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.

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References

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  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]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [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]

2012

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]

2011

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]

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]

2010

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

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]

2009

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]

2008

2007

2006

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]

2005

2003

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

1991

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

Ahn, T. J.

Ayotte, N.

Azaa, J.

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

Azaña, J.

Chu, S. T.

Dong, J.

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]

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]

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]

Doucet, S.

Ferrera, M.

Fu, S.

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]

Huang, D.

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]

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]

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]

Krcmarík, D.

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]

Kulishov, M.

LaRochelle, S.

Little, B. E.

Liu, D.

Liu, F.

Lu, L.

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]

Luo, B.

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]

McCloskey, D. N.

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

Morandotti, R.

Moss, D. J.

Park, Y.

Qiang, L.

Qiu, M.

Razzari, L.

Shum, P.

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]

Slavík, R.

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]

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]

Su, Y.

Venema, L.

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

Wang, T.

Wu, J.

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]

Xu, J.

Ye, T.

Zhang, L.

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]

Zhang, X.

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]

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]

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]

Zhang, Y.

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]

Zhang, Z.

Frontiers of Optoelectronics

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]

Hist. Theory

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

IEEE Photon. Technol. Lett.

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

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]

Nat Commun

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]

Nature

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

Opt. Commun.

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]

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]

Opt. Express

Opt. Lett.

Other

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

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Figures (6)

Fig. 1
Fig. 1

The schematic diagram of the computing system required for solving the constant-coefficient first-order differential equation.

Fig. 2
Fig. 2

Schematic diagram of the experimental scheme.

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.

Fig. 4
Fig. 4

Experimental setup for solving the constant-coefficient first-order linear ordinary differential equation.

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.

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.

Equations (8)

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dy(t) dt +ky(t)=x(t)
Y(ω)= 1 k+jω X(ω)
y(t)=u(t) e kt x(t)
Y(ω)= 1 k+jω X(ω)= X(ω) k n=0 ( jω k ) n
y( t )= 1 k n=0 ( 1 k ) n d n x(t) d t n
y 1 ( t )= 1 k x(t)
y 2 ( t )= 1 k (x(t) d y 1 ( t ) dt )= 1 k ( x(t) 1 k dx(t) dt )
y n ( t )= 1 k (x(t) d y n1 ( t ) dt )= 1 k ( x(t) 1 k dx(t) dt +...+ ( 1 k ) n1 d n1 x(t) d t n1 )

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