Tunable Photonic Hook Design Based on Anisotropic Cutting Liquid Crystal Microcylinder

Abstract

The selective control and manipulation of nanoparticles require developing and researching new methods for designing optical tweeters, mainly based on a photonic hooks (PHs) effect. This paper first proposes a tunable PH in which a structured beam illuminates an anisotropic cutting liquid crystal microcylinder based on the Finite-DifferenceTime-Domain (FDTD) method. The PHs generated by plane wave, Gaussian, and Bessel beam are analyzed and compared. The impact of beams and LC particle parameters on the PHs are discussed. Where the influence of the extraordinary refractive index ($n_{\mathrm{e}}$) on PHs is emphasized. Our results reveal that introducing birefringence can change the bending direction of PH. Besides, the maximum intensity of the PHs increases as $n_e$ increases regardless of the beam type. The PH generated by a plane wave has a higher maximum intensity and smaller FWHM than that generated by the Gaussian and Bessel beams. The smallest FWHM and maximum intensity of the PHs generated by the Gaussian falls between that generated by the plane wave and the Bessel beam. The PH generated by a Bessel beam has the minor maximum intensity and the largest FWHM. Still, it exceeds the diffraction limit and exhibits bending twice due to its self-recovery property. This paper provides a new way to modulate PH. This work offers novel theoretical models and the degree of freedom for the design of PHs, which is beneficial for the selective manipulation of nanoparticles. It has promising applications in Mesotronics and biomedicine.

1. Introduction

The optical tweezer [1] is an experimental technology that uses the interaction of light and matter to achieve particle trapping and manipulation. It was initially designed to facilitate the non-contact and non-destructive manipulation of tiny particles. Much research [2,3,4] has been conducted since Ashkin [5] proposed optical tweezers in the 1980s. Over the years, it played an important role in materials science [6,7], nanophotonics [8,9], and biomedicine [10,11]. Besides, it has been widely used in optical imaging [12], optical storage [13], and particle manipulation [14,15]. In recent years, a new manipulation method based on the photonic nanojet (PNJ) [16] has promoted the development of an optical tweezer. PNJ has the characteristics of much higher intensity than an incident wave, transverse half-height full width of subwavelength, and propagation to non-evanescent field region. It has shown high application value in the fields of super-resolution imaging [17,18], biochemical analysis [19], and nanoscale manipulation [20]. However, as an axial beam, its direction is limited to the axial, making it challenging to achieve selective particle manipulation. Airy beam [21,22] is a non-diffractive beam with self-recovery and self-bending characteristics. It can trap particles from any angle by passing obstacles. However, it is generated by bulky static or dynamic devices with demanding production conditions. It makes particle capture expensive.

In 2016, a curved PNJ was proposed and named Photonic Hook (PH) [23], which opens up new possibilities for achieving selective manipulation of particles. PH has the advantages of the Airy beam and PNJ, with high focusing, a multi-wavelength focal length distance, sub-diffraction full-width at half maximum (FWHM), limited diffraction, and self-bending characteristics [24]. Most importantly, it can detect and influence targets behind obstacles, and the achievement process is more straightforward than that of Airy beams. The research of PH has become a hot issue, and extensive achievements have been produced around it. A study [25] discusses the optical radiation force of PH generated by asymmetric cuboids, finding that PH can manipulate target particles behind obstacles. Following [23] in [26] describe the generation of PH based on a dielectric trapezoidal particle irradiated by a plane wave. Based on this model, a study successfully generated PH [27], confirming the existence of a PH experimentally for the first time. It makes the PH research from theoretical to application and provides strong support for the application of PH. Besides, the proposed concept of photonic hook scalpel [28], which uses an asymmetric fiber to generate PH at the tip, and the related results have broad application prospects in the development of laser surgery. Since then, a study [29] has experimentally generated a plasmonic PH using the dielectric Janus particles, further expanding the field of PH applications. With the further development of the research, a photonic hook generated by a multi-dielectric structure composed of periodically arranged scattering units was proposed [30], which simplified the PH generation process and provided innovative applications in the field of nanolithography and integrated optics. Recently, based on the optical Magnus effect, a novel method for generating PH using the interaction of plane waves and rotating particles has been proposed [31,32]. It illustrates the physical mechanism of PH production from an entirely new perspective. Up to now, the PH has generated considerable attention in diverse domains, such as optical nanomanipulation [33] and high-resolution imaging [34,35,36].

