Abstract

We discuss the self-imaging effect that occurs in a multimode planar x-ray waveguide (WG) with a nanometer vacuum gap, where an additional longitudinal periodicity has been imposed by a periodical structure (a micron scale step-like grating) on the reflecting sidewalls. Taking into account the general Montgomery conditions and the particular case of Talbot effect, we show that this additional longitudinal periodicity, if suitably designed, can filter out the asymmetric and the high order resonance modes, providing a coherent beam at the exit, even if the WG is illuminated by an incoherent source.

© 2011 Optical Society of America

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References

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  1. D. Marcuse, Theory of Dielectric Optical Waveguides(Academic, 1974).
  2. J. Miao, P. Charalambous, J. Kriz, and D. Sayre, Nature 400, 342 (1999).
    [CrossRef]
  3. K. Giewekemeyer, H. Neubauer, S. Kalbfleisch, S. P. Krüger, and T. Salditt, New J. Phys. 12, 035008 (2010).
    [CrossRef]
  4. L. De Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. Metzger, A. Guagliardi, A. Cedola, I. Burkeeva, and S. Lagomarsino, Phys. Rev. B 77, 081408 (2008).
    [CrossRef]
  5. W. D. Montgomery, J. Opt. Soc. Am. 57, 772 (1967).
    [CrossRef]
  6. H. F. Talbot, Philosophical Mag. 9, 401 (1836).
  7. J. Jahns and A. W. Lohmann, Appl. Opt. 48, 3438 (2009).
    [CrossRef] [PubMed]
  8. S. F. Helfert, B. Huneke, and J. Jahns, J. Eur. Opt. Soc. Rapid Pub. 4, 09031 (2009).
    [CrossRef]
  9. O. Bryngdahl, J. Opt. Soc. Am. 63, 41 (1973).
  10. R. Ulrich, Nouv. Rev. Opt. 6, 253 (1975).
    [CrossRef]
  11. I. Bukreeva, D. Pelliccia, A. Cedola, F. Scarinci, M. Ilie, C. Giannini, L. De Caro, and S. Lagomarsino, J. Synchrotron Radiat. 17, 61 (2010).
    [CrossRef]
  12. I. Bukreeva, A. Popov, D. Pelliccia, A. Cedola, S. Dabagov, and S. Lagomarsino, Phys. Rev. Lett. 97, 184801 (2006).
    [CrossRef] [PubMed]
  13. S. V. Kukhlevsky, X-Ray Spectrom. 32, 223 (2003).
    [CrossRef]
  14. T. Salditt, S. P. Krüger, C. Fuhse, and C. Bahtz, Phys. Rev. Lett. 100, 184801 (2008).
    [CrossRef] [PubMed]
  15. L. B. Soldano and E. C. M. Pennings, J. Lightwave Technol. 13, 615 (1995).
    [CrossRef]
  16. T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, Opt. Express 13, 6296 (2005).
    [CrossRef] [PubMed]
  17. Y. V. Kopylov, A. V. Popov, and A. V. Vinogradov, Opt. Commun. 118, 619 (1995).
    [CrossRef]
  18. S. V. Kukhlevsky and G. Nyitray, Opt. Commun. 218, 213 (2003).
    [CrossRef]
  19. A. W. Lohmann, H. Knuppertz, and J. Jahns, J. Opt. Soc. Am. A 22, 1500 (2005).
    [CrossRef]
  20. S. V. Kukhlevsky and L. Kozma, IEEE J. Sel. Top. Quantum Electron. 1, 941 (1995).
    [CrossRef]
  21. S. V. Kukhlevsky and G. Nyitray, Phys. Lett. A 291, 459 (2001).
    [CrossRef]

2010 (2)

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

2009 (2)

J. Jahns and A. W. Lohmann, Appl. Opt. 48, 3438 (2009).
[CrossRef] [PubMed]

S. F. Helfert, B. Huneke, and J. Jahns, J. Eur. Opt. Soc. Rapid Pub. 4, 09031 (2009).
[CrossRef]

2008 (2)

L. De Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. Metzger, A. Guagliardi, A. Cedola, I. Burkeeva, and S. Lagomarsino, Phys. Rev. B 77, 081408 (2008).
[CrossRef]

T. Salditt, S. P. Krüger, C. Fuhse, and C. Bahtz, Phys. Rev. Lett. 100, 184801 (2008).
[CrossRef] [PubMed]

2006 (1)

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

2005 (2)

