Abstract

The concepts of adaptive coordinates and adaptive spatial resolution significantly enhance the performance of Fourier Modal Method for the simulation of periodic photonic structures, especially metallo-dielectric systems. We present several approaches for constructing different types of analytical coordinate transformations that are applicable to a great variety of structures. In addition, we analyze these meshes with an emphasis on the resulting convergence characteristics. This allows us to formulate general guidelines for the choice of mesh type and mesh parameters.

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References

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  1. K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep.444, 101–202 (2007).
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  7. G. Granet, “Reformulation of the lamellar grating problem through the concept of adaptive spatial resolution,” J. Opt. Soc. Am. A16, 2510–2516 (1999).
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    [CrossRef]
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    [PubMed]
  10. S. Essig and K. Busch, “Generation of adaptive coordinates and their use in the Fourier Modal Method,” Opt. Express18, 23258–23274 (2010).
    [CrossRef] [PubMed]
  11. L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A5, 345–355 (2003).
    [CrossRef]
  12. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A14, 2758–2767 (1997).
    [CrossRef]
  13. T. Schuster, J. Ruoff, N. Kerwien, S. Rafler, and W. Osten, “Normal vector method for convergence improvement using the RCWA for crossed gratings,” J. Opt. Soc. Am. A24, 2880–2890 (2007).
    [CrossRef]
  14. A. W. Snyder and J. D. LoveOptical Waveguide Theory, (Chapman and Hall, 1983).
  15. A. Vial, A.-S. Grimault, D. Macías, D. Barchiesi, and M. Lamy de la Chapelle, “Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B71, 085416 (2005).
    [CrossRef]

2010

2009

2007

T. Schuster, J. Ruoff, N. Kerwien, S. Rafler, and W. Osten, “Normal vector method for convergence improvement using the RCWA for crossed gratings,” J. Opt. Soc. Am. A24, 2880–2890 (2007).
[CrossRef]

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep.444, 101–202 (2007).
[CrossRef]

2005

A. Vial, A.-S. Grimault, D. Macías, D. Barchiesi, and M. Lamy de la Chapelle, “Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B71, 085416 (2005).
[CrossRef]

2003

L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A5, 345–355 (2003).
[CrossRef]

2002

T. Vallius and M. Honkanen, “Reformulation of the Fourier modal method with adaptive spatial resolution: application to multilevel profiles,” Opt. Express10, 24–34 (2002).
[PubMed]

G. Granet and J.-P. Plumey, “Parametric formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. A4, S145–S149 (2002).
[CrossRef]

1999

1997

1996

Barchiesi, D.

A. Vial, A.-S. Grimault, D. Macías, D. Barchiesi, and M. Lamy de la Chapelle, “Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B71, 085416 (2005).
[CrossRef]

Busch, K.

S. Essig and K. Busch, “Generation of adaptive coordinates and their use in the Fourier Modal Method,” Opt. Express18, 23258–23274 (2010).
[CrossRef] [PubMed]

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep.444, 101–202 (2007).
[CrossRef]

Essig, S.

Giessen, H.

Gippius, N. A.

Granet, G.

Grimault, A.-S.

A. Vial, A.-S. Grimault, D. Macías, D. Barchiesi, and M. Lamy de la Chapelle, “Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B71, 085416 (2005).
[CrossRef]

Guizal, B.

Honkanen, M.

Kerwien, N.

Lalanne, P.

Lamy de la Chapelle, M.

A. Vial, A.-S. Grimault, D. Macías, D. Barchiesi, and M. Lamy de la Chapelle, “Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B71, 085416 (2005).
[CrossRef]

Li, L.

Linden, S.

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep.444, 101–202 (2007).
[CrossRef]

Love, J. D.

A. W. Snyder and J. D. LoveOptical Waveguide Theory, (Chapman and Hall, 1983).

Macías, D.

A. Vial, A.-S. Grimault, D. Macías, D. Barchiesi, and M. Lamy de la Chapelle, “Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B71, 085416 (2005).
[CrossRef]

Mingaleev, S. F.

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep.444, 101–202 (2007).
[CrossRef]

Morris, G. M.

Osten, W.

Plumey, J.-P.

