Abstract

We experimentally demonstrate an all-optical temporal computation scheme for solving 1st- and 2nd-order linear ordinary differential equations (ODEs) with tunable constant coefficients by using Fabry-Pérot semiconductor optical amplifiers (FP-SOAs). By changing the injection currents of FP-SOAs, the constant coefficients of the differential equations are practically tuned. A quite large constant coefficient tunable range from 0.0026/ps to 0.085/ps is achieved for the 1st-order differential equation. Moreover, the constant coefficient p of the 2nd-order ODE solver can be continuously tuned from 0.0216/ps to 0.158/ps, correspondingly with the constant coefficient q varying from 0.0000494/ps2 to 0.006205/ps2. Additionally, a theoretical model that combining the carrier density rate equation of the semiconductor optical amplifier (SOA) with the transfer function of the Fabry-Pérot (FP) cavity is exploited to analyze the solving processes. For both 1st- and 2nd-order solvers, excellent agreements between the numerical simulations and the experimental results are obtained. The FP-SOAs based all-optical differential-equation solvers can be easily integrated with other optical components based on InP/InGaAsP materials, such as laser, modulator, photodetector and waveguide, which can motivate the realization of the complicated optical computing on a single integrated chip.

© 2015 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]

2014 (3)

2013 (4)

S. Tan, L. Xiang, J. Zou, Q. Zhang, Z. Wu, Y. Yu, J. Dong, and X. Zhang, “High-order all-optical differential equation solver based on microring resonators,” Opt. Lett. 38(19), 3735–3738 (2013).
[Crossref] [PubMed]

F. Bontempi, S. Faralli, N. Andriolli, and G. Contestabile, “An InP Monolithically Integrated Unicast and Multicast Wavelength Converter,” IEEE Photon. Technol. Lett. 25(22), 2178–2181 (2013).
[Crossref]

M. Lu, H.-C. Park, A. Sivananthan, J. S. Parker, E. Bloch, L. A. Johansson, M. J. W. Rodwell, and L. A. Coldren, “Monolithic Integration of a High-Speed Widely Tunable Optical Coherent Receiver,” IEEE Photon. Technol. Lett. 25(11), 1077–1080 (2013).
[Crossref]

S. Tan, Z. Wu, L. Lei, S. Hu, J. Dong, and X. Zhang, “All-optical computation system for solving differential equations based on optical intensity differentiator,” Opt. Express 21(6), 7008–7013 (2013).
[Crossref] [PubMed]

2012 (1)

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

2011 (2)

2010 (2)

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), 29 (2010).
[Crossref] [PubMed]

J. Azaa, “Ultrafast analog all-optical signal processors based on fiber-grating devices,” IEEE Photon. J. 2(3), 359–386 (2010).
[Crossref]

2009 (2)

Y. Ding, X. Zhang, X. Zhang, and D. Huang, “Active microring optical integrator associated with electroabsorption modulators for high speed low light power loadable and erasable optical memory unit,” Opt. Express 17(15), 12835–12848 (2009).
[Crossref] [PubMed]

F. Wang, Y. Yu, X. Huang, and X. L. Zhang, “Single and Multiwavelength All-Optical Clock Recovery Using Fabry-Perot Semiconductor Optical Amplifier,” IEEE Photon. Technol. Lett. 21(16), 1109–1111 (2009).
[Crossref]

2008 (4)

2007 (2)

2006 (1)

2005 (1)

2003 (1)

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

1989 (1)

G. P. Agrawal and N. A. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25(11), 2297–2306 (1989).
[Crossref]

1987 (1)

J. C. Simon, “GaInAsP semiconductor laser amplifiers for single-mode fiber communications,” J. Lightwave Technol. 5(9), 1286–1295 (1987).
[Crossref]

1985 (1)

J. Buus and R. Plastow, “A Theoretical and Experimental Investigation of Fabry-Perot Semiconductor Laser Amplifiers,” IEEE J. Quantum Electron. 21(6), 614–618 (1985).
[Crossref]

Agrawal, G. P.

G. P. Agrawal and N. A. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25(11), 2297–2306 (1989).
[Crossref]

Ahn, T.-J.

Andriolli, N.

F. Bontempi, S. Faralli, N. Andriolli, and G. Contestabile, “An InP Monolithically Integrated Unicast and Multicast Wavelength Converter,” IEEE Photon. Technol. Lett. 25(22), 2178–2181 (2013).
[Crossref]

Asghari, M. H.

M. H. Asghari and J. Azaña, “Photonic integrator-based optical memory unit,” IEEE Photon. Technol. Lett. 23(4), 209–211 (2011).
[Crossref]

Ashrafi, R.

Ayotte, N.

Azaa, J.

J. Azaa, “Ultrafast analog all-optical signal processors based on fiber-grating devices,” IEEE Photon. J. 2(3), 359–386 (2010).
[Crossref]

Azaña, J.

