Optical pulling force upon elliptical cylinder nanoparticles in the infrared range

Mohsen Balaei; Tayebeh Naseri

doi:10.1364/OPTCON.451176

1. Introduction

Light as an electromagnetic radiation field carries energy and momentum. The exchange of the momentum with interacting particles may lead to an exerted radiation pressure [1,2]. Based on the nature of this force, it can be classified into three different categories; the gradient force, the scattering force, and the curl of spin angular momentum force which can be applied to manipulate atoms, molecules, and biological cells.

This approach proved significantly applicable in studying the properties of cell cytoplasm, stretching DNA chains, and measuring the stepping motion of an organism [3]. Moreover, the optical force has been used to implement atom cooling which has innumerable applications such as atom interferometers and observing Bose-Einstein condensation (BEC) [4,5], optical tweezers [6–8] and optical binding [9].

If the optical force on the heavy microscopic particles is exerted, the acceleration is virtually ignorable. On the other hand, this acceleration can be considerably large for small particles whose sizes are comparable with the incident light wavelength. It should be taken into consideration that a plane wave tends to push a particle in the forward direction. Under special circumstances, small particles can be accelerated in the opposite direction to the light-propagation direction due to the momentum conservation.

Recently, the optical pulling force (OPF) which can exert a force on an object against the direction of the applied light, has attracted broad attention because of its counterintuitive nature such as optical sorting [10], self-assembling, remote sampling [11], miniaturization of nanodevices [3], and enantioselective manipulation [13]. The optical pulling force can be achieved by a number of initiatives [12]; using the Bessel beam [14], Gaussian beam [15,16] and other tractor beams [16].

Apart from using one or more structured light beams, OPF is also achievable by using an object with proper optical features. The first structure is an optical gain medium [11,16]. The idea of negative radiation pressure using gain media, such as slabs, spheres, and deep subwavelength structures has been considerably investigated [17–20].

It has been further illustrated that the optical pressure on the anisotropic Rayleigh spheres is enhanced at the electric dipole resonance, and may also be enhanced by adjusting the anisotropic parameters. Quite the reverse, the optical forces on the anisotropic spheres might be considerably reduced for anisotropic spheres in the presence of electromagnetic transparency [21].

The momentum conservation law, however, introduces an essential constraint on the optical pressure to be repulsive in paraxial fields and strictly limits the capabilities of optomechanical control in preventing pulling force acting on nanoparticles and molecules. Applying the complex frequency plane reveals a fascinating potential to achieve a pulling force for a passive resonant object of any shape and composition, even in the paraxial approximation. This approach is illuminated on a dielectric Fabry–Perot cavity and a high-refractive-index dielectric nanoparticle [22].

Nanoparticles display unique properties that are quite different from those of individual atoms. In reality, the optical properties of the nanoparticles strongly depend on its shape. Because of their small size effect and tunable surface plasmon resonances, these particles yield a broad range of applications in chemical and biological sensors, plasmonic system and other fields of science and technology.

Manipulating and engineering different plasmon frequencies and consequently desired optical properties have been gained by playing with different shapes, as recently shown for nanowires, nanorings, nanorods, nanorices and nanostars [23].

In this study, we revisit the issue of optical pulling force by stepping out to the elliptic cylindrical geometry and investigating the its exceptional optical performance in the quasi-static approximation. The results show that tailoring of exposed nanoparticle allows enhancement of either the repulsive force or the pulling force of the presented structure in comparison with the conventional nanoparticle’s shapes. We adopt Maxwell’s stress tensor method to investigate the optical force with complex medium under the incident plane wave.

2. Theoretical model and methods

The physical model proposed in our work is schematically depicted in Fig. 1. An elliptic cylindrical nanoparticle consisting of two different media (with relative permittivity $\epsilon _1$ and $\epsilon _2$) immersed in the host medium with relative permittivity $\epsilon _h$ is illuminated by a linearly polarized and y-directed plane electromagnetic wave.

Fig. 1. Schematic of the structure of nanoparticle and the unit vectors of elliptical cylindrical coordinate system(on x-y plate).

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So as to evaluate the potentials and fields inside and outside this elliptic cylinder, elliptic coordinates are implemented for the solution of the problem. The most common definition of elliptic cylindrical coordinates is $(u,\nu, z)$ and the transformation equations from elliptic to Cartesian coordinates are given be:

(1)$$\begin{cases} x=a\cosh(u)\cos(\nu)\\ y=a\sinh(u)\sin(\nu)\\ z=z \end{cases}$$

where $u$ is a nonnegative real number and $\nu$ $\in$ $[0,2\pi ]$. Unit vectors in elliptic cylindrical coordinates are written as [24]:

(2)$$\begin{cases} \hat{u}=\frac{\sinh(u)\cos(\nu)}{\sqrt{\sinh^{2}(u)+sin^{2}(\nu)}}\hat{x}+\frac{\cosh(u)\sin(\nu)}{\sqrt{\sinh^{2}(u)+sin^{2}(\nu)}}\hat{y}\\ \\ \hat{\nu}={-}\frac{\cosh(u)\sin(\nu)}{\sqrt{\sinh^{2}(u)+sin^{2}(\nu)}}\hat{x}+\frac{\sinh(u)\cos(\nu)}{\sqrt{\sinh^{2}(u)+sin^{2}(\nu)}}\hat{y} \end{cases}$$

By assuming that $l_z>>l_x$ and $l_z>>l_y$ $\Rightarrow$

$\begin {cases} \frac {\partial \Phi }{\partial z}=0\\ \nabla ^{2}\Phi =\frac {1}{a^{2}(\sinh ^{2}(u)+\sin ^{2}(\nu ))}\big (\frac {\partial ^{2}}{\partial u^{2}}+\frac {\partial ^{2}}{\partial \nu ^{2}}\big )\Phi =0 \end {cases}$

General solution for electrical potential is:

\displaystyle\Phi=\sum_{n=0}^{\infty}\big[A_ncosh(nu)cos(n\nu)+B_ncosh(nu)sin(n\nu)\big]+\sum_{n=0}^{\infty}\big[C_nsinh(nu)cos(n\nu)+D_nsinh(nu)sin(n\nu)\big]

It must also satisfy Laplace’s equation boundary conditions.

(3)$$\begin{cases} \vec{E}_{out,2(u=\rho,\nu)}.\hat{\nu}=\vec{E}_{in,2(u=\rho,\nu)}.\hat{\nu} \\ \vec{E}_{in,2(u=0,\nu)}.\hat{\nu}=\vec{E}_{in,1(u=0,2\pi-\nu)}.\hat{\nu} \hspace{1cm} \textrm{(Electric field; the tangential component)}\\ \vec{E}_{in,1(u=\rho,\nu)}.\hat{\nu}=\vec{E}_{out,1(u=\rho,\nu)}.\hat{\nu} \end{cases}$$

(4)$$\begin{cases} \epsilon_h\vec{E}_{out,2(u=\rho,\nu)}.\hat{u}=\epsilon_2\vec{E}_{in,2(u=\rho,\nu)}.\hat{u} \\ \epsilon_2\vec{E}_{in,2(u=0,\nu)}.\hat{u}=\epsilon_1\vec{E}_{in,1(u=0,2\pi-\nu)}.\hat{u} \hspace{1cm} \textrm{(Displacement vector; the normal component)}\\ \epsilon_1\vec{E}_{in,1(u=\rho,\nu)}.\hat{u}=\epsilon_h\vec{E}_{out,1(u=\rho,\nu)}.\hat{u} \end{cases}$$

All requirements are accommodated by the choices, and suitable solutions for upper and lower parts of the elliptic cylinder have been calculated as:

(5)$$if\hspace{.25cm} 0<\nu<\pi: \hspace{.5cm} \begin{cases} \Phi_{out,2}={-}E_0a\sinh(u)\sin(\nu)-H\frac{E_0}{a}e^{{-}u}\sin(\nu)\\ \Phi_{in,2}={-}E_0a\frac{\epsilon_1}{\epsilon_2}A\sinh(u)\sin(\nu)+E_0aB\cosh(u)\sin(\nu) \end{cases}$$

(6)$$if\hspace{.25cm} \pi<\nu<2\pi: \hspace{.5cm} \begin{cases} \Phi_{out,1}={-}E_0a\sinh(u)\sin(\nu)-F\frac{E_0}{a}e^{{-}u}\sin(\nu)\\ \Phi_{in,1}={-}E_0aA\sinh(u)\sin(\nu)-E_0aB\cosh(u)\sin(\nu)\ \end{cases}$$

where the coefficients $A$ and $B$ may be determined:

(7)$$\begin{cases} A=\frac{[\sinh\rho+\cosh\rho][2\cosh\rho+\frac{(\epsilon_1+\epsilon_2)}{\epsilon_h}\sinh\rho]}{[\frac{\epsilon_1}{\epsilon_2}\sinh\rho+\frac{\epsilon_1}{\epsilon_h}\cosh\rho][\frac{\epsilon_1}{\epsilon_h}\sinh\rho+\cosh\rho]+[\sinh\rho+\frac{\epsilon_1}{\epsilon_h}\cosh\rho][\cosh\rho+\frac{\epsilon_2}{\epsilon_h}\sinh\rho]}\\ B=\frac{[\sinh\rho+\cosh\rho]\sinh\rho\frac{(\epsilon_1-\epsilon_2)}{\epsilon_2}}{[\frac{\epsilon_1}{\epsilon_2}\sinh\rho+\frac{\epsilon_1}{\epsilon_h}\cosh\rho][\frac{\epsilon_1}{\epsilon_h}\sinh\rho+\cosh\rho]+[\sinh\rho+\frac{\epsilon_1}{\epsilon_h}\cosh\rho][\cosh\rho+\frac{\epsilon_2}{\epsilon_h}\sinh\rho]} \end{cases}$$

According to the boundary conditions, the following electric fields can be obtained:

(8)$$\begin{cases} \vec{E_1}=\frac{E_0}{\sqrt{\sinh^{2}u+\sin^{2}\nu}}\big[(A\cosh u+B\sinh u)\sin\nu \hat{u}+(A\sinh u +B\cosh u)\cos\nu \hat{\nu}\big]\\ \vec{E_2}=\frac{E_0}{\sqrt{\sinh^{2}u+\sin^{2}\nu}}\big[(\frac{\epsilon_1}{\epsilon_2}A\cosh u-B\sinh u)\sin\nu \hat{u}+(\frac{\epsilon_1}{\epsilon_2}A\sinh u -B\cosh u)\cos\nu \hat{\nu}\big] \end{cases}$$

Based on Maxwell's equations, the magnetic fields can be expanded accordingly

(9)$$\vec{H}=\frac{1}{i\mu_0\mu\omega}\vec{\nabla}\times\vec{E} \Rightarrow \hspace{.5cm} \begin{cases} \vec{H_1}=\frac{E_0ke^{ikx}\hat{z}}{\mu_0\mu\omega}\big[A+\frac{\cosh u\sinh u}{\sinh^{2}u+sin^{2}\nu}B\big]\\ \vec{H_2}=\frac{E_0ke^{ikx}\hat{z}}{\mu_0\mu\omega}\big[\frac{\epsilon_1}{\epsilon_2}A-\frac{\cosh u\sinh u}{\sinh^{2}u+sin^{2}\nu}B\big] \end{cases}$$

The optical force on the particle is the integration over any surface S with the outward normal vector of the surface $n_j$ for the nanoparticle [25–27]

(10)$$F_i=\mathop{{\int\!\!\!\!\!\int}\mkern-21mu \bigcirc}{\sum_{j=1}^{3}T_{ij}n_jds}$$

The elements of Maxwell stress tensor can be expressed as:

$T_{ij}=\epsilon _0\epsilon E_iE_j+\mu _0 \mu H_iH_j-\frac {1}{2}(\mu _0 \mu \vec {H}.\vec {H}+\epsilon _0\epsilon \vec {E}.\vec {E})\delta _{ij}$

The optical force’s component in $u$ direction $(F_u)$ is:

(11)$$F_u=\mathop{{\int\!\!\!\!\!\int}\mkern-21mu \bigcirc}{( T_{uu}n_u+T_{u\nu}n_{\nu}+T_{uz}n_z)ds}$$

We can decompose the $F_u$ in X-direction and Y-direction. For this purpose, we can use scalar product of $\hat {u}$ and $\hat {x}$ on the integral as below:

(12)$$F_u^{x}=\mathop{{\int\!\!\!\!\!\int}\mkern-21mu \bigcirc}{\hat{u}.\hat{x}(T_{uu}n_u+T_{u\nu}n_{\nu}+T_{uz}n_z)ds}$$

In general to compute $F_i$ in direction $\hat {p}$:

(13)$$F_i^{p}=\mathop{{\int\!\!\!\!\!\int}\mkern-21mu \bigcirc}{\hat{i}.\hat{p}\sum_{j=1}^{3}T_{ij}n_jds}$$

where i=u, $\nu$ and z also $\hat {p}=\hat {x},\hat {y}$ and $\hat {z}$.

To obtain the total time-averaged optical force on the particle, the below theorem has been applied:

<A_{(t)}B_{(t)}>_T=\frac {1}{2}\Re (A_0B^{*}_0) \hspace{0.8cm} \textrm{if} \hspace{0.8cm} A_{(t)}=A_0e^{-i\omega t}+C.C \hspace{0.8cm} \textrm{and}\hspace{0.8cm} B_{(t)}=B_0e^{-i\omega t}+C.C

The bracket $< >$ corresponds to the time average over an optical cycle. By using Eq. (13) and supposing $\mu =\mu _0$, the total force component in cartesian coordinates can be calculated as:

(14)$$<F_x>_T=\frac{1}{4}al_z\epsilon_0 \sinh(\rho)\bigg(\Re(I_1)-\Re(I_2)-\frac{\mu_0}{\epsilon_0}\Re(I_3)+2\Re(I_4)+ \Re(I_5)-\Re(I_6)-\frac{\mu_0}{\epsilon_0}\Re(I_7)+2\Re(I_8)\bigg) \\$$

which:

\begin{cases} I_1=\int_{_0}^{{}^{\pi}}\epsilon_2\cos(\nu)|E_{2u}|^{2}d\nu \hspace{1cm} I_2=\int_{_0}^{{}^{\pi}}\epsilon_2\cos(\nu)|E_{2\nu}|^{2}d\nu\\ I_3=\int_{_0}^{{}^{\pi}}\cos(\nu)|H_{2}|^{2}d\nu \hspace{1.6cm} I_4=\int_{_0}^{{}^{\pi}}\epsilon_2\cos(\nu)E_{2u}E_{2\nu}^{*}d\nu\\ I_5=\int_{_{\pi}}^{{}^{2\pi}}\epsilon_1\cos(\nu)|E_{1u}|^{2}d\nu \hspace{1cm} I_6=\int_{\pi}^{{}^{2\pi}}\epsilon_1\cos(\nu)|E_{1\nu}|^{2}d\nu\\ I_7=\int_{_{\pi}}^{{}^{2\pi}}\cos(\nu)|H_{1}|^{2}d\nu \hspace{1.6cm} I_8=\int_{_{\pi}}^{{}^{2\pi}}\epsilon_1\cos(\nu)E_{1u}E_{1\nu}^{*}d\nu\\ \end{cases}

To calculate $<F_y>$ by Eq (13) must be considered that the surface $\rho =0$ has an important share (against to $<F_x>$) so we use $<F_{y,\rho \neq 0}>$ and $<F_{y,\rho =0}>$ notation to show the force on surface $\rho =0$ and other surfaces respectively.

(15)$$\displaystyle <F_y,\rho\neq 0>_T=\frac{1}{4}al_z\epsilon_0 \cosh(\rho)\bigg(\Re(I_1)-\Re(I_2)-\frac{\mu_0}{\epsilon_0}\Re(I_3)+2\Re(I_4)+ \Re(I_5)-\Re(I_6)-\frac{\mu_0}{\epsilon_0}\Re(I_7)+2\Re(I_8)\bigg)$$

which:

\begin{cases} I_1=\int_{_0}^{{}^{\pi}}\epsilon_2\sin(\nu)|E_{2u}|^{2}d\nu \hspace{1cm} I_2=\int_{_0}^{{}^{\pi}}\epsilon_2\sin(\nu)|E_{2\nu}|^{2}d\nu\\ I_3=\int_{_0}^{{}^{\pi}}\sin(\nu)|H_{2}|^{2}d\nu \hspace{1.6cm} I_4=\int_{_0}^{{}^{\pi}}\epsilon_2\sin(\nu)E_{2u}E_{2\nu}^{*}d\nu\\ I_5=\int_{_{\pi}}^{{}^{2\pi}}\epsilon_1\sin(\nu)|E_{1u}|^{2}d\nu \hspace{1cm} I_6=\int_{\pi}^{{}^{2\pi}}\epsilon_1\sin(\nu)|E_{1\nu}|^{2}d\nu\\ I_7=\int_{_{\pi}}^{{}^{2\pi}}\sin(\nu)|H_{1}|^{2}d\nu \hspace{1.6cm} I_8=\int_{_{\pi}}^{{}^{2\pi}}\epsilon_1\sin(\nu)E_{1u}E_{1\nu}^{*}d\nu\\ \end{cases}

(16)$$<F_y,\rho=0>_T=\frac{1}{4}al_z\epsilon_0 \bigg(\Re(I_1)-\Re(I_2)-\frac{\mu_0}{\epsilon_0}\Re(I_3)+2\Re(I_4)+ \Re(I_5)-\Re(I_6)-\frac{\mu_0}{\epsilon_0}\Re(I_7)+2\Re(I_8)\bigg) \\$$

which:

\begin{cases} I_1=\int_{_0}^{{}^{\pi}}\epsilon_2\sin(\nu)|E_{2u,\rho=0}|^{2}d\nu \hspace{1cm} I_2=\int_{_0}^{{}^{\pi}}\epsilon_2\sin(\nu)|E_{2\nu,\rho=0}|^{2}d\nu\\ I_3=\int_{_0}^{{}^{\pi}}\sin(\nu)|H_{2,\rho=0}|^{2}d\nu \hspace{1.6cm} I_4=\int_{_0}^{{}^{\pi}}\epsilon_2\sin(\nu)E_{2u,\rho=0}E_{2\nu,\rho=0}^{*}d\nu\\ I_5=\int_{_{\pi}}^{{}^{2\pi}}\epsilon_1\sin(\nu)|E_{1u,\rho=0}|^{2}d\nu \hspace{1cm} I_6=\int_{\pi}^{{}^{2\pi}}\epsilon_1\sin(\nu)|E_{1\nu,\rho=0}|^{2}d\nu\\ I_7=\int_{_{\pi}}^{{}^{2\pi}}\sin(\nu)|H_{1,\rho=0}|^{2}d\nu \hspace{1.6cm} I_8=\int_{_{\pi}}^{{}^{2\pi}}\epsilon_1\sin(\nu)E_{1u,\rho=0}E_{1\nu,\rho=0}^{*}d\nu\\ \end{cases}

So:

(17)$$<F_y>_T={<}F_y,\rho\neq 0>_T+{<}F_y,\rho=0>_T$$

By considering Eq (13) and this fact that the electric field has no components along z-axis and magnetic field strength has no component along x-axis or y-axis , can be understood that:

(18) $$<F_z>_T=0$$

3. Results and discussion

In order to validate our theoretical model, we utilize full wave simulations to evaluate the time-averaged electromagnetic force exerted on the elliptic cylinder nanoparticles illuminated by an arbitrarily optical field. We are now in a position to present numerical results. For simplicity, we assume both the background and the nanoparticle are non-magnetic with $\mu _1= \mu _2= \mu _h=1$. Dielectric function for SiC can be obtained via Drude-Lorentz’s permittivity model as below [28,29]:

(19)$$\epsilon_{SiC}{(\omega)}=\epsilon_{\infty}(\frac{\omega^{2}-\omega^{2}_l+i\gamma\omega}{\omega^{2}-\omega^{2}_t+i\gamma\omega})$$

where $\epsilon _{\infty }=6.7$, $\omega _l=1.827\times 10^{14}\frac {Rad}{s}$, $\omega _t=1.495\times 10^{14}\frac {Rad}{s}$ and $\gamma =0.9\times 10^{12}\frac {Rad}{s}$.

We study the impact of $\epsilon _1$ and $\epsilon _h$ on OPF in x-direction and y-direction exerted on the NP in Fig. 2.

Fig. 2. a) $\epsilon _h=1.5$ , b)$\epsilon _1=1.5$ , c) $\epsilon _h=1.5$ , d) $\epsilon _1=1.5$. $\lambda =11\mu m$, $L_x=50nm$ and $\epsilon _2=\epsilon _{SiC}$ are fixed for all cases.

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In Fig. 2(a) and Fig. 2(b), the half cylinder with complex susceptibility(SiC) in the nanoparticle can overcome the scattering forward force to obtain optical pulling force in both x and y directions and the loss is relatively small. The normalized optical forces $F_x/F0$ and $F_y/F0$ are plotted as a function of $\frac {L_y}{L_x}$($F_0\equiv \frac {1}{4}l_z\epsilon _0 E_0^{2}$).

