1. Introduction

Topology is a branch of mathematics, which concerns about the spatial property preserved under continuous deformation (stretching without tearing or gluing), and this property is topologically invariant [1]. Topology is relevant to physics in areas such as condensed matter physics, quantum field theory, cosmology and optical physics, where there are a class of structures or distributions sharing the common features [2]. Topology in optical physics mainly focuses on two subfields: topological photonics and singular optics [3]. In topological photonics, waveguide structures (photonic crystals or metamaterials) are designed to achieve the wave vector-space topology, which allows light to flow around large imperfections without back-reflection [4]. In singular optics, topological features of optical field itself and its organization of the local or global field morphology are studied [5, 6].

The topological features of optical fields are concentrated on optical singularities, which originate from the uncertainty of some parameters [6]. Optical singularities in an optical field are like the defects in a topological system, and the significance stems from the stability and persistence in time or space during propagation. The stability is from the local structure, which organizes the topology of the surrounding field and depends delicately on the symmetry of the system [7, 8]. In paraxial scalar optical fields with uniform polarization, there exist the phase singularities (also called optical vortices or wave dislocations), where the phase is undefined (phase defect) [8]. In paraxial vector optical fields (VOFs) with inhomogeneous polarization, there exist polarization singularities, which can be classified into three categories of C-points, L-lines and V-points. A C-point is an isolated point, in which the polarization state is circularly polarized and is surrounded by the elliptical polarization states [8–11]. Hence a C-point is an orientation defect of the elliptical polarization states with the same handedness and the undefined orientations. An L-line is a curve linked by continuous points, at anywhere the polarization state is linearly polarized, and on two sides of the curve the polarization states are elliptically polarized with opposite handedness [8–11]. So an L-line is handedness defect of the elliptical polarization states with the undefined handedness. A V-point is an isolated point surrounded by the linear polarization states with the undefined polarization directions [12–15].

Optical singularities have the topological property, because they can maintain their features invariably during the propagation [5]. The amplitude of phase singularity in the optical vortex is always zero, so the zero-amplitude point moves to form a dark line in space along the propagation direction. Polarization singularities have also the topological property, the C-points form C-lines and L-lines form L-surfaces during the propagation. Previous works made a lot of effort to understand two-dimensional topological structures of optical singularities [9–18]. It is very interesting to explore three-dimensional topological structures, e.g., the propagation dynamics of optical singularities [19–21].

Here we investigate the propagation dynamics of polarization singularities and their trajectories. We manipulate the symmetry of the optical field by shifting the polarization singularities away from the central axis, and observe the destruction of their topological properties. We build a model of paired centrosymmetric off-axis polarization singularities and then find that both can recover their topological property in the Fourier plane (reciprocal space), i.e. pseudo-topological property. The underlying physics is an invisible redistribution of the spin angular momentum (SAM) components (carrying orbital angular momentum, OAM) of polarization singularities, it is similar to the recombination of components in Talbot effect [22] that has been applied in the temporal cloaking [23]. Because each SAM component of polarization singularities carries opposite OAM states, the pseudo-topological properties may have important applications in optical encrypted communication and quantum information processing [5, 14, 24]. In addition, we complicate the centrosymmetric model and successfully create a series of optical fields that all contain the pseudo-topological properties, which are the Julia fractal vector optical fields (F-VOFs). We experimentally generated the Julia F-VOFs and verified the pseudo-topological properties.

2. Topological and pseudo-topological properties of the polarization singularities

Polarization singularities have the topological properties, because they can maintain their features invariably during the propagation. Each polarization singularity has two orthogonal SAM components, i.e., the right- and left-handed circular polarization, and this superposition mode is extrinsic in paraxial fields. Here we destroy the topological properties of polarization singularities by shifting them away from the central axis, and find that this destruction originates from the space separation of the SAM components. We build a model of paired centrosymmetric off-axis polarization singularities and achieve the recovery of the destroyed topological properties. This recovery is just an appearance, while there exists an invisible redistribution of both the SAM components and the OAM states, which illustrates the pseudo-topological properties.

2.1. Theory of the destruction of topological properties of polarization singularities

Polarization singularities exist in polarization-structured VOFs, which are easily designed and generated by the superposition of the right- and left-handed circularly polarized components carrying opposite phase distributions [5, 25–27], i.e.,

In the transverse plane of z = 0, the cylindrical-symmetry VOFs have only one singularity at $\left(x,y\right)=\left(0,0\right)$, which has the stable topological property. If shifting this singularity to $\left(x,y\right)=\left(\alpha ,\beta \right)$, in the vicinity of which, the optical field can be expressed as ${\mathbf{E}}^{\prime }\left(x,y\right)=E_{+}{ }^{\text{'}}\exp (-i{\varphi }^{\prime }){\widehat{\mathbf{e}}}_{+}+E_{-}{ }^{\text{'}}\exp (i{\varphi }^{\prime }){\widehat{\mathbf{e}}}_{-}$ in the local polar coordinate system with its origin at $\left(x,y\right)=\left(\alpha ,\beta \right)$, where $E_{+}{ }^{\text{'}}$ and $E_{-}{ }^{\text{'}}$ are the similar meanings to $E_{+}$ and $E_{-}$, ${\varphi }^{\prime }$ is the off-axis azimuthal angle as ${\varphi }^{\prime }\left(x,y,\alpha ,\beta \right)=\arctan \frac{x-\alpha }{y-\beta }$. When a point $\left(x,y\right)$ is close to the off-axis singularity at $\left(\alpha ,\beta \right)$, ${\varphi }^{\prime }$ can be approximately written, with a Maclaurin’s expansion, as

