## Abstract

We designed an electromagnetic refractor and a phase transformer using form-invariant coordinate transformation of Maxwell’s equations. The propagation direction of electromagnetic energy in these devices can be modulated as desired. Unlike the conventional dielectric refractor, electromagnetic fields at our refraction boundary do not conform to the Snell’s law in isotropic materials and the impedance at this boundary is matched which makes the reflection extremely low; and the transformation of the wave front from cylindrical to plane can be realized in the phase transformer with a slab structure. Two dimensional finite-element simulations were performed to confirm the theoretical results.

©2008 Optical Society of America

## 1. Introduction

Recently, the form-invariant coordinate transformation of Maxwell’s equations which can yield specific material has been employed to produce novel electromagnetic devices [1, 2]. Cylindrical, spherical and square cloaks [2–8] have been designed to realize invisibility by this approach. Also, a concentrators which can collect the energy to a small region and a rotation mapping of coordinate which made the information from outside appears as if it comes from a different angle, have been simulated [8, 9]. However, the design of these devices requires specific permittivity and permeability which are difficult to obtain in practice. Fortunately, metamaterial with tunable optical properties offers the potential to realize these kinds of anisotropic and inhomogeneous materials. And a lot of devices have been realized using this kind of structure. Cylindrical invisible cloak used in microwave frequency has been experimentally implemented and a structure of metamatrial for optical cloak has been reported [5, 7]. All these novel devices proved the coordinate transformation theory to be an effective way in controlling the electromagnetic wave.

In this paper, we propose another two electromagnetic structures, what we would call “refractor” and “phase transformer” using coordinate transformation. The propagation of electromagnetic wave can be modulated by these devices with some new phenomenon. In the following section, the designing approach of these devices is provided. Then, the simulation results are discussed and the advantages of our suggested structures compared with the conventional ones will be presented.

## 2. Principle

The coordinate transformation is a down-top procedure, which means that the structure is designed based on its given electromagnetic response. The interaction between the electromagnetic waves and the structure are determined by two factors. One is the material of the object; the other would be the shape of the object. The coordinate transformation method provides the way to design the new material parameters by changing the original shape of the object. A slab refractor and phase transformer are suggested in this way.

#### 2.1 Refractor

In order to control the propagation of the electromagnetic wave, a space transformation is needed as Fig. 1. The area enclosed by red lines is supposed to transmit the normal incident wave directly without deviation; the blue area is the region where we want to confine the propagation of the wave. A connection between these two areas can be built by applying coordinate transformation. So, the electromagnetic energy, which should propagate with no deviation, will deflect to the blue region.

The coordinate transformation equations for this design can be expressed as

The Jacobian transformation matrix between the transformed coordinate and the original coordinate is [2]

where *x ^{i}* denotes

*x*,

*y*,

*z*in original coordinate system,

*x*′ corresponds to

^{i}*x*′,

*y*′,

*z*′ in new coordinate system.

The permittivity and permeability tensors of transformed media become

A general transformation matrix is given as follows.

The matrix for permeability can be obtained with *μ ^{i′j′}*=

*ε*and

^{i′j′}*μ*=

*ε*. The original isotropic material changes to be anisotropy.

#### 2.2 Phase transformer

A conventional lens can modulate the phase by the specific shape such as convex lens and concave lens. However, the new phase transformer can realize this by the specific material. The space transformation for the design of this device is presented in Fig. 2. The rectangular area is the region where the wave front changes gradually from cylindrical to plane. The transformation of wave front III makes sure that the wave front becomes plane at the output boundary. The distance L between the output boundary and the source must be larger than *r*
_{2} so that the phase at the output boundary would be the same; width W must be larger than half of the height H for stability.

The coordinate transformation equations for phase transformer can be expressed as

Applying the same method as used in section 2.1, the material parameter for the phase transformer can be derived as

The tensor description can be completed by substituting *ε* with *μ* in Equation (6).

## 3. Simulation and discussion

#### 3.1 Refractor

Here, we set *θ* to be 45° for example. Four different incident conditions are presented in Fig. 3 performed by the finite-element method. A transverse-electric (TE) plane-wave is used, and the wavelength is specified to be 365nm, which is in the ultraviolet frequency range. The black line demonstrates the direction of the power flow. It is noted that the direction of Poynting vector deviates from the wave vector *k* in the metamaterial, in other words, the power flow direction is not normal to the wave front in this anisotropic structure.

In conventional refractors, the relationship between incident angle *θ*
_{1} and refraction angle *θ*
_{2} can be expressed as follows:

while in our designed refractor, we have

where *Δθ* is the designed deflection angle.

Figure 4 shows the relationship between the incident angle and the refraction angle in three cases. Incident angle in Fig. 4 changes from -90° to 90°, the designed deflection angle is 45°. For the first case (black line) of the conventional refractor, we set *n _{1}*=1.6,

*n*=1.4, and the same

_{2}*n*with

_{2}*n*=1 for the second case (blue line). The third case (red line) is our designed refractor.

_{1}Since the propagation direction of electromagnetic energy can be controlled, splitting can be realized by metamaterials combination. In Fig. 5, two different anisotropic materials are assembled together. The incident wave is normal to the *x* axis. The area for both refractors is 6µm wide and 3µm high. The distribution of Ez is showed in Fig. 5(a); the corresponding time-averaged energy density is presented in Fig. 5(b). Electromagnetic energy propagate straightly in region I and IV but turn left significantly in region II and turn right in region III because of the anisotropic material.

In a conventional refractor, refraction is always companied by reflection, so the backward scattering can not be neglected. In Fig. 5, we found that there is almost no backward scattering at the incident boundary because of the matched impedance. We integrate the power flow at the incident boundary in region I which is 5.249e-9 N/s and the output boundary in region IV which is 5.241e-9 N/s, the transmission effective is 99.8%. Some scattering is found in Fig. 5 because of the discontinuity of the boundary between the original space and the new space in Fig. 1.

#### 3.2 Phase transformer

A transverse-electric (TE) point source with wavelength of 365nm propagating along the *x* axis is used for the calculation. The area is 5µm wide and 3.5µm high enclosed by PML. Figure 6(a) demonstrates the distribution of *z* component of the electric field with the phase transformer in region II and Fig. 6(b) represents the distribution of Ez in free space. Figure 6(c) is the z component of electric field at *x*=4µm derived from Fig. 6(a) and Fig. 6(b), where the distribution of Ez is more uniform after the phase transformer than in free space.

It is noted that the wave front changes to be almost parallel to *y* axis after the phase transformer in Fig. 5(a). The scattering at the boundary between region II and III which is part of our current research cause the unsmooth of Ez in region II and III.

## 5. Conclusion

In this work, we have presented a new refractor and a phase transformer designed by the coordinate transformation theory. Both of them exhibit extraordinary phenomena. This approach provides a new way to control the propagation of electromagnetic waves and it can be extended to the design of new devices in further.

## Acknowledgments

This work was supported by 973 Program of China (No.2006CB302900) and 863 Program of China (2006AA04Z310).

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