Chapter 4 Experiment 2: Snell's Law of Refraction

Chapter 4

Experiment 2: Snell’s Law of Refraction

4.1 Introduction

In this and the following lab the light is viewed as a ray. A ray is a line that has an origin but does not have an end. Light is an electromagnetic disturbance and, as such, is described using Maxwell’s equations, which expresses the relationship between the electric and magnetic fields in an oscillating wave. Light propagates as a wave; yet, many optical phenomena can be explained by describing light in terms of rays. In the model for light, rays in a homogeneous medium travel in straight lines. This model is referred to as Geometric Optics and is a very elementary theory. In this theory light travels from its origin at a source in a straight line, unless it encounters a boundary to the medium. Beyond this boundary may be another medium which is distinguished by having a speed of light different from the original medium. In addition, light may be reflected at the boundary back into the original medium. A light ray that returns to the original medium is said to be “reflected”. A ray that passes into the other medium is said to be “refracted”. In most interactions between light and a boundary, both reflection and refraction occur. In order to frame laws that govern these phenomena we must define some terms. The boundary between two media is defined as a surface. The orientation of a surface at any specific point is characterized by a line perpendicular to the surface that we call the normal. A ray may encounter a boundary at any arbitrary incidence angle. The angle of incidence is measured with respect to the normal line. A reflected ray will have an angle of reflection that is also measured with respect to the normal. The refracted ray will be oriented by the angle of refraction measured between the ray and the normal to the surface.

Checkpoint For geometric Optics what assumption is made about the nature of light?

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What distinguishes the two media is that the speed of light is different from one medium to the other. We define the index of refraction n to be the measure of how much different the speed of light is in a certain medium from that of light through a vacuum. Light travels through a vacuum at 299,792,458 m/s. This speed is thought to be a universal constant and the highest speed allowed in nature as postulated in Einstein’s theory of Special Relativity. We use the symbol c to represent this speed. The index of refraction is a characteristic of the medium. It is the only thing that distinguishes one medium from another in geometric optics. It is defined as the ratio of the speed of light in a vacuum to the speed in a particular medium of interest, c c n = or v = . (4.1) v n Therefore, the value of the index of refraction is always greater than unity. Gasses have an index of refraction close to 1 (nair = 1.00028), while for water the index is about 1.33 and for plastic it is approximately 1.4. Depending on the type of glass the index of refraction of glass can vary from 1.5 to 1.7. Normally we might think that the index of refraction is a constant that is the same for all light. The index of refraction actually depends on the frequency (color) of the light wave to a small degree across the visible part of the spectrum and as such is different for different colors of light. The rules for reflection of light are:

1) The angle of incidence is equal to the angle of reflection,

0 θ1 = θ1 (4.2)

0 where θ1 is the angle of incidence and θ1 is the angle of the reflected ray that propagates in the same medium. (This is the commonly known rule, but this next rule is rarely stated though equally important) 2) The incident ray, the reflected ray, and the normal to the surface, all lie in the same plane.

Checkpoint What is the law of reflection?

We will not formally investigate these rules in this lab although you will be able to observe the phenomena of reflection as a side issue while performing this lab experiment. The rules for refraction are not so obvious although they where well known to the ancients.

1) The first rule is often cited as Snell’s Law; it is:

sin θ1 n2 = or n1 sin θ1 = n2 sin θ2. (4.3) sin θ2 n1

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where θ2 is the angle of refraction of the ray that is transmitted into the second medium. 2) The incident ray, the refracted ray and the normal to the surface, all lie in the same plane.

Checkpoint What is Snell’s Law? What phenomenon does Snell’s Law describe?