Under the linear approximation, when replacing the axisymmetric $1/\left(x^2+y^2\right)$ by a parameter t, we have ${\varphi }^{\prime }=\varphi +t\left(\alpha y-\beta x\right)$. Then the local optical field in the vicinity of $\left(x,y\right)=\left(\alpha ,\beta \right)$ can be rewritten as ${\mathbf{E}}^{\prime }\left(x,y\right)=E_{+}{ }^{\text{'}}\exp [-i\left(\varphi +{\alpha }^{\prime }y-{\beta }^{\prime }x\right)\left]{\widehat{\mathbf{e}}}_{+}+E_{-}{ }^{\text{'}}\exp [i\left(\varphi +{\alpha }^{\prime }y-{\beta }^{\prime }x\right)\right]{\widehat{\mathbf{e}}}_{-}$, where ${\alpha }^{\prime }=t\alpha$ and ${\beta }^{\prime }=t\beta$.

To explore the characteristics of off-axis polarization singularities at infinity, we can use the focal field as an equivalent, which is the Fourier transform of the input field under the paraxial approximation. For any VOF, the input field $\mathbf{E}=E_{+}{\widehat{\mathbf{e}}}_{+}+E_{-}{\widehat{\mathbf{e}}}_{-}$ corresponds to the focal field ${\mathbf{E}}_f=E_{f+}\left(x_f,y_f\right){\widehat{\mathbf{e}}}_{+}+E_{f-}\left(x_f,y_f\right){\widehat{\mathbf{e}}}_{-}$, in which $E_{\pm }$ and $E_{f\pm }$ represent for the complex amplitudes. For the off-axis case, the input field becomes ${\mathbf{E}}^{\prime }={E^{\prime }}_{+}\exp [-i\left({\alpha }^{\prime }y-{\beta }^{\prime }x\right)\left]{\widehat{\mathbf{e}}}_{+}+{E^{\prime }}_{-}\exp [i\left({\alpha }^{\prime }y-{\beta }^{\prime }x\right)\right]{\widehat{\mathbf{e}}}_{-}$. From the frequency shifting property of the Fourier transform, the corresponding focal field is

Now we can discuss the topological properties of the off-axis polarization singularities. In the focusing process, the central singularity and the field around it have the same shifting tendency upon Eq. (3). We can find from Eq. (1) that each polarization singularity can be decomposed into two pure phase singularities (phase vortices) based on the orthogonal SAM (right- and left-handed circularly-polarized) components, which can be seen as two eigenmodes. We can find from Eq. (3) that the right- and left-handed circularly-polarized phase singularities shift to ($-{\beta }^{\prime }$, ${\alpha }^{\prime }$) and (${\beta }^{\prime }$, $-{\alpha }^{\prime }$), respectively. As a result, the polarization singularities will no longer exist due to this separation, which means the destruction of topological properties.

2.2. Simulation of the destruction of topological properties of polarization singularities

To further understand the destruction of the topological properties of an off-axis polarization singularity, we investigate the details of focusing and restoring processes by using the beam propagation method [29], the results are verified again by the vectorial Rayleigh-Sommerfeld diffraction theory [30] and the experiments (see Sec. A in Appendix). We use the top-hat beam to avoid influences of the initial amplitude on the propagation. As mentioned above, the polarization singularities are decomposed into two eigen phase singularities. Figure 1 shows the propagation of the phase singularities of the right- (“$+$”, blue line) and left-handed (“−”, red line) circularly-polarized components of an off-axis polarization singularity at $\left(x,y\right)=\left(0,0.5\right)$, where x and y are normalized by the radius of the input field. The propagation in the z direction is from 0 to 2f (f is the focal length of the lens), which indicates the focusing and restoring processes. We can clearly see that two lines show the stable topological properties of the eigen phase singularities. At first, these two singularities overlap in space, but separate at $z=0.82f$. In the separation process, if the observation is along the +z axis, we can find that the phases ingularity of “$+$” rotates counterclockwise, whereas the phase singularity of “−” rotates clockwise. After passing the focal plane ($z=f$), these two singularities converge again.

 

Fig. 1 Trajectories of the eigen phase singularities of the right-handed (“+”) and left-handed (“−”) circularly-polarized components of an off-axis polarization singularity in the focusing and restoring processes. The optical field propagates along the +z axis. f is the focal length of the lens.

