Proposal for all-optical NOR gate using single quantum-dot semiconductor optical amplifier-based Mach–Zehnder interferometer

Optics Communications 285 (2012) 1710–1716

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Proposal for all-optical NOR gate using single quantum-dot semiconductor optical amplifier-based Mach–Zehnder interferometer E. Dimitriadou, K.E. Zoiros ⁎ Lightwave Communications Research Group, Department of Electrical and Computer Engineering, School of Engineering, Democritus University of Thrace, 12 Vas. Sofias Str., 67 100, Xanthi, Greece

a r t i c l e

i n f o

Article history: Received 27 August 2011 Received in revised form 8 November 2011 Accepted 28 November 2011 Available online 14 December 2011 Keywords: All-optical Boolean NOR logic Mach–Zehnder interferometer Modelling Modified Fredkin gate Quantum-dot semiconductor optical amplifier

a b s t r a c t The feasibility of realizing an all-optical NOR gate for 160 Gb/s return-to-zero data pulses using a single quantum-dot semiconductor optical amplifier (QD-SOA)-based Mach–Zehnder interferometer is theoretically investigated and demonstrated. The proposed scheme exploits a modified Fredkin gate driven not only by the pair of data streams between which the Boolean NOR function is executed but by the complement of one of these signals as well. A numerical simulation is conducted to evaluate the performance of the scheme against the extinction ratio and find for which choice of the critical data signals and QD-SOAs parameters this metric becomes acceptable. Provided that the specified requirements are satisfied, which is technologically feasible, the NOR gate can be realized for data signals of the same wavelength both with logical correctness and high quality. © 2011 Elsevier B.V. All rights reserved.

1. Introduction All-optical logic gates are fundamental units for realizing exclusively by means of light the signal processing functionalities required to fully exploit the great potential of optical fibers in the development of modern communications networks [1]. In particular the NOR is a universal gate that contributes decisively towards this goal, spanning from synthesizing any Boolean function [2] to building combinational photonic logic circuits [3] and managing packet contention [4] to bit-error monitoring [5], in the optical domain. As a result of its crucial multilateral role it has attracted intense research efforts, which have mainly relied on the nonlinear effects of semiconductor optical amplifiers (SOAs) [6–13]. However, the ultrafast capability of these approaches is inherently limited by the slow SOA gain recovery time, which compromises their performance. Although methods such as filter detuning [14,15], special modulation formats [16,17] and delayed interference [18,19] have been adopted to extend the speed of operation while mitigating the pattern-effect distortion, keeping pace with the excessively increasing single channel data rates [1] is a challenging task. On the other hand quantum-dot (QD) SOAs have compared to the conventional ones [20] an exceptional ultrafast response to launched data and exhibit a set of unique features [21]. Thus in this paper we propose to exploit this technology for executing Boolean NOR logic for return-to-zero (RZ) data pulses at 160 Gb/s by

⁎ Corresponding author. Tel.: + 30 25410 79 975; fax: + 30 25410 79 595. E-mail address: [email protected] (K.E. Zoiros). 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.11.122

incorporating these active devices in a Mach–Zehnder interferometer (MZI). This QD-SOA-assisted switching module has been predicted to be capable of Boolean NOR operation up to 250 Gb/s [22], but by cascading an OR and a NOT gate, which comes at the expense of an inevitable increase in complexity, power consumption, latency and footprint. However in order for a NOR gate to best serve the applications for which it is destined it would be preferable if it could be realized in a straightforward and accordingly more affordable manner by means of a single QD-SOA-based MZI switch, as it has been done for other all-optical logic functions [23–25]. For this purpose we have devised a circuit based on the MZI architecture recently exploited for the demonstration of a single modified Fredkin gate (MFG) [26] and on the fact that when conducting a logical operation between two data streams by means of a SOA-based interferometric structure the complement of the binary content of these signals can be used too as input to the configuration [27]. The feasibility of this design solution is theoretically explored and confirmed through numerical simulation, which aims at assessing the impact of the critical parameters on the chosen performance metric. Therefore the proposed scheme can constitute a possible option for the implementation of ultrafast NOR gate. 2. Concept of proposed NOR gate 2.1. Configuration and operation Fig. 1(a) shows the configuration of the proposed NOR gate. It is based on the MZI architecture, which in its general form has two inputs, C and D, and two outputs, R and S. A binary signal entering the

