Abstract

Recently, a paraxially individual far-field model was presented for the focusing and imaging analysis of pinhole photon sieves. By use of a local Taylor expansion of the integrated function of the Rayleigh–Sommerfeld diffraction formula, the small-size property of the individual pinholes, and the linear superposition principle, we extend this model to the nonparaxial case of high-numerical-aperture photon sieves. Some related problems, such as the validity range of this nonparaxial model and the selection conditions for the individual pinholes, are also discussed in detail.

© 2003 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.
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  5. E. H. Anderson, V. Boegli, L. P. Muray, “Electron beam lithography digital pattern generator and electronics for generalized curvilinear structures,” J. Vac. Sci. Technol. B 13, 2529–2534 (1995).
    [CrossRef]
  6. E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, D. Attwood, “Nanofabrication and diffractive optics for high-resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
    [CrossRef]
  7. L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, R. Seemann, “Sharper images by focusing soft X-rays with photon sieves,” Nature 414, 184–188 (2001).
    [CrossRef] [PubMed]
  8. G. E. Artzner, J. P. Delaboudinière, X. Y. Song, “Photon sieves as EUV telescopes for solar orbiter,” to be published in Innovative Telescopes and Instrumentation for Solar Astrophysics, S. L. Keil, S. V. Avakyan, S. I. Vavilov, eds., Proc. SPIE4853, 158–161 (2003).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  11. J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
    [CrossRef]
  12. C. J. R. Sheppard, M. Hrynevych, “Diffraction by a circular aperture: a generalization of Fresnel diffraction theory,” J. Opt. Soc. Am. A 9, 274–281 (1992).
    [CrossRef]
  13. See, for example, R. K. Tyson, Principles of Adaptive Optics, 2nd ed. (Academic, Boston, Mass., 1998), Subsect. 5.3.2.
  14. Y-T. Wang, Y. C. Pati, T. Kailath, “Depth of focus and the moment expansion,” Opt. Lett. 20, 1841–1843 (1995).
    [CrossRef] [PubMed]
  15. 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]
  16. M. Lax, W. H. Louisell, W. B. McKnight, “From Max-well to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [CrossRef]
  17. G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
    [CrossRef]
  18. Q. Cao, X. Deng, “Corrections to the paraxial approximation of an arbitrary free-propagation beam,” J. Opt. Soc. Am. A 15, 1144–1148 (1998).
    [CrossRef]
  19. Q. Cao, “Corrections to the paraxial approximation solutions in transversely nonuniform refractive-index media,” J. Opt. Soc. Am. A 16, 2494–2499 (1999).
    [CrossRef]

2002 (1)

2001 (2)

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

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]

2000 (1)

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, D. Attwood, “Nanofabrication and diffractive optics for high-resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[CrossRef]

1999 (1)

1998 (1)

1995 (2)

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

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

1992 (1)

1991 (1)

1983 (1)

G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

1981 (1)

1979 (1)

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[CrossRef]

1975 (1)

M. Lax, W. H. Louisell, W. B. McKnight, “From Max-well to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

1968 (1)

1967 (1)

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 sieves,” Nature 414, 184–188 (2001).
[CrossRef] [PubMed]

Agrawal, G. P.

G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

Anderson, E. H.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, D. Attwood, “Nanofabrication and diffractive optics for high-resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[CrossRef]

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

Arsenault, H.

Artzner, G. E.

G. E. Artzner, J. P. Delaboudinière, X. Y. Song, “Photon sieves as EUV telescopes for solar orbiter,” to be published in Innovative Telescopes and Instrumentation for Solar Astrophysics, S. L. Keil, S. V. Avakyan, S. I. Vavilov, eds., Proc. SPIE4853, 158–161 (2003).
[CrossRef]

Attwood, D.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, D. Attwood, “Nanofabrication and diffractive optics for high-resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[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 sieves,” Nature 414, 184–188 (2001).
[CrossRef] [PubMed]

Boegli, V.

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

Cai, A.

Cao, Q.

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]

Chao, W.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, D. Attwood, “Nanofabrication and diffractive optics for high-resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[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.

Delaboudinière, J. P.

G. E. Artzner, J. P. Delaboudinière, X. Y. Song, “Photon sieves as EUV telescopes for solar orbiter,” to be published in Innovative Telescopes and Instrumentation for Solar Astrophysics, S. L. Keil, S. V. Avakyan, S. I. Vavilov, eds., Proc. SPIE4853, 158–161 (2003).
[CrossRef]

Denbeaux, G.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, D. Attwood, “Nanofabrication and diffractive optics for high-resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[CrossRef]

Deng, X.

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 sieves,” Nature 414, 184–188 (2001).
[CrossRef] [PubMed]

Harteneck, B.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, D. Attwood, “Nanofabrication and diffractive optics for high-resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[CrossRef]

Harvey, J. E.

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[CrossRef]

Hrynevych, M.

Jahns, J.

Johnson, L.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, D. Attwood, “Nanofabrication and diffractive optics for high-resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[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 sieves,” Nature 414, 184–188 (2001).
[CrossRef] [PubMed]

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 sieves,” Nature 414, 184–188 (2001).
[CrossRef] [PubMed]

Lax, M.

G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

M. Lax, W. H. Louisell, W. B. McKnight, “From Max-well to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Louisell, W. H.

M. Lax, W. H. Louisell, W. B. McKnight, “From Max-well to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Lucero, A.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, D. Attwood, “Nanofabrication and diffractive optics for high-resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[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]

McKnight, W. B.

