Abstract

I measured the index of refraction in three glass shapes by means of four different methods, which were measuring the path of a laser through a square block, measuring total internal reflection in a semi-circular block, finding minimum deviation angle of a prism and measuring apparent thickness of the square block.  I found the indexes of refraction respectively to be 1.497 +/- 0.0738, 1.4688 +/- 0.00112, 1.35346 +/- 0.00415, and 1.528 +/- 0.011.  Thus I found that measuring the angle of total internal reflection offered the most precision in determining the value of the index of refraction.

Introduction

This experiment was conducted with the purpose of looking for methods of precisely determining the index of refraction of a glass object through simple means.  As stated above it essentially consisted of four separate experiments and then comparing the level of precision arising in each one.  The first method involved the use of a laser to trace the path of light into a square block and using the angle of refraction and the angle of reflection at the glass-air interface to compute the value of n (the index of refraction of the glass).  The second portion of this experiment used a semicircular glass block and a laser to calculate the angle of total internal reflection of the glass by adjusting the angle of the straight side of the block relative to that of the path of the laser until the laser no longer passed through the block, and from this angle calculating the value for n.  The third portion of this experiment was to find the minimum deviation angle of a glass triangular prism again using a laser.  During this experiment I adjusted the angle between the laser and the surface of the prism and followed the path of the laser on a backboard, determining the minimum deviation angle by looking for the angle at which the laser's direction of travel reversed relative to continued change of the interface angle.  In the last portion of the experiment the index of refraction was determined by using a microscope and determining the apparent height change caused by inserting a glass block between the microscope and a mark upon which it was focused.  The results of these seperate experiments are then compared in order to judge what method is best for measuring the index of refraction.

Experiment

Section 1:  The first portion of the experiment consisted of a laser, a horizontal cork board, and a vertical stopping board, mounted on a optics track, with a sheet of paper and a square glass block positioned on the cork board.  The items were positioned in such a way that the laser passed through the square block along its length, with the angle of interface between the block and the laser not perpendicular (Fig. 2).  Pins were then depressed in the cork board at key points along the path of the laser so one could determine the angles of refraction and reflection generated by the glass-air interfaces as shown below (Fig. 1).  Using this the path of the laser rays and the position of the block was traced using a sharp pencil.  Then the necessary angles and lengths were determined using a protractor and ruler respectively.  Since angle of incidence is equal to angle of reflection, a (see Fig. 1) was found by using the average of the reflection and incidence and combining the errors.  The thickness of the block was also measured using calipers.  The experiment was repeated for a total of three different angles of the glass-air interface.

Section 2:  In this portion of the experiment, a laser, semicircular glass block, rotating table with angular scale, and a stopping board were arranged on an optics track so as to allow the laser to pass into the curved surface of the block, through the center, and out the straight surface onto the stopping board (Fig. 3).  The block was centrally positioned on the rotating table, and thus allowed one to record the angle the laser made with the back of the block.  While rotating the block there existed some critical angle for which the glass would exhibit total internal reflection and thus the laser beam would not exit the block, this was measured by rotating the block in both directions and finding the point at which all the coherent light spot on the stopper board disappeared.  These angles were then recorded, and the critical angle was taken to be half of the range between the two measurements.

Section 3:  The third portion of the experiment used essentially the same set up as the second ecept that the semicircular block was replaced with a flat triangular prism arranged on the rotating stand in such a way that the sides of the triangle were perpendicular to the ground (Fig. 4).  The laser was passed through the prism, which was rotated on the stand in an attempt to locate the minimum deviation angle of the prism.  This was accomplished by first aligning the prism and the laser beam such that the beam would retroreflect back unto the beam generator, this was taken as the base angle, and then the prism was rotated so that the spot representing the laser's path on the blocking board moved along the board.  The necessary angle was found by locating the angle for which the direction of motion of the laser's path changed direction, the minimum deviation angle was thus found to be the difference in these two values.  The angle A, which was the angle of the corner of the prism was also found experimentally by recording the number of degrees passed through while moving from one retroreflection point to the next.

