Abstract

The perturbation method is combined with the Rigorous Coupled Wave Analysis (RCWA) to enhance its computational speed. In the original RCWA, a grating is approximated by a stack of lamellar gratings and the number of eigenvalue systems to be solved is equal to the number of subgratings. The perturbation method allows to derive the eigensolutions in many layers from the computed eigensolutions of a reference layer provided that the optical and geometrical parameters of these layers differ only slightly. A trapezoidal grating is considered to evaluate the performance of the method.

© 2010 Optical Society of America

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References

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  1. S. T. Peng, T. Tamir, and H. L. Bertoni, "Theory of periodic dielectric waveguides," IEEE Trans. Microw. Theory Tech. MTT-23, 123-133 (1975).
    [CrossRef]
  2. M. G. Moharam, and T. K. Gaylord, "Diffraction analysis of dielectric surface-relief gratings," J. Opt. Soc. Am. A 72, 1385-1392 (1982).
    [CrossRef]
  3. I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The dielectric lamellar diffraction grating," Opt. Acta (Lond.) 28, 413-428 (1981).
  4. I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The finitely conducting lamellar diffraction grating," Opt. Acta (Lond.) 28, 1087-1102 (1981).
  5. K. Knop, "Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves," J. Opt. Soc. Am. 68, 1206-1210 (1978).
    [CrossRef]
  6. Ph. Lalanne, and G. M. Morris, "Highly improved convergence of the coupled-wave method for TM-polarization," J. Opt. Soc. Am. A 13, 779-784 (1996).
    [CrossRef]
  7. G. Granet, and B. Guizal, "Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization," J. Opt. Soc. Am. A 13, 1019-1023 (1996).
    [CrossRef]
  8. L. Li, "Use of Fourier series in the analysis of discontinuous periodic structures," J. Opt. Soc. Am. A 13, 1870-1876 (1996).
    [CrossRef]
  9. L. Li, "Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity," J. Opt. Soc. Am. A 10, 2581-2591 (1993).
    [CrossRef]
  10. J. Y. Suratteau, M. Cadilhac, and R. Petit, "Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds," J. Opt. (Paris) 14, 273-288 (1983).
  11. D. M. Pai, and K. A. Awada, "Analysis of dielectric gratings of arbitrary profiles and thicknesses," J. Opt. Soc. Am. A 8, 755-762 (1991).
    [CrossRef]
  12. N. Chateau, and J.-P. Hugonin, "Algorithm for the rigorous coupled-wave analysis of grating diffraction," J. Opt. Soc. Am. A 11, 1321-1331 (1994).
    [CrossRef]
  13. N. P. van der Aa, and R. M. M. Mattheij, "Computing shape parameter sensitivity of the field of one-dimensional surface-relief gratings by using an analytical approach based on RCWA," J. Opt. Soc. Am. A 24, 2692-2700 (2007).
    [CrossRef]
  14. E. Popov, M. Nevière, B. Gralak, and G. Tayeb, "Staircase approximation validity for arbitrary-shaped gratings," J. Opt. Soc. Am. A 19, 33-42 (2002).
    [CrossRef]
  15. J. Chandezon, D. Maystre, and G. Raoult, "A new theoretical method for diffraction gratings and its numerical application," J. Opt. (Paris) 11, 235-241 (1980).
  16. L. Li, J. Chandezon, G. Granet, and J.-P. Plumey, "Rigorous and efficient grating-analysis method made easy for optical engineers," Appl. Opt. 38, 304-313 (1999).
    [CrossRef]
  17. T. Vallius, "Comparing the Fourier modal method with the C method: analysis of conducting multilevel gratings in TM polarization," J. Opt. Soc. Am. A 19, 1555-1562 (2002).
    [CrossRef]
  18. G. Granet, J. Chandezon, J.-P. Plumey, and K. Raniriharinosy, "Reformulation of the coordinate transformation method through the concept of adaptive spatial resolution. Application to trapezoidal gratings," J. Opt. Soc. Am. A 18, 2102-2108 (2001).
    [CrossRef]
  19. L. D. Landau, and E. M. Lifschitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed. Ed. Pergamon, New York, (1977).
  20. C. Cohen-Tannoudji, B. Diu, and F. Laloë, Mécanique quantique, T2, Ed. Hermann, Paris, (1994).
    [PubMed]

