Abstract

We present a new boundary integral formulation for time-harmonic wave diffraction from two-dimensional structures with many layers of arbitrary periodic shape, such as multilayer dielectric gratings in TM polarization. Our scheme is robust at all scattering parameters, unlike the conventional quasi-periodic Green’s function method which fails whenever any of the layers approaches a Wood anomaly. We achieve this by a decomposition into near- and far-field contributions. The former uses the free-space Green’s function in a second-kind integral equation on one period of the material interfaces and their immediate left and right neighbors; the latter uses proxy point sources and small least-squares solves (Schur complements) to represent the remaining contribution from distant copies. By using high-order discretization on interfaces (including those with corners), the number of unknowns per layer is kept small. We achieve overall linear complexity in the number of layers, by direct solution of the resulting block tridiagonal system. For device characterization we present an efficient method to sweep over multiple incident angles, and show a 25× speedup over solving each angle independently. We solve the scattering from a 1000-layer structure with 3 × 105 unknowns to 9-digit accuracy in 2.5 minutes on a desktop workstation.

© 2015 Optical Society of America

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References

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  1. M. D. Perry, R. D. Boyd, J. A. Britten, D. Decker, B. W. Shore, C. Shannon, and E. Shults, “High-efficiency multilayer dielectric diffraction gratings,” Opt. Lett. 20, 940–942 (1995).
    [Crossref] [PubMed]
  2. M. K. C. P. J. Barty, J. Britten, R. Beach, G. Beer, C. Brown, S. Bryan, J. Caird, T. Carlson, J. Crane, J. Dawson, A. Erlandson, D. Fittinghoff, M. Hermann, C. Hoaglan, A. Iyer, L. Ivanovic, I. Jovanovic, A. Komashko, O. Landen, Z. Liao, W. Molander, S. Mitchell, E. Moses, N. Nielsen, H.-H. Nguyen, J. Nissen, S. Payne, D. Pennington, L. Risinger, M. Rushford, K. Skulina, M. Spaeth, B. Stuart, G. Tietbohl, and B. Wattellier, “An overview of LLNL high-energy short-pulse technology for advanced radiography of laser fusion experiments,” Nuclear Fusion 44, S266 (2004).
    [Crossref]
  3. G. A. Kalinchenko and A. M. Lerer, “Wideband all-dielectric diffraction grating on chirped mirror,” J. Lightwave Technology 28, 2743–2749 (2010).
    [Crossref]
  4. H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nature Materials 9, 205–213 (2010).
    [Crossref] [PubMed]
  5. M. D. Kelzenberg, S. W. Boettcher, J. A. Petykiewicz, D. B. Turner-Evans, M. C. Putnam, E. L. Warren, J. M. Spurgeon, R. M. Briggs, N. S. Lewis, and H. A. Atwater, “Enhanced absorption and carrier collection in Si wire arrays for photovoltaic applications,” Nature Materials 9, 239–244 (2010).
    [Crossref] [PubMed]
  6. N. P. Sergeant, M. Agrawal, and P. Peumans, “High performance solar-selective absorbers using coated sub-wavelength gratings,” Opt. Express 18, 5525–5540 (2010).
    [Crossref] [PubMed]
  7. J. D. Joannopoulos, S. G. Johnson, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University, 2008), 2nd ed.
  8. R. Model, A. Rathsfeld, H. Gross, M. Wurm, and B. Bodermann, “A scatterometry inverse problem in optical mask metrology,” J. Phys.: Conf. Ser. 135, 012071 (2008).
  9. D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, vol. 93 of Applied Mathematical Sciences (Springer-Verlag, 1998), 2nd ed.
    [Crossref]
  10. A.-S. Bonnet-BenDhia and F. Starling, “Guided waves by electromagnetic gratings and non-uniqueness examples for the diffraction problem,” Math. Meth. Appl. Sci. 17, 305–338 (1994).
    [Crossref]
  11. J. A. Stratton, Electromagnetic Theory (John Wiley & Sons, 2007).
  12. J. D. Jackson, Classical Electrodynamics (Wiley, 1998), 3rd ed.
  13. R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–408 (1902).
    [Crossref]
  14. C. M. Linton and I. Thompson, “Resonant effects in scattering by periodic arrays,” Wave Motion 44, 165–175 (2007).
    [Crossref]
  15. A. H. Barnett and L. Greengard, “A new integral representation for quasi-periodic scattering problems in two dimensions,” BIT Numer. Math. 51, 67–90 (2011).
    [Crossref]
  16. S. Shipman, Resonant scattering by open periodic waveguides (Bentham Science Publishers, 2010), vol. 1 of Progress in Computational Physics (PiCP), pp. 7–50.
  17. G. L. Wojcik, J. Mould, E. Marx, and M. P. Davidson, “Numerical reference models for optical metrology simulation,” Proc. SPIE 1673, 06 (1992).
  18. A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 1995).
  19. H. Holter and H. Steyskal, “Some experiences from FDTD analysis of infinite and finite multi-octave phased arrays,” IEEE Trans. Antennae Prop. 50, 1725–1731 (2002).
    [Crossref]
  20. G. Bao and D. C. Dobson, “Modeling and optimal design of diffractive optical structures,” Surv. Math. Ind. 8, 37–62 (1998).
  21. J. Elschner, R. Hinder, and G. Schmidt, “Finite element solution of conical diffraction problems,” Adv. Comput. Math. 16, 139–156 (2002).
    [Crossref]
  22. I. M. Babuska and S. A. Sauter, “Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?” SIAM J. Numer. Anal. 34, 2392–2423 (1997).
  23. M. G. Moharam and T. G. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
    [Crossref]
  24. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [Crossref]
  25. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
    [Crossref]
  26. A. Lechleiter and D.-L. Nguyen, “A trigonometric Galerkin method for volume integral equations arising in TM grating scattering,” Adv. Comput. Math. 40, 1–25 (2014).
    [Crossref]
  27. M. H. Cho and W. Cai, “A parallel fast algorithm for computing the Helmholtz integral operator in 3-D layered media,” J. Comput. Phys. 231, 5910–5925 (2012).
    [Crossref]
  28. D. Y. K. Ko and J. R. Sambles, “Scattering matrix method for propagation of radiation in stratified media: attenuated total reflection studies of liquid crystals,” J. Opt. Soc. Am. A 5, 1863–1866 (1988).
    [Crossref]
  29. J. C. Nédélec and F. Starling, “Integral equation methods in a quasi-periodic diffraction problem for the time-harmonic Maxwell’s equations,” SIAM J. Math. Anal. 22, 1679–1701 (1991).
    [Crossref]
  30. A. Meier, T. Arens, S. N. Chandler-Wilde, and A. Kirsch, “A Nyström method for a class of integral equations on the real line with applications to scattering by diffraction gratings and rough surfaces,” J. Integral Equations Appl. 12, 281–321 (2000).
    [Crossref]
  31. R. Kress, “Boundary integral equations in time-harmonic acoustic scattering,” Mathl. Comput. Modelling 15, 229–243 (1991).
    [Crossref]
  32. S. Hao, A. H. Barnett, P. G. Martinsson, and P. Young, “High-order accurate Nyström discretization of integral equations with weakly singular kernels on smooth curves in the plane,” Adv. Comput. Math. 40, 245–272 (2014).
    [Crossref]
  33. T. Arens, “Scattering by biperiodic layered media: The integral equation approach,” Habilitation thesis, Karlsruhe (2010).
  34. M. J. Nicholas, “A higher order numerical method for 3-D doubly periodic electromagnetic scattering problems,” Commun. Math. Sci. 6, 669–694 (2008).
    [Crossref]
  35. Y. Otani and N. Nishimura, “A periodic FMM for Maxwell’s equations in 3D and its applications to problems related to photonic crystals,” J. Comput. Phys. 227, 4630–4652 (2008).
    [Crossref]
  36. O. P. Bruno and M. C. Haslam, “Efficient high-order evaluation of scattering by periodic surfaces: deep gratings, high frequencies, and glancing incidences,” J. Opt. Soc. Am. A 26, 658–668 (2009).
    [Crossref]
  37. A. Gillman and A. Barnett, “A fast direct solver for quasiperiodic scattering problems,” J. Comput. Phys. 248, 309–322 (2013).
    [Crossref]
  38. L. Greengard, K. L. Ho, and J.-Y. Lee, “A fast direct solver for scattering from periodic structures with multiple material interfaces in two dimensions,” J. Comput. Phys. 258, 738–751 (2014).
    [Crossref]
  39. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1964), 10th ed.
  40. C. M. Linton, “Lattice sums for the Helmholtz equation,” SIAM Review 52, 603–674 (2010).
    [Crossref]
  41. K. V. Horoshenkov and S. N. Chandler-Wilde, “Efficient calculation of two-dimensional periodic and waveguide acoustic Green’s functions,” J. Acoust. Soc. Amer. 111, 1610–1622 (2002).
    [Crossref]
  42. O. P. Bruno, S. Shipman, C. Turc, and S. Venakides, “Efficient evaluation of doubly periodic Green functions in 3D scattering, including Wood anomaly frequencies,” (2013). Preprint, arXiv:1307.1176v1.
  43. O. P. Bruno and B. Delourme, “Rapidly convergent two-dimensional quasi-periodic Gteen function throughout the spectrum-including Wold anomalies,” J. Comput. Phys. 262, 262–290 (2014).
    [Crossref]
  44. A. H. Barnett and L. Greengard, “A new integral representation for quasi-periodic fields and its application to two-dimensional band structure calculations,” J. Comput. Phys. 229, 6898–6914 (2010).
    [Crossref]
  45. A. Bogomolny, “Fundamental solutions method for elliptic boundary value problems,” SIAM Journal on Numerical Analysis 22, 644–669 (1985).
    [Crossref]
  46. A. H. Barnett and T. Betcke, “Stability and convergence of the Method of Fundamental Solutions for Helmholtz problems on analytic domains,” J. Comput. Phys. 227, 7003–7026 (2008).
    [Crossref]
  47. P. Martinsson and V. Rokhlin, “A fast direct solver for boundary integral equations in two dimensions,” J. Comp. Phys. 205, 1–23 (2005).
    [Crossref]
  48. N. A. Gumerov and R. Duraiswami, “A method to compute periodic sums,” J. Comput. Phys. 272, 307–326 (2014).
    [Crossref]
  49. C. Hafner, The Generalized Multipole Technique for Computational Electromagnetics (Artech House Books, 1990).
  50. L. Zhao and A. H. Barnett, “Robust and efficient solution of the drum problem via Nyström approximation of the Fredholm determinant,” (2014). arxiv:1406.5252, submitted to J. Comput. Phys.
  51. C. Müller, Foundations of the Mathematical Theory of Electromagnetic Waves (Springer-Verlag, 1969).
    [Crossref]
  52. V. Rokhlin, “Solution of acoustic scattering problems by means of second kind integral equations,” Wave Motion 5, 257–272 (1983).
    [Crossref]
  53. P. J. Davis and P. Rabinowitz, Methods of Numerical Integration (Academic, 1984).
  54. R. Kress, Linear Integral Equations, vol. 82 of Appl. Math. Sci. (Springer, 1999), 2nd ed.
    [Crossref]
  55. B. K. Alpert, “Hybrid Gauss-trapezoidal quadrature rules,” SIAM J. Sci. Comput. 20, 1551–1584 (1999).
    [Crossref]
  56. J. Lai, M. Kobayashi, and A. H. Barnett, “A fast solver for the scattering from a layered periodic structure with multi-particle inclusions,” (2014). In preparation.
  57. G. H. Golub and C. F. Van Loan, Matrix Computations, Johns Hopkins Studies in the Mathematical Sciences (Johns Hopkins University, 1996), 3rd ed.
  58. W. Y. Crutchfield, Z. Gimbutas, Gol, J. Huang, V. Rokhlin, N. Yarvin, and J. Zhao, Remarks on the implementation of the wideband FMM for the Helmholtz equation in two dimensions (Amer. Math. Soc., 2006), vol. 408 of Contemp. Math., pp. 99–110.
  59. A. H. Barnett and T. Betcke, “MPSpack: A MATLAB toolbox to solve Helmholtz PDE, wave scattering, and eigenvalue problems,” (2008–2014). http://code.google.com/p/mpspack .

