type-I intermittency) as shown in fig. 8. Figs. 8c and 8d also show that the period-3 window will not be strained out by the boundary crisis since its boundary does not collide with the coexisting unstable orbit. The width of the bias intensity between the destruction and creation of the chaotic attractors of the upper (or lower) branch is shown in fig. 9. From this figure, it is obvious that the width is linearly proportional to the bias intensity when the bias intensity is beyond the point that the crisis of creation is replaced by the intermittency. In summary, in the region ~< ‘a (= 3.22) shown
Volume 152, number
I,2
PHYSICS
0: 5.5
6.0
Fig. 6. Final output versus I2 (I, ) with I, (I*) ~4.8, B~0.2 b.
6.5
LETTERS
7 January
A
,L 5.5
I99 1
6.0
and o= -rt, where the chaotic boundary
intersects
the unstable
orbit at a and
_ (a) a6-
I
0
I
2
I
I
I
I
6
4
I
I
a
I
IO
+, (0)
Fig. 7. The final outputs ofthe system oz versus @I (a) and their basin (b) at I, = 4.8 and I,= 6.14. (a) The system exhibit three attractors, attractor A corresponds to both channels staying in the chaotic states, attractor L corresponds to the hopping solution and attractor P corresponds to both channels converging to a stable fixed point. (b) The initial value space, (g, (0), o*(O) ), is divided as 200-200 regular grid between the region of 0 d @,( 0) (and oz (0) ) d 10. q denotes the basin of A, denotes the basin of hopping solution (L) and 0 denotes the basins of P, respectively. One should note that the basin diagram is periodic in o, and & with a period of 271.
in fig. 2, there is only the symmetrical solution; when Z, I,,, there is no flip-flop operation using a positive optical pulse. In the region Z>Z,, a boundary crisis can be found within a bifurcation diagram as shown in fig. 6. It is to be noted that the hopping phenomena can still exist provided that the requirement of initial condition of @,(0) and &( 0), e.g. the same values as discussed above, is matching. When the
system is biased within a later pitchfork bifurcation regime, e.g. the second one, a periodic oscillation will develop in the outputs which will lead to a perioddoubling process as the bias intensity further increases. Similar results have also been found in the following two-dimensional coupled mappings: (i) coupled logistic mapping:
25
Volume 152, number
1.2
PHYXCS
5.5
LETTERS
A
7 January
I991
6.5
6.0
I,U,) IO,
I
6
6
+p#*) 4
0 5.5
6.0
65
I,U,)
I,(I,)
Fig.8.Finaloutputversus12
w 0
(I,)whereI,
4.60
Bias
26
relation
(I*)=431
for (a)and
(b),andI,
(1,)=4.82for(c)and
(d).
0.20
9 -d .?I s= 0.00 4.75
Fig. 9. The functional
6.5
6.0
5.5
4.65
4.90
Intensity
4.95
Il(or
5.00
I*)
5.05
for the width of the blank region AII, (or Al, ) versus bias intensity
5.
I, (or Iz) with B=0.2
and qJ,= -II’.
Volume
152, number
1,2
Y(n+l)=l-I,X(n)2; (ii) coupled element optical bistable Guassian transmission function: X(n+l)=L,
PHYSICS
(4) device with
exp{-a[Y(n)-X0]‘},
Y(n+1)=~2exp{-a[X(n)-Y0]2}.
