# Effect of the refractive index of a transparent layer coating a particle?

I am modelling a cell as a particle coated by a layer of a material whose refraction index might change. This layer is to resemble the membrane.

The purpose of the model is to simulate what happens when a laser beam meets the particle and is reflected back.

Unfortunately I am not sure on how to model the effect the refraction index of the layer. Specifically what happens after both blue arrows in the previous image.

I am lost regarding how to proceed, and my thesis supervisor, who is a doctor in Physic is unavailable for one month still.

How can the direction change of the laser be modelled?

• Snell's law? What more are you looking for? – Rob Jeffries Aug 24 '15 at 10:51
• Are you saying the refractive index changes inside the membrane, or is there a discontinuity at the surface of the membrane? If it's the latter, you should simply apply Snell's Law $\sin\theta_1/n_1 = \sin\theta_2/n_2$; if it changes internally, the net change of direction is still the same (by the time the index changes from $n_1$ to $n_2$ the direction will have changed as though the change was instantaneous) but the reflectivity will be different. Are you trying to understand the direction, intensity, displacement of the laser beam? Which is it? Don't want to answer the wrong question. – Floris Aug 24 '15 at 12:57
• @Floris the whole membrane has one refractive index, and there is a discontinuity at the surface of the membrane. And yes, I am trying to understand the direction, intensity, and displacement of the laser beam. – Keine Aug 24 '15 at 13:04
• IN addition to Snell's law, if the membrane thickness is on the order of the wavelength of your laser, you'll have to consider self-interference. See, e.g. Fabry-Perot etalon. – Carl Witthoft Aug 24 '15 at 13:12
• @CarlWitthoft, the laser wavelength is 780mn, the modelled membrane will have a thickness between 1 and 3 micrometers. – Keine Aug 24 '15 at 13:21

The diagram you want to use looks something like this:

Depending on how much attenuation there is in the membrane, you need to consider the potential of multiple reflections (or not).

I actually analyzed this problem in some depth - considering not only the intensity of reflections on the different surfaces, but also multiple reflections and even the effect of polarization. The intensity of the reflection of a laser off a refractive index discontinuity is definitely dependent on the polarization and should be considered in your model (even if you end up averaging over all polarization angles).

See whether the answer I wrote before addresses your question. If it does not, use the comment section for further clarifications.

• Great answer, thanks. Can the reflection off a a refractive discontinuity be disregarded?. My model includes a a way to find the reflected intensity of a laser beam off the cell surface (red volume in your graph) according to the incoming Intensity, Incoming-Angle, and Outgoing-Angle. ![Valid XHTML](http:i.imgur.com/lZ5RFej.png) In other words, if I can disregard E in the previous image, and only take into account the change of direction of the laser beam, I can calculate all other values. How can I estimate the error I would incurr by doing so? – Keine Aug 24 '15 at 15:20
• The error you incur will depend on the refractive index mismatch between $n_0$ and $n_1$ in my diagram. The full solution (no approximations / errors) is given in the linked answer. – Floris Aug 24 '15 at 15:22
• @Keine the reflectance at each interface (barring multiple reflection interference effects) follows a simple formula as provided at en.wikipedia.org/wiki/Fresnel_equations . You can use that to estimate the losses along the light rays' paths – Carl Witthoft Aug 24 '15 at 15:23
• Accckkthwwwpt Floris beat me to it :-) – Carl Witthoft Aug 24 '15 at 15:23
• Your laser is most likely linearly polarized. Whether it is S or P will depend on the setup: if normal to surface, direction of laser, and polarization vector are all in the same plane, it is S. See slide 4 at this link for a diagram. If it is "random" (or circularly polarized) you need to average the S and P results. – Floris Aug 25 '15 at 12:26