Abstract
Optical pulling forces exerted on small particles can be achieved by tailoring the properties of the electromagnetic field, the particles themselves, or the surrounding environment. However, the nonlinear optical effect of the surrounding environment has been largely neglected. Herein, we report the optical pulling forces on a Rayleigh particle immersed in a nonlinear optical liquid using high-repetition-rate femtosecond laser pulses. The analytic expression of time-averaged optical forces allows us to better understand the underlying mechanism of the particle transportation. It is shown that the two-photon absorption of the surrounding liquid gives rise to a negative radiation force. Transversely confined Rayleigh particles can be continuously dragged towards the light source during a pulling process.
1 Introduction
In the past 40 years, an optical tweezers technique has experienced impressive progress for trapping and manipulating small objects in air [1], in vacuum [2], [3], in liquid [4], [5], in solid [6], [7], or at interfaces [8], [9], [10], taking advantages of the optical forces produced by a tightly focused continuous-wave laser beam [4], [5]. This technique has become a very important tool in various disciplines, including optics [5], quantum physics [2], biological science [6], [11], and chemistry [12]. Recently, the optical tweezers technique has been extended by substituting a continuous-wave laser with high-repetition-rate femtosecond laser pulses [13], [14], [15], [16]. In the femtosecond laser tweezers, the high peak intensity of each laser pulse leads to instantaneous trapping of a particle and the high-repetition-rate ensures repetitive trapping by successive pulses. Interestingly, femtosecond laser pulses have revealed novel phenomena in the optical trapping of particles, such as the trapping split behavior in the process of capturing gold nanoparticles [13], directional ejection of optically trapped nanoparticles [16], [17], and immobilization dynamics of a single polystyrene sphere [15]. Moreover, it has been demonstrated that the nonlinear optical effects originating from the particle could modify the optical trapping potential [13], enable the realization of super-resolution optical manipulation [18], or enhance the optical force [14], [19], [20], [21]. It is noteworthy that the optical force arising from nonlinear polarization becomes significant and cannot be neglected if the trapped particles exhibit nonlinear optical effects [13], [18], [21], [22], [23], [24], [25], [26], [27]. However, the nonlinear optical response of the surrounding medium has never been considered before.
Theoretically, the optical forces exerted on a particle are divided into two parts: one is the gradient force, which is proportional to the gradient of intensity and drives the particle toward the equilibrium point [5]; the other is the radiation (i.e. scattering and absorption) force, which is proportional to the orbital part of the Poynting vector of the field and destabilizes the trap [28], [29], [30], [31]. In general, the radiation force acts as a pushing force (i.e. in the direction of a beam’s propagation) on a particle. However, the particle can be pulled by the negative radiation (scattering) force towards the light source. Recently, this counterintuitive optical pulling force has received great attention due to its academic interest and technological applications [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48].
The physical mechanisms of this pulling force can be exploited by the optical gradient force [33], backward scattering force [34], and/or negative photophoretic force [35], [36]. Through manipulation of the properties of the electromagnetic field, researchers realized the optical pulling force on particles with super-oscillating beams [37], solenoid beams [38], interference of multiple beams [34], nonparaxial gradientless beams [39], two interfering waves [40], etc. By contrast, the shape and material composition of a small particle have been modified to obtain the pulling force, including gain media [41], [42], microspheres with antireflection coatings [43], and chiral particles [44], [45]. In addition, optical pulling on particles can also be achieved through the use of surrounding media with specifically designed properties, such as resonator-waveguide systems [46], hyperbolic metamaterials [7], photonic crystal waveguides [47], and bilayer PT-symmetric structures [48].
In this work, we report the theoretical investigation of optical pulling forces on a dielectric Rayleigh particle immersed in nonlinear optical liquids (e.g. carbon disulfide) arising from focused femtosecond laser pulses. It is found that the two-photon absorption of the surrounding liquid is the predominant physical mechanism for the negative radiation force.
2 Theory
In the optical trapping experiment using high-repetition-rate femtosecond laser pulses, the pulse duration τF (FWHM) and repetition-rate v (i.e. the inverse of the pulse period T) for a commercial Ti:sapphire laser are typically about 100 fs and 80 MHz, respectively [13], [14], [15], [16]. Accordingly, the spectral bandwidth is so narrow that the pulsed laser can be regarded as a monochromatic field. Hence, the spatial and temporal characteristics of laser pulses can be treated independently. For laser pulses with a Gaussian temporal envelope, the time-harmonic electromagnetic waves are expressed as:
where
Now we consider a homogeneous, isotropic, non-magnetic spherical particle with a radius R and a permittivity
where
Under the excitation of ultrafast laser pulses, the time-averaged optical force acting on the Rayleigh particle (R≪≪λ) is [51]:
where * denotes the complex conjugate.
Substituting Eqs. (1)–(4) into Eq. (5) (see Section 1 in Supplementary Information for more details) we get:
where c and μ0 are the speed of light and permeability of light in vacuum, respectively.
Eq. (6), which is the basic result of the present work, gives the total optical forces as a sum of the gradient force and the radiation force on a nonlinear optical Rayleigh particle immersed in a nonlinear optical medium. Especially, Eq. (6) degenerates into the ones reported previously [21], [28], [30] for a Rayleigh particle without/with optical nonlinearity when the optical nonlinear response of the surrounding medium is neglected (i.e.
To trap and manipulate nanoparticles, an x-polarized Gaussian beam is focused by a high numerical-aperture (NA) objective lens (see Section 2 in Supplementary Information for more details). It is noteworthy that one determines
For the absorptive particle [i.e. Im(α)>0] with a refractive index larger than the surrounding medium [i.e. Re(α)>0], the backward gradient force can be larger or smaller than the positive radiation force, making it easy to realize the trapping or pushing of a high-refractive-index particle, respectively [5]. Interestingly, in order to produce a pulling force on the particle predominantly contributed by the negative radiation force [see Eq. (6)], the necessary requirement of Im(α)<0 should be satisfied. One could get Im(α)<0 for the gain particle immersed in a lossless surrounding medium [41], [42]. Alternatively, the negative radiation force can be realized for the particle immersed in a medium with nonlinear absorption, as we will demonstrate below.
3 Numerical simulations and discussions
First, we assume a silicon nitride (Si3N4) particle with the size of R=40 nm immersed in carbon disulfide (CS2). CS2 is chosen as a solvent because it is readily available. Moreover, it is well documented that CS2 exhibits a large refractive nonlinearity in the visible region and strong two-photon absorption in the short wavelength region. In addition, both CS2 and Si3N4 are highly transparent in the visible region. Accordingly, the linear absorption of the solution could be safety omitted. The linear and nonlinear optical parameters of both CS2 and Si3N4 within a wavelength range from 400 nm to 650 nm are taken from the reported ones (see Section 3 in Supplementary Information for more details).
Without loss of generality, we take the following parameters unless otherwise mentioned as NA=0.8, τF=100 fs, v=80 MHz, and the average power of laser pulses P=100 mW for numerical simulations. The optical forces on the Si3N4 nanoparticle immersed in CS2 solvent using tightly focused laser pulses are calculated with Eq. (6) and shown in Figure 1. Figure 1A illustrates the distributions of the longitudinal forces Fz on the particles produced by focused laser pulses on the z-axis (x=y=0) at different wavelengths from 400 nm to 650 nm. Here, positive (or negative) longitudinal forces mean that their direction is along the +z (or −z) direction. Obviously, the longitudinal forces exerted on the particle have different behaviors at different wavelengths. Within a wavelength range from 460 nm to 650 nm, the particle can be trapped at the focal plane due to the existence of the mechanical equilibrium point (e.g. Figure 1B1 and B2). However, in a short wavelength region (i.e. 400–460 nm), the particle will be pulled axially towards the light source owing to the strong negative longitudinal force. For an example, as shown in Figure 1B3, the longitudinal trapping force on the particle at 420 nm remains continuously negative during the pulling process. For the transverse force profiles, as shown in Figure 1C, the maximal y-axis force
To gain insight into the physical mechanisms for the pulling force of a nonlinear optical particle immersed in a nonlinear optical medium, we simulate the longitudinal force profiles on the z-axis and transverse force profiles on the y-axis for both the particle and solvent with or without optical nonlinearities at 420 nm. For the solvent without optical nonlinearity (i.e.
Since the power of the laser pulses plays an important role in the nonlinear optical effect and subsequently the optical force, we present in Figure 3A the power dependence of the longitudinal force at the focal point Fz(r=0) and transverse force
The Si3N4 particle in CS2 is just used as one example of many possible combinations for producing the optical pulling force. To pull the Rayleigh particle confined transversely, there are two necessary requirements to be satisfied: (i) the particle must be stably captured in the transverse plane, i.e. Re(α)>0; and (ii) the radiation force is negative and directs to the light source, i.e. Im(α)<0. For the sake of simplicity, we consider the Rayleigh particle with the effective permittivity of
In addition, we also investigate the influence of the solvent’s permittivity on the polarizability of Si3N4 nanoparticles at 420 nm. As mentioned above, for the fixed particle, the linear refraction index [i.e.
4 Conclusions
In summary, we studied the time-averaged optical forces exerted on a Rayleigh particle immersed in nonlinear optical solvent using high-repetition-rate ultrafast laser pulses. As an example, we investigated the characteristics of the three-dimensional optical forces for Si3N4 nanoparticles immersed in CS2 at different excitation wavelengths. Interestingly, it is shown that the Rayleigh particle confined transversely can be continuously pulled towards the light source during the pulling process at a specific wavelength (in this study 420 nm). The physical mechanism of the pulling forces is predominantly the negative radiation force originating from the two-photon absorption of the ambient liquid. Beyond the linear optics regime, the concept and results presented in this work provide a novel and practically nonlinear optical approach to manipulate optical forces, resulting in the particle transportation towards the light source.
Funding source: National Science Foundation of China
Award Identifier / Grant number: 11774055, 11474052, and 11504049
Funding source: Natural Science Foundation of Jiangsu Province
Award Identifier / Grant number: BK20171364
Funding source: National Key Basic Research Program of China
Award Identifier / Grant number: 2015CB352002
Funding statement: This work was financially supported by the National Science Foundation of China (Funder Id: http://dx.doi.org/10.13039/501100001809, Grant Nos: 11774055, 11474052, and 11504049), Natural Science Foundation of Jiangsu Province (Funder Id: http://dx.doi.org/10.13039/501100004608, BK20171364), and National Key Basic Research Program of China (2015CB352002).
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Supplementary Material
The online version of this article offers supplementary material (https://doi.org/10.1515/nanoph-2019-0095).
© 2019 Bing Gu, Yiping Cui and Qiwen Zhan et al., published by De Gruyter, Berlin/Boston
This work is licensed under the Creative Commons Attribution 4.0 Public License.