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Abstract
Topological structures currently are of special interest in the integration of singular optics and topological photonics. In this article, the topological flowers and spider webs, which got less attention comparing to the lemon-, star- structures of the same category, are investigated in 3D vector fields. We show that by strongly focusing higher-order singular beams, both the spin density (SD) vectors and the polarization states of the transverse fields on the focal plane exhibit flowers and spider webs structures in topology with 2|m − 1| folds/sectors (m is the beam order), and the topological structures of the SD vectors are demonstrated to have a 90°/|m − 1| rotation. On the other hand, the topological theory also needs to be developed according to the rapid growth of topological photonics. Here, by defining a ‘relative topological charge’, we have observed and analyzed the topological reactions of the loops (composed of SD singularities) rather than the ‘point-type’ singularities in conventional reactions. More specially, the ‘radial index’ and ‘azimuthal index’ are proposed to characterize the topological features of the flowers and spider webs, and have been verified that the ‘radial index’ is peculiar to 3D vector fields and proportional to 1/|m − 1| in general. Our work provides a way to describe the topological behaviors of groups of singularities and supplies new parameters for measuring the topological patterns in 3D vector fields, which will rich the topological theory and may have applications in topological photonics.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Singular optics is a branch of modern optics that mainly studies optical singularities and their topological structures [1,2], for instance optical vortices [3], polarization singularities [4] and polarization Möbius strips [5,6]. Since the seminal paper of Nye and Berry in 1974 [7], singular optics has been studied for nearly half a century and has offered fascinating applications ranging from classical optics, such as optical communications [8,9], optical manipulations [10,11] and optical imaging [12,13], to quantum optics, like quantum information processing [14,15]. Recently, the singular behaviors have been paid much attention in an emerging nano-field, named as topological photonics [16–18], which shows that the striking physical properties of light there, for example the bound states in continuum (BICs), can be understood in terms of their topology [19–22]. It has been found that the nature of the BICs is topological [23], i.e. they are actually a type of V-points (V type polarization singularities) in $k$-momentum space [24]. Explorations on the topological properties of light and their applications in photonic devices grow vigorously, and the synthesis of singular optics and topological photonics has developed and will continue to deep our understanding of optical phenomena in nano-scale [25].
The most well-known singularities in optics may be phase singularities and polarization singularities, with former originating from 2D scalar fields [1] and the latter representing the topological feature of 2D vector fields [4]. As the research goes to 3D vector fields, besides these two types of optical singularities, there also exist singularities of spin angular momentum density vectors (i.e. spin density vector, or SD vector as an abbreviation) [26–30], which describe the unique topological feature of 3D vector fields [31,32]. In 3D vector fields, the topological structures exhibit intriguing shapes, for instance the topological knots [33–37] evolved from the curves of singularities in 2D fields, and the polarization Möbius strips [5,6,28,38,39] corresponding to the star-, lemon-, monstar-structures around C-points in 2D vector fields. While, in 2D vector fields, there also exist other interesting topological structures: the topological flowers and spider webs [40–44]. They are formed in the higher-order singular beams consisting of only linear polarization with spatially varying orientation [45,46]. There is a V-point singularity in the beam center characterized by the Poincaré-Hopf (PH) index [40], which actually also determines the folds and sectors of the topological flowers and spider webs in 2D vector fields [41]. However, in most studies these intriguing topological structures are discussed in 2D vector fields. In 3D vector fields it is mainly illustrated that through tightly focusing the beams with the topological flowers or spider webs, the intensity landscapes can be tailored into the patterns with multi-folds [43,44]. Recently, more general singular vector fields and new types of hybrid vector fields developed from the (basic) singular beams are proposed [30,44,47], and many interesting topological phenomena have been found in these fields, such as the polarization singularity index of a generalized vector field and its (parity, symmetry, etc) properties [44], the C-points turned from the C-lines of the initial field at the focus [47]. There are still some questions: how is the polarization topological behavior of these topological flowers or spider webs in 3D vector fields? Whether can the SD singularities (which are unique to 3D vector fields) exhibit these interesting topological structures? What is the specialty of the topological flowers and spider webs there? To answer these questions is to unveil the topological properties of the flower and spider webs structures in 3D vector fields, and is one of the aims of this article. Furthermore, with the growth of applying of singular optics in nano-fields, the topological theory (of singular optics) itself also needs to be developed. Here, along with exploration of topological flower and spider webs in 3D vector fields, we will also deal with the following issues: Whether the new topological behaviors can be observed in 3D vector fields? Are the current topological quantities sufficient to describe the topological flowers or spider webs in 3D vector field? How to measure the new topological reactions? We believe that the solutions to these problems will rich the topological theory of singualr optics in 3D vector fields, and be useful in the synthesis of singular optics and topological photonics.
