Optical singularity states, which significantly affect propagation properties of light in free space or optical medium, can be geometrically classified into screw and edge types. These different types of singularity states do not exhibit direct connection, being decoupled from each other in the absence of external perturbations. Here we demonstrate a novel optical process in which a higher-order edge singularity state initially nested in the propagating Gaussian light field gradually involves into a screw singularity with a new-born topological charge determined by order of the edge state. The considered edge state comprises an equal superposition of oppositely charged vortex and antivortex modes. We theoretically and experimentally realize this edge-to-screw conversion process by introducing intrinsic vortex–antivortex interaction. We also present a geometrical representation for mapping this dynamical process, based on the higher-order orbital Poincaré sphere. Within this framework, the edge-to-screw conversion is explained by a mapping of state evolution from the equator to the north or south pole of the Poincaré sphere. Our demonstration provides a novel approach for manipulating singularity state by the intrinsic vortex–antivortex interactions. The presented phenomenon can be also generalized to other wave systems such as matter wave, water wave, and acoustic wave.
【AIGC One Sentence Reading】:This study reveals a novel optical process where a higher-order edge singularity transforms into a screw singularity, realized via vortex-antivortex interactions. The conversion is geometrically represented on the Poincaré sphere, offering a new method for manipulating singularity states in various wave systems.
【AIGC Short Abstract】:This study reveals a novel optical process where a higher-order edge singularity state transitions into a screw singularity, acquiring a new topological charge. This conversion is achieved through intrinsic vortex-antivortex interactions and visualized using the higher-order orbital Poincaré sphere. The demonstration offers a fresh approach to manipulate singularity states, with potential applications in various wave systems.
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1. INTRODUCTION
Phase singularity in the wavefront of a propagating wave is essential in determining its phase and amplitude structure, endowing the wave field with nontrivial semblances and natures [1–4]. Propagation properties of the phase singularities have been extensively studied in many physical systems including optics [5–9], Bose–Einstein condensate [10,11], and hydrodynamics [12]. Generally, the phase singularity is characterized by an abrupt phase change of wave at a region where both the real and imaginary parts of the wave field vanish.
In optics, the phase singularity states can be geometrically classified into screw and edge types [13,14]. The screw singularity state is closely related to an optical vortex with a screw-type phase distribution. Such a phase profile of a light beam exhibits a MathML phase change around an azimuthal angle, represented as MathML [15], where MathML denotes a topological charge of the screw phase state and MathML defines the azimuthal angle. The screw singularity leads to a doughnut-shaped spatial intensity distribution of a light field. Importantly, such a screw singularity state of light exhibits unique orbital angular momentum flow of photons [16], leading to many intriguing applications such as particle manipulations [17], high-capacity optical communications [18], and optical superresolution [19]. Unlike the screw singularity state, the edge singularity state is featured by a rapid phase change along a curved trajectory, which leads to a curve-shaped region where the intensity of light becomes zero. This kind of singularity state widely exists in spatially structured light fields including the Hermite–Gaussian light, petal light [20,21], elliptic and hyperbolic Ince–Gaussian light [22]. Most previous investigations reveal that there is no direct connection between the two types of singularity states. Nevertheless, Padgett and Courtial proposed an orbital Poincaré sphere, unifying representation of the edge and screw singularity states [23]. Specifically, the south and north poles of the Poincaré sphere represent the screw singularity states with canonical vortex and antivortex [see Figs. 1(b) and 1(h)], respectively, while the equator denotes straight-line (edge) singularity states, as shown in Figs. 1(d) and 1(e). Other points on the Poincaré sphere are defined as screw states with noncanonical vortices or antivortices. This geometrical representation is capable of mapping local singularity state evolution of structured light beams, in the presence of external perturbations [24].
Figure 1.Geometrical representations of the first- and second-order singularity states. The poles of the Poincaré sphere denote the screw singularity states with opposite topological charges, while the equator describes the edge states. (a), (b) The screw singularity states with positive topological charges. (g), (h) The screw singularity states with negative topological charges. (c)–(f) The first- and second-order edge states located at MathML [(c), (e)] and MathML [(d), (f)].
