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We experimentally demonstrate an all-optical temporal computation scheme for solving 1st- and 2nd-order linear ordinary differential equations (ODEs) with tunable constant coefficients by using Fabry-Pérot semiconductor optical amplifiers (FP-SOAs). By changing the injection currents of FP-SOAs, the constant coefficients of the differential equations are practically tuned. A quite large constant coefficient tunable range from 0.0026/ps to 0.085/ps is achieved for the 1st-order differential equation. Moreover, the constant coefficient p of the 2nd-order ODE solver can be continuously tuned from 0.0216/ps to 0.158/ps, correspondingly with the constant coefficient q varying from 0.0000494/ps2 to 0.006205/ps2. Additionally, a theoretical model that combining the carrier density rate equation of the semiconductor optical amplifier (SOA) with the transfer function of the Fabry-Pérot (FP) cavity is exploited to analyze the solving processes. For both 1st- and 2nd-order solvers, excellent agreements between the numerical simulations and the experimental results are obtained. The FP-SOAs based all-optical differential-equation solvers can be easily integrated with other optical components based on InP/InGaAsP materials, such as laser, modulator, photodetector and waveguide, which can motivate the realization of the complicated optical computing on a single integrated chip.
© 2015 Optical Society of America
1. Introduction
All-optical system for signal processing and computing is capable of overcoming the limitations of the traditional electronic system in term of processing speed and power consumption [1,2]. Just as the fundamental building blocks used in multi-functional electronic system, many equivalent devices in photonics have been proposed and demonstrated in recent years, including all-optical temporal differentiator [3–7] and all-optical temporal integrator [8–14]. Numbers of interesting applications based on these building blocks have been investigated, such as ultrafast optical pulse processing [3,4] and shaping [8], optical memory element [9,10] and real time optical computation of ordinary differential equations (ODEs) [15–21]. Among these applications, the ODE solver could be extremely useful, since the ODEs play an irreplaceable role in virtually any field of science or engineering, such as physics, chemistry, biology, economy and medical sciences [22].
Several schemes have been proposed to realize all-optical ODE solvers. The first one took advantage of an integrator based on fiber gratings [2,15]. The second one was based on an optical feedback loop [16], in which an optical intensity differentiator and a wavelength converter were needed. However, both these two schemes are incapable of being integrated due to their bulky and complex configurations. Recently, on-chip ODE solvers on the Si platform were also proposed [17–21]. It should be noted that the two of the most important figure-of-merits of ODE solvers are tunability and expandability, correspondingly to the ability of solving ODEs with variable constant coefficients and solving high-order linear ODEs respectively. However, only one specific 1st-order linear ODE with an invariable constant coefficient has been solved in [17,18]. Tan in our group [19] employed two cascaded micro-ring resonators (MRRs) on the Si based platform to solve the 2nd-order linear ODE, but only a pair values of constant coefficients (p and q) were demonstrated, and the tunability was not demonstrated. ODE solvers with tunable constant coefficients were further realized with a carrier injected silicon MRR [20] and a MRR integrated with two tunable interferometric couplers [21]. However, only 1st-order linear ODEs were experimentally solved in these two schemes, and the constant coefficient is only tuned from 0.038/ps to 0.082/ps in [20].
Here, based on the Fabry-Pérot semiconductor optical amplifier (FP-SOA), which has prominent advantages of tunable gain [23,24] and easy to be cascaded [25], we demonstrate the 1st- and 2nd-order linear ODE solvers with tunable constant coefficients. Note that these ODE solvers directly employ the transfer function of the FP-SOA, which are different from the traditional ODE solvers based on feedback loops [16,17]. When different currents are injected into FP-SOAs, the signal gains are changed, and thus the constant coefficients of the solved ODEs are tuned. Moreover, by cascading FP-SOAs, higher-order linear ODEs can also be solved. In order to analyze the FP-SOAs based ODE solvers, a theoretical model combining the carrier density rate equation of SOA with the transfer function of the FP cavity is constructed. For both 1st- and 2nd-order solvers, all the calculated output waveforms fit perfectly with the experimental results. Additionally, the FP-SOA is easily integrated with other active or passive components on the InP/InGaAsP platform [14,26,27], indicating the proposed scheme is potential to provide a path for realizing fully integrated optical computing system.
