## Matrix analysis of microring coupled-resonator optical waveguides

Optics Express, Vol. 12, Issue 1, pp. 90-103 (2004)

http://dx.doi.org/10.1364/OPEX.12.000090

Acrobat PDF (271 KB)

### Abstract

We use the coupling matrix formalism to investigate continuous-wave and pulse propagation through microring coupled-resonator optical waveguides (CROWs). The dispersion relation agrees with that derived using the tight-binding model in the limit of weak inter-resonator coupling. We obtain an analytical expression for pulse propagation through a semi-infinite CROW in the case of weak coupling which fully accounts for the nonlinear dispersive characteristics. We also show that intensity of a pulse in a CROW is enhanced by a factor inversely proportional to the inter-resonator coupling. In finite CROWs, anomalous dispersions allows for a pulse to propagate with a negative group velocity such that the output pulse appears to emerge before the input as in “superluminal” propagation. The matrix formalism is a powerful approach for microring CROWs since it can be applied to structures and geometries for which analyses with the commonly used tight-binding approach are not applicable.

© 2004 Optical Society of America

## 1. Introduction

1. 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]

2. Y. Xu, R. K. Lee, and A. Yariv, “Propagation and second-harmonic generation of electromagnetic waves in a coupled-resonator optical waveguide,” J. Opt. Soc. Am. B **77**, 387–400 (2000). [CrossRef]

3. N. Stefanou and A. Modinos, “Impurity bands in photonic insulators,” Phys. Rev. B **57**, 12 127–12 133 (1998). [CrossRef]

4. D. N. Christodoulides and N. K. Efremidis, “Discrete temporal solitons along a chain of nonlinear coupled microcavities embedded in photonic crystals,” Opt. Lett. **27**, 568–570 (2002). [CrossRef]

5. S. Mookherjea and A. Yariv, “Kerr-stabilized super-resonant modes in coupled-resonator optical waveguides,” Phys. Rev. E **66**, 046 610 (2002). [CrossRef]

6. J. E. Heebner and R. W. Boyd, “‘Slow’ and ‘fast’ light in resonator-coupled waveguides,” J. Mod. Opt. **49**, 2629–2636 (2002). [CrossRef]

*N*<5, have been proposed for optical filtering and modulation [7

7. C. K. Madsen, “General IIR optical filter design for WDM applications using all-pass filters,” IEEE J. Lightwave Technol. **18**, 860–868 (2000). [CrossRef]

8. G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. **37**, 525–532 (2001). [CrossRef]

9. B. E. Little, S. T. Chu, W. Pan, D. Ripin, T. Kaneko, Y. Kokubun, and E. Ippen, “Vertically coupled glass microring resonator channel dropping filters,” IEEE Photon. Technol. Lett. **11**, 215–217 (1999). [CrossRef]

*N*>10, can be regarded as a new type of waveguide termed Coupled-Resonator Optical Waveguide (CROW) with unique and controllable dispersion properties [3

3. N. Stefanou and A. Modinos, “Impurity bands in photonic insulators,” Phys. Rev. B **57**, 12 127–12 133 (1998). [CrossRef]

1. 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]

2. Y. Xu, R. K. Lee, and A. Yariv, “Propagation and second-harmonic generation of electromagnetic waves in a coupled-resonator optical waveguide,” J. Opt. Soc. Am. B **77**, 387–400 (2000). [CrossRef]

10. M. Bayindir, B. Temelkuran, and E. Ozbay, “Tight-binding description of the coupled defect modes in three-dimensional photonic crystals,” Phys. Rev. Lett. **84**, 2140–2143 (2000). [CrossRef] [PubMed]

1. 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]

**E**

_{K}of the CROW as a Bloch wave superposition of the individual resonator modes

**E**

_{Ω}[1

**24**, 711–713 (1999). [CrossRef]

*n*th resonator in the chain is centered at

*z*=

*n*Λ. Under the assumption of symmetric nearest neighbor coupling, the dispersion relation of the CROW is [1

**24**, 711–713 (1999). [CrossRef]

*κ*

_{1}are defined as

**r**) is the dielectric coefficient of the CROW and ε

_{0}(

**r**) is the dielectric coefficient of an individual resonator. Therefore, the coupling parameter

*κ*

_{1}represents the overlap of the modes of two neighboring resonators and Δα/2 gives the fractional self frequency shift of ω

_{K}.

