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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 3 — Jan. 30, 2012
  • pp: 2895–2904
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Radiation forces acting on a Rayleigh dielectric sphere produced by highly focused elegant Hermite-cosine-Gaussian beams

Zhirong Liu and Daomu Zhao  »View Author Affiliations


Optics Express, Vol. 20, Issue 3, pp. 2895-2904 (2012)
http://dx.doi.org/10.1364/OE.20.002895


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Abstract

We derive the analytical expression for the propagation of elegant Hermite-cosine-Gaussian (EHcosG) beams through a paraxial ABCD optical system and use it to study the radiation forces produced by highly focused this kind of beams acting on a Rayleigh dielectric sphere. Owing to the characteristics of focused EHcosG beams our analysis shows that it can be expected to simultaneously trap and manipulate dielectric spheres with low-refractive index at the focus point, and spheres with high-refractive index nearby the focus point. Finally, we discuss the stability conditions for effective trapping and manipulating the particle.

© 2012 OSA

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

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

], 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

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

9

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]

]. 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

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]

]. In order to stably capture particles the gradient forces must be greatly larger than the scattering force.

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

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

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]

]. Elegant Hermite Gaussian beams [25

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

27

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]

] and cosine-Gaussian beams [28

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]

] 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=±0.509μm nearby the focus point, and away from the focal plane the intensity distribution increases sharply and reaches a maximum at about z1=1.415μm 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
Epq(x1,y1;z=0)=G0Hp(x1w0)cos(Ωx1)Hq(y1w0)cos(Ωy1)exp(x12+y12w02),
(1)
where G0 denotes the electric-field strength at the beam-waist center OG(x1=y1=z=0), Hp and Hq represent the Hermite polynomials of orders p and q, respectively, w0 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)
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=1m1, (b) for various values of the cosine part associated parameter Ω with p=2. The other simulation parameter is selected as w0=1.0mm. (c) Schematic of an unapertured thin lens optical system, and in our case s=200mm and f=2mm are selected.
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)].

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

Epq(x1,y1;z=0)=G0Hp(x1w0)12[exp(iΩx1)+exp(iΩx1)]×Hq(y1w0)12[exp(iΩy1)+exp(iΩy1)]exp(x12+y12w02).
(2)

A(x)=12αexp(β124α)π1/2(11αw02)p/2Hp(β12αw011αw02)+12αexp(β224α)π1/2(11αw02)p/2Hp(β22αw011αw02),
(6)
B(y)=12αexp(β'124α)π1/2(11αw02)q/2Hq(β'12αw011αw02)+12αexp(β'224α)π1/2(11αw02)q/2Hq(β'22αw011αw02),
(7)
α=1w02+ikA2B,
(8)
β1=ikxB+iΩ,
(9)
β2=ikxBiΩ,
(10)
β'1=ikyB+iΩ,
(11)
β'2=ikyBiΩ.
(12)

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

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]

36

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

].

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
[ABCD]=[1z1+f01][101/f1][1s01]=[z1/fz1/fs+f+z11/f1s/f],
(13)
where s is the axial distance from the input plane to the thin lens, f is the focal length of the thin lens, and z1 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
Fig. 2 Intensity distributions of the EHcosG beams near the focal plane at different distances: (a) z1=0, (b) z1=1.415μm, (c) z1=2.83μm, and (d) z1=5.00μm. The other parameters are λ=1064nm,w0=1.0mm, p=2, Ω=2m−1, s=200mm, f=2mm, n1=1.592, and the input power of the beams is 100 mW.
). In the following simulations, the parameters are selected as: λ=1064nm,w0=1.0mm, p=2, Ω=2m−1, s=200mm f=2mm, 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=±0.509μm nearby the focus point. Away from the focal plane the intensity distribution increases sharply and reaches a maximum at about z1=1.415μm 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.

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., aλ/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 Fscat and the gradient force Fgrad, where Fgrad arises from the inhomogeneous field distribution [10

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]

,37

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

]. For Fscat, it is proportional to light intensity and is along the beam propagating direction. Then the scattering force is expressed as [10

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]

]
Fscat(r,z)=z^n2cCprI(r,z),
(14)
where z^ denotes the unit vector in the beam propagating direction, n2 represents the refractive index of the ambient, c=1/ε0μ0 is the speed of the light in vacuum, ε0 and μ0 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

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]

]

I(r,z)=z^n2ε0c2|E(r,z)|2.
(15)

For a small dielectric particle and in the Rayleigh regime, the particle scatters the light isotropically, and Cpr is equal to the scattering cross section Cscat and is given by [10

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]

]
Cpr=Cscat=83π(ka)4a2(m21m2+2)2,
(16)
where m=n1/n2 represents the relative index, n1 and a denote the refractive index and radius of the particle, respectively. For the gradient force Fgrad, it is produced by non-uniform electromagnetic fields, and its direction is along the gradient of light intensity. Therefore, the force Fgrad can be expressed as [10

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]

]

Fgrad(r,z)=2πn2a3c(m21m2+2)I(r,z).
(17)

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=40nm, and the refractive indices of two kinds of particles: n1=1.592 and n1=1.0, and the ambient with n2=1.332 (for example water).

