Why Do Pendry's Cloaking Equations Differ in My Calculations?

[andresordonez]

Hi, I'm reading this article (you may need to register to view it, the registration is free though).

http://www.sciencemag.org/content/312/5781/1780.full

(can I post a link to this article in Dropbox so that people reading this don't have to register without getting an infraction from the moderators??)

and I'm getting this:

[tex]
\epsilon'_{r'} = \epsilon \frac{R_2}{R_2-R_1} (r'-R_1)^2 \sin(\theta')
[/tex]
[tex]
\epsilon'_{\theta'} = \epsilon \frac{R_2}{R_2-R_1} \sin(\theta')
[/tex]
[tex]
\epsilon'_{\phi'} = \epsilon \frac{R_2}{R_2-R_1} \sin(\theta')
[/tex]

instead of equations (7) in Pendry's article:

[tex]
\epsilon'_{r'} = \frac{R_2}{R_2-R_1} \frac{(r'-R_1)^2}{r'}
[/tex]
[tex]
\epsilon'_{\theta'} = \frac{R_2}{R_2-R_1}
[/tex]
[tex]
\epsilon'_{\phi'} = \frac{R_2}{R_2-R_1}
[/tex]

The difference between these equations and the ones I get is not only the missing [tex]r'[/tex] and the extra [tex]sin(\theta')[/tex] but also the extra [tex]\epsilon[/tex]

This is what I'm doing. The new coordinates are given by equations (6):

[tex]
r^{\prime}=R_{1}+r\frac{\left(R_{2}-R_{1}\right)}{R_{2}}
[/tex]
[tex]
\theta^{\prime}=\theta
[/tex]
[tex]
\phi^{\prime}=\phi
[/tex]

The permittivity transforms according to:
[tex]
\epsilon_{r}^{\prime}=\epsilon\frac{Q_{\theta'}Q_{\phi'}}{Q_{r'}}
[/tex]
[tex]
\epsilon_{\theta}^{\prime}=\epsilon\frac{Q_{r'}Q_{\phi'}}{Q_{\theta'}}
[/tex]
[tex]
\epsilon_{\phi}^{\prime}=\epsilon\frac{Q_{r'}Q_{\theta'}}{Q_{\phi'}}
[/tex]

where [tex]Q_{u}[/tex] is given by:
[tex]
Q_u^2 = \left(\frac{\partial x}{\partial u}\right)^2 + \left(\frac{\partial y}{\partial u}\right)^2 + \left(\frac{\partial z}{\partial u}\right)^2
[/tex]

Then:
[tex]
Q_{r^{\prime}}^{2}=\left(\frac{\partial x}{\partial r^{\prime}}\right)^{2}+\left(\frac{\partial y}{\partial r^{\prime}}\right)^{2}+\left(\frac{\partial z}{\partial r^{\prime}}\right)^{2}
[/tex]
[tex]
\frac{\partial x}{\partial r^{\prime}}=\frac{\partial x}{\partial r}\frac{\partial r}{\partial r^{\prime}}+\frac{\partial x}{\partial\theta}\frac{\partial\theta}{\partial r^{\prime}}+\frac{\partial x}{\partial\phi}\frac{\partial\phi}{\partial r^{\prime}}=\frac{\partial x}{\partial r}\frac{\partial r}{\partial r^{\prime}}=\sin\theta\cos\phi\frac{R_{2}}{R_{2}-R_{1}}=\sin\theta^{\prime}\cos\phi^{\prime}\frac{R_{2}}{R_{2}-R_{1}}
[/tex]
[tex]
\frac{\partial y}{\partial r^{\prime}}=\frac{\partial y}{\partial r}\frac{\partial r}{\partial r^{\prime}}+\frac{\partial y}{\partial\theta}\frac{\partial\theta}{\partial r^{\prime}}+\frac{\partial y}{\partial\phi}\frac{\partial\phi}{\partial r^{\prime}}=\frac{\partial y}{\partial r}\frac{\partial r}{\partial r^{\prime}}=\sin\theta\sin\phi\frac{R_{2}}{R_{2}-R_{1}}=\sin\theta^{\prime}\sin\phi^{\prime}\frac{R_{2}}{R_{2}-R_{1}}
[/tex]
[tex]
\frac{\partial z}{\partial r^{\prime}}=\frac{\partial z}{\partial r}\frac{\partial r}{\partial r^{\prime}}+\frac{\partial z}{\partial\theta}\frac{\partial\theta}{\partial r^{\prime}}+\frac{\partial z}{\partial\phi}\frac{\partial\phi}{\partial r^{\prime}}=\frac{\partial z}{\partial r}\frac{\partial r}{\partial r^{\prime}}=\cos\theta\frac{R_{2}}{R_{2}-R_{1}}=\cos\theta^{\prime}\frac{R_{2}}{R_{2}-R_{1}}
[/tex]
[tex]
Q_{r^{\prime}}^{2}=\left(\frac{R_{2}}{R_{2}-R_{1}}\right)^{2}
[/tex]

