Longitudinal evolution of phase vortices generated by rotationally interleaved multi-spiral

Jinxin Wang; Xi Yang; Pengfei Li; Pengfei Li; Li Ma; Li Ma

doi:10.1364/OE.520505

1. Introduction

Optical vortices were discovered in 1974 [1] and created in 1989 [2], which have been developing for almost 30 years since the birth of optical vortices [3]. An optical vortex possesses a spiral wave front taking the form of $\textrm{exp} (il\varphi )$ and a phase singularity, where $\varphi$ is the azimuthal coordinate, and l is an integer quantum number and called the topological charge (TC), which have drawn much attention due to its carrying orbital angular momentum (OAM) [4–6]. The peculiar optical features of optical vortices have attracted increasing interest in fields such as micromanipulation [7], optical communication [8], optical encryption [9], optical tweezers [10], optical trapping [11], quantum science [12], and metasurfaces [13]. Thus, numerous studies have emerged on the generation [14–16], propagation [17–19] and detection [20–22] of vortex beams.

A considerable number of researches on optical vortices predominantly center on the generation and examination of topological charges at the focal plane or the other two-dimensional plane, such as measuring topological charges by interferential deconstruction [23], trapping particles by focused optical vortices [24], and mapping vortex lattices by metasurfaces [25]. However, there exists a notable gap of studies dedicated to investigate the longitudinal evolution of phase vortices, which is a matter of great significance in optical communication, encryption, particle capture, and related applications. Recently, significant research has delved into the transmission dynamics within nonlocal nonlinear media [26–28]. Additionally, several studies have investigated the variations in topological charges produced by single spiral over propagation distance [29,30]. While attentions have been directed towards the longitudinal properties in these notable works, a comprehensive exploration of the detailed longitudinal evolution properties of phase vortices, encompassing not only topological charges but also other relevant characteristics, remains conspicuously absent.

Structured spirals represent a viable method for generating optical vortices. Since the early 1990s, many approaches have been developed for producing optical vortex beams [31,32]. These include the uses of spiral phase plates (SPP) [33,34], computer-generated holograms (CGH) [35–37], cylindrical mirrors [38,39], Q-plates [40], and metasurfaces [41,42]. Nevertheless, the drawbacks for these methods are inevitable, such as the complex optical systems for CGH and cylindrical mirrors, hard fabrications for SPP and metasurface, and low resolutions for CGH. Among these, we introduce the spiral multi-pinhole plates as an important method, utilizing the spatial distributions of spiral pinholes to modulate the light field. Despite the inherent loss of partial incident energy, this method offers advantages such as cost-effectiveness, flexible adjustment, convenient production, and a high energy threshold. Consequently, it emerges as a significant alternative for optical vortex generation.

There exist various forms of spiral multi-pinhole plates, primarily depending on the arrangement of pinholes. While most researches focus on manipulating the TCs and OAM spectrum through the adjustment of spiral pinhole structures, it is crucial to consider the impact of radially asymmetric spiral on the generation of higher-order vortices. Notably, the centroids of vortices generated by a single spiral are farther from the origin of coordinates compared to those generated by multi-spiral [43]. Considering this, symmetric spiral arrays, specifically the rotationally interleaved multi-spiral, prove more suitable for generating high-order vortices. In contrast to previous notable studies focusing on the longitudinal properties based on a single spiral [29,30], the approach of rotationally interleaved multi-spiral for investigations of the longitudinal evolution of phase vortices has never been reported. Consequently, our study adopts this structure and further investigates the longitudinal properties of phase vortices.

In this work, we investigate the longitudinal variations of TCs generated by varying numbers of rotationally interleaved multi-spiral, concurrently examining the evolution of phase singularities. Initially, we provide a theoretical introduction to rotationally interleaved multi-spiral and obtain analytical expressions charactering the vortex field. Subsequently, through numerical simulations, we demonstrate the dependency of longitudinal TCs on both propagation distance and the number of multi-spiral. Additionally, we investigate the spatial variations of the generated phase singularities, denoted as average off-axis distance, via the propagation distance. In conclusion, we offer a succinct summary of our analytical results and research findings.

