In this paper we propose analyse the apodisation or windowing of the coupling coefficients in the unit cells of coupled resonator waveguide devices (CROWs) as a means to reduce the level of secondary sidelobes in the bandpass characteristic of their transfer functions. This technique is regularly employed in the design of digital filters and has been applied as well in the design of other photonic devices such as corrugated waveguide filters and fiber Bragg gratings. The apodisation of both Type-I and Type-II structures is discussed for several windowing functions.
©2007 Optical Society of America
There has been recently an increasing interest in the field of micro ring resonator based devices. On one hand, this interest is fuelled by the considerable development of different technologies in the field of integrated optics that allow the integration of single and multiple ring cavities in a variety of material substrates and configurations –. On the other hand, devices composed of single or multiple micro ring resonator cavities can be exploited in a wide variety of classical applications including, among others: channel filtering in WDM systems , linear and nonlinear digital optics , optical buffering  and modulation , dispersion compensation , switching . Furthermore, their range of applications can be extended to encompass the emergent field of quantum information processing .
Different architectures integrating multiple rings have been proposed that are capable of performing complex signal processing operations, including the well known side coupled integrated space sequence of resonators (SCISSORs) , coupled resonator optical waveguides (CROWs)  and 2D structures . Methods for the efficient analysis and synthesis of these structures have been developed by different groups that are based on techniques that are borrowed either from discrete time signal processing , microwave engineering , or solid state physics .
Coupled resonator optical waveguide (CROW) devices are particularly interesting for filtering applications , since their operation is similar to that of distributed feedback filters in the so called Type-I configuration  and to a stack of dielectric mirrors in the Type-II configuration . In fact, previously published contributions have demonstrated their periodic bandpass filtering behaviour . A common assumption in these works , is to take the CROW device as composed of a number of identical sections or unit cells that are cascaded to form the final device. In this case the final filter structure resembles an uniform filter and therefore strong sidelobes are obtained in the bandpass transfer functions. In this paper we propose the apodisation or windowing of the coupling coefficients in the unit cells as a means to reduce the level of secondary sidelobes. This technique is regularly employed in the design of digital filters  and has been applied as well in the design of other photonic devices such as corrugated waveguide filters  and fiber Bragg gratings .
The outline of the paper is as follows: we first describe and analyse in section 2 the apodisation of type-I CROWs giving the basic equations and considerations for their design. Results are provided that illustrate the secondary sidelobe reduction obtained for a variety of apodisation functions and comparison between the performance of several apodisation profiles is established based on the concept of the effective number of rings in the CROW structure. In section 3 a similar analysis is carried for Type-II CROW structures. In section 4 we provide the simulation results obtained using a Finite Difference Time Domain (FDTD) method that confirm the results obtained for a Type-II CROW. Finally section 5 provides the summary and conclusions.
2. Apodised type-I CROWs
We first consider Type-I CROWs as defined in . The general structure of the apodised Type-I CROWcomposed of N uncoupled rings with equal length Lc each one coupled to an in (upper) and a drop (lower) waveguide is shown in Fig. 1, where the individual unit cells are identified. Here we allow in each unit cell “i” different coupling values to the in and the drop waveguides. Also, the coupling values can change from one cell to another as a result of the apodization. We follow the same nomenclature as that employed in  for the cross and direct coupling parameters of each coupling region. Also shown in the figure is the ring separation parameter Lb and the electric field convention employed in the paper where the “+” superscript labels the fields propagating from left to right and the “-” superscript labels the fields propagating from right to left.
We use the transfer matrix method – to analyse the structure which is composed on N-1 unit cells and a closing ring cavity. The layouts of an arbitrary unit cell and the closing cavity are shown in Fig. 2.
The transfer matrices of the arbitrary unit cell MUCi (i=1,2,..N-1) and the closing cavity MCN are given respectively by:
and the physical parameters related to the cavity round trip phase shift and losses and the intercavity phase shift are given respectively by:
With α and β representing the waveguide attenuation and propagation constants.
The overall transfer matrix of the Type-I structure is then given by:
from which we get the relevant transmission and reflection transfer functions:
The apodisation is impressed in this case over the cross-coupled coefficients according to a specified window function  w(i) i=0,1,2…N-1. For type-I structures several options are possible: apodizing only the cross coupled coefficients in the in bus (κ 1i=κw(i)), apodizing only the cross coupled coefficients in the drop bus (κ 2i=κw(i)) or apodizing the cross coupled coefficients in both buses (κ 1i=κ 2i=κw(i)). In the following we consider the first and third cases and explore the effects of apodisation using standard windowing functions employed in signal processing applications .
The formulation above includes the general case of coupling losses. However, the simulations presented are for the case of a loss-less coupler, where t and κ are related through:
with K being the power coupling ratio of the coupler.
The left part shows the case of a single bus apodisation while the right part illustrates the effect of apodizing both buses.
The effect of sidelobe reduction due to the apodisation of the cross-coupling coefficients can be observed in both cases as compared to the case of no apodisation (G=0) which is also depicted in blue colour trace for reference. Higher reductions are obtained for the case where the cross coupled coefficients are apodised in both buses. In fact this result is obtained for the different apodisation windows that have been considered.