2. Operating principle
The constant coefficient 1st-order linear ODE is defined as
where is the input signal, is the output signal (equation solution), and k represents a positive constant of an arbitrary value referred as the constant coefficient.The differential equation Eq. (1) can be solved either in the time domain or in the frequency domain. Usually, an optical feedback loop with a differentiator or an integrator is required for all-optically solving Eq. (1) in the time domain. An all-optical ODE solver can also be demonstrated by simply taking advantage of the response of a specific device (an optical resonant cavity) in the frequency domain. The system transfer function of the 1st-order linear ODE solver is numerically achieved by using Fourier transform of Eq. (1):
where , and ω is the optical angular frequency. It can be seen that Eq. (2) represents the transfer function of a temporal integrator when k = 0 [12]. Therefore, an all-optical ODE solver is possible to be achieved by using an optical integrator operating without lossless condition, which means . An FP-SOA is a promising optical temporal integrator when it is operated at the exact lasing threshold condition (k = 0) [14,15], indicating the FP-SOA is capable of solving 1st-order ODE when it is operated slightly below the lasing threshold condition ().The transfer function of an FP-SOA can be expressed as [23]
where is the single pass gain of the FP-SOA. R1, R2 are the reflectivities of the input and output mirrors, respectively. is the optical confinement factor, is the gain coefficient related to the carrier density of the FP-SOA, represents the internal loss, and l is the cavity length. is the single pass time delay, ng is the group refractive index of the FP-SOA and c is the light speed in vacuum. When the angular frequency of the input lightwave (ω) is very close to one resonant frequency of the FP-SOA (ω0), we haveObviously, the transfer function (Eq. (4)) of an FP-SOA has the same formation with Eq. (2). Therefore, a 1st-order linear ODE can be solved by using a single FP-SOA, and the constant coefficient is replaced by:As an active component on the InP platform, the single pass gain (Gs) of the FP-SOA can be easily controlled by the injection current with a large tunable range [23,24]. Especially, the FP-SOA is capable of being operated at the exact lasing threshold condition (k = 0) when the injection current is large enough [14]. From Eq. (5), it is obvious that the constant coefficient (k) is changed with the Gs, which can be tuned in a large range by varying the injection current. The previously reported all-optical ODE solvers with tunable coefficients were meanly based on passive components on the Si platform [20,21], and the tunable ranges were relatively small due to the fact that there were no gains to compensate the losses. Consequently, the constant coefficients of the FP-SOAs based ODE solvers can be practically tuned in larger ranges.Since the linear ODE solvers based on integrators are linear devices, in general a Nth-order linear ODE can be solved by simply cascading N identical 1st-order linear ODE solvers, which are FP-SOAs here. Figure 1 shows the schematic diagram to solve a 2nd-order linear ODE by using two cascaded FP-SOAs. The inputs and outputs of the two FP-SOAs are governed by
where k1 and k2 are the constant coefficients of the first and second linear ODEs based on FP-SOAs, respectively. By simply inserting the from Eq. (7) into Eq. (6), we havewhere , . It is obvious from Eq. (8) that the output of the second FP-SOA is the solution of a 2nd-order linear ODE. Since the two cascaded FP-SOAs (FP-SOA1 and FP-SOA2) used in the 2nd-order ODE solver are separately controlled by different injection currents, the constant coefficients of the 2nd-order ODE can be flexibly tuned in a large range.3. Theoretical model and numerical analysis
Since the constant coefficient of the FP-SOA based ODE solver is determined by the single pass gain, the key to realize the ODE solver with tunable constant coefficients is to obtain the relation between the gain of SOA and the controllable parameter, which is injection current here. The gain characteristics of SOA have been investigated by numbers of theoretical models that employing the carrier density rate equation [28]:
where N is the carrier density, I is the injection current, e is the unit charge, and is the carrier lifetime. is the volume of the active region of SOA, w and d are the width and thickness of the active region, respectively. is the group velocity. is the gain coefficient, is the differential gain coefficient and N0 is the transparent carrier density. S is the photon density and can be expressed as , where A is the amplitude of the light signal, h is the Planck constant, and ν is the frequency of input light. From Eq. (9), we can see that when different currents are injected into the FP-SOA, the carrier density is changed to make the varied. Then, from the expression of Gs and Eq. (5), it is obvious that the constant coefficient is changed with the injection current.Based on Eqs. (5) and (9), we have built a model to theoretically solve the 1st- and 2nd-order linear ODEs with tunable constant coefficients by adjusting the injection currents of FP-SOAs. The main parameters of the two FP-SOAs that we used in the experiment are listed in Table 1.
