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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 9 — Apr. 27, 2009
  • pp: 7419–7430
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An integral equation based numerical solution for nanoparticles illuminated with collimated and focused light

Kürşat Şendur  »View Author Affiliations


Optics Express, Vol. 17, Issue 9, pp. 7419-7430 (2009)
http://dx.doi.org/10.1364/OE.17.007419


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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

Nano-optics is a rapidly growing field with a diverse set of existing and emerging practical applications. Near-field optical techniques that enhance localized surface plasmons may obtain intense optical spots beyond the diffraction limit for optical data storage [1

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

]. The magnetic storage industry is also interested in sub-wavelength optical spots for heat assisted magnetic recording to overcome the superparamagnetic limit [2-4

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]

]. The interaction of light with nanostructures reveals unique information about the structural and dynamic properties of matter, and is of great importance for biological and solid-state applications. Nano-optical transducers have been widely used in near-field scanning optical microscopy [5

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

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]

]. Hartschuh et al. [7

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]

] obtained 20 nm resolution images of carbon nanotubes using an apertureless configuration. The resolution and scanning time of the scanning near-field optical microscopes, however, are limited by the spot size and transmission efficiency of the nano-optical systems. Therefore, advances in nano-optical transducers benefit scanning near-field optical microscopes and applications that utilize these microscopes. In addition, intense sub-wavelength optical spots have potential applications in nanolithography [8

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]

] and bio-chemical sensing [9

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

]. All of these applications benefit from small optical spots. The transmission efficiency of nano-optical systems should also be maximized for practical applications since transmission efficiency of the nano-optical system will determine the data transfer rate of storage devices and scan times of near-field scanning microscopes.

Various parameters have to be optimized in order to achieve large transmission efficiency while keeping the optical spot size well below the diffraction limit. These parameters include not only geometry-dependent parameters and the material composition of the nano-optical transducer, but also source-dependent parameters, such as operational wavelength and the numerical aperture of the incident beam. Selecting an optimum set of parameters for a nano-optical transducer is important in achieving small spots and large transmission efficiencies. Optimizing the performance of nano-optical parameters requires modeling and simulation of these structures through 3-D full-wave solutions of Maxwell’s equations. An extensive parametric study of the aforementioned transducers requires efficient and accurate solutions of Maxwell’s equations.

2. Method of weighted residuals

A full-wave implementation of the method of weighted residuals (MWR) [21-25

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

], which is also known as the method of moments (MoM), has a number of advantages over FDTD and FEM for nano-optical system analysis. In MWR, the effects of the surrounding medium and boundaries are taken into account using a Green’s function formulation. Therefore, MWR requires only the discretization of the nano-optical transducer, whereas FDTD and FEM require the discretization of the entire computational space. Therefore, the resulting matrix equations of the MWR are smaller in size. An additional advantage of an integral equation based approach is the reduction of the additional error due to the discretization of the boundaries. In an integral equation based approach, the boundary conditions are handled in Green's function formulation; therefore, there is no additional error due to the discretization of the boundaries. In a differential equation based approach, such as FDTD and FEM, however, there is additional error introduced into the solution due to artificial boundary conditions. In addition, the integration of complicated excitation functions, such as focused beams in a dense medium, is easier in an integral equation based MWR compared to FDTD.

In this study, an integral equation based full-wave solution of Maxwell’s equation is developed. To discretize the integral equation into a matrix equation, we employ MWR in this work. In the rest of this section, the details of the solution are given when the incident beam is a collimated beam, which is approximated by a plane wave. In Sec. 4, the formulation will be extended to linearly and radially polarized focused incident beams, which are represented by the Richards-Wolf vector field equations.

