Trapping two types of particles using a focused partially coherent circular edge dislocations beam

Optics and Laser Technology 97 (2017) 191–197

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Trapping two types of particles using a focused partially coherent circular edge dislocations beam Hanghang Zhang a, Jinhong Li a,⇑, Ke Cheng b, Meiling Duan c,⇑, Zhifang Feng a a

Department of Physics, Taiyuan University of Science and Technology, Taiyuan 030024, China College of Optoelectronic Technology, Chengdu University of Information Technology, Chengdu 610225, China c Department of Physics, North University of China, Taiyuan 030051, China b

a r t i c l e

i n f o

Article history: Received 3 February 2017 Received in revised form 24 June 2017 Accepted 29 June 2017

Keywords: Radiation forces Circular edge dislocations beam Optical trapping

a b s t r a c t A focused partially coherent circular edge dislocations beam used to trap Rayleigh dielectric sphere with different refractive indices is studied. The dependence of radiation forces on the number of circular edge dislocations p, the spatial correlation length r0, relative refractive index nr, and particle radius a are analyzed and illustrated by numerical examples. It is shown that the focused partially coherent circular edge dislocations beam can be used to trap high index of refraction particles at focus F and bright ring R2, and simultaneously to capture low index of refraction particles at dark ring R1. It is much easier to capture the high index of refraction particles at focus F and the low index of refraction particles at dark ring as for the larger number of circular edge dislocations p and the spatial correlation length r0, therefore it is necessary to optimally choose on p and r0 for obtaining an optimal optical guiding. The ranges of the radius for two types of particles stably captured also have been determined. The obtained results are useful for analyzing the trapping efficiency of circular edge dislocations beams applied in micromanipulation technology. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Since Ashkin’s first observation of acceleration and trapping of particles by radiation forces from continuous-wave visible laser in 1970 [1], optical traps and manipulation have attracted intensive attention in lots of literature [2–6] due to their extensive applications in manipulating a wide variety of particles, including neutral atoms and molecules [7–12], microsized dielectric particles [13–15], uniaxial anisotropic sphere [16], metallic particles [17,18], and living biological cells [19–21]. By now, besides the Gaussian beams, many kinds of special beams are studied to trap particles. For example, the Bessel beam having the ability of selfreconstruction has been developed to manipulate particles in multiple axial sites [22]. Liu and Zhao numerically investigated the trapping effect of the focused generalized Multi-Gaussian Schell model beam at the focal plane [23]. Recently, the radiation force of modified circular Airy beams (MCAB) exerted on both a high index of refraction particle and a low index of refraction particle have been analyzed by Jiang et al., and it is shown that the two kinds of particles can be simultaneously stably trapped by MCAB at different positions [24]. In addition, it is noted that some other ⇑ Corresponding authors. E-mail addresses: [email protected] (J. Li), [email protected] (M. Duan). http://dx.doi.org/10.1016/j.optlastec.2017.06.025 0030-3992/Ó 2017 Elsevier Ltd. All rights reserved.

beams such as hollow Gaussian beams [25], Laguerre Gaussian beams [26–28], Lorentz-Gauss beams [29], the bottle beams [30], radially polarized beams [31,32], elegant Hermite-cosine Gaussian beams [33], elegant Hermite-cosh Gaussian beams [34] electromagnetic beams [35] and vortex beams [15,16,36,37] also have been explored. It was shown that laser beams with Gaussian-like intensity profile should be used to trap particles with refractive index np bigger than that nm of the ambient, namely, with relative refractive index nr = np/nm > 1, while laser beams with a hollowlike intensity profile are applicable to trap particles with relative refractive index nr < 1. Optical fields may contain wavefront dislocations, that is to say, points or lines or rings on the wavefront surface where the phase of the optical field is singular and its amplitude vanishes [38–43]. An edge dislocation is the p phase jump in the wave phase located along a line or ring in the transverse plane. According to the shape introduced by Soskin and Vasnetsov [40], there are mainly two types of edge dislocation: linear edge dislocation and circular edge dislocations. Vasnetsov et al. have analyzed how a circular edge dislocation of a wavefront can be created in an interference of two uniaxial Gaussian beams [44]. Phase singularities and spectral changes of higher-order Bessel–Gauss pulsed beams in free space have been studied in the literature [45], it is shown that some circular edge dislocations appear and the spectrum changes on

