Interference of speckle fields
The principle of holography was discovered by D. Gabor in 1947  even before the invention of the laser. Later, in 1962, it was generalized by Y. N. Denisyuk as a phenomenon of mapping the optical properties of an object in the wave field of the scattered radiation . The use of Helium-Neon laser  by E. Leith and J. Upatnieks in 1963 made it possible to fully demonstrate the remarkable properties of holographic images. Literally, optical holography began to develop rapidly immediately after this.
However, there was also a problem that has a fundamental basis – there always appeared the so-called laser spotting in the images. This unremovable noise was called speckle noise ("speckle" – grain, spot). The studies have shown that such noise is caused by the use of diffusely scattered coherent radiation. The reason for the the appearance of the speckle structure is the inevitable spatial modulation of the light fields due to the cross-interference of a set of radiation components.
However, as the studies in the field of holography of focused images have shown [4, 5], the outwardly chaotic structure of the speckle field is capable of carrying essential information about the investigated objects. In [6, 7], the different groups of researchers, independently of each other, revealed the effect of the formation of focused images of axial reconstructions by holograms obtained in diffusely scattered radiation. The reconstructed images had a number of curious properties. An important consequence of the applicability of holographic recording of the focused images of reference waves of arbitrary shape, including those diffusely scattered [8–10], in particular formed from the radiation scattered by the object , was the possibility of a peculiar degeneration of the reference wave. Its meaning lies in the fact that in diffuse-scattered coherent radiation, the focused images are recorded without a specially formed reference beam. The nature of such images, independent of the presence of a reference wave, was subsequently investigated and explained in [12, 13].
The result of the components interference of the diffusely scattered field (speckle structure) registered by the receiver is a low-frequency spatial carrier wave. It is necessary for the formation near the axis of the illuminating beam of the images found in [6, 7] and constituting the physical mechanism of speckle photography and speckle interferometry.
SPECKLE FIELD PHENOMENON
The practical use of laser radiation is accompanied by the formation of light fields that have a complex random spatial structure. Such fields, i. e. speckle fields, spontaneously arise when coherent radiation is scattered by any rough or essentially uneven surfaces, and also objects with an irregular amplitude or phase profile. Therefore, only specular reflection and transmission of laser radiation through objects that are uniformly transparent are not accompanied by the formation of speckle fields.
The random spatial modulation of the amplitude and phase of the speckle fields is the result of the coherent addition of independent contributions from various local sections (points) of the scattering surface or a volume partially illuminated by coherent radiation. In this case, for objects of any form, these contributions have random phase values that are different for different points in the speckle-field volume (Fig. 1).
The formation of speckle fields inevitably accompanies the solution of practical problems of coherent, nonlinear and atmospheric optics, holography and phase conjugation, holographic and speckle interferometry, and the use of multimode fiber-optical systems. In doing so, they generate a characteristic granular structure, which is superimposed on the resulting images, becoming a source of speckle noise.
However, on the other hand, in speckle interferometry, which has become a widespread and highly effective method of optical measurements, the speckle field is purposefully used as a carrier of information on the displacement and/or shape change of the object under study.
Interest in the study of the nature of speckle fields is due to two circumstances. On the one hand, there is the need of practice, since the interference of speckle fields is a physical mechanism of holographic and speckle-interferometry. Furthermore, in many practical problems of coherent optics, a superposition of the speckle field takes place, either with a smooth field or with another speckle field. On the other hand, peculiarity of the correlation properties of the speckle fields determines the appearance of many peculiar interference effects under their superposition, which determines the expediency of considering the interference of speckle fields as a full-fledged independent section of the theory of partial coherence.
After the first works in the field of applied speckle interferometry [14–16], the main attention of the researchers and engineers turned out to be paid to the field of various practical applications of the method, largely due to the simplicity and convenience of its implementation. Our research, on the contrary, was aimed at revealing and systematic study of the physical nature of phenomena directly related to the interference of speckle fields . A number of new effects were discovered, mainly due to the specificity of the phase distribution in such fields.
The speckle field is a complex volumetric distribution of fine-structure inhomogeneities, the so-called speckles. Within each of them, the phase has a constant value and varies randomly during the transition from one speckle to another (Fig. 2). Since the complex amplitude in each individual speckle is the sum of many small independent contributions from different points of the scattering object, the central limit theorem of probability theory is applicable to the resulting speckle field. According to this theorem, the complex amplitude of the resulting speckle field obeys Gaussian statistics.
