Abstract

In this paper, an exact three-dimensional transparent boundary condition for the parabolic wave equation in a rectangular computational domain is reported. It is a generalization of the well-known two-dimensional Basakov–Popov–Papadakis transparent boundary condition. It relates the boundary transversal derivative of the wave field at any given longitudinal position to the field values at all preceding computational steps. Several examples demonstrate propagation of light along simple structured optical fibers as well as in x-ray guiding structures. The proposed condition is simple and robust and can help to reduce the size of the computational domain considerably.

© 2011 Optical Society of America

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  1. V. A. Fock, Electromagnetic Diffraction and Propagation Problems (Pergamon, 1965).
  2. F. D. Tappert, “The parabolic approximation method,” Lect. Notes Phys. 70, 224–287 (1977).
    [CrossRef]
  3. J. S. Papadakis, “Exact nonreflecting boundary conditions for parabolic type approximations in underwater acoustics,” J. Comput. Acoust. 2, 83–98 (1994).
    [CrossRef]
  4. V. A. Baskakov and A. V. Popov, “Implementation of transparent boundaries for numerical solution of the Schrödinger equation,” Wave Motion 14, 123–128 (1991).
    [CrossRef]
  5. A. V. Popov, “Accurate modelling of transparent boundaries in quasi-optics,” Radio Sci. 31, 1781–1790 (1996).
    [CrossRef]
  6. D. Yevick, T. Friese, and F. Schmidt, “A comparison of transparent boundary conditions for the Fresnel equation,” J. Comput. Phys. 168, 433–444 (2001).
    [CrossRef]
  7. X. Antoine, A. Arnold, C. Besse, M. Ehrhardt, and A. Schädle, “A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations,” Commun. Comput. Phys. 4, 729–796 (2008).
  8. J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200(1994).
    [CrossRef]
  9. R. M. Feshchenko and A. V. Popov, “Exact transparent boundary condition for beam propagation in rectangular domain,” in Proceedings of 2010 12th International Conference on Transparent Optical Networks (ICTON) (IEEE, 2010), pp. 1–4.
    [CrossRef]
  10. Yu. 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]
  11. V. V. Aristov, M. V. Grigoriev, S. M. Kuznetsov, L. G. Shabelnikov, V. A. Yunkin, M. Hoffmann, and E. Voges, “X-ray focusing by planar parabolic refractive lenses made of silicon,” Opt. Commun. 177, 33–38 (2000).
    [CrossRef]
  12. S. Lagomarsino, I. Bukreeva, and A. Cedola, “Theoretical analysis of x-ray waveguides,” in Modern Developments in X-Ray and Neutron Optics, A. Erko, ed., Vol. 137 of Springer Series in Optical Sciences (Springer, 2008), pp. 91–111.
  13. I. Bukreeva, D. Pelliccia, A. Cedola, F. Scarinci, M. Ilie, C. Giannini, L. De Caro, and S. Lagomarsino, “Analysis of tapered front-coupling x-ray waveguides,” J. Synchrotron Radiat. 17, 61–68 (2009).
    [CrossRef] [PubMed]
  14. J. Vila-Comamala, K. Jefimovs, J. Raabe, B. Kaulich, and C. David, “Silicon Fresnel zone plates for high heat load x-ray microscopy,” Microelectron. Eng. 85, 1241–1244(2008).
    [CrossRef]
  15. B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interaction: photoabsorption, scattering, transmission and reflection at E=50–30,000eV, Z=1–92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
    [CrossRef]
  16. R. Jamier, P. Viale, S. Fevrier, J.-M. Blondy, S. L. Semjonov,M. E. Likhachev, M. M. Bubnov, E. M. Dianov, V. F. Khopin, M. Y. Salganskii, and A. N. Guryanov, “Depressed-index-core singlemode bandgap fiber with very large effective area,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, Technical Digest (CD) (Optical Society of America, 2006), paper OFC6.
    [PubMed]
  17. D. Yevick, J. Yu, and F. Schmidt, “Analytic studies of absorbing and impedance-matched boundary layers,” IEEE Photon. Technol. Lett. 9, 73–75 (1997).
    [CrossRef]

2009 (1)

I. Bukreeva, D. Pelliccia, A. Cedola, F. Scarinci, M. Ilie, C. Giannini, L. De Caro, and S. Lagomarsino, “Analysis of tapered front-coupling x-ray waveguides,” J. Synchrotron Radiat. 17, 61–68 (2009).
[CrossRef] [PubMed]

2008 (2)

J. Vila-Comamala, K. Jefimovs, J. Raabe, B. Kaulich, and C. David, “Silicon Fresnel zone plates for high heat load x-ray microscopy,” Microelectron. Eng. 85, 1241–1244(2008).
[CrossRef]

X. Antoine, A. Arnold, C. Besse, M. Ehrhardt, and A. Schädle, “A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations,” Commun. Comput. Phys. 4, 729–796 (2008).

