## Fano resonances in a multimode waveguide coupled to a high-Q silicon nitride ring resonator |

Optics Express, Vol. 22, Issue 6, pp. 6778-6790 (2014)

http://dx.doi.org/10.1364/OE.22.006778

Acrobat PDF (3800 KB)

### Abstract

Silicon nitride (Si_{3}N_{4}) optical ring resonators provide exceptional opportunities for low-loss integrated optics. Here we study the transmission through a multimode waveguide coupled to a Si_{3}N_{4} ring resonator. By coupling single-mode fibers to both input and output ports of the waveguide we selectively excite and probe combinations of modes in the waveguide. Strong asymmetric Fano resonances are observed and the degree of asymmetry can be tuned through the positions of the input and output fibers. The Fano resonance results from the interference between modes of the waveguide and light that couples resonantly to the ring resonator. We develop a theoretical model based on the coupled mode theory to describe the experimental results. The large extension of the optical modes out of the Si_{3}N_{4} core makes this system promising for sensing applications.

© 2014 Optical Society of America

## 1. Introduction

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

2. 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]

3. V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature **431**, 1081–1084 (2004). [CrossRef] [PubMed]

4. A. Ksendzov and Y. Lin, “Integrated optics ring-resonator sensors for protein detection,” Opt. Lett. **30**, 3344–3346 (2005). [CrossRef]

5. S. Fan, “Sharp asymmetric line shapes in side-coupled waveguide-cavity systems,” Appl. Phys. Lett. **80**, 908–910 (2002). [CrossRef]

6. C.-Y. Chao and L. J. Guo, “Biochemical sensors based on polymer microrings with sharp asymmetrical resonance,” Appl. Phys. Lett. **83**, 1527–1529 (2003). [CrossRef]

7. A. Chiba, H. Fujiwara, J. Hotta, S. Takeuchi, and K. Sasaki, “Fano resonance in a multimode tapered fiber coupled with a microspherical cavity,” Appl. Phys. Lett. **86**, 261106 (2005). [CrossRef]

9. A. C. Ruege and R. M. Reano, “Multimode waveguide-cavity sensor based on fringe visibility detection,” Opt. Express **17**, 4295–4305 (2009). [CrossRef] [PubMed]

10. B.-B. Li, Y.-F. Xiao, C.-L. Zou, Y.-C. Liu, X.-F. Jiang, Y.-L. Chen, Y. Li, and Q. Gong, “Experimental observation of Fano resonance in a single whispering-gallery microresonator,” Appl. Phys. Lett. **98**, 021116 (2011). [CrossRef]

11. Y.-F. Xiao, L. He, J. Zhu, and L. Yang, “Electromagnetically induced transparency-like effect in a single polydimethylsiloxane-coated silica microtoroid,” Appl. Phys. Lett. **94**, 231115 (2009). [CrossRef]

12. B.-B. Li, Y.-F. Xiao, C.-L. Zou, X.-F. Jiang, Y.-C. Liu, F.-W. Sun, Y. Li, and Q. Gong, “Experimental controlling of Fano resonance in indirectly coupled whispering-gallery microresonators,” Appl. Phys. Lett. **100**, 021108 (2012). [CrossRef]

7. A. Chiba, H. Fujiwara, J. Hotta, S. Takeuchi, and K. Sasaki, “Fano resonance in a multimode tapered fiber coupled with a microspherical cavity,” Appl. Phys. Lett. **86**, 261106 (2005). [CrossRef]

8. A. C. Ruege and R. M. Reano, “Sharp Fano resonances from a two-mode waveguide coupled to a single-mode ring resonator,” J. Lightwave Technol. **28**, 2964–2968 (2010). [CrossRef]

9. A. C. Ruege and R. M. Reano, “Multimode waveguide-cavity sensor based on fringe visibility detection,” Opt. Express **17**, 4295–4305 (2009). [CrossRef] [PubMed]

_{3}N

_{4}) core embedded in silicon dioxide (SiO

_{2}) cladding. The mode selectivity is achieved by positioning and butt-coupling single-mode fibers to the multimode waveguide at both ends. We observe the Fano resonance with various degree of asymmetry when the two fibers are independently scanned along the core width direction. Advantages and limitations of this system and the Fano resonance scheme for sensing applications are discussed.

## 2. Theoretical model

*A*and

_{j}*B*, one obtains an input-output relation of the form

_{j}*h*(

*x*,

*y*) couples to each of the Eigenmodes of the waveguide. The coupling efficiency to the

*j*th order Eigenmode of the waveguide

*η*is where (

_{j}*x′*,

*y′*) are the coordinates of the center of the input coupling fiber. These Eigenmode components of the waveguide propagate from the input port to the coupling region and gain different phase factors, therefore where

*P*

_{in}is the power through the input fiber,

*β′*is the propagation constant of the

_{j}*j*th order Eigenmode of the waveguide,

*L*

_{1}is the distance from the input port to the coupling region of the waveguide, and the waveguide is assumed to be lossless. The total field at the output port of the waveguide is where

*L*

_{2}is the distance from the coupling region to the output port of the waveguide. At the output port of the waveguide, a single-mode fiber with the same field profile as the input coupling fiber couples to the output field. The power through the output fiber

*P*

_{out}is then where (

*x″*,

*y″*) are the coordinates of the center of the output coupling fiber and

*S*(

_{j}*x″*,

*y″*)

*=*

*η*(

_{j}*x″*,

*y″*)exp(−

*iβ′*

_{j}L_{2}). Eventually, the power transmission through the input-output fibers is given by By using the explicit form of

*B*in Eq. (6) and

_{j}*β*in Eq. (5), the power transmission

_{k}*T*becomes where

*e*= 4

_{k}*π*

^{2}

*n*(

_{pk}*ν*)

*R/c*and Δ

*ν*is frequency detuning with respect to a reference frequency

*ν*

_{0}. Note that

*P*

_{in}cancels out in Eq. (12).

*j*th order Eigenmode of the ring resonator, resonances occur when with

*m*being an integer. The resonant frequency detunings of the

*j*th order Eigenmode of the ring resonator are The free spectral range (FSR) of the

*j*th order Eigenmode of the ring resonator is defined as the frequency difference between two adjacent resonant frequencies. If one assumes linear dispersion relations of the materials within one FSR (typically tens of GHz for a ring resonator), the FSR can be expressed as where

*n*(

_{gj}*ν*) =

*n*(

_{pj}*ν*) +

*ν*(d

*n*(

_{pj}*ν*)/d

*ν*) is the effective group index of the

*j*th order Eigenmode of the ring resonator at frequency

*ν*. In the experiment, we will use Eq. (15) to extract effective group indices from measured FSRs.

*j*th order Eigenmode Δ

*ν*

_{0j}, the terms for the other Eigenmodes in the summation in Eq. (12) are slowly varying functions with respect to Δ

*ν*and might be approximated as complex functions

*q*

_{k}_{(k≠j)}.

*q*is not a function of Δ

_{k}*ν*, but still a function of (

*x′*,

*y′*) and (

*x″*,

*y″*). Under this simplification, Eq. (12) reads where

*x′*,

*y′*) and (

*x″*,

*y″*). The exponential term in Eq. (16) can be further simplified by the Taylor expansion with respect to Δ

*ν*

_{0j}and keeping (Δ

*ν*− Δ

*ν*

_{0j}) up to the second power: where

*ζ*,

_{j}*α*,

_{j}*ϕ*and

_{j}*θ*are real and dimensionless functions of (

_{j}*x′*,

*y′*) and (

*x″*,

*y″*) that characterize the profile of the transmission spectra (see below). Their explicit expressions in terms of the coefficients in Eq. (16) are tedious and not given here. The expression of

*γ*is simple: Equation (17) will be used to analyze our experimental data. We note that Eq. (17) and Eq. (18) are also valid for the coupling of a multimode waveguide to a single-mode ring resonator.

