Calculation of trapping optical forces induced by a focused continuous Hermite Gaussian beam on a nano-dielectric spherical particle

An analytical expression for the intensity distribution of a focused continuous Hermite Gaussian beam after passing through a positive lens has been derived. Analytically, this intensity has been used to derive the gradient force acting on a nano-dielectric spherical particle . It is found that, the beam modes (p, l) have a direct influence on the trap stability, the number of trapping regions, the area of trapping zones and the particle size range.


Introduction
The mechanism of optical trapping and manipulation of mesoscopic particles using laser beams is essentially dependent on the forces of radiation pressure.Such optical forces originate from the exchange of light momentum between the light and trapped particle via a scattering process.The exerted optical forces on the particle have traditionally been decomposed into two components, namely the gradient and scattering forces.For the achievement of a stable trapping along the three dimensional directions, it is necessary that, the axial gradient component of the force pulling the particle towards the focal region must exceed the scattering component of the force pushing it away from that region [1][2][3][4][5][6].Undoubtedly, optical manipulation techniques have a diverse useful applications particularly in control methodology of small particles such as atoms, large molecules, small dielectric spheres in the size range of tens nanometers to tens of micrometers and even to biological particles such as viruses, single living cell, and organelles within cells [7][8][9][10][11][12][13][14].Besides the fundamental Gaussian beam (GB) [15][16][17][18], several special beams have been used for optical manipulation and trapping depending on each beam distinctive trapping feature.For example, Bessel beams [19], vortex beams [20,21], doughnut beams including hollow Gaussian beams [22] and elegant Laguerre-Gaussian beams [23], Laguerre-Gaussian beams [24,25], bottle beams [26], flat-topped beams [27] and Airy beams [28].Another type of important laser beams which has a rectangular symmetry along the propagation axis and has a useful applications is the Hermite Gaussian (HG) beam.These HG beams have really shown an extensive use in the field of optical manipulation and acceleration [29][30][31][32][33][34][35], free-space optical communication [36,37] and nonlinear optics [38,39].Since the last few years, the trapping characteristics of different Hermite Gaussian beams such as elegant Hermite-Cosine-Gaussian beam [40], Hermite Gaussian vortex beam [41], elegant third-order Hermite-Gaussian beams [42] and elegant Hermite-Cosh-Gaussian beams [43] have been theoretically studied.In contrast to the several studies previously made on the HG beam, and to the best of our knowledge, there is no any theoretical study has been conducted regarding the calculations of optical trapping forces induced by focused continuous Hermite Gaussian beam on a nano-dielectric spherical particle.Beside our study and particularly in the field of trapping, there is a theoretical study of a high frequency optical trap for atoms using Hermite-Gaussian beams performed in 2005 [30] in addition to the practical study of optical trapping and manipulation of light-absorbing particles by HG beam done in 2015 [32] only.In this paper, we have studied and calculated the optical trapping forces exerted by a focused continuous Hermite Gaussian beam on a nano-dielectric spherical particle.Originally, the paper was organized to include five sections.Section 2 contains the theoretical formulation in which we derive a theoretical expression for the intensity distribution and gradient force components.Section 3 contains the results and discussions in which the analysis of the optical gradient force was done and the effect of the beam modes (p, l) on the values of gradient force was discussed.Section 4 includes a discussion of the stability conditions for effective trapping and manipulating particles by HG beam.Eventually, the obtained conclusions are summarized in section 5.

Theoretical formulation
This section includes the theoretical formulation for HG beam interaction with particles.We first derive the analytical expression for the HG beam's focusing intensity distribution in section 2.1.Subsequently, we derive the gradient force components acting on a trapped particle in section 2.2.

Analytical derivation of the focusing intensity distribution of a Hermite Gaussian beam
Assuming that a nano-dielectric sphere, such as a silica nano-particle, with radius a and refractive index n p is immersed in a water with refractive index n m and illuminated by a linear-polarized Hermite Gaussian beam passing through a thin lens as schematically shown in figure (1) below.
It is well known that, the propagation of a general beam through a paraxial ABCD optical system is described by the Huygens-Fresnel integral of the form [44,45] where E 0 (x 0 , y 0 , 0) and E(x, y, z) denote the electric field components of the beam at the input plane (z = 0) and output plane (z), respectively, z is the axial distance between the two planes, k 2 = p l is the wave-number, λ is the wave length , (x 0 , y 0 ) and (x, y) denote the transverse coordinates at the input and output planes respectively.A, B, C and D are the transfer matrix elements of the optical system between the input and output planes.Let us consider the electric field of the beam at the input plane has the form [45] where H p (. ) and H l (. ) represent the Hermite polynomials of the p and l orders, G pl is a constant and w 0 is the beam waist of the fundamental Gaussian beam.By substituting equation (2) into equation (1), one can get ´--+ ´ For more simplicity, we assume that i

