OSA's Digital Library

Virtual Journal for Biomedical Optics

Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editor: Gregory W. Faris
  • Vol. 2, Iss. 3 — Mar. 7, 2007
« Show journal navigation

Improved scheme for accurate computation of high electric near-field gradients

Thomas Grosges, Houman Borouchaki, and Dominique Barchiesi  »View Author Affiliations


Optics Express, Vol. 15, Issue 3, pp. 1307-1321 (2007)
http://dx.doi.org/10.1364/OE.15.001307


View Full Text Article

Acrobat PDF (883 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

We present an improved adaptive mesh process that allows the accurate control of the numerical solution of interest derived from the solution of the partial differential equation. In the cases of two-dimensional studies, such an adaptive meshing is applied to compute phenomenon involving high field gradients in near-field (electric intensity, Poynting’s vector, optical forces,…). We show, that this improved scheme permits to decrease drastically the computationnal time and the memory requirements.

© 2007 Optical Society of America

1. Introduction

Since the advent of the near-field optical microscopy and its use for the characterization of nano-structures, several numerical methods have been developed for the modeling of the interactions between the electromagnetic field and the nanostructures which exhibit high gradients in the electric field. These gradients are of interest for the prediction of Raman or fluorescence in near-field. In principle, all the methods which make possible the computation of the optical near-field around metallic nanostructures illuminated by an external source. Nevertheless, only a few of them are able to deal with high local variations of the electromagnetic field [1

1. 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]

, 2

2. D. Barchiesi, B. Guizal, and T. Grosges, “Accuracy of local field enhancement models: toward predictive models?,” Appl. Phys. B ,84,55–60 (2006). [CrossRef]

]. This condition is directly connected with the numerical convergence of the algorithm and the discretization of the problem. Comparisons of various methods were reviewed in several papers [2

2. D. Barchiesi, B. Guizal, and T. Grosges, “Accuracy of local field enhancement models: toward predictive models?,” Appl. Phys. B ,84,55–60 (2006). [CrossRef]

, 3

3. D. Barchiesi, C. Girard, O.J.F. Martin, D. Van Labeke, and D. Courjon, “Computing the optical near-field distributions around complex subwavelength surface structures: A comparative study of different methods,” Phys. Rev. E 54,4285–4292 (1996). [CrossRef]

, 4

4. B. Guizal, D. Barchiesi, and D. Felbacq, “Electromagnetic beam diffraction by a finite lamellar structure,” J. Opt. Soc. Am. A 20,2274–2280 (2003). [CrossRef]

, 5

5. 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), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-21-8483. [CrossRef] [PubMed]

]. Among these different methods, the finite element method is a powerful tool that has proven its utility in the study of electromagnetic problems. In the present work, we compare the results to the Mie’s theory in the case of an infinite gold cylinder in order to improve the convergence of the numerical scheme to the solution of interest as the Poynting’s vector amplitude and the intensity of the electric field. We compare and discuss the advantages of the improved adaptive mesh scheme relatively to the classical adaptive remeshing scheme with h-method estimator. We show that the use of FEM with this improved adaptive meshing permits to decrease drastically the computational time and the memory requirement.

The organization of the paper is: Section 2 presents the formulation of the problem in near-field. Section 3 is devoted to the description of the improved adaptive mesh process, used to compute the total electromagnetic field and the Poynting’s vector amplitude. In Sec. 4, numerical results obtained with this improved mesh adaptation process in connection with FEM are presented and discussed before concluding in Sec. 5.

