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A note on double Floquet-Bloch transforms and the far-field asymptotics of Green's functions tailored to periodic structures (2402.08076v2)

Published 12 Feb 2024 in math.AP, math-ph, math.CV, and math.MP

Abstract: We propose a general procedure to study double integrals arising when considering wave propagation in periodic structures. This method, based on a complex deformation of the integration surface to bypass the integrands' singularities, is particularly efficient to estimate the Green's functions of such structures in the far field. We provide several illustrative examples and explicit asymptotic formulae. Special attention is devoted to the pathological cases of degeneracies, such as Dirac conical points for instance.

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References (51)
  1. L. Brillouin. Wave Propagation in Periodic Structures. McGraw-Hill, New York, 1946.
  2. V.M. Shalaev. Optical negative-index metamaterials. Nature Photonics, 1(1):41–48, 2007.
  3. Negative refractive index at optical wavelengths. Science, 315(5808):47–49, 2007.
  4. N.I. Zheludev. The road ahead for metamaterials. Science, 328(5978):582–583, 2010.
  5. Full-wave simulations of electromagnetic cloaking structures. Phys. Rev. E, 74(3):036621, 2006.
  6. Optical cloaking with metamaterials. Nature Photonics, 1(4):224–227, 2007.
  7. Mechanical metamaterials. Reports on Progress in Physics, 2023.
  8. Diffraction by a quarter-plane. Analytical continuation of spectral functions. Q. J. Mech. Appl. Math., 72(1):51–86, 2019.
  9. Sommerfeld-type integrals for discrete diffraction problems. Wave Motion, 97:102606, 2020.
  10. On the asymptotic properties of a canonical diffraction integral. Proc. R. Soc. A., 476:20200150, 2020.
  11. Analytical continuation of two-dimensional wave fields. Proc. R. Soc. A., 477(2245):20200681, 2021.
  12. A surprising observation in the quarter-plane diffraction problem. SIAM J. Appl. Math, 80(1):60–90, 2021.
  13. Vertex Green’s Functions of a Quarter-Plane: Links Between the Functional Equation, Additive Crossing and Lamé Functions. Q. J. Mech. Appl. Math., 74(3):251–295, 2021.
  14. Transient processes in a gas/plate structure in the case of light loading. Proc. R. Soc. A., 477(2253), 2021.
  15. Diffraction by a right-angled no-contrast penetrable wedge revisited: a double Wiener-Hopf approach. SIAM J. Appl. Math, 82(4):1495–1519, 2022.
  16. A contribution to the mathematical theory of diffraction: a note on double Fourier integrals. Q. J. Mech. Appl. Math., 76(1):1–47, 2022.
  17. Diffraction by a Right-Angled No-Contrast Penetrable Wedge: Analytical Continuation of Spectral Functions. Q. J. Mech. Appl. Math., 76(2):211–241, 2023.
  18. Diffraction by a right-angled no-contrast penetrable wedge: recovery of far-field asymptotics. Preprint, arXiv:2310.19892, 2023.
  19. P.A. Kuchment. Floquet theory for partial differential equations. Russian Mathematical Surveys, 37(4):1, 1982.
  20. S. Fliss. Analyse mathématique et numérique de problèmes de propagation des ondes dans des milieux périodiques infinis localement perturbés. PhD thesis, Ecole doctorale de l’Ecole Polytechnique (EDX) - ED 447, 2009.
  21. M.B. Radosz. The Principles of Limit Absorption and Limit Amplitude for Periodic Operators. PhD thesis, Karlsruher Instituts für Technologie, 2010.
  22. A. Lechleiter. The Floquet–Bloch transform and scattering from locally perturbed periodic surfaces. J. Math. Anal. Appl., 446(1):605–627, 2017.
  23. M.J. Lighthill. Studies on magneto-hydrodynamic waves and other anisotropic wave motions. Philos. Trans. R. Soc. A, 252(1014):397–430, 1960.
  24. M.J. Lighthill. Waves in Fluids. CUP, Cambridge, England, 1978.
  25. D.S. Jones. The Theory of Generalised Functions. CUP, Cambridge, 1982.
  26. Efficient extraction of resonant states in systems with defects. J. Comput. Phys., 477:111928, 2023.
  27. V. Laude. Principles and properties of phononic crystal waveguides. APL Materials, 9(8), 2021.
  28. The electronic properties of graphene. Rev. Mod. Phys., 81(1):109–162, 2009.
  29. M. Miniaci and R.K. Pal. Design of topological elastic waveguides. J. Appl. Phys., 130(14), 2021.
  30. Acoustic phase-reconstruction near the Dirac point of a triangular phononic crystal. Appl. Phys. Lett., 106(15):151906, 2015.
  31. Roton-maxon spectrum and stability of trapped dipolar bose-einstein condensates. Phys. Rev. Lett., 90(25):250403, 2003.
  32. Roton-like acoustical dispersion relations in 3d metamaterials. Nature Communications, 12(1), 2021.
  33. Robust collimated beaming in 3D acoustic sonic crystals. Preprint, arXiv:2301.09973, 2023.
  34. P.A. Martin. Discrete scattering theory: Green’s function for a square lattice. Wave Motion, 43(7):619–629, 2006.
  35. Asymptotics of dynamic lattice Green’s functions. Wave Motion, 67:15–31, 2016.
  36. High-frequency homogenization for periodic media. Proc. R. Soc. A., 466(2120):2341–2362, 2010.
  37. A rational framework for dynamic homogenization at finite wavelengths and frequencies. Proc. R. Soc. A., 475(2223):20180547, 2019.
  38. High-frequency homogenization in periodic media with imperfect interfaces. Proc. R. Soc. A., 476(2244):20200402, 2020.
  39. High-frequency homogenization for periodic dispersive media. Preprint, arXiv:2308.08559, 2023.
  40. B.V. Shabat. Introduction to complex analysis Part II. Functions of several variables. American Mathematical Society, 1992.
  41. C.J. Chapman. The wavenumber surface in blade-vortex interaction. In I.D. Abrahams, P.A. Martin, and M.J. Simon, editors, IUTAM symposium on diffraction and scattering in fluid mechanics and elasticity, Fluid Mechanics and Its Applications, pages 169–178. Springer, New York, NY, 2002.
  42. T. Kato. Perturbation theory for linear operators. Springer-Verlag, 1976.
  43. E.B. Davies. Linear Operators and their Spectra. CUP, 2007.
  44. Spectral study of the Laplace-Beltrami operator arising in the problem of acoustic wave scattering by a quarter-plane. Q. J. Mech. Appl. Math., 69(3):281–317, 2016.
  45. Integral representations of a pulsed signal in a waveguide. Acoustical Physics, 68(4):316–325, 2022.
  46. Saddle point method for transient processes in waveguides. J. Theor. Comput. Acoust., 30(04):2150018, 2022.
  47. Saddle point method interpretation of transient processes in car tires. Supercomputing Frontiers and Innovations, 10(1):31–45, 2023.
  48. C.H. Wilcox. Theory of bloch waves. J. Anal. Math, 33(1):146–167, 1978.
  49. Asymptotic analysis for periodic structures. North-Holland, 1978.
  50. F. Odeh and J.B. Keller. Partial Differential Equations with Periodic Coefficients and Bloch Waves in Crystals. J. Math. Phys., 5(11):1499–1504, 1964.
  51. D. Kapanadze. The far-field behaviour of green’s function for a triangular lattice and radiation conditions. Mathematical Methods in the Applied Sciences, 44(17):12746–12759, 2021.
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