Abstract

Schemes for focusing a hard x-ray beam to a small spot are described. The theoretical minimum spot size, assuming perfect mirror shape, is shown to be 4nm FWHM, independent of x-ray wavelength. This is less than the 10nm previously said to be the minimum achievable diffraction-limited x-ray spot size. While providing the penetrating power only possible with x rays, this approaches the resolution needed to image individual atoms or atomic layers. However, the perfect mirror assumption is physically unrealistic. This paper discusses the compensation of mirror shape errors by a corrector plate and shows that the tolerances for corrector plate shape are far looser than are tolerances for mirror shape. The full eventual success of achieving theoretical minimum resolution will require mirror shape precision considerably better than has been achieved at this time, though far looser than would be required for simpleminded paraboloidal focusing. Two variants of the scheme, subject to the same mathematical treatment, are described. (i) The “corrector plate” name is copied from the similarly functioning element of the same name in a Schmidt camera. The focusing is achieved using glancing, yet coherent, reflection from a high-Z paraboloidal mirror. The strategy is to obtain dominant focusing from reflection and to compensate with weak refractive focusing. The reflective focusing is strong and achromatic but insufficiently accurate. The refractive focusing is weak and chromatic but highly accurate. The corrector plate improves resolution the way eyeglasses help a person to see. It can, for example, be “fitted” the same trial-and-error way an optometrist establishes a prescription for glasses. Dimensional tolerances for the compensator are far looser than would be needed for a mirror to achieve the same resolution. Unlike compound refractive lenses, attenuation will be small, at least for wavelengths longer than 1Å, because the compensation layer is thin. (ii) For this variant, the corrector plate is a washer-shaped refractive or Fresnel lens, and the mirror is (theoretically) a perfect cone. All focusing is provided by the lens. Even though the cone provides no focusing, it improves the resolution by increasing the numerical aperture of the device. Compared to a paraboloidal shape, it is assumed that the conical shape can be more accurately fabricated. Of the two variants, only the first variant is, in principle, capable of achieving the theoretical minimum resolution. Configurations are suggested, in both case (i) and case (ii), that use currently possible construction precisions to produce resolutions better than have been achieved to date. However, both results will remain well above the theoretical minimum until fabrication techniques have been developed that provide greater precision than is possible at this time.

© 2009 Optical Society of America

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References

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  1. B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science 319, 810-813 (2008).
    [CrossRef] [PubMed]
  2. R. Medenwaldt and E. Uggerhøj, “Description of an x-ray microscope with 30 nm resolution,” Rev. Sci. Instrum. 69, 2974-2977 (1998).
    [CrossRef]
  3. V. V. Aristov, Yu A. Basov, S. V. Redkin, A. A. Snigirev, and V. A. Yunkin, “Bragg zone plates for hard x-ray focusing,” Nucl. Instrum. Methods Phys. Res. A 261, 72-74 (1987).
    [CrossRef]
  4. C. Bergemann, H. Keymeulen, and J. Vander Veen, “Focusing x-ray beams to nanometer dimensions,” Phys. Rev. Lett. 91, 204801 (2003).
    [CrossRef] [PubMed]
  5. D.Mills, ed., Third Generation Hard X-Ray Synchrotron Radiation Sources (Wiley-Interscience, 2002).
  6. S. Tamura, M. Yasumoto, N. Kamijo, Y. Suzuki, M. Awaji, A. Takeuchi, H. Takano, and K. Handa, “Development of a multilayer Fresnel zone plate for high-energy synchrotron radiation x-rays by DC sputtering deposition,” J. Synchrotron Radiat. 9, 154-159 (2002).
    [CrossRef] [PubMed]
  7. K. Evans-Lutterodt, A. Stein, J. Ablett, N. Bozovic, A. Taylor, and D. Tennant, “Using compound kinoform hard-x-ray lenses to exceed the critical angle limit,” Phys. Rev. Lett. 99, 134801 (2007).
    [CrossRef] [PubMed]
  8. D. Bilderback and S. Hubbard, “X-ray mirror reflectivities from 3.8 to 50 keV (3.3 to 0.25 Å Part II: Pt, Si, and other materials,” Nucl. Instrum. Methods Phys. Res. 195, 91-95(1982).
    [CrossRef]
  9. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).
  10. D. Jackson, Electrodynamics, 3rd ed. (Wiley, 1998), p. 490.
  11. D. Korsch, Reflective Optics (Academic, 1991).
  12. T. Kawai, K. Sawada, and Y. Takeuchi, “Ultra-precision micro structuring by means of mechanical machining” (IEEE, 2001), paper 0-7803-5998-4/01.
  13. J. Hubbell and S. Seltzer, X-Ray Attenuation Coefficients (NIST, 1996).
  14. B. X. Yang, “Fresnel and refractive lenses for x-rays,”Nucl. Instrum. Methods Phys. Res. A 328, 578-587(1993).
    [CrossRef]
  15. J. Yang, J. Deng, N. Chandrasekhar, and C. Joachim, “UHV-STM manipulation of single flat gold nano-islands for constructing interconnection nanopads on MoS2,” J. Phys. Conf. Ser. 61, 1288-1293 (2007).
    [CrossRef]
  16. X. Zeng, F. Duewer, M. Feser, C. Huang, A. Lyon, A. Tkachuk, and W. Yun, “Ellipsoidal and parabolic glass capillaries as condensers for x-ray microscopes,” Appl. Opt. 47, 2376-2381 (2008).
    [CrossRef] [PubMed]
  17. A. Snigirev, A. Bjeoumikhov, A. Erko, I. Snigireva, M. Grigoriev, V. Yunkin, M. Erko, and S. Bjeoumikhova, “Submicrometer hard x-ray focusing using a single-bounce ellipsoidal capillary combined with a Fresnel zone plate,” J. Synchrotron Radiat. 14, 227-228 (2007).
    [CrossRef] [PubMed]

