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SNELL'S LAW 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, which in the model for light travel in straight lines in a homogeneous medium. 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 of light with a boundary both reflection and refraction occur. In order to frame laws that govern these phenomena we must establish some definitions of 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. This line is referred to as 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 orientation with respect to the boundary that is delineated by an angle of reflection that is also measured with respect to the normal. So too, the refracted ray will be oriented with respect to the boundary by the angle of refraction measured between the ray and the normal to the surface. What distinguishes the two media is that the speed of light is different from one media 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 media from that of light through a vacuum. Light travels through a vacuum at 2.99792 x108 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 media. It is the only thing that distinguishes one media 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. n1 ≡ vo/v1 or n1 ≡ c/v1 Therefore, the value of the index of refraction is always greater than unity. Gasses have an index of refraction close to I (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 would think that the index of refraction is a constant with varying color of light where the color of light is an indication of the frequency of the light. The index of refraction actually depends on the frequency of the light wave to a small degree and as such is different for different colors of light. The rules for reflection of light are: a) The angle of incidence is equal to the angle of reflection. θ1 = θ1’ 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) b) The incident ray, the reflected ray, and the normal to the surface, all lie in a plane. 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. The first rule is often sited as Snell's Law. It is: a) sinθ1/sinθ2 = n2/n1 where θ2 is the angle of refraction of the ray that is transmitted into the second medium. Similar to the second rule for reflection we can say that the second rule for refraction is: b) The incident ray, the refracted ray and the normal to the surface, all lie in a plane. 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 travelling 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 90o 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 smallest angle for which a ray will be reflected 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 media to the index of the second media. If the second media is air (n = 1.00009), the sine of the angle is effectively the inverse of the index of refraction of the first material. Experiment 1: Index of Refraction In this experiment you will use a Light Ray Box. It consists of a light source and a Multi-slit Slide . The light source housing is mounted on 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 a Multi-slit Slide is a flat square piece of plastic with notches cut into each of the four edges. The a 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. Place the ray box on the end of a sheet of paper provided, such that the ray is projected onto the paper. Locate the oddly shaped tetrahedron lucite block. Place the block on the paper with the frosted side down. Aim the ray at one of the two parallel side of the block. Note that the ray emerges from the opposite side of the block in a direction parallel to the original ray but displaced slightly due to the refraction of the ray inside the block. You will now draw lines representing the incident and refracted rays so that you can measure the angle of incidence and refraction for several angles of incidence. Arrange the block for a specific angle of incidence. You will repeat the following steps through a series of five angles from a shallow angle to a steep angle of incidence. Using a pencil or pen, mark the location of the incident incident light ray θ2 refracted light ray ray at two points along the ray, such that its direction can be recreated later. It is advisable that one of the θ1 points be located at or near the incident surface. Next draw a line along the incident surface. Lastly, draw a point on the opposite side of the block where the ray emerges. These points should be sufficient to recreate the geometry of the refraction. Remove the block and draw lines representative of the incident and refracted rays, and a line representing the refracting surface. Use a protractor to measure the incident and refraction angles. Repeat the procedure for four more angles. Order the results in a table in your lab book. Plot the results as a function of increasing incident angle using the plotting program resident on the PC’s in the lab. This program is called GAX (Graphical Analysis). Your lab instructor will show you how to use the program, if you haven’t already used it in previous labs. Observe if there are any systematic changes in the index of refraction. To do this, 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 determine a standard deviation. The standard deviation can be obtained from the STATISTICS menu item in GAX. Note the result. Arrange the lucite block so Experiment 2: Total Internal Reflection that the single ray from the ray box hits the squared off end of the block. This is the side opposite the slanted side. Have the ray cross through the block and exit the slanted side. Observe the ray θ1 that exits this side as you turn the block to vary incident light ray θ2 the angle that the ray strikes the inside surface reflected ray of the slanted side, the incident angle θ1. You should also observe that there is a significant amount of intensity in the ray reflected off the inside surface. As you increase the incident angle the angle of refraction θ2will also refracted ray increase. Since θ2 will always be larger than θ1, there will come an angle of θ1 for which θ2 should exceed the maximum of 90o. At this point total reflection occurs and you should observe a noticeable increase in the intensity of the reflected ray. Adjust the block so that the refracted ray just skims the slanted surface of the block. Note any observations of interest as you do this operation. Decide on what position best represents the critical angle for the incident ray. Now mark the point where the incident ray enters the block, where the ray hits the slanted side and where the reflected ray emerges. Also draw a line along the edge of the slanted surface. Now, remove the block and using the points recreate the rays as they appeared inside the block and measure the angle of incidence. Use this angle to calculate the index of refraction for the lucite block and compare with the value you got in part one. Does it agree within the uncertainty of the first measurement as determined from the standard deviation? Comment. Estimate errors in the second measurement. 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 with an outer shell or cladding of a material with a lower index of refraction. 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 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. Observation 1: Optical 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 three centimeters 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 to always be in the realm of total reflection. Now, carefully rotate the incident end of the fiber to introduce and angle between the incident ray and the fiber. Observe that the spot on the paper has expanded to form a ring of light. Questions Following is a list of questions intended to help you prepare for this laboratory session. If you have read and understood this write-up, you should be able to answer most of these questions. Some of these questions may be asked in a quiz preceding the lab. • In optics angles are always measured with respect to what? • For geometric Optics what assumption is made about the nature of light? • If two adjacent media have the same index of refraction n, can you observe the phenomena of reflection or refraction? • What is the law of reflection? • What is Snell's Law? • What are the two conditions that allow total internal reflection to take place? • 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?