Therefore, the depth of random spatial modulation of the speckle-field amplitude can reach zero, and the intensity fluctuations have the same order as the average value of the intensity. This means that the contrast of the registered speckle structure, defined as the ratio of the standard deviation of the field intensity to its average value, is unity.
The characteristic dimensions of the spatial inhomogeneities of the speckle field, or, as they are usually called, the characteristic sizes of speckles, are determined by the irregular divergence of the scattered beam or by the width of its angular spectrum Δθ. The transverse size of speckles is usually determined by the relation σ⊥ ≈ λ / Δθ, longitudinal – by the ratio σ∥ ≈ λ / Δθ 2 (λ is the wavelength). Thus, with a three-dimensional view, the speckle field is a set of inhomogeneities elongated along the longitudinal axis, within which the phase can be regarded as a constant. This means that the characteristic size of the speckle field inhomogeneities can be considered as the region of its spatial correlation .
The speckle fields arising in the scattering of coherent laser radiation also themselves have a high degree of temporal and spatial coherence. This is easily seen by considering the superposition region of two speckle fields (two speckle field realizations from one source) having a deterministic (e. g., linear) relative phase shift. For this, e. g., the speckle fields can be directed to the superposition region at an angle to each other. Suppose that the spatial velocity of the phase change is large enough that several periods of regular phase modulation fit within the same speckle. Then the superposition speckle field will be modulated by a high contrast interference pattern in all cases when the optical path difference between the speckle fields does not exceed the coherence length of the laser source. In this case, the contrast (visibility) of the regular interference pattern will not depend on the change in the realizations of the speckle field. In particular, it does not change when superposition of different sections of the same speckle field is provided by its division along the wave front (e. g., performed by the Young interferometer scheme).
The phase of the regular interference pattern is constant within each speckle. Essential in such an interference experiment is the fact that during the transition from one speckle to the other (both in the transverse and in the longitudinal sections of the superposition speckle field) it undergoes a random change (leap). One can easily verify that an analogous result will also be observed for a superposition of a speckle field with a smooth field, when a random phase distribution in the speckle field causes random leaps in the phase of the interference pattern. With a relatively slow change in relative phase shift of two nonidentical speckle fields or speckle field and a smooth field, the period of the regular ("low-frequency") phase modulation is more than the characteristic dimension of speckle. Therefore, random phase leaps between fragments of the interference pattern inevitably lead only to a redistribution of the intensity in the speckle field, with no regular interference pattern observed .
A special situation occurs in the only case when conditions for the superposition of speckle fields, which are one and the same realization of the speckle field (identical speckle fields), are provided. In this case, they must be combined so that in a certain spatial region identical speckles overlap. This, then, is a superposition of identical speckle fields. Their mutual displacement in both the transverse and longitudinal directions does not exceed the corresponding sizes of the autocorrelation function of this realization of the speckle field.
The fact that in a mutual displacement exceeding the volume of the autocorrelation region, the identical speckle fields lose the property of mutual correlation and behave with interference in the same way as the different realizations of speckle fields, evidences the rarity and peculiarity of such situation. However, it is this rare situation that is of great practical importance, since all the methods of holographic and speckle-interferometry are based on its implementation. It should be noted that at the initial stage of the application of speckle interferometry methods there was an opinion that the so-called speckle-photography methods (this term was widely used in the literature and was later "replaced" by the concept of speckle interferometry) is based on a specific mechanism that is not connected with the principle of holography. However, further studies  made it possible to establish that both holographic and speckle interferometry are based on a common mechanism of interference of identical speckle fields
A fundamental feature of the interference of identical speckle fields consists in the fact that the phase of the regular interference pattern remains constant within the region of overlap of identical speckles in any section of the superposition speckle field. It is obvious that in this case the possibility of observing a regular interference pattern does not depend on the ratio of the characteristic speckle size and the period of regular modulation.
Therefore, it is in this and only in this case that a "low-frequency" interference pattern with a period exceeding the characteristic transverse size of the speckles can be observed. Obtaining and further analysis of such interference patterns is the task of holographic and speckle interferometry.
Thus, there are three main types of interference involving speckle fields:
• a speckle field with a smooth field;
• uncorrelated (non-identical) speckle fields;
• correlated (identical) speckle fields.
All these situations are often implemented in practice. Thus, the interference of a speckle field with a smooth field is typical for recording holograms of diffusely scattering or diffusely illuminated objects [3, 5]. Interference of uncorrelated speckle fields occurs in holographic recording of objects using reference waves of arbitrary shape [5, 8, 11]. Finally, the phenomenon of interference of correlated (identical) speckle fields underlies holographic and speckle interferometry.
