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非均匀波导中的最大声能流透射及鲁棒性分析

郭威 杨德森

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非均匀波导中的最大声能流透射及鲁棒性分析

郭威, 杨德森

Maximal transmission of acoustic energy flux in inhomogeneous waveguides and robustness analyses

Guo Wei, Yang De-Sen
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  • 以变截面含可穿透散射体波导为模型, 理论研究声波在非均匀波导中的最大透射问题. 通过耦合简正波理论构建模态域内透射矩阵和水平波数矩阵, 推导透射波能流的具体表达式, 分析任意入射波的能流透射率随频率的变化, 进而讨论任意给定频率下能够产生最大能流透射率的最佳入射波, 并给出数组全透射声场算例. 最佳入射波仅由可传播模态决定, 与衰逝模态无关. 利用衰逝模态不携带能流的特性, 讨论衰逝模态对产生能流最大透射声场的影响, 并分析最大能流透射的鲁棒性. 在频率满足一定条件时, 全透射声场可能表现出完美鲁棒性. 文中所述方法可延伸至多种非均匀波导以分析其中的能流最大透射问题.
    Inhomogeneity in a medium will cause wave scattering, influencing the transfer of energy or information. However, it is possible to prepare a prescribed wavefront which propagates through an inhomogeneous medium with unity flux-transmittance. This phenomenon is first predicted in the context of mesoscopic electron transport. Another remarkable phenomenon is the bimodal distribution of the transmission singular values, which implies that in a lossless medium the full solution space in the scattering region can be spanned only by open channels, which are completely transmitted, and closed channels, which are completely reflected. In mesoscopic physics, random-matrix theory is usually utilized to deal with the statistical properties of matrices with randomly distributed entries since the medium is assumed to be randomly fluctuating. In this paper, we propose a method of systematically studying the maximal flux transmission through an inhomogeneous acoustic waveguide. The model is chosen to be a waveguide with varying cross-sections and a penetrable scatterer, and the method is based on the coupled mode theory. This method can be used to analyze the frequency of nearly complete transmission for an arbitrary incident wave, and to analyze the incident wave that is able to generate the maximal flux-transmittance for any given frequency. We construct the transmission matrix and the horizontal wavenumber matrix by using orthonormal basis functions, and give the expression of flux-transmittance. Then the optimal incident wave which brings the maximal transmittance through the scattering region is derived based on singular value decomposition. The optimal incident waves are independent of the evanescent modes since evanescent modes do not transfer any energy. But the evanescent modes can give rise to the multivaluedness of wave solutions with complete flux transmission. Considering the fact that acoustic waveguides can naturally resist the influence of highly oscillating perturbations since most of them correspond to evanescent modes), the maximal flux transmission in waveguide is thus found to be highly robust. Especially at a specific frequency, the complete wave transmission has perfect robustness. This proposed method can be generalized to any other frequency, to other types of scatterers, or to other kinds of boundary conditions, and can provide guidance in designing acoustic metamaterials and in highly efficient communication.
      通信作者: 杨德森, dshyang@hrbeu.edu.cn
      Corresponding author: Yang De-Sen, dshyang@hrbeu.edu.cn
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    Nazarov Y V 1994 Phys. Rev. Lett. 73 134Google Scholar

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    Beenakker C W J 1997 Rev. Mod. Phys. 69 731Google Scholar

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    Vellekoop I M, Lagendijk A, Mosk A P 2010 Nat. Photonics 4 320Google Scholar

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    Popoff S M, Lerosey G, Carminati R, Fink M, Boccara A C, Gigan S 2010 Phys. Rev. Lett. 104 100601Google Scholar

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    Vellekoop I M, Mosk A P 2008 Phys. Rev. Lett. 101 120601Google Scholar

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    Choi W, Mosk A P, Park Q H, Choi W 2011 Phys. Rev. B 83 134207Google Scholar

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    Kim M, Choi Y, Yoon C, Choi W, Kim J, Park Q H, Choi W 2012 Nat. Photonics 6 581Google Scholar

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    Popoff S M, Goetschy A, Liew S F, Stone A D, Cao H 2014 Phys. Rev. Lett. 112 133903Google Scholar

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    Hao X, Martin-Rouault L, Cui M 2014 Sci. Rep. 4 5874

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    Gérardin B, Laurent J, Derode A, Prada C, Aubry A 2014 Phys. Rev. Lett. 113 173901Google Scholar

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    Liew S F, Popoff S M, Mosk A P, Vos W L, Cao H 2014 Phys. Rev. B 89 224202Google Scholar

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    Liew S F, Cao H 2015 Opt. Express 23 11043Google Scholar

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    Yamilov A, Petrenko S, Sarma R, Cao H 2016 Phys. Rev. B 93 100201Google Scholar

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    Wang Z, Chong Y D, Joannopoulos J D, Soljačić M 2009 Nature 461 772Google Scholar

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    Lu L, Joannopoulos J D, Soljačić M 2014 Nat. Photonics 8 821Google Scholar

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    Rahm M, Cummer S A, Schurig D, Pendry J B, Smith D R 2008 Phys. Rev. Lett. 100 063903Google Scholar

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    Yu N F, Capasso F 2013 Nat. Mater. 13 139

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    Asadchy V S, Faniayeu I A, Ra’di Y, Khakhomov S A, Semchenko I V, Tretyakov S A 2015 Phys. Rev. X 5 031005

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    Bonnet-Ben Dhia A S, Nazarov S A 2013 Acoust. Phys. 59 633Google Scholar

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    Bonnet-Ben Dhia A S, Chesnel L, Nazarov S A 2015 Inverse Prob. 31 045006Google Scholar

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    Bonnet-Ben Dhia A S, Lunéville E, Mbeutcha Y, Nazarov S A 2017 Math. Methods Appl. Sci. 40 335Google Scholar

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    Bonnet-Ben Dhia A S, Chesnel L, Nazarov S A 2018 J. Math. Pures Appl. 111 79Google Scholar

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    Derode A, Roux P, Fink M 1995 Phys. Rev. Lett. 75 4206Google Scholar

    [29]

    Fink M, Cassereau D, Derode A, Prada C, Roux P, Tanter M, Thomas J L, Wu F 2000 Rep. Prog. Phys. 63 1933Google Scholar

    [30]

    Lerosey G, Rosny J, Tourin A, Fink M 2007 Science 315 1120Google Scholar

    [31]

    Mosk A P, Lagendijk A, Lerosey G, Fink M 2012 Nat. Photonics 6 283Google Scholar

    [32]

    Ma F Y, Huang M, Xu Y C, Wu J H 2018 Sci. Rep. 8 5906Google Scholar

    [33]

    Chen J, Xiao J, Lisevych D, Shakouri A, Fan Z 2018 Nat. Commun. 9 4920Google Scholar

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    Li Y, Assouar B M 2016 Appl. Phys. Lett. 108 063502Google Scholar

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    Shen C, Cummer S A 2018 Phys. Rev. Appl. 9 054009Google Scholar

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    Maurel A, Mercier J F, Félix S 2014 J. Acoust. Soc. Am. 135 165Google Scholar

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    刘娟, 李琪 2021 物理学报 70 064301Google Scholar

    Liu J, Li Q 2021 Acta Phys. Sin. 70 064301Google Scholar

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    Lu Y Y, McLaughlin J R 1996 J. Acoust. Soc. Am. 100 1432Google Scholar

    [39]

    Lu Y Y 2005 J. Comput. Appl. Math. 173 247Google Scholar

    [40]

    Pagneux V 2010 J. Comput. Appl. Math. 234 1834Google Scholar

    [41]

    Bonnet-Ben Dhia A S, Chesnel L, Pagneux V 2018 Proc. R. Soc. London, Ser. A 474 20180050

    [42]

    Chéron É, Félix S, Pagneux V 2019 Phys. Rev. Lett. 122 125501Google Scholar

    [43]

    Guo W, Liu J, Bi W P, Aurégan Y, Pagneux V 2020 arXiv 2010.03646

    [44]

    Pagneux V, Maurel A 2002 P. Roc. Soc. A-Math. Phys. 458 1913Google Scholar

    [45]

    Li Q, Liu J, Guo W 2019 Chin. Phys. B 29 014303

    [46]

    Pagneux V, Maurel A 2006 Proc. R. Soc. A- Math. Phys. 462 1315

    [47]

    郭威, 杨德森 2020 物理学报 69 074301Google Scholar

    Guo W, Yang D S 2020 Acta Phys. Sin. 69 074301Google Scholar

  • 图 1  变截面含散射体非均匀波导示意图

    Fig. 1.  Configuration of the inhomogeneous waveguide with varying cross-sections and one scatterer.

