Multiple coupled microcavities have been widely utilized for optical communication systems [1,2,3,4,5,6,7,8,9]. For wavelength division multiplexing (WDM) techniques, optimal filter design is mandatory and flat responses are ultimately required [3]. Higher order coupled microcavities have been proposed as promising candidates for optical filtering. The dispersion analysis of large chain of microcavities has been recently introduced [5]. The application of multiple coupled microcavities can be extended to optical signal processing and routing. Systematic and rigorous design procedures are essential to obtain the required filter response especially with large number of microcavities.

The design of coupled resonators is usually done through the coupling parameters [10,11,12,13,14,15,16,17,18,19]. These parameters have a direct effect on the achievable transfer function of the filter. Once the coupling coefficient for each stage is determined, the geometrical specifications for each resonator are adjusted [11]. The design theory of coupled microcavities has been originally developed for microwave filters [20,21,22]. In the last decade, this work was extended to the design of optical filters [11] using Bragg gratings and ring resonators. In [11,12], the design procedure utilizes a transmission line equivalent network. Distributed capacitances and inductances are utilized to build a filter prototype which is mapped to the corresponding equivalent resonator design [21]. The method is used successfully for the design of Chebyshev-based filters [11]. Other filter types can be readily designed using different cascading configurations [12]. This design approach is limited to narrow band responses [21]. It is also based on add drop filters, where the output is only at the drop port of the coupled structure. Other approaches are developed for the optimal placement of zeros and poles of the filter transfer function [23] and [24]. These approaches are intended for the design of parallel and series coupled ring resonators [24]. They utilize resonators of equal coupling parameters to overcome the complexity of the design. Using these approaches, tapering of the coupling parameters is done intuitively [24]. This intuitive tuning is successful for a small set of cascaded series or parallel ring resonators. For higher order filters, optimal responses are not guaranteed.

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 [25], the microcavity coupling parameters are assumed to vary around known mean values. A perturbation theory is developed to propose a design optimization problem in the perturbation parameters. By dumping the higher order perturbation terms, we ignore the effect of multiple reflections among the rings introduced by the small adjustment of the coupling coefficients. This is a first order accurate approach as it takes into account only multiple reflections introduced by the zero order terms. The design problem is then formulated as a linear least square design problem. This linear least square problem has a unique solution that can be obtained in few seconds for tens of design parameters [26]. This can be contrasted with other approaches that converge to a local solution [14,15,16].

Another efficient approach for filter design using a large number of cascaded microcavities is based on linear phase filter (LPF) approximation [27]. An approximate objective function is exploited to solve the design as a linear program (LP) problem. This allows for fast and efficient solution of large scale problems. In addition, no initial design is required. The LP solver can find a feasible starting point by solving an initial feasibility problem. The computational time is less than one second for structures that contains up to 150 coupled microcavities.

This paper is organized as follows. The theory of cascaded series rings is summarized in Section 2. The perturbation approach is addressed in Section 3. Section 4 is dedicated for the linear phase filter approach. The conclusions are given in Section 5.

Complex ring configurations contain multiple ring to ring coupling stages. Each stage incorporates a directional coupler approximation. A single stage coupling is shown in Figure 1, where a, b, c, and d are the field values at the interfaces. The quantities σ and κ represent the through and cross-port field coupling parameters, respectively. The relationship between the fields can be summarized using the scattering matrix [28]:

    (1)

For the series-connected ring resonators shown in Figure 2, the a and b fields at the ith stage are related to the fields at the next stage through the relationship:

    (2)

where [28]

    (3)

Here, γ = θ − jα is the complex propagation factor inside curved structures. θ is the phase factor due to the propagation inside the ring and is given by:

    (4)

where f is the light frequency, n_eff is the real part effective waveguide index, and L_c is the cavity length of the resonator. The parameter α is the field attenuation coefficient.

For N resonator stages, the total transfer matrix is given by:

    (5)

The coupling parameters between the Nth ring and the output waveguide are σ_N+1 and κ_N+1. Utilizing b_N+1 = −σ_N+1a_N+1, the transfer function for the through port (reflectivity) R_out= b₁/a₁ is given by:

    (6)

Utilizing c_N+1 = jκ_N+1a_N+1, the transfer function for the drop port (transmissivity) defined as c_N+1/a₁ is given by:

    (7)

The calculated transfer functions in Equations (6) and (7) are utilized for analyzing coupled resonators of known ring coupling parameters at every stage σ_i, κ_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 [24]. Similar to the procedure of discrete digital filters design, we utilize Z domain representation of the optical filter response [24]. This is accomplished by utilizing z⁻¹ = e^−jθ, where z⁻¹ includes only the phase factor. All the transfer matrices are thus z-dependent and both R_out and T_out are transformed to the Z domain:

    (8)

The response of the coupled ring resonators follows the standard form of a linear discrete system (see Equation (8)). The calculated transfer function coefficients p_n and q_m controls the achievable response in terms of pass band and stop band quality and the extinction ratio as well. These coefficients are dependent on the ring coupling parameters (σ_i, κ_i) and cavity loss parameter defined as for all rings. Control of the waveguide to ring and ring to ring scattering parameters allows for a high degree of freedom to achieve different targeted filter responses. The Optical filter realization can be realized from either the through or the drop ports.

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 N cascaded coupled resonators, the coupling coefficient at each stage is perturbed round a mean value as:

    (9)

Substituting from Equation (9) in Equation (8), the polynomials A and B are of N + 1 order dependence on the individual perturbations δσ_i, ∀i. Assuming small perturbations δσ = [δσ₁, δσ₂, …, δσ_N+1]^T, higher order terms can be neglected. By ignoring the higher order perturbation terms, we neglect the effect of multiple reflections among the rings introduced by the small adjustment of the parameters σ. This is a first order accurate approach as it takes into account only multiple reflections introduced by the zero order terms . A linear approximation of the polynomials B(z) and A(z) dependence on the perturbed through port coupling δσ can thus be formulated. In this case we have:

    (10)

In Equation (10), c_i and d_i are the zero order terms in δσ that can be calculated by substituting with in Equation (8). The coefficients of the first order terms in δσ are represented by h_i and g_i, both are polynomials in .

In order to achieve a filter response with specific z-dependence, we match the ring filter coefficients (p, q) to the targeted filter coefficients (b, a). The following system of linear equations is thus constructed:

    (11)

In Equation (11), b and a are the vectors of the numerator and denominator polynomials of the target filter transfer function (8). The solution of the linear system of equation in Equation (11) produces the optimal δσ that satisfies the target filter transfer function.

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:

    (12)

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 ζ is the maximum allowable perturbation in the coupling parameters. This problem is convex and leads to a unique optimal design [26].

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 highly coupled microcavities, the through port coupling for each coupling stage (|σ_i| ≤ 1) is usually small. This allows for physically neglecting the higher order coupling between the microcavities as their amplitudes are proportional to σ_iσ_i+1. This assumption is mainly assumes that σ_iσ_i+1 « σ_i and can approximate the response to

    (15)

Notice that by neglecting the higher order coupling terms, and using the recursive formula (15), the polynomial A(z) in Equation (8) becomes unity. For low loss coupled microcavities, the recursive formula for the reflectivity at a the ith stage is:

    (16)

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 [30]. The symmetry assumption implies that:

    (17)

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

    (18)

From Equation (18), the through port transfer function can be represented as where is a real and periodic even function. It is thus sufficient to only consider .

The developed approximate transfer function is a linear phase filter formulation [30]. It transforms the design procedure into a convex optimization problem whose solution can be efficiently and accurately estimated for large number of coupled microcavities. The general formulation of the problem is

    (19)

In Equation (19), ω_s is the normalized stop band angular frequency, ω_p is the normalized pass band angular frequency, and ζ is the pass band ripples. The design parameters are the through coupling coefficients σ = In Equation (19), the last constraint is added to ensure the physical realization of the designable coupling coefficients.

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 [31]. One of advantages of this solver is the ability to find a feasible starting point. Thus, there is no need to supply an initial design which is considered as a significant advantage especially for problems with large number of 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 f_o around the central frequency f_o. The response of the through and drop port is given in Figure 8. The length of the cavity L_c is taken to be one quarter of the central wavelength. For the second example, the length of the cavity is taken to be half the central wavelength. Thus, the response is switched from band pass filter to band stop filter in the through port and vice versa in the drop port. The number of microcavities in this example is 30. The response in the through port has a rejection band of 0.4 f_o centered around f_o. This band has a flat response in the drop port as shown in Figure 9. For third example, a design of 150 rings is obtained for a flat response over 0.2 f_o and sharp roll off over 0.01 f_o from each side of the transmission band as shown in Figure 10. The computation time of this example is 1.2 s. However, solving the non convex minimax problem [30], the computational time for this example is 1.5 × 10⁵ 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 σ for all examples are given in Figure 11. A small change of 10% in the optimized parameters shifts the response by only 6%.

The computational time for different number of coupled rings is show in Figure 12. It is clear that this problem is linear programming and hence the computational time has small dependence on the number of rings.

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.