Rigorous full-wave calculation of optical forces on microparticles immersed in vector Pearcey beams

Xiaoyan Zhou; Shuxi Liu; Daomu Zhao

doi:10.1364/OE.491720

1. Introduction

Since the pioneering works by Ashkin [1–3], optical micromanipulation techniques are well established and have experienced intensive development in various disciplines [4–8], such as atomic physics, optics, and biological science. Nowadays, we can efficiently manipulate an object ranging from nanometers to tens of micrometers by multiple freedoms [9–13], including pushing, pulling, lateral shifting, rotating, and spinning. In recent years, structured light beams, as a powerful and versatile tool, have already proven their impact in the field of optical micromanipulations, as well as providing new insights into light-matter interactions [14,15].

One of the most important milestones in this realm is known as optical tweezer by using a single tightly focused Gaussian beam [1–3]. The generated strong gradient forces counteract the scattering forces in the propagation direction and thus a stable 3D optical trap can be formed in a tightly focused laser beam spot. In addition to the conventional Gaussian beams, diffraction-free Bessel beams have played a crucial role in the development of optical micromanipulation [16,17]. Researchers first demonstrated such beams can trap multiple particles along the beam axis, as well as transport these particles over long distances. Qiu et. al. and Chen et. al. successively proposed that a nonparaxial gradient-less Bessel beam is an excellent candidate for achieving optical pulling forces on an object, which is a counterintuitive notion and attracts the popular imagination [18,19]. However, the particle trajectories induced by these kinds of single-structured beams are straight and are lack of flexibility until the advent of Airy beams [20,21].

Similar to the Bessel modes [22,23], Airy beams exhibit properties of non-diffraction, and self-reconstruction. Remarkably, they accelerate in the transverse plane, resulting in a parabolic trajectory upon propagation. Because of this unique propagation, they are perfectly utilized to realize an optical “snow blower” by Dholakia et al. in 2008 [24]. Following this ground-breaking work, a vast number of studies have been reported to analyze the force distributions of particles placed in Airy beams or their variants [25–29]. For instance, morphing autofocusing Airy beams are explored to guide and transport microparticles along the primary rings [25,26]. Circular Airy vortex beams with autofocusing property are used to trap and rotate particles as well [27,28]. In view of the fact that the nonparaxial condition is necessary for a microscope system for optical tweezers, researchers conceived nonparaxial accelerating beams to realize a sharp bending and deliver particles to distances at steeper angles [29]. Moreover, optical pulling forces can arise from nonparaxial accelerating vector beams [30].

Pearcey beams, as another well-known example of caustic light fields, have been introduced in the past few years [31–33]. On the one hand, the light beams possess the form-invariance and self-healing properties upon propagation. On the other hand, auto-focusing and inversion effects of lights are found without the employment of any extra focusing optical elements. The beams show an “s-shaped” curved propagation trajectory. These intriguing performances can be perfectly inherited even under incoherence states [34]. Although, some investigations on optical micromanipulation with ring Pearcey beams or circular Pearcey-like vortex beams have been examined [33,35], the study on vector Pearcey beams is lacking. Consequently, the underlying physics is not transparent. We may wonder: how to model (conventional) Pearcey beams into an optical tweezer setup, where the conventional paraxial description of laser beams fails? Can this new Pearcey ramification retains the auto-focusing property? If so, will the beams trap particles into the auto-focusing point in 3D space or push them through the point and drive them forward along predesigned curved paths? Here, these fundamental questions will be clarified meticulously.

The paper is organized as follows. We first generalize the scalar Pearcey beams to vector counterparts, which are the exact solutions of the vector angular spectrum representation. Then, we present the rigorous full-wave solution for the optical force on spherical particles immersed in arbitrary vector beams with different polarization states, based on the generalized Lorenz-Mie theory and the Maxwell stress tensor approach [36–38]. A specific example of vector Pearcey beams is subsequently simulated and analyzed. We demonstrate the cases of dielectric microparticles of different sizes. Furthermore, we take the dielectric permittivity and magnetic permeability of particles into account. Finally, a summary and outlook are established.

2. Theory

2.1 Vector Pearcey beams

We start our analysis by considering the 3D Helmholtz equation [39]

(1)$$({\partial _{xx}} + {\partial _{yy}} + {\partial _{zz}} + {k^2}){\textbf E} = 0$$

where k is the wavenumber. The solution of the Helmholtz equation in free space can be described in terms of plane waves through its angular spectral function. We assume that vector electromagnetic fields propagate along the positive z-direction, without the time dependence ${e^{ - iwt}}$ throughout this paper. The vector electromagnetic fields of an arbitrary beam in a nonmagnetic and isotropic medium can be expressed rigorously in Cartesian vector components as [40]

