Abstract

The reflection efficiency of a lightly ruled glass grating, mainly in relation to its short-wavelength limit in the soft x-ray region, is discussed on the basis of the Rayleigh–Fano theory, which deals with Wood’s anomalies of optical gratings. The reflectances of waves diffracted by a glass grating are calculated under appropriately assumed conditions and the locations of the abrupt enhancement of reflection are derived. Theoretical predictions have been confirmed by a simulated experiment performed in the visible region.

© 1968 Optical Society of America

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References

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  1. G. Sprague, D. H. Tomboulian, and D. E. Bedo, J. Opt. Soc. Am. 45, 756 (1955); G. W. Stroke, in Encyclopedia of Physics29, S. Flügge Ed. (Springer-Verlag, Berlin, 1967), pp. 603–6.
    [Crossref]
  2. J. C. Miller, J. Opt. Soc. Am. 54, 353 (1964).
    [Crossref]
  3. P. Jaeglé, Compt. Rend. 259, 533, 4556 (1964); Thèses, Contribution à la Spectrographie dans l’Ultraviolet de la Région de Holweck (Université de Paris, 1965).
  4. A. P. Lukirskii and E. P. Savinov, Opt. Spektrosk. [Opt. Spektrosc.] 14, 147 (1963).
  5. T. Sagawa, Sci. Rept. Tôhoku Univ. (I)46, 119 (1962).
  6. T. Sagawa, Y. Iguchi, M. Sasanuma, A. Ejiri, S. Fujiwara, M. Yokota, S. Yamaguchi, M. Nakamura, T. Sasaki, and T. Oshio, J. Phys. Soc. Japan 21, 2602 (1966).
    [Crossref]
  7. Rayleigh, Proc. Roy. Soc. (London) A79, 399 (1907).
  8. U. Fano, Ann. Phys. 32, 393 (1938).
    [Crossref]
  9. R. W. Wood, Phil. Mag. 4, 396 (1902); Phys. Rev. 48, 928 (1935).
  10. A. Hessel and A. A. Oliner, Appl. Opt. 4, 1275 (1965).
    [Crossref]
  11. S. Fujiwara and Y. Iguchi, J. Opt. Soc. Am. 58, 361 (1968).
    [Crossref]
  12. In our expressions of boundary conditions, we assume that the whole grating surface is illuminated and that no part is shadowed by another part.
  13. The P and S components correspond to the components of the incident light with its electric vector parallel and perpendicular to the grooves, respectively.
  14. The spectral reflectances for the P component are almost the same as those of the S component, because the refractive index of glass is nearly equal to one in the soft x-ray region.
  15. G. Toraldo di Francia, La diffrazione della Luce (Edizioni Scientifiche Einaudi, Torino, Italia, 1958), p. 164; also, in La Théorie des Images Optiques, P. Fleury, A. Maréchal, and C. Anglande, Eds. (Éditions de la Revue d’Optique, Paris, 1949), p. 205.
  16. When a beam of soft x rays is incident from vacuum side on a glass grating, the total reflection phenomenon can occur, since the refractive index of glass is less than one in the soft x-ray region.
  17. The reflectances are normalized at the second enhancing angle; that is, at ϑc,0 for n>0 and ϑc,n for n<0.

1968 (1)

1966 (1)

T. Sagawa, Y. Iguchi, M. Sasanuma, A. Ejiri, S. Fujiwara, M. Yokota, S. Yamaguchi, M. Nakamura, T. Sasaki, and T. Oshio, J. Phys. Soc. Japan 21, 2602 (1966).
[Crossref]

1965 (1)

1964 (2)

J. C. Miller, J. Opt. Soc. Am. 54, 353 (1964).
[Crossref]

P. Jaeglé, Compt. Rend. 259, 533, 4556 (1964); Thèses, Contribution à la Spectrographie dans l’Ultraviolet de la Région de Holweck (Université de Paris, 1965).

1963 (1)

A. P. Lukirskii and E. P. Savinov, Opt. Spektrosk. [Opt. Spektrosc.] 14, 147 (1963).

1955 (1)

1938 (1)

U. Fano, Ann. Phys. 32, 393 (1938).
[Crossref]

1907 (1)

Rayleigh, Proc. Roy. Soc. (London) A79, 399 (1907).

1902 (1)

R. W. Wood, Phil. Mag. 4, 396 (1902); Phys. Rev. 48, 928 (1935).

Bedo, D. E.

Ejiri, A.

T. Sagawa, Y. Iguchi, M. Sasanuma, A. Ejiri, S. Fujiwara, M. Yokota, S. Yamaguchi, M. Nakamura, T. Sasaki, and T. Oshio, J. Phys. Soc. Japan 21, 2602 (1966).
[Crossref]

Fano, U.

