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Diffraction and Spatial Filtering P Diffraction and Spatial Filtering P. Bennett PHY452 "Optics" Concepts Fraunhofer and Fresnel Diffraction; Spatial Filtering; Geometric optics (lenses, conjugate object- image points, focal plane, magnification); Fast-Fourier Transform; Analytic functions for N-slit apertures. Background Reading “Diffraction and Fourier Optics” (RiceUnivOpticsLab.pdf). This is an excellent write-up, particularly for theory. We follow it closely. Diffraction Physics, J. M. Cowley - definitive reference for general principles and elegant presentation. Optics, text by E. Hecht - detailed reference document with much practical information. “Geometric optics” (image formation with lenses) and “diffraction” - any UG text. Edmund Optics catalog - excellent technical info, with online tutorials and prices! SpatialFilter.xlt and *.doc Fresnel diffraction calculations (XL template and help) Special Equipment and Skills Alignment of optical components; CCD camera; Laser (Edmund P54-025, 5mW, = 635nm). Precautions Never allow the laser beam into your eye, either directly (“looking” at it) or indirectly (by spurious reflection from metal, glass, etc). Even with viewing screens, the focused spot is extremely bright, so be sure to attenuate the beam to a comfortable viewing level at all times. This will save your eyes and also the detector! Do note exceed 5V for the laser diode ($600). Handle the optical components (lenses, pinhole, apertures, slits) carefully: always store them in the vertical mounts or the wooden rack. Never lay them flat on the table, since they scratch, clog or deform easily. Lens-cleaning tissues and an air brush are available for gentle cleaning. Vinyl gloves are available for handling (mounting, etc), if necessary. Check for scratches by reflections of a bright light. Do not touch the pinhole aperture with anything (fingers, lens, etc), since it is fragile and hard to clean. The overhead red lights may be useful to keep your vision “dark adapted”. (Pupils are insensitive to red, as any astronomer or amateur pilot knows). Background and Theory You will need a facile recall of “geometric optics” (refraction and reflection, ray tracing, etc) in order to set up the lenses for these experiments and to interpret your data. The essential points are summarized in Figure 1 below. Ray tracing locates the image position, with one ray passing undeflected through the lens center, and a second ray passing through the focus point f, emerging parallel to the optic axis. Object and image positions p and q are quantified by the lens equation as 1/f = 1/p + 1/q (lens equation) Eq. 1 Diffraction and Spatial Filter.doc p1/7 12/7/2010 Object Image Optic Axis p f q Fig. 1 Geometry for convex lens with real object and real image at distances p and q, respectively, measured from the lens center. Two special rays identify the image location by “ray tracing”. The image is real, and inverted with magnification M = -q/p. The plano-convex lens is oriented for similar entrance/exit refraction angles to minimize aberrations. You will also need to review “physical optics” (interference phenomena) to understand the diffraction behavior in this experiment. An excellent, concise description is given in the Rice University handout at http://www.owlnet.rice.edu/~dodds/Files332/fourier.pdf Following that reference, we consider the diffracted amplitude Ai (xi) in the image plane located a distance z downstream from an object plane with illumination amplitude Ao (xo), as shown in Figure 2. We adapt this reference slightly to handle the case of a point source on-axis a distance p before the object plane, and consider a 1D aperture (slit or edge) running along y. Fig. 2 Geometry of Fresnel integral in 2 dimensions. Huygen wavelets leave every point in the object plane with amplitude pattern Ao(x0, y0) and interfere to produce amplitude pattern Ai(xi, yi) in the image plane located a distance z away (after Hecht). One can then write the 1-dimensional Fresnel integral as ikx2 ik A( x ) ( phase ) A ( x )exp(0 ) exp( x x ) dx (Fresnel Integral) Eq. 2 ii 00 i 0 0 2 pz This is simply the Fourier Transform of the “illumination function” in square brackets, which includes a phase shift due to curvature of the illuminating wave from the point source. (The Rice ref assumes plane-wave illumination, or p-->inf). Diffraction and Spatial Filter.doc p2/7 12/7/2010 The Fresnel integral, in the limit of large p and z (thus [ ] is constant) yields Fraunhofer diffraction, given by ik A( x ) ( phase ) A ( x )exp( x x ) dx (Fraunhofer Diffraction) Eq. 3 ii 00L i 0 0 2 The measured intensity in