E x y z
For solving the integrals in equation (4) with respect to x¢ and y¢, we have to apply the integral formula of [46]  given as ( or y¢), y = (S x or S y ) and j = (p or l) in equation ( 4) and performing the integrals, the field E(x, y, z) at the output plane is found to be After lengthy algebra, we obtain ´-- = is the Rayleigh range, p determines the number of nodes along the x-direction, whereas l determines the number of nodes along the y-direction.In equation (1), A, B, C and D are given by [22] where f is the focal length, z 1 = z − f is the axial distance from the focal plane to the output plane, see figure (1).Considering the polarization of the electric field is along the x-direction, an associated magnetic field vector under the dipole approximation is given by , , 8 where c is the speed of light in vacuum, ò 0 and μ 0 are the dielectric constant and permeability in the vacuum respectively, e y ˆis the unit vector of the magnetic field direction.An important and measurable physical quantity in evaluating the optical force of the light is the irradiance at the position r = (x, y, z), which is defined as a time-averaged version of the Poynting vector and is given by To find the G 2 pl we have to normalize the irradiance using the relation P = ∫ A I dA.So that the irradiance of Hermite Gaussian beam becomes where P is the input power of the beam and e z ˆis a unit vector in the beam propagation direction.

Derivation of the gradient force components exerted by a Hermite Gaussian beam
To understand the interaction between the Hermite Gaussian beam and a trapped spherical particle, we will derive expressions for the individual components of the gradient force acting on the particle.This force arises due to the interaction of the non-uniform electric field of the HG beam with the induced dipole moment of the particle.The dipole moment itself is given by is the polarizability of a spherical particle in the Rayleigh regime and m n n p m = is the relative refractive index of the particle.The optical force exerted on the particle in the Rayleigh regime can be described by two components acting on the dipole, the first is the gradient force due to the Lorentz force acting on the dipole induced by the electromagnetic field.By using the electric dipole moment of equation (11) as an electrostatic analogy of the electromagnetic wave, a gradient force is defined by [16] The gradient force which the particle experiences in a steady state is the time-average version of equation (12) and given by By substituting equation (10) into equation (13), the expressions of the gradient force components will be: , , , 14 ´- where and y z , . From the above three equations ( 14)-( 16), it is clear that the gradient force consists of the three rectangular components which act as restoring forces directed towards the focal point in a case of m > 1.Therefore, the gradient force plays the key role for trapping the particle.The second force is the scattering force, which is much smaller than the gradient force for the small Rayleigh particles.This scattering force is given by [16] x y z e n c I x y z F , , , , , 17 where ka a m m is the cross section of the radiation pressure of the spherical particles in the Rayleigh regime.When substituting p = 0 and l = 0 in equations ( 14)-( 16), one can get the gradient force induced by Gaussian beam where which are identical with equations (5a-5c) previously obtained in [24].

Results and discussions
Complying with the previous theoretical formulation of optical forces and their frequent plotting, a number of interested results are analyzed and discussed as follows: First of all, we have to indicate that the selected parameters values used in the plots are: λ = 0.5145 μm, w 0 = 10 mm, f = 25 mm, P = 100 mW, n p = 1.59, n m = 1.332, a = 25 nm and 1 (p, l) 6 unless otherwise mentioned.Figure (2) represents the intensity behavior of HG beams along x − direction for different values of z 1 , p and l.In this figure, the overall result that can be seen is the decreasing of intensity with increasing the focal distance z 1 (the maximum value of beam intensity was observed at z 1 = 0).By increasing p value, the intensity decreases for different values of l and z 1 .The number of intensity peaks increase with increasing p values according to the relation: the number of peaks = p + 1, and consequently, it is anticipated that the number of trap regions increase.For example at p = 0, we found one intensity peak only at x = 0, while at p = 1, two intensity peaks are revealed on both sides of the center (x = 0) and so on.The increase of l values has no effect on the number of intensity peaks but really affect the intensity value.At the p-odd values a hole appears in the intensity curve at x = 0, while at p-even values a bright spot is observed at x = 0.The width of intensity peaks increase with increasing z 1 value and for this reason the side peaks displace toward increasing x values.In figure (2)) and for p = l = 0, the higher intensity value is realized at x = 0 for different values of z 1 .In figure 2(b-d), for different values of p, l and z 1 , the higher intensity values are displaced towards the increasing x values.The intensity value increases with increasing l-odd values and decreases with increasing the l-even values.
Figure (3) shows the change of transverse gradient force F gx along x − direction for different values of z 1 , p and l.Generally speaking, the transverse gradient force decreases with increasing z 1 values for different values of p and l.It is also recognized that, the F gx value starts to increase with increasing p value until it reaches a certain limit and tend to be constant or slightly increasing thereafter as shown in figures 3(a-c).The increase of l values cause a remarkable decrease in the F gx values as shown in figures 3(b, d).For different values of p, the number of trapping regions being revealed are equal to p + 1. Particularly and at p-even values, there is one effective trapping region at x = 0 and the rest of weaker trap regions are distributed along the center (x = 0) sides.While at p-odd values, there is no trap region at x = 0 but guiding forces are the dominant and the trap regions are distributed along the center (x = 0) sides.The stability of the trap regions becomes weaker as going far toward the increasing of x values.
Figure (4) is a graphical plot for the transverse gradient force F gx along z 1 and x directions.In this figure one can see that the value of F gx increases with increasing p values at constant l value as shown in figures 4(a-c).The F gx value increases with increasing l-odd values and decreases with increasing l-even values at a certain value of y as shown in figures 4(a, e, d, f).It can be also generally observed that all trapping regions are located along the transverse direction (x) at z 1 = 0 (the center of trapping zone) for different values of p and l, while the guiding forces (acceleration and deceleration forces) are acting along the axial direction (z 1 ) at different values of x depending on p and l values.On the other hand, it is clear that the number of trapping regions increase with increasing p values according to the relation (p + 1) as shown in figures 4(a-c).It can be also observed that the area of the trapping zone is affected by l values.Where at l-even values and for any value of p, the trap area becomes larger as shown in figures 4(a-c, e), but for any l-odd values, the trapping region becomes smaller as  shown in figures 4(d, f).In figures (4) and (5), the positive sign (+) indicates the trapping center while the horizontal and the vertical arrows point to the guiding forces along the axial (z 1 ) and transverse (x) directions.
Figure ( 5) is a plot showing the behavior of axial gradient force along both the transverse (x) and the axial (z 1 ) directions.From this figure, it is obvious that the axial gradient force F gz shows an increase with increasing p values at a certain y value as long as l remains constant.Moreover, it can be noted that, a slight increase is observed in the value of F gz at l-odd values, while at l-even values, remarkable decrease in the F gz value is revealed.Another observation is the area of trapping region which is being larger with increasing l-even values and being smaller at l-odd values.The number of trapping regions of the F gz is depending on the relation p + 1.It can be seen also that, the center of trapping regions along the axial direction (z 1 ) is located at z 1 = 0 for different values of p and l.It should be noted that, the scattering force values are much smaller than the F gx and F gz values.Also the increasing of the beam modes (p, l) cause a decrease in the scattering force value and this strengthen the trap stability.In figures (6)), and for p = l = 0, the higher value of F scat occurs at x = 0 for different values of z 1 .While in figures 6(b-d), the higher value of F scat is displaced toward the increasing of x values for different values of z 1 , p and l.
Figure (7) illustrates the relation between the maximum transverse and axial gradient forces with p values for different l values.In figure (7)), the maximum F gx values are effectively increasing between p = 0 and p = 1, after then the F m gx value starts to be slightly changed as it is being constant when p > 1 for different values of l.Also the maximum F gx decreases with increasing l values.This decrease exhibits a logarithmic gradual lowering with increasing l values.In figure (7)), it can be recognized that at l 2, the maximum F gz value starts to decrease between p = 0 and p = 1 after that the maximum F gz value shows a uniform increase at p > 1.At l = 3 the maximum F gz value decreases between p = 0 and p = 2 then the maximum F gz increases at p > 2 but not as much as those done with l < 3.As could be seen from the behavior of the displaced curves of the maximum F gz for different values of l and p, one can expect a gradual decrease in the maximum F gz with increasing p values after l value exceeds 3.

Trapping stability discussion
According to our analysis of the obtained results, it can be concluded that the radiation forces of the continuous focused Hermite Gaussian beams could be used for trapping and manipulating the Rayleigh dielectric particles.For the achievement of stably trap of particles, the axial gradient force F gz must exceed the scattering force F scat as being previously illustrated by figures (5) and (6).From the other point of view, we have to indicate that the Rayleigh dielectric particle usually suffer from the Brownian motion due to the thermal fluctuation from the ambient.Therefore, a proper trapping of the particle necessitate a presence of a deeply enough potential well induced by the gradient forces in order to overcome the kinetic energy of the trapping particle stem from the Brownian motion.The criteria of the trapping stability is given by the Boltzmann factor as follows [16 ( ) is the potential energy of the gradient force at r = 0, K B is the Boltzmann constant and T is the temperature of the ambient.As a result of the thermal fluctuation theorem of Einstein, the  Brownian force can be calculated by the following formula [43] F aK T 12 23 where κ is the viscosity of the ambient (e.g., water) which is κ = 7.977 × 10 −4 pa.s at the T = 300 °K. Figure (8) shows the comparison between the magnitude of all forces versus the particle radius a for different values of p and l, where F m gx is the maximum transverse gradient force, F m gz is the maximum axial gradient force and F m scat is the maximum scattering force.From this figure, it is evident that, there are two types of forces which cause a disturbance of trapping and manipulating of the particle.The first force represents the Brownian force and the second is the scattering force.In this figure, the control process (trapping and manipulating) of the particles of the maximum transverse gradient force occurs: in Figure 8(a) between the radius values a = 6 nm and a = 89 nm, in sub-Figure 8(b) between the radius values a = 7 nm and a = 95 nm, in sub-figure 8(c) between the radius values a = 8 nm and a = 97 nm and in figures 8(d) between the radius values a = 7 nm and a = 100 nm.On the other hand, the control process of the particles of the maximum axial gradient force occurs: in sub-Figure 8(a) between the radius values a = 25 nm and a = 28 nm, in sub-Figure 8(b) between the radius values a = 13 nm and a = 57 nm, in sub-figure 8(c) between the radius values a = 19 nm and a = 45 nm and in figures 8(d) between the radius values a = 13 nm and a = 66 nm.Generally speaking, it is necessary to clarify that the control process of the particles realized in the above explained cases for figure (8), are really occurred because F m gx and F m gz are larger than F B and F m scat within the indicated a values, where outside these indicated a values, the control process of the particles are not existing because F B or F m scat are larger than F m gx and F m gz .The difference between each two indicated a values constitutes the so-called particle size range Δa.It is clear that, Δa of the maximum transverse gradient force is larger than Δa of the maximum axial gradient force because F gx > F gz .The obtained particle size range applies only for the chosen parameters ( f = 25 mm and λ = 0.514 μm).In case of maximum transverse gradient force, the particle size range increases with increasing p and l-odd values.Where in the maximum axial gradient force, the particle size range increases with increasing of p-even and l-odd values.

Conclusion
On the light of the results obtained and discussed under sections 3 and 4, it can be concluded that: for different values of p and l, the higher intensity is observed to be located at z 1 = 0.The modes order (p, l) affect the values of the gradient forces (F gx , F gz ) as well as the scattering force and the beam intensity.The number of trapping regions are affected by increasing the p values.The values of F gx and F gz are influenced by the values of p and l modes.The particle size that can be trapped by F gx is in the range (6 − 100 nm) which agree with the result obtained by Harada and Asakura ( [16], figure (4)) who have calculated the gradient force by the generalized Lorenz-Mie theory and Rayleigh approximation.The particle size that can be trapped by the axial gradient force of HGB (F gz ) is in the range (13 − 66 nm), which be valid with the Rayleigh approximation.It is noted that, the most influence mode to achieve more stable trapping along x − direction is p > 1 and l = 0.
integral of equation (3) will become

Figure 1 .
Figure 1.Schematic Diagram of the focusing optical system radiating on a nano-dielectric particle for the case p = 3 and l = 1.

Figure 2 .
Figure 2. Shows the behavior of HGB intensity along x − direction for different values of z 1 , p and l at y = 0.1μm.

Figure 3 .Figure 4 .
Figure 3. Shows the change of transverse gradient force along x − direction for different values of z 1 , p and l at y = 0.1 μm

Figure ( 6 )
demonstrates the behavior of scattering force F scat along x − direction for different values of z 1 , p and l.The overall picture of this figure shows a decreasing of F scat with increasing of z 1 for different p and l values.

Figure 5 .
Figure 5. Shows the change of axial gradient force (F gz ) as a function of x − and z 1 at y = 0.1 μm.

Figure 6 .
Figure 6.Shows the change of scattering force (F scat ) along x − direction for different values of z 1 , p and l at y = 0.1 μm.

Figure 7 .
Figure 7. Shows the change of maximum transverse and axial gradient forces with p values for different values of l.