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MIT 2.71/2.710 Optics12/05/05 wk14-a-1Holography• Preamble: modulation and demodulation• The principle of wavefront reconstruction• The Leith-Upatnieks hologram• The Gabor hologram• Image locations and magnification• Holography of three-dimensional scenes• Transmission and reflection holograms• Rainbow hologramMIT 2.71/2.710 Optics12/05/05 wk14-a-2Modulation & Demodulation• Principle borrowed from radio telecommunications• Idea is to take baseband signal (e.g. speech, music, with maximum frequencies up to ~20kHz) and modulate it onto a carrier signal which is a simple tone at the frequency where the radio station emits, e.g. 104.3 MHz (that’s Boston’s WBCN station)• One of the benefits of modulation is that radio stations can be multiplexed by using a different emission frequency each• After selecting the desired station, the receiver follows a process of demodulation which recovers the basebandsignal and sends it to the speakers.MIT 2.71/2.710 Optics12/05/05 wk14-a-3Types of modulation• Amplitude modulation (AM)• Frequency modulation (FM)• Phase modulation (PM)• Digital methods (Amplitude Shift Keying – ASK, Frequency Shift Keying – FSK, Phase Shift Keying – PSK, etc.)used in radio at low frequencies only (“AM band” = 535kHz to 1.7MHz) ; as we will see, it is an almost-exact analog of holographydominant in commercial radio (“FM band” = 88MHz to 108MHz) ; there is an analog in optics, called “spectral holography,” but it is beyond the scope of the classMIT 2.71/2.710 Optics12/05/05 wk14-a-4Amplitude modulation()xf(baseband)()() ( )xuxfxfc2cos modulatedπ×==uc: carrier frequencyMIT 2.71/2.710 Optics12/05/05 wk14-a-5AM in the frequency domain()xf of spectrum()xf modulatedof spectrumMIT 2.71/2.710 Optics12/05/05 wk14-a-6AM in the frequency domain()xf of spectrum()xf modulatedof spectrum (zoom-in)(zoom-in)MIT 2.71/2.710 Optics12/05/05 wk14-a-7AM in the frequency domain() ()() ( ) ()[]() ()()[]()()[]cccc22c2121ee212cosccuuFuuFuuuuuFxfxuxfuFxfxuixui−+−==−+−∗→+×=→−δδπππmodulation in the space domainmodulation in the frequency domain:two replicas of the basebandspectrum, centered on the carrier frequencyMIT 2.71/2.710 Optics12/05/05 wk14-a-8Modulationsimple carrier tonemultiplication×()xf modulated()xuc2cosπ()xfMIT 2.71/2.710 Optics12/05/05 wk14-a-9Demodulationsimple carrier tonemultiplication×low-passfilter()xf modulated()xuc2cosπ()xfmust accommodatebaseband spectrumMIT 2.71/2.710 Optics12/05/05 wk14-a-10Demodulation()()xuxfc22cosπ×()()xuxfc22cos of spectrumπ×MIT 2.71/2.710 Optics12/05/05 wk14-a-11Demodulation()()xuxfc22cosπ×()()xuxfc22cos of spectrumπ×LP filter pass-bandMIT 2.71/2.710 Optics12/05/05 wk14-a-12The wavefront reconstruction problem• Wavefront is the amplitude (i.e. magnitude and phase) of the electric field as function of position• Traditional coherent imaging results in intensity images (because detectors do not respond fast enough at optical frequencies) → magnitude information is recovered but phase information is lost• Can we imprint intensity information on an optical wave? YES → photography (known since the 1840’s)• Can we imprint wavefront information on an optical wave? YES → holography (Gabor, late 1940s)MIT 2.71/2.710 Optics12/05/05 wk14-a-13Photography: recordingincident illumination(laser beam or white light)imaging systemfilm records intensity informationS2SMIT 2.71/2.710 Optics12/05/05 wk14-a-14Photography: reconstructing the intensityincident illumination(laser beam or white light)imaging systemat the image plane, an intensity pattern is formedthat replicates the originally recorded intensity2S2SMIT 2.71/2.710 Optics12/05/05 wk14-a-15Holography: recordingincident illumination(laser beam(laser beam♣♣))imaging systemfilm records the interference patternthe interference pattern(interferogram) of the object wavefrontand the reference wavefrontS2SR +reference beam(split from the same laser)R♣in general, the illumination must be quasi-monochromatic, and spatially mutually coherentwith the reference beam throughout the wavefrontMIT 2.71/2.710 Optics12/05/05 wk14-a-16Holography: reconstructing the wavefrontillumination:replicates the reference beamimaging systemR?what is the field at the image plane?2SR +MIT 2.71/2.710 Optics12/05/05 wk14-a-17Holography: reconstructing the wavefrontR?2SR +()=+++×=+×**222RSSRSRRSRR()*2222SRSRSRR +++×=The field being imaged is:nopMIT 2.71/2.710 Optics12/05/05 wk14-a-18Holography: reconstructing the wavefrontxuiR02eπ=()()*222200e 1e SSSxuixuiππ+++×=take the simplest possible reference wave, a plane wave:then the reconstructed field is:nopR?2SR +, spatial frequencyλθ00sin=uMIT 2.71/2.710 Optics12/05/05 wk14-a-19Holography: reconstructing the wavefrontfields departing from thehologramR2SR +nop()*220e SxuiπS()221e0Sxui+×πpropagatesat angle:0θ02θon-axisMIT 2.71/2.710 Optics12/05/05 wk14-a-20Holography: reconstructing the wavefrontfields departing from thehologramR2SR +nop()*220e SxuiπS()221e0Sxui+×π0θ02θon-axiswantedwantednot wantednot wantedMIT 2.71/2.710 Optics12/05/05 wk14-a-21Filtering the wavefront: bandlimited signal{}wSwS radius of circle within 0i.e. ,bandwidth has ≠ℑuvwMIT 2.71/2.710 Optics12/05/05 wk14-a-22Filtering the wavefront: bandlimited signal{}{}{}{} {}SSSSSwSℑ⊗ℑ=ℑ∗ℑ=ℑ+*22 because ,2bandwidth has 1 uvTerm n2wMIT 2.71/2.710 Optics12/05/05 wk14-a-23Filtering the wavefront: Fourier transform descriptionuvn2ww wu02u0opMIT 2.71/2.710 Optics12/05/05 wk14-a-24Filtering the wavefront: Fourier transform descriptionuvn2ww wu02u0oporiginalspectrumautocorrelation of theoriginal spectrumoriginal spectrum but phase--conjugated: inside-out, or “pseudo-scopic”MIT 2.71/2.710 Optics12/05/05 wk14-a-25Filtering the wavefront: Fourier transform descriptionuvn2ww wu02u0opwantedwantednotnotwantedwantedMIT 2.71/2.710 Optics12/05/05 wk14-a-26Filtering the wavefront: Fourier transform descriptionuwa low-pass filter ofpassband w or slightlygreater permits thedesired term o to pass, and eliminatesthe undesirable termsn and p.MIT 2.71/2.710 Optics12/05/05 wk14-a-27Holography: reconstructing the wavefrontillumination:replicates the reference beam4F system with Fourier plane filterRSthe field at the image planereplicates the original S stored in the hologram2SR +hologram:MIT 2.71/2.710 Optics12/05/05 wk14-a-28w2wFiltering the wavefront: Fourier transform descriptionuvnwu02u0opPotential


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