Experimental and Numerical Study of an Optoelectronics Flexible Logic Gate Using a Chaotic Doped Fiber Laser Experimental and Numerical Study of an Optoelectronics Flexible Logic Gate Using a Chaotic Doped Fiber Laser

In this chapter, we present the experimental and numerical study of an optoelectronics flexible logic gate using a chaotic erbium-doped fiber laser. The implementation consists of three elements: a chaotic erbium-doped fiber laser, a threshold controller, and the logic gate output. The output signal of the fiber laser is sent to the logic gate input as the threshold controller. Then, the threshold controller output signal is sent to the input of the logic gate and fed back to the fiber laser to control its dynamics. The logic gate output consists of a difference amplifier, which compares the signals sent by the threshold con- troller and the fiber laser, resulting in the logic output, which depends on an accessible parameter of the threshold controller. The dynamic logic gate using the fiber laser exhibits high ability in changing the logic gate type by modifying the threshold control parameter.


Introduction
An important advantage of erbium-doped fiber lasers (EDFLs) over other optical devices is a long interaction length of the pumping light with active ions that leads to a high gain and a single transversal-mode operation for a suitable choice of fiber parameters.The EDFL with coherent radiation at the wavelength of 1560 nm is an excellent device for applications in medicine, remote sensing, reflectometry, and all-optical fiber communications networks [1,2].Rare-doped fiber lasers subjected to external modulation from semiconductor pump lasers are known to exhibit chaotic dynamics [3][4][5][6][7][8][9][10][11][12].Besides, a very important advantage of the EDFL working in a chaotic regimen is its application to the development of basic logic gates [13], since it can process different logical gates and implements diverse arithmetic operations.The simplicity in switching chaotic EDFL between different logical operations makes this device more suitable for general proposes than traditional computer architecture with fixed wire hardware.Using a chaotic system as a computing device was proposed by Sinha and Ditto [14], who applied for this purpose a chaotic Chua's circuit with a simple threshold mechanism.After this pioneering work, chaotic computational elements received considerable attention from many researchers who developed new designs allowing higher capacity for universal general computing purposes enable to reproduce basic logic operations, such as AND, OR, NOT, XOR, NAND, and NOR [15][16][17][18][19][20][21][22][23][24][25][26].Likewise, a single chaotic element has the ability in reconfiguring into different logic gates through a threshold-based control [15,16].This device is also known as reconfigurable chaotic logic gate (RCLGs) and, due to its inherent nonlinear components, has advantages over standard programmable gate array elements [19] where reconfiguration is obtained by interchanging between multiple single-purpose gates.Also, discrete circuits working as RCLGs were proposed to reconfigure all logic gates [17,18].Additionally, reconfigurable chaotic logic gates arrays (RCGA), which morph between higher-order functions, such as those found in a typical arithmetic logic unit (ALU), were invented [20].Recently, some of the authors of this work proposed an optoelectronics flexible logic gate based on a fiber laser [27,28].
Here, we describe in detail the implementation of the optoelectronics flexible logic gate based on EDFL, which exploits the richness and complexity inherent to chaotic dynamics.Using a threshold controller, NOR and NAND logic operations are realized in the chaotic EDFL.This chapter is an extension of the article "Optoelectronic flexible logic gate based on a fiber laser.Eur.Phys.J. Special Topics.2014" [27].It is organized as follows.The theoretical model of the diode-pumped EDFL is described in Section 2. The experimental setup of the optical logic gate based on the EDFL is given in Section 3. Likewise, the discussion of theoretical and experimental results on the application of the NAND and NOR logic gates based on the EDFL as a function of the threshold controller is presented in Section 4. Finally, main conclusions are given in Section 5.

Theoretical arrangement
The EDFL is known to be extremely sensitive to external disturbances, which can destabilize its normal operation.This makes this device very promising for many applications where smallamplitude external modulation is required to control the laser dynamics.The mathematical model and experimental arrangement of the EDFL used in this work have been developed by Pisarchik et al. [6][7][8][9][10][11][12].

