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General Fresnel Vs. Fraunhofer Diffraction Several Coherent Phys 322 Chapter 10 Lecture 28 Diffraction Diffraction: general Fresnel vs. Fraunhofer diffraction Several coherent oscillators Single-slit diffraction Diffraction Grimaldi, 1600s: diffractio, deviation of light from linear propagation Diffraction is a consequence of multiple beam interference Diffraction Shadow of a Light does not hand always travel in a illuminated straight line. by a Helium- Neon laser It tends to bend around objects. This tendency is called diffraction. Any wave will do Shadow of this, including a zinc oxide matter waves and crystal acoustic waves. illuminated by electrons Diffraction of ocean water waves Ocean waves passing through slits in Tel Aviv, Israel Diffraction occurs for all waves, whatever the phenomenon. Radio waves diffract around mountains. When the wavelength is km long, a mountain peak is a very sharp edge! Another effect that occurs is scattering, so diffraction’s role is not obvious. Diffraction of a wave by a slit slit size Whether waves in water or electromagnetic radiation in air, passage through a slit yields a diffraction pattern that will appear more dramatic as the size of the slit approaches the wavelength of the wave. slit size slit size Why it’s hard to see diffraction Diffraction tends to cause ripples at edges. But poor source temporal or spatial coherence masks them. Example: a large spatially incoherent source (like the sun) casts blurry shadows, masking the diffraction ripples. Screen with hole Untilted rays yield a perfect shadow of the hole, but off-axis rays blur the shadow. A point source is required. Diffraction by an Edge x Transmission Even without a small slit, Light diffraction passing can be by edge strong. Simple propagation past an edge yields an unintuitive Electrons irradiance passing by an pattern. edge (MgO crystal) Huygens-Fresnel principle Every point of a wavefront serves as a source of a secondary spherical wave that has the same phase as the original wave at this point. Christiaan Huygens Augustin Fresnel The amplitude of the optical field at 1629 - 1695 1788-1827 any point beyond is the superposition of all secondary waves. Single slit diffraction? Expectation: Reality: No diffraction With diffraction Single slit and Huygens’s principle Waves bend around the edges Shadow of razor blade Single slit: diffraction minimum a 2 a a a sin when sin rays 1 and 1 2 22interfere destructively. Rays 2 and 2 also start a/2 apart and have the same path length difference. Under this condition, every ray originating in top half of the slit interferes destructively with the corresponding ray originating in the bottom half. 1st minimum at sin = /a Single slit: diffraction minimum a 4 a a a sin( ) When sin( ) rays 1 and 1 4 42 will interfere destructively. Rays 2 and 2 also start a/4 apart and have the same path length difference. Under this condition, every ray originating in top quarter of slit interferes destructively with the corresponding ray originating in second quarter. 2nd minimum at sin = 2/a Single slit: diffraction minimum Single slit diffraction minima: a sin = m (m=±1,±2,...) or sin = m/a a sin Narrower slits - broader maxima Note: minima only occur when a > Fresnel and Fraunhofer diffraction Fresnel diffraction: near-field Fraunhofer: far-field a R >> a2/ R - smallest of the distances from the aperture to the source or to the screen (plane waves) Fraunhofer vs. Fresnel Diffraction Close Far to the z from slit the slit Slit S P Incident wave Fresnel: near-field Fraunhofer: far-field Fraunhofer diffraction Fraunhofer (far-field) diffraction: Both the incoming and outgoing waves approach being planar. R >> a2/. R is the smaller of the two distances from the source to the aperture (with size a) and from the aperture to the observation point. Mathematical criteria: Path difference is a linear function of the aperture variables. P P S S a R2 a R1 asin Fraunhofer diffraction: single slit Fraunhofer diffraction: several coherent oscillators N rays meet far away and interfere Amplitudes are all about the same E0 ikr1t ikr2 t ikr3 t E E0 re E0 re E0 re ... ikr1t ik r2 r1 ik r3 r1 E E0 r e 1 e e ... kr2 r1 kd sin kr3 r1 2kd sin 2 2 N ikr1t i i i E E0 r e 1 e e ... e geometric series iN iN / 2 iN / 2 iN / 2 e 1 e e e i N 1 / 2 sinN / 2 e ei 1 ei / 2 ei / 2 ei / 2 sin / 2 Fraunhofer diffraction: several coherent oscillators r ikr t i N 1 / 2 sinN / 2 1 E E r e 1 e 0 sin / 2 distance to the point P from center of the slit: 1 R N 1 d sin r 2 1 kd sin sinN / 2 E E r eikRt 0 sin / 2 sin2 N / 2 Intensity: I I 0 sin2 / 2 sin2Nkd / 2 sin rapid I I 0 sin2kd / 2 sin slow Fraunhofer diffraction: several coherent oscillators r1 sin2Nkd / 2sin I I 0 sin2kd / 2 sin maxima: (/2)sinkdm m d sinm m 2 I I0N Need to know I0 due to a single source and I in a limit N Fraunhofer diffraction: line source Single oscillating source: source strength E E 0 sint kr r Source strength per unit length EL E y L i Ei sint kri ri N E y L i E sint kri i0 ri D / 2 D / 2 E sint kr E L sin t kr dy E dy L D / 2 r D / 2 r Single slit: Fraunhofer diffraction Intensity due to element dy: +b/2 r P E L dE sint kr dy y R r r R ysin -b/2 E b/ 2 E L sint kR ysin dy R b/ 2 kb / 2sin E b sin E L sint kR R 2 sin 2 I I 0 I0 sinc R >> b2/ Single slit: Fraunhofer diffraction I / I0 sinc2 Minima: =m, where m=±1, ±2… kb / 2sin kb / 2 sin m k 2 / bsin m Maxima: see graph .
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