Abstract

This paper considers a finite element approach to modeling electromagnetic waves in a periodic diffraction grating. In particular, an a priori error estimate associated with the α-quasi-periodic transformation is derived. This involves the solution of the associated Helmholtz problem being written as a product of eiαx and an unknown function called the α-quasi-periodic solution. To begin with, the well-posedness of the continuous problem is examined using a variational formulation. The problem is then discretized, and a rigorous a priori error estimate, which guarantees the uniqueness of this approximate solution, is derived. In previous studies, the continuity of the Dirichlet-to-Neumann map has simply been assumed and the dependency of the regularity constant on the system parameters, such as the wavenumber, has not been shown. To address this deficiency, in this paper an explicit dependence on the wavenumber and the degree of the polynomial basis in the a priori error estimate is obtained. Since the finite element method is well known for dealing with any geometries, comparison of numerical results obtained using the α-quasi-periodic transformation with a lattice sum technique is then presented.

© 2013 Optical Society of America

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  1. M. Peters, M. Rüdiger, B. Bläsi, and W. Platzer, “Electro-optical simulation of diffraction in solar cells,” Opt. Express 18, A584–A593 (2010).
    [CrossRef]
  2. R. A. Potyrailo, H. Ghiradella, A. Vertiatchikh, J. R. Cournoyer, K. Dovidenko, and E. Olson, “Morpho butterfly wing scales demonstrate highly selective vapour response,” Nat. Photonics 1, 123–128 (2007).
    [CrossRef]
  3. T. H. Jensen, M. Bech, O. Bunk, T. Donath, C. David, R. Feidenhans, and F. Pfeiffer, “Directional x-ray dark-field imaging,” Phys. Med. Biol 55, 3317–3323 (2010).
    [CrossRef]
  4. P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. A. McDonald, F. Marone, and M. Stampanoni, “Low-dose, simple, and fast grating-based x-ray phase-contrast imaging,” Proc. Natl. Acad. Sci. USA 107, 13576–13581 (2010).
    [CrossRef]
  5. G. Bao, L. Cowsar, and W. Masters, Mathematical Modeling in Optical Science, Vol. 22 of SIAM Frontiers in Applied Mathematics (Springer, 2001).
  6. G. Bao and A. Zhou, “Analysis of finite dimensional approximations to a class of partial differential equations,” Math. Methods Appl. Sci. 27, 2055–2066 (2004).
    [CrossRef]
  7. R. Petit, Electromagnetic Theory of Gratings (Springer, 1980).
  8. G. Bao, “Numerical analysis of diffraction by periodic structures: TM polarization,” Numer. Math. 75, 1–16 (1996).
    [CrossRef]
  9. G. Bao, Z. Chen, and H. Wu, “Adaptive finite element method for diffraction gratings,” J. Opt. Soc. Am. A 22, 1106–1114 (2005).
    [CrossRef]
  10. N. Lord, “Analysis of electromagnetic waves in a periodic diffraction grating using a priori error estimates and a dual weighted residual method,” Ph.D. thesis (University of Strathclyde, 2012) ( http://www.mathstat.strath.ac.uk/research/phd_mphil_theses ).
  11. N. Lord and A. J. Mulholland, “Analysis of the α,0-quasi periodic transformation for a periodic diffraction grating,” (Department of Mathematics and Statistics, University of Strathclyde, 2011) ( http://www.mathstat.strath.ac.uk/research/reports/2011 ).
  12. S. Chandler-Wilde, “Boundary value problems for the Helmholtz equation in a half-plane,” in Mathematical and Numerical Aspects of Wave Propagation (SIAM, 1995), pp. 188–197.
  13. G. Bao, Y. Cao, and H. Yang, “Numerical solution of diffraction problems by a least squares FEM,” Math. Methods Appl. Sci. 23, 1073–1092 (2000).
    [CrossRef]
  14. E. Wolf, ed., Rigorous Vector Theories of Diffraction Gratings, Vol. 21 of Progress in Optics (North-Holland, 1984), pp. 1–67.
  15. P. G. Ciarlet, The Finite Element Method for Elliptic Equations (North-Holland, 1978).
  16. J. M. Melenk and S. Sauter, “Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions,” Math. Comput. 79, 1871–1914 (2010).
    [CrossRef]
  17. A. Lechleiter and D. Nguyen, “Volume integral equations for scattering from anisotropic diffraction gratings,” Math. Methods Appl. Sci. 36, 262–274 (2012).
  18. S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Vol. 15 of Texts in Applied Mathematics (Springer, 2002).
  19. E. Kreyszig, Introductory Functional Analysis with Applications (Wiley, 1989).
  20. D. Maystre, Electromagnetic Theory of Gratings (Springer, 1980).
  21. L. F. Richardson, Measure and Integration. A Concise Introduction to Real Analysis (Wiley, 2009).
  22. D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Vol. 93 of Applied Mathematical Sciences (Springer-Verlag, 1992).
  23. A. Buffa, “Trace theorems for functional spaces related to Maxwell equations: an overview,” in Computational Electromagnetics, Vol. 28 of Lecture Notes in Computational Science and Engineering (Springer, 2002), pp. 23–34.
  24. Z. Ding, “A proof of the trace theorem of Sobolev spaces on Lipschitz domains,” Proc. Am. Math. Soc. 124, 591–601 (1996).
    [CrossRef]
  25. D. Braess, Finite Elements (Cambridge University, 1997).
  26. F. Ihlenburgh, Finite Element Analysis of Acoustic Scattering (Springer, 1998), Vol. 132.
  27. J. T. Oden and M. Ainsworth, A Posteriori Error Estimation in Finite Element Analysis (Wiley, 2000).
  28. K. Ito, Encyclopedic Dictionary of Mathematics, 2nd ed. (Massachusetts Institute of Technology, 1987).
  29. K. Yasumoto, T. Kushta, and H. Toyama, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603–2611 (2004).
    [CrossRef]
  30. R. E. Coath, “Investigating the use of replica Morpho butterfly scales for colour displays,” Society 5, 1–9 (2007) ( http://printfu.org/blue+morpho+didius+butterfly ).
  31. G. Berger, K. Müller, C. Denz, I. Földvári, and A. Péter, “Digital data storage in a phase-encoded holographic memory system: data quality and security,” Proc. SPIE 4988, 104–111 (2003).
    [CrossRef]