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The right- and left-handed circularly-polarized (two SAMs) components are indispensable to the polarization singularities. When the two eigen phase singularities overlap, the polarization singularities can maintain their features. When they separate, the polarization singularities will disappear, indicating that the topological properties are destroyed.

2.3. Pseudo-topological properties of centrosymmetric polarization singularities

An interesting phenomenon occurs when the VOF contains paired centrosymmetric polarization singularities, which can be realized by applying $\delta \left(x,y\right)={\varphi }_1+{\varphi }_2$, with ${\varphi }_1=\arctan \frac{x-\alpha }{y-\beta }$ and ${\varphi }_2=\arctan \frac{x+\alpha }{y+\beta }$. Paired centrosymmetric polarization singularities (points A and B) are located at $\left(x,y\right)=\left(\alpha ,\beta \right)$ and $\left(x,y\right)=\left(-\alpha ,-\beta \right)$, respectively. Point A (B) is composed of $A_{+}$ ($B_{+}$, right-handed circularly-polarized phase singularity) and $A_{-}$ ($B_{-}$, left-handed circularly-polarized phase singularity). We can find from Eq. (3) that in the focal field (or at infinity), $A_{+}$ and $B_{-}$ arrive at $\left(-{\beta }^{\prime },{\alpha }^{\prime }\right)$, while $A_{-}$ and $B_{+}$ arrive at $\left({\beta }^{\prime },-{\alpha }^{\prime }\right)$, making up paired new polarization singularities.

 

Fig. 2 Trajectories of the eigen phase singularities of the right-handed (“+”) and left-handed (“−”) circularly-polarized components of paired centrosymmetric off-axis polarization singularities in the focusing and restoring processes. A and B stand for different singularities. The optical field propagates along the +z axis. f is the focal length of the lens.

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Figure 2 shows the propagation of the eigen phase singularities of paired centrosymmetric off-axis polarization singularities at $\left(x,y\right)=\left(0,\pm 0.5\right)$, respectively. In the focusing process, $A_{+}$ and $B_{+}$ rotate counterclockwise, while $A_{-}$ and $B_{-}$ rotate clockwise, and the four have no interaction with each other. As a result, the polarization singularities disappear during propagation, but reappear when they are close to the focal plane due to new overlapping. Phase singularities $A_{+}$ and $B_{-}$ combine and form a new polarization singularity, while $A_{-}$ and $B_{+}$ lead to another one (see Sec. D in Appendix). This simulation result is identical to the former analytical theory.

In terms of optical fields containing polarization singularities, the polarization or SAM is an explicit parameter that can be detected, but the eigen phase singularities are not the case. More importantly, usually more attention is paid to the distributions of the input field and its focal field, so the propagation evolution between them is often ignored. If the input field has paired centrosymmetric polarization singularities, the corresponding focal field will also has paired centrosymmetric polarization singularities. Thus it will easily lead to a misunderstanding that these singularities have topological properties. In fact, the paired centrosymmetric off-axis polarization singularities have the invisible redistribution during the focusing process, so we should call it the pseudo-topological property.

2.4. Discussions

In the above sections, we used the word “polarization singularities”, and we do not distinguish the V-points and the C-points, because they have the same trajectories, we only show the final results. In fact, the pseudo-topological property originates from the different rotation directions of the eigen polarization states, i.e., the right- and left-handed circular polarizations. Taking the right-handed circularly-polarized component as an example, we can find from Eq. (3) that both the amplitude and phase of this component shift to $\left(-{\beta }^{\prime },{\alpha }^{\prime }\right)$ in the focal plane. We also examined that the amplitude and phase rotate synchronously during propagation. In the focal plane, when there occurs new overlapping of the eigen phase singularities, the polarization singularities are also recovered. Therefore, V-points and C-points have the pseudo-topological property.

As is well known, from the view of Fourier transformation, the input plane and the Fourier plane (i.e., the focal plane under the paraxial condition) correspond to the real and reciprocal spaces, respectively. During the whole focusing process, the VOF experiences an evolution from the real to reciprocal spaces. The centrosymmetric polarization singularities in the real space will disappear during propagation, and then reappear in the reciprocal space, showing a kind of destruction and recovery of the topological states, the underlying mechanism is an invisible redistribution of the eigen components. This phenomenon is like the Talbot effect [22], where the repetition of the image is only an apparent phenomenon. More importantly, the intrinsic components have a redistribution, which has been found to have unique applications in data temporal cloaking [23]. The schematic description of the pseudo-topological property is shown in Fig. 3.

 

Fig. 3 Schematic description of the pseudo-topological property of polarization singularities.

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For the VOFs containing paired centrosymmetric polarization singularities, if we want to transmit four messages, we can encode them with four eigen phase singularities and the focal field will carry encrypted information. In addition, the right- and left-handed circularly-polarized components of polarization singularities are two orthogonal SAM components, and the two eigen phase singularities correspond to two phase vortices, which carry OAMs [5, 14, 24]. The invisible redistribution in pseudo-topological polarization singularities are closely linked with both SAM and OAM states, so such an interesting phenomenon may have important applications in optical encryption and quantum information.

