Simulation of near-field plasmonic interactions

with a local approximation order DGTD method

Modulus of E field map in the bowtie antenna at t=12.3 fs, obtained with a DGTD-P₁P₃ method. Field values have been scaled to [0 , 10]

(:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/p_loc/bowtie_dt1.png

(:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/p_loc/nanolens_dt1.png

Several strategies can be considered to address this performance issue. The strategy discussed here relies on the use of non-uniform distribution of the polynomial order in the framework of a global time step DGDT method. By imposing low orders in small cells and high orders in large cells, it is possible to significantly alleviate both the global number of degrees of freedom and the time step restriction with a minimal impact on the method accuracy. Strategies exploiting locally adaptive (LA) formulations usually combine both h- and p- adaptivity in order to concentrate the computational effort in the areas of high field variations. Here, the adopted point of view is quite different: starting from a given mesh and an uniform distribution of the polynomial order k, the LA strategy exploits all the polynomial orders p, with p lesser or equal than k to obtain a solution of similar accuracy with a reduced computational cost.

http://www-sop.inria.fr/nachos/pics/results/p_loc/nanolens_mesh.png | Mesh setup for a metallic nanolens. The gray cells correspond to the metallic spheres, the blue cells to vacuum, while the red cells constitute the PML region. For this mesh, the ratio between the largest and the smallest edge length of the tetrahedral mesh is above 400.

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As a second application, we consider the computation of the extinction cross section of a metallic bowtie nanoantenna. These structures are actively studied for the very strong field enhancement they provide between the tips of the two triangular nanoparticles, which is known to be inversely proportionnal to the size of the gap. Hence, bowtie nanoantennas are good candidates for surface-enhanced Raman spectroscopy (SERS) applications. Recent advances in lithography techniques allowed the creation of structures with gaps as small as 3 nm, while the typical size of the full structure can get close to 200 nm. Additionally, realistic geometries of such antennas include small roundings at the edges and tips, whose typical size is between a few to a few tens of nanometers. In the present case, we consider a pair of 10 nm thick, equilateral prisms of edge length 100 nm, with a spacing gap of 3 nm. The rounding radius is 2 nm, and is uniform for all edges and tips. The material considered is gold, described by a Drude model.

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http://www-sop.inria.fr/nachos/pics/results/p_loc/bowtie3.png | Mesh setup for a bowtie nanoantenna. The gray cells correspond to the nanoantenna, the blue cells to vacuum, while the red cells constitute the PML region. For this mesh, the ratio between the largest and the smallest edge length of the tetrahedral mesh is close to 275.

http://www-sop.inria.fr/nachos/pics/result/p_loc/bowtie3.png | Mesh setup for a bowtie nanoantenna. The gray cells correspond to the nanoantenna, the blue cells to vacuum, while the red cells constitute the PML region. For this mesh, the ratio between the largest and the smallest edge length of the tetrahedral mesh is close to 275.

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As a second application, we consider the computation of the extinction cross section of a metallic bowtie nanoantenna. These structures are actively studied for the very strong field enhancement they provide between the tips of the two triangular nanoparticles, which is known to be inversely proportionnal to the size of the gap. Hence, bowtie nanoantennas are good candidates for surface-enhanced Raman spectroscopy (SERS) applications. Recent advances in lithography techniques allowed the creation of structures with gaps as small as 3 nm, while the typical size of the full structure can get close to 200 nm. Additionally, realistic geometries of such antennas include small roundings at the edges and tips, whose typical size is between a few to a few tens of nanometers. In the present case, we consider a pair of 10 nm thick, equilateral prisms of edge length 100 nm, with a spacing gap of 3 nm. The rounding radius is 2 nm, and is uniform for all edges and tips. The material considered is gold, described by a Drude model.

(:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/p_loc/nanolens_range.png

(:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/p_loc/bowtie_visu_P1P3_top.png

(:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/p_loc/nanolens_dt1.png

(:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/p_loc/bowtie_dt1.png

(:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/p_loc/nanolens_range.png

(:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/p_loc/nanolens_range.png

E_y field map in the nanolens device at t=10 fs. Field values have been scaled to [-15 , 15]

As a second application, we consider the computation of the extinction cross section of a metallic bowtie nanoantenna. These structures are actively studied for the very strong field enhancement they provide between the tips of the two triangular nanoparticles, which is known to be inversely proportionnal to the size of the gap. Hence, bowtie nanoantennas are good candidates for surface-enhanced Raman spectroscopy (SERS) applications. Recent advances in lithography techniques allowed the creation of structures with gaps as small as 3 nm, while the typical size of the full structure can get close to 200 nm. Additionally, realistic geometries of such antennas include small roundings at the edges and tips, whose typical size is between a few to a few tens of nanometers. In the present case, we consider a pair of 10 nm thick, equilateral prisms of edge length 100 nm, with a spacing gap of 3 nm. The rounding radius is 2 nm, and is uniform for all edges and tips. The material considered is gold, described by a Drude model.

