Exact Analytical Electromagnetic Wave Propagation In Practicable GLHUA Electromagnetic Invisible Outer Layer Cloak
Jianhua Li
Feng Xie
Lee Xie
Ganquan Xie
Ganquan Xie
It is totally different from the transform optics, by using GL no scattering modeling and inversion in [1-3], in this paper, we discover a new GLHUA outer annular layer electromagnetic (EM) invisible cloak with relative parameter not less than 1. The outer layer cloak can be any positive thickness. In GLHUA outer annular layer cloak, $R_1 \le r \le R_2 $,$R_1>0$,$R_2>R_1$, $\varepsilon _r = \mu _r = \frac{{R_2 ^2 {{r^2 $, $\varepsilon _\theta = \varepsilon _\phi = \mu _\theta = \mu _\phi = \frac{{R_2 - R_1 {{r - R_1 $. We discovered an exact analytical EM wave propagation solution of the full wave Maxwell EM equation in the GLHUA outer cloak. In GLHUA outer annular layer cloak, $R_1 \le r \le R_2 $, for the electric source located outside of GLHUA outer layer cloak, $r_s > R_2 $, the exact analytic radial EM wave field solution of Maxwell radial EM differential equation is \[ \begin{array{l \left[ {\begin{array{*{20c {E_r (\vec r,\vec r_s ) \\ {H_r (\vec r,\vec r) \\ \end{array \right] = \\ = \sum\limits_{l = 1^\infty {\left[ {\begin{array{*{20c {E_{r,l,1 (r,r_s ) \\ {H_{r,l,1 (r,r_s ) \\ \end{array \right] \cos \left( {k(R_2 - R_1 )\log \left( {\frac{{r - R_1 {{R_2 - R_1 \right) \right) \\ \sum\limits_{m = - l^l { \left[ {\begin{array{*{20c {D_e (\theta ,\phi ) \\ {D_h (\theta ,\phi ) \\ \end{array \right]Y_l^m (\theta ,\phi )Y_l^{m* (\theta _s ,\phi _s ) \\ + \sum\limits_{l = 1^\infty {\left[ {\begin{array{*{20c {E_{r,l,2 (r,r_s ) \\ {H_{r,l,2 (r,r_s ) \\ \end{array \right] \sin \left( {k(R_2 - R_1 )\log \left( {\frac{{r - R_1 {{R_2 - R_1 \right) \right) \\ \sum\limits_{m = - l^l { \left[ {\begin{array{*{20c {D_e (\theta ,\phi ) \\ {D_h (\theta ,\phi ) \\ \end{array \right]Y_l^m (\theta ,\phi )Y_l^{m* (\theta _s ,\phi _s ). \\ \end{array \] In the free space outside of the cloak, $r > R_2 $,$E_r (\vec r) = E_{b,r (\vec r)$ $H_r (\vec r) = H_{b,r (\vec r)$, on the boundary $r=R_2$, $E_r (\vec r)|_{r = R_2 = E_{b,r (\vec r)|_{r = R_2 $,$H_r (\vec r)|_{r = R_2 = H_{b,r (\vec r)|_{r = R_2 $, and the derivative ${\raise0.7ex\hbox{${\partial E_r (\vec r)$ \!\mathord{\left/ {\vphantom {{\partial E_r (\vec r) {\partial r\right.\kern-\nulldelimiterspace \!\lower0.7ex\hbox{${\partial r$|_{r = R_2 = {\raise0.7ex\hbox{${\partial E_{b,r (\vec r)$ \!\mathord{\left/ {\vphantom {{\partial E_{b,r (\vec r) {\partial r\right.\kern-\nulldelimiterspace \!\lower0.7ex\hbox{${\partial r$|_{r = R_2 $, ${\raise0.7ex\hbox{${\partial H_r (\vec r)$ \!\mathord{\left/ {\vphantom {{\partial H_r (\vec r) {\partial r\right.\kern-\nulldelimiterspace \!\lower0.7ex\hbox{${\partial r$|_{r = R_2 = {\raise0.7ex\hbox{${\partial H_{b,r (\vec r)$ \!\mathord{\left/ {\vphantom {{\partial H_{b,r (\vec r) {\partial r\right.\kern-\nulldelimiterspace \!\lower0.7ex\hbox{${\partial r$|_{r = R_2 $. The exact analytical radial EM wave solution show that there is no scattering from the GLHUA outer layer cloak to disturb the incident EM wave outside of the cloak in free space. The exact analytical radial EM wave solution show that the EM wave can not propagation penetrate into the sphere $r < R_1 $.In the free space inside of the concealment, $r < R_1 $,$E_r (\vec r) = 0$ $H_r (\vec r) = 0$. Therefore, GLHUA outer layer is practicable invisible cloak with cloaked concealment sphere $r < R_1 $. We discover a novel bridge mirage wave in the GLHUA outer layer cloak, $R_1 \le r \le R_2$, which is a bridge for the EM wave propagation without exceeding light speed propagation. We propose a GLHUA expansion method to find an exact analytical EM wave propagation in the GLHUA double cloak and other spherical annular layer cloak, and completely solved invisible cloak problem in the spherical annular devices. The idea and ingredients in this paper are breakthrough progress and different from all other cloak in other research publications. Our analytical EM wave in GLHUA cloak is undisputed evidence and rigorous proof to prove that the GLHUA double layer cloak is practicable invisible cloak without exceed light speed propagation. Our analytical EM wave in GLHUA cloak is undisputed evidence and rigorous proof to make that reader to understand our GLHUA invisible cloak. Copyright and patent of the GLLH EM cloaks and GL modeling and inversion methods are reserved by authors in GL Geophysical Laboratory. We accept ArXiv's copyright license policy. % \vskip0.5mm {DOI:~10.2529/PIERSxxxxxxxxxxxx %%\hskip1cm {\small (Received July 10, 2006, Published July %31, 2006)
GLHUAanEMC7212017arxiv.tex
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\title{Exact Analytical Electromagnetic Wave Propagation In
Practicable GLHUA Electromagnetic Invisible Outer Layer Cloak}% Force line breaks with \\

\author{Jianhua Li}
 \altaffiliation[Also at ]{GL Geophysical Laboratory, USA, glhua@glgeo.com}%Lines break automatically or can be forced with \\
\author{  Feng Xie, Lee Xie, Ganquan Xie}%
 %\email{GLHUA@GLGEO.COM}
\affiliation{%
GL Geophysical Laboratory, USA
%\textbackslash\textbackslash
}%

\hfill\break
\author{Ganquan Xie}
 %\homepage{http://www.Second.institution.edu/~Charlie.Author}
\affiliation{
Chinese Dayuling Supercomputational Sciences Center, China \\
Hunan Super Computational Sciences Society, China
%This line break forced% with \\
}%

\date{June 21, 2017}% It is always \today, today,
             %  but any date may be explicitly specified

\begin{abstract}
It is totally different from the transform optics, by using GL no scattering modeling and inversion in [1-3], in this paper, we discover a new GLHUA outer annular layer electromagnetic (EM) invisible cloak with relative parameter not less than 1. The outer layer cloak can be any positive thickness. In GLHUA outer annular layer cloak, $R_1  \le r \le R_2 $,$R_1>0$,$R_2>R_1$,
$\varepsilon _r  = \mu _r  = \frac{{R_2 ^2 }}{{r^2 }}$,
$\varepsilon _\theta   = \varepsilon _\phi   = \mu _\theta   = \mu _\phi   = \frac{{R_2  - R_1 }}{{r - R_1 }}$. We discovered an exact analytical EM wave propagation solution of the full wave Maxwell EM equation in the GLHUA outer cloak. In GLHUA outer annular layer cloak, $R_1  \le r \le R_2 $, for the electric source
located outside of GLHUA outer layer cloak, $r_s  > R_2 $,  the exact analytic radial EM wave field solution of Maxwell radial EM differential equation is
\[
\begin{array}{l}
 \left[ {\begin{array}{*{20}c}
   {E_r (\vec r,\vec r_s )}  \\
   {H_r (\vec r,\vec r)}  \\
\end{array}} \right] =  \\ 
  = \sum\limits_{l = 1}^\infty  {\left[ {\begin{array}{*{20}c}
   {E_{r,l,1} (r,r_s )}  \\
   {H_{r,l,1} (r,r_s )}  \\
\end{array}} \right]} \cos \left( {k(R_2  - R_1 )\log \left( {\frac{{r - R_1 }}{{R_2  - R_1 }}} \right)} \right) \\ 
 \sum\limits_{m =  - l}^l {} \left[ {\begin{array}{*{20}c}
   {D_e (\theta ,\phi )}  \\
   {D_h (\theta ,\phi )}  \\
\end{array}} \right]Y_l^m (\theta ,\phi )Y_l^{m*} (\theta _s ,\phi _s ) \\ 
  + \sum\limits_{l = 1}^\infty  {\left[ {\begin{array}{*{20}c}
   {E_{r,l,2} (r,r_s )}  \\
   {H_{r,l,2} (r,r_s )}  \\
\end{array}} \right]} \sin \left( {k(R_2  - R_1 )\log \left( {\frac{{r - R_1 }}{{R_2  - R_1 }}} \right)} \right) \\ 
 \sum\limits_{m =  - l}^l {} \left[ {\begin{array}{*{20}c}
   {D_e (\theta ,\phi )}  \\
   {D_h (\theta ,\phi )}  \\
\end{array}} \right]Y_l^m (\theta ,\phi )Y_l^{m*} (\theta _s ,\phi _s ). \\ 
 \end{array}
\]
In the free space outside of the cloak, $r > R_2 $,$E_r (\vec r) = E_{b,r} (\vec r)$ $H_r (\vec r) = H_{b,r} (\vec r)$, on the boundary $r=R_2$, $E_r (\vec r)|_{r = R_2 }  = E_{b,r} (\vec r)|_{r = R_2 } $,$H_r (\vec r)|_{r = R_2 }  = H_{b,r} (\vec r)|_{r = R_2 } $, and the derivative
${\raise0.7ex\hbox{${\partial E_r (\vec r)}$} \!\mathord{\left/
 {\vphantom {{\partial E_r (\vec r)} {\partial r}}}\right.\kern-\nulldelimiterspace}
\!\lower0.7ex\hbox{${\partial r}$}}|_{r = R_2 }  = {\raise0.7ex\hbox{${\partial E_{b,r} (\vec r)}$} \!\mathord{\left/
 {\vphantom {{\partial E_{b,r} (\vec r)} {\partial r}}}\right.\kern-\nulldelimiterspace}
\!\lower0.7ex\hbox{${\partial r}$}}|_{r = R_2 } $,
${\raise0.7ex\hbox{${\partial H_r (\vec r)}$} \!\mathord{\left/
 {\vphantom {{\partial H_r (\vec r)} {\partial r}}}\right.\kern-\nulldelimiterspace}
\!\lower0.7ex\hbox{${\partial r}$}}|_{r = R_2 }  = {\raise0.7ex\hbox{${\partial H_{b,r} (\vec r)}$} \!\mathord{\left/
 {\vphantom {{\partial H_{b,r} (\vec r)} {\partial r}}}\right.\kern-\nulldelimiterspace}
\!\lower0.7ex\hbox{${\partial r}$}}|_{r = R_2 } $. The exact analytical radial EM wave solution show that there is no scattering from the GLHUA outer layer cloak to disturb the incident EM wave outside of the cloak in free space. The exact analytical radial EM wave solution show that the EM wave can not propagation penetrate into the sphere $r < R_1 $.In the free space inside of the concealment, $r < R_1 $,$E_r (\vec r) = 0$ $H_r (\vec r) = 0$. Therefore, GLHUA outer layer is practicable invisible cloak with cloaked concealment sphere $r < R_1 $. We discover a novel bridge mirage wave in the GLHUA outer layer cloak, $R_1  \le r \le R_2$, which is a bridge for the EM wave propagation without exceeding light speed propagation. We propose a GLHUA expansion method to find an exact analytical EM wave propagation in the GLHUA double cloak and other spherical annular layer cloak, and completely solved invisible cloak problem in the spherical annular devices.
The idea and ingredients in this paper are breakthrough progress and different from all other cloak in other research publications. Our analytical EM wave in GLHUA cloak is undisputed evidence and rigorous proof to prove that the GLHUA double layer cloak is practicable invisible cloak without exceed light speed propagation. Our analytical EM wave in GLHUA cloak is undisputed evidence and rigorous proof to make that reader to understand our GLHUA invisible cloak.
Copyright and patent of the GLLH EM cloaks and GL modeling
and inversion methods are reserved by authors in GL Geophysical Laboratory.
We accept ArXiv's copyright license policy.
% \vskip0.5mm {\bf DOI}:~10.2529/PIERSxxxxxxxxxxxx
%%\hskip1cm {\small \it (Received July 10, 2006, Published July %31, 2006)}
\end{abstract}

