Abstract

Numerical methods of generating rough edges, surfaces, and volumes for subsequent simulations are commonly employed, but result in data with a variance that is downward biased from the desired value. Thus, it is highly desirable to quantify and to minimize this bias. Here, the degree of bias is determined through analytical derivations and numerical simulations as a function of the correlation length and the roughness exponent of several model power spectral density functions. The bias can be minimized by proper choice of grid size for a fixed number of data points, and this optimum grid size scales as the correlation length. The common approach of using a fixed grid size for such simulations leads to varying amounts of bias, which can easily be confounded with the physical effects being investigated.

© 2013 Optical Society of America

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References

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  1. P. Naulleau and J. Cain, “Experimental and model-based study of the robustness of line-edge roughness metric extraction in the presence of noise,” J. Vac. Sci. Technol. B. 25, 1647–1657 (2007).
    [CrossRef]
  2. V. Constantoudis, G. Patsis, L. Leunissen, and E. Gogolides, “Line edge roughness and critical dimension variation: fractal characterization and comparison using model functions,” J. Vac. Sci. Technol. B. 22, 1974–1981 (2004).
    [CrossRef]
  3. Z. Chen, L. Dai, and C. Jiang, “Nonlinear response of a silicon waveguide enhanced by a metal grating,” Appl. Opt. 51, 5752–5757 (2012).
    [CrossRef]
  4. A.-Q. Wang, L.-X. Guo, and C. Chai, “Fast numerical method for electromagnetic scattering from an object above a large-scale layered rough surface at large incident angle: vertical polarization,” Appl. Opt. 50, 500–508 (2011).
    [CrossRef]
  5. J. C. Novarini and J. W. Caruthers, “Numerical modelling of acoustic wave scattering from randomly rough surfaces: an image model,” J. Acoust. Soc. Am. 53, 876–884 (1973).
    [CrossRef]
  6. N. Garcia and E. Stoll, “Monte Carlo calculation for electromagnetic-wave scattering from random rough surfaces,” Phys. Rev. Lett. 52, 1798–1801 (1984).
    [CrossRef]
  7. E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83, 78–92 (1988).
    [CrossRef]
  8. C. A. Mack, “Reaction-diffusion power spectral density,” J. Microlithogr. Microfabr. Microsyst. 11, 043007 (2012).
    [CrossRef]
  9. C. A. Mack, “Correlated surface roughening during photoresist development,” Proc. SPIE 8325, 83250I (2012).
    [CrossRef]
  10. T. H. Naylor, J. L. Balintfy, D. S. Burdick, and K. Chu, Computer Simulation Techniques (Wiley, 1966), pp. 118–121.
  11. Y. Z. Hu and K. Tonder, “Simulation of 3-D random rough surface by 2-D digital filter and Fourier analysis,” Int. J. Mach. Tools Manuf. 32, 83–90 (1992).
    [CrossRef]
  12. L. Tsang, J. A. Kong, K.-H. Ding, and C. Ao, Scattering of Electromagnetic Waves—Numerical Simulations (Wiley, 2000).
  13. C. A. Mack, “Analytic form for the power spectral density in one, two, and three dimensions,” J. Microlithogr. Microfabr. Microsyst. 10, 040501 (2011).
    [CrossRef]
  14. J. A. Ogilvy and J. R. Foster, “Rough surfaces: Gaussian or exponential statistics?,” J. Phys. D. 22, 1243–1251(1989).
    [CrossRef]
  15. G. Palasantzas, “Roughness spectrum and surface width of self-affine fractal surfaces via the K-correlation model,” Phys. Rev. B. 48, 14472–14478 (1993).
    [CrossRef]
  16. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77: The Art of Scientific Computing, 2nd ed. (Cambridge University, 1992), p. 545.
  17. M. Matsumoto and T. Nishimura, “Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator,” ACM Trans. Model. Comput. Simul. 8, 3–30 (1998).
    [CrossRef]

2012 (3)

C. A. Mack, “Reaction-diffusion power spectral density,” J. Microlithogr. Microfabr. Microsyst. 11, 043007 (2012).
[CrossRef]

