We study numerically the dynamics of a vertical-cavity surface-emitting laser (VCSEL) with optical injection and show that the interplay of polarization bistability and noise yields a reliable logic output to two logic inputs. Specifically, by encoding the logic inputs in the strength of the light injected into the suppressed polarization mode of the VCSEL (the so-called ‘orthogonal’ injection), and by decoding the output logic response from the polarization state of the emitted light, we demonstrate an all-optical stochastic logic gate that exploits the ubiquitous presence of noise. It gives the correct logic output response for as short as 5 ns bit times when the dimensionless spontaneous emission coefficient, βsp, is within the range 10−4-10−1. Considering that typical values of βsp in semiconductor lasers are in the range 10−5-10−4, the VCSEL-based logic gate can be implemented with nowadays commercially available VCSELs, exploiting either their intrinsic noise, or external and background noise sources.
© 2012 OSA
In nonlinear systems the interplay between bistability and noise can result in non-trivial noise-induced effects which can be potentially exploited for applications [1–4]. A recent example is the numerical demonstration of a stochastic logic gate using a vertical-cavity surface-emitting laser (VCSEL) that exploits the interplay of polarization bistability, noise, and pump current modulation .
VCSELs emit linearly polarized light with the direction of the polarization along one of two orthogonal directions associated with crystalline or stress orientations. Some VCSELs display, when the pump current increases, a polarization switching (PS) to the orthogonal polarization. The PS is often accompanied by hysteresis  and it has been shown that the switching points and the size of the hysteresis region depend on the pump current sweep rate [6, 7].
When a VCSEL is subjected to optical injection, such that only the suppressed polarization receives injection, for appropriated injection parameters a polarization switching from the solitary laser polarization (in the following referred to as X) to the orthogonal one (Y) can also occur , either when the optical injection strength is increased [4, 9], or when the wavelength of the injected light is varied [10, 11]. With this configuration, which has been referred to as orthogonal injection, polarization bistability and hysteresis cycles have also been observed, that can be exploited for all-optical buffer memories [12, 13].
In the implementation of a VCSEL-based stochastic logic gate proposed in Ref , two logic inputs were encoded in an aperiodic three-level signal (i.e., the sum of two aperiodic square waves representing the two inputs) that was directly applied to the laser pump current, and the logic output was decoded from the polarization state of the emitted light (e.g., the output is a logic 0 if the laser emits the X polarization or a logic 1 if it emits the Y polarization). In Ref . it was shown that the laser gave the correct logic response with a probability that was controlled by the level of noise, and that was equal to one in a wide range of noise levels. This phenomenon, which has been referred to as logic stochastic resonance (LSR) , is receiving a lot of attention because it occurs in several natural stochastic bistable systems [14–16].
The aim of the present work is to demonstrate an all-optical implementation of the VCSEL-based logic gate. With this aim we use the spin-flip model  to simulate the polarization-resolved nonlinear dynamics of a VCSEL under orthogonal optical injection. We demonstrate the phenomenon of LSR, by which the laser gives the correct logic response (encoded in the polarization of the emitted light), to two logic inputs that are encoded in the injection strength. We find that the all-optical configuration can work almost one order of magnitude faster than the electro-optical counterpart described in Ref .
The model rate-equations are described in Sec. 2, and the results of the simulations are presented in Sec. 3. We show that for adequate parameters the polarization of the light emitted by the VCSEL switches between the X and Y polarizations in response to changes in the optical injection strength into the Y polarization. Three injection levels allow for codifying the two logic inputs and then obtaining the logic output of an OR gate (or of an AND gate) in the form of the emitted polarization, as displayed in Table 1 .
Key model parameters are the angular frequency detuning between the lasers, Δω, the strength of the optical power injected into Y polarization, Pinj = |Einj|2, the bit time Tbit, and the dimensionless spontaneous emission coefficient, βsp., which will also be referred to as the noise strength. We show that an optimal value of the detuning Δω allows operating the VCSEL-based logic gate with minimum switching injection power (i.e., minimum Pinj). We show that, as can be expected in a LSR phenomenon, an adequate value of βsp, allows for the correct operation of the logic gate, i.e., the laser emits the correct output polarization only when the noise strength is within a certain range of values, which depends on Tbit. If the noise is below this optimal range, errors occur in logic output, which are due to delays in the polarization switching; if the noise is stronger than the optimal range, errors can also occur due to the emission of both polarizations simultaneously. Typical values of the spontaneous emission coefficient in semiconductor media are of the order of 10−4-10−5 [17, 18], and our simulations demonstrate that the VCSEL-based logic gate can correctly process bits with a bit-time of the order of 4-5 ns when the noise strength is within the range 10−4-10−1. Therefore, the VCSEL-based logic gate proposed here could be implemented with nowadays commercially available VCSELs, exploiting either their intrinsic noise, or external and background noise sources.
