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

2. 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

3. (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

4. (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

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

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

7. 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

8. 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

9. 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. ...

10. (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

11. (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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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

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

28. (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