2008 (2)

B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science 319, 810-813 (2008).
[CrossRef] [PubMed]

X. Zeng, F. Duewer, M. Feser, C. Huang, A. Lyon, A. Tkachuk, and W. Yun, “Ellipsoidal and parabolic glass capillaries as condensers for x-ray microscopes,” Appl. Opt. 47, 2376-2381 (2008).
[CrossRef] [PubMed]

2007 (3)

K. Evans-Lutterodt, A. Stein, J. Ablett, N. Bozovic, A. Taylor, and D. Tennant, “Using compound kinoform hard-x-ray lenses to exceed the critical angle limit,” Phys. Rev. Lett. 99, 134801 (2007).
[CrossRef] [PubMed]

J. Yang, J. Deng, N. Chandrasekhar, and C. Joachim, “UHV-STM manipulation of single flat gold nano-islands for constructing interconnection nanopads on MoS2,” J. Phys. Conf. Ser. 61, 1288-1293 (2007).
[CrossRef]

A. Snigirev, A. Bjeoumikhov, A. Erko, I. Snigireva, M. Grigoriev, V. Yunkin, M. Erko, and S. Bjeoumikhova, “Submicrometer hard x-ray focusing using a single-bounce ellipsoidal capillary combined with a Fresnel zone plate,” J. Synchrotron Radiat. 14, 227-228 (2007).
[CrossRef] [PubMed]

2003 (1)

C. Bergemann, H. Keymeulen, and J. Vander Veen, “Focusing x-ray beams to nanometer dimensions,” Phys. Rev. Lett. 91, 204801 (2003).
[CrossRef] [PubMed]

2002 (1)

S. Tamura, M. Yasumoto, N. Kamijo, Y. Suzuki, M. Awaji, A. Takeuchi, H. Takano, and K. Handa, “Development of a multilayer Fresnel zone plate for high-energy synchrotron radiation x-rays by DC sputtering deposition,” J. Synchrotron Radiat. 9, 154-159 (2002).
[CrossRef] [PubMed]

1998 (1)

R. Medenwaldt and E. Uggerhøj, “Description of an x-ray microscope with 30 nm resolution,” Rev. Sci. Instrum. 69, 2974-2977 (1998).
[CrossRef]

1993 (1)

B. X. Yang, “Fresnel and refractive lenses for x-rays,”Nucl. Instrum. Methods Phys. Res. A 328, 578-587(1993).
[CrossRef]

1987 (1)

V. V. Aristov, Yu A. Basov, S. V. Redkin, A. A. Snigirev, and V. A. Yunkin, “Bragg zone plates for hard x-ray focusing,” Nucl. Instrum. Methods Phys. Res. A 261, 72-74 (1987).
[CrossRef]

1982 (1)

D. Bilderback and S. Hubbard, “X-ray mirror reflectivities from 3.8 to 50 keV (3.3 to 0.25 Å Part II: Pt, Si, and other materials,” Nucl. Instrum. Methods Phys. Res. 195, 91-95(1982).
[CrossRef]

Ablett, J.

