Abstract

Waveguides for short-wavelength x-rays have been successfully employed for microbeam and nanobeam production and microscopy experiments. The coherence of hard x-ray sources is generally poor, and therefore the spatial coherence filtering characteristics of waveguides have been attractive for high-resolution microscopy experiments. To quantify the spatial coherence filtering properties of a waveguide, we here report a theoretical study of the propagation of a partially coherent beam in a waveguide in the paraxial approximation. By propagating the cross-spectral density function associated with the partially coherent field, we quantify in detail the evolution of the spatial coherence as the beam proceeds along the waveguide. The propagation is efficiently accomplished using the communication-modes formalism. The generality of the approach makes it suitable to study more complex phenomena such as the second-order Talbot self-imaging effect and coherence revivals in waveguides. Numerical results are shown for waveguides illuminated by partially coherent hard x-rays.

© 2014 Optical Society of America

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    [CrossRef]

2013 (2)

2012 (1)

D. Pelliccia and D. M. Paganin, “Coherence vortices in vortex-free partially coherent x-ray fields,” Phys. Rev. A 86, 015802 (2012).
[CrossRef]

2011 (4)

I. Bukreeva, A. Cedola, A. Sorrentino, D. Pelliccia, V. Asadchikov, and S. Lagomarsino, “Resonance modes filtering in structured x-ray waveguides,” Opt. Lett. 36, 2602–2604 (2011).
[CrossRef]

M. Osterhoff and T. Salditt, “Coherence filtering of x-ray waveguides: analytical and numerical approach,” New J. Phys. 13, 103026 (2011).
[CrossRef]

V. Torres-Company, J. Lancis, H. Lajunen, and A. T. Friberg, “Coherence revivals in two-photon frequency combs,” Phys. Rev. A 84, 033830 (2011).
[CrossRef]

E. Poem and Y. Silberberg, “Photon correlations in multimode waveguides,” Phys. Rev. A 84, 041805 (2011).
[CrossRef]

2010 (4)

X. B. Song, J. Xiong, X. Zhang, and K. Wang, “Second-order Talbot self-imaging with pseudothermal light,” Phys. Rev. A 82, 033823 (2010).
[CrossRef]

K. H. Luo, X. H. Chen, Q. Liu, and L. A. Wu, “Nonlocal Talbot self-imaging with incoherent light,” Phys. Rev. A 82, 033803 (2010).
[CrossRef]

K. Giewekemeyer, H. Neubauer, S. Kalbfleisch, S. P. Krüger, and T. Salditt, “Holographic and diffractive x-ray imaging using waveguides as quasi-point sources,” New J. Phys. 12, 035008 (2010).
[CrossRef]

P. Martinsson, H. Lajunen, P. Ma, and A. T. Friberg, “Communication modes in vector diffraction,” Optik 121, 2087–2093 (2010).
[CrossRef]

2009 (2)

S. F. Helfert, B. Huneke, and J. Jahns, “Self-imaging effect in multimode waveguides with longitudinal periodicity,” J. Eur. Opt. Soc. Rapid Pub. 4, 09031 (2009).
[CrossRef]

K. H. Luo, J. M. Wen, X. H. Chen, Q. Liu, M. Xiao, and L. A. Wu, “Second-order Talbot effect with entangled photon pairs,” Phys. Rev. A 80, 043820 (2009).
[CrossRef]

2008 (2)

P. Martinsson, P. Ma, A. Burvall, and A. T. Friberg, “Communication modes in scalar diffraction,” Optik 119, 103–111 (2008).
[CrossRef]

L. De Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. Metzger, A. Guagliardi, A. Cedola, I. Burkeeva, and S. Lagomarsino, “In-line holography and coherent diffractive imaging with x-ray waveguides,” Phys. Rev. B 77, 081408R (2008).
[CrossRef]

2007 (4)

2006 (1)

C. Fuhse, C. Ollinger, and T. Salditt, “Waveguide-based off-axis holography with hard x-rays,” Phys. Rev. Lett. 97, 254801 (2006).
[CrossRef]

2004 (1)

2003 (2)

A. Thaning, P. Martinsson, M. Karelin, and A. T. Friberg, “Limits of diffractive optics by communication modes,” J. Opt. A 5, 153–158 (2003).
[CrossRef]

L. De Caro, C. Giannini, S. Di Fonzo, W. Yark, A. Cedola, and S. Lagomarsino, “Spatial coherence of x-ray planar waveguide exiting radiation,” Opt. Commun. 217, 31–45 (2003).
[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, “Propagation of a partially coherent focused X-ray beam within a planar X-ray waveguide,” J. Synchrotron Radiat. 9, 383–393 (2002).
[CrossRef]

2001 (2)

R. Pierri, A. Liseno, F. Soldovieri, and R. Solimene, “In-depth resolution for a strip source in the Fresnel zone,” J. Opt. Soc. Am. A 18, 352–359 (2001).
[CrossRef]

M. Berry, I. Marzoli, and W. Schleich, “Quantum carpets, carpets of light,” Phys. World 14, 39–44 (2001).

2000 (2)

1998 (1)

1996 (1)

S. Lagomarsino, W. Jark, S. Di Fonzo, A. Cedola, B. R. Müller, C. Riekel, and P. Engstrom, “Submicrometer x-ray beam production by a thin film waveguide,” J. Appl. Phys. 79, 4471–4473 (1996).
[CrossRef]

1995 (3)

Y. P. Feng, S. K. Sinha, E. E. Fullerton, G. Grübel, D. Abernathy, D. P. Siddons, and J. B. Hastings, “X-ray Fraunhofer diffraction patterns from a thin-film waveguide,” Appl. Phys. Lett. 67, 3647–3649 (1995).
[CrossRef]