2014 (5)

A. Lechleiter and D.-L. Nguyen, “A trigonometric Galerkin method for volume integral equations arising in TM grating scattering,” Adv. Comput. Math. 40, 1–25 (2014).
[Crossref]

S. Hao, A. H. Barnett, P. G. Martinsson, and P. Young, “High-order accurate Nyström discretization of integral equations with weakly singular kernels on smooth curves in the plane,” Adv. Comput. Math. 40, 245–272 (2014).
[Crossref]

L. Greengard, K. L. Ho, and J.-Y. Lee, “A fast direct solver for scattering from periodic structures with multiple material interfaces in two dimensions,” J. Comput. Phys. 258, 738–751 (2014).
[Crossref]

O. P. Bruno and B. Delourme, “Rapidly convergent two-dimensional quasi-periodic Gteen function throughout the spectrum-including Wold anomalies,” J. Comput. Phys. 262, 262–290 (2014).
[Crossref]

N. A. Gumerov and R. Duraiswami, “A method to compute periodic sums,” J. Comput. Phys. 272, 307–326 (2014).
[Crossref]

2013 (1)

A. Gillman and A. Barnett, “A fast direct solver for quasiperiodic scattering problems,” J. Comput. Phys. 248, 309–322 (2013).
[Crossref]

2012 (1)

M. H. Cho and W. Cai, “A parallel fast algorithm for computing the Helmholtz integral operator in 3-D layered media,” J. Comput. Phys. 231, 5910–5925 (2012).
[Crossref]

2011 (1)

A. H. Barnett and L. Greengard, “A new integral representation for quasi-periodic scattering problems in two dimensions,” BIT Numer. Math. 51, 67–90 (2011).
[Crossref]

2010 (6)

G. A. Kalinchenko and A. M. Lerer, “Wideband all-dielectric diffraction grating on chirped mirror,” J. Lightwave Technology 28, 2743–2749 (2010).
[Crossref]

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nature Materials 9, 205–213 (2010).
[Crossref] [PubMed]

M. D. Kelzenberg, S. W. Boettcher, J. A. Petykiewicz, D. B. Turner-Evans, M. C. Putnam, E. L. Warren, J. M. Spurgeon, R. M. Briggs, N. S. Lewis, and H. A. Atwater, “Enhanced absorption and carrier collection in Si wire arrays for photovoltaic applications,” Nature Materials 9, 239–244 (2010).
[Crossref] [PubMed]

N. P. Sergeant, M. Agrawal, and P. Peumans, “High performance solar-selective absorbers using coated sub-wavelength gratings,” Opt. Express 18, 5525–5540 (2010).
[Crossref] [PubMed]

A. H. Barnett and L. Greengard, “A new integral representation for quasi-periodic fields and its application to two-dimensional band structure calculations,” J. Comput. Phys. 229, 6898–6914 (2010).
[Crossref]

C. M. Linton, “Lattice sums for the Helmholtz equation,” SIAM Review 52, 603–674 (2010).
[Crossref]

2009 (1)

2008 (4)

M. J. Nicholas, “A higher order numerical method for 3-D doubly periodic electromagnetic scattering problems,” Commun. Math. Sci. 6, 669–694 (2008).
[Crossref]

Y. Otani and N. Nishimura, “A periodic FMM for Maxwell’s equations in 3D and its applications to problems related to photonic crystals,” J. Comput. Phys. 227, 4630–4652 (2008).
[Crossref]

A. H. Barnett and T. Betcke, “Stability and convergence of the Method of Fundamental Solutions for Helmholtz problems on analytic domains,” J. Comput. Phys. 227, 7003–7026 (2008).
[Crossref]

R. Model, A. Rathsfeld, H. Gross, M. Wurm, and B. Bodermann, “A scatterometry inverse problem in optical mask metrology,” J. Phys.: Conf. Ser. 135, 012071 (2008).

2007 (1)

C. M. Linton and I. Thompson, “Resonant effects in scattering by periodic arrays,” Wave Motion 44, 165–175 (2007).
[Crossref]

2005 (1)

P. Martinsson and V. Rokhlin, “A fast direct solver for boundary integral equations in two dimensions,” J. Comp. Phys. 205, 1–23 (2005).
[Crossref]

2004 (1)

M. K. C. P. J. Barty, J. Britten, R. Beach, G. Beer, C. Brown, S. Bryan, J. Caird, T. Carlson, J. Crane, J. Dawson, A. Erlandson, D. Fittinghoff, M. Hermann, C. Hoaglan, A. Iyer, L. Ivanovic, I. Jovanovic, A. Komashko, O. Landen, Z. Liao, W. Molander, S. Mitchell, E. Moses, N. Nielsen, H.-H. Nguyen, J. Nissen, S. Payne, D. Pennington, L. Risinger, M. Rushford, K. Skulina, M. Spaeth, B. Stuart, G. Tietbohl, and B. Wattellier, “An overview of LLNL high-energy short-pulse technology for advanced radiography of laser fusion experiments,” Nuclear Fusion 44, S266 (2004).
[Crossref]

2002 (3)

H. Holter and H. Steyskal, “Some experiences from FDTD analysis of infinite and finite multi-octave phased arrays,” IEEE Trans. Antennae Prop. 50, 1725–1731 (2002).
[Crossref]

J. Elschner, R. Hinder, and G. Schmidt, “Finite element solution of conical diffraction problems,” Adv. Comput. Math. 16, 139–156 (2002).
[Crossref]

K. V. Horoshenkov and S. N. Chandler-Wilde, “Efficient calculation of two-dimensional periodic and waveguide acoustic Green’s functions,” J. Acoust. Soc. Amer. 111, 1610–1622 (2002).
[Crossref]

2000 (1)

A. Meier, T. Arens, S. N. Chandler-Wilde, and A. Kirsch, “A Nyström method for a class of integral equations on the real line with applications to scattering by diffraction gratings and rough surfaces,” J. Integral Equations Appl. 12, 281–321 (2000).
[Crossref]

1999 (1)

B. K. Alpert, “Hybrid Gauss-trapezoidal quadrature rules,” SIAM J. Sci. Comput. 20, 1551–1584 (1999).
[Crossref]

1998 (1)

G. Bao and D. C. Dobson, “Modeling and optimal design of diffractive optical structures,” Surv. Math. Ind. 8, 37–62 (1998).

1997 (1)

I. M. Babuska and S. A. Sauter, “Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?” SIAM J. Numer. Anal. 34, 2392–2423 (1997).

1996 (2)

1995 (1)

1994 (1)

A.-S. Bonnet-BenDhia and F. Starling, “Guided waves by electromagnetic gratings and non-uniqueness examples for the diffraction problem,” Math. Meth. Appl. Sci. 17, 305–338 (1994).
[Crossref]

1992 (1)

G. L. Wojcik, J. Mould, E. Marx, and M. P. Davidson, “Numerical reference models for optical metrology simulation,” Proc. SPIE 1673, 06 (1992).

1991 (2)

R. Kress, “Boundary integral equations in time-harmonic acoustic scattering,” Mathl. Comput. Modelling 15, 229–243 (1991).
[Crossref]

J. C. Nédélec and F. Starling, “Integral equation methods in a quasi-periodic diffraction problem for the time-harmonic Maxwell’s equations,” SIAM J. Math. Anal. 22, 1679–1701 (1991).
[Crossref]

1988 (1)

1985 (1)

A. Bogomolny, “Fundamental solutions method for elliptic boundary value problems,” SIAM Journal on Numerical Analysis 22, 644–669 (1985).
[Crossref]

1983 (1)

V. Rokhlin, “Solution of acoustic scattering problems by means of second kind integral equations,” Wave Motion 5, 257–272 (1983).
[Crossref]

1981 (1)

1902 (1)

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–408 (1902).
[Crossref]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1964), 10th ed.

Agrawal, M.

Alpert, B. K.

B. K. Alpert, “Hybrid Gauss-trapezoidal quadrature rules,” SIAM J. Sci. Comput. 20, 1551–1584 (1999).
[Crossref]

Arens, T.

A. Meier, T. Arens, S. N. Chandler-Wilde, and A. Kirsch, “A Nyström method for a class of integral equations on the real line with applications to scattering by diffraction gratings and rough surfaces,” J. Integral Equations Appl. 12, 281–321 (2000).
[Crossref]

T. Arens, “Scattering by biperiodic layered media: The integral equation approach,” Habilitation thesis, Karlsruhe (2010).

Atwater, H. A.

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nature Materials 9, 205–213 (2010).
[Crossref] [PubMed]

M. D. Kelzenberg, S. W. Boettcher, J. A. Petykiewicz, D. B. Turner-Evans, M. C. Putnam, E. L. Warren, J. M. Spurgeon, R. M. Briggs, N. S. Lewis, and H. A. Atwater, “Enhanced absorption and carrier collection in Si wire arrays for photovoltaic applications,” Nature Materials 9, 239–244 (2010).
[Crossref] [PubMed]

Babuska, I. M.

I. M. Babuska and S. A. Sauter, “Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?” SIAM J. Numer. Anal. 34, 2392–2423 (1997).

Bao, G.

G. Bao and D. C. Dobson, “Modeling and optimal design of diffractive optical structures,” Surv. Math. Ind. 8, 37–62 (1998).

Barnett, A.

A. Gillman and A. Barnett, “A fast direct solver for quasiperiodic scattering problems,” J. Comput. Phys. 248, 309–322 (2013).
[Crossref]

Barnett, A. H.

S. Hao, A. H. Barnett, P. G. Martinsson, and P. Young, “High-order accurate Nyström discretization of integral equations with weakly singular kernels on smooth curves in the plane,” Adv. Comput. Math. 40, 245–272 (2014).
[Crossref]

A. H. Barnett and L. Greengard, “A new integral representation for quasi-periodic scattering problems in two dimensions,” BIT Numer. Math. 51, 67–90 (2011).
[Crossref]

A. H. Barnett and L. Greengard, “A new integral representation for quasi-periodic fields and its application to two-dimensional band structure calculations,” J. Comput. Phys. 229, 6898–6914 (2010).
[Crossref]

A. H. Barnett and T. Betcke, “Stability and convergence of the Method of Fundamental Solutions for Helmholtz problems on analytic domains,” J. Comput. Phys. 227, 7003–7026 (2008).
[Crossref]

L. Zhao and A. H. Barnett, “Robust and efficient solution of the drum problem via Nyström approximation of the Fredholm determinant,” (2014). arxiv:1406.5252, submitted to J. Comput. Phys.

J. Lai, M. Kobayashi, and A. H. Barnett, “A fast solver for the scattering from a layered periodic structure with multi-particle inclusions,” (2014). In preparation.

Barty, M. K. C. P. J.

M. K. C. P. J. Barty, J. Britten, R. Beach, G. Beer, C. Brown, S. Bryan, J. Caird, T. Carlson, J. Crane, J. Dawson, A. Erlandson, D. Fittinghoff, M. Hermann, C. Hoaglan, A. Iyer, L. Ivanovic, I. Jovanovic, A. Komashko, O. Landen, Z. Liao, W. Molander, S. Mitchell, E. Moses, N. Nielsen, H.-H. Nguyen, J. Nissen, S. Payne, D. Pennington, L. Risinger, M. Rushford, K. Skulina, M. Spaeth, B. Stuart, G. Tietbohl, and B. Wattellier, “An overview of LLNL high-energy short-pulse technology for advanced radiography of laser fusion experiments,” Nuclear Fusion 44, S266 (2004).
[Crossref]

Beach, R.

M. K. C. P. J. Barty, J. Britten, R. Beach, G. Beer, C. Brown, S. Bryan, J. Caird, T. Carlson, J. Crane, J. Dawson, A. Erlandson, D. Fittinghoff, M. Hermann, C. Hoaglan, A. Iyer, L. Ivanovic, I. Jovanovic, A. Komashko, O. Landen, Z. Liao, W. Molander, S. Mitchell, E. Moses, N. Nielsen, H.-H. Nguyen, J. Nissen, S. Payne, D. Pennington, L. Risinger, M. Rushford, K. Skulina, M. Spaeth, B. Stuart, G. Tietbohl, and B. Wattellier, “An overview of LLNL high-energy short-pulse technology for advanced radiography of laser fusion experiments,” Nuclear Fusion 44, S266 (2004).
[Crossref]

Beer, G.

M. K. C. P. J. Barty, J. Britten, R. Beach, G. Beer, C. Brown, S. Bryan, J. Caird, T. Carlson, J. Crane, J. Dawson, A. Erlandson, D. Fittinghoff, M. Hermann, C. Hoaglan, A. Iyer, L. Ivanovic, I. Jovanovic, A. Komashko, O. Landen, Z. Liao, W. Molander, S. Mitchell, E. Moses, N. Nielsen, H.-H. Nguyen, J. Nissen, S. Payne, D. Pennington, L. Risinger, M. Rushford, K. Skulina, M. Spaeth, B. Stuart, G. Tietbohl, and B. Wattellier, “An overview of LLNL high-energy short-pulse technology for advanced radiography of laser fusion experiments,” Nuclear Fusion 44, S266 (2004).
[Crossref]

Betcke, T.

A. H. Barnett and T. Betcke, “Stability and convergence of the Method of Fundamental Solutions for Helmholtz problems on analytic domains,” J. Comput. Phys. 227, 7003–7026 (2008).
[Crossref]

Bodermann, B.

R. Model, A. Rathsfeld, H. Gross, M. Wurm, and B. Bodermann, “A scatterometry inverse problem in optical mask metrology,” J. Phys.: Conf. Ser. 135, 012071 (2008).

Boettcher, S. W.

M. D. Kelzenberg, S. W. Boettcher, J. A. Petykiewicz, D. B. Turner-Evans, M. C. Putnam, E. L. Warren, J. M. Spurgeon, R. M. Briggs, N. S. Lewis, and H. A. Atwater, “Enhanced absorption and carrier collection in Si wire arrays for photovoltaic applications,” Nature Materials 9, 239–244 (2010).
[Crossref] [PubMed]

Bogomolny, A.

A. Bogomolny, “Fundamental solutions method for elliptic boundary value problems,” SIAM Journal on Numerical Analysis 22, 644–669 (1985).
[Crossref]

Bonnet-BenDhia, A.-S.

A.-S. Bonnet-BenDhia and F. Starling, “Guided waves by electromagnetic gratings and non-uniqueness examples for the diffraction problem,” Math. Meth. Appl. Sci. 17, 305–338 (1994).
[Crossref]

Boyd, R. D.

Briggs, R. M.

M. D. Kelzenberg, S. W. Boettcher, J. A. Petykiewicz, D. B. Turner-Evans, M. C. Putnam, E. L. Warren, J. M. Spurgeon, R. M. Briggs, N. S. Lewis, and H. A. Atwater, “Enhanced absorption and carrier collection in Si wire arrays for photovoltaic applications,” Nature Materials 9, 239–244 (2010).
[Crossref] [PubMed]

Britten, J.

