Abstract

We report the first experimental realization of an all-optical temporal integrator. The integrator is implemented using an all-fiber active (gain-assisted) filter based on superimposed fiber Bragg gratings made in an Er-Yb co-doped optical fiber that behaves like an ‘optical capacitor’. Functionality of this device was tested by integrating different optical pulses, with time duration down to 60 ps, and by integration of two consecutive pulses that had different relative phases, separated by up to 1 ns. The potential of the developed device for implementing all-optical computing systems for solving ordinary differential equations was also experimentally tested.

©2008 Optical Society of America

1. Introduction

All-optical circuits for computing, information processing, and networking can overcome the speed limitations of electronic-based systems [13]. However, in photonics, there are still no equivalents to some fundamental signal processing devices that form basic building blocks in electronic circuits, e.g. integrator. Thus, the design and demonstration of these fundamental photonic devices is a necessary first step towards the realization of all-optical processing circuits. Although several schemes for performing photonic integration have been previously proposed [48], a main challenge associated with their experimental implementation is that an advanced light-wave storage (capacitor-like) element [9] is necessary.

A photonic temporal integrator is a device that performs the cumulative time integral of the complex temporal envelope of an input arbitrary optical waveform [48]. Applications of the photonic integrator are in many fields; many of them can be found in direct analogy with applications of a photonic differentiator, which is its signal processing counterpart. The photonic temporal differentiator has been recently demonstrated [10,11] and has already proved to be very useful in a wide range of applications in ultrafast optical pulse processing [1113], shaping [14,15], and metrology [16,17]. Nonetheless, the importance of the optical integrator would go beyond these applications. For example, in direct analogy with the electronic-domain developments [18,19], a photonic temporal integrator is a key element for implementing all-optical analog computing systems. Analog computers have been shown to have a far superior performance, e.g. in terms of speed, as compared with conventional digital computers for a number of specific computing tasks, particularly for real-time solving of scientific and engineering problems that can be described by differential equations [18,20]. It is well known [18,19] that these problems can be solved directly in the analog domain in real time using a suitable combination of integrators, adders and multipliers. Because differentiators have significantly worse performance in terms of high-frequency noise, the use of integrators is strongly preferred [18,19]. The possibility of realizing these computations all-optically translates into a potential for speeds several orders of magnitude higher than with conventional electronic systems [1-3]. This perspective is particularly attractive when the solution is required in real-time, at speeds far beyond the reach of digital electronic computers, for immediate analysis or processing, e.g. in control/feedback applications requiring operation in picosecond time scales.

From basic signal processing theory [18] it follows that a temporal integrator can be implemented using a linear filtering device with a temporal impulse response h(t) proportional to the so-called unit step function u(t) :

h(t)u(t);u(t)=0fort<0,u(t)=1fort0.

where t is the time variable. Physically, this requires the use of a structure capable of storing an incoming time-varying waveform (e.g. electric field intensity) with an output being a continuous signal proportional to the sum of the total stored field at each instant of time. In electronics, this functionality is provided by a capacitor, which stores the electric charge proportionally to the sum of the incoming electric field. The integrated signal is then directly proportional to the voltage measured at the capacitor. The same principle cannot be directly transferred into photonics since complete stopping of photons would be necessary.

In the spectral domain, the transfer function H(ω) of an ideal photonic integrator is inversely-proportional to the base-band frequency (ω-ω 0), i.e. H(ω)={h(t)}∝1/j(ω-ω0)+π·δ(ω-ω0), where ℑ is the Fourier transform operator, δ is the Dirac delta-function, ω is the optical frequency and ω0 is the carrier frequency of the signal to be processed [46]. This implies that the transmission should be > 1 in the proximity of ω 0 and, ideally, it should become infinite at ω0.

The first suggestions of a photonic integrator [4,5] were based on a general feedback-based photonic filters, in which a gain element placed in the feedback loop made it possible to obtain transmission > 1 in the vicinity of the resonance (in practice, the transmission would not reach infinity at the resonance frequency due to the gain saturation of the considered gain medium). Although representing a significant step forward, being the first descriptions of a device that ‘integrates photons’, these original proposals did not evaluate any practical constraints in terms of required level of amplification versus achievable processing speed. Subsequently, three very interesting designs that already included discussions of practical limitations and trade-offs were proposed [68]. All of them, however, considered only passive resonant structures that do not allow to fulfill the condition of exact integration, as their transmission is limited to values ≤ 1. This has two main drawbacks. First, the energetic efficiency of the integration process is extremely low, for example an energetic efficiency of 0.1% is obtained in [6] (for processing error of 2%) and 0.3% in [7]. Secondly, the integration of long and narrow-bandwidth pulses suffers from high processing error as the filter transfer function around the resonant frequency (≤ 1) differs significantly from that of an ideal integrator (→ ∞). The first design [6] is based on a phase-shifted fiber Bragg grating (FBG), which physically forms a resonant cavity in which light resonates between two FBG reflectors that have relative π-phase shift. The phase shift ensures that the constructive resonance condition occurs at the center of the FBG reflection bandwidth. To obtain an acceptable processing error, numerical analysis that did not consider propagation loss [6] showed that each FBG reflector should provide a reflectivity of at least 99.99%. Obviously, this would be extremely challenging to achieve in practice, even considering other technologies of preparing the two-reflector resonator, e.g., Fabry-Perot (FP) resonator. The second suggested configuration [7] is formed by a single uniform FBG operating in reflection that provides a square-like impulse temporal response. A feature of this configuration that may be disadvantageous for some applications is that it provides the integrated signal two times: an ideal integrator provides a step-like impulse response (1) and thus the output of this integrator is the signal integral (processed by the rising step of the square-like impulse response) followed by time-inversed integral of the time-inverted signal (processed by the falling step of the square-like impulse response). The last proposed scheme is based on a specially-apodized FBG operating in reflection that provides a decreasing-exponential temporal impulse response [8]. Although this eliminates the presence of an undesired waveform outside the device’s operation time window present in the previous scheme, it is expected to be challenging to fabricate, as the designed grating modulation depth varies significantly over a very short length, requiring a very high level of photoinduced refractive index change at the very beginning of the FBG. In our work, a temporal integrator with notably relaxed fabrication constraints is suggested and fabricated by incorporating an active medium into a resonant cavity configuration similar to that described in [6]. The good performance of the prepared integrators allowed us to demonstrate their potential for all-optical computing applications, particularly for real time solving of differential equations of interest to a wide variety of engineering and scientific problems [20].

