Abstract

An expression is derived for the Bragg angle of synthetic multilayer x-ray reflectors. It displays the explicit dependences on not only the wavelength of the radiation and the period of the multilayer, but also on the ratio of the thicknesses of the two layers comprising one period. Absorptive dependence is also taken into account.

© 1989 Optical Society of America

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References

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  1. B. L. Henke, J. Y. Uejio, H. T. Yamada, R. E. Tackaberry, “Characterization of Multilayer X-Ray Analyzers: Models and Measurements,” Opt. Eng. 25, 937 (1986).
    [CrossRef]
  2. J. H. Underwood, T. W. Barbee, “Synthetic Multilayers as Bragg Diffractors for X-rays and Extreme Ultraviolet: Calculations of Performance,” AIP Conf. Proc. 75, 170 (1981).
    [CrossRef]
  3. T. W. Barbee, “Multilayers for X-Ray Optics,” Opt. Eng. 25, 898 (1986).
  4. A. H. Compton, S. K. Allison, X-Rays in Theory and Experiment (Van Nostrand, New York, 1935), p. 674.
  5. T. W. Barbee, “Sputtered Layered Synthetic Microstructure (LSM) Dispersion Elements,” AIP Conf. Proc. 75, 131 (1981).
    [CrossRef]
  6. A. E. Rosenbluth, J. M. Forsyth, “The Reflecting Properties of Soft X-ray Multilayers,” AIP Conf. Proc. 75, 280 (1981).
    [CrossRef]

1986 (2)

B. L. Henke, J. Y. Uejio, H. T. Yamada, R. E. Tackaberry, “Characterization of Multilayer X-Ray Analyzers: Models and Measurements,” Opt. Eng. 25, 937 (1986).
[CrossRef]

T. W. Barbee, “Multilayers for X-Ray Optics,” Opt. Eng. 25, 898 (1986).

1981 (3)

T. W. Barbee, “Sputtered Layered Synthetic Microstructure (LSM) Dispersion Elements,” AIP Conf. Proc. 75, 131 (1981).
[CrossRef]

A. E. Rosenbluth, J. M. Forsyth, “The Reflecting Properties of Soft X-ray Multilayers,” AIP Conf. Proc. 75, 280 (1981).
[CrossRef]

J. H. Underwood, T. W. Barbee, “Synthetic Multilayers as Bragg Diffractors for X-rays and Extreme Ultraviolet: Calculations of Performance,” AIP Conf. Proc. 75, 170 (1981).
[CrossRef]

Allison, S. K.

A. H. Compton, S. K. Allison, X-Rays in Theory and Experiment (Van Nostrand, New York, 1935), p. 674.

Barbee, T. W.

T. W. Barbee, “Multilayers for X-Ray Optics,” Opt. Eng. 25, 898 (1986).

T. W. Barbee, “Sputtered Layered Synthetic Microstructure (LSM) Dispersion Elements,” AIP Conf. Proc. 75, 131 (1981).
[CrossRef]

J. H. Underwood, T. W. Barbee, “Synthetic Multilayers as Bragg Diffractors for X-rays and Extreme Ultraviolet: Calculations of Performance,” AIP Conf. Proc. 75, 170 (1981).
[CrossRef]

Compton, A. H.

A. H. Compton, S. K. Allison, X-Rays in Theory and Experiment (Van Nostrand, New York, 1935), p. 674.

Forsyth, J. M.

A. E. Rosenbluth, J. M. Forsyth, “The Reflecting Properties of Soft X-ray Multilayers,” AIP Conf. Proc. 75, 280 (1981).
[CrossRef]

Henke, B. L.

B. L. Henke, J. Y. Uejio, H. T. Yamada, R. E. Tackaberry, “Characterization of Multilayer X-Ray Analyzers: Models and Measurements,” Opt. Eng. 25, 937 (1986).
[CrossRef]

Rosenbluth, A. E.

A. E. Rosenbluth, J. M. Forsyth, “The Reflecting Properties of Soft X-ray Multilayers,” AIP Conf. Proc. 75, 280 (1981).
[CrossRef]

Tackaberry, R. E.