2007 (1)

N. P. van der Aa, and R. M. M. Mattheij, "Computing shape parameter sensitivity of the field of one-dimensional surface-relief gratings by using an analytical approach based on RCWA," J. Opt. Soc. Am. A 24, 2692-2700 (2007).
[CrossRef]

2002 (2)

E. Popov, M. Nevière, B. Gralak, and G. Tayeb, "Staircase approximation validity for arbitrary-shaped gratings," J. Opt. Soc. Am. A 19, 33-42 (2002).
[CrossRef]

T. Vallius, "Comparing the Fourier modal method with the C method: analysis of conducting multilevel gratings in TM polarization," J. Opt. Soc. Am. A 19, 1555-1562 (2002).
[CrossRef]

2001 (1)

G. Granet, J. Chandezon, J.-P. Plumey, and K. Raniriharinosy, "Reformulation of the coordinate transformation method through the concept of adaptive spatial resolution. Application to trapezoidal gratings," J. Opt. Soc. Am. A 18, 2102-2108 (2001).
[CrossRef]

1999 (1)

L. Li, J. Chandezon, G. Granet, and J.-P. Plumey, "Rigorous and efficient grating-analysis method made easy for optical engineers," Appl. Opt. 38, 304-313 (1999).
[CrossRef]

1996 (3)

Ph. Lalanne, and G. M. Morris, "Highly improved convergence of the coupled-wave method for TM-polarization," J. Opt. Soc. Am. A 13, 779-784 (1996).
[CrossRef]

G. Granet, and B. Guizal, "Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization," J. Opt. Soc. Am. A 13, 1019-1023 (1996).
[CrossRef]

L. Li, "Use of Fourier series in the analysis of discontinuous periodic structures," J. Opt. Soc. Am. A 13, 1870-1876 (1996).
[CrossRef]

1994 (1)

N. Chateau, and J.-P. Hugonin, "Algorithm for the rigorous coupled-wave analysis of grating diffraction," J. Opt. Soc. Am. A 11, 1321-1331 (1994).
[CrossRef]

1993 (1)

L. Li, "Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity," J. Opt. Soc. Am. A 10, 2581-2591 (1993).
[CrossRef]

1991 (1)

D. M. Pai, and K. A. Awada, "Analysis of dielectric gratings of arbitrary profiles and thicknesses," J. Opt. Soc. Am. A 8, 755-762 (1991).
[CrossRef]

1983 (1)

J. Y. Suratteau, M. Cadilhac, and R. Petit, "Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds," J. Opt. (Paris) 14, 273-288 (1983).

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)

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The dielectric lamellar diffraction grating," Opt. Acta (Lond.) 28, 413-428 (1981).

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The finitely conducting lamellar diffraction grating," Opt. Acta (Lond.) 28, 1087-1102 (1981).

1980 (1)

J. Chandezon, D. Maystre, and G. Raoult, "A new theoretical method for diffraction gratings and its numerical application," J. Opt. (Paris) 11, 235-241 (1980).

1978 (1)

K. Knop, "Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves," J. Opt. Soc. Am. 68, 1206-1210 (1978).
[CrossRef]

1975 (1)

S. T. Peng, T. Tamir, and H. L. Bertoni, "Theory of periodic dielectric waveguides," IEEE Trans. Microw. Theory Tech. MTT-23, 123-133 (1975).
[CrossRef]

Adams, J. L.

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The finitely conducting lamellar diffraction grating," Opt. Acta (Lond.) 28, 1087-1102 (1981).

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The dielectric lamellar diffraction grating," Opt. Acta (Lond.) 28, 413-428 (1981).

Andrewartha, J. R.

