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November  2015, 9(4): 951-970. doi: 10.3934/ipi.2015.9.951

## The inverse electromagnetic scattering problem by a mixed impedance screen in chiral media

 1 Department of Mathematics, National and Kapodistrian University of Athens, Panepistimiopolis, 157 84 Athens, Greece, Greece 2 Department of Statistics and Insurance Science, University of Piraeus, 80 Karaoli and Dimitriou Str., 185 34 Piraeus, Greece

Received  December 2014 Revised  July 2015 Published  October 2015

In this paper the problem of scattering of time-harmonic electromagnetic waves by a mixed impedance scatterer in chiral media is considered. Our scatterer is a partially coated chiral screen, for which an impedance boundary condition on one side of its boundary, and a perfectly conducting boundary condition on the other side, is satisfied. The direct scattering problem for the modified Maxwell equations is formulated and the appropriate Sobolev space setting is considered. Issues of solvability due to uniqueness and existence are discussed. The corresponding inverse scattering problem is studied and uniqueness results concerning the mixed impedance screen are proved. Further, the shape reconstruction of the boundary of the partially coated screen is established. In particular, a chiral far-field operator is introduced and new results concerning its properties are proved. A modified linear sampling method based on a factorization of the chiral far-field operator, in order to reconstruct the screen is also presented. We end up with useful conclusions and remarks for screens in chiral media.
Citation: 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
##### References:
 [1] H. Ammari and J. C. Nedelec, Time-harmonic electromagnetic fields in chiral media,, in Modern Mathematical Methods in Diffraction Theory and its Applications in Engineering, (1997), 174. Google Scholar [2] H. Ammari and J. C. Nedelec, Time-harmonic electromagnetic fields in thin chiral curved layers,, SIAM J. Math. Anal., 29 (1998), 395. doi: 10.1137/S0036141096305504. Google Scholar [3] H. Ammari, K. Hamdache and J. C. Nedelec, Chirality in the Maxwell equations by the dipole approximation method,, SIAM J. Appl. Math., 59 (1999), 2045. doi: 10.1137/S0036139998334160. Google Scholar [4] C. E. Athanasiadis, P. A. Martin and I. G. Stratis, Electromagnetic scattering by a homogeneous chiral obstacle: Boundary integral equations and low-chirality approximations,, SIAM J. Appl. Math., 59 (1999), 1745. doi: 10.1137/S003613999833633X. Google Scholar [5] C. E. Athanasiadis, G. Costakis and I. G. Stratis, Electromagnetic scattering by a homogeneous chiral obstacle in a chiral environment,, IMA J. Appl. Math., 64 (2000), 245. doi: 10.1093/imamat/64.3.245. Google Scholar [6] C. E. Athanasiadis, G. Costakis and I. G. Stratis, Electromagnetic scattering by a perfectly conducting obstacle in a homogeneous chiral environment: Solvability and low-frequency theory,, Math. Meth. Appl. Sci., 25 (2002), 927. doi: 10.1002/mma.321. Google Scholar [7] C. E. Athanasiadis, On the far field patterns for electromagnetic scattering by a chiral obstacle in chiral environment,, J. Math. Anal. Appl., 309 (2005), 517. doi: 10.1016/j.jmaa.2004.09.058. Google Scholar [8] C. E. Athanasiadis and E. Kardasi, Beltrami Herglotz functions for electromagnetic scattering,, Appl. Anal., 84 (2005), 145. doi: 10.1080/00036810410001658188. Google Scholar [9] C. E. Athanasiadis and N. Berketis, Scattering relations for point-source excitation in chiral media,, Math. Meth. Appl. Sci., 29 (2006), 27. doi: 10.1002/mma.662. Google Scholar [10] C. E. Athanasiadis, D. Natrosvili, V. Sevroglou