2. Formulation of the problem in near-field

We study the particular case of 2D structures (nanowires, nanotubes, nanoslits…) considered infinite along z-direction. The different media are supposed to be non-magnetic and a p-polarized illuminating wave is used to produce resonances and localized plasmons. Assuming a timeharmonic dependance of the form exp(jωt) for the fields [6

6. M. Born and E. Wolf, Principle of Optics (Pergamon Press, Oxford, 1993).

], the partial differential equation (PDE) to be solved is the Helmholtz’ equation for the magnetic field Hz [7

7. J. Jin, The Finite Element Method in Electromagnetics (John Wiley and Sons, New York,1993).

]:

[·(1εr)+ω2c2]Hz=0inΩ,
(1)

where c is the velocity of light in vacuum and εr is the dimensionless relative complex permittivity of the considered medium (ε = ε 0 εr) which is a function of the spatial coordinates (x,y). The electric field E(x,y) is then deduced from the Maxwell’s equation:

Exy=jωε(×Hxy),
(2)

and the Poynting’s vector P(x,y) depends on both fields:

Pxy=12(Exy×H*xy),
(3)

3. The improved adaptive mesh method

In this section we will describe the principles of the proposed adaptive-mesh scheme to be employed in our numerical experiments. The improvement of this strategy with respect to the basic adaptive meshing approaches lies on the simultaneous utilisation of an adaptive meshing process and an a posteriori estimation of the interpolation error in the electromagnetic computation. These two features allow an important reduction in the number of computed nodes.

Fig. 1. Remeshing of one element e 1 1 into ek 2 for (a) the basic adaptation (k = 1−2, with 3 nodes), (b) the adaptation with the h-method interpolating the error on the PDE solution (Hz) and (c) the improved adaptive scheme. The adaptives scheme (b) and the improved one (c) using h-method produce more than two new elements (k = 1−4, with 5 nodes). Moreover, with the basic adaptation (a) and the a posteriori h-method (b), the error is estimated on the computed PDE solution Hik. This contrast to the improved scheme (c) for which Sϕki denotes the interpolation error of the solution of interest (ϕ = E or ϕ = P,‥) on the remeshing with respect to the threshold δϕi for the iterative step i.

In Fig. 1 we consider a one dimensional element e to outline the main differences between two widely employed adaptive meshing processes and our improved scheme. The basic adaptation process is shown in Fig. 1(a), it consists on dividing by two the length of segment e 1 1 by adding only one new node and evaluating the error on the PDE solution. The h-method estimator is illustrated in Fig. 1(b), and permits to divide the length of the segment with respect to a threshold δ for the error estimator based on the PDE solution (δH). The meshing process in Fig. 1(c) corresponds to the improved scheme proposed in this work and permits to divide the length of the segment with respect to a threshold δ for the error estimator based on the solution of interest (δE or δP,‥).

The strategy of the global adaptive process is governed by a map mesh element size that ensures the conformity with the underlying geometry domain and of the accuracy of the solution of interest. This size map results from an a posteriori estimation of the interpolation error on the field of interest (E, P,‥).

The main steps of the improved general adaptive remeshing scheme in a domain Ω defined from its boundary Γ can be summarized as follows:

  • 0 generation of a first coarse mesh (Ω) of Ω,
  • 1 Computation of the field Hz (PDE solution) by FEM on (Ω),
  • 2 Derivation of the solution of interest Sϕ (i.e. E or P,‥) on (Ω),
  • 3 Estimation of the physical error (i.e. computation of the interpolation error of the solution of interest Sϕ (ϕ being E or P,‥) based on the estimation of the discrete Hessian of Sϕ; definition of a physical size map (Ω) allowing to bound this error with a given threshold δϕ),
  • 4 remeshing of the domain conforming with the size map (Ω),
  • 5 loop to step 1 when threshold (δϕ) is not satisfied.