2. Y. Xu, R. K. Lee, and A. Yariv, “Propagation and second-harmonic generation of electromagnetic waves in a coupled-resonator optical waveguide,” J. Opt. Soc. Am. B **77**, 387–400 (2000). [CrossRef]

5. S. Mookherjea and A. Yariv, “Kerr-stabilized super-resonant modes in coupled-resonator optical waveguides,” Phys. Rev. E **66**, 046 610 (2002). [CrossRef]

12. K. Oda, N. Takato, and H. Toba, “A wide-FSR waveguide double-ring resonator for optical FDM transmission systems,” IEEE J. of Lightwave Technol. **9**, 728–736 (1991). [CrossRef]

13. 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]

*N*coupled ring resonators with input and output waveguides. Since the modal properties of ring resonators can be easily tailored and their fabrication technology is mature [14

14. J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P.-T. Ho, “Higher order filter response in coupled microring resonators,” IEEE Photon. Technol. Lett. **12**, 320–322 (2000). [CrossRef]

15. A. Melloni, R. Costa, P. Monguzzi, and M. Martinelli, “Ring-resonator filters in silicon oxynitride technology for dense wavelength-division multiplexing systems,” Opt. Lett. **28**, 1567–1569 (2003). [CrossRef] [PubMed]

## 2. Transfer matrix formalism

16. A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. **36**, 321–322 (2000). [CrossRef]

*t*and

*κ*are respectively the dimensionless transmission and coupling coefficients over the coupling length. The matrix is unitary and unimodular so that |

*t*|

^{2}+|

*κ*|

^{2}=1. Defining a vector with the different field components,

*R*is the ring radius and

*β*=

*n*(

*ω*)

*ω*/

*c*+

*iα*, where

*n*(

*ω*) is the frequency dependent effective index and α is the loss (or gain) per unit length in the ring. Combining (6) and (7), we have

## 3. CROW dispersion relation

*x*

_{n}is

*ϕ*is the azimuthal angle relative to the propagation direction in the counter-clockwise sense, and

*ρ*is the radial co-ordinate. For a mode of an infinite chain of ring resonators, the fields are periodic at the lattice constant, Λ. So applying Bloch’s theorem,

*K*is the CROW propagation constant. Combining this requirement with (8) leads to

*U*is the identity matrix.

*κ*)≫Re(

*κ*) for phase-matched coupling. We recall that at the resonant frequency of an individual resonator, Ω, Ω

*n*(Ω)

*R*/

*c*=

*m*, where

*m*is an integer, and

*n*(Ω) is the effective index at Ω. Therefore, approximating

*n*(Ω)≈

*n*(

*ω*

_{K}), we solve Eq. (11) to obtain

*κ*

_{2}≡Im(

*κ*)/(

*mπ*). The two dispersion relations corresponding to the ‘±’ coexist for an infinite structure to allow for both forward and backward wave propagation (i.e. positive and negative group velocities). Physically, for a finite structure without reflection and a unidirectional input as in Fig. 3, only the dispersion relation with the matching group and phase velocities as the input wave will be of significance.

*ωmπ*/Ω≪1, it is necessary that |

*κ*|≪1. This condition and the absence of all but the nearest neighbor coupling are thus the validity conditions for the tight-binding approximate result (13).

*κ*=-0.8

*i*and

*m*=100 as calculated using the “exact” form in Eq. (12) and the approximated form in Eq. (13). As

*ω*

_{K}/Ω increases, the exact dispersion relation deviates more significantly from the cosine form. For smaller values of

*κ*, the deviation from the cosine dispersion relation is reduced.

*K*). The eigenvalues are exp(-

*iK*

_{1}Λ)≡ξ

_{1}, exp[-

*i*(

*K*

_{1}Λ+

*π*)]≡-

*ξ*

_{1}, exp(-

*iK*

_{2}Λ)≡

*ξ*

_{2}, and exp[-

*i*(

*K*

_{2}Λ+

*π*)]≡-

*ξ*

_{2}. The corresponding (un-normalized) eigenvectors are

*κ*|≪1 and

*ω*≈Ω, such that

*γ*≈|

*κ*|≈0, ζ≈1, and

*ξ*

_{1}=

*ξ*

_{2}, the 4 eigenvectors reduce to 2 degenerate eigenvectors, representing the two different superpositions of the clockwise and counter-clockwise propagating waves in a single resonator:

*κ*|→0, the modes of the CROW are essentially the modes of the independent resonators, and as |

*κ*|→1, the microrings no longer act as resonators and the CROW modes are essentially conventional waveguide modes.

## 4. Finite CROWs and a travelling wave picture

*PQ*, we obtain an expression for the field components at the output of the CROW after N identical rings:

*P*

_{in}and

*P*

_{out}describe the coupling between the CROW and the input/output waveguides. For a single input to the waveguide, we set

*a*

_{N+1}=0. Therefore, the transfer functions at the “through” and “output” ports as shown in Fig. 3 are

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

*q̂*

_{ξ1}and

*q̂*

_{-ξ1}, or

*q̂*

_{ξ2}and

*q̂*

_{-ξ2}). The travelling wave is an eigenvector of (

*PQ*)

^{2}, and it is verified that the sense of propagation in the rings alternates between clockwise and counter-clockwise with each operation of ℙℚ, as depicted in Fig. 3. Therefore, taking the phase difference accumulated over two rings to be -2

*K*Λ, where

*K*is the Bloch wave vector, such that the phase difference between the output and the input is approximately -(

*N*-1)

*K*Λ, we can determine the CROW dispersion from the finite structure.

*i*. The rings are lossless, and their radius is 16.4

*µ*m.

*n*

_{eff}is taken to be constant and equal to 1.5. Figure 4 compares the dispersion relation extrapolated from the finite CROW with the dispersion relation of an infinite CROW as given by Eq. (12). The small amplitude ripples are manifested at the resonance frequencies of the finite structure. In the limit of an infinite number of resonators, the resonance peaks will be infinitesimally close to each other and the ripples will be smoothed out.

## 5. Pulse propagation

## 5.1. Semi-infinite case

*b*

_{1}in the first resonator is

*b*

_{1}(

*ω*)=-1/

*κ*

_{in}

*a*

_{0}(

*ω*), where

*κ*

_{in}is the coupling coefficient between the input waveguide and the first resonator. Since |

*κ*

_{in}|<1, the intensity of the field inside the CROW is higher than that of the input pulse by 1/|

*κ*

_{in}|

^{2}. This does not violate energy conservation, as the increased intensity is a consequence of the reduced group velocity and hence the spatial compression of the pulse inside the CROW. Using the dispersion relation in Eq. (13), the maximum group velocity in the CROW is

18. A. Melloni and F. Morichetti, “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. Quantum Electron. **35**, 365–379 (2003). [CrossRef]

*R*,

*S*can be expressed as

*κ*

_{in}=

*κ*, the intensity inside the rings is roughly enhanced by

*v*

_{g,max}[11], the energy velocity of a wave that is fully coupled into the semi-infinite CROW is proportional to |

*κ*|

^{2}. Hence, the intensity enhancement is proportional to the energy velocity reduction rather than the group velocity reduction.

19. J. E. Heeber, R. W. Boyd, and Q.-H. Park, “SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides,” J. Opt. Soc. Am. B **19**, 722–731 (2002). [CrossRef]

*κ*|

^{2}in the case of weak coupling. However, a CROW has the advantage that, even in the presence of loss, it is most transmitting for the frequencies of the CROW band, while a side-coupled resonator is most attenuating near the resonant frequency of the resonator.

*b*

_{1}is taken, 𝓔(

*t*,

*z*=0), can be expressed as the Fourier integral

*z*=

*N*Λ, each frequency component,

*b*

_{1}(

*ω*), acquires a phase shift of

*NK*Λ, so the field is

*K*(

*ω*) is given by the dispersion relation of the CROW, Eq. (13). Therefore, instead of integrating over frequency in (25), if we integrate over the half of the Brillouin zone that gives the appropriate group velocity (for example, the right half), we obtain

*x*=

*K*Λ, and invoking the Jacobi-Anger expansion [20

20. S. Mookherjea and A. Yariv, “Pulse propagation in a coupled-resonator optical waveguide to all orders of dispersion,” Phys. Rev. E **65**, 056 601 (2002). [CrossRef]

_{m,N}=

*x*sin(

*x*)cos(

*mx*)

*e*

^{-ixN}≠0 only for certain values of

*m*and

*N*:

*E*(

*t*,

*z*), such that 𝓔(

*t*,

*z*)=

*E*(

*t*,

*z*)

*e*

^{iΩt}, is given by the convolution integral

*J*

_{n}(

*t*) is only defined within |2

*πf*|≤1 [21

21. A. D. Poularikas, *The handbook af formulas and tables for signal processing* (IEEE Press, New York, 1998). [CrossRef]

*κ*

_{2}|)≤

*ω*≤Ω(1+|

*κ*

_{2}|).

*E*(

*t*,

*z*=0)=exp(-

*t*

^{2}/

*T*

^{2}) as calculated using Eq. (30). Figure 6(b) shows the numerical results obtained from the transfer matrices. The analytical solution is in excellent agreement with the fully numerical approach. As the pulse propagates, even though the main peak travels at the group velocity, the ripples develop only at the tail end of the pulse.

## 5.2. Finite case

*µ*m and

*n*

_{eff}=1.5. The inter-resonator coupling constant is -0.3

*i*and the coupling between the waveguides and CROW is -0.5

*i*. The transfer characteristics of this structure are shown in Fig. (7). We launch a 30.5ps (FWHM) long pulse centered at 1.55

*µ*m into the CROW.

*a*

_{n}or

*b*

_{n}. Figure 8 shows the evolution of the pulse through the CROW. Even though the output pulse is attenuated compared to the input, the field intensity inside the rings can be greater than the input, as in the case of the semi-infinite CROW. The intensity build-up is verified by a FDTD simulation discussed in Sect. 5.3. The significant increase in the intensity of the input pulse inside the CROWcan be used to enhance the strengths of nonlinear optical interactions. As noted earlier, we can account for loss (or gain) in our model by including an imaginary part to the propagation factor

*β*. We have found the transfer matrices give excellent agreement with experimental results [22

22. G. T. Paloczi, Y. Huang, A. Yariv, and S. Mookherjea, “Polymeric Mach-Zehnder interferometer using serially coupled microresonators,” Opt. Express **11**, 2666–2671 (2003). [CrossRef] [PubMed]

23. S. Longhi, M. Marano, M. Belmonte, and P. Laporta, “Superluminal pulse propagation in linear and nonlinear photonic grating structures,” IEEE. J. Sel. Top. Quantum Electron. **9**, 4–16 (2003). [CrossRef]

6. J. E. Heebner and R. W. Boyd, “‘Slow’ and ‘fast’ light in resonator-coupled waveguides,” J. Mod. Opt. **49**, 2629–2636 (2002). [CrossRef]

## 5.3. FDTD simulations

*µ*m wide. They are set in air and have an index of refraction of 3.5. The rings have a radius of 5

*µ*m, and the wavelength dependent effective index, as extrapolated from a separate FDTD simulation of the waveguides, is

*n*

_{eff}=3.617-0.5539λ. The coupling between the rings is -0.32

*i*, and the coupling between the rings and the waveguides is -0.4

*i*. A 2.4ps (FWHM) Gaussian pulse is launched into the system, and the fields at the through port, drop port, and inside the rings are monitored. We compare the transfer matrix method with the FDTD simulation in Fig. 10(a), showing that the approaches are in excellent agreement. The anomalous dispersion at the through port and the increase in intensity in the coupled rings are confirmed by the FDTD simulation and are evident in Fig. 10(b).

