Abstract

A full-wave approach to quantitative characterization of x-ray zone plate lenses is proposed. Distributed focusing efficiency η(z) of a multifocus optical element is defined as the energy flux through the Airy disk of a reference perfect lens with variable focal length z. Maxima of this function characterize diffraction efficiencies and spatial resolution of the zone plate foci. The parabolic wave equation is used to take into account diffraction effects inside the optical element. Rough and fuzzy interface models are introduced to describe realistic zone profiles. Numerical simulation reveals the limited capability of zone width reduction to improve the zone plate imaging performance. The possibilities of second-order focus enhancement by optimization of the zone plate thickness, line-to-space ratio, and zone tilt are studied numerically.

© 2002 Optical Society of America

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  1. D. T. Attwood, Soft X-Rays and Extreme Ultraviolet Radiation, Principles and Applications (Cambridge U. Press, Cambridge, N.Y., 1999).
  2. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1980).
  3. A. G. Michette, Optical Systems for Soft X-Rays (Plenum, New York, 1986).
  4. Yu. V. Kopylov, A. V. Popov, A. V. Vinogradov, “Application of the parabolic wave equation to x-ray diffraction optics,” Opt. Commun. 118, 619–636 (1995).
    [CrossRef]
  5. A. V. Vinogradov, A. V. Popov, Yu. V. Kopylov, A. N. Kurokhtin, Numerical Simulation of X-Ray Diffractive Optics (A&B Publishing House, Moscow, 1999).
  6. X-Ray Microscopy, Proceedings of the 6th International Conference (American Institute of Physics, Melville, N.Y., 2000).
  7. J. Maser, G. Schmahl, “Coupled wave description of the diffraction by zone plates with high aspect ratios,” Opt. Commun. 89, 355–362 (1992).
    [CrossRef]
  8. G. Schneider, “Zone plates with high efficiency in high orders of diffraction described by dynamical theory,” Appl. Phys. Lett. 71, 2242–2244 (1997).
    [CrossRef]
  9. J. Kirz, “Phase zone plates for x-ray and the extreme UV,” J. Opt. Soc. Am. 64, 301–309 (1974).
    [CrossRef]
  10. D. Tennant, S. Spector, A. Stein, C. Jacobsen, “Electron beam lithography of Fresnel zone plates using a rectilinear machine and trilayer resists,” in X-Ray Microscopy, Proceedings of the 6th International Conference (American Institute of Physics, Melville, N.Y., 2000), pp. 601–604.
  11. M. A. Leontovich, “A method to solve problems of electromagnetic wave propagation along the earth’s surface” (in Russian), Izv. Akad. Nauk USSR Ser. Fiz. 8(1), 16–22 (1944).
  12. V. A. Fock, Electromagnetic Diffraction and Propagation Problems (Pergamon, Oxford, UK, 1965).
  13. M. Levy, Parabolic Equation Method for Electromagnetic Wave Propagation (Institute of Electrical Engineers, London, 2000).
  14. V. E. Levashov, A. V. Vinogradov, “Analytical theory of the zone plates efficiency,” Phys. Rev. E 49, 5797–5803 (1994).
    [CrossRef]
  15. Yu. V. Kopylov, A. V. Popov, A. V. Vinogradov, “Diffraction phenomena inside thick Fresnel zone plates,” Radio Sci. 31, 1815–1822 (1996).
    [CrossRef]
  16. R. D. Richtmayer, K. W. Morton, Difference Methods for Initial Value Problems, 2nd ed. (Interscience, New York, 1967).
  17. P. V. Bliokh, A. A. Minakov, “Diffraction of light and lens effect of the stellar gravitational field,” Astrophys. Space Sci. 34, L7–L9 (1975).
    [CrossRef]
  18. W. Meyer-lise, G. Denbeaux, L. E. Johnson, W. Bates, A. Lucero, E. H. Anderson, “The high resolution x-ray microscope XM-1,” in X-Ray Microscopy, Proceedings of the 6th International Conference (American Institute of Physics, Melville, N.Y., 2000), pp. 129–134.
  19. G. Schneider, J. Maser, “Zone plates as imaging optics in high diffraction orders described by coupled wave theory,” in X-Ray Microscopy and Spectromicroscopy, Proceedings of the 5th International Conference on X-Ray Microscopy and Spectromicroscopy, (Springer-Verlag, Berlin, 1998), pp. IV-71–IV-76 (on CD).

