## Waveguide self-coupling based reconfigurable resonance structure for optical filtering and delay |

Optics Express, Vol. 19, Issue 9, pp. 8032-8044 (2011)

http://dx.doi.org/10.1364/OE.19.008032

Acrobat PDF (1777 KB)

### Abstract

We propose a novel waveguide self-coupling based reconfigurable resonance structure to work as a flat-top second-order tunable filter and a tunable delay line with low group delay dispersion. The high-order resonance features result from the mutual mode coupling between the clockwise and counter-clockwise resonance eigenmodes. The transfer-matrix method is used to theoretically analyze the device optical performances. The relations between the two embedded phase shifters for achieving flat-top filtering and group delay responses are given. As the coupled resonances are provided by only one physical resonator, the device is inherently more compact and resilient to fabrication errors compared to conventional microring resonators. Phase tuning for its reconfiguration is also simpler and more power-efficient.

© 2011 OSA

## 1. Introduction

1. Y. Vlasov, W. Green, and F. Xia, “High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks,” Nat. Photonics **2**(4), 242–246 (2008). [CrossRef]

2. B. Lee, A. Biberman, N. Sherwood-Droz, C. Poitras, M. Lipson, and K. Bergman, “High-speed 2 × 2 switch for multiwavelength siliconphotonic networks-on-chip,” J. Lightwave Technol. **27**(14), 2900–2907 (2009). [CrossRef]

3. K. Preston, S. Manipatruni, A. Gondarenko, C. B. Poitras, and M. Lipson, “Deposited silicon high-speed integrated electro-optic modulator,” Opt. Express **17**(7), 5118–5124 (2009). [CrossRef] [PubMed]

5. Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature **435**(7040), 325–327 (2005). [CrossRef] [PubMed]

6. B. Little, S. Chu, H. Haus, J. Foresi, and J. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. **15**(6), 998–1005 (1997). [CrossRef]

12. Q. Li, M. Soltani, S. Yegnanarayanan, and A. Adibi, “Design and demonstration of compact, wide bandwidth coupled-resonator filters on a siliconon- insulator platform,” Opt. Express **17**(4), 2247–2254 (2009). [CrossRef] [PubMed]

13. A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. **12**(10), 104008 (2010). [CrossRef]

19. J. Cardenas, M. A. Foster, N. Sherwood-Droz, C. B. Poitras, H. L. Lira, B. Zhang, A. L. Gaeta, J. B. Khurgin, P. Morton, and M. Lipson, “Wide-bandwidth continuously tunable optical delay line using silicon microring resonators,” Opt. Express **18**(25), 26525–26534 (2010). [CrossRef] [PubMed]

20. X. Zheng, I. Shubin, G. Li, T. Pinguet, A. Mekis, J. Yao, H. Thacker, Y. Luo, J. Costa, K. Raj, J. E. Cunningham, and A. V. Krishnamoorthy, “A tunable 1x4 silicon CMOS photonic wavelength multiplexer/demultiplexer for dense optical interconnects,” Opt. Express **18**(5), 5151–5160 (2010). [CrossRef] [PubMed]

21. A. Agarwal, P. Toliver, R. Menendez, S. Etemad, J. Jackel, J. Young, T. Banwell, B. E. Little, S. T. Chu, and W. Chen, “Fully programmable ring-resonator-based integrated photonic circuit for phase coherent applications,” J. Lightwave Technol. **24**(1), 77–87 (2006). [CrossRef]

6. B. Little, S. Chu, H. Haus, J. Foresi, and J. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. **15**(6), 998–1005 (1997). [CrossRef]

14. J. Heebner and R. Boyd, “'Slow' and 'fast' light in resonator-coupled waveguides,” J. Mod. Opt. **49**(14), 2629–2636 (2002). [CrossRef]