Section 4:  The final phase of this experiment used a microscope with adjustable height and focus to determine the apparent height of the same glass block used in section 1.  The microscope was focused on a mark made on its image table and this height was recorded using the scale available on the microscope, the glass block was then placed over the mark and the height of the microscope was repositioned, without changing the focus, until the image was again in focus in the microscope.  The difference between these measurements and additionally the thickness of the glass block as measured by calipers were then used to calculate the index of refraction of the block.

Results/Analysis

Section 1:  It is well known that when light goes from one material to another, its path is dictated by Snell's law that n₁*Sin(a) = n₂*Sin(b), where n₁ is the index of refraction of the material it is leaving, n₂ is the index of refraction of the material it is entering, a is the angle of incidence and b is the angle of refraction (See fig 1 for a and b position).  The index of refraction of a vacuum is defined to be exactly 1.  The index of refraction of air at STP is approximately 1.000293 (Serway, Physics for Scientists and Engineers 4th Edition).  Since all of these experiment were conducted at temperatures above 0°C, the air would necessarily be less dense and its index of refraction would be closer to 1.  Since the accuracy of these experiments is less than the difference between 1.000293 and 1, the index of refraction of the air in all that follows is assumed to be 1.  Therefore we get n = Sin(b)/Sin(a). The results obtained thus were :

Using a c^2 fit to a constant value the best approximation for n was found to be n = 1.4972 +/- 0.0738

With this data one can also find the expected deviation from a straight line (Fig. 1), S, and compare it with the measured deviation, with the results as follows :

As one can see unfortunately these quantities are not in horribly good agreement with the computed S being ~1.5 error terms away from the measured value in 2 out of 3 cases, this may mean that the approximation for the errors implicit in the measurement of the angles may be overly small, thus over constricting the error in the computation of S slightly.

Section 2:  Total internal reflection can occur when light tries to leave a medium of high index of refraction into a medium with a low index of refraction if the angle of incidence of the light onto the edge of that surface exceeds a certain critical value.  This angle is governed by Sin(q_tir) = n₀/n_tir, where q_tir denotes the critical angle for total internal reflection, n₀ is the index of refraction of the medium one is going into and n_tir is the index of refraction of the current medium.  Taking n₀ to be 1, for the same reasons as given above, we find, Sin(q_tir) = 1/n, where n is the index of refraction that we wish to find.

Section 3:  During this portion of the experiment the value of n is calculated from the minimum deviation angle of a prism.  One can show that n = Sin(q_m)/(Sin(A/2), where n is the index of refraction, q_m is the minimum deviation angle and A is the angle of the apex of the prism.  From this and the measured values for A and q_m using multiple measurements of each we find n to be as follows :

Unfortunately since the data for which apex of the prism was used for the measurement or which side q_m is the minimum deviation angle for were not recorded, these quantities are average across all three sides and may introduce and unknow uncertainty if the prism is not approximately symmetric.

Section 4:  In the final phase of this experiment the value for n of the glass block (used in section 1) was measured using a microscope to determine the apparent height deviation.  The relevant equation is that n = h/h', where h is the actual physical height of the prism, and h' is the apparent height of the prism.  The calculation resulted as follows :

This gives the mean value of n = 1.5278 +/- 0.0112.

We can note that this value of n corresponds well with the value of n measured in Section 1 for this same glass block.

Conclusion

From this experiment we find that the apparent best (most precise) mechanism for measuring the index of refraction of glass easily can be done by measuring the angle of total internal reflection.  However as this value of n was not measured for that particular block using any other means there is some concern that there may be systematic error introduced by the method in which one measures the point at which the glass begins to totally internally reflect, and the data gathered here prevents us from judging the potential extent of that error.  However this point appears to be readily reproducible if one can define the exact criterion for it.  The method of using a prism is of course limited by the need to have a specific shape of glass while the microscope method is less accurate than either it is relatively easy to perform and defines a good range into which results may be expected to fall.  The least precise method examined in this report was that of mapping the rays of reflection and refraction, however the accuracy of this method could be improved by increasing the length at which those rays are found.  These are just a variety of methods that can be used to give precise results for the index of refraction and with some experimentation (primarily cross checking and more accurate instruments) these methods could be refined to give even more precise data.  Unfortunately there is no way to check the accuracy of the methods of Section 2 and 3 as the value of n was computed only once for those objects.  The accuracy between the methods of Section 1 and 4 do appear to correlate well however.