American Institute of Mathematical Sciences

Advanced
  • Previous Article
    FEM-based discretization-invariant MCMC methods for PDE-constrained Bayesian inverse problems
  • IPI Home
  • This Issue
  • Next Article
    The localized basis functions for scalar and vector 3D tomography and their ray transforms
November  2016, 10(4): 915-941. doi: 10.3934/ipi.2016027

Imaging with electromagnetic waves in terminating waveguides

Liliana Borcea 1, and  Dinh-Liem Nguyen 1,

1. 

Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043, United States

Received  October 2015 Revised  July 2016 Published  October 2016

We study an inverse scattering problem for Maxwell's equations in terminating waveguides, where localized reflectors are to be imaged using a remote array of sensors. The array probes the waveguide with waves and measures the scattered returns. The mathematical formulation of the inverse scattering problem is based on the electromagnetic Lippmann-Schwinger integral equation and an explicit calculation of the Green tensor. The image formation is carried with reverse time migration and with $\ell_1$ optimization.
Keywords: imaging, terminating waveguide, Maxwell's equations., Electromagnetic, inverse scattering.
Mathematics Subject Classification: Primary: 35R30, 78A46; Secondary: 35Q6.
Citation: Liliana Borcea, Dinh-Liem Nguyen. Imaging with electromagnetic waves in terminating waveguides. Inverse Problems & Imaging, 2016, 10 (4) : 915-941. doi: 10.3934/ipi.2016027
References:
[1]

R. Alonso and L. Borcea, Electromagnetic wave propagation in random waveguides,, Multiscale Modeling & Simulation, 13 (2015), 847.  doi: 10.1137/130941936.  Google Scholar

[2]

T. Arens, D. Gintides and A. Lechleiter, Direct and inverse medium scattering problems in a planar 3D waveguide,, SIAM J. Appl. Math., 71 (2011), 753.  doi: 10.1137/100806333.  Google Scholar

[3]

A.-S. Bonnet-Bendhia and F. Starling, Guided waves by electromagnetic gratings and non-uniqueness examples for the diffraction problem,, Mathematical Methods in the Applied Sciences, 17 (1994), 305.  doi: 10.1002/mma.1670170502.  Google Scholar

[4]

L. Borcea, L. Issa and C. Tsogka, Source localization in random acoustic waveguides,, Multiscale Model. Simul., 8 (2010), 1981.  doi: 10.1137/100782711.  Google Scholar

[5]

L. Borcea and J. Garnier, Paraxial coupling of propagating modes in three-dimensional waveguides with random boundaries,, Multiscale Modeling & Simulation, 12 (2014), 832.  doi: 10.1137/12089747X.  Google Scholar

[6]

L. Bourgeois, F. L. Louër and E. Lunéville, On the use of Lamb modes in the linear sampling method for elastic waveguides,, Inverse Problems, 27 (2011).  doi: 10.1088/0266-5611/27/5/055001.  Google Scholar

[7]

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

[8]

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

[9]

L. Bourgeois and E. Lunéville, On the use of the linear sampling method to identify cracks in elastic waveguides,, Inverse Problems, 29 (2013).  doi: 10.1088/0266-5611/29/2/025017.  Google Scholar

[10]

S. Dediu and J. R. McLaughlin, Recovering inhomogeneities in a waveguide using eigensystem decomposition,, Inverse Problems, 22 (2006), 1227.  doi: 10.1088/0266-5611/22/4/007.  Google Scholar

[11]

L. Evans, Partial Differential Equations (Graduate Studies in Mathematics vol 19)(Providence, RI: American Mathematical Society),, Oxford University Press, (1998).   Google Scholar

[12]

L. Issa, Source Localization in Cluttered Acoustic Waveguides,, PhD thesis, (2010).   Google Scholar

[13]

J. D. Jackson, Classical Electrodynamics,, 2nd edition, (1975).   Google Scholar

[14]

A. K. Jordan and L. S. Tamil, Inverse scattering theory for optical waveguides and devices: Synthesis from rational and nonrational reflection coefficients,, Radio Science, 31 (1996), 1863.  doi: 10.1029/96RS02501.  Google Scholar

