Abstract

Using numerical solutions of Maxwell’s equations in conjunction with the Lorentz law of force, we compute the electromagnetic force distribution in and around a dielectric micro-sphere trapped by a focused laser beam. Dependence of the optical trap’s stiffness on the polarization state of the incident beam is analyzed for particles suspended in air or immersed in water, under conditions similar to those realized in practical optical tweezers. A comparison of the simulation results with available experimental data reveals the merit of one physical model relative to two competing models; the three models arise from different interpretations of the same physical picture.

© 2006 Optical Society of America

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References

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  1. M. Mansuripur, "Radiation Pressure and the linear momentum of the electromagnetic field," Opt. Express 12,5375-5401 (2004), http://www.opticsexpress.org/abstract.cfm?id=81636.
    [CrossRef] [PubMed]
  2. M. Mansuripur, A.R. Zakharian and J.V. Moloney, "Radiation Pressure on a dielectric wedge," Opt. Express 13,2064-2074 (2005), http://www.opticsexpress.org/abstract.cfm?id=83011.
    [CrossRef] [PubMed]
  3. M. Mansuripur, "Radiation Pressure and the linear momentum of light in dispersive dielectric media," Opt. Express 13,2245-2250 (2005), http://www.opticsexpress.org/abstract.cfm?id=83032.
    [CrossRef] [PubMed]
  4. M. Mansuripur, "Angular momentum of circularly polarized light in dielectric media," Opt. Express 13,5315-5324 (2005), http://www.opticsexpress.org/abstract.cfm?id=84895.
    [CrossRef] [PubMed]
  5. S. M. Barnett and R. Loudon, "On the electromagnetic force on a dielectric medium," submitted to J. Phys. B: At. Mol. Phys. (January 2006).
  6. A.R. Zakharian, M. Mansuripur and J.V. Moloney, "Radiation Pressure and the distribution of the electromagnetic force in dielectric media," Opt. Express 13,2321-2336 (2005), http://www.opticsexpress.org/abstract.cfm?id=83272.
    [CrossRef] [PubMed]
  7. R. Gauthier, "Computation of the optical trapping force using an FDTD based technique," Opt. Express 13,3707-3718 (2005), http://www.opticsexpress.org/abstract.cfm?id=83817.
    [CrossRef] [PubMed]
  8. A. Rohrbach and E.H.K. Stelzer, "Three-dimensional position detection of optically trapped dielectric particles," J. Appl. Phys. 91,5474-5488 (2002).
    [CrossRef]
  9. A. Rohrbach, "Stiffness of Optical Traps: Quantitative agreement between experiment and electromagnetic theory," Phys. Rev. Lett. 95,168102 (2005).
    [CrossRef] [PubMed]
  10. W.H. Wright, G.J. Sonek, and M.W. Berns, "Radiation trapping forces on microspheres with optical tweezers," Appl. Phys. Lett. 63,715-717 (1993).
    [CrossRef]
  11. W.H. Wright, G.J. Sonek, and M.W. Berns, "Parametric study of the forces on microspheres held by optical tweezers," Appl. Opt. 33,1735-1748 (1994).
    [CrossRef] [PubMed]
  12. A. Rohrbach and E. H. K. Stelzer, "Trapping forces, force constants, and potential depths for dielectric spheres in the presence of spherical aberrations," Appl. Opt. 41,2494-2507 (2002).
    [CrossRef] [PubMed]
  13. D. Ganic, X. Gan and M. Gu, "Exact radiation trapping force calculation based on vectorial diffraction theory," Opt. Express 12,2670-2675 (2004), http://www.opticsexpress.org/abstract.cfm?id=80240.
    [CrossRef] [PubMed]
  14. P.W. Barber and S.C. Hill, Light Scattering by Particles: Computational Methods (World Scientific Publishing Co. 1990).

2005

2004

2002

A. Rohrbach and E. H. K. Stelzer, "Trapping forces, force constants, and potential depths for dielectric spheres in the presence of spherical aberrations," Appl. Opt. 41,2494-2507 (2002).
[CrossRef] [PubMed]

A. Rohrbach and E.H.K. Stelzer, "Three-dimensional position detection of optically trapped dielectric particles," J. Appl. Phys. 91,5474-5488 (2002).
[CrossRef]

1994

1993

W.H. Wright, G.J. Sonek, and M.W. Berns, "Radiation trapping forces on microspheres with optical tweezers," Appl. Phys. Lett. 63,715-717 (1993).
[CrossRef]

Berns, M.W.

