## Synthesis of optical filters using microring resonators with ultra-large FSR |

Optics Express, Vol. 18, Issue 25, pp. 25936-25949 (2010)

http://dx.doi.org/10.1364/OE.18.025936

Acrobat PDF (1330 KB)

### Abstract

We propose a novel synthesis method for designing flexible, tunable non-periodic filters. It is based on a building block which is presented by first time for these purposes, being the poles position tuned by means of a coupling coefficient and an amplifier gain. We present the device, the design equations and a design of a filter with flat response, 54 dB of crosstalk and less than 1.3 dB of ripple to explain and validate the method. These filters can be used as part of optical cross-connects for selecting channels avoiding very restrictive free spectral ranges. Also we check the correct operation of the device, against fabrication tolerances of coupling coefficients. Frequency dependence of the transfer function poles as a unique feature improving the crosstalk of the device spectral response is also analyzed.

© 2010 OSA

## 1. Introduction

1. K. Jinguji and T. Yasui, “Synthesis of One-Input M-Output Optical FIR Lattice Circuits,” J. Lightwave Technol. **26**(7), 853–866 (2008). [CrossRef]

2. A. Melloni, “Synthesis of a parallel-coupled ring-resonator filter,” Opt. Lett. **26**(12), 917–919 (2001). [CrossRef]

3. V. Van, “Circuit-Based Method for Synthesizing Serially Coupled Microring Filters,” J. Lightwave Technol. **24**(7), 2912–2919 (2006). [CrossRef]

4. N. Ngo, “Synthesis of Tunable Optical Waveguide Filters Using Digital Signal Processing Technique,” J. Lightwave Technol. **24**(9), 3520–3531 (2006). [CrossRef]

6. K. Jinguji, “Synthesis of Coherent Two-Port Optical Delay-Line Circuit with Ring Waveguides,” J. Lightwave Technol. **14**(8), 1882–1898 (1996). [CrossRef]

7. K. Sasayama, M. Okuno, and K. Habara, “Coherent Optical Transversal Filter using silica-based single-mode waveguides,” Electron. Lett. **25**(22), 1508–1509 (1989). [CrossRef]

1. K. Jinguji and T. Yasui, “Synthesis of One-Input M-Output Optical FIR Lattice Circuits,” J. Lightwave Technol. **26**(7), 853–866 (2008). [CrossRef]

6. K. Jinguji, “Synthesis of Coherent Two-Port Optical Delay-Line Circuit with Ring Waveguides,” J. Lightwave Technol. **14**(8), 1882–1898 (1996). [CrossRef]

8. V. Van, “Dual-Mode Microring Reflection Filters,” J. Lightwave Technol. **25**(10), 3142–3150 (2007). [CrossRef]

9. S. Vargas and C. Vazquez, “Synthesis of Optical Filters Using Sagnac Interferometer in Ring Resonator,” IEEE Photon. Technol. Lett. **19**(23), 1877–1879 (2007). [CrossRef]

10. S. Xiao, M. H. Khan, H. Shen, and M. Qi, “Silicon-on-Insulator Microring Add-Drop Filters With Free Spectral Ranges Over 30 nm,” J. Lightwave Technol. **26**(2), 228–236 (2008). [CrossRef]

11. Q. Xu, D. Fattal, and R. G. Beausoleil, “Silicon microring resonators with 1.5-microm radius,” Opt. Express **16**(6), 4309–4315 (2008). [CrossRef] [PubMed]

11. Q. Xu, D. Fattal, and R. G. Beausoleil, “Silicon microring resonators with 1.5-microm radius,” Opt. Express **16**(6), 4309–4315 (2008). [CrossRef] [PubMed]

12. J. García, A. Martínez, and J. Martí, “Proposal of an OADM configuration with ultra-large FSR combining ring resonators and photonic bandgap structures,” Opt. Commun. **282**(9), 1771–1774 (2009). [CrossRef]

13. C. Vázquez and O. Schwelb, “Tunable, narrow-band, grating-assisted microring reflectors,” Opt. Commun. **281**(19), 4910–4916 (2008). [CrossRef]