2. Flowers and spider webs in higher-order singular vector beams
Singular vector beams refer to the linearly polarized beams with spatially varying orientation of the electric vectors, i.e. the beams with inhomogeneous linear polarization. These vector beams can carry vector singularity, the V-point, at which the electric vector is undefined. The V-point singularity usually is characterized by the Poincaré-Hopf (PH) index $\eta$, the conserved integer valued quantity with generic value $\eta =\pm 1$ [40]. The typical singular vector beams with V-point of $\eta =\pm 1$ are the (fundamental) radially polarized beam or the (fundamental) azimuthally polarized beam. The general form of these two types of beams at their profiles can be expressed by the following Jones matrix formulation [43,45,48]
The higher-order singular vector beams can exhibit topological flowers or spider web structures of the electric vector (or polarization) distributions, which are depicted in Fig. 1. Here $A(r)$ is assumed to be Gaussian, i.e. $A(r)=e^{-r^2/w^2}$, with $w$ the beam waist. $m$ is chosen as $\pm 2$, $\pm 3$ and $\pm 4$, and $o$ is set as $0$. Since the similarity of the higher-order azimuthally polarized beams and radially polarized beams, from here on we only consider the case with $o=0$. The blue arrows in Fig. 1 denote the electric vectors and the red curves depict the vector flow. It is seen that for $m\geq 2$, the electric vectors construct ‘flowers’, while for $m\leq -2$ the shape of ‘spider web’ is formed. The divisions between the petals of the flower or the sectors of the spider web are marked by dashed orange line, named as ‘D-lines’. The vectors along D-lines are radially oriented, while the other vectors are symmetric about the D-lines. The higher-order singular vector beam has $|m-1|$ D-lines, and $2|m-1|$ folds of the topological flowers or spider webs [40,41].
Fig. 1. The electric field vectors of higher-order singular vector beams. Here the order is selected as $m=\pm 2,\pm 3,\pm 4$. Flowers and spider webs are depicted by red flow lines.
3. Strongly focused higher-order singular vector beams
Suppose a higher-order vector beam descried by Eq. (1) is strongly focused in an aplanatic high numerical aperture (NA) system (see Fig. 2, the focal length is $f$ and the semi-aperture angle is $\alpha$). Assume that $A(r)$ is still Gaussian with $A(r)=e^{-r^2/w^2}$, and the entrance plane of the focusing system is coincident with the waist plane of the beam. By applying the Richards-Wolf vectorial diffraction theory [49], the electric field at observation point $P(\rho _s,\phi _s,z_s)$ in the focal region can be expressed as
Fig. 2. Constructing 3D vector fields by strongly focusing higher-order singular beams with $f$ the focal length and $\alpha$ the semi-aperture angle.
4. Flowers and spider webs in 3D vector fields
In this section, we will discuss the topological structures in this 3D vector field. First, the topological patterns of the polarization distribution in the focal plane is examined, then the spin density vectors and their topological structures are analyzed there. As it will be shown, in both two cases the topological flowers and spider webs are formed.
