During the past years, a special attention of the research and technology community has been paid to the structured light fields characterized by highly developed inhomogeneity of the amplitude, phase, polarization and spectral characteristics. This vibrant activity resulted in establishment of “structured light” as a new fruitful paradigm of physical optics [1–6]. The structured optical fields find interesting and productive applications in various branches of optical technologies, optical manipulations, optical communications and data processing as well as in optical diagnostics [7, 8].

Maybe, the most impressive feature of structured light fields is the existence of “singular” points (lines, surfaces, etc.) in space where certain parameters characterizing the field spatial structure (phase, ellipticity or helicity of polarization) are indeterminate. Despite the variability of types and sorts of optical singularities [9–22], they possess many similar features and are inherently interrelated (for example, the polarization singularities can be treated as phase singularities of separate orthogonal components of vector light beams [8, 16, 23]). The main common feature of optical singularities is their topological nature which makes the singularity stable with respect to small perturbations [24] and determines that each singularity qualitatively “organizes” the whole field in its vicinity, and different singularities combine and interact according to general laws. As a consequence, the “singular skeleton” (set of the field singularities with their positions and morphological characteristics [25–27]) represents a succinct characterization of the whole field [14, 15], which enables, for example, the economy encoding and representation of optical information [3]. In particular, the “singular” approach appears to be fruitful in the description and analysis of stochastic speckle fields [28] which frequently occur in problems of optical diagnostics of scattering objects and random surfaces [29].

The unique physical properties, great application potential as well as the vital interest of the community have inspired a series of consistent review publications treating various fundamental or applied aspects of the optical singularities (for example, Refs. [3–9, 13, 14, 17–20, 25, 30]). In this context, the present work pursues two main goals. The first one is to show how the optical singularities, even in their simplest scalar forms, illustrate the spectacular interrelations between the classical optics, non-linear optics, and quantum physics, up to the most fundamental ideas of quantum superpositions and multidimensional quantum states, and thus disclose the unity of the physical picture of the world. This task is quite compatible with the second goal, apparently much more utilitarian: description of some practical possibilities, offered by singular optics in analysis of chaotic speckle fields, and their use for reconstruction of scattering objects generating these fields. By the general approach, this paper adjoins our previous reviews on the adjacent topics [7, 8], and may be considered their further development and addition.

Due to their physical affinity, the main features of the optical singularities can be understood by considering the generic example of the point-like phase singularity; this is the topic of Section 2. The associated physical features: the wavefront dislocation, transverse energy circulation, specific mechanical properties (orbital angular momentum) are briefly analyzed, as well as the typical singularity-related manifestations in the non-linear optical phenomena (Section 3). Section 4 describes the singular photons in the context of quantum superpositions and quantum entanglement. Section 5 presents the concepts and approaches relevant for the stochastic speckle fields and their usage for the rough-object diagnostics. In Section 6, we briefly outline some interesting and important (in our opinion) features of singular optical fields and prospects of their studies and applications. The review contents are accomplished and summarized in Section 7.

Let us consider a scalar light field (properly, a paraxial beam with the spatially homogeneous polarization [3, 15]). The electric field of such a beam is described by the function ε ( r , ϕ ,   z ,   t ) = R e [ E ( r ,   ϕ ,   z ) exp ( i k z − i ω t ) ] (1) where (r, ϕ, z) is a cylindrical frame, ω is the radiation frequency, and k = 2π/λ is the wave number (λ is the wavelength). In the paraxial approximation [3, 15, 31], the complex amplitude E (r, ϕ, z) obeys the equation 1 r ∂ ∂ r ( r ∂ E ∂ r ) + 1 r 2 ∂ 2 E ∂ ϕ 2 + 2 i k ∂ E ∂ z = 0 (2) whose solutions can be represented in the form of azimuthal harmonics [9]. E ( r ,   ϕ ,   z ) = A ( r ,   z ) exp ( i l ϕ ) (3) where the coefficient l is called azimuthal mode index. The characteristic feature of the field (3) is the helical wavefront (WF) shape illustrated by Figure 1A (whence the term “screw WF dislocation” originates): after the round trip near the z-axis, the phase does not return to its initial value but changes by 2πl. Hence, the mode index l acquires the sense of the topological charge (TC) of the azimuthal mode (3) [9]. As the solution (3) of Eq. 2 should be unambiguous, once l ≠ 0, the wave amplitude A(r, z) = 0 at the axis, and only integer values of l are admissible (in practice, especially in cases of deliberate optical-field formation, beams of the form (3) with non-integer l can occur [32, 33] but these are “non-generic” and unstable: only exist in the initial cross section but destroy, with formation of a set of single-charge singularities, upon the beam propagation [34, 35].