1997 (1)

G. Schneider, “Zone plates with high efficiency in high orders of diffraction described by dynamical theory,” Appl. Phys. Lett. 71, 2242–2244 (1997).
[CrossRef]

1996 (1)

Yu. V. Kopylov, A. V. Popov, A. V. Vinogradov, “Diffraction phenomena inside thick Fresnel zone plates,” Radio Sci. 31, 1815–1822 (1996).
[CrossRef]

1995 (1)

Yu. V. Kopylov, A. V. Popov, A. V. Vinogradov, “Application of the parabolic wave equation to x-ray diffraction optics,” Opt. Commun. 118, 619–636 (1995).
[CrossRef]

1994 (1)

V. E. Levashov, A. V. Vinogradov, “Analytical theory of the zone plates efficiency,” Phys. Rev. E 49, 5797–5803 (1994).
[CrossRef]

1992 (1)

J. Maser, G. Schmahl, “Coupled wave description of the diffraction by zone plates with high aspect ratios,” Opt. Commun. 89, 355–362 (1992).
[CrossRef]

1975 (1)

P. V. Bliokh, A. A. Minakov, “Diffraction of light and lens effect of the stellar gravitational field,” Astrophys. Space Sci. 34, L7–L9 (1975).
[CrossRef]

1974 (1)

1944 (1)

M. A. Leontovich, “A method to solve problems of electromagnetic wave propagation along the earth’s surface” (in Russian), Izv. Akad. Nauk USSR Ser. Fiz. 8(1), 16–22 (1944).

Anderson, E. H.

W. Meyer-lise, G. Denbeaux, L. E. Johnson, W. Bates, A. Lucero, E. H. Anderson, “The high resolution x-ray microscope XM-1,” in X-Ray Microscopy, Proceedings of the 6th International Conference (American Institute of Physics, Melville, N.Y., 2000), pp. 129–134.

Attwood, D. T.

D. T. Attwood, Soft X-Rays and Extreme Ultraviolet Radiation, Principles and Applications (Cambridge U. Press, Cambridge, N.Y., 1999).

Bates, W.

W. Meyer-lise, G. Denbeaux, L. E. Johnson, W. Bates, A. Lucero, E. H. Anderson, “The high resolution x-ray microscope XM-1,” in X-Ray Microscopy, Proceedings of the 6th International Conference (American Institute of Physics, Melville, N.Y., 2000), pp. 129–134.

Bliokh, P. V.

P. V. Bliokh, A. A. Minakov, “Diffraction of light and lens effect of the stellar gravitational field,” Astrophys. Space Sci. 34, L7–L9 (1975).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1980).

Denbeaux, G.

W. Meyer-lise, G. Denbeaux, L. E. Johnson, W. Bates, A. Lucero, E. H. Anderson, “The high resolution x-ray microscope XM-1,” in X-Ray Microscopy, Proceedings of the 6th International Conference (American Institute of Physics, Melville, N.Y., 2000), pp. 129–134.

Fock, V. A.

V. A. Fock, Electromagnetic Diffraction and Propagation Problems (Pergamon, Oxford, UK, 1965).

Jacobsen, C.

D. Tennant, S. Spector, A. Stein, C. Jacobsen, “Electron beam lithography of Fresnel zone plates using a rectilinear machine and trilayer resists,” in X-Ray Microscopy, Proceedings of the 6th International Conference (American Institute of Physics, Melville, N.Y., 2000), pp. 601–604.

Johnson, L. E.

W. Meyer-lise, G. Denbeaux, L. E. Johnson, W. Bates, A. Lucero, E. H. Anderson, “The high resolution x-ray microscope XM-1,” in X-Ray Microscopy, Proceedings of the 6th International Conference (American Institute of Physics, Melville, N.Y., 2000), pp. 129–134.

Kirz, J.

Kopylov, Yu. V.

Yu. V. Kopylov, A. V. Popov, A. V. Vinogradov, “Diffraction phenomena inside thick Fresnel zone plates,” Radio Sci. 31, 1815–1822 (1996).
[CrossRef]

Yu. V. Kopylov, A. V. Popov, A. V. Vinogradov, “Application of the parabolic wave equation to x-ray diffraction optics,” Opt. Commun. 118, 619–636 (1995).
[CrossRef]

A. V. Vinogradov, A. V. Popov, Yu. V. Kopylov, A. N. Kurokhtin, Numerical Simulation of X-Ray Diffractive Optics (A&B Publishing House, Moscow, 1999).

Kurokhtin, A. N.