2. Theoretical analysis

We start our analysis by examining waves modulated by a single spiral. This spiral configuration consists of N small apertures arranged in a complete spiral, ensuring that azimuthal alterations along the spiral induce a radial length variation, leading to a 2π integer multiple phase changes along the optical axis direction. A single spiral can be written as [43]

(1)$$\begin{array}{{c}} {{\rho _n} = {{\left( {\rho_0^2 + \frac{{l{z_0}\lambda {\varphi_n}}}{\pi }} \right)}^{1/2}} = {\rho _0} + \frac{{l{z_0}\lambda }}{{2\pi {\rho _0}}}{\varphi _n}}\\ {{\varphi _n} = \frac{{2\pi (n - 1)}}{N}} \end{array},$$

where n represents the nth small hole in a spiral, ${\rho _0}$ is the initial radius, l is a specific TC generated at the distance of z₀ from the spiral plate, $\lambda$ is the wavelength of the monochromatic wave. According to the scalar diffraction theory, the complex amplitude of the diffracted light field generated by the nth small hole at distance z ($z \ge {z_0}$) is

(2)$$\begin{aligned} {E_n}(r,\theta ,z) &= \frac{{{\rho _0}}}{{iz\lambda }}\textrm{exp} \left[ {ik\left( {z + \frac{{{r^2} + \rho_0^2}}{{2z}}} \right)} \right]\\ &\quad\int_{{\rho _0}}^R {T({\rho _n},{\varphi _n})d{\rho _n}\int_0^{2\pi } {\textrm{exp} \left[ {il\frac{{{z_0}}}{z}{\varphi_n} - i\frac{{2\pi r{\rho_0}\cos ({\varphi_n} - \theta )}}{{\lambda z}}} \right]} } d{\varphi _n}, \end{aligned}$$

where T(ρ_n, φ_n) is the transmittance function of the spiral. Assuming ${\varphi _n} = {\phi _n} - \pi$, the above equation can be written as

(3)$$\begin{aligned} {E_n}(r,\theta ,z) &= \frac{{{\rho _0}}}{{iz\lambda }}\textrm{exp} \left[ {ik\left( {z + \frac{{{r^2} + \rho_0^2}}{{2z}}} \right)} \right]\textrm{exp} \left( { - il\pi \frac{{{z_0}}}{z}} \right)\int_{{\rho _0}}^R T ({\rho _n},{\phi _n} - \pi )d{\rho _n}\\ &\quad\int_0^{2\pi } {\textrm{exp} \left( {il\frac{{{z_0}}}{z}{\phi_n}} \right)\textrm{exp} \left[ {i\frac{{kr{\rho_0}\cos (\theta - {\phi_n})}}{z}} \right]} d{\phi _n}. \end{aligned}$$

Applying the Jacobi-Anger expansion

(4)$$\textrm{exp} (iz\cos \phi ) = \sum\limits_{p ={-} \infty }^\infty {{i^p}{J_p}(z)\textrm{exp} (ip\phi )} ,$$

Equation (3) can be expressed as follows

(5)$$\begin{aligned} {E_n}(r,\theta ,z) &= \frac{{{\rho _0}}}{{iz\lambda }}\textrm{exp} \left[ {ik\left( {z + \frac{{{r^2} + \rho_0^2}}{{2z}}} \right)} \right]\textrm{exp} \left( { - il\pi \frac{{{z_0}}}{z}} \right)\int_{{\rho _0}}^R T ({\rho _n},{\phi _n} - \pi )d{\rho _n}\\ &\quad\int_0^{2\pi } {\sum\limits_{p ={-} \infty }^\infty {{i^p}{J_p}\left( {\frac{{kr{\rho_0}}}{z}} \right)\textrm{exp} (ip\theta )\textrm{exp} \left[ {i\left( {l\frac{{{z_0}}}{z} - p} \right){\phi_n}} \right]} } d{\phi _n}. \end{aligned}$$

Accordingly, the final complex amplitude $E(r,\theta ,z)$ at the observation plane can be regarded as the superposition of the N sub light sources. E(r, θ, z) consequently gives form as