(5)
The second coupled mapping is used to describe the coupled element optical device formed by a hybrid optical bistable system incorporating an oscilloscope
[141. In this paper, the connection between the pitchfork bifurcation structure and the possibility of flipflop operation has been studied in a two-dimensional mapping which is used here as an approximation for a two-element optical bistable device with long delay time. For this mapping system, a pitchfork bifurcation structure does not ensure the flip-flop operation. The complexity that is found is different from the usual mappings, e.g. the logistic map and the one-dimensional Ikeda mapping [ 6 1. The differences are partially induced by the increased dimension of the system [ 121 and partially by the characteristics of the system considered. The occurrence of flip-flop operation for a pitchfork bifurcation in the long delay limit suggests that the desired flip-flop operation for ultra-fast optical signal processing may be realized when a finite delay exists in the device. To what extent the phenomena predicated here can be found in a more realistic approach is an unresolved question. Finally the reduction of the coupled delay-differential equations (eqs. ( 1) ) to the coupled two-dimensional mapping (eqs. (3 ) ) involves a singular limit. As mentioned before, to describe a real optical device with such a coupled mapping requires further justification. Nevertheless, the studies for single element bistable devices show that most of the results for long delay limit (i.e. T/r-+m) survive for T/ 72 10 [ 1O]. It seems worthwhile to confirm the predictions of the coupled mapping model using real systems with finite delay. Two possible approaches are proposed. A coupled hybrid electro-optic optical
LETTERS
A
7 January
199
1
bistable device can be set up to describe eqs. ( 1) and the long delay ( TB 7) can be realized by using a microcomputer delay line [ 15 ] or a long fiber loop [ 161. Another approach is to simulate the coupled delay-differential equations by electronic circuits [ 171 with a microcomputer delay line [ 151. An experiment using the second approach is currently pursued in our laboratory. C.-S. Wang, J.-T. Shy and Y.S. Gou would like to thank Drs. Y.H. Kao and J.C. Huang for useful discussions and Mr. N.-P. Chen for computer simulations. We also thank Dr. K. Otsuka for his kindness and preprints. This work was partially supported by the National Science Council of the Republic of China.
References [ I] H.M. Gibbs, Optical bistability:
controlling light with light (Academic Press, New York, 1985). [ 2 ] K. Otsuka, Electron. Lett. 24 ( 1988) 800. [3] K. Otsuka, Opt. Lett. 14 (1989) 72. (41 H.G. Schuster, Deterministic chaos (Physik-Verlag, Weinheim, 1984). [ 51 P. Berge, Y. Pomeau and C. Vidal, Order within chaos (Wiley, New York, 1984). [6] K. Ikeda, Opt. Commun. 30 (1979) 257. [ 7 ] C. Grebogi, E. Ott and J.A. Yorke, Physica D 7 ( 1983) 18 1. [8] K. Otsuka and K. Ikeda, Opt. Lett. 12 (1987) 599; Phys. Rev. Lett. 59 (1987) 194; Phys. Rev. A 39 (1989) 5209. [9] M.L. Berre, E. Ressayre, A. Tallet and H.M. Gibbs, Phys. Rev. Lett. 56 (1986) 274; R. Vallee and C. Marriott, Phys. Rev. A 39 ( 1989) 197. [IO] P. Nardone, P. Mandel and R. Kapral, Phys. Rev. A 33 (1986) 2465. [ 1 I ] F. Kaiser and D. Merkle, Phys. Lett. A I39 ( 1989) 133. [ 121 J.-L. Chem and J.K. McIver, Phys. Lett. A 142 (1989) 99. [ 131 H. Chat& and P. Manneville, Physica D 32 ( 1988) 409. [ 141 C.-S. Wang, C.-S. Lau, J.-T. Shy and Y.S. Gou, in preparation. [ I 5 ] H.M. Gibbs, F.A. Hopf, D.L. Kaplan and R.L. Shoemaker, Phys. Rev. Lett. 46 ( 198 1) 474; F.A. Hopf, D.L. Kaplan, H.M. Gibbs and R.L. Shoemaker, Phys. Rev. A 25 ( 1982) 2 172. [ 161 M.W. Derstine, H.M. Gibbs, F.A. Hopf and D.L. Kaplan, Phys. Rev. A 27 ( 1983) 3200. [ 17 ] Y.H. Kao, J.C. Huang and Y.S. Gou, J. Low Temp. Phys. 63 (1986) 287.
