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

Optics Express, Vol. 11, Issue 23, pp. 3160-3170 (2003)

http://dx.doi.org/10.1364/OE.11.003160

Acrobat PDF (2365 KB)

### 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

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]

## 2. Plane wave incident on a Debye material

_{∞}is the infinite frequency dielectric constant, ε

_{s}is the zero frequency dielectric constant, ω is the angular frequency, t

_{0}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],

**E⃑**

_{i}is the incident field,

**H⃑**

_{s}is the scattered H field, σ is the dc conductivity,

**ψ⃑**

_{n}≡

*χ*

^{m}in Eq. (3) can be updated recursively at each time step:

_{0}of free space, the recursive equation for updating the magnetic field is much simpler,

_{c}, to ensure stability of the calculation [5]. 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·t

_{c}and 0.95·t

_{c}is generally adequate, while Δt > 0.95t

_{c}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].

**k**

_{0}is the wavevector of the plane wave and

**p**is the point of calculation in the computation space. In polar coordinates,

_{0}. 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.

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

^{5}GHz. The bulk value for the refractive index of silver, n=0.14+i(4.523), was used [8]. Debye parameters for this refractive index are σ= 8.67651×10

^{6}(Ω m)

^{-1}, t

_{0}=6.29065×10

^{-15}s, ε

_{∞}=1, and ε

_{s}=-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 [5,9

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]

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

_{c}, and the calculation was run for 3000 time steps.

^{5}(Ωm)

^{-1}, t

_{0}=1.62952×10

^{-16}s, ε

_{∞}=10, and ε

_{s}=6.3262. The cell space was 200×200×200 cells and each cell was a 20 nm cube. The time step was 0.9 t

_{c}for a total of 500 time steps. The result was again compared to Mie theory in Fig. 2.

## 3. Plane wave incident on a Lorentz material

_{p}is the resonant frequency and Δ

_{p}is the damping coefficient. The corresponding time domain susceptibility function is [5]

*complex*values for χ

^{0}and

**ψ**

_{n}at each time step using Eqs. (4), (5), and (7), and insert the

*real*part of these numbers into Eq. (3).

_{0}=2.27412×10

^{15}Hz, δ=9.20695×10

^{13}Hz, ε

_{∞}=1, and ε

_{s}=2 corresponding to n=3.0+i(1.5) at a wavelength of 850 nm. The cell space is 200×200×200 with a 20 nm cubic cell. The calculation is run for 700 time steps with each step t=0.9 t

_{c}. The result is compared along the z axis to the Debye model and Mie theory in Fig. 3(b). In this example, the Debye model at only 500 time steps gives a better match to Mie theory than the Lorentz model at 700 time steps.

## 4. FDTD in dense media

_{0}of vacuum. If the computation space is embedded in a dense medium, ε

_{0}is simply replaced in every equation by ε

_{m}=n

^{2}·ε

_{0}, 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.

_{c}and 3000 time steps are calculated. The results are shown in Fig. 4.

^{7}(Ωm)

^{-1}, t

_{0}=7.17096×10

^{-15}s, ε

_{∞}=4, and ε

_{s}=-9120.07. The FDTD computational space is 100×100×100 cells, with a 5 nm cubic cell. The time step is 0.9 t

_{c}and 1500 time steps are taken. The FDTD calculation is compared to Mie theory in Fig. 5.

## 5. Focused field FDTD formulation

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]

^{iωt}is understood,

**a**(θ,ϕ) is a function describing the incident polarization,

**p**is the point of observation,

_{0}is the free space wavelength,

*ϕ*=tan

^{-1}(

*y*

_{p}

*/x*

_{p}).

^{iωt}. 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]. Therefore, the incident field must either be calculated separately for each component E

_{x}, E

_{y}, and E

_{z}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).

_{c}. The results are compared to Mie theory [15] in Fig. 6.

_{c}. The results are shown in Fig. 7.

## 6. Conclusion

## Appendix

## Acknowledgments

## 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. |

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. |

3. | I. K. Sendur and W. A. Challener, “Near-field radiation of bow-tie antennas and apertures at optical frequencies,” J. Microsc. |

4. | A. Taflove and S. C. Hagness, |

5. | K. S. Kunz and R. J. Luebbers, |

6. | G. Mie, “Beiträge zur optik truber medien, speziell kolloida ler metallösungen,” Ann. d. Physik |

7. | M. Born and E. Wolf, |

8. | D. W. Lynch and W. R. Hunter, “Silver (Ag)” in |

9. | G. Mur, “Absorbing boundary conditions for finite-difference approximation of the time-domain electromagnetic field equations,” IEEE Trans. Electromagn. Compat. |

10. | Z. P. Liao |

11. | J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. |

12. | E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. Roy Soc. London Ser. A |

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 |

14. | I. Ichimura, S. Hayashi, and G. S. Kino, “High-density optical recording using a solid immersion lens,” Appl. Opt. |

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

- 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).
- 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]
- 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]
- 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).
- G. Mie, "Beiträge zur optik truber medien, speziell kolloida ler metallösungen," Ann. d. Physik 25, 377 (1908). [CrossRef]
- 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).
- 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]
- Z. P. Liao et al., "A transmitting boundary for transient wave analyses," Scientia Sinica 27 1063-1076 (1984).
- J. P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys. 114, 185-200 (1994). [CrossRef]
- 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]
- 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]
- I. K. Sendur and W. A. Challener, "Interaction of focused beams with spherical nanoparticles," to be published.

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