Optics and Optical Design
Optical Interferometry - 08
Optical Interferometry
Preparation: Read about interference in chapter 2.5. Read about holography in chapter
4.5. Make the preparation problems at the last pages in this lab instruction and study the
theory below. It is also good if you read about the experiments before the lab. Then it is
easier to understand.
Introduction
Optical interferometry is part of the techniques for optical measurements which is a large
area in optics and consists mainly of spectrocopical methods. These can be used e.g. to
determine chemical and physical properties such as chemical composition and temperature
distribution in gases. For F4 there is a course, “atomic and molecular spectroscopy”,
where these types of measurements are studied in detail. Within medicin several optical
methods are used that handle the transport of light through highly scattering media such as
tissue. The course “medical optics” is within the photonics program where mathematical
methods are combined with optical measurements applied to medicine. In the course
“optics and optical communication” the optical properties of optical fibers are studied. We
will in this laboratory work restrict ourselves to methods where light is used to measure
deformations, displacements and refractive index distributions. It is often favourable to use
the light from a laser for this type of measurement. As the laser is a coherent light source it
is also important to investigate the coherence properties of this light source.
Part of this laboratory work has been a cooperation with other persons such as Bertil
Enquist and Razalina Brillante (Construction mechanics) and Håkan Bergström and Lars
Jönsson (the company Inspectra analys AB, www.inspectra.ideon.se).
Theory
The Michelson-interferometer
The coherence of laser light
The temporal coherence of a He-Ne laser is very long, often many kilometres. However
due to frequency beating between some frequencies obtained in the laser cavity (a volyme
limited by the two mirrors) the effective coherence length is reduced. The number of
frequencies is due to different longitudinal modes in the cavity. One longitudinal mode,
e.g. TEM00n is defined by the number of half wavelengths, n, that fits into the cavity
similarly to what is the case in a vibrating string. The cavity length of the HeNe-laser is
about 30 cm and n is then about 1 million. A small change in the number n mean a small
change in the wavelength and thus also the frequency.
The temporal coherence can be measured by studying the visibility of the interference
pattern obtained in a Michelson-interferometer (slightly modified version called a
Twyman-Green interferometer) see figur 1. Visibility is a value for the intensity-contrast
between dark and bright interference fringes:
I  I min
Visibility  max
I max  I min
The visibility is plotted as a function of the path difference between the two arms of the
interferometer (one of the mirrors is displaced). For a laser that has several longitudinal
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modes, the visibility curve will have maximas for path lengths that are multiples of the
double cavity length. The exact form of the curve is dependent on the number of modes
and their relative intensity.
We can think of the following experiment to simplify the understanding of the form of the
visibility curve: A laser cavity with the length L holds two modes, TEM00n and TEM00(n+1).
The light from this laser is introduced into a Michelson interferometer. We start with the
same distance in both arms. The mode TEM00n interferes with itself as well as TEM00(n+1).
As the two patterns are superimposed a high visibility is obtained. Now assume that we
move one of the mirrors a distance L/2 so that the path length increases with L. As before
the two modes interfere with themselves. However, now the two patterns are displaced
with half a wavelength or half a fringe. Where we have a bright fringe for TEM00n there is
a dark fringe for TEM00(n+1) and conversely, so the fringes are hardly seen. Thus the
visibility is low. If the mirror is moved another distance L/2 the two fringe pattern will
overlap in such a way that the visibility is high again. This means that the visibility curve
will repeat itself with the period 2L
Figur 1. Michelson interferometer. The two interfering (outgoing) beams or not exactly
parallel. That is why we obtain the fringe pattern. We record the pattern with a CCD
camera during the laboratory work.
It is important that the cavitity length is constant during the visibility mearsurement. A HeNe-laser has a rather stable cavitiy length after about one hour after turn on. A diode laser
has to be actively stabilized with a certain circuit to be constant for a long time. A diode
pumped laser (DPSS-laser) can have a coherence length of the order of 100 m. A HeNelaser with only one frequency (a mono-frequence laser) has a coherence length of many
kilometres.
