Abstract

The rigorous character of the differential method for diffraction gratings for TM polarization was questioned by Depine and Simon [ J. Opt. Soc. Am. A 4, 834 ( 1987)]. They believe that they have found an argument that explains the mismatch of the numerical results found for deep metallic gratings from both integral and differential method, and they think that this argument enables them to say that the differential theory is not a rigorous electromagnetic method. We show here that the proof of their argument is not established.

© 1988 Optical Society of America

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References

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  1. R. A. Depine, J. M. Simon, “Comparison between the differential and integral methods used to solve the grating problem in the H∥case,” J. Opt. Soc. A 4, 834–838 (1987).
    [CrossRef]
  2. D. Maystre, “A new general integral theory for dielectric coated gratings,” J. Opt. Soc. Am. 68, 490–495 (1978).
    [CrossRef]
  3. D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), p. 63.
    [CrossRef]
  4. D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1984), Vol. XXI.
    [CrossRef]
  5. M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–67 (1974).
    [CrossRef]
  6. P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), p. 101.
    [CrossRef]
  7. M. Nevière, H. Akhouayri, P. Vincent, R. Reinisch, “Surface enhanced second harmonic generation in dielectric deposited over a silver grating,” in Application and Theory of Periodic Structures, Diffraction Gratings and Moiré Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.814, 256–350 (1987).
  8. M. Nevière, “Sur un formalisme différentiel pour les problèmes de diffraction dans le domaine de résonance: applications à l’étude des réseaux optiques et de diverses structures périodiques,” Thèse de Doctorat d’Etat n. A.O. 11556 (Université d’Aix Marseille III, Marseille, France, 1975).
  9. P. Vincent, “New improvement of the differential formalism for high modulated gratings,” in Applications and Theory of Periodic Structures, Diffraction Gratings and Moiré Phenomena I, C. H. Chi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.240, 142–149 (1980).
  10. G. Tayeb, Laboratoire d’Optique Electromagnétique, Unité Associée au Centre National de la Recherche Scientifique No. 843, Faculté des Sciences et Techniques, Centre de Saint-Jérôme, 13397 Marseille Cedex 13, France (personal communication, 1986).
  11. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finite conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
    [CrossRef]
  12. M. Nevière, M. Cadilhac, R. Petit, “Applications of conformal mappings to the diffraction of electromagnetic waves by a grating,” IEEE Trans. Antennas Propag. AP-21, 37–46 (1973).
    [CrossRef]

1987 (1)

R. A. Depine, J. M. Simon, “Comparison between the differential and integral methods used to solve the grating problem in the H∥case,” J. Opt. Soc. A 4, 834–838 (1987).
[CrossRef]

1981 (1)

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

1978 (1)

1974 (1)

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–67 (1974).
[CrossRef]

1973 (1)

M. Nevière, M. Cadilhac, R. Petit, “Applications of conformal mappings to the diffraction of electromagnetic waves by a grating,” IEEE Trans. Antennas Propag. AP-21, 37–46 (1973).
[CrossRef]

Adams, J. L.

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

Akhouayri, H.

M. Nevière, H. Akhouayri, P. Vincent, R. Reinisch, “Surface enhanced second harmonic generation in dielectric deposited over a silver grating,” in Application and Theory of Periodic Structures, Diffraction Gratings and Moiré Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.814, 256–350 (1987).

Andrewartha, J. R.

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

Cadilhac, M.

M. Nevière, M. Cadilhac, R. Petit, “Applications of conformal mappings to the diffraction of electromagnetic waves by a grating,” IEEE Trans. Antennas Propag. AP-21, 37–46 (1973).
[CrossRef]

Craig, M. S.

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

Depine, R. A.

R. A. Depine, J. M. Simon, “Comparison between the differential and integral methods used to solve the grating problem in the H∥case,” J. Opt. Soc. A 4, 834–838 (1987).
[CrossRef]

Maystre, D.

D. Maystre, “A new general integral theory for dielectric coated gratings,” J. Opt. Soc. Am. 68, 490–495 (1978).
[CrossRef]

D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), p. 63.
[CrossRef]

D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1984), Vol. XXI.
[CrossRef]

McPhedran, R. C.

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

Nevière, M.

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–67 (1974).
[CrossRef]

M. Nevière, M. Cadilhac, R. Petit, “Applications of conformal mappings to the diffraction of electromagnetic waves by a grating,” IEEE Trans. Antennas Propag. AP-21, 37–46 (1973).
[CrossRef]

M. Nevière, H. Akhouayri, P. Vincent, R. Reinisch, “Surface enhanced second harmonic generation in dielectric deposited over a silver grating,” in Application and Theory of Periodic Structures, Diffraction Gratings and Moiré Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.814, 256–350 (1987).

M. Nevière, “Sur un formalisme différentiel pour les problèmes de diffraction dans le domaine de résonance: applications à l’étude des réseaux optiques et de diverses structures périodiques,” Thèse de Doctorat d’Etat n. A.O. 11556 (Université d’Aix Marseille III, Marseille, France, 1975).

Petit, R.

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–67 (1974).
[CrossRef]

M. Nevière, M. Cadilhac, R. Petit, “Applications of conformal mappings to the diffraction of electromagnetic waves by a grating,” IEEE Trans. Antennas Propag. AP-21, 37–46 (1973).
[CrossRef]

Reinisch, R.

M. Nevière, H. Akhouayri, P. Vincent, R. Reinisch, “Surface enhanced second harmonic generation in dielectric deposited over a silver grating,” in Application and Theory of Periodic Structures, Diffraction Gratings and Moiré Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.814, 256–350 (1987).

Simon, J. M.

R. A. Depine, J. M. Simon, “Comparison between the differential and integral methods used to solve the grating problem in the H∥case,” J. Opt. Soc. A 4, 834–838 (1987).
[CrossRef]

Tayeb, G.

G. Tayeb, Laboratoire d’Optique Electromagnétique, Unité Associée au Centre National de la Recherche Scientifique No. 843, Faculté des Sciences et Techniques, Centre de Saint-Jérôme, 13397 Marseille Cedex 13, France (personal communication, 1986).

Vincent, P.

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–67 (1974).
[CrossRef]

P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), p. 101.
[CrossRef]

M. Nevière, H. Akhouayri, P. Vincent, R. Reinisch, “Surface enhanced second harmonic generation in dielectric deposited over a silver grating,” in Application and Theory of Periodic Structures, Diffraction Gratings and Moiré Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.814, 256–350 (1987).

P. Vincent, “New improvement of the differential formalism for high modulated gratings,” in Applications and Theory of Periodic Structures, Diffraction Gratings and Moiré Phenomena I, C. H. Chi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.240, 142–149 (1980).

IEEE Trans. Antennas Propag. (1)

M. Nevière, M. Cadilhac, R. Petit, “Applications of conformal mappings to the diffraction of electromagnetic waves by a grating,” IEEE Trans. Antennas Propag. AP-21, 37–46 (1973).
[CrossRef]

J. Opt. Soc. A (1)

R. A. Depine, J. M. Simon, “Comparison between the differential and integral methods used to solve the grating problem in the H∥case,” J. Opt. Soc. A 4, 834–838 (1987).
[CrossRef]

J. Opt. Soc. Am. (1)

Nouv. Rev. Opt. (1)

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–67 (1974).
[CrossRef]

Opt. Acta (1)

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

Other (7)

P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), p. 101.
[CrossRef]

M. Nevière, H. Akhouayri, P. Vincent, R. Reinisch, “Surface enhanced second harmonic generation in dielectric deposited over a silver grating,” in Application and Theory of Periodic Structures, Diffraction Gratings and Moiré Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.814, 256–350 (1987).

M. Nevière, “Sur un formalisme différentiel pour les problèmes de diffraction dans le domaine de résonance: applications à l’étude des réseaux optiques et de diverses structures périodiques,” Thèse de Doctorat d’Etat n. A.O. 11556 (Université d’Aix Marseille III, Marseille, France, 1975).

P. Vincent, “New improvement of the differential formalism for high modulated gratings,” in Applications and Theory of Periodic Structures, Diffraction Gratings and Moiré Phenomena I, C. H. Chi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.240, 142–149 (1980).

G. Tayeb, Laboratoire d’Optique Electromagnétique, Unité Associée au Centre National de la Recherche Scientifique No. 843, Faculté des Sciences et Techniques, Centre de Saint-Jérôme, 13397 Marseille Cedex 13, France (personal communication, 1986).

D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), p. 63.
[CrossRef]

D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1984), Vol. XXI.
[CrossRef]

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Figures (1)

Fig. 1
Fig. 1

Schematic representation of a symmetric triangular grating.

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Table 1 Grating Efficiencies Calculated by the Differential Method with 100 Integration Points and Increasing Number of Fourier Coefficientsa

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Y ( x , y ) = n = + Y n ( y ) exp ( i n x ) ,
Y 0 ( y ) = 1 y / h , Y n ( y ) = ( 1 ) n + 1 n π sin ( n π y h ) for n 0 ,
d E n d y = F n ( y ) ,
d F n ( y ) d y = m = N + N υ n m ( y ) E m ( y ) .

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