## Coupling-induced resonance frequency shifts in coupled dielectric multi-cavity filters

Optics Express, Vol. 14, Issue 3, pp. 1208-1222 (2006)

http://dx.doi.org/10.1364/OE.14.001208

Acrobat PDF (1058 KB)

### Abstract

Coupling-induced resonance frequency shifts (CIFS) are theoretically described, and are found to be an important fundamental source of resonance frequency mismatch between coupled optical cavities that would be degenerate in isolation. Their deleterious effect on high-order resonant filter responses and complete correction by pre-distortion are described. Analysis of the physical effects contributing to CIFS shows that a positive index perturbation may bring about a resonance shift of either sign. Higher-order CIFS effects, the scaling of CIFS-caused impairment with finesse, FSR and index contrast, and the tolerability of frequency mismatch in telecom-grade filters are addressed. The results also suggest possible designs and applications for CIFS-free coupled-resonator systems.

© 2006 Optical Society of America

## 1. Introduction

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

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

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

10. M. J. Khan, C. Manolatou, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Mode-coupling analysis of multipole symmetric resonant add/drop filters,” IEEE J. Quantum Electron. **35**, 1451–1460 (1999). [CrossRef]

*effective uncoupled resonance frequencies*, leading to severe response impairments that must be corrected in design. In Ref. [13], we reported initial findings on coupling-induced resonance frequency shifts (CIFS). Here, we give the first more complete treatment and discussion. We first demonstrate the effect of CIFS on filter response by rigorous numerical eigenmode and finite-difference time-domain (FDTD) simulations. The simulations show that accounting for CIFS is necessary and sufficient for simplified transfer-matrix and coupled-mode models, supplemented by numerically computed coefficients, to accurately match full FDTD simulation results for a third-order microring-resonator filter. Next, we call upon coupled-mode theory to qualitatively explain the effect, and discuss the physical basis and rigorous evaluation of CIFS in traveling-wave and general resonators. Counter to what may be expected for positive-index perturbations, we show that CIFS can be of negative or positive sign. By compensating for the predicted shift, we verify that the ideal filter response may be recovered. We then analyze the scaling of the effect with index contrast, inter-cavity cross-coupling strength, bandwidth and free spectral range (FSR). Finally, we discuss CIFS-free coupled resonator designs and comment on generic methods of compensation for CIFS.

14. T. Barwicz, M. A. Popović, P. T. Rakich, M. R Watts, H. A. Haus, E. P. Ippen, and H. I. Smith, “Microring-resonator-based add-rop filters in SiN: fabrication and analysis,” Opt. Express **12**, 1437–1442 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-7-1437. [CrossRef] [PubMed]

15. M. A. Popović, M. R. Watts, T. Barwicz, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kärtner, and H. I. Smith, “High-index-contrast, wide-FSR microring-resonator filter design and realization with frequency-shift compensation,” in *Proc. Optical Fiber Comm. Conf*. (Optical Society of America, Washington, DC,2005).

*not*to a shift in frequency of an input signal.

## 2. CIFS and its effect in multipole resonators

_{10}-mode solution (cf. [17]). Coupling results in a frequency splitting ∆

*ω*[Fig. 1(b)] of the supermodes [16, 17]. It also causes a shift

*δω*in the mean supermode frequency. The latter may be interpreted as a coupling-induced shift in

*effective*uncoupled resonance frequencies, that we refer to as CIFS.

*a⃗*are associated with the resonant modes and normalized such that |

*a*|

_{n}^{2}is the energy of mode

*n*;

**ω**̿ is a diagonal matrix of uncoupled resonance frequencies, and matrix

**μ**̿ represents mode coupling. Off-diagonal elements

*μ*represent cavity-to-cavity coupling employed in engineering the supermodes. Diagonal elements

_{i,j}*μ*represent CIFS and modify ω̿ into effective cavity resonance frequencies. Solving (1) yields frequency splitting,

_{i,i}*μ*) is a second-order, self-coupling effect, as compared to the direct coupling

_{i,i}*μ*that is first-order, leads to frequency splitting [Fig. 1(b)] and sets the bandwidth. On the other hand, in

_{i,j}*spectral response*models, CIFS is contained linearly, as a modification of resonance frequency, while cross-coupling is squared [1

**15**, 998–1005 (1997). [CrossRef]

*effective uncoupled resonance frequencies*are again degenerate [Fig. 3(d)].

