Nonlinearity-Induced Multiplexed Optical Trapping and Manipulation

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Nonlinearity-induced multiplexed optical trapping and manipulation with femtosecond vector beams Yuquan Zhang, junfeng Shen, changjun min, Yunfeng Jin, Yuqiang Jiang, Jun Liu, Siwei Zhu, Yunlong Sheng, Anatoly V Zayats, and Xiaocong Yuan Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.8b01929 • Publication Date (Web): 08 Aug 2018 Downloaded from http://pubs.acs.org on August 9, 2018

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Nonlinearity-induced multiplexed optical trapping and manipulation with femtosecond vector beams

Yuquan Zhang1, Junfeng Shen2, Changjun Min1, Yunfeng Jin3, Yuqiang Jiang3, Jun Liu4, Siwei Zhu4, Yunlong Sheng5, Anatoly V. Zayats6,*, Xiaocong Yuan1,*

1

Nanophotonics Research Centre, Shenzhen University, Shenzhen 518060, China

2

College of Physics and Technology, Southwest Jiaotong University, Chengdu, 614200, China

3

State Key Laboratory of Molecular and Developmental Biology, Institute of Genetics and

Developmental Biology, Chinese Academy of Sciences, Beijing 100101, China 4

Institute of Oncology, Tianjin Union Medical Center, Tianjin 300121, China

5

Department of Physics, Physical Engineering and Optics, Center for Optics, Photonics and

Lasers (COPL), University Laval, Québec, Canada 6

Department of Physics, King’s College London, Strand, London WC2R 2LS, UK.

*Correspondence to [email protected] and [email protected]

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Optical trapping and manipulation of atoms, nanoparticles and biological entities is widely employed in quantum technology, biophysics and sensing. Single traps are typically achieved with linearly polarized light, while vortex beams form rotationally unstable symmetric traps. Here we demonstrate multiplexed optical traps reconfigurable with intensity and polarization of the trapping beam using intensitydependent polarizability of nanoparticles. Nonlinearity combined with a longitudinal field of focused femtosecond vortex beams results in a stable optical force potential with multiple traps, in striking contrast to linear trapping regime. The number of traps and their orientation can be controlled by the cylindrical vector beam order, polarization and intensity. The nonlinear trapping demonstrated here on the example of plasmonic nanoparticles opens up opportunities for deterministic trapping and polarizationcontrolled manipulation of multiple dielectric and semiconductor particles, atoms and biological objects since most of them exhibit a required intensity-dependent refractive index.

Keywords: optical trapping, Kerr nonlinearity, nanoparticles, cylindrical vector beams

Optical trapping originates from gradient and scattering forces induced by electromagnetic fields, such as focused laser beams or plasmonic waves, and has been extensively used as a powerful tool to study physical, chemical, and biological characteristics of individual trapped micro- and nano-objectsError! Reference source not found.-9. Until now, optical traps have been primarily achieved with linear optical effects of continuous-wave (CW) illumination. Even when pulsed beams have been employed, the optical force was still based on a traditional linear physical mechanism10. While single particle trapping can be readily realized with a simplest linearly polarized beam, multiple traps can be achieved either by scanning a trapping 2


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laser beam11, using diffractive optical elements12,13 or structured plasmonic surfaces14 to form multiple focuses with a single beam illumination. Multiple particle trapping has also been implemented with complex vortex beams15-17. These implementations however do not provide a stable azimuthal position of the trapped objects smaller than the beam diameter because of rotational symmetry of the trapping potential and angular momentum and, therefore, are not suitable for nanosized particles. At the same time, deterministic and stable trapping and manipulation of multiple nanoparticles is in high demand in sensing, nanofabrication and quantum technologies. The optical force depends on the refractive index of trapped particles, and, therefore, the balance of conventional gradient and scattering forces exerted on particles can be subverted by nonlinear interactions between a particle and a high intensity light in the presence of Kerr-type nonlinearity, a third-order nonlinearity manifested in the intensity dependent refractive index. Nonlinear trapping depends on polarization of the illuminating beam18 and, for linearly polarized beams, leads to simple splitting of the trapping potential19. Here we show that nonlinear trapping in combination with complex polarization states of generalized cylindrical vector beams (CVBs) can be used for deterministic trapping and manipulation of multiple particles. Femtosecond CVBs induce nonlinear trapping effects only in the certain locations across the beam profile due to the interplay of transverse and longitudinal components of the field. The number minima in the trapping potential, and, therefore, trapping sites, can be controlled by the order of the vector beam while their orientation by the polarization orientation, allowing manipulation of the trapped objects by varying the illuminating beam polarization. The mechanism of nonlinear trapping is also shown to be related to the longitudinal field components of the CVB beams. Because of a dominant role of the longitudinal field components in the nonlinear trapping mechanism, the proposed approach allows trapping of strongly scattering nano-objects which are difficult to 3