In general the path of a light ray is reversible in that if a light ray were to be reversed it would follow the same path. A ray traveling from a low index of refraction to a high index of refraction will experience a bending toward the normal. However a ray passing from a high index of refraction to a lower index will experience a bending away from the normal. The angle of refraction will be larger than the angle of incidence. So, what happens when the angle of refraction is greater than 90◦ for a given incidence angle. In this case light cannot be transmitted through the interface and as such it is reflected totally. The efficiency for this reflection is 99.99% (as compared to 95% for a typical silvered surface mirror). The largest angle for which a ray will be transmitted is the critical angle. One can show that the sine of this angle is the inverse of the ratio of the index of refraction of the first medium to the index of the second medium. If the second medium is air (n = 1.00028), the sine of the angle is effectively the reciprocal of the index of refraction of the first material.

Checkpoint In optics angles are always measured with respect to what?

4.2 The Experiment

The experiment consists of a single thin bundle of light rays exiting a light box. This ray will be incident upon a ‘D’ shaped dielectric so that we may deduce whether the laws of reflection and of refraction are obeyed by the interaction between the light and the object. A picture of the experiment is shown in Figure 4.1. We should recall that ‘dielectrics’ were placed between capacitor plates to increase capacitance and to insulate between the plates. Our refracting medium is a transparent dielectric.

4.2.1 Reflections and Refraction

In this experiment you will use the Light Ray Box shown on the right side of Figure 4.1. It consists of a light source and a Multi-slit Slide Set. The light source housing is mounted on

41 CHAPTER 4: EXPERIMENT 2 a colored plastic base in which it can slide back and forth. The utility of this feature will be explained in a future experiment. The Multi-slit Slide is a flat square piece of plastic or aluminum with notches cut into each of the four edges. The Multi-slit Slide slips into a slot on the end of the ray box to create rays of light. Choose the side with just one narrow notch and place that side down as you slip the a Multi-slit Slide into place. A single narrow beam should be observed emerging from the ray box. Also in this experiment you will use a turntable (goniometer) to orient the dielectric surface. The turntable has some friction with its stationary stand, so it is suggested that you spend several minutes practicing the act of changing angles before aligning the experiment to take your data. With some care you should be able to rotate the turntable and dielectric on its stand without sliding the stand on the tabletop. Once you can do this reliably, carefully place the dielectric ‘D’ on the turntable as the turntable markings indicate. As long as you do not suddenly move the turntable, friction will keep the dielectric on the turntable at this location and will allow you to rotate the dielectric while reading the angles from the turntable’s periphery. Figure 4.1 shows the experiment in progress. In the figure the light passes through the dielectric before striking the planar surface (the ‘relevant surface’ in the figure). Note that the ray strikes the planar surface precisely at the center of the turntable; this is also the pivot point for the turntable. As long as neither the ray box nor the turntable slides on the tabletop, this ray will always strike the relevant surface at the pivot point. In the figure it is easy to see that the incident ray is 40◦ from one side of the normal and the reflected ray is 40◦ from the opposite side of the same normal. The refracted ray has spread considerably (can you guess why?), but its refraction angle is about 73◦ (what is its uncertainty?). Is there any trend that you note regarding these refracted rays and how they spread?

4.2.2 Index of Refraction and the Law of Reflection

Now we need to align the ray from the ray box carefully before we begin taking data. Carefully rotate the turntable so that the flat side of the dielectric faces the ray box. Carefully slide and tilt the ray box so that the light strikes the dielectric precisely at the turntable pivot and such that both the incident light and the refracted light illuminates the protractor scale around the turntable’s periphery. Carefully rotate the turntable until the incident ray is at 0◦. At this angle only, the refracted ray should also illuminate the opposite side of the turntable at 0◦. Keeping the incident ray at 0◦, carefully adjust the ray box position to point the ray at the pivot point and/or carefully adjust the dielectric’s position on the turntable until the light ray passes straight across the turntable through the pivot point. This makes your apparatus ready to start taking data. Draw a nice table in your notebook (or execute Ga3 and prepare the columns and headers) to record the incident angle, the angle of reflection, the angle of refraction, and a calculated index of refraction. You will need 25-30 rows. Since neither 0◦ nor 90◦ will provide useful data, it is recommended that you begin with an incidence angle of 5◦ and that you increment by 10◦ between measurements. Carefully record the incident angle, the angle of reflection, and the angle of refraction. (How accurately can you measure these?) Always estimate the

