Abstract

We have generalized finite-difference (FD) simulations for time-dependent field propagation problems, in particular in view of ultra-short x-ray pulse propagation and dispersion. To this end, we first derive the stationary paraxial (parabolic) wave equation for the scalar field envelope in a more general manner than typically found in the literature. We then present an efficient FD implementation of propagators for different dimensionality for stationary field propagation, before we treat time-dependent problems by spectral decomposition, and suitable numerical sampling. We prove the validity of the numerical approach by comparison to analytical theory, using simple tractable propagation problems. Finally, we apply the framework to the problem of modal dispersion in X-ray waveguide. We show that X-ray waveguides can be considered as non-dispersive optical elements down to sub-femtosecond pulse width. Only when considering resonant absorption close to an X-ray absorption edge, we observe pronounced dispersion effects for experimentally achievable pulse widths. All code used for the work is made available as supplemental material.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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Generalized rectangular finite difference beam propagation method

Slawomir Sujecki
Appl. Opt. 47(23) 4280-4286 (2008)

References

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  27. C. K. Schmising, B. Pfau, M. Schneider, C. M. Günther, M. Giovannella, J. Perron, B. Vodungbo, L. Müller, F. Capotondi, E. Pedersoli, N. Mahne, J. Lüning, and S. Eisebitt, “Imaging Ultrafast Demagnetization Dynamics after a Spatially Localized Optical Excitation,” Phys. Rev. Lett. 112(21), 217203 (2014).
    [Crossref]
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    [Crossref]
  34. C. G. Gray and B.G. Nickel, “Debye potential representation of vector fields,” Am. J. Phys. 47(8), 736 (1979).

2017 (2)

Q. Zhong, M. Osterhoff, M. W. Wen, Z. S. Wang, and T. Salditt, “X-ray waveguide arrays: tailored near fields by multi-beam interference,” X-Ray Spectrom. 46(2), 107–115 (2017).
[Crossref]

L. Li, M. Wojcik, and C. Jacobsen, “Multislice does it all – calculating the performance of nanofocusing X-ray optics,” Opt. Express 25(3), 13107–13124 (2017).

2016 (2)

L. Samoylova, A. Buzmakov, O. Chubar O, and H. Sinn , “WavePropaGator: interactive framework for X-ray free-electron laser optics design and simulations,” J. Appl. Cryst. 49(4), 1347–1355 (2016).
[Crossref]

S. Hoffmann-Urlaub and T. Salditt, “Miniaturized beamsplitters realized by X-ray waveguides,” Acta Crystallographica Section A,  72(5), 515–522 (2016).
[Crossref]

2015 (2)

T. Salditt, S. Hoffmann, M. Vassholz, J. Haber, M. Osterhoff, and J. Hilhorst, “X-Ray Optics on a Chip: Guiding X Rays in Curved Channels, Phys. Rev. Lett. 115, 203902 (2015).
[Crossref]

I. Barke, H. Hartmann, D. Rupp, L. Flückiger, M. Sauppe, M. Adolph, S. Schorb, C. Bostedt, R. Treusch, C. Peltz, S. Bartling, T. Fennel, K. Meiwes-Broer, and T. Möller, “The 3D-architecture of individual free silver nanoparticles captured by X-ray scattering,” Nat. Commun. 6, 6187 (2015).
[Crossref] [PubMed]

2014 (2)

C. K. Schmising, B. Pfau, M. Schneider, C. M. Günther, M. Giovannella, J. Perron, B. Vodungbo, L. Müller, F. Capotondi, E. Pedersoli, N. Mahne, J. Lüning, and S. Eisebitt, “Imaging Ultrafast Demagnetization Dynamics after a Spatially Localized Optical Excitation,” Phys. Rev. Lett. 112(21), 217203 (2014).
[Crossref]

M. Ornigotti and A. Aiello, “The Hertz vector revisited: A simple physical picture,” J. Opt. 16(10), 105705 (2014).
[Crossref]

2010 (5)

A.W. Norfolk and E.J. Grace, “Reconstruction of optical fields with the Quasi-discrete Hankel transform,” Opt. Express 18(10), 10551–10556 (2010).
[Crossref] [PubMed]

R. Röhlsberger, K. Schlage, B. Sahoo, S. Couet, and R. Rüffer, “Collective Lamb Shift in Single-Photon Superradiance,” Science 328(5983), 1248–1251 (2010).
[Crossref] [PubMed]

H.N. Chapman and K.A. Nugent, “Coherent lensless x-ray imaging,” Nat. Photon 4(12), 833–839 (2010).
[Crossref]

A. Barty, “Time-resolved imaging using x-ray free electron lasers,” J. Phys. B: At. Mol. Opt. Phys. 43(19), 194014 (2010).
[Crossref]

S. P. Krüger, K. Giewekemeyer, S. Kalbfleisch, M. Bartels, H. Neubauer, and T. Salditt, “Sub-15 nm beam confinement by twocrossed x-ray waveguides,” Opt. Express 18(13), 13492–13501 (2010).
[Crossref] [PubMed]

2007 (1)

F. Pérez and B.E. Granger, “IPython: a System for Interactive Scientific Computing,” Computing in Science and Engineering 9(3), 21–29 (2007).
[Crossref]

2006 (1)

2005 (1)

C. Fuhse and T. Salditt, “Finite-difference field calculations for one-dimensionally confined X-ray waveguides,” Phys. B: Condensed Matter 357(1–2), 57–60 (2005).
[Crossref]

2004 (1)

A. Jarre, T. Salditt, T. Panzner, U. Pietsch, and F. Pfeiffer, “White beam x-ray waveguide optics,” Appl. Phys. Lett. 85(2),” 161–163 (2004).
[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 (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 Rad. 9, 383–393 (2002).
[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]

1979 (1)

C. G. Gray and B.G. Nickel, “Debye potential representation of vector fields,” Am. J. Phys. 47(8), 736 (1979).

1970 (1)

1959 (1)

E. Wolf, “A Scalar Representation of Electromagnetic Fields: II,” Proc. Phys. Soc. A,  74(3), 269 (1959).
[Crossref]

1953 (1)

H.S. Green and E. Wolf, “A Scalar Representation of Electromagnetic Fields,” Proc. Phys. Soc. A 66(12), 1129 (1953).
[Crossref]

1947 (1)

J. Crank and P. Nicolson, “A practical method for numerical evaluation of solutions of partial differential Eqs. of the heat-conduction type,” Math. Proc. Cam. Phil. Soc. 43(1), 50–67 (1947).
[Crossref]

Adolph, M.