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H. Zhang et al. / Optics and Laser Technology 97 (2017) 191–197

propagation. Therefore, in terms of the circular edge dislocations beam, as a kind of singular beam owning unique characteristic, it will be significant to study the trapping effect of such beam. This paper is devoted to study how to use a focused partially coherent circular edge dislocations beam to trap two types of particles. In Section 2, the analytical expression for the intensity of partially coherent circular edge dislocations beams via optical system is derived. The dependence of radiation forces on the number of circular edge dislocations p, the spatial correlation length r0 and relative refractive index nr are analyzed and illustrated by numerical examples in Section 3. The condition of the trapping stability is analyzed in Section 4. Finally, Section 5 provides the concluding remarks. 2. Intensity of partially coherent circular edge dislocations beams via optical system The initial field distribution of Laguerre–Gaussian (LG) beam at the source plane z = 0 can be expressed as [46,47]

Eðs; h; 0Þ ¼

pffiffiffi !l     s2 2s l 2s2 exp  expðilhÞ; Lp w0 w20 w20

ð1Þ

where s and h are the radial coordinate and azimuthal (angle) coordinate, respectively. w0 denotes the waist width, Llp denotes the Laguerre polynomial with mode orders p and l. LG beam is a typical mixed circular edge-screw dislocations beam, for l – 0 and p = 0, Eq. (1) reduces to the initial field distribution of screw dislocation beam (vortex beam), l is topological charge of vortex beam; for l = 0 and p – 0, Eq. (1) degenerates to the initial field distribution of circular edge dislocations beam, p is the number of circular edge dislocation; for l = 0 and p = 0, Eq. (1) degenerates to the initial field of fundamental Gaussian beam. Using the relations between Laguerre polynomial and Hermite polynomial [48]

expðilhÞsl Llp ðs2 Þ ¼

Based on the extended Huygens–Fresnel principle [50], under the paraxial approximation, the cross-spectral density function of partially coherent circular edge dislocations beam via the ABCD optical system can be expressed as  2 Z Z Z Z   k ikD 2 Wðq1 ; q2 ; zÞ ¼ ðq1  q22 Þ W 0 ðs1 ; s2 ;0Þ exp  2pB 2z  i ik h 2 2 Aðs1  s2 Þ  2ðq1x s1x þ q1y s1y  q2x s2x  q2y s2x Þ  exp  2z ð5Þ  ds1x ds1y ds2x ds2y ; where q1 = (q1x, q1y) and q2 = (q2x, q2y) denote the position vector at the z plane, k is the wave number related to the wavelength k0 of input laser by k = 2p/k0, A, B, C and D are the transfer matrix elements of the optical system. On substituting from Eq. (4) into Eq. (5), using the relation between the intensity and the cross-spectral density at any point of the output plane Iðq; zÞ ¼ Wðq; q; zÞ, we obtain the intensity of partially coherent circular edge dislocations beams via an ABCD optical system as follow

Iðqx ; qy ; zÞ ¼ E20

pffiffiffi ! pffiffiffi !   p   2sx 2sy ð1Þp X p s2 H exp  : H 2tr 2p2t w0 w0 w20 22p p! t¼0 t ð3Þ

Introducing the Schell model correlator [49], the cross-spectral density function of partially coherent circular edge dislocations beams at the source plane z = 0 is expressed as 

W 0 ðs1 ; s2 ; 0Þ ¼ hEðs1 ; 0Þ  Eðs2 ; 0Þi pffiffiffi ! pffiffiffi !   p X p  X p p 1 2s1x 2s2x 2 ¼ 4p E H2t1 H2t2 2 0 w w t t 0 0 2 ðp!Þ 1 2 t 1 ¼0t 2 ¼0 pffiffiffi ! pffiffiffi !  2  s þ s2 2s1y 2s2y  H2p2t1 H2p2t2 exp  1 2 2 w0 w0 w0 " # 2 ðs1  s2 Þ ;  exp  2r20 ð4Þ

r0 is the spatial correlation length.

