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Ibrahim and Al-Naimee Iraqi Journal of Science, 2016, Special Issue, Part B, pp: 328-340
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823
Complex Dynamics in incoherent source with ac- coupled optoelectronic
Feedback
Kejeen M. Ibrahim
1, Raied K. Jamal
1, Kais A. Al-Naimee
1, 2
1
Department of physics, college of science, University of Baghdad, Baghdad, Iraq 2
Istituto Nazionale di Ottica, Largo E.Fermi 6, 50125 Firenze, Italy.
Abstract:
The appearance of Mixed Mode Oscillations (MMOs) and chaotic
spiking in a Light Emitting Diode (LED) with optoelectronic feedback
theoretically and experimentally have been reported. The transition between
periodic and chaotic mixed-mode states has been investigated by varying
feedback strength. In incoherent semiconductor chaotically spiking attractors
with optoelectronic feedback have been observed to be the result of canard
phenomena in three-dimensional phase space (incomplete homoclinic
scenarios).
Keywords: Chaos, light-emitting diodes, Mixed Mode Oscillations,
optoelectronic feedback.
الديناميكيات المعقدة في مصدر غير متشاكه باقتران التغذية الكهروبصرية لمتيار المتناوب2, 1النعيمي, قيس عبد الستار 1, رائد كامل جمال 1*ابراهيم موسى ينژک
1العراق بغداد, جامعة بغداد, كمية العموم, الفيزياء,قسم
، فلورنس، ايطاليا52125 ،6 ، الركو فيرميالوطني للبصرياتالمعهد 1 ،2
الخالصةظهور متذبذات متعدد االنماط و الشواش في الثنائي الباعث لمضوء عن طريق التغذية العكسية
االنتقال بين الحالة النبضية و حالة شواش متعدد االنماط تم تحقيقهقد تم تحقيقه نظريا" وعمميا". الكهروضوئية( الشواش قد تم attractorمن خالل تغيير قوة التغذية العكسية. في شبه الموصل غير المتشاكه جاذب)
( في فضاء طور ثالثي االبعاد.canardمالحظته ليكون نتيجة لظاهرة )
Introduction Oscillatory dynamics in chemical, biological, and physical systems often takes the form of complex
temporal sequences known as mixed-mode oscillations (MMOs) [1]. The MMOs refers as complex
patterns that arise in dynamical systems, in which oscillations with different amplitudes are
interspersed. These amplitude regimes differ roughly by an order of magnitude. In each regime,
oscillations are created by a different mechanism and their amplitudes may have small variations.
These mechanisms govern the transition among regimes [2]. Additionally, MMOs have been observed
in laser systems, LEDs and in neurons [1, 3]. In a single LED, R. Meucci et al.(2012) demonstrate
numerically and experimentally the occurrence of complex sequences of periodic/chaotic Mixed Mode
Oscillations (MMOs) [4].The crucial element to induce MMOs in optoelectronic devices is the
existence of a threshold for light emission. In a LED the main recombination mechanism is the
ISSN: 0067-2904
Ibrahim and Al-Naimee Iraqi Journal of Science, 2016, Special Issue, Part B, pp: 328-340
823
spontaneous emission and the emitted light is simply proportional to the current through the junction.
However, as in electronic diodes, the current – voltage characteristics of a LED are highly nonlinear
and emission/conduction occurs only beyond a threshold voltage [5].In M.P. Hanias et al. (2011)
showed that the LED exposed chaotic behavior even if it works in its operation point, and the obtained
simulation results indicate that the proposed circuit can be used to generate chaotic signal, in a light
emitting manner, useful in code and decode applications. Hanias demonstrated that the simple
externally triggered optoelectronic circuit can be used in order first to generate chaotic voltage signals
and then to control the obtained chaotic signals by varying specific circuit parameters [6].Qualitatively
different mechanisms have been proposed to explain the generation of MMOs in other models [7].
These are: break-up of an invariant torus [8] and break-up (loss) of stability of a Shilnikov homoclinic
orbit [9, 10] subcritical Hopf-homoclinic bifurcation [7]. Hopfbifurcations in fast-slow systems of
ordinary differential equations can be associated with surprising rapid growth of periodic orbits. This
process is referred to as canard explosion [11].However, periodic-chaotic sequences and Farey
sequences of MMOs do not necessarily involve a torus or a homoclinic orbit, but can occur also
through the canard phenomenon [1]. Canards were first found in a study of the van der Pol system
using techniques from nonstandard analysis [12, 13], were first studied in 2D relaxation oscillators
[12, 14]. There, the nature of the classical canard phenomenon is the transition from a small amplitude
oscillatory state (STO) created in a Hopf bifurcation to a large amplitude relaxation oscillatory state
within an exponentially small range of a control parameter. This transition, also called canard
explosion, occurs through a sequence of canard cycles which can be asymptotically stable [7]. The
dynamics of LEDs can be typically described in terms of two coupled variables (intensity and carrier
density) evolving with very different characteristic time-scales. The introduction of a third degree of
freedom describing the AC-feedback loop, leads to a three-dimensional slow-fast system, displaying
complicated bifurcation sequences arising from the multiple time-scale competition between optical
intensity, carriers and the feedback nonlinear filter function. A similar scenario has been recently
observed in semiconductor lasers with optoelectronic feedback [15, 16].