## 2. Device

14. D. G. Rabus, M. Hamacher, U. Troppenz, and H. Heidrich, “High Q-Channel Dropping Filters Using Ring Resonators with Integrated SOAs,” IEEE Photon. Technol. Lett. **14**(10), 1442–1444 (2002). [CrossRef]

*L*, and the power coupling coefficient

_{b}*K*is different to 0.5, to assure the MI reflects some light. There is an optical isolator in the input port to avoid reflections [15

_{M}15. J. S. Yang, J. W. Roh, S. H. Ok, D. H. Woo, Y. T. Byun, W. Y. Lee, T. Mizumoto, and S. Lee, “An integrated optical waveguide isolator based on multimode interference by wafer direct bonding,” IEEE Trans. Magn. **41**(10), 3520–3522 (2005). [CrossRef]

16. W. Van Parys, D. Van Thourhout, R. Baets, B. Dagens, J. Decobert, O. Le Gouezigou, D. Make, and L. Lagae, “Amplifying Waveguide Optical Isolator with an Integrated electromagnet,” IEEE Photon. Technol. Lett. **19**(24), 1949–1951 (2007). [CrossRef]

*r(Ω)*, is the gratings reflectivity.

*z*domain, as explained in [17

17. C. Vazquez, S. Vargas, and J. M. S. Pena, “Sagnac Loop in Ring Resonators for Tunable Optical Filters,” J. Lightwave Technol. **23**(8), 2555–2567 (2005). [CrossRef]

*τ*has to be determined. This unitary delay is the total transit time in the RR. Which is given by

*τ = τ*, where

_{r}+ τ_{G}*τ*is the time delay due to waveguides length, including RR, couplers, amplifier and MI arms, and

_{r}*τ*is the gratings group delay. This group delay is the negative derivative of the diffraction gratings reflectivity phase

_{G}*ϕ*with respect to

_{G}(Ω)*Ω*. This phase, under certain conditions, see [18

18. H. Stoll, “Optimally Coupled, GaAs-Distributed Bragg Reflection Lasers,” IEEE Trans. Circ. Syst. **26**(12), 1065–1072 (1979). [CrossRef]

*k*is the corrugation coupling coefficient,

_{b}*α*the distributed loss coefficient, and

_{b}*n*the effective refractive index of gratings,

_{bef}*c*the vacuum light velocity, and

*Ω*the center frequency (rad/seg) of diffraction gratings spectral response.

_{G}*τ*. The gratings transfer function in the

*z*domain, can be considered as that of a device that introduces a fraction of the unitary time delay of

*τ*/

_{G}*τ*. Diffraction gratings transfer function is given by:where we have taken

*z = e*,

^{jΔΩτ}*ΔΩ*=

*2π(f – f*, being

_{G})*f*the center frequency of gratings transfer function.

_{G}*Ω*, equal to the resonance frequency of one of the longitudinal modes of the ring (with unitary delay

_{G}*τ*), it makes that

*z*in the rest of the transfer functions of the ring (

*z = e*) is equal to

^{jΩτ}*z*in the diffraction grating transfer function [see Eq. (5)].

17. C. Vazquez, S. Vargas, and J. M. S. Pena, “Sagnac Loop in Ring Resonators for Tunable Optical Filters,” J. Lightwave Technol. **23**(8), 2555–2567 (2005). [CrossRef]

*A*,

_{3}*A*, and

_{4a}*A*are given by:where

_{4b}*Z*and

_{c1}(z)*Z*are given by:

_{c2}(z)*α*is attenuation coefficient and

*L*is the waveguide lengths respectively.

_{r}*τ*and

_{M}*τ*, are the time delays due to the lengths

_{x}*L*and

_{M}*L*respectively.

_{x}*Z*and

_{p}*Z*) which will be described later.

_{p}*### 2.1 Output A_{3}

*A*, see Eq. (6), has two zeros. They are complex conjugated if the next relation is fulfilled:Otherwise, the zeros are real and different. The position of these zeros in the Z plane, near unitary circumference makes a sharper notch filter spectral response. Even the transfer function could have theoretically infinite crosstalk when placing them at the unitary circumference.

