Abstract

A numerical method for calculation of the electromagnetic field in two-dimensionally confined x-ray waveguides is presented. It is based on the parabolic wave equation, which is solved by means of a finite-difference scheme. The results are verified by a comparison to analytical theory, namely, Fresnel reflectivity and the weakly guiding optical fiber.

© 2006 Optical Society of America

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  1. E. Spiller and A. Segmüller, "Propagation of x rays in waveguides," Appl. Phys. Lett. 24, 60-61 (1973).
    [CrossRef]
  2. Y. P. Feng, S. K. Sinha, H. W. Deckman, J. B. Hastings, and D. P. Siddons, "X-ray flux enhancement in thin-film waveguides using resonant beam couplers," Phys. Rev. Lett. 71, 537-540 (1993).
    [CrossRef] [PubMed]
  3. M. J. Zwanenburg, J. F. Peters, J. H. H. Bongaerts, S. A. de Vries, D. L. Abernathy, and J. F. van der Veen, "Coherent propagation of x rays in a planar waveguide with a tunable air gap," Phys. Rev. Lett. 82, 1696-1699 (1999).
    [CrossRef]
  4. F. Pfeiffer, C. David, M. Burghammer, C. Riekel, and T. Salditt, "Two-dimensional x-ray waveguides and point sources," Science 297, 230-234 (2002).
    [CrossRef] [PubMed]
  5. A. Jarre, C. Fuhse, C. Ollinger, J. Seeger, R. Tucoulou, and T. Salditt, "Two-dimensional hard x-ray beam compression by combined focusing and waveguide optics," Phys. Rev. Lett. 94, 074801 (2005).
    [CrossRef] [PubMed]
  6. S. Lagomarsino, A. Cedola, P. Cloetens, S. D. Fonzo, W. Jark, S. Souillié, and C. Riekel, "Phase contrast hard x-ray microscopy with submicron resolution," Appl. Phys. Lett. 71, 2557-2559 (1997).
    [CrossRef]
  7. S. D. F. W. Jark, S. Lagomarsino, C. Giannini, L. D. Caro, A. Cedola, and M. Müller, "Non-destructive determination of local strain with 100-nanometre spatial resolution," Nature 403, 638-640 (2000).
    [CrossRef] [PubMed]
  8. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, 1974).
  9. M. J. Zwanenburg, J. H. H. Bongaerts, J. F. Peters, D. Riese, and J. F. van der Veen, "Focusing of coherent x-rays in a tapered planar waveguide," Physica B 283, 285-288 (2000).
    [CrossRef]
  10. C. Fuhse and T. Salditt, "Finite-difference field calculations for one-dimensionally confined x-ray waveguides," Physica B 357, 57-60 (2005).
    [CrossRef]
  11. R. Scarmozzino and R. M. Osgood, "Comparison of finite-difference and Fourier-transform solutions of the parabolic wave equation with emphasis on integrated-optics applications," J. Opt. Soc. Am. A 8, 724-731 (1991).
    [CrossRef]
  12. Y. V. Kopylov, A. V. Popov, and A. V. Vinogradov, "Application of the parabolic wave equation to x-ray diffraction optics," Opt. Commun. 118, 619-636 (1995).
    [CrossRef]
  13. J. H. H. Bongaerts, C. David, M. Drakopoulos, M. J. Zwanenburg, G. H. Wegdam, T. Lackner, H. Keymeulen, and J. F. van der Veen, "Propagation of a partially coherent focused x-ray beam within a planar x-ray waveguide," J. Synchrotron Radiat. 9, 383-393 (2002).
    [CrossRef]
  14. M. J. Zwanenburg, J. F. van der Veen, H. G. Ficke, and H. Neerings, "A planar x-ray waveguide with a tunable air gap for the structural investigation of confined fluids," Rev. Sci. Instrum. 71, 1723-1732 (2000).
    [CrossRef]
  15. D. Gloge, "Weakly guiding fibers," Appl. Opt. 10, 2252-2258 (1971).
    [CrossRef] [PubMed]
  16. C. Bergemann, H. Keymeulen, and J. F. van der Veen, "Focusing x-ray beams to nanometer dimensions," Phys. Rev. Lett. 91, 204801 (2003).
    [CrossRef] [PubMed]
  17. J. W. Thomas, Numerical Partial Differential Equations (Springer-Verlag, 1995).
  18. IDL is a registered trademark of Research Systems, Inc., Boulder, Colo.
  19. D. L. Windt, "IMD--Software for modeling the optical properties of multilayer films," Comput. Phys. 12, 360-370 (1998).
    [CrossRef]
  20. E. Snitzer, "Cylindrical dielectric waveguide modes," J. Opt. Soc. Am. 51, 491-498 (1961).
    [CrossRef]