4.1 Polarization flowers and spider webs
From Eqs. (3)–(9), we can easily find that in the focal plane ($z_s=0$), the field components $e_x$ and $e_y$ are always in phase or out of phase, i.e. the phase difference between these two components is $0$ or $\pi$. This implies that the transverse electric field $\bf {E_t}$ in the focal plane is linearly polarized (i.e. the linear polarization states of the transverse field). Figure 3 shows the distribution of the transverse electric field vectors on the focal plane, where the green arrows denote the vectors. Still the flow lines are painted in red and the dashed orange lines represent the divisions. By comparing Fig. 3 with Fig. 1, one can find that the transverse electric vectors in the focal plane have the same topological patterns of the initial incident higher-order singular vector beams. In other words, the polarization states of the transverse electric field at the focal plane also obey the rule (of the polarization distribution of the incident singular beams): when $m\geq 2$, they exhibit topological flowers, while when $m\leq -2$, they have the spider web shapes; the number of the flower petals or the web sectors is equal to $2|m-1|$. Note that the intensity pattern of the transverse field also has $2|m+1|$ or $2|m-1|$ folds (depending on the sign of $m$) in the focal plane and for more details one can go for [44].
Fig. 3. The transverse electric field vectors (or the polarization of the transverse electric field) at the focal plane. Here the order is selected as $m=\pm 2,\pm 3,\pm 4$, $\alpha =60^\circ$ and $f/w=2$.
Physically, the same topological patterns between the initial field and the focused field is caused by the focusing properties of the azimuthal polarization (and the radial polarization) and the stability of the topological structures. For a higher-order singular vector beam, the polarization states in the entrance plane are not purely radial or azimuthal. The longitudinal component ($z_s$ component) is generated from the radial polarized component through focusing, meanwhile the topological patterns of the radial polarized and the azimuthal polarized components are preserved. It means that during the focusing process, the radial component will subtract some part to becomes the longitudinal component, while the whole topological structures of the transverse field components are generally maintained. Thus, the transverse electric field vectors on the focal plane construct the same flowers or spider webs of the initial electric vectors. This also can be elucidated by writing the $\bf {E_t}$ into this form (according to Eqs. (3)–(9)):
Fig. 4. Absolution values of the weight functions $E_{W1}$ and $E_{W2}$ along the radial direction. Here the order is selected as $m=\pm 2,\pm 3,\pm 4$, $\alpha =60^\circ$ and $f/w=2$.
Note that the transverse electric vectors for $m=\pm 3$ and $\pm 4$ in Fig. 3 have the opposite orientation of their corresponding vectors in Fig. 1 due to the term ${\rm i}^{m+3}$ in front of the integral in Eqs. (12) and (13). From the topological view, the PH index $\eta$ does not change if all the vectors reversed their orientation, and physically since the electric field vector is time dependent, after a half period, the vector will reverse its direction if the electric field is locally linearly polarized (it is also worth noting that the orientation of the polarization is not time dependent).
4.2 SD vector flowers and spider webs
In above section, the polarization of the transverse electric field in the focal plane is discussed and it is found that they have the same topological structures (the same flower and spider-web patterns) of the incident field. While, we know that the ‘whole’ electric field actually also includes the longitudinal electric field, i.e. the polarization ellipses lie in 3D space. The 3D polarization states usually are not easy to discuss since their complicated distribution and more parameters in description. Instead, another physical quantity, the spin angular momentum density vector (also called spin density vector, abbreviated to SD vector) [26–29] will be used to discuss the polarization in 3D vector fields.
The SD vector not only expresses the density of the SAM, but also reflects the polarization states in 3D space, i.e. the absolute value of the SD vector describing the shape of the polarization ellipse and its direction representing the orientation of the polarization plane (more details about the SD vector and its relation to polarization ellipse can be seen in Supplement 1). Mathematically, the SD vector (of the electric field) $\textbf{s}_E$ can be written as [27–29]
As we discussed before, in the focal plane the $e_x$ and $e_y$ components are always in phase or out of phase, i.e. $\sin \phi _{yx}=0$. This indicates that the SD vectors are purely transverse in the focal plane, i.e. the ‘photonic wheels’ are formed [28,50–52]. The photonic wheels mean that the polarization ellipses there lie in the plane containing the propagation direction, i.e. the electric vectors spin around a transverse axis with time in analogy to rolling mechanical wheels (one can also refer to Supplement 1 for more information). Thus, the SD vectors can be drawn in a 2D transverse plane, which is shown in Fig. 5. Here the black arrows denote the SD vectors, and as before the red curves and the orange dashed lines represent the flow lines and the D-lines respectively. (Note that in this case, the energy flow also has interesting behavior, i.e., the toroidal energy flux occurs on the focal plane [30,46].)