N. Huang, M. Li, R. Ashrafi, L. Wang, X. Wang, J. Azaña, and N. Zhu, “Active Fabry-Perot cavity for photonic temporal integrator with ultra-long operation time window,” Opt. Express 22(3), 3105–3116 (2014).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

M. H. Asghari and J. Azaña, “Photonic integrator-based optical memory unit,” IEEE Photon. Technol. Lett. 23(4), 209–211 (2011).
[Crossref]

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), 29 (2010).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

J. Azaña, “Proposal of a uniform fiber Bragg grating as an ultrafast all-optical integrator,” Opt. Lett. 33(1), 4–6 (2008).
[Crossref] [PubMed]

Y. Park, T.-J. Ahn, Y. Dai, J. Yao, and J. Azaña, “All-optical temporal integration of ultrafast pulse waveforms,” Opt. Express 16(22), 17817–17825 (2008).
[Crossref] [PubMed]

N. K. Berger, B. Levit, B. Fischer, M. Kulishov, D. V. Plant, and J. Azaña, “Temporal differentiation of optical signals using a phase-shifted fiber Bragg grating,” Opt. Express 15(2), 371–381 (2007).
[Crossref] [PubMed]

R. Slavík, Y. Park, M. Kulishov, R. Morandotti, and J. Azaña, “Ultrafast all-optical differentiators,” Opt. Express 14(22), 10699–10707 (2006).
[Crossref] [PubMed]

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

Berger, N. K.

Bloch, E.

M. Lu, H.-C. Park, A. Sivananthan, J. S. Parker, E. Bloch, L. A. Johansson, M. J. W. Rodwell, and L. A. Coldren, “Monolithic Integration of a High-Speed Widely Tunable Optical Coherent Receiver,” IEEE Photon. Technol. Lett. 25(11), 1077–1080 (2013).
[Crossref]

Bontempi, F.

F. Bontempi, S. Faralli, N. Andriolli, and G. Contestabile, “An InP Monolithically Integrated Unicast and Multicast Wavelength Converter,” IEEE Photon. Technol. Lett. 25(22), 2178–2181 (2013).
[Crossref]

Buus, J.

J. Buus and R. Plastow, “A Theoretical and Experimental Investigation of Fabry-Perot Semiconductor Laser Amplifiers,” IEEE J. Quantum Electron. 21(6), 614–618 (1985).
[Crossref]

Cao, P.

Chen, J.

T. Yang, J. Dong, L. Lu, L. Zhou, A. Zheng, X. Zhang, and J. Chen, “All-optical differential equation solver with constant-coefficient tunable based on a single microring resonator,” Sci. Rep. 4, 5581 (2014).
[PubMed]

Chu, S. T.

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).
[Crossref] [PubMed]

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), 29 (2010).
[Crossref] [PubMed]

Coldren, L. A.

M. Lu, H.-C. Park, A. Sivananthan, J. S. Parker, E. Bloch, L. A. Johansson, M. J. W. Rodwell, and L. A. Coldren, “Monolithic Integration of a High-Speed Widely Tunable Optical Coherent Receiver,” IEEE Photon. Technol. Lett. 25(11), 1077–1080 (2013).
[Crossref]

Contestabile, G.

F. Bontempi, S. Faralli, N. Andriolli, and G. Contestabile, “An InP Monolithically Integrated Unicast and Multicast Wavelength Converter,” IEEE Photon. Technol. Lett. 25(22), 2178–2181 (2013).
[Crossref]

Dai, Y.

Ding, Y.

Dong, J.

Doucet, S.

Faralli, S.

F. Bontempi, S. Faralli, N. Andriolli, and G. Contestabile, “An InP Monolithically Integrated Unicast and Multicast Wavelength Converter,” IEEE Photon. Technol. Lett. 25(22), 2178–2181 (2013).
[Crossref]

Ferrera, M.

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).
[Crossref] [PubMed]

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), 29 (2010).
[Crossref] [PubMed]

Fischer, B.

Hu, S.

Hu, X.

Huang, D.

Huang, N.

Huang, X.

F. Wang, Y. Yu, X. Huang, and X. L. Zhang, “Single and Multiwavelength All-Optical Clock Recovery Using Fabry-Perot Semiconductor Optical Amplifier,” IEEE Photon. Technol. Lett. 21(16), 1109–1111 (2009).
[Crossref]

Jiang, X.

Johansson, L. A.

M. Lu, H.-C. Park, A. Sivananthan, J. S. Parker, E. Bloch, L. A. Johansson, M. J. W. Rodwell, and L. A. Coldren, “Monolithic Integration of a High-Speed Widely Tunable Optical Coherent Receiver,” IEEE Photon. Technol. Lett. 25(11), 1077–1080 (2013).
[Crossref]

Kulishov, M.

