As a general inference of the above presentation, one may note that, currently, the idea of metrological assessment of the statistical consistency, both in time and in space, for wave fields has been practically implemented. Flexible, reliable and attainable interference approaches for metrological evaluation of the field correlation moments (including the mixed moments, describing, e.g., correlations between the amplitudes and phases of the fields involved) of different orders have been developed [44].

The ideology underlying the design and composition of such metrological systems is based on the concept of phase object (PO). From now on, the term “PO” means a material object whose influence on the input radiation can be reduced to the phase modulation of the output (reflected or transmitted) beam; of course, this “pure” phase modulation can only be observed immediately after the light-object interaction, i.e., in the “boundary zone” [44, 45] (with further propagation, the phase modulations inevitably induce the amplitude inhomogeneities). In many cases the PO-induced phase transformations are random and are directly associated with the irregularities of the object transmission or reflection coefficients, which can be described in a unified way as E o u t x , y = T P O x , y E i n ;   T P O x , y = T P O exp i φ x , y , T P O ≡ 1 (9) where the input field amplitude E i n is supposed homogeneous, as well as the modulus of the transformation coefficient T _PO . Eqs. 9 show that the phase characteristics of the output radiation E o u t x , y are uniquely related to the phase characteristics of a random PO. Ultimately, the optical diagnostics of such a PO mainly requires a selection of appropriate methods for describing its phase structure; according to Eqs. 9, a complete description of such a random PO can be performed, akin to stochastic optical fields, via the probability density of the stochastic phase-modulation field φ(x, y) and its correlation and cross-correlation functions. Generally, Eqs. 9 constitute a basis for the flexible and efficient approach to description of the PO and the probing optical field interactions, which can be referred to as the “random phase screen model” [45, 46].

The concept of transient superposition was introduced for characterizing the phenomena obtainable with quasi-monochromatic beams of slightly different central frequencies [61]. Let us consider two waves for which the electric field values can be represented by Fourier integrals: E ∼ 1 R , t = ∫ − ∞ ∞ U 1 R , ω e − i ω t d ω ,   E ∼ 2 R , t = ∫ − ∞ ∞ U 2 R , ω e − i ω t d ω (14) (from now on, symbols with tilde “∼” denote “true” instantaneous characteristics, in contrast to the time-averaged ones which will be in the focus of further analysis). Here R = (x, y, z) ^T denotes the spatial coordinates (superscript “T” means the matrix transposition), the spectral densities can be presented in the form U 1 , 2 R , ω = A 1 , 2 R , ω exp i φ 1 , 2 R , ω where A 1 , 2 R , ω and φ 1 , 2 R , ω are real functions, and A 1 , 2 R , ω = A 1 , 2 R , − ω is an even while φ 1 , 2 R , ω = − φ 1 , 2 R , − ω is an odd function of ω (otherwise integrals 14 do not represent real-valued functions). Generally, the spectral densities in (14) satisfy the real-value conditions U 1 , 2 R , − ω = U 1 , 2 * R , ω . (15)

In turn, it is suitable to suppose the separability of the spatial and spectral arguments in A 1 , 2 r , ω , A 1 , 2 R , ω = a 1 , 2 R ρ 1 , 2 ω , (16) so the waves’ amplitudes are determined by a 1 , 2 R while the spectra ρ 1 , 2 ω are normalized by the condition ∫ − ∞ ∞ ρ 1 , 2 ω d ω = 1 . (17)

Additionally, to reflect the real situation of quasi-monochromatic waves, we suppose ω ¯ 2   –   ω ¯ 1 ≪ ω ¯ , δ ω 1 ≪ ω ¯ , δ ω 2 ≪ ω ¯ (18) where ω ¯ 1 , δω ₁ and ω ¯ 2 , δω ₂ are the central frequencies and widths of the spectra ρ 1 ω and ρ 2 ω , respectively, ω ¯ = | ω ¯ 2 + ω ¯ 1 |/2 is the mean central frequency of the waves (14). For determinacy, the assumption ω ¯ 2 > ω ¯ 1 is accepted in further reasonings.

We consider superposition of waves (14) that forms a resulting field E ∼ 1 R , t + E ∼ 2 R , t . Its observable characteristic is the intensity proportional to the energy density, I ∼ R , t = E ∼ 1 R , t 2 + E ∼ 2 R , t 2 + I ∼ 12 ,   I ∼ 12 = 2 E ∼ 1 R , t E ∼ 2 R , t . (19) According to the classic theory of the second-order coherence [61], the waves E ∼ 1 R , t and E ∼ 2 R , t are mutually incoherent. At the same time, under conditions (18), a certain statistic relation exists between the waves (14), which can be revealed by observations of the time-average behavior of the intensity (19). In general, expressions (19) contain the terms oscillating with zero or low frequencies (those are formed by the products of spectral components with positive and negative frequencies) and the rapidly oscillating ones in which spectral components with frequencies of the same sign are combined. The physically observable behavior is described by the quantities (19) averaged over the rapid oscillations.

