Optical Force on a Metal Nanorod Exerted by a Photonic Jet

In this article, we study the optical force exerted on nanorods. In recent years, the capture of micro-nanoparticles has been a frontier topic in optics. A Photonic Jet (PJ) is an emerging subwavelength beam with excellent application prospects. This paper studies the optical force exerted by photonic jets generated by a plane wave illuminating a Generalized Luneburg Lens (GLLs) on nanorods. In the framework of the dipole approximation, the optical force on the nanorods is studied. The electric field of the photonic jet is calculated by the open-source software package DDSCAT developed based on the Discrete Dipole Approximation (DDA). In this paper, the effects of the nanorods’ orientation and dielectric constant on the transverse force Fx and longitudinal force Fy are analyzed. Numerical results show that the maximum value of the positive force and the negative force are equal and appear alternately at the position of the photonic jet. Therefore, to capture anisotropic nanoscale-geometries (nanorods), it is necessary to adjust the position of GLLs continuously. It is worth emphasizing that manipulations with nanorods will make it possible to create new materials at the nanoscale.


Introduction
Optical capture by manipulating neutral atoms by lasers was first reported by Ashkin et al. in 1970 [1]. At present, optical manipulation can achieve the capture of living cells and organelles [2], DNA rotation [3], and chromosome surgery [4] with low damage, which is very important in modern medicine and biology. Later, optical capture was gradually applied to the manipulation of various nanomaterials. Nanomaterials and nanotechnology are already indispensable and important members of modern technology. For example, quantum dots play an important role in fluorescence detection, bioluminescence detection and other fields due to their unique photoelectric properties [5]. Semiconductor nanowires [6] have excellent performance in electronics and photonics due to their tunable direct bandgap and high carrier mobility. Anisotropic nanoscale-geometries, including nanorods [7], can be used in molecular imaging and photothermal cancer therapy. The capture and manipulation of nanomaterials are significant in the optoelectronic industry. Different arrangements of nanomaterials have different optical and physical properties [8], and optical force is an important way to realize the complex arrangement structure of nanomaterials. In 2006, Pelton et al. [9] reported three-dimensional optical capture of gold nanorods and enhanced optical force by longitudinal surface plasmon resonance. In 2008, Selhuber-Unkel [10] performed a quantitative analysis of the optical force on gold nanorods. The results showed that the interaction between an optical trap and a nanorod is related to the particle's polarizability. In 2010, Tong et al. [11] used a linearly polarized

Discrete Dipole Approximation
In the above, we briefly introduced the method of DDA to calculate near-field scattering. In this section, we will analyze the principle of DDA in detail. The electric dipole moment of a polarizable (the polarizability is α i ) dipole is P i = α i E i [30]. The total electric field at r i in the calculation area can be expressed as [31]: where E inc,i and E sca,i are the incident electric field and scattered electric field at r i . −A ij P j represents the scattering field generated by the dipole (at r j ) at r i . Therefore, Equation (1) can be expressed as: where, A ij [32] represents the influence of the dipole at position r j on the dipole at position where, k is the wave vector, r ij is the distance between point i and point j, andr ij is the unit vector from i to j. 1 3 is the identity matrix.r ijrij and 1 3 are shown in Equations (3) and (4) [33,34]. r x , r y , and r z are the coordinate components ofr ij along the x, y, and z directions, respectively. ∑ j =i A ij P j represents the scattered field at that point: x r x r y r x r z r y r x r 2 y r y r z r z r x r z r y r 2 z   Now, we can get the electric dipole moment at any position [35,36]: where [37]: where, α CM i is the Clausius-Mossotti polarizabilities (as shown in Equation (8)), m and n i are the refractive index and the dielectric constant at any position. d is the distance between dipoles, b 1 , b 2 , and b 3 are constants, k is the wave vector,â andê are the unit vectors of the incident direction and the polarization direction [38].
In this paper, n i = n 0 1 + f 2 GLLs − (r 0 /R) 2 1/2 / f GLLs , where, n 0 = 1. f GLLs are the focal length normalized radii of GLLs. It is unitless. r o is the radial coordinate (o = 1, 2, 3......, 30) and R is the maximum radius of the GLLs. Due to the limitation of memory and computing speed, we only divide the GLLs into 30 layers, and we will verify the correctness Nanomaterials 2022, 12, 251 4 of 15 of this model in Section 3.1. The design method and correctness verification of GLLs in DDSCAT have been completed in our previous work [39], and will not be described in this paper. Defining A jj ≡ α −1 j , Equation (6) has the following form [40]: In DDSCAT, the inner (original target sites j) and outer (vacuum sites i) fields of a particle will be represented separately as [41]: Finally, the electric dipole moment outside the particle is 0.