M. Lax, W. H. Louisell, W. B. McKnight, “From Max-well to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[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]

Mittra, R.

Muray, L. P.

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

Olynick, D. L.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, D. Attwood, “Nanofabrication and diffractive optics for high-resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[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 sieves,” Nature 414, 184–188 (2001).
[CrossRef] [PubMed]

Semonin, R. G.

Sheppard, C. J. R.

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 sieves,” Nature 414, 184–188 (2001).
[CrossRef] [PubMed]

Song, X. Y.

G. E. Artzner, J. P. Delaboudinière, X. Y. Song, “Photon sieves as EUV telescopes for solar orbiter,” to be published in Innovative Telescopes and Instrumentation for Solar Astrophysics, S. L. Keil, S. V. Avakyan, S. I. Vavilov, eds., Proc. SPIE4853, 158–161 (2003).
[CrossRef]

Southwell, W. H.

Stigliani, D. J.

Sun, J. A.

Tyson, R. K.

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

Veklerov, E.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, D. Attwood, “Nanofabrication and diffractive optics for high-resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[CrossRef]

Wang, Y-T.

Am. J. Phys. (1)

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[CrossRef]

J. Opt. Soc. Am. (3)

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

J. Vac. Sci. Technol. B (2)

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

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, D. Attwood, “Nanofabrication and diffractive optics for high-resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[CrossRef]

Nature (1)

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

Opt. Commun. (1)

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. (1)

Phys. Rev. A (2)

M. Lax, W. H. Louisell, W. B. McKnight, “From Max-well to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

Other (3)

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

G. E. Artzner, J. P. Delaboudinière, X. Y. Song, “Photon sieves as EUV telescopes for solar orbiter,” to be published in Innovative Telescopes and Instrumentation for Solar Astrophysics, S. L. Keil, S. V. Avakyan, S. I. Vavilov, eds., Proc. SPIE4853, 158–161 (2003).
[CrossRef]

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.

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Figures (4)

Fig. 1
Fig. 1

Schematic view of a photon sieve illuminated by an arbitrary scalar field.

Fig. 2
Fig. 2

Comparison between the far-field term F0(R) and Un(R)exp(-jkH), where Un(R) is the exact diffracted field distribution at the focal plane. The solid line is the real part of Un(R)exp(-jkH), and the asterisks plot the far-field term F0(R). The field values have been normalized according to F0(R=0)=1. See text for the concrete parameters. (a) For a pinhole corresponding to Nf=7.4×10-4; the negligible imaginary part of Un(R)exp(-jkH) is completely indistinguishable from zero in this case. (b) For a pinhole corresponding to Nf=0.06; the dashed curve is the small imaginary part of Un(R)exp(-jkH).

Fig. 3
Fig. 3

Comparison between the sum F0(R)+jF1(R) of the far-field term and the quasi-far-field term and Un(R)exp(-jkH) for a very large pinhole, where Un(R) is the exact diffracted field distribution at the focal plane. See text for the concrete parameters. The field values have been normalized according to F0(R=0)=1. (a) The solid line is the real part of Un(R)exp(-jkH) and the asterisks plot the far-field term F0(R). (b) The solid line is the imaginary part of Un(R)exp(-jkH), while the asterisks plot the quasi-far-field correction term F1(R).

Fig. 4
Fig. 4

Comparison between the sum F0(R)+jF1(R) of the far-field term and the quasi-far-field term and Un(R)exp(-jkH) for an extremely large pinhole. All notations are the same as those in Fig. 3. See text for the concrete parameters.

Equations (22)

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

Un(X, Y)=12πSnVn(x, y) exp(jkr01)r01qr01×-jk+1r01dxdy,
Un(X, Y)=1λSnVn(x, y)exp(jkr01) qr012dxdy,
r01=(q2+X2+Y2-2Xx-2Yy+x2+y2)1/2,
r01=(q2+R2)1/2+r2-2Xx-2Yy2(q2+R2)1/2,
Un(X, Y)=qλH2exp(jkH)SnVn(x, y)×expjk r2-2Xx-2Yy2Hdxdy,
Vn(x, y)=Anexp(jkLn)×exp{jk[gn(x-xn)+hn(y-yn)]},
Un(X, Y)=AnqλH2exp[jk(Ln+H)]Sn×expjk r2-2Xx-2Yy2Hdxdy,
Un(X, Y)=exp[jk(Ln+H)]F(X, Y),
F(X, Y)=kAnqH20anexpjk r22H×J0kρH rrdr,
F0(X, Y)=kAnan2qH2JinckanH ρ,
F1(X, Y)=k2Anan4q24H33J0kanH ρ+2J2kanH ρ-J4kanH ρ.
Un(X, Y)=kAnan2qH2exp[jk(Ln+H)]JinckanH ρ.
Un(0, 0)=kqAnan2Qn2exp[jk(Ln+Qn)]JinckanQn Rn,
k(Ln+Qn)=2mπ+const.,JinckanQn Rn>0,
k(Ln+Qn)=(2m+1)π+const.,JinckanQn Rn<0.
Un(0, 0)=kqAnan2Qn2exp[jk(Pn+Qn)]Jinckanfn rn,
w=πfnkrn.
Un(0, 0)dwJ1π2dw,
W(X, Y)jkU(X, Y)X.
W(X, Y)=n=1NWn(X, Y),
Wn(X, Y)=jkUn(X, Y)X,
Wn(X, Y)-X-xnH Un(X, Y).

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