I. Barke, H. Hartmann, D. Rupp, L. Flückiger, M. Sauppe, M. Adolph, S. Schorb, C. Bostedt, R. Treusch, C. Peltz, S. Bartling, T. Fennel, K. Meiwes-Broer, and T. Möller, “The 3D-architecture of individual free silver nanoparticles captured by X-ray scattering,” Nat. Commun. 6, 6187 (2015).
[Crossref] [PubMed]

Aiello, A.

M. Ornigotti and A. Aiello, “The Hertz vector revisited: A simple physical picture,” J. Opt. 16(10), 105705 (2014).
[Crossref]

Barke, I.

I. Barke, H. Hartmann, D. Rupp, L. Flückiger, M. Sauppe, M. Adolph, S. Schorb, C. Bostedt, R. Treusch, C. Peltz, S. Bartling, T. Fennel, K. Meiwes-Broer, and T. Möller, “The 3D-architecture of individual free silver nanoparticles captured by X-ray scattering,” Nat. Commun. 6, 6187 (2015).
[Crossref] [PubMed]

Bartels, M.

Bartling, S.

I. Barke, H. Hartmann, D. Rupp, L. Flückiger, M. Sauppe, M. Adolph, S. Schorb, C. Bostedt, R. Treusch, C. Peltz, S. Bartling, T. Fennel, K. Meiwes-Broer, and T. Möller, “The 3D-architecture of individual free silver nanoparticles captured by X-ray scattering,” Nat. Commun. 6, 6187 (2015).
[Crossref] [PubMed]

Barty, A.

A. Barty, “Time-resolved imaging using x-ray free electron lasers,” J. Phys. B: At. Mol. Opt. Phys. 43(19), 194014 (2010).
[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 Rad. 9, 383–393 (2002).
[Crossref]

Bostedt, C.

I. Barke, H. Hartmann, D. Rupp, L. Flückiger, M. Sauppe, M. Adolph, S. Schorb, C. Bostedt, R. Treusch, C. Peltz, S. Bartling, T. Fennel, K. Meiwes-Broer, and T. Möller, “The 3D-architecture of individual free silver nanoparticles captured by X-ray scattering,” Nat. Commun. 6, 6187 (2015).
[Crossref] [PubMed]

Brunetti, A.

T. Schoonjans, A. Brunetti, B. Golosio, M. Sanchez del Rio, A. Sole, C. Ferrero, and L. Vincze, “xraylib 3.1.0,” xraylib 3.1.0. Zenodo https://doi.org/10.5281/zenodo.12378 (2014).

Buzmakov, A.

L. Samoylova, A. Buzmakov, O. Chubar O, and H. Sinn , “WavePropaGator: interactive framework for X-ray free-electron laser optics design and simulations,” J. Appl. Cryst. 49(4), 1347–1355 (2016).
[Crossref]

Capotondi, F.

C. K. Schmising, B. Pfau, M. Schneider, C. M. Günther, M. Giovannella, J. Perron, B. Vodungbo, L. Müller, F. Capotondi, E. Pedersoli, N. Mahne, J. Lüning, and S. Eisebitt, “Imaging Ultrafast Demagnetization Dynamics after a Spatially Localized Optical Excitation,” Phys. Rev. Lett. 112(21), 217203 (2014).
[Crossref]

Chapman, H.N.

H.N. Chapman and K.A. Nugent, “Coherent lensless x-ray imaging,” Nat. Photon 4(12), 833–839 (2010).
[Crossref]

Chubar O, O.

L. Samoylova, A. Buzmakov, O. Chubar O, and H. Sinn , “WavePropaGator: interactive framework for X-ray free-electron laser optics design and simulations,” J. Appl. Cryst. 49(4), 1347–1355 (2016).
[Crossref]

Couet, S.

R. Röhlsberger, K. Schlage, B. Sahoo, S. Couet, and R. Rüffer, “Collective Lamb Shift in Single-Photon Superradiance,” Science 328(5983), 1248–1251 (2010).
[Crossref] [PubMed]

Crank, J.

J. Crank and P. Nicolson, “A practical method for numerical evaluation of solutions of partial differential Eqs. of the heat-conduction type,” Math. Proc. Cam. Phil. Soc. 43(1), 50–67 (1947).
[Crossref]

David, C.

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 Rad. 9, 383–393 (2002).
[Crossref]

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 Rad. 9, 383–393 (2002).
[Crossref]

Eisebitt, S.

C. K. Schmising, B. Pfau, M. Schneider, C. M. Günther, M. Giovannella, J. Perron, B. Vodungbo, L. Müller, F. Capotondi, E. Pedersoli, N. Mahne, J. Lüning, and S. Eisebitt, “Imaging Ultrafast Demagnetization Dynamics after a Spatially Localized Optical Excitation,” Phys. Rev. Lett. 112(21), 217203 (2014).
[Crossref]

Fennel, T.

I. Barke, H. Hartmann, D. Rupp, L. Flückiger, M. Sauppe, M. Adolph, S. Schorb, C. Bostedt, R. Treusch, C. Peltz, S. Bartling, T. Fennel, K. Meiwes-Broer, and T. Möller, “The 3D-architecture of individual free silver nanoparticles captured by X-ray scattering,” Nat. Commun. 6, 6187 (2015).
[Crossref] [PubMed]

Ferrero, C.

T. Schoonjans, A. Brunetti, B. Golosio, M. Sanchez del Rio, A. Sole, C. Ferrero, and L. Vincze, “xraylib 3.1.0,” xraylib 3.1.0. Zenodo https://doi.org/10.5281/zenodo.12378 (2014).

Flannery, B. P.

W. H. Press, S.A. Teukolsky, W.T. Vetterling, and B. P. Flannery, Numerical Recipes, 3rd ed. (Cambridge University, 2007).

Flückiger, L.

I. Barke, H. Hartmann, D. Rupp, L. Flückiger, M. Sauppe, M. Adolph, S. Schorb, C. Bostedt, R. Treusch, C. Peltz, S. Bartling, T. Fennel, K. Meiwes-Broer, and T. Möller, “The 3D-architecture of individual free silver nanoparticles captured by X-ray scattering,” Nat. Commun. 6, 6187 (2015).
[Crossref] [PubMed]

Fuhse, C.