The dynamical model
The simplest approach is to phenomenologically model the LED as an ideal p-n junction with a
uniform recombination region of cross-sectional area S and width ∆. The system dynamics is then
determined by three coupled variables, the carrier (electron) density N, the junction applied voltage Vd,
and the high-pass-filtered feedback voltage Vf, evolving with very different characteristic time scales.
(1)
( ) (2)
(3)
where γsp is the spontaneous emission rate, μ is the carrier mobility, Vbi is a built-in potential ,C is the
diode capacitance (here assumed to be voltage independent for simplicity), V0 is the dc bias voltage, R
is the current-limiting resistor, fF(Vf) is the feedback amplifier function, e is the electron charge, γf is a
cut off frequency, k is the photodetector responsivity , and is the photon density, which is assumed to
be linearly proportional to the carrier density Ǿ = ηN, where η is the LED quantum efficiency.
Equation (1) indicates that N in the active layer decreases due to radiative recombination and increases
with the forward injection current density J=eμN (Vd−Vbi)/∆. Nonradiative recombination and carrier
generation by optical absorption have been neglected since we checked that they do not significantly
change the dynamics. Equation (2) is the Kirchhoff law of the circuit (resistor-ideal diode) relating the
junction voltage Vd to the dc applied voltage V0 [the second term in Eq. (2) is the current flow across
the diode I = JS]. Equation (3) describes the nonlinear feedback loop where the voltage signal coming
from the detector kϕ is high-pass filtered and added to the dc bias through the amplifier function fF(Vf).
Consider just the solitary LED equations (1) and (2). A finite stationary carrier density, increasing
linearly with V0, is found only when Vd> Vth ≡ Vbi + γsp∆2 /μ; otherwise, the only stationary solution is
N=0 and Vd = V0 (zero current). Accordingly, light emission begins when the applied voltage V0
exceeds the threshold voltage Vth. By introducing the dimensionless variables x = eμRSN/∆, y = μ
(Vd−Vbi)/ ∆2γsp, and z = eμRS/kη∆ Vf − x and the time scale ť = γspt, so Eqs. (1), 2 and (3) become [1]:
ẋ = x1*(y1-1) + γ*y1 (4)
ẏ = -γ*(-δ+y1-a*(x1+z1)/ (1+s*(x1+z1)) +x1*y1) (5)
Ibrahim and Al-Naimee Iraqi Journal of Science, 2016, Special Issue, Part B, pp: 328-340
883
ż = -Ԑ*(z1+x1) (6)
x-the photon density, y - carrier density, z - high-pass filtered feedback current
Ԑ-feedback strength, δ-bias current.
Numerical simulations results
The numerical work is achieved by implement a dynamical model (i.e. Al-Naimee model) with the
use of fourth-order Runge-Kutta integration scheme, with time-step dt = 0.01. The total simulation
time chosen depends strongly on the magnitude of the temporal scales defined by the parameters γ, δ
and Ԑ.
It is noticed from Figure-1a, that the system periodic self-oscillations or Mixed Mode Oscillations
(MMOs) as the feedback strength decreased in which a number of large amplitude oscillations (L) is
followed by a number of small amplitude oscillations (S) to form a complex pattern (L1S1
L2 S2
Lnsn
) in
the time series. In our system L = 1.But S=1, 2, 3, 4 When Ԑ=0.00122, 0.0012, 0.00108, 0.00092
respectively. At Ԑ=0.00091, the system become chaotic. When Ԑ=0.000216 the system is period
doubling and it is periodic at Ԑ=0.0001. In these MMOs the ratio of L to S amplitudes is 1:1, 1:2, 1:3,
1:4 respectively.
In Figure-1b shows the corresponding FFT for these states where the distribution is decay
exponentially where many distinguished frequency could be seen. The small orbits of the attractor in
Figure-1c represent the low amplitude oscillations in the time series while the high amplitude spikes
have been represented as large orbits. Between the steady state and the chaotic spiking the system
passes through a cascade of period doubled and chaotic attractors of small amplitude. This is
illustrated by Figure-2, where a bifurcation diagram is computed from our system varying Ԑ over a
small interval contiguous to the initial Hopf bifurcation [17].
