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Abstract
A simple method for designing spatiotemporal coherency vortices (STCVs) and spatiotemporal dislocation curves (STDCs) is introduced by means of coherent-mode representation and Fourier transforms. A partially coherent pulsed beam is represented by an incoherent superposition of a Gaussian and a Hermite-Gaussian pulsed beam with different waist positions. It well demonstrates that there exist STCVs and STDCs in the space-time plane. The detailed numerical calculations are performed to address the dependence of waist distance of two modes, reference position, beam order, distribution of original spectrum, topological charge and mode weights ratio on the STCVs and STDCs. The physical interpretation behind numerical results is shown. A possible scheme for experimental synthesis of the STCVs is proposed. The obtained results may have potential applications in the fields of light-matter interaction, spatiotemporal spin-orbit angular momentum coupling and STCV-based optical trapping and optical manipulation.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optical fields carrying orbital angular momentum (OAM) have acquired and sustained, throughout the last three decades, an unprecedented attention from the scientific community, after Allen et al. introduced the Laguerre-Gaussian(LGpl) modes [1] with vortices of nonzero azimuthal index l. A classic optical vortex is pertinent to a beam-like field possessing a spiral phase in the transverse plane to the direction of propagation, say x-y plane, while the OAM vector is longitudinal, i.e., parallel to that direction. The beams carrying longitudinal OAM have already found numerous applications in classical and quantum optics, such as optical communications [2], optical tweezers [3], optical detection [4], quantum information processing and imaging [5,6], etc.
Recently, optical fields carrying transverse OAM, i.e., those in which the OAM vector is orthogonal to the direction of propagation has gained a great deal of attention. Optical vortices with transverse OAM were shown to exist in the x-t plane (or, equivalently, the x-z plane) and are referred to as spatiotemporal optical vortices (STOVs) [7]. While propagating, the STOVs undergo dispersion and diffraction effects simultaneously. The STOV is not specific to electromagnetic radiation: a number of natural aero- and hydro-dynamic phenomena such as tropical cyclones, moving tornadoes and human left ventricular blood flow are the examples of the mechanical analogues of the STOV. Bliokh and Nori pointed out that the STOV states could describe either photon or electron vortex states [8,9]. In 2016, Jhajj et al. presented the first experimental evidence of the STOVs and revealed that they constitute the fundamental element of the nonlinear collapse and are responsible for the subsequent propagation of short optical pulses in air [10]. More recently, the STOVs were generated experimentally in free space [11,12]. Bliokh then carried out an accurate theoretical analysis of such STOV states and their angular momentum properties [13]. At the same time, Huang et al. theoretically and experimentally demonstrated generation, propagation and diffraction of the STOVs by the use of 4f pulse shaper system [14,15]. Wan et al. transferred the STOV tube into a vortex ring, hence introducing toroidal vortices, through optical conformal mapping [16], and Chen et al. reported the spatiotemporal Bessel vortices with transverse OAM having topological charges beyond 100 [17]. More recently, Cao et al. showed the propagation of the STOVs through few-mode fiber and found the possible application in telecommunications [18]. However, all of the aforementioned investigations are confined to fully coherent pulsed sources.
On the other hand, it has also been realized that coherence of light plays the quintessential role for structured light propagation [19–21], especially in long-distance communications [22], optical trapping [23] and information encryption [24]. It is known that on interaction of deterministic light with optically fluctuating media, a classic optical vortex breaks down giving a rise to a multitude of coherency vortices with different indices [25]. It is also possible to synthesize light sources carrying a single or a finite set of coherence vortices with specified indices [26]. Theoretically, the coherency vortices are described as phase singularities of the second-order correlation functions [26–34]. Recently a unified approach for complete characterization of the OAM in stationary light beams via the coherence-OAM matrices whose elements represent self and joint radial correlations among the pairs of the OAM indices was achieved [35,36].
Meanwhile, optical vortices with transverse OAM embedded in partially coherent pulsed beams, are called spatiotemporal coherency vortices (STCVs) [37]. With the rapid development of ultrashort pulse technology, partially spectral or temporal coherence states in the random optical pulses have drawn widespread attention because of their groundbreaking applications in pulse shaping, ghost imaging, laser micromachining, medical diagnosis and treatment [38–50]. Till now, the STCVs have not been investigated thoroughly, with a few exceptions: in 2020, Hyde found out that the twisted space-frequency and space-time partially coherent pulsed sources could carry transverse OAM due to the statistical twists between their spatial and temporal dimensions [51]; in 2021, Mirando et al. demonstrated that a STCV can be generated from a pulsed source with partial temporal coherence and complete spatial coherence [52]; more recently, we investigated the coherence control of STCVs and found that source coherence makes it possible to control the dynamics of phase distributions and the positions of STCVs upon propagation [37]. However, these investigations of the STCVs relating to coherence still have some limitations. For instance, the currently known STCVs are limited to the transverse OAM with low topological charge values, l = 1 or 2. Moreover, their theoretical description remains far from trivial. Further, it is currently of considerable interest how to realize a STCV with an arbitrary value of topological charge in a simple manner.
In this paper we have explored a method for generating STCVs with the help of coherent-mode representation, which can readily deal with the high-order STCVs. We also have discovered the existence of spatiotemporal dislocation curve (STDC) in the x-t plane, in the partially coherent pulsed field. Usually, for stationary fully coherent light field, there are two kinds of pure dislocations. One is the screw dislocation, often called the optical vortex, which is a spiral phase ramp around a dark spot where the phase of the field is undefined and thus its amplitude is equal to zero; the other is the edge dislocation located along a line in the transverse x-y plane across which a π phase shift appears [53,54]. However, for partially coherent light field, the screw dislocation is often called the coherency vortex, which takes place at the position where the modulus of cross-spectral density of the field is equal to zero [55]. The edge dislocation also located along a line in the transverse x-y plane across which a π phase shift appears [56]. Analogous to the above definition, for partially coherent pulsed field, STDC take places in transverse x-t plane across which a π phase shift arise. Physically, along the STDC line, the modulus of mutual coherence function of partially coherent pulsed beam is equal to zero. In Section 2 we show the basic approach to modeling and present an analytical formula for describing the STCVs. The detailed numerical results and corresponding physical explanations are given in Section 3. Finally, in Section 4 we summarized our results and draw the main conclusion.
2. Theoretical approach
Consider a partially coherent pulsed beam, whose cross-spectral density can be expressed as an incoherent superposition of a number of coherent modes by means of eigenfunctions ϕmnk(r, ω, z; z0) [57], i.e.,
In time-space domain, the mutual coherence function of a partially coherent pulsed beam has form
Let us consider a STOV phase with topological charge q[11],
Thus, the mutual coherence function of a partially coherent pulsed beam with the single STCV phase of topological index qmnk becomes
From Eq. (15), one can predict the average intensity and the coherence state of a partially coherent pulsed beam according to the second-order coherence theory [58]. In our calculation, for simplicity, let us consider partially coherent pulsed beams consisting of only two modes, i.e., with a mutual coherence function given by the expression [28]
3. Generation of the STCVs and STDC
By use of Eqs. (16)-(20), the detailed numerical calculation related to the STCVs and spatiotemporal dislocation of partially coherent pulsed beam are shown for fixed x1, t1 and a variety of values of x2, t2, the calculation being done in a single z-plane, i.e., z1 = z2. The calculation parameters are z1 = z2 = 0, z0=-0.1 m, λ=800 nm, c = 3 × 108 m/s, w0 = σ=2 mm, T0 = Tc = 5fs, xs = 1 mm, τs = 1fs, x1 = 1 mm, t1 = 1fs, k = 0, q000 = qmnk = q = 1, λ1 + λ2 = 1, λ1 = 0.5, λ2 = 0.5 and m = 1 unless other values are specified.
Figure 1 shows the phase distribution of Γ′, curves of Re Γ′ = 0 (solid line) and Im Γ′ = 0 (dashed line), modulus distribution of Γ′ and spatiotemporal intensity I′ of a partially coherent pulsed beam with topological charge q = + 1 for different waist positions of two modes. One sees from Fig. 1(a) that when z0=-0.1 m, the STCV takes place in the spatiotemporal center (x2 = 0, t2 = 0), while the STDC also appears in the x2-t2 plane. The positions of STCV and STDC appear in the zeros of modulus of Γ′ rather than that of the spatiotemporal intensity (see Fig. 1(i) and Fig. 1(m)). From Eqs. (16) and (17), one can also see that the spatiotemporal intensity of a partially coherent pulsed beam with Eq. (16) is nonzero throughout the time-space plane because the mode ϕ000(r, ω, z; z0) of Eq. (17) has no zeros. These properties are similar to the case of coherent vortices in stationary partially beams [28]. When the distances between waist positions of the two modes become larger, the position of the STCV is always in the spatiotemporal center, but the STDCs gradually reduce, weakening and vanishing for negative t2 (see Figs. 1(f)-(h)). Further numerical calculation indicates that the STDCs disappear when z0<-3 m. This occurs because with increasing distance the coupling between two modes becomes weak. Nevertheless, the STCV always exists in the spatiotemporal center because the spatiotemporal vortex phase exists in modes 1 and mode 2, as shown in Eq. (16).
Fig. 1. (a)-(d) Phase distribution of Γ′, (e)-(h) level curves of Re Γ′ = 0 (solid line) and Im Γ′ = 0 (dashed line), (i)-(l) modulus distribution of Γ′ and (m)-(p) spatiotemporal intensity I′ of a partially coherent pulsed beam with topological charge q = + 1 for different waist positions of two modes.
Figure 2 gives the phase and modulus distributions of Γ′ of a partially coherent pulsed beam with topological charge q = + 1 for different reference positions x1, t1. As can be seen from Fig. 2, reference positions significantly affect the position of the STDC, but do not substantially influence the STCV itself. When the reference positions are gradually separated from the spatiotemporal center, the STDC gets closer to the STCV. The STDC and the STCV coincide when x1 = 10 mm, t1 = 10fs. At the same time, in the vicinity of the STDC, the modulus of Γ′ gets larger with increasing reference positions (see Figs. 2(d)-(f)).
Fig. 2. (a)-(c) Phase distribution and (d)-(f) Modulus distribution of Γ′ of a partially coherent pulsed beam with topological charge q = + 1 for different reference positions x1, t1.
Figure 3 shows the phase and the modulus distributions of Γ′ of a partially coherent pulsed beam with topological charge q = + 1 for different beam order values of mode 2, where the reference position is x1 = 10 mm, t1 = 10fs. From Figs. 3(a)-(f), there is no STDC when m = 0. There are 1, 2, 3, 4, 5 STDCs when beam order m = 1, 2, 3, 4, 5, respectively i.e., the number of the STDCs coincides with m. Also, the modulus of Γ′ distribution is almost symmetric about line x2 = 0. These results can be explained as follows: in Eq. (18) the second-mode eigenfunction ϕm0 k(r, ω, z; z0) includes Hermite polynomial Hm(·), which holds m solutions for equation Hm(x) = 0 corresponding to m STDCs. At the same time, Fig. 3 implies that the spatiotemporal dislocations result from the mode 2.
Fig. 3. (a)-(f) Phase distribution and (g)-(l) modulus distribution of Γ′ of a partially coherent pulsed beam with topological charge q = + 1 for different beam order of the mode 2.
Figure 4 presents the phase and modulus distributions of Γ′ of a partially coherent pulsed beam with topological charge q = + 1 for different values of parameter k. It is shown that k parameter has quite significant influence on the STDCs, which are cut off and become k + 1 segments according to the different k values. This is also a result of spatiotemporal coupling. This can be explained by use of Eq. (7). From Eq. (7), the original spectrum of the pulsed beams is $S(\omega ) = {|{{\phi_k}(\omega )} |^2}$ and the temporal intensity is $I(t) = {|{FT[{{\phi_k}(\omega )} ]} |^2}$.
Fig. 4. (a)-(c) Phase distribution and (d)-(f) Modulus distribution of Γ′ of a partially coherent pulsed beam with topological charge q = + 1 for different k parameter.
