Manipulation of metallic nanoparticle with evanescent vortex Bessel beam

Guanghao Rui; Xiaoyan Wang; Yiping Cui

doi:10.1364/OE.23.025707

1. Introduction

Since Ashkin and colleagues reported the first stable three-dimensional (3D) optical trapping, or optical tweezers, created using radiation pressure from a single focused laser beam, the capabilities of optical trapping have evolved from simple manipulation to the application of calibrated forces on, and the measurement of nanometer-level displacement of optically trapped objects [1–3 ]. In the early days, optical trapping has been mainly implemented in two size regimes: the sub-nanometer (e.g., cooling of atoms, ions and molecules) and micrometer scale (such as cells). Recently, new approaches have been developed to stably trap and manipulate mesoscopic objects, including metallic nanoparticles [4,5 ], carbon nanotubes [6,7 ] and quantum dots [8,9 ]. The continuous development of optical tweezers has revolutionized the experimental study of small particles and become an important tool for research in biology, physical chemistry and soft matter physics.

The unique size-dependent properties of metallic nanoparticles make them highly attractive in many areas from biology to electronics. For example, surface-enhanced Raman spectroscopy (SERS) takes advantage of the local field enhancement offered by optically resonant metallic nanoparticles to enhance the Raman signal and enable label-free detection of proteins, pollutants, and other molecules [10]. Due to the noncontact and “holding” nature, optical trapping is well suited to be combined with SERS, potentially enabling ultrasensitive molecular recognition in liquids. However, trapping metallic nanoparticle is challenging due to the strong scattering and absorption [11]. Several approaches were developed to increase the trapping efficiency of metallic nanoparticles [12–14 ]. For example, vectorial beams with spatially variant polarization or/and amplitude distributions were proposed to replace the conventional linear polarization as the illumination in optical tweezers [15–17 ]. Besides, it is shown recently that both the shape and the orientation of the metallic nanoparticle is a critical factor influencing its optical trapping [18].

Bessel beam is a non-diffracting beam solution of the free-space wave equation first introduced by Durnin et al. [19]. Since Bessel beams maintain their focus along the line, the position of manipulated object can be significantly varied, thus making micromanipulation systems flexible and more attractive for practical implementations [20,21 ]. The utilization of Bessel beams also opens new horizons in microporation [22], manipulation of micromachines [23] and micro-fabrication [24]. Different from zero-order Bessel beam that has a bright maximum in the center, high-order Bessel beam, which is one of the promising types of vortex optical beams, has doughnut shaped intensity profile with dark center while still possessing the non-diffraction and self-reconstructive properties. Except linear momentum, high-order Bessel beam possess orbital angular momentum (OAM), which is associated with the spatial field distribution in vortex beam [25]. The transfer of OAM to an object and its subsequent behavior has been investigated at the level of dielectric nanoparticles and particles much larger than the light wavelength within the scope of the ray optics [21,26–28 ]. Vortex Bessel beams can be generated from Gaussian beams or Laguerre-Gaussian beams after propagation through an axicon [25,29 ]. On the other hand, evanescent Bessel beam can be realized by transmitting the Bessel beam across a dielectric interface beyond the critical angel. However, these methods involve the use of bulky elements that are difficult to be integrated into a compact platform. There are alternative approaches to generate Bessel beams by cylindrical vector (CV) beams that have cylindrical symmetry in polarization distribution [30–32 ]. However, limited by the intrinsic focusing properties, conventional CV beams are only suitable for producing evanescent 0th and 1st order nonvortex Bessel beams. In this paper, we present a novel optical tweezers setup to trap and manipulate nanoparticles with evanescent vortex Bessel beam (EVBB). A versatile strategy is proposed that may be used to generate EVBB of any order by highly focusing a radially polarized vortex beam (RPVB) onto one-dimensional photonics band gap (1D PBG) structure. Moreover, we perform a theoretical study of the behavior of a single metallic nanoparticle in the EVBB. The feasibility of using zero-order evanescent Bessel beam to trap dielectric microparticles has been experimentally demonstrated by M. Gu et al [33], showing that the particles accumulate at the center of the focus under the radiation pressure. In this work we primarily dwell with the high-order EVBB that is generated with radially polarized light and its applications in manipulating metallic nanoparticle. High-order EVBB brings rich properties such as donut-shaped intensity pattern and OAM, leading to plenty of novel effects in trapping nanoparticles. Numerical simulation results show that the particle can be trapped near the surface and orbit along the high intensity ring of the EVBB. By tuning the order and the vortex handedness of the EVBB, the particle behavior can be holistically manipulated in terms of trajectory and rotating direction.

2. Results

2.1 Configuration of the optical tweezers

The proposed optical tweezers is schematically illustrated in Fig. 1 . The optical tweezers is constructed around a high numerical aperture (NA = 0.95n) aplanatic object lens. The main idea of the method is to build an EVBB that is beneficial for manipulating the behavior of the particles on a flat surface. As shown in Fig. 1, a RPVB with wavelength of 808 nm illuminates the pupil plane of an oil-immersion aplanatic lens to produce a spherical wave converging toward the last interface of the 1D PBG structure located at the focal plane. A circular photomask placed before the objective lens provides an annular illumination that blocks the incident light below the critical angle.

Fig. 1 Diagram of the proposed optical tweezers setup. An incident radially polarized vortex beam is highly focused by an objective lens onto a 1D PBG structure. Q(r, φ) is an observation point in the focal plane. The space between the lens and the PBG is filled with index-matching oil (n = 1.5).

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2.2 Focusing effect of RPVB

Since light wave can be interpreted quantum mechanically, photon can be viewed to carry both spin angular momentum (SAM) and OAM. The OAM arises from the spatial structure of the optical field and has a value of lħ per photon, where l is the topological charge (TC) that can take arbitrary integer values. A RPVB with TC of m can be expressed as:

{\vec{E}}_{i} (r, φ) = E_{0} e^{i m φ} {\hat{e}}_{r},

where E ₀ is the incident amplitude and φ is the azimuthal angle of the cylindrical coordinate system. When a RPVB is focused by high NA objective lens, the field distribution near the focus can be analyzed with the Richard-Wolf vectorial diffraction method as [34]:

\begin{array}{l} E_{r} (r, φ, z) = A \int_{θ_{\min}}^{θ_{\max}} P (θ) \sin θ \cos θ [J_{m + 1} (k r \sin θ) - J_{m - 1} (k r \sin θ)] \\ \times \exp (i k z \cos θ) \exp (i m φ) d θ, \end{array}

\begin{array}{l} E_{φ} (r, φ, z) = i A \int_{θ_{\min}}^{θ_{\max}} P (θ) \sin θ \cos θ [J_{m + 1} (k r \sin θ) + J_{m - 1} (k r \sin θ)] \\ \times \exp (i k z \cos θ) \exp (i m φ) d θ, \end{array}

E_{z} (r, φ, z) = i 2 A \int_{θ_{\min}}^{θ_{\max}} P (θ) \sin^{2} θ J_{m} (k r \sin θ) \times \exp (i k z \cos θ) \exp (i m φ) d θ,

where P(θ) is the pupil apodization function of the objective lens, θ_max is the maximal angle determined by the objective lens, and k is the wavenumber of the incident light in the medium. The constant A is given by A = πfl ₀/λ, where f is the focal length, λ is the wavelength of incident wave in the ambient environment, and l ₀ is associated with the laser beam power. From Eq. (2) and (3) , both radial and azimuthal components of the electric field near the focus have two terms. Therefore, the transverse electric field is a superposition of two different modes proportional to the (m-1)^th and (m + 1)^th order Bessel functions, respectively, except for the case of m = 0 when the illumination depicted by Eq. (1) is reduced to a nonvortex beam with radial polarization. For the radially polarized light without TC, E_φ vanishes and the transverse field near the focus is first-order Bessel function. For incident RPVB with m ≠ 0, it is interesting to notice that both of the two modes possess SAM and OAM simultaneously, with the same TC but different spins. When m = ± 1, the main lobe of the transverse electric field is related to zero-order Bessel function, corresponding to an intensity pattern with solid spot at the center. If m ≠ ± 1, a doughnut shaped profile can be resulted in the transverse plane perpendicular to the beam propagation. However, owing to the overlap between these two modes away from the main lobe, the transverse electric field exhibits hybrid polarization distribution and complicated phase patterns in the side lobes. Different from the transverse component, the longitudinal electric field only possess an m ^th order Bessel function with TC of m, which provide an opportunity to generate high-order Bessel beam carrying OAM through coupling the tightly focused RPVB with a resonant structure while depressing the magnitude of the transverse components.

2.3 Generation of EVBB

To design a resonant structure with high quality factor, 1D PBG is adopted that consists of 10 periods of alternating high and low index of refraction dielectric layers. Each period is comprised of two layers with GaP (n = 3.19) and Si₃N₄ (n = 2.01) and thickness of 220 nm and 120 nm, respectively. Attributed to the high density of state, the multi-layer structure is capable of providing gigantic transmission with band-edge resonance [35]. As shown in Fig. 2 , a delta-function like transmission coefficient with large resonant peak is obtained for PBG under radially polarized illumination. Such a sharp angular resonance combined with the rotational symmetry of the setup mimics an axicon for the generation of Bessel beams. Note that the transmission peak is located at θ_r = 63.08°, indicating that the incident wave experiences total internal reflection from the 1D PBG. The field builds up gradually within the stacked layer and is resonantly enhanced at the last interface of PBG structure. The field distributions just after the PBG can be expressed as:

\begin{array}{l} E_{r} (r, φ, z) = A \int_{θ_{\min}}^{θ_{\max}} P (θ) t (θ) \sin θ \cos θ [J_{m + 1} (k_{1} r \sin θ) - J_{m - 1} (k_{1} r \sin θ)] \\ \times \exp (i z \sqrt{k_{2}^{2} - k_{1}^{2} \sin^{2} θ}) \exp (i m φ) d θ, \end{array}

\begin{array}{l} E_{φ} (r, φ, z) = i A \int_{θ_{\min}}^{θ_{\max}} P (θ) t (θ) \sin θ \cos θ [J_{m + 1} (k_{1} r \sin θ) + J_{m - 1} (k_{1} r \sin θ)] \\ \times \exp (i z \sqrt{k_{2}^{2} - k_{1}^{2} \sin^{2} θ}) \exp (i m φ) d θ, \end{array}

\begin{array}{l} E_{z} (r, φ, z) = i 2 A \int_{θ_{\min}}^{θ_{\max}} P (θ) t (θ) \sin^{2} θ J_{m} (k_{1} r \sin θ) \\ \times \exp (i z \sqrt{k_{2}^{2} - k_{1}^{2} \sin^{2} θ}) \exp (i m φ) d θ, \end{array}

where t(θ) is the transmission coefficient of the PBG structure illuminated by radial polarization at incident angle of θ, k ₁ and k ₂ are the wave-vectors in the incident and exiting medium, respectively. Since the transmission coefficient can be approximated as t(θ) = t(θ_r)δ(θ-θ_r), Eq. (5)-(7) can be reduced to

\exp (i z \sqrt{k_{2}^{2} - k_{1}^{2} \sin^{2} θ})

multiplied by Bessel functions. Since

k_{2}^{2} - k_{1}^{2} \sin^{2} θ_{r} < 0

, only the evanescent field can penetrate into the existing medium. It is worthy of noting that a resonant dielectric waveguide is an alternative method for achieving enhanced evanescent waves [36]. However, it is not suitable for generated evanescent Bessel beam since it does not have the angular selectivity, which is the key to obtain Bessel beam from the radially polarized light.

Fig. 2 Transmission coefficient for radial polarized light incident on the 1D PBG.

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Numerical simulations based on Eq. (5)-(7) are presented in Fig. 3 for RPVB with m = 2. Figure 3(a) illustrates the field transverse profiles at the last interface of the PBG (z = 0). It can be seen that longitudinal term E_z dominates in the overall field distribution, thus ensuring the generation of 2nd order Bessel beam. The full-width-half-maximal (FWHM) of the primary ring and the dark center are calculated to be 0.33λ and 0.56λ, respectively. The field strength |E|² of the generated Bessel beam at different distances from the PBG and corresponding line-scans are shown in Fig. 3(b) and 3(c), respectively. The primary lobes nearly overlap with each other, indicating the non-diffraction properties of the Bessel beam [37]. Besides, one can also find that the field strength decreases with increasing propagation distance. This evanescent nature is clearly illustrated in Fig. 3(d) by plotting the peak intensity distribution along z axis. The decay length, which is defined by the propagation distance corresponding to the 1/e ² of the initial intensity, is estimated to be 0.41λ. Consequently, an evanescent 2nd order vortex Bessel beam can be obtained with the proposed setup when the illumination is RPVB with TC of 2.

Fig. 3 Numerical simulation results using vectorial diffraction theory for (a-d) m = 2 and (e-h) m = 3, respectively. (a, e) The longitudinal component |E_z|², radial component |E_r|² and azimuthal component |E_φ|² at the last interface of the 1D PBG for a RPVB illumination. (b, f) Total field strength |E|² at different distances from the last interface of the 1D PBG. (c, g) line-scans of Fig. 3(b) and 3(f) through the center. (d, h) |E|² along the z axis, showing the evanescent decay. Insets show the intensity decay in log scale.

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Similarly, by tuning the OAM of the incident RPVB, this setup enables generating EVBB of any order. As shown in Fig. 3(e)-3(f), an evanescent 3rd order vortex Bessel beam can be realized by RPVB with TC of 3. The FWHM of the primary ring, dark center and the decay length are measured to be 0.36λ, 0.86λ and 0.41λ, respectively.

2.4. Optical forces induced behavior of a metallic nanoparticle in evanescent vortex Bessel beam

As the result of conservation of angular momentum within a closed physical system, the OAM can be transferred from the light to matter, leading to a torque on the matter. This technique is christened optical spanner or wrench [2,38 ]. Assuming a metallic spherical Rayleigh particle with radius of a suspended in a medium with the dielectric constant of ε, its optical property can be described by the polarizability α [39]:

α = 4 π a^{3} \frac{ε_{m} (ω) - ε}{ε_{m} (ω) + 2 ε},

where ε_m is the relative permittivity of the metal from bulk material, and ω is the frequency. Within the field of the evanescent Bessel beam, the particle moves under the influence of time averaged light-induced forces and the optical forces exerted on the nanoparticle can be written as [39]:

\vec{F} = \frac{1}{4} ε_{0} Re {α} \nabla {| \vec{E} |}^{2} + \frac{n σ}{2 c} {\vec{E} \times {\vec{H}}^{*}} + \frac{σ}{2} Re {i \frac{ε_{0}}{k_{0}} (\vec{E} \cdot \nabla) {\vec{E}}^{*}},

where ε ₀ is the permittivity of free space,

σ = k Im (α)

is the total cross section of the particle, c is the speed of light, n is the refractive index of the surrounding medium and k ₀ is the wave-vector in free space. The first term is the gradient force that provides the 3D confinement in optical tweezers as long as it dominates the second and third terms. The second term is identified as the radiation pressure force proportional to the Poynting vector, and the third term is a force arising from the presence of spatial polarization gradients. The combination of the second and the third term gives the total scattering force.

To illustrate the ability of proposed optical tweezers in terms of optical trapping and manipulation, we assume a gold particle with radius of 50 nm is placed in the existing medium of the PBG structure. The index of refraction of gold is 0.16 + 4.97i at the wavelength of 808 nm. The input laser power is set to be 100 mW. Based on Eq. (9), the optical forces exerted on the gold nanoparticle can be numerically calculated and analyzed in cylindrical coordinates. Figure 4 shows the force acting upon a gold nanoparticle placed into the evanescent 2nd and 3rd order vortex Bessel beam. The radial component of the force is depicted in Fig. 4(a) and 4(b) as a function of the radial coordinates r. The particle is stably trapped by the gradient force at the radius of maximum light intensity. The corresponding equilibrium points are located at r_e = 0.47λ and 0.65λ, respectively. Figure 4(c) and 4(d) show the distribution of longitudinal force along z axis. Due to the evanescent decay of the generated Bessel beam, the total force keeps negative with a trapping length of about 5λ in the z direction, meaning that the particle within that range will be dragged towards the PBG structure. Since the particle is trapped off the beam axis, the inclination of the helical phase fronts and the corresponding momentum result in a tangential force, leading to the particle rotating around the beam axis. As shown in Fig. 4(e), the azimuthal force is constant thus the particle will orbit along the vortex ring. The direction of the orbital motion can be reversed by changing the handedness of the helical phase front. The torque of a transverse radiation pressure is written as

T_{z} (r, z = 0) = 2 r F_{t r a n s}^{s c a t} (r, φ, z = 0),

where

F_{t r a n s}^{s c a t}

denotes the scattering force in the transverse plane. Figure 4(f) illustrates the influence of the controllable factor m of the EVBB on T_z. It can be seen that the maximal magnitude of the T_z locates at the position of stably captured particle in the focal plane. Namely the transverse radiation pressure drives the trapped particle around the beam axis. Besides, the direction of T_z also depends on the TC of the incident vector field.

Fig. 4 Calculated optical forces on 50 nm (radius) gold nanoparticle in EVBB with different m. (a, c) Radial force along the r axis for m = 2 and 3. (b, d) Longitudinal force acting upon the same particle placed in the stable radial distance r denoted in Fig. 4. (a) and 4(c) (r = 0.47λ and 0.65λ, respectively) along the z axis for m = 2 and m = 3. (e) Azimuthal force acting upon the same particle placed in the stable radial distance r and (f) longitudinal component of torques for m = ± 2, ± 3. The gray and yellow bars denote the positions of stable captured particle at the focal plane with |m| = 2 and 3, respectively.

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Consequently, the numerical simulation results clearly demonstrate that the EVBB is capable of trapping metallic nanoparticle near the surface, and the transfer of extrinsic OAM from the light to the trapped particle drive its orbital motion. Moreover, the orbital rotating radius and direction of the trapped particle can be tuned with order and OAM of the Bessel beam, respectively. Theoretically the magnitude of T_z is proportional to linear velocity of the trapped particle and its corresponding angular velocity is determined by both the torque and the orbital trajectory. Note that the trajectory of the trapped particle will not vary with laser power, because a change in laser power only leads to a change of the magnitude of the optical force however the corresponding force distribution remains the same. It is also worthy of noting that the generated high-order Bessel beam is also useful to trap hollow particles and particles with a refractive index lower than the ambient.

2.5. Stability analysis

In some cases, the three-dimensional confinement of the gold particle requires the optical force is large enough to balance the gravity. The gravity of a gold nanoparticle with radius of 50 nm is calculated to be 1.01 × 10⁻¹⁷ N using the density of gold (~19.3g/cm³) and ignoring the buoyancy force, which is much smaller than the maximal optical force 1.25 × 10⁻¹¹ N (m = 2) and 1.02 × 10⁻¹¹ N (m = 3) in the longitudinal direction.

In addition, in order to form a stable trap, the potential well generated by the gradient force needs to be deep enough to overcome the kinetic energy of the particle in Brownian motion. This criterion can be expressed as R _thermal = exp(-U _max/k_BT)<<1, where $U_{\max} = ε_{0} Re (α) {| E |}_{\max}^{2} / 2$ is the potential depth [40], k_B is the Boltzmann constant, and T is the absolute temperature of the ambient. Assuming T = 300 K, R _thermal for the situations considered above are calculated to be 4.66 × 10⁻¹¹ (m = 2) and 2.18 × 10⁻¹¹ (m = 3).

Discussions above clearly demonstrated that stable 3D optical trapping can be formed for metallic nanoparticles from the force balance point of view. Besides, thermal mechanism in optical tweezers is another main reason that destabilizes the trap and a much more difficult factor to combat in conventional optical tweezers when the operating wavelength is close or at the resonant wavelength for the metallic nanoparticles [17,41 ], especially for the particles immersed in solution with low viscosity. For a gold nanoparticle immersed in water, the critical temperate of water is about 647 K [42], about which the water would evaporate to form the nucleation of vapour bubble, then the optical trap would be destroyed. The thermal effect is significant at the peak of the absorption cross section, which is about 532 nm for a gold nanoparticle with the radius of 50 nm. However, the thermal effect will also quickly decrease from the absorption peak wavelength. It is worthy of noting that the impact of optical overheating to the optical trap presented in this work is not as severe due to the low incident power and weak absorptivity. Since the radius of the particle is much smaller than the wavelength, the absorbed power by the particle can be estimated with dipole approximation as: P = α”ε ₀ ωE ²/2, where α” is the imaginary part of the polarizability of the particle, ε ₀ is the vacuum permittivity, ω is the angular frequency and E is the electric field at the equilibrium point. The temperature of water at the particle’s surface is calculated by COMSOL with the absorbed power P. Simulation results show that the gold nanoparticle can only be heated up to about 313 K, which is much smaller than the corresponding critical temperature.

3. Conclusions

In conclusion, we demonstrated a novel method to trap and manipulate metallic nanoparticle using an EVBB. A versatile method for generating EVBB of any order is proposed by focusing a RPVB onto a 1D PBG structure. We demonstrate that a metallic nanoparticle can be trapped near the surface of the PBG. Besides, it is shown that the EVBB can drive the orbital motion of the metallic nanoparticle trapped off the beam axis. The direction of the orbital motion is in accordance with the handedness of the helical phase front, and the orbital radius could be controlled with appropriate adjustments of the order of the Bessel beam. The non-contact tunable dynamic orbital motion of the trapped particle may offer a simple way to drive micro-fluidic flow. Furthermore, this technique can be extended to arbitrary nanoparticles and allow for non-contact rotational control of materials of interest to higher resolution imaging techniques.

Acknowledgments

This work is supported by the Fundamental Research Funds for the Central Universities (Grant No. 2242015KD001) and the National Key Basic Research Program of China (Grant No. 2015CB352002).

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Manipulation of metallic nanoparticle with evanescent vortex Bessel beam

Abstract

1. Introduction

2. Results

2.1 Configuration of the optical tweezers

2.2 Focusing effect of RPVB

2.3 Generation of EVBB

2.4. Optical forces induced behavior of a metallic nanoparticle in evanescent vortex Bessel beam

2.5. Stability analysis

3. Conclusions

Acknowledgments

References and links

Cited By

Figures (4)

Equations (10)

Optics Express