Two slit interference - Prelab questions

Transcription

1 Two slit interference - Prelab questions 1. Show that the intensity distribution given in equation 3 leads to bright and dark fringes at y = mλd/a and y = (m + 1/2) λd/a respectively, where m is an integer. 2. Derive equation 5 from the multiple slits distribution (Equation 7). 3. Given that the length of the bench is around 1 m, calculate the time it takes for a photon to traverse the length of the bench. Given this time-of-flight, calculate the average time between photons for a count rate of photons/second (as may correspond to the count rate of your experiment at the central maximum). What does this say about the number of photons present along the length of the bench at any point in time? 4. Guess/estimate what might happen in a three- or four-slit system. Do you expect the two-slit diffraction pattern overlaid on a single slit pattern, or something else entirely? 1

2 Two slit interference - Theory The double-slit experiment has its origins in an experiment performed in 1665 by Grimaldi, where he allowed light from the Sun to pass through two pinholes in an opaque screen. He had hoped to illustrate that when two circles of light overlap on a far screen, regions of darkness result. That he could not illustrate this was due to the lack of coherence in the source. The primary source of light, the Sun, was not an ideal point source, and thus lacked coherence. Young returned to the experiment in 1805, relying on knowledge of diffraction from a single slit to provide a spatially coherent primary source of light: the central maximum from the diffraction pattern that emerges from a single slit. The alternative is to use a laser source, which is spatially coherent by construction. When that was passed to two identical slits in close proximity, Young was able to observe interference fringes. The experiment gives rise to two interesting questions: what are the conditions of interference? And, more importantly, can it be done with a single photon? Consider the experimental configuration shown in Fig. 1. A single slit S, is illuminated by a light source. This single slit then acts as a source of spherical waves. If the distance d to the double slit is much greater than the separation a of the slits, d >> a, then the path difference for the light arriving at the double slits S 1 and S 2 will be minimal and the two slits act as sources of coherent spherical waves. Light travelling to point P in the image plane will have a path difference, r, which for small θ is given by r = aθ = ay D. (1) P S S 1 a r 1 r 2 y S 2 Δr d D Figure 1: Geometry for Young s double slit experiment. 2

3 The phase difference, δ, induced by this path difference is then ( ) 2π (ay ) δ = k r =, (2) λ D where λ is the wavelength of the light. The phase difference will modulate the intensity of the light measured at P due to the superposition of the light waves, with the resulting intensity distribution given by I = 2I 0 (1 + cos δ). (3) The spacing between successive bright or dark fringes is then y = λd a. (4) However, this interpretation does not take into account that the fringe pattern is modulated due to diffraction effects from the slits themselves. That was the main criticism of the experiment at the time for which Fresnel and Lloyd suggested variations using either a biprism (Fresnel) or a mirror (Lloyd) to create virtual, instead of real, sources. In this experiment, as we use real primary and secondary slits, we must take diffraction into account. The actual intensity distribution with angle is given by where I(θ) = 4I 0 ( sin 2 β β 2 α = (ka/2) sin θ and β = (kb/2) sin θ, ) cos 2 α, (5) with a and b being the distance between the two slits and the slit width, respectively. I 0 is the intensity at θ = 0. This is the reduction for N = 2 from the multiple slits diffraction irradiance pattern. A system with N slits of width b and separated by a distance a, may be expressed as follows E = C b/2 b/2 F (x) dx + C a+b/2 a b/2 F (x) dx + + C (N 1)a+b/2 (N 1)a b/2 F (x) dx, (6) where F (x) = sin [ωt k(r x sin θ)] and C is a constant related to the amplitude of the wave. This leads to the following expression for the intensity distribution at the image plane in the Fraunhofer limit (ie where the image plane is far from the slits, D > 2 a 2 /λ), I(θ) = I 0 ( sin β β where β = (πb/λ) sin θ, α = (πa/λ) sin θ, and I 0 = I(0)/N 2. ) 2 ( ) 2 sin Nα, (7) sin α The cos 2 α factor is the interference which is modulated by the sinc 2 β factor for single slit diffraction. 3

4 Two slit interference - Equipment The apparatus is from TeachSpin and is shown schematically in figure 2. Light sources Double slit and collimator Shutter Power Single slit and collimator Micrometers Detector box Figure 2: Schematic of the apparatus. (Not to scale.) The apparatus should be positioned with the detector box to the right. On left is the power source and two light sources, which consist of a red diode laser (5 mw, λ = 670±5 nm), and a low-intensity white lamp with green detachable filter. The filter limits the wavelength to 540 to 560 nm. When in use, the lamp will be positioned in front of the laser. This should be carefully pushed to the bottom of the track when the laser is used for calibration. The power is from a DC transformer and has controls which allow for switching between the laser and the lamp and the variation of the intensity of the lamp. You should find the apparatus with the top cover CLOSED. If this is not the case, CONSULT YOUR DEMONSTRATOR IMMEDIATELY. As shown in figure 2 the bench has a series of micrometers along its length. The first micrometer sits at the position of the double slit. This allows for the position of the double slit to be adjusted such that slits can be selectively blocked. At the right end of the apparatus is the detector box and second micrometer. The second micrometer controls the position of an aperture situated in front of the detectors and is calibrated in (true) mm. The detectors themselves are a photodiode, for use with the laser, and a photomultiplier tube (PMT) for single photon counting. The controls on the box govern the power for the PMT, as well as the outputs for both. The PMT is super-sensitive to photons and even a small amount of background light is enough to destroy the very thin layer of phosphor at the entrance window to the dynode chain. IMPORTANT! OPERATION OF THE APPARATUS. Also at the detector box is the shutter. It is the black cylinder with the cable coming out of it at the top of the box at the end of the optic bench. This switches between the two detectors. When the shutter is FULLY down, the PMT is blocked and the photodiode detector is exposed to the bench. When the shutter is up the reverse situation occurs. Along the top of the bench is a cover and this should always be in place when measuring the intensity pattern. WARNING! NEVER remove the bench cover when the shutter is UP. 4

5 Two slit interference - Procedure To open the track and remove components: 1. Make sure the shutter is down and the PMT bias turned off 2. Give each latch a slight lift and quarter-turn 3. Slowly slide the cover away and up from the detector box end. 4. You can now see the internal arrangement with the source slit and collimator sitting between the light sources and the first micrometer. 5. The slits and collimators are all magnetically mounted on their posts and may be removed safely using the tweezers provided. Using the Laser 1. You will note that the white light source/green filter may be raised and lowered in front of the laser. 2. Be careful when moving the lamp: it is on a delicate mount and any motion should be slow and deliberate. 3. Remove the single source slit and measure the slit width using the sliding microscope provided. Measure also the widths of the double slits and their separation. (Slit separation is conventionally defined as the distance between slit centres.) Be sure to note the errors of the measurements. Question 1 Can you think of any methods to better measure the widths of the slits? Discuss with your demonstrator. Alignment 1. Replace the slits in the appropriate orientation. 2. Begin by lowering the lamp out of the way of the laser. 3. Turn on the laser and position the source slit such that the beam reaches the double slit roughly centered. 4. Use the alignment screws to adjust the position of the laser beam until it is centered on the slit. 5. Adjustment should be performed such that maximum power reaches the single source slit and also the single-slit diffraction pattern should be projected centered on the double-slit. 5

