Abstract

We present an analytical approach to describe field propagation along a planar x-ray waveguide (WG) in the presence of absorption losses. The method utilizes the complete expression for the complex index of refraction in solving the Helmholtz equation describing the guided modes. In this way, the propagation modes for the WG are no longer imposed to be standing waves and the energy flow from the core to the cladding, a consequence of the absorption in the cladding, can be calculated. In addition, the method accurately describes the field coupling between a plane wave and the WG, reproducing the self-imaging phenomenon. The case of partially coherent illumination has also been calculated for a realistic laboratory x-ray source.

© 2013 Optical Society of America

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References

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  1. K. A. Nugent, Adv. Phys. 59, 1 (2010).
    [CrossRef]
  2. K. Giewekemeyer, H. Neubauer, S. Kalbfleisch, S. P. Kruger, and T. Salditt, New J. Phys. 12, 035008 (2010).
    [CrossRef]
  3. K. Giewekemeyer, S. P. Kruger, S. Kalbfleisch, M. Bartels, C. Beta, and T. Salditt, Phys. Rev. A 83, 023804 (2011).
    [CrossRef]
  4. D. Pelliccia, A. Sorrentino, I. Bukreeva, A. Cedola, F. Scarinci, M. Ilie, A. M. Gerardino, M. Fratini, and S. Lagomarsino, Opt. Express 18, 15998 (2010).
    [CrossRef]
  5. M. Osterhoff and T. Salditt, New J. Phys. 13, 103026 (2011).
    [CrossRef]
  6. J. H. H. Bongaerts, C. David, M. Drakopoulos, M. J. Zwanenburg, G. H. Wegdam, T. Lackner, H. Keymeulen, and J. F. van der Veen, J. Synchrotron Radiat. 9, 383 (2002).
    [CrossRef]
  7. I. Bukreeva, D. Pelliccia, A. Cedola, F. Scarinci, M. Ilie, C. Giannini, L. De Caro, and S. Lagomarsino, J. Synchrotron Radiat. 17, 61 (2009).
    [CrossRef]
  8. D. Pelliccia, I. Bukreeva, M. Ilie, W. Jark, A. Cedola, F. Scarinci, and S. Lagomarsino, Spectrochim. Acta Part B 62, 615 (2007).
    [CrossRef]
  9. I. Bukreeva, A. Popov, D. Pelliccia, A. Cedola, S. B. Dabagov, and S. Lagomarsino, Phys. Rev. Lett. 97, 184801 (2006).
    [CrossRef]
  10. C. Fuhse and T. Salditt, Opt. Commun. 265, 140 (2006).
    [CrossRef]
  11. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, 1974).
  12. J. A. Nelder and R. Mead, Comput. J. 7, 308 (1965).
    [CrossRef]
  13. O. Bryngdahl, J. Opt. Soc. Am. 63, 416 (1973).
    [CrossRef]
  14. I. Bukreeva, A. Cedola, A. Sorrentino, D. Pelliccia, V. Asadchikov, and S. Lagomarsino, Opt. Lett. 36, 2602 (2011).
    [CrossRef]
  15. M. Deutsch, G. Hölzer, J. Härtwig, J. Wolf, M. Fritsch, and E. Förster, Phys. Rev. A 51, 283 (1995).
    [CrossRef]

2011 (3)

M. Osterhoff and T. Salditt, New J. Phys. 13, 103026 (2011).
[CrossRef]

K. Giewekemeyer, S. P. Kruger, S. Kalbfleisch, M. Bartels, C. Beta, and T. Salditt, Phys. Rev. A 83, 023804 (2011).
[CrossRef]

I. Bukreeva, A. Cedola, A. Sorrentino, D. Pelliccia, V. Asadchikov, and S. Lagomarsino, Opt. Lett. 36, 2602 (2011).
[CrossRef]

2010 (3)

D. Pelliccia, A. Sorrentino, I. Bukreeva, A. Cedola, F. Scarinci, M. Ilie, A. M. Gerardino, M. Fratini, and S. Lagomarsino, Opt. Express 18, 15998 (2010).
[CrossRef]

K. A. Nugent, Adv. Phys. 59, 1 (2010).
[CrossRef]

K. Giewekemeyer, H. Neubauer, S. Kalbfleisch, S. P. Kruger, and T. Salditt, New J. Phys. 12, 035008 (2010).
[CrossRef]

2009 (1)