_{j}*α*,

_{j}*ϕ*and

_{j}*θ*in Eq. (17) are under-determined and extra constraint conditions can be added. In analogy to [14

_{j}14. S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A **20**, 569–572 (2003). [CrossRef]

*ϕ*< 1 and −1 <

_{j}*θ*< 1. The term in the square bracket of Eq. (17) describes a lineshape with amplitude in between zero and one. For

_{j}*θ*= 0 and

_{j}*ϕ*= 1, the lineshape is a normal Lorentzian dip, while for

_{j}*θ*= ±1 and

_{j}*ϕ*= 0, the lineshape turns into a Lorentzian peak. In between it gives an asymmetric lineshape, with maximum asymmetry for

_{j}*θ*result in mirror-symmetric lineshapes.

_{j}*α*and

_{j}*ζ*rescale the lineshape, but the degree of asymmetry remains the same. The coefficient

_{j}*θ*will be referred to as the Fano parameter, since its absolute value measures the degree of asymmetry of the lineshape and its sign determines one of the two mirror-symmetric lineshapes.

_{j}*γ*in Eq. (18) is a constant, while

_{j}*ζ*,

_{j}*α*,

_{j}*ϕ*and

_{j}*θ*vary with the input and output fiber positions. By tuning the input and output fiber positions with respect to the waveguide, it is possible to achieve

_{j}*θ*= 0 and

_{j}*ϕ*= 1 where the transmission lineshape of the

_{j}*j*th order Eigenmode of the ring resonator becomes a Lorentzian function. The quality factor of the Lorentzian lineshape is given by which is a function of internal and coupling losses and is independent of the input and output fiber positions. Consequently, when the input and output fiber positions are scanned with respect to the waveguide, the transmission lineshape varies, but the quality factor remains the same. Therefore Eq. (20) is also valid for the Fano resonance. We emphasize that the above assertion is based on the assumption that the resonances of the ring resonator are narrow enough to be well separated in frequency domain and is always valid for a single-mode ring resonator.

*θ*= 0 or 1 in Eq. (17)), the slope responsivity of the transmission spectrum defined as |

_{j}*dT/d*Δ

*ν*|

_{max}is

*α*, when Δ

_{j}/γ_{j}*ν*= Δ

*ν*

_{0j}. Thus for the same

*α*and

_{j}*γ*, the Fano resonance with the maximum asymmetry improves the slope responsivity by a factor of 1.54 compared to the Lorentzian resonance.

_{j}*A*=

_{j}*B*. Combining Eq. (7), Eq. (8) and Eq. (11), and using

_{j}*A*=

_{j}*B*, we obtain the off-resonance transmission for a given input and output positions of the fibers Since

_{j}*h*(

*x*,

*y*) is an even function and

*f*(

_{j}*x*,

*y*) is either an even or an odd function, it can be proved that

*T*

_{off}is invariant upon the transformation

*x′*↔

*x″*and

*x′*↔ −

*x″*. Hence

*x′*=

*x″*and

*x′*= −

*x″*are two symmetry axes of

*T*

_{off}in the coordinate system of (

*x″*,

*x′*). This property will be used in Section 5 to determine the central position of the waveguide.

## 3. Sample description

_{3}N

_{4}core, we assume that the modes of the ring resonator can be approximated by the modes of the straight waveguide, i.e.

