The theory of speckle dynamics in the image plane of a reflecting and thin transparent object is considered. It was assumed that the optical paths of the reflected and probing transparent object waves vary due to (1) translational motion, (2) oscillations with a period T, and (3) random relative displacements of pairs of scattering centers Δ u (reflecting object) and random changes in the refractive index Δ n (transparent object). The formulas relating the mean value, dispersion, and relaxation time of Δ u and Δ n values with the time-averaged radiation intensity at the observation point and the time autocorrelation function of this intensity are obtained. It is shown that at the averaging time multiple of T, the technique in real time allows to determine plastic deformations of the order of 10−3 on bases of the order of 10 microns, which is suitable for the control of elastic deformations on bases of the order of 100 microns. The possibilities of the method of averaged speckle images for the study of (1) features of the nucleation, start, and movement of the fatigue crack, and (2) the activity of living cells infected and not infected with the virus are demonstrated.
- high cycle fatigue
- live cell
- speckle dynamics
- estimation of time to failure
- crack nucleation
If a rough object is illuminated by coherent radiation, macroscopically homogeneous but microscopically inhomogeneous distribution of scattered radiation intensity emerges at some distance from the object and behind the lens that forms the object image. As the surface microrelief heights are random, the waves reflected from various microscopic areas of the surface have random amplitudes and phases. Mutual interference of these waves results in spotted or “speckle” structure of scattered radiation. When the object surface varies for some reason, the amplitudes and phases of the reflected waves change, so the speckle pattern also varies. Variation of wave amplitudes and phases can be caused by displacement of the object or its rigid rotation or by small variations in the distances between the area elements due to elastic or plastic strains of the surface. The reason may be a small or strong surface microrelief variation due to corrosion or loosening of the material with ionizing radiation, material fatigue, etc. The specified macroscopic or microscopic processes can occur simultaneously.
It is noteworthy that the speckle pattern can be formed when studying objects with a mirror surface by lighting them via mat glass. Phenomena occurring inside transparent objects can be studied in a similar way. Such objects are live cells cultured or precipitated on a transparent substrate. If there is an empty space, the cells can alter their shape and the intracellular processes randomly change the phases of the waves that have passed via the cell. So, by the character of the speckle pattern change, one can study and monitor the phenomena occurring on the surface or near the surface of reflecting objects or inside transparent media.
At present, one is familiar with application of variations, displacements, and interferences of speckle fields for the study of macroscopic phenomena, namely, for determination of transition, motion velocity, rotations, elastic or plastic strains of objects, gas, and fluid flows [1, 2, 3, 4, 5, 6, 7]. At the same time, the logic of speckle optics development and practical needs sets the task to analyze not only macroscopic but also microscopic processes occurring at the structural level. For example, a task like that appears in studies of the phenomena accompanying crack initiation in high-cycle fatigue of metallic materials as well as during the analysis of the processes occurring in the membranes and inside the cells of live systems. The rationale for such studies is related to development of a technique for assessing the remaining life of construction elements, and in case of biological objects, to individualized drug selection for a patient.
As the properties of materials are random at the structural level, the amplitudes and phases of the waves reflected from the object or that have passed through the object randomly vary in time and space. In the general case, the solution to the problem of establishing a relation between the parameters of wave phase dynamics and speckle dynamics is far from simple. Nevertheless, lately certain advances in solving such problems have been observed.
The objective of this publication is to familiarize the readers with the author’s recent developments in the theory and application of dynamic speckle interferometry. They were aimed at study of the processes occurring in technical materials in their high-cycle fatigue and also in live cells subject to some external effects. The rationale for the specified studies, for the theory of the techniques proposed, the conducted experiments and their practical prospects are discussed in brief. The advantage of the techniques under discussion over the conventional speckle holographic techniques is the possibility for real-time study of reversible and irreversible processes. Most part of this chapter is devoted to dynamic speckle interferometry of high-cycle fatigue. When high-cycle fatigue is studied, the technique permits determination of the limiting local microrelief and surface shapes variations (deformations) with high sensitivity. When the variation rate of the named values is monitored, the time to the fatigue crack start can be determined. When the reaction of cells to viruses and bacteria is studied, an opportunity for timely development of procedures preventing and blocking their development progress appears.
