logIE logEI y lum Fig. 3. Computed distributions of the electric field intensity E(units: [/mr)in the xz-and yz-planes near the focus of a 0.9NA, /=5.0mm, diffraction- limited objective. The 70=532nm plane-wave illuminating the entrance pupil of the lens is linearly polarized along the x-axis. Sx1 Sx10 0.604-020.00.20.40.60.6-040.20.00.20.40.6 x Jum x um Fig 4 Distribution of the Poynting vector S(units: [/m))in and around a glass micro-sphere (n= 1.5, d=460nm). The focused beam, obtained by sending a linearly-polarized plane-wave (polarization along y)through a 0. 9NA objective, propagates along the negative --axis. Sphere center offset from the focal point:(250,0, 50)nm. The(S, S_)vector-field is superimposed on the color-coded S. plot on the right-hand side if the laser power level is adjusted to a few microwatts, then the radiation pressure will work against the force of gravity to hold the bead in a stable trap along the :-axis )The computed anisotropy of this trap in the lateral direction is s)=1-(K/K,)=-015, where Kr and K, are the trap stiffness coefficients along the x-axis given by dFx/ ax for x- and y-polarized beams, respectively, Ref. [12]. The aforementioned s/ was computed at the x-offset value of 50nm, where z-offset A Oum is chosen to yield the maximum of Fr in the vicinity of the center of the small rectangles depicted in Fig. 5, where Fx is fairly insensitive to small variations of =. The computed stiffness anisotropy is plotted in Fig. 6 versus the particle diameter d for spherical #67575-$15.00USD Received 15 February 2006, revised 7 April 2006, accepted 10 April 2006 (C)2006OSA 17 April 2006/Vol 14, No 8/OPTICS EXPRESS 3665
Fig. 3. Computed distributions of the electric field intensity |E| 2 (units: [V2/m2]) in the xz- and yz-planes near the focus of a 0.9NA, f = 5.0mm, diffraction-limited objective. The λ0 = 532nm plane-wave illuminating the entrance pupil of the lens is linearly polarized along the x-axis. Fig. 4. Distribution of the Poynting vector S (units: [W/m2]) in and around a glass micro-sphere (n = 1.5, d = 460nm). The focused beam, obtained by sending a linearly-polarized plane-wave (polarization along y) through a 0.9NA objective, propagates along the negative z-axis. Sphere center offset from the focal point: (250,0,50)nm. The (Sx,Sz) vector-field is superimposed on the color-coded Sz plot on the right-hand side. if the laser power level is adjusted to a few microwatts, then the radiation pressure will work against the force of gravity to hold the bead in a stable trap along the z-axis.) The computed anisotropy of this trap in the lateral direction is sl = 1−(κx/κy) = −0.15, where κx and κy are the trap stiffness coefficients along the x-axis given by ∂Fx/∂x for x- and y-polarized beams, respectively, Ref. [12]. The aforementioned sl was computed at the x-offset value of 50nm, where z-offset ≈ 0μm is chosen to yield the maximum of Fx in the vicinity of the center of the small rectangles depicted in Fig. 5, where Fx is fairly insensitive to small variations of z. The computed stiffness anisotropy is plotted in Fig. 6 versus the particle diameter d for spherical #67575 - $15.00 USD Received 15 February 2006; revised 7 April 2006; accepted 10 April 2006 (C) 2006 OSA 17 April 2006 / Vol. 14, No. 8 / OPTICS EXPRESS 3665
beads having n=1.5, trapped under a 20=1064nm focused beam through a 0.9NA objective 00050.100150.200250.300050.100.150200250300050.100.15020025030 Fig. 5. Plots of the net force components(Fr, F) experienced by a glass micro-sphere(d 460nm, n=1.5) versus the offset from the focal point in the xs-plane. The incidence medium is air, no=532nm, the objective lens NA is 0.9, and the incident beams power is P= 1. oW. Top row: x-polarization, bottom row: y-polarization. The stiffness coefficients Kx, Ky are computed at the center of the small rectangles shown on the left-hand-side of the Fr plots 0=1064m,NA=0.9 -0.2 -0.4 micro-sphere diameter [um] Fig. 6. Computed trap stiffness anisotropy s)=1-(Kx/K, )versus particle diameter d, fo micro-spheres of refractive index n= 1.5 trapped in the air with a no= 1064nm laser beam focused through a 0. 9NA objective lens. The stiffness is computed at x-offset= 50nm, =-oftset Oum, where, for the chosen value of x-offset, the lateral trapping force F is at a maximum. For the offset ranges and particle diameters considered, the radiation force along the =-axis Fs, was found to be negative (i.e, inverted traps are necessary to achieve stable trapping). The #67575-$15.00USD Received 15 February 2006, revised 7 April 2006, accepted 10 April (C)2006OSA 17 April 2006/Vol 14, No 8/OPTICS EXPRESS
beads having n = 1.5, trapped under a λ 0 = 1064nm focused beam through a 0.9NA objective. Fx Fz (Fx,Fz) Fig. 5. Plots of the net force components (Fx,Fz) experienced by a glass micro-sphere (d = 460nm, n = 1.5) versus the offset from the focal point in the xz-plane. The incidence medium is air, λ0 = 532nm, the objective lens NA is 0.9, and the incident beam’s power is P = 1.0W. Top row: x-polarization, bottom row: y-polarization. The stiffness coefficients κx, κy are computed at the center of the small rectangles shown on the left-hand-side of the Fx plots. 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 micro-sphere diameter [μm] -0.4 -0.2 0 0.2 0.4 1 - κx /κy FDTD: n = 1.5, ninc = 1.0, λ0 = 1064nm, NA = 0.9 Cubic spline interpolation Fig. 6. Computed trap stiffness anisotropy sl = 1 − (κx/κy) versus particle diameter d, for micro-spheres of refractive index n = 1.5 trapped in the air with a λ0 = 1064nm laser beam focused through a 0.9NA objective lens. The stiffness is computed at x-offset = 50nm, z-offset ≈ 0μm, where, for the chosen value of x-offset, the lateral trapping force Fx is at a maximum. For the offset ranges and particle diameters considered, the radiation force along the z-axis, Fz, was found to be negative (i.e., inverted traps are necessary to achieve stable trapping). The #67575 - $15.00 USD Received 15 February 2006; revised 7 April 2006; accepted 10 April 2006 (C) 2006 OSA 17 April 2006 / Vol. 14, No. 8 / OPTICS EXPRESS 3666