The total electric field is a result of the interaction of an incident optical beam with a nanoparticle. The total electric field Etot(r⃗) is composed of two components

Etot(r)=Einc(r)+Escat(r)
(1)

where Einc (r⃗) and Escat(r⃗) 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 Escat(r⃗) in Eq. (1) represents the fields resulting from the interaction of the incident field Einc(r⃗) with the particles. In three-dimensional space, the scattered field Escat(r⃗) can be written as

Escat(r)=iωμ4πS′dS′G(r,r)·J(r)
(2)

where J⃗(r⃗) is the induced current over the particle, ω is the angular frequency, μ is the permeability, and

G(r,r)=[I+k2]eikrrrr
(3)

is the dyadic Green’s function in free space at point r⃗ due to a point source at point r⃗′ . By applying the boundary conditions at the surface of a conducting metal, the electric field integral equation is obtained as

Einctng(r)=t̂·iωμ4πS′dS′[I+k2]eikrrrr·J(r)
(4)

In Eq. (4), Einctng (r⃗) is the tangential component of the known incident electric field on the particle and the J⃗(r⃗) 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 J⃗(r⃗), we will expand it into a summation

J(r)j=1NIjbj(r)
(5)

where bj (r⃗) represents known basis functions with unknown coefficients Ij.

Mathematical representation of the triangular basis function associated with the nth edge is given as

bn(r)={ln2An+ρn+;rTn+ln2Anρn;rTn0;otherwise
(6)

where ln is the length of the edge, An + is the area of the triangle Tn +, and An - is the area of the triangle Tn -. In Fig. 2, two triangles Tn + and Tn - are the two triangles associated with the nth edge of the discretized particle.

Fig. 1. Example surface triangulation of (a) a sphere with a radius of R=350 nm and (b) a cube with a side length of L=200 nm. Triangulations are performed in Fortran and visualization performed via Matlab.

Substituting the expansion given in Eq. (5) back into the Eq. (4), and changing the order of the integration and summation we obtain

Einctng(r)j=1NIj{t̂·iωμ4πS′dS′[I+k2]eikrrrr·bj(r)}
(7)

Due to the approximation of the induced current with the summation in Eq. (5), there is a residual error in Eq. (7). The residual error in space can be written as

(r)=Einctng(r)+j=1NIj{t̂·iωμ4πS′dS′[I+k2]eikrrrr·bj(r)}
(8)

In the method of weighted residuals the error is distributed so that it is minimized in the minimum mean square sense. For this purpose, a new set of functions, known as weighting functions W⃗i(r⃗) are used. The residual error (r⃗) is distributed in space by equating the inner product of the residual error (r⃗) with the weighting function Wi(r⃗) to zero

(r),ωi(r)=Ω(r)·ωi(r)dr=0
(9)

The weighting function wi(r⃗) 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]

]. In this study, Galerkin’s weighting method is chosen, in which weighting functions are selected as identical to basis functions. Such a selection yields the best result in the minimum mean square sense.

Fig. 2. To discretize the induced current, triangular rooftop basis functions are used.

By placing the weighting functions into Eq. (9) we can obtain the resulting equations for the unknown coefficients of the basis functions. After mathematical manipulations, the result can be expressed as a system of linear equations as

Z·I¯=V¯
(10)

where Zi,j is the impedance matrix element on the ith row and jth column which is given as

Zi,j=iωμ4π[SdSwi(r)·S′dSeikrrrrbj(r)+SdSwi(r)·k2·S′dS′eikrrrrbj(r)]
(11)

and Vi is the excitation source element on the ith row given as

Vi=SdSwi(r)·Einc(r)
(12)

By solving the matrix equation in Eq. (10), we obtain the unknown coefficients of the basis functions in the induced current expansion in Eq. (5).

The matrix and vector elements in Eq. (11) are obtained using numerical integration techniques over triangular domains. An important issue in evaluating Eqs. (11) and (12) is the singularity in the kernel of the integrals. For the diagonal elements Zi,i, 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

SdSwi(r)·S′dSeikrrrrbj(r)=SdSwi(r)·(S′dSeikrr1rrbj(r)+S′dS1rrbj(r))
(13)

In this study, the singularity extraction technique for triangular domains [28

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]

] is employed to avoid numerical inaccuracy.