2012 (1)

A. Lechleiter and D. Nguyen, “Volume integral equations for scattering from anisotropic diffraction gratings,” Math. Methods Appl. Sci. 36, 262–274 (2012).

2010 (4)

J. M. Melenk and S. Sauter, “Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions,” Math. Comput. 79, 1871–1914 (2010).
[CrossRef]

T. H. Jensen, M. Bech, O. Bunk, T. Donath, C. David, R. Feidenhans, and F. Pfeiffer, “Directional x-ray dark-field imaging,” Phys. Med. Biol 55, 3317–3323 (2010).
[CrossRef]

P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. A. McDonald, F. Marone, and M. Stampanoni, “Low-dose, simple, and fast grating-based x-ray phase-contrast imaging,” Proc. Natl. Acad. Sci. USA 107, 13576–13581 (2010).
[CrossRef]

M. Peters, M. Rüdiger, B. Bläsi, and W. Platzer, “Electro-optical simulation of diffraction in solar cells,” Opt. Express 18, A584–A593 (2010).
[CrossRef]

2007 (2)

R. A. Potyrailo, H. Ghiradella, A. Vertiatchikh, J. R. Cournoyer, K. Dovidenko, and E. Olson, “Morpho butterfly wing scales demonstrate highly selective vapour response,” Nat. Photonics 1, 123–128 (2007).
[CrossRef]

R. E. Coath, “Investigating the use of replica Morpho butterfly scales for colour displays,” Society 5, 1–9 (2007) ( http://printfu.org/blue+morpho+didius+butterfly ).

2005 (1)

2004 (2)

G. Bao and A. Zhou, “Analysis of finite dimensional approximations to a class of partial differential equations,” Math. Methods Appl. Sci. 27, 2055–2066 (2004).
[CrossRef]

K. Yasumoto, T. Kushta, and H. Toyama, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603–2611 (2004).
[CrossRef]

2003 (1)

G. Berger, K. Müller, C. Denz, I. Földvári, and A. Péter, “Digital data storage in a phase-encoded holographic memory system: data quality and security,” Proc. SPIE 4988, 104–111 (2003).
[CrossRef]

2000 (1)

G. Bao, Y. Cao, and H. Yang, “Numerical solution of diffraction problems by a least squares FEM,” Math. Methods Appl. Sci. 23, 1073–1092 (2000).
[CrossRef]

1996 (2)

G. Bao, “Numerical analysis of diffraction by periodic structures: TM polarization,” Numer. Math. 75, 1–16 (1996).
[CrossRef]

Z. Ding, “A proof of the trace theorem of Sobolev spaces on Lipschitz domains,” Proc. Am. Math. Soc. 124, 591–601 (1996).
[CrossRef]

Ainsworth, M.

J. T. Oden and M. Ainsworth, A Posteriori Error Estimation in Finite Element Analysis (Wiley, 2000).

Bao, G.

G. Bao, Z. Chen, and H. Wu, “Adaptive finite element method for diffraction gratings,” J. Opt. Soc. Am. A 22, 1106–1114 (2005).
[CrossRef]

G. Bao and A. Zhou, “Analysis of finite dimensional approximations to a class of partial differential equations,” Math. Methods Appl. Sci. 27, 2055–2066 (2004).
[CrossRef]

G. Bao, Y. Cao, and H. Yang, “Numerical solution of diffraction problems by a least squares FEM,” Math. Methods Appl. Sci. 23, 1073–1092 (2000).
[CrossRef]

G. Bao, “Numerical analysis of diffraction by periodic structures: TM polarization,” Numer. Math. 75, 1–16 (1996).
[CrossRef]

G. Bao, L. Cowsar, and W. Masters, Mathematical Modeling in Optical Science, Vol. 22 of SIAM Frontiers in Applied Mathematics (Springer, 2001).

Bech, M.

T. H. Jensen, M. Bech, O. Bunk, T. Donath, C. David, R. Feidenhans, and F. Pfeiffer, “Directional x-ray dark-field imaging,” Phys. Med. Biol 55, 3317–3323 (2010).
[CrossRef]

Berger, G.

G. Berger, K. Müller, C. Denz, I. Földvári, and A. Péter, “Digital data storage in a phase-encoded holographic memory system: data quality and security,” Proc. SPIE 4988, 104–111 (2003).
[CrossRef]

Bläsi, B.

Braess, D.

D. Braess, Finite Elements (Cambridge University, 1997).

Brenner, S. C.

S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Vol. 15 of Texts in Applied Mathematics (Springer, 2002).

Buffa, A.

A. Buffa, “Trace theorems for functional spaces related to Maxwell equations: an overview,” in Computational Electromagnetics, Vol. 28 of Lecture Notes in Computational Science and Engineering (Springer, 2002), pp. 23–34.

