Abstract

In this paper, we propose and numerically investigate an all-optical temporal integrator based on a photonic crystal cavity. We show that an array of photonic crystal cavities enables high-order temporal integration. The effect of the value of the cavity’s free spectral range on the accuracy of the integration is considered. The influence of the coupling coefficients in the resonator array on the integration accuracy is demonstrated. A compact integrator based on a photonic crystal nanobeam cavity is designed, which allows high-precision integration of optical pulses of subpicosecond duration.

© 2014 Optical Society of America

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  1. H. J. Caulfield, S. Dolev, “Why future supercomputing requires optics,” Nat. Photonics 4(5), 261–263 (2010).
    [CrossRef]
  2. N. Quoc Ngo, “Design of an optical temporal integrator based on a phase-shifted fiber Bragg grating in transmission,” Opt. Lett. 32(20), 3020–3022 (2007).
    [CrossRef] [PubMed]
  3. M. Ferrera, Y. Park, L. Razzari, B. E. Little, S. T. Chu, R. Morandotti, D. J. Moss, J. Azaña, “On-chip CMOS-compatible all-optical integrator,” Nat. Commun. 1(3), 29 (2010).
    [CrossRef] [PubMed]
  4. A. Malacarne, R. Ashrafi, M. Li, S. LaRochelle, J. Yao, J. Azaña, “Single-shot photonic time-intensity integration based on a time-spectrum convolution system,” Opt. Lett. 37(8), 1355–1357 (2012).
    [CrossRef] [PubMed]
  5. Y. Park, T. J. Ahn, Y. Dai, J. Yao, J. Azaña, “All-optical temporal integration of ultrafast pulse waveforms,” Opt. Express 16(22), 17817–17825 (2008).
    [CrossRef] [PubMed]
  6. N. Q. Ngo, “Optical integrator for optical dark-soliton detection and pulse shaping,” Appl. Opt. 45(26), 6785–6791 (2006).
    [CrossRef] [PubMed]
  7. Y. Jin, P. Costanzo-Caso, S. Granieri, A. Siahmakoun, “Photonic integrator for A/D conversion,” Proc. SPIE 7797, 77970J (2010).
    [CrossRef]
  8. Y. Ding, X. Zhang, X. Zhang, 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]
  9. R. Slavík, Y. Park, N. Ayotte, S. Doucet, T. J. Ahn, S. LaRochelle, J. Azaña, “Photonic temporal integrator for all-optical computing,” Opt. Express 16(22), 18202–18214 (2008).
    [CrossRef] [PubMed]
  10. Y. Akahane, T. Asano, B.-S. Song, S. Noda, “Fine-tuned high-Q photonic-crystal nanocavity,” Opt. Express 13(4), 1202–1214 (2005).
    [CrossRef] [PubMed]
  11. P. Velha, J. C. Rodier, P. Lalanne, J. P. Hugonin, D. Peyrade, E. Picard, T. Charvolin, E. Hadji, “Ultra-high-reflectivity photonic-bandgap mirrors in a ridge SOI waveguide,” New J. Phys. 8(9), 204 (2006).
    [CrossRef]
  12. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).
  13. H. C. Liu, A. Yariv, “Synthesis of high-order bandpass filters based on coupled-resonator optical waveguides (CROWs),” Opt. Express 19(18), 17653–17668 (2011).
    [CrossRef] [PubMed]
  14. M. H. Asghari, J. Azaña, “On the design of efficient and accurate arbitrary-order temporal optical integrators using fiber bragg gratings,” J. Lightwave Technol. J. 27(17), 3888–3895 (2009).
    [CrossRef]
  15. N. L. Kazanskiy, P. G. Serafimovich, S. N. Khonina, “Use of photonic crystal cavities for temporal differentiation of optical signals,” Opt. Lett. 38(7), 1149–1151 (2013).
    [CrossRef] [PubMed]
  16. H. C. Liu, A. Yariv, “Designing coupled-resonator optical waveguides based on high-Q tapered grating-defect resonators,” Opt. Express 20(8), 9249–9263 (2012).
    [CrossRef] [PubMed]
  17. Q. Quan, M. Loncar, “Deterministic design of wavelength scale, ultra-high Q photonic crystal nanobeam cavities,” Opt. Express 19(5), 18529–18542 (2011).
    [CrossRef] [PubMed]
  18. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).
  19. D. L. Golovashkin, N. L. Kazanskiy, “Mesh domain decomposition in the finite-difference solution of Maxwell’s equations,” Opt. Mem. Neural Networks 18(3), 203–211 (2009).
    [CrossRef]
  20. M. H. Asghari, C. Wang, J. Yao, J. Azaña, “High-order passive photonic temporal integrators,” Opt. Lett. 35(8), 1191–1193 (2010).
    [CrossRef] [PubMed]
  21. N. Huang, M. Li, R. Ashrafi, L. Wang, X. Wang, J. Azaña, 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]
  22. G. Shambat, B. Ellis, J. Petykiewicz, M. Mayer, T. Sarmiento, J. Harris, E. E. Haller, J. Vuckovic, “Nanobeam photonic crystal cavity light-emitting diodes,” Appl. Phys. Lett. 99(7), 071105 (2011).
    [CrossRef]
  23. M. H. Asghari, J. Azaña, “Design of all-optical high-order temporal integrators based on multiple-phase-shifted Bragg gratings,” Opt. Express 16(15), 11459–11469 (2008).
    [CrossRef] [PubMed]

