## Optical near-field analysis of spherical metals: Application of the FDTD method combined with the ADE method

Optics Express, Vol. 15, Issue 18, pp. 11481-11491 (2007)

http://dx.doi.org/10.1364/OE.15.011481

Acrobat PDF (1221 KB)

### Abstract

The time-average energy density of the optical near-field generated around a metallic sphere is computed using the finite-difference time-domain method. To check the accuracy, the numerical results are compared with the rigorous solutions by Mie theory. The Lorentz-Drude model, which is coupled with Maxwell’s equation via motion equations of an electron, is applied to simulate the dispersion relation of metallic materials. The distributions of the optical near-filed generated around a metallic hemisphere and a metallic spheroid are also computed, and strong optical near-fields are obtained at the rim of them.

© 2007 Optical Society of America

## 1. Introduction

2. M. Ohtsu (Ed.), *Progress in Nano-Electro-Optics III-Industrial Applications and Dynamics of the Nano-Optical System* (Springer, 2004). [PubMed]

3. T. Matsumoto, T. Shimano, H. Saga, H. Sukeda, and M. Kiguchi, “Highly efficient probe with a wedge-shaped metallic plate for high density near-field optical recording,” J. Appl. Phys. **95**, 3901–3906 (2004). [CrossRef]

7. W. A. Challener, I. K. 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/abstract.cfm?URI=oe-11-23-3160 [CrossRef] [PubMed]

7. W. A. Challener, I. K. 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/abstract.cfm?URI=oe-11-23-3160 [CrossRef] [PubMed]

8. R. J. Zhu, J. Wang, and G. F. Jin, “Mie scattering calculation by FDTD employing a modified Debye model for Gold material,” Optik **116**, 419–422 (2005). [CrossRef]

10. A. D. Rakić, A. B. Djurišić, J. M. Elazar, and M. L. Majewski, “Optical Properties of Metallic Films for Vertical-Cavity Optoelectronic devices,” Appl. Opt. **37**, 5271–5283 (1998). [CrossRef]

11. T. Kashiwa and I. Fukai, “A treatment by FDTD method of dispersive characteristics associated with electronic polarization,” Microwave Optics Tech. Lett. **3**, 203–205 (1990). [CrossRef]

11. T. Kashiwa and I. Fukai, “A treatment by FDTD method of dispersive characteristics associated with electronic polarization,” Microwave Optics Tech. Lett. **3**, 203–205 (1990). [CrossRef]

13. J. A. Roden and S. D. Gedney, “Convolutional PML (CPML): An efficient FDTD implementation of the CFS-PML for arbitrary media,” Microwave Optical Tech. Lett. **27**, 334–339 (2000). [CrossRef]

13. J. A. Roden and S. D. Gedney, “Convolutional PML (CPML): An efficient FDTD implementation of the CFS-PML for arbitrary media,” Microwave Optical Tech. Lett. **27**, 334–339 (2000). [CrossRef]

## 2. Formulation

### 2.1 Optical near-field around a small sphere

*E*is given by:

^{(i)}*k*

_{0}=2

*π/λ(λ*: incident light wavelength), and

*E*

_{0}is the magnitude of the applied electric field.

*ux*is the unit vector of the x-direction. When the metallic sphere is small (

*k*≪1), the electromagnetic fields around the sphere (

_{0}a*r*>

*a*)

*E*^{(s)},

*Hθ*and

^{(s)}*H*are proportional to the electric dipole moment

_{ϕ}^{(s)}*p0*defined by

*p0*is enhanced when the light frequency satisfies the following equation:

*p0*depends on not only the frequency and the real part of

*ε*but also the shape of the object.

_{r}(ω)### 2.2 Dispersion relation of metals in the frequency domain

10. A. D. Rakić, A. B. Djurišić, J. M. Elazar, and M. L. Majewski, “Optical Properties of Metallic Films for Vertical-Cavity Optoelectronic devices,” Appl. Opt. **37**, 5271–5283 (1998). [CrossRef]

*ω*is the plasma frequency,

_{p}*K*in the symbol

*Σ*the number of oscillators with frequency

^{K}_{j=1}*ω*, and collision frequencies

_{j}*ν*and

_{0}*ν*and

_{j}. A_{0}*A*are constants which depend on the material [10

_{j}10. A. D. Rakić, A. B. Djurišić, J. M. Elazar, and M. L. Majewski, “Optical Properties of Metallic Films for Vertical-Cavity Optoelectronic devices,” Appl. Opt. **37**, 5271–5283 (1998). [CrossRef]

