## Abstract

The refractive indexes of 3 different glass lenses and water were determined using Newton’s rings interference pattern. The refractive index of each lens determined were as follows

nlens 1=1.329±0.008, nlens 2= 1.47±0.03, nlens 3=1.44±0.07 and the refractive index of water was found to be μ=1.26±0.03. The true value for the refractive indexes should be nglass=1.51947 for the glass the lenses were made from and μ=1.330 for water. This report gives an overview on how results were obtained and discusses reasons why the final values obtained were different to the expected values.

## Introduction

The purpose of this experiment was to determine the refractive index of 3 different glass lenses and water using an optical interference pattern. The interference pattern is known as ‘Newton’s rings’ and was first observed by Isaac Newton in 1717 [1]. This experiment also gives an insight into the wave behaviour of light, as it demonstrates that light will produce a visible interference pattern. The radius of curvature of a convex lens is radius of the sphere that would form if the curved face of a lens were extended. The focal length is the distance between the lens and the point at which light passing through the lens will focus. Initially both radius of convergence and focal length were determined to calculate final values for refractive index, these were then uses to calculate the refractive indexes. Refractive index is an important property of all transparent mediums and is defined as the ratio between the speed of light in a vacuum to the speed of light through the medium, in denser mediums light will propagate more slowly and therefore denser mediums will have larger RI values. Refractive index also tells us how light will bend as it travels through the medium[2]. Knowing the RI of different materials is important as it allows us to model how light is travelling in a medium.

## Theory

When two waves meet in space, they will interact; this is due to the principle of superposition: when two or more waves overlap, the resultant wave is the sum of the individual waves [2]. This is demonstrated in Figure 1 below.

The red and blue waves have overlapped at the same point in time and space, meaning the resultant displacement of the two waves at any point is the sum of the individual displacement of either wave. The green wave represents the resultant wave with amplitude A3=A1+A2.

With equipment set up as shown in figure 2, an interference pattern will form. The lens is assumed to be optically flat, meaning that the glass lens’ surface is considered flat to within a small fraction of the wavelength of incident monochromatic light [2]. Interference is observed because as monochromatic light from the light source travels through the lens, partial reflection will occur at the back of the glass lens and at the glass plate (illustrated in figure 3). This produces two coherent wave sources travelling away from the glass lens with a path difference, denoted Δl.

This path difference leads to an interference pattern shown in figure 4. What should also be taken into consideration is that when the light reflects at the air-glass plate boundary at the bottom, there will be a phase change of π radians. This is because the refractive index of the glass plate is larger than the refractive index of air. At the centre of the glass the air gap between the lens and plate is very small, meaning the two waves produced with a phase difference of π radians will have effectively no path difference. This means they will interfere destructively, such that a dark circle can be observed at the centre.

Figure 4: A picture of Newton’s rings [3]

If Δl is the path difference, then it can be denoted that:

Δl=2dn (1)

Where dn is the width of the gap between the glass plate and lens for fringe n. n represents the order in which the obdurate fringe is placed, with n=1 representing the innermost fringe and n increasing by 1 for every consecutive fringe moving outwards. The path difference of the fringe is related to n by equations 2 and 3:

2μdn=(n+1/2)λ0 (2) For bright fringes

2μdn= nλ0 (3) For dark fringes

Where μ is the refractive index of the medium between the glass lens and plate.

For fringe n, the relationship between it’s radius tn is and the gap width dn is as follows:

tn2=dn(2r0-dn) (4)

However as 2r0>>dn it is reasonable to approximate the equation as:

tn2 ≈ 2dnr0 (5)

Where r0 is the radius of curvature of the lens. By combining equations 4 and 5, dn can be eliminated to give equation 6 [4]:

2dn= tn2/r0 and μdn=(n+1/2)λ0

μtn2/r0 =(n+1/2)λ0 (6) For bright fringes

μtn2/r0 =nλ0. (7) For dark fringes

So a graph of tn2 against nλ0 will have a gradient m:

m=r0/μ (8)

Radii of curvature of a biconvex lens can also be related to the focal length and refractive index of the thin lens equation:

1/f=((nlens/nmedium)-1)(1/r0-1/r1) (9)

Where f represents focal length of the lens, nlens and nmedium represent the refractive indices of the lens and the medium surrounding the lens respectively. r0 and r1 represent the radii of convergence of either side of the lens. If the biconvex lens’ sides are symmetrical to one another, r0=-r1 and if the medium surrounding the lens is air, nmedium≈1. This can be substituted into equation 8 to give:

nlens =1+r0/2f (10)