2014 (1)

2013 (1)

2012 (2)

2011 (3)

2010 (4)

M. H. Asghari, C. Wang, J. Yao, J. Azaña, “High-order passive photonic temporal integrators,” Opt. Lett. 35(8), 1191–1193 (2010).
[CrossRef] [PubMed]

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

H. J. Caulfield, S. Dolev, “Why future supercomputing requires optics,” Nat. Photonics 4(5), 261–263 (2010).
[CrossRef]

Y. Jin, P. Costanzo-Caso, S. Granieri, A. Siahmakoun, “Photonic integrator for A/D conversion,” Proc. SPIE 7797, 77970J (2010).
[CrossRef]

2009 (3)

Y. Ding, X. Zhang, X. Zhang, 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]

M. H. Asghari, J. Azaña, “On the design of efficient and accurate arbitrary-order temporal optical integrators using fiber bragg gratings,” J. Lightwave Technol. J. 27(17), 3888–3895 (2009).
[CrossRef]

D. L. Golovashkin, N. L. Kazanskiy, “Mesh domain decomposition in the finite-difference solution of Maxwell’s equations,” Opt. Mem. Neural Networks 18(3), 203–211 (2009).
[CrossRef]

2008 (3)

2007 (1)

2006 (2)

N. Q. Ngo, “Optical integrator for optical dark-soliton detection and pulse shaping,” Appl. Opt. 45(26), 6785–6791 (2006).
[CrossRef] [PubMed]

P. Velha, J. C. Rodier, P. Lalanne, J. P. Hugonin, D. Peyrade, E. Picard, T. Charvolin, E. Hadji, “Ultra-high-reflectivity photonic-bandgap mirrors in a ridge SOI waveguide,” New J. Phys. 8(9), 204 (2006).
[CrossRef]

2005 (1)

Ahn, T. J.

Akahane, Y.

Asano, T.

Asghari, M. H.

Ashrafi, R.

Ayotte, N.

Azaña, J.

