Lab Report 3: Speckle Interferometry LIN PEI-YING, BAIG JOVERIA
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1 Lab Report 3: Speckle Interferometry LIN PEI-YING, BAIG JOVERIA
2 Abstract: Speckle interferometry (SI) has become a complete technique over the past couple of years and is widely used in many branches of physics. This experiment aims at understanding this concept of speckle interferometry and to illustrate the realization of an experimental setup that can visualize small deformations of a scattering object. Introduction: The speckle interferometry is an imaging technique that uses laser light to visualize and quantify the movement on the surface of an object between two states. Using this acquisition technique, its possible to monitor very finely object deformations over long periods. This technique relies on analyzing the phase of the speckle pattern by an interferometric method. The optical realization of this method can take several forms. Three of them are illustrated in Figure 1. In the first form, the laser light is used to illuminate an object and the observation is compared to a reference arm and recombined before it is fed in to the detector. During our experiment, we will be relying on this approach. Another approach is to illuminate two objects and feed the resulting beam to make an observation of a single object. Hence, by using the method described in our experiment, we are able to use the reference beam to code the phase of the speckle pattern and hence can be used in applications such as holography on a photographic film. Figure 1: Setups used for speckle interferometry Theoretical Background: The theoretical basis of speckle interferometry is based on the direct recording of a two beam interference pattern between a reference wave and a speckle wave which carried in itself the phase information of the speckle wave. An important aspect of interferometry is the smoothness of the interfering waves. Speckle waves, reflected or transmitted by rough diffusing surfaces exhibit rapid amplitude and phase fluctuations. Athough, formally the same two beam interference formula:
3 is applicable to this experiment, the resulting intensity has variable characteristics depending on the profile of the speckle. Hence, Speckle inteferometry can be defined as the set of techniques that aim at creating a two beam interference pattern involving at least one speckle wave and recording this to exploit the phase information present in the acquired data. We will now analyze the statistical and optical aspects of this technique: Statistical Aspects: Statistically, the equation described above can be used for several interpretations. The resulting intensity detected at the detector obeys different kind of statistical behavior. For a gaussian speckle field, the speckle phase is distributed uniformly over the 0 to 2π interval. In this context, it becomes relevant to introduce the first order statistics, given by the background and modulation. They are respectively defined as: This background and modulation determine the maximum and minimum and average of the possible realizations of the signal, thereby being considerably significant in practice, since they provide the best possible scenario for measurement of the interference signal. Similarly, the second order statistics such as the autocorrelation function which determines the roughness of the diffusing surface can also be a very useful quantity for optimal image acquisition. Optical Aspects: The output of a laser is divided to form a single illumination of an object surface and the other arm is used as a reference beam which are then recombined in front of the detector. The angle between the object and reference wave has to be kept as small as possible so that the resulting fringe pattern comprises of a carrier frequency that can be readily resolved by the photo detector. The phase of the speckle is modulated according to the object deformation. Figure 2 below shows how subsequent increase of the lens aperture leads to formation of smaller speckles which will also be experimentally verified in the next section. Figure 2: Effect of aperture size on speckle grain
4 Experimental Setup: The setup used for the experiment is illustrated in Figure 3 below: Figure 3: Experimental Setup for speckle interferometry The setup above comprises of a laser beam splitting in to two paths. The first path is used to illuminate the diffuser and the doublet gives the image of this object on to the CCD. The second path is used to serve as the reference beam. The laser beam is used because of its high coherence and intensity. The image is acquired by the use of an image acquisition card placed with the CCD. Experiment: 1. Alignment of the object path: According to the experimental setup, the object is a piezoelectric transducer (PZT). We want to measure its deformation when an electric field is applied on it. First, we block the reference beam and then align the illuminating path for the object. We direct aligned the illuminating beam such that the diffused light from the PZT is directed towards the pupil of the doublet. The focal length of doublet is 100 mm. The pupil is an iris diaphragm placed next to the doublet. The diameter of the diaphragm is adjustable and there is a fine scale on it. Then, we place the CCD camera and adjust it for the best position - focus. When choosing the best position for CCD, we should first make the iris diameter is open to maximum size and looking for the most clear image, that is the focal position of CCD.
