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Abstract
In this paper, partially coherent radially polarized vortex circular Airy beams (PCRPVCABs) are theoretically and experimentally studied for the first time. Comparing with partially coherent radially polarized circular Airy beam, the autofocusing ability of PCRPVCAB can be controlled or even enhanced by the topological charge of the vortex phase. Besides, we show a unique distribution of the degree of polarization, vortex-phase induced polarization transition, and the phenomenon of beam spot rotation after the PCRPVCAB passing through a polarizer. Our results make the PCRPVCAB a good candidate for optical micro-manipulation, free-space communications, formation of disordered optical lattices, etc.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Partially coherent beam [1–21] with controllable spatial coherence has been a topic of great interest in optics in the past decade due to their high directionality in optical communication [22,23] as well as the special advantages in material thermal processing and particle trapping [24–26]. Partially coherent vortex beam (PCVB), which carries orbital angular momentum, was introduced as a natural extension of coherent vortex beam [1,2,7,27,28], such beam displays unique correlation singularities or coherence vortices, which do not exist in a coherent vortex beam. The intensity profile of a PCVB can be modulated by its spatial coherence width, which is useful for particle trapping. Besides, the combination of spatial coherence and vortex phase provides PCVB the advantage of reducing turbulence-induced scintillation [27,29]. Recently, more and more attention is being paid to partially coherent vector beams [4,5,8,30–34], and their characterization, generation [35] and propagation have been studied extensively.
On the other hand, nonlinear beams with autofocusing and self-healing properties [36–42] have been widely studied in recent years. Circular Airy beam (CAB), also named abruptly autofocusing wave [38–42], has attracted great attention due to the potential value of its autofocusing property in optical micro-manipulation [43–45], as well as for generating light bullets and biomedical applications. By extending CAB to the partially coherent domain, Yunfeng Jiang [16] and Tong Li [46] proposed and experimentally generated partially coherent CAB, respectively. In 2020, Tong Li further proposed and experimentally generated partially coherent radially polarized CAB [47], which inherits autofocusing ability of radially polarized CAB and can create an optical potential well, whose depth can be adjusted by changing the coherence width.
In this paper, we investigate the properties of a partially coherent radially polarized CAB embedded with a vortex phase (i.e. PCRPVCAB). By using some identities of Bessel function, the quadruple Huygens-Fresnel integral for calculating the cross-spectral density matrix (CSDM) of a PCRPVCAB is analytically reduced to a double integral. Then we numerically illustrate the vortex phase-induced properties in detail, such as the enhancement of autofocusing ability, the unique distribution of degree of polarization (DoP), and the polarization transition. Furthermore, we report the method for generating PCRPVCAB experimentally and study its polarization characteristics. Our experimental results are consistent with the simulation.
2. Theory
In the source plane $z = 0$, the electric field of a coherent radially polarized CAB with $m$-order concentric vortex can be expressed as:
According to the identities of the Bessel function of the first kind $J_l(s)$ and modified Bessel function of the first kind $H_n(s)$, $e^{is\cos \varphi }=\sum _{l=-\infty }^{+\infty } i^l J_l(s)e^{il\varphi }$ and $H_n(s) = \frac {1}{2\pi }\int _{0}^{2\pi }e^{s\cos \theta }e^{in\theta } d\theta$ [49], we can obtain the integral of $\theta _1$ for the terms ($\cos \theta _1 e^{-im\theta _1}$) and ($\sin \theta _1 e^{-im\theta _1}$):
Here $m$ is the topological charge, $l$ is the number of the expansion item of $e^{is\cos \varphi }$. The detailed derivation of Eq. (5) is shown in Appendix A. In order to facilitate the calculation, we define $n = l\pm 1$, then the term $e^{i(l\pm 1-m)\theta _2}$ related to $\theta _2$ in Eq. (5) can be simplified as $e^{i(n-m)\theta _2}$. By applying $H_v\left (s\right ) = i^{-v} J_v\left (is\right )$, the integral of $\theta _2$ for the terms ($\cos \theta _2 e^{im\theta _2} e^{i(n-m)\theta _2}$) and ($\sin \theta _2 e^{im\theta _2}e^{i(n-m)\theta _2}$) can be obtained by:
Then the double integral for $\theta _1$ and $\theta _2$ of the elements of CSDM can be expressed as:
Notice that the term with large $l$ in Eq. (8) is nearly zero, and can be omitted. So we only calculate the term of $\left | l \right |\le 15$ for simplicity. According to Eq. (7) and Eq. (8), the characteristics of PCRPVCAB with $m$-order concentric vortex could be simulated and analyzed by a double integral:
In our simulation, we set $r_0=0.8\textrm {mm},a=0.1,w = 80\mu \textrm {m}$, and the wavelength is $1064\textrm {nm}$, the power of the beam is $10\textrm {mW}$.
