Radiation force of abruptly autofocusing Airy beams on a Rayleigh particle

Yunfeng Jiang; Kaikai Huang; Xuanhui Lu

doi:10.1364/OE.21.024413

1. Introduction

The circular Airy beam (CAB) has attract intensive interest due to its unique abruptly autofocusing property [1–3]. This beam abruptly focuses its energy right before the focal point while maintaining a low intensity profile until the very point. This feature is very useful in biomedical treatment, for only the intended area would be affected in the propagation direction. Up to now, the CAB with optical vortices [4, 5], and the radially polarized CAB [6] are proposed and observed in experiment. In optical trapping, the CAB may produce a greater gradient force on the particle, because the abruptly autofocusing property means a great intensity gradient in the focal region. The CAB has been used to trap and guide the micro particles in experiment [7]. But the advantages or the particularities of the optical trapping through CAB have not been detailedly analyzed.

Nowadays, the optical tweezers have been widely applied in many fields, since they are first invented in 1986 by Ashkin [8]. It now becomes an important tool to investigate the biological cells, DNA molecules, neutral atoms, and other particles [9–13]. The conventional optical tweezers are generally constructed with fundamental Gaussian beams (GB). Because of the gradient force of the Gaussian beam, the particle would be stably trapped at the beam waist [14]. Many researches show that other beams also can be used in optical manipulation [15]. For example, the Laguerre Gaussian beam can be used to rotate the particle [16]; the Bessel Gaussian beam can trap particles in multiple planes [17]; the Airy beam can be used to transversely clear the particles [18]. Besides, the radiation force of the cylindrical vector beam [19], Gaussian Schell model beam [20], Lorentz-Gaussian beam [21] and other beams [22–24] on the micro particles have also been studied in theory or in experiment. To our best knowledge, the radiation force of CAB on the Rayleigh particle has not been studied before.

In this paper, we investigate the propagation of the CAB under paraxial condition and non-paraxial condition. Because in most situations the light beams must be small enough to trap particles. So it is necessary to know whether the paraxial propagation expression is satisfied when the beam size is very small. The distributions of radiation forces on the Rayleigh particle demonstrate the unique trapping properties of CAB. Similar trapping can also be achieved with generalized abruptly autofocusing beams [25]. By analyzing the optical trap of CAB with different parameters and comparing it with that of the GB, some interesting and useful results are found in our investigations.

2. Non-paraxial propagation and paraxial propagation of CAB

The electric field of the circular airy beam (CAB) at the input plane is defined as [1]:

E (r, z = 0) = A i (\frac{r_{0} - r}{s}) \exp (a \frac{r_{0} - r}{s}),

where Ai(x) is the Airy function, r₀ is a parameter related with the radius of the beam at the initial plane, s is the radial scale, a is the decaying parameter. When r₀ is large enough, the CAB shows abruptly autofocusing properties while propagating in free space. The abruptly autofocuisng property is originated from the character of the Airy function. The radius of the CAB follows a parabolic trajectory and the intensity suddenly increases at the focus point.

At first, we’d like to investigate the non-paraxial effect on the abruptly autofocusing property of the CAB. For simplicity, we assume the incident light is linear polarized in the x-direction. The Rayleigh-Sommerfeld diffraction formula can be applied to analyze the propagation of the CAB in the free space. The electric field can be expressed as [26]:

\overset{⇀}{E} (r, φ, z) = E_{x} (r, z) \hat{x} + E_{z} (r, φ, z) \hat{z},

where E_x is not related with

φ

in our case, and it can be computed in terms of the Hankel transform pair [27]:

E_{x} (r, z) = 2 π \int_{0}^{\infty} \tilde{g} (k) J_{0} (2 π k r) e^{2 i π z k_{z}} k d k,

