## An integral equation based numerical solution for nanoparticles illuminated with collimated and focused light

Optics Express, Vol. 17, Issue 9, pp. 7419-7430 (2009)

http://dx.doi.org/10.1364/OE.17.007419

Acrobat PDF (471 KB)

### Abstract

To address the large number of parameters involved in nano-optical problems, a more efficient computational method is necessary. An integral equation based numerical solution is developed when the particles are illuminated with collimated and focused incident beams. The solution procedure uses the method of weighted residuals, in which the integral equation is reduced to a matrix equation and then solved for the unknown electric field distribution. In the solution procedure, the effects of the surrounding medium and boundaries are taken into account using a Green’s function formulation. Therefore, there is no additional error due to artificial boundary conditions unlike differential equation based techniques, such as finite difference time domain and finite element method. In this formulation, only the scattering nano-particle is discretized. Such an approach results in a lesser number of unknowns in the resulting matrix equation. The results are compared to the analytical Mie series solution for spherical particles, as well as to the finite element method for rectangular metallic particles. The Richards-Wolf vector field equations are combined with the integral equation based formulation to model the interaction of nanoparticles with linearly and radially polarized incident focused beams.

© 2009 Optical Society of America

## 1. Introduction

1. T. D. Milster, “Horizons for optical data storage,” Optics and Photonics News **16**, 28–32 (2005). [CrossRef]

2. R. E. Rottmayer, S. Batra, D. Buechel, W. A. Challener, J. Hohlfeld, Y. Kubota, L. Li, B. Lu, C. Mihalcea, K. Mountfield, K. Pelhos, C. Peng, T. Rausch, M. A. Seigler, D. Weller, and X. Yang, “Heat-assisted magnetic recording,” IEEE Trans. Magn. **42**, 2417–2421 (2006). [CrossRef]

5. D. W. Pohl, W. Denk, and M. Lanz, “Optical stethoscopy: image recording with resolution *λ*/20,” Appl. Phys. Lett. **44**, 651–653 (1984). [CrossRef]

6. A. Lewis, M. Isaacson, A. Harootunian, and A. Muray, “Development of a 500 Å spatial resolution light microscope,” Ultramicroscopy **13**, 227–231 (1984). [CrossRef]

7. A. Hartschuh, E. J. Sanchez, X. S. Xie, and L. Novonty, “High-resolution near-field Raman microscopy of single-walled carbon nanotubes,” Phys. Rev. Lett. **90**, 095503 (2003). [CrossRef] [PubMed]

8. L. Wang and X. Xu, “Numerical study of optical nanolithography using nanoscale bow-tie-shaped nano-apertures,” J. Microsc. **229**, 483–489 (2008). [CrossRef] [PubMed]

9. B. Liedberg, C. Nylander, and I. Lundstroem, “Surface plasmon resonance for gas detection and biosensing,” Sens. Actuators **4**, 299–304 (1983). [CrossRef]

## 2. Method of weighted residuals

10. W. A. Challener, I. K. Sendur, and C. Peng, “Scattered field formulation of finite difference time domain for a focused light beam in a dense media with lossy materials,” Opt. Express **11**, 3160–3170 (2003). [CrossRef] [PubMed]

04. K. Sendur, W. Challener, and C. Peng, “Ridge waveguide as a near field aperture for high density data storage,” J. Appl. Phys. **96**, 2743–2752 (2004). [CrossRef]

15. T. Grosges, A. Vial, and D. Barchiesi, “Models of near-field spectroscopic studies: comparison between finite-element and finite-difference methods,” Opt. Express **13**, 8483–8497 (2005). [CrossRef] [PubMed]

16. J. P. Kottmann and O. J. F. Martin, “Plasmon resonant coupling in metallic nanowires,” Opt. Express **8**, 655–663 (2001). [CrossRef] [PubMed]

21. J. H. Richmond, “Digital computer solutions of the rigorous equations for scattering problems,” Proc. IEEE **53**, 796–804 (1965). [CrossRef]

