Abstract

Recent developments in thin film technology have made possible the construction of multilayered thin film structures that act as efficient Bragg diffractors for x rays and extreme ultraviolet (EUV) radiation. These structures (which we term layered synthetic microstructures or LSMs) are analogous to multilayer interference filters for the visible spectral region and have important potential applications in many areas of x-ray/EUV instrumentation. In this paper the theory of x-ray diffraction by periodic structures is applied to LSMs, and approximate formulas for estimating their performance are presented. A more complete computation scheme based on optical multilayer theory is described, and it is shown that, by adjusting the refractive indices and thicknesses of the component layers, the diffracting properties may be tailored to specific applications. Finally, it is shown how the theory may be modified to take account of imperfections in the LSM structure and to compute the properties of nonperiodic structures.

© 1981 Optical Society of America

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References

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  1. H. Koeppe, Dissertation, Giessen (1929).
  2. W. Deubner, Ann. Phys. Leipzig 5, 261 (1930).
    [CrossRef]
  3. J. DuMond, J. P. Youtz, J. Appl. Phys. 11, 357 (1940).
    [CrossRef]
  4. J. Dinklage, R. Frerichs, J. Appl. Phys. 34, 2633 (1963).
    [CrossRef]
  5. J. Dinklage, J. Appl. Phys. 38, 3781 (1967).
    [CrossRef]
  6. E. Spiller, Appl. Phys. Lett. 20, 365 (1972).
    [CrossRef]
  7. E. Spiller, in Space Optics, Proceedings, ICO-IX, Santa Monica, 1972 (National Academy of Sciences, Washington, D.C., 1974), p. 525.
  8. E. Spiller, Appl. Opt. 15, 2333 (1976).
    [CrossRef] [PubMed]
  9. R.-P. Haelbich, C. Kunz, Opt. Commun. 17, 287 (1976).
    [CrossRef]
  10. T. W. Barbee, D. C. Keith, in “Workshop on Instrumentation for Synchrotron Radiation Research,” H. Winick, G. Brown, Eds, Stanford Synchrotron Radiation Laboratory Report 78/04 (May1978), p. III-36.
  11. T. W. Barbee, in Proceedings, Topical Conference on Low Energy X-ray Diagnostics, Monterey, Calif., 8–10 June 1980 (American Institute of Physics, New York, 1980).
  12. J. H. Underwood, T. W. Barbee, D. C. Keith, Proc. Soc. Photo-Opt. Instrum. Eng. 184, 123 (1979).
  13. W. H. Zachariasen, Theory of X-Ray Diffraction in Crystals (Wiley, New York, 1945).
  14. A. M. Saxena, B. P. Schoenborn, Acta. Crystallogr. Sect. A: 33, 805 (1977).
    [CrossRef]
  15. Note that Eqs. (8) and (10) imply that a correction for refraction must be applied to the Bragg Eq. (1). The diffraction pattern is symmetrical about y = 0 or θ=θmp, where θmp=θm+[2δ0/(sin2θ)].
  16. Based on a mean diffraction pattern (Ref. 13, p. 126ff.).
  17. L. G. Parratt, Phys. Rev. 95, 35A (1954).
    [CrossRef]
  18. A. V. Vinogradov, B. Ya. Zeldovich, Appl. Opt. 16, 89 (1977).
    [CrossRef] [PubMed]
  19. B. L. Henke, M. A. Tester, Adv. X-Ray Anal. 18, 76 (1975).
    [CrossRef]
  20. A. J. Burek, Space Sci. Instrum. 3, 53 (1976).
  21. D. L. McKenzie, P. B. Landecker, J. H. Underwood, Space Sci. Instrum. 2, 125 (1976).
  22. M. W. Charles, J. Appl. Phys. 42, 3329 (1971).
    [CrossRef]
  23. R. A. Simpson, IEEE Trans. Antennas Propag. AP-24, 17 (1976).
    [CrossRef]

1979 (1)