[15]

U. Kangro and R. Nicolaides, Divergence boundary conditions for vector helmholtz equations with divergence constraints,, ESAIM, 33 (1999), 479.  doi: 10.1051/m2an:1999148.  Google Scholar

[16]

A. Kirsch, An integral equation approach and the interior transmission problem for Maxwell's equations,, Inverse Probl. Imaging, 1 (2007), 159.  doi: 10.3934/ipi.2007.1.159.  Google Scholar

[17]

W. McLean, Strongly Elliptic Systems and Boundary Integral Operators,, Cambridge University Press, (2000).   Google Scholar

[18]

D. W. Mills and L. S. Tamil, Analysis of planar optical waveguides using scattering data,, J. Opt. Soc. Am. A, 9 (1992), 1769.  doi: 10.1364/JOSAA.9.001769.  Google Scholar

[19]

P. Monk, Finite Element Methods for Maxwell's Equations,, Oxford Science Publications, (2003).  doi: 10.1093/acprof:oso/9780198508885.001.0001.  Google Scholar

[20]

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

[21]

P. Roux and M. Fink, Time reversal in a waveguide: Study of the temporal and spatial focusing,, J. Acoust. Soc. Am., 107 (2000), 2418.  doi: 10.1121/1.428628.  Google Scholar

[22]

K. G. Sabra and D. R. Dowling, Blind deconvolution in ocean waveguides using artificial time reversal,, The Journal of the Acoustical Society of America, 116 (2004), 262.  doi: 10.1121/1.1751151.  Google Scholar

[23]

L. S. Tamil and A. K. Jordan, Spectral inverse scattering theory for inhomogeneous dielectric waveguides and devices,, Proceedings of the IEEE, 79 (1991), 1519.  doi: 10.1109/5.104226.  Google Scholar

[24]

C. Tsogka, D. A. Mitsoudis and S. Papadimitropoulos, Selective imaging of extended reflectors in two-dimensional waveguides,, SIAM Journal on Imaging Sciences, 6 (2013), 2714.  doi: 10.1137/130924238.  Google Scholar

[25]

Y. Xu, C. Matawa and W. Lin, Generalized dual space indicator method for underwater imaging,, Inverse Problems, 16 (2000), 1761.  doi: 10.1088/0266-5611/16/6/311.  Google Scholar

show all references

References:
[1]

R. Alonso and L. Borcea, Electromagnetic wave propagation in random waveguides,, Multiscale Modeling & Simulation, 13 (2015), 847.  doi: 10.1137/130941936.  Google Scholar

[2]

T. Arens, D. Gintides and A. Lechleiter, Direct and inverse medium scattering problems in a planar 3D waveguide,, SIAM J. Appl. Math., 71 (2011), 753.  doi: 10.1137/100806333.  Google Scholar

[3]

A.-S. Bonnet-Bendhia and F. Starling, Guided waves by electromagnetic gratings and non-uniqueness examples for the diffraction problem,, Mathematical Methods in the Applied Sciences, 17 (1994), 305.  doi: 10.1002/mma.1670170502.  Google Scholar

[4]

L. Borcea, L. Issa and C. Tsogka, Source localization in random acoustic waveguides,, Multiscale Model. Simul., 8 (2010), 1981.  doi: 10.1137/100782711.  Google Scholar

[5]

L. Borcea and J. Garnier, Paraxial coupling of propagating modes in three-dimensional waveguides with random boundaries,, Multiscale Modeling & Simulation, 12 (2014), 832.  doi: 10.1137/12089747X.  Google Scholar

[6]

L. Bourgeois, F. L. Louër and E. Lunéville, On the use of Lamb modes in the linear sampling method for elastic waveguides,, Inverse Problems, 27 (2011).  doi: 10.1088/0266-5611/27/5/055001.  Google Scholar

[7]

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

[8]

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

[9]

L. Bourgeois and E. Lunéville, On the use of the linear sampling method to identify cracks in elastic waveguides,, Inverse Problems, 29 (2013).  doi: 10.1088/0266-5611/29/2/025017.  Google Scholar

[10]