    图 2  平面波能流透射率随频率变化曲线

    Fig. 2.  Energy flux transmittance as a function of frequency when injecting a plane wave.

    图 3  (a) 平面波能流全透射声场($k = 0.95{\text{π}} $); (b) 平面波能流零透射声场($k = 0.735{\text{π}} $)

    Fig. 3.  (a) Wave field with unity energy flux transmittance generated by a plane wave ($k = 0.95{\text{π}} $); (b) wave field with zero energy flux transmittance generated by a plane wave ($k = 0.735{\text{π}} $).

    图 4  (a) Comsol计算平面波能流全透射声场($f = 712$ Hz); (b) Comsol计算平面波能流零透射声场($f = 551$ Hz)

    Fig. 4.  (a) Wave field calculated by Comsol with unity energy flux transmittance generated by a plane wave ($f = $$ 712$ Hz); (b) wave field calculated by Comsol with zero energy flux transmittance generated by a plane wave ($f = $$ 551$ Hz).

    图 5  (a) 奇异值平方分布; (b) 最佳入射波幅值分布; (c) 最佳入射波的模态展开系数; (d) 产生能流最大透射的声场. 波导参数与图2中使用的一致, 频率$k = 1.45{\text{π}} $

    Fig. 5.  (a) Distribution of squares of singular values; (b) modulus of the optimal incident wave; (c) expansion coefficients of the optimal incident wave; (d) wave field with the maximum energy flux transmittance. The geometry of the waveguide is same as that in Fig. 2, and the frequency is $k = 1.45{\text{π}} $.

    图 6  (a) 图3(a)情况下平面波叠加衰逝模态后的声场; (b) 图5(d)情况下最佳入射波叠加衰逝模态后的声场

    Fig. 6.  (a) Wave field generated by a plane wave mixed by evanescent modes in the case of Fig. 3(a); (b) wave field generated by the optimal incident wave mixed by evanescent modes in the case of Fig. 5(d).

    图 7  最佳入射波存在随机扰动时的声场

    Fig. 7.  Wave field generated by a disturbed optimal incident wave.

    图 8  最佳入射波存在随机干扰时能流透射率的 (a)期望和(b)相对标准差随频率的变化

    Fig. 8.  (a) Expectation and (b) relative standard deviation of the flux-transmittance when the optimal incident wave is randomly perturbed.

  • [1]

    Dorokhov O 1982 Solid State Commun. 44 915Google Scholar

    [2]

    Dorokhov O 1984 Solid State Commun. 51 381Google Scholar

    [3]

    Imry Y 1986 Europhys. Lett. 1 249Google Scholar

    [4]

    Pendry J B, MacKinnon A, Pretre A B 1990 Physica A 168 400Google Scholar

    [5]

    Mello P A, Pereyra P, Kumar N 1988 Ann. Phys. 181 290Google Scholar

    [6]

    Nazarov Y V 1994 Phys. Rev. Lett. 73 134Google Scholar

    [7]

    Beenakker C W J 1997 Rev. Mod. Phys. 69 731Google Scholar

    [8]

    Vellekoop I M, Lagendijk A, Mosk A P 2010 Nat. Photonics 4 320Google Scholar

    [9]

    Popoff S M, Lerosey G, Carminati R, Fink M, Boccara A C, Gigan S 2010 Phys. Rev. Lett. 104 100601Google Scholar

    [10]

    Vellekoop I M, Mosk A P 2008 Phys. Rev. Lett. 101 120601Google Scholar

    [11]

    Choi W, Mosk A P, Park Q H, Choi W 2011 Phys. Rev. B 83 134207Google Scholar

    [12]

    Kim M, Choi Y, Yoon C, Choi W, Kim J, Park Q H, Choi W 2012 Nat. Photonics 6 581Google Scholar

    [13]

    Popoff S M, Goetschy A, Liew S F, Stone A D, Cao H 2014 Phys. Rev. Lett. 112 133903Google Scholar

    [14]

    Hao X, Martin-Rouault L, Cui M 2014 Sci. Rep. 4 5874

    [15]

    Gérardin B, Laurent J, Derode A, Prada C, Aubry A 2014 Phys. Rev. Lett. 113 173901Google Scholar

    [16]

    Liew S F, Popoff S M, Mosk A P, Vos W L, Cao H 2014 Phys. Rev. B 89 224202Google Scholar

    [17]

