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Optics Express

  • Editor: Michael Duncan
  • Vol. 11, Iss. 23 — Nov. 17, 2003
  • pp: 3160–3170
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Scattered field formulation of finite difference time domain for a focused light beam in dense media with lossy materials

W. A. Challener, I. K. Sendur, and C. Peng  »View Author Affiliations


Optics Express, Vol. 11, Issue 23, pp. 3160-3170 (2003)
http://dx.doi.org/10.1364/OE.11.003160


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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

1. Introduction

The scattered field formalism for FDTD is presented by Kunz and Luebbers [5

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

]. In this formalism the incident electric field is specified throughout the computation space for the time duration of the calculation through a calculation which is separate from FDTD algorithm. This is particularly convenient if the incident field can be determined analytically. Dispersion errors in the incident field are not a factor for the scattered field approach. Another advantage of the scattered field approach is that the absorbing boundary conditions must only handle the scattered field rather than the total field, which is usually easier. Finally, in the case of a focused beam incident upon a small scattering object, the computation space in the scattered field formalism can be limited to just the volume surrounding the object. In the total field formalism the incident field must either be specified over all sides of the computation space, or if the incident field is just specified over one surface of the computation space this surface must be large enough to include the entire region over which there is an appreciable incident field strength, which may be considerably larger than the region in the immediate neighborhood of the scattering object and require correspondingly longer computation times.

The scattered field approach is straightforward for modeling dielectrics and perfect conductors, but is more complicated than the total field approach when modeling materials that exhibit a frequency-dependent permittivity or conductivity. As discussed in Chapter 8 of Ref. [5

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

], the scattered field formalism can be applied to frequency-dependent materials with a Debye or Lorentz dispersion in a computationally efficient manner. This reference considers the case of an electric field plane wave pulse. In this paper the discussion of Ref. [5

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

] is extended to include lossy materials that are illuminated with a plane wave and a focused optical beam. The case when the field is incident within an optically dense material is also considered.

2. Plane wave incident on a Debye material

A simple model of a frequency-dependent dielectric function which satisfies the Kramers-Kronig causality relations is the first order Debye equation [5

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

]. The FDTD approach for a plane wave incident upon a material whose dielectric constant is described by the Debye equation is considered in this section. The Debye equation for the dielectric constant is

ε=ε+εsε1+iωt0=ε+χ(ω)
(1)

where ε is the infinite frequency dielectric constant, εs is the zero frequency dielectric constant, ω is the angular frequency, t0 is the frequency-independent relaxation time, and χ(ω) is the frequency-dependent susceptibility. The Fourier transform of the susceptibility gives the corresponding time-dependent susceptibility function [5

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

],

χ(t)=12πeiωtχ(ω)dω=(εsεt0)·ett0U(t)
(2)

where U(t) is the Heavyside unit step function. Following the procedure and notation in Ref. [5

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

], the recursive equation for updating the scattered electric field Esn+1 at time step n+1 is obtained,

Esn+1=(1εε0+χ0ε0+σ·Δt){εε0Esn+ε0m=0n1Esnm·Δχm+Δt·×Hsn+12
σ·Δt·Ein+1ε0·Δt(ε1)·tEin+1ε0·Δt·t[Ein+1(t)*χ(t)]}
(3)

where E⃑ i is the incident field, H⃑ s is the scattered H field, σ is the dc conductivity,

χ0(εsε)(1eΔtt0),
(4)
Δχm=(εsε)(emΔtt0)(1eΔtt0)2,
(5)

and

t[Ein+1(t)*χ(t)]=tEin+1(tΛ)·χ(Λ)dΛ
=(εsεt0)t[ett0·0tEin+1(ξ)·eξt0dξ]
(6)

after the substitution ξ=t-Λ.

As described in Ref. [5

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

], the term ψ⃑ nm=0n1 Esnm·Δχm in Eq. (3) can be updated recursively at each time step:

Ψn=Esn·Δχ0+eΔtt0·Ψn1.
(7)

For nonmagnetic materials with a permeability µ0 of free space, the recursive equation for updating the magnetic field is much simpler,

Hsn+12=Δtμ0·×Esn.
(8)

When modeling absorptive materials such as metals or lossy dielectrics, the time step Δt is typically chosen to be slightly smaller than the Courant time step, tc, to ensure stability of the calculation [5

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

]. The Courant time step is determined by the dimensions of the Yee cell and the speed of light in vacuum. A value of Δt between 0.9·tc and 0.95·tc is generally adequate, while Δt > 0.95tc can often lead to divergences in the calculation. Using shorter time steps unnecessarily increases the computation time and can also lead to larger dispersion errors [5

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

].

