1. Introduction

Metallic Photonic Crystals (MPhCs) currently attract much attention because of the well-known strong reflectivity of metals at optical frequencies caused by surface-plasmon effects that occur in the optical regime and lead to flat-band modes [1], and because of the expectation that such photonic crystals might eventually be fabricated with low-cost technologies.

Embedding metallic wires into a dielectric background material or coating Dielectric Photonic Crystals (DPhCs) with metals are obvious extensions that are called Metallo- Dielectric Photonic Crystals (MDPhCs) [2]. In this paper, both pure MPhCs and MDPhCs are considered. For reasons of simplicity, only 2D, i.e. cylindrical structures are studied. This means that the off-plane loss of realistic structures with finite size is omitted.

Up to now, most of the papers dealing with metallic photonic crystals describe the dispersive characteristics, i.e., the frequency dependence of the permittivity by a Drude model that is known to be not very accurate, especially near those frequencies where plasmon effects occur [3]. In order to avoid inaccuracies caused by the Drude model a frequency domain approach has been used in the following investigation: The semi-analytic Multiple Multipole Program (MMP) implemented in the MaX-1 platform [4]. It allows the user to take advantage of measured data [5]. Furthermore, MMP provides highly accurate and reliable results.

DPhC filters have already been studied by several groups [6,7,8]. Often, the DPhC structure plays the role of a Bragg mirror, i.e., a nano cavity is embedded between two DPhC mirrors [7]. For this purpose, the PhC structure is operated within the Photonic Band Gap (PBG). DPhC filters may be embedded in conventional optical waveguides [9] or in PhC waveguides [10]. For the analysis of the DPhC filter characteristics one usually considers a simple DPhC slab that exhibits periodic symmetry in x direction and consists of a few layers of rods or holes in y direction [6,7,8]. In this paper, we consider such PhC slabs made of metallic rods in free space or embedded in a dielectric.

Obviously, any DPhC structure can be replaced by appropriate MPhC or MDPhC structures. When doing this, one should be aware of the losses that are substantial in metals and reduce the quality of the band gap. In order to keep the losses as small as possible, it is important to reduce the number of PhC layers as much as possible, which is also highly desirable from the miniaturization point of view. The densification of filter structures causes strong interactions of filter parts. This increases the complexity of the filter design. For the design of ultra-compact filters, standard design rules fail and one is forced to take advantage of numerical optimizers as demonstrated in the following.

Essentially, there are two different categories of numerical optimizers: deterministic and probabilistic or stochastic [11]. The former are much faster when started sufficiently close to the optimum but they usually are trapped near local optima and are not able to find the global optimum in complicated cases as in the PhC filter design. For defining a good start point for a MPhC filter that can be improved with a deterministic optimizer, we need experience that may be obtained from extensive numerical simulation of relatively simple structures as outlined in the following sections.

In order to reduce the computation time as much as possible, we take advantage of the Model-Based Parameter Estimation (MBPE) [12]. It is important to note that MBPE is originally designed for linear filters. When applied to MPhC structures, the pronounced frequency-dependence of the complex permittivity degrades the MBPE performance and requires the implementation of auxiliary routines that automatically subdivide the frequency range into reasonably small pieces. However, for the filter design at telecom wavelengths – where the frequency characteristics of low-loss metals are relatively smooth – no substantial problems with MBPE are encountered. The MMP-MBPE computation of the frequency dependence of an MPhC filter with 1000 frequency points and an accuracy better than 1% requires on the order of one minute on a personal computer, i.e., is sufficiently short even for stochastic optimizations with inefficient algorithms when there are no good start values for the initial filter design available.

2. Metallic Photonic Crystal Analysis

As for DPhCs the band diagrams of MPhCs and of MDPhCs provide a quick overview over the most important properties. As for DPhCs, the band diagrams of MPhCs and MDPhCs are usually normalized. Fig.1(a) shows the normalized band diagrams for a MPhC consisting of circular silver rods with different radii on a square lattice with lattice constant a. It is important to note that such band diagrams are only valid for a single lattice constant a because of the frequency dependence of the material properties (permittivity) and that the standard band diagrams do not contain any information on the material losses.

 

Fig. 1. (a) Band diagram for silver photonic crystals consisting of circular rods with radius r on a square lattice with lattice constant a=820nm, Ez-polarization (electric field parallel to the silver rods); black: r/a=0.2683; red: r/a=0.1114; blue: r/a=0.089. Green line: Wavelength 1.55μm. (b) Part of the band diagram of an infinite MPhC (left) and frequency dependence of the transmission coefficient of a 9 layer MPhC slab (right) for silver wires on a square lattice with a=820nm, r/a=0.0894, Ez-polarization. Green line: Wavelength 1.55μm.

