Radiation forces acting on a Rayleigh dielectric sphere produced by highly focused elegant Hermite-cosine-Gaussian beams

Zhirong Liu; Daomu Zhao

doi:10.1364/OE.20.002895

1. Introduction

Since Ashkin demonstrated the first practical laser traps and showed the use of radiation pressure to capture and manipulate micrometer sized particles [1], optical traps or tweezers have become powerful tools for the trapping and manipulating of various particles, such as micro-sized dielectric particles, neutral atoms, nonspherical particles, DNA molecules, living biological cells and metallic particles [2–9]. It is known that two types of radiation forces are identified in the optical tweezers: gradient force and scattering/absorption forces. The gradient force is proportional to the gradient of the square of the electric field (energy density) and is responsible to pull the particles towards the center of focus. The scattering/absorption force is due to the net momentum transfer caused by scattering/absorption of photons from the particles and tends to push the particles out of the focus, and destabilize the optical trap [10]. In order to stably capture particles the gradient forces must be greatly larger than the scattering force.

The conventional optical traps or tweezers, which are mainly constructed by highly focused fundamental Gaussian beams, are used to attract high-index particles (particles having a refractive index higher than the surrounding medium) into the bright focal region of the beam, and low-index particles (particles having a refractive index lower than the surrounding medium) are expelled from the beam, in contrast. Recently, some researchers have demonstrated that other beams such as bottle beams [11], zero-order Bessel beams [12], Hermite-Gaussian beams [13], Laguerre Gaussian beams [14], hollow Gaussian beams [15], pulsed Gaussian beams [16–19], radial polarized beams [8,20], and Lorentz-Gaussian beams [21] are also useful in trapping particles. Their trapping characteristics have been studied in detail and it has been found that the radiation forces produced by a laser beam are mainly related to the beam’s characteristic such as beam’s profile, coherence and polarization [21,22].

In recent years, the Hermite-sinusoidal-Gaussian (H-sin-G) beams, which are one of the solutions of the paraxial wave equation, have been introduced [23,24]. Elegant Hermite Gaussian beams [25–27] and cosine-Gaussian beams [28] have been investigated in detail, respectively. However, a more generalized case for elegant Hermite Gaussian beams and cosine-Gaussian beams, i.e., elegant Hermite-cosine-Gaussian (EHcosG) beams have been seldom mentioned, yet. In the present paper, the analytical expression for the propagation of EHcosG beams through a paraxial ABCD optical system is derived and used to study the radiation forces produced by highly focused EHcosG beams acting on a Rayleigh dielectric particle. Owing to the characteristics of focused EHcosG beams, which will be focused into a dark-centered beams at the focus point, and double-sharp-peaked distribution located at about $x = \pm 0.509$ $μm$ nearby the focus point, and away from the focal plane the intensity distribution increases sharply and reaches a maximum at about $z_{1} = 1.415 μm$ and then decreases along the optical axes, it is expected to simultaneously trap and manipulate particles with low-refractive index at the focus point and particles with high-refractive index nearby the focus point. Finally, the stability conditions for effective trapping and manipulating particles are analyzed.

2. Fields of EHcosG beams through a lens

In the rectangular coordinate system an EHcosG beam’s electric field distribution at the input plane $z = 0$ is defined by

E_{p q} (x_{1}, y_{1}; z = 0) = G_{0} H_{p} (\frac{x_{1}}{w_{0}}) \cos (Ω x_{1}) H_{q} (\frac{y_{1}}{w_{0}}) \cos (Ω y_{1}) \exp (- \frac{x_{1}^{2} + y_{1}^{2}}{w_{0}^{2}}),

where

G_{0}

denotes the electric-field strength at the beam-waist center

O_{G} (x_{1} = y_{1} = z = 0)

,

H_{p}

and

H_{q}

represent the Hermite polynomials of orders

p

and

q

, respectively,

w_{0}

is the waist width of the corresponding fundamental Gaussian beam, and

Ω

is the parameter associated with the cosine part, respectively. To visualize the shape of EHcosG beams characterized by Eq. (1) a preliminary demonstration is shown in Fig. 1(a) for various values of the mode index

p

and

q

(Here we assume

q = p

), and (b) the parameter of

Ω

. All of the curves have been normalized by a fixed input power 100 mW. It is evident from Fig. 1(a), (b) that, for the case

p = 0

, the intensity distribution reduces to the cosine-Gaussian distribution, while for

Ω = 0

the elegant Hermite Gaussian distribution. In addition, a main distribution peak can be found on z-axis for the case that

p

is taken as an even number or zero, while a dark-centered distribution on z-axis for the case that

p

is an odd number. Furthermore, the number side lobes increase and some of the energy is shifted to the outer lobes as

p

increases, while the intensity distribution is insensitive to the change of

Ω

[see Fig. 1(b)].

