Abstract

The B-spline modal method (BMM) as applied to lamellar gratings analysis is revisited, and a new implementation is presented. The main difference with our previous work is that we now take into account discontinuities by putting a spline with a degenerate knot on them. Our new approach is compared with other implementations of the BMM and is shown to be superior in terms of numerical convergence.

© 2014 Optical Society of America

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References

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  1. R. Petit, ed., Electromagnetic Theory of Gratings (Springer, 1980).
  2. E. Popov, ed., Gratings: Theory and Numeric Applications (Institut Fresnel, 2013), www.fresnel.fr .
  3. Y. K. Sirenko and G. Strom, eds., Modern Theory of Gratings (Springer, 2009).
  4. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewwartha, “The dielectric lamellar grating,” Opt. Acta 28, 413–428 (1981)
    [CrossRef]
  5. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewwartha, “The finitely conducting lamellar grating,” Optica Acta 28, 1087–1102 (1981).
  6. R. Harrington, “Matrix methods for field problem,” Proc. IEEE 55, 136–149 (1967).
    [CrossRef]
  7. M. G. Moharam and T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. A 72, 1385–1392 (1982).
    [CrossRef]
  8. R. H. Morf, “Exponentially convergent and numerically efficient solution of Maxwell’s equations for lamellar gratings,” J. Opt. Soc. Am. 12, 1043–1056 (1995).
  9. K. Edee, “Modal method based on subsectional Gegenbauer polynomilal expansion for lamellar grating,” J. Opt. Soc. Am. A 28, 2006–2013 (2011).
    [CrossRef]
  10. K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomilal expansion for lamellar gratings: weighting function, convergence and stability,” Prog. Electromagn. Res. 133, 17–35 (2013).
  11. D. Song, L. Yuan, and Y. Y. Lu, “Fourier-matching pseudo-spectral modal method for diffraction gratings,” J. Opt. Soc. Am. A 28, 613–620 (2011).
    [CrossRef]
  12. Y.-P. Chiou, W.-L. Yeh, and N.-Y. Shin, “Analysis of highly conducting lamellar gratings with multidomain pseudospectral method,” J. Lightwave Technol. 27, 5151–5159 (2009).
    [CrossRef]
  13. G. Granet, “Fourier-matching pseudospectral modal method for diffraction gratings: comment,” J. Opt. Soc. Am. A 29, 1843–1845 (2012).
    [CrossRef]
  14. K. Edee, P. Schiavone, and G. Granet, “Analysis of defect in extreme UV lithography mask using a modal method based on nodal B-spline expansion,” Jpn. J. Appl. Phys. 44, 6458–6462 (2005).
    [CrossRef]
  15. A.-M. Armeanu, K. Edee, G. Granet, and P. Schiavone, “Modal method based on spline expansion for the electromagnetic analysis of the lamellar grating,” Prog. Electromagn. Res. 106, 243–261 (2010).
    [CrossRef]
  16. P. Bouchon, F. Pardo, R. Hadar, and J. Pelouard, “Fast modal method for subwavelength gratings based on B-splines formulation,” J. Opt. Soc. Am. 27, 696702 (2010).
  17. M. Waltz, T. Zebrowski, J. Küchenmeister, and K. Bush, “B-spline modal method: a polynomial approach compared to the Fourier modal method,” Opt. Express 21, 14683–14697 (2013).
    [CrossRef]
  18. M. G. Cox, “The numerical evaluation of B-splines,” IMA J. Appl. Math. 10, 134–149 (1972).
    [CrossRef]
  19. C. De Boor, “On calculating with B-splines,” J. Approx. Theory 6, 50–62 (1972).
    [CrossRef]
  20. C. H. Sauvan, P. Lalanne, and J. P. Hugonin, “Truncation rules for modelling discontinuities with Galerkin method in electromagnetic theory,” Opt. Quantum Electron. 36, 271–284 (2004).
    [CrossRef]
  21. L. Li, “Use of Fourier series in the analysis of discontinuous periodic functions,” J. Opt. Soc. Am. 13, 1870–1876 (1996).
    [CrossRef]

2013 (2)

K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomilal expansion for lamellar gratings: weighting function, convergence and stability,” Prog. Electromagn. Res. 133, 17–35 (2013).