J. H. Underwood, T. W. Barbee, D. C. Keith, Proc. Soc. Photo-Opt. Instrum. Eng. 184, 123 (1979).

1977 (2)

A. M. Saxena, B. P. Schoenborn, Acta. Crystallogr. Sect. A: 33, 805 (1977).
[CrossRef]

A. V. Vinogradov, B. Ya. Zeldovich, Appl. Opt. 16, 89 (1977).
[CrossRef] [PubMed]

1976 (5)

A. J. Burek, Space Sci. Instrum. 3, 53 (1976).

D. L. McKenzie, P. B. Landecker, J. H. Underwood, Space Sci. Instrum. 2, 125 (1976).

R. A. Simpson, IEEE Trans. Antennas Propag. AP-24, 17 (1976).
[CrossRef]

E. Spiller, Appl. Opt. 15, 2333 (1976).
[CrossRef] [PubMed]

R.-P. Haelbich, C. Kunz, Opt. Commun. 17, 287 (1976).
[CrossRef]

1975 (1)

B. L. Henke, M. A. Tester, Adv. X-Ray Anal. 18, 76 (1975).
[CrossRef]

1972 (1)

E. Spiller, Appl. Phys. Lett. 20, 365 (1972).
[CrossRef]

1971 (1)

M. W. Charles, J. Appl. Phys. 42, 3329 (1971).
[CrossRef]

1967 (1)

J. Dinklage, J. Appl. Phys. 38, 3781 (1967).
[CrossRef]

1963 (1)

J. Dinklage, R. Frerichs, J. Appl. Phys. 34, 2633 (1963).
[CrossRef]

1954 (1)

L. G. Parratt, Phys. Rev. 95, 35A (1954).
[CrossRef]

1940 (1)

J. DuMond, J. P. Youtz, J. Appl. Phys. 11, 357 (1940).
[CrossRef]

1930 (1)

W. Deubner, Ann. Phys. Leipzig 5, 261 (1930).
[CrossRef]

Barbee, T. W.

J. H. Underwood, T. W. Barbee, D. C. Keith, Proc. Soc. Photo-Opt. Instrum. Eng. 184, 123 (1979).

T. W. Barbee, D. C. Keith, in “Workshop on Instrumentation for Synchrotron Radiation Research,” H. Winick, G. Brown, Eds, Stanford Synchrotron Radiation Laboratory Report 78/04 (May1978), p. III-36.

T. W. Barbee, in Proceedings, Topical Conference on Low Energy X-ray Diagnostics, Monterey, Calif., 8–10 June 1980 (American Institute of Physics, New York, 1980).

Burek, A. J.

A. J. Burek, Space Sci. Instrum. 3, 53 (1976).

Charles, M. W.

M. W. Charles, J. Appl. Phys. 42, 3329 (1971).
[CrossRef]

Deubner, W.

W. Deubner, Ann. Phys. Leipzig 5, 261 (1930).
[CrossRef]

Dinklage, J.

J. Dinklage, J. Appl. Phys. 38, 3781 (1967).
[CrossRef]

J. Dinklage, R. Frerichs, J. Appl. Phys. 34, 2633 (1963).
[CrossRef]

DuMond, J.

J. DuMond, J. P. Youtz, J. Appl. Phys. 11, 357 (1940).
[CrossRef]

Frerichs, R.

J. Dinklage, R. Frerichs, J. Appl. Phys. 34, 2633 (1963).
[CrossRef]

Haelbich, R.-P.

R.-P. Haelbich, C. Kunz, Opt. Commun. 17, 287 (1976).
[CrossRef]

Henke, B. L.

B. L. Henke, M. A. Tester, Adv. X-Ray Anal. 18, 76 (1975).
[CrossRef]

Keith, D. C.

