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**The imaging problem object imaging optics (lenses, etc.) image**

MIT 2.71/2.710 Optics 10/27/04 wk8-b-1

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**The imaging problem Illumination (coherent vs incoherent) image object**

imaging optics (lenses, etc.) free space free space MIT 2.71/2.710 Optics 10/27/04 wk8-b-2

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**(spatial) linear shift-invariant system**

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

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**(spatial) linear shift-invariant system**

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

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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 MIT 2.71/2.710 Optics 10/27/04 wk8-b-5

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

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**Spatial frequency representation (aka spatial frequency domain)**

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

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**Spatial frequency removal (aka spatial frequency domain)**

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

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**From space to spatial frequency:**

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

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**(aka spatial frequency domain)**

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

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**(aka spatial frequency domain)**

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

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**removed high-and low-frequency content (aka spatial frequency domain)**

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

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**Example: optical lithography**

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

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**Example: optical lithography**

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

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**The 2D Fourier integral (aka inverse Fourier transform) superposition**

sinusoids complex weight, expresses relative amplitude (magnitude & phase) of superposed sinusoids MIT 2.71/2.710 Optics 10/27/04 wk8-b-15

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**The 2D Fourier integral The complex weight coefficients G(u,v),**

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

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**2D Fourier transform pairs**

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

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**Space and spatial frequency representations SPATIAL FREQUENCY DOMAIN**

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

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**Fourier transform properties /1**

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

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**Fourier transform properties /2**

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

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**Fourier transform properties /3**

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

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**Fourier transform properties /4**

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

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**2D linear shift invariant systems**

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

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**2D linear shift invariant systems**

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

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**2D linear shift invariant systems**

input output convolution with impulse response Fourier transform transform Inverse Fourier are pair multiplication with transfer function MIT 2.71/2.710 Optics 10/27/04 wk8-b-25

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**Sampling space and frequency**

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

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**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 MIT 2.71/2.710 Optics 10/27/04 wk8-b-27

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**(aka spatial frequency domain)**

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

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