## 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.

The geometry here is unique in the sense that only one ring of the two rings is coupled to the buses. This arrangement differs from previous work in several ways. Although in [1], where a single ring couples to the cross-grid to generate the response, our architecture utilizes another ring to create its response. The structure in [2], on the other hand, has two rings, but they are not coupled directly to each other. The device studied in [3] however, employs two vertically coupled rings over a cross-grid. Although this structure appears very similar to ours and offers a different, albeit insightful, method to control the ring-ring interaction [4], it can be found that the two rings behave as one resonator and it is specifically designed to generate a superposition of the rings’ Lorentzian responses. It should be noted that similar responses to those displayed in [2–3] can be generated by our device and vice versa. And finally a system that uses three rings in the cross-grid arrangement, [5], has ring-ring interaction, but unlike our format, it has more than one ring interacting with the bus guides. All of the above configurations make use of a doubly terminated approach to obtain a desired spectral response via resonance, interference, or the Vernier effect. Here the secondary ring interacts only with the primary ring and this interaction takes place by way of a single coupling event.

## 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.

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

${E}_{{r}_{1}a}=-j{\kappa}_{a}{E}_{I}+{\tau}_{a}{E}_{{r}_{1}f}$ | (i) | ${E}_{{r}_{1}e}={\tau}_{\mathit{aa}}{E}_{{r}_{1}d}$ | (v) |

${E}_{{r}_{1}b}={e}^{\frac{3}{8}j\omega {T}_{r}}{e}^{-\frac{3}{16}{\alpha}_{r1}L}{E}_{{r}_{1}a}$ | (ii) | ${E}_{{r}_{1}f}={e}^{\frac{1}{4}j\omega {T}_{r}}{e}^{-\frac{1}{8}{\alpha}_{r1}L}{E}_{{r}_{1}e}$ | (vi) |

${E}_{{r}_{1}c}=-j{\kappa}_{b}{E}_{{r}_{2}b}+{\tau}_{b}{E}_{{r}_{1}b}$ | (iii) | ${E}_{{r}_{2}a}=-j{\kappa}_{b}{E}_{{r}_{1}b}+{\tau}_{b}{E}_{{r}_{2}b}$ | (vii) |

${E}_{{r}_{1}d}={e}^{\frac{3}{8}j\omega {T}_{r}}{e}^{-\frac{3}{16}{\alpha}_{r1}L}{E}_{{r}_{1}c}$ | (iv) | ${E}_{{r}_{2}b}={e}^{j\omega {T}_{r}}{e}^{-\frac{1}{2}{\alpha}_{r2}L}{E}_{{r}_{2}a}$ | (viii) |

where *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 ${E}_{{r}_{1}a}\dots {E}_{{r}_{2}d}$ 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 *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]. *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 ${\tau}_{a,b}=\sqrt{1-{\kappa}_{a,b}^{2}}$. As in traditional scattering-matrix theory, the analysis is assumed to be independent of the polarization of the E-field [6].

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

where

Figure 2 demonstrates the normalized unperturbed response of the Fig. 1 system. At the central wavelength, a traditional response is produced. A traditional response refers to the response of two parallel buses that interact through a series of rings; namely, a peak at the resonant wavelength with the power returning to zero until the next resonant peak. What is unique to this device is twofold: (1) the 2×2 provides unity power output from the through-port, a behavior opposite from the unity dropped power that occurs in the traditional parallel-bus-guide two-ring system or in the cross-grid bus-guide one-ring system, (2) the Fig.1 system generates an interesting spectral artifact outside of the traditional region of interest, namely, the dropping and through ports do not maintain their traditional values of unity and zero, respectively, over wavelength, but reverse after a characteristic spectral width as displayed in Fig. 2. Thus, we have a new double-pass or double-drop filter.

## 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.

Now in order to facilitate the analogy, a circuit referred to as a filter prototype needs to be constructed. A prototype filter is symbolic representation of all the elements of a circuit. Figure 3 depicts the prototype filter circuit of our system. Since in the optical domain, a termination refers to a coupling event other than a ring-to-ring action, we can logically say that circuit enclosed in the horizontal box is doubly terminated because it is assumed that there are two bus waveguides located next to the inverters ${K}_{1}^{D}$ and ${K}_{1}^{D}$ On the other hand, the vertically enclosed box only consists of one termination. Inverter ${K}_{1}^{S}$ can be seen as terminated with one of the bus guides, ${K}_{2}^{S}$ is a ring-ring interaction and therefore not a termination, and finally ${K}_{3}^{S}$ has nothing to terminate with. Therefore this circuit is singly terminated. These definitions and criteria are thoroughly developed for microwave theory in Mathaei and other texts on microwave filters [7–9] and specifically in Melloni’s text for the microwave-optical analogy [10].

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:

with the element values

(check⇒${g}_{N}^{S}$ =${\mathit{\text{Ng}}}_{1}^{S}$

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 ${K}_{1}^{S}$ 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 [11–12]. The method above, Eq.(9a–h), 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.

In accordance with Melloni’s derivation for a cascaded or doubly terminated system of rings, the expressions and simplifications for the impedance inverters themselves remain unchanged. Namely, the microwave concept of fractional bandwidth and high-order mode number are incorporated into the expressions via the notion that *FSR*_{q}
=*c*/(*Ln*_{eff}
)=*f*
_{0}/*M*_{q}
,with the length, *L*=*c*
*M*_{q}
/(*n*_{eff}
*f*
_{0}) and where *FSR*_{q}
, *f*
_{0} and *M*_{q}
are the rings’ free spectral range, central frequency and integer related to the mode number of the ring in question, respectively. Since we are consistently dealing with ring resonators, we will simply assume that *M*_{q}
is equal to the mode number of the ring (*m* of Eq. (15) in [6]). The junction parameter *ε*, a function of the passband tolerance, is equal to $\sqrt{2\left({A}_{m}-1\right)}$, where *A*_{m}
is the maximum accepted percentage of lost (or allowed) signal of the band *B* at the central frequency of the through (or drop) ports, all of which are illustrated in Fig. 2. This percentage parameter will predict precisely the output power ordinate at which the response will intersect in relation to the band. Along with *B*, the desired bandwidth at the preferred *A*_{m}
, this completes the transformation from the microwave theory to the photonic domain.

It is noteworthy that *B*, or more importantly, the resonance linewidth Δ*λ*, may be obtained from a graphical representation of the through-port and drop-port responses or via the previously stated drop- and through-field solutions, by the simple analytical argument of ${\mid \frac{{E}_{D}\left({\omega}_{\mathit{FWHM}}\right)}{{E}_{I}}\mid}^{2}=\frac{1}{2}{\mid \frac{{E}_{D}\left(\omega =2\pi \right)}{{E}_{I}}\mid}^{2}$ in which, once the equality is simplified and reduced, the solution yields the standard full width at half maximum, *ω*_{FWHM}
, not *B*(*A*_{m}
). In keeping with the convention of previous authors in the microring specialty, a conversion from frequency to wavelength will be introduced where

Although it is not explicitly utilized in these equations, it will be utilized in the plotting mechanism [13–14].

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, ${K}_{1}^{S}$, which can be interchanged with ${K}_{1}^{D}$.

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

with the element values

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].

## 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*=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 ${g}_{q}^{D}$
for *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.

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:

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.*
${a}_{q}^{S}$
, ${c}_{q}^{S}$
, ${a}_{q}^{D}$
, ${c}_{q}^{D}$
, 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 $\sqrt{2({A}_{m}-\left(\eta +1\right)}$. 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.

2) In order for the synthesis to hold, it was confirmed that each circuit must be dealt with independently. In the optical domain this is thought to consist of removing the influence that the element ${g}_{1}^{D}$ has in the *q*=2 element. Specifically the doubly dependent *ε*(since *N*^{D}
=1) is addressed by the ${\epsilon}^{1+\frac{1}{N}}$ term while all the other *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 ${\epsilon}^{\frac{2}{N}}$ term in all of the following elements.

3) The splitting coefficient, which is the leading term of all the elements, was intuitively thought to be $\frac{1}{2}$ 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 *A*_{m}
and Δ*λ*.

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

5) When *N*=6, an iterative search of coefficients revealed the ${g}_{4}^{F}$ coefficient. With simplifications, it was found to be a function of *N* as well.

6) The odd integer elements, ${g}_{3}^{F}$ and ${g}_{5}^{F}$, were found by symmetry arguments.

In the case of the dual-ring cross-connect arrangement, a pair order of zero, the ${K}_{N}^{D}$
will have a value *N*^{D}
=1 and separately the ${K}_{N}^{S}$
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:

The ${\epsilon}^{\frac{3}{2}}$ parameter is introduced in order to remove the *ε* dependency from ${g}_{1}^{S}$ while maintaining the $\sqrt[N]{\epsilon}$ reliance. The coefficient, $\frac{1}{4}$, 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 $\sqrt{2({A}_{m}-1)}$
. These values are then inserted into Eq. (14):

where

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 *K*_{q}
[10,8]:

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], 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.

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}=1.33*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.

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, *M*_{q}
=34, Δ*λ*=0.05 nm and *A*_{m}
=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 *A*_{m}
, 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 Δ*λ*=0.05 nm and *A*_{m}
will be 5.0. This will produce a through-port response which has intersections at *λ*=*λ*
_{0}±0.05 nm at |*E*_{T}
|^{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 |*E*_{D}
|^{2}=0.05. The details of this response are given in Tables 1 and 2 at *λ*
_{0}=1.33*µm* 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

In table 2 it is assumed that the operating wavelength, *λ*
_{0},of these devices is 1.33 *µm* where *n*=3.52 and *M*_{q}
=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.*
*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

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.

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