2003 (2)

S. V. Kukhlevsky and G. Nyitray, Opt. Commun. 218, 213 (2003).
[CrossRef]

S. V. Kukhlevsky, X-Ray Spectrom. 32, 223 (2003).
[CrossRef]

2001 (1)

S. V. Kukhlevsky and G. Nyitray, Phys. Lett. A 291, 459 (2001).
[CrossRef]

1999 (1)

J. Miao, P. Charalambous, J. Kriz, and D. Sayre, Nature 400, 342 (1999).
[CrossRef]

1995 (3)

L. B. Soldano and E. C. M. Pennings, J. Lightwave Technol. 13, 615 (1995).
[CrossRef]

Y. V. Kopylov, A. V. Popov, and A. V. Vinogradov, Opt. Commun. 118, 619 (1995).
[CrossRef]

S. V. Kukhlevsky and L. Kozma, IEEE J. Sel. Top. Quantum Electron. 1, 941 (1995).
[CrossRef]

1975 (1)

R. Ulrich, Nouv. Rev. Opt. 6, 253 (1975).
[CrossRef]

1974 (1)

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

1973 (1)

O. Bryngdahl, J. Opt. Soc. Am. 63, 41 (1973).

1967 (1)

1836 (1)

H. F. Talbot, Philosophical Mag. 9, 401 (1836).

Bahtz, C.

T. Salditt, S. P. Krüger, C. Fuhse, and C. Bahtz, Phys. Rev. Lett. 100, 184801 (2008).
[CrossRef] [PubMed]

Bryngdahl, O.

O. Bryngdahl, J. Opt. Soc. Am. 63, 41 (1973).

Bukreeva, I.

I. Bukreeva, D. Pelliccia, A. Cedola, F. Scarinci, M. Ilie, C. Giannini, L. De Caro, and S. Lagomarsino, J. Synchrotron Radiat. 17, 61 (2010).
[CrossRef]

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

Burkeeva, I.

L. De Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. Metzger, A. Guagliardi, A. Cedola, I. Burkeeva, and S. Lagomarsino, Phys. Rev. B 77, 081408 (2008).
[CrossRef]

Cedola, A.

I. Bukreeva, D. Pelliccia, A. Cedola, F. Scarinci, M. Ilie, C. Giannini, L. De Caro, and S. Lagomarsino, J. Synchrotron Radiat. 17, 61 (2010).
[CrossRef]

L. De Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. Metzger, A. Guagliardi, A. Cedola, I. Burkeeva, and S. Lagomarsino, Phys. Rev. B 77, 081408 (2008).
[CrossRef]

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

Charalambous, P.

J. Miao, P. Charalambous, J. Kriz, and D. Sayre, Nature 400, 342 (1999).
[CrossRef]

Cloetens, P.

Dabagov, S.

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

David, C.

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 (2010).
[CrossRef]

L. De Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. Metzger, A. Guagliardi, A. Cedola, I. Burkeeva, and S. Lagomarsino, Phys. Rev. B 77, 081408 (2008).
[CrossRef]

Diaz, A.

Fuhse, C.

T. Salditt, S. P. Krüger, C. Fuhse, and C. Bahtz, Phys. Rev. Lett. 100, 184801 (2008).
[CrossRef] [PubMed]

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 (2010).
[CrossRef]

L. De Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. Metzger, A. Guagliardi, A. Cedola, I. Burkeeva, and S. Lagomarsino, Phys. Rev. B 77, 081408 (2008).
[CrossRef]

Giewekemeyer, K.

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

Guagliardi, A.

L. De Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. Metzger, A. Guagliardi, A. Cedola, I. Burkeeva, and S. Lagomarsino, Phys. Rev. B 77, 081408 (2008).
[CrossRef]

Helfert, S. F.

S. F. Helfert, B. Huneke, and J. Jahns, J. Eur. Opt. Soc. Rapid Pub. 4, 09031 (2009).
[CrossRef]

Huneke, B.

S. F. Helfert, B. Huneke, and J. Jahns, J. Eur. Opt. Soc. Rapid Pub. 4, 09031 (2009).
[CrossRef]

Ilie, M.

I. Bukreeva, D. Pelliccia, A. Cedola, F. Scarinci, M. Ilie, C. Giannini, L. De Caro, and S. Lagomarsino, J. Synchrotron Radiat. 17, 61 (2010).
[CrossRef]

Jahns, J.

Kalbfleisch, S.

