SPECKLE INTERFEROMETRY AS A TOOL FOR FINE MEASUREMENTS

Transcription

1 SPECKLE INTERFEROMETRY AS A TOOL FOR FINE MEASUREMENTS By Amany Khalil Ibrahim Helwan University B.Sc. Physics & Computer Science, Faculty of Science, Helwan University (1999) SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE AT HELWAN UNIVERSITY AIN HELWAN, CAIRO FEBRUARY 2007

2 HELWAN UNIVRSITY FACULTY OF SCIENCE DEPARTMENT OF PHYSICS APPROVAL SHEET Name: Title: Degree: Amany Khalil Ibrahim Speckle Interferometry as a tool for fine measurements Master of Science (Physics) Supervision Committee Prof. Dr. Mohamed Mohamed Mansure El-Nicklawy Professor of Physics, Faculty of Science, Helwan University Prof. Dr. Amin Fahim Hassan Professor of Physics, Faculty of Science, Helwan University Dr. Nasser Abd El Hamid Moustafa Physics Department, Faculty of Science, Helwan University Dr. Ehab Abd El Rahamn Physics Department, Faculty of Science, Helwan University Higher Studies Thesis Approval Date: / / 2006 Faculty Council Approval: / / 2006 University Council Approval: / / 2006 ii

3 To my parents, sisters, brother, husband Maher and my beautiful son Ahmed iv

4 Contents Contents List of Figures Acknowledgements Summary Abstract v vii xiii xv xvii 1 Introduction Roughness What is roughness? Use of knowledge of roughness Main methods of roughness measurement General background about speckle and its use to study surface roughness The theoretical principles of speckle pattern Speckle metrology Holographic Interferometry Speckle Interferometry Speckle pattern photography(spp) Speckle pattern correlation interferometry(spc) An example of industrial application of speckle interferometry Study The Effect of Partially Coherent Light on The Visibility of Speckle Patterns Introduction Coherence Visibility v

5 2.4 The theoretical model Gaussian spectral line profile Lorentzian spectral line profile Results and discussion Experimental verification of theoretical model Conclusion Study The Effect of Some Parameters on The Visibility of Speckle Interferometry Introduction Theoretical principle of the double exposure speckle photography technique Experimental Setup Effect of displacement value on the visibility Effect of the surface roughness on the visibility Effect of the polarized light on the visibility Effect of the speckle size on the visibility Effect of the speckle type on the visibility Conclusion References 91 Arabic Summary 103 vi

6 List of Figures 1.1 Different forms of random roughness; (a) is a two-dimensionally rough surface, (b) has higher roughness when moving in x-direction Synthetic-aperture radar image of an urban area, dominated by speckle (courtesy of Howard Zebker, Stanford University) Ultrasound image (2.5 MHz) of the human liver showing speckle (courtesy of Graham Sommer, Stanford University) Light scattering from a rough surface Photograph of a speckle pattern The probability density function P I (I) as a function of the intensity I Objective speckle formation The relative number of fringes versus frequency Subjective speckle formations a) Schematic description of how speckles appear in a detector, b) a typical speckle pattern, and c) a random walk in the complex plane Interference fringes describing an in-plane rotation The Michelson arrangement of out-of-plane displacement sensitive speckle pattern correlation interferometry Studies of the lateral displacements of an object A Observation of the spectrum of negative H Example of an E.M. wave with a coherence time of approximately τ Construction of the theoretical model vii

7 2.3 The visibility of speckle patterns versus the spectral half width ν with spectral line shape as a parameter. The results are obtained considering the grain density to be = 10 6 in cm 2, the grain height to be 50µm The visibility of the speckle patterns versus the grain width for the case of Gaussian profile. The results are obtained for a grain height of 10µm and spectral half width of Hz The visibility of the speckle patterns versus the grain width for the case of Lorentzian profile. The results are obtained for a grain height of 10µm and spectral half width of Hz The visibility of the speckle patterns versus the grain height calculated for the case of Gaussian profile of δν = Hz. The results are obtained for a grain density of 10 6 in cm The visibility of the speckle patterns versus the grain height calculated for the case of Gaussian profile of δν = Hz. The results are obtained for a grain density of 10 6 in cm The visibility of the speckle patterns versus the grain height calculated for the case of Lorentzian profile of δν = Hz. The results are obtained for a grain density of 10 6 in cm The visibility of the speckle patterns versus the grain height calculated for the case of Lorentzian profile of δν = Hz. The results are obtained for a grain density of 10 6 in cm The experimental setup The obtained speckle patterns from partially coherent light The relation between speckle patterns contrast and surface roughness The experimental setup: C a condenser, L a convex lens, D a diffraction grating, RP a rectangular aperture, O The object The relation between speckle patterns contrast and surface roughness ν = Hz corresponding to L c = µm viii

8 2.15 The relation between speckle patterns contrast and surface roughness ν = Hz corresponding to L c = µm The relation between speckle patterns contrast and surface roughness at ν = Hz corresponding to L c = µm The relation between speckle patterns contrast and surface roughness at ν = Hz corresponding to L c = µm The relation between speckle patterns contrast and surface roughness with ν = , , and Hz as a parameter Effect of fringe visibility on relative errors in displacements for different number of fringes as a parameter The experimental setup: BE beam expander, L convex lens, O the object Spackle pattern before displacement Spackle pattern after displacement = 50µm The addition of two speckle patterns before and after displacement Young s fringes for a displacement = 50µm The plotting profile Young s fringes for a displacement = 150µm Young s fringes for a displacement = 250µm Young s fringes for a displacement = 350µm Young s fringes for a displacement = 500µm The relationship between the fringes visibility and displacement value for surface roughness 9.6µm Young s fringes of displacing an object with rms = 2.48µm a distance 50µm Young s fringes of displacing an object with rms = 6.3µm a distance 50µm ix

9 3.15 The relation between the visibility and displacement with different surface roughness as a parameter The visibility versus surface roughness for a constant displacement 50µm The experimental setup with polarizer P before the diffuser The relation between the visibility and displacement for surface of rms roughness = 9.6µm obtained by using polarized and unpolarized light The relation between the visibility and surface roughness for a 50µm displacement by using polarized and unpolarized light The Young s fringes for speckle size = 9.2µm obtained for a displacement of 100µm, z = 16.7cm, D = 1.4cm and surface roughness 9.6µm The Young s fringes for speckle size = 36µm obtained for a displacement of 100µm, z = 65.5cm,D = 1.4cm and surface roughness 9.6µm The Young s fringes for speckle size = µm obtained for a displacement of 100µm, z = 26.7cm, D = 2cm and surface roughness 9.6µm The Young s fringes for speckle size = µm obtained for a displacement of 100µm, z = 26.7cm,D = 1cm and surface roughness 9.6µm The relation between the speckle size and visibility at constant D = 1.4cm and different z values for surface roughness 9.6µm The relation between the speckle size and visibility at constant z = 26.7cm and different D values for surface roughness 9.6µm The experimental setup to obtain objective speckle patterns The experimental setup to obtain subjective speckle patterns, IS: imaging system (convex lens + aperture), u is the object distance and v is the image distance Young s fringes for displacement 100µm obtained from objective speckle patterns and speckle size =17.17µm Young s fringes for displacement 100µm obtained from subjective speckle patterns, speckle size =17.17µm and m = x

10 3.30 Young s fringes for displacement 100µm obtained from objective speckle patterns and speckle size =24.256µm Young s fringes for displacement 100µm obtained from subjective speckle patterns speckle size =24.256µm and m = Comparison between the two types (objective and subjective speckle patterns) for different displacements at m = Comparison between the two types (objective and subjective speckle patterns) for different displacements at m = xi

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12 Acknowledgements I kneel humbly to ALLA thanking HIM for showing me the right path, without HIS help my efforts would have gone astray. My sincere thanks are detected to Prof. Dr. M. M. El-Nicklawy, Professor of Physics, Faculty of science, Helwan University for his supervision, advice, encouragement and persistent motivation to complete this work. I am so grateful to Prof. Dr. A. F. Hassan, Professor of Physics, Faculty of Science, Helwan University for his supervision, continuous help, great assistance, discussion during this work and his fatherly attitude. I would like to express my sincere thanks to Dr. Nasser Abd El Hamid Moustafa Physics Department, Faculty of Science, Helwan University Dr. Ehab Abd El Rahamn Physics Department, Faculty of Science, Helwan University For continuous advices, encouragement and good guidance. I would like to extend my gratitude to Physics Department staff for their kind support, especially Mr. A. El Mahdy and Mr. H. Hashem. xiii

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14 Summary This thesis presents a theoretical and experimental study on the effects of some parameters influencing the speckle interferometry as a tool for fine distance measurements. It consists of three chapters. Chapter 1 contains an introduction to the basic information about surface roughness, speckle effect and speckle interferometry, so for instance the definitions of surface roughness, and a brief discussion of its different measurements methods and its importance are introduced. The speckle phenomena, definition and characterization of its types are also discussed in details. After that different speckle interferometry methods, speckle pattern photography and speckle pattern correlation interferometry, are defined and discussed. Finally an example of industrial application of speckle interferometry is introduced. Chapter 2 is devoted to study theoretical, the effects of some parameters on visibility of speckle pattern. For this propose, a theoretical model for a periodic rough surface was conssidered. Using this theoretical model,the effects of grain height,its density, the band width and spectral distribution of the line profile (Gaussian and Lorentzian) illuminating a rough surface on the visibility of speckle pattern are investigated. An experimental setup was constructed to study the effect of surface roughness and coherence of the illuminating light beam on the contrast of speckle pattern. The general behavior of the experimental results, which agree with published data, are compatible with the theoretical model. Chapter 3 includes experimental studies of some parameters affecting the visibility of the Young s fringes obtained from double exposure with lateral displacement xv

15 xvi technique. These parameters are the value of the displacement, the surface roughness of the object, the state of polarization of used light, speckle size and the type of speckle. Experimental results show that to obtain great visibility values some conditions must be satisfied. These conditions are: objective speckle pattern setup, polarized light source, smaller displacement value, object with high surface roughness ( ranging between 1.1µm to 9.6µm) and finally, the speckle size must be less than the displacement value.

