Polarised Light Evan Sheridan, Chris Kervick, Tom Power 11367741 October 22 2012 Abstract Properties of linear polarised light are investigated using Helium/Neon gas laser, polaroid,a silicon photodiode,a general purpose voltmeter (DVM) & a prism situated on a rotable mount. Malus Law is verified by successfully demonstrating a linear relationship between Intensity (I) & cos(φ) 2. Brewster s angle was successfully calculated to be 56 ±0.1 deg. Moreover, Brewster s angle was used to calculate the refractive index of glass,which was found to be 1.48 ± 0.03. Finally, experimental data agreed with the theoretical basis of the Fresnel equations. 1
1 Aims Verify Malus Law. Determine Brewster s angle & use this to calculate the refractive index of the prism. Investigate properties of & relations between reflectance R s, R p & angle of incidence. 2 Backround & Theory The oscillations of waves can either vibrate in the direction of propagation of the wave or they can vibrate in a superposition of a component perpendicular to the direction of propagation & a direction parallel to the propagation, i.e in the direction of the wave. A wave that vibrates solely in the perpendicular direction is called a transverse wave & a wave that vibrates parallel to the direction of propagation is called a longitudinal wave. Light can be described as an electromagnetic wave composed of an electric field component: E(x, t) = E 0 cos(kx ωt) & a magnetic field component that is not necessary for the discussion of polarisation. Polarisation is a property of all waves that do not oscillate longitudinally & it describes the orientation of the the vibration of the wave with respect to the direction of propagation. A polaroid is an object that can affect that polarisation of light. Essentially, it can act as an absorber of the electromagnetic wave depending on the orientation of the wave with respect to the polariser. A basic wire-grid polaroid is the ideal case to make unpolarised light linearly polarised. Basically, this polaroid can be made of grids of very thin stretched crystals that electrons are free to move in one direction. If the electromagnetic wave s orientation coincides with the grid then the dissipative work is done by the field on the electrons & that part of the wave isn t transmitted. If it isn t aligned with the grid then the wave doesn t interact & simply passes through. It is a fine distinction that the wave doesn t slip through the gaps but passes through without interacting with the material. Malus Law states that for linear polarised light we have the relation: I = I 0 cos 2 (φ) I=intensity outgoing light I 0 =intensity of incoming light φ=angle subtended between polarising axis & incoming light. The argument is as follows: We know I E 2 0 only the component parallel to the polarising axis will be transmitted I (E 0 cos(φ)) 2 I I 0 = E2 0 cos2 (φ) E 2 0 1
I = I 0 cos 2 (φ) Therefore, Malus Law tells us about how the intensity of linear polarised light changes when such light is passed through a polariser. Beyond Malus Law there are the Fresnal Equations. These equations describe how much light & more importantly what kind of polarisation of light is reflected & refracted when the light travels from one medium to another. The reflectance refers to the intensity of light reflected by a material. We have two kinds of reflectance : R s & R p. R s refers to s-polarised light, that is, light which is polarised perpendicular to the plane of incidence. Similarly, R p is light which is polarised parallel to the plane of incidence. From the equations for R p & R s we are led to an interesting phenomena whereby at a certain θ b, R p = 0. This θ b is know as Brewster s angle. Brwester s Angle occurs when the angle between the reflected ray & the refracted ray is π 2. We can derive it from Snell s Law: n 1 sin(θ b ) = n 2 sin(α) at 3 Experimental Method 3.1 Diagram α = π 2 θ b n 1 sin(θ b ) = n 2 cos(θ b ) tan(θ b ) = n 2 n 2 3.2 Verification of Malus Law In this part of the experiment we aim to verify Malus Law by measuring the intensity of light as a function of the angle of incidence. We set up the apparatus as in the diagram & do as following: Remove the polaroid & prism & align the laser beam with the photodiode detector so that the laser is in the centre of the photodiode s collector. 2