(6)$$\begin{aligned} E(r,\theta ,z) &= \sum\limits_{n = 1}^N {{E_n}(r,\theta ) = } \frac{{{\rho _0}}}{{iz\lambda }}\textrm{exp} \left[ {ik\left( {z + \frac{{{r^2} + \rho_0^2}}{{2z}}} \right)} \right]\textrm{exp} \left( { - il\pi \frac{{{z_0}}}{z}} \right)\int_{{\rho _0}}^R T ({\rho _n},{\phi _n} - \pi )d{\rho _n}\\ &\quad\int_0^{2\pi } {\sum\limits_{p ={-} \infty }^\infty {{i^p}{J_p}\left( {\frac{{kr{\rho_0}}}{z}} \right)\textrm{exp} (ip\theta )\sum\limits_{n = 1}^N {\textrm{exp} \left[ {i\left( {l\frac{{{z_0}}}{z} - p} \right){\phi_n}} \right]} } } d{\phi _n}. \end{aligned}$$

Substituting the expression of ${\phi _n} = {\varphi _n} + \pi$ and ${\varphi _n}$ in Eq. (1) into Eq. (6) and considering the integration of the radius $\rho$ as 1, Eq. (6) can be written as

(7)$$\begin{aligned} E(r,\theta ,z) &= \frac{{{\rho _0}}}{{iz\lambda }}\textrm{exp} \left[ {ik\left( {z + \frac{{{r^2} + \rho_0^2}}{{2z}}} \right)} \right]\textrm{exp} \left( { - il\pi \frac{{{z_0}}}{z}} \right)\\ &\quad\sum\limits_{p ={-} \infty }^\infty {{i^p}{J_p}\left( {\frac{{kr{\rho_0}}}{z}} \right)\textrm{exp} (ip\theta )\sum\limits_{n = 1}^N {\textrm{exp} \left\{ {i\left( {l\frac{{{z_0}}}{z} - p} \right)\left[ {\frac{{2\pi (n - 1)}}{N} + \pi } \right]} \right\}} } d{\phi _n}. \end{aligned}$$

A detailed discussion about Eq. (7) is as follows. If ${z_0}l/z$ takes an integer, the summation expression on the second line of Eq. (7) should be [44]

(8)$$\sum\limits_{n = 1}^N {\textrm{exp} \left\{ {i\left( {l\frac{{{z_0}}}{z} - p} \right)\left[ {\frac{{2\pi (n - 1)}}{N} + \pi } \right]} \right\}} = \left\{ \begin{array}{l} N,\textrm{ }\frac{{{z_0}}}{z}l = p\\ 0,\textrm{ }\frac{{{z_0}}}{z}l \ne p \end{array} \right..$$

As we set ${z_0}l/z = p$, Eq. (7) can be consequently simplified as

(9)$$E(r,\theta ,z) = \frac{{N{\rho _0}}}{{z\lambda }}{i^{p - 1}}\textrm{exp} \left[ {ik\left( {z + \frac{{{r^2} + \rho_0^2}}{{2z}}} \right)} \right]\textrm{exp} ({ - ip\pi } ){J_p}\left( {\frac{{kr{\rho_0}}}{z}} \right)\textrm{exp} (ip\theta ),$$

where ${J_p}(kr{\rho _0}/z)$ represents the doughnut-shaped intensity profile and $\textrm{exp} (ip\theta )$ is the helical phase distribution of the optical vortex with a TC of p. Equation (9) indicates that the generation of a vortex with a higher-order TC of l at the transmission distance z leads to the concurrent generation of derivative vortices, with TCs smaller than l as z increases. This reveals the longitudinal evolution of the variable TCs of the vortex generated by a single spiral.

If ${z_0}l/z$ takes a fraction, we set the ${z_0}l/z$ as

(10)$$\frac{{{z_0}}}{z}l = INT\left( {\frac{{{z_0}}}{z}l} \right) + MOD\left( {\frac{{{z_0}}}{z}l} \right),$$

where INT(…) is a downward rounding function and MOD(…) is the extraction of decimal function. Submitting Eq. (10) into Eq. (7) and giving $INT({{z_0}l/z} )= p$, Eq. (7) is given as

(11)$$\begin{aligned} E(r,\theta ,z) &= \frac{{{\rho _0}}}{{z\lambda }}{i^{p - 1}}\textrm{exp} \left[ {ik\left( {z + \frac{{{r^2} + \rho_0^2}}{{2z}}} \right)} \right]\textrm{exp} ({ - i\pi p} )\textrm{exp} \left[ { - i\pi \times MOD\left( {\frac{{{z_0}}}{z}l} \right)} \right]\\ &\quad{J_p}\left( {\frac{{kr{\rho_0}}}{z}} \right)\textrm{exp} (ip\theta )\sum\limits_{n = 1}^N {\textrm{exp} \left\{ {iMOD\left( {\frac{{{z_0}}}{z}l} \right)\left[ {\frac{{2\pi (n - 1)}}{N} + \pi } \right]} \right\}} . \end{aligned}$$

There, we set the fractional terms of phase distribution in Eq. (11) as

(12)$${\varphi _{per}} = \textrm{exp} \left[ { - i\pi \times MOD\left( {\frac{{{z_0}}}{z}l} \right)} \right]\sum\limits_{n = 1}^N {\textrm{exp} \left\{ {iMOD\left( {\frac{{{z_0}}}{z}l} \right)\left[ {\frac{{2\pi (n - 1)}}{N} + \pi } \right]} \right\}} ,$$

which indicating the phase change in azimuthal one cycle from 0 to $2\pi$. Thus, when ${z_0}l/z$ is a fraction, the phase distribution of the light field on the observation plane is accompanied by a phase disturbance φ_per on the basis of the integer order vortex phase exp(ipθ).

Subsequently, we will delve into the case of waves modulated by rotationally interleaved multi-spiral. Figure 1 illustrates the schematic diagram depicting the propagation of optical phase generated by rotationally interleaved multi-spiral. The red dashed box inserted in Fig. 1 shows the arrangement of single spiral pinholes. Owing to the requirements for multi-spiral to satisfy phase modulation principles and maintain geometric symmetry, the mathematical expression for rotationally interleaved multi-spiral, characterized with a designated number of spirals (M), is expressed as follows [43]:

(13)$$\begin{array}{{c}} {{\rho _{sn}} = {{\left( {\rho_0^2 + \frac{{l{z_0}\lambda {\varphi_{sn}}}}{\pi }} \right)}^{1/2}} = {\rho _0} + \frac{{l{z_0}\lambda }}{{2\pi {\rho _0}}}{\varphi _{sn}}}\\ {{\varphi _{sn}} = \frac{{2\pi (n - 1)}}{N} + \frac{{2\pi m(s - 1)}}{l}} \end{array},$$

where s indicates the sth spiral and $s \le M$, rotation angle is $2\pi m(s - 1)/l$, m refers to the introduced phase difference of $m \times 2\pi$ between the radial adjacent spirals, and M follows the relationship of $M = 2\pi l/2\pi m = l/m$. With formula derivations, the final light field can be written as

(14)$$\begin{aligned} {E_M}(r,\theta ,z) &= \frac{{{\rho _0}}}{{iz\lambda }}\textrm{exp} \left[ {ik\left( {z + \frac{{{r^2} + \rho_0^2}}{{2z}}} \right)} \right]\textrm{exp} \left( { - il\frac{{{z_0}}}{z}\pi } \right)\\ &\quad\sum\limits_{p ={-} \infty }^\infty {{i^p}{J_p}\left( {\frac{{kr{\rho_0}}}{z}} \right)} \textrm{exp} ({ip\theta } )\textrm{exp} \left[ {i\left( {p - \frac{{{z_0}}}{z}l} \right)\left( {\frac{{2\pi }}{N} + \frac{{2\pi m}}{l}} \right)} \right]\\ &\quad\sum\limits_{n = 1}^N {\textrm{exp} \left[ {i\frac{{2\pi n}}{N}\left( {\frac{{{z_0}}}{z}l - p} \right)} \right]} \sum\limits_{s = 1}^M {\textrm{exp} } \left[ {i\frac{{2\pi ms}}{l}\left( {\frac{{{z_0}}}{z}l - p} \right)} \right]. \end{aligned}$$

When ${z_0}l/z$ is an integer and expressed as p, we have

(15)$$\sum\limits_{n = 1}^N {\textrm{exp} \left[ {i\frac{{2\pi n}}{N}\left( {\frac{{{z_0}}}{z}l - p} \right)} \right]} \sum\limits_{s = 1}^M {\textrm{exp} } \left[ {i\frac{{2\pi ms}}{l}\left( {\frac{{{z_0}}}{z}l - p} \right)} \right] = MN.$$

Then, Eq. (14) can be written as

(16)$${E_M}(r,\theta ,z) = \frac{{MN{\rho _0}}}{{z\lambda }}{i^{p - 1}}\textrm{exp} \left[ {ik\left( {z + \frac{{{r^2} + \rho_0^2}}{{2z}}} \right)} \right]\textrm{exp} ({ - ip\pi } ){J_p}\left( {\frac{{kr{\rho_0}}}{z}} \right)\textrm{exp} ({ip\theta } ).$$

Equation (16) shows that if z₀l is divisible by z, integer order TC of $p = {z_0}l/z = ({z_0}m/z)M$ can be generated at the propagation distance z, which also indicates that the generated integer order vortex needs to be a multiple of the number of multi-spiral.