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The dielectric lamellar diffraction grating," Opt. Acta (Lond.) 28, 413-428 (1981).

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The finitely conducting lamellar diffraction grating," Opt. Acta (Lond.) 28, 1087-1102 (1981).

Awada, K. A.

D. M. Pai, and K. A. Awada, "Analysis of dielectric gratings of arbitrary profiles and thicknesses," J. Opt. Soc. Am. A 8, 755-762 (1991).
[CrossRef]

Bertoni, H. L.

S. T. Peng, T. Tamir, and H. L. Bertoni, "Theory of periodic dielectric waveguides," IEEE Trans. Microw. Theory Tech. MTT-23, 123-133 (1975).
[CrossRef]

Botten, I. C.

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The dielectric lamellar diffraction grating," Opt. Acta (Lond.) 28, 413-428 (1981).

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The finitely conducting lamellar diffraction grating," Opt. Acta (Lond.) 28, 1087-1102 (1981).

Cadilhac, M.

J. Y. Suratteau, M. Cadilhac, and R. Petit, "Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds," J. Opt. (Paris) 14, 273-288 (1983).

Chandezon, J.

G. Granet, J. Chandezon, J.-P. Plumey, and K. Raniriharinosy, "Reformulation of the coordinate transformation method through the concept of adaptive spatial resolution. Application to trapezoidal gratings," J. Opt. Soc. Am. A 18, 2102-2108 (2001).
[CrossRef]

L. Li, J. Chandezon, G. Granet, and J.-P. Plumey, "Rigorous and efficient grating-analysis method made easy for optical engineers," Appl. Opt. 38, 304-313 (1999).
[CrossRef]

J. Chandezon, D. Maystre, and G. Raoult, "A new theoretical method for diffraction gratings and its numerical application," J. Opt. (Paris) 11, 235-241 (1980).

Chateau, N.

N. Chateau, and J.-P. Hugonin, "Algorithm for the rigorous coupled-wave analysis of grating diffraction," J. Opt. Soc. Am. A 11, 1321-1331 (1994).
[CrossRef]

Craig, M. S.

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The dielectric lamellar diffraction grating," Opt. Acta (Lond.) 28, 413-428 (1981).

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The finitely conducting lamellar diffraction grating," Opt. Acta (Lond.) 28, 1087-1102 (1981).

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]

Gralak, B.

E. Popov, M. Nevière, B. Gralak, and G. Tayeb, "Staircase approximation validity for arbitrary-shaped gratings," J. Opt. Soc. Am. A 19, 33-42 (2002).
[CrossRef]

Granet, G.

G. Granet, J. Chandezon, J.-P. Plumey, and K. Raniriharinosy, "Reformulation of the coordinate transformation method through the concept of adaptive spatial resolution. Application to trapezoidal gratings," J. Opt. Soc. Am. A 18, 2102-2108 (2001).
[CrossRef]

L. Li, J. Chandezon, G. Granet, and J.-P. Plumey, "Rigorous and efficient grating-analysis method made easy for optical engineers," Appl. Opt. 38, 304-313 (1999).
[CrossRef]

G. Granet, and B. Guizal, "Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization," J. Opt. Soc. Am. A 13, 1019-1023 (1996).
[CrossRef]

Guizal, B.

G. Granet, and B. Guizal, "Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization," J. Opt. Soc. Am. A 13, 1019-1023 (1996).
[CrossRef]

Hugonin, J.-P.

N. Chateau, and J.-P. Hugonin, "Algorithm for the rigorous coupled-wave analysis of grating diffraction," J. Opt. Soc. Am. A 11, 1321-1331 (1994).
[CrossRef]

Knop, K.

K. Knop, "Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves," J. Opt. Soc. Am. 68, 1206-1210 (1978).
[CrossRef]

Lalanne, Ph.

Ph. Lalanne, and G. M. Morris, "Highly improved convergence of the coupled-wave method for TM-polarization," J. Opt. Soc. Am. A 13, 779-784 (1996).
[CrossRef]

Li, L.