P. Hlubina, “Coherence of light at the output of a fibre waveguide analysed in the space-frequency domain,” J. Mod. Opt. 42, 1407–1426 (1995).
[CrossRef]

P. Hlubina, “The mutual interference of modes of a few-mode fibre waveguide analysed in the frequency domain,” J. Mod. Opt. 42, 2385–2399 (1995).
[CrossRef]

1990 (1)

1987 (1)

E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature 326, 363–365 (1987).
[CrossRef]

1986 (1)

E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
[CrossRef]

1983 (1)

M. Bertero, C. de Mol, F. Gori, and L. Ronchi, “Number of degrees of freedom in inverse diffraction,” Opt. Acta 30, 1051–1065 (1983).
[CrossRef]

1982 (1)

1979 (3)

J. Jahns and A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1979).
[CrossRef]

R. Sudol and B. J. Thompson, “An explanation of the Lau effect based on coherence theory,” Opt. Commun. 31, 105–110 (1979).
[CrossRef]

1978 (1)

R. Ulrich and T. Kamiya, “Resolution of self-images in planar optical waveguides,” J. Opt. Soc. Am. A 68, 583–592 (1978).
[CrossRef]

1975 (1)

R. Ulrich, “Image formation by phase coincidences in optical waveguides,” Opt. Commun. 13, 259–264 (1975).
[CrossRef]

1948 (1)

E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. Phys. 437, 417–423 (1948).
[CrossRef]

Abernathy, D.

Y. P. Feng, S. K. Sinha, E. E. Fullerton, G. Grübel, D. Abernathy, D. P. Siddons, and J. B. Hastings, “X-ray Fraunhofer diffraction patterns from a thin-film waveguide,” Appl. Phys. Lett. 67, 3647–3649 (1995).
[CrossRef]

Agrawal, G. P.

Asadchikov, V.

Barrett, H. H.

H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley-Interscience, 2004).

Berry, M.

M. Berry, I. Marzoli, and W. Schleich, “Quantum carpets, carpets of light,” Phys. World 14, 39–44 (2001).

Bertero, M.

M. Bertero, C. de Mol, F. Gori, and L. Ronchi, “Number of degrees of freedom in inverse diffraction,” Opt. Acta 30, 1051–1065 (1983).
[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, “Propagation of a partially coherent focused X-ray beam within a planar X-ray waveguide,” J. Synchrotron Radiat. 9, 383–393 (2002).
[CrossRef]

Bukreeva, I.

I. Bukreeva, A. Cedola, A. Sorrentino, D. Pelliccia, V. Asadchikov, and S. Lagomarsino, “Resonance modes filtering in structured x-ray waveguides,” Opt. Lett. 36, 2602–2604 (2011).
[CrossRef]

D. Pelliccia, I. Bukreeva, M. Ilie, W. Jark, A. Cedola, F. Scarinci, and S. Lagomarsino, “Computer simulations and experimental results on air-gap X-ray waveguides,” Spectrochim. Acta B 62, 615–621 (2007).
[CrossRef]

Burkeeva, I.

L. De Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. Metzger, A. Guagliardi, A. Cedola, I. Burkeeva, and S. Lagomarsino, “In-line holography and coherent diffractive imaging with x-ray waveguides,” Phys. Rev. B 77, 081408R (2008).
[CrossRef]

Burvall, A.

Cedola, A.

I. Bukreeva, A. Cedola, A. Sorrentino, D. Pelliccia, V. Asadchikov, and S. Lagomarsino, “Resonance modes filtering in structured x-ray waveguides,” Opt. Lett. 36, 2602–2604 (2011).
[CrossRef]

L. De Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. Metzger, A. Guagliardi, A. Cedola, I. Burkeeva, and S. Lagomarsino, “In-line holography and coherent diffractive imaging with x-ray waveguides,” Phys. Rev. B 77, 081408R (2008).
[CrossRef]

D. Pelliccia, I. Bukreeva, M. Ilie, W. Jark, A. Cedola, F. Scarinci, and S. Lagomarsino, “Computer simulations and experimental results on air-gap X-ray waveguides,” Spectrochim. Acta B 62, 615–621 (2007).
[CrossRef]

L. De Caro, C. Giannini, S. Di Fonzo, W. Yark, A. Cedola, and S. Lagomarsino, “Spatial coherence of x-ray planar waveguide exiting radiation,” Opt. Commun. 217, 31–45 (2003).
[CrossRef]

S. Lagomarsino, W. Jark, S. Di Fonzo, A. Cedola, B. R. Müller, C. Riekel, and P. Engstrom, “Submicrometer x-ray beam production by a thin film waveguide,” J. Appl. Phys. 79, 4471–4473 (1996).
[CrossRef]

Chen, X. H.

K. H. Luo, X. H. Chen, Q. Liu, and L. A. Wu, “Nonlocal Talbot self-imaging with incoherent light,” Phys. Rev. A 82, 033803 (2010).
[CrossRef]

K. H. Luo, J. M. Wen, X. H. Chen, Q. Liu, M. Xiao, and L. A. Wu, “Second-order Talbot effect with entangled photon pairs,” Phys. Rev. A 80, 043820 (2009).
[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 Radiat. 9, 383–393 (2002).
[CrossRef]

De Caro, L.

L. De Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. Metzger, A. Guagliardi, A. Cedola, I. Burkeeva, and S. Lagomarsino, “In-line holography and coherent diffractive imaging with x-ray waveguides,” Phys. Rev. B 77, 081408R (2008).
[CrossRef]

L. De Caro, C. Giannini, S. Di Fonzo, W. Yark, A. Cedola, and S. Lagomarsino, “Spatial coherence of x-ray planar waveguide exiting radiation,” Opt. Commun. 217, 31–45 (2003).
[CrossRef]

de Mol, C.

