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Scattered field formulation of finite difference time domain for a focused light beam in dense media with lossy materials

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Abstract

Using the scattered field finite difference time domain (FDTD) formalism, equations for a plane wave incident from a dense medium onto lossy media are derived. The Richards-Wolf vector field equations are introduced into the scattered field FDTD formalism to model an incident focused beam. The results are compared to Mie theory scattering from spherical lossy dielectric and metallic spheres.

©2003 Optical Society of America

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Figures (7)

Fig. 1.
Fig. 1. |E|2 field intensity (incident plus scattered) in the xz plane at the center of a 100 nm silver sphere due to an incident plane wave propagating along the z axis and polarized along the x-axis with a wavelength of 700 nm calculated by (a) FDTD and (b) Mie theory.
Fig. 2.
Fig. 2. |E|2 field intensity (incident plus scattered) in the xz plane at the center of a 1 µm absorbing dielectric sphere in free space due to an incident plane wave with a wavelength of 850 nm. The field is calculated from (a) FDTD and (b) Mie theory.
Fig. 3.
Fig. 3. |E|2 field intensity (incident plus scattered) in the xz plane for a 1 µm lossy dielectric sphere with the same properties as Fig. 2. The FDTD calculation plotted in (a) uses the Lorentz dispersion relation. It is compared in (b) to the Debye dispersion relation and Mie theory along the z axis.
Fig. 4.
Fig. 4. |E|2 field intensity in the xz plane for a 100 nm spherical air bubble in a medium with refractive index=2 calculated by (a) FDTD and (b) Mie theory.
Fig. 5.
Fig. 5. |E|2 field intensity in the xz plane through the center of a 100 nm silver sphere in a medium with refractive index of 2 calculated by (a) FDTD and (b) Mie theory.
Fig. 6.
Fig. 6. FDTD and Mie theory calculation of the electric field intensity around a 100 nm silver sphere when illuminated by an x polarized focused beam with a half angle of 60° propagating in the -z direction. (a) is |Ex|2 for FDTD, (b) is |Ez|2 for FDTD, (c) is |Ex|2 for Mie theory, and (d) is |Ez|2 for Mie theory. The |Ey|2 component was negligible.
Fig. 7.
Fig. 7. FDTD and Mie theory calculation of the electric field intensity around a 500 nm silver sphere when illuminated by an x polarized focused beam with a half angle of 60° propagating in the -z direction. (a) |Ex|2 for FDTD, (b) |Ez|2 for FDTD, (c) |Ex|2 for Mie theory, and (d) |Ez|2 for Mie theory. The |Ey|2 component was negligible.

Equations (38)

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ε = ε + ε s ε 1 + i ω t 0 = ε + χ ( ω )
χ ( t ) = 1 2 π e i ω t χ ( ω ) d ω = ( ε s ε t 0 ) · e t t 0 U ( t )
E s n + 1 = ( 1 ε ε 0 + χ 0 ε 0 + σ · Δ t ) { ε ε 0 E s n + ε 0 m = 0 n 1 E s n m · Δ χ m + Δ t · × H s n + 1 2
σ · Δ t · E i n + 1 ε 0 · Δ t ( ε 1 ) · t E i n + 1 ε 0 · Δ t · t [ E i n + 1 ( t ) * χ ( t ) ] }
χ 0 ( ε s ε ) ( 1 e Δ t t 0 ) ,
Δ χ m = ( ε s ε ) ( e m Δ t t 0 ) ( 1 e Δ t t 0 ) 2 ,
t [ E i n + 1 ( t ) * χ ( t ) ] = t E i n + 1 ( t Λ ) · χ ( Λ ) d Λ
= ( ε s ε t 0 ) t [ e t t 0 · 0 t E i n + 1 ( ξ ) · e ξ t 0 d ξ ]
Ψ n = E s n · Δ χ 0 + e Δt t 0 · Ψ n 1 .
H s n + 1 2 = Δt μ 0 · × E s n .
E inc ( t ) = e i ω t i k 0 · p · U ( t )
k = k 0 ( sin θ cos ϕ x ̂ + sin θ sin ϕ y ̂ + cos θ z ̂ ) .
t [ E inc ( t ) * χ ( t ) ] = E 0 e i k 0 · p · ( ε s ε t 0 ) · t { e t t 0 0 t e ( i ω + 1 t 0 ) ξ d ξ }
= E 0 e i k 0 · p ( ε s ε 1 + i ω t 0 ) · [ i ω e i ω t + 1 t 0 e t t 0 ] .
ε ( ω ) = ε + ω p 2 ( ε s ε ) ω p 2 + 2 i ω δ p ω 2
χ p ( t ) = γ p e α p t sin ( β p t ) U ( t ) ,
α p χ p ,
β p ω p 2 δ p 2 ,
γ p ω p 2 ( ε s ε ) β p .
χ ̂ p ( t ) = i γ p e ( α p + i β p ) t U ( t )
χ p ( t ) = Re [ χ ̂ p ( t ) ] .
α p i β p 1 t 0
γ p i ( ε s ε ) t 0
χ ( t ) = γ p 2 i e t t 0 · U ( t ) γ p 2 i e t t 1 · U ( t ) χ 1 ( t ) χ 2 ( t )
t 1 1 α p + i β p .
t [ E inc ( t ) * χ ( t ) ] = E 0 e i k 0 · r p 2 i { ( t 0 γ p 1 + i ω t 0 ) · [ i ω e i ω t + 1 t 0 e t t 0 ]
( t 1 γ p 1 + i ω t 1 ) · [ i ω e i ω t + 1 t 1 e t t 1 ] } .
t [ E inc ( t ) * χ ( t ) ] = 1 2 ( E 0 γ p e i ω t i k 0 · r p ) [ ( ω t 0 1 + i ω t 0 ) ( ω t 1 1 + i ω t 1 ) ] .
e ( p ) = i λ 0 β d θ sin θ 0 2 π d φ a ( θ , φ ) e i k · p ,
p = ( x p , y p , z p ) = ( r p cos φ p , r p sin φ p , z p ) ,
k = 2 π λ 0 ( sin θ cos φ , sin θ sin φ , cos θ ) .
e A l = i l 1 [ 2 l + 1 l ( l + 1 ) ] · [ n s · ψ l ( k I a ) · ς l ( 1 ) ( k I a ) n s · ψ l ( k I a ) · ς l ( 1 ) ( k I a ) n s · ψ l ( k II a ) · ς l ( 1 ) ( k I a ) n a · ψ l ( k II a ) · ς l ( 1 ) ( k I a ) ]
m A l = i l 1 [ 2 l + 1 l ( l + 1 ) ] · [ n s · ψ l ( k I a ) · ς l ( 1 ) ( k I a ) n s · ψ l ( k I a ) · ς l ( 1 ) ( k I a ) n s · ψ l ( k II a ) · ς l ( 1 ) ( k I a ) n a · ψ l ( k II a ) · ς l ( 1 ) ( k I a ) ]
E r = cos φ ( k II ) 2 r 2 l = 1 l ( l + 1 ) e A l ψ l ( k II r ) P l ( 1 ) ( cos θ )
E θ = cos ϕ k II r l = 1 [ e A l ψ l ( k II r ) P l ( 1 ) ( cos θ ) sin θ + ( i sin θ ) m A l ψ l ( k II r ) P l ( 1 ) ( cos θ ) ]
E ϕ = sin ϕ k II r l = 1 [ e A l ψ l ( k II r ) P l ( 1 ) ( cos θ ) sin θ + i m A l ψ l ( k II r ) P l ( 1 ) ( cos θ ) sin θ ] .
[ x y z ] = [ cos 2 ϕ cos θ + sin 2 ϕ sin ϕ cos ϕ ( cos θ 1 ) cos ϕ sin θ sin ϕ cos ϕ ( cos θ 1 ) sin 2 ϕ cos θ + cos 2 ϕ sin ϕ sin θ cos ϕ sin θ sin ϕ sin θ cos θ ] · [ x y z ] .
[ E x E y E z ] = [ cos 2 ϕ cos θ + sin 2 ϕ sin ϕ cos ϕ ( cos θ 1 ) cos ϕ sin θ sin ϕ cos ϕ ( cos θ 1 ) sin 2 ϕ cos θ + cos 2 ϕ sin ϕ sin θ cos ϕ sin θ sin ϕ sin θ cos θ ] · [ E x E y E z ]
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