## Synthesis of coupled resonator optical waveguides by cavity aggregation

Optics Express, Vol. 18, Issue 2, pp. 1600-1606 (2010)

http://dx.doi.org/10.1364/OE.18.001600

Acrobat PDF (567 KB)

### Abstract

In this paper, the layer aggregation method is applied to coupled resonator optical waveguides. Starting from the frequency transfer function, the method yields the coupling constants between the resonators. The convergence of the algorithm developed is examined and the related parameters discussed.

© 2010 Optical Society of America

## 1. Introduction

1. J. Capmany and M. Muriel, “A new transfer matrix formalism for the analysis of fiber ring resonators: compound coupled structures for FDMA demultiplexing,” J. Lightwave Technol. **8**, 1904–1919 (1990). [CrossRef]

2. B. Little, S. Chu, H. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. **15**, 998–1005 (1997). [CrossRef]

3. C. Madsen and J. Zhao, “A general planar waveguide autoregressive optical filter,” J. Lightwave Technol. **14**, 437–447 (1996). [CrossRef]

4. V. Van, T. Ibrahim, P. Absil, F. Johnson, R. Grover, and P.-T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. **8**, 705–713 (2002). [CrossRef]

5. F. Xia, L. Sekaric, and Y. Vlasov, “Ultra compact optical buffers on a silicon chip,” Nat. Photonics **1**, 65–71 (2007). [CrossRef]

6. H. Tazawa and W. Steier, “Analysis of ring resonator-based traveling-wave modulators,” IEEE Photon. Technol. Lett. **18**, 211–213 (2006). [CrossRef]

7. C. Madsen, G. Lenz, A. Bruce, M. Cappuzzo, L. Gomez, and R. Scotti, “Integrated all-pass filters for tunable dispersion and dispersion slope compensation,” IEEE Photon. Technol. Lett. **11**, 1623–1625 (1999). [CrossRef]

8. B. Little, S. Chu, W. Pan, and Y. Kokubun, “Microring resonator arrays for VLSI photonics,” IEEE Photon. Technol. Lett. **12**, 323–325 (2000). [CrossRef]

9. H.-C. Ren, F. Vollmer, S. Arnold, and A. Libchaber, “High-Q microsphere biosensor - analysis for adsorption of rodlike bacteria,” Opt. Express **15**, 17410–17423 (2007). [CrossRef] [PubMed]

10. K. Vahala, “Optical microcavities,” Nature (London) **424**, 839–846 (2003). [CrossRef]

## 2. Layer aggregation method

25. J. Capmany, P. M.ñoz, J. D. Domenech, and M. A. Muriel, “Apodized coupled resonator waveguides,” Opt. Express **15**, 10196–10206 (2007). [CrossRef] [PubMed]

22. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. **24**, 711–713 (1999). [CrossRef]

*t*and

_{i}*κ*are the direct and cross-coupling field coefficients and the subscript

_{i}*i*= 0,1,…,

*N*refers to the couplers in the CROW device. A loss-less coupler is assumed hereafter, i.e. |

*t*|

_{i}^{2}+ |

*κ*|

_{i}^{2}= 1. In practical devices, all the resonators have the same perimeter. Hence, the impulse response of the filter is composed of impulses, or samples, of different amplitudes happening at times multiples of the ring resonator round trip time,

*T*. In this case it is convenient to use a discrete time notation [27], i.e.

*h*[

*n*] =

*h*(

*nT*). The layer aggregation method is based upon the analysis of the filter contributions to the impulse response time samples. The approach followed is similar to that developed for Fiber Bragg gratings in [17

17. J. Capmany, M. Muriel, and S. Sales, “Highly accurate synthesis of fiber and waveguide bragg gratings by an impedance reconstruction layer-aggregation method,” IEEE J. Quantum Electron. **43**, 889–898 (2007). [CrossRef]

*n*= 0 the impulse response is solely due to the first coupler of the device, while at time

*n*= 1 the response is due to the first two couplers:

*n*≥ 2,

*h*[

*n*] is formed by two contributions, one recursive and one non-recursive, as illustrated in Fig. 1(d) for

*n*= 2, with yellow and red lines. In this particular case, the recursive contribution is due to the first resonator (two turns, marked in red as ’x2’), while the non-recursive comes from the direct reflection from coupler number 2. Generalizing, for

*n*≥ 2:

*n*is given by the corresponding sample of the impulse response of CROW with

*n*-1 resonators, and the non-recursive part can be derived from the figures:

## 3. Reconstruction procedure

- Given
*H*[*k*], calculate its inverse discrete Fourier transform, IDFT,*h*[*n*] = IDFT{_{M}*H*[*k*]} - For every
*n*≥ 2 iterate using the set {*t*,_{i}*κ*)}_{i}*i*= 0, …,*n*- 1 and the method in [25] to find the impulse response of a CROW with25. J. Capmany, P. M.ñoz, J. D. Domenech, and M. A. Muriel, “Apodized coupled resonator waveguides,” Opt. Express

**15**, 10196–10206 (2007). [CrossRef] [PubMed]*n*- 1 rings, corresponding to*h*[_{r}*n*], and eqs. (3)(4) to obtain (*t*,_{n}*κ*)._{n}

## 4. Results and discussion

*K*= |

*κ*|

^{2}= 0.1 of

*N*= 5 and

*N*= 10 rings. The target transfer functions were calculated using the Transfer Matrix Method, TMM, from [25

**15**, 10196–10206 (2007). [CrossRef] [PubMed]

*N*= 10 sampled with 2

^{m}points (

*m*= 9 and

*m*= 14). The coupling constants

*K*,

_{i}*i*= 0,…,

*N*obtained through reconstruction are shown in Fig. 3(a) and 3(b), for 5 and 10 rings respectively, for different number of samples in

*H*(

*ω*).

_{d}*K*values, 0.1, 0.2 and 0.3, is shown in Fig. 3(c) and 3(d) for

*N*= 5 and

*N*= 10 rings respectively. The graphs show the value of the coupling constant for the last coupler in the CROW,

*K*. For 5 rings, convergence is achieved for

_{N}*K*= 0.2 and

*K*= 0.3 with

*m*≥ 10 and

*m*≥ 9 respectively, while

*K*= 0.1 requires

*m*≥ 11. The convergence worsens for 10 rings, where

*m*≥ 13 and

*m*≥ 12 are needed for

*K*= 0.2 and

*K*= 03 respectively, while

*K*= 0.1 needs

*m*> 14.

*t*values are modified following a weight (window) function. Starting with a nominal coupling constant

_{i}*K*= 0.1, a Hamming window with parameter

*H*= 0.2 was used as described in [25

**15**, 10196–10206 (2007). [CrossRef] [PubMed]

*K*= {0.5728, 0.3750, 0.1397, 0.1397, 0.3750, 0.5728} and

_{i}*K*= {0.5919, 0.5280, 0.4101, 0.2668, 0.1470, 0.1000, 0.1470, 0.2668, 0.4101, 0.5280, 0.5919} for 5 and 10 rings respectively. The TMM calculated target responses, sampled with the same points as in the uniform case of Fig. 2(a), are shown in Fig. 4(a).

_{i}*H*(

*ω*) exhibits less pronounced transitions between maxima and nulls. This smoothness is produced by the apodization of the power coupling constants amongst the resonators. Therefore, the situation in terms of signal sampling and Fourier transform, is more relaxed than in the previous example. The comparison of

_{d}*m*= 9 and

*m*= 14 in Fig. 4(a) reveals that

*m*= 9 is a good approximation of

*H*(

*ω*), far better than in the previous uniform example of Fig. 2(a).

_{d}*h*[

*n*], using

*m*= 9 and

*m*= 14, are of the order of 10

^{-14}. The convergence of all the coupling constants vs

*m*is shown in Fig. 4(c) and 4(d) for 5 and 10 rings respectively. As outlined before, lower sampling is required for the apodized cases, since for theses cases convergence is achieved for

*m*≥ 7 and

*m*≥ 9, for 5 and 10 rings respectively.

## 5. Conclusion

## Acknowledgement

## References and links

1. | J. Capmany and M. Muriel, “A new transfer matrix formalism for the analysis of fiber ring resonators: compound coupled structures for FDMA demultiplexing,” J. Lightwave Technol. |

2. | B. Little, S. Chu, H. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. |

3. | C. Madsen and J. Zhao, “A general planar waveguide autoregressive optical filter,” J. Lightwave Technol. |

4. | V. Van, T. Ibrahim, P. Absil, F. Johnson, R. Grover, and P.-T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. |

5. | F. Xia, L. Sekaric, and Y. Vlasov, “Ultra compact optical buffers on a silicon chip,” Nat. Photonics |

6. | H. Tazawa and W. Steier, “Analysis of ring resonator-based traveling-wave modulators,” IEEE Photon. Technol. Lett. |

7. | C. Madsen, G. Lenz, A. Bruce, M. Cappuzzo, L. Gomez, and R. Scotti, “Integrated all-pass filters for tunable dispersion and dispersion slope compensation,” IEEE Photon. Technol. Lett. |