*E*⃗

*(*

_{tot}*r*⃗) is composed of two components

*E*⃗

*(*

_{inc}*r*⃗) and

*E*⃗

*(*

_{scat}*r*⃗) are the incident and scattered electric field components, respectively. The incident electric field can be defined as the electric field propagating in space in the absence of a scattering object. The scattered electric field

*E*⃗

*(*

_{scat}*r*⃗) in Eq. (1) represents the fields resulting from the interaction of the incident field

*E*⃗

*(*

_{inc}*r*⃗) with the particles. In three-dimensional space, the scattered field

*E*⃗

*(*

_{scat}*r*⃗) can be written as

*J*⃗(

*r*⃗) is the induced current over the particle,

*ω*is the angular frequency,

*μ*is the permeability, and

*r*⃗ due to a point source at point

*r*⃗′ . By applying the boundary conditions at the surface of a conducting metal, the electric field integral equation is obtained as

*E*⃗

*(*

_{inc}^{tng}*r*⃗) is the tangential component of the known incident electric field on the particle and the

*J*⃗(

*r*⃗) is the unknown induced current on the nanoparticle. This is a Fredholm’s-type integral equation of the first kind since the unknown appears inside the integral. To solve Eq. (4) for

*J*⃗(

*r*⃗), we will expand it into a summation

*b*⃗

*(*

_{j}*r*⃗) represents known basis functions with unknown coefficients

*I*.

_{j}26. A. W. Glisson and D. R. Wilton, “Simple and efficient numerical methods for problems of electromagnetic radiation and scattering from surfaces,” IEEE Trans. Antennas Propag. **28**, 593–603 (1982). [CrossRef]

27. S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. **30**, 409–418 (1982). [CrossRef]

^{th}edge is given as

*l*is the length of the edge,

_{n}*A*

_{n}^{+}is the area of the triangle

*T*

_{n}^{+}, and

*A*

_{n}^{-}is the area of the triangle

*T*

_{n}^{-}. In Fig. 2, two triangles

*T*

_{n}^{+}and

*T*

_{n}^{-}are the two triangles associated with the n

^{th}edge of the discretized particle.

*W⃗*(

_{i}*r*⃗) are used. The residual error

*ℜ*(

*r*⃗) is distributed in space by equating the inner product of the residual error

*ℜ*(

*r*⃗) with the weighting function

*W*⃗

*(*

_{i}*r*⃗) to zero

*w*⃗

*(*

_{i}*r*⃗) in Eq. (9) can be selected in a number of different ways [23

23. R. F. Harrington, *Field Computation by Moment Methods*, (IEEE Press, New York, NY, 1993). [CrossRef]

*Z*is the impedance matrix element on the i

_{i,j}^{th}row and j

^{th}column which is given as

*V*is the excitation source element on the i

_{i}^{th}row given as

*Z*, the observation point and source point can be very close to each other or even coincide. In such instances, the numerical integration diverges, even though the integrals in Eq. (11) are integrable. To avoid this numerical problem, the singularity extraction technique is applied in Eq. (11). The integrals in Eq. (11) are divided into two parts: (1) the part that can be treated using the numerical integration, and (2) the part that is evaluated analytically. For example, the first term on the right hand side of Eq. (11), which has a first order singularity, can be separated into numerically- and analytically-treatable parts as

_{i,i}28. D. R. Wilton, S. M. Rao, A. W. Glisson, D. H. Schaubert, O. M. Al-Bundak, and C. M. Butler “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag. **33**, 276–281 (1984). [CrossRef]

28. D. R. Wilton, S. M. Rao, A. W. Glisson, D. H. Schaubert, O. M. Al-Bundak, and C. M. Butler “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag. **33**, 276–281 (1984). [CrossRef]

## 3. The interaction of metallic nanoparticles with a collimated beam

*z*direction, which is mathematically represented as

*E*⃗

*(*

_{inc}*r*⃗) and

*E*⃗

*(*

_{scat}*r*⃗) are the incident and scattered fields, respectively. The incident field for this problem is defined by Eq. (15) and the scattered field is obtained using a far-field approximation of Eq. (2). In the far-field region, the distance between the source and the observation point can be approximated by

