Advances in Speckle Metrology and Related Techniques E-Book

• Herausgeber: Wiley-VCH
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 533

Veröffentlichungsjahr: 2011

Beschreibung

Speckle metrology includes various optical techniques that are based on the speckle fields generated by reflection from a rough surface or by transmission through a rough diffuser. These techniques have proven to be very useful in testing different materials in a non-destructive way. They have changed dramatically during the last years due to the development of modern optical components, with faster and more powerful digital computers, and novel data processing approaches. This most up-to-date overview of the topic describes new techniques developed in the field of speckle metrology over the last decade, as well as applications to experimental mechanics, material science, optical testing, and fringe analysis.

Leseprobe

Contents

Cover

Related Titles

Title Page

Copyright

Dedication

Preface

List of Contributors

Chapter 1: Radial Speckle Interferometry and Applications

1.1 Introduction

1.2 Out-of-Plane Radial Measurement

1.3 In-Plane Measurement

1.4 Applications

1.5 Conclusions

References

Chapter 2: Depth-Resolved Displacement Field Measurement

2.1 Introduction

2.2 Low-Coherence Electronic Speckle Pattern Interferometry

2.3 Wavelength Scanning Interferometry

2.4 Spectral Optical Coherence Tomography

2.5 Tilt Scanning Interferometry

2.6 Depth-Resolved Techniques Viewed as Linear Filtering Operations

2.7 Phase Unwrapping in Three Dimensions

2.8 Concluding Remarks

Acknowledgments

References

Chapter 3: Single-Image Interferogram Demodulation

3.1 Introduction

3.2 The Fourier Spatial Demodulating Method

3.3 Linear Spatial Phase Shifting

3.4 Nonlinear Spatial Phase Shifting

3.5 Regularized Phase Tracking

3.6 Local Adaptive Robust Quadrature Filters

3.7 Single Interferogram Demodulation Using Fringe Orientation

3.8 Quadrature Operators

3.9 2D Steering of 1D Phase Shifting Algorithms

3.10 Conclusions

References

Chapter 4: Phase Evaluation in Temporal Speckle Pattern Interferometry Using Time–Frequency Methods

4.1 Introduction

4.2 The Temporal Speckle Pattern Interferometry Signal

4.3 The Temporal Fourier Transform Method

4.4 Time–Frequency Representations of the TSPI Signals

4.5 Concluding Remarks

References

Chapter 5: Optical Vortex Metrology

5.1 Introduction

5.2 Speckle and Optical Vortices

5.3 Core Structure of Optical Vortices

5.4 Principle of Optical Vortex Metrology

5.5 Some Applications

5.6 Conclusion

Acknowledgments

References

Chapter 6: Speckle Coding for Optical and Digital Data Security Applications

6.1 Introduction

6.2 Double Random Fourier Plane Encoding

6.3 Variants of the DRPE and Various Other Encryption Techniques

6.4 Attacks against Random Encoding

6.5 Speckle Coding for Optical and Digital Data Security

6.6 Encryption Using a Sandwich Phase Mask Made of Normal Speckle Patterns

6.7 Optical Encryption Using a Sandwich Phase Mask Made of Elongated Speckle Patterns

6.8 Speckles for Multiplexing in Encryption and Decryption

6.9 Multiplexing in Encryption Using Apertures in the FT Plane

6.10 Multiplexing by In-Plane Rotation of Sandwich Phase Diffuser and Aperture Systems

6.11 Speckles in Digital Fresnel Field Encryption

6.12 Conclusions

Acknowledgment

References

Index

Related Titles

Gläser, M., Kochsiek, M. (eds.)

Handbook of Metrology

2010

ISBN: 978-3-527-40666-1

Horn, A.

Ultra-fast Material Metrology

2009

ISBN: 978-3-527-40887-0

Gross, H. (ed.)

Handbook of Optical Systems

Volume 1: Fundamentals of Technical Optics

2005

ISBN: 978-3-527-40377-6

The Editor

Prof. Guillermo H. Kaufmann

Instituto de Fisica Rosario Universidad Nacional de Rosario Facultad de Ciencias Exactas e Ingeniería, Department of Physics and Centro Internacional Franco Argentino de Ciencias de la Información y de Sistemas Rosario, Argentina [email protected]

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Library of Congress Card No.: applied for

British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library.

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The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.d-nb.de.

© 2011 WILEY-VCH Verlag & Co. KGaA, Boschstr. 12, 69469 Weinheim, Germany

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Print ISBN: 978-3-527-40957-0

Epub ISBN: 978-3-527-63387-6

Acrobat Reader ISBN: 978-3-527-63386-9

To Carolina

Preface

The development of laser in the early 1960s generated many new research lines in the field of physical optics. The laser is a source that produces a beam of light that is intense, collimated, and coherent. However, the greatly increased coherency of laser light also leads to the appearance of other side effects such as the speckle phenomenon. People working with lasers quickly realized that the grainy appearance of rough objects when illuminated with coherent light was caused by an interference phenomenon. During the first few years after the invention of the laser, while much effort was carried out to minimize the effects of laser speckle, little was devoted to take the more positive path of putting it to good use. Perhaps the turning point came when it was realized that the light forming each individual speckle was fully coherent and it possessed a definite phase. This ability was recognized in 1968 and was quickly applied to the development of new laser-based techniques to measure displacements, deformations, and vibrations produced by rough objects.

The first multiauthor book on the subject (Speckle Metrology, edited by R.K. Erf) appeared in 1978. To fill the need for a book covering new aspects of speckles and novel topics such as fringe analysis and particle image velocimetry that had been mainly developed during the next decade, a new volume appeared in 1993 also entitled Speckle Metrology, edited by R. S. Sirohi. In the following years, the development of video cameras with higher resolution and high-speed data acquisition systems gave great impetus to speckle interferometry and related techniques such as digital speckle photography and digital holographic interferometry. These application-oriented techniques contributed to the publication of a new volume in 2001 edited by P.K. Rastogi, entitled Digital Speckle Pattern Interferometry and Related Techniques.

From the year 2000, new branches in speckle metrology have appeared, several new schemes have been proposed, and also the known approaches and techniques have been revisited and improved upon. The amount and the scope of these developments are reflected in very rich material, published in the specialized journals and presented each year at various international conferences. Therefore, time has come to review and sum up the most significant of these advances, and this book is the result of such efforts.

This book provides an up-to-date collection of the new material published in the field of speckle metrology and related techniques since 2000. Although there were several topics that could be included in such a book, we had to select only a few to keep its length reasonable. This means that other topics such as speckle techniques outside the visible part of the spectrum could not be treated. It is important to note that most of the selection of topics was carried out by taking into account that the? book should be useful for engineers, scientists, and graduate students who are engaged in the application of speckle techniques mainly to solve specific measurement problems in optical metrology, mechanical engineering, experimental mechanics, material science, nondestructive testing, and related fields or who are contemplating its use.