2. Model Equations
The spin-flip model (SFM) rate-equations, extended to take into account Y-polarized optical injection, are :
While the SFM model makes strong simplifications regarding the dynamics of real VCSELs, many detailed comparisons between experimental measurements and simulations [19–27] have demonstrated that the model can indeed successfully capture the main features of the polarization-resolved nonlinear dynamics of VCSELs (stochastic polarization switching, current-induced polarization switching, two-mode emission, bistability and hysteresis phenomena).
The optical power injected into the Y polarization is represented by Pinj = Einj2. The model equations are written in the reference frame of the injected field, and thus the detuning Δω is the difference between the optical frequency of the injected field and the frequency intermediate between the X and the Y polarization. Without optical injection and with γa = 0, the angular optical frequencies of the X and the Y polarizations are -γp and + γp respectively, and therefore, Δω = -γp (Δω = + γp) means that the injected field is resonant with the X (Y) polarized mode of the solitary VCSEL .
The two logic inputs are encoded in the injected power via a three-level signal, with the injection levels defined as follows. Let’s consider that Εinj(t) is the sum of two aperiodic square-waves, Ε1(t) + Ε2(t), that encode the two logic inputs. Since the logic inputs can be either 0 or 1, we have four distinct input sets: (0, 0), (0, 1), (1, 0), and (1, 1). Sets (0, 1) and (1, 0) give the same value of Εinj, and thus, the four input sets reduce to three Εinj values. Then, it is more convenient to introduce as parameters Εc and ΔΕ, such that the three injection levels are Εc-ΔΕ, Ec, and Εc + ΔΕ. For simplicity we chose Εc = ΔΕ, and therefore the 3 injection levels reduce to 0, ΔΕ, and 2ΔΕ. The duration of a bit is defined as Tbit = T1 + T2, where T1 is the time interval during which the injected power is constant and T2 is the duration of the ramp (up or down) to the injected power encoding next bit [with T2<<T1, as shown in Fig. 1(b) ].
The laser parameters used in the simulations, which are kept constant unless otherwise specifically indicated, are: k = 300 ns−1, µ = 2.5, α = 3, γa = 0.5 ns−1, γs = 50 ns−1, γp = 30 rad/ns, γn = 1ns−1. For these parameters the solitary laser emits the X polarization. Unless otherwise indicated, the injection parameters are Δω = 12 rad/ns, ΔE = 0.0015, Tbit = 5 ns, and T2 = 0.5 ns.
Figure 1 shows a typical time-trace of the X polarization (red solid line) when a three-level signal varies the injection strength (black dashed line). The dynamics is shown for three noise levels [βsp = 10−5 in Fig. 1(a), βsp = 0.1 in Fig. 1(c) and βsp = 1 in Fig. 1(e)]. Notice that the X polarization turns on when the injected signal is in level 1 (i.e., no optical injection) and turns off in the other two levels, when the Y polarization turns on. The intensity of the Y polarization (not shown for clarity) is either off (when the X polarization is on) or displays fast oscillations (when the X polarization is off).
In Fig. 1(a), the asterisks indicate the bits when the laser emits the wrong polarization. The criterion used to determine whether the laser emits the right polarization to a given input is as follows: The laser logic output response is considered correct if, when X is the polarization that has to be emitted according to Table 1, 80% or more of the total power is emitted in the X polarization; and in the bits when Y is the correct polarization, 20% or less of the total power is emitted in the X polarization. In other words, the response of the laser is determined only by detecting the intensity of the X polarization, being above or below a certain value. In Fig. 1(b) it can be observed that a wrong bit is due to the delay in the switching, as there are oscillations during the switching, but with larger noise the polarization switching is faster [Fig. 1(d)], but increasing still more the noise strength it plays a worst effect with delays in switching off the polarization mode [Fig. 1(f)].
Figure 2 displays the probability of success of the OR logic gate as a function of the bit time [Figs. 2(a) and 2(b)] and as a function of the noise strength [Figs. 2(c) and 2(d)]. The success probability was calculated over seven realizations of 500 bits and using 3 criteria for determining if the laser logic output response is correct: 90/10; 80/20 and 70/30, where the first number indicates the minimum percentage of light emitted in the X polarization during a bit, if X is the correct polarization, and the second number, the maximum percentage emitted in X if X is the wrong polarization.