_{3}*G*given by:The frequency dependence of the zero module through

*∣r(Ω)∣*, see Eq. (12), can be interpreted in the spectral response. It means that it is possible to have a different radial position of the zero for each FSR of the ring, by changing the module value and therefore the crosstalk (depending on the

*∣r(Ω)∣*value) in each period.

*∣r(Ω)∣*in the Eq. (12).

*K*and

_{i}*K*the notches of the spectral response, can be placed from –FSR/4 to + FSR/4, from the resonance frequencies of the longitudinal modes of the ring, which are

_{M}*f*, for integer

_{G}+ k × FSR*k*.

*K*tends to one. The phases of real zeros are zero, having the notch at the resonance frequencies of the longitudinal modes of the ring. In this case we use Eq. (15), with

_{i}*φ*to evaluate

_{z}= 0*∣r(Ω)∣*in Eqs. (12).

*K*from 0.05 to 0.95. The module of the zeros does not change, different to the phases, when they are complex conjugated. They have different values when they are real. In this simulation the

_{i}*∣r(Ω)∣*took a value of one, being these the zeros position for the FSRs inside the central lobe of the ideal DG. For FSRs outside the central lobe of the DG, the module of the zeros is less due to

*∣r(Ω)∣*values, ideally being equal to zero.

### 2.2 Output A_{4a}

*A*, is shown in Eq. (9). This output always has only one real positive zero that can be moved through the positive real axis of the Z plane, by increasing

_{4a}*G*. The value of

*G*that places this zero in the unitary circumference is given by:With this

*G*value, we have a spectral response of a notch filter with theoretically infinite crosstalk, located at the resonance frequencies of the longitudinal modes of the ring. This transfer function also has the term

*z*, that arises to take into account the delay between the input and output pulses, because of the transit time in

^{-τx/τ}*L*.

_{x}### 2.3 Output A_{4b}

*A*, see Eq. (10), has complex conjugated poles, and a zero at the origin of the Z plane. This zero is fixed, and only introduces a constant delay in the transfer function. Because of these characteristics, against to the outputs

_{4b}*A*and

_{4a}*A*;

_{3}*A*output transfer function can be used as an all pole function, for synthesizing filters with real coefficients.

_{4b}*∣Z*are frequency dependent, and different of zero only in the central lobe of the gratings spectral response, due to term

_{p}∣*∣r(Ω)∣*in Eq. (17). This spectral response of the diffraction grating must be as close as possible to an ideal passband filter, with a maximum reflectivity of one. The frequency dependence of the poles module can be interpreted in the same form as the zeros in the previous transfer functions. There is a different radial position of the poles, for each FSR. Increasing or decreasing the peaks of the spectral response for each FSR. Depending on how far the poles are from the unitary circumference for these FSR. The Eq. (19) gives the gain value that locates the

*∣Z*in the unitary circumference, which gives theoretically infinite crosstalk.The poles frequency dependence is basic for improving crosstalk as it is described in Section 5.

_{p}∣*φ*takes values between

_{p}*–π/2*to

*π/2*, meaning that only in this margin it is possible to synthesize filters with complex conjugated poles. This could seem a drawback, but by increasing the FSR of the filter it can be increased the bandwidth of the synthesized filter, while maintaining the desired shape. The peaks position in frequency are given by Eq. (15), changing

*φ*by

_{z}*φ*. And we must use these frequencies to evaluate

_{p}*∣r(Ω)∣*in Eq. (17), to find the poles module for each FSR.

*f*and the total delay time of the ring (

_{G})*τ*), by means of the electro-optic effect or thermo-optic effect, heating or injecting current in the gratings and in the ring length [19]. The only requirement we should care about is making

*f*to be located in one of the longitudinal resonant modes of the ring by any of the physical effects previously mentioned.