L. Li, J. Chandezon, G. Granet, and J.-P. Plumey, "Rigorous and efficient grating-analysis method made easy for optical engineers," Appl. Opt. 38, 304-313 (1999).
[CrossRef]

L. Li, "Use of Fourier series in the analysis of discontinuous periodic structures," J. Opt. Soc. Am. A 13, 1870-1876 (1996).
[CrossRef]

L. Li, "Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity," J. Opt. Soc. Am. A 10, 2581-2591 (1993).
[CrossRef]

Mattheij, R. M. M.

N. P. van der Aa, and R. M. M. Mattheij, "Computing shape parameter sensitivity of the field of one-dimensional surface-relief gratings by using an analytical approach based on RCWA," J. Opt. Soc. Am. A 24, 2692-2700 (2007).
[CrossRef]

Maystre, D.

J. Chandezon, D. Maystre, and G. Raoult, "A new theoretical method for diffraction gratings and its numerical application," J. Opt. (Paris) 11, 235-241 (1980).

McPhedran, R. C.

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The finitely conducting lamellar diffraction grating," Opt. Acta (Lond.) 28, 1087-1102 (1981).

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The dielectric lamellar diffraction grating," Opt. Acta (Lond.) 28, 413-428 (1981).

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]

Morris, G. M.

Ph. Lalanne, and G. M. Morris, "Highly improved convergence of the coupled-wave method for TM-polarization," J. Opt. Soc. Am. A 13, 779-784 (1996).
[CrossRef]

Nevière, M.

E. Popov, M. Nevière, B. Gralak, and G. Tayeb, "Staircase approximation validity for arbitrary-shaped gratings," J. Opt. Soc. Am. A 19, 33-42 (2002).
[CrossRef]

Pai, D. M.

D. M. Pai, and K. A. Awada, "Analysis of dielectric gratings of arbitrary profiles and thicknesses," J. Opt. Soc. Am. A 8, 755-762 (1991).
[CrossRef]

Peng, S. T.

S. T. Peng, T. Tamir, and H. L. Bertoni, "Theory of periodic dielectric waveguides," IEEE Trans. Microw. Theory Tech. MTT-23, 123-133 (1975).
[CrossRef]

Petit, R.

J. Y. Suratteau, M. Cadilhac, and R. Petit, "Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds," J. Opt. (Paris) 14, 273-288 (1983).

Plumey, J.-P.

G. Granet, J. Chandezon, J.-P. Plumey, and K. Raniriharinosy, "Reformulation of the coordinate transformation method through the concept of adaptive spatial resolution. Application to trapezoidal gratings," J. Opt. Soc. Am. A 18, 2102-2108 (2001).
[CrossRef]

L. Li, J. Chandezon, G. Granet, and J.-P. Plumey, "Rigorous and efficient grating-analysis method made easy for optical engineers," Appl. Opt. 38, 304-313 (1999).
[CrossRef]

Popov, E.

E. Popov, M. Nevière, B. Gralak, and G. Tayeb, "Staircase approximation validity for arbitrary-shaped gratings," J. Opt. Soc. Am. A 19, 33-42 (2002).
[CrossRef]

Raniriharinosy, K.

G. Granet, J. Chandezon, J.-P. Plumey, and K. Raniriharinosy, "Reformulation of the coordinate transformation method through the concept of adaptive spatial resolution. Application to trapezoidal gratings," J. Opt. Soc. Am. A 18, 2102-2108 (2001).
[CrossRef]

Raoult, G.

J. Chandezon, D. Maystre, and G. Raoult, "A new theoretical method for diffraction gratings and its numerical application," J. Opt. (Paris) 11, 235-241 (1980).

Suratteau, J. Y.

J. Y. Suratteau, M. Cadilhac, and R. Petit, "Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds," J. Opt. (Paris) 14, 273-288 (1983).

Tamir, T.

S. T. Peng, T. Tamir, and H. L. Bertoni, "Theory of periodic dielectric waveguides," IEEE Trans. Microw. Theory Tech. MTT-23, 123-133 (1975).
[CrossRef]

Tayeb, G.