2001 (1)

D. Yevick, T. Friese, and F. Schmidt, “A comparison of transparent boundary conditions for the Fresnel equation,” J. Comput. Phys. 168, 433–444 (2001).
[CrossRef]

2000 (1)

V. V. Aristov, M. V. Grigoriev, S. M. Kuznetsov, L. G. Shabelnikov, V. A. Yunkin, M. Hoffmann, and E. Voges, “X-ray focusing by planar parabolic refractive lenses made of silicon,” Opt. Commun. 177, 33–38 (2000).
[CrossRef]

1997 (1)

D. Yevick, J. Yu, and F. Schmidt, “Analytic studies of absorbing and impedance-matched boundary layers,” IEEE Photon. Technol. Lett. 9, 73–75 (1997).
[CrossRef]

1996 (1)

A. V. Popov, “Accurate modelling of transparent boundaries in quasi-optics,” Radio Sci. 31, 1781–1790 (1996).
[CrossRef]

1995 (1)

Yu. 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]

1994 (2)

J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200(1994).
[CrossRef]

J. S. Papadakis, “Exact nonreflecting boundary conditions for parabolic type approximations in underwater acoustics,” J. Comput. Acoust. 2, 83–98 (1994).
[CrossRef]

1993 (1)

B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interaction: photoabsorption, scattering, transmission and reflection at E=50–30,000eV, Z=1–92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
[CrossRef]

1991 (1)

V. A. Baskakov and A. V. Popov, “Implementation of transparent boundaries for numerical solution of the Schrödinger equation,” Wave Motion 14, 123–128 (1991).
[CrossRef]

1977 (1)

F. D. Tappert, “The parabolic approximation method,” Lect. Notes Phys. 70, 224–287 (1977).
[CrossRef]

Antoine, X.

X. Antoine, A. Arnold, C. Besse, M. Ehrhardt, and A. Schädle, “A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations,” Commun. Comput. Phys. 4, 729–796 (2008).

Aristov, V. V.

V. V. Aristov, M. V. Grigoriev, S. M. Kuznetsov, L. G. Shabelnikov, V. A. Yunkin, M. Hoffmann, and E. Voges, “X-ray focusing by planar parabolic refractive lenses made of silicon,” Opt. Commun. 177, 33–38 (2000).
[CrossRef]

Arnold, A.

X. Antoine, A. Arnold, C. Besse, M. Ehrhardt, and A. Schädle, “A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations,” Commun. Comput. Phys. 4, 729–796 (2008).

Baskakov, V. A.

V. A. Baskakov and A. V. Popov, “Implementation of transparent boundaries for numerical solution of the Schrödinger equation,” Wave Motion 14, 123–128 (1991).
[CrossRef]

Berenger, J.

J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200(1994).
[CrossRef]

Besse, C.

X. Antoine, A. Arnold, C. Besse, M. Ehrhardt, and A. Schädle, “A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations,” Commun. Comput. Phys. 4, 729–796 (2008).

Blondy, J.-M.

R. Jamier, P. Viale, S. Fevrier, J.-M. Blondy, S. L. Semjonov,M. E. Likhachev, M. M. Bubnov, E. M. Dianov, V. F. Khopin, M. Y. Salganskii, and A. N. Guryanov, “Depressed-index-core singlemode bandgap fiber with very large effective area,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, Technical Digest (CD) (Optical Society of America, 2006), paper OFC6.
[PubMed]

Bubnov, M. M.

R. Jamier, P. Viale, S. Fevrier, J.-M. Blondy, S. L. Semjonov,M. E. Likhachev, M. M. Bubnov, E. M. Dianov, V. F. Khopin, M. Y. Salganskii, and A. N. Guryanov, “Depressed-index-core singlemode bandgap fiber with very large effective area,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, Technical Digest (CD) (Optical Society of America, 2006), paper OFC6.
[PubMed]

Bukreeva, I.

I. Bukreeva, D. Pelliccia, A. Cedola, F. Scarinci, M. Ilie, C. Giannini, L. De Caro, and S. Lagomarsino, “Analysis of tapered front-coupling x-ray waveguides,” J. Synchrotron Radiat. 17, 61–68 (2009).
[CrossRef] [PubMed]

S. Lagomarsino, I. Bukreeva, and A. Cedola, “Theoretical analysis of x-ray waveguides,” in Modern Developments in X-Ray and Neutron Optics, A. Erko, ed., Vol. 137 of Springer Series in Optical Sciences (Springer, 2008), pp. 91–111.