C. A. Mack, “Correlated surface roughening during photoresist development,” Proc. SPIE 8325, 83250I (2012).
[CrossRef]

Z. Chen, L. Dai, and C. Jiang, “Nonlinear response of a silicon waveguide enhanced by a metal grating,” Appl. Opt. 51, 5752–5757 (2012).
[CrossRef]

2011 (2)

2007 (1)

P. Naulleau and J. Cain, “Experimental and model-based study of the robustness of line-edge roughness metric extraction in the presence of noise,” J. Vac. Sci. Technol. B. 25, 1647–1657 (2007).
[CrossRef]

2004 (1)

V. Constantoudis, G. Patsis, L. Leunissen, and E. Gogolides, “Line edge roughness and critical dimension variation: fractal characterization and comparison using model functions,” J. Vac. Sci. Technol. B. 22, 1974–1981 (2004).
[CrossRef]

1998 (1)

M. Matsumoto and T. Nishimura, “Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator,” ACM Trans. Model. Comput. Simul. 8, 3–30 (1998).
[CrossRef]

1993 (1)

G. Palasantzas, “Roughness spectrum and surface width of self-affine fractal surfaces via the K-correlation model,” Phys. Rev. B. 48, 14472–14478 (1993).
[CrossRef]

1992 (1)

Y. Z. Hu and K. Tonder, “Simulation of 3-D random rough surface by 2-D digital filter and Fourier analysis,” Int. J. Mach. Tools Manuf. 32, 83–90 (1992).
[CrossRef]

1989 (1)

J. A. Ogilvy and J. R. Foster, “Rough surfaces: Gaussian or exponential statistics?,” J. Phys. D. 22, 1243–1251(1989).
[CrossRef]

1988 (1)

E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83, 78–92 (1988).
[CrossRef]

1984 (1)

N. Garcia and E. Stoll, “Monte Carlo calculation for electromagnetic-wave scattering from random rough surfaces,” Phys. Rev. Lett. 52, 1798–1801 (1984).
[CrossRef]

1973 (1)

J. C. Novarini and J. W. Caruthers, “Numerical modelling of acoustic wave scattering from randomly rough surfaces: an image model,” J. Acoust. Soc. Am. 53, 876–884 (1973).
[CrossRef]

Ao, C.

L. Tsang, J. A. Kong, K.-H. Ding, and C. Ao, Scattering of Electromagnetic Waves—Numerical Simulations (Wiley, 2000).

Balintfy, J. L.

T. H. Naylor, J. L. Balintfy, D. S. Burdick, and K. Chu, Computer Simulation Techniques (Wiley, 1966), pp. 118–121.

Burdick, D. S.

T. H. Naylor, J. L. Balintfy, D. S. Burdick, and K. Chu, Computer Simulation Techniques (Wiley, 1966), pp. 118–121.

Cain, J.

P. Naulleau and J. Cain, “Experimental and model-based study of the robustness of line-edge roughness metric extraction in the presence of noise,” J. Vac. Sci. Technol. B. 25, 1647–1657 (2007).
[CrossRef]

Caruthers, J. W.

J. C. Novarini and J. W. Caruthers, “Numerical modelling of acoustic wave scattering from randomly rough surfaces: an image model,” J. Acoust. Soc. Am. 53, 876–884 (1973).
[CrossRef]

Chai, C.

Chen, Z.

Chu, K.

T. H. Naylor, J. L. Balintfy, D. S. Burdick, and K. Chu, Computer Simulation Techniques (Wiley, 1966), pp. 118–121.

Constantoudis, V.

V. Constantoudis, G. Patsis, L. Leunissen, and E. Gogolides, “Line edge roughness and critical dimension variation: fractal characterization and comparison using model functions,” J. Vac. Sci. Technol. B. 22, 1974–1981 (2004).
[CrossRef]

Dai, L.

Ding, K.-H.

L. Tsang, J. A. Kong, K.-H. Ding, and C. Ao, Scattering of Electromagnetic Waves—Numerical Simulations (Wiley, 2000).

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77: The Art of Scientific Computing, 2nd ed. (Cambridge University, 1992), p. 545.

Foster, J. R.

J. A. Ogilvy and J. R. Foster, “Rough surfaces: Gaussian or exponential statistics?,” J. Phys. D. 22, 1243–1251(1989).
[CrossRef]

Garcia, N.

N. Garcia and E. Stoll, “Monte Carlo calculation for electromagnetic-wave scattering from random rough surfaces,” Phys. Rev. Lett. 52, 1798–1801 (1984).
[CrossRef]

Gogolides, E.

V. Constantoudis, G. Patsis, L. Leunissen, and E. Gogolides, “Line edge roughness and critical dimension variation: fractal characterization and comparison using model functions,” J. Vac. Sci. Technol. B. 22, 1974–1981 (2004).
[CrossRef]

Guo, L.-X.

Hu, Y. Z.

Y. Z. Hu and K. Tonder, “Simulation of 3-D random rough surface by 2-D digital filter and Fourier analysis,” Int. J. Mach. Tools Manuf. 32, 83–90 (1992).
[CrossRef]

Jiang, C.

Kong, J. A.

L. Tsang, J. A. Kong, K.-H. Ding, and C. Ao, Scattering of Electromagnetic Waves—Numerical Simulations (Wiley, 2000).

Leunissen, L.

V. Constantoudis, G. Patsis, L. Leunissen, and E. Gogolides, “Line edge roughness and critical dimension variation: fractal characterization and comparison using model functions,” J. Vac. Sci. Technol. B. 22, 1974–1981 (2004).
[CrossRef]

Mack, C. A.

C. A. Mack, “Correlated surface roughening during photoresist development,” Proc. SPIE 8325, 83250I (2012).
[CrossRef]

C. A. Mack, “Reaction-diffusion power spectral density,” J. Microlithogr. Microfabr. Microsyst. 11, 043007 (2012).
[CrossRef]

C. A. Mack, “Analytic form for the power spectral density in one, two, and three dimensions,” J. Microlithogr. Microfabr. Microsyst. 10, 040501 (2011).
[CrossRef]

Matsumoto, M.

M. Matsumoto and T. Nishimura, “Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator,” ACM Trans. Model. Comput. Simul. 8, 3–30 (1998).
[CrossRef]

Naulleau, P.

P. Naulleau and J. Cain, “Experimental and model-based study of the robustness of line-edge roughness metric extraction in the presence of noise,” J. Vac. Sci. Technol. B. 25, 1647–1657 (2007).
[CrossRef]

Naylor, T. H.

T. H. Naylor, J. L. Balintfy, D. S. Burdick, and K. Chu, Computer Simulation Techniques (Wiley, 1966), pp. 118–121.

Nishimura, T.

M. Matsumoto and T. Nishimura, “Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator,” ACM Trans. Model. Comput. Simul. 8, 3–30 (1998).
[CrossRef]

Novarini, J. C.

J. C. Novarini and J. W. Caruthers, “Numerical modelling of acoustic wave scattering from randomly rough surfaces: an image model,” J. Acoust. Soc. Am. 53, 876–884 (1973).
[CrossRef]

Ogilvy, J. A.

J. A. Ogilvy and J. R. Foster, “Rough surfaces: Gaussian or exponential statistics?,” J. Phys. D. 22, 1243–1251(1989).
[CrossRef]

Palasantzas, G.

G. Palasantzas, “Roughness spectrum and surface width of self-affine fractal surfaces via the K-correlation model,” Phys. Rev. B. 48, 14472–14478 (1993).
[CrossRef]

Patsis, G.

V. Constantoudis, G. Patsis, L. Leunissen, and E. Gogolides, “Line edge roughness and critical dimension variation: fractal characterization and comparison using model functions,” J. Vac. Sci. Technol. B. 22, 1974–1981 (2004).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77: The Art of Scientific Computing, 2nd ed. (Cambridge University, 1992), p. 545.

Stoll, E.

