Spiral fractional vortex beams

Lixun Wu; Xuankai Feng; Zhongzheng Lin; Yuanhui Wen; Hongjia Chen; Yujie Chen; Siyuan Yu

doi:10.1364/OE.482361

1. Introduction

Optical vortex beams, carrying orbital angular momentum (OAM), were first demonstrated in 1992 [1], which can be quantitatively described by the spiral phase structure $\exp \left ( {i\ell \theta } \right )$, where $\theta$ is the azimuthal angle and $\ell$ represents the topological charge (TC). Due to intriguing properties of vortex beams such as phase singularity and orthogonal OAM states, it has shown promising potential applications in various fields including optical communication [2], optical manipulation [3,4], optical imaging [5,6] and quantum information [7]. In general, the value of TC ($\ell$) is an integer, and thus the spiral phase is an integer multiple of $2\pi$ corresponding to a single order of OAM states. However, TC ($\ell$) can also be taken as a non-integer, referring to fractional vortex beams that can be expressed as a series of coherent superposition integer OAM states with different weights [8]. The fractional vortex beam was first analyzed theoretically as propagating with an intricate-phase structure comprising chains of alternating charge vortices [9], followed by an experimental demonstration of the evolution of the vortex structure [10]. Since then, works have sprung up on the generation and the detection of fractional vortex beams [11–20], as well as those used to generalize the quantum theory of rotation angles [21] and to mimic the realization of the Hilbert’s Hotel paradox [22].

Fractional OAM is a useful complement to traditional integer OAM and has been showing various promising potential applications. Compared with the integer vortex beam that usually achieve a rotation of a particle on the light ring, the fractional vortex beam can use fractional OAM to either induce or hinder and even stop a particle’s rotation due to the unique radial opening ring intensity distribution, which has been demonstrated to have potential applications in the field of particle guidance or cell orientation [23,24]. The fractional vortex beam is available for studying high-dimensional quantum entanglement [25]. It has been theoretically demonstrated that half-integer OAM modes may transfer stably in specific anisotropic optical fibers [26]. The free space optical communication system based on fractional vortex modes has also been reported as one of candidates for increasing communication capacity [27]. The asymmetric intensity distribution of fractional vortex beams has been used for image processing [28].

Flexible conversion between different TC orders of OAM states is key to facilitate OAM-based applications. Among several developed approaches for OAM mode conversion, the use of optical coordinate transformation manifests its advantage with high efficiency while low complexity [29–33]. One conversion scheme harnessing the optical coordinate transformation for fractional vortex beams is to implement OAM mode multiplication and division by using a two-step log-polar coordinate transformation combined with fan-out elements [34,35], and another one is to apply a circular-sector transformation into a single optical element [36,37]. However, the limitation of the above two schemes is that only integer OAM mode conversion with integer factors can be realized. In 2020, Y. Wen et al. changed the angular phase gradient of the OAM beam via azimuth scaling based on spiral coordinate transformation [38], and the arbitrary multiplication and division of the OAM of light can thus be realized in theory. With that, a special fractional vortex beam with its cross-sectional spiral intensity distribution and a phase discontinuity in the radial direction can be generated when the resulting OAM has a non-integer TC, rather than an opening ring of the intensity and an azimuthal phase jump for any conventional fractional OAM modes. The rich physics behind such a novel spiral fractional vortex beam, as well as its connection to those conventional fractional vortex beams [8], is unclear yet, though they all carry non-integer OAM.

In this paper, we investigate the generation, propagation characteristics, and angular momentum density of the spiral fractional vortex beam and propose a novel scheme based on the spiral coordinate transformation by superimposing a spiral phase piecewise function to realize the conventional fractional vortex beam with an azimuthal phase jump at a certain polar angle, revealing the relationship between the spiral fractional vortex beam and its conventional counterpart. The optical intensity and phase distributions for both the spiral fractional vortex beam and the conventional one are demonstrated in simulation and experiment. The results show that for the spiral fractional vortex beam, the spiral intensity distribution of its cross-section will evolve into a focusing annular pattern, which is differentiated from the opening ring intensity distribution of the conventional fractional vortex beam. In addition, the conversion from the radial phase jump to the azimuthal phase jump developed in this work can theoretically, when sharing the same azimuthal scaling factor, be applicable to fractional OAM modes with different TC orders related to the given spiral phase piecewise function.