In Figs. 3(a)-(c)
Fig. 3 (a)-(c) The transverse gradient force produced by highly focused EHcosG beams at different position z1: (a) z1=0, (b) z1=1.415μm, (c) z1=2.83μm; (d)-(f) The longitudinal gradient force produced by highly focused EHcosG beams at different transverse position x: (d) x=0, (e) x=0.509μm and (f) x=1.18μm. Solid curves for the particles with n1=1.592, dashed curves for the particles with n1=1.0.
we plot the changes of the transverse gradient forces at different longitudinal position z1, and in Figs. 3(d)-(f) the longitudinal gradient forces at different transverse position x. Here, positive Fgrad,x means the direction of transverse gradient force is in the +x direction, and negative Fgrad,x means in the x direction, in contrast. Likewise, positive (or negative) Fgrad,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=±0.509μm 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.

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=|Fgrad,z|/|Fscat|1, where the ratio R is called the stability criterion. Figures 4(a), (b), (c), and (d)
Fig. 4 (a)-(d) The scattering force produced by highly focused EHcosG beams at different distance z1: (a) z1=0, (b) z1=1.415μm, (c) z1=2.83μm and (d) z1=5.00μm. Solid curves for the particles with n1=1.592, dashed curves for the particles with n1=1.0.
show the scattering forces at different places from the focus point. Compared with the longitudinal gradient forces at the same z1 [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
Fig. 5 Comparison of Fgrad,xm (solid black curve), Fgrad,zm (dashed red curve), Fscatm (dotted blue curve), Fb (dotted-dashed green curve) and Fg (dotted-dashed Brown curve) with different particles’ radius a, while the other parameters are w0=1mm, p=2, Ω=2m−1, f=2mm, s=200mm, n1=1.592, n2=1.332. Fgrad,xm, Fgrad,zm and Fscatm occur at (0.325μm,0.509μm,0), (0,0,0.679μm), (0.509μm,0.509μm,0), respectively.
, where Fgrad,xm is the maximum transverse gradient force, Fgrad,zm is the maximum longitudinal gradient force, Fscatm is the maximum scattering force, Fg is the gravity, andFb is the Brownian force, respectively. The Brownian force, which describes the influence of the Brownian motion, can be calculated by |Fb|=12πηakBT [38

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

], here η is the viscosity for water is 7.977×104Pas at the room temperature T=300K; a is the radius of particle and kB 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<40nm the disturbance is mainly from the Brownian motion, while the disturbance is mainly from the scattering force for a>40nm. After all, the stability criterion is valid for highly focused EHcosG beams to trap and manipulate the particles.

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

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

,3

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]

]:
Rthermal=exp(Umax/kBT)<<1,
(18)
where Umax=πε0n22a3|(m21)/(m2+2)||E|max2 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 (n1=1.592), the value of Rthermal at the maximum intensity position (x=0.509μm,y=0.509μm,z1=0) is about Rthermal0.001; for the low-index particles (n1=1.0), the value of Rthermal at the maximum intensity position is about Rthermal0.00008. 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 14nm<a<50nm 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]

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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]

OCIS Codes
(140.3300) Lasers and laser optics : Laser beam shaping
(140.7010) Lasers and laser optics : Laser trapping
(170.4520) Medical optics and biotechnology : Optical confinement and manipulation

ToC Category:
Optical Trapping and Manipulation

History
Original Manuscript: November 15, 2011
Revised Manuscript: December 14, 2011
Manuscript Accepted: December 14, 2011
Published: January 24, 2012

Virtual Issues
Vol. 7, Iss. 3 Virtual Journal for Biomedical Optics

Citation
Zhirong Liu and Daomu Zhao, "Radiation forces acting on a Rayleigh dielectric sphere produced by highly focused elegant Hermite-cosine-Gaussian beams," Opt. Express 20, 2895-2904 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-3-2895


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References

  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]
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