M. Waltz, T. Zebrowski, J. Küchenmeister, and K. Bush, “B-spline modal method: a polynomial approach compared to the Fourier modal method,” Opt. Express 21, 14683–14697 (2013).
[CrossRef]

2012 (1)

2011 (2)

2010 (2)

A.-M. Armeanu, K. Edee, G. Granet, and P. Schiavone, “Modal method based on spline expansion for the electromagnetic analysis of the lamellar grating,” Prog. Electromagn. Res. 106, 243–261 (2010).
[CrossRef]

P. Bouchon, F. Pardo, R. Hadar, and J. Pelouard, “Fast modal method for subwavelength gratings based on B-splines formulation,” J. Opt. Soc. Am. 27, 696702 (2010).

2009 (1)

2005 (1)

K. Edee, P. Schiavone, and G. Granet, “Analysis of defect in extreme UV lithography mask using a modal method based on nodal B-spline expansion,” Jpn. J. Appl. Phys. 44, 6458–6462 (2005).
[CrossRef]

2004 (1)

C. H. Sauvan, P. Lalanne, and J. P. Hugonin, “Truncation rules for modelling discontinuities with Galerkin method in electromagnetic theory,” Opt. Quantum Electron. 36, 271–284 (2004).
[CrossRef]

1996 (1)

L. Li, “Use of Fourier series in the analysis of discontinuous periodic functions,” J. Opt. Soc. Am. 13, 1870–1876 (1996).
[CrossRef]

1995 (1)

R. H. Morf, “Exponentially convergent and numerically efficient solution of Maxwell’s equations for lamellar gratings,” J. Opt. Soc. Am. 12, 1043–1056 (1995).

1982 (1)

M. G. Moharam and T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. A 72, 1385–1392 (1982).
[CrossRef]

1981 (2)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewwartha, “The dielectric lamellar grating,” Opt. Acta 28, 413–428 (1981)
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewwartha, “The finitely conducting lamellar grating,” Optica Acta 28, 1087–1102 (1981).

1972 (2)

M. G. Cox, “The numerical evaluation of B-splines,” IMA J. Appl. Math. 10, 134–149 (1972).
[CrossRef]

C. De Boor, “On calculating with B-splines,” J. Approx. Theory 6, 50–62 (1972).
[CrossRef]

1967 (1)

R. Harrington, “Matrix methods for field problem,” Proc. IEEE 55, 136–149 (1967).
[CrossRef]

Adams, J. L.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewwartha, “The dielectric lamellar grating,” Opt. Acta 28, 413–428 (1981)
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewwartha, “The finitely conducting lamellar grating,” Optica Acta 28, 1087–1102 (1981).

Andrewwartha, J. R.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewwartha, “The finitely conducting lamellar grating,” Optica Acta 28, 1087–1102 (1981).

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewwartha, “The dielectric lamellar grating,” Opt. Acta 28, 413–428 (1981)
[CrossRef]

Armeanu, A.-M.

A.-M. Armeanu, K. Edee, G. Granet, and P. Schiavone, “Modal method based on spline expansion for the electromagnetic analysis of the lamellar grating,” Prog. Electromagn. Res. 106, 243–261 (2010).
[CrossRef]

Botten, L. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewwartha, “The finitely conducting lamellar grating,” Optica Acta 28, 1087–1102 (1981).

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewwartha, “The dielectric lamellar grating,” Opt. Acta 28, 413–428 (1981)
[CrossRef]

Bouchon, P.

P. Bouchon, F. Pardo, R. Hadar, and J. Pelouard, “Fast modal method for subwavelength gratings based on B-splines formulation,” J. Opt. Soc. Am. 27, 696702 (2010).

Bush, K.

Chiou, Y.-P.

Cox, M. G.

M. G. Cox, “The numerical evaluation of B-splines,” IMA J. Appl. Math. 10, 134–149 (1972).
[CrossRef]

Craig, M. S.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewwartha, “The dielectric lamellar grating,” Opt. Acta 28, 413–428 (1981)
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewwartha, “The finitely conducting lamellar grating,” Optica Acta 28, 1087–1102 (1981).

De Boor, C.