J. H. Underwood, T. W. Barbee, D. C. Keith, Proc. Soc. Photo-Opt. Instrum. Eng. 184, 123 (1979).

T. W. Barbee, D. C. Keith, in “Workshop on Instrumentation for Synchrotron Radiation Research,” H. Winick, G. Brown, Eds, Stanford Synchrotron Radiation Laboratory Report 78/04 (May1978), p. III-36.

Koeppe, H.

H. Koeppe, Dissertation, Giessen (1929).

Kunz, C.

R.-P. Haelbich, C. Kunz, Opt. Commun. 17, 287 (1976).
[CrossRef]

Landecker, P. B.

D. L. McKenzie, P. B. Landecker, J. H. Underwood, Space Sci. Instrum. 2, 125 (1976).

McKenzie, D. L.

D. L. McKenzie, P. B. Landecker, J. H. Underwood, Space Sci. Instrum. 2, 125 (1976).

Parratt, L. G.

L. G. Parratt, Phys. Rev. 95, 35A (1954).
[CrossRef]

Saxena, A. M.

A. M. Saxena, B. P. Schoenborn, Acta. Crystallogr. Sect. A: 33, 805 (1977).
[CrossRef]

Schoenborn, B. P.

A. M. Saxena, B. P. Schoenborn, Acta. Crystallogr. Sect. A: 33, 805 (1977).
[CrossRef]

Simpson, R. A.

R. A. Simpson, IEEE Trans. Antennas Propag. AP-24, 17 (1976).
[CrossRef]

Spiller, E.

E. Spiller, Appl. Opt. 15, 2333 (1976).
[CrossRef] [PubMed]

E. Spiller, Appl. Phys. Lett. 20, 365 (1972).
[CrossRef]

E. Spiller, in Space Optics, Proceedings, ICO-IX, Santa Monica, 1972 (National Academy of Sciences, Washington, D.C., 1974), p. 525.

Tester, M. A.

B. L. Henke, M. A. Tester, Adv. X-Ray Anal. 18, 76 (1975).
[CrossRef]

Underwood, J. H.

J. H. Underwood, T. W. Barbee, D. C. Keith, Proc. Soc. Photo-Opt. Instrum. Eng. 184, 123 (1979).

D. L. McKenzie, P. B. Landecker, J. H. Underwood, Space Sci. Instrum. 2, 125 (1976).

Vinogradov, A. V.

Youtz, J. P.

J. DuMond, J. P. Youtz, J. Appl. Phys. 11, 357 (1940).
[CrossRef]

Zachariasen, W. H.

W. H. Zachariasen, Theory of X-Ray Diffraction in Crystals (Wiley, New York, 1945).

Zeldovich, B. Ya.

Acta. Crystallogr. Sect. A (1)

A. M. Saxena, B. P. Schoenborn, Acta. Crystallogr. Sect. A: 33, 805 (1977).
[CrossRef]

Adv. X-Ray Anal. (1)

B. L. Henke, M. A. Tester, Adv. X-Ray Anal. 18, 76 (1975).
[CrossRef]

Ann. Phys. Leipzig (1)

W. Deubner, Ann. Phys. Leipzig 5, 261 (1930).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

E. Spiller, Appl. Phys. Lett. 20, 365 (1972).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

R. A. Simpson, IEEE Trans. Antennas Propag. AP-24, 17 (1976).
[CrossRef]

J. Appl. Phys. (4)

M. W. Charles, J. Appl. Phys. 42, 3329 (1971).
[CrossRef]

J. DuMond, J. P. Youtz, J. Appl. Phys. 11, 357 (1940).
[CrossRef]

J. Dinklage, R. Frerichs, J. Appl. Phys. 34, 2633 (1963).
[CrossRef]

J. Dinklage, J. Appl. Phys. 38, 3781 (1967).
[CrossRef]

Opt. Commun. (1)

R.-P. Haelbich, C. Kunz, Opt. Commun. 17, 287 (1976).
[CrossRef]

Phys. Rev. (1)