N. Garcia and E. Stoll, “Monte Carlo calculation for electromagnetic-wave scattering from random rough surfaces,” Phys. Rev. Lett. 52, 1798–1801 (1984).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77: The Art of Scientific Computing, 2nd ed. (Cambridge University, 1992), p. 545.

Thorsos, E. I.

E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83, 78–92 (1988).
[CrossRef]

Tonder, K.

Y. Z. Hu and K. Tonder, “Simulation of 3-D random rough surface by 2-D digital filter and Fourier analysis,” Int. J. Mach. Tools Manuf. 32, 83–90 (1992).
[CrossRef]

Tsang, L.

L. Tsang, J. A. Kong, K.-H. Ding, and C. Ao, Scattering of Electromagnetic Waves—Numerical Simulations (Wiley, 2000).

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77: The Art of Scientific Computing, 2nd ed. (Cambridge University, 1992), p. 545.

Wang, A.-Q.

ACM Trans. Model. Comput. Simul. (1)

M. Matsumoto and T. Nishimura, “Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator,” ACM Trans. Model. Comput. Simul. 8, 3–30 (1998).
[CrossRef]

Appl. Opt. (2)

Int. J. Mach. Tools Manuf. (1)

Y. Z. Hu and K. Tonder, “Simulation of 3-D random rough surface by 2-D digital filter and Fourier analysis,” Int. J. Mach. Tools Manuf. 32, 83–90 (1992).
[CrossRef]

J. Acoust. Soc. Am. (2)

J. C. Novarini and J. W. Caruthers, “Numerical modelling of acoustic wave scattering from randomly rough surfaces: an image model,” J. Acoust. Soc. Am. 53, 876–884 (1973).
[CrossRef]

E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83, 78–92 (1988).
[CrossRef]

J. Microlithogr. Microfabr. Microsyst. (2)

C. A. Mack, “Reaction-diffusion power spectral density,” J. Microlithogr. Microfabr. Microsyst. 11, 043007 (2012).
[CrossRef]

C. A. Mack, “Analytic form for the power spectral density in one, two, and three dimensions,” J. Microlithogr. Microfabr. Microsyst. 10, 040501 (2011).
[CrossRef]

J. Phys. D. (1)

J. A. Ogilvy and J. R. Foster, “Rough surfaces: Gaussian or exponential statistics?,” J. Phys. D. 22, 1243–1251(1989).
[CrossRef]

J. Vac. Sci. Technol. B. (2)

P. Naulleau and J. Cain, “Experimental and model-based study of the robustness of line-edge roughness metric extraction in the presence of noise,” J. Vac. Sci. Technol. B. 25, 1647–1657 (2007).
[CrossRef]

V. Constantoudis, G. Patsis, L. Leunissen, and E. Gogolides, “Line edge roughness and critical dimension variation: fractal characterization and comparison using model functions,” J. Vac. Sci. Technol. B. 22, 1974–1981 (2004).
[CrossRef]

Phys. Rev. B. (1)

G. Palasantzas, “Roughness spectrum and surface width of self-affine fractal surfaces via the K-correlation model,” Phys. Rev. B. 48, 14472–14478 (1993).
[CrossRef]

Phys. Rev. Lett. (1)

N. Garcia and E. Stoll, “Monte Carlo calculation for electromagnetic-wave scattering from random rough surfaces,” Phys. Rev. Lett. 52, 1798–1801 (1984).
[CrossRef]

Proc. SPIE (1)

C. A. Mack, “Correlated surface roughening during photoresist development,” Proc. SPIE 8325, 83250I (2012).
[CrossRef]

Other (3)

T. H. Naylor, J. L. Balintfy, D. S. Burdick, and K. Chu, Computer Simulation Techniques (Wiley, 1966), pp. 118–121.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77: The Art of Scientific Computing, 2nd ed. (Cambridge University, 1992), p. 545.

L. Tsang, J. A. Kong, K.-H. Ding, and C. Ao, Scattering of Electromagnetic Waves—Numerical Simulations (Wiley, 2000).

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Figures (9)

Fig. 1.
Fig. 1.

For the 1D case, α = 0.5 , comparison of numerical simulations using the Thorsos method (symbols) to the predictions of the relevant equation from Table 1 (solid curves).

