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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 12 — Jun. 13, 2005
  • pp: 4439–4456
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Synthesis of dual-microring-resonator cross-connect filters

S. J. Emelett and R. A. Soref  »View Author Affiliations


Optics Express, Vol. 13, Issue 12, pp. 4439-4456 (2005)
http://dx.doi.org/10.1364/OPEX.13.004439


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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

A new type of resonant 2×2 optical filter is investigated in this paper. The structure has a crossbar geometry in which two bus-channel waveguides cross at 90°. This 2×2 inplane device, constructed typically in silicon-on-insulator, has lateral optical coupling of the waveguided microring resonators. The main advantage of these devices is that they are readily interconnected to create an N×N crossbar array, also known as an optical cross-connect, which makes filtering more versatile by utilizing both horizontal and vertical pathways for cascaded or multiplexed filtering.

2. 2-D optical scattering matrix theory

The backbone of our new inplane cross-connect can be described as two bus-channel waveguides arranged perpendicular to each other. At the intersection, a two-ring system is placed as shown in Fig. 1. Ring 1 is assumed to be coupled symmetrically to the drop- and through-waveguides and is also coupled to ring 2. Ring 2, which is identical to ring 1, is only coupled to ring 1, and is placed an optimized distance from ring 1. Ring 1 and ring 2 will also be known as the fixed and floating rings, respectively.

Fig. 1. Dual Microring Cross-Connect System. A signal enters at input EI , propagates down bus guide of width 2w to the bus-ring interaction region, and couples to ring 1 of radius R1 . The remaining uncoupled signal continues down bus guide to the through-port, ET . The coupled light propagates and resonates in ring 1, whose fields are dictated by Eqs. (i)(vi), and couples to ring 2 of radius R2 . The signal, which couples to ring 2, propagates and resonates in ring 2, whose fields are governed by (vii)–(viii). The signal then returns to ring 1, selectively couples to the drop guide, and finally exits the system via the drop guide ED .

Analysis begins with the E-field equations for the dual-ring cross-connect geometry:

Er1a=jκaEI+τaEr1f(i)Er1e=τaaEr1d(v)
Er1b=e38jωTre316αr1LEr1a(ii)Er1f=e14jωTre18αr1LEr1e(vi)
Er1c=jκbEr2b+τbEr1b(iii)Er2a=jκbEr1b+τbEr2b(vii)
Er1d=e38jωTre316αr1LEr1c(iv)Er2b=ejωTre12αr2LEr2a(viii)
ET=jκaEr1f+τaEI
(1)
ED=jκaaEr1d
(2)

where EI is the input field to the horizontal through-port bus waveguide, ET is the output field at the bus through-port, ED is the dropped field in the vertical bus waveguide, and Er1aEr2d are the fields at the respective points a…d in the corresponding rings. This nomenclature for the drop and through ports is the accepted and traditional one when dealing with cross-grid arrangements even though there may be discrepancies when compared to other optical filters. Due to the orientation of the rings, π/4 in reference to the intersection axis, the coefficients for the phase and absorption terms of the fixed ring bear the attributes of this geometry, while the floating ring logically does not. Although these coefficients are not completely necessary, they are used in order to clearly reveal the geometry. As is customary, back-reflections are neglected. The quantity Tr is the round trip signal time of the ring; Tr =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, neff . The quantity αr 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. κa and κaa are the coupling constants between the ring and bus guide, whose spacing are ta and taa , while κb is the coupling constant between the two rings, whose separation is tb . 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]

]. κa and κaa 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 αr 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 τa,b=1κa,b2. As in traditional scattering-matrix theory, the analysis is assumed to be independent of the polarization of the E-field [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]

].

The complete and expanded through-port and drop-port output-power expressions may be obtained by substituting (i) thru (viii) into Eq. (1) and (2) and simplifying the modulus squared to produce

ETEI2=Aατa+AΦΓaΓbτaaA(τa+κa2τaa+τa2τaa)τbAαA(1+τaτaa)τb+AΦτaτaaΓb2
(3)
EDEI2=Aκaκaa(AΦΓb+Aατb2)AαA(1+τaτaa)τb+AΦτaτaaΓb2
(4)

where

A=exp(αrL2+jωTr)
(5)
A=exp(αrL4+jωTr2)
(5a)
Aα=exp(αrL)
(6)
Aα=exp(αrL2)
(6a)
AΦ=exp(2jωTr)
(7)
AΦ=exp(jωTr)
(7a)
Γa,b=(κa,b2+τa,b2).
(8)

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

3.1 The singly and doubly terminated prototype circuits

By definition, an impedance inverter in a microwave or optical circuit imparts a π/2 phase shift, which can also be seen from the field equations that model the response of our 2×2 system. The impedance inverter is the analog of a coupling event. But by assuming that the present filter consists of impedance inverters which are innate to the resonators themselves, the cross connect may be called a direct-coupled resonator filter.

Fig. 2. Characteristic wavelength spectral response. The normalized through- and drop- port optical output power response of the cross-connect displays the general properties of the system. The quantities a and -a represent the desired half-width normalized wavelengths at Am while b and -b are the half-width at half maximum. The waveform shape is novel, as are the power asymptotes away from λ0 .

The impedance inverters expressions needed for designing a maximally flat or Butterworth response for the system, come from modifying the familiar microwave filter circuit expressions of Mathaei and the optical expressions of Melloni to obtain values for a singly terminated circuit which correspond to the vertically enclosed prototype filter depicted in Fig. 3:

K1S=πB2g1SFSR1
(9a)
KqS=πB2gqSgq1SFSRqFSRq1
(9b)
KN+1S=πB2gN+1SFSRN=0
(9c)

with the element values

g1S=a1SεN
(9d)
gqS=aqSaq1Scq1Sgq1SεN
(9e)
gN+1S=
(9f)

(check⇒gNS =Ng1S

aqS=sinπ2(2q1)N
(9g)
cqS=cos2(πq2N)
(9h)

where 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 K1S inverter) which interacts with the system. Its designed response is regularly solved for by the traditional signal-processing method which consists of 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]

12

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

]. The method above, Eq.(9ah), is not used to realize the desired response of a microring lattice filter, but will be used here in conjunction with the doubly terminated synthesis to realize one for the cross-grid system.

Fig. 3. Schematic of the cross-connect microring resonator prototype bandpass filter. The horizontal box represents the doubly terminated circuit, while the circuit vertically enclosed depicts the singly terminated filter in the hybrid synthesis. The first impedance inverters of both circuits are displayed as dotted boxes because they are not static in the filter synthesis. The physical positions of the two bus guides are understood to be located on both far ends of the horizontal box, but are not illustrated.

Δλλ022πcΔω.
(10)

Although it is not explicitly utilized in these equations, it will be utilized in the plotting mechanism [13

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

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]

].

Eq. (9c) indicates that the final inverter on the floating ring does not interact with any other entity. As can be seen in Fig. 3, the vertical box, the singly terminated prototype filter, has one termination, K1S, which can be interchanged with K1D.

If one now considers the prototype filter in the horizontal box, a simple one-ring doubly terminated system is encountered. This formation has the general characteristic equations of

K1D=πB2g1DFSR1
(11a)
KqD=πB2gqDgq1DFSRqFSRq1
(11b)
KN+1D=πB2gN+1DFSRN
(11c)

with the element values

gqD=2aqDεN
(11d)
aqD=sinπ2(2q1)N
(11e)

This arrangement is the traditional parallel-bus waveguide system with 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

The system here is clearly a combination of both singly and doubly terminated prototypes. By blending the two syntheses together in a fashion that collectively holds the fundamental elements of each, while allowing each its independence, a new procedure will be produced that yields the desired response. Although this course of action appears suspicious, further development will prove it to be trustworthy.

If one begins with the assumption that the regular order is always an even integer, i.e. N=2n , 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 ND =1 due to the innate geometry of the device. We therefore take the elemental value of gqD for ND =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.

In order to later validate the authenticity of the synthesis, the elements for a N=6 configuration will initially be solved. The aforementioned and other elements are as follows:

g1D=2a1DεND=2a1Dε
(12a)
g2F=12N+ηa2Sa1Sc1Sg1Sε1+1N
(12b)
g3F=1Na3Sa2Sc2Sg2Sε2N
(12c)
g4F=2N+1102a4Sa3Sc3Sg3Sε2N
(12d)
g5F=1Na5Sa4Sc4Sg4Sε2N
(12e)
gNF=N2N+ηaNSaN1ScN1SgN1Sε2N
(12f)
gN+1F=
(12g)

where 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. aqS , cqS , aqD , cqD , accordingly remain the same. We now introduce 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 2(Am(η+1). This puts limitations on the selection of minimum and maximum intersection points, but is essential to the soundness of the synthesis.

The general method of derivation of this synthesis is as follow:

1) A prototype circuit was created by utilizing the concepts of circuit terminations and resonators. It was then recognized that the system consisted of two types of circuits, singly and doubly terminated, combined in an unusual manner.

3) The splitting coefficient, which is the leading term of all the elements, was intuitively thought to be 12 from a symmetry standpoint or as a function of 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 Am and Δλ.

4) When designing the synthesis for N=4, it was understood that gN=4F must show agreement when N=2 gN=2F. With the application of another parameter, η, a workable synthesis was found. All other elements of the singly terminated synthesis remain the same.

6) The odd integer elements, g3F and g5F, were found by symmetry arguments.

In the case of the dual-ring cross-connect arrangement, a pair order of zero, the KND will have a value ND =1 and separately the KNS will be taken as 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:

g1D=2a1Dε
(13a)
g2F=14a2Sa1Dc1Sg1Dε32
(13b)
g3F=
(13c)

The ε32 parameter is introduced in order to remove the ε dependency from g1S while maintaining the εN reliance. The coefficient, 14, is introduced in order to account for the splitting which is a construct of the synthesis of the system. The junction parameter, according to the prescription, is 2(Am1) . These values are then inserted into Eq. (14):

K1D=πB2g1DFSR1
(14a)
K2F=πB2g2Fg1DFSR2FSR1
(14b)
K3F=πB2g3FFSR2=0
(14c)

where

a1D=sinπ2
(14d)
a2S=sin3π4
(14e)
c1S=cos2π4.
(14f)
Fig. 4. Depiction of regular orders of N=2, 4, and 6 or pair order η=0, 1, and 2 cross-grid systems.

The realization of the coupling coefficients between impedance inverters is obtained by equating the transfer matrix of the coupler to the transfer matrix of the impedance inverters Kq [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]

,8

8. R. E. Collins, Foundations for Microwave Engineering (McGraw-Hill, New York1966).

]:

κq=2KqKq2+1
(15)

Eq. (15) applies to all of the previously mentioned syntheses. The synthesis is completed and design of the system may ensue. The design procedure was outlined 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]

], and will be understood here as well to obtain the desired coupling coefficients.

4. Discussion and summary

It is worth noting that this method is a permutation of the traditional approach undertaken for filter synthesis. The present process was started with the understanding that this formulation has no obvious direct and customary analog from the microwave filter to the photonic disciplines. The present selection was initially tested to ensure that a true and a generally maximally flat solution is produced by generating spectral solutions for a single linewidth and junction parameter for higher regular-order systems, 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.

Fig. 5. A comparison of the passband characteristics for a system which consists of N=2, 4, and 6 rings. These through-port responses were obtained from the results of the synthesis prescribed in Eq. (12). A more box-like response is observed as N is increased.

Along with the analytical verification of this synthesis, a numeric assessment was conducted as well. This consisted of generating the spectral characteristic of this device via FDTD simulations. It was found that the numeric analysis does in fact generate a response that is similar to the shape of the predicted analytical responses. Specifically the doubly peaked drop and nearly continuous through responses were observed. Upon inspection of these results, it was found that the responses were shifted from the desired resonant wavelength as much as 2%. This result is most likely due to a discrepancy in the calculations utilizing the ring radius and the effective index of the guides. Also the linewidths were as much as two times larger than the desired ones and the maximum peaks and troughs of each port did not reach unity or null, respectively. These results can possibly be explained by an incorrect calculation of the guide separation via the coupling coefficients and not permitting the system to reach a state of equilibrium during the simulation. Although the numeric and analytic results do not exactly agree they still remove the notion that this response is in fact an artifact of the mathematics.

Another example of interest lies in the response of 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=1.33µm. 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.

Table 1(a). Coupling coefficients and of N=2, 4, and 6 systems.

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Table 1(b). Drop-port peaks wavelengths of N=2, 4, and 6 systems.

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In tables 1(a) and 1(b) it is assumed that the operating wavelength, λ 0, of these devices are 1.33 µm where n=3.52, Mq =34, Δλ=0.05 nm and Am =5.0. These results coincide with the normalized response displayed in Fig. 6.

The method of procuring different linewidths at different junctions is not directly correlated to standard protocols. For this reason additional precautions, in the form of examining a multitude of linewidths at an assortment of losses, were undertaken in order to further verify that the solution was in fact correct. A small portion of these simulations appear in Figs. 7 and 8 and are described in Table 2. Obtaining a desired full-width linewidth at an accepted junction requires that the preferred linewidth be halved and then inserted in the calculation. Therefore, one actually inserts the half-width linewidth. Where the linewidth is to be measured for the through-port, the maximum accepted percentage of loss Am , is then inserted. For instance, if the preferred junction is 5% of the maximum through-port value at a full-width linewidth of 0.1nm, then Δλ=0.05 nm and Am will be 5.0. This will produce a through-port response which has intersections at λ=λ 0±0.05 nm at |ET |2=0.95. The representation of this response is depicted in Fig. 7 and 8’s structure 1. The drop-port response would intersect at λ=λ 0±0.05 nm at |ED |2=0.05. The details of this response are given in Tables 1 and 2 at λ 0=1.33µm and are represented in a normalized fashion with Fig. 6. It is worth mentioning that at times, the actual intersection value differs from the desired coordinate

Table 2. Response parameters utilized in Figs. 7 and 8.

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In table 2 it is assumed that the operating wavelength, λ 0,of these devices is 1.33 µm where n=3.52 and Mq =34.

pair, but still remains in close proximity. This can be observed in Fig. 8 where structure 2 is precisely at its desired location, while structures 1 and 3 are slightly askew from their favorable locations. This divergence is most probably due to device limitations, which maybe be controlled by reevaluating the desired intersection points, i.e. Am 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, Mq , 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 Mq , 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(Mq ), 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 Mq , 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.

Fig. 6. The drop-port of the N=2, 4, and 6 cross-connect filter. The ability to specifically dictate the separation of the two central peaks is clearly displayed for pair order 0, 1, and 2. It is noteworthy that the displayed N=4, or η=1, response represents the highest order of peaks with the highest order of symmetry. In order to obtain the respective through-port responses, this figure is simply inverted and is displayed in the inset. It is understood that the value a is the half width of the desired response.
Fig. 7. The through-port responses of several N=2 filters with different linewidths and percentages of loss. The specifics of each device appear in Table 2.
Fig. 8. Detailed view of the responses depicted in Fig. 7. These responses, which coincide with Table 2, demonstrate the ability of the synthesis to realize a response that is subject to a desired linewidth and percentage of loss. These specific parameters and many others were used, along with the more box-like response displayed in Fig. 5, to add to validity of the synthesis.
Fig. 9. Tolerances of the dual-microring cross-connect filter. Structure 2 is displayed along with 5 and 10% deviations from the prescribed value of κ 2, as is presented in Table 2.

5. Conclusion

By following the prescribed synthesis, we have demonstrated a system that provides a unique response with respect to traditional microring designs. The synthesis, a microwave filter analogy, has been established to be accurate via both analytical and numeric methods and uses closed-form solutions. We found a new family of high-Q filters with double or quadruple passbands that may be useful for diverse applications such as wavelength-division multiplexing, wavelength demultiplexing, laser intercavity mode selection, and spectroscopic sensing.

The case of two rings is of particular interest because a 2×2 offers a highly sensitive and low-loss electrooptical spatial-routing switch (a “crosspoint switch”) when the two rings are perturbed, for example, by free-carrier injection. This switching will be the subject of a subsequent paper.

Appendix. Glossary of Symbols

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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. 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]

3.

Y. Kokubun, T. Kato, and S.T. Chu, “Box-Like Response of Microring Resonator Filter by Stacked Double-Ring Geometry,” IEICE Trans. Electron. E85-C,1018–1024, (2000).

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]

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]

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]

7.

G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters, Impedance Matching Networks and Coupling Structures (McGraw-Hill, New York1964), Chap. 4, 8, 11, 14.

8.

R. E. Collins, Foundations for Microwave Engineering (McGraw-Hill, New York1966).

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]

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]

12.

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

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]

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

  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]
  3. 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).
  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]
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  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]
  7. 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.
  8. R. E. Collins, Foundations for Microwave Engineering (McGraw-Hill, New York 1966).
  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]
  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]
  12. C. K . Madsen J.H. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach (Wiley & Sons, New York 1999).
  13. 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/>
  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]

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