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In this work, we proposed a novel strategy to manipulate the behavior of the metallic nanoparticle under the resonant condition by using engineered azimuthally polarized optical field. Through optimizing the spatial phase distribution of the illumination, the optical force can be tailored to support stable optical trapping while avoiding trap destabilization caused by optical overheating effect simultaneously. Besides, the resonant particle can be stably trapped at predefined location in 3 dimensional space, or revolves around the beam axis with characteristics that can be holistically controlled in terms of both trajectory and rotation direction. The technique demonstrated in this work may open up new avenues for optical manipulation.
© 2016 Optical Society of America
1. Introduction
When the size of matter is reduced from bulk to nanometer scale, new properties emerge in terms of optical, magnetic, electronic, and structural properties, make nanoparticles very promising for a wide range of applications from biology to electronics [1, 2 ]. Different from other nanoplatforms including semiconductor quantum dots and magnetic nanoparticles, plasmonic nanoparticles have attracted increased attentions in recent years due to various applications of plasmon resonance. For example, the local field enhancement offered by the resonant noble metal nanoparticles enable the amplifying of the Raman signal in surface-enhanced Raman scattering spectroscopy (SERS) [3]. Besides, gold nanoparticles have been brought to the forefront of cancer research in recent years due to their substantial heating they undergo when illuminated near the plasmon resonance [4–6 ]. In a wide range of applications involved metallic nanoparticles, it is often crucial to precisely and stably manipulate the particles, prorviding additional degree of freedom.
Ashkin and colleagues firstly report the optical control of particles by optical tweezers, which is constructed using radiation pressure from a single focused laser beam [7]. Since then, 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. Nowadays, optical tweezers have been widely used in various areas of science, particularly in physics, chemistry and biophysical studies [8]. The noncontact and holding nature of optical trapping make it very suitable to be combined with SERS spectroscopy and plasmonic photothermal therapy [9, 10 ]. A stable trapping occurs when the gradient force (also known as attractive force) overcomes the scattering force (also known as repulsive force), which can be easily achieved for dielectric particles. For metallic nanoparticle, however, stable 3 dimensional (3D) trapping is generally considered difficult due to strong scattering and absorption. Several special optical tweezers have been reported to increase trapping efficiency of metallic nanoparticles [11–13 ], such as replacing the conventional linearly polarized trapping laser with vectorial optical field [14–16 ], or plasmonic tweezers that utilizing the surface plasmon polaritions enhanced gradient force [17]. However, optical tweezers are not always successful in trapping metallic nanoparticles when trapping wavelength approaches the resonance of the nanoparticle due to two main reasons. Firstly, the induced polarization is enhanced at the resonant condition, leading to stronger scattering force. The resonantly enhanced scattering force pushes particles away from the focal spot of the trapping laser. Secondly, resonant metallic nanoparticles give rise to severe heating effects [18]. The high temperature of the nanoparticles may destabilize the optical trapping and then release the nanoparticles. This thermal effect remains the ultimate obstacle of realizing 3D stable trapping of metallic nanoparticles under resonance conditions.
With rapid advance in spatial engineering of light wave, optical complex field with unconventional spatial distributions in terms of amplitude, polarization and phase are rapidly becoming a current trend due to the possibility of exploring the fundamental physics with numerous potential applications including optical tweezers, microscopic and nanoscopic imaging, materials micromachining and processing, etc [19–22 ]. For example, “tractor beam” has been demonstrated by certain Bessel beams, which is capable of dragging the particles all the way towards the light source due to the generation of the negative scattering force [23, 24 ]. Optical complex field also show its potential in trapping particles under extreme conditions. With properly designed vectorial optical illumination, resonant metallic nanoparticles are possible to be trapped at the beam axis [16].
Besides amplitude and polarization, phase is another important property of light. Most researches dealt with optical beam with spatially homogeneous phase distribution. For these cases, the spatial dependence of phase in the beam cross section has been largely ignored. It is well known that light can be interpreted quantum mechanically thus can be viewed to carry orbital angular momentum (OAM). The properties of the beams of light with helical wavefront presents novel opportunities for scientific research and has been used as a useful tool in a variety of applications, ranging from optical spanner, optical tweezers, astronomy, quantum entanglement to microscopy etc [25]. For example, the helical modes of light can be focused into optical vortices that are capable of localizing and applying torques on trapped particles. The transfer of OAM to an object and its subsequent behavior has been investigated at the level of dielectric nanoparticles and particles [26–28 ]. Recently a novel method has been reported to dynamically manipulate the gold nanoparticle using vortex Bessel beam [29]. However, existing optical tweezers involved with vortex beam only apply to non-resonant condition meaning that the trapping laser wavelength is far from the resonance of the metallic particle. In this work we proposed a strategy to manipulate the behavior of the resonant metallic nanoparticle through sculpting the spatial phase distribution of an azimuthally polarized beam. The interaction between the nanoparticle and the engineered vectorial optical field enables an optical tweezers technique supporting not only stable 3D trapping but also a precise control of trajectory of the metallic nanoparticle even under the resonant condition. Additionally, the tailored focal field and optical force can be exploited to mitigate the thermal heating effect, overcoming the ultimate obstacle that prevents stable manipulation of resonant metallic nanoparticles.
2. Configuration of the optical tweezers
The proposed optical tweezers is schematically illustrated in Fig. 1 . The optical tweezers is constructed with a high numerical aperture (NA) objective lens that focuses the illumination as tightly as possible. The NA of the objective lens is chosen such that the maximum converging angle from the edge of the lens is 89°. A diffraction optical element (DOE) is placed in the pupil plane of the objective lens in order to tailor the spatial phase distribution of the illumination. The DOE is divided into two zones with different phase modulation functions. Considering an azimuthally polarized vortex beam (APVB) with topological charge (TC) of m normally illuminates the DOE, the electric field can be expressed as:
where E 0 is the incident amplitude and φ is the azimuthal angle of the cylindrical coordinate system. When an APVB is focused by high NA objective lens, the field distribution near the focus can be analyzed with the Richard-Wolf vectorial diffraction method as [30]: where Jn(r) is the nth order Bessel function of the first kind, M(θ) is the phase modulation of the DOE, θmax is the maximal angle determined by the NA of 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 the associated with the laser beam power. P(θ) is the pupil apodization function of the objective lens. For aplanatic objective lens obeys the Helmholtz condition, the apodization function is given by [31].
Fig. 1 Diagram of the proposed optical tweezers design. An incident azimuthally polarized beam is highly focused by an objective lens. A diffraction optical element is inserted at the entrance pupil plane of the objective lens.
For a metallic spherical Rayleigh particle suspended in a medium with the dielectric constant of ε, its optical property can be described by the polarizability α [32]:
where , εm(ω) is the relative permittivity of the metal from bulk material, a is the radius of the particle and ω is the frequency. Within the focal field of the APVB, the particle moves under the influence of the time averaged light-induced forces and the optical forces exerted on the nanoparticle can be expressed as [32]:where ε 0 is the permittivity of the free space, σ = kIm(α) is the total cross section of the particle, c is 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 is responsible to pull the particles towards the center of the focus. The combination of the second and the third terms gives the total scattering force, which are due to the net momentum transfer caused by scattering/absorption of photons from the particles, and the unique vector field distribution of the illumination, respectively. In cylindrical coordinate system, Eq. (5) can be rewritten as: The advantage of radial polarization in trapping metallic nanoparticle has been demonstrated both in theory and experiment [14, 15 ]. It is known that the axial field component of the tightly focused radial polarization does not contribute to the Poynting vector, therefore the radiation pressure force diminishes on the optical axis. However, if the incident light possesses helical phase front, the particle cannot be trapped at the beam axis due to the doughnut-shaped focal spot. Therefore, the radiation pressure force exerted on the particle is non-vanishing thus the advances from radial polarization starts to run out. In contrast, according to the unique focusing properties of APVB, the spin curl force does not exist on the optical axis due to the lack of longitudinal electric field (shown in Eq. (8)). Consequently, illumination with azimuthal polarization is suitable to manipulate the off-axial particles.Due to the conservation law of angular momentum in a closed physical system, the OAM of light will be transferred to the object, making it revolves around the beam axis. In order to limit the particle in a specific transverse plane, it is required to trap the particle along the beam axis. However, the creation of an equilibrium point in the longitudinal direction become difficult when the incident wavelength approaches the resonance of the particle. Therefore, the wavelength of the trapping laser beam is a vital parameter that determines the trapping efficiency for metallic nanoparticle. To illustrate the ability of the proposed optical tweezers, the most challenging situation is considered in this work, which is to dynamically control the motion of a metallic nanoparticle under the wavelength corresponding to the peak of absorption cross section (ACS). As shown in Fig. 2 , the scattering cross section (SCS) and ACS of a 50 nm radius gold nanoparticle immersed in water are peaked at 580 nm (also known as plasmonic resonance) and 532 nm, respectively. Compared with the plasmonic resonance, trapping metallic nanoparticle at its absorption resonant wavelength is much tougher. Firstly, the scattering force is maximized at the resonance of the ACS. It is because the scattering force will dramatically decrease from the absorption peak wavelength, which is corresponding to the maximum of the imaginary part of the polarizability. Secondly, the thermal effect is also significant at the peak of the ACS since physically the ACS is used to quantify the probability of a photon absorption. Therefore, from both force balance and the thermal effect points of view, 532 nm is the most challenging wavelength for manipulating a 50 radius gold nanoparticle. It is worthy of noting that the difference between the resonant wavelengths for SCS and ACS of a metallic nanoparticle will become negligible as the particle size decreases.
Fig. 2 (a) Scattering cross section and (b) absorption cross section of 50 nm radius gold nanoparticles in water.
2. Optical force induced behavior of a resonant metallic nanoparticle
To clearly illustrate the difficulty of manipulating resonant nanoparticle with conventional optical tweezers, the optical forces for the case of no pupil modulation are shown in Fig. 3 . In this case, the DOE shown in Fig. 1 is removed and the APVB with m = 2 is directly focused on the metallic nanoparticle. The index of gold is 0.467 + 2.4083i at the wavelength of 532 nm, and the laser power is assumed to be 100 mW. As shown in Fig. 3(c), a constant force is exerted on the particle along the azimuthal direction that drags the particle orbit around the beam axis with specific radius. The radius is determined by the position of the equilibrium point in the radial direction (shown in Fig. 3(b)). However, the particle will move far away from the light source with helical trajectory due to the lack of equilibrium point along the longitudinal direction. As shown in Fig. 3(a), the scattering force remains positive throughout the focal region and represents a dominating contribution to the total force. Note that the optical forces in the radial and azimuthal directions are calculated at z = 0 since there is no equilibrium position along the optical axis. The scattering forces in the longitudinal and azimuthal directions can be understood by the phase distributions shown in Fig. 3(e) and 3(f). It has been reported that the scattering force is proportional to the phase gradient of the electric field in the focal region [33]. Therefore, the phase curve with varying but positive slope or invariable slope give rise to positive scattering force with varying or constant magnitude, respectively.
Fig. 3 Optical forces on 50 nm (radius) gold nanoparticle produced by APVB with m = 2 at 532 nm using conventional trapping method. (a) Optical forces along the z-axis at the radial equilibrium point and φ = 0. (b) Optical forces along the r-axis at φ = 0 and z = 0. (c) Optical forces along the φ-axis at the radial equilibrium point and z = 0. (d) |E|2 distribution in the x-z plane at y = 0. (e) Eφ phase distribution along the z-axis at the radial equilibrium point and φ = 0. (f) Eφ phase distribution along the φ-axis at the radial equilibrium point and z = 0.
Since the scattering force cannot be depressed, a feasible way to overcome it is to alleviate its adverse impact to the optical trapping. For example, the direction of the scattering force can be changed to be against the power flow [23]. In other words, the spatial phase distribution of the illumination needs to be optimized for maximum trapping stability. The modulation function of the DOE is expressed as:
where Δϕ and θ 0 indicate the phase difference and the boundary between the inner and outer zones of the DOE, respectively. The intention of the two zones is to modulate the gradient force and scattering force separately, which are responsible by the circular and the annular domains of the DOE, respectively. In order to achieve a stable trapping for the case of APVB illumination with m = 2, the parameters of the DOE design are chosen to be Δϕ = 0.96π and θ 0 = 1.46 rad. Note that these parameters are optimized to achieve the maximal trapping stability. Due to the interference between the optical fields coming from different zones of the DOE, the symmetry of the intensity distribution near the focus breaks (shown in Fig. 4(d) ), leading to the redistribution of the gradient force. Besides, the phase distribution in the z axis experiences an obvious decline near the focus (shown in Fig. 4(e)). Note that only the phase of Eφ is considered because it dominates in the focal region. According to the relationship between the scattering force and the phase gradient of the electric field, negative scattering force is generated along the optical axis within the range corresponding to the phase curve with downward slope. As expected, the roles of the gradient force and the scattering force play in trapping metallic nanoparticle exchange within the focal region. The combination of the gradient force and the scattering force gives rise to an equilibrium point in the longitudinal direction (shown in Fig. 4(a)). In the transverse plane, there is still an equilibrium point along the radial direction as in normal optical tweezers (shown in Fig. 4(b)). Note that the non-vanishing scattering force is observed along the radial direction. It is because the location of the trapped particle along the optical axis is not exactly at the focus (z ≠ 0). The azimuthal force experienced by the nanoparticle is still constant (shown in Fig. 4(c)), which can be understood by the phase curve of Eφ with the invariable slope along the azimuthal direction (shown in Fig. 4(f)). Therefore, the resonant gold nanoparticle would rotate with fixed radius (r = 0.31λ) at a specific transverse plane (z = −0.54λ). Since the characteristics of the optical force strongly depends on the geometry of the illumination, the motion path of the nanoparticle can be freely adjusted by tuning the pupil function. For example, the orbital radius and the rotation direction of the particle are affected by the absolute value and the sign of the TC of the incident light, respectively. Figure 5 shows the optical forces when APVB with m = −3 is adopted. In this case Δϕ and θ 0 are optimized to be −0.63π and 1.47 rad, respectively. Similarly, the trajectory of the resonant gold nanoparticle is a circle at the plane of z = 0.03λ. Compared with the case shown in Fig. 4, the rotation direction is reversed due to the negative total force along the azimuthal direction. Besides, the rotation radius is increased to 0.52λ. It is worthy of noting that the DOE parameters used in Fig. 4 can still support a stable trap in this case, however the potential depth would be relatively decreased.Fig. 4 Optical forces on 50 nm (radius) gold nanoparticle at 532 nm using DOE with Δϕ = 0.96π and θ 0 = 1.46 rad for APVB with m = 2. (a) Optical forces along the z-axis at the radial equilibrium point and φ = 0. (b) Optical forces along the r-axis at the longitudinal equilibrium point and φ = 0. (c) Optical forces along the φ-axis at the longitudinal and radial equilibrium points. (d) |E|2 distribution in the x-z plane at y = 0. (e) Eφ phase distribution along the z-axis at the radial equilibrium point and φ = 0. (f) Eφ phase distribution along the φ-axis at the longitudinal and radial equilibrium points.
Fig. 5 Optical forces on 50 nm (radius) gold nanoparticle at 532 nm using DOE with Δϕ = −0.63π and θ 0 = 1.47 rad for APVB with m = −3. (a) Optical forces along the z-axis at the radial equilibrium point and φ = 0. (b) Optical forces along the r-axis at the longitudinal equilibrium point and φ = 0. (c) Optical forces along the φ-axis at the longitudinal and radial equilibrium points. (d) |E|2 distribution in the x-z plane at y = 0. (e) Eφ phase distribution along the z-axis at the radial equilibrium point and φ = 0. (f) Eφ phase distribution along the φ-axis at the longitudinal and radial equilibrium points.
Numerical calculations above clearly illustrate that the motion path of a metallic nanoparticle can be modulated effectively by adjusting the spatial phase distribution of an APVB. In some applications, it is meaningful to trap the revolving particles at specific off-axial positions, which requires equilibrium positions along the azimuthal direction. However, this cannot be satisfied by using an azimuthally polarized non-vortex beam (m = 0) as the illumination. For the case of m = 0, although the resonant nanoparticle can still be trapped in both longitudinal and radial directions, the particles do not experience any azimuthal forces, which can be explained by the flat phase distribution along the azimuthal direction. Therefore, the particle path is erratic azimuthally and changes frequently due to Brownian motion. According to the definition of the TC, which is the net change of phase in a circuit path enclosing the zero point, the TC remains zero if the phase of the light varies sinusoidal with respect to the azimuthal angle. However, this kind of optical complex field is found to support a precisely manipulation of the particle in 3D space. Considering an azimuthally polarized beam with inhomogeneous spatial phase distribution that can be written as:
The focal field distributions are expressed as: By substituting Eq. (11) and (12) into Eq. (6)-(8) , the optical forces for n = 4 are calculated and shown in Fig. 6 . In this case the Δϕ and θ 0 are optimized to be π and 1.44 rad. Similar to the cases of using APVB shown above, equilibrium points are still observed in both longitudinal and radial directions. It is interesting to notice that the phase curve of Eφ fluctuates as sine function azimuthally (shown in Fig. 6(d)), leading to the creation of equilibrium points along the azimuthal direction, the number of which is determined by |n|. Besides, the position of the equilibrium points can be changed by adjusting the initial phase of the illumination.Fig. 6 Optical forces on 50 nm (radius) gold nanoparticle at 532 nm using DOE with Δϕ = π and θ 0 = 1.44 rad for azimuthally polarized beam with sinusoidal varied phase (n = 4). (a) Optical forces along the z-axis at the radial and azimuthal equilibrium points. (b) Optical forces along the r-axis at the longitudinal and azimuthal equilibrium points. (c) Optical forces along the φ-axis at the longitudinal and radial equilibrium points. (d) Eφ phase distribution along the φ-axis at the longitudinal and radial equilibrium points.