Fig. 2.
Fig. 2.

For the 2D case, α = 0.5 , comparison of numerical simulations using the Thorsos method (symbols) to the predictions of the relevant equation from Table 1 (solid curves).

Fig. 3.
Fig. 3.

For the 3D case, α = 0.5 , comparison of numerical simulations using the Thorsos method (symbols) to the predictions of the relevant equation from Table 1 (solid curves).

Fig. 4.
Fig. 4.

For the 1D case, α = 1.0 , comparison of numerical simulations using the Thorsos method (symbols) to the predictions of the relevant equation from Table 1 (solid curves).

Fig. 5.
Fig. 5.

For the 1D case, N = 256 , comparison of numerical simulations using the Thorsos method (symbols) to the predictions of Eq. (31) (solid curves) for various values of the roughness exponent H .

Fig. 6.
Fig. 6.

Typical PSD taken from a generated rough edge ( σ = 5 nm , ξ = 10 nm , Δ x = 1 nm , and N = 1024 ): (a) one trial and (b) average of 100 trials. The input PSD is shown as the smooth red curve.

Fig. 7.
Fig. 7.

Convergence of the numerically generated 1D PSD to the input PSD function as a function of the number of trials being averaged together ( σ = 5 nm , ξ = 10 nm , Δ x = 1 nm , and N = 1 , 024 ). The standard 1 / M convergence trend is shown as the solid line, with simulations shown as the symbols.

Fig. 8.
Fig. 8.

Convergence of the numerically generated PSD to the input PSD function as a function of the number of trials being averaged together ( M ) and the number of points per side ( N ) for ξ = 10 nm and Δ x = 1 nm : (a) 2D case and (b) 3D case.

Fig. 9.
Fig. 9.

Linear interpolation error (relative RMS error of output PSD compared to the input PSD) when converting a 3D PSD onto a radial grid ( Δ x = 1 nm ).

Tables (4)

Tables Icon

Table 1. Relative Bias in the Variance ( ε rel ) Produced by the Thorsos Method

Tables Icon

Table 2. Simulation Grid Size that Minimizes the Relative Bias in the Variance Produced by the Thorsos Method, for a Given ξ and N

Tables Icon

Table 3. (Approximate) Minimum Possible Relative Bias in the Variance Produced by the Thorsos Method, for a Given N , When the Grid size is Set as in Table 2

Tables Icon

Table 4. Examples of the Minimimum Possible Relative Bias in the Variance Produced by the Thorsos Method (from Table 3), for Different Values of N

Equations (33)

Equations on this page are rendered with MathJax. Learn more.

