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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    [PubMed]
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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, 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]

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]

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)

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]

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]

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]

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]

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

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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