E. Popov, M. Nevière, B. Gralak, and G. Tayeb, "Staircase approximation validity for arbitrary-shaped gratings," J. Opt. Soc. Am. A 19, 33-42 (2002).
[CrossRef]

Vallius, T.

T. Vallius, "Comparing the Fourier modal method with the C method: analysis of conducting multilevel gratings in TM polarization," J. Opt. Soc. Am. A 19, 1555-1562 (2002).
[CrossRef]

van der Aa, N. P.

N. P. van der Aa, and R. M. M. Mattheij, "Computing shape parameter sensitivity of the field of one-dimensional surface-relief gratings by using an analytical approach based on RCWA," J. Opt. Soc. Am. A 24, 2692-2700 (2007).
[CrossRef]

Appl. Opt. (1)

L. Li, J. Chandezon, G. Granet, and J.-P. Plumey, "Rigorous and efficient grating-analysis method made easy for optical engineers," Appl. Opt. 38, 304-313 (1999).
[CrossRef]

IEEE Trans. Microw. Theory Tech. (1)

S. T. Peng, T. Tamir, and H. L. Bertoni, "Theory of periodic dielectric waveguides," IEEE Trans. Microw. Theory Tech. MTT-23, 123-133 (1975).
[CrossRef]

J. Opt. (Paris) (2)

J. Y. Suratteau, M. Cadilhac, and R. Petit, "Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds," J. Opt. (Paris) 14, 273-288 (1983).

J. Chandezon, D. Maystre, and G. Raoult, "A new theoretical method for diffraction gratings and its numerical application," J. Opt. (Paris) 11, 235-241 (1980).

J. Opt. Soc. Am. (1)

K. Knop, "Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves," J. Opt. Soc. Am. 68, 1206-1210 (1978).
[CrossRef]

J. Opt. Soc. Am. A (11)

Ph. Lalanne, and G. M. Morris, "Highly improved convergence of the coupled-wave method for TM-polarization," J. Opt. Soc. Am. A 13, 779-784 (1996).
[CrossRef]

G. Granet, and B. Guizal, "Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization," J. Opt. Soc. Am. A 13, 1019-1023 (1996).
[CrossRef]

L. Li, "Use of Fourier series in the analysis of discontinuous periodic structures," J. Opt. Soc. Am. A 13, 1870-1876 (1996).
[CrossRef]

L. Li, "Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity," J. Opt. Soc. Am. A 10, 2581-2591 (1993).
[CrossRef]

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

D. M. Pai, and K. A. Awada, "Analysis of dielectric gratings of arbitrary profiles and thicknesses," J. Opt. Soc. Am. A 8, 755-762 (1991).
[CrossRef]

N. Chateau, and J.-P. Hugonin, "Algorithm for the rigorous coupled-wave analysis of grating diffraction," J. Opt. Soc. Am. A 11, 1321-1331 (1994).
[CrossRef]

N. P. van der Aa, and R. M. M. Mattheij, "Computing shape parameter sensitivity of the field of one-dimensional surface-relief gratings by using an analytical approach based on RCWA," J. Opt. Soc. Am. A 24, 2692-2700 (2007).
[CrossRef]

E. Popov, M. Nevière, B. Gralak, and G. Tayeb, "Staircase approximation validity for arbitrary-shaped gratings," J. Opt. Soc. Am. A 19, 33-42 (2002).
[CrossRef]

T. Vallius, "Comparing the Fourier modal method with the C method: analysis of conducting multilevel gratings in TM polarization," J. Opt. Soc. Am. A 19, 1555-1562 (2002).
[CrossRef]

G. Granet, J. Chandezon, J.-P. Plumey, and K. Raniriharinosy, "Reformulation of the coordinate transformation method through the concept of adaptive spatial resolution. Application to trapezoidal gratings," J. Opt. Soc. Am. A 18, 2102-2108 (2001).
[CrossRef]

Opt. Acta (Lond.) (2)

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The dielectric lamellar diffraction grating," Opt. Acta (Lond.) 28, 413-428 (1981).

