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In this paper, the layer aggregation method is applied to coupled resonator optical waveguides. Starting from the frequency transfer function, the method yields the coupling constants between the resonators. The convergence of the algorithm developed is examined and the related parameters discussed.
©2010 Optical Society of America
1. Introduction
Waveguide photonic devices based on the interference of multiple scattered versions of the fundamental mode are instrumental for enabling a wide variety of signal processing functions, such as single and multiple channel filtering and interleaving in WDM systems [1,2], linear [3] and nonlinear [4] digital optics, optical buffering [5] and modulation [6], dispersion compensation [7] and switching [8] which are required in high speed optical communications. Furthermore, novel emerging applications in biophotonics [9] and quantum communications [10] are expected to benefit from the functionalities brought by these devices. Each particular application needs a specific device design which can be obtained by a suitable synthesis procedure which, taking as a starting point the impulse response or transfer function that is required, renders the values of the relevant parameters that characterize the device.
There is an extensive literature [11-17] related to the synthesis of distributed feedback devices and, in particular, of layer peeling/aggregation algorithms for fiber and waveguide Bragg gratings [11,12,14,17]. In these devices the scattering of the fundamental mode is produced in a continuous way along the entire device length. While these devices have widespread application and advantages, especially in fiber format, it has been recently shown by several researchers [18-22] that coupled resonator optical waveguide (CROW) structures are more suitable for small footprint integrated devices. In CROW devices the scattering of the fundamental mode is produced at discrete locations within the structure in coincidence with the positions of the directional couplers. Several algorithms have been proposed for CROW synthesis including, for instance, those based on Butterworth and Chebyshev responses [23] and those based on the definition of a polynomial whose roots are the zeros of the channel-dropping transmittance characteristic [24].
In this paper we propose a novel synthesis algorithm for CROW structures which is based on the layer aggregation technique [17] previously proposed for the synthesis of fiber Bragg grating filters. In this case the filter is reconstructed by successive cavity aggregations. Starting from the frequency transfer function, the method yields the coupling constants between the resonators. The convergence of the algorithm developed is examined and the related parameters discussed.
2. Layer aggregation method
The CROW is a periodic structure as shown in Fig. 1(a), composed of coupled resonators. The coupling constant can be equal for all the couplers (uniform CROW) or different by applying a windowing function (apodized CROW) as described in [25]. For the couplers, the well-known matrix [22] relating the input and output electromagnetic field amplitudes is used:
Fig. 1. CROW (a) naming convention, and layer aggregation for h[n] (b) n = 0, (c) n = 1 and (d) n ≥ 2
where ti and κi are the direct and cross-coupling field coefficients and the subscript i = 0,1,…,N refers to the couplers in the CROW device. A loss-less coupler is assumed hereafter, i.e. |ti|2 + |κi|2 = 1. In practical devices, all the resonators have the same perimeter. Hence, the impulse response of the filter is composed of impulses, or samples, of different amplitudes happening at times multiples of the ring resonator round trip time, T. In this case it is convenient to use a discrete time notation [27], i.e. h[n] = h(nT). The layer aggregation method is based upon the analysis of the filter contributions to the impulse response time samples. The approach followed is similar to that developed for Fiber Bragg gratings in [17]. Referring to Fig. 1(b) and 1(c), at time n = 0 the impulse response is solely due to the first coupler of the device, while at time n = 1 the response is due to the first two couplers:
In both cases the paths followed by the light within the CROWs are non-recursive, i.e. a waveguide section of the device is only traversed once. For n ≥ 2, h[n] is formed by two contributions, one recursive and one non-recursive, as illustrated in Fig. 1(d) for n = 2, with yellow and red lines. In this particular case, the recursive contribution is due to the first resonator (two turns, marked in red as ’x2’), while the non-recursive comes from the direct reflection from coupler number 2. Generalizing, for n ≥ 2:
Fig. 2. CROW (a) target TMM calculated response for m = 9 and m = 14 and (b) synthesized and target responses (inset transmission response).
where the recursive part at time n is given by the corresponding sample of the impulse response of CROW with n-1 resonators, and the non-recursive part can be derived from the figures:
3. Reconstruction procedure
The input for the reconstruction procedure described in this section is the (reflection) transfer function of a uniform, or apodized, CROW. The CROW response is periodic in frequency and the frequency spanning a single period is named free spectral range (FSR), which is inversely related to the round trip time T within the resonator. The transfer function can be expressed as a function of the radian frequency normalized by the round trip time, i.e. H(ωd) with ωd = ωT. Hence, ωd ∈ [0,2π[. The starting point for the reconstruction is a sampled version of H(ωd), H[k] for k = 0,..,M-1, where the index k corresponds to the set of discrete frequencies taken from ωd every ωd/2πM, and M a power of 2, m = log2 M, to use fast Fourier algorithms. The reconstruction steps are:
- Given H[k], calculate its inverse discrete Fourier transform, IDFT, h[n] = IDFTM {H[k]}
- For n = 0 and n = 1 use Eq. (2) to solve for (t 0, κ 0) and (t 1, κ 1).
Fig. 3. Reconstructed Ki values for (a) N = 5 and (b) N = 10 rings for a uniform CROW with K = 0.1, and KN for uniform CROWs with K =0.1, 0.2 and 0.3, all vs. IDFT number of points M used.