16 Abstract A double-exposure speckle photographic technique has been widely used in various fields of optical metrology, for example the displacement of a diffusing object. Therefore theoretical and experimental effects of some parameters affecting this technique as a tool for fine distance measurements are studied. Following this aim a theoretical model for periodic rough surface was constructed. This model allows the effect of grain height, its density, variable band width and spectral distribution of the illuminating radiation on the speckle pattern contrast to be studied. An experimental setup was constructed to study the effects of surface roughness and coherence on the contrast of speckle pattern. The general behavior of experimental results, which agree with published data, are compatible with the result of the theoretical model. Finally an experimental work was carried out to study the effects of the value of displacement, the surface roughness of the object, the state of polarization of the used light, speckle size and the type of speckle on the visibility. The experimental data show that using objective speckle pattern, high coherence polarized light source, high surface roughness ( ranging between 1.1µm to 9.6µm) and speckle size that is smaller than the displacement value give high visibility and consequently a good accuracy of displacement determination. Key words: Surface roughness, speckle, double-exposure speckle photographic technique. xvii

17 Chapter 1 Introduction 1.1 Roughness What is roughness? Roughness is some kind of description of the deviation of the surface heights from the mean height value [89]. There are several different parameters for describing surface quality. These parameters are derived specifically from profile measurements and therefore do not include any of the optical parameters available. Which parameter one uses is dependent upon the surface characteristic of most interest or importance in the given application. For instance [13], the average roughness denoted by R a is the average of the modulus of the surface heights relative to a mean surface height of zero. It is mostly used in the industry as a measure of surface finishing. Rms (root mean squar) roughness denoted by R q or σ is the rms value of the surface heights relative to a mean surface height of zero. It is used more in theoretical analysis because it is the standard deviation of surface heights. Other measures are related to such properties as the behaviour of the optical appearance of the surface under mechanical loads [68]. Roughness has been traditionally measured using a stylus whose output is directly related to the surface height. The statistics used for description of roughness can then 1

18 2 be determined by analysing the trace produced by the stylus. This implies that the surface under consideration is as rough in every direction (in a two-dimensional plane) as that measured, and in general this is not the case. A surface can have different roughnesses in different directions. For example, surfaces produced by the grinding process have very little roughness in one direction, but at ninety degrees to that direction the surface can be very rough by comparison. Figure (1.1) highlights this situation, surface (b) being typical of a grinding-produced surface. Figure 1.1: Different forms of random roughness; (a) is a two-dimensionally rough surface, (b) has higher roughness when moving in x-direction. Since the standard method of roughness measurement is the stylus profilometer, we are left with the understanding of roughness as being a purely one dimensional phenomenon, although in reality, it is not. It is therefore necessary to take this one-dimensionality into account when analysing roughness results and their meaning. Roughness is also a microtopographic feature and does not take into account the larger scale surface height variations known as waviness. It is a local phenomenon and loses meaning on scales larger than a few millimetres. This is the reason stylus profilometer measurements are usually taken over the order of one millimetre, such short distances minimising the effect of waviness. So, roughness is not an obvious property-as one would first imagine-but is intimately related to how the surface of

19 3 interest was measured and the measurement process Use of knowledge of roughness Knowledge of how rough something is tends to be useful usually in quality control of a manufacutred product. Production in businesses such as the paper industry, steel, aluminium milling, and engineering workshops needs to be of a certain standard. So there is a necessity to quantify the surface texture characteristics, and to determine what is of good standard and what is sub-standard. This is where surface roughness measurement is necessary and useful, forming an integral part of the overall production process. Surface roughness measurement is not only of use in industry, but also in other fields. Scientific equipment, such as that used in optics research, often requires certain surface characteristics. For instance, mirrors must be of a high quality since surface defects degrade optical signals and this is of concern in research. Overall, knowledge of the roughness of surfaces is only an indication of the desired surface qualities or optical appearance Main methods of roughness measurement There are two general categories for methods of surface roughness measurement; mechanical and optical [43, 78, 85]. The mechanical methods are usually variations on the stylus technique already briefly mentioned. The optical methods [77] are, for the most part, based on the scattering of light. The optical phenomena utilized include: reflectance in the specular direction, the total intensity of scattered light, the diffuseness of the angular scattering pattern, the speckle contrast, and the polarization [68]. Each phenomenon depends in some way on the surface roughness, forming the basis for surface inspection instruments.

20 4 1.2 General background about speckle and its use to study surface roughness Interference phenomena in scattered light, such as Fraunhofer s diffraction rings and Quételet s fringes, were of interest in the late nineteenth century. The first recorded laboratory observation of the speckle phenomenon appears to have been that by Exner [50] who studied Fraunhofer s diffraction rings, which are formed when coherent light is diffracted by randomly distributed particles of equal size. He sketched the radially fibrous structure of the light pattern produced by candle light, transmitted through a glass plate on which he had breathed. The superposition of a very large number of waves with random phase led to strong local fluctuations in intensity. The radial nature of the pattern caused by the non-monochromatic light source was replaced by a fine granular structure employing a red filter, creating a light source with lower spectral bandwidth [67]. In 1914 von Laue published a photograph of Fraunhofer s diffraction rings produced by light from a carbon arc lamp illuminating a glass plate covered with lycopodium powder [50]. The photograph shows the radially fibrous structure noted by Exner, and this feature was extensively discussed and reviewed in literature [33, 67]. With the invention of the laser in 1960, highly coherent light became available, and scientists began to study the phenomenon of speckle, and practical applications began to be reported in the literature. The surfaces of most materials are optically rough, i.e. the surface height variation is greater than one fourth of the wavelength of illuminating light. When light with a fair degree of spatial and temporal coherence is reflected from an optically rough surface, the light is scattered in all directions. The reflected waves created by different microscopic elements of the surface interfere and

21 5 produce random fluctuations in intensity with dark and bright spots. This intensity distribution has a characteristic granular structure and is called speckle pattern[33]. Speckle was also observed when laser light was transmitted through stationary diffusers, for the basic reason: the optical paths of different light rays passing through the transmissive object varied significantly in length on a scale of a wavelength. The speckle phenomenon thus appears frequently in optics; it is in fact the rule rather than the exception.for the same reason, speckle plays an important role in other fields where radiation is transmitted by or reflected from objects that are rough on the scale of a wavelength. Important cases include synthetic aperture radar imagery in the microwave region of the spectrum, and ultrasound medical imagery of organs in the human body. Figure (1.2) shows an image from a synthetic-aperture radar in which the speckle is quite visible, and figure (1.3) shows an ultrasound image of the human liver in which speckle is also evident. The speckle patterns contain information on surface characteristics, e.g. on surface roughness. They are random and can be described in statistical terms. Therefore speckles have the potential to be used for surface roughness measurements. Exact analogs of the speckle phenomenon appear in many other fields and applications[42]. For example, the squared magnitude of the finite-time Fourier transform of a sampled function of almost any random process shows fluctuations in the frequency domain that have the same first-order statistics as speckle. Anyone who has seen a speckle pattern in reality will have observed the curious granular appearance[50]. The speckle pattern moves if the head moves, and it seems impossible to focus the eye properly on the illuminated surface. If the speckle pattern is observed through a small hole, and if this hole is smaller than the pupil, the speckles

22 6 Figure 1.2: Synthetic-aperture radar image of an urban area, dominated by speckle (courtesy of Howard Zebker, Stanford University). Figure 1.3: Ultrasound image (2.5 MHz) of the human liver showing speckle (courtesy of Graham Sommer, Stanford University).