K. Evans-Lutterodt, A. Stein, J. Ablett, N. Bozovic, A. Taylor, and D. Tennant, “Using compound kinoform hard-x-ray lenses to exceed the critical angle limit,” Phys. Rev. Lett. 99, 134801 (2007).
[CrossRef] [PubMed]

Aristov, V. V.

V. V. Aristov, Yu A. Basov, S. V. Redkin, A. A. Snigirev, and V. A. Yunkin, “Bragg zone plates for hard x-ray focusing,” Nucl. Instrum. Methods Phys. Res. A 261, 72-74 (1987).
[CrossRef]

Awaji, M.

S. Tamura, M. Yasumoto, N. Kamijo, Y. Suzuki, M. Awaji, A. Takeuchi, H. Takano, and K. Handa, “Development of a multilayer Fresnel zone plate for high-energy synchrotron radiation x-rays by DC sputtering deposition,” J. Synchrotron Radiat. 9, 154-159 (2002).
[CrossRef] [PubMed]

Basov, Yu A.

V. V. Aristov, Yu A. Basov, S. V. Redkin, A. A. Snigirev, and V. A. Yunkin, “Bragg zone plates for hard x-ray focusing,” Nucl. Instrum. Methods Phys. Res. A 261, 72-74 (1987).
[CrossRef]

Bates, M.

B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science 319, 810-813 (2008).
[CrossRef] [PubMed]

Bergemann, C.

C. Bergemann, H. Keymeulen, and J. Vander Veen, “Focusing x-ray beams to nanometer dimensions,” Phys. Rev. Lett. 91, 204801 (2003).
[CrossRef] [PubMed]

Bilderback, D.

D. Bilderback and S. Hubbard, “X-ray mirror reflectivities from 3.8 to 50 keV (3.3 to 0.25 Å Part II: Pt, Si, and other materials,” Nucl. Instrum. Methods Phys. Res. 195, 91-95(1982).
[CrossRef]

Bjeoumikhov, A.

A. Snigirev, A. Bjeoumikhov, A. Erko, I. Snigireva, M. Grigoriev, V. Yunkin, M. Erko, and S. Bjeoumikhova, “Submicrometer hard x-ray focusing using a single-bounce ellipsoidal capillary combined with a Fresnel zone plate,” J. Synchrotron Radiat. 14, 227-228 (2007).
[CrossRef] [PubMed]

Bjeoumikhova, S.

A. Snigirev, A. Bjeoumikhov, A. Erko, I. Snigireva, M. Grigoriev, V. Yunkin, M. Erko, and S. Bjeoumikhova, “Submicrometer hard x-ray focusing using a single-bounce ellipsoidal capillary combined with a Fresnel zone plate,” J. Synchrotron Radiat. 14, 227-228 (2007).
[CrossRef] [PubMed]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Bozovic, N.

K. Evans-Lutterodt, A. Stein, J. Ablett, N. Bozovic, A. Taylor, and D. Tennant, “Using compound kinoform hard-x-ray lenses to exceed the critical angle limit,” Phys. Rev. Lett. 99, 134801 (2007).
[CrossRef] [PubMed]

Chandrasekhar, N.

J. Yang, J. Deng, N. Chandrasekhar, and C. Joachim, “UHV-STM manipulation of single flat gold nano-islands for constructing interconnection nanopads on MoS2,” J. Phys. Conf. Ser. 61, 1288-1293 (2007).
[CrossRef]

Deng, J.

J. Yang, J. Deng, N. Chandrasekhar, and C. Joachim, “UHV-STM manipulation of single flat gold nano-islands for constructing interconnection nanopads on MoS2,” J. Phys. Conf. Ser. 61, 1288-1293 (2007).
[CrossRef]

Duewer, F.

Erko, A.