The most important results are associated with the last (interference) term of (19), I _1,2. By using relations (15)–(18), its time-averaged value with sufficient accuracy can be described by equation I 12 = 2 a 1 R a 2 R ∫ 0 ω ¯ μ R , Δ ω e i Δ ω t + μ * R , Δ ω e − i Δ ω t d Δ ω (20) where μ R , Δ ω = ∫ 0 ∞ ρ 1 ω ρ 2 ω + Δ ω exp i φ 1 R , ω − i φ 2 R , ω + Δ ω d ω + ∫ 0 ∞ ρ 1 ω ρ 2 ω − Δ ω exp − i φ 1 R , ω + i φ 2 R , ω − Δ ω d ω . (21)

The quantity μ R , Δ ω characterizes the statistical and, possibly, regular interrelations (“coupling”) between the waves (14). The main properties of the transient superposition (20), (21) can be illustrated by the simple situation where both waves are quasi-monochromatic, ρ 1 , 2 ω = 1 2 δ ω − ω ¯ 1 , 2 + δ ω + ω ¯ 1 , 2 (22) and possess similar phase distributions such that φ 1 , 2 R , ω = φ 0 ω + ω c ζ R where the function ζ R characterizes the (common) wave-front shape, and φ 0 ω = τ 0 ω means that the initial phases of all spectral components are determined by linear phase retardations with respect to a certain initial moment of time common for both constituents. Under these conditions, I 12 R , t = 2 a 1 R a 2 R cos Δ ω ¯ t − ζ R c − τ 0 ,   Δ ω ¯ ≡ ω ¯ 2 − ω ¯ 1 (23) which supplies the usual expression of beatings observable in the superposition of waves with different frequencies. Eq. 23 describes the slowly-varying interference pattern moving with the velocity determined by the difference of the central frequencies.

The latter results of the previous Section testify that some essential features of the non-monochromatic optical fields can be understood via the simplified analysis of a bi-chromatic superposition containing only two frequencies (Eq. 22). In Section 4.1, the scalar field model was considered which is applicable to beams with a homogeneous linear polarization [58, 59]; now we address a bit more complex situation where the superposition includes two vector paraxial beams with arbitrary polarization in the transverse cross section.

In this case, it is suitable to start with the explicit expressions for bi-chromatic superposition that follow immediately from the vector analogs of Eqs. 14, 22: E ∼ r , z , t = E ∼ 1 r , z , t + E ∼ 2 r , z , t , E ∼ 1 , 2 r , z , t = Re E 1 , 2 r , z , t ,   E 1 , 2 r , z , t = u 1 , 2 r , z exp i k 1 , 2 s (24) where s = z − c t , c being the light velocity, r = (x, y) ^T is the transverse radius-vector, E 1 , 2 r , z , t is the complex vector corresponding to the positive-frequency part of the spectral expansion (14) (simplified form of the analytical signal [14]); the frequencies ω 1 , ω 2 coincide with the corresponding central frequencies ω ¯ 1 , ω ¯ 2 so there is no need for their separate notation, and the wavenumbers k _1,2 = ω _1,2/c. The transverse vector functions u 1 , 2 r , z are the slowly-varying paraxial complex amplitudes which can be expressed via the (x, y) components [58, 59]: u 1 , 2 = e x u 1 , 2 x + e y u 1 , 2 y (25) (e _x , e _y , and e _z are the unit vectors of a Cartesian frame). In turn, the complex amplitude components are characterized by their own amplitudes u 1 , 2 x r , z = a 1 , 2 x r , z exp i φ 1 , 2 x r , z ,   u 1 , 2 y r , z = a 1 , 2 y r , z exp i φ 1 , 2 y r , z . (26)

For illustration, we consider the field behavior in a single point of the beam cross section, which allows one to omit the coordinate dependence of (26). Then, Eqs 24, 26 determine the instantaneous behavior the electric field components. It is suitable to choose the time-scale origin so that φ 1 x = 0 at the considered point, and to characterize the polarization of each wave by the phase shift between the x- and y-components Δ φ 1 = φ 1 y − φ 1 x , Δ φ 2 = φ 2 y − φ 2 x . As a result, denoting the initial phase shift between the x-components as φ 2 x − φ 1 x = φ 21 , one obtains E x t = a 1 x ⁡ cos k 1 z − ω 1 t + a 2 x ⁡ cos k 2 z − ω 2 t + φ 21 , E y t = a 1 y ⁡ cos k 1 z − ω 1 t + Δ φ 1 + a 2 y ⁡ cos k 2 z − ω 2 t + Δ φ 2 + φ 21 . (27)

These relations describe the instantaneous behavior of the electric field in a bi-chromatic polarized wave [62–66]. In contrast to the usual linear or elliptic polarizations, observed in monochromatic fields, here the electric vector describes rather complex trajectories. If the frequencies ω ₁ and ω ₂ are commensurate, the trajectories are similar to the Lissajous figures [67] repeatedly reproduced with the beating period T = 2 π N 1 / ω 1 = 2 π N 2 / ω 2 where N ₁ and N ₂ are the smallest integer numbers satisfying the equality N 1 / ω 1 = N 2 / ω 2 . Their forms depend on the relations between the amplitudes, phases and frequencies of the composing waves (27), and can be rather intricate. Generally, these forms are characterized by the symmetric matrix M = m x x m x y m x y m y y = E x 2 E x E y E x E y E y 2 (28) where ⟨…⟩ means the average over the beating period (“moments of inertia tensor” of the Lissajous figure [62, 63]). This matrix is similar to the real coherence matrix (6) at τ = 0 and to the moment matrix in parametric characterization of the beam transverse profile [20, 68]. The Lissajous singularities appear as the generalizations of the usual monochromatic polarization singularities [69, 70] in points where the eigenvalues of the matrix (28) are equal and where at least one of the eigenvalues vanishes (analogs, respectively, of the C-points and s-contours [69, 70]). Such structures naturally appear in the processes of higher-harmonic generation where N ₁ = 1, N ₂ = 2, 3, … [62–64], and the involved beams are spatially inhomogeneous.

