Abstract

The Maxwell stress tensor and the distributed Lorentz force are applied to calculate forces on lossless media and are shown to be in excellent agreement. From the Maxwell stress tensor, we derive analytical formulae for the forces on both a half-space and a slab under plane wave incidence. It is shown that a normally incident plane wave pushes the slab in the wave propagation direction, while it pulls the half-space toward the incoming wave. Zero tangential force is derived at a boundary between two lossless media, regardless of incident angle. The distributed Lorentz force is applied to the slab in a direct way, while the half-space is dealt with by introducing a finite conductivity. In this regard, we show that the ohmic losses have to be properly accounted for, otherwise differing results are obtained. This contribution, together with a generalization of the formulation to magnetic materials, establishes the method on solid theoretical grounds. Agreement between the two methods is also demonstrated for the case of a 2-D circular dielectric particle.

© 2005 Optical Society of America

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References

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J. Mod. Opt. (1)

R. Loudon, �??Theory of radiation pressure on dielectric surfaces,�?? J. Mod. Opt. 49, 821-838 (2002).
[CrossRef]

Opt. Express (4)

Phys. Rev. A (1)

R. Loudon, S. M. Barnett, and C. Baxter, �??Radiation pressure and momentum transfer in dielectrics: the photon drag effect,�?? Phys. Rev. A 71, 063802 (2005).
[CrossRef]

Other (3)

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941), ISBN 0-07-062150-0.

J. A. Kong, Electromagnetic Wave Theory (EMW, 2005), ISBN 0-9668143-9-8.

L. Tsang, J. A. Kong, and K. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000), ISBN 0-471-38799-7.
[CrossRef]

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Figures (8)

Fig. 1.
Fig. 1.

A plane wave is incident onto a slab of thickness d and incident angle θ 0. The integration path for the application of the Maxwell stress tensor to calculate the force on a slab is shown by the dotted lines. The path is shrunk so that δz → 0. The integration is performed along the surface on both sides of a boundaries.

Fig. 2.
Fig. 2.

Force density from a normal incident wave onto a lossless slab as a function of slab thickness d. The free space wavelength is λ 0 = 640nm and εr = 4, μr = 1, Ei = 1. Shown are the forces calculated from the distributed Lorentz force (circles) and the Maxwell stress tensor (line). The background medium (region 0 and region 2) is free space.

Fig. 3.
Fig. 3.

Force density from an oblique incident wave on a quarter-wave slab (d = λ 1/4 = 80nm) as a function of incident angle θ 0. The free space wavelength is λ 0 = 640nm and εr = 1, μr = 4, Ei = 1. Shown are the forces calculated from the distributed Lorentz force (circles) and the Maxwell stress tensor (line). The background medium (region 0 and region 2) is free space.

Fig. 4.
Fig. 4.

Force density from a normal incident wave on a lossless slab (d = 80nm) as a function of the relative permittivity εr . The free space wavelength is λ 0 = 640nm and μr = 1,Ei = 1. Shown are the forces calculated from the distributed Lorentz force (circles) and the Maxwell stress tensor (line). The background media (region 0 and region 2) is free space.

Fig. 5.
Fig. 5.

Force density from a normal incident wave on a lossless half-space medium as calculated from the Maxwell stress tensor. The -directed force is shown as a function of the relative permittivity εr . The free space wavelength is λ 0 = 640nm and μr = 1, Ei = 1. The incident media (region 0) is free space. The dotted lines denote ±ε 0 Ei2 = 8.85 pN/m 2

Fig. 6.
Fig. 6.

The incident electric field magnitude [V/m] is due to three plane waves of free space wavelength λ 0 = 532 nm propagating at angles {π/2,7π/6, 11π/6}. The overlayed particle is a 2D polystyrene cylinder (εp = 2.56ε 0) of radius a = 0.3λ 0 with center position (x 0,y 0) = (0,100) [nm] embedded in water (εb = 1.69ε 0).

Fig. 7.
Fig. 7.

The total field is calculated from Mie theory due to the three incident plane waves on a polystyrene cylinder (εp = 2.56ε 0) of radius a = 0.3λ 0 with center position (x 0,y 0) = (0,100) [nm] embedded in water (εb = 1.69ε 0). Shown are (a) |z | [V/m], (b) |x | [10-3 H/m], and (c) |y | [10-3 H/m].

Fig. 8.
Fig. 8.

The distributed Lorentz force for a polystyrene cylinder (εp = 2.56ε 0) of radius a = 0.3λ 0 with center position (x 0,y 0) = (0,100) [nm] embedded in water (εb = 1.69ε 0) due to three incident plane waves are shown. The individual components are (a) fx [10-4 N/m 3] and (b) fy [10-4 N/m 3]. The total force is obtained by integrating the distributed Lorentz force inside the particle.

Equations (40)

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f ˉ ( r ˉ , t ) = G ˉ ( r ˉ , t ) t · T ˭ ( r ˉ , t ) ,
F ˉ ( t ) = t V dV G ˉ ( r ˉ , t ) S dS [ n ̂ · T ˭ ( r ˉ , t ) ] .
F ˉ = 1 2 Re { S dS [ n ̂ · T ˭ ( r ˉ ) ] } ,
T ˭ ( r ˉ ) = 1 2 ( D ˉ · E ˉ * + B ˉ * · H ˉ ) I ˭ D ˉ E ˉ * B ˉ * H ˉ ,
f ˉ = 1 2 Re { ρ e E ˉ * + J ˉ × B ˉ * + ρ m H ˉ * + M ˉ × D ˉ * } ,
P ˉ m = μ 0 ( μ r 1 ) H ˉ .
M ˉ = P ˉ m = μ 0 ( μ r 1 ) H ˉ .
J ˉ = P ˉ e = ε 0 ( ε r 1 ) E ˉ ,
ρ e = n ̂ · ( E ˉ 1 E ˉ 0 ) ε 0 ρ m = n ̂ · ( H ˉ 1 H ˉ 0 ) μ 0 ,
E ˉ i = y ̂ E i e ik 0 z z e i k x x ,
F ¯ = 1 2 Re { z ̂ · T ˭ ( z = 0 ) z ̂ · T ˭ ( z = 0 + ) + z ̂ · T ˭ ( z = d ) z ̂ · T ˭ ( z = d + ) } ,
z ̂ · T ˭ ( z = 0 + ) = z ̂ [ ε 1 2 E y ( z = 0 + ) 2 + μ 1 2 ( H x ( z = 0 + ) 2 H z ( z = 0 + ) 2 ) ]
+ x ̂ [ μ 1 H z ( z = 0 + ) H x * ( z = 0 + ) ] ,
z ̂ · T ˭ ( z = d ) = z ̂ [ ε 1 2 E y ( z = d ) 2 + μ 1 2 ( H x ( z = d ) 2 H z ( z = d ) 2 ) ]
+ x ̂ [ μ 1 H z ( z = d ) H x * ( z = d ) ] .
z ̂ · T ˭ ( z = 0 + ) = z ̂ · T ˭ ( z = d ) ,
F ˉ = 1 2 Re { z ̂ · T ˭ ( z = 0 ) z ̂ · T ˭ ( z = d + ) } .
F ˉ = z ̂ ε 0 2 E i 2 cos 2 θ 0 [ + R slab 2 T slab 2 ] ,
F ˉ i = z ̂ ε 0 2 E i 2 ,
F ˉ r = z ̂ ε 0 2 E i 2 R slab 2 ,
F ˉ t = z ̂ ε 0 2 E i 2 T slab 2 ,
θ b = tan 1 μ 1 μ 0 ,
F ¯ = 1 2 Re { z ̂ · T ˭ ( z = 0 ) z ̂ · T ˭ ( z = 0 + ) } .
z ̂ · T ˭ ( z = 0 ) = z ̂ [ ε 0 2 E y ( z = 0 ) 2 + μ 0 2 ( H x ( z = 0 ) 2 H z ( z = 0 ) 2 ) ]
+ x ̂ [ μ 0 H z ( z = 0 ) H x * ( z = 0 ) ] ,
z ̂ · T ˭ ( z = 0 + ) = z ̂ [ ε 1 2 E y ( z = 0 + ) 2 + μ 1 2 ( H x ( z = 0 + ) 2 H z ( z = 0 + ) 2 ) ]
+ x ̂ [ μ 1 H z ( z = 0 + ) H x * ( z = 0 + ) ] ,
F x = 1 2 Re { μ 0 H z ( z = 0 ) H x * ( z = 0 ) + μ 1 H z ( z = 0 + ) H x * ( z = 0 + ) } .
F x = 1 2 E i 2 k x ω 2 Re { ( k 0 z μ 0 ) * [ ( 1 + R hs ) ( 1 R hs ) * p 01 * T hs 2 ] }
p 01 = μ 0 k 1 z μ 1 k 0 z .
p 01 * T hs 2 = ( 1 + R hs ) ( 1 R hs ) *
F ˉ = z ̂ ε 0 2 E i 2 [ cos 2 θ 0 ( 1 + R hs 2 ) ε r cos 2 θ 1 T hs 2 ] ,
lim ε r F ̄ = z ̂ ε 0 E i 2 .
lim ε r 0 F ̄ = + z ̂ ε 0 E i 2 ,
P c = 1 2 Re { V dV J ˉ c ( r ˉ ) · E ˉ * ( r ˉ ) } ,
F ˉ = lim σ 0 1 2 Re { 0 dz [ J ˉ × B ˉ * z ̂ 1 v e ( J ˉ c · E ˉ * ) ] } ,
F z = ε 0 2 E i 2 ( 1 + R hs 2 ) lim σ 0 1 2 σ v e 2 k I E i 2 T hs 2 ,
v e 1 μ 1 ε 1 k I σ 2 μ 1 ε 1 .
F z = ε 0 2 E i 2 ( 1 + R hs 2 ) ε 1 2 E i 2 T hs 2
f ̄ = 1 2 Re { P ̄ e × μ 0 H * } ,

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