2. High-cycle fatigue: the problems and the rationale for the studies
When technical objects are exploited, their various elements are affected by alternate forces. As load cycle number
At present, various features of low-cycle fatigue are fairly well characterized [8, 9]. It is explained by the fact that in low-cycle fatigue, sufficient plastic deformations of the objects normally appear. That is why conventional approaches developed for devices exploited under the conditions of quasi-static loading are applicable for monitoring and estimating of the remaining object life.
The situation with high-cycle as well as giga-cycle fatigue is different. According to various literature sources, from 50 to 80% of equipment is destroyed due to high-cycle fatigue [10, 11, 12]. Despite research history [13, 14, 15, 16] and a great number of publications [17, 18], at present, there are no techniques for assessment and estimation of the remaining life of construction elements in their high-cycle fatigue that would meet the requirements of the engineering practice [11, 18]. According to [11, 18, 19], the situation has arisen due to absence of physical models of high-cycle fatigue. In the author’s opinion, this circumstance is in turn related to absence of physical monitoring techniques that would permit to record the features of local fatigue damage accumulation without interruption of exploitation or fatigue testing of various objects. Analysis of studies [20, 21, 22, 23, 24, 25] devoted to development of such methods shows that their practical application implies methodological issues.
Immediately after the creation of lasers and detection of spotted or speckle structure of scattered radiation, speckles were used to study fatigue phenomena [26, 27, 28]. However, because of high manpower effort and no monotone character of the recorded signal variation, the technique was not put to wide use.
The author learnt the difficulties of fatigue phenomena studies using conventional nondestructive testing techniques at first hand. Early in our research aimed at the study of high-cycle material fatigue, we used various modifications of optical, X-ray, magnetic, electric, and acoustic techniques (13 in all). The results obtained using these techniques were negative. The parameters of the signals either did not vary or the variations were at the same level as the hardware noise. As in high-cycle fatigue localization of fatigue damage takes place , it was supposed that the obtained negative result is related to a large base (averaging region) of the applied techniques. Hence, for measurements on a small base, we upgraded a speckle technique that was previously successfully used to monitor the damage in quasi-static deformation of specimens up to their fracture [29, 30].
Section 3 presents the results of the theoretical studies aimed at developing an optical technique intended to study the irreversible processes emerging while testing specimens for high-cycle fatigue. Section 4 discusses the experiments with high-cycle fatigue conducted using this technique.
This section discusses the results of theoretical underpinning of speckle techniques permitting the study of fatigue accumulation in periodical deformations of the objects. Sections 3.1 and 3.2 discuss the regularities of speckle dynamics observed while capturing the frames of the speckle image on fixed position of the object oscillations. Section 3.3 is dedicated to the theory of time-averaged speckle images.
3.1. Dynamic speckle interferometry of microscopic processes: reflecting object
Let the source of coherent radiation with wavelength λ located at point illuminate the point scattering centers located in some region S in plane (
A thin lens with focal distance
Let us take the relation between complex amplitude of light in proximity of point and at point in the same form as in :
where is the distribution intensity of the illuminating radiation, in the general case is the complex reflection coefficient that takes into account the proportion of the radiation going from point to point , is the vector targeted from point to point , is the vector targeted from point to point , is the vector of small displacement of the
Let us take arbitrary point and its conjugated point . It is known that a wave going from point forms an Airy pattern as the result of light diffraction on a diaphragm with diameter
Let us further assume that the region with radius
where and are unit vectors targeted from point to the radiation source and to the observer, respectively; , complex amplitude determines the expression preceding the summation sign; , is the relative displacement vector of the -th pair of scattering centers; and ,
Then, let us assume that process (
where and are values average by the ensemble of objects at time points and , respectively, and are variances of values at time points and , respectively, and is the correlation coefficient of phase differences at time points and .
If process is stationary, then = and = , and instead of (6), we have:
where is a normalized autocorrelation function of relative displacements,
At values that are small compared with λ, it is convenient to exclude the permanent part of , proceeding to a new normalized autocorrelation function . It is easy to show that at , we have . Therefore, the corresponding normalized spectral functions of intensity fluctuation and relative displacements are equal. The correlation (relaxation) times
At , establishing a relation between and is not a simple task. In , it was done for the case when function is a Gaussian function. It was shown that in this case, and are also Gaussian functions, with the function range at level 1/е is times as wide as that of function .
