# Determining path of light through a nanolens

In the case of light passing through stacked liquids, we can use a ray approximation of the path of light to determine the path taken through each layer of liquid using Snell's law. From all the examples I've seen, this is done when the lengths considered are larger than the wavelength of incoming light.

I'm working in a lab where are are creating a photothermal lens around nanoparticles for microscopy purposes. I'm curious if I can use the same "geometric optics" (i.e. with rays) approach to determine which path light will take as it passes through the lens. As an example, suppose we take the radius of the spherical lens to be 1 micrometers.

There seems to be no literature regarding when a geometric approach is wrong. At what point does using rays of light not give the right information (i.e. what lengths are we speaking of relative to the wavelength)?

• It seems like you got the right answer, already, but I would add one simple rule for scientific imaging: the geometric approach is ALWAYS wrong. Whether you are looking at microscopes or telescopes, the wave nature of light always sets the performance limits of your imaging system. If it doesn't, then you either don't have to care about imaging performance (e.g. in light collimation systems) or you have a poorly designed system. – CuriousOne Jul 3 '15 at 15:22

The ray theory of light is equivalent to the Eikonal Equation, which in turn is essentially a slowly varying envelope approximation to Maxwell's equations. If we write the electric and magnetic field vectors as $\mathbf{E}\left(\mathbf{r}\right) = \mathbf{e}\left(\mathbf{r}\right) e^{i\,\varphi\left(\mathbf{r}\right)}$, $\mathbf{H}\left(\mathbf{r}\right) = \mathbf{h}\left(\mathbf{r}\right) e^{i\,\varphi\left(\mathbf{r}\right)}$ ($\mathbf{e}(\mathbf{r})$ and $\mathbf{h}(\mathbf{r})$ are the "envelope" functions) and assume a monochromatic field (so the implicit time dependence is $e^{-i\,\omega\,t}$, then the conditions for soundness of the Eikonal equations are:

$$\left|\mathbf{e}\right|^{-1} \left|\nabla \times\mathbf{e}\right| \ll \left|\nabla \varphi\right| \approx \left|k\right|$$ $$\left|\mathbf{h}\right|^{-1} \left|\nabla \times\mathbf{h}\right| \ll \left|\nabla \varphi\right| \approx \left|k\right|$$

You can very quickly tell that your $1{\rm \mu\,m}$ microlens is going to impose a phasefront on the field that wholesale violates the above conditions.

In short, ray otpics: forget it!

I show how to derive these conditions in my answers here and here.

You are in a lucky position, though, as your lenses are not much bigger than a wavelength. Therefore, full numerical simulations using finite element version of Maxwell's Equations are more than practicable, because you can mesh your system with elements well smaller than a wavelength and still have a simulation that will be swift. So here's how I would be going about the problem.

1. You MAY get some joy from simple analytical models grounded on Mie theory of scatterers. Born and Wolf, Chapter 14 has everything you need to know about Mie theory: your lenses, as nominal spheres are very much the kind of thing described by Mie theory of scattering. I would imagine a Mathematical workbook grounded on Mie theory as a rough guide and sanity check as you build more sophisticated simulations.

2. MEEP, an MIT baby (with a LISP interface - of course: what else?) for Maxwell equation solution by finite difference time domain simulation, is widespread and, once you get up to speed, would be a powerful softtool for your problem. I believe there is a Python interface for it, which would be much more fitting to the problem than LISP.