Coordinate transformation makes perfect invisibility cloak with arbitrary shape
By investigating wave properties at cloak boundaries, invisibility cloaks with arbitrary shape constructed by general coordinate transformations are confirmed to be perfectly invisible to the external incident wave. The differences between line transformed cloaks and point transformed cloaks are discussed. The fields in the cloak medium are found analytically to be related to the fields in the original space via coordinate transformation functions. At the exterior boundary of the cloak, it is shown that no reflection is excited even though the permittivity and permeability do not always have a perfect matched layer form. While at the inner boundary, no reflection is excited either, and in particular no field can penetrate into the cloaked region. However, for the inner boundary of any line transformed cloak, the permittivity and permeability in a specific tangential direction are always required to be infinitely large. Furthermore, the field discontinuity at the inner boundary always exists; the surface current is induced to make this discontinuity self-consistent. For a point transformed cloak, it does not experience such problems. The tangential fields at the inner boundary are all zero, implying no field discontinuity exists.
The recent exciting development of invisibility cloaks, has attracted intense attentions and discussions -. Theoretically, the cloaks are constructed easily based on a coordinate transformation method as proposed in Ref. . The object inside the cloak is invisible to the outside observer, because the light is excluded from the object and the exterior field is not perturbed. The invisibility of the linearly radially transformed cylindrical and spherical cloaks has been confirmed by both numerical calculations [3, 4] and analytical solutions [5, 6]. Experimentally, the invisibility cloak with simplified material parameters has been implemented by Schurig et.al at the microwave frequency. Inspired by the idea of the invisibility cloak, some interesting applications, such as field concentration , field rotation , and electromagnetic wormholes , have been proposed.
Up to now, most of discussions on invisibility cloaks focus on the cylindrical and spherical cloaks produced by a coordinate transformation only in the radial direction. For instance, in Ref. , linearly radially transformed cylindrical and spherical cloaks are discussed in detail, and their invisibility is confirmed by ray tracing. In Refs.  and , the invisibility performances of such cylindrical and spherical cloaks are further confirmed by obtaining the exact fields in the cloak medium directly from Maxwell’s equations. In practice, it is sometimes desirable to have invisibility cloaks whose shapes are tailored for the objects to be cloaked. Thus, one needs to understand well about the properties of invisibility cloaks with arbitrary shape produced by general coordinate transformations. However, the investigations on an invisibility cloak with arbitrary shape are only seen in few papers [13, 14]. The mechanism why the invisibility of a general cloak produced by compressing space is ensured, is still unclear. In this paper, we investigate the electromagnetic (EM) properties of invisibility cloaks with arbitrary shape constructed by general coordinate transformations, and we confirm their perfect invisibility. To figure out invisibility cloaks’ main physical properties, we only focus on the ideal case without considering the practical implementation in this paper.
The paper is organized as follows. In section II, Maxwell’s equations in a curved coordinate system are derived. In section III, we show how to construct an invisibility cloak by compressing space in a general manner. In section IV, the wave behaviors and the medium properties at the exterior boundary of the cloak are investigated. In section V, we study the the wave behaviors and the medium properties at the inner boundary of the cloak. Through sections IV and V, the invisibility of cloaks with arbitrary shape is confirmed, and the fields in the cloak medium are derived with simple expressions. In section VI, the cloak parameters and the fields in the cloak are derived when the transformed space is described under arbitrary coordinate system. In section VII, two examples of invisibility cloaks, i.e., cylindrical and spherical invisibility cloaks are investigated. In section VII, the paper is summarized.
2 Maxwell’s equations in a curved coordinate system
Maxwell’s Equations in a Cartesian () space take the form as
Consider the transformation from cartesian space to an arbitrary curved space described by coordinates with
The length of a line element in the transformed space is given by , where the superscript denotes the transpose of matrix, and with
The volume of a space element is expressed as , where represents the determinant of . Here, it is noted that the way of describing the space transformation in this paper is similar as in Ref. , where the time transformation is also taken into account. The space-time metric tensor defined in Ref.  is in the present paper, where only space transformation is considered.
where the superscript ”-1” denotes the inverse of matrix.
The permittivity and permeability and in the Cartesian space are considered for a general case, i.e., they can be tensors. It is seen above that Maxwell’s equations in the curved space have the same form as in the Cartesian space. However, the definitions of the permittivity, permeability, current density, and electric charge density are different, as shown in Eqs. (7) and (8).
3 Construction of invisibility cloaks
To construct a cloak, one usually starts from compressing an enclosed space with the exterior boundary unchanged . As seen in Fig. 1, the region enclosed by boundary is compressed to the region bounded by the exterior boundary and the interior boundary . Such a space compression can be viewed as a certain coordinate transformation described by Eq. (3), which makes a connection between the points in the compressed space with coordinates and the points with Cartesian coordinates in the original space. The exterior boundary satisfies . Notice the interior boundary is obtained by blowing up a line or a point . Thus the cloaks can be divided into two classes: line transformed cloaks and point transformed cloaks. The compressed shaded region in Fig. 1 is the desired cloak.
