## Abstract

In this paper, a novel technique to set the coupling constant between cells of a coupled resonator optical waveguide (CROW) device, in order to tailor the filter response, is presented. The technique is demonstrated by simulation assuming a racetrack ring resonator geometry. It consists on changing the effective length of the coupling section by applying a longitudinal offset between the resonators. On the contrary, the conventional techniques are based in the transversal change of the distance between the ring resonators, in steps that are commonly below the current fabrication resolution step (nm scale), leading to strong restrictions in the designs. The proposed longitudinal offset technique allows a more precise control of the coupling and presents an increased robustness against the fabrication limitations, since the needed resolution step is two orders of magnitude higher. Both techniques are compared in terms of the transmission response of CROW devices, under finite fabrication resolution steps.

© 2009 Optical Society of America

## 1. Introduction

Coupled Resonator Optical Waveguide (CROW) devices [1] are very relevant structures for the processing of light [3–5], due to their resonant and nonlinear enhancement nature. Both linear and nonlinear operation regimes can be employed to implement a wide range of successful signal processing tasks. CROW based subsystems are envisaged to be critical in several photonics applications and industries. The filtering response of CROW devices is strongly dependent on the coupling constant K for all the couplers involved [6]. It is known that using the same K value for all the couplers produces filtering responses with significant side-lobes for the side-coupled integrated spaced sequence of resonators (SCISSOR) [2] or significant ripples in the pass-band for the direct coupled microrings (CROW) [1]. It is also known that the side-lobes/ripples can be reduced, and the pass/reject bands can be made wider, by apodizing the *K* value of each individual coupler in the structure, starting from a nominal *K* value (either increasing or decreasing it) [7].

A technique exists in which the adjustment of the *K* value is done by changing the distance between the waveguides on the coupler [8], *d* in Fig. 1. This is technologically challenging, since the resolution step required in the ’y’ direction [Fig. 1(a)] can be well below the fabrication step resolution which is currently on the order of tenths of nanometers.

In this document we propose a technique in which the coupling constant value K is changed by imposing a *longitudinal* offset between the coupled waveguides [Fig. 1(b)]. The offset shown in Fig. 1(b) is imposed in the longitudinal direction ’x’, while the distance between the waveguides, *d*, in the ’y’ direction is kept fixed in all the couplers of the CROW device. The offset implies a reduction in the coupler effective length. The technique allows for both increasing and decreasing the value of *K* starting from a nominal value as will be shown in next sections. Therefore, the offset technique presented is sufficient by itself for apodization, and optimized CROW’s can be produced with a fixed distance *d* between the rings, solely by changing the offsets.

## 2. Coupling constant control through the offset technique

The power exchange between waveguides in a parallel waveguides coupler can be obtained from the power coupling constant [10]:

where *L _{eff}* is the effective coupler length and

*L*=

_{b}*π*/(2

*κ*

_{12}) the beat length of the coupler, defined as the physical length where the power is completely transferred from one waveguide to another, and

*κ*

_{12}the coupling between the waveguides [10]. The combination of the coupler length and the distance between the waveguides provides different coupling constants and sets the range of variation of the coupling constant when changing the offset or the gap. The racetrack configuration with long coupling sections allows the employ of gaps big enough to avoid mode conversion losses [11].

The proposed longitudinal offset technique allows for both decreasing and increasing the value of *K*, starting from a nominal value. This is shown in Fig. 2(a) and Fig. 2(b) respectively. In both figures, *L*
^{0}
* _{e}* is the coupling effective length without offset, as in Fig. 1(a), while

*L*is the coupling effective length with offset, as in Fig. 1(b). In Fig. 2(a) the value of

^{off}_{e}*L*

^{0}

*is such that the starting value of*

_{e}*K*is placed in a positive slope region of the coupler transfer function vs. effective length. Therefore, a reduction in the effective length, down to

*L*, will imply an decrease in the

^{off}_{e}*K*value. In Fig. 2(b) the value of

*L*

^{0}

*is such that the starting value of*

_{e}*K*is placed in a negative slope region of the coupler transfer function vs. effective length. Therefore, a reduction in the effective length, down to

*L*, will imply an increase in the

^{off}_{e}*K*value. As the trend in integrated optics is the reduction of the devices footprint, in a normal situation the advice is to employ shorter couplers. However, to obtain high coupling values with short couplers, a small distance d between the waveguides is needed. With the negative slope technique

*K*=1 can be obtained ideally at the cost of incresed device footprint. However if some special applications require higher cavity lengths, i.e. for microwave filter applications, the election of longer coupling sections can be advantageous. Each of these two mentioned techniques is suited to each of the coupled resonator structures, either bus coupled (SCISSOR) or directly coupled (CROW), as already demonstrated in [7].