M. K. C. P. J. Barty, J. Britten, R. Beach, G. Beer, C. Brown, S. Bryan, J. Caird, T. Carlson, J. Crane, J. Dawson, A. Erlandson, D. Fittinghoff, M. Hermann, C. Hoaglan, A. Iyer, L. Ivanovic, I. Jovanovic, A. Komashko, O. Landen, Z. Liao, W. Molander, S. Mitchell, E. Moses, N. Nielsen, H.-H. Nguyen, J. Nissen, S. Payne, D. Pennington, L. Risinger, M. Rushford, K. Skulina, M. Spaeth, B. Stuart, G. Tietbohl, and B. Wattellier, “An overview of LLNL high-energy short-pulse technology for advanced radiography of laser fusion experiments,” Nuclear Fusion 44, S266 (2004).
[Crossref]

Britten, J. A.

Brown, C.

M. K. C. P. J. Barty, J. Britten, R. Beach, G. Beer, C. Brown, S. Bryan, J. Caird, T. Carlson, J. Crane, J. Dawson, A. Erlandson, D. Fittinghoff, M. Hermann, C. Hoaglan, A. Iyer, L. Ivanovic, I. Jovanovic, A. Komashko, O. Landen, Z. Liao, W. Molander, S. Mitchell, E. Moses, N. Nielsen, H.-H. Nguyen, J. Nissen, S. Payne, D. Pennington, L. Risinger, M. Rushford, K. Skulina, M. Spaeth, B. Stuart, G. Tietbohl, and B. Wattellier, “An overview of LLNL high-energy short-pulse technology for advanced radiography of laser fusion experiments,” Nuclear Fusion 44, S266 (2004).
[Crossref]

Bruno, O. P.

O. P. Bruno and B. Delourme, “Rapidly convergent two-dimensional quasi-periodic Gteen function throughout the spectrum-including Wold anomalies,” J. Comput. Phys. 262, 262–290 (2014).
[Crossref]

O. P. Bruno and M. C. Haslam, “Efficient high-order evaluation of scattering by periodic surfaces: deep gratings, high frequencies, and glancing incidences,” J. Opt. Soc. Am. A 26, 658–668 (2009).
[Crossref]

O. P. Bruno, S. Shipman, C. Turc, and S. Venakides, “Efficient evaluation of doubly periodic Green functions in 3D scattering, including Wood anomaly frequencies,” (2013). Preprint, arXiv:1307.1176v1.

Bryan, S.

M. K. C. P. J. Barty, J. Britten, R. Beach, G. Beer, C. Brown, S. Bryan, J. Caird, T. Carlson, J. Crane, J. Dawson, A. Erlandson, D. Fittinghoff, M. Hermann, C. Hoaglan, A. Iyer, L. Ivanovic, I. Jovanovic, A. Komashko, O. Landen, Z. Liao, W. Molander, S. Mitchell, E. Moses, N. Nielsen, H.-H. Nguyen, J. Nissen, S. Payne, D. Pennington, L. Risinger, M. Rushford, K. Skulina, M. Spaeth, B. Stuart, G. Tietbohl, and B. Wattellier, “An overview of LLNL high-energy short-pulse technology for advanced radiography of laser fusion experiments,” Nuclear Fusion 44, S266 (2004).
[Crossref]

Cai, W.

M. H. Cho and W. Cai, “A parallel fast algorithm for computing the Helmholtz integral operator in 3-D layered media,” J. Comput. Phys. 231, 5910–5925 (2012).
[Crossref]

Caird, J.

M. K. C. P. J. Barty, J. Britten, R. Beach, G. Beer, C. Brown, S. Bryan, J. Caird, T. Carlson, J. Crane, J. Dawson, A. Erlandson, D. Fittinghoff, M. Hermann, C. Hoaglan, A. Iyer, L. Ivanovic, I. Jovanovic, A. Komashko, O. Landen, Z. Liao, W. Molander, S. Mitchell, E. Moses, N. Nielsen, H.-H. Nguyen, J. Nissen, S. Payne, D. Pennington, L. Risinger, M. Rushford, K. Skulina, M. Spaeth, B. Stuart, G. Tietbohl, and B. Wattellier, “An overview of LLNL high-energy short-pulse technology for advanced radiography of laser fusion experiments,” Nuclear Fusion 44, S266 (2004).
[Crossref]

Carlson, T.

M. K. C. P. J. Barty, J. Britten, R. Beach, G. Beer, C. Brown, S. Bryan, J. Caird, T. Carlson, J. Crane, J. Dawson, A. Erlandson, D. Fittinghoff, M. Hermann, C. Hoaglan, A. Iyer, L. Ivanovic, I. Jovanovic, A. Komashko, O. Landen, Z. Liao, W. Molander, S. Mitchell, E. Moses, N. Nielsen, H.-H. Nguyen, J. Nissen, S. Payne, D. Pennington, L. Risinger, M. Rushford, K. Skulina, M. Spaeth, B. Stuart, G. Tietbohl, and B. Wattellier, “An overview of LLNL high-energy short-pulse technology for advanced radiography of laser fusion experiments,” Nuclear Fusion 44, S266 (2004).
[Crossref]

Chandler-Wilde, S. N.

K. V. Horoshenkov and S. N. Chandler-Wilde, “Efficient calculation of two-dimensional periodic and waveguide acoustic Green’s functions,” J. Acoust. Soc. Amer. 111, 1610–1622 (2002).
[Crossref]

A. Meier, T. Arens, S. N. Chandler-Wilde, and A. Kirsch, “A Nyström method for a class of integral equations on the real line with applications to scattering by diffraction gratings and rough surfaces,” J. Integral Equations Appl. 12, 281–321 (2000).
[Crossref]

Cho, M. H.

M. H. Cho and W. Cai, “A parallel fast algorithm for computing the Helmholtz integral operator in 3-D layered media,” J. Comput. Phys. 231, 5910–5925 (2012).
[Crossref]

Colton, D.

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, vol. 93 of Applied Mathematical Sciences (Springer-Verlag, 1998), 2nd ed.
[Crossref]

Crane, J.

M. K. C. P. J. Barty, J. Britten, R. Beach, G. Beer, C. Brown, S. Bryan, J. Caird, T. Carlson, J. Crane, J. Dawson, A. Erlandson, D. Fittinghoff, M. Hermann, C. Hoaglan, A. Iyer, L. Ivanovic, I. Jovanovic, A. Komashko, O. Landen, Z. Liao, W. Molander, S. Mitchell, E. Moses, N. Nielsen, H.-H. Nguyen, J. Nissen, S. Payne, D. Pennington, L. Risinger, M. Rushford, K. Skulina, M. Spaeth, B. Stuart, G. Tietbohl, and B. Wattellier, “An overview of LLNL high-energy short-pulse technology for advanced radiography of laser fusion experiments,” Nuclear Fusion 44, S266 (2004).
[Crossref]

Crutchfield, W. Y.

W. Y. Crutchfield, Z. Gimbutas, Gol, J. Huang, V. Rokhlin, N. Yarvin, and J. Zhao, Remarks on the implementation of the wideband FMM for the Helmholtz equation in two dimensions (Amer. Math. Soc., 2006), vol. 408 of Contemp. Math., pp. 99–110.

Davidson, M. P.

G. L. Wojcik, J. Mould, E. Marx, and M. P. Davidson, “Numerical reference models for optical metrology simulation,” Proc. SPIE 1673, 06 (1992).

Davis, P. J.

P. J. Davis and P. Rabinowitz, Methods of Numerical Integration (Academic, 1984).

Dawson, J.

M. K. C. P. J. Barty, J. Britten, R. Beach, G. Beer, C. Brown, S. Bryan, J. Caird, T. Carlson, J. Crane, J. Dawson, A. Erlandson, D. Fittinghoff, M. Hermann, C. Hoaglan, A. Iyer, L. Ivanovic, I. Jovanovic, A. Komashko, O. Landen, Z. Liao, W. Molander, S. Mitchell, E. Moses, N. Nielsen, H.-H. Nguyen, J. Nissen, S. Payne, D. Pennington, L. Risinger, M. Rushford, K. Skulina, M. Spaeth, B. Stuart, G. Tietbohl, and B. Wattellier, “An overview of LLNL high-energy short-pulse technology for advanced radiography of laser fusion experiments,” Nuclear Fusion 44, S266 (2004).
[Crossref]

Decker, D.

Delourme, B.

O. P. Bruno and B. Delourme, “Rapidly convergent two-dimensional quasi-periodic Gteen function throughout the spectrum-including Wold anomalies,” J. Comput. Phys. 262, 262–290 (2014).
[Crossref]

Dobson, D. C.

G. Bao and D. C. Dobson, “Modeling and optimal design of diffractive optical structures,” Surv. Math. Ind. 8, 37–62 (1998).

Duraiswami, R.

N. A. Gumerov and R. Duraiswami, “A method to compute periodic sums,” J. Comput. Phys. 272, 307–326 (2014).
[Crossref]

Elschner, J.

J. Elschner, R. Hinder, and G. Schmidt, “Finite element solution of conical diffraction problems,” Adv. Comput. Math. 16, 139–156 (2002).
[Crossref]

Erlandson, A.

M. K. C. P. J. Barty, J. Britten, R. Beach, G. Beer, C. Brown, S. Bryan, J. Caird, T. Carlson, J. Crane, J. Dawson, A. Erlandson, D. Fittinghoff, M. Hermann, C. Hoaglan, A. Iyer, L. Ivanovic, I. Jovanovic, A. Komashko, O. Landen, Z. Liao, W. Molander, S. Mitchell, E. Moses, N. Nielsen, H.-H. Nguyen, J. Nissen, S. Payne, D. Pennington, L. Risinger, M. Rushford, K. Skulina, M. Spaeth, B. Stuart, G. Tietbohl, and B. Wattellier, “An overview of LLNL high-energy short-pulse technology for advanced radiography of laser fusion experiments,” Nuclear Fusion 44, S266 (2004).
[Crossref]

Fittinghoff, D.

M. K. C. P. J. Barty, J. Britten, R. Beach, G. Beer, C. Brown, S. Bryan, J. Caird, T. Carlson, J. Crane, J. Dawson, A. Erlandson, D. Fittinghoff, M. Hermann, C. Hoaglan, A. Iyer, L. Ivanovic, I. Jovanovic, A. Komashko, O. Landen, Z. Liao, W. Molander, S. Mitchell, E. Moses, N. Nielsen, H.-H. Nguyen, J. Nissen, S. Payne, D. Pennington, L. Risinger, M. Rushford, K. Skulina, M. Spaeth, B. Stuart, G. Tietbohl, and B. Wattellier, “An overview of LLNL high-energy short-pulse technology for advanced radiography of laser fusion experiments,” Nuclear Fusion 44, S266 (2004).
[Crossref]

Gaylord, T. G.

Gillman, A.

A. Gillman and A. Barnett, “A fast direct solver for quasiperiodic scattering problems,” J. Comput. Phys. 248, 309–322 (2013).
[Crossref]

Gimbutas, Z.

W. Y. Crutchfield, Z. Gimbutas, Gol, J. Huang, V. Rokhlin, N. Yarvin, and J. Zhao, Remarks on the implementation of the wideband FMM for the Helmholtz equation in two dimensions (Amer. Math. Soc., 2006), vol. 408 of Contemp. Math., pp. 99–110.

Gol,

W. Y. Crutchfield, Z. Gimbutas, Gol, J. Huang, V. Rokhlin, N. Yarvin, and J. Zhao, Remarks on the implementation of the wideband FMM for the Helmholtz equation in two dimensions (Amer. Math. Soc., 2006), vol. 408 of Contemp. Math., pp. 99–110.

Golub, G. H.

G. H. Golub and C. F. Van Loan, Matrix Computations, Johns Hopkins Studies in the Mathematical Sciences (Johns Hopkins University, 1996), 3rd ed.

Greengard, L.

L. Greengard, K. L. Ho, and J.-Y. Lee, “A fast direct solver for scattering from periodic structures with multiple material interfaces in two dimensions,” J. Comput. Phys. 258, 738–751 (2014).
[Crossref]

A. H. Barnett and L. Greengard, “A new integral representation for quasi-periodic scattering problems in two dimensions,” BIT Numer. Math. 51, 67–90 (2011).
[Crossref]

A. H. Barnett and L. Greengard, “A new integral representation for quasi-periodic fields and its application to two-dimensional band structure calculations,” J. Comput. Phys. 229, 6898–6914 (2010).
[Crossref]

Gross, H.

R. Model, A. Rathsfeld, H. Gross, M. Wurm, and B. Bodermann, “A scatterometry inverse problem in optical mask metrology,” J. Phys.: Conf. Ser. 135, 012071 (2008).

Gumerov, N. A.

N. A. Gumerov and R. Duraiswami, “A method to compute periodic sums,” J. Comput. Phys. 272, 307–326 (2014).
[Crossref]

Hafner, C.

C. Hafner, The Generalized Multipole Technique for Computational Electromagnetics (Artech House Books, 1990).

Hao, S.

S. Hao, A. H. Barnett, P. G. Martinsson, and P. Young, “High-order accurate Nyström discretization of integral equations with weakly singular kernels on smooth curves in the plane,” Adv. Comput. Math. 40, 245–272 (2014).
[Crossref]

Haslam, M. C.

Hermann, M.