C. De Boor, “On calculating with B-splines,” J. Approx. Theory 6, 50–62 (1972).
[CrossRef]

Edee, K.

K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomilal expansion for lamellar gratings: weighting function, convergence and stability,” Prog. Electromagn. Res. 133, 17–35 (2013).

K. Edee, “Modal method based on subsectional Gegenbauer polynomilal expansion for lamellar grating,” J. Opt. Soc. Am. A 28, 2006–2013 (2011).
[CrossRef]

A.-M. Armeanu, K. Edee, G. Granet, and P. Schiavone, “Modal method based on spline expansion for the electromagnetic analysis of the lamellar grating,” Prog. Electromagn. Res. 106, 243–261 (2010).
[CrossRef]

K. Edee, P. Schiavone, and G. Granet, “Analysis of defect in extreme UV lithography mask using a modal method based on nodal B-spline expansion,” Jpn. J. Appl. Phys. 44, 6458–6462 (2005).
[CrossRef]

Fenniche, I.

K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomilal expansion for lamellar gratings: weighting function, convergence and stability,” Prog. Electromagn. Res. 133, 17–35 (2013).

Gaylord, T. K.

M. G. Moharam and T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. A 72, 1385–1392 (1982).
[CrossRef]

Granet, G.

K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomilal expansion for lamellar gratings: weighting function, convergence and stability,” Prog. Electromagn. Res. 133, 17–35 (2013).

G. Granet, “Fourier-matching pseudospectral modal method for diffraction gratings: comment,” J. Opt. Soc. Am. A 29, 1843–1845 (2012).
[CrossRef]

A.-M. Armeanu, K. Edee, G. Granet, and P. Schiavone, “Modal method based on spline expansion for the electromagnetic analysis of the lamellar grating,” Prog. Electromagn. Res. 106, 243–261 (2010).
[CrossRef]

K. Edee, P. Schiavone, and G. Granet, “Analysis of defect in extreme UV lithography mask using a modal method based on nodal B-spline expansion,” Jpn. J. Appl. Phys. 44, 6458–6462 (2005).
[CrossRef]

Guizal, B.

K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomilal expansion for lamellar gratings: weighting function, convergence and stability,” Prog. Electromagn. Res. 133, 17–35 (2013).

Hadar, R.

P. Bouchon, F. Pardo, R. Hadar, and J. Pelouard, “Fast modal method for subwavelength gratings based on B-splines formulation,” J. Opt. Soc. Am. 27, 696702 (2010).

Harrington, R.

R. Harrington, “Matrix methods for field problem,” Proc. IEEE 55, 136–149 (1967).
[CrossRef]

Hugonin, J. P.

C. H. Sauvan, P. Lalanne, and J. P. Hugonin, “Truncation rules for modelling discontinuities with Galerkin method in electromagnetic theory,” Opt. Quantum Electron. 36, 271–284 (2004).
[CrossRef]

Küchenmeister, J.

Lalanne, P.

C. H. Sauvan, P. Lalanne, and J. P. Hugonin, “Truncation rules for modelling discontinuities with Galerkin method in electromagnetic theory,” Opt. Quantum Electron. 36, 271–284 (2004).
[CrossRef]

Li, L.

L. Li, “Use of Fourier series in the analysis of discontinuous periodic functions,” J. Opt. Soc. Am. 13, 1870–1876 (1996).
[CrossRef]

Lu, Y. Y.

McPhedran, R. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewwartha, “The finitely conducting lamellar grating,” Optica Acta 28, 1087–1102 (1981).

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewwartha, “The dielectric lamellar grating,” Opt. Acta 28, 413–428 (1981)
[CrossRef]

Moharam, M. G.

M. G. Moharam and T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. A 72, 1385–1392 (1982).
[CrossRef]

Morf, R. H.

R. H. Morf, “Exponentially convergent and numerically efficient solution of Maxwell’s equations for lamellar gratings,” J. Opt. Soc. Am. 12, 1043–1056 (1995).

Pardo, F.

P. Bouchon, F. Pardo, R. Hadar, and J. Pelouard, “Fast modal method for subwavelength gratings based on B-splines formulation,” J. Opt. Soc. Am. 27, 696702 (2010).