3. Experimental generation of PCRPVCABs
Figure 1 shows our experimental setup for generating PCRPVCAB. A laser beam of $\lambda = 1064\textrm {nm}$ shaped by a polarization maintaining fiber (PMF) is focused onto the rotating ground-glass disk (RGGD, Daheng Optics, with a diameter of 50.8mm and a mesh number of 1500, the rotation velocity is 6000 rev/min) by lens $\textrm {L}_{1}$ with focal length $f_1=80\textrm {mm}$, producing a partially coherent beam with Gaussian correlations. Here $\textrm {L}_{1}$ is used to control the spot size on the RGGD, which determines the coherence width of the beam by varying the distance between $\textrm {L}_{1}$ and the RGGD. After passing through a circular aperture ($\textrm {CA}_1$), the beam is collimated by lens $\textrm {L}_{2}$ with focal length $f_2=50\textrm {mm}$, and splits into two beams by a polarization beam splitter (PBS). The reflected beam is used to measure the coherence width by using two-pinhole interference experiment [46,48], as shown in Fig. 1(a). According to the unified theory of polarization and coherence [46,48], the spectral visibility of fringes $\mathcal {V}$ is equal to the modulus of the spectral degree of coherence $\mu$, i.e.,
Then the coherence width $\delta$ can be calculated by:
where $d$ denotes the spacing between the two pinholes. In our experiment, $d = 2\textrm {mm}$ and the measured results of $\delta =2\textrm {mm}$ and $3\textrm {mm}$ are shown as examples in Fig. 2. The transmitted beam with horizontal polarization is used to generate PCRPVCABs. Since the phase-only spatial light modulator (SLM, BNS, $512 \times 512$ pixels, pixel pitch of $15 \times 15 \mu \textrm {m}^2$, response speed of the liquid crystal is 833Hz) can only modulate a vertically polarized beam, a half-wave plate ($\textrm {HP}_3$) is used to deflect the polarization of the incident beam to the vertical direction. So before illuminating the SLM, the beam becomes a linearly polarized plane wave with Gaussian statistics.A computer-generated hologram (CGH) encoded with the single-pixel checkerboard method [46,50] is loaded on the SLM. Figure 1(b) shows an example of CGH for generating the VCAB with $r_0 = 0.8$ mm, $w = 80\mu \textrm {m},a = 0.1$ and $m=0$. The modulated beam is reflected toward a $4f$ imaging system made up of a couple of identical lenses ($\textrm {L}_3$ and $\textrm {L}_4$) with $f = 250 \textrm {mm}$. The input plane of this optical system coincides with the SLM plane. According to [50], the targeted partially coherent vortex CAB will come out in the zero-order diffraction at the Fourier plane. A linear phase is introduced in the CGH to separate the modulated zero-order diffraction spot from the reflected central spot and a low-pass spatial filter that consists of an iris ($\textrm {CA}_2$) is responsible for selecting the zero-order. Another half-wave plate ($\textrm {HP}_4$) is used to deflect the polarization direction to the correct direction of the radial polarization converter (RPC). After passing through the RPC, the linearly polarized partially coherent VCAB becomes a partially coherent radially polarized VCAB. The RPC is just located at the output plane of the $4f$ system, which is the source plane of the generated PCRPVCABs. So a $2f$ imaging system made up of the lens $\textrm {L}_5$ with $f = 150 \textrm {mm}$ shown in Fig. 1(c) is used to observe the intensity distribution in the source plane, which means that the distances from $\textrm {L}_5$ to RPC and $\textrm {L}_5$ to beam profile analyzer (BPA) are both $300\textrm {mm}$. The beam profile is measured by a beam profile analyzer (BPA, Spiricon, SP620U, $1600 \times 1200$ pixels, pixel pitch of $4.4 \times 4.4 \mu \textrm {m}^2$, dynamic range of 62dB). When we observe the intensity distributions at distances different from the source plane, the $2f$ imaging system is removed, and the BPA is directly placed at the corresponding positions for observation. In our experiment, each intensity pattern of the PCRPVCAB is obtained by averaging 200 samples. In addition, apart from the divergence of the beam, we did not find any more meaningful characteristics of the intensity distribution in the far-field, so in the paper our focus is on the field distribution in the initial plane and autofocusing plane.