A. Snigirev, A. Bjeoumikhov, A. Erko, I. Snigireva, M. Grigoriev, V. Yunkin, M. Erko, and S. Bjeoumikhova, “Submicrometer hard x-ray focusing using a single-bounce ellipsoidal capillary combined with a Fresnel zone plate,” J. Synchrotron Radiat. 14, 227-228 (2007).
[CrossRef] [PubMed]

Erko, M.

A. Snigirev, A. Bjeoumikhov, A. Erko, I. Snigireva, M. Grigoriev, V. Yunkin, M. Erko, and S. Bjeoumikhova, “Submicrometer hard x-ray focusing using a single-bounce ellipsoidal capillary combined with a Fresnel zone plate,” J. Synchrotron Radiat. 14, 227-228 (2007).
[CrossRef] [PubMed]

Evans-Lutterodt, K.

K. Evans-Lutterodt, A. Stein, J. Ablett, N. Bozovic, A. Taylor, and D. Tennant, “Using compound kinoform hard-x-ray lenses to exceed the critical angle limit,” Phys. Rev. Lett. 99, 134801 (2007).
[CrossRef] [PubMed]

Feser, M.

Grigoriev, M.

A. Snigirev, A. Bjeoumikhov, A. Erko, I. Snigireva, M. Grigoriev, V. Yunkin, M. Erko, and S. Bjeoumikhova, “Submicrometer hard x-ray focusing using a single-bounce ellipsoidal capillary combined with a Fresnel zone plate,” J. Synchrotron Radiat. 14, 227-228 (2007).
[CrossRef] [PubMed]

Handa, K.

S. Tamura, M. Yasumoto, N. Kamijo, Y. Suzuki, M. Awaji, A. Takeuchi, H. Takano, and K. Handa, “Development of a multilayer Fresnel zone plate for high-energy synchrotron radiation x-rays by DC sputtering deposition,” J. Synchrotron Radiat. 9, 154-159 (2002).
[CrossRef] [PubMed]

Huang, B.

B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science 319, 810-813 (2008).
[CrossRef] [PubMed]

Huang, C.

Hubbard, S.

D. Bilderback and S. Hubbard, “X-ray mirror reflectivities from 3.8 to 50 keV (3.3 to 0.25 Å Part II: Pt, Si, and other materials,” Nucl. Instrum. Methods Phys. Res. 195, 91-95(1982).
[CrossRef]

Hubbell, J.

J. Hubbell and S. Seltzer, X-Ray Attenuation Coefficients (NIST, 1996).

Jackson, D.

D. Jackson, Electrodynamics, 3rd ed. (Wiley, 1998), p. 490.

Joachim, C.

J. Yang, J. Deng, N. Chandrasekhar, and C. Joachim, “UHV-STM manipulation of single flat gold nano-islands for constructing interconnection nanopads on MoS2,” J. Phys. Conf. Ser. 61, 1288-1293 (2007).
[CrossRef]

Kamijo, N.

S. Tamura, M. Yasumoto, N. Kamijo, Y. Suzuki, M. Awaji, A. Takeuchi, H. Takano, and K. Handa, “Development of a multilayer Fresnel zone plate for high-energy synchrotron radiation x-rays by DC sputtering deposition,” J. Synchrotron Radiat. 9, 154-159 (2002).
[CrossRef] [PubMed]

Kawai, T.

T. Kawai, K. Sawada, and Y. Takeuchi, “Ultra-precision micro structuring by means of mechanical machining” (IEEE, 2001), paper 0-7803-5998-4/01.

Keymeulen, H.

C. Bergemann, H. Keymeulen, and J. Vander Veen, “Focusing x-ray beams to nanometer dimensions,” Phys. Rev. Lett. 91, 204801 (2003).
[CrossRef] [PubMed]

Korsch, D.

D. Korsch, Reflective Optics (Academic, 1991).

Lyon, A.

Medenwaldt, R.

R. Medenwaldt and E. Uggerhøj, “Description of an x-ray microscope with 30 nm resolution,” Rev. Sci. Instrum. 69, 2974-2977 (1998).
[CrossRef]

Redkin, S. V.

V. V. Aristov, Yu A. Basov, S. V. Redkin, A. A. Snigirev, and V. A. Yunkin, “Bragg zone plates for hard x-ray focusing,” Nucl. Instrum. Methods Phys. Res. A 261, 72-74 (1987).
[CrossRef]

Sawada, K.