Pelouard, J.

P. Bouchon, F. Pardo, R. Hadar, and J. Pelouard, “Fast modal method for subwavelength gratings based on B-splines formulation,” J. Opt. Soc. Am. 27, 696702 (2010).

Sauvan, C. H.

C. H. Sauvan, P. Lalanne, and J. P. Hugonin, “Truncation rules for modelling discontinuities with Galerkin method in electromagnetic theory,” Opt. Quantum Electron. 36, 271–284 (2004).
[CrossRef]

Schiavone, P.

A.-M. Armeanu, K. Edee, G. Granet, and P. Schiavone, “Modal method based on spline expansion for the electromagnetic analysis of the lamellar grating,” Prog. Electromagn. Res. 106, 243–261 (2010).
[CrossRef]

K. Edee, P. Schiavone, and G. Granet, “Analysis of defect in extreme UV lithography mask using a modal method based on nodal B-spline expansion,” Jpn. J. Appl. Phys. 44, 6458–6462 (2005).
[CrossRef]

Shin, N.-Y.

Song, D.

Waltz, M.

Yeh, W.-L.

Yuan, L.

Zebrowski, T.

IMA J. Appl. Math. (1)

M. G. Cox, “The numerical evaluation of B-splines,” IMA J. Appl. Math. 10, 134–149 (1972).
[CrossRef]

J. Approx. Theory (1)

C. De Boor, “On calculating with B-splines,” J. Approx. Theory 6, 50–62 (1972).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. (3)

P. Bouchon, F. Pardo, R. Hadar, and J. Pelouard, “Fast modal method for subwavelength gratings based on B-splines formulation,” J. Opt. Soc. Am. 27, 696702 (2010).

R. H. Morf, “Exponentially convergent and numerically efficient solution of Maxwell’s equations for lamellar gratings,” J. Opt. Soc. Am. 12, 1043–1056 (1995).

L. Li, “Use of Fourier series in the analysis of discontinuous periodic functions,” J. Opt. Soc. Am. 13, 1870–1876 (1996).
[CrossRef]

J. Opt. Soc. Am. A (4)

Jpn. J. Appl. Phys. (1)

K. Edee, P. Schiavone, and G. Granet, “Analysis of defect in extreme UV lithography mask using a modal method based on nodal B-spline expansion,” Jpn. J. Appl. Phys. 44, 6458–6462 (2005).
[CrossRef]

Opt. Acta (1)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewwartha, “The dielectric lamellar grating,” Opt. Acta 28, 413–428 (1981)
[CrossRef]

Opt. Express (1)

Opt. Quantum Electron. (1)

C. H. Sauvan, P. Lalanne, and J. P. Hugonin, “Truncation rules for modelling discontinuities with Galerkin method in electromagnetic theory,” Opt. Quantum Electron. 36, 271–284 (2004).
[CrossRef]

Optica Acta (1)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewwartha, “The finitely conducting lamellar grating,” Optica Acta 28, 1087–1102 (1981).

Proc. IEEE (1)

R. Harrington, “Matrix methods for field problem,” Proc. IEEE 55, 136–149 (1967).
[CrossRef]

Prog. Electromagn. Res. (2)

K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomilal expansion for lamellar gratings: weighting function, convergence and stability,” Prog. Electromagn. Res. 133, 17–35 (2013).

A.-M. Armeanu, K. Edee, G. Granet, and P. Schiavone, “Modal method based on spline expansion for the electromagnetic analysis of the lamellar grating,” Prog. Electromagn. Res. 106, 243–261 (2010).
[CrossRef]

Other (3)

R. Petit, ed., Electromagnetic Theory of Gratings (Springer, 1980).

E. Popov, ed., Gratings: Theory and Numeric Applications (Institut Fresnel, 2013), www.fresnel.fr .

Y. K. Sirenko and G. Strom, eds., Modern Theory of Gratings (Springer, 2009).

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Figures (9)

Fig. 1.
Fig. 1.

Geometry of the problem.

Fig. 2.
Fig. 2.

(a) Single cubic B-spline basis function N0(u). Knots at u=1,2,3 mark transitions between polynomial segments of the function. (b) In the regular case, which has evenly spaced knots, each B-spline basis function is a translated copy of the previous one. Spline N2 overlaps two adjacent splines on the left and on the right. (c) Knot 4 is a three time degenerated knot.