The helical WF stipulates another important property of the azimuthal harmonics (3). In light beams, local directions of the energy flow are known to coincide with the WF normals [9, 15], which, for a helical WF, form 3D spiral lines; consequently, the energy propagates along the spiral-like trajectories skewed with respect to the beam axis [36]. Accordingly, there exists the transverse energy circulation near the phase singularity, which is the source of the orbital angular momentum (OAM) of light [3, 9, 15] and justifies the term “optical vortex” (OV) widely used for such singular structures [37].

A well-known example of solution to Equation 1 in the form of azimuthal harmonic is the family of Laguerre–Gaussian (LG) modes LG _p ^l of a stable laser cavity [31]. The LG modes are circularly symmetric, their intensity distribution consists of dark and bright rings centered at the axis of propagation z (Figure 1B [38]). The radial index p specifies the number of dark rings in the beam cross section, and does not affect the singular properties; so the condition p = 0 will be assumed below (upper row of Figure 1B). For the LG₀ ^l mode, the distribution of the complex amplitude has the form E LG ( r , ϕ , z ) = E s w 0 w ( z ) ( r w ( z ) ) | l | ⁡ exp ( − r 2 w 2 ( z ) ) × exp [ i ( k z + k r 2 2 R ( z ) + l ϕ − ( | l | + 1 ) arctan ( z z R ) ) ] (4) where z R = k w 0 2 2 ,   w 2 ( z ) = w 0 2 ( 1 + z 2 z R 2 ) ,   R ( z ) = z 2 + z R 2 z . (5)

The mechanical OAM L of such circularly-symmetric OV beams is described by the universal relation [39]. L = l W ω (6) where W is the light energy, i.e. each photon with the energy W = ℏ ω carries the OAM L p = l ℏ , ħ being the reduced Planck constant.

The spiral energy flows near the screw WF dislocation manifest themselves in many physical phenomena, for example, in photoinduced rotation of particles [40, 41]. But their direct observations in the OV edge-diffraction phenomena are especially simple and impressive [42–44] (Figures 2, 3). A spectacular picture of the intensity-profile evolution is shown in Figure 2 [43]. Here, the beam parameters and the observation plane are chosen such that the usual diffraction fringes are almost unnoticeable while the change of the beam transverse profile can be easily traced from the initial section immediately behind the screen (a) to the observation plane distanced from the screen by z = z _R (b). It can be seen that the initial pattern actually rotates in accordance with the direction of the transverse energy circulation, and one of the “edges” of the bright semi-ring penetrates into the shadow region behind the obstacle, while the other moves away from it. One might expect that the magnitude of the rotation increases with |l|, but in fact this is not the case; the larger initial slope of the vortex trajectories is compensated by an increase in the diffraction divergence, and beams with different |l| generally rotate the same way. However, in the case of large TCs, the bright semi-ring is located farther from the axis and is less distorted by diffraction, so the rotation itself is more noticeable.

Subsequent works [45–48] have shown that the usual edge diffraction offers rather informative and picturesque evidence of internal energy flows in beams with OV. Of particular interest is the behavior of the diffraction pattern when the screen covers an insignificant part of the beam “ring”, retaining the pattern of internal circulation as a whole (Figure 3A). The diffraction perturbation of the beam causes a shift of the amplitude zero (singularity “core”) relative to its initial axial location (in the case of the initial l-charged OV, |l| single-charged OVs are formed near the axis). Then, as the diffracted beam propagates, the displaced OVs migrate inside the beam “body” along helical trajectories that unwind in the direction opposite to the direction of energy circulation in the incident beam. At small distances behind the screen, the spirals unwind at a high rate, but with further propagation, the rotation slows down and stops in the far-field (Figure 3B). A similar OV migration occurs in a fixed section of the diffracted beam, when the edge of the screen moves perpendicular to its axis (Figures 3A, C)—a very spectacular illustration of the rotational properties of the field singularity.

The topological nature of the screw WF dislocation opens impressive possibilities for the data encoding and the information transfer [49–51]. The specific pattern of the transverse energy flows is responsible for another interesting phenomenon—rotational Doppler effect [52–59]. Indeed, the visible phase of the azimuthal harmonic (3) depends on the mutual angular positions of an observer and the beam with respect to the axis z (see Figure 1A). As a result, when the beam (or observer) rotates with the angular velocity Ω, the visible frequency changes by Δ ω = l Ω . This effect finds fruitful applications for the spectral analysis of light beams and the distant detection of the rotational motion of various objects [55, 58, 59], which are described in detail by recent reviews [59, 60]. For this reason, we do not dwell further upon the rotational Doppler shifts but proceed to some impressive non-linear effects involving the optical singularities.