EDFL theoretical model
Based on the power balance approach, we model diode-pumped EDFL dynamics by considering both the excited-state absorption (ESA) in erbium at the 1560-nm wavelength and the averaged population inversion along the pumped active fiber laser.The model addresses two evident factors, the ESA at the laser wavelength and the depleting of the pump wave at propagation along the active fiber, leading to undumped oscillations experimentally observed in the EDFL without external modulation [6,12].The energy-level diagram of the theoretical model used in this work is shown in Figure 1.Using a conventional system for EDFL balance Equations [6,7], which describe the variations of the intra-cavity laser power P (in units of s À1 ), that is, the sum of the contrapropagating waves' powers inside the cavity and the averaged population N (dimensionless variable) in the upper laser level "2," we can write EDFL equations as follows: where N can take values between 0 ≤ N ≤ 1 and is defined as as the upper laser-level population density "2," n 0 is the refractive index of an erbiumdoped fiber, and L is the length of the active fiber medium; σ 12 is the cross section of the absorption transition from the state "1" to the upper state "2," σ 12 is the stimulated cross section of the transition in return from the upper state "2" to the ground state "1," in is the photon round-trip time in the cavity (l 0 is the length intra-cavity tails of FBG couplers); α 0 ¼ N 0 σ 12 is the small-signal absorption of the erbium fiber at the laser wavelength (N 0 = N 1 + N 2 is the total erbium ions' populations density in the active fiber medium); is the cavity losses at threshold (γ 0 being the passive fiber losses, R B is the total reflection coefficient of the fiber Bragg grating (FBG) couplers); τ is the lifetime of erbium ions in the excited state "2 00 ; r ω is the factor addressing a match between the laser fundamental mode and erbium doped core volumes inside the active fiber, given as where r 0 is the fiber core radius and w 0 is the radius fundamental fiber mode.The spontaneous emission P sp into the fundamental laser mode is taken as Here, we assume that the erbium luminescence spectral bandwidth (λ g being the laser wavelength) is 10 À3 .P pump is the laser pump power given as where P p is the pump power at the fiber entrance and β ¼ αp α0 is the dimensionless coefficient that accounts for the ratio of absorption coefficients of the erbium fiber at pump wavelength λ p to that at laser wavelength λ g .Eqs. ( 1) and ( 2) describe the laser dynamics without external modulation.We add the modulation to the pump parameter as: where P 0 p is the laser pump power without modulation, A m and F m are the modulation amplitude and frequency, respectively.
The laser threshold is defined as ε = P p /P th , where is the threshold pump power, is threshold population of the level "2" and the radius of the pump beam w p = w 0 .In the numerical simulations, we choose the pump power P p = 7.4 Â 10 19 s À1 that yields the laser relaxation oscillation frequency f 0 ≈ 30 kHz.
In order to understand the dynamics of the EDFL, the bifurcation diagram of the local maxima of the laser power versus the pump modulation frequency F m is calculated.To perform numerical simulations, we normalize Eqs. ( 1) and (2) (as described in the appendix of reference [29]) and obtain the following equations: The constant parameters of Eqs. ( 8) and ( 9) are shown in Table 1 [30].
Figure 3 shows the numerical bifurcation diagram of the laser peak intensity versus the modulation frequency (0-20 kHz) for the 100% modulation depth (A m = 1).The laser dynamical behavior (periodic or chaotic) is determined by the modulation frequency.
In this work, we are interested in a chaotic regime.Figure 4 shows the times series corresponding to chaos for F m = 10 kHz.

Implementation of the optoelectronics flexible logic gate using the EDFL
Figure 5 shows the scheme of the proposed optoelectronics flexible logic gate using the EDFL.The reconfigurable logical gate contains two principal elements: a chaotic EDFL and a threshold controller.The dynamics behavior of the EDFL is described by the balance Eqs. ( 1) and ( 2).The threshold controller compares laser power P with value V T generated by the controller that releases output V T = E if P > E and V T = P otherwise, with E as threshold value.This output signal V T is added to the diode pump current P pump with a coupling coefficient K.The logic gate output subtracts V T from P yielding V 0 = P À V T .Next, we consider the laser and the controller models separately.