LaRochelle, S.

Lei, L.

Levit, B.

Li, M.

Little, B. E.

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).
[Crossref] [PubMed]

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), 29 (2010).
[Crossref] [PubMed]

Liu, D.

Liu, F.

Lu, L.

T. Yang, J. Dong, L. Lu, L. Zhou, A. Zheng, X. Zhang, and J. Chen, “All-optical differential equation solver with constant-coefficient tunable based on a single microring resonator,” Sci. Rep. 4, 5581 (2014).
[PubMed]

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

Lu, M.

M. Lu, H.-C. Park, A. Sivananthan, J. S. Parker, E. Bloch, L. A. Johansson, M. J. W. Rodwell, and L. A. Coldren, “Monolithic Integration of a High-Speed Widely Tunable Optical Coherent Receiver,” IEEE Photon. Technol. Lett. 25(11), 1077–1080 (2013).
[Crossref]

Morandotti, R.

Moss, D. J.

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).
[Crossref] [PubMed]

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), 29 (2010).
[Crossref] [PubMed]

Olsson, N. A.

G. P. Agrawal and N. A. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25(11), 2297–2306 (1989).
[Crossref]

Pan, T.

Park, H.-C.

M. Lu, H.-C. Park, A. Sivananthan, J. S. Parker, E. Bloch, L. A. Johansson, M. J. W. Rodwell, and L. A. Coldren, “Monolithic Integration of a High-Speed Widely Tunable Optical Coherent Receiver,” IEEE Photon. Technol. Lett. 25(11), 1077–1080 (2013).
[Crossref]

Park, Y.

Parker, J. S.

M. Lu, H.-C. Park, A. Sivananthan, J. S. Parker, E. Bloch, L. A. Johansson, M. J. W. Rodwell, and L. A. Coldren, “Monolithic Integration of a High-Speed Widely Tunable Optical Coherent Receiver,” IEEE Photon. Technol. Lett. 25(11), 1077–1080 (2013).
[Crossref]

Plant, D. V.

Plastow, R.

J. Buus and R. Plastow, “A Theoretical and Experimental Investigation of Fabry-Perot Semiconductor Laser Amplifiers,” IEEE J. Quantum Electron. 21(6), 614–618 (1985).
[Crossref]

Qiang, L.

Qiu, C.

Qiu, M.

Razzari, L.

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).
[Crossref] [PubMed]

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), 29 (2010).
[Crossref] [PubMed]

Rodwell, M. J. W.

M. Lu, H.-C. Park, A. Sivananthan, J. S. Parker, E. Bloch, L. A. Johansson, M. J. W. Rodwell, and L. A. Coldren, “Monolithic Integration of a High-Speed Widely Tunable Optical Coherent Receiver,” IEEE Photon. Technol. Lett. 25(11), 1077–1080 (2013).
[Crossref]

Simon, J. C.

J. C. Simon, “GaInAsP semiconductor laser amplifiers for single-mode fiber communications,” J. Lightwave Technol. 5(9), 1286–1295 (1987).
[Crossref]

Sivananthan, A.

M. Lu, H.-C. Park, A. Sivananthan, J. S. Parker, E. Bloch, L. A. Johansson, M. J. W. Rodwell, and L. A. Coldren, “Monolithic Integration of a High-Speed Widely Tunable Optical Coherent Receiver,” IEEE Photon. Technol. Lett. 25(11), 1077–1080 (2013).
[Crossref]

Slavík, R.

Su, Y.

Tan, S.

Tremblay, C.

Venema, L.

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

Wang, F.

F. Wang, Y. Yu, X. Huang, and X. L. Zhang, “Single and Multiwavelength All-Optical Clock Recovery Using Fabry-Perot Semiconductor Optical Amplifier,” IEEE Photon. Technol. Lett. 21(16), 1109–1111 (2009).
[Crossref]

Wang, L.

Wang, T.

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

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).
[Crossref] [PubMed]

Wang, X.

Wu, J.

Wu, Z.

Xiang, L.

Xu, J.

Yang, T.

T. Yang, J. Dong, L. Lu, L. Zhou, A. Zheng, X. Zhang, and J. Chen, “All-optical differential equation solver with constant-coefficient tunable based on a single microring resonator,” Sci. Rep. 4, 5581 (2014).
[PubMed]

Yang, Y.

Yao, J.

Ye, T.

Yu, Y.

S. Tan, L. Xiang, J. Zou, Q. Zhang, Z. Wu, Y. Yu, J. Dong, and X. Zhang, “High-order all-optical differential equation solver based on microring resonators,” Opt. Lett. 38(19), 3735–3738 (2013).
[Crossref] [PubMed]

F. Wang, Y. Yu, X. Huang, and X. L. Zhang, “Single and Multiwavelength All-Optical Clock Recovery Using Fabry-Perot Semiconductor Optical Amplifier,” IEEE Photon. Technol. Lett. 21(16), 1109–1111 (2009).
[Crossref]

Zhang, Q.