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trap otherwise due to scattering forces they experience. Since most of the materials exhibit Kerr-type nonlinear response under high peak intensity, short-pulse illumination, the developed trapping strategy can be used in numerous applications in sensing, biophysics, atom optics and optomechanics. A paraxial generalised CVB propagating in a z-direction (Fig. S1) has the transverse electric field in the x-y plane perpendicular to the propagation direction, described by

Em = cos(mφ + φ0 ) eˆ x + sin(mφ + φ0 ) eˆ y , where φ is the azimuthal angle, ϕ0 is a constant, eˆ x and eˆ y are the unit vectors in x- and y-directions, respectively, and the integer m is the

polarization order20. Any linear polarization in the transversal plane can be decomposed into two fundamental polarizations: radial and azimuthal. Varying the polarization order m may change the spatial distribution and the weights of the radial and azimuthal polarization components (see Supplementary Information). When CVB is focused, the radial polarization components produce strong longitudinal components Ez of the electrical field, while the azimuthal polarization components do not contribute to Ez 21-22. We used CVBs with polarization orders m = 0, 1, 2, and -1 corresponding to distinct complex distribution of the electrical field components (Fig. 1 and Fig. S2) to investigate the nonlinear effects in optical trapping of gold nanoparticles. In a low-intensity CW regime, when nonlinear effects in Au nanoparticles are negligible, the field profiles of all the studied beams provide a stable trapping in the central intensity minimum24 of the beam profile (Fig. 1 i-l; see Methods for the details of the experimental set-up). In the nonlinear regime, when high peak power femtosecond beam is capable to substantially modify the refractive index of the nanoparticles, the optical trapping potential is significantly altered as the refractive index depends on the position with respect to the field profile, leading to splitting of the trapping potential into stable multiple well-defined traps (Figs. 2 and 3). As one can see, the

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nanoparticles are trapped at the locations with the highest intensity of the longitudinal field Ez of the CVBs (Fig. 1 e-h). Considering the m=0 beam for instance, the double trapping potential is reconfigured to a single trap when the laser is switched from the pulsed mode with high peak power to CW mode with the same average power, removing nonlinear effects. When the laser is switched back to the pulsed mode, the gold nanoparticle trapped in the centre of the CW CVB beam is launched away escaping from the linear-regime trap, and the double trap reappears. In order to confirm the role of the longitudinal fields in the trapping and illustrate the potential for multiplex manipulation of the trapped multiple particles, the polarization distribution of the trapping beam was rotated without affecting the CVB order (see Methods). As the result, the Ez field distribution of the CVBs is rotated causing controlled rotation of the trapped nanoparticles (cf. Fig. 1 m,n, and c,d). Since the position of the trapped nonlinear particles is strongly dependent on the longitudinal Ez field pattern in the focus of the beam, multiple trapping sites can be formed and are tuneable by designing a polarization distribution of the pulsed CVB, allowing for multiplexed traps. Since the Ez component emerging due to the CVB focusing dominates the trapping force in the nonlinear regime, trapping stiffness can be enhanced by tighter focusing. The origin of the nonlinear optical forces and their relation to the longitudinal fields in the beam focus can be understood considering the polarization structure of the CVB and the Kerr-type nolinearity of the nanoparticles (Methods). The time-averaged force exerted on a small nanoparticle in a quasi-static limit ( ≪ ) is23-25 F =

 ε  1 σ σ 2 Re {α } ∇ E + Re E× H ∗ + Re i 0 ( E⋅∇ ) E*  4 2c 2  k0 

(

)

(1)

where, = f(|E|) and σ = f’(|E|) are the intensity-dependent polarizability and extinction cross-section of a nanoparticle, which for nonlinear particles depend on the field at the 5


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particle location, E and H are the electric and magnetic field of the trapping beam, c is the speed of light in vacuum, and ω is the angular frequency of the optical field. The first term is the force associated with the gradient of the electric field intensity, the middle term is the scattering force responsible for the radiation pressure directed along the direction of light propagation26, and the last term is the force arising from the gradients in the spatial distribution of the beam polarization24. This last term is usually neglected for low numerical aperture illumination and is zero for a homogeneously-polarized plane wave illumination27-28. In order to achieve trapping, the gradient force needs to provide a three-dimensional confinement potential that overcomes scattering forces29. For metallic particles, the scattering force is usually very strong because of large scattering and absorption coefficients; for this reason, such particles are more difficult to trap30. The scattering force described by the middle term of Eq. (1) is however absent along beam propagation direction for the longitudinal field components Ez. Although the transverse and longitudinal components overlap in space, the total scattering force along the propagation direction at the longitudinal field maximum is much smaller than that at the transverse field maximum31. The trapping becomes feasible as Ez provides a pure gradient force in the propagation direction that forms a sufficiently deep trapping potential. The latter can be achieved utilizing nonlinear response of nanoparticles induced by a pulsed trapping beam with sufficiently high peak power: due to the nonlinearity-induced changes of the polarizability and extinction, the trapping potential in the centre of the beam is effectively eliminated and enhanced in the longitudinal field maxima (Fig. 2). In the nonlinear regime, the pulsed illumination of plasmonic particles leads to the changes of their permittivity32 and, therefore, polarizability and extinction cross-section entering Eq. (1) (Fig. S4), resulting in the modifications of the trapping potential. Figure 2 shows the trapping potential and the electric field component distributions for the m=0 CVB 6


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in both linear and nonlinear regimes (See Methods for the details of the simulations). This allows to separate contributions of the longitudinal Ez and transverse Etr field components as well as the direction and magnitude of the gradient forces due to different field components (Fig. 2 b and d). In the presence of nonlinear effects, there is a force along the beam propagation at the centre of the beam, compared with the negative force in a linear regime in plane (III). Consequently, a nonlinear gradient force from Ez dominates the contribution to the trapping potential. As the result, the final force is directed towards the Ez focal spots in the nonlinear regime, yielding separated independent multiple trap sites. In a linear regime, the Ez field still generates a force and the trapping potential (Fig. 2 d), but it is smaller than by the trapping forces in the centre of the beam induced by Etr, whereas the influence of the Ez field leads to a well-known polarization-induced stiffness symmetry33. In the nonlinear regime, the central (Etr -related) trapping potential is eliminated, so that the Ez field component generates a dominant potential well for the optical traps. With the reduction of the peak power of the trapping beam, the nonlinear effects become smaller and the relative depth of the trapping potential associated with the longitudinal fields diminishes in comparison with the potential in the centre of the beam (Fig. 4 and S5). For the average powers below approximately 0.32 W, the trapping potential at the CVB axis associated with the transverse fields becomes dominant. The absolute depth of the trapping potential obviously also become smaller for lower trapping beam powers, so that the trapping forces are too weak to provide stable traps for both CW and femtosecond beams below approximately 0.95 W powers due to Browning motion in water (Supplementary Video S3). It should be noted that the temperature effects which may lead to damage and reshaping of the particles are related to average light power. With femtosecond beams, low average powers providing high peak powers can be used, actually reducing the unwanted heating and photon poisoning due to 7


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temperature dissipation between the pulses. For strongly absorbing metal particles, the damage and reshaping occur for powers in excess of tens of GW/cm2, not reached in the presented trapping measurements. Figure 3 illustrates the force distributions and trapping potential at the focus of higher-order CVBs in the nonlinear regime. It confirms that the correlation of the local potential energy minima with the distribution of the Ez components, while the total force depends on all the field components. The corresponding forces along the beam propagating direction induced behind the trapping plane are individually plotted in Fig. 3 g–i; negative force values identify a stable trapping potential at the Ez field maxima. Consequently, it is easy to understand that the distribution of the trapped nanoparticles changes simultaneously following a rotation of CVBs as observed in the experiments (Fig. 1). The nonlinear interaction under high peak powers provides the conditions necessary to form a required potential trapping potential due to the interplay between the transverse and longitudinal field distributions and induced nonlinear changes of polarizability of the nanoparticles. The forces at the beam centre are also modified by the nonlinearity switching from attractive at low intensities to repulsive at high intensities, greatly contributing to shaping the final trapping potential. The nonlinear optical forces enabled by pulsed cylindrical vector beams allows realisation of multiple stable optical traps which are readily reconfigurable by controlling polarization and intensity of the beams. The nonlinear light–matter interaction provides the conditions necessary to achieve gradient trapping force associated with the longitudinal field Ez position to form the splitting of a trap formed in the linear regime by the total field distribution of the trapping beam. The transverse field dominates trapping forces at low intensities while at high intensities the longitudinal field components determine the trapping potential. The described mechanism of nonlinear optical forces helps to improve and extend 8


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optical trapping functionalities for numerous applications involving small particles and their assemblies. As most materials exhibit the Kerr-type nonlinearity, the proposed trapping method can be universally employed in many different fields opening new possibilities for high precession manipulation of nanoparticles, including those that difficult to trap in a conventional way due to large scattering cross-section.

Methods. Experiment. The experimental setup is shown in Fig. S3. A Ti:sapphire laser light (a central wavelength of 840 nm) was used for trapping. The laser can be switched between CW and pulsed (100 fs pulse duration) regimes keeping the same average power. The laser beam was first expanded and collimated. The beam was linearly polarized after passing through the polarizing prism. The power was regulated by rotating the half-wave plate. The cascaded vortex retarder (VR, Thorlabs Inc., US) was used to produce a CVB of a required order. To control the orientation of the polarization of the higher order CVBs, the angle between two half-wave plates placed behind the VR was tuned without affecting the order of CVBs (cf. Figs. 1 m,n and 1 c,d). The obtained vector beam was then focused using an objective lens (50×, NA=0.65) onto a glass microtube cell from the side to trap Au nanoparticles. For trapping, Au nanoparticles (NanoSeedz Inc., China) with a diameter of 60 nm and a localised surface plasmon wavelength of 536 nm were diffused in water and injected into a square glass micro-tube (the wall thickness is 100 µm and its inner side length is 150 µm). To image nanoparticle location, a laser beam from a 532-nm CW laser was first expanded, collimated, and coupled to a single-mode fiber to illuminate the cell. Direct imaging of trapped particles was achieved using CCD cameras registering the 532-nm scattered signal from the nanoparticles after removing the trapping laser light with a set of filters.

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Trapping force modelling. The time-averaged force exerted on a small nanoparticle in a quasi-static limit ( ≪ ) was evaluated using standard approach24

〈〉 = Re(∑,, ∇ ∗ ), where is the polarizability of a nanoparticle, which for nonlinear particles depends on the local field at the particle location, and is the i-th component of the electric field of the trapping beam. Three-dimensional confining potential which defines the optical trap was evaluated via the time-averaged potential energy of a particle with polarizability P in the electric field E given by24 〈〉 = −〈P·E〉, where = is the induced polarizability of a nanoparticle which depends on the polarization of the electric field with χ being the susceptibility of a nanoparticle. Simulations of nonlinear effects. When nanoparticle is placed in a strong electromagnetic field, the Kerr-type nonlinearity influences its optical properties through both intensitydependent real and imaginary parts of the permittivity.32,34,35 In order to calculate the intensity dependent force and trapping potential, a split-step approach was employed. First, the optical fields of the CVBs at different positions in and around trapping plane are calculated as described in Supplementary Information. Next, the field induced polarizability of an Au nanoparticle is mapped across the focus area with a step of 2 nm taking into account the enhancement of the local fields due to the nanoparticle presence. Finally, the obtained nonlinear polarizability is used to calculate the forces and energy potential as described above. The intensity-dependent particle polarizability (α) and extinction cross-section (σext) determining trapping potential were evaluated in the quasi-static limit for small nanoparticles assuming that a medium surrounding a nanoparticle has a linear optical response with a permittivity ! as36 (||) = 4$%&

'( (|)* |)+', '( (|)* |)-',

and ./0 = .12300 + .351 =

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67 8

|(||)| +

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9 :; = > + (&) |? |2 is the intensity-dependent Au permittivity with ? being the local field at the nanoparticle position,32 > = - 25.096+2.045i is the linear permittivity of Au at a wavelength of 840 nm used for trapping,37 and (&) = (-76.8+4.3i) ×10-20 m2/V2 being the third-order nonlinear susceptibility of Au chosen from Ref. 38 as corresponding approximately to the wavelength and pulse duration used in the experiments (it was additionally checked that variations around this value does not qualitatively change the results apart from small variations in absolute values of the force and potential energy). The variations of both the real and imaginary parts of the nanoparticle polarizability generally follow the total field distribution of CVBs (cf. Figs. S4 and S2). This in turn results in the changes of the trapping potential and force induced by nonlinear effects (Figs. 2 and 3).

Acknowledgments

This work was supported, in part, by the National Natural Science Foundation of China under Grant numbers 61427819, 91750205, and 61605117. X.Y. acknowledges the support from the leading talents of Guangdong province program No. 00201505 and Science, and Technology Innovation Commission of Shenzhen No. KQTD2017033011044403 and ZDSYS201703031605029. Y.Z. acknowledges the support from Technology Innovation Commission of Shenzhen No. JCYJ2017818144338999. A.Z. acknowledges support from EPSRC (UK), ERC iCOMM project (789340), the Royal Society and the Wolfson Foundation. The authors appreciate valuable discussions and help from Dr. Zhenwei Xie and Dr. Ting Lei of Shenzhen University.