\pacs{13.40.-f, 41.20.-q, 41.20.jb,42.25.Bs}% PACS, the Physics and Astronomy
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\maketitle

\section{\label{sec:level1}INTRODUCTION} 
%First-level heading:\protect\\ 
%The line break was forced \lowercase{via}
%\textbackslash\textbackslash
By our GLHUA double layer cloak and GL no scattering modeling and inversion in our paper in arxiv[1-3][6][8], in this paper, we discover a new GLHUA outer annular layer EM invisible cloak with relative parameter not less than 1, the outer layer can be any thickness. In the outer annular layer cloak, $R_1  \le r \le R_2 $, the relative parameters,
$\varepsilon _r  = \mu _r  = \frac{{R_2 ^2 }}{{r^2 }}$.We propose a GLHUA expansion method to find an exact analytical EM wave propagation solution of Maxwell EM equation in the GLHUA double cloak. The analytical EM wave propagation in GLHUA double layer cloak is an exact analytical solution of the Maxwell EM wave equation in 3D sphere annular layer domain with GLHUA relative parameter materials. For simplicity, we call ��the analytical EM wave solution of 3D EM equation in GLHUA double layer cloak domain�� as ��exact analytical GLHUA EM wave��.  The exact analytical GLHUA EM wave is undisputed evidence and rigorous proof to prove GLHUA double cloak is the invisible cloak with relative parameter not less than 1 and without exceeding light speed propagation; The exact analytical GLHUA EM wave is undisputed evidence and rigorous proof to eliminate all criticism and suspicion on our GLHUA cloak, to make than reader understanding GLHUA cloak is right.
The exact analytical GLHUA EM wave is undisputed evidence and rigorous proof to prove GLHUA double cloak is new totally different invisible cloak and breakthrough progress from Pendry��s transform optics cloak [7] and all other cloak with infinite speed and exceeding light speed in other research publications.  The exact analytical GLHUA EM wave prove that the conventional Fermat principle and conventional approximate ray tracing method can not be used to study and explain GLHUA double layer cloak. The exact analytical GLHUA EM wave prove that the Dark box commercial software COMSOL can not be used to simulate 3D full wave propagation in GLHUA cloak. The exact analytical GLHUA EM wave prove that using above conventional method and wrong Dark box COMSOL to reject GLHUA double layer cloak that is wrong. The exact analytical GLHUA EM wave prove that Gunther Uhlmann is wrong and unreasonable to block our paper publication and making us suffered extremely life and research difficulties in our GLHUA cloak research in 17 years.  
   In 2000, in our Global Integral and Local Differential GILD EM modeling and inversion simulation [4], we observed a strange double layered cloak phenomena in Fig.2 in [5] and in Fig.11 of [5]. The double layered cloak phenomena was called ��GILD phenomena�� has been published in SEG Expanded Abstracts, Vol. 21 , No. 1, 692-695, 2002, by Jianhua Li at al. [5]. The double layer cloak ��GILD phenomena�� is nonzero variation solution imaging of the zero scattering inversion which cloaked the object of the scattering inversion imager. Since then, we created the Global and Local modeling and inversion [6] in 2006, and used it to create our novel GLLH [11] and GLHUA [1-3] invisible EM cloak. A ��New computational mirage ��has been published in PIERS abstract 2005 in Hang Zhou of China, 296[13].  In 2006, using transform optics, Pendry et al proposed first invisible cloak [7] with infinite speed and exceeding light speed propagation. Ulf Leonhardt et al. [27] proposed a cloak with finite speed based on a Euclid and non Euclid joint transform optics. However, ULF cloak [27] still has exceeding light speed difficulty and without theoretical full wave proof and without full wave simulation verification. In 2009, we published paper in arXiv[10], ��A Double Layer Electromagnetic Cloak And GL EM Modeling��, in this paper, the inner layer cloak with relative refractive index not less than 1 was proposed first in the world[10]. In 2010, first in the world, we published  ��GLLH EM Invisible Cloak with Novel Front Branching and without Exceed Light Speed Violation�� in ArXiv1005.3999 in the May of 2010 [11] and in PIERS 2010[12]. Next year, in 2011, Ulf Leonhardt et al. published paper ��Invisibility cloaking without superluminal propagation�� in Arxiv: 1105.0164v3 [14]. Ulf did write a complete review and references of cloak history in his paper[14], he did criticism for many papers about Pendry��s cloak and all published cloak papers and wrote that ��The fundamental problem is that perfect invisibility (Pendry cloak[7] )��requires that light should propagate in certain cloaking regions with a superluminal phase velocity that tends to infinity.�� In his paper [14] in 2011, Ulf cited our paper ArXiv:1005.3999 in 2010 as his reference [35] and wrote that ��The preprint [35], proposes a different method for cloaking without superluminal propagation.�� However, ULF and his colleagues did not really understand our GLLH cloak and our GL no scattering modeling and inversion. In ULF��s same paper in other Journal, ULF did not cite our paper in his reference. Up to now, there is no full wave theoretical analysis and no full wave computational simulation to verify ULF��s cloak. Pendry and ULF only used transform optics to create their cloak. Pendry and ULF and all other cloak researcher never used ��inversion�� for their cloaking research paper. In the world, from 2000 to now long cloaking research history, only we created GILD [4] and GL no scattering modeling and inversion method [4][5][6][8] and used it to discover double cloak phenomena in 2000, create GL double layer cloak and inner layer cloak without exceeding light speed in 2009[10], GLLH double with relative refractive index not less than 1 and without exceeding light speed in 2010 [11] first in the world, and practicable GLHUA double layer invisible cloak with relative EM parameter not less than 1 in 2016[1-3], and create an exact analytical EM wave in the GLHUA double layer cloak in this paper in 2017.

To find double layer invisible cloak with relative parameter not less than 1 is a scientific mythology and the dream of scientists. After 17 years of creation and effort, we created GLHUA double cloak and realized this dream. The new GLHUA double cloak and the analytical GLHUA EM wave propagation exactly and rigorouly proved that our GLHUA cloak class in [1-3] and GLLH cloak in [8] and GL double layer cloak in [9] are invisible cloak. Our GLHUA double layer cloak is a breakthrough scientific achievement of United America sciences, Chinese sciences and world��s sciences. 
 
We discovered GLHUA double cloak [1-3] and propose novel GLHUA expansion method to find analytic EM wave in the new GLHUA double layer cloak in section 3 of this paper, we detailed propose how do we find the analytic GLHUA EM wave in GLHUA outer layer cloak. On the outer spherical boundary of the GLHUA outer layer cloak, $r = R_2$, The analytical GLHUA EM wave and its first derivative are continuous connection with the incident EM wave and their derivative from source in outside of the cloak, $r_s  > R_2 $ that making there is no scattering from GLHUA cloak to disturbed the incident EM wave in outside of cloak. The analytical GLHUA EM wave show that in the any finite time the EM wave in GLHUA outer layer cloak can not be arrive to the inner spherical boundary,$r = R_1 $, that making the EM wave can not penetrate into the sphere $r < R_1 $. Therefore, the analytical GLHUA EM wave proved that GLHUA outer layer cloak is invisible cloak with sphere concealment $r < R_1 $. The analytical GLHUA EM wave in GLHUA inner layer cloak is proposed in the section 4. On the inner spherical boundary of the GLHUA inner layer cloak, $r = R_0$, the analytical GLHUA EM wave and its first derivative are continuous with the incident EM wave and their derivative wave excited in the concealment $r < R_0 $ that making there is no scattering from GLHUA cloak to disturbed the incident EM wave in the concealment $r < R_0$. The analytical GLHUA EM wave show that in the any finite time the EM wave in GLHUA inner layer cloak can not be arrive to the spherical boundary,$r = R_1 $, that making the EM wave can not propagate to outside of the boundary $r = R_1 $ Using the GLHUA expansion method, we proved existence of the analytical EM wave solution of GLLH cloak[8] and GL cloak [9]. Using the GLHUA expansion method, we find a mirage wave in the concealment of single layer cloak, if a special material is fill in the concealment.
\section {Practicable GLHUA outer layer cloakwith relative parameter not less than 1}
In this section, we created a novel practicable GLHUA outer annular layer EM invisible cloak with relative parameter not less than 1.

In the outer annular layer $R_1  \le r \le R_2 $ the radial relative electric permittivity and magnetic permeability are
\begin{equation}
\varepsilon _r (r) = \mu _r (r) = \frac{{R_2 ^2 }}{{r^2 }},   
\end{equation}
They are monotone decease function and
\begin{equation} 
 \frac{{R_2 ^2 }}{{R_1 ^2 }} \ge \varepsilon _r (r) = \mu _r (r) = \frac{{R_2 ^2 }}{{r^2 }} \ge 1,
\end{equation}
the angular relative electric permittivity and magnetic permeability are
\begin{equation} 
\varepsilon _\phi  (r) = \mu _\theta  (r) = \mu _\phi  (r) = \frac{{R_2  - R_1 }}{{r - R_1 }},
\end{equation} 
They are monotone decease function and 
\begin{equation} 
  \begin{array}{l}
 \infty  \ge \varepsilon _\theta  (r) = \varepsilon _\phi  (r) = \mu _\theta  (r) = \mu _\phi  (r) \\ 
  = \frac{{R_2  - R_1 }}{{r - R_1 }} \ge \frac{{R_2  - R_1 }}{{R_2  - R_1 }} = 1. \\ 
 \end{array}
\end{equation} 
From (1) and (3), on the boundary $r = R_2 $,The relative parameters in GLHUA outer layer cloak are equal to 1 and are continuous across the outer boundary $r = R_2 $. Summary, we have the following theorem 1.
\hfill\break \\

${\boldsymbol{Theorem \ 1:}}$ \  In GLHUA outer layer cloak, $R_1  \le r \le R_2 $ the radial relative electric permittivity and magnetic permeability in (1) and the angular relative electric permittivity and magnetic permeability in (3) are not less than 1 and are continuous across the outer boundary $r = R_2 $.
\\
\section {  Exact analytical EM wave propagation in the GLHUA outer layer cloak}

In this section, we create an analytical EM wave field propagation in the GLHUA outer cloak with relative parameters in (1) and (3). For simplicity, we create a new analytical expansion of the solution of the magnetic wave equation in the GLHUA outer layer cloak domain $R_1  \le r \le R_2 $. 

\subsection { The spherical harmonic expansion of incident radial GL magnetic wave field}
 
Suppose that the incident radial magnetic wave field ${H^b}_r$,

\begin{equation}
H^b {}_r = \cos \phi \frac{1}{r}\frac{{\partial g}}{{\partial \theta }} - \frac{{\cos \theta \sin \phi }}{{r\sin \theta }}\frac{{\partial g}}{{\partial \phi }} 
\end{equation}
is incident magnetic wave for GLHUA outer layer cloak, where    
\begin{equation}                  
g(\vec r,\vec r_s ) =  - \frac{1}{{4\pi }}\frac{{e^{  ik\left| {\vec r - \vec r_s } \right|} }}{{\left| {\vec r - \vec r_s } \right|}},
\end{equation}
$H^b_r$ is excited by the electric point source
\begin{equation}
\vec J_s  = \frac{1}{{r^2 \sin \theta }}\delta (r - r_s )\delta (\theta  - \theta _s )\delta (\phi  - \phi _s )\vec e_y ,
\end{equation}
located in the point $\vec r_s  = (r_s ,\theta _s ,\phi _s )$,$r_s  > R_{out}  > R_2 $. $H^b_r$ is the solution of the
magnetic wave equation
\begin{equation}
\begin{array}{l}
 \frac{\partial }{{\partial r}}\frac{1}{{\mu _\theta  }}\frac{{\partial \left( {r^2 \mu _r H_r } \right)}}{{\partial r}} + \frac{1}{{\sin \theta }}\frac{\partial }{{\partial \theta }}\sin \theta \frac{{\partial H_r }}{{\partial \theta }} \\ 
  + \frac{1}{{\sin ^2 \theta }}\frac{{\partial H_r }}{{\partial \phi ^2 }} + k^2 r^2 \varepsilon _\theta \mu _r  H_r  = 0, \\ 
 \end{array}
\end{equation}

in the annular layer domain $R_2  < r < R_{out} $. in free space, with relative parameters are equal to 1, $\mu _\theta   = \varepsilon _\theta   = 1$, $\mu _\phi   = \varepsilon _\phi   = 1$ $\mu _r  = \varepsilon _r  = 1$.

Define Global and Local - GL magnetic wave
       \begin{equation}      
      H(\vec r) = r^2 \mu _r H_r (\vec r),                               
       \end{equation}
The magnetic wave equation (8) is translated to the following GL magnetic wave equation

       \begin{equation}     
\begin{array}{l}
 \frac{\partial }{{\partial r}}\frac{1}{{\mu _\theta  }}\frac{{\partial H}}{{\partial r}} + \frac{1}{{r^2 \mu _r }}\frac{1}{{\sin \theta }}\frac{\partial }{{\partial \theta }}\sin \theta \frac{{\partial H}}{{\partial \theta }} \\ 
  + \frac{1}{{r^2 \mu _r }}\frac{1}{{\sin ^2 \theta }}\frac{{\partial H}}{{\partial \phi ^2 }} + k^2 \varepsilon _\theta  H = 0, \\ 
 \end{array}
       \end{equation}     
Using the GL electric wave field $E(\vec r) = r^2 \varepsilon _r E_r (\vec r),$ to replace GL magnetic wave field $H(\vec r) = r^2 \mu _r H_r (\vec r)$ in (10), the equation (10) become the GL electric wave equation.
The GL EM wave propagation in free space satisfies the equation (10) with relative parameter equal to 1; GL EM wave propagation in GLHUA outer layer cloak satisfies the equation (10) with GLHUA relative parameters not less than 1 in (1) and (3).

Define the operator 
\begin{equation}
\Re (\theta ,\phi ) = \frac{1}{{\sin \theta }}\frac{\partial }{{\partial \theta }}\sin \theta \frac{\partial }{{\partial \theta }} + \frac{1}{{\sin ^2 \theta }}\frac{\partial }{{\partial \phi ^2 }}
\end{equation}
\begin{equation}
D_h (\theta ,\phi ) = \cos \phi \frac{\partial }{{\partial \theta }} - \frac{{\cos \theta \sin \phi }}{{\sin \theta }}\frac{\partial }{{\partial \phi }}
\end{equation}
By direct calculation, we can prove
\begin{equation}
\Re (\theta ,\phi )D_h (\theta ,\phi ) = D_h (\theta ,\phi )\Re (\theta ,\phi ),
\end{equation}

The spherical harmonic expansion of the incident GL magnetic wave
$H^{(b)} (\vec r) = r^2 H^b {}_r(\vec r)$ is
\begin{equation}
\begin{array}{l}
 H^{(b)} (r,\theta ,\phi ) = r^2 H^{(b)} _r (r,\theta ,\phi ) \\ 
  =  - ik\sum\limits_{l = 1}^\infty  {rj_l (kr)} h^{(1)} (kr_s )\sum\limits_{m =  - l}^l {} D_h (\theta ,\phi )Y_l^m (\theta ,\phi )Y_l^{m*} (\theta _s ,\phi _s ) \\ 
  = \sum\limits_{l = 1}^\infty  {} H^{(b)} _l (r)\sum\limits_{m =  - l}^l {} D_h (\theta ,\phi )Y_l^m (\theta ,\phi )Y_l^{m*} (\theta _s ,\phi _s ), \\ 
 \end{array}
\end{equation}
Because the electric point source $(r_s ,\theta _s ,\phi _s )$ in outside of the GLHUA cloak, $r_s  > R_{out}  > R_2 $ and operator $D_h (\theta ,\phi )$ and $\Re (\theta ,\phi )$ can be commutative (13),
substitute (14) into the GL magnetic equation (10) in the sphere annular domain  ,without source, the equation (10) is split into the second ordinary differential equation for every $l$, $l = 1,2, \cdots$, as follows,
\begin{equation}
\frac{\partial }{{\partial r}}\frac{{\partial H^{(b)} _l (r)}}{{\partial r}} - \frac{{l(l + 1)}}{{r^2 }}H^{(b)} _l (r) + k^2 H^{(b)} _l (r) = 0,
\end{equation}

\subsection{Analytical EM Wave propagation in the GLHUA outer layer cloak}

We propose GLHUA expansion method to create analytic expansion of
the magnetic wave field in GLHUA outer layer cloak which satisfies the magnetic equation (10) in the outer layer domain with relative parameters (1) and (3). The GLHUA analytic magnetic wave field and incident magnetic wave and their derivative satisfy the boundary continuous condition across the boundary $r = R_2 $.

Substitute the spherical harmonic expansion of the magnetic wave field
   \begin{equation}    
 \begin{array}{l}
 H(r,\theta ,\phi ) =  \\ 
  = \sum\limits_{l = 1}^\infty  {H_l (r)} \sum\limits_{m =  - l}^l {} D_h (\theta ,\phi )Y_l^m (\theta ,\phi )Y_l^{m*} (\theta _s ,\phi _s ), \\ 
 \end{array}
\end{equation}
into the GL magnetic equation (10) in the GLHUA outer annular layer domain $R_1  \le r \le R_2 $, the equation (10) is split into the second ordinary differential equation for every $l$, $l = 1,2, \cdots$,  
\begin {equation}
\frac{\partial }{{\partial r}}\frac{1}{{\varepsilon _\theta  }}\frac{{\partial H_l (r)}}{{\partial r}} - \frac{{l(l + 1)}}{{\mu _r r^2 }}H_l (r) + k^2 \varepsilon _\theta  H_l (r) = 0,
\end{equation}
By comparison between magnetic equation (17) in GLHUA outer layer cloak and 
incident magnetic wave equation (15), the $l$ component magnetic field $H_l (r)$ and $l$ component incident magnetic field $H^{(b)} _l (r)$ are continuous across boundary $r = R_2 $,
\begin{equation}
H_l (R_2 ) = H^{(b)} _l (R_2 ),
\end{equation}
and their radial derivative are continuous across boundary $r = R_2 $,

\begin{equation}
\frac{\partial }{{\partial r}}H_l (R_2 ) = \frac{\partial }{{\partial r}}H^{(b)} _l (R_2 ),
\end{equation}
Substitute GLHUA cloak relative materials $\varepsilon _r (r) = \mu _r (r) = \frac{{R_2 ^2 }}{{r^2 }}$  in (1) and relative angular parameter in (3) into the equation (10), we have
\begin{equation}
\begin{array}{l}
\frac{\partial }{{\partial r}}\frac{{r - R_1 }}{{R_2  - R_1 }}\frac{{\partial H_l (r)}}{{\partial r}} \\
 - \frac{{l(l + 1)}}{{R_1 ^2 }}H_l (r) + k^2 \frac{{R_2  - R_1 }}{{r - R_1 }}H_l (r) = 0,
\end{array}
\end{equation}
We create GLHUA expansion of the $l$ component GL magnetic field
$H_l (r)$

\begin{equation}
\begin{array}{l}
 H_l (r) = g_{l,1} (r)\cos (k(R_2  - R_1 )\log (\frac{{r - R_1 }}{{R_2  - R_1 }})) \\ 
  + g_{l,2} (r)\sin (k(R_2  - R_1 )\log (\frac{{r - R_1 }}{{R_2  - R_1 }})), \\ 
 \end{array}
\end{equation}

By calculation,

\begin{equation}
\begin{array}{l}
 \frac{\partial }{{\partial r}}\frac{{r - R_1 }}{{R_2  - R_1 }}\frac{\partial }{{\partial r}}H_l (r) \\ 
  = \left( {\frac{\partial }{{\partial r}}\frac{{r - R_1 }}{{R_2  - R_1 }}\frac{\partial }{{\partial r}}g_{l,1} (r) + 2k\frac{\partial }{{\partial r}}g_{l,2} (r)} \right) \\ 
 \cos (k(R_2  - R_1 )\log (\frac{{r - R_1 }}{{R_2  - R_1 }})) \\ 
  + \left( {\frac{\partial }{{\partial r}}\frac{{r - R_1 }}{{R_2  - R_1 }}\frac{\partial }{{\partial r}}g_{l,2} (r) - 2k\frac{\partial }{{\partial r}}g_{l,1} (r)} \right) \\ 
 \sin (k(R_2  - R_1 )\log (\frac{{r - R_1 }}{{R_2  - R_1 }})) \\ 
  - k^2 g_{l,1} (r)\cos (k(R_2  - R_1 )\log (\frac{{r - R_1 }}{{R_2  - R_1 }}))\frac{{R_2  - R_1 }}{{r - R_1 }} \\ 
  - k^2 g_{l,2} (r)\sin (k(R_2  - R_1 )\log (\frac{{r - R_1 }}{{R_2  - R_1 }}))\frac{{R_2  - R_1 }}{{r - R_1 }}, \\ 
 \end{array}
\end{equation}


Substitute (21) and (22) into the equation (20), we obtain a new GLHUA two equation system

\begin{equation}
\begin{array}{l}
\frac{\partial }{{\partial r}}\frac{{r - R_1 }}{{R_2  - R_1 }}\frac{\partial }{{\partial r}}g_{l,1} (r) \\
+ 2k\frac{\partial }{{\partial r}}g_{l,2} (r) + \frac{{l(l + 1)}}{{R_2 ^2 }}g_{l,1} (r) = 0,
\end{array}
\end{equation}

\begin{equation}
\begin{array}{l}
\frac{\partial }{{\partial r}}\frac{{r - R_1 }}{{R_2  - R_1 }}\frac{\partial }{{\partial r}}g_{l,2} (r) \\
- 2k\frac{\partial }{{\partial r}}g_{l,1} (r) + \frac{{l(l + 1)}}{{R_2 ^2 }}g_{l,2} (r) = 0,
\end{array}
\end{equation}


We expand solution $(g_{l,1} (r),g_{l,2} (r))$ of the GLHUA two equation system (23) and (24) as analytical power series,
\begin{equation}
 g_{l,1} (r) = \sum\limits_{j = 0}^\infty  {a_{l,j} } (r - R_1 )^j ,
\end{equation}
\begin{equation}
g_{l,2} (r) = \sum\limits_{j = 0}^\infty  {b_{l,j} } (r - R_1 )^j ,
\end{equation}

By calculation,
\begin{equation}
\begin{array}{l}
 \frac{\partial }{{\partial r}}g_{l,1} (r) = \sum\limits_{j = 1}^\infty  {a_{l,j} } j(r - R_1 )^{j - 1}  \\ 
  = \sum\limits_{j = 0}^\infty  {a_{l,j + 1} } (j + 1)(r - R_1 )^j , \\ 
 \end{array}
\end{equation}

\begin{equation}
\begin{array}{l}
\frac{\partial }{{\partial r}}g_{l,2} (r) = \sum\limits_{j = 1}^\infty  {b_{l,j} } j(r - R_1 )^{j - 1}  \\ 
= \sum\limits_{j = 0}^\infty  {b_{l,j + 1} } (j + 1)(r - R_1 )^j ,
 \end{array}
\end{equation} 

\begin{equation} 
 \begin{array}{l}
 (r - R_1 )\frac{\partial }{{\partial r}}\frac{\partial }{{\partial r}}g_{l,1} (r) \\
= \sum\limits_{j = 2}^\infty  {a_{l,j} } j(j - 1)(r - R_1 )^{j - 1}  \\ 
  = \sum\limits_{j = 1}^\infty  {a_{l,j + 1} } (j + 1)j(r - R_1 )^j , \\ 
 \end{array}
\end{equation} 

\begin{equation} 
\begin{array}{l}
 (r - R_1 )\frac{\partial }{{\partial r}}\frac{\partial }{{\partial r}}g_{l,2} (r) \\
= \sum\limits_{j = 2}^\infty  {b_{l,j} } j(j - 1)(r - R_1 )^{j - 1}  \\ 
  = \sum\limits_{j = 1}^\infty  {b_{l,j + 1} } (j + 1)j(r - R_1 )^j ,
 \end{array}
\end{equation} 


Substitute (25-30) into GLHUA equation (23), we have
\begin{equation} 
\begin{array}{l}
 \frac{1}{{R_2  - R_1 }}\sum\limits_{j = 1}^\infty  {a_{l,j + 1} (j + 1)} j(r - R_1 )^j  \\ 
  + \frac{1}{{R_2  - R_1 }}\sum\limits_{j = 0}^\infty  {a_{l,j + 1} (j + 1)} (r - R_1 )^j  \\ 
  + 2k\sum\limits_{j = 0}^\infty  {b_{l,j + 1} (j + 1)} (r - R_1 )^j  \\ 
  - \frac{{l(l + 1)}}{{R_2 ^2 }}\sum\limits_{j = 0}^\infty  {a_{l,j} } (r - R_1 )^j = 0 \\ 
 \end{array}
\end{equation} 
Substitute (25-30) into GLHUA equation (24), we have

\begin{equation} 
\begin{array}{l}
 \frac{1}{{R_2  - R_1 }}\sum\limits_{j = 1}^\infty  {b_{l,j + 1} (j + 1)} j(r - R_1 )^j  \\ 
  + \frac{1}{{R_2  - R_1 }}\sum\limits_{j = 0}^\infty  {b_{l,j + 1} (j + 1)} (r - R_1 )^j  \\ 
  - 2k\sum\limits_{j = 0}^\infty  {a_{l,j + 1} (j + 1)} (r - R_1 )^j  \\ 
  - \frac{{l(l + 1)}}{{R_2 ^2 }}\sum\limits_{j = 0}^\infty  {b_{l,j} } (r - R_1 )^j  = 0, \\ 
 \end{array}
\end{equation} 

When  $j=0$ in (31), we have 
\begin{equation}
  \frac{1}{{R_2  - R_1 }}a_{l,1}  + 2kb_{l,1}  = \frac{{l(l + 1)}}{{R_2 ^2 }}a_{l,0} ,      
\end{equation}


When  $j=0$ in (32), we have

\begin{equation}
\frac{1}{{R_2  - R_1 }}b_{l,1}  - 2ka_{l,1}  = \frac{{l(l + 1)}}{{R_2 ^2 }}b_{l,0} ,
\end{equation} 

Solve the linear equation (33) and (34), we obtain
\begin{equation}
\begin{array}{l}
 \left[ {\begin{array}{*{20}c}
   {a_{l,1} }  \\
   {b_{l,1} }  \\
\end{array}} \right] = \frac{{l(l + 1)}}{{(1 + 4k(R_2  - R_1 )^2 )R_2^2 }} \\
\left[ {\begin{array}{*{20}c}
   {(R_2  - R_1 )} & { - 2k(R_2  - R_1 )^2 }  \\
   {2k(R_2  - R_1 )^2 } & {(R_2  - R_1 )}  \\
\end{array}} \right]\left[ {\begin{array}{*{20}c}
   {a_{l,0} }  \\
   {b_{l,0} }  \\
\end{array}} \right] \\ 
 \end{array}
\end{equation}
For every $j \ge 1$, in (31) and (32), we have
\begin{equation}
\begin{array}{l}
 \frac{1}{{R_2  - R_1 }}(j + 1)^2 a_{l,j + 1}  + 2k(j + 1)b_{l,j + 1}  \\ 
  = \frac{{l(l + 1)}}{{R_2 ^2 }}a_{l,j} , 
 \end{array}
\end{equation}
\begin{equation}
\begin{array}{l}
 \frac{1}{{R_2  - R_1 }}(j + 1)^2 b_{l,j + 1}  - 2k(j + 1)a_{l,j + 1}  \\ 
  = \frac{{l(l + 1)}}{{R_2 ^2 }}b_{l,j} , 
 \end{array}
\end{equation}

Solve the linear equation (36) and (37), we obtain

\begin{equation}
\begin{array}{l}
 \left[ {\begin{array}{*{20}c}
   {a_{l,j + 1} }  \\
   {b_{l,j + 1} }  \\
\end{array}} \right] = \frac{{l(l + 1)}}{{((j + 1)^3  + 4k(R_2  - R_1 )^2 (j + 1))R_2^2 }} \\ 
 \left[ {\begin{array}{*{20}c}
   {(j + 1)(R_2  - R_1 )} & { - 2k(R_2  - R_1 )^2 }  \\
   {2k(R_2  - R_1 )^2 } & {(j + 1)(R_2  - R_1 )}  \\
\end{array}} \right]\left[ {\begin{array}{*{20}c}
   {a_{l,j} }  \\
   {b_{l,j} }  \\
\end{array}} \right], \\ 
 \end{array}
\end{equation} 
There are two linearly independent vector function solution of the GLHUA two equations system (23) and (24), $(g_{l,1,p} (r),g_{l,2,p} (r)),p = 1,2,$ When  $a_{l,0}^{(1)}  = 1,b_{l,0}^{(1)}  = 0$ from (35) and (38), we obtain  
$(a_{l,j}^{(1)} ,b_{l,j}^{(1)} ),j = 1,2, \cdots$ and from analytical power series (25) and (26), we obtain the first solution of GLHUA quations (29) and (30),$(g_{l,1,1} (r),g_{l,2,1} (r))$  . When 
$a_{l,0}^{(2)}  = 0,b_{l,0}^{(2)}  = 1$ , from (35) and (38), we obtain 
$(a_{l,j}^{(2)} ,b_{l,j}^{(2)} ),j = 1,2, \cdots$, and from analytical power series (25) and (26), we obtain the second solution of GLHUA equations (23) and (24).$(g_{l,1,2} (r),g_{l,2,2} (r))$  ,for $l = 1,2, \cdots $ .


 For arbitrary two constants $A_1$   and  $A_2$, 
 the general solution of the GLHUA equation (23) and (24) is

\begin{equation} 
H_{l,1} (r) = A_1 g_{l,1,1} (r) + A_2 g_{l,1,2} (r),
\end{equation} 

\begin{equation} 
H_{l,2} (r) = A_1 g_{l,2,1} (r) + A_2 g_{l,2,2} (r),
\end{equation} 

Substitute (39) and (40) into (21), we obtain

\begin{equation} 
\begin{array}{l}
 H_l (r) \\ 
  = H_{l,1} (r)\cos \left( {k(R_2  - R_1 )\log \left( {\frac{{r - R_1 }}{{R_2  - R_1 }}} \right)} \right) \\ 
  + H_{l,2} (r)\sin \left( {k(R_2  - R_1 )\log \left( {\frac{{r - R_1 }}{{R_2  - R_1 }}} \right)} \right) \\ 
  = (A_1 g_{l,1,1} (r) + A_2 g_{l,1,2} (r)) \\ 
 \cos \left( {k(R_2  - R_1 )\log \left( {\frac{{r - R_1 }}{{R_2  - R_1 }}} \right)} \right) \\ 
  + (A_1 g_{l,2,1} (r) + A_2 g_{l,2,2} (r)) \\ 
 \sin \left( {k(R_2  - R_1 )\log \left( {\frac{{r - R_1 }}{{R_2  - R_1 }}} \right)} \right). \\ 
\end{array}    
\end{equation} 


From the continuous condition of  $H_l (r)$  and $H^{(b)}_l(r)$  on  $r = R_2$  (18) and their derivative continuous condition (19), we determinate the constants $A_1$   and  $A_2$ in (41). Substitute (41) into (16), we obtained the GL analytic magnetic wave $H(\vec r)$, in the GLHUA outer layer with relative parameter (1) and (3) not less than 1. 
Summary, we have proved the following theorem.
\hfill\break \\
\begin{figure}[b]
%\begin{m$$inipage}[t]{0.3\linewidth}
\centering
\includegraphics[width=0.86\linewidth,draft=false]{10-9.eps}
\caption{ (color online) 
the magnetic wave is being tangent to the outer boundary $r=R_2$ without any scattering. It is very strangle, there is a down crescent type GL wave in the upper part of the inner spherical surface, $r=R_1$ , and upward crescent type GL wave in the lower part of the same inner spherical surface, $r=R_1$. 
.}\label{fig1}
\end{figure}
\begin{figure}[b]
%\begin{minipage}[t]{0.3\linewidth}
\centering
\includegraphics[width=0.86\linewidth,draft=false]{10-31.eps}
\caption{ (color online) 
Magnetic wave propagation through GLHUA 
outer layer cloak, wave front at 31 step.}\label{fig2}
\end{figure}
\begin{figure}[b]
%\hskip5mm
%\begin{minipage}[t]{0.3\linewidth}
\centering
\includegraphics[width=0.86\linewidth,draft=false]{10-37.eps}
\caption{ (color online)
Magnetic wave propagation through GLHUA 
outer layer cloak, wave front at 37 step. }\label{fig3}
%\end{minipage}
\end{figure}
\begin{figure}[b]
%\hskip5mm
%\begin{minipage}[t]{0.3\linewidth}
\centering
\includegraphics[width=0.86\linewidth,draft=false]{10-43.eps}
\caption{ (color online)
Magnetic wave propagation through GLHUA outer layer cloak, wave front at 43 step. 
.}\label{fig4}
\end{figure}
\begin{figure}[h]
%\begin{minipage}[t]{0.3\linewidth}
\centering
\includegraphics[width=0.86\linewidth,draft=false]{10-49.eps}
\caption{ (color online) 
Magnetic wave propagation through GLHUA outer layer cloak, wave front at 49 step.}\label{fig5}
%\end{minipage}
\end{figure}
\begin{figure}[h]
%\hskip5mm
%\begin{minipage}[t]{0.3\linewidth}
\centering
\includegraphics[width=0.85\linewidth,draft=false]{10-55.eps}
\caption{ (color online) 
Magnetic wave propagation through GLHUA outer layer cloak, wave front at 55 step.}\label{fig6}
\end{figure}
${\boldsymbol{Theorem \ 2:}}$ \  In the outer annular layer cloak, $R_1  \le r \le R_2 $, the radial relative electric permittivity and magnetic permeability, 
$\varepsilon _r  = \mu _r  = \frac{{R_2 ^2 }}{{r^2 }}$ in (1), and the angular relative electric permittivity and magnetic permeability, 
$\varepsilon _\theta   = \varepsilon _\phi   = \mu _\theta   = \mu _\phi   = \frac{{R_2  - R_1 }}{{r - R_1 }},$ in (3).
The incident magnetic wave is excited by the electric point source in the point  $\vec r_s  = (r_s ,\theta _s ,\phi _s )$, $r_s  > R_{out}  > R_2 $ . The analytic GL radial magnetic wave field solution of 3D GL radial magnetic equation (10) and on the boundary $r=R_2$ with field and derivative continuous to the incident magnetic wave in (14) is

\begin{equation} 
\begin{array}{l}
 H(\vec r) =  \\ 
  = \sum\limits_{l = 1}^\infty  {H_{l,1} (r)} \cos \left( {k(R_2  - R_1 )\log \left( {\frac{{r - R_1 }}{{R_2  - R_1 }}} \right)} \right) \\ 
 \sum\limits_{m =  - l}^l {} D_h (\theta ,\phi )Y_l^m (\theta ,\phi )Y_l^{m*} (\theta _s ,\phi _s ) \\ 
  + \sum\limits_{l = 1}^\infty  {H_{l,2} (r)} \sin \left( {k(R_2  - R_1 )\log \left( {\frac{{r - R_1 }}{{R_2  - R_1 }}} \right)} \right) \\ 
 \sum\limits_{m =  - l}^l {} D_h (\theta ,\phi )Y_l^m (\theta ,\phi )Y_l^{m*} (\theta _s ,\phi _s ), \\ 
 \end{array}
\end{equation} 

