- Email: [email protected]
All optical AND and OR gate with excitable cavity soliton
Chaos, Solitons and Fractals 100 (2017) 100–104
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
All optical AND and OR gate with excitable cavity soliton H. Vahed∗, Z. Salehnezhad School of Engineering Emerging Technologies, University Of Tabriz, Tabriz 5166614761, Iran
a r t i c l e
i n f o
Article history: Received 16 October 2016 Accepted 1 May 2017
Keywords: Cavity soliton Cavity soliton laser Excitable Gate All optical
a b s t r a c t Cavity solitons (CSs) have an important role as information bits in all optical information processing. In this paper, we switched on a new type of CS as excitable cavity soliton in the cavity soliton laser (CSL). Also, we compared the behavior of excitable CS with self-pulsing and stable CSs by using of suitable values for system parameters and then we designed logical gates (AND, OR) with excitable CSs in the CSL. The capabilities of excitable CSs has been studied to improve possibilities for designing of all optical gates. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction Cavity solitons (CSs) are localized light peaks on a low-intensity, homogeneous background that can be controlled and produced with specified intensity and location in the transverse plane of cavity, orthogonal to the propagation direction of the beam [1]. Using CSs for information storage produce more compact optical schemes and each bit of information is represented by a CS [2].This is especially attractive in nonlinear photonics after CSs have been observed in semiconductor lasers [3]. The presence of a CS is considered as a bit ‘1’ and the absence of it’s as a bit ‘0’. Moreover computing by CSs is a way of achieving ultra-fast performance because photons travel faster than electrons and do not radiate energy, even at fast frequencies. Several researchers switched on CSs in the vertical cavity surface emitting semiconductor laser with holding beam [4–7]. In this system, semiconductor laser only acts as an amplifier and spatial modulator for the holding beam that injected from out and perpendicular to the cavity. Recently, switching on the CS in the vertical cavity surface emitting laser (VCSEL) without holding beam and with saturable absorber has been demonstrated experimentally and theoretically [8–10]. This system is known as a cavity soliton laser (CSL) [11]. The control and application of CSs studied at the CSL for optical computing [12–14]. When we remove the holding beam, it reduces spatial volume of system and increase flexibility of system. .In CSL, CSs are in darker background of the before state that is corresponding to spontaneous emission. In this case, difference of intensity between peak of CS and background ∗
Corresponding author. E-mail addresses: [email protected] (H. Vahed), [email protected] (Z. Salehnezhad). http://dx.doi.org/10.1016/j.chaos.2017.05.004 0960-0779/© 2017 Elsevier Ltd. All rights reserved.
will be optimized. Nowadays, three type of laser have been experimentally and theoretically realized that are included CSL with frequency selective feedback [15], CSL in face to face configuration [10] and monolithic CSL with saturable absorber [16]. The CSL with frequency selective feedback leads to first show of CSL in semiconductor system [15]. In 2008, P. Genevet et al. [10], are demonstrated experimentally realization of CSL by using of two mutually coupled micro resonators that one micro resonator is amplifier and other is saturable absorber. In previous studies, logical gates were found in the cavity with holding beam [17–19]. In ref. [20], A. Jacobo et al. consider a Kerr cavity driven by a broad holding beam and show how to use excitable regimes mediated by localized structures to perform AND, OR and NOT logical operations. Here, we focus to the excitable CS in the cavity with saturable absorber without holding beam. We switch on excitable CS and then it is compared with stable and self-pulsing CSs. Finally, the logic gates (AND and OR) has been proposed by using of excitable CSs. The CS behavior is dependent on “r” parameter that is ratio between the carrier lifetime in the active and passive medium [21]. By fixing all other parameters and different values of “r” parameter we can see different type of CSs in the forms of stable, self-pulsing and excitable CSs. Excitability is a concept arising originally from biology (e.g. in neuroscience), and found in a large variety of nonlinear system [22]. A system is said to be excitable if perturbation below a certain threshold decay exponentially while perturbations above induce a large response before going back to a resting state [20]. In this work, firstly we switch on excitable CS in the CSL. Then we compare the intensity evolution of stable, self-pulsing and excitable CSs. Finally, we propose to design an all optical gate based on the excitable CSs in the CSL .The paper is organized as follow. In Section 2 we introduce the model. In Section
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Fig. 1. Time evolution of the field intensity for three types of CSs. a) Stable CS r = 1.4, b) self-pulsing CS r = 1.7, c) excitable CS r = 2.
