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November  2016, 10(4): 915-941. doi: 10.3934/ipi.2016027

Imaging with electromagnetic waves in terminating waveguides

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.
Citation: Liliana Borcea, Dinh-Liem Nguyen. Imaging with electromagnetic waves in terminating waveguides. Inverse Problems and 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), 847-889. doi: 10.1137/130941936.

[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), 753-772. doi: 10.1137/100806333.

[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), 305-338. doi: 10.1002/mma.1670170502.

[4]

L. Borcea, L. Issa and C. Tsogka, Source localization in random acoustic waveguides, Multiscale Model. Simul., 8 (2010), 1981-2022. doi: 10.1137/100782711.

[5]

L. Borcea and J. Garnier, Paraxial coupling of propagating modes in three-dimensional waveguides with random boundaries, Multiscale Modeling & Simulation, 12 (2014), 832-878. doi: 10.1137/12089747X.

[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), 055001, 27pp. doi: 10.1088/0266-5611/27/5/055001.

[7]

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.

[8]

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.

[9]

L. Bourgeois and E. Lunéville, On the use of the linear sampling method to identify cracks in elastic waveguides, Inverse Problems, 29 (2013), 025017, 19pp. doi: 10.1088/0266-5611/29/2/025017.

[10]

S. Dediu and J. R. McLaughlin, Recovering inhomogeneities in a waveguide using eigensystem decomposition, Inverse Problems, 22 (2006), 1227-1246, URL http://stacks.iop.org/0266-5611/22/1227. doi: 10.1088/0266-5611/22/4/007.

[11]

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

[12]

L. Issa, Source Localization in Cluttered Acoustic Waveguides, PhD thesis, Rice University, 2010.

[13]

J. D. Jackson, Classical Electrodynamics, 2nd edition, Wiley New York etc., 1975.

[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), 1863-1876. doi: 10.1029/96RS02501.

[15]

U. Kangro and R. Nicolaides, Divergence boundary conditions for vector helmholtz equations with divergence constraints, ESAIM, Math. Model. Numer. Anal., 33 (1999), 479-492. doi: 10.1051/m2an:1999148.

[16]

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

[17]

W. McLean, Strongly Elliptic Systems and Boundary Integral Operators, Cambridge University Press, Cambridge, UK, 2000.

[18]

D. W. Mills and L. S. Tamil, Analysis of planar optical waveguides using scattering data, J. Opt. Soc. Am. A, 9 (1992), 1769-1778. doi: 10.1364/JOSAA.9.001769.

[19]

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

[20]

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

[21]

P. Roux and M. Fink, Time reversal in a waveguide: Study of the temporal and spatial focusing, J. Acoust. Soc. Am., 107 (2000), 2418-2429. doi: 10.1121/1.428628.

[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), 262-271. doi: 10.1121/1.1751151.

[23]

L. S. Tamil and A. K. Jordan, Spectral inverse scattering theory for inhomogeneous dielectric waveguides and devices, Proceedings of the IEEE, 79 (1991), 1519-1528. doi: 10.1109/5.104226.

[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), 2714-2739. doi: 10.1137/130924238.

[25]

Y. Xu, C. Matawa and W. Lin, Generalized dual space indicator method for underwater imaging, Inverse Problems, 16 (2000), 1761-1776. doi: 10.1088/0266-5611/16/6/311.

show all references

References:
[1]

R. Alonso and L. Borcea, Electromagnetic wave propagation in random waveguides, Multiscale Modeling & Simulation, 13 (2015), 847-889. doi: 10.1137/130941936.

[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), 753-772. doi: 10.1137/100806333.

[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), 305-338. doi: 10.1002/mma.1670170502.

[4]

L. Borcea, L. Issa and C. Tsogka, Source localization in random acoustic waveguides, Multiscale Model. Simul., 8 (2010), 1981-2022. doi: 10.1137/100782711.

[5]

L. Borcea and J. Garnier, Paraxial coupling of propagating modes in three-dimensional waveguides with random boundaries, Multiscale Modeling & Simulation, 12 (2014), 832-878. doi: 10.1137/12089747X.

[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), 055001, 27pp. doi: 10.1088/0266-5611/27/5/055001.

[7]

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.

[8]

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.

[9]

L. Bourgeois and E. Lunéville, On the use of the linear sampling method to identify cracks in elastic waveguides, Inverse Problems, 29 (2013), 025017, 19pp. doi: 10.1088/0266-5611/29/2/025017.

[10]

S. Dediu and J. R. McLaughlin, Recovering inhomogeneities in a waveguide using eigensystem decomposition, Inverse Problems, 22 (2006), 1227-1246, URL http://stacks.iop.org/0266-5611/22/1227. doi: 10.1088/0266-5611/22/4/007.

[11]

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

[12]

L. Issa, Source Localization in Cluttered Acoustic Waveguides, PhD thesis, Rice University, 2010.

[13]

J. D. Jackson, Classical Electrodynamics, 2nd edition, Wiley New York etc., 1975.

[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), 1863-1876. doi: 10.1029/96RS02501.

[15]

U. Kangro and R. Nicolaides, Divergence boundary conditions for vector helmholtz equations with divergence constraints, ESAIM, Math. Model. Numer. Anal., 33 (1999), 479-492. doi: 10.1051/m2an:1999148.

[16]

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

[17]

W. McLean, Strongly Elliptic Systems and Boundary Integral Operators, Cambridge University Press, Cambridge, UK, 2000.

[18]

D. W. Mills and L. S. Tamil, Analysis of planar optical waveguides using scattering data, J. Opt. Soc. Am. A, 9 (1992), 1769-1778. doi: 10.1364/JOSAA.9.001769.

[19]

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

[20]

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

[21]

P. Roux and M. Fink, Time reversal in a waveguide: Study of the temporal and spatial focusing, J. Acoust. Soc. Am., 107 (2000), 2418-2429. doi: 10.1121/1.428628.

[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), 262-271. doi: 10.1121/1.1751151.

[23]

L. S. Tamil and A. K. Jordan, Spectral inverse scattering theory for inhomogeneous dielectric waveguides and devices, Proceedings of the IEEE, 79 (1991), 1519-1528. doi: 10.1109/5.104226.

[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), 2714-2739. doi: 10.1137/130924238.

[25]

Y. Xu, C. Matawa and W. Lin, Generalized dual space indicator method for underwater imaging, Inverse Problems, 16 (2000), 1761-1776. doi: 10.1088/0266-5611/16/6/311.

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