N. Huang, M. Li, R. Ashrafi, L. Wang, X. Wang, J. Azaña, 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]

A. Malacarne, R. Ashrafi, M. Li, S. LaRochelle, J. Yao, J. Azaña, “Single-shot photonic time-intensity integration based on a time-spectrum convolution system,” Opt. Lett. 37(8), 1355–1357 (2012).
[CrossRef] [PubMed]

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

M. H. Asghari, C. Wang, J. Yao, J. Azaña, “High-order passive photonic temporal integrators,” Opt. Lett. 35(8), 1191–1193 (2010).
[CrossRef] [PubMed]

M. H. Asghari, J. Azaña, “On the design of efficient and accurate arbitrary-order temporal optical integrators using fiber bragg gratings,” J. Lightwave Technol. J. 27(17), 3888–3895 (2009).
[CrossRef]

R. Slavík, Y. Park, N. Ayotte, S. Doucet, T. J. Ahn, S. LaRochelle, J. Azaña, “Photonic temporal integrator for all-optical computing,” Opt. Express 16(22), 18202–18214 (2008).
[CrossRef] [PubMed]

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

M. H. Asghari, J. Azaña, “Design of all-optical high-order temporal integrators based on multiple-phase-shifted Bragg gratings,” Opt. Express 16(15), 11459–11469 (2008).
[CrossRef] [PubMed]

Caulfield, H. J.

H. J. Caulfield, S. Dolev, “Why future supercomputing requires optics,” Nat. Photonics 4(5), 261–263 (2010).
[CrossRef]

Charvolin, T.

P. Velha, J. C. Rodier, P. Lalanne, J. P. Hugonin, D. Peyrade, E. Picard, T. Charvolin, E. Hadji, “Ultra-high-reflectivity photonic-bandgap mirrors in a ridge SOI waveguide,” New J. Phys. 8(9), 204 (2006).
[CrossRef]

Chu, S. T.

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

Costanzo-Caso, P.

Y. Jin, P. Costanzo-Caso, S. Granieri, A. Siahmakoun, “Photonic integrator for A/D conversion,” Proc. SPIE 7797, 77970J (2010).
[CrossRef]

Dai, Y.

Ding, Y.

Dolev, S.

H. J. Caulfield, S. Dolev, “Why future supercomputing requires optics,” Nat. Photonics 4(5), 261–263 (2010).
[CrossRef]

Doucet, S.

Ellis, B.

G. Shambat, B. Ellis, J. Petykiewicz, M. Mayer, T. Sarmiento, J. Harris, E. E. Haller, J. Vuckovic, “Nanobeam photonic crystal cavity light-emitting diodes,” Appl. Phys. Lett. 99(7), 071105 (2011).
[CrossRef]

Ferrera, M.

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

Golovashkin, D. L.

D. L. Golovashkin, N. L. Kazanskiy, “Mesh domain decomposition in the finite-difference solution of Maxwell’s equations,” Opt. Mem. Neural Networks 18(3), 203–211 (2009).
[CrossRef]

Granieri, S.

Y. Jin, P. Costanzo-Caso, S. Granieri, A. Siahmakoun, “Photonic integrator for A/D conversion,” Proc. SPIE 7797, 77970J (2010).
[CrossRef]

Hadji, E.

P. Velha, J. C. Rodier, P. Lalanne, J. P. Hugonin, D. Peyrade, E. Picard, T. Charvolin, E. Hadji, “Ultra-high-reflectivity photonic-bandgap mirrors in a ridge SOI waveguide,” New J. Phys. 8(9), 204 (2006).
[CrossRef]

Haller, E. E.

G. Shambat, B. Ellis, J. Petykiewicz, M. Mayer, T. Sarmiento, J. Harris, E. E. Haller, J. Vuckovic, “Nanobeam photonic crystal cavity light-emitting diodes,” Appl. Phys. Lett. 99(7), 071105 (2011).
[CrossRef]

Harris, J.

G. Shambat, B. Ellis, J. Petykiewicz, M. Mayer, T. Sarmiento, J. Harris, E. E. Haller, J. Vuckovic, “Nanobeam photonic crystal cavity light-emitting diodes,” Appl. Phys. Lett. 99(7), 071105 (2011).
[CrossRef]

Huang, D.

Huang, N.

Hugonin, J. P.

P. Velha, J. C. Rodier, P. Lalanne, J. P. Hugonin, D. Peyrade, E. Picard, T. Charvolin, E. Hadji, “Ultra-high-reflectivity photonic-bandgap mirrors in a ridge SOI waveguide,” New J. Phys. 8(9), 204 (2006).
[CrossRef]

Jin, Y.

Y. Jin, P. Costanzo-Caso, S. Granieri, A. Siahmakoun, “Photonic integrator for A/D conversion,” Proc. SPIE 7797, 77970J (2010).
[CrossRef]

Kazanskiy, N. L.

N. L. Kazanskiy, P. G. Serafimovich, S. N. Khonina, “Use of photonic crystal cavities for temporal differentiation of optical signals,” Opt. Lett. 38(7), 1149–1151 (2013).
[CrossRef] [PubMed]

D. L. Golovashkin, N. L. Kazanskiy, “Mesh domain decomposition in the finite-difference solution of Maxwell’s equations,” Opt. Mem. Neural Networks 18(3), 203–211 (2009).
[CrossRef]

Khonina, S. N.

Lalanne, P.

P. Velha, J. C. Rodier, P. Lalanne, J. P. Hugonin, D. Peyrade, E. Picard, T. Charvolin, E. Hadji, “Ultra-high-reflectivity photonic-bandgap mirrors in a ridge SOI waveguide,” New J. Phys. 8(9), 204 (2006).
[CrossRef]

LaRochelle, S.

Li, M.

Little, B. E.

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

Liu, H. C.

Loncar, M.

Malacarne, A.

Mayer, M.

G. Shambat, B. Ellis, J. Petykiewicz, M. Mayer, T. Sarmiento, J. Harris, E. E. Haller, J. Vuckovic, “Nanobeam photonic crystal cavity light-emitting diodes,” Appl. Phys. Lett. 99(7), 071105 (2011).
[CrossRef]

Morandotti, R.

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

Moss, D. J.

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

Ngo, N. Q.

Noda, S.

Park, Y.

Petykiewicz, J.

G. Shambat, B. Ellis, J. Petykiewicz, M. Mayer, T. Sarmiento, J. Harris, E. E. Haller, J. Vuckovic, “Nanobeam photonic crystal cavity light-emitting diodes,” Appl. Phys. Lett. 99(7), 071105 (2011).
[CrossRef]

Peyrade, D.

P. Velha, J. C. Rodier, P. Lalanne, J. P. Hugonin, D. Peyrade, E. Picard, T. Charvolin, E. Hadji, “Ultra-high-reflectivity photonic-bandgap mirrors in a ridge SOI waveguide,” New J. Phys. 8(9), 204 (2006).
[CrossRef]

Picard, E.

P. Velha, J. C. Rodier, P. Lalanne, J. P. Hugonin, D. Peyrade, E. Picard, T. Charvolin, E. Hadji, “Ultra-high-reflectivity photonic-bandgap mirrors in a ridge SOI waveguide,” New J. Phys. 8(9), 204 (2006).
[CrossRef]

Quan, Q.

Quoc Ngo, N.

Razzari, L.

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

Rodier, J. C.

P. Velha, J. C. Rodier, P. Lalanne, J. P. Hugonin, D. Peyrade, E. Picard, T. Charvolin, E. Hadji, “Ultra-high-reflectivity photonic-bandgap mirrors in a ridge SOI waveguide,” New J. Phys. 8(9), 204 (2006).
[CrossRef]

Sarmiento, T.

G. Shambat, B. Ellis, J. Petykiewicz, M. Mayer, T. Sarmiento, J. Harris, E. E. Haller, J. Vuckovic, “Nanobeam photonic crystal cavity light-emitting diodes,” Appl. Phys. Lett. 99(7), 071105 (2011).
[CrossRef]

Serafimovich, P. G.

Shambat, G.

G. Shambat, B. Ellis, J. Petykiewicz, M. Mayer, T. Sarmiento, J. Harris, E. E. Haller, J. Vuckovic, “Nanobeam photonic crystal cavity light-emitting diodes,” Appl. Phys. Lett. 99(7), 071105 (2011).
[CrossRef]

Siahmakoun, A.