## 6. Comparison with Fabry-Perot resonators

*et al.*have recently demonstrated a CROW operating near 600nm using coupled Fabry-Perot resonators fabricated in a multi-layer fashion [24

24. M. Bayindir, S. Tanriseven, and E. Ozbay, “Propagation of light through localized coupled-cavity modes in one-dimensional photonic band-gap structures,” Appl. Phys. A **72**, 117–119 (2001). [CrossRef]

25. W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical-response of superlattices,” Phys. Rev. Lett. **58**, 160–163 (1987). [CrossRef] [PubMed]

26. C. M. de Sterke and J. E. Sipe, “Envelope-function approach for the electrodynamics of nonlinear periodic structures,” Phys. Rev. A **38**, 5149–5165 (1988). [CrossRef] [PubMed]

27. C. M. de Sterke, D. G. Salinas, and J. E. Sipe, “Coupled-mode theory for light propagation through deep nonlinear gratings,” Phys. Rev. E **54**, 1969–1989 (1996). [CrossRef]

28. B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. **76**, 1627–1630 (1996). [CrossRef] [PubMed]

29. D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. **62**, 1746–1749 (1989). [CrossRef] [PubMed]

*κ*corresponds to the transmission coefficient, while

*t*corresponds to the reflection coefficient. They can be calculated from the Fresnel coefficients or from an analysis of a Bragg stack in the case of a Fabry-Perot with Bragg end mirrors [11]. The two major differences between the Fabry-Perot resonator and the ring resonator are 1) the former can be fully described by a 2×2 transfer matrix while the latter requires a 4×4 matrix, and 2) the reflection and transmission coefficients may contain a real part. Therefore, in contrast to the ±cos(

*K*Λ) dependence in Eq. (13), there is only one dispersion curve for a Fabry-Perot CROW at a given frequency range and there may also be an additional phase shift in the cosine dependence.

22. G. T. Paloczi, Y. Huang, A. Yariv, and S. Mookherjea, “Polymeric Mach-Zehnder interferometer using serially coupled microresonators,” Opt. Express **11**, 2666–2671 (2003). [CrossRef] [PubMed]

30. Little Optics press release, “Higher order optical filters using microring resonators” (Little Optics, 2003), http://www.littleoptics.com/hofilter.pdf.

## 7. Conclusion

22. G. T. Paloczi, Y. Huang, A. Yariv, and S. Mookherjea, “Polymeric Mach-Zehnder interferometer using serially coupled microresonators,” Opt. Express **11**, 2666–2671 (2003). [CrossRef] [PubMed]

## Acknowledgments

## References and links

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

2. | Y. Xu, R. K. Lee, and A. Yariv, “Propagation and second-harmonic generation of electromagnetic waves in a coupled-resonator optical waveguide,” J. Opt. Soc. Am. B |

3. | N. Stefanou and A. Modinos, “Impurity bands in photonic insulators,” Phys. Rev. B |

4. | D. N. Christodoulides and N. K. Efremidis, “Discrete temporal solitons along a chain of nonlinear coupled microcavities embedded in photonic crystals,” Opt. Lett. |

5. | S. Mookherjea and A. Yariv, “Kerr-stabilized super-resonant modes in coupled-resonator optical waveguides,” Phys. Rev. E |

6. | J. E. Heebner and R. W. Boyd, “‘Slow’ and ‘fast’ light in resonator-coupled waveguides,” J. Mod. Opt. |

7. | C. K. Madsen, “General IIR optical filter design for WDM applications using all-pass filters,” IEEE J. Lightwave Technol. |

8. | G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. |

9. | B. E. Little, S. T. Chu, W. Pan, D. Ripin, T. Kaneko, Y. Kokubun, and E. Ippen, “Vertically coupled glass microring resonator channel dropping filters,” IEEE Photon. Technol. Lett. |

10. | M. Bayindir, B. Temelkuran, and E. Ozbay, “Tight-binding description of the coupled defect modes in three-dimensional photonic crystals,” Phys. Rev. Lett. |

11. | A. Yariv and P. Yeh, |

12. | K. Oda, N. Takato, and H. Toba, “A wide-FSR waveguide double-ring resonator for optical FDM transmission systems,” IEEE J. of Lightwave Technol. |

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

14. | J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P.-T. Ho, “Higher order filter response in coupled microring resonators,” IEEE Photon. Technol. Lett. |

15. | A. Melloni, R. Costa, P. Monguzzi, and M. Martinelli, “Ring-resonator filters in silicon oxynitride technology for dense wavelength-division multiplexing systems,” Opt. Lett. |

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

17. | B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filter,” IEEE J. Lightwave Technol. |

18. | A. Melloni and F. Morichetti, “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. Quantum Electron. |

19. | J. E. Heeber, R. W. Boyd, and Q.-H. Park, “SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides,” J. Opt. Soc. Am. B |

20. | S. Mookherjea and A. Yariv, “Pulse propagation in a coupled-resonator optical waveguide to all orders of dispersion,” Phys. Rev. E |

21. | A. D. Poularikas, |

22. | G. T. Paloczi, Y. Huang, A. Yariv, and S. Mookherjea, “Polymeric Mach-Zehnder interferometer using serially coupled microresonators,” Opt. Express |

23. | S. Longhi, M. Marano, M. Belmonte, and P. Laporta, “Superluminal pulse propagation in linear and nonlinear photonic grating structures,” IEEE. J. Sel. Top. Quantum Electron. |

24. | M. Bayindir, S. Tanriseven, and E. Ozbay, “Propagation of light through localized coupled-cavity modes in one-dimensional photonic band-gap structures,” Appl. Phys. A |

25. | W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical-response of superlattices,” Phys. Rev. Lett. |

26. | C. M. de Sterke and J. E. Sipe, “Envelope-function approach for the electrodynamics of nonlinear periodic structures,” Phys. Rev. A |

27. | C. M. de Sterke, D. G. Salinas, and J. E. Sipe, “Coupled-mode theory for light propagation through deep nonlinear gratings,” Phys. Rev. E |

28. | B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. |

29. | D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. |

30. | Little Optics press release, “Higher order optical filters using microring resonators” (Little Optics, 2003), http://www.littleoptics.com/hofilter.pdf. |

**OCIS Codes**

(230.3120) Optical devices : Integrated optics devices

(230.5750) Optical devices : Resonators

(230.7370) Optical devices : Waveguides

**ToC Category:**

Research Papers

**History**

Original Manuscript: October 14, 2003

Revised Manuscript: December 19, 2003

Published: January 12, 2004

**Citation**

Joyce Poon, Jacob Scheuer, Shayan Mookherjea, George Paloczi, Yanyi Huang, and Amnon Yariv, "Matrix analysis of microring coupled-resonator optical waveguides," Opt. Express **12**, 90-103 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-1-90

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### References

- 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]
- Y. Xu, R. K. Lee, and A. Yariv, �??Propagation and second-harmonic generation of electromagnetic waves in a coupled-resonator optical waveguide,�?? J. Opt. Soc. Am. B 77, 387�??400 (2000). [CrossRef]
- N. Stefanou and A. Modinos, �??Impurity bands in photonic insulators,�?? Phys. Rev. B 57, 12 127�??12 133 (1998). [CrossRef]
- D. N. Christodoulides and N. K. Efremidis, �??Discrete temporal solitons along a chain of nonlinear coupled microcavities embedded in photonic crystals,�?? Opt. Lett. 27, 568�??570 (2002). [CrossRef]
- S. Mookherjea and A. Yariv, �??Kerr-stabilized super-resonant modes in coupled-resonator optical waveguides,�?? Phys. Rev. E 66, 046 610 (2002). [CrossRef]
- J. E. Heebner and R. W. Boyd, �??�??Slow�?? and �??fast�?? light in resonator-coupled waveguides,�?? J. Mod. Opt. 49, 2629�??2636 (2002). [CrossRef]
- C. K. Madsen, �??General IIR optical filter design for WDM applications using all-pass filters,�?? IEEE J. Lightwave Technol. 18, 860�??868 (2000). [CrossRef]
- G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, �??Optical delay lines based on optical filters,�?? IEEE J. Quantum Electron. 37, 525�??532 (2001). [CrossRef]
- B. E. Little, S. T. Chu, W. Pan, D. Ripin, T. Kaneko, Y. Kokubun, and E. Ippen, �??Vertically coupled glass microring resonator channel dropping filters,�?? IEEE Photon. Technol. Lett. 11, 215�??217 (1999). [CrossRef]
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