Threshold controller
In our numerical simulations, we use the laser power P calculated by Eqs. ( 1), ( 2), and (6) as the input signal for the threshold controller.The output signal V T from the controller is used to control the diode pump current as: The threshold controller has two logic inputs 0 and 1, which generate the corresponding values I 1 and I 2 , where I 1,2 = 0 for input 0 and I 1,2 = V in otherwise, where V in is a certain value to define the threshold for E. A type of the logic gate is determined by a parameter V c .The procedure to obtain this parameter is explained in detail in section Results and Discussions.
The controller generates an initial value E defined by the inputs I 1 and I 2 being either 0 or V in and takes the value: Figure 5. Arrangement of the optoelectronics logic gate.E is the threshold controller, V c determines the logic response, I 1,2 is the logic input which takes the value of either V in or 0, V T is the output controller signal, P is the laser output intensity, P pump is the diode laser pump intensity, P p is the continuous component of the pumping, A m and F m are the modulation depth and frequency, and K is the gain factor.so that there are three possible options: The controller output is determined as: Figure 6.Electronic circuits of the threshold controller.
Experimental and Numerical Study of an Optoelectronics Flexible Logic Gate Using a Chaotic Doped Fiber Laser http://dx.doi.org/10.5772/intechopen.75466 where V T becomes the threshold signal.
Figure 6 shows the electronic circuits in the controller to generate E, V R , V 0, and P pump signals.
The electronic components used in the controller are presented in Table 2.

EDFL experimental arrangement
The experimental arrangement presented in Figure 7 consists of EDFL pumped by a laser diode (LD) from Thorlabs PL980 operating at 1560 and 977 nm, respectively.The Fabry-Perot fiber laser cavity with total length of 4.81 m is formed by an active, long EDFL of 88-cm length, and a 2.7μm core diameter, incorporating two fiber Bragg gratings (FBG1 and FBG2) with 0.288 and 0.544-nm full widths on half-magnitude bandwidth, having, respectively, -100% and -96%  reflectivity at the laser operating wavelength.A fiber laser formed by an erbium doped fiber (EDF) and two Bragg gratings (FBG1 and FBG2), is externally driven by the harmonic pump signal (through a sum circuit CI 741) applied to a diode pump laser (LD) current via a laser diode controller (LDC) from a wave function generator (WFG).A single-mode fiber is used to connect the optical components.
The current and temperature of the LD are controlled by a laser diode controller (LDC) (Thorlabs ITC510).The 145.5-mA pump current is selected to guarantee that the laser relaxation oscillation frequency is around F r = 30 kHz to provide a 20-mW power; which is above a 110-mA EDFL threshold current.A harmonic modulation signal A m sin 2πF m t ð Þ from wave function generator (WFG) (Tektronix AFG3102) is supplied to the diode pump current.The fiber laser output after passing through a polarization controller (P), wavelength division multiplexer (WDM), and an optical isolator (OI) is recorded with a photodiode (PD), and the electronic signal is compared with the signal generated by the threshold controller.The threshold controller with a summing circuit (CI 741) with dynamical control signal Vc and inputs logic signals I 1,2 controlled by a USB NI 6803, V T is a comparator circuit between laser intensity P and threshold controller E. The logic gate output V 0 is sent back to the driver (P pump ) of the EDFL to change its dynamics.The signals P from the EDFL, I 1,2 , V T , and V 0 from the threshold controller are analyzed with a multichannel oscilloscope.

