**2. Historical outlook**

The notion of coherence is the most fundamental concept of modern optics. As it has been shown by E. Wolf [5], this notion is intrinsically connected with other characteristics of light, such as intensity and polarization. One can distinguish between these characteristics mainly in didactic purposes. But in every practically important case we meet the problem of tight, inseparable interconnection of them. So, one cannot define coherence, in part aspiring to associate it with visibility of interference pattern, ignoring for that the states of polarization of superposed beams. Note, that attempts to explain the Young's interference experiment for "completely unpolarized" light lead sometimes to questionable conclusions [6]. At the same time, the most fundamental definition of polarized light is given just through the measure of mutual coherence of the orthogonally polarized components of a beam. At last, all three mentioned characteristics of a light beam are comprehensively expressed through known combinations of the Wolf's coherency matrix elements [1].

Incidentally, urge towards to associate coherence just with obvious interference (intensity modulation) effect does not always lead to true understanding the coherence phenomena. It is not enough that interference fringes are absent in superposition of completely mutually coherent but orthogonally polarized beams (it is well-known from the Fresnel-Arago laws and experiments). There are quite new concepts showing the absence of interference effect for superposing two waves of equal frequencies with strictly (*deterministically*) connected complex disturbances even with the same state of polarization. Refined example of this kind was given by L. Mandel in his concept of "anticoherence" [7]. In very simplified form, the Mandel's concept can be formulated in terms of conventional (static) holography.

As it is well known, a simple thin hologram reconstructs in plus-minus first diffraction orders two conjugate fields producing so-called main and conjugate images [8]. Having reliable techniques for optical phase conjugation, one can try to add two complex conjugated copies of the signal at one plane, to say at a plane of hologram. What is the result? Intensity of superposition of two such waves is determined as

inhomogeneously polarized fields. We represent the newest metrological tool connected with novel concept of optical currents (optical flows). Namely, we show that some intimate characteristics of complex optical fields with arbitrary degree of spatial coherence and arbitrary degree of polarization may be "deciphered" indirectly, by observation of the influence of such fields on embedded micto- and nanoparticles. This original metrological approach seems to be prospective for development of so-called optical traps and tweezers

Separate section of the Chapter is devoted to application of local Stokes-polarimetry in diagnostics of biological tissues, in the context of early (pre-clinical) diagnostics of some widespread diseases. We represent both experimental and data processing techniques

All considered metrological approaches and techniques are original, generated by recently

Some prospects of further investigations in the direction represented in this Chapter, as well as necessity and possible ways for overcoming some present shortcomings of optical

The notion of coherence is the most fundamental concept of modern optics. As it has been shown by E. Wolf [5], this notion is intrinsically connected with other characteristics of light, such as intensity and polarization. One can distinguish between these characteristics mainly in didactic purposes. But in every practically important case we meet the problem of tight, inseparable interconnection of them. So, one cannot define coherence, in part aspiring to associate it with visibility of interference pattern, ignoring for that the states of polarization of superposed beams. Note, that attempts to explain the Young's interference experiment for "completely unpolarized" light lead sometimes to questionable conclusions [6]. At the same time, the most fundamental definition of polarized light is given just through the measure of mutual coherence of the orthogonally polarized components of a beam. At last, all three mentioned characteristics of a light beam are comprehensively expressed through known

Incidentally, urge towards to associate coherence just with obvious interference (intensity modulation) effect does not always lead to true understanding the coherence phenomena. It is not enough that interference fringes are absent in superposition of completely mutually coherent but orthogonally polarized beams (it is well-known from the Fresnel-Arago laws and experiments). There are quite new concepts showing the absence of interference effect for superposing two waves of equal frequencies with strictly (*deterministically*) connected complex disturbances even with the same state of polarization. Refined example of this kind was given by L. Mandel in his concept of "anticoherence" [7]. In very simplified form, the

As it is well known, a simple thin hologram reconstructs in plus-minus first diffraction orders two conjugate fields producing so-called main and conjugate images [8]. Having reliable techniques for optical phase conjugation, one can try to add two complex conjugated copies of the signal at one plane, to say at a plane of hologram. What is the

Mandel's concept can be formulated in terms of conventional (static) holography.

result? Intensity of superposition of two such waves is determined as

metrology in the field of coherence and polarization, are outlined in the last section.

for manipulation of isolated particles of micro- and nanoscales.

leading to high-sensitive and reliable diagnostics.

combinations of the Wolf's coherency matrix elements [1].

by the members of our team.

