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December  2017, 11(6): 1091-1105. doi: 10.3934/ipi.2017050

Near-field imaging of sound-soft obstacles in periodic waveguides

Ming Li 1, and  Ruming Zhang 2,,

1. 

School of Mechanical Engineering, University of Shanghai for Science and Technology, 200093 Shanghai, China
2. 

Center for Industrial Mathematics, University of Bremen, 28359 Bremen, Germany

* Corresponding author: Ruming Zhang.

Received  January 2017 Revised  May 2017 Published  September 2017

Fund Project: The first author was supported by the Program for Fostering of Young Teachers in the Higher Education Institutions of Shanghai, China, No. ZZslg16032.
The second author was supported by the University of Bremen and the European Union FP7.

In this paper, we introduce a direct method for the inverse scattering problems in a periodic waveguide from near-field scattered data. The direct scattering problem is to simulate the point sources scattered by a sound-soft obstacle embedded in the periodic waveguide, and the aim of the inverse problem is to reconstruct the obstacle from the near-field data measured on line segments outside the obstacle. Firstly, we will approximate the scattered field by some solutions of a series of Dirichlet exterior problems, and then the shape of the obstacle can be deduced directly from the Dirichlet boundary condition. We will also show that the approximation procedure is reasonable as the solutions of the Dirichlet exterior problems are dense in the set of scattered fields. Finally, we will give several examples to show that this method works well for different periodic waveguides.

Keywords: Near-field imaging, direct method, periodic waveguide, least square method, limiting absorption principle.
Mathematics Subject Classification: Primary: 35R30, 35J05; Secondary: 78A46, 78M50.
Citation: Ming Li, Ruming Zhang. Near-field imaging of sound-soft obstacles in periodic waveguides. Inverse Problems & Imaging, 2017, 11 (6) : 1091-1105. doi: 10.3934/ipi.2017050
References:
[1]

T. Arens, D. Gintides and A. Lechleiter, Direct and inverse medium scattering in a three-dimensional homogeneous planar waveguide, SIAM J. Appl. Math., 71 (2011), 753-772.  doi: 10.1137/100806333.  Google Scholar

[2]

G. Bao, L. Cowsar and W. Masters, Mathematical Modeling in Optical Science, SIAM, Philadelphia, 2001. doi: 10.1137/1.9780898717594.  Google Scholar

[3]

L. Bourgeois and S. Fliss, On the identification of defects in a periodic waveguide from far field data, Inverse Problems, 30 (2014), 095004, 31pp. doi: 10.1088/0266-5611/30/9/095004.  Google Scholar

[4]

L. Bourgeois and E. Lunéville, The linear sampling method in a waveguide: A modal formulation, Inverse Problems, 24 (2008), 015018, 20pp. doi: 10.1088/0266-5611/24/1/015018.  Google Scholar

[5]

L. Bourgeois and E. Lunéville, On the use of sampling methods to identify cracks in acoustic waveguides, Inverse Problems, 28 (2012), 105011, 18pp. doi: 10.1088/0266-5611/28/10/105011.  Google Scholar

[6]

G. Bruckner and J. Elschner, A two-step algorithm for the reconstruction of perfectly reflecting periodic profiles, Inverse Problems, 19 (2003), 315-329.  doi: 10.1088/0266-5611/19/2/305.  Google Scholar

[7]

G. Bruckner and J. Elschner, The numerical solution of an inverse periodic transmission problem, Math. Meth. Appl. Sci., 28 (2005), 757-778.  doi: 10.1002/mma.588.  Google Scholar

[8]

C. Burkard and R. Potthast, A multi-section approach for rough surface reconstruction via the Kirsch-Kress scheme, Inverse Problems, 26 (2010), 045007, 23pp. doi: 10.1088/0266-5611/26/4/045007.  Google Scholar

[9]

D. Colton, R. Ewing and W. Rundell, Inverse Problems in Partial Differential Equation, SIAM, Phialdelphia, 1990.  Google Scholar

[10]

M. Ehrhardt, H. Han and C. Zheng, Numerical simulation of waves in periodic structures, Commun. Comput. Phys., 5 (2009), 849-870.   Google Scholar

[11]

M. Ehrhardt, J. Sun and C. Zheng, Evaluation of scattering operators for semi-infinite periodic arrays, Commun. Math. Sci., 7 (2009), 347-364.  doi: 10.4310/CMS.2009.v7.n2.a4.  Google Scholar

[12]

J. Elschner and G. Hu, An optimization method in inverse elastic scattering for one-dimensional grating profiles, Commun. Comput. Phys., 12 (2012), 1434-1460.  doi: 10.4208/cicp.220611.130112a.  Google Scholar

[13]

S. Fliss and P. Joly, Exact boundary conditions for time-harmonic wave propagation in locally perturbed periodic media, Appl. Numer. Math., 59 (2009), 2155-2178.  doi: 10.1016/j.apnum.2008.12.013.  Google Scholar

