Imaging with electromagnetic waves in terminating waveguides

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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

Abstract
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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 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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