## Spectral shift by half free-spectral-range for microring resonator employing the phase jump phenomenon in coupled-waveguide and application on all-microring wavelength interleaver

Optics Express, Vol. 17, Issue 10, pp. 7756-7770 (2009)

http://dx.doi.org/10.1364/OE.17.007756

Acrobat PDF (4792 KB)

### Abstract

Using coupled-mode theory, we have shown that there is a *π* phase jump between the input and the through/drop fields of a co-directional coupler when the gap width between the coupled-waveguides reaches certain values such that the length of the coupler equals to the odd integer (for through field) or even integer (for drop field) times of the Transfer Distance. We introduced an efficient numerical method based on combining the scattering matrix method and FDTD method for analyzing a microring that has material loss. By applying this method, we found that the phase jump phenomenon also occurs in a half-ring coupler when the gap width between the coupled half-ring waveguides reaches a critical value. We showed that, for a given operating bandwidth, it is important that the gap width between the rings has to be larger than a certain value in order to avoid the phase jump, or smaller in order to take advantage of the phase jump. Based on the phase jump phenomenon, we found that the through and the drop spectra of the single-arm and the double-arm microring can be manipulated to shift about one half free spectral range by selecting appropriate gap widths. A novel all-microring wavelength interleaver, based on the phase jump phenomenon, is proposed and numerically demonstrated.

© 2009 Optical Society of America

## 1. Introduction

1. M. C. M. Lee and M. C. Wu, “Tunable coupling regimes of silicon microdisk resonators using MEMS actuators,” Opt. Express **14**, 4703–4712 (2006). [CrossRef] [PubMed]

*π*/2 phase change for weak coupling and this phase change departs from

*π*/2 as the coupling conditions deviates from weak coupling causing the coupling-induced resonance frequency shifts (CIFS) for a resonator[3–5

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

*π*/2 in the CIFS effect is small but for telecom-grade devices where precise wavelength control is required, the CIFS effect has to be avoided or compensated[6

6. T. Barwicz, M. A. Popovic, P. T. Rakich, M. R. Watts, H. A. Haus, E. P. Ippen, and H. I. Smith, “Microring-resonator-based add-drop filters in SiN: fabrication and analysis,” Opt. Express **12**, 1437–1442 (2004). [CrossRef] [PubMed]

7. T. Barwicz, M. A. Popovic, M. R. Watts, P. T. Rakich, E. P. Ippen, and H. I. Smith, “Fabrication of add-drop filters based on frequency-matched microring resonators,” IEEE J. Lightwave Technol. **24**, 2207–2218 (2006). [CrossRef]

*π*phase jump, for the coupled waveguides when the gap width between the coupled waveguides changes. We demonstrated and analyzed in Sec. 2.1 that the phenomenon occurs in the conventional straight co-directional coupler, and we showed in Sec. 2.2 by numerical example that the phenomenon also occurs in half-ring coupler. Basic characteristics and discussions of the phenomenon with respect to the gap width variation were given. Spectral shift, based on the phase jump phenomenon, by half free spectral range (FSR) in a single-arm and a double-arm microring resonator were demonstrated in Sec. 2.3.

8. S. Cao, J. N. Damask, C. R. Doerr, L. Guiziou, G. Harvey, Y. Hibino, H. Li, S. Suzuki, K. Y. Wu, and P. Xie, “Interleaver technology: comparisons and applications requirements,” IEEE J. Lightwave Technol. **22**, 281–289 (2004). [CrossRef]

10. T. Mizuno, T. Kitoh, M. Oguma, Y. Inoue, T. Shibata, and H. Takahashi, “Uniform wavelength spacing Mach-Zehnder interference using phase-generating couplers,” IEEE J. Lightwave Technol. **24**, 3217–3226 (2006). [CrossRef]

11. K. Oda, N. Takato, H. Toba, and K. Nosu, “A wide-band guided-wave periodic multi/demultiplexer with a ring resonator for optical FDM transmission systems,” IEEE J. Lightwave Technol. **6**, 1016–1023 (1988). [CrossRef]

## 2. Theory

### 2.1. Symmetrical co-directional coupler

*L*is the coupling length,

*K*= (

_{a}*K*-

*CX*)/(1- ∣

*C*∣

^{2}), and Γ = (

*KC*-

*X*)/(1- ∣

*C*∣

^{2}).