2005 (2)

A. Jarre, C. Fuhse, C. Ollinger, J. Seeger, R. Tucoulou, and T. Salditt, "Two-dimensional hard x-ray beam compression by combined focusing and waveguide optics," Phys. Rev. Lett. 94, 074801 (2005).
[CrossRef] [PubMed]

C. Fuhse and T. Salditt, "Finite-difference field calculations for one-dimensionally confined x-ray waveguides," Physica B 357, 57-60 (2005).
[CrossRef]

2003 (1)

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

2002 (2)

J. H. H. Bongaerts, C. David, M. Drakopoulos, M. J. Zwanenburg, G. H. Wegdam, T. Lackner, H. Keymeulen, and J. F. van der Veen, "Propagation of a partially coherent focused x-ray beam within a planar x-ray waveguide," J. Synchrotron Radiat. 9, 383-393 (2002).
[CrossRef]

F. Pfeiffer, C. David, M. Burghammer, C. Riekel, and T. Salditt, "Two-dimensional x-ray waveguides and point sources," Science 297, 230-234 (2002).
[CrossRef] [PubMed]

2000 (3)

S. D. F. W. Jark, S. Lagomarsino, C. Giannini, L. D. Caro, A. Cedola, and M. Müller, "Non-destructive determination of local strain with 100-nanometre spatial resolution," Nature 403, 638-640 (2000).
[CrossRef] [PubMed]

M. J. Zwanenburg, J. H. H. Bongaerts, J. F. Peters, D. Riese, and J. F. van der Veen, "Focusing of coherent x-rays in a tapered planar waveguide," Physica B 283, 285-288 (2000).
[CrossRef]

M. J. Zwanenburg, J. F. van der Veen, H. G. Ficke, and H. Neerings, "A planar x-ray waveguide with a tunable air gap for the structural investigation of confined fluids," Rev. Sci. Instrum. 71, 1723-1732 (2000).
[CrossRef]

1999 (1)

M. J. Zwanenburg, J. F. Peters, J. H. H. Bongaerts, S. A. de Vries, D. L. Abernathy, and J. F. van der Veen, "Coherent propagation of x rays in a planar waveguide with a tunable air gap," Phys. Rev. Lett. 82, 1696-1699 (1999).
[CrossRef]

1998 (1)

D. L. Windt, "IMD--Software for modeling the optical properties of multilayer films," Comput. Phys. 12, 360-370 (1998).
[CrossRef]

1997 (1)

S. Lagomarsino, A. Cedola, P. Cloetens, S. D. Fonzo, W. Jark, S. Souillié, and C. Riekel, "Phase contrast hard x-ray microscopy with submicron resolution," Appl. Phys. Lett. 71, 2557-2559 (1997).
[CrossRef]

1995 (1)

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

1993 (1)

Y. P. Feng, S. K. Sinha, H. W. Deckman, J. B. Hastings, and D. P. Siddons, "X-ray flux enhancement in thin-film waveguides using resonant beam couplers," Phys. Rev. Lett. 71, 537-540 (1993).
[CrossRef] [PubMed]

1991 (1)

1973 (1)

E. Spiller and A. Segmüller, "Propagation of x rays in waveguides," Appl. Phys. Lett. 24, 60-61 (1973).
[CrossRef]

1971 (1)

1961 (1)

Abernathy, D. L.

M. J. Zwanenburg, J. F. Peters, J. H. H. Bongaerts, S. A. de Vries, D. L. Abernathy, and J. F. van der Veen, "Coherent propagation of x rays in a planar waveguide with a tunable air gap," Phys. Rev. Lett. 82, 1696-1699 (1999).
[CrossRef]

Bergemann, C.

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

Bongaerts, J. H. H.

