## Abstract

An effective method for designing wave shape transformers (WSTs) is investigated by adopting the coordinate transformation theory. Following this method, the devices employed to transform electromagnetic (EM) wave fronts from one style with arbitrary shape and size to another style, can be designed. To verify this method, three examples in 2D spaces are also presented. Compared with the methods proposed in other literatures, this method offers the general procedure in designing WSTs, and thus is of great importance for the potential and practical applications possessed by such kinds of devices.

©2008 Optical Society of America

## 1. Introduction

Based on the form-invariant of Maxwell’s equations, the coordinate transformation theory is stated by Pendry *et al*.[1–4], which is firstly adopted to design invisibility cloaks [1, 5–10]. Besides, with the increasing interests of this method, several other electromagnetic (EM) devices with new functionalities have been suggested: beam shifters [3] translate the incoming wave in the direction perpendicular to the propagating direction without altering the shapes of the wave fronts; concentrators [4, 11–13] collect the energy to a small region; field rotators [13, 14] rotate the fields in a region and thus made the information from the outside appears as if it comes from a different angle; photon funnels [15] are used as the devices to compress wave beams from large width to the thin, and phase transformers [16, 17], transform the wave front shape from one style to another, for example, cylindrical waves to plane waves.

Among these devices mentioned above, photon funnels and phase transformers can be classified as wave shape transformers (WSTs). A WST is a device that can transform wave fronts from one style (having the certain shape and size) to another style. Such a device is popular in conventional fields and has many potential and practical applications. The literatures [15–17] have discussed these devices and shown several examples, including devices that transform the wave shapes from cylindrical to plane, and devices that shift wave beams along parallel directions. However, these discussions were limited to the simple cases that the wave shapes are cylindrical or plane. In the more general cases, if we consider the complicated situations that to transform a wave front from arbitrary shapes to another arbitrary style, what the method and procedures should we adopt to design the corresponding WSTs? In this paper, we will introduce a general method of designing WSTs. By adopting this method, a WST that transform wave fronts with arbitrary shape and size to another arbitrary style can be designed.

## 2. The general method

In the free space, waves propagate perpendicular to the phase fronts, thus the wave shape and size of a non-plane wave will be changed momentarily. If a device is adopted to limit the propagation of a wave in one region and to transform the wave shape to another style at the moment the wave leaves, the device is used as a WST. Geometrically speaking, such a device causes the spatial deformation by mapping the position points traced by the wave propagating in the free space to the points within the device region. Therefore the constitutive parameters (permittivity and permeability) in the device region can be established by using the coordinate transformation theory.

We start our discussions by considering the situation that an EM wave propagates in the free space, and at one moment, is incident onto a WST with a certain phase front. After propagating in the WST, this wave changes its phase front to another style. As shown in Fig. 1, we call these two phase fronts as the original front and the new front, and denote them by S_{1} and S_{2}, respectively. In our scheme, we limit our discussions to the cases that the wave propagates in the right half space, thus the shapes of these two fronts, surfaces S_{1} and S_{2}, are assumed to be arbitrary under the limitation that both the two surfaces must intersect only once with an arbitrary line that is along the x-axis and within the WST region. In the following derivation, we suppose the wave propagates along the x-axis within the WST region, and that surfaces S_{1} and S_{2} have the general expressions in Cartesian systems as follows:

Shown by Fig. 1, during a given time interval, an arbitrary point A on S_{1}, will move to P, then to C, when the wave propagates in the free space; and will move to P′, then to B, when the wave propagates in the WST region. As a result, after the time interval, the set of point C forms the imaginary wave front in the free space, and the set of point B the transformed wave front in the WST region ∑. Geometrically, there must be a unique mapping between P and P′ as well as C and B, and the mapping can be also used as the description of the spatial deformation. To mathematically describe such a mapping, we define four distance variables: *l _{p}*=|

**AP**|,

*l*′=|

_{p}**AP**′|,

*L*=|

**AC**|,

*L*′=|

**AB**|, and adopt the following expression:

Equation (3) establishes a linear mapping between P and P′ by proportionally changing *l _{p}* to

*l*′, and the validity is obvious. First of all, the mapping expressed by Eq. (3) is homeomorphous; secondly, when the point P tends to C, the point P′ correspondingly tends to B, which means that the mapping has no singularity; thirdly, from the viewpoint of physics, Eq. (3) prescribes the relationship between the points (denoted by P) traced by the imaginary waves in the old space and the points (denoted by P′) traced by the real waves in the transformed space, which well meets the requirement of the coordinate transformation theory [4].