M. Bertero, C. de Mol, F. Gori, and L. Ronchi, “Number of degrees of freedom in inverse diffraction,” Opt. Acta 30, 1051–1065 (1983).
[CrossRef]

Di Fonzo, S.

L. De Caro, C. Giannini, S. Di Fonzo, W. Yark, A. Cedola, and S. Lagomarsino, “Spatial coherence of x-ray planar waveguide exiting radiation,” Opt. Commun. 217, 31–45 (2003).
[CrossRef]

S. Lagomarsino, W. Jark, S. Di Fonzo, A. Cedola, B. R. Müller, C. Riekel, and P. Engstrom, “Submicrometer x-ray beam production by a thin film waveguide,” J. Appl. Phys. 79, 4471–4473 (1996).
[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 Radiat. 9, 383–393 (2002).
[CrossRef]

Engstrom, P.

S. Lagomarsino, W. Jark, S. Di Fonzo, A. Cedola, B. R. Müller, C. Riekel, and P. Engstrom, “Submicrometer x-ray beam production by a thin film waveguide,” J. Appl. Phys. 79, 4471–4473 (1996).
[CrossRef]

Feng, Y. P.

Y. P. Feng, S. K. Sinha, E. E. Fullerton, G. Grübel, D. Abernathy, D. P. Siddons, and J. B. Hastings, “X-ray Fraunhofer diffraction patterns from a thin-film waveguide,” Appl. Phys. Lett. 67, 3647–3649 (1995).
[CrossRef]

Friberg, A. T.

V. Torres-Company, J. Lancis, H. Lajunen, and A. T. Friberg, “Coherence revivals in two-photon frequency combs,” Phys. Rev. A 84, 033830 (2011).
[CrossRef]

P. Martinsson, H. Lajunen, P. Ma, and A. T. Friberg, “Communication modes in vector diffraction,” Optik 121, 2087–2093 (2010).
[CrossRef]

P. Martinsson, P. Ma, A. Burvall, and A. T. Friberg, “Communication modes in scalar diffraction,” Optik 119, 103–111 (2008).
[CrossRef]

A. Burvall, P. Martinsson, and A. T. Friberg, “Communication modes in large-aperture approximation,” Opt. Lett. 32, 611–613 (2007).
[CrossRef]

P. Martinsson, H. Lajunen, and A. T. Friberg, “Communication modes with partially coherent fields,” J. Opt. Soc. Am. A 24, 3336–3342 (2007).
[CrossRef]

A. Burvall, P. Martinsson, and A. T. Friberg, “Communication modes applied to axicons,” Opt. Express 12, 377–383 (2004).
[CrossRef]

A. Thaning, P. Martinsson, M. Karelin, and A. T. Friberg, “Limits of diffractive optics by communication modes,” J. Opt. A 5, 153–158 (2003).
[CrossRef]

Fuhse, C.

C. Fuhse, C. Ollinger, and T. Salditt, “Waveguide-based off-axis holography with hard x-rays,” Phys. Rev. Lett. 97, 254801 (2006).
[CrossRef]

Fullerton, E. E.

Y. P. Feng, S. K. Sinha, E. E. Fullerton, G. Grübel, D. Abernathy, D. P. Siddons, and J. B. Hastings, “X-ray Fraunhofer diffraction patterns from a thin-film waveguide,” Appl. Phys. Lett. 67, 3647–3649 (1995).
[CrossRef]

Gamliel, A.

Giannini, C.

L. De Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. Metzger, A. Guagliardi, A. Cedola, I. Burkeeva, and S. Lagomarsino, “In-line holography and coherent diffractive imaging with x-ray waveguides,” Phys. Rev. B 77, 081408R (2008).
[CrossRef]

L. De Caro, C. Giannini, S. Di Fonzo, W. Yark, A. Cedola, and S. Lagomarsino, “Spatial coherence of x-ray planar waveguide exiting radiation,” Opt. Commun. 217, 31–45 (2003).
[CrossRef]

Giewekemeyer, K.

K. Giewekemeyer, H. Neubauer, S. Kalbfleisch, S. P. Krüger, and T. Salditt, “Holographic and diffractive x-ray imaging using waveguides as quasi-point sources,” New J. Phys. 12, 035008 (2010).
[CrossRef]

Gori, F.

M. Bertero, C. de Mol, F. Gori, and L. Ronchi, “Number of degrees of freedom in inverse diffraction,” Opt. Acta 30, 1051–1065 (1983).
[CrossRef]

F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1979).
[CrossRef]

Grübel, G.

Y. P. Feng, S. K. Sinha, E. E. Fullerton, G. Grübel, D. Abernathy, D. P. Siddons, and J. B. Hastings, “X-ray Fraunhofer diffraction patterns from a thin-film waveguide,” Appl. Phys. Lett. 67, 3647–3649 (1995).
[CrossRef]

Guagliardi, A.

L. De Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. Metzger, A. Guagliardi, A. Cedola, I. Burkeeva, and S. Lagomarsino, “In-line holography and coherent diffractive imaging with x-ray waveguides,” Phys. Rev. B 77, 081408R (2008).
[CrossRef]

Hastings, J. B.

Y. P. Feng, S. K. Sinha, E. E. Fullerton, G. Grübel, D. Abernathy, D. P. Siddons, and J. B. Hastings, “X-ray Fraunhofer diffraction patterns from a thin-film waveguide,” Appl. Phys. Lett. 67, 3647–3649 (1995).
[CrossRef]

Helfert, S. F.