*A*=0 (

_{j}*j*=1,2,…,

*K*) in Eq. (5),

*ε*(

_{r}*ω*) becomes the relative permittivity of the D model, and the material is composed of free electrons. When

*A*is zero,

_{0}*ε*becomes the relative permittivity of the L model, and the material consists of bound electrons. Figure 1 shows the relative permittivity of gold parameterized by the LD model (

_{r}(ω)*K*=5) [10,20,21] against wavelengths. The solid curves (——), the dashed curves (–––-), and the dashed-dotted curves (–.–.–) are the numerical results of the LD model, the D model, and the L model, respectively. The circles (○ ○ ○) are the experimental data from Johnson and Christy [22

22. P. B. Johnson and R.W. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B **6**, 4370–4379 (1972). [CrossRef]

### 2.3 FDTD formulation for the LD model

23. M. Bordovsky, P. Catrysse, S. Dods, M. Freitas, J. Klein, L. Kotacka, V. Tzolov, I. Uzunov, and J. Zhang, “Waveguide design, modeling, and optimization - from photonic nano-devices to integrated photonic circuits,” Proc. SPIE **5355**, 65–80 (2004). [CrossRef]

*t*is the increment of the time domain. The finite difference equations for the electromagnetic field in the metal are expressed as:

**H**^{n+1/2},

*,*

**E**^{n+1}*and*

**J**_{j}^{n+1}*. If the medium is characterized by the D model,*

**P**_{j}^{n+1}*γ*=0 and

_{j}*vanishes from Eqs. (6) and (9). The stability requirement for the difference equations is*

**P**_{j}^{n}*cΔt*<1/{1/(Δ

*x*)

^{2}+1/(Δ

*y*)

^{2}+1/(Δ

*z*)

^{2}}

^{1/2}as if the medium were free space, where

*c*is the velocity of light, Δ

*x*, Δ

*y*, and Δ

*z*are the increments of the spatial coordinates.

*and*

**E***are the total electromagnetic field outside the object,*

**H***and*

**E**^{(i)}*are the incident electromagnetic field (incident light). Each component of the electromagnetic field is located at different positions in the Yee cell, and there is a half-time-step difference between*

**H**^{(i)}**and**

*E***. We take space and time averages as follows:**

*H**w*〉

*, the total time-average energy density in a steady state, we examined the following two methods;*

_{N}*w*〉

*is computed using the steady state of*

_{N}*and*

**E***.*

**H***w*〉

*[12].*

_{N}## 3. Numerical results

*a*,

*z*<0 and the flat surface on the

*x-y*plane,

*a*in the

*x*-direction and a radius

*b*in the

*y*-direction (

*a*>

*b*). In Fig. 2, the symbol O is the origin of the coordinate system and the symbol

*P*is the observation point.

### 3.1 Comparisons of the FDTD results with Mie solutions

*a*=50.0 nm is analyzed to compare the numerical results by the FDTD method with the Mie solutions.

*w*〉

*near the sphere as a function of the distance*

_{N}*r*(>

*a*). The observation points are on the

*x*-axis (

*θ*=90°,

*ϕ*=0°). In Fig. 3, the circles (○ ○ ○) and the rectangles (□ □ □) indicate the FDTD results with the cell size Δ=Δ

*x*=Δ

*y*=Δ

*z*=2.5 nm and Δ=1.25 nm, respectively. The solid line (——) is the Mie solution. The incident light wavelength

*λ*is 550 nm. In the range of 57.5 nm <

*r*<70.0 nm, the relative error defined by |Mie solution — FDTD result|/|Mie solution| is less than 2.5 % when Δ=2.5 nm. However, the accuracy of the FDTD results decreases as the observation point r approaches the surface of the sphere. The reasons for that are caused by:

*r*=55.0 nm.

*w*〉

*is less than 2.1 % at both*

_{N}*r*=55.0 nm and

*r*=60.0 nm. We used a multiprocessor (Opteron 270, 2.0 GHz) workstation to solve the problem. When Δ=1.25 nm, the computation time was about 150 hours and the required memory size was about 1.6 GB. When Δ=2.5 nm, they were 20 hours and 310 MB, respectively.

*a*=50 nm,

*λ*=550 nm,

*θ*=90°,

*ϕ*=0°.

*w*〉

*near the gold sphere and the incident light wavelength*

_{N}*λ*. The solid line is the Mie solution. The circles and the rectangles indicate the FDTD results when Δ=2.5 nm and 5.0 nm, respectively. These results tell us the following:

*w*〉

*of both the FDTD results and the Mie solution reach the maximum at*

_{N}*λ*=550 nm.