*f*(

_{j}*x*,

*y*) ≃

*g*(

_{j}*x*,

*y*). If one assumes that the refractive index of the thin Si

_{3}N

_{4}core is equal to that of bulk amorphous Si

_{3}N

_{4}(

*n*= 1.99) and that the refractive index of the SiO

_{2}cladding is 1.45 at 978 nm wavelength, the waveguide modes can be calculated by the finite element method (FEM) using COMSOL (COMSOL Inc.). The FEM simulation shows two transverse magnetic (TM, polarized along the core thickness, or

*y*-direction) modes (TM

_{0}and TM

_{1}) with effective phase indices of 1.4576 and 1.4530 and three transverse electric (TE, polarized along the core width, or

*x*-direction) modes (TE

_{0}, TE

_{1}and TE

_{2}) with effective phase indices of 1.4768, 1.4703, 1.4600, respectively. We note that the effective phase indices of the fundamental TM and TE modes calculated by the effective index method [17] are 1.4578 and 1.4770, respectively, in good agreement with the FEM simulations. The possibility to use the effective index method greatly facilitates design and optimization of such waveguides with a high aspect-ratio. The calculated mode profiles are shown in Fig. 2. Due to much smaller dimension of the Si

_{3}N

_{4}core in the

*y*-direction compared to the wavelength, the waveguide shows single-mode properties in the

*y*-direction and multimode properties in the

*x*-direction.

_{3}N

_{4}the effective group indices are also calculated by the FEM. The results of the two TM modes are 1.4895 and 1.4866 and those of the three TE modes are 1.5478, 1.5502, 1.5504, respectively. Nevertheless, we do not expect a detailed agreement of the calculated effective group indices with the experiment because the actual effective group indices depend on the stress-optic effect and changes in Si

_{3}N

_{4}refractive index and thickness due to the annealing process [18

18. J. F. Bauters, M. J. R. Heck, D. D. John, J. S. Barton, C. M. Bruinink, A. Leinse, R. G. Heideman, D. J. Blumenthal, and J. E. Bowers, “Planar waveguides with less than 0.1 dB/m propagation loss fabricated with wafer bonding,” Opt. Express **19**, 24090–24101 (2011). [CrossRef] [PubMed]

## 4. Experimental method

*μ*m at 980 nm wavelength are used to butt-couple light in and out of the waveguide structure. To this end, both fibers are mounted on

*xyz*-translation stages. The two translation stages are motorized in the

*x*-direction to scan along the core of the waveguide.

*V*to control the frequency detuning and records the signals on PD1, PD2 and PD3 as functions of

_{S}*V*. The laser frequency detuning Δ

_{S}*ν*as a function of the voltage

*V*can be well approximated by a quadratic relation. The coefficients of this quadratic function can be found by fitting to the scan voltages corresponding to maximum transmission of the FPI, using known FSR of 4.00 GHz of the FPI. From this transmission spectra as a function of the frequency detuning of the laser are obtained.

_{S}## 5. Results and discussion

_{3}N

_{4}core caused by the annealing process [18

18. J. F. Bauters, M. J. R. Heck, D. D. John, J. S. Barton, C. M. Bruinink, A. Leinse, R. G. Heideman, D. J. Blumenthal, and J. E. Bowers, “Planar waveguides with less than 0.1 dB/m propagation loss fabricated with wafer bonding,” Opt. Express **19**, 24090–24101 (2011). [CrossRef] [PubMed]

*p*,

*ν*

_{0j}and

*d*are real coefficients,

_{j}*b*are complex coefficients, and

_{j}*e*are fixed parameters calculated by the effective phase indices. The three resonances demonstrate different degree of asymmetry in their lineshapes. Although Eq. (12) is derived under the assumption that only modes with one polarization are excited, we can prove that fitting Eq. (12) to the transmission data with modes in two orthogonal polarizations gives correct parameters

_{j}*a*and hence correct quality factors. The quality factors that can be extracted from the fit (see Eq. (20), where

_{j}t′_{j}*a*≡

_{j}t′_{j}*d*) are (8.1 ± 0.2)×10

_{j}^{6}, (1.9±0.1)×10

^{7}and (6.9±0.2)×10

^{6}for resonances 1, 2 and 3, respectively. We attribute resonance 1 to the TM

_{0}mode (mode profile in Fig. 2(b)) and resonances 2 and 3 to the TE

_{0}mode (lower coupling and bending losses, therefore higher quality factor; mode profile in Fig. 2(d)) and TE

_{1}mode (mode profile in Fig. 2(e)), respectively. Resonances of other modes can not be recognized in the transmission measurements due to their low contrasts.

