1. 1. FT of delta function, :
Go backwards with inverse:
2. FT of complex exponential:
, which is a delta function at positive wo. So
is represented by a positive frequency in the FT, because in optics, we like to use phase functions of
to represent a wave moving in the positive direction when k is positive. So
is “normal” and
appears at negative frequency. This is enforced in the text by the choice of the FT to be
rather than:
, the most common choice for FTs.

2. Reading Quiz
Which is true for functions f1 and f2 and their FT (ignoring constant factors)?
Convolution(FT1,FT2) = Convolution(f1,f2)
FT(f1f2)=Convolution(FT1,FT2)
FT(Convolution(f1+f2))=FT1 + FT2

3. Reading Quiz
I read the assignment for at least 20 min.
yes no

4. Write and keep on the board
Animations and compare cross correlation
Write and keep on the board

5. Announcements
Please return polaroid slides

6. Reading Quiz
The Fourier transform of a convolution of functions is a _____ of functions
a. sum
b. product
c. convolution

7. Reading Quiz Q3. The width of a convolution of two functions is always
a. equal to the width of the narrower function
b. equal to the width of the broader function
c. equal to or less than the width of the narrower function
d. equal to or greater than the width of the broader function

8. Fourier theory

9. Carrier frequency-envelope principle
Optical pulses are often a steady (“carrier”) wave at multiplied by an envelope function
11 waves
101 waves
The FT f(w) is the FT of _____ centered at _____.
The width Dw is the width of _____

10. Which pulse f(t) will have f(w) centered around the highest frequency?
a) b) c)
Which f(t) will have the greatest width Dw in f(w) around its central frequency?

11. Fourier theory and delta functions

12. Fourier theory and delta functions
Or, for any two variables u,v:
Explain conceptually!

13. Fourier theory and delta functions
So appears at a negative frequency in the FT (because of our convention of as a “typical” wave at frequency w).

14. Convolutions
A convolution of two functions is the area under the product of the two functions, which changes as the one of the functions is shifted.
more correct:
“…as one of the functions (flipped about the origin) is shifted.” (But we can ignore flipping with symmetric functions)

15. Sketch the convolution of the two functions
(vertically displaced for display).
I got it mostly right
I tried, but got it mostly wrong

16. Sketch the convolution of the two functions
(vertically displaced for display).
I got it mostly right
I tried, but got it mostly wrong

17. Convolution Theorems Take transform inverses of above equations!
products in one “space” become convolutions in the other, and vice versa.
belongs on the product side

18. Prove carrier-frequency envelope principle
If a pulse is a steady (“carrier”) wave multiplied by an envelope function g(t)
…the FT f(w) is the FT of the envelope, g(w), but centered at ± wo. the width Dw is the width of g(w).

19. Sketch
You should be able to approach this in a couple of ways

20. A gaussian pulse is g(t) with FT given by g(w)
Find the complex FT of
First write the cos as a sum of exponentials
I got it mostly right
I tried, but got it mostly wrong

21. The FT of a single square pulse g(t) centered at t = 0 is
Using convolutions, find the FT of the double square pulse, each of width t, centered at –t1, and t1
I got it mostly right
I tried, but got it mostly wrong

22. Sketch FT(g(t)h(t)).
g(w)
h(w)