Bunk, O.

T. H. Jensen, M. Bech, O. Bunk, T. Donath, C. David, R. Feidenhans, and F. Pfeiffer, “Directional x-ray dark-field imaging,” Phys. Med. Biol 55, 3317–3323 (2010).
[CrossRef]

Cao, Y.

G. Bao, Y. Cao, and H. Yang, “Numerical solution of diffraction problems by a least squares FEM,” Math. Methods Appl. Sci. 23, 1073–1092 (2000).
[CrossRef]

Chandler-Wilde, S.

S. Chandler-Wilde, “Boundary value problems for the Helmholtz equation in a half-plane,” in Mathematical and Numerical Aspects of Wave Propagation (SIAM, 1995), pp. 188–197.

Chen, Z.

Ciarlet, P. G.

P. G. Ciarlet, The Finite Element Method for Elliptic Equations (North-Holland, 1978).

Coath, R. E.

R. E. Coath, “Investigating the use of replica Morpho butterfly scales for colour displays,” Society 5, 1–9 (2007) ( http://printfu.org/blue+morpho+didius+butterfly ).

Colton, D.

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Vol. 93 of Applied Mathematical Sciences (Springer-Verlag, 1992).

Cournoyer, J. R.

R. A. Potyrailo, H. Ghiradella, A. Vertiatchikh, J. R. Cournoyer, K. Dovidenko, and E. Olson, “Morpho butterfly wing scales demonstrate highly selective vapour response,” Nat. Photonics 1, 123–128 (2007).
[CrossRef]

Cowsar, L.

G. Bao, L. Cowsar, and W. Masters, Mathematical Modeling in Optical Science, Vol. 22 of SIAM Frontiers in Applied Mathematics (Springer, 2001).

David, C.

T. H. Jensen, M. Bech, O. Bunk, T. Donath, C. David, R. Feidenhans, and F. Pfeiffer, “Directional x-ray dark-field imaging,” Phys. Med. Biol 55, 3317–3323 (2010).
[CrossRef]

Denz, C.

G. Berger, K. Müller, C. Denz, I. Földvári, and A. Péter, “Digital data storage in a phase-encoded holographic memory system: data quality and security,” Proc. SPIE 4988, 104–111 (2003).
[CrossRef]

Ding, Z.

Z. Ding, “A proof of the trace theorem of Sobolev spaces on Lipschitz domains,” Proc. Am. Math. Soc. 124, 591–601 (1996).
[CrossRef]

Donath, T.

T. H. Jensen, M. Bech, O. Bunk, T. Donath, C. David, R. Feidenhans, and F. Pfeiffer, “Directional x-ray dark-field imaging,” Phys. Med. Biol 55, 3317–3323 (2010).
[CrossRef]

Dovidenko, K.

R. A. Potyrailo, H. Ghiradella, A. Vertiatchikh, J. R. Cournoyer, K. Dovidenko, and E. Olson, “Morpho butterfly wing scales demonstrate highly selective vapour response,” Nat. Photonics 1, 123–128 (2007).
[CrossRef]

Feidenhans, R.

T. H. Jensen, M. Bech, O. Bunk, T. Donath, C. David, R. Feidenhans, and F. Pfeiffer, “Directional x-ray dark-field imaging,” Phys. Med. Biol 55, 3317–3323 (2010).
[CrossRef]

Földvári, I.

G. Berger, K. Müller, C. Denz, I. Földvári, and A. Péter, “Digital data storage in a phase-encoded holographic memory system: data quality and security,” Proc. SPIE 4988, 104–111 (2003).
[CrossRef]

Ghiradella, H.

R. A. Potyrailo, H. Ghiradella, A. Vertiatchikh, J. R. Cournoyer, K. Dovidenko, and E. Olson, “Morpho butterfly wing scales demonstrate highly selective vapour response,” Nat. Photonics 1, 123–128 (2007).
[CrossRef]

Ihlenburgh, F.

F. Ihlenburgh, Finite Element Analysis of Acoustic Scattering (Springer, 1998), Vol. 132.

Ito, K.

K. Ito, Encyclopedic Dictionary of Mathematics, 2nd ed. (Massachusetts Institute of Technology, 1987).

Jensen, T. H.

T. H. Jensen, M. Bech, O. Bunk, T. Donath, C. David, R. Feidenhans, and F. Pfeiffer, “Directional x-ray dark-field imaging,” Phys. Med. Biol 55, 3317–3323 (2010).
[CrossRef]

Kress, R.

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Vol. 93 of Applied Mathematical Sciences (Springer-Verlag, 1992).

Kreyszig, E.

E. Kreyszig, Introductory Functional Analysis with Applications (Wiley, 1989).

Kushta, T.

K. Yasumoto, T. Kushta, and H. Toyama, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603–2611 (2004).
[CrossRef]

Lechleiter, A.

A. Lechleiter and D. Nguyen, “Volume integral equations for scattering from anisotropic diffraction gratings,” Math. Methods Appl. Sci. 36, 262–274 (2012).

Liu, X.

P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. A. McDonald, F. Marone, and M. Stampanoni, “Low-dose, simple, and fast grating-based x-ray phase-contrast imaging,” Proc. Natl. Acad. Sci. USA 107, 13576–13581 (2010).
[CrossRef]

Liu, Y.

P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. A. McDonald, F. Marone, and M. Stampanoni, “Low-dose, simple, and fast grating-based x-ray phase-contrast imaging,” Proc. Natl. Acad. Sci. USA 107, 13576–13581 (2010).
[CrossRef]

Lord, N.