The book is organized into six chapters. The measurement of radial deformations in cylinders and the evaluation of residual stresses are important engineering problems that are better understood when they are analyzed in polar coordinates. Chapter describes latest digital speckle pattern interferometry (DSPI) systems, which are sensitive to polar coordinates. First, the authors present several configurations based on the use of conical mirrors mounted on piezoelectric transducers to allow the application of phase shifting algorithms for evaluating the phase distribution. A more recently developed interferometer based on the use of a diffractive optical element, which overcomes some limitations of earlier configurations, is also described. Finally, these systems are applied to the measurement of translations, mechanical stresses, and residual stresses. Application examples spread all over the chapter show the extent to which radial speckle interferometry has been developed into a powerful tool for industrial measurements.

Chapter 2 analyzes different approaches to measure internal displacement and strain fields within a weakly scattering material. These techniques have many potential applications that range from the development of failure mechanisms in different media to the detection of retinal disease. First, these authors analyze low coherent interferometry (LCI) that involves illumination of semitransparent scattering test objects with a broadband source and scanning the sample through the required depth range. After describing some limitations of the latter approach, the authors present two recently developed techniques called wavelength scanning interferometry (WSI) and tilt scanning interferometry (TSI), which have some practical advantages over LCI, the most important being an improved signal-to-noise ratio. In WSI, temporal sequences of speckle interferograms are recorded while the wavelength of the laser is tuned at a constant rate, so this technique needs a special light source. TSI is based on tilting an illuminating single wavelength beam during the acquisition of an image sequence, a procedure that provides the necessary depth-dependent phase shifts that allow the reconstruction of the object structure and its internal displacements. A theoretical framework is presented to allow the visualization of the spatial resolution and displacement component measured by any of the techniques presented in this chapter. The chapter concludes with a section on recent developments on phase unwrapping in three dimensions.

Chapter 3 presents different spatial methods to evaluate phase distributions from single-image interferograms. This chapter analyzes not only the different approaches that can be used when a spatial carrier is introduced in the interferometric data but also the more difficult task of automatic demodulation of a single interferogram containing no carrier. First, the authors review the well-known Fourier transform method to recover the phase of a single interferogram with a spatial carrier frequency and also different spatial phase shifting algorithms. They also describe various asynchronous algorithms that do not require the knowledge of the carrier frequency. In the rest of the chapter, the authors present several techniques to recover the modulating phase from a single-image interferogram without a carrier. Among the various approaches that are described, we can mention the regularized phase tracking technique and also local adaptable robust quadrature filters that do not require previous fringe normalization. They also describe single-interferogram demodulation methods based on the determination of fringe orientation, on the vortex transform, and on a general n-dimensional quadrature transform.

Chapter 4 reviews temporal speckle pattern interferometry (TSPI) and discusses several approaches that can be used to analyze the recorded data. TSPI was mainly developed during the past decade to measure the temporal evolution of low-speed dynamic deformation fields. It is based on the analysis of a time series of speckle interferograms, which codes the temporal and spatial phase changes produced during the dynamic deformation of the object. In TSPI, the optical phase distribution is extracted from the speckle intensity at each pixel independent of all the other pixels in the image, so that phase unwrapping is also performed as a function of time. As temporal phase unwrapping involves only 1D signals, this procedure is generally much easier to carry out than spatial 2D unwrapping. Until recently, the most common phase recovery technique used in TSPI was the Fourier transform due to its simplicity and short computational time. However, TSPI signals frequently present nonmodulated pixels, modulation loss and noise that affect the bias, and the modulation intensity terms of the signals to be analyzed. This chapter also describes more robust phase recovery methods that were mainly applied in the past decade to analyze TSPI signals. The numerical algorithms described in this chapter include 1D approaches based on the Fourier transform, the windowed Fourier transform, wavelet transforms, the S-transform, quadratic time–frequency distributions, and the empirical mode decomposition and the Hilbert transform. Two-dimensional and 3D approaches based on the windowed Fourier and directional wavelet transforms are also described.

One of the features that characterize a speckle field is the presence of points where both the real and the imaginary parts of the complex amplitude are equal to zero. Therefore, the intensity at this discrete number of points is also zero and their phase is not defined. Phase singularities or optical vortices can be associated with a sign or charge depending on whether the phase rotates clockwise around them or not. Chapter 5 presents some latest work on the application of optical vortices in optical metrology. These techniques are based on the fact that phase singularities are well-defined geometrical points with unique core structures and spatial configuration, which serve as unique fingerprints and present valuable information as identifiable markers. The authors also present various applications of the so-called optical vortex metrology, such as the measurement of nanometric displacements that use the information on the locations of phase singularities before and after the introduction of a displacement. Other applications presented in this chapter include the determination of rotational displacements and the use of optical vortices both for fluid mechanical investigations and for tracking the dynamics of a biological specimen.

Finally, due to the growing importance of security applications, Chapter 6 deals with the application of speckles for coding optical and digital data. In an encryption system, the information is encoded in such a way that the original information is revealed only by applying the correct key. To realize this aspect of security need, most of the optical architectures use random phase masks for coding and encryption. The phase masks made using speckle patterns work well as random phase masks, and have been used in optical and digital encoding of information. This chapter presents a broad review of various coding techniques used for optical and digital data security applications. Various speckle coding techniques are then discussed with an emphasis on the work carried out by the authors' group. This includes methods for preparation of speckle masks and techniques for their easy alignment, the use of elongated speckle patterns and various multiplexing techniques. The chapter also includes a large number of references that will be very useful for the reader interested in the implementation of these approaches.

To conclude, I would like to thank the authors of the different chapters for their contributions and cooperation. I also wish to thank Valerie Molière of Wiley-VCH Verlag GmbH for inviting me to edit this book and to Anja Tschörtner from the same editorial department for her help and support.

Last but not least, I am grateful to my wife Carolina who has tolerated me with patience over the past difficult year while I was recovering from various health problems.

Guillermo H. Kaufmann

Rosario, Argentina

July 2010

List of Contributors

Armando Albertazzi Gonçalves Jr.

