Abstract

Numerical evidence is presented that shows that, for metallic lamellar gratings in TM polarization, the coupled-wave method formulated by Moharam and Gaylord [ J. Opt. Soc. Am. A 3, 1780 ( 1986)] converges slowly. (In some cases, for achieving a relative error of less than 1% in diffraction efficiencies, the number of spatial harmonics retained in the computation must be much greater than 100.) By classification of the modal methods for analyzing diffraction gratings into two distinct categories, the cause for the slow convergence is analyzed and attributed to the use of Fourier expansions to represent the permittivity and the electromagnetic fields in the grating region. The eigenvalues and the eigenfunctions of the modal fields in the grating region, whose accurate determination is crucial to the success of the coupled-wave method, are shown to converge slowly as a result of the use of these Fourier expansions. Despite its versatility and simplicity, the coupled-wave method should be used with caution for metallic surface-relief gratings in TM polarization.

© 1993 Optical Society of America

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References

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  1. M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,”J. Opt. Soc. Am. 72, 1385–1392 (1982).
    [CrossRef]
  2. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of grating diffraction—E-mode polarization and losses,”J. Opt. Soc. Am. 73, 451–455 (1983).
    [CrossRef]
  3. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of metallic surface-relief gratings,” J. Opt. Soc. Am. A 3, 1780–1787 (1986).
    [CrossRef]
  4. J. Yamakita, K. Rokushima, “Modal expansion method for dielectric gratings with rectangular grooves,” in Application, Theory, and Fabrication of Periodic Structures, Diffraction Gratings, and Moiré Phenomena II, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.503, 239–243 (1984).
    [CrossRef]
  5. T. Schimert, R. Magnusson, “Diffraction from metal strip gratings with high spatial frequency in the infrared spectral region,” J. Opt. Soc. Am. A 7, 1719–1722 (1990).
    [CrossRef]
  6. M. G. Moharam, T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar-grating diffraction,”J. Opt. Soc. Am. 73, 1105–1112 (1983).
    [CrossRef]
  7. M. G. Moharam, Center for Research in Electro-Optics and Lasers, University of Central Florida, Orlando, Fla. 32826 (personal communication, 1991).
  8. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
    [CrossRef]
  9. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
    [CrossRef]
  10. L. C. Botten, M. S. Craig, R. C. McPhedran, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103–1106 (1981).
    [CrossRef]
  11. J. Y. Suratteau, M. Cadilhac, R. Petit, “Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds,” J. Optics (Paris) 14, 273–288 (1983).
    [CrossRef]
  12. L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. (to be published).
  13. R. E. Collin, “Reflection and transmission at a slotted dielectric interface,” Can. J. Phys. 34, 398–411 (1956).
    [CrossRef]
  14. P. Sheng, R. S. Stepleman, P. N. Sanda, “Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907–2916 (1982).
    [CrossRef]
  15. S. T. Peng, “Rigorous formulation of scattering and guidance by dielectric grating waveguides: general case of oblique incidence,” J. Opt. Soc. Am. A 6, 1869–1883 (1989).
    [CrossRef]
  16. T. Tamir, H. C. Wang, A. A. Oliner, “Wave propagation in sinusoidally stratified dielectric media,” IEEE Trans. Microwave Theory Tech. MTT-12, 323–335 (1964).
    [CrossRef]
  17. R. S. Chu, J. A. Kong, “Modal theory of spatially periodic media,” IEEE Trans. Microwave Theory Tech. MTT-25, 18–24 (1977).
  18. C. B. Burckhardt, “Diffraction of a plane wave at a sinusoidally stratified dielectric grating,”J. Opt. Soc. Am. 56, 1502–1509 (1966).
    [CrossRef]
  19. K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,”J. Opt. Soc. Am. 68, 1206–1210 (1978).
    [CrossRef]

1990 (1)

1989 (1)

1986 (1)

1983 (3)

1982 (2)

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

P. Sheng, R. S. Stepleman, P. N. Sanda, “Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907–2916 (1982).
[CrossRef]

1981 (3)

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

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103–1106 (1981).
[CrossRef]

1978 (1)

1977 (1)

R. S. Chu, J. A. Kong, “Modal theory of spatially periodic media,” IEEE Trans. Microwave Theory Tech. MTT-25, 18–24 (1977).