G. Granet and J.-P. Plumey, “Parametric formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. A4, S145–S149 (2002).
[CrossRef]

Rafler, S.

Ruoff, J.

Schuster, T.

Snyder, A. W.

A. W. Snyder and J. D. LoveOptical Waveguide Theory, (Chapman and Hall, 1983).

Tikhodeev, S. G.

Tkeshelashvili, L.

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep.444, 101–202 (2007).
[CrossRef]

Vallius, T.

Vial, A.

A. Vial, A.-S. Grimault, D. Macías, D. Barchiesi, and M. Lamy de la Chapelle, “Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B71, 085416 (2005).
[CrossRef]

von Freymann, G.

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep.444, 101–202 (2007).
[CrossRef]

Wegener, M.

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep.444, 101–202 (2007).
[CrossRef]

Weiss, T.

J. Opt. A

G. Granet and J.-P. Plumey, “Parametric formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. A4, S145–S149 (2002).
[CrossRef]

L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A5, 345–355 (2003).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Express

Phys. Rep.

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep.444, 101–202 (2007).
[CrossRef]

Phys. Rev. B

A. Vial, A.-S. Grimault, D. Macías, D. Barchiesi, and M. Lamy de la Chapelle, “Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B71, 085416 (2005).
[CrossRef]

Other

A. W. Snyder and J. D. LoveOptical Waveguide Theory, (Chapman and Hall, 1983).

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Figures (25)

Fig. 1
Fig. 1

Schematic view of a typical system of interest in FMM computations. The system is finite in 3-direction and periodic with respect to the 12 plane. kin denotes the wave vector of an incident plane wave. Ox̄123 describes a Cartesian coordinate system.

Fig. 2
Fig. 2

Illustration of the effect of adaptive coordinates. Panel (a) depicts the permittivity distribution for a fiber with radius 0.3 centered in a square unit cell within a Cartesian coordinate system. Panel (b) displays a section of the nondifferentiable fiber mesh from Ref. [6]. Some points where the mesh is nondifferentiable have been marked with green, dashed circles. The point where the coordinate lines of the mesh are parallel in 1- and 2-direction is marked with a red circle. Panel (c) displays the g ε 11-component of the effective permittivity tensor that is obtained when the adaptive coordinates from (b) are applied to (a). In the transformed space, the effective permittivity tensor is fully grid-aligned but the numerical values at the rectangle edges are several orders of magnitude larger than 4. The color scale has been saturated at 4 in order to show more features.

Fig. 3
Fig. 3

Layout of a complex unit cell for the illustration of mesh construction. A crescent with rounded tips is described by four circles—a large, outer circle (blue, radius r1), a small, inner circle (red, radius r2), and two tiny circles at the crescent tips (radii r3 and r4). To define the orientation of the crescent within the unit cell, the centers (M1, N1) and (M2, N2) of the inner and outer circles need to be specified. The centers (M3, N3) and (M4, N4) of the tiny circles are uniquely determined, once the centers of the inner and outer circles and the radii of all four circles are specified.

Fig. 4
Fig. 4

Illustration of a linear transition from Cartesian coordinate lines to curved coordinate lines via the linear transition function LT. The straight coordinate lines a and b are, respectively, mapped onto the green and blue bent coordinate lines. The mapped lines in between (marked with crosses) are given by the linear transition between the bent outer lines.

Fig. 5
Fig. 5

Illustration of how to mesh complex structures. Specific coordinate lines that pass through the structure’s characteristic points are selected (panel (a)) and subsequently mapped onto the structure’s surface.

Fig. 6
Fig. 6

Nondifferentiable meshes for a crescent with parameters r1 = 0.25, r2 = 0.2, r3 = r4 = 0.02, M1 = 0.45, N1 = 0.38, M2 = M1, N2 = 0.5884, L = 0.8. Panel (a) displays the basic analytical mesh. Panel (b) depicts a mesh where an additional coordinate line compression using the parameters G = 0.05, Δx1 = 0.4, x ¯ 1 1 = x P 1, x ¯ 2 1 = x Q 1, Δx2 = 0.4, x ¯ 1 2 = x R 2, x ¯ 2 2 = x P 2 has been applied (see section 8 for details). Panel (c) features a close-up of (a) of the left crescent tip.

