Strictly conformal transformation optics for directivity enhancement and unidirectional cloaking of a cylindrical wire antenna
用于圆柱形线天线的方向性增强和单向隐形的严格共形变换光学器件

Since Leonhardt and Pendry discovered a transformation-optical solution for the problem of invisibility^1,2, the theory of transformation optics (TO) has caught the interest of scientists. TO recipe incorporates a geometrical interpretation of the device’s function into a calculated material. This theory allowed researchers to derive the material that could realize a specific functionality, essentially an inverse problem. Many fabulous devices have been created using this theory like field rotators^3,4,5, field concentrators^6,7, carpet cloaks^{8,9,10,11,12,13,14,15,16}, polarization splitters^{4,17,18,19,20}, beam expanders and squeezers^21,22, directivity enhancers^{23,24,25,26,27,28,29}, and waveguide couplers^{30,31,32,33,34,35,36,37}. TO has also been used for compressing lenses^{38,39,40,41,42,43}.
自从 Leonhardt 和 Pendry 发现了解决隐形问题的变换光学解决方案^{1 , 2}以来，变换光学 (TO) 理论引起了科学家的兴趣。 TO 配方将设备功能的几何解释融入到计算材料中。该理论使研究人员能够推导出可以实现特定功能的材料，本质上是一个逆问题。使用这个理论已经创造了许多令人难以置信的设备，如场旋转器^{3 , 4 , 5} ，场集中器^{6 , 7} ，地毯斗篷^{8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16} ，偏振分束器^{4 , 17 , 18 , 19 , 20} , 扩束镜^{压缩器21、22 ，方向性增强器23、24、25、26、27、28、29 ，以及波导耦合器30、31、32、33、34、35、36、37 。​​} TO^{还被用于压缩透镜38、39、40、41、42、43 。}

Despite being an interesting systematic approach and providing a mathematical path for the design of functional devices, the material calculated by TO is often difficult to realize. In general, the constitutive parameters can possess extreme permittivity and permeability values and a high degree of anisotropy. Attaining anisotropy is quite challenging since the available natural materials exhibit low anisotropy levels. Metamaterial unit cells can provide higher anisotropy at the expense of introducing loss and limiting the bandwidth. The type of transformation used has a significant impact on material complexity. For instance, linear transformations result in a homogeneous anisotropic medium^{12,15,18,19,30,31,33,44,45}, whereas using quasi-conformal and conformal transformations minimizes and eliminates the anisotropy, respectively^{1,8,23,24,25,26,27,28,29,35,41,46,47,48}.
尽管这是一种有趣的系统方法，并为功能器件的设计提供了数学路径，但 TO 计算的材料通常难以实现。一般来说，本构参数可以具有极端的介电常数和磁导率值以及高度的各向异性。获得各向异性非常具有挑战性，因为可用的天然材料表现出较低的各向异性水平。超材料晶胞可以提供更高的各向异性，但代价是引入损耗和限制带宽。所使用的转换类型对材料的复杂性有重大影响。例如，线性变换产生均匀各向异性介质^{12 , 15 , 18 , 19 , 30 , 31 , 33 , 44 , 45} ，而使用准共形和共形变换分别最小化和消除各向异性^{1 , 8 , 23 , 24 , 25 , 26 , 27 、 28、29、35、41、46、47、48 。​​​​}

For the case of an omnidirectional cloak, the medium possesses singular constitutive parameters near the concealment region². The singularity embedded in omnidirectional cloaks stems from using a point-transformed mapping in 3D and a line-transformed mapping in 2D. As a result, a new class of unidirectional invisibility cloaks was introduced, capable of providing invisibility for one incident angle. The mapping is plane-transformed for such unidirectional cloaks, which can lead to material singularity mitigation. Unidirectional cloaks can be designed using linear transformations^49,50,51,52, non-linear transformations^49,53,54, and conformal transformations^{1,55,56,57,58,59,60}. A cloak prototype was fabricated using metamaterial cells⁵¹ and metal 3D printing⁵². Employing metasurfaces for the realization of the structure is also an active area of research^61,62.
对于全向斗篷的情况，介质在隐藏区域²附近具有奇异的本构参数。全向斗篷中嵌入的奇点源于使用 3D 中的点变换映射和 2D 中的线变换映射。因此，一种新型单向隐形斗篷被引入，能够为一个入射角提供隐形。对于这种单向斗篷，映射是平面变换的，这可以减少材料奇点。单向斗篷可以使用^{线性变换49、50、51、52 、非线性变换49、53、54}和^{共形变换1、55、56、57、58、59、60来设计。}使用超材料单元⁵¹和金属3D打印⁵²制造斗篷原型。采用超表面来实现结构也是一个活跃的研究领域^{61 , 62} 。

On the other hand, conformal and quasi-conformal transformations have been proven to help improve the directivity of a radiating source. Researchers have demonstrated that they can flatten the phase fronts of the cylindrical wave emanating from a point source by using (quasi-) conformal mappings^23,24,25,26. Following a similar approach, a conformal transformation was numerically calculated to minimize the phase error at the horn antenna aperture, leading to a directive beam^27,28,63.
另一方面，共形和准共形变换已被证明有助于提高辐射源的方向性。研究人员已经证明，他们可以通过使用（准）共形映射^{23 、 24 、 25 、 26}来平坦化从点源发出的柱面波的相前。按照类似的方法，对共形变换进行数值计算，以最小化喇叭天线孔径处的相位误差，从而产生^{定向波束27、28、63} 。

An interesting TO application is the case of a bi-functional device that provides both unidirectional cloaking and directivity enhancement. Recently, an anisotropic solution was proposed by numerically solving Laplace’s equation⁶⁴. The designed metamaterial shell could cloak the wire antenna for an incident TM wave and enhance the directivity of the wire antenna that emanated TE waves. The same research group recently proposed an isotropic solution using numerical quasi-conformal transformation⁶⁵. The resulting isotropic magnetic transformation medium could be used to unidirectionally cloak the loop antenna in the presence of an incident TM wave and to enhance the directivity of the antenna. Since the incident wave and the radiating wave from the loop antenna were TM polarized, an isotropic magnetic material was used. The magnetic medium was realized using printed SRR and meander line structures, limiting the bandwidth.
一个有趣的 TO 应用是提供单向隐形和方向性增强的双功能设备。最近，通过数值求解拉普拉斯方程⁶⁴提出了各向异性解。设计的超材料外壳可以遮盖线状天线的入射TM波，并增强线状天线发射TE波的方向性。同一研究小组最近提出了使用数值准共形变换的各向同性解决方案⁶⁵ 。由此产生的各向同性磁变换介质可用于在存在入射TM波的情况下单向遮盖环形天线并增强天线的方向性。由于环形天线的入射波和辐射波是TM偏振的，因此使用各向同性磁性材料。磁介质是使用印刷 SRR 和曲折线结构实现的，限制了带宽。

