# Representing a 3D system in 2D (Electromagnetic modelling)

Ok so I'm a complete beginner in computational modelling (I use analytical methods of physics typically) but I would like to model an anisotropic, aperiodic (but not random) finite array of metallic nanorods, all of which have their long axis orientated in the same direction as eachother, but this direction can be completely arbitrary.

What I am interested in is the charge oscillations in each nanorod (localised surface plasmons) which can electromagnetically couple to neighbouring nanorods and generate a collective charge density wave (plasmon/polariton) which can in principle propagate through the system. For example I could impose a periodic driving of the charge density in one nanorod at a particular frequency and see how it decomposes into the different allowed momentum states of the system and propagates. I want to have a simulation where I can see the propagation of these waves.

The problem is that because of the aperiodicity I can't use periodic boundary conditions. I believe this means that a full 3D electromagnetic simulation is too computationally heavy (I have a good spec iMac but nothing better). So I assume I have to do a 2D model, but I have no idea if this makes any sense for my system and how to implement. These nanorods in principle can have any orientation in 3-dimensions and the EM field is intrinsically 3D so how can I map this problem to 2D. How reliable is this approach for modelling the reality?

I'm hoping for some suggestions on what software to use and how to get started and perhaps some insight into this 3D to 2D mapping problem. I have access to COMSOl but happy to try other (free) software.

• If you are used with analytical model. What would you do analytically? – nicoguaro Mar 9 '16 at 15:04
• Model the charge oscillation of each nanorod as a point dipole, work in the quasistatic limit, neglect retardation, assume nearest neighbour dipoles are aligned, only take into account nearest neighbours. This is brutal enough simplification that I can fully solve for the eigenmodes etc. and yes I could make a simulation using these simplifications but it would be nice to see a more realistic electromagnetic simulation showing the same qualitative results. – Tom Mar 9 '16 at 15:09
• Can you separate the equations for the in-plane and out-plane fields? Because, in that case you would have a 2D problem that you can solve using the FEM and the other dimension is solved analytically. – nicoguaro Mar 9 '16 at 15:20
• Hmm I'm not sure. I'm also not sure exactly what you mean and how it would work/make sense to solve out-of-plane analytically and in-plane numerically. In a perfect world with infinite computation power I would just tell the machine the geometry and material of each nanorod in 3D, tell it how they're arranged, tell it about Maxwell's equations and then solve. But I'm struggling to see how to simplify this problem. I can kind of see that if they were orientated out of the plane then maybe 2D makes sense but with an arbitrary orientation I'm not sure. I realise I'm rambling! – Tom Mar 9 '16 at 15:27
• If you have a problem of the form $u(x,y,z) = Z(z) U(x,y)$ for Maxwell equations then you can solve in that way. I don't see what is the difference about the orientation, just change the plane that you are solving for. – nicoguaro Mar 9 '16 at 15:30