8. | B. Little, S. Chu, W. Pan, and Y. Kokubun, “Microring resonator arrays for VLSI photonics,” IEEE Photon. Technol. Lett. |

9. | H.-C. Ren, F. Vollmer, S. Arnold, and A. Libchaber, “High-Q microsphere biosensor - analysis for adsorption of rodlike bacteria,” Opt. Express |

10. | K. Vahala, “Optical microcavities,” Nature (London) |

11. | E. Peral, J. Capmany, and J. Marti, “Iterative solution to the Gel’fand-Levitan-Marchenko coupled equations and application to synthesis of fiber gratings,” IEEE J. Quantum Electron. |

12. | R. Feced, M. Zervas, and M. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. |

13. | L. Poladian, “Simple grating synthesis algorithm,” Opt. Lett. |

14. | J. Skaar, L. Wang, and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. |

15. | J. Skaar, L. Wang, and T. Erdogan, “Synthesis of thick optical thin-film filters with a layer-peeling inverse-scattering algorithm,” Appl. Opt. |

16. | A. Rosenthal and M. Horowitz, “Inverse scattering algorithm for reconstructing strongly reflecting fiber bragg gratings,” IEEE J. Quantum Electron. |

17. | J. Capmany, M. Muriel, and S. Sales, “Highly accurate synthesis of fiber and waveguide bragg gratings by an impedance reconstruction layer-aggregation method,” IEEE J. Quantum Electron. |

18. | J. E. Heebner, P. Chak, S. Pereira, J. E. Sipe, and R. W. Boyd, “Distributed and localized feedback in microres-onator sequences for linear and nonlinear optics,” J. Opt. Soc. Am. B |

19. | J. Poon, J. Scheuer, S. Mookherjea, G. Paloczi, Y. Huang, and A. Yariv, “Matrix analysis of microring coupled-resonator optical waveguides,” Opt. Express |

20. | Y. Landobasa, S. Darmawan, and M.-K. Chin, “Matrix analysis of 2-D microresonator lattice optical filters,” IEEE J. Quantum Electron. |

21. | D. L. MacFarlane and E. M. Dowling, “Z-domain techniques in the analysis of Fabry-Perot étalons and multilayer structures,” J. Opt. Soc. Am. A |

22. | A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. |

23. | A. Melloni and M. Martinelli, “Synthesis of direct-coupled-resonators bandpass filters for WDM systems,” J. Lightwave Technol. |

24. | R. Orta, P. Savi, R. Tascone, and D. Trinchero, “Synthesis of multiple-ring-resonator filters for optical systems,” IEEE Photon. Technol. Lett. |

25. | J. Capmany, P. M.ñoz, J. D. Domenech, and M. A. Muriel, “Apodized coupled resonator waveguides,” Opt. Express |

26. | A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,”, Electron. Lett. |

27. | A. V. Oppenheim, R. W. Schafer, and J. R. Buck, |

**OCIS Codes**

(130.2790) Integrated optics : Guided waves

(130.3120) Integrated optics : Integrated optics devices

(230.5750) Optical devices : Resonators

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: November 19, 2009

Revised Manuscript: January 8, 2010

Manuscript Accepted: January 9, 2010

Published: January 13, 2010

**Citation**

Pascual Muñoz, José David Doménech, and José Capmany, "Synthesis of coupled resonator optical waveguides by cavity aggregation," Opt. Express **18**, 1600-1606 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-2-1600