The most well-known method for deriving the quasi-conformal mapping between physical and virtual spaces is to solve Laplace’s equation with proper Dirichlet and Neumann boundary conditions. The level of anisotropy neglected by pursuing this method is proportional to the conformal module mismatch between the two spaces. If the difference between the conformal modules of two spaces is significant, the transformation calculated by numerically solving Laplace’s equation will deviate from being quasi-conformal. In such cases, using the strictly conformal TO formula is questionable, both mathematically and physically. In the case of carpet cloaks, for example, even a minor conformal module mismatch has been shown to cause a lateral shift¹¹. For the device providing unidirectional cloaking and directivity enhancement, the boundary condition associated with the numerical method procedure does not lead to conformal module matching⁶⁵. As the radius of the concealment region increases, the conformal module mismatch increases, and numerical calculation alone can not remedy this defect. A strictly conformal transformation can fix this issue, leading to a purely conformal solution.
推导物理空间和虚拟空间之间的拟共形映射的最著名的方法是用适当的狄利克雷和诺依曼边界条件求解拉普拉斯方程。采用这种方法所忽略的各向异性水平与两个空间之间的共形模块失配成正比。如果两个空间的共形模差异很大，则通过数值求解拉普拉斯方程计算出的变换将偏离拟共形。在这种情况下，使用严格共形的 TO 公式在数学和物理上都是有问题的。例如，在地毯斗篷的情况下，即使很小的保形模块不匹配也已被证明会导致横向移位¹¹ 。对于提供单向隐形和方向性增强的设备，与数值方法过程相关联的边界条件不会导致共形模块匹配⁶⁵ 。随着隐藏区域半径的增加，共形模块失配增加，仅靠数值计算无法弥补这一缺陷。严格的共形变换可以解决这个问题，从而得到纯粹的共形解决方案。

A closed-form, strictly conformal map between the doubly connected physical and virtual spaces is established here, leading to a perfect conformal module match. The resulting isotropic dielectric shell cloaks the wire antenna from an impinging TM wave and enhances the wire antennas’s directivity. Unidirectional invisibility is obtained by keeping the outer boundary of virtual and physical space alike and mapping the inner cylindrical wire to a slit. The latter also bilaterally improves the directivity. The refractive index is then modified and discretized to investigate the practicality of the device. The device’s unidirectional cloaking property is discussed from two distinct perspectives in the “Discussion” section. Ray-tracing and full-wave simulations are carried out using COMSOL to confirm the device’s functionality.
这里在双重连接的物理空间和虚拟空间之间建立了封闭形式的、严格共形的映射，从而实现了完美的共形模块匹配。由此产生的各向同性介电壳可以使线状天线免受 TM 波的影响，并增强线状天线的方向性。通过保持虚拟空间和物理空间的外边界相似并将内部圆柱形线映射到狭缝来获得单向隐形。后者还双边地提高了方向性。然后修改折射率并离散化以研究该装置的实用性。 “讨论”部分从两个不同的角度讨论了该设备的单向隐形特性。使用 COMSOL 进行光线追踪和全波模拟来确认设备的功能。

Here, the virtual and physical spaces are related to each other by a conformal map. The virtual and physical spaces are denoted by the w-plane, which has Cartesian coordinates of (u, v), and the z-plane, which has Cartesian coordinates of (x, y). Assuming the complex variables $w=u+\mathrm{i}v$ and $z=x+\mathrm{i}y$, the conformal mapping from the physical space to the virtual space is expressed by the analytic function $w=f(z)$. In such a scenario, if the virtual space is filled with the refractive index of n(u, v), the refractive index of the physical space is derived using the following well-known TO formula¹:
在这里，虚拟空间和物理空间通过共形映射相互关联。虚拟空间和物理空间由w平面和 z 平面表示，w 平面具有笛卡尔坐标 ( u , v )， z平面具有笛卡尔坐标 ( x , y )。假设复杂变量 $w=u+\mathrm{i}v$ 和 $z=x+\mathrm{i}y$ ，从物理空间到虚拟空间的共形映射由解析函数表示 $w=f(z)$ 。在这种情况下，如果虚拟空间充满折射率n ( u , v )，则使用以下众所周知的TO公式¹导出物理空间的折射率：

where n(u, v) is the refractive index of the virtual space. Here, the refractive index of $n(u,v)=1$ is assumed.
其中n ( u , v ) 是虚拟空间的折射率。这里，折射率为 $n(u,v)=1$ 假设。

Figure 1 depicts the schematics of virtual and physical spaces. An annulus with the inner radius of $\rho$ and the outer radius of one in the physical space is mapped to the doubly connected circular virtual space furnished with a slit $-L\le w\le L$ along the u-axis.
图1描绘了虚拟空间和物理空间的示意图。内半径为的圆环 $\rho$ 物理空间中的一个的外半径被映射到带有狭缝的双连通圆形虚拟空间 $-L\le w\le L$ 沿u轴。

Schematics of the (a) physical space, and (b) virtual space for the directivity enhancement and cloaking of a cylindrical wire antenna.
用于圆柱形线天线的方向性增强和隐形的 ( a ) 物理空间和 ( b ) 虚拟空间的示意图。

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For the case of 2D simply connected shapes, Riemann’s mapping theorem implies that one can find a conformal mapping between two desired non-empty shapes. This theorem has been the critical factor for all transformation-optical researchers that employ conformal mappings. However, for the conformal mapping of 2D doubly connected regions, there exists another exciting theorem. After adapting the theorem to our case, it reads: “let D be a non-degenerate doubly connected region, then there exists a unique real number $\rho$, $0<\rho <1$, such that there exists a one-to-one analytic function f(z) that maps the annulus $R_u$: $\rho <\left|z\right|<1$ onto D. If the outer boundaries correspond to each other, then f(z) is determined up to a rotation of the annulus”. The unique value ${\rho }^{-1}$ is referred to as the conformal modulus of D⁶⁶. This means that for a given doubly connected virtual space, there exists a unique annulus with a specified inner radius of $\rho$ and an outer radius of 1.
对于二维简单连接形状的情况，黎曼映射定理意味着可以在两个所需的非空形状之间找到共形映射。该定理一直是所有采用共形映射的变换光学研究人员的关键因素。然而，对于二维双连通区域的共形映射，还存在另一个令人兴奋的定理。将定理应用到我们的情况后，它写道：“设D为非简并双连通区域，则存在唯一实数 $\rho$ , $0<\rho <1$ ，这样存在映射环面的一对一解析函数f ( z ) $R_u$ : $\rho <\left|z\right|<1$ 到D上。如果外边界彼此对应，则f ( z ) 取决于环面的旋转”。独特的价值 ${\rho }^{-1}$ 被称为D ⁶⁶的共形模量。这意味着对于给定的双连通虚拟空间，存在一个唯一的圆环，其指定内半径为 $\rho$ 外半径为1。

There are four critical points to note regarding the configuration in Fig. 1:
关于图1中的配置，有四个关键点需要注意：

1. 1.

