Chapter 2. Interference
PI1. Cos-Waves depending on Space and Time.(see p.80)
Look at FileFig.(I1COSWS) and plot 3 cos-waves with the same wavelength  and
time periode T.
a. Plot “snap shots” of cos[2(x/ - ti/T) + i)] for three values of ti and i and
depending on x = 0...10. Give an interpretation of the values of the ordinate at a specific
value of xi.
b. Plot cos[2(xi/ - t/T) + i)] for three values of xi and i. depending on t = 0...10.
Give an interpretation of the value of the ordinate at a specific value of ti.
PI2. Superposition of two cos-waves with fixed Optical path Difference.(see p.82)
Look at FileFig.(I2COSSUPS) with six plots of the amplitude of the superposition of
two cosine waves with a fixed optical path difference as function of the space
coordinate for fixed time. Consider now with a fixed value for 1
u1( x)
1
2. A . cos 2.  .
2. 
. cos 2.  . x

t1
T
1
2.  .
2. 
2
Make a choice of , T, and t1 and choose x1. Use the same range for x and t and get the
same graph for v1(t) after changing appropriate quantities.
PI3. 3-D Graph of Maxima and Minima.(see p.84)
Look at FileFig.(I3COSGRA) and modify the graph such that one sees only one
wavelength over the range of the graph. (The second graph may not show zero if it’s
scale is much different from the first.)
PI4. Average( see p.85)
Make a choice of all constants and show with a numerical calculation that one has for
the average
T
2
av = 1/T  [cos{( 2(x/ -t/T) -2(/2)/ }]2 dt = ½
0
PI5. Fresnel’s Mirrors.(see p.93)
Go to FileFig. (I7FREMIRS) and use for the calculation of IF 1(Y) the values:  = .0002
Rad,  = .0005 mm, b = 1000mm, f = 5000mm, and Y from –10 mm to 10 mm. Use c
= b cos. Since c, f, and b are fixed, fringe separation can only be modified by
changing  or .
a. How is the pattern changing when changing . How much must  be changed to
double the separation of the maxima or to make them one half.
b. How is the pattern changing when changing . How much must  be changed to
double the separation of the maxima or to make them one half.
c. What angle  has to be chosen to have the maxima separated by 500 microns when
using X-rays of 500 Angstrom.
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PI6. Young’s and Lloyd Experiment.(see p.93)
Look at FileFig.(I8FRYOLOS) and consider the intensities of Lloyd’s experiment
IY(Y) = cos((a/ )(Y/X)), and Young’s experiment IL(Y) ) = cos( (a/)(Y/X) + /2).
a. The phase shift of  or /2 appears in the formula as /2. How comes that ?
b. How is the separation of the maxima changing when making the separation of the
source points larger?
c. How is the separation of the maxima changing when making the separation of the
source points smaller?
d. Show with numerical examples that there are no changes of the fringe pattern when
keeping the product (a/ ) or (a/ )(Y/X) constant. Note that Y/X corresponds to the
angle under which we observe the maxima.
PI7. Plane parallel Plate in two Beam Interferometry with different Refractive
Indices.(see p.95)
A plane parallel plate of thickness D has refractive index n2. The medium of the
incident light is n1 and of the transmitted light n3. The incident light is from the top.
Consider only two waves reflected from the interfaces n1 to n2 and n2 to n3.
Calculate the optical path difference and phase difference for different refractive
indices.
Example: For n1 = 1.5, n2 = 1, and n3 = 1.5
 = 2D +/2
 =(2/)( 2D +/2) = 2( 2D/) +
and for n1 = 1, n2 = 1.22, and n3 =1.5
 = 2D
 =(2/)( 2D) = 2( 2D/)
Make a list for
a. n1 = 1,
n2 = 1.5
b. n1 = 1.5
n2 = 1
c. n1 = 1.22
n2 = 1.5
n3 = 1.22
n3 = 1.22
n3 = 1.22
PI8. Plane parallel Plate in two Beam Interferometry.(see p.97)
Consider FileFig.(I9PLANS) and derive a formula for the wavelength of the maxima
for a plate of thickness D and refractive index n2. Verify the formula using the graph.
PI9. Wedge shaped Film.(see p.99)
Go to FileFig.(I10WEDGES) and assume (all in mm) x = 0 xf = 100,  = .0005 and
that the height at xf is 10.
a. Calculated , it should be 0.00005 and count the total number of fringes in the
range from x = 0 to xf.
b. Choose two points, x1 and x2, and calculate the heights, h(x1) and h(x2). Call the
difference [h(x1)- h(x2] = h.
i). Calculating the number of fringes from point x1 to x2 by dividing h by /2.
Why? Compare with the graph.
ii). The fringes in the interval from x1 to x2 are a fraction of the total number of
fringes. In the same way, the fringes in h are the same fraction of the fringes in xf =
2
10.
c. What is the separation of fringes when using light of wavelength 0.0005 mm and
placing an object with a diameter of 10 microns on one side of the plates of length d =
10 cm ?
PI10. Newton’s Rings.(see p.101)
Go to FileFig.(I11NEWTONS) and calculate from the interference patter the radius of
curvature R of the lens.
From the wedge, FileFig .(I10WEDGES), we expect that one has for the mth ring
(fringe) a height in the air gap of m/2. Derive (m/2 – R)2 = R2 – rm2 for the
calculation of rm or R.
Calculate a set of rm for:  = .0005 mm and R = 5 meter and use n = 1 in the air gap.
PI11. Ring pattern and Michelson interferometer.(see p.107)
Go to FileFig.(I13MICHANS) and write the formula in degrees. Assume  = -90 to 90
and (in microns)  = 0.5 and D = 1.