Although extensive research [37,38,39,40] has been carried out on the PNJ generated by the interaction between particles and various structured beams [41,42,43], the research on PH mainly focuses on the scattering of plane waves by particles. Previous studies on PNJs found that the particle and incident beam parameters closely affected the near field. Studying the scattering of particles to the structured beam can provide more modulation degrees of freedom and is expected to achieve selective manipulation of particles. Recently, a study [44] proposed a method to generate a twin photonic hook by irradiating an ellipsoid particle with a Bessel beam. Another research [45] explored the PH produced from a spherical particle illuminated by a Bessel–Gaussian beam. This research concludes that the effective length, intensity, and FWHM of PH can be modulated by varying the beam parameters. Therefore, it is vital for the design of PHs optical tweezers [25,46] to introduce structured beams into the PHs system. It will facilitate the development of a new generation of optical tweezers.

To the best of our comprehension, most of the current research is based on isotropic particles [47,48,49]. Compared with isotropic particles, the shape and refractive index of anisotropic particles [50,51] show asymmetry, which can generate ample torque through the light field to achieve accurate directional control. In addition, the optical anisotropy of anisotropic particles can be designed and controlled for optical detection and directional trap of particles. A study proposed controlled photonic hooks produced by a typical shape composed of uniaxial crystals [52]. It is found that active switching between photon nanojets and PH can be achieved in a variety of structures by changing the polarization state of the incident field. The PHs generation and modulation scheme proposed in this study can be extended to more anisotropic materials. Liquid crystal (LC) is the uniaxial anisotropic material with the lowest symmetry in optics and was used to produce Airy family beams [53,54,55]. Introducing LC dispersion into a polymer or fluid medium results in the spontaneous creation of droplets on a micrometer scale. These droplets exhibit a distinct molecular arrangement [56,57,58] in tangential and radial directions, which express significant advantages in the optical field. It has been verified that the tunable PNJ [59,60,61] is produced by changing the refractive index distribution of LC materials by applying an electric or light field. A conclusion is given that the horizontal and vertical displacements of PNJ depend strongly on the director of the LC. In contrast, LC materials have not been introduced into PH studies.

In this paper, we aim to study the critical properties of the PH for an optical tweeter using the scattering of LC to structured beams and further explore them [62]. A model is proposed using the scattering of a cutting liquid crystal cylinder to beams. The PHs generated by plane wave, Gaussian beam, and Bessel beam are discussed and compared. The characteristics of PHs, including FWHM, bending angle, and effective intensity, are analyzed. Furthermore, we conduct numerical investigations into the scattering characteristics of PH by incorporating the effects of birefringence. The results of this work provide a novel method to design PH, theoretical guidance, and a numerical model for the design of a new generation of optical tweezers.

The overall layout of this article is as follows. Section 2 presents the numerical model of a cutting LC cylinder illuminated by a beam based on the Finite-DifferenceTime-Domain (FDTD) method. The expressions of a plane wave, Gaussian beam, and Bessel beam are provided. The theoretical framework to generate the PHs has also been built. Based on this model, the PHs generated by the scattering of cutting the LC cylinder to plane wave, Gaussian beam, and Bessel beam are calculated in Section 3. The characteristics of PHs generated by various beams are compared. The impact of the beam and particle parameters on PHs is discussed in detail. The anisotropy refractive index of LC is emphasized. In the last section, the findings are summarized, and the potential application areas of this work are indicated. This work paves the way for developing novel optical devices with promising applications in optical manipulation, photoetch, and biomedicine.

2. Materials and Methods

Figure 1 presents the FDTD model that incident wave propagates along the x-axis and illuminates on cutting microcylinder [63] made from an anisotropic liquid crystal. In Figure 1, a perfectly matched layer with scattering boundaries fully absorbs outward waves and configures the polarization as transverse electric (TE). The simulated area has an area of $14\times 14$ ${\mathsf{\mu }\mathrm{m}}^2$, the corresponding number of data grids is $700\times 700$. This means that the size of the grid is 0.02 $\mathsf{\mu }$m (about $\lambda$/32). Thus, the simulation results are sufficiently accurate. Assuming a circular microcylinder with a radius of 2 $\mathsf{\mu }$m, infinitely long, is cut at a distance of 1 $\mathsf{\mu }$m from the center and filled with LC. During practical experiments, the cutting microcylinder can be securely affixed to a gold base. Gold is chosen for its excellent conductivity and reflectivity [64,65], enabling adequate reflection of incoming plane waves and reducing interference during the propagation process.