N. Lord, “Analysis of electromagnetic waves in a periodic diffraction grating using a priori error estimates and a dual weighted residual method,” Ph.D. thesis (University of Strathclyde, 2012) ( http://www.mathstat.strath.ac.uk/research/phd_mphil_theses ).

N. Lord and A. J. Mulholland, “Analysis of the α,0-quasi periodic transformation for a periodic diffraction grating,” (Department of Mathematics and Statistics, University of Strathclyde, 2011) ( http://www.mathstat.strath.ac.uk/research/reports/2011 ).

Marone, F.

P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. A. McDonald, F. Marone, and M. Stampanoni, “Low-dose, simple, and fast grating-based x-ray phase-contrast imaging,” Proc. Natl. Acad. Sci. USA 107, 13576–13581 (2010).
[CrossRef]

Masters, W.

G. Bao, L. Cowsar, and W. Masters, Mathematical Modeling in Optical Science, Vol. 22 of SIAM Frontiers in Applied Mathematics (Springer, 2001).

Maystre, D.

D. Maystre, Electromagnetic Theory of Gratings (Springer, 1980).

McDonald, S. A.

P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. A. McDonald, F. Marone, and M. Stampanoni, “Low-dose, simple, and fast grating-based x-ray phase-contrast imaging,” Proc. Natl. Acad. Sci. USA 107, 13576–13581 (2010).
[CrossRef]

Melenk, J. M.

J. M. Melenk and S. Sauter, “Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions,” Math. Comput. 79, 1871–1914 (2010).
[CrossRef]

Mulholland, A. J.

N. Lord and A. J. Mulholland, “Analysis of the α,0-quasi periodic transformation for a periodic diffraction grating,” (Department of Mathematics and Statistics, University of Strathclyde, 2011) ( http://www.mathstat.strath.ac.uk/research/reports/2011 ).

Müller, K.

G. Berger, K. Müller, C. Denz, I. Földvári, and A. Péter, “Digital data storage in a phase-encoded holographic memory system: data quality and security,” Proc. SPIE 4988, 104–111 (2003).
[CrossRef]

Nguyen, D.

A. Lechleiter and D. Nguyen, “Volume integral equations for scattering from anisotropic diffraction gratings,” Math. Methods Appl. Sci. 36, 262–274 (2012).

Oden, J. T.

J. T. Oden and M. Ainsworth, A Posteriori Error Estimation in Finite Element Analysis (Wiley, 2000).

Olson, E.

R. A. Potyrailo, H. Ghiradella, A. Vertiatchikh, J. R. Cournoyer, K. Dovidenko, and E. Olson, “Morpho butterfly wing scales demonstrate highly selective vapour response,” Nat. Photonics 1, 123–128 (2007).
[CrossRef]

Péter, A.

G. Berger, K. Müller, C. Denz, I. Földvári, and A. Péter, “Digital data storage in a phase-encoded holographic memory system: data quality and security,” Proc. SPIE 4988, 104–111 (2003).
[CrossRef]

Peters, M.

Petit, R.

R. Petit, Electromagnetic Theory of Gratings (Springer, 1980).

Pfeiffer, F.

T. H. Jensen, M. Bech, O. Bunk, T. Donath, C. David, R. Feidenhans, and F. Pfeiffer, “Directional x-ray dark-field imaging,” Phys. Med. Biol 55, 3317–3323 (2010).
[CrossRef]

Platzer, W.

Potyrailo, R. A.

R. A. Potyrailo, H. Ghiradella, A. Vertiatchikh, J. R. Cournoyer, K. Dovidenko, and E. Olson, “Morpho butterfly wing scales demonstrate highly selective vapour response,” Nat. Photonics 1, 123–128 (2007).
[CrossRef]

Richardson, L. F.

L. F. Richardson, Measure and Integration. A Concise Introduction to Real Analysis (Wiley, 2009).

Rüdiger, M.

Sauter, S.

J. M. Melenk and S. Sauter, “Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions,” Math. Comput. 79, 1871–1914 (2010).
[CrossRef]

Scott, L. R.

S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Vol. 15 of Texts in Applied Mathematics (Springer, 2002).

Stampanoni, M.

P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. A. McDonald, F. Marone, and M. Stampanoni, “Low-dose, simple, and fast grating-based x-ray phase-contrast imaging,” Proc. Natl. Acad. Sci. USA 107, 13576–13581 (2010).
[CrossRef]

Toyama, H.

K. Yasumoto, T. Kushta, and H. Toyama, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603–2611 (2004).
[CrossRef]

Vertiatchikh, A.

R. A. Potyrailo, H. Ghiradella, A. Vertiatchikh, J. R. Cournoyer, K. Dovidenko, and E. Olson, “Morpho butterfly wing scales demonstrate highly selective vapour response,” Nat. Photonics 1, 123–128 (2007).
[CrossRef]

Wang, Z.

P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. A. McDonald, F. Marone, and M. Stampanoni, “Low-dose, simple, and fast grating-based x-ray phase-contrast imaging,” Proc. Natl. Acad. Sci. USA 107, 13576–13581 (2010).
[CrossRef]

Wu, H.

Wu, Z.

P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. A. McDonald, F. Marone, and M. Stampanoni, “Low-dose, simple, and fast grating-based x-ray phase-contrast imaging,” Proc. Natl. Acad. Sci. USA 107, 13576–13581 (2010).
[CrossRef]

Yang, H.

G. Bao, Y. Cao, and H. Yang, “Numerical solution of diffraction problems by a least squares FEM,” Math. Methods Appl. Sci. 23, 1073–1092 (2000).
[CrossRef]

Yasumoto, K.