Naturally, the study of phase singularities was extended to nonlinear optics, primarily to active schemes, e.g. in photorefractive laser oscillators. When the angular aperture of the cavity was compressed to the level of modes with the lowest transverse indices, a dynamic pattern of nucleation of pairs (dipoles) of OVs was observed in the output radiation [61]. The effect was then considered in the passive scheme of an induced nonlinear lens, where it was accompanied by the appearance of a closed spatial dislocation line in the form of a “seam on a tennis ball” and a quadrupole of vortices in the beam cross section [62]. In the stationary case, such closed and open (going into the far-field zone) “dark lines” of dislocations wrap around regions with the field intensity maxima, forming three-dimensional cells, and in the non-stationary case of dynamic light scattering processes, they create a “light boiling” structure, up to optical turbulence [63–65].

A number of new effects accompany the singular beams’ propagation in non-linear media. As was shown in [66], the non-linear medium asymmetry (astigmatism) destroys the OV with |l| > 1 (this effect is in parallel to the high-order OV decomposition in linear astigmatic systems [67, 68]); however, the instability of the cubic non-linear medium influences the first-order singularity also. Evidently, each singularity contains an inevitable amplitude zero, and the results of the beam self-focusing in the medium with cubic nonlinearity (positive non-linear refractive index) was unclear. A series of research works was undertaken that established the azimuthal instability of the bright ring of an OV beam and its decomposition into separate solitons [69].

On the other hand, a defocusing cubic medium also creates conditions for the OV instability and generation of “vortex” solitons [70, 71]. In the work [72], a beam with the edge WF dislocation is considered. As is shown in Figure 4, with growing non-linearity the initial edge dislocation (black line) acquires waviness and ultimately breaks up into separate “dark” solitons.

Obviously, the nature of the screw WF dislocation determines the pattern of the second-harmonic generation with pumping by an OV beam in a medium with quadratic nonlinearity. In the case of an initial TC l, a vortex with a TC of 2l should appear in the second-harmonic wave. However, the emergence of two spatially separated intensity zeros was experimentally recorded when the frequency of the OV beam with the TC l = 1 is doubled, which means the influence of the “irrotational” component in the beam (possibly, as a consequence of astigmatism) [73].

Further detailed analysis of the second-harmonic generation has shown that, as the intensity increases, the ring profile of the OV beam (like those shown in Figure 1B, top row) becomes unstable in a medium with quadratic nonlinearity, and the bright ring breaks up into several soliton-like beams of doubled optical frequency; moreover, the initial OV beam of a unit TC generates three separate beams [74]. The OAM conservation law “works” in this situation too, forcing the “combined” intensity profile of three beams to rotate as a whole in the transverse plane as they propagate.

A particularly interesting manifestation of nonlinear optical interactions involving OV beams is the generation of singular vortex fields in the process of spontaneous parametric scattering (SPS) [75–79, 82]. In the first experiments [75], azimuthal harmonics in the form of Bessel beams arose as a result of the amplification of quantum noise in a lithium triborate (LBO) crystal 15 mm long under pulse pumping (1 ps, 30 GW/cm², 527 nm, a focused Gaussian beam with a half-maximum diameter of 61 μm). Under these conditions, signal beams were observed with a wavelength of 960 nm; in 45% of cases they had a profile described by the zero-order Bessel function (non-vortex), and in 36% vortex Bessel beams with TCs l = ± 1 appeared. In other cases, the signal wave had a complex speckle structure without apparent regularity. Unfortunately, in [75], the result of the parametric amplification depended decisively on the random pattern of “seed” noises, and it was not possible to achieve stable and reproducible generation of vortex modes.

