RADIATION FORCES OF A CYLINDRICAL BEAM L. Ouahid 1 , M. Boustimi 2 , F. Khannous 1 , H. Nebdi 1 *, A. Belafhal 1 1 Laboratory of Nuclear, Atomic and Molecular Physics Department of Physics, Faculty of Sciences, University Chouaïb Doukkali, P. B.: 20, 24000. El Jadida , Morocco 2 Department of Physics. College of Applied Sciences. Umm Al-Qura University PO Box:715. Makkah. Kingdom of Saudi Arabia * Corresponding author. E-mail: nebdi.h@ucd.ac.ma -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0 2 4 6 8 x 10 -32 x(m) Fscat,x (arb.un) -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0 2 4 6 8 x 10 -33 l=1 l=10 Abstract The radiation force [1] on a dielectric sphere produced by highly focused Quadratic Bessel-Gauss (QBG) beam, in the Rayleigh scattering regime, is theoretically investigated. Numerical results demonstrate that the focused QBG beam can be used to trap and guide the particles with the refractive index less than of the ambient. The radiation force caused by the less–order focused QBG beam has been studied under different input parameters and different focus lengths of thin lens. The trapping stability is also discussed. QBG Beams The field distribution of the QBG beam is expressed as [2]: (b) (a) -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 -1 -0.5 0 0.5 1 x 10 -14 x(m) Fgrad,x (N) -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 -1 -0.5 0 0.5 1 x 10 -15 l=1 l=10 0.5 1 N) 0.5 1 l=1 l=10 ( 29 (29 (29 (29 (29 - + × + - ′ = 2 2 2 2 2 2 2 2 2 0 2 2 2 2 1 exp 2 exp , , z r J z r B A z z B kD i l kz i z C z r U l R l ω μ ω ω π φ ω ω φ where ω(z) is the spot size of the above beam as a function of the system parameters, and given by: Radiation force For this family beams, we have calculated analytically the scattering force F scat and the gradient one F grad , which are given respectively as follows: Where: (d) (e) (c) -0.03 -0.02 -0.01 0 0.01 0.02 0.03 -6 -4 -2 0 2 4 6 x 10 -12 x(m) Fgrad,x (N) w0=0.3mm w0=0.4mm w0=0.5mm -0.03 -0.02 -0.01 0 0.01 0.02 0.03 -1.5 -1 -0.5 0 0.5 1 1.5 x 10 -11 x(m) Fgrad,x (N) f=3m f=5m f=6m Fig.2 : (a)The scattering force for both highly focused high-order(l=10) and low-order (l=1) QBG beam at the focus plane,. (b) The transverse gradient force for the highly focused low-order QBG beam at focus plane ( f=5 m, w 0 =0.5 mm). (c) The longitudinal gradient force along the z-axis. -4 -3 -2 -1 0 1 2 3 4 x 10 -6 -1 -0.5 0 x(m) Fgrad,z (N -4 -3 -2 -1 0 1 2 3 4 x 10 -6 -1 -0.5 0 Fig. 2: (d) : The gradient force for different input beam waists, where l=1. (e) : The transverse gradient force for different input focus length where, w 0 =0.5 mm. ( 29 2 2 2 0 2 () 1 2 R R B AB z A i z z ω ω μ = - + + 2 2 2 2 2 4 + + - + + - - = s z s z ξ χ R z 1 2 + = μ ξ ( 29 ( 29 ( 29 ( 29 (29 (29 k l l R R R R R scat e z r J z r J z AB z B A z B A r z AB z B A C m m a k n z r F + + - + - - × + + - ′ + - = 2 * 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 