1. Appendix A : Fourier transform

2. Appendix : Linear systems

3. Appendix : Shift-invariant systems

4. Impulse-response function : h(t) Transfer function : H(n)
Impulse-response function (the response of the system to an impulse, or a point, at the input)
Transfer function (the response to spatial harmonic functions)

5. Appendix : Transfer function

6. 4. Fourier Optics

7. 2D Fourier transform Plane wave In a plane (e.g. z=0)
(harmonic function)
One-to-one correspondence because
with
“+”, forward
“-” , backward

8. Superposition of plane waves
In general (superposition integral of harmonic functions)
The transmitted wave is a superposition of plane wave
Superposition of plane waves

9. Spatial harmonic function and plane wave

10. Spatial spectral analysis

11. Spatial frequency and propagation angle

12. Amplitude modulation

13. B. Transfer Function of Free Space
Impulse response function h
Transfer function H

14. Transfer Function of Free Space
: evanescent wave
We may therefore regard 1/l as the cutoff spatial frequency (the spatial bandwidth) of the system

15. Fresnel approximation
Fresnel approximation for transfer function of free space
If a is the largest radial distance in the output plane, the largest angle qm  a/d, and
we may write

16. Input - Output Relation

17. Impulse-Response Function of Free Space
= Inverse Fourier transform of the transfer function
Paraboloidal wave
Free-Space Propagation as a Convolution

18. Huygens-Fresnel Principle and the impulse-response function
The Huygens-Fresnel principle states that each point on a wavefront generates a spherical wave. The envelope of these secondary waves constitutes a new wavefront.
Their superposition constitutes the wave in another plane.
The system’s impulse-response function for propagation between the planes z = 0 and z = d is
In the paraxial approximation, the spherical wave is approximated by the paraboloidal wave.
 Our derivation of the impulse response function is therefore consistent with the H.-F. principle.

19. Within the Fresnel approximation, there are two approaches to
In summary:
Within the Fresnel approximation, there are two approaches to
determining the complex amplitude g(x, y) in the output plane,
given the complex amplitude f(x, y) in the input plane:
Space-domain approach
in which the input wave is expanded in terms of paraboloidal elementary waves
Frequency-domain approach
in which the input wave is expanded as a sum of plane waves.

20. 4.2 Optical Fourier transform
A. Fourier Transform in the Far Field
If the propagation distance d is sufficiently long,
the only plane wave that contributes to the complex amplitude at a point (x, y) in the output plane
is the wave with direction making angles
Proof!

21. Proof : for Fraunhofer approximation
If f(x, y) is confined to a small area of radius b, and if the distance d is sufficiently large so that the Fresnel number is small,
Condition of Validity of Fraunhofer Approximation
when the Fresnel number

23. Three configurations Input placed Input placed Input placed
back focal plane
Input placed
against lens
Input placed
in front of lens
Input placed
behind lens
 Phase representation of a thin lens in paraxial approximation

24. (a) The input placed directly against the lens
Pupil function ; Ul
Ul’
From the Fresnel approximation when d = f ,
Fourier transform
Quadratic phase factor

25. If d = f (b) The input placed in front of the lens
Exact Fourier transform !

26. (c) The input placed behind the lens
Scaleable Fourier transform !
As d reduces, the scale of the transform is made smaller.

27. In summary, convex lens can perform Fourier transformation
The intensity at the back focal plane of the lens is therefore proportional to the squared absolute value of the Fourier transform of the complex amplitude of the wave at the input plane, regardless of the distance d.

28. Diffraction of Light
Regimes of Diffraction

29. A. Fraunhofer diffraction
Aperture function : b d

30. Radius : Note, for focusing Gaussian beam
with an infinitely large lens,
Radius :

31. B. Fresnel diffraction

32. Spatial filtering in 4-f system

33. Impulse-response function is
Transfer Function of the 4-f Spatial Filter With Mask Transmittance p(x, y) :
The transfer function has the same shape as the pupil function.
Impulse-response function is

35. 4.5 Holography If the reference wave is a uniform plane wave,
Original object wave!!

38. Off-axis holography ( qmin  qs/2 )  2qs  Spreading-angle width : qs
Ambiguity term
Assume that the object wave has a complex amplitude

39. Fourier-transform holography

40. Holographic spatial filters
IFT
Called “ Vander Lugt filter” or “Vander Lugt correlator”

41. Volume holography THICK Recording medium Transmission hologram :
Reflection hologram :

42. Volume holographic gratingkg
Grating vector kr
kg = k0 - kr k0
Grating period
L = 2p/ |kg|
Proof !!

43. Volume holographic grating = Bragg grating
Bragg condition :

44. “Holographic data storage prepares for the real world”
Laser Focus World – October 2003
200-Gbyte capacity in disk form factor
100 Mbyte/s data-transfer rate
“Holographic storage drives such as this prototype from Aprilis
are expected to become commercially available for write-once-read-many
(WORM) applications in 2005.”

45. C. Single-lens imaging system
Impulse response function
At the aperture plane :
Assume d1 = f
Beyond the lens :

46. Single-lens imaging system
Transfer function

47. Imaging property of a convex lens
From an input point S to the output point P ;
magnification
Fig. 1.22, Iizuka

48. Diffraction-limited imaging of a convex lens
From a finite-sized square aperture of dimension a x a
to near the output point P ;