I. Bukreeva, D. Pelliccia, A. Cedola, F. Scarinci, M. Ilie, C. Giannini, L. De Caro, and S. Lagomarsino, J. Synchrotron Radiat. 17, 61 (2009).
[CrossRef]

2007 (1)

D. Pelliccia, I. Bukreeva, M. Ilie, W. Jark, A. Cedola, F. Scarinci, and S. Lagomarsino, Spectrochim. Acta Part B 62, 615 (2007).
[CrossRef]

2006 (2)

I. Bukreeva, A. Popov, D. Pelliccia, A. Cedola, S. B. Dabagov, and S. Lagomarsino, Phys. Rev. Lett. 97, 184801 (2006).
[CrossRef]

C. Fuhse and T. Salditt, Opt. Commun. 265, 140 (2006).
[CrossRef]

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, J. Synchrotron Radiat. 9, 383 (2002).
[CrossRef]

1995 (1)

M. Deutsch, G. Hölzer, J. Härtwig, J. Wolf, M. Fritsch, and E. Förster, Phys. Rev. A 51, 283 (1995).
[CrossRef]

1973 (1)

1965 (1)

J. A. Nelder and R. Mead, Comput. J. 7, 308 (1965).
[CrossRef]

Asadchikov, V.

Bartels, M.

K. Giewekemeyer, S. P. Kruger, S. Kalbfleisch, M. Bartels, C. Beta, and T. Salditt, Phys. Rev. A 83, 023804 (2011).
[CrossRef]

Beta, C.

K. Giewekemeyer, S. P. Kruger, S. Kalbfleisch, M. Bartels, C. Beta, and T. Salditt, Phys. Rev. A 83, 023804 (2011).
[CrossRef]

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, J. Synchrotron Radiat. 9, 383 (2002).
[CrossRef]

Bryngdahl, O.

Bukreeva, I.

I. Bukreeva, A. Cedola, A. Sorrentino, D. Pelliccia, V. Asadchikov, and S. Lagomarsino, Opt. Lett. 36, 2602 (2011).
[CrossRef]

D. Pelliccia, A. Sorrentino, I. Bukreeva, A. Cedola, F. Scarinci, M. Ilie, A. M. Gerardino, M. Fratini, and S. Lagomarsino, Opt. Express 18, 15998 (2010).
[CrossRef]

I. Bukreeva, D. Pelliccia, A. Cedola, F. Scarinci, M. Ilie, C. Giannini, L. De Caro, and S. Lagomarsino, J. Synchrotron Radiat. 17, 61 (2009).
[CrossRef]

D. Pelliccia, I. Bukreeva, M. Ilie, W. Jark, A. Cedola, F. Scarinci, and S. Lagomarsino, Spectrochim. Acta Part B 62, 615 (2007).
[CrossRef]

I. Bukreeva, A. Popov, D. Pelliccia, A. Cedola, S. B. Dabagov, and S. Lagomarsino, Phys. Rev. Lett. 97, 184801 (2006).
[CrossRef]

Cedola, A.

I. Bukreeva, A. Cedola, A. Sorrentino, D. Pelliccia, V. Asadchikov, and S. Lagomarsino, Opt. Lett. 36, 2602 (2011).
[CrossRef]

D. Pelliccia, A. Sorrentino, I. Bukreeva, A. Cedola, F. Scarinci, M. Ilie, A. M. Gerardino, M. Fratini, and S. Lagomarsino, Opt. Express 18, 15998 (2010).
[CrossRef]

I. Bukreeva, D. Pelliccia, A. Cedola, F. Scarinci, M. Ilie, C. Giannini, L. De Caro, and S. Lagomarsino, J. Synchrotron Radiat. 17, 61 (2009).
[CrossRef]

D. Pelliccia, I. Bukreeva, M. Ilie, W. Jark, A. Cedola, F. Scarinci, and S. Lagomarsino, Spectrochim. Acta Part B 62, 615 (2007).
[CrossRef]

I. Bukreeva, A. Popov, D. Pelliccia, A. Cedola, S. B. Dabagov, and S. Lagomarsino, Phys. Rev. Lett. 97, 184801 (2006).
[CrossRef]

Dabagov, S. B.