_{G}*e*multiplying each pole (where

^{j2πfΔ}*f*is the difference between

_{Δ}*f*and the frequency of the nearest resonant mode of the ring). This term changes the poles position in the Z plane, rotating them

_{G}*2πf*, and increasing

_{Δ}·τ radians*f*the peaks position in the spectral response, from its ideal position. The module of the poles can also be affected if there is a change in

_{Δ}Hz*∣r(Ω)∣*. Finally, if the spectral response of the DG reflectivity is like an ideal bandpass filter; the spectral response will only be displaced in frequency.

## 3. Synthesis Equations

9. S. Vargas and C. Vazquez, “Synthesis of Optical Filters Using Sagnac Interferometer in Ring Resonator,” IEEE Photon. Technol. Lett. **19**(23), 1877–1879 (2007). [CrossRef]

*A*, an ARR is used as in [9

_{4b}9. S. Vargas and C. Vazquez, “Synthesis of Optical Filters Using Sagnac Interferometer in Ring Resonator,” IEEE Photon. Technol. Lett. **19**(23), 1877–1879 (2007). [CrossRef]

1. K. Jinguji and T. Yasui, “Synthesis of One-Input M-Output Optical FIR Lattice Circuits,” J. Lightwave Technol. **26**(7), 853–866 (2008). [CrossRef]

*A*) used for compensating the insertion loss due to the MIARR, ARR stages, and by the spectral response normalization step, in the algorithm of synthesizing the zeros by the lattice filter stage.

**19**(23), 1877–1879 (2007). [CrossRef]

*G*, only appears in the poles module, making this parameter, ideal to be used for tuning the radial position of the poles. This selection is reinforced by the fact that the gratings must have a fix spectral response, and that

*K*and

_{i}*K*contribute to the loss introduced by the filter, see numerator of Eq. (10). These parameters are not suitable for tuning this radial location.

_{o}*K*and

_{i}*K*values, filter loss are minimized once

_{o}*G*is fixed. The following coupling coefficients are obtained: The value of

*G*is chosen arbitrary; but taking into account that it must be greater than a minimum value (

*G*), to have positive

_{min}*K*and

_{i}*K*values. This minimum value is given by:Also we have to take into account that, as

_{o}*G*increases

*χ*decreases, which decreases the filter loss, see Eq. (20) and (21) and numerator of Eq. (10). Anyway, this loss can be compensated with the external gain stage

*A*.

*K*is used to change

_{M}*φ*, see Eq. (18), to the value needed to synthesize the desired poles. The value of

_{p}*K*, for a given pole phase, is given by:In Eq. (24), the positive or the negative sign, can be selected indistinctly, the change in the selection only sums π rads in the phase response, see Eq. (10).

_{M}## 4. Example and Discussion

*Hd[z]*) to be synthesized is given by:This is a fourth order transfer function with two pairs of complex conjugated poles, as shown in Table 2 , which means two stages of the MIARR are needed. The excess loss coefficient

*γ*,

_{i}*γ*and

_{o}*γ*for the three couplers, are equal to 0.025 (0.1 dB/coupler) and the waveguide losses are 2 dB/cm [20

_{M}20. Y. A. Vlasov and S. J. McNab, “Losses in single-mode silicon-on-insulator strip waveguides and bends,” Opt. Express **12**(8), 1622–1631 (2004). [CrossRef] [PubMed]

21. M. Gnan, S. Thoms, D. S. Macintyre, R. M. De La Rue, and M. Sorel, “Fabrication of low-loss photonic wires in silicon-on-insulator using hydrogen silsesquioxane electron-beam resis,” Electron. Lett. **44**(2), 115–116 (2008). [CrossRef]

*τ*0.75

_{M}=*τ*and

*τ*0.5

_{x}=*τ*. The waveguide effective refractive index

*n*is 3.5, the parameters

_{ef}*n*and

_{bef}, α_{b}*k*, of all the diffraction gratings used in the MIARR stages, are 3.6, 0 m

_{b}^{−1}and 480 m

^{−1}, respectively. And the ratio FSR to full width at half maximum (FWHM) of gratings is chosen equal to 0.28.