*K*,

*C*, and

*X*are defined as:

**E**

_{1}and

**E**

_{2}are the mode fields of the waveguide 1 and 2, respectively.

*L*=

_{c}*π*/(2

*K*) is defined as the Transfer Distance. 2

_{a}*L*is the coupling length for one complete cycle of power exchange.

_{c}*K*is the coupling coefficient,

*C*represents the cross-coupling coefficient, and

*X*is the self-coupling coefficient of the coupled modes for the symmetrical co-directional coupler. When

**B**

_{0}=0, the input-output relationship is:

*C*and

*X*are equal to zero, Γ=0,

*K*=

_{a}*K*, and

*L*=

_{c}*π*/(2

*K*), the matrix elements are given as:

*K*,

*C*,

*X*and hence

*K*, Γ and

_{a}*L*are varied accordingly, and up to a certain gap width the cosine and the sine function in Eq. (2a)–(2d) change sign, a

_{c}*π*phase jump occur in

**s**

_{11},

**s**

_{21},

**s**

_{12}and

**s**

_{22}. From the definition of

*L*and Eq. (2a)–(2d), the phase jump occurs at the gap width for which

_{c}*n*= odd integer for

**s**

_{11}and

**s**

_{22},

*n*= even integer for

**s**

_{12}and

**s**

_{21}. The underlying physics of the

*π*phase jump is fundamentally due to the phase accumulation of the back and the forth coupling each contributes

*π*/2 phase,

*n*+ 1 is the number of the forth and the back coupling within

*L*. Notice that for

**B**

_{0}=0, the phase difference between

**A**

_{0}and

**A**

_{1}is

*ϕ*

_{11}, and the phase difference between

**A**

_{0}and

**B**

_{1}is

*ϕ*

_{21}, according to Eq. (6) and (7).

*L*decreases with the gap width as expected, secondly, the

_{c}*π*phase jump occurred for wCMT as well as for rCMT as predicted, there is a phase difference of Γ

*L*between wCMT and rCMT consistent with Eq.(2a)–(2d), thirdly, the magnitude of the matrix element has a minimum at the gap width where there is phase jump, and fourthly, only limited number of phase jump occurred as the gap width is varying; the odd order phase jump occurs for

*ϕ*

_{11}and the even order phase jump occurs for

*ϕ*

_{21}. According to Eq. (5) mathematically, the phase jump will keep on occurring with

*n*>4 but at negative gap width that is not realistic and the CMT would not be valid, therefore, only limited number of phase jump occurs for a given coupling length

*L*. When

*L*increases, the first-order jump (

*n*=1) occurs at larger gap width and more higher order jumps occur at smaller gap width.

### 2.2. Symmetrical half-ring coupler

*S*-matrix. Similar method was given by Ref. 3

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

**A**

_{0},

**A**

_{1}and

**B**

_{1}. The matrix elements of

*S*-matrix,

**s**

_{11}and

**s**

_{21}, are then obtained through Eq. (6) and (7). Step 3 and 4 are repeated with launching M2 at plane 3,

**s**

_{22}and

**s**

_{12}are obtained accordingly. A numerical example, referring to Fig. 3, was carried out with the following parameters:

*n*=2.5,

_{core}*n*=1.5, waveguide width=300 nm,

_{cladding}*λ*=1550 nm, material loss coefficient

*α*=0 dB/cm and outer radius

_{mat}*R*=12.15

*μ*m. There was only one mode for the ring waveguide and it was a TE-like mode. The results are shown in Figs. 4(a) and 4(b). There is a noticeable phase protruding about 0.1

*π*in

*ϕ*

_{11}right below the gap width of the phase jump, no satisfactory explanation could be given for the protrude phase. Nevertheless, it is clear that the phase jump indeed occurs in the half-ring coupler and its magnitude is close to

*π*. The phase jump occurred at the minimum

**s**

_{11}and maximum

**s**

_{21}. There is only one phase jump for 011 and no phase jump for 021 in this case. The criteria for the phase jump to occur in the half-ring coupler could be modified from Eq. (10) as

*L*=

_{eff}*n L*

_{c,eff}, where

*L*is the effective coupling length and

_{eff}*L*

_{c,eff}is the effective Transfer Distance, and

*n*=1 in this particular example. For half-ring coupler with larger radius of curvature with the same gap width, we expect that

*L*would be larger and

_{eff}*L*would be smaller such that the number of phase jump would be increased as the gap width is varying.