## 3. Coupling constant sensitivity analysis

The forthcoming analysis is based on the fabrication imperfections in Silicon-On-Insulator (SOI) technology using photolithographic processes. The resolution of the lithography machines is of tens of nanometers but can be increased by post-processing the mask [12]. An important characteristic of the photolithographic processes is the exposure dose. In the fabrication process, this dose can be changed progressively from left to right of the wafer covering a deviation of ±20 nm from the nominal thickness of the features in the mask. In this section, a detailed analysis of the coupling in the parallel waveguide couplers employed for the CROW structures will be shown. The coupling values have been determined numerically with the Beam Propagation Method (BPM) through RSoft’s Beamprop software [13]. All the simulations have been carried out in 3 dimensions with a spatial resolution of 10×10×10 nm. The simulated layerstack recreates SOITEC wafers of Silicon-On-Insulator employed in the ePIX-fab SOI fabrication platform [14], 220 nm height Silicon core on top of 2 micrometers of *SiO*
_{2}. The couplers lengths shown in the next sections have been chosen to show two different case studies: for the first case, the coupler length is 53.3*µm* and the distance between the waveguides is *d*=150*nm*, hence the starting *K* value is at a negative slope over coupler transfer function, and a longitudinal offset implies an increase in the coupling constant; for the second case a coupler length of 20*µm* and waveguide separation of *d*=100*nm* are used, being the starting *K* value at a positive slope over coupler transfer function from Eq. (1), and hence an offset implies a decrease in the coupling constant.

## 3.1. Gap coupling control CROWs

The coupling constant *K* values obtained through BPM simulations of parallel waveguide couplers, with nominal lengths 53.3*µm* and 20*µm*, are shown in Figs. 3(a) and 3(b) respectively. The 3D graph shows the value of *K* versus two parameters, waveguide center-to-center distance, *d*
_{0} and waveguide width, *w*. In photolithographic fabrication processes, the waveguide width can be changed from a nominal value, 500 nm in our study, changing the exposure dose [12].

From the graphs and Eq. (1), for a fixed waveguide width, an increase of *d*
_{0} (reduction of **κ**
_{12}), results in a lower value of *K* for the same coupler length, *L _{c}*. The graphs also show, a significant variation in

*K*for changes in

*d*

_{0}of tenths of nanometers. Approximately a 10 % change in

*K*for 20 nm can be derived from the plots. This argument can be reversed to state that a small change in

*K*, for instance 1 % change, requires a very small change in

*d*

_{0}(in the order of nanometers), that might not be attainable by some fabrication processes.

On the other hand, for a fixed distance between waveguides, *d*
_{0}, a variation of the waveguide width, due to an over/down-exposure [12], does not result in significant changes in *K*. This can be explained through Eq. (1) again, since as the waveguide width changes *w*, the overlap of the fields, *κ*
_{12} remains similar, for this very particular case.

Therefore, in this kind of couplers, to accurately set the desired coupling constant changing the gap, a resolution of nanometers in the fabrication process is needed.

## 3.2. Offset coupling control CROWs

A BPM analysis of the parallel waveguides coupler is also performed for couplers in which *d*
_{0} is kept constant, but with a longitudinal offset between the coupler arms. The results are shown in Figs. 3(c) and 3(d), for (*L _{c}*=53.3

*µm*,gap=0.15

*µm*) and (

*L*=20

_{c}*µm*,gap=0.10

*µm*) respectively.

In this case, the variation of the coupling constant is controlled by changing the waveguides relative longitudinal position in steps of micrometers or of hundreds of nanometers. From the graph, a variation of 10 % in *K* requires a step of approximately 2 *µm*. Reversing the argument, fine control of *K*, for instance a change of 1 %, can be obtained by an offset step of 200 nm.

Note that depending on the coupler design, an increase or decrease of *K* is obtained when applying an offset between the coupler arms, as seen in Figs. 3(c) (*K* increases with offset) and 3(d)(*K* decreases with offset).

Therefore, in this kind of couplers, to accurately set the desired coupling constant changing the offset, a resolution of hundreds of nanometers in the fabrication process is needed. Moreover with proper design the offset can help to increase or decrease the value of *K*.

## 3.3. Comparison gap vs. offset coupling control

The comparison of both techniques shows that the step size to control *K* is two orders of magnitude higher in the offset technique. This leads to the following relevant conclusions:

1. To change *K* a given amount, higher precision is required in the gap technique.

2. Once the minimum achievable gap of a given fabrication process is known, the way to increase *K* in the gap technique is to layout longer couplers. This approach entails larger ring perimeters and the consequent change in the FSR. The change in the FSR can be avoided employing smaller bend radii but at the cost of increased losses in the bends.

3. With proper coupler design, the offset technique allows for both increasing and decreasing the value of *K* from a nominal value, for a fixed coupler length and gap.

