DETERMINATION OF BREWSTER S ANGLE FOR GLASS AND PLASTIC USING A POLARIZED MONOCHROMATIC LIGHT SOURCE Utsav Hanspal Physics Honors Research Paper Dr. Watson
Hanspal 2 INTRODUCTION When light moves between two media of differing refractive index (n), some of the light is reflected from the surface of the denser material. This reflected ray s intensity changes with changes in the incident angle (θ i ) of light on the new medium. At one particular angle of incidence light with a particular polarization is not reflected at all. This loss in light intensity is due to polarization by reflection and the angle of incidence for which there is no reflected ray is called the Brewster's angle θ B (also known as the Polarization angle). This phenomenon of polarization by reflection is illustrated in the figure below. Figure1. Polarization by reflection and Brewster's angle (θ B ) Theoretically, polarization by reflection can be explained as follows When the incident ray of light crosses the interface, the light is temporarily absorbed by the atoms in the second medium. Electrons in these energy rich atoms vibrate by oscillating back and forth in the direction of the electric field vectors (shown in the key of figure 1) in the refracted ray, perpendicular to the direction the refracted light is traveling. The light is re-emitted by the atoms to form both the reflected and refracted rays. The fraction of the incident light that is reflected depends on both the angle of incidence and the polarization direction of the incident light. The functions that describe the reflection of light polarized parallel and perpendicular to the plane of incidence are
Hanspal 3 called the Fresnel Equations. According to the Fresnel Law when light moves from a medium of a given refractive index (n 1 ) into a second medium with refractive index (n 2 ), both reflection and refraction of the light may occur. This can be explained with the aid of a diagram, as shown in figure 2 below: Figure2. Illustration of Fresnel s Law: the incident light ray PO strikes at point O the interface between two media of refractive indexes n 1 and n 2. Part of the ray is reflected as ray OQ and part refracted as ray OS. The angles that the incident, reflected and refracted rays make to the normal of the interface are given as θ i, θ r and θ t, respectively. law: The relationship between these angles is given by the law of reflection also called Snell's n 1 sin θ i = n 2 sin θ t At Brewster s angle, the reflected and refracted ray are perpendicular to each other (the angle of 90º indicates the reflected light is completely polarized parallel to the interface). Therefore the sum of θ i + θ t = 90º (Refer to figure 2). Or θ t = 90º - θ i. Incorporating this fact into Snell s Law and rearranging it, we get: n 1 sin θ i = n 2 cos θ i Which implies that, tan θ i = n 2 / n 1
B Hanspal 4 This tangent angle in fact gives the value for Brewster s angle (θ B ), therefore, on final rearrangements we get the final equation to be: It is important to note that the perpendicular component of polarization is almost always reflected more strongly than the parallel component (see figure 3). Figure 3 also shows that for one angle of incidence, called Brewster s angle, none of the parallel polarization is reflected. Figure3. Components of polarization, parallel and perpendicular to the plane of incidence When light comes in at the Brewster s angle the reflected wave has no electric field vectors parallel to the refracted ray. This is because the electrons do not oscillate along the direction of the propagation of the wave (as can be seen from figure 1, light is not produced in the direction of oscillation of electron, but every other angle, being maximum in the perpendicular direction). Therefore, the reflected wave also has no electric field vectors parallel to the reflected ray, because that is the direction of propagation of the wave. The only direction possible is perpendicular to the direction of propagation of the wave. Thus, when an already polarized light is incident on a surface of different refractive index than the medium of the ray of light, no reflected ray is produced (Hecht 348-350).