_{c,eff}*λ*. For longer wavelength, the evanescent field extends further out, a shorter

_{c}*L*

_{c,eff}is expected, and more phase jump would occur at longer wavelengths. The coefficient of

*K*in Eq. (3) is inversely proportional to

*λ*, the cosine and the sine function in Eq. (2a)–(2d) change sign whenever the wavelength changes by the order of magnitude of

*L*, we therefore assert that the wavelength difference between two adjacent phase jump of the same order would be in the order of

*L*for half-ring coupler.

_{eff}*λ*vs. gap width, g, and the result is shown in Fig. 6. Several important aspects with respect to the phase jump phenomenon, when the device is operated within a certain bandwidth, are implied: firstly, if one wants to avoid the phase jump, the gap width has to be larger than a certain value determined by the upper bound of the operating bandwidth, in the region of g

_{c}_{III}as shown in Fig. 6 for C-band operation as an example, secondly, if one wants to utilize the phase jump for all operating wavelength, the gap width has to be smaller than a certain value determined by the lower bound of the operating bandwidth, in the region of g

_{I}, and thirdly, if one wants to utilize the phase jump within the operating wavelength range, then the gap width has to be controlled and varied within a certain range g

_{II}determined by the upper and the lower bound of the operating bandwidth.

### 2.3. Single-arm and double-arm microring

_{C}at the phase jump, g

_{L}when there is no phase jump, and g

_{S}when there is phase jump. For a single-arm microring with g

_{S}gap width, the through field undergoes

*π*phase jump and the field that is coupled back to the bus after one round trip in the ring undergoes twice

*π*/2 phase change from the coupling, the over-all through field at resonance is in constructive interference in contrast to that of the microring with g

_{L}gap width where the over-all through field at resonance is in destructive interference, therefore, we expect that there will be a spectral shift between the microrings with g

_{L}and g

_{S}gap width, the amount of the spectral shift will be around half the free spectral range (FSR). For a double-arm microring, we define that the microring with both gap width is g

_{L}or g

_{S}as type I, and the microring with one gap is g

_{L}and the other gap is g

_{S}as type II. For type I, the field that couples back to the input bus after one round trip in the ring undergoes one phase jump (or no phase jump) same as the through field such that the over-all through field at resonance is in destructive interference due to twice

*π*/2 phase change from the coupling, therefore, we expect that there is no spectral shift, neither in the through port nor in the drop port for type I, with respect to the single-arm microring with g

_{L}. For type II, the field that couples back to the input bus after one round trip in the ring undergoes one phase jump (or no phase jump) that is opposite to the through field such that the over-all through field at resonance is in constructive interference, therefore, we expect that there is a half FSR spectral shift both in the through port and in the drop port for type II.

*β*matching. The spectrum at the through port

*I*and the spectrum at the drop port

_{through}*I*were calculated by the analytical expressions, Eq. (A.5), Eq. (A.14), and Eq. (A.15), as given in Appendix A by using the elements of the

_{drop}*S*-matrix for the half-ring coupler. Numerical measurement on the

*S*-matrix elements of the half-ring coupler as introduced in Sec. 2.2 was performed first, and then the measured matrix elements were substituted into the analytical expressions in Appendix A to obtain the

*I*and the

_{through}*I*spectra. We have also applied the FDTD method to obtain

_{drop}*I*and

_{through}*I*spectra in order to verify the validity of the analytical expressions. Same parameters for the half-ring coupler in Sec. 2.2 were used here except that the material power loss coefficient was chosen to be 50 dB/cm for the microring in order to, firstly, reduce the computational time for the FDTD method and secondly, profound spectra that is close to critical coupling could be obtained.

_{drop}*S*-matrix elements, black solid lines. Both methods gave identical results except that the dip depths are slightly different due to the limited FDTD computational time. Fig. 7(a) shows

*I*spectrum for a single-arm microring with g

_{through}_{L}=350 nm, i.e. no phase jump. Fig. 7(b) shows

*I*spectrum for the single-arm microring with g

_{through}_{S}=20 nm, i.e. with phase jump. It is apparent that the spectra are shifted with each other by approximately half the FSR.