INTERACTION OF SPECKLE-FIELDS AND SPECKLE FINE STRUCTURE
Over time, the physical unity of holographic and speckle interferometry, which we specified in [5, 17], became obvious. They are two related means of providing interference of identical speckle fields. In both cases, the signal of the measurement information is the low-frequency interferogram modulated by speckles, formed in the superposition region, as a rule, of two identical speckle fields under conditions when the period of this interferogram exceeds the speckle’s characteristic size.
Here it is necessary to emphasize that when receiving and interpreting such interferograms, an important role is played by taking into account the regularities of the localization of interference fringes. The effect of the localization of the fringes is directly related to the magnitude of the coherence volume and consists in the formation in the superposition zone of a limited region where the visibility of the interference fringes has a maximum value. This phenomenon is well known in classical interferometry with extended thermal sources giving radiation with essentially limited spatial coherence.
When laser radiation is used in the case of superposition of smooth beams, the fringe localization region, as a rule, occupies the entire superposition zone and is limited only by the coherence length of the source. As for the methods of holographic and speckle interferometry, when a superposition of identical speckle fields having a limited region of spatial correlation is provided, the phenomenon of localization of fringes is as typical as for interferometry with extended thermal sources. However, it should be noted that the approach adopted in classical interferometry and based on the geometric theory of localization does not give a sufficiently complete and physically adequate description of the distribution of the visibility of interference fringes when studying the phenomenon of fringe localization in the case of interference of speckle fields .
Therefore, in our works, considerable attention has been paid to the development of a new approach to explaining the regularities of localization and distribution of the visibility of interference fringes during superposition of speckle fields, based on the identification of the role of speckles as areas of correlation of fields: their size, shape, and fine structure. Within the framework of this approach, a number of previously unknown effects were discovered due to the fine structure of the speckle fields, which, as a rule, does not manifest itself in direct observation, but has a significant effect on the results of the experiments. Fig. 3 clearly shows the asymmetry of the visibility curve in the region of interference fringe localization caused by a linear increase in the transverse dimensions of the speckles as they move away from the plane of the specklegram.
The practical use of these effects, coupled with the study of the imaging properties of the spectrograms, [5, 17] made it possible to develop original methods for interference optical measurements , and to develop an interesting direction of optical information processing, i. e. image subtraction .
In the first approximation, the speckle field can be considered as a complex interference structure with an irregular distribution of amplitude and phase due to a randomly varying phase difference between the coherently interacting field components. In this case, the elements of the speckle structure look like spots with a uniform amplitude distribution, within each of which the phase has a deterministic value. Meanwhile, diffraction image of the point source has a coherent, as is known, the fine structure due to the presence of secondary maxima of the amplitude and sign of the phase change at the transition from one to another of such maximum.
In the experiments aimed at detecting fine transverse speckle patterns , supervisory system with the entrance pupil of different shapes and sizes were used defining speckle size and shape, respectively. The variation of degree of mutual speckle displacement provides a rotational shift of the scattering surface between the two exposures. As expected, the increase in the relative aperture size resulted in a decrease in the area where the contrast of the speckle and holographic interferogram of the rotational shear was close to unity. However, in addition the oscillations of the visibility of the interference fringes as the distance from the center of rotation were clearly observed. Naturally, when using a circular aperture, the visibility in secondary maxima was significantly lower than in the main one. Thus, transverse shear of fringes by half a period was observed in the transition from one maximum to another, which corresponds to a phase sign change.
Note that the presence of such visibility oscillations drawn attention of several authors. However, the nature of the phenomenon remains unclear. The fact is that when using the circular aperture, it was not possible to detect more than one Bessel function secondary maximum of the first kind of first order due to substantial visibility drop in subsequent secondary maxima of the oscillating interferogram (Fig. 4).
Meanwhile, the hypothesis about the nature of this effect was convincingly confirmed using more complex aperture shapes, forming much more intensive secondary maxima. In particular, in [5, 17], we used a ring aperture and two or more holes of different shapes, allowing not only to demonstrate numerous secondary maxima, but also to measure the distribution of the visibility. The measurement results showed good agreement with the theoretical and calculated data. Fig. 4 clearly shows the oscillation curve of the visibility of interference fringes, and their failure by a half period when passing through the region of zero value of the visibility function. The distribution of the values the visibility function and the effect of the interference fringes dephasing is shown in Fig.5.