I. Bukreeva, A. Popov, D. Pelliccia, A. Cedola, S. B. Dabagov, and S. Lagomarsino, Phys. Rev. Lett. 97, 184801 (2006).
[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, J. Synchrotron Radiat. 9, 383 (2002).
[CrossRef]

De Caro, L.

I. Bukreeva, D. Pelliccia, A. Cedola, F. Scarinci, M. Ilie, C. Giannini, L. De Caro, and S. Lagomarsino, J. Synchrotron Radiat. 17, 61 (2009).
[CrossRef]

Deutsch, M.

M. Deutsch, G. Hölzer, J. Härtwig, J. Wolf, M. Fritsch, and E. Förster, Phys. Rev. A 51, 283 (1995).
[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, J. Synchrotron Radiat. 9, 383 (2002).
[CrossRef]

Förster, E.

M. Deutsch, G. Hölzer, J. Härtwig, J. Wolf, M. Fritsch, and E. Förster, Phys. Rev. A 51, 283 (1995).
[CrossRef]

Fratini, M.

Fritsch, M.

M. Deutsch, G. Hölzer, J. Härtwig, J. Wolf, M. Fritsch, and E. Förster, Phys. Rev. A 51, 283 (1995).
[CrossRef]

Fuhse, C.

C. Fuhse and T. Salditt, Opt. Commun. 265, 140 (2006).
[CrossRef]

Gerardino, A. M.

Giannini, C.

I. Bukreeva, D. Pelliccia, A. Cedola, F. Scarinci, M. Ilie, C. Giannini, L. De Caro, and S. Lagomarsino, J. Synchrotron Radiat. 17, 61 (2009).
[CrossRef]

Giewekemeyer, K.

K. Giewekemeyer, S. P. Kruger, S. Kalbfleisch, M. Bartels, C. Beta, and T. Salditt, Phys. Rev. A 83, 023804 (2011).
[CrossRef]

K. Giewekemeyer, H. Neubauer, S. Kalbfleisch, S. P. Kruger, and T. Salditt, New J. Phys. 12, 035008 (2010).
[CrossRef]

Härtwig, J.

M. Deutsch, G. Hölzer, J. Härtwig, J. Wolf, M. Fritsch, and E. Förster, Phys. Rev. A 51, 283 (1995).
[CrossRef]

Hölzer, G.

M. Deutsch, G. Hölzer, J. Härtwig, J. Wolf, M. Fritsch, and E. Förster, Phys. Rev. A 51, 283 (1995).
[CrossRef]

Ilie, M.

D. Pelliccia, A. Sorrentino, I. Bukreeva, A. Cedola, F. Scarinci, M. Ilie, A. M. Gerardino, M. Fratini, and S. Lagomarsino, Opt. Express 18, 15998 (2010).
[CrossRef]

I. Bukreeva, D. Pelliccia, A. Cedola, F. Scarinci, M. Ilie, C. Giannini, L. De Caro, and S. Lagomarsino, J. Synchrotron Radiat. 17, 61 (2009).
[CrossRef]

D. Pelliccia, I. Bukreeva, M. Ilie, W. Jark, A. Cedola, F. Scarinci, and S. Lagomarsino, Spectrochim. Acta Part B 62, 615 (2007).
[CrossRef]

Jark, W.

D. Pelliccia, I. Bukreeva, M. Ilie, W. Jark, A. Cedola, F. Scarinci, and S. Lagomarsino, Spectrochim. Acta Part B 62, 615 (2007).
[CrossRef]

Kalbfleisch, S.

K. Giewekemeyer, S. P. Kruger, S. Kalbfleisch, M. Bartels, C. Beta, and T. Salditt, Phys. Rev. A 83, 023804 (2011).
[CrossRef]

K. Giewekemeyer, H. Neubauer, S. Kalbfleisch, S. P. Kruger, and T. Salditt, New J. Phys. 12, 035008 (2010).
[CrossRef]

Keymeulen, 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, J. Synchrotron Radiat. 9, 383 (2002).
[CrossRef]

Kruger, S. P.

K. Giewekemeyer, S. P. Kruger, S. Kalbfleisch, M. Bartels, C. Beta, and T. Salditt, Phys. Rev. A 83, 023804 (2011).
[CrossRef]

K. Giewekemeyer, H. Neubauer, S. Kalbfleisch, S. P. Kruger, and T. Salditt, New J. Phys. 12, 035008 (2010).
[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, J. Synchrotron Radiat. 9, 383 (2002).
[CrossRef]

Lagomarsino, S.