3.2. Theory of speckle dynamics of microscopic processes: transparent object
Let us discuss an optical system (Figure 2) that forms an image of a thin transparent (phase) object. Let the source of coherent light 1 illuminate thin diffuser 2 consisting of point stationary diffusers chaotically located near plane (
are the optical path lengths of the
3.3. Theory of time averaging method for speckle images
The results of the theory discussed above in Sections 3.1 and 3.2 were used in studies of irreversible processes arising in metals with their high-cycle fatigue [31, 34] and in live cells . In the cases when spatially homogeneous and temporally stationary random variation of optical wave paths occurred at the structural level, good coincidence between the theory and the experiment was observed.
However, the drawback of the theories discussed in Sections 3.1 and 3.2 is the difficulty of application in the cases when random variations of the wave phases occur due to various simultaneous processes occurring at different rates.
For example, in high-cycle metal fatigue, the phase of the wave reflected from part of the surface can vary as a result of translational motion of the object, its elastic or plastic deformation, phase transformation, and formation of microcracks.
When a live cell is studied, the wave phase can vary due to diffusion of the substances through the membrane, endocytosis (capture of large particles due to local cell shape variation), and protein synthesis as well as cell motion.
To overcome the drawback named in [32, 36, 37], the theory was upgraded. The idea consisted in application of the time averaging procedure to speckle dynamics. If the characteristic time
Let us discuss the results of the theoretical studies, first for a reflecting object, and then for a transparent object. In , from the model of scattering reflecting object discussed in Section 3.1, a problem concerning speckle dynamics in the image plane of a flat surface performs a complex motion. It was supposed that the scattering centers located in plane (
where are constants,
A similar problem concerning speckle dynamics in the image plane of a transparent object was solved in . An optical system presented in Figure 2 was discussed. It was supposed that object 3, located near thin diffuser 2, is a thin transparent plate whose lateral surfaces are parallel to plane (
Now, in formula (12), the value
If the roughness is homogeneous, i.e., , then instead of (12), we have:
is a normalized temporal autocorrelation function of radiation intensity corresponding to the translational motion of the plate, and is a normalized temporal autocorrelation function of phase difference of wave pairs changing in time as a result of movement of a rough transparent plate.
Let us note that the theory of dynamics of time-averaged speckles in the image plane of a thin transparent object in the absence of its displacement and oscillation was discussed in .
3.4. Discussion of theoretical results
Therefore, if translational motion of a reflecting or transparent object is absent, the averaging time is divisible by the cyclic loading period, and there are no irreversible deformations in the object, then, according to Eqs. (9)–(11), the observer in the image plane will see a pattern of averaged speckles invariable in time. If irreversible processes that alter the optical paths of the waves emerge in a small region of the object at some phase of oscillation, then the speckle pattern in the conjugated region will change. As the formulas for normalized autocorrelation functions (6) and (11) at coincide, the pattern of averaged speckles can be regarded as a speckle pattern of some stationary object. In case of the emergence of irreversible processes that alter the shape of the reflecting object at the structural and/or macroscopic level, or the density of the transparent object, the speckle pattern of such a “stationary” object will vary.
For the reflecting object, value
4.1. Dynamic speckle interferometry of flat specimens in periodic bending
In our first experiment, the results of theoretical analysis presented in Section 3.1 were applied to study fatigue phenomena emerging in high-cycle fatigue of medium-carbon steel 50 . The scheme of specimen loading is presented in Figure 3; the dimensions and shapes of the specimen as well as the speckle image of the control area are shown in Figure 4. Before testing, the sample was subjected to fine grinding and annealing. A 2 mm thick flat specimen was loaded with 50 Hz frequency; the number of cycles reached 1,200,000; and maximum cycle amplitude σmax varied from 0.2 to 0.82 σ02, where σ02 is the flow limit of steel 50. The surface area near the maximum stresses was illuminated by a laser module with wavelength λ = 655 nm and 20-mW power. Speckle images with magnification m = 0.1 were captured at a certain phase of the object oscillation and entered into a computer with a frequency about 10 Hz. The minimum speckle size 2
where the angle brackets denote arithmetic mean values in the named object ensemble, and σ1 and σ2 are standard deviations of values
Figure 5 presents a typical dependencies
Analysis of the experimental data obtained in the sections of stationary surface relief variations showed the following. When σmax varies from 0.2σ02 to 0.82 σ02, correlation time
In many cases, we recorded very complex dependences
4.2. Application of time-averaged speckles
We used the technique of time-averaged speckles to study the micro- and macro-variations of the surface shape in high-cycle fatigue of constructive and model materials. The main objective of the conducted research work was to clear up the matter of what happens in the material during crack initiation.