   The continuity of the transformation at the outer rim is sufficient to achieve the reflectionless property at the outer contour⁶⁷. Hence, the outer boundary of both physical and virtual spaces is chosen alike. In the “Discussion” section, this point is discussed in greater detail.
   外缘处的变换的连续性足以实现外轮廓⁶⁷处的无反射特性。因此，物理空间和虚拟空间的外边界的选择是相似的。在“讨论”部分，更详细地讨论了这一点。

2. 2.

   There are two main options to improve the directivity of a cylindrical wire. Either one should map the outer contour of the virtual space to a square or a rectangle in the physical space, which is in contrast with the invisibility requirement, or one can transform the inner cylindrical wire in the physical space into a slit in the virtual space. An observer from the outside will perceive the electrical current placed on the inner cylinder of physical space as an electrical current excited on a slit. As a result, the directivity is enhanced since an electrical current excited on a slit radiates mainly along the direction normal to the slit, leading to two directive beams.
   改善圆柱形导线的方向性有两种主要选择。要么将虚拟空间的外轮廓映射到物理空间中的正方形或长方形，这与隐形要求相反，要么将物理空间中的内部圆柱线转变为虚拟空间中的狭缝。来自外部的观察者会将物理空间内圆柱体上的电流感知为狭缝上激发的电流。结果，由于在狭缝上激发的电流主要沿着垂直于狭缝的方向辐射，因此增强了方向性，从而产生两个定向光束。

3. 3.

   Based on the first and second points, the outside observer looking along the x-axis sees the perfect electric conductor (PEC) cylinder as the PEC slit. If the wave’s polarization is TM (magnetic field along the z-axis), the slit (or wire) will not disturb the incident TM wave and will eventually become invisible. Hence unidirectional cloaking can be achieved for the TM polarization. In the case of TE polarization (electric field along the z-axis), a perfect magnetic conductor (PMC) cylinder achieves invisibility⁵⁵.
   基于第一点和第二点，沿x轴观察的外部观察者将完美电导体 (PEC) 圆柱体视为 PEC 狭缝。如果波的偏振是TM（沿z轴的磁场），狭缝（或线）将不会干扰入射的TM波并最终变得不可见。因此，TM 偏振可以实现单向隐身。在 TE 极化（沿z轴的电场）的情况下，完美磁导体 (PMC) 圆柱体实现了不可见性⁵⁵ 。

4. 4.

   The shell is designed to increase the wire’s directivity, which emits TE polarized waves when excited by an electrical current. The dielectric medium designed by conformal transformation for this polarization is entirely efficient. However, for cloaking, TM polarization is involved. Conformal mapping results in a fully magnetic medium for this polarization, and obtaining a wideband magnetic response is difficult⁶⁵. Although, if the structure’s size is electrically large, as it is in the geometrical optics (GO) regime, the dielectric medium designed based on the TE assumption works well for TM⁶⁸.
   外壳的设计是为了增加导线的方向性，当电流激励时，导线会发射 TE 极化波。通过共形变换为这种极化设计的介电介质是完全有效的。然而，对于隐身来说，涉及TM极化。共形映射导致这种极化产生全磁介质，并且获得宽带磁响应很困难⁶⁵ 。不过，如果结构的电尺寸较大，如在几何光学 (GO) 体系中，则基于 TE 假设设计的介电介质对于 TM ⁶⁸效果很好。

It is worth noting that the propagation direction that the cylindrical wire is invisible for (along the x-axis) is perpendicular to the direction of the directivity enhancement (along the $\pm y$-axis) in our configuration.
值得注意的是，圆柱线不可见的传播方向（沿x轴）垂直于方向性增强的方向（沿 $\pm y$ -axis）在我们的配置中。

A direct calculation of the mapping depicted in Fig. 1 is not possible. The mapping should be calculated by splitting the problem into a collection of conformal transformations. In this process, some preliminary conformal maps should be introduced. The fundamental analytic functions employed in the following calculations are available in the conformal mapping handbooks^69,70.
无法直接计算图1中描绘的映射。应通过将问题分解为保角变换的集合来计算映射。在此过程中，应引入一些初步的等角图。以下计算中采用的基本分析函数可在共形映射^{手册69、70}中找到。

The conformal map from a rectangle to the upper half-space is introduced first. This mapping will be used later in the process. Figure 2 depicts this case.
首先介绍从矩形到上半空间的共形映射。该映射稍后将在该过程中使用。图2描述了这种情况。

Schematic of the conformal mapping from a rectangle in the z-plane to the upper half of the w-plane⁴¹.
从z平面中的矩形到w平面⁴¹的上半部的共形映射的示意图。

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Vertices $z=\pm K,\pm K+i{K}^{\prime }$ are mapped to the vertices $w=\pm 1,\pm k^{-1}(0<k<1)$. Table 1 summarizes the associated vertices.
顶点 $z=\pm K,\pm K+i{K}^{\prime }$ 被映射到顶点 $w=\pm 1,\pm k^{-1}(0<k<1)$ 。表1总结了相关的顶点。

The analytic function $w=\mathrm{s}\mathrm{n}(z,k)$ known as the elliptic sine function provides such a transformation, where k is the modulus of the elliptical integrals. K(k) and $K^{\prime }(k)$ are the complete and complementary complete elliptical integrals of the first with the $K^{\prime }(k)=K(\sqrt{1-k^2})$ property.
解析函数 $w=\mathrm{s}\mathrm{n}(z,k)$ 称为椭圆正弦函数的函数提供了这样的变换，其中k是椭圆积分的模。 K ( k ) 和 $K^{\prime }(k)$ 是第一个的完整且互补的完整椭圆积分 $K^{\prime }(k)=K(\sqrt{1-k^2})$ 财产。