K. Yasumoto, T. Kushta, and H. Toyama, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603–2611 (2004).
[CrossRef]

Zhang, K.

P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. A. McDonald, F. Marone, and M. Stampanoni, “Low-dose, simple, and fast grating-based x-ray phase-contrast imaging,” Proc. Natl. Acad. Sci. USA 107, 13576–13581 (2010).
[CrossRef]

Zhou, A.

G. Bao and A. Zhou, “Analysis of finite dimensional approximations to a class of partial differential equations,” Math. Methods Appl. Sci. 27, 2055–2066 (2004).
[CrossRef]

Zhu, P.

P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. A. McDonald, F. Marone, and M. Stampanoni, “Low-dose, simple, and fast grating-based x-ray phase-contrast imaging,” Proc. Natl. Acad. Sci. USA 107, 13576–13581 (2010).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

K. Yasumoto, T. Kushta, and H. Toyama, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603–2611 (2004).
[CrossRef]

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

Math. Comput. (1)

J. M. Melenk and S. Sauter, “Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions,” Math. Comput. 79, 1871–1914 (2010).
[CrossRef]

Math. Methods Appl. Sci. (3)

A. Lechleiter and D. Nguyen, “Volume integral equations for scattering from anisotropic diffraction gratings,” Math. Methods Appl. Sci. 36, 262–274 (2012).

G. Bao and A. Zhou, “Analysis of finite dimensional approximations to a class of partial differential equations,” Math. Methods Appl. Sci. 27, 2055–2066 (2004).
[CrossRef]

G. Bao, Y. Cao, and H. Yang, “Numerical solution of diffraction problems by a least squares FEM,” Math. Methods Appl. Sci. 23, 1073–1092 (2000).
[CrossRef]

Nat. Photonics (1)

R. A. Potyrailo, H. Ghiradella, A. Vertiatchikh, J. R. Cournoyer, K. Dovidenko, and E. Olson, “Morpho butterfly wing scales demonstrate highly selective vapour response,” Nat. Photonics 1, 123–128 (2007).
[CrossRef]

Numer. Math. (1)

G. Bao, “Numerical analysis of diffraction by periodic structures: TM polarization,” Numer. Math. 75, 1–16 (1996).
[CrossRef]

Opt. Express (1)

Phys. Med. Biol (1)

T. H. Jensen, M. Bech, O. Bunk, T. Donath, C. David, R. Feidenhans, and F. Pfeiffer, “Directional x-ray dark-field imaging,” Phys. Med. Biol 55, 3317–3323 (2010).
[CrossRef]

Proc. Am. Math. Soc. (1)

Z. Ding, “A proof of the trace theorem of Sobolev spaces on Lipschitz domains,” Proc. Am. Math. Soc. 124, 591–601 (1996).
[CrossRef]

Proc. Natl. Acad. Sci. USA (1)

P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. A. McDonald, F. Marone, and M. Stampanoni, “Low-dose, simple, and fast grating-based x-ray phase-contrast imaging,” Proc. Natl. Acad. Sci. USA 107, 13576–13581 (2010).
[CrossRef]

Proc. SPIE (1)

G. Berger, K. Müller, C. Denz, I. Földvári, and A. Péter, “Digital data storage in a phase-encoded holographic memory system: data quality and security,” Proc. SPIE 4988, 104–111 (2003).
[CrossRef]

Society (1)

R. E. Coath, “Investigating the use of replica Morpho butterfly scales for colour displays,” Society 5, 1–9 (2007) ( http://printfu.org/blue+morpho+didius+butterfly ).

Other (17)

E. Wolf, ed., Rigorous Vector Theories of Diffraction Gratings, Vol. 21 of Progress in Optics (North-Holland, 1984), pp. 1–67.

P. G. Ciarlet, The Finite Element Method for Elliptic Equations (North-Holland, 1978).

D. Braess, Finite Elements (Cambridge University, 1997).

F. Ihlenburgh, Finite Element Analysis of Acoustic Scattering (Springer, 1998), Vol. 132.

J. T. Oden and M. Ainsworth, A Posteriori Error Estimation in Finite Element Analysis (Wiley, 2000).

K. Ito, Encyclopedic Dictionary of Mathematics, 2nd ed. (Massachusetts Institute of Technology, 1987).

N. Lord, “Analysis of electromagnetic waves in a periodic diffraction grating using a priori error estimates and a dual weighted residual method,” Ph.D. thesis (University of Strathclyde, 2012) ( http://www.mathstat.strath.ac.uk/research/phd_mphil_theses ).

N. Lord and A. J. Mulholland, “Analysis of the α,0-quasi periodic transformation for a periodic diffraction grating,” (Department of Mathematics and Statistics, University of Strathclyde, 2011) ( http://www.mathstat.strath.ac.uk/research/reports/2011 ).

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Figures (3)

Fig. 1.
Fig. 1.

Diagram showing the truncated periodic grating domain. Define Ω1 to be the region above the scattering region {(x,y):0x<d,byB}, and the substrate Ω2 to be {(x,y):0x<d,Byb}.

Fig. 2.
Fig. 2.

(a) Double layered dielectric transmitting cylinders. (b) Comparison between the reflection efficiency of order 0 from the α-quasi-periodic method (full line) and the lattice sum technique (dots) [29] for dielectric transmitting cylinders for the TM case [see (a)]. The reflection efficiency is shown as a function of the ratio of the wavelength of the incident field (λ) to the lattice period (d).