## 3. Physical sources and the sign of CIFS in general resonators

*a⃗(t)*, in a suitably formulated CMT (Appendix B), may be described by a variation of the coupled-mode equations of the form of eqn. (1):

**W**̿ is the energy non-orthogonality (basis mode overlap) matrix, and

**M**̿ is a customary coupling overlap matrix that describes the interaction. Matrices

**W**̿ and

**M**̿ depend on the particular formulation of CMT as reduced from Maxwell’s equations (c.f. [17]; another is described in Appendix B). Regardless of the particular formulation of the CMT, the matrix

**μ̿**represents a total effective coupling matrix with respect to mode amplitudes taking energy non-orthogonality into account. For example, in the case of two coupled resonators as in Fig. 1, the CIFS for the resonator associated with amplitude

*a*of two total modes is, according to Eq. (3) (Fig. 3):

_{1}*δ*

*ω*

_{1}= 0 and the resonators oscillate at their uncoupled natural frequencies in ω̿. With coupling present, the CIFS is generally non-zero. We briefly consider the physical interpretation of (4). For a basis of orthogonal modes (e.g. a lone resonator perturbed by a nearby dielectric object with no relevant modes of its own)

*W*=

_{12}*W*= 0, and from (4) the frequency shift is negative since

_{21}*M*/

_{ii}*W*is positive definite in the lossless case. This is an intuitive result if one considers the wave equation or its stationary integral for frequency (lossless case) [20]:

_{ii}*M*(source of power exchange) is a large term relative to self-coupling

_{21}*M*, the net CIFS could be found positive. This initially unintuitive result is more easily understood in the spatial-propagation picture in Sec. 4, specific to traveling-wave resonators.

_{11}21. H. A. Haus, W. P. Huang, and A. W. Snyder, “Coupled-mode formulations,” Opt. Lett. **14**, 1222–1224 (1989). [CrossRef] [PubMed]

21. H. A. Haus, W. P. Huang, and A. W. Snyder, “Coupled-mode formulations,” Opt. Lett. **14**, 1222–1224 (1989). [CrossRef] [PubMed]

22. C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. **35**, 1322–1331 (1999). [CrossRef]

05. M. Lončar, T. Yoshie, Y. Qiu, P. Gogna, and A. Scherer, “Low-threshold photonic crystal laser,” in Proc. SPIE **5000**, 16–26 (2003). [CrossRef]

07. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, “Theoretical analysis of channel drop tunneling processes,” Phys. Rev. B **59**, 15882–15892 (1999). [CrossRef]

24. J. Scheuer and A. Yariv, “Two-dimensional optical ring resonators based on radial Bragg resonance,” Opt. Lett. **28**, 1528–1530 (2003). [CrossRef] [PubMed]

22. C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. **35**, 1322–1331 (1999). [CrossRef]

## 4. Coupling-induced frequency shifts in traveling-wave-resonator structures

**15**, 998–1005 (1997). [CrossRef]

12. R. Orta, P. Savi, R. Tascone, and D. Trinchero, “Synthesis of multiple-ring-resonator filters for optical systems,” IEEE Photonics Technol. Lett. **7**, 1447–1449 (1995). [CrossRef]

*L*is the round-trip cavity length and

*n*is the group index of the traveling-wave mode. The phase perturbation ∆

_{g}*ϕ*is explicitly manifest in the scattering or S-matrix,

**U**̿, of a lumped, point-interaction description of the coupler in Fig. 6(a) (note that in fact

**U**̿ is an off-diagonal 2×2 submatrix of the total 4-port scattering matrix, since reflection and backward transmission are neglected). In general, the interaction is distributed over an extended length of propagation where the fields in the traveling wave cavity and nearby coupled structure are “within reach” of each other’s evanescent tails. The net effect on relevant modes may be evaluated with respect to reference planes [1–4

