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Radiation pressure and the distribution of electromagnetic force in dielectric media

Armis R. Zakharian, Masud Mansuripur, and Jerome V. Moloney

Open Access Open Access

Abstract

Using the Finite-Difference-Time-Domain (FDTD) method, we compute the electromagnetic field distribution in and around dielectric media of various shapes and optical properties. With the aid of the constitutive relations, we proceed to compute the bound charge and bound current densities, then employ the Lorentz law of force to determine the distribution of force density within the regions of interest. For a few simple cases where analytical solutions exist, these solutions are found to be in complete agreement with our numerical results. We also analyze the distribution of fields and forces in more complex systems, and discuss the relevance of our findings to experimental observations. In particular, we demonstrate the single-beam trapping of a dielectric micro-sphere immersed in a liquid under conditions that are typical of optical tweezers.

©2005 Optical Society of America

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Figures (12)
Fig. 1.
Fig. 1. Computed force density Fz (per unit cross-sectional area) versus z inside a dielectric slab illuminated with a normally incident plane-wave (λo=0.64 µm). The slab, suspended in free-space, has n=2.0, d=110nm. The incident beam propagates along the negative z-axis.

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Fig. 2.
Fig. 2. Time-snapshots of the Ez component of a p-polarized plane-wave, λ o=0.65 µm, incident at θ=50° on a semi-infinite dielectric of refractive index n=3.4, located in the region z<0. (a) Abrupt transition of the refractive index at z=0. (b) Linear transition of the refractive index from n o=1.0 to n 1=3.4 over a 40 nm-thick region.

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Fig. 3.
Fig. 3. One-dimensional Gaussian beam in a homogeneous medium of refractive index n=2.0. The beam waist is at z=0.3µm, and the beam’s propagation direction is along the negative z-axis. (a) Time snapshot of Hx distribution for p-light, with superposed arrows depicting the (Ey, Ez ) vector field. (b) Time snapshot of Ex distribution for s-light, with the (Hy, Hz ) vector field superposed. (c) Force density distribution of the Fy component in the case of p-polarization, with the (Fy, Fz ) vector field superposed. (d) Distribution of the force density component Fy for the s-polarized beam, with the (Fy, Fz ) vector field superposed.

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Fig. 4.
Fig. 4. Time snapshots of the field components for a linearly-polarized wave (λo=0.65µm) having a top-hat cross-sectional profile with smooth edges, incident at θ inc=50° from free-space onto a semi-infinite dielectric of refractive index ns =2.0, located in the half-space z<0. (Left) Magnetic field Hx in the case of p-polarization; the superposed arrows represent the electric-field (Ey, Ez ). (Right) Electric field Ex in the case of s-polarization; the superposed arrows represent the magnetic-field (Hy, Hz ).

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Fig. 5.
Fig. 5. Left column: Fy and Fz force density plots for the p-polarized beam of Fig. 4(a). Right column: Fy and Fz force fields for the s-polarized beam of Fig. 4(b). The free-space/dielectric interface is not shown to exclude from the color scale the region of high force density due to the induced surface charges in the case of p-polarization.

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Fig. 6.
Fig. 6. Left to right: time-snapshot plots of Hx, Ey, Ez components of a p-polarized, one-dimensional Gaussian beam (λo=0.65 µm, FWHM=0.5 µm), propagating in free-space in the negative z-direction; the beam’s waist is at z=0.5 µm.

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Fig. 7.
Fig. 7. Gaussian beam of Fig. 6 in the presence of a glass cylinder having rc =0.2 µm, nc=2.0, centered at (y, z)=(0.2 µm, 0). Left column, top to bottom: distributions of |Hx |, Fy, Fz for p-light. Force density profiles include the contributions by surface charges. Right column, top to bottom: distributions of |Ex |, Fy, Fz for s-light. The field plots at the top are shown over a large region; dashed lines indicate the boundary of the cylinder.