0 2 2 2 2 2 2 2 2 2 2 2 2 6 4 0 2 1 2 4 1 1 2 exp 4 1 2 1 3 4 , ω μ ω μ μ μ ω μ ε π r (29 (29 (29 ( 29 (29 (29 ( 29 [ ] (29 (29 Λ + ′ + + + - + - + - + + + + - - - + + + - × + - ′ + - = - - - - k l l l l R k l l grad e r z r J z r J B A f s B f z r z r J z r J f s A B f z z B B A f s B f z e z r J z r J B A r B A r C m m a n r F 2 1 2 2 2 * 2 2 2 2 2 2 2 2 1 2 2 2 2 2 0 2 2 2 * 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 * 2 2 2 2 2 2 2 2 2 2 0 2 2 1 2 2 2 0 2 2 2 2 2 3 0 2 1 2 2 1 2 1 4 1 2 4 2 exp 2 1 2 1 χ μ ω μ ω μ χ χ χ ξ χ ξ ω ω μ ω μ χ ξ ξ σ ω μ ω μ χ ξ ω χ χ ξ ω ε π r Fig.2: Schematic and matrix transfert of the used optical system. Conclusion In this work, we have studied analytically and numerically the radiation force on a dielectric sphere produced by highly focused QBG beam in the Rayleigh scattering regime. We have also investigated how the other different input parameters such as different focus lengths of thin lens, and waist can affect the trapping and manipulating the particle. Our numerical results demonstrate that the focused QBG beam can be used to trap and guide the particles with the refractive index less than of the ambient . References [1] . Ashkin, Phys. Rev. Lett. 24 (1970) 156.of C. Zhao, L. Wang and X. Lu, optik 119 (2008) 477-480. [2] A. Belafhal, L. Dalil-Essakali, Opt. Commun. 177 (2000) 181-188. [3] M. Kercher,The Scattering of Light and Other Electromagnetic Radiation, Academic Press, New York, 1969. [4]Y. Harada, T. Asakura, Radiation forces on a dielectric sphere in the Rayleigh scattering regime, Opt. Commun.124 (1996) 529–541. [5]Radiation forces of highly focused Bessel–Gaussian beams on a dielectric sphere C. Zhao, L. Wang_, Xuanhui Lu. Optik 119 (2008) 477–480. (g) (f) 2 3 4 5 6 7 8 9 10 x 10 -8 10 -20 10 -18 10 -16 10 -14 10 -12 10 -10 magnitude de force(N) Particule Radius a(m) Fgrad,x (l=10) Fgrad,x (l=1) Fg 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 10 -13 10 -12 10 -11 10 -10 magnitude de force(N) n1 Fgrad,x (l=10) Fgrad,x (l=1) Fig. 3 : (f) Comparison of F m grad,x ,and F g for different radius a, while the other parameters are n 1 =1.33 (g) : Influence of the Bessel function order to the F grad,x versus the refractive index,where a=50nm. ( z=0, f=5 m , w0=0.5 mm). + - - + + - - = 1 1 1 1 1 f s f z f s f z f z D C B A 2 2 2 4 + + - + + - - = z f z f s f z z z f z f s f z R ξ χ (29 (29 (29 (29 (29 (29 - Λ + - Λ = Λ + - + - 2 * 2 2 1 2 2 * 2 2 1 2 2 2 2 2 2 2 2 2 1 2 2 2 2 1 2 2 * 2 2 2 1 2 2 2 z r J z r J z r J z r J z r J z r J r l l l l l l ω μ ω μ ω μ ω μ ω μ ω μ μ (29 (29 (29 (29 (29 (29 (29 (29 - + - = + - * + - 2 * 2 2 1 2 2 * 2 2 1 2 2 2 2 2 2 2 2 2 1 2 2 2 2 1 2 2 * 2 2 2 2 2 1 1 z r J z r J z r J z z r J z r J z r J z r l l l l l l ω μ ω μ ω μ ω ω μ ω μ ω μ ω μ σ 2 2 2 2 2 0 1 2 1 1 2 - + - + - + - + + - - - = Λ AB z i B A f s A f B z i f s B f z R R ξ ξ ω * Λ = Λ 1 2