Enhancement of trapping efficiency by utilizing a hollow sinh-gaussian beam

ABSTRACT Propagation properties and optical forces upon a Rayleigh dielectric sphere for a newly proposed hollow sinh-Gaussian beam (HsGB) are intensively investigated. In view of the

targeted laser beam’s unique tight focusing properties that a significantly sharp, peak-centered, and adjustable intensity distribution would be produced in the focal vicinity, the tightly

focused HsGB could be exploited to trap and manipulate nano-sized dielectric spheres with high-refractive index in the focal region. The interesting and meaningful features for the novel

HsGB mainly include that, compared with the conventional fundamental Gaussian beams under the same optical power, the tightly focused HsGB has much higher intensity gradient and deeper

potential well through optimizing targeted laser beam’s parameters. Theretofore, the novel HsGB optical tweezers could drastically enhance its trapping efficiency. Finally, the trapping

stability conditions are discussed in detail. The analytical and numerical results obtained here could provide a directive suggestion for researchers in optimizing experimental parameters in

constructing a novel HsGB tweezers and making use of a HsGB. SIMILAR CONTENT BEING VIEWED BY OTHERS SIMULTANEOUS AND INDEPENDENT CAPTURE OF MULTIPLE RAYLEIGH DIELECTRIC NANOSPHERES WITH

SINE-MODULATED GAUSSIAN BEAMS Article Open access 08 January 2021 ENHANCEMENT OF OPTICAL LEVITATION WITH HYPERBOLIC METAMATERIALS Article Open access 19 January 2024 MITIGATING STIMULATED

BRILLOUIN SCATTERING IN MULTIMODE FIBERS WITH FOCUSED OUTPUT VIA WAVEFRONT SHAPING Article Open access 13 November 2023 INTRODUCTION Recently, dark-hollow beams (DHBs) have attracted

considerable research interest and extensive attention due to its interesting characteristics and potential usefulness in application fields mainly including atomic optics, optical

communication, and optical manipulation1,2,3. Till now, several methods have been applied to generate DHBs4,5,6,7,8, and meanwhile, several theoretical models have been proposed to describe

DHBs9,10,11. In 2012, Sun _et al_. introduced a novel mathematical model called hollow sinh-Gaussian beams (HsGBs) to depict DHBs, and the targeted laser beams’ propagation characteristics

in free space were also examined12. Moreover, owing to the HsGBs’ special properties, especially for its central hollow region, they further predicted that HsGBs might find potential

applications in atom guiding and trapping12. In recent years, several works on tight focusing and application of various polarized HsGBs’ were reported13,14,15,16. For instance, Lin _et al_.

considered the radially polarized HsGBs’ tight focusing performance and showed that high beam quality and subwavelength focusing could be achieved, which was expected be applied to achieve

focusing with superresolution13; Liu _et al_. examined the tightly focused azimuthally polarized HsGBs and found that a steeper depleting pattern and an exciting pattern without side lobes

could be produced, which was helpful to achieve super resolution in STED microscopy14; Senthilkumar _et al_. numerically investigated the tight focusing properties of spirally polarized

HsGBs and revealed that many novel focal patterns including flattop profile, focal hole axially separated focal spots and focal spot with long focal depth could be evolved, which tunable

focal patterns were expected to be useful for optical manipulation of micro particles15; Zou _et al_. studied the propagation properties of a single HsGB and their interactions through the

quadratic-index medium16. As we know that photon carries both linear and angular momentums. Through momentum exchange light field would exert radiation (optical) force and torque on objects

it encounters. Since the first demonstration of acceleration and trapping of particles by radiation pressure17, especially a single-beam gradient force optical trap (i.e., optical tweezers)

for dielectric particles was first observed in experiment18, researches on optical tweezers have made great progress. Through the unremitting efforts of scientific researchers, nowadays,

optical tweezers have become a powerful and indispensable tool in the following areas such as physics, chemistry, and biology19. Researchers have revealed that optical forces are closely

related to the structure of the focused laser fields. And till now, a great variety of structured light fields have been generated and tailored to manipulate various tiny

objects19,20,21,22,23,24,25,26,27,28. In this work, we first derived the propagation formula for the HsGBs through an ABCD optical system, and then analyzed the HsGBs’ tight focusing

properties. Subsequently, we exploited the highly focused HsGBs to trap and manipulate nano-sized dielectric spheres with high-refractive index in the focal region. The interesting and

meaningful features for the novel HsGBs mainly included that, compared with the conventional fundamental Gaussian beams under the same optical power, the tightly focused HsGBs have much

higher intensity gradient and deeper potential well through optimizing targeted laser beam’s parameters. Theretofore, the novel HsGB optical tweezers can greatly enhance its trapping

efficiency. Finally, we analyzed the trapping stability conditions. PROPAGATION OF HSGBS THROUGH AN ABCD OPTICAL SYSTEM According to the novel mathematical model of dark-hollow beams

proposed by Sun _et al_., the electric field of HsGBs in the original plane _z_ = 0 takes the form12 $${E}_{n}(r,0)={G}_{0}{\sinh }^{n}(\frac{r}{{w}_{0}})\exp

(-\frac{{r}^{2}}{{w}_{0}^{2}}).$$ (1) In Eq. (1), _G_0 denotes a constant related to the laser beams power _P_0, and _n_ (_n_ = 0, 1, 2, ⋅⋅⋅) represents the order of HsGBs. Here, it is noted

that the central dark size and the construction of HsGBs could conveniently be controlled by the laser beams order. Also, one can note that, for the case _n_ = 0, Eq. (1) represents the

fundamental Gaussian beams with waist size _w_0; for the case _n_ ≥ 1, Eq. (1) stands for the so-called hollow sinh-Gaussian beams (HsGBs). It is noted from Eq. (1) that HsGBs could be

considered as superposition of a series of eccentric Gaussian beam, of which characteristic would be determined by the order _n_ and waist size _w_0. In order to visualize the profiles of

HsGBs, Fig. 1 illustrates the contour graphs of the normalized intensity distribution of HsGBs with several orders n. All the contour graphs and curves in Fig. 1 have been normalized by a

fixed optical power _P_0 = 100 mW. It is clearly seen from Fig. 1 that, for the case n = 0, the irradiance profile of fundamental Gaussian beam presents a peak-centered configuration; for

the case n ≥ 1, the irradiance profile of HsGB presents a single bright ring configuration, and furthermore, the central dark size increases as n increases. Therefore, by selecting

appropriate laser beams order, one may obtain HsGBs with ideal intensity distribution. For the convenience of integration, Eq. (1) was to be rewritten in the form $${E}_{n}(r,0)={G}_{0}\sum

_{m=0}^{n}{a}_{m}{b}_{m}\exp [\frac{-{(r+{c}_{m})}^{2}}{{w}_{0}^{2}}].$$ (2) In Eq. (2), several coefficients are, respectively, given by $${a}_{m}={(-1)}^{m}{2}^{-n}(\begin{array}{c}n\\

m\end{array}),\,{b}_{m}=\exp [(m-\frac{n}{2}{)}^{2}],\,{c}_{m}={w}_{0}(m-\frac{n}{2}),$$ (3) where \((\begin{array}{c}n\\ m\end{array})\) is a binomial coefficient, and −_c__m_ denotes a

center parameter. Within the framework of the paraxial approximation, propagation of laser beams through a paraxial ABCD optical system could be described by the generalized Huygens-Fresnel

diffraction integral, known as Collins formula, which takes the following form in a cylindrical coordinate system29 $$\begin{array}{c}{E}_{n}(r,z)=\frac{i}{\lambda B}\exp (\,-\,ikz)\\

\,\,\,\,\times {\int }_{0}^{2\pi }{\int }_{0}^{\infty }{E}_{n}(r^{\prime} ,0)\exp \{-\frac{ik}{2B}[A{r}^{^{\prime} 2}-2rr^{\prime} \cos (\theta -\theta ^{\prime} )+D{r}^{2}]\}r^{\prime}

dr^{\prime} d\theta ^{\prime} .\end{array}$$ (4) In Eq. (4), _E__n_(_r_′, 0) and _E__n_(_r_, _z_) denote the electric fields in the input and output planes, respectively. _r_′, _θ_′ and _r_,

_θ_ are the radial and azimuthal angle coordinates in the input and output planes, respectively. _z_ is the axial distance between the input and output planes along the optical axis. _k_ is

the wave number related to the wavelength _λ_ by _k_ = 2_π_/_λ_. A, B, C and D are the transfer matrix elements of the optical system. Substituting Eqs (2) and (3) into Eq. (4), and

recalling the following integral formulae30: $${J}_{0}(t)=\frac{1}{2\pi }{\int }_{0}^{2\pi }\exp (it\,\cos \,\phi )d\phi ,$$ (5) $$\begin{array}{rcl}{\int }_{0}^{\infty }{t}^{\mu }\exp

(\,-\,{a}^{2}{t}^{2}){J}_{v}(pt)dt & = & \Gamma (\frac{\mu +v+1}{2})\frac{{p}^{v}}{{2}^{v+1}{a}^{\mu +v+1}\Gamma (v+1)}\\ & & \times \,{}_{1}F_{1}(\frac{\mu

+v+1}{2},v+1;-\,\frac{{p}^{2}}{4{a}^{2}}),\end{array}$$ (6) where _J__v_(_x_) stands for the _v_-order Bessel function of the first kind, _Γ_(_x_) denotes the gamma function, and 1_F_1(_a_,

_b_; _x_) is the confluent hypergeometric function, after tedious but straightforward integration, one could get $$\begin{array}{c}{E}_{n}(r,z)=\frac{i{G}_{0}k}{2B}\exp (\,-\,ikz)\exp

(-\frac{ikD{r}^{2}}{2B})\\ \,\,\,\,\times \sum _{m=0}^{n}\sum _{s=0}^{\infty }{(-1)}^{m}{2}^{-n}(\begin{array}{c}n\\ m\end{array})\frac{{(n-2m)}^{s}}{{w}_{0}^{s}\cdot s!}\\ \,\,\,\,\times

\Gamma (1+\frac{s}{2})\cdot {(\frac{1}{{w}_{0}^{2}}+\frac{ikA}{2B})}^{-1-s/2}\cdot

{}_{1}F_{1}(1+\frac{s}{2},1;-\,\frac{{(\frac{kr}{2B})}^{2}}{(\frac{1}{{w}_{0}^{2}}+\frac{ikA}{2B})}).\end{array}$$ (7) Equation (7) is the general propagation formula for HsGBs through an

ABCD optical system, by which the novel laser beams propagation and transformation through various optical systems could be treated conveniently. TIGHT FOCUSING PROPERTIES OF HSGBS In order

to investigate HsGBs’ tight focusing properties, let us consider the laser beams propagate through a thin lens system as shown in Fig. 2. According to the principle of Matrix Optics31, the

transfer matrix between the input and output planes could be given by $$[\begin{array}{cc}A & B\\ C & D\end{array}]=[\begin{array}{cc}1 & f+\Delta z\\ 0 &

1\end{array}][\begin{array}{cc}1 & 0\\ -1/f & 1\end{array}]=[\begin{array}{cc}1-z/f & z\\ -1/f & 1\end{array}].$$ (8) In Eq. (8) _f_ is the focal length, Δ_z_ is the distance

between the geometrical focus and the output plane, and _z_ = _f_ + Δ_z_ is the axial distance from the output (or reference) plane to the input plane. Substituting Eq. (8) into Eq. (7),

one could obtain the normalized intensity distribution of HsGBs through a lens optical system. In Fig. 2, the left and right plots show the intensity distributions of the 3rd-order HsGBs in

the input and output planes, respectively. It is clearly seen from Fig. 2 that through the focusing system the original millimeter-sized and hollow laser beam turns to a submicro-sized and

peak-centered configuration. To further examine the HsGB’s tight focusing properties in detail, Fig. 3 plots the intensity distribution of different ordered HsGBs near the focus both in the

transverse plane (a) Δ_z_ = 0, and the longitudinal plane (b) _x_ = 0. The parameters are, respectively, selected as: _λ_0 = 0.514 μm, _w_0 = 10 mm, and _f_ = 25 mm. For comparison

convenience, the orders _n_ of HsGBs are set as 0 (i.e., fundamental Gaussian beams), 1, 3 and 5, respectively. From Fig. 3, it is obvious that both the transverse (see Fig. 3(a)) and the

longitudinal (see Fig. 3(b)) intensity distribution presents a significantly sharp and peak-centered configuration. Moreover, the focusing intensity distribution would be sensitively

adjusted by the HsGBs’ order: increase the HsGBs’ order, and the peaks become dramatically sharper. Due to these focusing properties, the highly focused HsGBs might be exploited to trap and

manipulate nano-sized dielectric particles with high-refractive index in the focal region. OPTICAL FORCES ON A RAYLEIGH DIELECTRIC SPHERE PRODUCED BY TIGHTLY FOCUSED HSGBS In this section,

we investigate the optical forces exerted on a nano-sized dielectric sphere produced by the tightly focused HsGBs. For simplicity, the radius of sphere is assumed to be much smaller than the

laser beam wavelength (_a_ ≤ _λ_/20). In this situation, the Rayleigh approximation is valid to deal with the scattering between the focused laser field and the sphere, and the optical

forces exerted on the objects mainly include gradient force \({\mathop{F}\limits^{\rightharpoonup }}_{grad}\) and scattering force \({\mathop{F}\limits^{\rightharpoonup }}_{scat}\).

According to the refs20,32, the gradient force and the scattering force are, respectively, defined by $${\mathop{F}\limits^{\rightharpoonup }}_{grad}(r,z)=\frac{2\pi

{n}_{2}{a}^{3}}{c}(\frac{{n}_{1,2}^{2}-1}{{n}_{1,2}^{2}+2})\nabla I(r,z),$$ (9) $${\mathop{F}\limits^{\rightharpoonup }}_{scat}(r,z)=\hat{z}\frac{{n}_{2}}{c}{C}_{pr}I(r,z),$$ (10) with

$$I(r,z)=\frac{{n}_{2}{\varepsilon }_{0}c}{2}{|E(r,z)|}^{2},$$ (11) $${C}_{pr}={C}_{scat}=\frac{8}{3}\pi {(ka)}^{4}{a}^{2}{(\frac{{n}_{1,2}^{2}-1}{{n}_{1,2}^{2}+2})}^{2},$$ (12) where _a_ is

the sphere’s radius, _n_1,2 = _n_1/_n_2 stands for the relative index with _n_1 and _n_2, respectively, representing the refractive index of the sphere and the ambient, and c is the light

field’s propagation speed in vacuum. It is pointed out that _C__pr_ denotes the cross section for the optical pressure of the sphere, which is assumed to be equal to the scattering cross

section _C__scat_ for a dielectric sphere in the Rayleigh regime. Using Eqs (7–12), one could calculate the optical forces acting upon a Rayleigh dielectric sphere produced by highly focused

HsGBs. Without loss of generality, in the following calculations we select the radius of sphere _a_ = _λ_/20 = 25 nm, the high-refractive index of the nano-sized particle _n_1 = 1.59 (i.e.,

glass), and the index of surrounding medium _n_2 = 1.33 (i.e., water). Figure 4(a–c) illustrates the transverse gradient force \({\mathop{F}\limits^{\rightharpoonup }}_{grad,x}\), and Fig. 