Here, we briefly consider the instantaneous field in a single point; in this case, Eqs 27 allow to disclose the formation mechanism of the time-averaged characteristics in the general polychromatic case [71]. Some numerical results are presented in Figure 4, which illustrates the case of superposition of beams with orthogonal linear (circular) polarizations. This situation is free from the limitations associated with the harmonic generation, and the conditions N ₁ = 4, N ₂–N ₁ = 1 (ω ₂/ω ₁ = 1.25) are accepted. Accordingly, the electric-field-vector motion is periodic with the period depending of the frequency difference, T = 2 π / ω 2 − ω 1 . The electric-field behavior illustrated by Figure 4 represents a sort of beatings whose specific features are dictated by the vector nature of electromagnetic field. However, this motion cannot be treated as slow oscillations because there are many “fast” details inside this “slow” period.

Notably, in Figures 4A, B, the electric vector always rotates in the positive (counter-clockwise) direction while in Figures 4C, D, the rotation handedness changes inside the single period (in points A and B). This corresponds to the zero average handedness of the electric field rotation, which is an example of the Lissajous singularity [62, 63] analogous to that realized at points of s-contours in monochromatic fields [1, 59, 70]. The complicated electric-vector trajectories associated with arbitrary bi-chromatic and, generally, polychromatic fields [71, 72] cause the complex states of polarization whose systematic studies are yet at the early stage. Specific analytical instruments for their description are being developed, in particular, the bi-chromatic Stokes parameters and the polarization matrices (Eq. 28 and its 3D generalization [63, 73]), the technique based on the time-dependent modified Jones vector [72], etc. Such fields show interesting properties, especially in the tightly-focused state; for example, their spin AM and the spin vector, indicating the circulation handedness and the axis, around which the electric field circulates, may be different, and thus supply independent characterization of the field [73]. Also, the Lissajous figures supply impressive manifestations of the SU (2) symmetry group transformations in optics [74]. It may be expected that the purposeful creation and application of desirable electric-vector patterns, akin to those depicted in Figure 4, will be helpful for realization of specific fine features in the light-matter interaction, with applications for optical manipulation and data processing techniques.

In the previous Sections, the light fields description was mainly based on the non-observable amplitudes and phases; the only observable characteristic was the intensity—the light energy density averaged over the rapid oscillations. However, the light fields can be fruitfully and instructively characterized by the internal energy flows which form a physically meaningful and application-oriented framework for the optical field characterization [58, 59]. Their importance for monochromatic fields is obvious and well recognized; now we briefly outline their generalizations for polychromatic (at the first stage, bi-chromatic) light fields [57].

Generally, the field dynamical properties are characterized by energy flow density (Poynting vector) P whose instant value P is determined by equation [14] P ∼ = c 4 π E ∼ × H ∼ (29) where H ∼ is the instantaneous magnetic-field vector. In a paraxial beam satisfying Eqs. 24, the complex vectors characterizing the transverse electric and magnetic fields (see the comments to Eqs. 24) are determined by the standard relations [58, 59] E = e x E x + e y E y ,   H = e x H x + e y H y = − e x E y + e y E x . (30)

Additionally, there is a weak longitudinal field [58, 59] described by equations E z = E 1 z + E 2 z ,   H z = H 1 z + H 2 z (31) where E 1 , 2 z = i k 1 , 2 ∂ u 1 , 2 x ∂ x + ∂ u 1 , 2 y ∂ y exp i k 1 , 2 s ,   H 1 , 2 z = i k 1 , 2 ∂ u 1 , 2 x ∂ y − ∂ u 1 , 2 y ∂ x exp i k 1 , 2 s (32) (definitions of Eqs 24, 25 are used). Relations (30)–(32) determine the complex positive-frequency quantities similar to the 3^rd Eq. 24; the instantaneous values are determined akin to the 2^nd Eq. 24.

For paraxial fields, the Poynting vector (29) can be decomposed into the longitudinal (main) and transverse (internal) parts, P ∼ = P ∼ z + P ∼ ⊥ . The explicit expressions for the summands of this equation follow directly from Eqs 29–32. However, their interpretation depends on the assumed conditions of observation; the general approach here is the same as in the intensity analysis of the superposition of partially-coherent monochromatic components (Section 4.1). Results of observation are determined by the observation time in comparison with the period of beatings T = 2 π / ∆ ω ¯ = λ 2 2 / c ∆ λ ¯ + λ 2 / c , where λ ₂ is the shorter wavelength corresponding to ω ₂, and ∆ λ ¯ = λ 1 − λ 2 . Here, the first summand can be identified with the coherence time, the second one is the wave 2 oscillation period. Naturally, under the typical conditions (18), the period of beatings practically coincides with the coherence time.