B. L. Henke, J. Y. Uejio, H. T. Yamada, R. E. Tackaberry, “Characterization of Multilayer X-Ray Analyzers: Models and Measurements,” Opt. Eng. 25, 937 (1986).
[CrossRef]

Uejio, J. Y.

B. L. Henke, J. Y. Uejio, H. T. Yamada, R. E. Tackaberry, “Characterization of Multilayer X-Ray Analyzers: Models and Measurements,” Opt. Eng. 25, 937 (1986).
[CrossRef]

Underwood, J. H.

J. H. Underwood, T. W. Barbee, “Synthetic Multilayers as Bragg Diffractors for X-rays and Extreme Ultraviolet: Calculations of Performance,” AIP Conf. Proc. 75, 170 (1981).
[CrossRef]

Yamada, H. T.

B. L. Henke, J. Y. Uejio, H. T. Yamada, R. E. Tackaberry, “Characterization of Multilayer X-Ray Analyzers: Models and Measurements,” Opt. Eng. 25, 937 (1986).
[CrossRef]

AIP Conf. Proc. (3)

J. H. Underwood, T. W. Barbee, “Synthetic Multilayers as Bragg Diffractors for X-rays and Extreme Ultraviolet: Calculations of Performance,” AIP Conf. Proc. 75, 170 (1981).
[CrossRef]

T. W. Barbee, “Sputtered Layered Synthetic Microstructure (LSM) Dispersion Elements,” AIP Conf. Proc. 75, 131 (1981).
[CrossRef]

A. E. Rosenbluth, J. M. Forsyth, “The Reflecting Properties of Soft X-ray Multilayers,” AIP Conf. Proc. 75, 280 (1981).
[CrossRef]

Opt. Eng. (2)

T. W. Barbee, “Multilayers for X-Ray Optics,” Opt. Eng. 25, 898 (1986).

B. L. Henke, J. Y. Uejio, H. T. Yamada, R. E. Tackaberry, “Characterization of Multilayer X-Ray Analyzers: Models and Measurements,” Opt. Eng. 25, 937 (1986).
[CrossRef]

Other (1)

A. H. Compton, S. K. Allison, X-Rays in Theory and Experiment (Van Nostrand, New York, 1935), p. 674.

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Figures (2)

Fig. 1
Fig. 1

Plane wave impinging on a multilayer at a grazing angle equal to the Bragg angle θb. The high and low Z layers have refractive indices n1 and n2 and thicknesses d1 and d2, respectively.

Fig. 2
Fig. 2

Reflectivity at 1.54 Å as a function of grazing angle for a tungsten–silicon multilayer with d1 = 5 Å and d2 = 30 Å. The value of θb calculated using Eqs. (6) and (8) is 22.5804 mrad, while the peak reflectivity occurs at 22.4336 mrad.

Equations (8)

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sin θ b = λ 2 4 d 2 + δ ( 2 δ ) ,
Δ φ = 2 k 0 n 1 d 1   cos θ 1 + 2 k 0 n 2 d 2   cos θ 2 = 2 k 0 ( n 1 d 1   cos θ 1 + n 2 d 2   cos θ 2 ) ,
Δ φ = 2 k 0 ( d 1 n 1   cos θ 1 + d 2 n 2 2 n 1 2   sin 2 θ 1 ) .
Δ l = Δ φ k 0
m λ = 2 ( n 1 d 1   cos θ 1 + d 2 n 2 2 n 1 2   sin 2 θ 1 ) ,
cos θ 1 = 1 2 n 1  ( d 1 2 d 2 2 1 ) [ λ d 1 d 2 2 λ 2 d 2 2 + 4 ( n 2 2 n 1 2 ) ( d 1 2 d 2 2 1 ) ] ,
cos θ 1 = d 2 λ n 1 ( λ 2 4 d 2 2 + n 1 2 n 2 2 ) .
sin θ b = 1 n 1 2 ( 1 cos 2 θ 1 ) .

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