## Synthesis of dual-microring-resonator cross-connect filters

Optics Express, Vol. 13, Issue 12, pp. 4439-4456 (2005)

http://dx.doi.org/10.1364/OPEX.13.004439

Acrobat PDF (2044 KB)

### Abstract

A new type of resonant, waveguided, 2×2 cross-connect optical filter is proposed and synthesized using a microwave filter analog. The optical passbands of the device are determined using 2D scattering matrix theory and the desired response is generated via a synthesis for a combined singly and doubly terminated circuit. This synthesis realizes the microring coupling coefficients necessary for maximally flat infrared spectral response. Closed-form analytical solutions are presented. Devices containing two, four, and six microrings were investigated.

© 2005 Optical Society of America

## 1. Introduction

1. S. T. Chu, B.E. Little, W. Pan, T. Kaneko, 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]

2. S. T. Chu, B.E. Little, W. Pan, T. Kaneko, and Y. Kokubun, “Second-Order Filter Response from Parallel Coupled Glass Microring Resonators,” IEEE Photon Technol. Lett. **11**,1426–1428, (1999). [CrossRef]

4. W. K. Burns and A. F. Milton
, “Waveguides Transitions and Junctions,” in *Guided-Wave Optoelectronics*-Second Edition,
T. Tamir, ed. (Springer-Verlag, Brooklyn, New York, 1990). [CrossRef]

2. S. T. Chu, B.E. Little, W. Pan, T. Kaneko, and Y. Kokubun, “Second-Order Filter Response from Parallel Coupled Glass Microring Resonators,” IEEE Photon Technol. Lett. **11**,1426–1428, (1999). [CrossRef]

5. Y. Yanagase, S. Suzuki, Y. Kokubun, and S. T. Chu, “Box-Like Filter Response and Expansion of FSR by a Vertically Triple Coupled Microring Resonator Filter,” J. Lightwave Technol. **20**, 1525–1529, (2002). [CrossRef]

## 2. 2-D optical scattering matrix theory

(i) | (v) | ||

(ii) | (vi) | ||

(iii) | (vii) | ||

(iv) | (viii) |

*E*

_{I}is the input field to the horizontal through-port bus waveguide,

*E*

_{T}is the output field at the bus through-port,

*E*

_{D}is the dropped field in the vertical bus waveguide, and

*T*

_{r}is the round trip signal time of the ring;

*T*

_{r}=

*Ln*/

*c*, where

*L*is the circumference of the ring,

*n*is the refractive index in all portions of the ring and

*c*is the speed of light in vacuum. It is assumed that the refractive index,

*n*, is very similar to the waveguides’ effective index,

*n*

_{eff}. The quantity

*α*

_{r}is the initial or static absorption coefficient of the ring material. Our theory allows for intrinsic waveguide losses to be accounted for by using a complex index definition,

*n*+

*jκ̄*. However when calculations are presented later, we have chosen to simplify the analysis by assuming an initially lossless system in which

*κ̄*=0. For the moment, static loss is neglected in order to concentrate upon the essential characteristics and synthesis of this device. There is no need to distinguish between the first and second rings because the rings are identical, but the distinction for the field’s nomenclature is left on in Fig. 1 and the field equations in order to verify the direction of propagation.

*κ*

_{a}and

*κ*

_{aa}are the coupling constants between the ring and bus guide, whose spacing are

*t*

_{a}and

*t*

_{aa}, while

*κ*

_{b}is the coupling constant between the two rings, whose separation is

*t*

_{b}. These coupling coefficients are explained in [6

6. S. J. Emelett and R. A. Soref, “Design and Simulation of Silicon Microring Optical Routing Switches,” J. Lightwave Technol. **23**, 1800–1807, (2005). [CrossRef]

*κ*

_{a}and

*κ*

_{aa}are the same for the through and drop waveguides, but once again are represented in order to aid in future designs and to facilitate propagation-direction determination. This concept holds true for the dealing with

*α*

_{r}as well, but also maintaining the loss dependency in these equation permits others to readily insert the loss into their calculations. And finally, τ

_{a,b}is calculated from the notion of lossless coupling, namely

6. S. J. Emelett and R. A. Soref, “Design and Simulation of Silicon Microring Optical Routing Switches,” J. Lightwave Technol. **23**, 1800–1807, (2005). [CrossRef]

## 3. Microwave circuit analog of the coupled ring system - filter synthesis

### 3.1 The singly and doubly terminated prototype circuits

9. S. B. Cohn, “Direct-Coupled-Resonator Filters,” Proc. IRE.187–195, Feb. (1957). [CrossRef]

10. A. Melloni and M. Martinelli, “Synthesis of Direct-Coupled-Resonators Bandpass Filters for WDM Systems,” J. Lightwave Technol. **20**, 296–303, (2002). [CrossRef]

*q*represents the specific element number and

*N*is the order of the system,

*i.e*. the total number of rings. This system is conventionally referred to as a microring lattice filter. In the optical domain it is traditionally constructed of

*N*rings with only one bus waveguide (located at the

*z*transforms and synthesis algorithms which produce the zeros and poles of the systems that in turn generate the response [11

11. C. K. Madsen and G. Lenz, “Optical All-Pass Filters for Phase Response Design with Applications for Dispersion Compensation,” IEEE Photon. Technol. Lett. **10**, 994–996, (1998). [CrossRef]

13. R. Grover, *Indium Phosphide based optical micro-ring resonators*. Ph.D.thesis, Univ. of Maryland, College Park, Maryland, U.S.A., (2003), http://www.enee.umd.edu/research/microphotonics.

14. B.E. Little, S.T. Chu, H.A. Haus, J. Foresi, and J.-P Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. **15**, 998–1005, (1992). [CrossRef]

*N*rings. In the case represented in Fig. 3 it can be seen that

*N*=1. This system can also be designed with the aforementioned

*z*-transform approach or with the just-mentioned Melloni’s method. Another technique which provides a Butterworth response is elucidated in Little’s manuscript where he calls for a ratio of the fractional power coupled to zero the power loss ratio in polynomial form. When Melloni’s synthesis results are tested against this ratio, the anticipated excellent agreement is found [14

14. B.E. Little, S.T. Chu, H.A. Haus, J. Foresi, and J.-P Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. **15**, 998–1005, (1992). [CrossRef]

## 3.2 The cross-connect prototype circuit synthesis and realization

*i.e*.

*N*=2

*n*

^{′}, where

*n*

^{′}is an integer, then the spectral response displayed in Fig. 2 can be expected. Regular order refers to the total number of rings in the system, not just one part of the circuit. If the regular order is an odd integer, the following procedure does not apply and the characteristics of the device and synthesis are lost. In the doubly terminated circuit elements, whose realization is depicted in the horizontal box of Fig. 3, we can find the circuit’s order is always

*N*

^{D}=1 due to the innate geometry of the device. We therefore take the elemental value of

*N*

^{D}=1. Progressing to the intersection of the boxes in Fig. 3 or the unifying element of the synthesis, we see that the circuit now belongs to both the singly and doubly terminated prototypes. Therefore, the synthesis must reflect this change and will persist through the rest of the synthesis. For this reason, a traditional recursion relation or generating function cannot be created. This property of the synthesis runs in accordance with the microwave theory.

*N*=6 configuration will initially be solved. The aforementioned and other elements are as follows:

*N*is the regular order of the system-the total number of rings,

*η*is the pair order in the system,

*e.g.*if

*N*=2, 4, or 6 as shown in Fig. 4, then

*η*=0,1 or 2, and the elemental constants,

*i.e.*

*F*to signify the singly terminated circuit comprising the elements pertaining to the floating ring(s) segment. The nomenclature is changed because the overall circuit is no longer a strictly singly terminated circuit but one with a contributions from both the vertical and horizontal circuits depicted in Fig. 3. Along with the changes to the elements, the junction parameter

*ε*is altered to

*q*=2 element. Specifically the doubly dependent

*ε*(since

*N*

^{D}=1) is addressed by the

*q*=1 terms are maintained along with the aforementioned singly dependent ones. This trend of removing the dependency of the previous element is maintained by the

*N*. Numeric trials resulted in the realization that it is a function of the order. The trials consisted of modeling the system with the newly created coupling coefficients and obtaining a convergence to the desired

*A*

_{m}and Δ

*λ*.

*N*=4, it was understood that

*N*=2

*η*, a workable synthesis was found. All other elements of the singly terminated synthesis remain the same.

*N*=6, an iterative search of coefficients revealed the

*N*as well.

*N*

^{D}=1 and separately the

*N*=2, as has been previously prescribed. We can see that the system can be thought of as a doubly terminated circuit with the floating ring inverter as an additional term in the synthesis. Generating the elements in the previously stated fashion of Eq. (12a, 12b, 12g) yields:

*ε*dependency from

*K*

_{q}[10

10. A. Melloni and M. Martinelli, “Synthesis of Direct-Coupled-Resonators Bandpass Filters for WDM Systems,” J. Lightwave Technol. **20**, 296–303, (2002). [CrossRef]

6. S. J. Emelett and R. A. Soref, “Design and Simulation of Silicon Microring Optical Routing Switches,” J. Lightwave Technol. **23**, 1800–1807, (2005). [CrossRef]

## 4. Discussion and summary

*i.e.*

*N*=4 and 6. The field equations for the four- and six-ring systems were solved by the previously utilized scattering matrix approach and simplified with the commercial software

*Mathematica.*The

*N*=2, 4, and 6 results were then plotted concurrently to verify that as the regular order number increased, the spectral response of the system produced a more “box-like” characteristic form. Figure 5 clearly confirms that as the regular order number is increased, this synthesis generates the anticipated response, namely an increasingly rectangular shape.

*N*>2 systems, which are responses that we can predict using the present synthesis. For instance, when

*N*=4, the dropping port produces four peaks as shown in Fig. 6, but this pattern unfortunately terminates at this number,

*i.e.*four-peaked response for

*N*=6, 8, 10, etc., and also coincides with the most symmetrical responses for

*N*>2. Fig. 6 also displays the

*N*=2 and 6 responses and their ability to be controlled by the synthesis. The only part that is being controlled though is the spacing of the two central peaks. The specifics of the quartet peak effect are presented in Table 1 in relation to

*λ*

_{0}=1.33

*µm*. These number were obtained by entering the design parameters of the device (which are located below the tables) into Eq.(12) and Eq.(15). This table includes the wavelength locations of unity output in the passband peaks of the drop-port filter characteristic These wavelengths correspond to the notch zeros in the through-port filter response which is displayed in normalized fashion in the inset of Fig. 6. These results demonstrate the aforementioned notion that the

*N*=4 response is the most symmetric of the

*N*>2 responses. The present synthesis does not offer a means for obtaining equal wavelength separations between quartet peaks when

*N*is 4 or greater.

*λ*

_{0}, of these devices are 1.33

*µm*where

*n*=3.52,

*M*

_{q}=34, Δ

*λ*=0.05 nm and

*A*

_{m}=5.0. These results coincide with the normalized response displayed in Fig. 6.

*λ*

_{0},of these devices is 1.33

*µm*where

*n*=3.52 and

*M*

_{q}=34.

*i.e.*

*A*

_{m}and

*Δλ*, and/or reconfiguring the ring radii, but also may arise from the conversion from the frequency to wavelength domains, or possibly an intrinsic discrepancy of the synthesis itself may play a role. Although the discrepancies that appear in Fig. 8 are not drastic, when a significant inconsistency occurs it can be corrected by the previously mentioned reassessment of the intersection points and ring radii. This precisely consists of selecting another mode number,

*M*

_{q}, and re-computing the entire synthesis in light of this change. Usually only a very a small change can be obtained in this manner, unless the desired intersection is beyond the possibilities of the device. This is due to the fact that the synthesis is quite insensitive to

*M*

_{q}, as long as the device in question is possible of obtaining such responses. Once the device in question is out of range of the desired response, the influence of increasing or decreasing the radius,

*R*(

*M*

_{q}), has a more dramatic effect. If the preferred location is still not obtained, then selecting a different co-ordinate pair near,

*i.e.*an intersection point above, below, to the left and/or the right, the desired location that should fulfill the required position when the response actually meets the former intersection point. This point is chosen and the synthesis is reexamined until a suitable convergence is reached. These severe steps are usually not needed as long as the synthesis’s demands do not reach beyond what is logically and physically plausible. Examining Fig. 8’s devices with these techniques, structures 1 and 3 can be brought up to a perfect intersection if that is desired, but for all practical purposes these results serve their purposes. Although only a very small selection of specific radii and responses are presented for a single wavelength, it can be found that this synthesis is in fact wavelength independent and the role of the device’s dimensions, which are a function of

*M*

_{q}, are more concerned with the “tuning” procedure, device capabilities, and the actual fabrication limitations and not specifically with the limitations of the synthesis. This fact is supported and impressions of the device’s tolerances are given in Fig. 9. It displays Fig. 8’s Structure 2 and plots of a 5% and a 10% increase and decrease in the ring-ring coupling coefficient,

*κ*

_{2}.