*G*for each stage; see Eq. (23), in this case 1.0817 and 0.9215 respectively. After an arbitrary value of

_{min}*G*greater than

*G*in the two stages is chosen; in this case

_{min}*G*is selected to be equal to 2.1. Knowing the complex conjugated poles, modules and phases, and using Eqs. (20)–(24) the values of

*K*,

_{i}*K*, and

_{o}*K*, for each stage are calculated. The loss introduced by the two stages filter, are then calculated, being 0.0052 for

_{M}*G = 2.1*, here we can iteratively increase

*G*, repeating the processes until the losses are 0.0067, the same value of the numerator of Eq. (25), being not necessary the gain stage

*A*to compensate the losses. The final values are summarized in Table 2. Another approximation is to use the minimum possible gain in the rings, see Eq. (23), and to compensate the losses by means of the external stage

*A*. In this new approach, the found values are summarized in Table 3 . Remember

*G > G*, not equal.In this case the loss is 199.5 × 10

_{min}^{−6}, then a gain

*A*of 33 should be used, a little more than 30 dB. These two solutions are extreme, but tailored intermediate solutions can be found, showing the flexibility of the design method.

*Hs[z])*of the two stages MIARR filter, is shown in Fig. 4 . A non-periodic transfer function is obtained, synthesized inside the diffraction grating bandwidth, also shown in this figure, with the shape of the desired transfer function

*Hd[z]*given in Eq. (25), which is also shown in this figure.

12. J. García, A. Martínez, and J. Martí, “Proposal of an OADM configuration with ultra-large FSR combining ring resonators and photonic bandgap structures,” Opt. Commun. **282**(9), 1771–1774 (2009). [CrossRef]

*Hs[z]*.

## 5. Variation of coupling coefficients and poles frequency dependence

### 5.1 Variation of coupling coefficients

*Ks*) with a uniform distribution between ± 5% of the theoretically calculated values. In Fig. 5 , are shown the eleven spectral results after doing this, ten with randomly generated

*Ks*, and one with the ideal

*Ks*. The ripple changes to 6 dB in the worst case and the crosstalk in less than 8 dB, see Fig. 5, maintaining a crosstalk at least of 46.5 dB, a very good merit figure.

### 5.2 Poles frequency dependence

22. J. Proakis, and D. Manolakis, *Digital Signal Processing*, (Prentice Hall 2006). [PubMed]

22. J. Proakis, and D. Manolakis, *Digital Signal Processing*, (Prentice Hall 2006). [PubMed]

*φ*changed by the poles phase

_{z}*φ*, see Eq. (18). Now when we calculate the spectral response we change

_{p}*z*by

*e*and rotate the phasor

^{jΩτ}*e*over the unitary circumference, calculating the transfer function value for each frequency. The position of the poles is the same in each rotation, which means peaks of the same amplitude and in the same frequency location by period. Then a periodic spectral response is generated.

^{jΩτ}*f*. As we can see the poles module diminishes once you move from the 0 FSR to greater (or smaller) FSR. This diminishing e.g. in the first stage pole module, from the FSR 0 to FSR 1, could represent a peak variation of 11 dB.

_{G}## 6. Conclusion

## Acknowledgments

## References and links

1. | K. Jinguji and T. Yasui, “Synthesis of One-Input M-Output Optical FIR Lattice Circuits,” J. Lightwave Technol. |

2. | A. Melloni, “Synthesis of a parallel-coupled ring-resonator filter,” Opt. Lett. |

3. | V. Van, “Circuit-Based Method for Synthesizing Serially Coupled Microring Filters,” J. Lightwave Technol. |

4. | N. Ngo, “Synthesis of Tunable Optical Waveguide Filters Using Digital Signal Processing Technique,” J. Lightwave Technol. |

5. | C. K. Madsen, “General IIR Optical Filter Design for WDM Applications Using All-Pass Filters,” J. Lightwave Technol. |

6. | K. Jinguji, “Synthesis of Coherent Two-Port Optical Delay-Line Circuit with Ring Waveguides,” J. Lightwave Technol. |

7. | K. Sasayama, M. Okuno, and K. Habara, “Coherent Optical Transversal Filter using silica-based single-mode waveguides,” Electron. Lett. |

8. | V. Van, “Dual-Mode Microring Reflection Filters,” J. Lightwave Technol. |

9. | S. Vargas and C. Vazquez, “Synthesis of Optical Filters Using Sagnac Interferometer in Ring Resonator,” IEEE Photon. Technol. Lett. |

10. | S. Xiao, M. H. Khan, H. Shen, and M. Qi, “Silicon-on-Insulator Microring Add-Drop Filters With Free Spectral Ranges Over 30 nm,” J. Lightwave Technol. |

11. | Q. Xu, D. Fattal, and R. G. Beausoleil, “Silicon microring resonators with 1.5-microm radius,” Opt. Express |

12. | J. García, A. Martínez, and J. Martí, “Proposal of an OADM configuration with ultra-large FSR combining ring resonators and photonic bandgap structures,” Opt. Commun. |

13. | C. Vázquez and O. Schwelb, “Tunable, narrow-band, grating-assisted microring reflectors,” Opt. Commun. |

14. | D. G. Rabus, M. Hamacher, U. Troppenz, and H. Heidrich, “High Q-Channel Dropping Filters Using Ring Resonators with Integrated SOAs,” IEEE Photon. Technol. Lett. |

15. | J. S. Yang, J. W. Roh, S. H. Ok, D. H. Woo, Y. T. Byun, W. Y. Lee, T. Mizumoto, and S. Lee, “An integrated optical waveguide isolator based on multimode interference by wafer direct bonding,” IEEE Trans. Magn. |

16. | W. Van Parys, D. Van Thourhout, R. Baets, B. Dagens, J. Decobert, O. Le Gouezigou, D. Make, and L. Lagae, “Amplifying Waveguide Optical Isolator with an Integrated electromagnet,” IEEE Photon. Technol. Lett. |

17. | C. Vazquez, S. Vargas, and J. M. S. Pena, “Sagnac Loop in Ring Resonators for Tunable Optical Filters,” J. Lightwave Technol. |

18. | H. Stoll, “Optimally Coupled, GaAs-Distributed Bragg Reflection Lasers,” IEEE Trans. Circ. Syst. |

19. | Q. Xu, I. Cremmos, O. Schwelb, and N. Uzunogly, |

20. | Y. A. Vlasov and S. J. McNab, “Losses in single-mode silicon-on-insulator strip waveguides and bends,” Opt. Express |

21. | M. Gnan, S. Thoms, D. S. Macintyre, R. M. De La Rue, and M. Sorel, “Fabrication of low-loss photonic wires in silicon-on-insulator using hydrogen silsesquioxane electron-beam resis,” Electron. Lett. |

22. | J. Proakis, and D. Manolakis, |

**OCIS Codes**

(050.2770) Diffraction and gratings : Gratings

(230.3990) Optical devices : Micro-optical devices

(350.2460) Other areas of optics : Filters, interference

**ToC Category:**

Optical Devices

**History**

Original Manuscript: October 26, 2010

Manuscript Accepted: November 17, 2010

Published: November 26, 2010

**Citation**

Salvador Vargas and Carmen Vazquez, "Synthesis of optical filters using microring resonators with ultra-large FSR," Opt. Express **18**, 25936-25949 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-25-25936

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### References

- K. Jinguji and T. Yasui, “Synthesis of One-Input M-Output Optical FIR Lattice Circuits,” J. Lightwave Technol. 26(7), 853–866 (2008). [CrossRef]
- A. Melloni, “Synthesis of a parallel-coupled ring-resonator filter,” Opt. Lett. 26(12), 917–919 (2001). [CrossRef]
- V. Van, “Circuit-Based Method for Synthesizing Serially Coupled Microring Filters,” J. Lightwave Technol. 24(7), 2912–2919 (2006). [CrossRef]
- N. Ngo, “Synthesis of Tunable Optical Waveguide Filters Using Digital Signal Processing Technique,” J. Lightwave Technol. 24(9), 3520–3531 (2006). [CrossRef]
- C. K. Madsen, “General IIR Optical Filter Design for WDM Applications Using All-Pass Filters,” J. Lightwave Technol. 18(6), 860–868 (2000). [CrossRef]
- K. Jinguji, “Synthesis of Coherent Two-Port Optical Delay-Line Circuit with Ring Waveguides,” J. Lightwave Technol. 14(8), 1882–1898 (1996). [CrossRef]
- K. Sasayama, M. Okuno, and K. Habara, “Coherent Optical Transversal Filter using silica-based single-mode waveguides,” Electron. Lett. 25(22), 1508–1509 (1989). [CrossRef]
- V. Van, “Dual-Mode Microring Reflection Filters,” J. Lightwave Technol. 25(10), 3142–3150 (2007). [CrossRef]
- S. Vargas and C. Vazquez, “Synthesis of Optical Filters Using Sagnac Interferometer in Ring Resonator,” IEEE Photon. Technol. Lett. 19(23), 1877–1879 (2007). [CrossRef]
- S. Xiao, M. H. Khan, H. Shen, and M. Qi, “Silicon-on-Insulator Microring Add-Drop Filters With Free Spectral Ranges Over 30 nm,” J. Lightwave Technol. 26(2), 228–236 (2008). [CrossRef]
- Q. Xu, D. Fattal, and R. G. Beausoleil, “Silicon microring resonators with 1.5-microm radius,” Opt. Express 16(6), 4309–4315 (2008). [CrossRef] [PubMed]
- J. García, A. Martínez, and J. Martí, “Proposal of an OADM configuration with ultra-large FSR combining ring resonators and photonic bandgap structures,” Opt. Commun. 282(9), 1771–1774 (2009). [CrossRef]
- C. Vázquez and O. Schwelb, “Tunable, narrow-band, grating-assisted microring reflectors,” Opt. Commun. 281(19), 4910–4916 (2008). [CrossRef]
- D. G. Rabus, M. Hamacher, U. Troppenz, and H. Heidrich, “High Q-Channel Dropping Filters Using Ring Resonators with Integrated SOAs,” IEEE Photon. Technol. Lett. 14(10), 1442–1444 (2002). [CrossRef]
- J. S. Yang, J. W. Roh, S. H. Ok, D. H. Woo, Y. T. Byun, W. Y. Lee, T. Mizumoto, and S. Lee, “An integrated optical waveguide isolator based on multimode interference by wafer direct bonding,” IEEE Trans. Magn. 41(10), 3520–3522 (2005). [CrossRef]
- W. Van Parys, D. Van Thourhout, R. Baets, B. Dagens, J. Decobert, O. Le Gouezigou, D. Make, and L. Lagae, “Amplifying Waveguide Optical Isolator with an Integrated electromagnet,” IEEE Photon. Technol. Lett. 19(24), 1949–1951 (2007). [CrossRef]
- C. Vazquez, S. Vargas, and J. M. S. Pena, “Sagnac Loop in Ring Resonators for Tunable Optical Filters,” J. Lightwave Technol. 23(8), 2555–2567 (2005). [CrossRef]
- H. Stoll, “Optimally Coupled, GaAs-Distributed Bragg Reflection Lasers,” IEEE Trans. Circ. Syst. 26(12), 1065–1072 (1979). [CrossRef]
- Q. Xu, I. Cremmos, O. Schwelb, and N. Uzunogly, Photonic Microresonator Research and Applications (Springer New York 2010), Chap. 9.
- Y. A. Vlasov and S. J. McNab, “Losses in single-mode silicon-on-insulator strip waveguides and bends,” Opt. Express 12(8), 1622–1631 (2004). [CrossRef] [PubMed]
- M. Gnan, S. Thoms, D. S. Macintyre, R. M. De La Rue, and M. Sorel, “Fabrication of low-loss photonic wires in silicon-on-insulator using hydrogen silsesquioxane electron-beam resis,” Electron. Lett. 44(2), 115–116 (2008). [CrossRef]
- J. Proakis, and D. Manolakis, Digital Signal Processing, (Prentice Hall 2006). [PubMed]

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