Radiation force acting on a Rayleigh dielectric sphere produced by Whittaker-Gaussian beams

Optics and Laser Technology 107 (2018) 239–243

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Radiation force acting on a Rayleigh dielectric sphere produced by Whittaker-Gaussian beams Bin Tang a,⇑, Kai Chen a, Li Huang a, Xin Zhou b,⇑, Xianzhong Lang a a b

School of Mathematics & Physics, Changzhou University, Changzhou 213164, China School of Sciences, Hunan University of Technology, Zhuzhou 412008, China

a r t i c l e

i n f o

Article history: Received 21 December 2017 Received in revised form 10 April 2018 Accepted 13 May 2018

a b s t r a c t Optical trapping and manipulating of micro particles have attracted extensive interests due to the advantages of being noncontact and noninvasive. In this work, the field distribution of Whittaker-Gaussian (WG) beams propagating through a lens and the radiation force acting on the Rayleigh particle are investigated numerically and theoretically. The results show that the WG beams can trap the particles with both high and low index of refractive near the focus. The influences of optical parameters on the radiation forces are analyzed in detail. Furthermore, the conditions for trapping stability are also discussed. Ó 2018 Elsevier Ltd. All rights reserved.

1. Introduction Optical trapping and manipulating of micron-sized particles have attracted intensive attention since the seminal work by Ashkin who successfully captured a dielectric sphere by using a single laser beam [1]. This work made a pioneering contribution to the development of optical tweezer technology due to the advantages of being noncontact and noninvasive [2–5]. Nowadays, optical tweezer is widely used in biomedical, atomic physics, biophysics and other related fields, which can capture and manipulate various objects, such as atoms [6], molecules [7], DNA molecules [8], and living cells or organelles [9]. As a consequence, optical tweezers have been developed into one of the most promising tools in micromanipulation from trapping to rotating and sorting. It is known that the mechanical action of light on particles is the consequence of exchange of momentum and energy between photons and particles. Qiu et al. also addressed the problems of how the field, the mechanical momentum, the optical force and the stability of the negative force are related to the momentum transfer process [10]. In recent years, many researches have demonstrated that different particles can be trapped and manipulated by using different types of focused laser beams. For example, the particles with refractive index larger than the ambient medium can be trapped by flat-topped beams [11], Lorentz-Gaussian beam [12], Gaussian Shell-model beams [13]. By contrast, particles with refractive index smaller than the ambient medium can be trapped by dark hollow beams [14,15]. In addition, many beams can be used to trap two

⇑ Corresponding authors. E-mail addresses: [email protected] (B. Tang), [email protected] (X. Zhou). https://doi.org/10.1016/j.optlastec.2018.05.026 0030-3992/Ó 2018 Elsevier Ltd. All rights reserved.

types of particles with different refractive index such as bottle beams [16,17], Hermite-cosine-Gaussian beams [18], LaguerreGaussian beams [19,20], partially coherent cosine-Gaussiancorrelated Schell-model beam [21], and beams generated by Gaussian mirror resonator [22]. Also, the trapping characteristics of some other beams such as the radially and azimuthally polarized beams [23,24], Airy beams [25] and pulsed Gaussian beams [26], Bessel beams [27], and gradientless light beams [28] have been explored by using the Rayleigh scattering theory. Recently, a novel kind of beam named Whittaker-Gaussian (WG) beam [29] was introduced by D. Lopez-Mago et al as the general solution of the paraxial wave equation in circular cylindrical coordinates, which constitutes a special case of the general circular beams [30]. Actually, the complex amplitude of the circular beams can be described by either the confluent hypergeometric function or the Whittaker functions, and are characterized by several independent parameters. Up to now, people have studied the quality factor, the kurtosis parameter [29] as well as the influence of atmospheric turbulence on the transmission of orbital angular momentum [31]. However, to our knowledge, the optical radiation forces of the WG beams on Rayleigh dielectric particles have not been reported elsewhere. In this work, the analytical expression for the WG beams propagating through the paraxial ABCD system is derived. The radiation forces of the WG beams acting on a Rayleigh dielectric particle are calculated numerically and theoretically based on the Rayleigh scattering theory. The influences of optical parameters on the radiation forces are analyzed. In addition, the conditions of stable trapping are also discussed under the Rayleigh approximation. The results show that the WG beam can be used to trap and manipulate

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the particles with both high and low refractive index stably on the focus plane. Our results would have potential applications in optical trapping and manipulation. 2. Field distribution of a focused Whittaker-Gaussian beam In the cylindrical coordinate system, the electric field distribution of a Whittaker-Gaussian (WG) beam at the incident plane (z = 0) is given by [29]:



 r0 2 ir 0 2 Eðr 0 ; h0 Þ ¼ exp  2 exp  2 w0 4W 0
 m=2 ir 0 muþ1 ir 0 2 ; ð1Þ  exp ðimh0 Þð ; m þ 1; Þ F 1 2 2W 0 2 1 2W 0 2

where w0 is the beam waist, u and m represent complex continuous radial order and integer angular mode number, respectively. W 0 is a parameter related with the beam and it represents the scale of conP ðaÞn n stant phase. 1F1(a,b,x) = 1 n¼0 n!ðbÞ x is the confluent hypergeometric n

function. In the paraxial approximation, propagation expression of the WG beam passing through the ABCD optical system can be calculated by using the Collins formula, which takes the form as follows [22]:

ik 2pB

Eðr; h; zÞ ¼

Z

1

Z 2p

Eðr 0 ; h0 Þ  0  ik  exp  ½Ar 0 2  2r 0 rcosðh0  hÞ þ Dr 2  r 0 dr0 dh0 ; 2B 1

ð2Þ where k = 2p/k is the wave number, A, B, C and D are the transfer matrix elements of the paraxial optical system. By applying the following integral formula [32]:

Z

1 0





x exp bx J m ðcxÞdx
 m þ u þ 1 1 ¼ cm bðuþmþ1Þ=2 2m1 C C ð m þ 1Þ 1 2
 mþuþ1 c2 ; ; m þ 1;   F1 2 4b u

Z 2p exp 0

2

ð3Þ


 ikqr kqr n cosðu  hÞ exp ðinuÞdu ¼ i 2pJ n exp ðinhÞ; B B ð4Þ

where CðxÞ is Gamma function and Jt ðÞ is the tth-order of the first kind Bessel function. After tedious but straightforward calculations, we can get the analytical formula of the WG beam through the paraxial ABCD optical system as follows

Eðr; h; zÞ ¼



m=2
m ik i kr exp  Dr 2 exp ðimhÞ 2 B 2B B 2W 0 muþ1 n 1 X i 2 n ðbÞðmþnþ1Þ 2m1 Cðm þ n  2 n ðm þ 1Þ ð2W Þ n! n n¼0 0 ! 2 ðkrÞ 1 þ 1ÞC ðm þ 1Þ1 F 1 m þ n þ 1; m þ 1;  2 ; 4B b i

mþ1

k

where b ¼ w1 2 þ 0



Ak 2B

þ 4W1

0

2

A B C D






¼

ðz=f Þs þ f þ z

1=f

1  s=f

 ;

3. Radiation forces of the focused Whittaker-Gaussian beam In this section, we mainly discuss the radiation forces acting on dielectric particles produced by the focused WG beams. For the convenience of calculation, we assume that the particle is a sphere and the radius a of the particle is much smaller than the wavelength of the incident light, i.e., ak. The Rayleigh approximation is applicable under this circumstance. The radiation forces exerting on the particle in the Rayleigh regime include the scattering force and the gradient force. The scattering force F scat is proportional to the light intensity, which can be expressed as [33]:

* * F scat ðr; zÞ ¼ e z nm aIðr; zÞ=c;

ð7Þ

* where e z is the unity vector along the direction of beam propagation, nm is the refractive index of the ambient, c is the speed of the light in vacuum. Iðr; zÞ is the intensity of the focused WG beam, and

a¼ ð5Þ

 i.

z=f

the output plane. In the following calculations, we choose k = 1.06 mm, f = 20 mm, s = 200 mm, u = 3 and the input power of the WG beams is assumed to be 1 W. These values keep unchanged unless otherwise explained. Substituting Eq. (6) into Eq. (5), we can obtain the intensity distribution of a focused WG beam with different value of m and n at the focus plane (z = 0 lm) as shown in Fig. 2, where n is defined as n = w0 =W 0 . From Fig. 2, one can see that there exists a peak for the intensity distribution of the GB beam when m = 0, and the intensity increases with increasing of the ration. However, the intensity distribution of the WG beam takes on a hollow Gaussian-like distribution that has two peaks at the focus plane when m = 1 and m = 2, respectively, as shown in Fig. 2(b) and (c) . In this case, the light intensity decreases with increasing of the ratio n. Due to these special properties of the focused WG beams, one can expect that it is useful for trapping and manipulating the microscopic particles by the focused WG beams.

it is given by Iðr; zÞ ¼ nm e0 cjEðr; zÞj2 =2, and a is defined as the scattering coefficient [33]:

For the apertureless lens system as shown in Fig. 1, the optical matrix between the input plane and the output plane is given by




Fig. 1. Schematic of an apertureless thin lens optical system.

ð6Þ

where s is the distance from the input plane to the lens, f is the focal length of the lens and z is the distance between the focal plane and

128p5 a6 3k

4






r2  1 2 ; r2 þ 2

ð8Þ

where r ¼ np =nm , np denotes the refractive index of the particle. The * gradient force F grad is induced by inhomogeneous optical field, and its direction is along the gradient of light intensity. The gradient * force F grad is given by [33]: By using Eqs. (7)–(9), we can calculate the radiation forces acting on a Rayleigh dielectric sphere produced by the focused WG beam. Without loss of generality, we choose the radius of the particle a = 50 nm, the refractive index of the ambient nm = 1.33 (e.g. water), and the refractive indices of two kinds of particles: np = 1.59 (e.g. glass particles in the water) or np = 1 (e.g. the bubbles

B. Tang et al. / Optics and Laser Technology 107 (2018) 239–243

241

Fig. 2. Intensity distribution of focused WG beams with different values of n at the focus plane. (a) m = 0, (b) m = 1, and (c) m = 2.

in the water) in the following calculations, which always keep unchanged unless otherwise stated. Fig. 3 shows the transverse gradient force of focused WG beams with different values of m and n. Here, the positive value of Fgrad;x * the transverse gradient force means the direction of F grad is along the positive direction of the x-axis. On the contrary, the negative value of Fgrad;x the transverse gradient force signifies the direction * of F grad is along the negative direction of the x-axis. From Fig. 3 (a1), it is clearly seen that there is one stable equilibrium point for the particles with r > 1, and the magnitude of the gradient force increases with increasing of the value of n. Meanwhile, one can see from Fig. 3(b2) and (c2) that there exists one stable equilibrium point for the particles with r < 1. Further, the magnitude of the gradient force decreases with increasing of the value of n. It can be seen from Section 2 that there exists a sharp peak for the focused WG beam when m = 0, which degenerates to the Gaussian beam for trapping particles with high refractive index. When the value of m equals to 1, the WG beam evolves into a hollow beam which can trap particles with low refractive index. With increasing of the value of m, the beam keeps the hollow Gaussian beam-like shape and the magnitudes of optical forces get small gradually. Therefore, we can use the focused WG beam to trap or manipulate

simultaneously the particles with both r > 1 and r < 1 at the focus plane by changing the parameter m. Also, we can improve the capturing stability of the particles with both high and low refractive index by controlling the value of n. To investigate the influences of the optical parameters (e.g. f and np) on the radiation forces, Fig. 4 depicts the evolutions of optical radiation forces of the WG beam when taking different values. From Fig. 4(a) and (b), it can be found that the trapping stability reduces gradually when the value of f changes from 20 mm to 24 mm for both high and low refractive index. Fig. 4(c) and (d) shows the gradient forces acting on particles with different values of the refractive index. One can see that the trapping stability increases with increasing the value of np for the particles with high refractive index. On the contrary, the trapping stability increases with decreasing of the value of np for the particles with low refractive index. 4. Discussion on trapping stability From the above discussion, one can see that the radiation forces of the focused WG beam may be used to trap and manipulate the Rayleigh dielectric spheres. It’s well known that some conditions are required for capturing the particles stably under the Rayleigh

Fig. 3. Gradient forces of the focused WG beams at the focus plane with different values of n.

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Fig. 4. Gradient forces of focused WG beams for different values of np and f at the focus plane.

Fig. 5. Scattering forces of the focused WG beams with different values of n on the focus plane. (a) m = 0, (b) m = 1, and (c) m = 2.

approximation. Firstly, the gradient force must be larger than the scattering force, i.e. R ¼ F grad =jF scat j P 1, where the ratio R is defined as the stability criterion. We give in Fig. 5 the scattering forces of focused WG beams acting on particles with low refractive index np = 1. Compared with Fig. 3, one can easily find that the magnitude of gradient forces is much larger than the gradient forces. Secondly, the gradient force must overcome the effect of Brownian motion. In order to trap the particles stably, the potential well generated by the gradient force is required to overcome the kinetic energy of the particles. This condition can be expressed by using Boltzmann factor [1]: Rthermal ¼ expðU max =kB TÞ  1, 2 2 ! where U max ¼ pnm 2 e0 a3 rr2 1  E represents the maximum of þ2 the potential well, kB is the Boltzmann constant, and the laboratory temperature T takes the value of 300 K. For the particles with np = 1.59, the value of Rthermal on the focus plane is about Rthermal 3.79  1016. By contrast, for the particles with np = 1.00, the value of Rthermal on the focus plane is about Rthermal  1.8  10-3. Thus it can

be seen all the values of Boltzmann factors on the focus plane are extremely small. Therefore, the Brownian motion can be overcome or ignored in our case. Lastly, the gradient force must overcome the gravity of the particle. We can calculate that the order of magnitude of the particle gravity is 1018 N assuming that the density of particles is 2  103 kg/m3. From Fig. 3, we find that the magnitude of the gradient forces is 1012 N. Therefore, the effect of gravity forces of particles is negligible. Above all, the particle can be effectively confined by the focused WG beam. 5. Conclusions In conclusion, we numerically and theoretically studied the radiation force produced by Whittaker-Gaussian (WG) beams acting on a Rayleigh dielectric sphere. The results show that the focused WG beam can trap the particles with both high and low index of refractive at the focus plane. The influences of optical parameters on the radiation forces are also analyzed in detail.

B. Tang et al. / Optics and Laser Technology 107 (2018) 239–243

Meanwhile, the conditions for trapping stability are discussed in this paper. Acknowledgements This Science Science Science

work was sponsored by the Qing Lan Project, Natural Foundation of Jiangsu Province (BK20141169), Natural Foundation of China (11604094, 51501018) and Natural Foundation of Hunan Province (2017JJ2068), China.

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