By comparing Fig. 5 with Figs. 1 and 3, we can get: $(a)$ the SD vectors in the focal plane, similar to the electric vectors also can construct the topological flower and the spider web patterns; $(b)$ these topological patterns of the SD vectors have the same shapes of those for the incident electric field, i.e. the flower shape for $m \geq 2$ and the spider web shape for $m\leq -2$; $(c)$ the special part is that the topological patterns of the SD vectors in Fig. 5 have a rotation from those in Figs. 1 and 3. In the following we will demonstrate that the value of this rotation is $90^\circ /|m-1|$.
Since $\sin \phi _{yx}=0$, Eq. (15) can be re-written as
SD singularity. From Fig. 5, it is also found that first, the vectors at the two sides of the D-lines have the opposite directions, which is quite different from that in both Fig. 1 and Fig. 3; second, the background of plots are painted by two colors (yellow and pink), and the vectors in the same color of a fold have the same flow, while they reverse their directions by passing through a color-band, for instance in the upper space of the plot for $m=2$, the vectors in the first yellow-band (in the most inner circle) rotate in clockwise, while they rotate counterclockwise in the first pink-band (the one next to the most inner circle) and they reverse their directions again in the second yellow-band (the one next to the first pink-band but with a bigger radius). (Note that the backgrounds of the plots for $m=-3,-4$ are not painted in color because in the plotting range the vectors in the same fold do not change their flow direction.) This direction reverse is caused by the ‘SD singularity’, a feature singularity in 3D vector fields [31,32]. Along D-lines and the interfaces of two colors (yellow and pink), there are SD singularities, i.e. the SD singularities compose the D-lines and the interfaces of different colored bands. There are two kinds of SD singularities: one is the SD vector singularity at which the SD is zero and the direction of the SD vector is undefined, thus the SD vector will reverse its direction by passing through this singularity; another is named as the SD phase singularity, and here (i.e. in the focal plane) the SD phase singularity is a phase singularity of the complex SD field [31],
which means that when $|s^{(x)}_E|=|s^{(y)}_E|=0$, the SD phase singularity occurs. It has been verified that in a field with ‘photonic wheels’ (such as the current field in the focal plane) these two kinds of singularities are identical to each other [31,32], which means in this field if a point is a phase singularity of the complex SD field, it is also the SD vector singularity. Thus here we just use the SD singularity to represent both kinds of singularities. The contour plots of the phase of the complex SD field $\textbf{s}^{(xy)}_E$ for two typical cases $m=2,-3$ are shown in Fig. 6(a) and (c) respectively, where the white lines denote the lines of the SD singularities. There are two types of the white lines: the straight ones are actually the D-lines, and the white loops are the interfaces of different colored bands shown in Fig. 5. It can be seen that there is a $\pi$ phase shift by passing through a line of the SD singularities (corresponding to the direction reverse of the SD vectors). For $m=2$, there are $1$ D-line (the straight white line) and $11$ loops of SD singularities (the white loops); while for $m=-3$, there are only $4$ D-lines (no straight white lines), which is the reason for the no-colored background of $m=-3$ in Fig. 5. In Fig. 5, since some colored bands are too narrow and the number of the vectors are not enough, the direction reverse is not very clear to be seen. Here much more vectors are drawn in Fig. 6(b) to illustrate this reverse at the SD singularities (with vectors in same color denoting the same flow direction). This also indicates that it is more convenient to identify the SD singularity by using the phase plots (rather than the vector plots). From here on, we will always use the phase plots to discuss the topological behaviors with SD singularities.Fig. 6. SD vectors and SD singularity. The parameters are the same as in Fig. 3. (a) and (c) are the phase plots of the complex SD fields corresponding to the vector plots in Fig. 5 with the same order. There are two types of curves with SD singularities: one is the D-lines denoted by white straight lines in (a) and (c), and the other is the loops denoted by white circles. (b) illustrates the reverse of the SD vectors at the loops with SD singularities.