**2. Historical outlook** 

$$I = < \left| a \exp\left[i\left(\alpha t - \mathbf{k}\mathbf{r} + \varphi\_1\right)\right] + b \exp\left[-i\left(\alpha t - \mathbf{k}\mathbf{r} + \varphi\_2\right)\right]\right|^2>\tag{1}$$

where *a* and *b* are amplitudes of two conjugated waves that are believed constants (stationary optical fields [9]), 2 *T* is angular frequency of oscillations (*T* being a time period) that is the same for both superposed counterparts, **k** is the wave vector, **r** is the position vector of the observation point, 1,2 are the initial phases of two waves, and ... denotes time averaging. It is easy to see that, in contrast with common interference, the temporal multiplier does not vanish in this expression:

$$I = a^2 + b^2 + 2ab < \cos\left[2\left(\alpha t - \mathbf{k}\mathbf{r}\right) + \left(\varphi\_1 - \varphi\_2\right)\right] > \equiv a^2 + b^2 \,\,,\tag{2}$$

so that the averaging results in vanishing the "interference" term. This conclusion is illustrated in Fig. 1, from which one can see that summation of two complex conjugated beams, *u* and \* *u* , everywhere along the Arrow of time gives rise to the real value of constant magnitude. Thus, interference between deterministically, unambiguously connected two waves of equal frequency and identical state of polarization is absent. Of course, if one implements phase conjugation of one of two waves figuring in Eq. (1), interference effect will be provided due to compensation of the temporal multiplier and, as a consequence, inefficiency of time averaging.

Fig. 1. Superposition of two complex conjugated replicas of a signal results in a real signal of constant amplitude along the Arrow of time, without interference effect.

What is less discussed within the framework of classical (wave) theory of partial coherence is the influence of an "observer" on coherent properties of studied beams. This problem is quite typical for quantum optics [10 - 12] (as well as for quantum physics in general [13]). But the results obtained along last quarter of a century compel to take into account influence of the conditions of observation and detector characteristics on evaluation of the coherence of light in classical approach also. Let us give some arguments for this point.

When we consider interference of two waves with close but non-equal frequencies (1 2 , ), we observe moving intensity fringes. This effect is widely used in the optical heterodyning (optical nonlinear mixing) technique. If the "register" is stationary and possesses low temporal resolution, averaging over large enough temporal interval results in smoothing of interference fringes, so that visibility 0 *V* ; we conclude that two waves are mutually incoherent. But, on the other hand, using a "register" with higher temporal resolution, which moves in the direction and with speed of propagation of moving intensity waves, we recognize the same two waves as mutually coherent that *form* observable interference fringes in moving coordinates! To say, one can register a hologram form such waves at the properly moving "register" (though this task is not simple in practice).

Once more example related to the problem of interest is the influence of *spectral* resolution (or, more strictly, inhomogeneous spectral sensitivity) of the detector on our conclusions concerning *spatial* coherence of the elaborated optical field. If one uses in the classical Young's two-pinhole interference arrangement a source with uniform spectral density, then a detector with uniform spectral sensitivity "evaluates" the field as completely spatially incoherent for arbitrary sampling points at the beam cross-section due to superposing numerous scaled in wavelength spectral interference patterns, so that minima of intensity (of the field of homogeneous spectral distribution!) are absent in the resulting pattern. *Such detector is "blind" to spatial coherence of such optical field*. Nevertheless, the human eye detects spatial coherence of a field due to non-uniform spectral sensitivity of visual receptors and inhomogeneous spatial distribution of *colors* over analyzed field. Of course, the same is true for evaluation of *temporal* coherence, to say, in the arrangement of the Newton's rings in white light.

Less trivial case has been considered by Wolf [14, 15, see also numerous useful references herein] in the context of induced spectral changes resulting in remarkable transformations of temporal coherence of polychromatic optical fields due to the presence of material *intermediary*, as diffraction or scattering object.

The next, and more closer to our consideration, example *pseudodepolarization* [16] (in modern terminology, "global" depolarization [6]) resulting from stationary scattering of laser radiation in multiply scattering media, such as turbid media, multi-mode waveguides, the most of natural objects, including biological ones. Here the role of detector becomes fundamental. Really, the universal approach to determine all polarization characteristics of a field (both the state of polarization and the degree of polarization [17]) consists in Stokespolarimetry of the analyzed field. For that, Stokes-polarimetric analysis gives quite different results for local and "global" (space-averaged) measurements. So, the point-wise measuring Stokes parameters shows complete (unity) degree of polarization, but the state of polarization changes from point to point. Space averaging over ten and more speckles shows seeming depolarization. This case is the central subject of study of vector singular optics [18].

The mentioned examples illustrate, in part, importance of taking into account of temporal and spatial resolution of the used detector, as well as a choice of the coordinates (stationary or moving) for metrological estimation of coherence and polarization of light, and even understanding (*definition*) of these phenomena.

Let us briefly highlight another two positions that are important for our consideration. Firstly, as it has been pointed out by I. Freund [19], we do not ready to investigate experimentally the problem of coherence and polarization of optical light in general, 3D

heterodyning (optical nonlinear mixing) technique. If the "register" is stationary and possesses low temporal resolution, averaging over large enough temporal interval results in smoothing of interference fringes, so that visibility 0 *V* ; we conclude that two waves are mutually incoherent. But, on the other hand, using a "register" with higher temporal resolution, which moves in the direction and with speed of propagation of moving intensity waves, we recognize the same two waves as mutually coherent that *form* observable interference fringes in moving coordinates! To say, one can register a hologram form such

Once more example related to the problem of interest is the influence of *spectral* resolution (or, more strictly, inhomogeneous spectral sensitivity) of the detector on our conclusions concerning *spatial* coherence of the elaborated optical field. If one uses in the classical Young's two-pinhole interference arrangement a source with uniform spectral density, then a detector with uniform spectral sensitivity "evaluates" the field as completely spatially incoherent for arbitrary sampling points at the beam cross-section due to superposing numerous scaled in wavelength spectral interference patterns, so that minima of intensity (of the field of homogeneous spectral distribution!) are absent in the resulting pattern. *Such detector is "blind" to spatial coherence of such optical field*. Nevertheless, the human eye detects spatial coherence of a field due to non-uniform spectral sensitivity of visual receptors and inhomogeneous spatial distribution of *colors* over analyzed field. Of course, the same is true for evaluation of *temporal* coherence, to say, in the arrangement of the Newton's rings in

Less trivial case has been considered by Wolf [14, 15, see also numerous useful references herein] in the context of induced spectral changes resulting in remarkable transformations of temporal coherence of polychromatic optical fields due to the presence of material

The next, and more closer to our consideration, example *pseudodepolarization* [16] (in modern terminology, "global" depolarization [6]) resulting from stationary scattering of laser radiation in multiply scattering media, such as turbid media, multi-mode waveguides, the most of natural objects, including biological ones. Here the role of detector becomes fundamental. Really, the universal approach to determine all polarization characteristics of a field (both the state of polarization and the degree of polarization [17]) consists in Stokespolarimetry of the analyzed field. For that, Stokes-polarimetric analysis gives quite different results for local and "global" (space-averaged) measurements. So, the point-wise measuring Stokes parameters shows complete (unity) degree of polarization, but the state of polarization changes from point to point. Space averaging over ten and more speckles shows seeming depolarization. This case is the central subject of study of vector singular

The mentioned examples illustrate, in part, importance of taking into account of temporal and spatial resolution of the used detector, as well as a choice of the coordinates (stationary or moving) for metrological estimation of coherence and polarization of light, and even

Let us briefly highlight another two positions that are important for our consideration. Firstly, as it has been pointed out by I. Freund [19], we do not ready to investigate experimentally the problem of coherence and polarization of optical light in general, 3D

waves at the properly moving "register" (though this task is not simple in practice).

white light.

optics [18].

*intermediary*, as diffraction or scattering object.

understanding (*definition*) of these phenomena.

case, when paraxial approximation violates and one cannot neglect any of three Cartesian coordinates for description of behaviour of the electric vector. As an example, Freund references the study of polarization of relict – cosmic microwave radiation (CMR) [20] that is believed to be almost *isotropic* (nondirectional) [21]. Measuring the Stokes parameters for such radiation is, to all appearance, not well-grounded. It is the same that measuring the Stokes parameters in various directions from interior of a cloud of light-scattering small particles, what approach is not generally accepted [22, 23]. Nevertheless, it is the only what we have!

The second example concerns directly to one of the problem discussed in the following sections of this overview, *viz.* so-called *optical currents (flows)* [24]. Though it is prematurely now to solve comprehensively this problem, especially in experimental aspect, it is clear that micro- or nanoparticles serving for diagnostics of inhomogeneously polarized and partially coherent optical field [25 - 30] affect this field as absorbing and retransmitting particles with their own characteristics, so that the state of a field, in general, changes under influence of such secondary radiators.

Pronouncing call of the times in the topic under consideration consists in involving the ideas, approaches and techniques of Singular Optics [2]. It is seen, in part, from recent important review [3] devoted to the structure of partially coherent optical fields. Investigations in this scientific domain have been actually stimulated, in part, by papers [31 - 34]. For that, many usual and new results of the theory of partial coherence and partial polarization become essentially urgent just in the singular optics concept. On the other hand, there are good reasons to wait that attracting the fundamentals of the theory of partial coherence will lead to development of new practical applications of singular optics. So, two mentioned areas are of "mutual benefits".

Moreover, as it has been argued in papers [30, 35, 36], "*Usual beam parameters either characterize a beam 'in a whole' (power, momentum, beam size and divergence angle) or describe its 'shape' via certain spatial distributions (amplitude, phase, polarization state, etc.)… Usual beam parameters provide only rough and, sometimes, distorted picture of internal processes that constitute a real 'inner life' of a light beam. These processes are related to the fundamental dynamical and geometrical aspects of light fields, and are associated with the permanent energy redistribution inside the beam 'body', which underlies the beam evolution and transformations. The internal energy flows provide a natural and efficient way for 'peering' into the light fields and studying their most intimate and deep features.*" It is of interest, in the context of this review, to correlate this statement with the Wolf's methodology of observable quantities that is the most influential concept of physical optics since 1954: "optics of observable quantities, such as correlation functions and averaged in time intensities" [17]. Paradoxial contradiction between two undoubtedly true statements is apparent. Really, this contradiction is just eliminated as one takes into account that internal energy flows ("optical currents" [24]) may be revealed only by carrying out the experiments with observable quantities. Similarly, vibrational phase [18] of an optical wave is unobservable, while its difference with a phase of coherent reference phase is liable to registration just as the phase of the mutual coherence function of two waves. More globally, two mentioned approaches are complementary, being intimately interconnected, similarly to approaches of statistical physics and thermodynamics.

To facilitate understanding the material of the following sections, let us briefly remind the main light beam characteristics involved in our consideration. The key notion of *geometrical optics* is a ray; a bundle of rays constitutes an optical beam. The only information provided by rays is the direction of propagation of light. More diverse, more fundamental and more informative parameters of light are derived within the framework of wave optics. There are wave amplitude and phase (both quantities are unobservable directly though lying in the basis of theory [17]); polarization that is determined by the amplitude ratio and phase difference of orthogonal components of a beam; the degree of polarization is determined by correlations between these components; coherence parameters characterizing concordance of light disturbances at two (or more) spatialtemporal points of light field. These base characteristics defined in seminal book [17] as well as in numerous other handbooks in optics, are added by the Poynting vector determining the direction of propagation of light beams in terms of electrodynamics, and by more complex polarization parameters than simple azimuth of polarization and ellipticity inherent in elementary wave optics. In our study, much attention will be paid to the Stokes parameters, polarization parameters intrinsically connected with both energy and coherent characteristics of light beams. These notions will be specified in the corresponding sections of the chapter.

Organization of the review is not quite usual. The most of review papers written up to now on the topic of interest are of theoretical nature, sometimes with valued simulation background. Relatively less attention is paid to experimental aspects of the problem. Partially, this gap is filled by recent books and book chapters written by the authors of this review [37, 38]. But the scope of experimental results rapidly grows, and this outlook does not repeat our previous papers. So, we represent here several independent experimental current views on the problem of metrology of coherent and polarization properties of optical fields, with especial accent on singular optical prerequisites and consequences of our experimental approaches. In spite of forced incompleteness of this consideration, the authors hope that it would stimulate further experimental investigations in this field and lead to broadening of practically significant applications of *Singular Optics of Partially Coherent and Inhomogeneously Polarized Optical Fields*.
