Diffraction and Interference

Transcription

1 Experiment #32 Diffraction and Interference Goals: Perform a quantitative investigation of two-slit interference Explore use of a photodiode to measure light intensity References 1. I. G. Main, Vibrations and Waves in Physics (Cambridge University Press, 1994) 2. E. Hecht, Optics, 4th Edition (Addison Wesley, 2002) 3. Appendix to Expt 48, Semiconductor Devices 4. The Double Slit Experiment, Physics World, 2002, Introduction Diffraction and Interference All waves, including light waves, obey the principle of superposition, which states that when several effects occur simultaneously, their net effect is the sum of the individual effects. This leads to the interesting phenomena of interference and diffraction. To illustrate these phenomena, you will investigate interference of a wave passing through two slits. This experiment was first successfully performed by Thomas Young around He was able to use his results to demonstrate that light did behave as a wave and used the result to measure the wavelength of light. In 1974, Claus Jönsson performed the experiment with electrons to demonstrate the wave nature of the electrons. Since then particle interference has been demonstrated with neutrons, atoms and molecules as large as carbon-60 and carbon-70. A simple schematic of this situation is shown in Fig. 1. A plane wave, whose wavefronts are perpendicular to the direction of propagation shown on the diagram, is incident normally on a screen with two apertures that take the form of long narrow slits parallel to each other and perpendicular to the plane of the diagram. The superposition of the waves from the two slits is observed a large distance L >> d from the screen, so that we can consider superposition of plane waves at the observation screen. These two waves can be described by two fields E 1 = A 1 (θ) cos() (1) E 2 = A 2 (θ) cos( + α), (2) 1

2 Incident wave d θ d sinθ L Figure 1: Diagram showing interference of a wave passing through two slits. where ω is the frequency of the light, t is time, and α is the time-independent phase difference between the two fields. If the two fields are in-phase, or coherent, at the source, the phase difference is due only to the difference in pathlengths travelled by the two waves. As shown in Fig. 1, the pathlength distance at an observation angle θ is given by d sin θ, where d is the spacing between the slits. This small difference in the distances travelled by the waves from the two slits introduces a phase difference α (θ) α (θ) = k d sin θ, (3) where k = 2π/λ is the magnitude of the wavevector and λ is the wavelength of the light in the medium. In the limit of L >> d, we can also assume that the amplitudes A 1 and A 2 of the superimposed waves are equal, although these amplitudes will, in general, depend on θ. To combine the field components E 1 and E 2, we add their phasors as if they were vectors. Then the field of the combined wave can be calculated using a vector diagram as shown in Fig. 2. The resultant field is given by where the resultant amplitude is given by E = E 1 + E 2 (4) = A 1 (θ) cos() + A 2 (θ) cos( + α) (5) = A(θ) cos( + δ) (6) A = 2 A 0 (θ) cos α(θ) 2, (7) where A 0 (θ) = A 1 (θ) = A 2 (θ). In most cases it is the intensity rather than the field that is detected, where the intensity of an electromagnetic wave is the field squared. For the field given in Eq. 7, the amplitude of the intensity observed in the direction given by θ depends on the square of the amplitude 2

3 (a) (b) A sin(+δ) A α A 2 A 2 sin(+α) ½ A ½ A ½ α A o α δ A 1 A 1 sin() ½ α A o Figure 2: Vector diagrams for calculating the amplitude of superimposed waves. The general case is shown in part (a) and the case with two equal amplitudes A 1 = A 2 is shown in part (b). of the wave radiated by each slit and a term that depends on the path length difference Ψ 2 and is given by I (θ) = 4 A 2 0 (θ) Ψ 2 (θ). (8) If we assume that the amplitude of the wave radiated by each slit varies slowly with θ, the interference pattern will be dominated by the behaviour of the path-length dependent term Ψ 2, where ( ) 1 Ψ 2 (θ) = 4 cos 2 2 k d sin θ, (9) which has maxima in directions given by sin θ = n λ d (n = 0, 1, 2,...). (10) The Teachspin Two-Slit Interference Apparatus To measure two-slit interference, you will use the Teachspin Two-Slit Interference apparatus. A photo is shown in Fig. 3. The apparatus consists of a long rectangular metal assembly, with a detection box containing a photodiode (PD) and a photomultiplier tube (PMT) at one end. The apparatus should be oriented so that the detector box sits to the right hand side. The PMT is extremely sensitive to light so make sure that the shutter that protects it is closed at all times. This is the case when the rod which projects from the top of the detector box is pushed all the way down. Exposing the PMT to room light could damage it. Inside the metal assembly are several optical components shown schematically in Fig. 4: 3