S. F. Helfert, B. Huneke, and J. Jahns, “Self-imaging effect in multimode waveguides with longitudinal periodicity,” J. Eur. Opt. Soc. Rapid Pub. 4, 09031 (2009).
[CrossRef]

Hlubina, P.

P. Hlubina, “Coherence of light at the output of a fibre waveguide analysed in the space-frequency domain,” J. Mod. Opt. 42, 1407–1426 (1995).
[CrossRef]

P. Hlubina, “The mutual interference of modes of a few-mode fibre waveguide analysed in the frequency domain,” J. Mod. Opt. 42, 2385–2399 (1995).
[CrossRef]

Huneke, B.

S. F. Helfert, B. Huneke, and J. Jahns, “Self-imaging effect in multimode waveguides with longitudinal periodicity,” J. Eur. Opt. Soc. Rapid Pub. 4, 09031 (2009).
[CrossRef]

Ilie, M.

D. Pelliccia, I. Bukreeva, M. Ilie, W. Jark, A. Cedola, F. Scarinci, and S. Lagomarsino, “Computer simulations and experimental results on air-gap X-ray waveguides,” Spectrochim. Acta B 62, 615–621 (2007).
[CrossRef]

Jahns, J.

S. F. Helfert, B. Huneke, and J. Jahns, “Self-imaging effect in multimode waveguides with longitudinal periodicity,” J. Eur. Opt. Soc. Rapid Pub. 4, 09031 (2009).
[CrossRef]

J. Jahns and A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

Jark, W.

D. Pelliccia, I. Bukreeva, M. Ilie, W. Jark, A. Cedola, F. Scarinci, and S. Lagomarsino, “Computer simulations and experimental results on air-gap X-ray waveguides,” Spectrochim. Acta B 62, 615–621 (2007).
[CrossRef]

S. Lagomarsino, W. Jark, S. Di Fonzo, A. Cedola, B. R. Müller, C. Riekel, and P. Engstrom, “Submicrometer x-ray beam production by a thin film waveguide,” J. Appl. Phys. 79, 4471–4473 (1996).
[CrossRef]

Kalbfleisch, S.

K. Giewekemeyer, H. Neubauer, S. Kalbfleisch, S. P. Krüger, and T. Salditt, “Holographic and diffractive x-ray imaging using waveguides as quasi-point sources,” New J. Phys. 12, 035008 (2010).
[CrossRef]

Kamiya, T.

R. Ulrich and T. Kamiya, “Resolution of self-images in planar optical waveguides,” J. Opt. Soc. Am. A 68, 583–592 (1978).
[CrossRef]

Karelin, M.

A. Thaning, P. Martinsson, M. Karelin, and A. T. Friberg, “Limits of diffractive optics by communication modes,” J. Opt. A 5, 153–158 (2003).
[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, “Propagation of a partially coherent focused X-ray beam within a planar X-ray waveguide,” J. Synchrotron Radiat. 9, 383–393 (2002).
[CrossRef]

Krüger, S. P.

K. Giewekemeyer, H. Neubauer, S. Kalbfleisch, S. P. Krüger, and T. Salditt, “Holographic and diffractive x-ray imaging using waveguides as quasi-point sources,” 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, “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.

I. Bukreeva, A. Cedola, A. Sorrentino, D. Pelliccia, V. Asadchikov, and S. Lagomarsino, “Resonance modes filtering in structured x-ray waveguides,” Opt. Lett. 36, 2602–2604 (2011).
[CrossRef]

L. De Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. Metzger, A. Guagliardi, A. Cedola, I. Burkeeva, and S. Lagomarsino, “In-line holography and coherent diffractive imaging with x-ray waveguides,” Phys. Rev. B 77, 081408R (2008).
[CrossRef]

D. Pelliccia, I. Bukreeva, M. Ilie, W. Jark, A. Cedola, F. Scarinci, and S. Lagomarsino, “Computer simulations and experimental results on air-gap X-ray waveguides,” Spectrochim. Acta B 62, 615–621 (2007).
[CrossRef]

L. De Caro, C. Giannini, S. Di Fonzo, W. Yark, A. Cedola, and S. Lagomarsino, “Spatial coherence of x-ray planar waveguide exiting radiation,” Opt. Commun. 217, 31–45 (2003).
[CrossRef]

S. Lagomarsino, W. Jark, S. Di Fonzo, A. Cedola, B. R. Müller, C. Riekel, and P. Engstrom, “Submicrometer x-ray beam production by a thin film waveguide,” J. Appl. Phys. 79, 4471–4473 (1996).
[CrossRef]

Lajunen, H.

V. Torres-Company, J. Lancis, H. Lajunen, and A. T. Friberg, “Coherence revivals in two-photon frequency combs,” Phys. Rev. A 84, 033830 (2011).
[CrossRef]

P. Martinsson, H. Lajunen, P. Ma, and A. T. Friberg, “Communication modes in vector diffraction,” Optik 121, 2087–2093 (2010).
[CrossRef]

P. Martinsson, H. Lajunen, and A. T. Friberg, “Communication modes with partially coherent fields,” J. Opt. Soc. Am. A 24, 3336–3342 (2007).
[CrossRef]

Lancis, J.

V. Torres-Company, J. Lancis, H. Lajunen, and A. T. Friberg, “Coherence revivals in two-photon frequency combs,” Phys. Rev. A 84, 033830 (2011).
[CrossRef]

Lau, E.

E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. Phys. 437, 417–423 (1948).
[CrossRef]

Liseno, A.

Liu, Q.

K. H. Luo, X. H. Chen, Q. Liu, and L. A. Wu, “Nonlocal Talbot self-imaging with incoherent light,” Phys. Rev. A 82, 033803 (2010).
[CrossRef]

K. H. Luo, J. M. Wen, X. H. Chen, Q. Liu, M. Xiao, and L. A. Wu, “Second-order Talbot effect with entangled photon pairs,” Phys. Rev. A 80, 043820 (2009).
[CrossRef]

Lohmann, A. W.

