## Efficient wavelength multiplexers based on asymmetric response filters |

Optics Express, Vol. 21, Issue 9, pp. 10903-10916 (2013)

http://dx.doi.org/10.1364/OE.21.010903

Acrobat PDF (1318 KB)

### Abstract

We propose integrated photonic wavelength multiplexers based on serially cascaded channel add-drop filters with an asymmetric frequency response. By utilizing the through-port rejection of the previous channel to advantage, the asymmetric response provides optimal rejection of the adjacent channels at each wavelength channel. We show theoretically the basic requirements to realize an asymmetric filter response, and propose and evaluate the possible implementations using coupled resonators. For one implementation, we provide detailed design formulas based on a coupled-mode theory model, and more generally we provide broad guidelines that enumerate all structures that can provide asymmetric passbands in the context of a pole-zero design approach to engineering the device response. Using second-order microring resonator filter stages as an example, we show that the asymmetric multiplexer can provide 2.4 times higher channel packing (bandwidth) density than a multiplexer using the same order stages (number of resonators) using conventional all-pole maximally-flat designs. We also address the sensitivities and constraints of various implementations of our proposed approach, as it affects their applicability to CMOS photonic interconnects.

© 2013 OSA

## 1. Introduction

1. X. Zheng, J. Lexau, Y. Luo, H. Thacker, T. Pinguet, A. Mekis, G. Li, J. Shi, P. Amberg, N. Pinckney, K. Raj, R. Ho, J. E. Cunningham, and A. V. Krishnamoorthy, “Ultra-low-energy all-CMOS modulator integrated with driver,” Opt. Express **18**, 3059–3070 (2010) [CrossRef] [PubMed] .

7. B. Moss, C. Sun, M. Georgas, J. Shainline, J. Orcutt, J. Leu, M. Wade, H. Li, R. Ram, M. Popovic, and V. Stojanovic, “A 1.23 pJ/bit 2.5Gb/s monolithically-integrated optical carrier-injection ring modulator and all-digital driver circuit in commercial 45nm SOI,” in International Solid-State Circuits Conference (2013), pp. 126–127.

8. S. Beamer, K. Asanović, C. Batten, A. Joshi, and V. Stojanović, “Designing multi-socket systems using silicon photonics,” in *Proceedings of the 23rd International Conference on Supercomputing* (ACM, New York, NY, USA, 2009), ICS ’09, pp. 521–522 [CrossRef] .

9. D. A. B. Miller, “Device requirements for optical interconnects to silicon chips,” Proceedings of the IEEE **97**, 1166–1185 (2009) [CrossRef] .

5. J. S. Orcutt, A. Khilo, C. W. Holzwarth, M. A. Popović, H. Li, J. Sun, T. Bonifield, R. Hollingsworth, F. X. Kärtner, H. I. Smith, V. Stojanović, and R. J. Ram, “Nanophotonic integration in state-of-the-art CMOS foundries,” Opt. Express **19**, 2335–2346 (2011) [CrossRef] [PubMed] .

7. B. Moss, C. Sun, M. Georgas, J. Shainline, J. Orcutt, J. Leu, M. Wade, H. Li, R. Ram, M. Popovic, and V. Stojanovic, “A 1.23 pJ/bit 2.5Gb/s monolithically-integrated optical carrier-injection ring modulator and all-digital driver circuit in commercial 45nm SOI,” in International Solid-State Circuits Conference (2013), pp. 126–127.

10. C. Batten, A. Joshi, J. Orcutt, A. Khilo, B. Moss, C. Holzwarth, M. Popovic, H. Li, H. Smith, J. Hoyt, F. Kartner, R. Ram, V. Stojanovic, and K. Asanovic, “Building many-core processor-to-DRAM networks with monolithic CMOS silicon photonics,” Micro, IEEE **29**, 8–21 (2009) [CrossRef] .

11. A. Prabhu and V. Van, “Realization of asymmetric optical filters using asynhcronous coupled-microring resonators,” Opt. Express **15**, 9645–9658 (2007) [CrossRef] [PubMed] .

15. R. Kurzrok, “General three-resonator filters in waveguide,” IEEE Trans. Microwave Theory Tech. **14**, 46–47 (1966) [CrossRef] .