23 7 appear larger. Speckles observed on an imaging system, e.g. with the eye or a camera, are called image speckles or subjective speckles [34]. For speckle patterns collected on a screen, the speckle size depends on the area of the surface that is illuminated. The smaller this area, the larger are the speckles. This type of speckles is called far-field speckles or objective speckles. The surface roughness is obtained from the speckle by studying the correlation between two speckle patterns obtained from the surface under consideration, either by changing the orientation of the laser beam or by changing the wavelength of the laser beam [53]. Other techniques that relate the speckle contrast to the surface roughness and to the spatial or temporal coherence of the source have been proposed [89]. 1.3 The theoretical principles of speckle pattern In figure (1.4), light is incident on, and scattered from, a rough surface of height variations greater than the wavelength λ of the light. As is shown from the figure, light is scattered in all directions [44]. These scattered waves interfere and form an interference pattern consisting of dark and bright spots or speckles which are randomly distributed in space. In white light illumination, this effect is scarcely observable owing to lack of spatial and temporal coherence. Applying laser light, however, gives the scattered light a characteristic granular appearance as shown in the image of a speckle pattern in figure (1.5). It is easily realized that the light field at a specific point in a speckle pattern must be the sum of a large number N of components representing the light from all points on the scattering surface. The complex amplitude at point in a speckle pattern can therefore be written as u = 1 N u k = 1 N U k e iφ k (1.3.1) N N k=1 k=1

24 8 By assuming that: (1) the amplitude and phase of each component are statistically independent, and (2) the phases φ k are uniformly distributed over all values between π and +π, [44] has shown that the complex amplitude u will obey Gaussian statistics. Figure 1.4: Light scattering from a rough surface Further he has shown that the probability density function P I for the intensity in a speckle pattern is given as P I (I) = 1 ( I exp I ) I (1.3.2) where I is the mean intensity. The intensity of a speckle pattern thus obeys negative exponential statistics. Figure (1.6) shows a plot of P I (I). From this plot we see that the most probable intensity value is zero, that is, black. A measure of the contrast in a speckle pattern is the ratio C = σ I / I, where σ I is the standard deviation of the intensity given by σ 2 I = I 2 = (I I ) 2 = I 2 2 I I + I 2 = I 2 I 2 (1.3.3)

25 9 Figure 1.5: Photograph of a speckle pattern Figure 1.6: The probability density function P I (I) as a function of the intensity I

26 10 where the brackets denote mean values. By using I 2 = 0 P I (I)I 2 di = 2 I 2 (1.3.4) we find the contrast C in a speckle pattern to be unity. From figure (1.5) we see that the size of the bright and dark spots varies. To find a representative value of the speckle size, consider figure (1.7), where a rough surface is illuminated by laser light over an area having circular cross-section of diameter D [44, 75]. The resulting so-called objective speckle pattern is observed on a screen S at a distance z from the scattering surface. For simplicity, we consider only the y-dependence of the intensity. An arbitrary point P on the screen will receive light contributions from all points on the scattering surface. Figure 1.7: Objective speckle formation Let us assume that the speckle pattern at P is a superposition of the fringe patterns formed by light scattered from all point pairs on the surface. Any two points separated by a distance l will give rise to fringes of fundamental spatial frequency ν = l/(λz). The fringes of highest fundamental spatial frequency ν max will be formed by the two

27 11 edge points, for which ν max = D λz (1.3.5) The period of this pattern is a measure of the smallest objective speckle size σ o which therefore is σ o = 1.22λz D (1.3.6) For smaller separations l, there will be a large number of point pairs giving rise to fringes of the corresponding spatial frequency. The number of point pairs separated by l is proportional to (D l). Since the various fringe patterns have random individual phases they will add incoherently. The contribution of each frequency to the total intensity will therefore be proportional to the corresponding number of pairs of scattering points. Since this number is proportional to (D l), which in turn is proportional to (ν max ν), the relative number of fringes versus frequency, i.e. the spatial frequency spectrum will be linear, as shown in figure (1.8). Figure (1.9) shows the same situation as in figure (1.7) except that the scattering surface now is imaged on to a screen by means of a lens L. The calculation of the size of the resulting so-called subjective speckles is analogous to the calculation of the objective speckle size. Here the cross-section of the illuminated area has to be exchanged by the diameter of the imaging lens. The subjective speckle size σ s therefore is given as σ s = 2.4λb D (1.3.7) where b is the image distance and D is the diameter of the lens. By introducing the aperture number F = f D (1.3.8)

28 12 Figure 1.8: The relative number of fringes versus frequency. Figure 1.9: Subjective speckle formations

29 13 where f is the focal length, we get σ s = (1 + m)λf (1.3.9) where m = (b f)/f is the magnification of the imaging system. From this equation we see that the speckle size increases with decreasing the aperture (increasing aperture number). This can be easily verified by stopping down the eye aperture when looking at a speckle pattern. Speckle formation in imaging cannot be explained by means of geometrical optics which predicts that a point in the object is imaged to a point in the image. The field at a point in the image plane therefore should receive contributions only from the conjugate object point, thus preventing the interference with light from other points on the object surface. However, even an ideal lens will not image a point into a point but merely form an intensity distribution around the geometrical image point due to diffraction of the lens aperture. This is indicated in figure (1.9). It is therefore possible for contributions from various points on the object to interfere so as to form a speckle pattern in the image plane. 1.4 Speckle metrology Speckle metrology is an important and growing part of optical measuring techniques in experimental mechanics [3]. If the object is imaged, each point P on the detector will gain contribution of light coming from a coherence volume, determined by at least the Airy spot and the roughness of the surface. A summation of these wave packages that illuminate point P is illustrated in figure (1.10c), describing a so-called random walk. As long as the complex amplitude A follows the statistics indicated in the figure, the speckle field

30 14 is fully developed. The observed speckle pattern could be thought of as a fingerprint of the illuminated area in the sense that the observed pattern is unique for the microstructure of the specific surface area. Another area will give rise to a totally different random speckle pattern. Figure 1.10: a) Schematic description of how speckles appear in a detector, b) a typical speckle pattern, and c) a random walk in the complex plane. When the surface area is moved or deformed, the observed speckles in the image plane will also move accordingly. This is the reason why speckle correlation techniques are so good for determining in-plane motions of an object. The size of the speckles will affect how sensitive the system is to decorrelate due to large movements of the object. The most important statistical characteristic of laser speckle is its size. The speckle size gives the magnitude of the detector plane areas that can be expected to have coherent properties. The fact that the speckle pattern is coherent within these limits is necessary for speckle interferometry to work. The speckle size along the propagation direction is found by [3]: σ z = 8λf 2, where f is the effective F number of the imaging system,meaning that the speckle size is much larger in the z-direction than in the x,y-directions and has the shape

31 15 of a cigar. 1.5 Holographic Interferometry In holographic interferometry, invented by Stetson and Powell in 1965, two holographic recordings were made on the same photographic plate, one obtained before object deformation and one after [3]. In reconstructing the hologram, the difference between the object fields is seen as interference fringes covering the object. Drawbacks of the technique were the time-consuming for the wet processing of the photographic plates and due to the interferograms being sometimes difficult to interpret, specialists were often needed. In the 1970s, electronic recording media started to replace photographic plates; today, the CCD-detector is in common use [30]. The spatial resolution of electronic recording media was, however, not nearly as high as for holographic plates. Electronic techniques were initially named Electronic Speckle Pattern Interferometry (ESPI) [61], which was an analogue technique. Reconstruction of the object fields is now made numerically in the computer and displayed on a TV monitor, meaning that optical reconstruction is not longer necessary. This way, both the recording and processing of the measured data could all be done digital and the technique was also named: Digital Speckle Pattern Interferometry (DSPI), TV holography [86] and simply speckle interferometry (SI) [29]. 1.6 Speckle Interferometry Speckle interferometry systems study 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

32 16 by the speckles to determine the deformation or (displacement) of the object. Classical interferometry obtain the shape of a polished part by comparing the deformed reflected wave with a plane reference wave, no speckles are present. The difference between the waves will give rise to interference fringes describing the shape of the object, i.e. only one interferogram is needed. 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 a second interferogram after the object has deformed or (displaced) in some way is captured. The fringes obtained when these two interferograms are compared describe the deformation of the object. The reference wave can be either a smooth wave (as in out-of-plane setups) or a speckle pattern (used in shearography and in-plane set-ups), as long as it is constant in time. For a speckle interferometer, a motion or a deformation of the objects surface will introduce a change in the intensity (or phase) of the individual speckles. This change can be measured and is often visualised as interference fringes figure (1.11) forming black lines covering the surface of the object. These lines connect points on the objects surface that were given an equal amount of deformation or rigid body motion, according to certain rules concerning illumination and observation directions. If the deformation or (displacement) is too large, the fringes become too dense and may vanish due to speckle decorrelation. Two main techniques are grouped within the general classification of speckle pattern interferometry[59, 71].These are: a. speckle pattern photography; and b. speckle pattern correlation interferometry.

33 Figure 1.11: Interference fringes describing an in-plane rotation 17

34 18 In both of these a fringe pattern is derived from an optically rough surface observed in its original and displaced positions. Two major factors which distinguish speckle pattern photography from speckle correlation techniques are the difference in displacement magnitude sensitivity and the absence of a reference beam at the recording stage Speckle pattern photography(spp) Speckle pattern photography(spp), also called electronic speckle photography (ESP), digital speckle photography (DSP), is a technique used to determine displacement fields [3]. Before computers were used to record images, photographic plates were exposed by two speckle fields (one before and one after the object was displaced). The obtained specklegram is illuminated by a narrow laser beam a diffraction halo modulated with Youngs fringes appear. The frequency of these fringes carries information about the displacement between the two exposures. The first computer based speckle photography system was presented in the early 1980s by Peters and Ranson.The images are now recorded by a CCD-detector and are stored on separate frames in the computer. Since images before and after displacing are stored on different frames, it is possible to use beside convolution integration and cross-correlation, the Fourier transform algorithm to determine Youngs fringes. The result is obtained much faster since no film processing or reconstruction with laser light is needed. Discussions on the subject of speckle photography can be found, for example, in Burch and Tokarski (1968), Dainty (1975), Erf (1978), Fourney (1978), Hung (1978) and Jones and Wykes (1989).