S. Dediu and J. R. McLaughlin, Recovering inhomogeneities in a waveguide using eigensystem decomposition,, Inverse Problems, 22 (2006), 1227.  doi: 10.1088/0266-5611/22/4/007.  Google Scholar

[11]

L. Evans, Partial Differential Equations (Graduate Studies in Mathematics vol 19)(Providence, RI: American Mathematical Society),, Oxford University Press, (1998).   Google Scholar

[12]

L. Issa, Source Localization in Cluttered Acoustic Waveguides,, PhD thesis, (2010).   Google Scholar

[13]

J. D. Jackson, Classical Electrodynamics,, 2nd edition, (1975).   Google Scholar

[14]

A. K. Jordan and L. S. Tamil, Inverse scattering theory for optical waveguides and devices: Synthesis from rational and nonrational reflection coefficients,, Radio Science, 31 (1996), 1863.  doi: 10.1029/96RS02501.  Google Scholar

[15]

U. Kangro and R. Nicolaides, Divergence boundary conditions for vector helmholtz equations with divergence constraints,, ESAIM, 33 (1999), 479.  doi: 10.1051/m2an:1999148.  Google Scholar

[16]

A. Kirsch, An integral equation approach and the interior transmission problem for Maxwell's equations,, Inverse Probl. Imaging, 1 (2007), 159.  doi: 10.3934/ipi.2007.1.159.  Google Scholar

[17]

W. McLean, Strongly Elliptic Systems and Boundary Integral Operators,, Cambridge University Press, (2000).   Google Scholar

[18]

D. W. Mills and L. S. Tamil, Analysis of planar optical waveguides using scattering data,, J. Opt. Soc. Am. A, 9 (1992), 1769.  doi: 10.1364/JOSAA.9.001769.  Google Scholar

[19]

P. Monk, Finite Element Methods for Maxwell's Equations,, Oxford Science Publications, (2003).  doi: 10.1093/acprof:oso/9780198508885.001.0001.  Google Scholar

[20]

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

[21]

P. Roux and M. Fink, Time reversal in a waveguide: Study of the temporal and spatial focusing,, J. Acoust. Soc. Am., 107 (2000), 2418.  doi: 10.1121/1.428628.  Google Scholar

[22]

K. G. Sabra and D. R. Dowling, Blind deconvolution in ocean waveguides using artificial time reversal,, The Journal of the Acoustical Society of America, 116 (2004), 262.  doi: 10.1121/1.1751151.  Google Scholar

[23]

L. S. Tamil and A. K. Jordan, Spectral inverse scattering theory for inhomogeneous dielectric waveguides and devices,, Proceedings of the IEEE, 79 (1991), 1519.  doi: 10.1109/5.104226.  Google Scholar

[24]

C. Tsogka, D. A. Mitsoudis and S. Papadimitropoulos, Selective imaging of extended reflectors in two-dimensional waveguides,, SIAM Journal on Imaging Sciences, 6 (2013), 2714.  doi: 10.1137/130924238.  Google Scholar

[25]

Y. Xu, C. Matawa and W. Lin, Generalized dual space indicator method for underwater imaging,, Inverse Problems, 16 (2000), 1761.  doi: 10.1088/0266-5611/16/6/311.  Google Scholar

[1]

Christodoulos E. Athanasiadis, Vassilios Sevroglou, Konstantinos I. Skourogiannis. The inverse electromagnetic scattering problem by a mixed impedance screen in chiral media. Inverse Problems & Imaging, 2015, 9 (4) : 951-970. doi: 10.3934/ipi.2015.9.951
[2]

Frederic Weidling, Thorsten Hohage. Variational source conditions and stability estimates for inverse electromagnetic medium scattering problems. Inverse Problems & Imaging, 2017, 11 (1) : 203-220. doi: 10.3934/ipi.2017010
[3]

Tan Bui-Thanh, Omar Ghattas. Analysis of the Hessian for inverse scattering problems. Part III: Inverse medium scattering of electromagnetic waves in three dimensions. Inverse Problems & Imaging, 2013, 7 (4) : 1139-1155. doi: 10.3934/ipi.2013.7.1139
[4]