Where operator
\begin{equation} 
D_h (\theta ,\phi ) = \cos \phi \frac{\partial }{{\partial \theta }} - \frac{{\cos \theta \sin \phi }}{{\sin \theta }}\frac{\partial }{{\partial \phi }},
\end{equation} 
\begin{equation}
\left[ {\begin{array}{*{20}c}
   {H_{l,1} (r)}  \\
   {H_{l,2} (r)}  \\
\end{array}} \right] = \left[ {\begin{array}{*{20}c}
   {g_{l,1,1} (r)} & {g_{l,1,2} (r)}  \\
   {g_{l,2,1} (r)} & {g_{l,2,2} (r)}  \\
\end{array}} \right]\left[ {\begin{array}{*{20}c}
   {A_{l,1} }  \\
   {A_{l,2} }  \\
\end{array}} \right],            
\end{equation}

\begin{equation} 
\begin{array}{l}
 \left[ {\begin{array}{*{20}c}
   {g_{l,1,1} (r)} & {g_{l,1,2} (r)}  \\
   {g_{l,2,1} (r)} & {g_{l,2,2} (r)}  \\
\end{array}} \right] =  \\ 
  = \sum\limits_{j = 0}^\infty  {} \left[ {\begin{array}{*{20}c}
   {a_{l,j}^{(1)} } & {b_{l,j}^{(1)} }  \\
   {a_{l,j}^{(2)} } & {b_{l,j}^{(2)} }  \\
\end{array}} \right](r - R_1 )^j , 
 \end{array}
\end{equation}
\begin{equation}
\begin{array}{l}
 \left[ {\begin{array}{*{20}c}
   {a_{l,j + 1}^{(1)} } & {b_{l,j + 1}^{(1)} }  \\
   {a_{l,j + 1}^{(2)} } & {b_{l,j + 1}^{(2)} }  \\
\end{array}} \right] = \frac{{l(l + 1)}}{{((j + 1)^3  + 4k(R_2  - R_1 )^2 (j + 1))R_2^2 }} \\ 
 \left[ {\begin{array}{*{20}c}
   {(j + 1)(R_2  - R_1 )} & { - 2k(R_2  - R_1 )^2 }  \\
   {2k(R_2  - R_1 )^2 } & {(j + 1)(R_2  - R_1 )}  \\
\end{array}} \right]\left[ {\begin{array}{*{20}c}
   {a_{l,j}^{(1)} } & {b_{l,j}^{(1)} }  \\
   {a_{l,j}^{(2)} } & {b_{l,j}^{(2)} }  \\
\end{array}} \right], \\ 
 \end{array}
\end{equation}
\begin{equation}
\left[ {\begin{array}{*{20}c}
   {a_{l,0}^{(1)} } & {b_{l,0}^{(1)} }  \\
   {a_{l,0}^{(2)} } & {b_{l,0}^{(2)} }  \\
\end{array}} \right] = \left[ {\begin{array}{*{20}c}
   1 & 0  \\
   0 & 1  \\
\end{array}} \right],
\end{equation}
\begin{equation}
\begin{array}{l}
 \left[ {\begin{array}{*{20}c}
   {g_{l,1,1} (R_2 )} & {g_{l,1,2} (R_2 )}  \\
   {\left( {\frac{{\partial g_{l,1,1} }}{{\partial r}} + kg_{l,2,1} } \right)|(R_2 )} & {\left( {\frac{{\partial g_{l,1,2} }}{{\partial r}} + kg_{l,2,2} } \right)|(R_2 )}  \\
\end{array}} \right]\left[ {\begin{array}{*{20}c}
   {A_{l,1} }  \\
   {A_{l,2} }  \\
\end{array}} \right] \\ 
  = \left[ {\begin{array}{*{20}c}
   {H^{(b)} _l (R_2 )}  \\
   {\frac{\partial }{{\partial r}}H^{(b)} _l (R_2 )}  \\
\end{array}} \right], \\ 
 \end{array}
 \end{equation}
 \begin{equation}      
\begin{array}{l}
 \left[ {\begin{array}{*{20}c}
   {H^{(b)} _l (R_2 )}  \\
   {\frac{\partial }{{\partial r}}H^{(b)} _l (R_2 )}  \\
\end{array}} \right] =  \\ 
 \left[ {\begin{array}{*{20}c}
   { - ikR_2 j_l (kR_2 )h^{(1)} (kr_s )}  \\
   { - ik((l + 1)j_l (kR_2 ) - kR_2 j_{l + 1} (kR_2 ))h_l ^{(1)} (kr_s )}  \\
\end{array}} \right], \\ 
 \end{array}
 \end{equation}
The $j_l (kr)$ is spherical Bessel function, $h^{(1)} _l (kr_s ) = j_l (kr_s ) + iy_l (kr_s )$  is first type spherical Hankel function,$y_l (kr)$  is spherical Neumann function.
Now, we prove that GLHUA exact analytical radial magnetic wave (42) does satisfy the GL radial magnetic equation (10).

Substitute the new GLHUA outer layer invisible cloak (1) and (3) into the GL
radial magnetic differential equation (10), we obtain GL radial magnetic eqution in GLHUA 
cloak domain 
\begin{equation}
\begin{array}{l}
 \frac{\partial }{{\partial r}}\frac{{r - R_1 }}{{R_2  - R_1 }}\frac{\partial }{{\partial r}}H(\vec r) \\ 
  + \frac{1}{{R_2 ^2 }}\left( {\frac{1}{{\sin \theta }}\frac{\partial }{{\partial \theta }}\sin \theta \frac{\partial }{{\partial \theta }}} \right)H\left( {\vec r} \right) \\ 
  + \frac{1}{{R_2 ^2 }}\frac{1}{{\sin ^2 \theta }}\frac{{\partial ^2 }}{{\partial ^2 \phi }}H\left( {\vec r} \right) \\ 
  + k^2 \frac{{R_2  - R_1 }}{{r - R_1 }}H(\vec r) = 0, \\ 
 \end{array}
\end{equation}      
Substitute the GLHUA analytical EM wave (42) into the equation (50),and
By definition of operator $\Re (\theta ,\phi )$ in (11) and definition of operator $D_h (\theta ,\phi )$
in (12), and $\Re (\theta ,\phi )D_h (\theta ,\phi ) = D_h (\theta ,\phi )\Re (\theta ,\phi )$ in (13), we find that
\begin{equation}
 \begin{array}{l}
 \frac{\partial }{{\partial r}}\frac{{r - R_1 }}{{R_2  - R_1 }}\frac{\partial }{{\partial r}}H(\vec r) \\ 
  + \frac{1}{{R_2 ^2 }}\left( {\frac{1}{{\sin \theta }}\frac{\partial }{{\partial \theta }}\sin \theta \frac{\partial }{{\partial \theta }}} \right)H\left( {\vec r} \right) \\ 
  + \frac{1}{{R_2 ^2 }}\frac{1}{{\sin ^2 \theta }}\frac{{\partial ^2 }}{{\partial ^2 \phi }}H\left( {\vec r} \right) \\ 
  + k^2 \frac{{R_2  - R_1 }}{{r - R_1 }}H(\vec r) \\ 
  = \sum\limits_{l = 1}^\infty  {\left( {\frac{\partial }{{\partial r}}\frac{{r - R_1 }}{{R_2  - R_1 }}\frac{\partial }{{\partial r}}H_{l,1} (r)} \right.}  \\ 
 \left. { + 2k\frac{\partial }{{\partial r}}H_{l,2} (r) - \frac{{l(l + 1)}}{{R_2 ^2 }}H_{l,1} (r)} \right) \\ 
 \cos \left( {k(R_2  - R_1 )\log \left( {\frac{{r - R_1 }}{{R_2  - R_1 }}} \right)} \right) \\ 
 \sum\limits_{m =  - l}^l {} D_h (\theta ,\phi )Y_l^m (\theta ,\phi )Y_l^{m*} (\theta _s ,\phi _s ) \\ 
  + \sum\limits_{l = 1}^\infty  {\left( {\frac{\partial }{{\partial r}}\frac{{r - R_1 }}{{R_2  - R_1 }}\frac{\partial }{{\partial r}}H_{l,2} (r)} \right.}  \\ 
 \left. { - 2k\frac{\partial }{{\partial r}}H_{l,1} (r) - \frac{{l(l + 1)}}{{R_2 ^2 }}H_{l,2} (r)} \right) \\ 
 \sin \left( {k(R_2  - R_1 )\log \left( {\frac{{r - R_1 }}{{R_2  - R_1 }}} \right)} \right) \\ 
 \sum\limits_{m =  - l}^l {} D_h (\theta ,\phi )Y_l^m (\theta ,\phi )Y_l^{m*} (\theta _s ,\phi _s ), \\ 
 \end{array}
\end{equation}
By (45)-(47), $g_{l,1,1} $ and $g_{l,2,1} $ satisfy the GLHUA two equation system (23) and (24)
With initial $(1,0)$,
$g_{l,1,2} $ and $g_{l,2,2}$ satisfy the GLHUA two equation system (23) and (24)
With initial $(0,1)$.,
By (44), we know that $H_{l,1} $ and $H_{l,2} $ satisfy the GLHUA two equation system (23) and (24). Substitute $H_{l,1} $ and $H_{l,2} $  in (44) into (51). We have proved that exact analytical GL radial magnetic $H(\vec r)$ in (44) is solution of the GL radial magnetic equation (1). Based on the
Condition (48) and (49), on the outer spherical boundary $r=R_2$, The magnetic $H(\vec r)$ and its derivative satisfy the continuous boundary conditions (18) and (19). Therefore, the theorem 9 is
proved. 

Suppose that the incident electric wave is excited by the electric point source (11) located in the point  $\vec r_s  = (r_s ,\theta _s ,\phi _s )$ ,$r_s  > R_{out}  > R_2 $. by same analytic process, from the continuous of the incident GL electric wave and its derivative on the boundary, $r = R_2 $ , we can create the exact GL analytic electric wave in the GLHUA outer layer.$E(\vec r)$  .
Suppose that the incident EM wave is plane wave generated by the plane
wave  $\vec p(\vec r) = e^{i\left| k \right|r\cos (\vec k \cdot \vec r)} \vec e$ , by same analytic process in section 3, from the continuous of incident GL EM plane wave and its derivative on the boundary, $r = R_2 $ , we can create the exact GL analytic EM plane wave in the GLHUA outer layer. $E(\vec r)$  
$H(\vec r)$in (48).
Finally, we obtain the analytic radial electric wave 
 \begin{equation}
E_r (\vec r) = \frac{1}{{R_2^2 }}E(\vec r),
 \end{equation}
and the analytic radial magnetic wave 
   \begin{equation}
H_r (\vec r) = \frac{1}{{R_2^2 }}H(\vec r),
 \end{equation}                            
which satisfy the radial Maxwell EM equation (10) in the GLHUA outer layer cloak with relative parameters in (1) and (3) not less than 1.
Summary, we have the following 
\begin{figure}[h]
\centering
\includegraphics[width=0.85\linewidth,draft=false]{10-61.eps}
\caption{ (color online)
Magnetic wave propagation through GLHUA outer layer cloak, wave front at 61 step.
}\label{fig7}
%\end{minipage}
\end{figure}
\begin{figure}[h]
%\begin{minipage}[t]{0.3\linewidth}
\centering
\includegraphics[width=0.85\linewidth,draft=false]{10-67.eps}
\caption{ (color online) 
Magnetic wave propagation through GLHUA 
outer layer cloak, wave front at 67 step.
}\label{fig8}
%\end{minipage}
%\hskip5mm
\end{figure}
\begin{figure}[h]
%\begin{minipage}[t]{0.3\linewidth}
\centering
\includegraphics[width=0.85\linewidth,draft=false]{10-73.eps}
\caption{ (color online) 
Magnetic wave propagation through GLHUA 
outer layer cloak, wave front at 73 step.
 }\label{fig9}
%\end{minipage}
%\hskip5mm
%begin{minipage}[t]{0.3\linewidth}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.85\linewidth,draft=false]{10-79.eps}
\caption{ (color online) 
Magnetic wave propagation through GLHUA 
outer layer cloak, wave front at 79 step.
}\label{fig10}
%\end{minipage}
\end{figure}

${\boldsymbol{Theorem \ 3:}}$ \  In the outer annular layer cloak, $R_1  \le r \le R_2 $, the radial relative electric permittivity and magnetic permeability, 
$\varepsilon _r  = \mu _r  = \frac{{R_2 ^2 }}{{r^2 }}$ in (1), and the angular relative electric permittivity and magnetic permeability, 
$\varepsilon _\theta   = \varepsilon _\phi   = \mu _\theta   = \mu _\phi   = \frac{{R_2  - R_1 }}{{r - R_1 }},$ in (3).
The incident magnetic wave is excited by the electric point source in the point  $\vec r_s  = (r_s ,\theta _s ,\phi _s )$, $r_s  > R_{out}  > R_2 $ . The analytic GL radial electromagnetic wave field solution of 3D Maxwell radial electromagnetic equation in GLHUA outer layer cloak domain and generated by the incident EM wave on the boundary $r=R_2$ is

 \begin{equation}
\begin{array}{l}
 \left[ {\begin{array}{*{20}c}
   {E_r (\vec r)}  \\
   {H_r (\vec r)}  \\
\end{array}} \right] =  \\ 
  = \sum\limits_{l = 1}^\infty  {\left[ {\begin{array}{*{20}c}
   {E_{r,l,1} (\vec r)}  \\
   {H_{r,l,1} (\vec r)}  \\
\end{array}} \right]} \cos \left( {k(R_2  - R_1 )\log \left( {\frac{{r - R_1 }}{{R_2  - R_1 }}} \right)} \right) \\ 
 \sum\limits_{m =  - l}^l {} \left[ {\begin{array}{*{20}c}
   {D_e (\theta ,\phi )}  \\
   {D_h (\theta ,\phi )}  \\
\end{array}} \right]Y_l^m (\theta ,\phi )Y_l^{m*} (\theta _s ,\phi _s ) \\ 
  + \sum\limits_{l = 1}^\infty  {\left[ {\begin{array}{*{20}c}
   {E_{r,l,2} (\vec r)}  \\
   {H_{r,l,2} (\vec r)}  \\
\end{array}} \right]} \sin \left( {k(R_2  - R_1 )\log \left( {\frac{{r - R_1 }}{{R_2  - R_1 }}} \right)} \right) \\ 
 \sum\limits_{m =  - l}^l {} \left[ {\begin{array}{*{20}c}
   {D_e (\theta ,\phi )}  \\
   {D_h (\theta ,\phi )}  \\
\end{array}} \right]Y_l^m (\theta ,\phi )Y_l^{m*} (\theta _s ,\phi _s ), \\ 
 \end{array}
  \end{equation}
\begin{figure}[h]
\centering
\includegraphics[width=0.85\linewidth,draft=false]{10-85.eps}
\caption{ (color online) 
Magnetic wave propagation through GLHUA 
outer layer cloak, wave front at 85 step.
   }\label{fig14}
%\end{minipage}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.85\linewidth,draft=false]{10-91.eps}
\caption{ (color online) 
Magnetic wave propagation through GLHUA 
outer layer cloak, wave front at 91 step.
 }\label{fig15}
%\end{minipage}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.86\linewidth,draft=false]{10-97.eps}
\caption{ (color online) 
Magnetic wave propagation through GLHUA 
outer layer cloak, wave front at 97 step.
 }\label{fig16}
%\end{minipage}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.85\linewidth,draft=false]{10-99.eps}
\caption{ (color online) 
Magnetic wave propagation through GLHUA outer layer cloak, wave front at 99 step. 
}\label{fig17}
%\end{minipage}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.86\linewidth,draft=false]{10-100.eps}
\caption{ (color online) 
Magnetic wave propagation through GLHUA outer layer cloak, wave front at 100 step.
 }\label{fig18}
%\end{minipage}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.86\linewidth,draft=false]{10-101.eps}
\caption{ (color online) 
Magnetic wave propagation through GLHUA outer layer cloak, wave front at 101 step.
 }\label{fig19}
%\end{minipage}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.86\linewidth,draft=false]{10-103.eps}
\caption{ (color online) 
Magnetic wave propagation through GLHUA outer layer cloak, wave front at 103 step..
}\label{fig31}
%\end{minipage}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.86\linewidth,draft=false]{10-115.eps}
\caption{ (color online) 
Magnetic wave propagation through GLHUA outer layer cloak, wave front at 115 step.
}\label{fig31}
%\end{minipage}
\end{figure}


In the outside of the cloak,$r > R_2 $,$E_r (\vec r) = E_{b,r} (\vec r)$, $H_r (\vec r) = H_{b,r} (\vec r)$, on the boundary $r = R_2 $, 
$E_r (\vec r)|_{r = R_2 }  = E_{b,r} (\vec r)|_{r = R_2 } $,
$H_r (\vec r)|_{r = R_2 }  = H_{b,r} (\vec r)|_{r = R_2 } $
, 
  \begin{equation}
{\raise0.7ex\hbox{${\partial E_r (\vec r)}$} \!\mathord{\left/
 {\vphantom {{\partial E_r (\vec r)} {\partial r}}}\right.\kern-\nulldelimiterspace}
\!\lower0.7ex\hbox{${\partial r}$}}|_{r = R_2 }  = {\raise0.7ex\hbox{${\partial E_{b,r} (\vec r)}$} \!\mathord{\left/
 {\vphantom {{\partial E_{b,r} (\vec r)} {\partial r}}}\right.\kern-\nulldelimiterspace}
\!\lower0.7ex\hbox{${\partial r}$}}|_{r = R_2 } 
  \end{equation}
  \begin{equation}
{\raise0.7ex\hbox{${\partial H_r (\vec r)}$} \!\mathord{\left/
 {\vphantom {{\partial H_r (\vec r)} {\partial r}}}\right.\kern-\nulldelimiterspace}
\!\lower0.7ex\hbox{${\partial r}$}}|_{r = R_2 }  = {\raise0.7ex\hbox{${\partial H_{b,r} (\vec r)}$} \!\mathord{\left/
 {\vphantom {{\partial H_{b,r} (\vec r)} {\partial r}}}\right.\kern-\nulldelimiterspace}
\!\lower0.7ex\hbox{${\partial r}$}}|_{r = R_2 } ,
  \end{equation}
 The exact analytical radial EM wave solution show that there is no scattering from the GLHUA outer layer cloak to disturb the incident EM wave in free space outside of the cloak.
%\end{document}
Proof: Our theorem 3 is proved by using similar proof in theorem 2.
\section {Simulation of the exact analytic magnetic wave propagation through the GLHUA outer layer cloak}
\begin{figure}[h]
\centering
\includegraphics[width=0.86\linewidth,draft=false]{10-121.eps}
\caption{ (color online) 
Magnetic wave propagation through GLHUA outer layer cloak, wave front at 121 step.
}\label{fig31}
%\end{minipage}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.86\linewidth,draft=false]{10-127.eps}
\caption{ (color online) 
Magnetic wave propagation through GLHUA outer layer cloak, wave front at 127 step.
}\label{fig31}
%\end{minipage}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.86\linewidth,draft=false]{10-133.eps}
\caption{ (color online) 
Magnetic wave propagation through GLHUA outer layer cloak, wave front at 133 step.
}\label{fig31}
%\end{minipage}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.86\linewidth,draft=false]{10-139.eps}
\caption{ (color online) 
Magnetic wave propagation through GLHUA outer layer cloak, wave front at 139 step.
}\label{fig31}
%\end{minipage}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.86\linewidth,draft=false]{10-145.eps}
\caption{ (color online) 
Magnetic wave propagation through GLHUA outer layer cloak, wave front at 145 step.
}\label{fig31}
%\end{minipage}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.86\linewidth,draft=false]{10-151.eps}
\caption{ (color online) 
Magnetic wave propagation through GLHUA outer layer cloak, wave front at 151 step.
}\label{fig31}
%\end{minipage}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.86\linewidth,draft=false]{10-157.eps}
\caption{ (color online) 
Magnetic wave propagation through GLHUA outer layer cloak, wave front at 157 step.
}\label{fig25}
%\end{minipage}
\end{figure}

In GLHUA outer cloak, $R_1  \le r \le R_2 $, the incident EM wave field is excited by the electric current point source located in $\vec r_s  = (13.0m,0.5\pi ,0)$, the electric current point source is
\begin{equation}
S(\vec r,\vec r_s ) = e^{ - i\omega t} \delta (\vec r - \vec r_s )\vec e_y ,
\end{equation}
where
\begin{equation}
\begin{array}{l}
 \vec e = \vec e_y  = (0,1,0) \\ 
  = \left( {\begin{array}{*{20}c}
   {\sin \theta \sin \phi } & {\cos \theta \sin \phi } & {\cos \phi }  \\
\end{array}} \right), \\ 
 \end{array}
\end{equation}
In the free space $r>R_2$, the angular frequency $\omega  = 2\pi f$, $f = 0.{\rm 1669555} \times {\rm 10}^{\rm 9} Hz$, 

  In GLHUA outer cloak, $R_1  \le r \le R_2 $, $R_1  = 0.66m$,$R_2  = 1.0m$, the incident radial magnetic wave excited by the above electric current source is
                  \begin{equation}
H^b {}_r = \cos \phi \frac{1}{r}\frac{{\partial g}}{{\partial \theta }} - \frac{{\cos \theta \sin \phi }}{{r\sin \theta }}\frac{{\partial g}}{{\partial \phi }},
\end{equation}

The GLHUA exact analytical EM wave propagation show that on the inner spherical surface  boundary $r=R_1$, the GLHUA cloak material does generate a strange wave. We call the strange wave as ��GLHUA Mirage wave��, for simply, we call it "GL wave". In Figure 1 (9), the magnetic wave is being tangent to the outer boundary $r=R_2$ without any scattering. It is very strangle, there is a down crescent type GL wave in the upper part of the inner spherical surface, $r=R_1$ , and upward crescent type GL wave in the lower part of the same inner spherical surface, $r=R_1$. In the next step, the magnetic wave smoothly enter the outer annular layer $R_1  < r < R_2 ,$ is presented in Figure 2,(31) The upper and lower two crescent type GL wave attached on the inner spherical surface, $r=R_1$ , maintain the shape and propagation from left to right; along the same inner
spherical surface, a red small part of the arc GL arc wave generated; The wave propagation phase image in figure 3(43) is similar with the image in figure 2., the magnetic wave in the GLHUA outer layer, the red arc and crescent GL wave on the inner spherical surface
are propagation to left, the arc and crescent GL wave phase shape is growing along on
the same inner spherical surface, $r=R_1$, and the arc GL wave is catching to crescent
GL wave. In figure 4 (55), the wave propagation phase image is similar with the image in figure 3, the phase image in figure 4 is propagated to left of image in figure 3, the upper and lower crescent GL wave along on the inner spherical surface, $r=R_1$ is connected
into double crescent GL wave. 
\begin{figure}[h]
\centering
\includegraphics[width=0.86\linewidth,draft=false]{10-163.eps}
\caption{ (color online) 
Magnetic wave propagation through GLHUA outer layer cloak, wave front at 163 step.
}\label{fig26}
%\end{minipage}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.86\linewidth,draft=false]{10-169.eps}
\caption{ (color online) 
Magnetic wave propagation through GLHUA outer layer cloak, wave front at 169 step.
}\label{fig27}
%\end{minipage}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.86\linewidth,draft=false]{10-175.eps}
\caption{ (color online) 
Magnetic wave propagation through GLHUA outer layer cloak, wave front at 175 step.
}\label{fig28}
%\end{minipage}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.86\linewidth,draft=false]{10-176.eps}
\caption{ (color online) 
Magnetic wave propagation through GLHUA outer layer cloak, wave front at 176 step.
}\label{fig29}
%\end{minipage}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.86\linewidth,draft=false]{10-181.eps}
\caption{ (color online) 
Magnetic wave propagation through GLHUA outer layer cloak, wave front at 181 step.
}\label{fig30}
%\end{minipage}
\end{figure}

The phase image in figure 5(91) is propagated to left of image in figure 4. In the figure 5, the double crescent GL wave is growing up to a spherical surface GL wave, which is connected with the arc GL wave, the magnetic wave in GLHUA outer layer is propagating to left of it in figure 4, follow and closed to the GL wave along on the inner spherical surface, $r=R_1$. In the figure 6(99), in the left side of the inner spherical surface, the spherical surface GL wave is growing up to like a wave front on the left side of the inner spherical surface, in the right side of the inner spherical surface, the arc GL wave is growing as half spherical surface GL wave which phase is right part of the inner spherical surface. The right half spherical surface GL wave is connected with the left side spherical surface GL wave. the magnetic wave in GLHUA outer layer is propagating to left of it in figure 5, follow and more closed to the GL wave along on the inner spherical surface, $r=R_1$. The phase image in figure 7(100) is similar with the phase image in figure 6 and is propagated to left of image in figure 6. The magnetic wave propagation in figure 7 is important step. In the figure 7, the magnetic wave in right side of inner spherical surface meets the spherical surface GL wave in the left side of the inner spherical surface. In figure 8 (101), the magnetic wave use the left spherical GL wave as wave front for new propagation, the original wave front of the magnetic wave is disengaged and as new spherical surface GL wave attached in the right side of the inner spherical surface. The magnetic wave change wave front in the figure 7 shows that the magnetic wave front propagation in the GLHA outer layer cloak is discontinuous and the ray is also discontinuous. The phase image in figure 9(103) is similar with the phase image in figure 8 and is propagated to left of image in figure 8. In the next magnetic wave propagation step in the figure 9, 10, 11,��, let P1 be cross point of the wave front and equatorial and P2 be cross point of the outer spherical surface, $r=R_2$ and equatorial. 
 
