## Coupled-resonator optical waveguides for temporal integration of optical signals |

Optics Express, Vol. 22, Issue 11, pp. 14004-14013 (2014)

http://dx.doi.org/10.1364/OE.22.014004

Acrobat PDF (2328 KB)

### 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

## 1. Introduction

1. H. J. Caulfield and 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, and 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, and 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, and 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, and A. Siahmakoun, “Photonic integrator for A/D conversion,” Proc. SPIE **7797**, 77970J (2010). [CrossRef]

8. 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]

9. 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]

10. Y. Akahane, T. Asano, B.-S. Song, and 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, and E. Hadji, “Ultra-high-reflectivity photonic-bandgap mirrors in a ridge SOI waveguide,” New J. Phys. **8**(9), 204 (2006). [CrossRef]

## 2. Problem statement

*a*,

_{i}*i*= [1,

*N*] is the complex amplitude of the resonant mode in the

*i*-th resonator;

*i*= [1,

*N*], are the left and right coupling coefficients of the

*i*-th resonator, respectively;

*r*,

_{i}*i*= [1,

*N*] is the energy loss of the

*i*-th resonator to the exterior space; and

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

**M**.

*N*= 1, Eq. (3) is reduced to the form

*u*(

*t*) is the Heaviside step function.

*N*= 2, Eq. (3) can be written as

14. M. H. Asghari and 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]

*ω*

_{0}is written as

*ω*

_{0}, which is the pole of Eq. (14). This causes a large error when pulses with a narrow spectral range are integrated. The difference between Eqs. (4) and (14) near

*ω*

_{0}, and thus the integration error, increase with decreasing

*Q*-factor of the resonator

*Q*-factors of 10

^{4}, 10

^{5}and 10

^{6}. Figure 4 shows examples of the input test pulses. This is the original Gaussian pulse with a duration of 1 ps and its first three derivatives. Thus, the first-order integrator restores a Gaussian pulse from its first derivative. The second- and third-order integrators must restore the Gaussian pulse from the second and third derivatives, respectively.

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

*ω*

_{0}is zero for the ideal differentiator function [

*ω*

_{0}quantitatively but not qualitatively. Therefore, differentiation of signals with a narrow spectral range by a high-

*Q*resonator does not lead to errors like those shown in Fig. 3.

*ω*

_{0}vary with the type of resonator. For example, for a ring resonator, the error of integration at high frequencies is determined by the free spectral range (FSR) and depends in particular on the radius of the resonator. For PC resonators, the error of integration at high frequencies is influenced by the PC bandgap and neighboring resonant modes.

*Q*-factor of the resonator is 5 × 10

^{4}. The resonators with FSR values of 12 nm (RMSE = 14%) and 15 nm (RMSE = 6%) integrate the 1 ps impulse with large distortions. High-quality integration was achieved at an FSR of 25 nm. Similarly, a good quality of integration for 150 fs impulse is clearly achieved for FSR values as large as 120 nm (RMSE = 4%). The development of ring resonators with such a large FSR is associated with certain difficulties. PC resonators are a suitable candidate for the integration of subpicosecond optical pulses.

## 3. Example of integrator based on PC nanobeam cavity

10. Y. Akahane, T. Asano, B.-S. Song, and 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, and E. Hadji, “Ultra-high-reflectivity photonic-bandgap mirrors in a ridge SOI waveguide,” New J. Phys. **8**(9), 204 (2006). [CrossRef]

16. H. C. Liu and 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*

_{1}and

*Q*

_{2,}the coupling coefficient iswhere

*Q*

_{0}is the

*Q*-factor of the resonator containing only the hole defect zone (

*ω*

_{0}is the resonant frequency corresponding to the Bragg wavelength. The value of

*a*can be approximated from the calculation of the

*Q*-factor of a single resonator with different values of

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

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

19. D. L. Golovashkin and 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]

16. H. C. Liu and 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*-factor of the resonator is 3.6 × 10

^{4}, and

^{3}for a wavelength of 1.57 µm. The second- and third-order integrators have dimensions of 12.0 × 0.5 × 0.2 µm

^{3}and 18.0 × 0.5 × 0.2 µm

^{3}, respectively. Thus, these integrators are at least 10 times more compact than any of those previously suggested.