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Our goal is to design a band pass filter with low transmission loss at the telecom wavelength 1.55μm. By properly selecting the lattice constant a and the wire radius r, one can easily design the MPhC in such a way that 1.55μm is either in the fundamental band gap that extends from zero frequency to some cutoff frequency for the first mode or in one of the higher band gaps. The green line in Fig.1 represents a wavelength of 1.55μm which is inside the fundamental band gap for the lattice constant a=820nm when the radius of the silver wires is large enough. One therefore could use such a MPhC as a Bragg mirror and embed an appropriate nano cavity between two identical MPhC pieces. Of course, these MPhC pieces must be of finite size, i.e., MPhC slabs with a finite number of layers. For a finite MPhC slab one can easily compute the frequency dependence of the transmission coefficient T for an incident plane wave. As one can see from Fig. 1(b), the band diagram of the perfect MPhC and the frequency dependence of T are strongly correlated when the number of layers is large enough.

In those areas of the frequency spectrum where a mode between the Γ and X points is present, one observes a useful behavior of T(ω), similar to a higher order bandpass filter. The number of peaks with high transmission increases with the number of MPhC layers. In order to obtain a band pass filter one therefore might design a single MPhC in such a way that the band pass area is simply the frequency range of the first mode between the Γ and X points, i.e., one essentially designs a PhC operating between two band gaps. For obtaining a band pass around 1.55μm wavelength, i.e., 193.55THz, one could reduce the radii of the silver rods or reduce the lattice constant. However, from Fig. 1(b) one can see that such a band pass filter shows not a very good performance. In order to improve the filter performance, numerical optimizers are applied in the following.

3. Deterministic filter optimization

Instead of cascading different PhC slabs and finding ways to reduce undesired interactions between the slabs, a PhC filter is designed that consists of N strongly interacting 1-layer slabs. This may also be considered as a single N-layer slab with arbitrary radii to be optimized. For reasons of simplicity and in order to keep the computation time short, only N=5 is considered in the following. Increasing the number of layers is similar to increasing the filter order in classical filter design: With a relatively low order, no high-performance filter can be achieved. At first sight, one might therefore conclude that N=5 is far from being enough for obtaining a reasonable band pass filter with desired center frequency and band width. It will be demonstrated below that the filters with N=5 layers can be optimized in such a way that 1) the desired center frequency is obtained precisely, 2) the insertion loss is reasonably small, 3) and that the bandwidth may be modified within some limits.

Note that one may not only optimize the radii of the N PhC layers but also the locations of the rods or even the shape and material properties of the rods. This increases the search space and offers the chance of finding better or more compact filters at the price of longer computation time. However, for a first exploratory study it is reasonable to start with low numbers of model parameters to be optimized. Therefore, we assume that the rods are not displaced and have circular cross sections, i.e., we only optimize N=5 radii r_i. Assuming symmetry of the structure, i.e., r₄=r₂ and r₅=r₁, further simplifies the optimization, that is the optimization is carried out in the 3-dimensional parameter space r_i, where i=1,2,3. Starting with a reasonable initial guess – obtained from the analysis of various 5 layer structures – the optimization procedure presented in [13] finds the filter response shown in Fig. 2(a).

 

Fig. 2. Transmission characteristics for MPhC band pass filters, operating at 1.55μm; 5 layers of silver rods; lattice constant a=820nm. (a): version optimized for maximum transmission at 1.55μm; radii of the wires: r₁=r₅=91nm, r₂=r₄=73nm, r₃=220nm. The colored curves show the transmission response when all radii are simultaneously decreased by 1, 2, 3, 4nm, respectively. Inset: calculation unit in MMP. (b): Radii r₁=r₅=89.8nm, r₃=207.44nm, r₂=r₄ tuned from 20nm (dark color) to 200nm (bright color).

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As one can see, the optimization mainly shifts the center frequency to the desired value. The bandwidth remains almost the same and the insertion loss becomes slightly higher. From Fig. 2(a) one may understand why it is possible to mainly shift the center frequency: When the radii of all rods are simultaneously increased or decreased, one essentially observes a shift of the center frequency with a slight change of insertion loss and width of the pass band. When only a symmetric pair of rod radii is modified, one observes a much more drastic change of the filter characteristic (see Fig. 2(b)). For small radii of the four outer rods, no pass-band ripple is observed. The center frequency shifts towards higher frequencies with increasing radii and the bandwidth is increased. At some size of the radii, substantial pass-band ripple is observed. The dependence of the filter characteristic on the center rod is similar but moves in opposite direction: For an increasing radius of the center rod, the center frequency shifts toward lower frequencies. This allows one to tune both the center frequency and the band width. Of course, this optimization has some limitations. For filters with wider bandwidths and low pass band ripple as well as for filters with narrower band widths and low insertion loss, the optimization space with N=3 (5 symmetric circular rods on a equidistant lattice) is too small. Note that shifting the center frequency is relatively easy within the frequency range considered in Fig. 2. For stronger frequency shifts, not only the rod radii but also the lattice constant must be modified.