I. Bukreeva, A. Cedola, A. Sorrentino, D. Pelliccia, V. Asadchikov, and S. Lagomarsino, Opt. Lett. 36, 2602 (2011).
[CrossRef]

D. Pelliccia, A. Sorrentino, I. Bukreeva, A. Cedola, F. Scarinci, M. Ilie, A. M. Gerardino, M. Fratini, and S. Lagomarsino, Opt. Express 18, 15998 (2010).
[CrossRef]

I. Bukreeva, D. Pelliccia, A. Cedola, F. Scarinci, M. Ilie, C. Giannini, L. De Caro, and S. Lagomarsino, J. Synchrotron Radiat. 17, 61 (2009).
[CrossRef]

D. Pelliccia, I. Bukreeva, M. Ilie, W. Jark, A. Cedola, F. Scarinci, and S. Lagomarsino, Spectrochim. Acta Part B 62, 615 (2007).
[CrossRef]

I. Bukreeva, A. Popov, D. Pelliccia, A. Cedola, S. B. Dabagov, and S. Lagomarsino, Phys. Rev. Lett. 97, 184801 (2006).
[CrossRef]

Marcuse, D.

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

Mead, R.

J. A. Nelder and R. Mead, Comput. J. 7, 308 (1965).
[CrossRef]

Nelder, J. A.

J. A. Nelder and R. Mead, Comput. J. 7, 308 (1965).
[CrossRef]

Neubauer, H.

K. Giewekemeyer, H. Neubauer, S. Kalbfleisch, S. P. Kruger, and T. Salditt, New J. Phys. 12, 035008 (2010).
[CrossRef]

Nugent, K. A.

K. A. Nugent, Adv. Phys. 59, 1 (2010).
[CrossRef]

Osterhoff, M.

M. Osterhoff and T. Salditt, New J. Phys. 13, 103026 (2011).
[CrossRef]

Pelliccia, D.

I. Bukreeva, A. Cedola, A. Sorrentino, D. Pelliccia, V. Asadchikov, and S. Lagomarsino, Opt. Lett. 36, 2602 (2011).
[CrossRef]

D. Pelliccia, A. Sorrentino, I. Bukreeva, A. Cedola, F. Scarinci, M. Ilie, A. M. Gerardino, M. Fratini, and S. Lagomarsino, Opt. Express 18, 15998 (2010).
[CrossRef]

I. Bukreeva, D. Pelliccia, A. Cedola, F. Scarinci, M. Ilie, C. Giannini, L. De Caro, and S. Lagomarsino, J. Synchrotron Radiat. 17, 61 (2009).
[CrossRef]

D. Pelliccia, I. Bukreeva, M. Ilie, W. Jark, A. Cedola, F. Scarinci, and S. Lagomarsino, Spectrochim. Acta Part B 62, 615 (2007).
[CrossRef]

I. Bukreeva, A. Popov, D. Pelliccia, A. Cedola, S. B. Dabagov, and S. Lagomarsino, Phys. Rev. Lett. 97, 184801 (2006).
[CrossRef]

Popov, A.

I. Bukreeva, A. Popov, D. Pelliccia, A. Cedola, S. B. Dabagov, and S. Lagomarsino, Phys. Rev. Lett. 97, 184801 (2006).
[CrossRef]

Salditt, T.

M. Osterhoff and T. Salditt, New J. Phys. 13, 103026 (2011).
[CrossRef]

K. Giewekemeyer, S. P. Kruger, S. Kalbfleisch, M. Bartels, C. Beta, and T. Salditt, Phys. Rev. A 83, 023804 (2011).
[CrossRef]

K. Giewekemeyer, H. Neubauer, S. Kalbfleisch, S. P. Kruger, and T. Salditt, New J. Phys. 12, 035008 (2010).
[CrossRef]

C. Fuhse and T. Salditt, Opt. Commun. 265, 140 (2006).
[CrossRef]

Scarinci, F.