(2)$${{\textbf E}_{\textrm{inc}}}(x,y,z) = \int_{{k_{{x_{\min }}}}}^{{k_{{x_{\max }}}}} {\int_{{k_{{y_{\min }}}}}^{{k_{{y_{\max }}}}} {\tilde{{\bf P}}({k_x},{k_y}){e^{i[{k_x}(x - {x_0}) + {k_y}(y - {y_0}) + {k_z}(z - {z_0})]}}d{k_x}d} {k_y}} ,$$

where $(x,y,z)$ is an arbitrary point in the real space.$({k_x},{k_y},{k_z}) = {\bf k}$ are the wave vectors and satisfy the expression ${k^2} = {k_x}^2 + {k_y}^2 + {k_z}^2. k = 2\pi {n_m}/\lambda $ denotes the wavenumber with ${n_m}$ and $\lambda$ being the refractive index of the background medium and the optical wavelength in vacuum, respectively. In spherical coordinates, the wave vectors can be represented as ${k_x} = k\sin \alpha \cos \beta ,\textrm{ }{k_y} = k\sin \alpha \sin \beta ,$ and ${k_z} = k\cos \alpha $ with $\alpha (\beta )$ being the polar (azimuthal) angle. $({x_0},{y_0},{z_0})$ is the focal point of the light fields. It is worth noting that our investigation only concerns the far field propagation, thus evanescent wave components are suppressed. The vector angular spectrum function $\tilde{{\bf P}}({k_x},{k_y})$ takes the form [30,41]

(3)$$\tilde{{\bf P}}({k_x},{k_y}) = \frac{{{E_0}}}{{4{\pi ^2}}}\textrm{P}({k_x},{k_y}){\bf Q}(\alpha ,\beta ),$$

where ${E_0}$ is the characteristic amplitude, $\textrm{P}({k_x},{k_y})$ is the spectrum distribution describing the profile of the incident beams, and ${\bf Q}(\alpha ,\beta )$ refers to a complex vector function denoting the polarization states of light waves, written as

(4)$${\bf Q}(\alpha ,\beta ) = \left[ \begin{array}{l} {p_x}(\cos \alpha {\cos^2}\beta + {\sin^2}\beta ) + {p_y}(\cos \alpha - 1)\sin \beta \cos \beta \\ {p_x}(\cos \alpha - 1)\sin \beta \cos \beta + {p_y}(\cos \alpha {\sin^2}\beta + {\cos^2}\beta )\\ \textrm{ } - {p_x}\sin \alpha \cos \beta - {p_y}\sin \alpha \sin \beta \end{array} \right]\left\{ \begin{array}{l} {{\boldsymbol e}_{\boldsymbol x}}\\ {{\boldsymbol e}_{\boldsymbol y}}\\ {{\boldsymbol e}_{\boldsymbol z}} \end{array} \right\},$$

with $({p_x},{p_y})$ and $\textrm{(}{{\boldsymbol e}_{\boldsymbol x}},{{\boldsymbol e}_y},{{\boldsymbol e}_z}\textrm{)}$ respectively being the polarization factors and the unit vectors in a Cartesian coordinate system. Typically, $({p_x},{p_y}) = (1,0)$ for x-linear polarization, $({p_x},{p_y}) = (1, \pm i)$ for circular polarization, $({p_x},{p_y}) = (\cos \beta ,\sin \beta )$ for radial polarization (TM mode), and $({p_x},{p_y}) = ( - \sin \beta ,\cos \beta )$ for azimuthal polarization (TE mode). The initial polarization is set in terms of specific consideration. According to the relation between electric and magnetic fields, the evolution of radiation magnetic fields is expressed as [39]

(5)$${{\bf H}_{\textrm{inc}}}(x,y,z){\bf = }\frac{1}{{\omega \mu }}{\bf k} \times {{\bf E}_{\textrm{inc}}}(x,y,z), $$

where $\omega $ is the angular frequency of the light beams and $\mu $ is the permeability of the background medium.

Let us now recall the spectrum distribution of the Pearcey function [31,34], which reads

(6)$${\rm P}({k_x},{k_y}) = {e^{i{k_x}^4}}\delta (\frac{{{k_x}^2}}{{2q}} - {k_y}).$$

Here, $\delta ({\cdot} )$ denotes the Dirac delta function and q is the dimensionless semi-latus rectum, which denotes the distance of the focus from the directrix for a standard parabola equation. On substituting from Eq. (6) into Eq. (3), the vector angular spectrum function of the vector Pearcey beams can be derived:

(7)$$\tilde{{\bf P}}({k_x},{k_y}) = \frac{{{E_0}}}{{4{\pi ^2}}}{e^{i{k_x}^4}}\delta (\frac{{{k_x}^2}}{{2q}} - {k_y}){\bf Q}(\alpha ,\beta ).$$

Substituting Eq. (7) into Eq. (2) and Eq. (5), the electromagnetic field distribution of the vector Pearcey beams is readily obtained. It needs to be emphasized that the electromagnetic fields of vector Pearcey beams with different polarizations are solved based on the Maxwell’s equations, which are rigorous beyond the conventional paraxial approximation.