Fig. 1 (a), (b) Intensity distribution of an EHcosG beam in the $x$ direction at the $z = 0$ plane (a) for various values of the order of the Hermite polynomial $p$ (Here we assume $q = p$ ) with $Ω = 1 m^{- 1}$ , (b) for various values of the cosine part associated parameter $Ω$ with $p = 2$ . The other simulation parameter is selected as $w_{0} = 1.0 mm$ . (c) Schematic of an unapertured thin lens optical system, and in our case $s = 200$ mm and $f = 2$ mm are selected.

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On the other hand, Eq. (1) can be rewritten in the form

\begin{array}{l} E_{p q} (x_{1}, y_{1}; z = 0) = G_{0} H_{p} (\frac{x_{1}}{w_{0}}) \frac{1}{2} [\exp (i Ω x_{1}) + \exp (- i Ω x_{1})] \\ \times H_{q} (\frac{y_{1}}{w_{0}}) \frac{1}{2} [\exp (i Ω y_{1}) + \exp (- i Ω y_{1})] \exp (- \frac{x_{1}^{2} + y_{1}^{2}}{w_{0}^{2}}) . \end{array}

Equation (2) indicates that the EHcosG beams can be generated in laboratory by superposition of two decentered elegant Hermite Gaussian beams as cosine one is just the superposition of the two angular spectra. Within the framework of paraxial approximation, the propagation expression for an EHcosG beam through the first-order ABCD optical system can be expressed as [29,30]

\begin{array}{l} E_{p q} (x, y; z) = - \frac{i}{λ B} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} E_{p q} (x_{1}, y_{1}; z = 0) \\ \times \exp {- \frac{i k}{2 B} [A (x_{1}^{2} + y_{1}^{2}) - 2 (x x_{1} + y y_{1}) + D (x^{2} + y {}^{2})]} d x_{1} d y_{1}, \end{array}

where

k

is the wave number related to the wavelength

λ

by

k = 2 π / λ

,

(x_{1}, y_{1})

and

(x, y)

denote the transverse coordinates at the input and output planes, respectively. A, B and D are the transfer matrix elements of the optical system between the input and output planes, and here an unimportant phase factor is omitted for simplicity. On substituting from Eq. (2) into Eq. (3) and recalling the following integral formula [31]

\int_{- \infty}^{\infty} H_{n} (α x) \exp [- {(x - y)}^{2}] d x = π^{1 / 2} {(1 - α^{2})}^{n / 2} H_{n} (\frac{α y}{\sqrt{1 - α^{2}}}),

after tedious integral calculations, one obtains

E_{p q} (x, y; z) = \frac{i G_{0}}{λ B} \exp [- \frac{i k}{2 B} D (x^{2} + y {}^{2})] A (x) B (y),

where

\begin{array}{l} A (x) = \frac{1}{2 \sqrt{α}} \exp (\frac{β_{1}^{2}}{4 α}) π^{1 / 2} (^{1 - \frac{1}{α w_{0}^{2}}) p / 2} H_{p} (\frac{\frac{β_{1}}{2 α w_{0}}}{\sqrt{1 - \frac{1}{α w_{0}^{2}}}}) \\ + \frac{1}{2 \sqrt{α}} \exp (\frac{β_{2}^{2}}{4 α}) π^{1 / 2} (^{1 - \frac{1}{α w_{0}^{2}}) p / 2} H_{p} (\frac{\frac{β_{2}}{2 α w_{0}}}{\sqrt{1 - \frac{1}{α w_{0}^{2}}}}), \end{array}