*ϕ*

_{21}is not exactly

*π*/2 for g

_{S}=20 nm; weak coupling assumption is not exactly applied to g

_{S}=20 nm. This assertion can be confirmed from Fig. 4(b) and from the non-zero Γ

*L*in Fig. 2(c) for the co-directional coupler. This is basically the coupling-induced resonance frequency shift (CIFS) phenomenon when weak coupling assumption is not exactly applied [3

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

_{L}=350 nm and g

_{S}=20 nm. Again, the resonance frequencies are slightly different due to the CIFS effect. However, approximately half FSR spectral shift between the type I and the type II are clearly shown. Without the CIFS effect, the spectral shift should be exactly half FSR.

*ϕ*

_{12}and

*ϕ*

_{21}, if there is any, does not cause spectral shift, since the field undergoes both couple-in and couple-out that accumulates phase jump twice to yield 2

*π*phase change.

*and g*

_{L}_{S}in order to obtain the half FSR spectral shift. Multi-ring device of this type can be referred to as the generalized type II in contrast to the generalized type I in which all gap widths are g

_{L}or g

_{S}.

_{C}, referring to Fig. 2(c), g

_{L}and g

_{S}are not restricted to the opposite side of a particular g

_{C}, it is only required that one lies in the phase jump region and the other one lies in the region where there is no phase jump.

## 3. Application on wavelength interleaver

*S*-matrix, including its magnitude and phase, for each half-ring coupler in Fig. 9 can be numerically measured as introduced in Sec. 2.2, and then substituting into Eq. (A.14) and Eq. (A.15), applying the method twice, all the output spectra can be obtained. The values for the relevant parameters of the waveguide in Fig. 9 were the same as those for the waveguides in Sec. 2.3 with g

_{L}=350 nm and g

_{S}=20 nm. The results are shown in Figs. 10(a)–10(c). It is apparent that the spectra of the drop ports were separated approximately half-way to each other.

16. M. A. Popovic, T. Barwicz, M. R. Watts, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kartner, and H. I. Smith, “Multistage high-order microring-resonator add-drop filters,” Opt. Lett. **31**, 2571–2573 (2006). [CrossRef] [PubMed]

_{S}that was 20 nm. The CIFS effect needs to be avoided or reduced as much as possible such that the type I and the type II has the FSR-matching, i.e. resonant frequencies located at exactly half FSR of each other, in order to interleave the equally spaced input channels. There are several solutions to the CIFS problem: firstly, it is feasible to optimize the 3 dB bandwidth of the multi-ring interleaver such that the FSR mismatch falls within the 3 dB bandwidth, secondly, type I with all g

_{L}is preferred than that with all g

_{S}, since the CIFS is negligible for large gap width microring, thirdly, adjusting the radius of curvature or the material refractive index for each type microring to match the FSR is also feasible, fourthly, using microrings with larger radius of curvature; since microring with larger

*R*has longer effective coupling length (

*L*) that yields more phase jump and the first-order jump would occur at larger gap width with longer effective coupling length as was discussed in Sec. 2.1; we can select g

_{eff}_{L}and g

_{S}that are around the first-order g

_{C}that is large such that the CIFS effect could be minimized. The interleaver for dense channel spacing signal also calls for rings with large radius of curvature, for example: for rings with

*R*= 12.15

*μ*m as in the previous numerical examples, FSR= 13 nm; and for rings with

*R*=115

*μ*m, FSR=1.6 nm, 0.8 nm channel spacing signal could be interleaved. We would therefore expect that the CIFS effect could be greatly reduced for interleaver that targets DWDM application. We did not use

*R*=115

*μ*m ring for numerical example in this paper was due to the limited computational time available. Race-track microring with long coupling length and hence larger gap widths also serves the purpose for reducing the CIFS effect.