Federal University of Santa Catarina

Department of Mechanical Engineering

Laboratory of Metrology and Automation

UFSC Campus

88040-970 Florianopolis

Brazil

Julio Estrada

Centro de Investigaciones en Optica

Loma del Bosque 115

Colonia Lomas del Campestre

37150 Leon

Mexico

Alejandro Federico

Instituto Nacional de Tecnología Industrial

Electrónica e Informática

Av. General Paz 5445

B1650KNA San Martín

Argentina

Steen G. Hanson

Technical University of Denmark

Department of Photonics Engineering

DTU Fotonik, Frederiksborgvej 399

Building 128

P.O. Box 49

4000 Roskilde

Denmark

Jonathan M. Huntley

Loughborough University

Wolfson School of Mechanical and Manufacturing Engineering

Ashby Road

Loughborough LE11 3TU

United Kingdom

Guillermo H. Kaufmann

Instituto de Fisica Rosario

Blvd. 27 de Febrero 210 bis

S2000EZP Rosario

Argentina

Arvind Kumar

Indian Institute of Technology Delhi

Department of Physics

Hauz Khas

New Delhi 110016

India

Antonio Quiroga

Universidad Complutense de Madrid

Facultad de Ciencias Fisicas

Departamento de Optica

Cuidad Universitaria

28040 Madrid

Spain

Pablo D. Ruiz

Loughborough University

Wolfson School of Mechanical and Manufacturing Engineering

Ashby Road

Loughborough LE11 3TU

United Kingdom

Manuel Servin

Centro de Investigaciones en Optica

Loma del Bosque 115

Colonia Lomas del Campestre

37150 Leon

Mexico

Kehar Singh

Indian Institute of Technology Delhi

Department of Physics

Hauz Khas

New Delhi 110016

India

Madan Singh

Instruments Design, Development and Facility Centre

Staff Road

Ambala 133001

India

Mitsuo Takeda

The University of Electro-Communications

Department of Information and Communication Engineering

Laboratory for Information Photonics and Wave Signal Processing

1-5-1, Chofugaoka, Chofu

182-8585 Tokyo

Japan

Matías R. Viotti

Federal University of Santa Catarina

Department of Mechanical Engineering

Laboratory of Metrology and Automation

UFSC Campus

88040-970 Florianopolis

Brazil

Wei Wang

Heriot-Watt University

School of Engineering and Physical Sciences

Department of Mechanical Engineering

Edinburgh EH14 4AS

United Kingdom

Chapter 1

Radial Speckle Interferometry and Applications

Armando Albertazzi Gonçalves Jr. and Matías R. Viotti

1.1 Introduction

The invention of laser in the 1960s led to the development of sources of light with a high degree of coherence and allowed to see a new effect with a grainy aspect, which appeared when optically rough surfaces were illuminated with a laser light. This effect was called speckle effect characterized by a random distribution of the scattered light.

After the advent of laser sources, this effect was considered a mere nuisance, mainly for holography techniques. Nevertheless, important research efforts began in the late 1960s and early 1970s, focusing on the development of new methods for performing high-sensitivity measurements on diffusely reflecting surfaces. These efforts paved the way for the development of electronic speckle pattern interferometry (ESPI), the basic principle of which was to combine speckle interferometry with electronic detection and processing. ESPI avoided the awkward and high time-consuming need for film processing, thus allowing real-time measurement of the object. However, first results were a bit discouraging due to low detector resolution, low sensitivity, and high signal-to-noise ratio.

Constant advances in technology, particularly with respect to high resolution and speed data acquisition systems, and software development for data processing allowed linking, first, vacuum-tube television cameras or, until today, CCD or CMOS cameras to a host computer in order to acquire a digital image of the surface illuminated with laser light. Advances in data transmission enabled to directly link cameras to the computer (IEEE-1394 interface) and transmit digital images without extra elements to digitize the acquired image (such as the well-known frame grabbers). Because of the use of both digital images and processing techniques, ESPI was called DSPI (digital speckle pattern interferometry).

Nowadays, there are a large number of interferometric systems that allow to monitor a large variety of physical parameters. They can be mainly grouped in two families: (i) interferometers with sensitivity to out-of-plane displacements and (ii) interferometers with in-plane sensitivity. Several approaches can be put in these two families. Among them, radial interferometers can be highlighted, a special class of interferometers that are able to measure in polar or cylindrical coordinates. Radial out-of-plane interferometers are very convenient for some engineering applications dealing with measurement of deformations in pipes, bearings, and other cylinders. Since radial in-plane interferometers can be made in a robust and compact way, they are also of great engineering interest as they allow small interferometers to perform measurements outside the laboratory. This kind of interferometers will be discussed in the following sections. Section 1.2 will describe radial out-of-plane interferometers to measure internal and external cylinders. In-plane interferometers will be discussed in Section 1.3, showing two different configurations. Finally, Section 1.4 will show some applications of in-plane radial interferometers.

1.2 Out-of-Plane Radial Measurement

Perhaps the simplest way to measure the out-of-plane displacement component on a surface is by illuminating it and viewing it in the normal direction. Figure 1.1 shows a possible configuration for out-of-plane measurement in Cartesian coordinates. The laser light is expanded and collimated by the lens and is directed to a partial plane mirror that splits the laser light into two beams. Part of the light is deflected to the right and illuminates the rough surface to be measured, which scatters the light forming a speckle pattern. The other part is transmitted through the partial plane mirror and illuminates a rough surface that produces another speckle pattern, which is taken as a reference. The camera captures both images of the measured surface, viewed through the partial plane mirror, and the image of the reference surface reflected by the partial plane mirror. The resulting image shows the coherent interference of the two speckle patterns emerging from both surfaces. A piezo translator (PZT) is used to move the reference surface in a submicrometric range to produce controlled phase shifts.

Figure 1.1 A typical optical setup to obtain out-of-plane sensitivity.

The sensitivity direction of this configuration is represented by the vector drawn on the surface to be measured. It is computed by the vector addition of two unitary vectors pointing to the illumination source and to the camera pupil center, respectively. In this case, since both are practically aligned with the z-axis, the sensitivity vector is also almost aligned with the z-axis and its magnitude is very close to 2.0. For the case of illumination with collimated light and imaging through telecentric lenses, the sensitivity vector is equal to 2.0 and perfectly parallel to the z-axis. Therefore, in this case the sensitivity vector has a component only along the z-axis and it is given by Equation 1.1:

(1.1)

The out-of-plane displacement component w along the z-axis between two object states can be computed from the measured phase difference Δϕ by Equation 1.2:

(1.2)

In some cases where noncollimated illumination is used or nontelecentric imaging is involved, Equation 1.1 has to be modified to accomplish for a small amount of in-plane sensitivity. Those cases are discussed in Refs [1, 2].