3. Julia set and pseudo-topological Julia F-VOFs

The pseudo-topological property originates from the propagating coincidence of the paired centrosymmetric off-axis polarization singularities. Can this simple model be generalized or is it universal? Does this pseudo-topological property still exist when there are more singularities? Besides we also need a series of complicated VOFs that all contain the pseudo-topological property. So we have designed theoretically and generated experimentally the Julia F-VOFs and verified their pseudo-topological property.

3.1. Design method of the Julia F-VOFs

We find a complicated centrosymmetric model in the mathematical fractal concept of Julia set, which is named after the French mathematician Julia who investigated its property circa 1915 and culminated in his famous paper in 1918 [31]. Julia set is connected with the complex quadratic polynomials $f_c\left(z\right)=z^2+c$ and is given by $J\left(f_c\right)=\left\{z\in C:\forall n\in N,\left|f_c^n\left(z\right)\right|\le 2\right\}$, where c is a complex parameter and $f_c^n\left(z\right)$ is the nth iteration (or nth hierarchy) of $f_c\left(z\right)$.

To generate a Julia fractal set, we need four steps: (1) Choose a complex number c; (2) Let the initial value z₀ be the set of points in the complex plane as $z_0=x+iy$; (3) Start the iteration as $z_{n+1}=z_n^2+c$; (4) Select the area satisfying $\left|z_n\right|\le 2$, which is a Julia set. We can design different kinds of Julia sets by choosing different c. The fractal has a characteristic that is so-called self-similarity. The fractal geometries of three typical Julia sets can befound from Sec. B in Appendix.

Julia sets have a common important feature which is centrosymmetry. Based on this, we successfully combine the concept of Julia set with polarization singularities to create the pseudo-topological Julia F-VOFs. By setting $\delta \left(x,y\right)=\arg (z_n)$ and $E_{+}=E_{-}$ in Eq. (1), we can obtain a kind of multiple V-points Julia F-VOFs, where $\arg (z_n)$ is the argument of Julia set in the complex plane. Of course, by setting $\delta \left(x,y\right)=\arg (z_n)$ and $E_{+}/E_{-}=\left|z_n\right|$, we can obtain another kind of multiple C-points Julia F-VOFs, where $\left|z_n\right|$ is the modulus of Julia set in the complex plane.

3.2. Generation of the Julia F-VOFs

To generate the Julia F-VOFs in experiment, we use a spatial light modulator (SLM) and a 4f system [27]. The experimental configuration used here is very similar to one used in [27] (see Sec. A in Appendix). The holographic grating (hologram) displayed on SLM is a two-dimentional (2D) grating, which has an amplitude transmission function

Simulated and measured results of multiple V-points Julia F-VOFs can be found in Fig. 4, which are in fact the local linearly polarized Julia F-VOFs, and the intensity patterns of the x-components show the distributions of the polarization states. As the iteration or hierarchy of the Julia sets increases, i.e., n becomes bigger, these V-points will arrange in a special shape exhibiting the geometry of Julia set, which depends on c.

 

Fig. 4 Multiple V-points Julia F-VOFs. The first (third) row shows the simulated x-components with c = 0.5 (c = 0.5 exp (iπ/4), and the second (fourth) row shows the corresponding measured results. All pictures have the same dimension of 3.42 × 3.42 mm².

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For multiple C-points Julia F-VOFs, the simulated and measured results can be found in Fig. 5, which is in fact a kind of complicated Poincaré fields [25]. Besides the x-components, the Stokes parameter S₃ shows the distribution of the ellipticity of polarization states [5]. We can find that the C-points arrange in the geometry of Julia set. More interestingly, the L-lines also appear, surrounding the C-points and forming a clearer geometry of Julia set.

 

Fig. 5 Multiple C-points Julia F-VOFs with c = 0.5 exp (iπ/4). The first (second) row shows the simulated (measured) x-components, and the third (fourth) row shows the simulated (measured) Stokes parameter S₃. All pictures have the same dimension of 3.42 × 3.42 mm².

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3.3. Locate the polarization singularities

Julia F-VOFs have $2^n$ off-axis polarization singularities, whose locations can be controlled. We can determine the locations of polarization singularities by solving the equation sets: $\left|E_x\right|=0$ and $\left|E_y\right|=0$ [5] (see Sec. F in Appendix).

 

Fig. 6 Total intensity and intensities of right- (E₊) and left-handed circular polarization components (E₋) of Julia F-VOF with c = 0.5 and n = 1 in the focusing and restoring processes, with different propagation distance z. f is the focal length of the lens. The eigen phase and polarization singularities are marked by the white and black dots, respectively. The dash lines are guidelines which help to identify the rotation of these singularities. Pictures with no marks mean the disappearance of polarization singularities. All pictures in the first four columns have the same dimension of 2.6 × 2.6 mm², while those in other columns have the same dimension of 170 × 170 μm².