To bound the deviation η between the exact solution S and the physical solution S associated with the mesh (Ω) (obtained by finite element computation or derivation), an indirect approach is used. This consists in defining S˜ an interpolation of S on mesh (which can be a piecewise linear or quadratic function, depending on the degree of the elements of ) and η˜ the deviation between S and S˜. Such an interpolation error e˜ can be used to bound the gap between the solution obtained numerically and the unknown exact solution:

ηCη˜
(4)

where ∥·∥ is a norm and C a constant independent from . In other words, the finite element error is bounded by the interpolation error of the solution [9

9. P. Houston, I. Perugia, and D. Schotzau, “Energy norm a posteriori error estimation for mixed discontinuous Galerkin approximations of the Maxwell operator,” Comput. Methods Appl. Mech. Eng. 194,499–510 (2005). [CrossRef]

, 10

10. D. Pardo, L. Demkowicz, C. Torre-Verdìn, and L. Tabarovsky, “A goal-oriented hp-adaptive nite element method with electromagnetic applications. Part I: Electrostatics,” Int. J. Numer. Methods Eng. 65,1269–1309 (2005). [CrossRef]

, 11

11. D. Xue and L. Demkowicz, “Modeling of electromagnetic absorption/scattering problems on curvilinear geometries using hp finite/infinite element method,” Finite Elem. Anal. Design 42,570–579 (2006). [CrossRef]

, 12

12. H. Borouchaki, P. Lang, A. Cherouat, and K. Saanouni, “Adaptive remeshing in large plastic strain with damage,” Int. J. Numer. Methods Eng. 63,1–36 (2005). [CrossRef]

, 14

14. M. Ainsworth and J.T. Oden, “A posteriori error estimation in finite element analysis,” Comput. Methods Appl. Mech. Eng. 142,1–88 (1997). [CrossRef]

, 15

15. R. Radovitzky and M. Ortiz, “Error estimation and adaptive meshing in strongly non-linear dynamic problems,” Comput. Methods Appl. Mech. Eng. 172,203–240 (1999). [CrossRef]

]. Therefore, the problem is reduced to the characterization of those meshes on which the interpolation error is limited by a threshold [13

13. M. Berzins, “Mesh quality: a function of geometry, error estimates or both?,” Eng. Comput. 15,236–247 (1999). [CrossRef]

]. Also, the examination of a ‘measurement’ of the interpolation error makes it possible to obtain constraints on the size of the mesh element and the interpolation error gives the way to quantify the degree of ‘regularity’ of the exact solution S. The remeshing process consists in building a new mesh of Ω by using the interpolation error achieved on the solution of interest Sϕ.

The resolution method, based on the local deviation of the solution surface, gives a bound locally in the vicinity of each mesh vertex. This deviation can be quantified simply by considering the Hessian along the normal to the surface (i.e. the second fundamental form of the surface). Let P be a vertex of the solution surface. Locally, in the vicinity of P, this surface has a parametric representation ψ(u,ν), (u,ν) being the parameters, with P=ψ(0,0). Applying the Taylor expansion of order two to ψ in the vicinity of P, we obtain:

ψuν=P+ψuu+ψνν+12(ψuuʺu2+2ψʺ+ψννʺν2)+o(u2+ν2)ê
(5)

where the derivative of ψ are evaluated at (0,0) and ê = (1,1,1). If n(P) denotes the normal to the surface at P, then the quantity 〈n(p)∣(ψ(u,ν)−ψ(0,0))〉 (〈·∣·〉 being the scalar product) represents the gap point ψ(u,ν) to the tangent plane at P, and can be written as:

12(n(P)ψuuʺu2+2n(P)ψʺ+n(P)ψννʺ)+o(u2+ν2)
(6)

which is proportional to the second fundamental form of the surface for u 2+ν 2 small enough. Hence, to measure the local deviation of the surface, one only has to consider, for each node of the surface, the maximal deviation from the adjacent nodes to the tangent plane at this node. This size at each node of (Ω) is then defined as being proportional to the inverse of this deviation [12

12. H. Borouchaki, P. Lang, A. Cherouat, and K. Saanouni, “Adaptive remeshing in large plastic strain with damage,” Int. J. Numer. Methods Eng. 63,1–36 (2005). [CrossRef]

]. Formally, the size associated with a node w of (Ω) can be written as:

hϕ(w)=δϕη(w,S(w))
(7)

where η(w,S(w)) denotes this new deviation. The size map ϕ associated with the vertices of (Ω) is deduced.