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### References

- J. Capmany and M. Muriel, "A new transfer matrix formalism for the analysis of fiber ring resonators: compound coupled structures for FDMA demultiplexing," J. Lightwave Technol. 8, 1904-1919 (1990). [CrossRef]
- B. Little, S. Chu, H. Haus, J. Foresi, and J.-P. Laine, "Microring resonator channel dropping filters," J. Lightwave Technol. 15, 998-1005 (1997). [CrossRef]
- C. Madsen and J. Zhao, "A general planar waveguide autoregressive optical filter," J. Lightwave Technol. 14, 437-447 (1996). [CrossRef]
- V. Van, T. Ibrahim, P. Absil, F. Johnson, R. Grover, and P.-T. Ho, "Optical signal processing using nonlinear semiconductor microring resonators," IEEE J. Sel. Top. Quantum Electron. 8, 705-713 (2002). [CrossRef]
- F. Xia, L. Sekaric, and Y. Vlasov, "Ultra compact optical buffers on a silicon chip," Nat. Photonics 1, 65-71 (2007). [CrossRef]
- H. Tazawa and W. Steier, "Analysis of ring resonator-based traveling-wave modulators," IEEE Photon. Technol. Lett. 18, 211-213 (2006). [CrossRef]
- C. Madsen, G. Lenz, A. Bruce, M. Cappuzzo, L. Gomez, and R. Scotti, "Integrated all-pass filters for tunable dispersion and dispersion slope compensation," IEEE Photon. Technol. Lett. 11, 1623-1625 (1999). [CrossRef]
- B. Little, S. Chu, W. Pan, and Y. Kokubun, "Microring resonator arrays for VLSI photonics," IEEE Photon. Technol. Lett. 12, 323-325 (2000). [CrossRef]
- H.-C. Ren, F. Vollmer, S. Arnold, and A. Libchaber, "High-Q microsphere biosensor - analysis for adsorption of rod like bacteria," Opt. Express 15, 17410-17423 (2007). [CrossRef] [PubMed]
- K. Vahala, "Optical microcavities," Nature (London) 424, 839-846 (2003). [CrossRef]
- E. Peral, J. Capmany, and J. Marti, "Iterative solution to the Gel’fand-Levitan-Marchenko coupled equations and application to synthesis of fiber gratings," IEEE J. Quantum Electron. 32, 2078-2084 (1996). [CrossRef]
- R. Feced, M. Zervas, and M. Muriel, "An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings," IEEE J. Quantum Electron. 35, 1105-1115 (1999). [CrossRef]
- L. Poladian, "Simple grating synthesis algorithm," Opt. Lett. 25, 787-789 (2000). [CrossRef]
- J. Skaar, L. Wang, and T. Erdogan, "On the synthesis of fiber Bragg gratings by layer peeling," IEEE J. Quantum Electron. 37, 165-173 (2001). [CrossRef]
- J. Skaar, L. Wang, and T. Erdogan, "Synthesis of thick optical thin-film filters with a layer-peeling inverse scattering algorithm," Appl. Opt. 40, 2183-2189 (2001). [CrossRef]
- A. Rosenthal and M. Horowitz, "Inverse scattering algorithm for reconstructing strongly reflecting fiber bragg gratings," IEEE J. Quantum Electron. 39, 1018-1026 (2003). [CrossRef]
- J. Capmany, M. Muriel, and S. Sales, "Highly accurate synthesis of fiber and waveguide bragg gratings by an impedance reconstruction layer-aggregation method," IEEE J. Quantum Electron. 43, 889-898 (2007). [CrossRef]
- J. E. Heebner, P. Chak, S. Pereira, J. E. Sipe, and R. W. Boyd, "Distributed and localized feedback in microresonator sequences for linear and nonlinear optics," J. Opt. Soc. Am. B 21, 1818-1832 (2004). [CrossRef]
- J. Poon, J. Scheuer, S. Mookherjea, G. Paloczi, Y. Huang, and A. Yariv, "Matrix analysis of microring coupled resonator optical waveguides," Opt. Express 12, 90-103 (2004). [CrossRef] [PubMed]
- Y. Landobasa, S. Darmawan, and M.-K. Chin, "Matrix analysis of 2-D microresonator lattice optical filters," IEEE J. Quantum Electron. 41, 1410-1418 (2005). [CrossRef]
- D. L. MacFarlane and E. M. Dowling, "Z-domain techniques in the analysis of Fabry-Perot ´etalons and multilayer structures," J. Opt. Soc. Am. A 11, 236-245 (1994). [CrossRef]
- A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, "Coupled-resonator optical waveguide: a proposal and analysis," Opt. Lett. 24, 711-713 (1999). [CrossRef]
- A. Melloni and M. Martinelli, "Synthesis of direct-coupled-resonators bandpass filters for WDM systems," J. Lightwave Technol. 20, 296-303 (2002). [CrossRef]
- R. Orta, P. Savi, R. Tascone, and D. Trinchero, "Synthesis of multiple-ring-resonator filters for optical systems," IEEE Photon. Technol. Lett. 7, 1447-1449 (1995). [CrossRef]
- J. Capmany, P. Muñoz, J. D. Domenech, and M. A. Muriel, "Apodized coupled resonator waveguides," Opt. Express 15, 10196-10206 (2007). [CrossRef] [PubMed]
- A. Yariv, "Universal relations for coupling of optical power between microresonators and dielectric waveguides," Electron. Lett. 36, 321-322 (2000). [CrossRef]
- A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-time signal processing, Signal Processing Series, 2nd ed. (Prentice-Hall International, 1999).

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