_{p}In Eq. (3), parameter *L* is the wave path traveled by the wave in the free space during the given time interval. Since waves propagate at the same speed in the free space, *L* must be a constant for arbitrary starting point A in S_{1}. In principle, *L* can be an arbitrary constant, larger or smaller than the length of the WST. But in practice, we usually assign *L* with several wavelengths and make it close to the length of the WST, for that such a choice avoids the too sharp deformation when we carry out the coordinate transformation and thus obtain a comparatively smooth distribution of the constitutive parameters. In one word, choosing of parameter *L* as well as the length of the WST only limits to the requirements of applications and the fabrication level.

Then, we need to express *l _{p}*,

*l*′ and

_{p}*L′*with the coordinate variables, and by solving Eqs. (1–3), to obtain the exact relationship between P and P′. Supposing the coordinate of P is (

*x*,

_{p}*y*,

_{p}*z*), the line that includes P is written as:

_{p}where (*k _{x}*,

*k*,

_{y}*k*) is the line direction vector. Shown in Fig. 1, A is the intersection point of line AP and surface S

_{z}_{1}, so the coordinate (

*x*,

_{a}*y*,

_{a}*z*) is involved in the simultaneous Eqs. (1, 4). Obviously, (

_{a}*x*,

_{a}*y*,

_{a}*z*) and (

_{a}*k*,

_{x}*k*,

_{y}*k*) are associated to each other. By further considering the principle that when a wave propagates in the free space, the direction AP must be perpendicular to the tangent plane of the front surface S

_{z}_{1}at A, we thus have

where, *f _{i}*(

*x*,

_{a}*y*,

_{a}*z*) is the partial differential coefficient of function

_{a}*f*(

*x*,

*y*,

*z*) cited in Eq. (1). Equation (5) together with (1, 4) determine (

*x*,

_{a}*y*,

_{a}*z*) uniquely. Thereupon, the coordinate of B, (

_{a}*x*,

_{b}*y*,

_{b}*z*), can be also obtained by substituting

_{b}*y*=

*y*and

_{a}*z*=

*z*into Eq. (2).

_{a}Supposing (*x _{p′}*,

*y*,

_{p′}*z*) is the coordinate of P′, we have:

_{p′}Solving Eqs. (3, 6), we get:

Equation (7) is a general formula derived from Eq. (3), describing the geometrical mapping between the points traced by waves propagating in the free space and in the WST region.

Finally, we can use the method prescribed by Rahm *et al*. [4] to obtain the relative permittivity and permeability of a WST via

Where, *α*(*x _{p}*,

*y*,

_{p}*z*) (before transformation, especially in the free space, it is equal to unit matrix) and

_{p}*α*(

*x*,

_{p′}*y*,

_{p′}*z*) (after transformation), respectively, denote the permittivity or permeability tensors;

_{p′}*Q*is the Jacobian transformation matrix between (

*x*,

_{p}*y*,

_{p}*z*) and (

_{p}*x*,

_{p′}*y*,

_{p′}*z*) with its elements defined as

_{p′}According to Eqs. (6, 7), Eq. (9) is thus expressed as

$${Q}_{\lambda \eta}=\frac{\partial {\lambda}_{p\prime}}{\partial {\eta}_{p}}=\frac{\partial {\lambda}_{a}}{\partial {\eta}_{p}},\lambda =y,z;\eta =x,y,z.$$

In Eq. (10), *∂l _{p}*/

*∂η*and

_{p}*∂λ*/

_{a}*∂η*are determined by Eqs. (1–5), and should be solved concretely when Eqs. (1, 2) are given. By now, the general method of designing WSTs that transform wave fronts between two arbitrary shapes has been established. Especially, Eq. (8) offers a general form in obtaining the relative permittivity and permeability, so, what we need to do in special designing is mainly to establish the Jacobian transformation matrix

_{p}*Q*according to Eq. (10), and then to apply the matrix

*Q*to Eq. (8) to obtain the constitutive parameters.

Note that, we have investigated the above method by using the coordinate transformation theory which is based on the form-invariant of Maxwell’s equations. In the derivation, the frequency dispersion is not involved, so the results are wavelength-independent.

## 3. Examples in 2D spaces

#### 3.1 Convex cylindrical waves to concave cylindrical waves

Transforming a convex cylindrical wave to a concave cylindrical wave, the corresponding spatial deformation occurs in 2D spaces. Shown in Fig. 2(a), the centers of the original front circle and the new front circle are located at (*x*
_{1},0) and (*x*
_{2},0), respectively, and the functions of these two circles are

After a simple deduction according to the method introduced in section 2, we obtain:

$${x}_{b}={R}_{2}\mathrm{cos}{\theta}_{2}+{x}_{2},{y}_{b}={y}_{a},$$

$${l}_{b}=\left({x}_{p}-{x}_{a}\right)\sqrt{1+{\mathrm{tan}}^{2}{\theta}_{1}},$$

where,

As the results,

$$\frac{\partial {\theta}_{2}}{\partial {\eta}_{p}}=\frac{-{R}_{1}\mathrm{cos}{\theta}_{1}}{\sqrt{{R}_{2}^{2}-{R}_{1}^{2}{\mathrm{sin}}^{2}{\theta}_{1}}}\times \frac{\partial {\theta}_{1}}{\partial {\eta}_{p}},$$

$$\frac{\partial {x}_{a}}{\partial {\eta}_{p}}=-{R}_{1}\mathrm{sin}{\theta}_{1}\frac{\partial {\theta}_{1}}{\partial {\eta}_{p}},\frac{\partial {y}_{a}}{\partial {\eta}_{p}}={R}_{1}\mathrm{cos}{\theta}_{1}\frac{\partial {\theta}_{1}}{\partial {\eta}_{p}},$$

$$\frac{\partial {x}_{b}}{\partial {\eta}_{p}}=-{R}_{2}\mathrm{sin}{\theta}_{2}\frac{\partial {\theta}_{2}}{\partial {\eta}_{p}},$$

$$\frac{\partial {l}_{p}}{\partial {\eta}_{p}}=\left({x}_{p}-{x}_{a}\right)\frac{1}{\sqrt{1+{\mathrm{tan}}^{2}{\theta}_{1}}}\frac{\mathrm{tan}{\theta}_{1}}{{\mathrm{cos}}^{2}{\theta}_{1}}\frac{\partial {\theta}_{1}}{\partial {\eta}_{p}}$$

$$+\left(s+\frac{\partial {x}_{a}}{\partial {\eta}_{p}}\right)\sqrt{1+{\mathrm{tan}}^{2}{\theta}_{1}}\xb7s=\{\begin{array}{c}1,\eta =x\\ 0,\eta =y\end{array}$$

Substituting Eqs. (6, 12–14) into (10), we obtain the Jacobian transformation matrix with the elements as follows:

$$+\left({R}_{2}\mathrm{cos}{\theta}_{2}-{R}_{1}\mathrm{cos}{\theta}_{1}+{x}_{2}-{x}_{1}\right)[\left({x}_{p}-{R}_{1}\mathrm{cos}{\theta}_{1}-{x}_{1}\right)\frac{1}{\sqrt{1+{\mathrm{tan}}^{2}{\theta}_{1}}}\frac{\mathrm{tan}{\theta}_{1}}{{\mathrm{cos}}^{2}{\theta}_{1}}\frac{\partial {\theta}_{1}}{\partial {\eta}_{p}}$$

$$+\left(s-{R}_{1}\mathrm{sin}{\theta}_{1}\frac{\partial {\theta}_{1}}{\partial {\eta}_{p}}\right)\sqrt{1+{\mathrm{tan}}^{2}{\theta}_{1}}]\}+{R}_{1}\mathrm{sin}{\theta}_{1}\frac{\partial {\theta}_{1}}{\partial {\eta}_{p}},s=\{\begin{array}{c}1,\eta =x\\ 0,\eta =y\end{array}$$

$${Q}_{y\eta}={R}_{1}\mathrm{cos}{\theta}_{1}\frac{\partial {\theta}_{1}}{\partial {\eta}_{p}},\eta =x,y.$$

Where, *∂θ*
_{1}/*∂η*
* _{p}* is defined in Eq. (14).