Another approach was realized in [76–79, 82] where the SPS process was used for creation and investigation of multidimensional entangled states of vortex photons. Therewith, the structures of the idle and signal beams were not studied immediately but the stress was made on the non-local connection between them. In these experiments [79], the pump radiation from an Ar laser (λ = 351 nm) entered the anisotropic crystal BBO (barium beta-borate) where the signal and idle waves with equal wavelengths 702 nm were generated, propagating at angles 4° with respect to the pump beam (see Figure 5). Herewith, each pump photon generates two scattered ones which are in the single entangled quantum state. Denoting the photon state described by the LG₀ ^l mode (4) as | l 〉 , and if the pump wave is a Gaussian beam with zero OAM (Equation 4 with l = 0), this entangled state can be represented as | ψ 〉 = C 0,0 | 0 〉 1 | 0 〉 2 + C 1 , − 1 | 1 〉 1 | − 1 〉 + C − 1,1 | − 1 〉 1 | 1 〉 2 + C 2 , − 2 | 2 〉 1 | − 2 〉 2 + C − 2,2 | − 2 〉 1 | 2 〉 2 + . . . (7) where | l 1 〉 1 | l 2 〉 2 is the two-photon state in which the idle wave (channel 1 in Figure 5) is described by the LG 0 l 1 mode, and the signal wave (channel 2)—by the LG 0 l 2 mode (strictly speaking, in the expansion (7), modes with non-zero radial indices p should also be present but their contribution is relatively small).

Thus, the photons obtained after the parametric conversion have no definite OAM and no definite TC; however, the subsequent measurement of the TC of one of them, which gave, for example, the value l ₁ = 1 for an idle wave, leads to the reduction of state (7): from the entire infinite sum, only the term remains with | 1 〉 1 | − 1 〉 2 . Consequently, the signal photon also acquires a definite TC l ₂ = −1, and this occurs instantly (“teleportation”), despite the fact that at the moment of measurement it can be spatially removed from the idle one by a macroscopic distance (“non-locality”). Without going into details, we can see here how quantum mechanical effects “work” with OV states obtained from a completely classical pump field.

In the setup of Figure 5, each of the photons enters the mode detector, which consists of a hologram with the groove bifurcation (“fork”) and a single-mode optical fiber. Such holograms are generally used for the generation of optical singularities [9, 25, 83, 84]; here they operate in the “reverse mode.” The fact is that the hologram, which is designed to generate an OV beam with azimuthal index l from the initial Gaussian beam with a smooth WF, will create the same Gaussian beam if it is illuminated (while maintaining the other conditions) by an OV beam with index − l. In its turn, the next element of the mode detector, a single-mode fiber, has selective sensitivity specifically to a Gaussian beam: only a Gaussian beam obtained after a hologram can “penetrate” a single-mode fiber (in other cases, higher LG modes are obtained at the hologram output, whose size of the spatial distribution does not satisfy the fiber excitation conditions). Therefore, the appropriate choice of the hologram in the signal or idle channel allows one to purposefully “check” the presence of a photon state with one or another TC value in the superposition (7).

The connection between photons in channels 1 and 2 is detected via analyzing the coincidences of the detector signals in both channels in the photon counting mode. Figure 6A convincingly demonstrates that when measurements show the presence of a photon with TC l ₁ in channel 1, then a photon with l ₂ = –l ₁ is formed in channel 2.

The most convincing evidence for the entanglement of photon states is obtained when not “pure” LG modes are detected in the channels, but their superpositions. For example, state (7) can be represented in the form | ψ 〉 = ( a 0 | 0 〉 1 + a 1 | 1 〉 1 ) ( a 0 C 0,0 | 0 〉 2 + a 1 C 1 , − 1 | − 1 〉 2 ) + ( a 1 | 0 〉 1 − a 0 | 1 〉 1 ) ( a 1 C 0,0 | 0 〉 2 − a 0 C 1 , − 1 | − 1 〉 2 ) + C − 1,1 | − 1 〉 1 | 1 〉 2 + C 2 , − 2 | 2 〉 1 | − 2 〉 2 + C − 2,2 | − 2 〉 1 | 2 〉 2 + . . . , (8) where a ₀ and a ₁ are real numbers, and the normalization condition a 0 2 + a 1 2 = 1 is required (here only the two first summands of (7) are transformed, the rest of the terms remain unchanged). The expression ( a 0 | 0 〉 1 + a 1 | 1 〉 1 ) in the first summand represents the superposition of a Gaussian and LG 0 1 beams, for which the intensity distribution possesses the shape of a ring with the amplitude zero displaced from the axis [9], see Figure 6B.

Such a superposition is realized in channel 1 when the hologram center slightly shifts from the propagation axis of photon 1 (in Figure 6B, from top to bottom, three stages of such a shift are shown: with growing shift, the “ring” transforms into “crescent”). After passing the hologram, the state (8) reduces to the first summand, i.e. the photon 2 appears in the superposition state too, ( a 0 C 0,0 | 0 〉 2 + a 1 C 1 , − 1 | − 1 〉 2 ) , but here a Gaussian beam is combined with the mode LG 0 − 1 . For detection of this superposition, the hologram is removed in channel 2, and the mode detector scans the transverse intensity distribution, in order to determine the position of the second-photon amplitude zero with respect to the beam center. The coincidence calculation shows that, indeed, the photon in channel 2 is also in the state of superposition of the LG and Gaussian modes (Figure 6B).