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Modulus of E field map in the nanolens device at t=12.3 fs, obtained with a DGTD-P₁P₃ method. Field values have been scaled to [0 , 10]

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(:table border='0' width='100%' align='center' cellspacing='1px':) (:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/p_loc/bowtie_dt0.png (:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/p_loc/bowtie_dt1.png (:table border='0' width='100%' align='center' cellspacing='1px':) (:tableend:)

Left figure : polynomial order repartition for the bowtie antenna mesh with respect to time-step for the P₁P₃ case. Right figure : extinction cross section of the bowtie nanoantenna obtained with DGTD-P₁, DGTD-P₃ and DGTD-P₁P₃ methods. Less than 2 % of relative error is observed between DGTD-P₃ and DGTD-P₁P₃ computations, for a speedup factor of 2.0

(:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/p_loc/bowtie_visu_P1P3.png (:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/p_loc/bowtie_visu_P1P3_top.png

(:table border='0' width='100%' align='center' cellspacing='1px':) (:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/p_loc/bowtie_range.png (:tableend:)

Modulus of E field map in the nanolens device at t=12.3 fs, obtained with a DGTD-P₁P₃ method. Field values have been scaled to [0 , 10]

(:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/p_loc/bowtie_visu_P1P3.png (:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/p_loc/bowtie_visu_P1P3_top.png (:tableend:)

E field map in the nanolens device at t=12.3 fs, obtained with a DGTD-P₁P₃ method

(:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/p_loc/bowtie_visu_P1P3.png (:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/p_loc/bowtie_visu_P1P3_top.png

(:table border='0' width='100%' align='center' cellspacing='1px':)

(:table border='0' width='100%' align='center' cellspacing='1px':) (:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/p_loc/bowtie_visu_P1P3.png (:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/p_loc/bowtie_visu_P1P3_top.png (:tableend:)

http://www-sop.inria.fr/nachos/pics/results/p_loc/bowtie3.png |

http://www-sop.inria.fr/nachos/pics/results/p_loc/bowtie3.png |

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http://www-sop.inria.fr/nachos/pics/results/p_loc/bowtie_mesh.png | Mesh setup for a bowtie nanoantenna. The gray cells correspond to the nanoantenna, the blue cells to vacuum, while the red cells constitute the PML region. For this mesh, the ratio between the largest and the smallest edge length of the tetrahedral mesh is close to 275.

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Left figure : polynomial order repartition for the nanolens mesh with respect to time-step for the P₁P₃ case. Right figure : field enhancement in the vicinity of the smallest sphere of a self-similar nanolens obtained with DGTD-P₁, DGTD-P₃ and DGTD-P₁P₃ methods. Less than 1 % of relative error is observed between DGTD-P₃ and DGTD-P₁P₃ computations, for a speedup factor of 2.6

Left figure : polynomial order repartition for the nanolens mesh with respect to time-step for the P₁P₃ case

Left figure : polynomial order repartition for the nanolens mesh with respect to time-step for the P₁P₃ case. The red elements correspond to P₁ approximation, the green ones to P₂, and the gray ones to P₃

Right figure : field enhancement in the vicinity of the smallest sphere of a self-similar nanolens obtained with DGTD-P₁, DGTD-P₃ and DGTD-P₁P₃ methods. Less than 1 % of relative error is observed between DGTD-P₃ and DGTD-P₁P₃ computations, for a speedup factor of 2.6

Left figure : polynomial order repartition for the nanolens mesh with respect to time-step for the P₁P₃ case. The red elements correspond to P₁ approximation, the green ones to P₂, and the gray ones to P₃.

Left figure : polynomial order repartition for the nanolens mesh with respect to time-step for the P₁P₃ case. The red elements correspond to P₁ approximation, the green ones to P₂, and the gray ones to P₃.