Due to the optimization of the position of the new vertex in respect to the a posteriori interpolation error achieved on the field of interest, the improved adaptive remeshing procedure permits to reduce the number of the iterations and to control the accuracy of the solution. That contrasts to the basic adaptive process where two loop sequences were necessary; the first one on the error on the PDE solution and the second one on the error on the field of interest (E, P,…). In near-field physics, the physical phenomenon of interest are those implying a strong variation of the electric field E (or the Poynting’s vector amplitude ∣P∣ or the electromagnetic force F), while the variations of the magnetic field H (the PDE solution) are smooth. In contrast with the classical remeshing procedure and/or the classical adaptive meshing scheme with h-method, the proposed adaptive scheme optimizes the mesh only in the regions where the solution of interest (E, or ∣P∣, or F,…) presents high local variations. Moreover, it permits to control a priori the error on the solution of interest (E, or ∣P∣, or F,…) and not on the PDE solution (H), as it is done with the classical error estimator.

4. Applications to high gradient of the electric field computation

At this stage, we use an example to illustrate the computation of the the electromagnetic field with the improved adaptive mesh scheme. We compare the result of our numerical experiments that we have obtained with a classical remeshing process with h-method (a posteriori error estimation on the PDE solution Hz). In the following, the classical remeshing process with h-method is denoted as classical remeshing process.

4.1. Case of the gold nano-cylinder

It consists in the computation of the electric and magnetic field intensities; E and H, around an infinite circular-cylinder of gold with radius a = 15 nm, illuminated with a p-polarized wave at λ = 660 nm, as depicted in Fig. 2. The harmonic solution of the problem (Eq. 1) enables the use of the experimental values of the relative permittivities of the materials (εr = −13.6969 + j1.0356) and a non-regular non-Cartesian adaptive mesh to reproduce the curvature of the nanostructure. The FEM adaptive mesh is realized employing the BL2D-V2 bidimensional mesher [16

16. P Laug and H Borouchaki 2003 “BL2D-V2: mailleur bidimensionnel adaptatif,” Report INRIA RT-0275http://www-rocq1.inria.fr/gamma/cdrom/www/bl2d-v2/INDEX.html

] to match closely the geometry of the object. This simple case is studied through a comparison of the results obtained employing FEM with a classical adaptive meshing, with the improved adaptive mesh scheme and with the analytical solution computed with the Mie’s Theory [17

17. G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25,377–445 (1908). [CrossRef]

, 18

18. H. Du, “Mie-scattering calculation,” Appl. Opt. 43,1951–1956 (2004). [CrossRef] [PubMed]

].

Figures 3 show the evolution of the normalized intensity of the electric field ∣E2/∣E02 for the starting coarse mesh (see Fig. 3(a,b) for the classical remeshing and the improved remeshing, respectively), the first and last remeshing (see Fig. 3(c) and Fig. 3(e) for the classical remeshing, Fig. 3(d) and Fig. 3(f) for the improved adaptive mesh scheme). The computational times and the number of nodes at each iterative step for the two remeshing procedures are given in Table 1. This computation time can strongly vary, depending on the resonant behavior of the diffracting structure [19

19. C. Ropers, D.J. Park, G. Stibenz, G. Steinmeyer, J. Kim, D.S. Kim, and C. Lienau, “Femtosecond Light Transmission and Subradiant Damping in Plasmonic Crystals,” Phys. Rev. Lett. 94,113901–4 (2005). [CrossRef] [PubMed]

]. Furthermore, the computation time of FEM strongly depends on the number of mesh refinements and of the method used to inverse the matrix obtained from the variational scheme. Therefore the present CPU times are indicative instead of absolute values. In this paper, we use an optimized LU decomposition for sparse matrix [20

20. T.A. Davis and I.S. Duff, “A combined unifrontal multifrontal method for unsymmetric sparse matrices,” ACM T. Math Software 25,1–20 (1999). [CrossRef]

] or, for larger matrices, an iterative biconjugate stabilized gradient with hermitian preconditioning.