5 we switch on excitable CSs in the CSL and then we compare the intensity evolution of stable, self-pulsing and excitable CSs. In Section 4 we propose to design an optical logic gate based on excitable CSs in the CSL and find conditions for realization of it. 2. Dynamical equations of CSL The dynamics of VCSEL with saturable absorber that work as a CSL are described by a set of equations [9]:
F˙ = (1 − iα )D + (1 − iβ )d − 1 + i∇⊥2 F D˙ = b1
μ − D 1 + |F |2 − BD2
d˙ = b2 −γ − d 1 + s|F |2 − Bd2
(1-a) (1-b)
(1-c)
F is the slowly varying amplitude of the electric field and D and d are population variables related to the carrier densities in the active and passive materials, respectively. The parameters α
and b1 (β and b2 ) are the line width enhancement factor and the ratio of the photon lifetime to the carrier lifetime in the active (passive) material, μ is the pump parameter of the active material, γ measures absorption in the passive material and s is the saturation parameter. Time is scaled to the photon lifetime in empty cavity (∼3 ps) and space to diffraction length (∼5 μm). These three equations are dimensionless. For more details on the definition of the parameters and the variables see [8]. In this paper, we consider the set of parameters B = 0.1, β = 1, γ = 2and s = 1 [9]. To study dynamical behaviors of system, (1-a) to (1-c) are solved by use of split-step method with periodic boundary conditions in two-dimensional spatial network 128 × 128. In this method, we are separating the algebraic and Laplacian terms on the right-hand part of the equations. The algebraic term is integrated using a Runge–Kutta algorithm, while for the Laplacian operator a two-dimensional Fast Fourier transform is adopted. We select a time step 0.01 and a space step 0.25 to ensure stability of convergence of the algorithm (δ t2 ≤ δ s2 /4). For simple presentation, we select the center of transverse plane as
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Fig. 2. The excitable CS peak intensity versus the injected Gaussian beam amplitude.
point (0, 0) like the Cartesian coordinate. Therefore, the x and y coordinates can change between −64,−63,…, 0,…,63,64. 3. Switch ON CSs and comparison We switch on CSs by using of the semi coherent injection. In order to switch on a CS we added in (1-a) the term [23]: 2 2 2 Fin j = F0 eiϕ e[−(x−x0 ) −(y−y0 ) /2w ]
(2)
F0 is the amplitude of Gaussian pulse and ϕ and w are the phase and width for Gaussian pulse, respectively. This pulse injected in the point of (x0 ,y0 ). The duration of injection would be tinj . Therefore, we inject a Gaussian beam in center of the cavity transverse plane for 50 time units. The Gaussian beam parameters are F0 = 0.28, w = 4 and ϕ = 0. The pump parameter of active material has been selected μ = 4.9. We can switch on different types of CSs depend on the parameter ‘r’ [14, 21]. We choose three different values of r that r = b2 /b1 . Our simulations show that we have stable CS, self-pulsing CS and Excitable CS for r = 1.4, r = 1.7 and r = 2 respectively. In Fig. 1, we show and compare the field intensity evolution for three types of CSs (stable, self-pulsing and excitable CSs). Firstly, the field intensity is in noise level at t = 0. At t = 50 t.u. we inject the semi coherent injection during tin j = 50t.u. and the field intensity grows to values much larger than those of the stable CS. When the injection beam switched off, the field intensity falls to value comparable with noise level. Then, the field intensity (also, the populations) try to recovers the stationary value through the damped oscillations. Lastly, we have a stable CS. For excitable CS, the behavior is as same as the stable CS in the injection moment. But, we cannot see the initial peak of injection because the field intensity is very large and broad for the excitable CS. For creation of self-pulsing CS, we change the parameter r for the stable CS and the stable CS converts to self-pulsing CS. As shown in Fig. 1(a), stable CS remains stable for long time and the final amplitude is small (about 2 or 3) for the stable CS. In self-pulsing CS, the CS position is fixed on the injection point but the amplitude of electrical field oscillates with time. Firstly, the amplitude oscillation of the self-pulsing CS was irregular but we see the regular oscillation with fixed period and frequency by passing of time (Fig. 1(b)). The peak of self-pulsing CS gets fixed on a certain value that is bigger than the peak of stable CS. Finally,