After averaging of the fast-varying terms (oscillation frequencies 2ω ₁, 2ω ₂ and ω 1 + ω 2 ), the slowly-varying part of the longitudinal flow density P _z can be found from equations P z = e z P z ,   P z = c 8 π u 1 2 + u 2 2 + u 1 ⋅ u 2 * e − i Δ k ⋅ s + u 2 ⋅ u 1 * e i Δ k ⋅ s (33) where Δ k = k 2 − k 1 = Δ ω ¯ / c (see Eq. 23; in this Section, due to identity of ω 1 and ω ¯ 1 , ω 2 and ω ¯ 2 , Δ ω ¯ = ω 2 − ω 1 ). The longitudinal flow (33) is proportional to the superposition intensity; the expression in parentheses can be derived from Eqs. 14–16 upon the condition (22). The transverse Poynting vector (TPV) distribution is more interesting. It is associated with the longitudinal field (31), (32), and its value follows from the expression P ∼ ⊥ = c 4 π − e x Re E 1 z + E 2 z ⋅ Re H 1 y + H 2 y − e y Re E 1 x + E 2 x ⋅ Re H 1 z + H 2 z (34) whence, after omitting the rapidly oscillating terms proportional to exp ± 2 i k 1 , 2 s and exp ± i k 1 + k 2 s , one obtains the time-averaged expression P ⊥ = P O ⊥ + P S + P int . (35)

In Eq. 35, the first term characterizes the summary orbital (canonical) energy flow [58, 59] of both frequency components P O ⊥ = c 8 π 1 k 1 Im u 1 x * ∇ ⊥ u 1 x + u 1 y * ∇ ⊥ u 1 y + 1 k 2 Im u 2 x * ∇ ⊥ u 2 x + u 2 y * ∇ ⊥ u 2 y = c 8 π 1 k 1 a 1 x 2 ∇ ⊥ φ 1 x + a 1 y 2 ∇ ⊥ φ 1 y + 1 k 2 a 2 x 2 ∇ ⊥ φ 2 x + a 2 y 2 ∇ ⊥ φ 2 y (36) where the representation (26) has been used, ∇ ⊥ = e x ∂ / ∂ x + e y ∂ / ∂ y is the transverse gradient operator. Likewise, the second term of (35) expresses the summary spin flow [58, 59], P S = − c 8 π ∇ ∅ 1 k 1 Im u 1 y * u 1 x + 1 k 2 Im u 2 y * u 2 x = − c 8 π ∇ ∅ 1 k 1 a 1 x a 1 y ⁡ sin φ 1 x − φ 1 y + 1 k 2 a 2 x a 2 y ⁡ sin φ 2 x − φ 2 y (37) where ∇ ∅ is the “skew gradient” operator, ∇ ∅ = − e z × ∇ ⊥ = e x ∂ ∂ y − e y ∂ ∂ x .

The contributions (36) and (37) are time-independent, while the last term of (35) describes the slowly varying part, emerging due to interference between the monochromatic components E ∼ 1 , 2 r , z , t of Eqs. 24, oscillating in space and time with frequencies Δk and Δ ω ¯ , correspondingly: P int = c 8 π Im 1 k 1 u 2 y * ∇ ⊥ u 1 y + u 2 x * ∇ ⊥ u 1 x e − i Δ k ⋅ s + 1 k 2 u 1 y * ∇ ⊥ u 2 y + u 1 x * ∇ ⊥ u 2 x e i Δ k ⋅ s + 1 k 1 u 2 x * ∇ ∅ u 1 y − u 2 y * ∇ ∅ u 1 x e − i Δ k ⋅ s + 1 k 2 u 1 x * ∇ ∅ u 2 y − u 1 y * ∇ ∅ u 2 x e i Δ k ⋅ s . (38)

The first and second lines of (38) unite the terms with identical polarizations of the waves 1 and 2; on the contrary, each term in the third and fourth lines combines orthogonal polarizations of the composing waves. For this reason, the first- and second-line terms are non-zero in case when both waves are polarized identically, and vanish if the waves 1 and 2 are polarized orthogonally. The third and fourth lines vanishe if both waves have identical linear polarizations and do not vanish if these identical polarizations are circular. One can treat the first-line contributions as the two-frequency modification of the orbital flow, whereas the second line performs the similar spin-flow corrections.

In turn, Eq. 35 and its constituents determine the longitudinal angular momentum (AM) density [1, 58, 59] of the beam as L z = 1 c 2 x − x c P y − y − y c P x (39) where x c , y c specify the reference point with respect to which the AM is defined. The local behavior of L _z is associated with the peculiar points in the TPV distribution. In particular, this distribution may possess singularities where the TPV magnitude is zero and its direction is indeterminate [1, 58, 59]. In this case, the TPV circulation (vortex) is observed near the singularity, which indicates the presence of a non-zero AM (39) with respect to the singular point in this area. In a non-monochromatic wave, the TPV pattern is non-stationary, and its evolution in time discloses physical mechanism of the formation and evolution of the TPV singularities.