3. Trapping stability analysis
To evaluate the stability of the optical trapping, the potential depth is numerically calculated for regions near the equilibrium point. The potential depth can be estimated by the optical force along the trapping length direction, which is expressed as . The trapping length is defined as the range within which the particles are pulled towards the equilibrium point. Traditionally an optical trap with potential depth larger than kBT can be considered as stable [34]. The temperature T is taken to be 20°C (≈293 K). For the cases of APVB with m = 2 and m = −3, the potential depths are 37 × kBT and 25 × kBT in the longitudinal direction and 11 × kBT and 5 × kBT in the radial direction, respectively. For incident beam with sinusoidal varied phase, the potential depths are 43 × kBT, 15 × kBT and 20 × kBT in longitudinal, radial and azimuthal directions, respectively. Note that the equilibrium locations along the azimuthal direction have the same potential depth.
The calculations above show that metallic nanoparticle can be stably trapped and manipulated at the absorption resonance from the force balance point of view. Besides, the thermal mechanism is another factor that may destabilize the optical tweezers [18], especially for resonant metallic nanoparticle immersed in liquid with low viscosity. For a gold nanoparticle immersed in water, if the temperature of water near the particle surface is above the critical temperature (about 647K [35]), the water will evaporate and form the nucleation of vapor bubble, leading to the escape of the particle from the trap. Due to the high magnitude of the polarizability near the resonance, the optical forces involved in the optical trapping can be significantly enhanced. Therefore, the thermal effect is possible to be mitigated by reducing the input power while maintaining the stability of the optical trapping. Assuming the gold nanoparticle reaches the critical temperature, the maximum allowable input power is found to be nearly 26 mW, which is simulated by an optic-thermal coupling model built with COMSOL Multiphysics. For the cases of vortex beam illumination with m = 2 and −3, the corresponding potential depths are calculated to be 10 × kBT and 7 × kBT along axial axis and 3 × kBT and 2 × kBT radial axis, respectively. For the case of azimuthally polarized light with sinusoidal varied phase, the potential depths along the longitudinal, radial and azimuthal directions are calculated to be 11 × kBT, 4 × kBT and 5 × kBT, respectively. Please note that the temperature T used here is 647 K. The results clearly demonstrate that the overheating effect can be avoided in this novel optical tweezers via elaborately adjusting the input laser. Meanwhile, the optical force is still large enough to support a stable optical trapping for resonant metallic nanoparticle.
4. Conclusions
In conclusion, we proposed and numerically demonstrated a novel optical tweezers design that is capable of manipulating metallic nanoparticle under the most challenging conditions. By tailoring the spatial phase distribution of the azimuthally polarized light, the characteristics of the optical forces can be altered to enable the adjustment of the motion path of the resonant gold nanoparticle. An APVB with complex phase distribution is capable of making the particle revolves around the beam axis in a specific transverse plane. The trajectory of the particle in terms of rotation radius and the direction of rotation can be holistically controlled by modulating the TC of the illumination. Besides, the particle can be stably trapped off the beam axis, which is realized by introducing a sinusoidal varied phase to the azimuthally polarized non-vortex beam. More importantly, the overheating phenomenon can be avoided by elaborately adjusting the input power. The technique presented in this work can be easily adapted for the other kinds of metallic and semiconducting nanoparticles by adjusting the phase modulation applied to the illumination. Although an objective lens with maximum refracted angle 89° is necessary to realize the trapping in the water, it is still within the capabilities of the existing total internal reflection fluorescence microscopy. It should be noted that the requirements for the objective lens is less stringent if the nanoparticles are to be manipulated in air, in which case a NA less than 0.98 is sufficient. This versatile trapping method may open up new avenues for optical manipulation and their applications in various scientific fields.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (Grant Nos: 11504049, 11474052), Natural Science Foundation of Jiangsu Province (Grant No: BK20150593) and the Fundamental Research Funds for the Central Universities (Grant No. 2242015KD001).
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