[14]

P. Joly, J. Li and S. Fliss, Exact boundary conditions for periodic waveguides containing a local perturbation, Commun. Comput. Phys., 1 (2006), 945-973.   Google Scholar

[15]

A. Kirsch and R. Kress, On an integral equation of the first kind in inverse acoustic scattering, in Inverse Problems, Oberwolfach, 77 (1986), 93-102.  doi: 10.1007/978-3-0348-7014-6_6.  Google Scholar

[16]

A. Kirsch and R. Kress, A numerical method for an inverse scattering problem, Academic Press, Boston, 4 (1987), 279-290.   Google Scholar

[17]

A. Kirsch and R. Kress, An optimisition method in inverse acoustic scattering, in Boundary Elements IX, Stuttgart, 3 (1987), 3-18.   Google Scholar

[18]

A. Kirsch, R. Kress, P. Monk and A. Zinn, Two methods for solving the inverse acoustic scattering problem, Inverse Problems, 4 (1988), 749-770.  doi: 10.1088/0266-5611/4/3/013.  Google Scholar

[19]

J. Li, G. Sun and B. Zhang, The Kirsch-Kress method for inverse scattering by infinite locally rough interfaces, Appl. Ana., 96 (2017), 85-107.  doi: 10.1080/00036811.2016.1192141.  Google Scholar

[20]

J. Li, H. Liu and J. Zou, Locating multiple multiscale acoustic scatterers, Multiscale Model. Simul., 12 (2014), 927-952.  doi: 10.1137/13093409X.  Google Scholar

[21]

J. Li, H. Liu and Q. Wang, Locating multiple multiscale electromagnetic scatterers by a single far-field measurement, SIAM J. Imaging Sciences, 6 (2013), 2285-2309.  doi: 10.1137/130920356.  Google Scholar

[22]

J. Li, P. Li, H. Liu and X. Liu, Recovering multiscale buried anomalies in a two-layered medium, Inverse Problems, 31 (2015), 105006, 28pp. doi: 10.1088/0266-5611/31/10/105006.  Google Scholar

[23]

P. Monk and V. Selgas, Sampling type methods for an inverse waveguide problem, Inverse Problems and Imaging, 6 (2012), 709-747.  doi: 10.3934/ipi.2012.6.709.  Google Scholar

[24]

K. Sakoda, Optical Properties of Photonic Crystals, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-662-14324-7.  Google Scholar

[25]

J. Sun and C. Zheng, Numerical scattering analysis of TE plane waves by a metallic diffraction grating with local defects, J. Opt. Soc. Am. A., 26 (2009), 156-162.  doi: 10.1364/JOSAA.26.000156.  Google Scholar

[26]

J. Sun and C. Zheng, Reconstruction of obstacles embedded in waveguides, Contemporary Mathematics, 586 (2013), 341-350.  doi: 10.1090/conm/586/11652.  Google Scholar

show all references

References:
[1]

T. Arens, D. Gintides and A. Lechleiter, Direct and inverse medium scattering in a three-dimensional homogeneous planar waveguide, SIAM J. Appl. Math., 71 (2011), 753-772.  doi: 10.1137/100806333.  Google Scholar

[2]

G. Bao, L. Cowsar and W. Masters, Mathematical Modeling in Optical Science, SIAM, Philadelphia, 2001. doi: 10.1137/1.9780898717594.  Google Scholar

[3]

L. Bourgeois and S. Fliss, On the identification of defects in a periodic waveguide from far field data, Inverse Problems, 30 (2014), 095004, 31pp. doi: 10.1088/0266-5611/30/9/095004.  Google Scholar

[4]

L. Bourgeois and E. Lunéville, The linear sampling method in a waveguide: A modal formulation, Inverse Problems, 24 (2008), 015018, 20pp. doi: 10.1088/0266-5611/24/1/015018.  Google Scholar

[5]

L. Bourgeois and E. Lunéville, On the use of sampling methods to identify cracks in acoustic waveguides, Inverse Problems, 28 (2012), 105011, 18pp. doi: 10.1088/0266-5611/28/10/105011.  Google Scholar

[6]

G. Bruckner and J. Elschner, A two-step algorithm for the reconstruction of perfectly reflecting periodic profiles, Inverse Problems, 19 (2003), 315-329.  doi: 10.1088/0266-5611/19/2/305.  Google Scholar

[7]

G. Bruckner and J. Elschner, The numerical solution of an inverse periodic transmission problem, Math. Meth. Appl. Sci., 28 (2005), 757-778.  doi: 10.1002/mma.588.  Google Scholar

[8]

C. Burkard and R. Potthast, A multi-section approach for rough surface reconstruction via the Kirsch-Kress scheme, Inverse Problems, 26 (2010), 045007, 23pp. doi: 10.1088/0266-5611/26/4/045007.  Google Scholar

[9]