2. Theory and analysis

The azimuth-scaling spiral transformation is performed by mapping the spiral coordinates of the input and output planes to achieve the scaling of the spiral phase gradient of OAM modes. The whole transformation process can be implemented by two phase mask plates. The first phase plate is for performing the spiral coordinate transformation, referred to as the optical transformer, and the second one for correcting the phase distortion induced by free-space propagation, referred to as the phase corrector. Assuming that the position parameter along the logarithmic spiral of the input and output OAM modes are $\left ( {{r_1},{\theta _1}} \right )$ and $\left ( {{r_2},{\theta _2}} \right )$, respectively, the corresponding coordinate mapping relationship can be expressed as

(1)$${\theta _1} = \theta + 2m\pi,\,{\rm{ }}{\theta _2} = {\theta _1}/n,\,{\rm{ }}{r_1} = {r_0}\exp \left( {a{\theta _1}} \right),\,{\rm{ }}{r_2} = r_0^{1 + \frac{1}{n}}r_1^{ - \frac{1}{n}} = {r_0}\exp \left( { - a{\theta _2}} \right),$$

where $\theta$ is the polar angle, $m$ an integer to represent the number of spiral turns, $n$ the scaling factor, ${r_0}$ the radial position at $\theta = 0$, and $a$ the parameter of the logarithmic spiral. From Eq. (1), we can notice that the spiral on the output plane will reverse the rotation after the spiral coordinate transformation mapping. Two phase masks can be expressed as [38]:

(2)$$Q({r_1},{\theta _1}) = \frac{k}{d}\left\{ {r_0^{1 + {n^{ - 1}}}\frac{{r_1^{1 - {n^{ - 1}}}}}{{1 - {n^{ - 1}}}}\cos \left[ {\left( {1 - {n^{ - 1}}} \right){\theta _1}} \right]} \right\} - \frac{{kr_1^2}}{{2d}},$$

(3)$$P\left( {{r_2},{\theta _2}} \right) ={-} \frac{k}{d}\left\{ {\frac{{r_0^{\left( {1 + n} \right)}r_2^{1 - n}}}{{n - 1}}\cos \left[ {\left( {n - 1} \right){\theta _2}} \right]} \right\} - \frac{{kr_2^2}}{{2d}},$$

where $k = \frac {{2\pi }}{\lambda }$ is the wave number and $d$ is a distance between the two phase planes. The incident OAM mode is selected as the perfect vortex beam, and the field distribution in the initial plane can be described as [39]:

(4)$${U_0\left( {r,{\theta}} \right) = \exp \left[ { - \frac{{{{\left( {r - {r_{\max }}} \right)}^2}}}{{w_0^2}}} \right]\exp \left( {i\ell{\theta}} \right)},$$

where ${r_{\max }}$ is a parameter determining the beam radius, and ${w_0}$ is the beam waist. In order to clearly illustrate, we investigate the propagation dynamics, the complex amplitude distributions and the angular momentum density of the output fractional and integer OAM modes obtained by the spiral transformation in free space using the angular spectrum diffraction integral [40]. Considering an $\vec{x}$-polarized field $\vec {A} = U(x,y,z) \text{exp}[i(kz - {\omega} t)] \hat{e}_x$, where $U(x,y,z)$ is the output field distribution by the spiral transformation, the time-averaged angular momentum density can be defined as [41]

(5)$$\left\langle {\vec J} \right\rangle = \vec r \times {\varepsilon _0}\left\langle {\vec E \times \vec B} \right\rangle = \vec r \times \frac{{{\varepsilon _0}}}{2}\left[ {i\omega \left( {U{\nabla _ \bot }{U^*} - {U^*}{\nabla _ \bot }U} \right) + 2\omega k{{\left| U \right|}^2}{{\hat e}_z}} \right],$$

where ${\nabla _ \bot } = \frac {\partial }{{\partial x}}{\hat e_x} + \frac {\partial }{{\partial y}}{\hat e_y},$ $\varepsilon _0$ is the electric permittivity of vacuum, $\omega =ck$ is the angular frequency, ${{U^*}}$ is the conjugate of $U$. The parameters for the simulation results were selected as ${r_{\max }} = 300\, \mu m$, ${w_0} = 100\, \mu m$, ${r_0} = 320 \, \mu m$, $a = \ln \left ( {1.2} \right )/2\pi$, and $d = 4.95\, mm$ in Fig. 1.