Under the theoretical framework of FDTD [66,67,68,69], the Maxwell curl equations for linear, isotropic, non-dispersive, and lossy materials are as follows:

$${\displaystyle \frac{\partial \mathbf{H}}{\partial t}}=-{\displaystyle \frac{1}{\mu }}\nabla \times \mathbf{E}-{\displaystyle \frac{1}{\mu }}\left({\mathbf{M}}_{\mathrm{source}}+{\sigma }^{\ast }\mathbf{H}\right)$$

(1)

$${\displaystyle \frac{\partial \mathbf{E}}{\partial t}}={\displaystyle \frac{1}{\epsilon }}\nabla \times \mathbf{H}-{\displaystyle \frac{1}{\epsilon }}\left({\mathbf{J}}_{\mathrm{source}}+\sigma \mathbf{E}\right)$$

(2)

where $\sigma$ and ${\sigma }^{\ast }$ represents the electrical conductivity and effective magnetic loss, respectively. $\epsilon$ and $\mu$ are the permittivity and permeability. The current density ($\mathbf{J}$) and the effective magnetic current density ($\mathbf{M}$) can be considered as independent sources of electric (${\mathbf{J}}_{\mathrm{source}}$) and magnetic (${\mathbf{M}}_{\mathrm{source}}$) field energy.

In the Cartesian coordinate system, the coupled scalar expressions of Equations (1) and (2) are expanded as follows.

$${\displaystyle \frac{\partial H_x}{\partial t}}={\displaystyle \frac{1}{\mu }}\left[{\displaystyle \frac{\partial E_y}{\partial z}}-{\displaystyle \frac{\partial E_z}{\partial y}}-\left(M_{\mathrm{source}{}_x}+{\sigma }^{\ast }{H}_x\right)\right]$$

(3)

$${\displaystyle \frac{\partial H_y}{\partial t}}={\displaystyle \frac{1}{\mu }}\left[{\displaystyle \frac{\partial E_z}{\partial x}}-{\displaystyle \frac{\partial E_x}{\partial z}}-\left(M_{\mathrm{source}{}_y}+{\sigma }^{\ast }{H}_y\right)\right]$$

(4)

$${\displaystyle \frac{\partial H_z}{\partial t}}={\displaystyle \frac{1}{\mu }}\left[{\displaystyle \frac{\partial E_x}{\partial y}}-{\displaystyle \frac{\partial E_y}{\partial x}}-\left(M_{\mathrm{source}{}_z}+{\sigma }^{\ast }{H}_z\right)\right]$$

(5)

$${\displaystyle \frac{\partial E_x}{\partial t}}={\displaystyle \frac{1}{\epsilon }}\left[{\displaystyle \frac{\partial H_z}{\partial y}}-{\displaystyle \frac{\partial H_y}{\partial z}}-\left(J_{\mathrm{source}{}_x}+\sigma E_x\right)\right]$$

(6)

$${\displaystyle \frac{\partial E_y}{\partial t}}={\displaystyle \frac{1}{\epsilon }}\left[{\displaystyle \frac{\partial H_x}{\partial z}}-{\displaystyle \frac{\partial H_z}{\partial x}}-\left(J_{\mathrm{source}{}_y}+\sigma E_y\right)\right]$$

(7)

$${\displaystyle \frac{\partial E_z}{\partial t}}={\displaystyle \frac{1}{\epsilon }}\left[{\displaystyle \frac{\partial H_y}{\partial x}}-{\displaystyle \frac{\partial H_x}{\partial y}}-\left(J_{\mathrm{source}{}_z}+\sigma E_z\right)\right]$$

(8)

Equations (3)–(8) are the partial differential equations that electromagnetic waves interact with general three-dimensional objects in the FDTD numerical algorithm.

As everyone knows, the analytic expression of the incident plane wave in the x direction is expressed as

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \mathbf{E}\left(x\right)={\mathbf{E}}_{\mathbf{0}}exp\left(jkx\right)\end{array}\end{array}$$

(9)

${\mathbf{E}}_{\mathbf{0}}$ is the electric field complex amplitude respect with polarization, and $\left|{\mathbf{E}}_{\mathbf{0}}\right|=1$. $k=2\pi n/\lambda$ is the wave number, where $\lambda$ and n are the free space wavelengths and the ambient medium refractive index.

For the fundamental mode Gaussian beam, the analytic expression can be expressed as [66,67]:

$$\mathbf{E}(\rho ,x)={\mathbf{E}}_{\mathbf{0}}{\displaystyle \frac{w_0}{w\left(x\right)}}exp\left({\displaystyle \frac{-{\rho }^2}{w{\left(x\right)}^2}}\right)exp\left(jk{\displaystyle \frac{{\rho }^2}{2R\left(x\right)}}\right)exp\left(jkx\right)$$

(10)

where $\rho =\sqrt{x^2+y^2}$ and z represent the radial and axial distances from the observation point to the beam focus. $w\left(x\right)=w_0\sqrt{1+{\left(x/X_0\right)}^2}$ is the beam waist radius where the amplitude of the electric field decreases along the beam axis to $1/e$ of its value. Where, $w_0$ is the initial beam waist radius, and $R\left(x\right)=X_0\left({\displaystyle \frac{x}{X_0}}+{\displaystyle \frac{X_0}{x}}\right)$ is the curvature radius of the beam wavefront at position x. ${\mathrm{X}}_0={\displaystyle \frac{1}{2}}k{w}_0^2={\displaystyle \frac{\pi w_0^2}{\lambda }}$ is the Rayleigh dimension.

