Abstract

An electromagnetic analysis of diffraction by Fresnel linear zone plates ruled into a homogeneous or a stratified media is developed. It takes advantage of the new possibilities brought by the introduction of the R-matrix propagation algorithm into the theory of lamellar gratings, namely, the large increase of the stability domain of the numerical results. Thus a linear Fresnel zone plate can be delt with as a period of such a grating, and the diffracted field can be computed for both TE and TM polarizations. Numerical examples show the convergence of the method and study the focusing properties of positive and negative zone plates. The influence of different parameters such as incidence or number of lines is pointed out.

© 1995 Optical Society of America

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References

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  1. V. V. Aristov, A. I. Erko, V. V. Martinov, “Principles of Bragg–Fresnel multilayer optics,” Rev. Phys. Appl. 23, 1623–1630 (1988);S. Babin, A. Erko, “Fabrication of diffractive x-ray optical elements,” Nucl. Instrum. Methods Phys. Res. A 282, 529–535 (1989).
    [CrossRef]
  2. J. Maser, “Evaluation of the efficiency of zones plates with high aspect ratios by application of coupled wave theory,” in X-ray Microscopy III, A. G. Michette, G. R. Morrison, G. J. Buckley, eds. (Springer-Verlag, Berlin, 1992), pp. 104–106.
    [CrossRef]
  3. A. Sammar, J.-M. André, “Diffraction of multilayer gratings and zone plates in the x-ray region using the Born approximation,” J. Opt. Soc. Am. A 10, 600–613 (1993).
    [CrossRef]
  4. A. Sammar, J.-M. André, “Dynamical theory of stratified Fresnel linear zone plates,” J. Opt. Soc. Am. A 10, 2324–2337 (1993);A. Mirone, M. Idir, P. Dhez, G. Soullie, A. Erko, “Dynamical theory for Bragg–Fresnel multilayer lenses for X-UV and X-ray range,” Opt. Commun. 111, 191–198 (1994).
    [CrossRef]
  5. R. Petit, Electromagnetic Theory of Gratings, (Springer-Verlag, Berlin, 1980).
    [CrossRef]
  6. J. T. Sheridan, C. J. R. Sheppard, “Coherent imaging of thick fine isolated structures,” J. Opt. Soc. Am. A 10, 614–632 (1993).
    [CrossRef]
  7. L. Li, “A multilayer modal method for diffraction gratings of arbitrary profile, depth, and conductivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
    [CrossRef]
  8. F. Montiel, M. Nevière, “Differential theory of gratings: extension to deep gratings of arbitrary profile and permittivity through the R-matrix propagation algorithm,” J. Opt. Soc. Am. A 12, 3241–3250 (1994).
    [CrossRef]
  9. A. E. Sammar, “Etude théorique et expérimentale de systèmes optiques interférentiels dispersifs et focalisants pour la spectroscopie et l’imagerie x,” Ph.D. dissertation (Université Pierre et Marie Curie, Paris, May6, 1993).
  10. J. P. Hugonin, R. Petit, “A numerical study of the problem of diffraction at a non-periodic obstacle,” Opt. Commun. 20, 360–364 (1977);J. P. Hugonin, R. Petit, “Theoretical and numerical study of a locally deformed stratified medium,” J. Opt. Soc. Am. 71, 664–674 (1981).
    [CrossRef]
  11. M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications ὰ l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).
    [CrossRef]
  12. M. Nevière, J. Flamand, “Electromagnetic theory as it applies to x-ray and XUV gratings,” Nucl. Instrum. Methods 172, 273–279 (1980).
    [CrossRef]
  13. M. Nevière, J. Flamand, J. M. Lerner, “Optimization of gratings for soft x-ray monochromators,” Nucl. Instrum. Methods 195, 183–189 (1982).
    [CrossRef]
  14. P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980).
    [CrossRef]
  15. J. W. Goodman, Introduction ὰ l’Optique de Fourier et à l’Holographie (Masson, Paris, 1972), p. 31

1994 (1)

F. Montiel, M. Nevière, “Differential theory of gratings: extension to deep gratings of arbitrary profile and permittivity through the R-matrix propagation algorithm,” J. Opt. Soc. Am. A 12, 3241–3250 (1994).
[CrossRef]

1993 (4)

1988 (1)

V. V. Aristov, A. I. Erko, V. V. Martinov, “Principles of Bragg–Fresnel multilayer optics,” Rev. Phys. Appl. 23, 1623–1630 (1988);S. Babin, A. Erko, “Fabrication of diffractive x-ray optical elements,” Nucl. Instrum. Methods Phys. Res. A 282, 529–535 (1989).
[CrossRef]

1982 (1)

M. Nevière, J. Flamand, J. M. Lerner, “Optimization of gratings for soft x-ray monochromators,” Nucl. Instrum. Methods 195, 183–189 (1982).
[CrossRef]

1980 (1)

M. Nevière, J. Flamand, “Electromagnetic theory as it applies to x-ray and XUV gratings,” Nucl. Instrum. Methods 172, 273–279 (1980).
[CrossRef]

1977 (1)

J. P. Hugonin, R. Petit, “A numerical study of the problem of diffraction at a non-periodic obstacle,” Opt. Commun. 20, 360–364 (1977);J. P. Hugonin, R. Petit, “Theoretical and numerical study of a locally deformed stratified medium,” J. Opt. Soc. Am. 71, 664–674 (1981).
[CrossRef]

1974 (1)

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications ὰ l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).
[CrossRef]

André, J.-M.