4(d–f) the longitudinal gradient force \({\mathop{F}\limits^{\rightharpoonup }}_{grad,z}\) exerted upon a Rayleigh dielectric sphere for the focused different ordered HsGBs in the focal

vicinity. It is clearly found from Fig. 4 that one stable equilibrium point is distributed at the focus both in the transverse and longitudinal directions for high-refractive-index (_n_1,2 

> 1) spheres (see Fig. 4(a,d)). It indicates that the focused HsGBs could be exploited to trap and manipulate spheres with refractive index larger than that of the surrounding medium.

Moreover, from Fig. 4(a,d), one could also find that both the transverse and longitudinal gradient forces become much greater and sharper (i.e., the optical trap stiffness, which is defined

by differentiating the gradient forces, is enhanced) when increase the HsGBs’ order _n_, which indicates that high-ordered HsGBs could greatly improve the optical tweezers’ optical trapping

efficiency. Meanwhile, the optical trapping range would decrease when the HsGBs’ order _n_ increases. In addition, it should be noted that the HsGB’s optical forces would be influenced by

several factors, such as the laser beam power _P_0, laser beam size _w_0, focal length _f_, particle’s size _a_, and the relative refractive index _n_1,2 = _n_1/_n_2, etc. It is easily

obtained from Eqs (9) and (10) that, the optical forces would increase when increase the laser beam power or reduce the laser beam size. To investigate the influence of the rest factors on

the HsGB’s optical forces, Fig. 5 presents the changes of the transverse gradient forces (a–c) and the longitudinal gradient forces (d–f) for several values of focal length _f_, particle’s

size _a_, and particle’s refractive index _n_1. It is clearly seen from Fig. 5(a,d) that the shorter the focal length is, the larger the optical forces would become. And it could also be

found that when increase the particle’s size (see Fig. 5(b,e)), or increases the gap between the refractive index of the particle and that of the surrounding medium (see Fig. 5(c,f)), the

HsGB’s optical forces would increase, and meanwhile, the optical trap stiffness would be significantly improved. In other words, shorter focal length, bigger particle size, and greater gap

between the refractive index _n_1 of the particle and the refractive index _n_2 of the surrounding medium are all favorable factors for the optical trapping. TRAPPING STABILITY ANALYSIS

Through the above analysis, one may have an image that tightly focused HsGBs could be exploited to trap and manipulate high-refractive-index Rayleigh dielectric spheres in the focal regime.

In order to capture or manipulate a Rayleigh particle stably, there are still two necessary factors that the optical manipulation system has to fulfill18

$$R=|{\overrightarrow{F}}_{grad,z}|/|{\mathop{F}\limits^{\rightharpoonup }}_{scat}|\ge 1,$$ (13) $${R}_{thermal}=\exp (\,-\,{U}_{{\rm{\max }}}/{k}_{B}T)\ll 1,$$ (14) Equation (13) stands for

the first necessary factor, which indicates that the maximum axial gradient force should be greater than the maximum scattering force, where R is defined as the stability criterion. Figure 

6(a–c) shows the scattering force exerted upon a Rayleigh dielectric sphere for the focused different ordered HsGBs near the focus at several planes. Comparing the longitudinal gradient

force \({\mathop{F}\limits^{\rightharpoonup }}_{grad,z}\) (see Fig. 5(d–f)) and the scattering force \({\mathop{F}\limits^{\rightharpoonup }}_{scat}\) (see Fig. 6(a–c)) at the same position,

the magnitude of \({\mathop{F}\limits^{\rightharpoonup }}_{grad,z}\) is much greater (about 19-times greater) than that of \({\mathop{F}\limits^{\rightharpoonup }}_{scat}\). Since

\({\mathop{F}\limits^{\rightharpoonup }}_{grad,z}\) is dominated in the _z_ direction, the total longitudinal optical forces (i.e., \({\mathop{F}\limits^{\rightharpoonup }}_{grad,z}\) + 

\({\mathop{F}\limits^{\rightharpoonup }}_{scat}\)) would presents the restoring force characteristics, which could be verified by Fig. 6(d–f). Hence, the first stability criterion, i.e., Eq.