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Fig. 1. NOR gate proposed configuration (a) and truth table (b).

setup from inputs C and D is split via the symmetric input coupler C1 into a pair of equal parts, which travel separated along the identical QD-SOAs located in their path. At the same time two data-carrying signals, A and B, enter the upper and lower MZI arms, respectively, through a wavelength selective coupler (WSC). It is assumed that A and B are at least an order of magnitude stronger than C and D while their wavelength detuning from C and D in the 1550 nm region is less than the homogeneous broadening of QD-SOA1 and 2, respectively [23–25]. In this manner they are capable of effectively modifying the nonlinear optical properties of the respective QD-SOAs and imparting a phase difference between the copies of both C and D. When the divided replicas of C and D recombine at coupler C2 they interfere and the logical outcome that is produced at output port R is determined by two factors [26]. First, by the amount of the relative phase shift they have acquired, which in turn depends on the combination of the binary content of A and B. Second, by the availability of signals C and D which undergo the change of the QD-SOAs gain dynamics by A and B. Thus, for the case of input C only, if both A and B are ‘on’ or ‘off’ the initial phase balance between the components of C is preserved and the result is ‘0’, whereas if either A or B is ‘on’ this balance is broken and the result is ‘1’. Therefore this mode of operation corresponds to the XOR logic executed between A and B [23–25]. However, for the case of input D only, the result of the same physical behavior of the circuit is the logical conjugate of that obtained before, or equivalently that of XNOR logic between A and B, since for a signal coming from input D, R is the cross output port where destructive switching occurs [27]. Consequently in the presence of both inputs C and D the logical result at R is obtained from the superposition of the previous individual cases, as expressed by the Boolean algebraic equation [26] R ¼ C ðA⊕BÞ þ DðA⊙BÞ

ð1Þ

where the symbols ⊕ and ⊙ denote modulo-2 addition and its inverse, respectively. Now since A and B are by default the signals between which it is our intention to execute NOR logic, this means that we cannot intervene on them. Nevertheless we have the freedom to determine which signal from C and D will suffer the impact of A and B acting on the corresponding QD-SOAs, and what binary form of it would be most suitable in the context of the pursued goal. For this purpose we replace in (1) the four possible logical pairs of A and B trying to find if there is a condition in terms of C and D that permits to realize the truth table of A NOR B shown in Fig. 1(b). By doing this we obtain the values of C and D shown in the two columns in the middle of Fig. 1(b), where the symbol ‘X’ denotes the ‘don't care’ term. From these we deduce that Boolean NOR logic can actually be achieved with the proposed scheme provided that C = 0 and D = Ā. In fact by substituting the specified C