*w*〉

*near the gold sphere against the angle*

_{N}*θ*. The observation points are fixed at 10 nm from the surface of the sphere and laid on the

*x-z*plane (

*ϕ*=0°). The incident light wavelength is 550 nm. The rectangles, the circles, and the filled circles show the FDTD results with the radius

*a*=20.0 nm and the cell size Δ=1.0 nm,

*a*=50.0 nm and Δ=2.0 nm, and

*a*=100 nm and Δ=2.0 nm, respectively. The dotted lines are the Mie solutions. The energy density 〈

*w*〉

*reaches its maximum about at*

_{N}*θ*=90° when

*a*=20.0 nm. If

*k*≪1 (

_{0}r*r*≥

*a*), 〈

*w*〉

*is given by:*

_{N}*w*〉

*attains the maximum at*

_{N}*θ*=90° and

*ϕ*=0°. The maximum of 〈

*w*〉

*generated by the 50-nm sphere is larger than that by the 20-nm and the 100-nm ones. Since*

_{N}*k*=2

_{0}a*π a/λ≈*0.57 is not much smaller than unity when

*a*=50.0 nm and

*λ*=550 nm, Eq. (14) is inapplicable and the effect of size must be considered. The FDTD results are in good agreement with the Mie solutions. The relative errors at the maximum of 〈

*w*〉

*are 3.5 % for*

_{N}*a*=20.0 nm when Δ/

*λ*=1.818×10

^{-3}, 0.86 % for

*a*=50.0 nm and 0.58 % for

*a*=100.0 nm when Δ/

*λ*=3.64×10

^{-3}.

### 3.2 Optical near-field around a hemisphere and a spheroid

*w*〉

*generated around a silver hemisphere with*

_{N}*a*radius a=50.0 nm (see Fig. 2 (b)). The observation plane corresponds to the

*x-z*plane at

*y*=0. The incident light wavelength

*λ*is 500 nm. As shown in Fig. 6, a strong optical near-field is generated at the rim of the hemisphere. The maximum of 〈

*w*〉

*appears around at*

_{N}*θ*=90°,

*ϕ*=0° and

*θ*=90°,

*〈*=180°which correspond to the direction of the polarization of the incident electric filed. Silver, λ=500 nm, a=50 nm, Δ=2.0 nm, y=0 nm.

*w*〉

*and the incident light wavelength*

_{N}*λ*. The observation point is placed at 10 nm from the surface of the hemisphere and on the

*x*-axis (

*θ*=90°,

*ϕ*=0°). The solid curve (——), the dashed curve (–––-), and the circles (○ ○ ○) represent the energy density 〈

*w*〉

*generated by the metallic hemisphere made of gold, silver, and perfect conductor, respectively. The following features of those curves will be noted:*

_{N}*w*〉

*for the gold sphere:*

_{N}*w*〉

*by the hemisphere is about 2.0 times as large as that by the sphere (see Fig. 4).*

_{N}*w*〉

*for the hemisphere is*

_{N}*λ*=580 nm, which is shifted toward longer than that for the sphere.

*w*〉

*for the silver hemisphere is about 2.1 times as large as that for the gold one. The energy loss in silver becomes smaller than that in gold because the imaginary part of relative permittivity of silver is Im[*

_{N}*ε*(

_{r}*ω*)]=0.93 at

*λ*=580 nm, whereas that of gold is Im[

*ε*(

_{r}*ω*)]=2.15.

*w*〉

*around a silver spheroid with radii*

_{N}*a*=50.0 nm and

*b*=40.0 nm (see Fig. 2 (c)). The observation plane corresponds to the

*x-z*plane at

*y*=0. The incident light wavelength

*λ*is 500 nm. The intensity of 〈

*w*〉

*increases at the projections of the spheroid as well as the hemisphere.*

_{N}*w*〉

*and the incident light wavelength*

_{N}*λ*. The observation point is placed at 10 nm from the surface of the spheroid and on the

*x*-axis (

*θ*=90°,

*ϕ*=0°). The solid curve, the dashed curve, and the circles represent the energy density 〈

*w*〉

*generated by the metallic spheroid made of gold, silver, and perfect conductor, respectively. The distribution of 〈*

_{N}*w*〉

*by the spheroid attains the maximum at*

_{N}*θ*=90° and

*ϕ*=0°, 180°, because the major axis of the spheroid corresponds with the polarized direction of the incident light. The results obtained from Fig. 9 are as follows:

*w*〉

*for the gold sphere:*

_{N}*w*〉

*by the spheroid is about 1.5 times as large as that by the sphere (see Fig. 4).*

_{N}*w*〉

*for the spheroid is*

_{N}*λ*=570 nm, which is shifted toward longer than that wavelength for the sphere.

