Design of rotators based on coordinate transformation

Abstract

Traditional cloak and rotator are obtained from mapping in single direction, either radial or tangential direction. However, this article proposed several novel structures obtained from mapping along multidirection, all of above designs are based on coordinate transformation theory. Those structures include “rotational cloak”, “rotational concentrator”, and so on, which are different from traditional structures. This paper discusses the optical properties of these novel structures; moreover, we have investigated the effects of various parameters on their properties, such as virtual radius and initial angle. Also, some interesting conclusions are drawn: 1) the energy convergent capability increases with virtual radius; 2) rotational angle of power flow direction depends on the value of initial angle and the location of optical source;, 3) the permittivity and permeability of transformed material may be negative when virtual radius a greater than outer radius b, it is our well-known left-handed material.

Key words： rotator; coordinate transformation; rotational structure mapping; energy distribution

Introduction

Recently, invisible cloaking has received much attention since Pendry et al. [1] proposed an interesting idea of using coordinate transformation, which squeezes space from a volume into a shell with the exterior boundary unchanged. Optical wave is excluded from interior subdomain without perturbing the exterior field. There is no reflection at the exterior and interior boundaries [2]; hence, the incoming electromagnetic wave cannot penetrate into the cloaked region and will be bent smoothly in the cloak region. The two-dimensional (2D) cylindrical invisibility cloak has been verified by Schurig et al. [3] experimentally at the microwave frequency. At the same time, 2D cloaks with other cross section have been proposed, such as cylindrical cloaks [4-7], elliptic cloaks [8-10], polygonal cloaks [11,12], heart-shaped cloaks [13], and arbitrarily shaped cloaks [2,14-17]. Inspired by the idea of the invisible cloak, some other interesting devices have been proposed, such as concentrator [18-23], rotator [24], anti-cloak [25], and waveguide [11,26].

All the previous research works on coordinate transformation either map along radial direction to get a cloak and concentrator [18] or tangential direction to get a rotator [24]. In this paper, a simple 2D cylindrical structure is investigated by means of mapping along both radial and tangential directions. Material properties can be deduced from geometric transformation. The transmission behaviors of electromagnetic wave propagating through this structure and the energy distributions are analyzed in this paper.

Principle

We use the schematic diagram of 2D transformation structure in cylindrical coordinate system, which is shown in Fig. 1. The whole transformation space is divided into two regions: region I-the core material layer (0≤r<b); region II-the circular layer (b≤r<c).

Fig.1 Cross section of transformation region (including three parts separated by three circles with radius of a, b, and c, respectively)

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To redistribute the field, it is necessary to divide the whole transformation process into two steps: the first step is to compress the space within 0≤r<a into a circle with a radius of b; the second is to expand the space between a and c to region II. Meanwhile, the rotating angle θ of region II is the function of {\theta }_{\mathrm{0}}{\displaystyle \frac{c-r}{c-a}}. This transformation is continuous to free space at r=c, so there is no reflection at exterior boundary. The process can be mathematically described as

(1)

(2)

where r', θ', and z'are contracted cylindrical coordinates r, θ, and z, respectively. a(0≤a<∞) is a virtual radius; in other words, a is the radius of the inner cylinder in the original coordinate space. b and c are the radii of the inner and outer cylinders in the transformed coordinate space, respectively.

The Jacobian transformation matrix [1] in transformed space is given by Eq. (2) as

(3)A_{r^{\prime }r}=\left[\begin{array}{lll}{\displaystyle \frac{\partial r^{\prime }}{\partial r}}& {\displaystyle \frac{\partial r^{\prime }}{\partial \theta }}& {\displaystyle \frac{\partial r^{\prime }}{\partial z}}\\ {\displaystyle \frac{\partial {\theta }^{\prime }}{\partial r}}& {\displaystyle \frac{\partial {\theta }^{\prime }}{\partial \theta }}& {\displaystyle \frac{\partial {\theta }^{\prime }}{\partial z}}\\ {\displaystyle \frac{\partial z^{\prime }}{\partial r}}& {\displaystyle \frac{\partial z^{\prime }}{\partial \theta }}& {\displaystyle \frac{\partial z^{\prime }}{\partial z}}\end{array}\right],

so the Jacobian transformation matrixes of Eq. (1) in regions I and II can be expressed as

(4)

(5)

respectively. According to tensor property, the Jacobian transformation matrix in Cartesian coordinate system can be obtained from the following equation:

(6){\mathit{A}}_{x^{\prime }x}={\mathit{A}}_{x^{\prime }{r}^{\prime }}{\mathit{A}}_{r^{\prime }r}{\mathit{A}}_{rx},

where

(7){\mathit{A}}_{x^{\prime }{r}^{\prime }}=\left[\begin{array}{lll}\cos {\theta }^{\prime }& -r^{\prime }\sin {\theta }^{\prime }& \mathrm{0}\\ \sin {\theta }^{\prime }& r^{\prime }\cos {\theta }^{\prime }& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{1}\end{array}\right],

(8){\mathit{A}}_{rx}=\left[\begin{array}{lll}\cos \theta & \sin \theta & \mathrm{0}\\ -\sin \theta /r& \cos \theta /r& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{1}\end{array}\right].