35 Speckle pattern correlation interferometry(spc) Speckle pattern correlation interferometry, was described initially by Leendertz and indeed it was the need to overcome some of the inherent problems of holographic interferometry [71]. Groh had used the relocated negative of an image-plane speckle pattern as a shadow mask as a means of detecting fatigue cracks. At about the same time Maron independently performed a similar experiment with the introduction of an off-axis reference beam. In both cases the overall change in the correlation of the live and recorded speckle patterns was recorded during the development of a fatigue crack. Leendertz showed that the individual intensities of image-plane speckle could be made to vary cyclically for a given direction of object motion if they interfered with a reference beam of specific geometry. Such a pattern is obtained by correlating the image-plane intensity distribution of the surface in its displaced and undisplaced positions. When this is done the directional sensitivity of the fringes is a function of the reference-beam geometry, which may be arranged to give out-of-plane or in-plane displacement sensitivity or out-of-plane displacement gradient sensitivity. In early experiments correlation fringes were observed by the superposition of a negative of the undisplaced speckle pattern upon a positive of the displaced-state speckle pattern. The principle of speckle pattern correlation fringe formation Consider the interferometer shown in figure (1.12) [71]. A plane wavefront U 0 is split into two components of equal intensity by the beamspliter B. These illuminate the optically rough surfaces D 1 and D 2.These wavefronts scattered from D 1 and D 2

36 20 interfere on recombination at B and are recorded in the image plane of the lensaperture combination L. The intensity distribution in that plane will consist of the interference pattern formed between the image-plane speckle patterns of D 1 and D 2 as seen in the dashed position. Let U 1 = u 1 expiψ 1 and U 2 = u 2 expiψ 2 be the complex amplitudes of these wavefronts where u 1,u 2 and ψ 1,ψ 2 correspond respectively to the randomly varying amplitude and phase of the individual image plane speckles. The intensity of a given point in the image plane will be I 1 where I 1 = I 1 + I I 1 I 2 cos Ψ (1.6.1) and I 1 = U 1 U 1 I 2 = U 2 U 2 Ψ = ψ 1 ψ 2 When D 1 is displased a distance d 1 parallel to the surface-normal, the resultant phase change is given by φ(d 1 ) = 4πd 1 /λ (1.6.2) This will change the intensity at the point to I 2 where I 2 = I 1 + I I 1 I 2 cos(ψ + φ(d 1 )) (1.6.3) the correlation coefficient, ρ(i 1, I 2 ) of I 1, I 2 ρ(i 1, I 2 ) = I 1 I 2 I 1 I 2 ( I 2 1 I 1 2 ) 1 2( I 2 2 I 2 2 ) 1 2 is calculated, and it is shown that when

37 21 φ = 2nπ (1.6.4) I 1 and I 2 have maximum correlation when I 1 = I 2. The correlation coefficient is zero, i.e. I 1 and I 2 become uncorrelated when φ = (2n + 1)π (1.6.5) Thus, using equation (1.6.2), the maximum correlation occurs along lines where d 1 = 1 nλ (1.6.6) 2 and minimum correlation exists where d 1 = 1 2 (n + 1 )λ (1.6.7) 2 so that the variation in correlation represents the variation in d 1, the normal displacement of the object surface.

38 22 Figure 1.12: The Michelson arrangement of out-of-plane displacement sensitive speckle pattern correlation interferometry

39 An example of industrial application of speckle interferometry Because of the simplicity of techniques founded on laser speckle, numerous applications are possible [53]. One example is the study of strains in joints of precast reinforced concrete elements. These elements, such as large panels,boards, and supporting walls, are widely used in the construction of buildings. The assembly of these elements, together with the joints that make up the structures, plays an important role in the rigidity of the structures. Displacement measurements made by conventional means do not yield sufficient information about the global behavior of the joints. Speckle interferometry leads to a satisfactory solution. The principle of the technique is that described in figure (1.16). The experiment is carried out on real joints 2-m long, and not on models. Because of the large surface to be illuminated, a powerful laser must be used. The laser beam goes through anamorphic optics consisting of two cylindrical lenses that spread out the beam only on the surface of the joint. Two exposures are made on the same photographic plate, the first with an unloaded join and the second with the load added. After development, the negative is observed with the apparatus shown in figure (1.17). For measurement purposes, it is preferable to study the negative point by point. Displacements between 0.04mm and a few millimeters are easily done. The relative displacements of various parts of the structure under study may thus be determined. The uniformity of the distribution of the load may also be verified.

40 24 Figure 1.13: Studies of the lateral displacements of an object A Figure 1.14: Observation of the spectrum of negative H

41 Chapter 2 Study The Effect of Partially Coherent Light on The Visibility of Speckle Patterns 2.1 Introduction It is evident that speckle patterns appear in the diffraction and image planes when a diffuse object is illuminated by more or less coherent light. This chapter study the effect of partially coherent light on visibility of speckle patterns. To study this effect, a theoretical model for a periodic rough surface was constructed. An equation for the visibility of speckle patterns is investigated by using partially coherent light. 2.2 Coherence For any electromagnetic (E.M) wave, one can introduce two concepts of coherence, namely spatial and temporal coherence [62]. To define spatial coherence, let us consider two points P 1 and P 2 which, at time t = 0, lie on the same wavefront of some given E.M. wave and let E 1 (t) and E 2 (t) be the corresponding electric fields at these points. By definition, the difference between the 25

42 26 phases of the two fields at time t = 0 is zero. Now, if this difference remains zero at any time t > 0, we will say that there is a perfect coherence between the two points. If this occurs for any two points of the E.M. wavefront, we will say that the wave has perfect spatial coherence. In practice, for any point P 1, the point P 2 must lie within some finite area around P 1 if we want to have a good phase correlation. In this case we will say that the wave has a partial spatial coherence and, for any point P, we can introduce a suitably defined coherence area S c (P). To define temporal coherence, we now consider the electric field of the E.M. wave at a given point P, at times t and t+τ. If, for a given time delay τ, the phase difference between the two field values remains the same for any time t, we will say that there is temporal coherence over a time τ. If this occurs for any value of τ, the E.M. wave will be said to have perfect time coherence. If this occurs for a time delay τ such that 0 < τ < τ 0, the wave will be said to have partial temporal coherence, with a coherence time equal to τ 0. An example of an E.M. wave with a coherence time equal to τ 0 is shown in Figure (2.1). This shows a sinusoidal electric field undergoing phase jumps at time intervals equal to τ 0. We see that the concept of temporal coherence is directly connected with that of monochromaticity. We will in fact show, although this is already obvious from the example shown in Figure (2.1), that an E.M. wave with a coherence time τ 0 has a bandwidth ν 1/τ 0.

43 27 Figure 2.1: Example of an E.M. wave with a coherence time of approximately τ Visibility The definition of visibility of interference pattern [44] V = I max I min I max + I min (2.3.1) where I max and I min are two neighboring maxima and minima of the interference pattern described by the following equation I = I 1 + I I 1 I 2 γ(τ) cos( φ) (2.3.2) γ(τ) is the degree of coherence of the light beam. Since cos( φ) varies between +1 and -1 we have which, when substituted into equation (2.3.1), gives I max = I 1 + I I 1 I 2 γ(τ) (2.3.3) I min = I 1 + I 2 2 I 1 I 2 γ(τ) (2.3.4) V = 2 I 1 I 2 γ(τ) I 1 + I 2 (2.3.5)

44 28 For two waves of equal intensity, I 1 = I 2, equation (2.3.5) becomes V = γ(τ) (2.3.6) which shows that in this case γ(τ) is exactly equal to the visibility. γ(τ) is termed the complex degree of coherence and is a measure of the ability of the two wave fields to interfere. 2.4 The theoretical model Figure 2.2: Construction of the theoretical model

45 29 Suppose a grain of dimension (a, b) is located on the surface of the transparent diffuser figure (2.2), where its center having the coordinates (x i, y i, 0). The wave function scattered from an element of dimensions (dx,dy, 0) located at position (x,y, 0) and illuminating a point P of coordinates (X,Y, Z) on a screen placed a distance Z apart from the diffuser is given by Ψ = a o r ei(wt kr) dxdy (2.4.1) where a o /r is the scattered amplitude per unit area from the diffuser, r 2 = (X x) 2 + (Y y) 2 + Z 2 = X 2 + Y 2 + Z 2 2Xx 2Y y + x 2 + y 2 (2.4.2) Let the grain under study is of dimensions a, b and position (x i,y i, 0) R 2 = (X x i ) 2 + (Y y i ) 2 + Z 2 = X 2 + Y 2 + Z 2 2Xx i 2Y y i + x 2 i + y 2 i (2.4.3) For X x and x i and Y y and y i we get: or r 2 = R 2 2X(x x i ) 2Y (y y i ) (2.4.4) [ r = R 1 2X(x x i) 2Y (y y ] 1/2 i) (2.4.5) R 2 R 2 Using the Bernwlli inequality, one gets (1 + ζ) n 1 + nζ 1! for ζ 1. Thus r can be written as [ r R 1 X(x x i) Y (y y ] i) R 2 R 2 r R X(x x i) R Y (y y i) R (2.4.6) (2.4.7)