Fig. 7
Fig. 7

Characteristic points, selected coordinate lines, and nondifferentiable mesh for a step-index fiber. The outer radius is 0.3 and the inner radius is 0.1. The circles are centered in (0.45,0.4) and the unit cell is [0, 1] × [0, 0.8].

Fig. 8
Fig. 8

Illustration of the construction principle for smoothed meshes. Panel (a) depicts how the nondifferentiable transition from (Cartesian) characteristic line to a circle arc (as described in section 3) is smoothed by a (differentiable) parabola. The parabola and the circle arc meet at the point ū = x + τ with smoothing parameter τ. Straight line and parabola meet at a which also depends on τ. Panels (b) and (c) show how the characteristic coordinate lines passing through P, Q, R and S are mapped.

Fig. 9
Fig. 9

Smoothed meshes for a circular structure (radius r = 0.3) centered in a square unit cell. Panels (a) and (b) depict meshes with smoothing parameters τ = 0.07 and τ = 0.035, respectively. Panel (c) shows a close-up of the mesh in (b). See text for details.

Fig. 10
Fig. 10

Illustration of the construction principle for differentiable meshes. Panel (a) shows the partitioning of the unit cell and panel (b) depicts the resulting differentiable mapping.

Fig. 11
Fig. 11

Differentiable meshes for a circular structure (radius r = 0.3) centered in a square unit cell. Panels (a) and (b) depict meshes with smoothing parameters τ = 0.07 and τ = 0.035, respectively. Panel (c) shows a close-up of the mesh in (b). See text for details.

Fig. 12
Fig. 12

g ε 11-component of the effective permittivity for different meshes for a circular structure (radius r = 0.3) that is centered in a square unit cell. Panels (a), (b), and (c) show the transformed permittivity for the nondifferentiable, the smoothed, and the differentiable meshes of Figs. 2(b), 9(b), and 11(b), respectively. For a sampling with 1024×1024 points, the values at the corners of the rectangle for the nondifferentiable mesh exceed 5000. However, we have saturated the color scale at 4 in order to be able to display more features. The smoothed and differentiable meshes have been constructed with a smoothing parameter τ = 0.035. All plots show the sector [0.2, 0.8] × [0.2, 0.8] of the [0, 1] × [0, 1] unit cell.

Fig. 13
Fig. 13

Construction points and mesh for an elliptical structure that is not aligned with the axes of the square array associated with the unit cell. This example further illustrates how our construction principle may lead to aperiodic meshes. See text for further details.

Fig. 14
Fig. 14

Illustration of different approaches for enforcing periodicity of meshes. Panels (a) and (b) depict the method of adding linear transitions to the outer edge of the unit cell. Panel (c) illustrates the method of the mirror structure. Color has been added to mark the actual structure. See text for further details.

Fig. 15
Fig. 15

Analysis of a square array of silicon waveguides with trapezoidal cross section using the mirror-structure approach for periodic mesh construction. Panel (a) depicts a schematic of the system. Panels (b) and (c) show, respectively, the periodic mesh for the trapezoidal structure and the absolute value of the transverse electric field distribution of the fundamental mode on a logarithmic scale. There are no discernable detrimental effects due to the mirror structure that is used to ensure periodicity of the mesh. The yellow color in panel (b) and the white trapezoid in panel (c) have been added to guide the eye.

Fig. 16
Fig. 16

Illustration of the meshing in nonrectangular unit cells. Panel (a) depicts the structure in ordinary (Cartesian) space—a circular rod in a hexagonal lattice. Panel (b) shows the transformed structure when the nonrectangular unit cell is mapped onto the rectangular unit cell—the circle turns into an ellipse. Panel (c) shows the mesh in the original unit cell that is obtained by meshing the ellipse and mapping this mesh back into ordinary (Cartesian) space. The parameters are r = 0.2, α = 30° and a [0, 1] × [0, 1] unit cell.