C. Fuhse and Tim Salditt, “Finite-difference field calculations for two-dimensionally confined x-ray waveguides,” Appl. Opt. 45(19), 4603–4608 (2006).
[Crossref] [PubMed]

C. Fuhse and T. Salditt, “Finite-difference field calculations for one-dimensionally confined X-ray waveguides,” Phys. B: Condensed Matter 357(1–2), 57–60 (2005).
[Crossref]

Giewekemeyer, K.

Giovannella, M.

C. K. Schmising, B. Pfau, M. Schneider, C. M. Günther, M. Giovannella, J. Perron, B. Vodungbo, L. Müller, F. Capotondi, E. Pedersoli, N. Mahne, J. Lüning, and S. Eisebitt, “Imaging Ultrafast Demagnetization Dynamics after a Spatially Localized Optical Excitation,” Phys. Rev. Lett. 112(21), 217203 (2014).
[Crossref]

Golosio, B.

T. Schoonjans, A. Brunetti, B. Golosio, M. Sanchez del Rio, A. Sole, C. Ferrero, and L. Vincze, “xraylib 3.1.0,” xraylib 3.1.0. Zenodo https://doi.org/10.5281/zenodo.12378 (2014).

Goodman, J.W.

J.W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Grace, E.J.

Granger, B.E.

F. Pérez and B.E. Granger, “IPython: a System for Interactive Scientific Computing,” Computing in Science and Engineering 9(3), 21–29 (2007).
[Crossref]

Gray, C. G.

C. G. Gray and B.G. Nickel, “Debye potential representation of vector fields,” Am. J. Phys. 47(8), 736 (1979).

Green, H.S.

H.S. Green and E. Wolf, “A Scalar Representation of Electromagnetic Fields,” Proc. Phys. Soc. A 66(12), 1129 (1953).
[Crossref]

Günther, C. M.

C. K. Schmising, B. Pfau, M. Schneider, C. M. Günther, M. Giovannella, J. Perron, B. Vodungbo, L. Müller, F. Capotondi, E. Pedersoli, N. Mahne, J. Lüning, and S. Eisebitt, “Imaging Ultrafast Demagnetization Dynamics after a Spatially Localized Optical Excitation,” Phys. Rev. Lett. 112(21), 217203 (2014).
[Crossref]

Haber, J.

T. Salditt, S. Hoffmann, M. Vassholz, J. Haber, M. Osterhoff, and J. Hilhorst, “X-Ray Optics on a Chip: Guiding X Rays in Curved Channels, Phys. Rev. Lett. 115, 203902 (2015).
[Crossref]

Hartmann, H.

I. Barke, H. Hartmann, D. Rupp, L. Flückiger, M. Sauppe, M. Adolph, S. Schorb, C. Bostedt, R. Treusch, C. Peltz, S. Bartling, T. Fennel, K. Meiwes-Broer, and T. Möller, “The 3D-architecture of individual free silver nanoparticles captured by X-ray scattering,” Nat. Commun. 6, 6187 (2015).
[Crossref] [PubMed]

Hilhorst, J.

T. Salditt, S. Hoffmann, M. Vassholz, J. Haber, M. Osterhoff, and J. Hilhorst, “X-Ray Optics on a Chip: Guiding X Rays in Curved Channels, Phys. Rev. Lett. 115, 203902 (2015).
[Crossref]

Hoffmann, S.

T. Salditt, S. Hoffmann, M. Vassholz, J. Haber, M. Osterhoff, and J. Hilhorst, “X-Ray Optics on a Chip: Guiding X Rays in Curved Channels, Phys. Rev. Lett. 115, 203902 (2015).
[Crossref]

Hoffmann-Urlaub, S.

S. Hoffmann-Urlaub and T. Salditt, “Miniaturized beamsplitters realized by X-ray waveguides,” Acta Crystallographica Section A,  72(5), 515–522 (2016).
[Crossref]

Husakou, A.

A. Husakou, “Nichtlineare Phaenomene spektral ultrabreiter Strahlung in Photonischen Kristallfasern und Hohlen Wellenleitern,” Ph.D. dissertation, Freie Universität Berlin, Germany (2002).

Jacobsen, C.

Jarre, A.

A. Jarre, T. Salditt, T. Panzner, U. Pietsch, and F. Pfeiffer, “White beam x-ray waveguide optics,” Appl. Phys. Lett. 85(2),” 161–163 (2004).
[Crossref]

Kalbfleisch, S.

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 Rad. 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]

Krüger, S. P.

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 Rad. 9, 383–393 (2002).
[Crossref]

Li, L.

Lüning, J.

C. K. Schmising, B. Pfau, M. Schneider, C. M. Günther, M. Giovannella, J. Perron, B. Vodungbo, L. Müller, F. Capotondi, E. Pedersoli, N. Mahne, J. Lüning, and S. Eisebitt, “Imaging Ultrafast Demagnetization Dynamics after a Spatially Localized Optical Excitation,” Phys. Rev. Lett. 112(21), 217203 (2014).
[Crossref]

Mahne, N.

C. K. Schmising, B. Pfau, M. Schneider, C. M. Günther, M. Giovannella, J. Perron, B. Vodungbo, L. Müller, F. Capotondi, E. Pedersoli, N. Mahne, J. Lüning, and S. Eisebitt, “Imaging Ultrafast Demagnetization Dynamics after a Spatially Localized Optical Excitation,” Phys. Rev. Lett. 112(21), 217203 (2014).
[Crossref]

Marathay, A. S.

Meiwes-Broer, K.

I. Barke, H. Hartmann, D. Rupp, L. Flückiger, M. Sauppe, M. Adolph, S. Schorb, C. Bostedt, R. Treusch, C. Peltz, S. Bartling, T. Fennel, K. Meiwes-Broer, and T. Möller, “The 3D-architecture of individual free silver nanoparticles captured by X-ray scattering,” Nat. Commun. 6, 6187 (2015).
[Crossref] [PubMed]

Möller, T.

I. Barke, H. Hartmann, D. Rupp, L. Flückiger, M. Sauppe, M. Adolph, S. Schorb, C. Bostedt, R. Treusch, C. Peltz, S. Bartling, T. Fennel, K. Meiwes-Broer, and T. Möller, “The 3D-architecture of individual free silver nanoparticles captured by X-ray scattering,” Nat. Commun. 6, 6187 (2015).
[Crossref] [PubMed]

Müller, L.

C. K. Schmising, B. Pfau, M. Schneider, C. M. Günther, M. Giovannella, J. Perron, B. Vodungbo, L. Müller, F. Capotondi, E. Pedersoli, N. Mahne, J. Lüning, and S. Eisebitt, “Imaging Ultrafast Demagnetization Dynamics after a Spatially Localized Optical Excitation,” Phys. Rev. Lett. 112(21), 217203 (2014).
[Crossref]

Neubauer, H.