Beams made from pipe steel 09Г2С loaded by three-point bending were used in the experiment. The shape and dimensions of the specimens as well as the position of the bumps are presented in Figure 6. A Charpy notch was made in the beam to localize the crack initiation spot. The surface of the beams was subject to fine grinding, and then they were annealed in vacuum. After annealing, the surface of some specimens was polished. After polishing, roughness parameter
The specimens were tested in a resonance-type machine of MIKROTRON (RUMUL) type with near-100 Hz frequency and a 0.1 load ratio. The value of the maximum force exerted (1.1 kN) was selected experimentally from the condition of emergence of a 0.1-mm long crack after hundreds of thousands of loading cycles. Variation of resonance frequency by 10% usually preceded the emergence of a crack of this length.
The scheme of the optical setup applied for recording of the average speckle images is presented in Figure 7. The optical setup was located on the platform of the testing machine. Object 4 was illuminated by beam 2 from laser module 1 of a KLM-H650-40-5 type with the wavelength of 0.65 μm and 40-mW power. As speckles do not emerge when a mirror surface is illuminated, mat glass 3 was put into the lighting beam to form speckle fields when specimens with the polished surface were tested. The speckle image was recorded in the specimen image plane. The magnification of the optical system equaled 0.7. The diaphragm size of lens 5 was selected so that the minimum speckle size slightly exceeded that of a photocell in the photocell array of the TV camera 6. A monochrome VIDEOSCAN-415M-USB TV camera with an array containing 782 × 582 photocells of 8.3 × 8.3 μm size was used in the experiments. Averaging time
Figure 8 presents distribution of correlation coefficient
The first image was obtained at loading cycle number
Figure 9 presents three-dimensional surface profiles near the notch at
As it is seen from Figure 9, at 92,000 cycles, two zones emerged in front of the notch. The first zone is a pitch about 0.5 μm deep with the diameter of 500 μm. The pitch center was at the distance of about 250 μm from the notch tip. Besides, a second small zone of about 50 × 100-μm size where a fatigue crack initiated emerged immediately at the notch tip. This area is shown by an arrow in Figure 9b. This zone consisted of irregularities of 5- to 1-μm transverse size and the height of scores of nanometers.