The meaning of radial out-of-plane measurement here is related to the measurement of the displacement component normal to a cylindrical surface or, in other words, in the direction of the radius of the cylinder. As usual in cylindrical coordinates, a positive radial out-of-plane displacement increases the value of the radius. Radial out-of-plane displacement components are very important in engineering applications. They are responsible for the diameter and form deviations of cylindrical surfaces, which are very closely connected to the technical performance of cylindrical parts. Therefore, sometimes they are referred to as radial out-of-plane deformations. Since the measured quantity is the displacement field between two object states, the expression radial out-of-plane displacement is preferred in this chapter.

Pure radial out-of-plane displacement measurement can be accomplished only by DSPI using special optics. The main idea is to use optical elements to promote illumination and viewing directions that result in radial sensitivity. This section presents possible configurations for three application classes: short internal cylinders, long internal cylinders, and external cylinders.

1.2.1 Radial Deformation Measurement of Short Internal Cylinders

To measure the radial out-of-plane displacement component, special optical elements are required. Ideally, it should optically transform Cartesian coordinates into cylindrical ones. In 1991, Gilbert and Matthys [2, 3] used two panoramic annular lenses to obtain out-of-plane radial sensitivity. This special lens produces a 360° panoramic view of the scene. When introduced inside a cylinder, such lenses image the inner surface of the cylinder from a near-radial direction. They used two lenses: one panoramic annular lens to illuminate the inner surface of the cylinder in a near-radial direction and another one in the opposite side for imaging. The measurement was possible in a cylindrical ring region between both lenses.

Another possibility to produce radial sensitivity is by using conical mirrors. Figure 1.2 shows the very interesting optical transformation produced by a 45° conical mirror when it is introduced inside an inner cylindrical surface and is aligned with the cylinder axis. When viewed from left to right, the inner surface of the cylinder is reflected on the conical mirror surface all the way around 360°, producing a panoramic image. If the observer is far enough, the inner cylindrical surface is optically transformed into a virtual flat disk. Therefore, the out-of plane displacement component of this virtual flat disk corresponds to the radial out-of-plane displacement component.

Figure 1.2 Optical transformation produced by a conical mirror placed inside a cylindrical surface.

Figure 1.3 shows a possible optical setup to measure the radial out-of-plane displacement component of an inner cylinder. A 45° conical mirror is placed inside the internal cylindrical surface to be measured and is aligned to the cylinder axis. Laser light is collimated and split by a partial mirror into two beams: the active and the reference beams. The active beam is deflected toward the conical mirror. The light that reaches the conical mirror is deflected toward the internal surface of the inner cylinder and reaches it orthogonally, producing a speckle field. The light coming back from the speckle field of the cylindrical surface is reflected back by the conical mirror, goes through the partial plane mirror, and is imaged by the camera lens. The reference beam reaches the reference surface, produces a speckle field, and is reflected back to the partial plane mirror and imaged by the camera lens at the same time. The two speckle fields imaged by the camera lens interfere coherently, and the resulting intensities are grabbed by the camera and digitally processed. A piezoelectric translator is placed behind the reference surface to displace it and apply phase shifting to improve image processing capabilities.

Figure 1.3 Basic configuration for radial out-of-plane displacement measurements of short cylinders using a 45° conical mirror.

If collimated light is used for illumination and a telecentric imaging system is used, or the camera is far enough, the sensitivity vector is always radial and with constant magnitude equal to 2.0. The radial out-of-plane displacement component ur between two object states is computed for each point on the measured region from the phase difference Δϕ by Equation 1.3:

(1.3)

The measurement depth along the cylinder axis is limited by the conical mirror dimensions. Since the conical mirror angle is 45°, its radius cannot be greater than the inner cylinder radius, which makes the maximum theoretically possible measurement depth to be equal to the conical mirror radius.

In practice, the measurement depth along the cylinder axis is smaller. The image reflected by the conical mirror becomes very compressed near the conical mirror vertex, which reduces the lateral resolution of the reflected image by an unacceptable level. Therefore, the practical measuring limit is about two-thirds of the conical mirror radius. The inner third of the image of the virtual flat disk is not used at all.

In order to reconstruct the radial out-of-plane displacement field on the cylindrical surface, and to present the results in an appropriate way, a numerical mapping can be applied. Figure 1.4a represents the camera view. The gray area corresponds to the measurement region on the cylindrical surface. A point P in such image corresponds to a defined position in the cylindrical surface, as shown in Figure 1.4b. The geometrical mapping is straightforward and can be done by the set of Equation 1.4:

Figure 1.4 Relationship between the virtual flat disk (a) and the cylindrical surface (b).

(1.4)

where X, Y, and Z are Cartesian coordinates of points on the cylindrical surface, RC is the reconstructed cylinder radius, x and y are Cartesian coordinates in the image plane, is the polar radius in the image plane, is the polar angle in both image plane and cylindrical coordinates, ri is the inner radius of the region of interest in the image plane, and M is a calibration constant related to image magnification.

In most engineering applications, only the radial deformation of the cylindrical surface is of interest since it produces form deviations. However, in practice, it is almost inevitable that some amounts of rigid body motion – translations and rotations – are superimposed onto the radial deformation component. That comes from the limited stiffness of the mechanical fixture that is unable to keep the conical mirror and/or the cylindrical part to be measured unchanged in the exact place. Fortunately, it is possible to compensate small translations and tilts with the help of software.

A small amount Δx of lateral translation in the X-direction in the cylinder to be measured will produce radial displacement components δr that are not constant in all directions, but depend upon the cosine of the polar angle θ. It is given by

(1.5)

where δr is the radial displacement, Δx is the amount of lateral displacement in the X-direction, r is the radius, and θ is the polar angle. Note that δr depends on cos(θ) and that the coefficient of cos(θ) is the translation amount Δx.

The amount of rigid body translations in both X- and Y-axes in a given cross section can be determined from the Fourier series coefficients. To do that, the radial displacement field ur must be determined all the way around 360° along a circle that corresponds to such section as a function of the polar angle θ. The amount of translation can be computed by the first-order Fourier coefficients

(1.6)

where Δx and Δy are the rigid body translation components in the X- and Y-axes, respectively, and ur(θ) is the radial displacement component all the way around this section.

The above procedure can be repeated for each section of the conical mirror. It is then possible to compute the mean translation components for each different section along the cylindrical surface. Here, if all translations have the same value and direction, it means that only rigid body translation is present. If not, a relative rotation between the mirror and the cylinder axis happens and/or there is a kind of bending of the cylinder axis due to deformation.

If it is possible to connect all different rigid body translation vector ends of each cylinder section by the same straight line, it means that a rigid body rotation is present. In order to quantify the amount of rotation, one can apply linear regression for all Δx and another linear regression for Δy for all sections. The obtained slope is related to the rotation components of xz and yz planes. Then, these rotation values can be used to mathematically compensate that undesirable effect. It is important to make it clear that even if the rotation is superimposed onto any other kind of displacement pattern, this procedure can quantify and remove only the rigid body rotations and displacement components, without affecting or distorting the remaining displacement field.