T. Kawai, K. Sawada, and Y. Takeuchi, “Ultra-precision micro structuring by means of mechanical machining” (IEEE, 2001), paper 0-7803-5998-4/01.

Seltzer, S.

J. Hubbell and S. Seltzer, X-Ray Attenuation Coefficients (NIST, 1996).

Snigirev, A.

A. Snigirev, A. Bjeoumikhov, A. Erko, I. Snigireva, M. Grigoriev, V. Yunkin, M. Erko, and S. Bjeoumikhova, “Submicrometer hard x-ray focusing using a single-bounce ellipsoidal capillary combined with a Fresnel zone plate,” J. Synchrotron Radiat. 14, 227-228 (2007).
[CrossRef] [PubMed]

Snigirev, A. A.

V. V. Aristov, Yu A. Basov, S. V. Redkin, A. A. Snigirev, and V. A. Yunkin, “Bragg zone plates for hard x-ray focusing,” Nucl. Instrum. Methods Phys. Res. A 261, 72-74 (1987).
[CrossRef]

Snigireva, I.

A. Snigirev, A. Bjeoumikhov, A. Erko, I. Snigireva, M. Grigoriev, V. Yunkin, M. Erko, and S. Bjeoumikhova, “Submicrometer hard x-ray focusing using a single-bounce ellipsoidal capillary combined with a Fresnel zone plate,” J. Synchrotron Radiat. 14, 227-228 (2007).
[CrossRef] [PubMed]

Stein, A.

K. Evans-Lutterodt, A. Stein, J. Ablett, N. Bozovic, A. Taylor, and D. Tennant, “Using compound kinoform hard-x-ray lenses to exceed the critical angle limit,” Phys. Rev. Lett. 99, 134801 (2007).
[CrossRef] [PubMed]

Suzuki, Y.

S. Tamura, M. Yasumoto, N. Kamijo, Y. Suzuki, M. Awaji, A. Takeuchi, H. Takano, and K. Handa, “Development of a multilayer Fresnel zone plate for high-energy synchrotron radiation x-rays by DC sputtering deposition,” J. Synchrotron Radiat. 9, 154-159 (2002).
[CrossRef] [PubMed]

Takano, H.

S. Tamura, M. Yasumoto, N. Kamijo, Y. Suzuki, M. Awaji, A. Takeuchi, H. Takano, and K. Handa, “Development of a multilayer Fresnel zone plate for high-energy synchrotron radiation x-rays by DC sputtering deposition,” J. Synchrotron Radiat. 9, 154-159 (2002).
[CrossRef] [PubMed]

Takeuchi, A.

S. Tamura, M. Yasumoto, N. Kamijo, Y. Suzuki, M. Awaji, A. Takeuchi, H. Takano, and K. Handa, “Development of a multilayer Fresnel zone plate for high-energy synchrotron radiation x-rays by DC sputtering deposition,” J. Synchrotron Radiat. 9, 154-159 (2002).
[CrossRef] [PubMed]

Takeuchi, Y.

T. Kawai, K. Sawada, and Y. Takeuchi, “Ultra-precision micro structuring by means of mechanical machining” (IEEE, 2001), paper 0-7803-5998-4/01.

Tamura, S.

S. Tamura, M. Yasumoto, N. Kamijo, Y. Suzuki, M. Awaji, A. Takeuchi, H. Takano, and K. Handa, “Development of a multilayer Fresnel zone plate for high-energy synchrotron radiation x-rays by DC sputtering deposition,” J. Synchrotron Radiat. 9, 154-159 (2002).
[CrossRef] [PubMed]

Taylor, A.

K. Evans-Lutterodt, A. Stein, J. Ablett, N. Bozovic, A. Taylor, and D. Tennant, “Using compound kinoform hard-x-ray lenses to exceed the critical angle limit,” Phys. Rev. Lett. 99, 134801 (2007).
[CrossRef] [PubMed]

Tennant, D.

K. Evans-Lutterodt, A. Stein, J. Ablett, N. Bozovic, A. Taylor, and D. Tennant, “Using compound kinoform hard-x-ray lenses to exceed the critical angle limit,” Phys. Rev. Lett. 99, 134801 (2007).
[CrossRef] [PubMed]

Tkachuk, A.

