Lab Report 2: Speckle

Submitted by: Group G2 - Joveria Baig and Pei-Ying Lin

Introduction:

Speckle is a random noise pattern, which generated by spatial interference of coherent light. That is,

when a coherent wave illuminates a scattering surface, we can obtain the speckle pattern. It appears in

the form of a randomly arranged set of grains on a dark background. Grains in a speckle pattern are not

faithful images of grains that may exist in the random medium. Speckle arises when imaging grains that

are not resolved by the imaging instrument, or when light is observed in an arbitrary plane that is not

conjugated to the grainy object. In addition, speckle may be observed in the image of a inhomogeneous

medium, when the grains of the medium are not resolved, or in a diffraction figure. The inhomogeneous

medium is often called diffuser. These two kinds of speckle have quite different properties. Coherence

also has a major impact on the phenomenon. In this laboratory work, we will define the difference of

strong and weak diffusers and investigate its properties related to statistical optics.

Theoretical Background:

The statistical properties of the scattering object have an influence in the illuminated area where the

speckle pattern is observable.

1. Roughness amplitude: a strong diffuser with mean value, gives a fully formed speckle. This

is because the pupil can be given by

Here, is physical pupil and is diffuser

Hence, for , it is strong diffuser, and also can be regarded as fully formed speckle. This means

there is no bright peak at the center of the observation plate.

2. Roughness autocorrelation width: with our definition of the roughness autocorrelation

for a circular object of scatter, the autocorrelation would be

with the width equal to

Hence, the larger the angle , the smaller will be the illumination width on the screen and vice versa.

Also, the shape of the aperture stop has an influence on the size of the speckle grains. That is, the

smaller the numerical aperture, the larger the speckle grains.

3. Roughness autocorrelation width of a scattering object: we assume a very simple model (rectangular

function) for the autocorrelation function of the scattering object.

Autocorrelation function gives the characteristic size of the grain of speckle. The explanation is the grain

size is based on the spatial autocorrelation in an image varies with grain size. We can imagine that, the

autocorrelation can be regarded as the correlation between two rectangular regions in an image,

measured by calculating the correlation between intensity of each pixel in one rectangular region with

the pixel in the corresponding location in the second rectangular region. And the spatial autocorrelation

is the correlation between these two sets of values. The value of the correlation approaches uniform

where the offset between the rectangular regions is small relative to grain size and approaches zero

where the offset approaches the size of the largest grains. By calculating the spatial correlation at a

variety of offsets, say, the distances between the two rectangular regions, produces a curve that

describes the correlation as a function of distance. This autocorrelation curve depicts that this statistical

approach has been used to characterize and model data, even measure the size of objects in images.

The scattering object is illuminated uniformly over a disk (with diameter d), and we want to describe the

speckle pattern obtained at a distance D form this scattering object like the experimental setup shows.

The characteristic size of speckle grains is related to d, the diameter of the illuminated disk, and to D the

distance to the scattering object. The reason like follows. Each point in the image can be regarded like

illuminated by a small area in the object. The size of this area is determined by the simply model of

diffraction-limited resolution of the lens, which is given by the Airy function. The light at neighbor points

in the image also has been scattered from areas. However, two points in the image, which are

illuminated by areas in the object, which are separated by the diameter of the Airy disk, have light

intensities that are not related. Therefore, the change in speckle size with lens aperture can be

observed, for example, through a very small hole, the speckles will increase significantly in size.

Experimental verification of general properties of speckle:

An experiment to test for the speckles formed by various different diffusers of varying properties was

performed with the setup shown in figure 1.

The speckle pattern for the case of varying roughness of a diffuser and for different types of diffusers i.e.

strong diffuser and weak diffuser were tested with the setup above. The speckle observed with the

weak diffuser showed a clear bright peak at the center of the plate as expected, confirming the

theoretical claim made earlier. With the strong diffuser, no bright peak was seen in the center. In

addition, by varying the roughness amplitude of the diffuser, it was observed that a strongly scattering

diffuser with a greater roughness amplitude gave a fully developed speckle. In addition, varying the

diffuser roughness resulted in speckle occupying a larger or smaller portion of the screen which will be

analyzed in detail later in the section on roughness autocorrelation.

Study of Speckle in the Fourier Plane

Airy disk diffraction pattern observed on a CCD:

In the setup shown above, the diffuser was removed and a small iris

diaphragm was placed approximately 7 cm from the CCD. After adding

optical filters and adjusting the polarizer, the diffraction airy spot was

observed on the image acquired by the CCD. The sampling period is equal

to one pixel of a CCD array which is approximately 8.3 µm. The diffraction

spot viewed on the monitor is shown in Figure 2-a, along with a horizontal

profile of the diffraction spot fitted with an airy disk function (Figure 2-b).