10. M. S. Dahlem, C. W. Holzwarth, A. Khilo, F. X. Kärtner, H. I. Smith, and E. P. Ippen, “Reconfigurable multi-channel second-order silicon microring-resonator filterbanks for on-chip WDM systems,” Opt. Express **19**(1), 306–316 (2011). [CrossRef] [PubMed]

12. Q. Li, M. Soltani, S. Yegnanarayanan, and A. Adibi, “Design and demonstration of compact, wide bandwidth coupled-resonator filters on a siliconon- insulator platform,” Opt. Express **17**(4), 2247–2254 (2009). [CrossRef] [PubMed]

19. J. Cardenas, M. A. Foster, N. Sherwood-Droz, C. B. Poitras, H. L. Lira, B. Zhang, A. L. Gaeta, J. B. Khurgin, P. Morton, and M. Lipson, “Wide-bandwidth continuously tunable optical delay line using silicon microring resonators,” Opt. Express **18**(25), 26525–26534 (2010). [CrossRef] [PubMed]

22. J. B. Khurgin, “Expanding the bandwidth of slow-light photonic devices based on coupled resonators,” Opt. Lett. **30**(5), 513–515 (2005). [CrossRef] [PubMed]

23. J. B. Khurgin and P. A. Morton, “Tunable wideband optical delay line based on balanced coupled resonator structures,” Opt. Lett. **34**(17), 2655–2657 (2009). [CrossRef] [PubMed]

24. L. Tobing, S. Darmawan, D. Lim, M. Chin, and T. Mei, “Relaxation of Critical Coupling Condition and Characterization of Coupling-Induced Frequency Shift in Two-Ring Structures,” IEEE J. Sel. Top. Quantum Electron. **16**(1), 77–84 (2010). [CrossRef]

25. M. Popovic, C. Manolatou, and M. Watts, “Coupling-induced resonance frequency shifts in coupled dielectric multi-cavity filters,” Opt. Express **14**(3), 1208–1222 (2006). [CrossRef] [PubMed]

15. Q. Li, Z. Zhang, J. Wang, M. Qiu, and Y. Su, “Fast light in silicon ring resonator with resonance-splitting,” Opt. Express **17**(2), 933–940 (2009). [CrossRef] [PubMed]

26. B. E. Little, J. P. Laine, and S. T. Chu, “Surface-roughness-induced contradirectional coupling in ring and disk resonators,” Opt. Lett. **22**(1), 4–6 (1997). [CrossRef] [PubMed]

28. F. Morichetti, A. Canciamilla, M. Martinelli, A. Samarelli, R. De La Rue, M. Sorel, and A. Melloni, “Coherent backscattering in optical microring resonators,” Appl. Phys. Lett. **96**(8), 081112 (2010). [CrossRef]

## 2. Device structure

29. L. Liao, D. Samara-Rubio, M. Morse, A. Liu, D. Hodge, D. Rubin, U. Keil, and T. Franck, “High speed silicon Mach-Zehnder modulator,” Opt. Express **13**(8), 3129–3135 (2005). [CrossRef] [PubMed]

## 3. Modeling

*i.e.*,

*i.e.*,

*t*=

_{a}^{2}*t*= 0.5, and the transmission and reflection functions are expressed as (in z-domain): where

_{b}^{2}*ϕ*is the round-trip phase change,

*L*=

*L*+

_{1}*L*+

_{2}*L*is the round-trip physical length,

_{3}*t*= cos(Δ

_{io}*ϕ*) and

_{io}*κ*= sin(Δ

_{io}*ϕ*) are the resonator input/output effective transmission and coupling coefficients,

_{io}*t*= cos(Δ

_{x}*ϕ*) and

_{x}*κ*= sin(Δ

_{x}*ϕ*) are the resonator effective mutual-mode transmission and coupling coefficients,

_{x}*ϕ*and Δ

_{io}*ϕ*determine the resonator input/output and crossing coupling strengths, respectively. In silicon, phase shifts can be obtained by using the free carrier plasma dispersion effect [31

_{x}31. R. Soref and B. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. **23**(1), 123–129 (1987). [CrossRef]

## 4. Analysis and discussion

*ϕ*and Δ

_{io}*ϕ*. The resonance structure can be reconfigured for different applications by tuning these two parameters. In the following subsections, we first consider several limiting cases and then analyze its optical performances as a tunable filter and a delay line. The device tolerance to structural parameters and temperature variations is discussed in the last subsection.

