Abstract

We propose and experimentally demonstrate a feasible integrated scheme to solve all-optical differential equations using microring resonators (MRRs) that is capable of solving first- and second-order linear ordinary differential equations with different constant coefficients. Employing two cascaded MRRs with different radii, an excellent agreement between the numerical simulation and the experimental results is obtained. Due to the inherent merits of silicon-based devices for all-optical computing, such as low power consumption, small size, and high speed, this finding may motivate the development of integrated optical signal processors and further extend optical computing technologies.

© 2013 Optical Society of America

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2013 (1)

2012 (2)

L. Lei, J. Dong, Y. Yu, S. Tan, and X. Zhang, J. Lightwave Technol. 30, 3532 (2012).
[CrossRef]

L. Lu, J. Wu, T. Wang, and Y. Su, Front. Optoelectron. 5, 99 (2012).
[CrossRef]

2010 (4)

Y. Park, M. H. Asghari, R. Helsten, and J. Azana, IEEE Photon. J. 2, 1040 (2010).
[CrossRef]

H. J. Caulfield and S. Dolev, Nat. Photonics 4, 261 (2010).
[CrossRef]

M. Ferrera, Y. Park, L. Razzari, B. Little, S. Chu, R. Morandotti, D. Moss, and J. Azaña, Nat. Commun. 1, 29 (2010).
[CrossRef]

M. Asghari, C. Wang, J. Yao, and J. Azaña, Opt. Lett. 35, 1191 (2010).
[CrossRef]

2008 (5)

2007 (1)

2005 (1)

1991 (1)

D. N. McCloskey, Hist. Theory 30, 21 (1991).
[CrossRef]

1987 (1)

S. Smith, A. Walker, F. Tooley, and B. Wherrett, Nature 325, 27 (1987).
[CrossRef]

Ahn, T.-J.

Asghari, M.

Asghari, M. H.

Y. Park, M. H. Asghari, R. Helsten, and J. Azana, IEEE Photon. J. 2, 1040 (2010).
[CrossRef]

M. H. Asghari and J. Azaña, Opt. Lett. 33, 1548 (2008).
[CrossRef]

Ayotte, N.

Azana, J.

Y. Park, M. H. Asghari, R. Helsten, and J. Azana, IEEE Photon. J. 2, 1040 (2010).
[CrossRef]

Azaña, J.

Caulfield, H. J.

H. J. Caulfield and S. Dolev, Nat. Photonics 4, 261 (2010).
[CrossRef]

Chen, W.

B. Little, S. Chu, W. Chen, J. Hryniewicz, D. Gill, O. King, F. Johnson, R. Davidson, K. Donovan, and W. Chen, ECOC 2008: 34th European Conference on Optical Communication (IEEE, 2008), pp. 1–2.

B. Little, S. Chu, W. Chen, J. Hryniewicz, D. Gill, O. King, F. Johnson, R. Davidson, K. Donovan, and W. Chen, ECOC 2008: 34th European Conference on Optical Communication (IEEE, 2008), pp. 1–2.

Chu, S.

M. Ferrera, Y. Park, L. Razzari, B. Little, S. Chu, R. Morandotti, D. Moss, and J. Azaña, Nat. Commun. 1, 29 (2010).
[CrossRef]

M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. Sipe, S. Chu, B. Little, and D. Moss, Nat. Photonics 2, 737 (2008).
[CrossRef]

B. Little, S. Chu, W. Chen, J. Hryniewicz, D. Gill, O. King, F. Johnson, R. Davidson, K. Donovan, and W. Chen, ECOC 2008: 34th European Conference on Optical Communication (IEEE, 2008), pp. 1–2.

Dainese, M.

Davidson, R.

B. Little, S. Chu, W. Chen, J. Hryniewicz, D. Gill, O. King, F. Johnson, R. Davidson, K. Donovan, and W. Chen, ECOC 2008: 34th European Conference on Optical Communication (IEEE, 2008), pp. 1–2.

Dolev, S.

H. J. Caulfield and S. Dolev, Nat. Photonics 4, 261 (2010).
[CrossRef]

Dong, J.

Donovan, K.

B. Little, S. Chu, W. Chen, J. Hryniewicz, D. Gill, O. King, F. Johnson, R. Davidson, K. Donovan, and W. Chen, ECOC 2008: 34th European Conference on Optical Communication (IEEE, 2008), pp. 1–2.

Doucet, S.

Duchesne, D.

M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. Sipe, S. Chu, B. Little, and D. Moss, Nat. Photonics 2, 737 (2008).
[CrossRef]

Ferrera, M.

M. Ferrera, Y. Park, L. Razzari, B. Little, S. Chu, R. Morandotti, D. Moss, and J. Azaña, Nat. Commun. 1, 29 (2010).
[CrossRef]

M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. Sipe, S. Chu, B. Little, and D. Moss, Nat. Photonics 2, 737 (2008).
[CrossRef]

Gill, D.

B. Little, S. Chu, W. Chen, J. Hryniewicz, D. Gill, O. King, F. Johnson, R. Davidson, K. Donovan, and W. Chen, ECOC 2008: 34th European Conference on Optical Communication (IEEE, 2008), pp. 1–2.

Helsten, R.

Y. Park, M. H. Asghari, R. Helsten, and J. Azana, IEEE Photon. J. 2, 1040 (2010).
[CrossRef]

Hryniewicz, J.

B. Little, S. Chu, W. Chen, J. Hryniewicz, D. Gill, O. King, F. Johnson, R. Davidson, K. Donovan, and W. Chen, ECOC 2008: 34th European Conference on Optical Communication (IEEE, 2008), pp. 1–2.

Hu, S.

Huang, D.

Johnson, F.

B. Little, S. Chu, W. Chen, J. Hryniewicz, D. Gill, O. King, F. Johnson, R. Davidson, K. Donovan, and W. Chen, ECOC 2008: 34th European Conference on Optical Communication (IEEE, 2008), pp. 1–2.

King, O.

B. Little, S. Chu, W. Chen, J. Hryniewicz, D. Gill, O. King, F. Johnson, R. Davidson, K. Donovan, and W. Chen, ECOC 2008: 34th European Conference on Optical Communication (IEEE, 2008), pp. 1–2.

Kulishov, M.

LaRochelle, S.

Lei, L.

Liscidini, M.

M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. Sipe, S. Chu, B. Little, and D. Moss, Nat. Photonics 2, 737 (2008).
[CrossRef]

Little, B.

M. Ferrera, Y. Park, L. Razzari, B. Little, S. Chu, R. Morandotti, D. Moss, and J. Azaña, Nat. Commun. 1, 29 (2010).
[CrossRef]

M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. Sipe, S. Chu, B. Little, and D. Moss, Nat. Photonics 2, 737 (2008).
[CrossRef]

B. Little, S. Chu, W. Chen, J. Hryniewicz, D. Gill, O. King, F. Johnson, R. Davidson, K. Donovan, and W. Chen, ECOC 2008: 34th European Conference on Optical Communication (IEEE, 2008), pp. 1–2.

Liu, D.

Liu, F.

Lu, L.

L. Lu, J. Wu, T. Wang, and Y. Su, Front. Optoelectron. 5, 99 (2012).
[CrossRef]

McCloskey, D. N.

D. N. McCloskey, Hist. Theory 30, 21 (1991).
[CrossRef]

Morandotti, R.

M. Ferrera, Y. Park, L. Razzari, B. Little, S. Chu, R. Morandotti, D. Moss, and J. Azaña, Nat. Commun. 1, 29 (2010).
[CrossRef]

M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. Sipe, S. Chu, B. Little, and D. Moss, Nat. Photonics 2, 737 (2008).
[CrossRef]

Moss, D.

M. Ferrera, Y. Park, L. Razzari, B. Little, S. Chu, R. Morandotti, D. Moss, and J. Azaña, Nat. Commun. 1, 29 (2010).
[CrossRef]

M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. Sipe, S. Chu, B. Little, and D. Moss, Nat. Photonics 2, 737 (2008).
[CrossRef]

Oppenheim, A.

A. Oppenheim and A. Willsky, Signals and Systems, 2nd ed. (Prentice-Hall, 1997), p. 997.

Park, Y.

Y. Park, M. H. Asghari, R. Helsten, and J. Azana, IEEE Photon. J. 2, 1040 (2010).
[CrossRef]

M. Ferrera, Y. Park, L. Razzari, B. Little, S. Chu, R. Morandotti, D. Moss, and J. Azaña, Nat. Commun. 1, 29 (2010).
[CrossRef]

R. Slavík, Y. Park, N. Ayotte, S. Doucet, T.-J. Ahn, S. LaRochelle, and J. Azaña, Opt. Express 16, 18202 (2008).
[CrossRef]

Qiang, L.

Qiu, M.

Rabus, D. G.

D. G. Rabus, Integrated Ring Resonators: The Compendium, Vol. 127 of Springer Series in Optical Sciences (Springer, 2007).

Razzari, L.

M. Ferrera, Y. Park, L. Razzari, B. Little, S. Chu, R. Morandotti, D. Moss, and J. Azaña, Nat. Commun. 1, 29 (2010).
[CrossRef]

M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. Sipe, S. Chu, B. Little, and D. Moss, Nat. Photonics 2, 737 (2008).
[CrossRef]

Sipe, J.

M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. Sipe, S. Chu, B. Little, and D. Moss, Nat. Photonics 2, 737 (2008).
[CrossRef]

Slavík, R.

Smith, S.

S. Smith, A. Walker, F. Tooley, and B. Wherrett, Nature 325, 27 (1987).
[CrossRef]

Su, Y.

Tan, S.

Tooley, F.

S. Smith, A. Walker, F. Tooley, and B. Wherrett, Nature 325, 27 (1987).
[CrossRef]

Walker, A.

S. Smith, A. Walker, F. Tooley, and B. Wherrett, Nature 325, 27 (1987).
[CrossRef]

Wang, C.

Wang, T.

Wherrett, B.