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

Knuppertz, H.

Kopylov, Y. V.

Y. V. Kopylov, A. V. Popov, and A. V. Vinogradov, Opt. Commun. 118, 619 (1995).
[CrossRef]

Kozma, L.

S. V. Kukhlevsky and L. Kozma, IEEE J. Sel. Top. Quantum Electron. 1, 941 (1995).
[CrossRef]

Kriz, J.

J. Miao, P. Charalambous, J. Kriz, and D. Sayre, Nature 400, 342 (1999).
[CrossRef]

Krüger, S. P.

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

T. Salditt, S. P. Krüger, C. Fuhse, and C. Bahtz, Phys. Rev. Lett. 100, 184801 (2008).
[CrossRef] [PubMed]

Kukhlevsky, S. V.

S. V. Kukhlevsky, X-Ray Spectrom. 32, 223 (2003).
[CrossRef]

S. V. Kukhlevsky and G. Nyitray, Opt. Commun. 218, 213 (2003).
[CrossRef]

S. V. Kukhlevsky and G. Nyitray, Phys. Lett. A 291, 459 (2001).
[CrossRef]

S. V. Kukhlevsky and L. Kozma, IEEE J. Sel. Top. Quantum Electron. 1, 941 (1995).
[CrossRef]

Lagomarsino, S.

I. Bukreeva, D. Pelliccia, A. Cedola, F. Scarinci, M. Ilie, C. Giannini, L. De Caro, and S. Lagomarsino, J. Synchrotron Radiat. 17, 61 (2010).
[CrossRef]

L. De Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. Metzger, A. Guagliardi, A. Cedola, I. Burkeeva, and S. Lagomarsino, Phys. Rev. B 77, 081408 (2008).
[CrossRef]

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

Lohmann, A. W.

Marcuse, D.

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

Metzger, T. H.

L. De Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. Metzger, A. Guagliardi, A. Cedola, I. Burkeeva, and S. Lagomarsino, Phys. Rev. B 77, 081408 (2008).
[CrossRef]

Miao, J.

J. Miao, P. Charalambous, J. Kriz, and D. Sayre, Nature 400, 342 (1999).
[CrossRef]

Mocuta, C.

L. De Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. Metzger, A. Guagliardi, A. Cedola, I. Burkeeva, and S. Lagomarsino, Phys. Rev. B 77, 081408 (2008).
[CrossRef]

Montgomery, W. D.

Neubauer, H.

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

Nyitray, G.

S. V. Kukhlevsky and G. Nyitray, Opt. Commun. 218, 213 (2003).
[CrossRef]

S. V. Kukhlevsky and G. Nyitray, Phys. Lett. A 291, 459 (2001).
[CrossRef]

Pelliccia, D.

I. Bukreeva, D. Pelliccia, A. Cedola, F. Scarinci, M. Ilie, C. Giannini, L. De Caro, and S. Lagomarsino, J. Synchrotron Radiat. 17, 61 (2010).
[CrossRef]

L. De Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. Metzger, A. Guagliardi, A. Cedola, I. Burkeeva, and S. Lagomarsino, Phys. Rev. B 77, 081408 (2008).
[CrossRef]

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

Pennings, E. C. M.

L. B. Soldano and E. C. M. Pennings, J. Lightwave Technol. 13, 615 (1995).
[CrossRef]

Pfeiffer, F.

Popov, A.

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

Popov, A. V.

Y. V. Kopylov, A. V. Popov, and A. V. Vinogradov, Opt. Commun. 118, 619 (1995).
[CrossRef]

Salditt, T.

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

T. Salditt, S. P. Krüger, C. Fuhse, and C. Bahtz, Phys. Rev. Lett. 100, 184801 (2008).
[CrossRef] [PubMed]

Sayre, D.

J. Miao, P. Charalambous, J. Kriz, and D. Sayre, Nature 400, 342 (1999).
[CrossRef]

Scarinci, F.

I. Bukreeva, D. Pelliccia, A. Cedola, F. Scarinci, M. Ilie, C. Giannini, L. De Caro, and S. Lagomarsino, J. Synchrotron Radiat. 17, 61 (2010).
[CrossRef]

Soldano, L. B.

L. B. Soldano and E. C. M. Pennings, J. Lightwave Technol. 13, 615 (1995).
[CrossRef]

Stampanoni, M.

Talbot, H. F.

H. F. Talbot, Philosophical Mag. 9, 401 (1836).

Ulrich, R.