As a general conclusion, the offset technique presents more relaxed design and fabrication requirements than the gap technique. These conclusions are for the couplers themselves, however it is important to study the impact of these techniques and limitations on the spectral response of CROW devices.

## 4. Implications in the CROW spectral response

#### 4.1. Spectral response and targeted K values

The frequency response of several CROWs is calculated in this section, using the Transfer Matrix Method [15, 16], in order to compare the performance of both presented techniques. Two CROW devices, with 3 and 5 racetracks, for each coupling control technique, are studied. The value of the coupling constants has been extracted from the BPM simulations shown in Fig. 3. The transmission transfer functions, i.e. output on the opposite side of injection, are shown in Fig. 4 for 3 and 5 rings devices, both for the uniform and apodized cases. The spectra shown are limited to a single Free Spectral Range (FSR) in normalized units of *δ*/2*π*, being:

with *L _{cav}* the optical path length of a single cavity and

*n*the mode effective index of the waveguides. The nominal

_{e}*K*value, for the uniform device, is

*K*=0.2. This value is changed in the 4 couplers (3 rings device) and 6 couplers (5 rings device), following a Hamming window law [7] with window parameter

*H*=0.25, for the apodized cases. On CROWdevices the direct coupling coefficient is apodized, therefore after applying the Hamming window the coupling constant values are higher than the nominal value, as detailed in [7].

The *K* values are summarized in Tables 1 and 2, along with the gaps/offsets required by design to obtain the apodized responses depicted in Fig. 4 (dashed lines). These *K* values are for a nominal waveguide width of 500 nm that only allows one mode to propagate. The propagation losses have been estimated to be about 2*dB*/*cm* with negligible losses in bends since *R*>2*µm* [9]. The offsets and gaps presented have been extracted also from the BPM simulations data for both coupler lengths of 53.3*µm* and 20*µm*, Fig 1. The spectra shown in Fig. 4 are given for the *K _{n}* values in Tables 1 and 2, regardless of the apodization technique used.

## 4.2. Sensitivity to finite resolution in fabrication

The sensitivity of the spectral response to finite resolution in the fabrication process, for the couplers analyzed in Section 3, is presented. In practice the fabrication process, either photo or electron-beam lithography, has a finite step resolution. The spectral response was calculated for gaps/offsets rounded in 10 and 20 nm steps. Hence, the *K _{n}* values are recalculated using Fig. 3, assuming the discrete values of

*d*

_{0}/offset.

The results are shown in Figs. 5 and 6 for the 3 and 5 racetracks CROWs respectively. In the first example shown in Fig. 5(a), the 3-ring CROW responses where the coupler length is fixed to 20 *µ*m are plotted. The spectra show the ideal response, already depicted in Fig. 4, and the apodized responses where the gaps/offsets have been rounded in 10 and 20 nm steps. The

’ideal’ responses in Figs. 5 and 6 refer to the dashed plots in Fig.4. While the offset apodized response is completely overlapped with the ideal response, the gap apodized responses are deviated from the ideal one. As another example, the 5-ring CROW responses, using couplers of *L _{c}*=53.3

*µ*m, with a finite step in setting

*d*

_{0}and the offsets in Table 2(b), are shown in Fig. 6(b). As in all the examples from Figs. 5 and 6, the

*K*values used are for gaps and offsets rounded in 10 and 20 nm steps. For instance, if the finite step is 20 nm, the gaps in Table 1(b) will be 120, 160, 200, 200, 160, 120 and

_{n}*K*(n=1,2,3,4,5,6) 0.7385, 0.4669, 0.1942, 0.1942, 0.4669, 0.7385. Note that the offset technique spectra and the 10 nm step overlap in Fig. 6(b), while a step of 20 nm shows a spectral response already deviated from the targeted.

_{n}Regarding the delay response, *τ _{d}* of the CROW, it can be obtained using the following expression:

where Tc is the round trip time in a single ring and *ϕ*(*δ*) is the phase of the transmision response. Figures 7 and 8 show the normalized Group Delay of the examples shown above.

In view of these results, the offset apodization technique appears to be more robust than the gap apodization technique in terms of the required spectral sensitivity of the CROW devices.

## 5. Conclusions

A novel technique for controlling the coupling between ring resonator in CROW devices has been presented. Employing this technique, an apodized CROWcan be fabricated using identical fixed distances between rings and obtain the final design solely by changing the longitudinal offset. In the simulation examples provided, this technique requires two orders of magnitude less fabrication precision to set the coupling constant values, and under finite resolution steps in fabrication processes, clearly outperforms the gap technique.

## Acknowledgements

This work has been funded through the Spanish Plan Nacional de I+D+i 2008-2011 project TEC2008-06145/TEC CROWN and by the Generalitat Valenciana through projects PROME-TEO/2008/092 and GV/2007/240 APRIL.

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