A. V. Vinogradov, A. V. Popov, Yu. V. Kopylov, A. N. Kurokhtin, Numerical Simulation of X-Ray Diffractive Optics (A&B Publishing House, Moscow, 1999).

Leontovich, M. A.

M. A. Leontovich, “A method to solve problems of electromagnetic wave propagation along the earth’s surface” (in Russian), Izv. Akad. Nauk USSR Ser. Fiz. 8(1), 16–22 (1944).

Levashov, V. E.

V. E. Levashov, A. V. Vinogradov, “Analytical theory of the zone plates efficiency,” Phys. Rev. E 49, 5797–5803 (1994).
[CrossRef]

Levy, M.

M. Levy, Parabolic Equation Method for Electromagnetic Wave Propagation (Institute of Electrical Engineers, London, 2000).

Lucero, A.

W. Meyer-lise, G. Denbeaux, L. E. Johnson, W. Bates, A. Lucero, E. H. Anderson, “The high resolution x-ray microscope XM-1,” in X-Ray Microscopy, Proceedings of the 6th International Conference (American Institute of Physics, Melville, N.Y., 2000), pp. 129–134.

Maser, J.

J. Maser, G. Schmahl, “Coupled wave description of the diffraction by zone plates with high aspect ratios,” Opt. Commun. 89, 355–362 (1992).
[CrossRef]

G. Schneider, J. Maser, “Zone plates as imaging optics in high diffraction orders described by coupled wave theory,” in X-Ray Microscopy and Spectromicroscopy, Proceedings of the 5th International Conference on X-Ray Microscopy and Spectromicroscopy, (Springer-Verlag, Berlin, 1998), pp. IV-71–IV-76 (on CD).

Meyer-lise, W.

W. Meyer-lise, G. Denbeaux, L. E. Johnson, W. Bates, A. Lucero, E. H. Anderson, “The high resolution x-ray microscope XM-1,” in X-Ray Microscopy, Proceedings of the 6th International Conference (American Institute of Physics, Melville, N.Y., 2000), pp. 129–134.

Michette, A. G.

A. G. Michette, Optical Systems for Soft X-Rays (Plenum, New York, 1986).

Minakov, A. A.

P. V. Bliokh, A. A. Minakov, “Diffraction of light and lens effect of the stellar gravitational field,” Astrophys. Space Sci. 34, L7–L9 (1975).
[CrossRef]

Morton, K. W.

R. D. Richtmayer, K. W. Morton, Difference Methods for Initial Value Problems, 2nd ed. (Interscience, New York, 1967).

Popov, A. V.

Yu. V. Kopylov, A. V. Popov, A. V. Vinogradov, “Diffraction phenomena inside thick Fresnel zone plates,” Radio Sci. 31, 1815–1822 (1996).
[CrossRef]

Yu. V. Kopylov, A. V. Popov, A. V. Vinogradov, “Application of the parabolic wave equation to x-ray diffraction optics,” Opt. Commun. 118, 619–636 (1995).
[CrossRef]

A. V. Vinogradov, A. V. Popov, Yu. V. Kopylov, A. N. Kurokhtin, Numerical Simulation of X-Ray Diffractive Optics (A&B Publishing House, Moscow, 1999).

Richtmayer, R. D.

R. D. Richtmayer, K. W. Morton, Difference Methods for Initial Value Problems, 2nd ed. (Interscience, New York, 1967).

Schmahl, G.

J. Maser, G. Schmahl, “Coupled wave description of the diffraction by zone plates with high aspect ratios,” Opt. Commun. 89, 355–362 (1992).
[CrossRef]

Schneider, G.

G. Schneider, “Zone plates with high efficiency in high orders of diffraction described by dynamical theory,” Appl. Phys. Lett. 71, 2242–2244 (1997).
[CrossRef]

G. Schneider, J. Maser, “Zone plates as imaging optics in high diffraction orders described by coupled wave theory,” in X-Ray Microscopy and Spectromicroscopy, Proceedings of the 5th International Conference on X-Ray Microscopy and Spectromicroscopy, (Springer-Verlag, Berlin, 1998), pp. IV-71–IV-76 (on CD).

Spector, S.

D. Tennant, S. Spector, A. Stein, C. Jacobsen, “Electron beam lithography of Fresnel zone plates using a rectilinear machine and trilayer resists,” in X-Ray Microscopy, Proceedings of the 6th International Conference (American Institute of Physics, Melville, N.Y., 2000), pp. 601–604.

Stein, A.