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and D in Eq. (1) and using the laws of Boolean algebra we verify ― 
AB þ A B B  ¼A  ¼A that R ¼ A þ B, which by definition equals A NOR B. This suggests that in order for the employed QD-SOA-based MZI to be configured as NOR gate for data A and B it is necessary that we launch from its input D the complement of data A while the other input C must be null. In practice the required complement of data A can be generated from the original data as described in the next subsection. It is noteworthy that unlike the NOR gate implementation with the MFG in [26] our proposed scheme does not require to combine A and B and inject them together into the same active element. In contrast A and B are inserted independently into a separate QD-SOA device, which offers some significant practical benefits. First, we can use the same wavelength for A and B to discriminate them from D inside the MZI, which reduces the required number of different laser sources to two against three in the MFG. This possibility translates in turn into an important saving in hardware resources, while [28] it contributes to the execution of NOR logic with reduced cost and complexity. Second, since each QD-SOA is now driven with only one strong signal, we can control and keep the level of light intensity going into it to a reasonable level. In this manner we can lessen the burden that would be imposed instead on its gain dynamics by the excessive sum of the power of two strong data signals, like A and B, if they were simultaneously launched into it, which could impair its performance and in particular its ultrafast capability [29]. 2.2. Generation of data A complement As it was highlighted in the previous subsection, the implementation of the proposed NOR gate scheme critically relies on the availability of the complement of data signal A. In order to generate this indispensable signal from the original data we cannot use another interferometric switch, like the QD-SOA-based MZI, since this would oppose the target objective of our work, which is to design the NOR gate by means of only one such module. Therefore it is necessary to adopt other ways for inverting signal A when it contains a sequence of RZ data pulses at 160 Gb/s bit rate. In this context there are three different technological options for realizing Boolean NOT operation on a data signal with such characteristics. More specifically, these include 1) Semiconductor optical amplifiers (SOAs) assisted by optical filtering, 2) Periodically poled lithium-niobate (PPLN) waveguides and 3) Highly nonlinear fibers (HLNF). Of these solutions the first one allows to employ a common SOA-based platform for the NOT plus NOR combination, which in turn can render the design of the whole system more functional, controllable, scalable and amenable to integration. Nevertheless each one has its own exceptional advantages and constitutes a potential candidate in the effort to obtain logical data inversion in the best possible way. For this reason there is no absolute criterion based on which we could favor one over the others and an unambiguous decision would require to make a thorough comparison between them. This task is however outside the scope of our paper, but the essence is that there is more than a single choice according to the needs of the specific application that the proposed NOR gate scheme is destined to serve. Thus in the following we provide for each considered nonlinear medium concise information about the configuration, operation and parameters of the corresponding circuit in which it is incorporated for achieving the desired ultrafast logical inversion: 1) Semiconductor optical amplifiers (SOAs) assisted by optical filtering This technological option exploits the attractive features of SOAs, either conventional or QD, to execute the Boolean NOT function by means of cross-gain modulation (XGM), which is simple and straightforward to implement according to the pump-probe configuration shown in Fig. 2(a) [30]. The operation of the scheme is as follows. Two signals, the pump (A) and the probe (B) at wavelengths λA and λB, respectively, are combined and enter the (QD-) SOA

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Fig. 2. Generation of data A complement using (a) (QD-) SOA assisted by optical filtering [32,33], and the principle of operation in (b) [31], (c) Periodically poled lithium niobate (PPLN) waveguide [36–38] and (d) Highly nonlinear fiber (HLNF) [40,41]. OBPF: Optical bandpass filter.

propagating in the same direction. The pump consists of data pulses and perturbs the (QD-) SOA gain in a way inverse to its binary content so that its complement is mapped on the probe at wavelength λB, which is selected using an optical bandpass filter (OBPF). The outcome of this process is logically described by the function ĀB [31], which indicates that the setup could be configured as NOT gate for data signal A to be used then as input D in our case. This can be achieved at 160 Gb/s by exploiting the fact that the pump pulses modulate not only the (QD-) SOA gain but its refractive index as well. As a consequence the transmitted probe acquires a chirp whose dynamics evolve on a shorter time scale than the SOA gain recovery. Thus by properly detuning the OBPF from the central probe wavelength it is possible to select the part of the broadened probe spectrum related to this fast chirp and enhance significantly the (QD-) SOA response. This in turn enables to overcome the speed limitations with the associated pattern-dependent performance degradation in conventional SOAs [32] and improve the efficiency of XGM and the integrity of the derived signal in QD-SOAs [33]. In the first case the potential of this technique for error-free operation was demonstrated [32] for 160 Gb/s RZ data pulses having a full-width at half-maximum (FWHM) pulsewidth of 1.9 ps, very close to those in our simulation, an average power of 4.8 mW and located at 1549.98 nm. Also the SOA was biased at 250 mA and the OBPF had a 1.4 nm bandwidth and was detuned to the lower wavelength side (blue shift) by 1.23 nm. In the second case, owing to the inherent capability of QD-SOAs to handle more conveniently ultrafast data pulses, the same functionality as in ref. [32] has been recently shown for relaxed requirements for the filter's shape and detuning [33]. This has been done for 160 Gb/s RZ data pulses of 2.5 ps FWHM (again quite close to those in our simulation), 4.5 dBm mean power and positioned at 1543 nm. The SOA was a 6.15-mm long device consisting of columnar QDs, so that it can be made polarization-independent, was biased at 2 A and had a small signal gain and output saturation power of the order of 25 dB and 19 dBm, respectively, around 1550 nm. The conducted eye diagram measurements have revealed that by slightly blue-shifting a common type filter of 1 nm FWHM by 0.35 nm versus the probe carrier then the output signal exhibits a very high Q2-factor of 19 dB, an improved extinction ratio, a suppressed noise level and a high power of 13 dBm, which can be controlled by a variable optical attenuator and reduced to the level required for further use. These promising results obtained for both cases suggest that the configuration of Fig. 2(a) could be well employed for realizing the logical inversion of data