Aristov, V. V.

V. V. Aristov, A. I. Erko, V. V. Martinov, “Principles of Bragg–Fresnel multilayer optics,” Rev. Phys. Appl. 23, 1623–1630 (1988);S. Babin, A. Erko, “Fabrication of diffractive x-ray optical elements,” Nucl. Instrum. Methods Phys. Res. A 282, 529–535 (1989).
[CrossRef]

Erko, A. I.

V. V. Aristov, A. I. Erko, V. V. Martinov, “Principles of Bragg–Fresnel multilayer optics,” Rev. Phys. Appl. 23, 1623–1630 (1988);S. Babin, A. Erko, “Fabrication of diffractive x-ray optical elements,” Nucl. Instrum. Methods Phys. Res. A 282, 529–535 (1989).
[CrossRef]

Flamand, J.

M. Nevière, J. Flamand, J. M. Lerner, “Optimization of gratings for soft x-ray monochromators,” Nucl. Instrum. Methods 195, 183–189 (1982).
[CrossRef]

M. Nevière, J. Flamand, “Electromagnetic theory as it applies to x-ray and XUV gratings,” Nucl. Instrum. Methods 172, 273–279 (1980).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction ὰ l’Optique de Fourier et à l’Holographie (Masson, Paris, 1972), p. 31

Hugonin, J. P.

J. P. Hugonin, R. Petit, “A numerical study of the problem of diffraction at a non-periodic obstacle,” Opt. Commun. 20, 360–364 (1977);J. P. Hugonin, R. Petit, “Theoretical and numerical study of a locally deformed stratified medium,” J. Opt. Soc. Am. 71, 664–674 (1981).
[CrossRef]

Lerner, J. M.

M. Nevière, J. Flamand, J. M. Lerner, “Optimization of gratings for soft x-ray monochromators,” Nucl. Instrum. Methods 195, 183–189 (1982).
[CrossRef]

Li, L.

Martinov, V. V.

V. V. Aristov, A. I. Erko, V. V. Martinov, “Principles of Bragg–Fresnel multilayer optics,” Rev. Phys. Appl. 23, 1623–1630 (1988);S. Babin, A. Erko, “Fabrication of diffractive x-ray optical elements,” Nucl. Instrum. Methods Phys. Res. A 282, 529–535 (1989).
[CrossRef]

Maser, J.

J. Maser, “Evaluation of the efficiency of zones plates with high aspect ratios by application of coupled wave theory,” in X-ray Microscopy III, A. G. Michette, G. R. Morrison, G. J. Buckley, eds. (Springer-Verlag, Berlin, 1992), pp. 104–106.
[CrossRef]

Montiel, F.

F. Montiel, M. Nevière, “Differential theory of gratings: extension to deep gratings of arbitrary profile and permittivity through the R-matrix propagation algorithm,” J. Opt. Soc. Am. A 12, 3241–3250 (1994).
[CrossRef]

Nevière, M.

F. Montiel, M. Nevière, “Differential theory of gratings: extension to deep gratings of arbitrary profile and permittivity through the R-matrix propagation algorithm,” J. Opt. Soc. Am. A 12, 3241–3250 (1994).
[CrossRef]

M. Nevière, J. Flamand, J. M. Lerner, “Optimization of gratings for soft x-ray monochromators,” Nucl. Instrum. Methods 195, 183–189 (1982).
[CrossRef]

M. Nevière, J. Flamand, “Electromagnetic theory as it applies to x-ray and XUV gratings,” Nucl. Instrum. Methods 172, 273–279 (1980).
[CrossRef]

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications ὰ l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).
[CrossRef]

Petit, R.

J. P. Hugonin, R. Petit, “A numerical study of the problem of diffraction at a non-periodic obstacle,” Opt. Commun. 20, 360–364 (1977);J. P. Hugonin, R. Petit, “Theoretical and numerical study of a locally deformed stratified medium,” J. Opt. Soc. Am. 71, 664–674 (1981).
[CrossRef]

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications ὰ l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).
[CrossRef]

R. Petit, Electromagnetic Theory of Gratings, (Springer-Verlag, Berlin, 1980).
[CrossRef]

Sammar, A.

Sammar, A. E.

A. E. Sammar, “Etude théorique et expérimentale de systèmes optiques interférentiels dispersifs et focalisants pour la spectroscopie et l’imagerie x,” Ph.D. dissertation (Université Pierre et Marie Curie, Paris, May6, 1993).

Sheppard, C. J. R.

Sheridan, J. T.

Vincent, P.