2pB

  p X p  X p p Q 1Q 2; 2 4p t t 2 ðp!Þ t1 ¼0t2 ¼0 1 2 1

ð6Þ

d

½ 2 t1 X 2t 2 X X

Q1 ¼

2t2

ðd

ð2t 1 2c1 þd2c2 Þ t 2

Þð1Þc1 þc2 ð2iÞ

2

c1 ¼0 d¼0 c2 ¼0

 t2 ð2t 1 Þ! d! p 2  pffiffiffiffiffiffiffi 1  2 c1 !ð2t1  2c1 Þ! c2 !ðd  2c2 Þ! w0 M 1 M1 pffiffiffi!2t1 2c1 2  d2c2  d2c 2 2 2 2 2  1 w0 M1 w0 w0 M1 r20 "  ,sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#   1 ikqx 2 iGx  H2t2 d  1 2 H2t1 2c1 þd2c2 pffiffiffiffiffiffiffi w0 M1 B w0 M 1 2 M2 " # !   2 2 2t 2c þd2c2 þ1 1 ikqx Gx  1 12 exp exp ; ð7aÞ  M2 4M 1 B 4M 2 

ð2Þ

Eðs; z ¼ 0Þ ¼ E0

2

k

where

   p l l ð1Þp XX r p H2tþlr ðsx ÞH2p2tþr ðsy Þ; i t r 22pþl n! t¼0 r¼0

   p l with Hn being the Hermite polynomial of order n, being t r binomial coefficients, the initial field distribution of circular edge dislocations beam can be expressed in following alternative form in Cartesian coordinates



½ 21   ½pt X2 X X1 2p2t d

Q2 ¼

e1 ¼0 d1 ¼0 e2 ¼0

2p  2t 2 d1



ð2p2t1 2e1 þd1 2e2 Þ t 2 p

ð1Þe1 þe2 ð2iÞ

2

 pt2 ð2p  2t 1 Þ! d1 ! p 2  pffiffiffiffiffiffiffi 1  2 e1 !ð2p  2t 1  2e1 Þ! e2 !ðd1  2e2 Þ! w0 M 1 M1 pffiffiffi!2p2t1 2e1 2  d1 2e2  d1 2e 2 2 2 2 2  1 w0 M 1 w0 w0 M1 r20 "     sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# ikqy 1 2 iGy  H2p2t2 d1  H2p2t1 2e1 þd1 2e2 pffiffiffiffiffiffiffi = 1 2 w0 M1 B w0 M 1 2 M2 " !  2 # 2 2p2t 1 2e1 þd1 2e2 þ1 Gy ikqy 1  2 exp ; ð7bÞ  M2 exp B 4M2 4M1 

M1 ¼

1 1 ikA þ  ; w20 2r20 2B

ð7cÞ

M2 ¼

1 1 ikA 1  þ þ ; w20 2r20 2B 4M1 r40

ð7dÞ

Gx ¼

ikqx ikqx :  B 2M 1 Br20

ð7eÞ

Due to the symmetry, Gy is obtained by replacing qx with qy in Gx. Let us now consider the partially coherent circular edge dislocations beam passing through a thin lens as shown in Fig. 1. The transfer matrix for this optical system is give by

193

sy(mm)

ρy(μm)

H. Zhang et al. / Optics and Laser Technology 97 (2017) 191–197

ρx(μm)

sx(mm) Fig. 1. Schematic of the focusing optical system.



A

B

C

D



 ¼¼

1 z1 0

1



1

0

1=f

1



 ¼

 z ; 1

z1 =f 1=f

ð8Þ

where z = z1 + f is the axial distance from the input plane to the output (reference) plane, f is the focal length of the thin lens, z1 is the axial distance from the focus plane to the output plane and whose value is negative (or positive) when the particle located on the left (right) side of the focal plane. Substituting Eq. (8) into Eqs. (6) and (7), we could obtain the intensity distribution of a partially coherent circular edge dislocations beam through a lens optical system. In our calculation, the values of the parameters are taken as E20 = 2  1012 V2/m2, k0 = 632.8 nm, f = 5 mm and w0 = 0.5 mm, unless otherwise stated in the text. Fig. 1 gives the schematic of the focusing optical system for partially coherent circular edge dislocations beam. Left and right plot is the intensity distributions of the partially coherent circular edge dislocations beam with p = 1 at the input and focal plane, respectively. The point F is focus of the thin lens. Fig. 1 displays that the intensity distribution of the partially coherent circular edge dislocations beam has an intense intensity maximum at the centre and a dark ring near the centre at the focal plane. Owing to this special focus characteristic, we believe that the partially coherent circular edge dislocations beam has the ability to trap particles with different refractive indices at the focal plane.

Fig. 2 gives the intensity distribution at the focal plane of the focused partially coherent circular edge dislocations beam for the different number of circular edge dislocation p and spatial correlation length r0. The calculation parameters are p = 1 in Fig. 2(a), p = 3 in Fig. 2(b), p = 0, 1, 3 in Fig. 2(c) and r0 = 0.5, 1, 10 mm in Fig. 2(d). Fig. 2(a) shows that there exists central intensity maximum and one dark ring near the centre intensity distribution for partially coherent circular edge dislocations beam with p = 1 at the focal plane, owing to this special focusing characteristic, we believe that the partially coherent circular edge dislocations beam can trap particles with high index of refraction at the focus and those with low index of refraction at the dark ring. From Fig. 2(b) we can see that intensity distribution has a central intensity maximum and three dark rings for partially coherent circular edge dislocations beam with p = 3, thus it can be seen that the number of dark rings equals to the number of circular edge dislocation. Fig. 2(c) indicates that the central intensity distribution becomes more and more sharp as p increases, when p = 0, intensity distribution only takes on central intensity maximum and no dark rings, i.e., isn’t able to trap particles with low index of refraction for partially coherent circular edge dislocations beam with p = 0 at focal plane. From Fig. 2(d) we find that the central intensity maximum increases as spatial correlation length r0 increases. The two intensity minimum increases as r0 decreases, and when correlation

(a)

ρy(μm)

ρy(μm)

(b)

ρx(μm)

ρx(μm)

5

0 -6

σ0=0.5mm

(d)

p=0 p=1 p=3

Intensity 1016V2/m2

Intensity 1016V2/m2

(c) 10

σ0=1mm

10

σ0=10mm 5

0 -4

-2

0

ρx μm

2

4

6

-4

-2

0

2

4

ρx μm

Fig. 2. Intensity distributions of focused partially coherent circular edge dislocations beam at the focal plane.

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H. Zhang et al. / Optics and Laser Technology 97 (2017) 191–197

length r0 = 0.5 mm, the two intensity minimum vanish, i.e., the dark rings vanish, therefore, isn’t able to trap particles with low index of refraction as forr0  0.5 mm.

a  k. In this case, the particle could be treated as a point dipole, and the Rayleigh scattering model is used to calculate the radiation forces on the particle of focused partially coherent circular edge dislocations beam. There are two kinds of radiation forces: scattering force and gradient force. According the Rayleigh scattering theory, the scattering force and gradient force are expressed as [14]

3. Radiation forces produced by the focused partially coherent circular edge dislocations beam

FScat ¼ ez

We assume that the particle is Rayleigh dielectric sphere, its radius a is sufficiently smaller than the wavelength of light k, i.e.,

 2 128nm p5 a6 n2r  1 Iðqx ; qy ; zÞ; n2r þ 2 3ck4

8 4

z1=0

(a)

-2

0

2

ρx μm

FGrad,x 10 PN

2

ρy

(c)

0 -4 ρ x=-2.468μm

ρx

-4

ρx=1.426μm

-2

0

ρx μm

2

4

nr>1 ρx=0μm ρx=2.468μm

(d)

10

F

nr<1

-8

4

FGrad,z(PN)

FScat(PN)

2

-4

nr>1

4

nr<1

0

z1=0

(b) nr>1

ð9aÞ

0

-10 R1=1.426μm

(e)

3

nr>1 ρx=0μm ρx=2.468μm

10

FGrad,z(PN)

FGrad,z(PN)+FScat(PN)

-100

R2=2.468μm

0

-10 -100

-50

0

z1 μm

(f)

50

100

nr<1 ρx=1.426μm

0

-3 -50

0

50

z1 μm

FGrad,z+FScat(PN)

3

10

-100

-50

0

z1 μm

50

100

nr<1 ρx=1.426μm

(g)

0

-3 -100

-50

0

50

10

z1 μm Fig. 3. Radiation forces of focused partially coherent circular edge dislocations beam on the high index of refraction particles (nr > 1) and low index of refraction particles (nr < 1) (a) FScat at the focal plane, (b) FGrad,x at the focal plane, (c) stable equilibrium regions at the focal plane, (d) FGrad,z at the points of qx = 0 and 2.468 lm, (e) FGrad,z + FScat at the points of qx = 0 and 2.468 lm, (f) FGrad,z at the point of qx = 1.426 lm, (g) FGrad,z + FScat at the point of qx=1.426 lm.

195

H. Zhang et al. / Optics and Laser Technology 97 (2017) 191–197

FGrad

  2pnm a3 n2r  1 ¼ rIðqx ; qy ; zÞ; n2r þ 2 c

of refraction particles in R2 ring, similarly the particle is forced to shift to the R1 ring. Fig. 3(d) and (e) shows that the direction of the axial radiation force FGrad,z + FScat points to the Focus F and the R2 ring for high index of refraction particles near to the Focus F or the R2 ring. Therefore, Focus F and R2 ring are the stable equilibrium regions, high index of refraction particles can be stably captured in these regions. From Fig. 3(f) and (g), we know that the direction of the axial radiation force points to the R1 ring for low index of refraction particles near to the R1 ring. i.e., R1 ring is an equilibrium ring for low index of refraction particles, the particles can be stably trapped at R1 ring. Therefore the focused partially coherent circular edge dislocations beam can simultaneously trap or manipulate two kinds of particles. The effect of the number of circular edge dislocations p and the spatial correlation length r0 of the focused partially coherent circular edge dislocations beams on the radiation forces exerted on the high index of refraction particles are plotted in Fig. 4, the calculation parameters are the same as Fig. 3. From Fig.4(a) and (b) it is found that as the number of circular edge dislocations p increases, the transverse gradient force FGrad,x will increase, the trapping equilibrium region will narrow, and the axial radiation forces FGrad,z + FScat is almost not changed. From Fig. 4(c) and (d) we see that both the transverse gradient force FGrad,x and the axial radiation forces FGrad,z + FScat will increase with the increment of the spatial correlation length r0. Therefore, the bigger the number of circular edge dislocations p is, the bigger the spatial correlation length r0 is, the better the trapping stability of the high index of refraction particles at the focus. Fig. 5 depicts the influence of the number of circular edge dislocations p and the spatial correlation length r0 of the focused partially coherent circular edge dislocations beams on the radiation forces exerted on the low index of refraction particles, the calculation parameters are the same as Fig. 3. From Fig. 5(a) and (b) it can be found that the partially coherent circular edge dislocations beam with p = 0 isn’t able to trap particles with low index of

ð9bÞ

where ez is a unity vector along with beam propagation, nr = np/nm specifies the relative refractive index, np and nm denote refractive indices of the particle and ambient, respectively, c is the speed of light in a vacuum, a is radius of the particle, and k = k0/nm. It can be found that scattering force FScat is along the direction of beam propagation and is proportional to the beam intensity, while FGrad is along the gradient of the light intensity. It is well known that the gradient force including transverse gradient force FGrad,x (FGrad,y) along the x-y plane and axial gradient force FGrad,z along the z direction. The changes of the radiation forces of the focused partially coherent circular edge dislocations beam acting on the particle with high index of refraction (nr > 1) and low index of refraction particle (nr < 1) at the focus (F point), the sites of intensity minimum (R1 ring) and intensity maximum (R2 ring) are depicted in Fig. 3, p = 1, r0 = 10 mm, nm = 1.332 (water), np1 = 1 (air bubble), np2 = 1.592 (glass), and other calculation parameters are the same as in Fig. 2. From Figs. 3(a) and 2(d) we see that FScat is proportional to the beam intensity and the biggest scattering force acted on particle is at the focus, the physical reason can be explained from Eq. (9a) evidently. Note that the sign of the gradient force denotes the direction of the force: positive (or negative) FGrad,x represents that the direction of transverse gradient force is along the +x (or x) direction; for the positive (or negative) FGrad,z, it means that the axial gradient force is along the +z (or z) direction. From Fig. 3 (b) and (c), we can find that the direction of the transverse gradient force is towards the Focus F for high index of refraction particles in R1 ring at the focal plane, thus the particle is guided to move towards the Focus F; the direction of the transverse gradient force points to the R2 ring for high index of refraction particles outside R1 ring, the particle is leaded to shift to the R2 ring; the direction of the transverse gradient force points to the R1 ring for high index