1.2.2 Radial Deformation Measurement of Long Internal Cylinders

There are a large number of practical applications where longer cylinders have to be measured. For these cases, the configuration present in the previous section is limited by the maximum measurement depth of two-thirds of the conical mirror radius. One possibility would be to measure the cylinder deformations in a piecewise manner. The idea is to divide the cylinder in few virtual sections and measure each of them sequentially. The data are separately processed and then stitched together to produce the total results. However, this approach requires an excellent loading repeatability, very stable experimental conditions, and is an intensively time-consuming procedure. Consequently, this piecewise approach is not practical.

In most engineering applications, the deformation of cylindrical surfaces does not need to be known for each point on the surface. It could be good enough to measure the deformation field in few separate measurement rings, each one in a different section. Therefore, the idea of a piecewise measurement comes back, but it must be done simultaneously.

A special design of a stepped 45° conical mirror can be used to make possible the simultaneous measurement of radial out-of-plane displacements of long inner cylinders [4]. The main idea is presented in Figure 1.5. The continuous 45° conical mirror is replaced for a stepped version. In this figure, four conical sections are separated apart by three cylindrical connecting rods. Each conical section of the stepped mirror reflects the collimated light and forms a measurement ring where the radial out-of-plane measurement is done. The gap between each conical section of the stepped conical mirror is not measured at all. In practice, the lack of this information is not important in most applications where the radial deformations fields are quite smooth. In these cases, the information in few equally spaced sections is sufficient to describe the main behavior of the cylindrical part from the engineering point of view. Only four measurement zones are represented in the figure for simplicity. In practice, a larger number of measurement zones can be achieved.

Figure 1.5 Basic configuration for radial out-of-plane displacement measurements of long cylinders using a stepped 45° conical mirror.

Figure 1.6 shows an example of an actual stepped conical mirror with seven measuring zones. It was designed for a specific application, which required the length to be about 34 mm. It was machined in copper in a high-precision diamond turning machine and a layer of titanium was applied to increase the reflectivity and to protect the reflecting surface against mechanical damages. The reflecting areas are oriented at 45° with respect to the mirror axis. The regions in between the reflecting areas have a negative conical angle due to the geometry constraints of the available diamond tool used in the machining process. In practice, it is not possible to use the first conical section for measurement since it is too small and the lateral resolution of the image reflected on that area is unacceptably poor.

Figure 1.6 Actual view of the seven sections of a stepped 45° conical mirror.

The stepped conical mirror of Figure 1.6 was used in a configuration similar to Figure 1.5 to measure the deformations of an inner cylinder of a hermetic gas compressor used in domestic refrigerators. The goal was to study the effects of tightening the four clamping bolts, shown in Figure 1.7 under the four vertical arrows, on the shape of the inner cylinder of the compressor. A set of four 90° phase-shifted images was acquired with an equal initial torque level applied to all bolts. The corresponding phase pattern was stored as the reference phase pattern. After that, the final torque level was applied to the four bolts and another sequence of four 90° phase-shifted images was acquired and the loaded phase pattern was computed and stored.

Figure 1.7 Deformation of the inner cylinder of a hermetic gas compressor was measured after tightening four bolts.

The resulting phase difference can be seen in Figure 1.8. The top left side of the figure shows the natural image. Seven annular regions can be distinguished, each one corresponding to each conical mirror section and to the radial displacement field of a different section in the inner cylinder. Fringe discontinuities can be present between neighbor annular regions since there is no surface continuity between them. A polar to Cartesian mapping was first applied to extract data. The resulting image is shown on the right-hand side of the figure. The horizontal axis corresponds to the polar angle. The vertical axis is related to the radius, which is connected to the axial position of the measured ring. Seven horizontal stripes are visible in this image. The first one in the bottom corresponds to the first section on the nose of the conical mirror. The poor lateral resolution of this stripe is evident in this image. Finally, the bottom left image is the low-pass filtered version of the previous image.

Figure 1.8 Phase difference pattern on the stepped conical mirror surface. Top left: the original image. Right: after a polar to Cartesian mapping. Bottom left: low-pass filtered version.

One line was extracted from the center of each stripe and processed. The radial displacement field for six sections was computed. The results are shown in Figure 1.9. Figure 1.9a shows a polar diagram of all sections. The scale division is 1.0 µm. A 3D representation of the deformed cylinder is presented in Figure 1.9b on a much exaggerated scale. This analysis is very useful in engineering for understanding the optimization of the design for stiffness of high-precision cylindrical surfaces.

Figure 1.9 (a and b) Measurement results for the deformation of the inner cylindrical surface.

1.2.3 Radial Deformation Measurement of External Cylinders

Radial out-of-plane displacement components can also be measured on external cylindrical surfaces by DSPI. The main idea is represented in Figure 1.10: an internal 45° conical mirror produces an appropriate optical transformation that maps the external cylindrical surface into a flat virtual disk. The ray diagram in Figure 1.10a makes it clear that parallel rays are reflected by the conical mirror and are transformed in radial rays. Figure 1.10b shows an example of a small piston inside a 45° inner conical mirror. The central part shows the upper part (top) of the piston. The cylindrical surface is reflected on the conical mirror and is transformed into a flat disk. The two lateral circular bearings (pinholes) are also visible on the virtual disk area and are distorted due to the reflection on the conical mirror surface.

Figure 1.10 (a and b) Optical transformation produced in a cylindrical surface due to an internal 45° conical mirror.

The DSPI interferometer to measure the radial out-of-plane displacement component is schematically shown in Figure 1.11. The part to be measured is placed and aligned in a 45° external conical mirror. To measure only the radial out-of-plane component, the angle of the conical mirror should be 45° and both illumination source and viewing directions must come from infinity. That can be obtained with collimated illumination and a telecentric imaging system. However, if the diameter of the conical mirror is quite large, collimated illumination and telecentric imaging costs become prohibitive. For these cases, the configuration of Figure 1.12 is feasible since some degree of axial sensitivity is tolerated. Alternatively, to obtain pure radial sensitivity to measure large cylinders, the 45° conical mirror of Figure 1.12 can be replaced by a quasi-conical mirror with curved reflecting surface calculated in such a way to reflect the diverging light coming from a point source like it was a collimated (plane) wavefront and to generate radial illumination and viewing on the cylindrical surface. However, the manufacturing of such special curved mirror can be very expensive.