4 Figure 3: Photo of Teachspin Two-Slit Interference Apparatus Figure 4: Schematic of Teachspin Two-Slit Interference Apparatus At the left end of the apparatus are two light sources, a red laser diode and a greenfiltered light bulb, as well as controls for the light sources. The laser diode is in a small box and the light bulb is in a cylindrical holder to the right of the laser diode. You will only use the laser diode in this experiment. A single slit, the source slit, is used to clean up the incident light; In the centre is an assembly in which a screen containing a double slit and a slit blocker can be placed. The position of the slit blocker can be adjusted with a micrometer; The detector box is at the right end of the apparatus. In front of the detector is there is a detector slit. The position of the detector slit can be adjusted with a micrometer. The intensity detected by the photodiode can be measured by connecting a voltmeter to the photodiode output. All of the slits are positioned on magnetic mounts. 4

5 Photodiodes Photodiodes are sensitive light detection devices. They are semiconductor devices based on a p-n junction, that is a junction between two regions of semiconductor material where one region is doped with a donor impurity and the other is doped with an acceptor impurity. When a photon of sufficient energy hits the junction, it excites an electron and produces a photocurrent. The current produced is proportional to the light intensity incident on the device. While they have lower sensitivity than a photomultiplier tube, they are robust and of lower cost. Prelab Questions 1. Derive the position-dependence of the intensity pattern quoted in Eq The derivation of Eq. 9 assumes that the amplitude from each slit is independent of θ. This is true for an infinitely-narrow slit. What happens if the slit has finite width? 3. How should the interference pattern for a double-slit change as the slit spacing increases? As the slits become wider? Apparatus HeNe laser metre stick or tape measure Teachspin Two-Slit Interference Apparatus laser diode source and detector slits (narrow, both are the same) slit blocker aperture (wide) double slit (choice of 3 different ones) photodiode multimeter travelling microscope Experiment Part I: Qualitative observations. You derived a mathematical relationship between the separation between the two slits, their distance to the screen and the positions of the bright and dark bands on the screen. Use the HeNe laser, the metre stick and the 3 double slits provided to explore this relationship qualitatively. Record enough data to allow you to estimate the slit separation and slid width. 5

6 Part II: Quantitative observations. Use the Teachspin apparatus to investigate the doubleslit interference pattern for one of the 3 double slits provided. Pick one and explain why you chose that particular one. Please take care handling the slits and other components of the apparatus. 1. Before starting your measurements, the apparatus needs to be aligned. For best results, all slits should be positioned with their long edges set to be accurately vertical. In this experiment, you will use the laser light source and the photodiode detector. Before starting, remove all of the slits. Turn on the laser diode. The laser should be aligned so that the beam travels down the optical path, through all of the circular apertures, and is eventually centred on the circular aperture in front of the detector. Use a small piece of card stock to check that this is the case. If it is not, please contact an instructor. Replace the detector slit. Mount the source slit and align it to maximize light through the source slit and centre the transmitted light on the circular aperture in front of the detector. Light passing through this narrow source slit will undergo single-slit diffraction and you can follow this process by moving your viewing card downstream. You will see the beam spread out horizontally, reaching a width of about 1 cm by the time it reaches the middle of the apparatus. Compare the appearance of the beam at the detector with and without the source slit. Why use the source slit? Mount the double slit and align it, maximizing the amount of light through the slit. Put your viewing card downstream of the two-slit structure and observe the interference pattern. Mount the slit-blocker on the holder just downstream of the double slit and align it so that the light from both slits emerges. 2. Record the interference pattern of the double slit. Move the detector slit across the front of the photodiode using the translation stage and record the photodiode voltage. Note: it is always good practice to turn a micrometer screw in the same direction when taking a reading. This avoids problems with backlash in the screw system, always present to some degree. 3. Fit Eq. 9 to the data for the double-slit interference pattern. Does this provide a good model for the data? If not, why not? 4. Use the slit-blocker, just upstream of the double-slit system, to block the light coming from either of the two slits. This slit-blocker is controlled by the micrometer screw on the front centre of the apparatus. Record the intensity as a function of distance perpendicular to the slits for first one slit and then the other. How will the intensity transmitted through each aperture affect the interference pattern that you recorded? 6

7 5. Plot all three intensity functions on the same graph. 6. Modify your model function to account for the shape of the apertures and fit to your data to obtain the slit width and slit spacing. 7. Measure the slit width and slit spacing with the travelling microscope. Compare to the results of fitting the model function to the data. BJF/2007-2, NA/2010 7