Cedola, A.

I. Bukreeva, D. Pelliccia, A. Cedola, F. Scarinci, M. Ilie, C. Giannini, L. De Caro, and S. Lagomarsino, “Analysis of tapered front-coupling x-ray waveguides,” J. Synchrotron Radiat. 17, 61–68 (2009).
[CrossRef] [PubMed]

S. Lagomarsino, I. Bukreeva, and A. Cedola, “Theoretical analysis of x-ray waveguides,” in Modern Developments in X-Ray and Neutron Optics, A. Erko, ed., Vol. 137 of Springer Series in Optical Sciences (Springer, 2008), pp. 91–111.

David, C.

J. Vila-Comamala, K. Jefimovs, J. Raabe, B. Kaulich, and C. David, “Silicon Fresnel zone plates for high heat load x-ray microscopy,” Microelectron. Eng. 85, 1241–1244(2008).
[CrossRef]

Davis, J. C.

B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interaction: photoabsorption, scattering, transmission and reflection at E=50–30,000eV, Z=1–92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
[CrossRef]

De Caro, L.

I. Bukreeva, D. Pelliccia, A. Cedola, F. Scarinci, M. Ilie, C. Giannini, L. De Caro, and S. Lagomarsino, “Analysis of tapered front-coupling x-ray waveguides,” J. Synchrotron Radiat. 17, 61–68 (2009).
[CrossRef] [PubMed]

Dianov, E. M.

R. Jamier, P. Viale, S. Fevrier, J.-M. Blondy, S. L. Semjonov,M. E. Likhachev, M. M. Bubnov, E. M. Dianov, V. F. Khopin, M. Y. Salganskii, and A. N. Guryanov, “Depressed-index-core singlemode bandgap fiber with very large effective area,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, Technical Digest (CD) (Optical Society of America, 2006), paper OFC6.
[PubMed]

Ehrhardt, M.

X. Antoine, A. Arnold, C. Besse, M. Ehrhardt, and A. Schädle, “A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations,” Commun. Comput. Phys. 4, 729–796 (2008).

Feshchenko, R. M.

R. M. Feshchenko and A. V. Popov, “Exact transparent boundary condition for beam propagation in rectangular domain,” in Proceedings of 2010 12th International Conference on Transparent Optical Networks (ICTON) (IEEE, 2010), pp. 1–4.
[CrossRef]

Fevrier, S.

R. Jamier, P. Viale, S. Fevrier, J.-M. Blondy, S. L. Semjonov,M. E. Likhachev, M. M. Bubnov, E. M. Dianov, V. F. Khopin, M. Y. Salganskii, and A. N. Guryanov, “Depressed-index-core singlemode bandgap fiber with very large effective area,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, Technical Digest (CD) (Optical Society of America, 2006), paper OFC6.
[PubMed]

Fock, V. A.

V. A. Fock, Electromagnetic Diffraction and Propagation Problems (Pergamon, 1965).

Friese, T.

D. Yevick, T. Friese, and F. Schmidt, “A comparison of transparent boundary conditions for the Fresnel equation,” J. Comput. Phys. 168, 433–444 (2001).
[CrossRef]

Giannini, C.

I. Bukreeva, D. Pelliccia, A. Cedola, F. Scarinci, M. Ilie, C. Giannini, L. De Caro, and S. Lagomarsino, “Analysis of tapered front-coupling x-ray waveguides,” J. Synchrotron Radiat. 17, 61–68 (2009).
[CrossRef] [PubMed]

Grigoriev, M. V.

V. V. Aristov, M. V. Grigoriev, S. M. Kuznetsov, L. G. Shabelnikov, V. A. Yunkin, M. Hoffmann, and E. Voges, “X-ray focusing by planar parabolic refractive lenses made of silicon,” Opt. Commun. 177, 33–38 (2000).
[CrossRef]

Gullikson, E. M.

B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interaction: photoabsorption, scattering, transmission and reflection at E=50–30,000eV, Z=1–92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
[CrossRef]

Guryanov, A. N.

R. Jamier, P. Viale, S. Fevrier, J.-M. Blondy, S. L. Semjonov,M. E. Likhachev, M. M. Bubnov, E. M. Dianov, V. F. Khopin, M. Y. Salganskii, and A. N. Guryanov, “Depressed-index-core singlemode bandgap fiber with very large effective area,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, Technical Digest (CD) (Optical Society of America, 2006), paper OFC6.
[PubMed]

Henke, B. L.