Figure 5 shows the normalized spectrum of the source and normalized temporal intensity corresponding to a partially coherent pulsed beam for different values of k. S0(ω) and I0(t) indicate the maxima of S(ω) and I(t), respectively. As can be seen from Fig. 5, different values of k lead to different original spectrum distributions (Fig. 5(a)-(c)), and different temporal intensity distributions (Fig. 5(d)-(f)). From Fig. 5(d)-(f), there is a zero intensity value at t = 0 for k = 1, while there are two zero intensity values at t=±2.7fs for k = 2, and three zero values at t = 0, t=±4.7fs for k = 3. The positions where the temporal intensities are zero correspond to the positions where the STDCs are cut off in the Fig. 4(a)-(c), which indicates a spatiotemporal coupling.
Fig. 5. (a)-(c) Normalized spectrum of the source and (d)-(f) normalized temporal intensity corresponding to partially coherent pulsed beams with different k.
Figure 6 shows the phase and modulus distributions of Γ′ of a partially coherent pulsed beam for different topological charge q. As can be seen in Fig. 6, the phase of STCV rotates along a closed counterclockwise path and increases by 2π for q=+1. While the phase of STCV increases by 4π and 6π for q=+2 and q=+3, respectively. However, the phase of STCV rotates along a closed clockwise path and increases 2π, 4π, 6π for q = -1, q = -2, q = -3, respectively. The topological charge q has no noticeable influence on the STDCs. From Eqs. (6)(g)-(i), the size of vortex core of STCV increases with increasing |q|.
Fig. 6. (a)-(f) Phase distribution and (g)-(i) modulus distribution of Γ′ of a partially coherent pulsed beam for different topological charge.
Figure 7 shows the phase distribution of Γ′, level curves of Re Γ′ = 0 (solid line) and Im Γ′ = 0 (dashed line) of a partially coherent pulsed beam with topological charge q = + 1 for different mode weights. Figure 7(a)-(f) and Fig. 6(a) suggest that the STDC gets farther from the spatiotemporal center with increasing mode weight ratio λ1/λ2, and disappears when the mode weight ratio λ1/λ2 becomes large enough. These results imply that the STDC depends on the effect of mode 2, which degrades with increase of λ1/λ2. However, if we increase the effect of mode 2, i.e., decrease λ1/λ2, shown in Fig. 7(g)-(l), the STDCs always exist and are near the STCV. The two coincide in x2 coordinate when the mode weight λ1/λ2 is small enough. In addition, the value of λ1/λ2 is closely related with the coherence state of the partially coherent pulsed beam. Namely, the pulsed beam becomes fully coherent when either λ1/λ2 = ∞ or 0, otherwise, it acquires a partially coherent state. In general, the mode weight ratio λ1/λ2 implies the influence of coherence state on the STDC and the STCV, including the spatial and temporal coherence counterparts. These qualitative results have a good agreement with our previous research [37].
Fig. 7. (a)-(c), (g)-(i) Phase distribution of Γ′, (d)-(f), (j)-(l), level curves of Re Γ′ = 0 (solid line) and Im Γ′ = 0 (dashed line) of a partially coherent pulsed beam with topological charge q = + 1 for different mode weights ratio.
The above results are confined to the same topological charge q for different modes. Figure 8 shows the phase distribution of Γ′ of a partially coherent pulsed beam for different topological charges q000 and q100 corresponding to mode 1 and mode 2, respectively. The calculation parameters are (a) q000 = 0, q100 = 1; (b) q000 = 1, q100 = 2; (c) q000 = 2, q100 = 3; (d) q000 = 3, q100 = 4; (e) q000 = 0, q100 = 2; (f) q000 = 1, q100 = 3; (g) q000 = 2, q100 = 4; (h) q000 = 3, q100 = 5. Other parameters are the same as Fig. 1. As can be seen form Fig. 8(a), there are no the STCV and the STDC compared with Fig. 1(a). For larger q value, there is one STCV in the spatiotemporal center and the phase of STCV increases by 2π, 4π and 6π for Fig. 8(b), Fig. 8(c) and Fig. 8(d), respectively. From Fig. 8(e), there are two STCVs near the spatiotemporal center and the STDC appears at the line x2 = 0. For larger q value, i.e., Fig. 8(f), one more STCV with topological charge +1 emerges at the spatiotemporal center. While, with the increase of q, there are one STCV with topological charge +2 and one STCV with topological charge +3 at the spatiotemporal center for Fig. 8(g) and Fig. 8(h), respectively. These results can be explained as follows. For example, when q000 = 2 and q100 = 4, Eq. (16) can be re-expressed as
Fig. 8. Phase distribution of Γ′ of a partially coherent pulsed beam for different topological charges q000 and q100 corresponding to mode 1 and mode 2, respectively. (a) q000 = 0, q100 = 1; (b) q000 = 1, q100 = 2; (c) q000 = 2, q100 = 3; (d) q000 = 3, q100 = 4; (e) q000 = 0, q100 = 2; (f) q000 = 1, q100 = 3; (g) q000 = 2, q100 = 4; (h) q000 = 3, q100 = 5.
