## Abstract

We propose and experimentally demonstrate an all-optical differentiator-based computation system used for solving constant-coefficient first-order linear ordinary differential equations. It consists of an all-optical intensity differentiator and a wavelength converter, both based on a semiconductor optical amplifier (SOA) and an optical filter (OF). The equation is solved for various values of the constant-coefficient and two considered input waveforms, namely, super-Gaussian and Gaussian signals. An excellent agreement between the numerical simulation and the experimental results is obtained.

©2013 Optical Society of America

## 1. Introduction

All-optical systems can overcome the optical-electrical-optical (O/E/O) conversion speed limitation of traditional electronic-based systems and decrease the power consumption greatly when applied to signal processing, computing and telecommunications [1]. In analogy with the development in electronic domain, many equivalent devices in photonic domain have been proposed, including all-optical differentiators [2, 3] and all-optical integrators [4–6]. With the assistance of these devices, more complicated optical computing can be realized, such as solving differential equations. Differential equations model a wide variety of basic engineering systems and physical phenomena [7], including temperature diffusion processes, physical problems of motion subject to acceleration inputs and frictional forces, response of different RC circuits and so on. In order to solve differential equations in photonic domain, an all-optical differentiator or integrator is indispensable. To realize all-optical integrators, solutions have been proposed using active or passive resonant cavities [4–6], leading to a main drawback that there’s a fundamental limitation on the operation frequency bandwidth, which is inherently limited by the characteristic FSR of the used cavity [8]. For all-optical differentiators, there are two different kinds, namely, the optical intensity differentiator and field differentiator [9]. The latter provides the differentiation of the optical signal’s complex field envelop (including the amplitude and phase), bringing about the demand of realizing interference in the loop required for solving differential equations, which is fairly difficult in all-optical fiber systems.

There are several schemes proposed to solve differential equations, utilizing photonic temporal integrator based on fiber gratings [4] or silicon micro-ring resonator [10]. For the integrator-based scheme, due to the intrinsic limited time window of an optical integrator, it has a limitation in the processing bandwidth (speed); meanwhile the currently proposed micro-ring based differential-equation solver lacks of tunability and flexibility, which can only be used to solve one specific differential equation with an invariable constant-coefficient. To the best of our knowledge, there are no reported schemes using all-optical intensity differentiator to solve constant-coefficient first-order linear ordinary differential equations with variable constant-coefficient and experimentally demonstrate it.

In this letter, we propose and experimentally demonstrate an all-optical system to solve ordinary differential equation and investigate the performance of the proposed scheme. Based on an all-optical intensity differentiator and a wavelength converter, an excellent agreement with theoretical results can be obtained experimentally.

## 2. Principle of operation

The constant-coefficient first-order linear ordinary differential equation can be expressed as:

where x(t) represents the input signal, y(t) is the equation solution (output signal) and k denotes a positive constant of an arbitrary value.Numerically using Fourier transformation, the solution in spectral domain can be described by:

Therefore, in time domain:

where u(t) represents the unit step function.Meanwhile the solution can also be derived using Taylor expansion:

It should be noticed that the sequence can be convergent only when$\left|-j\omega /k\right|<1$. Therefore, for different input signals, there’s a threshold of k (k_{th}) below which the solution cannot be obtained in practice. According to our simulation using parameter sweep, for 40 Gb/s input super-Gaussian and Gaussian signals, k_{th} is 2/ps and 0.7/ps, respectively.

The schematic diagram of the computing system [11] required for solving the constant-coefficient first-order differential equation is shown in Fig. 1 . The operation process can be explained as below: for the first circle, only the input signal x(t) transmits in the loop, so y(t) should be expressed by:

for the second circle, the negative first-order differentiation of x(t) along with x(t) circulate in the loop, y(t) can be given by:The corresponding schematic diagram of the experiment is illustrated in Fig. 2 , a continuous-wave probe light at ${\lambda}_{p}$ (generated from LD2) is launched into the first SOA with another control signal at ${\lambda}_{c}$ (illustrated in Fig. 2 as the blue Gaussian pulse x(t)) simultaneously [2]. Due to the cross-phase modulation (XPM) in SOA and frequency discrimination of the OF, the positive differentiation of the input signal (${\lambda}_{c}$) can be obtained at the wavelength of ${\lambda}_{p}$ (the red positive monocycle signal after the optical differentiator in Fig. 2). In order to circulate in the loop, the second SOA cascaded with an OF served as an inverted wavelength converter [12] is needed. Consequently, the negative differentiation of the input signal is carried out at the wavelength of ${\lambda}_{c}$(the blue negative monocycle signal after the wavelength converter in Fig. 2) and then circulates in the loop along with the original input signal. As the circulation goes and finally stabilizes, the solution of the equation can be obtained at the output. To obtain different value of the constant-coefficient, the coupling efficiency of the couplers demonstrated in Fig. 2 needs to be changed.

## 3. Numerical analysis

Using super-Gaussian (m = 3) and Gaussian pulse sequence at 40 Gb/s as the input x(t), respectively, according to Eq. (3), the theoretical convolution solutions of Eq. (1) are depicted in Fig. 3
. For convenience of comparison, the maximum values of the solutions are normalized. From the left side of Eq. (1), dt in the differentiation term is of the unit of “10^{−12} sec”; therefore, to match with the unit and the order of magnitude of the differentiation term, the unit of k in the second term should set to “10^{12} /sec”, which equals to “/ps”. It should be noted that although such high value of k cannot be obtained in practice, Eq. (1) can still be solved in experiment. Because for practical differentiators, the transfer function $H(\omega )$equals to$\tau \cdot j\omega $ (such as the differentiator based on micro-ring resonator [13]),rather than exactly equals to$j\omega $, where $\tau $compensates the big order of magnitude induced by dt; so in experiment, the constant-coefficient k doesn’t need to be k/ps, enabling the realization of solving differential equations practically.

According to the numerical results in Fig. 3, different solutions are calculated for different positive values of k. When k>k_{th}, the solution can be obtained practically but it resembles the input signal and doesn't change greatly with the variation of k; whereas for k≤k_{th}, the solution changes obviously while k changes, meanwhile it’s also easy to distinguish between the solution and the input signal, however, it can only be obtained theoretically.

## 4. Experimental results and discussion

The experimental setup of the scheme is shown in Fig. 4 : LD1 and LD2 generate CW beams at ${\lambda}_{c}$ and${\lambda}_{p}$ for control and probe light, respectively. ${\lambda}_{c}$ is modulated by a transmitter that consists of two cascaded Mach-Zehnder modulators (MZM) driven by data and clock signal generated from a bit pattern generator (BPG). For solving differential equations, it’s the pulse shape that should be paid attention to. Therefore, in order to observe the results more clearly, data signals provided by the BPG are set to be coding sequence with a period of ‘10000000’. Two kinds of data signals are generated by the transmitter, super-Gaussian (m = 3) and Gaussian (nearly transform-limited) signals. The attenuators (ATT) and EDFAs are used to adjust the optical power, cooperating with the couplers to change the value of the constant k. The optical delay line (ODL) is exploited to synchronize input signal and the differentiation signal meanwhile eliminates their coherence. The two SOAs are designed for nonlinear optical signal processing biased at 210mA. ${\lambda}_{c}$and${\lambda}_{p}$are set to 1530.5nm and 1562nm, while the power of the them injected into SOA1 is 10.69dBm (11.72mW) and 2.13dBm (1.63mW), respectively, in order to reduce the impact of XGM effect in the first SOA [2]. Filter1 is a Gaussian-type bandpass filter (BPF) with 3dB bandwidth of 0.3nm; filter2 is a narrow bandpass filter with 3dB bandwidth of 1nm [12]. By properly tuning the two filters, positive differentiation signals at${\lambda}_{p}$and inverted wavelength conversion signal at${\lambda}_{c}$can be obtained after the two filters, respectively, thus ensuring the realization of solving the differential equation with positive values of k. The temporal waveform of the final results is analyzed through a communication signal analyzer (CSA).

The measured results in terms of power intensity are shown in Fig. 5
. For comparison, the theoretically calculated results (yellow dot line) are also given in the same plot in which the calculated results show an excellent agreement with the measured solutions. The time scale of the input pulse is 100ps/div, but here only one pulse is shown which occupies almost a quarter of one unit in the image; while the time scale of the output signals is 6ps/div. Since the solutions obtained in the experiment meet the condition that k>k_{th}; according to our simulation, under this condition, the solutions don't change obviously while k changes, so only one solution is shown in Fig. 5 for a specific value of k for input Gaussian and super-Gaussian pulse.

Moreover, for k≤k_{th}, the solution demonstrated by Eq. (5) is not convergent, so real solutions cannot be obtained in practice. Therefore, to observe obvious changes of the solutions and compare solutions for different values of k, only the first and second terms of the right side of Eq. (5), namely, the input signal and the first-order differentiation signal are obtained and superposed as the approximate solutions. By adjusting the power of the differentiation signal, different solutions for different values of k while k≤k_{th} are illustrated in Fig. 6
along with the theoretically calculated results (yellow dot line), which still show the rough profiles of the real solutions. The time scale of the pulses is the same as that of Fig. 5(a). It should be noted that due to the direct current (DC) term [14] induced by the differentiation process, the experimental solution is actually the superposition of the real solution and a direct current component.

## 5. Conclusion

We have introduced and experimentally demonstrated a general design for solving the first-order linear ordinary differential equation. This signal computing functionality can be implemented using an all-optical intensity differentiator and a wavelength converter. Our experimental results show an excellent agreement with the numerical simulations for different kinds of input signals and different values of the constant-coefficient. This scheme can be applied in ultrahigh-speed all-optical signal processing systems and has the potential to be expanded for more complicated computing functionality.

## Acknowledgments

This work is supported by the National Science Foundation for Distinguished Young Scholars of China (Grand No.61125501) and the National Basic Research Program of China (Grant No. 2011CB301704).

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