B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interaction: photoabsorption, scattering, transmission and reflection at E=50–30,000eV, Z=1–92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
[CrossRef]

Hoffmann, M.

V. V. Aristov, M. V. Grigoriev, S. M. Kuznetsov, L. G. Shabelnikov, V. A. Yunkin, M. Hoffmann, and E. Voges, “X-ray focusing by planar parabolic refractive lenses made of silicon,” Opt. Commun. 177, 33–38 (2000).
[CrossRef]

Ilie, M.

I. Bukreeva, D. Pelliccia, A. Cedola, F. Scarinci, M. Ilie, C. Giannini, L. De Caro, and S. Lagomarsino, “Analysis of tapered front-coupling x-ray waveguides,” J. Synchrotron Radiat. 17, 61–68 (2009).
[CrossRef] [PubMed]

Jamier, R.

R. Jamier, P. Viale, S. Fevrier, J.-M. Blondy, S. L. Semjonov,M. E. Likhachev, M. M. Bubnov, E. M. Dianov, V. F. Khopin, M. Y. Salganskii, and A. N. Guryanov, “Depressed-index-core singlemode bandgap fiber with very large effective area,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, Technical Digest (CD) (Optical Society of America, 2006), paper OFC6.
[PubMed]

Jefimovs, K.

J. Vila-Comamala, K. Jefimovs, J. Raabe, B. Kaulich, and C. David, “Silicon Fresnel zone plates for high heat load x-ray microscopy,” Microelectron. Eng. 85, 1241–1244(2008).
[CrossRef]

Kaulich, B.

J. Vila-Comamala, K. Jefimovs, J. Raabe, B. Kaulich, and C. David, “Silicon Fresnel zone plates for high heat load x-ray microscopy,” Microelectron. Eng. 85, 1241–1244(2008).
[CrossRef]

Khopin, V. F.

R. Jamier, P. Viale, S. Fevrier, J.-M. Blondy, S. L. Semjonov,M. E. Likhachev, M. M. Bubnov, E. M. Dianov, V. F. Khopin, M. Y. Salganskii, and A. N. Guryanov, “Depressed-index-core singlemode bandgap fiber with very large effective area,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, Technical Digest (CD) (Optical Society of America, 2006), paper OFC6.
[PubMed]

Kopylov, Yu. V.

Yu. 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]

Kuznetsov, S. M.

V. V. Aristov, M. V. Grigoriev, S. M. Kuznetsov, L. G. Shabelnikov, V. A. Yunkin, M. Hoffmann, and E. Voges, “X-ray focusing by planar parabolic refractive lenses made of silicon,” Opt. Commun. 177, 33–38 (2000).
[CrossRef]

Lagomarsino, S.

I. Bukreeva, D. Pelliccia, A. Cedola, F. Scarinci, M. Ilie, C. Giannini, L. De Caro, and S. Lagomarsino, “Analysis of tapered front-coupling x-ray waveguides,” J. Synchrotron Radiat. 17, 61–68 (2009).
[CrossRef] [PubMed]

S. Lagomarsino, I. Bukreeva, and A. Cedola, “Theoretical analysis of x-ray waveguides,” in Modern Developments in X-Ray and Neutron Optics, A. Erko, ed., Vol. 137 of Springer Series in Optical Sciences (Springer, 2008), pp. 91–111.

Likhachev, M. E.

R. Jamier, P. Viale, S. Fevrier, J.-M. Blondy, S. L. Semjonov,M. E. Likhachev, M. M. Bubnov, E. M. Dianov, V. F. Khopin, M. Y. Salganskii, and A. N. Guryanov, “Depressed-index-core singlemode bandgap fiber with very large effective area,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, Technical Digest (CD) (Optical Society of America, 2006), paper OFC6.
[PubMed]

Papadakis, J. S.

J. S. Papadakis, “Exact nonreflecting boundary conditions for parabolic type approximations in underwater acoustics,” J. Comput. Acoust. 2, 83–98 (1994).
[CrossRef]

Pelliccia, D.

I. Bukreeva, D. Pelliccia, A. Cedola, F. Scarinci, M. Ilie, C. Giannini, L. De Caro, and S. Lagomarsino, “Analysis of tapered front-coupling x-ray waveguides,” J. Synchrotron Radiat. 17, 61–68 (2009).
[CrossRef] [PubMed]

Popov, A. V.