Fig. 17
Fig. 17

Illustration of how to shift the compression function from Ref. [9]. In all panels x is plotted on the horizontal axis and on the vertical axis. Panel (a) depicts the original compression function. Panel (b) shows how the function is shifted to larger values. The point of intersection where the transformed coordinate leaves the unit cell is denoted by x̃. In panel (c) the compression function is shifted by exactly this amount to the right. The piece that is outside the unit cell in panel (c) is attached to the left side due to periodicity. Panel (d) shows how the equidistant coordinate lines (vertical) are mapped onto compressed coordinate lines (horizontal).

Fig. 18
Fig. 18

Illustration how two one-dimensional compression functions are combined to a two-dimensional compressed mesh. The compression in (a) is applied in horizontal direction and the compression depicted in (b) (with a constant density between 1 and 2) is applied in the vertical direction. Panel (c) depicts the resulting mesh.

Fig. 19
Fig. 19

Convergence characteristics of different meshes for computing the effective index of refraction of guided modes in a low-index fiber. Panel (a) displays the convergence characteristics of Cartesian, nondifferentiable, smoothed, and differentiable meshes without coordinate line compression. Panels (b) to (d) depict the corresponding dependence of the relative error (color-coded) on the compression parameters G and Δx for a fixed number of 997 plane waves. The same compression function is applied in 1- and 2-direction. All computations have been performed on a grid with 1024 × 1024 sampling points in transformed space. Panel (b) displays the results for the nondifferentiable mesh. Panels (c) and (d) depict results for the differentiable mesh with τ = 0.002 and τ = 0.015, respectively. The sketches in panel (d) depict compression functions for four specific pairs of G and Δx. The color bar of panel (d) also applies to panels (b) and (c). See the text for further details.

Fig. 20
Fig. 20

Dependence of the relative error (color-coded) for a low-index fiber on the compression parameters G and Δx for 293 planes waves and 1024 × 1024 discretization points in transformed space (cf. Fig. 19 for the results of the same system with 997 plane waves). Panel (a) depicts the results for the nondifferentiable mesh and panels (b) and (c) depict the results for the differentiable mesh with τ = 0.002 and τ = 0.015, respectively. The color bar of panel (c) also applies to panels (b) and (c). See the text for further details.

Fig. 21
Fig. 21

Dependence of the relative error (color-coded) for a high-index fiber on the compression parameters G and Δx for 997 planes waves and 1024 × 1024 discretization points (cf. Fig. 19 for the results of a low-index fiber with the same number of plane waves). Panel (a) depicts the results for the nondifferentiable mesh and panels (b) and (c) depict the results for the differentiable mesh with τ = 0.002 and τ = 0.015, respectively. The color bar of panel (c) also applies to panels (b) and (c). See the text for further details.

Fig. 22
Fig. 22

Dependence of the relative error (color-coded) for a low-index fiber on the compression parameters G and Δx for 997 planes waves and 1000 × 1000 discretization points (cf. Fig. 19 for the results of the same system with 1024×1024 discretization points). Panel (a) depicts the results for the nondifferentiable mesh and panels (b) and (c) depict the results for the differentiable mesh with τ = 0.002 and τ = 0.015, respectively. The color bar of panel (c) also applies to panels (b) and (c). The area within the white box in panel (a) is selected for further detailed investigation in Fig. 23. See the text for further details.

Fig. 23
Fig. 23

Detailed investigation of the error associated with nondifferentiable and differentiable meshes. Panel (a) represents at blow-up of the results highlighted in the white box of Fig. 22(a). The maximum relative error of the first ten guided eigenmodes of a low-index fiber have been obtained with 997 plane waves and 1000 × 1000 real-space points in steps of 0.001 for G and 0.00005 for Δx. Panel (b) depicts a line-cut through (a) at G = 0.165 and the relative error of each of the first ten guided eigenmodes together with the distance of the structure’s physical surface to the numerical surface. Panel (c) illustrates the definition of the numerical surface as the middle between the two coordinate lines closest to the physical surface. Panel (d) displays the computed distance between numerical and physical surface for the same parameter range that is used in (a). Panel (e) displays the results of a similar computation as shown in (b), this time using the differentiable mesh with τ = 0.002 and G = 0.165. See text for further details.