\begin{array}{l} B (y) = \frac{1}{2 \sqrt{α}} \exp (\frac{β'_{1}^{2}}{4 α}) π^{1 / 2} (^{1 - \frac{1}{α w_{0}^{2}}) q / 2} H_{q} (\frac{\frac{β'_{1}}{2 α w_{0}}}{\sqrt{1 - \frac{1}{α w_{0}^{2}}}}) \\ + \frac{1}{2 \sqrt{α}} \exp (\frac{β'_{2}^{2}}{4 α}) π^{1 / 2} (^{1 - \frac{1}{α w_{0}^{2}}) q / 2} H_{q} (\frac{\frac{β'_{2}}{2 α w_{0}}}{\sqrt{1 - \frac{1}{α w_{0}^{2}}}}), \end{array}

α = \frac{1}{w_{0}^{2}} + \frac{i k A}{2 B},

β_{1} = \frac{i k x}{B} + i Ω,

β_{2} = \frac{i k x}{B} - i Ω,

β'_{1} = \frac{i k y}{B} + i Ω,

β'_{2} = \frac{i k y}{B} - i Ω .

It should be pointed out that Eqs. (5)-(12) are only valid for the field distribution of the EHcosG beams passing through an unapertured first-order ABCD optical system. For the optical system with an aperture, the effect of the aperture must be included, for example, like the Refs [32–36].

Now we consider the EHcosG beams propagating through an unapertured lens system as shown in Fig. 1(c). The transfer matrix for such a lens system is given by

[\begin{matrix} A & B \\ C & D \end{matrix}] = [\begin{matrix} 1 & z_{1} + f \\ 0 & 1 \end{matrix}] [\begin{matrix} 1 & 0 \\ - 1 / f & 1 \end{matrix}] [\begin{matrix} 1 & s \\ 0 & 1 \end{matrix}] = [\begin{matrix} - z_{1} / f & � - z_{1} / f � s + f + z_{1} \\ - 1 / f & 1 - s / f \end{matrix}],

where

s

is the axial distance from the input plane to the thin lens,

f

is the focal length of the thin lens, and

z_{1}

is the axial distance from the focal plane to the output plane. The point

F

in Fig. 1(c) is the geometrical focus point. On substituting from Eq. (13) into Eqs. (5)-(12), one can obtain the normalized intensity distribution of an EHcosG beam passing through the optical system (see Fig. 2 ). In the following simulations, the parameters are selected as:

λ = 1064

nm,

w_{0} = 1.0

mm,

p = 2

,

Ω = 2

m⁻¹,

s = 200

mm

f = 2

mm, and the input power of the EHcosG beams is assumed to be 100 mW. From Fig. 2 we find that the intensity at the focal plane has a dark-centered distribution at the focus point, and double sharp peaks located at about

x = \pm 0.509

μm

nearby the focus point. Away from the focal plane the intensity distribution increases sharply and reaches a maximum at about

z_{1} = 1.415 μm

and then decreases along the optical axes [see Fig. 2(b), (c) and (d)]. Owing to these focusing characteristics, one can expect that it is useful for the highly focused EHcosG beams to trap and manipulate particles with different refractive indexes.

Fig. 2 Intensity distributions of the EHcosG beams near the focal plane at different distances: (a) $z_{1} = 0$ , (b) $z_{1} = 1.415 μm$ , (c) $z_{1} = 2.83 μm$ , and (d) $z_{1} = 5.00 μm$ . The other parameters are $λ = 1064$ nm, $w_{0} = 1.0$ mm, $p = 2$ , $Ω = 2$ m⁻¹, $s = 200$ mm, $f = 2$ mm, $n_{1} = 1.592$ , and the input power of the beams is 100 mW.