Nickel, B.G.

C. G. Gray and B.G. Nickel, “Debye potential representation of vector fields,” Am. J. Phys. 47(8), 736 (1979).

Nicolson, P.

J. Crank and P. Nicolson, “A practical method for numerical evaluation of solutions of partial differential Eqs. of the heat-conduction type,” Math. Proc. Cam. Phil. Soc. 43(1), 50–67 (1947).
[Crossref]

Norfolk, A.W.

Nugent, K.A.

H.N. Chapman and K.A. Nugent, “Coherent lensless x-ray imaging,” Nat. Photon 4(12), 833–839 (2010).
[Crossref]

Ornigotti, M.

M. Ornigotti and A. Aiello, “The Hertz vector revisited: A simple physical picture,” J. Opt. 16(10), 105705 (2014).
[Crossref]

Osterhoff, M.

Q. Zhong, M. Osterhoff, M. W. Wen, Z. S. Wang, and T. Salditt, “X-ray waveguide arrays: tailored near fields by multi-beam interference,” X-Ray Spectrom. 46(2), 107–115 (2017).
[Crossref]

T. Salditt, S. Hoffmann, M. Vassholz, J. Haber, M. Osterhoff, and J. Hilhorst, “X-Ray Optics on a Chip: Guiding X Rays in Curved Channels, Phys. Rev. Lett. 115, 203902 (2015).
[Crossref]

Paganin, D. M.

D. M. Paganin, Coherent X-ray Optics (Oxford University, 2006).
[Crossref]

Panzner, T.

A. Jarre, T. Salditt, T. Panzner, U. Pietsch, and F. Pfeiffer, “White beam x-ray waveguide optics,” Appl. Phys. Lett. 85(2),” 161–163 (2004).
[Crossref]

Parrent, G. B.

Pedersoli, E.

C. K. Schmising, B. Pfau, M. Schneider, C. M. Günther, M. Giovannella, J. Perron, B. Vodungbo, L. Müller, F. Capotondi, E. Pedersoli, N. Mahne, J. Lüning, and S. Eisebitt, “Imaging Ultrafast Demagnetization Dynamics after a Spatially Localized Optical Excitation,” Phys. Rev. Lett. 112(21), 217203 (2014).
[Crossref]

Peltz, C.

I. Barke, H. Hartmann, D. Rupp, L. Flückiger, M. Sauppe, M. Adolph, S. Schorb, C. Bostedt, R. Treusch, C. Peltz, S. Bartling, T. Fennel, K. Meiwes-Broer, and T. Möller, “The 3D-architecture of individual free silver nanoparticles captured by X-ray scattering,” Nat. Commun. 6, 6187 (2015).
[Crossref] [PubMed]

Pérez, F.

F. Pérez and B.E. Granger, “IPython: a System for Interactive Scientific Computing,” Computing in Science and Engineering 9(3), 21–29 (2007).
[Crossref]

Perron, J.

C. K. Schmising, B. Pfau, M. Schneider, C. M. Günther, M. Giovannella, J. Perron, B. Vodungbo, L. Müller, F. Capotondi, E. Pedersoli, N. Mahne, J. Lüning, and S. Eisebitt, “Imaging Ultrafast Demagnetization Dynamics after a Spatially Localized Optical Excitation,” Phys. Rev. Lett. 112(21), 217203 (2014).
[Crossref]

Pfau, B.

C. K. Schmising, B. Pfau, M. Schneider, C. M. Günther, M. Giovannella, J. Perron, B. Vodungbo, L. Müller, F. Capotondi, E. Pedersoli, N. Mahne, J. Lüning, and S. Eisebitt, “Imaging Ultrafast Demagnetization Dynamics after a Spatially Localized Optical Excitation,” Phys. Rev. Lett. 112(21), 217203 (2014).
[Crossref]

Pfeiffer, F.

A. Jarre, T. Salditt, T. Panzner, U. Pietsch, and F. Pfeiffer, “White beam x-ray waveguide optics,” Appl. Phys. Lett. 85(2),” 161–163 (2004).
[Crossref]

Pietsch, U.

A. Jarre, T. Salditt, T. Panzner, U. Pietsch, and F. Pfeiffer, “White beam x-ray waveguide optics,” Appl. Phys. Lett. 85(2),” 161–163 (2004).
[Crossref]

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]

Press, W. H.

W. H. Press, S.A. Teukolsky, W.T. Vetterling, and B. P. Flannery, Numerical Recipes, 3rd ed. (Cambridge University, 2007).

Röhlsberger, R.

R. Röhlsberger, K. Schlage, B. Sahoo, S. Couet, and R. Rüffer, “Collective Lamb Shift in Single-Photon Superradiance,” Science 328(5983), 1248–1251 (2010).
[Crossref] [PubMed]

Rüffer, R.

R. Röhlsberger, K. Schlage, B. Sahoo, S. Couet, and R. Rüffer, “Collective Lamb Shift in Single-Photon Superradiance,” Science 328(5983), 1248–1251 (2010).
[Crossref] [PubMed]

Rupp, D.

I. Barke, H. Hartmann, D. Rupp, L. Flückiger, M. Sauppe, M. Adolph, S. Schorb, C. Bostedt, R. Treusch, C. Peltz, S. Bartling, T. Fennel, K. Meiwes-Broer, and T. Möller, “The 3D-architecture of individual free silver nanoparticles captured by X-ray scattering,” Nat. Commun. 6, 6187 (2015).
[Crossref] [PubMed]

Sahoo, B.

R. Röhlsberger, K. Schlage, B. Sahoo, S. Couet, and R. Rüffer, “Collective Lamb Shift in Single-Photon Superradiance,” Science 328(5983), 1248–1251 (2010).
[Crossref] [PubMed]

Salditt, T.