Fig. 2. Geometry of the study for the infinite circular-cylinder along the z-axis illuminated by a p-polarized incident wave of vector k.

Table 1. Number of nodes and computational time (in s) for each mesh step for the classical remeshing and the improved adaptive remeshing process.

table-icon
View This Table

The evolution of the computational mesh as a function of the adaptive remeshing scheme (see Fig. 4(a,c,e) for the classical remeshing and Fig. 4(b,d,f) for the improved remeshing). In contrast with a classical mesher, the refinement and the ‘regularity’ is obvious in the case of the improved meshing (Fig. 4(b,d,f)), which permits to describe accurately the shape of the field intensity. Moreover, the adaptation of the mesh to the solution of interest (i.e. intensity of the electric field) clearly appears in Fig. 5(a) with the improved remeshing scheme in contrast to a classical remeshing process (Fig. 5(b)).

We also show in Fig. 6(a-f) the evolution of the normalized intensity of the magnetic field ∣H2/∣H02 as function of the remeshing for the both the classical and the improved adaptive process. The starting coarse mesh and two successive remeshing are presented in Fig. 6(a,c,e) for the classical remeshing and in Fig. 6(b,d,f) for the improved remeshing. In contrast to the strong variations of the intensity and shape of the electric field, the intensity and shape of the magnetic field are smoothly varying. Therefore an optimization of the mesh seems to be less critical when only the magnetic field is the solution of interest. However, this is rarely the case and that clearly shows the adavantage of an adaptive meshing process based on the solution of interest (E, P) instead of the computational variable H (the PDE solution).

Fig. 3. Normalized intensity of the electric field ∣E2/∣E02 in the xy-plane computed by the FEM with the classical adaptive remeshing (a,c,e) and the improved adaptive scheme (b,d,f). The number of nodes are (a) Nnodes = 892, (b) Nnodes = 768, (c) Nnodes = 4818, (d) Nnodes = 5556, (e) Nnodes = 71260 and (f) Nnodes = 31669, respectively.
Fig. 4. Computational mesh for the classical adaptive remeshing (a,c,e) and the improved adaptive scheme (b,d,f).
Fig. 5. Last computational mesh for (a) the improved adaptive scheme and (b) the classical adaptive remeshing. The adaptation to the solution of interest (intensity of the electric field) clearly appears with the improved remeshing.
Fig. 6. Normalized intensity of the magnetic field ∣H2/∣H02 in the xy-plane computed by the FEM with classical adaptive remeshing (a,c,e) and improved adaptive scheme (b,d,f). The number of nodes are (a) Nnodes = 892, (b) Nnodes = 768, (c) Nnodes = 4818, (d) Nnodes = 5556, (e) Nnodes = 71260 and (f) Nnodes = 31669, respectively.
Fig. 7. Evolution of the normalized intensity of the electric and magnetic field as a function of the mesh refinement (a,c) and (b,d) the errors, relatively to Mie computation, as a function of the distance from the center of the nano-object (radius a = 15 nm) along the x-axis for successive mesh refinements with the classical and the improved adaptive remeshing.

