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Transcript
Physics 316 B2 Experiment B2: Monochromatic Fringes with the Fresnel Biprism Reference: Hecht (4th ed.) p. 398 The wave nature of light was established in the early part of the 19th century, although many interference and diffraction phenomena had long been known. Fresnel's experiment with the biprism was one of the earliest experiments to yield values for the wavelength of light. As you will find, accurate values do not usually emerge, but the experiment does provide some useful experiences in careful alignment and fine adjustment. The basic concept is simple: monochromatic light from a source passes through a slit S. Light passing through the two halves of the biprism is deviated so that it seems as though there are two (‘virtual') slits, S’ and S", and the resulting interference pattern is the same as for two slits. path difference s' ! a " r2 s " s" ! r1 r0 interference fringes x = ave. spacing r Figure 1 The wavelength can be calculated from the average fringe separation (x), in terms of the various distances (ro and r): 2r ( n " 1)# x [1] ! = 0 n = refractive index of prism (r + r0 ) Apparatus: Biprism (make a note of its reference number) Slit Laser (He-Ne) Measuring microscope (optional) Beam expander lens (Addendum) Condensing lens (Addendum) Mercury lamp with green filter (wavelength 5461 Å, Addendum) 1 Revised 3/7/08 Physics 316 B2 Procedure Set up the biprism as shown in the diagram (Fig. A1). The biprism is to be mounted on the special holder that permits rotation of the biprism in its own plane. Accurate alignment of the edge of the biprism with the slit (S) is critical to obtain the best fringes. Align the laser, filter, slit (S) and biprism and look, by eye, for fringes. You should be able to see them, and if you use a simple magnifying lens, they should be easily seen. Arrange the separation of the components so that you obtain a large number of fringes when you view them from across the room. Start with the observing position no more than 100 cm from the prism. When you have adjusted to get sharp fringes, increase the prism-observing distance to about 2m, making whatever adjustments are needed to maintain fringe quality. Measurement of the wavelength requires knowledge of the biprisim angle. This is not easy to measure accurately because it is so small; several methods are outlined in different optics books, but we have developed our method which is easy to use and gives accurate results. To measure the distance between components several meters apart, ask the T.A. for a steel tape measure. ! am am e r be lase ected b fl e r and " reflected, refracted beam Figure 2 Measurement of the biprism angles, using a He-Ne laser beam The biprisms are not usually symmetrical, so you will need to measure both angles and use the average in the calculation of the wavelength. For a beam of light at normal incidence (Fig. 2), one reflected beam will return to the laser, but the other reflected beam (from the back surface) will be refracted and will emerge at angle β = 2nα (since α, β are small angles). 2 Revised 3/7/08 Physics 316 B2 Mount the biprism in a lens-holder, so that you can rotate it and vary the angle of incidence easily. Reflected spots can be located on the wall on the far side of the room, and the angle then determined by measuring the distances between the reflected spots and the distance apart of the components. Using the laser, the angle of minimum deviation can be found: Figure 3 ! + D$ sin"# 2 % and with α, D small angles, we have With n = ! sin" $ # 2% " ! + D$ # 2 % D n= & 1+ "! $ ! # 2% We then have two equations: n = 1 + [2] D and ! = 2n" . ! Estimate the standard errors in n and α. (NOTE: angles α, β, D are in radians in this calculation.) Solve for n and α using the measured values of β and D. Separation of the virtual slits is a = 2r0 (n - 1)α. Note on approximations: Hecht (p. 439 problem 9.15), asks you to investigate the conditions under which the approximation, path difference = r1 - r2 ! asin " , is valid, in the derivation of the expression for the average fringe spacing. This approximation requires that r >> a 2 ! where r = the distance between biprism and fringes, and a = the separation between virtual slits. Typically, we find a ! 0.5cm , and with ! " 5 # 10 $5 cm , we find that we need r ! 5000cm , which can not be met, unless a is very much reduced. Closer examination shows that this requirement comes from calculation of the path difference between rays to a single fringe, from the two virtual slits, and this yields 3 Revised 3/7/08 Physics 316 B2 a2 !r = asin" # cos2 " 2r " = mean angle to point of observation [3] Failure to meet the condition r >> a 2 ! means that !r turns out negative! Further examination, however, shows that this is not the way to consider the problem, but that the approximation is indeed valid: " There will be a dark fringe when !r = n (n odd) 2 So ! "2 2 [4] n = " sin# $ cos # 2 2r whence ' dn 2 $ #2 [5] = &# cos ! + cos! sin ! ) d! " % r ( since θ is small we put cos θ =1 ' dn 2 $ #2 = &# + sin! ) d! " % r ( [6] thus $ ' ) " & 1 d! = & # ) dn 2# & 1+ sin! ) % r ( The angle between adjacent dark fringes (dθ) is given by dn = 2: " # d! = %& 1 $ sin ! '( # r whence " ! = " d# $% 1 + sin# &' r ! 0.5 ! Typically ~ ~ 10 "3 so ignore the sin " term r 500 r ! " # d$ # x & = [2r0 (n ! 1)" ]% ( as before. $ r + r0 ' [7] [8] [9] [10] [11] Calculations: Once you have the optical components aligned and have fringes ready to measure, make a rough measurement, to check that you are obtaining reasonable values. 4 Revised 3/7/08 Physics 316 B2 Also, make a rough estimate of the percentage error that is attainable in the various measurements. This will guide you to the precision needed in measuring the large distance, r. From a consideration of the error propagation, you can see where it is necessary to be very careful, and where apparently inaccurate measurements can still be tolerated. In obtaining the best estimate of the wavelength, you will need to calculate the average fringe spacing. Do this by plotting a graph of fringe position vs. fringe number, and fitting a least squares line. Why is this better than taking a straight average of the measured spacings? Qualitative demonstration: The Fresnel biprism lends itself well to a demonstration that interference fringes cannot be obtained when the two beams of light do not have the same polarization. As you have carried out the experiment, the two beams (coming through the two halves of the biprism) have random polarization since they both originate in the mercury lamp. You are provided with two pieces of Polaroid. Attach a piece of masking tape to each as a marker, and by rotating one Polaroid relative to the other, find out which relative orientation produces total extinction. You can then label the polaroids to indicate the directions of their optic axes. Using tape and some cardboard, mount the two polaroids as shown in the diagram and then hold them over the biprism, so that one of the emerging beams will be polarized in one plane, and the other beam has its polarization axis at right angles to the first. The fringe pattern should disappear—you should see a generally illuminated field of view. Remove the polaroids, and the fringes should reappear. Figure 4 5 Revised 3/7/08