J. Jahns and A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

Luo, K. H.

K. H. Luo, X. H. Chen, Q. Liu, and L. A. Wu, “Nonlocal Talbot self-imaging with incoherent light,” Phys. Rev. A 82, 033803 (2010).
[CrossRef]

K. H. Luo, J. M. Wen, X. H. Chen, Q. Liu, M. Xiao, and L. A. Wu, “Second-order Talbot effect with entangled photon pairs,” Phys. Rev. A 80, 043820 (2009).
[CrossRef]

Ma, P.

P. Martinsson, H. Lajunen, P. Ma, and A. T. Friberg, “Communication modes in vector diffraction,” Optik 121, 2087–2093 (2010).
[CrossRef]

P. Martinsson, P. Ma, A. Burvall, and A. T. Friberg, “Communication modes in scalar diffraction,” Optik 119, 103–111 (2008).
[CrossRef]

Marcuse, D.

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

Martinsson, P.

P. Martinsson, H. Lajunen, P. Ma, and A. T. Friberg, “Communication modes in vector diffraction,” Optik 121, 2087–2093 (2010).
[CrossRef]

P. Martinsson, P. Ma, A. Burvall, and A. T. Friberg, “Communication modes in scalar diffraction,” Optik 119, 103–111 (2008).
[CrossRef]

A. Burvall, P. Martinsson, and A. T. Friberg, “Communication modes in large-aperture approximation,” Opt. Lett. 32, 611–613 (2007).
[CrossRef]

P. Martinsson, H. Lajunen, and A. T. Friberg, “Communication modes with partially coherent fields,” J. Opt. Soc. Am. A 24, 3336–3342 (2007).
[CrossRef]

A. Burvall, P. Martinsson, and A. T. Friberg, “Communication modes applied to axicons,” Opt. Express 12, 377–383 (2004).
[CrossRef]

A. Thaning, P. Martinsson, M. Karelin, and A. T. Friberg, “Limits of diffractive optics by communication modes,” J. Opt. A 5, 153–158 (2003).
[CrossRef]

Marzoli, I.

M. Berry, I. Marzoli, and W. Schleich, “Quantum carpets, carpets of light,” Phys. World 14, 39–44 (2001).

Metzger, T. H.

L. De Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. Metzger, A. Guagliardi, A. Cedola, I. Burkeeva, and S. Lagomarsino, “In-line holography and coherent diffractive imaging with x-ray waveguides,” Phys. Rev. B 77, 081408R (2008).
[CrossRef]

Miller, D. A. B.

Mocuta, C.

L. De Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. Metzger, A. Guagliardi, A. Cedola, I. Burkeeva, and S. Lagomarsino, “In-line holography and coherent diffractive imaging with x-ray waveguides,” Phys. Rev. B 77, 081408R (2008).
[CrossRef]

Müller, B. R.

S. Lagomarsino, W. Jark, S. Di Fonzo, A. Cedola, B. R. Müller, C. Riekel, and P. Engstrom, “Submicrometer x-ray beam production by a thin film waveguide,” J. Appl. Phys. 79, 4471–4473 (1996).
[CrossRef]

Myers, K. J.

H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley-Interscience, 2004).

Neubauer, H.

K. Giewekemeyer, H. Neubauer, S. Kalbfleisch, S. P. Krüger, and T. Salditt, “Holographic and diffractive x-ray imaging using waveguides as quasi-point sources,” New J. Phys. 12, 035008 (2010).
[CrossRef]

Ollinger, C.

C. Fuhse, C. Ollinger, and T. Salditt, “Waveguide-based off-axis holography with hard x-rays,” Phys. Rev. Lett. 97, 254801 (2006).
[CrossRef]

Osterhoff, M.

M. Osterhoff and T. Salditt, “Coherence filtering of x-ray waveguides: analytical and numerical approach,” New J. Phys. 13, 103026 (2011).
[CrossRef]

Paganin, D. M.

Pelliccia, D.

K. J. Tsanaktsidis, D. M. Paganin, and D. Pelliccia, “Analytical description of partially coherent propagation and absorption losses in x-ray planar waveguides,” Opt. Lett. 38, 1808–1810 (2013).
[CrossRef]

D. Pelliccia and D. M. Paganin, “Coherence vortices in vortex-free partially coherent x-ray fields,” Phys. Rev. A 86, 015802 (2012).
[CrossRef]

I. Bukreeva, A. Cedola, A. Sorrentino, D. Pelliccia, V. Asadchikov, and S. Lagomarsino, “Resonance modes filtering in structured x-ray waveguides,” Opt. Lett. 36, 2602–2604 (2011).
[CrossRef]

L. De Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. Metzger, A. Guagliardi, A. Cedola, I. Burkeeva, and S. Lagomarsino, “In-line holography and coherent diffractive imaging with x-ray waveguides,” Phys. Rev. B 77, 081408R (2008).
[CrossRef]

D. Pelliccia, I. Bukreeva, M. Ilie, W. Jark, A. Cedola, F. Scarinci, and S. Lagomarsino, “Computer simulations and experimental results on air-gap X-ray waveguides,” Spectrochim. Acta B 62, 615–621 (2007).
[CrossRef]

Pierri, R.

Piestun, R.

Poem, E.

E. Poem and Y. Silberberg, “Photon correlations in multimode waveguides,” Phys. Rev. A 84, 041805 (2011).
[CrossRef]

Riekel, C.