46 30 The integration of the wave amplitude given by equation (2.7), over the entire area of a grain possioned on (x i,y i, 0) gives the resultant wave amplitude emitted from it and reaching point P on the screen as A g1 e iφ i = xi + a yi + b 2 2 x i a y 2 i b 2 a o r ei(ωt kr) dxdy (2.4.8) A g1 e iφ i = a xi + a yi + b 2 2 o R ei(ωt kr) e i kx(x x i ) R e i ky (y y i ) R dxdy (2.4.9) x i a y 2 i b 2 A g1 e iφ i = a o R ei(ωt kr) [e i kx(x x i ) R i kx R A g1 e iφ i = a oab R ei(ωt kr) ] xi + a o 2 x i a o 2 [ sin( kxa 2R ) kxa 2R [e i ky (y y i ) R i ky R ] yi + a o 2 ][ sin( ky b 2R ) ky b 2R y i a o 2 (2.4.10) ] (2.4.11) Let now another type of grain of thickness δz w.r.t the first one be present on the diffuser, the wave function emitted from this grain will have a phase difference δφ, with respect to the first one given by: δφ = kδr(µ 1) = kzδz(µ 1) R Where µ is the refracted index of used diffuser. Since R δr the amplitude is considered to be constant. Thus the wave amplitude reaching the point P from the second grain type will be given by: A g2 e iφ i = a oab R ei(ωt kr δφ) with R 2 = X 2 + Y 2 + Z 2 2Xx i 2Y y i = R 2 o [ sin( kxa 2R ) kxa 2R [ 1 2Xx i R 2 o ] [ sin( ky b 2R ) ky b 2R Y y i R 2 o ] ] (2.4.12) where R 2 o = X 2 + Y 2 + Z 2. Using the binomial expansion theorem (1 + x) n = 1 + nx 1! + n(n 1) 2! x one gets after excluding the higher terms R R o Xx i R o Y y i R o.

47 31 Let now an N N number of grains of the first type per unit area be resident on the diffuser, thus the resultant wave amplitude reaching the point P on the screen will be given by: A g1 e (iφ 1) = a oab R e i k a Ro e i k b Ro [ sin( kxa 2 X N l=1 N 2 Y l=1 2R ) kxa 2R ][ sin( ky b 2R ) ky b 2R e i k Ro 2a(l 1)X ] e i(ωt kro) e i k Ro 2b(l 1)Y (2.4.13) A g1 e (iφ 1) = a oab R [ sin( kxa 2R ) kxa 2R ][ sin( ky b 2R ) ky b 2R ] e i(ωt kro) e i k Ro ( ax 2 + by 2 ) [1 + e iδ 1 + e 2iδ e i(n 1)δ 1 ] [1 + e iδ 2 + e 2iδ e i(n 1)δ 2 ] (2.4.14) where δ 1 = k R o (2a)X, and δ 2 = k R o (2b)Y A g1 e (iφ 1) = a oab R [ sin( kxa 2R ) kxa 2R e i k Ro ( ax+by 2 )1 e inδ 1 1 e iδ 1 ][ sin( ky b 2R ) ky b 2R ] 1 einδ 2 1 e iδ 2 e i(ωt kro) (2.4.15) Similarly for the second type of the grains of number N N per unit area, the resultant wave reaching the point P will be given by: [ ][ A g2 e (iφ 2) = a oab sin( kxa sin( ky b R 2R ) kxa 2R 2R ) ky b 2R e iδφ e i k Ro (3(aX+bY ) 2 ) 1 einδ 1 1 e iδ 1 ] e i(ωt kro) 1 einδ 2 1 e iδ 2 (2.4.16)

48 32 From equations and , it is seen that the phase difference between the two resultant waves (φ 2 φ 1 ) is given by: (φ 2 φ 1 ) = δφ + k R o (ax + by ) = kzδz(µ 1) R + k R o (ax + by ) (2.4.17) Since A g1 2 = A g2 2 = ( ) [ 2 ao ab sin( kxa R ( sin( Nδ 1 2 ) sin( δ 1 2 ) ) 2R kxa 2R ) 2 ( sin( Nδ 2 ] 2 [ sin( ky b 2 ) sin( δ 2 2 ) ) 2R ky b 2R ] 2 ) 2 (2.4.18) the resultant intensity at P will be given by: I = A g1 2 + A g A g1 A g2 cos(φ 2 φ 1 ) (2.4.19) with A g1 = A g2 = A equation (2.27) becomes I = 2 A 2 [1 + cos(φ 2 φ 1 )] (2.4.20) For Xx i R o and Y y i R o R o it follows R R o and (φ 2 φ 1 ) = kzδz(µ 1) R + k (ax + by ) (2.4.21) R Setting Y = 0, A 2 can be rewrite in the form ( ) [ ] 2 2 ( A 2 ao ab sin( δ 14 ) sin( Nδ 1 = ) ) 2 2 R δ 14 (2.4.22) sin( δ 1 2 ) With sin( δ 1 4 ) sin( δ 1 2 ), therefore A 2 can be written as ( ) [ 2 A 2 ao ab sin( Nδ 1 = ) ] 2 2 R δ 14 (2.4.23)

49 33 with sin(θ) = θ θ3 3! + θ5 5! θ7 7! +... cos(θ) = 1 θ2 2! + θ4 4! θ6 6! +... ( ) [ 2 A 2 ao ab Nδ1 (Nδ 1/2) 3 + (Nδ 1/2) ! 5! = R δ 14 ] 2 ( ) [ 2 A 2 ao ab Nδ1 = (1 (Nδ 1/2) 2 + (Nδ 1/2) ) 2 3! 5! R δ 14 ] 2 With cos( Nδ 1 2 ) 1 (Nδ 1/2) 2 3! ( ao ab A 2 R + (Nδ 1/2) 4 5! +... ) 2 4N 2 (cos( Nδ 1 2 ))2 (2.4.24) Thus the intensity can be write as setting ( ) 2 ao ab I = 2 4N 2 (cos( Nδ 1 R 2 ))2 [1 + cos(φ 1 φ 2 )] (2.4.25) φ 1 φ 2 = mν (2.4.26) where and m = ( 2π c )[ ] ((Zδz(µ 1)) + (ax)) R (2.4.27) δ 1 = gν (2.4.28) where g = ( 2π c )[ ] 2aX R (2.4.29)

50 34 ( ) 2 ao ab I = 8 N 2 (cos( Ngν R 2 ))2 [1 + cos(mν)] (2.4.30) In the forgoing derivation, monochromatic irradiation of the rough surface was considered. In the paragraph to follow, the rough surface will be assumed to be non-monochromatic having different spectral distribution Gaussian spectral line profile Assuming a symmetrical spectral line profile around ν o having a half width ν and given by the g(ν) = a o (ν) 2 = a g (ν) 2 e α(ν νo)2 (2.4.31) to irradiation the rough surface of the target where a g (ν) 2 = α π The intensity on the screen might be given by : I G = and α = 4ln2 ( ν) 2. Ig(ν)dν (2.4.32) ( ) 2 ao ab I G = 8 N 2 (cos( Ngν R 2 ))2 [1 + cos(mν)] a g (ν) 2 e α(ν νo)2 dν (2.4.33) Setting C G = 4 a g (ν) ( 2 a oab ) 2 R N 2, (cos( Ngν 2 ))2 = 1 [1 + cos(ngν)], 2 ν ν o = x and dν = dx I G = C G [1 + cos(ng(x + ν o ))][1 + cos(m(x + ν o ))]e α(x)2 dx (2.4.34) I G = C G [1+cos(Ng(x+ν o ))+cos(m(x+ν o ))+cos(ng(x+ν o )) cos(m(x+ν o ))]e α(x)2 dx (2.4.35)

51 35 I G = C G [I 1 + I 2 + I 3 + I 4 ] (2.4.36) where I 1 = e α(x)2 dx = π α (2.4.37) I 2 = cos(m(x + ν o ))e α(x)2 dx (2.4.38) I 2 = [cos(mx) cos(mν o ) sin(mx)sin(mν o )]e αx2 dx I 2 = cos(mx) cos(mν o )e α(x)2 dx sin(mx)sin(mν o )]e α(x)2 dx If f(x) is symmetric about the origin ν o and f(x) is an even function, thus f(x)sin(bx)dx = 0 and I 2 can be written as: sin(mν o ) sin(mx)e α(x)2 dx = 0 I 2 = cos(mν o ) cos(mx)e α(x)2 dx (2.4.39) From [25] 0 e α2 x 2 cos(bx)dx = π 2 2α e b 4α 2 π I 2 = α cos(mν o)e m 2 4α (2.4.40)