U. Fano, Ann. Phys. 32, 393 (1938).
[Crossref]

Fujiwara, S.

S. Fujiwara and Y. Iguchi, J. Opt. Soc. Am. 58, 361 (1968).
[Crossref]

T. Sagawa, Y. Iguchi, M. Sasanuma, A. Ejiri, S. Fujiwara, M. Yokota, S. Yamaguchi, M. Nakamura, T. Sasaki, and T. Oshio, J. Phys. Soc. Japan 21, 2602 (1966).
[Crossref]

Hessel, A.

Iguchi, Y.

S. Fujiwara and Y. Iguchi, J. Opt. Soc. Am. 58, 361 (1968).
[Crossref]

T. Sagawa, Y. Iguchi, M. Sasanuma, A. Ejiri, S. Fujiwara, M. Yokota, S. Yamaguchi, M. Nakamura, T. Sasaki, and T. Oshio, J. Phys. Soc. Japan 21, 2602 (1966).
[Crossref]

Jaeglé, P.

P. Jaeglé, Compt. Rend. 259, 533, 4556 (1964); Thèses, Contribution à la Spectrographie dans l’Ultraviolet de la Région de Holweck (Université de Paris, 1965).

Lukirskii, A. P.

A. P. Lukirskii and E. P. Savinov, Opt. Spektrosk. [Opt. Spektrosc.] 14, 147 (1963).

Miller, J. C.

Nakamura, M.

T. Sagawa, Y. Iguchi, M. Sasanuma, A. Ejiri, S. Fujiwara, M. Yokota, S. Yamaguchi, M. Nakamura, T. Sasaki, and T. Oshio, J. Phys. Soc. Japan 21, 2602 (1966).
[Crossref]

Oliner, A. A.

Oshio, T.

T. Sagawa, Y. Iguchi, M. Sasanuma, A. Ejiri, S. Fujiwara, M. Yokota, S. Yamaguchi, M. Nakamura, T. Sasaki, and T. Oshio, J. Phys. Soc. Japan 21, 2602 (1966).
[Crossref]

Rayleigh,

Rayleigh, Proc. Roy. Soc. (London) A79, 399 (1907).

Sagawa, T.

T. Sagawa, Y. Iguchi, M. Sasanuma, A. Ejiri, S. Fujiwara, M. Yokota, S. Yamaguchi, M. Nakamura, T. Sasaki, and T. Oshio, J. Phys. Soc. Japan 21, 2602 (1966).
[Crossref]

T. Sagawa, Sci. Rept. Tôhoku Univ. (I)46, 119 (1962).

Sasaki, T.

T. Sagawa, Y. Iguchi, M. Sasanuma, A. Ejiri, S. Fujiwara, M. Yokota, S. Yamaguchi, M. Nakamura, T. Sasaki, and T. Oshio, J. Phys. Soc. Japan 21, 2602 (1966).
[Crossref]

Sasanuma, M.

T. Sagawa, Y. Iguchi, M. Sasanuma, A. Ejiri, S. Fujiwara, M. Yokota, S. Yamaguchi, M. Nakamura, T. Sasaki, and T. Oshio, J. Phys. Soc. Japan 21, 2602 (1966).
[Crossref]

Savinov, E. P.

A. P. Lukirskii and E. P. Savinov, Opt. Spektrosk. [Opt. Spektrosc.] 14, 147 (1963).

Sprague, G.

Tomboulian, D. H.

Toraldo di Francia, G.

G. Toraldo di Francia, La diffrazione della Luce (Edizioni Scientifiche Einaudi, Torino, Italia, 1958), p. 164; also, in La Théorie des Images Optiques, P. Fleury, A. Maréchal, and C. Anglande, Eds. (Éditions de la Revue d’Optique, Paris, 1949), p. 205.

Wood, R. W.

R. W. Wood, Phil. Mag. 4, 396 (1902); Phys. Rev. 48, 928 (1935).

Yamaguchi, S.

T. Sagawa, Y. Iguchi, M. Sasanuma, A. Ejiri, S. Fujiwara, M. Yokota, S. Yamaguchi, M. Nakamura, T. Sasaki, and T. Oshio, J. Phys. Soc. Japan 21, 2602 (1966).
[Crossref]

Yokota, M.

T. Sagawa, Y. Iguchi, M. Sasanuma, A. Ejiri, S. Fujiwara, M. Yokota, S. Yamaguchi, M. Nakamura, T. Sasaki, and T. Oshio, J. Phys. Soc. Japan 21, 2602 (1966).
[Crossref]

Ann. Phys. (1)

U. Fano, Ann. Phys. 32, 393 (1938).
[Crossref]

Appl. Opt. (1)

Compt. Rend. (1)

P. Jaeglé, Compt. Rend. 259, 533, 4556 (1964); Thèses, Contribution à la Spectrographie dans l’Ultraviolet de la Région de Holweck (Université de Paris, 1965).

