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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References

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  1. K. S. Yee, �??Numerical solution of initial boundary value problems involving Maxwell�??s equations in isotropic media,�?? IEEE Trans. Antennas Prop. AP-14, 302-307 (1966).
  2. W. A. Challener, T. W. McDaniel, C. D. Mihalcea, K. R. Mountfield, K. Pelhos and I. K. Sendur, �??Light Delivery Techniques for Heat-Assisted Magnetic Recording,�?? Jpn. J. Appl. Phys. 42, 981-988 (2003).
    [CrossRef]
  3. I. K. Sendur and W. A. Challener, "Near-field radiation of bow-tie antennas and apertures at optical frequencies," J. Microsc. 210, 279-283 (2003).
    [CrossRef] [PubMed]
  4. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method 2nd ed. (Artech House, Boston, 2000).
  5. K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, Boca Raton, 1993).
  6. G. Mie, "Beiträge zur optik truber medien, speziell kolloida ler metallösungen," Ann. d. Physik 25, 377 (1908).
    [CrossRef]
  7. M. Born and E. Wolf, Principles of Optics 5th ed. (Pergamon Press, Oxford, 1975), section 13.5.
  8. D. W. Lynch and W. R. Hunter, "Silver (Ag)" in Handbook of the Optical Constants of Solids, E. D. Palik, ed. (Academic, San Diego, 1998).
  9. G. Mur, "Absorbing boundary conditions for finite-difference approximation of the time-domain electromagnetic field equations," IEEE Trans. Electromagn. Compat. 23, 1073-1077 (1981).
    [CrossRef]
  10. Z. P. Liao et al., "A transmitting boundary for transient wave analyses," Scientia Sinica 27 1063-1076 (1984).
  11. J. P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys. 114, 185-200 (1994).
    [CrossRef]
  12. E. Wolf, �??Electromagnetic diffraction in optical systems I. An integral representation of the image field,�?? Proc. Roy Soc. London Ser. A 253, 349-357 (1959).
    [CrossRef]
  13. B. Richards and E. Wolf, �??Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,�?? Proc. Roy Soc. London Ser. A 253, 358-379 (1959).
    [CrossRef]
  14. I. Ichimura, S. Hayashi and G. S. Kino, �??High-density optical recording using a solid immersion lens,�?? Appl. Opt. 36, 4339-4348 (1997).
    [CrossRef] [PubMed]
  15. I. K. Sendur and W. A. Challener, "Interaction of focused beams with spherical nanoparticles," to be published.

Ann. d. Physik (1)

G. Mie, "Beiträge zur optik truber medien, speziell kolloida ler metallösungen," Ann. d. Physik 25, 377 (1908).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Antennas Prop. (1)

K. S. Yee, �??Numerical solution of initial boundary value problems involving Maxwell�??s equations in isotropic media,�?? IEEE Trans. Antennas Prop. AP-14, 302-307 (1966).

IEEE Trans. Electromagn. Compat. (1)

G. Mur, "Absorbing boundary conditions for finite-difference approximation of the time-domain electromagnetic field equations," IEEE Trans. Electromagn. Compat. 23, 1073-1077 (1981).
[CrossRef]

J. Comput. Phys. (1)

J. P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys. 114, 185-200 (1994).
[CrossRef]

J. Microsc. (1)

I. K. Sendur and W. A. Challener, "Near-field radiation of bow-tie antennas and apertures at optical frequencies," J. Microsc. 210, 279-283 (2003).
[CrossRef] [PubMed]

Jpn. J. Appl. Phys. (1)

W. A. Challener, T. W. McDaniel, C. D. Mihalcea, K. R. Mountfield, K. Pelhos and I. K. Sendur, �??Light Delivery Techniques for Heat-Assisted Magnetic Recording,�?? Jpn. J. Appl. Phys. 42, 981-988 (2003).
[CrossRef]

Proc. Roy Soc. London Ser. A (2)

E. Wolf, �??Electromagnetic diffraction in optical systems I. An integral representation of the image field,�?? Proc. Roy Soc. London Ser. A 253, 349-357 (1959).
[CrossRef]

B. Richards and E. Wolf, �??Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,�?? Proc. Roy Soc. London Ser. A 253, 358-379 (1959).
[CrossRef]

Scientia Sinica (1)

Z. P. Liao et al., "A transmitting boundary for transient wave analyses," Scientia Sinica 27 1063-1076 (1984).

Other (5)

M. Born and E. Wolf, Principles of Optics 5th ed. (Pergamon Press, Oxford, 1975), section 13.5.

D. W. Lynch and W. R. Hunter, "Silver (Ag)" in Handbook of the Optical Constants of Solids, E. D. Palik, ed. (Academic, San Diego, 1998).

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method 2nd ed. (Artech House, Boston, 2000).

K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, Boca Raton, 1993).

I. K. Sendur and W. A. Challener, "Interaction of focused beams with spherical nanoparticles," to be published.

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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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