W.H. Wright, G.J. Sonek, and M.W. Berns, "Parametric study of the forces on microspheres held by optical tweezers," Appl. Opt. 33,1735-1748 (1994).
[CrossRef] [PubMed]

W.H. Wright, G.J. Sonek, and M.W. Berns, "Radiation trapping forces on microspheres with optical tweezers," Appl. Phys. Lett. 63,715-717 (1993).
[CrossRef]

Gan, X.

Ganic, D.

Gauthier, R.

Gu, M.

Mansuripur, M.

Moloney, J.V.

Rohrbach, A.

A. Rohrbach, "Stiffness of Optical Traps: Quantitative agreement between experiment and electromagnetic theory," Phys. Rev. Lett. 95,168102 (2005).
[CrossRef] [PubMed]

A. Rohrbach and E. H. K. Stelzer, "Trapping forces, force constants, and potential depths for dielectric spheres in the presence of spherical aberrations," Appl. Opt. 41,2494-2507 (2002).
[CrossRef] [PubMed]

A. Rohrbach and E.H.K. Stelzer, "Three-dimensional position detection of optically trapped dielectric particles," J. Appl. Phys. 91,5474-5488 (2002).
[CrossRef]

Sonek, G.J.

W.H. Wright, G.J. Sonek, and M.W. Berns, "Parametric study of the forces on microspheres held by optical tweezers," Appl. Opt. 33,1735-1748 (1994).
[CrossRef] [PubMed]

W.H. Wright, G.J. Sonek, and M.W. Berns, "Radiation trapping forces on microspheres with optical tweezers," Appl. Phys. Lett. 63,715-717 (1993).
[CrossRef]

Stelzer, E. H. K.

Stelzer, E.H.K.

A. Rohrbach and E.H.K. Stelzer, "Three-dimensional position detection of optically trapped dielectric particles," J. Appl. Phys. 91,5474-5488 (2002).
[CrossRef]

Wright, W.H.

W.H. Wright, G.J. Sonek, and M.W. Berns, "Parametric study of the forces on microspheres held by optical tweezers," Appl. Opt. 33,1735-1748 (1994).
[CrossRef] [PubMed]

W.H. Wright, G.J. Sonek, and M.W. Berns, "Radiation trapping forces on microspheres with optical tweezers," Appl. Phys. Lett. 63,715-717 (1993).
[CrossRef]

Zakharian, A.R.

Appl. Opt.

Appl. Phys. Lett.

W.H. Wright, G.J. Sonek, and M.W. Berns, "Radiation trapping forces on microspheres with optical tweezers," Appl. Phys. Lett. 63,715-717 (1993).
[CrossRef]

J. Appl. Phys.

A. Rohrbach and E.H.K. Stelzer, "Three-dimensional position detection of optically trapped dielectric particles," J. Appl. Phys. 91,5474-5488 (2002).
[CrossRef]

Opt. Express

Phys. Rev. Lett.

A. Rohrbach, "Stiffness of Optical Traps: Quantitative agreement between experiment and electromagnetic theory," Phys. Rev. Lett. 95,168102 (2005).
[CrossRef] [PubMed]

Other

S. M. Barnett and R. Loudon, "On the electromagnetic force on a dielectric medium," submitted to J. Phys. B: At. Mol. Phys. (January 2006).

P.W. Barber and S.C. Hill, Light Scattering by Particles: Computational Methods (World Scientific Publishing Co. 1990).

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Figures (14)

Fig. 1.
Fig. 1.

Surface force integrated over the left-half of the cylinder as function of the cylinder’s refractive index ncyl . The incident plane-wave has vacuum wavelength λ0 = 0.65μm, and the incidence medium’s refractive index is ninc = 1.0 (solid lines) or ninc = 1.3 (dashed lines). The exact results of the Lorentz-Mie theory (circles) are compared with those of our FDTD-based method (triangles and crosses). In the case of cylinder immersed in water, the charges induced at the solid-liquid interface were lumped together, then subjected to the average E-field at the interface; in other words, no attempts were made in these calculations to distinguish the force of the light’s E-field on the solid surface from that on the adjacent liquid. For each FDTD simulation the grid resolution Δ is indicated.

Fig. 2.
Fig. 2.

Dielectric prism of refractive index n immersed in a host medium of refractive index ninc .

Fig. 3.
Fig. 3.

Computed distributions of the electric field intensity |E|2 (units: [V 2/m 2]) 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.
Fig. 4.

Distribution of the Poynting vector S (units: [W /m 2]) 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.

Fig. 5.
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.

Fig. 6.
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.

Fig. 7.
Fig. 7.