27
PHYSICS LETTERS A
7 January 1991
Pitchfork bifurcation, crisis, and flip-flop operation in a two-element optical bistable device Ching-Sheu Wang a Jyh-Long Chern b Jow-Tsong Shy Yih-Shun Gou a and John K. Mclver b b
Institute ofOptical-Electronics, National Chiao-Tung University, Hsinchu, Taiwan 30050, ROC Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM 87131, USA Department ofPhysics, National TsingHua University, Hsinchu, Taiwan 30043, ROC
Received 24 April 1990; revised manuscript received 29 August 1990; accepted for publication 8 November 1990 Communicated by A.P. Fordy
Flip-flop operation and the pitchfork bifurcation in a coupled element optical bistable device are studied. In the long delay limit, the dynamics is described by a two-dimensional mapping. Within a pitchfork bifurcation structure, a flip-flop operation is found. However, this flip-flop operation cannot be realized when a boundary crisis appears due to the increase of the input intensity.
All-optical flip-flop operation is a key function for ultrafast optical signal processing. In normal optical bistable devices [1], bistability is realized through an “S-shaped” hysteresis. However, the switching time from up to down will include the switchingtime of the bias laser because there is no “negative” optical pulse to switch down. Therefore, utilizing such an S-shaped hysteresis limits possible application on ultrafast optical signal processing. Recently Otsuka proposed an all-optical flip-flop operation utilizing the pitchfork bifurcation structure in a two-element optical bistable device and showed that several ultrafast all-optical binary logic functions can be realized [2,3]. Because of its potential applications, it is worthwhile to further investigate the connection between the bifurcation structure and the flip-flop operation in all-optical systems. The analysis by Otsuka is based on a short transit time, or short delay limit. Because there is a delay in a real device, it must be considered. In general, a delay will increase the dimension of the system, and may also destabilize the possible final steady states, i.e. it may destroy the pitchfork bifurcation structure which has been used to realize the desired all-optical flip-flop operation. In this paper, we study the flip-flop operation of Otsuka but with a long delay approximation. The dy0375-960l/91/$ 03.50 © 1991
—
namics is then described by a two-dimensional mapping. In the mapping form it is not obvious that within a pitchfork bifurcation structure a flip-flop operation can be realized for the output. Instead such a structure may exhibit a periodic oscillation similar to that which occurs in the logistic mapping [4,5] and the one-dimensional Ikeda mapping [61. The dynamics that occurs in the short delay limit, which is described by a set of differential equations [21, and the long delay limit are expected to be dramatically different. Unlike the one-dimensional mapping case, it will be shown that the flip-flop operation can be created when the system is derived within the regime where a pitchfork bifurcation structure exists. As the input intensity increases, the desired flip-flop operation may disappear even though the system is still within the pitchfork bifurcation structure. The termination ofthe flip-flop operation is related to the onset of a boundary crisis which is generated by the collision between a chaotic attractor and the coexisting unstable orbit [7]. Such a feature is dramatically different from that exhibited in the short delay limit [21. Following the approach of Otsuka and Ikeda [81, the dynamical equations of a ring cavity containing
Elsevier Science Publishers B.V. (North-Holland)
21
Volume 152, number 1,2
PHYSICS LETTERS A
coupled nonlinear elements as shown in fig. 1 can be described as follows [2,3]:
two
El(l)=A!
+BE2(t—T)
exp{i[Ø2(t)+Ø0]}
,
E2(t)=A2+BE~(t—T)exp{i[01(t)+Øo]}, rØ~(t)=—Ø(t)+ E1(t—T)H, T02(t)=
—02(t)+
1E2(l—T)
(1)
2
02(1) =
7 January 1991
I E2 ( t T) —
(2)
2
After rescaling the time by the delay T and considering a small coupling (or a large dissipation), i.e. B<< 1, and A~,B=O(l),eqs. (2) can be further reduced to a two-dimensional mapping as follows: 2cos[Ø Ø,(n+l)=11+2B(I12)” 2(n)+Ø0] 2cos[Ø Ø2(n+ 1 )=12+2B(1211 )“ 1(n)+Ø0] (3) 2 and ~ (m=l, 2). Symbolically can write eqs. (3) as X(n+l)=F(X(n)). Here we Om= IEmI The stationary solutions X” of eqs. (3) obey r=F(r), and period-n stationary solutions obey ~ (~)=F(F(...F(r)...)). Linear stability analysis shows that the period-n stationary solutions will be stable when they satisfy detiJi <1 with J,~=ôF(n)(X~”)/öX~.To describe a real optical device with such a coupled mapping requires justifi.
