## Abstract

We report on a mode adapter that transitions light between two SOI waveguides having different widths. The device has been designed using a two-dimensional embedded coordinate transformation and consists of a thin sheet of a controlled anisotropic material directly placed on top of the Si slab. We demonstrate that this layer effectively controls the flow of energy propagating in the Si slab by performing three-dimensional full-wave simulations. The proposed geometry can be implemented with planar optical metamaterials for various applications in guided optics.

© 2010 OSA

## 1. Introduction

Metamaterials are artificial homogeneous media with a much wider range of electromagnetic properties than those found in conventional materials [1–6]. Their inner structure consists of a compact arrangement of sub-wavelength inclusions that, when properly designed, provide a wealth of new material parameters such as negative refractive indices and exotic permittivity and permeability tensors [5,6]. The unique possibilities offered by metamaterials have sparked a great deal of interest because they can serve as powerful components and devices, including “perfect” lenses with a resolution beyond the diffraction limit [7,8], active modulators [9,10], compact polarizers [11] and cloaks capable of concealing a given region of space from electromagnetic detection [12,13].

So far, many breakthroughs have been made in the microwave and THz regime but a considerable effort is currently under way to expand the metamaterial approach to infrared and visible wavelengths [14–23]. Although spectacular progress is being made, current optical metamaterials suffer from significant limitations compared to their counterparts operating at lower frequencies. First, a number of attractive properties are difficult to obtain without the use of metallic metamaterials, causing severe performance issues due to the high absorption coefficient of metals in the infrared and visible region of the spectrum. Second, fabricating artificial media made of sub-wavelength elements represents a huge technological challenge. At optical wavelengths, the size of the metamaterial unit cells must be on the order of a few tens of nanometers, which can only been accomplished with nanofabrication tools such as electron beam lithography or nanoimprint lithography. Since these technologies are optimized for planar geometries, most optical metamaterials fabricated to date are (quasi) two-dimensional structures rather than real bulk artificial media.

These limitations suggest that new strategies must be found to fully exploit the metamaterial approach at optical wavelengths. One possible route that is currently actively investigated relies on dielectric photonic crystals operating in the homogenization regime [18–21]. For example, two teams independently used an aperiodic photonic crystal etched in a SOI substrate as a gradient index medium capable of concealing a defect on an otherwise flat reflecting wall [18,19]. These structures are neither resonant nor metallic so they can operate over a large bandwidth with virtually no material losses. Interestingly, these metamaterials based on the photonic crystal geometry are planar structures, yet they provide useful functionalities because they are not used in free space but rather in a guided optics configuration.

Here we present a different approach to use planar metamaterials with arbitrary permittivity and permeability tensors in guided optics. In contrast to Refs. [18–21], we demonstrate that the artificial structure does not need to occupy the entire thickness of the guiding layer. In fact, we show that it is sufficient to place a very thin layer of planar metamaterial on top of the waveguide to gain control over the signal even though the main part of the energy does not propagate through it. To illustrate these claims, we design a mode adapter to transition the energy from a large SOI ridge waveguide to a narrower one. The taper has been devised using the method of transformation optics. Although we simply considered a 2D transformation, we show that this structure can effectively act upon the three dimensional flow of light guided by the SOI structure.

## 2. Geometry

Figure 1 shows the structure under consideration. It consists of two SOI ridge waveguides separated by a central region where the top ridge has been replaced by an artificial layer acting as a mode adapter. As shown in the cross-sectional view of Fig. 1(b), the Si thickness is 200 nm within the waveguides and 170 nm everywhere else. The artificial layer has a length l = 5µm, and the same transverse dimensions as those of the input ridge, that is, a thickness of 30 nm and a width of 3 µm. The width of the output waveguide is 0.6 µm. The device operates at the wavelength λ = 1.55 µm.

## 3. Simulations and design

Given their transverse dimensions, the output waveguide is a single-mode structure while the input waveguide supports higher-order modes that will not be considered in this study. Figure 2 shows the transverse profile of the input and output fundamental modes computed with a commercial eigenmode solver. The modes are almost TE-polarized, with the E field mainly oriented along the y-axis and most of the energy propagating within the Si slab. Our aim is to ease the transition between the two waveguides by gradually reducing the width of the input mode with the adapter shown in Fig. 1.

