Numerical modeling of optical levitation and trapping of the “stuck” particles with a pulsed optical tweezers Jian-liao Deng, Qing Wei, and Yu-zhu Wang Key Laboratory for Quantum Optics, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China
[email protected]
Yong-qing Li Department of Physics, East Carolina University, Greenville, North Carolina, 27858-4353, USA
Abstract: We present the theoretical analysis and the numerical modeling of optical levitation and trapping of the stuck particles with a pulsed optical tweezers. In our model, a pulsed laser was used to generate a large gradient force within a short duration that overcame the adhesive interaction between the stuck particles and the surface; and then a low power continuous–wave (cw) laser was used to capture the levitated particle. We describe the gradient force generated by the pulsed optical tweezers and model the binding interaction between the stuck beads and glass surface by the dominative van der Waals force with a randomly distributed binding strength. We numerically calculate the single pulse levitation efficiency for polystyrene beads as the function of the pulse energy, the axial displacement from the surface to the pulsed laser focus and the pulse duration. The result of our numerical modeling is qualitatively consistent with the experimental result.
©2005 Optical Society of America OCIS codes: (170.4520) optical confinement and manipulation; (140.7010) trapping
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
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Received 24 March 2005; revised 25 April 2005; accepted 2 May 2005
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14. C. A. Xie, M. A. Dinno, and Y. Q. Li, “Near-infrared Raman spectroscopy of single optically trapped biological cells,” Opt. Lett. 27, 249-251 (2002). 15. C. A. Xie, and Y. Q. Li, “Raman spectra and optical trapping of highly refractive and nontransparent particles,” Appl. Phys. Lett. 81, 951-953 (2002). 16. C. A. Xie, and Y. Q. Li, “Confocal micro-Raman spectroscopy of single biological cells using optical trapping and shifted excitation difference techniques,” J. Appl. Phys. 93, 2982-2986 (2003) 17. J. L. Deng, Q. Wei, M. H. Zhang, Y. Z. Wang, and Y. Q. Li, “Study of the effect of alcohol on single human red blood cells using near-infrared laser tweezers Raman spectroscopy,” J. Raman Spectrosc. 36, 257-261 (2005). 18. B. Agate, C. T. A. Brown, W. Sibbett, and K. Dholakia, “Femtosecond optical tweezers for in-situ control of two-photon fluorescence,” Opt. Express, 12, 3011-3017 (2004). 19. A. A. Ambardekar, and Y. Q. Li, Optical levitation and manipulation of stuck particles with pulsed optical tweezers, Opt. Lett. (in print) 20. T. G. M. van de Ven. Colloidal Hydrodynamics, (Academic Press, San Diego, 1989). 21. J. Happel, and H. Brenner, Low Reynolds Number Hydrodynamics (Prentice-Hall, Englewood Cliffs, NJ, 1965).
1.
Introduction
Optical tweezers is a single-beam gradient force optical trap, formed by tightly focusing a laser beam with an objective lens of high numerical aperture (NA). It has become a powerful tool for trapping and manipulation of dielectric and biological micron-sized particles typically using continuous wave (cw) laser[1-4]. It has been routinely applied to manipulate living cells, bacteria, viruses, chromosomes and other organelles [5-6], and recently been applied to the study of molecular motors at the single-molecule level [4,7] the physics of colloid systems [8, 9], the mechanical properties of polymers [10-11], and the control of optically trapped structures [12-13]. The combination of near-infrared Raman spectroscopy with optical tweezers allows characterizing optically trapped living cells and other particles [14-17]. Agate and co-workers have demonstrated that femtosecond optical tweezers are just as effective as cw optical tweezers [18]. The trapping force generated by the cw optical tweezers is typically in the order of 10-12 N [4]. This weak force is efficient to confine micro-particles suspended in liquids, but not sufficient to levitate the particles that are stuck on the glass surface, where they have to overcome the binding force. Recently Ambardekar and co-worker have demonstrated that both the stuck dielectric and biological micron-sized particles can be levitated and manipulated with a pulsed optical tweezers [19]. In their experiment, an infrared pulse laser at 1.06 μm was used to generate a large gradient force (up to 10-9 N) within a short duration (~45 μs) that overcame the adhesive interaction between the stuck particle and the surface; and then a low-power cw diode laser at 785 nm was used for trapping and manipulating the levitated particle. Here, we present a theoretical analysis and the numerical modeling of optical levitation and trapping of stuck particles with a pulsed optical tweezers. We describe the radiation force generated by the pulsed optical tweezers as a pulsed gradient force and describe the binding interaction between the stuck beads and glass surface as the dominative van der Waals force with a randomly distributed binding strength. The equation of motion for the bead will be used to describe the trajectory of the bead’s position as the stuck bead is detached from the surface and moves to the trap center. We will calculate the single pulse levitation efficiency for polystyrene beads as the functions of the pulse energy, the axial displacement from the surface to the pulsed laser focus and the pulse duration by solving the equation of motion for the polystyrene bead. 2.
