In class, we sketched graphs of light intensity vs position on the screen for both single and double slit interference patterns, as well as for the wave nature of the electric and magnetic fields produced by light as it propagates. We may not all be good artists, and when in a hurry in class, we might not do the best job of copying sketches like these from the board Neatly reproduce each of these three sketches, properly labelling all parts. You may use a computer, if you do all of the work yourself. (A mark of zero will be given to all parties involved if any work is not original.) Make sure each drawing is properly titled and labelled In particular, on the sketch for double slit interference, label those aspects of the graph that are the result of double slit interference as DS, and those that are the result of single slit interference as SS.

5.8 Polarization (and B) oscillates I to z source Light is a transverse wave. Unlike most waves, which require a medium, light requires no medium or transmission, but is merely the oscillation of the combined electric and magnetic fields in the space through which the light wave propagates. Consider three mutually perpendicular axes, x, y and z, shown below. unpolarized light X A polarizer is a material that will transmit light which oscillates only in a particular direction. The two grey pieces of plastic that you held up to the light in class were both polarizing filters. V x 8 X B polarization axis B polarized light transmitted If z is the direction of propagation of the light wave, then at any point in space where the wave passes, the electric and magnetic fields oscillate in the xy plane, perpendicular to each other, polarizing filter such that at all times, & x B points in the direction of propagation (z). To see this oscillation as a wave, consider a series of points in space in the direction of propagation. Thus, a second polarizer, with polarizing axis perpendicular to the first, can eliminate all transmission of light X (oscillates in xz plane) At all times, x B points polarization axis along the z axis B Z (oscillates in yz plane) no light (8 = 0) Light can also be polarized by: reflection from smooth surfaces The direction of the oscillation of & (and thus B) is determined by the direction of oscillation of the scattering from particles in the atmosphere charged particle(s) that emitted the radiation. A typical light source has charged particles passing through a "bi-refractive" crystal, one that has different indices of refraction (electrons) oscillating in all possible directions, producing light that travels in all directions, and for light polarized in different directions whose electric and magnetic fields oscillate in all possible directions as the light propagates. This is called unpolarized light.This still gives the formula This can also be arranged to give the locations on the screen (relative to the centre at the interference pattern) of the dark fringes )tL x], : (n - a)? Specically, the distances from the centre of the interference pattern to the first four dark fringes are ,'=J J=ivxs=,x4=l Zn" 2 Thus. the constant distance between successive dark fringes in the interference pattern is Ax.2it AL 2 d 7 . . . . M- This is even valid for the two dark fringes that straddle the centre, each "2d- from centre, M. resulting in a distance between them of the same AX : ? . Rearranging this gives a relatively easy way to experimentally determine the wavelength of light Shown on the next page are images I was able to obtain in the classroom 01 a double slit pattern. You will notice that in addition to the many small bright and dark fringes, there is an overall modulation to the pattern where it is brightest in the centre, then it gets dark, before getting brighteragain, and thus continuing to get dark then bright. as ittades away off to either side. The explanation for this overall modulation will come in a future lesson. (Note: There should be no need to print the page with the images on it. It will use up a lot of ink for no great prot.) Divide the beam into two equal sections, as shown. Any ray in the lower half can be 5.7 Single Slit Interference matched up with a corresponding ray from the upper half that has travelled one half wavelength further by the time it reaches the screen. Thus, each pair of rays meets Consider light of wavelength A incident on a single slit of width w. An interference pattern is completely out of phase, and cancels out. As a result, complete destructive interference produced on a screen a distance L away. Because of diffraction, light will travel out from behind will occur at this location on the screen. The angular position of this first minimum is the slit in all directions. Because of the width of the slit, some of the light will travel further before reaching a particular spot on the screen than light from other parts of the slit. sin 0 A In the following series of diagrams, all parallel rays of light are supposedly travelling from different portions of the slit to the same part of the interference pattern. Thus, they are not strictly parallel, c) Secondary Maximum but if the screen is far enough away, the approximation is valid. Increase 0 further, so that the path difference between the top of the beam and the bottom a) Central Maximum is now one and a half wavelengths. All light travelling nearly straight out from the slit to converge on the centre of the screen travels nearly the same distance. Thus, all parts of the beam arrive in phase, and interfere 3A constructively, producing a bright area at the centre of the pattern, called the central e maximum. If 0 is the angular position of this maximum, then light from source W sin 8 = 0 nearly parallel rays of light converge and interfere large distance away on screen light from source -S. Now, divide the beam into three equal sections. The bottom two thirds of the beam will effectively cancel, as shown in section b) above. This leaves one third of the beam un- cancelled. Thus, at this location on the screen there is a secondary maximum, roughly one-third as bright as the central maximum. The angular position is given by sin 0 = - 3A 2 w b) First Minimum d) Second Minimum Consider light travelling from different parts of the slit to converge at a spot on the screen with angular position 0, such that light from the upper part of the slit travels a full Once again, increase the angle 0 until the maximum path difference is two wavelengths. wavelength further than light from the bottom of the slit, as shown. 21 light from source N2 light from sourceDivide the beam into four equal sections. Cancellation occurs as before. both inthe upper two sections. and in the lower two sections, with nothing left over. Thus, at this location, there is another dark spot, or minimum, given by sine : E W In general. the maxima in light intensity on the screen occur when . 