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MIT 2.71/2.710 Optics 10/27/04 wk8-b-1 The imaging problem object imaging optics (lenses, etc.) image

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MIT 2.71/2.710 Optics 10/27/04 wk8-b-2 The imaging problem Illumination (coherent vs incoherent) object image imaging optics (lenses, etc.) free space

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MIT 2.71/2.710 Optics 10/27/04 wk8-b-3 The imaging problem Illumination (coherent vs incoherent) object image (spatial) linear shift-invariant system

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MIT 2.71/2.710 Optics 10/27/04 wk8-b-4 The imaging problem object image (spatial) linear shift-invariant system

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MIT 2.71/2.710 Optics 10/27/04 wk8-b-5 Our approach Today: – linear shift invariant (LSI) systems in the space/spatial frequency domains – mathematical properties of Fourier transforms Monday: – free space propagation: Fresnel and Fraunhofer diffraction Wednesday: – examples of Fraunhofer diffraction: amplitude and phase diffraction gratings – wave description of light propagation through a lens – Fourier transformation and imaging using lenses

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MIT 2.71/2.710 Optics 10/27/04 wk8-b-6 Spatial filtering

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MIT 2.71/2.710 Optics 10/27/04 wk8-b-7 Spatial frequency representation space domain 3 sinusoids Fourier domain (aka spatial frequency domain)

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MIT 2.71/2.710 Optics 10/27/04 wk8-b-8 Spatial frequency removal Fourier domain (aka spatial frequency domain) space domain 2 sinusoids (1 removed)

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MIT 2.71/2.710 Optics 10/27/04 wk8-b-9 From space to spatial frequency: 2D 2D Fourier analysis Can I express an arbitrary g(x,y) as a superposition of sinusoids?... etc....

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MIT 2.71/2.710 Optics 10/27/04 wk8-b-10 Spatial frequency representation space domain g(x,y) Fourier domain (aka spatial frequency domain)

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MIT 2.71/2.710 Optics 10/27/04 wk8-b-11 Low-pass filtering space domain Fourier domain (aka spatial frequency domain) removed high-frequency content

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MIT 2.71/2.710 Optics 10/27/04 wk8-b-12 Band-pass filtering removed high-and low-frequency content space domain Fourier domain (aka spatial frequency domain)

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MIT 2.71/2.710 Optics 10/27/04 wk8-b-13 Example: optical lithography Original nested Ls original pattern (nested Ls) mild low-pass filtering Notice: (i) blurring at the edges (ii) ringing

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MIT 2.71/2.710 Optics 10/27/04 wk8-b-14 Example: optical lithography Original nested Ls original pattern (nested Ls) severe low-pass filtering Notice: (i) blurring at the edges (ii) ringing

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MIT 2.71/2.710 Optics 10/27/04 wk8-b-15 2D The 2D Fourier integral (aka inverse Fourier transform) superpositionsinusoids complex weight, expresses relative amplitude (magnitude & phase) of superposed sinusoids

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MIT 2.71/2.710 Optics 10/27/04 wk8-b-16 2D The 2D Fourier integral The complex weight coefficients G(u,v), Fourier transform Aka Fourier transform of g(x,y) are calculated from the integral (1D so we can draw it easily... )

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MIT 2.71/2.710 Optics 10/27/04 wk8-b-17 2D 2D Fourier transform pairs Image removed due to copyright concerns (from Goodman, Introduction to Fourier Optics, page 14)

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MIT 2.71/2.710 Optics 10/27/04 wk8-b-18 Space and spatial frequency representations SPACE DOMAIN 2D 2D Fourier transform 2D 2D Fourier integral aka 2D inverse 2D Fourier transform SPATIAL FREQUENCY DOMAIN

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MIT 2.71/2.710 Optics 10/27/04 wk8-b-19 Fourier transform properties /1 Fourier transforms and the delta function Linearity of Fourier transforms if and then for any pair of complex numbers

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MIT 2.71/2.710 Optics 10/27/04 wk8-b-20 Fourier transform properties /2 Let Shift theorem (space frequency) Shift theorem (frequency space) Scaling theorem

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MIT 2.71/2.710 Optics 10/27/04 wk8-b-21 Fourier transform properties /3 Let and Convolution theorem (space frequency) Convolution theorem (frequency space) Let

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MIT 2.71/2.710 Optics 10/27/04 wk8-b-22 Fourier transform properties /4 Let and Let Correlation theorem (space frequency) Correlation theorem (frequency space)

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MIT 2.71/2.710 Optics 10/27/04 wk8-b-23 2D 2D linear shift invariant systems input output impulse response convolution with impulse response transfer function multiplication with transfer function Fourier transform Inverse Fourier transform

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MIT 2.71/2.710 Optics 10/27/04 wk8-b-24 2D 2D linear shift invariant systems SPACE DOMAIN SPATIAL FREQUENCY DOMAIN input output Fourier transform Inverse Fourier transform impulse response convolution with impulse response transfer function multiplication with transfer function

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MIT 2.71/2.710 Optics 10/27/04 wk8-b-25 2D 2D linear shift invariant systems input output Fourier transform Inverse Fourier transform impulse response convolution with impulse response transfer function multiplication with transfer function arepair

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MIT 2.71/2.710 Optics 10/27/04 wk8-b-26 Sampling space and frequency pixel size field size space domain spatial frequency domain Nyquist relationships: frequency resolution

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MIT 2.71/2.710 Optics 10/27/04 wk8-b-27 The Space–Bandwidth Product Nyquist relationships: from space spatial frequency domain: from spatial frequency space domain: : 1D Space–Bandwidth Product (SBP) aka number of pixels in the space domain

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MIT 2.71/2.710 Optics 10/27/04 wk8-b-28 SBP: example space domain Fourier domain (aka spatial frequency domain)

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