Laser Self-Trapping in Optical Tweezers for Nonlinear Particles

: The optical tweezers are used to trap the particles embedded in a suitable fluid. The optical trap efficiency is significantly enhanced for nonlinear particleswhich response to the Kerr effect. The optical transverse gradient force makes these particles’ mass density in trapping region increasing, and the Kerr medium can be created. When the laser Gaussian beam propagates through it, the self-focusing, and consequentlyself-trappingcan appear. In this paper, a model describing the laser self-trapping in nonlinear particle solution of optical tweezers is proposed. The expressions for the Kerr effect, effective refractive index of nonlinear particle solution and the intensity distribution of reshaped Gaussian laser beam are derived, and the self-trapping of laser beam is numerically investigated. Finally, the guide properties of nonlinear particles-filled trapping region and guiding condition are analysed and discussed.


Introduction
A.Ashkin has shown that a dielectric particle having refractive index larger than that of its embedding fluid will be trapped in the focus of laser Gaussian beam [1]. Theparticle'soptical trap efficiency Qis higher when the refractive index of embedding fluid is lower [2,3,4], orthe refractive index of trapped particle is higher. Consequently, the optical tweezers are efficient to trap the Kerr particle [5,6,7,8].This is easily resolved for a single particle in the thin fluid where having no influence of nonlinear particle solution on the laser beam. However,if the nonlinear particle solution is thick enough, the density of the trapped nonlinear particle will become more and more increasing in the trapping region along laser axisdue to action of transverse optical forces, so called the trapping cylinder [9]. Therefore,not only does nonlinear particle solution's mass refractive indexincrease in the trapping cylinder [10]but also itsnonlinearity. Consequently, the nonlinear particle solutionwill become the graded index (GRIN) medium. The Gaussian laser beam propagating through GRIN medium will be self-focused [11], and then self-trapped in the small trapping cylinder [12]. This phenomenonis similar to that the laser beam propagating through a self-written optical waveguide in a solid photopolymer material volume [13],Alkalineearth atoms [14], and thermophoresis [15]. Specially, the self-trapping has been experimentally observed in the human red blood cellssuspension by the propagation of laser beam, seen as nonlinear medium [16].Lamhot and his co-workers also investigated the optical soliton beam in nanoparticle suspension by virtue of thermophoresis [15].In this paper, we propose the model of optical tweezers by using the thicknonlinear particle solution. The expressions ofthe reshaped laser beam andeffective refractive index of nonlinear particle solution are also derived. The selffocusing and self-trapping of laser beam in the trappingregion are numerically calculated byusing the iteration method. Finally,the guide properties of trapping region in optical tweezers, guiding condition are analyzed and discussed.

Principle model for simulation
Assuming thatthe optical tweezersforourinvestigation model isillustrated in Fig.1. An incoming laser Gaussian beam (ILGB) with peak intensity 0 I and radius of beam waist of 0 W propagates throughthe thick chamber of nonlinear particle solution. The nonlinear particle has radius a , linear index p n , nonlinear refractive coefficient 2 n embedded in the fluid of lower refractive index f n . Under the action of optical transverse gradient force F  , nonlinear particles are pulled to the laser axis and hold in the trapping region with radius 0 /2 W [9]. This makes the mass density p m of nonlinear particles in the trapping region and the effective refractive index of nonlinear particle solution eff n increase [10]. The refractive index of nonlinear particle is directly proportional to ILGB's intensity. The effective refractive index of nonlinear particle solution in the trapping region is graded and reduced from laser axis, i.e., the trapping region becomes a GRIN one.

Fig. 1
The sketch of optical tweezers to trap nonlinear particles embedded in fluid for self-trapping simulation.
Every differential GRIN cylinder with the thickness of 2 da  ( Fig.1) will operate as the nonlinear thin lens (NTL) and on the contrary, NTL also reshape the ILGB. The self-focusing appears continuously through NTL. Consequently, the beam waist of the reshaped laser Gaussian beam (RLGB)decreases, the mass density of nonlinear particles in the differential trapping cylinderincreases, and the effective refractive index of nonlinear particle solutionincreases. The self-focusing and increasing of divergence angle 0 of RLGB are simultaneously occurred by the beam waist's decrease. If both processes are in balance, thespatial optical soliton will appear [13,15,17], meansthat the RLGB will be self-trapped in the center of the trapped region.

Theoretical background 3.1. Effective refractive index in nonlinear particle solution
Consider a solution of nonlinear particles embedded in the fluid with certain mass density. Under the optical tweezers, a numberof the nonlinear particlesm is trapped inthe differential trapping cylinder with thickness 2 d na  , the mass densities of particles p m and fluid f m in the differential trapping cylinder(DTC) can be derived as: are total volumes of particles, fluid and the differential trapping cylinder, respectively.Using approximation of mass refractive index for themulti-component mixtures [10], the effective refractive index eff n of nonlinear particle solution in the differential trapping cylindercan be derived as: Consider the intensity distribution of ILGB is [18]: Whennonlinear particle solution is irradiated by ILGB, its effective refractive index contribution will be radial-graded and Eq. (3) will be modified as: is the nonlinear coefficient of nonlinear particle. We consider the laser wavelength to be shorter than the radius of beam waist. The mass density of particles in the trapping region will increase if all of particles are trapped and directly pulled to laser beam's axis, which is called the trapping condition of optical tweezers. This condition will be always satisfied if these particles on the edge of beam's waist W0 (see Fig.1) (where the laser intensity and its gradient are the smallest) are trapped. That means that the optical force acting these particles must be larger than 1pN [1]. Using Eq. (4) and Eq. (5) and Eq. 5 in Ref. 3 we obtained the trapped condition as following: Considering the trapped particles are pulled in the region near laser beam's axis. This means that their positions are approximately 0 W   . Assuming that the differential trapping cylinder is placed at the waist of ILGB and 0 dz  (see Fig.1), the function of intensity radial distribution in the input surface of differential trapping cylinder can be simplified as follows: 22 00 22 00 where eff n describes the radial distribution of the effective refractive index in differential trapping cylinder and to be a function of radial radius  , beam waist 0 W , i.e. particle'smass density p m . Therefore, it is similar to the index change of thermophoresis irradiated by laser beam in work [15] as the function of temperature and particle concentration. Consequently, we can substitute eff n into nonlinear paraxial wave equation in work [15] to calculate the optical spatial soliton beam.In this paper, we consider the trapping region of optical tweezers as a consecutive series of NTLs and use iteration method to calculate the change of each NTL's focal length of and laser beam's waist. The intensity distribution and self-trapping of RLGB is also shown and discussed.

The focal of NTL and intensity distribution of RLGB
The effective refractive index in Eq.(9) can be simplified as follows:  With refractive index given in Eq.(10), the differential trapping cylinder will become NTL with focal length given as follows [18]: The ILGB will be reshaped to RLGB when it propagates through the NTL [18]. Its intensity distribution is given by: is the radius of laser beam at z; is the radius of reshaped beam waist placed at the output surface of differential trapping cylinder given by: is the Rayleigh range.

Simulation procedure
Firstly, we check the trapping condition using Eqs. (6), (7) with an collection of parameters. The self-trapping process will be numerically observed if this trapping condition satisfied. To observe the self-trapping process in optical tweezers, we use the solution of nonlinear particles embedded in fluid, the simulation scheme given in Fig.2 above. The self-trapping is related to the change of beam waist 0 W , focal length of NTL nl f in z. We simulated in a selfconsistent manner: we use an initial prediction to find , then substituting again to Eqs. (13) and (16) to find the next ones.
The calculating process is iterated until the divergence and focusing angles are close one to each other. The simulation procedure is given in Fig.2.

Results and discussion
We consider an optical tweezers using ILGB with

  
, that means the trapping condition of optical tweezers to be satisfied and all particles locating inside the beam waist W0 are also trapped and pulled in beam's axis. These make the mass density of particles inside beam waist increase and the self-focusing appears. Trapping and selffocusing makes the mass density of particles increase and the self-focusing more powerfully. Consequently, the laser beam will be trapped inside the cylinder of solution with nonlinear particles. Secondly, we numerically calculate the laser self-trapping process. Simulation process was done until i=16 at which  The self-trapping simulation in nonlinear particle solution are shown in Fig.3. We see that the ILGB (Fig.3a)isself-trapped (Fig.3b) intrapping region at where its divergence and focusing (angles) effects are in balance (see Fig.4). When the balance of divergence and focusing effects occurs, the longitudinal gradient of laser intensity reduces, and then the distribution of the longitudinal gradient force also reduces, that has been shown in works [3,8,12,23].Moreover, the spatial optical soliton appears and propagates continuously.Our result in Fig.3 is similar to that obtained by Li [13]for the self-trapping of optical beam in the self-written optical waveguide in a solid photopolymer material.   5 shows RLGB's beam waist shortening along the trapping region. Since the particle mass density increases, the effective refractive index in nonlinear particle solutionalso increases (Fig.6). at boundary amongdifferential trapping cylindersshown in Fig.8 and Fig.9, respectively.

Conclusion
We have theoretically shown that the Gaussian laser beam used for optical tweezers can be selftrapped in trapping region by Kerr effect in nonlinear particle solution. The self-trapping of Gaussian beam in nonlinear particle solution has been numerically calculated, the guide properties of the trapping region in optical tweezers is also analyzed and discussed. Theseresults showthat the Gaussian beam for optical tweezersnot only traps nonlinear particles, but also be reshaped and self-trapped. Finally, our results may hinta new study for increasing and stabilizing the density ofnonlinear biomedial cells in experimental observation [16].