Figure 1.11 Basic configuration for pure radial out-of-plane displacement measurements of external cylinders using a 45° internal conical mirror, collimated light, and telecentric imaging.

Figure 1.12 Basic configuration for quasi-radial out-of-plane displacement measurements of large external cylinders using a 45° internal conical mirror.

The configuration of Figure 1.12 was used to measure the thermal deformation of an automotive engine piston [37]. It is made of aluminum and has some steel inserts used to control the thermal deformation and the shape of the engine piston at high temperatures. The way both materials interact and the resulting deformation mechanism were of interest in this investigation.

A large stainless steel conical mirror was used and the engine piston was mounted inside it. Electrical wires were wrapped in the groove of the first piston ring for heating the piston close to its crown. Controlled current levels were applied for heating the piston incrementally. Figure 1.13a shows the camera view of the piston inside the conical mirror. The groove of the first ring was filled with heating wires and covered with thermal paste. The next two grooves are clearly visible as darker circular lines near the maximum diameter. The pinhole of the piston looks distorted due to reflection in the conical mirror. A set of four 90° phase-shifted images was first acquired and the reference phase pattern was computed and stored. A controlled current was applied in order to increase the piston temperature to about 1 K. After the temperature stabilized, another series of four 90° phase-shifted images were acquired and another phase pattern computed and stored. The phase difference pattern is shown in Figure 1.13b. From the phase difference pattern, it is possible to see that the shape deviation is much stronger in the central part of the image, which corresponds to the bottom of the piston, and less intense near the crown. This happens due to the presence of the steel inserts located somewhere between the crown and the bottom of the piston. This effect can be clearly seen after extracting and analyzing the behavior of the four sections represented in Figure 1.14. The section represented in polar coordinates in Figure 1.14a was extracted from the bottom of the piston, where strong shape deformations are present. The sections in Figure 1.14b–d are located closer to the piston crown, where the shape deformations are smaller. Finally, a 3D plot of the deformed piston is represented on a much exaggerated scale in Figure 1.15. The piston crown is located in the left part of the figure.

Figure 1.13 (a) Camera view of the engine piston reflected by the conical mirror. (b) The phase difference pattern after heating the engine piston.

Figure 1.14 (a–d) Polar graphics of the thermal deformations of four sections of the engine piston after heating.

Figure 1.15 3D representation of the thermal deformation of the engine piston after heating.

1.3 In-Plane Measurement

Optical configurations for measuring in-plane displacements are usually based on the two-beam illumination arrangement first described by Leendertz in 1970 [5]. These interferometers are generally capable of measuring the displacement component, which is coincident with the in-plane direction.

Figure 1.16 shows the basic setup for this kind of interferometer. Two expanded and eventually collimated beams illuminate the object surface forming two angles with the direction of illumination, namely, and . Thus, two speckle distributions coming from the object surface, with their respective sensitivity vectors and , interfere in the imaging plane of the camera. The change in the speckle phase will be [1]

(1.7)

where represents the resultant sensitivity vector obtained from the subtraction between the sensitivity vectors from every beam and it becomes perpendicular to the -direction of observation when . In this case, if the illumination vectors are in the plane, the net sensitivity can be expressed as [1]

(1.8)

where is the component of the sensitivity vector along the -direction and is the wavelength of light source. According to this equation, it is noted that can be changed in order to adjust the sensitivity of the interferometer from zero (illumination perpendicular to the object surface) to a maximum limit value of (illumination parallel to the object surface).

Figure 1.16 Optical setup to obtain in-plane sensitivity.

To obtain the phase difference for two object states, Equation 1.8 should be substituted into Equation 1.7:

(1.9)

where is the component of the displacement field along the -direction.

For this kind of interferometer, maximum visibility of subtraction fringes will be obtained when the optical system correctly resolves every speckle produced by the scattering surface and the ratio between both illumination beams intensities is equal to 1 [6].

Figure 1.17 shows a drawing of a conventional in-plane digital speckle pattern interferometer with symmetrical dual-beam illumination. According to this figure, two expanders are used to illuminate the object. As the distance between the object and the expander lens is a hundred times larger than the measurement region, the variation in the sensitivity vector across the field of view can be considered negligible.

Figure 1.17 Dual-beam illumination interferometer. LA, He–Ne laser; BM, beam splitter; M1 and M2, mirrors; PZT, piezoelectric-driven mirror; L, lens; CCD, camera; CU, control unit; PC, personal computer; TS, test specimen.

In practical situations, three-dimensional displacement fields are frequently separated in one component normal to the surface to be measured and two components along the tangential direction. For a plane or smooth surface, the former will be known as the out-of-plane displacement component and latter ones as in-plane components. In-plane displacements are more interesting mainly for engineering applications where the main task is to determine strain and stress fields applied in mechanical parts when their integrity has to be evaluated. Nowadays, electrical strain gauges are the most widely used devices in industrial and academic laboratories to monitor strain and stress fields [7]. Even though portability, robustness, accuracy, and range of measurement of strain gauges have been firmly established, their installation is time consuming and requires skills and aptitude of a well-trained technician.

The interferometer shown in Figure 1.17 presents sensitivity in only one direction (1D sensitivity). An important requirement in many engineering measurements is to simultaneously compute both in-plane components [1] necessary to measure in two determined directions (2D sensitivity). These systems are made of two interferometers sensitive to two orthogonal displacement directions and are based on polarization discrimination methods by using a polarizing beam splitter that splits the laser beam into two orthogonal linearly polarized beams [8, 9]. Thus, it is possible to simultaneously measure both displacement components. Two drawbacks can be found for this approach, namely, (i) test surface can appreciably depolarize the two orthogonal polarized dual-beam illumination sets causing cross interference between them and (ii) optical setup becomes more bulky and complex. References [10, 11] have managed to deal with these limitations by developing a novel double illumination DSPI system. This interferometer presents an optical arrangement that gives radial in-plane sensitivity and its first version will be described in detail in the following section.

1.3.1 Configuration Using Conical Mirrors

Figure 1.18 shows a cross section of the interferometer used to obtain radial in-plane sensitivity [10–12]. The most important component is a conical mirror that is positioned close to the specimen surface. This figure also displays two particular light rays chosen from the collimated illumination source. Each light ray is reflected by the conical mirror surface toward a point P over the specimen surface, reaching it with the same incidence angle. The illumination directions are indicated by the unitary vectors nA and nB and the sensitivity direction is given by the vector k obtained from the subtraction of both unitary vectors. As the angle is the same for both light rays, in-plane sensitivity is reached at point P. Over the same cross section and for any other point over the specimen surface, it can be verified that there is only one couple of light rays that merge at that point. Also, in the cross section shown in Figure 1.18, the incidence angle is always the same for every point over the specimen surface and symmetric with respect to the mirror axis. By taking into account unitary vectors and by comparing Figures 1.16 and 1.18, the reader can note similarities in both configurations. As a consequence, if the direction of the normal of the specimen surface and the axis of the conical mirror are parallel to each other, then nA and nB will have the same angle. Therefore, the sensitivity vector k will be parallel to the specimen surface and in-plane sensitivity will be obtained.