Table 1. Parameters of the two FP-SOAs
Although both the Gaussian and super-Gaussian pulses can be used as the input signal of in Eq. (1), the difference between the input and output signals is more observable for the super-Gaussian pulse [19]. Thus the super-Gaussian pulse is employed as the input signal for both simulation and experiment. Figure 2 shows the measured input super-Gaussian waveform (green curve), in which the blue dashed line is the fitted waveform used as the input waveform in the numerical analysis. The super-Gaussian pulse at a repetition rate of 2.5 GHz has a peak power of 0.5mW. The calculated constant coefficients of the two 1st-order linear ODEs versus injection currents are depicted in Figs. 3(a) and 3(b). It can be seen that the constant coefficients are continuously changed with the injection currents, which indicating the constant coefficients can be continuously tuned. Since the gains of FP-SOAs increase with the injection currents, the constant coefficients decrease with the injection currents, which are evidenced by Figs. 3(a) and 3(b). It should be noted that the injection current of FP-SOA2 is quite lower than that of FP-SOA1. Actually, FP-SOA2 shows an excellent performance in the experimental test, and a high extinction ratio of 45 dB at the wavelength of 1551.2 nm is obtained with a low injection current of 42 mA.
Fig. 2 Measured input super-Gaussian waveform (green curve), the blue dashed line is the fitted waveform, which is the input waveform in the numerical analysis.
Fig. 3 The calculated constant coefficients of the 1st-order ODE solvers based on FP-SOAs with different injection currents: (a) for FP-SOA1, (b) for FP-SOA2. (c) and (d) are the simulated output waveforms of the 1st-order ODE solvers based on FP-SOA1 and FP-SOA2 with different injection currents, respectively. (e) is the simulated output waveforms of the 2nd-order ODE solver based on cascaded FP-SOAs with different injection currents combinations.
In order to achieve observable difference between the output waveforms with different constant coefficients, several specific injection currents are chosen for FP-SOAs. When the injection current of FP-SOA1 is chosen as 120 mA, 150 mA, 160 mA and 165 mA, corresponding to the constant coefficient of 0.073/ps, 0.051/ps, 0.031/ps and 0.019/ps respectively, the simulated output waveforms are illustrated in Fig. 3(c). When the injection current of FP-SOA2 is chosen as 20 mA, 40 mA, 45 mA and 48 mA, corresponding to the constant coefficient of 0.085/ps, 0.043/ps, 0.019/ps and 0.0026/ps respectively, the simulated output waveforms are illustrated in Fig. 3(d). From Figs. 3(c) and 3(d), it is obvious that the output waveforms are distorted comparing with the input waveform and are broadened when the injection currents increase for the two FP-SOAs. Specifically, when the injection current of FP-SOA2 is higher than 48 mA, the constant coefficient k is very close to zero, indicating that FP-SOA2 with high injection current is capable of acting as an integrator.
Next, the 2nd-order linear ODE with tunable constant coefficients is numerically solved by cascading FP-SOA1 and FP-SOA2 (As shown in Fig. 1). The constant coefficients p and q are controlled by the injection currents of the two FP-SOAs. Any value of p in the range of 0.0216~0.158/ps and the corresponding q can be obtained in this scheme. Here four specific combinations of the injection currents are chosen for the two FP-SOAs. They are: 120 & 20 mA (representing the injection currents of FP-SOA1 and FP-SOA2 are 120 mA and 20 mA respectively), 150 & 40 mA, 160 & 45 mA, 165 & 48 mA; correspondingly with the constant coefficients of 0.158/ps & 0.006205/ps2 (representing p = 0.158/ps and q = 0.006205/ps2), 0.094/ps & 0.002193/ps2, 0.05/ps & 0.000589/ps2, 0.0216/ps & 0.0000494/ps2, respectively. The corresponding simulated output waveforms are shown in Fig. 3(e), which theoretically confirm the feasibility of using cascaded FP-SOAs to solve all-optical high-order ODEs with tunable constant coefficients.
