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:

<br /> \epsilon&#039;_{r&#039;} = \epsilon \frac{R_2}{R_2-R_1} (r&#039;-R_1)^2 \sin(\theta&#039;)<br />
<br /> \epsilon&#039;_{\theta&#039;} = \epsilon \frac{R_2}{R_2-R_1} \sin(\theta&#039;)<br />
<br /> \epsilon&#039;_{\phi&#039;} = \epsilon \frac{R_2}{R_2-R_1} \sin(\theta&#039;)<br />

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

<br /> \epsilon&#039;_{r&#039;} = \frac{R_2}{R_2-R_1} \frac{(r&#039;-R_1)^2}{r&#039;}<br />
<br /> \epsilon&#039;_{\theta&#039;} = \frac{R_2}{R_2-R_1}<br />
<br /> \epsilon&#039;_{\phi&#039;} = \frac{R_2}{R_2-R_1}<br />

The difference between these equations and the ones I get is not only the missing r&#039; and the extra sin(\theta&#039;) but also the extra \epsilon

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

<br /> r^{\prime}=R_{1}+r\frac{\left(R_{2}-R_{1}\right)}{R_{2}}<br />
<br /> \theta^{\prime}=\theta<br />
<br /> \phi^{\prime}=\phi<br />

The permittivity transforms according to:
<br /> \epsilon_{r}^{\prime}=\epsilon\frac{Q_{\theta&#039;}Q_{\phi&#039;}}{Q_{r&#039;}}<br />
<br /> \epsilon_{\theta}^{\prime}=\epsilon\frac{Q_{r&#039;}Q_{\phi&#039;}}{Q_{\theta&#039;}}<br />
<br /> \epsilon_{\phi}^{\prime}=\epsilon\frac{Q_{r&#039;}Q_{\theta&#039;}}{Q_{\phi&#039;}}<br />

where Q_{u} is given by:
<br /> 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<br />

Then:
<br /> 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}<br />
<br /> \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}}<br />
<br /> \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}}<br />
<br /> \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}}<br />
<br /> Q_{r^{\prime}}^{2}=\left(\frac{R_{2}}{R_{2}-R_{1}}\right)^{2}<br />

<br /> 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}<br />
<br /> \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}<br />
<br /> \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}<br />
<br /> \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}<br />
<br /> Q_{\theta^{\prime}}^{2}=\left[\frac{R_{2}}{R_{2}-R_{1}}\left(r^{\prime}-R_{1}\right)\right]^{2}<br />

<br /> 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}<br />
<br /> \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}<br />
<br /> \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}<br />
<br /> \frac{\partial z}{\partial\phi^{\prime}}=\frac{\partial z}{\partial\phi}=0<br />
<br /> Q_{\phi^{\prime}}^{2}=\left[\frac{R_{2}}{R_{2}-R_{1}}\left(r^{\prime}-R_{1}\right)\right]^{2}\sin^{2}\theta^{\prime}<br />

Finally:
<br /> \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}<br />
<br /> \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}<br />
<br /> \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}<br />

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 \epsilon (relative permittivity) is just because in the paper it is assumed to be 1 (vacuum or air)