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

Optics Express, Vol. 15, Issue 3, pp. 1307-1321 (2007)

http://dx.doi.org/10.1364/OE.15.001307

Acrobat PDF (883 KB)

### 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

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]

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]

*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.

## 2. Formulation of the problem in near-field

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

*H*[7]:

_{z}*c*is the velocity of light in vacuum and

*ε*is the dimensionless relative complex permittivity of the considered medium (

_{r}*ε*=

*ε*

_{0}

*ε*) which is a function of the spatial coordinates (

_{r}*x*,

*y*). The electric field

**E**(

*x*,

*y*) is then deduced from the Maxwell’s equation:

**P**(

*x*,

*y*) depends on both fields:

**H**) governs the remeshing process. Therefore a loss of accuracy is expected on

**E**, especially if it exhibits high gradients. Moreover, in near-field computations, the discontinuity of the electric components normal to the material surfaces makes such gradients to appear in distinct zones where

**H**is varying. In contrast to previous works where the

*a posteriori*error estimator is achieved on the PDE solution [8

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]

*a posteriori*estimation of the interpolation error on the solution of interest (

**E**or

**P**) derivated from the PDE solution (

*H*).

## 3. The improved adaptive mesh method

*a posteriori*estimation of the interpolation error in the electromagnetic computation. These two features allow an important reduction in the number of computed nodes.

*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 (

*δ*). 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

_{H}*δ*for the error estimator based on the solution of interest (

*δ*or

_{E}*δ*,‥).

_{P}*a posteriori*estimation of the interpolation error on the field of interest (

**E**,

*P*,‥).

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

*η*between the exact solution

_{ℱ}*S*and the physical solution

*S*associated with the mesh

_{ℱ}*ℱ*(Ω) (obtained by finite element computation or derivation), an indirect approach is used. This consists in defining

*S*˜

_{ℱ}an interpolation of

*S*on mesh

*ℱ*(which can be a piecewise linear or quadratic function, depending on the degree of the elements of

*ℱ*) and

*η*˜

_{ℱ}the deviation between

*S*and

*S*˜

_{ℱ}. Such an interpolation error

*e*˜

_{ℱ}can be used to bound the gap between the solution obtained numerically and the unknown exact solution:

*C*a constant independent from

*ℱ*. 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. 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]

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]

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

*S*. The remeshing process consists in building a new mesh of Ω by using the interpolation error achieved on the solution of interest

*S*.

_{ϕ}*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:

*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:

*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

*ℱ*(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]

*w*of

*ℱ*(Ω) can be written as:

*w*,

*S*(

*w*)) denotes this new deviation. The size map

*ℋ*

_{ϕ}associated with the vertices of

*ℱ*(Ω) is deduced.

*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

*h*-method (

*a posteriori*error estimation on the PDE solution

*H*). In the following, the classical remeshing process with

_{z}*h*-method is denoted as classical remeshing process.

### 4.1. Case of the gold nano-cylinder

**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 (

*ε*= −13.6969 +

_{r}*j*1.0356) 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-0275*http://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]

**E**∣

^{2}/∣

**E**∣

_{0}^{2}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]

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]

**H**∣

^{2}/∣

**H**∣

_{0}^{2}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).

*H*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.∥

_{z}*δ*∥

_{ϕ}_{∞}≤ 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.

*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

*δ*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.

_{ϕ}### 4.2. Case of the gold nano-square

*a*= 15 nm deposited on a glass substrate (

*ε*= 2.25) and illuminated by a p-polarized incident field under an angle

_{r}*θ*=60

*°*along the y-axis. Strong confinement of the energy (Poynting’s vector amplitude) can be observed at the edges of the square section. Like in the previous example, the use of an optimized mesh refinement processing is crucial to assure the convergence of the numerical result. The used computational domain Ω consists in a square-box of 600 nm × 600 nm and the first step consists in defining a coarse mesh (

*N*= 670) at the boundaries of the subdomains (nanoparticle, substrate).

_{nodes}## 5. Conclusion

*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

## 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. |

2. | D. Barchiesi, B. Guizal, and T. Grosges, “Accuracy of local field enhancement models: toward predictive models?,” Appl. Phys. B , |

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 |

4. | B. Guizal, D. Barchiesi, and D. Felbacq, “Electromagnetic beam diffraction by a finite lamellar structure,” J. Opt. Soc. Am. A |

5. | T. Grosges, A. Vial, and D. Barchiesi, “Models of near-field spectroscopic studies: comparison
between Finite-Element and Finite-Difference methods,” Opt. Express |

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

7. | J. Jin, |

8. | P. Ingelström and A. Bondeson, “Goal-Oriented error estimation and h-adaptivity for Maxwell’s equations,” Comput. Methods Appl. Mech. Eng. |

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. |

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. |

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 |

12. | H. Borouchaki, P. Lang, A. Cherouat, and K. Saanouni, “Adaptive remeshing in large plastic strain with damage,” Int. J. Numer. Methods Eng. |

13. | M. Berzins, “Mesh quality: a function of geometry, error estimates or both?,” Eng. Comput. |

14. | M. Ainsworth and J.T. Oden, “A posteriori error estimation in finite element analysis,” Comput. Methods Appl. Mech. Eng. |

15. | R. Radovitzky and M. Ortiz, “Error estimation and adaptive meshing in strongly non-linear dynamic problems,” Comput. Methods Appl. Mech. Eng. |

16. | P Laug and H Borouchaki 2003 “BL2D-V2: mailleur bidimensionnel adaptatif,” |

17. | G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. |

18. | H. Du, “Mie-scattering calculation,” Appl. Opt. |

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. |

20. | T.A. Davis and I.S. Duff, “A combined unifrontal multifrontal method for unsymmetric sparse matrices,” ACM T. Math Software |

**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/oe/abstract.cfm?URI=oe-15-3-1307

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### References

- 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]
- D. Barchiesi, B. Guizal and T. Grosges, "Accuracy of local field enhancement models: toward predictive models?" Appl. Phys. B, 84, 55-60 (2006). [CrossRef]
- 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]
- 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]
- 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]
- M. Born, and E. Wolf, Principle of Optics (Pergamon Press, Oxford, 1993).
- J. Jin, The Finite Element Method in Electromagnetics (John Wiley and Sons, New York, 1993).
- 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]
- 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]
- 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]
- 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]
- 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]
- M. Berzins, "Mesh quality: a function of geometry, error estimates or both?" Eng. Comput. 15, 236-247 (1999). [CrossRef]
- M. Ainsworth and J. T. Oden, "A posteriori error estimation in finite element analysis," Comput. Methods Appl. Mech. Eng. 142, 1-88 (1997). [CrossRef]
- 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]
- 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).
- G. Mie, "Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen," Ann. Phys. 25, 377-445 (1908). [CrossRef]
- H. Du, "Mie-scattering calculation," Appl. Opt. 43, 1951-1956 (2004). [CrossRef] [PubMed]
- 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]
- 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]

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