the excitable CS has a peak that is varying larger than stable and self-pulsing CSs (about 120) (Fig. 1(c)). Our simulation show that the excitable CS peak is independent of injected Gaussian beam amplitude (Fig. 2). But the delay time between the beginning of injection and formation of the maximum amplitude peak is dependent on the Gaussian beam amplitude (Fig. 3). 4. Logical gates AND, OR A logical gate is a device that performs a certain Boolean logic operation on one or more logical input and produces a signal logical output. The relationship between input and output signals are displayed in what are known as truth tables [24]. Now we want to use the excitable CS to creating of a logical gate. We try to find the necessary conditions for realizing of the logic gate in CSL. For this goal we consider the presence of an excitable Cs as a bit ‘1’ and the absence of the excitable CS is as a bit ‘0 . Inputs of logic gate can be created by injecting of the Gaussian beam in the transverse plane of cavity. When two excitable CS are placed in suitable distance of each other’s, the mutual interaction between CSs creates output. In plots we consider the intensity value below 1 as a bit ‘0 and the intensity value over than 10 with significant amplitude as a bit ‘1’. Moreover, we consider three excitable CSs in a linear diagonal arrangement of transverse plane, whose position is fixed by the Gaussian beam. The two outer CSs will represent the input signals, while the central CS will be the output signal. When we change the excitability threshold and distance of inputs, the central CS (output) switch on in response to two inputs. Therefore, the central excitable CS (output) can be switched depend on the two outer diagonal CS (inputs). Finally, different logical operation can be achieved and we will have all optical logic gates by using of excitable CS. For example, when we inject two outer excitable CS (two input are 1), an excitable CS (output is 1) switched on in central of cavity transverse plane and we will have a AND logic gate. Fig. 4 shows this logical operation for AND gate. We injected two Gaussian beam with F0 = 0.35, w = 4 and φ = 0 in positions of (62,−62) and (−62, 62). We plot the spatial profile of intensity variable along the diagonal direction of transverse plane at time of 60 t.u. in Fig. 5. This profile has one central peak as output of AND logic gate. For OR logic gate, we inject one excitable CS and a weak injection as two inputs for logic gate (two input are 1 and 0) and
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Fig. 3. The delay time in formation of the excitable CS versus the injected Gaussian beam amplitude.
Fig. 4. Snapshots of the AND gate with two inputs ‘1’.This generates a ‘1’ bit in the output. The snapshots are ordered from left to right.
Fig. 5. Profile of intensity variable along the diagonal direction of transverse plane at time of 60 t.u.
Fig. 6. Snapshots of the OR gate with one input ‘1’and other one input 0. This generates a ‘1’ bit at the output. The snapshots are ordered from left to right.
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Fig. 7. Profile of intensity variable along the diagonal direction of transverse plane at time of 90 t.u.
then an excitable CS (output is 1) switched on in central of cavity transverse plane. Fig. 6 shows this logical operation for OR gate. We inject two Gaussian beam with F0 = 0.35, w = 4 and φ = 0 (this input is 1) and F0 = 0.001, w = 4 and φ = 0 (this input is 0) in positions of (42,−42) and (−42, 42) respectively. In Fig. 4 we show the operation of an OR gate when one inputs is ‘1 and the other ‘0 leading to a ‘1 bit at the output (center). The spatial profile of intensity variable has been plotted along the diagonal direction of transverse plane at time of 90 t.u. in Fig. 5. This profile has one central peak as output of OR logic gate. In Figs. 5 and 7, the peak of excitable CS does not corresponds with the point of (0,0). When we switch on two excitable CS as inputs for our gate, we create two very deep dips in the population profile. The population transfer between these dips causes to formation of central excitable CS. We can create the final excitable Cs in favorite position with suitable parameter for initial injection beams. Also, we can control the position of final excitable CS with controlling of the noise level. The other logic operations can be created in CSL easily. For example, the case (1, 0) can be confirmed in the logic gate AND. Because of symmetry, the output result would be same for couples cases (for example (0, 1) and (1, 0) inputs). The case of (1, 1) input would have a similar result for