Fig. 1. Experimental setup for generating PCRPVCABs. (a) Two-pinhole interference experimental setup for measuring the coherence width; (b) The CGH for generating the VCAB with $r_0 = 0.8\textrm {mm}, w = 80\mu \textrm {m},a = 0.1$ and $m=0$; (c) $2f$ imaging system for observing the intensity distribution in the source plane. M, mirror; PMF, polarization-maintaining fiber; $\textrm {L}_1\sim \textrm {L}_6$, thin lens; RGGD, rotating ground-glass disk; $\textrm {CA}$, circular aperture, which is used as low-pass filter; HP, half-wave plate; PBS, polarization beam splitter; SLM, phase-only spatial light modulator; RPC, radial polarization converter; BPA, beam profile analyzer.
Fig. 2. Two examples of the two-pinhole interference fringes for measuring the coherence width of the beam before it illuminating the spatial light modulator. (a) $\delta =2\textrm {mm}$ and (b) $\delta = 3 \textrm {mm}$.
4. Results and discussions
Figure 3 shows the variation of the normalized intensity of the PCRPVCABs with different coherence width $\delta$ versus the propagation distance $z$. Table 1 shows both the simulation and experimental results of the autofocusing ability, i.e., $I_m/I_0$, of the PCRPVCABs with different topological charge $m$ and coherence width $\delta$, where $I_m$ is the maximum intensity at the autofocusing plane, and $I_0$ is the maximum intensity in the source plane. Notice that $\delta =$ Inf denotes the fully coherent radially polarized vortex CAB (RPVCAB). From Fig. 3 and Table 1, it can be seen that the autofocusing ability of the PCRPVCAB decreases as $\delta$ decreases for the same $m$, but the autofocusing focal length remains $25cm$ and is independent with both coherence width $\delta$ and topological charge $m$. We find that the focal position $z_f$ of the PCRPVCAB can be calculated by:
where $\lambda =1064 \textrm {nm}$ is the wavelength, $R_0$ is the radius of the first ring of the PCRPVCAB, which is $0.874\textrm {mm}$ in this paper and slightly larger than $r_0$ [42]. Furthermore, the relationship between autofocusing ability and topological charge is unique. Under the simulation parameters we set, the autofocusing ability of the PCRPVCAB with $\delta > 1\textrm {mm}$ is the strongest for $m=1$, while for $\delta = 1\textrm {mm}$ and $\delta = 0.5\textrm {mm}$, the autofocusing ability decreases with the increase of the topological charge. This tendency of autofocusing ability to change with coherence width and topological charge can be explained by decomposing PCRPVCAB into completely polarized and completely unpolarized parts. The autofocusing ability of the PCRPVCABs is mainly contributed by the completely polarized part. The completely unpolarized part diverges with propagation and does not have the autofocusing ability. For $\delta = 2\textrm {mm}, 3\textrm {mm}$ and Inf, vortex with $m=1$ can cause an increase in the autofocusing ability due to that the topological charge eliminates the polarization singularity caused by the radial polarization, which autofocuses the energy to a smaller focus. As the coherence width decreases, the energy of the completely unpolarized part will gradually increase. When the coherence width is less than $1 \textrm {mm}$, the peak intensity of the completely unpolarized part in the autofocus plane approaches or even exceeds that of the completely polarized part. The autofocusing ability promoted by $m = 1$ cannot make up for the loss of the autofocusing ability caused by the divergence of unpolarized part. More details are shown in Fig. 13. The above characteristics reveal that we can control the autofocusing ability of PCRPVCAB by the tunable coherence width and topological charge, which makes PCRPVCAB a good candidate for the application of optical trapping and biomedical nursing by further reducing the light intensity before the target, so as to avoid the damage and disturbance caused by the beam with high intensity.