2. Principle of operation

A schematic of the idea for implementing the photonic temporal integrator is shown in Fig. 1. Let us assume a general FP cavity composed of two identical mirrors, each characterized by a field reflectivity r defined as the ratio of the reflected and incident field amplitudes, |r| ≤ 1, separated by a distance L. The net gain in the cavity medium, defined as the round-trip field amplitude gain that excludes loss due to the mirrors, is given by γ with γ<1 for loss and γ>1 for gain. It can be easily proved [21] that for such FP cavity, the temporal impulse response is, within a certain fraction of the FP free spectral range:

h(t)exp(kt)u(t),

where k=(1/T)ln(r2γ) and T is the round-trip propagation time in the FP cavity (T=2Ln/c, with n being the cavity refractive index and c being the speed of light in vacuum). Simply, the signal stored in a FP cavity is leaking out following an exponential time variation. Comparing the impulse response of the FP cavity, Eq. (2), with that of an ideal integrator, Eq. (1), we infer that the FP cavity would behave as a temporal integrator when r2γ=1. This condition, known also as the lasing threshold condition [22], means that the loss associated with the reflections in the FP mirrors are perfectly compensated for by the net gain in the cavity. At this condition, the field of an incoming ultrashort impulse will be stored and subsequently delivered from the FP cavity with a constant flow (Fig.1), exactly as required for optical temporal integration. It should be noted that there is a fundamental limitation in terms of the fastest temporal feature of the input waveform that can be processed with a resonant cavity-based optical integrator; this limitation is given by the spectral range over which FP cavity provides the temporal impulse response given by (2). For pulses shorter than the round-trip propagation time in the cavity, T, the signal would be released in the form of discrete impulses, temporally spaced by T. In practice, only temporal features longer than ≈5T will be integrated with small-enough processing error [6], i.e. the integrator processing bandwidth is limited to ≈1/5 of the integrator free spectral range.

 figure: Fig. 1.

Fig. 1. Concept diagram of the proposed photonic temporal integrator. The integrator is implemented using two superimposed fiber Bragg gratings (acting as a resonant cavity) permanently photo-inscribed in an Er-Yb co-doped optical fiber that provides optical gain. The gain level is controlled via power of the optical pump (980-nm laser diode). The inset shows the measured (circles) and numerically calculated (solid, blue curve) integrator spectral transfer function. For comparison, the spectral transfer function of an ideal integrator is also shown (solid, red curve).

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In addition of the already-mentioned significantly increased energetic efficiency of the integration process, the use of an active filtering configuration provides an important advantage with the possibility of adjusting the filtering characteristics through tuning of the optical pumping. This latter feature is particularly interesting for solving differential equations as will be discussed in a few examples presented below. A disadvantage of using the gain medium is the presence of spontaneous emission, which generates noise.

3. Implementation and fabrication

Different technological approaches could be considered for implementing the proposed integrator concept [9,22,23]. Our implementation is based on optical waveguide (all-fiber) technology [22]. Here, the two FP mirrors can be realized using two FBGs created by a periodic change of the refractive index along the direction of light propagation within a single-mode optical fiber [24]. The length of the two FBG-made FP mirrors limits the minimum achievable FP cavity length and thus also the cavity round-trip time T that determines the integrator processing speed. To get T<10 ps with corresponding processing bandwidth of tens of GHz, we need a mirror spacing of 1 mm or less when considering a cavity with refractive index of 1.45 corresponding to silica optical fibers. Such short FBGs are difficult to make with the required level of reflectivity. This limitation, however, can be overcome by using spatially overlapped (superimposed) FBGs that are slightly chirped (the grating period is varied along the propagation axis in the fibre core) [25], forming a distributed resonant cavity with a length not limited by the desired round-trip time T. Such structure made in an active fiber was already reported to reach the laser threshold condition [22]. Detailed description and theoretical analysis of this structure can be found in [22].

The two superimposed chirped FBGs were photoinscribed into a specialty high-gain (~40 dB/m) fiber co-doped with Er and Yb with photosensitive inner cladding (ErYb-302, INO, Canada) loaded with Deuterium. Grating writing was done by scanning a chirped phase mask [24] (period chirp of 0.5 nm/cm) with a laser beam from a frequency-doubled Argon ion cw laser (244 nm and power of 50 mW) [24]. Each FBG was fabricated by a single phase mask exposure. Between the two exposures, the phase mask was moved with respect to the fiber by the required effective cavity length. An ≈ 8-cm long active fiber section was spliced with passive single-mode fibers. We prepared various samples of an optical integrator design with the round-trip propagation time T=10 ps that required a 1-mm shift of the phase mask between the two consecutive fiber exposures. The FBGs were typically 3-4 cm long depending on the used FBG apodization profile that was of tanh shape. We analyzed the structures theoretically using the theoretical model developed in ref. [22] and found that the FBGs’ reflectivities necessary to fulfill the lasing threshold condition, r2γ=1, were r=99.9% corresponding to a power transmission of -27 dB. Obviously, using fibers with even higher active ions doping level (resulting in higher net round-trip gain γ) would further reduce the reflectivity needed. For shorter round-trip times, the net round-trip gain γ would be smaller, as it is given by the product of the fiber gain factor and the cavity length. Consequently, the FBGs reflectivities would need to be increased or a fiber with a higher gain factor should be used. Alternative technologies, such as ring resonators in InP [9] with diameters down to 10 µm, would allow the realization of round-trip times as fast as about 0.3 ps corresponding to processing speeds up to ≈650 GHz. Furthermore these InP rings have typical gain around 30 dB/mm that is three orders of magnitude higher than in the used active fiber.

The theoretically expected and measured spectral filtering responses for one of the realized temporal integrators are shown in Fig. 1 together with the filtering transfer function of an ideal integrator. The spectral transfer function was measured using an Optical Vector Analyzer (OVA from LUNA Technologies, U.S.A.). As the specialty fiber had a slightly elliptical core, the realized component was slightly birefringent. Thus, the experiment was set in such a way that the processed light propagated through the component with its polarization aligned along one of the principal axes of the integrator. Clearly, the resolution of the OVA (1.5 pm) was insufficient to measure properly the central part of the transmission peak, which had a full width at half maximum (FWHM), according to the simulated data, of 0.2 pm. Despite this fact, we were still able to measure about 9 dB of amplification while the offresonance amplification was measured to be 4 dB. It is believed that considerably higher values of the peak amplification could have been measured using an instrument with sufficiently high spectral resolution (the theoretically-predicted value was 23 dB as shown in the inset of Fig. 1). From the spectral filtering responses shown in Fig. 1, we estimated that the theoretical and experimental bandwidth over which the implemented integrator can operate was about 20 GHz, which corresponds to temporal features of approximately 5T=50 ps. The observable slight oscillations in the experimental data are believed to be artifacts of the measurement method used for the device characterization.