Displacement measurements
In a holographic setup it is possible to measure displacements of three-dimensional diffuce
objects. This will be discussed below. For nearly plane and strongly reflecting objects a
simpler setup with a Michelson interferomer can be used (see figure 2). As the light source
is a coherent laser the emitted beams will interfere consecutively contructively and
destructively (we see bright and darc “fringes) dependent on the local height of the object.
If the mirror in figure 2 is translated upwards (in the figure) a distance of a fourth of a
wavelength, the path length change will be half a wavelength for the light. This will cause
the bright and dark fringes to change place. The conclusion is that the surface structure can
be determined.
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Figur 2. Michelson Interferometer. The fringes can be understood as height curves in
relation to the surface of the reflecting object.
Speckle photography
When the displacement of an object is parallel with the observation direction measurement
using holographic interferometry is the best choise. However, if the displacement is
perpendicular to this direction then this method is very insensitive. This is true especially
if the distance to the object is long. An alternative method in this case is speckle
photography.
Speckles
Upon inspection of a hologram, or a diffuse object illuminated by laser light, the image
appears grainy. The image seems to be covered with small, randomly scattered light and
dark areas, or speckles. These speckles are seen independently of where the eye is focused,
and when the observer moves, the speckles move in relation to the object being observed.
The speckles arise due to the use of coherent light. Three properties of speckles are of
interest, the intensity, the speckle contrast and the characteristic size of a speckle.
Laser
Detector
Observation
Plane
Scattering
surface
Figure 3. Observation of speckles near an object.
It is rather easy to explain the formation of speckles. Each part of an object will scatter
light to the observer or to a detector, as in figure 3. Due to the fact that we use a laser as
the light source, the light from each point of the object will interfere with the light from all
other points. When a detector, or the eye of an observer, is moved along the observation
plane a random pattern of interference fringes is seen. This is the speckle pattern. The
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randomness is due to the unevenness of the observed surface as the phase of the scattered
light is dependent on the local height. This means that the total vector U that represents the
amplitude of the electric field is the sum of many contributions (figure 4).
Im
U
Re
Figure 4. The total electric vector U is a sum of randomly oriented vectors with
different phase and amplitude.
The intensity is proportional to the length of the U-vector squared and the intensity
distribution is easily derived mathematically (solution to a “random-walk” problem). The
result is that the most probable intensity is zero and the contrast is 1. The latter means that
the standard deviation of the intensity is as big as the mean intensity. This high contrast is
the reason for that we experience the speckle pattern as very irritating. The mean size of
the speckles bs as observed in the image plane of a lens with the diameter D is given by the
following relation:
   z
bs  122
. 
,
 D 
where z is the distance between the image plane and the lens, se also figure5.
Figur 5. Imaging of speckles with a lens.
Displacement measurement with speckle photography
There are two different ways to do the measurement of displacements perpendicular to the
observation direction:
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* Double exposure and illumination afterwards with coherent light. Double
exposure means that the same photographic film is exposed twice and
developped.
* Two exposures are made with a CCD-camera and the images are then correlated.
In the second method the the object is displaced between the two exposures. By making a
twodimensional correlation between the two images, or between parts of them, the
displacement vector can be determined. If the whole object is moved uniformely good
correlation is obtained even if the whole image is used in the correlation. However, if
small parts of the object have moved in a different way, the image has to be devided into
subimages which have to be correlated individually.
In the following there is a short discussion about correlation to increase the understanding:
Two one-dimensional functions f(x) and g(x), noise function (or speckles along a line), are
correlated with each other se figure 6. Practically this means that one function is moved
over the other and a (correlation-) number C describes how much the functions seams to
be alike. When one function is translated slightly,  x , along the x-axis and a new
correlation number C ( x ) is calculated again. This procedure is repeated for all possible
displacements and the result can be something like shown in figure 6.
A mathematic description of the cross correlation C between two function f(x) and g(x) for
different  x is given by
C x  

 f x g x  x dx

a)
Figur 6.
b)
The correlation plotted as a function of relative displacement
Note that the scales are different in vertical direction
a) Correlation between two different noise functions
b) Correlation between a noise function with itself
(this is called an autocorrelation).
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During the lab works we will correlate two images. This means that we will make a twodimensiional correlation. The principle is the same as for one-dimensional correlation.
Two images are correlated, see figure 7. Instead of combining two function we now
combine two images and the similarity (the correlation number) is calculated. If the
images not are similar, the correlation number is low, white colour in figure 8, and where
the images are close in shape the correlation number is big, black in figure 8.
a)
b)
Figur 7. a) Orginal ”F”
b)The ”F” is translated some pixels to the right, x, and some vertically, y.
Figur 8. Slightly exaggerated result of a twodimensional correlation between the figure
7a) and b). Dark colour means a big value for the correlation.
Speckle photography is a technique for the measurement of translation, deformation,
rotation and vibration as long as it is parallel to the surface of the object. The precision of
this method is not as high as for holographic interferometry but on the other hand the
condition for mechanical stability is not so stringent.
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The Moiré method
The appearance of a moiré patterns is a common phenomena. If one looks at two grating
patterns on OH-slides placed on top of each other, there is a pattern of dark and bright
fringes superposed on the pattern of high frequency that is due to the two grating. This low
frequency pattern is the moiré fringes. Moiré fringes can be used to show threedimensional contours and they can also be used to detect deformation in a working engine.
Figur 9. Here we see moiré fringes with crossing gratings (left) and with gratings of
different frequencies (right).
Figur 9 illustrates how moiré fringes are obtained when two grating patterns are placed on
top of each other. In the case to the left there is an angle  between the gratings and in the
case to the right the period is different, g1 < g2.
Looking at the figure to the right it is clear that there is one more line from the finer
grating than from the coarser which means:
G  n  g2   n  1  g1
where n is the number of lines and G is the distance given in the figure. By eliminating n
we get:
1
1
1


G g1 g 2
And by changing from period to spatial frequencies (lines/mm) we have:
F  f1  f 2
If f1= 2 lines/mm and f2=1,5 lines/mm we get F=0,5 lines/mm.
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Shadow moiré used for contouring
Figur 10. Shadow moiré.
Allthough there are lots of different moiré techniques we will here discuss shadow moiré
as this is a method used in the lab works. In this case two grating patterns are projected on
the object surface by shining light from a light source through a grating as seen in figure
10. The observer looks from another point and sees two grating patterns. One is the grating
and the other is the shadow pattern. If the two gratings have different frequencies a moiré
pattern is observed. If e.g. the object has valleys and tops the shadow pattern will have a
varying frequency.
In the figure there is one line more between A and C than between A and C’ (4 in stead of
3). This means that the observer sees a moiré fringe (from bright to dark and then bright
again) between these points. The number of moiré fringes that we see between two points
as in this case between A and C is due to the variation of height of the object. The number
of fringes Nz, between A and another point E is a measure of the distance, W, between the
grating and the object (see the equation withing the figure). For the point E Nz = 2 in stead
of 1 in the point C.
Note that if the denominatior, tan   tan   , is not constant for all x-values the
measurement will be more difficult than nessesary. Think of how you can avoid this
difficulty (preparation problem 2). In figure 11 is seen an example of shadow moiré of a
coin.
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Figur 11. Shadow moiré of a coin
Moiré techniques in the plastering industry
When producing plasterboard it is more and more important to control the quality and
select the boards. Thus it is important to have effective measurement techniques. It is not
economical to have a person to inspect the boards in the production. Here the moiré
technique offers a fast and simple way of controlling the plasterboard. Both bending and
local deformations can be detected. The quality control can be made with a computer
system where each board is given a quality mark. This has been implemented on a factory
in Sweden where shadow moiré is used to check plasterboards directly online. Some
centimetres above the plasterboard is positioned a transparent plate with a grating with
some lines/mm. A strong linear light source projects the grating pattern on the board and
the detection of the moiré patterns is performed with one or many video cameras. These
are linked with a personal computer that analyses the video signal in real time.
