Simultaneous and independent capture of multiple Rayleigh dielectric nanospheres with sine-modulated Gaussian beams

This study investigates the propagation properties and radiation forces on Rayleigh dielectric particles produced by novel sine-modulated Gaussian beams (SMGBs) because of the unique focusing properties of four independent light intensity distribution centers and possessing many deep potential wells in the output plane of the target laser. The described beams can concurrently capture and manipulate multiple Rayleigh dielectric spheres with high refractive indices without disturbing each other at the focus plane. Spheres with a low refractive index can be guided or confined in the focus but cannot be stably trapped in this single beam trap. Simulation results demonstrate that the focused SMGBs can be used to trap particle in different planes by increasing the sine-modulate coefficient g. The conditions for effective and stable capture of high-index particles and the threshold of detectable radius are determined at the end of this study.


Propagation of SMGBs through an ABCD system
In the rectangular coordinate system, the electric field of SMGBs at the origin plane ( z 1 = 0 ) takes the form where E 0 denotes a constant related to the laser beams power P. q 1 is a complex parameter of the incident Gaussian beam, g is the modulation coefficient related to the sine and can take on any value other than zero, and w 0 represents the waist size of the corresponding normal Gaussian beam.
Using the extended Huygens-Fresnel diffraction integral in the paraxial approximation, we can determine the electric field of SMGBs using an ABCD optical system We probe the focusing properties of SMGBs by considering the beam propagation through a lens system (Fig. 1). The transfer matrix for this system is given by the reference 33 , where z is the longitudinal coordinate at the beginning of the focusing lens, z = f + δz. δz is the distance from the focal point on the axis, and f is the focal length of the thin lens. We assume that P is the incident light power. Therefore, we can also obtain the initial value of the electric field.

Figure 1.
An illustration of the focusing optical system. (a) shows the schematic of the SMGBs, where z is the longitudinal coordinate at the beginning of the focusing lens, z = f + δz. δz is the distance from the focal point on the axis, and f is the focal length of the thin lens. (b,c) demonstrate the normalized intensity distribution of SMGBs and Gaussian beams at different distance values of δz , respectively. Other parameters are = 1.064 µm , w 0 = 5 mm, f = 5 mm , P = 4 w and the sine-modulation coefficient g = 1.  Figure 1 shows the intensity distribution of SMGBs through a lens system to visualize the profiles of SMGBs. Figure 1a illustrates the focusing optical system of SMGBs with four independent focus points at both the initial and focal planes. Figure 1b,c describe the dependence of the intensity distribution of the SMGBs and Gaussian beams on the beam index δz , respectively, further examining the SMGBs' focusing properties in detail. Both Gaussian beams and SMGBs' intensity amplitudes decrease with the increase in the distance from the focal point δz . Gaussian beams have only one light intensity center in one direction, whereas SMGBs has two. Because of the symmetrical distribution of SMGBs in the focal plane, we could have four independent potential wells of the target laser. As the distance δz of SMGBs decreases, the dark area in the middle increases and the four light centers move further from the focus point. Because of this special focusing characteristic, we assume that SMGBs can trap multiple particles with different refractive indices at the focal plane.

Optical forces on a Rayleigh dielectric sphere produced by SMGBs
This section demonstrates the radiation forces exerted on a nanosphere produced by SMGBs. Since the radius of Rayleigh's sphere is much smaller than the light wavelength (a ≤ /20) , it is the electric dipole of the light field. Here, according to Harada and Asakura, two types of optical forces act on the sphere: the scattering force F Scat and the gradient force F Grad . The scattering force arises from the light scattering by the dipole and travels along the beam propagation direction, which is proportional to the intensity of the beam. And the nonuniform electromagnetic field on the dipole that acts as the restoring forces responsible for drawing the particles back to the beam center produces the gradient. The scattering and the gradient force are defined by 34,35 with where − → e z is a unit vector in the beam propagation direction, c is the speed of the light field in a vacuum, and C pr equals the scattering cross section C Scat for a dielectric sphere in the Rayleigh regime. In the following simulations, we select the radius of nanosphere as a = 20 nm which is within the Rayleigh range for the wavelength we consider. n r = n p /n m is a relative index with n p and n m , respectively, representing the refractive index of the sphere as n p = 1 (air bubble), n p = 1.59 (glass), and ambient as n m = 1.33 (water) 36 . Figure 2 shows the simulated summation of the radiation force field exerted on the nanospheres at the focal plane. The gradient arrows of the gradient field graph are toward the centers, whereas the directions and lengths www.nature.com/scientificreports/ of the arrows represent the directions and magnitudes of the resultant forces (Fig. 2b). The position of trapped particles slightly deviates from the equilibrium point. The gradient force is proportional to the intensity gradient and points to the focus, which is the direction of the intensity gradient maximum. However, the gradient force is nearly zero at the focus point (Fig. 2). The scattering force is proportional to the optical intensity and points toward beam propagation. In the focus point, the gradient force is small; therefore, the optimum position for the trapped particles is slightly shifted from the focus. Therefore, in the following article, we also consider the influence of scattering at different planes. Because of the four focal points in the intensity distribution from the contour plot in Fig. 2a, SMGBs can independently and simultaneously trap and manipulate multiple particles over other types of beams. Based on the theoretical analysis above, the gradient and scattering forces are calculated (Figs. 3, 4). The sign of the gradient force indicates the direction of the force. The transverse gradient force for positive F Grad,x is along the +x direction and that for negative F Grad,x is along the −x direction. Similarly, for positive (negative) F Grad,z , the longitudinal gradient force is in the +z(−z) direction. The scattering force is always along the +z direction.
In Fig. 3, we depict the transverse gradient forces along the x direction. Figure 3b,c are the exploded views of Fig. 3a. Figure 3b shows that a stable equilibrium point a 1 toward the transverse gradient force exists for particles  www.nature.com/scientificreports/ with a low refractive index. However, high refractive index particles have two stable equilibrium points a 2 and a 3 at x = ±0.265 µm . From Figs. 3a,c, 4c,f, one can find that the high-index particles have four stable equilibrium points (x = ±0.265 µm, y = ±0.265 µm) in the focal plane. Note that multiple stable equilibrium points exist for high-index nanospheres. Therefore, the results show that the focused SMGBs can trap or manipulate multiple particles. Figures 3a,b and 4a,c show that low-index particles have one stable equilibrium point at the focus in the transverse direction, whereas Fig. 4b shows a red channel along the z-axis in the intensity distribution, indicating that the axial gradient force always equals zero when x = 0, y = 0 . Therefore, low-index particles can be confined or guided in the focus (2D trap) but cannot be stably trapped in a single SMGB. We can add two orthogonal laser beams to form a three-dimension trap to stably trap the particles. Figure 4d-f show the scattering force at different propagation distances from the focus point. We find that the scattering force is negligible compared to the transverse and longitudinal gradient forces (Figs. 3a, 4a,c).
In Fig. 5, we plot the changes of the longitudinal gradient forces of the high-index particles at one of the equilibrium points (x = 0.265 µm, y = 0.265 µm ) of the focal plane with different sine-modulation coefficient g . From the longitudinal gradient forces, one can find that there is one stable equilibrium at g = 1 , as g increases, it is observed that there appear two stable equilibrium points. In this case, we can use the focused SMGBs to trap spheres in different plane.
To further study the influences of beams' optical parameters generated by SMGBs on the transverse gradient force, Fig. 6a-c illustrate the radiation forces exerted on the high-index particles for different values of radius a , the beam waist w 0 , and the distance δz from the focus point. Figure 6d-f demonstrate the radiation forces  www.nature.com/scientificreports/ exerted on the low-index microspheres. The numerical results show that the gradient force F Grad is related to the beam waist w 0 , and F Grad increases by orders of magnitude as the beam waist w 0 increases from 5 mm to 1 5 mm . However, F Grad is inversely proportional to the distance δz between the particle and focal plane. And the gradient potential well is wider with an increase in distance δz. The gradient force increases as the particle radius increases. Figure 6a-c show that there are two stable equilibrium points in the focus plane for the highindex particles. In Fig. 6d-f, the trend of the gradient force of changes of the low-index particle is the same as the high-index particles.

Trapping stability analysis
The above discussion confirms that the radiation forces produced by SMGBs can trap and manipulate Rayleigh dielectric particles of n r > 1 . To stably trap spheres under the Rayleigh scattering regime, two conditions must be considered. The potential well of the gradient force should be deep enough to overcome the kinetic energy contributed by Brownian motion because of the thermal fluctuation in the surrounding environment. This stability condition is represented by the Boltzmann factor 6,34 where k B is the Boltzmann constant and T = 300K is adopted as the absolute temperature of the particle environment. U max is the maximum depth of the potential well, which is written as 5 For the high refractive index particles (n p = 1.59) , where the sine-modulation coefficient g = 1 at one of the maximum intensity position x = 0.265 µm, y = 0.265 µm, δz = 0 and R thermal = 0.0012 , the value of R thermal at g = 4 is about R thermal = 0.0197 . The Boltzmann factor values around the focal plane are sufficiently small. Therefore, the particles can overcome Brownian motion and stably captured by SMGBs. Two forces are toward light transmission: the scattering force proportional to the intensity of the incident light in the light propagation directions and the gradient force proportional to the intensity gradient along the light gradient direction. Therefore, the second condition is the stability criterion R , stating that the backward longitudinal gradient force should be greater than the forward scattering force. According to Einstein's fluctuation-dissipation theorem, the magnitude of Brownian force can be expressed by where η = 7.977 × 10 − 4 Pa s is the viscosity of water at T = 300 K 19 . Figure 7 demonstrates that the magnitudes of all forces include the maximum axial gradient force F m Grad,z at x, y = 0.265 µm , the maximum transverse gradient force F m Grad,x , the maximum scattering force F m Scat,x , and the Brownian force F b versus the radius a on the high-index microspheres, respectively. The x and z subscripts in each term represent the force component in the coordinate axes. For the high-index particles, when a > 4.431 nm, the longitudinal gradient force is greater than the Brownian force in Fig. 7. Therefore, the Brownian force does not affect the particles, and we can stably capture multiple microspheres in the focal plane.

Conclusion
In this study, we raise and confirm that SMGBs can improve the efficiency of trapping Rayleigh dielectric nanospheres and that SMGBs can simultaneously trap up to four nanospheres on the focal plane due to the unique property of having four individual centers of light intensity distribution. First, we theoretically calculated the Collins integration and a paraxial ABCD optical system. The propagation properties of the electrical field profile  www.nature.com/scientificreports/ and the gradient and scattering forces of focused SMGBs are graphically studied. The study further analyzes the stability criterion and the Boltzmann factor of SMGBs. Based on a case study of trapping particles in water, which undergoes Brownian motion because of thermal fluctuation, the axial gradient and Brownian forces of the SMGBs are compared. The results show that SMGBs can successfully trap Rayleigh dielectric particles with a radius as small as 4.431 nm. The proposed technique of using controllable SMGBs to trap Rayleigh dielectric spheres has a potential for optical micromanipulation. However, applying SMGBs under different conditions must be researched, which will require much work in the future.