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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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Presentation on theme: "MIT 2.71/2.710 Optics 10/27/04 wk8-b-1 The imaging problem object imaging optics (lenses, etc.) image."— Presentation transcript:

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

2 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

3 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

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

5 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

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

7 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)

8 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)

9 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....

10 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)

11 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

12 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)

13 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

14 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

15 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

16 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... )

17 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)

18 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

19 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

20 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

21 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

22 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)

23 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

24 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

25 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

26 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

27 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

28 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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