Uggerhøj, E.

R. Medenwaldt and E. Uggerhøj, “Description of an x-ray microscope with 30 nm resolution,” Rev. Sci. Instrum. 69, 2974-2977 (1998).
[CrossRef]

Vander Veen, J.

C. Bergemann, H. Keymeulen, and J. Vander Veen, “Focusing x-ray beams to nanometer dimensions,” Phys. Rev. Lett. 91, 204801 (2003).
[CrossRef] [PubMed]

Wang, W.

B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science 319, 810-813 (2008).
[CrossRef] [PubMed]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Yang, B. X.

B. X. Yang, “Fresnel and refractive lenses for x-rays,”Nucl. Instrum. Methods Phys. Res. A 328, 578-587(1993).
[CrossRef]

Yang, J.

J. Yang, J. Deng, N. Chandrasekhar, and C. Joachim, “UHV-STM manipulation of single flat gold nano-islands for constructing interconnection nanopads on MoS2,” J. Phys. Conf. Ser. 61, 1288-1293 (2007).
[CrossRef]

Yasumoto, M.

S. Tamura, M. Yasumoto, N. Kamijo, Y. Suzuki, M. Awaji, A. Takeuchi, H. Takano, and K. Handa, “Development of a multilayer Fresnel zone plate for high-energy synchrotron radiation x-rays by DC sputtering deposition,” J. Synchrotron Radiat. 9, 154-159 (2002).
[CrossRef] [PubMed]

Yun, W.

Yunkin, V.

A. Snigirev, A. Bjeoumikhov, A. Erko, I. Snigireva, M. Grigoriev, V. Yunkin, M. Erko, and S. Bjeoumikhova, “Submicrometer hard x-ray focusing using a single-bounce ellipsoidal capillary combined with a Fresnel zone plate,” J. Synchrotron Radiat. 14, 227-228 (2007).
[CrossRef] [PubMed]

Yunkin, V. A.

V. V. Aristov, Yu A. Basov, S. V. Redkin, A. A. Snigirev, and V. A. Yunkin, “Bragg zone plates for hard x-ray focusing,” Nucl. Instrum. Methods Phys. Res. A 261, 72-74 (1987).
[CrossRef]

Zeng, X.

Zhuang, X.

B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science 319, 810-813 (2008).
[CrossRef] [PubMed]

Appl. Opt. (1)

J. Phys. Conf. Ser. (1)

J. Yang, J. Deng, N. Chandrasekhar, and C. Joachim, “UHV-STM manipulation of single flat gold nano-islands for constructing interconnection nanopads on MoS2,” J. Phys. Conf. Ser. 61, 1288-1293 (2007).
[CrossRef]

J. Synchrotron Radiat. (2)

S. Tamura, M. Yasumoto, N. Kamijo, Y. Suzuki, M. Awaji, A. Takeuchi, H. Takano, and K. Handa, “Development of a multilayer Fresnel zone plate for high-energy synchrotron radiation x-rays by DC sputtering deposition,” J. Synchrotron Radiat. 9, 154-159 (2002).
[CrossRef] [PubMed]

A. Snigirev, A. Bjeoumikhov, A. Erko, I. Snigireva, M. Grigoriev, V. Yunkin, M. Erko, and S. Bjeoumikhova, “Submicrometer hard x-ray focusing using a single-bounce ellipsoidal capillary combined with a Fresnel zone plate,” J. Synchrotron Radiat. 14, 227-228 (2007).
[CrossRef] [PubMed]

Nucl. Instrum. Methods Phys. Res. (1)

D. Bilderback and S. Hubbard, “X-ray mirror reflectivities from 3.8 to 50 keV (3.3 to 0.25 Å Part II: Pt, Si, and other materials,” Nucl. Instrum. Methods Phys. Res. 195, 91-95(1982).
[CrossRef]

Nucl. Instrum. Methods Phys. Res. A (2)

V. V. Aristov, Yu A. Basov, S. V. Redkin, A. A. Snigirev, and V. A. Yunkin, “Bragg zone plates for hard x-ray focusing,” Nucl. Instrum. Methods Phys. Res. A 261, 72-74 (1987).
[CrossRef]