J. H. H. Bongaerts, C. David, M. Drakopoulos, M. J. Zwanenburg, G. H. Wegdam, T. Lackner, H. Keymeulen, and J. F. van der Veen, "Propagation of a partially coherent focused x-ray beam within a planar x-ray waveguide," J. Synchrotron Radiat. 9, 383-393 (2002).
[CrossRef]

M. J. Zwanenburg, J. H. H. Bongaerts, J. F. Peters, D. Riese, and J. F. van der Veen, "Focusing of coherent x-rays in a tapered planar waveguide," Physica B 283, 285-288 (2000).
[CrossRef]

M. J. Zwanenburg, J. F. Peters, J. H. H. Bongaerts, S. A. de Vries, D. L. Abernathy, and J. F. van der Veen, "Coherent propagation of x rays in a planar waveguide with a tunable air gap," Phys. Rev. Lett. 82, 1696-1699 (1999).
[CrossRef]

Burghammer, M.

F. Pfeiffer, C. David, M. Burghammer, C. Riekel, and T. Salditt, "Two-dimensional x-ray waveguides and point sources," Science 297, 230-234 (2002).
[CrossRef] [PubMed]

Caro, L. D.

S. D. F. W. Jark, S. Lagomarsino, C. Giannini, L. D. Caro, A. Cedola, and M. Müller, "Non-destructive determination of local strain with 100-nanometre spatial resolution," Nature 403, 638-640 (2000).
[CrossRef] [PubMed]

Cedola, A.

S. D. F. W. Jark, S. Lagomarsino, C. Giannini, L. D. Caro, A. Cedola, and M. Müller, "Non-destructive determination of local strain with 100-nanometre spatial resolution," Nature 403, 638-640 (2000).
[CrossRef] [PubMed]

S. Lagomarsino, A. Cedola, P. Cloetens, S. D. Fonzo, W. Jark, S. Souillié, and C. Riekel, "Phase contrast hard x-ray microscopy with submicron resolution," Appl. Phys. Lett. 71, 2557-2559 (1997).
[CrossRef]

Cloetens, P.

S. Lagomarsino, A. Cedola, P. Cloetens, S. D. Fonzo, W. Jark, S. Souillié, and C. Riekel, "Phase contrast hard x-ray microscopy with submicron resolution," Appl. Phys. Lett. 71, 2557-2559 (1997).
[CrossRef]

David, C.

F. Pfeiffer, C. David, M. Burghammer, C. Riekel, and T. Salditt, "Two-dimensional x-ray waveguides and point sources," Science 297, 230-234 (2002).
[CrossRef] [PubMed]

J. H. H. Bongaerts, C. David, M. Drakopoulos, M. J. Zwanenburg, G. H. Wegdam, T. Lackner, H. Keymeulen, and J. F. van der Veen, "Propagation of a partially coherent focused x-ray beam within a planar x-ray waveguide," J. Synchrotron Radiat. 9, 383-393 (2002).
[CrossRef]

de Vries, S. A.

M. J. Zwanenburg, J. F. Peters, J. H. H. Bongaerts, S. A. de Vries, D. L. Abernathy, and J. F. van der Veen, "Coherent propagation of x rays in a planar waveguide with a tunable air gap," Phys. Rev. Lett. 82, 1696-1699 (1999).
[CrossRef]

Deckman, H. W.

Y. P. Feng, S. K. Sinha, H. W. Deckman, J. B. Hastings, and D. P. Siddons, "X-ray flux enhancement in thin-film waveguides using resonant beam couplers," Phys. Rev. Lett. 71, 537-540 (1993).
[CrossRef] [PubMed]

Drakopoulos, M.

J. H. H. Bongaerts, C. David, M. Drakopoulos, M. J. Zwanenburg, G. H. Wegdam, T. Lackner, H. Keymeulen, and J. F. van der Veen, "Propagation of a partially coherent focused x-ray beam within a planar x-ray waveguide," J. Synchrotron Radiat. 9, 383-393 (2002).
[CrossRef]

Feng, Y. P.

Y. P. Feng, S. K. Sinha, H. W. Deckman, J. B. Hastings, and D. P. Siddons, "X-ray flux enhancement in thin-film waveguides using resonant beam couplers," Phys. Rev. Lett. 71, 537-540 (1993).
[CrossRef] [PubMed]

Ficke, H. G.

M. J. Zwanenburg, J. F. van der Veen, H. G. Ficke, and H. Neerings, "A planar x-ray waveguide with a tunable air gap for the structural investigation of confined fluids," Rev. Sci. Instrum. 71, 1723-1732 (2000).
[CrossRef]

Fonzo, S. D.