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The finitely conducting lamellar diffraction grating," Opt. Acta (Lond.) 28, 1087-1102 (1981).

Other (2)

L. D. Landau, and E. M. Lifschitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed. Ed. Pergamon, New York, (1977).

C. Cohen-Tannoudji, B. Diu, and F. Laloë, Mécanique quantique, T2, Ed. Hermann, Paris, (1994).
[PubMed]

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Figures (3)

Fig. 1
Fig. 1

The grating configuration: two periods are depicted.

Fig. 2
Fig. 2

The staircase approximation: the grating is replaced by a stack of lamellar gratings.

Fig. 3
Fig. 3

Surface grating made of a stack of two dielectric trapezoidal elements. Numerical parameters: λ = 240 nm, d = 480 nm, b1 = 240 nm, b2 = 120 nm, b3 = 80 nm, h1 = 80 nm, h2 = 200 nm, θ = 75°, n1 = n3 = 1.51309, n2 = 1.

Tables (8)

Tables Icon

Table 1 Convergence of the RCWA with respect of the truncation order M Results for the grating described by Figure 3 with (Nc = 220)

Tables Icon

Table 2 Convergence of the RCWA with respect of the number of layers Nc Results for the grating described by Figure 3 with 30 Fourier harmonics (M = 30)

Tables Icon

Table 3 Results for the grating described by Figure 3 with 220 layers (Nc = 220) for each trapezoidal element and 30 Fourier harmonics (M = 30)

Tables Icon

Table 4 Results for the grating described by Figure 3 with 220 layers (Nc = 220) for each trapezoidal element and 30 Fourier harmonics (M = 30)

Tables Icon

Table 5 Results for the grating described by Figure 3 with 220 layers (Nc = 220) for each trapezoidal element and 30 Fourier harmonics (M = 30)

Tables Icon

Table 6 Results for the grating described by Figure 3 with 220 layers (Nc = 220) for each trapezoidal element and 30 Fourier harmonics (M = 30)

Tables Icon

Table 7 Results for the grating described by Figure 3 with 220 layers (Nc = 220) for each trapezoidal element and 30 Fourier harmonics (M = 30)

Tables Icon

Table 8 Results for the grating described by Figure 3 with 220 layers (Nc = 220) for each trapezoidal element and 30 Fourier harmonics (M = 30)

Equations (38)