It is convenient to express the time dependence of the electric field as a complex quantity. The amplitude of the plane wave is zero for t≤0,

Einc(t)=eiωtik0·p·U(t)
(9)

where k 0 is the wavevector of the plane wave and p is the point of calculation in the computation space. In polar coordinates,

k=k0(sinθcosϕx̂+sinθsinϕŷ+cosθẑ).
(10)

The actual field amplitude can be obtained by choosing either the real or imaginary part of Eq. (9).

In the scattered field formulation of Eq. (3) a convolution integral of the incident field must also be computed. For a sinusoidal field,

t[Einc(t)*χ(t)]=E0eik0·p·(εsεt0)·t{ett00te(iω+1t0)ξdξ}
(11)
=E0eik0·p(εsε1+iωt0)·[iωeiωt+1t0ett0].
(12)

In the limit of infinite time, the second term in the square bracket of Eq. (12) goes to zero, leaving only the steady state result from the first term. Therefore, one criterion for reaching the steady state result in the computation is t≫t0. In fact, since the calculation must necessarily be run until the steady state is reached, this second term can be eliminated from the calculation a priori which in turn helps to reduce the computation time.

Plane wave scattering by a metallic sphere can be computed analytically by Mie theory [6

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

,7

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

] (also, see appendix). The scattering from a 100 nm diameter silver sphere was calculated using both FDTD and Mie theory for comparison as shown in Fig. 1. The wavelength was 700 nm corresponding to a frequency of 4.28275×105 GHz. The bulk value for the refractive index of silver, n=0.14+i(4.523), was used [8

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).

]. Debye parameters for this refractive index are σ= 8.67651×106(Ω m)-1, t0=6.29065×10-15 s, ε=1, and εs=-6163.4. The cell space was 200 by 200 by 200 cells, and each cell was a 2 nm cube. Second order Mur boundary conditions were used on the faces and first order Mur boundary conditions on the edges [5

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

,9

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]

], although there are many other boundary condition options available which may further improve the accuracy of the calculation [4

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

,10

10. Z. P. Liaoet al., “A transmitting boundary for transient wave analyses,” Scientia Sinica 271063–1076 (1984).

,11

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

]. However, the scattered field FDTD formulation described in this paper is independent of the particular boundary conditions chosen. The incident field was polarized along the x-axis and propagated in the +z direction. The time step was 0.9 tc, and the calculation was run for 3000 time steps.

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.

The scattering from a 1 µm lossy dielectric sphere with a refractive index n=3.0+i(1.5) in free space at a wavelength of 850 nm was calculated as a second example. The Debye parameters were σ=1.99621×105 (Ωm)-1, t0=1.62952×10-16 s, ε=10, and εs=6.3262. The cell space was 200×200×200 cells and each cell was a 20 nm cube. The time step was 0.9 tc for a total of 500 time steps. The result was again compared to Mie theory in 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.

3. Plane wave incident on a Lorentz material

A very similar formalism can be applied to dispersive materials that obey a Lorentz dispersion equation,

ε(ω)=ε+ωp2(εsε)ωp2+2iωδpω2
(13)

where ωp is the resonant frequency and Δp is the damping coefficient. The corresponding time domain susceptibility function is [5

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

]

χp(t)=γpeαptsin(βpt)U(t),
(14)

where

αpχp,
(15)
βpωp2δp2,
(16)

and

γpωp2(εsε)βp.
(17)

As described in Ref. [5

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

], a complex susceptibility function can be defined,

χ̂p(t)=iγpe(αp+iβp)tU(t)
(18)

such that

χp(t)=Re[χ̂p(t)].
(19)

If we make the following substitutions into Eq. (17)

αpiβp1t0
(20)

and

γpi(εsε)t0
(21)

then it is in the same form as that derived for Debye materials in Eq. (2). To use the Debye equation format, we calculate the complex values for χ0 and ψ n at each time step using Eqs. (4), (5), and (7), and insert the real part of these numbers into Eq. (3).