In order to test the remeshing, we focus on the region close to the nano-object where high gradient of the electromagnetic field occurs. We compute the variation of the normalized electromagnetic intensity at short distances of detection from the center of the cylinder along the x-axis where the contribution of the scattered field is supposed to be important. In this region, a fine meshing is necessary for FEM to accurately describe the intensity variations (as illustrated in Fig. 3). Consequently, the remeshing is adapted to the electric field computation (Poynting’s vector amplitude, force vector amplitude,…) instead of the Hz amplitude. As shown in Table 1, the remeshing process permits to decrease drastically the total number of nodes. We can also note that a distinction must be done between: ‘convergence of the numerical method’ and ‘convergence to the physical solution’. We consider that the result of the computation has converged to the physical solution when the relative error between the result and the analytical solution (when it exists) is smaller than a maximal tolerance (e.g.∥δϕ ≤ 0.5%). when no analytical solution exists, the relative error can be computed by comparing the solutions obtained with two successive mesh refinements or by the interpolation error estimator of the remeshing procedure.

We show in Fig. 7(a,c) the evolution of the normalized (a) electric and (c) magnetic intensity computed with FEM, for various distance from the center of the nano-object (radius a = 15 nm) along the x-axis, for successive mesh refinements with classical and improved adaptive remeshing. Figure 7(b,d) shows the error relatively to the Mie’s results as a function of the distance and the mesh refinements. Mie and FEM computations are performed using the same non-Cartesian mesh. In contrast to the classical remeshing, which need four iterations (0-step and four iterative steps), only two improved adaptive remeshing are necessary to stabilize the solution and to assure a convergence of the numerical results to the physical solution (with an relative error δϕ around 0.5%, see Fig. 7(b)). The smallest cell size is less than 0.03 nm in the region where high gradient of field occurs.

4.2. Case of the gold nano-square

Fig. 8. Geometry of the study for the infinite square-cylinder along the z-axis.

Finally, we present the evolution of the amplitude around the nanostructure. Figure 10(a) shows the amplitude along the x-axis, at the interface between the substrate and the nanostructure, and Fig. 10(b) is along the y-axis (i.e. on the lateral face of the square-cylinder), for various mesh refinements. The topic is to compute the value of the Poynting’s vector amplitude (or intensity, force,‥) and to describe accurately the shape of the physical field in the vicinity of the edges of the object. The convergence and the stabilization to the solution as a function of the remeshing clearly appears.

Fig. 9. Normalized amplitude of the Poynting’s vector ∣P∣/∣P0∣ computed by the FEM with improved adaptive mesh scheme and the mesh view for (a,b) Nnodes = 3371, (c,d) Nnodes = 13205 and (e,f) Nnodes = 78175, respectively.
Fig. 10. Evolution of the intensity as a function of the distance along (a) the x-axis and (b) the y-axis, for various mesh refinements.

5. Conclusion

In this paper we have presented an improved adaptive mesh process based on the a posteriori error indicator estimation on the solution. This approach is used in connexion with the numerical method FEM for the computation of the intensity of the total electromagnetic field and/or the Poynting’s vector amplitude in context of near-field enhancement. We have shown that a sufficient grid size refinement must be considered to assure the convergence and stabilization of the solution, particularly in the regions where a strong confinement of the light around the nanostructure takes place. The a posteriori error estimator is achieved on the field of interest (E, P). The main advantage of this a posteriori error estimator is to decrease the remeshing number on the PDE solution and to control the error on the solution. We have shown that an adaptive mesh scheme in connection with FEM is suitable to describe accurately the complex geometries and to represent accurate map of the intensity. This improved procedure accelerates drastically the convergence of the solution and minimizes both the memory requirement and the computational time. Moreover, its extension and its application to problems in three dimensions is direct.

Acknowledgments

Authors gratefully thank Dr D. Macìas for fruitful discussions on the improvement of the paper.

References and links

1.

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]

2.

D. Barchiesi, B. Guizal, and T. Grosges, “Accuracy of local field enhancement models: toward predictive models?,” Appl. Phys. B ,84,55–60 (2006). [CrossRef]

3.

D. Barchiesi, C. Girard, O.J.F. Martin, D. Van Labeke, and D. Courjon, “Computing the optical near-field distributions around complex subwavelength surface structures: A comparative study of different methods,” Phys. Rev. E 54,4285–4292 (1996). [CrossRef]

4.

