Complex Dynamics in incoherent source with ac- coupled ... · Complex Dynamics in incoherent source with ac- coupled optoelectronic Feedback Kejeen M. Ibrahim 1, Raied K. Jamal ,

Ibrahim and Al-Naimee Iraqi Journal of Science, 2016, Special Issue, Part B, pp: 328-340

__________________________ Email: [email protected]*

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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


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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)


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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.


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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


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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=0.0001 periodic


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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=0.0001 periodic


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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


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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.


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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


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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


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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


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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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Documents

Complex Dynamics in incoherent source with ac- coupled ... · Complex Dynamics in incoherent source with ac- coupled optoelectronic Feedback Kejeen M. Ibrahim 1, Raied K. Jamal ,