It is worth mentioning that if the modulus k is known, then the aspect ratio of the rectangle $K(k)/K^{\prime }(k)$ is uniquely determined, and vice versa. It is helpful to define the intermediate parameter $q=e^{-\pi K^{\prime }(k)/K(k)}$, which depends on the aspect ratio and modulus k. Modulus k can be expressed as an infinite series of q following the below formula:
值得一提的是，如果模数k已知，则矩形的长宽比 $K(k)/K^{\prime }(k)$ 是唯一确定的，反之亦然。定义中间参数很有帮助 $q=e^{-\pi K^{\prime }(k)/K(k)}$ ，这取决于纵横比和模量k 。模k可以表示为q的无穷级数，遵循以下公式：

The problem here is finding the analytic function $w=f(z)$ that maps the interior of the physical space’s annulus $\rho <\left|z\right|<1$ to the interior of the virtual space’s unity circle $|w|=1$, furnished with the slit $-L\le w\le L$. Applying the symmetry principle to the unity circle $\left|z\right|=1$, we conclude that the analytic function $w=f(z)$ also maps the annulus $\rho <\left|z\right|<{\rho }^{-1}$ to the entire w-plane, including the $-\mathrm{\infty }\le w\le -L^{-1},-L\le w\le L$, and $L^{-1}\le w\le \mathrm{\infty }$ slits. Both domains have symmetry with respect to the real axis. Hence, if a mapping is found that can map the upper half of the annulus $\rho <\left|z\right|<{\rho }^{-1}$ to the upper half of the w-plane, it is identical with $w=f(z)$ based on the symmetry principle, provided that it maps the points according to Table 2.
这里的问题是找到解析函数 $w=f(z)$ 映射物理空间环的内部 $\rho <\left|z\right|<1$ 到虚拟空间统一圆的内部 $|w|=1$ ，配有狭缝 $-L\le w\le L$ 。将对称原理应用于单位圆 $\left|z\right|=1$ ，我们得出解析函数 $w=f(z)$ 还绘制了环面 $\rho <\left|z\right|<{\rho }^{-1}$ 到整个w平面，包括 $-\mathrm{\infty }\le w\le -L^{-1},-L\le w\le L$ ， 和 $L^{-1}\le w\le \mathrm{\infty }$ 开缝。两个域都相对于实轴对称。因此，如果找到可以映射环面上半部分的映射 $\rho <\left|z\right|<{\rho }^{-1}$ 到w平面的上半部分，它与 $w=f(z)$ 基于对称原理，只要按照表2映射点即可。

The procedure for finding the required conformal map is depicted in Fig. 3, where the unknown mapping is derived indirectly using two known conformal maps.
寻找所需共形图的过程如图3所示，其中使用两个已知共形图间接导出未知映射。

Schematic of the procedure for deriving the conformal map $w=f(z)$.
导出共形图的过程示意图 $w=f(z)$ 。

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The problem, by far, includes finding a transformation that maps the upper half of the annulus $\rho <\left|z\right|<{\rho }^{-1}$ to the upper half of the w-plane. On the other hand, the transformation from a rectangle to the upper half of the w-plane has already been derived (see Fig. 2). Hence, finding a conformal map between the rectangle and the upper half of the annulus $\rho <\left|z\right|<{\rho }^{-1}$ is the key to finding the solution.
到目前为止，问题包括找到映射环面上半部分的变换 $\rho <\left|z\right|<{\rho }^{-1}$ 到w平面的上半部分。另一方面，从矩形到w平面上半部分的变换已经导出（见图2 ）。因此，找到矩形和环面上半部分之间的等角映射 $\rho <\left|z\right|<{\rho }^{-1}$ 是找到解决方案的关键。

The exponential function $z=i\rho e^{-i\eta }$ maps the rectangular region $-\pi /2<\mathrm{R}\mathrm{e}\{\eta \}<\pi /2,$ $0<\mathrm{I}\mathrm{m}\{\eta \}<-2\log \rho$ onto the upper half of the annulus $\rho <\left|z\right|<{\rho }^{-1}$. Table 3 summarizes the points involved in the mapping.
指数函数 $z=i\rho e^{-i\eta }$ 绘制矩形区域 $-\pi /2<\mathrm{R}\mathrm{e}\{\eta \}<\pi /2,$ $0<\mathrm{I}\mathrm{m}\{\eta \}<-2\log \rho$ 到环面的上半部分 $\rho <\left|z\right|<{\rho }^{-1}$ 。表3总结了映射中涉及的点。

The vertices and their mapped counterparts in Fig. 3 are labeled with numbers according to Tables 2 and 3.
图3中的顶点及其映射对应物根据表2和表3用数字标记。

Based on Fig. 3, the analytic function $w=f(i\rho e^{-i\eta })$ maps the rectangle in the $\eta$-plane to the upper half of the w-plane. Based on Tables 2 and 3, the correspondence in Table 4 is established.
根据图3 ，解析函数 $w=f(i\rho e^{-i\eta })$ 映射矩形 $\eta$ -平面到w平面的上半部分。根据表2和表3 ，建立表4的对应关系。

Comparing Tables 1 and 4, leads to the following equalities:
比较表1和表 4 ，得出以下等式：

In conclusion, the following analytic function $w=f(z)$ maps the interior of the physical space’s annulus $\rho <\left|z\right|<1$ to the interior of the virtual space’s unity circle $|w|=1$ furnished with the slit $-L\le w\le L$:
综上，可得如下解析函数 $w=f(z)$ 映射物理空间环面的内部 $\rho <\left|z\right|<1$ 到虚拟空间统一圆的内部 $|w|=1$ 配有狭缝 $-L\le w\le L$ :

where the q parameter associated with the above conformal mapping is $q=e^{-\pi K^{\prime }(k)/K(k)}={\rho }^4$ based on Eq. (3). According to Eq. (2) and the $k=L^2$ relation, the following formula relates the half-length of the virtual space’s slit L to the inner radius of the physical space’s annulus $\rho$:
其中与上述共角映射相关的q参数是 $q=e^{-\pi K^{\prime }(k)/K(k)}={\rho }^4$ 基于方程。 （ 3 ）。根据方程。 ( 2 ) 和 $k=L^2$ 关系式，虚拟空间狭缝L的半长与物理空间环面内半径的关系式如下： $\rho$ :

Following Eq. (1), the refractive index of the physical space is derived:
遵循等式。 ( 1 )、物理空间的折射率推导：

where cn and dn are the elliptic cosine and the delta amplitude functions, respectively.
其中 cn 和 dn 分别是椭圆余弦和 delta 幅度函数。