Authors contributions

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X.Y. initiated and supervised the work. Y.J., J.L. and S.Z. participated in the design of the experiments. Y.Z. and Y.J. carried out the experiments. J.S., C.M. and Y.S. performed the modelling. Y.Z., X.Y. and A.Z. interpreted the results. All authors contributed to the discussion of the results and preparation of the manuscript.

Data availability The data access statement: all the data supporting this research are provided in full in the results section and in supplementary information. The data supporting the plots within this paper are available from the corresponding authors on reasonable request. Correspondence and requests for materials should be addressed to A.Z. or X.Y.

Competing financial interests The authors declare no competing financial interests.

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Figure 1. Nonlinear trapping of gold nanoparticles under various polarization conditions. (a-d) The polarization in the x-y plane (black arrows), the total |E|2 and the longitudinal |Ez|2 spatial field distributions for the generalized CVBs with m = 0 (a), 1 (a), 2 (c), and -1 (d). (e-h) The image of the 60 nm gold nanoparticles trapped with femtosecond pulses in the nonlinear regime: multiple nanoparticles trapped apart from the case of CVB with m=1 in a pattern determined by the longitudinal component distribution. (i-l) The image of the 60 nm gold nanoparticles trapped in the linear (CW) regime: only one gold particle is trapped for all polarization conditions in a pattern determined by the total field distribution. (m, n) The |Ez|2 spatial distributions (left panels) for the generalized CVBs with m = 2 (m) and -1 (n) with polarization rotated by π/2 and π/4 compared to (c) and (d), respectively, and the associated images of the trapped particles (right panels): as the orientation of the |Ez|2 distribution rotates, the pattern of trapped nanoparticles rotates accordingly. The scale bars are 1 µm. The crosses indicate the center of the focused beam.

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Figure 2. The trapping potential and force distributions for a gold nanoparticle for CVB with m=0. (a, c) The trapping potential and (b, d) the forces acting on a nanoparticle in the nonlinear (a, b) and linear (b, d) trapping regimes. The background colours indicate the distribution of the electric field |Ez| (purple) and |Etr| (yellow). The direction and the relative magnitude of the total force are shown near the field maxima of |Ez| (black arrows) and |Etr| (red arrows) in front of (I, z= -200 nm), in (II, z=0), and behind (III, z=200 nm) the trapping plane. The blue arrows indicate the direction and magnitude of the in-plane (perpendicular to the beam propagation direction) gradient forces due to the |Ez| field variations. A pulsed trapping beam has a peak power of 1.785×105 W (average power is 1.54 W, pulse width is 100 fs). The CW trapping beam has the same average power (1.54 W). The trapping potential is normalized on the trapping intensity I0 and kT (k is the Boltzmann constant and T=300 K).

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Figure 3. The trapping potential and force distributions in the nonlinear regime for a gold nanoparticle induced by pulsed CVBs of m= 1, 2 and -1 orders. (a, b, c) The trapping potential (3D profiles and 2D projections) in the trapping plane which is generated primarily by the Ez field distribution of the beams. (d-f) Spatial variation of the gradient forces in the trapping plane: the colour scale indicates the magnitude of the force and the arrows indicate its direction. (g-i) Distributions of the forces at the distance 200 nm behind the trapping plane: note that the force is negative (directed towards the trapping plane) at the positions of the traps formed by the longitudinal Ez field of CVBs. The distributions plotted for CVBs with m= 1 (a, d, g), 2 (b, e, h), and -1 (c, f, i). A pulsed trapping beam has a peak power of 1.785×105 W (average power is 1.54 W, pulse width is 100 fs).

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Figure 4. The intensity dependence of the nonlinear trapping potential. The dependence of the trapping potential depth in the longitudinal field maxima and in the centre of the m=0 femtosecond trapping beam on the incident peak power (plotted from the data in Fig. S5). All other parameters are as in Fig. 2.

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TOC Figure:

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Figure 1. Nonlinear trapping of gold nanoparticles under various polarization conditions. 174x131mm (300 x 300 DPI)


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Figure 2. The trapping potential and force distributions for a gold nanoparticle for CVB with m=0. 211x122mm (300 x 300 DPI)


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Figure 3. The trapping potential and force distributions in the nonlinear regime for a gold nanoparticle induced by pulsed CVBs of m= 1, 2 and -1 orders. 169x110mm (300 x 300 DPI)


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Figure 4. The intensity dependence of the nonlinear trapping potential. 170x115mm (300 x 300 DPI)


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TOC 200x133mm (300 x 300 DPI)


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Nonlinearity-Induced Multiplexed Optical Trapping and Manipulation

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