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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 6 — Mar. 25, 2013
  • pp: 7008–7013
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All-optical computation system for solving differential equations based on optical intensity differentiator

Sisi Tan, Zhao Wu, Lei Lei, Shoujin Hu, Jianji Dong, and Xinliang Zhang  »View Author Affiliations


Optics Express, Vol. 21, Issue 6, pp. 7008-7013 (2013)
http://dx.doi.org/10.1364/OE.21.007008


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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 OSA

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

1. L. Venema, “Photonic technologies,” Nature 424(6950), 809 (2003). [CrossRef]

]. In analogy with the development in electronic domain, many equivalent devices in photonic domain have been proposed, including all-optical differentiators [2

2. 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(13), 1872–1874 (2007). [CrossRef] [PubMed]

, 3

3. M. Kulishov and J. Azaña, “Long-period fiber gratings as ultrafast optical differentiators,” Opt. Lett. 30(20), 2700–2702 (2005). [CrossRef] [PubMed]

] and all-optical integrators [4

4. R. Slavík, Y. Park, N. Ayotte, S. Doucet, T. J. Ahn, S. LaRochelle, and J. Azaña, “Photonic temporal integrator for all-optical computing,” Opt. Express 16(22), 18202–18214 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-22-18202. [CrossRef] [PubMed]

6

6. M. Ferrera, Y. Park, L. Razzari, B. E. Little, S. T. Chu, R. Morandotti, D. J. Moss, and J. Azaña, “All-optical 1st and 2nd order integration on a chip,” Opt. Express 19(23), 23153–23161 (2011), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-19-23-23153. [CrossRef] [PubMed]

]. 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

7. D. N. McCloskey, “History, differential equations, and the problem of narration,” Hist. Theory 30(1), 21–36 (1991). [CrossRef]

], 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

4. R. Slavík, Y. Park, N. Ayotte, S. Doucet, T. J. Ahn, S. LaRochelle, and J. Azaña, “Photonic temporal integrator for all-optical computing,” Opt. Express 16(22), 18202–18214 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-22-18202. [CrossRef] [PubMed]

6

6. M. Ferrera, Y. Park, L. Razzari, B. E. Little, S. T. Chu, R. Morandotti, D. J. Moss, and J. Azaña, “All-optical 1st and 2nd order integration on a chip,” Opt. Express 19(23), 23153–23161 (2011), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-19-23-23153. [CrossRef] [PubMed]

], 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

8. J. Azaa, “Ultrafast analog all-optical signal processors based on fiber-grating devices,” IEEE Photon. Technol. Lett. 2, 359–386 (2010).

]. For all-optical differentiators, there are two different kinds, namely, the optical intensity differentiator and field differentiator [9

9. R. Slavík, Y. Park, D. Krčmařík, and J. Azaña, “Stable all-fiber photonic temporal differentiator using a long-period fiber grating interferometer,” Opt. Commun. 282(12), 2339–2342 (2009). [CrossRef]

]. 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

4. R. Slavík, Y. Park, N. Ayotte, S. Doucet, T. J. Ahn, S. LaRochelle, and J. Azaña, “Photonic temporal integrator for all-optical computing,” Opt. Express 16(22), 18202–18214 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-22-18202. [CrossRef] [PubMed]

] or silicon micro-ring resonator [10

10. L. Lu, J. Wu, T. Wang, and Y. Su, “Compact all-optical differential-equation solver based on silicon microring resonator,” Frontiers of Optoelectronics 5(1), 99–106 (2012). [CrossRef]

]. 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:
dy(t)dt+ky(t)=x(t)
(1)
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:

Y(ω)=1k+jωX(ω)
(2)

Therefore, in time domain:
y(t)=u(t)ektx(t)
(3)
where u(t) represents the unit step function.

Meanwhile the solution can also be derived using Taylor expansion:
Y(ω)=1k+jωX(ω)=X(ω)kn=0(jωk)n
(4)
Using inverse Fourier transformation, the temporal solution will be:

y(t)=1kn=0(1k)ndnx(t)dtn
(5)

It should be noticed that the sequence can be convergent only when|jω/k|<1. Therefore, for different input signals, there’s a threshold of k (kth) 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, kth is 2/ps and 0.7/ps, respectively.

The schematic diagram of the computing system [11

11. A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signal and System (Prentice Hall, 1996).

] required for solving the constant-coefficient first-order differential equation is shown in Fig. 1
Fig. 1 The schematic diagram of the computing system required for solving the constant-coefficient first-order differential equation.
. 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:
y1(t)=1kx(t)
(6)
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:
y2(t)=1k(x(t)dy1(t)dt)=1k(x(t)1kdx(t)dt)
(7)
as the circulation goes and finally stabilizes, the solution of the differential equation can be obtained at the output:
yn(t)=1k(x(t)dyn1(t)dt)=1k(x(t)1kdx(t)dt+...+(1k)n1dn1x(t)dtn1)
(8)
in agreement with the theoretically derived solution in Eq. (5) using Taylor expansion. Equation (8) can also be used to explain the condition of convergence for k: for every one more circle the signal transmits in the loop, there’s one more high order differentiation added to the output. In order to obtain the solution when the whole system reaches a steady state, the added high order differentiation should be diminishing with the differentiation order, which leads to the condition of convergence for k as discussed above.

