We are asked to make a qualitative sketch of a Fresnel diffraction pattern of a

Question

We are asked to make a qualitative sketch of a Fresnel diffraction pattern of a wide slit, using a Cornu spiral. Please explain the thought process as well. Thanks.

Explanation / Answer

A. Start the LabWindows/CVI DAQ program Diffn3 (shortcut on the desktop). Turn on the electronics rack power and Initialize the DAQ in that order. Use the HELP pull-down menu (at the LHS top of blue taskbar of the Diffn3 program) to familiarize yourself with the features of the Diffn3 program. Move the photodiode detector to the x = 2.5 cm position (middle of its range of travel) using the “GoTo X = Xuser” button. Adjust the position of the red He-Ne laser so that the unobstructed laser beam falls precisely on the entrance aperture of the optical fiber head (the far end of which is coupled to the solid-state silicon photodiode and photodiode preamplifier. {n.b. If the computer should crash, turn the power to the electronics rack off before hard-rebooting the computer.) B. Fraunhofer Diffraction: Place one of the slides containing single and multiple slits on the holder close to the laser, approximately 1 m from the detector. This approximates the conditions for Fraunhofer diffraction. Carefully position the slide such that the resultant interference/diffraction pattern is horizontal. Observe this with a screen, and make a qualitative sketch of the pattern. It is very important to insure that the optic fiber intercepts the diffraction pattern symmetrically at both limits of travel (x = 0 and x = 5 cm). Tilt the slide with the slits if necessary to make the pattern exactly parallel to the translator travel, with approximately equal intensities at x = 0 cm and x = 5. Use the Manual ADC Scan button at each of these locations to verify the amount of light at each location. When ready to take data, click on the “START SCAN” button. A pop-up window will ask you to temporarily cover the aperture of optical fiber detector aperture with the flap that is attached on the optical fiber head in order to take an ADC pedestal reading. Immediately after this reading has been taken, another pop-up window will ask you to uncover the optical fiber detector head to begin taking data. After the data-taking scan has completed, examine the four on-line plots, then, if good, save your data to a text-format file for off-line analysis. Note that the photodiode detector current is IPD = -K( - ), where K = 500 (nA/Volt) is the transfer function of the photodiode current => voltage preamp. Scan at least one multiple slit and also a single slit having the same width as the slits comprising the multislit pattern. Make hardcopies of the photodiode current vs. position plots (both linear and logarithmic). Save your data to the hard drive or to a diskette. Please remove/delete these data files when you have completed the D-2 experiment C. Fresnel Diffraction: The Fresnel formalism is valid for any position of the light source and photodetector relative to an obstacle. It is greatly simplified if the source is at infinity; i.e., when the incident rays of light are parallel to each other. Again use the red He-Ne laser for this portion of the experiment. There are three possible experimental procedures: 1. Ideally, this experiment should be performed with incident plane-waves. This can be achieved by passing the laser beam through a microscope objective in order to focus it to a point, and then expanding the laser beam by placing that point at the focus of a positive lens. In practice, this is difficult (but possible) to achieve, requiring extreme care in the alignment of the optical elements. 2. Place half of a beam expander (basically a lens and collimator) and expand the laser beam to make a spot several centimeters wide at the position of the optical fiber detector, centered on the detector aperture when it is positioned at x = 2.5 cm. This will, of course, place the source point at the focus of the diverging lens. Measure the size of the spot at several positions and deduce the effect source position. 3. Use one of the single slits from part A (for example 0.04 mm) to spread the laser beam across the face of the detector. The source is then at the position of the slit. Place the slide holder approximately 1 m from the optical fiber detector, and measure that distance{ = the distance from the source (`, the negative focal point of the diverging lens, or the slit position, depending on the method used)}. It is useful to temporarily minimize the Diffn3 DAQ program and run the Mathcad program “Fresnel.mcd” (shortcut on the desktop) for the semi-infinite plane and thin obstacle (see below) using the actual separation distances, the laser wavelength and obstacle dimensions. This will give you a sense of the scan width to use. Note that the Fresnel.mcd program assumes an incident plane wave. It is also necessary to make a geometric correction if the source is not located at ` (see below). Make a scan of that range with no obstacle in place. Chose a scan range (selectable from a pull-down menu: “Change X_Lo, X_Hi Limits”) such that the scan is centered on the profile of laser beam. Semi-infinite plane: Intercept the light with the edge of a razor blade (epoxied into a slide mount). Position the razor blade such that the edge of its shadow falls on the aperture of the fiber optic detector when positioned at x = 2.5 cm. Note that if you position the blade such that the shadow region is scanned last (i.e. x > 2.5 cm) then your plots will resemble the calculated curve (see below). Scan over a sufficient range to cover the intensity pattern. Narrow obstacles: Repeat the above procedure with one or several slide mounts holding wires. Record the distance from each obstacle to detector, and the diameter of the wire, and make copies of your plots A. Fraunhofer Diffraction In the Fraunhofer limit, the observed intensity from an array of slits of width a, separated from each other by a distance d, is the product of the intensity distribution from N coherent sources multiplied by the diffraction pattern from a single slit. At a distance r from the slit pattern, the intensity at a point x on the screen is governed by the angle ? = tan -1 (x/r). The interference pattern is, in turn, determined by the phase angle a = pdsin?/?, where ? is the wavelength of the light passing through the slit arrangement. Similarly, the diffraction pattern is determined by ß = pa sin?/?. The net result is that: I(x) = I1(sinß/ß) 2 (sin (Na)/sina) 2 , where I1 is the intensity of light from a single slit. The LabWindows/CVI program Nslit (see shortcut on the desktop) computes this function for any choice of a, d, N and r. Alternatively, the Mathcad program Slits.mcd performs the same function. Use one or both of these programs to make a quantitative comparison with your Fraunhofer data, including positions of maxima and minima, and relative intensities. B. Fresnel Diffraction The analysis of Fresnel diffraction patterns is significantly more complicated. So long as the source is at infinity (parallel incident rays), the pattern depends on a single parameter u x x r ( ) / through the two Fresnel integrals: / = 2 0 1 2 b ? g ( ) ( ) 2 2 0 0 ( ) cos 2 and ( ) sin 2 u u C u t dt S u t dt = = p p ? ? . The traditional way to analyze Fresnel patterns is by graphical means associated with the Cornu Spiral: a plot of S(u) vs. C(u) for -` ^

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