Y. Jin, P. Costanzo-Caso, S. Granieri, A. Siahmakoun, “Photonic integrator for A/D conversion,” Proc. SPIE 7797, 77970J (2010).
[CrossRef]

Slavík, R.

Song, B.-S.

Velha, P.

P. Velha, J. C. Rodier, P. Lalanne, J. P. Hugonin, D. Peyrade, E. Picard, T. Charvolin, E. Hadji, “Ultra-high-reflectivity photonic-bandgap mirrors in a ridge SOI waveguide,” New J. Phys. 8(9), 204 (2006).
[CrossRef]

Vuckovic, J.

G. Shambat, B. Ellis, J. Petykiewicz, M. Mayer, T. Sarmiento, J. Harris, E. E. Haller, J. Vuckovic, “Nanobeam photonic crystal cavity light-emitting diodes,” Appl. Phys. Lett. 99(7), 071105 (2011).
[CrossRef]

Wang, C.

Wang, L.

Wang, X.

Yao, J.

Yariv, A.

Zhang, X.

Zhu, N.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

G. Shambat, B. Ellis, J. Petykiewicz, M. Mayer, T. Sarmiento, J. Harris, E. E. Haller, J. Vuckovic, “Nanobeam photonic crystal cavity light-emitting diodes,” Appl. Phys. Lett. 99(7), 071105 (2011).
[CrossRef]

J. Lightwave Technol. J. (1)

M. H. Asghari, J. Azaña, “On the design of efficient and accurate arbitrary-order temporal optical integrators using fiber bragg gratings,” J. Lightwave Technol. J. 27(17), 3888–3895 (2009).
[CrossRef]

Nat. Commun. (1)

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

Nat. Photonics (1)

H. J. Caulfield, S. Dolev, “Why future supercomputing requires optics,” Nat. Photonics 4(5), 261–263 (2010).
[CrossRef]

New J. Phys. (1)

P. Velha, J. C. Rodier, P. Lalanne, J. P. Hugonin, D. Peyrade, E. Picard, T. Charvolin, E. Hadji, “Ultra-high-reflectivity photonic-bandgap mirrors in a ridge SOI waveguide,” New J. Phys. 8(9), 204 (2006).
[CrossRef]

Opt. Express (9)

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

H. C. Liu, A. Yariv, “Synthesis of high-order bandpass filters based on coupled-resonator optical waveguides (CROWs),” Opt. Express 19(18), 17653–17668 (2011).
[CrossRef] [PubMed]

H. C. Liu, A. Yariv, “Designing coupled-resonator optical waveguides based on high-Q tapered grating-defect resonators,” Opt. Express 20(8), 9249–9263 (2012).
[CrossRef] [PubMed]

Q. Quan, M. Loncar, “Deterministic design of wavelength scale, ultra-high Q photonic crystal nanobeam cavities,” Opt. Express 19(5), 18529–18542 (2011).
[CrossRef] [PubMed]

Y. Ding, X. Zhang, X. Zhang, 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]

R. Slavík, Y. Park, N. Ayotte, S. Doucet, T. J. Ahn, S. LaRochelle, J. Azaña, “Photonic temporal integrator for all-optical computing,” Opt. Express 16(22), 18202–18214 (2008).
[CrossRef] [PubMed]

Y. Akahane, T. Asano, B.-S. Song, S. Noda, “Fine-tuned high-Q photonic-crystal nanocavity,” Opt. Express 13(4), 1202–1214 (2005).
[CrossRef] [PubMed]

M. H. Asghari, J. Azaña, “Design of all-optical high-order temporal integrators based on multiple-phase-shifted Bragg gratings,” Opt. Express 16(15), 11459–11469 (2008).
[CrossRef] [PubMed]

N. Huang, M. Li, R. Ashrafi, L. Wang, X. Wang, J. Azaña, 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]

Opt. Lett. (4)

Opt. Mem. Neural Networks (1)

D. L. Golovashkin, N. L. Kazanskiy, “Mesh domain decomposition in the finite-difference solution of Maxwell’s equations,” Opt. Mem. Neural Networks 18(3), 203–211 (2009).
[CrossRef]

Proc. SPIE (1)

Y. Jin, P. Costanzo-Caso, S. Granieri, A. Siahmakoun, “Photonic integrator for A/D conversion,” Proc. SPIE 7797, 77970J (2010).
[CrossRef]

Other (2)

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

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Figures (10)

Fig. 1
Fig. 1

Scheme of coupled-resonator optical waveguide.

