Abstract

Recently, a new class of diffractive optical element called a photon sieve, which consists of a great number of pinholes, was developed for the focusing and imaging of soft x rays. In terms of the closed-form formula for the far field of individual pinholes and the linear superposition principle, we present a simple yet accurate analytical model for the focusing of the pinhole photon sieve. This model is applicable to arbitrary paraxial illumination with arbitrary complex amplitude distribution at the photon sieve plane. We check the validity range of this model by comparing it with the exact Fresnel diffraction integral. Some special problems, such as the individual quasi-far-field correction for very large pinholes and the related phase shift induced by this correction, are also discussed.

© 2002 Optical Society of America

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References

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  1. G. Schmahl, D. Rudolph, P. Guttmann, O. Christ, “Zone plates for x-ray microscopy,” in X-Ray Microscopy, G. Schmahl, D. Rudolph, eds. (Springer-Verlag, Berlin, 1984), Vol. 43, pp. 63–74.
  2. E. H. Anderson, V. Boegli, L. P. Muray, “Electron beam lithography digital pattern generator and electronics forgeneralized curvilinear structures,” J. Vac. Sci. Technol. B 13, 2529–2534 (1995).
    [CrossRef]
  3. L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, R. Seemann, “Sharper images by focusing soft x-rays with photon sieve,” Nature (London) 414, 184–188 (2001).
    [CrossRef]
  4. See, for example, R. K. Tyson, Principles of Adaptive Optics, 2nd ed. (Academic, Boston, Mass., 1998), Subsect. 5.3.2.
  5. Y-T. Wang, Y. C. Pati, T. Kailath, “Depth of focus and the moment expansion,” Opt. Lett. 20, 1841–1843 (1995).
    [CrossRef] [PubMed]
  6. J. I. Garcı́a-Sucerquia, R. Castañeda, F. F. Medina, G. Matteucci, “Distinguishing between Fraunhofer and Fresnel diffraction by the Young’s experiment,” Opt. Commun. 200, 15–22 (2001).
    [CrossRef]
  7. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Wiley, New York, 1972).

2001

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

J. I. Garcı́a-Sucerquia, R. Castañeda, F. F. Medina, G. Matteucci, “Distinguishing between Fraunhofer and Fresnel diffraction by the Young’s experiment,” Opt. Commun. 200, 15–22 (2001).
[CrossRef]

1995

Y-T. Wang, Y. C. Pati, T. Kailath, “Depth of focus and the moment expansion,” Opt. Lett. 20, 1841–1843 (1995).
[CrossRef] [PubMed]

E. H. Anderson, V. Boegli, L. P. Muray, “Electron beam lithography digital pattern generator and electronics forgeneralized curvilinear structures,” J. Vac. Sci. Technol. B 13, 2529–2534 (1995).
[CrossRef]

Adelung, R.

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

Anderson, E. H.

E. H. Anderson, V. Boegli, L. P. Muray, “Electron beam lithography digital pattern generator and electronics forgeneralized curvilinear structures,” J. Vac. Sci. Technol. B 13, 2529–2534 (1995).
[CrossRef]

Berndt, R.

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

Boegli, V.

E. H. Anderson, V. Boegli, L. P. Muray, “Electron beam lithography digital pattern generator and electronics forgeneralized curvilinear structures,” J. Vac. Sci. Technol. B 13, 2529–2534 (1995).
[CrossRef]

Castañeda, R.

J. I. Garcı́a-Sucerquia, R. Castañeda, F. F. Medina, G. Matteucci, “Distinguishing between Fraunhofer and Fresnel diffraction by the Young’s experiment,” Opt. Commun. 200, 15–22 (2001).
[CrossRef]

Christ, O.

G. Schmahl, D. Rudolph, P. Guttmann, O. Christ, “Zone plates for x-ray microscopy,” in X-Ray Microscopy, G. Schmahl, D. Rudolph, eds. (Springer-Verlag, Berlin, 1984), Vol. 43, pp. 63–74.

Garci´a-Sucerquia, J. I.

J. I. Garcı́a-Sucerquia, R. Castañeda, F. F. Medina, G. Matteucci, “Distinguishing between Fraunhofer and Fresnel diffraction by the Young’s experiment,” Opt. Commun. 200, 15–22 (2001).
[CrossRef]

Guttmann, P.

G. Schmahl, D. Rudolph, P. Guttmann, O. Christ, “Zone plates for x-ray microscopy,” in X-Ray Microscopy, G. Schmahl, D. Rudolph, eds. (Springer-Verlag, Berlin, 1984), Vol. 43, pp. 63–74.

Harm, S.

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

Johnson, R. L.

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

Kailath, T.

Kipp, L.

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

Matteucci, G.

J. I. Garcı́a-Sucerquia, R. Castañeda, F. F. Medina, G. Matteucci, “Distinguishing between Fraunhofer and Fresnel diffraction by the Young’s experiment,” Opt. Commun. 200, 15–22 (2001).
[CrossRef]

Medina, F. F.

J. I. Garcı́a-Sucerquia, R. Castañeda, F. F. Medina, G. Matteucci, “Distinguishing between Fraunhofer and Fresnel diffraction by the Young’s experiment,” Opt. Commun. 200, 15–22 (2001).
[CrossRef]

Muray, L. P.

E. H. Anderson, V. Boegli, L. P. Muray, “Electron beam lithography digital pattern generator and electronics forgeneralized curvilinear structures,” J. Vac. Sci. Technol. B 13, 2529–2534 (1995).
[CrossRef]

Pati, Y. C.

Rudolph, D.