and I. G. Stratis, An application of the reciprocity gap functional to inverse mixed impedance problems in elasticity,, Inverse Problems, 26 (2010). doi: 10.1088/0266-5611/26/8/085011. Google Scholar [11] C. E. Athanasiadis, D. Natrosvili, V. Sevroglou and I. G. Stratis, A boundary integral equations approach for direct mixed impedance problems in elasticity,, J. Integral Equations Appl., 23 (2011), 183. doi: 10.1216/JIE-2011-23-2-183. Google Scholar [12] C. E. Athanasiadis, V. Sevroglou and K. I. Skourogiannis, The direct electromagnetic scattering problem by a mixed impedance screen in chiral media,, Appl. Anal., 91 (2012), 2083. doi: 10.1080/00036811.2011.584183. Google Scholar [13] C. E. Athanasiadis, D. Natrosvili, V. Sevroglou and I. G. Stratis, A mixed impedance scattering problem for partially coated obstacles in two-dimensional linear elasticity,, in The Thirteenth International Conference on Integral Methods in Science and Engineering, (). Google Scholar [14] F. Cakoni, D. Colton and E. Darringrand, The inverse electromagnetic scattering problem for screens,, Inverse Problems, 19 (2003), 627. doi: 10.1088/0266-5611/19/3/310. Google Scholar [15] F. Cakoni and D. Colton, Qualitative Methods in Inverse Electromagnetic Scattering Theory,, Springer-Verlag, (2005). Google Scholar [16] F. Cakoni and E. Darringrand, The inverse electromagnetic scattering problem for a mixed boundary value problem for screens,, J. Comput. Appl. Math., 174 (2005), 251. doi: 10.1016/j.cam.2004.04.012. Google Scholar [17] F. Collino, M. Fares and H. Haddar, Numerical and analytical studies of the linear sampling method in electromagnetic scattering problems,, Inverse Problems, 19 (2003), 1279. doi: 10.1088/0266-5611/19/6/004. Google Scholar [18] D. Colton and R. Kress, Integral Equation Methods in Scattering Theory,, John Wiley & Sons, (1983). Google Scholar [19] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory,, Springer-Verlag, (1992). doi: 10.1007/978-3-662-02835-3. Google Scholar [20] D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region,, Inverse Problems, 12 (1996), 383. doi: 10.1088/0266-5611/12/4/003. Google Scholar [21] D. Colton, M. Pianna and R. Potthast, A simple method using Morozov's discrepancy principle for solving inverse scattering problem,, Inverse Problems, 13 (1997), 1477. doi: 10.1088/0266-5611/13/6/005. Google Scholar [22] D. Colton, H. Haddar and M. Pianna, The linear sampling method in inverese electromagnetic scattering theory,, Inverse Problems, 19 (2003). doi: 10.1088/0266-5611/19/6/057. Google Scholar [23] D. L. Jaggard, X. Sun and N. Engheta, Canonical sources and duality in chiral media,, IEEE Trans. Antennas and Propagation, 36 (1988), 1007. doi: 10.1109/8.7205. Google Scholar [24] D. L. Jaggard and N. Engheta, Chirality in electrodynamics: Modeling and applications,, in Directions in Electromagnetic Wave Modelling (eds. H. L. Bertoni and L. B. Felsen), (1991). Google Scholar [25] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems,, Oxford University Press, (2008). Google Scholar [26] A. Lakhtakia, V. K. Varadan and V. V. Varadan, Time-harmonic Electromagnetic Fields in Chiral Media,, Lecture Notes in Physics, (1989). Google Scholar [27] A. Lakhtakia, V. K. Varadan and V. V. Varadan, Surface integral equations for scattering by PEC scatterers in isotropic chiral media,, Internat. J. Engrg. Sci., 29 (1991), 179. doi: 10.1016/0020-7225(91)90014-T. Google Scholar [28] A. Lakhtakia, Beltrami Fields in Chiral Media,, World Scientific, (1994). Google Scholar [29] I. V. Lindell, A. H. Sihvola, S. A. Tretyakov and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media,, Artech