Let Q1 be cross point of the wave front and outer spherical surface $r=R_2$ and Q2 be the cross point of line $z = z_{q_1 }$ of Q1 and line $x = x_{p_2 } $ of P2. Let d1 be distance between P1 and P2 , d2 be the distance between Q1 and Q2 . In the figure 9, 10, 11,��,the magnetic wave is propagating forward to left, on the magnetic wave front, d1 is minimal and phase velocity at point P1 on the wave front is also minimal; . The d2 is maximum and phase velocity at point Q1 on the wave front is also maximum; the d1 < d2, the difference of d2- d1 in monotone decreasing. In the figure 17 , d1=d2=0. the every point on the magnetic wave front are propagating out of the GLHUA cloak simultaneously, the magnetic wave is recovered to the incident wave in free space at same time. In the figure 18, the magnetic wave front is recovered incident wave front in free space that show GLHUA outer layer cloak is invisible. In the figure 9, 10, 11,��, along inner spherical surface, $r=R_1$, the upper GL wave is propagating counterclockwise and the lower GL wave is propagating clockwise with some discontinuous.  In figure 9 (103), the right GL wave attached on the right inner spherical surface is deformed. In figure 10 (127), the right GL wave attached on the right inner spherical surface is more deformed to double crescent, and left arc GL wave is disengaged from the right GL wave. In figure 11(145), the right GL wave attached on the right inner spherical surface is more deformed to double crescent, and left arc GL wave is disengaged from the right GL wave and is more discontinuous. In figure 12 (151), the right GL wave attached on the right inner spherical surface is discontinuously split to the upper crescent GL wave and lower crescent GL wave. In figure 13 (163). along inner spherical surface, $r=R_1$, the upper crescent GL wave and arc GL wave is propagating counterclockwise and the lower crescent GL wave and arc GL wave is propagating clockwise with some discontinuous. The phase image of the magnetic wave front and GL wave in figure 14 (175) is similar with the phase image in the figure 13, moreover, the left arc GL wave is shrink to very short. The new magnetic wave front is appear in right side in the new period.  The distance between two magnetic wave front is wave length  . 
\begin{figure}[h]
\centering
\includegraphics[width=0.86\linewidth,draft=false]{10-185.eps}
\caption{ (color online) 
Magnetic wave propagation through GLHUA outer layer cloak, wave front at 185 step.
}\label{fig31}
%\end{minipage}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.86\linewidth,draft=false]{10-186.eps}
\caption{ (color online) 
Magnetic wave propagation through GLHUA outer layer cloak, wave front at 186 step.
}\label{fig32}
%\end{minipage}
\end{figure}

The phase Image of the magnetic wave front and GL wave along the inner spherical surface, $r=R_1$ in figure 15(176) and figure 16 (186) is same as phase image in figure 1 and figure 2, respectively, Summary, the incident magnetic wave in free space never be disturbed by GLHUA outer cloak, GLHUA cloak is invisible; The magnetic wave can not propagating penetrated into inside of the sphere $r < R_1$.The phase velocity of magnetic wave propagation is less than light speed. The exact analytic GLHUA wave propagation simulation shows that the GL wave along the inner spherical surface $r=R_1$ is generated by the GLHUA outer cloak material. The GL wave is a bridge for the EM wave propagation through the GLHUA outer cloak without exceeding light speed. The GL wave is a cloaking making that the EM wave propagation can not penetrate into the concealment. In the figure 33 (187) and figure 34 (193), the wave front, GL mirage wave, and next wave front are continuous propagation from right to left.In figure 35 (194), the wave front arrived the left side of the outer boundary $r=R_2$,
and it has been tangent to the left side of the outer boundary $r=R_2$.
The GL mirage wave is propagation from right to left around the inner boundary
$r=R_1$. The phase image of the wave front and GL mirage wave is like as in Figure 2.  In  figure 36 (198) and figure 37 (200), the wave front is propagation outside of the sphere, $r> R_2$, and recovered to incident magnetic wave. The GL mirage wave around inner sphere $r=R_1$ and next wave front are continuos propagation from right to left, the GL mirage wave will be
bridge for next wave front.
 Summary, the incident magnetic wave in free space never be disturbed by GLHUA outer cloak, GLHUA cloak is invisible; The magnetic wave can not propagating penetrated into inside of the sphere $r < R_1$.The phase velocity of magnetic wave propagation is less than light speed. The exact analytic GLHUA wave propagation simulation shows that the GL wave along the inner spherical surface $r=R_1$ is generated by the GLHUA outer cloak material. The GL wave is a bridge for the EM wave propagation through the GLHUA outer cloak without exceeding light speed. The GL wave is a cloaking making that the EM wave propagation can not penetrate into the concealment.
When $R_1 > 0.8 R_2$, the figure 38 show that the GL mirage crescent wave disappear, the GL mirage wave become GL ARC migrage on the inner boundary $r=R_1$ that is bridge for EM wave propagation in GLHUA invisible cloak
without exceeding light speed. 
\section{Discussion and Conclusion}

Substitute the inverse transformation to incident analytical EM wave, we can obtain the analytical EM wave propagation in the $0$ to $R_1$ spherical radial transformation cloak. It is totally different from transformation optic,  without using transformation, our GLHUA analytical expansion can also be directly used to find analytical EM wave
propagation in the $0$ to $R_1$ spherical radial transformation cloak. For example, we propose 
GLHUA analytical expansion method to find analytical EM wave propagation in Pendry cloak.

\subsection{GLHUA analytical expansion method to find analytical EM wave in Pendry Cloak}

Substitute relative EM parameters in Pendry cloak
\begin{equation}
\varepsilon _r  = \mu _r  = \frac{{R_2 }}{{R_2  - R_1 }}\frac{{(r - R_1 ){}^2}}{{r^2 }},
\end{equation}

\begin{equation}
\varepsilon _\theta   = \mu _\theta   = \varepsilon _\phi   = \mu _\phi   = \frac{{R_2 }}{{R_2  - R_1 }},
\end{equation}        
into the equation (17), the (17) become to
\begin{equation}
\begin{array}{l}
 \frac{\partial }{{\partial r}}(\frac{{R_2  - R_1 }}{{R_2 }}\frac{{\partial H_l (r)}}{{\partial r}}) \\ 
  - \frac{{R_2  - R_1 }}{{R_2 }}\frac{{l(l + 1)}}{{(r - R_1 )^2 }}H_l (r) \\ 
  + k^2 \frac{{R_2 }}{{R_2  - R_1 }}H_l (r) = 0, \\ 
 \end{array}
\end{equation}
We create GLHUA analytical expansion of radial magnetic wave $l$ component $H_{l,p}$,
$p=1,2$,  

\begin{equation}
\begin{array}{l}
 H_{l,p}  = H_{l,1,p} (r)\cos \left( {k\frac{{R_2 (r - R_1 }}{{R_2  - R_1 }}} \right) \\ 
  + H_{l,2,p} (r)\sin \left( {k\frac{{R_2 (r - R_1 }}{{R_2  - R_1 }}} \right), \\ 
 \end{array}
\end{equation}
Substitute (63) into (62), we have novel GLHUA dual two expansion equations,$p=1,2$,

\begin{equation}
\begin{array}{l}
 \frac{\partial }{{\partial r}}(\frac{{R_2  - R_1 }}{{R_2 }}\frac{{\partial H_{l,1,p} (r)}}{{\partial r}}) + 2k\frac{{\partial H_{l,2,p} (r)}}{{\partial r}} \\ 
  = \frac{{R_2  - R_1 }}{{R_2 }}l(l + 1)\frac{1}{{(r - R_1 )^2 }}H_{l,1,p} (r), \\ 
 \end{array}
\end{equation}

\begin{equation}
\begin{array}{l}
 \frac{\partial }{{\partial r}}(\frac{{R_2  - R_1 }}{{R_2 }}\frac{{\partial H_{l,2,p} (r)}}{{\partial r}}) - 2k\frac{{\partial H_{l,1,p} (r)}}{{\partial r}} \\ 
  = \frac{{R_2  - R_1 }}{{R_2 }}l(l + 1)\frac{1}{{(r - R_1 )^2 }}H_{l,2,p} (r), \\ 
 \end{array}
\end{equation}
%\subsection {GLHUA analytical expansion method to find analytical EM wave in %Pendry Cloak}

We propose negative power series expansion 
  \begin{equation}
 H_{l,1,p} (r) = \sum\limits_{j = 0}^\infty  {a_{l,j,p} } (r - R_1 )^{ - j} ,
\end{equation}
\begin{equation}
H_{l,2,p} (r) = \sum\limits_{j = 0}^\infty  {b_{l,j,p} } (r - R_1 )^{ - j} ,
\end{equation}
where $p=1,2$, when $p=1$ , $a_{l,0,p}=1$, $b_{l,0,p}=0$, when $p=2$ , $a_{l,0,p}=0$, $b_{l,0,p}=1$.  Substitute (66) and (67) into GLHUA dual equations (64) and (65), we obtain

\begin{equation}
\begin{array}{l}
 H_{l,1}  = H_{l,1,1} (r)\cos \left( {k\frac{{R_2 (r - R_1 )}}{{R_2  - R_1 }}} \right) \\ 
  + H_{l,2,1} (r)\sin \left( {k\frac{{R_2 (r - R_1 )}}{{R_2  - R_1 }}} \right) \\ 
  = \frac{{R_2 (r - R_1 )}}{{R_2  - R_1 }}j_l \left( {k\frac{{R_2 (r - R_1 )}}{{R_2  - R_1 }}} \right), \\ 
 \end{array}
\end{equation}
\begin{equation}
\begin{array}{l}
 H_{l,2}  = H_{l,1,2} (r)\cos \left( {k\frac{{R_2 (r - R_1 )}}{{R_2  - R_1 }}} \right) \\ 
  + H_{l,2,2} (r)\sin \left( {k\frac{{R_2 (r - R_1 )}}{{R_2  - R_1 }}} \right) \\ 
  = \frac{{R_2 (r - R_1 )}}{{R_2  - R_1 }}n_l \left( {k\frac{{R_2 (r - R_1 )}}{{R_2  - R_1 }}} \right), \\ 
 \end{array}
\end{equation}

             For arbitrary constants $A$ and $B$,
\begin{equation}
\begin{array}{l}
 H_l (r) = AH_{l,1} (r) + BH_{l,2} (r) \\ 
  = A\frac{{R_2 (r - R_1 )}}{{R_2  - R_1 }}j_l \left( {k\frac{{R_2 (r - R_1 )}}{{R_2  - R_1 }}} \right) \\ 
  + B\frac{{R_2 (r - R_1 )}}{{R_2  - R_1 }}n_l \left( {k\frac{{R_2 (r - R_1 )}}{{R_2  - R_1 }}} \right). \\ 
 \end{array}
\end{equation}
From the continuous boundary condition on $r=R_2$, (18) and (19) in this paper, we obtained
$B=0$, 
\begin{equation}
\begin{array}{l}
H_l (r) =  \\
- ik\frac{{R_2 (r - R_1 )}}{{R_2  - R_1 }}j_l \left( {k\frac{{R_2 (r - R_1 )}}{{R_2  - R_1 }}} \right){h^(1)}_l (kr_s ),
\end{array}
\end{equation}
\hfill\break \\
${\boldsymbol{Theorem \ 4:}}$ \ In the spherical annular layer $R_1  \le r \le R_2 $, the EM relative parameters is
presented by (60) and (61), the electric point delta source is denoted by (7) in this paper, then analytical
radial GL magnetic wave propagation in Pendry cloak is 
\begin{equation}
\begin{array}{l}
 H(\vec r) =  \\ 
  =  - ik\sum\limits_{l = 1}^\infty  {\frac{{R_2 (r - R_1 )}}{{R_2  - R}}} j_l \left( {k\frac{{R_2 (r - R_1 )}}{{(R_2  - R_1 )}}} \right)h^{(1)} _l (kr_s ) \\ 
 \sum\limits_{m =  - l}^l {} D_h (\theta ,\phi )Y_l^m (\theta ,\phi )Y_l^{m*} (\theta _s ,\phi _s ) \\ 
  = \sum\limits_{l = 1}^\infty  {} H^{(b)} _l \left( {k\frac{{R_2 (r - R_1 )}}{{(R_2  - R_1 )}}} \right) \\ 
 \sum\limits_{m =  - l}^l {} D_h (\theta ,\phi )Y_l^m (\theta ,\phi )Y_l^{m*} (\theta _s ,\phi _s ) \\ 
 \end{array}
\end{equation}
\begin{equation}
\begin{array}{l}
 H_r (\vec r) =  \\ 
  =  - ik\sum\limits_{l = 1}^\infty  {\frac{1}{{(r - R)}}} j_l \left( {k\frac{{R_2 (r - R_1 )}}{{(R_2  - R_1 )}}} \right)h^{(1)} _l (kr_s ) \\ 
 \sum\limits_{m =  - l}^l {} D_h (\theta ,\phi )Y_l^m (\theta ,\phi )Y_l^{m*} (\theta _s ,\phi _s ) \\ 
  = \frac{{R_2  - R_1 }}{{R_2 }}\frac{1}{{(r - R_1 )^2 }}\sum\limits_{l = 1}^\infty  {} H^{(b)} _l \left( {k\frac{{R_2 (r - R_1 )}}{{(R_2  - R_1 )}}} \right) \\ 
 \sum\limits_{m =  - l}^l {} D_h (\theta ,\phi )Y_l^m (\theta ,\phi )Y_l^{m*} (\theta _s ,\phi _s ). \\ 
 \end{array} 
\end{equation}      
\hfill\break \\

