## Abstract

Modeling and simulation results on new, resonant, waveguided 2 × 2 switches and 1 × 1 modulators are presented here. The devices employ two coupled microrings: one fixed and one floating. The fixed ring is coupled to bus waveguides that are crossed or are locally parallel. Electrooptic and thermooptic switching at *λ* = 1.55 μm are investigated. A novel peaks-and-valley spectral response allows low-power switching with low crosstalk and low insertion loss. Complete switching is attained when the complex index of both rings is perturbed by Δ*n* ~ 4 × 10^{-4}. The modulator’s optical output power is a linear function of Δ*n* over three to five decades of Δ*n*.

©2005 Optical Society of America

## 1. Introduction

This paper presents new designs for resonant, waveguided electrooptic and thermooptic 2 × 2 switches and 1 × 1 modulators. Each inplane device contains two identical microring resonators [1] that are coupled to each other. The novelty comes from the fact that one of the two rings is not coupled to any bus channel waveguide, whereas the second ring is coupled to two buses [2]. This floating-ring approach differs from the previous 2 × 2 switches and 1 × 1 modulating devices investigated in [3–8] wherein all the rings had bus coupling. The floating-ring switch has an unusual spectral response comprised of two pass bands and one stop band. This response gives rise to novel turn-on curves for switching and modulation that exhibit high sensitivity to electrooptic perturbation of the rings (low-power, low-crosstalk operation is predicted.) We discuss inplane coupling of the non-floating ring to two intersecting channel waveguides in a cross-grid configuration convenient for matrix switching. We also describe a device in which the non-floating ring couples to two locally parallel non-intersecting buses. Many 2 × 2 interconnection geometries are thereby feasible.

Generally, these active resonant devices are part of the emerging trend of microphotonic integration upon silicon, wherein the interconnected waveguided components comprise a highly functional chip-scale optical network amenable to optoelectronic integration upon CMOS. Advances in microring device design will allow the ring diameter to be reduced, in some cases, from 50 wavelengths to just a few wavelengths, thus enabling a higher packing density of components and promoting a “Moore’s Law for Photonics.”

Theory of the drop-port and through-port outputs is based in this paper upon our use of microwave filter analogs [9] to analyze the filter response of high-Q floating-ring devices in their OFF and ON states, where OFF can refer to a particular index bias point. From this analysis, we predict practical behavior: highly linear 1 × 1 modulation and complete 2 × 2 switching-- both of which are a very sensitive function of refractive-index change in the two rings.

## 2. Dual microring resonator cross connect response theory

In the 2D scattering-matrix approach, the equations for the dropped power and through power of the 2 × 2 cross-connect system in its initial state have been developed in [9] and the results are cited here:

Due to the very limited space to place the rings, contacts, and doping implants inside of the cross-grid, another orientation that is more experimentally appropriate was sought. By simply inspecting the prototype circuit, Fig. 3 of [9], another arrangement was devised. Through analysis, we have found that a dual microring device with locally parallel bus waveguides will produce the desired response, Fig. 2. This geometry allows optical interconnection of many 2 × 2s without waveguide crossovers as in the permutation-matrix switch. Embarking on the same approach that was done in [9], field equations for all the pertinent positions were derived and simplified. It was found that the through equation, Eq. (1), was identical, but the drop expression differed from Eq. (2) by a factor of $\mathrm{exp}\left(\frac{{\alpha}_{r}L}{8}-\frac{\mathit{j\omega}{T}_{r}}{4}\right).$

This artifact is believed to be generated by the discrepancy of the locations of the drop and through coupling events in each system. We can see that in the traditional cross-grid, Fig. 1, there exists a separation of *π*/2 on the ring, whereas the alternative geometry has a separation of *π* and their difference results in the aforementioned factor of *π*/2. When the coupling coefficients that were obtained from the synthesis in [9] are inserted into the new response equation and a lossless system is assumed, an exact match with the cross-grid, Fig. 1, response was found. This is sensible because without loss, there is nothing real to contribute to the change of the response. On the other hand, if losses *α*
_{r} are included, then a shift of Eq. (2) takes place as a function of *exp* (*α*
_{r}/4). In order to remove this discrepancy, the two bus guides can be orientated to be separated by *π*/2, but for all reasonable values of loss, the system remains nearly identical. The validity of this solution was proven in a similar manner as the original response, namely a close inspection of several response that have different intersection parameters and over an assortment of different mode numbers. The details of these results are beyond the scope of this text. Even though we propose this architecture for experimental realization, when we refer to the cross-grid arrangement it is understood that both designs are to be considered.

As in most ring-bus devices, the coupling coefficients play an integral role in the response. But we will show that here the coefficients are even more scrutinized. A graphical representation of the tolerances is given in Fig. 9 of [9]. Therefore, even though we propose using Silicon, lower-index materials that have a more relaxed loss assessment and fabrication tolerances might be chosen initially in order to facilitate the validity of the design.

## 3. Switch and modulator synthesis, characteristics, and classifications

#### 3.1 Maximally flat, triangular and spike responses

Unlike the response that is sought in [9], *i.e*. one in which maximizes the number of derivatives that are zero around a given ordinate position over some desired linewidth, a different response is preferred when realizing a cross-grid modulator or switch. The central reasons behind the absence of utility of the maximally flat characteristic for the cross-grid switch/modulator is two-fold:

- The traditional parallel guide system [1] has a perfectly symmetric response centered on the resonance wavelength, while the cross-grid arrangement [9] generates the same ordinate bilateral symmetry, but not identically shaped peaks.
- The unorthodox end product of the cross-grid array system must be considered differently than the traditional parallel guide system because the perturbation must encounter several characteristics of the response, as opposed to just one.

Therefore, we must propose a different method to obtain a response that is more applicable to the system under investigation.

When considering the systems spectral characteristics with respect to switching and modulation we must first define a simple method that can be used to conceptualize the phenomena at hand. This technique essentially involves visualizing the plotted response translating left or right, depending on the nature of the perturbation. As the index is altered, via free-carrier plasma dispersion or thermooptic effects as is outlined in [3], a new characteristic switching curve is generated by the intersection of the vertical central wavelength line and the respective curves. With close inspection one can easily surmise the devices switching or modulation properties. This procedure will be conducted formally in the following subsection.

If we now consider this procedure applied to the maximally flat response, Fig. 2 of [9], we find a response that is far from ideal. If we begin by shifting the curve, n.b. movement can occur either to the left or right because of translational invariance of the response, we have a large throughput and small drop response followed by an abrupt reverse in values. The reason behind this rapid turn around is solely due to the synthesis’s innate quality of having the steepest possible sidewalls. The reversed response is short-lived because the peaks of the drop port are quite narrow in relation to the throughput port, even if the throughput is designed to be relatively wide. The response then returns to its original state, namely a strong throughput port and weak dropping port. The main problem with this type of response is the incredibly slender perturbation widow that must be maintained in order to obtain a reasonably performing switch or modulator. Although, having the ability to dictate a specific linewidth of a device is quite desirable, this type of response which may be useful for some purposes, will be altered to meet the constraints of this specific system. By focusing on the two main problems, enumerated above, a response that serves its purpose as a switch or modulator will be proposed.

By taking the original synthesis of [9], we can begin to modify it in order to gain insight into the problem at hand. The elements of the *N*=2 system are as follows:

where

These values are then inserted into

and

and finally into

to obtain the coupling coefficients. The junction parameter, *ε*, is set equal to the *N* = 2 (and *η* = 0) prescription of $\sqrt{2\left({A}_{m}-1\right)}.$ The microwave concept of fractional bandwidth and high-order mode number are incorporated into the expressions via the notion that
*FSR _{q}* =

*c*/(

*L n*) =

*f*

_{0}/

*M*, with the length,

_{q}*L*=

*cM*/(

_{q}*n f*) and where

_{0}*FSR*,

_{q}*f*

_{0}and

*M*are the rings’ free spectral range, central frequency and integer related to the mode number of the ring in question, respectively.

_{q}With all the parameters defined, a description of the dependence on certain variables to the response of the system is required in order to gain insight into the solution of the problem. Neglecting the coupling coefficients, which are multivariable functions, the length, *L*, is the only other parameter that can be altered without changing materials or waveguide profiles. If one considers keeping the fixed ring, *R _{I}*, at some length,

*L*,and deviating the floating ring,

_{I}*R*, by increments of multiple wavelengths, one finds that the maximally flat response begins to wane. This fading of the maximally flat response is soon overtaken by a more locally symmetric one. A locally symmetric response refers to an increased likeness of all three peaks and valleys, as well as the continued bilateral symmetry. This effect is caused by the phase shift that ensues because the path lengths are no longer equal and the interference that arises is no longer strictly predictable from filter theory. Since this altering of the response logically occurs because the system has been corrupted, one infers that similar modification to the synthesis should deliver comparable results. But as was dictated earlier, a system with equally sized rings is desired.

_{2}Instead of modifying the path lengths of the rings, which will introduce the unwanted Vernier effect, another method that will produce a very similar effect is proposed. If a system is to be designed according to the given synthesis, it must have some definitive value for the mode number of the rings, *M _{q}* above or in

Since this quantity, *M*
_{q}, plays a role in the synthesis and is a function of *R*, varying it in a fashion that is similar to a path length increase should produce a similar result. But unlike the aforementioned concept of actually increasing the radius, or path length, of said ring, we shall disrupt the coupling coefficient, which in principle, via the logic of the above synthesis, are equivalent. If one attempts to simply disturb the maximally flat response by selecting various values of mode numbers, one finds that there is a preferred value to generate a curve that will prove to be productive in switching and modulating applications. This relation is:

where *M _{1}* and

*M*are equal. What is suggested by this statement is that the ring 1 coupling coefficients remains the same,

_{2}*i.e*. the traditional synthesis calculation previously mentioned, while the coefficient for ring 2 is calculated by evaluating the synthesis with a value of $\frac{1}{4}{M}_{q}.$ Although the requirement of

*M*having to be an integer (in this case one divisible by four) may put a limitation on possible ring radii selection, it will be shown that this constraint provides a well-proportioned characteristic. Fig. 3 displays the response of this alteration to the synthesis, the triangular response. Additionally, we will find that when perturbations are applied to a device with this geometry, it generates a behavior very effective for modulation purposes. Although this technique produces a most desirable modulating, response, it has the unattractive quality of having little control over the attainable linewidth. Since the linewidth is reduced by this method, a larger linewidth and junction parameter can originally be prescribed, but then device limitations begin to surface.

_{q}On the other hand , further modification of the synthesis and therefore the local symmetry of the response will bring a slightly more dramatic change to the response and a more applicable signature for switching. This relation is achieved and maximized by

Once again, this means that a value of 2*M _{1}* and $\frac{1}{4}{M}_{2}.$ are inserted into the synthesis to obtain the necessary coupling coefficients, where

*M*

_{1}=

*M*

_{2}. For instance, if one chooses rings with radii

*R*, where

*R*=

*R*

_{1}=

*R*

_{2}, and

*R*relates to

*M*in Eq. 11. If we now arbitrarily set

_{q}*M*= 56, then the synthesis is conducted with

_{q}*κ*

_{1}(

*M*

_{1}=112) and

*κ*

_{2}(

*M*

_{2}= 14), even though the ring radii remain in agreement with Eq. (20) where

*M*= 56 . This behavior is displayed in Fig. 4.

_{q}#### 3.2 Index biases and perturbations

The spectral responses displayed in Figs. 5, 6 and 7 are expected for an unperturbed device having a maximally flat, triangular, and spike responses, respectively, where we then define the ordinate line at *λ*/*λ*
_{0}= 1 as the symmetric, or resonant, reference. For later clarification we will also refer to this as the A-bias. The remaining position on the plots is the B-bias, or the mid-range bias. In order to distinguish from the left and right points we assign *l* and *d*, respectively. These sites will serve as initial perturbation positions and are always viewed in relation from the symmetric point, A.

In order to alter the index of the rings we will follow the same methodology as was done in [3]. We define the index of the rings as

where *δn* is the infinitesimal index term which is needed to obtain an output power value at some reference or bias point , *e.g*. A or B, Δ*n* is variational index parameter that will be scanned to obtain a characteristic perturbative switching/modulation response, *κ*¯ is the extinction coefficient and is defined by $\frac{\mathit{\alpha \lambda}}{4\pi}$ where *α* is the absorption coefficient, *δκ*¯ is the infinitesimal extinction coefficient which will be included in the calculation for obtaining some specific bias position and is defined as $\frac{\mathit{\delta \alpha \lambda}}{4\pi}$ where *δα* is the infinitesimal absorption coefficient that will be included and finally Δ*κ*¯ is the variational extinction coefficient that
will be scanned for a response and is defined as $\frac{\Delta \mathit{\alpha \lambda}}{4\pi}$ where Δ*α* is the variational absorption coefficient. Drawing upon the same conditions, that the effective index is approximately the same as the refractive index, it is proposed that a variation in the effective index is approximately the same as a variation in the refractive index.

When calculating the turn-on characteristics of the switch and in order to maintain a one-variable perturbation we shall define

and

as perturbation parameters, which simplifies Eq.(23) into

As before, we will assume a lossless material because we are interested in the overall characteristics of the device and not the limitations of material system, so we set *κ*¯ = 0 . The methods of obtaining *r _{δ}* and

*r*are developed in [3]. This method will allow us to alter the index to the specific bias point and then continue with the sweeping index scan. It should also be noted that the calculations included a continuous reevaluation of the coupling coefficients for each step of index perturbation. This step insured a realistic switching behavior because the unperturbed guides and the perturbed guides are constantly changing their coupling.

_{Δ}In order to simply define and represent the bias points more easily, a relationship was devised from the previous expression. This requires inspecting the first several terms and their respective signs. For instance, if the *l*·B, *i.e*. the left-hand B point, of Figs 5, 6, or 7 is desired, then we want to induce a positive index change to translate the curve to the right to have the *l*·B line up to the central wavelength. We will call that action “positive” and represent it as +B. Then if a negative perturbative index scan, *i.e*. translating the curve to the left to obtain a switching/modulating relation, is desired from the initial *l*·B, it will be referred to as +B -. The use of the directional prefix would be redundant because only one bias point B can be reached with a positive index change. We can see this nomenclature is compatible with Eq.(23a). For instance, if one replaces the first *δn* with the bias point that is wanted and then introduces the initial and scanning index parameters, *i.e*. the sign of the index perturbation, then this naming relation is actualized. Now recalling that our index perturbation can be designed to any phenomena or material system, we must consider the significance of the signs and value of the parameter. If the sign is positive, then in centrosymmetric Silicon we are referring to the thermooptic effect or to the Kerr effect and in both cases *r*=0. On the other hand , if the sign is negative then we will infer either that: 1) the disturbance is either free-carrier plasma dispersion (FCPD) related and the value of *r* is dependent on concentration, material and wavelength [3], which is a function of the applied bias or dependent on an optical flux if the device is optically controlled or 2) is caused by a polymeric tuning method and we will assume takes place without loss, *r*=0 [10]. By considering another example, when the central wavelength reference point, A, is desired, then there is no initial perturbation needed so *δn* = 0. If a thermooptic sweep response were desired then the system would be represented as A+. On the other hand, if a FCPD relation is called for, then A – would symbolize an -Δ*n*.

For our final scenario, if the initial point is *d*·B needed, then we must displace the curve to the right. But unlike a positive-index translation, there is a nonzero *r*-value if FCPD or optical flooding is utilized, but if a polymeric tuning technique is used we can assume that *r*=0. But by using a symmetry argument we can infer that this translation can be viewed as a thermooptic perturbation reversed because of the aforementioned translational invariance.

Once a complete system is configured, a method to obtain the necessary index displacement is required. This technique requires either graphically or numerically obtaining *λ _{δ}* , the wavelength value at the specific bias point. Eq.(20) is then modified to justify this alteration to the index while maintaining the integrity of the system:

where *n _{δ}* is the new index,

*n*+

*δn*, and

*M*is held constant. It should also be noted that this equation brings to light the notion that it is possible to simply utilize

_{q}*λ*as the signal and maintain a similar result without the need for an ancillary index perturbation. Depending on the system at hand, the precise amount of perturbation that is required may make it more feasible to tune the laser source to the reference/bias point rather than tuning the entire system via index alteration and then sweeping through some index range.

_{δ}The Q, or quality, factor of a resonator is a measure of the sharpness of the resonance. Traditionally, it can be defined as *λ*
_{0} / Δ*λ* where *λ*
_{0} is the central wavelength and Δ*λ* is the linewidth at half maximum. This is derived as a ratio of the stored energy over power loss during one cycle [1,11,12]. But since we have a doubly peaked dropping port where there is a zero response at resonance, it must be verified what is actually being measured. We found, by using the energy definition of Q, we have derived the relation:

where *α*
_{d}=*α*
_{r1}+*α*
_{r2} + *α*
_{through} + *α*
_{drop} + *α*
_{ring-ring}, *λ _{B}* and Δ

*λ*are the distributed loss, the wavelength at the B point and the linewidth at half maximum of the B point peak, respectively. It is assumed that

_{B}*α*

_{r1}=

*α*

_{r2}= 0, and

*α*

_{through},

*α*

_{drop}and

*α*

_{ring-ring}can be found via

*exp*(-

*αL*) = -

*τ*

^{2}= 1-

*κ*

^{2}, respectively for all the coupling events. But due to the singly terminated nature of the second ring, the loss procured from it should be neglected because what ever is coupled out is coupled back in and therefore

*α*

_{ring-ring}=0. This expression, (27), will also generate a Q for a single ring system as well, but by assuming that the path length is 2 L =4

*π*R, a relation that generates the desired Q for the dual-ring system is achieved. A pathlength increase can be justified because in the derivation, a single ring was assumed. By doubling the pathlength, we include the fact that there is another ring while maintaining the difference between the traditional parallel guide system [1,3], which would call for a superposition of each resonator in the derivation because each ring is coupled to a bus and to each other. The fact that this circuit has a singly terminated resonator also supports the validity of this argument. This derived quantity deviates from the numerically obtained ratio by less than 0.01%.

## 4. Results and discussion

Looking at the triangular-response switching in Fig. 8, we find that complete 2 × 2 switching happens at Δ*n* = 2 × 10^{-4} for the A- and A+ biases, for the +B- bias and for the -B+ bias. In these four cases, this Δ*n* value must be “accurately applied” to produce the spike valleys in Fig. 8, otherwise the switching is incomplete. For the thermooptic A+ bias, the crosstalk is -40 dB, compared to -23 dB in the free-carrier A- case. For the A+ and A- biases, the output power of the through port goes from a high value to a low value during switching, whereas, in +B- and -B+ cases, it is the dropping-port output power that swings from high to low. Looking at Fig. 6, if the bias is B on the right-side drop-port spectral peak, the 2 × 2 switching occurs only for index decrease in the rings. If the bias is B on the left-side drop peak, then 2 × 2 switching occurs for an index increase.

Let us now consider the linearity of a 1 × 1 modulator that is depicted in the Fig. 8 result. There are two choices here: to use the through-port output as the modulator’s output port (with the drop and add ports dangling), or to use the drop-port output as modulator output (with through and add dangling. We find that the 1 × 1 linearity is excellent as follows. If we choose the A+ or A- bias in Fig. 8, then the through-port intensity goes from “dark to light” with linear response over three decades of Δ*n*. Alternatively, for the drop port output, if any of the four +B-, +B+, -B-, and -B+ biases are utilized, then the drop intensity goes from “dark to light” with 4.5 decades of linearity versus Δ*n*, and this Δ*n* in turn is presumably a linear function of the applied voltage or current.

Turning now to Fig. 9 for the spike-spectrum device, we find that the 2 × 2 switching has the same general shapes as in Fig. 8 during A and B bias, except that in Fig. 9, the optical crosstalk is -30 dB for FCPD (A- bias) and -50 dB for thermooptic control (A+ bias). Here, complete 2 × 2 switching occurs at the same applied Δ*n* as in Fig. 8. The 2 × 2 device of Fig 9 switches only for +B- bias (-Δ*n*) and -B+ bias (+ Δ*n* ) as in Fig. 8. If we examine the linearity of 1 × 1 modulation curves are in Fig. 9, the results are optimum in the sense that the 1 × 1 device is linear over five decades of Δ*n* when any one of the four B biases is selected.

The physical reason for the linear modulation characteristic is that the triangle and spike spectral features have steep and flat “side walls” that are shown in Figs. 6 and 7. The output power rides up or down this flat wall as Δ*n* is turned on. The 1 × 1 modulator can indeed give “light to dark” modulation, but in that case, linearity is a bit harder to achieve. Light to dark is found for any of the six initial biases in Figs. 8 and 9. Although “dark to light” is obtained for the range 10^{-6} < Δ*n* < 10^{-4}, the “light to dark” result requires that Δ*n* swings from an initial condition Δ*n* = 5 × 10^{-4} applied, to Δ*n* = 5 × 10^{-2} applied, as illustrated in Figs. 8 and 9. These results show that the drop-port modulators and through-port modulators are available, depending upon A or B bias. In Ref 1, the 1 × 1 “light to dark” operation is somewhat easier, requiring in that case a swing from 10^{-4} to 3 × 10^{-3} for A bias.

Generally, the Q of these SOI planar, resonant double-ring 2 × 2 switches and 1 × 1modulators is in the range of 5000 to 10,000 which is believed to be practical. The 2 × 2s are available in a crossed-grid and locally parallel bus waveguide geometry shown here in Figs. 1 and 2, and the dimensions of this switch are scaleable for operation at the mid-wave and longwave infrared as well as the near infrared. The 2 × 2 turn-on characteristics are identical for the crossed and parallel bus geometries. This new “floating ring” switch has a unique spectral response consisting of two valleys and one peak at the through-port. Although the Q is not extremely high, the linewidth of the central peak is only ~ 0.2 nm which imposes a tuning accuracy upon the laser source, or requires some “trimming” of this ring-mode resonance wavelength to hit the laser wavelength using, for example, polymeric “loading” (cladding) on the rings.

## Acknowledgments

The authors wish to thank the Air Force Office of Scientific Research (Dr. Gernot Pomrenke) for sponsorship of this inhouse research at AFRL/SNHC.

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