As a function of the bit time, in Figs. 2(a) and 2(b) one can observe that for the less strict criterion (70/30, blue line) the probability grows to one for smaller bit times as compared to the more restrictive criteria (80/20 and 90/10, black and red lines respectively). One can also observe that the minimum bit time, with the 80/20 criterion as employed in Ref , is about 5-7 ns depending on the noise strength, which is significantly lower than that found in , where the logic inputs were applied as a direct modulation of the pump current, which is was about 20-30 ns.
As a function of the noise strength, one can observe in Figs. 2(c) and 2(d) that, for the 70/30 and 80/20 criteria, the probability of a correct response has a wide flat region where P = 1. Considering the criteria 80/20, in Fig. 2(c) the plateau with P = 1 occurs for βsp = 10−4 – 0.1, which clearly demonstrates the robusness of the correct operation of the logic gate.
Figure 3 displays the probability of correct logic operation represented with color code, in a two-dimensional parameter space: as function of the noise strength and the bit time, in Figs. 3(a) and 3(b); and as a function of the bit time and the injection strength, in Figs. 3(c) and 3(d). Figures 3(a) and 3(b) are done with two injection strengths (respectively ΔE = 0.0015 and ΔE = 0.005), and Figs. 3(c) and 3(d), with two noise levels (βsp = 10−4 and 0.1 respectively).
The squares in Fig. 3(a) indicate the parameters used in Figs. 1(a), 1(c) and 1(e), and show that for a noise level of βsp = 10−5 the minimum bit time with probability of correct operation equal to 1 is 7 ns, counterintuitively with βsp = 0.01 the minimum bit time decreases to 4 ns, and thus noise can improve the speed of the logic gate. Figure 3(b) shows that, with large enough optical injection, correct operation of the logic gate can be obtained even when the laser is under the influence of very strong noise, higher than βsp = 1 [notice the difference in the right-top corner of Figs. 3(a) and 3(b)].
In Figs. 3(c) and 3(d), which display the success probability as a function of the bit time and ΔE, for weak (βsp = 10−4) and for strong (βsp = 0.1) noise respectively, one can observe that if the injection strength is below a threshold at about ΔE = 0.001, the laser does not give the correct logic response, regardless of the bit time or the noise strength. The region with probability equal to 1 is wider in Fig. 3(d) (starting with bit times as low as 3 ns) than in Fig. 3(c) (that starts at 6 ns), because moderated noise levels (βsp up to 0.01) are beneficial by reducing the switching delay and allowing for correct operation with shorter bit times.
In order to demonstrate that the correct operation does not require a fine tuning of the laser parameters, in Fig. 4 we present the success probability as a function of various parameters: in captions (a) and (b), as a function of the injection strength and the frequency detuning, for two bit times (7 and 5 ns); in caption (c), as a function of the birefringence and spin-flip parameters, and in caption (d) as a function of the dicroism and the pump current parameters. In Fig. 4(a) where we used a bit time of 7 ns, there is a wide red area where the success probability if equal to 1, but with a bit time of 5 ns [Fig. 4(b)], this red region becomes more narrow. One can also notice this region is very similar to the stable injection-locking region in a laser with cw optical injection. Since γp = 30 rad/ns and the optical frequency of the Y polarization ≈γp, in the red region the injected field is resonant with the Y polarization).
Figures 4(c) and 4(d) show that the correct operation of the logic gate is robust to variations of the birefringence, dicroism, the spin-flip rate and the pump current parameters, as a wide red region is observed corresponding to a success probability equal to one.
We have shown that in a VCSEL with orthogonal optical injection, there is a wide region of noise strengths where the probability of success of logic gate operation function is equal to 1. We also demonstrated that intrinsic laser noise and external noise can be exploited for improving the performance of the logic gate, giving a reliable and correct logic response, evenly robust to variations of the laser parameters. For too low or too high noise strength, some mistakes occur (see Fig. 1), which are due to either to turn-on or to turn-off delays. In the opto-electronic stochastic logic gate proposed in Ref , the minimum bit time for successful operation was about 30-40 ns, whereas in the optical implementation proposed here, the probability equals 1 for much shorter bit times, as 5-7 ns (see Fig. 3). Therefore, the all-optical implementation presents a clear advantage for applications where noise is unavoidable and fast switching is required.
This research was supported in part by the Academia Program of the ICREA foundation, Generalitat de Catalunya; the Air Force Office of Scientific Research through project FA8655-12-1-2140, the Spanish Ministerio de Ciencia e Innovacion through project FIS2009-13360-C03-02, by the Agencia de Gestio d'Ajuts Universitaris i de Recerca, Generalitat de Catalunya, through project 2009 SGR 1168.
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