Eqs 36–38 illustrate the general scheme of interactions (or “coupling”) between the separate monochromatic contributions of a polychromatic field. Each pair of discrete spectral component forms its own set of orbital, spin and interference contributions which, in complex, produce a rather complicated picture of the internal energy flows. However, the resulting observed picture depends on the observation time. For example, in the special situation where the polychromatic field is formed by the equidistant frequency comb ([2], part 3), not only the TPV pattern but also the intensity and phase distributions show periodic variations, including the beam profile expansions and contractions, rotations (near the wave-packet “center of gravity” x c , y c , see Eq. 39) and revolutions (around the propagation axis so that x c = 0 , y c = 0 in Eq. 39) [76, 77] with associated two forms of the orbital AM [2, 78], etc.

In the limit case, when the observation time exceeds the maximum period of oscillations, the observed TPV structure is determined exclusively by the time-independent terms (36) and (37), which describe the simple sum of contributions caused by each spectral component. Obviously, this conclusion can be extended to arbitrary polychromatic field: for large enough observation times the resulting TPV structure looks as a direct superposition of the TPVs of all spectral components, P x , y = ∫ 0 ∞ P x , y ω d ω

Accordingly, the AM density can also be found via the simple summation of expressions like (39): L z = ∫ 0 ∞ L z ω d ω .

When the observation period approaches the beating periods, then the temporal variations, whose periods are determined by differences between the spectral-component frequencies, become noticeable. This case is described by Eqs 36, 37 supplemented with the interference contributions (38). The most interesting (although hardly observable) situations occur if an observer can resolve the time intervals “inside” the beating period. Then, the complex non-stationary TPV behavior can be realized (see, e.g., Figure 5 [79]). The instantaneous structures with strong TPV circulation and, consequently, with high AM may exist (Figures 5A, B). However, just the opposite TPV circulations occur in other moments inside the beating period, and for larger observation times the structures disappear.

Nevertheless, the instantaneous vortices similar to those depicted in Figure 5, like other optical vortices known from the literature [1, 58, 59, 69, 80–82], can be used for implementing the light-induced mechanical action on small particles, optical trapping and micromanipulation.

First of all, we emphasize that in the ST optics, as well as for the stationary fields, the main way for extraction the optical-field parameters is the interference with a properly chosen reference wave. However, in case of ultra-short (femtosecond) pulses, the interference requires some special precautions. The main one is that the whole measurement process should be completed within the limited spatial and temporal duration of a single pulse (see [2], part 14); the second precaution is that both the object and reference waves are essentially polychromatic, and the interference pattern contains a “mixture” of overlapping monochromatic contributions, which implies the additional task of its deciphering and interpretation. These factors have led to the development of “single-shot spectral interference” (SSSI) schemes [84, 85] where the reference beam E r e f and probe beam E pr in are formed from (or are governed by) the same initial laser pulse that is used for generation of the object (sample) beam E S .

In the arrangement of Figure 6A, the interference occurs in the 0.5 mm thick fused-silica “witness plate” where the Kerr effect produces the interference pattern. Regarding the regime, the interference signal is generated in the form E pr out I ∼ χ 3 E S E S * E pr in or E pr out P ∼ χ 3 E S E i * E pr in , χ 3 being the non-linear susceptibility (in the latter case, the auxiliary reference pulse E i with the same central wavelength and bandwidth 2 nm is applied). The resulting probe pulse E pr out is analyzed in the imaging spectrometer which forms a set of images of separate spectral components of the observed pulse thus enabling the instantaneous detection of the spectrally-resolved spatial distributions, whence the time-resolved pulse shape can be extracted via the Fourier transform. The data of E pr out I give access to the transient intensity of the sample pulse E S , whereas E pr out P contains its transient phase in the form of time-dependent grating ∼ I S x , t cos   k ¯ x ⁡ sin ⁡ θ + Δ φ x , t where k ¯ = ω ¯ / c is the central wavenumber, θ is the angle between the main axis and the auxiliary reference beam E i (see Figure 6 where θ = 6° [85]).

Another version of SSSI presented in Figure 6B involves only linear interactions [2, 86]. Here, the reference and sample (tested) pulses are collimated and spectrally resolved in the horizontal direction by the diffraction grating. The cylindrical lens projects the spectral distribution onto the camera input. In the vertical dimension, the reference and sample pulses’ trajectories cross at a small angle so that their images overlap at the camera thus forming the interference fringes. In this simple configuration, the spatial resolution is not high and is limited to the vertical direction only, but, using the beam-splitter, the same procedure can be simultaneously applied to the orthogonal transverse direction. Additionally, several copies of the sample pulse, with prescribed transverse shifts with respect to the reference position, can be analyzed simultaneously to improve resolution and signal-to-noise ratio.

In this way, the SSSI enables to obtain a spatially and spectrally resolved map of the sample-pulse field distribution. However, achievement of high spatial and spectral (temporal) resolution on a single-shot basis is still a challenge and its practical realization meets many difficulties, despite the crucial importance for high-power and low-repetition-rate laser systems.

Upon the conditions of Ref. [85], the SSSI is used for the experimental characterization of the ST optical vortex (STOV), which is a representative of a new family of singular ST light fields. Due to their unique physical properties, uniting the essential ST coupling with expressive topological and singular-optics features, the STOVs are in the focus of the most scrupulous and permanently growing attention during the last years [2, 3, 5, 85, 87–99]. Prior to discuss the physical properties and manifestations of the STOVs, we briefly outline their formal description.