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3. Radiation forces produced by the EHcosG beams

For the sake of simplicity, we assume that the radius $a$ of the particle is much smaller than the wavelength of the laser, i.e., $a \leq λ / 20$ quantitatively. In this case, the Rayleigh approximation is applicable and the particle can be seen as a point dipole. Under this approximation, the radiation forces include the scattering force ${\overset{⇀}{F}}_{s c a t}$ and the gradient force ${\overset{⇀}{F}}_{g r a d}$ , where ${\overset{⇀}{F}}_{g r a d}$ arises from the inhomogeneous field distribution [10,37]. For ${\overset{⇀}{F}}_{s c a t}$ , it is proportional to light intensity and is along the beam propagating direction. Then the scattering force is expressed as [10]

{\overset{⇀}{F}}_{s c a t} (r, z) = \hat{z} \frac{n_{2}}{c} C_{p r} I (r, z),

where

\hat{z}

denotes the unit vector in the beam propagating direction,

n_{2}

represents the refractive index of the ambient,

c = 1 / \sqrt{ε_{0} μ_{0}}

is the speed of the light in vacuum,

ε_{0}

and

μ_{0}

denote the dielectric constant and the magnetic permeability in the vacuum, respectively. Here,

I (r, z)

is defined as a time-averaged version of the Poynting vector and is given by [10]

I (r, z) = \hat{z} \frac{n_{2} ε_{0} c}{2} {| E (r, z) |}^{2} .

For a small dielectric particle and in the Rayleigh regime, the particle scatters the light isotropically, and $C_{p r}$ is equal to the scattering cross section $C_{s c a t}$ and is given by [10]

C_{p r} = C_{s c a t} = \frac{8}{3} π {(k a)}^{4} a^{2} {(\frac{m^{2} - 1}{m^{2} + 2})}^{2},

where

m = n_{1} / n_{2}

represents the relative index,

n_{1}

and

a

denote the refractive index and radius of the particle, respectively. For the gradient force

F_{g r a d}

, it is produced by non-uniform electromagnetic fields, and its direction is along the gradient of light intensity. Therefore, the force

F_{g r a d}

can be expressed as [10]

F_{g r a d} (r, z) = \frac{2 π n_{2} a^{3}}{c} (\frac{m^{2} - 1}{m^{2} + 2}) \nabla I (r, z) .

By using Eqs. (14)-(17), one can calculate the radiation forces acting on a Rayleigh dielectric sphere produced by focused EHcosG beams. Without loss of generality, in the following simulations we select the radius of the particle $a = 40$ nm, and the refractive indices of two kinds of particles: $n_{1} = 1.592$ and $n_{1} = 1.0$ , and the ambient with $n_{2} = 1.332$ (for example water).

In Figs. 3(a)-(c) we plot the changes of the transverse gradient forces at different longitudinal position $z_{1}$ , and in Figs. 3(d)-(f) the longitudinal gradient forces at different transverse position $x$ . Here, positive $F_{g r a d, x}$ means the direction of transverse gradient force is in the $+ x$ direction, and negative $F_{g r a d, x}$ means in the $- x$ direction, in contrast. Likewise, positive (or negative) $F_{g r a d, z}$ means the direction of longitudinal gradient force is in the $+ z$ (or $- z$ ) direction. From Figs. 3(a), (d), and (e) it is evident that there exists one stable equilibrium point at the focus point for the particles with $m < 1$ , and still there are two stable equilibrium points at about $x = \pm 0.509 μm$ for particles with $m > 1$ . It means that one can use the highly focused EHcosG beams to trap or manipulate the particles with $m < 1$ at the focus point and simultaneously trap or manipulate the particle with $m > 1$ nearby the focus point. From Section 2, we have found that the focused beams have a special dark-centered configuration at the focus point and two sharp peaks located nearby, so it indicates that the special focused beam may be used to simultaneously trap or manipulate particles with $m < 1$ at the focus point and $m < 1$ nearby the focus point. From Figs. 3(b), (c), (e) and (f) we can see there are equilibrium points for particles with $m > 1$ nearby the focus point, which confirms our conclusion.

Fig. 3 (a)-(c) The transverse gradient force produced by highly focused EHcosG beams at different position $z_{1}$ : (a) $z_{1} = 0$ , (b) $z_{1} = 1.415 μm$ , (c) $z_{1} = 2.83 μm$ ; (d)-(f) The longitudinal gradient force produced by highly focused EHcosG beams at different transverse position $x$ : (d) $x = 0$ , (e) $x = 0.509 μm$ and (f) $x = 1.18 μm$ . Solid curves for the particles with $n_{1} = 1.592$ , dashed curves for the particles with $n_{1} = 1.0$ .

