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

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

Where

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

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 $\eta =2k_{\text{im}}=4\pi n_{\mathrm{f}}\beta /\lambda$ 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 $z=0\mu$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 $r=w_0/\sqrt{2}$.

 

Fig. 1 (a) Schematic diagram of the gradient and scattering forces caused by a time-averaged optical vortex. Bold black arrows are the force vectors and narrow black arrows are the corresponding r and z components. Bold blue arrows are Poynting vectors and narrow blue arrows are the corresponding θ and z components. Radial dependence of (b) gradient force for the r component and the z component and (c) scattering force for the θ component and the z component in Eqs. (13) and (15) at $z=0\mu$m with $n_{\mathbf{f}}=1.49$, $n_{\mathbf{p}}=1.52$, $w_0=2\mu$m, $d=50$ nm, and $\eta =9.24\times {10}^3$ m⁻¹. Each force is normalized by that maximum magnitude.

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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 $z=0\mu$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 $z=0\mu$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 theθ 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.

Table 1. Physical properties of NOA65 used in computer simulation.

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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 $z=0\mu$m.

 

Fig. 2 Schematic diagram of numerical system. UV optical vortex is focused onto the liquid resin and a twisted fiber is generated at the focal plane and pushed forward. Coarse-grained particles are generated at the strong-intensity ring on the focal plane. The particles are launched from this ring by optical forces one after another.

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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:

 

Fig. 3 Unsteady velocities of a coarse-grained particle with specific weights ${\gamma }_{\mathbf{s}}=1,2,5$, and $10$. (a) r-directional velocity, v_r, and (b) z-directional velocity, v_z. Red, green, blue, and yellow solid lines are results for $m{a}_{\mathbf{s}}=1,2,5$, and $10$, respectively. Black dashed line indicates the constant velocity at which the motion of the particle is overdamped. Each velocity is normalized by that terminal value.

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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:

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 $\left(r,z\right)=\left(w_0/\sqrt{2},0\right)$.

 

Fig. 4 Distributions of light intensity in the rz plane for absorption coefficients of (a) η = 0, (b) $\eta =9.24\times {10}^2$, (c) $\eta =9.24\times {10}^3$, and (d) $\eta =9.24\times {10}^4$ m⁻¹. Each intensity is normalized by the point of maximum value placed at $\left(r,z\right)=\left(w_0/\sqrt{2},0\right)$.

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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/m² at $r=0\mu$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 $z=0\mu$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 $z=100\mu$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 μm, 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 $z=0\text{ μ}$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 d³, and $F_{\text{scat},z}$ is proportional to d⁶. 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.

 

Fig. 5 z directional optical forces acting on coarse-grained particles with various diameters. Where, each force is normalized by $F_{\mathbf{s}\mathbf{c}\mathbf{a}\mathbf{t},z}$ in $d=50$ nm. Black solid line represents scattering force $F_{\mathbf{s}\mathbf{c}\mathbf{a}\mathbf{t},z}$ in Eq. (20). Red dashed line, green dotted line, blue dash-dot line, and yellow dash-dot-dot line represent gradient forces $F_{\mathbf{g}\mathbf{r}\mathbf{a}\mathbf{d},z}$ in Eq. (19) for $\eta =9.24\times {10}^4$, $9.24\times {10}^3$, $9.24\times {10}^2$, and $9.24\times {10}^1$ m⁻¹, respectively. The inset graph has a single logarithmic expression on the vertical axis. In this graph, each force is represented by its absolute value. ’A’ is the intersection between the black solid and blue dash-dot lines and ’B’ is that between the black solid and green dotted lines.

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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.

 

Fig. 6 Dynamics of coarse-grained particles with diameter $d=50$ nm subjected to optical forces created by an optical vortex. The absorption coefficient is $\eta =9.24\times {10}^3$ m⁻¹. For gray and white structures, the initial rotational position is θ = 0 and $\theta =\pi$ rad, respectively. (a) Three-dimensional representation projected onto (b) the xy plane and (c) the yz plane. In these figures, the z directional representation is compressed by 100 fold.

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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 μm, as shown in Fig. 6(b), and the period of the helix is evaluated to be 300 μm, 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.

 

Fig. 7 yz projection of helical orbit formed by coarse-grained particles with diameters of (a) $d=25$ nm, (b) $d=30$ nm, (c) $d=40$ nm, and (d) $d=50$ nm. The absorption coefficient is $\eta =9.24\times {10}^3$ m⁻¹. The z directional representation is compressed 100 fold. The intervals between vertical lines show the period of the helical orbit.

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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 E₀, 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.

 

Fig. 8 Diameter dependence of the period of the helical orbit for various absorption coefficients. Red dashed line shows $\eta =9.24\times {10}^3$ m⁻¹, green dotted line shows $\eta =9.24\times {10}^2$ m⁻¹, blue dash-dot line shows $\eta =9.24\times {10}^1$ m⁻¹, and black solid line shows $\eta \to 0$ m⁻¹ . ’A’ and ’B’ are the same as shown in Fig. 5.

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For $\eta \to 0$ m⁻¹, L_p approach asymptotically a constant value near 290 μm, 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 $\check{a}repsilo{n}_{\mathrm{f}}$. 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.

 

Fig. 9 z directional forces and period of helical structures for wavelengths λ = 250, $300$, and $350$ nm. (a) Logarithmic expression of z directional forces acting on coarse-grained particles with various diameters. Where, each force is normalized by $F_{\mathbf{s}\mathbf{c}\mathbf{a}\mathbf{t},z}$ in $d=50$ nm. Black solid line represents scattering force $F_{\mathbf{s}\mathbf{c}\mathbf{a}\mathbf{t},z}$ shown in Eq. (20). Red dashed line, green dotted line, and blue dash-dot line represent gradient forces $F_{\mathbf{g}\mathbf{r}\mathbf{a}\mathbf{d},z}$ shown in Eq. (19) for $\eta =9.24\times {10}^4$, $9.24\times {10}^3$, and $9.24\times {10}^2$ m⁻¹, respectively. (b) Diameter dependence of the period of helical structures for various absorption coefficients. Red dashed line shows $\eta =9.24\times {10}^4$ m⁻¹, green dotted line shows $a=9.24\times {10}^3$ m⁻¹, blue dash-dot line shows $\eta =9.24\times {10}^2$ m⁻¹, and black solid line shows $\eta \to 0$ m⁻¹.

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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 μm 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 μm [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.