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OSE 6211 Fourier Optics
Spring 2009
Mid Term Exam 1
Tuesday, Feb 24
This is a closed notes, closed book exam. Answer all questions. Try to use
drawings & figures to help explain your answers. Clearly state all your
assumptions, definitions and approximations. The number of points for each
question is given in italics.
Tables of Fourier transforms and theorems are appended to this exam paper.
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Name: ___________________________________
Question Score
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- Sketch the periodic function, f 1 (x), and find its Fourier series representation where:
L x L
L x L
L x L
f x
1 ( ) ,^ period 2L
(25 points)
- Now consider a truncated version of the function f 1 (x), from question 1:
x N L
f x x N L
f x
1 2 ,
where, N is a positive integer. Sketch this function for N = 3. Find and sketch the Fourier
transform of this function.
(25 points)
- In class, we saw that we can model pulse propagation in a material with dispersion as
follows:
ωτ ω
ω τ ω i d
ik z E z E ⎟
∞
− ∞
( , ) ( 0 , )exp
2 2
Now let’s assume that the input pulse is already chirped: i.e.
2 0
2
0 2
( 0 , ) exp
τ
τ τ
iC E E.
(a) Is the duration of the pulse affected if the chirp parameter, C, is varied?
(b) Is the spectral width of the pulse affected if the chirp parameter, C, is varied?
The pulse travels through a dispersive medium with GVD parameter, k 2 :
(c) Show that, for the right sign combinations of C and k 2 , the pulse duration can be
shortened after propagating a certain distance.
(d) If k 2 = 25 ps
2 /km, τ 0 = 100 fs, and C = -2, then find the distance the pulse must
propagate to reach the minimum pulse width. By how much is the pulse width reduced?
(25 points)