Figure 1.18 Cross section of the upper and lower parts of the conical mirror to show the radial in-plane sensitivity of the interferometer.

The above description can be extended to any other cross sections of the conical mirror. If the central point is kept out from this analysis, it can be demonstrated that each point of the specimen surface is illuminated by only one pair of the light rays. As both rays are coplanar with the mirror axis and symmetrically oriented to it, a full 360° radial in-plane sensitivity is obtained for a circular region over the specimen.

A practical configuration of the radial in-plane interferometer is shown in Figure 1.19. The light from a diode laser is expanded and collimated via two convergent lenses and the collimated beam is reflected toward the conical mirror by a mirror that forms a 45° angle with the axis of the conical mirror. The central hole placed on this mirror prevents the laser light from directly reaching the sample surface having triple illumination and provides a viewing window for the CCD camera.

Figure 1.19 Optical arrangement of the radial in-plane interferometer.

The intensity of the light is not constant over the whole circular illuminated area on the specimen surface and it is particularly higher at the central point because it receives light contribution from all cross sections. As a result, a very bright spot will be visible in the central part of the circular measurement region and consequently fringe quality will be reduced. To reduce this effect, the conical mirror is formed by two parts with a small gap between them. The distance of this gap is adapted in such a way that the light rays reflected at the center are blocked. Thus, a small circular shadow is created in the center of the illuminated area and fringe blurring is avoided.

As can be seen from Figure 1.19, for each point over the specimen the two rays of the double illumination originate from the reflection of the upper and lower parts of the conical mirror. A piezo translator was used to join the upper part of the conical mirror, so that its lower part is fixed while the upper part is mobile. As a consequence, the PZT moves the upper part of the conical mirror along its axial direction and the gap between both parts is increased. Then, a small optical path change between both light rays that intersect on each point is produced and the PZT device allows the introduction of a phase shift to evaluate the optical phase distribution by means of any phase shifting algorithm [13].

Due to the use of collimated light, it can be verified that the optical path change is exactly the same for each point of the illuminated surface. The relation between the displacement of the piezoelectric transducer and the optical path change is given by the following equation [12, 14]:

(1.10)

where is the angle between the conical mirror axis and its surface in any cross section.

Finally, the radial in-plane displacement field can be calculated from the optical phase distribution [1]:

(1.11)

where is the wavelength of the laser and is the angle between the illumination direction and the normal direction of the specimen surface.

1.3.2 Configuration Using a Diffractive Optical Element

Two main drawbacks can be identified in the setup shown in Figure 1.19: (i) it uses a high-quality conical mirror that is quite expensive and (ii) it requires wavelength stabilization of the laser used as light source, which cannot be easily achieved for a compact and cheap diode laser. As a consequence, applications outside the laboratory can be difficult or even unfeasible.

As it is well known, diffractive structures can separate white light into its spectrum of colors. However, if the incident light is monochromatic, the grating will generate an array of regularly spaced beams in order to split and shape the wavefront beam [15]. The diffraction angle of the spaced beams is given by the well-known grating equation [15, 16]

(1.12)

where is the period of the grating structure and is the diffraction angle for the order m. From this equation, it is clear that the orders −1 and +1 have symmetrical angles with the incident rays.

The recent development of microlithography manufacturing allowed the production of diffractive optical elements (DOEs). The ability to manufacture diffraction gratings with a large variety of geometries and configurations made possible the development of a new and flexible family of optical elements with tailor-made functions. Diffractive lenses, beam splitters, and diffractive shaping optics are some examples of the many possibilities. A special diffractive optical element can be designed to achieve radial in-plane sensitivity with DSPI. It is made as a circular diffraction grating with a binary profile and a constant pitch pr as shown in Figure 1.20. Its geometry is like a disk with a clear aperture in the center.

Figure 1.20 Cross section of the diffractive optical element showing radial in-plane sensitivity.

If an axis-symmetric circular binary DOE (see Figure 1.20) is used instead of conical mirrors, a double illuminated circular area with radial in-plane sensitivity will also be achieved [17, 18]. The symmetry of the orders −1 and +1 will produce double illumination with symmetrical angles, which produces radial in-plane sensitivity. Some advantages can be found by comparing DOE and conical mirror usage: (i) due to advances in microlithography techniques, DOE manufacturing has reached a certain maturity that makes it less expensive than special fabricated conical mirrors, and (ii) because of dual-beam illumination setup, interferometer sensitivity is independent of the wavelength of the laser used as the light source, which will be discussed next.

By considering Equation 1.11, the corresponding fringe equation is as follows:

(1.13)

According to Equation 1.13, sensitivity of the method would change if angle or the wavelength of the light source is modified. For example, if angle is increased, sensitivity would also increase.

By observing Figure 1.20, it is evident that the diffraction angle and the angle between the direction of illumination and the normal to the specimen surface () have the same magnitude. Thus, . By substituting Equation 1.12 in Equation 1.11aa and by considering the first-order diffraction (m = 1)

(1.14)

In the same way, the corresponding fringe equation will be

(1.15)

Equations 1.14 and 1.15 show that the relationship between the displacement field and the optical phase distribution depends only on the period of the grating of the DOE and not on laser wavelength. This particular and curious effect can be understood through the following explanation: when wavelength of the illumination source increases/decreases, sine function of the diffraction angle decreases/increases by the same amount (see Equation 1.13). As is divided by in Equation 1.11, the ratio between them will be constant.

Reference [18] compares the influence on the sensitivity of the interferometer when a DOE is used instead of conical mirror. According to Viotti et al, when the setup shown in Figure 1.17 is used with a red light source or with a green one, phase maps obtained with the green laser had approximately 1.5 more fringes compared to those obtained with the red laser. Figure 1.21a shows a phase map obtained for a red light source and Figure 1.21b shows a phase map obtained for green light for the same displacement field.

Figure 1.21 Phase maps obtained by using the radial in-plane interferometer with conical mirror for wavelength light source of (a) 658 nm and (b) 532 nm [18].

On the other hand, Figure 1.22a and b shows the phase maps for the same displacement field obtained by using the diffractive optical element instead of the conical mirror. As the figure shows, it can be noted that fringe amounts are the same for both. Thus, Figure 1.22a and b clearly confirms the result obtained in Equation 1.14.