4. Experimental results: integrator properties

To evaluate the performance of the developed integrator, we carried out two basic experiments. The first one demonstrates the basic integration property, i.e. evaluation of a time cumulative integral of a simple optical pulse. The second one demonstrates the coherent operation of the integrator, i.e., that the device operates on both the amplitude and the phase of the processed signal. In both experiments, the slight exponential decay of the integrated signal was caused by pumping slightly below the lasing threshold condition (r2γ=1). The decay time τ was typically 5 ns. The input power was low enough to prevent gain saturation that would otherwise affect the gain factor γ and thus “break” the integration condition of r2γ=1. A polarization controller was placed in front of the integrator to eliminate the effect of the slight photoinduced birefringence of the realized integrators.

The set-up for the first experiment is shown in Fig. 2. First, we prepared individual pulses with a FWHM time width of 140 and 60 ps, respectively. These pulses were generated by modulating a 5-mW CW tunable laser tuned to the integrator resonance wavelength through a Mach-Zehnder intensity modulator. The modulator was driven by an electric pulse generator (Model 3600 Impulse generator, Picosecond Pulse Lab. Inc., Boulder, CO, U.S.A) with a repetition rate of 200 MHz and a pulse width (FWHM) of 70 ps. To obtain 60-ps and 140-ps optical pulses, 35-GHz and 2.5-GHz 3-dB bandwidth modulators were used, respectively. The repetition rate was further reduced to 2 MHz using a square-like gating from an electric programmable waveform generator (Model AWG710B, Tektronix Inc., U.S.A.). In our experiment, the carrier frequency of the input signal was set to match the central frequency of the integrator. However, in practice, the central frequency of the integrator (resonant cavity) could be tuned, e.g., via temperature control. The expected and measured results in terms of power intensity are shown in Fig. 3. The input and output signals from the integrator were captured by a 20-GHz bandwidth photodiode and viewed with a sampling oscilloscope. For comparison, we numerically calculated the time integral of the measured input pulse fields. The intensity of the resulting integrated signal (square of the numerically obtained time integral) is also given in the same plot showing an excellent agreement between the calculated and measured time integrals. These results confirm that the integrator is fully capable of processing temporal features as fast as 60 ps FWHM, corresponding to a full signal bandwidth over 20 GHz.

 figure: Fig. 2.

Fig. 2. Experimental setup for the integration of pulses generated using electro-optically modulated signal. TL: Tunable laser, ISO: optical isolator, PC: polarization controller, IMOD: Optical intensity modulator, PPG: Picosecond electric pulse generator with 70 ps FWHM time width, AWG: Electric arbitrary waveform generator with 500 MHz bandwidth, Pump: 980-nm semiconductor pump laser, Amp: Erbium-doped fiber amplifier, OS: Optical sampler (photoreceiver).

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 figure: Fig. 3.

Fig. 3. Experimental results demonstrating time-domain integration of a single optical Gaussian pulse for two different input pulse FWHM time widths ((a) 140 ps and (b) 60 ps). The temporal optical intensity of the input pulse (orange curve) and the integrator output (green curve) are captured using a 20-GHz photoreceiver. For comparison, the square of the numerically calculated time cumulative integral of the measured input pulse field (square root of the measured temporal intensity profile) is also shown (yellow curve).

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In the second experiment, we used a double-pulse waveform composed of two replicas of the same optical pulse as an input signal. For inter-pulse relative phases fixed to zero, inphase pulses, we would expect the time cumulative integral to look like two steps with each corresponding to the integral of one pulse, separated by the pulses’ relative time delay. However, for inter-pulse relative phases fixed to π, out-of-phase pulses, the second step should have the opposite direction, forming thus a square-like temporal waveform, the length of which would be given by the relative time delay. This phenomenon is schematically shown in Fig. 4(a). The set-up for implementation of this experiment is shown in Fig. 4(b). The individual optical pulses were generated by a passively mode-locked fiber laser (Pritel Inc., Naperville, IL, U.S.A.), operating at a repetition rate of 16.7 MHz, followed by a 0.4-nm Gaussian-shape band-pass optical filter, which resulted in individual pulses with a FWHM time width of ≈ 14 ps. The pulse wavelength was tuned to coincide with the integrator resonance wavelength. The pulse replicas were obtained using a Michelson interferometer adjusted to achieve different inter-pulse delays by coarse moving mirror alignment and different inter-pulse relative phases by fine alignment of the moving mirror. The relative phase was fixed to either zero (in-phase pulses) or π (out-of-phase pulses). For the shortest time delay (170 ps), we confirmed the relative phase of the two pulses by measuring the optical spectrum of the double-pulse structure. For in-phase pulses, the spectrum had a maximum at its centre, whereas for the out-of-phase pulses, there was a minimum. The two pulse replicas were polarized along one of the integrator’s birefringence polarization axis by adjusting a polarization controller. The integrated output was amplified by an Er-doped fiber amplifier and detected by a 20-GHz photodiode and a sampling oscilloscope.

 figure: Fig. 4.

Fig. 4. (a) Diagram showing all-optical integration of two consecutive optical pulses with different relative phases. For relative phases of 0 (in-phase – the field amplitudes are of the same sign, red curves) and π (out-of-phase – the field amplitudes are of opposite signs, blue curves), the time integral is expected to be a double step-rising waveform, and a flat-top waveform, respectively. (b) Experimental setup for the double pulse integration. 14-ps pulses are generated from the FFL followed by an optical bandpass filter (0.4 nm 3dB-bandwidth). Time-delayed pulse replicas are made by using a fiber-coupled Michelson interferometer. For the shortest time delay (170 ps), we confirmed the relative phase of the two pulses measuring the optical spectrum of the double-pulse structure, which is shown in the inset: for in-phase pulses (red line), the spectrum has a maximum at the integrator central frequency, while for the out-of-phase pulses (blue line), there is a minimum. The output spectrum is shown as green line.

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The results of this experiment are shown in Fig. 5. As expected, the integrator simply summed up the area under the two waveforms in time when the two pulses had no phase difference, resulting in two consecutive steps respectively corresponding to the integration of the leading pulse and the subsequently added integration of the trailing pulse. The integral of the two identical pulses is two times the integral of the single pulse, which is four times in intensity – consequently, the measured second step in the integral should be three times higher than the first one. In contrast, when the pulses were out of phase, the second optical pulse compensated for the cumulative time integral of the first pulse, leading to a square-like output time profile with a duration fixed by the input inter-pulse delay. In both experiments, the slight exponential decay of the integrated signal was caused by a slight net round-trip loss within the cavity. The observed spikes are believed to be artifacts caused by the limited bandwidth of the photoreceiver with respect to the bandwidth of the used pulses. Nevertheless, these results confirm that the developed integrator operates on the complex temporal envelope (amplitude and phase) of the optical signals to be processed.

 figure: Fig. 5.