D. Tennant, S. Spector, A. Stein, C. Jacobsen, “Electron beam lithography of Fresnel zone plates using a rectilinear machine and trilayer resists,” in X-Ray Microscopy, Proceedings of the 6th International Conference (American Institute of Physics, Melville, N.Y., 2000), pp. 601–604.

Tennant, D.

D. Tennant, S. Spector, A. Stein, C. Jacobsen, “Electron beam lithography of Fresnel zone plates using a rectilinear machine and trilayer resists,” in X-Ray Microscopy, Proceedings of the 6th International Conference (American Institute of Physics, Melville, N.Y., 2000), pp. 601–604.

Vinogradov, A. V.

Yu. V. Kopylov, A. V. Popov, A. V. Vinogradov, “Diffraction phenomena inside thick Fresnel zone plates,” Radio Sci. 31, 1815–1822 (1996).
[CrossRef]

Yu. V. Kopylov, A. V. Popov, A. V. Vinogradov, “Application of the parabolic wave equation to x-ray diffraction optics,” Opt. Commun. 118, 619–636 (1995).
[CrossRef]

V. E. Levashov, A. V. Vinogradov, “Analytical theory of the zone plates efficiency,” Phys. Rev. E 49, 5797–5803 (1994).
[CrossRef]

A. V. Vinogradov, A. V. Popov, Yu. V. Kopylov, A. N. Kurokhtin, Numerical Simulation of X-Ray Diffractive Optics (A&B Publishing House, Moscow, 1999).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1980).

Appl. Phys. Lett. (1)

G. Schneider, “Zone plates with high efficiency in high orders of diffraction described by dynamical theory,” Appl. Phys. Lett. 71, 2242–2244 (1997).
[CrossRef]

Astrophys. Space Sci. (1)

P. V. Bliokh, A. A. Minakov, “Diffraction of light and lens effect of the stellar gravitational field,” Astrophys. Space Sci. 34, L7–L9 (1975).
[CrossRef]

Izv. Akad. Nauk USSR Ser. Fiz. (1)

M. A. Leontovich, “A method to solve problems of electromagnetic wave propagation along the earth’s surface” (in Russian), Izv. Akad. Nauk USSR Ser. Fiz. 8(1), 16–22 (1944).

J. Opt. Soc. Am. (1)

Opt. Commun. (2)

Yu. V. Kopylov, A. V. Popov, A. V. Vinogradov, “Application of the parabolic wave equation to x-ray diffraction optics,” Opt. Commun. 118, 619–636 (1995).
[CrossRef]

J. Maser, G. Schmahl, “Coupled wave description of the diffraction by zone plates with high aspect ratios,” Opt. Commun. 89, 355–362 (1992).
[CrossRef]

Phys. Rev. E (1)

V. E. Levashov, A. V. Vinogradov, “Analytical theory of the zone plates efficiency,” Phys. Rev. E 49, 5797–5803 (1994).
[CrossRef]

Radio Sci. (1)

Yu. V. Kopylov, A. V. Popov, A. V. Vinogradov, “Diffraction phenomena inside thick Fresnel zone plates,” Radio Sci. 31, 1815–1822 (1996).
[CrossRef]

Other (11)

R. D. Richtmayer, K. W. Morton, Difference Methods for Initial Value Problems, 2nd ed. (Interscience, New York, 1967).

W. Meyer-lise, G. Denbeaux, L. E. Johnson, W. Bates, A. Lucero, E. H. Anderson, “The high resolution x-ray microscope XM-1,” in X-Ray Microscopy, Proceedings of the 6th International Conference (American Institute of Physics, Melville, N.Y., 2000), pp. 129–134.

G. Schneider, J. Maser, “Zone plates as imaging optics in high diffraction orders described by coupled wave theory,” in X-Ray Microscopy and Spectromicroscopy, Proceedings of the 5th International Conference on X-Ray Microscopy and Spectromicroscopy, (Springer-Verlag, Berlin, 1998), pp. IV-71–IV-76 (on CD).

A. V. Vinogradov, A. V. Popov, Yu. V. Kopylov, A. N. Kurokhtin, Numerical Simulation of X-Ray Diffractive Optics (A&B Publishing House, Moscow, 1999).

X-Ray Microscopy, Proceedings of the 6th International Conference (American Institute of Physics, Melville, N.Y., 2000).

D. T. Attwood, Soft X-Rays and Extreme Ultraviolet Radiation, Principles and Applications (Cambridge U. Press, Cambridge, N.Y., 1999).

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1980).