Holographic interferometry
In the text book there is discussed how it is possible to obtain a three-dimensional image
when illuminating a hologram. An important point is that this image is such an exact
replica of the original object that it is impossible to find the difference between the laser
illuminated object and the reconstructed image. This is used in real time holography where
the hologram is repositioned exactly where it was taken. If this is done correctly the light
from the real object and the image created by the hologram will interfere. However, there
will be no fringes until we move or deform the object or the hologram. As in ordinary
interferometry we can detect changes of the order of the wavelength of the light. Another
interferometric method is double exposed holograms. In this case we record two images
on the holographic plate. When this plate is illuminated after development we can say that
we recreate two light waves that interfere. We can also say that we destroy the first image
at some areas where there has been a change in optical path length of an odd number of
half wavelengths. When observing such a hologram we can make measurements with the
accuracy of the order of the wavelength of the light used.
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Laboratory experiments
- Remember to bring all information with you that you need for the report.
- Store all images that you want to use in your report
Lab work - Moiré
On top of the plasterboard there is a glass plate with a grating pattern (2 lines/mm). In one
of the preparation problems you had to find out how to position the camera and the light
source in a suitable way. In this simple geometry all plane surfaces will be free from moiré
fringes, which will make our anlysis of deformation easier. Position your camera and light
source in this way. Use the program in the computer to display and store the images. You
have to make some adjustments to position the source in relation to the camera to avoid
reflexes and obtain a good image of the fringes.
On the plaster board you can see deformations. Take a picture of the moiré fringes for the
biggest deformation. Store the image. Determine the scale factor according to preparation
problem 2. Using this scale factor you can determine the depth of the deformation
according to the equation in figure 10.
At the long sides of a plasterboard, there is a slope that facilitate the puttying see figure 12.
Take an image of the moiré fringes at the slope. Calibrate the imaging system by taking an
image of a ruler that is on the slope. If you have changed the scale factor in relation to the
previous measurement, you have to determine it again. Calculate the angle of the slope .
Figur 12. Part of a plasterboard with the slope for puttying
Lab work – Speckle correlation
A speckle pattern is only formed in coherent light. It is smeared out if the light source is
incoherent like an ordinary white light source. To better understand the speckle
phenomena we will compare how subjective and objective speckles are seen by the eye.
Subjective speckles are observed when we look at a laser spot on a wall. How are the
speckles moving when we turn our head sideways or vertically? Test this effect with and
without spectacles.Why are the speckles moving as they do? What happens when we look
on the speckles through a small hole (form a hole with your fingers)? We can create
objective speckles by illuminating a surface with a laser and then look att the pattern
obtained from the interference of light from this secondary source. What is the difference
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between subjective and objective speckles. How can we change the size of the objective
speckles.
We will now use speckle correlation to measure small displacements. By recording the
speckle pattern before and after a translation we can measure the displacement. The
program “corrdemo.m” shows how the correlation is performed.
A piece of rubber is placed in a holder where we can measure the effect of shearing.
Illuminate the rubber piece homogenouosly with laser light. Use the program
“takeoneimage.m” to take an image of the object.
Adjust the CCD camera and the objective in such a way that nearly the whole rubber piece
is imaged (but not more). Check that the object is sharply imaged. It can be a bit difficult
to judge if the diffuce surface is sharply imaged so it is wise to use a thin paper with some
text on it that is positioned close to the rubber piece.
It is important that the speckle size is adjusted in relation to the pixelsize in the images.
What relation would you have so that the speckle size is not restricting the accuracy of the
measurement.
From the preparation problem we remember how we could calculate the speckle size in
the image plane of the camera (the CCD chip). To be able to compare the speckle size
with the pixel size, it is suitable to determine how long the distance will be in the image
that corresponds to a pixel. How can we determine this distance without measuring
directly on the CCD chip (compare with preparation problem 5)? Note! It can be a bit
tricky to determine the exact positon of the lens but the position of the CCD chip is well
known (it is close to the screw on the side of the camera that is closest to the objective).
We also need to know the focal length and the distance between the CCD chip and the
rubber piece to solve the problem. After your calculation you know the speckle size that is
appropriate. How can you now adjust the speckle size? Make you adjustment so that you
obtain the correct speckle size.
Now you can press the plate for the weights and see on the screen what happens to the
speckles. Remember that the setup is sensitive, so it is wise not the touch the table
between the exposures. It is also important to handle the weight carefully when you put
them on top of the plate.