Right figure : field enhancement in the vicinity of the smallest sphere of a self-similar nanolens obtained with DGTD-P₁, DGTD-P₃ and DGTD-P₁P₃ methods. Less than 1 % of relative error is observed between DGTD-P₃ and DGTD-P₁P₃ computations, for a speedup factor of 2.6

Right figure: field enhancement in the vicinity of the smallest sphere of a self-similar nanolens obtained with DGTD-P₁, DGTD-P₃ and DGTD-P₁P₃ methods. Less than 1 % of relative error is observed between DGTD-P₃ and DGTD-P₁P₃ computations, for a speedup factor of 2.6

(:table border='0' width='100%' align='center' cellspacing='1px':)

Right figure: field enhancement in the vicinity of the smallest sphere of a self-similar nanolens} obtained with DGTD-P₁, DGTD-P₃ and DGTD-P₁P₃ methods. Less than 1 % of relative error is observed between DGTD-P₃ and DGTD-P₁P₃ computations, for a speedup factor of 2.6

(:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/p_loc/nanolens_dt0.png (:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/p_loc/nanolens_dt1.png

(:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/p_loc/nanolens_dt0.png (:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/p_loc/nanolens_dt1.png

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(:table border='0' width='100%' align='center' cellspacing='1px':) (:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/p_loc/nanolens_dt0.png (:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/p_loc/nanolens_dt1.png (:tableend:)

(:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/p_loc/nanolens_visu_P1.png (:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/p_loc/nanolens_visu_P3.png

(:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/p_loc/nanolens_visu_P1P3.png

E_y field map in the nanolens device at t=10 fs

(:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/p_loc/nanolens_visu_P1.png (:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/p_loc/nanolens_visu_P3.png (:cellnr align='center':) DGTD-P₁ method (:cell align='center':) DGTD-P₃ method (:tableend:) (:table border='0' width='100%' align='center' cellspacing='1px':)

(:cellnr align='center':) DGTD-P₁P₃ method

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(:table border='0' width='100%' align='center' cellspacing='1px':) (:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/p_loc/nanolens_visu_P1.png (:cellnr align='center':) DGTD method with affine elements (:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/p_loc/nanolens_visu_P1P3.png (:cellnr align='center':) DGTD method with curvilinear elements (:tableend:)

http://www-sop.inria.fr/nachos/pics/results/p_loc/nanolens_mesh.png | Mesh setup for a metallic nanolens. The gray cells correspond to the metallic spheres, the blue cells to vacuum, while the red cells constitute the PML region. For this mesh, the ratio between the largest and the smallest edge length of the tetrahedral mesh is above 400.

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%lfloat text-align=center width=300px http://www-sop.inria.fr/nachos/pics/results/p_loc/nanolens_mesh.png | Mesh setup for a metallic nanolens. The gray cells correspond to the metallic spheres, the blue cells to vacuum, while the red cells constitute the PML region. For this mesh, the ratio between the largest and the smallest edge length of the tetrahedral mesh is above 400.%

Mesh setup for a metallic nanolens. The gray cells correspond to the metallic spheres, the blue cells to vacuum, while the red cells constitute the PML region. For this mesh, the ratio between the largest and the smallest edge length of the tetrahedral mesh is above 400.\\

To overcome the limitation of the diffraction limit, it is possible to exploit the focusing effect provided by coupled surface plasmons. A typical nanolens is composed of a chain of metallic nanoparticles (nanospheres being the most common) of decreasing size, aligned with the polarization direction of the incident field. When the nanospheres are of significantly different sizes, the local field enhancement of the first particle is not perturbed by the second one because of its small relative size. As a result, the locally enhanced field of the first particle acts as an incident field for the second particle, resulting in a second enhancement, and so on. Eventually, the strongest enhancement is obtained in the gap between the two smaller particles.

http://www-sop.inria.fr/nachos/pics/results/p_loc/nanolens_mesh.png | Mesh setup for a metallic nanolens. The gray cells correspond to the metallic spheres, the blue cells to vacuum, while the red cells constitute the PML region. For this mesh, the ratio between the largest and the smallest edge length of the tetrahedral mesh is above 400.

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http://www-sop.inria.fr/nachos/pics/results/p_loc/nanolens_mesh.png | Mesh setup for a metallic nanolens. The gray cells correspond to the metallic spheres, the blue cells to vacuum, while the red cells constitute the PML region. For this mesh, the ratio between the largest and the smallest edge length of the tetrahedral mesh is above 400.

http://www-sop.inria.fr/nachos/pics/results/p_loc/nanolens_mesh.png | Mesh setup for a metallic nanolens. The gray cells correspond to the metallic spheres, the blue cells to vacuum, while the red cells constitute the PML region. For this mesh, the ratio between the largest and the smallest edge length of the tetrahedral mesh is above 400.

http://www-sop.inria.fr/nachos/pics/results/p_loc/nanolens_mesh.png | Mesh setup for a metallic nanolens. The gray cells correspond to the metallic spheres, the blue cells to vacuum, while the red cells constitute the PML region. For this mesh, the ratio between the largest and the smallest edge length of the tetrahedral mesh is above 400.

width=320px% http://www-sop.inria.fr/nachos/pics/results/p_loc/nanolens_mesh.png | Mesh setup for a metallic nanolens. The gray cells correspond to the metallic spheres, the blue cells to vacuum, while the red cells constitute the PML region. For this mesh, the ratio between the largest and the smallest edge length of the tetrahedral mesh is above 400.

