Abstract

A conventional Fresnel zone plate (FZP) consists of concentric rings with an alternating binary transmission of zero and one. In an azimuthally structured Fresnel zone plate (aFZP), the light transmission of the transparent zones is modulated in the azimuthal direction, too. The resulting structure is of interest for extreme ultraviolet and x-ray imaging, in particular, because of its improved mechanical stability as compared to the simple ring structure of an FZP. Here, we present an analysis of the optical performance of the aFZP based on scalar diffraction theory and show numerical results for the light distribution in the focal plane. These will be complemented by calculations of the optical transfer function.

© 2014 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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2010

2005

G. Andersen, “Large optical photon sieves,” Opt. Lett. 30, 2976–2978 (2005).
[CrossRef]

W. Chao, B. D. Harteneck, J. A. Liddle, and D. T. Attwood, “Soft x-ray microscopy at a spatial resolution better than 15 nm,” Nature 435, 1210–1213 (2005).
[CrossRef]

2003

2002

2001

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, “Sharper images by focusing soft x-rays with photon sieves,” Nature 414, 184–188 (2001).
[CrossRef]

1996

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, “A compound refractive lens for focusing high-energy x-rays,” Nature 384, 49–51 (1996).
[CrossRef]

1994

J. Als-Nielsen, D. Jacquemain, K. Kjaer, F. Leveiller, M. Lahav, and L. Leiserowitz, “Principles and applications of grazing incidence x-ray and neutron scattering from ordered molecular monolayers at the air-water interface,” Phys. Rep. 246, 251–313 (1994).
[CrossRef]

1990

1985

A. Heuberger, “X-ray lithography,” J. Vac. Sci. Technol. B 6, 107–121 (1985).
[CrossRef]

1969

G. Schmahl and D. Rudolph, “High-power zone plates as image forming systems for soft x-rays,” Optik 29, 577–585 (1969).

1964

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1964).

Adelung, R.

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, “Sharper images by focusing soft x-rays with photon sieves,” Nature 414, 184–188 (2001).
[CrossRef]

Als-Nielsen, J.

J. Als-Nielsen, D. Jacquemain, K. Kjaer, F. Leveiller, M. Lahav, and L. Leiserowitz, “Principles and applications of grazing incidence x-ray and neutron scattering from ordered molecular monolayers at the air-water interface,” Phys. Rep. 246, 251–313 (1994).
[CrossRef]

Andersen, G.

Attwood, D. T.

W. Chao, B. D. Harteneck, J. A. Liddle, and D. T. Attwood, “Soft x-ray microscopy at a spatial resolution better than 15 nm,” Nature 435, 1210–1213 (2005).
[CrossRef]

D. T. Attwood, Soft X-Rays and Extreme Ultraviolet Radiation: Principles and Applications (Cambridge University, 1999).

Berndt, R.

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, “Sharper images by focusing soft x-rays with photon sieves,” Nature 414, 184–188 (2001).
[CrossRef]

Cao, Q.

Chao, W.

W. Chao, B. D. Harteneck, J. A. Liddle, and D. T. Attwood, “Soft x-ray microscopy at a spatial resolution better than 15 nm,” Nature 435, 1210–1213 (2005).
[CrossRef]

Ezoe, Y.

Harm, S.

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, “Sharper images by focusing soft x-rays with photon sieves,” Nature 414, 184–188 (2001).
[CrossRef]

Harteneck, B. D.

W. Chao, B. D. Harteneck, J. A. Liddle, and D. T. Attwood, “Soft x-ray microscopy at a spatial resolution better than 15 nm,” Nature 435, 1210–1213 (2005).
[CrossRef]

Hayashi, T.

Helfert, S.

J. Jahns and S. Helfert, Introduction to Micro- and Nanooptics (VCH-Wiley, 2012).

Heuberger, A.

A. Heuberger, “X-ray lithography,” J. Vac. Sci. Technol. B 6, 107–121 (1985).
[CrossRef]

Jacquemain, D.