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The complex number c in Julia set and Julia F-VOFs is the same one, and it can be written as $c=a\exp (ib)$, where a and b are real numbers. When n = 0, we can obtain the solution of $\left(x,y\right)=\left(0,0\right)$, which means that no matter what the value of c is, any Julia F-VOF has only one singularity at $\left(x,y\right)=\left(0,0\right)$. When n = 1, we can obtain two solutions:

When $n\ge 3$, we will obtain a $2^n$ equation set, which has $2^n$ solutions. So, any Julia F-VOF with an iterations n (or nth hierarchy) has $2^n$ singularities. Although the equation set becomes more complicated, the calculation can be still done if needed. In particular, Julia F-VOF with c = 0 and arbitrary n will degenerate into high-order cylindrical symmetry VOF with topological charge of $2^n$, and has only one singularity at the origin of coordinate system. It is of great importance to locate the polarization singularities, because the polarization distribution of Julia F-VOFs is various, and the propagation dynamics of which is quite complex. In view of the topological property, however, by tracking the trajectory of every singularity, we can always find the lines. The direct design of polarization singularities may help us to find novel topological structures and properties [15, 21], and our experimental setup can meet all requirements.

3.4. Verify the pseudo-topological property of polarization singularities

To verify the pseudo-topological property of polarization singularities, we need to investigate the propagation dynamics near the focal field (infinity) of Julia F-VOFs. So we use a lens to focus the optical fields and move the CCD continuously to perform the experimental verifications. We extract the right- and left-handed circularly polarized components from the total optical field by blocking the left- and right-handed one (see Sec. A in Appendix).

In general, to locate the singularities in experiment, we need to measure both the intensity and phase distributions of the optical field [15, 19, 21]. It is feasible when $z<0.5f$, but we meet some difficulties as the light spot becomes too small when $z>0.5f$. Also, the optical field has special intensity distribution, so it is hard to add another plane wave to interfere with it, as a result, the phase distribution is very difficult to be detected. Fortunately, we find that the measurement of intensity is enough to locate the singularities. The experimental results of Julia F-VOF with $c=0.5$ and n = 1 are shown in Fig. 6 (corresponding to the trajectories in Fig. 2). The eigen phase and polarization singularities are marked by the white and blackdots, respectively. The dash lines are guidelines which help to identify the rotation of these singularities. Some pictures have no marks, meaning that the polarization singularities exhibit the destruction of topological property, due to the space separation of the eigen phase singularities. We can see clearly that the phase singularities of the right-handed circularly-polarized component (“$+$”) rotates counterclockwise, while that of the left-handed one (“−”) rotates clockwise, which is the same as the above theoretical analysis and simulations.

Furthermore, we investigate the pseudo-topological property of Julia F-VOF with $c=0.5$ and n = 3 by simulation, and the propagation of the eigen phase singularities of the right-handed (“$+$”) and left-handed (“−”) circularly-polarized components are shown in Fig. 7. We can see clearly the redistribution of components in the focal field, indicating the pseudo-topological property. These results are also verified in experiment (see Sec. E in Appendix).

 

Fig. 7 Trajectories of the eigen phase singularities of the right-handed (“+”) and left-handed (“−”) circularly-polarized components of polarization singularities in Julia F-VOF (c = 0.5 and n = 3) in the focusing and restoring processes. A, B, ... , H stand for different singularities. The optical fields propagate along the +z axis. f is the focal length of the lens.

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4. Conclusion

We explore the propagation dynamics of paired centrosymmetric polarization singularities and reveal their pseudo-topological property. By creating and generating the Julia F-VOFs, we experimentally verify the pseudo-topology. We destroy the topological property of polarization singularities by shifting them away from the central axis. The underlying physics is to destroy the global symmetry of the optical fields, but retain the local symmetry. The destroyed topological property can be recovered by adding two-fold rotational symmetry to the optical fields, i.e., introducing another centrosymmetric singularity. This recovery is not real, so it is called the pseudo-topological property, which is related to an invisible redistribution of SAM and OAM states. It may have potential applications in optical encrypted communication and quantum information processing.

The topological properties of polarization singularities have attracted growing interest in recent years, but the most focus on the theoretical study due to the difficulty of generation. The existing controllable method for generating multiple polarization singularities is only the multi-beam interference, which is lack of flexibility [32]. Using the SLM, we experimentally generate arbitrary multiple polarization singularities with very high quality, and the Julia F-VOFs are taken as examples. This experimental method can be further generalized to allow richer topological dynamics of more complicated structures.

Appendix

A. Experimental details for generating the Julia F-VOFs

To generate the Julia F-VOFs in experiment, we use a spatial light modulator (SLM) and a 4f system, the experimental setup is the same as that in Ref. [33]. A linearly-polarized laser beam at a wavelength of 532 nm is collimated by a pair of lenses (L1 and L2) to illuminates the SLM located at the input plane of the 4f system composed of a pair of lenses (L3 and L4). The designed holographic grating (hologram) displayed at the SLM has the amplitude transmission function as follows

(7)$$t\left(x,y\right)=\frac{1}{2}+\frac{1}{4}\left[E_{+}\text{cos }(2\pi f_{0}x-\delta )+E_{-}\text{cos }(2\pi f_{0}y+\delta )\right],$$

where f₀ is the spatial frequency of the grating.

