Abstract

We report on the optimization of staircase grating profiles for the case of absorbing grating materials. Using a simple numerical algorithm, we determined the grating parameters, maximizing the first-order diffraction efficiency for different numbers of staircase steps. The results show that there is a significant difference between the staircase profiles for nonnegligible and negligible absorption. The obtained solutions are of importance for diffractive optics in the soft-x-ray and extreme-ultraviolet ranges.

© 2003 Optical Society of America

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References

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  1. H. P. Herzig, ed., Micro-optics (Taylor & Francis, London, 1998).
  2. R. Tatchyn, P. L. Csonka, and I. Lindau, J. Opt. Soc. Am. 72, 1630–1639 (1982).
  3. M. B. Stern, in Micro-optics, H. P. Herzig, ed. (Taylor & Francis, London, 1998), pp. 53–86.
  4. E. Di Fabrizio, F. Romanato, M. Gentili, S. Cabrini, B. Kaulich, J. Susini, and R. Barrett, Nature 401, 895–898 (1999).
    [CrossRef]
  5. W. Yun, B. Lai, A. A. Krasnoperova, E. Di Fabrizio, Z. Cai, F. Cerrina, Z. Chen, M. Gentili, and E. Gluskin, Rev. Sci. Instrum. 70, 3537–3541 (1999).
    [CrossRef]
  6. B. Nöhammer, J. Hoszowska, H. P. Herzig, and C. David, presented at the X-Ray Microscopy Conference 2002, Grenoble, France, July 29–August 2, 2002.
  7. E. Aarts and J. K. Lenstra, Local Search in Combinatorial Optimization (Wiley, Chichester, U.K., 1997).

Stern, M. B.

M. B. Stern, in Micro-optics, H. P. Herzig, ed. (Taylor & Francis, London, 1998), pp. 53–86.

Other

H. P. Herzig, ed., Micro-optics (Taylor & Francis, London, 1998).

R. Tatchyn, P. L. Csonka, and I. Lindau, J. Opt. Soc. Am. 72, 1630–1639 (1982).

M. B. Stern, in Micro-optics, H. P. Herzig, ed. (Taylor & Francis, London, 1998), pp. 53–86.

E. Di Fabrizio, F. Romanato, M. Gentili, S. Cabrini, B. Kaulich, J. Susini, and R. Barrett, Nature 401, 895–898 (1999).
[CrossRef]

W. Yun, B. Lai, A. A. Krasnoperova, E. Di Fabrizio, Z. Cai, F. Cerrina, Z. Chen, M. Gentili, and E. Gluskin, Rev. Sci. Instrum. 70, 3537–3541 (1999).
[CrossRef]

B. Nöhammer, J. Hoszowska, H. P. Herzig, and C. David, presented at the X-Ray Microscopy Conference 2002, Grenoble, France, July 29–August 2, 2002.

E. Aarts and J. K. Lenstra, Local Search in Combinatorial Optimization (Wiley, Chichester, U.K., 1997).

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Figures (5)

Fig. 1
Fig. 1

(A) Design of a transmission grating maximizing the first-order diffraction efficiency in the case of absorbing grating material (solid lines) and in the case of negligible absorption (dashed lines). (B) Correlation between free design parameter b1 and the optical properties of the grating material (quantified by δ/β).

Fig. 2
Fig. 2

General form of a staircase profile enabling the optimization of the first-order diffraction efficiency.

Fig. 3
Fig. 3

Optimum normalized widths w̲i (A) and heights h̲i (B) of a four-step staircase profile as a function of δ/β of the grating material.

Fig. 4
Fig. 4

First-order diffraction efficiency of a four-step profile (solid curve) featuring optimal values for the heights and widths of the steps. In addition, the diffraction efficiency of the ideal continuous profile (see Fig. 1) and a four-step profile optimized for negligible absorption are depicted.

Fig. 5
Fig. 5

First-order diffraction efficiency for different optimized grating designs.

Equations (3)

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η=i=1Nϕi2,
ϕi=exp-2πh̲iβ/δexp2πih̲i-x̲isinπw̲i/π,
h̲i=hiδ/λ,    w̲i=wi/b,    x̲i=xi/b.

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