6 Question 2 Look at where exactly the photons originate in your light source, and where the photons are incident on the detector. Should you include the size of source and detector in your error? 6. The collimator before the double slit should be positioned such that the vertical edges are parallel with the images of the two slits. This can be checked by observing the light distribution on a white card downstream of the double-slit. Adjustment of the micrometer in the double-slit area will translate the slit stage and collimator relative to each other; this can be used to block the passage of light from slits. 7. Slight adjustment of the orientations of the slit stage or collimator are required until the blocking of a slit is observed on the card as a knife-edge, rather than a gradual vertical decrease. Question 3 Are photons diffracting from the front or the back of the slit? If you had a very thick slit would you consider the entry or exit of the photons the diffraction plane? Measurement 1. Using the white card, observe the image of the two slits just beyond the collimator. 2. If the system is correctly aligned, you should see the double-slit interference pattern at the far end of the bench at the shutter s position. 3. Adjust the double-slit micrometer and make brief observations (diagrams are useful!) on the images at the shutter for both one-slit and two-slit interference. 4. Remember to set the slit back to its two slit interference position before moving on. Question 4 What happens when you switch from one to two slits (and vice-versa) to the pattern at the shutter? Can you observe two-slit interference by eye? You will note that at the shutter is another slit, of the same width as the source. This acts as an aperture for the detector, allowing for sampling the intensity pattern as one moves the micrometer. Do NOT touch this slit or adjust its position in its holder. Measuring the intensity distribution You are now in a position to measure the intensity distribution of the pattern using the photodiode. The photodiode itself is a 1 cm 2 solid-state photodiode and generates a current upon illumination by a light source. 1. Now replace the bench cover and close the four latches. The cable emerging from the top of the shutter carries the current from the photodiode and should be connected to the INPUT of the photodiode on the detector box. The OUTPUT line on the photodetector box carries the voltage as converted from the photodiode s current. 6

7 2. You may use the digital multimeter (on the 2V scale) to measure that voltage, which is proportional to the intensity of the light being sampled. 3. With the cover on and the laser off, measure the zero offset of the diode. 4. Turn the laser on, and position the first micrometer such that it is in the position where light from both slits is allowed to emerge. 5. Starting at the zero position of the second micrometer, measure the intensity at regular intervals between 0 and 10 mm. 6. Be sure to choose a suitable interval such that any pattern is adequately resolved. You may enter the data directly into Excel. Question 5 If you were provided with a clean 2-slit interference pattern, rather than measuring one yourself, what properties of the system could you determine? Single photon experiment At this point, we now looking to count single photons. We will investigate whether sending individual photons from a low intensity monochromatic source towards a double-slit produces observable interference effects for this set-up. Measurement of each slice of the intensity distribution is now performed over a fixed time period to allow for an accumulated pattern of single photon detector counts to be analysed. The PMT should be configured and measurements taken: 1. With the PMT and laser power OFF, the shutter DOWN and the PMT dial turned down to 0, remove the cover and raise the lamp to be in line with the laser. 2. When in operation, the light from the lamp should traverse the same path as the light from the laser did in the previous run. 3. The lamp intensity should not be raised to a high level; generally it should be kept below Be careful not to disturb the slits or collimators as they should now be aligned. Once the lamp is on and the filter is in place, replace the cover. 5. The output of the PMT should be connected to an oscilloscope and a digital counter. With the shutter closed, turn on the high-voltage toggle switch and observe on the oscilloscope how the pulses change as the PMT accelerating voltage is slowly increased. Once 4-5 is reached on the dial, higher pulses should be seen on the oscilloscope and a count-rate of 0-10 photon events per second recorded on the counter. At this point, the voltage is set correctly. 6. Raise the shutter. The count rate should now be much higher; check that this varies in an appropriate way as the bulb intensity is adjusted, bearing in mind that you should not go well above 6. (This isn t Spinal Tap...). 7

8 7. Connect the output of the PMT discriminator to the oscilloscope to observe these pulses simultaneously with the analog pulses. The discriminator generates a pulse each time a PMT pulse exceeds an adjustable threshold. Adjust the discriminator level so that one signal is produced per analog pulse above 50 mv from the PMT. The PMT accelerating voltage may need to be increased to achieve this. Adjust the discriminator level such that a high count rate occurs (this may be up to 10 3 /second). 8. Perform the same measurements as in the earlier section, only this time measuring actual photons in 10-second intervals using the counter. Take several background measurements and find an average. Analyse the data as before, making any necessary adjustments to the parameters given the change in the light source conditions. Question 6 Using Excel, plot the intensity measurements obtained. Fit the data to the assumed intensity distribution for double-slit diffraction patterns using the Solver excel add-in with parameters obtained from measurement of the slit widths and separations. Compare the fitted values to those expected. Comment briefly on your findings. Question 7 How do you expect the width of the maximum to change with the wavelength of the incident light? Is this what you observe? Question 8 What do you conclude about the nature of the interference after completing this experiment? 8