J. Als-Nielsen, D. Jacquemain, K. Kjaer, F. Leveiller, M. Lahav, and L. Leiserowitz, “Principles and applications of grazing incidence x-ray and neutron scattering from ordered molecular monolayers at the air-water interface,” Phys. Rep. 246, 251–313 (1994).
[CrossRef]

Jahns, J.

Johnson, R. L.

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, “Sharper images by focusing soft x-rays with photon sieves,” Nature 414, 184–188 (2001).
[CrossRef]

Kipp, L.

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, “Sharper images by focusing soft x-rays with photon sieves,” Nature 414, 184–188 (2001).
[CrossRef]

Kjaer, K.

J. Als-Nielsen, D. Jacquemain, K. Kjaer, F. Leveiller, M. Lahav, and L. Leiserowitz, “Principles and applications of grazing incidence x-ray and neutron scattering from ordered molecular monolayers at the air-water interface,” Phys. Rep. 246, 251–313 (1994).
[CrossRef]

Kohn, V.

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, “A compound refractive lens for focusing high-energy x-rays,” Nature 384, 49–51 (1996).
[CrossRef]

Koshiishi, M.

Lahav, M.

J. Als-Nielsen, D. Jacquemain, K. Kjaer, F. Leveiller, M. Lahav, and L. Leiserowitz, “Principles and applications of grazing incidence x-ray and neutron scattering from ordered molecular monolayers at the air-water interface,” Phys. Rep. 246, 251–313 (1994).
[CrossRef]

Leiserowitz, L.

J. Als-Nielsen, D. Jacquemain, K. Kjaer, F. Leveiller, M. Lahav, and L. Leiserowitz, “Principles and applications of grazing incidence x-ray and neutron scattering from ordered molecular monolayers at the air-water interface,” Phys. Rep. 246, 251–313 (1994).
[CrossRef]

Lengeler, B.

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, “A compound refractive lens for focusing high-energy x-rays,” Nature 384, 49–51 (1996).
[CrossRef]

Leveiller, F.

J. Als-Nielsen, D. Jacquemain, K. Kjaer, F. Leveiller, M. Lahav, and L. Leiserowitz, “Principles and applications of grazing incidence x-ray and neutron scattering from ordered molecular monolayers at the air-water interface,” Phys. Rep. 246, 251–313 (1994).
[CrossRef]

Liddle, J. A.

W. Chao, B. D. Harteneck, J. A. Liddle, and D. T. Attwood, “Soft x-ray microscopy at a spatial resolution better than 15 nm,” Nature 435, 1210–1213 (2005).
[CrossRef]

Lohmann, A. W.

A. W. Lohmann, Optical Information Processing (TU Ilmenau University, 2006).

Maeda, R.

Maeda, Y.

McCutchen, C. W.

Mita, M.

Mitsuda, K.

Mitsuishi, I.

Rudolph, D.

G. Schmahl and D. Rudolph, “High-power zone plates as image forming systems for soft x-rays,” Optik 29, 577–585 (1969).

Schmahl, G.

G. Schmahl and D. Rudolph, “High-power zone plates as image forming systems for soft x-rays,” Optik 29, 577–585 (1969).

Seemann, R.

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, “Sharper images by focusing soft x-rays with photon sieves,” Nature 414, 184–188 (2001).
[CrossRef]

Shirata, T.

Skibowski, M.

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, “Sharper images by focusing soft x-rays with photon sieves,” Nature 414, 184–188 (2001).
[CrossRef]

Snigirev, A.

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, “A compound refractive lens for focusing high-energy x-rays,” Nature 384, 49–51 (1996).
[CrossRef]

Snigireva, I.

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, “A compound refractive lens for focusing high-energy x-rays,” Nature 384, 49–51 (1996).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1964).

Takano, T.

Walker, S. J.