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4. Analysis of trapping stability

In the above discussion, our analysis shows that the radiation forces of focused EHcosG beams may be used to trap and manipulate the Rayleigh dielectric particles. In order to stably trap particles, firstly the longitudinal gradient force must be greatly larger than the scattering force, i.e. $R = | F_{g r a d, z} | / | F_{s c a t} | \geq 1$ , where the ratio $R$ is called the stability criterion. Figures 4(a), (b), (c), and (d) show the scattering forces at different places from the focus point. Compared with the longitudinal gradient forces at the same $z_{1}$ [see Figs. 3(d), (e) and (f)], the magnitude of the scattering forces is much smaller (about 10 times smaller) than the longitudinal gradient forces near the focal plane. For convenient comparison within these forces in the system, magnitude of all forces versus particle’s radius $a$ is plotted in Fig. 5 , where $F_{g r a d, x}^{m}$ is the maximum transverse gradient force, $F_{g r a d, z}^{m}$ is the maximum longitudinal gradient force, $F_{s c a t}^{m}$ is the maximum scattering force, $F_{g}$ is the gravity, and $F_{b}$ is the Brownian force, respectively. The Brownian force, which describes the influence of the Brownian motion, can be calculated by $| F_{b} | = \sqrt{12 π η a k_{B} T}$ [38], here $η$ is the viscosity for water is $7.977 \times 10^{- 4}$ $P a \cdot s$ at the room temperature $T = 300 K$ ; $a$ is the radius of particle and $k_{B}$ is the Boltzmann constant. It is evident from Fig. 5 that the gravity of the particle could be neglected comparing with the gradient force, and it is further found that for the case $a < 40 nm$ the disturbance is mainly from the Brownian motion, while the disturbance is mainly from the scattering force for $a > 40 nm$ . After all, the stability criterion is valid for highly focused EHcosG beams to trap and manipulate the particles.

Fig. 4 (a)-(d) The scattering force produced by highly focused EHcosG beams at different distance $z_{1}$ : (a) $z_{1} = 0$ , (b) $z_{1} = 1.415 μm$ , (c) $z_{1} = 2.83 μm$ and (d) $z_{1} = 5.00 μm$ . Solid curves for the particles with $n_{1} = 1.592$ , dashed curves for the particles with $n_{1} = 1.0$ .

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Fig. 5 Comparison of $F_{g r a d, x}^{m}$ (solid black curve), $F_{g r a d, z}^{m}$ (dashed red curve), $F_{s c a t}^{m}$ (dotted blue curve), $F_{b}$ (dotted-dashed green curve) and $F_{g}$ (dotted-dashed Brown curve) with different particles’ radius $a$ , while the other parameters are $w_{0} = 1$ mm, $p = 2$ , $Ω = 2$ m⁻¹, $f = 2$ mm, $s = 200$ mm, $n_{1} = 1.592$ , $n_{2} = 1.332$ . $F_{g r a d, x}^{m}$ , $F_{g r a d, z}^{m}$ and $F_{s c a t}^{m}$ occur at (0.325μm,0.509μm,0), (0,0,0.679μm), (0.509μm,0.509μm,0), respectively.

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Another factor due to the Brownian motion will also strongly affect the trapping stability when the particles are very small. To trap the particle potential well, which is induced by the radiation forces, must be deep enough to overcome the kinetic energy of the particle due to thermal fluctuation. This condition can be given by using the Boltzmann factor [2,3]:

R_{t h e r m a l} = \exp (- U_{\max} / k_{B} T) < < 1,

where

U_{\max} = π ε_{0} n_{2}^{2} a^{3} | (m^{2} - 1) / (m^{2} + 2) | \cdot | \overset{⇀}{E} |_{\max}^{2}

is the maximum depth of the potential well, and

T

is the absolute temperature of the ambient. In the above numerical examples, at room temperature of 300 K, for the high-index particles

(n_{1} = 1.592)

, the value of

R_{t h e r m a l}

at the maximum intensity position

(x = 0.509 μm, y = 0.509 μm, z_{1} = 0)

is about

R_{t h e r m a l} \approx 0.001

; for the low-index particles

(n_{1} = 1.0)

, the value of

R_{t h e r m a l}

at the maximum intensity position is about

R_{t h e r m a l} \approx 0.00008

. Obviously, all the Boltzmann factors near the focus point are extremely small. Therefore in our case the Brownian motion can be overcome and the particles can be stably trapped by highly focused EHcosG beams. In all, in our case the particles with

14 nm < a < 50 nm

can be stably trapped and manipulated.