Figure 10 presents joint graphs showing altered relief heights
The scan line of the profilometer passed through the pit center, and value
Comparison of surface shape variation within the pit with variations of value
As seen from Figure 9, about half of the crack formation region lies within the pit, and the other half is beyond the pit. For the latter section, the measurement of equaled 0.8. Suppose in (11) if the cosine equals 1, , and = 0, we obtain that the roughness parameter
The measurement of
According to the profilometer data, variation of value
Figure 12 presents typical dependences of
Analysis of three-dimensional profiles of the surfaces of two specimens showed that they were the same in the qualitative sense. The surface spots for plotting of dependences
It is seen in Figure 12 that notable deviations of dependences 4 and 5 from the horizontal line start practically simultaneously at 70,000–80,000 cycles. This suggests that the formation of irregularities in the small zone and the contractions probably start simultaneously. Above, it was shown that a decline of value
4.3. Studying fatigue of Plexiglas
The experiments conducted with Plexiglas of “ACRUMA” brand aimed at studying the peculiarities of fatigue damage accumulation in the volume of the specimen shown in Figure 6. The specimens were made from a plate 5-mm thick. The protective film preserving the polished surfaces was taken off before the fatigue test. The equipment shown in Figure 7 was used to record the speckle images. The specimen was illuminated through mat glass, the sounding waves passed through the specimen. The observation direction was selected either along the normal or at the angle of 30° to the normal. In the latter case, irreversible processes in the depth of the specimen could be controlled by distribution of value
4.4. Dynamic speckle interferometry of intracellular processes
In our early experiments studying the processes occurring inside live cells, we used the formulas discussed in Section 3.2 of the theory. The main long-term objective of the undertaken studies was search of approaches that would permit creation of an optical technique and a device studying the processes in the live cell membranes. The first objective of the experiments was search of cell metabolism parameters. We used an optical setup whose scheme corresponded to the optical scheme discussed in the theory (Figure 2). The photo of the setup is shown in Figure 13. A semi-conductor laser module with the wavelength λ = 532 nm and 20-mW power as well as the TV camera discussed above in Section 4.2 were used. The setup was placed into a thermostat that maintained the temperature of 36 ± 0.1°С. A transparent cuvette with two glass supporters in a horizontal position in the nutrient solution was fixed on a small table near the mat glass. The first supporter contained a monolayer of cultured cells; the second one was cell-free. Speckle images of the supporters are shown in Figure 14. The light lines in the picture show the typical sections selected for determination of value
While processing the films of 20- to 40-s duration recorded for several hours, we detected various types of dependences . Along with the graphs similar to those presented in Figure 15, we obtained dependences that did not level off as well as graphs that had rather a composite view. All-day graphs of dependences at a 0.5-h pitch were reproducible in about 50% of the cases. Analysis of obtained dependences showed that their complicated character is probably connected to the presence of several processes altering the phases of the sounding waves at different rates. In this regard, we used a time-averaged speckle technique discussed in Section 3.3. Application of this technique enabled us to obtain well-reproducible data. Figure 16 presents typical dependences of the time-averaged digital value of intensity corresponding to the nutrient solution, to a cell in the nutrient solution and to a herpes simplex virus-infected cell in the solution. The dependences are taken from our paper . Analysis of the dependences belonging to the latter type showed that variation features of value well corresponds to the stages of virus development in cells. Figure 17 presents typical dependences corresponding to the nutrient solution, to the cells in the nutrient solution, and to the herpes simplex virus-infected cells in the nutrient medium.
The shown dependences were well-reproducible when a monolayer of cultured cells of various cell lines was infected with herpes virus.
Figure 18 presents joint dependences of and temperature
As the metabolic processes are manifested more distinctly when the temperature rises, the above data presented in  were used to substantiate application of value as the parameter that characterizes the activity of cultured cells.
5. Discussion of results and prospects for further studies
5.1. Studying high-cycle fatigue of metals
The conducted theoretical and experimental research showed that the dynamic variant of speckle interferometry can be used for quantitative evaluation of irreversible displacements and deformations occurring in metals with high-cycle fatigue on the 10-μm order bases. The peculiarity of this evaluation is that application of time-average speckles makes it can be conducted in real time, i.e., without interruption of cyclic deformations (Figure 19).
The technique is fairly simple, because it does not require any synchronization of the load value applied to the object and the frame capture moment. It is characterized by high accuracy and sensitivity.
In , it was shown that when a 10 × 10-pixel fragment is selected at the variation of
High sensitivity of the speckle technique to deformations is based on the phenomenon of multiple-wave interference with the same initial phases. Speckles emerge as the result of multiple-wave interference with random initial phases. So the emergence of multiple-ray interference is in many ways similar to wave interference in a diffraction grid is not evident. This question was briefly discussed in [34, 40]. Because of the significance of the matter, let us dwell upon it. Let there be a great number of point diffusers on the surface in a region of diameter with the center at point
The conducted experiments showed the possibility in principle to determine the limiting roughness parameter variation , of value for the reflecting object and the refraction index of a transparent object corresponding to the crack start. By monitoring the variation rate of these parameters on the bases of the 10–100 μm order, one can assess the approximate time to macro-fracture. It is noteworthy that in practice, not always is there an opportunity to illuminate the object under control with laser radiation, and the conventional nondestructive testing techniques are poorly adapted for measurement on such bases. The discussed speckle method could be used as the tool for upgrading of the conventional testing techniques. The technique is convenient for target detection in search of local irreversible deformations and calibration of other techniques.