The inverse of the mapping in Eq. (4) can be derived using the definition of the $\mathrm{s}{\mathrm{n}}^{-1}$ function.
等式中映射的逆。 ( 4 ) 可以利用定义推导出 $\mathrm{s}{\mathrm{n}}^{-1}$ 功能。

The above integral can be solved numerically or by using “InverseJacobiSN” in Mathematica⁷¹.
上述积分可以通过数值求解或使用Mathematica ⁷¹中的“InverseJacobiSN”来求解。

Here, the radius of the wire antenna equals $\rho =0.1$ m. The corresponding slit length of $2L=0.4$ m is calculated following Eq. (5). Figure 4 depicts the Cartesian grids inside the virtual space and their mapped counterparts in the physical space. The v-constant grids representing a set of rays propagating along the u-axis in the virtual space are guided around the wire as they are mapped to the physical space. Hence, the unidirectional cloaking effect is to be expected. Furthermore, the mapped u-constant grids near the wire are perpendicular to the circumference of the wire. As a result, it is expected that rays emanating from the wire perpendicular to its boundary will be gradually focused along the $\pm y$ direction by the transformation medium, resulting in two directive beams. Following Eq. (6), the refractive index is calculated and illustrated in Fig. 5.
这里，线天线的半径等于 $\rho =0.1$ 米。对应的狭缝长度为 $2L=0.4$ m 的计算公式如下： （ 5 ）。图4描绘了虚拟空间内的笛卡尔网格及其在物理空间中的映射对应部分。 v常数网格表示一组在虚拟空间中沿u轴传播的光线，当它们映射到物理空间时，它们会围绕导线被引导。因此，单向隐形效应是可以预期的。此外，线附近映射的u常数网格垂直于线的圆周。因此，预计从垂直于其边界的线发出的光线将逐渐沿着线聚焦 $\pm y$ 方向由变换介质决定，产生两个定向光束。遵循等式。由式( 6 )计算出折射率，如图5所示。

Grid transformation between the virtual and physical spaces. (a) Cartesian grids in the virtual space, and (b) the corresponding mapped ones in the physical space.
虚拟空间和物理空间之间的网格转换。 ( a ) 虚拟空间中的笛卡尔网格，( b ) 物理空间中相应的映射网格。

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The refractive index profile of the physical space in Fig. 1a with the inner radius of $\rho =0.1$ m.
图1a中内半径为的物理空间的折射率分布 $\rho =0.1$ 米。

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Ray-tracing and full-wave simulations are carried out to verify the of the device’s functionality. The ray-tracing simulations assess the response of the device in the GO regime. First, 60 parallel rays are launched along the x-axis from the left side of the simulation domain to evaluate the cloaking functionality. The results are depicted in Fig. 6a, where the color expression represents the optical path length in meters. It is seen that the rays are guided around the wire, implying that the cylindrical wire is unidirectionally invisible. To verify the directivity enhancement, 38 equally-spaced rays are generated in the direction perpendicular to the boundary of the wire. The results are illustrated in Fig. 6b, where the rays generated from the wire’s surface are gathered along the $\pm y$ direction by the dielectric shell. This confirms the directivity improvement as the omnidirectional pattern of the cylindrical wire changes to a pattern with two beams along the $\pm y$ direction.
进行射线追踪和全波模拟以验证设备的功能。光线追踪模拟评估了设备在 GO 状态下的响应。首先，从模拟域左侧沿x轴发射 60 条平行光线，以评估隐形功能。结果如图6a所示，其中颜色表达式代表光路长度（以米为单位）。可以看出，光线围绕金属丝被引导，这意味着圆柱形金属丝是​​单向不可见的。为了验证方向性增强，在垂直于导线边界的方向上生成 38 条等距射线。结果如图6b所示，其中从导线表面产生的射线沿着 $\pm y$ 介电壳的方向。这证实了方向性的改善，因为圆柱形导线的全向模式变为沿线方向有两个光束的模式。 $\pm y$ 方向。

Ray-tracing simulation of the physical space with the refractive index depicted in Fig. 5. (a) The cloaking effect for 60 parallel rays launched along the x-axis. (b) The directivity enhancement effect for 38 equally-spaced rays launched from the inner rim. The color expression presents the optical path length in meters.
具有如图5所示折射率的物理空间的射线追踪模拟。 ( a ) 沿x轴发射的 60 条平行光线的隐身效果。 ( b )从内缘发射的38条等距射线的方向性增强效果。颜色表达式表示光路长度（以米为单位）。

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The full-wave simulations are carried out at the frequency of 3 GHz. The directivity enhancement is investigated by exciting the cylindrical wire with an out-of-plane electrical current density $J_z$. Such a current produces TE waves. The amplitude of the electric field $E_z$ and the far-field pattern with and without the dielectric shell are depicted in Fig. 7. The directivity enhancement is seen by comparing the results with the omnidirectionally radiating bare wire.
全波仿真在 3 GHz 频率下进行。通过用面外电流密度激励圆柱形导线来研究方向性增强 $J_z$ 。这样的电流产生TE波。电场幅度 $E_z$ 具有和不具有介电壳的远场方向图如图7所示。通过与全向辐射裸线的结果比较可以看出方向性的增强。

The cylindrical wire enclosed by the refractive index profile depicted in Fig. 5 and excited by the uniform electric current density $J_z$ at 3 GHz. (a) The amplitude of $E_z$, and (b) the normalized far-field pattern with and without the dielectric shell.
由图5所示的折射率分布包围并由均匀电流密度激励的圆柱形导线 $J_z$ 在 3 GHz 时。 ( a )振幅 $E_z$ ，和 ( b ) 具有和不具有介电壳的归一化远场方向图。

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To investigate the cloaking functionality, a TM Gaussian wave that propagates along the x-axis illuminates the structure from the left boundary. The magnetic field amplitude $H_z$ is depicted in Fig. 8 for three cases: the bare wire, and the cases where the refractive index is realized by pure permittivity and permeability. As mentioned in “Introduction”, the pure permittivity performs similarly to the pure permeability if the structure is electrically large. However, perfect cloaking is obtained for the case where pure permeability is employed. The pure permeability case is simulated only for comparison’s sake.
为了研究隐身功能，沿x轴传播的 TM 高斯波从左边界照亮结构。磁场幅度 $H_z$ 图8描绘了三种情况：裸线，以及折射率由纯介电常数和磁导率实现的情况。正如“简介”中提到的，如果结构电性较大，则纯介电常数的表现与纯磁导率类似。然而，在采用纯渗透性的情况下，可以获得完美的隐形效果。模拟纯渗透率情况只是为了比较。