Classical correlation would give a pattern of coincidences that is simply a mixture of Gaussian and LG modes: the intensity minimum would remain at the beam center, but would be smeared, and the intensity would be everywhere greater than zero.

Multidimensional entangled quantum states are of interest for fundamental research of the quantum physics foundations, in particular, for testing Bell’s inequalities [77–79, 82]. On the other hand, they make it possible to create quantum memory cells with more than two states, which leads to an increase in the speed and reliability of information processing, and thus are promising for numerous applications in the field of quantum cryptography, encoding, quantum communication networks, and quantum computers [80–82]. Pioneering experiments with entangled OV photons [79, 80, 82] were awarded the 2022 Nobel Prize.

A curious modification of the concept of entangled vortex photons was realized in the “non-local OV” [85, 86]. In this case, the OV is observed via the correlations of photons produced from SPS, and the phase singularity appears in a nonlocal coordinate plane where one dimension is the usual coordinate of one photon whereas the second dimension corresponds to the transverse momentum of the second photon. This idea demonstrates the power and flexibility of the quantum-mechanical concepts and supplies their pictorial realization with intuitively clear macroscopic objects.

We hope that the above sections supply a representative exposition of some not very well known properties of beams with screw WF dislocations. Their contents illustrate a relatively small part of the huge massive of facts and concepts accumulated since the OV discovery [21, 22, 37], and the works in this direction are still growing like an avalanche up to the present time. Some additional aspects will be presented in further sections; a representative picture of current development of this fascinating and productive field of research can be traced with the help of periodic special issues [87–90] and in other reviews, for example Refs. [5, 6, 30, 33].

Meantime, we proceed to the next step of the present review, which addresses the fields with multiple singularities formed by the coherent light scattering by various random objects, and their applications for the optical-diagnostic purposes.

In this Section, we consider some additional examples illustrating the physical productivity and practical relevance of the ideas and concepts of singular optics. As the first such example we mention the group of phenomena explicitly demonstrating the peculiar internal energy flows in the beams with OV [138–147]. In Section 2, we discussed such manifestations in the edge-diffraction phenomena but really, the OV-caused “disbalance” of energy flows comes to light in every situation where its symmetry is broken [68, 148]. This is the case, for example, if an “oblique” section of the OV beam is important. This “orbital” analogue of the geometric spin Hall effect [149] manifests itself in the deformations of the transverse beam profile and in the corresponding shifts of the beam “center of gravity” when the beam experiences reflection or refraction at a plane interface [140–146]. The effect is especially expressive in case of non-specular reflection near the critical angle of total reflection [145], or when the beam undergoes a grating diffraction into a highly non-geometric order [138, 139]. This “orbital” Hall effect expresses the interaction between the “intrinsic” and “extrinsic” degrees of freedom of a singular light beam, which are discussed in much detail in a series of topical reviews (see, e.g., Ref. [150]).

Another important enhancement of the phase-singularity concepts is coupled with their penetration into the near-field optics, especially, into the vibrant domain of near-surface evanescent waves [151–156]. In works [151, 152], the singular evanescent wave is excited during the total internal reflection of an OV-carrying LG mode (4). The resulting field in the low-index region possesses vortex properties: it has well-defined OAM, residing in an azimuthal phase relative to the propagation direction of the internally reflected light. Such surface modes are characterized by a small mode volume, they can strongly couple to atomic or molecular systems in the vicinity of the surface. In case of counter-propagating, symmetrically incident LG _p ^l modes, efficient 3D optical traps with parameters adjustable via the LG mode indices, can be realized.

