E. K. A. ADVANCED PHYSICS LABORATORY PHYSICS 3081, 4051 FRAUNHOFER DIFFRACTION

Transcription

1 E. K. A. ADVANCED PHYSICS LABORATORY PHYSICS 3081, 4051 FRAUNHOFER DIFFRACTION References for Fraunhofer Diffraction 1. Jenkins and White Fundamentals of Optics. Chapters on Fraunhofer diffraction and resolving power of instruments. Other texts on optics may also be consulted. Part 1. Use the He Ne laser and photometer to observe the diffraction pattern from single slit and multi slit systems and to measure in detail the pattern (intensity versus transverse position) for a 5 slit system. Note that the wavelength of the laser light is 6328 Å Part 2. Use a grating spectrometer to measure accurately the wavelengths of the visible lines in the mercury spectrum and to measure the resolving power of the spectrometer. Apparatus mw Helium Neon laser meter Optical Bench (magnetic) and holders. 3. Slides with electroformed slit openings, in single and multi slit patterns. 4. Photometer (with a range of 0.1 to 1000 Lux, full scale) with an optical fiber input. The resolution of the 1 mm diameter fiber can be reduced by a 0.1 mm collimating slit. The collimator and fiber are mounted on a carriage which can be moved transversely in 0.01 mm steps. Part 1. Procedure 1. Make qualitative observations of the patterns on a distant white screen for several

2 single slit and multiple slit slides. Note the number of subsidiary maxima (where present) between the main interference maxima, and record which of the main maxima are almost missing, because of the interference modulation of the single slit diffraction pattern. 2. For the 5 slit slide, use the photometer to record a careful scan of the intensity versus transverse position at the end of the optical bench. In order to accomplish this: (a) Place the laser at one end of the bench. Place the carriage which is holding the fiber and collimating slit at the other end of the bench, with the transverse motion near one limit of its travel. (b) With both slits removed, use the photometer to adjust the laser so that it aims directly at the fiber. (c) Place the 0.1mm collimating slit on the carriage, and use the photometer to adjust the horizontal position of the slit to be centered on the fiber. (d) Place the 5 slit slide directly in front of the laser, and adjust its horizontal position in the laser beam until the central bright fringe of the interference pattern is incident on the collimating slit at the other end of the bench. (e) Record the distance from the 5 slit slide to the collimating slit. (f) Move the carriage transversely, taking photometer readings every 0.1 mm for the full range of the travel. In order to determine the position and the peak reading of the central maximum, make sure that you take a few readings on the other side of the maximum before moving through the full pattern on one side. Analysis of the Five Slit Pattern Your data consist of N points at which you have measured the intensity I i at tranverse position x i. In the reference you will find an expression for the intensity as a function of angle, the slit width and spacing for a 5 slit screen: I = I 0 f(b, d, θ). Convert the x i to θ i

3 using the distance from the 5-slit screen to the collimating slit. Define the calculated value of I at each point Îi = I 0 f(b, d, θ i ). The deviation of the calulated value from the measured value is Îi I i. For any choice of the parameters I 0, b and d, a measure of the goodness of fit is χ 2 = N (Îi I i ) 2. i=1 (χ 2 is placed in quotes since the deviations are not normalized in the usual way.) Write a computer program (spreadsheet can also be used) which can calculate χ 2 from your data points for a given value of I 0, b and d. Determine your initial choice for b from the positions of the principal maxima; your initial choice for d from the missing maxima; your initial for I 0 to make the Îi = I i at the central maximum. We are going to do a graphical miniminzation of χ 2 to determine the best values of I 0, b and d. Plot χ 2 vs b for values in the vicinity of your initial choice. The best value of b is that which gives the minimum value of χ 2. Do the same for I 0 and d. Using these best values as starting values, repeat the procedure a second time. Measurement of Wavelengths with Grating Use a divided circle spectrometer readable to 10 seconds of arc, and a reflecting diffraction grating to measure the wave lengths of the visible mercury spectrum lines with about 0.01% accuracy and compare with accepted wave length values. The resolving power of the system for the yellow Hg doublet is also measured and compared with the expected value. Apparatus, The Diffraction Grating The grating used is a reflection grating of about 1200 lines/mm. CAUTION The grating surface should never be touched by the fingers or any other object. The grating is mounted in a recessed holder on a 3 legged table which fits on the central table of the

