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Help me with Pendry's cloaking article equations

  1. Jan 2, 2012 #1
    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!
     
  2. jcsd
  3. Jan 2, 2012 #2
    That must have taken some time to type. sorry no actual help from me here.
     
  4. Jan 2, 2012 #3
    @bm0p700f:

    not more time than with a pencil, check this out: http://www.lyx.org/
     
  5. Jan 3, 2012 #4
    Well, the extra [tex] \epsilon [/tex] (relative permittivity) is just because in the paper it is assumed to be 1 (vacuum or air)
     
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