8

2

0 -4 -8

-4

-2

0 ρx μm

2

2

FGrad,x 10 PN

0 -3

-4

-2

0

ρ x μm

p=0 p=1 p=3

0

-10 -100

4

σ0=0.5mm σ0=1mm σ0=10mm

z1=0

3

-6

10

-50

0

50

100

z1 μm

6

(c)

ρx=0

(b) FGrad,z+FScat(PN)

4 FGrad,x 10 PN

p=0 p=1 p=3

z1=0

2

4

FGrad,z+FScat(PN)

(a)

10

ρx=0 σ0=0.5mm σ0=1mm σ0=10mm

(d)

0

-10 -100

-50

0

50

100

z1 μm

Fig. 4. Effect of (a) (b) the number of circular edge dislocations and (c) (d) the spatial correlation length on the radiation forces exerted on the high index of refraction particles. (a) and (c) FGrad,x at the focal plane, (b) and (d) FGrad,z + FScat at the focus.

H. Zhang et al. / Optics and Laser Technology 97 (2017) 191–197

10

(a)

5

2

FGrad,x 10 PN

4

p=0 p=1 p=3

z1=0

FGrad,z+FScat(PN)

196

0 -5

ρx=0.921μm

ρx=-1.426μm

-10

-4

-2

0

2

p=3, ρx=0.921μm

2 0 -2 -4

4

p=1, ρx=1.426μm

(b)

-100

-50

0

ρx μm σ0=0.5mm σ0=1mm σ0=10mm

z1=0

0 -3

ρx=1.426μm ρx=-1.626μm

-6

-4

-2

0

2

4 FGrad,z+FScat(PN)

(c)

3

2

FGrad,x 10 PN

6

50

100

z1(μm) σ0=1mm,

(d)

ρx=1.626μm

2 0 σ0=10mm,

-2

ρx=1.426μm

-4

4

-100

-50

0

ρx μm

50

100

z1(μm)

Fig. 5. Effect of (a) (b) the number of circular edge dislocations and (c) (d) the spatial correlation length on the radiation forces exerted on the low index of refraction particles. (a) and (c) FGrad,x at the focal plane, (b) and (d) FGrad,z + FScat at the different points of qx.

0

0

10

(a)

(b)

-10

10

-20

10

m

-14

10

m

FGrad,x

FGrad,z

FScat

FB

-18

-30

10

10

0

0

2

10

4

Magnitude Force(N)

Magnitude Force(N)

10

-10

10

-20

10

30

40

a(nm)

m

m

FGrad,x

FGrad,z

FScat

FB

-18

-30

20

-14

10

10

10

0

0

2

10

4

20

30

40

a(nm)

m Fig. 6. Comparisons of the variations of Fm Grad;x , FGrad;z , FScat and F B versus the particles’ radius under the condition p = 1 and r0 = 10 mm. (a) nr > 1, (b) nr < 1.

refraction, which is consistent with the conclusion come to in Fig. 2 (c). As the p increases, the FGrad,x will increase and the FGrad,z + FScat is nearly not changed. From Fig. 5(c) and (d) we see that the focused partially coherent circular edge dislocations beam isn’t able to trap particles with low index of refraction when r0  0.5 mm, which is consistent with the conclusion obtained from Fig. 2(d). The FGrad,x and the FGrad,z + FScat will increase with increasing the r0. Fig. 5(a) and (c) infer that the positions at which the particles of low index of refraction are stably trapped will be different for the different p and r0. Therefore, it will become much easier to capture the particles at the intensity minimum in the focal plane as for the larger values of p and r0. 4. Analysis of trapping stability In the above analysis, it is shown that the radiation force produced by the focused partially coherent circular edge dislocations beams can be used to trap Rayleigh dielectric particles with different refractive indices. Under the Rayleigh approximation the axial gradient force must be greatly larger than scattering force in order