Q. Zhong, M. Osterhoff, M. W. Wen, Z. S. Wang, and T. Salditt, “X-ray waveguide arrays: tailored near fields by multi-beam interference,” X-Ray Spectrom. 46(2), 107–115 (2017).
[Crossref]

S. Hoffmann-Urlaub and T. Salditt, “Miniaturized beamsplitters realized by X-ray waveguides,” Acta Crystallographica Section A,  72(5), 515–522 (2016).
[Crossref]

T. Salditt, S. Hoffmann, M. Vassholz, J. Haber, M. Osterhoff, and J. Hilhorst, “X-Ray Optics on a Chip: Guiding X Rays in Curved Channels, Phys. Rev. Lett. 115, 203902 (2015).
[Crossref]

S. P. Krüger, K. Giewekemeyer, S. Kalbfleisch, M. Bartels, H. Neubauer, and T. Salditt, “Sub-15 nm beam confinement by twocrossed x-ray waveguides,” Opt. Express 18(13), 13492–13501 (2010).
[Crossref] [PubMed]

C. Fuhse and T. Salditt, “Finite-difference field calculations for one-dimensionally confined X-ray waveguides,” Phys. B: Condensed Matter 357(1–2), 57–60 (2005).
[Crossref]

A. Jarre, T. Salditt, T. Panzner, U. Pietsch, and F. Pfeiffer, “White beam x-ray waveguide optics,” Appl. Phys. Lett. 85(2),” 161–163 (2004).
[Crossref]

Salditt, Tim

Saleh, B.E.A

B.E.A Saleh and M.C. Teich, “Fundamentals of photonics,” Wiley (1991).
[Crossref]

Samoylova, L.

L. Samoylova, A. Buzmakov, O. Chubar O, and H. Sinn , “WavePropaGator: interactive framework for X-ray free-electron laser optics design and simulations,” J. Appl. Cryst. 49(4), 1347–1355 (2016).
[Crossref]

Sanchez del Rio, M.

T. Schoonjans, A. Brunetti, B. Golosio, M. Sanchez del Rio, A. Sole, C. Ferrero, and L. Vincze, “xraylib 3.1.0,” xraylib 3.1.0. Zenodo https://doi.org/10.5281/zenodo.12378 (2014).

Sauppe, M.

I. Barke, H. Hartmann, D. Rupp, L. Flückiger, M. Sauppe, M. Adolph, S. Schorb, C. Bostedt, R. Treusch, C. Peltz, S. Bartling, T. Fennel, K. Meiwes-Broer, and T. Möller, “The 3D-architecture of individual free silver nanoparticles captured by X-ray scattering,” Nat. Commun. 6, 6187 (2015).
[Crossref] [PubMed]

Schlage, K.

R. Röhlsberger, K. Schlage, B. Sahoo, S. Couet, and R. Rüffer, “Collective Lamb Shift in Single-Photon Superradiance,” Science 328(5983), 1248–1251 (2010).
[Crossref] [PubMed]

Schmidt, J.D.

J.D. Schmidt, “Numerical Simulation of Optical Wave Propagation with Examples in MATLAB,” SPIE monographs (2010).

Schmising, C. K.

C. K. Schmising, B. Pfau, M. Schneider, C. M. Günther, M. Giovannella, J. Perron, B. Vodungbo, L. Müller, F. Capotondi, E. Pedersoli, N. Mahne, J. Lüning, and S. Eisebitt, “Imaging Ultrafast Demagnetization Dynamics after a Spatially Localized Optical Excitation,” Phys. Rev. Lett. 112(21), 217203 (2014).
[Crossref]

Schneider, M.

C. K. Schmising, B. Pfau, M. Schneider, C. M. Günther, M. Giovannella, J. Perron, B. Vodungbo, L. Müller, F. Capotondi, E. Pedersoli, N. Mahne, J. Lüning, and S. Eisebitt, “Imaging Ultrafast Demagnetization Dynamics after a Spatially Localized Optical Excitation,” Phys. Rev. Lett. 112(21), 217203 (2014).
[Crossref]

Schoonjans, T.

T. Schoonjans, A. Brunetti, B. Golosio, M. Sanchez del Rio, A. Sole, C. Ferrero, and L. Vincze, “xraylib 3.1.0,” xraylib 3.1.0. Zenodo https://doi.org/10.5281/zenodo.12378 (2014).

Schorb, S.

I. Barke, H. Hartmann, D. Rupp, L. Flückiger, M. Sauppe, M. Adolph, S. Schorb, C. Bostedt, R. Treusch, C. Peltz, S. Bartling, T. Fennel, K. Meiwes-Broer, and T. Möller, “The 3D-architecture of individual free silver nanoparticles captured by X-ray scattering,” Nat. Commun. 6, 6187 (2015).
[Crossref] [PubMed]

Sinn, H.

L. Samoylova, A. Buzmakov, O. Chubar O, and H. Sinn , “WavePropaGator: interactive framework for X-ray free-electron laser optics design and simulations,” J. Appl. Cryst. 49(4), 1347–1355 (2016).
[Crossref]

Sole, A.

T. Schoonjans, A. Brunetti, B. Golosio, M. Sanchez del Rio, A. Sole, C. Ferrero, and L. Vincze, “xraylib 3.1.0,” xraylib 3.1.0. Zenodo https://doi.org/10.5281/zenodo.12378 (2014).

Teich, M.C.

B.E.A Saleh and M.C. Teich, “Fundamentals of photonics,” Wiley (1991).
[Crossref]

Teukolsky, S.A.

W. H. Press, S.A. Teukolsky, W.T. Vetterling, and B. P. Flannery, Numerical Recipes, 3rd ed. (Cambridge University, 2007).

Treusch, R.

I. Barke, H. Hartmann, D. Rupp, L. Flückiger, M. Sauppe, M. Adolph, S. Schorb, C. Bostedt, R. Treusch, C. Peltz, S. Bartling, T. Fennel, K. Meiwes-Broer, and T. Möller, “The 3D-architecture of individual free silver nanoparticles captured by X-ray scattering,” Nat. Commun. 6, 6187 (2015).
[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 Rad. 9, 383–393 (2002).
[Crossref]

Vassholz, M.

T. Salditt, S. Hoffmann, M. Vassholz, J. Haber, M. Osterhoff, and J. Hilhorst, “X-Ray Optics on a Chip: Guiding X Rays in Curved Channels, Phys. Rev. Lett. 115, 203902 (2015).
[Crossref]

Vetterling, W.T.

W. H. Press, S.A. Teukolsky, W.T. Vetterling, and B. P. Flannery, Numerical Recipes, 3rd ed. (Cambridge University, 2007).

Vincze, L.

T. Schoonjans, A. Brunetti, B. Golosio, M. Sanchez del Rio, A. Sole, C. Ferrero, and L. Vincze, “xraylib 3.1.0,” xraylib 3.1.0. Zenodo https://doi.org/10.5281/zenodo.12378 (2014).

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]

Vodungbo, B.