Yamasaki, N. Y.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Vac. Sci. Technol. B

A. Heuberger, “X-ray lithography,” J. Vac. Sci. Technol. B 6, 107–121 (1985).
[CrossRef]

Nature

W. Chao, B. D. Harteneck, J. A. Liddle, and D. T. Attwood, “Soft x-ray microscopy at a spatial resolution better than 15 nm,” Nature 435, 1210–1213 (2005).
[CrossRef]

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, “A compound refractive lens for focusing high-energy x-rays,” Nature 384, 49–51 (1996).
[CrossRef]

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, “Sharper images by focusing soft x-rays with photon sieves,” Nature 414, 184–188 (2001).
[CrossRef]

Opt. Lett.

Optik

G. Schmahl and D. Rudolph, “High-power zone plates as image forming systems for soft x-rays,” Optik 29, 577–585 (1969).

Phys. Rep.

J. Als-Nielsen, D. Jacquemain, K. Kjaer, F. Leveiller, M. Lahav, and L. Leiserowitz, “Principles and applications of grazing incidence x-ray and neutron scattering from ordered molecular monolayers at the air-water interface,” Phys. Rep. 246, 251–313 (1994).
[CrossRef]

Other

D. T. Attwood, Soft X-Rays and Extreme Ultraviolet Radiation: Principles and Applications (Cambridge University, 1999).

A. W. Lohmann, Optical Information Processing (TU Ilmenau University, 2006).

J. Jahns and S. Helfert, Introduction to Micro- and Nanooptics (VCH-Wiley, 2012).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1964).

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Figures (12)

Fig. 1.
Fig. 1.

Various diffractive elements used for focusing: (a) conventional FZP, (b) photon sieve, and (c) azimuthally structured FZP.

Fig. 2.
Fig. 2.

(a) Transmission function of conventional FZP in radial direction shown as a function of r02. Azimuthal modulation of a single ring is shown (b) for Cartesian coordinates and (c) along the azimuthal coordinate ϕ for r0=const.K, number of openings of the ring (later referred to as Km for the mth ring); ϕl and ϕu, lower and upper coordinates for single azimuthal opening; Δϕ, width of the opening; ϕs, random azimuthal offset of a ring.

Fig. 3.
Fig. 3.

(a) Notation used for polar coordinates in object and observation plane. (b) Setup considered consisting of aFZP illuminated by a plane wave of wavelength λ. The focus is generated at a distance z=r12/2λ from the aFZP.

Fig. 4.
Fig. 4.

Conventional FZP: (a) transmission function in (x0,y0) for M=5. (b) FZP shown in (r02,ϕ) diagram. (c) 2D amplitude distribution in the focal plane (f, focal length; D, diameter of the FZP: D=2R). (d) Optical transfer function as autocorrelation function in (r02,ϕ)-coordinates.

Fig. 5.
Fig. 5.

Conventional FZP: (a) normalized intensities in the focal plane, calculated for finite ring widths (solid line) and infinitesimal ring widths (dashed line). (b) Difference of intensities. Here M=5.

Fig. 6.
Fig. 6.

Conventional FZP: (a) normalized intensities in the focal plane, calculated for finite ring widths (solid line) and infinitesimal ring widths (dashed line). (b) Difference of intensities. Here M=50.

Fig. 7.
Fig. 7.

Error ΔImax as a function of the number of rings M.

Fig. 8.
Fig. 8.

Case 1: aFZP in (x0,y0) for M=5 rings, K1=11 openings, and ΔK=0. (b) aFZP shown in (r02,ϕ) diagram. (c) 2D amplitude distribution in focal plane. (d) Autocorrelation in (r02,ϕ).

Fig. 9.
Fig. 9.

Case 2: aFZP in (x0,y0) for M=5 rings, K1=11 openings, and ΔK=2. (b) aFZP shown in (r02,ϕ) diagram. (c) 2D amplitude distribution in focal plane. (d) Autocorrelation in (r02,ϕ).

Fig. 10.
Fig. 10.

