Imaging Systems Laboratory II. Laboratory 8: The Michelson Interferometer / Diffraction April 30 & May 02, 2002

Transcription

1 Imaging Systems Laboratory II Laboratory 8: The Michelson Interferometer / Diffraction April 30 & May 02, 2002 Abstract. In the last lab, you saw that coherent light from two different locations in an optical beam could be combined after having traveled along two different paths. The recombined light exhibited sinusoidal fringes whose spatial frequency depended on the difference in angle of the light beams when recombined. The regular variation in relative phase of the light beams resulted in constructive interference (when the relative phase difference is 2πn, where n is an integer) and destructive interference where the relative phase difference is 2πn + π. In this laboratory, you will extend this idea in two ways, first by looking at a Michelson Interferometer, which generates circular fringes and is an example of a division of amplitude interferometer, and second by studying the diffraction effects associated with more complicated apertures. Lab reports will be due one week after you collect your data. 1 Michelson Theory The "ease" with which fringes can be created and viewed is a function of the coherence of the source, which is inversely proportional to the range of temporal frequencies the source emits. Light from a laser is all of the "same" wavelength (actually a very narrow range of λ), while white light contains all frequencies from the blue (λ=400nm) to the red (λ=700nm). Thus it is much easier to create fringes with a laser than with white light. The coherence length of the source can be increased by spectral filtering to remove much of the spectrum (and thus much of the available intensity). The Michelson interferometer is probably the most famous example of an interferometer that splits the beam by dividing the amplitude of the wave at a certain surface. This surface is known as the beamsplitter consists of a half-silvered mirror which reflects some of the light and transmits the rest. 2 Part I Procedure The Michelson interferometer apparatus is shown below. Unfortunately, we only have one setup, so we will have this in one lab, and take turns using it. Meanwhile, you ll be working with your Part II set up in another of the small labs. The light is split into two beams by the beamsplitter, which is "half-silvered" on the "inside" so that light is reflected after transiting through the glass (careful the diagram is a bit misleading in this regard). The two beams emerging from the beam splitter travel perpendicular paths to mirrors where they are directed back to be recombined at the beamsplitter surface. One beam can be tilted by an adjustable mirror and the length of the other path may be changed by a translatable mirror. Note that the path length is changed by TWICE the

2 translation distance of the mirror. A compensator (a plain piece of glass of the same thickness as the beamsplitter) is placed in the beam that was transmitted by beamsplitter. The light transits the compensator twice, and thus the path length traveled in glass by light in this arm is identical to that traveled in glass in the arm that is deflected through 90 degrees by the beamsplitter. This equalization of the path length traveled simplifies (somewhat) the use of the interferometer with a "broad-band" white-light source (which we will not use here). The compensator is not really necessary when a laser is used, because the coherence length is much longer than the path-length difference. The first step is to bring the two beams in to rough alignment. Do this by directing the laser beam through the (empty) filter holder and into the interferometer, and observing the spot pattern produced at the observation plane (a nearby wall is fine for this). Adjust the tiltable mirror until the two main spots appear nearly coincident. Now insert a lens into the beam to expand the laser beam into a spherical wave. The mirrors generate images of the source of the spherical wave, and the resulting spherical wavefronts superpose. The squared magnitude of the "spatial modulation" of the superposition is the visible fringe pattern. The drawings below give an idea of the form of the fringes that will be visible for various configurations of the two mirrors when used with spherical waves. (To observe these patterns, you will probably need to dim the lights.) Figure 1. The Michelson Interferometer.

3 Figure 2. Typical interference patterns obtainable with the Michelson Interferometer. The left pattern is well-centered. Complete the following: 1. Describe the procedure required to center the fringe pattern what type of adjustments were required, once you inserted the lens into the beam, and why do these adjustments allow you to center the pattern? (A sequence of drawings may help the instructor to understand your answer.) 2. Once you have obtained circular fringes that are approximately centered, move the translatable mirror to increase or decrease the optical path difference. (Be sure that you know in which sense you are changing the path length of the direct beam!). Note the "direction" that the fringes move; in other words, do they "appear" from or "disappear" into the center as the path length difference increases? Explain. 3. Place a piece of plastic wrap in one arm of the interferometer. Describe and explain its effect. 4. Put a source of heat in or under one of the arms of the interferometer; your hand will work (if you are warmblooded!), but a more intense source such as a soldering iron works better. Note the effect. 3 Diffraction Theory Coherence, which is a very important concept in understanding both interference and diffraction, refers to the definite relationship of the phases measured at different points in the wave; in other words, the phase of the sinusoidal electric fields is rigidly deterministic. Coherence has two flavors: spatial and temporal. For spatially coherent