A. V. Popov, “Accurate modelling of transparent boundaries in quasi-optics,” Radio Sci. 31, 1781–1790 (1996).
[CrossRef]

Yu. 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]

V. A. Baskakov and A. V. Popov, “Implementation of transparent boundaries for numerical solution of the Schrödinger equation,” Wave Motion 14, 123–128 (1991).
[CrossRef]

R. M. Feshchenko and A. V. Popov, “Exact transparent boundary condition for beam propagation in rectangular domain,” in Proceedings of 2010 12th International Conference on Transparent Optical Networks (ICTON) (IEEE, 2010), pp. 1–4.
[CrossRef]

Raabe, J.

J. Vila-Comamala, K. Jefimovs, J. Raabe, B. Kaulich, and C. David, “Silicon Fresnel zone plates for high heat load x-ray microscopy,” Microelectron. Eng. 85, 1241–1244(2008).
[CrossRef]

Salganskii, M. Y.

R. Jamier, P. Viale, S. Fevrier, J.-M. Blondy, S. L. Semjonov,M. E. Likhachev, M. M. Bubnov, E. M. Dianov, V. F. Khopin, M. Y. Salganskii, and A. N. Guryanov, “Depressed-index-core singlemode bandgap fiber with very large effective area,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, Technical Digest (CD) (Optical Society of America, 2006), paper OFC6.
[PubMed]

Scarinci, F.

I. Bukreeva, D. Pelliccia, A. Cedola, F. Scarinci, M. Ilie, C. Giannini, L. De Caro, and S. Lagomarsino, “Analysis of tapered front-coupling x-ray waveguides,” J. Synchrotron Radiat. 17, 61–68 (2009).
[CrossRef] [PubMed]

Schädle, A.

X. Antoine, A. Arnold, C. Besse, M. Ehrhardt, and A. Schädle, “A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations,” Commun. Comput. Phys. 4, 729–796 (2008).

Schmidt, F.

D. Yevick, T. Friese, and F. Schmidt, “A comparison of transparent boundary conditions for the Fresnel equation,” J. Comput. Phys. 168, 433–444 (2001).
[CrossRef]

D. Yevick, J. Yu, and F. Schmidt, “Analytic studies of absorbing and impedance-matched boundary layers,” IEEE Photon. Technol. Lett. 9, 73–75 (1997).
[CrossRef]

Semjonov, S. L.

R. Jamier, P. Viale, S. Fevrier, J.-M. Blondy, S. L. Semjonov,M. E. Likhachev, M. M. Bubnov, E. M. Dianov, V. F. Khopin, M. Y. Salganskii, and A. N. Guryanov, “Depressed-index-core singlemode bandgap fiber with very large effective area,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, Technical Digest (CD) (Optical Society of America, 2006), paper OFC6.
[PubMed]

Shabelnikov, L. G.

V. V. Aristov, M. V. Grigoriev, S. M. Kuznetsov, L. G. Shabelnikov, V. A. Yunkin, M. Hoffmann, and E. Voges, “X-ray focusing by planar parabolic refractive lenses made of silicon,” Opt. Commun. 177, 33–38 (2000).
[CrossRef]

Tappert, F. D.

F. D. Tappert, “The parabolic approximation method,” Lect. Notes Phys. 70, 224–287 (1977).
[CrossRef]

Viale, P.

R. Jamier, P. Viale, S. Fevrier, J.-M. Blondy, S. L. Semjonov,M. E. Likhachev, M. M. Bubnov, E. M. Dianov, V. F. Khopin, M. Y. Salganskii, and A. N. Guryanov, “Depressed-index-core singlemode bandgap fiber with very large effective area,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, Technical Digest (CD) (Optical Society of America, 2006), paper OFC6.
[PubMed]

Vila-Comamala, J.

J. Vila-Comamala, K. Jefimovs, J. Raabe, B. Kaulich, and C. David, “Silicon Fresnel zone plates for high heat load x-ray microscopy,” Microelectron. Eng. 85, 1241–1244(2008).
[CrossRef]

Vinogradov, A. V.

Yu. 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]

Voges, E.

V. V. Aristov, M. V. Grigoriev, S. M. Kuznetsov, L. G. Shabelnikov, V. A. Yunkin, M. Hoffmann, and E. Voges, “X-ray focusing by planar parabolic refractive lenses made of silicon,” Opt. Commun. 177, 33–38 (2000).
[CrossRef]

Yevick, D.

D. Yevick, T. Friese, and F. Schmidt, “A comparison of transparent boundary conditions for the Fresnel equation,” J. Comput. Phys. 168, 433–444 (2001).
[CrossRef]

D. Yevick, J. Yu, and F. Schmidt, “Analytic studies of absorbing and impedance-matched boundary layers,” IEEE Photon. Technol. Lett. 9, 73–75 (1997).
[CrossRef]

Yu, J.