L. G. Parratt, Phys. Rev. 95, 35A (1954).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

J. H. Underwood, T. W. Barbee, D. C. Keith, Proc. Soc. Photo-Opt. Instrum. Eng. 184, 123 (1979).

Space Sci. Instrum. (2)

A. J. Burek, Space Sci. Instrum. 3, 53 (1976).

D. L. McKenzie, P. B. Landecker, J. H. Underwood, Space Sci. Instrum. 2, 125 (1976).

Other (7)

H. Koeppe, Dissertation, Giessen (1929).

W. H. Zachariasen, Theory of X-Ray Diffraction in Crystals (Wiley, New York, 1945).

Note that Eqs. (8) and (10) imply that a correction for refraction must be applied to the Bragg Eq. (1). The diffraction pattern is symmetrical about y = 0 or θ=θmp, where θmp=θm+[2δ0/(sin2θ)].

Based on a mean diffraction pattern (Ref. 13, p. 126ff.).

T. W. Barbee, D. C. Keith, in “Workshop on Instrumentation for Synchrotron Radiation Research,” H. Winick, G. Brown, Eds, Stanford Synchrotron Radiation Laboratory Report 78/04 (May1978), p. III-36.

T. W. Barbee, in Proceedings, Topical Conference on Low Energy X-ray Diagnostics, Monterey, Calif., 8–10 June 1980 (American Institute of Physics, New York, 1980).

E. Spiller, in Space Optics, Proceedings, ICO-IX, Santa Monica, 1972 (National Academy of Sciences, Washington, D.C., 1974), p. 525.

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Figures (9)

Fig. 1
Fig. 1

Diffraction of x rays by a layered synthetic microstructure. Coordinate system used in calculating the properties of the structure is shown; y coordinate extends out of the plane of the figure.

Fig. 2
Fig. 2

Reflectivity vs glancing angle for a tungsten–carbon LSM of fifteen layer pairs: dw = 12.4 Å; dc = 20.0 Å. This curve is computed for a wavelength λ = 2.263 Å, for which the following optical constants were used: tungsten, δ = 9.09 × 10−5, β = 2.08 × 10−5; carbon, δ = 1.38 × 10−5, β = 1.64 × 10−8.

Fig. 3
Fig. 3

Peak reflectivity of an LSM as a function of A2: curve K, kinematical approximation; curve D, dynamical approximation (no absorption). These solid curves apply to any LSM structure. Lower curves are computed, using the full theory of Sec. III for a tungsten–carbon LSM with dw = dc = 10 Å. A was varied by changing N.

Fig. 4
Fig. 4

Reflectivity curves computed for an LSM consisting of 101 layer pairs of tungsten and carbon at four wavelengths and for unpolarized radiation. θ1 indicates the position of the first-order peak predicted from the Bragg equation using 2d = 40.0 Å. Angular difference between θ1 and the peak in R is the refraction correction.

Fig. 5
Fig. 5

First-order integrated reflectivities of typical multilayer structures and crystals in the soft x-ray region. Solid curves are calculations based on dynamical theory. LSMs are those referred to in the text and in Figs. 2 and 6. Pb Soap is a thick lead stearate multilayer.19 Curves for thallium acid phthalate (TlAP) and potassium acid phthalate (KAP) are Darwin-Prins calculations of Burek.20 Experimental points for KAP (circles) were taken from Burek,20 for TIAP (triangles) from McKenzie et al.,21 and for the optimized lead stearate multilayer (squares) from Charles.22

Fig. 6
Fig. 6

First-order resolving power of crystals and multilayers. Experimental points for lead sterate (squares) were taken from Charles.22 Curves for KAP and TlAP were computed by Burek20 using the Darwins-Prins theory.

Fig. 7
Fig. 7

Reflectivity curves computed at four wavelengths for an Al–B LSM of 1000 layer pairs; these curves are for unpolarized radiation. θ1 is the first-order Bragg angle computed using 2d = 40.00 Å.