House, (1994). Google Scholar [30] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations,, Cambridge University Press, (2000). Google Scholar [31] P. Monk, Finite Element Methods for Maxwell's Equations,, Oxford University Press, (2003). doi: 10.1093/acprof:oso/9780198508885.001.0001. Google Scholar [32] G. F. Roach, I. G. Stratis and A. N. Yannacopoulos, Mathematical Analysis of Deterministic and Stochastic Problems in Complex Media Electromagnetics,, Princeton Series in Apllied Mathematics, (2012). Google Scholar [33] E. P. Stephan, Boundary integral equations for screen problems in $\mathbbR^3$,, Integral Equations Operator Theory, 10 (1987), 236. doi: 10.1007/BF01199079. Google Scholar [34] C. T. Tai, Dyadic Green Functions in Electromagnetic Theory,, $2^{nd}$ edition, (1994). Google Scholar

show all references

##### References:
 [1] H. Ammari and J. C. Nedelec, Time-harmonic electromagnetic fields in chiral media,, in Modern Mathematical Methods in Diffraction Theory and its Applications in Engineering, (1997), 174. Google Scholar [2] H. Ammari and J. C. Nedelec, Time-harmonic electromagnetic fields in thin chiral curved layers,, SIAM J. Math. Anal., 29 (1998), 395. doi: 10.1137/S0036141096305504. Google Scholar [3] H. Ammari, K. Hamdache and J. C. Nedelec, Chirality in the Maxwell equations by the dipole approximation method,, SIAM J. Appl. Math., 59 (1999), 2045. doi: 10.1137/S0036139998334160. Google Scholar [4] C. E. Athanasiadis, P. A. Martin and I. G. Stratis, Electromagnetic scattering by a homogeneous chiral obstacle: Boundary integral equations and low-chirality approximations,, SIAM J. Appl. Math., 59 (1999), 1745. doi: 10.1137/S003613999833633X. Google Scholar [5] C. E. Athanasiadis, G. Costakis and I. G. Stratis, Electromagnetic scattering by a homogeneous chiral obstacle in a chiral environment,, IMA J. Appl. Math., 64 (2000), 245. doi: 10.1093/imamat/64.3.245. Google Scholar [6] C. E. Athanasiadis, G. Costakis and I. G. Stratis, Electromagnetic scattering by a perfectly conducting obstacle in a homogeneous chiral environment: Solvability and low-frequency theory,, Math. Meth. Appl. Sci., 25 (2002), 927. doi: 10.1002/mma.321. Google Scholar [7] C. E. Athanasiadis, On the far field patterns for electromagnetic scattering by a chiral obstacle in chiral environment,, J. Math. Anal. Appl., 309 (2005), 517. doi: 10.1016/j.jmaa.2004.09.058. Google Scholar [8] C. E. Athanasiadis and E. Kardasi, Beltrami Herglotz functions for electromagnetic scattering,, Appl. Anal., 84 (2005), 145. doi: 10.1080/00036810410001658188. Google Scholar [9] C. E. Athanasiadis and N. Berketis, Scattering relations for point-source excitation in chiral media,, Math. Meth. Appl. Sci., 29 (2006), 27. doi: 10.1002/mma.662. Google Scholar [10] C. E. Athanasiadis, D. Natrosvili, V. Sevroglou and I. G. Stratis, An application of the reciprocity gap functional to inverse mixed impedance problems in elasticity,, Inverse Problems, 26 (2010). doi: 10.1088/0266-5611/26/8/085011. Google Scholar [11] C. E. Athanasiadis, D. Natrosvili, V. Sevroglou and I. G. Stratis, A boundary integral equations approach for direct mixed impedance problems in elasticity,, J. Integral Equations Appl., 23 (2011), 183. doi: 10.1216/JIE-2011-23-2-183. Google Scholar [12] C. E. Athanasiadis, V. Sevroglou and K. I. Skourogiannis, The direct electromagnetic scattering problem by a mixed impedance screen in chiral media,, Appl. Anal., 91 (2012), 2083. doi: 10.1080/00036811.2011.584183. Google Scholar [13] C. E. Athanasiadis, D. Natrosvili, V. Sevroglou and I. G. Stratis, A mixed impedance scattering problem for partially coated obstacles in two-dimensional