D. Colton, R. Ewing and W. Rundell, Inverse Problems in Partial Differential Equation, SIAM, Phialdelphia, 1990.  Google Scholar

[10]

M. Ehrhardt, H. Han and C. Zheng, Numerical simulation of waves in periodic structures, Commun. Comput. Phys., 5 (2009), 849-870.   Google Scholar

[11]

M. Ehrhardt, J. Sun and C. Zheng, Evaluation of scattering operators for semi-infinite periodic arrays, Commun. Math. Sci., 7 (2009), 347-364.  doi: 10.4310/CMS.2009.v7.n2.a4.  Google Scholar

[12]

J. Elschner and G. Hu, An optimization method in inverse elastic scattering for one-dimensional grating profiles, Commun. Comput. Phys., 12 (2012), 1434-1460.  doi: 10.4208/cicp.220611.130112a.  Google Scholar

[13]

S. Fliss and P. Joly, Exact boundary conditions for time-harmonic wave propagation in locally perturbed periodic media, Appl. Numer. Math., 59 (2009), 2155-2178.  doi: 10.1016/j.apnum.2008.12.013.  Google Scholar

[14]

P. Joly, J. Li and S. Fliss, Exact boundary conditions for periodic waveguides containing a local perturbation, Commun. Comput. Phys., 1 (2006), 945-973.   Google Scholar

[15]

A. Kirsch and R. Kress, On an integral equation of the first kind in inverse acoustic scattering, in Inverse Problems, Oberwolfach, 77 (1986), 93-102.  doi: 10.1007/978-3-0348-7014-6_6.  Google Scholar

[16]

A. Kirsch and R. Kress, A numerical method for an inverse scattering problem, Academic Press, Boston, 4 (1987), 279-290.   Google Scholar

[17]

A. Kirsch and R. Kress, An optimisition method in inverse acoustic scattering, in Boundary Elements IX, Stuttgart, 3 (1987), 3-18.   Google Scholar

[18]

A. Kirsch, R. Kress, P. Monk and A. Zinn, Two methods for solving the inverse acoustic scattering problem, Inverse Problems, 4 (1988), 749-770.  doi: 10.1088/0266-5611/4/3/013.  Google Scholar

[19]

J. Li, G. Sun and B. Zhang, The Kirsch-Kress method for inverse scattering by infinite locally rough interfaces, Appl. Ana., 96 (2017), 85-107.  doi: 10.1080/00036811.2016.1192141.  Google Scholar

[20]

J. Li, H. Liu and J. Zou, Locating multiple multiscale acoustic scatterers, Multiscale Model. Simul., 12 (2014), 927-952.  doi: 10.1137/13093409X.  Google Scholar

[21]

J. Li, H. Liu and Q. Wang, Locating multiple multiscale electromagnetic scatterers by a single far-field measurement, SIAM J. Imaging Sciences, 6 (2013), 2285-2309.  doi: 10.1137/130920356.  Google Scholar

[22]

J. Li, P. Li, H. Liu and X. Liu, Recovering multiscale buried anomalies in a two-layered medium, Inverse Problems, 31 (2015), 105006, 28pp. doi: 10.1088/0266-5611/31/10/105006.  Google Scholar

[23]

P. Monk and V. Selgas, Sampling type methods for an inverse waveguide problem, Inverse Problems and Imaging, 6 (2012), 709-747.  doi: 10.3934/ipi.2012.6.709.  Google Scholar

[24]

K. Sakoda, Optical Properties of Photonic Crystals, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-662-14324-7.  Google Scholar

[25]

J. Sun and C. Zheng, Numerical scattering analysis of TE plane waves by a metallic diffraction grating with local defects, J. Opt. Soc. Am. A., 26 (2009), 156-162.  doi: 10.1364/JOSAA.26.000156.  Google Scholar

[26]

J. Sun and C. Zheng, Reconstruction of obstacles embedded in waveguides, Contemporary Mathematics, 586 (2013), 341-350.  doi: 10.1090/conm/586/11652.  Google Scholar

Figure 1.  The scattering problem in the periodic waveguide
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Figure 2.  Direct method for inverse scattering problems.
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Figure 3.  A periodic half guide.
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Figure 4.  Waveguide 1
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Figure 5.  Waveguide 2
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Figure 6.  Four scatterers
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Figure 7.  (a)-(b): numerical result for scatter 1 with waveguides
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Figure 8.  (a)-(b): numerical result for scatter 2 with waveguides
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Figure 9.  (a)-(b): numerical result for scatter 3 with waveguides
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Figure 10.  (a)-(b): numerical result for scatter 4 with waveguides
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Figure 11.  (a)-(b): numerical result for scatter 1 with waveguides
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Figure 12.  (a)-(b): numerical result for scatter 2 with waveguides
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Figure 13.  (a)-(b): numerical result for scatter 3 with waveguides
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Figure 14.  (a)-(b): numerical result for scatter 4 with waveguides
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