Chapter 4. Coherence
PC1. Two Source Points.(see p.185)
Look at FileFig.(C1COH2S) and the graphs for I1() + II1()
I1   
d
sin     sin   
 

 d   sin    


 

II1   
2
2
a
 cos      sin      
 

d
sin      sin      
 

  d    sin      


 

2
2
2
a
 cos      sin     sin  1   
 

2
a. Obtain from FileFig.(D10FAGRDSLS) the second formula by using N = 2.
Change  = s/Z and observe the disappearance and reappearance of fringes.
b. Study the disappearance and reappearance of fringes depending on a.
c. Study the disappearance and reappearance of fringes depending on d, and compare.
PC2. Extended Source.(see p.190)
Look at FileFig.(C2COHEX) and the graphs for
1



I1 k  


0
2
d
sin    sin   k  sin    
2
 
  cos  a   sin     sin    d

k

2
 

 d   sin    sin    

k

 

Change  = s/Z and observe the disappearance and reappearance of fringes.
a. Study the disappearance and reappearance of fringes depending on a.
b. Study the disappearance and reappearance of fringes depending on d and explain.
PC3. Visibility.(see p.194)
Calculate for the visibility of an extended source the integration of
V ={cos(a/X)(Y’)}
and obtain
Vis = {sin(a/X)(Y’)}/ (a/X)(Y’) 
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PC4. Caparison of Visibilities.(see p.194)
Calculate the visibility of two point sources
Vis = cos(a/X)(Y') 
the one dimensional extended source
Vis = {sin(a/X)(Y’)}/ (a/X)(Y’) 
and the round source.
Vis = [J1 (2(aR/X))]/(aR/X) 
The formula for the round source is obtained similar to the discussion of slit and round
opening in FileFig.(D2FASLITS) and (D7FARON3DS),
Make all formulas depending on s/Z and use (all in mm) a = 1, Z = 9000, and = .0005.
a. Determine the distance s when the fringes disappear for the first time (look at the
graph) and compare with the calculations of PC1 and PC2.
b. Vary a and  and find out how the visibility depends on these parameters.
PC5. Calculation of the Visibility for Fresnel’s Mirror Interferometer.(see p.193 and
93)
Fresnel mirror interferometer has been discussed in the chapter on interference, see
FileFig.(I7FREMIRS). The general set-up and the coordinates are shown in Fig.2.8, and
the intensity is given in eq.2.35 as
I(Y) = Io[cos{(Y2bsin)/[(c+f)λ]}]2
This can be written with a = 2bsin and X =(c+f) as
I(Y) = Io[cos{(aY)/(Xλ)}]2 and for the visibility one has from eq. 4.14
V = cos({(a)/λ}Y’/X] 
The optical path difference is  = Y’a/X
a. Relate  to the angle , on which the separation of the sources depends, and make a
sketch indicating Y’ and Y’/X . For  = 0 we have a maximum.
b. Find the first minimum and call Y’/X = Y’/(c+f) = γ and give the corresponding value
of sin.
PC6. Michelson Stellar Interferometer.(see p.195)
The Michelson stellar interferometer was discussed for the determination of the distance
of two source points. Formulate the same experiment for the determination of the length
of a line source. Write a file for a numerical demonstration.
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PC7. Quasimonochromatic Light.(see p.198)
Quasimonochromatic light and the length of the wave train.
2
A(Y) =  cos( a Y/X) d
1
Make a graph of the superposition of amplitudes of the wavelength interval 2 - 1 for a
band width in the spectral region of 0.4 to 0.8 ·10-3mm (visible), 8 to 12·10-3mm
(infrared) and 50 to 300·10-3mm (far infrared). Express the corresponding frequency band
widths in cm -1 and calculate the corresponding coherence lengths.
PC8. Quasimonochromatic Light and Interferogram.(see p.199)
Quasimonochromatic light is used by the Michelson interferometer for Fourier transform
spectroscopy.
Go to FileFig.(C6SUPERS) and get the superposition of the amplitudes of wavelength
from 1 to10 as
10
A(Y) =  [cos(2 x/)]2d
1
The superposition of the intensities is then
10
I(Y) =  [cos(2 x/)]2d
1
a. Make a graph of both for x = -50 to 50.
b. Apply the formula (cos α)2 = ½(1 + cos 2 α) to the integral of the intensity and make a
plot again.
The graphs of the intensity are of course the same, but one depends linear on the cosfunction while the other quadratic. The linear dependence is of use for Fourier transform
spectroscopy, see Section 9.2.3.
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