Fig. 3.
Fig. 3.

(a) Transmitting dielectric lamellar grating. (b) Error in computing R0 using the α-quasi-periodic method with respect to the adaptive method in [9].

Equations (126)

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k(x,y)={k1R,for(x,y){0xd,yB},k1R,for(x,y)Ω1,k0C,for(x,y)Ω0\Ω3,k3C,for(x,y)Ω3,k2C,for(x,y)Ω2,k2C,for(x,y){0xd,yB},
U(x,y)=eiαxUα(x,y).
Lα#s([0,d])={gLs([0,d]):g(d)=eiαdg(0)},H#s([0,d])={gHs([0,d]):g(d)=g(0)},Hα#s([0,d])={gHs([0,d]):g(d)=eiαdg(0)},
H#s(Ω)={fHs(Ω):f(d,y)=f(0,y),y[B,B]},Hα#s(Ω)={fHs(Ω):f(d,y)=eiαdf(0,y),y[B,B]},
vH2=|v|H1(Ϝ)2+k2vL2(Ϝ)2.
inf(1,1k)vHvH1(Ϝ)sup(1,1k)vH
v=eiαxvα.
vαH#l(Ω)l2lvHα#l(Ω)
vαL#2(Ω)=vLα#2(Ω).
Tf(x)=T±f(x)=nZiβjnf(nα)(±B)einαx,
zn=arg(kj2nα2),
R(Tg,g)Γ±=dnZsin(zn/2)|βjn||g(nα)|20I(Tg,g)Γ±=dnZcos(zn/2)|βjn||g(nα)|20}.
·(1k2(x,y)U(x,y))+U(x,y)=0,(x,y)Ω,
nU(x,y)|Γ+=T+U+g(x)xΓ+,
nU(x,y)|Γ=TUxΓ,
T+U(x)=nZiβ1nU(nα)(B)einαx,TU(x)=nZiβ2nU(nα)(B)einαx,g(x)=2iβ10eiβ10B+iαx,
Tαf(x)=T±αf(x)=nZiβjnf(n)(±B)ei2πndx,
f(n)(±B)=1d0df(x,±B)ei2πnx/ddx.
TfHα#12(Γ±)2C2sup(|kj2|,1)fHα#12(Γ±)2
|(Tf,g)Γ±|dCsup(|kj|,1)fHα#12(Γ±)gHα#12(Γ±)
|βjn|2{C2|kj2|,if|kj2|>nα2,C2nα2if|kj2|<nα2.
·(1k2(x,y)U(x,y))+U(x,y)=f(x,y),
DγUL2(Ω)Creg(1+CsC(k0,k3))k|γ|1fL2(Ω),
U(x,y)=SGj(xx0,yy0)f(x0,y0)dx0dy0
Gj(x,y)=12dnZcjneinαx+iβjn|y|iβjn+12dnZdjneinαxiβjn|y|iβjn,
Imn(y,y0)=R(cjneiβjn|yy0|βjn+djneiβjn|yy0|βjn)f(mα)(y0)dy0,
Imn(y,y0)=0ify0y,
Imn(y,y)=1βjn(cjn+djn)f(mα)(y)ify0=y.
U(x,y)=1d[0,d]×RnZeinα(xx0)(cjneiβjn|yy0|2iβjn+djneiβjn|yy0|2iβjn)×mZeimαx0f(mα)(y0)dx0dy0.
U(x,y)=1dnZ[0,d]×Reinα(xx0)(cjneiβjn|yy0|2iβjn+djneiβjn|yy0|2iβjn)×mZeimαx0f(mα)(y0)dx0dy0
=1dm,nZ[0,d]einα(xx0)2ieimαx0Imn(y,y0)dx0.
U(x,y)=nZeinα(x)2iInn(y,y0).
U(x,y)=nZeinα(x)2i(1βjn(cjn+djn)f(nα)(y))
UL2(S)supnZ,j{|cjn|,|djn|}|βjn|nZeinαxf(nα)(y)L2(S).
UL2(Ω)supj{0,1,2,3},nZ{|cjn|,|djn|}|βjn|f(x,y)L2(Ω).
xγ1U=1dm,nZ0d(inα)γ1einα(xx0)eimα(x0),R(cjneiβjn|yy0|+djneiβjn|yy0|2iβjn)f(mα)(y0)dy0dx0,
xγ1UL2(S)1dsupnZnαγ1m,nZ0deinα(xx0)eimα(x0),R(cjneiβjn|yy0|+djneiβjn|yy0|2iβjn)f(mα)(y0)dy0dx0L2(S).
xγ1UL2(Ω)supnZnαγ1UL2(Ω).
yγ2U=1dm,nZ0deinα(xx0)eimα(x0)R(iβjn)γ2cjneiβjn|yy0|+(iβjn)γ2djneiβjn|yy0|2iβjnf(mα)(y0)dy0dx0+p=2γ2[nZ(cjneiβjn|yy0|+djneiβjn|yy0|)2iβjnf(nα)(y0)]SjSlf(nα)(y0),
|[nZ(cjneiβjn|yy0|+djneiβjn|yy0|2iβjn)f(nα)(y0)]SjSl|supnZ,l{0,1,2,3}(|βjn|p2|cjneiβjn|yy0||)nZ|f(nα)(y0)|+supnZ,l{0,1,2,3}(|βjn|p2|djneiβjn|yy0||)nZ|f(nα)(y0)|.
|e±iβjn|yy0||=1,
|eiβjn|yy0||1,
|eiβjn|yy0||esin(zn/2)kjsup{2πN0d,|α|}/krefsup{y0,y}Ω3|yy0|,
|[nZ(cjneiβjn|yy0|+djneiβjn|yy0|2iβjn)f(nα)(y0)]SjSl|supnZ,j{0,1,2,3}|βjn|p2(|cjn|,|djn|)sup(esin(zn/2)|kj|N0krefsup{y0,y}S0S3|yy0|,1)nZ|f(nα)(y0)|supnZ,j{0,1,2,3}Cs0(esin(zn/2)|kj|N0krefsup{y0,y}S0S3|yy0|,1)|βjn|p2fL2(S).
yγ2UL2(S)supnZβjnγ2UL2(S)+supnZ,j{0,1,2,3}Cs0(esinzn/2|kj|N0krefsup{y0,y}S0S3|yy0|,1)×(p=2γ2|βjn|p2)fL2(S)supnZβjnγ2UL2(S)+supnZ,j{0,1,2,3}Cs0(esinzn/2|kj|N0krefsup{y0,y}S0S3|yy0|,1)×(γ21)|βjn|γ22fL2(S).
xγ1UL2(S)supnZnαγ1supj{0,1,2,3},nZ{|cjn|,|djn|}|βjn|f(x,y)L2(Ω)supj{0,1,2,3},nZkγ11|sinθn||cosθn|γ1{|cjn|,|djn|}f(x,y)L2(Ω).
yγ2UL2(S)supj{0,1,2,3},nZ{|cjn|,|djn|}(k|eizn/2cosθn|)γ21f(x,y)L2(Ω)+C(k0,k3)CssupnZ,j|βjn|γ22fL2(S)supnZkγ21(|cosθn|)γ21fL2(S)+C(k0,k3)CssupnZ,j(k|eizn/2cosθn|)γ21fL2(S)
Creg=supnZ{|sinθn|γ1|cosθn|,|cosθn|γ21},
yγ2UL2(S)(Cregkγ21+Cregkγ21C(k0,k3)Cs)fL2(S)Creg(1+CsC(k0,k3))kγ21fL2(S).
yγ2UL2(Ω)Creg(1+CsC(k0,k3))kγ21fL2(Ω).
α·(1k2(x,y)αUα(x,y))+Uα(x,y)=0,(x,y)Ω
(T+αn)Uα=2iβ10eiβ10B,onΓ+,(Tαn)Uα=0,onΓ.
Uα(d,y)=Uα(0,y),y[B,B],
U=(eiαxUα)=(eiαx)Uα+eiαxUα=[iαeiαx0]Uα+eiαxUα.
Ω.(1k2U)v¯+ΩUv¯=0.
a(U,v)=(1k2U,v)Ω(U,v)Ω(1k2T±U,v)Γ±,
(f,v)Γ+=Γ+2iβ10k12ei(αxβ10B)v¯,
a(U,v)=(f,v)Γ+.
a(Uα,vα)=(1k2Uα,vα)Ω((1α2k2)Uα,vα)Ωiα(1k2xUα,vα)Ω+iα(1k2Uα,xvα)Ω(1k2T±αUα,vα)Γ±,(fα,vα)Γ+=Γ+2iβ10k12eiβ10Bv¯α.
a(Uα,vα)=(fα,vα)Γ+.
|(1k2U,v)Ω|1kref2Ω|U.v¯|dxdy,1kref2ULα#2(Ω)vLα#2(Ω)
|(U,v)Ω|ULα#2(Ω)vLα#2(Ω).
|Γ±T±Uv¯dx|2C2d2(|kj2|ULα#2(Ω)2+UHα#1(Ω)2)vHα#1(Ω)2,
|Γ±1k2T±Uv¯dx|Cd1kref2(|kj2|ULα#2(Ω)2+UHα#1(Ω)2)1/2vHα#1(Ω).
|a(U,v)|1kref2|U|Hα#1(Ω)|v|Hα#1(Ω)+ULα#2(Ω)vLα#2(Ω)+Cd1kref2(|kj2|ULα#2(Ω)2+UHα#1(Ω)2)vHα#1(Ω)2,
|a(U,v)|C0sup(1,1kref2,|k2|kref2)UHα#1(Ω)vHα#1(Ω).
|a(U,U)|+Ω|U|2=|Ω1k2|U|2Γ±1k2TUU¯|||Ω1k2|U|2||Γ±1k2TUU¯|||k2Cdsup(|kj|,1)/kref2|UHα#1(Ω)2
|a(U,U)+ULα#2(Ω)2|M1UHα#1(Ω)2.
α.(1k2αUα)+Uα=fα,inΩ,(T+αn)Uα=0,onΓ+,(Tαn)Uα=0,onΓ.
UαHCstabfαL#2(Ω),
UαH2=|Uα|H#1(Ω)2+k2UαLα#2(Ω)2.
UαH222UHα#1(Ω)2+k2ULα#2(Ω)2.
infψX{vψLα#2(Ω)+hpvψLα#2(Ω)+(hp)12vψLα#2(Γ±)+hpvψHα#12(Γ±)}C(hp)lvHα#l(Ω).
a(Uh,ϕ)=(f,ϕ)Γ+
|a(U,v)|CcUHvH
|a(U,v)|1kref2|U|Hα#1(Ω)|v|Hα#1(Ω)+ULα#2(Ω)vLα#2(Ω)+Cdkref2UHvH.
|a(U,v)|1kref2(|U|Hα#1(Ω)|v|Hα#1(Ω)+k2ULα#2(Ω)vLα#2(Ω)+CdUHvH).
|U|Hα#1(Ω)|v|Hα#1(Ω)+k2ULα#2(Ω)vLα#2(Ω)UHvH
|a(U,v)|CcUHvH,
1R(k2)|U|Hα#1(Ω)2ULα#2(Ω)2|a(U,U)|+|(1k2T±U,U)Γ±|
(1k2U,U)Ω(U,U)Ω=a(U,U)+(1k2T±U,U)Γ±.
|R(1k2U,U)Ω(U,U)Ω||a(U,U)+(1k2T±U,U)Γ±|.
|R(1k2U,U)Ω|(U,U)Ω|a(U,U)|+|(1k2T±U,U)Γ±|.
|R(1k2U,U)Ω|1R(k2)|U|Hα#1(Ω)2.
ehLα#2(Ω)C1ehH.
.(1k2w)+w=ϕ(x,y)Ω,(T±*nw)=0onΓ±,
ehLα#2(Ω)=supϕC(Ω)|a(eh,wψ)|ϕLα#2(Ω)
|a(eh,wψ)|=|(1k2eh,(wψ))Ω(eh,wψ)Ω(1k2T±eh,wψ)Γ±|1kref2(|eh|Hα#1(Ω)|wψ|Hα#1(Ω)+k2ehLα#2(Ω)wψLα#2(Ω)+CdehHwψHα#1(Ω))
|a(eh,wψ)|(Cd+1)1kref2ehHwψHα#1(Ω).
|a(eh,wψ)|C(Cd+1)1kref2hpehHwHα#2(Ω).
wHα#2(Ω)ϕLα#2(Ω)(1+CsC(k0,k3))Cregk=Cstabk.
ehLα#2(Ω)CCstabhkpkref2(Cd+1)ehH=C1ehH.
eαhH4ckC4(2Cd+1)UαψαH,
eαhL#2(Ω)2ckC4C1(2Cd+1)UαψαH,
1R(k2)(|eh|Hα#1(Ω)2R(k2)ehLα#2(Ω)2)|a(eh,Uψ)|+|(1k2T±eh,eh)Γ±|.
1R(k2)(|eh|Hα#1(Ω)2k2ehLα#2(Ω)2)CcehHUψH+Cdkref2ehHUψH
1R(k2)(|eh|Hα#1(Ω)2k2ehLα#2(Ω)2)2Cd+1kref2ehHUψH.
|eh|Hα#1(Ω)2kehLα#2(Ω)ehHck(2Cd+1)ehHUψH.
ehH2kC1ehHck(2Cd+1)UψH.
ehHckC4(2Cd+1)UψH.
eαhH4ckC4(2Cd+1)UαψαH.
ehLα#2(Ω)ckC4C1(2Cd+1)UψH.
eαhL#2(Ω)2ckC4C1(2Cd+1)UαψαH.
T±αMUαh(x)=n=MMiβjnUαh(n)(±B)ei2πndx.
eαM=UαUαM.
eαMH4ckdC4(CUαψαH+e(Bb)cmin(M|α|)2kj2UαHα#12(Γ1,±)),eαML#2(Ω)2ckdC1C4(CUαψαH+e(Bb)cmin(M|α|)2kj2UαHα#12(Γ1,±)),
aM(UM,v)=(f,v)Γ+
aM(UM,v)=(1k2UM,v)Ω(UM,v)Ω(1k2T±MUM,v)Γ±,(f,v)Γ+=(2iβ10k12ei(αxβ10B),v)Γ+,
a(U,v)aM(UM,v)=0.
(1k2eM,v)Ω(eM,v)Ω(1k2T±MeM,v)Γ±=(1k2(T±T±M)U,v)Γ±
aM(eM,v)=(1k2(T±T±M)U,v)Γ±.
|((T±T±M)U,v)Γ±|de(Bb)cmin(M|α|)2kj2UHα#12(Γ1,±)vHα#12(Γ±)
1R(k2)(|eM|Hα#1(Ω)2k2eMLα#2(Ω)2)|(1k2T±MeM,eM)Γ±|+|aM(eM,Uψ)|
1R(k2)(eMH2kC1eMH)Cdkref2UψH+dkref2e(Bb)sin(zn/2)(M|α|)2kj2UHα#12(Γ1,±).
eMHckC4d(CUψH+e(Bb)cmin(M|α|)2kj2UHα#12(Γ1,±))
eαMH4ckC4d(CUαψαH+e(Bb)cmin(M|α|)2kj2UαHα#12(Γ1,±)).
eαMLα#2(Ω)2dC4ckC1(CUαψαH+e(Bb)cmin(M|α|)2kj2UαHα#12(Γ1,±)).
eαH4ckC4(3Cd+1)UαψαH+4ckC4de(Bb)cmin(M|α|)2kj2UαH#12(Γ1,±),
eαL#2(Ω)2ckC4C1(3Cd+1)UαψαH+2dckC4C1e(Bb)cmin(M|α|)2kj2UαH#12(Γ1,±),
1R(k2)|ehM|Hα#1(Ω)2ehMLα#2(Ω)2|aM(ehM,ehM)|+|(1k2T±ehM,ehM)Γ±|.
1R(k2)ehMH22k2ehMLα#2(Ω)2|aM(ehM,ehM)|+1kref2|(T±MehM,ehM)|.
ehMH2C1kehMHck(2Cd+1)UMψH,
ehMHckC4(2Cd+1)UMψH.
ehMHckC4(2Cd+1)UψH.
eαhMH4ckC4(2Cd+1)UαψαH.
eαhML#2(Ω)2ckC4C1(2Cd+1)UαψαH.

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