Abstract

The reflection efficiencies for the various orders of diffraction produced by a grating are calculated under the following assumptions. The grating is for the most part flat, but with a fraction of the periodicity interval a described by a profile function z=f(x). The reflection and transmission efficiencies are calculated as a series expansion in . Numerical calculations have been performed for the case of ultrasoft x rays incident at near grazing angles, using a complex index of refraction η=1−δ0λ2+0λ3.25 or tabulated values. No effects of shadowing by the grating profile have been considered, and the numerical work has been done for a symmetrical triangular trench incised in the flat surface.

© 1964 Optical Society of America

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References

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  1. J. W. Strutt ( Rayleigh), The Theory of Sound (Dover Publications, Inc., New York, 1896), Vol. II, Sec. 272a.
  2. See, for example: W. C. Meecham, J. Appl. Phys. 27, 361 (1956), or I. Abubakar, Proc. Cambridge Phil. Soc. 59, 231 (1963). Compare also: G. Sprague, D. H. Tomboulian, and D. E. Bedo, J. Opt. Soc. Am. 45, 756 (1955). The results of this last paper, when applied to a situation with the parameters of our surface, are at considerable variance with the results obtained herein.
    [Crossref]
  3. C. Eckart, Phys. Rev. 44, 12 (1933).
    [Crossref]
  4. B. A. Lippman, J. Opt. Soc. Am. 43, 408 (1953).
    [Crossref]
  5. B. L. Henke and J. C. Miller, “Ultrasoft X-Ray Interaction Coefficients,” , August1959.
  6. A. P. Lukiriskii and E. P. Savinov, Opt. Spectry. 14, 147 (1963) [Opt. i Spektroskopiya 14, 285 (1963)].
  7. See, for example, the gratings reported upon by H. A. Kirkpatrick, J. Quant. Spectry. Radiative Transfer2, 715–724 (1962). A private communication from this author also suggests that the values quoted are within the bounds of actual grating parameters.
    [Crossref]

1963 (1)

A. P. Lukiriskii and E. P. Savinov, Opt. Spectry. 14, 147 (1963) [Opt. i Spektroskopiya 14, 285 (1963)].

1956 (1)

See, for example: W. C. Meecham, J. Appl. Phys. 27, 361 (1956), or I. Abubakar, Proc. Cambridge Phil. Soc. 59, 231 (1963). Compare also: G. Sprague, D. H. Tomboulian, and D. E. Bedo, J. Opt. Soc. Am. 45, 756 (1955). The results of this last paper, when applied to a situation with the parameters of our surface, are at considerable variance with the results obtained herein.
[Crossref]

1953 (1)

1933 (1)

C. Eckart, Phys. Rev. 44, 12 (1933).
[Crossref]

Eckart, C.

C. Eckart, Phys. Rev. 44, 12 (1933).
[Crossref]

Henke, B. L.

B. L. Henke and J. C. Miller, “Ultrasoft X-Ray Interaction Coefficients,” , August1959.

Kirkpatrick, H. A.

See, for example, the gratings reported upon by H. A. Kirkpatrick, J. Quant. Spectry. Radiative Transfer2, 715–724 (1962). A private communication from this author also suggests that the values quoted are within the bounds of actual grating parameters.
[Crossref]

Lippman, B. A.

Lukiriskii, A. P.

A. P. Lukiriskii and E. P. Savinov, Opt. Spectry. 14, 147 (1963) [Opt. i Spektroskopiya 14, 285 (1963)].

Meecham, W. C.

See, for example: W. C. Meecham, J. Appl. Phys. 27, 361 (1956), or I. Abubakar, Proc. Cambridge Phil. Soc. 59, 231 (1963). Compare also: G. Sprague, D. H. Tomboulian, and D. E. Bedo, J. Opt. Soc. Am. 45, 756 (1955). The results of this last paper, when applied to a situation with the parameters of our surface, are at considerable variance with the results obtained herein.
[Crossref]

Miller, J. C.

B. L. Henke and J. C. Miller, “Ultrasoft X-Ray Interaction Coefficients,” , August1959.

Rayleigh,

J. W. Strutt ( Rayleigh), The Theory of Sound (Dover Publications, Inc., New York, 1896), Vol. II, Sec. 272a.

Savinov, E. P.

A. P. Lukiriskii and E. P. Savinov, Opt. Spectry. 14, 147 (1963) [Opt. i Spektroskopiya 14, 285 (1963)].

Strutt, J. W.

J. W. Strutt ( Rayleigh), The Theory of Sound (Dover Publications, Inc., New York, 1896), Vol. II, Sec. 272a.

J. Appl. Phys. (1)

See, for example: W. C. Meecham, J. Appl. Phys. 27, 361 (1956), or I. Abubakar, Proc. Cambridge Phil. Soc. 59, 231 (1963). Compare also: G. Sprague, D. H. Tomboulian, and D. E. Bedo, J. Opt. Soc. Am. 45, 756 (1955). The results of this last paper, when applied to a situation with the parameters of our surface, are at considerable variance with the results obtained herein.
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Spectry. (1)

A. P. Lukiriskii and E. P. Savinov, Opt. Spectry. 14, 147 (1963) [Opt. i Spektroskopiya 14, 285 (1963)].

Phys. Rev. (1)

C. Eckart, Phys. Rev. 44, 12 (1933).
[Crossref]

Other (3)

B. L. Henke and J. C. Miller, “Ultrasoft X-Ray Interaction Coefficients,” , August1959.

See, for example, the gratings reported upon by H. A. Kirkpatrick, J. Quant. Spectry. Radiative Transfer2, 715–724 (1962). A private communication from this author also suggests that the values quoted are within the bounds of actual grating parameters.
[Crossref]

J. W. Strutt ( Rayleigh), The Theory of Sound (Dover Publications, Inc., New York, 1896), Vol. II, Sec. 272a.

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Figures (5)

Fig. 1
Fig. 1

Grating profile in the general case showing the various parameters used. Material with complex index of refraction η occupies the region z<f(x).