Fig. 5. Experimental results demonstrating time-domain integration of double optical pulses. (a) Time-domain optical intensity of the input signal - two 14-ps consecutive optical pulse with various inter-pulse delays (170 ps, 500 ps, and 1000ps). The integrated output depends on the relative phase of the two pulses: (b) integration for in-phase pulses; (c) integration for out-ofphase pulses.

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Finally, we performed an experiment that combines both basic properties shown above and should serve as an example of an arbitrary waveform integration. In this experiment, we processed two consecutive flat-top pulses set in-phase and out-of-phase. The experimental setup is shown in Fig. 2 – an electric arbitrary waveform generator (AWG) with 500 MHz bandwidth that was programmed to produce the desired waveform (two flat-top pulses of 1 and 2 ns duration, respectively, set in-phase or out-of-phase) was used to drive the modulator. The optical integration of a single flat-top pulse should result in a simple rising ramp, duration of which corresponds to the input pulse duration. The results of the carried-out experiments are shown in Fig. 6: the left panel shows the results when the two pulses are in phase, while the right panel shows the out-of-phase setting. The input electric waveform is shown first (blue), followed by the input optical intensity waveform (red). Finally, the output optical intensity waveform (yellow) is shown. As anticipated for the in-phase setting, two summed rising ramps are present at the output. These two ramps are distinguished by their different time widths; ~2 ns and ~1 ns, respectively. The out-of-phase output is formed by a positive slope ramp with a duration corresponding to the first flat-top pulse (~2 ns) followed by a negative slope ramp with duration corresponding to the second opposite-phase flat-top pulse (~1 ns).

 figure: Fig. 6.

Fig. 6. Experimental results for the optical integration of two flat-top pulses set in-phase and out-of-phase. Left: in-phase; Right: out-of-phase. The input electric waveform: blue; the modulated optical intensity waveform: red, and the output optical intensity waveform: yellow.

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5. Experimental results: all-optical computing of differential equations

To give a simple application example of the developed photonic temporal integrator, Fig. 7(a) shows an integrator-based computation system [18] used for solving the following constantcoefficient first-order linear differential equation:

dy(t)dt+ky(t)=x(t)

where x(t) represents the input signal, y(t) is the equation solution (output signal), and k represents a complex constant of an arbitrary value. The simple differential equation Eq. (3) actually models a wide variety of basic engineering systems and physical phenomena [20], including problems of motion subject to acceleration inputs and frictional forces, temperature diffusion processes, response of different RC circuits, population dynamics in biology and economy, etc. As anticipated, the key element in the resulting computation system is a temporal integrator. Interestingly, the photonic device demonstrated here is governed by the general differential equation Eq. (3), with k being a real-valued constant, k=(1/T)ln(r2γ). This is associated with the fact that our integrator design itself is based on a feedback configuration (FP resonant cavity) such as that of the computing system required for solving Eq. (3), see schematic in Fig. 7(a). As demonstrated above, the device behaves as a temporal integrator when it is operated at the exact lasing threshold conditions (k=0). Moreover, the demonstrated device is capable of solving Eq. (3) when it is operated either slightly below the lasing threshold condition (k > 0) or slightly above the threshold condition (k < 0). Figs. 7(b)-(c) represent the solution of Eq. (3) as computed experimentally with our photonic device for two different input signals, i.e. an ultrashort temporal impulse (Fig. 7(b)) directly generated from our mode-locked fiber laser, and a constant excitation over a limited temporal window (2.9 ns, Fig. 7(c)) obtained by modulating a cw laser light with an electrical square-like pulse prepared using the previously-specified AWG. In the two cases, the differential equation was experimentally solved for different positive values of the variable k, which have been varied by simply adjusting the optical pumping power in the active medium. For comparison, Eq. (3) was also numerically solved for the two considered input waveforms and various values of k. In the first experiment (results shown in Fig. 7(b)), the input ultrashort pulse can be considered as a temporal delta function, x(t)=δ(t), and the system response can be analytically calculated from Eq. (3) resulting in the equation anticipated for a general FP cavity, y(t)=h(t)=exp(-kt)u(t). The intensity of this analytical temporal impulse response is depicted in Fig. 7(b) for the different evaluated values of the constant k, showing an excellent agreement with the experimental results. The general solution of Eq. (3) is obtained as the numerical convolution of the considered input signal (e.g. flat-top waveform used in the second experiment) with the analytical temporal impulse response h(t) defined above. The intensity of the numerically obtained solutions of Eq. (3) using the squared-root of the measured flat-top waveform as the input signal are plotted in Fig. 7(c) for the two different evaluated k values, confirming again the excellent agreement between the theoretical and experimental solutions. The excellent agreement between the experimental and numerical solutions of Eq. (3) proves the potential of the demonstrated photonic temporal integrator for all-optical computing of differential equations.

 figure: Fig. 7.

Fig. 7. (a) Schematic diagram of an integrator-based optical computing system designed for solving the first-order linear ordinary differential equation (ODE) defined in the figure. The two graphs at the bottom show the experimental (solid curves) and numerical (circles) solutions of the ODE for two different input optical signals: (b) an input ultrashort temporal impulse (FWHM time-width=60 ps) and (c) a constant excitation over a limited temporal window (2.9-ns long square-like pulse). In each case, the ODE is solved for different positive values of the parameter k.

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6. Conclusions

We have reported the first experimental demonstration of a photonic temporal integrator capable of performing the time-domain cumulative integral of an input arbitrary optical waveform. We have shown that this fundamental signal processing functionality can be implemented in the optical domain using an active resonant cavity operated at the exact lasing threshold condition. This development is of particular interest for the implementation of ultrahigh-speed all-optical computers since the time integrator is a key device in an analog computing system devoted to solving differential equations in real time. This important feature has been illustrated by successfully solving a basic, general differential equation using the developed experimental integrator. As an additional advantage, the photonic integrator demonstrated here is based on a compact and robust waveguide (all-fiber) implementation, which could be readily incorporated in future integrated photonic circuits.

Acknowledgments

This research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC), by the Grant Agency of AS of the Czech Republic (contract no. KJB200670601), by the Czech Science Foundation (contract no. GA102/07/0999) and the Canada Research Chair in Optical fibre communications and components. We thank to Prof. M. Rochette and Prof. L. Chen from McGill University for lending us the picoseconds pulse generator and the tunable narrow-bandpass filter.

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19. K. OgataModern Control Engineering4th ed.(Prentice Hall, Upper Saddle River, NJ, USA2001).