Table 1. Autofocusing ability $I_m/I_0$ of the PCRPVCABs with different topological charge $m$ and coherence width $\delta$, where $I_m$ is the maximum intensity at the autofocusing plane located at $z=0.25m$, and $I_0$ is the maximum intensity in the source plane. White cells: simulation results; Gray cells: experimental results.
All PCRPVCABs with different coherence width and topological charge have the same intensity distribution in the source plane, as shown in Fig. 4(a1). Figure 4(a2) and Fig. 4(a3) show the intensity associated with the x- and y-components of the electric field, $I_x$ and $I_y$, which verify that the beams in the source plane is radially polarized. Figures 4(A1)–4(A3) depict the experimentally generated PCRPVCAB. Our experimental results are consistent with the simulation results.
Fig. 3. Normalized maximum intensity of the PCRPVCABs with different topological charge $m = 0,1,2,3$ as functions of propagation distance $z$. (a) $\delta = 0.5\textrm {mm}$; (b) $\delta = 1\textrm {mm}$; (c) $\delta = 2\textrm {mm}$; (d) $\delta = 3\textrm {mm}$; (e) $\delta =$ Inf.
Fig. 4. Intensity distribution $I$ and the intensity associated with the x- and y-components of the electric field, $I_x$ and $I_y$, in the source plane. (a1)-(a3): simulation results; (A1)-(A3): experimental results. The line graphs denote the intensity distributions along the cross line $y=0$. Each picture is independently normalized and the size is $3.6\times 3.6\textrm {mm}^2$.
For comparison, we first studied the intensity distribution of the fully coherent RPVCABs with $m = 1, 2$ and $3$ in the autofocusing plane. The related simulation and experimental results are shown in Fig. 5. Figure 6 shows the intensity distributions $I$, $I_x$ and $I_y$ in the autofocusing plane of the PCRPVCABs with $\delta = 0.5\textrm {mm}, 1\textrm {mm}, 2\textrm {mm},3\textrm {mm}$ for $m = 1$. The corresponding experimental results are shown in Fig. 7. One can see that, for $m=1$, there is no singularity caused by the radial polarization or the vortex phase in the center of the focus, which is much different from the partially coherent radially polarized CABs demonstrated in Ref. [47]. The outer lobes become more and more indistinguishable as $\delta$ decreases. For $\delta = 0.5\textrm {mm},1\textrm {mm}$, only a circular spot with an approximate Gaussian profile can be seen, as shown in Fig. 6(a1), Fig. 6(a2), Fig. 7(A1) and Fig. 7(A2).
Fig. 5. Autofocusing intensity distributions $I$, $I_x$ and $I_y$ of fully coherent RPVCABs with different topological charge $m = 1, 2, 3$. The line graphs denote the intensity distributions along the cross line $y=0$. (a)-(c):simulation results; (A)-(C):experimental results. Each picture is independently normalized and the size is $1\times 1\textrm {mm}^2$.
Fig. 6. Simulation intensity distributions $I$, $I_x$ and $I_y$ of PCRPVCAB in the autofocusing plane with $m=1$. (a1)-(c1) $\delta = 0.5\textrm {mm}$; (a2)-(c2) $\delta = 1\textrm {mm}$; (a3)-(c3) $\delta = 2\textrm {mm}$; (a4)-(c4) $\delta = 3\textrm {mm}$. The line graphs denote the intensity distributions along the cross line $y=0$. Each picture is independently normalized and the size is $1\times 1\textrm {mm}^2$.
Fig. 7. The corresponding experimental patterns for Fig. 6. Each picture is independently normalized and the size is $1\times 1\textrm {mm}^2$.
Figure 8 shows the simulation intensity distributions $I$, $I_x$ and $I_y$ in the autofocusing plane of the PCRPVCABs with several different combinations of parameters $(\delta, m)$. The corresponding experimental patterns are shown in Fig. 9. Comparing Fig. 8 with Figs. 5–7, one can find that, for $m > 1$, there is a dark notch in the center of the focus caused by the polarization singularity and the vortex phase, which becomes more blurred due to the finite coherence width, i.e., the intensity at the central point becomes higher as $\delta$ decreases. From Fig. 8, one can see that, when the coherence width $\delta$ remains the same, as the topological charge $m$ increases, the size of the dark notch gradually increases, but the intensity at the center of dark notch gradually decreases. For example, when $\delta =1\textrm {mm}$, in the case of $m = 2$, the dark notch completely disappears, and the the beam spot has an approximate Gaussian profile, but in the case of $m = 3$, there is still a blurred dark notch in the center, as shown in Fig. 8(a3) and Fig. 8(a4). The above results indicate that PCRPVCAB possessing autofocusing ability can create an optical potential well with adjustable depth at the center of the beam by adjusting the coherence width or topological charge. Furthermore, for PCRPVCAB, there is an interesting phenomenon, that is, the long axis of the components $I_x$ and $I_y$ orient vertically and horizontally, respectively. For $m=1$, the focus possesses one main lobe, which splits into two for $m>1$. The two ends of the lobe are connected, not completely separate, and the side lobes on the periphery also rotate slightly. The above phenomenon is different from that of traditional coherent vortex beams and the PCVB. This feature shows good potential for the fields of optical trapping and micro-manipulating.
Fig. 8. Simulation intensity distributions $I$, $I_x$ and $I_y$ of PCRPVCAB in the autofocusing plane with several different combinations of parameters $(\delta, m)$. The line graphs denote the intensity distributions along the cross line $y=0$. Each picture is independently normalized and the size is $1\times 1\textrm {mm}^2$.
Fig. 9. The corresponding experimental patterns for Fig. 8. Each picture is independently normalized and the size is $1\times 1\textrm {mm}^2$.
To explain the intensity distribution of the PCRPVCABs, we calculate in Fig. 10 the polarization state of PCRPVCABs in the autofocusing plane [48] for different values of the topological charge $m$ and coherence width $\delta$, and in Fig. 11 the variation of the state of polarization of a PCRPVCAB at several propagation distances for different topological charge $m$ with $\delta = 2\textrm {mm}$. Notice that, according to [47], the state of polarization of a partially coherent radially polarized CAB ($m = 0$) remains invariant on propagation and always displays radial polarization. From Fig. 10 and Fig. 11, we can see that the vortex phase induces dramatic changes of the state of polarization on propagation, e.g., radial polarization disappears and the distribution of the polarization state varies along propagation axis, which is named polarization transition [51,52]. For $m > 0$, the state of polarization displays left-handed elliptical polarization around the beam center and right-handed elliptical polarization outside of the beam center. The area, where the state of polarization displays left-handed elliptical polarization around the beam center, increases during propagation or when the coherence width $\delta$ decreases. In particular, for the beam with $m = 1$, the area is much larger than that of the beam with $m \ne 1$. Furthermore, the polarization ellipse rotates clockwise, which explains why the long axis of the brightest main lobe for the $I_x$ and $I_y$ patterns oriented vertically and horizontally, respectively, and the two ends of the lobe are connected, not completely separate, as shown in Figs. 5–9. In addition, for the beam with $m < 0$, the handedness of the polarization ellipse is reversed, and the polarization ellipse rotates anti-clockwise on propagation. These results are not shown in the figure due to space limitations. Thus modulating the vortex phase provides us with a way for modulating the polarization state of a PCRPVCAB, which is beneficial to the application of the beam in the field of forming adjustable disordered optical lattices. It is well known that the formation of optical lattice comes from the standing wave field formed by interfering laser beams in the same polarization. Compared with PCRPCAB, PCRPVCAB has an advantage in the formation of disordered optical lattices [53–55]. For PCRPCAB studied in Ref. [47], its completely polarized component is always radially polarized during propagation, which means that only the non-polarized component can be used to modulate the lattice depth for a disordered optical lattice. However, due to its unique polarization characteristics, PCRPVCAB can form a more complex disordered optical lattice characterized by local disorder and global order, i.e., the states of adjacent lattice sites are different, but the optical lattices follow the distribution law of the polarization state of the beam in the global view.