\tilde{g} (k) = 2 π \int_{0}^{\infty} E (r, 0) J_{0} (2 π k r) r d r,

where

λ

is the wavelength, k is the spatial frequency,

k_{z} = \sqrt{λ^{- 2} - k^{2}}

. The expression of E_z in cylindrical coordinates is [26]:

\begin{array}{l} E_{z} (r, φ, z) = - \int_{0}^{2 π} \int_{0}^{\infty} \frac{k \cos θ}{k_{z}} \tilde{g} (k) e^{i 2 π [k_{z} z + k r \cos (θ - φ)]} k d k d θ \\ = - \int_{0}^{\infty} \frac{k^{2}}{2 k_{z}} \tilde{g} (k) e^{i 2 π k_{z} z} (\int_{0}^{2 π} e^{i 2 π k r \cos (θ - φ) + i θ} d θ + \int_{0}^{2 π} e^{i 2 π k r \cos (θ - φ) - i θ} d θ) d k \\ = - i 2 π \int_{0}^{\infty} \frac{k^{2}}{k_{z}} \tilde{g} (k) e^{i 2 π k_{z} z} J_{1} (2 π k r) d k \cdot \cos φ, \end{array}

where the integral representation of Bessel function,

J_{n} (t) = \frac{i^{- n}}{π} \int_{0}^{π} e^{i t \cos θ} \cos (n θ) d θ,

is used [28]. The numerical computation of Eq. (3)-(5) is complicated, and some numerical methods could be used [29, 30]. But under paraxial approximation, we can obtain an analytic expression for the propagation of the CAB at the beam center (r = 0) which is useful in calculating the distribution of the radiation force along the beam axis.

Under the paraxial approximation, the component of E_z in Eq. (2) can be neglected. The propagation of the CAB can be calculated from the Collins formula [31]:

E (r, z) = \frac{2 i π}{λ z} \exp (- \frac{i π r^{2}}{2 λ z}) \int_{0}^{\infty} A i (\frac{r_{0} - r^{'}}{s}) \exp (a \frac{r_{0} - r^{'}}{s}) \exp (\frac{- i π {r^{'}}^{2}}{2 λ z}) J_{0} (\frac{2 π r r^{'}}{λ z}) r^{'} d r^{'},

where r and r′ are coordinates in the output plane and the input plane, respectively. When r₀ islarge enough, the integral of Eq. (6) can be extended to

- \infty

without affecting the result. So the electric field in the beam center can be expressed as follows:

E (0, z) = \frac{2 i π}{λ z} \int_{- \infty}^{\infty} A i (\frac{r_{0} - r^{'}}{s}) \exp (a \frac{r_{0} - r^{'}}{s}) \exp (\frac{- i π {r^{'}}^{2}}{λ z}) r^{'} d r^{'} .

Applying the integral formulae of the Airy transform [32],

φ_{β} (y) = \frac{1}{β} \int_{- \infty}^{\infty} e^{- x^{2}} A i (\frac{y - x}{β}) d x = \frac{\sqrt{π}}{β} \exp [\frac{1}{4 β^{3}} (y + \frac{1}{24 β^{3}})] A i (\frac{y}{β} + \frac{1}{16 β^{4}}),

\frac{1}{β} \int_{- \infty}^{\infty} x e^{- x^{2}} A i (\frac{y - x}{β}) d x = y φ_{β} (y) + α^{3} {φ^{″}}_{β} (y),

we can obtain:

E (0, z) = \sqrt{π} \exp (P_{1}) [A i (P_{2}) (- \frac{1}{4 s^{3} c^{3 / 2}} - b c^{1 / 2}) - \frac{A i^{'} (P_{2})}{s c^{1 / 2}}] \exp (- c b^{2} + \frac{a r_{0}}{s}),

where

A i^{'} (x)

is the derivative of the Airy function,

b = - \frac{i λ a z}{2 π s},

c = \frac{i π}{λ z},

\begin{array}{l} P_{1} = \frac{r_{0} + b}{4 c s^{3}} + \frac{1}{96 c^{3} s^{6}}, \\ P_{2} = \frac{r_{0} + b}{s} + \frac{1}{16 c^{2} s^{4}} . \end{array}