In practical designs, the converse mapping from P′ to P is required, so, we should treat P′ as the known point and assign *y*=*y _{p′}*. Substituting the results into (11), we get

and,

$${y}_{p}=\left({x}_{p}-{x}_{1}\right)\mathrm{tan}{\theta}_{1}.$$

Equations (16, 17) are necessary in establishing the final form of Eq. (15). Once the final form of Eq. (15) is offered, the relative permittivity and permeability of the WST can be easily obtained by solving Eq. (8).

#### 3.2 Cylindrical waves to plane waves

Similarly, transforming a cylindrical wave to a plane wave also occurs in 2D spaces. Shown in Fig. 2(b), the front surface functions are

$${S}_{2}:y=k\left(x-{x}_{2}\right),$$

where k is the slope of S_{2}.

Solving equations (3–10, 18), all variables except *θ*
_{2} and (*x _{b}*,

*y*), have the same results as that in Eqs. (12–14). Here,

_{b}*θ*

_{2}is not cited, and (

*x*,

_{b}*y*) has another form:

_{b}Accordingly, the partial differential coefficient of *x _{b}* is

Substituting Eqs. (6, 12–14, 19 and 20) into (10), we obtain the Jacobian transformation matrix with the elements as follows:

$$+\left(\frac{{R}_{1}\mathrm{sin}{\theta}_{1}+k{x}_{2}}{k}-{R}_{1}\mathrm{cos}{\theta}_{1}-{x}_{1}\right)[\left({x}_{p}-{R}_{1}\mathrm{cos}{\theta}_{1}-{x}_{1}\right)\frac{1}{\sqrt{1+{\mathrm{tan}}^{2}{\theta}_{1}}}\frac{\mathrm{tan}{\theta}_{1}}{{\mathrm{cos}}^{2}{\theta}_{1}}\frac{\partial {\theta}_{1}}{\partial {\eta}_{p}}$$

$$+\left(s-{R}_{1}\mathrm{sin}{\theta}_{1}\frac{\partial {\theta}_{1}}{\partial {\eta}_{p}}\right)\sqrt{1+{\mathrm{tan}}^{2}{\theta}_{1}}]\}+{R}_{1}\mathrm{sin}{\theta}_{1}\frac{\partial {\theta}_{1}}{\partial {\eta}_{p}},s=\{\begin{array}{c}1,\eta =x\\ 0,\eta =y\end{array}$$

$${Q}_{y\eta}={R}_{1}\mathrm{cos}{\theta}_{1}\frac{\partial {\theta}_{1}}{\partial {\eta}_{p}},\eta =x,y.$$

Where,

$${\theta}_{1}=\mathrm{arcsin}\frac{{y}_{p\text{'}}}{{R}_{1}},$$

$${x}_{p}=L\frac{{x}_{p\text{'}}-{R}_{1}\mathrm{cos}{\theta}_{1}-{x}_{1}}{\frac{{R}_{1}\mathrm{sin}{\theta}_{1}+k{x}_{2}}{k}-{R}_{1}\mathrm{cos}{\theta}_{1}-{x}_{1}}\frac{1}{\sqrt{1+{\mathrm{tan}}^{2}{\theta}_{1}}}+{R}_{1}\mathrm{cos}{\theta}_{1}+{x}_{1},$$

$${y}_{p}=\left({x}_{p}-{x}_{1}\right)\mathrm{tan}{\theta}_{1}.$$

#### 3.3 Photon funnels

A photon funnel can be employed as a device to compress EM wave beams, which is in fact a wave shape transformer. The method we introduced in section 2 is also available in designing such devices, yet the special procedure should be adjusted slightly. In this example, we consider a 2D photon funnel, which is used to compress a plane wave beam. For the sake of simplicity, we suppose that wave beams propagate along the x-axis and are compressed in the y-axis direction. Shown in Fig. 3, a right traveling wave beam will be compressed from the width of |*y*
_{1}| to the width of |*y*
_{2}|.