    Liew S F, Cao H 2015 Opt. Express 23 11043Google Scholar

    [18]

    Yamilov A, Petrenko S, Sarma R, Cao H 2016 Phys. Rev. B 93 100201Google Scholar

    [19]

    Wang Z, Chong Y D, Joannopoulos J D, Soljačić M 2009 Nature 461 772Google Scholar

    [20]

    Lu L, Joannopoulos J D, Soljačić M 2014 Nat. Photonics 8 821Google Scholar

    [21]

    Rahm M, Cummer S A, Schurig D, Pendry J B, Smith D R 2008 Phys. Rev. Lett. 100 063903Google Scholar

    [22]

    Yu N F, Capasso F 2013 Nat. Mater. 13 139

    [23]

    Asadchy V S, Faniayeu I A, Ra’di Y, Khakhomov S A, Semchenko I V, Tretyakov S A 2015 Phys. Rev. X 5 031005

    [24]

    Bonnet-Ben Dhia A S, Nazarov S A 2013 Acoust. Phys. 59 633Google Scholar

    [25]

    Bonnet-Ben Dhia A S, Chesnel L, Nazarov S A 2015 Inverse Prob. 31 045006Google Scholar

    [26]

    Bonnet-Ben Dhia A S, Lunéville E, Mbeutcha Y, Nazarov S A 2017 Math. Methods Appl. Sci. 40 335Google Scholar

    [27]

    Bonnet-Ben Dhia A S, Chesnel L, Nazarov S A 2018 J. Math. Pures Appl. 111 79Google Scholar

    [28]

    Derode A, Roux P, Fink M 1995 Phys. Rev. Lett. 75 4206Google Scholar

    [29]

    Fink M, Cassereau D, Derode A, Prada C, Roux P, Tanter M, Thomas J L, Wu F 2000 Rep. Prog. Phys. 63 1933Google Scholar

    [30]

    Lerosey G, Rosny J, Tourin A, Fink M 2007 Science 315 1120Google Scholar

    [31]

    Mosk A P, Lagendijk A, Lerosey G, Fink M 2012 Nat. Photonics 6 283Google Scholar

    [32]

    Ma F Y, Huang M, Xu Y C, Wu J H 2018 Sci. Rep. 8 5906Google Scholar

    [33]

    Chen J, Xiao J, Lisevych D, Shakouri A, Fan Z 2018 Nat. Commun. 9 4920Google Scholar

    [34]

    Li Y, Assouar B M 2016 Appl. Phys. Lett. 108 063502Google Scholar

    [35]

    Shen C, Cummer S A 2018 Phys. Rev. Appl. 9 054009Google Scholar

    [36]

    Maurel A, Mercier J F, Félix S 2014 J. Acoust. Soc. Am. 135 165Google Scholar

    [37]

    刘娟, 李琪 2021 物理学报 70 064301Google Scholar

    Liu J, Li Q 2021 Acta Phys. Sin. 70 064301Google Scholar

    [38]

    Lu Y Y, McLaughlin J R 1996 J. Acoust. Soc. Am. 100 1432Google Scholar

    [39]

    Lu Y Y 2005 J. Comput. Appl. Math. 173 247Google Scholar

    [40]

    Pagneux V 2010 J. Comput. Appl. Math. 234 1834Google Scholar

    [41]

    Bonnet-Ben Dhia A S, Chesnel L, Pagneux V 2018 Proc. R. Soc. London, Ser. A 474 20180050

    [42]

    Chéron É, Félix S, Pagneux V 2019 Phys. Rev. Lett. 122 125501Google Scholar

    [43]

    Guo W, Liu J, Bi W P, Aurégan Y, Pagneux V 2020 arXiv 2010.03646

    [44]

    Pagneux V, Maurel A 2002 P. Roc. Soc. A-Math. Phys. 458 1913Google Scholar

    [45]

    Li Q, Liu J, Guo W 2019 Chin. Phys. B 29 014303

    [46]

    Pagneux V, Maurel A 2006 Proc. R. Soc. A- Math. Phys. 462 1315

    [47]

    郭威, 杨德森 2020 物理学报 69 074301Google Scholar

    Guo W, Yang D S 2020 Acta Phys. Sin. 69 074301Google Scholar

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出版历程
  • 收稿日期:  2021-03-14
  • 修回日期:  2021-04-14
  • 上网日期:  2021-06-07
  • 刊出日期:  2021-09-05

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