_{x}### 4.1 limiting cases

*ϕ*= 0 (or

_{io}*t*= 1), the amplitude responses for the transmission and reflection ports are |

_{io}*H*(

_{t}*z*)| = |

*t*| and |

_{x}*H*(

_{r}*z*)| = |

*κ*|, which implies light is partially reflected (without forming resonances) and its reflectivity is determined by the cross coupling coefficient. When Δ

_{x}*ϕ*varied from 0 to

_{x}*π*/2, the reflection increases from zero to full reflection.

*ϕ*= 0 (or

_{ x}*t*= 1), the crossing coupling is at its maximum and the light just crosses over without inducing backward coupling. The transmission function can then be simplified towhich is essentially the transfer function for a microring notch filter cascaded with a waveguide. Resonances occur at

_{x}*ϕ*= 2

*mπ*(

*m*is an integer) and the effective resonator round-trip length is

*L*. The waveguide-resonator coupling coefficient is

*κ*, which can be controlled by tuning Δ

_{io}*ϕ*. A typical transmission spectrum is shown in Fig. 2(a) . As the phase changes rapidly around resonance, a large group delay can be obtained at the resonance frequency. Group delay is the first order derivative of the phase with respect to angular frequency. Group delay can be positive or negative, corresponding to slow light (pulse delay) or fast light (pulse advance) phenomenon. Letting Φ be the phase of the transmission function, then the group delay is

_{x}*τ*= -dΦ /d

_{g}*ω*= -(dΦ /d

*ϕ*)

*T*, where

_{R},*T*= (

_{R}*n*+

_{eff}*ω*d

*n*/d

_{eff}*ω*)

*L*/

*c*=

*n*/

_{g}L*c*is the light round-trip transmission time,

*n*is the waveguide group refractive index, and

_{g}*ω*is the angular frequency. Figure 2(b) shows the normalized group delay (

*τ*/

_{g}*T*) for the transmission port. As only a single resonance is formed in the resonator, the intensity and group delay spectra both exhibit sharp peaks.

_{R}*ϕ*=

_{io}*π*/2 (or

*t*= 0), the device transfer functions then become which are essentially the transfer functions for an add-drop filter (ADF) cascaded with a waveguide (reflection port has an extra phase shift Δ

_{io}*ϕ*). Resonances occur at

_{io}*ϕ*=

*mπ*-

*π*/2. Its free spectral range (FSR) is a half smaller than that in the previous case. It means that in this configuration the resonator effective circumference is 2

*L*. Figures 2(c) and 2(d) show the intensity and group delay spectra for a typical set of parameters.

*ϕ*=

_{x}*π*/2 (or

*t*= 0), all the light is reflected back at the crossing coupler and the device is essentially composed by two separate Sagnac loop mirrors. If the input and output couplers are ideal 3-dB couplers, then all the light can be reflected back to the input-port.

_{x}### 4.2 Tunable filter

*ϕ*and Δ

_{x}*ϕ*are small, the mutual mode coupling between the CW and CCW resonance modes and the coupling to the external access waveguide are both weak, and then we can configure the device as a second-order filter. The complex poles of the transfer equations Eqs. (6) and (7) are essentially the complex eigenmodes of the device. The complex phase of the pole can be written as

_{io}*ϕ*and

_{p0}*γ*are associated with the real-valued resonance frequency and linewidth:

*i.e.*, 2

*ϕ*<

_{p0}*γ*, these two modes are regarded as undistinguishable. Under weak coupling (