Fig. 1. (a) The propagation dynamics of the output OAM modes with the same scaling factor of n=1/2 and different input TC $(\ell =3, 4, 5)$. (b) The complex amplitude distributions of the output OAM modes. (c) The angular momentum density distributions (background) and the directions (arrows) for the transverse angular momentum density of the output OAM modes.

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The output OAM mode generated by the azimuth-scaling spiral transformation has the TC $(n\ell )$, which is determined by the scaling factor $n$ and the incident OAM $(\ell )$. The conversion between arbitrary integer input and output OAM modes can be achieved when the resulting TC $(n\ell )$ is an integer. However, the OAM modes generated by the spiral transformation when the resulting OAM is a non-integer TC are interesting as well, though being different from the integer OAM mode with a spiral phase structure defined in polar coordinates. As shown in Fig. 1(b) for the case n=1/2 with different incident TC $(\ell =3, 4, 5)$, the resulting fractional OAM modes generated in the initial plane $(z=0)$ have the spiral intensity distributions and radial phase jumps, which we refer to as spiral fractional vortex beams. This novel fractional vortex beam occurs due to the output phase distribution varying continuously along the spiral strip, which leads to the phase discontinuity at the boundary of the spiral strip when the resulting OAM is a non-integer. It can easily see from Fig. 1(a) that the spiral fractional vortex beam will be focusing and remain at a certain focal length during the propagation, which is quite different from the integer vortex beam. Examples are shown in Fig. 1(b) at the position $(z=15\, cm)$, the intensity distribution of the spiral fractional vortex beam evolves into a circular pattern and the phase distribution of the spiral fractional vortex beam is unstable during its propagation. It is shown that the phase distribution of the focusing annular pattern will transform into the phase distribution of the lower-order integer vortex beam. The angular momentum densities and directions of the output OAM modes by spiral transformation are illustrated in Fig. 1(c). It is easily observed that the angular momentum density distribution of the spiral fractional vortex beam at the initial plane $(z=0)$ is similar to that of the integer vortex beam, which demonstrates it also carries orbital angular momentum. Intriguingly, the angular momentum density of the spiral fractional vortex beam at the focus position $(z = 15\, cm)$ is mainly distributed in the outer ring where the intensity distribution is weak. The angular momentum density of the spiral fractional vortex beam gradually flows from the main lobe to the side lobe during propagation, which is caused by its phase discontinuity in the radial direction.

In order to further reveal the relationship between spiral fractional vortex beams and conventional fractional vortex beams, we propose a novel scheme based on the spiral coordinate transformation by superimposing a spiral phase piecewise function to convert the radial phase jump to the azimuthal phase jump. The spiral mapping relationship between the input and output planes of the partial spiral strip is shown in Fig. 2(a), given that the incident OAM mode with TC of $(\ell =1)$ and the scaling factor of $n=3/2$, as an example. It is noted that the phase information of the three-turn spiral extracted from the input plane is compressed to two turns and rotated inward by the spiral transformation. As shown in Fig. 2(a), a single spiral transformation leads to the special phase distribution of the spiral fractional vortex beam having the phase discontinuity in the radial direction. The phase information extracted from the incident integer OAM along the spiral is angular phase continuity, and the phase distribution of the resulting fractional OAM generated by applying the azimuth-scaling spiral transformation has a single-turn phase information that is not an integer multiple of $2\pi$ but is still angular phase continuity. Therefore, the position of the phase jump is transferred to the radial direction. The phase distribution of the conventional fractional vortex beam around the singularity shows the azimuthal phase jump at a certain polar angle but radial phase continuity, which forms a notched dark line at the position of the phase jump in the light field. By analogy, the radial phase jump in the phase distribution of the spiral fractional vortex beam also causes a dark field distribution at the position of the phase jump, forming a special spiral intensity distribution. Here, we superimpose a $\pi$ phase shift on the partial position of the input spiral strip [Fig. 2(a)], so that the spiral phase information extracted from the input plane produces an azimuthal phase jump. In this way, the phase distribution of the output spiral strip is adjusted to the radial phase continuity as the conventional fractional vortex beam. Further, we formulate the above phase superposition as a spiral phase piecewise function and expand this process to other areas to cover the entire light field. The spiral phase piecewise function is added to the optical transformer in order to modify the phase distribution of the input integer OAM mode. The phase function of the phase corrector does not require modification because the phase shift induced by the propagation is unchanged. The spiral phase piecewise function can be written as

(6)$$f({\theta _1}) = \left\{ {\begin{array}{cc} {2\pi \ell} & {0 \le {\theta _1}\bmod 2\pi tn < 2\pi n}\\ {2\pi \left( {1 - n} \right)\ell} & {2\pi n \le {\theta _1}\bmod 2\pi tn < 4\pi n}\\ {2\pi \left( {1 - 2n} \right)\ell} & {4\pi n \le {\theta _1}\bmod 2\pi tn < 6\pi n}\\ {\cdots} & {\cdots}\\ {2\pi \left( {1 - \left( {t - 1} \right)n} \right)\ell} & {2\pi \left( {t - 1} \right)n \le {\theta _1}\bmod 2\pi tn < 2\pi tn} \end{array}} \right\},$$

where $n$ is the scaling factor that is a fraction, with $t$ the denominator of $n$, and $\ell$ the TC. We can find that the input TC orders $\ell$, $\ell +t$, and $\ell +2t$ have the same representation in the normalized piecewise function in Eq. (6). It demonstrates that for incident OAM modes with integer TC orders $\ell +Mt$, where $M$ is an arbitrary integer, conventional fractional vortex beams can be converted by superimposing a spiral phase piecewise function with the same scaling factor $n$. In other words, a spiral phase piecewise function can be used for establishing the relationship between spiral fractional vortex beams and conventional fractional vortex beams for different TC orders, rather than just a one-to-one correspondence. Being as exemplified visualization, Fig. 2(b) shows the phase distributions of the spiral phase piecewise functions with different scaling factors.

Fig. 2. (a) The generation scheme for two types of fractional vortex beams with scaling factor of $n=3/2$. (b) The phase distributions of spiral phase piecewise functions with different scaling factors of $n=1/2, 4/5, 3/2, 4/3$.

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In order to verify the validity of the proposed scheme, we investigate the complex amplitude distributions and OAM power spectra for both the spiral fractional OAM mode and the conventional one with different scaling factors in simulations. Figure 3 compares the complex amplitude distributions of the two types of fractional vortex beams with the same TC order. The field distributions of spiral fractional vortex beams show the spiral optical intensity distributions and radial phase jumps in Fig. 3(a). It is worth mentioning that the spiral intensity distribution is most obvious when the spiral fractional OAM mode is half-integer. As shown in Fig. 3(b), the phase distributions of conventional vortex beams exhibit the azimuthal phase jumps, which lead to the radial opening ring intensity distributions. More intriguingly, the radially notched dark field of the conventional fractional vortex beam is most distinct with a half-integer TC. The OAM power spectra show that the transformed beam with fractional topological charge can also be decomposed into a superposition of integer OAM modes with different weights, which is similar to the characteristics of the conventional fractional OAM states. It is also further demonstrated that the scheme of superimposing the spiral phase piecewise function on the spiral transformation can convert the radial phase jump to the azimuthal phase jump. Besides, the spiral phase piecewise function can be a bridge to establish the connection between spiral fractional vortex beams and conventional fractional vortex beams.

Fig. 3. The complex amplitude distributions and OAM power spectra of (a) spiral fractional vortex beams and (b) conventional fractional vortex beams, both with the input OAM mode [$\ell =1$, ${r_{\max }} = 300\, \mu m$, ${w_0} = 100\, \mu m$] and scaling factors of $n=1/2, 4/5, 4/3, 3/2$.

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Moreover, we analyzed the far-field behavior of two types of fractional vortex beams. Figure 4 shows the far-field intensity, phase distribution and OAM power spectra of the two types of fractional vortex beams with the same TC order. It can be seen that the far-field distribution of the conventional fractional vortex beam generated by superimposing the spiral phase piecewise function on spiral transformation shows common features with the previously reported work [42,43]. The topological properties of the conventional fractional vortex beam are illustrated by the difference between the intensity and phase distribution in the near field and far fields, respectively. The intensity distribution of the conventional fractional vortex beam changes from the original homogenous notched annular pattern to a special symmetric petal-like spot shown in Fig. 4(a1). In contrast, the far-field intensity distribution of the spiral fractional vortex beam is much akin to that of the near field. It is worth mentioning that, as shown in Fig. 5(b1), when the topological charge is larger than a half-integer, the intensity distribution is closer to the ring pattern, and when it is smaller than a half-integer, the intensity distribution shows a clear spiral intensity distribution. The far-field phase distributions of the spiral fractional vortex beam also keep the radial phase jumps shown in Fig. 4(b2). By comparing Fig. 3 and Fig. 4, the OAM power spectra of two types of fractional vortex beams are almost the same for both the near and far fields.

Fig. 4. The far-field intensity, phase distributions, and OAM power spectra of (a1)-(a3) conventional fractional vortex beams and (b1)-(b3) spiral fractional vortex beams. All parameters are selected the same as those in Fig. 3.

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Fig. 5. (a) The complex amplitude distributions, and (b) OAM power spectra of the spiral fractional vortex beam at different transverse planes with the input OAM mode [$\ell =1$, ${r_{\max }} = 300\, \mu m$, ${w_0} = 100\, \mu m$] and scaling factors of $n= 3/2$.

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To observe the evolution of the non-integer topological charge during propagation, we investigate the complex amplitude distribution and OAM power spectra of a spiral fractional vortex beam in different transverse planes. As shown in Fig. 5(b), we notice that the spiral phase of the main lobe of the spiral fractional vortex beam will evolve during propagation, of which auto-focusing propagation property attributes to. Nevertheless, our results demonstrate that the OAM power spectrum of the spiral fractional vortex beam does not evolve with different propagation distances shown in Fig. 5(a) because of OAM conservation during propagation, which is consistent with the conclusion reached in the literature [44].

In addition, we further investigated the propagation characteristics of the spiral fractional vortex beam generated bwith different parameters. We select three representative parameters to demonstrate their effect on the propagation dynamics of spiral fractional vortex beams. Figure 6 shows the side view of the propagation dynamics and the normalized maximum intensity distributions of spiral fractional vortex beams in free space generated with different parameters. The transformed beam by applying the azimuth-scaling spiral transformation requires a low-pass spatial filtering system to extract the OAM modes in the spatial frequency domain. The parameters $f_1$ and $f_2$ are the focal lengths of the two thin lenses in the $4f$ low-pass spatial filtering system, respectively. As shown in Fig. 6(a), the transformed beams with different fractional topological charges propagate with different focusing ring widths. The parameter $r_0$ has the possibility to control the focal depth of the spiral fractional vortex beam during propagation [Fig. 6(b)]. It is worth mentioning that the parameters $f_1$ and $f_2$ not only control the spot size of the transformed beam but also have the ability to manipulate the focus position and focal depth of the beam during propagation [Fig. 6(c)].

Fig. 6. The propagation dynamics and the maximum intensity distributions of spiral fractional vortex beams with different parameters. The parameters are chosen as (a) $n\ell =3/2, 5/2, 7/2$, $r_0=320\, \mu m$, $f_1=45\, mm,\, f_2=100\, mm$, (b) $n\ell =3/2$, $r_0=280,\,310,\, 340\, \mu m$, $f_1=45\, mm,\, f_2=100\, mm$, and (c) $n\ell =3/2$, $r_0=320\, \mu m$, $f_1=45\, mm,\, f_2=75, 100, 125\, mm$.

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3. Experimental demonstration and discussions

The optical intensity distributions of these two types of fractional vortex beams have been further verified by experiments using a diffractive optical element (DOE) device [38]. The fabrication process of the DOE device mainly includes maskless optical lithography and reactive ion etching. The phase distributions of two masks can be implemented via etching different pixelated depths patterned on a double-sided quartz plate for engineering the incident optical wavefront. The parameters of the DOE device are selected as the quartz thickness $d = 4.95\, mm$, the refractive index $n _0 = 1.44$, the pixel size $p = 3\, \mu m$, and the wavelength $\lambda = 1550\, n m$. Figure 7(a) shows an illustration of the fabricated device for generating two types of fractional vortex beams. The phase distributions of the optical transformers for two types of fractional OAM mode conversions are shown in Fig. 7(b). Figure 7(c) shows the schematic of the experimental setup for device characterization. The phase masks loaded on the spatial light modulator are computer-generated holograms for complex amplitude modulation. The $4f$ filter system is used to select the positive first-order fringe of the beam passing through the Fourier plane, producing an input perfect vortex beam. After passing through the fabricated DOE device, the output fractional OAM mode obtained by low-pass spatial filtering of the output of the spiral transformation stage is observed by the camera.

Fig. 7. (a) Several sets of the spiral fractional OAM and conventional fractional OAM mode conversion devices with different scaling factors of $n=1/2, 3/2, 4/3$. (b) The phase distributions of the optical transformers for the spiral fractional OAM mode (left) and the conventional one (right) with the scaling factor of $n=3/2$. (c) Experimental setup of the specific optical system. BE, beam expander; HWP, half wave plate; SLM, spatial light modulator; BS, beam splitter; M, mirror; four thin lenses L1, L2 (${f_0} = 10\, cm$), L3 (${f_1} = 4.5\, cm$), L4 (${f_2} = 10\, cm$); CA, circular aperture; CCD, charge-coupled device (camera).

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The experimental results of two types of fractional OAM modes with different scaling factors are shown in Fig. 8. We can see in Fig. 8(a), that the spiral fractional OAM modes generated by the spiral transformation show spiral optical intensity distributions similar to the previous simulation results. In contrast, the conventional fractional vortex beams present radial opening ring intensity distributions, as expected, by superimposing spiral phase piecewise functions based on spiral transformation [Fig. 8(b)]. The experimental results thus verify the feasibility of the proposed scheme, which converts the radial phase jump to the azimuthal phase jump and reveals the connection between the spiral fractional vortex beam and its conventional counterpart.

Fig. 8. Experimental results of the intensity distributions of (a) the spiral fractional OAM modes, and (b) the conventional fractional OAM modes, both with ${r_{\max }} = 377\, \mu m$, ${w_0} = 87\, \mu m$, different input OAM modes [$\ell =3, 3, 4$], and scaling factors of $n=1/2, 3/2, 4/3$.

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Finally, we investigate the propagation properties of two types of fractional vortex beams in simulation and experiment. The parameters of the incident perfect vortex beam for simulation and experiment are ${r_{\max }} = 300\, \mu m$, ${w_0} = 100\, \mu m$, $\ell =3$ and the scaling factor is $n=1/2$. Figure 9(a) shows the propagation dynamics of the spiral fractional vortex beam generated by a single spiral transformation at a certain distance. We can see that the spiral fractional vortex beam with a spiral intensity distribution will be focusing during its propagation. When propagating to the position $z=10\, cm$, the focused spot shows a circular intensity distribution as presented in Figs. 9(b) and 9(c). The experimental results are in close agreement with the corresponding simulation results. The propagation dynamics of the conventional fractional vortex beam converted by superimposing the spiral phase piecewise function using spiral transformation is shown in Fig. 9(d). We can see from Fig. 9(e) that the phase distributions of the conventional fractional vortex beam show the azimuthal phase jumps. As shown in Fig. 9(f), the conventional fractional vortex beam can keep a radial opening ring intensity distribution over a certain propagation distance. The feasibility of this new scheme for converting the radial phase jump to the azimuthal phase jump is well demonstrated both in simulation and experiment.

Fig. 9. (a) The propagation dynamics of the spiral fractional OAM mode. (d) The propagation dynamics of the conventional fractional OAM mode. (b) and (e) Simulation results of the transverse complex amplitude distribution at the corresponding positions marked by the red lines in (a) and (d). (c) and (f) Experimental results of the transverse intensity distribution at different positions are consistent with the simulation.

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4. Conclusions

In conclusion, we analyze the generation, propagation characteristics, and angular momentum density of the spiral fractional vortex beam and propose a novel scheme by superimposing the spiral phase piecewise function on spiral transformation to convert the radial phase jump to the azimuthal phase jump, highlighting the connection between the spiral vortex beam and its conventional one. The focus position and focal depth of the spiral fractional vortex beam can be controlled by selecting different parameters. The spiral fractional vortex beam with a phase discontinuity in the radial direction is generated by a single spiral transformation, which can carry orbital angular momentum and evolve into a focusing annular pattern during its propagation in simulations and experimental results. In contrast, the conventional fractional vortex beam generated by superimposing the spiral phase piecewise function on spiral transformation can sustain a radial opening ring distribution over a certain propagation distance. Our work opens an avenue for fractional vortex beams which may lead to practical applications in optical information processing and particle manipulation.

Funding

National Key Research and Development Program of China (2019YFA0706302); Basic and Applied Basic Research Foundation of Guangdong Province (2021B1515020093, 2021B1515120057); Local Innovative and Research Teams Project of Guangdong Pearl River Talents Program (2017BT01X121).

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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Spiral fractional vortex beams

Abstract

1. Introduction

2. Theory and analysis

3. Experimental demonstration and discussions

4. Conclusions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (9)

Equations (6)

Optics Express