Fig. 10. The polarization state of PCRPVCABs at the autofocusing plane for different values of the topological charge $m$ and the coherence width $\delta$. The normalized intensity profile underneath each polarization pattern. The blue (or yellow) ellipsoid with counterclockwise arrow (or clockwise arrow) denotes left- (or right-) handed elliptical polarization. The unit of each picture is $0.1 \textrm {mm}$.
Fig. 11. Variation of the polarization state of a PCRPVCAB at several propagation distances for different topological charge m with $\delta = 2\textrm {mm}$. The normalized intensity profile underneath each polarization pattern. The blue (or yellow) ellipsoid with counterclockwise arrow (or clockwise arrow) denotes left- (or right-) handed elliptical polarization. The unit of each picture is $0.1 \textrm {mm}$.
To study the characteristics of the degree of polarization (DoP), we calculate the DoP (cross line, $y = 0$) in the autofocusing plane of the PCRPVCABs with different topological charge for $\delta =1\textrm {mm}$ and $3\textrm {mm}$, as shown in Fig. 12(a) and 12(b). Compared with the DoP distribution of the beam with $m=0$ (black line in Fig. 12) [47], the DoP singularity in the center disappears, and its value decreases as the coherence width decreases. It is noted that, when $\delta = 1\textrm {mm}$, the value of DoP at the center for $m=1$ is smaller than that for $m = 2$ and $3$. To explain this phenomenon, we decompose the PCRPVCABs into a completely polarized part $I_p$ and a completely unpolarized part $I_u$ according to the unified theory of polarization and coherence [48], as shown in Fig. 13. Figures 13(A) and 13(B) show our experimental results of the $I_p$ and $I_u$, which are obtained by the four Stokes parameters ($S_0, S_1, S_2, S_3$), i.e., $I_p = (S_1^2+S_2^2+S_3^2)^{1/2}$ and $I_u = S_0-I_p$ [56]. Notice that the DoP can be obtained by $\textrm {DoP} = I_p/S_0$. It can be seen that the vortex phase produces a dark notch in the beam center of the intensity distribution $I_u$. Also, the area of the dark notch increases with the increase of the topological charge, and the dark notch becomes blurred or even indistinguishable with the decrease of the coherence width. In addition, for $m=1$, the intensity distribution $I_p$ is a small solid spot. For $m \ne 1$, a dark notch appears in the center of the spot, which also becomes blurred with the decrease of the coherence width. Even for the case with $\delta = 0.5\textrm {mm}, m=3$, the central dark notch disappears completely. From Fig. 13, one can see that as the coherence width decreases, both the completely polarized and unpolarized parts become more divergent, and the energy of the unpolarized part gradually increases, and its peak intensity at $\delta =0.5\textrm {mm}$ even exceeds that of the completely polarized part. This also explains the dependence of the autofocusing ability on the coherence width and topological charge shown in Table 1.
Fig. 12. Degree of polarization (cross line, $y = 0$) in the autofocusing plane of the PCRPVCABs with different topological charge for (a) $\delta =1\textrm {mm}$ and (b) $\delta = 3\textrm {mm}$.
Fig. 13. Decomposition of the intensity distributions for the PCRPVCABs with several different combinations of parameters $(\delta, m)$. The line graphs denote the intensity distributions along the cross line $y=0$. $I_p$: completely polarized part; $I_u$: completely unpolarized part; (a)(b): simulation results, (A)(B): experimental results. The size of each picture is $1\times 1\textrm {mm}^2$.