Similar singular structures can be realized in the surface plasmon-polariton (SPP) waves supported by a metal-dielectric interface [2, 3]; in particular, this situation offers advantages of remarkably higher light intensity due to plasmonic enhancement [133]. In [153, 154], the radially propagating vortex SPP was excited due to coupling of a circularly polarized laser beam via a coaxial ring-like aperture in the gold film. As a result, near the metal-vacuum interface, the surface wave appears with the electric field components E pr ( r ) = 0 ,   E p ϕ ( r ) = i σ u p ( r ) e i σ ϕ ,   E p z ( r ) = − u p ( r ) e i σ ϕ (23) (r and ϕ are the polar coordinates in the interface plane, z-axis is directed along its normal) where u p ( r ) = A e − κ z H 1 ( 1 ) ( k r ) H 1 ( 1 ) ( k r 0 ) ,   r ≥ r 0 , (24) r ₀ is the aperture radius, σ = ±1 is the incident wave helicity, κ specifies the SPP near-interface confinement, and H 1 ( 1 ) is the Hankel function [157]; the constants κ and A are determined by the incident laser beam and by the excitation geometry. The field (23), (24) is characterized by the helical phase but, in contrast to the usual OVs (3), where the helicity “evolves” in the cross-section plane, now the plane (r, ϕ) is the propagation plane. The phase dislocation “strength” (TC) is determined by the incident field polarization helicity σ. In the field (23), (24), the energy flow propagates within the (r, ϕ) plane along the spiral lines; the helical structure can be easily observed in the near field by means of interference, which is especially evident when multiple vortex SPPs are formed simultaneously [155, 156]. The phase terms σϕ in Eq. 22 are analogous to the spiral phase acquired by electron waves scattered by a cylinder containing a magnetic flux (optical analog of the Aharonov—Bohm effect [153, 158]). The formation of such surface waves can be treated as a sort of spin-orbital interaction, highly sensitive to the incident beam polarization, which offers a perfect quantum weak-measurement tool with a built-in post-selection in the SPP mode [154]. The vortex SPPs of this type show valuable abilities in observation fine light-matter interaction effects and characterizing the SPP-supporting interface topology. At this point, we should note that, generally, the SPP fields serve as efficient instruments for ultra-sensitive testing the surface properties, up to detection of single molecules or atomic-size defects [159, 160], as well as the surface roughness [161–163]. In this context, the unique properties of the singular SPPs similar to those described by Eqs 23, 24 may open new prospects due to their specific topological nature and peculiar polarization features.

Very interesting possibilities arise from the ideas of “coherence vortices” (CV)—“hidden” phase singularities which exist in the correlation function rather than in the immediately observable field distribution (1) [164–183]. Generally, usual partially-coherent paraxial fields are spatially inhomogeneous and are characterized by the two-point correlation function F ( r 1 , r 2 ) = 〈 E * ( r 1 ) E ( r 2 ) 〉 〈 I ( r 1 ) 〉 1 / 2 〈 I ( r 2 ) 〉 1 / 2 (25) where I ( r ) = | E ( r ) | 2 is the local intensity, and angular brackets denote the ensemble average (this formula can be reduced to (14) if the field statistical properties in points r ₁ and r ₂ are identical, and F ( r 1 , r 2 ) depends only on the difference ρ = r ₁—r ₂). Actually, the coherence function (25) is four-dimensional [166, 167] but it can be characterized via its 2D projections. The most common situations are the following: 1) one point (say, r ₁) is fixed and the correlation function (25) depends only on r ₂ [165, 174–176], and 2) points r ₁ and r ₂ are interrelated such that r ₁ =—r ₂ [166, 167, 176]. The CVs are the singularities of the “reduced” 2D correlation functions F ( c o n s t , r 2 ) and F ( r 2 , − r 2 ) which may show the typical singular behavior in the r ₂-plane while the observable complex amplitude distribution ⟨E (r ₂)⟩ is everywhere regular at any moment of time.

For example, in the “wandering beam” model, the propagation of a LG₀ ^l beam (4) is considered while its axis position in the transverse plane is a random function, and the cross-correlation function of Eq. 25 can be determined as 〈 E * ( r 1 ) E ( r 2 ) 〉 = ∫ E * ( r 1 − r ′ ) E ( r 2 − r ′ ) q ( r ′ ) d 2 r ′ ,   q ( r ) = 1 π ρ c 2 exp ( − r 2 ρ c 2 ) (26) where ρ _c is the measure of the transverse coherence. With the help of this model, it is demonstrated for a time-invariant linear optical system that there exists a definite connection between the usual OVs (phase singularities of the field amplitude), which appear when the system is illuminated by spatially coherent light, and the CVs of the function F ( c o n s t , r 2 ) which appear when it is illuminated by partially coherent light [173–176]. Usual OV beams can evolve into CVs when the degree of coherence falls down (ρ _c → 0): according to the conservation laws, the TC, associated with the phase singularity, “moves” from the field to the coherence function, as well as the OAM does.

When the projection 2) is used, the correlation function (25) F ( r 2 , − r 2 ) contains annular ring-like edge phase dislocations [9] with the configurations depending on the input partially-coherent LG _p ^l beam characteristics [176–179]. The singularity of the function F ( r 2 , − r 2 ) may exist even in a non-vortex beam (l = 0) due to the non-zero radial index [178]. Interestingly, the number of ring dislocations in the far-field correlation function equals to 2p + |l| for the low-coherence cases. This fact may offer efficient means for measuring the magnitude and sign of TC of partially coherent OVs [178, 182], which would be particularly useful in atmospheric laser communication.