5. Conclusion

In summary, we have derived the analytical expression for the propagation of the EHcosG beams through a paraxial ABCD optical system and used it to study the radiation forces produced by highly focused this kind of beams in the Rayleigh scattering regime. Owing to the characteristics of focused EHcosG beams, which have a dark-centered configuration at the focus point and a double-peak configuration nearby the focus point, it is expected to simultaneously trap and manipulate particles with low-refractive index at the focus point and particles with high-refractive index nearby the focus point. Finally, the conditions for effective trapping and manipulating the particle have been analyzed. Our results are interesting and useful for particle trapping and manipulating.

Acknowledgment

This work was supported by the National Natural Science Foundation of China (NSFC) (11074219 and 10874150), and the Zhejiang Provincial Natural Science Foundation of China (R1090168).

References and links

1. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970). [CrossRef]

2. A. Ashkin, “Trapping of atoms by resonance radiation pressure,” Phys. Rev. Lett. 40(12), 729–732 (1978). [CrossRef]

3. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11(5), 288–290 (1986). [CrossRef] [PubMed]

4. L. Oroszi, P. Galajda, H. Kirei, S. Bottka, and P. Ormos, “Direct measurement of torque in an optical trap and its application to double-strand DNA,” Phys. Rev. Lett. 97(5), 058301 (2006). [CrossRef] [PubMed]

5. K. Taguchi, H. Ueno, T. Hiramatsu, and M. Ikeda, “Optical trapping of dielectric particle and biological cell using optical ﬁbre,” Electron. Lett. 33(5), 413–414 (1997). [CrossRef]

6. A. Ashkin, “History of optical trapping and manipulation of small-neutral particle, atoms, and molecules,” IEEE J. Sel. Top. Quantum Electron. 6(6), 841–856 (2000). [CrossRef]

7. K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75(9), 2787–2809 (2004). [CrossRef] [PubMed]

8. Q. W. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004). [CrossRef] [PubMed]

9. M. Dienerowitz, M. Mazilu, P. J. Reece, T. F. Krauss, and K. Dholakia, “Optical vortex trap for resonant confinement of metal nanoparticles,” Opt. Express 16(7), 4991–4999 (2008). [CrossRef] [PubMed]

10. Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5-6), 529–541 (1996). [CrossRef]

11. C. H. Chen, P. T. Tai, and W. F. Hsieh, “Bottle beam from a bare laser for single-beam trapping,” Appl. Opt. 43(32), 6001–6006 (2004). [CrossRef] [PubMed]

12. V. Garcés-Chávez, D. Roskey, M. D. Summers, H. Melville, D. McGloin, E. M. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85(18), 4001–4003 (2004). [CrossRef]

13. T. P. Meyrath, F. Schreck, J. L. Hanssen, C. S. Chuu, and M. G. Raizen, “A high frequency optical trap for atoms using Hermite-Gaussian beams,” Opt. Express 13(8), 2843–2851 (2005). [CrossRef] [PubMed]

14. M. Bhattacharya and P. Meystre, “Using a Laguerre-Gaussian beam to trap and cool the rotational motion of a mirror,” Phys. Rev. Lett. 99(15), 153603 (2007). [CrossRef] [PubMed]

15. C. L. Zhao, L. G. Wang, and X. H. Lu, “Radiation forces on a dielectric sphere produced by highly focused hollow Gaussian beams,” Phys. Lett. A 363(5-6), 502–506 (2007). [CrossRef]

16. H. Little, C. T. A. Brown, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, “Optical guiding of microscopic particles in femtosecond and continuous wave Bessel light beams,” Opt. Express 12(11), 2560–2565 (2004). [CrossRef] [PubMed]

17. A. A. Ambardekar and Y. Q. Li, “Optical levitation and manipulation of stuck particles with pulsed optical tweezers,” Opt. Lett. 30(14), 1797–1799 (2005). [CrossRef] [PubMed]