S. Lagomarsino, W. Jark, S. Di Fonzo, A. Cedola, B. R. Müller, C. Riekel, and P. Engstrom, “Submicrometer x-ray beam production by a thin film waveguide,” J. Appl. Phys. 79, 4471–4473 (1996).
[CrossRef]

Ronchi, L.

M. Bertero, C. de Mol, F. Gori, and L. Ronchi, “Number of degrees of freedom in inverse diffraction,” Opt. Acta 30, 1051–1065 (1983).
[CrossRef]

Salditt, T.

M. Osterhoff and T. Salditt, “Coherence filtering of x-ray waveguides: analytical and numerical approach,” New J. Phys. 13, 103026 (2011).
[CrossRef]

K. Giewekemeyer, H. Neubauer, S. Kalbfleisch, S. P. Krüger, and T. Salditt, “Holographic and diffractive x-ray imaging using waveguides as quasi-point sources,” New J. Phys. 12, 035008 (2010).
[CrossRef]

C. Fuhse, C. Ollinger, and T. Salditt, “Waveguide-based off-axis holography with hard x-rays,” Phys. Rev. Lett. 97, 254801 (2006).
[CrossRef]

Scarinci, F.

D. Pelliccia, I. Bukreeva, M. Ilie, W. Jark, A. Cedola, F. Scarinci, and S. Lagomarsino, “Computer simulations and experimental results on air-gap X-ray waveguides,” Spectrochim. Acta B 62, 615–621 (2007).
[CrossRef]

Schleich, W.

M. Berry, I. Marzoli, and W. Schleich, “Quantum carpets, carpets of light,” Phys. World 14, 39–44 (2001).

Siddons, D. P.

Y. P. Feng, S. K. Sinha, E. E. Fullerton, G. Grübel, D. Abernathy, D. P. Siddons, and J. B. Hastings, “X-ray Fraunhofer diffraction patterns from a thin-film waveguide,” Appl. Phys. Lett. 67, 3647–3649 (1995).
[CrossRef]

Silberberg, Y.

E. Poem and Y. Silberberg, “Photon correlations in multimode waveguides,” Phys. Rev. A 84, 041805 (2011).
[CrossRef]

Sinha, S. K.

Y. P. Feng, S. K. Sinha, E. E. Fullerton, G. Grübel, D. Abernathy, D. P. Siddons, and J. B. Hastings, “X-ray Fraunhofer diffraction patterns from a thin-film waveguide,” Appl. Phys. Lett. 67, 3647–3649 (1995).
[CrossRef]

Soldovieri, F.

Solimene, R.

Song, X. B.

X. B. Song, J. Xiong, X. Zhang, and K. Wang, “Second-order Talbot self-imaging with pseudothermal light,” Phys. Rev. A 82, 033823 (2010).
[CrossRef]

Sorrentino, A.

Sudol, R.

R. Sudol and B. J. Thompson, “An explanation of the Lau effect based on coherence theory,” Opt. Commun. 31, 105–110 (1979).
[CrossRef]

Thaning, A.

A. Thaning, P. Martinsson, M. Karelin, and A. T. Friberg, “Limits of diffractive optics by communication modes,” J. Opt. A 5, 153–158 (2003).
[CrossRef]

Thompson, B. J.

R. Sudol and B. J. Thompson, “An explanation of the Lau effect based on coherence theory,” Opt. Commun. 31, 105–110 (1979).
[CrossRef]

Torres-Company, V.

V. Torres-Company, J. Lancis, H. Lajunen, and A. T. Friberg, “Coherence revivals in two-photon frequency combs,” Phys. Rev. A 84, 033830 (2011).
[CrossRef]

Tsanaktsidis, K. J.

Ulrich, R.

R. Ulrich and T. Kamiya, “Resolution of self-images in planar optical waveguides,” J. Opt. Soc. Am. A 68, 583–592 (1978).
[CrossRef]

R. Ulrich, “Image formation by phase coincidences in optical waveguides,” Opt. Commun. 13, 259–264 (1975).
[CrossRef]

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, “Propagation of a partially coherent focused X-ray beam within a planar X-ray waveguide,” J. Synchrotron Radiat. 9, 383–393 (2002).
[CrossRef]

Wang, K.

X. B. Song, J. Xiong, X. Zhang, and K. Wang, “Second-order Talbot self-imaging with pseudothermal light,” Phys. Rev. A 82, 033823 (2010).
[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]

Wen, J. M.

K. H. Luo, J. M. Wen, X. H. Chen, Q. Liu, M. Xiao, and L. A. Wu, “Second-order Talbot effect with entangled photon pairs,” Phys. Rev. A 80, 043820 (2009).
[CrossRef]

Wolf, E.

E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature 326, 363–365 (1987).
[CrossRef]

E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
[CrossRef]

E. Wolf, “New theory of partial coherence in the space–frequency domain. Part I: spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
[CrossRef]

Wu, L. A.

K. H. Luo, X. H. Chen, Q. Liu, and L. A. Wu, “Nonlocal Talbot self-imaging with incoherent light,” Phys. Rev. A 82, 033803 (2010).
[CrossRef]

K. H. Luo, J. M. Wen, X. H. Chen, Q. Liu, M. Xiao, and L. A. Wu, “Second-order Talbot effect with entangled photon pairs,” Phys. Rev. A 80, 043820 (2009).
[CrossRef]

Xiao, M.

K. H. Luo, J. M. Wen, X. H. Chen, Q. Liu, M. Xiao, and L. A. Wu, “Second-order Talbot effect with entangled photon pairs,” Phys. Rev. A 80, 043820 (2009).
[CrossRef]

Xiong, J.

X. B. Song, J. Xiong, X. Zhang, and K. Wang, “Second-order Talbot self-imaging with pseudothermal light,” Phys. Rev. A 82, 033823 (2010).
[CrossRef]

Yark, W.