The underlying physics is not difficult to conceive. In an optical medium with complex electric susceptibility, the total light momentum can be enhanced when stimulated emission occurs. Due to linear momentum conservation, the object will experience a pulling force. There are two plasmonic resonant modes, i.e., resonant long-wavelength $\lambda _+$ ; and the short-wavelength $\lambda _-$. The two figures, Fig. 2(a) and 2(b), show red shift in the resonance frequency which is accompanied by increasing the permittivity of the dielectric nanoparticle $\epsilon _2$ in both and $F_y$. The increase in the resonant magnitudes in the first peak in $F_x$ and the second peak in the second peak in $F_y$ result from decreasing the permittivity of the dielectric half of nanoparticle. The resonance wavelength is more sensitive to the permittivity of $\epsilon _2$ and there is a red shift in this trend.

As it is shown in Fig. 3, the sign of optical force (pulling or pushing force) is closely intertwined with the location of the hemispheres in elliptical nanoparticle. Since the electrical field is applied in the y-direction, closer hemisphere has a formative impact on optical pressure imposing the NP. The scenario is drastically changed when one changes the location of SiC (with complex susceptibility) hemisphere from up to button or visa versa.

Fig. 3. Normalized optical force on $\boldsymbol {a)}$ elliptical cylinder with $L_y/L_x=0.5$ and $\boldsymbol {b)}$ circular cylinder nanoparticle($L_y/L_x=1$). Fixed parameter is $\epsilon _h = 1.5$.

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Next, we consider the geometry parameter of the elliptical cylinder $\alpha = ly/lx$. In Fig. 4(a) and 4(b), we investigate the influence of $\alpha$ in the optical force in both x and y directions. As shown in Fig. 4(a), by increasing $\alpha$, the Fig. 3. normalized OPF in x-direction is considerably enhanced in general but the enhancement trend at a resonance wavelength $\lambda = 10.5\mu m$ is obvious. In Fig. 4(b), the OPF augmentation in y-direction is ten times larger than in x-direction.

Fig. 4. Normalized optical force on elliptic cylinder nanoparticle as a function of incident wavelength with and $L_y/L_X$ $\boldsymbol {a)}$ in x-direction $\boldsymbol {b)}$ in y-direction. Other fixed parameters are $\epsilon _h =1.5$, $\epsilon _1 = 1.5$ and $\epsilon _2 = \epsilon _{SiC}{(\omega )}$ and $L_x=50nm$

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Local field of this NP is illustrated in Fig. 5. by increasing the permittivity of upper hemisphere while the rest of parameters are fixed, the local field distribution would be debilitated considerably (Fig. 5(a), 5(b), and 5(c)). It can also be inferred for $\alpha = 1$ or a conventional cylinder (Fig. 5(d), 5(e), and 5(f)) the decline in local field is much more noticeable as is is found to be decreased with increasing $\epsilon _2$. It is worth mentioning that in the elliptical cylinder the total electric field in the vicinity of NP and consequently the experienced optical pressure is of significance. To put it differently, this can confirm the idea underlying this work. By having the profound insight of physics, such a feature indicates the possibility of a controlled particle sorting according to particle geometry.

Fig. 5. Distributions of local electric fields inside and outside nanoparticles for different values of $\epsilon _2$ and $\alpha$. $\epsilon _h = 2.5$(streamlines exhibit electric field lines)

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In this work, the optical force is considered merely on a single nanoparticle, and it would be a significant topic to study optical force on composites of these nanoparticles.

4. Conclusion

In conclusion, we investigate the optical force on a NP with elliptical cylinder shape based on Maxwell’s stress tensor method. Furthermore, the physical and geometrical parameters would provide the an opportunity for the tunable interaction between the incident field and NP. The flexibility and tunablility of the OPF would broaden the potential applications in optical separation, sorting and transporting of elliptical cylinder nanoparticles. The optical pulling force in y-direction can be further enhanced as the $\alpha$ increases to a certain extent at about $\alpha =0.68$. In addition, at a given incident wavelength, the NP may be pulled backward by pulling force or pushed forward by the pulling force or pushed forward by the pushing force, which is depending on the where the SiC hemisphere(with complex susceptibility) is located in the NP that provides a powerful way to select nanoparticle’s materials.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Optical pulling force upon elliptical cylinder nanoparticles in the infrared range

Abstract

1. Introduction

2. Theoretical model and methods

3. Results and discussion

4. Conclusion

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Equations (24)

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