(a) A linearly-polarized Gaussian beam having wavelength λ0=532nm and 1/e (amplitude) radius r 0= 4.0mm is focused through a 1.4NA oil immersion objective lens. The lens has focal length f=3.0mm and aperture radius (at the entrance pupil) Ra =2.85mm; the refractive index of the immersion oil is 1.47. A glass plate of the same index as the oil separates the oil from the water (nwater =1.33), where the focused spot is used to trap various dielectric beads. The marginal rays are lost by total internal reflection at the glass-water interface; there is also some degree of apodization due to Fresnel reflection at this interface, but the phase aberrations induced by the transition from oil/glass to water are ignored in our calculations. The neglect of the spherical aberrations thus induced is justifiable, so long as the trap is not too far from the oil/water interface, Ref. [12]. (b)-(d) Logarithmic plots of the intensity distribution for x-, y-, and z-components of the E-field at the focal plane. The relative peak intensities are |Ex |2:|Ey |2: |Ez |2 ≈ 1000 : 9 : 200. The total intensity distribution at the focal plane, I(x,y) = |Ex |2 +|Ey |2 + |Ez |2, (not shown) is elongated in the x-direction; the full-width at half-maximum intensity (FWHM) of the focused spot is 300nm along x and 196nm along y. The transmission efficiency of the oil/glass-to-water interface is 92.6%; that is, the overall loss of optical power to total-internal and Fresnel reflections at this interface is 7.4%.

Fig. 8.
Fig. 8.

Computed distributions of the electric field intensity |E|2 (units: [V 2/m 2]) in the xz-and xz-planes near the focus of a λ0 = 532nm beam in water. The focused spot is obtained by sending an x-polarized plane-wave through an oil-immersion ≈ 1.4NA objective. The oil and water are separated by a thin glass slide, index-matched to the immersion oil, Fig. 7.

Fig. 9.
Fig. 9.

Plots of the net force components (Fx ,Fz ), computed with Method I, for a glass micro-bead (d = 460nm, n = 1.5) versus the offset from the focal point. The host medium is water (ninc = 1.33), λ0 = 532nm, the objective lens NA is ≈ 1.4, and the incident beam’s power is P = 1.0W. Top row: x-polarization, bottom row: y-polarization. The non-trapping behavior of x-polarized beam, which is contrary to experimental observations, indicates the invalidity of Method I used in these calculations.

Fig. 10.
Fig. 10.

Plots of the net force components (Fx ,Fz ), computed with Method II, for a glass micro-bead (d = 460nm, n = 1.5) versus the offset from the focal point. The host medium is water (ninc = 1.33), λ0 = 532nm, the objective lens NA is ≈ 1.4, and the incident beam’s power is P = 1.0W. Top row: x-polarization, bottom row: y-polarization. The lateral trapping force Fx at the center of the small rectangle in the case of x-polarization is weaker than the corresponding Fx in the case of y-polarization (ratio = 0.92).

Fig. 11.
Fig. 11.

Plots of the net force components (Fx ,Fz ), computed with Method III, for a glass micro-bead (d = 460nm, n = 1.5) versus the offset from the focal point. The host medium is water (ninc = 1.33), λ0 = 532nm, the objective lens NA is ≈ 1.4, and the incident beam’s power is P = 1.0W. Top row: x-polarization, bottom row: y-polarization. The lateral trapping force Fx at the center of the small rectangle in the case of x-polarization is stronger than the corresponding Fx in the case of y-polarization (ratio = 1.2).

Fig. 12.
Fig. 12.

Trap stiffness anisotropy sl = 1 - (κx /κy ) versus particle diameter d, computed with method II for polystyrene micro-beads (n = 1.57) trapped in water (ninc = 1.33) under a λ0 = 1064nm laser beam focused through an oil-immersion ≈ 14NA objective lens. Numerically, κx and κy were evaluated with a forward finite-difference approximation at x-offset = 50 - 100nm, offset discretization Δx = 50nm, and z-offset = z 0 where Fz (x = 0,z 0) = 0. The computed value of z 0 for each particle is listed on the right-hand side. The solid triangles (green) represent experimental data obtained with a system similar to that depicted in Fig. 7, operating at λ0 = 1064nm, and suffering from chromatic (and possibly spherical) aberrations. The solid circles (blue) represent experimental data from Table II of Rohrbach Ref. [9]. Although Rohrbach uses a 1.2NA water immersion objective in his experiments, the comparison is warranted here because our simulated 1.4NA focused beam, upon transmission from oil to water, loses its marginal rays and acquires characteristics that are not too far from those of a 1.2NA diffraction-limited focused spot. (Our experimental methodology is similar to that of Rohrbach, the only difference being that we use the oil-immersion objective depicted in Fig. 7(a), whereas Rohrbach uses a water-immersion objective.)