to ensure that the single-beam gradient force trap particles stably, i.e. satisfying R = |FGrad,z|/|FScat| 1, and this ratio R is termed the stability criterion. Another factor strongly affecting the trapping stability is the Brownian motion when the particles are very small. To trap the particle, the gradient force should overcome the Brownian force Fb = (12pgakbT)1/2 [51], where g is viscosity of the ambient and g = 7.977  104 pa s for water at the temperature T = 300 K, kb denotes the Boltzman constant, and a is the radius of the particle. The magnitudes of all the forces including the maximum transm verse gradient force Fm Grad;x , the maximum axial gradient force FGrad;z , the maximum scattering force FScat and the Brownian force FB versus the particles’ radius for nr > 1 and nr < 1 are represented in Fig. 6, respectively, the calculation parameters are the same as Fig. 3. From Fig. 6(a) it is found that when a < 0.2 nm the Brownian force is larger than the axial gradient force, on the other hand, when a > 20.6 nm the scattering force is larger than the axial gradient force, therefore, high index of refraction particles whose radius is in the range of 0.2 nm < a < 20.6 nm can be stably trapped at the focus under the condition that p = 1 and r0 = 10 mm.

H. Zhang et al. / Optics and Laser Technology 97 (2017) 191–197

Likewise, from Fig. 6(b) we see that the low index of refraction particles with 0.4 nm < a < 11.5 nm can be stably captured at the R1 ring at the focal plane (Fig. 3(c)). 5. Conclusions In this paper, by using the extended Huygens–Fresnel principle and Rayleigh scattering regime, the analytical expressions for the intensity and radiation forces of focused partially coherent circular edge dislocations beams have been derived, and used to study the optical trapping effect of focused partially coherent circular edge dislocations beams acting on dielectric sphere with different refractive indices. It is shown that the radiation forces acting on the particle depend on the number of circular edge dislocations p, the spatial correlation length r0 and relative refractive index nr. The focused partially coherent circular edge dislocations beam can be used to trap high index of refraction particles at focus F and bright ring R2, and simultaneously capture low index of refraction particles at dark ring R1. As the number of circular edge dislocations p increases, the transverse gradient force FGrad,x increase both for high index of refraction particles at focus F and for low index of refraction particles at dark ring R1, whereas the axial radiation forces FGrad,z + FScat is almost not changed. With increasing the spatial correlation length r0, the radiation force FGrad + FScat will increase for high index of refraction particles at focus F and low index of refraction particles at dark ring R1. Therefore, it is much easier to capture the high index of refraction particles at focus F and the low index of refraction particles at dark ring (the minimum intensity) as for the larger values of p and r0. The limits of the radius for two types of particles stably captured are determined. The results obtained in this paper will provide the valuable information for trapping and manipulating the Rayleigh particles using circular edge dislocations beams, which may be applied in biotechnology, nanotechnology and so on. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Nos. 61405136 and 61505075), and the Applied Basic Research Foundation of Shanxi Province, China (Grant No. 201601D021019). References [1] A. Ashkin, Phys. Rev. Lett. 24 (1970) 156. [2] A. Ashkin, Biophys. J. 61 (1992) 569. [3] J.P. Home, D. Hanneke, J.D. Jost, D. Leibfried, D.J. Wineland, New J. Phys. 13 (2011) 073026.