4 light, the phase difference φ = φ 1 - φ 2 of the electric field measured at the same time at any two points in space separated by a vector distance r remains constant for all times. If the phase difference measured at the same location at two different times separated by t = t 1 - t 2 is the same for all points in space, then the light is temporally coherent. Light from a laser may be considered to be both spatially and temporally coherent. The properties of coherent light allow phase differences of light that has traveled different paths to be made visible, since the phase difference is constant with time. For example, the examples considered in the first interference lab may be described in terms of the twoaperture experiment originally performed by Thomas Young. The observed patterns were variants of the sinusoidal fringe pattern he first observed. Patterns of light and dark fringes also may be observed in the irradiance patterns generated from large single apertures in coherent light. These effects are lumped into the single category of diffraction, and are due to time-invariant constructive and destructive interference of coherent light emerging from different points in the same aperture. The phase difference of light from different points in the same "large" source can be seen as a similar pattern of dark and bright fringes, though not (usually) with sinusoidal spacing. In other words, the single large aperture (or multiple large apertures) may be considered as a collection of a very large (infinite) number of infinitesimal contiguous point sources, each emitting spherical waves of light with the same wavelength in phase. Since all sources emit the same frequency, the summation at any observation point will oscillate with that same frequency. However, light from different points in the aperture(s) combine at an observation point after traveling different distances, and thus will have different phases. The vector sum of the amplitudes (magnitude and phase) create bright fringes at locations where they add in phase (at least approximately), and create dark fringes where the magnitude of the sum is zero. Diffraction patterns created by one object and viewed at observation planes located at different distances will have very different character. If the distance between source and observation plane is small, we see "near-field" or "Fresnel" diffraction, while at large distances the pattern is "far-field" or "Fraunhofer" diffraction. In Fresnel diffraction, the "shape" of the aperture is still visible (circular or rectangular hole, etc.), though with fringing appearing at the edges. In Fresnel diffraction, the size of the irradiance pattern is proportional to the size of the aperture. In Fraunhofer diffraction, the pattern of the diffracted light does NOT resemble the aperture and the size of the observed pattern is INVERSELY proportional to the size of the diffracting structure. In other words, two diffracting objects of the same shape but different sizes will generate Fraunhofer diffraction patterns of the same shape but reciprocal sizes; the larger object yields a smaller (though brighter) diffraction pattern. The mathematical relation between the shape and size of the output relative to that of the input is a Fourier transform. For a particular aperture shape f(α,β), the output Fraunhofer diffraction pattern has irradiance proportional to

5 ( ( xα + yβ ) 2πi λr I x, y) = f ( α, β ) e dαdβ, where R is the distance between the aperture and the screen and the integral is just the Fourier transform of f(α,β). 2 4 Part II Procedure Equipment: He:Ne Laser, spatial filter, beam expanding and collimating lenses, set of Metrologic slides to serve as diffracting objects. 1. Set up the experimental bench as in Figure 1 with the observing screen fairly close to the aperture (Fresnel diffraction). Measure and record the relevant distances. A number of "apertures" are available for use, including single and multiple slits of different spacings, single and multiple circular apertures, needles (both tips and eyes), razor blades, etc. In addition, aluminum foil and needles are available to make your own apertures. a. Begin with a one of the larger single slits or circular apertures. Note the form of the diffraction pattern. For example, sketch its form and note the sizes/locations of any features. For a slit or circular aperture, you should note "light" and "dark" regions in the pattern; measure the positions of some maxima and minima (about 3). Use the data to derive a scale of the spatial oscillation rate of the pattern. Sketch the pattern including the scale. b. Increase the distance between the screen and the diffracting object. Repeat the measurements of part a. What is the relation between the change in distance to the observation screen and the change in scale of the pattern? c. Repeat parts (a) and (b) using a knife edge as the object. Sketch the pattern observed. You will see that the intensity distribution near the edge of the geometric shadow is not a sharp transition, but rather an undulatory pattern; a magnifying lens or microscope may be helpful to view the pattern.

6 Figure 3. The experimental set-up for Fresnel diffraction. 2. Now observe the diffraction pattern "far" from the aperture (Fraunhofer diffraction). Since our labs are small, you will need to insert a lens after the diffracting aperture to bring the far-field pattern in to a convenient distance for study. Use the same set-up for all measurements and try to take an image of at least one of your diffraction patterns with the CCD camera, clearly showing fringes, as evidence of the nonrectilinear propagation of light. a. Observe Fraunhofer diffraction from apertures of the same shape but different sizes; these are available in the slide set. Measure the size of any observable features and repeat this measurements using the other slits and then the other apertures. What is the influence of the physical dimension of the diffracting objects on the pattern? b. Repeat the procedure using a periodic structure (diffraction grid or grating) as the object. What is special in its pattern? c. Now overlay a periodic structure (grid) with a circular aperture and observe the pattern. The "overlaying" computes the product ("modulation") of the two apertures. Describe the resulting diffraction pattern in terms of the two patterns obtained individually with each mask. d. For an aperture of a known fixed (small) size, estimate the location of the "boundary" between the Fresnel and Fraunhofer diffraction regions. Record and justify your measurement. e. Examine the image and diffraction pattern of a slide of Albert Einstein or other halftone image. Note the features of the diffraction pattern and relate them to the features of the transparency.

7 f. Now, remove your focusing lens and examine the pattern generated by a "Fresnel Zone Plate" (FZP) at different screen distances. The FZP is available in the slide set; it is a circular grating whose spacing decreases with increasing distance from the center. Sketch a "side view" of the transmission of the FZP and note that there is no hole in the center of the plate. Now, for sufficiently large distances, you should see that a bright point appears at the center of the pattern on the screen. Examine the character of this spot closely as you change the location of the screen. Describe how the spot changes at different distances. If time permits, you might also want to overlap another transparency (such as a circular aperture) and the FZP and record the result.