Fig. A1.
Fig. A1.

Examples of electrostatic field distributions in two dielectric media. On the left-hand side, the medium’s dielectric constant ε(x) varies along the x-axis while the electric displacement D = D 0 remains constant. On the right-hand side,ε is constant within the crescent-shaped medium (r 0 < r < r 1, θ 0 < θ < θ 1), while the azimuthally oriented D(r) = (D 0 r 0/r) θ ^ decreases with the inverse of the radius r.

Fig. A2.
Fig. A2.

When the object is surrounded by a medium other than the free space, a narrow gap may be imagined to exist between the object and its surroundings, so that, within the gap, the polarization density P(r) may be set to zero. The integration boundary is then placed within the gap to ensure that the integrated force densities corresponding to Eqs. (A1) and (A2) lead to the same total force.

Tables (2)

Tables Icon

Table 1. Definitions of the surface charge and surface force densities for methods I,II and III.

Tables Icon

Table 2. Radiation force on the prism of Fig. 2, computed with the exact method and with the FDTD simulations. A p-polarized (Hx ,Ey ,Ez ) Gaussian beam illuminates the prism at Brewster’s angle θB . The net radiation force exerted on the prism is denoted by (Fy ,Fz ). The value of the mesh parameter Δ used in each simulation is indicated.

Equations (19)

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F 1 ( r ) = ( · P ) E + ( P t ) × B
F 2 ( r ) = ( P · ) E + ( P t ) × B
F 2 ( r ) = 1 4 ε 0 ( ε 1 ) ( E x 2 + E y 2 + E z 2 ) .
F 1 x surface ( x = 0 ) = 1 2 P x ( 0 ) [ E 0 + E x ( 0 ) ] = ( D 0 2 ε 0 ) [ 1 + 1 ε ( 0 ) ] P x ( 0 ) ; x = 0
F 1 x bulk ( x ) = E x ( x ) d P x ( x ) dx = ( 1 2 ε 0 ) d [ D 0 P x ( x ) ] 2 dx ; 0 < x < L
F 1 x surface ( x = L ) = ( D 0 2 ε 0 ) [ 1 + 1 ε ( L ) ] P x ( L ) ; x = L
F 2 x surface ( x = 0 ) = 1 2 P x ( 0 ) [ E x ( 0 ) E 0 ] = ( D 0 2 ε 0 ) [ 1 1 ε ( 0 ) ] P x ( 0 ) ; x = 0
F 2 x s u r f a c e ( x ) = P x ( x ) d E x ( x ) / d x = ( 1 / 2 ε 0 ) d P x 2 ( x ) / d x ; 0 < x < L
F 2 x surface ( x = L ) = ( D 0 2 ε 0 ) [ 1 1 ε ( L ) ] P x ( L ) ; x = L
F 2 surface ( r , θ = θ 0 , 1 ) = ± ( 1 2 ε 0 ) ( 1 1 ε 2 ) ( D 0 r 0 r ) 2 θ ̂ ; θ = θ 0 , θ 1
F 2 surface ( r , θ = θ 0 , 1 ) = ± ( 1 2 ε 0 ) ( 1 1 ε ) 2 ( D 0 r 0 r ) 2 θ ̂ ; θ = θ 0 , θ 1
F 2 x bulk ( r , θ ) = ( 1 ε 0 ε ) ( 1 1 ε ) ( D 0 2 r 0 2 r 3 ) r ̂ ; θ 0 < θ < θ 1
F total = ( 1 ε 0 ) ( 1 1 ε 2 ) ( D 0 r 0 r ) 2 sin [ ( θ 1 θ 0 ) 2 ] r ̂ .
ΔF ( r ) = F 2 ( r ) F 1 ( r ) = ( P ) E + ( P ) E = ( P x E ) x + ( P y E ) y + ( P z E ) z
Δ T total = r × ΔF ( r ) dxdydz
= r × [ ( P x E ) + x + ( P y E ) y + ( P z E ) z ] dxdydz
= { [ y ( P y E z ) y z ( P z E y ) z ] x ̂ + [ z ( P z E x ) z x ( P x E z ) x ] y ̂ + [ x ( P x E y ) x y ( P y E y ) z ] z ̂ dxdydz
= [ ( P y E z P z E z ) x ̂ + ( P z E x P x E z ) y ̂ + ( P x E y P y E x ) z ̂ ] dxdydz
= ( E × P ) dxdydz

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