Fig. 24
Fig. 24

Convergence characteristics of different meshes for a square array of metallic cylinders of finite height. Panel (a) shows the transmittance, reflectance, and absorbance spectra that have been computed using 1750 plane waves and the nondifferentiable mesh. Our data agree well with those of Ref. [6] for this system. Panel (b) depicts the convergence characteristics (in terms of plane waves) of the reflectance for different meshes at the resonance frequency of 829 nm. The data in panels (a) and (b) have been obtained by using the same compression function with parameters G = 0.02 and Δx = 0.5 and a real space discretization of 1024 × 1024 points.

Fig. 25
Fig. 25

Dependence of the on-resonance reflectance from a square array of finite-height metallic cylinders on the compression parameters G and Δx for different meshes. Panel (a) shows the results when using the nondifferentiable mesh and panel (b) shows the results when using the differentiable mesh with τ = 0.005. All computations have been performed with 997 plane waves with a real space grid of 1024×1024 sampling points. The parameter G has been stepped in 0.005 intervals and Δx in 0.0025 intervals. The color scale was saturated at 0.81. See text for further details.

Equations (56)

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x ¯ 1 = x ¯ 1 ( x 1 , x 2 ) ,
x ¯ 2 = x ¯ 2 ( x 1 , x 2 ) ,
x ¯ 3 = x 3 .
ξ ρ α τ σ E τ = i k 0 g μ ρ σ H σ ,
ξ ρ σ τ σ H τ = i k 0 g ε ρ σ E σ .
g ρ σ = x ρ x ¯ τ x σ x ¯ τ ,
ε ρ σ = x ρ x ¯ τ x σ x ¯ χ ε ¯ τ χ ,
x 1 x ¯ 1 = ( x ¯ 1 x 1 x ¯ 2 x 2 x ¯ 1 x 2 x ¯ 2 x 1 ) 1 = | J | 1 x ¯ 2 x 2 .
( x 1 M i ) 2 + ( x 2 N i ) 2 = r i 2 , i = 1 , , 4 ,
N 3 = ( r 2 + r 3 ) 2 ( r 1 r 3 ) 2 + N 1 2 N 2 2 2 N 1 2 N 2 , M 3 = M 1 ( r 1 r 3 ) 2 ( N 3 N 1 ) 2 .
x P 1 = M 3 r 3 cos ( arctan ( N 3 N 1 M 1 M 3 ) ) , x P 2 = N 3 + r 3 2 ( x P 1 M 3 ) 2 ,
x D 1 = M 3 + r 3 cos ( arctan ( N 2 N 3 M 2 M 3 ) ) , x D 2 = N 3 + r 3 2 ( x D 1 M 3 ) 2 .
L T ( c , c ¯ , d , d ¯ , x ) = d ¯ c ¯ d c x + c ¯ c d ¯ c ¯ d c
R = ( x P 1 , N 1 r 1 2 ( x R 1 M 1 ) 2 ) , Q = ( 2 M 1 x P 1 , x P 2 ) , S = ( x Q 1 , x R 2 ) .