where A.,w E
2
1,{(l—R)icn2[l—exp(_~,I)/ii]}~ (m= 1,2) is the normalized amplitude of the incident bias laser light from the mth channel and B (RR 2x 0)’~’ exp( —p1/2) indicates the coupling between the elements. Here E,~is the electric field of the incident bias laser light from the mth channel, ‘i is the absorption rate, n 2 is the nonlinear refractive-index coefficient, R and R0 are the reflectivities of the mirrors M~,M3 and M2, M4 respectively, K is the wave number and I is the medium length. Meanwhile, the variable 0,,, (m = 1, 2) denotes the phase shift across the mth medium, 0,,, is the time derivative of 0~~~ Oo is the linear phase across each element, t is the medium response time, and T=L/c (2L is the total length of the ring cavity, c is the velocity of light) is the transit time through each element. The differential terms in eqs. (1) can be omitted in the long delay limit, i.e. t/T—*0. In this case, the dynamical equations obey
cation as it does for the one-element case [9,10]. Nevertheless, a recent study suggests that such a mapping approach can provide information about the dynamics in a more realistic situation [11]. First we will consider the system at the same bias intensities, i.e. i~ = ‘2 = Fig. 2 follows from eqs. (3), for B=0.2, Oo= —it and with the condition that the last value of iteration for a control parameter is the ~.
10
E~(t)=A~ +BE2(t—T) exp{i[02(t)+0o]}
2
_
E2(t)=A2+BE1(l—T) Oi(t)= 1E1(t—T)1
5
/ I
Ei.
I
Ia
M
1J~______ 1
/11 I
exp{i[Ø~(t)+Øo]}
~
lb
.2
0
2
3.220
I 4.793
6I
8I
10
Bias Intensity 1:11 12 M 4
M3
Fig. 1. The conceptual geometry of a two-element optical bistable device.
22
Fig. 2. Final outputs 0,,, versus bias intensity I (=Ii =12) with B=0.2 and Oo= —it and under a slow input-sweeping process. The same figure can be obtained using equal initial conditions, e.g. 0~(0)‘~02(0) 0, for each!.
Volume 152, number 1,2
PHYSICS LETTERS A
initial value of the next parameter and the number of time steps of the iteration is large enough to obtam the final states, i.e. the system is under a slowly sweeping input process. Here, 0~and 02 are equal at any bias intensities j=ji =12. In fact, this situation is exactly equivalent to the one-dimensional Ikeda mapping and the transition scenario is following a period-doubling cascade. However, as the input intensities ‘=11=12 biased within the period-doubling regime, the stationary solution of the system shows a pitchfork bifurcation under proper initial condition for 0 and 02, i.e. the symmetric solution (0l=02) bifurcates to an asymmetrical solution. In the short delay limit [2,3] when the system is biased within the first pitchfork bifurcation regime, the possibility that each element can be set to the upper or lower state of the bifurcated branch is dependent on the negligible unbalance in I or Oo for two elements. For the mapping approach, the output of the system depends on the initial values Ø~(0)and 02(0). There are two modes for the outputs of the system, one is the asymmetrical solution that each element stays at the upper or lower state separately and the other is a “hopping” solution in which the outputs of both channels are the same and hop between upper and lower states together. These two modes are distinguished by the initial conditions of 0~(0) and 02(0). When the initial condition is 0l(0)=02(0)=0, the system output is always in the hopping mode. Such a phenomenon is a general feature of a higher dimensional mapping [12]. In order to investigate the ‘2flip-flop operation, 2) versus (Ii) when