*w*〉

*for the silver spheroid is about 1.6 times as large as that for the gold one. The reason is the same as (H-2).*

_{N}*w*〉

*for the silver spheroid is smaller than that for the silver hemisphere.*

_{N}*w*〉

*and the incident light wavelength*

_{N}*λ*for the gold spheroids with

*b/a*=0.4, 0.6, 0.8, and 1.0. The observation point and the cell size are the same as Fig. 9 except the cell size is 2.0 nm when the radius

*b*is 20.0 nm. From these results, the maximum of 〈

*w*〉

*increases and appears at a longer light wavelength as*

_{N}*b/a*becomes smaller.

## 4 Conclusions

*b/a*) becomes smaller.

## Acknowledgments

## References and links

1. | M Ohtsu and K. Kobayashi, |

2. | M. Ohtsu (Ed.), |

3. | T. Matsumoto, T. Shimano, H. Saga, H. Sukeda, and M. Kiguchi, “Highly efficient probe with a wedge-shaped metallic plate for high density near-field optical recording,” J. Appl. Phys. |

4. | T. MatsumotoM. Ohtsu, ed. (Springer, 2004). |

5. | T. Uno, |

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

7. | W. A. Challener, I. K. 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 |

8. | R. J. Zhu, J. Wang, and G. F. Jin, “Mie scattering calculation by FDTD employing a modified Debye model for Gold material,” Optik |

9. | H. Tamaru and K. Miyano, “Localized Surface Plasmon Resonances of Metal Nanoparticles: Numerical Simulations and Their Experimental Verification,” Kogaku Japanese Journal of Optics |

10. | A. D. Rakić, A. B. Djurišić, J. M. Elazar, and M. L. Majewski, “Optical Properties of Metallic Films for Vertical-Cavity Optoelectronic devices,” Appl. Opt. |

11. | T. Kashiwa and I. Fukai, “A treatment by FDTD method of dispersive characteristics associated with electronic polarization,” Microwave Optics Tech. Lett. |

12. | T. Yamaguchi, T. Yamasaki, and T. Hinata, “FDTD Analysis of Optical Near-field Around a Metallic Sphere,” in |

13. | J. A. Roden and S. D. Gedney, “Convolutional PML (CPML): An efficient FDTD implementation of the CFS-PML for arbitrary media,” Microwave Optical Tech. Lett. |

14. | J. De Moerloose and M. A. Stuchly, “Behavior of Berenger’s ABC for evanescent waves,” IEEE Microwave Guided Wave Lett. |

15. | J. P. Bérenger, “Improved PML for the FDTD Solution of Wave-Structure Interaction Problems,” IEEE Trans. Antennas Propag. |

16. | J. P. Bérenger, “Evanescent Waves in PML’s: Origin of the Numerical Reflection in Wave-Structure Interaction Problems,” IEEE Trans. Antennas Propag. |

17. | T. Yamaguchi, “Numerical Analysis of Pulse Reflection from Anisotropic Dielectric Layer,” Special Issue of Nihon Univ. CST 2006 Annual Conf. Short Note |

18. | J. A. Stratton, |

19. | T. Hosono, |

20. | M. A. Ordal, L. L. Long, R. J. Bell, S. E. Bell, R. R. Bell, R. W. Alexander. Jr., and C. A. Ward, “Optical properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared,” Appl. Opt. |

21. | J. B. Judkins and R. W. Ziolkowski, “Finite-difference time-domain modeling of nonperfectly conducting metallic thin-film gratings,” J. Opt. Soc. Am. A |

22. | P. B. Johnson and R.W. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B |

23. | M. Bordovsky, P. Catrysse, S. Dods, M. Freitas, J. Klein, L. Kotacka, V. Tzolov, I. Uzunov, and J. Zhang, “Waveguide design, modeling, and optimization - from photonic nano-devices to integrated photonic circuits,” Proc. SPIE |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(240.6680) Optics at surfaces : Surface plasmons

(260.2030) Physical optics : Dispersion

(260.2110) Physical optics : Electromagnetic optics

(290.4020) Scattering : Mie theory

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: June 26, 2007

Revised Manuscript: August 17, 2007

Manuscript Accepted: August 18, 2007

Published: August 24, 2007

**Virtual Issues**

Vol. 2, Iss. 10 *Virtual Journal for Biomedical Optics*

**Citation**

Takashi Yamaguchi and Takashi Hinata, "Optical near-field analysis of spherical metals: Application of the FDTD method combined with the ADE method," Opt. Express **15**, 11481-11491 (2007)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-18-11481