4. Stochastic optimization

A good initial guess is essential for the deterministic optimization. When this is not available, the optimizer may converge toward a local optimum that is far away from the desired goal. The animated Fig. 3(a) shows several filter responses for randomly defined radii. As one can see, the variety of responses is huge. Random search is certainly a very inefficient optimization technique but the resulting animation provides a quick overview over the complexity of the solutions. Since only a small fraction of possible designs is in the vicinity of the desired filter response, it is a time-consuming task for a stochastic optimizer to find a reasonable solution. This also holds for advanced stochastic optimizers such as Genetic Algorithms (GA) and Evolutionary Strategies (ES). However, as soon as the stochastic optimization has found a roughly acceptable design, tone may pass it to a deterministic optimizer that converges towards the nearest local optimum much faster than ES or GA. The speed of the convergence of a simple deterministic optimizer is illustrated in Fig. 3(b). Thus, the main purpose of the stochastic optimizer is to roughly explore a huge search space and to find initial designs that are promising but not yet good enough, whereas a deterministic optimizer best is applied for the fine tuning.

 

Fig. 3. Transmission coefficient versus frequency for various MPhC structures with 5 layers of silver rods. (a): (50kB movie) random search; (b): (42kB movie) deterministic search, starting at a reasonable initial guess. The desired filter characteristic is indicated by the red and green lines: For the frequency ranges marked red/green line, maximum/minimum transmission is desired. Radii for the optimal solution: r₁=r₅=89.8nm, r₃=207.4nm, r₂=r₄=87.9nm.

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For the stochastic search, the variety of possible characteristics requires a more advanced definition of the fitness. Figure 3 shows several sections of the frequency range with different goals. For the pass-band range – illustrated with the red line – one wants to obtain maximum transmission, whereas minimum transmission is desired for the range illustrated by the green line, i.e., one has a multi-objective optimization problem. The fitness definition therefore requires integration of the transmission coefficient over the red range, integration of the reflection coefficient over the green range and proper weighting.

5. Dielectric background – MDPhC filters

MPhCs consist of metallic rods in free space. Although it is possible to grow nano wires on some substrate in such a way that a pure MPhC is obtained, it is much more likely that a fabrication process will first start with a dielectric substrate containing holes that are filled by metal in a second step. Incidentally, the dielectric background material might also belong to a conventional Integrated Optics (IO) chip. For example, one could insert a small MDPhC filter consisting of only a few metallic wires inside conventional IO waveguide.

Although the dielectric background properties considerably affect the band diagram and filter properties, the design of MDPhC filters can be carried out exactly as for the MPhC case.

First, an initial guess is found either from experience, trial and error, or stochastic search. Deterministic optimization then quickly leads to the desired optimum – provided that such an optimum exists, which essentially means that the number of model parameters N and the search space are big enough. The dielectric background material essentially allows one to reduce the size of the filter structure. At the same time, the contrast of the permittivity |ϵ_metal|/|ϵ_background| is reduced which in turn can deteriorate the filter performance.

The animated Fig. 4 shows the optimization process for two MDPhC filters consisting of 5 silver rods in dielectrics with slightly different permittivities. The lattice constant was reduced to 500nm because the wavelength in such a dielectric medium is shorter than in free space. Thus, the size of this filter is shorter than the MPhC filter considered before. One can see that the width of the pass band is increased. This has two reasons: First, it is more difficult to obtain narrow band pass filters when the contrast of the permittivity is reduced as mentioned above. Second, the fitness definition of the optimization procedure affects not only the center frequency but also the bandwidth. Here, the fitness definition widens the bandwidth, i.e., the goal of the optimization was not to find a minimum bandwidth solution. Note that such wide-band solutions are difficult to obtain with the concept of a cavity embedded between two PhC layers acting as mirrors.

 

Fig. 4. Transmission coefficient versus frequency. Optimization of two MDPhC structures with 5 layers of silver rods, lattice constant a=500nm. Left hand side: (37kB movie) relative permittivity of the background material ϵ_B=2.0; right hand side: (31kB movie) ϵ_B=2.5. Radii for the optimal solution with different background material: (1) ϵ_B=2.0: r₁=r₅=49.70nm, r₃=87.70nm, r₂=r₄=20.00nm; (2) ϵ_B=2.5: r₁=r₅=42.80nm, r₃=123.70nm, r₂=r₄=64.00nm.

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6. Summary

Metallic and metallo-dielectric photonic crystal slabs have been analyzed using the highly accurate MMP solver that can easily account for material losses and dispersion. In order to speed up the computation, MMP has been combined with MBPE that was originally designed for the efficient analysis of linear microwave filters. Ultra-compact band pass filters with a total length of 4100nm for the MPhC case (without background material) and 2500nm for the MDPhC case (with glass as background material) have been designed. In the design process both experience and stochastic search algorithms were used for finding an initial model that was then optimized with efficient deterministic optimization procedures. It has been demonstrated that the center frequency of the filter can be shifted to any desired frequency by optimizing the wire radii, provided that a reasonable lattice constant is used. Furthermore, the bandwidth may be tuned within some limits – that depend on the number of PhC layers. Finally, the insertion loss and pass-band ripple can be kept reasonably small. Note that the material loss in the wires is not significant because the number of wires is small.