## 2. Coupling of modes in time model

16. H. Haus and W. Huang, “Coupled-mode theory,” Proceedings of the IEEE **79**, 1505–1518 (1991) [CrossRef] .

*N*-order system. The specific device we are interested in is a 2

^{th}*-order implementation for which we rigorously derive the CMT model and solve the full design equations.*

^{nd}*per*mode, i.e. per FSR of the system. We assume a narrowband approximation, i.e. that the passband is much smaller than the FSR, so that the adjacent azimuthal modes do not contribute to the same passband. However, these constraints are artificial and the same approach can be applied if a single cavity is used to supply multiple resonances that contribute to the passband, for example.

### 2.1. Designing a response with one transmission zero at finite detuning from the passband center

*S*

_{j,input},

*j*∈ {thru,drop}. In general, the number of finite transmission zeros in each s-parameter is equal to

*N*, the resonant order of the system, minus the

*minimum*number of resonators that must be traversed from input to output [13]. Using this rule as a guide, the circuit shown in Fig. 1(b) can achieve the desired transmission response. Specifically, the resonant order of the system is

*N*= 2, and the minimum number of resonators that the light must pass through is one to the drop port, and zero to the through port.

17. R. Collin, *Foundations for Microwave Engineering*, IEEE Press Series on Electromagnetic Wave Theory (John Wiley & Sons, 2001) [CrossRef] .

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

### 2.2. Approximate design equations for an N^{th}-order system

*N*order resonant system of serially coupled resonators can be written as

^{th}*N*first order differential equations, where

*a*is the energy amplitude in the

_{k}*k*

^{th}ring,

*ω*is the resonant frequency of the

_{k}*k*

^{th}ring,

*r*are the decay rates to the input bus and drop bus, respectively,

_{i,d}*μ*is the energy coupling rate to the

_{kl}*k*

^{th}ring from the

*l*

^{th}ring,

*s*is the amplitude of the input wave, and

_{i}*s*is the amplitude of the drop-port output wave. If the resonant frequencies of all rings are set equal, as is the case for typical square passband responses, the desired filter shape is synthesized through choice of the ring-ring couplings,

_{d}*μ*, and the input and drop port decay rates,

_{kl}*r*and

_{i}*r*.

_{d}*μ*

_{(N)(N+1)}

*a*

_{N}_{+1}|/|

*μ*

_{(N)(N−1)}

*a*

_{N}_{−1}| << 1). This is because off-resonance the rings do not like to exchange energy (i.e. when the detuning is much larger than the coupling rate [19]), so coupling from ring 1 to ring 2 is weak, and back from ring 2 back to ring 1 is weaker still because it is a second-order effect in the detuning-induced suppression of coupling. Hence, in the coupling equations we can assume dominant coupling from the ring energy amplitude that is closer to the input bus. This simplifies Eq. 1 by completely decoupling the equations, and should perfectly recover the response in the off-resonant wings of the passband (only). Our goal is to design a circuit to achieve one finite transmission zero in the drop port of the device. Figure 2(a) shows an extension of Fig. 1(b) to achieve this for increasing order microring filters. In all of these filters, bypassing the

*N*

^{th}ring with a tap at the (

*N*− 1)

^{th}ring coupled directly to the drop port enables the asymmetric response by ensuring a single drop-port response function zero. The weaker the tap coupling, the further detuned the transmission zero is from the passband. In the limit of zero tap coupling to the (

*N*− 1)

^{th}ring, the standard symmetric response is recovered.

*N*− 1)

^{th}ring. After further making the foregoing off-resonant approximation, i.e. that the energy amplitude

*a*is excited primarily by the previous energy amplitude

_{k}*a*

_{k}_{−1}, Eqs. 1 can be simplified to

*r*is the decay rate to the tap port,

_{t}*ϕ*is the propagation phase accumulated in the interference arm, and

*s′*[see Fig. 3(a)] is given by The output wave,

_{d}*s*, can be then be found from Letting

_{d}*d/dt*→

*jω*to solve for the steady state frequency response of the system, Eqs. 2–4 can be solved for the transfer function,

*S*

_{d,i}(

*ω*) ≡

*s*/

_{d}*s*(valid off resonance) The root of the numerator in Eq. 5 gives the frequency position of the transmission zero which, since it is off resonant, can be found from this approximate model. Setting the imaginary part of the root to zero to place the transmission zero on the real frequency axis and introducing

_{i}*δω*as the desired detuning from the passband (resonant) frequency to the transmission zero, two simple design equations can be derived that give the phase delay needed in the interference arm [see Fig. 1(b)] as well as the decay rate to the tap port: The remaining decay rates and ring-ring couplings can be taken from the standard all-pole design synthesis techniques [18

_{zd}18. B. Little, S. Chu, H. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” IEEE J. Lightwave Technol. **15**, 998–1005 (1997) [CrossRef] .