linear elasticity,, in The Thirteenth International Conference on Integral Methods in Science and Engineering, (). Google Scholar [14] F. Cakoni, D. Colton and E. Darringrand, The inverse electromagnetic scattering problem for screens,, Inverse Problems, 19 (2003), 627. doi: 10.1088/0266-5611/19/3/310. Google Scholar [15] F. Cakoni and D. Colton, Qualitative Methods in Inverse Electromagnetic Scattering Theory,, Springer-Verlag, (2005). Google Scholar [16] F. Cakoni and E. Darringrand, The inverse electromagnetic scattering problem for a mixed boundary value problem for screens,, J. Comput. Appl. Math., 174 (2005), 251. doi: 10.1016/j.cam.2004.04.012. Google Scholar [17] F. Collino, M. Fares and H. Haddar, Numerical and analytical studies of the linear sampling method in electromagnetic scattering problems,, Inverse Problems, 19 (2003), 1279. doi: 10.1088/0266-5611/19/6/004. Google Scholar [18] D. Colton and R. Kress, Integral Equation Methods in Scattering Theory,, John Wiley & Sons, (1983). Google Scholar [19] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory,, Springer-Verlag, (1992). doi: 10.1007/978-3-662-02835-3. Google Scholar [20] D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region,, Inverse Problems, 12 (1996), 383. doi: 10.1088/0266-5611/12/4/003. Google Scholar [21] D. Colton, M. Pianna and R. Potthast, A simple method using Morozov's discrepancy principle for solving inverse scattering problem,, Inverse Problems, 13 (1997), 1477. doi: 10.1088/0266-5611/13/6/005. Google Scholar [22] D. Colton, H. Haddar and M. Pianna, The linear sampling method in inverese electromagnetic scattering theory,, Inverse Problems, 19 (2003). doi: 10.1088/0266-5611/19/6/057. Google Scholar [23] D. L. Jaggard, X. Sun and N. Engheta, Canonical sources and duality in chiral media,, IEEE Trans. Antennas and Propagation, 36 (1988), 1007. doi: 10.1109/8.7205. Google Scholar [24] D. L. Jaggard and N. Engheta, Chirality in electrodynamics: Modeling and applications,, in Directions in Electromagnetic Wave Modelling (eds. H. L. Bertoni and L. B. Felsen), (1991). Google Scholar [25] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems,, Oxford University Press, (2008). Google Scholar [26] A. Lakhtakia, V. K. Varadan and V. V. Varadan, Time-harmonic Electromagnetic Fields in Chiral Media,, Lecture Notes in Physics, (1989). Google Scholar [27] A. Lakhtakia, V. K. Varadan and V. V. Varadan, Surface integral equations for scattering by PEC scatterers in isotropic chiral media,, Internat. J. Engrg. Sci., 29 (1991), 179. doi: 10.1016/0020-7225(91)90014-T. Google Scholar [28] A. Lakhtakia, Beltrami Fields in Chiral Media,, World Scientific, (1994). Google Scholar [29] I. V. Lindell, A. H. Sihvola, S. A. Tretyakov and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media,, Artech House, (1994). Google Scholar [30] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations,, Cambridge University Press, (2000). Google Scholar [31] P. Monk, Finite Element Methods for Maxwell's Equations,, Oxford University Press, (2003). doi: 10.1093/acprof:oso/9780198508885.001.0001. Google Scholar [32] G. F. Roach, I. G. Stratis and A. N. Yannacopoulos, Mathematical Analysis of Deterministic and Stochastic Problems in Complex Media Electromagnetics,, Princeton Series in Apllied Mathematics, (2012). Google Scholar [33] E. P. Stephan, Boundary integral equations for screen problems in $\mathbbR^3$,, Integral Equations Operator Theory, 10 (1987), 236. doi: 10.1007/BF01199079. Google Scholar [34] C. T. Tai, Dyadic Green Functions in Electromagnetic Theory,, $2^{nd}$ edition, (1994). Google Scholar
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