We want to assign integer numbers to the maxima. For the maximum at  = 0 we have
from 2D = m that m = 4. To find out if the next maximum has a higher or lower m,
plot the function y = [2D cos(2/360)]/ depending on  and read off the numbers for
m. Note, the numbers of the maxima are in reversed order compared to Young’s
experiment.
PI12. Plane parallel plate.(see p.108-111)
Derive the transmitted and reflected intensity for a plane parallel plate having inside the
refractive index n2 and on the outside two different refractive indices n1 and n3. (Result
is given in FileFig.(I14PLANIDS).
PI13. Interference of white light on a thin film.(see p.113)
Go to FileFig.(I15PLANIDS) and consider a thin glass plate of thickness D in
microns. The refractive indices are n1 = 1, n2 = 1.5, n3 = 1 and for the wavelength
range assume  = 0.2 to 1 in microns.
Change the thickness D such that the last maximum has wavelength
a.  = 0.4 m (blue)
b.  = 0.5 m (green)
c.  = 0.7 m (red)
and give the thickness of D in terms of the wavelength.
PI14. Reflection and Transmission coefficients.(p.111-112)
Reflection and transmission coefficients as use for the calculation of the formulas of
intensity reflection and transmission of the plane parallel plate.
We have used the scalar wave theory which can not give us expression for r12
depending on the angle of incidence and refraction, and the indices for the media 1
and 2. From Maxwell’s theory we have the result that the light is polarized and the
amplitude reflection coefficient for the case that the light is polarized parallel to
the plane of incidence is
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r ║ = - (n2 cos  - n1 cos “)/(n2 cos  + n1 cos “)
and that the light is polarized perpendicular to the plane of incidence
r ┴ = (n1 cos  - n2 cos “)/(n1 cos  + n2 cos “)
The light is incident in medium with refractive index n1 at an angle  and is
refracted into medium with index n2 at the refraction angle ”. For r21 the light is
incident in medium with refractive index n2 at an angle ” and is refracted into
medium with index n1 at the refraction angle .
a. Show that we have r12 = - r21 and therefore we may write r12 2 = r212 = R
b. We define 12 = T12 and 21 = T21 and have from energy conservation
that R + T12 = 1 and R +T21 = 1. Show that 12 = 21 .
c. Show that we have 1221 = 1- R.
PI15. Plane parallel plate, graphs depending on wavelength.(see p.113)
Go to FileFig.(I15PLANDIS) and look at the transmitted intensity T as function of
the wavelength for normal incidence. Use for the thickness D = 0.002, and for the
wavelength range 0.0004 to 0.002. Assume that the refractive index of the plate is
larger than the refractive indices on the outside.
a. The separation of the fringes is getting smaller or larger , when one
makes the thickness larger ?
b. Extend the wavelength range to “see” the last fringe, and find the relation of the
wavelength to the thickness of the plate.
c.. How much is the intensity changing when changing the refractive index from n2 =
1.5 to Silicon with n = 3.4.
PI16. Dependence on angle of the Finesse.(p.117).
The resolving power of a Fabry-Perot is characterized by the finesse πg/2 = 2r/(1r)2. Make a graph of g = 2r/(1-r)2 for the range of angles from 0 to 10o. The
reflection coefficients are given in Chapter 5, in FileFig.(M2frn2l) when the second
medium has higher index and (M3frn2s) when the second medium has lower
refractive index. Use indices n = 1 and 1.5.
a. Second medium at higher index (M2frn2l).
Plot gp() = 2 rp()/(1-rp()2) and similarly gs() on the same graph and give the
percentage change compared to the value at  = 0.
b. Second medium at lower index (M3frn2l).
Plot gp() = 2 rp()/(1-rp()2) and similarly gs() on the same graph and give the
percentage change compared to the value at  = 0.
c. Compare and make your comment.
PI17. Plane parallel plate with mirror surface.
Consider a plane parallel plate with the second surface being a mirror. Show that
fringes may not be observed because the light of destructive interference can not
“escape”..
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PI18. Fabry-Perot.(see p.115)
Go to FileFig.(I16FABRYS) and look at three graphs depending on thickness D
from 0 to 11 and  = 0.1 , each for a separate value of r. The band width is defined
as bw = 2/g with g = 2r/(1-r2).
a. Use make a graph of bw for r = 0.5 to 0.99. Determine the values of r for bw =
0.02 and find the corresponding r for a two times smaller band width. Make graphs
of the peaks, determine the band width from the grap, and compare with “factor 2”.
b. Go to FileFig.(I17FABRLS) do the same for peaks depending on wavelength .
PI19.Interference with an array of source points.(see p.122)
Go to FileFig.(I20ARRAYS) and look at a graph for interference, observed when using
N sources at distance a. Because the angle is small, one may use  = X/Y, where X is
the distance of the observation screen and Y the position of the interference fringe on
the screen.
The graphs depend on the angle , the angle which the parallel beams make with the
normal of the observation screen. Use for the range of the angle 0 to 0.5, a = 0.1 and 
= 0.0005, and N = 5.
a. What is the effect of changing N.?
b. How do the maxima change when changing a.?
c. How do the maxima change when changing .?
d. What are the changes when keeping a/ constant.?
PI20. Summation of random phase angles.(see p.125)
Go to FileFig.(I21RANDS) for calculation of the superposition of waves, having a
random value of phase angles. The angles are counted with respect to a reference wave,
for which we may set the phase angle to zero.
a. Consider a summation of real terms
y = (1/N) sin(2i)
where the summation is from i = 0 to N.
How large have one to make N to have y smaller than 0.01?
b. Repeat the same calculation for a complex term ei2i.
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