z i = j = M M w j η i j .
w j = FT { PSD ( k ) } .
z ( x n ) = μ z + 1 L j = N / 2 N / 2 1 F ( f j ) e i 2 π f j x n ,
F ( f j ) = LPSD ( f j ) { ( η 1 + i η 2 ) / 2 , j 0 , ± N / 2 η 1 , j = 0 , ± N / 2 ,
Re { F ( f x , f y ) } = Re { F ( f x , f y ) } , Re { F ( f x , f y ) } = Re { F ( f x , f y ) } , Im { F ( f x , f y ) } = Im { F ( f x , f y ) } , Im { F ( f x , f y ) } = Im { F ( f x , f y ) } .
Re { F ( f x , f y , f z ) } = Re { F ( f x , f y , f z ) } , Re { F ( f x , f y , f z ) } = Re { F ( f x , f y , f z ) } , Re { F ( f x , f y , f z ) } = Re { F ( f x , f y , f z ) } , Re { F ( f x , f y , f z ) } = Re { F ( f x , f y , f z ) } , Im { F ( f x , f y , f z ) } = Im { F ( f x , f y , f z ) } , Im { F ( f x , f y , f z ) } = Im { F ( f x , f y , f z ) } , Im { F ( f x , f y , f z ) } = Im { F ( f x , f y , f z ) } , Im { F ( f x , f y , f z ) } = Im { F ( f x , f y , f z ) } .
z ( x n ) = μ z + 1 L j = N / 2 N / 2 1 γ j PSD ( f j ) e i 2 π j ( n / N ) ,
γ j = { ( η 1 + i η 2 ) / 2 , j 0 , N / 2 η 1 , j = 0 , N / 2 .
z = μ z ,
( z ( x n ) μ z ) 2 = 1 L j = N / 2 N / 2 1 k = N / 2 N / 2 1 γ j γ k PSD ( f k ) PSD ( f j ) e i 2 π ( j + k ) ( n / N ) .
( z μ z ) 2 = 1 L j = N / 2 N / 2 1 PSD ( f j ) .
( z μ z ) 2 = 1 L 2 j = N / 2 N / 2 1 k = N / 2 N / 2 1 PSD ( f j , f k ) .
( z z ¯ ) 2 = 1 L j = N / 2 N / 2 1 PSD ( f j ) PSD ( 0 ) L .
σ sample 2 = ( z z ¯ ) 2 ( N N 1 ) ( 1 L j = N / 2 N / 2 1 PSD ( f j ) PSD ( 0 ) L ) ( 1 + 1 N ) .
σ 2 = PSD ( f ) d f .
R ˜ ( x ) = e ( x / ξ ) 2 α ,
α = 0.5 : R ( x ) = σ 2 e | x | / ξ , PSD ( f ) = 2 σ 2 ξ 1 + ( 2 π f ξ ) 2 , α = 1.0 : R ( x ) = σ 2 e ( x / ξ ) 2 , PSD ( f ) = π σ 2 ξ e ( π f ξ ) 2 ,
ε d = PSD ( f ) d f 1 L j = PSD ( f j ) .
ε d = σ 2 [ 1 coth ( L 2 ξ ) ] .
ε d 2 σ 2 e L / ξ .
ε hi = PSD ( f ) d f f N f N PSD ( f ) d f = 2 f N PSD ( f ) d f ,
ε hi = σ 2 [ 1 2 π tan 1 ( π ξ Δ x ) ] .
ε hi σ 2 ( 2 π 2 ) ( Δ x ξ ) .
ε lo = PSD ( 0 ) L = σ 2 ( 2 ξ L ) .
σ sample 2 σ 2 [ 1 + 2 e L / ξ ( 2 π 2 ) ( Δ x ξ ) 2 ( ξ L ) + 1 N ] .
ε rel = σ 2 σ sample 2 σ 2 ,
ε d 2 σ 2 e ( L / ξ ) 2 .
ε hi = σ 2 erfc ( π ξ 2 Δ x ) ( 2 π 3 / 2 ) ( Δ x ξ ) e ( π ξ 2 Δ x ) 2 .
PSD ( f ) = PSD ( 0 ) [ 1 + ( 2 π f ξ ) 2 ] H + d / 2 ,
1 D : PSD ( 0 ) = 2 σ 2 ξ ( π Γ ( H + 1 2 ) Γ ( H ) ) , 2 D : PSD ( 0 ) = 2 π σ 2 ξ 2 ( 2 H ) , 3 D : PSD ( 0 ) = 8 π σ 2 ξ 3 ( π Γ ( H + 3 2 ) Γ ( H ) ) .
1 D : ε rel ( Γ ( H + 1 2 ) π Γ ( H + 1 ) ) ( Δ x π ξ ) 2 H + 2 ( π Γ ( H + 1 2 ) Γ ( H ) ) ( ξ L ) 1 N 2 D : ε rel ( 2 2 π ) ( Δ x π ξ ) 2 H + 2 π ( 2 H ) ( ξ L ) 2 1 N 2 3 D : ε rel ( 2 + 2 π ) ( Δ x π ξ ) 2 H + 8 π ( π Γ ( H + 3 2 ) Γ ( H ) ) ( ξ L ) 3 1 N 3 .
Δ x π ξ N 1 / ( 1 + 2 H ) .
ε rel , min 2 ( Γ ( H + 3 2 ) π Γ ( H + 1 ) ) ( 1 N 2 H / ( 1 + 2 H ) ) .

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