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Computation of the optical trapping force using an FDTD based technique

Robert C. Gauthier

Open Access Open Access

Abstract

The computation details related to computing the optical radiation pressure force on various objects using a 2-D grid FDTD algorithm are presented. The technique is based on propagating the electric and magnetic fields through the grid and determining the changes in the optical energy flow with and without the trap object(s) in the system. Traces displayed indicate that the optical forces and FDTD predicted object behavior are in agreement with published experiments and also determined through other computation techniques. We show computation results for a high and low dielectric disc and thin walled shell. The FDTD technique for computing the light-particle force interaction may be employed in all regimes relating particle dimensions to source wavelength. The algorithm presented here can be easily extended to 3-D and include torque computation algorithms, thus providing a highly flexible and universally useable computation engine.

©2005 Optical Society of America

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Figures (8)
Fig. 1.
Fig. 1. (X, Y) computation domain showing a single sphere axially offset and displaced in the divergent region of the source plane. The computation domain is discretized into an (I, J) grid. The boundary is composed of a perfectly matched layer 15 grid points wide.

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Fig. 2.
Fig. 2. (a)Efficiency factor Qx for radial force on a high dielectric sphere in water. (square-1 µm, circle-4 µm from minimum waist). (b)Efficiency factor Qy for the axial force on the same sphere. Sphere and beam parameters defined in text. The FDTD analysis predicts that axial trapping is expected and that the sphere is pushed in the direction of beam propagation.

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Fig. 3.
Fig. 3. (a) Efficiency factor Qx for radial force on a low dielectric sphere in water. (square-1 µm, circle-4 µm from minimum waist). (b) Efficiency factor Qy for the axial force on the same sphere. Sphere and beam parameters defined in text. The FDTD analysis correctly predicts that the sphere is pushed out of the beam and along the beam propagation direction.

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Fig. 4.
Fig. 4. FDTD computation domain traces. Center trace is a gray scale representation of the electric field plotted for each (I, J) grid point. White represents high electric field values. In this trace a plane wave is incident from the left onto a positive lens (large circle) and is focused before diverging. The top trace shows the electric field profile down the Y-axis centerline. The envelope maximum of the E field profile corresponds to the focal point of the lens and can be quantified knowing the grid dimensions or source wavelength in the various mediums. The lower trace is a top plot of the E field profile through the focal region. From this trace the minimum waist can be determined as well the level of aberrations present in the optical trap system.

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Fig. 5.
Fig. 5. Efficiency factor Qy for the high dielectric disc axially aligned with the beam center and propagated through the focal region of the beam. Focusing system and E field profile shown in figure 4. In the minimum waist region Qy is less than zero indicating that laser trapping can be accomplished with these beam-object-focusing-system parameters.

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Fig. 6.
Fig. 6. Radial trapping efficiency (Qx) for the thick walled dielectric shell. The central region has 1.00 index, the shell has 1.45, and the ambient index is 1.33. The shell is pushed out of axial alignment with the beam but may be axially trapped, offset from the beam axis as indicated in the figure.

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Fig. 7.
Fig. 7. (a) FDTD field profile for the dual beam counter-propagating trap configuration. Left and right Gaussian beams have been propagated a short distance in the system. Beam on the left is focusing after passing through the lens while the beam on the right is still diverging. (b) Axial force profile obtained when the beam separation corresponds to 4F. The mid point axial location corresponds to an unstable equilibrium for this sphere between these beams. (c) Axial force profile obtained when the beam separation corresponds to 6F. The mid point axial location corresponds to a stable equilibrium for this sphere between these beams.

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Fig. 8.
Fig. 8. FDTD E-field trace of the fiber-to-fiber coupling through a dual beam trapped ball lens. The FDTD trapping computation engine developed permits the optical system to be modeled, the field and beams propagated and the resultant E fields permit the determination of the coupling efficiency.

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Equations (6)

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F x = Δ S x c L , Δ S x = S xo − S x
F y = Δ S y c L , Δ S y = S yo − S y
A xy = exp [ − 2 ( x − x c W 0 ) 2 ]
E z ( x , y c ) = E z ( x , y c ) + A xy sin ( ω t )
H x ( x , y c ) = H x ( x , y c ) + A xy sin ( ω t )
Q x , y = c F x , y n s P
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