Fatigue experiments improved understanding of the processes occurring in high-cycle fatigue of materials. For example, the pins that we observed at the notch tip (shown by the arrow in Figure 9b) turned out to be pieces of iron carbide. We managed to identify them using Raman scattering . This fact speaks for the significance of heat generation accounting in high-cycle fatigue. The experiments also showed that irreversible processes in the small region close to the notch emerge at the early stages of the fatigue. If inconsistent local deformations emerge at the stage of loading increase, then in arbitrary unloading residual compressive stress must inevitably appear. Thus, the stress and deformation fields at the notch tip will vary considerably with progress of the fatigue. In this context, analysis of not only local plastic, but also of periodically varying elastic deformations is important.
The opportunity for application of the elastic deformation control technique that we have discussed was substantiated in . Deformation of the 10−3 order corresponds to values of the 1-nm order and of the 10-μm order. Plastic deformations 10−3 up are more characteristic for most of the conventional constructive materials. If we increase the measurement base by an order, we will proceed into the range of elastic deformations of the 10−4 order. It is deformations of this order that emerge in constructions during exploitation. In , we conducted a successful pilot model experiment recording values and
The dependence of value
In the conducted fatigue experiments, we used an optical system that permits determination mean values, variance, and relaxation time of -component of vector . However, the measurements of the above values for other components of vector are of practical interest. Development of such a technique is the subject of our further research.
5.2. Study of the processes inside live cells
The matters of the accuracy and sensitivity of the technique, the peculiarities of speckle image variations, contribution assessment for values and to variation of value
We underpinned and approved a cell activity parameter , or variance of the difference in the optical path of the cell-sounding waves is such a parameter. However, it is not yet clear what constituents of metabolism affect variation of value and to what extent. The study of this problem is the subject of our further research.
Obtaining data by averaging by the cell thickness is a drawback of this technique. Still, the logic of the technique development and the practical needs set the task of determining the mean value, variance, and relaxation time of the medium refraction index in every small section of the cell. The author reported about the ways to solve this problem at two conferences [44, 45]. Theoretical and experimental underpinning of a speckle tomography for the live cell that would permit a solution to this problem is the subject of our further research.
This work discussed theoretical and experimental underpinning of an interference technique that permits studying irreversible processes occurring near the surface of reflecting objects and inside thin transparent objects by variation of speckle images. The author’s scientific interests lie in the sphere of studying longevity of living and nonliving matter. So, the research targets were specimens made from constructive materials tested for high-cycle fatigue as well as cultured live cells.
The theoretically obtained formulas established the relationship among the parameters characterizing variation of optical wave paths in small sections of an object and the parameters characterizing variation of speckles in the conjugated region. Such parameters for a reflecting object were the mean value, variance, the relaxation time of difference in displacement of scattering centers (points of the surface), time-averaged radiation intensity , and correlation coefficient
For the variant, when variation of in time is a random stationary process, we obtained a relation between the normalized spectrum of value and the normalized fluctuation spectrum of .
In experiments set up for testing of the theory, speckle dynamics was created by displacement of a rough transparent plate, by variation of shape, roughness, and the refraction index of metal and Plexiglas specimens in their high-cycle fatigue. Good quantitative coincidence of the theory and the experiment was shown.
It was shown that by way of speckle time-averaging the technique permits real-time determination of the 1-nm-order measurements of on the 10-μm order on bases Δ
It was demonstrated that with increasing base Δ
It was also shown that using this technique, one can determine the limiting variations of roughness parameter Ra, value ε and refraction index n that correspond to the crack start. Therefore, knowing the limiting measurements of these parameters and monitoring their variation rate while exploiting the part, one could in principle assess the time to its macroscopic fracture. Development of such a technique can be the subject of further research.
When the technique was applied for studying live cells, it was shown that the variance of value can be used as a live cell activity parameter. It was also shown that dependences and
The author would like to thank the young colleagues and students I. Kamantsev, N. Drukarenko, Yu. Mikhailova, and K. Myznov for their help in conducting experiments.