23. M. H. Asghari and 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]

## 4. Conclusion

## Acknowledgments

## References and links

1. | H. J. Caulfield and S. Dolev, “Why future supercomputing requires optics,” Nat. Photonics |

2. | N. Quoc Ngo, “Design of an optical temporal integrator based on a phase-shifted fiber Bragg grating in transmission,” Opt. Lett. |

3. | 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. |

4. | A. Malacarne, R. Ashrafi, M. Li, S. LaRochelle, J. Yao, and J. Azaña, “Single-shot photonic time-intensity integration based on a time-spectrum convolution system,” Opt. Lett. |

5. | Y. Park, T. J. Ahn, Y. Dai, J. Yao, and J. Azaña, “All-optical temporal integration of ultrafast pulse waveforms,” Opt. Express |

6. | N. Q. Ngo, “Optical integrator for optical dark-soliton detection and pulse shaping,” Appl. Opt. |

7. | Y. Jin, P. Costanzo-Caso, S. Granieri, and A. Siahmakoun, “Photonic integrator for A/D conversion,” Proc. SPIE |

8. | 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 |

9. | 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 |

10. | Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “Fine-tuned high-Q photonic-crystal nanocavity,” Opt. Express |

11. | P. Velha, J. C. Rodier, P. Lalanne, J. P. Hugonin, D. Peyrade, E. Picard, T. Charvolin, and E. Hadji, “Ultra-high-reflectivity photonic-bandgap mirrors in a ridge SOI waveguide,” New J. Phys. |

12. | H. A. Haus, |

13. | H. C. Liu and A. Yariv, “Synthesis of high-order bandpass filters based on coupled-resonator optical waveguides (CROWs),” Opt. Express |

14. | M. H. Asghari and J. Azaña, “On the design of efficient and accurate arbitrary-order temporal optical integrators using fiber bragg gratings,” J. Lightwave Technol. J. |

15. | N. L. Kazanskiy, P. G. Serafimovich, and S. N. Khonina, “Use of photonic crystal cavities for temporal differentiation of optical signals,” Opt. Lett. |

16. | H. C. Liu and A. Yariv, “Designing coupled-resonator optical waveguides based on high-Q tapered grating-defect resonators,” Opt. Express |

17. | Q. Quan and M. Loncar, “Deterministic design of wavelength scale, ultra-high Q photonic crystal nanobeam cavities,” Opt. Express |

18. | A. Taflove and S. C. Hagness, |

19. | D. L. Golovashkin and N. L. Kazanskiy, “Mesh domain decomposition in the finite-difference solution of Maxwell’s equations,” Opt. Mem. Neural Networks |

20. | M. H. Asghari, C. Wang, J. Yao, and J. Azaña, “High-order passive photonic temporal integrators,” Opt. Lett. |

21. | 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. | G. Shambat, B. Ellis, J. Petykiewicz, M. Mayer, T. Sarmiento, J. Harris, E. E. Haller, and J. Vuckovic, “Nanobeam photonic crystal cavity light-emitting diodes,” Appl. Phys. Lett. |

23. | M. H. Asghari and J. Azaña, “Design of all-optical high-order temporal integrators based on multiple-phase-shifted Bragg gratings,” Opt. Express |

**OCIS Codes**

(130.3120) Integrated optics : Integrated optics devices

(200.4560) Optics in computing : Optical data processing

(320.7085) Ultrafast optics : Ultrafast information processing

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: April 23, 2014

Revised Manuscript: May 22, 2014

Manuscript Accepted: May 22, 2014

Published: May 30, 2014

**Citation**

Nikolay L. Kazanskiy and Pavel G. Serafimovich, "Coupled-resonator optical waveguides for temporal integration of optical signals," Opt. Express **22**, 14004-14013 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-11-14004

Sort: Year | Journal | Reset

### References

- H. J. Caulfield, S. Dolev, “Why future supercomputing requires optics,” Nat. Photonics 4(5), 261–263 (2010). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- N. Q. Ngo, “Optical integrator for optical dark-soliton detection and pulse shaping,” Appl. Opt. 45(26), 6785–6791 (2006). [CrossRef] [PubMed]
- Y. Jin, P. Costanzo-Caso, S. Granieri, A. Siahmakoun, “Photonic integrator for A/D conversion,” Proc. SPIE 7797, 77970J (2010). [CrossRef]
- 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]
- 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]
- H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).
- 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]
- 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]
- 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]
- 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]
- A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).
- 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]
- M. H. Asghari, C. Wang, J. Yao, J. Azaña, “High-order passive photonic temporal integrators,” Opt. Lett. 35(8), 1191–1193 (2010). [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]
- 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]
- 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]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.