20. A. Melloni and M. Martinelli, “Synthesis of direct-coupled-resonators bandpass filters for WDM systems,” IEEE J. Lightwave Technol. **20**, 296 (2002) [CrossRef] .

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

*-order pole-zero filter using Eqs. 6 and 7 with a design zero location of*

^{nd}*δω*/

*r*= 10.

_{i}### 2.3. Rigorous solution of the 2^{nd}-order filter synthesis problem

*d/dt*→

*jω*, a transmission matrix

**T̳**

_{3×3}, where

*s⃗*

_{−}=

**T̳**

_{3×3}

*·s⃗*

_{+}, can be derived, The coupling arm can now be connected. That is, port

*s′*is connected to

_{d}*s′*after a finite propagation distance. For our 3-port model, this eliminates one input and output port, and reduces the model to a 2 × 2 T-matrix, described by where where

_{a}*ϕ*(≡

*βL*) is the phase accumulated in the coupling arm. The reduced transmission matrix is We are interested in the through port response,

*T*

_{11}, and the drop port response,

*T*

_{21}, given by

*δω′*is the relative detuning of the two rings such that their resonances are at

*ω*

_{1}=

*ω*

_{0}+

*δω′*and

*ω*

_{2}=

*ω*

_{0}−

*δω′*.

*T*

_{11}is set to 0 and solved for

*δω*, giving This equation produces two constraints that must be satisfied in order to have real zeros in the through port: The constraint in Eq. 19 simplifies to which leads to a design equation that determines

*δω′*:

*T*

_{21}(

*δω*), given by Eq. 16. Its transmission zero is given by Since

*ϕ*is not yet determined, this equation for the zero location can be interpreted graphically as a root locus in the complex-

*δω*plane. For various

*ϕ*, the zero is located on a circle with radius

*δω′*,

*r*), as shown in Fig. 3(b). This interpretation leads to a simple design equation for

_{d}*ϕ*, the phase accumulated in the interference arm, to place the zero on the real frequency axis: as well as for the tap coupling,

*r*, At this point, we have fixed all degrees of freedom of the model. The total list of parameters of the device relevant in our synthesis includes

_{t}*r*,

_{i}*μ*,

*δω*,

_{zd}*r*,

_{t}*r*,

_{d}*δω′*and

*ϕ*. The first three (

*r*,

_{i}*μ*,

*δω*) are chosen to be inputs to the model. The choice of

_{zd}*r*and

_{i}*μ*largely determines the passband shape (maximally flat, equiripple, bandwidth, etc.), and these exist in all-pole (serially-coupled) ring filters. Detuning

*δω*is the desired location of the drop port zero. Without loss of generality, here, we choose

_{zd}*r*and

_{i}*μ*to be those of an all-pole 2

*-order Butterworth filter. This is given by*

^{nd}*r*=

_{i}*μ*[18

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

20. A. Melloni and M. Martinelli, “Synthesis of direct-coupled-resonators bandpass filters for WDM systems,” IEEE J. Lightwave Technol. **20**, 296 (2002) [CrossRef] .

*r*,

_{t}*r*,

_{d}*δω′*and

*ϕ*using the derived expressions. Figure 4 shows the transmission of a representative pole-zero filter, and a 2

*-order Butterworth filter of equal 3 dB bandwidth for comparison.*

^{nd}*r*and

_{i}*μ*no longer give exactly the bandwidth and passband ripple that a Butterworth or Chebyshev design sets for them, but they are close enough for all practical designs that they can either be used as is or adjusted slightly to get the exact desired parameters. The fixing of the zeros ensures that the passband is not distorted and has (in principle) complete dropping of wavelengths in the passband.