Case 3: aFZP in (x0,y0) for M=5 rings, K1=11 openings, ΔK=0, and a random azimuthal phase added. (b) aFZP shown in (r02,ϕ) diagram. (c) 2D amplitude distribution in focal plane. (d) Autocorrelation in (r02,ϕ).

Fig. 11.
Fig. 11.

Case 4: aFZP in (x0,y0) for M=5 rings, K1=11 openings, ΔK=2, and a random azimuthal phase added. (b) aFZP shown in (r02,ϕ) diagram. (c) 2D amplitude distribution in focal plane. (d) Autocorrelation in (r02,ϕ).

Fig. 12.
Fig. 12.

Large aFZP: intensity plots for (a) M=10 rings, K1=20 openings, and ΔK=2; and (b) M=50 rings, K1=10 openings, and ΔK=1. In both cases, ϕs=0. Gray curves show the intensity of a focal plot generated by a conventional FZP.

Tables (1)

Tables Icon

Table 1. Four Design Variations

Equations (26)

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g(r02)=m=1Mvm(r02)
vm(r02)=rect[r02[m(1/2)]r12r12/2].
wm(ϕ)=k=1Kmrect[ϕk(2π/Km)ϕsπ/Km].
ga(r02,ϕ)=m=1Mvm(r02)wm(ϕ).
u(x,y,z)eikzλzu0(x0,y0)eik2z[(x+x0)2+(y+y0)2]dx0dy0.
x0=r0cosϕundy0=r0sinϕ
x=rcosθundy=rsinθ.
u(r,θ,z)=eikzr02=0ϕ=02πga(r02,ϕ)eiπλz(r02+r2)ei2πrr0λzcos(ϕθ)dϕdr02.
u(r,θ,z)=eikzm=1Mr02=0vm(r02)Wm(r,θ)ei2πr022λzdr02.
Wm(r,θ,z)=ϕ=02πwm(ϕ)ei2πrr0λzcos(ϕθ)dϕ.
Wm(r,z)=2πJ0(2πrr0λz).
u(r,z=f)=(2πR2)ei2πr2λfJ1(2πRr/λf)2πRr/λf.
OTF(r02,ϕ)=r02ϕga(r02,ϕ)ga(r02r02,ϕϕ)dr02dϕr02ϕga2(r02,ϕ)dr02dϕ.
vm(r02)=δ(r02rm2)
u(r,θ,z)m=1Mr0=0δ(r02rm2)Wm(r,θ,z)ei2πr022λzdr02.
u(r,θ,z)m=1Meiπλzmr12Wm(rm,θ,z).
u(r,θ,z=f)m=1MWm(rm,θ,f).
u(r,θ,z=f)2πm=1MJ0(2πrmrλz).
Wm=k=1Kmϕl(k)ϕu(k)ei2πrrmλzcos(ϕθ)dϕ.
ei2πrrmλzcosφ=J0(2πrrmλz)+2n=1inJn(2πrrmλz)cosφ.
φlφuei2πrrmλzcosφdφ=J0(2πrrmλz)Δφ+φlφu2n=1inJn(2πrrmλz)cosφdφ=J0(2πrrmλz)Δφ+2n=1innJn(2πrrmλz)[sin(nφu)sin(nφl)].
Wm=k=1Km[J0(2πrrmλz)πKm+2n=1innJn(2πrrmλz){sin(nφu(k))sin(n(φl(k))}].
Wm=πJ0(2πrrmλz)+2k=1Kmn=1innJn(2πrrmλz){sin[n(ϕu(k)θ)]sin[n(ϕl(k)θ)]}.
WmπJ0(2πrrmλz)+2k=1Kmn=1KminnJn(2πrrmλz){sin[n(ϕu(k)θ)]sin[n(ϕl(k)θ)]}.
u(r,θ,z)m=1Mei2πmr122λzWm(r,θ,z).
u(r,θ,z=f)m=1MWm(r,θ,f),

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