The corresponding schematic diagram of the experiment is illustrated in Fig. 2
Fig. 2 Schematic diagram of the experimental scheme.
, a continuous-wave probe light at λp (generated from LD2) is launched into the first SOA with another control signal at λc (illustrated in Fig. 2 as the blue Gaussian pulse x(t)) simultaneously [2

2. 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(13), 1872–1874 (2007). [CrossRef] [PubMed]

]. Due to the cross-phase modulation (XPM) in SOA and frequency discrimination of the OF, the positive differentiation of the input signal (λc) can be obtained at the wavelength of λp (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

12. J. Dong, S. Fu, X. Zhang, P. Shum, L. Zhang, and D. Huang, “Analytical solution for SOA-based all-optical wavelength conversion using transient cross-phase modulation,” IEEE Photon. Technol. Lett. 18(24), 2554–2556 (2006). [CrossRef]

] is needed. Consequently, the negative differentiation of the input signal is carried out at the wavelength of λc(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
Fig. 3 (a) theoretical solutions with different positive values of k for input 40Gb/s super-Gaussian pulse (m = 3); (b) theoretical solutions with different positive values of k for input 40Gb/s Gaussian (nearly transform-limited) pulse.
. 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 “1012 /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(ω)equals toτjω (such as the differentiator based on micro-ring resonator [13

13. F. Liu, T. Wang, L. Qiang, T. Ye, Z. Zhang, M. Qiu, and Y. Su, “Compact optical temporal differentiator based on silicon microring resonator,” Opt. Express 16(20), 15880–15886 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-20-15880. [CrossRef] [PubMed]

]),rather than exactly equals tojω, where τcompensates 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>kth, 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≤kth, 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 measured results in terms of power intensity are shown in Fig. 5
Fig. 5 Experimental results demonstrating solutions of the first-order linear ordinary differential equation. (a1) input Gaussian (nearly transform-limited) pulse; (a2) experimental (green line) and simulated convolution solution (yellow dot line) for Gaussian input when k = 1/ps; (b1) input super-Gaussian pulse (m = 3); (b2) experimental (green line) and simulated convolution solution (yellow dot line) solution for super-Gaussian input when k = 2.5/ps.
. 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>kth; 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≤kth, 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≤kth are illustrated in Fig. 6
Fig. 6 The experimental superposed results of the input signal and the first-order differentiation signal (green line) along with simulated convolution solutions (yellow dot line) for different values of k with the input Gaussian signal:(a) k = 0.1/ps; (b)k = 0.2/ps; (c)k = 0.5/ps.
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

14. J. Dong, B. Luo, Y. Zhang, D. Huang, and X. Zhang, “Reconfigurable photonic differentiators based on all-optical phase modulation and linear filtering,” Opt. Commun. 284(24), 5792–5797 (2011). [CrossRef]

] 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).

References and links

1.

L. Venema, “Photonic technologies,” Nature 424(6950), 809 (2003). [CrossRef]

2.

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(13), 1872–1874 (2007). [CrossRef] [PubMed]

3.

M. Kulishov and J. Azaña, “Long-period fiber gratings as ultrafast optical differentiators,” Opt. Lett. 30(20), 2700–2702 (2005). [CrossRef] [PubMed]

4.

R. Slavík, Y. Park, N. Ayotte, S. Doucet, T. J. Ahn, S. LaRochelle, and J. Azaña, “Photonic temporal integrator for all-optical computing,” Opt. Express 16(22), 18202–18214 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-22-18202. [CrossRef] [PubMed]

5.

M. Ferrera, Y. Park, L. Razzari, B. E. Little, S. T. Chu, R. Morandotti, D. J. Moss, and J. Azaña, “On-chip CMOS-compatible all-optical integrator,” Nat Commun 1(3), 1–29 (2010). [CrossRef] [PubMed]

6.

M. Ferrera, Y. Park, L. Razzari, B. E. Little, S. T. Chu, R. Morandotti, D. J. Moss, and J. Azaña, “All-optical 1st and 2nd order integration on a chip,” Opt. Express 19(23), 23153–23161 (2011), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-19-23-23153. [CrossRef] [PubMed]

7.

D. N. McCloskey, “History, differential equations, and the problem of narration,” Hist. Theory 30(1), 21–36 (1991). [CrossRef]

8.

J. Azaa, “Ultrafast analog all-optical signal processors based on fiber-grating devices,” IEEE Photon. Technol. Lett. 2, 359–386 (2010).

9.