S. Lagomarsino, A. Cedola, P. Cloetens, S. D. Fonzo, W. Jark, S. Souillié, and C. Riekel, "Phase contrast hard x-ray microscopy with submicron resolution," Appl. Phys. Lett. 71, 2557-2559 (1997).
[CrossRef]

Fuhse, C.

A. Jarre, C. Fuhse, C. Ollinger, J. Seeger, R. Tucoulou, and T. Salditt, "Two-dimensional hard x-ray beam compression by combined focusing and waveguide optics," Phys. Rev. Lett. 94, 074801 (2005).
[CrossRef] [PubMed]

C. Fuhse and T. Salditt, "Finite-difference field calculations for one-dimensionally confined x-ray waveguides," Physica B 357, 57-60 (2005).
[CrossRef]

Giannini, C.

S. D. F. W. Jark, S. Lagomarsino, C. Giannini, L. D. Caro, A. Cedola, and M. Müller, "Non-destructive determination of local strain with 100-nanometre spatial resolution," Nature 403, 638-640 (2000).
[CrossRef] [PubMed]

Gloge, D.

Hastings, J. B.

Y. P. Feng, S. K. Sinha, H. W. Deckman, J. B. Hastings, and D. P. Siddons, "X-ray flux enhancement in thin-film waveguides using resonant beam couplers," Phys. Rev. Lett. 71, 537-540 (1993).
[CrossRef] [PubMed]

Jark, S. D. F. W.

S. D. F. W. Jark, S. Lagomarsino, C. Giannini, L. D. Caro, A. Cedola, and M. Müller, "Non-destructive determination of local strain with 100-nanometre spatial resolution," Nature 403, 638-640 (2000).
[CrossRef] [PubMed]

Jark, W.

S. Lagomarsino, A. Cedola, P. Cloetens, S. D. Fonzo, W. Jark, S. Souillié, and C. Riekel, "Phase contrast hard x-ray microscopy with submicron resolution," Appl. Phys. Lett. 71, 2557-2559 (1997).
[CrossRef]

Jarre, A.

A. Jarre, C. Fuhse, C. Ollinger, J. Seeger, R. Tucoulou, and T. Salditt, "Two-dimensional hard x-ray beam compression by combined focusing and waveguide optics," Phys. Rev. Lett. 94, 074801 (2005).
[CrossRef] [PubMed]

Keymeulen, H.

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

J. H. H. Bongaerts, C. David, M. Drakopoulos, M. J. Zwanenburg, G. H. Wegdam, T. Lackner, H. Keymeulen, and J. F. van der Veen, "Propagation of a partially coherent focused x-ray beam within a planar x-ray waveguide," J. Synchrotron Radiat. 9, 383-393 (2002).
[CrossRef]

Kopylov, Y. V.

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

Lackner, T.

J. H. H. Bongaerts, C. David, M. Drakopoulos, M. J. Zwanenburg, G. H. Wegdam, T. Lackner, H. Keymeulen, and J. F. van der Veen, "Propagation of a partially coherent focused x-ray beam within a planar x-ray waveguide," J. Synchrotron Radiat. 9, 383-393 (2002).
[CrossRef]

Lagomarsino, S.

S. D. F. W. Jark, S. Lagomarsino, C. Giannini, L. D. Caro, A. Cedola, and M. Müller, "Non-destructive determination of local strain with 100-nanometre spatial resolution," Nature 403, 638-640 (2000).
[CrossRef] [PubMed]

S. Lagomarsino, A. Cedola, P. Cloetens, S. D. Fonzo, W. Jark, S. Souillié, and C. Riekel, "Phase contrast hard x-ray microscopy with submicron resolution," Appl. Phys. Lett. 71, 2557-2559 (1997).
[CrossRef]

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, 1974).

Müller, M.

S. D. F. W. Jark, S. Lagomarsino, C. Giannini, L. D. Caro, A. Cedola, and M. Müller, "Non-destructive determination of local strain with 100-nanometre spatial resolution," Nature 403, 638-640 (2000).
[CrossRef] [PubMed]

Neerings, H.