the AND and OR logic gates. We can express the results in terms of new parameter as exciting energy. This energy would be defined as E = |Fin j |2 × tin j for semicoherent injection. Therefore, this energy depends on the amplitude of injection beam, duration of injection and distance between CSs. It’s clear if change one of these parameters we must change other two parameters to compensation of necessary energy. In the proposed gate we used the excitable CS. Also, these gates operate in CSL with saturable absorber without holding beam. Therefore, this system is better than the previous system (with holding Beam). Because, we have a simpler and more compact system without holding beam that is easier to control and prevent of imposition significant energy to the system. As a result we expect thermal effects haven’t significant effect on our system. We studied OR and AND logic gate in our system. Two outer injections are inputs and the output forms itself in central of transverse plane of cavity. The input’s power and distance of two inputs are such that excitable excursion of inputs are enough to excite the output. The inputs show an excitable excursion which triggers an excitable excursion of the output. 5. Conclusions In this paper, we have switched excitable CS in the CSL with saturable absorber by selecting of the suitable parameter and then we have compared the time evolution of intensity for excitable,
stable and self-pulsing CSs. Our simulations show that the peak of excitable CS is independent of Gaussian beam amplitude, but it is dependent on r parameter. If we increase the parameter r, the peak of excitable CS increases too. We have proposed all optical logic gate AND and OR by using of the excitable CS. These logic gates can be used to perform all optical computing. The physical mechanism underlying the logic gates is interaction between the excitable CSs. the strength of interaction depend on the distance of two outer injection (inputs) and power of the injected Gaussian beams. Therefore, the truth of the AND and OR Logic gates can be performed by suitable selection of these parameters. References [1] Lugiato LA. IEEE J Quantum Electron 2003;39:193. [2] Babland S, Tredicce JR, Brambilla M, Lugiato LA, Balle S, Giudici M, et al. Nature 2002;419:669. [3] Brambilla M, Lugiato LA, Prati F, Spinelli L, Firth WJ. Phys Rev Lett 1997;79:2042. [4] Spinelli L, Tissoni G, Brambilla M, Prati F, Lugiato LA. Phys Rev A 1998;58:2542–59. [5] Hachair X, Barland S, Furfaro L, Giudici M, Balle S, Tredicce J, et al. Phys Rev A 2004;69:043817. [6] Hachair X, Furfaro L, Javaloyes J, Giudici M, Balle S, Tredicce J, et al. Phys Rev A 2005;72:013815. [7] Hachair X, Pedaci F, Caboche E, Barland S, Giudici M, Tredicce JR, et al. IEEE J Sel Top Quantum Electron 2006;12:339–51. [8] Bache M, Prati F, Tissoni G, Kheradmand R, Lugiato LA, Protsenko I, et al. Appl Phys B 2005;81:913–20. [9] Prati F, Caccia P, Tissoni G, Lugiato LA, Aghdami K, Tajalli H. Appl Phys B 2007;88:405–10. [10] Genevet P, Barland S, Giudici M, Tredicce JR. Phys Rev Lett 2008;101:123905. [11] Tissoni G, Aghdami K, Prati F, Brambilla M, Lugiato LA. Cavity soliton laser based on a VCSEL with saturable absorber. In: Localized states in physics: solitons and patterns. Berlin: Springer; 2011. p. 187–211. [12] Vahed H, Prati F, Turconi M, Barlan S, Tissoni G. Phil Trans R Soc A 2014;372:20140016. [13] Vahed H, Prati F, Tissoni G, Lugiato LA, Tajalli H. Eur Phys J D 2012;66:148–53. [14] Vahed H. IEEE Photonic Tech Lett 2015;27:2019–22. [15] Tanguy Y, Ackemann T, Firth WJ, Jager R. Phys Rev Lett 20 08;10 0:013907. [16] Elsass T, Gauthron K, Beaudoin G, Sagnes I, Kuszelewicz R, Barbay S. Appl Phys B 2010;98:327–31. [17] Gomila D, Jacobo A, Matias M, Colet P. Logical operations using excitable cavity solitons. Advanced photonics & renewable energy, OSA technical digest (CD). Optical Society of America; 2010. paper NME64. [18] Jacobo A, Gomila D, Colet P, Matias M. All optical logical operations using excitable cavity solitons. In: Photonics society winter topicals meeting series (WTM). IEEE; 2010. p. 122–3. [19] Eslami M, Kheradmand R. Opt Rev 2012;19:242–6. [20] Jacobo A, Gomila D, Colet P, Matias M. New J Phys 2012;14:013040. [21] Prati F, Tissoni G, Lugiato LA, Aghdami KM, Brambilla M. Eur Phys J D 2010;59:73–9. [22] Winfree AT. The geometry of biological time. 2nd ed. New York: Springer; 2001. [23] Aghdami K, Prati F, Caccia P, Tissoni G, Lugiato LA, Kheradmand R, et al. Eur Phys J D 2008;47:447. [24] Hunsperger RG. Integrated optics. 6th ed. Berlin: Springer; 2009.