Fig. 2
Fig. 2

Result of first-order integration of optical pulse with duration of 1 ps by resonators with Q-factors of 3 × 104 and 5 × 104.

Fig. 3
Fig. 3

Result of integration by resonators with Q-factors of 104,105 and 106. Input optical signal is first derivative of Gaussian pulse with duration of 100 ps.

Fig. 4
Fig. 4

Gaussian pulse with duration of 1 ps and its first three derivatives.

Fig. 5
Fig. 5

RMSE of integration of the first derivative of a Gaussian pulses with a durations of 1 ps and 150 fs for different FSR values. The Q-factor of the resonator is 5 × 104.

Fig. 6
Fig. 6

Schemes of (a) PC nanobeam cavity and (b) array of two such cavities.

Fig. 7
Fig. 7

(a) The amplitude and (b) phase of the frequency response of the integrator.

Fig. 8
Fig. 8

Results of integration of corresponding derivatives of Gaussian pulse with duration of 150 fs by (a) one PC resonator, (b) an array of two PC resonators, and (c) an array of three PC resonators and (d) the result of integration of the third derivative of a Gaussian pulse with a duration of 200 fs by an array of three PC resonators. Q-factor of resonators is 3.6 × 104.

Fig. 9
Fig. 9

TF of three-resonator array for two variants of the coupling coefficients: (a) amplitude of the TF, (b) phase of the TF.

Fig. 10
Fig. 10

Results of third-order integration for two values of the coupling coefficients.

Tables (1)

Tables Icon

Table 1 Geometric Parameters of PC Resonator Shown in Fig. 6(a)

Equations (18)

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

Ma= p in ,
where M=( s 1 + κ 0 i κ 1 0 0 0 i κ 1 s 2 i κ 2 0 0 0 i κ 2 s 3 0 0 i κ N2 0 0 0 0 s N1 i κ N1 0 0 0 i κ N1 s N + κ N ).
T N ( s ) p tr p in = 2 ( i ) N1 κ 0 κ N κ 1 κ 2 κ N1 det(M) ,
T 1 ( s )= 2 κ 0 s+2 κ 0
E( x,t )= P in ( tx/ v g )exp( i m 0 xi ω 0 t )= = R( ω ω 0 )exp( im( ω )xiωt )dω , ,
P tr ( t )= R( ω )H( ω )exp( iωt )dω = = P in ( t )h( t ), ,
h 1 ( t )= κ 0 exp( κ 0 t )u(t),
P tr ( t )= κ 0 t P in ( T )exp( i κ 0 ( tT ) )dT .
T 2 ( s )= i2 κ 0 κ 1 ( s+ κ 0 ) 2 + κ 1 2 .
T 3 ( s )= 2 κ 0 κ 1 κ 2 s ( s+ κ 0 ) 2 +( s+ κ 0 )( κ 1 2 + κ 2 2 ) .
h 2 ( t )= κ 0 ( κ 0 +i κ 1 )t ( 2i κ 1 t 1 )u(t).
h 2 ( t )2i κ 0 κ 1 texp( κ 0 t )u(t).
P tr ( t )~i κ 0 κ 1 T 2 = t T 1 = T 2 P in ( T 1 ) i κ 0 ( t T 1 ) d T 1 d T 2 .
H int ( s )=1/s .
H df ( s )= s s+ κ 0 .
T 1 2 ( s )= 2 κ 0 s+2 κ 0 2 κ 0 s+2 κ 0 .
κ= ω 0 4 Q 1 Q 2 = ω 0 4 Q 0 a n reg ,
κ 0 = κ N = κ j ,j=1,N1.

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