**15**, 998–1005 (1997). [CrossRef]

**U**̿. Then, propagation along the remainder of the structure in the interaction region is treated as that in the uncoupled structures. This partitioning, without loss of generality, simplifies analysis to that of isolated resonators, plus lumped point interactions. The lumped interaction matrix displays only the frequency dependence of the coupling interaction itself (and not of the propagation/dynamical phase), as seen in Fig. 6(b). In addition, loss due to propagation (e.g. in bent waveguides constituting the rings) and any excess loss caused by the interaction are separated. In our simulations, we obtain the point S-matrix

**U**̿ by evaluating the S-matrix with respect to reference planes 1-4 of the coupled structure in Fig. 6(a), as well as that of the uncoupled ring and bus waveguides. The latter permit normalizing the uncoupled propagation phase and loss out of the former to obtain a point scattering S-matrix

**U**̿ [referenced to the red plane in Fig. 6(a)] which represents the perturbation due to interaction.

**U**̿ with four degrees of freedom – a power coupling ratio κ, and three phases

*θ*

_{o}, θ_{1}θ_{2}:*b⃗*=

**U**̿ ∙

*a⃗*,

*b⃗*= [

*b*

_{2},

*b*

_{3}]

^{T},

*a⃗*≡ [

*a*

_{1}, a

_{4}]

^{T}[Fig. 6(a)] and

**U**̿ is referenced to a single input-output reference plane. The unitary condition,

**U**̿

^{†}

**U**̿ =

**I**̿, alone requires that the phases obey

*κ*→ 0 and phases

*θ*

_{0,1,2}→ 0, such that

**U**̿ →

**I**̿ (identity matrix) and uncoupled behavior is recovered. In the presence of coupling,

*κ*represents power coupling and

*θ*

_{0}±

*θ*

_{1}are two independent coupling-induced phase shifts on either side due to interaction, which translate to frequency shifts in resonant elements. For a coupler that is symmetric about a (horizontal) axis separating the input and output ports [Fig. 6(a)], reciprocity combined with the geometric symmetry requires equal off-diagonal elements, i.e.

*θ*

_{2}→ 0, reducing the number of free parameters to three. But, the two diagonal coupling-induced phase shifts remain independently determined, permitting two different CIFS values for resonators on each side of the coupler.

**U**̿ for the ring and bus waveguide structures used in the filter example of Fig. 2. A narrower coupling gap of 100nm is used in order to amplify and make clearly visible the phase shifts due to interaction. Figure 6(c) shows power coupling (cross state ~60%) and its wavelength dependence due to change in mode confinement. The total output power sums to >99.9% for even smaller coupling gaps and supports this discussion using the unitary matrix model. Figure 2(b) shows the phase shifts, which by inspection can be seen to, and were verified to, obey condition (8). The cross-state phase shifts are equal and near 90° as expected. However, it is interesting to note that the bar-state phase shifts on the bus and ring sides are large and of opposite sign (“anomalous” and normal, respectively). This is a manifestation of the multiple contributions to the CIFS, discussed in the temporal resonance picture previously and briefly further discussed below. It implies that, for example, when coupling two resonators of different radii such as in a Vernier scheme (c.f. [1

**15**, 998–1005 (1997). [CrossRef]

12. R. Orta, P. Savi, R. Tascone, and D. Trinchero, “Synthesis of multiple-ring-resonator filters for optical systems,” IEEE Photonics Technol. Lett. **7**, 1447–1449 (1995). [CrossRef]

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

**U**̿ of the directional coupler, including CIFS. This is a practical approach for resonant filter design.

### 4.1. Coupling of modes in space picture

*β*of waveguide

_{i}*i*is modified to

*=*β ¯

_{i}*β*+

_{i}*δ*

*β*by the diagonal term

_{i}*δ*

*β*of a coupling matrix analogous to

_{i}**μ**̿ in the temporal system of Eqs. (1) or (3), e.g.:

*δ*

*β*is given for mode 1 of two modes and may in general vary in value along the propagation direction,

_{1}**K**̿ is the coupling overlap matrix and

**P**̿ is the non-orthogonality matrix as in [17]. For perfectly synchronous couplers, this is the only coupling-induced phase contribution and the accumulated phase is the integral of

*δ*

*β*(

_{i}*z*) along the propagation direction over the interaction region, with CIFS given by (6). In general, at least parts of a coupling region may be non-synchronous, and then additional phase is accumulated in each waveguide due to beating in the mismatched coupler. The total accumulated phase is obtained by considering the phase of the total integrated CMT solution [16] for bar-state propagation through a coupler. For a uniform coupler along the propagation direction, the total phase is

*δ*≡ (

β ¯

_{2}-

β ¯

_{1})/2,

*κ*̅

^{2}≡

*κ*̅

_{21}

*κ*̅

_{12}. Non-orthogonal couplings

*κ*̅

_{21},

*κ*̅

_{12}are off-diagonal elements of

**P**̿

^{-1}∙

**K**̿. In gradual couplers where synchronism, propagation constant and coupling vary with length along the propagation direction, the terms in (10) contain integrals with respect to distance. The three phase terms on the right-hand side of (10) are due to the uncoupled

*β*, the coupling-induced

*β*shift, and the non-synchronous slip phase. The latter two terms result from coupling and contribute a CIFS according to (6). The third term may be significant when asynchronous (asymmetric or bent) and interferometric couplers are employed. For synchronous (

*δ*= 0), strongly asynchronous (|

*δ*|≫|

*κ*̅|), or optically short (|

*κ*̅|

*z*≪ 1) couplers, no significant amount of power is coupled over and back with phase slip, and the second term in (10) is the dominant contribution.

*K*is the usual positive effective index (negative CIFS) contribution due to the presence of the high index adjacent bus waveguide or resonator. The second term in (9) that gives a positive CIFS contribution can be understood by considering two weakly guided TE coupled slabs of half the width necessary to cut off the second guided mode. At zero wall-to-wall spacing (strong coupling regime), the two guides merge and the antisymmetric mode becomes cut off, while the symmetric mode morphs into the fundamental guided mode. In approaching this situation, clearly the antisymmetric mode’s effective index drops much faster than that of the symmetric mode rises. This example clarifies the negative contribution to the average effective index of the two supermodes, seen in Fig. 4 and Fig. 6, which for synchronous coupled waveguides corresponds to a

_{11}*δβ*< 0 in (9), or a CIFS > 0.

## 5. Compensation of CIFS in design

## 6. Higher-order CIFS effects

## 7. CIFS scaling laws and CIFS-free resonator configurations

**15**, 998–1005 (1997). [CrossRef]

*N*> 1.

*independent*of bandwidth and FSR (globally scaled for all cavities). For cavity-cavity coupling it scales as

*E*-field overlaps in [17]) and the CMT model (3), and assuming evanescently coupled generic cavities. We retain only the basic index perturbation term

*M*/

_{11}*W*in

_{11}*μ*[Eq. (4)], disregarding the other terms as they are of same order. These contributions may cancel to make

_{11}*μ*smaller or zero in magnitude [Fig. 4(b)], but the first term gives a reasonable scaling for extremal values.

_{11}*M*, depends on the overlap of the evanescent tail of cavity 1 field squared over the second cavity’s core. Cavity-cavity coupling

_{11}*M*, on the other hand, is first-order in the evanescent tail of the first cavity. Thus

_{21}*M*~

_{11}*M*as gap is varied. This leads to the dependence (11). A similar consideration, and the different scaling of bandwidth with μ

_{21}^{2}_{i}leads to the different conclusion for cavity-bus coupling. More generally, the CIFS for a cavity scales inversely with the FSR of adjacent cavities, but does not scale with FSR changes of the cavity itself (for constant bandwidth and coupling geometry).

*1*/

*L*, the effective coupling length to either a cavity or bus waveguide,

_{eff}*M*~

_{11}*M*as gap is varied, while both

_{21}^{2}*M*are linear in interaction length (area). For straight parallel directional couplers,

_{11},M_{21}*L*is the length. For curved couplers (such as ring-bus waveguide couplers), the diverging coupler’s curvature may be represented as a single effective radius

_{eff}*1*/

*R*=

_{eff}*1*/

*R*+

_{1}*1*/

*R*[1

_{2}**15**, 998–1005 (1997). [CrossRef]

*n*in HIC means that a smaller microring radius is required for a given FSR (

_{g}*FSR*=

*c*/2

*π*

*Rn*), leading to a shorter stronger coupler, and greater CBR.

_{g}*μ*~

_{i,j}*μ*for flat-top filters, we may conclude that the cavities that are coupled to access waveguides contribute a greater CIFS than “interior” cavities coupled only to other adjacent cavities. From (11), this will be particularly true in high-finesse filters. To first order, compensating only the outermost rings may thus be sufficient. In our example (Fig. 7), only the central ring was compensated instead.

^{2}_{i}28. M. M. Lee and M. C. Wu, “MEMS-actuated microdisk resonators with variable power coupling ratios,” IEEE Photon. Technol. Lett. **17**, 1034–1036 (2005). [CrossRef]

## 8. Conclusions

*1/finesse*due to cavity-cavity coupling. Theory and rigorous simulations show that CIFS can be positive or negative depending on the dominant of a number of contributing factors including index perturbation, mode non-orthogonality and mode field distortion. Standing- and traveling-wave pictures of the CIFS were considered. A perturbative lossless directional coupler model contains all necessary, and just enough, degrees of freedom to describe CIFS in traveling-wave cavities. A typical microring-resonator filter without CIFS compensation had a severely distorted simulated response. With CIFS compensation, the ideal synthesized response was recovered and verified by simulation. Therefore, CIFS must be rigorously taken into account in filter design, in cases where the resonator frequencies cannot be individually tuned post-fabrication. CIFS-free resonators may be engineered to enable applications such as MEMS-actuated coupling strength control via gap change, without shifting the resonance.

## Appendix A. Simulation methods and case study details

**15**, 998–1005 (1997). [CrossRef]

14. T. Barwicz, M. A. Popović, P. T. Rakich, M. R Watts, H. A. Haus, E. P. Ippen, and H. I. Smith, “Microring-resonator-based add-rop filters in SiN: fabrication and analysis,” Opt. Express **12**, 1437–1442 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-7-1437. [CrossRef] [PubMed]

15. M. A. Popović, M. R. Watts, T. Barwicz, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kärtner, and H. I. Smith, “High-index-contrast, wide-FSR microring-resonator filter design and realization with frequency-shift compensation,” in *Proc. Optical Fiber Comm. Conf*. (Optical Society of America, Washington, DC,2005).

^{9}.

## Appendix B. Vector-field coupled mode theory in time derivation

*H*-fields of the uncoupled resonator modes. The

*E*-fields may be derived from the curl equation of the total system. A choice of

*H*-field basis set then ensures that Gauss’ law is preserved in the supermode trial fields of the coupled configuration. Starting from the vector-wave equation for the total

*H*field,

*n*as the distribution

## Acknowledgments

## References and links:

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

02. | S. Blair and Y. Chen, “Resonant-enhanced evanescent-wave fluorescence biosensing with cylindrical optical cavities,” Appl. Opt. |

03. | P. P. Absil, J. V. Hryniewicz, B. E. Little, P. S. Cho, R A. Wilson, L. G. Joneckis, and P.-T. Ho, “Wavelength conversion in GaAs microring resonators,” Opt. Lett. |

04. | B. Liu, A. Shakouri, and J. E. Bowers, “Passive microring-resonator-coupled lasers,” Appl. Phys. Lett. |

05. | M. Lončar, T. Yoshie, Y. Qiu, P. Gogna, and A. Scherer, “Low-threshold photonic crystal laser,” in Proc. SPIE |

06. | C. K. Madsen and J. H. Zhao, |

07. | S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, “Theoretical analysis of channel drop tunneling processes,” Phys. Rev. B |

08. | H. A. Haus, B. E. Little, M. A. Popović, S. T. Chu, M. R. Watts, and C. Manolatou, “Optical resonators and filters,” in |

09. | H. A. Haus, “Microwaves and Photonics,” in |

10. | M. J. Khan, C. Manolatou, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Mode-coupling analysis of multipole symmetric resonant add/drop filters,” IEEE J. Quantum Electron. |

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

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

13. | C. Manolatou, M. A. Popović, P. T. Rakich, T. Barwicz, H. A. Haus, and E. P. Ippen, “Spectral anomalies due to coupling-induced frequency shifts in dielectric coupled-resonator filters,” in |

14. | T. Barwicz, M. A. Popović, P. T. Rakich, M. R Watts, H. A. Haus, E. P. Ippen, and H. I. Smith, “Microring-resonator-based add-rop filters in SiN: fabrication and analysis,” Opt. Express |

15. | M. A. Popović, M. R. Watts, T. Barwicz, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kärtner, and H. I. Smith, “High-index-contrast, wide-FSR microring-resonator filter design and realization with frequency-shift compensation,” in |

16. | H. A. Haus, |

17. | H. A. Haus and W.-P. Huang, “Coupled-mode theory,” in |

18. | T. Barwicz, M. A. Popović, M. R. Watts, P. T. Rakich, E. P. Ippen, and H. I. Smith, “Fabrication of add-drop filters based on frequency-matched microring resonators,” submitted to J. Lightwave Technol. |

19. | M. Popović, “Complex-frequency leaky mode computations using PML boundary layers for dielectric resonant structures,” in |

20. | A. D. Berk, “Variational principles for electromagnetic resonators and waveguides,” IRE Trans. Antennas Propag., April 1956, pp. 104–111. |

21. | H. A. Haus, W. P. Huang, and A. W. Snyder, “Coupled-mode formulations,” Opt. Lett. |

22. | C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. |

23. | M. J. Khan, M. Lim, C. Joyner, T. Murphy, H. A. Haus, and H. I. Smith, “Integrated Bragg grating structures,” in |

24. | J. Scheuer and A. Yariv, “Two-dimensional optical ring resonators based on radial Bragg resonance,” Opt. Lett. |

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

26. | S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, “Effect of a layered environment on the complex natural frequencies of 2D WGM dielectric-ring resonators,” J. Lightwave Technol. |

27. | B. E. Little, J.-P. Laine, and S. T. Chu, “Surface-roughness-induced contradirectional coupling in ring and disk resonators,” Opt. Lett. |

28. | M. M. Lee and M. C. Wu, “MEMS-actuated microdisk resonators with variable power coupling ratios,” IEEE Photon. Technol. Lett. |

**OCIS Codes**

(130.3120) Integrated optics : Integrated optics devices

(230.5750) Optical devices : Resonators

**ToC Category:**

Optical Devices

**History**

Original Manuscript: November 28, 2005

Revised Manuscript: January 20, 2006

Manuscript Accepted: January 23, 2006

Published: February 6, 2006

**Citation**

Miloš Popovic, Christina Manolatou, and Michael Watts, "Coupling-induced resonance frequency shifts in coupled dielectric multi-cavity filters," Opt. Express **14**, 1208-1222 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-3-1208

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### References

- B. E. Little, S. T. Chu, H. A. Haus, J. Foresi and J.-P. Laine, "Microring resonator channel dropping filters," J. Lightwave Technol. 15, 998-1005 (1997). [CrossRef]
- S. Blair and Y. Chen, "Resonant-enhanced evanescent-wave fluorescence biosensing with cylindrical optical cavities," Appl. Opt. 40, 570-582 (2001). [CrossRef]
- P. P. Absil, J. V. Hryniewicz, B. E. Little, P. S. Cho, R. A. Wilson, L. G. Joneckis and P.-T. Ho, "Wavelength conversion in GaAs microring resonators," Opt. Lett. 25, 554-556 (2000). [CrossRef]
- B. Liu, A. Shakouri and J. E. Bowers, "Passive microring-resonator-coupled lasers," Appl. Phys. Lett. 79, 3561-3563 (2001). [CrossRef]
- M. Lončar, T. Yoshie, Y. Qiu, P. Gogna and A. Scherer, "Low-threshold photonic crystal laser," in Proc. SPIE5000, 16-26 (2003). [CrossRef]
- C. K. Madsen and J. H. Zhao, Optical filter design and analysis: a signal processing approach (Wiley, 1999).
- S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou and H. A. Haus, "Theoretical analysis of channel drop tunneling processes," Phys. Rev. B 59, 15882-15892 (1999). [CrossRef]
- H. A. Haus, B. E. Little, M. A. Popović, S. T. Chu, M. R. Watts and C. Manolatou, "Optical resonators and filters," in Optical Microcavities, K. Vahala, ed. (World Scientific, Singapore, 2004).
- H. A. Haus, "Microwaves and Photonics," in OSA TOPS 23 Symposium on Electro-Optics: Present and Future, H.A. Haus, ed., (Optical Society of America, Washington, DC, 1998), pp. 2-8.
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