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### References

- M Ohtsu and K. Kobayashi, Optical Near Field (Springer-Verlag, Berlin, 2004).
- M. Ohtsu, ed., Progress in Nano-Electro-Optics III-Industrial Applications and Dynamics of the Nano-Optical System (Springer, 2004). [PubMed]
- T. Matsumoto, T. Shimano, H. Saga, H. Sukeda, and M. Kiguchi, "Highly efficient probe with a wedge-shaped metallic plate for high density near-field optical recording," J. Appl. Phys. 95, 3901-3906 (2004). [CrossRef]
- T. Matsumoto, "Near-Field Optical-Head Technology for High-Density, Near-Field Optical Recording," in Progress in Nano-Electro-Optics III, M. Ohtsu, ed., (Springer, 2004).
- T. Uno, Finite Difference Time Domain Method for Electromagnetic Field and Antennas, in Japanese (Corona Publishing Co., Ltd, 1998).
- A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Third Edition (Artech House, 2005).
- W. A. Challener, I. K. 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). [CrossRef] [PubMed]
- R. J. Zhu, J. Wang, and G. F. Jin, "Mie scattering calculation by FDTD employing a modified Debye model for Gold material," Optik 116, 419-422 (2005). [CrossRef]
- H. Tamaru and K. Miyano, "Localized Surface Plasmon Resonances of Metal Nanoparticles: Numerical Simulations and Their Experimental Verification," Kogaku Japanese Journal of Optics 33, 165-170 (2004).
- A. D. Rakiæ, A. B. Djurišiæ, J. M. Elazar, and M. L. Majewski, "Optical Properties of Metallic Films for Vertical-Cavity Optoelectronic devices," Appl. Opt. 37, 5271-5283 (1998). [CrossRef]
- T. Kashiwa and I. Fukai, "A treatment by FDTD method of dispersive characteristics associated with electronic polarization," Microwave Optics Tech. Lett. 3, 203-205 (1990). [CrossRef]
- T. Yamaguchi, T. Yamasaki, and T. Hinata, "FDTD Analysis of Optical Near-field Around a Metallic Sphere," in The Papers of Technical Meeting on Electromagnetic Theory (IEE, Japan, 2006), pp. 143-148.
- J. A. Roden and S. D. Gedney, "Convolutional PML (CPML): An efficient FDTD implementation of the CFS-PML for arbitrary media," Microwave Optical Tech. Lett. 27, 334-339 (2000). [CrossRef]
- J. De Moerloose and M. A. Stuchly, "Behavior of Berenger’s ABC for evanescent waves," IEEE Microwave Guided Wave Lett. 5, 344-346 (1995). [CrossRef]
- J. P. Bérenger, "Improved PML for the FDTD Solution of Wave-Structure Interaction Problems," IEEE Trans. Antennas Propag. 45, 466-473 (1997). [CrossRef]
- J. P. Bérenger, "Evanescent Waves in PML’s: Origin of the Numerical Reflection in Wave-Structure Interaction Problems," IEEE Trans. Antennas Propag. 47, 1497-1503 (1999). [CrossRef]
- T. Yamaguchi, "Numerical Analysis of Pulse Reflection from Anisotropic Dielectric Layer," Special Issue of Nihon Univ. CST 2006 Annual Conf. Short Note 1, 113-116 (2007).
- J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).
- T. Hosono, The Foundation of Electromagnetic Wave Theory, in Japanese (Shoko-do, 1973).
- M. A. Ordal, L. L. Long, R. J. Bell, S. E. Bell, R. R. Bell, R. W. Alexander. Jr., and C. A. Ward, "Optical properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared," Appl. Opt. 22, 1099-1119 (1983). [CrossRef] [PubMed]
- J. B. Judkins and R. W. Ziolkowski, "Finite-difference time-domain modeling of nonperfectly conducting metallic thin-film gratings," J. Opt. Soc. Am. A 12, 1974-1983 (1995). [CrossRef]
- P. B. Johnson and R.W. Christy, "Optical Constants of the Noble Metals," Phys. Rev. B 6, 4370-4379 (1972). [CrossRef]
- M. Bordovsky, P. Catrysse, S. Dods, M. Freitas, J. Klein, L. Kotacka, V. Tzolov, I. Uzunov, and J. Zhang, "Waveguide design, modeling, and optimization - from photonic nano-devices to integrated photonic circuits," Proc. SPIE 5355, 65-80 (2004). [CrossRef]

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