S. Smith, A. Walker, F. Tooley, and B. Wherrett, Nature 325, 27 (1987).
[CrossRef]

Willsky, A.

A. Oppenheim and A. Willsky, Signals and Systems, 2nd ed. (Prentice-Hall, 1997), p. 997.

Wosinski, L.

Wu, J.

L. Lu, J. Wu, T. Wang, and Y. Su, Front. Optoelectron. 5, 99 (2012).
[CrossRef]

Wu, Z.

Xu, J.

Yang, Z.

M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. Sipe, S. Chu, B. Little, and D. Moss, Nat. Photonics 2, 737 (2008).
[CrossRef]

Yao, J.

Ye, T.

Yu, Y.

Zhang, X.

Zhang, Z.

Front. Optoelectron. (1)

L. Lu, J. Wu, T. Wang, and Y. Su, Front. Optoelectron. 5, 99 (2012).
[CrossRef]

Hist. Theory (1)

D. N. McCloskey, Hist. Theory 30, 21 (1991).
[CrossRef]

IEEE Photon. J. (1)

Y. Park, M. H. Asghari, R. Helsten, and J. Azana, IEEE Photon. J. 2, 1040 (2010).
[CrossRef]

J. Lightwave Technol. (1)

Nat. Commun. (1)

M. Ferrera, Y. Park, L. Razzari, B. Little, S. Chu, R. Morandotti, D. Moss, and J. Azaña, Nat. Commun. 1, 29 (2010).
[CrossRef]

Nat. Photonics (2)

H. J. Caulfield and S. Dolev, Nat. Photonics 4, 261 (2010).
[CrossRef]

M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. Sipe, S. Chu, B. Little, and D. Moss, Nat. Photonics 2, 737 (2008).
[CrossRef]

Nature (1)

S. Smith, A. Walker, F. Tooley, and B. Wherrett, Nature 325, 27 (1987).
[CrossRef]

Opt. Express (4)

Opt. Lett. (4)

Other (3)

B. Little, S. Chu, W. Chen, J. Hryniewicz, D. Gill, O. King, F. Johnson, R. Davidson, K. Donovan, and W. Chen, ECOC 2008: 34th European Conference on Optical Communication (IEEE, 2008), pp. 1–2.

A. Oppenheim and A. Willsky, Signals and Systems, 2nd ed. (Prentice-Hall, 1997), p. 997.

D. G. Rabus, Integrated Ring Resonators: The Compendium, Vol. 127 of Springer Series in Optical Sciences (Springer, 2007).

Supplementary Material (1)

» Media 1: MOV (1443 KB)     

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Figures (5)

Fig. 1.
Fig. 1.

(a) First-order linear ODE solver based on an add–drop MRR (Media 1); (b) first-order {blue [top left x(t) and middle y1(t)] and red [middle x(t) and bottom y2(t) arrows]} and second-order {black [top x(t) and bottom y(t)] arrows} linear ODE solver based on two cascaded MRRs; and (c) first-order {blue [left top x(t), middle y1(t)], red [middle right x(t) and bottom right y2(t)], and magenta [bottom left x(t)] arrows} and third-order {black [bottom y(t)] arrows} linear ODE solver based on three cascaded MRRs.

Fig. 2.
Fig. 2.

(a) Schematic of the cascaded MRRs composed of two add–drop MRRs with different radii and Q factors (vertical grating couplers not shown); (b) SEM picture of the cascaded MRRs, and the scale bar represents 200 μm; (c) SEM picture of the coupling region between the ring and the straight waveguide, and the scale bar represents 500 nm; and (d) measured spectra of each MRR and the cascaded ones, employing ports 1 and 2 for MRR1 (green/top solid line), ports 5 and 6 for MRR2 (blue/middle solid line) and ports 3 and 4 for the cascaded MRRs (black/bottom solid line)

Fig. 3.
Fig. 3.

Experimental setup for the high-order all-optical ODE solver with the on-chip MRRs.

Fig. 4.
Fig. 4.

Theoretical solutions of Eq. (1) with different constant coefficients for two kinds of input pulses: (a) Gaussian input; (b) super-Gaussian input; and (c) experimental input signal.

Fig. 5.
Fig. 5.

Measured amplitude spectra, simulated phase spectra, measured output waveforms (turquoise/solid line), and corresponding simulated ideal results (red/dotted line) of the MRR-based all-optical ODE solver: (a) amplitude spectrum, (b) phase spectrum, and (c) temporal waveform for MRR1; (d) amplitude spectrum, (e) phase spectrum, and (f) temporal waveform for MRR2; (g) amplitude spectrum, (h) phase spectrum, and (j) temporal waveform for the cascaded MRRs.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

dy(t)dt+ky(t)=x(t),
H(ω)=1jω+k,
T(ω)=4τcoτtr/[j(ωω0)+1τ],
dy1(t)dt+k1y1(t)=x(t),
dy2(t)dt+k2y2(t)=y1(t).
d2y(t)dt2+a·dy(t)dt+by(t)=x(t),
d3y(t)dt3+a·d2y(t)dt2+b·dy(t)dt+cy(t)=x(t),

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