*y*-direction (see Fig. 2), moving the coupling fibers along the

*y*-direction changes the overall coupling efficiency, but hardly affects the relative weight of different modes excited and collected in the waveguide given that the confinement in the

*y*-direction only very weakly depends on the mode. As a consequence, we observe that the transmission lineshape and Fano resonances are independent of the fiber positions in the

*y*-direction. Therefore in the experiment both of the input and output fibers are aligned to the center of the waveguide in the

*y*-direction (

*y*=

*y′*= 0) by maximizing the transmission amplitude and are only scanned in the

*x*-direction.

*T*/d

*ν*|

_{max}) of the four transmission spectra are 4.23, 4.90, 5.35 and 0.22 GHz

^{−1}, respectively. From the fits, the Fano parameter

*θ*is extracted and the off-resonant transmission

*T*

_{off}is obtained as

*T*

_{off}as a function of the input and output fiber positions (

*x′*and

*x″*, respectively) on a logarithmic scale. By using the symmetric property of

*T*

_{off}(Eq. (21)), two symmetry axes,

*x′*=

*x″*and

*x′*= −

*x″*, are determined and shown as dashed lines in Fig. 7. The intersection of the two symmetry axes corresponds to the center of the waveguide and is defined as the origin of the input and output fiber positions (

*x′*=

*x″*= 0). Close to the edges of Fig. 7, the TM

_{1}mode of the waveguide is more strongly excited or more efficiently collected because of its larger spatial extent. Consequently, valley I can be understood as follows: the TM

_{0}mode is weakly excited, while the output fiber only couples to this mode (the TM

_{1}mode is orthogonal to the fiber mode). Similarly, at valley II, only the TM

_{0}mode is strongly excited while the output fiber couples weakly to this mode.

## 6. Potential application

_{3}N

_{4}core (see Fig. 2). By using thin top cladding, the mode field of the ring resonator can interact with the environment very efficiently. We have fabricated low-loss waveguides and ring resonators with thin top cladding as an intermediate step before wafer bonding [18

18. J. F. Bauters, M. J. R. Heck, D. D. John, J. S. Barton, C. M. Bruinink, A. Leinse, R. G. Heideman, D. J. Blumenthal, and J. E. Bowers, “Planar waveguides with less than 0.1 dB/m propagation loss fabricated with wafer bonding,” Opt. Express **19**, 24090–24101 (2011). [CrossRef] [PubMed]

^{7}) has the lowest transmission contrast. Analysis of losses of resonance 2 compared with resonances 1 and 3 indicates that resonance 2 is over-coupled to the waveguide. Therefore by increasing the coupling distance between the waveguide and the ring resonator in the fabrication process, resonance 2 can be optimized to critical coupling with the maximum contrast and a higher quality factor (> 1.9 × 10

^{7}).

^{−1}, while the maximum slope responsivity of the system is 34.37 GHz

^{−1}obtained from a symmetric lineshape when both the input and output fibers are aligned to the center of the waveguide. However one should realize that the light that is not recorded in the Fano resonance is still available and also carries information on the resonance of the ring resonator. A more detailed analysis considering detecting all the output power of the waveguide is needed to investigate whether there can be an overall gain in sensitivity compared to a single-mode-waveguide-ring-resonator configuration. However this is beyond the scope of this article.

## 7. Conclusion

_{3}N

_{4}waveguide coupled to a high-Q Si

_{3}N

_{4}ring resonator at 978 nm wavelength. By scanning the positions of single-mode fibers at both input and output ports of the waveguide, we selectively excite and probe combinations of the modes in the waveguide. Fano resonances are observed as a function of the input and output fiber positions, which originates from the interference between the modes in the waveguide after mixing by the ring resonator. We developed a theoretical model based on the coupled mode theory to describe the system. We also showed that the effective index method is a good approximation in calculating the modes of our high-aspect-ratio Si

_{3}N

_{4}waveguide. With the large spatial extent of the mode out of the Si

_{3}N

_{4}core, our Si

_{3}N

_{4}ring resonator is of interest for sensing applications.

## Acknowledgments

## References and links

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

2. | 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. |

3. | V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature |

4. | A. Ksendzov and Y. Lin, “Integrated optics ring-resonator sensors for protein detection,” Opt. Lett. |

5. | S. Fan, “Sharp asymmetric line shapes in side-coupled waveguide-cavity systems,” Appl. Phys. Lett. |

6. | C.-Y. Chao and L. J. Guo, “Biochemical sensors based on polymer microrings with sharp asymmetrical resonance,” Appl. Phys. Lett. |

7. | A. Chiba, H. Fujiwara, J. Hotta, S. Takeuchi, and K. Sasaki, “Fano resonance in a multimode tapered fiber coupled with a microspherical cavity,” Appl. Phys. Lett. |

8. | A. C. Ruege and R. M. Reano, “Sharp Fano resonances from a two-mode waveguide coupled to a single-mode ring resonator,” J. Lightwave Technol. |

9. | A. C. Ruege and R. M. Reano, “Multimode waveguide-cavity sensor based on fringe visibility detection,” Opt. Express |

10. | B.-B. Li, Y.-F. Xiao, C.-L. Zou, Y.-C. Liu, X.-F. Jiang, Y.-L. Chen, Y. Li, and Q. Gong, “Experimental observation of Fano resonance in a single whispering-gallery microresonator,” Appl. Phys. Lett. |

11. | Y.-F. Xiao, L. He, J. Zhu, and L. Yang, “Electromagnetically induced transparency-like effect in a single polydimethylsiloxane-coated silica microtoroid,” Appl. Phys. Lett. |

12. | B.-B. Li, Y.-F. Xiao, C.-L. Zou, X.-F. Jiang, Y.-C. Liu, F.-W. Sun, Y. Li, and Q. Gong, “Experimental controlling of Fano resonance in indirectly coupled whispering-gallery microresonators,” Appl. Phys. Lett. |

13. | P. Rabiei and W. H. Steier, “Polymer microring resonators,” in |

14. | S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A |

15. | J. F. Bauters, M. J. R. Heck, D. John, D. Dai, M.-C. Tien, J. S. Barton, A. Leinse, R. G. Heideman, D. J. Blumenthal, and J. E. Bowers, “Ultra-low-loss high-aspect-ratio Si |

16. | M.-C. Tien, J. F. Bauters, M. J. R. Heck, D. T. Spencer, D. J. Blumenthal, and J. E. Bowers, “Ultra-high quality factor planar Si |

17. | R. M. Knox and P. P. Toulios, “Integrated circuits for the millimeter through optical frequency range,” in |

18. | J. F. Bauters, M. J. R. Heck, D. D. John, J. S. Barton, C. M. Bruinink, A. Leinse, R. G. Heideman, D. J. Blumenthal, and J. E. Bowers, “Planar waveguides with less than 0.1 dB/m propagation loss fabricated with wafer bonding,” Opt. Express |

**OCIS Codes**

(060.2310) Fiber optics and optical communications : Fiber optics

(130.0130) Integrated optics : Integrated optics

(230.5750) Optical devices : Resonators

(230.7370) Optical devices : Waveguides

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: December 16, 2013

Revised Manuscript: March 5, 2014

Manuscript Accepted: March 7, 2014

Published: March 17, 2014

**Citation**

Dapeng Ding, Michiel J. A. de Dood, Jared F. Bauters, Martijn J. R. Heck, John E. Bowers, and Dirk Bouwmeester, "Fano resonances in a multimode waveguide coupled to a high-Q silicon nitride ring resonator," Opt. Express **22**, 6778-6790 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-6-6778

