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Simple and efficient approaches for filter design at optical frequencies using a large number of coupled microcavities are proposed. The design problem is formulated as an optimization problem with a unique global solution. Various efficient filter designs are obtained at both the drop and through ports. Our approaches are illustrated through a number of examples.

Multiple coupled microcavities have been widely utilized for optical communication systems [

The design of coupled resonators is usually done through the coupling parameters [

We discuss in this review two simple structured optimization methods for the design optimization of coupled microcavities. Though these techniques are simple, they can design filters with tens of coupled microcavities in few seconds. These techniques are based on adopting simplified transfer functions that transform the nonlinear optimization problem of highly coupled parameters to a linear optimization problem with a global solution. In the first technique [

Another efficient approach for filter design using a large number of cascaded microcavities is based on linear phase filter (LPF) approximation [

This paper is organized as follows. The theory of cascaded series rings is summarized in

Complex ring configurations contain multiple ring to ring coupling stages. Each stage incorporates a directional coupler approximation. A single stage coupling is shown in

Coupled ring resonators.

For the series-connected ring resonators shown in

where [

Here,

where _{eff}_{c}

The structure of cascaded coupled ring resonators.

For

The coupling parameters between the _{N}_{+1} and _{N}_{+1}. Utilizing _{N}_{+1} = _{N}_{+1}_{N}_{+1}, the transfer function for the through port (reflectivity) _{out}_{1}_{1} is given by:

Utilizing _{N}_{+1} = _{N}_{+1}_{N}_{+1}, the transfer function for the drop port (transmissivity) defined as _{N}_{+1}/_{1} is given by:

The calculated transfer functions in Equations (6) and (7) are utilized for analyzing coupled resonators of known ring coupling parameters at every stage _{i}_{i}

It is a common procedure to transform a linear system to a suitable frequency domain representation. This representation can be further utilized to simplify both the analysis and synthesis of the cascaded system [^{−1} = e^{−jθ}, where ^{−1} includes only the phase factor. All the transfer matrices are thus _{out}_{out}

The response of the coupled ring resonators follows the standard form of a linear discrete system (see Equation (8)). The calculated transfer function coefficients _{n}_{m}_{i}_{i}

The filter coefficients in Equation (8) are of nonlinear dependence on the coupling parameters. Those nonlinear terms are the contribution of multiple coupling among the cascaded resonators. For large values of coupled parameters, we cannot neglect multiple coupling. Utilizing the perturbation theory, a simple formulation to reduce the complexity of the filter transfer function is developed. For global optimization of cascaded series rings filters, robust optimization techniques can be incorporated to provide optimal filter designs. For the proposed approach, the target design has a set of mean coupling parameters. Then we formulate a modified design optimization problem to estimate the required perturbation of the coupling parameters around the mean to achieve the desired filter response. The overall coupling coefficients are thus calculated. This approach does not neglect the multiple coupling effects and take into consideration the unavoidable resonator losses.

For

Substituting from Equation (9) in Equation (8), the polynomials _{i}_{1}, _{2}, …, _{N}_{+1}]^{T}

In Equation (10), _{i}_{i}_{i}_{i}

In order to achieve a filter response with specific z-dependence, we match the ring filter coefficients (

In Equation (11),

The system of linear Equation (11) is overdetermined. However, a least squares solution leads to the optimal perturbation parameters. The design problem can thus be cast as a constrained optimization problem with a quadratic objective function over linear constraints:

In Equation (12), the first constraint is placed to ensure the total through coupling parameter less than unity. The second constraint imposes a trust region for the perturbation model. The parameter

For lossy structures, the total loss of the ring modifies the optimization problem constraints. This can be taken into consideration by a direct modification of the system coefficients in Equations (10) and (11). The same system of equations is solved to get the perturbation in the coupling coefficients in the presence of the losses.

Our formulation can be contrasted with other conventional nonlinear least square problems. These approaches have complex dependence on parameters, computationally expensive and their solution is not globally optimal.

The perturbation technique is verified through the design of fifth order and tenth order optical filters using series connected ring resonators. We carry out optimization for both lossless and lossy structures. The optimization algorithm achieves the required response efficiently within the trusted perturbation region. Our algorithm is also applied to a set of lossy structures to predict the change in the achievable design with a loss increase.

For an ideal target response, we have applied our approach to the design of the fifth order drop filter [

(

Our algorithm is applied to extract the vector of perturbation in the through coupling parameters (^{T}

Our technique is also utilized to design a number of ten series connected rings. For this example, the targeted filter has small pass band ripples with an increased normalized bandwidth and improved steep filter pass band to stop band transition. The targeted filter design is proposed using matlab digital filter design functions [

The target filter is a Chebyshev type I filter with a bandwidth of one third of the free spectral range and minimal pass band ripples. The (cheby1) matlab function [^{T}

The coupling coefficients for the optimized tenth order optical filter utilizing the perturbation approach.

(

For this problem, the number of optimization parameters is 11 representing the through coupling coefficient _{i}^{T}

To illustrate the accuracy and correctness of the perturbation approach, we also compare our results to that is in [^{T}

We utilize our approach for the design of the same target filter. The average couplings ^{T}^{T }

The achieved fifth order filter response utilizing the perturbation approach for both lossless and lossy structures as compared to the target filter.

To further illustrate the universality of our approach, we design the same ideal target filter utilizing lossy ring resonators with a maximum power loss factor (1 − τ^{2}) of 15.4%. Utilizing our approach, the same filter response can be achieved except for magnitude scaling due to losses. In ^{2}) = 15.4%) are larger than the ideal case as shown in

The optimal coupling coefficients for the optimized filter for both lossless and lossy case as compared to the coefficients predicted in [

For highly coupled microcavities, the through port coupling for each coupling stage (|_{i}_{i}_{i}_{+1}. This assumption is mainly assumes that _{i}_{i}_{+1} « _{i}

Notice that by neglecting the higher order coupling terms, and using the recursive formula (15), the polynomial

For symmetric coupled microcavities, the through coupling coefficients are symmetric around the middle coefficient. This is a practical assumption which allows for having a linear phase filter response [

The total through port coupling can be represented in terms of the normalized angular frequency (

From Equation (18), the through port transfer function can be represented as

The developed approximate transfer function is a linear phase filter formulation [

In Equation (19), _{s}_{p}

The proposed technique is exploited in filter design problems. For this purpose, the interior point-based solver, SeDuMi, is used for solving the linear programming problem [

The drop and through response of the optimized structure with 36 microcavities [

For the first example, a filter response with minimum ripples in the passband is obtained. The number of microcavities is 36. The pass band is 1.0 _{o}_{o}_{c}_{o}_{o}_{o}_{o}^{5} s for a starting point in the middle of the feasible domain. This comparison is performed on a 2.2 GHz dual core processor computer with 2.0 GB of RAM. The optimized values of

The computational time for different number of coupled rings is show in

The drop and through response of the optimized structure with 30 microcavities [

The drop and through response of the optimized structure with 150 microcavities [

The coupling coefficient of the optimized design of both the 30, 36 and 150 cascaded microcavities [

The computational time for different number of microcavities.

Two novel design procedures for filters with large numbers of microcavities were reviewed. These procedures are efficient and simple. Both approaches exploit convex optimization techniques for formulating the design problem. This formulation allows for fast and accurate solution of the design problem. The accuracy and the efficiency of these approaches allow for solving design problems with few hundred of variables in less than one second.

_{2}microring resonator optical channel dropping filters