L. De Caro, C. Giannini, S. Di Fonzo, W. Yark, A. Cedola, and S. Lagomarsino, “Spatial coherence of x-ray planar waveguide exiting radiation,” Opt. Commun. 217, 31–45 (2003).
[CrossRef]

Zhang, X.

X. B. Song, J. Xiong, X. Zhang, and K. Wang, “Second-order Talbot self-imaging with pseudothermal light,” Phys. Rev. A 82, 033823 (2010).
[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]

Ann. Phys. (1)

E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. Phys. 437, 417–423 (1948).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

Y. P. Feng, S. K. Sinha, E. E. Fullerton, G. Grübel, D. Abernathy, D. P. Siddons, and J. B. Hastings, “X-ray Fraunhofer diffraction patterns from a thin-film waveguide,” Appl. Phys. Lett. 67, 3647–3649 (1995).
[CrossRef]

J. Appl. Phys. (1)

S. Lagomarsino, W. Jark, S. Di Fonzo, A. Cedola, B. R. Müller, C. Riekel, and P. Engstrom, “Submicrometer x-ray beam production by a thin film waveguide,” J. Appl. Phys. 79, 4471–4473 (1996).
[CrossRef]

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

S. F. Helfert, B. Huneke, and J. Jahns, “Self-imaging effect in multimode waveguides with longitudinal periodicity,” J. Eur. Opt. Soc. Rapid Pub. 4, 09031 (2009).
[CrossRef]

J. Mod. Opt. (2)

P. Hlubina, “Coherence of light at the output of a fibre waveguide analysed in the space-frequency domain,” J. Mod. Opt. 42, 1407–1426 (1995).
[CrossRef]

P. Hlubina, “The mutual interference of modes of a few-mode fibre waveguide analysed in the frequency domain,” J. Mod. Opt. 42, 2385–2399 (1995).
[CrossRef]

J. Opt. A (1)

A. Thaning, P. Martinsson, M. Karelin, and A. T. Friberg, “Limits of diffractive optics by communication modes,” J. Opt. A 5, 153–158 (2003).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (6)

J. Opt. Soc. Am. B (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)

E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature 326, 363–365 (1987).
[CrossRef]

New J. Phys. (2)

K. Giewekemeyer, H. Neubauer, S. Kalbfleisch, S. P. Krüger, and T. Salditt, “Holographic and diffractive x-ray imaging using waveguides as quasi-point sources,” New J. Phys. 12, 035008 (2010).
[CrossRef]

M. Osterhoff and T. Salditt, “Coherence filtering of x-ray waveguides: analytical and numerical approach,” New J. Phys. 13, 103026 (2011).
[CrossRef]

Opt. Acta (1)

M. Bertero, C. de Mol, F. Gori, and L. Ronchi, “Number of degrees of freedom in inverse diffraction,” Opt. Acta 30, 1051–1065 (1983).
[CrossRef]

Opt. Commun. (5)

L. De Caro, C. Giannini, S. Di Fonzo, W. Yark, A. Cedola, and S. Lagomarsino, “Spatial coherence of x-ray planar waveguide exiting radiation,” Opt. Commun. 217, 31–45 (2003).
[CrossRef]

J. Jahns and A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1979).
[CrossRef]

R. Sudol and B. J. Thompson, “An explanation of the Lau effect based on coherence theory,” Opt. Commun. 31, 105–110 (1979).
[CrossRef]

R. Ulrich, “Image formation by phase coincidences in optical waveguides,” Opt. Commun. 13, 259–264 (1975).
[CrossRef]

Opt. Express (1)

Opt. Lett. (4)

Optik (2)

P. Martinsson, P. Ma, A. Burvall, and A. T. Friberg, “Communication modes in scalar diffraction,” Optik 119, 103–111 (2008).
[CrossRef]

P. Martinsson, H. Lajunen, P. Ma, and A. T. Friberg, “Communication modes in vector diffraction,” Optik 121, 2087–2093 (2010).
[CrossRef]

Phys. Rev. A (6)

D. Pelliccia and D. M. Paganin, “Coherence vortices in vortex-free partially coherent x-ray fields,” Phys. Rev. A 86, 015802 (2012).
[CrossRef]

K. H. Luo, J. M. Wen, X. H. Chen, Q. Liu, M. Xiao, and L. A. Wu, “Second-order Talbot effect with entangled photon pairs,” Phys. Rev. A 80, 043820 (2009).
[CrossRef]

X. B. Song, J. Xiong, X. Zhang, and K. Wang, “Second-order Talbot self-imaging with pseudothermal light,” Phys. Rev. A 82, 033823 (2010).
[CrossRef]

K. H. Luo, X. H. Chen, Q. Liu, and L. A. Wu, “Nonlocal Talbot self-imaging with incoherent light,” Phys. Rev. A 82, 033803 (2010).
[CrossRef]

V. Torres-Company, J. Lancis, H. Lajunen, and A. T. Friberg, “Coherence revivals in two-photon frequency combs,” Phys. Rev. A 84, 033830 (2011).
[CrossRef]

E. Poem and Y. Silberberg, “Photon correlations in multimode waveguides,” Phys. Rev. A 84, 041805 (2011).
[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, “In-line holography and coherent diffractive imaging with x-ray waveguides,” Phys. Rev. B 77, 081408R (2008).
[CrossRef]

Phys. Rev. Lett. (2)

C. Fuhse, C. Ollinger, and T. Salditt, “Waveguide-based off-axis holography with hard x-rays,” Phys. Rev. Lett. 97, 254801 (2006).
[CrossRef]

E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
[CrossRef]

Phys. World (1)

M. Berry, I. Marzoli, and W. Schleich, “Quantum carpets, carpets of light,” Phys. World 14, 39–44 (2001).

Spectrochim. Acta B (1)

D. Pelliccia, I. Bukreeva, M. Ilie, W. Jark, A. Cedola, F. Scarinci, and S. Lagomarsino, “Computer simulations and experimental results on air-gap X-ray waveguides,” Spectrochim. Acta B 62, 615–621 (2007).
[CrossRef]

Other (2)

H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley-Interscience, 2004).