For the Hamming window , the following apodisation function is implemented:
where the case H=0 is equivalent to no apodisation. Again, sidelobe reduction is observed for both cases with a better performance for the case where the cross coupled coefficients are apodised in both buses. For the Kaiser apodisation window :
Again, sidelobe reduction is observed for both cases with a better performance for the case where the cross coupled coefficients are apodised in both buses. Apparently, the Kaiser window function provides the best performance regarding sidelobe suppression. However, one must be careful when comparing the performance of the different apodisation functions as it will now be explained. Apodising the coss coupling coefficients is in effect equivalent to reducing the number of rings in the CROW waveguide. The argument is equivalent to that provided in  to demonstrate that the effective length of an apodised grating is lower than that of an uniform device. To compare the performance of the different apodisation windows one has to choose a reference metric for the effective number of rings of the CROW device. Then one should compare apodised CROW structures with the same number of effective rings.
We define the effective number of rings in the apodised TypeI CROW structure by:
The effective number of rings for the type-I apodised CROWS in Figs. 3, 4 and 5 is displayed in the upper part of each graph. Obviously, for uniform (i.e non apodised) CROWs N=Neff (in this case Neff=10). In each case, as the apodisation parameter is increased the value of Neff decreases.
In Fig. 6, we compare the performance of the three windows for a CROW device with (N=10), Lb=0.5 ·Lc and the same effective number of rings Neff=6.9.
It can be observed that the performance of the three apodisation windows is quite similar when the number of effective rings is the same.
Regarding the delay response, τd of the CROW, it can be obtained using the following expression, where Tc is the round trip time in a single ring:
where ϕ(δ) is the phase of the reflection response in Eq. 11. The results are shown in Fig. 7 for Gaussian and Kaiser apodisation. As expected, more delay on the centre of the pass band is obtained for smaller bandwidth responses, i.e. a bigger degree of apodisation, which is in good agreement with resonator theory shown elsewhere.
3. Apodised type-II CROWs
In this section, Type-II CROWs as defined in  are analyzed. The structure consists on a set of coupled rings between two regular waveguides, as shown in Fig. 8. An N ring Type-II structure is the composed of N-1 unit cells closed by an opening and a closing section which connect them to the input and output waveguides respectively. Figure 9 shows the layouts of the unit cell and the input and output closing sections. The unit cell transfer matrix is, using the same symbol convention as above:
The matrix corresponding to the input and output coupling sections of the CROW, (Fig. 9) opening and closing sections respectively, OS and CS, are the following:
The response of Type-I and Type-II CROWs are complementary, that is, while for Type-I the transmission and reflection from Eq. 11 are band-reject and band-pass respectively, for Type-II the transmission and reflection are band-pass and band-reject.
In the case of Type-II CROWs the apodisation is impressed on the direct coupling coefficients, ti, therefore ti=tw(i), where the window functions are given by Eqs. 14–16. The results are shown in Fig. 10. The apodisation reduces the ripples in the passbands, at a cost of an increase in the filter bandwidth. In Fig. 10-(a) comparison of the three windowing functions for an fixed effective number of rings, (Eq. 16), of 6.6 is shown.
4. FDTD simulation vs. analytical expressions
The previous analytical model has been checked against numerical simulations performed with the finite-difference time-domain (FDTD) method , using a freely available software package with subpixel smoothing for increased accuracy . The simulations were performed for micro rings on InP deep-etch waveguides technology . Simulations were performed in 2D, by reducing the vertical dimension using the effective index method . Therefore waveguides and rings are of w=0.3 microns width, ring radius is set to R=5 microns and the effective index is neff=3.271. In order to set the precise coupling between the micro rings, and between micro rings and input/output straight waveguides, simulations to determine the lateral distance for a given coupling were performed, and the results are shown in Fig.10. The results show the coupling decreases exponentially with the lateral distance, in good agreement with theoretical analysis . Using the coupling vs. lateral distance relation from Fig. 11, a type II CROW consisting on 6 micro rings, and coupling apodised with a Hamming window (H=0.12), was simulated. The result is compared with the analytical model in Fig. 11. The simulated response matches the model at the location of the peaks on the top of the response. The mismatch is due to radiation losses, that are not accounted for in the model.
5. Summary and conclusions
We have proposed and analysed the apodisation or windowing of the coupling coefficients in the unit cells of coupled resonator waveguide devices (CROWs) as a means to reduce the level of secondary sidelobes in the bandpass characteristic of their transfer functions. This technique which is regularly employed in the design of digital filters has also been applied as well in the past for the design of other photonic devices such as corrugated waveguide filters and fiber Bragg gratings. The apodisation of both Type-I and Type-II structures have been discussed for several windowing functions and sidelobe suppression in both structures demonstrated. In Type-I structures the cross coupling coefficient is apodised and windowing can be applied either to one bus or the two in the structure with the second option giving an increased performance in terms of secondary sidelobe suppression. The effectiveness of the different windowing functions must be compared using an independent metric and for this purpose the effective number of rings has been defined. Windows with a similar value of effective number of rings in the CROW structure yield a similar performance as far as sidelobe suppression is concerned. Finally, we have presented simulation results for a Type-II structure computed using FDTD method that accurately match the analytical results.
This work has been partially funded by the Generalitat Valenciana through project GV/2007/240 APRIL (Active Passive Rings Integrated and Layered), and through the Spanish Plan Nacional de I+D+i projects TEC2004-04754-C03-02 and TEC2007-68065-C03-02. J.D. Domenech wishes to acknowledge a collaboration grant through the project GV/2007/240 APRIL.
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