C. K. Schmising, B. Pfau, M. Schneider, C. M. Günther, M. Giovannella, J. Perron, B. Vodungbo, L. Müller, F. Capotondi, E. Pedersoli, N. Mahne, J. Lüning, and S. Eisebitt, “Imaging Ultrafast Demagnetization Dynamics after a Spatially Localized Optical Excitation,” Phys. Rev. Lett. 112(21), 217203 (2014).
[Crossref]

Wang, Z. S.

Q. Zhong, M. Osterhoff, M. W. Wen, Z. S. Wang, and T. Salditt, “X-ray waveguide arrays: tailored near fields by multi-beam interference,” X-Ray Spectrom. 46(2), 107–115 (2017).
[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 Rad. 9, 383–393 (2002).
[Crossref]

Wen, M. W.

Q. Zhong, M. Osterhoff, M. W. Wen, Z. S. Wang, and T. Salditt, “X-ray waveguide arrays: tailored near fields by multi-beam interference,” X-Ray Spectrom. 46(2), 107–115 (2017).
[Crossref]

Wojcik, M.

Wolf, E.

E. Wolf, “A Scalar Representation of Electromagnetic Fields: II,” Proc. Phys. Soc. A,  74(3), 269 (1959).
[Crossref]

H.S. Green and E. Wolf, “A Scalar Representation of Electromagnetic Fields,” Proc. Phys. Soc. A 66(12), 1129 (1953).
[Crossref]

Zhong, Q.

Q. Zhong, M. Osterhoff, M. W. Wen, Z. S. Wang, and T. Salditt, “X-ray waveguide arrays: tailored near fields by multi-beam interference,” X-Ray Spectrom. 46(2), 107–115 (2017).
[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 Rad. 9, 383–393 (2002).
[Crossref]

Acta Crystallographica Section A (1)

S. Hoffmann-Urlaub and T. Salditt, “Miniaturized beamsplitters realized by X-ray waveguides,” Acta Crystallographica Section A,  72(5), 515–522 (2016).
[Crossref]

Am. J. Phys. (1)

C. G. Gray and B.G. Nickel, “Debye potential representation of vector fields,” Am. J. Phys. 47(8), 736 (1979).

Appl. Opt. (1)

Appl. Phys. Lett. (1)

A. Jarre, T. Salditt, T. Panzner, U. Pietsch, and F. Pfeiffer, “White beam x-ray waveguide optics,” Appl. Phys. Lett. 85(2),” 161–163 (2004).
[Crossref]

Computing in Science and Engineering (1)

F. Pérez and B.E. Granger, “IPython: a System for Interactive Scientific Computing,” Computing in Science and Engineering 9(3), 21–29 (2007).
[Crossref]

J. Appl. Cryst. (1)

L. Samoylova, A. Buzmakov, O. Chubar O, and H. Sinn , “WavePropaGator: interactive framework for X-ray free-electron laser optics design and simulations,” J. Appl. Cryst. 49(4), 1347–1355 (2016).
[Crossref]

J. Opt. (1)

M. Ornigotti and A. Aiello, “The Hertz vector revisited: A simple physical picture,” J. Opt. 16(10), 105705 (2014).
[Crossref]

J. Opt. Soc. Am. (1)

J. Phys. B: At. Mol. Opt. Phys. (1)

A. Barty, “Time-resolved imaging using x-ray free electron lasers,” J. Phys. B: At. Mol. Opt. Phys. 43(19), 194014 (2010).
[Crossref]

J. Synchrotron Rad. (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 Rad. 9, 383–393 (2002).
[Crossref]

Math. Proc. Cam. Phil. Soc. (1)

J. Crank and P. Nicolson, “A practical method for numerical evaluation of solutions of partial differential Eqs. of the heat-conduction type,” Math. Proc. Cam. Phil. Soc. 43(1), 50–67 (1947).
[Crossref]

Nat. Commun. (1)

I. Barke, H. Hartmann, D. Rupp, L. Flückiger, M. Sauppe, M. Adolph, S. Schorb, C. Bostedt, R. Treusch, C. Peltz, S. Bartling, T. Fennel, K. Meiwes-Broer, and T. Möller, “The 3D-architecture of individual free silver nanoparticles captured by X-ray scattering,” Nat. Commun. 6, 6187 (2015).
[Crossref] [PubMed]

Nat. Photon (1)

H.N. Chapman and K.A. Nugent, “Coherent lensless x-ray imaging,” Nat. Photon 4(12), 833–839 (2010).
[Crossref]

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]

Opt. Express (3)

Phys. B: Condensed Matter (1)

C. Fuhse and T. Salditt, “Finite-difference field calculations for one-dimensionally confined X-ray waveguides,” Phys. B: Condensed Matter 357(1–2), 57–60 (2005).
[Crossref]

Phys. Rev. Lett. (3)

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]

T. Salditt, S. Hoffmann, M. Vassholz, J. Haber, M. Osterhoff, and J. Hilhorst, “X-Ray Optics on a Chip: Guiding X Rays in Curved Channels, Phys. Rev. Lett. 115, 203902 (2015).
[Crossref]

C. K. Schmising, B. Pfau, M. Schneider, C. M. Günther, M. Giovannella, J. Perron, B. Vodungbo, L. Müller, F. Capotondi, E. Pedersoli, N. Mahne, J. Lüning, and S. Eisebitt, “Imaging Ultrafast Demagnetization Dynamics after a Spatially Localized Optical Excitation,” Phys. Rev. Lett. 112(21), 217203 (2014).
[Crossref]

Proc. Phys. Soc. A (2)

H.S. Green and E. Wolf, “A Scalar Representation of Electromagnetic Fields,” Proc. Phys. Soc. A 66(12), 1129 (1953).
[Crossref]

E. Wolf, “A Scalar Representation of Electromagnetic Fields: II,” Proc. Phys. Soc. A,  74(3), 269 (1959).
[Crossref]

Science (1)

R. Röhlsberger, K. Schlage, B. Sahoo, S. Couet, and R. Rüffer, “Collective Lamb Shift in Single-Photon Superradiance,” Science 328(5983), 1248–1251 (2010).
[Crossref] [PubMed]

X-Ray Spectrom. (1)

Q. Zhong, M. Osterhoff, M. W. Wen, Z. S. Wang, and T. Salditt, “X-ray waveguide arrays: tailored near fields by multi-beam interference,” X-Ray Spectrom. 46(2), 107–115 (2017).
[Crossref]

Other (9)

J.W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

J.D. Schmidt, “Numerical Simulation of Optical Wave Propagation with Examples in MATLAB,” SPIE monographs (2010).