Giovanni Bozza, Massimo Brignone, Matteo Pastorino, Andrea Randazzo, Michele Piana. Imaging of unknown targets inside inhomogeneous backgrounds by means of qualitative inverse scattering. Inverse Problems & Imaging, 2009, 3 (2) : 231-241. doi: 10.3934/ipi.2009.3.231
[5]

Masaru Ikehata. The enclosure method for inverse obstacle scattering using a single electromagnetic wave in time domain. Inverse Problems & Imaging, 2016, 10 (1) : 131-163. doi: 10.3934/ipi.2016.10.131
[6]

Shitao Liu, Roberto Triggiani. Determining damping and potential coefficients of an inverse problem for a system of two coupled hyperbolic equations. Part I: Global uniqueness. Conference Publications, 2011, 2011 (Special) : 1001-1014. doi: 10.3934/proc.2011.2011.1001
[7]

Keji Liu. Scattering by impenetrable scatterer in a stratified ocean waveguide. Inverse Problems & Imaging, 2019, 13 (6) : 1349-1365. doi: 10.3934/ipi.2019059
[8]

Jun Lai, Ming Li, Peijun Li, Wei Li. A fast direct imaging method for the inverse obstacle scattering problem with nonlinear point scatterers. Inverse Problems & Imaging, 2018, 12 (3) : 635-665. doi: 10.3934/ipi.2018027
[9]

Roland Griesmaier. Reciprocity gap music imaging for an inverse scattering problem in two-layered media. Inverse Problems & Imaging, 2009, 3 (3) : 389-403. doi: 10.3934/ipi.2009.3.389
[10]

Zhiming Chen, Shaofeng Fang, Guanghui Huang. A direct imaging method for the half-space inverse scattering problem with phaseless data. Inverse Problems & Imaging, 2017, 11 (5) : 901-916. doi: 10.3934/ipi.2017042
[11]

W. Wei, H. M. Yin. Global solvability for a singular nonlinear Maxwell's equations. Communications on Pure & Applied Analysis, 2005, 4 (2) : 431-444. doi: 10.3934/cpaa.2005.4.431
[12]

Björn Birnir, Niklas Wellander. Homogenized Maxwell's equations; A model for ceramic varistors. Discrete & Continuous Dynamical Systems - B, 2006, 6 (2) : 257-272. doi: 10.3934/dcdsb.2006.6.257
[13]

Peter Monk, Virginia Selgas. Sampling type methods for an inverse waveguide problem. Inverse Problems & Imaging, 2012, 6 (4) : 709-747. doi: 10.3934/ipi.2012.6.709
[14]

Kaitlyn (Voccola) Muller. SAR correlation imaging and anisotropic scattering. Inverse Problems & Imaging, 2018, 12 (3) : 697-731. doi: 10.3934/ipi.2018030
[15]

M. Eller. On boundary regularity of solutions to Maxwell's equations with a homogeneous conservative boundary condition. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 473-481. doi: 10.3934/dcdss.2009.2.473
[16]

Oleg Yu. Imanuvilov, Masahiro Yamamoto. Calderón problem for Maxwell's equations in cylindrical domain. Inverse Problems & Imaging, 2014, 8 (4) : 1117-1137. doi: 10.3934/ipi.2014.8.1117
[17]

B. L. G. Jonsson. Wave splitting of Maxwell's equations with anisotropic heterogeneous constitutive relations. Inverse Problems & Imaging, 2009, 3 (3) : 405-452. doi: 10.3934/ipi.2009.3.405
[18]

Andreas Kirsch. An integral equation approach and the interior transmission problem for Maxwell's equations. Inverse Problems & Imaging, 2007, 1 (1) : 159-179. doi: 10.3934/ipi.2007.1.159
[19]

Cleverson R. da Luz, Gustavo Alberto Perla Menzala. Uniform stabilization of anisotropic Maxwell's equations with boundary dissipation. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 547-558. doi: 10.3934/dcdss.2009.2.547
[20]

Gang Bao, Bin Hu, Peijun Li, Jue Wang. Analysis of time-domain Maxwell's equations in biperiodic structures. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 259-286. doi: 10.3934/dcdsb.2019181

2018 Impact Factor: 1.469

Tools

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences