Isolated Slit and Groove

This tutorial example follows a benchmark setup investigated by P. Lalanne et al. [1]. Performance of FEM for the same setup has also been demonstrated [2]. The benchmark setup consists of computing the near field in an isolated (i.e., non-periodic) pattern illuminated by a plane wave. The geometry consists of an isolated, sub-wavelength slit in a silver film on a substrate with a neighboring, parallel groove in the silver film. This setup is illuminated by a plane wave at perpendicular incidence from above and with in-plane electric field polarization (resp. out-of-plane magnetic field polarization). The energy flux of light transmitted through the slit to a detector region placed a specific distance below the slit is detected and normalized to the energy flux through the slit, computed in a second simulation where the groove is not present. Due to the geometrical, source and material properties plasmonic effects lead to a very critical dependence of normalized transmission on the physical parameters. This makes accurate computation of normalized transmission a challenging benchmark problem.

Triangular meshes of the slit-groove setup (left) and the slit setup (right) used for normalization. Grey: Silver film, blue: substrate, red: detector region, green: air. Note the prerefinement of the mesh at metal corners:
mesh1 mesh2
Computed field intensities, with (right) and without (left) parts of the FEM mesh (top row: \log(I), bottom row: \Re(H_z), pseudo color scale):
f1 f2
f3 f4

Definition of the geometry (please note CornerRefinement in the definitions for slit and groove):

  • layout.jcm [ASCII]

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    Layout2D {
  UnitOfLength = 1e-09
  MeshOptions {
    MinimumMeshAngle = 28
    MaximumSideLength = 50
  }
  Objects {       
    Parallelogram {
      Name = "CD"
      DomainId = 1
      Priority = ComputationalDomain
      Width = 900
      Height = 950
      Boundary {
        Class = Transparent
      }
    }
    Parallelogram {
      Name = "Substrate"
      DomainId = 2
      Priority = 3
      Width = 900
      Height = 400
      Port = South
      Alignment {
        Parent {
          Domain = "CD"
          Port = South
        }
        Orientation = Parallel
        Displacement = [0 0]
      }
    }
    Parallelogram {
      Name = "Ag"
      DomainId = 3
      Priority = 2
      Width = 900
      Height = 400
      Port = South
      Alignment {
        Parent {
          Domain = "Substrate"
          Port = North
        }
        Orientation = AntiParallel
        Displacement = [0 0]
      }
    } 
    Parallelogram {
      Name = "Groove"
      DomainId = 1
      Priority = 4
      Width = 100
      Height = 100
      Port = North
      Alignment {
        Parent {
          Domain = "Ag"
          Port = North
        }
        Orientation = Parallel
        Displacement = [250 0]
      }
      MeshOptions {
        CornerRefinement {
          MaximumSideLength = 0.1
          Progression = 3
        }
      }
    } 
    Parallelogram {
      Name = "Slit"
      DomainId = 1
      Priority = 4
      Width = 100
      Height = 400
      Port = North
      Alignment {
        Parent {
          Domain = "Ag"
          Port = North
        }
        Orientation = Parallel
        Displacement = [-250 0]
      }
      MeshOptions {
        CornerRefinement {
          MaximumSideLength = 0.1
          Progression = 3
        }
      }
    }
    Parallelogram {
      Name = "Detector"
      DomainId = 4
      Priority = 4
      Width = 200
      Height = 20
      Port = South
      Alignment {
        Parent {
          Domain = "Slit"
          Port = South
        }
        Orientation = Parallel
        Displacement = [0 -400]
      }
    }
  }
}

Definition of material properties:

  • materials.jcm [ASCII]

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Material {
  Name = "Air"
  DomainId = 1
  RelPermittivity = 1
  RelPermeability = 1.0
}
Material {
  Name = "Glass"
  DomainId = 2
  RelPermittivity = 2.25
  RelPermeability = 1.0
}
Material {
  Name = "Ag"
  DomainId = 3
  RelPermittivity = (-33.22, 1.17)
  RelPermeability = 1.0
}
Material {
  Name = "Detector_Glass"
  DomainId = 4
  RelPermittivity = 2.25
  RelPermeability = 1.0
}

Definition of source properties:

  • sources.jcm [ASCII]

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    SourceBag {
  Source {
    MagneticFieldStrength {
      PlaneWave {
        Incidence = FromAbove
        3DTo2D = yes
        ThetaPhi = [0 0]
	Lambda0 = 8.52e-07
        SP = [1 0]
      }
    }
  }
}

Alternatively, the source could be defined as plane wave with ElectricFieldStrength and P polarization.

Project type, accuracy settings and post-process definitions:

  • project.jcmp [ASCII]

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    Project {
  InfoLevel = -1
  StorageFormat = Binary
  Electromagnetics {
    TimeHarmonic {
      Scattering {
        FieldComponents = Magnetic
        Accuracy {
          FiniteElementDegree = 3
          Precision = 0.001
          Refinement {
            MaxNumberSteps = 0
          }
        }
      }
    }
  }
}

PostProcess {
  FluxIntegration {
    FieldBagPath = "project_results/fieldbag.jcm"
    OutputFileName = "project_results/flux4.jcm"
    OutputQuantity = ElectromagneticFieldEnergyFlux
    InterfaceType = ExteriorDomain
    DomainIdPairs = [4 4]
  }
}

The post-process computes the energy flux from the domain with DomainId = 4 to the adjacent exterior domain with DomainId = 4.

The data_analysis folder also contains a script for performing simulations with and without the groove (for normalizing the energy flux, according to the benchmark problem). The script also allows to change numerical and physical project parameters, e.g., for checking the accuracy. Some exemplary field distributions are displayed below.

[1]P. Lalanne, et al., Numerical analysis of a slit-groove diffraction problem, JEOS - Rapid 2, 07022 (2007).
[2]S. Burger, et al., Finite-element based electromagnetic field simulations: Benchmark results for isolated structures, Proc. SPIE 8880, 88801Z (2013), http://arxiv.org/pdf/1310.2732