T. Arens, S. N. Chandler-wilde, and J. A. Desanto, On integral equation and least squares methods for scattering by diffraction gratings, Communications In Computational Physics, vol.1, issue.6, pp.1010-1042, 2006.

T. Arens, K. Sandfort, S. Schmitt, and A. Lechleiter, Analysing Ewald's method for the evaluation of Green's functions for periodic media, IMA Journal of Applied Mathematics, vol.78, issue.3, pp.405-431, 2013.
DOI : 10.1093/imamat/hxr057

A. Barnett and L. Greengard, A new integral representation for quasi-periodic fields and its application to two-dimensional band structure calculations, Journal of Computational Physics, vol.229, issue.19, pp.6898-6914, 2010.
DOI : 10.1016/j.jcp.2010.05.029

A. Barnett and L. Greengard, A new integral representation for quasi-periodic scattering problems in two dimensions, BIT Numerical Mathematics, vol.26, issue.1, pp.67-90, 2011.
DOI : 10.1007/s10543-010-0297-x

A. Bonnet-bendhia and F. Starling, Guided waves by electromagnetic gratings and non-uniqueness examples for the diffraction problem, Mathematical Methods in the Applied Sciences, vol.12, issue.5, pp.305-338, 1994.
DOI : 10.1002/mma.1670170502

O. P. Bruno and B. Delourme, All-frequency Green's functions, Wood anomalies and uniqueness proofs in periodic rough-surface scattering, 2013.

O. P. Bruno and M. C. Haslam, Efficient high-order evaluation of scattering by periodic surfaces: deep gratings, high frequencies, and glancing incidences, Journal of the Optical Society of America A, vol.26, issue.3, pp.658-668, 2009.
DOI : 10.1364/JOSAA.26.000658

O. P. Bruno and M. C. Haslam, Efficient high-order evaluation of scattering by periodic surfaces: vector-parametric gratings and geometric singularities, Waves in Random and Complex Media, pp.530-550, 2010.
DOI : 10.1006/acha.1993.1006

O. P. Bruno and L. A. Kunyansky, A Fast, High-Order Algorithm for the Solution of Surface Scattering Problems: Basic Implementation, Tests, and Applications, Journal of Computational Physics, vol.169, issue.1, pp.80-110, 2001.
DOI : 10.1006/jcph.2001.6714

O. P. Bruno and F. Reitich, Synopsis, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, vol.82, issue.3-4, pp.3-4317, 1992.
DOI : 10.1017/S0305004100046570

O. P. Bruno and F. Reitich, Numerical solution of diffraction problems: a method of variation of boundaries III Doubly periodic gratings, Journal of the Optical Society of America A, vol.10, issue.12, pp.2551-2562, 1993.
DOI : 10.1364/JOSAA.10.002551

O. P. Bruno, S. P. Shipman, C. Turc, and S. Venakides, Efficient evaluation of doubly periodic Green functions in 3d scattering, including Wood anomaly frequencies. In preparation, Preliminary version available at arXiv: http, 2013.

F. Capolino, D. R. Wilton, and W. A. Johnson, Efficient computation of the 3D Green???s function for the Helmholtz operator for a linear array of point sources using the Ewald method, Journal of Computational Physics, vol.223, issue.1, pp.250-261, 2007.
DOI : 10.1016/j.jcp.2006.09.013

S. N. Chandler-wilde and D. C. Hothersall, Efficient calculation of the green function for acoustic propagation above a homogeneous impedance plane, Journal of Sound and Vibration, vol.180, issue.5, pp.705-724, 1995.
DOI : 10.1006/jsvi.1995.0110

S. N. Chandler-wilde and C. R. Ross, Scattering by Rough Surfaces: the Dirichlet Problem for the Helmholtz Equation in a Non-locally Perturbed Half-plane, Mathematical Methods in the Applied Sciences, vol.9, issue.12, pp.959-976, 1996.
DOI : 10.1002/(SICI)1099-1476(199608)19:12<959::AID-MMA806>3.0.CO;2-R

D. Colton and R. Kress, Inverse Acoustic And Electromagnetic Scattering Theory, Applied Mathematical Sciences, vol.93, 1998.

A. D. Craik, Prehistory of Fa???????? di Bruno's Formula, The American Mathematical Monthly, vol.112, issue.2, pp.119-130, 2005.
DOI : 10.2307/30037410

J. A. Desanto, G. Erdmann, W. Hereman, and M. Misra, Theoretical and computational aspects of scattering from rough surfaces: one-dimensional perfectly reflecting surfaces, Waves in Random Media, vol.5, issue.4, pp.385-414, 1998.
DOI : 10.1088/0959-7174/6/2/004

A. Dienstfrey, F. Hang, and J. Huang, Lattice sums and the two-dimensional, periodic Green's function for the Helmholtz equation, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.457, issue.2005, pp.45767-85, 2001.
DOI : 10.1098/rspa.2000.0656

R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Computers in Physics, vol.3, issue.5, 1994.
DOI : 10.1063/1.4822863