Fig. 8
Fig. 8

Variation of refractive index through several LSM layers: bold line, ideal variation with infinitely sharp gradients at the interfaces; dashed line, model in which the index varies according to a cosine function over a transition distance dtr. In the center of the figure, this smooth function has been approximated by ten thin sublayers of constant refractive index.

Fig. 9
Fig. 9

Peak reflectivity R m p plotted as a function of dtr for the first 3 orders of reflection of Al radiation (λ = 8.34 Å) from a W/C LSM (dW = 24.3 Å, dC = 33.4 Å).

Equations (37)

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m λ = 2 d sin θ m ,
n ^ = 1 - δ - i β ,
f ^ = f 0 + Δ f + i Δ f ,
δ = N r e λ 2 2 π ( Z + Δ f ) ,
β = N r e λ 2 2 π ( Δ f ) = μ λ 4 π ,
ϕ = N [ ( Z + Δ f ) 2 + Δ f 2 ] 1 / 2 r e .
F ( θ ) = 0 d ϕ ( z ) exp ( i Q z ) d z ,
I ( θ ) I 0 = { y 2 + ( y 2 - 1 ) cot 2 [ A ( y 2 - 1 ) 1 / 2 ] } - 1 ,
A = 2 N d K m F ( θ m )
y = π m N 2 A sin 2 θ m [ ( θ - θ m ) sin 2 θ m - 2 δ 0 ] .
δ 0 = r e λ 2 2 π j N ¯ j ( Z + Δ f ) j ,
R m p = tanh 2 A .
R m I = I ( θ ) I 0 d ( θ - θ m ) ,
R m I = tan θ m m N ( A tanh A ) .
R m p = 1 ,
R m I = A tan θ m m N .
Δ θ m 1 / 2 2.3 π A tan θ m m N ,             λ Δ λ π m N 2.3 A .
I ( θ ) I 0 ( sin A y y ) 2 ,
R m p = A 2 ,
R m I = A 2 tan θ m m N ,
Δ θ m 1 / 2 tan θ m m N ,             λ Δ λ m N .
F ( θ ) = 2 Q [ ϕ A 2 ( 1 - cos Q d A ) + ϕ B 2 ( 1 - cos Q d B ) + ϕ A ϕ B ( cos Q d A + cos Q d B - cos Q d - 1 ) ] 1 / 2 ,
F ( θ ) = 2 Q sin ( Q d 4 ) ( ϕ A 2 + ϕ B 2 + 2 ϕ A ϕ B cos Q d 2 ) 1 / 2 .
F ( θ m ) = d π m sin ( m π d A d ) ( ϕ A - ϕ B ) .
F ( θ m ) = d 4 ( ϕ A - ϕ B ) for m = 1 , = 0 for m 1.
N eff = π m 2 / [ K d 2 sin ( m π d A / d ) ( ϕ A - ϕ B ) ] .
F j , j + 1 σ = ( E j R E j ) σ = g j - g j + 1 g j + g j + 1 ,
F j , j + 1 π = ( E j R E j ) π = g i / n ^ j 2 - g j + 1 / n ^ j + 1 2 g j / n ^ j 2 + g j + 1 / n ^ j + 1 2 ,
g j = ( n ^ j 2 - cos 2 θ ) 1 / 2 .
g j = ( θ 2 - 2 δ j - 2 i β j ) 1 / 2 .
R j , j + 1 = a j ( E j R / E j ) ,
a j = exp ( - i π λ g j d j )
R j , j + 1 = a j 4 ( R j + 1 , j + 2 + F j , j + 1 R j + 1 , j + 2 F j , j + 1 + 1 ) .
I ( θ ) I 0 = ( R 12 ) 2 .
R m I = 2 d 2 K π m 3 sin ( m π d A d ) ( ϕ A - ϕ B ) tan θ tanh A .
tan π ( d A d ) opt = π [ ( d A d ) opt + β B β A - β B ] .
Δ θ m 1 / 2 = 2.3 π R m I ,

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