P. Piot, PHYS 630 – Fall 2008Modulation and Demodulation• Principle borrow from radio telecommunication– Take a wide band signal [few Hz to kHz]– Modulate it with a single frequency “the carrier signal”which is the frequency associated to a given station– Emit the signal– Given the station frequency, a receiver follows a demodulationprocedure that reconstructs the audio signal• Type of modulation– Amplitude Modulation -- analog to holography– Frequency Modulation -- analog to spectral holography– Phase Modulation– Digital methodsP. Piot, PHYS 630 – Fall 2008Amplitude Modulation I• Consider f(x) the “baseband” function• The modulated function is going to be f(x)cos(2πf x)carrier frequencyf(x) cos(2πf x)f(x)P. Piot, PHYS 630 – Fall 2008Amplitude Modulation II• Consider f(x) the “baseband” function• The modulated function is going to be f(x)cos(2πf x)carrier frequencyP. Piot, PHYS 630 – Fall 2008Demodulation• To demodulate:– Multiply modulatedsignal by cos(2πf x)– Use a low pass filterto cut high frequenciescarrier frequencyLF filterP. Piot, PHYS 630 – Fall 2008Optical analogy• Usually a wave is described by a complex amplitude, I.e. anamplitude and phase information• Usually a detector, e.g. a CCD camera, cannot detect the phase(time scale associated to optical phenomena is ~10 fs too fast!!).Only intensity can be measured• So amplitude information can be recovered but not phaseinformations• Can we record the phase information of a wave?– Yes: this is holography• Holography enables the reconstruction of a wave…P. Piot, PHYS 630 – Fall 2008Intensity recording• Recording the intensity of awave on a photographic emul-sion allows the reconstructionof the intensity ONLYP. Piot, PHYS 630 – Fall 2008Holography I• Mix the original wave (object wave) with a reference wave• Record the sum intensityreferenceobjectHologram: a record of the sum intensity of U0 and UrP. Piot, PHYS 630 – Fall 2008Holography II• To decode the information stored on the hologram: use the samereference wave and illuminate the hologram• The wave generated in the hologram plane is:• Need to separate the relevant information…hologramhologramobjectconjugated objectreference waveP. Piot, PHYS 630 – Fall 2008Example of a plane object wave• Specialize to the case of a uniform plane wave for the referencewave the “normalized” amplitude downstream of the hologram is• Take the special case of a plane object wavehologramobjectconjugateP. Piot, PHYS 630 – Fall 2008Example of a plane spherical wave• Still consider a uniform plane wave for the reference wave• Take the special case of a spherical object wavewhereconjugateobjectP. Piot, PHYS 630 – Fall 2008Off-axis Holography• One mean of separating the three term in the “reconstructed” wave is to usean off-axis reference waveThe reconstructed wave isWe must haveambiguityconjugateobjectmaximum frequencyassociated to f(x,y)P. Piot, PHYS 630 – Fall 2008Fourier Holography• One can compute optically the Fourier transform of an object andencode it on an hologram• Then do an inverse Fourier transform of the reconstructed wave• Can use this technique to do spatial filtering (Vander Lugt holography)conjugateobjectf(x,y)f(x,y)referencereferenceP. Piot, PHYS 630 – Fall 2008Practical implementation• Hologram generation• Note: One can also create“virtual” objects by digitallygenerating the hologram• Object restoration(Images from Wikipedia)P. Piot, PHYS 630 – Fall 2008Analogy between AM &
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