M. J. Zwanenburg, J. F. van der Veen, H. G. Ficke, and H. Neerings, "A planar x-ray waveguide with a tunable air gap for the structural investigation of confined fluids," Rev. Sci. Instrum. 71, 1723-1732 (2000).
[CrossRef]

Ollinger, C.

A. Jarre, C. Fuhse, C. Ollinger, J. Seeger, R. Tucoulou, and T. Salditt, "Two-dimensional hard x-ray beam compression by combined focusing and waveguide optics," Phys. Rev. Lett. 94, 074801 (2005).
[CrossRef] [PubMed]

Osgood, R. M.

Peters, J. F.

M. J. Zwanenburg, J. H. H. Bongaerts, J. F. Peters, D. Riese, and J. F. van der Veen, "Focusing of coherent x-rays in a tapered planar waveguide," Physica B 283, 285-288 (2000).
[CrossRef]

M. J. Zwanenburg, J. F. Peters, J. H. H. Bongaerts, S. A. de Vries, D. L. Abernathy, and J. F. van der Veen, "Coherent propagation of x rays in a planar waveguide with a tunable air gap," Phys. Rev. Lett. 82, 1696-1699 (1999).
[CrossRef]

Pfeiffer, F.

F. Pfeiffer, C. David, M. Burghammer, C. Riekel, and T. Salditt, "Two-dimensional x-ray waveguides and point sources," Science 297, 230-234 (2002).
[CrossRef] [PubMed]

Popov, A. V.

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

Riekel, C.

F. Pfeiffer, C. David, M. Burghammer, C. Riekel, and T. Salditt, "Two-dimensional x-ray waveguides and point sources," Science 297, 230-234 (2002).
[CrossRef] [PubMed]

S. Lagomarsino, A. Cedola, P. Cloetens, S. D. Fonzo, W. Jark, S. Souillié, and C. Riekel, "Phase contrast hard x-ray microscopy with submicron resolution," Appl. Phys. Lett. 71, 2557-2559 (1997).
[CrossRef]

Riese, D.

M. J. Zwanenburg, J. H. H. Bongaerts, J. F. Peters, D. Riese, and J. F. van der Veen, "Focusing of coherent x-rays in a tapered planar waveguide," Physica B 283, 285-288 (2000).
[CrossRef]

Salditt, T.

C. Fuhse and T. Salditt, "Finite-difference field calculations for one-dimensionally confined x-ray waveguides," Physica B 357, 57-60 (2005).
[CrossRef]

A. Jarre, C. Fuhse, C. Ollinger, J. Seeger, R. Tucoulou, and T. Salditt, "Two-dimensional hard x-ray beam compression by combined focusing and waveguide optics," Phys. Rev. Lett. 94, 074801 (2005).
[CrossRef] [PubMed]

F. Pfeiffer, C. David, M. Burghammer, C. Riekel, and T. Salditt, "Two-dimensional x-ray waveguides and point sources," Science 297, 230-234 (2002).
[CrossRef] [PubMed]

Scarmozzino, R.

Seeger, J.

A. Jarre, C. Fuhse, C. Ollinger, J. Seeger, R. Tucoulou, and T. Salditt, "Two-dimensional hard x-ray beam compression by combined focusing and waveguide optics," Phys. Rev. Lett. 94, 074801 (2005).
[CrossRef] [PubMed]

Segmüller, A.

E. Spiller and A. Segmüller, "Propagation of x rays in waveguides," Appl. Phys. Lett. 24, 60-61 (1973).
[CrossRef]

Siddons, D. P.

Y. P. Feng, S. K. Sinha, H. W. Deckman, J. B. Hastings, and D. P. Siddons, "X-ray flux enhancement in thin-film waveguides using resonant beam couplers," Phys. Rev. Lett. 71, 537-540 (1993).
[CrossRef] [PubMed]

Sinha, S. K.

Y. P. Feng, S. K. Sinha, H. W. Deckman, J. B. Hastings, and D. P. Siddons, "X-ray flux enhancement in thin-film waveguides using resonant beam couplers," Phys. Rev. Lett. 71, 537-540 (1993).
[CrossRef] [PubMed]

Snitzer, E.

Souillié, S.

S. Lagomarsino, A. Cedola, P. Cloetens, S. D. Fonzo, W. Jark, S. Souillié, and C. Riekel, "Phase contrast hard x-ray microscopy with submicron resolution," Appl. Phys. Lett. 71, 2557-2559 (1997).
[CrossRef]

Spiller, E.