2.2 Optical force based on electromagnetic scattering theory

In this section, we set out the general equations describing optical forces within the framework of electromagnetic scattering theory. To simplify the optical force calculation, the origin of the coordinate system is located at a particle center. Based on the full-wave generalized Lorenz-Mie theory, the incident electromagnetic fields of arbitrary beams can be expanded in terms of vector spherical wave functions ${\bf (N}_{mn}^{(1)}(k,{\bf r}),{\bf M}_{mn}^{(1)}(k,{\bf r}))$[37]:

(8)$${{\bf E}_{\textrm{inc}}}(r,\theta ,\phi ) ={-} i\sum\limits_{n = 1}^\infty {\sum\limits_{m ={-} n}^n {{E_{mn}}} } [{p_{mn}}{\bf N}_{mn}^{(1)}(k,{\bf r}) + {q_{mn}}{\bf M}_{mn}^{(1)}(k,{\bf r})],$$

(9)$${{\bf H}_{\textrm{inc}}}(r,\theta ,\phi ) ={-} \frac{k}{{\omega \mu }}\sum\limits_{n = 1}^\infty {\sum\limits_{m ={-} n}^n {{E_{mn}}} } [{q_{mn}}{\bf N}_{mn}^{(1)}(k,{\bf r}) + {p_{mn}}{\bf M}_{mn}^{(1)}(k,{\bf r})],$$

where $(r,\theta ,\phi )$ is the spherical coordinates corresponding to the Cartesian coordinates $(x,y,z)$. The coefficient factor ${E_{mn}} = {E_0}{i^n}{\gamma _{mn}}$ with ${\gamma _{mn}} = {[\frac{{(2n + 1)(n - m)!}}{{n(n + 1)(n + m)!}}]^{1/2}}$. Partial-wave expansion coefficients (or beam-shape coefficients) ${p_{mn}}$ and ${q_{mn}}$ are obtained from

(10)$${p_{mn}} = \frac{{kr}}{{{j_n}(kr)}}\int_{\phi = 0}^{2\pi } {\int_{\theta = 0}^\pi {[{{\bf e}_r} \cdot {{\bf E}_{inc}}(r,\theta ,\phi )]} } {\mathrm{{\cal F}}_{mn}}(\theta ,\phi )\sin \theta d\theta d\phi ,$$

(11)$$i{q_{mn}} ={-} \frac{{Zkr}}{{{j_n}(kr)}}\int_{\phi = 0}^{2\pi } {\int_{\theta = 0}^\pi {[{{\bf e}_r} \cdot {{\bf H}_{inc}}(r,\theta ,\phi )]} } {\mathrm{{\cal F}}_{mn}}(\theta ,\phi )\sin \theta d\theta d\phi ,$$

with the functions

(12)$${\mathrm{{\cal F}}_{mn}}(\theta ,\phi ) = \frac{{{i^{1 - n}}}}{{{E_0}\sqrt {4\pi n(n + 1)} }}Y_{mn}^\ast (\theta ,\phi ),$$

and

(13)$$Y_{mn}^\ast (\theta ,\phi ) = \sqrt {\frac{{(2n + 1)(n - m)!}}{{4\pi (n + m)!}}} P_n^m(\cos \theta ){e^{ - im\phi }},$$

where ${j_n}({\cdot} )$ is the spherical Bessel function of order n; $Z = \sqrt {\mu /\varepsilon } $ denotes the impedance defined by the permeability $\mu $ and the permittivity $\varepsilon $ in the background medium; ${{\bf e}_r}$ is the radial unit vector in the spherical coordinate system. $Y_{mn}^\ast (\theta ,\phi )$ stands for the complex conjugate of the spherical harmonic function with $P_n^m(\cos \theta )$ being the associated Legendre function of the first kind. With the help of a complicated set of algebra, it then follows that the partial-wave expansion coefficients can be simplified into

(14)$$\begin{aligned} {p_{mn}} &= \frac{{{\gamma _{mn}}}}{{4{\pi ^2}}}\int_{{k_{{x_{\min }}}}}^{{k_{{x_{\max }}}}} {\int_{{k_{{y_{\min }}}}}^{{k_{{y_{\max }}}}} {{\rm P}({k_x},{k_y}){e^{ - im\beta }}{e^{ - i({k_x}{x_0} + {k_y}{y_0} + {k_z}{z_0})}}} } \\ &\textrm{ }[i({p_x}\sin \beta - {p_y}\cos \beta ){\pi _{mn}}(\cos \alpha ) + ({p_x}\cos \beta + {p_y}\sin \beta ){\tau _{mn}}(\cos \alpha )]d{k_x}d{k_y}, \end{aligned}$$

(15)$$\begin{aligned} {q_{mn}} &= \frac{{{\gamma _{mn}}}}{{4{\pi ^2}}}\int_{{k_{{x_{\min }}}}}^{{k_{{x_{\max }}}}} {\int_{{k_{{y_{\min }}}}}^{{k_{{y_{\max }}}}} {{\rm P}({k_x},{k_y}){e^{ - im\beta }}{e^{ - i({k_x}{x_0} + {k_y}{y_0} + {k_z}{z_0})}}} } \\ &\textrm{ }[({p_x}\cos \beta + {p_y}\sin \beta ){\pi _{mn}}(\cos \alpha ) + i({p_x}\sin \beta - {p_y}\cos \beta ){\tau _{mn}}(\cos \alpha )]d{k_x}d{k_y}, \end{aligned}$$

where ${\pi _{mn}}(\cos \alpha ) = mP_n^m(\cos \alpha )/\sin \alpha$ and ${\tau _{mn}}(\cos \alpha ) = dP_n^m(\cos \alpha )/d\alpha$ are two auxiliary functions. One can see that the expansion coefficients ${q_{mn}}$ can be obtained from the coefficients ${p_{mn}}$ by exchanging these two auxiliary functions.

The scattered electromagnetic fields are also described via the vector spherical wave functions ${\bf (N}_{mn}^{(3)}(k,{\bf r}),{\bf M}_{mn}^{(3)}(k,{\bf r}))$[37]:

(16)$${{\bf E}_{\textrm{sca}}}(r,\theta ,\phi ) = i\sum\limits_{n = 1}^\infty {\sum\limits_{m ={-} n}^n {{E_{mn}}} } [{a_{mn}}{\bf N}_{mn}^{(3)}(k,{\bf r}) + {b_{mn}}{\bf M}_{mn}^{(3)}(k,{\bf r})],$$

(17)$${{\bf H}_{\textrm{sca}}}(r,\theta ,\phi ) = \frac{k}{{\omega \mu }}\sum\limits_{n = 1}^\infty {\sum\limits_{m ={-} n}^n {{E_{mn}}} } [{b_{mn}}{\bf N}_{mn}^{(3)}(k,{\bf r}) + {a_{mn}}{\bf M}_{mn}^{(3)}(k,{\bf r})],$$

where the partial-wave expansion coefficients of the scattered fields ${a_{mn}}$ and ${b_{mn}}$ are given by ${a_{mn}} = {a_n}{p_{mn}}$ and ${b_{mn}} = {b_n}{q_{mn}}$, with ${a_n}$ and ${b_n}$ being the Mie coefficients. Hereto, we have deduced the generalized expression of partial-wave expansion coefficients of arbitrary beams at arbitrary polarization states, which plays an important role in evaluating the optical forces for a particle illuminated by light beams. Especially, the partial-wave expansion coefficients of incident vector Pearcey beams can be easily obtained by substituting Eq. (6) into Eq. (14) and Eq. (15).

Outside the particle, the time-averaged Maxwell stress tensor can be determined by the incident and scattered fields,

(18)$$\left\langle {{\bf \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over T} }} \right\rangle {\bf = }\frac{1}{2}\textrm{Re} [\varepsilon {{\bf E}_{\textrm{tot}}}{\bf E}_{\textrm{tot}}^\mathrm{\ast } + \mu {{\bf H}_{\textrm{tot}}}{\bf H}_{\textrm{tot}}^\mathrm{\ast } - \frac{1}{2}(\varepsilon {{\bf E}_{\textrm{tot}}} \cdot {\bf E}_{\textrm{tot}}^\mathrm{\ast } + \mu {{\bf H}_{\textrm{tot}}} \cdot {\bf H}_{\textrm{tot}}^\mathrm{\ast }){\bf \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over I} }],$$

where ${{\bf E}_{\textrm{tot}}}{\bf = }{{\bf E}_{\textrm{inc}}} + {{\bf E}_{\textrm{sca}}}$ and ${{\bf H}_{\textrm{tot}}}{\bf = }{{\bf H}_{\textrm{inc}}} + {{\bf H}_{\textrm{sca}}}$ are total electric and magnetic fields, respectively. ${\bf \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over I} }$ is a unit tensor. By integrating the Maxwell stress tensor over a closed surface, the time-averaged optical force acting on the particle can be evaluated as [42,43],

(19)$${\bf F = }\oiint\nolimits_s {\hat{{\bf n}}} \cdot \left\langle {{\bf \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over T} }} \right\rangle dS.$$

Thus, the three Cartesian components of optical forces in a lossless background medium read

(20)$${F_x} = \textrm{Re} [{F_1}],\textrm{ }{F_y} = {\mathop{\rm Im}\nolimits} [{F_1}],\textrm{ }{F_z} = \textrm{Re} [{F_2}],$$

where

(21)$$\begin{array}{l} {F_1} = \frac{{2\pi \varepsilon }}{{{k^2}}}{E_0}^2\sum\limits_{n = 1}^\infty {\sum\limits_{m ={-} n}^n {[{c_{11}}F_1^{(1)} - {c_{12}}F_1^{(2)} + {c_{13}}F_1^{(3)}]} } ,\\ {F_2} ={-} \frac{{4\pi \varepsilon }}{{{k^2}}}{E_0}^2\sum\limits_{n = 1}^\infty {\sum\limits_{m ={-} n}^n {[{c_{21}}F_2^{(1)}\textrm{ + }{c_{22}}F_2^{(2)}]} } , \end{array}$$

with the coefficients

(22)$$\begin{array}{l} {c_{11}} = {[\frac{{(n - m)(n + m + 1)}}{{{n^2}{{(n + 1)}^2}}}]^{1/2}},\\ {c_{12}} = {[\frac{{n(n + 2)(n + m + 1)(n + m + 2)}}{{{{(n + 1)}^2}(2n + 1)(2n + 3)}}]^{1/2}},\\ {c_{13}} = {[\frac{{n(n + 2)(n - m)(n - m + 1)}}{{{{(n + 1)}^2}(2n + 1)(2n + 3)}}]^{1/2}},\\ {c_{21}} = {[\frac{{n(n + 2)(n - m + 1)(n + m + 1)}}{{{{(n + 1)}^2}(2n + 1)(2n + 3)}}]^{1/2}},\\ {c_{22}} = \frac{m}{{n(n + 1)}}, \end{array}$$

and

(23)$$\begin{array}{l} F_1^{(1)} = {{\tilde{a}}_{mn}}\tilde{b}_{{m_1}n}^\ast{+} {{\tilde{b}}_{mn}}\tilde{a}_{{m_1}n}^\ast{-} {{\tilde{p}}_{mn}}\tilde{q}_{{m_1}n}^\ast{-} {{\tilde{q}}_{mn}}\tilde{p}_{{m_1}n}^\ast ,\\ F_1^{(2)} = {{\tilde{a}}_{mn}}\tilde{a}_{{m_1}{n_1}}^\ast{+} {{\tilde{b}}_{mn}}\tilde{b}_{{m_1}{n_1}}^\ast{-} {{\tilde{p}}_{mn}}\tilde{p}_{{m_1}{n_1}}^\ast{-} {{\tilde{q}}_{mn}}\tilde{q}_{{m_1}{n_1}}^\ast ,\\ F_1^{(3)} = {{\tilde{a}}_{m{n_1}}}\tilde{a}_{{m_1}n}^\ast{+} {{\tilde{b}}_{m{n_1}}}\tilde{b}_{{m_1}n}^\ast{-} {{\tilde{p}}_{m{n_1}}}\tilde{p}_{{m_1}n}^\ast{-} {{\tilde{q}}_{m{n_1}}}\tilde{q}_{{m_1}n}^\ast ,\\ F_2^{(1)} = {{\tilde{a}}_{mn}}\tilde{a}_{m{n_1}}^\ast{+} {{\tilde{b}}_{mn}}\tilde{b}_{m{n_1}}^\ast{-} {{\tilde{p}}_{mn}}\tilde{p}_{m{n_1}}^\ast{-} {{\tilde{q}}_{mn}}\tilde{q}_{m{n_1}}^\ast ,\\ F_2^{(2)} = {{\tilde{a}}_{mn}}\tilde{b}_{mn}^\ast{-} {{\tilde{p}}_{mn}}\tilde{q}_{mn}^\ast , \end{array}$$

where the asterisk superscript denotes the complex conjugate. The indices ${m_{1\textrm{ }}} = m\textrm{ } + \textrm{ }1$, ${n_1} = \textrm{ }n\textrm{ } + \textrm{ }1$, and

(24)$$\begin{array}{l} {{\tilde{a}}_{mn}} = {a_{mn}} - \frac{1}{2}{p_{mn}},{{\tilde{p}}_{mn}} = \frac{1}{2}{p_{mn}},\\ {{\tilde{b}}_{mn}} = {b_{mn}} - \frac{1}{2}{q_{mn}},{{\tilde{q}}_{mn}} = \frac{1}{2}{q_{mn}}. \end{array}$$

3. Results and discussion

With the above approach, we will compute the optical forces of vector Pearcey beams exerting on a dielectric microsphere suspended in water. In our analysis, the power of the incident vector Pearcey beams is set to be $P = 100\textrm{mW}$ as a typical example, corresponding to ${E_0} \approx 2.42 \times {10^{ - 4}}\textrm{V} \cdot {\textrm{m}^{\textrm{ - 1}}}$. We first investigate the case of a spherical Mie particle, whose dimensions are comparable to the wavelength. The particle is made of polystyrene with radius ${r_s} = 0.5\mathrm{\mu}\textrm{m}$, relative permittivity ${\varepsilon _r} = 2.53$, relative permeability ${\mu _r} = 1$, and mass density $\rho = 1050\textrm{kg} \cdot {\textrm{m}^{\textrm{ - 3}}}$. In this situation, the gravity (about 5.39fN), the buoyancy (about 5.13fN), and the Brownian force (about 7.89fN at the temperature $T = 300\textrm{K}$) are small enough to be ignored in comparison with the transverse and longitudinal optical forces exerted by vector Pearcey beams.

To overview the distribution of optical forces, it is necessary for us to demonstrate the main propagation properties of vector Pearcey beams first. Usually, the liquid-crystal-based spatial light modulator is employed to generate the light beams by modulating the domain of spatial frequency. Due to the polarization-dependent feature of a liquid crystal, we consider the linear polarization (x-linear polarization) in our studies. Figures 1(a1)-(c1) display the propagation dynamics of vector Pearcey beams with $q = 0.5$. It can be found evidently that the beams will autofocus into an intensive spot at the auto-focusing distance $z = {z_f} \approx 440/k$. Since there exists a singularity in both x and y axes at $z = {z_f}$, the beams are highly symmetry about the auto-focusing spot. The properties of auto-focusing performance and inversion effect can be perfectly inherited under the nonparaxial approximation. Therefore, we will explore the optical force at these representative propagation distances $z = 0$, $z = {z_f}$, and $z = 2{z_f}$ in detail. The distribution of the optical force in the x-y plane is visualized in Figs. 1(a1)-(c1) with the magnitude and direction characterized by the lengths and the directions of white arrowheads, respectively. The image in the upper right corner is a partial magnification. It can be clearly observed that the white arrowheads point to the location of the peak intensity of the main and nearby side lobes. There are equilibrium positions close to the peak intensity, providing the possibility of trapping particles transversely. We indeed notice that the light beams with x-linear polarization are symmetrical about the y axis, the transverse trapping position of the particle always appears on this axis where ${F_y}$ is zero. Figures 1(a2)-(c2) present the characteristics of optical force along the x axis at the trapping position near the main lobe. It is obvious that ${F_y}$ is always symmetric while ${F_x}$ is always antisymmetric. The optical force along the y axis is shown in Figs. 1(a3)-(c3). Along this axis, ${F_y}$ exhibits an oscillating distribution, which matches the pattern of the light intensity. ${F_x}$ tends to zero. At the initial plane $z = 0$, ${F_y}$ is prominent and the force pointing north (${F_y}\mathrm{\ < }0$) is stronger than the force pointing south (${F_y}\mathrm{\ > }0$) in the periphery of the main and side lobes (Fig. 1(a3)). On account of the effects of nonparaxiality, ${F_y}$ is not strictly antisymmetric at the auto-focusing distance $z = {z_f}$ (Fig. 1(b3)). Due to the inversion effect of the beams, the force pointing south (${F_y} > 0$) is stronger than that pointing north (${F_y} < 0$) at $z = 2{z_f}$ (Fig. 1(c3)). During propagation, the longitudinal optical force at these transverse tapping positions is always positive, as depicted in Figs. 1(a4)-(c4) by solid lines. This means that the vector Pearcey beams with auto-focusing property cannot trap dielectric microparticles of ${r_s} = 0.5\mathrm{\mu}\textrm{m}$ longitudinally, and these particles always move in the positive z-direction. Considering the absorbability of the particle, a small imaginary part should be introduced into the relative permittivity of the microsphere. Referring to the literature [19], we denote ${\mathop{\rm Im}\nolimits} \{ {\varepsilon _r}\} = 0.01$. Corresponding calculated results plotted by dashed lines, are overlapped in Figs. 1(a4)-(c4). Compared with the lossless case, the longitudinal force ${F_z}$ becomes larger, while the transverse forces (${F_x}$, ${F_y}$) remain the same. The core reasons can be interpreted: the microparticle absorbs more forward photons, weakening the momentum of forward scattered photons.

Fig. 1. Optical force experienced by a polystyrene particle of ${r_s} = 0.5\mathrm{\mu}\textrm{m}$ in the vector Pearcey beams. From left to right each panel corresponds to different distances $z = 0$, $z = {z_f}$, and $z = 2{z_f}$, respectively. (a1)-(c1) The transverse intensity profiles; (a2)-(c2) The transverse force along a horizontal line which goes through the trapping position on the y axis; (a3)-(c3) and (a4)-(c4) The transverse and longitudinal forces along the y axis. The white arrowheads in the first row denote the magnitude and direction of the force distribution; the black dots in the second and third rows mark the transverse trapping positions; the associated beam profiles (shaded areas) in the bottom row are overlapped for reference. Solid and dashed curved lines in second to forth rows correspond to ${\varepsilon _r} = 2.53$ and ${\varepsilon _r} = 2.53 + 0.01i$, respectively.

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In order to give a more vivid illustration, we exhibit the propagation trajectory of the vector Pearcey beams in Fig. 2(a). The distribution of optical forces (${F_y}$, ${F_z}$) near the auto-focusing point in the y-z plane is marked with white arrowheads. The longitudinal scattering force is superior to the longitudinal gradient force in spite of the strong field intensity near the auto-focusing point. Hence, the Mie particle is trapped transversely near the auto-focusing point and simultaneously pushed through the point rather than be trapped therein. To give a clear picture of optical manipulation, we show that the field profiles corresponding to incident vector Pearcey beam and “s-shaped” trajectory in Fig. 2(b). The trapped particles are transported along the curved path, circumventing the obstacles. Note that the pink spheres indicate obstacles. Furthermore, we study the optical force behaviors of dielectric Mie particles with different sizes at the auto-focusing plane. In Fig. 2(c), we present the transverse force ${F_y}$ along the y axis for polystyrene microparticles with radius ${r_s} = 0.3\mathrm{\mu}\textrm{m},\textrm{ }0.5\mathrm{\mu}\textrm{m},1\mathrm{\mu}\textrm{m},$ and $2\mathrm{\mu}\textrm{m}$, respectively. With particle sizes increasing, a stable transverse trapping position on the y axis is observed throughout. The maximum of ${F_y}$ goes up, reaching to nearly 300pN when the radius is $2\mathrm{\mu}\textrm{m}$. It can be seen that the larger the radius of the microparticle, the deeper the trapping potential well of the optical force, and the stronger the trapping stability of the microparticle. In addition, the longitudinal force ${F_z}$ along the y axis is plotted in Fig. 2(d). The beam profiles (shaded areas) are overlapped in the same box for reference. ${F_z}$ keeps positive and the Mie particle is always pushed through the auto-focusing point. Interestingly, for ${r_s} = 1\mathrm{\mu}\textrm{m}$ and $2\mathrm{\mu}\textrm{m}$, the force distribution is asymmetric and has a dip at the location of peak intensity. This is because the incident field excites multistage radiation interference. When the particle size is large, the phase interference between the multipoles is constructive at the location of peak intensity, leading to more photons being scattered forward by the particle and thus the force forward is reduced.

Fig. 2. (a) Simulation results of propagation trajectory of the beam propagation. The white arrowheads denote the distribution of optical forces. (b) Due to the curved optical manipulation traces of vector Pearcey beams, the trapped particles are able to circumvent obstacles. The pink spheres indicate obstacles. (c) and (d) ${F_y}$ and ${F_z}$ experienced by Mie particles with different radiuses along the y axis at the autofocusing plane $z = {z_f}$.

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Next, we investigate the case of particle size smaller than the Mie particle. In this scenario, a Rayleigh polystyrene microsphere with ${r_s} \le 0.05\lambda $ is chosen as an example. Figure 3(a) presents the longitudinal force distribution at the lateral trapping position in different depths of focus, correspondingly. It is clearly observed that when the particle size pertains to region I, the longitudinal force ${F_z}$ is always positive, meaning the particle is pushed forwards along the propagation direction. However, the situation becomes complex when the size belongs to region II. A conversion between positive and negative longitudinal forces occurs, which signifies that there are several inflection points ${F_z} = 0$ in the focusing depths. As the size of particle continues to decrease, the longitudinal force ${F_z}$ is negative, as shown in region III.

Fig. 3. (a) Optical force ${F_z}$ experienced by a Rayleigh polystyrene particle with different sizes in the vector Pearcey beams along the propagation direction. (b) Diagram of the force ${F_z}$ as a function of the dielectric permittivity and magnetic permeability with parameters $k{r_s} = \pi /20$ at $z = 1.2{z_f}$.

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In addition to the particle size, the dielectric permittivity and magnetic permeability, as the main parameters of particles, also play an important impact on the optical force. The calculated optical force in arbitrary units as a function of relative permittivity and relative permeability is shown in Fig. 3(b) with $k{r_s} = \pi /20$ at propagation distance $z = 1.2{z_f}$. White regions indicate positive optical force and coloured regions indicate the parameter space where the optical force is negative. Perfect symmetry occurs in this case, because when we exchange relative permittivity and relative permeability, we also exchange E- and H-fields. Here, different classes of particles are sorted by vector Pearcey beams, some are moved towards the auto-focusing point, and some are transported far away.

4. Conclusion

In conclusion, we present the electromagnetic fields of vector Pearcey beams based on the Maxwell’s equations, which are rigorous beyond the paraxial approximation and can be realized in a microscope system setup for optical tweezers. These beams maintain the inherent properties of autofocusing performance and inversion effect. Furthermore, we deduce a rigorous solution to evaluate the optical forces when a spherical particle of an arbitrary radius is illuminated by beams with different polarizations, based on the generalized Lorenz-Mie theory and Maxwell stress tensor approach. This is a general method which is suitable for any nonparaxial beams. Specifically, we analyze the force exerted by vector Peacey beams on a microparticle. The results demonstrate that, the trapped dielectric Mie particle will be pushed through the auto-focusing point, instead of being trapped near the point in three dimensions. Eventually, the particle will be transversely trapped near the beam axis and simultaneously transported along a curved trajectory based on the longitudinal optical forces. For Rayleigh particles, particle size, permittivity and permeability are the vital and coordinated factors, resulting in the particles being pushed through or trapped near the auto-focusing point. These fruitful dynamics might provide an additional degree of freedom for optical transport and trapping of microparticles.

Funding

National Natural Science Foundation of China (12174338, 11874321).

Acknowledgment

We are grateful for discussions with Prof. Chengwei Qiu and Dr. Xiaozhou Pan in National University of Singapore and for the help from Dr. Hao Wu in Nankai University.

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.

References

1. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970). [CrossRef]

2. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11(5), 288–290 (1986). [CrossRef]

3. A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235(4795), 1517–1520 (1987). [CrossRef]

4. K. Svoboda, C. F. Schmidt, B. J. Schnapp, and S. M. Block, “Direct observation of kinesin stepping by optical trapping interferometry,” Nature 365(6448), 721–727 (1993). [CrossRef]

5. M. L. Juan, R. Gordon, Y. Pang, F. Eftekhari, and R. Quidant, “Self-induced back-action optical trapping of dielectric nanoparticles,” Nat. Phys. 5(12), 915–919 (2009). [CrossRef]

6. J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011). [CrossRef]

7. M. Koch and A. Rohrbach, “Object-adapted optical trapping and shape-tracking of energy-switching helical bacteria,” Nat. Photonics 7(9), 680–690 (2013). [CrossRef]

8. L. R. Liu, J. D. Hood, Y. Yu, J. T. Zhang, N. R. Hutzler, T. Rosenband, and K. K. Ni, “Building one molecule from a reservoir of two atoms,” Science 360(6391), 900–903 (2018). [CrossRef]

9. C. Zensen, N. Villadsen, F. Winterer, S. R. Keiding, and T. Lohmüller, “Pushing nanoparticles with light-a femtonewton resolved measurement of optical scattering forces,” APL Photonics 1(2), 026102 (2016). [CrossRef]

10. A. Dogariu, S. Sukhov, and J. J. Sáenz, “Optically induced ‘negative forces’,” Nat. Photonics 7(1), 24–27 (2013). [CrossRef]

11. J. Ahn, Z. Xu, J. Bang, Y. Deng, T. M. Hoang, Q. Han, R. Ma, and T. Li, “Optically levitated nanodumbbell torsion balance and GHz nanomechanical rotor,” Phys. Rev. Lett. 121(3), 033603 (2018). [CrossRef]

12. R. Reimann, M. Doderer, E. Hebestreit, R. Diehl, M. Frimmer, D. Windey, F. Tebbenjohanns, and L. Novotny, “GHz rotation of an optically trapped nanoparticle in vacuum,” Phys. Rev. Lett. 121(3), 033602 (2018). [CrossRef]

13. M. P. J. Lavery, F. C Speirits, S. M Barnett, and M. J Padgett, “Detection of a spinning object using light's orbital angular momentum,” Science 341(6145), 537–540 (2013). [CrossRef]

14. E. Ottea and C. Denz, “Optical trapping gets structure: Structured light for advanced optical manipulation,” Appl. Phys. Rev. 7(4), 041308 (2020). [CrossRef]

15. Y. Yang, Y. Ren, M. Chen, Y. Arita, and C. Rosales-Guzmáne, “Optical trapping with structured light: a review,” Adv. Photonics 3(03), 034001 (2021). [CrossRef]

16. V. G. Chavez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a selfreconstructing light beam,” Nature 419(6903), 145–147 (2002). [CrossRef]

17. K. Dholakia and W. Lee, “Optical trapping takes shape: The use of structured light fields,” Adv. At., Mol., Opt. Phys. 56, 261–337 (2008). [CrossRef]

18. A. Novitsky, C. Qiu, and H. Wang, “Single gradient less light beam drags particles as tractor beams,” Phys. Rev. Lett. 107(20), 203601 (2011). [CrossRef]

19. J. Chen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics 5(9), 531–534 (2011). [CrossRef]

20. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979 (2007). [CrossRef]

21. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007). [CrossRef]

22. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4(4), 651–654 (1987). [CrossRef]

23. J. Durnin, J. J. Miceli Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987). [CrossRef]

24. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008). [CrossRef]

25. I. Chremmos, P. Zhang, J. Prakash, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Fourier-space generation of abruptly autofocusing beams and optical bottle beams,” Opt. Lett. 36(18), 3675 (2011). [CrossRef]

26. P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Trapping and guiding microparticles with morphing autofocusing Airy beams,” Opt. Lett. 36(15), 2883 (2011). [CrossRef]

27. S. Liu, M. Wang, P. Li, P. Zhang, and J. Zhao, “Abrupt polarization transition of vector autofocusing Airy beams,” Opt. Lett. 38(14), 2416 (2013). [CrossRef]

28. M. Chen, S. Huang, X. Liu, Y. Chen, and W. Shao, “Optical trapping and rotating of micro-particles using the circular Airy vortex beams,” Appl. Phys. B 125(10), 184 (2019). [CrossRef]

29. W. Lu, H. Chen, S. Liu, and Z. Lin, “Rigorous full-wave calculation of optical forces on dielectric and metallic microparticles immersed in a vector Airy beam,” Opt. Express 25(19), 23238 (2017). [CrossRef]

30. H. Wu, X. Zhang, P. Zhang, P. Jia, Z. Wang, Y. Hu, Z. Chen, and J. Xu, “Optical pulling force arising from nonparaxial accelerating beams,” Phys. Rev. A 103(5), 053511 (2021). [CrossRef]

31. J. D. Ring, J. Lindberg, A. Mourka, M. Mazilu, K. Dholakia, and M. R. Dennis, “Auto-focusing and self-healing of Pearcey beams,” Opt. Express 20(17), 18955 (2012). [CrossRef]

32. D. M. Deng, C. D. Chen, X. Zhao, B. Chen, X. Peng, and Y. S. Zheng, “Virtual source of a Pearcey beam,” Opt. Lett. 39(9), 2703 (2014). [CrossRef]

33. X. Zhou, Z. Pang, and D. Zhao, “Generalized ring Pearcey beams with tunable autofocusing properties,” Ann. Phys. 533(7), 2100110 (2021). [CrossRef]

34. X. Zhou, Z. Pang, and D. Zhao, “Partially coherent Pearcey-Gauss beams,” Opt. Lett. 45(19), 5496 (2020). [CrossRef]

35. D. Xu, Z. Mo, J. Jiang, H. Huang, Q. Wei, Y. Wu, X. Wang, Z. Liang, H. Yang, H. Chen, H. Huang, H. Liu, D. Deng, and L. Shui, “Guiding particles along arbitrary trajectories by circular Pearcey-like vortex beams,” Phys. Rev. A 106(1), 013509 (2022). [CrossRef]

36. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley and Sons: New York, 1983).

37. G. Gouesbet and G. Gréhan, Generalized Lorenz-Mie theories (Springer: Berlin, 2011).

38. P. H. Jones, O. M. Marago, and G. Volpe, Optical Tweezers Principles and Applications (Cambridge University, 2015).

39. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

40. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2006).

41. J. Chen, J. Ng, P. Wang, and Z. Lin, “Analytical partial wave expansion of vector Bessel beam and its application to optical binding,” Opt. Lett. 35(10), 1674 (2010). [CrossRef]

42. J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley and Sons, 1999).

43. A. Zangwill, Modern Electrodynamics (Cambridge University Press, 2012).

Rigorous full-wave calculation of optical forces on microparticles immersed in vector Pearcey beams

Abstract

1. Introduction

2. Theory

2.1 Vector Pearcey beams

2.2 Optical force based on electromagnetic scattering theory

3. Results and discussion

4. Conclusion

Funding

Acknowledgment

Disclosures

Data availability

References

Data availability

Cited By

Figures (3)

Equations (24)

Optics Express