In Fig. 3, once (*x*
_{1}, *y*
_{1}) and (*x*
_{2}, *y*
_{2}) are given, the top boundary of the WST is determined by

where *k*=(*y*
_{2}-*y*
_{1})/(*x*
_{2}-*x*
_{1}). Since the spatial compression is in the y-axis direction, the variables cited in Eq. (3) should be redefined: *l _{p′}*=|AP′|=

*y*|,

_{p′}*l*=|AP|=|

_{p}*y*,

_{p|}*L*′=|AB|=|

*y*, and

_{b}*L*=|AC|=|

*y*|. Then, according to Eq. (3), we have

_{c}where *y _{b}*=

*kx*-

_{p}*kx*

_{1}+

*y*

_{1}. As the results,

Finally, we get the Jacobian transformation matrix as follows:

## 4. Full-wave simulations and discussions

Full-wave finite-element simulations by using COMSOL Multiphysics software were performed to verify our conclusions in section 3. Figs. 4 and 5 display the cuts in the x-y plane of the simulation results using TE-mode harmonic waves. The computation domains, ∑(∑_{1} and ∑_{2} in Fig. 5(b)), are configured by media with the parameters (permittivity and permeability) governed by the equations deduced in section 3, and other regions are assumed to be the free space. The inner boundaries in Fig. 4 are set to be continuous. Whereas in Fig. 5, the funnel′s boundaries are set to be perfect electric conducting (PEC), by doing so, the compression performance can be seen clearly.

Figure 4(a) shows a WST through which a convex cylindrical wave with radius of *R*
_{1} is transformed to be a concave cylindrical wave with radius of *R*
_{2}. Such a device is actually a perfect concave lens. Comparing with this WST, a conventional concave lens will produce large image dispersing when the flare angle of the incident wave is large, whereas this WST can overcome such limits. Theoretically, the flare angle of the incident beam into this WST can be large to*π*. Another application of this WST is long-distance transmitting and focusing of energy when the radius of the second cylindrical surface is large enough.

In Fig. 4(b), we show a device through which a cylindrical wave is transformed to a plane wave. Obviously, such a device provides the capability of transforming the wave shape between cylindrical and plane, and thus can be used as perfect convex lens. We notice that, this case was already discussed by Lin and Jiang *et al*.[16, 17], however, our design is more general and more effective. For that, under the control of our designed WST, the direction of the transformed plane wave can be of any angles with respect to the main direction (the x-axis), which therefore has the capability to easily control the beam directions.

Figure 5 shows the performance of photon funnels. In (a), a plane wave beam with large width is compressed to a thin beam without changing the wave shape, and therefore, the wave energy is concentrated. In principle, such compressions and concentrations are perfect compared with conventional devices. Another application of such devices is to enlarge wave beams when the wave is incident onto the right end (see Fig. 5(a)), in other words, such photon funnels can be used as perfect magnifiers and microscopes.

What is shown in Fig. 5(b) is the complex of a wave transformer (cylindrical to plane) and a photon funnel. Firstly, a cylindrical wave is transformed to a plane wave. Then, the plane wave with large width is compressed to a thin beam. As a whole, a cylindrical wave is collimated and compressed to be a thin beam. This complex is in fact a complicated photon funnel which is applied to cylindrical waves.

## 5. Conclusions

By using the coordinate transformation theory in designing the desired WST, the derived relative permeability and permittivity have the tensor form, and some elements have the values varying from near zero to the higher. When the values of the relative permeability and permittivity fall into (0, 1), the conventional materials can not be applied any more, therefore we should rely on the meta-materials. Besides, the constitutive parameters of the WST are position-dependent, and even may be highly anisotropic for some complicated WSTs. Fortunately, the recently developed meta-material techniques, for instance, the split ring resonator (SRR) [18, 19] and the dielectric resonator (DR) [20], can be used to achieve the materials with the values of their effective relative permeability and permittivity below one. Adopting meta-materials, it is possible to manufacture some simple WSTs supposing that their constitutive parameters are not highly anisotropic and the distributions are comparatively uniform. However, in the complicated cases, there are difficulties in practical applications of the WSTs at the present technique situations. So the practical WSTs will depend on the further developments of meta-materials.

In summary, we have investigated the general method of designing WSTs to transform EM wave fronts from one style with arbitrary shape and size to another style, and presented some examples as the verification. Though the examples have limited to some special conditions, the generalization of this method is obvious according to the analysis in section 2.

For instance, a WST to transform the wave front from ellipsoid-shaped surface to paraboloid-shaped surface can be designed as well. Also, our method is available in cases with acoustic waves and the other types of waves.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant Nos.50632030, 10474077 and 60871027) and the 973-project of the Ministry of Science and Technology of China (Grant No. 2009CB613306). A Shaanxi National Science Foundation, also supported this work.

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