For the zero-order Bessel beam, the analytic expression is expressed as follows [70]

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \mathbf{E}\left(x\right)={\mathbf{E}}_{\mathbf{0}}{J}_0\left(k_{\perp }\rho \right)exp\left(j{k}_{\Vert }x\right)\end{array}\end{array}$$

(11)

where $\rho$ and $\phi$ are the radial and angular coordinates of the field point. $k_{\perp }$ and $k_{\Vert }$ are the transverse and axial components of the wave vector k, respectively. $x=\rho cos\phi$ and $y=\rho sin\phi$, z is the axial coordinate of the field point. $J_0$ is the zero-order Bessel function.

To the anisotropic materials, the permittivity $\tilde{\epsilon }(x,y)$ [58,71] is defined as follows

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \tilde{\epsilon }(x,y)=\left[\begin{array}{ccc}{\epsilon }_0+\Delta \epsilon {sin}^2\theta (x,y)& -\Delta \epsilon sin\theta (x,y)cos\theta (x,y)& 0\\ -\Delta \epsilon sin\theta (x,y)cos\theta (x,y)& {\epsilon }_0+\Delta \epsilon {cos}^2\theta (x,y)& 0\\ 0& 0& {\epsilon }_0\end{array}\right]\end{array}\end{array}$$

(12)

In Equation (11), $\Delta \epsilon ={\epsilon }_e-{\epsilon }_o$ is the refractive index difference between the extraordinary and ordinary permittivities. Where $n_o$ and $n_e$ denote the refractive indices of the material along the ordinary and extraordinary directions, respectively. In addition, angle $\theta (x,y)$ is the principal axis alignment of the LC molecules concerning the y-axis. To simplify the analysis, the $xy$ plane is set as the isotropic plane, and the specific principal axis direction is the z direction, as shown in Figure 1. Thus, the LC molecules inside the cylindrical structure are tangentially aligned parallel to the surface. This assumption facilitates the introduction of optical anisotropy and allows us to verify its influence on electromagnetic fields.

The scattering models of the plane wave, Gaussian beam, and Bessel beam illuminating an anisotropic particle are established, focusing our attention on the PHs generated by the cutting LC microcylinder. Figure 2 is the schematic diagram of the PH. The start point, inflection point, and end point define it. Locating them is the key to calculating the bending angle. The inflection point is the position exhibiting the highest electromagnetic field intensity. The start point corresponds to the maximum intensity on the particle surface. The position of the extent of optical intensity decay in the PH determines the end point. It is the location where the optical intensity decreases to $I_{max}/e$, where e is the natural constant. Thus, the bending angle is determined by calculating the angular difference between the vectors extending from the inflection point to the end point and from the inflection point to the start point.

3. Results

In this section, we perform numerical calculations to determine the PH formed by illuminating cutting microcylinder particles with various beam types, including the plane wave, Gaussian beam, and Bessel beam. The research is based on this ideal condition: the liquid crystal is confined to an infinitely thin and rigid film. The numerical results discuss the impact of beam and particle parameters on PHs, focusing on the anisotropic particles. The characteristics of PH, including maximum intensity, bending angle, and FWHM, are examined in detail. We present explicit definitions for the terminology employed in the calculations. FWHM is defined as the specific threshold at which it signifies half of the maximum intensity of the PH. At the threshold level, a horizontal line intersects the intensity curve at two locations, and the horizontal separation between these points corresponds to the FWHM. The electric field intensity of the PH is defined as ${\left|\mathbf{E}\right|}^2/\phantom{{\left|\mathbf{E}\right|}^2{\left|max\left({\mathbf{E}}_{inc}\right)\right|}^2}{\left|max\left({\mathbf{E}}_{inc}\right)\right|}^2$, so the unit is “arb. units” (“a.u.”). Moreover, the refractive index of the environment is set $n=1.33$ [72,73], and the wavelength of incident light was $632.8$ nm. The ordinary refractive index is $n_o=1.5$, but the extraordinary refractive index $n_e$ changes from $1.4$ to $2.0$. This paper uses TE polarized wave (only the x-direction of the electric field is considered) incident particles as an example to explore the scattering of cutting LC microcylinders to different beams. The research method of the TM case is similar to that of the TE case, but we do not elaborate on it in detail there.

