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Fresnel and Fraunhofer Diffraction

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Fresnel and Fraunhofer Diffraction Diffraction • Interference with more than 2 beams – 3, 4, 5 beams – Large number of beams • Diffraction gratings – Equation –Uses • Diffraction by an aperture – Huygen’s principle again, Fresnel zones, Arago’s spot – Qualitative effects, changes with propagation distance – Fresnel number again – Imaging with an optical system, near and far field – Fraunhofer diffraction of slits and circular apertures – Resolution of optical systems • Diffraction of a laser beam LASERS 51 April 03 Interference from multiple apertures L Bright fringes when OPD=nλ 40 x nLλ d x = d Intensity source OPD two slits position on screen screen Complete destructive interference halfway between OPD 1 OPD 1=nλ, OPD 2=2nλ OPD 2 all three waves40 interfere constructively d Intensity source three position on screen equally spaced slits screen OPD 2=nλ, n odd outer slits constructively interfere middle slit gives secondary maxima LASERS 51 April 03 Diffraction from multiple apertures • Fringes not sinusoidal for more than two slits 2 slits • Main peak gets narrower – Center location obeys same 3 slits equation • Secondary maxima appear 4 slits between main peaks – The more slits, the more 5 slits secondary maxima – The more slits, the weaker the secondary maxima become • Diffraction grating – many slits, very narrow spacing – Main peaks become narrow and widely spaced – Secondary peaks are too small to observe LASERS 51 April 03 Reflection and transmission gratings • Transmission grating – many closely spaced slits • Reflection grating – many closely spaced reflecting regions Input screen wave path length to Input observation point opaque Huygens wave transmitting wavelets opening wavelets path length to observation point screen absorbing reflecting Transmission grating Reflection grating LASERS 51 April 03 Grating equation – transmission grating with normal incidence Diffracted light Θ pλ input d sinθ = d l Except for not making a • Θd is angle of diffracted ray small angle approximation, • λ is wavelength this is identical to formula • l is spacing between slits for location of maxima in multiple slit problem earlier • p is order of diffraction LASERS 51 April 03 Diffraction gratings – general incidence angle pλ • Grating equation sinθ − sinθ = d i l l=distance between grooves (grating spacing) Θi Θ =incidence angle (measured from normal) i Θd Θd=diffraction angle (measured from normal) p=integer (order of diffraction) • Same formula whether it’s a transmission or reflection grating – n=0 gives straight line propagation (for transmission grating) or law of reflection (for reflection grating) LASERS 51 April 03 Intensities of orders – allowed orders • Diffraction angle can be found only for certain values of p strong diffracted weak diffracted order order – If sin(Θd) is not input between –1 and 1, beam there is no allowed Θd • Intensity of other orders are different depending on wavelength, incidence angle, and construction of grating • Grating may be blazed to make Blazed grating a particular order more intense than others – angles of orders unaffected by blazing LASERS 51 April 03 Grating constant (groove density) vs. distance between grooves • Usually the spacing between grooves for a grating is not given – Density of grooves (lines/mm) is given instead 1 – g = l – Grating equation can be written in terms of grating constant sin(Θd ) − sin(Θi ) = pgλ LASERS 51 April 03 2nd Diffraction grating - applications order 1st order • Spectroscopy grating – Separate colors, similar to negative prism orders • Laser tuning Littrow mounting – input – narrow band mirror and output angles identical – Select a single line of λ Θ 2sin()Θ = multiline laser d – Select frequency in a grating tunable laser • Pulse stretching and compression – Different colors travel different path lengths two identical LASERS 51 gratings April 03 Fabry-Perot Interferometer Input transmitted through first mirror Beam is partially reflected and partially transmitted at each mirror Transmitted All transmitted beams interfere with each other field Reflected All reflected beams interfere with field Partially each other reflecting OPD depends on mirror mirrors separation • Multiple beam interference – division of amplitude – As in the diffraction grating, the lines become narrow as more beams interfere LASERS 51 April 03 Fabry-Perot Interferometer 1 free spectral range, Linewidth= fsr fsr*finesse transmission 0 frequency or wavelength • Transmission changes with frequency – Can be very narrow range where transmission is high • Width characterized by finesse • Finesse is larger for higher reflectivity mirrors – Transmission peaks are evenly spaced • Spacing called “Free spectral range” • Controlled by distance between mirrors, fsr=c/(2L) • Applications – Measurement of laser linewidth or other spectra – Narrowing laser line LASERS 51 April 03 Diffraction at an aperture—observations Light Aperture through aperture on screen downstream • A careful observation of the light transmitted by an aperture reveals a fringe structure not predicted by geometrical optics • Light is observed in what should be the shadow region LASERS 51 April 03 Pattern on screen at various distances Near Field Intermediate field 2.5mm 2500 mm Immediately 25 mm from screen, 250 mm pattern doesn’t behind screen bright fringes just light penetrates inside edges into shadow closely resemble region mase Far field – at a large enough distance shape of pattern no longer changes but it gets bigger with larger distance. Symmetry of original mask still is evident. LASERS 51 April 03 Huygens-Fresnel diffraction screen observing with screen aperture Point Wavelets source generated in hole • Each wavelet illuminates the observing screen • The amplitudes produced by the various waves at the observing screen can add with different phases • Final result obtained by taking square of all amplitudes added up – Zero in shadow area – Non-zero in illuminated area LASERS 51 April 03 Fresnel zones • Incident wave propagating to right • What is the field at an observation point a b + λ/2 observation distance of b away? First Fresnel point • Start by drawing a sphere with radius zone b+λ/2 • Region of wave cut out by this sphere is b the first Fresnel zone • All the Huygens wavelets in this first incident Fresnel zone arrive at the observation wavefront point approximately in phase • Call field amplitude at observation point due to wavelets in first Fresnel zone, A1 LASERS 51 April 03 Fresnel’s zones – continued • Divide incident wave into additional Fresnel zones by drawing circles with radii, observation b +λ b+2λ/2, b+3λ/2, etc. b +λ/2 point • Wavelets from any one zone are approximately in phase b at observation point – out of phase with wavelets from a neighboring zone incident • Each zone has nearly same area wavefront • Field at observation point due to second Fresnel zone is A2, etc. • All zones must add up to the uniform field that we must have at the observation point LASERS 51 April 03 Adding up contributions from Fresnel zones •A1, the amplitude due to the first zone and A2, the amplitude from the second zone, are out of phase (destructive interference) –A2 is slightly smaller than A1 due to area and distance • The total amplitude if found by adding contributions of all Fresnel zones … A=A1-A2+A3-A4+ minus signs because the amplitudes are out of phase amplitudes slowly decrease So far this is a complex way of showing an obvious fact. LASERS 51 April 03 Diffraction from circular apertures • What happens if an aperture the diameter of the first Fresnel zone is inserted in the beam? • Amplitude is twice as high as before inserting aperture!! – Intensity four times as large observation b +λ • This only applies to b +λ/2 point intensity on axis b incident wavefront Blocking two Fresnel zones gives almost zero intensity on axis!! LASERS 51 April 03 Fresnel diffraction by a circular aperture • Suppose aperture size and observation distance chosen so that aperture allows just light from first Fresnel zone to pass – Only the term A1 will contribute – Amplitude will be twice as large as case with no aperture! • If distance or aperture size changed so two Fresnel zones are passed, then there is a dark central spot – alternate dark and light spots along axis – circular fringes off the axis LASERS 51 April 03 Fresnel diffraction by circular obstacle— Arago’s spot • Construct Fresnel zones just as before except start with first zone beginning at edge of aperture • Carrying out the same reasoning b observation as before, we find that the point intensity on axis (in the geometrical shadow) is just what b+λ/2 it would be in the absence of the obstacle • Predicted by Poisson from incident Fresnel’s work, observed by wavefront Arago (1818) LASERS 51 April 03 Character of diffraction for different locations of observation screen • Close to diffracting screen (near field) – Intensity pattern closely resembles shape of aperture, just like you would expect from geometrical optics – Close examination of edges reveals some fringes • Farther from screen (intermediate) – Fringes more pronounced, extend into center of bright region – General shape of bright region still roughly resembles geometrical shadow, but edges very fuzzy • Large distance from diffracting screen (far field) – Fringe pattern gets larger – bears little resemblance to shape of aperture (except symmetries) – Small features in hole lead to larger features in diffraction pattern – Shape of pattern doesn’t change with further increase in distance, LASERSbut 51 it continues to get larger April
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