Il (Ia)we i5 plot Om (i.e. Em! fixed at a value within the first bifurcated regime of fig. 2. A typical case for I~(‘2) =4.5 is shown in fig. 3, where the points U and L correspond to the upper and lower states at 1= 4.5. If the system is in the “hopping” mode, i.e. Ø~and 02 are the same and Osciliate, we can set one element to the upper state by applying a trigger pulse with proper intensity and duration to the other channel. This is shown in fig. 4. In fig. 5 it is shown that switchingbetween the upper and lower states (i.e. flip-flop operation) can be realized by applying a trigger optical pulse to the element which is in the lower state. This can be understood by looking at fig. 3. Notice that in fig. 3, in the region 5<12 (1 ) <6 the hysteresis of the output is
7 January 1991 10
(a)
-
unstable stable
~--
u / N
~
5
/ //
~
- -
~ I
I
0
2
I
i2u
4
I
)6
8
10
10
(b)
U
~~-._--~ A
5
/3
- - -
I
-~
,‘‘
----~‘.--“
~1 I 00
‘ I
10
Fig. 3. Final outputs versus ‘2 (Ii) where I (12)=4.5; (a) 02 (0) versus 12 (Ii); (b) Oi (02) versus 12 (I~); B= 0.2 and 00= — It.
connected with a portion of the period-doubling cascade. Fig. 3 also indicates that an all-optical flip-flop operation is possible when the two bias intensities are different and ‘2 (Ii) is within the hysteresis region. A stronger coupling can make the width of the hysteresis loop larger and therefore relaxing the constraints on the performance of an all-optical flip-flop operation. A larger bias intensity I~(Ii) also increases the width of the hysteresis loop for switching. In fact, when the two bias intensities are different and within the hysteresis region, the behavior of the outputs of the coupled system are similar to that at the same bias intensities. Both “asymmetrical” and 23
Volume 152, number 1,2
trigger I
PHYSICS LETTERS A
7 January 1991
struction and creation of chaotic attractors with their basins occurs, and the crisis observed here belongs to the boundary crisis [7]. The typical transition scenario of crisis for I~(‘2) =4.8 is shown in fig. 6. The chaotic boundary intersects with the unstable orbit at points a and b. When the input intensity is
~
trIgger 2
increased to a, the coexisting three different attractor basins suddenly collapse into a single attractor basin, i.e. the strange attractor of the upper (or lower) branch is destroyed. The chaotic attractor is
~ (fl)
LiJ1~
~
2(n)
5020
5000
5040
TIme Step n Fig. 4. The system outputs 0 (02) change from hopping mode into the lower (upper) state with a trigger pulse. The intensity of the trigger is 1.5 and its duration is 9 iterative cycles. Here the bias intensity is 4.5.
interrupted before the input intensity is increased to b. At point b, the strange attractor of the upper (or lower) branch suddenly appears and the system recovers to coexisting three different attractors. It is interesting to note that the hysteresis structure is interrupted by the crisis and the all optical flip-flop operation of this coupled system is not possible. The attractors and their corresponding basins for .
(1k) biased just below the crisis of sudden destruction of chaotic attractors are shown in figs. 7a and 7b respectively. Fig. 7a shows that the system ex‘2
tn gen I g
____________________________________
trigger 2
I
___________________________________ 5000
5200
5400
TIme
5600
5800
6000
Step n
Fig. 5. Flip-flop operation through a trigger pulse injection; assuming B=0.2, Oo= —it, and the bias intensity is 4.5. The intensity of the trigger is 1.5 and its duration is 9 iterative cycles.