3.1. Plane Wave

Before discussing the PH produced by the interaction between structured beams and cutting LC microcylinders, pay attention to the results generated by the plane wave, as shown in Figure 3. The first column is the electric field intensity a plane wave exerts on the cutting LC microcylinders with different $n_e$. As $n_e$ changes, the shape and characteristics of PHs continue to change, including effective length, max intensity, FWHM, and bending angle. The effective length of PHs gradually decreases as $n_e$ increases. On the contrary, during this process, the maximum intensity of the PHs gradually increases. This phenomenon is because the particles converge on the plane wave more significantly as $n_e$ increases, resulting in greater intensity. However, it is precisely because more beams are concentrated near the particle surface that the effective length of the PH becomes shorter. Therefore, during the experiment, one must choose the suitable refractive index according to the purpose of the experiment to obtain the PH with a longer effective length or greater intensity.

Moreover, compared with the first column, it is found that with the change of $n_e$, the bending direction of the PH also changes, which is meaningful for selective manipulation of particles. To further explore how the characteristic of PH changes with $n_e$, columns 2 to 4 are calculated as auxiliary graphs to the first column. The second column is the three-dimensional figures of PH, where all the leading and side lobes of PH in the $xy$ plane are shown. According to the three-dimensional figures, the energy flow of the electric field near the particle is demonstrated, and the formation process of the PHs is expressed. The third and fourth columns are the electric field intensity of PH in the y and x directions, with the inflection point as the reference point, respectively. With these graphs, we can intuitively observe the change of PH intensity along the axis as $n_e$ changes and further explore the influence of $n_e$ on the FWHM and bending angle of PHs. In the third column, $n_e$ affects the main lobe and the side lobes of PHs. The inflection point of the main lobe is away from the $y=0$ axis as $n_e$ increases. This shows from the side that the bending degree of PH is gradually increasing. Besides, the FWHM has also become narrower during this process. Therefore, a conclusion is proposed that a PH with a narrower FMHM can be obtained by increasing $n_e$. It has a crucial role in nanoparticle manipulation. Although we can preliminarily obtain the trend of the effective length of the photonic hook with the increase in $n_e$ according to the results of the first column, we still calculate the figures of the fourth column to verify this conclusion. Comparing the figures in the fourth column, it is found that the length of PH decreases as $n_e$ increases.

To further verify the correctness of conclusions drawn from Figure 3, the PH characteristics are quantitatively calculated in

Table 1. The comparison of the parameters of the PH generated by cutting microcylinder particles with different extraordinary refractive indices using plane waves.

${\mathit{n}}_{\mathbf{e}}$	Max Intensity (a.u.)	FWHM	Bending Angle
1.4	3.36	1.26$\lambda$	−11.86°
1.5	5.27	0.66$\lambda$	−20.91°
1.6	5.45	0.47$\lambda$	+20.86°
1.7	6.07	0.53$\lambda$	+18.91°
1.8	6.63	0.43$\lambda$	+21.55°
1.9	7.80	0.37$\lambda$	+25.46°

. Parameter settings are consistent with those shown in Figure 3. The Max Intensity in the second column corresponds to the intensity of the inflection point. Its values are $3.36$, $5.27$, $5.45$, $6.07$, $6.63$, and $7.80$ when $n_e$ change from $1.4$ to $1.9$. On the contrary, the FWHM of PHs are $1.26\lambda$, $0.66\lambda$, $0.47\lambda$, $0.53\lambda$, $0.43\lambda$, and $0.37\lambda$. The FWHM of PHs has narrowed by almost half in this process. This is consistent with our conclusion in Figure 3. The results in Figure 3 do not intuitively yield the bending angle as one of the most important parameters of the PH, so modulation of the bending angle is a big problem for PH optical tweezers. We can not intuitively obtain the bending angle from the results in Figure 3. We give the changing bending angle trend as $n_e$ in the fourth column of Table 1. The bending angle is negative from $n_e=1.4$ to $n_e=1.5$. It means that the PHs bend towards the negative direction of the y plane. The bending angle is positive when $n_e$ changes from 1.6 to 1.9. Thus, the PHs bend towards the positive direction of the y plane. In this process, the bending angle increases gradually as $n_e$ increases. According to Table 1, it can be observed that the FWHM of the PH is less than 0.5 $\lambda$, which means it can break the diffraction limit. This indicates that the PH has a high application value.

To further describe the causes and characteristics of PHs generated by a cutting LC cylinder, the energy distribution is calculated in Figure 4. It can be seen that the energy is incident from the left side of the particle, through the convergence of the particles, and mainly distributed in the front of the particle in Figure 4a–d. At the front of the particle, the direction of energy flows from both sides, but the energy distribution on both sides is not uniform. More energy is concentrated under the $y<0$ plane. This is the physical mechanism of PH formation determined by the shape of the cutting LC cylinder. Comparing Figure 4a–d, when $n_e$ = 1.4 and $n_e$ = 1.5, energy is concentrated towards the plane y > 0. When $n_e$ = 1.6 and $n_e$ = 1.7, energy is focused on the plane y < 0. This corresponds to the results in Figure 3. The direction of energy flow determines the bending direction of the PHs. Besides, the forward energy of the particle is closer to the particle as $n_e$ increases. The bending angle of PHs also becomes bigger. It also explains the reason for the narrowing of FWHM: with the rise of $n_e$, the position of the inflection point moves forward correspondingly, and more energy is gathered, which further causes the narrowing of FWHM.

Next, let us reveal the continuous change rule of PH characteristics as $n_e$ changes. Figure 5a–c elaborates the variations of bending angle, inflection point intensity, and FWHM of PHs as $n_e$ increases, respectively. We are only concerned with the values here. Thus, we ignore the ± of the bending angle in the drawing. When $n_e<1.55$, the bending angle is negative. Otherwise, the bending angle is positive. This is because with the increase in $n_e$, the convergence of particles on the beam becomes more assertive, and the focus of PHs moves to the particle surface, resulting in the change of its bending characteristics. We chose the step size of $n_e$ to be 0.01 because, under this step size, we can preliminarily obtain the characteristic on PHs and exclude the mutative effect of exceptional $n_e$ on PHs as much as possible.

In Figure 5a, it is observed that the overall trend of the bending angle is upward in waves. The maximum value of 26.95° is achieved at $n_{\mathrm{e}}$ = 1.94, while the minimum value of 7.35° is achieved at $n_{\mathrm{e}}$ = 1.56. The bimodal phenomenon explains the reason for the wavy trend. As shown in Figure 3j,n,r,v, there are two peaks with similar intensity close together. As $n_e$ increases, the position of the inflection point (maximum peak) moves back and forth between these two peaks. Thus, the bending angle shows a wave-rising trend as $n_e$ increases. In summary, increasing the extraordinary refractive index results in the PH with larger bending angles. In the experiment, we can obtain a smooth trend by suppressing the forward peaks of adjacent peaks. Figure 5b calculates the changing trend of the inflection point electric field intensity as $n_e$. The inflection point intensity gradually increases as $n_e$ increases. The minimum value and maximum value are 3.35 and 8.37, corresponding to $n_{\mathrm{e}}$ = 1.4 and $n_{\mathrm{e}}$ = 1.94. Thus, PHs with strong intensity can be obtained by increasing the extraordinary refractive index in the experiment. It has great potential applications in optical information storage and lithography. Figure 5c gives the impact of $n_e$ on the FWHM. It can be seen that the general tendency of FWHM to change with $n_e$ is gradually decreasing. However, there is a jump peak from $n_e$ = 1.64 to $n_e$ = 1.65. The bimodal phenomenon explains it, as shown in Figure 6. When $n_e$ is less than 1.64, the inflection point coincides with the forward peak of the adjacent peaks. In contrast, when $n_e$ is greater than 1.64, the inflection point coincides with the backward peak of the adjacent peaks. Since the FWHM of the backward peak is always greater than that of the forward peak, the FWHM mutates when $n_e$ goes from 1.64 to 1.65. Therefore, within the range of $n_{\mathrm{e}}$ from 1.4 to 2, the intensity of the PH gradually increases, the FWHM gradually decreases, and the bending angle fluctuates, but the overall trend is upward. This is beneficial for generating PH with dimensions smaller than the diffraction limit and with practical value.

3.2. Gaussian Beam

Having discussed plane waves, we focus on the most common beam in real life, the Gaussian beams. This section discusses the influence $n_e$ exerts on the PH generated by the Gaussian beam. Similar to the analytical process of the plane wave case, the characteristics of PH caused by the Gaussian beam are emphasized, including the bending angle, the electric field intensity, and the FWHM. In the calculation results, a cutting LC microcylinder is illuminated by a Gaussian beam with waist radius $w_0=2$ $\mathsf{\mu }$m propagating along the x-axis. Figure 7 is the schematic of the PH that a Gaussian beam interacts with a cutting LC microcylinder. However, the scattering results of particles at different locations in the Gaussian beam are different. In this paper, the particles are placed behind the waist of the Gaussian beam to compare with the plane wave and Bessel beam incident model of cutting liquid crystal cylinder, in which the initial position of the incident light is $x=-6\mathsf{\mu }$m. The scattering of Gaussian beams by particles at different beam positions has been extensively studied, which is not the focus of this paper, and we will build an experimental platform to explore it in the future.