Put the polaroid back in place & rotate it until the maximum intensity is transmitted, i.e rotate it until the DVM reaches a max value. To make sure this is the maxmimum, rotate 10 counter-clockwise from the maximum & 10 clockwise from the maximum & the reading on the DVM should be the same each time if we have chosen the correct maximum. Now rotate 180 in either clockwise or counter-clockwise direction from maximum & every 10 record the intensity reading on the DVM. The intensity should read 0 after a rotation through 90. 3.3 Measurement of R s & R p For R s : Rotate polaroid to get maximum intensity of light I 0. Replace the prism & zero the rotable mount. Rotate the prism until the reflected light coincides with the light in the polaroid. Rotate the prism through 90 using the rotable mount & also rotate the photodiode with it & record R s off the DMV every 10. Do this up to 90 if possible. For R p : Place the half-wave plate between the laser & the polaroid to get R p polarised light. Repeat the same method as above. Brewster s angle will be observed when the reading on the DMV is zero. 4 Results & Analysis 4.1 Malus Law Plotting the intensity of light transmitted through the polariser as a function of the angle the polariser is inclined at we clearly see that the polaroid behaves exactly as we predicted. Malus Law states I = I 0 cos(φ) 2, therefore after we rotate through an angle of π 2 we expect that the intensity should read zero which it does & we also witness a clear sinusoidal relationship. 3
The above plot clearly illustrates a linear relationship between Intensity (I) & cos(φ)2. Our initial intensity was recorded to be : 0.39 ±0.01 V & the above graph has slope : m = I0 = I cos(φ))2 which is calculated to be : 0.32 ±0.2 V which is within experimental error & Malus Law is thus verified. 4.2 Rs & Rp 4
The above plot plots both R p &R s as functions or φ. Analysis of the plot corresponds with theory. We expect that the R p light should read zero intensity at some φ. From the graph Brewster s angle is calculated to be: 56.01 ± 0.01. Using the argument that tan(θ b ) = n 2 n 1 with tan(θ b )n 1 = n 2 n air = 1 n 2 = n glass = 1.48 ± 0.03 δt an(x) = ( f x δx)2 +... = sec 4 (x)δx 2 = sec 2 (x)δx 5 Discussion & Conclusions Malus Law was verified within experimental error. However, it took more than one data set to actually successfully verify the law because of the sensitivity of the apparatus to external conditions. Care with equipment during experimentation is essentially for any experiment, however, this experiment moreso. One recommendation would be to eliminate as much light as possible from the surrounding error so as not to affect the readings on the DVM. Moreover, aligning the laser beam with the centre of the photodiode was often overlooked & resulted in incorrect results. Although Brewster s angle was correctly obtained to be 56 ±0.01, which is the general accepted value, quite a few considerations must be taken into account. Once again, the apparatus is quite tricky & it may take some time to get the correct configuration. In doing this, it is noteworthy to make sure that the photodiode isn t picking up any superfluous reflections from the prism. By this, quite a few times a refracted ray would be internally reflected & be collected by the photodiode. Sometimes two beams would show & it would be difficult to discern what ray was the correct one. It was concluded that the one that fitted the data the best (the internally refracted ray didn t fit the data) would be chosen. It must be noted that Brewster s angle was not exactly confirmed because the polariser was not 100 % efficient. Therefore, the dot did not entirely vanish but there did seem to be a vanishing point whereby the dot was most faint & intensity at a minimum. Perhaps a more efficient polariser would ensure that Brewster s angle be calculated more accurately. If unpolarised light were to shine on the prism at Brewster s angle it would be expected that the prism itself would act as a polaroid & reflect s-polarised light only because as we know from the Fresnel equations & experimentation that R p (θ b ) = 0. The linearly polarised light behaved as expected when measurements for R p & R s were made. The theoretical expectations were made using the Fresnel equations. It is expected that R s light will never read zero intensity as the angle is increased & we of course witnessed Brewster s angle which is predicted by the Fresnel equation for light polarised parallel to the plane of incidence- R p light. 5.1 References Optics-Hecht. 5