E. Spiller and A. Segmüller, "Propagation of x rays in waveguides," Appl. Phys. Lett. 24, 60-61 (1973).
[CrossRef]

Thomas, J. W.

J. W. Thomas, Numerical Partial Differential Equations (Springer-Verlag, 1995).

Tucoulou, R.

A. Jarre, C. Fuhse, C. Ollinger, J. Seeger, R. Tucoulou, and T. Salditt, "Two-dimensional hard x-ray beam compression by combined focusing and waveguide optics," Phys. Rev. Lett. 94, 074801 (2005).
[CrossRef] [PubMed]

van der Veen, J. F.

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

J. H. H. Bongaerts, C. David, M. Drakopoulos, M. J. Zwanenburg, G. H. Wegdam, T. Lackner, H. Keymeulen, and J. F. van der Veen, "Propagation of a partially coherent focused x-ray beam within a planar x-ray waveguide," J. Synchrotron Radiat. 9, 383-393 (2002).
[CrossRef]

M. J. Zwanenburg, J. F. van der Veen, H. G. Ficke, and H. Neerings, "A planar x-ray waveguide with a tunable air gap for the structural investigation of confined fluids," Rev. Sci. Instrum. 71, 1723-1732 (2000).
[CrossRef]

M. J. Zwanenburg, J. H. H. Bongaerts, J. F. Peters, D. Riese, and J. F. van der Veen, "Focusing of coherent x-rays in a tapered planar waveguide," Physica B 283, 285-288 (2000).
[CrossRef]

M. J. Zwanenburg, J. F. Peters, J. H. H. Bongaerts, S. A. de Vries, D. L. Abernathy, and J. F. van der Veen, "Coherent propagation of x rays in a planar waveguide with a tunable air gap," Phys. Rev. Lett. 82, 1696-1699 (1999).
[CrossRef]

Vinogradov, A. V.

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

Wegdam, G. H.

J. H. H. Bongaerts, C. David, M. Drakopoulos, M. J. Zwanenburg, G. H. Wegdam, T. Lackner, H. Keymeulen, and J. F. van der Veen, "Propagation of a partially coherent focused x-ray beam within a planar x-ray waveguide," J. Synchrotron Radiat. 9, 383-393 (2002).
[CrossRef]

Windt, D. L.

D. L. Windt, "IMD--Software for modeling the optical properties of multilayer films," Comput. Phys. 12, 360-370 (1998).
[CrossRef]

Zwanenburg, M. J.

J. H. H. Bongaerts, C. David, M. Drakopoulos, M. J. Zwanenburg, G. H. Wegdam, T. Lackner, H. Keymeulen, and J. F. van der Veen, "Propagation of a partially coherent focused x-ray beam within a planar x-ray waveguide," J. Synchrotron Radiat. 9, 383-393 (2002).
[CrossRef]

M. J. Zwanenburg, J. H. H. Bongaerts, J. F. Peters, D. Riese, and J. F. van der Veen, "Focusing of coherent x-rays in a tapered planar waveguide," Physica B 283, 285-288 (2000).
[CrossRef]

M. J. Zwanenburg, J. F. van der Veen, H. G. Ficke, and H. Neerings, "A planar x-ray waveguide with a tunable air gap for the structural investigation of confined fluids," Rev. Sci. Instrum. 71, 1723-1732 (2000).
[CrossRef]

M. J. Zwanenburg, J. F. Peters, J. H. H. Bongaerts, S. A. de Vries, D. L. Abernathy, and J. F. van der Veen, "Coherent propagation of x rays in a planar waveguide with a tunable air gap," Phys. Rev. Lett. 82, 1696-1699 (1999).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

E. Spiller and A. Segmüller, "Propagation of x rays in waveguides," Appl. Phys. Lett. 24, 60-61 (1973).
[CrossRef]

S. Lagomarsino, A. Cedola, P. Cloetens, S. D. Fonzo, W. Jark, S. Souillié, and C. Riekel, "Phase contrast hard x-ray microscopy with submicron resolution," Appl. Phys. Lett. 71, 2557-2559 (1997).
[CrossRef]

Comput. Phys. (1)

D. L. Windt, "IMD--Software for modeling the optical properties of multilayer films," Comput. Phys. 12, 360-370 (1998).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. Synchrotron Radiat. (1)