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Relevant Surface Dielectric

Incident Refracted Ray Ray Ray Box Reflected Ray

Figure 4.1: A photograph of the light ray box, dielectric object, and turntable that we will use to investigate ray optics.

angle using the center of the ray. Compute the index of refraction using sin θ n = air (4.4) sin θdielectric

where θair is on the flat side of the dielectric and θdielectric is on the round side of the dielectric. For now keep four significant digits in n. The website offers a template for Vernier Software’s Graphical Analysis 3.4 (Ga3) pro- gram. Your lab instructor will show you how to use the program if you haven’t already used it in previous labs. Plot the index of refraction for the Lucite plastic versus an integer index. Students may also choose to use Excel R for this purpose. Observe if there are any systematic changes in the index of refraction. Before disassem- bling your experiment, carefully repeat any data points that seem to vary substantially from the observed trend. To determine whether systematic changes are significant, one must have an idea of how much deviation in the index can be attributed to random variations. One way to assess this is to find an average value for the index, and from that to determine a standard deviation. The average and standard deviation can be obtained from the Ana- lyze/Statistics menu item in Ga3. You will need to draw a box around your data points before the Statistics will be active. Review Section 2.6.1 for strategies to specify statistical measurements objectively.

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Checkpoint Do all of your reflected rays obey the Law of Reflection?

4.2.3 Total Internal Reflection

Do you have any incident rays that do not have refracted rays? Is this consistent with Snell’s law? We would like to observe this “total internal reflection” in more detail. First, where does the energy in the incident light ray go if it is not transmitted through the surface? Do you observe anything that supports your hypothesis? Second, we want to measure this “critical angle” of incidence where the slightest rotation results in refracted light. The symmetry of the planar surface indicates that there should be two critical angles in 360◦ of rotation. Measure both of them and verify that they are the same but on opposite sides of the normal. Also observe the angles of the reflected rays; does the Law of Refection suggest a way to use these reflections to improve your critical angle measurements? If you were diving and looking up, this would be the opening angle of a “critical cone” outside of which you could not see through the surface. Use these critical angle measurements to calculate the index of refraction, n 1 n = air ≈ , sin θc sin θc and specify the mean and standard error (see Section 2.6.1). Compare with the value you got in Section 4.2.2. Does it agree within the uncertainty of the first measurement as determined from the standard error? Might the error in your measurement of critical angle be large enough to explain any disagreement? Comment on this in your Analysis.

Checkpoint If two adjacent media have the same index of refraction, n, can you observe the phenomena of reflection or refraction?

Checkpoint What is a critical angle? What are the two conditions that allow total internal reflection to take place?

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4.2.4 Optical Fiber

One of the more recent and ubiquitous applications of total internal reflection is the optical fiber. Fiber optics has revolutionized the communications industry. A fiber consists of a long thin thread of glass called the core surrounded by an outer shell of a material with a lower index of refraction called the cladding. Light entering the end of the core of an optical fiber is transmitted or piped to the other end with very little loss in intensity even though the fiber may be bent in a circular shape. The thinness of the fiber and the lower index of refraction in the cladding ensures that light will always strike the fiber side at an angle greater than the critical angle for total internal reflection and be totally reflected back into the fiber. Thus the light bounces off the inside as it caroms down the length of the fiber following every bend or twist. Light is composed of thousands of discernible wavelengths (colors) and each color can be modulated to communicate many billions of bits of information per second. Intelligible voice in one telephone call requires only 50,000 bits per second. One fiber can communicate hundreds of thousands of phone calls. More complex information requires more bandwidth and thus fewer channels can be communicated on a fiber. Find the optical fiber among the items on your lab table. Be careful in handling the fiber. It is, after all, a piece of glass and will break if bent too sharply. Align one end of the fiber with the ray emerging from the ray box. Observe the light emerging from the other end of the fiber. It will look like a bright pinpoint of light. Point this end of the fiber vertically down onto a sheet of paper on the table with the end of the fiber held about five millimeters above the table. You should see a bright spot on the paper where the light rays coming from the fiber hit the paper. The light has been channeled through the fiber even around the bends as the ray bounces off the inside surface of the fiber, possibly many times. The light persists with little loss of intensity because of the great efficiency of these reflections that are at such steep angles to the surface normal as always to be in the regime of total reflection. Now, carefully rotate the incident end of the fiber to introduce an angle between the incident ray and the fiber end’s normal. Observe that the spot on the paper has expanded to form a ring of light. Note these observations and try to explain them in your Data.