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(:table border='0' width='90%' align='center' cellspacing='1px':) (:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/p_loc/nanolens_mesh.png (:tableend:)

Mesh setup for a metallic nanolens. The gray cells correspond to the metallic spheres, the blue cells to vacuum, while the red cells constitute the PML region. For this mesh, the ratio between the largest and the smallest edge length of the tetrahedral mesh is above 400.

To overcome this limitation, several strategies can be considered. The strategy discussed here relies on the use of non-uniform distribution of the polynomial order in the framework of a global time step DGDT method. By imposing low orders in small cells and high orders in large cells, it is possible to significantly alleviate both the global number of degrees of freedom and the time step restriction with a minimal impact on the method accuracy. Strategies exploiting locally adaptive (LA) formulations usually combine both h- and p- adaptivity in order to concentrate the computational effort in the areas of high field variations. Here, the adopted point of view is quite different: starting from a given mesh and an uniform distribution of the polynomial order k, the LA strategy exploits all the polynomial orders p, with p lesser or equal than k to obtain a solution of similar accuracy with a reduced computational cost.

To overcome this limitation, several strategies can be considered. The strategy discussed here relies on the use of non-uniform distribution of the polynomial order in the framework of a global time step DGDT method. By imposing low orders in small cells and high orders in large cells, it is possible to significantly alleviate both the global number of degrees of freedom and the time step restriction with a minimal impact on the method accuracy. Strategies exploiting locally adaptive (LA) formulations usually combine both h- and p- adaptivity in order to concentrate the computational effort in the areas of high field variations. Here, the adopted point of view is quite different: starting from a given mesh and an uniform distribution of the polynomial order k, the LA strategy exploits all the polynomial orders p, with p lesser or equal than k to obtain a solution of similar accuracy with a reduced computational cost.

combine both h and p- adaptivity in order to concentrate the computational effort in the areas of high field variations. Here, the adopted point of view is quite different: starting from a given mesh and an uniform distribution of the polynomial order k, the LA strategy exploits all the polynomial orders p, with p lesser or equal than k to obtain a solution of similar accuracy with a reduced computational cost.

In most of the existing works on the development of high order DGTD methods for the numerical modeling of light/matter interactions on the nanoscale, the formulation of the method is derived assuming a uniform distribution of the polynomial order to the cells of the underlying mesh. However, in the case of a mesh showing large variations in cell size, the time step imposed by the smallest cells can be a serious hindrance when trying to exploit high approximation orders. Indeed, a potentially large part of the CPU time is spent in the update of the physical field inside small cells where high polynomial orders might not be essential, while they are necessary in the larger cells.

To overcome this limitation, several strategies can be considered. The strategy discussed here relies on the use of non-uniform distribution of the polynomial order in the framework of a global time step DGDT method. By imposing low orders in small cells and high orders in large cells, it is possible to significantly alleviate both the global number of degrees of freedom and the time step restriction with a minimal impact on the method accuracy. Strategies exploiting locally adaptive (LA) formulations usually combine both h- and p-adaptivity in order to concentrate the computational effort in the areas of high field variations. Here, the adopted point of view is quite different: starting from a given mesh and an uniform distribution of the polynomial order k, the LA strategy exploits all the polynomial orders p, with p lesser or equal than k to obtain a solution of similar accuracy with a reduced computational cost.

(:linebreaks:)

In most of the existing works on the development of high order DGTD methods for the numerical modeling of light/matter interactions on the nanoscale, the formulation of the method is derived assuming a uniform distribution of the polynomial order to the cells of the underlying mesh. However, in the case of a mesh showing large variations in cell size, the time step imposed by the smallest cells can be a serious hindrance when trying to exploit high approximation orders. Indeed, a potentially large part of the CPU time is spent in the update of the physical field inside small cells where high polynomial orders might not be essential, while they are necessary in the larger cells.

(:title Simulation of near-field plasmonic interactions\\\

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