Optical Force on a Nanorod
In the framework of the dipole approximation, the polarizability of arbitrarily oriented nanorod is put into the optical force equation as follows [42]: where, V is the volume of nanorods, ε m and ε d are the dielectric functions of metal and medium, respectively. The polarizability tensor of the particle is [29]: The coordinate system inside the particle is (ξ,η,ζ). The schematic diagram of the polarization distribution and orientation of the nanorods is shown in Figure 1. α z and α t represent the longitudinal (parallel to the axis) and transverse (parallel to the bottom surface) polarizability of the nanorod, respectively. Both the ξ and η axes are parallel to the bottom surface of the nanorod, so the polarizability along these two directions is the same. The rotation matrix connecting the two systems is [29,43]: The polarizability tensor of the rotated nanorod in the system of coordinates xyz is then done by the matrix product [44]:Â =R −1αR (15) and the nanorod dipole moment p excited by the total electric field E at the nanorod position r = (x, y, z) is: Finally, the optical force on the nanorods is [45]: (a) Nanorod coordinate system (b) Nanorod orientation The polarizability tensor of the rotated nanorod in the system of coordinates xyz is then done by the matrix product [44]: and the nanorod dipole moment p excited by the total electric field E at the nanorod position ( , , ) r x y z = is: Finally, the optical force on the nanorods is [45]: By rewriting p as: Equation (17) can be expanded to: In this paper, we will analyze the x F , longitudinal force y F and z F exerted by the photonic jet on a nanorod.

Results
This section will analyze the simulation results. We use DDSCAT to generate a GLLs containing 30 layers with different refractive indices. The internal structure of GLLs is Figure 1. (a,b) respectively show the nanorod's coordinate system (ξ,η,ζ) and the angle between the axis (ξ) of the nanorod and the (x,y,z) coordinate system (θ 0 , ϕ 0 ). θ 0 represents the angle between the axis of the nanorod and the z-axis, and ϕ 0 represents the angle between the projection of the axis of the nanorod on the xoy plane and the x-axis.
By rewriting p as: Equation (17) can be expanded to: In this paper, we will analyze the F x , longitudinal force F y and F z exerted by the photonic jet on a nanorod.

Results
This section will analyze the simulation results. We use DDSCAT to generate a GLLs containing 30 layers with different refractive indices. The internal structure of GLLs is shown in Figure 2. The thickness of each layer is almost the same, and the refractive index of each layer can be calculated by n i = n 0 1 + f 2 GLLs − (r 0 /R) 2 1/2 / f GLLs . In all numerical simulations, the maximum radius R and focal length f GLLs of GLLs are 2 µm and 1.2 a.u., and the wavelength will be introduced separately in each section. The center of mass of GLLs is at (0, 0, 0), and the plane wave always propagates along +x and is polarized along the y-direction. The reason for choosing these parameters is to ensure correctness (compared with Mie theoretical calculation results) and a better jet effect. The nanoparticles are gold nanorods, and their polarizability is given in Equations (11)- (13). As the calculation framework in this paper is dipole approximation, the volume of nanoparticles must meet specific requirements, and the effective radius of nanorods must meet certain requirements 2πa e f f 1. In this paper, we only discuss the effect of orientation and dielectric constant of nanorod on F x , F y , and F z . Both electric field (a.u.) and optical force (in N) are displayed in the xoy plane and the positions of GLLs are marked with white circles. When the electric field is shown separately, the value is |E|/|E 0 |. For convenience, we will use |E| to represent the value. Because the photonic jet is a superposition of the incident and scattering fields on the shadow side of the particle, we must know the ratio of the field in the area of the photonic jet to the incident field. There are two other important parameters for PJ, which are the focal length f and the full width at half maxima (FWHM) at the focal point. These two parameters describe the distance from the focal point of the PJ to the outer surface of the GLLs and the width where the intensity is half the intensity of the focal point, respectively. They respectively represent the horizontal and vertical scales of the PJ. When the FWHM is less than half of the wavelength, the PJ may break the diffraction limit, so these two scales tend to be measured in wavelength λ. We use r1…r28 to represent concentric rings from 1 to 28 layers, R represents the maximum radius of GLLs, and fGLLs represents the focal length of GLLs. In the following, we first compare the photonic jet of GLLs under the DDA framework with that under the Mie theory framework. Here, we will only show the comparison results of photonic jet with wavelength 0.5 µm and 0.6328 µm. Then we analyzed the influence of the orientation of the nanorods on the optical force, and finally analyzed the influence of the wavelength on the optical force.
This section may be divided by subheadings. It should provide a concise and precise description of the experimental results, their interpretation, as well as the experimental conclusions that can be drawn.

Numerical Validation
In this section, we calculate the photonic jet generated by a plane wave irradiating a GLLs respectively through DDA and Mie theory and compare the results. The schematic diagram of the PJ irradiating nanorod is shown in Figure 3. DDSCAT and Jan Schäfer [46][47][48][49][50]'s Mie theory software were used, and the two results were normalized by I norm = (I i − I min )/(I max − I min ) [51], where, I i , I min and I max are the intensity at any position, the maximum value, and the minimum value, respectively. The error of all comparisons is within one-thousandth. As shown in Figure 4, the wavelengths are 0.5 ( Figure 4a) and 0.6328 µm (Figure 4b) respectively, the maximum radius of GLLs is 2 µm (i.e., Mie size parameter q ∼20), and the focal length is 1.2 a.u.. We use r1…r28 to represent concentric rings from 1 to 28 layers, R represents the maximum radius of GLLs, and fGLLs represents the focal length of GLLs.    respectively. The first picture in each row is the intensity distribution along the x-axis through the focal point of the photonic jet. The second picture is also the intensity distribution but along the yaxis through the focal point.

Orientation of Nanorods
In this section, we will analyze the influence of nanorod's orientation on the optical force. As mentioned earlier, the focal length GLLs f and the maximum radius of GLLs are 1.2 and 2 μm. The incident wavelength is 0.6328 μm, the dielectric constant of the gold nanorod is −11.740 + 1.2611i, and the volume is 4.2 × 10 −3 μm 3 . We will first show the electric field of the photonic jet under this set of parameters, as shown in Figure 5. The intensity of the focal point of the photonic jet is 51.7 a.u., the focal length f is 0.62 λ, and the FWHM is 0.5 λ. Next, we will use two subsections to discuss the influence of the orientation 0 ϕ and 0 θ of the nanorods on the optical force. The optical force in this paper has several discrete regions. For convenience, we define the independent region of optical force as discrete optical force, as shown in Figure  6. Figure 6a,b show two kinds of discrete optical forces, respectively. The first picture in each row is the intensity distribution along the x-axis through the focal point of the photonic jet. The second picture is also the intensity distribution but along the y-axis through the focal point.

Orientation of Nanorods
In this section, we will analyze the influence of nanorod's orientation on the optical force. As mentioned earlier, the focal length f GLLs and the maximum radius of GLLs are 1.2 and 2 µm. The incident wavelength is 0.6328 µm, the dielectric constant of the gold nanorod is −11.740 + 1.2611i, and the volume is 4.2 × 10 −3 µm 3 . We will first show the electric field of the photonic jet under this set of parameters, as shown in Figure 5. The intensity of the focal point of the photonic jet is 51.7 a.u., the focal length f is 0.62 λ, and the FW HM is 0.5 λ. Next, we will use two subsections to discuss the influence of the orientation ϕ 0 and θ 0 of the nanorods on the optical force.  respectively. The first picture in each row is the intensity distribution along the x-axis through the focal point of the photonic jet. The second picture is also the intensity distribution but along the yaxis through the focal point.

Orientation of Nanorods
In this section, we will analyze the influence of nanorod's orientation on the optical force. As mentioned earlier, the focal length GLLs f and the maximum radius of GLLs are 1.2 and 2 μm. The incident wavelength is 0.6328 μm, the dielectric constant of the gold nanorod is −11.740 + 1.2611i, and the volume is 4.2 × 10 −3 μm 3 . We will first show the electric field of the photonic jet under this set of parameters, as shown in Figure 5. The intensity of the focal point of the photonic jet is 51.7 a.u., the focal length f is 0.62 λ, and the FWHM is 0.5 λ. Next, we will use two subsections to discuss the influence of the orientation 0 ϕ and 0 θ of the nanorods on the optical force. The optical force in this paper has several discrete regions. For convenience, we define the independent region of optical force as discrete optical force, as shown in Figure  6. Figure 6a,b show two kinds of discrete optical forces, respectively. The optical force in this paper has several discrete regions. For convenience, we define the independent region of optical force as discrete optical force, as shown in Figure 6. Figure 6a,b show two kinds of discrete optical forces, respectively.  ϕ =°°°°. The reason why we choose these angles is because of the unique properties of the discrete optical force at these angles. There are significant differences in the size and shape of these discrete optical forces. Of course, we have calculated the optical forces at all angles. However, the distribution patterns of these optical forces at these four angles is the most representative. We will explain their unique properties one by one. When 0 ϕ is one of these four cases, the optical force (especially y F ) has a unique and typical distribution The force on one side is more potent than the other side. Such particles may always move to one side in the y-direction. At this time, we can also move GLLs along the x-direction so that the nanorods always receive the same magnitude and opposite optical forces on both sides of y. This allows the nanorods to be in a state of dynamic equilibrium and to be stably captured in the y-direction. Figure 8 shows the variation of the maximum and minimum values of x F , y F , and z F with 0 ϕ . As shown in Figure 8a

ϕ 0 Orientation of Nanorods
In this section, we will study the influence of the change of ϕ 0 on the optical force (θ 0 = 90 • ). We respectively show F x and F y when ϕ 0 = 0 • , 90 • , 99 • , and 105 • . The reason why we choose these angles is because of the unique properties of the discrete optical force at these angles. There are significant differences in the size and shape of these discrete optical forces. Of course, we have calculated the optical forces at all angles. However, the distribution patterns of these optical forces at these four angles is the most representative. We will explain their unique properties one by one. When ϕ 0 is one of these four cases, the optical force (especially F y ) has a unique and typical distribution pattern of positive and negative forces. When ϕ 0 = 0 • , F y is almost perfectly symmetric with respect to y = 0, including the magnitude and scope of positive and negative forces. When ϕ 0 = 90 • , the F y in the y > 0 area is significantly smaller than the F y in the y < 0 area (reversed when ϕ 0 = 105 • ). When ϕ 0 = 99 • , the scope of positive force increases significantly, and the negative force converges on the particle surface. Still, the maximum value of the negative force is higher than that of the positive force. Because of the uniqueness of the optical forces in these four angles, we decided to show them, as shown in Figure 7a-d. When ϕ 0 is equal to other values, the distribution pattern of positive and negative forces is always similar to the above four cases, so it will not be shown. In increasing ϕ 0 from 0 to 180 degrees, the change of F x is mainly reflected in the intensity of the force. At the position of the PJ, the optical force is symmetrical about y = 0, and the positive force and the negative force alternately appear. The maximum value of the positive force and the negative force are almost equal, and both have a clear action area. Therefore, when capturing nanorods with different orientations in the light propagation direction (x), the negative force can always be exerted on the nanorods by accurately moving the GLLs to achieve the purpose of capture. F y has different directions of action on both sides of y = 0, and the positive and negative forces also appear alternately along +x. Therefore, the nanorods oscillate repeatedly on both sides of y = 0. However, when ϕ 0 ≥ 45 • , the positive and negative forces on both sides of y = 0 appear asymmetry. The force on one side is more potent than the other side. Such particles may always move to one side in the y-direction. At this time, we can also move GLLs along the x-direction so that the nanorods always receive the same magnitude and opposite optical forces on both sides of y. This allows the nanorods to be in a state of dynamic equilibrium and to be stably captured in the y-direction. Figure 8 shows the variation of the maximum and minimum values of F x , F y , and F z with ϕ 0 . As shown in Figure 8a,b we calculate the change rules of the maximum and minimum values of F x , F y , and F z in the process of ϕ 0 increasing from 0 • to 180 • . Since the light propagates in the x direction and θ 0 = 90 • , the radius of the bottom surface of the nanorod is small, so the intensity of F z is relatively tiny compared to F x and F y . Through research, it can be Nanomaterials 2022, 12, 251 9 of 15 found that the maximum and minimum values of F x and F y have precisely the same trend with the increase of ϕ 0 . The optical force decreases first, reaching a minimum at 39 • . Then increase, get a local maximum at 69 • , and drop immediately. When ϕ 0 = 99 • , it comes the local minimum value and increases. Reach the maximum value when ϕ 0 = 159 • . In summary, different ϕ 0 will cause the force on nanorods to be very different, so we can propose targeted solutions for particles with different ϕ 0 to achieve stable capture. so that the nanorods always receive the same magnitude and opposite optical forces on both sides of y. This allows the nanorods to be in a state of dynamic equilibrium and to be stably captured in the y-direction. Figure 8 shows the variation of the maximum and minimum values of
Similarly, these angles were chosen to demonstrate the unique properties of discrete optical forces and the distribution patterns of positive and negative forces. The distribution pattern of positive and negative forces determines the direction of force and the way to capture the particles. When 0 0 θ =°, the discrete optical force of y F is almost perfectly symmetric with respect to y = 0. When 0 30 θ =°, the discrete optical force of y F in the 0 y < area is smaller than the y F in the 0 y > area. When 0 60 θ =°, the scope of negative force increases significantly, and the positive force converges on the particle surface. Still, the maximum value of the positive force is higher than that of the negative force. When 0 180 θ =°, the range of negative force is greater than that of positive force, but the maximum value of negative force is smaller. Because of the uniqueness of the optical forces in these four angles, we decided to show them, as shown in Figure 9a

θ 0 Orientation of Nanorods
In this section, we will discuss the influence of θ 0 on optical forces (ϕ 0 = 45 • ). As shown in Figure 9a-d, we offer F x and F y at θ 0 = 0 • , 30 • , 60 • , and 180 • , respectively. Similarly, these angles were chosen to demonstrate the unique properties of discrete optical forces and the distribution patterns of positive and negative forces. The distribution pattern of positive and negative forces determines the direction of force and the way to capture the particles. When θ 0 = 0 • , the discrete optical force of F y is almost perfectly symmetric with respect to y = 0. When θ 0 = 30 • , the discrete optical force of F y in the y < 0 area is smaller than the F y in the y > 0 area. When θ 0 = 60 • , the scope of negative force increases significantly, and the positive force converges on the particle surface. Still, the maximum value of the positive force is higher than that of the negative force. When θ 0 = 180 • , the range of negative force is greater than that of positive force, but the maximum value of negative force is smaller. Because of the uniqueness of the optical forces in these four angles, we decided to show them, as shown in Figure 9a-d. When θ 0 is other values, the distribution pattern of optical force is similar to the above four situations, and the repeated content will not be shown in this article. θ 0 in 0 • − 42 • , F x and F y are almost all symmetric with respect to y = 0, with only variations in the intensity of the force. Still, the range of F y changes significantly around 30 • , 60 • and 180 • , as shown in Figure 9b-d. F x is not symmetric about y = 0. The range of the F x 's negative force is significantly increased. The wide distribution of negative forces allows particles to be captured more easily. Figure 10 shows the variation of the maximum and minimum values of F x , F y , and F z with θ 0 . As shown in Figure 10a,b, we studied the maximum values of the positive and negative forces of F x , F y , and F z in the range of 0 • ≤ θ 0 ≤ 180 • . The numerical results show that the maximum values of F x and F y decrease first, then increase and finally decrease again with the increase of θ 0 . When θ 0 = 120 • , F z is a local minimum value. Otherwise, the trend of F z increasing with θ 0 is basically the same as that of F x . The maximum value of positive force and negative force are almost equal. Under the same parameters, the area where the maximum positive force and the negative force are located in the photon ejection is fixed. By moving the GLLs, the nanorods can be accurately captured.
decrease first, then increase and finally decrease again with the increase of 0  . When 0 120   , z F is a local minimum value. Otherwise, the trend of z F increasing with 0  is basically the same as that of x F . The maximum value of positive force and negative force are almost equal. Under the same parameters, the area where the maximum positive force and the negative force are located in the photon ejection is fixed. By moving the GLLs, the nanorods can be accurately captured.

Dielectric Constant
In this section, we discuss the effects of the real ( 1 ε ) and imaginary ( 2 ε ) parts of the nanorod's dielectric constant on the optical force, as shown in Figure 11. The GLLs f and R of GLLs are 1.2 and 2 μm. The wavelength is 0.6328 μm, and the volume of the nanorod is 4.2 × 10 −3 μm 3 . In Figure 11a

Dielectric Constant
In this section, we discuss the effects of the real (ε 1 ) and imaginary (ε 2 ) parts of the nanorod's dielectric constant on the optical force, as shown in Figure 11. The f GLLs and R of GLLs are 1.2 and 2 µm. The wavelength is 0.6328 µm, and the volume of the nanorod is 4.2 × 10 −3 µm 3 . In Figure 11a,b we show the maximum and minimum optical force changes with the real part (ε 1 ) of the dielectric constant when the imaginary part ε 2 of the nanorod's dielectric constant is 1, respectively. From the figures, we can find that the optical force first increases and then decreases with the increase of ε 1 . When ε 1 = −2, the optical force has the local maximum value. From Figure 11c,d, we can find that as ε 2 increases, the optical force gradually increases.

Conclusions
In this paper, we studied the optical forces x F and y F exerted by a photonic jet on nanorods under the framework of the dipole approximation. The PJ is generated by a plane wave irradiating a GLLs, and the numerical result is calculated by DDSCAT. We

Conclusions
In this paper, we studied the optical forces F x and F y exerted by a photonic jet on nanorods under the framework of the dipole approximation. The PJ is generated by a plane wave irradiating a GLLs, and the numerical result is calculated by DDSCAT. We divided GLLs into 30 layers of concentric rings with different refractive indices. We compare the calculation results of DDSCAT with the results of the Mie theory to validate the PJ's electric field. Then we use the PJ to illuminate the nanorods and study the optical forces F x and F y on the nanorods. Of course, the size of nanorods must satisfy the Rayleigh approximation. The effects of the orientation and dielectric constant of nanorods on the optical force are investigated. The angle between the axis of the nanorod and the z-axis is represented by θ 0 , and the angle between the projection of the axis on the xoy plane and the x-axis is defined by ϕ 0 . We first discuss ϕ 0 's effect on F x and F y by making θ 0 = 90 • (the axis of the nanorods is in the xoy plane). We find that F x is always symmetric about the y-axis, and F y is not symmetric about the y-axis, only around ϕ 0 = 90 • and ϕ 0 = 105 • . In contrast, the intensity of F z can be ignored since the maximum value of the positive and negative optical forces are almost equal. It is necessary to constantly adjust the position of the GLLs when capturing these oriented nanorods so that the nanorods are always in the area of negative or positive forces. Then we investigate the effect of θ 0 on optical force when ϕ 0 = 45 • . The numerical results show that F x is asymmetric about the y-axis near θ 0 = 45 • and 60 • , and the area of the negative force of F y increases significantly (including θ 0 = 180 • ), and the positive force converges on the surface of GLLs. Similarly, the maximum values of the positive and negative optical forces are equal. Due to the wider distribution of negative forces at certain angles, it is easier to adjust GLLs to capture nanorods. Finally, we discussed the effect of the dielectric constant of the nanorods on the optical force. Under the same conditions, a larger dielectric constant generally leads to a powerful optical force. In this paper, the optical force exerted by the photonic jet on elongated particles in the form of nanorods instead of spherical nanoparticles with different orientations and materials (dielectric constant) is studied. The numerical results describe the direction and magnitude of the optical force, which makes it possible to directionally manipulate the nanorods. Note that the presented results will be valid not only for GLLs, but also for other particles [52] that form a photonic jet with similar characteristics. These results are expected to provide theoretically support for the manipulation of nanorods and the arrangement of nanoarrays.

Conflicts of Interest:
The authors declare no conflict of interest.

Abbreviations
The following abbreviations are used in this manuscript: PJ Photonic jet GLLs Generalized Luneburg Lens DDA Discrete Dipole Approximation FWHM Full Width at Half Maxima