Chapter 18 & 19: Speckle Patterns & Their Properties.
Speckle 2
Sample problems
18-19 -S1
Formally the equation of formation of interferometric fringes in speckle patterns is the same than
that of smooth wave fronts. Also the final outcome yields similar results (See figure 18.19).Both
equations have similar forms. Going back to section 18, the final intensity, equation (18.45), can
be put under the form,
I fi  I1  I 2  2 I1I 2 cos 
(a)
What are the main differences between the two processes?
Solution to 18-19 –S1
One has the same equation of interference of two coherent wave fronts; I1 and I2 are the
intensities of each wave fronts and ϕ their relative phase. There is an obvious difference that can
be seen in Figure 18.19.(a) One has a uniform distribution of gray levels, the other has a mottled
appearance due to the speckle; (b) the visibility is also different. Mathematically the difference is
that the above equation for smooth wave fronts applies everywhere in the interference space
while in the case of speckles is a point function (is valid in a very small volume) in the region of
interference. In-plane displacements of the smooth wave fronts with respect to the reference
wave front leaves the interference pattern unchanged, for speckle patterns an in-plane
displacement causes the loss of the interference phenomenon. This is the decorrelation property
of speckle patterns.
18-19 -S2
From sample problem 18-19 -S1 one can conclude that speckle interferometry (SI) is a form of
interferometry that make possible to extend classical interferometry based on smooth wave fronts
to wave fronts with a random structure. Since double beam interferometry implies two wave
fronts, how many combination we may have?
Solution 18-19 –S2
In problem S1 the case of two speckle wave fronts has been considered. A second possible case
is the interference of a smooth wave front and a speckle wave front. The answer to this question
is in Section18.2.2 and is graphically represented in Figure 18.11. When a speckle pattern is
mixed with a smooth wave front, another speckle pattern is produced.
18-19 -S3
What is the relationship between a random surface and the generated speckle pattern?
Solution to 18-19 –S3
The answer to this question falls within the field of statistical analysis. A partial answer is given
in graphical form in Figure 18.1. Figure 18.1(a) represents a random rough surface, figure
18.1(b) represents a rough surface but it is not random surface. Looking from the point of view
of a mathematical description a random surface can be thought of as the superposition of a very
large set of diffraction gratings of different periodicities and orientations, figure 18.1 (a). The
correlation function can be utilized to characterize a random surface,
G( x, y) 
1
z( x' , y' ) z( x' x, y' y) dx' dy'
A A
Where z(x,y) is the function that provides the surface heights and A is the area of integration. For
a random surface the statistical distribution of heights is a circular Gaussian distribution similar
to that shown in Figure 18.10 for the light intensities. This means that if we consider a point on
the surface and we take points at a distance r of this point the cross-correlation goes rapidly to
zero. If we consider Figure 18.1(b) the cross-correlation will show a periodic component.
Practically it means if I take the FT of the speckle pattern of a random surface I will observe a set
of random spots representing the periodicity and orientation of the equivalent sinusoidal gratings
forming the surface. If I compute the FT of a speckle pattern of a rough surface of the type
represented by Figure 18.1 (b), I will observe beside the random components a stronger
diffraction spot corresponding to the periodic component resulting from the machining process.
This frequency can be utilized as a grating engraved on the surface of the specimen, thus
allowing the interferometric moiré fringes to be observed.
18-19 -S4
From sample problem 18-19 -S2 we know that mixing a smooth wave front with a speckle wave
front generates another speckle wave front. This is exactly what occurs when one makes a
hologram of a rough surface body and a reference wave front. Could we call this mixing of wave
fronts a case of speckle interferometry?
Solution to 18-19 –S4
It falls within a case of interferometry but according with established nomenclature it is not a
case of SI. The formation of the fringes depends on the presence of a reference beam that is not
present in SI. The SI utilizes two wavefronts that generate the fringes. In holographic
interferometry the reconstruction of the surface before and after deformation generates the
fringes. The reconstruction of the interfering surface requires a reference beam that forms part of
the holographic process. This difference becomes clear when reviewing Section 21.8 and the
explanation of the formation of the isothetic lines illustrated in Figure 21.20.
18-19 -S5
Mixing two wave fronts coming from the undeformed and deformed configurations of a surface
we can get holographic interferograms. What is the relationship of this process with SI?
Solution to 18-19 –S5
According to the historical process of development between SI and holographic interferometry
(HI), HI preceded SI and SI development where influenced by HI. The term SI is applied to 2-D
surfaces and one can consider SI as a rough wave front version of moiré interferometry. One can
consider this nomenclature as robust enough to separate SI and HI. From the practical point of
view it means that in SI is missing the smooth reference beam that is the core root of HI and
makes possible the deformation analysis of 3-D surfaces.
18-19 -S6
What is the significance of the first order and second order statistics in SI?
Solution to 18-19 –S6
The first order statistic defines the intensity distribution that appears in Sample Problem 18-19 S1. This equation can be written, (18.36).
IP  I0 P 1  Vcs x, y cos  (b)
The above expression does not only provide an understanding of the formation of the fringes but
is a practical guide to select parameters that optimize the visibility of the fringes that is the signal
to noise ratio. The first order statistics deal directly with the irradiance of the speckle fields and
the effect of the different superposition of speckle fields. The second order statistics defines the
region where the interferometric effect can take place and the effect of displacements in the
formation of the interference fringes. All these factors influence the selection of the best possible
conditions to produce interference effects, selection of intensities of the component wavefronts
and lens aperture. The term I0 P depends fundamentally on the first order statistics. The term
Vcs x, y  depends fundamentally on the second order statistics.