4. Results and discussion
As a first example, the reconstruction is applied for the target transfer functions corresponding to uniform CROW devices with power coupling constant K = |κ|2 = 0.1 of N = 5 and N = 10 rings. The target transfer functions were calculated using the Transfer Matrix Method, TMM, from [25]. Figure 2(a) shows the response for N = 10 sampled with 2m points (m = 9 and m = 14). The coupling constants Ki, i = 0,…,N obtained through reconstruction are shown in Fig. 3(a) and 3(b), for 5 and 10 rings respectively, for different number of samples in H(ωd).
The reconstruction procedure converges when more samples are used, and for smaller M when less rings are used. This can be understood from Fig. 2(a), where the differences in |H(ωd)| are shown for m = 9 and m = 14. A small number of samples do not reproduce the transitions from peaks to nulls in |H(ωd)|. The steepness of these transitions is related to K and N, being smoother for bigger K and/or when apodization is used [25] and for smaller number of rings. Therefore, all the results presented are worst case. The differences between samples of h[n], using m = 9 and m = 14, are of the order of 10-4.
The situation is analogue to sampling of continuous time signals, where a proper sampling frequency must be used according to the Nyquist criteria. Down sampling will lead to errors in estimating the spectral components of the sampled signal. In the procedure presented in this paper, the start point is a continuous spectrum |H(ωd)| which is discretized for computational reasons, and the end point is the inverse (discrete) Fourier transform of it, yielding the filter impulse response h[n], that is used for the calculation of the power coupling constants amongst the resonators. Note that all this is ultimately related to Eq. (4), from which (tn, κn) are calculated for every n, tn = (-1)n(h[n] - hr[n]) /∏i=0 n-1|κi|2, so the imprecision calculating h[n] and hr[n] using the TMM and the IDFTM, besides the multiplication of small K values in the denominator, can make the reconstruction procedure divergent.
Fig. 4. Hamming (H=0.2) apodized CROW with nominal K = 0.1 (a) target TMM calculated response for m = 9 and m = 14, (b) synthesized and target responses (inset transmission response) and reconstructed Ki values for (c) N = 5 and (d) N = 10 rings, all vs. IDFT number of points M used.
The convergence of the reconstruction for uniform CROWs with different K values, 0.1, 0.2 and 0.3, is shown in Fig. 3(c) and 3(d) for N = 5 and N = 10 rings respectively. The graphs show the value of the coupling constant for the last coupler in the CROW, KN. For 5 rings, convergence is achieved for K = 0.2 and K = 0.3 with m ≥ 10 and m ≥ 9 respectively, while K = 0.1 requires m ≥ 11. The convergence worsens for 10 rings, where m ≥ 13 and m ≥ 12 are needed for K = 0.2 and K = 03 respectively, while K = 0.1 needs m > 14.
The reconstruction was also applied to a CROW device with apodized coefficients, where the ti values are modified following a weight (window) function. Starting with a nominal coupling constant K = 0.1, a Hamming window with parameter H = 0.2 was used as described in [25]. Hence, the coupler values for the target response calculation using the TMM are:
that correspond to Ki = {0.5728, 0.3750, 0.1397, 0.1397, 0.3750, 0.5728} and Ki = {0.5919, 0.5280, 0.4101, 0.2668, 0.1470, 0.1000, 0.1470, 0.2668, 0.4101, 0.5280, 0.5919} for 5 and 10 rings respectively. The TMM calculated target responses, sampled with the same points as in the uniform case of Fig. 2(a), are shown in Fig. 4(a).
Compared to the previous example, the targeted spectral response H(ωd) exhibits less pronounced transitions between maxima and nulls. This smoothness is produced by the apodization of the power coupling constants amongst the resonators. Therefore, the situation in terms of signal sampling and Fourier transform, is more relaxed than in the previous example. The comparison of m = 9 and m = 14 in Fig. 4(a) reveals that m = 9 is a good approximation of H(ωd), far better than in the previous uniform example of Fig. 2(a).
Consequently, this results in less error on the impulse response approximation, since the differences between the samples of h[n], using m = 9 and m = 14, are of the order of 10-14. The convergence of all the coupling constants vs m is shown in Fig. 4(c) and 4(d) for 5 and 10 rings respectively. As outlined before, lower sampling is required for the apodized cases, since for theses cases convergence is achieved for m ≥ 7 and m ≥ 9, for 5 and 10 rings respectively.
Finally, the responses calculated using the reconstructed coupling coefficients and the TMM, are compared with the targeted responses in Figs. 2(b) and 4(b). An inset shows also the reconstructed transmission response. In both cases, uniform and apodized, the match between the target and reconstructed responses is excellent.
5. Conclusion
In this paper, the layer aggregation method for the synthesis of multilayer structures has been adapted to coupled resonator optical waveguides. A procedure to link the contribution of each cavity to the impulse response at every time has been derived. The procedure has been applied to the synthesis of CROWs, starting from targeted transfer functions. The procedure convergence has been examined in terms of the related parameters. An excellent match between the targeted and synthesized responses has been demonstrated for worst case convergence parameters.
Acknowledgement
The authors acknowledge financial support from the Spanish MICINN project TEC2008-06145 and the GVA through project PROMETEO/2008/092. J.D. Doménech acknowledges the FPI research grant BES-2009-018381.
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