Save images made without load and with 500g, 700g and 1000g load.
Start matlab and correlate the images with the program ”corr.m”. Explain the result! The
program displays the displacement in the unit “pixels”. How long is the displacement of
the rubber piece if the resulting displacement is on pixel?
Lab work - Interferometry
Visibility curve for a HeNe-lasers
Here we will make a setup of a Michelson interferometer and observe part of the visibility
curve for the HeNe-laser.
Make the setup of the interferometer and align it with the help of a small aperture. Check
that the two arms are equal in distance. Look at the interference fringes on a distant screen.
How can you change the direction of and distance between the fringes? Explain how you
can do that. The next step is to see what happens if the length of one arm of the
interferometer is changed. Study the visibility and make a plot as a function of the path
length difference. Explain why the visibility curve has the form it has.
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Interference in white light
We will now use a Michelson interferometer with a diffuse source. In this case it is
possible to observe fringes in white light.
Start with light from a sodium lamp as light source. Adjust the right mirror so that you
obtain circular fringes. If you then move the mirror away from or against the beam splitter
you will see that the fringes will change in size. Try to obtain as broad fringes as possible.
When you have found a position where the fringes are as broad as possible you change the
light source to a white light source. See if you can find fringes for white light. Why are the
fringes coloured? Introduce a glass plate in the arm where you have the right mirror. See if
you can observe fringes again. How much and in what direction shall you move the
mirror? Measure the distance you have moved the mirror. Discuss the result.
Interference in sodium light
When you adjusted the interferometer using sodium light you may have observed that the
contrast of the fringes changed. This is due to that the light from the light source consists
of light with different wavelengths. Measure the distance between the two positions where
you have minimum visibility on both sides of the position where the path length difference
is zero. Compare your results with measurements of the wavelength difference between
the two lines.
Lab work - Holographic interferometry
Examination of double exposed hologram
Study the metal holder for the three bars. Each bar can be deformed in relation to its initial
position by letting a weight pull the metal string that is attached on the back side and runs
over a pulley.
Study the double exposed hologram showing the three metal bars.
1: Which of the outer bars has been deformed the most.
2: Why are the fringes slightly leaning?
3: Why are the fringes at constant distance at the upper part on the outer bars while the
distance is varying at the bottom?
4: Try to explain the fringe pattern on the bar in the middle.
5: When the double exposed hologram of a bar was made there was no force on the bar in
one of the exposures. Can you find out which?
6: Study a hologram of a sugar solution. This is also a double exposure between which
there has been a change of the refractive index between exposures. Where has the
refractive index changed most?
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Preparation problems
1. A shadow moiré setup is shown in figure 13. Calculate the scale factor
g
tan   tan  
at the studied deformation.
Figur 18. Setup for shadow moiré.
2. Study the section on moiré technique, especially figure 10 (or 13). How shall you
position the camera (the observer) and the light source so that a flat object will not give
any fringes? An alternative formulation is: How should the camaer and light source be
g
positioned to that the relation
in figure 13 is constant all over the
tan   tan  
object (for all x)?
3. Explain the difference between moiré fringes and interference fringes.
4. A laser with several longitudinal modes has a cavity length of 30 cm. It illuminates a
Michelson interferometer. What distance must you move one of the mirrors in the
interferometer to go from one visibility maximum to the next?
5. Suppose that we will make a speckle photo with an objective on the CCD camera that
has the focal length f = 16 mm. The distance between the object and the CCD chip is L
= 100 mm and the light source is a red HeNe-laser.
a) What is the distance between the object and the lens and between the lens and the
CCD chip, respectively?
b) What is the magnification from object plan (the object) to the image plane (CCD
chip)?
c) If a distance of 11 mm on the object is imaged on 480 pixels in the final image, how
long is the distance that corresponds to one pixel in the image plane (CCD chip)?
d) What will be the size of speckles in the image plane if the camera has an f-number
of 8? (This means that f/D = 8)
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Figur 13. Two 1D-funktions f and g that we want to correlate with each other.
6. Make a drawing of the correlation between the functions f(x) and g(x) which are
defined in figure 13. What can we conclude from the correlation?
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