“hopping” solutions are possible depending on the initial conditions of Ø~(0)and 02(0); however, the upper and lower states of one output are different from the other output. When the input intensity 11 (Ia) is set between the first bifurcation regime of fig. 2 and beyond 4.793, a collision between a chaotic attractor and coexisting unstable orbit can be observed. Therefore, as the bias intensity ‘2 (Ii) is increased, the crisis of sudden de24
hibits three attractors, attractorA corresponds to both channels staying in the chaotic branches, attractor L corresponds to the hopping solution and attractor P corresponds to both channels converging to a stable fixed point. Actually, the attractor of L is the lowestdimensional spatio-temporal intermittency [13]. As the bias intensity ‘2 (Ii) reaches a point such that the attractors A and L touch the basin of attractor P the boundary crisis happens and both attractors A and L are destroyed. It is worthwhile to note that when the bias intensity 1~(12) is increased beyond a certain value between 4.805 and 4.81 the boundary crisis of sudden creation of chaotic attractors will be replaced by a tangent bifurcation (i.e. type-I intermittency) as shown in fig. 8. Figs. 8c and 8d also show that the period-3 window will not be strained out by the boundary crisis since its boundary does not collide with the coexisting unstable orbit. The width of the bias intensity between the destruction and creation of the chaotic attractors of the upper (or lower) branch is shown in fig. 9. From this figure, it is obvious that the width is linearly proportional to the bias intensity when the bias intensity is beyond the point that the crisis of creation is replaced by the intermittency. In summary, in the region ~< ‘a (= 3.22) shown
Volume 152, number
I,2
PHYSICS
0: 5.5
6.0
Fig. 6. Final output versus I2 (I, ) with I, (I*) ~4.8, B~0.2 b.
6.5
LETTERS
7 January
A
,L 5.5
I99 1
6.0
and o= -rt, where the chaotic boundary
intersects
the unstable
orbit at a and
_ (a) a6-
I
0
I
2
I
I
I
I
6
4
I
I
a
I
IO
+, (0)
Fig. 7. The final outputs ofthe system oz versus @I (a) and their basin (b) at I, = 4.8 and I,= 6.14. (a) The system exhibit three attractors, attractor A corresponds to both channels staying in the chaotic states, attractor L corresponds to the hopping solution and attractor P corresponds to both channels converging to a stable fixed point. (b) The initial value space, (g, (0), o*(O) ), is divided as 200-200 regular grid between the region of 0 d @,( 0) (and oz (0) ) d 10. q denotes the basin of A, denotes the basin of hopping solution (L) and 0 denotes the basins of P, respectively. One should note that the basin diagram is periodic in o, and & with a period of 271.
in fig. 2, there is only the symmetrical solution; when Z,
system is biased within a later pitchfork bifurcation regime, e.g. the second one, a periodic oscillation will develop in the outputs which will lead to a perioddoubling process as the bias intensity further increases. Similar results have also been found in the following two-dimensional coupled mappings: (i) coupled logistic mapping:
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Volume 152, number
1.2
PHYXCS
5.5
LETTERS
A
7 January
I991
6.5
6.0
I,U,) IO,
I
6
6
+p#*) 4
0 5.5
6.0
65
I,U,)
I,(I,)
Fig.8.Finaloutputversus12
w 0
(I,)whereI,
4.60
Bias
26
relation
(I*)=431
for (a)and
(b),andI,
(1,)=4.82for(c)and
(d).
0.20
9 -d .?I s= 0.00 4.75
Fig. 9. The functional
6.5
6.0
5.5
4.65
4.90
Intensity
4.95
Il(or
5.00
I*)
5.05
for the width of the blank region AII, (or Al, ) versus bias intensity
5.
I, (or Iz) with B=0.2
and qJ,= -II’.
Volume
152, number
1,2
Y(n+l)=l-I,X(n)2; (ii) coupled element optical bistable Guassian transmission function: X(n+l)=L,
PHYSICS
(4) device with
exp{-a[Y(n)-X0]‘},
Y(n+1)=~2exp{-a[X(n)-Y0]2}.