*t*≈ 1 and

_{io}*t*≈ 1), the condition for the undistinguishable eigenmodes can be approximated to

_{x}*k*and

_{io}*k*should satisfy a more stringent requirement. Stating from the pole and zero positions and their function on reflection spectrum, we deduce the relation between the poles and zeros required to generate a flat-top passband. The reflection transfer function has two conjugate poles (

_{x}*p*and

_{0}*p*) and two conjugate zeros (

_{0}**z*and

_{0}*z*):

_{0}**at*, but the zeros have a larger imaginary part than the poles have, meaning that the zeros are located outside the poles in the pole-zero diagram,

_{x}t_{io}*i.e.*,

*ϕ*>

_{z0}*ϕ*. Thus, the addition of zeros makes the reflection passband smoother and also drop faster. To get an exact flat passband, the second-order derivative of the intensity transfer function should be zero at the uncoupled resonance frequency,

_{p0}*i.e.*,

*d*

^{2}[|

*H*(

_{r}*z*)|

^{2}]/

*dϕ*

^{2}= 0 at

*ϕ*= 0. After mathematical treatment, we get the following condition for the flat-top passband response: Equation (16) states the relationship between the poles and zeros, which can be converted to an expression for tuning parameters Δ

*ϕ*and Δ

_{io}*ϕ*.

_{x}*ϕ*and Δ

_{io}*ϕ*for various loss factors to satisfy the flat-top passband response. The addition of the waveguide propagation loss up-shifts the Δ

_{x}*ϕ*versus Δ

_{x}*ϕ*curve, which means strong mutual mode coupling is needed to compensate for the loss-induced resonance broadening. The solid curves in Fig. 3(b) show the reflection spectra of our device for three different Δ

_{io}*ϕ*’s under the flat-top condition. The intensity remains almost flat between the pole frequencies. Beyond the pole frequencies, the intensity drops rapidly and reaches the minimum around the zero frequencies. Beyond the zeros, the intensity increases slightly. To compare with a conventional microring ADF, we also plot the drop spectra of a double-ring ADF (dotted curves) with its waveguide-ring and ring-ring coupling coefficients being

_{io}*κ*and

_{io}*κ*, respectively. From the comparison, we see that our device has a faster roll-off but a lower out-of-band rejection ratio because of the excess zeros in the transfer function.

_{x}*ϕ*, larger waveguide loss slightly increases the reflection bandwidth but also deteriorates the filtering performance in terms of insertion loss and cross-talk level as seen from Figs. 3(c) and 3(d). For narrow-bandwidth filters (Δ

_{io}*ϕ*is small), loss can more significantly degrade its performances, because a narrow bandwidth means a high-Q resonance and hence it is more sensitive to loss.

_{io}### 4.3 Tunable delay line

*t*in Eq. (7), the reflection group delay at the uncoupled resonance frequency (the operating frequency) can be expressed aswhere

_{wg}*r*= Re(

_{0}*z*) = Re(

_{0}*p*) =

_{0}*at*is the real part of the zeros and poles. Note that |

_{io}t_{x}*p*| < |

_{0}*z*|, and therefore, the group delay is always positive. Similarly, group delay can also be achieved from the transmission port. In the following analysis, we only focus on the reflection port, as it has a low insertion loss and can simultaneously perform channel dropping and delay functions.

_{0}*i.e.*,

*d*

^{3}Φ

_{r}(

*ϕ*)/

*dϕ*

^{3}= 0 at

*ϕ*= 0, where Φ

_{r}(

*ϕ*) is the phase of

*H*(

_{r}*z*). From Eq. (7) and after some mathematical manipulation, we get the condition for the flat-top group delay response:For the lossless case (

*a*= 1), the above equation can be simplified to

*p*with Eq. (14), Eq. (19) becomesIn terms of Δ

_{0}*ϕ*and Δ

_{x}*ϕ*, Eq. (20) can be rewritten asHence, for low loss and high optical delay cases, Δ

_{io}*ϕ*and Δ

_{x}*ϕ*have a simple relation given by Eq. (21).