Fig. 3.
Fig. 3.

Illustration of two typical B-splines expansion scheme used to approximate a function and its derivative with end conditions and a discontinuous derivative at x=1/2.

Fig. 4.
Fig. 4.

Absolute error in the approximation of function |cos(πx)| with 32 cubic splines and a distribution of splines similar to that of Fig. 3(a). With the same number of splines and a degenerated knot at x=0.5 the error is less than 107.

Fig. 5.
Fig. 5.

Comparison in the accuracy of the approximation of the derivative of |cos(πx)| when a degenerated knot or a regular knot is put at x=1/2.

Fig. 6.
Fig. 6.

Convergence of the eigenvalue with the smallest imaginary part of a metallic grating with parameters c=.5d d/λ=1, ϵ1=ϵ21=1, ϵ22=ϵ3=(0.22i*6.71)2, θ=30.

Fig. 7.
Fig. 7.

Convergence of the zeroth reflected order of a metallic grating with the same parameters as in Fig. 6.

Fig. 8.
Fig. 8.

Relative error of an eigenvalue as a function of the number of basis functions that are qudratic and cubic B-splines, respectively. The exact value of the eigenvalue is such that (β/k)2=1.590243492557222.

Fig. 9.
Fig. 9.

Convergence plot of the absolute error of reflectance calculation with the BMM using quadratic and cubic B-splines. The assumed exact value is taken from [17] and is R=0.04228344. The asymptotic convergence behavior is fitted as Nr with r=1.7, r=1.5 for quadratic and cubic splines, respectively. This asympptotic behavior is only reached for a number of knots greater than 100, which corresponds approximately to 5 knots per wavelength.

Equations (25)

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

ϵ2(x)={ϵ21if0x<cϵ22ifcx<d,
ψl(x,y)=q(Alq+exp(iβlqy)+Alqexp(iβlqy))ϕlq(x),
Ll(x)ϕlq(x)=βlq2ϕlq(x)ϕlq(x+d)=τϕlq(x)pldϕlqdx(c)=pl+dϕlqdx(c+),
Ll={d2dx2+k2ϵl(x)forTEpolarizationϵl(x)ddx1ϵl(x)ddx+k2ϵl(x)forTMpolarization.
βlq={βlq2ifβlq2R+iβlq2ifβlq2Rβlq2with negative imaginary part ifβlq2C.
Rq=β1qβ10|A1q+|2,
Tq={β3qβ10|A3q|2forTEpolarizationϵ1ϵ3β3qβ10|A3q|2forTMpolarization,
Ni,0(u)={1ifui<u<ui+10otherwise,
Ni,l(u)=uuiui+l1uiNi,l1+ui+luui+lui+1Ni+1,l1,
f(x)=|cos(πx)|
f(x)={πsin(πx)if0x<1/2πsin(πx)if1/2x<1.
f(x)=n=0MfnNn(x)f(x)=n=0MfnNn(x).
fn=DfnD=R1L.
Rmn=01Nm(x)Nn(x)dxLmn=01Nm(x)Nn(x)dx.
yEz=iωBxBx=μ0HxxEz=iωByBy=μ0HyxHyyHx=iωDzDz=ϵ0ϵ(x)Ez,
yHz=iωDxDx=ϵ0ϵ(x)ExxHz=iωDyDy=ϵ0ϵ(x)EyxEyyEx=iωBzBz=μ0Hz.
Ex=1ϵ0ϵ(x)Dx,
Dy=ϵ0ϵ(x)Ey,
Dz=ϵ0ϵ(x)Ez.
S=n=0MSnNn.
Dx=ϵ0ϵxExwithϵx=R1/ϵ1RDy=ϵ0ϵyEywithϵy=R1RϵDz=ϵ0ϵzEzwithϵz=R1Rϵ.
Rϵmn=0dϵ(x)Nm(x)Nn(x)dx
R1/ϵmn=0d1ϵ(x)Nm(x)Nn(x)dx.
Le=DD+k2ϵz,
Lh=ϵxDϵy1D+k2ϵx.

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