18. J. L. Deng, Q. Wei, Y. Z. Wang, and Y. Q. Li, “Numerical modeling of optical levitation and trapping of the “stuck” particles with a pulsed optical tweezers,” Opt. Express 13(10), 3673–3680 (2005). [CrossRef] [PubMed]

19. L. G. Wang and C. L. Zhao, “Dynamic radiation force of a pulsed gaussian beam acting on rayleigh dielectric sphere,” Opt. Express 15(17), 10615–10621 (2007). [CrossRef] [PubMed]

20. Y. J. Zhang, B. F. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010). [CrossRef]

21. Y. F. Jiang, K. K. Huang, and X. H. Lu, “Radiation force of highly focused Lorentz-Gauss beams on a Rayleigh particle,” Opt. Express 19(10), 9708–9713 (2011). [CrossRef] [PubMed]

22. L. G. Wang, C. L. Zhao, L. Q. Wang, X. H. Lu, and S. Y. Zhu, “Effect of spatial coherence on radiation forces acting on a Rayleigh dielectric sphere,” Opt. Lett. 32(11), 1393–1395 (2007). [CrossRef] [PubMed]

23. L. W. Casperson and A. A. Tovar, “Hermite–sinusoidal-Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A 15(4), 954–961 (1998). [CrossRef]

24. A. A. Tovar and L. W. Casperson, “Production and propagation of Hermite-sinusoidal-Gaussian laser beams,” J. Opt. Soc. Am. A 15(9), 2425–2432 (1998). [CrossRef] [PubMed]

25. A. E. Siegman, “Hermite-gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. Am. 63(9), 1093–1094 (1973). [CrossRef]

26. A. E. Siegman, Lasers (University Science Books, Sausalito, Ca, 1986).

27. B. Lü and H. Ma, “A comparative study of elegant and standard Hermite-Gaussian beams,” Opt. Commun. 174(1-4), 99–104 (2000). [CrossRef]

28. H. T. Eyyuboğlu and Y. Baykal, “Analysis of reciprocity of cos-Gaussian and cosh-Gaussian laser beams in a turbulent atmosphere,” Opt. Express 12(20), 4659–4674 (2004). [CrossRef] [PubMed]

29. S. A. Collins Jr., “Lens-system diffraction integral written in terms of Matrix Optics,” J. Opt. Soc. Am. 60(9), 1168–1177 (1970). [CrossRef]

30. S. Wang and D. Zhao, Matrix Optics (CHEP-Springer, Beijing, 2000).

31. A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transfoms (McGraw-Hill, New York, 1954).

32. D. Zhao, H. Mao, W. Zhang, and S. Wang, “Propagation of off-axial Hermite-cosine-Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun. 224(1-3), 5–12 (2003). [CrossRef]

33. D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236(4-6), 225–235 (2004). [CrossRef]

34. Z. Mei and D. Zhao, “Propagation of Laguerre-Gaussian and elegant Laguerre-Gaussian beams in apertured fractional Hankel transform systems,” J. Opt. Soc. Am. A 21(12), 2375–2381 (2004). [CrossRef] [PubMed]

35. H. Mao and D. Zhao, “Different models for a hard-aperture function and corresponding approximate analytical propagation equations of a Gaussian beam through an apertured optical system,” J. Opt. Soc. Am. A 22(4), 647–653 (2005). [CrossRef] [PubMed]

36. D. Deng, “Propagation of elegant Hermite cosine Gaussian laser beams,” Opt. Commun. 259(2), 409–414 (2006). [CrossRef]

37. J. P. Gordon, “Radiation forces and momenta in dielectric media,” Phys. Rev. A 8(1), 14–21 (1973). [CrossRef]

38. K. Okamoto and S. Kawata, “Radiation force exerted on subwavelength particles near a nanoaperture,” Phys. Rev. Lett. 83(22), 4534–4537 (1999). [CrossRef]

Radiation forces acting on a Rayleigh dielectric sphere produced by highly focused elegant Hermite-cosine-Gaussian beams

Abstract

1. Introduction

2. Fields of EHcosG beams through a lens

3. Radiation forces produced by the EHcosG beams

4. Analysis of trapping stability

5. Conclusion

Acknowledgment

References and links

Cited By

Figures (5)

Equations (18)

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