52 36 Similar to I 2, I 3 which is given by I 3 = cos(ng(x + ν o ))e α(x)2 dx (2.4.41) have the solution π I 3 = α cos(ngν o)e (Ng) 2 4α (2.4.42) The last term I 4 which is given by I 4 = can be written after considers cos(m(x + ν o )) cos(ng(x + ν o ))e α(x)2 dx (2.4.43) cos a cos b = 1 [cos(a b) + cos(a + b)] 2 as: I 4 = 1 2 [cos((m + Ng)(x + ν o)) + cos((m Ng)(x + ν o ))]e α(x)2 dx I 4 = 1 [ cos((m + Ng)(x + ν o ))e α(x)2 dx + 2 and This gives cos((m + Ng)(x + ν o ))e α(x)2 dx = cos((m Ng)(x + ν o ))e α(x)2 dx = ] cos((m Ng)(x + ν o ))e α(x)2 dx π α cos((m + Ng)ν o)e ((m+ng)2 )/4α π α cos((m Ng)ν o)e ((m Ng)2 )/4α I 4 = 1 2 [ ] π π α cos((m + Ng)ν o)e ((m+ng)2 )/4α + α cos((m Ng)ν o)e ((m Ng)2 )/4α (2.4.44)

53 37 Thus I G can be write as I G = C G π α (1 + cos(mν o)e m2 4α + cos(ngνo )e (Ng)2 4α (2.4.45) [cos((m + Ng)ν o)e ((m+ng)2 )/4α + cos((m Ng)ν o )e ((m Ng)2 )/4α ]) For Ng m, cos((m + Ng)ν o )e ((m+ng)2 )/4α cos((m Ng)ν o )e ((m Ng)2 )/4α cos(mν o )e (m)2 4α and I G can then be written as π I G = C G α [1 + 2 cos(mν o)e m2 4α + cos(ngνo )e (Ng)2 4α ] (2.4.46) I GMax = C G π α [1 + 2 cos(mν o)e m2 4α + e (Ng) 2 4α ] I GMin = C G π α [1 + 2 cos(mν o)e m2 4α e (Ng) 2 4α ] at certain δz and varying X values the visibility is determinate according to equation (2.3.1) V G = e (Ng)2 4α [1 + (2 cos(mν o )e m2 4α )] (2.4.47) Lorentzian spectral line profile Considering a symmetrical spectral line profile around ν o following a Lorentzian profile function where a L (ν o ) 2 = 2 π ν The intensity distribution is a o (ν) 2 = a L (ν o ) ( ν ν o ν/2 )2 (2.4.48) I L = I a o (ν) 2 dν (2.4.49)

54 38 1 I L = C L [1 + cos(ng(x + ν o ))][1 + cos(m(x + ν o ))] 1 + ( x (2.4.50) )2dx ν/2 where C L = 4 a L (ν) 2 a oab R N2 1 I L = C L [1+cos(Ng(x+ν o ))+cos(m(x+ν o ))+cos(ng(x+ν o )) cos(m(x+ν o ))] 1 + ( x )2dx ν/2 (2.4.51) with I L = C L [I 1 + I 2 + I 3 + I 4 ] (2.4.52) where I 1 = 1 π ν 1 + ( x = )2dx 2 ν/2 (2.4.53) I 2 = I 2 = 1 cos(m(x + ν o )) 1 + ( x (2.4.54) )2dx ν/2 1 [cos(mx) cos(mν o ) sin(mx)sin(mν o )] 1 + ( x )2dx ν/2 I 2 = 1 cos(mx) cos(mν o ) 1 + ( x )2dx ν/2 1 sin(mx)sin(mν o ) 1 + ( x )2dx ν/2 if f(x) is symmetric about the origin ν o and f(x) is an even function, then f(x)sin(bx)dx = 0 Then I 2 can be written as: sin(mν o ) I 2 = cos(mν o ) 1 sin(mx) 1 + ( x )2dx = 0 ν/2 1 cos(mx) 1 + ( x (2.4.55) )2dx ν/2

55 39 Since it follows 0 1 π ν 1 + ( x cos(bx)dx = )2 2 e b 2 (2.4.56) ν/2 I 2 = π ν 2 Following the same procedure given above, it follows with I 3 = cos(mν o )e m ν 2 (2.4.57) 1 cos(ng(x + ν o )) 1 + ( x (2.4.58) )2dx ν/2 For I 4 which is given by one gets I 4 = I 4 = I 3 = π ν 2 cos(ngν o )e Ng ν 2 (2.4.59) 1 cos(ng(x + ν o )) cos(m(x + ν o )) 1 + ( x (2.4.60) )2dx ν/2 1 2 [cos((m + Ng)(x + ν 1 o)) + cos((m Ng)(x + ν o ))] 1 + ( x )2dx ν/2 = 1 2 [ 1 cos((ng+m)(x+ν o )) 1 + ( x )2dx+ ν/2 1 cos((ng m)(x+ν o )) 1 + ( x )2dx] ν/2 cos((m + Ng)(x + ν o )) 1 π ν 1 + ( x = )2dx 2 ν/2 cos(((m + Ng))ν o )e (m+ng) ν 2 cos((m Ng)(x + ν o )) 1 π ν 1 + ( x = )2dx 2 ν/2 cos((m Ng)ν o )e (m Ng) ν 2

56 40 I 4 = 1 2 [π ν 2 cos((m + Ng)ν o )e (m+ng) ν 2 + π ν cos((m Ng)ν o )e (m Ng) ν 2 ] 2 (2.4.61) I L = C L [I 1 + I 2 + I 3 + I 4 ] (2.4.62) Then I L can be write as I L = C L π ν 2 [1 + cos(mν o)e m ν 2 + cos(ngν o )e Ng ν 2 (2.4.63) [cos((m + Ng)ν o)e (m+ng) ν 2 + cos((m Ng)ν o )e (m Ng) ν 2 ]] For Ng m, cos((m + Ng)ν o )e (m+ng) ν 2 cos((m + Ng)ν o )e (m Ng) ν 2 cos(mν o )e m ν 2 then I L can be written as I L = C L π ν 2 [1 + 2 cos(mν o)e m ν 2 + cos(ngν o )e Ng ν 2 ] (2.4.64) I LMax = C L π α [1 + 2 cos(mν o)e m ν 2 + e Ng ν 2 ] I LMin = C L π α [1 + 2 cos(mν o)e m ν 2 e Ng ν 2 ] For a certain δz and varying X values the visibility is determinate according to equation (2.3.1) V L = e Ng ν 2 [1 + (2 cos(mν o )e m ν 2 )] (2.4.65)

57 Results and discussion In this part the effect of the grain height(surface roughness), the spectral half width and the density of the grains on the visibility is studied for the two of types of the spectral distributions namely the Gaussian and Lorentzian profiles. Computation Equations (2.4.47) and (2.4.65) were programmed using MATLAB. Since the visibility was developed for great distance between the diffuser and the screen, this distance was chosen in the computation to be in order of meter. The computation was carried out considering beside the Gaussian and Lorentzian amplitude distribution of different half widthes, different grain densities, grain heights and the following values for the parameters affecting the visibility. The distance between the diffuser and screen Z = 400cm. The area of diffuser is constant = 1cm 2. a. Effect of the spectral profile type and its half width on the visibility Figure (2.3) shows the obtained results between the visibility of speckle patterns and the spectral half width in case of Gaussian and Lorentzian profile. The calculation was carried out considering the grain density and height to be 10 6 in cm 2 and 50µm respectively. From the figure it is evident that the visibility decreases with increasing the half width. This behavior is due to the fact that by increasing the half width the degree of coherence and consequently the coherence length of the light beam will be reduced and since the speckle pattern is a a phenomena based on superposition of waves that must have a phase correlation in order to give the granulate behavior, therefore its visibility will decrease by increasing the half width of the illuminating

58 42 beam. That the visibility of Gaussian distribution is greater than that obtained from Lorentzian one considers the same half width is due to the higher effective spectral band width in the Lorentzian distribution which is larger than that in case of the Gaussian one. This fact leads to reduced phase correlation between the superimposed waves. The obtained behavior is in agreement with experimental data given in [20,21]. Figure 2.3: The visibility of speckle patterns versus the spectral half width ν with spectral line shape as a parameter. The results are obtained considering the grain density to be = 10 6 in cm 2, the grain height to be 50µm.

59 43 b. Effect of the density of the grains on the visibility At a grain height of 10µm, a spectral half width of Hz and varying grain width, equations (2.4.47) and (2.4.65) were computed to study the effect of the grain density on the visibility of speckle patterns. Figures (2.4) and (2.5) show the dependence of the visibility of speckle patterns on the grain width in case of Gaussian and Lorentzian profile respectively. It is evident that the greater density of the grains yields higher visibility of the speckle patterns. This behavior can be interpreted by the following. By increasing the density the randomization in difference between the interfering beam become smaller leading to increase the visibility.

60 44 Figure 2.4: The visibility of the speckle patterns versus the grain width for the case of Gaussian profile. The results are obtained for a grain height of 10µm and spectral half width of Hz.