The hologram is a two-dimensional grating carrying the space-variant phase δ and will diffract the incoming optical field into four diffraction orders (±1st orders in both the x and y dimensions). The two $+$1st orders in the x and y dimensions are selected by a spatial filter (SF) located in the Fourier plane of the 4f system. Two λ/4 wave plates are placed in the Fourier plane of the 4f system convert the two linearly-polarized $+$1st orders into the right- and left-handed circular polarizations, as the two bases for generating the Julia F-VOFs. The right- and left-handed circularly-polarized $+$1st orders are combined by the Ronchi grating located at the output plane of the 4f system, and then the generated Julia F-VOFs can be described by Eq. (1) of the main text. One should point out that the spatial geometry of the Julia F-VOFs depends on the spatial distribution of the additional space-variant phase $\delta \left(x,y\right)$.

B. Fractal geometries of typical Julia sets

To intuitively illuminate the fractal feature, here we provide fractal geometries of three typical Julia sets with different values of c. (a) $c=0.285+0.01i$, (b) $c=-0.4+0.6i$, (c) $c=-0.8i$ and the iteration (or hierarchy) $n=100$, as shown in Fig. 8.

 

Fig. 8 Fractal geometries of three typical Julia sets with different values of c. (a) c = 0.285 + 0.01i, (b) c = −0.4 + 0.6i, (c) c = −0.8i and the iteration (or hierarchy) n = 100.

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C. Expression of polarization singularities

One may have a question about the expression of the polarization singularities used in the main manuscript, i.e.,

(8)$$\mathbf{E}\left(x,y\right)=E_{+}\text{exp }[-i\delta \left(x,y\right)\left]{\widehat{\mathbf{e}}}_{+}+E_{-}\text{exp }[i\delta \left(x,y\right)\right]{\widehat{\mathbf{e}}}_{-},$$

indicating that any polarization singularity is a superposition of right- and left-handed circularly polarized fields (${\widehat{\mathbf{e}}}_{+}$ and ${\widehat{\mathbf{e}}}_{-}$) carrying opposite space-variant phases of $-\delta \left(x,y\right)$ and $\delta \left(x,y\right)$. $E_{+}$ and $E_{-}$ ($\left|E_{+}{|}^2+\right|E_{-}{|}^2=1$) are the space-invariant amplitudes [5, 26].

However, the more widely used expression of polarization singularities is

(9)$$\tilde{\mathbf{E}}\left(x,y\right)=E_{+}\text{exp }[-i\theta \left(x,y\right)]{\widehat{\mathbf{e}}}_{+}+E_{-}{\widehat{\mathbf{e}}}_{-},$$

indicating a superposition of a right-handed circularly polarized field (${\widehat{\mathbf{e}}}_{+}$) carrying space-variant phase of $-\theta \left(x,y\right)$ and a left-handed circularly polarized plane wave (${\widehat{\mathbf{e}}}_{-}$) [5, 11]. If $\mathbf{E}\left(x,y\right)$ and $\tilde{\mathbf{E}}\left(x,y\right)$ represent for the same polarization singularities, we can easily find the relation of $\theta \left(x,y\right)=2\delta \left(x,y\right)$. So, Eq. (9) can be further deduced as

(10)$$\tilde{\mathbf{E}}\left(x,y\right)=\text{exp }[-i\delta \left(x,y\right)]\mathbf{E}\left(x,y\right).$$

Clearly, $\tilde{\mathbf{E}}\left(x,y\right)$ has an additional phase than $\mathbf{E}\left(x,y\right)$, which is called the dynamical phase [34]. If we only care about the polarization distributions, Eqs. (8) and (9) have no difference, because the dynamical phase can be ignored. Other theoretical works prefer to use Eq. (9) for simplicity.

However, if we investigate the propagation dynamics of polarization singularities, the polarization distributions of the optical field must be clean, which means that no additional phase or amplitude is allowed. If we use polarization conversion devices (such as q-plates [35, 36]) to generate the polarization singularities, $\tilde{\mathbf{E}}\left(x,y\right)$ will have no additional dynamical phase. However, our experiments are based on the superposition of right- and left-handed circularly polarized fields carrying opposite space-variant phases because our method is flexible, so we use Eq. (8) or Eq. (1) in the main text.

The stability of the polarization singularities comes from their local topological structures, i.e., the surrounding polarization distributions [8, 34]. In these two expressions, the surrounding polarization distributions of the polarization singularities are completely the same. Unique difference is that, $\mathbf{E}\left(x,y\right)$ has zero amplitudes in the center, but $\tilde{\mathbf{E}}\left(x,y\right)$ is nonzero. Now that both expressions have the same topological structure, we use $\mathbf{E}\left(x,y\right)$ to easily see the singularities as dark spots.