Azimuthal index and radial index. Figures 5 and 6 show two types of SD singularity curves, the D-lines and the loops. Based on our previous analysis for both the incident field and the transverse part of polarization at the focal plane, we can deduce from Figs. 5, 6 with Eq. (18) that the SD singularities along the D-lines are caused by the $e_x$ and the $e_y$ components, the inherent characteristic of the incident beam; whereas the SD singularities located at the loops are determined by the $e_z$ component, the feature of the 3D vector field. The D-lines divide the topological pattern into $2|m-1|$ petals (or sectors) of the flower (or spider web) azimuthally, which indeed play a role of ‘azimuthal index’ (similar to the the ‘azimuthal index’ of a LG beam); whereas the loops of SD singularities divide the topological pattern into ‘strips’ of circles radially, which accordingly work as a ‘radial index’. Then we can get a conclusion: in a 2D singular vector field, there is an ‘azimuthal index’ which makes topological shape exhibit folds azimuthally, while in a 3D singular vector field, besides the ‘azimuthal index’ there also exists a ‘radial index’ dividing the topological shape into ‘strips’ radially. Here we use $\sigma _{\rm az}$ and $\sigma _{\rm ra}$ to represent the ‘azimuthal index’ and the ‘radial index’ respectively. So, $\sigma _{\rm az}=2|m-1|$ while the value of $\sigma _{\rm ra}$ or the number of the loops (of SD singularities) does not seem to have an obvious relation with the beam order $m$ (see Figs. 5 and 6). For instance, in Fig. 6 there is no loop for $m=-3$ (i.e. $\sigma _{\rm ra}=0$) in the observed region, while in the same region there are $11$ loops for $m=2$ (i.e. $\sigma _{\rm ra}=11$). Also Figs. 5 and 6 show that the loops are not distributed evenly. In the following, we will discuss the factors affecting the ‘radial index’ and the distribution of these ‘strips’.
Relative topological charge. Before analyzing the factors for the ‘radial index’ and the distribution of the ‘strips’, let us first define a ‘topological quantity’ of the loop. The topological charge/index of a point-singularity is well-defined by its topological structure surrounding, thus according to the conventional rule of the singular optics [41,53], we cannot specify the topological charge/index of a closed curve composed by singular points, such as the loop of SD singularities. While, the peculiarity of the loops in the current situation is that they are concentric circles and the phase around them have symmetric distribution. The topological behaviors of the loops essentially is a general form of the topological behaviors of the point-singularities. So that, we can define a ‘relative topological charge’ to describe the topological characteristic of the loop with SD singularities. As is shown in Fig. 7, if along an arbitrary radial direction (for instance the black arrow in Fig. 7) the phase passing though the loop $l_1$ (at a point-singularity $p_1$) has a $+\pi$ increase, the relative topological charge of the loop is defined as $+1$; whereas if the phase has a $-\pi$ change, the relative topological charge of the loop is $-1$. According to this definition, the loops $l_1$, $l_3$, $l_5$, $l_7$ have charge $+1$ separately, and the charge for each loop $l_2$, $l_4$, $l_6$ and $l_8$ is $-1$. Here the ‘relative’ means that the charge of a loop is dependent on the radial direction and relative to its neighboring loops. Along a different radial direction, the relative topological charge of a loop may be changed, but the relative relations of the charges between these loops remain the same.
Fig. 7. Definition of the relative topological charges of the loops. $p_1$ denotes the point-singularity located at the loop (of SD singularities) $l_1$. Based on this definition, the topological reaction between the loops can be measured.