B. Guizal, D. Barchiesi, and D. Felbacq, “Electromagnetic beam diffraction by a finite lamellar structure,” J. Opt. Soc. Am. A 20,2274–2280 (2003). [CrossRef]

5.

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), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-21-8483. [CrossRef] [PubMed]

6.

M. Born and E. Wolf, Principle of Optics (Pergamon Press, Oxford, 1993).

7.

J. Jin, The Finite Element Method in Electromagnetics (John Wiley and Sons, New York,1993).

8.

P. Ingelström and A. Bondeson, “Goal-Oriented error estimation and h-adaptivity for Maxwell’s equations,” Comput. Methods Appl. Mech. Eng. 192,2597’2616 (2003). [CrossRef]

9.

P. Houston, I. Perugia, and D. Schotzau, “Energy norm a posteriori error estimation for mixed discontinuous Galerkin approximations of the Maxwell operator,” Comput. Methods Appl. Mech. Eng. 194,499–510 (2005). [CrossRef]

10.

D. Pardo, L. Demkowicz, C. Torre-Verdìn, and L. Tabarovsky, “A goal-oriented hp-adaptive nite element method with electromagnetic applications. Part I: Electrostatics,” Int. J. Numer. Methods Eng. 65,1269–1309 (2005). [CrossRef]

11.

D. Xue and L. Demkowicz, “Modeling of electromagnetic absorption/scattering problems on curvilinear geometries using hp finite/infinite element method,” Finite Elem. Anal. Design 42,570–579 (2006). [CrossRef]

12.

H. Borouchaki, P. Lang, A. Cherouat, and K. Saanouni, “Adaptive remeshing in large plastic strain with damage,” Int. J. Numer. Methods Eng. 63,1–36 (2005). [CrossRef]

13.

M. Berzins, “Mesh quality: a function of geometry, error estimates or both?,” Eng. Comput. 15,236–247 (1999). [CrossRef]

14.

M. Ainsworth and J.T. Oden, “A posteriori error estimation in finite element analysis,” Comput. Methods Appl. Mech. Eng. 142,1–88 (1997). [CrossRef]

15.

R. Radovitzky and M. Ortiz, “Error estimation and adaptive meshing in strongly non-linear dynamic problems,” Comput. Methods Appl. Mech. Eng. 172,203–240 (1999). [CrossRef]

16.

P Laug and H Borouchaki 2003 “BL2D-V2: mailleur bidimensionnel adaptatif,” Report INRIA RT-0275http://www-rocq1.inria.fr/gamma/cdrom/www/bl2d-v2/INDEX.html

17.

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25,377–445 (1908). [CrossRef]

18.

H. Du, “Mie-scattering calculation,” Appl. Opt. 43,1951–1956 (2004). [CrossRef] [PubMed]

19.

C. Ropers, D.J. Park, G. Stibenz, G. Steinmeyer, J. Kim, D.S. Kim, and C. Lienau, “Femtosecond Light Transmission and Subradiant Damping in Plasmonic Crystals,” Phys. Rev. Lett. 94,113901–4 (2005). [CrossRef] [PubMed]

20.

T.A. Davis and I.S. Duff, “A combined unifrontal multifrontal method for unsymmetric sparse matrices,” ACM T. Math Software 25,1–20 (1999). [CrossRef]

OCIS Codes
(000.4430) General : Numerical approximation and analysis
(260.2110) Physical optics : Electromagnetic optics
(290.4020) Scattering : Mie theory

ToC Category:
Physical Optics

History
Original Manuscript: November 9, 2006
Revised Manuscript: January 8, 2007
Manuscript Accepted: January 8, 2007
Published: February 5, 2007

Virtual Issues
Vol. 2, Iss. 3 Virtual Journal for Biomedical Optics

Citation
Thomas Grosges, Houman Borouchaki, and Dominique Barchiesi, "Improved scheme for accurate computation of high electric near-field gradients," Opt. Express 15, 1307-1321 (2007)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-3-1307