20. G.F. SimmonsDifferential Equations with Applications and Historical Notes, 2nd ed. (McGraw-Hill, New York, USA1991).

21. C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach (John Wiley & Sons, New York, USA1999).

22. G. Brochu, S. LaRochelle, and R. Slavík, “Modeling and experimental demonstration of ultracompact multiwavelength distributed Fabry-Perot fiber lasers,” J. Lightwave Technol. 23, 44–53 (2005). [CrossRef]  

23. Y. Barbarin, S. Anantathanasarn, E. A. J. M. Bente, Y. S. Oei, M. K. Smit, and R. Notzel, “1.55 µm range InAs-InP (100) quantum-dot Fabry-Perot and ring lasers using narrow deeply etched ridge waveguides,” IEEE Photon. Technol. Lett. 18, 2644–2646 (2006). [CrossRef]  

24. R. Kashyap, Fiber Bragg Gratings (Academic Press, San Diego, 1999).

25. G. E. Town, K. Sugden, J. A. R. Williams, I. Bennion, and S. B. Poole, “Wide-band Fabry-Perot-like filters in optical fiber,” IEEE Photon. Technol. Lett. 7, 78–80 (1995). [CrossRef]  

References

  • View by:

  1. L. Venema, “Photonics Technologies,” Nature Insight 424, No. 6950 (2003).
  2. C. K. Madsen, D. Dragoman, and J. Azaña (editors), Special Issue on “Signal Analysis Tools for Optical Signal Processing,” EURASIP J. Appl. Signal Proc. 10, 1449–1623 (2005).
    [Crossref]
  3. J. Azaña, C. K. Madsen, K. Takiguchi, and G. Cincontti (editors), Special Issue on “Optical Signal Processing,” IEEE/OSA J. Lightwave Technol. 24, 2484–2767 (2006).
    [Crossref]
  4. N. Q. Ngo and L. N. Binh, “Optical realization of Newton-Cotes-Based Integrators for Dark Soliton Generation,” IEEE/OSA J. Lightwave Technol. 24, 563–572 (2006).
    [Crossref]
  5. N. Q. Ngo, “Optical integrator for optical dark-soliton detection and pulse shaping,” Appl. Opt. 45, 6785–6791 (2006).
    [Crossref] [PubMed]
  6. N. Q. Ngo, “Design of an optical temporal integrator based on a phase-shifted fiber Bragg grating in transmission,” Opt. Lett. 32, 3020–3022 (2007).
    [Crossref]
  7. J. Azaña, “Proposal of a uniform fiber Bragg grating as an ultrafast all-optical integrator,” Opt. Lett. 33, 4–6 (2008).
    [Crossref]
  8. M. A. Preciado and M. A. Muriel, “Ultrafast all-optical integrator based on a fiber Bragg grating: proposal and design,” Opt. Lett. 33, 1348–1350 (2008).
    [Crossref] [PubMed]
  9. M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oel, H. Binsma, G.-D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
    [Crossref] [PubMed]
  10. M. Kulishov and J. Azaña, “Long-period fiber gratings as ultrafast optical differentiators,” Opt. Lett. 30, 2700–2702 (2005).
    [Crossref] [PubMed]
  11. R. Slavík, Y. Park, M. Kulishov, R. Morandotti, and J. Azaña, “Ultrafast all-optical differentiators,” Opt. Express 14, 10699–10707 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-22-10699.
    [Crossref] [PubMed]
  12. P. Yao, F. Zeng, and Q. Wang, “Photonic generation of Ultra-Wideband signals,” J. Lightwave Technol. 25, 3219–3235 (2007).
  13. J. Xu, X. Zhang, J. Dong, D. Liu, and D. Huang, “High-speed all-optical differentiator based on a semiconductor optical amplifier and an optical filter,” Opt. Lett. 32, 1872–1874 (2007).
    [Crossref] [PubMed]
  14. Y. Park, M. Kulishov, R. Slavík, and J. Azaña, “Picosecond and sub-picosecond flat-top pulse generation using uniform long-period fiber grating,” Opt. Express, 14, 12671–12678 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-26-12670.
    [Crossref]
  15. R. Slavík, L. K. Oxenløwe, M. Galili, H. C. H. Mulvad, Y. Park, J. Azaña, and P. Jeppesen, “Demultiplexing of 320 and 640 Gbit/s OTDM data using ultrashort flat-top pulses,” IEEE Photon. Technol. Lett. 19, 1855–1857 (2007).
    [Crossref]
  16. F. Li, Y. Park, and J. Azaña, “Complete temporal pulse characterization based on phase reconstruction using optical ultrafast differentiation (PROUD),” Opt. Lett. 32, 3364–3366 (2007).
    [Crossref] [PubMed]
  17. F. Li, Y. Park, and J. Azaña, “Precise and simple group delay measurement of dispersive devices based on ultrafast optical differentiation,” in Proc. of OFC/NFOEC’08, Paper OWD5, 2008.
  18. A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and Systems, 2nd ed. (Prentice Hall, Upper Saddle River, NJ, USA1996).
  19. K. OgataModern Control Engineering4th ed.(Prentice Hall, Upper Saddle River, NJ, USA2001).
  20. G.F. SimmonsDifferential Equations with Applications and Historical Notes, 2nd ed. (McGraw-Hill, New York, USA1991).
  21. C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach (John Wiley & Sons, New York, USA1999).
  22. G. Brochu, S. LaRochelle, and R. Slavík, “Modeling and experimental demonstration of ultracompact multiwavelength distributed Fabry-Perot fiber lasers,” J. Lightwave Technol. 23, 44–53 (2005).
    [Crossref]
  23. Y. Barbarin, S. Anantathanasarn, E. A. J. M. Bente, Y. S. Oei, M. K. Smit, and R. Notzel, “1.55 µm range InAs-InP (100) quantum-dot Fabry-Perot and ring lasers using narrow deeply etched ridge waveguides,” IEEE Photon. Technol. Lett. 18, 2644–2646 (2006).
    [Crossref]
  24. R. Kashyap, Fiber Bragg Gratings (Academic Press, San Diego, 1999).
  25. G. E. Town, K. Sugden, J. A. R. Williams, I. Bennion, and S. B. Poole, “Wide-band Fabry-Perot-like filters in optical fiber,” IEEE Photon. Technol. Lett. 7, 78–80 (1995).
    [Crossref]

2008 (3)

2007 (5)

2006 (6)

J. Azaña, C. K. Madsen, K. Takiguchi, and G. Cincontti (editors), Special Issue on “Optical Signal Processing,” IEEE/OSA J. Lightwave Technol. 24, 2484–2767 (2006).
[Crossref]

N. Q. Ngo and L. N. Binh, “Optical realization of Newton-Cotes-Based Integrators for Dark Soliton Generation,” IEEE/OSA J. Lightwave Technol. 24, 563–572 (2006).
[Crossref]

N. Q. Ngo, “Optical integrator for optical dark-soliton detection and pulse shaping,” Appl. Opt. 45, 6785–6791 (2006).
[Crossref] [PubMed]

Y. Park, M. Kulishov, R. Slavík, and J. Azaña, “Picosecond and sub-picosecond flat-top pulse generation using uniform long-period fiber grating,” Opt. Express, 14, 12671–12678 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-26-12670.
[Crossref]

Y. Barbarin, S. Anantathanasarn, E. A. J. M. Bente, Y. S. Oei, M. K. Smit, and R. Notzel, “1.55 µm range InAs-InP (100) quantum-dot Fabry-Perot and ring lasers using narrow deeply etched ridge waveguides,” IEEE Photon. Technol. Lett. 18, 2644–2646 (2006).
[Crossref]

R. Slavík, Y. Park, M. Kulishov, R. Morandotti, and J. Azaña, “Ultrafast all-optical differentiators,” Opt. Express 14, 10699–10707 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-22-10699.
[Crossref] [PubMed]

2005 (3)

2004 (1)

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oel, H. Binsma, G.-D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[Crossref] [PubMed]

2003 (1)

L. Venema, “Photonics Technologies,” Nature Insight 424, No. 6950 (2003).

1995 (1)

G. E. Town, K. Sugden, J. A. R. Williams, I. Bennion, and S. B. Poole, “Wide-band Fabry-Perot-like filters in optical fiber,” IEEE Photon. Technol. Lett. 7, 78–80 (1995).
[Crossref]

Anantathanasarn, S.

Y. Barbarin, S. Anantathanasarn, E. A. J. M. Bente, Y. S. Oei, M. K. Smit, and R. Notzel, “1.55 µm range InAs-InP (100) quantum-dot Fabry-Perot and ring lasers using narrow deeply etched ridge waveguides,” IEEE Photon. Technol. Lett. 18, 2644–2646 (2006).
[Crossref]

Azaña, J.

F. Li, Y. Park, and J. Azaña, “Precise and simple group delay measurement of dispersive devices based on ultrafast optical differentiation,” in Proc. of OFC/NFOEC’08, Paper OWD5, 2008.

J. Azaña, “Proposal of a uniform fiber Bragg grating as an ultrafast all-optical integrator,” Opt. Lett. 33, 4–6 (2008).
[Crossref]

F. Li, Y. Park, and J. Azaña, “Complete temporal pulse characterization based on phase reconstruction using optical ultrafast differentiation (PROUD),” Opt. Lett. 32, 3364–3366 (2007).
[Crossref] [PubMed]

R. Slavík, L. K. Oxenløwe, M. Galili, H. C. H. Mulvad, Y. Park, J. Azaña, and P. Jeppesen, “Demultiplexing of 320 and 640 Gbit/s OTDM data using ultrashort flat-top pulses,” IEEE Photon. Technol. Lett. 19, 1855–1857 (2007).
[Crossref]

Y. Park, M. Kulishov, R. Slavík, and J. Azaña, “Picosecond and sub-picosecond flat-top pulse generation using uniform long-period fiber grating,” Opt. Express, 14, 12671–12678 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-26-12670.
[Crossref]

R. Slavík, Y. Park, M. Kulishov, R. Morandotti, and J. Azaña, “Ultrafast all-optical differentiators,” Opt. Express 14, 10699–10707 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-22-10699.
[Crossref] [PubMed]

M. Kulishov and J. Azaña, “Long-period fiber gratings as ultrafast optical differentiators,” Opt. Lett. 30, 2700–2702 (2005).
[Crossref] [PubMed]

Barbarin, Y.

Y. Barbarin, S. Anantathanasarn, E. A. J. M. Bente, Y. S. Oei, M. K. Smit, and R. Notzel, “1.55 µm range InAs-InP (100) quantum-dot Fabry-Perot and ring lasers using narrow deeply etched ridge waveguides,” IEEE Photon. Technol. Lett. 18, 2644–2646 (2006).
[Crossref]

Bennion, I.

G. E. Town, K. Sugden, J. A. R. Williams, I. Bennion, and S. B. Poole, “Wide-band Fabry-Perot-like filters in optical fiber,” IEEE Photon. Technol. Lett. 7, 78–80 (1995).
[Crossref]

Bente, E. A. J. M.

Y. Barbarin, S. Anantathanasarn, E. A. J. M. Bente, Y. S. Oei, M. K. Smit, and R. Notzel, “1.55 µm range InAs-InP (100) quantum-dot Fabry-Perot and ring lasers using narrow deeply etched ridge waveguides,” IEEE Photon. Technol. Lett. 18, 2644–2646 (2006).
[Crossref]

Binh, L. N.

N. Q. Ngo and L. N. Binh, “Optical realization of Newton-Cotes-Based Integrators for Dark Soliton Generation,” IEEE/OSA J. Lightwave Technol. 24, 563–572 (2006).
[Crossref]

Binsma, H.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oel, H. Binsma, G.-D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[Crossref] [PubMed]

Brochu, G.

De Vries, T.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oel, H. Binsma, G.-D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[Crossref] [PubMed]

Den Besten, J. H.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oel, H. Binsma, G.-D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[Crossref] [PubMed]

Dong, J.

Dorren, H. J. S.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oel, H. Binsma, G.-D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[Crossref] [PubMed]

Galili, M.

R. Slavík, L. K. Oxenløwe, M. Galili, H. C. H. Mulvad, Y. Park, J. Azaña, and P. Jeppesen, “Demultiplexing of 320 and 640 Gbit/s OTDM data using ultrashort flat-top pulses,” IEEE Photon. Technol. Lett. 19, 1855–1857 (2007).
[Crossref]

Hill, M. T.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oel, H. Binsma, G.-D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[Crossref] [PubMed]

Huang, D.

Jeppesen, P.

R. Slavík, L. K. Oxenløwe, M. Galili, H. C. H. Mulvad, Y. Park, J. Azaña, and P. Jeppesen, “Demultiplexing of 320 and 640 Gbit/s OTDM data using ultrashort flat-top pulses,” IEEE Photon. Technol. Lett. 19, 1855–1857 (2007).
[Crossref]

Kashyap, R.

R. Kashyap, Fiber Bragg Gratings (Academic Press, San Diego, 1999).

Khoe, G.-D.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oel, H. Binsma, G.-D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[Crossref] [PubMed]

Kulishov, M.

LaRochelle, S.

Leijtens, X. J. M.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oel, H. Binsma, G.-D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[Crossref] [PubMed]

Li, F.

F. Li, Y. Park, and J. Azaña, “Precise and simple group delay measurement of dispersive devices based on ultrafast optical differentiation,” in Proc. of OFC/NFOEC’08, Paper OWD5, 2008.