D. M. Paganin, Coherent X-ray Optics (Oxford University, 2006).
[Crossref]

W. H. Press, S.A. Teukolsky, W.T. Vetterling, and B. P. Flannery, Numerical Recipes, 3rd ed. (Cambridge University, 2007).

B.E.A Saleh and M.C. Teich, “Fundamentals of photonics,” Wiley (1991).
[Crossref]

T. Schoonjans, A. Brunetti, B. Golosio, M. Sanchez del Rio, A. Sole, C. Ferrero, and L. Vincze, “xraylib 3.1.0,” xraylib 3.1.0. Zenodo https://doi.org/10.5281/zenodo.12378 (2014).

L. Melchior and T. Salditt, “Finite difference methods for stationary and time-dependent x-ray propagation - Simulations and Figures,” Figshare (2017) [retrieved 16 November 2017], https://doi.org/10.6084/m9.figshare.5449939.

L. Melchior, “PyPropagate on Github,” Github (2017) [retrieved 27 September 2017], https://github.com/TheLartians/PyPropagate .

A. Husakou, “Nichtlineare Phaenomene spektral ultrabreiter Strahlung in Photonischen Kristallfasern und Hohlen Wellenleitern,” Ph.D. dissertation, Freie Universität Berlin, Germany (2002).

Supplementary Material (1)

NameDescription
» Code 1       PyPropagate simulation code

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Figures (8)

Fig. 1
Fig. 1 Comparison of different propagation algorithms for free space propagation and propagation in cylindrical waveguides. Optical intensity is normalized by |ψ0|2 and color coded, while the time-averaged direction of energy flow is indicated by white streamlines in the lower half of the images. Left: For free space propagation, we set the initial field distribution of a two or three dimensional monochromatic Gaussian beam with 12 keV photon energy, a waist size of σr = 0.25 μm, at 1 mm from the focus. The step sizes are chosen as Δx = Δy = Δr = 10 nm and Δz = 400 μm. At these step sizes the results of Fresnel propagation agree well with the analytical solution, while finite difference algorithms produce significant numerical errors. Right: For the waveguide propagation we choose a plane wave initial condition. The simulated waveguide consists of a vacuum core and a Germanium cladding at room temperature. The diameter is chosen as 50 nm and the initial beam is Gaussian distributed with a waist size of σr = 100 nm. The analytical solution is obtained by eigenmode projection onto guiding modes that do not dissipate into the cladding. Therefore modes that rapidly dissipate into the cladding present in numerical simulations are not taken into account. The step sizes of the numerical propagators are Δx = Δy = Δr = 0.7 nm and Δz = 0.8 μm. The waveguide edge is indicated by a dashed white line in the upper image half. At these step sizes multislice methods produce significant numerical artifacts visible as horizontal lines that prevent accurate determination of energy flow. Finite difference methods produce significantly better results that are nearly identical with simulations of much smaller step sizes.
Fig. 2
Fig. 2 Convergence behaviour of different propagation algorithms for free space propagation (a)–(b) and propagation in matter (c)–(d). The largest magnitude of the difference of the propagated field envelope up from the analytical solution ua normalized by the largest magnitude of the analytical solution is shown as a function of the number of propagation steps Nz. The step sizes are chosen as Δx = Δy = Δr ≈ 1 nm (a), 10 nm (b), 0.1 nm (c), 1 nm (d) and Δz = sz/Nz with the propagation distance sz = 20 mm (a)–(b), 1 mm (c)–(d). The Gaussian beam and waveguide parameters are identical to the parameters given in Fig. 1. For Gaussian beams the analytical solutions are exact, while the analytical solution for waveguides is based on eigenmode projection. To avoid bias caused by the non-guiding modes, for (c) and (d) the maximum is taken after propagating 0.5 mm inside the waveguide. While accuracy of Fresnel methods is independent of the propagation step size for free space propagation, they are outperformed by finite difference methods when propagating in matter. For cylindrically symmetric problems, the cylindrical symmetric propagators perform almost equally well as the 3D propagators at a significantly reduced computational cost.
Fig. 3
Fig. 3 Above: normalized intensity distribution of a 7.9 keV plane wave in a tantalum curved slab waveguide with 200 nm diameter and a radius of curvature of 40 mm. The waveguide edges are illustrated by dashed white lines. The step sizes are Δx = 350 pm and Δz = 50 nm. Center: intensity distribution from above simulation, straightened by a coordinate system that curves with the waveguide. Below: the intensity distribution of the curved coordinate system obtained directly by piecewise paraxial propagation. Due to the straightened geometry the size of the simulation box can be chosen much smaller with identical computational effort, resulting in a step size of Δx′ = 40 pm and thus improved accuracy.
Fig. 4
Fig. 4 (a) Schematic illustration of the angled pulse test-case. Two Gaussian pulses start at z = 0, one propagating downward at an angle α, and reaching the intersection point at a corresponding time delay. (b) Normalized locally averaged instantaneous intensity of the pulsed Gaussian beams at z = l1 as a function of x and t′, confirming the expected delay. The dashed line indicates the expected intersection time t′ = τ′. (c,d) Time-averaged normalized intensity around the intersection point, for (c) two monochromatic Gaussian beams, and (d) the two periodic Gaussian pulses with pulse duration of 0.3 fs. Due to the different path lengths, the pulses in (d) do not interfere, as the angled pulse reaches the center 1.2 fs after the straight pulse.
Fig. 5
Fig. 5 Left: The refractive index of water n is approximated by n′, the Taylor series of first order around E0, to match the analytical approximation. Right: Ratio of the pulse duration Σt and the initial pulse duration σt as a function of propagation distance for a Gaussian pulse in water at E0 = 12 keV base photon energy and FWHMt = 10 as. Analytical values calculated as in [21].
Fig. 6
Fig. 6 Left axis: real and imaginary part of the refractive index of silicon. Right axis: Spectrum of the Gaussian pulse with base energy E0 = 12 kev and FWHMt = 5 as pulse duration.
Fig. 7
Fig. 7 (a,b,c) Normalized instantaneous intensity of the 5 attosecond beam for different propagation distances z ≈ 0.5 mm (a), z ≈ 2 mm (b), z ≈ 4 mm (c) in the silicon slab waveguide of 100 nm diameter. The waveguide’s edges are indicated by black lines. (d,e) Using the stretched time coordinate t′, we can track individual pulses as they propagate through the whole system, shown in locally averaged intensity for different time points t′ (d) and again averaged over the waveguide cross-section for all time-points t′ (e) to accurately determine the group velocity. The time-points of (d) are indicated in (e) by striped black lines. The separation of the wave packets into groups traveling with different velocities is clearly visible.
Fig. 8
Fig. 8 (a) Left axis: real and imaginary part of the refractive index of nickel around the K absorption edge. The data is taken from xraylib [22]. Right axis: normalized initial spectrum of the 300 as pulse with a base energy of E0 ≈ 8.33 keV. (b) Normalized spectrum and (c) locally centered and averaged intensity (right) of an 300 as pulse with a base energy of E0 ≈ 8.33 keV propagating in nickel at the propagation distances z = 0 and z = 42 mm. (d–f) Normalized pulse width Σt(z) and normalized maximal intensity A(z) of 300 as pulses with a base energy of E0 = 8.3328 keV in homogeneous nickel (d), a 100 nm Va slab waveguide (e) and cylindrical waveguide (f) in a nickel cladding. The pulse widths and maximum intensities are determined by fitting Voigt profiles to the locally averaged intensity profiles. The sudden rise in pulse intensity at the waveguide entrance is due to propagation of the initial plane wave from outside into the waveguide.