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
All optical AND and OR gate with excitable cavity soliton H. Vahed∗, Z. Salehnezhad School of Engineering Emerging Technologies, University Of Tabriz, Tabriz 5166614761, Iran
a r t i c l e
i n f o
Article history: Received 16 October 2016 Accepted 1 May 2017
Keywords: Cavity soliton Cavity soliton laser Excitable Gate All optical
a b s t r a c t Cavity solitons (CSs) have an important role as information bits in all optical information processing. In this paper, we switched on a new type of CS as excitable cavity soliton in the cavity soliton laser (CSL). Also, we compared the behavior of excitable CS with self-pulsing and stable CSs by using of suitable values for system parameters and then we designed logical gates (AND, OR) with excitable CSs in the CSL. The capabilities of excitable CSs has been studied to improve possibilities for designing of all optical gates. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction Cavity solitons (CSs) are localized light peaks on a low-intensity, homogeneous background that can be controlled and produced with specified intensity and location in the transverse plane of cavity, orthogonal to the propagation direction of the beam [1]. Using CSs for information storage produce more compact optical schemes and each bit of information is represented by a CS [2].This is especially attractive in nonlinear photonics after CSs have been observed in semiconductor lasers [3]. The presence of a CS is considered as a bit ‘1’ and the absence of it’s as a bit ‘0’. Moreover computing by CSs is a way of achieving ultra-fast performance because photons travel faster than electrons and do not radiate energy, even at fast frequencies. Several researchers switched on CSs in the vertical cavity surface emitting semiconductor laser with holding beam [4–7]. In this system, semiconductor laser only acts as an amplifier and spatial modulator for the holding beam that injected from out and perpendicular to the cavity. Recently, switching on the CS in the vertical cavity surface emitting laser (VCSEL) without holding beam and with saturable absorber has been demonstrated experimentally and theoretically [8–10]. This system is known as a cavity soliton laser (CSL) [11]. The control and application of CSs studied at the CSL for optical computing [12–14]. When we remove the holding beam, it reduces spatial volume of system and increase flexibility of system. .In CSL, CSs are in darker background of the before state that is corresponding to spontaneous emission. In this case, difference of intensity between peak of CS and background ∗
Corresponding author. E-mail addresses: [email protected] (H. Vahed), [email protected] (Z. Salehnezhad). http://dx.doi.org/10.1016/j.chaos.2017.05.004 0960-0779/© 2017 Elsevier Ltd. All rights reserved.
will be optimized. Nowadays, three type of laser have been experimentally and theoretically realized that are included CSL with frequency selective feedback [15], CSL in face to face configuration [10] and monolithic CSL with saturable absorber [16]. The CSL with frequency selective feedback leads to first show of CSL in semiconductor system [15]. In 2008, P. Genevet et al. [10], are demonstrated experimentally realization of CSL by using of two mutually coupled micro resonators that one micro resonator is amplifier and other is saturable absorber. In previous studies, logical gates were found in the cavity with holding beam [17–19]. In ref. [20], A. Jacobo et al. consider a Kerr cavity driven by a broad holding beam and show how to use excitable regimes mediated by localized structures to perform AND, OR and NOT logical operations. Here, we focus to the excitable CS in the cavity with saturable absorber without holding beam. We switch on excitable CS and then it is compared with stable and self-pulsing CSs. Finally, the logic gates (AND and OR) has been proposed by using of excitable CSs. The CS behavior is dependent on “r” parameter that is ratio between the carrier lifetime in the active and passive medium [21]. By fixing all other parameters and different values of “r” parameter we can see different type of CSs in the forms of stable, self-pulsing and excitable CSs. Excitability is a concept arising originally from biology (e.g. in neuroscience), and found in a large variety of nonlinear system [22]. A system is said to be excitable if perturbation below a certain threshold decay exponentially while perturbations above induce a large response before going back to a resting state [20]. In this work, firstly we switch on excitable CS in the CSL. Then we compare the intensity evolution of stable, self-pulsing and excitable CSs. Finally, we propose to design an all optical gate based on the excitable CSs in the CSL .The paper is organized as follow. In Section 2 we introduce the model. In Section
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Fig. 1. Time evolution of the field intensity for three types of CSs. a) Stable CS r = 1.4, b) self-pulsing CS r = 1.7, c) excitable CS r = 2.
5 we switch on excitable CSs in the CSL and then we compare the intensity evolution of stable, self-pulsing and excitable CSs. In Section 4 we propose to design an optical logic gate based on excitable CSs in the CSL and find conditions for realization of it. 2. Dynamical equations of CSL The dynamics of VCSEL with saturable absorber that work as a CSL are described by a set of equations [9]:
F˙ = (1 − iα )D + (1 − iβ )d − 1 + i∇⊥2 F D˙ = b1
μ − D 1 + |F |2 − BD2
d˙ = b2 −γ − d 1 + s|F |2 − Bd2
(1-a) (1-b)
(1-c)
F is the slowly varying amplitude of the electric field and D and d are population variables related to the carrier densities in the active and passive materials, respectively. The parameters α
and b1 (β and b2 ) are the line width enhancement factor and the ratio of the photon lifetime to the carrier lifetime in the active (passive) material, μ is the pump parameter of the active material, γ measures absorption in the passive material and s is the saturation parameter. Time is scaled to the photon lifetime in empty cavity (∼3 ps) and space to diffraction length (∼5 μm). These three equations are dimensionless. For more details on the definition of the parameters and the variables see [8]. In this paper, we consider the set of parameters B = 0.1, β = 1, γ = 2and s = 1 [9]. To study dynamical behaviors of system, (1-a) to (1-c) are solved by use of split-step method with periodic boundary conditions in two-dimensional spatial network 128 × 128. In this method, we are separating the algebraic and Laplacian terms on the right-hand part of the equations. The algebraic term is integrated using a Runge–Kutta algorithm, while for the Laplacian operator a two-dimensional Fast Fourier transform is adopted. We select a time step 0.01 and a space step 0.25 to ensure stability of convergence of the algorithm (δ t2 ≤ δ s2 /4). For simple presentation, we select the center of transverse plane as
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Fig. 2. The excitable CS peak intensity versus the injected Gaussian beam amplitude.
point (0, 0) like the Cartesian coordinate. Therefore, the x and y coordinates can change between −64,−63,…, 0,…,63,64. 3. Switch ON CSs and comparison We switch on CSs by using of the semi coherent injection. In order to switch on a CS we added in (1-a) the term [23]: 2 2 2 Fin j = F0 eiϕ e[−(x−x0 ) −(y−y0 ) /2w ]
(2)
F0 is the amplitude of Gaussian pulse and ϕ and w are the phase and width for Gaussian pulse, respectively. This pulse injected in the point of (x0 ,y0 ). The duration of injection would be tinj . Therefore, we inject a Gaussian beam in center of the cavity transverse plane for 50 time units. The Gaussian beam parameters are F0 = 0.28, w = 4 and ϕ = 0. The pump parameter of active material has been selected μ = 4.9. We can switch on different types of CSs depend on the parameter ‘r’ [14, 21]. We choose three different values of r that r = b2 /b1 . Our simulations show that we have stable CS, self-pulsing CS and Excitable CS for r = 1.4, r = 1.7 and r = 2 respectively. In Fig. 1, we show and compare the field intensity evolution for three types of CSs (stable, self-pulsing and excitable CSs). Firstly, the field intensity is in noise level at t = 0. At t = 50 t.u. we inject the semi coherent injection during tin j = 50t.u. and the field intensity grows to values much larger than those of the stable CS. When the injection beam switched off, the field intensity falls to value comparable with noise level. Then, the field intensity (also, the populations) try to recovers the stationary value through the damped oscillations. Lastly, we have a stable CS. For excitable CS, the behavior is as same as the stable CS in the injection moment. But, we cannot see the initial peak of injection because the field intensity is very large and broad for the excitable CS. For creation of self-pulsing CS, we change the parameter r for the stable CS and the stable CS converts to self-pulsing CS. As shown in Fig. 1(a), stable CS remains stable for long time and the final amplitude is small (about 2 or 3) for the stable CS. In self-pulsing CS, the CS position is fixed on the injection point but the amplitude of electrical field oscillates with time. Firstly, the amplitude oscillation of the self-pulsing CS was irregular but we see the regular oscillation with fixed period and frequency by passing of time (Fig. 1(b)). The peak of self-pulsing CS gets fixed on a certain value that is bigger than the peak of stable CS. Finally,