3 A 5 it sine :0,#,.... m\" 2 w 2 w and the minima will occur when sine", : % 2H? 33-- As before, for small values of 6, sins - tans = 5 where y" is the distance from the centre of the screen to the n"' minimum. (Note: yfmm this note and x from the double slit note are measured in the same direction, and the two different letters are used to indicate features of the Single and doubie slit interference patterns respectively. ) Combining these last two expressions gives the actual distances from the centre to the first few minima to be A L : 2 A L 3AL 4AL Y. : W V2 W y; : W Y4 : W There is a consistent distance between the minima of Aye W Note that torthe locations ofthe two minima that straddle the centre, there is no factorot V2 in the formula that there was for the double slit pattern. Thus, the central bright region is. in fact twice as wide as all of the other maxima, being 2 Ay. Set a sketch of light intensity versus position on the screen forthe pattern produced by a single slit would appear as follows: First Minimum Second Minimu \\ b by AV Ay Ay Ay Ay Position Secondary Maximum Tertiary Maximum When there are two slits, each of width w, with a distance d between their centres, then the resuiting pattern is a combination (or convolution) of the single slit pattern and the predicted double slit pattern. The single slit pattern determines the overall brightness, while the double slit pattern determines the specific oscillation of brightand dark fringes, intensity Central Maximum \\_ /' Bright Fringes First Minimum \\ Dark Fringes Secondary Maximum \\ i Second Minimum \\ i Tertiary Maximum\\ ii \\_ . - I ) Position Ay'Ay'Ay'Ax IAXI The math resulting from this experiment is largely identical to that of Lesson 5.3, because this 5.5 Young's Double Slit Experiment experiment is also effectively two point interference. We do, however, redefine, or at least re- describe, a few of the variables If light is a wave, then light from two sources should interfere, and many experiments were done to try to detect this interference. However, these experiments were doomed to failure because x, is still the distance from a point on the n" nodal line to the perpendicular bisector of the those doing them didn't know enough about the production or nature of the (supposed) light sources, but it is now also the distance measured on the screen from the centre of the waves. Most sources of light at the time were incandescent, that is light produced by hot objects. interference pattern to the n" nodal line (specifically, x2 is shown in the diagram below, Such light is incoherent. It is a combination of light waves from many vibrating electrons, with no labelled as x, for generality.) consistent direction of oscillation or emission. Additionally, the frequency of visible light is in the vicinity of 10" Hz, and the light from two unrelated sources will have an inconsistent phase 0, is still the angle from the centre of the pattern to the n" nodal line, but it is now also the relationship that shifts from constructive to destructive, more or less randomly, with a frequency angle between the centre of the interference pattern and the n" dark fringe, as viewed from of about this value. As a result, the human eye will be entirely incapable of detecting an the location of the slits. interference pattern, because the blended light from the two different incoherent sources will be a continual ultra-high frequency blur of changing patterns. It will appear uniform and constant - ie. without pattern. Thomas Young's brilliant idea was to use a single source of light, and pass it through two very narrow side by side openings (or slits) in an opaque barrier. He placed a white screen behind the source barrier for the light to shine on. If light really was a wave, then two things should happen: Light should diffract outward from each of the two slits, as if from point sources. ii) The light from these two sources would always be in phase, since they were, in intensity effect, the same source, duplicated. Thus, an interference pattern exactly like that seen in Lesson 5.3 on Two Point Interference should be produced. Since light can't be seen while it is traversing space, but only when it is incident upon a surface, and L has changed slightly. In Lesson 5.3, it was the diagonal line, the hypotenuse in the then reflected toward our eyes, this interference pattern would not be visible, were triangle shown above. It is now the horizontal line, the straight line distance from the slits it not for the screen that it is projected onto. In practice, this means that, rather than to the screen. This is a much more easily measured quantity, and, since it turns out that seeing a series of concentric circles and nodal lines, like we drew in Lesson 5.3, we the angles are very small, not appreciably different from the diagonal length of the would see a series of bright and dark spots (called fringes) where the constructive hypotenuse. and destructive interference occurs on the screen (the dark fringes being where the nodal lines intersect the screen.) Thus, the equation sin On = (n - 1/2) 4 It took Young some time to get his experiment to bear fruit. His early trials had slits that were too wide to produce any significant diffraction, too far apart to produce distinguishable nodal lines, and he also used a white light source. By narrowing the slits, moving them closer together, and by placing a filter over the light, limiting it to one colour (retrospectively now understood to be one is still fully applicable. From this new diagram, however, L in = tan On , rather than sin 0,, as it wavelength) he was finally able to achieve the results that he predicted, effectively demonstrating conclusively that light behaves as a wave. was in Lesson 5.3. However, for small enough angles, tan 0, = sin 0,. (Go ahead, calculate the an and sine of 10 for yourself, and see. Then try again with 0.1. In each case, the tan and sine This diagram illustrates the basic experiment. A of the angles are almost indistinguishable, especially so for 0. 1".) screen Intensity portion of the (invisible) two point interference bright fringes source pattern produced by the diffraction of the light from the slits is shown. A graph is shown to indicate the Thus , 'n = tan 0, = sin 0, , and we may still say relative intensity of light versus position on the dark fringes screen. The high values of intensity correspond to barrier constructive interference (bright fringes), while the low values correspond to destructive interference Xn = (n - 1/2) Position (dark fringes), where the nodal lines touch the screen. provided x, is small compared to L (i.e. 0, is small.)