Manipulation of resonant metallic nanoparticle using 4Pi focusing system

Xiaoyan Wang; Guanghao Rui; Liping Gong; Bing Gu; Yiping Cui

doi:10.1364/OE.24.024143

1. Introduction

In 1986, Ashkin and his colleges firstly utilized a single tightly focused laser beam to trap individual nano- and microsized dielectric particles in three-dimensions (3D) [1]. Nowadays this technique is known as optical tweezers and has become an important tool for research in the fields of biology, physical chemistry and soft matter physics [2]. The mechanical stability of a nanoparticle is the competition between two types of optical force generated by laser beam: gradient force and scattering force. Gradient force arises from induced polarization in dielectric and points towards high intensity focal volume. Radiation pressure is another mechanical force exerted on particles due to the conservation of momentum during absorption and scattering of photons. Different with gradient force, scattering force usually has a negative impact on optical tweezers since it points out of the focal region. Consequently, it is necessary to have dominating gradient force in order to obtain a stable optical trapping.

Due to the plasmonic properties such as field enhancement, Raman scattering and local heating, metallic nanoparticles are highly attractive in many areas from biology to electronics [3, 4]. The noncontact and holding nature of optical tweezers make it highly suitable to manipulate metallic nanoparticles so as to apply those exotic properties in free solution and in absence of substrate, which have a wide range of applications such as surface-enhanced Raman scattering and photothermal therapy [5, 6]. Different from dielectric nanoparticles, trapping metallic nanoparticles are generally considered to be difficult owing to the strong absorption and scattering of light close to their localized surface plasmon resonance (LSPR). Although it has been experimentally demonstrated that metallic nanoparticles can be stably trapped when the trapping wavelength is sufficiently away from the LSPR of the particles [7], it is still tough to trap metallic nanoparticle at wavelength close to LSPR since the optical trapping can be easily destroyed by the dramatically increased scattering force and severe optical heating effect [8]. With rapid advance in spatial engineering of light wave, generation of optical fields with unconventional spatial distribution in terms of phase, amplitude and polarization becomes possible. These novel optical beams with unconventional features have also been discovered to enhance the performance of the optical tweezers [9–11]. For example, negative scattering force pointing against the optical power flow was reported for certain Bessel beam that allows “tractor beam” drags particles towards the light source [12, 13]. In addition, taking advantage of the subtle balance between gradient force and scattering force, axial equilibrium point can be created by tailoring the spatial distribution of a vectorial optical illumination, enabling 3D trapping and manipulating of resonant metallic nanoparticles [14]. However, applying these techniques into practice remains difficult because the required spatial amplitude distribution is still challenging for existing optical modulation technology. Besides, the scattering force would become non-conservative due to the asymmetric focal pattern caused by the introduction of the multi-zone phase plate. Consequently, the particles can only be trapped from one side of the focal field. In this work, we proposed a novel optical tweezers constructed around a 4Pi microscopy, which was developed to improve the axial resolution by coherent interference of the two counter-propagating focused wave-fronts [15–18]. The radially polarized light can be tightly focused into spherical spots with diffraction-limit size, enhanced intensity point spread function and equal three-dimensional spatial resolution, leading to greatly enhanced gradient force. Besides, there is no contribution of the axial scattering force to the particle since it is canceled by the symmetry of the 4Pi focusing system. The interactions between the metallic nanoparticle and the engineered focal field enable an optical tweezers enabling not only stable 3D trapping but also a precise control of the motion trajectory of the metallic nanoparticles even under the resonant condition. Additionally, since the optical force is remarkably enhanced due to higher focusing efficiency and magnitude of the polarizability near the resonance, the optical thermal effect can be relieved via reducing the input laser power, overcoming the ultimate obstacle that prevents stable manipulation of resonant metallic nanoparticles.

2. Configuration of the optical tweezers

As shown in Fig. 1, the proposed optical tweezers is constructed around a 4Pi microscopy, which is consisted of two high numerical aperture (NA) aplanatic lens that focus the light as tight as possible. Two counter-propagating beams with radial polarization normally illuminate the focusing system, whose instantaneous electric field vectors are represented by the arrows in Fig. 1. The electric field of the input beam can be expressed as:

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

where E₀ is the incident amplitude, r and ϕ are the radius and azimuthal angle of the cylindrical coordinate system, and m is the topological charge (TC) of the light. The electric field in the vicinity of the focus of a high NA lens for radially polarized vortex beam can be calculated using Richard-Wolf vectorial method [19]:

\vec{E} (r, ϕ, z) = i^{m} A \int_{θ_{\min}}^{θ_{\max}} P (θ) \sin θ e^{i k z \cos θ} e^{i m ϕ} {\begin{cases} \cos θ [J_{m + 1} (k r \sin θ) - J_{m - 1} (k r \sin θ)] {\vec{e}}_{r} \\ - i \cos θ [J_{m + 1} (k r \sin θ) + J_{m - 1} (k r \sin θ)] {\vec{e}}_{ϕ} \\ - 2 i \sin θ J_{m} (k r \sin θ) {\vec{e}}_{z} \end{cases}} d θ,

where θ_max and θ_min are the maximum and minimum angles determined by the NA of the objective lens, respectively. J_m(r) is the mth order Bessel function of the first kind. The constant A is given by A = πfl₀/λ, where f is the focal length, λ is the wavelength of incident wave in the ambient environment, and l₀ is the associated with the laser beam power. P(θ) is the pupil apodization function of the objective lens. For aplanatic objective lens obeys sine condition, the apodization function is given by P(θ) = cos^1/2θ. From Eq. (2), both radial and azimuthal components of the electric field near the focus have two terms. Therefore, the total focal field is a superposition of three different modes proportional to (m-1)^th, m^th and (m + 1)^th order Bessel functions, respectively, except for m = 0 when the light depicted by Eq. (1) is reduced to a radial polarization with flat wavefront, in which case the focal field only contains the zeroth- and first-order Bessel beams for longitudinal and radial components, respectively. For radially polarized beam with spiral phase wavefront, notice that not only radial and longitudinal modes but also azimuthal modes exist in the focal volume. For a 4Pi focusing system, the electric field near the common focus can be expressed as:

{\vec{E}}_{f} (r, ϕ, z) = {\vec{E}}_{L} (r, ϕ, z) + {\vec{E}}_{R} (r, ϕ, - z),

where E_L and E_R are the electric field focused by the left and right objective, respectively.

Fig. 1 The diagram of the optical tweezers using 4Pi focusing system.

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Assuming a gold nanoparticle suspended in the space between the two objective lens, where is filled with medium with dielectric constant of ε, its optical property can be described by the polarizability α [20]:

α = \frac{α_{0}}{1 - i α_{0} k^{3} / (6 π)},

where α₀ = 4πa³[ε_m(ω)-ε]/[ε_m(ω) + 2ε], ε_m(ω) is the relative permittivity of the metal from bulk material, a is the radius of the particle and ω is the frequency. The motion behavior of the nanoparticle is influenced by the time averaged force induced by the focal field E_f. The optical forces exerted on the nanoparticle can be classified into gradient force and scattering force expressed as [20]:

{\vec{F}}_{g r a d} = \frac{1}{4} ε_{0} Re (α) \nabla {| \vec{E} |}^{2},

{\vec{F}}_{s c a t} = \frac{n σ}{c} 〈 S 〉 - \frac{ε_{0} σ}{2 k_{0}} Im [(\vec{E} \cdot \nabla) {\vec{E}}^{*}],

where S is the Poynting vector depicting the intensity and direction of the energy flux, ε_m is the permittivity of the free space, c is the light speed, n is the refractive index of the surrounding ambient, k₀ is the wave-vector in free space, σ = kIm(α) is the total cross section of the particle. In cylindrical coordinate system, the optical force can be expressed as:

F_{r} = \frac{1}{4} ε_{0} Re (α) \nabla_{r} {| \vec{E} |}^{2} + \frac{n σ}{2 c} Re (E_{ϕ} H_{z}^{*} - E_{z} H_{ϕ}^{*}) - \frac{ε_{0} σ}{2 k_{0}} Im (E_{r} \frac{\partial E_{r}^{*}}{\partial r} + \frac{E_{ϕ}}{r} \frac{\partial E_{r}^{*}}{\partial ϕ} + E_{z} \frac{\partial E_{r}^{*}}{\partial z}),

F_{ϕ} = \frac{1}{4} ε_{0} Re (α) \nabla_{ϕ} {| \vec{E} |}^{2} + \frac{n σ}{2 c} Re (E_{z} H_{r}^{*} - E_{r} H_{z}^{*}) - \frac{ε_{0} σ}{2 k_{0}} Im (E_{r} \frac{\partial E_{ϕ}^{*}}{\partial r} + \frac{E_{ϕ}}{r} \frac{\partial E_{ϕ}^{*}}{\partial ϕ} + E_{z} \frac{\partial E_{ϕ}^{*}}{\partial z}),

F_{z} = \frac{1}{4} ε_{0} Re (α) \nabla_{z} {| \vec{E} |}^{2} + \frac{n σ}{2 c} Re (E_{r} H_{ϕ}^{*} - E_{ϕ} H_{r}^{*}) - \frac{ε_{0} σ}{2 k_{0}} Im (E_{r} \frac{\partial E_{z}^{*}}{\partial r} + \frac{E_{ϕ}}{r} \frac{\partial E_{z}^{*}}{\partial ϕ} + E_{z} \frac{\partial E_{z}^{*}}{\partial z}) .

3. Trapping metallic nanoparticle under the resonant condition

In this work, the most challenging situation is considered to illustrate the ability of the proposed optical tweezers, namely to trap and manipulate resonant metallic nanoparticle in 3D. Assuming a gold nanoparticle with radius of 50 nm immersed in water, the trapping wavelength is chosen to be 532 nm, which gives rise to extremely strong scattering force and severe thermal effect that may destabilize the optical trapping. The advantage of radial polarization in trapping off-resonant metallic nanoparticle has been both theoretically and experimentally demonstrated [21, 22]. The unique features of tightly focused radial polarization such as extremely strong axial component and absence of axial Poynting vector provide larger gradient force and smaller scattering force. However, as the trapping wavelength approaches the resonance of the nanoparticle, the scattering force become comparable to or even larger than the gradient force owing to the unique vector fielddistribution of radial polarization [23]. Considering a radially polarized non-vortex beam (m = 0) with wavelength of 532 nm and power of 2mW is focused by an objective lens with maximum focusing angle of 89°, the features of the optical force exerted on a resonant metallic nanoparticle can be understood by the intensity and phase distributions of the focal field (shown in Figs. 2(a) and 2(b)). Since the gradient force and scattering force are proportional to the intensity gradient and phase gradient of the focal field, respectively [24], the elliptical shaped intensity pattern leads to asymmetric potential depth in transversal and longitudinal directions, and positive axial scattering force is generated in the focal volume since the slope of E_z keeps positive. In this case, there is an equilibrium point in radial direction at r = 0 (shown in Fig. 2(c)), however, the particle will be strongly pushed away from the light source due to the dominating positive scattering force along the optical axis (shown in Fig. 2(d)).

Fig. 2 (a) Intensity distribution in the vicinity of focal point for tightly focused radially polarized non-vortex beam. (b) Phase distribution of E_z in the vicinity of focal point along z-axis. Optical forces exerted on 50 nm (radius) resonant gold nanoparticle along (c) radial and (d) longitudinal axes.

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Different from focusing system with only one objective lens, 4Pi microscopy utilizes two counter-propagating beams to create various interference field in the focal region, providingmore degrees of freedom to tailor the optical force. For example, a spherical focusing spot can be obtained when the illumination is radial polarization with m = 0 (shown in Fig. 3(a)). The corresponding line-scans of the axial and transversal energy density distributions are shown in Fig. 3(b), demonstrating that the axial and transversal focal spot sizes are nearly equal with size mismatch less than 0.0015λ. The FWHM spot size is calculated to be 0.4423λ, corresponding to a focal volume of 0.0453λ³. Note that the power of each input light is set to be 1 mW to keep the total power as the same as the case of Fig. 2. As a fair comparison, the optical forces exerted on the resonant gold nanoparticle are calculated and illustrated in Figs. 3(c) and 3(d). Clearly both the axial and radial gradient forces are enhanced because the energy is highly concentrated into spherical spot with diffraction-limit size in 4Pi focusing system. Moreover, the axial scattering force is canceled due to the counter-propagating light beam, leading to the formation of an equilibrium point along the longitudinal axis at z = 0. To better evaluate the stability of the optical trapping, the potential depth is numerical estimated as $U = - \int F \cdot d s$ . Traditionally an optical trap with potential depth U larger than k_BT can be considered as stable. In this case, the potential depths are calculated to be both 30 × k_BT in the transversal and longitudinal directions.

Fig. 3 (a) Intensity distribution in the vicinity of focal point for radially polarized non-vortex beam focused by 4Pi focusing system. (b) Line-scans of corresponding axial and transversal intensity distributions. Optical forces exerted on 50 nm (radius) resonant gold nanoparticle along (c) radial and (d) longitudinal axes.

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In addition to the non-vortex beam, light wave possessing spiral wavefront has attracted increasing attention recently. The interaction between object and the vortex beam enables the transfer of orbital angular momentum (OAM) from the light to the object, making the object to revolve around the beam axis [25–28]. The proposed optical tweezers is also capable to rotate metallic nanoparticle under the resonant condition, which can be realized by introducing the vortex beam as the light source. Figure 4(a) shows the intensity distribution in the z-x plane at y = 0 for radially polarized illumination with m = 2. Due to the phase singularity of the OAM mode, the focal field is no longer spherical but doughnut with a dark center. As shown in Fig. 4(b), the particle would be trapped off-axially at location corresponding to the peak intensity in the transversal plane. Similar to the case of non-vortex beam, the position of the equilibrium point in the z axis is still at the focal plane due to the lack of scattering force (shown in Fig. 4(c)). In the azimuthal direction, there is no gradient force since the intensity pattern is axisymmetric. However, a constant scattering force is exerted on the particle, which drags the particle to revolve around the beam axis (shown in Fig. 4(d)). The stability of the optical trapping can be demonstrated by the stability analysis showing that potential depths of 5.7 × k_BT and 5.4 × k_BT can be achieved in radial and axial directions, respectively. To better illustrate the motion of the particle, the total force is projected in the x-y plane and shown in the inset of Fig. 4(d). It is known that the rotation radius is determined by the position of the equilibrium point in the radial direction, which can be adjusted by the absolute value of the TC of the vortex beam. Besides, the rotation direction can also be changed by switching the sign of the TC of the illumination.

Fig. 4 (a) Intensity distribution in the vicinity of focal point for radially polarized vortex beam with m = 2 focused by 4Pi focusing system. Optical forces exerted on 50 nm (radius) resonant gold nanoparticle along (b) radial, (c) longitudinal and (d) azimuthal axes. Inset: The projection of optical force in x-y plane. The arrow and circle indicate the direction of the optical force and the contour of the intensity distribution, respectively.

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4. Manipulation of resonant metallic nanoparticle with controllable motion trajectory

So far it has been demonstrated that the proposed optical tweezers is capable of trapping resonant metallic nanoparticle. Once the particle is trapped in the focal region, its relative position can be changed by shifting the position of the sample or the input laser via a translation stage. However, in some specific applications it is desirable to move the particle without mechanical disturbance, which also can be realized by the proposed optical tweezers. The wave-vector of the converging beam can be expressed as:

\vec{k} = k [\begin{array}{l} \sin θ \cos φ {\vec{e}}_{x} \\ \sin θ \sin φ {\vec{e}}_{y} \\ \cos θ {\vec{e}}_{z} \end{array}],

Assuming the particle is expected to shift from the origin to P(x₀, y₀, z₀), additional phase is required to be introduced to the input light:

η (θ, φ) = e^{- i \vec{k} \cdot \vec{P}} = e^{- i k (x_{0} \cdot \sin θ \cos φ + y_{0} \cdot \sin θ \sin φ + z_{0} \cdot \cos θ)} .

For objective lens obeys sine condition, the converging angle can be expressed as θ = sin⁻¹(ρ × NA/R/n_t), where ρ is the radial coordinate in the incident space, R is the radius of the pupil, and n_t is the refractive index of the ambient medium. The focal field in Cartesian coordinate can be expressed as:

\vec{E} = - \frac{i A}{π} \int_{0}^{θ_{\max}} \int_{0}^{2 π} P (θ) η (θ, φ) \sin θ e^{i m φ} [\begin{array}{l} \cos θ \cos φ {\vec{e}}_{x} \\ \cos θ \sin φ {\vec{e}}_{y} \\ \sin θ {\vec{e}}_{z} \end{array}] e^{i k [z \cos θ + r \sin θ \cos (φ - ϕ)]} d φ d θ .

It is worthy of noting that the phase modulations applied on the right- and left-propagating light beams are

η_{1} (θ, φ) = e^{- i \vec{k} \cdot \vec{P}}

and

η_{2} (θ, φ) = e^{i \vec{k} \cdot \vec{P}}

respectively. As an example, the peak location is moved to P = (5λ, 10λ, 2λ) and the corresponding intensity distribution is shown in Figs. 5(a) and 5(b). Compared with Figs. 3(a) and 3(b), both the shape and the intensity of the spherical spot keep the same. Consequently, the particle can still be stably trapped except that the location of the equilibrium point is changed accordingly. Note that this method only support to shift particle with limited distance that is determined by the distribution of the trapping potential. The single maximum displacement is estimated to be 1λ from the potential well shown in Fig. 5(c). However, complex motion trajectory can be achieved by segment movement. As the spiral pattern shown in Fig. 5(d), the particle will follow this designed path by switching the applied phase modulation accordingly.

Fig. 5 Intensity distribution in the (a) x-y plane and (b) z-x plane of a spherical spot centered at P = (5λ, 10λ, 2λ). (c) Distributions of potential depth along x and z axes. (d) Motion trajectory of a particle following a spiral path in the x-y plane. The direction of the pulling force is indicated by the arrows.

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5. Manipulation of multiple resonant metallic nanoparticles

In previous calculations, objective lens with NA close to 1 is used in order to obtain a single spherical focal spot efficiently. Such high NA can be achieved with reflective type of parabolic objectives. However, the requirement for objective lens can be less stringent for specific applications. For example, multiple focused spots are necessary in some cases such as to control multiple nanoparticles simultaneously, which can be realized by adjusting the NA of the objective lenses. If the maximum focusing angle of the objective lens is reduced to θ_max = 71.8° (NA = 0.95 objective lens), multiple focal spots can be generated however the spot is elongated along the transverse direction with size mismatch of about 0.14λ. The size mismatch can be reduced by properly choosing the minimum converging angle θ_min of the focusing system by blocking the central region of the incoming beam with an opaque disk. Figure 6(a) shows the intensity distribution for θ_min of 39.5°. There are three spherical focal spots in thefocal region with average size of about 0.56λ. As shown in Fig. 6(b), although the peak intensities of the side lobes are nearly half of that of the main lobe, the induced optical force is still large enough to support a stable 3D trapping with multiple equilibrium locations (shown in Figs. 6(c) and 6(d)). Besides, the number of the spherical spot can be adjusted by carefully choosing the parameters of the 4Pi microscopy. As shown in Fig. 7, five and nine spherical focal spots are generated with focusing condition of (θ_min = 44.9°_, θ_max = 64.2°) and (θ_min = 49.5°_, θ_max = 58.2°), respectively, providing an effective tool for multiple particles trapping, delivering and self-assembling.

Fig. 6 (a) Intensity distribution in the vicinity of focal point for radially polarized non-vortex beam focused by 4Pi focusing system with (θ_min = 39.5°, θ_max = 71.8°). (b) Line-scan of corresponding axial intensity distribution. (c) Optical forces exerted on 50 nm (radius) resonant gold nanoparticle along longitudinal axis. (d) Distribution of potential depth along z axis.

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Fig. 7 Intensity distribution in the vicinity of focal point for radially polarized non-vortex beam focused by 4Pi focusing system with (a) (θ_min = 44.9°_, θ_max = 64.2°) and (b) (θ_min = 49.5°_, θ_max = 58.2°).

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6. Thermal stability analysis

Discussion above clearly demonstrated that stable 3D optical trapping and manipulation can be formed for metallic nanoparticles at the resonant condition from the force balance point of view. Besides, thermal mechanism is another factor that may destabilize the optical tweezers. It has been reported that the temperature of water near the surface of a gold nanoparticle must be kept below 647K [29]. Otherwise the optical trap would be destroyed due to the formation of vapor bubble, even though the potential depth is large enough to sustain a stable trapping. Due to the high absorptivity of the resonant metallic nanoparticle within the focal volume of the 4Pi focusing system, thermal effect can be mitigated by reducing the input power while maintaining adequate potential depth. By solving an optic-thermal coupling model built with COMSOL Multiphysics [14], the maximum input power is simulated to be nearly 0.2 mW and 1 mW for radially polarized illumination with m = 0 and m = 2, and the corresponding potential depths are calculated to be 3.0 × k_BT and 2.7 × k_BT along axial axis and 3.0 × k_BT and 2.8 × k_BT along radial axis, respectively. Consequently, optical overheating in this novel optical tweezers can be avoided while maintaining deep enough trapping potential, enabling stable trapping of metallic nanoparticle under the most challenging condition.

7. Conclusions

In conclusion, we proposed and numerically demonstrated a novel optical tweezers using 4Pi focusing system. By focusing the radially polarized light with two high NA aplanatic objective lenses, specific number of diffraction-limit spherical spots can be obtained in the focal volume by modulating the focusing condition. The counter-propagating light waves enable the stable trap of the metallic nanoparticles even under the resonant condition due to the absence of the axial scattering force. By adjusting the OAM mode of the illumination, resonant metallic nanoparticle can be driven to rotate around the beam axis with controllable rotation direction and rotation radius. Besides, complex motion trajectory can be realized for resonant metallic nanoparticle by introducing additional gradient phase modulation. Moreover, the overheating effect can be avoided by elaborately adjusting the input power. Note that the homogeneous environment is vital to achieve the proposed optical tweezers since the interference pattern would be disturbed by inhomogeneous environment. Besides, the optical heating effect would not be as severe if the particle is trapped in air, or the particle is imbedded in other medium with large viscosity or high critical temperature for bubble formation. This versatile trapping method can be easily adapted for other kinds of metallic and semiconductor nanoparticles, opening up new avenues for optical trapping and their applications in various scientific fields.

Funding

National Natural Science Foundation of China (11504049, 11474052); Natural Science Foundation of Jiangsu Province (BK20150593); National Key Basic Research Program of China (2015CB352002); Fundamental Research Funds for the Central Universities (2242015KD001).

References and links

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Manipulation of resonant metallic nanoparticle using 4Pi focusing system

Abstract

1. Introduction

2. Configuration of the optical tweezers

3. Trapping metallic nanoparticle under the resonant condition

4. Manipulation of resonant metallic nanoparticle with controllable motion trajectory

5. Manipulation of multiple resonant metallic nanoparticles

6. Thermal stability analysis

7. Conclusions

Funding

References and links

Cited By

Figures (7)

Equations (12)

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