Hanspal 5 Natural light, which is randomly polarized as shown in figure 4, can be represented by components of polarization parallel and perpendicular to the plane of incidence. If natural light is incident on a dielectric surface, the Fresnel equations describe the reflection for each of the polarization components. Thus, natural light is always at least partially polarized upon reflection. The objectives of this lab were to determine the Brewster s angle for glass and plastic. Figure4. Polarization of natural light PROCEDURE A piece of glass was setup such that at 90 on the protractor, the incident ray (from a red He Laser source) hit the surface perpendicularly. The protractor was then rotated to angle values less than 90. This alignment of the incoming laser and the protractor ensured that the angle on the protractor equaled the angle of the incident ray, thus gave the value of the incidence angle (See figure 5). The reflected ray was directed to a fiber optic light intensity sensor which measured the intensity of the reflected light. The angle range was limited between 12 and 80 because of the expanse of the fiber optic light intensity sensor holder. The same setup and procedure was repeated for plastic. Figure5. Experimental apparatus
Hanspal 6 RESULTS: Table1. Data collected for each material at two angles of polarization Angle (º) Glass Intensity (0º) Glass Intensity (90º) Glass Intensity (Var.) Plastic Intensity (0º) Plastic Intensity (90º) 78.5 2.2 8.2 2.3 77 2.2 8.1 76 2.2 8.2 2.2 2.2 75 1.2 2.2 8.1 2.2 2.1 74 2.2 2.2 73 1.1 2.1 7.9 2.2 72 1.2 2.2 7.9 2.2 2.2 71 1.2 2.1 7.9 2.2 70 2.2 7.9 2.0 69 2.2 7.9 1.8 68 1 2.1 7.9 2.3 1.8 66 2.2 7.3 2.7 2.0 65 1 2.0 6.8 3.0 2.0 64 2.0 6.4 3.0 1.6 63 2.0 6.4 3.2 1.4 62 1.2 1.9 6.2 3.6 1.2 61 1.8 6.0 3.6 1.2 60 1.4 1.4 59 1.6 5.8 3.6 1.4 58 3.4 1.6 57 1.4 5.2 3.4 1.4 56 1.3 1.3 5.2 3.5 55 1.3 4.9 3.3 1.3 54 1.5 1.25 4.2 3.3 53 1.2 3.8 3.3 1.1 52 1.6 3.6 3.9 0.8 51 3.4 4.2 0.8 50 1.7 3.2 4.8 0.8 49 5.2 0.7 48 2.1 2.6 5.4 0.6 47 2.2 6.1 0.6 46 0.6 1.8 0.5 45 2.4 0.5 1.8 7.0 0.4 44 0.4 1.5 0.3 43 2.6 0.4 1.2 7.4 0.2 41 0.3 0.8 7.8 0.2 40 2.8 0.2 0.6 7.8 0.2 39 0.2 0.5 7.7 0.1 38 0.1 0.3 7.6 0.05 37 2.8 0.05 0.2 0 36 0 0.1 0 35 2.8 0 0.1 8.4 0 34 0 0 0 33 3.0 0 0 9.0 0 32 0.05 0 31 0.05 0.2 9.8 30 3.2 0.1 0.3 9.8 0.05 28 0.2 0.6 10.9 0.2 27 4.4 26 0.3 1.2 12.7 0.6 25 4.7 24 1.2 1.9 14.3 1.0 23 5.0 22 2 4.4 16.4 1.8 21 5.6 20 6.6 3.1 7.4 17.2 3.3 18 7.3 4.8 9.1 19.5 5.2 16 8.6 10.3 21.3 6.6 15 9.8 13.2 23.5 8.47 14 9.8 14.9 26.2 13 9.8 16.6 11.0 12 19.8 14.0
Hanspal 7 As can be seen from figure 6, there was no point of zero intensity or Brewster s angle for glass when light with 0 polarization is incident on it. Figure6. Determination of Brewster s angle for glass with 0 polarization 12 Intensity (Arbitrary Units) 10 8 6 4 2 0 10 20 30 40 50 60 70 80 Angle (Degrees) It can also be clearly seen from the data for glass incident with p-polarized (90º) light in table 1 and figure 7 that Brewster s angle for this given piece of glass lies in the range of 33-36 with a mean of 34.5. Figure7. Determination of Brewster s angle for glass with 90 polarization 12 Intensity (Arbitrary Units). 10 8 6 4 2 0 0 10 20 30 40 50 60 70 80 90-2 Angle (Degrees)
