The Lens makers equation

This one is not perfect but its pretty good combining ray drawing, simulation and practical work to make an interesting investigation. When I have finished the pages on applying the IA criteria I will show how they apply by adding comments to this lab in the same way that it was done in the student samples for the old IA.

Introduction

The power (1/f) of a thin lens is related to the curvature of its faces (R1 and R2) and the refractive index (n) of the material by the lens equation.[1]

Where R is positive if the centre of the circle is to the right and negative if to the left as shown below.
Note: this equation is only valid if the surrounding medium is air.

In this investigation the validity of this equation will be investigated by ray drawing, simulation and experiment with jelly lenses.

Geometric drawing

First I will consider a glass convex lens (n=1.5) with both sides having radius of curvature = 10cm
According to the lens makers equation

To do the drawing I used SMART notes software since this has easy to use rulers and protractors. This was not as easy as expected since the formula is for thin lenses the rays passing through each surface were quite close together making the angles difficult to measure accurately.

The diagram was started by drawing the two circles of equal size then overlapping them to make the lens. To find the principal focus the path of a ray parallel to the axis as it passed in and out of the lens. This was achieved by measuring the angle of incidence at the first face with a protractor then applying Snell’s law to find the angle of refraction which was used to plot the path through the lens. Snell’s law was then applied to the second face and the final path plotted. The result is shown below.

The blue lines are the normal to the two faces and the red lines are the light ray. The incident ray is parallel to the axis at some arbitrary distance from it. The angle of incidence at the first face was measured to be 15°.

Applying Snell’s law at the first face:

i2 = 10°

By drawing the ray and measuring the angles the angle of incidence at the second face was found to be 19°.

Applying Snell’s law at the second face:

i4 = 29°
The final position of the principal focus was found by continuing the final ray until it met the axis. This was found to be at 11.4 cm from the first face however the focal length is measured from the centre of the lens so this gives a value f = 10.4 cm
Uncertainties: All angles were measured to the nearest degree so the angle of incidence could have been a maximum of 15.5°. This would give an angle of refraction , i2 = 10.25° this would not give a measurable difference in the refracted ray so would not give a different path through the lens. To get some idea of the uncertainty the drawing was redone with the radii set at 10.1mm and the incident ray at 14.5°. The result was a focal length of 11 cm as shown below

Reducing the radii by 1mm gives a shorter focal length of 9.5 cm

So making small mistakes in the measurement of lengths and angles could give rise to a difference of around ±0.5 cm in the final result. So I can conclude that within the uncertainties of my measurement the lens makers equation agrees with my drawing.

Algodoo simulation

Algodoo is a simulation programme in which the optical properties of objects can be varied. To make a lens two circles are made with the same radius. They are them made to overlap and the shapes combined with the intersect option. When the original circles are deleted there remains a lens shaped piece of material. It is important to know the radius of the circles this is displayed as they are made; they are both set to be 10m. A 10m long bar is also drawn to use as a measuring stick.

By adding a laser beam the optical properties of the lens can be investigated. At first the beam simply reflected from the surface so the opacity was set in the appearance menu until the ray could be seen to refract through the lens. Changing the refractive index in the properties menu altered the path of the ray, it was set at 1.5. To measure the position of the focal point relative to the 10cm predicted by the lens makers equation the stick was moved into position along the axis of the lens.

As can be seen the simulation gives very close agreement to the lensmakers equation. Using the simulation it is very easy to add more lasers to see if the equation holds for all rays parallel to the axis.

From this image it can be seen that the rays furthest from the axis are brought to a focus much closer than 10m, this is the cause of spherical aberration that can be reduced by stopping down the lens with a small aperture thereby taking away the outer rays.

Using this simulation it is also possible to produce assymetric lenses such as the one below with a front surface of radius 20m and a back surface of 10m. According to the equation this one should have focal length given by:

The simulation again agrees with the formula for rays that are close to the axis.

Using the simulation we can also test the effect of changing the refractive index of the surroundings. If we surround the lens with a material of refractive index 1.3 (water) then the focal length is much longer as can be seen below.

According to the equation

The simulation gives a value of 33m which is in good agreement.

The lens makers equation is only valid for thin lenses, this can be tested in Aglodoo by using a fat lens.

Here it can be seen that the focal length is longer than predicted but not by much. If the lens were much fatter then the distance from the centre to the predicted focal point would be inside the lens and therefore not valid.