The amplitude of $H_z$ for a TM Gaussian wave illuminating the wire antenna covered with (a) no cloak, (b) a pure dielectric cloak, and (c) a pure magnetic cloak.
振幅为 $H_z$ TM 高斯波照射线状天线，天线覆盖有 ( a ) 无斗篷、( b ) 纯电介质斗篷和 ( c ) 纯磁斗篷。

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To bring the refractive index profile in Fig. 5 one step closer to realization, we have changed the background refractive index to 2 and replaced the below unity values with one. Figure 9 depicts the modified refractive index.
为了使图5中的折射率分布更接近实现，我们将背景折射率更改为 2，并将以下单位值替换为 1。图9描绘了修改后的折射率。

The modified refractive index profile in Fig. 5. The background refractive index is changed to 2 and the superluminal index values are replaced with a unity refractive index.
修改后的折射率分布如图5所示。背景折射率更改为 2，超光速折射率值替换为单位折射率。

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Figure 10 illustrates the ray-tracing results for the refractive index profile shown in Fig. 9, where the color expression represents the optical path length.
图10示出了图9所示的折射率分布的光线追踪结果，其中颜色表达式代表光路长度。

Ray-tracing simulation of the physical space with the refractive index depicted in Fig. 9. (a) The cloaking effect for 60 parallel rays launched along the x-axis. (b) The directivity enhancement effect for 36 equally-spaced rays launched from the inner rim. The color expression presents the optical path length in meters.
具有图9所示折射率的物理空间的射线追踪模拟。 ( a ) 沿x轴发射的 60 条平行光线的隐身效果。 ( b )从内缘发射的36条等距射线的方向性增强效果。颜色表达式表示光路长度（以米为单位）。

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The full-wave simulation results are depicted in Fig. 11. To illuminate the structure, a TM Gaussian wave propagating along the x-axis is excited from the left boundary. The magnetic field amplitude $H_z$ is depicted in Fig. 11a. Also, the directivity enhancement functionality is shown by exciting the cylindrical wire with an electrical current density $J_z$. The amplitude of the electric field $E_z$ and the far-field pattern with and without the dielectric shell are depicted in Fig. 11b,c.
全波模拟结果如图11所示。为了照亮该结构，从左边界激发沿x轴传播的 TM 高斯波。磁场幅度 $H_z$ 如图11a所示。此外，通过用电流密度激励圆柱形导线来显示方向性增强功能 $J_z$ 。电场幅度 $E_z$ 有和没有电介质壳的远场方向图如图11b 、c 所示。

Full-wave simulation results for the cylindrical wire enclosed by the refractive index depicted in Fig. 9. (a) The amplitude of $H_z$ for a TM Gaussian wave illuminating the structure. (b) The amplitude of $E_z$ for the wire excited by the uniform electric current density $J_z$ at 3 GHz. (c) The normalized far-field pattern with and without the dielectric shell.
由折射率包围的圆柱形导线的全波模拟结果如图9所示。 ( a )振幅 $H_z$ 用于照亮结构的 TM 高斯波。 ( b ) 的幅值 $E_z$ 对于均匀电流密度激励的导线 $J_z$ 在 3 GHz 时。 ( c ) 有和没有电介质壳的归一化远场方向图。

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The ray-tracing and full-wave simulation results for the dielectric shell in Fig. 9 show that the device maintains its functionality after manipulating the refractive index.
图9中介电壳的射线追踪和全波模拟结果表明，该器件在操纵折射率后仍保持其功能。

Next, the discretized refractive index is examined. A hexagonal lattice is selected to effectively tessellate the refractive index profile. Figure 12a depicts the arrangement of the hexagonal cells with the side length of $L_{\mathrm{C}\mathrm{e}\mathrm{l}\mathrm{l}}$. Two cases are investigated where the side length of the hexagons $L_{\mathrm{C}\mathrm{e}\mathrm{l}\mathrm{l}}$ equals $\lambda /5$ and $\lambda /10$ for the frequency of 10 GHz. The sampling points (centers of the hexagon cells) are illustrated in Figs. 12b,c. Based on the refractive index profile and the local wavelength, both samplings are coarse near the cylindrical wire.
接下来，检查离散化的折射率。选择六方晶格以有效地镶嵌折射率分布。图12a描绘了边长为 的六边形单元的排列 $L_{\mathrm{C}\mathrm{e}\mathrm{l}\mathrm{l}}$ 。研究了两种情况，其中六边形的边长 $L_{\mathrm{C}\mathrm{e}\mathrm{l}\mathrm{l}}$ 等于 $\lambda /5$ 和 $\lambda /10$ 频率为 10 GHz。采样点（六边形单元的中心）如图 1 和 2 所示。 12 b、c。根据折射率分布和局部波长，两个采样在圆柱形线附近都是粗糙的。

Discretization of refractive index profile in Fig. 9. (a) The hexagonal lattice configuration. (b) Sampling points for $L_{\mathrm{C}\mathrm{e}\mathrm{l}\mathrm{l}}=\lambda /5$. (c) Sampling points for $L_{\mathrm{C}\mathrm{e}\mathrm{l}\mathrm{l}}=\lambda /10$.
图9中折射率分布的离散化。 ( a ) 六方晶格结构。 ( b ) 采样点 $L_{\mathrm{C}\mathrm{e}\mathrm{l}\mathrm{l}}=\lambda /5$ 。 ( c ) 采样点 $L_{\mathrm{C}\mathrm{e}\mathrm{l}\mathrm{l}}=\lambda /10$ 。

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Full-wave simulations are carried out to compare the results of the sampled shell depicted in Figs. 12b,c, with the ones illustrated in Fig. 11a,b. Results are presented in Fig. 13.
进行全波模拟以比较图 2 和 3 中所示的采样壳的结果。图12b 、c与图11a 、b所示的相同。结果如图13所示。