Checkpoint In the case of fiber optics do you expect the core or the cladding to have a greater value for the index of refraction? Why are optical fibers immune to electrical noise?

4.2.5 Lenticular Lenses

Most of us have walked past displays at the movie theater or the video store and noticed that the image suddenly changed as we walked past. Some of us were very curious how this magic happened. Before we leave today we would like to dispel some of the mystery behind

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this marvelous invention.

(a) (b) purple lenticular lens is seen blue red is is seen seen blue

lenticular print red

Figure 4.2: A lenticular lens placed on a lenticular print in (a) refracts the light from parallel strips from the print into a common direction in (b) that can be seen by a passerby. As the viewer passes, he sees adjacent parallel strips showing a different image.

Place the flat side of the dielectric against the slit in the ray box and note what happens to the projected ray as you slowly slide it across the slit. We see that the ray gets deflected by the curved surface of the block. When the ray is on the left side of center it gets deflected to the right and when the ray is on the right side of center it gets deflected to the left. Therefore, someone standing on the right will see the light emitted from the left side of center and as he walks toward center the light he sees is emitted from closer to center. Now imagine that the block has much smaller radius, is much longer, and that there are hundreds placed side by side. This is illustrated in Figure 4.2. Figure 4.2(a) shows a 3D perspective drawing of an abbreviated lenticular print and lens. An end view of the system is shown in Figure 4.2(b). A viewer standing on the left side of Figure 4.2(b) would see a field of pure blue; but as he walks to the right, the field would slowly change to purple when he is in the center and finally to red when he is at the right side. Each segment of the lenticular lens yields one horizontal pixel. Instead of the solid color shown in the figure, the vertical strip’s color can be varied to match the corresponding image. Similarly, the adjacent strip’s color is varied to match the adjacent image. It is in this way that each image is formed and separated from the other images. By projecting one image toward the viewer’s left eye and the same scene from a shifted perspective toward the viewer’s right eye, the lenticular lens can also simulate a 3D image. In this case, however, the scene cannot be smoothly changed as the viewer moves across the display; it is necessary that the adjacent image is intended for the other eye. Photoshop R , GIMP, and other image processing applications have “banding” plugins to generate suitable lenticular prints from an ordered series of images. Additionally, lenticular

46 CHAPTER 4: EXPERIMENT 2 lenses may be purchased from Amazon.com and ebay. It would be wise to consider the de- sired product (http://www.microlens.com/pages/choosing_right_lens.htm) before wasting too much time figuring this out for yourself.

4.3 Analysis

Verify the self-consistency of Snell’s law by comparing the two indices of refraction measured in Section 4.2.2 and Section 4.2.3. Use the strategy in Section 2.9. What have you noticed while performing the experiment that might (probably does?) contribute to the difference between these measurements? Might these other sources of errors be large enough to explain your differences?

4.4 Conclusions

What have you measured that you and/or your science peers might need to use in the future? What physical relations do your data support? Contradict? Which are not satisfactorily tested? Communicate with complete sentences and define all symbols. (Or better yet, label your equations above and simply name them here in your Conclusions.) How might this experiment be improved? If you were to repeat the experiment, what would you try to do differently and why? What applications might benefit from your observations?

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