## 3. Design of a serial demultiplexer based on asymmetric response stages

*-order Butterworth response on the left side of the passband, and it rolls off much faster on the right side between the center frequency and the zero location. To the right of the zero, there is again an increase in transmission. If this increase in transmission is detrimental in a particular application, e.g. as a crosstalk level, the designer must be able to set bounds on the maximum tolerable out-of-band transmission. In general, there is a tradeoff between how close a zero is to the passband (allowing a sharper rolloff) and the worst-case off-resonant rejection out of band. For the purposes of a serial demultiplexer, this will affect the adjacent channel rejection. The zero location in Fig. 4 was chosen to ensure a minimum 20 dB adjacent channel rejection.*

^{nd}*T*

_{21}|

^{2}and |

*T*

_{31}|

^{2}, and Channel 2 drop port, |

*T*

_{61}|

^{2}= |

*T*

_{64}

*T*

_{21}|

^{2}, where |

*T*

_{64}|

^{2}(and |

*T*

_{31}|

^{2}) has the response shown in Fig. 5(b). The through port response of the Channel 1 filter shapes the left side of the drop port response at the Channel 2 filter. This outcome is achieved when the channel spacing is set equal to the detuning of the zero from the passband center.

*-order Butterworth filter achieves a channel spacing of 106GHz, i.e. bandwidth utilization under 19%. The pole-zero filter bank gives a 2.4 times denser channel packing, i.e. higher bandwidth density, with no increase in filter order. Figures 6(a) and 6(b) show the responses of the example pole-zero and Butterworth demultiplexers based on second-order filter stages for comparison. It should be noted that although the pole-zero filters were derived from the Butterworth design, the transmission zero causes there to be a slight ripple in the passband. Comparing the pole-zero filter to a Chebyshev filter with approximately the same ripple, the channel packing is about 1.8 times denser using a pole-zero filter bank compared to a Chebyshev filter bank.*

^{nd}## 4. Alternative topologies for a filter response with two poles and one zero

5. J. S. Orcutt, A. Khilo, C. W. Holzwarth, M. A. Popović, H. Li, J. Sun, T. Bonifield, R. Hollingsworth, F. X. Kärtner, H. I. Smith, V. Stojanović, and R. J. Ram, “Nanophotonic integration in state-of-the-art CMOS foundries,” Opt. Express **19**, 2335–2346 (2011) [CrossRef] [PubMed] .

6. J. S. Orcutt, B. Moss, C. Sun, J. Leu, M. Georgas, J. Shainline, E. Zgraggen, H. Li, J. Sun, M. Weaver, S. Urošević, M. Popović, R. J. Ram, and V. Stojanović, “Open foundry platform for high-performance electronic-photonic integration,” Opt. Express **20**, 12222–12232 (2012) [CrossRef] [PubMed] .

## 5. Conclusion

## Acknowledgments

## References and links

1. | X. Zheng, J. Lexau, Y. Luo, H. Thacker, T. Pinguet, A. Mekis, G. Li, J. Shi, P. Amberg, N. Pinckney, K. Raj, R. Ho, J. E. Cunningham, and A. V. Krishnamoorthy, “Ultra-low-energy all-CMOS modulator integrated with driver,” Opt. Express |

2. | G. Li, X. Zheng, J. Yao, H. Thacker, I. Shubin, Y. Luo, K. Raj, J. E. Cunningham, and A. V. Krishnamoorthy, “25Gb/s 1V-driving CMOS ring modulator with integrated thermal tuning,” Opt. Express |

3. | J. C. Rosenberg, W. M. J. Green, S. Assefa, D. M. Gill, T. Barwicz, M. Yang, S. M. Shank, and Y. A. Vlasov, “A 25 Gbps silicon microring modulator based on an interleaved junction,” Opt. Express |

4. | N. Sherwood-Droz and M. Lipson, “Scalable 3D dense integration of photonics on bulk silicon,” Opt. Express |

5. | J. S. Orcutt, A. Khilo, C. W. Holzwarth, M. A. Popović, H. Li, J. Sun, T. Bonifield, R. Hollingsworth, F. X. Kärtner, H. I. Smith, V. Stojanović, and R. J. Ram, “Nanophotonic integration in state-of-the-art CMOS foundries,” Opt. Express |

6. | J. S. Orcutt, B. Moss, C. Sun, J. Leu, M. Georgas, J. Shainline, E. Zgraggen, H. Li, J. Sun, M. Weaver, S. Urošević, M. Popović, R. J. Ram, and V. Stojanović, “Open foundry platform for high-performance electronic-photonic integration,” Opt. Express |

7. | B. Moss, C. Sun, M. Georgas, J. Shainline, J. Orcutt, J. Leu, M. Wade, H. Li, R. Ram, M. Popovic, and V. Stojanovic, “A 1.23 pJ/bit 2.5Gb/s monolithically-integrated optical carrier-injection ring modulator and all-digital driver circuit in commercial 45nm SOI,” in International Solid-State Circuits Conference (2013), pp. 126–127. |

8. | S. Beamer, K. Asanović, C. Batten, A. Joshi, and V. Stojanović, “Designing multi-socket systems using silicon photonics,” in |

9. | D. A. B. Miller, “Device requirements for optical interconnects to silicon chips,” Proceedings of the IEEE |

10. | C. Batten, A. Joshi, J. Orcutt, A. Khilo, B. Moss, C. Holzwarth, M. Popovic, H. Li, H. Smith, J. Hoyt, F. Kartner, R. Ram, V. Stojanovic, and K. Asanovic, “Building many-core processor-to-DRAM networks with monolithic CMOS silicon photonics,” Micro, IEEE |

11. | A. Prabhu and V. Van, “Realization of asymmetric optical filters using asynhcronous coupled-microring resonators,” Opt. Express |

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

13. | M. Popovic, “Theory and design of high-index-contrast microphotonic circuits,” Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA (2007). |

14. | M. Popovic, “Sharply-defined optical filters and dispersionless delay lines based on loop-coupled resonators and ‘negative’ coupling,” in |

15. | R. Kurzrok, “General three-resonator filters in waveguide,” IEEE Trans. Microwave Theory Tech. |

16. | H. Haus and W. Huang, “Coupled-mode theory,” Proceedings of the IEEE |

17. | R. Collin, |

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

19. | H. A. Haus, |

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

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

22. | W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. |

23. | B. Little, S. Chu, J. Hryniewicz, and P. Absil, “Filter synthesis for periodically coupled microring resonators,” Opt. Lett. |

24. | S. Chu, B. Little, W. Pan, T. Kaneko, S. Sato, and Y. Kokubun, “An eight-channel add-drop filter using vertically coupled microring resonators over a cross grid,” IEEE Photon. Technol. Lett. |

25. | Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, and M. Lipson, “Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency,” Phys. Rev. Lett. |

26. | N. Sherwood-Droz, H. Wang, L. Chen, B. Lee, A. Biberman, K. Bergman, and M. Lipson, “Optical 4×4 hitless silicon router for optical networks-on-chip (NoC),” Opt. Express |

27. | S. Emelett and R. Soref, “Synthesis of dual-microring-resonator cross-connect filters,” Opt. Express |

**OCIS Codes**

(130.3120) Integrated optics : Integrated optics devices

(250.5300) Optoelectronics : Photonic integrated circuits

(130.7408) Integrated optics : Wavelength filtering devices

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: February 27, 2013

Revised Manuscript: April 20, 2013

Manuscript Accepted: April 22, 2013

Published: April 26, 2013

**Citation**

Mark T. Wade and Miloš A. Popović, "Efficient wavelength multiplexers based on asymmetric response filters," Opt. Express **21**, 10903-10916 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-9-10903