## 5. Conclusion

## References and Links

1. | S. T. Chu, B.E. Little, W. Pan, T. Kaneko, and Y. Kokubun, “An Eight-Channel Add-Drop Filter Using Vertically Coupled Microring Resonators over a Cross Grid,” IEEE Photon Technol. Lett. |

2. | S. T. Chu, B.E. Little, W. Pan, T. Kaneko, and Y. Kokubun, “Second-Order Filter Response from Parallel Coupled Glass Microring Resonators,” IEEE Photon Technol. Lett. |

3. | Y. Kokubun, T. Kato, and S.T. Chu, “Box-Like Response of Microring Resonator Filter by Stacked Double-Ring Geometry,” IEICE Trans. Electron. |

4. | W. K. Burns and A. F. Milton
, “Waveguides Transitions and Junctions,” in |

5. | Y. Yanagase, S. Suzuki, Y. Kokubun, and S. T. Chu, “Box-Like Filter Response and Expansion of FSR by a Vertically Triple Coupled Microring Resonator Filter,” J. Lightwave Technol. |

6. | S. J. Emelett and R. A. Soref, “Design and Simulation of Silicon Microring Optical Routing Switches,” J. Lightwave Technol. |

7. | G. L. Matthaei, L. Young, and E. M. T. Jones, |

8. | R. E. Collins, |

9. | S. B. Cohn, “Direct-Coupled-Resonator Filters,” Proc. IRE.187–195, Feb. (1957). [CrossRef] |

10. | A. Melloni and M. Martinelli, “Synthesis of Direct-Coupled-Resonators Bandpass Filters for WDM Systems,” J. Lightwave Technol. |

11. | C. K. Madsen and G. Lenz, “Optical All-Pass Filters for Phase Response Design with Applications for Dispersion Compensation,” IEEE Photon. Technol. Lett. |

12. | C. K. Madsen and J.H. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach (Wiley & Sons, New York1999). |

13. | R. Grover, |

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

**OCIS Codes**

(230.0230) Optical devices : Optical devices

(250.5300) Optoelectronics : Photonic integrated circuits

(350.2460) Other areas of optics : Filters, interference

(350.3950) Other areas of optics : Micro-optics

**ToC Category:**

Research Papers

**History**

Original Manuscript: May 2, 2005

Revised Manuscript: May 27, 2005

Published: June 13, 2005

**Citation**

S. Emelett and R. Soref, "Synthesis of dual-microring-resonator cross-connect filters," Opt. Express **13**, 4439-4456 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-12-4439

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### References

- S. T. Chu, B.E. Little, W. Pan, T. Kaneko 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]
- S. T. Chu, B.E. Little, W. Pan, T. Kaneko and Y. Kokubun, �?? Second-Order Filter Response from Parallel Coupled Glass Microring Resonators,�?? IEEE Photon Technol. Lett. 11,1426-1428, (1999). [CrossRef]
- Y. Kokubun, T. Kato, S.T. Chu, �??Box-Like Response of Microring Resonator Filter by Stacked Double-Ring Geometry,�?? IEICE Trans. Electron. E85-C,1018-1024, (2000).
- W. K. Burns and A. F. Milton, �??Waveguides Transitions and Junctions,�?? in Guided-Wave Optoelectronics-Second Edition, T. Tamir, ed. (Springer-Verlag, Brooklyn, New York, 1990). [CrossRef]
- Y. Yanagase, S. Suzuki, Y. Kokubun, S. T. Chu, �??Box-Like Filter Response and Expansion of FSR by a Vertically Triple Coupled Microring Resonator Filter,�?? J. Lightwave Technol. 20, 1525-1529, (2002). [CrossRef]
- S. J. Emelett and R. A. Soref, �??Design and Simulation of Silicon Microring Optical Routing Switches,�?? J. Lightwave Technol. 23, 1800-1807, (2005). [CrossRef]
- G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters, Impedance Matching Networks and Coupling Structures (McGraw-Hill, New York 1964), Chap. 4, 8, 11, 14.
- R. E. Collins, Foundations for Microwave Engineering (McGraw-Hill, New York 1966).
- S. B. Cohn, �??Direct-Coupled-Resonator Filters,�?? Proc. IRE. 187-195, Feb. (1957). [CrossRef]
- A. Melloni and M. Martinelli, �??Synthesis of Direct-Coupled-Resonators Bandpass Filters for WDM Systems,�?? J. Lightwave Technol. 20, 296-303, (2002). [CrossRef]
- C. K . Madsen and G. Lenz, �??Optical All-Pass Filters for Phase Response Design with Applications for Dispersion Compensation,�?? IEEE Photon. Technol. Lett. 10, 994-996, (1998). [CrossRef]
- C. K . Madsen J.H. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach (Wiley & Sons, New York 1999).
- R. Grover, Indium Phosphide based optical micro-ring resonators. Ph.D.thesis, Univ. of Maryland, College Park, Maryland, U.S.A., (2003), <a href= " http://www.enee.umd.edu/research/microphotonics.">http://www.enee.umd.edu/research/microphotonics.<a/>
- B.E. Little, S.T. Chu, H.A. Haus, J. Foresi and J.-P Laine, �??Microring resonator channel dropping filters,�?? J. Lightwave Technol. 15, 998-1005, (1992). [CrossRef]

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