Fig. 1. Schematic representation illustrating the propagation of optical vortices and the corresponding phase evolution generated by rotationally interleaved multi-spiral.

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When ${z_0}l/z$ is a fraction, submitting Eq. (10) into Eq. (14) and setting $p = INT({{z_0}l/z} )$, Eq. (14) can be written as

(17)$${E_M}(r,\theta ,z) = \frac{{{\rho _0}}}{{z\lambda }}{i^{p - 1}}\textrm{exp} \left[ {ik\left( {z + \frac{{{r^2} + \rho_0^2}}{{2z}}} \right)} \right]\textrm{exp} \left( { - il\frac{{{z_0}}}{z}\pi } \right){J_p}\left( {\frac{{kr{\rho_0}}}{z}} \right)\textrm{exp} ({ip\theta } ){\varphi ^{\prime}_{per}},$$

where

(18)$$\begin{aligned} {{\varphi ^{\prime}}_{per}} &= \textrm{exp} \left[ { - i \times MOD\left( {\frac{{{z_0}}}{z}l} \right)\left( {\frac{{2\pi }}{N} + \frac{{2\pi m}}{l}} \right)} \right] \times \\ &\quad\sum\limits_{n = 1}^N {\textrm{exp} \left[ {i\frac{{2\pi n}}{N} \times MOD\left( {\frac{{{z_0}}}{z}l} \right)} \right]} \sum\limits_{s = 1}^M {\textrm{exp} \left[ {i\frac{{2\pi ms}}{l} \times MOD\left( {\frac{{{z_0}}}{z}l} \right)} \right]} . \end{aligned}$$

Equation (17) indicates that when ${z_0}l/z$ is not an integer, it cannot generate integer order TC and the phase distribution is phase perturbation ${\varphi ^{\prime}_{per}}$ added to an integer order vortex ${z_0}l/z = ({{z_0}m/z} )M$.

It is important to note that, theoretically, a single spiral has the capability to generate a vortex with TC linearly decreasing along the transmission direction. However, the TCs generated by rotationally interleaved multi-spiral are subject to constraints imposed by the number of multi-spiral and present the characteristic of a step-wise reduction trend. The longitudinal evolution of phase vortices in this multi-spiral configuration are more intricate compared to those arising from a single spiral.

Moreover, based on above theoretical analyses, we can logically infer the characteristics of light fields when the transmission distance extends to be a large value as $z \gg {z_0}$. Consequently, as the parameter p will approach the value of $p = INT({{z_0}l/z} )\approx 0$, the vortex beam, whether generated by a single spiral and multi-spiral, exhibits concentric circular distributions as intensity profile represented by J₀(…) and phase profile being $\textrm{exp} [{ik{r^2}/({2z} )} ]$.

3. Numerical simulations and discussion

Based on the theoretical analysis presented in Section 2, we further perform the numerical simulations to discuss the longitudinal evolution of the optical phase distributions generated by rotationally interleaved multi-spiral. We selected the vortices with the TCs of l = 6 and l = 4 to investigate the evolutional properties along the propagation direction, as shown in Figs. 2 and 3. It should be noted that the vortices with specific topological charges are predetermined according to the spiral structures as expressed in Eqs. (1) and (13). For the following numerical examples, the relevant default parameters are set as: $\lambda = 532.9\textrm{nm},$ z₀= 1 m, ${\rho _0} = 3\textrm{mm},$ N = 72, and d = 0.015 mm, where d is the radius size of the small hole.

Fig. 2. The phase distribution of generated TC of l = 6 beam at different propagation distances using single spiral and rotationally interleaved multi-spiral with M = 2, respectively.

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Fig. 3. The phase distribution of generated TC of l = 4 beam at different propagation distances using rotationally interleaved multi-spiral with M = 2 and 4, respectively.

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Figure 2 shows the longitudinal evolution of the phase vortices with TCs specified as l = 6. The optical masks are presented on the right side, while the intensity patterns of the vortices generated at the specified distances z₀= 1 m with a topological charge l = 6 are shown on their left side. The vortex patterns produced by a single spiral with M = 1 are displayed above the axis, while the cases generated by multiple spirals with M = 2 are presented below the axis. During the analyses of longitudinal evolutions, the topological charge is determined by the loop integration of $l = \frac{1}{{2\pi }} \times \oint_C {\nabla \phi (\boldsymbol{r} )dr}$, indicating the phase distributions around the center, where C is a closed loop surrounding the singularities [3]. The phase vortices in the case of M = 1 hold the TCs of l = 6, 5, 4, 3, 2 and 1 at the propagation distances of z = 1 m, 1.2 m, 1.5 m, 2 m, 3 m and 6 m, respectively. It is evident that the TCs are linearly varied with the propagation distance. While the phase vortices in the case of M = 2, the TCs are l = 6, 4 and 2 at propagation distance z = 1 m, 1.5 m and 3 m, respectively. This observation presents the step-wise reduction trend in TCs with the propagation distance increase, resulting from the limitation of the multi-spiral numbers. In the process of step-wise TCs reduction, it notes that the phase singularities are first separated with rotation and then aggregated into new integer order TCs. Noticeably, at other propagation distances except for the specific ones, the phase vortices exhibit the spatially separated phase singularities and incomplete circular intensity profiles, which are consequently excluded from consideration.

The further simulated results are phase vortices with TCs of l = 4 generated by the rotationally interleaved multi-spiral of M = 2 and M = 4. Figure 3 shows the corresponding phase distributions at different propagation distances. The optical masks are depicted on the right side, while the adjacent positions on the left display the intensity patterns of the vortices at a distance of z₀= 1 m, each corresponding to a topological charge of l = 4. When the number of multi-spiral is M = 2, the generated vortices carry the TCs of 4 and 2 at propagation distances of z = 1 m and 2 m, respectively. Then the two singularities move away from the center after a propagation distance of z = 2 m. When the number of multi-spiral is M = 4, the generated phase vortex shows exclusively TC of l = 4 at the propagation distance of z = 1 m. The singularities initially rotate around the center and subsequently separate spatially with the propagation distance increase. The simulation results indicate that, in the longitudinal transmission direction, the generated TCs depend on both the propagation distance and the number of rotationally interleaved spirals, which is consistent with the analysis in the Section 2. It demonstrates that the generated TCs are multiples of the number of multi-spiral. While the more multi-spiral there are, the fewer vortices generated in the longitudinal direction.

The changes of the generated TCs are essentially resulted from the variations of the spatial distributions of the singularities. As the propagation distance increases, the spatial singularities accordingly change distributions, with associated singularities shift outward to be annihilation. Meanwhile, the number of the multi-spiral additionally restricts the quantity of the singularities, ensuring the consequent TCs as multiples of spirals. From the simulated phase diagrams as mentioned above, it can be intuitively seen that the phase singularities rotate and separate symmetrically around the center. To investigate the spatially variational distributions of the phase singularities, we introduce the average distance d_avg here as a parameter for further discussion. The d_avg refers to the average distance of all phase singularities to the center. Figure 4 shows the relationship between the average distance d_avg and the propagation distance, with the distance is taken as the logarithm of the x-axis. Figure 4(a) represents the longitudinal evolution of the average distance d_avg when generating TCs of 1, 2, 3, 4, and 6 using a single spiral. It finds that d_avg initially fluctuates and eventually increases linearly. The larger the TC generated, the more tortuous of the curve becomes, while the minimum corresponds exactly to the integer order TC. This also indicates that when the TC is an integer order, phase singularities are closest to the center. Figures. 4(b) and 4(c) display the average distance d_avg that varies with the propagation distance when using different rotationally interleaved multi-spiral to generate different TCs. When the average distance d_avg is minimum, it corresponds to integer order topological charges, which are the same as the TCs generated by a single spiral. When the number of multi-spiral M is equal to the generated TC of l, no other TCs are generated. Thus, the average distances d_avg are generally linear variation, as shown in Fig. 4(c).

Fig. 4. The relationship between the average distance d_avg and the propagation distance: (a) M = 1, l = 1, 2, 3, 4, 6, (b) M = 2, l = 2, 4, 6, (c) M = l = 1, 2, 3, 4, 6, respectively.

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Moreover, we consider one-dimensional cross-sectional lines of normalized intensity corresponding to the phase vortices in Figs. 2 and 3, as shown in Fig. 5. In case of a single spiral of M = 1 creating a TC of l = 6, the maximum intensity values linearly decrease as the propagation distance increase, as shown by the distinct color curves in Fig. 5(a). In the instances of multi-spiral of M = 2 generating TCs of l = 6 and l = 4, the intensity profiles exhibit fluctuant variations. Specifically, the first two maxima in Fig. 5(b) correspond to vortices observed at distances of z = 1 m and 1.5 m, while during the range of 1m < z < 1.5 m the intensities oscillate to the lower values. Similarly, the first two maxima in Fig. 5(c) coincide with vortices observed at distances of z = 1 m and 1.8 m, but the smaller intensities appear in the range of 1m < z < 1.8 m. This phenomenon is concurrently influenced by the designed TCs and propagation distances, which are distinctly contrast with the case of linear variations depicted in Fig. 5(a). Meanwhile, as the number of multi-spirals is M = 4 to generate a TC of l = 4, the intensity undergoes a rapid decline with the increase in propagation distance, as shown in Fig. 5(d). Additionally, the beam profile exhibits a swift departure from the characteristic hollow shape, observed notably at the distance of z = 1.4 m. It's notable that the peak heights in Fig. 5(a) differ between both sides, whereas in Figs. 5(b)–5(d), the peaks maintain consistent heights. This phenomenon arises from the structural geometric symmetries inherent in the spiral structures, wherein the multi-spiral configuration notably preserves centrical circular symmetry [43]. These findings demonstrate that the intensities generated by a single spiral exhibit a linear decrease as the propagation distance increases. In contrast, the vortices generated by the multi-spiral exhibit fluctuating variations along the distance z. Overall, the vortex intensities are subject to the combined influences of the generated TCs and propagation distances.

Fig. 5. Corresponding one-dimensional cross-sectional lines of normalized intensity at different propagation distances when (a) M = 1, l = 6, (b) M = 2, l = 6, (c) M = 2, l = 4, (d) M = 4, l = 4, respectively.

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The above simulation results demonstrated that the TCs of the phase vortices generated by rotationally interleaved multi-spirals present variation with respect to propagation distance. The TCs generated during the transmission process depend on both the number of multi-spirals and propagation distance. These results are consistent with the theoretical analyses in the Section 2. We additionally study the longitudinal evolution of the spatial phase singularities and intensity profiles, revealing a robust correlation with the generated TCs and propagation distance.

Optical vortices are well known that they have widespread applications across various fields, primarily due to special spatial intensity, phase, and OAM [3]. Specifically, by utilizing the intensity gradient force and OAM multiplexing, vortices play a pivotal role in optical particle manipulations and communication systems. Our investigations reveal that phase vortices, generated through rotationally interleaved multi-spiral, exhibit longitudinal variations in both TCs and singularities, thereby offering promising potential applications in optical tweezers and communication technologies.

4. Conclusion

In this study, we focus on the longitudinal evolution of phase vortices generated by rotationally interleaved multi-spiral and investigate the variations of the resulted TCs with the propagation distance increases. The theoretical analyses indicate that the TCs of phase vortices generated by multi-spiral are subject to the number of multi-spiral. Simulations of the different vortices, produced by diverse numbers of multi-spiral, demonstrate the joint influences of the propagation distance and the spiral number on the TCs and the spatial phase singularities in the longitudinal transmission direction. In the transmission direction, the TCs present the characteristic of a step-wise reduction trend, while the average off-axis distance d_avg of the phase singularities incrementally increases with the propagation distance, accompanying with a decline in intensity profiles. These results not only contribute valuable insights into capturing particles in the longitudinal direction but also hold potential relevance for research in vortex beam-related fields.

Funding

National Natural Science Foundation of China (11904211); Applied Basic Research Project of Shanxi Province, China (20210302124701); Technology Innovation Center Program of Changzhi (2022cx002).

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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Longitudinal evolution of phase vortices generated by rotationally interleaved multi-spiral

Abstract

1. Introduction

2. Theoretical analysis

3. Numerical simulations and discussion

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Equations (18)

Optics Express