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### References

- X. Zheng, J. Lexau, Y. Luo, H. Thacker, T. Pinguet, A. Mekis, G. Li, J. Shi, P. Amberg, N. Pinckney, K. Raj, R. Ho, J. E. Cunningham, and A. V. Krishnamoorthy, “Ultra-low-energy all-CMOS modulator integrated with driver,” Opt. Express18, 3059–3070 (2010). [CrossRef] [PubMed]
- G. Li, X. Zheng, J. Yao, H. Thacker, I. Shubin, Y. Luo, K. Raj, J. E. Cunningham, and A. V. Krishnamoorthy, “25Gb/s 1V-driving CMOS ring modulator with integrated thermal tuning,” Opt. Express19, 20435–20443 (2011). [CrossRef] [PubMed]
- J. C. Rosenberg, W. M. J. Green, S. Assefa, D. M. Gill, T. Barwicz, M. Yang, S. M. Shank, and Y. A. Vlasov, “A 25 Gbps silicon microring modulator based on an interleaved junction,” Opt. Express20, 26411–26423 (2012). [CrossRef] [PubMed]
- N. Sherwood-Droz and M. Lipson, “Scalable 3D dense integration of photonics on bulk silicon,” Opt. Express19, 17758–17765 (2011). [CrossRef] [PubMed]
- J. S. Orcutt, A. Khilo, C. W. Holzwarth, M. A. Popović, H. Li, J. Sun, T. Bonifield, R. Hollingsworth, F. X. Kärtner, H. I. Smith, V. Stojanović, and R. J. Ram, “Nanophotonic integration in state-of-the-art CMOS foundries,” Opt. Express19, 2335–2346 (2011). [CrossRef] [PubMed]
- J. S. Orcutt, B. Moss, C. Sun, J. Leu, M. Georgas, J. Shainline, E. Zgraggen, H. Li, J. Sun, M. Weaver, S. Urošević, M. Popović, R. J. Ram, and V. Stojanović, “Open foundry platform for high-performance electronic-photonic integration,” Opt. Express20, 12222–12232 (2012). [CrossRef] [PubMed]
- B. Moss, C. Sun, M. Georgas, J. Shainline, J. Orcutt, J. Leu, M. Wade, H. Li, R. Ram, M. Popovic, and V. Stojanovic, “A 1.23 pJ/bit 2.5Gb/s monolithically-integrated optical carrier-injection ring modulator and all-digital driver circuit in commercial 45nm SOI,” in International Solid-State Circuits Conference (2013), pp. 126–127.
- S. Beamer, K. Asanović, C. Batten, A. Joshi, and V. Stojanović, “Designing multi-socket systems using silicon photonics,” in Proceedings of the 23rd International Conference on Supercomputing (ACM, New York, NY, USA, 2009), ICS ’09, pp. 521–522. [CrossRef]
- D. A. B. Miller, “Device requirements for optical interconnects to silicon chips,” Proceedings of the IEEE97, 1166–1185 (2009). [CrossRef]
- C. Batten, A. Joshi, J. Orcutt, A. Khilo, B. Moss, C. Holzwarth, M. Popovic, H. Li, H. Smith, J. Hoyt, F. Kartner, R. Ram, V. Stojanovic, and K. Asanovic, “Building many-core processor-to-DRAM networks with monolithic CMOS silicon photonics,” Micro, IEEE29, 8–21 (2009). [CrossRef]
- A. Prabhu and V. Van, “Realization of asymmetric optical filters using asynhcronous coupled-microring resonators,” Opt. Express15, 9645–9658 (2007). [CrossRef] [PubMed]
- C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach (Wiley-Interscience, 1999). [CrossRef]
- M. Popovic, “Theory and design of high-index-contrast microphotonic circuits,” Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA (2007).
- M. Popovic, “Sharply-defined optical filters and dispersionless delay lines based on loop-coupled resonators and ‘negative’ coupling,” in IEEE Conference on Lasers and Electro-Optics (2007), pp. 1–2. Paper CthP6.
- R. Kurzrok, “General three-resonator filters in waveguide,” IEEE Trans. Microwave Theory Tech.14, 46–47 (1966). [CrossRef]
- H. Haus and W. Huang, “Coupled-mode theory,” Proceedings of the IEEE79, 1505–1518 (1991). [CrossRef]
- R. Collin, Foundations for Microwave Engineering, IEEE Press Series on Electromagnetic Wave Theory (John Wiley & Sons, 2001). [CrossRef]
- B. Little, S. Chu, H. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” IEEE J. Lightwave Technol.15, 998–1005 (1997). [CrossRef]
- H. A. Haus, Waves and Fields in Optoelectronics, Solid state physical electronics series (Prentice-Hall, 1984).
- A. Melloni and M. Martinelli, “Synthesis of direct-coupled-resonators bandpass filters for WDM systems,” IEEE J. Lightwave Technol.20, 296 (2002). [CrossRef]
- R. Orta, P. Savi, R. Tascone, and D. Trinchero, “Synthesis of multiple-ring-resonator filters for optical systems,” IEEE Photon. Technol. Lett.7, 1447–1449 (1995). [CrossRef]
- W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron.40, 1511–1518 (2004). [CrossRef]
- B. Little, S. Chu, J. Hryniewicz, and P. Absil, “Filter synthesis for periodically coupled microring resonators,” Opt. Lett.25, 344–346 (2000). [CrossRef]
- S. Chu, B. Little, W. Pan, T. Kaneko, S. Sato, and Y. Kokubun, “An eight-channel add-drop filter using vertically coupled microring resonators over a cross grid,” IEEE Photon. Technol. Lett.11, 691–693 (1999). [CrossRef]
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