The permittivity and permeability tensors of the cloak in Cartesian coordinate system are given in Eq. (7). It seems that the cloak medium is very complex, whose permittivity and permeability are tensors and their values vary with the spatial location. However, the eigen functions of the wave equations in the cloak medium are quite simple, which relate with the eigen function of the uncompressed space by Eq. (9).
In order to achieve the invisibility, the cloak should be able to exclude light from a protected object without perturbing the exterior field. Thus for the above cloak, at the exterior boundary , external incident light should excite no reflection. While at the interior boundary , no reflection is either excited, and light can’t penetrate into the cloaked region. In the following sections, we will prove the invisibility of the cloak by investigating the wave behaviors at the cloak’s exterior and inner boundaries. For the simplicity of our discussions and considering the practical application, the invisibility cloak is considered to be placed in air. Then the permittivity and permeability of the cloak in Eq. (7) will be simplified to
which is the same as proposed in Refs.  and . The cloak is considered to be lossless at the working frequency.
4 Cloak’s exterior boundary
In this section, we will prove that no reflection is excited at the exterior boundary. The transmitted electric field and magnetic field without interacting with the inner boundary are expressed as
where and represent the electric and magnetic fields of the external incident waves. According to Eq. (9), it is easily seen that the fields expressed in Eq. (11) satisfy Maxwell’s equations in the cloak medium. Thus in order to prove no reflection excited at , one only needs to confirm that tangential components of () and () keep continuous across .
Decompose and into and , where the subscripts represent ’s normal direction pointing outward from the cloak; and represent ’s two tangential directions, which are vertical with each other. Thus Eq. (11) can also expressed as
where , , represent the unit vectors in , , and directions, respectively.
At the exterior boundary , , , and . So () characterize the exterior boundary . Therefore, it is obvious that the vectors () lie in the same line as the normal direction of , where , , and . For the special case when , can be expressed as , i.e., the vector with the magnitude in the direction. Therefore, on can be expressed as
where and ; when the direction of is as the same as the direction, and if the direction of is opposite to the direction. Substituting Eq. (13) into Eq. (12) and noticing that , , and are orthogonal with each other, it is easily obtained that at
which indicates that the tangential components of () and () are continuous across . Thus, it is proved that no reflection is excited at the exterior boundary.
Consider the permittivity and permeability at for the transformed cloak. It should be noticed that no flection excited at the exterior boundary does not imply that the exterior boundary is a perfectly matched layer (PML), where the permittivity and permeability at have the PML form with the principle axes in , and directions, respectively. We find that parameters at have the PML form only when is a symmetry matrix. Observing Eq. (13), we have and , indicating that and are the eigen vectors of with the same eigen value . Considering is a symmetry matrix, we can know the other eigen vector of is with eigen value . Thus, , , and are eigen vectors of , with eigen values , , and , respectively. Thus based on Eq. (10), and for a symmetric can be expressed as
where the diagonal elements correspond to the principle axes , , and , respectively. The radially transformed cylindrical and spherical cloaks fall into this category.
5 Cloak’s inner boundary
In this section, we will prove that at the inner boundary , no reflection is excited and no field can penetrate into the cloaked region. As discussed in section III, the inner boundary is constructed by blowing up a line or a point, as seen in Fig. 2(a) and (b). So in the following, two cases that: (1) line transformed cloaks, (2) point transformed cloaks, will be discussed separately.
5.1 Case (1): line transformed cloaks
Assume that , and characterize the line, which is mapped to the inner boundary . We have and , and at . Each point on the line maps to a closed curve on . The parameter can be expressed as a function of with . is the gradient of , which points in the direction of the greatest increase rate of . For , we have , where . Thus, and have the same direction.
For the ease of our discussion, we again decompose the incident fields at the inner boundary as , , where the subscripts denotes ’s normal direction, which points outward from the cloaked region; and denote the tangential directions of , with vertical with the plane determined by two vectors in and directions, and vertical with . Since varies on the surface , the direction of denoted by should not be parallel to ’s normal direction. Thus, the plane determined by the vectors in and directions always exists. The , , and directions are unique, as illustrated in Fig. 2(a). and at can also be expressed in Eq. (12), however, with the different definitions of , and . Since () characterize the inner boundary , characterize the normal direction of . Then at can be written as
where when the direction of is as the same as direction, and when the direction of is opposite to direction.
Notice that is orthogonal with both and , where denotes the unit the vector in the direction of . Substituting Eq. (18) into Eq. (12), it is easily derive that
However, the other components of fields are not zero. In particular,
To further investigate how the waves interact with the inner boundary, the values of the permittivity and permeability at are needed. Observing expressed in Eq. (18), it is easily obtained that , indicating that . Thus one of ’s eigen vectors is with the eigen value , implying . Because is a symmetry matrix, the other two eigen vectors denoted by and should be orthogonal to each other and in plane. The corresponding eigen values are denoted by and , respectively, with
where , .