Now we can compare the non-paraxial results calculated by Eq. (2) with the paraxial results calculated by Eq. (10). In our calculation, we assume a = 0.08, $λ$ = 1064 nm, r₀ = 10s, $ξ = λ z / 2 π s^{2}$ , the incident power is 1W. The parameter s is usually considered as a proportional coefficient. It does not affect the beam shape or the propagation characteristics of the CAB under the paraxial approximation,as we can see in Fig. 1. As Fig. 1(a) shows, when s = 5 $μ m$ , the paraxial result calculated by Eq. (10) and the non-paraxial result calculated by Eq. (2) agree quite closely. The CAB shows abruptly autofocusing property at about $ξ = 5.8$ , and the intensity reaches its maximum value at $ξ = 6.6$ . When s is too small, the paraxial result does not agree with the non-paraxial result, as we can see in Fig. 1(b). When s = 1 $μ m$ , the non-paraxial result shows that the abruptly autofocusing property of the CAB along the beam axis becomes weak, which indicates that the beam size of the abruptly autofocusing Airy beam is generally limited in practice.

Fig. 1 Comparison about the intensity of the CAB in the beam center of the paraxial and the non-paraxial propagation results. (a) s = 5 $μ m$ , (b) s = 1 $μ m$ .

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2. Radiation force of CAB exerted on a Rayleigh particle

As is well known, the Rayleigh dielectric particle can be considered as a point dipole in the light fields.And the polarisability $α$ is [33]:

α = 4 π R^{3} \frac{ε_{p} - ε_{m}}{ε_{p} + 2 ε_{m}},

where R is the radius of the particle,

ε_{p}

and

ε_{m}

are dielectric functions of the particle and the medium surrounding the particle, respectively. So the gradient force

{\overset{⇀}{F}}_{g}

and the scattering force

F_{s}

can be calculated by [14, 34]:

{\overset{⇀}{F}}_{g} = \frac{1}{4} ε_{0} ε_{m} Re (α) \nabla | {\overset{⇀}{E}}^{2} |,

F_{s} = \frac{ε_{0} ε_{m}^{3} k_{0}^{4}}{12 π} | α^{2} | | {\overset{⇀}{E}}^{2} |,

where

ε_{0}

is the dielectric constant in vacuum, k₀ is the vacuum wave number.

In our calculation, we choose s = 5 $μ m$ . So the paraxial expression of Eq. (10) can be used to calculate the longitudinal radiation force of the CAB at the beam center. However, Eq. (2) should also be used when we want to calculate the transverse radiation force of CAB. We assume R = 20 nm, the refractive index of the surrounding medium n_m = 1.33 (i.e., water). Two kinds of particles are discussed below, whose refractive index is n_p = 1.59 (i.e., glass micro particle) or n_p = 1.00 (i.e., air bubble), so $ε_{p} = n_{p}^{2}$ , $ε_{m} = n_{m}^{2}$ .