[tex]
Q_{\theta^{\prime}}^{2}=\left(\frac{\partial x}{\partial\theta^{\prime}}\right)^{2}+\left(\frac{\partial y}{\partial\theta^{\prime}}\right)^{2}+\left(\frac{\partial z}{\partial\theta^{\prime}}\right)^{2}
[/tex]
[tex]
\frac{\partial x}{\partial\theta^{\prime}}=\frac{\partial x}{\partial\theta}=r\cos\theta\cos\phi=\frac{R_{2}}{R_{2}-R_{1}}\left(r^{\prime}-R_{1}\right)\cos\theta^{\prime}\cos\phi^{\prime}
[/tex]
[tex]
\frac{\partial y}{\partial\theta^{\prime}}=\frac{\partial y}{\partial\theta}=r\cos\theta\sin\phi=\frac{R_{2}}{R_{2}-R_{1}}\left(r^{\prime}-R_{1}\right)\cos\theta^{\prime}\sin\phi^{\prime}
[/tex]
[tex]
\frac{\partial z}{\partial\theta^{\prime}}=\frac{\partial z}{\partial\theta}=-r\sin\theta=-\frac{R_{2}}{R_{2}-R_{1}}\left(r^{\prime}-R_{1}\right)\sin\theta^{\prime}
[/tex]
[tex]
Q_{\theta^{\prime}}^{2}=\left[\frac{R_{2}}{R_{2}-R_{1}}\left(r^{\prime}-R_{1}\right)\right]^{2}
[/tex]

[tex]
Q_{\phi^{\prime}}^{2}=\left(\frac{\partial x}{\partial\phi^{\prime}}\right)^{2}+\left(\frac{\partial y}{\partial\phi^{\prime}}\right)^{2}+\left(\frac{\partial z}{\partial\phi^{\prime}}\right)^{2}
[/tex]
[tex]
\frac{\partial x}{\partial\phi^{\prime}}=\frac{\partial x}{\partial\phi}=-r\sin\theta\sin\phi=-\frac{R_{2}}{R_{2}-R_{1}}\left(r^{\prime}-R_{1}\right)\sin\theta^{\prime}\sin\phi^{\prime}
[/tex]
[tex]
\frac{\partial y}{\partial\phi^{\prime}}=\frac{\partial y}{\partial\phi}=r\sin\theta\cos\phi=\frac{R_{2}}{R_{2}-R_{1}}\left(r^{\prime}-R_{1}\right)\sin\theta^{\prime}\cos\phi^{\prime}
[/tex]
[tex]
\frac{\partial z}{\partial\phi^{\prime}}=\frac{\partial z}{\partial\phi}=0
[/tex]
[tex]
Q_{\phi^{\prime}}^{2}=\left[\frac{R_{2}}{R_{2}-R_{1}}\left(r^{\prime}-R_{1}\right)\right]^{2}\sin^{2}\theta^{\prime}
[/tex]

Finally:
[tex]
\epsilon_{r^{\prime}}=\epsilon\frac{\left[\frac{R_{2}}{R_{2}-R_{1}}\left(r^{\prime}-R_{1}\right)\right]^{2}\sin\theta^{\prime}}{\frac{R_{2}}{R_{2}-R_{1}}}=\epsilon\frac{R_{2}}{R_{2}-R_{1}}\left(r^{\prime}-R_{1}\right)^{2}\sin\theta^{\prime}
[/tex]
[tex]
\epsilon_{\theta^{\prime}}=\epsilon\frac{\left(\frac{R_{2}}{R_{2}-R_{1}}\right)^{2}\left(r^{\prime}-R_{1}\right)\sin\theta^{\prime}}{\frac{R_{2}}{R_{2}-R_{1}}\left(r^{\prime}-R_{1}\right)}=\epsilon\frac{R_{2}}{R_{2}-R_{1}}\sin\theta^{\prime}
[/tex]
[tex]
\epsilon_{\phi^{\prime}}=\epsilon\frac{\left(\frac{R_{2}}{R_{2}-R_{1}}\right)^{2}\left(r^{\prime}-R_{1}\right)}{\frac{R_{2}}{R_{2}-R_{1}}\left(r^{\prime}-R_{1}\right)\sin\theta^{\prime}}=\epsilon\frac{R_{2}}{R_{2}-R_{1}}\csc\theta^{\prime}
[/tex]

Any kind of help is more than welcome!

 

[bm0p700f]

That must have taken some time to type. sorry no actual help from me here.

 

[andresordonez]

@bm0p700f:

not more time than with a pencil, check this out: http://www.lyx.org/

 

[andresordonez]

Well, the extra [tex] \epsilon [/tex] (relative permittivity) is just because in the paper it is assumed to be 1 (vacuum or air)