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Fig. 8.
Fig. 8. Cross-sectional plots of force-density component distributions (color contours) through the center of dielectric sphere (rs =1.0 µm, ns =1.6), centered at (x, y, z)=(0.25µm, 0, 0) and immersed in a medium of index n o=1.3. The arrows show the projection of the force density vector in the cross-sectional plane, e.g., plots on the yz-plane show the (Fy, Fz ) vector field. Top to bottom: Fx, Fy, Fz . Left to right: xy-plane, xz-plane, yz-plane. The circularly-polarized incident beam is focused at (x, y, z)=(0, 0, 0.7µm) through a 1.25NA immersion objective.

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Fig. 9.
Fig. 9. Cross-sectional profiles of the integrated force-density components along the direction perpendicular to each cross-section. (Left) Distribution of ∫ F z (x, y, z) dz in the central xy-plane. (Middle) Distribution of ∫ Fy (x, y, z) dy in the central xz-plane. (Right) Distribution of ∫ F x (x, y, z) dx in the central yz-plane. The range of integration includes the charges on the surface of the sphere, but excludes any forces that might be exerted on the surrounding liquid.

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Fig. 10.
Fig. 10. Computed force components Fx (left column) and Fz (right column) versus the vertical displacement Δz of the focal point from the sphere center (Δz>0 when the beam is focused above the sphere center). Fx and Fz are in units of pico Newtons. Each colored curve represents a fixed lateral displacement Δx of the center of the particle from the optical axis of the objective lens. The beam is focused in the liquid through a 1.25NA immersion lens. The polarization state of the 0.5 mW plane wave (λo=0.65µm) entering the objective’s pupil is: circular (top row), linear along the x-axis (middle row), linear along the y-axis (bottom row).

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Fig. 11.
Fig. 11. Plots of field (Ex, Hx ) and force-density (Fy, Fz ) distribution when the Gaussian beam of Fig. 3 is incident from the free-space onto the edge (i.e., side-wall) of a dielectric half-slab. Left column: p-polarization; right column: s-polarization. Both the beam center and the slab-edge are at y=0. The top row shows profiles of Hx (p-light) and Ex (s-light) over a large area that includes the slab’s edge (dashed white lines) as well as the surrounding free-space. The color-coded force-density plots of Fy (second row) and Fz (third row) also show the vector field (Fy, Fz ) as superposed arrows. In the p-polarized case the color scale has been adjusted to exclude high force values (white) at the edges/corners of the half-slab.

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Fig. 12.
Fig. 12. Force density integrals, ∫ Fy (y, z) dz (black) and ∫ Fz (y, z) dz (red), for the half-slab of Fig. 11 over a range of the z-axis that includes the top and bottom facets of the slab. The z-integrated force is plotted versus the y-coordinate over the entire illuminated region including the slab’s vertical side-wall located at y=0. The incident beam is p-polarized on the left- and s-polarized on the right-hand-side.

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Equations (11)

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F = ρ b E + J b × B ,
J ̲ b = ∂ P ̲ ⁄ ∂ t = ε o ( ε − 1 ) ∂ E ̲ ⁄ ∂ t .
∇ × H ̲ = σ E ̲ + ∂ D ̲ ⁄ ∂ t .
ε o ∂ E ̲ ⁄ ∂ t = ∇ × H ̲ − ( σ E ̲ + ∂ P ̲ ⁄ ∂ t ) = ∇ × H ̲ − J ̲ ̂ ,
J ̲ ̂ = ∇ × H ̲ − ε o ∂ E ̲ ⁄ ∂ t .
ε 0 ε ∞ ∂ E ̅ ⁄ ∂ t = ∇ × H ̅ − σ E ̅ − J ̅ p ,
J ̅ ̂ = ( σ E ̅ + J ̅ p ) ⁄ ε ∞ + ( 1 − 1 ⁄ ε ∞ ) ∇ × H ̅ .
< F ̅ > = ( 1 ⁄ T ) ∫ 0 T ( E ̅ ∇ · ε o E ̅ + J ̂ ̅ × μ o H ̅ ) d t .
F ( edge ) = 1 4 ε o ( ε − 1 ) ∣ E o ∣ 2 .
S z ( y , z = 0.5 μ m ) = 1 2 E o H o exp [ − 2 ( y ⁄ y o ) 2 ] .
∫ S z ( y , z = 0.5 μ m ) d x = π ⁄ 8 E o H o y o = 0.5 × 10 − 3 W ⁄ m .
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