Figure 2 shows the distributions of the longitudinal and transverse radiation force when n_p = 1.59. In order to understand the advantages of the CAB over the GB in optical trapping, we assume the initial radius (i.e., r₀) and the focal length (i.e., z_f, the distance between the initial plane and the waist plane) of the GB are the same as the CAB. The scattering force F_s is directed to the propagation direction, and the gradient force F_g is directed to the equilibrium point in Fig. 2. From Figs. 2(a)-2(c), we can see that the particle could be trapped at the focus point (z_f = 975 $μ m$ ) by the longitudinal gradient force. And there are several stable equilibrium points in the distribution of the sum of the longitudinal gradient force and scattering force (Fig. 2(c)), which indicates that the CAB can trap particles in multiple planes. Typically speaking, for the first two equilibrium points z_a = 981 $μ m$ , z_c = 1076 $μ m$ , and the zero point between them, z_b = 1038 $μ m$ , the particle can be longitudinally trapped at z_a and z_c, but cannot be trapped at z_b. Note that the position of z_a is slightly shifted forward from the focus point, because of the influence of the scattering force. Comparing with GB, the longitudinal gradient force of CAB is increased, the scattering force of CAB is decreased. In our case, the GB cannot trap the particle at the beam waist, because the backward gradient force is not large enough to overcome the forward scattering force (Fig. 2(c)). Figures 2(d)-2(f) show that the particle can be transversely trapped at z_a and z_c, but cannot be transverse trapped at z_b by CAB. The transverse gradient force decreases as z increases. The gradient force generated from CAB is also larger than that generated from GB at the same position. However, the transverse gradient force cannot attract particles at the beam center when z = z_b, which is different from GB. Comparing with GB, the gradient force of CAB is larger and the trapping region of CAB is smaller, as is shown in Figs. 2(a) and 2(d), which indicates that the stiffness of the optical trap formed by CAB is greater. So the particle will be trapped more stably by CAB.

Fig. 2 The distribution of the radiation force on the Rayleigh particle with n_p = 1.59. (a) The longitudinal gradient force; (b) the scattering force, (c) the sum of the gradient force and the scattering force, the inset figure shows the positions of z_a, z_b and z_c. (d) The transverse gradient force at z_a = 981 $μ m$ ; (e) the transverse gradient force at z_b = 1038 $μ m$ , (f) the transverse gradient force of z_c = 1076 $μ m$ .

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The distribution of the radiation force for the particle with n_p = 1.00 are shown in Fig. 3. In contrary to the case of n_p = 1.59, this particle could be longitudinally trapped at z_b = 1038 $μ m$ , but could not be trapped at z_a and z_c. Although there is equilibrium point at the beam center in Fig. 3(b), yet the gradient force is too small to overcome the Brownian motion, as we will show below. So the particle could not be trapped at z_b.

Fig. 3 The distribution of the radiation force on the Rayleigh particle with n_p = 1.00. (a) The sum of the gradient force and the scattering force. (b) The transverse gradient force at z_b = 1038 $μ m$ .

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The radiation force can be modulated by controlling corresponding parameters of the CAB. Figure 4(a) shows that when a becomes smaller, the longitudinal radiation force (i.e., the sum of the longitudinal gradient force and the scattering force) for each optical trap will become larger. So the number of effective optical traps formed along the beam axis will be increased. However, the trapping positions are not changed with a. The transverse gradient force also increases with a, as Fig. 4(b) shows. Figure 4(c) shows that the parameter r₀ is related with positions of the optical traps, the trapping positions would be moved backward as r₀ decreases. The transverse gradient force also increases as r₀ decreases (Fig. 4(d)). However, r₀ should not be too small, because the CAB possesses the abruptly autofocusing property only when r₀ is large enough [1].

Fig. 4 The distributions of radiation forces with different parameters exerted on the Rayleigh particle with n_p = 1.59. (a)The longitudinal radiation forces with different a, while s = 5um, r₀ = 50um; (b) the transverse gradient forces at the first equilibrium point in Fig. 4(a). (c) The longitudinal radiation forces with different r₀, while a = 0.08, s = 5um; (d) the transverse gradient forces at the first equilibrium points in Fig. 4(c). (e) The longitudinal gradient forces with different s, while a = 0.08, r₀ = 50um; (f) the transvers gradient forces at the first equilibrium points in Fig. 4(e).

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The distribution of radiation force will be stretched or compressed by varying s, because s is a proportional coefficient. When s increases, the positions of the optical traps will be moved forward, the longitudinal radiation force for each optical trap is decreased and the corresponding trapping region is broadened obviously, as Fig. 4(e) shows. The transverse gradient force also increases as s decreases (Fig. 4(f)). However, s should not be too small because the abruptly autofocusing property of CAB will disappear as Fig. 1(b) shows.

