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Effect of diffraction 1 Fraunhofer diffraction The math is simplified if the rays are parallel. This is called “Fraunhofer diffraction.” The text achieves this condition by making the source and the screen “far away” from the slit. Converging lenses can be used to achieve the condition in a practical way. 2 The general case, illustrated here, where the rays are not parallel, is called “Fresnel diffraction.” We do not deal with this case. 3 4 5 6 To calculate the intensity at the screen, contributions from each small part (x in the figure) of the slit are added. Then the finite sum is converted to an integral by letting x go to zero. 7 Relative intensity in single-slit diffraction for three different values of a/. The narrower the slit the wider the central diffraction maximum. For the same slit width, the relative intensity for two different wavelengths. Since the central maximum for B is wider, we can see that B > A,. 8 Homework Problem • For Fraunhofer diffraction, the intensity of the center of the central maximum decreases if the slit size is reduced. This is clear from the figure. If the slit is narrower, fewer rays (representing the paths of Huygen’s wavelets) reach P0. • The first minimum occurs for a sin = , so when a is reduced is increased. • Thus, the answer is A. The intensity equation may be misinterpreted to imply that intensity at =0 is independent of a, so C may seem correct. It is not. 9 Diffraction imposes a fundamental limit on optical devices, since the light from two separate objects cannot be distinguished (or “resolved”) if the central diffraction maxima of the light from two sources overlap. 10 Light enters the human eye through the pupil, which is a circular aperture of diameter around 2 mm. Diffraction therefore limits the ability to resolve distance objects. Applying Rayleigh’s criterion, two “dots” (as shown) cannot be resolved if (=D/L) is less than R = 1.22 /d, where d is the diameter of the pupil. Note: the index of refraction inside the eye is similar to that of water (n = 1.33), so the wavelength inside the eye is less than outside. It is the wavelength inside that matters since the diffraction takes place inside the eye. Taking = 550 nm, n = 1.33, and d = 2 mm as typical values, R is about 0.025 rad. So, two objects 1 m away from you cannot be resolved by the eye if they are less than 0.25 mm apart. This is not the result of defective eyesight. It is the result of diffraction. 11 Example: The aperture of a telescope has diameter 6.0 cm. For white light, take the wavelength to be 550 nm. For this telescope, what is the minimum angular separation between objects that can be resolved according to Rayleigh’s criterion? What minimum distance does this imply between resolvable objects on the Moon? Note: the Earth’s atmosphere limits the angular resolution of a telescope to no better than 1 arc second (i.e., 2 rad/(360x60x60) = 5 x 106 rad). 12 Active and Adaptive Optics Material properties limit the size of precision mirrors for telescopes (the largest is 200 inches, 5.1 m). “Active optics” overcomes this by making a larger mirror out of smaller mirrors that can be independently aimed, achieving the overall effect of a bigger mirror. As a result, 8 m mirrors are in use. “Adaptive optics” addresses the limit from atmospheric turbulence. A “wavefront sensor” detects the distortions, a computer calculates corrections, and the mirror segments are rapidly adjusted (in milliseconds) to correct for the distortions. 13 Intensity from double-slit interference if diffraction is ignored (corresponding to the case of vanishingly narrow slits, a<<). Single-slit diffraction. Double-slit interference including diffraction. 14 Double-slit Intensity I() for three different slit widths (shown) for slit separation d = 50. 15 Diffraction grating (showing five slits) 16 The lines formed by a diffraction grating become narrower if the number of rulings is increased. Image on screen for monochromatic source 17 Grating spectrometer diffraction grating 18 Atoms emit light when excited. The light is emitted at specific wavelengths that are characteristic of the particular atom. Thus, a grating spectrometer can be used to identify atoms from their emission spectrum. Hydrogen emits light at four visible wavelengths. Hydrogen lines from a grating spectrometer (m=3 is not shown, for clarity, since the lines overlap with m=2 and m=4) 19 X-ray diffraction 20 21 22 23