18-19 -S7
Does the term objective vs. subjective speckle indicate an important distinction in SI?
Solution to 18-19 –S7
The important factors in both cases are the same, the estimation of the dimension of what can be
called of the average size of the region of interference analyzed in the second order statistics.
Hence the distinction is more an operative definition to extend derivations done in one case the
other case but still the two cases are governed by similar rules.
18-19 –S8
What is the importance of the distinction between resolved and unresolved speckle patterns?
Solution to 18-19 –S8
In the initial development of SI high resolution emulsions were utilized, the same emulsions used
in HI. During this period of time the fundamental first and second order statistics where
developed. Hence there was a direct relationship between experimental applications and the
mathematical models developed to understand the rules governing the pattern formation. Soon
after video cameras were utilized to record speckle patterns and they were followed by CD and
CMOS cameras. The first approach utilized was to increase the size of the speckles to make
possible the resolution of speckles; the rule was one speckle, one pixel of the camera sensor. It
took some time to realize that this condition was very restrictive and posed limitation to the
practical use of speckle interferometry particularly it has to be applied to large surfaces. The low
level of efficiency in the utilization light intensity resulting this rule provided an incentive to
move to the integration rule that is more than one pixel in one sensor. The required laser power
became beyond the possibilities of practical applications. Hence SI moved to the partially
resolved speckles regime.
18-19 – S9
What are the basic aspects of the integration of pixels? What connection exists between resolved
pixel regime SI and integrated pixels SI?
Solution to 18-19 – S9
The equation (a) of sample problem 18-19 –S1 is still valid in the integrated regime with the
same interpretation resulting from the first and second order statistics, this time on the basis of
the integrated regime. To be more specific the developments of Chapter 18.5 and the concept of
m cells in a pixel where m represents the statistically independent subareas (dark squares in
Figure 18.17) compared to the light square, pixel can be utilized. For m=1 we have the initially
developed first order statistics, Figure 18.18 and the verification shown in 18.26. Basically the
problem reduces to the ratio of the resulting background illumination I0 P and the visibility
term Vcs x, y  , that is the signal to noise ratio. This ratio depends on the optimization of the wave
field intensities that and the selection of the best lens aperture. Equation (a) of sample problem
18.19 –S1 can be written as,
I fi  I0  I M cos  (c)
The ratio IM/ I0 determines the quality of the signal. I0 is the integrated background resulting
from the addition of m cells, IM is the integrated modulation term also resulting from the addition
of m cells. Figure 18.26 shows experimental results that indicate how the probability density of
the intensity distribution is reduced as the aperture of the lens increases. In summary there is a
parallel between the integrated regime and the classical resolves SI. This parallel can be used to
design a given set up.
The main variables are: available laser power, aperture of the lens. Selecting an aperture and
knowing the pixel size it is possible to estimate the number of speckles per pixel. Having a
measure of the number of speckles per pixel and plots of the type shown in Figure 18.26 one can
see if it is possible to have enough modulation to get adequate fringe visibility. Knowing the
laser power and the area to be illuminated one can make an estimate of the average intensity and
this average intensity can be related to the sensor response for the particular wavelength selected.
All the above tools serve as a guide but actual experimental measurements will dictate the final
selection of variables. A rule of thumb is that as the area to be observed increases it is necessary
to increase the lens aperture. The average intensity of the object should be measured and the
response of the camera to this intensity should also be measured. Over saturated pixels should be
avoided according to what is stated in Section 18.9, the average pixel modulation decreases as
1
m with saturation and increases as
m without saturation.
Problems to solve
18-19.1
Following the reasoning of sample problem 18-19 –S7 what are the basic variables that are
common in the two cases.
18-19.2
Provide an idea of the dimensions of the regions where the interference effect can take place.
Hint: Look at the developments of Chapter18.4
18-19.3
In your own words explain why you think that the concept of sensitivity vector is important.
18-19.4
Suppose that you have to measure the in-plane displacements of a large surface 3m × 4m. What
kind of optical arrangement you will select.
Hint: Utilize the sensitivity vector concept to get your answer.
18-19.5
What will happen in the formation of the pattern in the out-of-plane interferometer if you replace
the rough surface by a mirror?
18-19.6
Design a contouring interferometer with double illumination assuming that the object is large
enough that it is not practical to use collimated illumination.
Hint: Utilize the sensitivity vector concept to get your answer.
18-19.7
Figure P18.19.1 was obtained with the method described in Section 19.3. The full field of view
is 60 mm. The disk diameter is 50 mm, the thickness t=4 mm. The optical parameters are given
in Section 19.3, then the picture is equivalent to a moiré pattern taken with a pitch p=16.1 μm.
The original picture was taken with high resolution film. The pictures were digitized with 2024 ×
2024 pixels. Verify that the displacement field corresponds to the disk under diametrical
compression. Utilize a method of your choice to retrieve the vertical displacement information at
different points of the horizontal axis. A similar disk was utilized to get the pattern of problem
14.1. Utilizing several sections: What is the relationship of the loads applied to the two disks?
Figure P18.19.1. Disk under diametrical compression u and v patterns.
18-19.8
Follow the Holostrain manual process to process a speckle pattern of a disk under diametrical
compression.