5 2. Average diameter of the speckle grains: Here we show our experimentally results that the characteristic size of grains of speckle depends only on the diameter of the diaphragm. When measuring approximately the average diameter of the speckle grains in the image in units of pixels. We can see the relationship between the numerical aperture image and the characteristic size of speckle grains. The numerical aperture increases, the small grain size of speckle decreases as shown in Fig.4. Fig. 4. The diagrams show with different diameter of diaphragm from small to large. Left are CCD image and right are using MATLAB functions to see its approximate grain sizes
6 Second, we deduced from that measurement an approximate value of the diameter of the speckle grains (the sampling size, P e is defined by the size of the CCD pixels: P e We found that the measure limitation is controlled by pixel of CCD. Fig.5. Autocorrelation achieved by the MATLAB code snippet on the left column, and right side if for auto correlation profile. This diagram 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.
7 3. Study of the sampling of the speckle pattern by the pixels of the CCD detector: The calculation of the 2D Fourier transform of the illumination gives us information about the effect of that sampling. Calculate the Fourier transform at the center of the image of the speckle pattern for different diameters of the iris diaphragm. N e means number of points used to calculate the Fast Fourier Transform. P e is the size of one CCD pixel. The inverse of the whole pixels on the CCD means a pixel on the spectrum. Hence we can make a pixel on the spectrum corresponds to the following interpretation: Here, P e represent the size of one CCD pixel. According to the following pictures, it also convinced us that the Numerical aperture increases, the speckle grains become smaller, proving the theory described earlier. Fig.6. This diagram shows CCD profiles with different NA from small to large aperture from left to right, top to bottom, separately 1mm, 3mm, 10mm and 15 mm.
8 In addition, the width of this Fourier Transform is determined by the cut-off frequency of the theoretical modulation transfer function of the objective, which is equal to Here is the numerical aperture in the image space. We can regard an array pixel width corresponds to a cut-off frequency since it can be treated as a limited boarder. In most cases, attempting to resolve beyond this limit is impossible. In the following, we will acquire the speckle pattern in the image of the diffusing object. The diameter we choose for the iris diaphragm is around 1-3mm to make us increase the size of speckle grains. 4. Speckle interferometry Study of the phase of the speckle grains: First, for the interferometry, we should check the alignment of the reference beam. We set up the reference path and adjust the spatial filtering of the laser beam through the pinhole and check the position of the beam splitter cube so that the reference beam is centered on the CCD detector. Place and adjust one or more neutral density filters to maximize the contrast of the interferences between the reference beam and the object path. Each speckle grain has a phase that is approximately constant over its surface since it is coherence volume. However this phase varies randomly from grain to the next. The CCD detector, like any other optical detector, is sensitive only to the illumination, not to the phase. But through the interference with a reference beam, we have access to the phase of the speckle pattern. Reduce the diameter of the iris diaphragm in order to increase the size of the speckle grains and observe the speckle pattern in the presence of the reference beam. Adjust the equilibrium between the intensities of the two paths using the neutral density filters because when observing the clear interference fringes, we should check the intensities are almost similar. Show by changing the voltage applied to the piezoelectric ceramic that the intensity of each speckle grain varies very rapidly in the presence of the reference wave. Show that this is not the case in the absence of the reference wave. 5. Speckle interferometry Study of the object deformation: In this part, we record the fringe patterns with deformation of different PZT voltages. For a speckle interferometer, a motion or a deformation of the object s surface will introduce a change in the intensity (or phase) of the individual speckles. This change can be measured and is often visualized as interference fringes forming black lines covering the surface of the object. These lines connect points on the object s surface that were given an equal amount of deformation or rigid body motion.
9 Fig.7. These diagrams show that we apply different PZT voltage to make deformation of object s surface. The left is applying to 2V and the right is for 8V. If the deformation is too large, the fringes become too dense and may disappear. Interferometric methods are therefore well suited to measure small deformations, i.e. smaller than a laser speckle. We could say that speckle correlation and speckle interferometric methods complement each other in this respect. Also, we observed that the relation of the PZT voltage verse fringe is linearly proportional. If we apply to large voltage, the rings become more dense and concentrated. 6. Conclusion: In this laboratory work, we know that speckle interferometry systems can be used for studying the phase information of the speckles. As opposed to classical interferometry where optically smooth surfaces are studied and no speckle pattern appears, speckle interferometry uses the phase information carried by the speckles to determine the deformation of the object. In speckle interferometry optically rough surfaces are studied and therefore the interference pattern obtained when the reflected wave and the reference wave interfere will be a random speckle pattern with varying phase and amplitude. Therefore, in speckle interferometry the fringes obtained when these two interferograms are compared describe the deformation of the object. The reference wave can be either a smooth wave or a speckle pattern, as long as it is constant in time.
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