D. Yevick, J. Yu, and F. Schmidt, “Analytic studies of absorbing and impedance-matched boundary layers,” IEEE Photon. Technol. Lett. 9, 73–75 (1997).
[CrossRef]

Yunkin, V. A.

V. V. Aristov, M. V. Grigoriev, S. M. Kuznetsov, L. G. Shabelnikov, V. A. Yunkin, M. Hoffmann, and E. Voges, “X-ray focusing by planar parabolic refractive lenses made of silicon,” Opt. Commun. 177, 33–38 (2000).
[CrossRef]

At. Data Nucl. Data Tables (1)

B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interaction: photoabsorption, scattering, transmission and reflection at E=50–30,000eV, Z=1–92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
[CrossRef]

Commun. Comput. Phys. (1)

X. Antoine, A. Arnold, C. Besse, M. Ehrhardt, and A. Schädle, “A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations,” Commun. Comput. Phys. 4, 729–796 (2008).

IEEE Photon. Technol. Lett. (1)

D. Yevick, J. Yu, and F. Schmidt, “Analytic studies of absorbing and impedance-matched boundary layers,” IEEE Photon. Technol. Lett. 9, 73–75 (1997).
[CrossRef]

J. Comput. Acoust. (1)

J. S. Papadakis, “Exact nonreflecting boundary conditions for parabolic type approximations in underwater acoustics,” J. Comput. Acoust. 2, 83–98 (1994).
[CrossRef]

J. Comput. Phys. (2)

J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200(1994).
[CrossRef]

D. Yevick, T. Friese, and F. Schmidt, “A comparison of transparent boundary conditions for the Fresnel equation,” J. Comput. Phys. 168, 433–444 (2001).
[CrossRef]

J. Synchrotron Radiat. (1)

I. Bukreeva, D. Pelliccia, A. Cedola, F. Scarinci, M. Ilie, C. Giannini, L. De Caro, and S. Lagomarsino, “Analysis of tapered front-coupling x-ray waveguides,” J. Synchrotron Radiat. 17, 61–68 (2009).
[CrossRef] [PubMed]

Lect. Notes Phys. (1)

F. D. Tappert, “The parabolic approximation method,” Lect. Notes Phys. 70, 224–287 (1977).
[CrossRef]

Microelectron. Eng. (1)

J. Vila-Comamala, K. Jefimovs, J. Raabe, B. Kaulich, and C. David, “Silicon Fresnel zone plates for high heat load x-ray microscopy,” Microelectron. Eng. 85, 1241–1244(2008).
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Opt. Commun. (2)

Yu. 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).
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V. V. Aristov, M. V. Grigoriev, S. M. Kuznetsov, L. G. Shabelnikov, V. A. Yunkin, M. Hoffmann, and E. Voges, “X-ray focusing by planar parabolic refractive lenses made of silicon,” Opt. Commun. 177, 33–38 (2000).
[CrossRef]

Radio Sci. (1)

A. V. Popov, “Accurate modelling of transparent boundaries in quasi-optics,” Radio Sci. 31, 1781–1790 (1996).
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Wave Motion (1)

V. A. Baskakov and A. V. Popov, “Implementation of transparent boundaries for numerical solution of the Schrödinger equation,” Wave Motion 14, 123–128 (1991).
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Other (4)

V. A. Fock, Electromagnetic Diffraction and Propagation Problems (Pergamon, 1965).

R. M. Feshchenko and A. V. Popov, “Exact transparent boundary condition for beam propagation in rectangular domain,” in Proceedings of 2010 12th International Conference on Transparent Optical Networks (ICTON) (IEEE, 2010), pp. 1–4.
[CrossRef]

S. Lagomarsino, I. Bukreeva, and A. Cedola, “Theoretical analysis of x-ray waveguides,” in Modern Developments in X-Ray and Neutron Optics, A. Erko, ed., Vol. 137 of Springer Series in Optical Sciences (Springer, 2008), pp. 91–111.

R. Jamier, P. Viale, S. Fevrier, J.-M. Blondy, S. L. Semjonov,M. E. Likhachev, M. M. Bubnov, E. M. Dianov, V. F. Khopin, M. Y. Salganskii, and A. N. Guryanov, “Depressed-index-core singlemode bandgap fiber with very large effective area,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, Technical Digest (CD) (Optical Society of America, 2006), paper OFC6.
[PubMed]

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Figures (5)

Fig. 1
Fig. 1

Silicon rod with radius 70 nm is illuminated by a plane x-ray wave incident orthogonally on its front surface. A cylindrical refracted wave propagates freely through the outer border of an RCD. That the border does not influence its propagation means that the border is transparent.