In this paper, we reveal for the first time to our knowledge that a center-packed higher-order edge singularity state initially embedded in the propagating Gaussian field gradually involves into a screw singularity state with a nontrivial topological charge, in the presence of the intrinsic orbit–orbit interaction. The higher-order edge state comprises an equal superposition of two vortex modes with opposite topological charges. We thus introduce an off-center-packed vortex or antivortex array circulating around the edge state, generating nontrivial interactions between the vortices and antivortices, i.e., the intrinsic orbit–orbit interaction [25]. Such an interaction process has induced interesting phenomena including the vortex–antivortex annihilation, repulsion [25,26], and turbulence state [27]. Here we demonstrate that this important process leads to a new phenomenon of edge-to-screw singularity state conversions. To explain this intriguing effect, we present a theoretical model describing the orbit–orbit interactions as well as the resultant edge-to-screw conversion. It is relevant to stress that, by designing vectorial structured light beam with specific singularity state, the edge-to-screw conversion appears with the aid of the spin–orbit interaction [28]. However, the presented edge-to-screw conversion, which relies on the intrinsic orbit–orbit interaction [29], is obviously different from that reported in Ref. [28].
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2. THEORY FOR GEOMETRICAL REPRESENTATION OF EDGE-TO-SCREW STATE CONVERSION
We start by considering an arbitrary singularity state as a superposition of two antipodal higher-order basis MathML, in a normalized form, written as MathMLwhere the antipodal basis is defined as MathML, featuring the canonical vortex and antivortex with topological charge of MathML. Here MathML and MathML denote the transverse coordinates. The parameters MathML and MathML denote the propagation-dependent polar and azimuthal angles of the orbital Poincaré sphere (Fig. 1), built by three Stokes parameters as follows: MathMLwhere MathML (MathML) represents the MathML-component of the Pauli matrix vector defined in the Cartesian coordinate system. The orbital Poincaré sphere demonstrated here is analogous to the polarization Poincaré sphere. The main difference is that the orbital Poincaré sphere is defined in the higher-order framework, describing the topological singularity states of the structured light beam.
Figure 1 illustrates typical singularity states of the light beam represented by the first-order (MathML) and second-order (MathML) Poincaré spheres, with inserts showing their phase and intensity distributions. By definition, MathML represent two pure oppositely helical singularity states located at the north and south poles of the orbital Poincaré sphere; see Figs. 1(a), 1(b) and 1(g), 1(h), respectively. These pure screw singularity states result in light modes having the Laguerre–Gaussian profile. The singularity states positioned at the equator are a coherent superposition of MathML and MathML with equal weight, featured by the high-order Hermite–Gaussian edges, as evident from the multi-petal light patterns separated by dark lines [Figs. 1(c)–1(f)]. Other points on the Poincaré sphere are associated with the screw states with noncanonical vortices or antivortices. Using this framework, we can geometrically represent variations of the singularity states and light patterns with the parameters MathML and MathML. These Poincaré-sphere mappings allow us to visualize singularity state evolution and the resultant edge-to-screw state conversion, in the presence of the intrinsic orbit–orbit interaction.
For this purpose, we demonstrate the intrinsic orbit–orbit interaction process in a spatially structured light, which comprises a center-packed singularity state and an appropriately designed off-center vortex structure as perturbations. In general, the center-packed singularity state displays trivial evolution dynamics in the absence of external perturbations, being invariant with propagation distance. By contrast, the off-center-packed vortices and antivortices in the host light field exhibit nontrivial dynamics, owing to their significant interactive behaviors among them. Thus, it is possible to realize an edge-to-screw state conversion in the presence of the intrinsic orbit–orbit interaction. To demonstrate this idea, we consider an off-center vortex array MathML (here MathML) as the perturbation, initially introduced to the edge singularity state MathML. Note that MathML and MathML are nested into the Gaussian envelope: MathML, where MathML denotes the envelope width. In this case, a general theoretical model for the orbit–orbit interaction process can be initialized by the following light field: MathML
Since the singularity state MathML can be regarded as a superposition of differently charged vortices and antivortices, the introduction of MathML leads to the nontrivial intrinsic orbit–orbit interactions. The dynamical orbit–orbit interaction process along propagation distance MathML can be revealed, according to the Helmholtz wave equation MathML, where MathML is the free-space wavenumber with MathML being the wavelength. However, solving a general form of the Helmholtz wave equation for the orbit–orbit interaction is challenging owing to the complexity of the vortex structure MathML.
The interaction process can be significantly simplified by considering MathML as a circular vortex array, expressed as MathML
Such a vortex structure MathML contains MathML vortices with spatial positions located at (MathML), where MathML denotes the radius of a circle at which the vortices initially locate. The azimuthal angle of these vortices is set as MathML, such that these vortices with identical topological charge symmetrically surround the beam center. Note that an off-center-packed circular antivortex array can be realized simply by taking a conjugate form of the MathML. Figure 2(a) depicts the designed spatial distributions of the vortices (marked as red points) at three different cases of higher-order edge singularity states (indicated by the blue lines). For example, for the first-order (MathML) scenario, the edge state is influenced by the two symmetrical vortices with identical topological charge.
Figure 2.Theoretical demonstrations of the edge-to-screw conversions. (a) The designed structured light state comprising a higher-order edge state (blue curves) and an off-center-packed circular vortex array (red dots), for three different cases of topological orders. (b)–(e) Geometrical mappings of the first- and second-order edge-to-screw conversion onto the first [(b), (c)] and second [(d), (e)] order orbital Poincaré sphere. The state evolutions induced by the circular vortex MathML [(b), (d)] and antivortex MathML [(c), (e)] arrays are considered. The propagation distance is set to 400 mm. Three different values of MathML are examined, as indicated in the panels.
We then derive a theoretical model for the edge-to-screw singularity state conversion. Without loss of generality, we set the initial edge state as MathML, i.e., MathML. Adopting the approach of Ref. [25], we derive the following equations which govern the propagation dynamics of the complex light field: MathMLwith MathMLHere MathML represents the propagating field of the Gaussian envelope after a distance of MathML. MathML is a constant, and MathML with MathML denoting the Rayleigh range. MathML and MathML are MathML-dependent polynomials, with a relevant coefficient MathML. The terms MathML and MathML describe dynamical behaviors of the vortices and antivortices in the evolving host field MathML. The term MathML includes a product of two independent factors, accompanied by a MathML-dependent coefficient MathML. The first factor represents independent evolution of the individual off-center vortices initially introduced by MathML. It suggests that the off-center vortices can be maintained during propagation, but their positions vary with distance. The second factor in MathML represents the on-axis higher-order edge singularity state, which is propagation-invariant. It means that MathML is a noninteractive term, which cannot induce a change of singularity state during beam propagation. Indeed, the noninteractive term occurring in other vortex structures has shown independent evolution of individual vortices and antivortices [26,30]. By contrast, the term MathML includes a product of two propagation-dependent polynomials, MathML and MathML, representing a nontrivial coupling between the vortices [MathML] and antivortices [MathML]. Such a product signifies the intrinsic orbit–orbit interactions, leading to a conversion of the edge singularity state toward the screw singularity one. It is emphasized that without the perturbation (i.e., MathML), the initial singularity state MathML is maintained during beam propagation, indicated by the term MathML.
To investigate how the MathML affects the singularity state, Eq. (5) is expanded into a set of Taylor series around an origin position (here we consider the beam center, i.e., MathML and MathML). We extract the lowest-order term, yielding a MathML-dependent singularity state, in a normalized form, written as MathMLwhere the coefficients MathML and MathML are contributed from the noninteractive term MathML. In contrast, the coefficient MathML is imparted by the interactive term MathML. Equation (7) clearly shows that it is the coefficient MathML, caused by the propagation-dependent orbit–orbit interaction term MathML, that alters the initially balanced weights on MathML and MathML, and gives rise to a significant modification to the original singularity state. These coefficients are strongly dependent on the circular vortex structure parameter MathML, as well as the MathML-dependent term MathML. Hence, they also allow us to engineer the orbit–orbit interaction, leading to a controllable edge-to-screw singularity state conversion. Equation (7) is connected to Eq. (1), indicating that the variation of the coefficients MathML, MathML, and MathML with propagation distance can be geometrically represented onto the orbital Poincaré sphere.
According to the derived theoretical model, we reveal dynamical evolutions of the higher-order edge singularity states. Figure 2(b) depicts three different cases of evolution processes of the singularity state, starting from equator of the first-order orbital Poincaré sphere to the north hemisphere, when the propagation distance is gradually increasing from MathML to 400 mm. Here the Gaussian envelope width is fixed to MathML. Such processes suggest a conversion from the edge singularity state to the screw one. It is shown that the evolution trajectories are strongly relevant to the radius MathML of the circular vortex array. The locations of the final singularity states are determined by MathML: a smaller value of MathML leads to a final state closer to the north pole of the Poincaré sphere (corresponding to the canonical vortex state). Interestingly, reversing the topological charge of the circular vortex array [i.e., by using V(MathML)] while keeping other parameters unchanged, we generate an opposite process; see the theoretical results in Fig. 2(c). Figures 2(d) and 2(e) illustrate similar edge-to-screw conversion phenomena under the perturbations of MathML and MathML, in the case of the second-order Poincaré sphere.
In addition, we mention that increasing the value of MathML leads to weakening the orbit–orbit interaction strength. This is because the orbit–orbit coupling effect becomes weaker when the distance between adjacent vortices and antivortices is larger. We further derive a critical radius MathML of the circular vortex array at which the intrinsic orbit–orbit interaction becomes negligible. To this end, we set the propagation distance to infinity (MathML) and consider the final singularity state still being at the equator of the orbital Poincaré sphere. Accordingly, we obtain a critical condition from Eq. (7) expressed as MathML. Based on this relation, we obtain MathML
Equation (8) shows that MathML is determined by the Gaussian waist MathML and order number MathML. An effective orbit–orbit interaction takes place when MathML. Similar implementations can be applied to the circular antivortex structure MathML which results in opposite evolution paths.
3. EXPERIMENTAL VERIFICATIONS
We present experimental evidence for the predicted edge-to-screw singularity state conversion. We experimentally generate the structured light comprising a designed edge state and a circular off-center-packed vortex array as perturbation, using an experimental setup shown in Fig. 3(a). A linearly polarized He–Ne laser working with a wavelength of MathML is appropriately expanded and collimated through a beam expander (BE). Then the beam is divided into two paths after passing the first beam splitter (BS1). One of the split beams is regarded as a reference beam which is used to interfere with the desired light field generated by the phase-only spatial light modulator (Holoeye LETO II SLM, MathML, pixel size 6.4 μm). Another beam passes through the second BS and impinges on the SLM. The reflected light beam is modulated by a phase-only hologram loaded onto the SLM. The modulated beam is then injected into a lens with a focal length of MathML, which performs the optical Fourier transformation of the phase-only hologram. In the experiment, the incident laser beam should be linearly polarized along the horizontal direction of the SLM, in which case we obtain the most efficient modulation of the incident beam. The designed light pattern is generated at the focal plane (MathML) and propagates along the distance MathML. The propagating light pattern is superimposed by the reference beam at BS3. Note that an iris is utilized to isolate the unwanted diffractive orders of the hologram. The selected light beam, together with the interference pattern, is recorded by the charge-coupled device (CCD).
Figure 3.(a) Experimental setup. BE, beam expander; P, polarizer; BS, beam splitter; SLM, spatial light modulator; CCD, charge-coupled device; M, mirror. The laser was working at the wavelength of MathML. The iris was used to isolate the unwanted orders of diffraction from the SLM. (b), (c) The computer-generated phase-only holograms for observing the first-order (MathML) (b) and second-order (MathML) (c) edge-to-screw state conversions, based on Eqs. (9) and (10).
The phase-only hologram for the SLM requires to encode both the phase and amplitude information of the interactive vortices and antivortices. We achieve the required phase mask using an encoding technique previously proposed by Bolduc et al. [31]. To do so, we consider an analytical solution for the overall light field which includes the singularity state and the perturbation of MathML [or MathML]. In the Fourier domain, the initial light field can be represented as MathML, where MathML and MathML denote the spatial frequencies in the Fourier domain corresponding to the MathML and MathML coordinates in the real space. Here MathML and MathML represent the amplitude and phase of MathML, respectively. The resultant phase-only holographic mask is then expressed as MathMLwhere MathMLHere the operator MathML represents a modulo function, and MathML. Note that the hologram includes a blazed grating MathML (MathML being the grating period), which is utilized to diffract the target light field into the first-order component of the hologram. To carefully separate the desired field from the 0th-order diffraction, in the experiment, the period of the blazed grating in the MathML direction is carefully chosen as MathML. Based on the encoding technique, we generate the phase-only holograms, e.g., see Figs. 3(b) and 3(c) for the designed structured light fields in the first- (MathML) and second-order (MathML) frameworks, respectively.
We experimentally demonstrate the edge-to-screw singularity state conversion. Here we only consider the circular vortex array MathML as perturbation. The antivortex array MathML demonstrates similar results (see Fig. 2) that will not be shown here in experiment. Table 1 shows the calculated critical radius of the circular vortex array, as well as the corresponding radius used in the experiments, for different topological orders. All these experiments were performed with an identical Gaussian envelope whose width was set as MathML. To cohere with the theoretical analysis, the beam propagation distance is restricted at MathML. With these settings, Fig. 4(b1) presents a measurement of the generated first-order structured light field at the focal plane (MathML), clearly showing two symmetrically distributed vortices and one dark line across the beam center (the first-order edge state), while Fig. 4(b2) presents a developed light field measured at MathML. To characterize the singularity state at the beam center, we record the plane-wave interference pattern at the same distance, with the result depicted in Fig. 4(b3). A clear dislocation in the interference fringes is observed, suggesting a generation of the screw singularity state with charge one topology. Such an edge-to-screw singularity state conversion signifies the occurrence of the nontrivial orbit–orbit interaction during beam propagation. These experimental observations are in accordance with the theoretical results, correspondingly shown in Figs. 4(a1)–4(a3), confirming the edge-to-screw conversion.
Critical Structure-Parameters of Different Ordersa
MathML
1
2
3
5
10
MathML (μm)
250
391
494
654
942
MathML (μm)
200
300
400
550
770
MathML denotes the singularity order; MathML and MathML denote the respective critical structure-parameter and experimental configuration. By considering MathML, 2, 3, 5, and 10, and setting the same width of Gaussian background MathML, the results of MathML are obtained based on Eq. (8).
Figure 4.Observations of the first- [(a), (b)] and second-order [(c), (d)] edge-to-screw singularity state conversions. Panels (a1)–(d1) denote the petal-like structured light fields measured at the focal plane MathML. Panels (a2)–(d2) present measurements of the propagating light fields recorded at MathML. Panels (a3)–(d3) show the plane-wave interference patterns of the singularity states at MathML. (a), (c) The theoretical results; (b), (d) the corresponding experimental measurements. Panels in the first and second columns share the same scale bar: 0.5 mm. Scale bars in (a3), (b3) and (c3), (d3) are 0.1 mm and 0.13 mm.
