1. Introduction

Transformation optics (TO) has been recently established as a new method in optical design. In particular, it makes it possible to demonstrate some very novel applications, such as an invisibility cloak [1,2]. Many different optical devices have been theoretically designed as well [1–20]. Most of the TO devices can be achieved with anisotropic transformation media, which are in turn realized through metamaterials. Because of the ease of fabrication, most of the experimental studies were performed in the microwave regime using metamaterials with artificial metallic resonating atoms [21–25]. On the other hand, demonstrations at much smaller wavelengths are actually quite challenging because of the stronger absorption of metallic elements and the larger difficulty in fabricating macroscopic samples with subwavelength building blocks. To tackle these problems, there are approaches involving simplification of the transformation mappings (with reduced or limited functionalities). This leads to transformation media with smaller index variations and constant anisotropy. Cloaking at infrared and at visible frequencies has been successfully demonstrated with this approach [26–32]. More recently, photonic crystals (PhCs) have been proposed as an alternative means to construct transformation optical devices within the eikonal limit. A one-dimensional photonic crystal from different dielectrics is proposed to approximate the effective medium of a TO invisibility cloak [33]. Although PhCs technically work in the frequency regime with wavelengths comparable to the lattice constant instead of the effective medium regime, they are found reliable in many different situations where the refractive index plays a more dominant role than the impedance. For example, we can also use PhC to mimic metamaterials for negative refraction, to construct zero-averaged-index photonic band gaps, and very recently a homogenized zero refractive index material [34–37]. Because the unit cell is large with respect to the wavelength, the sample can be potentially made larger, making fabrication easier. Moreover, wave propagation within the media has a very low loss without employing resonating metallic artificial atoms. The PhC approach, together with its associated wave phenomena, in approximating a low-loss effective medium, is thus worthwhile for further exploration. It also has the potential to enrich the design paradigm of integrated optical circuits with PhCs [38]. In this paper, we propose a transformation approach based on a PhC to achieve TO devices. Instead of searching for different PhCs to give a range of different equifrequency contours (EFC) in approximating a specific effective medium profile of a TO device [33], our approach directly generates the TO device by transforming a regular PhC to a curved and gradient PhC. Once the function of the regular PhC is prescribed, e.g., a specific ray trajectory, a wavefront modification, etc., the curved PhC allows wave propagation in a geometrical transformed fashion. Thus, it acts as a TO device to control the propagation of Bloch waves inside. An additional advantage is that we are no longer restricted by the conventional elliptical or hyperbolic-type EFCs given by an effective medium. In the following sections, we will demonstrate our approach on a single unit cell, then on the whole PhC, and finally demonstrate two TO devices, a wave compressor and a bending waveguide.

2. Transforming photonic crystals

First, we use a schematic diagram to reveal our approach in obtaining a transformation optical device with a curved PhC. Figure 1(a) shows the regular PhC in the virtual space before transformation. The dashed lines denote the unit cell boundaries, while the solid lines represent the specific structure within each unit cell. A blue arrow is also drawn to represent a Bloch wave traveling with a fixed wave vector within the crystal. After the regular PhC is transformed to a curved one in Fig. 1(b), the Bloch wave is traveling along a bent trajectory and is actually governed by the curvature of the transformed PhC. In other words, we are controlling the Bloch waves instead of the plane waves in the effective medium regime.

 

Fig. 1 Transforming (a) a regular PhC in virtual space to (b) a curved PhC in physical space. The black dashed lines outline the boundary of the unit cells of the PhC; the black solid lines outline specific structures within each unit cell. The blue arrows indicate the wave propagation paths.

Download Full Size | PDF

The regular PhC is transformed to the curved PhC through a global mapping in one step. However, it is easier to start our discussion at the level of a single unit cell. Figure 2(a) shows a single unit cell of lattice constant a in the regular PhC in Cartesian coordinate ($x\text{'}$,$y\text{'}$,$z\text{'}$). It is transformed to a curved unit cell in cylindrical coordinate (r,θ,z), shown in Fig. 2(b). In the current example, the coordinate mapping (concerning a particular unit cell) can be written as

 

Fig. 2 Transforming (a) a regular unit cell to (b) the curved unit cell. The lattice constant of the regular PhC unit cell is a. The curved PhC unit cell (composed of isotropic dielectric materials) spans an angle of $\Delta \theta =\pi /3$ with an inner radius $r_1=0.455a$ and an outer radius $r_2=1.455a$. The curved unit cell has a circular cylinder with radius $r_d=0.2a$ and permittivity $12.5{\epsilon }_b$ in background permittivity${\epsilon }_b=1$. The color map shows the E-field pattern at a normalized frequency $\omega a/\left(2\pi c\right)=0.59$.

Download Full Size | PDF

which is general enough for our consideration. Suppose we concentrate on the E-polarization (E-field pointing out of plane) and restrict ourselves to a dielectrics-only PhC $\epsilon (r,\theta )$ in the physical space for ease of fabrication. According to the TO theory, the coordinate mapping induces a relationship between the regular PhC (${\mu }_{x\text{'}}(x\text{'},y\text{'})$,${\mu }_{y\text{'}}(x\text{'},y\text{'})$,${\epsilon }_{z\text{'}}(x\text{'},y\text{'})$) and the curved PhC $\epsilon (r,\theta )$ through

As an example, we choose the curved unit cell as a cylindrical sector spanning an angle of $\Delta \theta =\pi /3$ with inner radius $r_1=0.455a$ and outer radius $r_2=1.455a$. It has a background permittivity ${\epsilon }_b=1$ and consists of a circular cylinder of radius $r_d=0.2a$ and permittivity $12.5{\epsilon }_b$ in the middle of the unit cell. The material parameters of the regular (square) unit cell can be obtained through Eq. (2) with $R=0.955a$ and $g(r)=r$ (this choice of mapping is just for ease of demonstration). The geometric parameters are chosen to satisfy $R\Delta \theta =a$ so that it is a square unit cell in the virtual space. The curved unit cell holds the simpler materials, i.e., dielectrics, while the regular unit cell is square in shape (easier for analysis) but with anisotropic materials in general. We also note that the shape of the cylinder inside the regular unit cell is not circular anymore. Under such a coordinate transformation, the regular and the curved unit cells should be equivalent to each other under TO. In particular, for a same set of Floquet boundary conditions applying on the two pairs of edges, the two unit cells give the same set of eigenmode frequencies (band structure). The color map in Fig. 2 shows the E-field pattern of the fourth mode for zero phase change in the radial direction (or $x\text{'}$-direction in virtual space) and a $\pi /3$ phase change in the angular direction (or $y\text{'}$-direction in the virtual space). The two eigenmodes are calculated by using the finite-element method with the commercial package COMSOL Multiphysics. Both of them appear at the same normalized frequency $\omega a/\left(2\pi c\right)=0.59$, while the field patterns are just equivalent to each other according to the coordinate transformation in Eq. (1). This equivalence allows us to define the EFCs (i.e., the refractive indices, which control how light propagates in the virtual space) in the virtual space and also the effective indices in the physical space after transformation.

Next, we describe the transformation of the whole PhC starting from the established transformation of a single unit cell. There are many different ways to transform the periodic PhC into the curved PhC in physical space. We therefore need to generate a family of different unit cells in the physical space that are equivalent to the same unit cell in the virtual space. Here, our approach is to simultaneously apply geometric scaling and permittivity scaling to a single unit cell in the physical space by

which can be regarded as the scaling law of the curved photonic crystal and can be proved from TO theory with geometrical scaling. Therefore, the resultant curved PhC is periodic in the angular direction, while in the radial direction each ring of thickness $\Delta r$ is geometrically scaled down by a constant factor $s=r_2/r_1$, and ${\epsilon }_b$ is scaled up by a constant factor $s^2$ from its neighboring outer ring. Based on this principle, we can design the TO devices through the scaling of the curved unit cells. The global mapping (written as $g(r)$ again) that corresponds to this scaling process can be written as $g^{\prime }(r)=R/r$, where $R\Delta \theta =a$ to ensure our square regular unit cell.

Furthermore, the above analysis can also be easily adapted to the H-polarization for dielectric-only PhCs by

For the scaling law, technically speaking, we should scale the magnetic permeability from TO theory. However, we scale the permittivity, instead of the magnetic permeability, by the same amount. This has the advantage that the EFC still remains unchanged, which is a requirement for how the wave propagates, but the whole curved PhC still consists of only dielectrics for ease of fabrication. Therefore, we use the scaling law specified by Eq. (3) for both polarizations, but we have to keep in mind that there is such a reduced-parameter approximation in the H-polarization. Although we have discussed the scaling law according to the mapping in Eq. (1), the scaling law can also be used in other kinds of mappings. For example, in conformal mappings of moderate curvatures, different unit cells in the physical space can be regarded as similar in shape, and the materials inside can be obtained through the scaling law, except that $\Delta r$ now generally means the size of each cell.

3. Transformation optical devices with curved PhC

Now, we can use the scaling approach to design transformation optical devices. We begin with the bending waveguide, which has been investigated by other transformation approaches with effective media [10–12]. It serves as a simple example (with the schematic also shown in Fig. 1) to demonstrate how a curved PhC with scaling law can be used for wave manipulation. Figure 3 (a) shows a bending waveguide (for H-polarization) constructed by a curved PhC in which a Bloch wave can make a turn of 180 degrees. The whole waveguide consists of three sections enclosed by perfect electric conductors (PEC). The initial and final sections are two rectangular air-filled waveguides with height $3.21a$. The waveguide in between is bounded by two half circles of radii $3.66a$ and $6.87a$ and is filled with the curved PhC of four layers of unit cells. Each unit cell of the outermost layer has a thickness of a in the radial direction and spans an angle of $\Delta \theta =9^{\text{o}}$. It consists of a circular cylinder of radius $0.275a$ and permittivity $4{\epsilon }_b$ in a background permittivity ${\epsilon }_b=1$. The unit cells in the inner three layers are consecutively geometrically scaled down by a constant factor $s=1.17$ with permittivities scaled up by $s^2$ according to our scheme. The index distribution of the whole bending waveguide is shown in Fig. 3(a), ranging from 1 to 3.2. Based on Eqs. (1) and (4), the geometric and constitutive parameters of the virtual regular unit cell can be derived. The working frequency is chosen at $\omega a/\left(2\pi c\right)=0.686$. The EFC at this frequency is shown as the black solid line in Fig. 3 (b), which can approximate a circle (red dashed lines), so that the virtual regular PhC can approximate an effectively isotropic medium. We note that our material (PhC) design comes from transforming physical to virtual space, while the device design comes from transforming virtual to physical space. An input plane wave is then fed into the upper port of the waveguide with the curved PhC, and the simulated H-field pattern is shown in Fig. 3 (c); the plane wave couples to the Bloch wave inside the curved PhC, propagates along a bent path to turn 180 degrees, and exits at the lower port with high transmission and an elapsed phase about 4π.

 

Fig. 3 (a) A curved PhC bending waveguide with inner radius 3.66a and outer radius 6.87a. Each unit cell of the outermost layer spans an angle of $\Delta \theta =9^{\text{o}}$ with thickness a in the r-direction. It consists of a circular cylinder with radius $r_d=0.275a$ and permittivity $4{\epsilon }_b$ in the background permittivity ${\epsilon }_b=1$. The scale factor is $s=1.17$. (b) EFC (black line) of the regular PhC at normalized frequency $\omega a/\left(2\pi c\right)=0.686$. The red dashed line represents the EFC of the corresponding isotropic effective medium. (c) H-field pattern at the same normalized frequency for a plane wave incident on the upper port. (d) H-field pattern for the corresponding effective medium ${\epsilon }_{eff}=n_{reg}{a}^2/{\left(r\Delta \theta \right)}^2$and ${\mu }_{eff}=n_{reg}$, where $n_{reg}=-0.146$. All length scales are normalized to a.

Download Full Size | PDF

To gain further understanding of the curved PhC, we investigate its effective medium representation. The size of the dispersion surface can be fitted by $ka=n_{reg}\omega a/c$, giving rise to $n_{reg}=-0.146$. The minus sign indicates a negative group velocity (the size of the EFC decreases with an increase of frequency). However, only the magnitude but not the minus sign is relevant to our investigation here, since we are exploring only the normal incidence but not the effect of negative refraction (which can be potentially useful in future investigations). We have also chosen a case in which there is nearly no reflection from the regular PhC so that we can simply set the effective impedance to $Z_{reg}=1$ as an approximation. It corresponds to an effective medium ${\mu }_{z\text{'}}={\epsilon }_{x\text{'}}={\epsilon }_{y\text{'}}=n_{reg}$, which the regular PhC is mimicking. It induces, through TO with Eq. (1) and $g^{\prime }\left(r\right)=R/r$, the effective medium in the physical space to be governed by ${\mu }_z^{\left(eff\right)}=n_{reg}{a}^2/{\left(r\Delta \theta \right)}^2$ and ${\epsilon }_r^{\left(eff\right)}={\epsilon }_{\theta }^{\left(eff\right)}=n_{reg}$. However, since only the permittivity of the curved PhC is scaled as mentioned, the effective medium is therefore governed by

The same waveguide is then simulated again but with the effective medium instead of the curved PhC. The result shown in Fig. 3 (d) with the effective medium agrees very well with the simulation with the curved PhC with a total phase elapse around $4\pi$ at the U-turn and with high transmission. For both simulations, the wavefronts at the entrance and exit of the waveguides are not completely flat owing to the approximation we introduced for H-polarization, that is, that we scale the permittivity instead of the magnetic permeability. As a result, we can conclude that the curved PhC in Fig. 3 (a) can mimic the gradient index profile of a bending waveguide.

An advantage of transforming PhCs, instead of effective media, is that more general shapes for the EFCs are available apart from ellipses and hyperbolas. Here, we would like to use a curved PhC to design a wave compressor [13] by exploring the use of a PhC to mimic a very anisotropic effective medium, for E-polarization in our case. This is particular difficult in the effective medium approach without using artificial magnetic resonances, which are usually associated with high absorption loss. Figure 4 (a) shows the working principle of the wave compressor. Here, we transform a regular PhC with very flat EFC, or equivalently very large anisotropy, although the EFC is not an ellipse anymore. The Bloch waves inside are propagating only along the horizontal direction for a range of different incidence angles θ due to the flat EFC. The curved PhC (circular in shape) with inner radius $R_1$ and outer radius $R_2$ is also shown in the same figure. The Bloch waves inside can only travel along the radial direction. By choosing a permittivity ${(R_2/R_1)}^2$ for the output medium, every incident horizontal ray to the curved PhC also exits in the horizontal direction (blue arrows). This is due to the conservation of angular momentum for the traveling wave, similar to an electromagnetic hyperlens. Because this picture holds for every horizontal ray, the device thus compresses a wide beam to a narrow beam, acting like a wave compressor.

 

Fig. 4 (a) Transforming a regular PhC with flat EFC to achieve a wave compressor. (b) Index profile of the wave compressor. The compressor has inner radius $R_1=10.5a$ and outer radius $R_2=19.6a$; see text for details of the configuration. (c) EFC of the regular PhC at normalized frequency $\omega a/\left(2\pi c\right)=0.58$. (d) E-field pattern when a Gaussian beam of width 10.3a is incident from the left. The beam is compressed to a width of 5.5a at the exit. (e) E-filed pattern of the same compressor with the corresponding effective medium, ${\epsilon }_{eff}=n_{reg}{a}^2/{\left(r\Delta \theta \right)}^2$, ${\mu }_{\theta }^{\left(eff\right)}=n_{reg}$, and ${\mu }_r^{\left(eff\right)}=\infty$, where $n_{reg}=-0.261$. (f) The normalized E-field amplitude distribution of the beam along the y-direction. The blue solid line represents the distribution of the incident beam. The black dashed line represents the ideal exit beam profile with compressed ratio $R_2/R_1$, and the red solid line represents the actual distribution of the exit beam from the simulation. All length scales are normalized to a.

Download Full Size | PDF

Figure 4(b) shows a closer look at the wave compressor. The curved PhC is bounded by two circles of radii $R_1=10.5a$ and $R_2=19.6a$ and is filled with 12 layers of unit cells. Each unit cell in the outermost layer has a thickness a in the radial direction and spans an angle of $\Delta \theta =3^{\text{o}}$. It consists of a cylinder in a shape of a small sector (looks like a bar in Fig. 4 (b), has thickness $0.6a$ in radial direction and spans an angle of $0.9°$) of permittivity $5{\epsilon }_b$ in a background permittivity ${\epsilon }_b=1$. The inner 11 layers are consecutively geometrically scaled down by $s=1.05$ with permittivities scaled up by $s^2$ according to our scheme. The color map shows the refractive index profile ranging from 1 to 4. Again, we transform the unit cell at the outermost layer to the regular unit cell and solve for the EFC, which is shown in Fig. 4(c) at a normalized frequency $\omega a/\left(2\pi c\right)=0.58$ for the E-polarization. We have chosen such a configuration because its flatness of EFC and also because of the very small reflection for the regular PhC; i.e., it is impedance matched to vacuum. We can therefore assign an effective medium ${\epsilon }_{z\text{'}}={\mu }_{y\text{'}}=n_{reg}$ and ${\mu }_{x\text{'}}=\infty$ where $n_{reg}=-0.261$, which the regular PhC is mimicking. Now, a Gaussian beam of width $10.3a$ impinges (from the left hand side) on the wave compressor horizontally. Figure 4 (d) shows the simulated E-field pattern. The red-dashed lines indicate the edge of the beam profile, as predicted by the ray picture in Fig. 4(a). The agreement is very good, showing that the beam is now compressed in the vertical direction. We emphasize that the regular PhC is transformed to the curved PhC directly without the need of an effective medium representation in the physical space. However, we can still work out the effective medium and investigate how faithfully the curved PhC is mimicking it. From TO theory with Eq. (1) and $g^{\prime }\left(r\right)=R/r$, the effective medium in the physical space is induced form the one in the virtual space and is governed by

The same compressor is then simulated again with the effective medium instead of the curved PhC. The E-field profile is shown in Fig. 4(e), which agrees very well with the simulation with the curved PhC; even the phase information in the compressor looks almost the same as the one in Fig. 4(d). According to the discussion of Fig. 4(a), the wave compressor has a compression ratio $R_2/R_1$ for the size of the beam. Figure 4(f) compares the beam width of the exit beam along the vertical direction. The blue solid line shows the normalized E-field amplitude distribution of the incident beam for comparison. The black dashed line shows the ideal exit beam if it is compressed by a ratio of $R_2/R_1=1.87$. The red solid line is the normalized E-filed amplitude distribution of the exit beam from the simulation with the curved PhC, which is nearly the same as the ideal case. Since the beam width is compressed, the power of the beam will be enhanced with the same ratio. We have calculated the power at the center of the exit beam, it is enhanced by a ratio around 1.87 also. Therefore the very anisotropic curved PhC in this example works as a wave compressor.

4 Conclusion

In this work, transformation optical devices can be designed by transforming a regular photonic crystal to a curved and gradient one. We established a scaling law for curved photonic crystals, which can be used to construct the whole transformation medium. The approach will be useful in constructing transformation media requiring refractive indices smaller than 1, being negative or highly anisotropic, which can be made from low-loss dielectric-only photonic crystals. In particular, we can construct a low-loss very anisotropic medium for E-polarization, which is difficult using conventional effective medium. A bending waveguide and a wave compressor are thus demonstrated in order to control the propagation of Bloch waves.