(5)
The second coupled mapping is used to describe the coupled element optical device formed by a hybrid optical bistable system incorporating an oscilloscope
[141. In this paper, the connection between the pitchfork bifurcation structure and the possibility of flipflop operation has been studied in a two-dimensional mapping which is used here as an approximation for a two-element optical bistable device with long delay time. For this mapping system, a pitchfork bifurcation structure does not ensure the flip-flop operation. The complexity that is found is different from the usual mappings, e.g. the logistic map and the one-dimensional Ikeda mapping [ 6 1. The differences are partially induced by the increased dimension of the system [ 121 and partially by the characteristics of the system considered. The occurrence of flip-flop operation for a pitchfork bifurcation in the long delay limit suggests that the desired flip-flop operation for ultra-fast optical signal processing may be realized when a finite delay exists in the device. To what extent the phenomena predicated here can be found in a more realistic approach is an unresolved question. Finally the reduction of the coupled delay-differential equations (eqs. ( 1) ) to the coupled two-dimensional mapping (eqs. (3 ) ) involves a singular limit. As mentioned before, to describe a real optical device with such a coupled mapping requires further justification. Nevertheless, the studies for single element bistable devices show that most of the results for long delay limit (i.e. T/r-+m) survive for T/ 72 10 [ 1O]. It seems worthwhile to confirm the predictions of the coupled mapping model using real systems with finite delay. Two possible approaches are proposed. A coupled hybrid electro-optic optical
LETTERS
A
7 January
199
1
bistable device can be set up to describe eqs. ( 1) and the long delay ( TB 7) can be realized by using a microcomputer delay line [ 15 ] or a long fiber loop [ 161. Another approach is to simulate the coupled delay-differential equations by electronic circuits [ 171 with a microcomputer delay line [ 151. An experiment using the second approach is currently pursued in our laboratory. C.-S. Wang, J.-T. Shy and Y.S. Gou would like to thank Drs. Y.H. Kao and J.C. Huang for useful discussions and Mr. N.-P. Chen for computer simulations. We also thank Dr. K. Otsuka for his kindness and preprints. This work was partially supported by the National Science Council of the Republic of China.
References [ I] H.M. Gibbs, Optical bistability:
controlling light with light (Academic Press, New York, 1985). [ 2 ] K. Otsuka, Electron. Lett. 24 ( 1988) 800. [3] K. Otsuka, Opt. Lett. 14 (1989) 72. (41 H.G. Schuster, Deterministic chaos (Physik-Verlag, Weinheim, 1984). [ 51 P. Berge, Y. Pomeau and C. Vidal, Order within chaos (Wiley, New York, 1984). [6] K. Ikeda, Opt. Commun. 30 (1979) 257. [ 7 ] C. Grebogi, E. Ott and J.A. Yorke, Physica D 7 ( 1983) 18 1. [8] K. Otsuka and K. Ikeda, Opt. Lett. 12 (1987) 599; Phys. Rev. Lett. 59 (1987) 194; Phys. Rev. A 39 (1989) 5209. [9] M.L. Berre, E. Ressayre, A. Tallet and H.M. Gibbs, Phys. Rev. Lett. 56 (1986) 274; R. Vallee and C. Marriott, Phys. Rev. A 39 ( 1989) 197. [IO] P. Nardone, P. Mandel and R. Kapral, Phys. Rev. A 33 (1986) 2465. [ 1 I ] F. Kaiser and D. Merkle, Phys. Lett. A I39 ( 1989) 133. [ 121 J.-L. Chem and J.K. McIver, Phys. Lett. A 142 (1989) 99. [ 131 H. Chat& and P. Manneville, Physica D 32 ( 1988) 409. [ 141 C.-S. Wang, C.-S. Lau, J.-T. Shy and Y.S. Gou, in preparation. [ I 5 ] H.M. Gibbs, F.A. Hopf, D.L. Kaplan and R.L. Shoemaker, Phys. Rev. Lett. 46 ( 198 1) 474; F.A. Hopf, D.L. Kaplan, H.M. Gibbs and R.L. Shoemaker, Phys. Rev. A 25 ( 1982) 2 172. [ 161 M.W. Derstine, H.M. Gibbs, F.A. Hopf and D.L. Kaplan, Phys. Rev. A 27 ( 1983) 3200. [ 17 ] Y.H. Kao, J.C. Huang and Y.S. Gou, J. Low Temp. Phys. 63 (1986) 287.
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