J. Opt. Soc. Am. (3)

J. Phys. Soc. Japan (1)

T. Sagawa, Y. Iguchi, M. Sasanuma, A. Ejiri, S. Fujiwara, M. Yokota, S. Yamaguchi, M. Nakamura, T. Sasaki, and T. Oshio, J. Phys. Soc. Japan 21, 2602 (1966).
[Crossref]

Opt. Spektrosk. [Opt. Spektrosc.] (1)

A. P. Lukirskii and E. P. Savinov, Opt. Spektrosk. [Opt. Spektrosc.] 14, 147 (1963).

Phil. Mag. (1)

R. W. Wood, Phil. Mag. 4, 396 (1902); Phys. Rev. 48, 928 (1935).

Proc. Roy. Soc. (London) (1)

Rayleigh, Proc. Roy. Soc. (London) A79, 399 (1907).

Other (7)

T. Sagawa, Sci. Rept. Tôhoku Univ. (I)46, 119 (1962).

In our expressions of boundary conditions, we assume that the whole grating surface is illuminated and that no part is shadowed by another part.

The P and S components correspond to the components of the incident light with its electric vector parallel and perpendicular to the grooves, respectively.

The spectral reflectances for the P component are almost the same as those of the S component, because the refractive index of glass is nearly equal to one in the soft x-ray region.

G. Toraldo di Francia, La diffrazione della Luce (Edizioni Scientifiche Einaudi, Torino, Italia, 1958), p. 164; also, in La Théorie des Images Optiques, P. Fleury, A. Maréchal, and C. Anglande, Eds. (Éditions de la Revue d’Optique, Paris, 1949), p. 205.

When a beam of soft x rays is incident from vacuum side on a glass grating, the total reflection phenomenon can occur, since the refractive index of glass is less than one in the soft x-ray region.

The reflectances are normalized at the second enhancing angle; that is, at ϑc,0 for n>0 and ϑc,n for n<0.

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Figures (5)

Fig. 1a
Fig. 1a

Designation of spectral orders for various diffracted waves and of angles measured from the grating surface.

Fig. 1b
Fig. 1b

Profile of the rectangular grating with various parameters.

Fig. 2
Fig. 2

Curves showing the calculated spectral reflectances for the S component of waves diffracted by the lightly ruled glass grating (grating constant a = 1 μ, groove width b = 0.2 μ, groove depth δ = 25 Å, refractive index η = 1 − 2.70 × 10−6λ2, angle of grazing incidence ϑ = 7°).

Fig. 3
Fig. 3

Diagram showing the principle of the experiment. The glass grating G is cemented to the glass prism P.

Fig. 4
Fig. 4

Curves showing the calculated (——) and measured (—●—) reflectances for both P and S components of waves diffracted by the glass grating, as functions of the angle of grazing incidence [grating constant a≈1/180 mm, groove width b≈1/540 mm, refractive index η = 1.768 (for λ = 0.546 μ)].

Tables (1)

Tables Icon

Table I Values of the critical wavelength λc,n for a glass grating (grating constant a = 1 μ, angle of grazing incidence ϑ = 7°, refractive index η = 1 − 2.70 × 10−6λ2).

Equations (6)

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B n = B n ( 0 ) + σ B n ( 1 ) + σ 2 B n ( 2 ) + .
{ B 0 ( 0 ) = ( sin ϑ - g sin ξ 0 ) / D 0             B n ( 0 ) = 0 ( n 0 ) , B 0 ( 1 ) = 0 ,             B n ( 1 ) = - 2 i ζ n sin ϑ ( P n / D 0 D n ) ( n 0 ) , B n ( 2 ) = - ( sin ϑ / D 0 D n ) × [ g ( 1 - η 2 ) ( sin ξ 0 - sin ξ n ) m ζ m ζ n - m + m ( 2 ζ m ζ n - m / D m ) { sin ψ m ( sin ψ m + g sin ξ n ) P m - g η sin ξ m ( sin ξ m - sin ξ n ) Q m } ] ,
ζ n = - sin ( π n b / a ) / ( π n b / a ) , { D 0 = sin ϑ + g sin ξ 0 D n = sin ψ n + g sin ξ n ,
{ P n = ( 1 - g η ) + ( g - η ) [ ( 1 / η ) cos 2 ϑ - g sin ξ 0 sin ξ n ] Q n = ( 1 - g η ) + ( g - η ) [ ( 1 / η ) cos 2 ϑ + sin ξ 0 sin ψ n ] .
R n = ( sin ψ n / sin ϑ ) B n 2 .
cos ψ n = η cos ξ n = cos ϑ + ( n λ / a ) ,