## 4. Conclusions

## Appendix A: Analytical expressions for the through and the drop spectra of a single-arm and a double-arm microring resonator based on the *S*-matrix method

### A.1 Through spectrum of a single-arm microring resonator

*α*is the power loss coefficient of the ring mode, and

_{ring}*β*is the propagation constant of the ring mode. Substituted Eq. (A.3) into Eq. (A.2) then (A.1), we obtain:

_{ring}*I*/

_{through}*I*= ∣

_{i}**A**

^{2}

_{1}∣/∣

**A**

^{2}

_{0}∣. From Eq. (A.4), we obtain

*I*/

_{through}*I*, recast into the form of a standard Fabry-Perot cavity:

_{i}*π*√

*r*/(1-

*r*)

### A.2 Through and drop spectra of a double-arm microring resonator

*S*matrices,

*S*

^{(1)}and

*S*

^{(2)}, for the double-arm microring:

**B**

_{0},

**B**

_{1},

**A**

_{0}, and

**C**

_{0}can be obtained :

*I*/

_{through}*I*= ∣

_{i}**A**

_{1}∣

^{2}/∣

**A**

_{0}∣

^{2}and

*I*/

_{drop}*I*= ∣

_{i}**C**

_{0}∣

^{2}/∣

**A**

_{0}∣

^{2}respectively. From Eq. (A.11), and Eq. (A.13) we obtain

*I*/

_{through}*I*and

_{i}*I*/

_{drop}*I*, recast into the form of a standard Fabry-Perot cavity:

_{i}*π*√

*r*/(1-

*r*)

## References and links

1. | M. C. M. Lee and M. C. Wu, “Tunable coupling regimes of silicon microdisk resonators using MEMS actuators,” Opt. Express |

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

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

4. | S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, “Effect of a layered environment on the complex nature frequencies of two-dimensional WGM dieletric-ring resonators,” IEEE J. Lightwave Technol. |

5. | O. Schwelb, “On the nature of resonance splitting in coupled multiring optical resonators,” Opt. Commun. |

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

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

8. | S. Cao, J. N. Damask, C. R. Doerr, L. Guiziou, G. Harvey, Y. Hibino, H. Li, S. Suzuki, K. Y. Wu, and P. Xie, “Interleaver technology: comparisons and applications requirements,” IEEE J. Lightwave Technol. |

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

10. | T. Mizuno, T. Kitoh, M. Oguma, Y. Inoue, T. Shibata, and H. Takahashi, “Uniform wavelength spacing Mach-Zehnder interference using phase-generating couplers,” IEEE J. Lightwave Technol. |

11. | K. Oda, N. Takato, H. Toba, and K. Nosu, “A wide-band guided-wave periodic multi/demultiplexer with a ring resonator for optical FDM transmission systems,” IEEE J. Lightwave Technol. |

12. | M. Kohtoku, S. Oku, Y. Kadota, and Y. Yoshikuni, “200-GHz FSR periodic multi/demultiplexer with flattened transmission and rejection band by using a Mach-Zehnder interference with a ring resonator,” IEEE Photon. Technol. Lett. |

13. | Z. Wang, S. J. Chang, C. Y. Ni, and Y. J. Chen, “A high-performance ultracompact optical interleaver based on double-ring assisted Mach-Zehnder interferometer,” IEEE Photon. Technol. Lett. |

14. | J. Song, Q. Fang, S. H. Tao, M. B. Yu, G. Q. Lo, and D. L. Kwong, “Passive ring-assisted Mach-Zehnder interleaver on silicon-on-insulator,” Opt. Express |

15. | C. K. Okamoto, |

16. | M. A. Popovic, T. Barwicz, M. R. Watts, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kartner, and H. I. Smith, “Multistage high-order microring-resonator add-drop filters,” Opt. Lett. |

17. | S. Xiao, M. H. Khan, H. Shen, and M. Qi, “A highly compact third-order silicon microring add-drop filter with a very large free spectral range, a flat passband and a low delay dispersion,” Opt. Express |

18. | S. Xiao, M. H. Khan, H. Shen, and M. Qi, “Silicon-on-insulator microring add-drop filters with free spectral ranges over 30 nm,” IEEE J. Lightwave Technol. |

19. | B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. |

**OCIS Codes**

(130.1750) Integrated optics : Components

(130.2790) Integrated optics : Guided waves

(130.3120) Integrated optics : Integrated optics devices

(230.3120) Optical devices : Integrated optics devices

(230.4555) Optical devices : Coupled resonators

(130.7408) Integrated optics : Wavelength filtering devices

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: February 25, 2009

Revised Manuscript: April 3, 2009

Manuscript Accepted: April 19, 2009

Published: April 27, 2009

**Citation**

Chih T. Shih and Shiuh Chao, "Spectral shift by half free-spectral-range for microring resonator employing the phase jump phenomenon in coupled-waveguide and application on all-microring wavelength interleaver," Opt. Express **17**, 7756-7770 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-10-7756