Fig. 2
Fig. 2

The structure of the various diffracted orders with angles considerably exaggerated for purposes of clarity.

Fig. 3
Fig. 3

Grating profile for detailed calculations of Bn(1) with silicon dioxide as the grating material.

Fig. 4
Fig. 4

First- and second-order reflection efficiencies in percent versus wavelength in angstroms for the silicon dioxide grating with profile as shown in Fig. 3. The experimental points shown are taken from Ref. 6.

Fig. 5
Fig. 5

First- and second-order reflection efficiencies in percent versus incidence angle in degrees for the SiO2 grating with profile as shown in Fig. 3. The values shown are for fixed wavelength of 40 Å.

Equations (31)

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( 2 u / x 2 ) + ( 2 u / z 2 ) + k 2 u = 0 ,
u ( i ) ( x , z ) = exp [ i ( ω / c ) ( x cos ϑ - z sin ϑ ) ] .
u ( r ) ( x , z ) = n B n exp [ i ( ω / c ) ( x cos ψ n + z sin ψ n ) ] ,
u ( t ) ( x , z ) = C m exp [ i ( η ω / c ) ( x cos ξ n - z sin ξ n ) ] ,
u ( i ) [ x , f ( x ) ] + u ( r ) [ x , f ( x ) ] = u ( t ) [ x , f ( x ) ] .
cos ψ n = η cos ξ n = cos ϑ + n λ / a ,
H tan = ( i c / ω μ ) ( E y / ν ) ,
E tan = ( i μ c / ω η 2 ) ( H y / ν ) ,
ϕ ( i ) ( x ) = exp [ - i ( ω / c ) f ( x ) sin ϑ ] ,
E n ( r ) ( x ) = exp [ i 2 π n x / a + i ( ω / c ) f ( x ) sin ψ n ] ,
E n ( t ) ( x ) = exp [ i 2 π n x / a - i ( η ω / c ) f ( x ) sin ξ n ] .
ϕ ( i ) ( x ) + n B n E n ( r ) ( x ) - n C n E n ( t ) ( x ) = 0 ,
ϕ ( i ) ( x ) sin ( ϑ - ζ ) - n B n E n ( r ) ( x ) sin ( ψ n + ζ ) - n g C n E n ( t ) ( x ) sin ( ξ n - ζ ) = 0 ,
f ( x ) = - ( x + a - w 2 ) tan γ :             - a - w 2 < x < 0 , f ( x ) = ( x - a - w 2 ) tan γ :             0 < x < a - w 2 , f ( x ) = 0 :             ( a - w ) / 2 < x < ( a + w ) / 2 ,
D n 0 ( 0 , sin ϑ ) + n D n n ( 0 , - sin ψ n ) B n = n D n n ( 0 , η sin ξ n ) C n ,
D n n ( γ , Γ ) = δ n n + [ L n n ( Γ ) cos γ - n n ] ,
n n = 1 π ( n - n ) sin π ( n - n )
L n n ( Γ ) = 0 1 d z exp [ - i ( 2 π Γ / λ ) f ( a z / 2 ) ] cos π ( n - n ) z .
D n 0 ( γ , sin ϑ ) sin ϑ - n D n n ( γ , - sin ψ n ) B n sin ψ n = n g D n n ( γ , η sin ξ n ) C n sin ξ n .
M n n ( Γ ) = i sin γ 0 1 d z × exp [ - i ( 2 π Γ / λ ) f ( a z / 2 ) ] sin π ( n - n ) z
B n = sin ϑ - g sin ξ 0 sin ϑ + g sin ξ 0 δ n 0 + B n ( 1 ) , C n = 2 sin ϑ sin ϑ + g sin ξ 0 δ n 0 + C n ( 1 ) .
B n ( 1 ) [ sin ψ n + g sin ξ n ] = ( sin ϑ cos γ - g sin ξ n ) L n 0 ( sin ϑ ) - sin ϑ - g sin ξ 0 sin ϑ + g sin ξ 0 ( sin ϑ cos γ - g sin ξ n ) L n 0 ( - sin ϑ ) - 2 sin ϑ sin ϑ + g sin ξ 0 ( g sin ξ 0 cos γ - g sin ξ n ) × L n 0 ( η sin ξ 0 ) .
sin ψ n = { 1 - [ cos ϑ + n ( λ / a ) ] 2 } 1 2
sin ψ n = i { [ cos ϑ + n ( λ / a ) ] 2 - 1 } 1 2
g sin ξ n = { g 2 - [ cos ϑ + n ( λ / a ) ] 2 } 1 2 .
L n 0 ( Γ ) = 0 1 d z e i α z Γ cos π n ( 1 - z ) ,
L n 0 ( Γ ) = ( 1 / i α Γ ) ( e i α Γ - 1 ) .
r u ( n ) = ( 2 / 2 ) [ B ˆ n ( 1 ) 2 + B ˜ n ( 1 ) 2 ] ( sin ψ n / sin ϑ ) ,
r T E ( n ) = 2 B ˆ n ( 1 ) 2 ( sin ψ n / sin ϑ ) ;             r T H ( n ) = 2 B ˜ n ( 1 ) 2 ( sin ψ n / sin ϑ ) .
η sin ξ n = { ( 1 - δ ) 2 - β 2 - [ cos ϑ + n ( λ / a ) ] 2 + i 2 β ( 1 - δ ) } 1 2 ,
( 1 - δ ) 2 - β 2 - [ cos ϑ + n ( λ / a ) ] 2 = 0.