D. Simulation method and simulation results

We use two methods for simulation: the beam propagation method [29] and the vectorial Rayleigh-Sommerfeld diffraction theory [30]. Both the two methods show the same results. We add the phase factor of the lens to the original optical field, and then let it propagate. We can obtain directly any cross section of the optical field at any z, where z is the propagation distance normalized by the focal length f of the focusing lens. We extract the phase and amplitude of the right- and left-handed circularly polarized components of the optical field to locate the singularities. The orientation distribution of polarization ellipses of the optical field containing two centrosymmetric off-axis polarization singularities is shown in Fig. 9, and several phase distributions are shown in Fig. 10. These two figures can give help on understanding the pseudo-topology.

 

Fig. 9 Distributions of the polarization ellipse orientation of the optical feild with two centrosymmetric off-axis polarization singularities. (a) is the input field, and (b) is the corresponding focal field. The polarization singularities are marked by the circles, and A₊, A₋, B₊, and B₋ are their eigen components. The superscript “+” (“−”) indicates the right-handed (left-handed) circular polarization.

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Fig. 10 Phase distributions of right- (E₊) and left-handed circular polarization components (E₋) of two centrosymmetric off-axis polarization singularities in the focusing and restoring process, with different propagation distance z, and f is the focal length. A and B stand for different singularities. The dimension of the first column is normalized to be 1 × 1, and the dimension of the second column is 0.5 × 0.5, the dimensions of the rest are all 0.12 × 0.12, relatively.

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E. Additional experimental results

Here we show some added experimental results. Figure 11 is the case of a single off-axis polarization singularity, corresponding to Fig. 1 in the main text. Figure 12 is Julia F-VOF with $c=0.5$ and n = 3, corresponding to Fig. 7 in the main text. The eigen phase and polarization singularities are marked by the white and black dots, respectively. The dash lines are guidelines which help to identify the rotation of these singularities. Some pictures have no marks, meaning that there occurs the destruction of the topological property of polarization singularities, due to the space separation of the eigen phase singularities. The experiments verified our theory and simulation.

 

Fig. 11 Total intensity and intensities of right-handed (E₊) and left-handed (E₋) circularly-polarized components of the optical field with an off-axis polarization singularity at different propagation distance z (f is the focal length of the lens). The phase and polarization singularities are marked by the white and black dots, respectively. Dash lines are guidelines which help to identify the rotation of these singularities. Images with no marks mean the disappearance of polarization singularities. The images in first four columns have dimensions of 2.6 × 2.6 mm², while those in other columns have dimensions of 170×170 μm².

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Fig. 12 Total intensity and intensities of right-handed (E₊) and left-handed (E₋) circularly-polarized components of Julia F-VOF with c = 0.5 and n = 3 at different propagation distance z (f is the focal length of the lens). The phase and polarization singularities are marked by the white and black dots, respectively. Dash lines are guidelines which help to identify the rotation of these singularities. Images with no marks mean the disappearance of polarization singularities. The images in first four columns have dimensions of 2.6 × 2.6 mm², while those in other columns have dimensions of 290 × 290 μm².

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F. Details for locating the polarization singularities

Julia F-VOFs have $2^n$ off-axis polarization singularities, whose locations can be controlled. We can determine the locations of polarization singularities by solving the equation sets: $\left|E_x\right|=0$ and $\left|E_y\right|=0$.

The complex number c in Julia set and Julia F-VOFs is the same one, and it can be written as $c=a\exp (ib)$, where a and b are real numbers.

Due to the zero-intensity feature of singularity, to calculate the locations of singularities, we must solve the equation set

(11)$$\{\begin{array}{l}\left|E_x\right|=0,\hfill \\ \left|E_y\right|=0.\hfill \end{array}$$

When n = 0, the expression of Julia F-VOF in Eq. (1) can be simplified as $\mathbf{E}=\cos \varphi {\widehat{\mathbf{e}}}_x+\sin \varphi {\widehat{\mathbf{e}}}_y$, where ϕ is the azimuthal angle in the polar coordinate. Compared with Eq. (11), we can easily find the solution of $\left(x,y\right)=\left(0,0\right)$, meaning that no matter what the value of c is, Julia F-VOF has only one singularity at $\left(x,y\right)=\left(0,0\right)$. In this case, Julia F-VOF is just a radially polarized VOF.

When n = 1, from Eqs. (8) and (11), we can obtain

(12)$$\{\begin{array}{ll}\hfill & 4x^2-4y^2+a\text{cos }b=0,\hfill \\ \hfill & 8xy+a\text{sin }b=0.\hfill \end{array}$$

This quadratic parameterized equation set has two solutions

(13)$$\{\begin{array}{ll}\left(x,y\right)\hfill & =\left(\sqrt{a}\text{sin }(b/2)/2,-\sqrt{a}\text{cos }(b/2)/2\right),\hfill \\ \left(x,y\right)\hfill & =\left(-\sqrt{a}\text{sin }(b/2)/2,\sqrt{a}\text{cos }(b/2)/2\right).\hfill \end{array}$$

Equation (13) shows the locations of singularities of Julia F-VOF with n = 1. In this case, Julia F-VOF has two singularities.