the excitable CS has a peak that is varying larger than stable and self-pulsing CSs (about 120) (Fig. 1(c)). Our simulation show that the excitable CS peak is independent of injected Gaussian beam amplitude (Fig. 2). But the delay time between the beginning of injection and formation of the maximum amplitude peak is dependent on the Gaussian beam amplitude (Fig. 3). 4. Logical gates AND, OR A logical gate is a device that performs a certain Boolean logic operation on one or more logical input and produces a signal logical output. The relationship between input and output signals are displayed in what are known as truth tables [24]. Now we want to use the excitable CS to creating of a logical gate. We try to find the necessary conditions for realizing of the logic gate in CSL. For this goal we consider the presence of an excitable Cs as a bit ‘1’ and the absence of the excitable CS is as a bit ‘0 . Inputs of logic gate can be created by injecting of the Gaussian beam in the transverse plane of cavity. When two excitable CS are placed in suitable distance of each other’s, the mutual interaction between CSs creates output. In plots we consider the intensity value below 1 as a bit ‘0 and the intensity value over than 10 with significant amplitude as a bit ‘1’. Moreover, we consider three excitable CSs in a linear diagonal arrangement of transverse plane, whose position is fixed by the Gaussian beam. The two outer CSs will represent the input signals, while the central CS will be the output signal. When we change the excitability threshold and distance of inputs, the central CS (output) switch on in response to two inputs. Therefore, the central excitable CS (output) can be switched depend on the two outer diagonal CS (inputs). Finally, different logical operation can be achieved and we will have all optical logic gates by using of excitable CS. For example, when we inject two outer excitable CS (two input are 1), an excitable CS (output is 1) switched on in central of cavity transverse plane and we will have a AND logic gate. Fig. 4 shows this logical operation for AND gate. We injected two Gaussian beam with F0 = 0.35, w = 4 and φ = 0 in positions of (62,−62) and (−62, 62). We plot the spatial profile of intensity variable along the diagonal direction of transverse plane at time of 60 t.u. in Fig. 5. This profile has one central peak as output of AND logic gate. For OR logic gate, we inject one excitable CS and a weak injection as two inputs for logic gate (two input are 1 and 0) and
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Fig. 3. The delay time in formation of the excitable CS versus the injected Gaussian beam amplitude.
Fig. 4. Snapshots of the AND gate with two inputs ‘1’.This generates a ‘1’ bit in the output. The snapshots are ordered from left to right.
Fig. 5. Profile of intensity variable along the diagonal direction of transverse plane at time of 60 t.u.
Fig. 6. Snapshots of the OR gate with one input ‘1’and other one input 0. This generates a ‘1’ bit at the output. The snapshots are ordered from left to right.
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Fig. 7. Profile of intensity variable along the diagonal direction of transverse plane at time of 90 t.u.
then an excitable CS (output is 1) switched on in central of cavity transverse plane. Fig. 6 shows this logical operation for OR gate. We inject two Gaussian beam with F0 = 0.35, w = 4 and φ = 0 (this input is 1) and F0 = 0.001, w = 4 and φ = 0 (this input is 0) in positions of (42,−42) and (−42, 42) respectively. In Fig. 4 we show the operation of an OR gate when one inputs is ‘1 and the other ‘0 leading to a ‘1 bit at the output (center). The spatial profile of intensity variable has been plotted along the diagonal direction of transverse plane at time of 90 t.u. in Fig. 5. This profile has one central peak as output of OR logic gate. In Figs. 5 and 7, the peak of excitable CS does not corresponds with the point of (0,0). When we switch on two excitable CS as inputs for our gate, we create two very deep dips in the population profile. The population transfer between these dips causes to formation of central excitable CS. We can create the final excitable Cs in favorite position with suitable parameter for initial injection beams. Also, we can control the position of final excitable CS with controlling of the noise level. The other logic operations can be created in CSL easily. For example, the case (1, 0) can be confirmed in the logic gate AND. Because of symmetry, the output result would be same for couples cases (for example (0, 1) and (1, 0) inputs). The case of (1, 1) input would have a similar result