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A High-Accuracy Mode Solver for Acoustic Scattering by a Periodic Array of Axially Symmetric Obstacles

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Abstract

This paper is concerned with guided modes of an acoustic wave propagation problem on a periodic array of axially symmetric obstacles. A guided mode refers to a quasi-periodic eigenfield that propagates along the obstacles but decays exponentially away from them in the absence of incidences. Thus, the problem can be studied in an unbound unit cell due to the quasi-periodicity. We truncate the unit cell onto a cylinder enclosing the interior obstacle in terms of utilizing Rayleigh’s expansion to design an exact condition on the lateral boundary. We derive a new boundary integral equation (BIE) only involving the free-space Green function on the boundary of each homogeneous region within the cylinder. Due to the axial symmetry of the boundaries, each BIE is decoupled via the Fourier transform to curve BIEs and they are discretized with high-accuracy quadratures. With the lateral boundary condition and the side quasi-periodic condition, the discretized BIEs lead to a homogeneous linear system governing the propagation constant of a guided mode at a given frequency. The propagation constant is determined by enforcing that the coefficient matrix is singular. The accuracy of the proposed method is demonstrated by a number of examples even when the obstacles have sharp edges or corners.

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References

  1. Achenbach, J.D., Kitahara, M.: Reflection and transmission of an obliquely incident wave by an array of spherical cavities. J. Acoust. Soc. Am. 80(4), 1209–1214 (1986)

    Article  Google Scholar 

  2. Bao, G.: Finite element approximation of time harmonic waves in periodic structures. SIAM J. Numer. Anal. 32(4), 1155–1169 (1995)

    Article  MathSciNet  Google Scholar 

  3. Bao, G., Li, P.: Maxwell’s Equations in Periodic Structures. Applied Mathematical Sciences. Springer, Singapore (2021)

    Google Scholar 

  4. Bonnet-Bendhia, A., Starling, F.: Guided waves by electromagnetic gratings and non-uniqueness examples for the diffraction problem. Math. Methods Appl. Sci. 17(5), 305–338 (1994)

    Article  MathSciNet  Google Scholar 

  5. Bremer, J., Gimbutas, Z., Rokhlin, V.: A nonlinear optimization procedure for generalized gaussian quadratures. SIAM J. Sci. Comput. 32(4), 1761–1788 (2010)

    Article  MathSciNet  Google Scholar 

  6. Bruno, O.P., Fernandez-Lado, A.G.: On the evaluation of quasi-periodic green functions and wave-scattering at and around Rayleigh-wood anomalies. J. Comput. Phys. 410, 109352 (2020)

    Article  MathSciNet  Google Scholar 

  7. Bulgakov, E.N., Maksimov, D.N.: Optical response induced by bound states in the continuum in arrays of dielectric spheres. J. Opt. Soc. Am. B 35(10), 2443–2452 (2018)

    Article  Google Scholar 

  8. Boisvert, R., Olver, F., Lozier, D., Clark, C.: The NIST Handbook of Mathematical Functions. Cambridge University Press, New York (2010)

  9. Cheng, H., Crutchfield, W.Y., Doery, M., Greengard, L.: Fast, accurate integral equation methods for the analysis of photonic crystal fibers I: theory. Opt. Express 12(16), 3791–3805 (2004)

    Article  Google Scholar 

  10. Colton, D., Kress, R.: Inverse Acoustic and Electromagnetic Scattering Theory. Applied Mathematical Sciences. Springer, Berlin (2013)

    Book  Google Scholar 

  11. Ebbesen, T.W., Lezec, H.J., Ghaemi, H.F., Thio, T., Wolff, P.A.: Extraordinary optical transmission through sub-wavelength hole arrays. Nature 391(6668), 667–669 (1998)

    Article  Google Scholar 

  12. Evans, D.V., Linton, C.M.: Edge waves along periodic coastlines. Q. J. Mech. Appl. Mech. 46(4), 643–656 (1993)

    Article  MathSciNet  Google Scholar 

  13. Helsing, J., Karlsson, A.: An explicit kernel-split panel-based nyström scheme for integral equations on axially symmetric surfaces. J. Comput. Phys. 272, 686–703 (2014)

    Article  MathSciNet  Google Scholar 

  14. Janning, D.S., Munk, B.A.: Effects of surface waves on the currents of truncated periodic arrays. IEEE Trans. Antennas Propag. 50(9), 1254–1265 (2002)

    Article  Google Scholar 

  15. Kleemann, B.H.: Fast integral methods for integrated optical systems simulations: a review. In: Smith, D.G., Wyrowski, F., Erdmann, A. (eds.) Optical Systems Design 2015: Computational Optics, Volume 9630 of Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, p. 96300Q (2015)

  16. Kress, R.: Linear Integral Equations, vol. 82, 3rd edn. Springer, Berlin (2014)

    Book  Google Scholar 

  17. Lai, J., O’Neil, M.: An FFT-accelerated direct solver for electromagnetic scattering from penetrable axisymmetric objects. J. Comput. Phys. 390, 152–174 (2019)

    Article  MathSciNet  Google Scholar 

  18. Li, C., Zhou, S., Liu, T., Xiao, S.: Symmetry-protected bound states in the continuum supported by all-dielectric metasurfaces. Phys. Rev. A 100, 063803 (2019)

    Article  Google Scholar 

  19. Li, L.: Use of Fourier series in the analysis of discontinuous periodic structures. J. Opt. Soc. Am. A 13(9), 1870–1876 (1996)

    Article  Google Scholar 

  20. Linton, C., Zalipaev, V., Thompson, I.: Electromagnetic guided waves on linear arrays of spheres. Wave Motion 50, 29–40 (2013)