(13) is fulfilled well. Secondly, to stably trap or manipulate a particle, the potential well, which is generated by the gradient force and defined by \({U}_{{\rm{\max }}}=\pi {\varepsilon

}_{0}{n}_{2}^{2}{a}^{3}|({n}_{1,2}^{2}-1)/({n}_{1,2}^{2}+2)|\cdot |{E}_{{\rm{\max }}}{|}^{2}\), should surpass the particle’s kinetic energy. So, the second stability criterion could be

expressed by Eq. (14), where _k__B_ denotes the Boltzmann factor and _T_ is the absolute temperature of the ambient. In our numerical examples, at room temperature of 300 K, for the

high-refractive index particle (_n_1 = 1.59), the value of _R__thermal_ at the maximum intensity position (_x_ = 0, _y_ = 0, Δ_z_ = 0) is about _R__thermal_ ≈ 0.055. Theretofore, the second

stability criterion, i.e., Eq. (14) is also fulfilled very well. In all, through the above analyzing we could draw a conclusion that in order to obtain better trapping efficiency by HsGBs

tweezers, one could select higher-order HsGBs and higher relative refractive index dielectric spheres with radius _a_ < 25 nm. CONCLUSIONS Based on the generalized Huygens-Fresnel

diffraction integral, we first derived the analytical formula for a newly proposed hollow sinh-Gaussian beam (HsGB) propagating through an ABCD optical system, and then analyzed its tight

focusing. Due to the targeted laser beam’s unique tight focusing properties that a significantly sharp, peak-centered, and adjustable intensity distribution would be produced in the focal

vicinity, subsequently, then we exploited the highly focused HsGBs to trap and manipulate nano-sized dielectric spheres with high-refractive index in the focal region. The interesting and

meaningful features for the novel HsGBs mainly include that, compared with the conventional fundamental Gaussian beams under the same optical power, the tightly focused HsGBs have much

higher intensity gradient and deeper potential well through optimizing targeted laser beam’s parameters. Theretofore, the novel HsGB optical tweezers can greatly enhance its trapping

efficiency. Finally, we analyzed the trapping stability conditions. The analytical and numerical results obtained here could provide a directive suggestion for researchers in optimizing

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Optik_ 89, 174–180 (1992). CAS  Google Scholar  Download references ACKNOWLEDGEMENTS This work was supported by the National Natural Science Foundation of China (Grant Nos 11764015 and

11864013), and by the Scientific Project of Jiangxi Education Department of China (Grant No. GJJ170361). AUTHOR INFORMATION AUTHORS AND AFFILIATIONS * Department of Applied Physics, East

China Jiaotong University, Nanchang, 330013, China Zhirong Liu, Xun Wang & Kelin Hang Authors * Zhirong Liu View author publications You can also search for this author inPubMed Google

Scholar * Xun Wang View author publications You can also search for this author inPubMed Google Scholar * Kelin Hang View author publications You can also search for this author inPubMed 

Google Scholar CONTRIBUTIONS Z.L. supervised the project and conceived the idea. X.W. wrote the manuscript. K.H. performed the numerical simulations and discussed the results. All authors

discussed the results and reviewed the manuscript. CORRESPONDING AUTHOR Correspondence to Zhirong Liu. ETHICS DECLARATIONS COMPETING INTERESTS The authors declare no competing interests.

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Enhancement of trapping efficiency by utilizing a hollow sinh-Gaussian beam. _Sci Rep_ 9, 10187 (2019). https://doi.org/10.1038/s41598-019-46716-5 Download citation * Received: 25 February

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