F. Li, Y. Park, and J. Azaña, “Complete temporal pulse characterization based on phase reconstruction using optical ultrafast differentiation (PROUD),” Opt. Lett. 32, 3364–3366 (2007).
[Crossref] [PubMed]

Liu, D.

Madsen, C. K.

C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach (John Wiley & Sons, New York, USA1999).

Morandotti, R.

Mulvad, H. C. H.

R. Slavík, L. K. Oxenløwe, M. Galili, H. C. H. Mulvad, Y. Park, J. Azaña, and P. Jeppesen, “Demultiplexing of 320 and 640 Gbit/s OTDM data using ultrashort flat-top pulses,” IEEE Photon. Technol. Lett. 19, 1855–1857 (2007).
[Crossref]

Muriel, M. A.

Nawab, S. H.

A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and Systems, 2nd ed. (Prentice Hall, Upper Saddle River, NJ, USA1996).

Ngo, N. Q.

Notzel, R.

Y. Barbarin, S. Anantathanasarn, E. A. J. M. Bente, Y. S. Oei, M. K. Smit, and R. Notzel, “1.55 µm range InAs-InP (100) quantum-dot Fabry-Perot and ring lasers using narrow deeply etched ridge waveguides,” IEEE Photon. Technol. Lett. 18, 2644–2646 (2006).
[Crossref]

Oei, Y. S.

Y. Barbarin, S. Anantathanasarn, E. A. J. M. Bente, Y. S. Oei, M. K. Smit, and R. Notzel, “1.55 µm range InAs-InP (100) quantum-dot Fabry-Perot and ring lasers using narrow deeply etched ridge waveguides,” IEEE Photon. Technol. Lett. 18, 2644–2646 (2006).
[Crossref]

Oel, Y.-S.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oel, H. Binsma, G.-D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[Crossref] [PubMed]

Ogata, K.

K. OgataModern Control Engineering4th ed.(Prentice Hall, Upper Saddle River, NJ, USA2001).

Oppenheim, A. V.

A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and Systems, 2nd ed. (Prentice Hall, Upper Saddle River, NJ, USA1996).

Oxenløwe, L. K.

R. Slavík, L. K. Oxenløwe, M. Galili, H. C. H. Mulvad, Y. Park, J. Azaña, and P. Jeppesen, “Demultiplexing of 320 and 640 Gbit/s OTDM data using ultrashort flat-top pulses,” IEEE Photon. Technol. Lett. 19, 1855–1857 (2007).
[Crossref]

Park, Y.

F. Li, Y. Park, and J. Azaña, “Precise and simple group delay measurement of dispersive devices based on ultrafast optical differentiation,” in Proc. of OFC/NFOEC’08, Paper OWD5, 2008.

R. Slavík, L. K. Oxenløwe, M. Galili, H. C. H. Mulvad, Y. Park, J. Azaña, and P. Jeppesen, “Demultiplexing of 320 and 640 Gbit/s OTDM data using ultrashort flat-top pulses,” IEEE Photon. Technol. Lett. 19, 1855–1857 (2007).
[Crossref]

F. Li, Y. Park, and J. Azaña, “Complete temporal pulse characterization based on phase reconstruction using optical ultrafast differentiation (PROUD),” Opt. Lett. 32, 3364–3366 (2007).
[Crossref] [PubMed]

R. Slavík, Y. Park, M. Kulishov, R. Morandotti, and J. Azaña, “Ultrafast all-optical differentiators,” Opt. Express 14, 10699–10707 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-22-10699.
[Crossref] [PubMed]

Y. Park, M. Kulishov, R. Slavík, and J. Azaña, “Picosecond and sub-picosecond flat-top pulse generation using uniform long-period fiber grating,” Opt. Express, 14, 12671–12678 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-26-12670.
[Crossref]

Poole, S. B.

G. E. Town, K. Sugden, J. A. R. Williams, I. Bennion, and S. B. Poole, “Wide-band Fabry-Perot-like filters in optical fiber,” IEEE Photon. Technol. Lett. 7, 78–80 (1995).
[Crossref]

Preciado, M. A.

Simmons, G.F.

G.F. SimmonsDifferential Equations with Applications and Historical Notes, 2nd ed. (McGraw-Hill, New York, USA1991).

Slavík, R.

R. Slavík, L. K. Oxenløwe, M. Galili, H. C. H. Mulvad, Y. Park, J. Azaña, and P. Jeppesen, “Demultiplexing of 320 and 640 Gbit/s OTDM data using ultrashort flat-top pulses,” IEEE Photon. Technol. Lett. 19, 1855–1857 (2007).
[Crossref]

Y. Park, M. Kulishov, R. Slavík, and J. Azaña, “Picosecond and sub-picosecond flat-top pulse generation using uniform long-period fiber grating,” Opt. Express, 14, 12671–12678 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-26-12670.
[Crossref]

R. Slavík, Y. Park, M. Kulishov, R. Morandotti, and J. Azaña, “Ultrafast all-optical differentiators,” Opt. Express 14, 10699–10707 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-22-10699.
[Crossref] [PubMed]

G. Brochu, S. LaRochelle, and R. Slavík, “Modeling and experimental demonstration of ultracompact multiwavelength distributed Fabry-Perot fiber lasers,” J. Lightwave Technol. 23, 44–53 (2005).
[Crossref]

Smalbrugge, B.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oel, H. Binsma, G.-D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[Crossref] [PubMed]

Smit, M. K.

Y. Barbarin, S. Anantathanasarn, E. A. J. M. Bente, Y. S. Oei, M. K. Smit, and R. Notzel, “1.55 µm range InAs-InP (100) quantum-dot Fabry-Perot and ring lasers using narrow deeply etched ridge waveguides,” IEEE Photon. Technol. Lett. 18, 2644–2646 (2006).
[Crossref]

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oel, H. Binsma, G.-D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[Crossref] [PubMed]

Sugden, K.

G. E. Town, K. Sugden, J. A. R. Williams, I. Bennion, and S. B. Poole, “Wide-band Fabry-Perot-like filters in optical fiber,” IEEE Photon. Technol. Lett. 7, 78–80 (1995).
[Crossref]

Town, G. E.

G. E. Town, K. Sugden, J. A. R. Williams, I. Bennion, and S. B. Poole, “Wide-band Fabry-Perot-like filters in optical fiber,” IEEE Photon. Technol. Lett. 7, 78–80 (1995).
[Crossref]

Venema, L.

L. Venema, “Photonics Technologies,” Nature Insight 424, No. 6950 (2003).

Wang, Q.

Williams, J. A. R.

G. E. Town, K. Sugden, J. A. R. Williams, I. Bennion, and S. B. Poole, “Wide-band Fabry-Perot-like filters in optical fiber,” IEEE Photon. Technol. Lett. 7, 78–80 (1995).
[Crossref]

Willsky, A. S.

A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and Systems, 2nd ed. (Prentice Hall, Upper Saddle River, NJ, USA1996).

Xu, J.

Yao, P.

Zeng, F.

Zhang, X.

Zhao, J. H.

C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach (John Wiley & Sons, New York, USA1999).

Appl. Opt. (1)

EURASIP J. Appl. Signal Proc. (1)

C. K. Madsen, D. Dragoman, and J. Azaña (editors), Special Issue on “Signal Analysis Tools for Optical Signal Processing,” EURASIP J. Appl. Signal Proc. 10, 1449–1623 (2005).
[Crossref]

IEEE Photon. Technol. Lett. (3)

R. Slavík, L. K. Oxenløwe, M. Galili, H. C. H. Mulvad, Y. Park, J. Azaña, and P. Jeppesen, “Demultiplexing of 320 and 640 Gbit/s OTDM data using ultrashort flat-top pulses,” IEEE Photon. Technol. Lett. 19, 1855–1857 (2007).
[Crossref]

Y. Barbarin, S. Anantathanasarn, E. A. J. M. Bente, Y. S. Oei, M. K. Smit, and R. Notzel, “1.55 µm range InAs-InP (100) quantum-dot Fabry-Perot and ring lasers using narrow deeply etched ridge waveguides,” IEEE Photon. Technol. Lett. 18, 2644–2646 (2006).
[Crossref]

G. E. Town, K. Sugden, J. A. R. Williams, I. Bennion, and S. B. Poole, “Wide-band Fabry-Perot-like filters in optical fiber,” IEEE Photon. Technol. Lett. 7, 78–80 (1995).
[Crossref]

IEEE/OSA J. Lightwave Technol. (2)

J. Azaña, C. K. Madsen, K. Takiguchi, and G. Cincontti (editors), Special Issue on “Optical Signal Processing,” IEEE/OSA J. Lightwave Technol. 24, 2484–2767 (2006).
[Crossref]

N. Q. Ngo and L. N. Binh, “Optical realization of Newton-Cotes-Based Integrators for Dark Soliton Generation,” IEEE/OSA J. Lightwave Technol. 24, 563–572 (2006).
[Crossref]

J. Lightwave Technol. (2)

Nature (1)

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oel, H. Binsma, G.-D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[Crossref] [PubMed]

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L. Venema, “Photonics Technologies,” Nature Insight 424, No. 6950 (2003).

Opt. Express (1)

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Y. Park, M. Kulishov, R. Slavík, and J. Azaña, “Picosecond and sub-picosecond flat-top pulse generation using uniform long-period fiber grating,” Opt. Express, 14, 12671–12678 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-26-12670.
[Crossref]

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Figures (7)

Fig. 1.
Fig. 1. Concept diagram of the proposed photonic temporal integrator. The integrator is implemented using two superimposed fiber Bragg gratings (acting as a resonant cavity) permanently photo-inscribed in an Er-Yb co-doped optical fiber that provides optical gain. The gain level is controlled via power of the optical pump (980-nm laser diode). The inset shows the measured (circles) and numerically calculated (solid, blue curve) integrator spectral transfer function. For comparison, the spectral transfer function of an ideal integrator is also shown (solid, red curve).
Fig. 2.
Fig. 2. Experimental setup for the integration of pulses generated using electro-optically modulated signal. TL: Tunable laser, ISO: optical isolator, PC: polarization controller, IMOD: Optical intensity modulator, PPG: Picosecond electric pulse generator with 70 ps FWHM time width, AWG: Electric arbitrary waveform generator with 500 MHz bandwidth, Pump: 980-nm semiconductor pump laser, Amp: Erbium-doped fiber amplifier, OS: Optical sampler (photoreceiver).
Fig. 3.
Fig. 3. Experimental results demonstrating time-domain integration of a single optical Gaussian pulse for two different input pulse FWHM time widths ((a) 140 ps and (b) 60 ps). The temporal optical intensity of the input pulse (orange curve) and the integrator output (green curve) are captured using a 20-GHz photoreceiver. For comparison, the square of the numerically calculated time cumulative integral of the measured input pulse field (square root of the measured temporal intensity profile) is also shown (yellow curve).
Fig. 4.
Fig. 4. (a) Diagram showing all-optical integration of two consecutive optical pulses with different relative phases. For relative phases of 0 (in-phase – the field amplitudes are of the same sign, red curves) and π (out-of-phase – the field amplitudes are of opposite signs, blue curves), the time integral is expected to be a double step-rising waveform, and a flat-top waveform, respectively. (b) Experimental setup for the double pulse integration. 14-ps pulses are generated from the FFL followed by an optical bandpass filter (0.4 nm 3dB-bandwidth). Time-delayed pulse replicas are made by using a fiber-coupled Michelson interferometer. For the shortest time delay (170 ps), we confirmed the relative phase of the two pulses measuring the optical spectrum of the double-pulse structure, which is shown in the inset: for in-phase pulses (red line), the spectrum has a maximum at the integrator central frequency, while for the out-of-phase pulses (blue line), there is a minimum. The output spectrum is shown as green line.
Fig. 5.
Fig. 5. Experimental results demonstrating time-domain integration of double optical pulses. (a) Time-domain optical intensity of the input signal - two 14-ps consecutive optical pulse with various inter-pulse delays (170 ps, 500 ps, and 1000ps). The integrated output depends on the relative phase of the two pulses: (b) integration for in-phase pulses; (c) integration for out-ofphase pulses.
Fig. 6.
Fig. 6. Experimental results for the optical integration of two flat-top pulses set in-phase and out-of-phase. Left: in-phase; Right: out-of-phase. The input electric waveform: blue; the modulated optical intensity waveform: red, and the output optical intensity waveform: yellow.
Fig. 7.
Fig. 7. (a) Schematic diagram of an integrator-based optical computing system designed for solving the first-order linear ordinary differential equation (ODE) defined in the figure. The two graphs at the bottom show the experimental (solid curves) and numerical (circles) solutions of the ODE for two different input optical signals: (b) an input ultrashort temporal impulse (FWHM time-width=60 ps) and (c) a constant excitation over a limited temporal window (2.9-ns long square-like pulse). In each case, the ODE is solved for different positive values of the parameter k.

Equations (3)

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h ( t ) u ( t ) ; u ( t ) = 0 for t < 0 , u ( t ) = 1 for t 0 .
h ( t ) exp ( k t ) u ( t ) ,
dy ( t ) dt + ky ( t ) = x ( t )

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