61 Figure 2.5: The visibility of the speckle patterns versus the grain width for the case of Lorentzian profile. The results are obtained for a grain height of 10µm and spectral half width of Hz. 45

62 46 c. Effect of the grain height on the visibility By keeping grain width equals 10µm and varying the grain height, equations (2.4.47) and (2.4.65) were computed to study the effect of the grain height on the visibility of the speckle patterns. Figures (2.6) and (2.7) show the dependence of the visibility of speckle patterns on the grain height in case of Gaussian spectral distribution of δν = and δν = respectively. Figures (2.8) and (2.9) show the same but for the case of Lorentzian spectral distribution of δν = and δν = respectively. It is evident that by increasing the half width of the radiation the visibility of speckle patterns decreases. It is due to the inverse dependence of the degree of coherence of the light beam and its spectral half width. The behavior of the dependence of the visibility of the speckle pattern on the grain height of the roughness, as shown in the figures, can be explained through the fact that an infinite extended polished surface reflects the incident wave in a defined direction and therefore the obtained pattern has a visibility equals to zero. If the surface has a finite area the visibility takes values other than zero because of the diffraction. By increasing the roughness scattering begins to take place and the superposition of the scattered waves leads to the appearance of speckle which is still superimposed with a direct reflected part of the radiation. This directed reflected part will be reduced by increasing the roughness leading to an increasing of the visibility. Further increase of the roughness heights leads to a path difference in the order of the coherence length and therefore to a drop of the visibility. This behavior is clearly seen in the figures. That further increase of the roughness heights leads to an increase in the number of scattered beams and inturn the number of interfering beams which rise the sharpness and contrast of the speckle pattern and therefore its visibility increases. As the interfering beams reach

63 47 its maximum effective number the visibility tends to be constant. Figure 2.6: The visibility of the speckle patterns versus the grain height calculated for the case of Gaussian profile of δν = Hz. The results are obtained for a grain density of 10 6 in cm 2.

64 48 Figure 2.7: The visibility of the speckle patterns versus the grain height calculated for the case of Gaussian profile of δν = Hz. The results are obtained for a grain density of 10 6 in cm 2.

65 Figure 2.8: The visibility of the speckle patterns versus the grain height calculated for the case of Lorentzian profile of δν = Hz. The results are obtained for a grain density of 10 6 in cm 2. 49

66 50 Figure 2.9: The visibility of the speckle patterns versus the grain height calculated for the case of Lorentzian profile of δν = Hz. The results are obtained for a grain density of 10 6 in cm 2.

67 Experimental verification of theoretical model To verify the theoretical model an experiment [22, 23] was carried out using a partially coherent light obtained from a mercury lamp. As shown in figure (2.10), the Figure 2.10: The experimental setup. light from the primary source was, after being filtered, focussed by a condenser lens L o onto a small pinhole P o which acts as a secondary point source. That small pinhole was also situated at the focal plane of the collimating lens L 1. The collimated light illuminates a diffusely transmitting object where its image was produced at the plane P 2, by a double imaging system consisting of two lenses L 2 and L 3 together with an aperture P 1. The image at the plane P 2 showing the speckle intensity distribution was captured by a webcam (digital camera). Six transmitting objects, having different values of surface roughnesses, prepared as test samples were used. These are glasses plates polished by using emery powders. In order to find the relation between the speckle intensities fluctuation and the surface roughness, the surface roughness of these objects was precisely mechanically measured beforehand by using a stylus instrument. Figure (2.11) shows the obtained speckle patterns from partially coherent

68 52 light. By using software program (ImageJ program), the values of the mean intensity Figure 2.11: The obtained speckle patterns from partially coherent light. I and the standard deviation σ I of speckle patterns was obtained and substituted in the contrast equation C = σ I / I (2.5.1) This procedure was repeated for the six objects. Figure (2.12) shows the dependence of the speckle patterns contrast on the surface roughness obtained from the two spectral lines of the Hg lamp (the green and yellow line).

69 Figure 2.12: The relation between speckle patterns contrast and surface roughness 53

70 54 To increase the rang of spectral half width, we used the setup in figure (2.14). Figure 2.13: The experimental setup: C a condenser, L a convex lens, D a diffraction grating, RP a rectangular aperture, O The object A Halogen lamp was focussed by a condenser C onto a small pinhole which acts as a secondary point source. The small pinhole was also situated at the focal plane of the collimating lens L. The collimated light illuminates a diffraction grating D. The spectrum of a Halogen lamp appears beyond the diffraction grating. By using a rectangular aperture, we can select bands with different widthes. For each band width we repeat the previous experimental procedures. Figures (3.15), (3.16),(3.17), (3.18) show the dependence of the speckle patterns contrast on the surface roughness obtained for spectral half width equal Hz, Hz, Hz and Hz respectively. Figure (3.19) shows the dependence of the speckle patterns contrast on the surface roughness for different spectral half widthes.

71 Figure 2.14: The relation between speckle patterns contrast and surface roughness ν = Hz corresponding to L c = µm. 55

72 56 Figure 2.15: The relation between speckle patterns contrast and surface roughness ν = Hz corresponding to L c = µm.

73 Figure 2.16: The relation between speckle patterns contrast and surface roughness at ν = Hz corresponding to L c = µm. 57

74 58 Figure 2.17: The relation between speckle patterns contrast and surface roughness at ν = Hz corresponding to L c = µm.

75 Figure 2.18: The relation between speckle patterns contrast and surface roughness with ν = , , and Hz as a parameter. 59

76 Conclusion From figures (2.12, 2.14 and 2.15) it is evident that since the coherence length of the used waves were greater than the greatest roughness height, therefore the contrast increases with increasing the hight of the roughness without droping in its value at a certain value of the roughness height. This behavior is in agreement with the theoretical results demonstrated in figures (2.6 to 2.9) and that polished in the lecture [7, 21, 22]. As the coherence length was decreased and become smaller than the roughness, the contrast began to decrease at greater values of the roughness heights as seen from figures (2.16 and 2.17). Figure (2.18) represents the contrast as a function of the roughness height with the half width of the illuminating beam as a parameter.

77 Chapter 3 Study The Effect of Some Parameters on The Visibility of Speckle Interferometry 3.1 Introduction The speckle photography method (section 1.6.1) appears to hold considerable promise regarding the determination of displacements and strains in static and dynamic problems[15]. This method, based on the speckle effect, stems from the classical paper of Burch and Tokkarski [38]. It has been exploited by several authors for the measurement of in-plane displacements, out -of-plane tilts, and nowadays there is an extensive literature concerning its utility in different metrological problems[72]. It is well known that speckle photographs which give low contrast fringes can be found in practical applications of this technique, a fact that is due to speckle decorrelation effects associated in excessive movement of the surface in the line-of-sight direction and to lens aberrations. Therefore, in these cases the diffraction halo will also contribute to a small shift in fringe minima. Thus a significant errors in displacements appear. These errors are functions of fringe number and fringe visibility, and they 61

78 62 increase as the latter decrease as shown in figure(3.1)[15]. Therefore if the visibility of the fringes is high, this means that we can determine the displacement with high accuracy. Thus we study some parameters which acts on the visibility of the fringes and consequently on the accuracy of displacement measurements. These parameters are the surface roughness of the object, the effect of state of polarization of the light, speckle size and the type of speckle. Figure 3.1: Effect of fringe visibility on relative errors in displacements for different number of fringes as a parameter

79 Theoretical principle of the double exposure speckle photography technique One approach is to observe the fringe visibility of the test surface using the double exposure technique. Let the test surface in plane (x,y) be illuminated with a laser beam and the speckle pattern is recorded on a photographic plate H [53]. The speckle on H is described by a function D(x,y),which represents the intensity of the light on H. The transmitted amplitude t(x,y) of the negative H is given by t(x,y) = a bd(x,y) (3.2.1) where a and b are two characteristic constants of the photographic emulsion used. When two equal exposures are made, for the object, which is shifted by a distance x 0 between the exposures, the final recorded D t (x,y) is the sum of recorded D values in each exposure and is equal to D t (x,y) = D(x,y) + D(x x 0,y) (3.2.2) transmission of the negative is given by: t(x,y) = a bd(x,y) [δ(x,y) + δ(x x 0,y)] (3.2.3) When the negative H, which is placed in the front of the focal plane of a lens, is illuminated with a collimated beam of wavelength λ, the Fourier transform of the amplitude transmission t(x,y) of H is obtained in the back focal plane of the lens (Fourier transform plane) and is given by t(u,v) = aδ(u,v) b D(u,v)[1 + exp(iπux 0 /λ)] (3.2.4) Where the tilde variables represent the Fourier transform and (u, v) are the angular coordinates of a point in the focal plane. The first term aδ(u,v) is the direct part of

80 64 the spectrum; it represents homogenous illumination which can be attributed to point source located at infinity when diffraction effects are neglected. D(u, v) in the second term is the Fourier transform of D(x,y). It is modulated by [1+exp(iπux 0 /λ)]. Since contains fine structures, its transform spreads out considerably in the focal plane. Like D(x,y), the transform has speckle patterns. If the DC part of the spectrum is neglected, then the intensity in the Fourier transform plane is I = D(u,v) exp(iπux 0 /λ) = D(u,v) 2 cos 2 (πux 0 /λ)) (3.2.5) The diffused background D(u,v) 2 is modulated by cos 2 (πux 0 /λ), which represents Young s fringes. The angular distance between two consecutive bright (or dark) fringes is equal to λ/x Experimental Setup Figure (3.2) shows the experimental arrangement [64]. Spatially coherent unpolarized light from a He-Ne laser source (0.95mW) was, after being expanded and collimated by a lens, employed for illumination. The test surfaces were prepared by polishing the test glasses with emery powders. The roughness of the test surface was precisely mechanically measured beforehand by using a stylus instrument. The sensor of the webcam camera has received directly the two speckle pattern before and after displacing the object (objective speckle patterns section 1.3) by means of a precision micrometer. 3.4 Effect of displacement value on the visibility To study the effect of displacement value on the visibility, the obtained signals before and after displacement were digitally stored on the Hard Disk (H.D) of a pc-computer,

81 Figure 3.2: The experimental setup: BE beam expander, L convex lens, O the object. 65

82 66 figures (3.3) and (3.4). Figure 3.3: Spackle pattern before displacement Figure 3.4: Spackle pattern after displacement = 50µm The Fourier transform of the added doubly exposure speckle pattern (figure (3.5)) was calculated by means of software program ImageJ program. Figure (3.6) shows a sample of the obtained Fourier transform for displacing a rough surface of rms (root mean squar) roughness of 9.6µm a distance of 50µm.