Hanspal 8 From the data for glass incident with random intensity p-polarized light in table 1 and figure 8 that Brewster s angle for this given piece of glass lies in the range of 32-34 with a mean of 33. Figure8. Brewster s angle for glass with 90 polarized light with variable intensity 12 Intensity (Arbitrary Units). 10 8 6 4 2 0 10 20 30 40 50 60 70 80 Angle (Degrees) Similar to the glass case in figure 6, there was no point of zero intensity or Brewster s angle for plastic when light with 0 polarization is incident on it (see figure 9). Figure9. Determination of Brewster s angle for plastic with 0 polarization 25 Intensity (Arbitrary Units). 20 15 10 5 0 0 10 20 30 40 50 60 70 80 Angle (Degrees)
Hanspal 9 It can be observed in figure 10 that Brewster s angle for plastic lies in the range of 33-37 with a mean of 35. Figure10. Determination of Brewster s angle for plastic with 90 polarization 18 Intensity (Arbitrary Units). 16 14 12 10 8 6 4 2 0 0 10 20 30 40 50 60 70 80 90 Angle (Degrees) DISCUSSION The following values for Brewster s angle for glass and plastic were determined from the results represented numerically and graphically above: Table6. Brewster s angle value for glass and plastic for various polarizations MATERIAL BREWSTER S ANGLE RANGE Glass (0 polarization) N/A -- Glass (90 polarization) 55.5 ±1.5 Glass (90 polarization, variable intensity) 57 ±1 Plastic (0 polarization) N/A -- Plastic (90 polarization) 55 ±2 Using Brewster s Law and theoretical refractive indices of glass and plastic, the following θ B values were calculated: P.T.O Using refractive index of Crown Glass 1 (common glass) = tan -1 [1.52(n 2 )/1.00(n 1 )] = 56º Using refractive index of Plastic 1 = tan -1 [1.55 (n 2 ) / 1.00 (n 1 )] = 57º 1 It is important to note, that there are a variety of glass and plastic refractive indices depending upon the exact chemical composition, therefore, the materials used in the lab could have had a different refractive index since their exact chemical composition was unknown (Wood).
Hanspal 10 Table7. Percent errors in the experimental results MATERIAL EXP. VALUE TH. VALUE PERCENT ERROR Glass 56º 56º 0% Plastic 55º 57º 3.5% Thus, the value for the Brewster s angle for each substance could have been within a range of various angles depending on the chemical composition of the material. For instance, the range of glass refractive indices ranges from 1.474 for Pyrex glass to 2.04 for Arsenic tri-sulfide glass, giving a range of Brewster s angle between 55.8º to 63.8º. Similarly the range for plastic s refractive indices varies between 1.46 and 1.55. This gives a Brewster s angle range from 55.6º to 57º. From the conclusions made above, no clear evidence of the material identity can be determined from the value of the Brewster s angle since there is no significant difference in the values of the Brewster s angles for glass and plastic and in some cases their values overlap, depending upon their chemical compositions. However, an assured conclusion that can be made is the fact that light intensity does not, in any significant way, affect the Brewster s angle. This also confirms the existence of Brewster s angle, for both glass and plastic, which means that at a specific angle of incidence of p-polarized light there is indeed no reflection, which also proves polarization through reflection.
Hanspal 11 BIBLIOGRAPHY Hecht, Eugene. Optics. 4. San Francisco: Addison Wesley, 2002. Wood, Robin. "Refraction Index of Various Substances for 3D modelers." 18 March 2007. Robin Wood. 22 Mar 2008 <http://robinwood.com/catalog/technical/gen3dtuts/gen3dpages/refractionindexlist.html>.