1966 (1)

1964 (1)

T. Tamir, H. C. Wang, A. A. Oliner, “Wave propagation in sinusoidally stratified dielectric media,” IEEE Trans. Microwave Theory Tech. MTT-12, 323–335 (1964).
[CrossRef]

1956 (1)

R. E. Collin, “Reflection and transmission at a slotted dielectric interface,” Can. J. Phys. 34, 398–411 (1956).
[CrossRef]

Adams, J. L.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

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

Andrewartha, J. R.

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

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Botten, L. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

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

L. C. Botten, M. S. Craig, R. C. McPhedran, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103–1106 (1981).
[CrossRef]

Burckhardt, C. B.

Cadilhac, M.

J. Y. Suratteau, M. Cadilhac, R. Petit, “Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds,” J. Optics (Paris) 14, 273–288 (1983).
[CrossRef]

Chu, R. S.

R. S. Chu, J. A. Kong, “Modal theory of spatially periodic media,” IEEE Trans. Microwave Theory Tech. MTT-25, 18–24 (1977).

Collin, R. E.

R. E. Collin, “Reflection and transmission at a slotted dielectric interface,” Can. J. Phys. 34, 398–411 (1956).
[CrossRef]

Craig, M. S.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103–1106 (1981).
[CrossRef]

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

Gaylord, T. K.

Knop, K.

Kong, J. A.

R. S. Chu, J. A. Kong, “Modal theory of spatially periodic media,” IEEE Trans. Microwave Theory Tech. MTT-25, 18–24 (1977).

Li, L.

L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. (to be published).

Magnusson, R.

McPhedran, R. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103–1106 (1981).
[CrossRef]

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

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Moharam, M. G.

Oliner, A. A.

T. Tamir, H. C. Wang, A. A. Oliner, “Wave propagation in sinusoidally stratified dielectric media,” IEEE Trans. Microwave Theory Tech. MTT-12, 323–335 (1964).
[CrossRef]

Peng, S. T.

Petit, R.

J. Y. Suratteau, M. Cadilhac, R. Petit, “Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds,” J. Optics (Paris) 14, 273–288 (1983).
[CrossRef]

Rokushima, K.

J. Yamakita, K. Rokushima, “Modal expansion method for dielectric gratings with rectangular grooves,” in Application, Theory, and Fabrication of Periodic Structures, Diffraction Gratings, and Moiré Phenomena II, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.503, 239–243 (1984).
[CrossRef]

Sanda, P. N.

P. Sheng, R. S. Stepleman, P. N. Sanda, “Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907–2916 (1982).
[CrossRef]

Schimert, T.

Sheng, P.

P. Sheng, R. S. Stepleman, P. N. Sanda, “Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907–2916 (1982).
[CrossRef]

Stepleman, R. S.

P. Sheng, R. S. Stepleman, P. N. Sanda, “Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907–2916 (1982).
[CrossRef]

Suratteau, J. Y.

J. Y. Suratteau, M. Cadilhac, R. Petit, “Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds,” J. Optics (Paris) 14, 273–288 (1983).
[CrossRef]

Tamir, T.

T. Tamir, H. C. Wang, A. A. Oliner, “Wave propagation in sinusoidally stratified dielectric media,” IEEE Trans. Microwave Theory Tech. MTT-12, 323–335 (1964).
[CrossRef]

Wang, H. C.

T. Tamir, H. C. Wang, A. A. Oliner, “Wave propagation in sinusoidally stratified dielectric media,” IEEE Trans. Microwave Theory Tech. MTT-12, 323–335 (1964).
[CrossRef]

Yamakita, J.

J. Yamakita, K. Rokushima, “Modal expansion method for dielectric gratings with rectangular grooves,” in Application, Theory, and Fabrication of Periodic Structures, Diffraction Gratings, and Moiré Phenomena II, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.503, 239–243 (1984).
[CrossRef]

Can. J. Phys. (1)

R. E. Collin, “Reflection and transmission at a slotted dielectric interface,” Can. J. Phys. 34, 398–411 (1956).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

T. Tamir, H. C. Wang, A. A. Oliner, “Wave propagation in sinusoidally stratified dielectric media,” IEEE Trans. Microwave Theory Tech. MTT-12, 323–335 (1964).
[CrossRef]

R. S. Chu, J. A. Kong, “Modal theory of spatially periodic media,” IEEE Trans. Microwave Theory Tech. MTT-25, 18–24 (1977).

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (3)

J. Optics (Paris) (1)

J. Y. Suratteau, M. Cadilhac, R. Petit, “Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds,” J. Optics (Paris) 14, 273–288 (1983).
[CrossRef]

Opt. Acta (3)

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

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103–1106 (1981).
[CrossRef]

Phys. Rev. B (1)

P. Sheng, R. S. Stepleman, P. N. Sanda, “Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907–2916 (1982).
[CrossRef]

Other (3)

J. Yamakita, K. Rokushima, “Modal expansion method for dielectric gratings with rectangular grooves,” in Application, Theory, and Fabrication of Periodic Structures, Diffraction Gratings, and Moiré Phenomena II, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.503, 239–243 (1984).
[CrossRef]

M. G. Moharam, Center for Research in Electro-Optics and Lasers, University of Central Florida, Orlando, Fla. 32826 (personal communication, 1991).

L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. (to be published).

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Figures (6)

Fig. 1
Fig. 1

The grating geometry and the diffraction configuration used for studying convergence of the CWM. The values of the parameters are incident angle θ = 30°, wavelength λ = 1.0 μm, grating period d = 1.0 μm, groove depth h = 1.0 μm, and duty cycle d1/d = 0.5.

Fig. 2
Fig. 2

Convergence of diffraction efficiencies for TE polarization computed by the CWM (hollow symbols) and the MMME (filled symbols).

Fig. 3
Fig. 3

Convergence of (a) the negative first-order and (b) the zeroth-order diffraction efficiencies for TM polarization computed by the CWM (hollow symbols) and the MMME (filled symbols).

Fig. 4
Fig. 4

The original discontinuous permittivity function (a) is changed to a continuous one (b) when it is represented by a truncated Fourier expansion (15 = 2 × 7 + 1 terms are used).

Fig. 5
Fig. 5

Convergence of the tenth eigenvalue, −ρ10, for TE polarization. Circles represent the real part, triangles rrepresent the imaginary part, and dashed lines mark the exact values.

Fig. 6
Fig. 6

Convergence of the ninth eigenvalue, −ρ9, for TM polarization. Circles represent the real part, triangles represent the imaginary part, and dashed lines mark the exact values.

Tables (1)

Tables Icon

Table 1 Numerical Values of TM Diffraction Efficiencies of the Grating Shown in Fig. 1 Computed at Two Truncation Orders with the CWM and the MMME

Equations (15)

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

F ( x , y ) = m ( a m cos λ m y + b m sin λ m y ) u m ( x ) ,
L u m ( x ) = ρ m u m ( x ) , u m ( x + d ) = exp ( i α 0 d ) u m ( x ) .
ρ m = λ m 2 .
u m ( x ) = n c m n exp ( i α n x ) ,
α n = α 0 + 2 π n / d .
F ( x , y ) = n s n ( y ) exp ( i α n x ) ,
s n ( y ) = c n exp ( i λ y ) ,
A ν = ρ ν
B ξ = λ ξ ,
( x ) = { 1 x < d 1 / 2 2 d 1 / 2 < x < d / 2 .
n = O ( 1 / n ) .
U n = - d / 2 d / 2 u ( x ) exp ( - i α n x ) d x ,
U n = O ( 1 / n 3 ) ,             TE ,
U n = O ( 1 / n 2 ) ,             TM .
[ 1 d u d x ] + = [ 1 d u d x ] - ,             TM ,

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