x ¯ 2 ( x 1 , x 2 ) = x 2 , ( x 1 , x 2 ) .
x ¯ 1 ( x 1 , x 2 ) = L T ( 0 , 0 map left edge onto itself , x P 1 , M 1 r 1 2 ( x 2 N 1 ) 2 map P R coordinate line on circle arc , x 1 ) ( 13 ) = M 1 r 1 2 ( x 2 N 1 ) 2 x 1 x P 1 , ( x 1 , x 2 ) .
C A L / R ( i , x 2 ) = M i r i 2 ( x 2 N i ) 2 , C A T / B ( i , x 1 ) = N i ± r i 2 ( x 1 M i ) 2 , i = 1 , , 4.
x ¯ 1 ( x 1 , x 2 ) = { x 1 ( x 1 , x 2 ) , , , , , L T ( 0 , 0 , x P 1 , C A L ( 1 , x 2 ) , x 1 ) ( x 1 , x 2 ) L T ( x P 1 , C A L ( 1 , x 2 ) , x Q 1 , C A R ( 1 , x 2 ) , x 1 ) ( x 1 , x 2 ) L T ( x Q 1 , C A R ( 1 , x 2 ) , 1 , 1 , x 1 ) ( x 1 , x 2 ) .
x ¯ 2 ( x 1 , x 2 ) = { x 2 ( x 1 , x 2 ) , , , , , L T ( L , L , x P 2 , C A T ( 3 , x 1 ) , x 2 ) x 1 x D 1 L T ( L , L , x P 2 , C A B ( 2 , x 1 ) , x 2 ) x 1 ( x D 1 , x E 1 ) L T ( L , L , x P 2 , C A T ( 4 , x 1 ) , x 2 ) x E 1 x 1 } ( x 1 , x 2 ) L T ( x P 2 , C A T ( 3 , x 1 ) , x R 2 , C A B ( 1 , x 1 ) , x 2 ) x 1 x D 1 L T ( x P 2 , C A B ( 2 , x 1 ) , x R 2 , C A B ( 1 , x 2 ) , x 2 ) x 1 ( x D 1 , x E 1 ) L T ( x P 2 , C A T ( 4 , x 1 ) , x R 2 , C A B ( 1 , x 1 ) , x 2 ) x E 1 x 1 } ( x 1 , x 2 ) L T ( x R 2 , C A B ( 1 , x 1 ) , 0 , 0 , x 2 ) ( x 1 , x 2 ) .
g ( x 1 , a , b , x ) = b ( x 1 a ) 2 + x
a = 2 u ¯ 0.5 ( r 2 ( u ¯ 0.5 ) 2 + 0.5 x ) r 2 ( u ¯ 0.5 ) 2 + u ¯ ,
b = 1 4 ( u ¯ 0.5 ) 2 [ ( r 2 ( u ¯ 0.5 ) 2 + 0.5 x ) ( r 2 ( u ¯ 0.5 ) 2 ) ] 1 .
x ¯ 2 ( x 1 , x 2 ) = { x 2 , x 1 [ 0 , a ) ( 1 a , 1 ] , x 2 [ 0 , 1 ] L T ( 0 , 0 , x , g ( x 1 , a , b , x ) , x 2 ) , x 1 [ a , u ¯ ] L T ( 0 , 0 , x , C A ( x 1 ) , x 2 ) x 1 ( u ¯ , 1 u ¯ ) L T ( 0 , 0 , x , g ( x 1 , 1 a , b , x ) , x 2 ) , x 1 [ 1 u ¯ , 1 a ] } x 2 [ 0 , x ] L T ( x , g ( x 1 , a , b , x ) , x + , g ( x 1 , a , b , x + ) , x 2 ) , x 1 [ a , u ¯ ] L T ( x , C A ( x 1 ) , x + , C A + ( x 1 ) , x 2 ) x 1 ( u ¯ , 1 u ¯ ) L T ( x , g ( x 1 , 1 a , b , x ) , x + , g ( x 1 , 1 a , b , x + ) , x 2 ) , x 1 [ 1 u ¯ , 1 a ] } x 2 ( x , x + ) L T ( x + , g ( x 1 , a , b , x + ) , 1 , 1 , x 2 ) , x 1 [ a , u ¯ ] L T ( x + , C A + ( x 1 ) , 1 , 1 , x 2 ) , x 1 ( u ¯ , 1 u ¯ ) L T ( x + , g ( x 1 , 1 a , b , x + ) , 1 , 1 , x 2 ) , x 1 [ 1 u ¯ , 1 a ] } x 2 [ x + , 1 ]