R. Slavík, Y. Park, D. Krčmařík, and J. Azaña, “Stable all-fiber photonic temporal differentiator using a long-period fiber grating interferometer,” Opt. Commun. 282(12), 2339–2342 (2009). [CrossRef]

10.

L. Lu, J. Wu, T. Wang, and Y. Su, “Compact all-optical differential-equation solver based on silicon microring resonator,” Frontiers of Optoelectronics 5(1), 99–106 (2012). [CrossRef]

11.

A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signal and System (Prentice Hall, 1996).

12.

J. Dong, S. Fu, X. Zhang, P. Shum, L. Zhang, and D. Huang, “Analytical solution for SOA-based all-optical wavelength conversion using transient cross-phase modulation,” IEEE Photon. Technol. Lett. 18(24), 2554–2556 (2006). [CrossRef]

13.

F. Liu, T. Wang, L. Qiang, T. Ye, Z. Zhang, M. Qiu, and Y. Su, “Compact optical temporal differentiator based on silicon microring resonator,” Opt. Express 16(20), 15880–15886 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-20-15880. [CrossRef] [PubMed]

14.

J. Dong, B. Luo, Y. Zhang, D. Huang, and X. Zhang, “Reconfigurable photonic differentiators based on all-optical phase modulation and linear filtering,” Opt. Commun. 284(24), 5792–5797 (2011). [CrossRef]

OCIS Codes
(070.1170) Fourier optics and signal processing : Analog optical signal processing
(250.5980) Optoelectronics : Semiconductor optical amplifiers

ToC Category:
Optics in Computing

History
Original Manuscript: October 8, 2012
Revised Manuscript: January 11, 2013
Manuscript Accepted: January 18, 2013
Published: March 13, 2013

Citation
Sisi Tan, Zhao Wu, Lei Lei, Shoujin Hu, Jianji Dong, and Xinliang Zhang, "All-optical computation system for solving differential equations based on optical intensity differentiator," Opt. Express 21, 7008-7013 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-6-7008


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References

  1. L. Venema, “Photonic technologies,” Nature 424(6950), 809 (2003). [CrossRef]
  2. 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(13), 1872–1874 (2007). [CrossRef] [PubMed]
  3. M. Kulishov and J. Azaña, “Long-period fiber gratings as ultrafast optical differentiators,” Opt. Lett. 30(20), 2700–2702 (2005). [CrossRef] [PubMed]
  4. R. Slavík, Y. Park, N. Ayotte, S. Doucet, T. J. Ahn, S. LaRochelle, and J. Azaña, “Photonic temporal integrator for all-optical computing,” Opt. Express 16(22), 18202–18214 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-22-18202 . [CrossRef] [PubMed]
  5. M. Ferrera, Y. Park, L. Razzari, B. E. Little, S. T. Chu, R. Morandotti, D. J. Moss, and J. Azaña, “On-chip CMOS-compatible all-optical integrator,” Nat Commun 1(3), 1–29 (2010). [CrossRef] [PubMed]
  6. M. Ferrera, Y. Park, L. Razzari, B. E. Little, S. T. Chu, R. Morandotti, D. J. Moss, and J. Azaña, “All-optical 1st and 2nd order integration on a chip,” Opt. Express 19(23), 23153–23161 (2011), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-19-23-23153 . [CrossRef] [PubMed]
  7. D. N. McCloskey, “History, differential equations, and the problem of narration,” Hist. Theory 30(1), 21–36 (1991). [CrossRef]
  8. J. Azaa, “Ultrafast analog all-optical signal processors based on fiber-grating devices,” IEEE Photon. Technol. Lett. 2, 359–386 (2010).
  9. R. Slavík, Y. Park, D. Kr?ma?ík, and J. Azaña, “Stable all-fiber photonic temporal differentiator using a long-period fiber grating interferometer,” Opt. Commun. 282(12), 2339–2342 (2009). [CrossRef]
  10. L. Lu, J. Wu, T. Wang, and Y. Su, “Compact all-optical differential-equation solver based on silicon microring resonator,” Frontiers of Optoelectronics 5(1), 99–106 (2012). [CrossRef]
  11. A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signal and System (Prentice Hall, 1996).
  12. J. Dong, S. Fu, X. Zhang, P. Shum, L. Zhang, and D. Huang, “Analytical solution for SOA-based all-optical wavelength conversion using transient cross-phase modulation,” IEEE Photon. Technol. Lett. 18(24), 2554–2556 (2006). [CrossRef]
  13. F. Liu, T. Wang, L. Qiang, T. Ye, Z. Zhang, M. Qiu, and Y. Su, “Compact optical temporal differentiator based on silicon microring resonator,” Opt. Express 16(20), 15880–15886 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-20-15880 . [CrossRef] [PubMed]
  14. J. Dong, B. Luo, Y. Zhang, D. Huang, and X. Zhang, “Reconfigurable photonic differentiators based on all-optical phase modulation and linear filtering,” Opt. Commun. 284(24), 5792–5797 (2011). [CrossRef]

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