Aim: To estimate the refractive index of a glass slide by analysing the magnitude and polarisation of reflected light from the optical interface. Introduction: In a study of optics, an important quantity of materials is the refractive index. The refractive index of an optical medium is defined as the ratio of the speed of light in vacuum and the phase velocity of the medium (Hecht 1979):

It also allows one to relate the optical properties of a medium to its dielectric properties (Fox 2006). The deviation of a monochromatic incident beam into an optical material is described by Snells Law:

Fresnel Equations describe the behaviour of light moving between two media of differing refractive indices:Estimation of refractive index by analysis of interface behaviour in response to incident light

Cameron Knox 11377987

These equations describe the fraction of reflected and refracted light at an interface and the phase shift induced by surface interactions of the reflected light. The reflection and transmission are related by the following through simple or conservation of energy arguments:

It can be shown that for a particular interface, the Brewsters Angle is the angle of incidence which results in the least reflection from the interface of p-type polarisation:

This is somewhat useful for transparent materials, but this is not particularly useful for opaque or complex materials such as multi-layer thin films as there may be several minimum points of reflection. Ellipsometry offers a way to measure the phase and amplitude of reflected light at an interface. The refractive index can be determined by finding the ellipsometric parameters in the relation:

Where psi represents the ratio of reflectances and delta represents the phase difference of the reflected light. These are determined by determining the Fourier Coefficients of the intensity expression:

And the refractive index is given by:

MethodsTwo methods were used in determining the refractive index of the sample: Angle resolved remittance EllipsometryBoth methods are somewhat advantageous as they do not require the measurement of the transmitted beam, they are non-contact and non-destructive. This make the techniques useful for determining the refractive index of complex media or thin films as it is not always practical to measure refracted light. Not only this, the non-destructive and non-contact features have developed interest in medical and biological fields where this is important. One medical example is the surface characterisation of polymers (Werner& Jacobasch, 1999)Angle resolved remittance utilises a setup as shown in figure 1. The incident beam is polarised to either s or p linear polarisation with a linear polariser. The reflection is then measured with a lux meter. Reflection is measured for a range of angles, about 10 i89. The starting angle will often be limited by the size of the apparatus. A profile is developed by plotting the relative reflection (Er/Ei) against the angle of incidence. This process is repeated for the opposite polarisation. Figure 1

Ellipsometry involved measuring the phase change of reflected light from the surface as well as the relative strength of the reflected light. The experimental setup is similar to what is shown in figure 2. Rather than changing the incident angle, a rotating analysing polariser determines the polarising angle of the reflected light. The initial polariser is set to 45, which is comprised of a superposition of p and s polarised light (Feynman, 1963). As s and p light interact differently with the surface (a change in the wave phase due to electric field interactions with the surface), one would expect the reflection to be attenuated in a certain direction, giving rise to an elliptical phasor diagram. Figure 2

Results and AnalysisThe results were analysed using a Matlab script written by the subject lecturer Matt Arnold. The scripts use a non-linear regression fit to estimate the refractive index based upon Fresnels equations and the ellipsometric parameters. The angle resolved remittance script was run a few times, using the output as the new guess for the next exectution. By using this method, the uncertainties in the estimated refractive index were minimised. The estimated refractive index was determined to be 1.53+0.07i 0.01+0.02i, assuming that the refractive index of air is 1. This would appear to be a reasonable result, as common or crown glass is cited to have a refractive index of about 1.523 (Opticians Friend, 2005). The remittance profile is shown in figure 3. Figure 3

The imaginary component of the refractive index would appear to be rather high. Most citations (Schubert, 2004) dont quote an imaginary value for glass, and it is evident from everyday experience that glass doesnt absorb much visible light at all. For this reason it would be reasonable to suggest that the uncertainty in the imaginary component is much higher than calculated by the script.When the orientation of the light is near normal to the interface, the reflection of s and p polarised light is practically equal. As the incidence changes, the amounts become distinguishable due to the electric field alignment having a significant interaction with the surface. As the incidence becomes much steeper (towards a glancing angle), the interactions tend to favour reflection.Figure 4

Even though the laser that were used are quite stable, there is some fluctuation in the over the period of measurement. This could be improved quite simply by taking the measurements faster. However this would introduce errors into the angle of incidence. I would suggest a completely automated data logging system, such as a LABView program. This could take many more data points and do such over a much shorter period to ensure that the laser power didnt deviate too much. As can be seen from figure 4, the Brewster angle occurs at approximately 56.8. This correlates with a calculated estimate refractive index of 1.528 from equation 7. For the ellipsometric data it was initially expected that the reflected beam polarisation would shift in angle, but due to the presence of both p and s light, the intensity of this beam would never quite go to zero. As can be seen in figure 5, this isnt a correct assumption. The polar plot of the intensity clearly shows a drop to zero. The polar plots also show that the p-s axis (x-y) has been rotated. The trend indicates that the degree of rotation of the polarisation axis gets smaller towards larger incident angles. As the measurements were taken close to Brewsters angle, one would expect the reflected light to be made up of mostly s-type as the p-type would be almost completely transmitted through the glass. This would suggest that the reflected beam is linearly polarised in the s (rotated s) axis. This occurs with elliptical light, at a phase difference of =0, , 2 etc, which is now not really elliptical, but linear. Elliptical light is a general case of the other common polarisations, circular and linear (Hecht, 1979). Figure 5

The analysis script didnt work out so well, however the ellipsometric parameters seemed to calculate a reasonable answer of n=1.5126 using an uncertainty of approximately 10%. This is relatively close to the value estimated by the angle resolved remittance. Figure 6 shows the fit generated from the ellipsometric parameters. Figure 6

The experiment may have been more accurate by analysing the reflected flux for incident p and s separately and averaging the components. A better understanding of the concept may have been gained from repeating the experiment for other materials, such as those with a significant imaginary part of the refractive index. Like in the angle resolved remittance, an automated system may have improved the accuracy of the measurements by minimising the change in flux of the laser.

Reference List1) E. Hecht, A. Zajac (1979). Optics. Addison-Wesley.2) Fox, Mark (2006). Quantum Optics: An Introduction. Oxford University Press.3) E. F. Schubert (2004). Materials-Refractive-index-and-extinction-coefficient.pdf. Internet www page @ URL: http://homepages.rpi.edu/~schubert/Educational-resources/Materials-Refractive-index-and-extinction-coefficient.pdf (07/09/2014)4) Opticians Friend (2005). Refractive Indices and Lens Materials. Internet www page @ URL: http://www.opticiansfriend.com/lenses.html (01/09/2014)5) Werner, Jacobasch (1999). Surface characterization of polymers for medical devices, The International Journal of Artificial Organs, 22(3):160-1766) R. Feynman (1963). Feynman Lectures on Physics Volume I: Mainly Mechanics, Radiation and Heat. Basic Books.7) M. Arnold lecture notes and lab manuals