for the AND and OR logic gates. We can express the results in terms of new parameter as exciting energy. This energy would be defined as E = |Fin j |2 × tin j for semicoherent injection. Therefore, this energy depends on the amplitude of injection beam, duration of injection and distance between CSs. It’s clear if change one of these parameters we must change other two parameters to compensation of necessary energy. In the proposed gate we used the excitable CS. Also, these gates operate in CSL with saturable absorber without holding beam. Therefore, this system is better than the previous system (with holding Beam). Because, we have a simpler and more compact system without holding beam that is easier to control and prevent of imposition significant energy to the system. As a result we expect thermal effects haven’t significant effect on our system. We studied OR and AND logic gate in our system. Two outer injections are inputs and the output forms itself in central of transverse plane of cavity. The input’s power and distance of two inputs are such that excitable excursion of inputs are enough to excite the output. The inputs show an excitable excursion which triggers an excitable excursion of the output. 5. Conclusions In this paper, we have switched excitable CS in the CSL with saturable absorber by selecting of the suitable parameter and then we have compared the time evolution of intensity for excitable,
stable and self-pulsing CSs. Our simulations show that the peak of excitable CS is independent of Gaussian beam amplitude, but it is dependent on r parameter. If we increase the parameter r, the peak of excitable CS increases too. We have proposed all optical logic gate AND and OR by using of the excitable CS. These logic gates can be used to perform all optical computing. The physical mechanism underlying the logic gates is interaction between the excitable CSs. the strength of interaction depend on the distance of two outer injection (inputs) and power of the injected Gaussian beams. Therefore, the truth of the AND and OR Logic gates can be performed by suitable selection of these parameters. References [1] Lugiato LA. IEEE J Quantum Electron 2003;39:193. [2] Babland S, Tredicce JR, Brambilla M, Lugiato LA, Balle S, Giudici M, et al. Nature 2002;419:669. [3] Brambilla M, Lugiato LA, Prati F, Spinelli L, Firth WJ. Phys Rev Lett 1997;79:2042. [4] Spinelli L, Tissoni G, Brambilla M, Prati F, Lugiato LA. Phys Rev A 1998;58:2542–59. [5] Hachair X, Barland S, Furfaro L, Giudici M, Balle S, Tredicce J, et al. Phys Rev A 2004;69:043817. [6] Hachair X, Furfaro L, Javaloyes J, Giudici M, Balle S, Tredicce J, et al. Phys Rev A 2005;72:013815. [7] Hachair X, Pedaci F, Caboche E, Barland S, Giudici M, Tredicce JR, et al. IEEE J Sel Top Quantum Electron 2006;12:339–51. [8] Bache M, Prati F, Tissoni G, Kheradmand R, Lugiato LA, Protsenko I, et al. Appl Phys B 2005;81:913–20. [9] Prati F, Caccia P, Tissoni G, Lugiato LA, Aghdami K, Tajalli H. Appl Phys B 2007;88:405–10. [10] Genevet P, Barland S, Giudici M, Tredicce JR. Phys Rev Lett 2008;101:123905. [11] Tissoni G, Aghdami K, Prati F, Brambilla M, Lugiato LA. Cavity soliton laser based on a VCSEL with saturable absorber. In: Localized states in physics: solitons and patterns. Berlin: Springer; 2011. p. 187–211. [12] Vahed H, Prati F, Turconi M, Barlan S, Tissoni G. Phil Trans R Soc A 2014;372:20140016. [13] Vahed H, Prati F, Tissoni G, Lugiato LA, Tajalli H. Eur Phys J D 2012;66:148–53. [14] Vahed H. IEEE Photonic Tech Lett 2015;27:2019–22. [15] Tanguy Y, Ackemann T, Firth WJ, Jager R. Phys Rev Lett 20 08;10 0:013907. [16] Elsass T, Gauthron K, Beaudoin G, Sagnes I, Kuszelewicz R, Barbay S. Appl Phys B 2010;98:327–31. [17] Gomila D, Jacobo A, Matias M, Colet P. Logical operations using excitable cavity solitons. Advanced photonics & renewable energy, OSA technical digest (CD). Optical Society of America; 2010. paper NME64. [18] Jacobo A, Gomila D, Colet P, Matias M. All optical logical operations using excitable cavity solitons. In: Photonics society winter topicals meeting series (WTM). IEEE; 2010. p. 122–3. [19] Eslami M, Kheradmand R. Opt Rev 2012;19:242–6. [20] Jacobo A, Gomila D, Colet P, Matias M. New J Phys 2012;14:013040. [21] Prati F, Tissoni G, Lugiato LA, Aghdami KM, Brambilla M. Eur Phys J D 2010;59:73–9. [22] Winfree AT. The geometry of biological time. 2nd ed. New York: Springer; 2001. [23] Aghdami K, Prati F, Caccia P, Tissoni G, Lugiato LA, Kheradmand R, et al. Eur Phys J D 2008;47:447. [24] Hunsperger RG. Integrated optics. 6th ed. Berlin: Springer; 2009.