Theory
2.1 Pulsed optical tweezers The pulsed optical tweezers is composed of two parts: the cw trapping system and the pulsed levitation system. The focus of the pulsed laser beam is at the same position as the focus of the cw trapping beam. The configuration is shown in Fig. 1. The polystyrene bead radius is a, and the distance from the bottom of bead to the surface is h, which is very small when the #6964 - $15.00 US
(C) 2005 OSA
Received 24 March 2005; revised 25 April 2005; accepted 2 May 2005
16 May 2005 / Vol. 13, No. 10 / OPTICS EXPRESS 3674
bead is “stuck” on the surface. In order to detach the stuck bead from the surface with the pulsed tweezers, the focus of the both pulsed and cw beam is initially adjusted to have an axial displacement z0 from the surface of the bottom glass plate. Then, a pulse is fired to generate large gradient force acting on the stuck bead within the pulse duration. This pulsed gradient force will break the binding interaction between the bead and the surface so that the stuck bead is levitated and moves into the focus of the cw beam. (a) Polystyrene bead (b)
z z0
h
Glass surface
10 μm
Fig. 2. (a) The beads were stuck on the surface. (b) The marked bead was levitated with a pulse and moved to the focus.
Fig. 1. Schematics of the pulsed optical tweezers
Figure 2 shows the image of a 2-μm polystyrene bead that sticks on the glass surface before the application of a pulse (the image was defocused with an axial distance z) and the image of the levitated bead that was detached with the application of a pulse and moved to the focus of the cw beam [19]. 2.2 Gradient forces In order to describe the equation of motion for the bead, we begin with modeling the radiation forces generated by the cw and the pulsed beams that were applied to the bead. Assume that the beam intensity of the focused laser beams can be described with a Guassian expression.
I (x, y, z, t) = I0 +
2 2U ω0
τ π ω(z)
ω02
ω(z)
2
( (
( (
)
2
)
exp(−(t τ ) ) exp − 2 x2 + y 2 / ω(z) 2
2
)
exp − 2 x2 + y2 / ω( z)
2
)
,
(1)
where the first term describes the cw beam with an intensity I0 and the second term describes the pulse beam with the single pulse energy U per unit cross-section area, the pulse duration τ. Here we assume that the pulse beam is completely overlapped with the cw beam and they have the same spatial intensity distribution. ω(z) is the radius of the beam waist at point z along the light propagation and it can be expressed as
(
ω ( z ) = ω 0 1 + ( z / z s )2
)
1/ 2
,
(2)
where ω0 is the radius of the beam waist at the point z=0 and zs=πω0 /λ0 is a distance parameter. The radiation force acting on the bead includes the scattering force Fscatt and gradient force Fgrad. In the levitation experiment with pulsed optical tweezers, the stuck bead is not at the focus of the pulsed beam but with an axial displacement. Thus, the scattering force can be neglected comparing to the gradient force. Ashkin has calculated the forces of a single-beam gradient laser trap on a dielectric sphere in the ray optical regime [3]. Here, we use an approximate expression for the gradient force. It is known that the gradient force arises 2
#6964 - $15.00 US
(C) 2005 OSA
Received 24 March 2005; revised 25 April 2005; accepted 2 May 2005
16 May 2005 / Vol. 13, No. 10 / OPTICS EXPRESS 3675
from intensity gradient of the inhomogeneous field, Fgrad ∝∇I(x,y,z) [4]. Here, we approximately express the gradient force acting on the bead that is located on the beam axis (with x=0 and y=0), produced by the cw beam as
Fcw = −k z
z
(1 + (z / z ) )
2 2
,
(3a)
s
where kz is the spring constant and z the position deviation of the bead center from the beam focus (z=h+a-z0). The express in Eq. (3a) is consistent with the harmonic approximation of an optical trap, in which the restoring force acting on the bead is described as –kzz for z