Figures 4(d1)–4(d3) illustrate similar edge-to-screw conversion in the second-order (MathML) framework. Similarly, Fig. 4(d1) depicts the intensity distribution of the generated structured light at MathML. Based on the designed hologram [Fig. 3(c)], we succeeded to observe four vortices spatially distributed on a circular ring and the second-order edge singularity state indicated by two dark lines across the beam center. Figure 4(d2) presents a measurement of the propagated light field at MathML, clearly showing the screw singularity state at the beam center, as judged from the measured interference fringes in Fig. 4(d3). Interestingly, the dislocations in the fringes suggest that the second-order edge-to-screw conversion results in an on-axis vortex with a topological charge of MathML, which is different from the first-order case. The presented experiments in Figs. 4(d1)–4(d3) agree well with our theoretical results shown in Figs. 4(c1)–4(c3). Both the theoretical and experimental outcomes in Fig. 4 indicate observations of the intrinsic vortex–antivortex interactions and reveal a strong relationship between the edge-to-screw conversion and the topological order MathML.
To illustrate the dependence of the edge-to-screw conversion on the topological order, we demonstrate further results with higher-order topological orders including the third, fifth, and tenth orders, with results illustrated in Fig. 5. Relevant experimental parameters can be found in Table 1. The associated structured field distributions are generated and measured at the focal plane MathML; see corresponding intensity distributions in Figs. 5(a)–5(c), respectively. These generated light patterns include MathML circular vortices and an MathMLth-order edge state at the beam center. These initial measurements show good agreement with the theoretical results; see Figs. 5(d)–5(f). Owing to the significant orbit–orbit interaction, the propagated fields recorded at MathML induce final screw singularity states, as confirmed by their corresponding interference patterns displayed in Figs. 5(g)–5(i), respectively. Clearly, the final states are the higher-order optical vortex, exhibiting nontrivial orbital angular momentum. To quantitatively identify the topological charges of the resulting screw states, we implement the following procedures. First, we find out two continuous fringes running from the top to bottom side of the interference patterns [marked by black dashed curves in Figs. 5(g)–5(i)]. We then count the fringes at the top and bottom parts of the interference patterns. The difference in the number of fringes is the topological charge of the screw singularity state. Accordingly, for the third-, fifth-, and tenth-order edge-to-screw conversion, we obtain the final screw singularity states with topological charges of MathML, 5, and 10, respectively. Figure 6 further shows both theoretically and experimentally the linear relationship between the new-born topological charge in the screw singularity states and the order of the edge states.
Figure 5.Observations of the higher-order edge-to-screw state conversions. (a)–(c) The experimentally generated higher-order edge states at MathML with imbedded circular vortex array for three different orders: (a) MathML, (b) MathML, and (c) MathML. The scale bars in (a), (b) and (c) are 0.47 mm and 0.63 mm, respectively. (d)–(f) The theoretical results corresponding to (a)–(c). (g)–(i) The plane-wave interference patterns experimentally measured at MathML. The scale bars: (g) 0.2 mm, (h) 0.25 mm, and (i) 0.35 mm. (j)–(l) The theoretical outcomes corresponding to (g)–(i).
Figure 6.The linear relationship between topological orders of higher-order edge state and the topological charge of the resultant screw singularity state. The black-rectangle scatters and red line represent experimental and theoretical results, respectively. MathML denotes the order of the edge state while MathML is the new-born topological charge.
In summary, we have demonstrated both theoretically and experimentally an optical process of edge-to-screw singularity state conversion, in the presence of intrinsic orbit–orbit interactions. This process gives rise to the generation of vortex light modes with a helical wavefront controlled by the incident edge states. We have presented a theoretical model, allowing to geometrically represent such an intrinsic edge-to-screw state conversion onto the higher-order orbital Poincaré sphere. Particularly, we have demonstrated using circular vortex or antivortex array to control the singularity state evolution trajectory based on the Poincaré-sphere representation, leading to the controllable edge-to-screw conversions. We mention that controlling singularity states of light has drawn considerable attention [3,32]. Our demonstrations based on the intrinsic orbit–orbit interactions provide a novel approach to manipulate the singularity state of light. Our results also offer a new way to the generation of the higher-order helical modes of light, which might find potential applications, e.g., in optical communication [18,33,34] and particle manipulation [17,35]. Owing to wave similarities, the edge-to-screw conversion phenomena demonstrated in this work could be generalized to other physical wave systems such as matter waves [36–39], water waves [40,41], and acoustic waves [42].