Jelly lenses

Making lenses with Algodoo is easy but in practice it’s more difficult, in an attempt to reproduce the same conditions as in the simulation I decided to make lenses out of jelly[2]. Using Jelly I could form lens shapes by cutting with a circular pastry cutter of radius 4cm but before doing this I measured the refractive index of the jelly. This turned out to be quite difficult due the fact that the jelly was not totally transparent so tracing the path of the ray was problematic. I started with a laser and but when I photographed the beam the photo was over exposed however if a halogen ray lamp was used a reasonable photograph could be taken. The photograph was then analysed in notebook.

i = 54°
r = 34°

n = sin 54°/sin 34° = 1.45

Uncertainties: The uncertainty in measuring the angles is mainly due to the thickness of the ray and the difficulty in deciding its exact path, this gives an uncertainty of about ±2° so the maximum possible value for n = sin 56°/sin32° = 1.56 This is an uncertainty of ± 0.1 So n = 1.5 ± 0.1.

The radius of each face of the jelly lens was 4cm so according to the lens equation the focal length should equal 4cm. This has an uncertainty of approximately ±1 cm calculating by substituting the largest and smallest values of n into the lens makers equation. The path of a ray through the lens is shown below.

Here the scale has been adjusted using the length of the lens (4cm) as a reference. The focal length is 4.7cm which is a bit longer than predicted although within the uncertainty quoted. One reason for the difference could be that curvature of the lens faces changed when it was moved from one place to another. This was checked by placing circles of radius 4cm next to the image.

This shows that the lens in fact kept its shape fairly well although the quite blurry nature of the image doesn’t enable a very exact measurement.

Conclusion

The aim of this investigation was to find out the extent to which the lens makers equation can be used to calculate the focal length of a lens. Drawing rays and applying Snell’s law gave close correlation with prediction as did experiments with the simulation software algodoo. This is as expected for thin lenses since the lens makers equation is derived by applying Snell’s law to the spherical surfaces with the approximation that for small angles sinθ ≈ θ. [3] Experiments with thicker lenses and rays far from the axis showed the limitations of the equation. Due to quite large uncertainties the experiments with jelly lenses were far from conclusive but did show that the lens did behave roughly as predicted.

Discussion and evaluation

As mentioned the lens makers equation is based on the assumption that sinθ ≈ θ for small angles, it is therefore quite interesting to note that there was close correlation in the ray drawing exercise even though the angles weren’t so very small the biggest angle being 29° which is 0.51 rads. The sin of this is 0.48 so the approximation actually applies for quite big angles.

It has been stated may times that the lens makers equation only applies to thin lenses. A thin lens is one where the distance from the centre to the focal point is approximately the same as the distance from one surface to the focal point. This certainly wasn’t the case in the thick example in Algodoo but the focal length wasn’t very far from predicted. By treating a thick lens as two lenses it is possible to derive a “thick lens equation” called Gullstrand’s equation[4].

One of the biggest problems with the use of jelly was not the making of the lens, which turned out quite well, but the fact that the jelly was rather cloudy. This made ray tracing quite difficult leading to be uncertainties in the values obtained. In further experiments it would be better to make clear jelly from gelatine rather than using shop bought raspberry flavour.

The refractive index of jelly was measured by photographing the ray passing through a rectangular block. This was not very accurate due to the quality of the photograph and the width of the lines. An alternative method was also attempted where the position of the rays was marked with a pencil. These results were not used since the uncertainties were so big that the spread of values was from 1 - 2.5. To reduce the uncertainties in n a camera with higher resolution could be used combined with a light source with a finer beam.

I had originally thought of using a laser as the light source but it was very difficult to photograph (as it was so bright) and spread out rather a lot when passing through the jelly. This was probably due to scattering by small particles suspended in the jelly, if this is the case the same thing must be taking place with the halogen source but the intensity is so much smaller that it was not observable.

An alternative way of measuring n could be to use the reflected light. When light is reflected of a medium such as jelly it undergoes full polarisation at an angle of incidence given by sini = n (Brewster’s angle)[5]. This method has the advantage of using the more intense reflected light that could be projected onto a screen and the top surface of the jelly which is very flat.

One final observation is that on closer scrutiny of the photograph, the ray coming into the jelly lens is not quite parallel to the axis. The lens should be rotated anticlockwise about 5°, this would cause the focal point to be closer to the lens more in line with the prediction.

[1] http://hyperphysics.phy-astr.gsu.edu

[2] Jelly is a desert made of gelatine, in this case rasberry flavoured.

[3] http://www.math.ubc.ca/~cass/courses/m309-01a/chu/MirrorsLenses/refraction-curved.htm

[4] http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/gullstrand.html#c1

[5] http://www.phy.davidson.edu/fachome/dmb/EdibleOpticalMaterials/find_n.htm

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