x ¯ 1 ( x 1 , x 2 ) = x 1 , x ¯ 2 ( x 1 , x 2 ) = x 2 , ( x 1 , x 2 ) ,
x ¯ 1 ( x 1 , x 2 ) = x 1 x ( 0.5 r 2 ( x 2 0.5 ) 2 ) , x ¯ 2 ( x 1 , x 2 ) = x 2 , ( x 1 , x 2 )
x ¯ 1 ( x 1 , x 2 ) = 2 x 1 1 x + x r 2 ( x 2 0.5 ) 2 + 1 2 , x ¯ 2 ( x 1 , x 2 ) = x ¯ 1 ( x 2 , x 1 ) , ( x 1 , x 2 ) .
r 2 = ( x ¯ 1 x 0 1 ) 2 δ + ( x ¯ 2 x 0 2 ) 2 γ
x ¯ 1 ( x 1 , x 2 ) = b ( x 1 ) ( x 2 a ) 2 + x 1 , x ¯ 2 ( x 1 , x 2 ) = x 2 , ( x 1 , x 2 ) .
b ( x 1 ) = 1 4 ( u ¯ = x 0 2 ) 2 δ γ 2 ( r 2 γ ( u ¯ x 0 2 ) 2 ) ( 1 δ ( r 2 γ ( u ¯ x 0 2 ) 2 ) + x 0 1 x 1 ) .
x ¯ 1 ( a , x 2 ) ( 25 ) = a x ( 0.5 r 2 ( x 2 0.5 ) 2 ) , x ¯ 1 ( u ¯ , x 2 ) ( 26 ) = 2 u ¯ 1 x + x r 2 ( x 2 0.5 ) 2 + 0.5 ,
x ¯ 1 x 1 | ( a , x 2 ) ( 25 ) = 1 x ( 0.5 r 2 ( x 2 0.5 ) 2 ) , x ¯ 1 x 1 | ( u ¯ , x 2 ) ( 26 ) = 2 r 2 ( x 2 0.5 ) 2 x + x , x ¯ 1 x 2 | ( a , x 2 ) ( 25 ) = a x x 2 0.5 r 2 ( x 2 0.5 ) 2 , x ¯ 1 x 2 | ( u ¯ , x 2 ) ( 26 ) = 2 u ¯ 1 x + x x 2 0.5 r 2 ( x 2 0.5 ) 2 .
x ¯ 1 ( x 1 , x 2 ) = x ¯ 1 ( a , x 2 ) x 1 u ¯ a u ¯ + x ¯ 1 ( u ¯ , x 2 ) x 1 a u ¯ a + f ( x 1 , x 2 ) ( x 1 u ¯ ) ( x 1 a ) .
x ¯ 1 x 1 ( x 1 , x 2 ) = x ¯ 1 ( a , x 2 ) a u ¯ + x ¯ 1 ( u ¯ , x 2 ) u ¯ a + f x 1 ( x 1 , x 2 ) ( x 1 u ¯ ) ( x 1 a ) + f ( x 1 , x 2 ) ( x 1 a ) + f ( x 1 , x 2 ) ( x 1 u ¯ ) .
f ( u ¯ , x 2 ) = 1 u ¯ a ( x ¯ 1 ( a , x 2 ) a u ¯ x ¯ 1 ( u ¯ , x 2 ) u ¯ a + x ¯ 1 x 1 | ( u ¯ , x 2 ) ) , f ( a , x 2 ) = 1 a u ¯ ( x ¯ 1 ( a , x 2 ) a u ¯ x ¯ 1 ( u ¯ , x 2 ) u ¯ a + x ¯ 1 x 1 | ( a , x 2 ) ) .
f ( x 1 , x 2 ) = f ( u ¯ , x 2 ) f ( a , x 2 ) u ¯ a x 1 + f ( u ¯ , x 2 ) f ( u ¯ , x 2 ) f ( a , x 2 ) u ¯ a u ¯ .
x ¯ 2 ( x 1 , x 2 ) = b ( x 2 ) ( x 1 a ) 2 + x 2 .
x 0 1 = 0.5 , x 0 2 = 0.5 , γ = ( x + x 2 x 2 1 ) 2 , δ = 1.
( x ¯ 1 ( x 1 , x 2 ) , x ¯ 2 ( x 1 , x 2 ) ) = ( x ¯ 2 ( x 2 , x 1 ) , x ¯ 1 ( x 2 , x 1 ) ) .
x ¯ 1 ( a , x 2 ) ( 28 ) = b ( a ) ( x 2 a ) 2 + a , with b ( a ) ( 29 ) = 1 4 a x v ¯ 2 ( s + 0.5 x ) s 2 , x ¯ 1 ( u ¯ , x 2 ) ( 36 ) = b ( u ¯ ) ( x 2 a ) 2 + u ¯ , with b ( u ¯ ) ( 37 ) = 1 2 v ¯ 3 ( s 0.5 Δ ) Δ s 2 ,