_{io}*ϕ*on Δ

_{x}*ϕ*for different loss factors based on Eq. (18). Δ

_{io}*ϕ*is almost linearly proportional to Δ

_{x}*ϕ*in a small range of interest, which implies that Δ

_{io}*ϕ*and Δ

_{x}*ϕ*can be simultaneously tuned by one external voltage as long as their corresponding active region lengths are in scale. This can greatly simply the drive circuit for group delay tuning. Figure 4(b) shows the insertion loss changes as a function of Δ

_{io}*ϕ*. Higher waveguide loss increases the insertion loss, similar to the flat-top filter case. Yet distinct from the previous case, larger Δ

_{io}*ϕ*can also lead to a higher insertion loss, more evident for the lossless case (

_{io}*a*= 1), as not all light is reflected back at the operating frequency. Figures 4(c) and 4(d) show the group delay bandwidth (approximately the filter bandwidth) and group delay (at the operating frequency) change as functions of Δ

*ϕ*. A large group delay can be obtained by using smaller Δ

_{io}*ϕ*and Δ

_{io}*ϕ*yet with a sacrifice of smaller bandwidth, limited by the so-called delay-bandwidth product in resonance systems [32

_{x}32. G. Lenz, B. Eggleton, C. Madsen, and R. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. **37**(4), 525–532 (2001). [CrossRef]

*ϕ*and Δ

_{io}*ϕ*be small, the loss should also be kept low enough.

_{x}*ϕ*= 0.06

_{io}*π*, 0.08

*π*and 0.1

*π*. As the third-order GDD is eliminated at the operating frequency, the group delay is almost flat within its bandwidth, and it can be continuously tuned for a relatively large range. For instance, if the device size is

*L*=

*L*+

_{1}*L*+

_{2}*L*= 60 μm, the waveguide group refractive index

_{3}*n*= 4.0, and the loss factor

_{g}*a*= 0.997 (corresponding to a 5 dB/cm average waveguide propagation loss), then FSR ≈ 10 nm and

*T*≈ 0.8 ps at 1.55 μm wavelength, and the group delay thus can be continuously tuned for 28 ps with its bandwidth kept >10 GHz.

_{R}31. R. Soref and B. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. **23**(1), 123–129 (1987). [CrossRef]

*e.g.*, waveguide

*L*) in our structure, the power consumption spent on the frequency tuning is also one half smaller for our structure.

_{1}### 4.4 Tolerance to structural parameters

*ϕ*= 0.08π and Δ

_{io}*ϕ*= 0.011π and vary the input and output coupling ratios. Figures 6(a) -6(c) show the reflection spectra for various combinations of

_{x}*t*

_{a}^{2}and

*t*

_{b}^{2}. When the input and output coupling ratios are balanced,

*i.e.*,

*t*

_{a}^{2}=

*t*

_{b}^{2}, the filter spectrum is always symmetric, as shown in Fig. 6(a). The imperfection of the input and output couplers (non-ideal 3-dB couplers) causes the filter passband rounded and broadened. If only one of the couplers is imperfect, then the passband becomes slightly rounded and inclines towards one side, with its central wavelength shifts a little bit, as shown in Fig. 6(b). The most interesting combination is when the couplers are complementary,

*i.e.*,

*t*

_{a}^{2}+

*t*

_{b}^{2}= 1. In this case, the passband top remains almost flat while it inclines towards one side, as shown in Fig. 6(c). From these plots, it can be seen that if the variation of

*t*

_{a}^{2}and

*t*

_{b}^{2}is within ±0.1, the high-order filtering performance is still reasonably good.