Figure 8 calculates the PHs caused by the Gaussian beam. The first column is the electric field intensity with different $n_e$. Like the plane wave case, the first column shows that a Gaussian beam interacts with the cutting LC microcylinder. A PH appears on the back side of the particle. The second column is the three-dimensional figures of PHs in the $xy$ plane, which are further calculated to elucidate the characteristics of PHs. The third and fourth columns are the electric field intensity of PH in the y and x directions. Comparing with Figure 3 and Figure 8, the intensity of PHs in Figure 8 is less than in Figure 3 under the same $n_e$. The intensity of PHs still increases as $n_e$ increases, and the effective length of PHs decreases as $n_e$ increases. However, more secondary peaks appear in the x direction when a Gaussian beam illuminates the cutting LC microcylinder compared with the plane wave case comparing with the fourth column of Figure 3 and Figure 8. The intensity of secondary peaks is closer to that of the main peak. This makes it more difficult to choose the inflection point and further causes difficulties in determining the bending angle. Besides, the FWHM of PHs in Figure 8 is wider than that in Figure 3. It is related to the waist radius of the Gaussian beam.

To further compare the characteristics of PH generated by the plane wave and Gaussian beam,

Table 2. The comparison of the parameters of the PH generated by cutting microcylinder particles with different extraordinary refractive indices using a Gaussian beam.

${\mathit{n}}_{\mathbf{e}}$	Max Intensity (a.u.)	FWHM	Bending Angle
1.4	2.37	1.36$\lambda$	−7.27°
1.5	3.35	0.82$\lambda$	−18.11°
1.6	3.64	0.75$\lambda$	+8.11°
1.7	4.24	0.65$\lambda$	+15.54°
1.8	4.36	0.53$\lambda$	+18.16°
1.9	4.92	0.60$\lambda$	+13.90°

is given. It can be seen that the intensity of the inflection point is 2.37, 3.35, 3.64, 4.24, 4.36, and 4.92. The trend of FWHM changing with $n_e$ is the opposite to the trend of inflection point intensity. The FWHM is $1.36\lambda$, $0.82\lambda$, $0.75\lambda$, $0.65\lambda$, $0.53\lambda$, and $0.6\lambda$. Thus, a conclusion is provided that the incident wave source also determines the characteristics of PHs generated by the cutting LC microcylinder. Compared with the Gaussian beam, the plane wave acting on cutting LC microcylinders is more likely to produce PHs with higher intensity inflection points and narrower FWHM. Of course, compared with the plane wave, the Gaussian beam has more tuning regimes, which is also extremely important for generating tunable PHs. However, the trend of bending angle changing with $n_e$ cannot be obtained from Table 2. Thus, Figure 9 is calculated to explore the influence of $n_e$ on PHs.

The intensity distribution of the inflection point is shown in Figure 9b, where the minimum value of 2.36 is achieved at $n_{\mathrm{e}}$ = 1.4, and the maximum value of 5.26 is achieved at $n_{\mathrm{e}}$ = 2.0. It with the conclusions drawn from Figure 9 and Table 2; the intensity of the inflection point increases as $n_e$ increases. When it comes to the FWHM, its values decrease initially and then fluctuate around the diffraction limit as $n_{\mathrm{e}}$ increases, as shown in Figure 9c. As shown in the third and fourth columns of Figure 8, as well as Table 2, the peak values of transverse light field intensity and axial light field intensity of the PH increase with the increase in $n_{\mathrm{e}}$. As shown in Figure 9c, the FWHM values decrease initially and then fluctuate around the diffraction limit as $n_{\mathrm{e}}$ increases. The maximum value 1.35 $\lambda$ is achieved at $n_{\mathrm{e}}$ = 1.4, and the minimum value 0.41$\lambda$ is achieved at $n_{\mathrm{e}}$ = 1.92. However, the bending angle fluctuates as $n_{\mathrm{e}}$ increases. Its minimum value of 1.99° is achieved at $n_{\mathrm{e}}$ = 1.41, and the maximum value of 20.72° is achieved at $n_{\mathrm{e}}$ = 1.93. The reason for the fluctuation change is the same as the case of the plane wave above, which is caused by the bimodal phenomenon. A more general rule can be obtained in the experiment by suppressing the bimodal phenomenon. Here, we calculate the bending angle according to the definition of the inflection point in this paper.

3.3. Bessel Beam

After discussing the characteristics of PHs, such as how plane waves and Gaussian beams interact with the cutting LC microcylinder, let us focus on Bessel beams. This section discusses the scattering characteristics of the cutting LC microcylinder to the zero-order Bessel beam. The specific principle is shown in Figure 10, where the generated PH is on the right side of the anisotropic cutting microcylinder particle, and the incident beam on the left side is a Bessel beam.