B. X. Yang, “Fresnel and refractive lenses for x-rays,”Nucl. Instrum. Methods Phys. Res. A 328, 578-587(1993).
[CrossRef]

Phys. Rev. Lett. (2)

K. Evans-Lutterodt, A. Stein, J. Ablett, N. Bozovic, A. Taylor, and D. Tennant, “Using compound kinoform hard-x-ray lenses to exceed the critical angle limit,” Phys. Rev. Lett. 99, 134801 (2007).
[CrossRef] [PubMed]

C. Bergemann, H. Keymeulen, and J. Vander Veen, “Focusing x-ray beams to nanometer dimensions,” Phys. Rev. Lett. 91, 204801 (2003).
[CrossRef] [PubMed]

Rev. Sci. Instrum. (1)

R. Medenwaldt and E. Uggerhøj, “Description of an x-ray microscope with 30 nm resolution,” Rev. Sci. Instrum. 69, 2974-2977 (1998).
[CrossRef]

Science (1)

B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science 319, 810-813 (2008).
[CrossRef] [PubMed]

Other (6)

D.Mills, ed., Third Generation Hard X-Ray Synchrotron Radiation Sources (Wiley-Interscience, 2002).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

D. Jackson, Electrodynamics, 3rd ed. (Wiley, 1998), p. 490.

D. Korsch, Reflective Optics (Academic, 1991).

T. Kawai, K. Sawada, and Y. Takeuchi, “Ultra-precision micro structuring by means of mechanical machining” (IEEE, 2001), paper 0-7803-5998-4/01.

J. Hubbell and S. Seltzer, X-Ray Attenuation Coefficients (NIST, 1996).

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Figures (6)

Fig. 1
Fig. 1

Ray propagation through an ideal paraboloidal mirror. For the application described in Subsection 4F, v 0 = 110 μm , w 0 = 78 mm , w F = 39 mm , and m = 0.00141 .

Fig. 2
Fig. 2

(a) Excess path length (compared to the path length of central ray through P 0 ) of a ray reflected from a mirror having deviant profile. The dashed curve shows the ideal paraboloidal profile. The bold curve shows the deviant profile. Thickness w m of the compensating layer may turn out negative, in which case | w m | would be subtracted from a uniform disk (not shown) of the same material. (b) Detail view of deflection in compensating wedge. (c) Pictorial demonstration of Eq. (27).

Fig. 3
Fig. 3

Energy dependence of total absorption coefficient μ for beryllium. An empirical fit, accurate to approximately 5%, is indicated. Also shown (same scale) is the critical Fresnel number N 0 and the (valid for 1 < E γ < 10 keV ) approximation 5 E γ 2 [ ke V 2 ] .

Fig. 4
Fig. 4

Toolkit of trim corrector plates to be used for prescribing the parameters needed to compensate individual mirrors. These are just cartoons. The corrector plates for mirrors with today’s tolerances (looking more like craggy mountains than rolling hills) will have to be made using modern microfabrication techniques.

Fig. 5
Fig. 5

Illustration of the use of a compensator to convert a conical mirror into a focusing device. The corrector plate consists of one or the other of (a) refractive lens option or (b) zone plate option for the application described in Subsection 5B, R 0 = 290 μm , w C w F = 158 mm , r max = 15 μm , and θ z p 0.003 . Dashed curves indicate the apparent, mirror-reflected optics. The drawing is not accurate.

Fig. 6
Fig. 6

Fresnel ring pattern for compensating conical mirror into high NA focusing device. The deviations from R 0 of the ring radii are given by r n = 2 n ( w C w F ) λ .

Tables (2)

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Table 1 Typical Parameter Values for E γ = 6 keV , E γ = 18 keV , and E γ = 60 keV a

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Table 2 Mirror and Compensator Parameters a

Equations (43)