The singularity in the second term on the right hand side of Eq. (11) is third order, which is more difficult to handle analytically compared to a first order singularity. However, for the triangular rooftop basis functions used in this study, the second term on the right hand side of Eq. (11) can be simplified as

SdSwi(r)·k2S′dSeikrrrrbj(r)=SdS(·wi(r))S′dSeikrrrr(·bj(r))
(14)

which has a first order singularity and is handled with the formulations given in the literature [28

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

Einc=x̂eikz
(15)

The results of the integral equation based solution are compared to the results of the analytical Mie series solution for spherical particles and the finite element method for rectangular metallic particles [29

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

].

To compare the results, the radar cross sections of particles are calculated for different cross sections of the far-field. The radar cross section is defined as

RCS=limr4πr2Es2Ei2
(16)

where Einc(r⃗) and Escat(r⃗) 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

rrrr̂·r
(17)

Substituting Eq. (17) back into Eq. (2), the scattered field in the far-zone can be written as

EscatFFiωμ4πeikrrS′J(r)eikr̂·rdS′
(18)
Fig. 3. A comparison of the MWR results with the Mie series solution for the RCS of a conducting sphere with a radius of 140 nm. The operating wavelength is 700 nm. θ and ϕ components of the radar cross section are plotted on various cuts: (a) RCSθ as a function of ϕ on θ=90° cut, (b) RCSθ as a function of θ on ϕ=0° cut, (c) RCSϕ as a function of ϕ on θ=90° cut, and (d) RCSϕ as a function of θ on ϕ=90° cut.
Fig. 4. Percent relative error of MWR results in Fig. 3 compared to the Mie series solution for : (a) RCSθ on θ=90° cut, (b) RCSθ on ϕ=0° cut, (c) RCSϕ on θ=90° cut, and (d) RCSϕ on ϕ=90° cut.
Fig. 5. A comparison of the MWR results with the Mie series solution for the RCS of a conducting sphere with a radius of 350 nm. The operating wavelength is 700 nm. θ and ϕ components of the radar cross section are plotted on various cuts: (a) RCSθ as a function of ϕ on θ=90° cut, (b) RCSθ as a function of θ on ϕ=0° cut, (c) RCSϕ as a function of ϕ on θ=90° cut, and (d) RCSϕ as a function of θ on ϕ=0° cut.
Fig. 6. Percent relative error of MWR results in Fig. 5 compared to the Mie series solution for : (a) RCSθ on θ=90° cut, (b) RCSθ on ϕ=0° cut, (c) RCSϕ on θ=90° cut, and (d) RCSϕ on ϕ=90° cut.

Substituting the coefficients obtained from the solution of the matrix equation into the induced current expansion in Eqs. (5) and (18), and changing the order of the integration and summation, the expression for the far-zone scattered field can be simplified as

EscatFFiωμ4πeikrrn=1NαnS′bn(r)eikr̂·rdS′
(19)

Using Eqs. (15), (16), and (19), the scattering cross section of various particles now can be obtained.

In Fig. 3, the radar cross section of a sphere with a radius of 140 nm is presented to compare MWR results with the analytical Mie series solution. The operating wavelength of the laser source is 700 nm. A comparison of the MWR results with the analytical Mie series solution shows a good agreement between the results. The percent relative error between the MWR results and Mie series solution is presented for the 140 nm particle in Fig. 4. The results suggest that the error is smaller than 1.7 % on various cuts. The main source of this error is the discretization of the nanoparticle. This error can be further reduced by increasing the number of unknowns. A similar comparison is provided in Fig. 5 for a larger sphere with a radius of 350 nm. The results in Fig. 5 also show a good agreement between the results. The relative error between the results is presented in Fig. 6. The results suggest that the maximum error for this case is about 2.5 %.

Fig. 7. A comparison of the FEM and MWR results for the radar cross section of a conducting cube with a side length of 200 nm. The operating wavelength for the incident beam is 700 nm. θ component of the radar cross section is plotted on various ϕ cuts: (a) RCSθ as a function of θ on ϕ=0° cut, (b) RCSθ as a function of θ on ϕ=90° cut.

The method in this study is capable of addressing the near-field computations. Once the unknown coefficients in Eq. (5) are calculated, these equations can be substituted back into the electric field integral to calculate the near-field distributions.