A. G. Michette, Optical Systems for Soft X-Rays (Plenum, New York, 1986).

D. Tennant, S. Spector, A. Stein, C. Jacobsen, “Electron beam lithography of Fresnel zone plates using a rectilinear machine and trilayer resists,” in X-Ray Microscopy, Proceedings of the 6th International Conference (American Institute of Physics, Melville, N.Y., 2000), pp. 601–604.

V. A. Fock, Electromagnetic Diffraction and Propagation Problems (Pergamon, Oxford, UK, 1965).

M. Levy, Parabolic Equation Method for Electromagnetic Wave Propagation (Institute of Electrical Engineers, London, 2000).

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Figures (14)

Fig. 1
Fig. 1

Intensity distribution over one period of an (a) idealized, (b) realistic rough, and (c) fuzzy zone plate.

Fig. 2
Fig. 2

Half a period fragment of output field amplitude corresponding to Fig. 1 (solid curve, sharp; dots, rough; dashed curve, fuzzy interface).

Fig. 3
Fig. 3

Segment of fuzzy zone plate profile; fuzzy layer thickness a=1 nm, zone width Δr=10 nm.

Fig. 4
Fig. 4

Airy disk r=δ(z)=3.83z/kA of the reference perfect lens used in the definition of DFE.

Fig. 5
Fig. 5

DFE η(z) of a multifocus optical element (zone plate).

Fig. 6
Fig. 6

DFE for two zone plates: solid curve, exact zone law [Eq. (1)]; dashed curve, simplified Fresnel law rn=nλf.

Fig. 7
Fig. 7

First-order focusing efficiency as a function of zone plate radius: circles, idealized zone profile; squares, fuzzy profile; dashed curves, efficiency for thinner zone plates.  

Fig. 8
Fig. 8

Partial diffraction efficiency to second order of a nickel zone plate as a function of radius and zone plate thickness for λ=2.4 nm, f2=225 μm, L:S=1:1.

Fig. 9
Fig. 9

Second-order focus efficiency η2 of a nickel zone plate as a function of zone filling coefficient α=L/d; λ=2.4 nm, f2=225 μm, zone plate thickness b=480 nm.

Fig. 10
Fig. 10

Sketch of a tilted zone plate: ψ=ψ(r) is variable tilt angle; dashed curves show optimal thickness b(r).

Fig. 11
Fig. 11

Local grating efficiency to the second order (coupled-wave calculation). The bold curve shows the optimum geometrical thickness.

Fig. 12
Fig. 12

Sketch of a bent zone plate.

Fig. 13
Fig. 13

Geometrical illustration of the optimal thickness for a tilted zone plate.

Fig. 14
Fig. 14

Partial diffraction efficiencies obtained from the zone plate output-field analysis for a constant-thickness zone plate (squares) and its optimized variable-thickness modification (circles).

Tables (1)

Tables Icon

Table 1 Dependence of Second-Order Focusing Efficiency η2 on Fuzzy Layer Thickness a for a Constant Thickness Zone Plate and Its Variable-Thickness Modification

Equations (17)

Equations on this page are rendered with MathJax. Learn more.

rn=[nλf+(nλ/2)2]1/2,
dn=rn+1-rn-1λf/n(n1).
2ik uz+2ux2+2uy2+k2(-1)u=0,
2ik umn+1-umnτ
+um+1n+1-2umn+1+um-1n+1+um+1n-2umn+um-1n2h2
+um+1n+1-um-1n+1+um+1n-um-1n4mh2+k2(mn+1/2-1) 
×umn+1+umn2=0,
u(r, z)kizexp[ik(r2/2z)A]0u0(ρ)J0krz ρ×expikρ22z-ρ48z3ρdρ,
η(z)=2.39A20δ(z)|u(r, z)|2rdr.
dηdz2.39 kγA3z2Im0Au0(ρ)J1γ ρA×expikρ22z-ρ48z3ρ2dρ0Au0*(σ)J0γ σA×exp-ikσ22z-σ48z3σdσ
u0(ρ)A(ρ)exp{ik[S(ρ)-ρ2/2 f+ρ4/8f3]},
u0(ρ)=U0[ρ,t(ρ)]=m=-Cm(ρ)exp[-imt(ρ)],
t(ρ)=πn=kf(1+ρ2/f2-l)k(ρ2/2 f-ρ4/8f3)
ψm(r)=ϕm(r)/2mr/2 f.
bopt(r)2λf2(1+L/S)mr2.
b=b0,r<R0,
b=b0(R0/r)2,r>R0.

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