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δ m 1 n m 1.3 × 10 11 ρ m λ 2 .
δ Pt 4.0 × 10 3 ke V 2 E γ 2 .
θ max 70 keV . mr E γ λ × 0.056 r / nm .
Q ( v sin ϕ , v cos ϕ , w ) , I ( 0 , v I , w F + w I ) .
Δ l = ( w w F w I ) 2 + v 2 2 v v I cos ϕ + v I 2 ( w w F ) w I + v 2 2 v v I cos ϕ v I 2 2 ( w w F ) .
d 2 E E 0 | S = ρ d ϕ d ρ .
d 2 E / d ρ E 0 | I = i k ρ r e i k Δ l d ϕ 2 π = i k ρ r exp ( i k ( w I + v 2 v I 2 2 ( w w F ) ) ) exp ( i k v v I cos ϕ w w F ) d ϕ 2 π ,
1 2 π 0 2 π d ϕ e i C cos ϕ = J 0 ( C ) .
d E / d ρ E 0 | I i k ρ r exp ( i k ( w I + v 2 v I 2 2 ( w w F ) ) ) J 0 ( k v v I w w F ) .
FWHM v I 0 = 2.405 w w F v λ 2 π 0.191 λ m .
FWHM min 0.191 λ θ max = 3.4 nm .
w ( ρ ) = w ¯ ( ρ ) + Δ w ( ρ ) ,
u = x , v = v 0 + y , w = w 0 + z .
w ¯ ( u , v ) = w F + w 0 v 0 u 2 + v 2 + w 0 2 v 0 2 ( u 2 + v 2 ) .
x = ξ + cos α cos γ ( z S ) , y = η + cos β cos γ ( z S ) , z < S .
cos α n = w u 1 + w u 2 + w v 2 , cos β n = w v 1 + w u 2 + w v 2 , cos γ n = 1 1 + w u 2 + w v 2 ,
cos ϵ = cos α cos α n + cos β cos β n + cos γ cos γ n .
cos α p = cos α 2 cos ϵ cos α n , cos β p = cos β 2 cos ϵ cos β n , cos γ p = cos γ 2 cos ϵ cos γ n ,
ξ = x cos α p cos γ p ( z + S ) , η = v 0 + y cos β p cos γ p ( z + S ) .
Δ l before ( v ) = P F ( P R + P 0 F ) + ( 1 cos Δ β ) T P ( w 0 w F + z ) 2 + ( v 0 + y ) 2 ) ( z + ( w 0 w F ) 2 + v 0 2 ) + Δ β 2 2 ( S z ) ) ,
w m ( v ) δ m = Δ l before ( v ) ,
Δ l measured = Δ l before ± σ l , metrology .
w m = 1 δ m ( Δ l before ± σ l , metrology ) ± σ l , fabrication ,
Δ l effective = Δ l before ( Δ l before ± σ l , metrology ) ± σ l , fabrication δ m = ± σ l , metrology 2 + σ l , fabrication 2 δ m 2 ± σ l , metrology .
δ Be = 3.41 × 10 4 keV 2 E γ 2 .
sin ψ = ( 1 δ Be ) sin ( ψ + | Δ ψ | ) sin ψ + | Δ ψ | cos ψ δ Be sin ψ ,
d w Be d v Δ β δ Be ,
n F = Δ l before λ .
w Be , max = Δ n F λ δ Be = ( 3640 nm / keV ) Δ n F E γ .
Δ n F = σ l , fabrication λ 10 nm λ .
w Be , max = σ l , fabrication δ Be .
n m = 1 δ m + i β m .
N 0 = δ m 2 π β m = 2 δ m μ λ .
N 0 { 5 ke V 2 E γ 2 for 1 < E γ < 10 keV 500 for E γ > 10 keV .
Δ N F N 0 = σ l , fabrication / λ ( 5 ke V 2 ) E γ 2 ( 0.13 keV / nm ) σ l , fabrication E γ .
d w Be d v 4 π δ Be σ l , fabrication λ w , imp . , ( e . g . , = 4 π 10 5 10 8 10 2 = 1.2   at   6 keV ) .
NA R 0 + 2 r max w C w F .
Δ ρ | washer = 0.61 λ NA = 0.61 λ ( w C w F ) R 0 + 2 r max .
R n = R 0 + 2 n ( w C w F ) λ , n = 0 , ± 1 , ± 2 , , ± n max ,
Δ R n = Δ r n = ( w C w F ) λ 2 n .
Δ ρ | disk = 1.22 Δ r max = 1.22 ( w C w F ) λ 2 n max .
Δ ρ | washer Δ ρ | disk = 0.61 λ ( w C w F ) R 0 + 2 r max 1.22 ( w C w F ) λ 2 n max r max 2 R 0 .
r n 2 = r 0 2 + n λ f ,

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