*ϕ*= 0.08π and Δ

_{io}*ϕ*= 0.007π. The imperfection of the couplers also degrades the delay line performance with a similar trend as in the filter case. For the complimentary combination of

_{x}*t*

_{a}^{2}and

*t*

_{b}^{2}(Fig. 6(f)), the larger the difference between

*t*

_{a}^{2}and

*t*

_{b}^{2}is, the sharper the group delay curve becomes. As the passband inclines to one side, the corresponding transition band becomes narrower (faster roll-off) and thus the phase change becomes more rapid, leading to a high group delay. It can be seen that the group delay is still within an acceptable level, if the coupling ratio change is within ±0.1.

*κ*to only value in a limited range, or in other words,

_{x}*κ*cannot be very small and thus the filter and delay line bandwidth cannot be very narrow. However, as long as Eqs. (16) and (18) are satisfied, the flat-top passband and group delay responses still can be obtained.

_{x}33. L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. **13**(4), 615–627 (1995). [CrossRef]

*n*and the coupling coefficients. Due to the symmetry of the structure, the interference paths (except the resonance loop back interference) are all balanced, and therefore, the change in

_{eff}*n*causes the intensity and group delay spectra to only experience a shift without any change in their profiles. The coupling change has a more negative effect on the device performances as discussed previously. For a typical silicon wire waveguide-based 3-dB directional coupler, our simulation based on beam propagation method (BPM) shows that temperature increases by 10 °C, coupling ratio only changes by <1%, which has an almost negligible effect according to Fig. 6. To stabilize the device over temperature variation, athermal waveguide design can be employed [34

_{eff}34. L. Zhou, K. Okamoto, and S. Yoo, “Athermalizing and trimming of slotted silicon microring resonators with UV-sensitive PMMA upper-cladding,” IEEE Photon. Technol. Lett. **21**(17), 1175–1177 (2009). [CrossRef]

## 5. Conclusion

## Acknowledgement

## References and links

1. | Y. Vlasov, W. Green, and F. Xia, “High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks,” Nat. Photonics |

2. | B. Lee, A. Biberman, N. Sherwood-Droz, C. Poitras, M. Lipson, and K. Bergman, “High-speed 2 × 2 switch for multiwavelength siliconphotonic networks-on-chip,” J. Lightwave Technol. |

3. | K. Preston, S. Manipatruni, A. Gondarenko, C. B. Poitras, and M. Lipson, “Deposited silicon high-speed integrated electro-optic modulator,” Opt. Express |

4. | G. Reed, G. Mashanovich, F. Gardes, and D. Thomson, “Silicon optical modulators,” Nat. Photonics |

5. | Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature |

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

7. | L. Zhou and A. W. Poon, “Fano resonance-based electrically reconfigurable add-drop filters in silicon microring resonator-coupled Mach-Zehnder interferometers,” Opt. Lett. |

8. | L. Zhou and A. W. Poon, “Electrically reconfigurable silicon microring resonator-based filter with waveguide-coupled feedback,” Opt. Express |

9. | J. E. Cunningham, I. Shubin, X. Zheng, T. Pinguet, A. Mekis, Y. Luo, H. Thacker, G. Li, J. Yao, K. Raj, and A. V. Krishnamoorthy, “Highly-efficient thermally-tuned resonant optical filters,” Opt. Express |

10. | M. S. Dahlem, C. W. Holzwarth, A. Khilo, F. X. Kärtner, H. I. Smith, and E. P. Ippen, “Reconfigurable multi-channel second-order silicon microring-resonator filterbanks for on-chip WDM systems,” Opt. Express |

11. | P. Dong, N. N. Feng, D. Feng, W. Qian, H. Liang, D. C. Lee, B. J. Luff, T. Banwell, A. Agarwal, P. Toliver, R. Menendez, T. K. Woodward, and M. Asghari, “GHz-bandwidth optical filters based on high-order silicon ring resonators,” Opt. Express |

12. | Q. Li, M. Soltani, S. Yegnanarayanan, and A. Adibi, “Design and demonstration of compact, wide bandwidth coupled-resonator filters on a siliconon- insulator platform,” Opt. Express |