4 spectroscope. The grating is used on a divided circle spectrometer which has various adjustments with which the student should become familiar. (The instructor should be consulted.) 1. The collimator slit width and focus can be adjusted. 2. The telescope has an angular aim adjustment system permitting angles to be read at any of four vernier stations, in degrees, minutes, and nearest 10 seconds of arc. An illuminated magnifier is provided. For gross angle changes a clamping knob is loosened and the telescope is rotated about the vertical spectrometer axis. For fine adjustments the clamp is tightened and the fine motion screw is used (over its useful range). The subsequent adjustments are done as follows: 1. If a white sheet is held tilted for illumination in front of the telescope objective, the field is illuminated and the pointer (dark) is visible. Adjust the eyepiece focus for sharp focus on the pointer. 2. Focus the telescope on a distant object. (Do not make further changes of the telescope focus adjustments.) 3. Set the collimator slit width to a small value, turn on the Hg arc and position it so maximum intensity is seen looking into the center of the collimator lens (with the telescope 90 away). Set the telescope to read exactly on one of the verniers. Now loosen the divided circle clamp so the telescope and divided circle rotate together and set the telescope to view the forward direction towards the slit. (A dark blue glass, etc. may be used to reduce the glare and minimum slit width used.) Without disturbing the 180 reading, position the telescope so the pointer is centered on the image of the collimator slit. (Now do not touch the lower controls that move both telescope and divided circle. All subsequent angle changes should leave Only loosen the clamp screws a small amount or things come apart and need repair.

5 the divided circle unmoved, so just the telescope and verniers rotate.) Now focus the collimator for sharp focus. If you cannot reach the adjustment while looking through the telescope, make changes and then view, until best focus seems to have been obtained. 4. Place the reflection grating on its 3 leg table on the spectrometer table with the legs in the three depressions spaced 120 provided. (The grating table should now be very close to level.) Similar clamps when loosened, etc. permit rotation of the spectrometer center table. (Do not turn using the grating table as this may cause it to fall off (i.e., grab lower)). The telescope should still be at the 180 position for viewing the slit if the grating were removed.) 5. With the room lights off, the first order reflected spectrum should be visible looking at the grating center with the head at 30 to 80 from the collimator (eyes at collimator height). The collimator slit should be a little open from its minimum position. At least 5 violet lines λ < 4400 Å should be seen, four blue starting with the 4916 Å line, the strong green line at Å and several before it, the strong and Å yellow pair and a few weaker yellow lines and five or six red lines. 6. The telescope viewing is done with the telescope both left of the collimator and right of the collimator. It is best to use the vernier which comes about 180 from the collimator in each case (not the same for L and R) or for the original 180 setting. Now set the grating almost perpendicular to the collimator. For a better setting measure the angle of bright third red line (beyond the yellow lines) on each side and evaluate (left) and (right) from the collimator. Set the telescope at the mean and use the center table fine angular adjustment to bring the red line to the pointer position. Check that it is at the same viewing on the other side of the collimator. A single thickness of kleenex tissue over the objective lens also effectively reduces excess brightness.

6 7. Note the angular position to the nearest 10 sec of arc when each spectral line is centered on the pointer. (Note: for valid lines the line bottoms and tops are all at common heights in the view. If lines appear having different top or bottom positions, ignore them as spurious. It is possible to see the second order spectra of the first two violet lines on each side at 80. In this case 2λ = d sin θ. If you can observe them, do so! Measure all valid lines on each side and then remove the grating and check the (180 ) angle reading of forward view of the collimator slit to see that it is not very different from the start value. Work fast enough to complete all (left and right) measurements during the period. Analysis For each spectral line evaluate (right) and (left) from the collimator and the average. For a few main lines over the spectrum evaluate the grating spacing d such that λ = d sin θ. Use the average of these d values to calculate the wavelengths of the other lines. Your table should list these values (measured), reference table values, and the differences and discuss the degree of agreement or disagreement. Measurement of the Resolving Power of the Grating Spectrometer Using the Yellow Mercury Doublet The resolving power of optical instruments is ultimately limited by diffraction. For the grating spectrometer the limiting aperture is the 3 cm diameter telescope lens. Place a variable slit over the telescope lens and reduce the width until the yellow lines broaden and merge. Open back the slit so that intensity patterns corresponding to 2, 3 and 4 times the Rayleigh criterion are observed. Measure the corresponding slit widths. Calculate the corresponding spectrometer resolving power and compare with theory.