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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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MULT. INTEGERS 1. IF THE SIGNS ARE THE SAME THE ANSWER IS POSITIVE 2. IF THE SIGNS ARE DIFFERENT THE ANSWER IS NEGATIVE.

MULT. INTEGERS 1. IF THE SIGNS ARE THE SAME THE ANSWER IS POSITIVE 2. IF THE SIGNS ARE DIFFERENT THE ANSWER IS NEGATIVE.

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