Re: Cloaking
Hi iaia!
Ok, I understand - you are not doing the nonmagnetic cloak and the paper from APL (Cai and Shalaev) you mentioned only because it deals with the second order mapping.
Yes, you need to substract the incident field (plane wave) from the total field and only then to integrate. This substraction is the same as in the example from the Comsol's EM user guide (the one from which I guess you took the parameters for PML) - the scattering on a dielectric scatterer or however it is called.
Both mu and epsilon are anisotropic and both are equal, yes. I guess you know that in the 2D geometry (i.e. when all the quantities are invariant along the z-axis) where the anisotropy can be described by diagonal tensors (like here in the cylindrical coordinates) the TE and TM modes are decoupled. In other words, you can define TE and TM waves. For TE waves only eps_z (because E has only a z component) and mu_xx,mu_xy,mu_yx,mu_yy (because H has components only in the x-y plane) matter. The other components of the eps and mu tensors do not infuence the TE wave. For TM waves it's mu_z and eps_xx,yx,xy,yy. This is exactly what Comsol offers you to set in these simulations. In other words: you don't have to use the hybrid mode. I don't say it's not good - that's some other thing, I'll have to sit down and read a bit and then I will tell you. Right now, I can't do that.
Ok, back to mapping. Do you understand me what I mean by the mapping? You can put it this way: the mapping is the thing that determines the parameters of the cloak. The number of possible realizations of a cloak with given parameters (R1 - the inner radius and R2 - the outer radius) is the same as a number of functions that map some parameter rho to the radial coordinate r. Ok, these mappings have to satisfy some basic assumptions, but my point here is that there is infinitely many mappings. The cylindrical cloak from the PRE paper (Cummer, the first reported simulation) is the simplest case - for a linear mapping. In the APL paper they discuss I think general mappings and particularly the second order mapping. Theoretically, all these mappings should give ideal cloaks, however it's not so simple because any realistic cloak, including the one implemented in a numerical simulation in Comsol is not perfect. I think there are different kinds of imperfections, but I wouldn't go into that now. So, again, we come to the point that weird things can happen if you use nonlinear mappings. I already mentioned you that I noticed that a exponential-like mapping leads to funny behaviour.
I hope that now you understand that there is no reason to think that the cloak discussed in the APL paper (second order mapping) should give the same results as the first order cloak (i.e. mapping) from the PRE paper. My advise is that you establish a firm criteria for the cloak quality (like the scattering cross section) and that you then compare different cloaks. However, I must warn you that it is highly probable that the mere mesh implementation will lead to very different results. You have to figure out some way to be systematic about it. Perhaps you should read the Phys. Rev. papers published recently on the cylindrical cloak:
Phys. Rev. Lett. 99 113903 (Z. Ruan et al.)
Phys. Rev. B 76 121101 (B. Zhang et al.)
There you go
I could have already written a paper on cloaks
Regards
Added after 37 seconds:
can you access these papers? do you want me to send them to you?