Topological reaction of the loops and the convergence of light. Here we will show that the number of the loops of SD singularities (or the ‘radial index’, $\sigma _{\rm ra}$) and their distribution are determined by two opposite effects: the topological annihilation of the loops (${\rm Eff}_{t}$) and the convergence of light (${\rm Eff}_{c}$). Figure 8 shows the contour plots of the phase of the complex SD field $\textbf{s}^{(xy)}_E$ with different values of the semi-aperture angle $\alpha$ for $m=+2$. Here the radial direction is chosen to make the topological charge of the innermost loop ($l_1$) be $+1$ (see Fig. 8(a)). Then the topological charge of the loop $l_2$ is $-1$, and the topological charge of $l_3$ is $+1$, and so on. As the semi-aperture angle $\alpha$ increases, the loops move closer to the center due to the convergence of light, which can be seen clearly in the plots with $\alpha =30^\circ$, $40^\circ$, $50^\circ$ and $60^\circ$. When $\alpha$ gets bigger (from $60^\circ$, $65^\circ$ to $70^\circ$), the topological annihilation takes place. The loop $l_1$ with charge $+1$ annihilates with the loop $l_2$ with charge $-1$. Thus at $\alpha =70^\circ$, the innermost loop becomes $l_3$ with its radius bigger than that at $\alpha =30^\circ$. So it seems that the effect of the convergence (${\rm Eff}_{c}$) pulls the loops towards the center of the field, whereas the topological annihilation (${\rm Eff}_{t}$) pushes the loops leave the center .
Fig. 8. Topological annihilation of the loops and the convergence of focusing system ($m=+2$). Here first four loops $l_j$ ($j=1,2,\ldots, 4$) are marked out. These two effects ${\rm Eff}_{c}$, ${\rm Eff}_{t}$ can be seen clearly in this process, where ${\rm Eff}_{c}>{\rm Eff}_{t}$.
These two effects are more obvious to be seen in Fig. 9, where $m=-3$. Since the symmetric distribution of the SD complex field in the focal plane, here only a quarter of the region is shown. When $\alpha$ increases from $15^\circ$ to $28^\circ$, the loops move closer to the center (i.e. ${\rm Eff}_{c}$ is dominant), while as $\alpha$ increases from $28^\circ$ to $30^\circ$, the topological annihilation happens for the loops $l_1$ and $l_2$, then the innermost loop changes into $l_3$. As $\alpha$ continues to grow, the effect of the convergence (${\rm Eff}_{c}$) makes more loops flock to the center, at the same time the effect of the topological annihilation (${\rm Eff}_{t}$) lets the loops go far from the center. It also seems in this case ($m=-3$), the latter effect is stronger than the former (${\rm Eff}_{t}>{\rm Eff}_{c}$), so that when $\alpha$ arrives at $68^\circ$ the innermost loop is $l_{15}$ (i.e. $7$ pairs of the loops have been annihilated) and it radius is much bigger than that in the plot of $\alpha =15^\circ$.
Fig. 9. Topological annihilation of the loops and the convergence of focusing system ($m=-3$). These two effects ${\rm Eff}_{c}$, ${\rm Eff}_{t}$ can be seen clearly in this process, where ${\rm Eff}_{c}<{\rm Eff}_{t}$.
The positions of the loops along the radial distance $\rho _s$ are depicted for $m=3$, $4$, $-2$ and $-4$ in Figs. 10 and 11, where the ‘star’ on the $\rho _s$ axis denoting the positions of the loops. It can be seen that the convergence of light has almost the same effect on all the cases, since as $\alpha$ increases to $80^\circ$ there are same number of loops (i.e. $20$ loops) emerged into the observed region ($0<\rho _s<11 \lambda$) except the case of $m=-4$ (in which all the loops have been annihilated). In contrast, the topological annihilation works differently in these cases. The ${\rm Eff}_{t}$ is much stronger in $m=-4$, where all the loops have been annihilated in the range $0<\rho _s<11 \lambda$ when $\alpha =80^\circ$, whereas the ${\rm Eff}_{t}$ is weaker in $m=3$, since only $3$ pairs of loops have been annihilated in the same situation. Through observing Figs. 10 and 11 with Figs. 8 and 9, we find a rule that the effect of the topological annihilation is proportional to the number of the folds (or the number of the D-Lines), that is
thus, it follows:Fig. 10. Topological annihilation of the loops and the convergence of focusing system depicted along the radial distance $\rho _s$ in a relative wide range ($m=3,4$).