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[4] M. Koch, A. Rohrbach, Nat. Photon. 6 (2012) 680. [5] M. Woerdemann, C. Alpmann, M. Esseling, C. Denz, Laser Photonics Rev. 7 (2013) 839. [6] S. Xia, D. Song, Y. Zong, L. Tang, Z. Chen, Opt. Express 23 (2015) 4397. [7] A. Ashkin, Phys. Rev. Lett. 40 (1978) 729. [8] S. Chu, J.E. Bjorkholm, A. Ashkin, A. Cable, Phys. Rev. Lett. 57 (1986) 314. [9] M.D. Wang, M.J. Schnitzer, H. Yin, R. Landick, J. Gelles, S.M. Block, Science 282 (1998) 902. [10] A.D. Mehta, M. Rief, J.A. Spudich, D.A. Smith, R.M. Simmons, Science 283 (1999) 1689. [11] N. Calander, M. Willander, Phys. Rev. Lett. 89 (2002) 143603. [12] D.D. Yavuz, P.B. Kulatunga, E. Urban, T.A. Johnson, N. Proite, T. Henage, T.G. Walker, M. Saffman, Phys. Rev. Lett. 96 (2006) 063001. [13] A. Ashkin, J.M. Dziedzic, J.E. Bjorkholm, S. Chu, Opt. Lett. 11 (1986) 288. [14] Y. Harada, T. Asakura, Opt. Commun. 124 (1996) 529. [15] M. Šiler, P. Jákl, O. Brzobohaty´, P. Zemánek, Opt. Express 20 (2012) 24304. [16] T. Qu, Z. Wu, Q. Shang, Z. Li, L. Bai, L. Gong, J. Quant. Spectrosc. Radiat. Transf. 162 (2015) 103. [17] Q. Zhan, Opt. Express 12 (2004) 3377. [18] M. Dienerowitz, M. Mazilu, P.J. Reece, T.F. Krauss, K. Dholakia, Opt. Express 16 (2008) 4991. [19] J. Dai, M.P. Sheetz, Biophys. J. 68 (1995) 988. [20] F. Mao, Q. Xing, K. Wang, L. Lang, Z. Wang, L. Chai, Q. Wang, Opt. Commun. 256 (2005) 358. [21] Y. Chang, L. Hsu, S. Chi, Opt. Commun. 246 (2005) 97. [22] V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, K. Dholakia, Nature 419 (2002) 145. [23] X. Liu, D. Zhao, Opt. Commun. 354 (2015) 250. [24] Y. Jiang, Z. Cao, H. Shao, W. Zheng, B. Zeng, X. Lu, Opt. Express 24 (2016) 18072. [25] C. Zhao, L. Wang, X. Lu, Phys. Lett. A 363 (2007) 502. [26] C. Zhao, Y. Cai, Opt. Lett. 36 (2011) 2251. [27] H. Chai, L. Wang, Micron 43 (2012) 887. [28] H. Yu, W. She, Phys. Rev. A 92 (2015) 023844. [29] Y. Jiang, K. Huang, X. Lu, Opt. Express 19 (2011) 9708. [30] C. Chen, P. Tai, W. Hsieh, Appl. Opt. 43 (2004) 6001. [31] Y. Zhang, B. Ding, T. Suyama, Phys. Rev. A 81 (2010) 023831. [32] Y. Zhuang, Y. Zhang, B. Ding, T. Suyama, Opt. Commun. 284 (2011) 1734. [33] Z. Liu, D. Zhao, Opt. Express 20 (2012) 2895. [34] Z. Liu, K. Huang, D. Zhao, Opt. Lasers Eng. 51 (2013) 761. _ [35] I. Bialynicki-Birula, T. Radozycki, Phys. Rev. A 93 (2016) 063402. [36] A.S. Reyna, G. Boudebs, B.A. Malomed, C.B. de Araújo, Phys. Rev. A 93 (2016) 013840. [37] J. Shu, Z. Chen, J. Pu, J. Opt. Soc. Am. A 30 (2013) 916. [38] J. Nye, M. Berry, Proc. R. Soc. Lond. A 336 (1974) 165. [39] G. Indebetouw, J. Mod. Opt. 40 (1993) 73. [40] M.S. Soskin, M.V. Vasnetsov, Prog. Opt. 42 (2001) 219. [41] J. Li, J. Zeng, M. Duan, Opt. Express 23 (2015) 11556. [42] J. Li, W. Wang, M. Duan, J. Wei, Opt. Express 24 (2016) 20413. [43] J. Li, J. Zeng, Opt. Commun. 383 (2017) 341. [44] M. Vasnetsov, V. Gorshkov, I. Marienko, M. Soskin, Opt. Spectrosc. 88 (2000) 260. [45] C. Ding, L. Pan, B. Lü, J. Opt. Soc. Am. A 26 (2009) 2654. [46] E. Zauderer, J. Opt. Soc. Am. A 3 (1986) 465. [47] J. Qu, Y. Zhong, Z. Cui, Y. Cai, Opt. Commun. 283 (2010) 2772. [48] I. Kimel, L.R. Elias, IEEE J. Quantum Elect. 29 (1993) 2562. [49] M. Zahid, M.S. Zubairy, Opt. Commun. 70 (1989) 361. [50] L.C. Andrews, R.L. Phillips, Laser Beam Propagation through Random Media, SPIE, Bellingham, 2005. [51] K. Okamoto, S. Kawata, Phys. Rev. Lett. 83 (1999) 4534.