J. H. H. Bongaerts, C. David, M. Drakopoulos, M. J. Zwanenburg, G. H. Wegdam, T. Lackner, H. Keymeulen, and J. F. van der Veen, "Propagation of a partially coherent focused x-ray beam within a planar x-ray waveguide," J. Synchrotron Radiat. 9, 383-393 (2002).
[CrossRef]

Nature (1)

S. D. F. W. Jark, S. Lagomarsino, C. Giannini, L. D. Caro, A. Cedola, and M. Müller, "Non-destructive determination of local strain with 100-nanometre spatial resolution," Nature 403, 638-640 (2000).
[CrossRef] [PubMed]

Opt. Commun. (1)

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

Phys. Rev. Lett. (4)

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

Y. P. Feng, S. K. Sinha, H. W. Deckman, J. B. Hastings, and D. P. Siddons, "X-ray flux enhancement in thin-film waveguides using resonant beam couplers," Phys. Rev. Lett. 71, 537-540 (1993).
[CrossRef] [PubMed]

M. J. Zwanenburg, J. F. Peters, J. H. H. Bongaerts, S. A. de Vries, D. L. Abernathy, and J. F. van der Veen, "Coherent propagation of x rays in a planar waveguide with a tunable air gap," Phys. Rev. Lett. 82, 1696-1699 (1999).
[CrossRef]

A. Jarre, C. Fuhse, C. Ollinger, J. Seeger, R. Tucoulou, and T. Salditt, "Two-dimensional hard x-ray beam compression by combined focusing and waveguide optics," Phys. Rev. Lett. 94, 074801 (2005).
[CrossRef] [PubMed]

Physica B (2)

M. J. Zwanenburg, J. H. H. Bongaerts, J. F. Peters, D. Riese, and J. F. van der Veen, "Focusing of coherent x-rays in a tapered planar waveguide," Physica B 283, 285-288 (2000).
[CrossRef]

C. Fuhse and T. Salditt, "Finite-difference field calculations for one-dimensionally confined x-ray waveguides," Physica B 357, 57-60 (2005).
[CrossRef]

Rev. Sci. Instrum. (1)

M. J. Zwanenburg, J. F. van der Veen, H. G. Ficke, and H. Neerings, "A planar x-ray waveguide with a tunable air gap for the structural investigation of confined fluids," Rev. Sci. Instrum. 71, 1723-1732 (2000).
[CrossRef]

Science (1)

F. Pfeiffer, C. David, M. Burghammer, C. Riekel, and T. Salditt, "Two-dimensional x-ray waveguides and point sources," Science 297, 230-234 (2002).
[CrossRef] [PubMed]

Other (3)

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, 1974).

J. W. Thomas, Numerical Partial Differential Equations (Springer-Verlag, 1995).

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Figures (3)

Fig. 1
Fig. 1

The grid, on which the electromagnetic field is calculated, is equidistantly spaced by Δx, Δy, and Δz in the respective directions. The wave vector of the incident plane wave is assumed to be parallel to the x axis.

Fig. 2
Fig. 2

Reflectivity of a Gaussian-shaped beam on a Ni surface calculated with the finite-difference (FD) approach and compared to the Fresnel solution (grazing angle 0.2°, no surface roughness). In the regime of total external reflection the curves are in excellent agreement. For higher photon energies the FD approach yields a higher reflectance.

Fig. 3
Fig. 3

Field intensities in circular waveguides calculated with the FD approach and the weakly guiding fiber approximation are practically the same. Calculations are shown for a vacuum guiding core in Ge with radii of 12, 16, 24, and 40 nm. The photon energy is 12.5 keV, and the number of excited modes rises from 1 to 3. (Color bars indicate |ψ|2∕|E 0|2 in percentages.)