both cases is given by I(xi) = |Ai(xi)| , so (phase) does not matter. The Fraunhofer pattern for a few special cases yields analytic expressions, as listed below, and shown in Figure 3. We assume small angles, such that sin() ~ ~ xi/L. 2 sin( )kw sin kwxi Ix10(i ) I , where . (1-slit, width w) Eq. 4a 22L 2 sin( )2 kd sin kdxi Ix20(i ) 4 I cos ( ), where . (2-slits, spacing d) Eq. 4b 22L 2 2 sin( ) sin(N ) IxNi() I0 . (N-slits) Eq. 4c sin( ) 2 2sinJka1 IIc () 0 (Circular aperture) Eq. 4d ka sin Sin()/ is the well-known sinc() function, which has value 1 at the origin, zeros at m,0 = ±m, and maxima near n,max ~ ±(2n+1)/2. The N-slit pattern shows simple, but important scaling: central max I(0) ~ N2, width (FWHM of central max) ~ 1/N and integrated intensity (central max) ~ N. The 2-slit pattern may be recognized as the N-slit pattern with N=2. The last result is for a circular aperture of radius a, where J1 is the first-order Bessel function. The pattern is circular, and we may use sin ~ ~ r/L. The central maximum is known as the “Airy disk” and has radius (to the first minimum) given by ~1.22 . 1 2a 100.0 1 slit 10.0 2 slit 4 slit 1.0 I/I0 0.0 0.5 1.0 1.5 2.0 0.1 0.0 theta/pi Fig 3 Fraunhofer diffraction patterns for 1, 2 and 4 slits with equal width and spacing. Intensity is normalized to that for a single slit. Diffraction and Spatial Filter.doc p3/7 12/7/2010 It is useful to consider the elegant "Convolution theorem", which may be written as Ffx ()*() gx FuGu () (). (Convolution Thm) Eq. 5 In words, “the (Fourier) transform of the convolution of two functions is the product of their transforms”. As an example, the 2-slit object may be considered as the convolution product of a single slit f(x) with delta functions g(x) = (x-s/2) + (x+s/2). The transform G(u) is exp(i)+exp(-i) = 2cos(). This envelope function G(u) multiplies the single-slit function F(u). That is, F(u) is the “envelope function” for the rapidly changing G(u), as can be seen in Figure 3. For moderate values for z or p, the Fresnel integral does not yield an analytic solution. In the old days (before desktop computers), one would work from a graphical solution in reduced coordinates, in the form of the famous Cornu spiral. This antique student-torture device will be relegated to the bottom drawer in the lab where the slide rules are kept. You will simply compute the integral numerically, using Matlab, or XL (can use SpatialFilterTemplate.xls). Fraunhofer Diffraction We begin with a simple setup to observe Fraunhofer diffraction, as shown in Figure 4. All optical components (laser, lenses, slits, detector) must all be carefully aligned so that light passes exactly through the center at normal incidence (adjust 3 translations and 1 rotation). It will be useful to mark the position of each support stand using a removable white tag. Objects can be moved out of the beam by lifting the stand and placing it on the end of the rail. It is useful to “follow the beam” along the rail with a small index card, as you adjust each component. Many lenses are “Plano Convex”, meaning “flat-curved (positive)”. This design reduces spherical aberration by making entrance and exit refractions nearly equal (place them with the flat side facing the close-focus). The laser is fixed on one end of the optical rail, and determines the overall height of all components. Laser CCD Camera MO CL Slits View-screen L z MO = Microscope Objective (20x) CL: Condenser Lens (3.5” w/ variable iris) Slits: Pasco selector wheel or variable-slit (for Fresnel) Fig. 4 Setup for Fraunhofer and Fresnel diffraction. Place the view-screen about 1 meter from the laser, and set the height to match the laser. Remove the MO from the laser-mount and adjust the laser tilt to center the beam on the view- screen. Leave the pinhole out (in its protective box) for now. Place the Condenser Lens (CL) on the rail and align it to give a level beam focused at the detector screen (almost infinity). Finally, place the slit assembly just after CL (distance does not matter, since the beam is essentially parallel). Strictly speaking, Fraunhofer diffraction requires a parallel exit beam (focus at infinity), but the small convergence used here will hardly affect the interference, and will provide Diffraction and Spatial Filter.doc p4/7 12/7/2010 well-defined patterns at the screen. Be sure to take one calibration image (ruler or grid) to define the "camera length" for your screen (pixels per cm). Camera setup (see operation notes in lab) The CCD camera includes a 105mm "macro" lens, which allows for close-up images with minimal distortions over a large field of view.
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