A system transition from one type of behavior to another depending on the value of a set of
important parameters like (Ԑ) value .When the (Ԑ) value is changing from 0 to maximum value 2.5˟10-
4 the amplitude of x scale in time series divide in three parts. The first one is curve line where Ԑ value
is starting from 1˟10-5
to 1.75˟10-4
that called regular oscillation. The second part called period
doubling that is starting from 1.75˟10-4
to 2.3˟10-4
. Finally, the chaos behavior region is starting
from2.3˟10-4
to 2.5˟10-4
as shown in Figure-2.
Ibrahim and Al-Naimee Iraqi Journal of Science, 2016, Special Issue, Part B, pp: 328-340
883
Figure 1- Numerical simulations of Al-Naimee model, a-Time series at different Ԑ value but γ=0.01,
S=0.2, a=1 and δ=2.75.
(a) Ԑ=0.00122 L=1, S=1 Ԑ=0.0012 L=1, S=2
Ԑ=0.00108 L=1,S=3 Ԑ=0.00092 L=1, S=4
Ԑ=0.00091 chaotic system Ԑ=0.000216 period doubling
Ԑ=0.0001 periodic
Ibrahim and Al-Naimee Iraqi Journal of Science, 2016, Special Issue, Part B, pp: 328-340
882
Figure 1- Numerical simulations of Al-Naimee model, b-The corresponding FFT at different Ԑ value
but γ=0.01, S=0.2, a=1 and δ=2.75.
(b) Ԑ=0.00122 L=1, S=1 Ԑ=0.0012 L=1, S=2
Ԑ=0.00108 L=1, S=3 Ԑ=0.00092 L=1, S=4
Ԑ=0.00091 chaotic system Ԑ=0.000216 period doubling
Ԑ=0.0001 periodic
Ibrahim and Al-Naimee Iraqi Journal of Science, 2016, Special Issue, Part B, pp: 328-340
888
Figure 1 -Numerical simulations of Al-Naimee model, c-The corresponding attractor by drawing the
relationship between photon density equation (x1) vs. population inversion equation (y1) at different Ԑ
value but γ=0.01, S=0.2, a=1 and δ=2.75.
(c) Ԑ=0.00122 L=1, S=1 Ԑ=0.0012 L=1, S=2
Ԑ=0.00108 L=1,S=3 Ԑ=0.00092 L=1, S=4
Ԑ=0.00091 chaotic system Ԑ=0.000216 period doubling
Ԑ=0.0001 periodic
Ibrahim and Al-Naimee Iraqi Journal of Science, 2016, Special Issue, Part B, pp: 328-340
883
0.0 5.0x10-5
1.0x10-4
1.5x10-4
2.0x10-4
2.5x10-4
1.7
1.8
1.9
2.0
2.1
periodic period doubling
chaotic
? ? ? ?
Inte
nsi
ty
Figure 2 - The Bifurcation diagram for the model equations. The photon density as a function of
feedback strength Ԑ for the dc bias current value δ=2.75. The other parameters are: γ = 0.01, s=0.2,
a=1.
These mixtures can form periodic sequences following the Farey arithmetic and plotting of a
suitably defined winding number R against the bifurcation parameter leads to a devil’s staircase [1]
where the winding number can calculated by R = L/ (L+S), L is the number of large amplitude
oscillations in a single periodic pattern, S is the number of small amplitude oscillations .Devil's
staircase is the picturesque used to describe the intricate, often fractal, structure of such staircases [18].
The complete bifurcation diagram corresponding to the mixed-mode wave forms can be represented
by plotting the winding number as a function of the control parameter Ԑ Figure-3. In our system we
have L = 1 and hence states 10, 1
1, 1
2, 1
3, and 1
4 have R equal to 1, 1/2, 1/3, 1/4, and 1/5, respectively.
The typical sequence of MMOs states is observed as Ԑ decreased.
Figure 3-The winding number R as a function of feedback strength Ԑ.
Experimental setup and results
The experimental setup is illustrated in Figure-4. We consider a closed-loop optical system,
consisting of a semiconductor device with AC-coupled nonlinear optoelectronic feedback. The device
is consisting of a gallium arsenide infrared emitting diode optically coupled to a monolithic silicon
phototransistor detector. The output light is sent to a photodetector producing a current proportional to
the optical intensity. The corresponding signal is sent to a variable gain amplifier characterized by a
nonlinear transfer function of the form f (w) = Aw/ (1 + sw), where A is the amplifier gain and s a
saturation coefficient, and then fed back to the injection current of the LED. The feedback strength is
determined by the amplifier gain, while its high-pass frequency cutoff can be varied (between 1 Hz
Ibrahim and Al-Naimee Iraqi Journal of Science, 2016, Special Issue, Part B, pp: 328-340
883
and 100 KHz) by means of a tunable high-pass filter. External signals or control bias can be added to
the LED pumping current.
