1. Chapter 11 Fourier optics
April 15,17 Fourier transform
11.2 Fourier transforms
One-dimensional transforms
Complex exponential representation:
Fourier transform
Inverse Fourier transform
In time domain:
Note: Other alternative definitions exist.
Proof:
Notations:
The fancy “F” (script MT) is hard to type.

2. Example: Fourier transform of the Gaussian function
1) The Fourier transform of a Gaussian function is again a Gaussian function (self Fourier transform functions).
2) Standard deviations (where f (x) drops to e-1/2):

3. 11.2.2 Two-dimensional transformsx y a

4. J0(u)
J1(u) u

6. Read: Ch11: 1-2
Homework: Ch11: 2,3
Due: April 26

7. Sifting property of the Dirac delta function:
April 19 Dirac delta function
The Dirac delta function
Dirac delta function: A sharp distribution function that satisfies: 1 x
d (x)
A way to show delta function
Sifting property of the Dirac delta function:
If we shift the origin, then 1 x
d (x- x0) x0

8. Delta sequence: A sequence that approaches the delta function when the distribution is gradually narrowed.
The 2D delta function:
The Fourier representation of delta function:

9. Displacement and phase shift:
The Fourier transform of a function displaced in space is the transform of the undisplaced function multiplied by a linear phase factor.
Proof:

10. Fourier transform of some functions: (constants, delta functions, combs, sines and cosines):
f (x)
F (k)
f (x)
F(k) 1 2p … x k x k
f (x)
F (k)
f (x)
F(k) 1 1 … x k x k

11. A (k)
f (x) k x
f (x)
B (k) k x
f (x)
A (k) x k
f (x)
B (k) x k

12. Read: Ch11: 2
Homework: Ch11: 4,8,10,11,12,17
Due: April 26

13. Point-spread function
April 22, 24 Convolution theorem
11.3 Optical applications
Linear systems
Linear system: Suppose an object f (y, z) passing through an optical system results in an image g(Y, Z), the system is linear if
1) af (y, z)  ag(Y, Z),
2) af1(y, z) + bf2(y, z)  ag1(Y, Z) + bg2(Y, Z).
We now consider the case of 1) incoherent light (intensity addible), and 2) MT = +1.
The flux density arriving at the image point (Y, Z) from dydz is y
I0 (y,z) z Y
Ii (Y,Z) Z
Point-spread function

14. Example: y
I0 (y,z) z Y
Ii (Y,Z) Z
The point-spread function is the irradiance produced by the system with an input point source. In the diffraction-limited case with no aberration, the point-spread function is the Airy distribution function.
The image is the superposition of the point-spread function, weighted by the source radiant fluxes. y
I0 (y,z) z Y
Ii (Y,Z) Z

15. Space invariance:
Shifting the object will only cause the shift of the image:

16. 11.3.3 The convolution integral Convolution integral:
The convolution integral of two functions f (x) and h(x) is x
f (x)
h (x)
h (X-x) X
f (x)h (x)
Symbol: g(X) = f(x)h(x)
Example 1: The convolution of a triangular function and a narrow Gaussian function.
Question: What is f(x-a)h(x-b)?
Answer:

17. Example 2: The convolution of two square functions.
f (x)
h (x)
h (X-x) X
f (x)h (x)
The convolution theorem:
Proof:
Example: f (x) and h(x) are square functions.

18. Frequency convolution theorem:
Please prove it.
Example: Transform of a Gaussian wave packet.
Transfer functions:
Optical transfer function T (OTF)
Modulation transfer function M (MTF)
Phase transfer function F (PTF)

19. Read: Ch11: 3
Homework: Ch11: 18,24,27,28,29,34,35
Due: May 3

20. April 26,29 Fourier methods in diffraction theory
Fraunhofer diffraction:
dydz
P(Y,Z) r R x y z Y Z X
Aperture function: The field distribution over the aperture: A(y, z) = A0(y, z) exp[if (y, z)]
Each image point corresponds to a spatial frequency.
The field distribution of the Fraunhofer diffraction pattern is the Fourier transform of the aperture function:

21. The double slit (with finite width):
The single slit: z
A (z)
b/2
-b/2 kZ
E (kZ)
2p/b
Rectangular aperture:
Fraunhofer-: The light interferes destructively here.
Fourier-: The source has no spatial frequency here.
The double slit (with finite width):
f (z)
h (z)
g (z)  = z z z
-b/2
b/2
-a/2
a/2
-a/2
a/2
F (kZ)
H (kZ)
G (kZ) × = kZ kZ kZ

22. Apodization: Removing the secondary maximum of a diffraction pattern.
Three slits:
|F (kZ)|2
F (kZ)
f (z) z kZ kZ -a a
Apodization: Removing the secondary maximum of a diffraction pattern.
Rectangular aperture  sinc function  secondary maxima.
Circular aperture  Bessel function (Airy pattern)  secondary maxima.
Gaussian aperture  Gaussian function  no secondary maxima.
f (z) kZ
F (kZ) z

23. = The Fourier transform of an individual aperture
Array theorem:
The Fraunhofer diffraction pattern from an array of identical apertures
= The Fourier transform of an individual aperture
× The Fourier transform of a set of point sources arrayed in the same manner.
Convolution theorem z z =  y y y
Example: The double slit (with finite width).

24. Read: Ch11: 3
Homework: Ch11: 37,38,40
Due: May 3

25. May 1 Spectra and correlation
Considering a laser pulse described by E(t) = f (t).
The temporal radiant flux is The total energy is
Parseval ( , French mathematician)’s formula:
If F(w) =F{f(t)}, then
|F(w)|2 is the power spectrum.
Unitarity of Fourier transform

26. Application: Lorentzian profile:
f (t)
Application: Lorentzian profile: g w w0

27. Nature line width: The frequency bandwidth caused by the finite lifetime of the excited states.
Line broadening mechanisms:
Natural broadening: Governed by the uncertainty principle. Lorentzian profile.
Doppler broadening: The light emitted will be red or blue shifted depending on the velocity of the emitting atoms relative to the observer. Gaussian profile.
Pressure broadening: The collision with other atoms interrupts the emission process. Lorentzian profile.
Autocorrelation:
The autocorrelation of f (t) is
Wiener-Khintchine theorem:
Determining the spectrum by autocorrelation:
Symbol: cff (t) = f (t)f (t).
Prove: Let h(t) = f *(-t), then

28. Cross Correlation:
The cross correlation of f (t) and h(t) is
Symbol: cfh(t) = f(t)h(t).
For real functions,
Properties (please prove them):
1) f (t)h(t) = f *(-t)h(t). Can be treated as the definition of cross correlation.
2) If f is an even function, then f (t)h(t) = f (t)h(t).
3) Cross-correlation theorem: F{f(t)h(t)} = F{f (t)}* ·F{h(t)}
Applications of cross correlation:
Optical pattern recognition
Optical character recognition
Rotational fitting in laser spectroscopy

29. Read: Ch11: 3
Homework: Ch11: 39,49,50
Due: May 8

30. All the world’s a stage,
and all the men and women merely players.
William Shakespeare