Figure 1.22 Phase maps obtained by using the radial in-plane interferometer with DOE for wavelength light source of (a) 658 nm and (b) 532 nm [18].

As shown in Figure 1.19, a similar optical arrangement can be built in order to integrate the diffractive optical element. This new practical configuration of the radial in-plane interferometer is shown in Figure 1.23. The light from a diode laser (L) is expanded by a plane concave lens (E). Then, it passes through the elliptical hole of the mirror M1, which forms a 45° angle with the axis of the DOE, illuminating mirrors M2 and M3 and being reflected back to the mirror M1. Thus, the central hole placed at M1 allows that the light coming from the laser source reaches mirrors M2 and M3. In addition, this hole has other functions, namely, (i) to prevent the laser light from directly reaching the specimen surface having triple illumination and (ii) to provide a viewing window for the CCD camera. Mirror M1 directs the expanded laser light to the lens (CL) in order to obtain an annular collimated beam. Finally, the light is diffracted by the DOE mainly in the first diffraction order toward the specimen surface. Residual nondiffracted light or light from higher diffraction is not considered a problem since this kind of light is not directed to the central measuring area on the specimen surface.

Figure 1.23 Optical arrangement of the radial in-plane interferometer with DOE.

M2 and M3 are two special circular mirrors. The former is joined to a piezoelectric actuator (PZT) and the later has a circular hole with a diameter slightly larger than the diameter of M2. Mirror M3 is fixed while M2 is mobile. The PZT actuator moves the mirror M2 along its axial direction generating a relative phase difference between the beam reflected by M2 (central beam) and the one reflected by M3 (external beam). The boundary between both beams is indicated in Figure 1.23 with dashed lines. According to this figure, it is possible to see that every point over the illuminated area receives one ray coming from M2 and other one from M3. Thus, PZT enables the introduction of a phase shift to calculate the optical phase distribution by means of phase shifting algorithms.

As stated before, the intensity of light is not constant over the whole circular illuminated area on the specimen surface and it is particularly higher at the central point because it receives light contribution from all cross sections. As a result, a very bright spot will be visible in the central part of the circular measurement region and consequently fringe quality will be reduced. For this reason, the outlier diameter of mirror M2 and the diameter of central hole of M3 are computed obtaining a gap of about 1.0 mm and blocking the light rays reflected to the center of the measurement area.

1.4 Applications

1.4.1 Translation and Mechanical Stress Measurements

The polar radial displacement field measured in a circular region provides sufficient information to characterize the mean level of both rigid body translations and strains or stresses that occur in that region. For uniform displacement, strain, or stress fields, the complete determination of the associated parameters is almost a straightforward process [19, 20]. In this section, rigid body computation will be analyzed. Mechanical stress field computation will be considered in the next section.

If a uniform in-plane translation is applied on the specimen surface, the following radial displacement field is developed:

(1.16)

where ur is the radial component of the in-plane displacement, ut is the amount of uniform translation, α is the angle that defines the translation direction, and r and θ are polar coordinates. Readers can note that the displacement field does not depend on the radius r at all.

When a uniform stress field is applied to the measured region, the radial in-plane displacement field can be derived from the linear strain–displacement or stress–displacement relations. Usually x and y Cartesian coordinates are used to describe strain or stress states. Since the radial in-plane speckle interferometer measures polar coordinates, the strain and stress states are better described in terms of the principal axes 1 and 2, where the strains and stresses assume the maximum and minimum values, respectively. If is the angle that the principal axis 1 forms with the x-axis, the in-plane radial displacement field is related to the principal strain and stress components by the following equations [21]:

(1.17)

(1.18)

where and are the principal strains, and are the principal stresses, and are the material's Young modulus and Poisson ratio, respectively, and is the principal angle.

Figure 1.24 shows two examples of interferograms obtained with the radial in-plane speckle interferometer. The phase difference patterns correspond to the radial displacement component. Figure 1.24a corresponds to a displacement pattern of pure translation of about ut = 1.5 µm in the direction of α = 120° with the horizontal axis. Note that the fringes caused by pure translation are straight lines pointing to the polar origin. This behavior is predicted by Equation 1.16 since the radial displacement component is independent of the radius r. The phase difference pattern of Figure 1.24b is due to a single stress state of about 40 MPa applied in a steel specimen in the vertical direction. Note that, due to Poisson's effect, the number of fringes in the vertical axis is about three times larger than that in the horizontal one.

Figure 1.24 Two wrapped phase maps obtained with the radial in-plane speckle interferometer: (a) is due to pure translation and (b) is due to a uniaxial stress field applied in the vertical direction.

In order to quantify the rigid body translations or mechanical stress fields from the measured radial in-plane displacement field, two approaches can be used, namely, (i) the Fourier approach or (ii) the least squares one.

The former uses data of a single sampling circle, concentric with the polar origin, and the latter uses the whole image.

For the Fourier approach, a finite number of regularly spaced sampling points can be extracted from the same circular line all the way around 360°. From this data set, the first three Fourier series coefficients are computed by Equation 1.19. To determine the amount of translation ut, it is necessary to compute the sine and cosine components and the total magnitude of the first Fourier series coefficient by [21]

(1.19)

where rs is the sampling radius, HnS(rs) and HnC(rs) are, respectively, the sine and cosine component of the nth Fourier series coefficient, and HnS(rs) is the total magnitude of the nth harmonic. As a singular case, readers can note that if n = 0, components H0S(rs) = 0 and H0C(rs) = H0(rs) will be equal to the mean value of ur(rs, θ) along the sampling radius rs.

To compute the translation component ut, Equation 1.16 can be expanded to

(1.20)

In this case, only the first harmonic is present. The translation amount ut and its direction α can be computed from the first Fourier series coefficient by

(1.21)

In the same way, the cos term of Equation 1.18 can be expanded to obtain

(1.22)

As stated before, it is possible to verify that the principal stresses and direction can be determined from the zero- and second-order Fourier coefficients by

(1.23)

In practical situations, it is very usual that both stresses and rigid body translations appear mixed up in the same interferogram. They can be measured simultaneously and computed independently since different Fourier series coefficients are involved and the terms of a Fourier series are mutually orthogonal.

The other approach is based on the least squares method. In this approach, a set of experimental data is sampled from the measured displacement field. No particular sampling strategy is required, but it is a good practice to select sampling points regularly distributed over all measured region. The sampled data are fitted to a mathematical model by least squares. An appropriate mathematical model can be obtained by adding and rewriting Equations 1.20 and 1.22:

(1.24)

Terms K0R, K1C, K1S, K2C, and K2S are easily identified by comparison with Equations 1.21 and 1.23. K0