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### References

- M. C. M. Lee and M. C. Wu, "Tunable coupling regimes of silicon microdisk resonators using MEMS actuators," Opt. Express 14, 4703-4712 (2006). [CrossRef] [PubMed]
- A. Yariv and P. Yeh, Photonics:optical electronics in modern communications (Oxford University Press Inc., 2007), pp. 184-189.
- M. A. Popovic, C. Manolatou, and M. R. Watts, "Coupled-induced resonance frequency shifts in coupled dielectric multi-cavity filters," Opt. Express 14, 1208-1222 (2006). [CrossRef] [PubMed]
- S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, "Effect of a layered environment on the complex nature frequencies of two-dimensional WGM dieletric-ring resonators," IEEE J. Lightwave Technol. 20, 1563-1572 (2002). [CrossRef]
- O. Schwelb, "On the nature of resonance splitting in coupled multiring optical resonators," Opt. Commun. 281, 1065-1071 (2008). [CrossRef]
- T. Barwicz, M. A. Popovic, P. T. Rakich, M. R. Watts, H. A. Haus, E. P. Ippen, and H. I. Smith, "Microringresonator-based add-drop filters in SiN: fabrication and analysis," Opt. Express 12, 1437-1442 (2004). [CrossRef] [PubMed]
- T. Barwicz, M. A. Popovic, M. R. Watts, P. T. Rakich, E. P. Ippen, and H. I. Smith, "Fabrication of add-drop filters based on frequency-matched microring resonators," IEEE J. Lightwave Technol. 24, 2207-2218 (2006). [CrossRef]
- S. Cao, J. N. Damask, C. R. Doerr, L. Guiziou, G. Harvey, Y. Hibino, H. Li, S. Suzuki, K. Y. Wu, and P. Xie, "Interleaver technology: comparisons and applications requirements," IEEE J. Lightwave Technol. 22, 281-289 (2004). [CrossRef]
- C. K. Madsen and J. H. Zhao, Optical filter design and analysis: a signal processing approach (John Willey & Sons Inc., 1999), pp. 165-177.
- T. Mizuno, T. Kitoh, M. Oguma, Y. Inoue, T. Shibata and H. Takahashi, "Uniform wavelength spacing Mach-Zehnder interference using phase-generating couplers," IEEE J. Lightwave Technol. 24, 3217-3226 (2006). [CrossRef]
- K. Oda, N. Takato, H. Toba, and K. Nosu, "A wide-band guided-wave periodic multi/demultiplexer with a ring resonator for optical FDM transmission systems," IEEE J. Lightwave Technol. 6, 1016-1023 (1988). [CrossRef]
- M. Kohtoku, S. Oku, Y. Kadota, and Y. Yoshikuni, "200-GHz FSR periodic multi/demultiplexer with flattened transmission and rejection band by using a Mach-Zehnder interference with a ring resonator," IEEE Photon. Technol. Lett. 12, 1174-1176 (2000). [CrossRef]
- Z. Wang, S. J. Chang, C. Y. Ni, and Y. J. Chen, "A high-performance ultracompact optical interleaver based on double-ring assisted Mach-Zehnder interferometer," IEEE Photon. Technol. Lett. 19, 1072-1074 (2007). [CrossRef]
- J. Song, Q. Fang, S. H. Tao, M. B. Yu, G. Q. Lo, and D. L. Kwong, "Passive ring-assisted Mach-Zehnder interleaver on silicon-on-insulator," Opt. Express 16, 8359-8365 (2008). [CrossRef] [PubMed]
- C. K. Okamoto, Fundamentals of Optical Waveguides (Academic Press, 2000), Chap. 4.
- M. A. Popovic, T. Barwicz, M. R. Watts, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kartner, and H. I. Smith, "Multistage high-order microring-resonator add-drop filters," Opt. Lett. 31, 2571-2573 (2006). [CrossRef] [PubMed]
- S. Xiao, M. H. Khan, H. Shen, and M. Qi, "A highly compact third-order silicon microring add-drop filter with a very large free spectral range, a flat passband and a low delay dispersion," Opt. Express 15, 14765-14771 (2007). [CrossRef] [PubMed]
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