Based on the beam-wander model, investigations of the partially coherent LG beam propagation for any radial and azimuthal indices and at any propagation distance have been performed in [183]. It was shown that, as the coherence decreases, the correlation function acquires increasing number of the single-charged vortex-type singularities, and this effect depends on the radial index of the input LG beam. These observations indicate that a proper choice of randomization is favorable for the CVs’ detection and open new possibilities to sort photons not only by their TC but also by radial indices. During the beam propagation, the CVs exhibit “self-healing” properties, which are interpreted as a van Cittert–Zernike-style [28] evolution that depends strongly on the manner in which the beam is randomized. Though the total OAM of the beam is conserved, different distributions of the OAM density can be realized by adjusting the input radial index and propagation distance. These features can be applied in optical communications employing both azimuthal and radial orders of the OV beams as well as for the fine tuning of the rotation of particles trapped in OV beams [183].

Besides the beam-wander conditions, the CVs can be generated when an OV beam passes a random scatterer (e.g., rotating ground-glass disc) [179, 180], or due to special randomization of multiple partially-coherent “source” beams regularly arranged over the input plane [181]. In the arrangement of [179], the “regular” large-scale singularity is “hidden” inside the visually chaotic speckle structure but can be recovered via the correlation analysis. Notably, such CV structures are “robust”: After the beam passes through an obstacle which apparently blocks a noticeable part of the beam profile, its coherence function experiences essential changes but with further beam propagation, these changes disappear and the CV is restored [179, 180]. Together with the self-healing properties mentioned in the above paragraph [183], this fact is illustration of the important general feature of the CV structures: their high stability and low sensitivity to external perturbations, even when compared with the usual OVs. Remarkably, this stability can be enhanced by additional randomization and/or decrease of coherence of the input laser radiation [179, 181, 183].

All these facts illustrate the exclusive features of CVs as specific topological entities carrying information in the correlation degree of freedom, and testify for their bright prospects in applications for the data encoding, optical communication as well as in formation of structured fields with special configurations for optical trapping and manipulation. Remarkably, despite the impressive and sometimes counter-intuitive properties, the CVs are not so exotic as it seems at first glance: the correlation functions of black-body radiation are known to possess an infinite number of phase singularities (related to the zeros of the spherical Bessel functions [184]).

Since the early days of the “singular era” in optics, optical communications and data processing remain among the most important fields of application [49–51]. In the recent years, employment of the structured-light concepts opens up new possibilities in fundamental applications, including image visualization, increasing the throughput of communication systems through mode-separation multiplexing, high-dimensional quantum cryptography and the creation of multidimensional quantum encryption systems [80, 81, 185–188]. However, the usual optical encryption protocols have been primarily based on the first-order field characteristics, which are strongly affected by interference effects and make the systems unstable because of light–matter interaction. This defect is avoided in an alternative optical-encryption protocol [189] whereby the information is encoded into the second-order spatial coherence distribution of a structured random light beam via a generalized van Cittert–Zernike theorem. The new approach has two key advantages over its conventional counterparts: 1) the complexity of measuring the spatial coherence distribution of light enhances the encryption protocol security, and 2) the relative insensitivity of the second-order statistical characteristics of light to environmental noise makes the protocol robust against the environmental fluctuations, e.g., the atmospheric turbulence.

Optical singularities are essential elements of multiple applications of structured light for precision material handling. Manipulating the amplitude, intensity, phase or polarization leads to new fundamental implementations for solving a significant range of problems, such as optical communication technology, data security in information optics, material nanotechnology, etc. The main principles of optical manipulation [152, 190–192] are the same as were outlined in Section 5.5, see Figures 15C–E (a particle is “kept” inside the light-intensity minimum or maximum) but, due to the implementation facility and flexibility of control, specially tailored structured light fields [193–195] realize the “smart” optical-trapping technologies enabling the material engineering at the atomic level. The modern optical traps minimize the photon-scattering and thermal effects and use the coherence as an additional control channel. They find applications in the control of cold atoms, manipulation of the quantum states of the degenerate gases, generation of non-conventional states of the matter waves, etc.