Tables (2)

Tables Icon

Table 1 Overview of the propagators implemented by PyPropagate. The algorithms are abbreviated as Crank-Nicolson method (CN), Alternating Direction Implicit method (ADI), Fast Fourier Transform based multislice method (FFT) and Discrete Hankel transform with resampling [20] based multislice method (DHT). The boundary conditions at the simulation box edges differ depending on the used propagation scheme. As an addition to the time complexity, an example runtime is shown, where the simulation box is divided into a grid with Nx = Ny = 1024 or Nr = 1024 and Nz = 1000 nodes. The example runtime has been determined on a 2.7 GHz dual-core consumer notebook from 2011.

Tables Icon

Table 2 Group velocity and pulse duration for the first three symmetrical guided modes in the 100 nm diameter silicon waveguide for a Gaussian pulse with FWHMt = 5 as. The modes denoted as (slab) and (cyl) correspond to slab and cylindrical geometry, respectively.

Equations (35)

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u z = 1 2 i k Δ u i k n 2 1 2 u .
2 ψ ˜ z 2 + ( k 2 n 2 k 2 ) ψ ˜ = ( 2 z 2 + β 2 ) ψ ˜ = 0 ,
( z + i β ) ( z i β ) ψ ˜ = ( z i β ) ( z + i β ) ψ ˜ = 0 .
ψ ˜ z = i β ψ ˜ ψ ˜ z = i β ψ ˜
β = k 2 n 2 k 2 k n k 2 2 k n .
ψ z = 1 2 i k n Δ ψ i k n ψ .
u z = 1 2 i k n Δ u i k ( n 1 ) u .
ψ Gauss = ψ 0 σ r 2 w r ( z ) 2 exp ( i k n z r 2 2 w r ( z ) 2 ) ,
u ω 0 ( t ) = ω 1 [ u ( ω + ω 0 ) e i k z 1 { ω ω 0 } ] ( t ) .
I inst c 0 A 2 = 2 c 0 ( u ω 0 ( t ) e i ( ω 0 t k 0 z ) ) 2 = 2 c 0 ( ( u ω 0 ( t ) ) cos ( ω 0 t k 0 z ) ( u ω 0 ( t ) ) sin ( ω 0 t k 0 z ) ) 2 .
I inst , avg c 0 | u ω 0 | 2 .
u ω 0 ( t ) = ω 1 [ u ( ω + ω 0 ) e i k z 1 { ω ω 0 } ] ( t + a z / c ) = ω 1 [ u ( ω + ω 0 ) e i k z / s 1 { ω ω 0 } ] ( t ) .
v g = Δ z Δ t = Δ z Δ t + a Δ z c = c a + c / v g ,
E ^ = t [ E ( ω ) ] = 1 2 π E e i ω t d t .
Δ E ^ + k 2 n 2 E ^ = 0 ,
E ^ = 0 and B ^ = 1 i ω × E ^ .
E ^ = i k × ψ .
E ^ = i k ( × ψ ) = 0 , Δ E ^ = i k Δ ( × ψ ) = i k × ( Δ ψ ) = n 2 k 2 E ^ .
Δ ψ + k 2 n 2 ψ = 0 ,
E ^ = i k ( ψ ) × e p and B ^ = 1 c ( 1 k 2 ( e p ψ ) + n 2 ψ e p ) .
ψ = u e i n k x ,
S = μ 0 1 E × B ,
E = ω 1 [ E ^ ( t ) ] = 2 π i ( E ^ ( ω i ) ) cos ( ω i ) + ( E ^ ( ω i ) ) sin ( ω i ) ,
S = 1 μ 0 π i ( E ^ ( ω i ) ) × ( B ^ ( ω i ) ) + ( E ^ ( ω i ) ) × ( B ^ ( ω i ) ) .
( e p ψ ) | n 2 k 2 ψ | ,
( E ^ ) × ( B ^ ) = ( i k ψ × e p ) × ( 1 c ( 1 k 2 ( e p ψ ) + n 2 ψ e p ) ) = 1 c k ( ψ + e p ) × ( 1 k 2 ( e p ψ ) + n 2 ψ e p ) 1 ω ( ψ × e p ) × ( n 2 ψ e p ) ( n 2 ) ω ( ψ ) ( ( ψ × e p ) × e p ) = ( n 2 ) ω ( ψ ) ( ψ )
( E ^ ) × ( B ^ ) ( n ) 2 ω ( ψ ) ( ψ ) ,
( ψ ) ( ψ ) ( ψ ) ( ψ ) = | ψ | 2 arg ( ψ ) .
S c 0 π i ( n ( ω i ) ) 2 | ψ ( ω i ) | 2 arg ( ψ ( ω i ) ) k i ,
S c 0 π i ( n ( ω i ) ) 3 | ψ ( ω i ) | 2 arg ( ψ ( ω i ) ) arg ( ψ ( ω i ) ) .
I = S c 0 π ( n ) 3 | ψ | 2
A : = ω 1 [ ψ ] ( t ) ,
A : = ω 1 [ ψ 1 { ω 0 } ] ( t ) = 1 2 π 0 ψ e i ω t d ω ,
A = ω 1 [ ψ ( t ) ] = 2 ( A ) .
u ω 0 ( t ) = ω 1 [ u ( ω + ω 0 ) e i k z 1 { ω ω 0 } ] ( t ) ,

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