To experts in EM theory.
I’m fairly certain that for a biaxial material, conservation of energy restricts the imaginary parts of the the permittivity and permeability tensors. Specifically, let us assume the exp{+jwt} time harmonic convention and, for a biaxial material, define the permittivity tensor by diag(exx,eyy,ezz) where exx = exx’ – j*exx’’, eyy = eyy’ – j*eyy’’, and ezz = ezz’ – j*ezz’’. Then, it’s probably well known that conservation of energy requires exx’’ >=0, eyy’’ >=0, and ezz’’ >=0. Similarly for the permeability tensor.
More generally, I'm guessing that for general 3x3 permittivity and permeability tensors characterizing an anisotropic media, conservation of energy places the same restriction the imaginary parts of their eigenvalues. Is this correct?
Most importantly, I'm looking for a reference that explicitly states what restrictions (like the above) on material parameter tensors result from physical considerations such as conservation of energy. Hopefully, such a reference will verify my guess concerning eigenvalues of the material parameter tensors.
Thank you in advance for any light you can shed on this issue.
Greg.
First, there are 9 numbers in a tensor, but there are only three degrees of freedom. That is, there are really on three numbers in a tensor. We live in a three-dimensional world so we can only have different material properties in three different directions. The other numbers in a tensor arise only when the material is being analyzed in a coordinate system that does not align with the principle axes of the anisotropic material. In other words, the tensors have been rotated in some manner. This makes the tensors always symmetric about their diagonal.
The imaginary component is not restricted unless you make further simplifications, such as being a passive medium. Depending on what sign convention you are working with, that will force the imaginary components to be all negative or all positive so that it conveys loss and not gain. Some more special cases arise when the material is gyrotropic, or chiral.
Here are some references for you. The first one is a book chapter that I wrote and is intended to be an introduction to some very advanced topics, but I step through anisotropy, tensors, rotation, bianisotropy, and related topics I think (hope) pretty well.
Raymond C. Rumpf "Engineering the Dispersion and Anisotropy of Periodic Electromagnetic Structures," Solid State Physics, Vol. 66, pp. 213-300, 2015.
Eroglu, Abdullah. Wave propagation and radiation in gyrotropic and anisotropic media. Springer Science & Business Media, 2010.
Mackay, Tom G. Electromagnetic anisotropy and bianisotropy: a field guide. World Scientific, 2009.
Hope this helps!
First, there are 9 numbers in a tensor, but there are only three degrees of freedom. That is, there are really on three numbers in a tensor. We live in a three-dimensional world so we can only have different material properties in three different directions. The other numbers in a tensor arise only when the material is being analyzed in a coordinate system that does not align with the principle axes of the anisotropic material. In other words, the tensors have been rotated in some manner. This makes the tensors always symmetric about their diagonal.
The imaginary component is not restricted unless you make further simplifications, such as being a passive medium. Depending on what sign convention you are working with, that will force the imaginary components to be all negative or all positive so that it conveys loss and not gain. Some more special cases arise when the material is gyrotropic, or chiral.
Here are some references for you. The first one is a book chapter that I wrote and is intended to be an introduction to some very advanced topics, but I step through anisotropy, tensors, rotation, bianisotropy, and related topics I think (hope) pretty well.
Raymond C. Rumpf "Engineering the Dispersion and Anisotropy of Periodic Electromagnetic Structures," Solid State Physics, Vol. 66, pp. 213-300, 2015.
Eroglu, Abdullah. Wave propagation and radiation in gyrotropic and anisotropic media. Springer Science & Business Media, 2010.
Mackay, Tom G. Electromagnetic anisotropy and bianisotropy: a field guide. World Scientific, 2009.
Hope this helps!
Dear Dr. Wilson,
One paper which I have used in the past is
Jin Au Kong , Theorems of bianisotropic media, Proceedings of the IEEE, Sept. 1972
It contains the restrictions that you mention due to energy conservation. It also contains the ones due to reciprocity, which could apply to your cases as well since most media are reciprocal.
Ivan
The following papers may offer some assistance:
Babak Alavikia, Ali Kabiri, and Omar M. Ramahi, "Poynting theorem constraints on the signs of the imaginary parts of the electromagnetic constitutive parameters," Journal of the Optical Society of America A, vol. 32, pp. 522-532, 2015.
Ricardo A. Depine and Akhlesh Lakhtakia, "Poynting theorem constraints on the signs of the imaginary parts of the electromagnetic constitutive parameters: comment," Journal of the Optical Society of America A, vol. 32, pp. 1564-1565, 2015.
Babak Alavikia, Ali Kabiri, and Omar M. Ramahi, "Poynting theorem constraints on the signs of the imaginary parts of the electromagnetic constitutive parameters: reply," Journal of the Optical Society of America A, vol. 32, pp. 1566-1566, 2015.
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