This procedure cannot be used for the convolution term in Eq. (3), however. Because the electric field is obtained by taking the real or imaginary part of a complex quantity, the convolution term must be calculated for the real susceptibility function of Eq. (13). The Lorentz susceptibility function can be rewritten as a sum of two terms which are both in the same form as the Debye susceptibility function,

χ(t)=γp2iett0·U(t)γp2iett1·U(t)χ1(t)χ2(t)
(22)

where

t11αp+iβp.
(23)

By comparison to the Debye result for the convolution term in Eq. (12), one immediately obtains

t[Einc(t)*χ(t)]=E0eik0·rp2i{(t0γp1+iωt0)·[iωeiωt+1t0ett0]
(t1γp1+iωt1)·[iωeiωt+1t1ett1]}.
(24)

Again, assuming that the calculation is properly run to the steady state result, the second term in each of the square brackets can be neglected, so the result simplifies to

t[Einc(t)*χ(t)]=12(E0γpeiωtik0·rp)[(ωt01+iωt0)(ωt11+iωt1)].
(25)

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.

4. FDTD in dense media

In optical measurements it is often the case that the light propagates towards a scattering object within a dielectric medium other than air or vacuum. All of the FDTD equations have been formulated in terms of the permittivity ε0 of vacuum. If the computation space is embedded in a dense medium, ε0 is simply replaced in every equation by εm=n2·ε0, the permittivity of the medium. The Courant time step, however, is still calculated for free space for which the speed of light is maximized and the time step minimized to ensure convergence of the calculation even when parts of the computation space still include vacuum.

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.

An example is that of a 100 nm air bubble with n=1.0 embedded in a medium with n=2. A plane wave is incident with a free space wavelength of 700 nm. The computation space is 200×200×200 cells and each cell is a 2.5 nm cube. The plane wave propagates in the +z direction and is polarized along the x axis. Each time step is 0.9 tc and 3000 time steps are calculated. The results are shown in Fig. 4.

A second example is for light incident from a dielectric medium with n=2 onto an embedded 100 nm silver sphere. The incident plane wave has a free space wavelength of 700 nm. The sphere is modeled with a Debye dispersion, and the parameters are σ=1.12658×107(Ωm)-1, t0=7.17096×10-15 s, ε=4, and εs=-9120.07. The FDTD computational space is 100×100×100 cells, with a 5 nm cubic cell. The time step is 0.9 tc and 1500 time steps are taken. The FDTD calculation is compared to Mie theory in 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.

5. Focused field FDTD formulation

Wolf and Richards presented a method for calculating the electric field semi-analytically near the focus of an aplanatic lens [12

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

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]

]. The total electric field in the neighborhood of the focus is given by [14

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]

]

e(p)=iλ0βdθsinθ02πdφa(θ,φ)eik·p,
(26)

where the time dependence eiωt is understood, a(θ,ϕ) is a function describing the incident polarization, p is the point of observation,

p=(xp,yp,zp)=(rpcosφp,rpsinφp,zp),
(27)

and

k=2πλ0(sinθcosφ,sinθsinφ,cosθ).
(28)

λ0 is the free space wavelength, rp=xp2+yp2 , and ϕ=tan-1(yp/xp ).

Equation (26) for the focused field can be used directly within the scattered field formalism of FDTD. As was done for plane waves, the field is turned on throughout the computation space at time t=0. The incident field is thus determined by its steady state value as calculated by the Richards and Wolf theory multiplied by eiωt. The spatial dependence of the incident field need only be calculated once prior to beginning the FDTD calculation and can be used repeatedly for additional FDTD calculations on other scattering objects illuminated by the same incident focused field. One subtlety involved in calculating the incident field is that the x, y, and z coordinates of the electric field components within the Yee cell are offset in different directions from the origin of the cell [5

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

]. Therefore, the incident field must either be calculated separately for each component Ex, Ey, and Ez in the focused beam at each cell in the computation space, or the incident field can be calculated once for each cell and interpolated to provide the field amplitudes within the cells, which is the approach taken here. The time-dependent part of the convolution term is given by Eqs. (12) and (25).

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.

An example for the focused beam similar to that of Fig. 1 with a 100 nm silver sphere in air is now considered. The Debye parameters of silver are the same as for Fig. 1. The incident beam at a wavelength of 700 nm is linearly polarized along the x-axis prior to focusing and propagates in the -z direction. The incident field amplitude is normalized to unity at the focus. It is focused at the center of the sphere and has a half angle of 60°. The computation space is 200×200×200 cells and each cell is a 2 nm cube. The steady state results were obtained after 3000 time steps where each step was 0.9 tc. The results are compared to Mie theory [15

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

] in Fig. 6.

