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4591X_PROJ_Zill.qxd 1/29/06 7:24 PM Page xxx PROJECT FOR SECTION 15.4 © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION Fraunhofer Diffraction by a Circular Aperture Y M © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION S X ρ θ P ψ w L O O Lens © Jones & Bartlett Learning, LLC Anton M. Jopko, Ph.D. Figure 2 NOT SALE OR DISTRIBUTION Department of Physics and FOR Astronomy, McMaster University © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION lens, and its origin is where all the light from the star would appear in the absence of diffraction. Because of diffraction, however, some light also appears at P. Point The stars©inJones the sky & areBartlett an enormous distance from us, P is a general point but very closeLearning, to O, being only Learning, LLC © Jones & Bartlett LLCarc so they can be considered point sources of light. If you seconds away. NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION look at such a star through a telescope, you might expect In Figure 2, we have merged the aperture and the to see just another point of light, albeit a much brighter lens, because in practice the edge of the lens also defines one. However, this is not the case. Because it is a wave, the aperture. Because of the circular symmetry of the light is diffracted as it passes through the circular aperlens and the diffraction pattern, it is very desirable to ture&ofBartlett the telescope so that theLLC light is spread out over a convert to polar coordinates. a wave be emitted at a © Jones Learning, © Jones & Bartlett Learning,LetLLC small fuzzy region that we call the diffraction pattern. point S in the lens with coordinates (X, Y) or 1r, u2 and NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION This project will investigate the shape of the diffraction arrive at P with coordinates (L, M) or angular coordipattern for light passing through a circular aperture of ranates 1w, c2 . Then X r cos u, Y r sin u, and dius R. L w cos c, and M w sin c. Here r is the radial disFor simplicity, we assume the light is of one wavetance from the center of the lens to the source S of the length l, or color. This light has the form of a spherical © Jones & Bartlett Learning, LLC Jones Learning, LLC emitted wave and u is its © polar angle; & w Bartlett is the angular wave front near the star, but by the time it reaches us the c radius of P and is its polar angle. NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION wave front forms a plane wave. All points on the wave The waves emitted at the aperture are in phase and front have the same phase. We now point the telescope have the same amplitude, but they all travel different with its circular aperture and lens directly at the star so distances to point P so they become out of phase there. that the plane wave fronts are incident from the left, as The intensity of light at P will be proportional to the in Figure©1.Jones & Bartlett Learning, LLC Jones & Bartlett LLC square of©the resultant amplitude Learning, of all waves arriving there. We now need to calculate this resultant amplitude NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION Y by taking into account the waves’ phase differences. M We define the wave number of the incident and emitX Lens P ted waves to be k 2p>l. Then according to Principles of Optics, seventh edition, by Born and Wolf, the resultL ant & amplitude P from all the emitted waves in the © Jones & Bartlett © Jones Bartlettat Learning, LLC O Learning, LLC aperture is just the Fourier transform of the aperture: NOT FOR SALE OR DISTRIBUTION O NOT FOR SALE OR DISTRIBUTION U1P2 C Aperture radius R ik1LXMY2 e dXdY aperture Figure 1 Diffraction of light where C is a constant, proportional in part to the bright© Jones & Bartlett Learning, LLC © Jones Bartlett LLC ness of the star. The intensity at P will&then be givenLearning, by From Huygen’s principle, each pointSALE in the opening NOTpattern FOR for SALE ORasDISTRIBUTION NOT FOR OR DISTRIBUTION U1P22. This is the diffraction the star a of the circular aperture emits a wave in all directions. function of the angular radius w. Fraunhofer diffraction requires that the waves leave the aperture in a nearly parallel bundle traveling toward a Related Problems very distant point P. The only purpose of the lens is to 1. Show© that the resultant amplitude at P usingLLC the two form a © point image&ofBartlett this parallel bundle at LLC a much Jones Learning, Jones & Bartlett Learning, systems of polar coordinates can be written closer distance to the aperture. Diffraction would still NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION occur even without the lens. The dotted line joining the R 2p two origins is also the axis of the aperture and lens. The eikrw cos 1uc2rdudr U1P2 C LM system of coordinates is in the focal plane of the 0 0 © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION xxx © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION PROJECT FOR SECTION 15.4 Fraunhofer Diffraction by a Circular Aperture © Jones & Bartlett Learning, LLC. NOT FOR SALE OR DISTRIBUTION. 4591X_PROJ_Zill.qxd 1/29/06 7:24 PM Page xxxi 2. Using the identity 7. What is the value of the smallest nonzero root of J1? Using l 550 nm, R = 10 cm, and the smallest root just found, Learning, calculate the angular © Jones & Bartlett LLC radius w (in arc sec© Jones & Bartlett onds) of the central diffraction disk. NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION 2 J1 1kRw2 where Jn is the Bessel function of the first kind, show 8. Draw a plot of as a function of kRw as well kRw that the resultant amplitude reduces to as the intensity, its square. The diffraction pattern of R central disk surrounded by U1P2 ©2pC dr Learning, LLC the star consists of a©bright JonesJ0 1krw2r & Bartlett Jones & Bartlett LLC several thin, faint concentric rings. This disk isLearning, named 0 NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION the Airy disk in honor of G. B. Airy, who was the first for any c. We choose c 0. (This expression is also to calculate the diffraction pattern of a circular aperture known as a Hankel transform of a circular aperture.) in 1826. 3. Using the recurrence relation 9. What happens to the angular width of the diffraction pattern if the radius R of the aperture is doubled? © Jonesd & n1 Bartlett Learning, LLC © Jones & Bartlett Learning, LLC 3u Jn1 1u2 4 un1 Jn 1u2, 10. What happens the angular of the diffraction du SALE OR DISTRIBUTION NOT FOR NOT FORtoSALE OR width DISTRIBUTION pattern if the wavelength of the light is doubled? show that 11. What happens to the angular width of the diffraction x pattern if the focal length of the lens is doubled? u J0 1u2 du x J1 1x2 12. Suppose that a circular aperture has the shape of an an0 © Jones & Bartlett Learning, LLC © Jones &nulus Bartlett Learning, LLC with inner radius a and outer radius b. Find U(P). 2J 1kRw2 1 NOT FOR4. SALE ORU1P2 DISTRIBUTION ORisDISTRIBUTION (This result of practical importance because reflectShow that . Therefore the in-NOT FOR SALE CsR2 kRw ing telescopes almost always have an obstruction in the tensity is given by central part of the aperture.) 13. Suppose that the annulus in Problem 12 is very narrow 2 J11kRw 2 small butLearning, not in¢a being U 1P2©2 c d I0 Jones & Bartlett Learning, LLC such that b a ¢a ©, with Jones & Bartlett LLC kRw finitesimal. Show then that the approximate resultant FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION amplitude is givenNOT by U1P2 C12pa¢a2J0 1kwa2 . 2 J1 1kRw2 [Hint: Interpret the result U(P) from Problem 12 as an 5. What is lim ? wS0 kRw d1u J1 1u2 2 approximation for u J0 1u2 with u kwa.] 6. What is the physical significance of I0? du in a ina Learning, eix cos LLC e da Jn 1x2, 2p 0 2p v © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION PROJECT FOR SECTION 15.4 Fraunhofer Diffraction by a Circular Aperture © Jones & Bartlett Learning, LLC. NOT FOR SALE OR DISTRIBUTION. xxxi