Zhang, X.

Zhang, X. L.

F. Wang, Y. Yu, X. Huang, and X. L. Zhang, “Single and Multiwavelength All-Optical Clock Recovery Using Fabry-Perot Semiconductor Optical Amplifier,” IEEE Photon. Technol. Lett. 21(16), 1109–1111 (2009).
[Crossref]

Zhang, Z.

Zheng, A.

T. Yang, J. Dong, L. Lu, L. Zhou, A. Zheng, X. Zhang, and J. Chen, “All-optical differential equation solver with constant-coefficient tunable based on a single microring resonator,” Sci. Rep. 4, 5581 (2014).
[PubMed]

Zhou, L.

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Figures (9)

Fig. 1
Fig. 1 Schematic illustration of the 2nd-order ODE solver based on two cascaded FP-SOAs.
Fig. 2
Fig. 2 Measured input super-Gaussian waveform (green curve), the blue dashed line is the fitted waveform, which is the input waveform in the numerical analysis.
Fig. 3
Fig. 3 The calculated constant coefficients of the 1st-order ODE solvers based on FP-SOAs with different injection currents: (a) for FP-SOA1, (b) for FP-SOA2. (c) and (d) are the simulated output waveforms of the 1st-order ODE solvers based on FP-SOA1 and FP-SOA2 with different injection currents, respectively. (e) is the simulated output waveforms of the 2nd-order ODE solver based on cascaded FP-SOAs with different injection currents combinations.
Fig. 4
Fig. 4 Experimental setup for the tunable 1st-order ODE solver based on FP-SOA.
Fig. 5
Fig. 5 (a) is the measured ASE spectra of the FP-SOAs, the black curve is for FP-SOA1 and the blue curve is for FP SOA2. (b) is the enlarged ASE spectra near the wavelength of 1558.15 nm of FP-SOA1 (black curve) and FP-SOA2 (blue curve). (c) is the measured amplitude spectrum of FP-SOA1 (green curve), simulated amplitude spectrum (red dashed line) and phase spectrum (blue dashed line) of an ideal ODE solver. (d) is the measured amplitude spectrum of FP-SOA2 (yellow curve), simulated amplitude spectrum (red dashed line) and phase spectrum (blue dashed line) of an ideal ODE solver.
Fig. 6
Fig. 6 Measured output waveforms (green curves) and simulated results (blue dashed lines) of the 1st-order ODE solver based on FP-SOA1 with different injection currents: (a) 120 mA, (b) 140 mA, (c) 150 mA, (d) 155 mA, (e) 160 mA, (f) 165 mA.
Fig. 7
Fig. 7 Measured output waveforms (green curves) and simulated results (blue dashed lines) of the 1st-order ODE solver based on FP-SOA2 with different injection currents: (a) 20 mA, (b) 35mA, (c) 40 mA, (d) 42.5 mA, (e) 45 mA, (f) 48 mA.
Fig. 8
Fig. 8 Experimental setup for the tunable 2nd-order ODE solver based on cascaded FP-SOAs.
Fig. 9
Fig. 9 Measured output waveforms (green curves) and simulated results (blue dashed lines) of the 2nd-order ODE solver based on cascaded FP-SOAs with different injection currents combinations: (a) 120 & 20 mA, (b) 140 & 35mA, (c) 150 & 40 mA, (d) 155 & 42.5 mA, (e) 160 & 45 mA, (f) 165 & 48 mA.

Tables (1)

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Table 1 Parameters of the two FP-SOAs

Equations (9)

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dy( t ) dt +ky( t )=x( t )
H( ω )= 1 jω+k
T( ω )= ( 1 R 1 )( 1 R 2 ) G s e jωτ 1 R 1 R 2 G s e j2ωτ
T( ω )= ( 1 R 1 )( 1 R 2 ) G s e j( ω ω 0 )τ R 1 R 2 G s e j( ω ω 0 )τ ( 1 R 1 )( 1 R 2 ) G s 1 R 1 R 2 G s +j(1+ R 1 R 2 G s )(ω ω 0 )τ 1 j(ω ω 0 )+ 1 R 1 R 2 G s (1+ R 1 R 2 G s )τ
k= ( 1 R 1 R 2 G s ) / ( (1+ R 1 R 2 G s )τ )
d y 1 ( t ) dt + k 1 y 1 ( t )=x( t )
dy( t ) dt + k 2 y( t )= y 1 ( t )
d 2 y( t ) d t 2 +p dy( t ) dt +qy( t )=x( t )
N t = I eV N τ c ν g g N S

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