M. K. C. P. J. Barty, J. Britten, R. Beach, G. Beer, C. Brown, S. Bryan, J. Caird, T. Carlson, J. Crane, J. Dawson, A. Erlandson, D. Fittinghoff, M. Hermann, C. Hoaglan, A. Iyer, L. Ivanovic, I. Jovanovic, A. Komashko, O. Landen, Z. Liao, W. Molander, S. Mitchell, E. Moses, N. Nielsen, H.-H. Nguyen, J. Nissen, S. Payne, D. Pennington, L. Risinger, M. Rushford, K. Skulina, M. Spaeth, B. Stuart, G. Tietbohl, and B. Wattellier, “An overview of LLNL high-energy short-pulse technology for advanced radiography of laser fusion experiments,” Nuclear Fusion 44, S266 (2004).
[Crossref]

Hinder, R.

J. Elschner, R. Hinder, and G. Schmidt, “Finite element solution of conical diffraction problems,” Adv. Comput. Math. 16, 139–156 (2002).
[Crossref]

Ho, K. L.

L. Greengard, K. L. Ho, and J.-Y. Lee, “A fast direct solver for scattering from periodic structures with multiple material interfaces in two dimensions,” J. Comput. Phys. 258, 738–751 (2014).
[Crossref]

Hoaglan, C.

M. K. C. P. J. Barty, J. Britten, R. Beach, G. Beer, C. Brown, S. Bryan, J. Caird, T. Carlson, J. Crane, J. Dawson, A. Erlandson, D. Fittinghoff, M. Hermann, C. Hoaglan, A. Iyer, L. Ivanovic, I. Jovanovic, A. Komashko, O. Landen, Z. Liao, W. Molander, S. Mitchell, E. Moses, N. Nielsen, H.-H. Nguyen, J. Nissen, S. Payne, D. Pennington, L. Risinger, M. Rushford, K. Skulina, M. Spaeth, B. Stuart, G. Tietbohl, and B. Wattellier, “An overview of LLNL high-energy short-pulse technology for advanced radiography of laser fusion experiments,” Nuclear Fusion 44, S266 (2004).
[Crossref]

Holter, H.

H. Holter and H. Steyskal, “Some experiences from FDTD analysis of infinite and finite multi-octave phased arrays,” IEEE Trans. Antennae Prop. 50, 1725–1731 (2002).
[Crossref]

Horoshenkov, K. V.

K. V. Horoshenkov and S. N. Chandler-Wilde, “Efficient calculation of two-dimensional periodic and waveguide acoustic Green’s functions,” J. Acoust. Soc. Amer. 111, 1610–1622 (2002).
[Crossref]

Huang, J.

W. Y. Crutchfield, Z. Gimbutas, Gol, J. Huang, V. Rokhlin, N. Yarvin, and J. Zhao, Remarks on the implementation of the wideband FMM for the Helmholtz equation in two dimensions (Amer. Math. Soc., 2006), vol. 408 of Contemp. Math., pp. 99–110.

Ivanovic, L.

M. K. C. P. J. Barty, J. Britten, R. Beach, G. Beer, C. Brown, S. Bryan, J. Caird, T. Carlson, J. Crane, J. Dawson, A. Erlandson, D. Fittinghoff, M. Hermann, C. Hoaglan, A. Iyer, L. Ivanovic, I. Jovanovic, A. Komashko, O. Landen, Z. Liao, W. Molander, S. Mitchell, E. Moses, N. Nielsen, H.-H. Nguyen, J. Nissen, S. Payne, D. Pennington, L. Risinger, M. Rushford, K. Skulina, M. Spaeth, B. Stuart, G. Tietbohl, and B. Wattellier, “An overview of LLNL high-energy short-pulse technology for advanced radiography of laser fusion experiments,” Nuclear Fusion 44, S266 (2004).
[Crossref]

Iyer, A.

M. K. C. P. J. Barty, J. Britten, R. Beach, G. Beer, C. Brown, S. Bryan, J. Caird, T. Carlson, J. Crane, J. Dawson, A. Erlandson, D. Fittinghoff, M. Hermann, C. Hoaglan, A. Iyer, L. Ivanovic, I. Jovanovic, A. Komashko, O. Landen, Z. Liao, W. Molander, S. Mitchell, E. Moses, N. Nielsen, H.-H. Nguyen, J. Nissen, S. Payne, D. Pennington, L. Risinger, M. Rushford, K. Skulina, M. Spaeth, B. Stuart, G. Tietbohl, and B. Wattellier, “An overview of LLNL high-energy short-pulse technology for advanced radiography of laser fusion experiments,” Nuclear Fusion 44, S266 (2004).
[Crossref]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1998), 3rd ed.

Joannopoulos, J. D.

J. D. Joannopoulos, S. G. Johnson, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University, 2008), 2nd ed.

Johnson, S. G.

J. D. Joannopoulos, S. G. Johnson, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University, 2008), 2nd ed.

Jovanovic, I.

M. K. C. P. J. Barty, J. Britten, R. Beach, G. Beer, C. Brown, S. Bryan, J. Caird, T. Carlson, J. Crane, J. Dawson, A. Erlandson, D. Fittinghoff, M. Hermann, C. Hoaglan, A. Iyer, L. Ivanovic, I. Jovanovic, A. Komashko, O. Landen, Z. Liao, W. Molander, S. Mitchell, E. Moses, N. Nielsen, H.-H. Nguyen, J. Nissen, S. Payne, D. Pennington, L. Risinger, M. Rushford, K. Skulina, M. Spaeth, B. Stuart, G. Tietbohl, and B. Wattellier, “An overview of LLNL high-energy short-pulse technology for advanced radiography of laser fusion experiments,” Nuclear Fusion 44, S266 (2004).
[Crossref]

Kalinchenko, G. A.

G. A. Kalinchenko and A. M. Lerer, “Wideband all-dielectric diffraction grating on chirped mirror,” J. Lightwave Technology 28, 2743–2749 (2010).
[Crossref]

Kelzenberg, M. D.

M. D. Kelzenberg, S. W. Boettcher, J. A. Petykiewicz, D. B. Turner-Evans, M. C. Putnam, E. L. Warren, J. M. Spurgeon, R. M. Briggs, N. S. Lewis, and H. A. Atwater, “Enhanced absorption and carrier collection in Si wire arrays for photovoltaic applications,” Nature Materials 9, 239–244 (2010).
[Crossref] [PubMed]

Kirsch, A.

A. Meier, T. Arens, S. N. Chandler-Wilde, and A. Kirsch, “A Nyström method for a class of integral equations on the real line with applications to scattering by diffraction gratings and rough surfaces,” J. Integral Equations Appl. 12, 281–321 (2000).
[Crossref]

Ko, D. Y. K.

Kobayashi, M.

J. Lai, M. Kobayashi, and A. H. Barnett, “A fast solver for the scattering from a layered periodic structure with multi-particle inclusions,” (2014). In preparation.

Komashko, A.

M. K. C. P. J. Barty, J. Britten, R. Beach, G. Beer, C. Brown, S. Bryan, J. Caird, T. Carlson, J. Crane, J. Dawson, A. Erlandson, D. Fittinghoff, M. Hermann, C. Hoaglan, A. Iyer, L. Ivanovic, I. Jovanovic, A. Komashko, O. Landen, Z. Liao, W. Molander, S. Mitchell, E. Moses, N. Nielsen, H.-H. Nguyen, J. Nissen, S. Payne, D. Pennington, L. Risinger, M. Rushford, K. Skulina, M. Spaeth, B. Stuart, G. Tietbohl, and B. Wattellier, “An overview of LLNL high-energy short-pulse technology for advanced radiography of laser fusion experiments,” Nuclear Fusion 44, S266 (2004).
[Crossref]

Kress, R.

R. Kress, “Boundary integral equations in time-harmonic acoustic scattering,” Mathl. Comput. Modelling 15, 229–243 (1991).
[Crossref]

R. Kress, Linear Integral Equations, vol. 82 of Appl. Math. Sci. (Springer, 1999), 2nd ed.
[Crossref]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, vol. 93 of Applied Mathematical Sciences (Springer-Verlag, 1998), 2nd ed.
[Crossref]

Lai, J.

J. Lai, M. Kobayashi, and A. H. Barnett, “A fast solver for the scattering from a layered periodic structure with multi-particle inclusions,” (2014). In preparation.

Landen, O.

M. K. C. P. J. Barty, J. Britten, R. Beach, G. Beer, C. Brown, S. Bryan, J. Caird, T. Carlson, J. Crane, J. Dawson, A. Erlandson, D. Fittinghoff, M. Hermann, C. Hoaglan, A. Iyer, L. Ivanovic, I. Jovanovic, A. Komashko, O. Landen, Z. Liao, W. Molander, S. Mitchell, E. Moses, N. Nielsen, H.-H. Nguyen, J. Nissen, S. Payne, D. Pennington, L. Risinger, M. Rushford, K. Skulina, M. Spaeth, B. Stuart, G. Tietbohl, and B. Wattellier, “An overview of LLNL high-energy short-pulse technology for advanced radiography of laser fusion experiments,” Nuclear Fusion 44, S266 (2004).
[Crossref]

Lechleiter, A.

A. Lechleiter and D.-L. Nguyen, “A trigonometric Galerkin method for volume integral equations arising in TM grating scattering,” Adv. Comput. Math. 40, 1–25 (2014).
[Crossref]

Lee, J.-Y.

L. Greengard, K. L. Ho, and J.-Y. Lee, “A fast direct solver for scattering from periodic structures with multiple material interfaces in two dimensions,” J. Comput. Phys. 258, 738–751 (2014).
[Crossref]

Lerer, A. M.

G. A. Kalinchenko and A. M. Lerer, “Wideband all-dielectric diffraction grating on chirped mirror,” J. Lightwave Technology 28, 2743–2749 (2010).
[Crossref]

Lewis, N. S.

M. D. Kelzenberg, S. W. Boettcher, J. A. Petykiewicz, D. B. Turner-Evans, M. C. Putnam, E. L. Warren, J. M. Spurgeon, R. M. Briggs, N. S. Lewis, and H. A. Atwater, “Enhanced absorption and carrier collection in Si wire arrays for photovoltaic applications,” Nature Materials 9, 239–244 (2010).
[Crossref] [PubMed]

Li, L.

Liao, Z.

M. K. C. P. J. Barty, J. Britten, R. Beach, G. Beer, C. Brown, S. Bryan, J. Caird, T. Carlson, J. Crane, J. Dawson, A. Erlandson, D. Fittinghoff, M. Hermann, C. Hoaglan, A. Iyer, L. Ivanovic, I. Jovanovic, A. Komashko, O. Landen, Z. Liao, W. Molander, S. Mitchell, E. Moses, N. Nielsen, H.-H. Nguyen, J. Nissen, S. Payne, D. Pennington, L. Risinger, M. Rushford, K. Skulina, M. Spaeth, B. Stuart, G. Tietbohl, and B. Wattellier, “An overview of LLNL high-energy short-pulse technology for advanced radiography of laser fusion experiments,” Nuclear Fusion 44, S266 (2004).
[Crossref]

Linton, C. M.

C. M. Linton, “Lattice sums for the Helmholtz equation,” SIAM Review 52, 603–674 (2010).
[Crossref]

C. M. Linton and I. Thompson, “Resonant effects in scattering by periodic arrays,” Wave Motion 44, 165–175 (2007).
[Crossref]

Martinsson, P.

P. Martinsson and V. Rokhlin, “A fast direct solver for boundary integral equations in two dimensions,” J. Comp. Phys. 205, 1–23 (2005).
[Crossref]

Martinsson, P. G.

S. Hao, A. H. Barnett, P. G. Martinsson, and P. Young, “High-order accurate Nyström discretization of integral equations with weakly singular kernels on smooth curves in the plane,” Adv. Comput. Math. 40, 245–272 (2014).
[Crossref]

Marx, E.

G. L. Wojcik, J. Mould, E. Marx, and M. P. Davidson, “Numerical reference models for optical metrology simulation,” Proc. SPIE 1673, 06 (1992).

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University, 2008), 2nd ed.

Meier, A.

A. Meier, T. Arens, S. N. Chandler-Wilde, and A. Kirsch, “A Nyström method for a class of integral equations on the real line with applications to scattering by diffraction gratings and rough surfaces,” J. Integral Equations Appl. 12, 281–321 (2000).
[Crossref]

Mitchell, S.

M. K. C. P. J. Barty, J. Britten, R. Beach, G. Beer, C. Brown, S. Bryan, J. Caird, T. Carlson, J. Crane, J. Dawson, A. Erlandson, D. Fittinghoff, M. Hermann, C. Hoaglan, A. Iyer, L. Ivanovic, I. Jovanovic, A. Komashko, O. Landen, Z. Liao, W. Molander, S. Mitchell, E. Moses, N. Nielsen, H.-H. Nguyen, J. Nissen, S. Payne, D. Pennington, L. Risinger, M. Rushford, K. Skulina, M. Spaeth, B. Stuart, G. Tietbohl, and B. Wattellier, “An overview of LLNL high-energy short-pulse technology for advanced radiography of laser fusion experiments,” Nuclear Fusion 44, S266 (2004).
[Crossref]

Model, R.

R. Model, A. Rathsfeld, H. Gross, M. Wurm, and B. Bodermann, “A scatterometry inverse problem in optical mask metrology,” J. Phys.: Conf. Ser. 135, 012071 (2008).

Moharam, M. G.

Molander, W.