## Method

### Determining the radius of fringes:

The equipment was set up as shown in Figure 2, the monochromatic light source used was a low pressure sodium-vapour lamp, which emits effectively monochromatically at a wavelength of λ0=589.3nm. When switched on, the interference pattern formed could be observed through the travelling microscope. Initially, the experiment was carried out with air as the medium between the lens and the plate, such that μ≈1. Initially, crosshairs were centred on the central dark fringe. The vernier calliper was then adjusted until the crosshairs touched the edge of the first dark fringe (n=1). The position of the microscope was recorded and then repeated up to n=7. After this the microscope was moved in the opposite direction to record the inner position of the ring on the other side. By subtracting the inner position from the outer position, the diameter of each ring 2tn could be determined and hence by halving this, radius tn could be recorded. Data was recorded for 3 different lenses and also repeated for lens 1 in water. Graphs of tn2 against nλ0 were plotted and fitted using origin with gradients equal to r0/μ . For data collected using air as the medium, μ ≈1 so gradient is just equal to radius of curvature.

### Determining focal length of lens:

Equipment was set up as shown in figure 5, distance between each lens and the screen was adjusted until the light from the torch was focused on the screen. This could be observed as the light on the screen would form a sharp point, where light passing through the lens converged. The distance between the lens and the screen was then measured using the metre ruler, and focal length was recorded.

## Results

Lens 3

Lens 2

Lens 1

Using the gradient of each line on figure 6, radius of curvatures were determined as r0=0.408±0.004m for lens 1, r0=0.184±0.005m for lens 2 and r0=0.26±0.02m for lens 3. The values for the focal length (f) for each lens were respectively found to be f=0.6200±0.0005m,

f=0.1950±0.0005 and f=0.3000±0.0005.

Once radius of curvature for lens 1 was determined, it was used along with the gradient of figure 7 to obtain the refractive index of water, this was found to be μ=1.26±0.03. For each lens equation 10 was used to calculate refractive index and compared to the known value for the refractive index of the lens made from N-BK7 crown glass. The refractive index of this glass is nglass=1.51947[5]. The refractive index of each lens were found to be nlens 1=1.329±0.008,

nlens 2= 1.47±0.03 and nlens 3=1.44±0.07. Error on the focal length was taken as half the smallest division on the ruler(0.5cm), tn error was found using the error on the microscope position readings. Error on the gradient was found using origin, and partial differentiation of equations 8 and 10 were used to determine the errors on r0 and RI of each lens.

## Discussion

The values for the refractive index of glass in each lens obtained were: nlens 1=1.329±0.008,

nlens 2= 1.47±0.03 and nlens 3=1.44±0.07. The true value of the refractive index of the glass the lenses were made from was: nglass=1.51947. This does not lie within the error bounds for any of these results, with all results being smaller than expected. The value for the refractive index of water was determined to be: μ=1.26±0.03. The true value of the refractive index of water was μ=0.1330 [6], again this is larger than the result of the experiment and doesn’t lie within the error bounds.

A possible source of error could have been a parallax error on reading the position of the travelling microscope, this would lead to incorrect values for the radius of fringes tn and hence incorrect gradient values on the graphs and refractive index values. One way this could be avoided would be by taking multiple errors for position and recording the mean value and standard error on the mean. Alternatively position of microscope readings could be taken digitally, by setting up a digital micrometer along the travelling microscope. As all the values were smaller than expected, it suggests there may have been a systematic error, for example when recording the focal length the length where the light is focused on the screen may have been wrongly interpreted each time. Had values of focal length been systematically measured at larger lengths each time than the actual value for the focal length then according to equation 10 the values for the calculated refractive index would all be smaller than their true values. Another possible systematic error may have been in measuring values of tn, it is possible that the the radius of each fringe had been measured too far in. This would lead to smaller tn2 values and hence a smaller gradient on the graphs, meaning radius of curvature calculated would be smaller then expected, which would account for the smaller than expected refractive indices.

Measuring the focal length also had a lot of potential for human error as it was done by hand and measured with a meter ruler by eye. This would lead to a large random error which should have been taken into account in error calculations. A better approach to finding the focal length would be to use a clamp and stand to hold the lens and torch firmly in place. Whilst there was a lot of room for random errors in this investigation it is much more likely that the errors were systematic, due to the fact that all results were smaller than their expected values.

## Conclusion

In conclusion the values for the refractive index of each lens obtained from this experiment were nlens 1=1.329±0.008, nlens 2= 1.47±0.03 and nlens 3=1.44±0.07, and the value for the refractive index of water obtained was μ=1.26±0.03. The expected values were nglass=1.51947 and μ=1.26±0.03. The difference in values is most likely due to a systematic error, should the experiment be repeated these could be eliminated to obtain more accurate results.

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