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications ὰ l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).
[CrossRef]

P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980).
[CrossRef]

J. Opt. Soc. Am. A (5)

Nouv. Rev. Opt. (1)

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications ὰ l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).
[CrossRef]

Nucl. Instrum. Methods (2)

M. Nevière, J. Flamand, “Electromagnetic theory as it applies to x-ray and XUV gratings,” Nucl. Instrum. Methods 172, 273–279 (1980).
[CrossRef]

M. Nevière, J. Flamand, J. M. Lerner, “Optimization of gratings for soft x-ray monochromators,” Nucl. Instrum. Methods 195, 183–189 (1982).
[CrossRef]

Opt. Commun. (1)

J. P. Hugonin, R. Petit, “A numerical study of the problem of diffraction at a non-periodic obstacle,” Opt. Commun. 20, 360–364 (1977);J. P. Hugonin, R. Petit, “Theoretical and numerical study of a locally deformed stratified medium,” J. Opt. Soc. Am. 71, 664–674 (1981).
[CrossRef]

Rev. Phys. Appl. (1)

V. V. Aristov, A. I. Erko, V. V. Martinov, “Principles of Bragg–Fresnel multilayer optics,” Rev. Phys. Appl. 23, 1623–1630 (1988);S. Babin, A. Erko, “Fabrication of diffractive x-ray optical elements,” Nucl. Instrum. Methods Phys. Res. A 282, 529–535 (1989).
[CrossRef]

Other (5)

J. Maser, “Evaluation of the efficiency of zones plates with high aspect ratios by application of coupled wave theory,” in X-ray Microscopy III, A. G. Michette, G. R. Morrison, G. J. Buckley, eds. (Springer-Verlag, Berlin, 1992), pp. 104–106.
[CrossRef]

R. Petit, Electromagnetic Theory of Gratings, (Springer-Verlag, Berlin, 1980).
[CrossRef]

A. E. Sammar, “Etude théorique et expérimentale de systèmes optiques interférentiels dispersifs et focalisants pour la spectroscopie et l’imagerie x,” Ph.D. dissertation (Université Pierre et Marie Curie, Paris, May6, 1993).

P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980).
[CrossRef]

J. W. Goodman, Introduction ὰ l’Optique de Fourier et à l’Holographie (Masson, Paris, 1972), p. 31

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Figures (12)

Fig. 1
Fig. 1

Schematic representation of a Bragg–Fresnel linear zone plate and notation.

Fig. 2
Fig. 2

Schematic representation of a Bragg–Fresnel zone plate grating.

Fig. 3
Fig. 3

Near-field map for increasing values of ρ; x is expressed in millimeters; the incident plane wave is assumed to have unit amplitude in this and all following figures.

Fig. 4
Fig. 4

Evolution of the average intensity I as a function of incidence θ0 (in degrees) for the zone plate of Fig. 3, with ρ = 1.25, near Bragg incidence.

Fig. 5
Fig. 5

Behavior of |B(k)| as function of the diffraction angle θk in degrees, near the Bragg angle for the zone plate of Fig. 4.

Fig. 6
Fig. 6

Reconstruction of the near-field intensity map through the value of B(k) using 500 values of k and θk ∈ [89.035°−89.055°] for the zone plate of Fig. 4.

Fig. 7
Fig. 7

Same as Fig. 6 but for a much wider range of x: (a) Nk = 500, (b) Nk = 3000, (c) Nk = 5000; the only exact peak is indicated by an arrow; all other maximums are introduced by the use of a finite value (Nk) of k.

Fig. 8
Fig. 8

Field-intensity map as a function of radial coordinate in the specular direction for the zone plate of Fig. 4 and for values of Nk.

Fig. 9
Fig. 9

Field intensity as a function of radial coordinate in the specular direction for a zone plate with L = 8 (instead of 2), Nk = 500, and two values of the number N of Fourier coefficients.

Fig. 10
Fig. 10

Same as Fig. 9 but for N = 48 and different values of L: (a) L = 2, (b) L = 4, (c) L = 6.

Fig. 11
Fig. 11

Influence of polarization of the results of Fig. 8.

Fig. 12
Fig. 12

Focusing capabilities of Fresnel zone plates, with the same parameters as in Figs. 9 and 10 but for negative zone plates. (a) L = 2, (b) L = 4, (c) L = 6, (d) L = 8.

Equations (9)

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x n = n a ,
cos θ 0 = λ / ( 2 D ) ,
E G ( x ) = 1 2 π + B ( k ) exp ( ikx ) d k ,
B ( k ) = + E G ( x ) exp ( ikx ) d x .
B ( k ) = d / 2 + d / 2 E G ( x ) exp ( ikx ) d x .
E ( x , y ) = 1 2 π + B ( k ) exp { i [ k x + β ( k ) ( y h ) ] } d k × [ exp ( i α x ) ] ,
β 2 ( k ) = ( 2 π / λ ) 2 ( α + k ) 2 , Im [ β ( k ) ] + Re [ β ( k ) ] > 0 .
( 2 π / λ ) sin θ k = k + α ,
sin θ k = sin θ 0 + k λ / ( 2 π ) .

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