N. Guérin, S. Enoch, and G. Tayeb, Combined Method for the Computation of the Doubly Periodic Green's Function, Journal of Electromagnetic Waves and Applications, vol.10, issue.2, pp.205-221, 2001.
DOI : 10.1163/156939301X01363

A. Kirsch, Diffraction by periodic structures, Inverse problems in mathematical physics, pp.87-102, 1992.
DOI : 10.1007/3-540-57195-7_11

A. Kirsch, Uniqueness theorems in inverse scattering theory for periodic structures, Inverse Problems, vol.10, issue.1, pp.145-152, 1994.
DOI : 10.1088/0266-5611/10/1/011

R. Kress, Linear Integral Equations, Applied Mathematical Sciences, vol.82, 1999.

H. Kurkcu and F. Reitich, Stable and efficient evaluation of periodized Green???s functions for the Helmholtz equation at high frequencies, Journal of Computational Physics, vol.228, issue.1, pp.75-95, 2009.
DOI : 10.1016/j.jcp.2008.08.021

N. N. Lebedev, Special Functions And Their Applications Revised English edition, 1965.

C. Linton, The Green's function for the two-dimensional Helmholtz equation in periodic domains, Journal of Engineering Mathematics, vol.33, issue.4, pp.377-402, 1998.
DOI : 10.1023/A:1004377501747

C. Linton and I. Thompson, One- and two-dimensional lattice sums for the three-dimensional Helmholtz equation, Journal of Computational Physics, vol.228, issue.6, pp.1815-1829, 2009.
DOI : 10.1016/j.jcp.2008.11.013

C. M. Linton, Lattice Sums for the Helmholtz Equation, SIAM Review, vol.52, issue.4, pp.630-674, 2010.
DOI : 10.1137/09075130X

E. G. Loewen and E. Popov, Diffraction gratings and applications, 1997.

E. Martensen, ??ber eine Methode zum r??umlichen Neumannschen Problem mit einer Anwendung f??r torusartige Berandungen, Acta Mathematica, vol.109, issue.0, pp.75-135, 1007.
DOI : 10.1007/BF02391810

A. W. Mathis and A. F. Peterson, A comparison of acceleration procedures for the two-dimensional periodic Green's function, IEEE Transactions on Antennas and Propagation, vol.44, issue.4, pp.567-571, 1996.
DOI : 10.1109/8.489309

D. Maystre, I Rigorous Vector Theories of Diffraction Gratings, Progress in optics, pp.1-67, 1984.
DOI : 10.1016/S0079-6638(08)70121-5

D. Maystre, Theory of Wood???s Anomalies, Plasmonics, pp.39-83, 2012.
DOI : 10.1007/978-3-642-28079-5_2

J. A. Monro and J. , A Super-Algebraically Convergent, Windowing-Based Approach to the Evaluation of Scattering from Periodic Rough Surfaces, 2007.

A. Moroz, Exponentially convergent lattice sums, Optics Letters, vol.26, issue.15, pp.1119-1121, 2001.
DOI : 10.1364/OL.26.001119

J. Nédélec and F. Starling, Integral Equation Methods in a Quasi-Periodic Diffraction Problem for the Time-Harmonic Maxwell???s Equations, SIAM Journal on Mathematical Analysis, vol.22, issue.6, pp.1679-1701, 1991.
DOI : 10.1137/0522104

M. Nevière and E. Popov, Light propagation in periodic media: differential theory and design, 2003.

N. A. Nicorovici and R. C. Mcphedran, Lattice sums for off-axis electromagnetic scattering by gratings, Physical Review E, vol.50, issue.4, pp.3143-3160, 1994.
DOI : 10.1103/PhysRevE.50.3143

N. A. Nicorovici, R. C. Mcphedran, and R. Petit, Efficient calculation of the Green???s function for electromagnetic scattering by gratings, Physical Review E, vol.49, issue.5, pp.4563-4577, 1994.
DOI : 10.1103/PhysRevE.49.4563

E. J. Nyström, ??ber Die Praktische Aufl??sung von Integralgleichungen mit Anwendungen auf Randwertaufgaben, Acta Mathematica, vol.54, issue.0, pp.185-204, 1930.
DOI : 10.1007/BF02547521

R. Petit, Electromagnetic Theory Of Gratings, volume 22 of Topics in Current Physics, 1980.

G. R. Valenzuela, Theories for the interaction of electromagnetic and oceanic waves?a review. Boundary- Layer Meteorology, pp.61-85, 1978.

M. E. Veysoglu, H. A. Yueh, R. T. Shin, and J. A. Kong, Polarimetric passive remote sensing of periodic surfaces, Journal of Electromagnetic Waves and Applications, vol.5, issue.3, pp.267-280, 1991.

B. Zhang and S. N. Chandler-wilde, Integral equation methods for scattering by infinite rough surfaces, Mathematical Methods in the Applied Sciences, vol.245, issue.6, pp.463-488, 2003.
DOI : 10.1002/mma.361