When n = 2, from Eqs. (8) and (11), we can obtain

(14)$$\{\begin{array}{l}64\left(x^2-y^2\right)xy\text{cos }b-16\left(x^4-6x^2{y}^2+y^4\right)\text{sin }b+16axy+a^2\text{sin }b=0,\hfill \\ 64\left(x^2-y^2\right)xy\text{sin }b+16\left(x^4-6x^2{y}^2+y^4\right)\text{cos }b+8a\left(x^2-y^2\right)+a^2\text{cos }b+a=0.\hfill \end{array}$$

This quartic parameterized equation set has four solutions

(15)$$\{\begin{array}{ll}\left(x,y\right)\hfill & =\left(\sqrt{\left(\sqrt{a}{\sigma }_1-{\sigma }_2\right)/8},-\sqrt{\left(\sqrt{a}{\sigma }_1+{\sigma }_2\right)/8}\right),\hfill \\ \left(x,y\right)\hfill & =\left(-\sqrt{\left(\sqrt{a}{\sigma }_1-{\sigma }_2\right)/8},\sqrt{\left(\sqrt{a}{\sigma }_1+{\sigma }_2\right)/8}\right),\hfill \\ \left(x,y\right)\hfill & =\left(\sqrt{\left(\sqrt{a}{\sigma }_3-{\sigma }_4\right)/8},\sqrt{\left(\sqrt{a}{\sigma }_3+{\sigma }_4\right)/8}\right),\hfill \\ \left(x,y\right)\hfill & =\left(-\sqrt{\left(\sqrt{a}{\sigma }_3-{\sigma }_4\right)/8},-\sqrt{\left(\sqrt{a}{\sigma }_3+{\sigma }_4\right)/8}\right),\hfill \end{array}$$

with

(16)$$\{\begin{array}{ll}{\sigma }_1\hfill & =\sqrt{a+1+2\sqrt{a}\text{sin }(b/2)},\hfill \\ {\sigma }_2\hfill & =a\text{cos }b-\sqrt{a}\text{sin }(b/2),\hfill \\ {\sigma }_3\hfill & =\sqrt{a+1-2\sqrt{a}\text{sin }(b/2)},\hfill \\ {\sigma }_4\hfill & =a\text{cos }b+\sqrt{a}\text{sin }(b/2).\hfill \end{array}$$

Equation (15) shows the locations of four singularities in Julia F-VOF with n = 2.

When $n\ge 3$, from Eqs. (8) and (11), we will obtain a $2^n$ parameterized equation set, which have $2^n$ solutions. So, Julia F-VOF with an iteration n (or with nth hierarchy) has $2^n$ singularities. To get their locations we need to solve the $2^n$ parameterized equation set, which is hard to give the analytical solutions, while we easily numerically solve it. For instance, for Julia F-VOF with $c=0.5$ (i.e. $a=0.5$ and b = 0) and n = 3, the locations of its eight singularities are

(17)$$\{\begin{array}{ll}\left(x,y\right)\hfill & =\left(0.1983,0.5209\right),\hfill \\ \left(x,y\right)\hfill & =\left(0.1983,-0.5209\right),\hfill \\ \left(x,y\right)\hfill & =\left(-0.1983,0.5209\right),\hfill \\ \left(x,y\right)\hfill & =\left(-0.1983,-0.5209\right),\hfill \\ \left(x,y\right)\hfill & =\left(0.3077,0.3357\right),\hfill \\ \left(x,y\right)\hfill & =\left(0.3077,-0.3357\right),\hfill \\ \left(x,y\right)\hfill & =\left(-0.3077,0.3357\right),\hfill \\ \left(x,y\right)\hfill & =\left(-0.3077,-0.3357\right).\hfill \end{array}$$

In particular, the Julia F-VOF with c = 0 and arbitrary n will degenerate into high-order cylindrical symmetry VOF with topological charge of $2^n$, and has only one singularity at the origin of coordinate system.

Consequently, the nth-hierarchy Julia F-VOF (with ireations n) has singularities of $2^n$, their locations can be flexibly designed by adjusting a and b. Parameter a determines the distance of singularities from the origin of coordinate system, while parameter b dominates the rotation angle. It is of great importance to locate the singularities, because the diffraction or focusing behavior of F-VOF is quite complex and the design of F-VOF with specific phase and polarization distribution is also various. In the view of topological property, however, tracking the trajectory of every singularity, we can always find the invariants, which may be dark points, lines, or holes, although they may be intertwined together. The direct design of singularity may open a door to shape the focal field. We believe that it will have more important applications such as lithography, optical trapping, material processing and so on.

Funding

National Key R&D Program of China (2017YFA0303800, 2017YFA0303700); National Natural Science Foundation of China (11534006, 11774183, 11674184); Natural Science Foundation of Tianjin (16JCZDJC31300); 111 Project (B07013).

Acknowledgments

We acknowledge the support by Collaborative Innovation Center of Extreme Optics.

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