13. | A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. |

14. | J. Heebner and R. Boyd, “'Slow' and 'fast' light in resonator-coupled waveguides,” J. Mod. Opt. |

15. | Q. Li, Z. Zhang, J. Wang, M. Qiu, and Y. Su, “Fast light in silicon ring resonator with resonance-splitting,” Opt. Express |

16. | X. Luo, H. Chen, and A. W. Poon, “Electro-optical tunable time delay and advance in silicon microring resonators,” Opt. Lett. |

17. | A. Melloni, F. Morichetti, C. Ferrari, and M. Martinelli, “Continuously tunable 1 byte delay in coupled-resonator optical waveguides,” Opt. Lett. |

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

19. | J. Cardenas, M. A. Foster, N. Sherwood-Droz, C. B. Poitras, H. L. Lira, B. Zhang, A. L. Gaeta, J. B. Khurgin, P. Morton, and M. Lipson, “Wide-bandwidth continuously tunable optical delay line using silicon microring resonators,” Opt. Express |

20. | X. Zheng, I. Shubin, G. Li, T. Pinguet, A. Mekis, J. Yao, H. Thacker, Y. Luo, J. Costa, K. Raj, J. E. Cunningham, and A. V. Krishnamoorthy, “A tunable 1x4 silicon CMOS photonic wavelength multiplexer/demultiplexer for dense optical interconnects,” Opt. Express |

21. | A. Agarwal, P. Toliver, R. Menendez, S. Etemad, J. Jackel, J. Young, T. Banwell, B. E. Little, S. T. Chu, and W. Chen, “Fully programmable ring-resonator-based integrated photonic circuit for phase coherent applications,” J. Lightwave Technol. |

22. | J. B. Khurgin, “Expanding the bandwidth of slow-light photonic devices based on coupled resonators,” Opt. Lett. |

23. | J. B. Khurgin and P. A. Morton, “Tunable wideband optical delay line based on balanced coupled resonator structures,” Opt. Lett. |

24. | L. Tobing, S. Darmawan, D. Lim, M. Chin, and T. Mei, “Relaxation of Critical Coupling Condition and Characterization of Coupling-Induced Frequency Shift in Two-Ring Structures,” IEEE J. Sel. Top. Quantum Electron. |

25. | M. Popovic, C. Manolatou, and M. Watts, “Coupling-induced resonance frequency shifts in coupled dielectric multi-cavity filters,” Opt. Express |

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

27. | B. E. Little, S. T. Chu, and H. A. Haus, “Second-order filtering and sensing with partially coupled traveling waves in a single resonator,” Opt. Lett. |

28. | F. Morichetti, A. Canciamilla, M. Martinelli, A. Samarelli, R. De La Rue, M. Sorel, and A. Melloni, “Coherent backscattering in optical microring resonators,” Appl. Phys. Lett. |

29. | L. Liao, D. Samara-Rubio, M. Morse, A. Liu, D. Hodge, D. Rubin, U. Keil, and T. Franck, “High speed silicon Mach-Zehnder modulator,” Opt. Express |

30. | C. Madsen, and J. Zhao, |

31. | R. Soref and B. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. |

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

33. | L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. |

34. | L. Zhou, K. Okamoto, and S. Yoo, “Athermalizing and trimming of slotted silicon microring resonators with UV-sensitive PMMA upper-cladding,” IEEE Photon. Technol. Lett. |

**OCIS Codes**

(230.3120) Optical devices : Integrated optics devices

(230.5750) Optical devices : Resonators

(130.7408) Integrated optics : Wavelength filtering devices

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: January 13, 2011

Revised Manuscript: April 1, 2011

Manuscript Accepted: April 1, 2011

Published: April 12, 2011

**Citation**

Linjie Zhou, Tong Ye, and Jianping Chen, "Waveguide self-coupling based reconfigurable resonance structure for optical filtering and delay," Opt. Express **19**, 8032-8044 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-9-8032

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