To discuss the scattering characteristics that the Bessel beam exerts on a cutting LC microcylinder, the scattering intensity is calculated in Figure 11. The first to the fourth columns are the electric field intensity, the three-dimensional electric field intensity, and the electric field intensity in the y and x directions, respectively. The intensity of PHs generated by the interaction between the Bessel beam and an LC microcylinder gradually increases as $n_e$ increases in Figure 11. The general trend is consistent with plane waves and Gaussian beams producing PHs. However, the intensity of PHs generated by the Bessel beam is less than that produced by the plane wave and Gaussian beam comparing Figure 3, Figure 8 and Figure 11. In the second and fourth columns of Figure 11, the bimodal phenomenon still exists when a Bessel beam illuminates to a cutting LC microcylinder. Also, the adjacent peaks are more intense than Figure 3 and Figure 8. This creates some difficulties in determining the inflection point. Besides, the PH is formed by the forward scattering of particles and the interference of the self-healing Bessel beam. This is why the shape of PHs in the front of the particle is not a whole. Thus, the PH that appears in front of the particle bends twice.

To further elucidate the characteristics of the PHs formed by the Bessel beam acting on particles,

Table 3. The comparison of the parameters of the PH generated by cutting microcylinder particles with different extraordinary refractive indices using a zero-order Bessel beam.

${\mathit{n}}_{\mathbf{e}}$	Max Intensity (a.u.)	FWHM	Bending Angle
1.4	1.66	1.20$\lambda$	−18.35°
1.5	2.07	0.88$\lambda$	−12.51°
1.6	2.31	0.90$\lambda$	+22.02°
1.7	2.55	0.64$\lambda$	+11.55°
1.8	2.93	0.52$\lambda$	+16.19°
1.9	2.84	0.63$\lambda$	+13.26°

is given. As we observe in Figure 11, the electric field intensity at the inflection point gradually increases as $n_e$ increases. However, the influence of $n_e$ on FWHM cannot be judged in Table 3, although FWHM tends to narrow with the rise of $n_e$. Regarding the influence of $n_e$ on the bending angle, there is no obvious rule in Table 3. Thus, Figure 12 is calculated to explore how $n_e$ affects the characteristics of PHs. Figure 12a–c are the changing trend of bending angle, electric field intensity of inflection point, and FWHM as $n_e$ changes. Because of the self-restoring property of the Bessel beam and the influence of bimodal phenomena, the changing trend of the bending angle is chaotic with the increase in $n_e$, as shown in Figure 12a. We only found that the maximum value of 22.02° is achieved at $n_{\mathrm{e}}$ = 1.60, and the minimum value of 4.77° is achieved at $n_{\mathrm{e}}$ = 1.63. On the contrary, the electric field intensity of the inflection point and FWHM present a more obvious rule with the increase in $n_e$. In Figure 12b, the intensity of the inflection point shows a more robust trend as $n_e$ increases, although there is some volatility between $n_e$ = 1.84 and $n_e$ = 1.90. In Figure 12c, as $n_e$ increases, the overall trend of FWHM is gradually narrowing. However, there are some abrupt peaks in the range 1.53 < $n_e$ < 1.59 and 1.86 < $n_e$ < 1.92. It results from the self-restoring effect of the Bessel beam and the bimodal phenomenon.

4. Conclusions

The FDTD method was employed in this study to compute the PH characteristics (maximum intensity, bending angle, and FWHM) of anisotropic cutting microcylinder particles under different structured beam illuminations (plane wave, Gaussian beam, zero-order Bessel beam). The results show that without changing the beam properties, the PH generated by incident light can be influenced by changing the $n_{\mathrm{e}}$ of the particles. The paper analyzes the influence of $n_{\mathrm{e}}$ on PH characteristics, which contributes to increasing the degree of freedom in PH modulation. We also observed that introducing birefringence can change the bending direction of PH. The results show that regardless of the beam type, as the $n_{\mathrm{e}}$ increases, the maximum intensity of the PH also increases. The overall trend of the FWHM initially decreases and then enters a fluctuating state with no apparent regularity in the bending angle. The PH generated by a plane wave has a higher maximum intensity and smaller FWHM, which is beneficial for applications in super-resolution imaging. The PH generated by a Gaussian beam has a lower maximum intensity than the plane wave but higher than the Bessel beam, and the smallest FWHM falls between the two. However, its length is longer, which is beneficial for applications in optical manipulation. The PH generated by a Bessel beam has the minor maximum intensity and the largest FWHM. Still, it exceeds the diffraction limit and exhibits bending twice due to its self-recovery property. This paper provides a new way to modulate PH. Besides, it also provides insights into the applications of different beams in PH generation and offers novel models for designing PH optical tweezers. The results of this paper have promising application prospects in nanoparticle manipulation, optical detection, and biomedicine.