As can be seen from Eq. (21) that there one STCV outside of the parentheses at the spatiotemporal center with topological charge +2. Inside the parentheses, there are the superposition of mode 1 and mode 2. The mode 2 has another STCV with topological charge +2. Equation (2) can be also regarded as a modulation of mode 1 and mode 2 on the STCV outside of the parentheses.
Last, an illustrating schematic of the experimental setup for generating STCVs are presented in Fig. 9. A pulsed beam, may be dispersed along the horizontal direction by a grating, and collimated by a cylindrical lens. A spatial light modulator can be employed, loaded with the desired holograms to shape the dispersed pulsed beam. The produced beam can then be converged by the second cylindrical lens and a grating, which then produces a STCV. Further experimental details belong to Ref. [12]. The key operational difference of our method is in the design of the holograms. Considering that STCVs are produced via incoherent superposition of two (or more) modes, here we first introduce the random numbers into both modes and sum them. By refreshing the random numbers, we realize the incoherent superposition of the modes. Considering that the commercial SLMs are normally phase-type and our approach involves combining the STCVs with two or more modes [complex functions, see Eqs. (15) or (16)], we prefer the protocol for creating holograms described by Frobes et al [60–62]. It can encode the complex amplitude function into the phase-type holograms [refer to Refs. [60–62] for more details].
Fig. 9. Schematic diagram of experimental generation of the STCVs. The distance between all adjacent optical devices is the focal length of the cylindrical lens in the horizontal direction.
4. Conclusion
With the help of the coherent-mode representation and Fourier transforms, we have introduced a straightforward, both theoretically and practically, method for generating STCVs and STDCs. The method involves a partially coherent pulsed beam represented by the incoherent superposition of a Gaussian pulsed beam and a Hermite-Gaussian pulsed beam with different waist positions. The detailed numerical calculations are shown to address the dependence of waist distance z0 of two modes, reference position (x1, t1), beam order m, distribution of original spectrum related to the k parameter, topological charge q and mode weights ratio λ1/λ2 on the STCVs and spatiotemporal dislocation. It is revealed that, for the same topological charge q corresponding to the mode 1 and mode 2, there exist the STCVs and the STDCs in the space-time plane. The phase distribution and the core size of STCVs vary with increasing topological charge |q|. The spatiotemporal dislocation lines vanish when |z0| or λ1/λ2 are large enough. There are m STDCs corresponding to the beam order m of the Hermite-Gaussian mode. The original spectrum distribution plays a crucial role on the STDC which is cut into k + 1 segments according to different k values related to the original spectrum. When the reference position is far from the spatiotemporal center or λ1/λ2 decreases within ratio 2, the STDCs always exist and move towards the STCV, and the two finally coincide when the reference position is far enough or λ1/λ2 is small enough. On the other hand, for the different topological charge q values, corresponding to the mode 1 and mode 2, the STCVs and the STDCs simultaneously exists in the space-time plane only for a larger topological charge difference (q100-q000 ≥ 2). In addition, a simple scheme of the experimental setup for generating the STCVs and STDCs is proposed. The results obtained in this paper may have potential applications in the fields of light-matter interaction, spatiotemporal spin-orbit angular momentum coupling and STCV-based optical trapping and optical manipulation.
Funding
National Natural Science Foundation of China (12174171, 12004220); Central Plains Talents Program of Henan (ZYYCYU202012144); Natural Science Foundation of Henan Province (232102210169); Excellent Overseas Visiting Scholar Program of Henan (026); Agence Nationale de la Recherche (ANR-19-QUAN-0001-04); University of Miami via the Cooper Fellowship.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are available from the corresponding authors upon reasonable request.
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