x ¯ 1 x 2 | ( a , x 2 ) ( 28 ) = 2 b ( a ) ( x 2 a ) , x ¯ 1 x 1 | ( a , x 2 ) ( 28 ) = 1 4 v ¯ 2 ( x 2 a ) 2 ( s + 0.5 x ) x s 2 + 1 , x ¯ 2 x 2 | ( u ¯ , x 2 ) ( 36 ) = 2 b ( u ¯ ) ( x 2 a ) , x ¯ 1 x 1 | ( u ¯ , x 2 ) ( 36 ) = 1 2 v ¯ 2 ( x 2 a ) 2 ( s 0.5 Δ ) Δ s 2 + 1.
0.5 r = x ¯ 1 ( α , 0.5 ) , ( α , 0.5 )
x α = x rec + y rec sin α x rec = x α y α sin α cos α , y α = y rec cos α y rec = y α 1 cos α .
Circle α = { ( x , y ) : ( x 1 2 ( 1 + sin α ) ) 2 + ( y 1 2 cos α ) 2 = r 2 } ,
Circle rec ( 42 ) = { ( x y sin α cos α , y 1 cos α ) : ( x 1 2 ( 1 + sin α ) ) 2 + ( y 1 2 cos α ) 2 = r 2 } , = { ( x , y ) : ( x + y sin α 1 2 ( 1 + sin α ) ) 2 + ( y cos α 1 2 cos α ) 2 = r 2 } .
( x x 0 y y 0 ) ( cos ϕ sin ϕ sin ϕ cos ϕ ) ( c 2 0 0 d 2 ) ( cos ϕ sin ϕ sin ϕ cos ϕ ) ( x x 0 y y 0 ) = 1.
c = r ( 1 sin α ) 1 / 2 , d = r ( 1 + sin α ) 1 / 2 , ϕ = 45 ° , ( x 0 , y 0 ) = ( 0.5 , 0.5 ) .
x ¯ ( x ) = α + β x + γ 2 π sin ( 2 π x x l 1 x l x l 1 ) , x [ x l 1 , x l ] , l = 2 , , n
α = x l x ¯ l 1 x l 1 x ¯ l x l x l 1 , β = x ¯ l x ¯ l 1 x l x l 1 , γ = ( x l x l 1 ) G ( x ¯ l x ¯ l 1 ) ,
x [ 0 , Δ x ] : x ¯ ( x ) = x ¯ 2 x ¯ 1 Δ x x + G Δ x ( x ¯ 2 x ¯ 1 ) 2 π sin ( 2 π x Δ x ) ,
x [ Δ x , L ] : x ¯ ( x ) = α ˜ + β ˜ x + γ ˜ sin ( 2 π x Δ x L Δ x ) .
α ˜ = L ( x ¯ 2 x ¯ 1 ) Δ x L L Δ x , β ˜ = L ( x ¯ 2 x ¯ 1 ) L Δ x , γ ˜ = G ( L Δ x ) ( L ( x ¯ 2 x ¯ 1 ) ) 2 π .
L = α ˜ + β ˜ x ˜ + γ ˜ sin ( 2 π x ˜ Δ x L Δ x ) + x ¯ 1
x [ 0 , L ˜ ] : x ¯ ( x ) = α ˜ + β ˜ ( x + x ˜ ) + γ ˜ sin ( 2 π x + x ˜ Δ x L Δ x ) + x ¯ 1 L ,
x ( L ˜ , L ˜ + Δ x ) : x ¯ ( x ) = x ¯ 2 x ¯ 1 Δ x ( x L ˜ ) + G Δ x ( x ¯ 2 x ¯ 2 ) 2 π sin ( 2 π x L ˜ Δ x ) + x ¯ 1 ,
x [ L ˜ + Δ x , L ] : x ¯ ( x ) = α ˜ + β ˜ ( x L ˜ ) + γ ˜ sin ( 2 π x L ˜ Δ x L Δ x ) + x ¯ 1 .
error = max { | n eff , analyt , 1 st mode n eff , num , 1 st mode | n eff , analyt , 1 st mode , , | n eff , analyt , 10 th mode n eff , num , 10 th mode | n eff , analyt , 10 th mode } ,

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