signal A. However both schemes have used a continuous-wave (CW) beam for the probe aiming at achieving ultrafast all-optical wavelength conversion, and as a by-product the output signal is a mirror image of the original RZ data in the time domain, which cannot be used as input D in our proposed circuit for the NOR gate since all driving signals must have the same polarity. What we want instead is to obtain the exact logical complement of the data signal while preserving its initial format and coding. This issue can be resolved if the CW probe is replaced by a synchronized clock signal comprising of a train of periodic pulses [34]. In this manner the XGM process takes place according to the graphical representation in Fig. 2(b) [31], which eventually results in the desired logical inversion of data signal A. Therefore if the peak power of the RZ Gaussian-profile clock pulses is adjusted to match the average power, to which it is linked through the pulse duty cycle [35], of the probe used in the two reported filter-assisted XGM-based schemes [32,33] then the inverted data signal should be generated with similarly high performance and be available for input D of our proposed QD-SOA-based MZI NOR gate implementation. 2) Periodically poled lithium niobate (PPLN) waveguides This technological option exploits the high nonlinear efficiency, ultrafast response time, negligible spontaneous emission noise and complete bit rate transparency of parametric processes in compact PPLN waveguide devices to obtain logic operations, like the NOT, at data speeds of 160 Gb/s [36–38]. This is achieved by means of a combination of sum frequency generation (SFG) and pump depletion nonlinear mechanisms, which manifest in the generic two-pump configuration whose simplified block diagram is shown in Fig. 2(c). The operation of the scheme is as follows. Two synchronized input signals A and B comprising of RZ data pulses and whose wavelengths λA and λB are located in the 1550 nm band are injected into the PPLN waveguide. These pumps interact nonlinearly and SFG occurs under the quasiphase matching condition that can be set through a temperature controller attached to the waveguide [38]. If both A and B are at a high logical level, i.e. A = ‘1’ and B = ‘1’, then during the SFG process they are depleted and suppressed at the output, which is equivalent to a logical ‘0’, whilst in all other combinations of logical cases where SFG does not occur they retain their initial level [38]. This essentially means that the output data bit at λB is ‘1’ only when input A and B data bits are ‘0’ and ‘1’, respectively, which is logically expressed by the function ĀB. Consequently if signal B is a clock pulse stream so that it is continuously held to

E. Dimitriadou, K.E. Zoiros / Optics Communications 285 (2012) 1710–1716

a logical ‘1’ then we can implement the desired NOT gate for signal A after the corresponding wavelength is selected with an OBPF. By following some basic design rules extracted by carrying out a theoretical analysis [37] and using approximately PA = 5 W (by deduction to hyperbolic secant-shaped pulses, having 2 ps FWHM and a practically ideal extinction ratio (ER) as in our case), λA = 1550 nm, PB = PA·λA/λB, λB = 1538 nm and a 15 mm long PPLN together with the other operational characteristics of this device detailed in ref. [36] it has been shown through simulations that it is possible to successfully realize at 160 Gb/s the most complex of the logic operations, namely the XOR. This potential has been demonstrated experimentally as well through bit error rate measurements [38]. Thus by keeping signal B ‘on’ this gate can be degenerated to the simpler NOT with equally high performance in terms of the Q-factor and ER so that it could be readily used to invert data signal A, as required in our proposed scheme. 3) Highly nonlinear fibers (HLNF) This technological option exploits the almost immediate response of the Kerr effect in a fiber of high nonlinear coefficient constructed by special materials, such as chalcogenide glasses [39], to achieve in a simple and flexible manner ultrafast all-optical logic, including NOT, by means of cross-phase modulation (XPM) [40,41] according to the generic scheme shown in Fig. 2(d). The operation of the scheme is as follows. Two strong, data-modulated signals and a weaker probe clock signal, which are located at different wavelengths, λA, λB and λP, respectively, in the C-band, are inserted in a highly nonlinear fiber (HNLF). Due to the fiber nonlinearity these signals interact so that the probe acquires a time-dependent nonlinear phase shift through XPM and accordingly a chirp is imposed on it given by the sum of the separate contributions induced from each data signal [40]. As a result the spectrum of the probe is broadened to an extent that depends on whether none, one or two of the data signals are present. Thus by varying their power level and selecting or removing the proper spectral slice by tuning analogously an OBPF we can realize different logic functions. Especially for the NOT case that concerns us, only one data signal is present at the input of the fiber and the OBPF is positioned precisely at the original probe's wavelength. By conducting numerical simulation [40] and according to the detailed analysis and reasoning in [41] it has been predicted that it would be indeed possible to achieve logical inversion of RZ data at 160 Gb/s with a Q-factor and a contrast ratio of the order of 7 and 10 dB, respectively, for a HLNF having length 125 m, nonlinear parameter 34 W − 1 km− 1, loss 0.9 dB/km, zerodispersion wavelength 1550 nm, dispersion slope 0.03 ps/nm 2/km, and for data/probe RZ Gaussian-shaped pulses having power and wavelength characteristics of PA = 200 mW (by deduction to the 2 ps FWHM used in our simulation and assuming that the average power is the same), λA = 1525 nm, PP = 15 mW(with the same rationale as before), λP = 1560 nm, with the former signal advancing the latter by 0.25 ps. Therefore if the above circuit is designed according to these realistic values of the involved parameters then when a RZ data signal A at 160 Gb/s repetition rate is inserted then a high quality logical complement of it can be generated and used to drive input D in our proposed scheme. 3. Simulation In order to investigate the feasibility of the proposed NOR gate we have simulated its operation, starting from the power, PR (t), that emerges at port R of the QD-SOA-based MZI, when it receives a signal, whose power is PD (t), from input D. This is given by [26]

P R ðt Þ ¼

  

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 α G ðt Þ P D ðt Þ G1 ðt Þ þ G2 ðt Þ þ 2 G1 ðt ÞG2 ðt Þ  cos − LEF ln 1 2 4 G 2 ðt Þ

ð2Þ

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This relationship indicates that the calculation of the target power function requires the knowledge of the time-dependent gains experienced by the split constituents of D in QD-SOAs 1 and 2, G1 (t) and G2 (t), respectively, for a given QD-SOAs linewidth enhancement factor, αLEF. For this purpose we have exploited a comprehensive model that has been widely adopted by the scientific community for analyzing the performance of QD-SOA-based MZI all-optical gates [23–25]. This accounts for the propagation of a strong optical signal, like A and B in our case, through a QD-SOA (assumed for simplicity to have ideal facet reflectivity and negligible amplified spontaneous emission [23,25]) according to the photon rate Eq. (3) cited below [23–25], and for the concomitant change of the QD-SOA dynamics described by the 3-level rate equations for the electron transitions between the wetting layer (WL), excited state (ES) and ground state (GS), which are given by (4–6) [42], respectively: ∂SA;B ¼ gSA;B −aint SA;B ∂z