D. Pelliccia, A. Sorrentino, I. Bukreeva, A. Cedola, F. Scarinci, M. Ilie, A. M. Gerardino, M. Fratini, and S. Lagomarsino, Opt. Express 18, 15998 (2010).
[CrossRef]

I. Bukreeva, D. Pelliccia, A. Cedola, F. Scarinci, M. Ilie, C. Giannini, L. De Caro, and S. Lagomarsino, J. Synchrotron Radiat. 17, 61 (2009).
[CrossRef]

D. Pelliccia, I. Bukreeva, M. Ilie, W. Jark, A. Cedola, F. Scarinci, and S. Lagomarsino, Spectrochim. Acta Part B 62, 615 (2007).
[CrossRef]

Sorrentino, A.

van der Veen, J. F.

J. H. H. Bongaerts, C. David, M. Drakopoulos, M. J. Zwanenburg, G. H. Wegdam, T. Lackner, H. Keymeulen, and J. F. van der Veen, J. Synchrotron Radiat. 9, 383 (2002).
[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, J. Synchrotron Radiat. 9, 383 (2002).
[CrossRef]

Wolf, J.

M. Deutsch, G. Hölzer, J. Härtwig, J. Wolf, M. Fritsch, and E. Förster, Phys. Rev. A 51, 283 (1995).
[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, J. Synchrotron Radiat. 9, 383 (2002).
[CrossRef]

Adv. Phys. (1)

K. A. Nugent, Adv. Phys. 59, 1 (2010).
[CrossRef]

Comput. J. (1)

J. A. Nelder and R. Mead, Comput. J. 7, 308 (1965).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Synchrotron Radiat. (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, J. Synchrotron Radiat. 9, 383 (2002).
[CrossRef]

I. Bukreeva, D. Pelliccia, A. Cedola, F. Scarinci, M. Ilie, C. Giannini, L. De Caro, and S. Lagomarsino, J. Synchrotron Radiat. 17, 61 (2009).
[CrossRef]

New J. Phys. (2)

M. Osterhoff and T. Salditt, New J. Phys. 13, 103026 (2011).
[CrossRef]

K. Giewekemeyer, H. Neubauer, S. Kalbfleisch, S. P. Kruger, and T. Salditt, New J. Phys. 12, 035008 (2010).
[CrossRef]

Opt. Commun. (1)

C. Fuhse and T. Salditt, Opt. Commun. 265, 140 (2006).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. A (2)

K. Giewekemeyer, S. P. Kruger, S. Kalbfleisch, M. Bartels, C. Beta, and T. Salditt, Phys. Rev. A 83, 023804 (2011).
[CrossRef]

M. Deutsch, G. Hölzer, J. Härtwig, J. Wolf, M. Fritsch, and E. Förster, Phys. Rev. A 51, 283 (1995).
[CrossRef]

Phys. Rev. Lett. (1)

I. Bukreeva, A. Popov, D. Pelliccia, A. Cedola, S. B. Dabagov, and S. Lagomarsino, Phys. Rev. Lett. 97, 184801 (2006).
[CrossRef]

Spectrochim. Acta Part B (1)

D. Pelliccia, I. Bukreeva, M. Ilie, W. Jark, A. Cedola, F. Scarinci, and S. Lagomarsino, Spectrochim. Acta Part B 62, 615 (2007).
[CrossRef]

Other (1)

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

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Figures (2)

Fig. 1.
Fig. 1.

(a) Calculated intensity for the second (dark blue) and fifth (red) order in the present case, compared with the approximated method of [8] (green and light blue, respectively). (b) Energy flow associated with the second-order transverse field.

Fig. 2.
Fig. 2.

Calculated intensity distribution (arb. units) at the beginning of a WG for different incident fields. (a) Plane wave at normal incidence. (b) Plane wave with incidence angle 0.8 mrad with respect to the normal. (c) Plane wave with incidence angle 3 mrad with respect to the normal. (d) Polychromatic source at normal incidence. (e) Extended monochromatic source at normal incidence. (f) Extended polychromatic source at normal incidence.

Equations (6)

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

Φ(x,z)=mcmϕm(x,z),
cm=Φin(x,0)ϕm*(x,0)dx.
ϕm(x,z)=ψm(x)eikzz.
ψm(x)={Aexp(γm(xd/2))xd/2Bexp(ikxmx)+Cexp(ikxmx)d/2<x<d/2Dexp(γm(x+d/2))xd/2
det[1eikxmd2eikxmd200eikxmd2eikxmd21γmikxmeikxmd2ikxmeikxmd200ikxmeikxmd2ikxmeikxmd2γm]=0.
Sm(x)=I(x)φm(x)x

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