4. Experimental demonstration
The experimental setup for the proposed 1st-order linear ODE solver with FP-SOA is shown in Fig. 4. A continuous wave (CW) light with a wavelength of 1558.15nm is emitted by a tunable laser source (TLS). A Mach-Zehnder modulator (MZM) is driven by 10 Gb/s self-coded data signal from a bit pattern generator (BPG), to modulate this CW light. In order to better observe the output waveforms, the data signals are coded with a bit sequence of “1000” [19]. Consequently, a repetition rate of 2.5 GHz is achieved for the output light pulse of the MZM. Subsequently, an erbium-doped fiber amplifier (EDFA) is employed to compensate the loss induced by the MZM, and a following attenuator (ATT) is used to adjust the optical power that injected into the FP-SOA to a relatively small value, such as −9 dBm. Since the FP-SOA is a polarization sensitive device, a polarization controller (PC) is required to control the polarization of the optical pulse. Additionally, an adjustable constant-current source is utilized to supply different currents to the FP-SOA. Then the amplified spontaneous emission (ASE) noise of the FP-SOA is suppressed by a tunable Gaussian-type bandpass filter (TBPF) with a 3dB bandwidth of 0.3 nm and a center wavelength of 1558.15 nm. Another EDFA and ATT are employed to control the output optical power. Finally, the temporal output waveform of the FP-SOA is analyzed through a communication signal analyzer (CSA). Different output waveforms can be observed by varying the injection current of the FP-SOA, indicating that the 1st-order linear ODE with tunable constant coefficient can be solved by a single FP-SOA.
The measured ASE spectra of each FP-SOA are shown in Fig. 5. For comparison, the amplitude responses of ideal 1st-order linear ODE solvers are also illustrated in Figs. 5(c) and 5(d) (red dashed line, defined by Eq. (2)). It can be seen that the transfer functions of FP-SOAs and ideal ODE solvers are closely approximated. As is shown in Fig. 5(b), although the free spectral ranges of the two FP-SOAs are different, there are peaks in the spectra of the two FP-SOAs that perfectly aligned. The wavelength of two aligned peaks (in the ellipse with the brown dashed line shown in Fig. 5(b)) is 1558.15 nm, which is aligned with the signal wavelength. The 3-dB bandwidths of the two peaks are 0.046 nm and 0.051 nm respectively. Consequently, the two FP-SOAs can be cascaded to solve 2nd-order ODE operating at this wavelength. It should be noted that the resonant frequency of the FP-SOA varies with the injection current due to the carrier effect, and also varies with temperature due to the thermal optical effect. In our experiment, there are a thermoelectric cooler (TEC) and a thermistor in each of FP-SOAs. Therefore, temperatures of the FP-SOAs can be tuned and controlled separately to keep the resonant frequencies unchanged when varying the injection currents.
Fig. 5 (a) is the measured ASE spectra of the FP-SOAs, the black curve is for FP-SOA1 and the blue curve is for FP SOA2. (b) is the enlarged ASE spectra near the wavelength of 1558.15 nm of FP-SOA1 (black curve) and FP-SOA2 (blue curve). (c) is the measured amplitude spectrum of FP-SOA1 (green curve), simulated amplitude spectrum (red dashed line) and phase spectrum (blue dashed line) of an ideal ODE solver. (d) is the measured amplitude spectrum of FP-SOA2 (yellow curve), simulated amplitude spectrum (red dashed line) and phase spectrum (blue dashed line) of an ideal ODE solver.
As mentioned in the numerical analysis section, the measured input super-Gaussian waveform with a repetition of 2.5 GHz is shown in Fig. 2. For FP-SOA1, when the injection current is changed to 120 mA, 140 mA, 150 mA, 155 mA, 160 mA and 165 mA, correspondingly with the constant coefficient of 0.073/ps, 0.063/ps, 0.051/ps, 0.042/ps, 0.031/ps and 0.019/ps respectively, and the measured output waveforms are shown in Figs. 6(a)–6(f). It should be stressed that the constant coefficients under different injection currents are calculated from the proposed model rather than fitted from the measured output waveforms. The blue dashed lines in Fig. 6 are theoretical output waveforms achieved by the proposed model. Obviously, there are excellent agreements between the calculated results and the experimental solutions for the ODE solver based on FP-SOA1. Moreover, since the gain of FP-SOA increases with the injection current, the constant coefficient decreases with the current, which is evidenced as the output waveform is broadened when the current increases. Then FP-SOA1 is replaced by FP-SOA2, and the injection current is set as 20 mA, 35 mA, 40 mA, 42.5 mA, 45 mA and 48 mA, corresponding to the constant coefficient of 0.085/ps, 0.061/ps, 0.043/ps, 0.032/ps, 0.019/ps and 0.0026/ps respectively. The measured output waveforms are shown in Figs. 7(a)–7(f), in which the blue dashed lines are the theoretical output waveforms. It can be seen that the measured output waveforms also closely approximate the theoretical ones for the ODE solver based on FP-SOA2. Consequently,the feasibility and repeatability of using FP-SOA to solve all-optical 1st-order linear ODE with tunable constant coefficient are proved. The excellent agreements between the theoretical and experimental results for both FP-SOAs confirm that the proposed theoretical model is capable of analyzing any all-optical 1st-order linear ODE solver based on FP-SOA. Moreover, the constant coefficient tuning range of the demonstrated 1st-order ODE solver is much larger than that in [20], where the constant coefficient is only tuned from 0.038/ps to 0.082/ps.