\subsection { Comparison GLHUA cloak and Pendry Cloak}
In GLHUA invisible cloak (1) and (3), the refractive index  $n = \sqrt {\varepsilon _\theta  \mu _r } $   is large than 1 and going to infinity when $r$ going to $R_1$. The phase velocity of EM wave propagation in GLHUA cloak is less than light speed and going to zero when $r$ going to $R_1$. GLHUA invisible cloak is practicable. In Pendry invisible cloak (60) and (61) , the refractive index  $n = \sqrt {\varepsilon _\theta  \mu _r } $   is less than 1 and going to zero when $r$ going to $R_1$. The phase velocity of EM wave propagation in Pendry cloak is exceeding light speed and going to infinity when $r$ going to $R_1$. Pendry invisible cloak is very difficulty practicable or can not be practicable.The analytical Magnetic wave in GLHUA cloak (54) in theorem 3 and analytical Magnetic  wave in Pendry cloak (73) in theorem 4 are 
totally different.Any annular layer cloaking by transformation will make the phase velocity exceeding light speed and tends to infinity in $r=R_1$. 
GLHUA invisible cloak overcome the fundamental difficuties of exceeding light
speed and tends to infinity in transformation cloak and in Pendry cloak.
Our GLHUA analytical expansion method can be used to find analytical EM
wave in transformation cloak directly without using transformation.
\subsection {Analytical EM wave in GLHUA cloak is undisputed evidence to
prove that GLHUA cloak is practicable invisible cloak and super scirnces is being born}
Our analytical EM wave in GLHUA cloak is undisputed evidence and rigorous proof to prove that the GLHUA double layer cloak is practicable invisible cloak without exceed light speed propagation. Our analytical EM wave in GLHUA cloak is undisputed evidence and rigorous proof to prove GLHUA double layer invisible cloak and their theoretical proof in  are right. 

GL simulation shows that there are novel GLHUA mirage bridge which is generated by the
GLHUA outer layer invisible cloak materials in (1) and (3). The GLHUA mirage bridge make the GLHUA analytical EM wave propagation without exceeding light speed. The GLHUA mirage bridge  is obvious for $R_1=0.5R_2$ or $R_1=0.66R_2$,
for $R_> 0.8R_2$, the GLHUA mirage bridge wave become to the arc  GLHUA mirage wave on the inner spherical layer $r=R_1$. 

\begin{figure}[h]
\centering
\includegraphics[width=0.86\linewidth,draft=false]{10-187.eps}
\caption{ (color online) 
Magnetic wave propagation through GLHUA outer layer cloak, wave front at 187 step.
}\label{fig33}
%\end{minipage}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.86\linewidth,draft=false]{10-193.eps}
\caption{ (color online) 
Magnetic wave propagation through GLHUA outer layer cloak, wave front at 193 step.
}\label{fig34}
%\end{minipage}
\end{figure}  

\begin{figure}[h]
\centering
\includegraphics[width=0.86\linewidth,draft=false]{10-194.eps}
\caption{ (color online) 
Magnetic wave propagation through GLHUA outer layer cloak, wave front at 194 step.
}\label{fig35}
%\end{minipage}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.86\linewidth,draft=false]{10-198.eps}
\caption{ (color online) 
Magnetic wave propagation through GLHUA outer layer cloak, wave front at 198 step.
}\label{fig36}
%\end{minipage}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.86\linewidth,draft=false]{10-200.eps}
\caption{ (color online) 
Magnetic wave propagation through GLHUA outer layer cloak, wave front at 200 step.
}\label{fig37}
%\end{minipage}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.86\linewidth,draft=false]{08.eps}
\caption{ (color online) 
When $R_1 =0.8 R_2$, the GL mirage cresent wave is disppear, the GL mirage ARC wave on  the inner sphrical boundary $r=R_1$ is bridge for EM wave propagation
in GLHUA outer layer cloak without exceeding light speed.
}\label{fig37}
%\end{minipage}
\end{figure}

 \begin{equation}
\begin{array}{l}
 \varepsilon _r  = 1, \\ 
 \varepsilon _\theta   = \varepsilon _\phi   = \frac{1}{2}\left( {\left( {\frac{{r - R_2 }}{{R_3  - R_2 }}} \right)^\alpha   + \left( {\frac{{R_3  - R_2 }}{{r - R_2 }}} \right)^\alpha  } \right), \\ 
 \end{array}
\end{equation}

In our paper [5-6][11-12][15], we proposed GL no scattering modeling and inversion to create practicable GLHUA and GLLH two class of the invisible cloak. Our GLHUA cloak by GL no scattering modeling and inversion is different from GLLH cloak in 2010. In 1971, we used no scattering inversion idea to construct a novel 3D 20 nodes high accurate curve element and developed 3D finite element method first in China and discovered super convergence of 3D iso parameter finite element first in the world [26][27]. GILD and GL scattering and no scattering modeling and inversion idea and method play very important role for research works in invisible cloak. When Pendry cloak in 2006, we had been working in no scattering modeling and inversion for 6 years since 2000. Our idea and method are different from Pendry and other cloak research.

Because there is no experiment to study the complete invisible cloak, therefore the full wave theoretical proof and full wave computational simulation are necessary to verify the invisible clock. GLHUA invisible cloak and GLLH invisible cloak have been verified by the full wave theoretical proof and full wave simulation using GL full wave no scattering modeling and inversion in [1]. 
GLHUA double cloak and mirage in [13] show that GL no scattering modeling and inversion is powerful method to make practicable invisible cloak and to investigate new invisible natural science Inventor Easton LaChappelles mind control robot hands is beginning to investigate visible thinking science and visible social Science. In current general science, visible natural science and invisible thinking science and social science are main object. Now invisible natural science and invisible thinking science and visible social science will be main object for a novel super science. Our GLHUA practicable double cloak and Easton LaChappelles mind control robot hands show that the new novel super science is being born.

\begin{acknowledgments}
We wish to acknowledge the support of the GL Geophysical Laboratory and thank the GLGEO Laboratory to approve the paper
publication. Authors thank to Professor P. D. Lax for his concern and encouragements  Authors thank to Dr. Michael Oristaglio and Professor You Zhong Guo for their encouragments
\end{acknowledgments}

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\end{thebibliography}
%=============================================

\end{document}


1706.10147.extracted_bib
{1 Jianhua,Li, F. Xie, Lee Xie, G. Xie, GLHUA Electromagnetic Invisible Double Layer Cloak With Relative Parameters Not Less Than 1 and GL No Scattering Inversion, arXiv:1612.02857.
1706.10147.extracted_bib
{2 Jianhua,Li, F. Xie, Lee Xie, G. Xie,The Theoretical Proof for GLHUA EM Invisible Double Layer Cloak By Using GL No Scattering Modeling and Inversion arXiv:1612.00534
1706.10147.extracted_bib
{3 Jianhua,Li, F. Xie, Lee Xie, G. Xie, ELECTROMAGNETIC WAVE PROPAGATION IN GLHUA INVISIBLE SPHERE BY GL NO SCATERING FULL WAVE MODELING AND INVERSION3 arXiv:1612.0258
1706.10147.extracted_bib
{4 Xie,~G., J.H.~Li, E. ~Majer, D. ~Zuo, M. ~Oristaglio ``3-D electromagnetic modeling and nonlinear inversion," Geophysics, Vol.~65, No.~3, 804--822, 2000.
1706.10147.extracted_bib
{5 Jianhua,Li, G. Xie, C, Lin, J. Liu, SEG, Expanded Abstracts, 21, 1, 692-695, (2002), https://www.onepetro.org/conference-paper/SEG-2002-0692
1706.10147.extracted_bib
{6 Xie,~G., F.~Xie, L.~Xie, and J.~Li, ``New GL method and its advantages for resolving historical diffculties," Progress In Electromagnetics Research, PIER~63, 141--152, 2006.
1706.10147.extracted_bib
{7 Pendry, J. B., D. Schurig, and D. R. Smith, Science Express, 312, 1780, (2006.)
1706.10147.extracted_bib
{8 Xie,~G., J.~Li, L.~Xie, and F.~Xie, ``GL metro carlo EM inversion," Journal of Electromagnetic Waves and Applications, Vol.~20, No.~14, 1991--2000, 2006.
1706.10147.extracted_bib
{9 G.Xie, J. H. Li, L. Xie, F. Xie, arXiv:0904.3040 (2009).
1706.10147.extracted_bib
{10 G. Xie, J.H. Li, F. Xie, L. Xie, arXiv:0904.3168 (2009).
1706.10147.extracted_bib
{11 G. Xie, J. Li, L. Xie, F. Xie, arXiv:1005.3999, (2010).
1706.10147.extracted_bib
{12 G. Xie, J. Li, L. Xie, and F. Xie, PIERS Proc., 260 - 265, July 5-8, Cambridge USA, (2010)
1706.10147.extracted_bib
{13 F. Xie, L. Xie, PIERS 2005 Abstract, 296, (2005)
1706.10147.extracted_bib
{14 Perczel, T. T. and U. Leonhardt, arXiv:1105.0164,(2011).
1706.10147.extracted_bib
{15 Jianhua Li, G. Xie, L. Xie, F. Xie, and H. Zhou, PIERS Proc.18-26, Suzhou, China, (2011).
1706.10147.extracted_bib
{16 Jianhua Li, G. Xie, F. Xie, and L. Xie, PIERS Proc., 66-72, Moscow, Russia, (2009)
1706.10147.extracted_bib
{17 G. Xie, J. Li, L. Xie, and F. Xie, PIERS Proc, Suzhou, China, 870. (2011)
1706.10147.extracted_bib
{18 G. Xie, J. H. Li, F. Xie, L. Xie, arXiv:0907.0858 (2009).
1706.10147.extracted_bib
{19 G. Xie, J. H. Li, L. Xie, F. Xie, arXiv:1002.4249, (2010).
1706.10147.extracted_bib
{20 G. Xie, J. Li, F. Xie, L. Xie, PIERS Proc, 335-243, Moscow, Russia, (2009).
1706.10147.extracted_bib
{21 G. Xie, J.H. Li, F. Xie, L. Xie, PIERS Proc, 623 - 630, Xian, China (2010)
1706.10147.extracted_bib
{22 G. Xie, J.H. Li, L. Xie, F. Xie, PIERS Proc, 181 KL, MALAYSIA, (2012)
1706.10147.extracted_bib
{23 Jianhua Li, G. Xie, L. Xie, and F. Xie, PIERS Abstract, 273,Moscow, Russia, 2012,
1706.10147.extracted_bib
{24 G. Xie and J. Li, Science In China, vol. 31, no.12, 1989.
1706.10147.extracted_bib
{25 G. Xie and J. Li, Science In China, vol. 3, no.12, 1988.
1706.10147.extracted_bib
{26 G. Xie, Mathematical Practice and Knowledge,1, 1, 28�?1, 1975,
1706.10147.extracted_bib
{27 U. Leonhardt and T Tyc, Sciences 323 (2009).
1706.10147.extracted_bib
{28 M. Krizek, International journal of numerical analysis and modeling, 2(1): 43, 56 2005
1706.10147.extracted_bib
{29 H. Chen, B. Wu, B. Zhang, A. Kong, PRL, 99, 063903 (2007).