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MIT 2.71/2.710 Optics 10/25/04 wk8-a-1 The spatial frequency domain

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MIT 2.71/2.710 Optics 10/25/04 wk8-a-2 Recall: plane wave propagation path delay increases linearly with x plane of observation

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MIT 2.71/2.710 Optics 10/25/04 wk8-a-3 Spatial frequency angle of propagation? plane of observation electric field

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MIT 2.71/2.710 Optics 10/25/04 wk8-a-4 Spatial frequency angle of propagation? plane of observation

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MIT 2.71/2.710 Optics 10/25/04 wk8-a-5 The cross-section of the optical field with the optical axis is a sinusoid of the form Spatial frequency angle of propagation? i.e.it looks like where spatial frequency is called the spatial frequency

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MIT 2.71/2.710 Optics 10/25/04 wk8-a-6 2D sinusoids

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MIT 2.71/2.710 Optics 10/25/04 wk8-a-7 2D sinusoids min max

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MIT 2.71/2.710 Optics 10/25/04 wk8-a-8 2D sinusoids min max

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MIT 2.71/2.710 Optics 10/25/04 wk8-a-9 2D sinusoids min max

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MIT 2.71/2.710 Optics 10/25/04 wk8-a-10 Periodic Grating /1: vertical Space domain Frequency (Fourier) domain

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Periodic Grating /2: tilted Space domain Frequency (Fourier) domain MIT 2.71/2.710 Optics 10/25/04 wk8-a-11

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Superposition: multiple gratings Space domain Frequency (Fourier) domain MIT 2.71/2.710 Optics 10/25/04 wk8-a-12

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MIT 2.71/2.710 Optics 10/25/04 wk8-a-13 More superpositions Space domain Frequency (Fourier) domain

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MIT 2.71/2.710 Optics 10/25/04 wk8-a-14 Size of object vsfrequency content Space domain Frequency (Fourier) domain

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MIT 2.71/2.710 Optics 10/25/04 wk8-a-15 Superimposing shifted objects Space domain Frequency (Fourier) domain

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MIT 2.71/2.710 Optics 10/25/04 wk8-a-16 Linear shift invariant systems in the time domain

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MIT 2.71/2.710 Optics 10/25/04 wk8-a-17 Linear shift invariant systems

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MIT 2.71/2.710 Optics 10/25/04 wk8-a-18 Linear shift invariant systems impulse excitation shift invariance

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MIT 2.71/2.710 Optics 10/25/04 wk8-a-19 Linear shift invariant systems impulse excitation linearity

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MIT 2.71/2.710 Optics 10/25/04 wk8-a-20 Linear shift invariant systems arbitrary sifting property of the δ-functionconvolution integral

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MIT 2.71/2.710 Optics 10/25/04 wk8-a-21 Linear shift invariant systems sinusoid attenuation phase delay transfer function same frequency

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MIT 2.71/2.710 Optics 10/25/04 wk8-a-22 From time to frequency: Fourier analysis Can I express an arbitrary g(t) as a superposition of sinusoids?

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MIT 2.71/2.710 Optics 10/25/04 wk8-a-23 The Fourier integral inverse Fourier transform (aka inverse Fourier transform) superposition sinusoids complex weight, expresses relative amplitude (magnitude & phase) of superposed sinusoids

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MIT 2.71/2.710 Optics 10/25/04 wk8-a-24 The Fourier transform The complex weight coefficients G(ν), akaFourier transform Fourier transform of g(t) are calculated from the integral

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MIT 2.71/2.710 Optics 10/25/04 wk8-a-25 Fourier transform pairs

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MIT 2.71/2.710 Optics 10/25/04 wk8-a-26 2Dpairs 2D Fourier transform pairs Image removed due to copyright concerns (from Goodman, Introduction to Fourier Optics, page 14)

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Overview from last week Optical systems act as linear shift-invariant (LSI) filters (we have not yet seen why) Analysis tool for LSI filters: Fourier transform.

Overview from last week Optical systems act as linear shift-invariant (LSI) filters (we have not yet seen why) Analysis tool for LSI filters: Fourier transform.

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