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Final Exam October 2, 2008
2:00-3:
Rules:
- Closed book, closed notes- only pen/ pencil and one sheet of paper allowed
- Multiple choice set 1: one and only one answer is correct- circle the corresponding letter
- Show all derivations for Problems 2 and 3. Use verso and additional sheets if necessary.
Name:
1. Multiple choice questions (6x5=30%)
a. Considering a real function x and positive constant a , the following is true for a linear operator L
A. L(ax)=aL(x/a)
B. L(x)=aL(x/a)
C. L(ax)=aL(ax)
D. L(x)=aL(ax)
b. Considering two real functions x and y , the following is true for a linear operator L
A. L[(x+y)
2 ]=L[x
2 ]+L[y
2 ]
B. L[(x+y) 2 ]=L[x 2 ]-L[y 2 ]
C. L[(x+y)
2 ]=L[x
2 ]+L[y
2 ]+2L[xy]
D. L[(x+y) 2 ]=L[x 2 ]+L[y 2 ]+2L(x)L(y)
c. The second order differential operator, [ ]
2
2 x
, has the following form in the frequency domain ( q
denotes the angular frequency and i = − 1 )
A. iq [ ]
B. − iq [ ]
C. [ ]
2 q
D. [ ]
2 − q
d. Given a general function f and the Dirac-delta function δ , the following expression is true
A. f ( ) x = f ( x ') δ( x − x ')
B. f^ ( ) x^^ δ(^ x^ x^ ')^ dx '
∞
−∞
C. f ( ) x f ( x ' x ) δ ( x x ') dx '
∞
−∞
D. f ( ) x f ( x ') δ ( x x ') dx '
∞
−∞
e. If f(x) and g(x) are two functions, F(k) and G(k) their respective Fourier transforms, the following is
true ( ℑ stands for the Fourier operator, ∨ denotes convolution operation, a and b are positive constants):
A. ℑ[ f ( x / a ) ∨ g bx ( )] =( a / b F ak G k ) ( ) ( / b )
B. ℑ[ f ( x / a ) ∨ g bx ( )] =( ab F ak G k ) ( ) ( / b )
C. ℑ[ f ( x / a ) ∨ g bx ( )] =( ab F k ) ( / a G kb ) ( )
D. ℑ[ f ( x / a ) ∨ g bx ( ) (^) ] = F ak G k ( ) ( / b )
f. A function has the form
if x f x if x
⎧^ −^ −^ <^ <
⎩ <^ <
and is plotted below.
The following is true about its Fourier transform F:
A.
sin( ) ( ) 10
k F k k
B.
sin( ) /2 / ( ) 10 ( )
k ik ik F k e e k
− = +
C. F is a real function
D. F(0)=
f(x)
‐ 1 x
1
‐ 10
10
e) Ignoring reflections, what is the power density at the image plane?
f) Light travels 8 minutes from Sun to Earth. What is the transverse magnification?
f) The droplet starts to evaporate until it disappears. Sketch the power density at the image during evaporation.
3. In a Young’s double slit experiment, the distance between the slits
is a=0.5 mm and the wavelength of the incident light is λ =0.
microns. (30%)
a) If the desired fringe period is 1 mm at the screen, what is the necessary
screen distance L? Assume the slit width is small compared with a.
b) If a thin plate of glass (n=1.5) of thickness 0.1 mm is placed over one of the slits, what is the resulting lateral
fringe displacement at the screen?
c) What is the interference pattern (irradiance distribution) that results from a and b, if the slit width is equal to
a/2. Sketch the two irradiance distributions.
k
L
a