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. 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]
  2. D. Barchiesi, B. Guizal and T. Grosges, "Accuracy of local field enhancement models: toward predictive models?" Appl. Phys. B,  84, 55-60 (2006). [CrossRef]
  3. D. Barchiesi, C. Girard, O. J. F. Martin, D. Van Labeke and D. Courjon, "Computing the optical near-field distributions around complex subwavelength surface structures: A comparative study of different methods," Phys. Rev. E 54, 4285-4292 (1996). [CrossRef]
  4. B. Guizal, D. Barchiesi and D. Felbacq, "Electromagnetic beam diffraction by a finite lamellar structure," J. Opt. Soc. Am. A 20, 2274-2280 (2003). [CrossRef]
  5. 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]
  6. M. Born, and E. Wolf, Principle of Optics (Pergamon Press, Oxford, 1993).
  7. J. Jin, The Finite Element Method in Electromagnetics (John Wiley and Sons, New York, 1993).
  8. P. Ingelström and A. Bondeson, "Goal-oriented error estimation and h-adaptivity for Maxwell’s equations," Comput. Methods Appl. Mech. Eng. 192, 2597-2616 (2003). [CrossRef]
  9. P. Houston, I. Perugia, and D. Schotzau, "Energy norm a posteriori error estimation for mixed discontinuous Galerkin approximations of the Maxwell operator," Comput. Methods Appl. Mech. Eng. 194, 499-510 (2005). [CrossRef]
  10. D. Pardo, L. Demkowicz, C. Torre-Verdìn and L. Tabarovsky, "A goal-oriented hp-adaptive nite element method with electromagnetic applications. Part I: Electrostatics," Int. J. Numer. Methods Eng. 65, 1269-1309 (2005). [CrossRef]
  11. D. Xue and L. Demkowicz, "Modeling of electromagnetic absorption/scattering problems on curvilinear geometries using hp finite/infinite element method," Finite Elem. Anal. Design 42, 570-579 (2006). [CrossRef]
  12. H. Borouchaki, P. Lang, A. Cherouat and K. Saanouni, "Adaptive remeshing in large plastic strain with damage," Int. J. Numer. Methods Eng. 63, 1-36 (2005). [CrossRef]
  13. M. Berzins, "Mesh quality: a function of geometry, error estimates or both?" Eng. Comput. 15, 236-247 (1999). [CrossRef]
  14. M. Ainsworth and J. T. Oden, "A posteriori error estimation in finite element analysis," Comput. Methods Appl. Mech. Eng. 142, 1-88 (1997). [CrossRef]
  15. R. Radovitzky and M. Ortiz, "Error estimation and adaptive meshing in strongly non-linear dynamic problems," Comput. Methods Appl. Mech. Eng. 172, 203-240 (1999). [CrossRef]
  16. P. Laug and H. Borouchaki, "BL2D-V2: mailleur bidimensionnel adaptatif," Report INRIA RT-0275http://www-rocq1.inria.fr/gamma/cdrom/www/bl2d-v2/INDEX.html (2003).
  17. G. Mie, "Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen," Ann. Phys. 25, 377-445 (1908). [CrossRef]
  18. H. Du, "Mie-scattering calculation," Appl. Opt. 43, 1951-1956 (2004). [CrossRef] [PubMed]
  19. C. Ropers, D. J. Park, G. Stibenz, G. Steinmeyer, J. Kim, D. S. Kim, and C. Lienau, "Femtosecond light transmission and subradiant damping in Plasmonic Crystals," Phys. Rev. Lett. 94, 113901-4 (2005). [CrossRef] [PubMed]
  20. T. A. Davis and I. S. Duff, "A combined unifrontal multifrontal method for unsymmetric sparse matrices," ACM Trans.Math Softw. 25, 1-20 (1999). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited