05-3: Imaging Systems Laboratory II Laboratory : Snell s Law, Dispersion and the Prism March 9 &, 00 Abstract. This laboratory exercise will demonstrate two basic properties of the way light interacts with matter. The first is the law of refraction (Snell s Law), which tells us that light rays will suffer a change in direction when they cross a boundary between two indices of refraction. The second is the phenomenon of dispersion, which is the variation of the index of refraction with wavelength. To investigate both effects, we will use prisms made of three different materials: glass, water and mineral oil. Lab write-ups are due March 6, 00 for the Tuesday lab section and March 8, 00 for the Thursday section. I. Theory In the lecture part of the course, we learned that a light ray incident on an interface between two transparent materials is refracted at an angle determined by the incident angle, θ, (measured from the normal to the interface) and indices of refraction, n and n, of the two materials. This relationship is known as Snell s Law, and can be written in the form n sin θ = nsin θ, () where n and n are the indices of refraction for the two media and θ and θ are the angles that the incident and transmitted rays make with the normal to the surface, respectively. The situation is shown in Figure below. n n θ θ Figure : A ray traversing the boundary of two indices of refraction.
We also saw that when one applies Snell s law to the case of a prism, it can be shown by way of geometrical arguments that for a ray that passes through one face of the prism and emerges from another face without reflections, the total deviation of the ray from its original path is given by the formula ( n sin θi cos sin θ ) θ + arcsin sin, () = i i where is the total deviation angle of the ray, is the apex angle between the entrance and exit face, n is the index of refraction of the prism material, and θ i is the angle of incidence on the entrance face. Figure illustrates this situation. θ i Figure : A triangular prism with apex angle. Figure 3: Typical dependence of the deviation angle,, as a function of the angle of incidence, θ I. In this case, n=.5 and =60 degrees.
For a given prism, and n are fixed, so the above expression for is a function only of the angle of incidence on the entrance face, i.e. = (θ i ). This function generally has a minimum at some incident angle that depends on the index of refraction for the prism material, so that for example, a curve of as a function of θ i typically looks like Figure 3 above. This minimum should occur when the ray s path is parallel to the base of the prism, for an isosceles-triangular prism (like the one shown in Figure ). If the minimum angle of deviation min can be measured for a given prism, it follows from Snell s law and the above equation that ( ) min + () sin n =. (3) sin In the above discussion, it is assumed that the index of refraction is constant. This assumption is perfectly valid for considering what happens to a single wavelength (i.e. color) as it travels through the prism, but we have also learned that the index of refraction is a function of wavelength. This effect has to do with how individual atoms respond as electromagnetic waves pass by, a process known as non-resonant scattering, but the end result is that most materials used in optical systems show a slight increase in the index of refraction for light of lower wavelength. Figure 4 shows the wavelength dependence of the index of refraction for several different substances. The unit of wavelength used is the Angstrom (Å), where Å = 0-0 m. Figure 4: Dependence of the index of refraction on wavelength for several different transparent substances. 3
II. Experimental Set-Up To get ready to do the experiments below, you will need () a glass prism, () one of the prisms made of glass slides that can be filled with water or mineral oil, (3) a rotary table or angle sheet on which to place the prisms to change the orientation easily, and (4) a laser. Once you have all of these things, set the prism on the rotary table and arrange the laser so you can point the laser beam through the prism easily, as in Figure 5. DO NOT LOOK DIRECTLY AT THE LASER BEAM. The prism should be able to rotate through a large range in the angle of incidence on the input face. Use a book with a white piece of paper taped to it or a piece of white paper taped to the wall as a screen. Make sure before taking any data that the laser spot remains on your screen as you rotate the prism. laser C prism A B Rotary table screen Figure 5: Experimental set-up for this laboratory exercise. III. Procedure Start with the glass prism, and complete the following steps: ) Measure the distance from the center of the rotary table to the screen as best you can. Try taking 3 to 5 independent measurements and averaging the result. What is the standard deviation, σ, of the result? Use σ / N as an estimate of the uncertainty in your distance measure. ) Mark the position of the laser spot on the screen in the case where no prism is present. This will give you the position of the undeviated beam. 4
3) Measure the apex angle of the prism. You can draw lines parallel to the faces of the prism, and extend these lines with a ruler to make the measurement easier to do with a protractor. 4) Place the prism in the center of the rotary table or angle sheet. Mark the position of the laser spot for different incident angles for as large a range as you can measure. Take at least 8 data points, recording the value of the incident angle (as judged by the position of the rotary table) and the distance from the undeviated spot position. 5) Convert the distances from the previous step to deviation angles (θ i ). This can be done by noting that dist(ab)/dist(ac) = tan, (4) where dist(ab) is the distance from A to B in Figure 5 and dist(ac) is the distance from A to C, both of which you have measured. 6) Now remove your glass prism and fill the hollow prism made from the microscope slides with the distilled water provided. Repeat steps through 5. 7) Next try steps through 5 with after filling the microscope-slide prism with mineral oil. 8) Finally, go back to the glass prism and use one of the fiber-fed light sources and a slit instead of the laser. Put the fiber source at least a couple of feet away from the prism. Make a slit that has width of a couple of millimeters and length of about cm. Measure the angle of incidence as best you can, and then measure the deviation for three different colors: red, green or yellow, and blue. If you have trouble, you may want to consider holding filters from the optics kit in front of the slit to cut out the unwanted colors. In any case, you will probably have to turn the lights off and shield your screen from the ambient light to be able to see the slit on your screen. IV. Analysis In your lab write-up, be sure to complete the following: ) For each of the three prisms, graph the deviation angle as a function of the incident angle on the input face of the prism, θ i. Find the minimum deviation angle, min and estimate the uncertainty in this quantity. ) Using Formula 3 on page 3, calculate the index of refraction for the glass, water, and mineral oil used, based on your value for min and in each case. 3) From your measurements in Step 8 above, derive the index of refraction for red (λ 700nm), green (λ 550nm) and blue (λ 450nm) light. Do your data show an increase in n as the wavelength is reduced, as in Figure 4? The laser is also red (λ = 63.8nm). How well does your value of n agree between the laser measurements and the red filter measurement? 5