5. Summary
In conclusion, we analytically reduce the quadruple Huygens-Fresnel integral for calculating the cross spectral density matrix of the PCRPVCAB to a double integral, and both theoretically and experimentally study the statistical properties of PCRPVCAB in detail. Our experimental results are consistent with the simulation results. Our results show that PCRPVCAB inherits the autofocusing property of CAB. Its autofocusing ability can be controlled by adjusting the coherence width and topological charge, but the autofocusing focal length is independent with the two parameters. Besides, the vortex phase induces rotation of the beam spot and the beam profile of a PCRPVCAB in the autofocusing plane can be shaped by modulating the topological charge, which is useful for optical trapping, particle rotating [44,57] and atomic cooling [58]. Furthermore, we also study the polarization transition induced by the vortex phase, that is, radial polarization disappears and the distribution of the polarization state varies along propagation axis, which can explain why the long axis of the components $I_x$ and $I_y$ oriented vertically and horizontally, respectively. This polarization characteristic may be useful for the detection of phase object and the formation of disordered optical lattices.
Appendix
Appendix A: Integral of $\theta _1$
In this appendix, we show in detail the process of calculating the integral of $\theta _1$, i.e., the derivation of Eq. (5). Take the integral of the term $\cos \theta _1 e^{-im\theta _1}$ as an example, which is described as:
By using Euler’s formula $\cos \theta = \frac {1}{2}\left ( e^{i\theta }+e^{-i\theta }\right )$, Eq. (13) can be rewritten as:
Then according to the identity of the Bessel function of the first kind $J_l(s)$, i.e. $e^{is\cos \varphi }=\mathop\sum\limits _{l=-\infty }^{+\infty } i^l J_l(s)e^{il\varphi }$ [49], Eq. (14) can be rewritten as:
Finally, by using the identity of modified Bessel function of the first kind $H_n(s)$, i,e., $H_n(s) = \frac {1}{2\pi }\int _{0}^{2\pi }e^{s\cos \theta }e^{in\theta } d\theta$ [49], we can obtain the integral of $\theta _1$ for the term ($\cos \theta _1 e^{-im\theta _1}$) shown in Eq. (5a):
Notice that $m$ is the topological charge, $l$ is the number of the expansion item of $e^{is\cos \varphi }$. Similarly, by applying the Euler’s formula $\sin \theta = \frac {1}{2i}\left ( e^{i\theta }-e^{-i\theta }\right )$, the result of the integral of $\theta _1$ for the term ($\sin \theta _1 e^{-im\theta _1}$), i.e., Eq. (5b), can be obtained.
Appendix B: Integral of $\theta _2$
In this appendix, we show in detail the process of calculating the integral of $\theta _2$ shown in Eq. (6). In order to facilitate the calculation, we define $n = l\pm 1$, then the terms $e^{i(l\pm 1-m)\theta _2}$ related to $\theta _2$ in Eq. (5) can be simplified as $e^{i(n-m)\theta _2}$. Take the integral of the term $\cos \theta _2 e^{im\theta _2} e^{i(n-m)\theta _2}$ as an example, which is described as:
By using Euler’s formula $\cos \theta = \frac {1}{2}\left ( e^{i\theta }+e^{-i\theta }\right )$, Eq. (17) becomes:
Then by using the identity of modified Bessel function of the first kind $H_n(s)$, i,e., $H_n(s) = \frac {1}{2\pi }\int _{0}^{2\pi }e^{s\cos \theta }e^{in\theta } d\theta$ [49], we can obtain:
Finally, by applying $H_v\left (s\right ) = i^{-v} J_v\left (is\right )$, the integral of $\theta _2$ for the term ($\cos \theta _2 e^{im\theta _2} e^{i(n-m)\theta _2}$) shown in Eq. (19) can be rewritten as:
Funding
National Key Research and Development Program of China (2017YFA0304202); National Natural Science Foundation of China (11804298, 11474254); Fundamental Research Funds for the Central Universities of China. (2016XZZX004-01, 2017QN81005).
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.
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