ð3Þ

NQ h ∂Nw J N ð1−hÞ N − w þ − w ¼ eLw τw2 Lw τ 2w τwR ∂t

ð4Þ

∂h Lw Nw ð1−hÞ h ð1−f Þh f ð1−hÞ − − þ ¼ NQ τw2 τ 2w τ21 τ12 ∂t

ð5Þ

2

gLw SA;B c ∂f ð1−f Þh fð1−hÞ f − − − ¼ pffiffiffiffiffi τ21 τ12 εr τ 1R NQ ∂t

ð6Þ

where z is the longitudinal direction along the QD-SOAs length, L, i.e. z = 0 stands for the input and z = L for the output facet of each QDSOA, and t is the local time measured in a coordinate system moving pffiffiffiffiffi with the pulse group velocity, V g ¼ c= εr , where c is the speed of light in vacuum and εr is the QD-SOAs material permittivity. The functions involved in the derivatives above are, the photon density of data signals A and B, SA,B = SA = SB, which is related to their power, PA,B = PA = PB, via SA,B = SA,B(z, t) = PA,B(z, t)/(AeffVghv) (where Aeff is the effective cross-section of the QD-SOAs and hν the photon energy), the electron density in WL, Nw, and the electron occupation probability in the ES and GS, h and f, respectively. Also g = gmax (2f − 1), where gmax is the maximum modal gain, aint is the material absorption coefficient, J the injection current density, e the electron charge, τw2 the electron relaxation time from the WL to the ES, NQ the surface density of QDs, Lw the effective thickness of the active layer, τ2w the electron escape time from the ES to the WL, τwR the spontaneous radiative lifetime in the WL, τ21 the electron relaxation time from the ES to the GS, τ12 the electron escape time from the GS to the ES and τ1R the spontaneous radiative lifetime in the QDs. The system of coupled Eqs. (3)–(6) is numerically solved in a stepwise manner for pulses of data A and B that belong to a 160 Gb/s RZ pseudo-random binary sequence (PRBS) of word length 2 7–1 and have a Gaussian power profile, PA, B(0, t) = Ppeakexp[− 4ln2(t/ TFWHM) 2], where Ppeak is their peak power and TFWHM = 2 ps is their FWHM. For this purpose each pulse of A and B is sampled over its period at discrete intervals while the correspondingly driven QD-SOA is divided into uniform spatial segments. Then the 4th order Runge– Kutta method is applied on the created spatio-temporal grid in order to find the amplification factors in both MZI arms, which by definition are G1(t) = SA(L, t)/SA(0, t) and G2(t) = SB(L, t)/SB(0, t). This procedure is followed for typical QD-SOA structure fixed parameters values taken from the literature on other QD-SOA-based interferometric gates [23–25], which include gmax = 14 cm − 1, aint = 2 cm − 1, τw2 = 3 ps, NQ = 5 × 10 10cm − 2, Lw = 0.25 μm, τ2w = 1 ns, τwR = 0.2 ns, τ21 = 0.16 ps, τ12 = 1.2 ps and τ1R = 0.4 ns. Finally G1(t)and G2(t) are replaced in (2) to find the light intensity at output port R for a signal D having the same shape and temporal

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characteristics as data A, B, and αLEF = 4.5, which lies in the range of the typical values for practical QD-SOAs with gain in the 1550 nm window [43]. 4. Results and discussion The evaluation of the performance of the scheme requires to select the appropriate metric. This can be done by considering from a qualitative perspective the logical outcome at port R of the MZI with respect to the truth table of the NOR gate (Fig. 1(b)). More specifically, for (A, B) = (1, 0) and (1, 1), and since D ¼A¼ 0, then PD(t) = 0 also, and according to (2) R becomes zero, which corresponds to fully extinguished spaces. On the other hand, for (A, B) = (0, 0), G1(t) ≈G2(t), and since D ¼A¼ 1, R is maximized, which corresponds to marks of uniform amplitude. Finally, and according to the principle of operation of the employed MZI, the impact of (A, B) = (0, 1) lies in between the previous cases. Therefore the reliable assessment of the performance of the proposed NOR gate dictates to compare the level of the marks and the spaces that occur at the exit of the NOR gate due to the logical pairs (A, B) = (0, 0) and (A, B) = (0, 1). The most suitable metric for this purpose is the extinction ratio between the minimum and maximum peak power of the marks and 1 0 spaces, Pmin and Pmax , respectively, which is defined as

Fig. 4. Variation of extinction ratio versus peak data power for different QD-SOAs lengths and current density 3.5 kA/cm2.

In order for the ‘1's to be unambiguously distinguished from the ‘0's and the NOR gate to be exploited in real applications with no operational constraints the ER must be over 10 dB [44]. The satisfaction of this requirement depends in turn critically on the peak power of data signals A and B as well as on the QD-SOAs length and injected current density. Thus in order to check if it is possible to achieve the specified minimum of the ER we have investigated and assessed the impact of these parameters on the ER by observing and interpreting the curves derived from the numerical simulation. These are shown in Figs. 3–5, where the composite form of the first two of them is attributed to the physical correlation between the involved parameters. Fig. 3 depicts the ER variation against the peak power of the data signals, Ppeak, for three different current density values and the QDSOAs length set to 4 mm. The ER is increased with power reaching its target minimum for a threshold of approximately 10 dBm, which is common for all curves. However the attained maximum is shifted to the right as the current density becomes larger. From a physical perspective this happens because the current density determines

the power required to alter the optical properties of a QD-SOA and properly saturate its gain, and the higher it is the more power is necessary for this purpose [45]. This fact also explains that as we move well enough into the falling slope of the curves a larger current density is necessary to enhance the ER and hence improve performance for a given power. After having increased up to a certain point, the ER is then reduced but the power margin within which it remains tolerable is broadened as more carriers are supplied, up to approximately 3 dB. This in turn allows to select the peak data power from a wider range of permissible values, which potentially offers greater flexibility in the design of the NOR gate. Following Fig. 3 and its implications, J = 3.5 kA/cm 2 is accordingly chosen to monitor the dependence of the ER on the QD-SOAs length, which is illustrated in Fig. 4 for a triplet of cases. As before the curves exhibit a bell-like shape, which is attributed to the different level of saturation that the QD-SOAs are brought into as the peak power of signals A and B is altered. This in turn results in respective excursions of the phase difference induced on the clock replicas relative to the borderline of π, thus affecting likewise the magnitude of switching at port R and consequently the form of the ER therein, which is dropped on either side of a maximum. The latter, on the other hand, is shifted to the left as the QD-SOAs length is increased, and hence a longer device is required to obtain the specified minimum of the ER with smaller power, which is desirable from a practical standpoint as it can be provided in a more affordable way. This is explained by the fact that under a given current density the saturation power of a

Fig. 3. Variation of extinction ratio versus peak data power for different current densities and QD-SOAs length 4 mm.

Fig. 5. Variation of extinction ratio versus current density for peak data power 7.5 dBm and QD-SOAs length 4.5 mm.

ERðdBÞ ¼ 10log

P 1min P 0max

! ð7Þ

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QD-SOA is decreased with the increase of its length [45], which means that it can be saturated more easily or equivalently with lower power. The primary concern for realizing the NOR gate by providing the lowest possible peak data power subsequently justifies to set Ppeak = 7.5 dBm, for which the ER threshold of 10 dB is surpassed for QD-SOAs length 4.5 mm. Then it is necessary to simultaneously adjust the current density so that the joined combination of the involved parameters ensures good performance. This is done with the help of Fig. 5, where it is observed that the ER is sharply increased with a large slope for small current densities, and after exceeding its defined minimum it becomes almost independent on this parameter because there is a redundancy of supplied carriers and the QD-SOAs have been sufficiently saturated. For the current density minimum integer value of 2 kA/cm 2, or equivalently for a bias current of 270 mA, which is reasonable for practical QD-SOA devices, it is guaranteed that the ER is acceptable. From the analytical interpretation of Figs. 3–5 it is deduced that a suitable selection of the involved critical parameters is P2 peak = 7.5 dBm, L = 4.5 mm and J = 2 kA/cm . With this choice the NOR logic can be executed between the indicative 16-bit-long segments inside the PRBS of data A and B shown in Fig. 6(a) and (b), respectively, with an adequate ER of 10.4 dB, which is reflected on the profile of the obtained pulses in Fig. 6(c). In fact the largest amount of power emerges at the bit slots of A and B that are both empty, as expected according to the NOR truth table. Furthermore, the obtained marks have the same height while the spaces are strongly suppressed or fully extinguished when they originate from the binary data pairs (A, B) = (0, 1) or (1, 0), (1, 1), respectively. These observations designate that the outcome of the NOR gate is logically correct and of high quality. Moreover, in direct analogy to these results, the corresponding pseudo-eye diagram [46] depicted in Fig. 7(b) is open and resembles that of the original data in Fig. 7(a), with the marks having a negligible [47] peak difference of 0.19 dB and the spaces a single envelope. Finally, the aforementioned choice of critical parameters determines the QD-SOAs operating point. This is defined by the extent, Gsat, that the gain of each QD-SOA drops from the unsaturated level and by how much, ΔP, the required peak data power deviates from the 3 dB saturation input power under pulsed mode of operation. Thus by plotting the gain response of each QD-SOA to the corresponding data pattern of Fig. 6 and the gain variation as a function of the launched peak data power, it can be found that Gsat ≈ 5.3 dB from Fig. 8 and ΔP ≈ 4 dB from Fig. 9. Notably in this saturation regime the variation of the gain of each mark is quite uniform and independent of the values of the preceding alternating data bits, which enables to achieve pattern-free NOR operation to a good extent as well, in consistency with Fig. 7(b).

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Fig. 6. Simulation results for C = 0, D = Ā, and specified suitable choice of critical parameters: (a) data stream A, (b) data stream B, (c) logical outcome of A NOR B operation.

metric to be acceptable the peak data power must be 4 dB higher than the 3 dB saturation input power of the QD-SOAs, which additionally must be of medium length and biased with a moderate current. Provided that these requirements are satisfied, which is technologically feasible, the proposed scheme enables to realize the NOR gate both with logical correctness and high quality. This is achieved in a straightforward, affordable and efficient manner between data signals of the same wavelength, which helps reduce complexity and impose less strain on the QD-SOAs gain dynamics. Therefore and owing

5. Conclusion In conclusion the feasibility of realizing an ultrafast all-optical NOR gate with a single QD-SOA-based MZI has been theoretically investigated and demonstrated. The proposed scheme exploits a modified Fredkin gate configured in such way that among its four available inputs, two receive the data signals between which the considered Boolean logic is executed while the third, which is switched at the crossed output port as a result of this function, carries the complement of the first data input, and the last one is null. The operation of the scheme was numerically simulated by taking into account the propagation of data pulses through a QD-SOA and the modification of the optical properties of the latter by the former, in the context of an interferometric structure. In this manner a set of curves was obtained for the variation of the extinction ratio against the critical data signals and QD-SOAs parameters. From their analytical interpretation it has been specified that in order for the defined performance

Fig. 7. Simulated pseudo-eye diagrams of (a) data A and B, (b) A NOR B operation that corresponds to Fig. 6(c).

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Fig. 8. Gain response of (a) QD-SOA1 and (b) QD-SOA2 in upper and lower MZI arms to data streams A and B of Fig. 6(a) and (b), respectively.

Fig. 9. Variation of QD-SOAs gain versus peak data power for specified suitable choice of other critical parameters.

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