As a second example, a focused beam incident upon a 500 nm silver sphere is considered. The wavelength is 700 nm and the incident beam is polarized along the x axis prior to focusing. The same Debye parameters are chosen for silver. The half angle for the focused beam is still 60°. The cell space is 200×200×200 cells and each cell is a 5 nm cube. The steady state results were obtained after 1500 time steps where each step was 0.9 tc. The results are shown in 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.

6. Conclusion

A scattered field FDTD formalism has been developed for modeling plane waves or focused beams incident upon Debye or Lorentz scattering objects. The Richards-Wolf vector field formalism is used to compute the incident field for focused beams. The surrounding medium may be free space or a dielectric with n>1. Several examples have been calculated for beams incident upon metallic and lossy dielectric spheres and compared to Mie theory with good agreement.

Appendix

Mie theory calculations were based on the derivation in Ref. [7

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

]. This reference presents the field equations for the scattered fields outside of the sphere but does not complete the derivation for the fields within the sphere. Continuing with the notation in this reference,

eAl=il1[2l+1l(l+1)]·[ns·ψl(kIa)·ςl(1)(kIa)ns·ψl(kIa)·ςl(1)(kIa)ns·ψl(kIIa)·ςl(1)(kIa)na·ψl(kIIa)·ςl(1)(kIa)]
(A1)
mAl=il1[2l+1l(l+1)]·[ns·ψl(kIa)·ςl(1)(kIa)ns·ψl(kIa)·ςl(1)(kIa)ns·ψl(kIIa)·ςl(1)(kIa)na·ψl(kIIa)·ςl(1)(kIa)]
(A2)

and within the sphere,

Er=cosφ(kII)2r2l=1l(l+1)eAlψl(kIIr)Pl(1)(cosθ)
(A3)
Eθ=cosϕkIIrl=1[eAlψl(kIIr)Pl(1)(cosθ)sinθ+(isinθ)mAlψl(kIIr)Pl(1)(cosθ)]
(A4)
Eϕ=sinϕkIIrl=1[eAlψl(kIIr)Pl(1)(cosθ)sinθ+imAlψl(kIIr)Pl(1)(cosθ)sinθ].
(A5)

Calculations of a focused incident beam using Mie theory were performed by summing the scattered fields from incident plane waves. The incident plane waves were spaced at 2° intervals along the polar direction and 6° intervals along the azimuthal direction. For a specific incident plane wave, the coordinates of the point (x,y,z) were first transformed into a coordinate system for which the plane wave was propagating along the z′-axis and polarized along the x′-axis. The transformation matrix is

[xyz]=[cos2ϕcosθ+sin2ϕsinϕcosϕ(cosθ1)cosϕsinθsinϕcosϕ(cosθ1)sin2ϕcosθ+cos2ϕsinϕsinθcosϕsinθsinϕsinθcosθ]·[xyz].
(A6)

After the scattered field was calculated at this point, the incident plus scattered field components were transformed back into the original coordinate system according to

[ExEyEz]=[cos2ϕcosθ+sin2ϕsinϕcosϕ(cosθ1)cosϕsinθsinϕcosϕ(cosθ1)sin2ϕcosθ+cos2ϕsinϕsinθcosϕsinθsinϕsinθcosθ]·[ExEyEz]
(A7)

and added to the incident plus scattered fields from the other plane waves.

Acknowledgments

This work was performed as part of the Information Storage Industry Consortium (INSIC) program in Heat Assisted Magnetic Recording (HAMR), with the support of the U. S. Department of Commerce, National Institute of Standards and Technology, Advanced Technology Program, Cooperative Agreement Number 70NANB1H3056.

References and links

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. Liaoet al., “A transmitting boundary for transient wave analyses,” Scientia Sinica 271063–1076 (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.

OCIS Codes
(000.4430) General : Numerical approximation and analysis
(290.4020) Scattering : Mie theory

ToC Category:
Research Papers

History
Original Manuscript: September 29, 2003
Revised Manuscript: November 7, 2003
Published: November 17, 2003

Citation
W. Challener, I. Sendur, and C. Peng, "Scattered field formulation of finite difference time domain for a focused light beam in dense media with lossy materials," Opt. Express 11, 3160-3170 (2003)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-23-3160


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References

  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]
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