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### References

- F. Xia, L. Sekaric, Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1, 65–71 (2007). [CrossRef]
- B. E. Little, S. T. Chu, W. Pan, D. Ripin, T. Kaneko, Y. Kokubun, E. Ippen, “Vertically coupled glass microring resonator channel dropping filters,” IEEE Photon. Technol. Lett. 11, 215–217 (1999). [CrossRef]
- V. R. Almeida, C. A. Barrios, R. R. Panepucci, M. Lipson, “All-optical control of light on a silicon chip,” Nature 431, 1081–1084 (2004). [CrossRef] [PubMed]
- A. Ksendzov, Y. Lin, “Integrated optics ring-resonator sensors for protein detection,” Opt. Lett. 30, 3344–3346 (2005). [CrossRef]
- S. Fan, “Sharp asymmetric line shapes in side-coupled waveguide-cavity systems,” Appl. Phys. Lett. 80, 908–910 (2002). [CrossRef]
- C.-Y. Chao, L. J. Guo, “Biochemical sensors based on polymer microrings with sharp asymmetrical resonance,” Appl. Phys. Lett. 83, 1527–1529 (2003). [CrossRef]
- A. Chiba, H. Fujiwara, J. Hotta, S. Takeuchi, K. Sasaki, “Fano resonance in a multimode tapered fiber coupled with a microspherical cavity,” Appl. Phys. Lett. 86, 261106 (2005). [CrossRef]
- A. C. Ruege, R. M. Reano, “Sharp Fano resonances from a two-mode waveguide coupled to a single-mode ring resonator,” J. Lightwave Technol. 28, 2964–2968 (2010). [CrossRef]
- A. C. Ruege, R. M. Reano, “Multimode waveguide-cavity sensor based on fringe visibility detection,” Opt. Express 17, 4295–4305 (2009). [CrossRef] [PubMed]
- B.-B. Li, Y.-F. Xiao, C.-L. Zou, Y.-C. Liu, X.-F. Jiang, Y.-L. Chen, Y. Li, Q. Gong, “Experimental observation of Fano resonance in a single whispering-gallery microresonator,” Appl. Phys. Lett. 98, 021116 (2011). [CrossRef]
- Y.-F. Xiao, L. He, J. Zhu, L. Yang, “Electromagnetically induced transparency-like effect in a single polydimethylsiloxane-coated silica microtoroid,” Appl. Phys. Lett. 94, 231115 (2009). [CrossRef]
- B.-B. Li, Y.-F. Xiao, C.-L. Zou, X.-F. Jiang, Y.-C. Liu, F.-W. Sun, Y. Li, Q. Gong, “Experimental controlling of Fano resonance in indirectly coupled whispering-gallery microresonators,” Appl. Phys. Lett. 100, 021108 (2012). [CrossRef]
- P. Rabiei, W. H. Steier, “Polymer microring resonators,” in Optical Microcavities, K. Vahala, eds. (World Scientific, 2005), pp. 321–324.
- S. Fan, W. Suh, J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569–572 (2003). [CrossRef]
- J. F. Bauters, M. J. R. Heck, D. John, D. Dai, M.-C. Tien, J. S. Barton, A. Leinse, R. G. Heideman, D. J. Blumenthal, J. E. Bowers, “Ultra-low-loss high-aspect-ratio Si3N4 waveguides,” Opt. Express 19, 3163–3174 (2011). [CrossRef] [PubMed]
- M.-C. Tien, J. F. Bauters, M. J. R. Heck, D. T. Spencer, D. J. Blumenthal, J. E. Bowers, “Ultra-high quality factor planar Si3N4 ring resonators on Si substrates,” Opt. Express 19, 13551–13556 (2011). [CrossRef] [PubMed]
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