G. Schmahl, D. Rudolph, P. Guttmann, O. Christ, “Zone plates for x-ray microscopy,” in X-Ray Microscopy, G. Schmahl, D. Rudolph, eds. (Springer-Verlag, Berlin, 1984), Vol. 43, pp. 63–74.

Schmahl, G.

G. Schmahl, D. Rudolph, P. Guttmann, O. Christ, “Zone plates for x-ray microscopy,” in X-Ray Microscopy, G. Schmahl, D. Rudolph, eds. (Springer-Verlag, Berlin, 1984), Vol. 43, pp. 63–74.

Seemann, R.

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

Skibowski, M.

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

Tyson, R. K.

See, for example, R. K. Tyson, Principles of Adaptive Optics, 2nd ed. (Academic, Boston, Mass., 1998), Subsect. 5.3.2.

Wang, Y-T.

J. Vac. Sci. Technol. B

E. H. Anderson, V. Boegli, L. P. Muray, “Electron beam lithography digital pattern generator and electronics forgeneralized curvilinear structures,” J. Vac. Sci. Technol. B 13, 2529–2534 (1995).
[CrossRef]

Nature (London)

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

Opt. Commun.

J. I. Garcı́a-Sucerquia, R. Castañeda, F. F. Medina, G. Matteucci, “Distinguishing between Fraunhofer and Fresnel diffraction by the Young’s experiment,” Opt. Commun. 200, 15–22 (2001).
[CrossRef]

Opt. Lett.

Other

G. Schmahl, D. Rudolph, P. Guttmann, O. Christ, “Zone plates for x-ray microscopy,” in X-Ray Microscopy, G. Schmahl, D. Rudolph, eds. (Springer-Verlag, Berlin, 1984), Vol. 43, pp. 63–74.

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Wiley, New York, 1972).

See, for example, R. K. Tyson, Principles of Adaptive Optics, 2nd ed. (Academic, Boston, Mass., 1998), Subsect. 5.3.2.

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Figures (4)

Fig. 1
Fig. 1

Schematic view of a general photon sieve with arbitrary paraxial illumination.

Fig. 2
Fig. 2

Analytical description of the detailed change of the field value Un(0, 0) at the focal point with the increase of the ratio d/w. The center of the pinhole is located at the center of a white zone.

Fig. 3
Fig. 3

Comparison between the individual far-field term F0(ρ2) and the exact numerical integral F(ρ2). The solid line is the real part of F(ρ2), and the asterisks are the far-field term F0(ρ2). The Fresnel number Nf is chosen such that Nf=0.05. The field values have been normalized according to F0(ρ2=0)=1. The negligible imaginary part of the exact numerical integral F(ρ2) is also shown as the dashed curve.

Fig. 4
Fig. 4

Comparison between the individual quasi-far-field approximation [i.e., the sum of the far-field term F0(ρ2) and the quasi-far-field correction term jF1(ρ2)] and the exact numerical integral F(ρ2). The Fresnel number Nf is chosen such that Nf=0.2. The field values have been normalized according to F0(ρ2=0)=1. (a) The solid line is the real part of F(ρ2), and the asterisks are the far-field term F0(ρ2). (b) The solid line is the imaginary part of F(ρ2), and the asterisks are the quasi-far-field correction term F1(ρ2).

Equations (34)

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Un(X, Y)=1λq-Vn(x, y)×expjk (X-x)2+(Y-y)22qdxdy,
Vn(x, y)=Anexp(jkLn)exp{jk[gn(x-xn)+hn(y-yn)]},
Un(X, Y)
=AnλqexpjkLn+R22q×Snexpjk r2-2Xx-2Yy2qdxdy,
Un(X, Y)=2AnexpjkLn+R22qF(ρ),
F(ρ)=πλq0anexpjk r22qJ0kρq rrdr,
F0(ρ)=NfJinckanq ρ,
F1(ρ)=Nf2123J0kanq ρ+2J2kanq ρ-J4kanq ρ,
Un(X, Y)=2NfAnexpjkLn+R22qJinckanq ρ.
Un(0, 0)=2NfAnexpjkLn+rn22qJinckanq Rn,
kLn+rn22q=2mπ,Jinckanq Rn>0,
kLn+rn22q=(2m+1)π,Jinckanq Rn<0.
Un(0, 0)=2NfAnexpjkrn22 fJinckanf rn,
kanrnf=3.832, 7.016, 10.173, 13.324, .
krnf=πw.
dw=2.44, 4.47, 6.48, 8.48,,
Un(0, 0)dw J1π2dw.
F(ρ2)=Nf01exp(jNfρ12)J0(2Nfρ1ρ2)ρ1dρ1,
F0(ρ2)=NfJinc(2Nfρ2),
F1(ρ2)=Nf212 [3J0(2Nfρ2)+2J2(2Nfρ2)-J4(2Nfρ2)].
kLn+rn22q+φ(Rn)=2mπ,
0anexpjk r22qC(ρ, r)rdr,
C(ρ, r)=02πexp-jkρrcos ϕ cos θ+sin ϕ sin θqdθ.
C(ρ, r)=02πexp-j kρrqcos θdθ.
J0(-v)=12π02πexp(-jv cos θ)dθ
C(ρ, r)=2πJ0kρq r.
F0(ρ)=πλq0anJ0kρq rrdr,
F1(ρ)=πλq0ankr22q J0kρq rrdr.
F0(ρ)=Nfu20uvJ0(v)dv,
F1(ρ)=Nf2u40uv3J0(v)dv,
F0(ρ)=NfJ1(u)u,
F1(ρ)=Nf2u22J0(u)+uJ1(u)-4 J1(u)u.
F1(ρ)=Nf2J1(u)u-2uJ2(u)u.
F1(ρ)=Nf22J0(u)+J2(u)-J1(u)u-J3(u)u.

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