In our cylindrical case, the elastic properties [15] of the transformed medium are described by the matrix

(9)\mathit{ϵ}{\mathit{\text{'}}}_{\overrightarrow{x}}=\mathit{\mu }\mathit{\text{'}}={\displaystyle \frac{{\mathit{A}}_{x^{\prime }x}\mathit{ϵ}{\mathit{A}}_{x^{\prime }x}^{\mathrm{T}}}{\left|{\mathit{A}}_{x^{\prime }x}\right|}}.

Supposing the original space is isotropic, the relative permittivity tensor of cylindrical model is simpler expressed in cylindrical system instead of Cartesian system. In region II, the components of transformed media’s relative permittivity \mathit{ϵ}{\mathit{\text{'}}}_{\overrightarrow{r}} can be simplified as follows:

(10)$$\begin{gathered} \{{ϵ}_{r^{\prime }{r}^{\prime }}={\displaystyle \frac{r^{\prime }-h}{r^{\prime }}}, \\ {ϵ}_{{\theta }^{\prime }{r}^{\prime }}={\displaystyle \frac{{\theta }_{\mathrm{0}}(r^{\prime }-h)}{(a-c)k}}, \\ {ϵ}_{{\theta }^{\prime }{\theta }^{\prime }}={\displaystyle \frac{k^{\mathrm{2}}{(a-c)}^{\mathrm{2}}+{\theta }_{\mathrm{0}}^{\mathrm{2}}{(r^{\prime }-h)}^{\mathrm{2}}}{k^{\mathrm{2}}{(a-c)}^{\mathrm{2}}(r^{\prime }-h)}}{r}^{\prime }, \\ {ϵ}_{z^{\prime }{z}^{\prime }}={\displaystyle \frac{r^{\prime }-h}{k^{\mathrm{2}}{r}^{\prime }}}, \\ {ϵ}_{r^{\prime }{\theta }^{\prime }}={ϵ}_{{\theta }^{\prime }{z}^{\prime }}={ϵ}_{z^{\prime }{r}^{\prime }}={ϵ}_{r^{\prime }{z}^{\prime }}=\mathrm{0}, \end{gathered}$$

where

h={\displaystyle \frac{c(b-a)}{c-a}},

k={\displaystyle \frac{c-b}{c-a}}.

However, it is inconvenient for our simulations to express permittivity in cylindrical system, so we had better transform BoldItalic_r and BoldItalic_r back to Cartesian coordinate system through the following orthogonal transformation:

(11){\mathit{ϵ}}_{\overrightarrow{x}}=\mathit{e}{\mathit{ϵ}}_{\overrightarrow{r}}{\mathit{e}}^{\mathrm{T}},

where

\mathit{e}=\left[\begin{array}{lll}\cos \theta & -\sin \theta & \mathrm{0}\\ \sin \theta & \cos \theta & \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{1}\end{array}\right].

For region II (b≤r<c), in Cartesian coordinate system, the relative permittivity can be expressed as

(12)\mathit{ϵ}=\left[\begin{array}{lll}\mathrm{1}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{1}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& a^{\mathrm{2}}/b^{\mathrm{2}}\end{array}\right].

Specially, supposing the original space is free space, the relative permittivity tensor is equal to third-order unit matrix; the coordinate transformation

r^{\prime }={\displaystyle \frac{c-b}{c-a}}r+{\displaystyle \frac{c(b-a)}{c-a}}={\displaystyle \frac{c-b}{c}}r+b

has included the whole transformed region 0≤r<c while a=0, so the coordinate transformation

r^{\prime }={\displaystyle \frac{b}{a}}r

in the region within radius r=a is invalid. The result of this coordinate transformation is to expand one point in electromagnetic space to an extended volume with interior radius r=b in physical space. Because the external boundary of region II keeps unchanged (there is no tensile deformation or torsional deformation), the material parameters are continuous to outer free space r≥c. The region II is not a free space any longer, it is distorted. However, the external region of the transformation region remains a free space.

