# American Institute of Mathematical Sciences

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

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

 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.

Citation: Ming Li, Ruming Zhang. Near-field imaging of sound-soft obstacles in periodic waveguides. Inverse Problems and 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. [2] G. Bao, L. Cowsar and W. Masters, Mathematical Modeling in Optical Science, SIAM, Philadelphia, 2001. doi: 10.1137/1.9780898717594. [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. [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. [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. [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. [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. [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. [9] D. Colton, R. Ewing and W. Rundell, Inverse Problems in Partial Differential Equation, SIAM, Phialdelphia, 1990. [10] M. Ehrhardt, H. Han and C. Zheng, Numerical simulation of waves in periodic structures, Commun. Comput. Phys., 5 (2009), 849-870. [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. [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. [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. [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. [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. [16] A. Kirsch and R. Kress, A numerical method for an inverse scattering problem, Academic Press, Boston, 4 (1987), 279-290. [17] A. Kirsch and R. Kress, An optimisition method in inverse acoustic scattering, in Boundary Elements IX, Stuttgart, 3 (1987), 3-18. [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. [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. [20] J. Li, H. Liu and J. Zou, Locating multiple multiscale acoustic scatterers, Multiscale Model. Simul., 12 (2014), 927-952.  doi: 10.1137/13093409X. [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. [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. [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. [24] K. Sakoda, Optical Properties of Photonic Crystals, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-662-14324-7. [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. [26] J. Sun and C. Zheng, Reconstruction of obstacles embedded in waveguides, Contemporary Mathematics, 586 (2013), 341-350.  doi: 10.1090/conm/586/11652.

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. [2] G. Bao, L. Cowsar and W. Masters, Mathematical Modeling in Optical Science, SIAM, Philadelphia, 2001. doi: 10.1137/1.9780898717594. [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. [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. [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. [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. [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. [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. [9] D. Colton, R. Ewing and W. Rundell, Inverse Problems in Partial Differential Equation, SIAM, Phialdelphia, 1990. [10] M. Ehrhardt, H. Han and C. Zheng, Numerical simulation of waves in periodic structures, Commun. Comput. Phys., 5 (2009), 849-870. [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. [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. [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. [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. [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. [16] A. Kirsch and R. Kress, A numerical method for an inverse scattering problem, Academic Press, Boston, 4 (1987), 279-290. [17] A. Kirsch and R. Kress, An optimisition method in inverse acoustic scattering, in Boundary Elements IX, Stuttgart, 3 (1987), 3-18. [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. [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. [20] J. Li, H. Liu and J. Zou, Locating multiple multiscale acoustic scatterers, Multiscale Model. Simul., 12 (2014), 927-952.  doi: 10.1137/13093409X. [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. [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. [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. [24] K. Sakoda, Optical Properties of Photonic Crystals, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-662-14324-7. [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. [26] J. Sun and C. Zheng, Reconstruction of obstacles embedded in waveguides, Contemporary Mathematics, 586 (2013), 341-350.  doi: 10.1090/conm/586/11652.
The scattering problem in the periodic waveguide
Direct method for inverse scattering problems.
A periodic half guide.
Waveguide 1
Waveguide 2
Four scatterers
(a)-(b): numerical result for scatter 1 with waveguides
(a)-(b): numerical result for scatter 2 with waveguides
(a)-(b): numerical result for scatter 3 with waveguides
(a)-(b): numerical result for scatter 4 with waveguides
(a)-(b): numerical result for scatter 1 with waveguides
(a)-(b): numerical result for scatter 2 with waveguides
(a)-(b): numerical result for scatter 3 with waveguides
(a)-(b): numerical result for scatter 4 with waveguides
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