Fig. 2
Fig. 2

Propagation of the x-ray radiation in a hollow waveguide made inside a silicon slab and shaped like a square prism, which is illuminated by a Gaussian beam incident on its front surface. Each side of the square is 50 nm in length. The propagating modes show (2,2) symmetry.

Fig. 3
Fig. 3

Propagation of the x-ray radiation in a hexagonally shaped waveguide made of six cylindrical rods ( radii = 20 nm ), which is illuminated by a plane wave incident on its front surface from the left to right at small angle ( 0.001 rad ). Two modes—a symmetrical and an antisymmetrical—propagate in the central hollow core of the structure, resulting in oscillations of the intensity due to the modes’ interference.

Fig. 4
Fig. 4

Propagation of the optical radiation in a microstructured fiber. The structure is illuminated by a Gaussian beam incident on its front surface. There is an interesting snowflake pattern of intensity visible in the third graph, which arises because the radiation leaks through the gaps between the rods.

Fig. 5
Fig. 5

Propagation of Gaussian beams in free space. The curves show the difference V [see Eq. (35)] between the numerical and exact solutions for a number of values of parameters τ, h, and ψ. The maximum is caused by the discretization error.

Equations (38)

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2 i k u z + 2 u x 2 = 0 .
2 i k p F ( p ) + 2 F ( p ) x 2 = 0 ,
u ( z , x ) = 1 2 π i c i c + i exp ( p z ) C ± ( p ) exp ( ± i 2 i k p x ) d p ,
u x = ± i 2 i k 2 π i c i c + i exp ( p z ) p C ± ( p ) exp ( i 2 i k p x ) d p = ± i 2 i k 2 π i z c i c + i exp ( p z ) 1 p C ± ( p ) exp ( i 2 i k p x ) d p = 2 k π i z 0 z u ( ζ , ± a ) z ζ d ζ .
2 i k u z + 2 u x 2 + 2 u y 2 = 0 ,
U ( z , μ , x , y ) = + u ( μ , η , y ) Γ ( z μ , x η ) d η , Γ ( s , t ) = i k 2 π s exp ( i k t 2 2 s ) ,
2 i k U μ = 2 i k + u ( μ , η , y ) Γ ( z μ , x η ) μ d η + 2 i k + u ( μ , η , y ) μ Γ ( z μ , x η ) d η = 2 i k + u ( μ , η , y ) Γ ( z μ , x η ) z d η + 2 u ( μ , η , y ) η 2 Γ ( z μ , x η ) d η + 2 u ( μ , η , y ) y 2 Γ ( z μ , x η ) d η = 2 i k U z 2 U x 2 2 U y 2 = 2 U y 2 .
2 i k U μ + 2 U y 2 = 0 .
U ( z , z , x , y ) = u ( z , x , y ) .
U x = 2 k π i z 0 z U ( ζ , μ , ± a , y ) z ζ d ζ .
u x = 2 k π i z 0 z U ( ζ , μ , ± a , y ) z ζ d ζ | z = μ .
| x | < a , | y | < b .
2 i k u z + 2 u x 2 + 2 u y 2 + k 2 α u = 0 , α = ε ( z , x , y ) 1 .
2 i k u m , s n + 1 u m , s n τ = k 2 α m , s n α m , s n + 1 u m , s n + 1 + α m , s n u m , s n 2 + u m + 1 , s n + 1 2 u m , s n + 1 + u m 1 , s n + 1 2 h 2 + u m + 1 , s n 2 u m , s n + u m 1 , s n 2 h 2 + u m , s + 1 n + 1 2 u m , s n + 1 + u m , s 1 n + 1 2 h 2 + u m , s + 1 n 2 u m , s n + u m , s 1 n 2 h 2 ,
u m , s n = u ( τ n , h m , h s ) , α m , s n = α ( τ n , h m , h s ) 0 n N τ , 1 m , s N 1 ,