Simulation results and analysis

In the mathematical procedures above, there is no approximation on systems of Eqs. (10) and (12), which are strict analytic results. In our simulations, finite element method is used to simulate the designed structures. The relative size of the designed model is set for inner radius b=0.15 μm, exterior radius c=0.3 μm, and operating frequency f=1.5×10¹⁵ Hz; the incident field is Gaussian beam input from port 1. We set perfectly matched layers (PMLs) at the top and bottom boundaries to absorb electromagnetic wave and avoid reflection. The schematic diagram of simulation is shown in Fig. 2.

Fig.2 Configuration of computational domain

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Figure 3 shows the simulation results for mapping along both radial and tangential directions with different virtual radius and initial angles. All of the following lengths are in units of microns and no longer repeat. The right color bar indicates the normalized electric field intensity. The initial angle of Figs. 3(a) to 3(d) is θ₀=π/2, and θ₀=0 for Figs. 3(e) to 3(h). So Figs. 3(a) to 3(d) and Figs. 3(e) to 3(h) are the rotational and traditional structures, respectively. Those traditional structures are given so as to show the influence of initial angle θ₀.

Fig.3 Simulation results of mapping with plane wave incident case (white lines indicate power flow direction from left to right). (a) a=0.00 μm, θ₀=π/2; (b) a=0.05 μm, θ₀=π/2; (c) a=0.25 μm, θ₀=π/2; (d) a=0.45 μm, θ₀=π/2; (e) a=0.00 μm, θ₀=0; (f) a=0.05 μm, θ₀=0; (g) a=0.25 μm, θ₀=0; (h) a=0.45 μm, θ₀=0

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As shown in Fig. 3, the properties of these transformation structures depend on parameters a and θ₀. For example, Figs. 3(a) and 3(e) are “rotational cloak” and “traditional cloak”, respectively, so the electromagnetic field is totally excluded from region I (0≤r<b) to region II (b≤r<c), and in this case, the virtual radius a=0 μm; when 0≤a<b, as shown in Figs. 3(b) and 3(f), there is partial electromagnetic field penetrated into region I; Figs. 3(c) and 3(g) are “rotational concentrator” and “traditional concentrator”, respectively, most of the electromagnetic field is concentrated into region I from region II while b≤a<c; as Figs. 3(d) and 3(h) show, their energy convergent capability is much stronger than any other traditional concentrators. From the discussion above, we may safely draw several conclusions: 1) the energy convergent capability increases with virtual radius, which can be further verified from the following discussion; 2) the total rotation angle of power flow direction in process through region II is θ₀. Compared with traditional cloaks, the main differences are the power flow direction and the convergent capability

Figure 4 is the energy density distribution according to Figs. 3(e) to 3(h), respectively. As shown in Fig. 4, as the coordinate transformation and transform region are axisymmetric, the energy density distribution from (-0.4, 0.0) to (0.4, 0.0) is enough to illustrate the distribution of the whole region. Figure 4 shows once again that the electromagnetic energy gradually sank into the central region with a increasing from 0.00 to 0.45 μm and uniformly distributed over the central region. Most energy is excluded into region II and concentrated around external boundary (r=c) while 0≤a<b. Particularly, the electromagnetic energy is totally excluded from region I to II while a=0, and it is discontinuous at r=b and r=c. The energy density of region II becomes larger with the increase of r and then sharply drops down at external boundary. However, most energy is concentrated in and uniformly distributed over the central cylinder while a>b. Specially, as shown in Fig. 3(d), the power density of region II (b≤r<c) is negative, indicating that the Poynting vector is opposite with the wave vector, which is a fundamental characteristic of left-handed material (ϵ<0, μ<0), whose vectors BoldItalic, BoldItalic, and BoldItalic form a left-handed helix [27].

Fig.4 Energy density distribution according to Figs. 3(e)-3(h), respectively. (a) a=0.00 μm; (b) a=0.05 μm; (c) a=0.25 μm; (d) a=0.45 μm

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Conclusion

In conclusion, the rotational and nonrotational structures are discussed, which depend on the range of initial angle θ₀ and virtual radius a. The analytical result shows that the energy convergent capability increases with virtual radius a, whether for rotational or nonrotational structure. Meanwhile, electromagnetic energy is uniformly distributed over the central cylinder, and the rotation structure has no effect on energy density distribution. However, the electromagnetic field outside the transformation region keeps intact. The power flow direction will rotate θ₀ around origin and then rotate θ₀ clockwise; so the total rotation angle is 0 while the optical source lies outside the transformation region; however, we can imagine if light source is placed inside the inner cylinder region, the total angle is equal to θ₀. Specially, the power density of region II is negative, indicating that the Poynting vector and wave vector are antiparallel, which is a fundamental characteristic of the left-handed material.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 50735007 and 60707006).

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