Figure - 4 Sketch of the experimental setup for LED.
Figure-5 is set of graphs representing the dynamic behavior of the experimental setup when the bias
current is fixed at (0.4712) mA.The column (a) in Figure-5 shows the time series when the beginning
of the behavior of MMOs appears. The increasing of feedback strength to 0.0046 is lead to MMOs of
order 11 as shown in Figure-5a (1). The MMOs of order 1
2 is appeared at 0.0513 as shown in Figure-5a
(2), and the MMOs transition between 12and 1
3 at 0.056, as shown Figure-5a (3), and the MMOs of
order 14 at 0.0653, as shown Figure-5a (4). Then the MMOs of order 1
5 at 0.093, as shown in Figure-
5a (5), and so on. The column (b) in Figure-5 demonstrates the decay exponentially of the
corresponding FFT , and column (c) in Figure-5 demonstrates the corresponding attractor which shows
the beginning of the existence of one large phase-space orbit with high amplitude (L), and small
phase-space orbit with low amplitude (S) where used the impeded technique to plot it.
Ibrahim and Al-Naimee Iraqi Journal of Science, 2016, Special Issue, Part B, pp: 328-340
883
Figure 5 - (a) Experimental time series for a single LED with optoelectronic feedback as amplifier
gain is increased:(a-1)G=0.0046,(a-2) G=0.0513,(a-3)G=0.056,(a-4)G=0.0653,(a-5)G=0.093.
(a) 1
2
3
4
5
Ibrahim and Al-Naimee Iraqi Journal of Science, 2016, Special Issue, Part B, pp: 328-340
883
Figure 5-(b) the corresponding FFT, (1)G=0.0046,(2)G=0.0513,(3)G=0.056,(4)G=0.0653,(5)G=0.093
(b) 1
2
3
4
5
Ibrahim and Al-Naimee Iraqi Journal of Science, 2016, Special Issue, Part B, pp: 328-340
883
Fig.5. (c) the corresponding attractor. (1)G=0.0046, (2) G=0.0513, (3) G=0.056, (4) G=0.0653, (5)
G=0.093.
The system periodic self-oscillations or (MMOs) as the feedback strength increased and by fixing
the bias current. In correspondence to the large pulses, each oscillation period can be decomposed into
a sequence of periods of slow motion, near extrema, separated by fast relaxations between them. This
behavior (relaxation oscillations) is typical of slow–fast systems. To examine the influence of the
gradually increasing in the feedback strength on the output dynamics of the incoherent source (LED)
while the DC bias current is kept constant, the bifurcation diagram is sketched for increment values of
feedback strength as illustrated in Figure (6).
0.00 0.05 0.10 0.15 0.20 0.25 0.30
-5
0
amp.
feedback strength
Figure 6 -(C)Experimental bifurcation diagram (maxima of photon densities vs. feedback strength
variation).
(C ( 1
2
3
4
5
Ibrahim and Al-Naimee Iraqi Journal of Science, 2016, Special Issue, Part B, pp: 328-340
883
In contrast to the numerical, as the feedback strength is increased, we observe the complete sequence
of transitions going from the 11 state to1
7, thus reproducing qualitatively the numerical results.
However, the detailed bifurcation diagram in terms of the winding number R reveals an extraordinary
complexity as shown in Figure-7. The typical sequence of MMOs states that is observed as feedback
strength increased is: 11 (R= 1/2) and 1
2 (R = 1/3) and 1
3 (R = 1/4) and 1
4 (R = 1/5) and 1
5(R=1/6) and
16(R=1/7) finally 1
7 (R=1/8). In the transition between these states, random concatenations of the
adjacent patterns are observed.
0.00 0.05 0.10 0.15 0.20 0.25 0.30
0.0
0.2
0.4
0.6
0.8
1.0
R
v
11
12
14
15
16
17
Figure 7 -Plot of the winding number R as a function Vn.
Conclusion
We demonstrate numerically and experimentally the appearance of complex periodic mixed-mode
oscillations in a light-emitting diode with optoelectronic feedback. Numerically, the transitions from
periodic to period doubling, MMOs and chaotic states, by varying feedback strength and fixing bias
current have been noticed. The chaotic behavior could be studied in terms of FFT and attractor
corresponding to time series. It is noticed that the system periodic self-oscillations or Mixed Mode
Oscillations (MMOs) as the feedback strength decreased. Experimentally, Mixed Mode Oscillations
(MMOs) as the feedback strength increased have been noticed. There is a difference between
numerical and experimental due to the feedback is negative in experimental.
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