To this purpose we start with considering the simplest Gaussian ST wave packet [100]. In the scalar paraxial approximation, its electric field distribution can be presented as E ST 0 = u ST 0 e i k ¯ s where s = z − c t , u ST 0 = ⁡ exp − s 2 2 ζ 2 u 00 H G = A b π exp − x 2 + y 2 2 b 2 − s 2 2 ζ 2 + i k ¯ x 2 + y 2 2 R − i χ , (40) A is the normalization constant, and u 00 H G x , y , z denotes the Gaussian complex amplitude distribution (zero-order Hermite-Gaussian mode) [58, 59, 80, 101]. The packet (40) propagates along the longitudinal axis z as a usual Gaussian beam, and is additionally modulated by the longitudinal Gaussian envelope exp − s 2 / 2 ζ 2 , with the length ζ = cτ and duration τ. Upon propagation, coefficients of Eq. 40 vary according to the known Gaussian-beam rules [58, 80] b = b 0 1 + z 2 z R 2 ,   R = z 2 + z R 2 z , χ = ⁡ arctan z z R ,   z R = k ¯ b 0 2 (41) where b ₀ is the beam waist radius, and it is supposed that the beam waist is situated at z = 0. Based on the wave packet (40), the usual (longitudinal) optical vortex (OV) can be constructed as u XY 1 = y b e − 2 i χ − i σ x b e − 2 i χ u ST 0 = A b π y b e − 2 i χ − i σ x b e − 2 i χ exp − x 2 + y 2 2 b 2 + i k ¯ 2 x 2 + y 2 R − s 2 2 ζ 2 (42) where σ = ±1 denotes the OV sign. Quite similarly, the STOV appears if the pre-exponential multiplier involves the spatial transverse (say, x) and longitudinal (temporal) s coordinates: u ST 1 = s ζ + i σ x b e − i χ u ST 0 = A b π s ζ + i σ x b e − i χ exp − x 2 + y 2 2 b 2 − s 2 2 ζ 2 + i k ¯ x 2 + y 2 2 R − i χ . (43)

Like Eq. 42, this expression is a solution to the paraxial wave equation but its z-dependent evolution differs in some important aspects: in (42), the coefficients of x and y in the first parentheses evolve identically while in (43), the coefficient of x, exp (–iχ)/b varies according to the rules (41) but the coefficient of s remains constant. This stipulates peculiar features of the STOV propagation which are discussed below.

The properties of the STOV (43) are illustrated in Figure 7. The intensity distribution in the (x, s) plane, calculated for the case ζ = b ₀, σ = +1 and z = 0 (Figure 7A) is doughnut-shaped, with bright ring and dark spot in the center. Moreover, the isolated amplitude zero in point (s = 0, x = 0) is coupled with the phase singularity: the field phase is indeterminate at s = 0, x = 0, and grows by 2π upon the circulation near this point (Figure 7B). These features resemble the field pattern of a circular OV (42) in the transverse (x, y) plane depicted in Figure 7C for a comparison. The 3D spatial profile (instantaneous intensity distribution) of the STOV forms, generally, a toroidal structure that can be illustrated by the equal-intensity surfaces (Figures 7D–F). For the STOV (43), the toroid is situated in the longitudinal plane (s, x) containing the propagation axis [90, 95], while a light pulse with the conventional transverse OV forms a similar toroid in the (x, y) plane illustrated by Figure 7D; the “depth” of the latter toroid (its size along the s-direction) is determined by the pulse duration ζ/c.

The energy flows in the STOV are determined by Eq. 43 and the scalar versions of Eqs 33, 36 with k ₁ = k ₂ = k ¯ : P z = c w = c 8 π s ζ 2 + x b 2 + 2 σ s x ζ b sin ⁡ χ u ST 0 2 (44) (w is the STOV energy density, u ST 0 2 is determined by Eq. 40); P x = c 2 8 π ω ¯ σ s b ζ cos ⁡ χ ⋅ u ST 0 2 + x R P z = c 8 π k ¯ σ s ζ b 0 b 2 1 + 2 x 2 b 0 2 1 − b 0 2 b 2 + k ¯ x R s 2 ζ 2 + x 2 b 2 u ST 0 2 ,   P y = y R P z . (45)

The TPV pattern in Figure 7A shows a sort of circulatory energy flow associated with the corresponding orbital angular momentum (OAM) of the STOV [90, 94–96]. It also resembles the OAM of a conventional OV (Figure 7C) but is directed orthogonally to the beam propagation (Figure 7E). Additional distinctions follow from the fact that the STOV (43) contains only the x-directed TPV component in the (s, x) plane. In contrast to the conventional circular OV (Figure 7C), in the STOV, a certain “imbalance” exists in the transverse energy flows between the regions s > 0 and s < 0, which is well seen in Figure 7A and Eqs. 45. Due to this imbalance, the spatial configuration of the STOV does not preserve the circular symmetry and changes in the course of propagation (see Figure 8).

As a result, the “perfect” circular STOV is only realized in a single cross section (under conditions of Figure 8, this is the waist section but proper adjustment of the parameters ζ, b ₀ and χ enables to get the circular STOV in any longitudinal location up to the far field [85]). In other cross sections, the intensity pattern is typical for the anisotropic (“non-canonical”) OV [20, 21, 101]. Ultimately, the energy-flow imbalance causes the intensity profile distortion, and the two-lobe structure appears. The whole evolution of the propagating STOV looks as a sort of rotation. The sense of this rotation is dictated by the sign of the STOV topological charge which is positive in Figure 8.

A STOV with the ring-like structure in a certain cross section z = z ₁ is described similarly to (43) with the only difference that the parameter ζ in the pre-exponential parentheses is replaced by the complex quantity ζ c = ζ e i χ z 1 so that u ST 1 = s ζ e − i χ z 1 + i σ x b e − i χ u ST 0 . (46) In general case, when b ≠ ζ , the STOV is not circular even in the ring-like cross section but shows a certain anisotropy, similar to the astigmatic longitudinal OVs [101].

Like in the case of conventional OVs [58, 59, 80], the STOVs of any integer order l (also called topological charge) may exist, in which the phase increment on a circulation near the vortex center is 2πl. For a higher-order STOV, at least in a single cross section z = z ₁ the complex amplitude distribution can be represented as u ST l x , y , s , z 1 = s ζ + i σ x b l e − i χ z 1 u ST 0 x , y , s , z 1 (47) where l = σ|l|. However, in other cross sections this simple form is destroyed: the pre-exponential binomial must be expanded in a series in degrees of (s/ζ), (x/b), and each summand evolves in its own way [94, 101]. This fact stipulates a rather rich and non-trivial picture of the STOV profile evolution, and enables purposeful formation of a necessary profile (e.g., ring-like one) at any specific position along the propagation axis up to the far field [85, 94].

The STOVs (43), (46), (47) considered so far are oriented such that their intensity toroids and the energy circulation are concluded within the plane (z, x) (see Figures 7A–E, 8). However, the STOVs with other toroid orientations may also exist [92]. For example, the following function is the solution of the ST paraxial wave equation [100]: u ST 11 = A b π α s ζ + β y b e − i χ + i σ x b e − i χ exp − x 2 + y 2 2 b 2 − s 2 2 ζ 2 + i k ¯ x 2 + y 2 2 R − i χ . (48) This STOV propagates along axis s but its equal-intensity toroid is adjusted along the plane (x, w) which is inclined at an angle ϑ (Figure 7F); the OAM orientation in the (z, y) plane is characterized by the angle ϑ _OAM which, generally, differs from ϑ. For example, if b _y = ζ, χ _y = 0 (conditions of Figure 7F) and α = β = 1/√2, ϑ = π/4. Practically, STOVs of arbitrary orientation can be realized [92].

It was already mentioned in comments to Eqs. 45 that the specific TPV distribution in the plane (x, s) is coupled with the transverse OAM with respect to any y-oriented axis. It is suitable to consider the OAM defined with respect to the moving axis (x = 0, s = 0) crossing the wave-packet center. Thus, the y-component of the Poynting vector gives no contribution, and the OAM density can be determined as L y = 1 c 2 s P x − x P z . (49) (cf. Eq. 39). The total OAM of the STOV is obtained via the integration of (49) over dxdyds. In this procedure, the term x P z gives a zero contribution due to the symmetry of expression (44), and the total OAM can be calculated as Λ y = ∫ L y   d x d y d s = ∫ s P x   d x d y d s = A 2 b 0 8 π 2 ω ¯ b 4 σ ζ ∫ s 2 1 + 2 x 2 b 0 2 1 − b 0 2 b 2   × exp − x 2 + y 2 b 2 − s 2 ζ 2 d x d y d s = A 2 σ 16 π ω ¯   ζ 2 b 0 . (50)

In agreement with the angular momentum conservation, the result does not depend on z, despite that the pulse configuration changes rather impressively (as is seen, for example, in Figure 8). The quantity (50) depends on a set of the STOV parameters. However, like the longitudinal OAM of the conventional OV beams, the transverse OAM expresses the deep topological properties of the field which are largely “masked” by the specific parameters of the beam shape. To disclose this topological essence, the numerical OAM value (50) should be normalized by the beam energy. The total STOV energy is determined by the energy density distribution (44), W = ∫ w x , y , z , s d x d y d s = A 2 8 π ζ (51) which, in view of Eq. 50, determines the OAM per unit energy of the STOV [95, 96] Λ y W = σ 2 ω ¯ ζ b 0 = σ 2 γ ω ¯ (52) where γ = ζ / b 0 is the parameter of the STOV ellipticity (anisotropy). This result can be immediately generalized to the higher-order STOV whose transverse OAM obeys the condition Λ y W = l 2 ω ¯ ζ b 0 = l 2 γ ω ¯ .

It is instructive to compare this transverse OAM with the longitudinal OAM Λ _z of the conventional OV. In view of the generally astigmatic character of the STOV, for comparison we consider the light pulse with the astigmatic transverse OV, for which the longitudinal OAM Λ _z , normalized per unit energy, obeys the relation [96, 101] Λ z W = l 2 ω ¯ γ x y + γ x y − 1 (53) where γ x y = b 0 y / b 0 x is the ratio of the orthogonal beam-waist dimensions. Remarkably, in the symmetric case, when in Eq. 52 γ = 1 , and in Eq. 53 γ x y = 1 , the simple correspondence takes place: Λ y W = l 2 ω ¯ = 1 2 Λ z W . (54)

This relation was the subject of a controversial discussion [95, 96] but it finds a simple qualitative support in juxtaposition of the corresponding energy flow patterns presented in Figures 7A, C [95]. The energy circulation in the conventional OV (Figure 7C) is “complete” and contains the “full” circulation including the contributions along both orthogonal transverse components while in the STOV field, only the ± x-oriented circulation contributions are present, so the circulation loses a half of its “complete” value.

There are several prospective approaches to the practical STOV generation discussed in literature [2, 91–93]. Conceptually, all of them employ one or several “source” light pulses without special structure, usually obtained from the mode-locked laser, which undergo certain structuring manipulations. In this regard, the most direct method is based on the superposition of properly prepared and phase-shifted non-vortex pulses, for example, those described by the first and second summands in pre-exponential parentheses of Eq. 43 [87]. In principle, this approach is applicable for obtaining any of the complicated STOV structures, e.g., characterized by Eqs 47, 48, as well as by their generalizations. But it requires preparing a number of special light pulses with prescribed configurations and their precise alignment, which is practically difficult.

Another group of approaches involves manipulations with the STOV Fourier-spectra, U k x , k y , k = ∫ u x , y , s exp − i k x x + k y y − i k − k ¯ s d x d y d s (55) where k = ω/c is the wavenumber corresponding to the spectral frequency ω, and u x , y , s is the complex amplitude; here we conventionally consider the STOV characteristics at a certain fixed longitudinal position z = const, and the fourth argument z is omitted. For example, in application to arbitrarily oriented STOV (48), U k x , k y , k = 2 3 / 2 π A ζ b − i ζ α k − k ¯ − i β k y b + σ k x b × ⁡ exp − 1 2 k x 2 + k y 2 b 2 − 1 2 ζ 2 k − k ¯ 2 . (56) Such spectral density in the Fourier space k x , k y , k can be obtained if a Gaussian pulse (e.g., that described by Eq. 40) passes an optical system with transmission function T k x , k y , k ∝ − i ζ α k − k ¯ − i β k y b + σ k x b . (57)

A sort of such transformation is implemented in the 2D pulse shaper [85, 90] (Figures 9A, B) where the general principle of “structuring light in time” ([3], part 21) is realized. Namely, the input dispersion element performs “spectrum-to-space” transformation such that different spectral components are spatially separated; ideally, each spectral component enters its own spatial channel. Then, in each channel, the prescribed modulation of the spatial (amplitude and phase) and, if appropriate, polarization distributions is executed, after which the spectral channels are recombined by another dispersion element operating in the inverse mode. The schemes of Figures 9A, B employ the simplified version of this principle. The dispersion elements are diffraction gratings that disperse the input wave packet components with different values of k along the horizontal direction. Then, in the cylindrical-lens focal plane, the complex amplitude distribution is formed proportional to U i n k x , k –Fourier spectrum of the input wave packet u i n x , s . In this plane, the distribution U i n k x , k is transformed: in Figure 9A, due to reflection at the programmable spatial light modulator (SLM), in Figure 9B, upon transmission through the phase plate. The output field recombination is performed by the same (in the reflection scheme Figure 9A) or similar (in Figure 9B) cylindrical lens and grating. In Figures 9A, B, the Fourier-plane modulation introduces the spiral phase distribution which, in vertical direction, is accepted by the spatial k _x -components, but its horizontal “part” is imparted to the temporal-frequency components. Accordingly, the transmission ∝ − i ζ α k − k ¯ + σ k x b is approximately realized in the Fourier plane (cf. Eq. 57), which forms the longitudinal STOV with the transverse OAM at the shaper output. Alternatively, a π-step phase mask can be placed in the Fourier plane. Depending on the mask orientation angle α, the two-lobe structure is realized at the shaper output (see the leftmost or rightmost images of Figure 8), which produces the ring-like STOV in the far field [85]. This method is rather flexible for generation of STOVs with variable positive or negative topological charges and prescribed ring-like structure localization, dictated by the phase mask or the SLM loading.

But the most universal approaches for the STOV generation involve the specially designed metasurfaces [92]. The main element of the corresponding arrangement (Figure 9C) is the photonic-crystal slab furnished with the grating formed of material with permittivity ε = 12 (the grating profile is shown by the yellow inset). The whole system is polarization-sensitive and is placed between the polarizers with prescribed input and output polarization. With specially adjusted sizes w ₁, w ₂, h ₁, h ₂ (see Figure 9C), very narrow Fano resonance is excited in the slab due to which its transmission for normally incident light of the central frequency ω ¯ = c k ¯ vanishes, T 0 , 0 , k ¯ = 0 , but for small deviations it can be expressed via the Taylor expansion T k x , k y , k ∝ a k k − k ¯ + a x k x (58) with, generally, complex coefficients a _k and a _x . The phase difference between a _k and a _x is determined by the slab properties and can be adjusted to π/2, which realizes the transmission function (57) with β = 0 responsible for the longitudinal STOV (43). The terms proportional to k _y appear in the Taylor expansion if the slab is slightly tilted around the x- or y-axes, and in this manner the STOV with arbitrary orientation (see, e.g., Eq. 48) can be produced [92].