3. Analysis of trapping stability

There are several necessary conditions for stably trapping in optical traps by CAB. First the backward longitudinal gradient force must be large enough to overcome the forward scattering force, which is satisfied as we can see in Figs. 2(c), 3(a) and 4. Second, the longitudinal gradient force must be large enough to overcome the influences of the buoyancy and the gravity. In our investigations, the difference between the buoyancy and the gravity for the glass micro particle or the air bubble is about 10⁻⁷ pN, which is smaller than maximum value of longitudinal radiation force.

The third condition is that the potential well of the gradient force must be larger than the kinetic energy of the Brownian particle. It is judged by the Boltzmann factor as follows [8, 14]:

R = \exp (- U / k_{B} T) ≪ 1,

where

U = \frac{1}{4} ε_{0} ε_{m} Re (α) \cdot Δ (E^{2})

is the potential energy of the gradient force at the trap position. k_B is the Boltzmann constant, and T is the temperature of the medium. We assume T = 300 K in our calculation.

Δ (E^{2})

denotes the intensity difference related with the potential well of the gradient force. For the CAB, the values of

Δ (E^{2})

in the longitudinal direction and in the transverse direction are not equivalent, which can be known from the intensity distribution. For simplicity, we only analyze the optical traps formed in Figs. 2 and 3. For the glass particle trapped at z_a, R_b for the longitudinal gradient force and the transverse gradient force is about

3.7 \times 10^{- 11}

and

5.1 \times 10^{- 12}

, respectively. For the glass particle trapped at z_c, R_b for the longitudinal gradient force and the transverse gradient force is about

3.9 \times 10^{- 4}

and

4.5 \times 10^{- 5}

, respectively. These values are far less than 1. So the glass particle could be stably trapped at z_a and z_c. But for the air bubble trapped at z_b, R_b for the longitudinal gradient force is

2.4 \times 10^{- 5}

and for the transverse gradient force is 0.59, which approaches 1. So this particle cannot be stably trapped by the transverse gradient force of CAB, because of the influence of the Brownian motion.

4. Conclusions

In conclusions, we have studied the radiation force of CAB exerted on a dielectric spherical particle in the Rayleigh scattering regime. The numerical results show that the CAB can be used to trap the particle whose refractive index is larger than the surrounding medium. The particle can be trapped at different positions along the beam axis, at the focus or the position some distance behind the focus. Comparing with the conventional optical trap with GB, the longitudinal gradient force and the transverse gradient force are both increased by using the CAB, when the initial radius, the focus length, and the incident power of the two beams are the same. Because of the unique abruptly autofocusing property of CAB, the trap stiffness of the optical trap formed by CAB is larger than that of the GB. For the particle whose refractive index is smaller than the surrounding medium, the longitudinal trapping by CAB is stable, but the transverse trapping is not stable because of the influence of the Brownian motion. The trapping properties of CAB can be controlled by corresponding parameters of the CAB. The value of the longitudinal radiation force can be controlled by varying a or s, the positions of the optical traps along the beam axis can be controlled by varying r₀ or s, and the trapping range can be effectively controlled by s. In most cases, the decrease of the beam size is helpful to increase the gradient force. But for the optical trap formed by CAB, the abruptly autofocusing property will disappear when the beam size is too small. We believe our investigation results are useful in optical micromanipulation or other fields.

Acknowledgment

This work was supported by the National Basic Research Program of China (Grant No.2012CB921602), National Nature Science Foundation of China (Grant No.10974177, 10874012) and the program of International S&T Cooperation of China (Grant No.2010DFA04690).

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Radiation force of abruptly autofocusing Airy beams on a Rayleigh particle

Abstract

1. Introduction

2. Non-paraxial propagation and paraxial propagation of CAB

2. Radiation force of CAB exerted on a Rayleigh particle

3. Analysis of trapping stability

4. Conclusions

Acknowledgment

References and links

Cited By

Figures (4)

Equations (16)

Optics Express