4. Linearly and radially polarized focused beam

Richards and Wolf developed a method for calculating the electric field semi-analytically near the focus of an aplanatic lens [30

30. 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).

, 31

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 253, 358–379 (1959).

]. Using the Richards-Wolf method, we can obtain both transverse and longitudinal components near the focus for both linear and radial polarizations. The total electric field in the vicinity of the focus is given as

Einc(r)=iλ0αsinθ02πa(θ,ϕ)eik·r
(20)

where α is the half angle of the beam. In Eq. (20), a⃗(θ,ϕ) is the weighting vector for a plane wave incident from the θ, ϕ direction. Here it should be noted that a⃗(θ, ϕ) is a polarization dependent quantity, which is given as

a(θ,ϕ)=[cosθcosθcos2ϕ+sin2ϕcosϕsinϕcosϕsinϕsinθcosϕ]cosθ
(21)
a(θ,ϕ)=[cosθcosϕcosθsinϕsinθ]cosθ
(22)

for linear and radial polarizations, respectively. The interaction of a focused beam with linear or radial polarization can be obtained by substituting Eq. (20) back into Eq. (12).

In Fig. 8, various components of the near-field radiation from a sphere are plotted when the incident beam is a linearly polarized focused beam obtained from an optical lens system with a numerical aperture of 0.85. The operating frequency is 700 nm. The results are plotted for spherical particles with radii 70 and 140 nm. The Ex and Ez components are plotted on the ϕ = π/2 cut as a function of θ. For small spheres, the Ex component has a maximum at θ = π/2. As the spherical particle gets larger, we observe a shift of the location at which the Ex 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 Ez component, as shown in Fig. 8.

Fig. 8. Electric field components when a linearly polarized focused beam of light interacts with spheres of various sizes. The linearly polarized focused beam is obtained from an optical lens system with a numerical aperture of 0.85. The operating frequency is 700 nm.
Fig. 9. Electric field components when a radially polarized focused beam of light interacts with spheres of various sizes. The radially polarized focused beam is obtained from an optical lens system with a numerical aperture of 0.85. The operating frequency is 700 nm.

5. Conclusion

In this work, an integral equation based numerical solution was developed. The formulations for both plane waves and focused beams were given. For focused beams, the Richards-Wolf vector field equations were combined with the integral equation based formulation to model both linearly and radially polarized focused beams. The results of the integral equation based solution were compared to the results of the analytical Mie series solution for spherical particles and the finite element method for rectangular metallic particles. The methods showed a good agreement.

Acknowledgments

This work was performed with the support of the European Community Marie Curie International Reintegration Grant (IRG) Agreement Number MIRG-CT-2007-203690.

References and links

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]

3.

T. W. McDaniel, W. A. Challener, and K. Sendur, “Issues in heat-assisted perpendicular recording,” IEEE Trans. Magn. 39, 1972–1979 (2003). [CrossRef]

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]

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]

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]

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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]

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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]

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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]

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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]

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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]

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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]

21.

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

22.

R. F. Harrington, “Matrix methods for field problems,” Proc. IEEE 55, 136–149 (1967). [CrossRef]

23.

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

24.

E. K. Miller, L. Medgyesi-Mitschang, and E. H. Newman, Eds., Computational Electromagnetics (IEEE Press, New York, NY, 1992).

25.

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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]

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]

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 253, 349–357 (1959).

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 253, 358–379 (1959).

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

  1. T. D. Milster, "Horizons for optical data storage," Opt. 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]
  3. T. W. McDaniel, W. A. Challener, and K. Sendur, "Issues in heat-assisted perpendicular recording," IEEE Trans. Magn. 39, 1972-1979 (2003). [CrossRef]
  4. 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]
  5. D. W. Pohl, W. Denk, and M. Lanz, "Optical stethoscopy: image recording with resolution ?/20," Appl. Phys. Lett. 44, 651-653 (1984). [CrossRef]
  6. A. Lewis, M. Isaacson, A. Harootunian, and A. Muray, "Development of a 500 ? 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]
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