The need for new ultra-compact structured-light sources gave impetus to the development of artificial optical materials, including metamaterials and metasurfaces, which determined the great enhancement of the means for purposeful light-field engineering [196–201]. The discovery of toroidal optical dipole traps made it possible to realize the conditions for confining a super-fluid Bose–Einstein condensate by introducing a weak radial barrier with tunable coupling [202]. The natural next step is associated with the possibility of manipulating de Broglie atomic waves by analogy with the manipulation of light waves in optics. Accordingly, the singular-optics ideas can be generalized to the matter-wave optics [203], as a complex of concepts and tools for manipulating the amplitude and phase of the atomic and electron waves.

We cannot exhaustively describe this fascinating emerging field within the limited frame of this review but merely mention that it opens new and very promising ways in the material science and optics. The central concept of the singular atom optics is the “electron vortex” [204–210] being the electron-wave analog of the usual screw WF dislocation (azimuthal harmonic (3)). With all the precautions caused by the different physical nature (electric charge instead of neutrality, half-integer spin, obeying the Dirac equation [211] instead of the Helmholtz one (2)), the electron vortex carries distinct characteristic features of the wave singularity stipulated by its topological nature. Just like an OV, an electron vortex is a spiral de Broglie wave carrying a quantized OAM. However, unlike the photon OAM, electronic OAM can directly excite dipole transitions in atoms due to the Coulomb interaction, thereby realizing new applications in nanooptics also, as well as participate in specific electromagnetic interactions [209, 210]. Waves of electronic matter have the ability of coherent transformation, with the possibility of creating elements analogous to holographic diffractive optics. Following the optical analogy, electron vortices are promising candidate for the data encoding and qubits-based quantum memory elements.

To sum up, the principles of the wave diagnostic and engineering capabilities, worked out on the example of optical singularities, are intensively transferring to the wave fields of other physical nature [30, 212, 213]. In these new fields, they form powerful grounds and promising prospects for impressive applications in the huge area of science and technology, from the subatomic scales to the biological cells and to the world of galaxies [30, 214, 215].

In this review, we tried to expose some selected features of singular optics which seem to us the most relevant and interesting in both fundamental and applied aspects. To this purpose, the generic properties of point-like phase singularities in scalar fields (OVs) are discussed in detail. This example is especially useful and demonstrative as it enables, via simple and intuitively clear models, to show the common features of vortex motions in different physical systems, from light fields to tornado storms and spiral galaxies [7, 25] and thus effectively exposes the unity of physical world and the general character of physical laws. In particular, the OV diffraction properties are discussed which reveal the singularity-related energy circulation in light beams (Section 2). The peculiarities of the OV-beams’ non-linear behavior, associated with their specific intensity profiles, are described in Section 3. On the other hand, the spontaneous parametric down-conversion with participation of the OV photons (Section 4) supplies a picturesque and instructive illustration of quantum entanglement but also opens impressive possibilities in advanced data encoding, quantum encryption, communication and computing.

In Section 5, the main attention is paid to the stochastic speckle fields that are known [92–97] to contain multiple optical singularities, which, due to their topological nature, form coherent and interrelated networks (“singular skeletons”) and characterize the optical field “as a whole”. In this Section, the principles of the statistical characterization of random singular fields are outlined; the specific features of the fields produced by fractal and non-fractal random scatterers are discussed, as well as the possibilities for their recognition via optical diagnostic means. Simultaneously, the methods of the singular-skeleton detection, and of the combined employment of correlation-optics and singular-optics approaches for the practical field diagnostics are presented in Section 5.3 and Section 5.5. In particular, they supply new solutions to the famous “phase problem” in optics [112, 114]: non-interference recovering the “full” field information from the immediately observed intensity distribution.

Section 6 offers a brief outlook of the singularity-associated problems and knowledges that are the subjects of continuing discussions and investigations. In particular, it describes the singularity-induced beam shifts, stipulated by the internal energy flows [138, 140, 145]; phase singularities and vortex-like structures in the surface evanescent waves [151, 153]; “coherence vortices” inherent in the field coherence function rather than in the complex amplitude distribution “per se” [175]. Advanced applications of the singular optical fields in optical communication systems and optical manipulation techniques are discussed. Finally, possible extrapolations of the singular-optics ideas and concepts on wave fields of other physical nature, such as acoustic waves, and, especially, matter waves and electron beams in quantum mechanics, are characterized in brief.

As a concluding remark, we should emphasize that it is impossible to exhaustively describe the development of concepts and applications associated with optical singularities, even restricted to the simplest case of phase dislocations in scalar fields, within the framework of one review. We believe that the data and knowledges, presented above, fairly reflect the current state of the art in singular optics but these are inevitably restricted by the experiences and interests of the authors. Many further impressive results and ideas can be found in other collections, for example, in topical special issues [87–90], recent reviews [5, 6, 30, 33], etc.