Calculation of material properties in transformation media

[radiofeda]

Hi everybody,

I'm focusing on meta-materials. I have recently read Schurig's paper "http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-14-21-9794"). In the article, the components of the permittivity tensor are given by
$$\varepsilon^{i'j'} = \left|\rm{det}(\Lambda^{i'}_{i})\right|^{-1} \Lambda^{i'}_{i} \Lambda^{j'}_{j} \varepsilon^{ij}$$
where the Jacobian matrix
$$\Lambda_{\alpha}^{\alpha'} = \frac{\partial x^{\alpha'}}{\partial x^{\alpha}}$$
and the roman indices run from1 to 3, for the three spatial coordinates, as is standard practice.

Working out the algebra, the components of the permittivity (permeability) tensor can be obtained by
$$\left(\varepsilon^{i'j'}\right) = \left|\rm{det}\left(\Lambda\right)\right|^{-1}\Lambda^T \Lambda$$
where $$\Lambda$$ is a matrix, which components are the counterpart of the contravariant coefficients $$\Lambda_{\alpha}^{\alpha'}$$.

For cylindrical cloak, the components of the transformation matrix are
$$\left(\Lambda^{i'}_{j}\right) = \left( \begin{array}{ccc} \frac{\rho'}{\rho}-\frac{ax^2}{\rho^3} & -\frac{axy}{\rho^3} & 0 \\ -\frac{ayx}{\rho^3} & \frac{\rho'}{\rho}-\frac{ay^2}{\rho^3} & 0 \\ 0 & 0 & 1 \\ \end{array} \right)$$
It is easy to find the material properties. For instance, the z component of the permittivity tensor is
$$\varepsilon_z = \varepsilon^{3,3} = \frac{\rho^2}{\rho'(\rho'-a)} = \frac{1}{\left|\rm{det}\left(\Lambda\right)\right|}$$

However, in the paper "http://pre.aps.org/abstract/PRE/v74/i3/e036621" ), the components of the relative permittivity and permeability tensor specified in cylindrical coordinates are given
$$\varepsilon_z = \mu_z = \left(\frac{b}{b-a}\right)^2 \frac{\rho-a}{\rho}$$

It can be seen that the two formula are not equal obviously. And the other nonzero components of the permittivity and permeability tensor are not equal too.

I have deduced the formulas for many times. Depressingly, I can not figure out the problem. Could somebody please give me some comments on the calculation of material properties in transformation optics.

Regards.

 

[Andy Resnick]

I just glanced through the papers, but I wonder if you are looking in the wrong place: see, for example, eqns 20-22 and 29-30 in the OE paper.

 

[radiofeda]

I just glanced through the papers, but I wonder if you are looking in the wrong place: see, for example, eqns 20-22 and 29-30 in the OE paper.

I see. However, it seems not easy to obtain eq.(29) from eq.(26) by applying eq.(6) or (7).