Fig. 6 Measured output waveforms (green curves) and simulated results (blue dashed lines) of the 1st-order ODE solver based on FP-SOA1 with different injection currents: (a) 120 mA, (b) 140 mA, (c) 150 mA, (d) 155 mA, (e) 160 mA, (f) 165 mA.
Fig. 7 Measured output waveforms (green curves) and simulated results (blue dashed lines) of the 1st-order ODE solver based on FP-SOA2 with different injection currents: (a) 20 mA, (b) 35mA, (c) 40 mA, (d) 42.5 mA, (e) 45 mA, (f) 48 mA.
Then, by cascading FP-SOA1 and FP-SOA2, an all-optical 2nd-order linear ODE solver is built. The setup is shown in Fig. 8. Comparing with the setup for the 1st-order ODE solver (Fig. 4), the setup for the 2nd-order ODE solver has five additional devices: an FP-SOA (FP-SOA2), an optical bandpass filter to suppress the ASE noise, an EDFA and ATT to control the optical power and a PC to control the polarization. The constant coefficients (p and q) of the 2nd-order ODE are practically changed by varying the injection currents of the two FP-SOAs. Six specific and representative combinations of the injection currents are chosen in the experiment. They are: 120 & 20 mA, 140 & 35 mA, 150 & 40 mA, 155 & 42.5 mA, 160 & 45 mA, 165 & 48 mA; the corresponding constant coefficients are 0.158/ps & 0.006205/ps2, 0.124/ps & 0.003843/ps2, 0.094/ps & 0.002193/ps2, 0.074/ps & 0.001344/ps2, 0.05/ps & 0.000589/ps2, 0.0216/ps & 0.0000494/ps2, respectively. The measured output waveforms (green curves) under different injection currents combinations are shown in Fig. 9, along with the theoretical solutions (blue dashed lines) obtained by the proposed model. It is clear that all the experimental and theoretical output waveforms are perfectly matched, which confirm that high order linear ODEs with various values of constant coefficients can be solved by cascaded FP-SOAs.
Fig. 9 Measured output waveforms (green curves) and simulated results (blue dashed lines) of the 2nd-order ODE solver based on cascaded FP-SOAs with different injection currents combinations: (a) 120 & 20 mA, (b) 140 & 35mA, (c) 150 & 40 mA, (d) 155 & 42.5 mA, (e) 160 & 45 mA, (f) 165 & 48 mA.
5. Conclusion
In summary, all-optical 1st- and 2nd-order linear ODE solvers with tunable constant coefficients based on FP-SOAs have been proposed and experimentally demonstrated. By simply varying the injection currents, the gains of FP-SOAs are changed, resulting in variable constant coefficients of the ODE solvers. Specifically, a quite large constant coefficient tuning range of 0.082/ps have been achieved for the 1st-order ODE solver. Moreover, the constant coefficient p of the 2nd-order ODE solver is varied continuously from 0.0216/ps to 0.158/ps, correspondingly with the constant coefficient q varying from 0.0000494/ps2 to 0.006205/ps2. A theoretical model based on the carrier density rate equation of SOA and the transfer function of the FP cavity has been built to analyze the proposed scheme. For both 1st- and 2nd-order ODE solvers, excellent agreements between simulated and the experimental results have been obtained. The FP-SOAs based all-optical ODE solvers can be easily integrated with other optical components based on InP/InGaAsP. Consequently, with the impressive merits of tunability, expandability, and ease-of-integration, the proposed scheme can motivate the development of the monolithically integrated optical computing system.
Acknowledgments
This work was supported by the National Basic Research Program of China (Grant No. 2011CB301704), the National Science Fund for Distinguished Young Scholars (No. 61125501), the NSFC Major International Joint Research Project (No. 61320106016) and Scientific and Technological Innovation Cross Team of Chinese Academy of Sciences.
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