Full-wave simulation results for the cylindrical wire enclosed by the discretized refractive index. (a) Discretized refractive index profile, (b) the amplitude of $H_z$ for a TM Gaussian wave illuminating the structure, and (c) the amplitude of $E_z$ for the wire excited by the uniform electric current density $J_z$ at 3 GHz for $L_{\mathrm{C}\mathrm{e}\mathrm{l}\mathrm{l}}=\lambda /5$. (d) Discretized refractive index profile, (e) the amplitude of $H_z$ for a TM Gaussian wave illuminating the structure, and (f) the amplitude of $E_z$ for the wire excited by the uniform electric current density $J_z$ at 3 GHz for $L_{\mathrm{C}\mathrm{e}\mathrm{l}\mathrm{l}}=\lambda /10$.
由离散折射率包围的圆柱形导线的全波模拟结果。 ( a ) 离散折射率分布，( b ) 振幅 $H_z$ 对于照射结构的 TM 高斯波，以及 ( c ) 的振幅 $E_z$ 对于均匀电流密度激励的导线 $J_z$ 在 3 GHz 时 $L_{\mathrm{C}\mathrm{e}\mathrm{l}\mathrm{l}}=\lambda /5$ 。 ( d ) 离散折射率分布，( e ) 振幅 $H_z$ 对于照射结构的 TM 高斯波，以及 ( f ) 的振幅 $E_z$ 对于均匀电流密度激励的导线 $J_z$ 在 3 GHz 时 $L_{\mathrm{C}\mathrm{e}\mathrm{l}\mathrm{l}}=\lambda /10$ 。

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It is seen that the cloaking and the directivity enhancing properties degrade for the $L_{\mathrm{C}\mathrm{e}\mathrm{l}\mathrm{l}}=\lambda /5$ case since the sampling is very coarse. However, the results obtained for the $L_{\mathrm{C}\mathrm{e}\mathrm{l}\mathrm{l}}=\lambda /10$ case are almost identical to the Fig. 11a,b results where the refractive index is continuous.
可以看出，隐形和方向性增强特性随着 $L_{\mathrm{C}\mathrm{e}\mathrm{l}\mathrm{l}}=\lambda /5$ 由于采样非常粗略。然而，获得的结果为 $L_{\mathrm{C}\mathrm{e}\mathrm{l}\mathrm{l}}=\lambda /10$ 情况几乎与折射率连续的图11a 、b 结果相同。

It is worth investigating the perfectness of the invisibility provided by the dielectric shell. The proposed conformal map transforms the circular outer boundaries of the physical space and the virtual space to each other. This aspect contributes to maintaining the map’s continuity at the outer boundary and, as a result, avoiding reflections. However, this effect is inherently not perfect. Based on the uniqueness theorem for analytic functions, if the values of two conformal maps on a specific boundary (outer rims of both spaces) equal each other, the two conformal maps are equal (Theorem 10.39)⁷². It means that the perfect transformation continuity requirement for cloaking $(x^{\prime }=x,y^{\prime }=y,z^{\prime }=z)$, forces the conformal map to be a unity transformation, which is not intended. In fact, if the radial and angular coordinates of the virtual and physical spaces are denoted as $(R,\psi )$ and $(r,\varphi )$, the conformal mapping provided in Eq. (4) only ensures the perfect equality of the radial components at the outer contour $(R=r)$. Due to the uniqueness theorem, the angular components can not be equal simultaneously $(\psi \ne \varphi )$. As the radius of the wire increases, the angular components deviate more.
介电壳提供的隐形性的完美程度值得研究。所提出的共形图将物理空间和虚拟空间的圆形外边界相互转换。此方面有助于保持地图在外部边界的连续性，从而避免反射。然而，这种效果本质上并不完美。根据解析函数的唯一性定理，如果特定边界（两个空间的外缘）上的两个共形映射的值彼此相等，则这两个共形映射相等（定理10.39） ⁷² 。这意味着隐身的完美变换连续性要求 $(x^{\prime }=x,y^{\prime }=y,z^{\prime }=z)$ ，强制等角贴图进行统一变换，这不是有意的。事实上，如果虚拟空间和物理空间的径向坐标和角坐标表示为 $(R,\psi )$ 和 $(r,\varphi )$ ，等式中提供的共形映射。 ( 4 )仅保证外轮廓处的径向分量完全相等 $(R=r)$ 。由于唯一性定理，角度分量不能同时相等 $(\psi \ne \varphi )$ 。随着线的半径增加，角度分量偏差更大。

Figure 14 plots $\psi$ versus $\varphi$ for the inner radius values of $\rho =0.1,0.2$, and 0.3 m. It is seen that the angular coordinate equality between the two spaces is quite good for a small inner radius value. This equality becomes increasingly absolute as $\rho$ converges to zero, making the physical and virtual spaces similar, representing a unity transformation.
图14绘图 $\psi$ 相对 $\varphi$ 对于内半径值 $\rho =0.1,0.2$ 和 0.3 m。可以看出，对于较小的内半径值，两个空间之间的角坐标相等性相当好。这种平等变得越来越绝对 $\rho$ 收敛到零，使物理空间和虚拟空间相似，代表统一变换。

The angular coordinate of the virtual space $\psi$ versus the angular coordinate of the physical space $\varphi$ on the physical space’s outer contour for $\rho =0.1,0.2$, and 0.3 m.
虚拟空间的角坐标 $\psi$ 与物理空间的角坐标 $\varphi$ 在物理空间的外轮廓上 $\rho =0.1,0.2$ 和 0.3 m。

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This perfectness of the cloaking can be studied from a different point of view. A refractive index of one $(\left|f^{\prime }(z)\right|=1)$ on the outer boundary is sufficient for avoiding reflections. Here, the complex variables $z=x+iy=r{e}^{i\text{φ}}$ and $w=f(r{e}^{i\text{φ}})=u+iv=R{e}^{i\psi }$ describe the physical and virtual spaces, respectively. Following Eq. (1), the refractive index formula in the polar coordinates follows the below relation:
隐形衣的完美性可以从不同的角度来研究。折射率为一 $(\left|f^{\prime }(z)\right|=1)$ 在外边界上足以避免反射。这里，复杂变量 $z=x+iy=r{e}^{i\text{φ}}$ 和 $w=f(r{e}^{i\text{φ}})=u+iv=R{e}^{i\psi }$ 分别描述物理空间和虚拟空间。遵循等式。 ( 1 ) 极坐标下的折射率公式如下：

where the subscripts denote partial derivatives and the Cauchy–Riemann conditions in polar coordinates $u_r=v_{\text{φ}}/r,v_r=-u_{\text{φ}}/r$ are considered. The proposed conformal mapping ensures $R=r$ equality at the outer rim. Hence, the refractive index at the exterior border of the cloak $(R=r=1)$ equals $\sqrt{R_r^2+{\psi }_r^2}$ or $\sqrt{R_{\text{φ}}^2+{\psi }_{\text{φ}}^2}$ equivalently. Figure 15 plots the refractive index of the physical space’s outer contour for the inner radius values of $\rho =0.1,0.2$, and 0.3 m. It is seen that the refractive index begins to deviate from unity as the inner radius increases.
其中下标表示偏导数和极坐标中的柯西-黎曼条件 $u_r=v_{\text{φ}}/r,v_r=-u_{\text{φ}}/r$ 被考虑。所提出的共形映射确保 $R=r$ 外缘平等。因此，斗篷外缘的折射率 $(R=r=1)$ 等于 $\sqrt{R_r^2+{\psi }_r^2}$ 或者 $\sqrt{R_{\text{φ}}^2+{\psi }_{\text{φ}}^2}$ 同等地。图15绘制了内半径值的物理空间外轮廓的折射率 $\rho =0.1,0.2$ 和 0.3 m。可以看出，随着内半径的增加，折射率开始偏离统一。

The refractive index of the physical space’s outer contour versus $\varphi$ for $\rho =0.1,0.2$, and 0.3 m.
物理空间外轮廓的折射率与 $\varphi$ 为了 $\rho =0.1,0.2$ 和 0.3 m。

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To verify the perfectness of the unidirectional cloaking, the structure from the simulation section is illuminated by a TM Gaussian wave that propagates along the x-axis. The refractive index is realized by a pure permeability to eliminate any imperfection from using a pure dielectric material for the TM wave. Figure 16 shows the results for the cases in which the inner radius values of $\rho =0.2$, and 0.3 m are chosen, resulting in excellent cloaking. The case with the inner radius of $\rho =0.1$ m has already been depicted in Fig. 8c.
为了验证单向隐身的完美性，模拟部分的结构由沿x轴传播的 TM 高斯波照亮。折射率是通过纯磁导率实现的，以消除使用纯电介质材料产生 TM 波的任何缺陷。图16显示了内半径值为 $\rho =0.2$ 选择 、 和 0.3 m，从而获得出色的隐形效果。内半径为的情况 $\rho =0.1$ m 已在图8c中描述。

A TM Gaussian wave illuminates the wire antenna enclosed by a pure magnetic cloak. (a) $\rho =0.2$ m, and (b) $\rho =0.3$ m.
TM 高斯波照亮由纯磁斗篷封闭的线状天线。 （一） $\rho =0.2$ 米，和（ b ） $\rho =0.3$ 米。

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It’s also worth looking into the Zhukovsky mapping⁵⁶. The Zhukovsky transformation $w=f(z)=z+{\rho }^2/z$ maps the inner wire with the radius of $\rho$ in the physical space to a slit with the half-length of $L=2\rho$ in the virtual space. In addition, the outer contour of physical space $|z|=1$ is mapped to an ellipse with major and minor axes equal to $1+{\rho }^2$ and $1-{\rho }^2$, respectively. As $\rho$ increases, the physical space’s circular outer boundary transforms into an ellipse in the virtual space with a larger axial ratio. Hence, the outer boundaries of physical and virtual spaces become less similar.
朱可夫斯基映射也值得研究⁵⁶ 。朱可夫斯基变换 $w=f(z)=z+{\rho }^2/z$ 绘制半径为 的内线 $\rho$ 在物理空间中形成一条半长的狭缝 $L=2\rho$ 在虚拟空间中。此外，物理空间的外轮廓 $|z|=1$ 映射到一个椭圆，长轴和短轴等于 $1+{\rho }^2$ 和 $1-{\rho }^2$ ， 分别。作为 $\rho$ 增大，物理空间的圆形外边界在虚拟空间中转变为轴比更大的椭圆。因此，物理空间和虚拟空间的外部边界变得不太相似。

Comparing the slit half-length for the Zhukovsky case $L=2\rho$ with Eq. (5), it is clear that Eq. (5) contains an additional product series term. The value of the series decreases as $\rho$ increases, and it approaches one as $\rho$ approaches zero. In conclusion, for the mapping problem depicted in Fig. 1, the conformal mapping of Eq. (4) converges to the Zhukovsky mapping for very small values of $\rho$.
朱可夫斯基案例的狭缝半长比较 $L=2\rho$ 与方程。 ( 5 ) 显然，式(5) ( 5 ) 包含附加产品系列术语。该系列的价值随着 $\rho$ 增加，并且它接近 1 $\rho$ 趋近于零。总之，对于图1所示的映射问题，方程的共角映射为： ( 4 ) 对于非常小的值收敛到 Zhukovsky 映射 $\rho$ 。

As the theory implies and as seen in the cases of the Zhukovsky mapping and the conformal mapping presented in Eq. (4), for a specific doubly connected physical space (annulus shape in this work), there exists a unique value for the slit length embedded in the virtual space. The value of slit length is directly related to the conformal module of the physical space ${\rho }^{-1}$. The numerical method presented in the literature employs Dirichlet and Neumann boundary conditions on the physical space’s inner and outer boundaries to ensure the continuity of the transformation on the outer rim and the orthogonality of the meshes⁶⁵. However, the boundary conditions cannot establish any correspondence between ${\rho }^{-1}$ and the slit length. The method presented in the literature offers no solution for conformal module matching. As a result, as the radius of the wire increases, the numerically calculated transformation deviates substantially from being conformal.
正如该理论所暗示的那样，正如在朱可夫斯基映射和等式中提出的共角映射的情况中所看到的那样。 ( 4 )、对于特定的双连通物理空间(本文中的环形形状)，嵌入虚拟空间的狭缝长度存在唯一值。狭缝长度的大小与物理空间的共形模量直接相关 ${\rho }^{-1}$ 。文献中提出的数值方法在物理空间的内外边界上采用狄利克雷和诺依曼边界条件，以保证外缘上变换的连续性和网格的正交性⁶⁵ 。然而，边界条件不能建立任何对应关系 ${\rho }^{-1}$ 和狭缝长度。文献中提出的方法没有提供共形模块匹配的解决方案。结果，随着线的半径增加，数值计算的变换显着偏离共形。

The doubly connected annulus in the physical space is mapped to a circle furnished with a slit using a closed-form, strictly conformal map. Such a mapping provides unidirectional cloaking while improving the directivity of a cylindrical wire antenna. The conformal mapping leads to an isotropic dielectric material. Both ray-tracing and full-wave simulations confirm the proposed design method.
使用封闭形式的严格共形映射将物理空间中的双重连接环映射到带有狭缝的圆。这种映射提供了单向隐形，同时提高了圆柱形线天线的方向性。共形映射产生各向同性介电材料。射线追踪和全波模拟都证实了所提出的设计方法。