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

TE : ( x x + y y + k 2 ν 2 ( x , y ) ) E z = 0 ,
TM : ( x 1 ν 2 ( x , y ) x + y 1 ν 2 ( x , y ) y + k 2 ) H z = 0 .
ν j ( x ) = { n 1 , x [ d l j 2 , d + l j 2 ] , n 2 , x [ 0 , d l j 2 ] [ d + l j 2 , d ] .
F ( x , y ) = 1 k 2 y 2 F ( x , y ) ,
TE : = 1 k 2 x 2 + ν 2 ( x ) ,
TM : = 1 k 2 ν 2 ( x ) x 1 ν 2 ( x ) x + ν 2 ( x ) .
F ( x , y ) = ψ ( x ) e ± i k r y ,
ψ ( x ) = r 2 ψ ( x ) ,
ψ ( x + d ) = e i k α d ψ ( x ) .
Im ( r ) < 0 or r > 0 if r is real .
TE : < f , g > E = 1 d x 0 x 0 + d f ¯ ( x ) g ( x ) d x ,
TM : < f , g > H = 1 d x 0 x 0 + d 1 ν 2 ( x ) f ¯ ( x ) g ( x ) d x .
TE : = 1 k 2 d 2 d x 2 + ν 2 ¯ ( x ) ,
TM : = 1 k 2 ν 2 ¯ ( x ) d d x 1 ν 2 ¯ ( x ) d d x + ν 2 ¯ ( x ) ,
f ( x ) = m = 1 a m ψ m ( x ) , where a m = < ψ m , f > .
F ( x , y ) = m = 1 ( A m e i k r m y + B m e i k r m y ) ψ m ( x ) .
ψ m ( x ) = n = n = ψ m n e n ( x ) ,
e n ( x ) = e i k α n x , α n = sin θ + n λ d .
TE : L E = α α ,
TM : L H = ˜ 1 [ I α 1 α ] .
F ( x , y ) = m = 1 m = 2 N + 1 ( A m e i k r m y + B m e i k r m y ) ψ m ( x ) ,
ψ m ( x ) = n = N n = N ψ n m e n ( x ) .
( 0 ) ψ p ( 0 ) ( x ) = λ p ( 0 ) ψ p ( 0 ) ( x )
( 0 ) = 𝒫 = ζ 𝒫 ˜ , ζ 1
λ p = n ζ n λ p ( n ) , ψ p = n ζ n ψ p ( n ) .
( ( 0 ) + ζ 𝒫 ˜ ) n ζ n ψ p ( n ) = m , n ζ m + n λ p ( m ) ψ p ( n ) .
( 0 ) ψ p ( 0 ) = λ p ( 0 ) ψ p ( 0 ) ,
( 0 ) ψ p ( 1 ) + 𝒫 ˜ ψ p ( 0 ) = λ p ( 0 ) ψ p ( 1 ) + λ p ( 1 ) ψ p ( 0 ) ,
( 0 ) ψ p ( 2 ) + 𝒫 ˜ ψ p ( 1 ) = λ p ( 0 ) ψ p ( 2 ) + λ p ( 1 ) ψ p ( 1 ) + λ p ( 2 ) ψ p ( 0 ) .
< ψ p , ψ p > = 1 < ψ p ( 0 ) , ψ p > real .
< ψ p ( 0 ) , ψ p ( 0 ) > = 1 ,
< ψ p ( 0 ) , ψ p ( 1 ) > = < ψ p ( 1 ) , ψ p ( 0 ) > = 0 ,
< ψ p ( 0 ) , ψ p ( 2 ) > = < ψ p ( 2 ) , ψ p ( 0 ) > = 1 2 < ψ p ( 1 ) , ψ p ( 1 ) > .
λ p = λ p ( 0 ) + < ψ p ( 0 ) , 𝒫 ψ p ( 0 ) > ,
ψ p = ψ p ( 0 ) + Σ q p < ψ q ( 0 ) , 𝒫 ψ p ( 0 ) > λ p ( 0 ) λ q ( 0 ) .
λ p = λ p ( 0 ) + < ψ p ( 0 ) , 𝒫 ψ p ( 0 ) > + Σ q p < ψ q ( 0 ) , 𝒫 ψ p ( 0 ) > < ψ p ( 0 ) , 𝒫 ψ q ( 0 ) > λ p ( 0 ) λ q ( 0 )
ψ p = ψ p ( 0 ) + Σ q p < ψ q ( 0 ) , 𝒫 ψ p ( 0 ) > λ p ( 0 ) λ q ( 0 ) ψ q ( 0 ) + Σ q p Σ l p < ψ l ( 0 ) , 𝒫 ψ p ( 0 ) > < ψ q ( 0 ) , 𝒫 ψ l ( 0 ) > ( λ p ( 0 ) λ l ( 0 ) ) ( λ p 0 λ q ( 0 ) ) ψ q ( 0 ) Σ q p < ψ p ( 0 ) , 𝒫 ψ p ( 0 ) > < ψ q ( 0 ) , 𝒫 ψ p ( 0 ) > ( λ p ( 0 ) λ q ( 0 ) ) 2 ψ q ( 0 ) 1 2 Σ q p < ψ p ( 0 ) , 𝒫 ψ q ( 0 ) > < ψ q ( 0 ) , 𝒫 ψ p ( 0 ) > ( λ p ( 0 ) λ q ( 0 ) ) 2 ψ q ( 0 ) .
ξ ( n p ) = Int ( log 10 | eff RCWA eff PM ( n p ) | ) ,

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