To design the mode adapter, we use the technique of transformation optics [24,25]. The first step of this powerful method consists in conceptually warping space so as to force light to follow a desired path. Here we consider an embedded coordinate transformation [26,27] restricted to the surface occupied by the mode adapter on top of the Si slab (Fig. 1). As shown in Fig. 3 , our transformation has two effects: (i) it isotropically compresses the metric with respect to the surrounding Cartesian space, resulting in a much denser coordinate grid in both x and y directions; and (ii) it bends and brings the y-coordinate lines closer together using an exponential transform [28]. Notice that the upper and lower exponential curves are not connected to the edges of the output waveguide: in fact, the exact width of the tapered space must take into account the fact that the lateral extension of the mode can be larger than the physical width of the waveguides (cf. Fig. 2). The second step of the method exploits the invariance properties inherent to Maxwell’s equations to reformulate the problem in terms of material parameters instead of the transformed spatial coordinates [24,25]. The resulting medium is anisotropic with gradients in its permittivity ε and permeability µ:

where ε_{Si}= 11.8 is the permittivity of Si, and

*f*and $\overline{\overline{\theta}}$ are related to the transformation steps (i) and (ii), respectively. The parameter

*f*is a simple scaling constant that corresponds to the compressed metric within the transformation optical region. $\overline{\overline{\theta}}$ is a tensor which can be expressed as a function of the length

*l*of the transformation optical layer and the ratio γ of the widths of the output and input modes:

A slightly different version of Eqs. (1) and (2) can be found in [28] where they have been devised in the context of two-dimensional tapers. The different components of Eq. (2) are plotted in [28]. θ_{xx} and θ_{zz} vary between 1 and 5.5, θ_{yy} vary between 1 and 14 while the off-diagonal elements vary between −8 and + 8. We ascribe these material parameters to the three-dimensional artificial layer of Fig. 1. Since the transformation only affects the x-y plane, ε and µ do not vary along the third (z) direction over the entire thickness of the transformation optical medium. We emphasize that the function of the artificial layer considered in this study is not simply to control the wave propagation in the plane of the transformation. Rather, our goal is to show that this layer can effectively control the shape and properties of the three-dimensional mode guided by the SOI structure.

Note that, in general, this class of embedded transformations does not yield to reflectionless components because the boundaries of the transformation optical region are not included in the calculation of the material parameters [26,27]. However, reflections can be suppressed if the in-plane and normal components of the metric tensor are conserved across the boundaries. In our case, it would be possible to satisfy this condition at either the input or the output of the mode adapter by choosing a scaling factor *f* that smoothly connects the coordinate lines in the transformation optical region to that of the outside medium. However, for reasons that will become apparent in the following sections, we consider the situation shown in Fig. 3 where abrupt topological discontinuities exists at the boundaries of the mode adapter because the coordinate grid is much denser than everywhere else. In other words, the values of the material parameters are very high within the mode adapter, resulting in a significant index contrast between this layer and the underlying Si slab.

## 4. Results

To simulate the structure shown in Fig. 1, we perform 3D full-wave simulations using commercial finite element code (Comsol Multiphysics). The fundamental mode of the input waveguide is launched using a wave port. We also take advantage of the symmetry to ease the computational effort by simulating only one half of the structure. All the results shown here have been obtained at the wavelength λ = 1.55 µm.

We first begin with the case of a coordinate transformation with a scaling factor *f* = 5 in Eq. (1). Figure 4
shows a map of the electric field visualized in the horizontal (x-y) propagation plane located halfway through the central Si slab. The plot reveals that the incoming mode becomes increasingly narrower and similar to the electromagnetic field of the output mode. Although this result validates the transformation considered in the previous section, we emphasize that the field of Fig. 4 is visualized in a plane located *under* the thin layer of transformation optical material. To gain insight into this behavior, we have plotted the transverse power of the mode at different cross-sections along the geometry. The results, shown in Fig. 5
, indicate that the propagation is governed by two successive mechanisms. Near the entrance of the coupler, the mode appears predominantly guided by the transformation optical layer [Figs. 5(a) and 5(b)]. In fact, the trend shown by these two plots is modulated by an oscillatory exchange of energy between the Si slab and the transformation optical region, suggesting that the two layers interact by directional coupling. Near the end of the device, however, these oscillations disappear. The mode becomes asymmetric, with most of its energy being confined along the top surface of the Si slab, just under the thin sheet of transformation optical layer [Figs. 5(c) and 5(d)]. This change in the propagation characteristics can be attributed to the gradients in the material parameters of the mode adapter. Their values become much higher than those of the Si slab and Figs. 5(c) and 5(d) suggest that the resulting index mismatch between the two layers is high enough to trap and guide the mode at their interface.