u m + 1 , s n + 1 u m , s + 1 n + 1 + C m , s n + 1 u m , s n + 1 u m 1 , s n + 1 u m , s 1 n + 1 = u m + 1 , s n + u m , s + 1 n C ˜ m , s n u m , s n + u m 1 , s n + u m , s 1 n ,
C m , s n = 4 k 2 h 2 α m , s n 4 i k h 2 / τ , C ˜ m , s n = 4 k 2 h 2 α m , s n + 4 i k h 2 / τ .
u N , s n + 1 u N 2 , s n + 1 2 h + 2 σ ( u N 1 , s n + 1 l = 1 n γ l U N 1 , s n + 1 l , n + 1 ) , x = a ,
u 2 , s n + 1 u 0 , s n + 1 2 h 2 σ ( u 1 , s n + 1 l = 1 n γ l U 1 , s n + 1 l , n + 1 ) , x = a ,
u m , N n + 1 u m , N 2 n + 1 2 h + 2 σ ( u m , N 1 n + 1 l = 1 n γ l U m , N 1 n + 1 , n + 1 l ) , y = a ,
u m , 2 n + 1 u m , 0 n + 1 2 h 2 σ ( u m , 1 n + 1 l = 1 n γ l U m , 1 n + 1 , n + 1 l ) , y = a ,
γ l = 2 ( l + 1 + l ) ( l + l 1 ) ( l + 1 + l 1 ) , σ = 2 k i π τ , U m , s n , p = U ( τ n , τ p , h m , h s ) , u m , s n = U m , s n , n .
U N 1 , s + 1 p , n + 1 + B U N 1 , s p , n + 1 U N 1 , s 1 p , n + 1 = U N 1 , s + 1 p , n B ˜ U N 1 , s p , n + U N 1 , s 1 p , n ,
U 1 , s + 1 p , n + 1 + B U 1 , s p , n + 1 U 1 , s 1 p , n + 1 = U 1 , s + 1 p , n B ˜ U 1 , s p , n + U 1 , s 1 p , n ,
U m + 1 , N 1 n + 1 , p + B U m , N 1 n + 1 , p U m 1 , N 1 n + 1 , p = U m + 1 , N 1 n , p B ˜ U m , N 1 n , p + U m 1 , N 1 n , p ,
U m + 1 , 1 n + 1 , p + B U m , 1 n + 1 , p U m 1 , 1 n + 1 , p = U m + 1 , 1 n , p B ˜ U m , 1 n , p + U m 1 , 1 n , p ,
B = 2 4 i k h 2 / τ , B ˜ = 2 + 4 i k h 2 / τ , 0 p n .
U N 1 , N p , n + 1 U N 1 , N 2 p , n + 1 2 h + 2 σ ( U N 1 , N 1 p , n + 1 l = 1 n γ l U N 1 , N 1 p , n + 1 l ) , x = a , y = a ,
U N 1 , 2 p , n + 1 U N 1 , 0 p , n + 1 2 h 2 σ ( U N 1 , 1 p , n + 1 l = 1 n γ l U N 1 , 1 p , n + 1 l ) , x = a , y = a ,
U 1 , N p , n + 1 U 1 , N 2 p , n + 1 2 h + 2 σ ( U 1 , N 1 p , n + 1 l = 1 n γ l U 1 , N 1 p , n + 1 l ) , x = a , y = a ,
U 1 , 2 p , n + 1 U 1 , 0 p , n + 1 2 h 2 σ ( U 1 , 1 p , n + 1 l = 1 n γ l U 1 , 1 p , n + 1 l ) , x = a , y = a .
U N , N 1 n + 1 , p U N 2 , N 1 n + 1 , p 2 h + 2 σ ( U N 1 , N 1 n + 1 , p l = 1 n γ l U N 1 , N 1 n + 1 l , p ) , y = a , x = a ,
U 2 , N 1 n + 1 , p U 0 , N 1 n + 1 , p 2 h 2 σ ( U 1 , N 1 n + 1 , p l = 1 n γ l U 1 , N 1 n + 1 l , p ) , y = a , x = a ,
U N , 1 n + 1 , p U N 2 , 1 n + 1 , p 2 h + 2 σ ( U N 1 , 1 n + 1 , p l = 1 n γ l U N 1 , 1 n + 1 l , p ) , y = a , x = a ,
U 2 , 1 n + 1 , p U 0 , 1 n + 1 , p 2 h 2 σ ( U 1 , 1 n + 1 , p l = 1 n γ l U 1 , 1 n + 1 l , p ) , y = a , x = a .
u 0 = { 0 , r > r 0 nm exp ( r 2 w 2 ) , r r 0 nm ,
u g ( x , y , z ) = w 2 w 2 + i 2 z / k exp [ i k sin ψ ( ( x z sin ψ cos α / 2 ) + ( y z sin ψ sin α / 2 ) ) ] × exp [ ( ( x z sin ψ cos α ) 2 + ( y z sin ψ sin α ) 2 ) / ( w 2 + i 2 z / k ) ] ,
V ( τ n ) = m , s = 0 N | u m , s n u g ( τ n , h m , h s ) | 2 / m , s = 0 N | u g ( τ n , h m , h s ) | 2 .

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