Numerical results
In order to use the arrangement of the optoelectronics logic gate shown in Figure 5, it is necessary to determine V c and V in signals and find the required logic gates NAND or NOR.The value of V c was gradually changed (À20 V < V c < 2 V) and for each value of V c the value of V in was changed (2 V < V in < 20 V). Figure 8 shows the values of V c versus V in which we use to determine the logical gates NAND and NOR.If we set the parameter V in = 10 V and V c varies from À1 to À9 V, we get the NOR gate; but if V c changes from À11 to À20 V, the NAND gate is used.The numerical results of NOR and NAND operations of the reconfigurable dynamic logic gate Eqs. ( 1), (2), and ( 10)-( 13) are shown in Figure 9 for A m = 1 V and F m = 10 kHz.
For the time interval from t = 0 to 6.5 ms, we have a NOR logic gate, where the signal from V c = À 3 V to V in = 15 V produces three different combinations of the threshold controller V T as For the time interval from t = 6.5 to t = 13 ms, Figure 9 shows a NAND logic gate, where the signal from V c = À20 V to V in = 15 V produces three different combinations of the threshold controller V T as

Experimental results
Similar to the results of the numerical simulations, a change was made in the parameters for V c versus V in to determine required NAND or NOR logic gates.Figure 10 shows the values of V c   from t = 8 to t = 16 ms for V c = À220 mV.By comparing Figure 9 with Figure 11, we can see a good agreement between the numerical and experimental results.

Conclusions
In this chapter, we have described the implementation of an optoelectronics logic gate based on a diode-pumped EDFL.We have demonstrated good functionality of our system for NOR and NAND logic operations, taking advantage of optical chaos and a threshold controller.The system was controlled by a split signal from the threshold controller, allowing the diode pump laser to mismatch between the output threshold controller signal and the output EDFL signal.The numerical results obtained from the EDFL equations have displayed good agreement with the experimental results.We have demonstrated that the chaotic dynamic behavior of the diode-pumped EDFL and the electronic threshold controller can be successfully used to obtain NAND or NOR logic gates to be constructive bricks of different logic systems.The main contribution of the developed optoelectronics logic gate is addressed in optical computing.The proposed device is more adaptable and faster than a conventional wired hardware, since it can be implemented as an arithmetic processing unit or an optical memory RAM.
Experimental and Numerical Study of an Optoelectronics Flexible Logic Gate Using a Chaotic Doped Fiber Laser http://dx.doi.org/10.5772/intechopen.75466

Figure 2
Figure2presents the time series of the laser intensity at the following driven frequencies: (a) F m = 3 kHz, the laser behavior is chaotic, (b) F m = 4 kHz, the EDFL response is a period 4, (c) F m = 3 kHz, the EDFL response is a period 3, (d) F m = 7 kHz, the EDFL response is a period 2, (e) F m = 10 KHz, chaos, (f) F m = 15 kHz and (g) F m = 20 kHz, a period 1 with decreasing amplitude as the modulation frequency is increased.

Figure 3 .
Figure 3. Numerical bifurcation diagram of laser peak intensity versus modulation frequency (F m ) for A m = 1.

Figure 4 .
Figure 4. Time series of laser intensity P for F m = 10 kHz and A m = 1.

Figure 7 .
Figure 7. Experimental scheme of the optoelectronics logical gate based on EDFL.

Figure 8 .
Figure 8. Diagram of values for V c and V in to determine the logic gate type, either NAND or NOR.

Figure 10 .
Figure 10.Diagram of values for V c and V in to determine the logic gate type, either NAND or NOR.

Figure 11 and
Figure 11 and Table 3 show the experimental results of the dynamic NOR and NAND logic operations for A m = 700 mV, F m = 15 kHz, and V in = 200 mV.The NOR gate corresponds to the time series from t = 0 ms to t = 8 ms for V c = ©40 mV, and for the NAND gate for the time series

Table 2 .
Parameters for electronic components of circuits shown in Figure6.

Table 3 .
Experimental data for implementation of NOR and NAND optoelectronics logical gates.academic groups of the CULAGOS with orientation in optoelectronics, Agreement RG/019/ 2016-UdeG, Mexico.