Equations (31)

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Δ ψ + k 2 n 2 ( x , y , z ) ψ = 0 ,
ψ ( x , y , z ) = u ( x , y , z ) exp ( i k x ) .
2 i k u x + ( 2 y 2 + 2 z 2 ) u + k 2 ( n 2 1 ) u = 0.
A := i 2 k ,
F ( x , y , z ) := i k 2 [ n 2 ( x , y , z ) 1 ] ,
u x = A ( 2 u y 2 + 2 u z 2 ) + F ( x , y , z ) u .
u jk n := u ( x n , y j , z k ) ,
x n := n Δ x , n = 0 , 1 2 , 1 , 1 1 2 , 2 , , M x ,
y j := j Δ x , j = 0 , 1 , 2 , , M y ,
z k := k Δ z , k = 0 , 1 , 2 , , M z .
u ( x n , y j , z k ) = E 0 exp [ i ( n cl 1 ) k x n ]     for   y { 0 , M y }   or   z { 0 , M z } ,
u jk n + 1 / 2 u jk n Δ x / 2 = A Δ y 2 δ y 2 u jk n + 1 / 2 + A Δ z 2 δ z 2 u jk n + 1 2 ( F jk n u jk n + F jk n + 1 / 2 u jk n + 1 / 2 ) ,
δ y 2 u jk n := u j - 1 k n 2 u jk n + u j + 1 k n ,
δ z 2 u jk n := u jk - 1 n 2 u jk n + u jk + 1 n ,
F jk n := F ( x n , y j , z k ) .
r y := A Δ x Δ y 2 , r z := A Δ x Δ z 2 , C jk n := F jk n Δ x 4 ,
( 1 r y 2 δ y 2 C jk n + 1 / 2 ) u jk n + 1 / 2 = ( 1 + r z 2 δ z 2 + C jk n ) u jk n .
[ B 1 Θ Θ B 2 Θ Θ B M z 1 ] [ u 1 n + 1 / 2 u M z - 1 n + 1 / 2 ] = [ r 1 . . r M z 1 ] ,
B k = [ 1 + r y C 1 k n + 1 / 2 r y 2 0 r y 2 1 + r y C 2 k n + 1 / 2 r y 2 0 0 r y 2 1 + r y C M y - 1 k n + 1 / 2 ] ,
u k n 1 / 2 = [ u 1 k n + 1 / 2 u M y - 1 , k n + 1 / 2 ] ,     r k = [ ( 1 + r z 2 δ z 2 + C 1 k n ) u 1 k n + r y 2 u 0 k n + 1 / 2 ( 1 + r z 2 δ z 2 + C 2 k n ) u 2 k n ... ( 1 + r z 2 δ z 2 + C M y - 2 k n ) u M y - 2 k n ( 1 + r z 2 δ z 2 + C M y - 1 k n ) u M y - 1 k n + r y 2 u M y k n + 1 / 2 ] .
B k u k n + 1 / 2 = r k , k = 1 , , M z 1 .
u jk n + 1 u jk n + 1 / 2 Δ x / 2 = A Δ y 2 δ y 2 u jk n + 1 / 2 + A Δ z 2 δ z 2 u k n + 1 + 1 2 ( F ij n + 1 / 2 u ij n + 1 / 2 + F ij n + 1 u ij n + 1 ) .
B j v j n + 1 = r j , j = 1 , , M y 1 ,
B j = [ 1 + r z C j 1 n + 1 r z 2 0 r z 2 1 + r z C j 2 n + 1 r z 2 0 0 r z 2 1 + r z C jM z - 1 n + 1 ] ,
v j n + 1 = [ u j 1 n + 1 . . u jM z - 1 n + 1 ] ,    
r j = [ ( 1 + r y 2 δ y 2 + C j 1 n + 1 / 2 ) u j 1 n + 1 / 2 + r z 2 u j 0 n + 1 ( 1 + r y 2 δ y 2 + C j 2 n + 1 / 2 ) u j 2 n + 1 / 2 ( 1 + r y 2 δ y 2 + C jM z - 2 n + 1 / 2 ) u jM z - 2 n + 1 / 2 ( 1 + r y 2 δ y 2 + C jM z - 1 n + 1 / 2 ) u jM z - 1 n + 1 / 2 + r z 2 u jM z n + 1 ] .
ψ 0 m ( r ) = A m { J 0 ( u m r / a ) J 0 ( u m ) , r < a K 0 ( w m r / a ) K 0 ( w m ) , r a .
u m J 1 ( u m ) J 0 ( u m ) = w m K 1 ( w m ) K 0 ( w m ) ,
ψ ( r , x ) = m c m ψ 0 m ( r ) exp ( i β m x ) exp ( μ m x / 2 ) .
c m = 2 π 0 ψ 0 m ( r ) E 0 r d r
μ m = 2 π 0 | ψ 0 m ( r ) | 2 μ ( r ) r d r .

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