Since , . Observing Eq. (10), it is obtained that . Therefore, it is known that , and are the principle axes of the cloaked medium at with and expressed as
where the diagonal elements correspond to the principle axes ,, and , respectively. Since , we have , indicating the cloaked medium at is isotropic in plane. Therefore, and can be considered as the principle axes with . For and , it is seen that they have infinitely large values. Thus, the inner boundary operates similarly as a combination of the PEC (perfect electric conductor) and PMC (perfect magnetic conductor), which can support both electric and magnetic surface displacement currents in direction[8, 12, 15]. In order to have zero reflection at , the boundary conditions at this PEC and PMC combined layer require that the incident electric (magnetic) fields in direction and normal electric (magnetic) displacement fields are all zero. From Eq. (20), we have . Since , it is obtained that . Therefore, it is achieved that no reflection is excited at , and and expressed in Eq. (11) are just the total fields in the cloak medium. The PEC and PMC combined layer guarantees that no field can penetrate into the cloaked region. It is worth noting that the induced displacement surface currents in direction make and at down to zero. However, and are not zero at the location approaching in the cloak medium. Thus and are discontinuous across the inner boundary [8, 12, 15].
5.2 Case (2): point transformed cloaks
In this case, a point with the coordinate maps to the inner boundary. At , we have , , and . The incident electric and magnetic fields at the inner boundary can be decomposed into and , where the definition of the subscript denotes ’s normal direction, which direct outward from the cloaked region; and represent ’s two tangential directions, which are vertical with each other, as shown in Fig. 2(b). Consider at the inner boundary, which can be expressed as
where . Then substituting Eq. (27) into Eq. (11), we derive that at
Unlike the case (1), in this case tangential fields are all zero, implying that no field discontinuity exists at .
Analyzing similarly as in the case (1), we obtain that is eigen vector of with the eigen value . While the other two eigen vectors are and with the corresponding eigen values , indicating . Considering , we have . Therefore, , , and are principle axes of the cloak medium at with and given as
where the diagonal elements are in the principle axes , , and , respectively. Since , . Considering that and tangential components of incident fields at are zero, it can be conclude that no reflection is excited at , and no field penetrates into the cloaked region. The fields expressed in Eq. (11) are the total fields in the cloak medium.
In the above sections, it has been proved that no reflection is excited at both the exterior boundary and the inner boundary of the cloak, and no field can penetrate into the cloaked region. Therefore, the invisibility of invisibility cloaks with arbitrary shape constructed by general coordinate transformations is confirmed.
6 Transformation under arbitrary coordinate system
The cloak parameters and the fields inside the cloak are expressed in Eqs. (10) and (11), respectively. These results are expressed under the Cartesian coordinate system, i.e., representing Cartesian coordinates in the transformed space. However sometimes, it is much easier to discuss cloaks under other coordinate systems, such as the cylindrical cloak under the cylindrical coordinate system . Thus, it is necessary to obtain the corresponding expressions for cloak parameters and the fields in a cloak under an arbitrary coordinate system, which has also been discussed in Ref. .
Consider coordinate transformation, where denotes the coordinates of an arbitrary coordinate system in the transformed space. The spatial metric tensor of such coordinate system is , where can be obtained easily by considering the relationship between Cartesian coordinate system and this arbitrary coordinate system. For the cylindrical coordinate system and spherical coordinate system, the metric tensors are and , respectively. The spatial metric tensor of is expressed as , where is shown in Eq. (4). Assuming that are the corresponding Cartesian coordinates of in the transformed space, the spatial metric tensor of is then obtained as , where , and represents expressed under the coordinates . Thus, from Eq. (10), the permittivity and permeability of the cloak in Cartesian coordinates are obtained as . Then expressing and in coordinate system, we easily have
where with , and . As an example, for the cylindrical coordinate system.
Consider the fields in the cloak. It is easy to know that the fields expressed in Cartesian coordinate system are and . Thus, the fields expressed in the coordinate system are expressed as following
where represents expressed under the coordinates and , with ; and represent incident electrical and magnetic field vectors expressed under the coordinate system.
If denotes the corresponding coordinates of the coordinate system in the original space, then can be written as , where
Then Eqs. (34) and (35) can be expressed as
7 Examples: cylindrical and spherical cloaks
In this section, based on the results obtained above, the well known radially transformed cylindrical and spherical cloaks will be discussed as examples.
7.1 Cylindrical cloaks
A two-dimensional cylindrical cloak is constructed by compressing EM fields in a cylindrical region into a concentric cylindrical shell . Its inner boundary is blown up by a straight line. Thus, a cylindrical cloak is actually a line transformed cloak. Here consider a generalized coordinate transformation that with and , while and are kept unchanged. Thus, and defined in the above section are and , respectively, which indicates that and . and are and , respectively. is equal to . Substituting these expressions into Eq. (37), the permittivity and permeability of the cloak expressed in cylindrical coordinate system are obtained easily
It is seen that at the exterior boundary , the cloak medium has the PML form with and , which results from the symmetry of , which can be calculated easily.
Consider the fields and incident upon the cloak. It is derived that . Then, the fields in the cloaked medium can be obtained directly from Eq. (38)