A detailed study of the visibility of the oscillation effects and localization of the interference fringes in the superimposed speckle fields was carried out in . In these studies, it was shown that for observing the interferogram it was necessary that the magnitude of the relative speckle displacement when recording is not resolved by the observation optical system and does not exceed the volume of the spatial coherence of the radiation used to obtain both holographic and speckle interferograms. It was found that the region of localization of the fringes are usually located outside the specklegram plane.
An interesting consequence of the speckle fine structure was the discovery of the effect of interference fringes branching in the region of their localization . It was possible to establish a link between the branch points and the zero points of the visibility complex amplitude function, which allows to determine the spatial location of these points and their degree of localization. It should be noted that the effect of the interference fringes branching is observed in both transverse and longitudinal sections of the superimposed speckle fields. Branching fringes can be observed in different types of mutual displacement of identical speckle fields, as well as non-identical superposition of speckle fields.
As it turned out, by using spatially modulated reference waves in holographic interferometry, the presence of points with zero intensity in the speckle fields (so-called dislocation speckle fields) causes a distortion of the reconstructed microstructure (secondary) speckle field even with the use of the reconstructing wave, identical to a reference one. As a result, there is a partial correlation of the interfering speckle fields, which in particular affects the visibility incidence of regular interference patterns.
No less interesting results were obtained in the study of the fine structure of the longitudinal speckles . It was found that when registering specklegram in the Fourier plane, the longitudinal translational displacement of the object results in only radial displacement of speckles that provides the identity conditions of the recorded speckle fields. Therefore, in the field scattered by a specklegram a superposition of identical secondary speckles occurs, thereby the obtained speckle interferograms have high contrast.
However, the greatest interest in this part of the research, in our opinion, is a confirmation of the hypothesis about the presence of longitudinal fine structure of the speckle fields. First manifestation of such a structure was specifically found and studied in detail in works which can be found in the review . Indeed, the longitudinal autocorrelation function of the speckle field is characterized by the presence of secondary maxima, which relative magnitude is determined by of the parameters of the aperture forming speckle field structure.
Longitudinal displacement of a diffusely scattering object leads to additional radially-symmetrical lateral displacement of speckles, or similarly, to the displacement of the maximum cross-correlation function, and also to change its characteristic appearance with respect to the form of the autocorrelation function. The magnitude of this displacement increases linearly to the specklegram periphery that determines the said change. Therefore, with increasing mutual longitudinal displacement of speckle fields, the cross-correlation function decreases initially as expected, but then it significantly increases, as secondary maxima of the identical longitudinal displacing speckles manifest themselves. The experiments to obtain speckle interferograms of longitudinal displacement with a volumetric registration of speckle fields fully confirmed the character of spatial orientation of speckles in the speckle field volume.
A thorough study of speckle longitudinal fine structure confirmed the fact that when the interference of identical longitudinally displaced speckle fields occurs, periodic change in the position and magnitude of maxima of cross-correlation function is observed due to the presence of secondary maxima longitudinal autocorrelation function. As a result, depending on the mutual displacement of speckle fields, the visibility of speckle interferograms changes in a predictable manner, wherein with definite values of the magnitude of this displacement, because of the spatial superposition of different maxima of the identical speckles fine structure, the characteristic dephasing of the visibility of interference fringes is observed.
It should be noted that the study of the phase characteristics of speckle fields is of great importance because, while remaining concealed at both the direct observation of speckle fields and the results of the interference, they have the most significant influence on the result of observation.
In particular, contrary to the concept that the phase of the developed speckle field is a random variable uniformly distributed in space, we have shown  that for symmetric apertures, used as sources of speckle fields, the phase difference in neighboring speckles has distinctly irregular probability density. This fact should also not be ignored when working with speckle fields.
Because of the wide application of lasers in various fields of science, engineering and technology, a wide range of professionals will always face the manifestation of the speckle effect and the phenomena associated with the superposition and interference of the speckle fields. The nature and regularity of such events, of course, should be taken into account when interpreting the results of many applied research. Moreover, as the digital techniques are introduced to coherent optics, new opportunities associated with the development of holographic and digital speckle pattern interferometry emerge (see, e. g., [18. – 19]). Computer optical experiment with the recording and reconstruction of the scattered object light field, opening interesting new aspects of research of the phenomenon of interference, nonetheless, does not change the basic mechanism of interaction of speckle fields, therefore it can be assumed that the results of research cycle, an overview of which is set out in this paper, will remain relevant in the modern world.