    Article  MathSciNet  Google Scholar 

  21. Liu, Y., Barnett, A.H.: Efficient numerical solution of acoustic scattering from doubly-periodic arrays of axisymmetric objects. J. Comput. Phys. 324, 226–245 (2016)

    Article  MathSciNet  Google Scholar 

  22. Liu, Z., Chan, C.T., Sheng, P., Goertzen, A.L., Page, J.H.: Elastic wave scattering by periodic structures of spherical objects: theory and experiment. Phys. Rev. B 62, 2446–2457 (2000)

    Article  Google Scholar 

  23. Lu, W., Lu, Y.Y.: Efficient boundary integral equation method for photonic crystal fibers. J. Lightwave Technol. 20(11), 1610–1616 (2012)

    Article  Google Scholar 

  24. Lu, W., Lu, Y.Y.: Efficient high order waveguide mode solvers based on boundary integral equations. J. Comput. Phys. 272, 507–525 (2014)

    Article  MathSciNet  Google Scholar 

  25. Lu, W., Lu, Y.Y.: High order integral equation method for diffraction gratings. J. Opt. Soc. Am. A 29(5), 734–740 (2012)

    Article  Google Scholar 

  26. Lu, W., Lu, Y.Y.: Waveguide mode solver based on Neumann-to-Dirichlet operators and boundary integral equations. J. Comput. Phys. 231(4), 1360–1371 (2012)

    Article  MathSciNet  Google Scholar 

  27. Lu, W., Lu, Y.Y., Qian, J.: Perfectly matched layer boundary integral equation method for wave scattering in a layered medium. SIAM J. Appl. Math. 78(1), 246–265 (2018)

    Article  MathSciNet  Google Scholar 

  28. Marinica, D.C., Borisov, A.G., Shabanov, S.V.: Bound states in the continuum in photonics. Phys. Rev. Lett. 100, 183902 (2008)

    Article  Google Scholar 

  29. Porter, R., Evans, D.V.: Rayleigh Bloch surface waves along periodic gratings and their connection with trapped modes in waveguides. J. Fluid Mech. 386(1), 233–258 (1999)

    Article  MathSciNet  Google Scholar 

  30. Shore, R.A., Yaghjian, A.D.: Travelling electromagnetic waves on linear periodic arrays of lossless spheres. Electron. Lett. 41, 578–580 (2005)

    Article  Google Scholar 

  31. Shore, R.A., Yaghjian, A.D.: Traveling waves on two- and three-dimensional periodic arrays of lossless scatterers. Radio Sci. 42(6), 1–40 (2007)

    Article  Google Scholar 

  32. Thompson, I., Linton, C.M.: Guided surface waves on one- and two-dimensional arrays of spheres. SIAM J. Appl. Math. 70(8), 2975–2995 (2010)

    Article  MathSciNet  Google Scholar 

  33. Twersky, V.: Multiple scattering of sound by a periodic line of obstacles. J. Acoust. Soc. Am. 53(1), 96–112 (1973)

    Article  Google Scholar 

  34. Twersky, V.: Lattice sums and scattering coefficients for the rectangular planar array. J. Math. Phys. 16(3), 644–657 (1975)

    Article  Google Scholar 

  35. Twersky, V.: Low frequency coupling in the planar rectangular lattice. J. Math. Phys. 16(3), 658–666 (1975)

    Article  Google Scholar 

  36. Twersky, V.: Multiple scattering of waves by the doubly periodic planar array of obstacles. J. Math. Phys. 16(3), 633–643 (1975)

    Article  Google Scholar 

  37. Vaishnav, J.Y., Walls, J.D., Apratim, M., Heller, E.J.: Matter-wave scattering and guiding by atomic arrays. Phys. Rev. A 76, 013620 (2007)

    Article  Google Scholar 

  38. Wu, B., Cho, M.H.: Robust fast direct integral equation solver for three-dimensional doubly periodic scattering problems with a large number of layers. J. Comput. Phys. 495(C), 112573 (2024)

    MathSciNet  Google Scholar 

  39. Young, P., Hao, S., Martinsson, P.G.: A high-order Nyström discretization scheme for boundary integral equations defined on rotationally symmetric surfaces. J. Comput. Phys. 231(11), 4142–4159 (2012)

    Article  MathSciNet  Google Scholar 

  40. Yuan, L., Lu, Y.Y.: Bound states in the continuum on periodic structures: perturbation theory and robustness. Opt. Lett. 42(21), 4490–4493 (2017)

    Article  Google Scholar 

  41. Zhou, J., Lu, W.: Numerical analysis of resonances by a slab of subwavelength slits by Fourier-matching method. SIAM J. Numer. Anal. 59(4), 2106–2137 (2021)

    Article  MathSciNet  Google Scholar 

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Funding

W.L. is partially supported by National Key Research and Development Program of China (Grant No. 2023YFA1009100), NSFC Grant 12174310 and a Key Project of Joint Funds For Regional Innovation and Development (U21A20425).

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Authors and Affiliations

  1. School of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, China

    Hangya Wang & Wangtao Lu

  2. Institute of Fundamental and Transdisciplinary Research, Zhejiang University, Hangzhou, 310027, China

    Wangtao Lu

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  1. Hangya Wang
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Correspondence to Wangtao Lu.

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Wang, H., Lu, W. A High-Accuracy Mode Solver for Acoustic Scattering by a Periodic Array of Axially Symmetric Obstacles. J Sci Comput 101, 23 (2024). https://doi.org/10.1007/s10915-024-02659-2

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  • DOI: https://doi.org/10.1007/s10915-024-02659-2

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