R. Ulrich, Nouv. Rev. Opt. 6, 253 (1975).
[CrossRef]

Vinogradov, A. V.

Y. V. Kopylov, A. V. Popov, and A. V. Vinogradov, Opt. Commun. 118, 619 (1995).
[CrossRef]

Weitkamp, T.

Ziegler, E.

Appl. Opt. (1)

IEEE J. Sel. Top. Quantum Electron. (1)

S. V. Kukhlevsky and L. Kozma, IEEE J. Sel. Top. Quantum Electron. 1, 941 (1995).
[CrossRef]

J. Eur. Opt. Soc. Rapid Pub. (1)

S. F. Helfert, B. Huneke, and J. Jahns, J. Eur. Opt. Soc. Rapid Pub. 4, 09031 (2009).
[CrossRef]

J. Lightwave Technol. (1)

L. B. Soldano and E. C. M. Pennings, J. Lightwave Technol. 13, 615 (1995).
[CrossRef]

J. Opt. Soc. Am. (2)

O. Bryngdahl, J. Opt. Soc. Am. 63, 41 (1973).

W. D. Montgomery, J. Opt. Soc. Am. 57, 772 (1967).
[CrossRef]

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

J. Synchrotron Radiat. (1)

I. Bukreeva, D. Pelliccia, A. Cedola, F. Scarinci, M. Ilie, C. Giannini, L. De Caro, and S. Lagomarsino, J. Synchrotron Radiat. 17, 61 (2010).
[CrossRef]

Nature (1)

J. Miao, P. Charalambous, J. Kriz, and D. Sayre, Nature 400, 342 (1999).
[CrossRef]

New J. Phys. (1)

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

Nouv. Rev. Opt. (1)

R. Ulrich, Nouv. Rev. Opt. 6, 253 (1975).
[CrossRef]

Opt. Commun. (2)

Y. V. Kopylov, A. V. Popov, and A. V. Vinogradov, Opt. Commun. 118, 619 (1995).
[CrossRef]

S. V. Kukhlevsky and G. Nyitray, Opt. Commun. 218, 213 (2003).
[CrossRef]

Opt. Express (1)

Philosophical Mag. (1)

H. F. Talbot, Philosophical Mag. 9, 401 (1836).

Phys. Lett. A (1)

S. V. Kukhlevsky and G. Nyitray, Phys. Lett. A 291, 459 (2001).
[CrossRef]

Phys. Rev. B (1)

L. De Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. Metzger, A. Guagliardi, A. Cedola, I. Burkeeva, and S. Lagomarsino, Phys. Rev. B 77, 081408 (2008).
[CrossRef]

Phys. Rev. Lett. (2)

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

T. Salditt, S. P. Krüger, C. Fuhse, and C. Bahtz, Phys. Rev. Lett. 100, 184801 (2008).
[CrossRef] [PubMed]

X-Ray Spectrom. (1)

S. V. Kukhlevsky, X-Ray Spectrom. 32, 223 (2003).
[CrossRef]

Other (1)

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

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Figures (3)

Fig. 1
Fig. 1

Step-index multimode WG of vacuum guiding layer width d and cladding refractive index n = 1 δ + i γ .

Fig. 2
Fig. 2

(a) Simulated x-ray intensity distribution (a) behind a phase grating with a period of D = 2 d = 200 nm , λ = 0.154 nm , 0.5 duty cycle, and Π phase shift and (b) in FCWG with a vacuum gap of d = D / 2 . (c) Lateral intensity distribution at odd (dashed curve) and at the even (solid curve) fractional Talbot distance.

Fig. 3
Fig. 3

Intensity distribution in the guiding layer of the WG for coherent source [(a) plane WG and (b) structured WG] and for incoherent source [asymmetric excitation; (c) plane WG and (d) structured WG]. PWE numerical solution.

Equations (5)

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

U ( x , 0 ) = m = 1 m max a m cos ( k x x ) exp ( i γ m z ) ,
k z m = 2 π ( 1 λ ) 2 ( m p x ) 2 2 π ( 1 λ m 2 z T ) , m = 0 , ± 1 , ± 2 , ,
u z ( x ) = u z + p z ( x ) u 0 ( x ) = m A m exp ( i k x m x ) .
k x l = 2 π ( 1 λ ) 2 ( l p z ) 2 l = ± 1 , ± 2 , .
( k x m ) 2 + ( k z 1 ) 2 = ( 2 π λ ) 2 .

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