I've been looking at the field of metamaterials for quite a while, now. Most of it is predicated on the assumption (which is empirically demonstrable) that electromagnetic waves, when they reach a structure significantly beneath the wavelength of the light, 'feel' the material as being a homogeneous medium, with an effective refractive index. However, I'm having some difficulty understanding this. I often see equations such as those for plane waves (see below). However, I don't see how these provide any hints towards my problem about resolution into a homogenous medium, or any upper bound on the structure size of said medium. $$\boldsymbol{E}(\boldsymbol{r},t)=\boldsymbol{E_0}cos(\omega t-\boldsymbol{k}\cdot \boldsymbol{r} + \phi _0)$$ $$\boldsymbol{B}(\boldsymbol{r},t)=\boldsymbol{B_0}cos(\omega t-\boldsymbol{k}\cdot \boldsymbol{r} + \phi _0)$$ I've also seen several other equations which simply appear to treat light waves as existing at a single point, and not actually taking up any space.

I've taken a look at the diffraction limit, but I can't seem to find any logical reasoning for how this could be applicable (as its derivation appears to be based around the distance between the feature and an objective - well, here there is no objective), as well as applying the Heisenberg uncertainty principle (which either leads me in rather cyclic logic, or which references the patterns observed on a screen - which I again, don't see how it helps).

In short, I understand how a the fields of a light-wave interact with a barrier, but I can't find a reason why these interactions should somehow be bounded, in terms of their range, by the wavelength enough resolve periodic subwavelength structures into an effective homogenised medium.

Thank you for your time, and please alert me in the comments if I've phrased this badly!

  • $\begingroup$ I think it mainly has to do with the fact that the derivation of index of refractions assumes an homogeneous field across the (meta)-atom: it means that only the field at the point is important and not its gradient or any higher order spatial derivative. If the field varies to much across the (meta)-atom, you need (at first approximation) to consider its gradient in addition to its mean value and it complicates a lot the model. The goal of meta-atoms is to be able to assign to it a effective permittivity and permeability that would give a specific index of refraction in the same way as a atom. $\endgroup$
    – EigenDavid
    Commented Jun 28, 2018 at 9:56
  • $\begingroup$ Look at the Lorentz oscillator model for the refractive index, as well as the dipolar approximation. You can also google "metasurface homogenization", there are a few results that should help you $\endgroup$
    – EigenDavid
    Commented Jun 28, 2018 at 9:57
  • $\begingroup$ I've taken a look at the Lorentz Oscillator model - to my understanding, it basically just uses the assumption of a time-varying EM field - but I don't see how that proves the fact that light produces an EM field external to the photon itself - can't find much in the way of formulae which include fields external to the point of a photon, without including the entire universe in a plane-wave style approximation. $\endgroup$ Commented Jun 28, 2018 at 10:00
  • $\begingroup$ I'll check out metasurface homogenisation, though! $\endgroup$ Commented Jun 28, 2018 at 10:00
  • $\begingroup$ I suppose a good answer to my question could be simply: Is there an equation for EM waves which predicts subwavelength effects, such as surface homogenisation. $\endgroup$ Commented Jun 28, 2018 at 10:02


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