M. K. C. P. J. Barty, J. Britten, R. Beach, G. Beer, C. Brown, S. Bryan, J. Caird, T. Carlson, J. Crane, J. Dawson, A. Erlandson, D. Fittinghoff, M. Hermann, C. Hoaglan, A. Iyer, L. Ivanovic, I. Jovanovic, A. Komashko, O. Landen, Z. Liao, W. Molander, S. Mitchell, E. Moses, N. Nielsen, H.-H. Nguyen, J. Nissen, S. Payne, D. Pennington, L. Risinger, M. Rushford, K. Skulina, M. Spaeth, B. Stuart, G. Tietbohl, and B. Wattellier, “An overview of LLNL high-energy short-pulse technology for advanced radiography of laser fusion experiments,” Nuclear Fusion 44, S266 (2004).
[Crossref]

Moses, E.

M. K. C. P. J. Barty, J. Britten, R. Beach, G. Beer, C. Brown, S. Bryan, J. Caird, T. Carlson, J. Crane, J. Dawson, A. Erlandson, D. Fittinghoff, M. Hermann, C. Hoaglan, A. Iyer, L. Ivanovic, I. Jovanovic, A. Komashko, O. Landen, Z. Liao, W. Molander, S. Mitchell, E. Moses, N. Nielsen, H.-H. Nguyen, J. Nissen, S. Payne, D. Pennington, L. Risinger, M. Rushford, K. Skulina, M. Spaeth, B. Stuart, G. Tietbohl, and B. Wattellier, “An overview of LLNL high-energy short-pulse technology for advanced radiography of laser fusion experiments,” Nuclear Fusion 44, S266 (2004).
[Crossref]

Mould, J.

G. L. Wojcik, J. Mould, E. Marx, and M. P. Davidson, “Numerical reference models for optical metrology simulation,” Proc. SPIE 1673, 06 (1992).

Müller, C.

C. Müller, Foundations of the Mathematical Theory of Electromagnetic Waves (Springer-Verlag, 1969).
[Crossref]

Nédélec, J. C.

J. C. Nédélec and F. Starling, “Integral equation methods in a quasi-periodic diffraction problem for the time-harmonic Maxwell’s equations,” SIAM J. Math. Anal. 22, 1679–1701 (1991).
[Crossref]

Nguyen, D.-L.

A. Lechleiter and D.-L. Nguyen, “A trigonometric Galerkin method for volume integral equations arising in TM grating scattering,” Adv. Comput. Math. 40, 1–25 (2014).
[Crossref]

Nguyen, H.-H.

M. K. C. P. J. Barty, J. Britten, R. Beach, G. Beer, C. Brown, S. Bryan, J. Caird, T. Carlson, J. Crane, J. Dawson, A. Erlandson, D. Fittinghoff, M. Hermann, C. Hoaglan, A. Iyer, L. Ivanovic, I. Jovanovic, A. Komashko, O. Landen, Z. Liao, W. Molander, S. Mitchell, E. Moses, N. Nielsen, H.-H. Nguyen, J. Nissen, S. Payne, D. Pennington, L. Risinger, M. Rushford, K. Skulina, M. Spaeth, B. Stuart, G. Tietbohl, and B. Wattellier, “An overview of LLNL high-energy short-pulse technology for advanced radiography of laser fusion experiments,” Nuclear Fusion 44, S266 (2004).
[Crossref]

Nicholas, M. J.

M. J. Nicholas, “A higher order numerical method for 3-D doubly periodic electromagnetic scattering problems,” Commun. Math. Sci. 6, 669–694 (2008).
[Crossref]

Nielsen, N.

M. K. C. P. J. Barty, J. Britten, R. Beach, G. Beer, C. Brown, S. Bryan, J. Caird, T. Carlson, J. Crane, J. Dawson, A. Erlandson, D. Fittinghoff, M. Hermann, C. Hoaglan, A. Iyer, L. Ivanovic, I. Jovanovic, A. Komashko, O. Landen, Z. Liao, W. Molander, S. Mitchell, E. Moses, N. Nielsen, H.-H. Nguyen, J. Nissen, S. Payne, D. Pennington, L. Risinger, M. Rushford, K. Skulina, M. Spaeth, B. Stuart, G. Tietbohl, and B. Wattellier, “An overview of LLNL high-energy short-pulse technology for advanced radiography of laser fusion experiments,” Nuclear Fusion 44, S266 (2004).
[Crossref]

Nishimura, N.

Y. Otani and N. Nishimura, “A periodic FMM for Maxwell’s equations in 3D and its applications to problems related to photonic crystals,” J. Comput. Phys. 227, 4630–4652 (2008).
[Crossref]

Nissen, J.

M. K. C. P. J. Barty, J. Britten, R. Beach, G. Beer, C. Brown, S. Bryan, J. Caird, T. Carlson, J. Crane, J. Dawson, A. Erlandson, D. Fittinghoff, M. Hermann, C. Hoaglan, A. Iyer, L. Ivanovic, I. Jovanovic, A. Komashko, O. Landen, Z. Liao, W. Molander, S. Mitchell, E. Moses, N. Nielsen, H.-H. Nguyen, J. Nissen, S. Payne, D. Pennington, L. Risinger, M. Rushford, K. Skulina, M. Spaeth, B. Stuart, G. Tietbohl, and B. Wattellier, “An overview of LLNL high-energy short-pulse technology for advanced radiography of laser fusion experiments,” Nuclear Fusion 44, S266 (2004).
[Crossref]

Otani, Y.

Y. Otani and N. Nishimura, “A periodic FMM for Maxwell’s equations in 3D and its applications to problems related to photonic crystals,” J. Comput. Phys. 227, 4630–4652 (2008).
[Crossref]

Payne, S.

M. K. C. P. J. Barty, J. Britten, R. Beach, G. Beer, C. Brown, S. Bryan, J. Caird, T. Carlson, J. Crane, J. Dawson, A. Erlandson, D. Fittinghoff, M. Hermann, C. Hoaglan, A. Iyer, L. Ivanovic, I. Jovanovic, A. Komashko, O. Landen, Z. Liao, W. Molander, S. Mitchell, E. Moses, N. Nielsen, H.-H. Nguyen, J. Nissen, S. Payne, D. Pennington, L. Risinger, M. Rushford, K. Skulina, M. Spaeth, B. Stuart, G. Tietbohl, and B. Wattellier, “An overview of LLNL high-energy short-pulse technology for advanced radiography of laser fusion experiments,” Nuclear Fusion 44, S266 (2004).
[Crossref]

Pennington, D.

M. K. C. P. J. Barty, J. Britten, R. Beach, G. Beer, C. Brown, S. Bryan, J. Caird, T. Carlson, J. Crane, J. Dawson, A. Erlandson, D. Fittinghoff, M. Hermann, C. Hoaglan, A. Iyer, L. Ivanovic, I. Jovanovic, A. Komashko, O. Landen, Z. Liao, W. Molander, S. Mitchell, E. Moses, N. Nielsen, H.-H. Nguyen, J. Nissen, S. Payne, D. Pennington, L. Risinger, M. Rushford, K. Skulina, M. Spaeth, B. Stuart, G. Tietbohl, and B. Wattellier, “An overview of LLNL high-energy short-pulse technology for advanced radiography of laser fusion experiments,” Nuclear Fusion 44, S266 (2004).
[Crossref]

Perry, M. D.

Petykiewicz, J. A.

M. D. Kelzenberg, S. W. Boettcher, J. A. Petykiewicz, D. B. Turner-Evans, M. C. Putnam, E. L. Warren, J. M. Spurgeon, R. M. Briggs, N. S. Lewis, and H. A. Atwater, “Enhanced absorption and carrier collection in Si wire arrays for photovoltaic applications,” Nature Materials 9, 239–244 (2010).
[Crossref] [PubMed]

Peumans, P.

Polman, A.

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nature Materials 9, 205–213 (2010).
[Crossref] [PubMed]

Putnam, M. C.

M. D. Kelzenberg, S. W. Boettcher, J. A. Petykiewicz, D. B. Turner-Evans, M. C. Putnam, E. L. Warren, J. M. Spurgeon, R. M. Briggs, N. S. Lewis, and H. A. Atwater, “Enhanced absorption and carrier collection in Si wire arrays for photovoltaic applications,” Nature Materials 9, 239–244 (2010).
[Crossref] [PubMed]

Rabinowitz, P.

P. J. Davis and P. Rabinowitz, Methods of Numerical Integration (Academic, 1984).

Rathsfeld, A.

R. Model, A. Rathsfeld, H. Gross, M. Wurm, and B. Bodermann, “A scatterometry inverse problem in optical mask metrology,” J. Phys.: Conf. Ser. 135, 012071 (2008).

Risinger, L.

M. K. C. P. J. Barty, J. Britten, R. Beach, G. Beer, C. Brown, S. Bryan, J. Caird, T. Carlson, J. Crane, J. Dawson, A. Erlandson, D. Fittinghoff, M. Hermann, C. Hoaglan, A. Iyer, L. Ivanovic, I. Jovanovic, A. Komashko, O. Landen, Z. Liao, W. Molander, S. Mitchell, E. Moses, N. Nielsen, H.-H. Nguyen, J. Nissen, S. Payne, D. Pennington, L. Risinger, M. Rushford, K. Skulina, M. Spaeth, B. Stuart, G. Tietbohl, and B. Wattellier, “An overview of LLNL high-energy short-pulse technology for advanced radiography of laser fusion experiments,” Nuclear Fusion 44, S266 (2004).
[Crossref]

Rokhlin, V.

P. Martinsson and V. Rokhlin, “A fast direct solver for boundary integral equations in two dimensions,” J. Comp. Phys. 205, 1–23 (2005).
[Crossref]

V. Rokhlin, “Solution of acoustic scattering problems by means of second kind integral equations,” Wave Motion 5, 257–272 (1983).
[Crossref]

W. Y. Crutchfield, Z. Gimbutas, Gol, J. Huang, V. Rokhlin, N. Yarvin, and J. Zhao, Remarks on the implementation of the wideband FMM for the Helmholtz equation in two dimensions (Amer. Math. Soc., 2006), vol. 408 of Contemp. Math., pp. 99–110.

Rushford, M.

M. K. C. P. J. Barty, J. Britten, R. Beach, G. Beer, C. Brown, S. Bryan, J. Caird, T. Carlson, J. Crane, J. Dawson, A. Erlandson, D. Fittinghoff, M. Hermann, C. Hoaglan, A. Iyer, L. Ivanovic, I. Jovanovic, A. Komashko, O. Landen, Z. Liao, W. Molander, S. Mitchell, E. Moses, N. Nielsen, H.-H. Nguyen, J. Nissen, S. Payne, D. Pennington, L. Risinger, M. Rushford, K. Skulina, M. Spaeth, B. Stuart, G. Tietbohl, and B. Wattellier, “An overview of LLNL high-energy short-pulse technology for advanced radiography of laser fusion experiments,” Nuclear Fusion 44, S266 (2004).
[Crossref]

Sambles, J. R.

Sauter, S. A.

I. M. Babuska and S. A. Sauter, “Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?” SIAM J. Numer. Anal. 34, 2392–2423 (1997).

Schmidt, G.

J. Elschner, R. Hinder, and G. Schmidt, “Finite element solution of conical diffraction problems,” Adv. Comput. Math. 16, 139–156 (2002).
[Crossref]

Sergeant, N. P.

Shannon, C.

Shipman, S.

S. Shipman, Resonant scattering by open periodic waveguides (Bentham Science Publishers, 2010), vol. 1 of Progress in Computational Physics (PiCP), pp. 7–50.

O. P. Bruno, S. Shipman, C. Turc, and S. Venakides, “Efficient evaluation of doubly periodic Green functions in 3D scattering, including Wood anomaly frequencies,” (2013). Preprint, arXiv:1307.1176v1.

Shore, B. W.

Shults, E.

Skulina, K.

M. K. C. P. J. Barty, J. Britten, R. Beach, G. Beer, C. Brown, S. Bryan, J. Caird, T. Carlson, J. Crane, J. Dawson, A. Erlandson, D. Fittinghoff, M. Hermann, C. Hoaglan, A. Iyer, L. Ivanovic, I. Jovanovic, A. Komashko, O. Landen, Z. Liao, W. Molander, S. Mitchell, E. Moses, N. Nielsen, H.-H. Nguyen, J. Nissen, S. Payne, D. Pennington, L. Risinger, M. Rushford, K. Skulina, M. Spaeth, B. Stuart, G. Tietbohl, and B. Wattellier, “An overview of LLNL high-energy short-pulse technology for advanced radiography of laser fusion experiments,” Nuclear Fusion 44, S266 (2004).
[Crossref]

Spaeth, M.

M. K. C. P. J. Barty, J. Britten, R. Beach, G. Beer, C. Brown, S. Bryan, J. Caird, T. Carlson, J. Crane, J. Dawson, A. Erlandson, D. Fittinghoff, M. Hermann, C. Hoaglan, A. Iyer, L. Ivanovic, I. Jovanovic, A. Komashko, O. Landen, Z. Liao, W. Molander, S. Mitchell, E. Moses, N. Nielsen, H.-H. Nguyen, J. Nissen, S. Payne, D. Pennington, L. Risinger, M. Rushford, K. Skulina, M. Spaeth, B. Stuart, G. Tietbohl, and B. Wattellier, “An overview of LLNL high-energy short-pulse technology for advanced radiography of laser fusion experiments,” Nuclear Fusion 44, S266 (2004).
[Crossref]

Spurgeon, J. M.

M. D. Kelzenberg, S. W. Boettcher, J. A. Petykiewicz, D. B. Turner-Evans, M. C. Putnam, E. L. Warren, J. M. Spurgeon, R. M. Briggs, N. S. Lewis, and H. A. Atwater, “Enhanced absorption and carrier collection in Si wire arrays for photovoltaic applications,” Nature Materials 9, 239–244 (2010).
[Crossref] [PubMed]

Starling, F.

A.-S. Bonnet-BenDhia and F. Starling, “Guided waves by electromagnetic gratings and non-uniqueness examples for the diffraction problem,” Math. Meth. Appl. Sci. 17, 305–338 (1994).
[Crossref]

J. C. Nédélec and F. Starling, “Integral equation methods in a quasi-periodic diffraction problem for the time-harmonic Maxwell’s equations,” SIAM J. Math. Anal. 22, 1679–1701 (1991).
[Crossref]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1964), 10th ed.

Steyskal, H.

H. Holter and H. Steyskal, “Some experiences from FDTD analysis of infinite and finite multi-octave phased arrays,” IEEE Trans. Antennae Prop. 50, 1725–1731 (2002).
[Crossref]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (John Wiley & Sons, 2007).

Stuart, B.

M. K. C. P. J. Barty, J. Britten, R. Beach, G. Beer, C. Brown, S. Bryan, J. Caird, T. Carlson, J. Crane, J. Dawson, A. Erlandson, D. Fittinghoff, M. Hermann, C. Hoaglan, A. Iyer, L. Ivanovic, I. Jovanovic, A. Komashko, O. Landen, Z. Liao, W. Molander, S. Mitchell, E. Moses, N. Nielsen, H.-H. Nguyen, J. Nissen, S. Payne, D. Pennington, L. Risinger, M. Rushford, K. Skulina, M. Spaeth, B. Stuart, G. Tietbohl, and B. Wattellier, “An overview of LLNL high-energy short-pulse technology for advanced radiography of laser fusion experiments,” Nuclear Fusion 44, S266 (2004).
[Crossref]

Taflove, A.

A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 1995).

Thompson, I.

C. M. Linton and I. Thompson, “Resonant effects in scattering by periodic arrays,” Wave Motion 44, 165–175 (2007).
[Crossref]

Tietbohl, G.

M. K. C. P. J. Barty, J. Britten, R. Beach, G. Beer, C. Brown, S. Bryan, J. Caird, T. Carlson, J. Crane, J. Dawson, A. Erlandson, D. Fittinghoff, M. Hermann, C. Hoaglan, A. Iyer, L. Ivanovic, I. Jovanovic, A. Komashko, O. Landen, Z. Liao, W. Molander, S. Mitchell, E. Moses, N. Nielsen, H.-H. Nguyen, J. Nissen, S. Payne, D. Pennington, L. Risinger, M. Rushford, K. Skulina, M. Spaeth, B. Stuart, G. Tietbohl, and B. Wattellier, “An overview of LLNL high-energy short-pulse technology for advanced radiography of laser fusion experiments,” Nuclear Fusion 44, S266 (2004).
[Crossref]

Turc, C.

O. P. Bruno, S. Shipman, C. Turc, and S. Venakides, “Efficient evaluation of doubly periodic Green functions in 3D scattering, including Wood anomaly frequencies,” (2013). Preprint, arXiv:1307.1176v1.

Turner-Evans, D. B.

M. D. Kelzenberg, S. W. Boettcher, J. A. Petykiewicz, D. B. Turner-Evans, M. C. Putnam, E. L. Warren, J. M. Spurgeon, R. M. Briggs, N. S. Lewis, and H. A. Atwater, “Enhanced absorption and carrier collection in Si wire arrays for photovoltaic applications,” Nature Materials 9, 239–244 (2010).
[Crossref] [PubMed]

Van Loan, C. F.

G. H. Golub and C. F. Van Loan, Matrix Computations, Johns Hopkins Studies in the Mathematical Sciences (Johns Hopkins University, 1996), 3rd ed.

Venakides, S.

O. P. Bruno, S. Shipman, C. Turc, and S. Venakides, “Efficient evaluation of doubly periodic Green functions in 3D scattering, including Wood anomaly frequencies,” (2013). Preprint, arXiv:1307.1176v1.

Warren, E. L.

M. D. Kelzenberg, S. W. Boettcher, J. A. Petykiewicz, D. B. Turner-Evans, M. C. Putnam, E. L. Warren, J. M. Spurgeon, R. M. Briggs, N. S. Lewis, and H. A. Atwater, “Enhanced absorption and carrier collection in Si wire arrays for photovoltaic applications,” Nature Materials 9, 239–244 (2010).
[Crossref] [PubMed]

Wattellier, B.

M. K. C. P. J. Barty, J. Britten, R. Beach, G. Beer, C. Brown, S. Bryan, J. Caird, T. Carlson, J. Crane, J. Dawson, A. Erlandson, D. Fittinghoff, M. Hermann, C. Hoaglan, A. Iyer, L. Ivanovic, I. Jovanovic, A. Komashko, O. Landen, Z. Liao, W. Molander, S. Mitchell, E. Moses, N. Nielsen, H.-H. Nguyen, J. Nissen, S. Payne, D. Pennington, L. Risinger, M. Rushford, K. Skulina, M. Spaeth, B. Stuart, G. Tietbohl, and B. Wattellier, “An overview of LLNL high-energy short-pulse technology for advanced radiography of laser fusion experiments,” Nuclear Fusion 44, S266 (2004).
[Crossref]

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University, 2008), 2nd ed.

Wojcik, G. L.

G. L. Wojcik, J. Mould, E. Marx, and M. P. Davidson, “Numerical reference models for optical metrology simulation,” Proc. SPIE 1673, 06 (1992).

Wood, R. W.

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–408 (1902).
[Crossref]

Wurm, M.

R. Model, A. Rathsfeld, H. Gross, M. Wurm, and B. Bodermann, “A scatterometry inverse problem in optical mask metrology,” J. Phys.: Conf. Ser. 135, 012071 (2008).

Yarvin, N.

W. Y. Crutchfield, Z. Gimbutas, Gol, J. Huang, V. Rokhlin, N. Yarvin, and J. Zhao, Remarks on the implementation of the wideband FMM for the Helmholtz equation in two dimensions (Amer. Math. Soc., 2006), vol. 408 of Contemp. Math., pp. 99–110.

Young, P.

S. Hao, A. H. Barnett, P. G. Martinsson, and P. Young, “High-order accurate Nyström discretization of integral equations with weakly singular kernels on smooth curves in the plane,” Adv. Comput. Math. 40, 245–272 (2014).
[Crossref]

Zhao, J.

W. Y. Crutchfield, Z. Gimbutas, Gol, J. Huang, V. Rokhlin, N. Yarvin, and J. Zhao, Remarks on the implementation of the wideband FMM for the Helmholtz equation in two dimensions (Amer. Math. Soc., 2006), vol. 408 of Contemp. Math., pp. 99–110.

Zhao, L.

L. Zhao and A. H. Barnett, “Robust and efficient solution of the drum problem via Nyström approximation of the Fredholm determinant,” (2014). arxiv:1406.5252, submitted to J. Comput. Phys.

Adv. Comput. Math. (3)

J. Elschner, R. Hinder, and G. Schmidt, “Finite element solution of conical diffraction problems,” Adv. Comput. Math. 16, 139–156 (2002).
[Crossref]

A. Lechleiter and D.-L. Nguyen, “A trigonometric Galerkin method for volume integral equations arising in TM grating scattering,” Adv. Comput. Math. 40, 1–25 (2014).
[Crossref]

S. Hao, A. H. Barnett, P. G. Martinsson, and P. Young, “High-order accurate Nyström discretization of integral equations with weakly singular kernels on smooth curves in the plane,” Adv. Comput. Math. 40, 245–272 (2014).
[Crossref]

BIT Numer. Math. (1)

A. H. Barnett and L. Greengard, “A new integral representation for quasi-periodic scattering problems in two dimensions,” BIT Numer. Math. 51, 67–90 (2011).
[Crossref]

Commun. Math. Sci. (1)

M. J. Nicholas, “A higher order numerical method for 3-D doubly periodic electromagnetic scattering problems,” Commun. Math. Sci. 6, 669–694 (2008).
[Crossref]

IEEE Trans. Antennae Prop. (1)

H. Holter and H. Steyskal, “Some experiences from FDTD analysis of infinite and finite multi-octave phased arrays,” IEEE Trans. Antennae Prop. 50, 1725–1731 (2002).
[Crossref]

J. Acoust. Soc. Amer. (1)

K. V. Horoshenkov and S. N. Chandler-Wilde, “Efficient calculation of two-dimensional periodic and waveguide acoustic Green’s functions,” J. Acoust. Soc. Amer. 111, 1610–1622 (2002).
[Crossref]

J. Comp. Phys. (1)

P. Martinsson and V. Rokhlin, “A fast direct solver for boundary integral equations in two dimensions,” J. Comp. Phys. 205, 1–23 (2005).
[Crossref]

J. Comput. Phys. (8)

N. A. Gumerov and R. Duraiswami, “A method to compute periodic sums,” J. Comput. Phys. 272, 307–326 (2014).
[Crossref]

A. Gillman and A. Barnett, “A fast direct solver for quasiperiodic scattering problems,” J. Comput. Phys. 248, 309–322 (2013).
[Crossref]

L. Greengard, K. L. Ho, and J.-Y. Lee, “A fast direct solver for scattering from periodic structures with multiple material interfaces in two dimensions,” J. Comput. Phys. 258, 738–751 (2014).
[Crossref]

O. P. Bruno and B. Delourme, “Rapidly convergent two-dimensional quasi-periodic Gteen function throughout the spectrum-including Wold anomalies,” J. Comput. Phys. 262, 262–290 (2014).
[Crossref]

A. H. Barnett and L. Greengard, “A new integral representation for quasi-periodic fields and its application to two-dimensional band structure calculations,” J. Comput. Phys. 229, 6898–6914 (2010).
[Crossref]

A. H. Barnett and T. Betcke, “Stability and convergence of the Method of Fundamental Solutions for Helmholtz problems on analytic domains,” J. Comput. Phys. 227, 7003–7026 (2008).
[Crossref]

M. H. Cho and W. Cai, “A parallel fast algorithm for computing the Helmholtz integral operator in 3-D layered media,” J. Comput. Phys. 231, 5910–5925 (2012).
[Crossref]

Y. Otani and N. Nishimura, “A periodic FMM for Maxwell’s equations in 3D and its applications to problems related to photonic crystals,” J. Comput. Phys. 227, 4630–4652 (2008).
[Crossref]

J. Integral Equations Appl. (1)

A. Meier, T. Arens, S. N. Chandler-Wilde, and A. Kirsch, “A Nyström method for a class of integral equations on the real line with applications to scattering by diffraction gratings and rough surfaces,” J. Integral Equations Appl. 12, 281–321 (2000).
[Crossref]

J. Lightwave Technology (1)

G. A. Kalinchenko and A. M. Lerer, “Wideband all-dielectric diffraction grating on chirped mirror,” J. Lightwave Technology 28, 2743–2749 (2010).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

J. Phys.: Conf. Ser. (1)

R. Model, A. Rathsfeld, H. Gross, M. Wurm, and B. Bodermann, “A scatterometry inverse problem in optical mask metrology,” J. Phys.: Conf. Ser. 135, 012071 (2008).

Math. Meth. Appl. Sci. (1)

A.-S. Bonnet-BenDhia and F. Starling, “Guided waves by electromagnetic gratings and non-uniqueness examples for the diffraction problem,” Math. Meth. Appl. Sci. 17, 305–338 (1994).
[Crossref]

Mathl. Comput. Modelling (1)

R. Kress, “Boundary integral equations in time-harmonic acoustic scattering,” Mathl. Comput. Modelling 15, 229–243 (1991).
[Crossref]

Nature Materials (2)

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nature Materials 9, 205–213 (2010).
[Crossref] [PubMed]

M. D. Kelzenberg, S. W. Boettcher, J. A. Petykiewicz, D. B. Turner-Evans, M. C. Putnam, E. L. Warren, J. M. Spurgeon, R. M. Briggs, N. S. Lewis, and H. A. Atwater, “Enhanced absorption and carrier collection in Si wire arrays for photovoltaic applications,” Nature Materials 9, 239–244 (2010).
[Crossref] [PubMed]

Nuclear Fusion (1)

M. K. C. P. J. Barty, J. Britten, R. Beach, G. Beer, C. Brown, S. Bryan, J. Caird, T. Carlson, J. Crane, J. Dawson, A. Erlandson, D. Fittinghoff, M. Hermann, C. Hoaglan, A. Iyer, L. Ivanovic, I. Jovanovic, A. Komashko, O. Landen, Z. Liao, W. Molander, S. Mitchell, E. Moses, N. Nielsen, H.-H. Nguyen, J. Nissen, S. Payne, D. Pennington, L. Risinger, M. Rushford, K. Skulina, M. Spaeth, B. Stuart, G. Tietbohl, and B. Wattellier, “An overview of LLNL high-energy short-pulse technology for advanced radiography of laser fusion experiments,” Nuclear Fusion 44, S266 (2004).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Philos. Mag. (1)

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–408 (1902).
[Crossref]

Proc. SPIE (1)

G. L. Wojcik, J. Mould, E. Marx, and M. P. Davidson, “Numerical reference models for optical metrology simulation,” Proc. SPIE 1673, 06 (1992).

SIAM J. Math. Anal. (1)

J. C. Nédélec and F. Starling, “Integral equation methods in a quasi-periodic diffraction problem for the time-harmonic Maxwell’s equations,” SIAM J. Math. Anal. 22, 1679–1701 (1991).
[Crossref]

SIAM J. Numer. Anal. (1)

I. M. Babuska and S. A. Sauter, “Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?” SIAM J. Numer. Anal. 34, 2392–2423 (1997).

SIAM J. Sci. Comput. (1)

B. K. Alpert, “Hybrid Gauss-trapezoidal quadrature rules,” SIAM J. Sci. Comput. 20, 1551–1584 (1999).
[Crossref]

SIAM Journal on Numerical Analysis (1)

A. Bogomolny, “Fundamental solutions method for elliptic boundary value problems,” SIAM Journal on Numerical Analysis 22, 644–669 (1985).
[Crossref]

SIAM Review (1)

C. M. Linton, “Lattice sums for the Helmholtz equation,” SIAM Review 52, 603–674 (2010).
[Crossref]

Surv. Math. Ind. (1)

G. Bao and D. C. Dobson, “Modeling and optimal design of diffractive optical structures,” Surv. Math. Ind. 8, 37–62 (1998).

Wave Motion (2)

C. M. Linton and I. Thompson, “Resonant effects in scattering by periodic arrays,” Wave Motion 44, 165–175 (2007).
[Crossref]

V. Rokhlin, “Solution of acoustic scattering problems by means of second kind integral equations,” Wave Motion 5, 257–272 (1983).
[Crossref]

Other (18)

P. J. Davis and P. Rabinowitz, Methods of Numerical Integration (Academic, 1984).

R. Kress, Linear Integral Equations, vol. 82 of Appl. Math. Sci. (Springer, 1999), 2nd ed.
[Crossref]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1964), 10th ed.

O. P. Bruno, S. Shipman, C. Turc, and S. Venakides, “Efficient evaluation of doubly periodic Green functions in 3D scattering, including Wood anomaly frequencies,” (2013). Preprint, arXiv:1307.1176v1.

J. Lai, M. Kobayashi, and A. H. Barnett, “A fast solver for the scattering from a layered periodic structure with multi-particle inclusions,” (2014). In preparation.

G. H. Golub and C. F. Van Loan, Matrix Computations, Johns Hopkins Studies in the Mathematical Sciences (Johns Hopkins University, 1996), 3rd ed.

W. Y. Crutchfield, Z. Gimbutas, Gol, J. Huang, V. Rokhlin, N. Yarvin, and J. Zhao, Remarks on the implementation of the wideband FMM for the Helmholtz equation in two dimensions (Amer. Math. Soc., 2006), vol. 408 of Contemp. Math., pp. 99–110.

A. H. Barnett and T. Betcke, “MPSpack: A MATLAB toolbox to solve Helmholtz PDE, wave scattering, and eigenvalue problems,” (2008–2014). http://code.google.com/p/mpspack .

C. Hafner, The Generalized Multipole Technique for Computational Electromagnetics (Artech House Books, 1990).

L. Zhao and A. H. Barnett, “Robust and efficient solution of the drum problem via Nyström approximation of the Fredholm determinant,” (2014). arxiv:1406.5252, submitted to J. Comput. Phys.

C. Müller, Foundations of the Mathematical Theory of Electromagnetic Waves (Springer-Verlag, 1969).
[Crossref]

J. A. Stratton, Electromagnetic Theory (John Wiley & Sons, 2007).

J. D. Jackson, Classical Electrodynamics (Wiley, 1998), 3rd ed.

A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 1995).

S. Shipman, Resonant scattering by open periodic waveguides (Bentham Science Publishers, 2010), vol. 1 of Progress in Computational Physics (PiCP), pp. 7–50.

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, vol. 93 of Applied Mathematical Sciences (Springer-Verlag, 1998), 2nd ed.
[Crossref]

J. D. Joannopoulos, S. G. Johnson, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University, 2008), 2nd ed.

T. Arens, “Scattering by biperiodic layered media: The integral equation approach,” Habilitation thesis, Karlsruhe (2010).

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Figures (7)

Fig. 1
Fig. 1 (a) Geometry of scattering problem. The periodicity is in the horizontal direction, with one period lying between the vertical blue dotted lines. Γi for i = 1,...,I are the material interfaces. The medium is uniform in the ith layer, which lies above the ith interface and has wavenumber ki. Our algorithm also uses: Li and Ri which are the left and right walls of one period Ωi of the ith layer, Pi the proxy circle for this layer, and U, D the upper and lower fictitious interfaces (at y = yU and y = yD) where the radiation condition is applied. (b) Zoom of the top part of the geometry, showing quadrature nodes for Nyström method and collocation (for clarity, less nodes are shown than actually used), including the near-field neighboring copies.
Fig. 2
Fig. 2 Convergence of u(0.15, 0.6) and flux error for 30 sine interfaces (see the inset in (b)) with ω = 10 and random εi between 1 and 2. (a) Convergence in N, number of nodes per sine interface (blue square) and flux error (red triangle) while P = 110. (b) Convergence in P (blue square) and flux error (red triangle), fixing N = 70 per sine interface. All other parameters are fixed at Mw = 110, M = 100, K = 10, and R = 2. (Color online.)
Fig. 3
Fig. 3 Convergence of u(0.15, 0.6) and flux error for 30 mixed sine and rectangle interfaces (see the inset in (c)) with ω = 10 and random εi between 1 and 2. (a) Convergence in N on sine interface (blue square) and flux error (red triangle), while N = 110 on each line segment of rectangle interfaces, and P = 110. (b) Convergence in N on each line segment of rectangle interfaces (blue square) and flux error (red triangle), while N = 70 on sine interfaces and P = 110. (c) Convergence in P (blue square) and flux error (red triangle), while N = 70 on sine, N = 110 on rectangle interfaces. All other parameters are fixed at Mw = 110, M = 120, K = 20, and R = 2. (Color online.)
Fig. 4
Fig. 4 (a) The 100-interface structure tested at a Wood anomaly for the top layer. (b) Real part of total field u + uinc in the rectangles drawn in (a), for ω = 9π, θinc = −cos−1(1 − 2π/ω), ε1 = 1 and all other εi are randomly chosen between 1 and 2. Ni = 260 on sines, 100 × 2 on triangles, 90 × 5 on rectangles, Mw = 120, M = 60, P = 120, and K = 10. Flux error is 5×10−10, total solution time 35 sec (not including field evaluation). (Color online.)
Fig. 5
Fig. 5 (a) First 30 interfaces of the 1000-interface structure used in Tables 1 and 2. (b) Real part of the total field u + uinc in the rectangle drawn in (a): ω = 40, θinc = −π/5, ε1 = 1, and all other εi randomly chosen between 1 and 2. Flux error is 4.7 × 10−8. (Color online.)
Fig. 6
Fig. 6 (a) 1000-interface structure consisting of 7 complex-shaped interfaces on top of 993 sine interfaces. (b) Real part of total field u + uinc in the rectangular region drawn in (a): ω = 40, θinc = −π/4, ε1 = 1 and all other εi are chosen randomly between 1 and 3. N1 = 160 × 6, N2 = 160 × 6, N3 = 250, N4 = 180 × 2, N5 = 160 × 3, N6 = 160 × 5, and N7 = 300 on the first 7 interfaces. Ni = 300 for the rest of the sine interfaces. Mw = 130, M = 80, P = 150, K = 20, R = 1.5. Flux error is 7 × 10−9, total matrix filling time 192 sec, Schur complement 107 sec, block matrix solve 103 sec, total memory used 28 GB. Field evaluation (1000 × 1000 grid points) took 446 sec. (Color online.)
Fig. 7
Fig. 7 (a) 30-interface structure. Reflection (blue solid line) and transmission (red dashed line) as a function of incident angle from −π to 0 for: (b) ω = 2 with periodic ε = {1, 4, 1, 4,···, 4, 1, 4, 1}, average flux error 8.3 × 10−9; (c) ω = 2 with ε1 = 1 and all other εi random between 1 and 4, average flux error 1.3 × 10−10; (d) same structure as (b) but ω = 10, average flux error 4.7 × 10−7; and (e) same structure as (c) but ω = 10, average flux error 4.1 × 10−10. (Color online.)

Tables (2)

Tables Icon

Table 1 CPU time, memory, and flux error: ω = 5 (period is 0.8λ in vacuum), ε1 = 1 and all other εi are random between 1 and 2, θinc = −π/5, Ni = 70 on sine, Ni = 100×2 on triangle, and Ni = 100×5 on rectangle interfaces, Mw = 120, M = 60, P = 60, K = 20, and R = 2.

Tables Icon

Table 2 CPU time, memory, and flux error: ω = 40 (period is 6.4λ in vacuum), ε1 = 1 and all other εi are random between 1 and 2, θinc = −π/5, Ni = 180 on sine, Ni = 150 × 2 on triangle, and Ni = 340 × 5 on rectangle interfaces, Mw = 120, M = 60, P = 160, K = 20, and R = 2.

Equations (84)

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u inc ( r ) = { e i k r , r Ω 1 0 , otherwise
α : = e i d k 1 cos θ inc .
Δ u i ( r ) + k i 2 u i ( r ) = 0 , r = ( x , y ) Ω i
u 1 u 2 = u inc and u 1 n u 2 n = u inc n on Γ 1 ,
u i u i + 1 = 0 and u i n u i + 1 n = 0 on Γ i , i = 2 , 3 , I .
u i ( x + d , y ) = α u i ( x , y ) , for all ( x , y ) 2 .
u 1 ( x , y ) = n a n U e i κ n x e i k n U ( y y U ) for y y U ,
u I + 1 ( x , y ) = n a n D e i κ n x e i k n D ( y + y D ) for y y D ,
κ n : = k 1 cos θ inc + 2 π n d ,
× E = i ω μ H ,
× H = i ω ε E .
G i ( r , r ) : = i 4 H 0 ( 1 ) ( k i r r )
G qp i ( r , r ) : = l α l G i ( r , r + l d ) , where d : = ( d , 0 ) ,
( 𝒮 W i σ ) ( r ) : = W G i ( r , r ) σ ( r ) d s r , ( 𝒟 W i τ ) ( r ) : = W G i n ( r , r ) τ ( r ) d s r , r 2
( 𝒮 ˜ W i σ ) ( r ) : = l = 1 1 α l W G i ( r + r + l d ) σ ( r ) d s r ,
( 𝒟 ˜ W i τ ) ( r ) : = l = 1 1 α l W G i n ( r , r + l d ) τ ( r ) d s r .
ϕ p i ( r ) : = G i n p ( r , y p i ) + i k i G i ( r , y p i ) , r Ω i , p = 1 , , P
u 1 = 𝒟 ˜ Γ 1 1 τ 1 + 𝒮 ˜ Γ 1 1 σ 1 + p = 1 P c p 1 ϕ p 1
u i = 𝒟 ˜ Γ i 1 i τ i 1 + 𝒮 ˜ Γ i 1 i σ i 1 + 𝒟 ˜ Γ i i τ i + 𝒮 ˜ Γ i i σ i + p = 1 P c p i ϕ p i , i = 2 , 3 , , I
u I + 1 = 𝒟 ˜ Γ I I + 1 τ I + 𝒮 ˜ Γ I I + 1 σ I + p = 1 P c p I + 1 ϕ p I + 1
( S ˜ V , W i σ ) ( r ) : = l = 1 1 α l W G i ( r , r + l d ) σ ( r ) d s r ,
( D ˜ V , W i τ ) ( r ) : = l = 1 1 α l W G i n ( r , r + l d ) τ ( r ) d s r ,
( D ˜ V , W i , * σ ) ( r ) : = l = 1 1 α l W G i n ( r , r + l d ) σ ( r ) d s r ,
( T ˜ V , W i τ ) ( r ) : = l = 1 1 α l W 2 G i n n ( r , r + l d ) τ ( r ) d s r .
u 1 = 1 2 τ 1 + D ˜ Γ 1 , Γ 1 1 τ 1 + S ˜ Γ 1 , Γ 1 1 σ 1 + p = 1 P c p 1 ϕ p 1 , on Γ 1
u 2 = 1 2 τ 1 + D ˜ Γ 1 , Γ 1 2 τ 1 + S ˜ Γ 1 , Γ 1 2 σ 1 + D ˜ Γ 1 , Γ 2 2 τ 2 + S ˜ Γ 1 , Γ 2 2 σ 2 + p = 1 P c p 2 ϕ p 2 , on Γ 1
u 1 n = T ˜ Γ 1 , Γ 1 1 τ 1 + 1 2 σ 1 + D ˜ Γ 1 , Γ 1 1 , * σ 1 + p = 1 P c p 1 ϕ p 1 n , on Γ 1
u 2 n = T ˜ Γ 1 , Γ 1 2 τ 1 1 2 σ 1 + D ˜ Γ 1 , Γ 1 2 , * σ 1 + T ˜ Γ 1 , Γ 2 2 τ 2 + D ˜ Γ 1 , Γ 2 2 , * σ 2 + p = 1 P c p 2 ϕ p 2 n , on Γ 1 .
τ 1 + ( D ˜ Γ 1 , Γ 1 1 D ˜ Γ 1 , Γ 1 2 ) τ 1 + ( S ˜ Γ 1 , Γ 1 1 S ˜ Γ 1 , Γ 1 2 ) σ 1 D ˜ Γ 1 , Γ 2 2 τ 2 S ˜ Γ 1 , Γ 2 2 σ 2 + p = 1 P ( c p 1 ϕ p 1 c p 2 ϕ p 2 ) | Γ 1 = u inc | Γ 1 ,
( T ˜ Γ 1 , Γ 1 1 T ˜ Γ 1 , Γ 1 2 ) τ 1 + σ 1 + ( D ˜ Γ 1 , Γ 1 1 , * D ˜ Γ 1 , Γ 1 2 , * ) σ 1 T ˜ Γ 1 , Γ 2 2 τ 2 D ˜ Γ 1 , Γ 2 2 , * σ 2 + p = 1 P ( c p 1 ϕ p 1 n c p 2 ϕ p 2 n ) | Γ 1 = u inc n | Γ 1 .
τ i + ( D ˜ Γ i , Γ i i D ˜ Γ i , Γ i i + 1 ) τ i + ( S ˜ Γ i , Γ i i S ˜ Γ i , Γ i i + 1 ) σ i + D ˜ Γ i , Γ i 1 i τ i 1 + S ˜ Γ i , Γ i 1 i σ i 1 D ˜ Γ i , Γ i + 1 i + 1 τ i + 1 S ˜ Γ i , Γ i + 1 i + 1 σ i + 1 + p = 1 P ( c p i ϕ p i c p i + 1 ϕ p i + 1 ) | Γ i = 0 ,
( T ˜ Γ i , Γ i i T ˜ Γ i , Γ i i + 1 ) τ i + σ i + ( D ˜ Γ i , Γ i i , * D ˜ Γ i , Γ i i + 1 , * ) σ i + T ˜ Γ i , Γ i 1 i τ i 1 + D ˜ Γ i , Γ i 1 i , * σ i 1 T ˜ Γ i , Γ i + 1 i + 1 τ i + 1 D ˜ Γ i , Γ i + 1 i + 1 , * σ i + 1 + p = 1 P ( c p i ϕ p i n c p i + 1 ϕ p i + 1 n ) | Γ i = 0 .
τ I + ( D ˜ Γ I , Γ I I D ˜ Γ I , Γ I I + 1 ) τ I + ( S ˜ Γ I , Γ I I S ˜ Γ I , Γ I I + 1 ) σ I + D ˜ Γ I , Γ I 1 I τ I 1 + S ˜ Γ I , Γ I 1 I σ I 1 + p = 1 P ( c p I ϕ p I c p I + 1 ϕ p I + 1 ) | Γ I = 0 ,
( T ˜ Γ I , Γ I I T ˜ Γ I , Γ I I + 1 ) τ I + σ I + ( D ˜ Γ I , Γ I I , * D ˜ Γ I , Γ I I + 1 , * ) σ I + T ˜ Γ I , Γ I 1 I τ I 1 D ˜ Γ I , Γ I + 1 I , * σ I 1 + p = 1 P ( c p I ϕ p I n c p I + 1 ϕ p I + 1 n ) | Γ I = 0 .
η : = [ η 1 , η 2 , , η I ] T , where η i : = [ τ i σ i ] , i = 1 , 2 , , I .
c = [ c 1 , c 2 , , c I + 1 ] T , f = [ u inc | Γ 1 , u inc n | Γ 1 , 0 , , 0 ] T .
A η + Bc = f ,
A i , i = [ I + ( D ˜ Γ i , Γ i i D ˜ Γ i , Γ i i + 1 ) ( S ˜ Γ i , Γ i i S ˜ Γ i , Γ i i + 1 ) ( T ˜ Γ i , Γ i i T ˜ Γ i , Γ i i + 1 ) I + ( D ˜ Γ i , Γ i i , * D ˜ Γ i , Γ i i + 1 , * ) ] , i = 1 , 2 , , I , A i , i + 1 = [ D ˜ Γ i , Γ i + 1 i + 1 S ˜ Γ i , Γ i + 1 i + 1 T ˜ Γ i , Γ i + 1 i + 1 D ˜ Γ i , Γ i + 1 i + 1 , * ] , i = 1 , 2 , , I 1 , A i , i 1 = [ D ˜ Γ i , Γ i 1 i S ˜ Γ i , Γ i 1 i T ˜ Γ i , Γ i 1 i D ˜ Γ i , Γ i 1 i , * ] , i = 2 , 3 , , I ,
B i , i = [ ϕ 1 i | Γ i , , ϕ P i | Γ i ϕ 1 i n | Γ i , , ϕ P i n | Γ i ] , B i , i + 1 = [ ϕ 1 i + 1 | Γ i , , ϕ P i + 1 | Γ i ϕ 1 i + 1 n | Γ i , , ϕ P i + 1 n | Γ i ] , i = 1 , 2 , I
α 1 u 1 | R 1 u 1 | L 1 = α 1 ( D ˜ R 1 , Γ 1 1 τ 1 + S ˜ R 1 , Γ 1 1 σ 1 + p = 1 P c p 1 ϕ p 1 | R 1 ) ( D ˜ L 1 , Γ 1 1 τ 1 + S ˜ L 1 , Γ 1 1 σ 1 + p = 1 P c p 1 ϕ p 1 | L 1 ) = ( α 2 D R 1 + d , Γ 1 1 α D L 1 d , Γ 1 1 ) τ 1 + ( α 2 S R 1 + d , Γ 1 1 α S L 1 d , Γ 1 1 ) σ 1 + p = 1 P ( α 1 ϕ p 1 | R 1 ϕ p 1 | L 1 ) c p 1
C η + Qc = 0
C i , i = [ α 2 D R i + d , Γ i i α D L i d , Γ i i α 2 S R i + d , Γ i i α S L i d , Γ i i α 2 T R i + d , Γ i i α T L i d , Γ i i α 2 D R i + d , Γ i i , * α D L i d , Γ i i , * ]
C i , i 1 = [ α 2 D R i + d , Γ i 1 i α D L i d , Γ i 1 i α 2 S R i + d , Γ i 1 i α S L i d , Γ i 1 i α 2 T R i + d , Γ i 1 i α T L i d , Γ i 1 i α 2 D R i + d , Γ i 1 i , * α D L i d , Γ i 1 i , * ]
Q i , i = : Q i = [ α 1 ϕ 1 i | R i ϕ 1 i | L i , , α 1 ϕ P i | R i ϕ P i | L i α 1 ϕ 1 i n | R i ϕ 1 i n | L i , , α 1 ϕ P i n | R i ϕ P i n | L i ] for i = 1 , 2 , , I + 1 .
D ˜ U , Γ 1 1 τ 1 + S ˜ U , Γ 1 1 σ 1 + p = 1 P ϕ P 1 | U c p 1 n a n U e i κ n x = 0 .
T ˜ U , Γ 1 1 τ 1 + D ˜ U , Γ 1 1 , * σ 1 + p = 1 P ϕ p 1 n | U c p 1 n a n U i k n U e i κ n x = 0 .
a = [ a U , a D ] T = [ a K U , , a K U , a K D , , a K D ] T .
Z η + Vc + Wa = 0 ,
Z = [ Z U 0 0 0 0 Z D ] , V = [ V U 0 0 0 0 V D ] , W = [ W U 0 0 W D ] ,
Z U = [ D ˜ U , Γ 1 1 S ˜ U , Γ 1 1 T ˜ U , Γ 1 1 D ˜ U , Γ 1 1 , * ] , Z D = [ D ˜ D , Γ I I + 1 S ˜ D , Γ I I + 1 T ˜ D , Γ I I + 1 D ˜ D , Γ I I + 1 , * ] ,
V U = [ ϕ 1 1 | U , , ϕ P 1 | U ϕ 1 1 n | U , , ϕ P 1 n | U ] , V D = [ ϕ 1 I + 1 | D , , ϕ P I + 1 | D ϕ 1 I + 1 n | D , , ϕ P I + 1 n | D ] ,
W U = [ e i κ K x | U , , e i κ K x | U i k K U e i κ K x | U , , i k K U e i κ K x | U ] ,
W D = [ e i κ K x | D , , e i κ K x | D i k K D e i κ K x | D , , i k K D e i κ K x | D ] .
[ A B 0 C Q 0 Z V W ] [ η c a ] = [ f 0 0 ] ,
( Q i ) m p = { α 1 ϕ p i ( x m i + d ) ϕ p i ( x m i ) , m = 1 , , M w , p = 1 , , P α 1 ϕ p i n ( x m M w i + d ) ϕ p i n ( x m M w i ) , m = M w + 1 , , 2 M w , p = 1 , , P
Γ i f ( r ) d s r j = 1 N i f ( z j i ) w j i
w ( s ) = 2 π v ( s ) q v ( s ) q + v ( 2 π s ) q , where v ( s ) = ( 1 q 1 2 ) ( π s π ) 3 + 1 q s π π + 1 2 , 0 s < 2 π ,
( B i , i ) j p = [ ϕ p i ( z j i ) , j = 1 , , N i , p = 1 , , P ϕ p i n ( z j N i i ) , j = N i + 1 , , 2 N i , p = 1 , , P
N den : = 2 i = 1 I N i ,
𝒩 = N den + I P + 2 ( 2 K + 1 ) .
x = [ η , c 1 , a U , c 2 , c 3 , c I 1 , c I , c I + 1 , a D ] T = [ η , x 1 , x 2 , , x I + 1 ] T
x 1 : = [ c 1 , a U ] T , x i : = c i , i = 2 , 3 , , I , x I + 1 : = [ c I + 1 , a D ] T .
[ B 1 , 1 B 1 , 2 0 0 0 B 2 , 2 B 2 , 3 0 A 0 0 B 3 , 3 0 0 0 0 B I 1 , I + 1 0 0 0 B I , I + 1 C 1 , 1 0 0 0 0 Q 1 0 0 0 C 2 , 1 C 2 , 2 0 0 0 0 Q 2 0 0 0 C 3 , 2 C 3 , 3 0 0 0 0 Q 3 0 0 0 0 C I , I 1 C I , I 0 0 0 0 0 0 0 0 0 C I + 1 , I 0 0 0 Q I + 1 ] x = [ f 0 0 0 0 0 0 0 0 0 ] .
B 1 , 1 = [ B 1 , 1 0 ] , B I , I + 1 = [ B I , I + 1 0 ] , C 1 , 1 = [ C 1 , 1 Z U ] , C I + 1 , I = [ C I + 1 , I Z U ] , Q 1 = [ Q 1 0 V U W U ] , Q I + 1 = [ Q I + 1 0 V D W D ] .
A 1 , 1 η 1 + A 1 , 2 η 2 + B 1 , 1 x 1 + B 1 , 2 x 2 = f .
C 1 , 1 η 1 + Q 1 x 1 = 0 , C 2 , 1 η 1 + C 2 , 2 η 2 + Q 2 x 2 = 0 .
( A 1 , 1 B 1 , 1 Q 1 C 1 , 1 B 1 , 2 Q 2 C 2 , 1 ) η 1 + ( A 1 , 2 B 1 , 2 Q 2 C 2 , 2 ) η 2 = f ,
( A 2 , 1 B 2 , 2 Q 2 C 2 , 1 ) η 1 + ( A 2 , 2 B 2 , 2 Q 2 C 2 , 2 B 2 , 3 Q 3 C 3 , 2 ) η 2 + ( A 2 , 3 B 2 , 3 Q 3 C 3 , 3 ) η 3 = 0 .
[ A 1 , 1 A 1 , 2 0 0 0 0 0 A 2 , 1 A 2 , 2 A 2 , 3 0 0 0 0 0 A 3 , 2 A 3 , 3 A 3 , 4 0 0 0 0 0 0 0 A I 1 , I 2 A I 1 , I 1 A I 1 , I 0 0 0 0 0 A I , I 1 A I , I ] [ η 1 η 2 η I 1 η I ] = [ f 0 0 0 0 ] ,
A 1 , 1 = A 1 , 1 B 1 , 1 Q 1 C 1 , 1 B 1 , 2 Q 2 C 2 , 1 ,
A i , i = A i , i B i , i Q i C i , i B i , i + 1 Q i + 1 C i + 1 , i , i = 2 , 3 , , I 1 ,
A I , I = A I , I B I , I Q I C I , I B I , I + 1 Q I + 1 C I + 1 , I ,
A i , i + 1 = A i , i + 1 B i , i + 1 Q i + 1 C i + 1 , i + 1 , i = 1 , 2 , , I 1 ,
A i , i 1 = A i , i 1 B i , i Q i C i , i 1 , i = 2 , 3 , , I .
A ˜ i , i = A i , i A i , i 1 ( A ˜ i 1 , i 1 ) 1 A i 1 , i f ˜ i = f i ( A ˜ i 1 , i 1 ) 1 A i 1 , i f ˜ i .
A ˜ i , i η i = f ˜ i A i , i + 1 η i + 1 .
x 1 = Q 1 C 1 , 1 η 1 ,
x i = Q i [ C i , i 1 C i , i ] [ η i 1 η i ] , i = 2 , 3 , , I ,
x I + 1 = Q I + 1 C I + 1 , I η I .
( 𝒟 V , W i τ ) ( r ) = l = 1 1 α l W G i n ( r + r + l d ) τ ( r ) d s r .
W G i n ( r , r + l d ) τ ( r ) d s r
k n U > 0 k n U | a n U | 2 + k n D > 0 k n D | a n D | 2 = k 1 cos θ inc .
Flux error : = | k n U > 0 k n U | a n U | 2 + k n D > 0 k n D | a n D | 2 k 1 cos θ inc k 1 cos θ inc | .
T ( θ inc ) : = k n D > 0 k n D | a n D | 2 k 1 cos θ inc , R ( θ inc ) : = k n U > 0 k n U | a n U | 2 k 1 cos θ inc ,

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