Taken together, the plots shown in Fig. 4 and Fig. 5 demonstrate that the transverse width of the mode decreases monotonously even though the interactions between the Si slab and the transformation optical region vary considerably from one side of the device to the other. However, the vertical asymmetry of the mode at the end of the mode adapter [Fig. 5(d)] is detrimental because its field pattern has a poor overlap with the mode supported by the output waveguide (Fig. 2). As a result, the transmission coefficient between the input and the output waveguides is only 17% due to unwanted insertion losses at the exit of the mode adapter.

To improve the design, one must adjust the interactions between the Si slab and the transformation optical region so as to minimize the mode asymmetry at the end of the device. Here we do not attempt to leverage all aspects of these interactions but note that they can be tuned without affecting the exponential transform by modifying the scaling factor *f* in the permittivity given by Eq. (1). As explained in section 3, this proportionality constant controls the mismatch between the material parameters of the transformation optical medium and those of the SOI waveguide. Since the mode asymmetry is induced by the high mismatch between the two layers, the value of *f* must be decreased to improve the transmission.

It should be reminded that the mode asymmetry only occurs towards the end of the mode adapter and that the Si slab and the transformation optical layer exchange energy by directional coupling in the first half of the device [Figs. 5(a) and 5(b)]. Consequently, reducing the value of *f* in Eq. (1) is also an opportunity to optimize the propagation at the entrance of the structure. In practice, it is desirable to minimize the amount of power coupled to the transformation optical layer because it will be implemented with optical metamaterials with potentially high material losses. By taking these considerations into account, one can increase the transmission coefficient up to 45%, as shown in Fig. 6
where the transmission is plotted as a function of *f*.

Figure 7
shows the evolution of the mode for the most favorable structure (scaling factor *f*≈2). It can be seen that we have been able to optimize the propagation in the first half of the coupler because the signal remains predominantly guided by the Si slab [Fig. 7(a)–7(c)]. In addition, the mode is reshaped without loosing its original symmetry. On the other hand, the shape of the mode is still not optimal at the end of the structure. As shown in Fig. 7(d), the vertical confinement has decreased, yielding a better overlap with the output waveguide, but the mode remains asymmetric and guided along the top surface of the Si slab.

Due to this persistent field asymmetry, the structure cannot compete with conventional tapers in its present form—in comparison, we calculated that the coupling coefficient can reach 80% by simply narrowing the top Si ridge. Yet, we have shown that the transmission can be optimized to a surprisingly good degree by simply adjusting a proportionality factor in the material parameters of the mode adapter. This result provides hope that a more sophisticated coordinate transformation—one that would not be purely devised in two dimensions but that would also take the three dimensional nature of the mode into account—would yield even better results.

## 5. Conclusion

We have used the technique of embedded coordinate transformations to design a waveguide mode adapter on SOI. Although the performances of the device can be improved, the important point of this study is that a planar sheet of transformation medium directly placed on top of an optical waveguide is capable of modifying the properties of the guided signal. This result has two interesting implications. First, the proposed geometry is compatible with current nanofabrication techniques—since the transformation optical region is very thin, the fabrication process would only require one single layer of lithographically patterned elements. Secondly, we have seen that the signal can be forced to propagate mainly in the optical waveguide (Fig. 7) so any material losses associated to the metamaterial structure would be significantly mitigated. These conclusions can be extended to other waveguide geometries as well as other types of transformations. In particular, this approach should be especially useful for complex manipulations of light that cannot be achieved with traditional means.

## Acknowledgment

This work was supported by the French Agence Nationale pour la Recherche (ANR Metaphotonique, contract number 7452RA09).

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