83 67 Figure 3.5: The addition of two speckle patterns before and after displacement Figure 3.6: Young s fringes for a displacement = 50µm

84 68 The figure shows, as expected, the fringes of Young double slit [70]. By varying the displacement of the same sample the visibility for each displacement was calculate according to the relation, V = I max I min I max + I min (3.4.1) where I max and I min were obtained from the plotted profile of the Young s fringes given in (figure 3.7). Figures (3.8), (3.9),(3.10) and (3.11)show the obtained Fourier Figure 3.7: The plotting profile transform for displacing a rough surface of rms roughness of 9.6µm a distance of 150µm, 250µm, 350µm and 500µm respectively. Figure (3.12) shows the visibility as a function of the displacement for a surface having roughness of rms 9.6µm. The figure shows that the visibility decrease as the displacement increases. This behavior is due to the decrease of the fringe spacing by increasing the displacement of the object as seen from figures (3.8 to 3.11). According to [71] when the Young s fringe spacing becomes comparable to the speckle size, the fringe visibility decreases and drops to zero when they become equal. AS the

85 69 Figure 3.8: Young s fringes for a displacement = 150µm Figure 3.9: Young s fringes for a displacement = 250µm

86 70 Figure 3.10: Young s fringes for a displacement = 350µm Figure 3.11: Young s fringes for a displacement = 500µm

87 71 displacement increases the autocorrelation function between the two sets of speckle patterns decrease due to the spatial coherence. Figure 3.12: The relationship between the fringes visibility and displacement value for surface roughness 9.6µm

88 Effect of the surface roughness on the visibility To study the effect of surface roughness on the accuracy of the displacement,we repeat the steps in the section 3.3 for different surface roughness. Figures (3.13) and (3.14) show the obtained Young s fringes for surface roughness 2.48µm and 6.3µm at displacement 50µm respectively. Figure (3.15) shows the relation of figure (3.12) with different roughness as a parameter. From the figure, it is evident, that the accuracy of the determination of the displacement increases by decreases the displacement and increases the surface roughness. Figure (3.16) shows the relation between the visibility and the surface roughness for a displacement of 50µm. This behavior can be explained as follows. In case of a surface without roughness with infinite dimension, the transmitted wave is a plane one as the illuminating wave. This will lead to a humongous illumination and the visibility is in this case will be equal to zero. As the dimension of the surface become finite, one gets the diffraction patterns of a single obstacle, which shows a bad visibility as the dimensions are great compared to the wave length of the illuminating radiation. By introducing roughness on the surface more and more, spherical waves will be transmitted as the rms of the roughness increases. These waves will superimpose to give great difference between maximum and minimum leading to increase of the visibility [56].

89 73 Figure 3.13: Young s fringes of displacing an object with rms = 2.48µm a distance 50µm Figure 3.14: Young s fringes of displacing an object with rms = 6.3µm a distance 50µm

90 74 Figure 3.15: The relation between the visibility and displacement with different surface roughness as a parameter.

91 Figure 3.16: The visibility versus surface roughness for a constant displacement 50µm 75

92 Effect of the polarized light on the visibility To study the effect of polarized light on the visibility, at the same speckle size, the setup represented in figure (3.2) was used but with polarizer P placed before the object O as shown in figure (3.17). Figure 3.17: The experimental setup with polarizer P before the diffuser. Figure (3.18) shows the relation between the visibility and the displacement for a sample of rms roughness of 9.6µm by using polarized and unpolarized light. Figure (3.19) shows the relation between the visibility and surface roughness for a 50µm displacement by using polarized and unpolarized light. From these figures it is evident that the polarized light improves the visibility and hence the accuracy of displacement measurements. This behavior can be explained in view of the fact that interfering waves of the same plane of polarization improves the interference phenomena.

93 Figure 3.18: The relation between the visibility and displacement for surface of rms roughness = 9.6µm obtained by using polarized and unpolarized light 77

94 78 Figure 3.19: The relation between the visibility and surface roughness for a 50µm displacement by using polarized and unpolarized light

95 Effect of the speckle size on the visibility To study the effect of speckle size on the visibility, the setup in figure (3.17) was used. As mentioned previously in section(1.3), the objective speckle size is equal σ o = 1.22λz D (3.7.1) To change the speckle size, firstly set the diameter of illuminated area constant D = 1.4cm and change the distance between the test object and the camera z taking into consideration that, the speckle size must be less than the displacement value which in this case is constant and equals to 100µm. Figures (3.20)and (3.21) show the obtained Young s fringes for different values of speckle size 9.2µm and 36µm respectively. Secondly set the distance between the test object and the camera constant and equals to z = 26.7cm and change the diameter of the illuminating area D. Figures (3.22)and (3.23) show the obtained Young s fringes for different values of speckle size µm and 20.61µm respectively. Figures (3.24) and (3.25) show the relation between the speckle size and the visibility. From the figures it is evident that the visibility is better for greater speckle size. Both of the speckle size and their adjacent distances increase. It leads to decrease the effective number of speckles which play a role in the interference phenomena. Inturn the autocorrelation function between these less number of speckles will increase which improves the visibility

96 80 Figure 3.20: The Young s fringes for speckle size = 9.2µm obtained for a displacement of 100µm, z = 16.7cm, D = 1.4cm and surface roughness 9.6µm Figure 3.21: The Young s fringes for speckle size = 36µm obtained for a displacement of 100µm, z = 65.5cm,D = 1.4cm and surface roughness 9.6µm

97 81 Figure 3.22: The Young s fringes for speckle size = µm obtained for a displacement of 100µm, z = 26.7cm, D = 2cm and surface roughness 9.6µm Figure 3.23: The Young s fringes for speckle size = µm obtained for a displacement of 100µm, z = 26.7cm,D = 1cm and surface roughness 9.6µm

98 82 Figure 3.24: The relation between the speckle size and visibility at constant D = 1.4cm and different z values for surface roughness 9.6µm

99 Figure 3.25: The relation between the speckle size and visibility at constant z = 26.7cm and different D values for surface roughness 9.6µm 83

100 Effect of the speckle type on the visibility As mentioned previously in section (1.3), there are two types of speckle, objective and subjective speckle. To study the effect of the two types on the visibility, the two setups represented in figures (3.26), (3.27) were used. To study the effect of Figure 3.26: The experimental setup to obtain objective speckle patterns Figure 3.27: The experimental setup to obtain subjective speckle patterns, IS: imaging system (convex lens + aperture), u is the object distance and v is the image distance. two speckle patterns on the visibility when the magnification factor m of the imaging system in subjective speckle pattern setup equals to 1 where m = v u (3.8.1) v and u are the image distance and object distance respectively.

101 85 The speckle size of subjective speckle patterns was calculated according to the following equation σ s = 2.4λv a (3.8.2) to give σ s = 17.15µm, where a the diameter of the viewing lens aperture. Adjust the objective speckle patterns setup to give the same speckle size. Figures (3.28) and (3.29) show the Young s fringes from the objective and subjective speckle patterns for m = 1 and a displacement of 100µm respectively. Repeat the previous procedure for a magnification factor equals to 2. In this case the speckle size will be equal to µm. Figures (3.30) and (3.31) show the Young s fringes from the objective and subjective speckle patterns for m = 2 and a displacement of 100µm respectively.

102 86 Figure 3.28: Young s fringes for displacement 100µm obtained from objective speckle patterns and speckle size =17.17µm Figure 3.29: Young s fringes for displacement 100µm obtained from subjective speckle patterns, speckle size =17.17µm and m = 1

103 87 Figure 3.30: Young s fringes for displacement 100µm obtained from objective speckle patterns and speckle size =24.256µm. Figure 3.31: Young s fringes for displacement 100µm obtained from subjective speckle patterns speckle size =24.256µm and m = 2 Figures (3.32) and (3.33) show a comparison between the two types for different displacements at m = 1 and m = 2 respectively. From the figures, it is evident that the objective speckle patterns gives high visibility than the subjective speckle patterns.