visible light - Relation between wavelength and system size

We always say that when a given light wave interacts with atoms bound in a molecule, only waves with wavelength close to the inter-atomic-spacing are able to probe the system. In other context (macroscopic oscillations in a system), one also talks about the wavelength of some fluctuation in a system being larger than the system size, in which case such fluctuations are omitted/ignored.

Questions:

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  What is it that links the wavelength of a wave to its interaction with a system? Be it acoustic waves or EM. Physical intuition would be greatly appreciated, but please don't hesitate showing the math behind it as well, if you see it fit!

  
  


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  How does one go about quantifying such problems? i.e. if I have $\lambda_1$ slightly larger than system size $d$, or less larger, how do I conclude whether to consider such oscillations in the system or not?

  
  

Answer

In principle, a wave of any size will interact with a system of any size. The question should therefore be posed differently: how is the interaction of the two affected by their relative size?

Let's take the simple example of scatter. You are familiar (whether you know it or not) with Rayleigh scatter - it's an elastic light scattering phenomenon that makes the sky blue. The Rayleigh scatter cross section (effective probability of interaction) is given by

$$\sigma = \frac{2\pi^4}{3}\frac{d^6}{\lambda^4}\left(\frac{n^2-1}{n^2+2}\right)^2$$

In this equation, $d$ is the diameter of the particle, $n$ is its refractive index, and $\lambda$ is the wavelength of the scattered light. This expression applies when $d<<\lambda$ - typically 1/10th or smaller. So right there we have an interaction that occurs with a wave that's much bigger than the "system" (the particle, in this case).

As wavelengths become shorter, the scattering mechanism is better described as Mie scatter (wavelength on the same order as the particle) - this is characterized by resonances, meaning that some sizes of particles will scatter better than others, but it's not a monotonic relationship (like for Rayleigh scatter).

At even shorter wavelengths, light (or other waves, e.g. acoustic waves) start to behave more "normally" - that is the regime you usually think about when you are talking about direct visualization, optical microscopy, ultrasound imaging etc.

But just because the interactions of long waves with small objects don't easily make pretty images doesn't mean they are not happening - the physics may be a bit harder, and the interaction more statistical and less deterministic, but nonetheless - they do interact.