## Abstract

Optical vortices, which carry orbital angular momentum, have attracted much attention in various research fields, such as materials processing, chirality control, and particle manipulation. A recent study experimentally confirmed that twisted fibers of polymerized photocurable resins with a constant period can be formed via irradiation by an optical vortex. It is suspected that this phenomenon is caused by the projection of the angular momentum of an optical vortex to the photocurable resin. The detailed mechanism of the growth of such peculiar fibers has not yet been clarified. In this study, which focuses on one aspect of polymerized structure formation, we develop a coarse-grained particle model in which the particle dynamics in the framework of the Rayleigh scattering theory involving light absorption is theoretically simulated. The period of the twisted fibers expressed using the coarse-grained particles is found to be in reasonable agreement with experimental values and independent of the input power of the laser. In addition, the shape of the polymerized fibers can be controlled by modulating the degree of light absorption.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Numerous studies have investigated optical vortices and their generation methods [1, 2] ever since Allen and his collaborators found that optical vortices have a polarization-independent orbital angular momentum [3, 4]. According to Maxwell’s electromagnetic field theory, light carries energy and momentum. It is also known that a mechanical torque causes the transfer of the angular momentum of a circularly polarized light beam [5]. Although both effects contribute to optical vortices [6], light beams with orbital angular momentum have the ring-like transverse intensity distribution and the helical wavefront due to which the light energy propagates along spiral orbits. Swartzlander Jr. [7] and his co-workers [8, 9] experimentally demonstrated the holographic formation of optical vortices and microparticle manipulation. A variety of fundamental properties and applications related to optical vortices have been presented [10–12].

It has recently been reported that twisted plastic fibers can be made by irradiating an optical vortex laser beam in a photocurable resin, whose chirality is modulated by the orbital angular momentum of light [13, 14]. In addition to plastic fibers, metallic and monocrystalline silicon needles have been produced using optical vortices [15, 16]. The transfer of orbital angular momentum from the laser beam to the liquid resin is suspected to cause the formation of such peculiar fibers. However, it was found that the scale of the periodic structure is quite different from the wavelength of light [14]. In one study, the orbital angular momentum of an optical vortex was transferred to nanoparticles that were manipulated by a surface plasmon enhanced with metallic nano-objects [17]. In this case, the optical vortex induced the swirling of nearby fields and then the electromagnetic fields that attracted nanoparticles caused particle transport. As a result, the particles showed a periodical swirling motion whose period did not agree with the frequency of light. The aforementioned optical phenomena are expected to open the door to new research fields in optics, materials science, and interdisciplinary areas. However, the detailed process of the transfer of the orbital angular momentum of light to materials have not been clarified yet.

As a theoretical model of photopolymerized twisted-fiber formation, the optical soliton [18] is believed to have an important role in the phenomenon. In a recent report, Lee et al. proposed the effect of optical vortex soliton on the helical fiber formation in a photocurable resin [19]. They stated that the refractive index of a photocurable resin was changed by the self-focusing of incident beams due to non-linear optical effects, which caused twisted-fiber formation according to the propagation of light. However, the primitive growth mechanism of solidified polymers has not yet been clarified. In this study, we theoretically discuss a nanoparticle generation process induced by the irradiation of optical vortex, where the orbital angular momentum of a Laguerre-Gaussian mode affects a driving force of nanoparticles cured at an interface of different indices of refraction. The present model suggests that the light absorption of the liquid resin has a possibility of force generation on photopolymerized nanoparitcles that may result in helical fiber structures by irradiating optical vortex. As shown in a previous report, irradiation of ultraviolet (UV) light in a photocurable resin is suspected to trigger a photopolymerization reaction that causes the formation of an interface between a liquid resin and a solid polymer. The polymerization causes a difference in the indices of refraction between the two phases. The optical forces acting on a solid body can be evaluated using Rayleigh scattering theory [20–23], where the effects of light absorption on a photocurable initiator that triggers the photochemical reaction are expressed by complex indices of refraction. Herein, we develop a coarse-grained model [24–26] to discuss the trajectories of polymerized nanoparticles in the primitive process of photopolymerization by irradiating optical vortices. The optical forces caused by an optical vortex act on the coarse-grained bead, enhancing the transfer of orbital angular momentum. The coarse-grained bead is launched onto a helical orbit by these forces, which represents the characteristic structure of twisted fiber. In particular, due to the complex part of the index of refraction, the gradient force suppresses the growth of twisted fibers.

As a consequence, the angular momentum of Laguerre-Gaussian-mode light is projected onto a twisted fiber, which is affected by the scattering and gradient forces of an optical vortex. It has been suggested that the period of the helical structure is determined by the ratio of both forces and is also affected by the complex component of the index of refraction that causes light absorption.

## 2. Theoretical model and numerical method

#### 2.1. Theoretical model of optical vortex

We assume that nanoparticles are generated when a photocurable liquid resin is cured. Gradient and scattering forces are induced when a laser beam hits the interface between liquid and solid phases with different refractive indices. A nanoparticle generated on the focal plane is pushed forward by optical forces; particles are successively generated and launched in the liquid. A resin fiber is elongated by repeating this process.

#### 2.2. Decay of optical intensity by absorption

Herein, we model the condition where radiated electromagnetic fields are partly absorbed when UV light is irradiated in a liquid resin. In such a case, the index of refraction is expressed by a complex number as

where*n*

_{re}and

*n*

_{im}are the real and imaginary parts of the index of refraction, respectively, and

*i*

is the imaginary unit. Here, we model the condition ${n}_{\text{re}}\gg {n}_{\text{im}}$. When an electromagnetic field propagates along the *z* direction in a medium, the corresponding wave vector is expressed as

*ω*and

*c*are the angular frequency and the speed of light, respectively. Considering Eq. (1), we construct Maxwell’s equations for propagating electromagnetic fields. In cylindrical coordinates, by applying the paraxial approximation, for the case of the Laguerre-Gaussian mode, we obtain the electric field

**E**and magnetic field

**H**. Note that

**E**and

**H**for any transversely limited light beam have not only transverse but also longitudinal components because of the Maxwell equations $\nabla \cdot \mathbf{E}=\nabla \cdot \mathbf{H}=0$. For Laguerre-Gaussian-mode light, these longitudinal components are related with the helical wavefront shape. In particular, the zeroth radial and the first azimuthal mode are expressed as follows:

Where

*ε*

_{0}is vacuum permittivity, and

*w*

_{0}is the beam waist that gives the Rayleigh range as follows: and the other functions are defined as and

Therefore, the light intensity **I** is derived from the time average of Poynting vector $\mathbf{S}\left(r,\theta ,z,t\right)\equiv \mathbf{E}\left(r,\theta ,z,t\right)\times \mathbf{H}\left(r,\theta ,z,t\right)$ as

*t*_{p} is the period of **S**, and ${\mathbf{I}}^{\prime}$ the light intensity without light absorption, as shown below. As described above, the laser beam propagates in a helical orbit because **I** has both azimuthal and axial components.

#### 2.3. Forces of optical vortex acting on a coarse-grained particle

The optical forces acting on the nucleated photocurable resin caused by the irradiation of Laguerre-Gaussian-mode UV light are reflected by the behavior of coarse-grained particles. When the diameter *d* of coarse-grained nanoparticles is much smaller than the wavelength *λ* of light, the behavior of the particles obeys Rayleigh scattering theory. In this study, we apply this theory to the particles in the range of $d/\lambda \lesssim 0.1$ based on our previous experimental works using Gaussian beams [27, 28] and another literatures [21, 22]. In this theory, the optical forces consist of the gradient force **F**_{grad} and the scattering force **F**_{scat}. **F**_{grad} is caused by the presence of adielectric particle in the gradient of time-averaged electric fields. Based on the paraxial approximation, the square of the *z* component of **E** expressed by Eq. (5) is much smaller than the other components and is thus ignored. The curvature of the wavefront along the *z*-axis, $R\left(z\right)$, is sufficiently large to satisfy the relation $\left({k}_{\text{im}}{r}^{2}\right)/R\left(z\right)\approx 0$. Furthermore, considering absorption, **F**_{grad} is expressed by the following equation

*α*is the polarizability of the coarse-grained dielectric particle that is represented by the relative dielectric constant of the particle

*ε*

_{p}and the relative dielectric constant of the resin

*ε*

_{f}as follows:

*ε*_{p} and *ε*_{f} are expressed by the complex index of refraction, such that ${\epsilon}_{\mathrm{r}}={n}_{\mathrm{r}}^{2}{\epsilon}_{0}$, where the subscript r is replaced by p for particle or f for liquid resin. *n*_{p} and *n*_{f}, which have their real and imaginary parts in the same manner as *n* in Eq. (1), are governed by the real part *n*_{re} that is always much larger than *n*_{im} as previously mentioned. Hereafter, we replace *n*_{re} for the liquid resin by *n*_{f} and define $\eta =2{k}_{\text{im}}=4\pi {n}_{\mathrm{f}}\beta /\lambda $ that is the absorption coefficient for the decay of light intensity as $\mathbf{I}={\mathbf{I}}^{\prime}\text{exp}\left(-\eta z\right)$. *η* depends on the wavelength *λ* and the degree of light absorption $\beta ={n}_{\text{im}}/{n}_{\text{re}}$. In this study, *β* is a parameter that is sufficiently smaller than 1 and in the range from $2.00\times {10}^{-6}$ to $2.00\times {10}^{-3}$. When we set the value of ${n}_{\mathrm{f}}=1.49$ as a physical property of NOA65 that is a photocurable resin used in a previous study [14], *η* is in the range from $9.24\times {10}^{1}$ m^{−1} to $9.24\times {10}^{4}$ m^{−1}. A coarse-grained particle is trapped near the ring with the strongest light intensity and forced to move in the negative *z* direction by **F**_{grad}, as shown in Fig. 1(a). Figure 1(b) shows the radial dependence of **F**_{grad} at $z=0\text{}\mu $m for the *r* component, ${F}_{\text{grad},r}$, and the *z* component, ${F}_{\text{grad},z}$. Note that ${F}_{\text{grad},r}$ and ${F}_{\text{grad},z}$ are normalized by maximum values of $\left|{F}_{\text{grad},r}\right|$ and $\left|{F}_{\text{grad},z}\right|$. It is found that the magnitude of ${F}_{\text{grad},z}$ has a maximum value at $r={w}_{0}/\sqrt{2}$.

In Eq. (13), the first and second terms of the *z* component correspond to the focusing and absorption of light, respectively. The former attracts a particle to the focal plane and the latter draws a particle in the negative *z* direction.

A photocurable resin consists of photocurable initiators and polymerizable monomers. When UV light is irradiated into the resin, the initiator is firstly excited by the absorption of light and triggers the polymerization process. That is, the light absorption is mainly caused by the photoinitiator before the excitation of monomers. The photoinitiator is assumed to be only in the liquid resin but not included in the polymerized particles. Thus, we do not consider the absorption of the particles. This is a reason why the absorption force is not taken into account for the polymerized particles. Based on this framework, *F*_{scat} can be calculated as follows:

*C*_{scat} is the scattering cross-section [22], and ${k}_{\mathrm{f}}=2\pi {n}_{\mathrm{f}}/\lambda $ is the wave number of light in the liquid resin. As a consequence, the scattering force consists of both *θ* and *z* components, as shown in Fig. 1(a), and thus a coarse-grained particle receives the force along a helical orbit. Figure 1(c) shows the radial dependence of **F**_{scat} at $z=0\text{}\mu $m for the *θ* component, ${F}_{\text{scat},\theta}$, and the *z* component, ${F}_{\text{scat},z}$. Note that ${F}_{\text{scat},\theta}$ and ${F}_{\text{scat},z}$ are normalized by maximum values of $\left|{F}_{\text{scat},\theta}\right|$ and $\left|{F}_{\text{scat},z}\right|$, respectively. The maximum values of the *θ* component of **F**_{scat} on $z=0\text{}\mu $m, ${F}_{\text{scat},\theta}$, and the *z* component, ${F}_{\text{scat},z}$, are placed at $r={w}_{0}/2$ and ${w}_{0}/\sqrt{2}$, respectively. In particular, the ratio of the*θ* component to the *z* component is $1/\left({k}_{\mathrm{f}}r\right)$. A coarse-grained particle is pushed out at a constant angle, being bounded near $r={w}_{0}/\sqrt{2}$ due to the gradient force.

#### 2.4. Coarse-grained model of photocurable resin

In the computer simulation, a continuous photopolymerization process of NOA65 monomers, whose properties are listed in Table 1, is simulated.

NOA65 includes benzophenone as a photocurable initiator and has a specific absorption coefficient *η* [29]. The indices of refraction for the liquid and the polymerized solid are $1.49$ and $1.52$, respectively [14], where UV light with a $405$-nm wavelength was used to induce the photoreaction. As shown in Fig. 2, a coarse-grained particle represents a polymerized product generated in a circular area where the intensity of light becomes the strongest on the focal plane at $z=0\text{}\mu $m.

In this simulation, a coarse-grained particle, which represents a tiny fragment of the polymerized resin, is thrown into a spiral orbit directed by **F**_{grad} and **F**_{scat}. Fragments are continuously generated at the focal plane and pushed forward, following previously launched fragments, in the optical vortex. Here, it is assumed that the photoreaction is limited to the focal plane on which the light intensity is strongest. With this continuous process, photocured particles are assumed to form a polymerized twisted object. In the actual system, the liquid resin is solidified by the optical vortex and as a result, a twisted helical object forms as the photoreaction proceeds.

The motion of a coarse-grained particle that obeys Newton’s equation of motion is expressed as follows:

*v*is the velocity of a coarse-grained particle. The friction caused by Stokes’ drag is expressed as $\xi v=3\pi \mu dv$. The mass of a coarse-grained particle is $M={\gamma}_{\mathrm{s}}{\rho}_{\mathrm{w}}\pi {d}^{3}/6$, where

*γ*

_{s}is the specific weight of NOA65 as a solid, and ${\rho}_{\mathrm{w}}=1.0\times {10}^{3}$ kg/m

^{3}is the density of water. We simulate the motion of particles for ${\gamma}_{\mathrm{s}}=1,2,5,$ and $10$, where

*γ*

_{s}is unclear in the real system. Figure 3 shows the velocity of a particle in the optical vortex computed with a time step of $\mathrm{\Delta}t=1\times {10}^{-14}$ s, where Figs. 3(a) and 3(b) show the

*r*component,

*v*, and the

_{r}*z*component,

*v*, respectively.

_{z}In Fig. 3, the velocity responses considering the acceleration according to Eq. (17) are shown by solid lines, which depend on *γ*_{s}. The overdamped condition, in which particles are always transported at terminal velocity, is described as follows:

*v*and

_{r}*v*sufficiently converge to the terminal velocity before $t=1\times {10}^{-11}$ s. This result indicates that the effect of the inertial force is negligibly small in Eq. (17) due to the small mass, even though acceleration may not be negligible in the optical vortex. In this study, the transport of the coarse-grained particles is modeled using the overdamped equation shown in Eq. (18) with a time step of $\mathrm{\Delta}t=1\times {10}^{-8}$ s, which was determined to be a suitable value for simulating the growth mechanism of the photocured object. Details of this kind of computer simulation can be found in the literature [26].

_{z}## 3. Results and discussion

The distributions of the light intensity of the Laguerre-Gaussian mode are analyzed in the *rz* plane for various values of *η*, as shown in Fig. 4, where the intensity was evaluated based on the magnitude $\left|\mathbf{I}\right|$ expressed by Eq. (11). Each intensity is normalized by the point of maximum value placed at $\left(r,z\right)=\left({w}_{0}/\sqrt{2},0\right)$.

Figure 4 shows the results for various values of *η*. In these results, $\left|\mathbf{I}\right|$ is radially distributed perpendicular to the *z*-axis; it becomes 0 W/m^{2} at $r=0\text{}\mu $m and reaches its maximum near $r=w\left(z\right)/\sqrt{2}$, which is typical for the Laguerre-Gaussian mode. When a light is focused on a plane at $z=0\text{}\mu $m, the intensity is gradually reduced with further beam propagation, broadening the radial distribution as a function of *z*, as shown in Eq. (11). In the present case, it is hypothesized that the absorption reduces the intensity due to momentum transfer from the light to the photocurable resin. Figure 4(b) shows the result of light absorption for $\eta =9.24\times {10}^{2}$ m^{−1}; this result is very similar to that for *η* = 0 m^{−1}, shown in Fig. 4(a). For $\eta =9.24\times {10}^{3}$ m^{−1}, shown in Fig. 4(c), the absorption clearly weakens the intensity of light near the focal plane compared with the results in Fig. 4(a). The intensity is radially distributed with *z* and still appears at $z=100\text{}\mu $m. For $\eta =9.24\times {10}^{4}$ m^{−1}, shown in Fig. 4(d), the light intensity seems to be limited to very near the focus point, within 20 *μ*m, along the *z*-axis. For this case, it is predicted that the UV light is strongly absorbed in the photocurable resin and thus the electromagnetic field that causes an optical vortex is strong only near the focus point. If photoreaction occurs in the region which has significant intensity, an expected shape of polymerized fibers becomes the same as the intensity field shown in Fig. 4. That is, the diameter of the fiber is broaden and the length is restricted. However, actual fibers obtained in experiments does not have such a trend. These results suggest that the absorption of UV light enhances the photoreaction that polymerizes the liquid resin and that the optical vortex simultaneously transfers its angular momentum to the produced objects. Due to the absorption, the optical vortex acts on the polymerized objects within a limited region near the focus point.

In the present model, it is assumed that the monomers of a photocurable resin are polymerized by UV light irradiation. The polymerization and subsequent growth process are modeled using coarse-grained particles whose behavior is influenced by a Laguerre-Gaussian-mode optical vortex. When optical forces act on the coarse-grained particles at a focal spot where the light intensity is strongest, such that $r={w}_{0}/\sqrt{2}$ at $z=0\text{\mu}$m, the *z* components of the gradient force ${F}_{\text{grad},z}$ and the scattering force ${F}_{\text{scat},z}$ are given by Eq. (13) and Eq. (15), respectively. For better understanding in later discussion of this study, we represent ${F}_{\text{grad},z}$ and ${F}_{\text{scat},z}$ in $\left(r,z\right)=\left({w}_{0}/\sqrt{2},0\right)$ as follows:

${F}_{\text{grad},z}$ and ${F}_{\text{scat},z}$ act in the negative and positive directions along the *z*-axis, respectively. When the magnitude of ${F}_{\text{scat},z}$ is greater than that of ${F}_{\text{grad},z}$, a coarse-grained particle may be launched into a helical orbit according to the corresponding Laguerre-Gaussian mode. We focus on two parameters, namely *η* and *d*, which were not uniquely determined in experiments. Other known physical properties are listed in Table 1. ${F}_{\text{grad},z}$ depends on *η* and *d*^{3}, and ${F}_{\text{scat},z}$ is proportional to *d*^{6}. The *d* dependence of ${F}_{\text{scat},z}$ and ${F}_{\text{grad},z}$ for various values of *η* is shown in Fig. 5. Where, each force is normalized by ${F}_{\text{scat},z}$ in $d=50$ nm.

As shown in the inset of Fig. 5, $\left|{F}_{\text{scat},z}\right|$ (solid line) crosses $\left|{F}_{\text{grad},z}\right|$ for $\eta =9.24\times {10}^{2}$ and $9.24\times {10}^{3}$ m^{−1} at points ’A’ and ’B’, respectively. $\left|{F}_{\text{grad},z}\right|$ is above $\left|{F}_{\text{scat},z}\right|$ for $\eta =9.24\times {10}^{4}$ m^{−1} and below it for $\eta =9.24\times {10}^{1}$ m^{−1} in the range of $10\le d\le 50$ nm. When *η* has infinitely small value: $\eta \to 0$ m^{−1} (Not shown in figure), $\left|{F}_{\text{grad},z}\right|$ approaches asymptotically a constant (0 N). These results mean that the coarse-grained particles are successively launched into a helical orbit when the relation $\left|{F}_{\text{scat},z}\right|>\left|{F}_{\text{grad},z}\right|$ is satisfied. For this condition, we performed a computation using coarse-grained particles with $d=50$ nm in the solution for $\eta =9.24\times {10}^{3}$ m^{−1}. A coarse-grained particle was generated on the focal plane and was launched into a helical orbit.

Figure 6(a) shows a three-dimensional view of particles subjected to the optical forces created by an optical vortex, and Figs. 6(b) and 6(c) show the corresponding projections onto the *xy* and *yz* planes, respectively. In the aforementioned scenario, the coarse-grained particles are bound to the focal plane at a radius of $r={w}_{0}/\sqrt{2}$, where the light intensity is strongest. Gray- and white-colored helices consisting of coarse-grained particles represent typical orbits that are initially placed at angles of *θ* = 0 and *π* rad, respectively. The diameter of the orbit is approximately 3 *μ*m, as shown in Fig. 6(b), and the period of the helix is evaluated to be 300 *μ*m, as shown in Fig. 6(c). The trends of the coarse-grained model correspond to those of a photocurable twisted fiber with a constant diameter and period. In experiments [14], a fiber with a 4-μm diameter and a 60-μm long period of the helix was obtained; these dimensions are consistent with the our computational results from the conditions of $d=27$ nm and $\eta =1\times {10}^{4}$ m^{−1} [30].

For $\eta =9.24\times {10}^{3}$ m^{−1}, coarse-grained particles with $d>24$ nm are expected to propagate in the positive *z* direction, as shown in Fig. 5. The trajectories of particles with diameters of $d=25$, $30$, $40$, and $50$ nm, initially located at *θ* = 0 rad, are shown in Fig. 7.

The particles are launched into the helical orbit in these conditions. The period of the helix depends on *d*. When the transport of coarse-grained particles is overdamped, the velocity is determined by the optical forces. Therefore, the period of a helical orbit can be evaluated in terms of the ratio of the magnitude of the scattering force to that of the gradient force. The *z* component of the forces is expressed as ${F}_{\text{scat},z}+{F}_{\text{grad},z}$ and the *θ* component is given by ${F}_{\text{scat},0a}$ as follows:

The radius of a helix is determined by $r={w}_{0}/\sqrt{2}$ for coarse-grained particles trapped by ${F}_{\text{grad},r}$. The period of the helix, *L*_{p}, is expressed as

In this study, we consider *L*_{p} to be a function of *d* and *η*, with the other properties fixed. Note that the laser power, related to *E*_{0}, does not appear in this equation. That is, the laser power has no effect on the helical structure. On the other hand, it is suspected that the laser power influences other factors, such as the photochemical reaction rate outside Rayleigh scattering theory, which relates to the time scale of the fiber formation. This point has not been correctly treated in this study and remains to be solved in future work. As shown in Fig. 7, *L*_{p} grows with increasing *d* and approaches to asymptotic value with the rate of ${d}^{-3}$, as expressed by Eq. (22). Figure 8 shows *L*_{p} as a function of *η* for various *d* values, with the other parameters fixed.

For $\eta \to 0$ m^{−1}, *L*_{p} approach asymptotically a constant value near 290 *μ*m, corresponding to ${L}_{\mathrm{p}}=\pi {k}_{\mathrm{f}}{w}_{0}^{2}$. For $\eta =9.24\times {10}^{1}$ m^{−1}, the *d* dependence of *L*_{p} is in the range of $10\le d\le 50$ nm. The lower limit originates the magnitude ${F}_{\text{grad},z}$ being lower than that of ${F}_{\text{scat},z}$, and the upper limit can be derived from Rayleigh scattering theory $d\ll \lambda $, as shown in Fig. 5. For $\eta =9.24\times {10}^{2}$ and $9.24\times {10}^{3}$ m^{−1}, the *d* dependence of *L*_{p} is in the range of $\left|{F}_{\text{scat},z}\right|\ge \left|{F}_{\text{grad},z}\right|$, whose critical points are defined as ’A’ and ’B’ in Fig. 5.

Figure 9 shows the *λ* dependence of *L*_{p} for *λ* = 250, $300$, and $350$ nm. Where, the change in the indices of refraction with respect to the wavelength is not considered. In *L*_{p} shown in Eq. (22), the parameters which vary with the wavelength *λ* are *η*, *k*_{f}, *ε*_{p} and $\stackrel{\u02c7}{a}repsilo{n}_{\mathrm{f}}$. In general, wavelength dependence of *n*_{p} and *n*_{f} included in *ε*_{p} and *ε*_{f} are smaller than that of *k*_{f}. The former is lower than $O\left({10}^{-1}\right)$ and the latter is the rate of ${\lambda}^{-1}$. When *λ* changes, the effect on *L*_{p} from ${k}_{\mathrm{f}}^{-3}$ may be larger than that from $\left({\epsilon}_{\mathrm{p}}+2{\epsilon}_{\mathrm{f}}\right)/\left({\epsilon}_{\mathrm{p}}-{\epsilon}_{\mathrm{f}}\right)$. That from *η* is much larger than both components. Thus, wavelength dependence of *L*_{p} with several *η* values shown in Fig. 9 is treated as the first order approximation. The *z* component of optical forces is changed by *λ*, which results in a variety of *L*_{p} values, where the reciprocal of *λ* is proportional to *k*_{f}. In Eqs. (19) and (20), *k*_{f} is included in ${F}_{\text{scat},z}$, which is proportional to ${k}_{\mathrm{f}}^{4}$. Therefore, ${F}_{\text{scat},z}$ increases with decreasing *λ*.

For $\lambda =250,300$, and $350$ nm, $\left|{F}_{\text{scat},z}\right|$ crosses $\left|{F}_{\text{grad},z}\right|$ with $\eta =9.24\times {10}^{3}$ and $9.24\times {10}^{4}$ m^{−1} in the range of $10\le d\le 50$ nm. As a result, $\left|{F}_{\text{scat},z}\right|$ and $\left|{F}_{\text{grad},z}\right|$ crosses at larger *d* as *λ* increases. Thus, the period of the helical structure with $\eta =9.24\times {10}^{4}$ m^{−1} varies with *d*, as shown in Fig. 9(b). *L*_{p} becomes longer with decreasing *λ*. In Eq. (22), the first term represents the asymptotic value of *L*_{p} as $d\to \infty $, and the second term represents the asymptotic rate of *L*_{p} to *d*. In Figs. 9(b), the period for $\eta \to 0$ m^{−1} is the asymptotic limit derived from the first term in Eq. (22); it increases from 340 to 470 *μ*m when *λ* decreases from 350 to 250 nm. The slope of the curves as a function of *d* changes the period with $\eta =9.24\times {10}^{2}$, $9.24\times {10}^{3}$, and $9.24\times {10}^{4}$ m^{−1} and becomes smaller with decreasing *λ*, which is caused by the second term in Eq. (22). Therefore, the change in wavelength affects the value of *L*_{p}. The absorption coefficient is also an important factor in the formation of helically twisted fibers.

In the actual system, thermal fluctuations, which cause the randomness, are not ignored especially in liquids. It is suspected that nuclei of photopolymerized polymers randomly generated near a focal plane, from small clusters and polymerize to form twisted fiber structures. In this study, we demonstrated an ideal case of the particle generation, tracking the trajectory affected by the optical vortex at the initial growth process of fiber structures. The present result is expected to be the first step to clarify the complicated system involving photochemical reactions in the optical vortex fields.

## 4. Conclusions

This study proposed a theoretical model of twisted fiber formation in a photocurable resin via UV Laguerre-Gaussian-mode optical vortices. We assumed that polymerized objects are generated on the focal plane and such products were modeled as coarse-grained particles to apply Rayleigh scattering theory with the light absorption effect due to complex indices of refraction. The coarse-grained particles were launched into a helical orbit when the *z* component of the scattering force exceeded that of the gradient force. The period of helical orbits includes an experimentally determined value, namely 60 *μ*m [14, 30]. The present model suggests that the light absorption of the liquid resin has a possibility of force generation on photopolymerized nanoparticles that may contribute to the whole process of helical fiber formation by optical vortex solitons [19]. When we changed the absorption coefficient or the wavelength of the optical vortices, the diameter dependence of the aforementioned periods was modulated. The input power of laser did not affect the structure of the twisted fibers. This fact implies that the light absorption by the photocurable resin and the wavelength of UV optical vortex have an important role in determining the shape of the polymerized twisted fibers.

## Funding

Japan Society for the Promotion of Science (18J10338, 18H05242, JP16H06504).

## Acknowledgement

The present study was supported by Grant-in-Aid for the Japan Society for the Promotion of Science (JSPS) Research Fellow, JSPS KAKENHI 18H05242 for Scientific Research (S), and JSPS KAKENHI Grant Number JP16H06504 in Scientific Research on Innovative Areas “Nano-Material Optical-Manipulation.”

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