Fig. 11. Topological annihilation of the loops and the convergence of focusing system depicted along the radial distance $\rho _s$ in a relative wide range ($m=-2,-4$).
Fig. 12. Variation of the innermost radius $r_{\rm inner}$ with the semi-aperture angle $\alpha$. A quantitative way to show the joint effect of the topological annihilation and the convergence of light.
In summary, we discuss the topological flowers and spider webs of the SD vectors in this part. First, it is analytically demonstrated that the SD vectors in the focal plane have the same topological patterns of both the input electric field and the transverse part of the polarization except a rotation with value $90^\circ /|m-1|$. Second, through introducing the SD singularity, we propose a ‘radial index’, $\sigma _{\rm ra}$ describing the topological feature of the SD vectors, and illustrate that in a 3D vector field the topological pattern cannot only exhibit folds or sectors of the flowers or the spider-webs determined by the ‘azimuthal index’ ($\sigma _{\rm az}$), but also exist strips of circles radially characterized by the ‘radial index’ ($\sigma _{\rm ra}$),. At last, by defining a ‘relative topological charge’, the two opposite effects (the convergence of light effect ${\rm Eff}_{c}$ and the topological annihilation effect ${\rm Eff}_{t}$) determining the ‘radial index’ (i.e. the distribution of the ‘strips’) are discussed, and it is found the ‘radial index’ is coupling with the ‘azimuthal index’ by the relation $\sigma _{\rm ra} \propto 1/|m-1| \propto 1/\sigma _{\rm az}$.
5. Conclusions
The topological flowers and spider webs, as well as their properties are studied in 3D vector fields. By strongly focusing the higher-order singular vector beams, the 3D vector fields containing the topological flowers and spider webs of both the (transverse) polarization distribution and the SD vector field in the focal plane are constructed. We demonstrate analytically that these topological structures in both cases exhibit the same patterns and also obey the same rule, as: for $m\geq 2$ ($m$ is the order of the singular beam), the topological flowers are formed; for $m\leq -2$, the topological spider webs are wove; while the number of the D-lines is $|m-1|$ and the folds/sectors of the topological flowers/spider webs are $2|m-1|$. However, the topological patterns of the SD vectors have a rotation difference from that of the polarization distribution, and the value of the rotation is calculated as $90^\circ /|m-1|$. More specially, two indices describing the topological features of the flowers and spider webs in optical vector fields are proposed: the ‘azimuthal index’ ($\sigma _{\rm az}$) and the ‘radial index’ ($\sigma _{\rm ra}$). The ‘azimuthal index’ characterizes the folds/sectors of the topological flowers/spider webs azimuthally and is equal to $2|m-1|$, which actually features the conventional topological patterns in 2D vector fields. While, when a vector field is 3D, the ‘azimuthal index’ no longer suffices, and there needs an additional index, the ‘radial index’ which characterizes the strips of the topological patterns radially. The value of the ‘radial index’ is not determinate, but depends on two opposite effects, the convergence of light and the topological annihilation. Through define a ‘relative topological charge’, the interactions of these two effect are analyzed. Our results show that the ‘radial index’ is coupling with the ‘azimuthal index’ with the relation $\sigma _{\rm ra} \propto 1/|m-1| \propto 1/\sigma _{\rm az}$.
Our work not only presents interesting topological structures in 3D vector fields, but also shows a new type of topological behaviors, the topological annihilation of the SD singularities loops (i.e. a general form of topological reactions for point singularities) through defining a relative topological charge. The ‘radial index’ and ‘azimuthal index’, as well as the relative topological charge proposed in this article provide new parameters for characterizing the topological behaviors in both 2D and 3D vector fields. The properties of the topological structures and the related rules will be helpful in scenarios with higher-order singular beams, for instance in optical tweezers. Our findings enrich the topological theory of singular optics in 3D vector fields, and may have implications in topological related research areas, such as the BICs and topological metaphotonics.
Funding
National Natural Science Foundation of China (11974281, 12104283); Fundamental Research Funds for the Central Universities (GK202103021).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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