## 4. Linearly and radially polarized focused beam

*α*is the half angle of the beam. In Eq. (20),

*a*⃗(

*θ*,

*ϕ*) is the weighting vector for a plane wave incident from the

*θ*,

*ϕ*direction. Here it should be noted that

*a*⃗(

*θ*,

*ϕ*) is a polarization dependent quantity, which is given as

*E*and E

_{x}_{z}components are plotted on the

*ϕ*=

*π*/2 cut as a function of

*θ*. For small spheres, the

*E*component has a maximum at

_{x}*θ*=

*π*/2. As the spherical particle gets larger, we observe a shift of the location at which the

*E*component has a maximum field. This is due to the increased interaction between a larger sphere and a wider range of angular components of a focused beam. As the size of the spherical particle gets larger, the particle interacts more with components that are incident to larger angles. A similar shift is also observed in the

_{x}*E*component, as shown in Fig. 8.

_{z}*E*and

_{x}*E*components are plotted on the

_{z}*ϕ*=

*π*/2 cut as a function of

*θ*. The incident beam parameters are identical to the previous set of results with the exception that a radial polarization is used instead of a linear polarization. Contrary to the results in Fig. 8,

*E*shows a minimum at

_{x}*θ*=

*π*/2 in Fig. 9. This is due to the difference in the strength of various components of the linearly and radially polarized incident focused beams. For the linearly polarized focused wave, the

*x*-component of the electric field is much stronger than the other two components. The radially polarized wave, on the other hand, has a strong

*z*-component in the focal region.

## 5. Conclusion

## Acknowledgments

## References and links

1. | T. D. Milster, “Horizons for optical data storage,” Optics and Photonics News |

2. | R. E. Rottmayer, S. Batra, D. Buechel, W. A. Challener, J. Hohlfeld, Y. Kubota, L. Li, B. Lu, C. Mihalcea, K. Mountfield, K. Pelhos, C. Peng, T. Rausch, M. A. Seigler, D. Weller, and X. Yang, “Heat-assisted magnetic recording,” IEEE Trans. Magn. |

3. | T. W. McDaniel, W. A. Challener, and K. Sendur, “Issues in heat-assisted perpendicular recording,” IEEE Trans. Magn. |

04. | K. Sendur, W. Challener, and C. Peng, “Ridge waveguide as a near field aperture for high density data storage,” J. Appl. Phys. |

5. | D. W. Pohl, W. Denk, and M. Lanz, “Optical stethoscopy: image recording with resolution |

6. | A. Lewis, M. Isaacson, A. Harootunian, and A. Muray, “Development of a 500 Å spatial resolution light microscope,” Ultramicroscopy |

7. | A. Hartschuh, E. J. Sanchez, X. S. Xie, and L. Novonty, “High-resolution near-field Raman microscopy of single-walled carbon nanotubes,” Phys. Rev. Lett. |

8. | L. Wang and X. Xu, “Numerical study of optical nanolithography using nanoscale bow-tie-shaped nano-apertures,” J. Microsc. |

9. | B. Liedberg, C. Nylander, and I. Lundstroem, “Surface plasmon resonance for gas detection and biosensing,” Sens. Actuators |

10. | W. A. Challener, I. K. Sendur, and C. Peng, “Scattered field formulation of finite difference time domain for a focused light beam in a dense media with lossy materials,” Opt. Express |

11. | J. T. II Krug, E. J. Sánchez, and X. S. Xie, “Design of near-field probes with optimal field enhancement by finite difference time domain electromagnetic simulation,” J. Chem. Phys. |

12. | T. Yamaguchi, “Finite-difference time-domain analysis of hemi-teardrop-shaped near-field optical probe,” Electron. Lett. |

13. | T. Yamaguchi and T. Hinata, “Optical near-field analysis of spherical metals: Application of the FDTD method combined with the ADE method,” Opt. Express |

14. | L. Liu and S. He, “Design of metal-cladded near-field fiber probes with a dispersive body-of-revolution finite-difference time-domain method,” Appl. Opt. |

15. | T. Grosges, A. Vial, and D. Barchiesi, “Models of near-field spectroscopic studies: comparison between finite-element and finite-difference methods,” Opt. Express |

16. | J. P. Kottmann and O. J. F. Martin, “Plasmon resonant coupling in metallic nanowires,” Opt. Express |

17. | J. P. Kottmann and O. J. F. Martin, “Accurate Solution of the Volume Integral Equation for High-Permittivity Scatterers,” IEEE Trans. Antennas Propag. |

18. | J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, “Dramatic localized electromagnetic enhancement in plasmon resonant nanowires,” Chem. Phys. Lett. |

19. | J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, “Field polarization and polarization charge distributions in plasmon resonant nanoparticles,” New J. Phys. |

20. | J. Jung and T. Sondergaard, “Green’s function surface integral equation method for theoretical analysis of scatterers close to a metal interface,” Phys. Rev. B |

21. | J. H. Richmond, “Digital computer solutions of the rigorous equations for scattering problems,” Proc. IEEE |

22. | R. F. Harrington, “Matrix methods for field problems,” Proc. IEEE |

23. | R. F. Harrington, |

24. | E. K. Miller, L. Medgyesi-Mitschang, and E. H. Newman, Eds., |

25. | R. C. Hansen, Ed., |

26. | A. W. Glisson and D. R. Wilton, “Simple and efficient numerical methods for problems of electromagnetic radiation and scattering from surfaces,” IEEE Trans. Antennas Propag. |

27. | S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. |

28. | D. R. Wilton, S. M. Rao, A. W. Glisson, D. H. Schaubert, O. M. Al-Bundak, and C. M. Butler “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag. |

29. | All the FEM calculations in this report are performed with High Frequency Structure Simulator (HFSSTM) from Ansoft Inc. |

30. | E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Philos. Trans. R. Soc. London Ser. A |

31. | B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. tructure of the image field in an aplanatic system,” Philos. Trans. R. Soc. London Ser. A |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

**ToC Category:**

Scattering

**History**

Original Manuscript: December 19, 2008

Revised Manuscript: March 6, 2009

Manuscript Accepted: March 21, 2009

Published: April 21, 2009

**Citation**

Kursat Sendur, "An integral equation based numerical solution
for nanoparticles illuminated with collimated
and focused light," Opt. Express **17**, 7419-7430 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-9-7419