The numerical aperture of the system was deduced by a simple calculation

of:

D

Laser source

Polarizer

Pinhole as a spatial filter

Doublet

Diffuser Screen or CCD

Figure 1: Experimental setup

Figure 3: Measurement of Numerical Aperture

The numerical aperture of the system is calculated to be 0.003. Hence the diffraction limit of the system

is given by

:

Case of a weak diffuser

A weak diffuser is now placed in the setup. Figure 4 shows the speckle profiles observed for varying

aperture sizes fitted with the airy pattern. As can be seen from these horizontal profiles of the speckles,

as the aperture size increases, and hence the Numerical aperture increases, the speckle grains become

smaller, proving the theory described earlier. Hence, the speckle for a wider aperture width shows a

smaller grain size in the speckle as shown in figure 4.

Figure 2: Airy Diffraction spot observed in the Fourier plane. (left): figure 2 a: Spot observed on the monitor. (Right): figure 2b

Figure 4: Horizontal profile for speckles observed with varying aperture sizes. (Top left): Speckle with aperture

size = 3mm. (Top Right): with aperture size: 5 mm. (Bottom left): Aperture size: 2mm. (Bottom Right): Aperture

size: 1 mm

Fully Developed Speckle pattern in the Fourier plane

The next step in the experiment involved a precise measurement of the speckle size by measurement of

the roughness auto correlation. The auto correlation function is defined by:

Hence, the autocorrelation in the time domain represents a convolution of the function on itself or the

integral of the overlap of the function with its shifted version. Hence, in the Fourier plane, this would

represent a multiplication of the Fourier transform of the function with a Fourier transform of its

conjugate. This is achieved by the MATLAB code snippet. Figure 5 shows the result of the auto

correlation of the speckle

pattern. As expected, there is

a high intensity at the center

of the image diminishing

away from the center.

The autocorrelation function

of the speckle is shown for

the case of a weak diffuser

and a strong diffuser in Figure

6. The auto correlation width

measured from the figure for

the case of a weak diffuser

using the full width half

maximum corresponds to a

value of 112 µm which is very

close to the airy spot

diffraction limit of 106µm

calculated above.

Displacement of speckle pattern in the Fourier plane:

When the diffuser is translated, the speckle pattern does not move. This is consistent with the

theoretical explanation since the Fourier transform properties imply that a shift in the object plane will

only correspond to an addition of phase in the image plane without bringing about a displacement of

the Fourier speckle.

Figure 5: Speckle autocorrelation

Figure 6: Autocorrelation profile for two diffusers. (Top):

Profile for a strong diffuser. (Bottom): profile for a weak

diffuser

Speckle in an arbitrary plane

The speckle pattern is now observed in an

arbitrary plane and hence no longer obeys the

Fourier properties. When the doublet is

removed from the setup and the diffuser is

translated, the speckle moves with the

translation of the diffuser. This can be

qualitatively explained by the Thales theorem

which states that the ratio of displacements is

equal to the ratio of the distances to the

pinhole. Hence, a cross correlation image of

the speckle with its displaced version is shown

in figure 7. As expected, the highest intensity

is not seen this time in the center but at a

certain translation which can be used to

determine the motion of the speckle.

Speckle in the image of a scattering object:

In the next section of this experiment, a slide

was placed next to a rotating diffuser to

obtain an image of the slide with the diffuser

rotating. When the diffuser is rotated at a

greater velocity, the image acquired is clearer

since the CCD can acquire greater information

from the frames per second that it acquires,

thereby giving a much clearer image. These

images are shown in figure 8.

Conclusion

In the course of these experiments, we were able to prove that the statistical properties of the

scattering object affect the speckle pattern that is observed. The object with a greater degree of

scattering i.e. with higher roughness amplitude gives a fully developed speckle. In addition, the

roughness autocorrelation width gives a measurement for the grain size of the speckle and the smaller

the autocorrelation width, the greater is the illuminated area where the speckle is observable. The

shape of the aperture also affects the speckle grain. However, all these claims are generally justified in

the Fourier plane where the spherical wave fronts converge. The analysis in an arbitrary plane was also

presented. Therefore, we can conclude that speckle from a scattering object can be used to improve

image quality.

Figure 7: Cross correlation of images obtained for displaced diffusers in arbitrary plane

Figure 8: Image obtained with slides placed with diffusers. (left): stationary weak

diffuser. (right): rotating diffuser