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

Supplementary Material (1)

» Media 1: AVI (4880 KB)     

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Figures (8)

Fig. 1.
Fig. 1.

Geometry of the coupling and propagation of the incident field in the waveguide.

Fig. 2.
Fig. 2.

Calculations for a waveguide with d=100nm, θ0=1.6mrad, and σ=2.0mrad. (a) Total intensity, integrated over the transverse direction and normalized to its initial value, plotted as a function of the propagation distance z along the waveguide. (b) Corresponding plot of the effective degree of coherence μ2. (c) Intensity map I(x,z) showing the gradual disappearance of higher-order modal components.

Fig. 3.
Fig. 3.

(a) Modulus (arb. units) and (b) phase (rad) of the CSD at the entrance of the waveguide (z=0mm). (c) Modulus and (d) phase of the CSD at z=10mm. The parameters used here are the same as for Fig. 1. (e) Bar plot of the coupling coefficients as a function of the mode order at the waveguide entrance (z=0mm, blue bars) and at z=10mm (red bars) (Media 1).

Fig. 4.
Fig. 4.

Calculations for a waveguide with d=100nm, θ0=1.6mrad, and σ=0.3mrad. (a) Total intensity, integrated over the transverse direction and normalized to its initial value, plotted as a function of the propagation distance z along the waveguide. (b) Corresponding plot of the effective degree of coherence μ2. (c) Intensity map I(x,z).

Fig. 5.
Fig. 5.

Bar plot of the coupling coefficients as a function of the mode order. High-divergence case (σ=2mrad, blue bars) and low divergence (σ=0.3mrad, red bars).

Fig. 6.
Fig. 6.

Two-dimensional intensity maps I(x,z) calculated for symmetric illumination. Waveguide and incident beam parameters are d=100nm, θ0=0rad, and σ=0.5mrad, respectively. (a) Case of negligible absorption. (b) Realistic case of non-negligible absorption.

Fig. 7.
Fig. 7.

(a) Modulus and (b) phase map of the function W(x1=d/4,x2=x,z1=z,z2=z), calculated for symmetric illumination and negligible absorption. (c) Modulus and (d) phase of the same function calculated for non-negligible absorption. (e) Line profiles along x=0 of the phase map in panels (b) and (d), black and red lines, respectively. Calculation parameters are the same as for Fig. 6.

Fig. 8.
Fig. 8.

(a) Modulus and (b) phase map of the function W(x1=d/4,x2=3d/4,z1,z2), calculated for symmetric illumination. Waveguide and incident beam parameters are d=100nm, θ0=0rad, and σ=0.5mrad, respectively. (c) Modulus and (d) phase of the function W(x1=d/4,x2=d/2,z1,z2) calculated with the same parameters.

Equations (28)

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U2(x,z=z2)=G(x,X)U1(X,z=z1)dX.
G(x,X)=mgmφm(x)ψm*(X).
G(x,X)ψm(X)dX=gmφm(x),
G*(x,X)φm(x)dx=gm*ψm(X).
W(x1,x2,z1,z2,ω)=U*(x1,z1,ω)U(x2,z2,ω).
Ψml(X1,X2)=ψm*(X1)ψl(X2),
Φml(x1x2)=ϕm*(x1)ϕl(x2)
U(x,z)=mcmϕm(x)eikzmz.
cm=0dUin(x,z=0)ϕm*(x)dx,
U(X,z1)=mcmϕm(X)eikzmz1,
U(x,z2)=mcmϕm(x)eikzmz2.
0dG(x,X)ϕm(X)dX=eikzm(z2z1)ϕm(x),
W(x1,z1,x2,z2)=m,nSmnϕm*(x1)ϕn(x2)×ei(kznz2kzmz1),
Smn=cm*cn.
Φmn(x1,x2)=ϕm*(x1)ϕn(x2)
μz(x1,x2)=W(x1,x2,z,z)W(x1,x1,z,z)W(x2,x2,z,z),
μ02=m,n|Smn|2(mSmm)2.
μz2=m,n|gm|2|gn|2|Smn|2(m|gm|2Smm)2.
Uθ(x,z=0)=exp(ik(xd/2)sinθ).
W(x1,x2,z1,z2)=Uθ*(x1,z1)Uθ(x2,z2)θ.
W(x1,x2,z1,z2)=π+πh(θ)Uθ*(x1,z1)Uθ(x2,z2)dθ,
h(θ)=12πσexp((θθ0)22σ2).
zSI=deff2λ,
Δd=λπnR21nR2λπ12δ2δ.
ϕm(x)={eγmxx<0cos(κmx)+γmκmsin(κmx)0xdχmeγm(xd)x>d,
tan(κmd)=2γmκmκm2γm2.
cm=cm(θ)=cm(1)(θ)+cm(2)(θ)+cm(3)(θ),
cm(1)(θ)=eikd2sinθγm*iksinθ,cm(2)(θ)=i2eikd2sinθ[(1+iγm*κm*)eid(κm*+ksinθ)1κm*+ksinθ+(1iγm*κm*)eid(κm*ksinθ)1κm*ksinθ],cm(3)(θ)=χm*eikd2sinθγm*+iksinθ.

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