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### References

- T. D. Milster, "Horizons for optical data storage," Opt. Photonics News 16, 28-32 (2005). [CrossRef]
- R. E. Rottmayer, S. Batra, D. Buechel, W. A. Challener, J. Hohlfeld, Y. Kubota, L. Li, B. Lu, C. Mihalcea, K. Mountfield, K. Pelhos, C. Peng, T. Rausch, M. A. Seigler, D. Weller, and X. Yang, "Heat-assisted magnetic recording," IEEE Trans. Magn. 42, 2417-2421 (2006). [CrossRef]
- T. W. McDaniel, W. A. Challener, and K. Sendur, "Issues in heat-assisted perpendicular recording," IEEE Trans. Magn. 39, 1972-1979 (2003). [CrossRef]
- K. Sendur, W. Challener, and C. Peng, "Ridge waveguide as a near field aperture for high density data storage," J. Appl. Phys. 96, 2743-2752 (2004). [CrossRef]
- D. W. Pohl, W. Denk, and M. Lanz, "Optical stethoscopy: image recording with resolution ?/20," Appl. Phys. Lett. 44, 651-653 (1984). [CrossRef]
- A. Lewis, M. Isaacson, A. Harootunian, and A. Muray, "Development of a 500 ? spatial resolution light microscope," Ultramicroscopy 13, 227-231 (1984). [CrossRef]
- A. Hartschuh, E. J. Sanchez, X. S. Xie, and L. Novonty, "High-resolution near-field Raman microscopy of single-walled carbon nanotubes," Phys. Rev. Lett. 90, 095503 (2003). [CrossRef] [PubMed]
- L. Wang and X. Xu, "Numerical study of optical nanolithography using nanoscale bow-tie-shaped nano-apertures," J. Microsc. 229, 483-489 (2008). [CrossRef] [PubMed]
- B. Liedberg, C. Nylander, I. Lundstroem, "Surface plasmon resonance for gas detection and biosensing," Sens. Actuators 4, 299-304 (1983). [CrossRef]
- W. A. Challener, I. K. Sendur, and C. Peng, "Scattered field formulation of finite difference time domain for a focused light beam in a dense media with lossy materials," Opt. Express 11, 3160-3170 (2003). [CrossRef] [PubMed]
- J. T. II Krug, E. J. Sánchez, and X. S. Xie, "Design of near-field probes with optimal field enhancement by finite difference time domain electromagnetic simulation," J. Chem. Phys. 116, 10895 (2002). [CrossRef]
- T. Yamaguchi, "Finite-difference time-domain analysis of hemi-teardrop-shaped near-field optical probe," Electron. Lett. 44, 4455427 (2008). [CrossRef]
- T. Yamaguchi and T. Hinata, "Optical near-field analysis of spherical metals: Application of the FDTD method combined with the ADE method," Opt. Express 15, 11481-11491 (2007). [CrossRef] [PubMed]
- L. Liu and S. He, "Design of metal-cladded near-field fiber probes with a dispersive body-of-revolution finite-difference time-domain method," Appl. Opt. 44, 3429-3437 (2005). [CrossRef] [PubMed]
- T. Grosges, A. Vial, and D. Barchiesi, "Models of near-field spectroscopic studies: comparison between finite-element and finite-difference methods," Opt. Express 13, 8483-8497 (2005). [CrossRef] [PubMed]
- J. P. Kottmann and O. J. F. Martin, "Plasmon resonant coupling in metallic nanowires," Opt. Express 8, 655-663 (2001). [CrossRef] [PubMed]
- J. P. Kottmann and O. J. F. Martin, "Accurate Solution of the Volume Integral Equation for High-Permittivity Scatterers," IEEE Trans. Antennas Propag. 48, 1719-1726 (2000). [CrossRef]
- J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, "Dramatic localized electromagnetic enhancement in plasmon resonant nanowires," Chem. Phys. Lett. 341, 1-6 (2001). [CrossRef]
- J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, "Field polarization and polarization charge distributions in plasmon resonant nanoparticles," New J. Phys. 2, 27 (2000). [CrossRef]
- J. Jung and T. Sondergaard, "Green’s function surface integral equation method for theoretical analysis of scatterers close to a metal interface," Phys. Rev. B 77, 245310 (2008). [CrossRef]
- J. H. Richmond, "Digital computer solutions of the rigorous equations for scattering problems," Proc. IEEE 53, 796-804 (1965). [CrossRef]
- R. F. Harrington, "Matrix methods for field problems," Proc. IEEE 55, 136-149 (1967). [CrossRef]
- R. F. Harrington, Field Computation by Moment Methods, (IEEE Press, New York, NY, 1993). [CrossRef]
- E. K. Miller, L. Medgyesi-Mitschang, and E. H. Newman, Eds., Computational Electromagnetics (IEEE Press, New York, NY, 1992).
- R. C. Hansen, ed., Moment Methods in Antennas and Scattering, (Artech, Boston, MA, 1990).
- A. W. Glisson and D. R. Wilton, "Simple and efficient numerical methods for problems of electromagnetic radiation and scattering from surfaces," IEEE Trans. Antennas Propag. 28, 593-603 (1982). [CrossRef]
- S. M. Rao, D. R. Wilton, and A. W. Glisson, "Electromagnetic scattering by surfaces of arbitrary shape," IEEE Trans. Antennas Propag. 30, 409-418 (1982). [CrossRef]
- D. R. Wilton, S. M. Rao, A. W. Glisson, D. H. Schaubert, O. M. Al-Bundak, and C. M. Butler "Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains," IEEE Trans. Antennas Propag. 33, 276-281 (1984). [CrossRef]
- All the FEM calculations in this report are performed with High Frequency Structure Simulator (HFSSTM) from Ansoft Inc.
- Q2. E. Wolf, "Electromagnetic diffraction in optical systems I. An integral representation of the image field," Philos. Trans. R. Soc. London Ser. A 253, 349-357 (1959).
- Q3. B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system," Philos. Trans. R. Soc. London Ser. A 253, 358-379 (1959).

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