May  2016, 10(2): 519-547. doi: 10.3934/ipi.2016010

The factorization method for the Drude-Born-Fedorov model for periodic chiral structures

1. 

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

Received  November 2014 Published  May 2016

We consider the electromagnetic inverse scattering problem for the Drude-Born-Fedorov model for periodic chiral structures known as chiral gratings both in $\mathbb{R}^2$ and $\mathbb{R}^3$. The Factorization method is studied as an analytical as well as a numerical tool for solving this inverse problem. The method constructs a simple criterion for characterizing shape of the periodic scatterer which leads to a fast imaging algorithm. This criterion is necessary and sufficient which gives a uniqueness result in shape reconstruction of the scatterer. The required data consists of certain components of Rayleigh sequences of (measured) scattered fields caused by plane incident electromagnetic waves. We propose in this electromagnetic plane wave setting a rigorous analysis for the Factorization method. Numerical examples in two and three dimensions are also presented for showing the efficiency of the method.
Citation: Dinh-Liem Nguyen. The factorization method for the Drude-Born-Fedorov model for periodic chiral structures. Inverse Problems and Imaging, 2016, 10 (2) : 519-547. doi: 10.3934/ipi.2016010
References:
[1]

H. Ammari and G. Bao, Maxwell's equations in periodic chiral structures, Math. Nachr., 251 (2003), 3-18. doi: 10.1002/mana.200310026.

[2]

H. Ammari, K. Hamdache and J.-C. Nédélec, Chirality in the Maxwell equations by the dipole approximation, SIAM J. Appl. Math., 59 (1999), 2045-2059. doi: 10.1137/S0036139998334160.

[3]

T. Arens and N. Grinberg, A complete factorization method for scattering by periodic structures, Computing, 75 (2005), 111-132. doi: 10.1007/s00607-004-0092-0.

[4]

T. Arens, Scattering by biperiodic layered media: The integral equation approach, Habilitation Thesis, Universität Karlsruhe, 2010.

[5]

C. Athanasiadis, G. Costakis and I. G. Stratis, Electromagnetic scattering by a homogeneous chiral obstacle in a chiral environment, SIAM J. Appl. Math., 64 (2000), 245-258. doi: 10.1093/imamat/64.3.245.

[6]

C. Athanasiadis and E. Kardasi, Inverse electromagnetic scattering by a perfect conductor in a chiral environment, J. Inv. Ill-Posed Problems, 16 (2008), 1-18. doi: 10.1515/jiip.2008.001.

[7]

C. Athanasiadis and I. G. Stratis, Uniqueness of the inverse scattering problem by a chiral obstacle, Int. J. Appl. Electromagn. Mech., 9 (1998), 123-133.

[8]

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.

[9]

A. Boutet de Monvel and D. Shepelsky, Direct and inverse scattering problem for a stratified nonreciprocal chiral medium, Inverse Problems, 13 (1997), 239-251. doi: 10.1088/0266-5611/13/2/004.

[10]

A. Boutet de Monvel and D. Shepelsky, A frequency-domain inverse problem for a dispersive stratified chiral medium, J. Math. Phys., 41 (2000), 6116-6129. doi: 10.1063/1.1286052.

[11]

V. M. Churikov, V. I. Kopp and A. Z. Genack, Chiral diffraction gratings in twisted microstructured fibers, Opt. Lett., 35 (2010), p342. doi: 10.1364/OL.35.000342.

[12]

P. Ciarlet Jr. and G. Legendre, Well-posedness of the Drude-Born-Fedorov model for chiral media, Math. Models Methods Appl. Sci., 17 (2007), 461-484. doi: 10.1142/S0218202507001991.

[13]

P.-H. Cocquet, P.-A. Mazet and V. Mouysset, On well-posedness of some homogenized Drude-Born-Fedorov systems on a bounded domain and applications to metamaterials, C. R. Math., 349 (2011), 99-103. doi: 10.1016/j.crma.2010.11.019.

[14]

D. Colton, J. Coyle and P. Monk, Recent developments in inverse acoustic scattering theory, SIAM Review, 42 (2000), 369-414. doi: 10.1137/S0036144500367337.

[15]

K. M. Flood and D. L. Jaggard, Band-gap structure for periodic chiral media, J. Opt. Soc. Am. A, 13 (1996), p1395. doi: 10.1364/JOSAA.13.001395.

[16]

T. Gerlach, The two-dimensional electromagnetic inverse scattering problem for chiral media, Inverse Problems, 15 (1999), 1663-1675. doi: 10.1088/0266-5611/15/6/315.

[17]

B. Gompf, J. Braun, T. Weiss, H. Giessen, M. Dressel and U. Hübner, Periodic nanostructures: Spatial dispersion mimics chirality, Phys. Rev. Lett., 106 (2011), 185501. doi: 10.1103/PhysRevLett.106.185501.

[18]

S. Guenneau and F. Zolla, Homogenization of 3D finite chiral photonic crystals, Physica B: Condensed Matter, 394 (2007), 145-147. doi: 10.1016/j.physb.2006.12.021.

[19]

S. Heumann, The Factorization Method for Inverse Scattering from Chiral Media, PhD thesis, Karlsruher Institut für Technologie, 2012.

[20]

D. L. Jaggard, N. Engheta, M. W. Kowarz, P. Pelet, J. C. Liu and Y. Kim, Periodic chiral structures, IEEE Trans. Antennas Propagat., 37 (1989), 1447-1452. doi: 10.1109/8.43564.

[21]

A. Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator, Inverse Problems, 14 (1998), 1489-1512. doi: 10.1088/0266-5611/14/6/009.

[22]

A. Kirsch, The factorization method for Maxwell's equations, Inverse Problems, 20 (2004), S117-S134. doi: 10.1088/0266-5611/20/6/S08.

[23]

A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford Lecture Series in Mathematics and its Applications 36, Oxford University Press, 2008.

[24]

V. I. Kopp and A. Z. Genack, Twist defect in chiral photonic structures, Phys. Rev. Lett., 89 (2002), 033901. doi: 10.1103/PhysRevLett.89.033901.

[25]

V. I. Kopp and A. Z. Genack, Chiral fibres: Adding twist, Nature Photonics, 5 (2011), 470-472. doi: 10.1038/nphoton.2011.158.

[26]

V. V. Kravchenko and H. Oviedo, Time-dependent electromagnetic fields in chiral media, J. Phys. A: Math. Theor., 43 (2010), 455213, 9pp. doi: 10.1088/1751-8113/43/45/455213.

[27]

A. Lakhtakia, V. V. Varadan and V. K. Varadan, Scattering by periodic achiral-chiral interfaces, J. Opt. Soc. Am. A, 6 (1989), 1675-1681; erratum 7, 951 (1990). doi: 10.1364/JOSAA.6.001675.

[28]

A. Lechleiter, The Factorization method is independent of transmission eigenvalues, Inverse Probl. Imag., 3 (2009), 123-138. doi: 10.3934/ipi.2009.3.123.

[29]

A. Lechleiter and D.-L. Nguyen, Factorization method for electromagnetic inverse scattering from biperiodic structures, SIAM J. Imaging Sci., 6 (2013), 1111-1139. doi: 10.1137/120903968.

[30]

A. Lechleiter and D.-L. Nguyen, A trigonometric Galerkin method for volume integral equations arising in TM grating scattering, Adv. Comput. Math., 40 (2014), 1-25. doi: 10.1007/s10444-013-9295-2.

[31]

S. Li, An inverse problem for Maxwell's equations in bi-isotropic media, SIAM J. Math. Anal., 37 (2005), 1027-1043. doi: 10.1137/S003614100444366X.

[32]

S. R. McDowall, Boundary determination of material parameters from electromagnetic boundary information, Inverse Problems, 13 (1997), 153-163. doi: 10.1088/0266-5611/13/1/012.

[33]

S. R. McDowall, An electromagnetic inverse problem in chiral media, T. Am. Math. Soc., 352 (2000), 2993-3013. doi: 10.1090/S0002-9947-00-02518-6.

[34]

S. Nicaise, Time domain study of the Drude-Born-Fedorov model for a class of heterogeneous chiral materials, Math. Meth. Appl. Sci., 36 (2013), 794-813. doi: 10.1002/mma.2627.

[35]

P. Ola, Boundary integral equations for the scattering of electromagnetic waves by a homogeneous chiral obstacle, J. Math. Phys., 35 (1994), 3969-3980. doi: 10.1063/1.530836.

[36]

R. Potthast and I. G. Stratis, On the domain derivative for scattering by impenetrable obstacles in chiral media, IMA J. Appl. Math., 68 (2003), 621-635. doi: 10.1093/imamat/68.6.621.

[37]

G. F. Roach, I. G. Stratis and A. N. Yannacopoulos, Mathematical Analysis of Deterministic and Stochastic Problems in Complex Media Electromagnetics, Princeton Series in Applied Mathematics, Princeton University Press, 2012.

[38]

C. Sabah and H. G. Roskos, Design of a terahertz polarization rotator based on a periodic sequence of chiral-metamaterial and dielectric slabs, Prog. Electromagn. Res., 124 (2012), 301-314. doi: 10.2528/PIER11112605.

[39]

K. Sandfort, The factorization method for inverse scattering from periodic inhomogeneous media, PhD thesis, Karlsruher Institut für Technologie, 2010, URL http://digbib.ubka.uni-karlsruhe.de/volltexte/1000019400.

[40]

G. Schmidt, On the diffraction by biperiodic anisotropic structures, Appl. Anal., 82 (2003), 75-92. doi: 10.1080/0003681031000068275.

[41]

I. G. Stratis, Electromagnetic scattering problems in chiral media: A review, Electromagnetics, 19 (1999), 547-562. doi: 10.1080/02726349908908673.

show all references

References:
[1]

H. Ammari and G. Bao, Maxwell's equations in periodic chiral structures, Math. Nachr., 251 (2003), 3-18. doi: 10.1002/mana.200310026.

[2]

H. Ammari, K. Hamdache and J.-C. Nédélec, Chirality in the Maxwell equations by the dipole approximation, SIAM J. Appl. Math., 59 (1999), 2045-2059. doi: 10.1137/S0036139998334160.

[3]

T. Arens and N. Grinberg, A complete factorization method for scattering by periodic structures, Computing, 75 (2005), 111-132. doi: 10.1007/s00607-004-0092-0.

[4]

T. Arens, Scattering by biperiodic layered media: The integral equation approach, Habilitation Thesis, Universität Karlsruhe, 2010.

[5]

C. Athanasiadis, G. Costakis and I. G. Stratis, Electromagnetic scattering by a homogeneous chiral obstacle in a chiral environment, SIAM J. Appl. Math., 64 (2000), 245-258. doi: 10.1093/imamat/64.3.245.

[6]

C. Athanasiadis and E. Kardasi, Inverse electromagnetic scattering by a perfect conductor in a chiral environment, J. Inv. Ill-Posed Problems, 16 (2008), 1-18. doi: 10.1515/jiip.2008.001.

[7]

C. Athanasiadis and I. G. Stratis, Uniqueness of the inverse scattering problem by a chiral obstacle, Int. J. Appl. Electromagn. Mech., 9 (1998), 123-133.

[8]

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.

[9]

A. Boutet de Monvel and D. Shepelsky, Direct and inverse scattering problem for a stratified nonreciprocal chiral medium, Inverse Problems, 13 (1997), 239-251. doi: 10.1088/0266-5611/13/2/004.

[10]

A. Boutet de Monvel and D. Shepelsky, A frequency-domain inverse problem for a dispersive stratified chiral medium, J. Math. Phys., 41 (2000), 6116-6129. doi: 10.1063/1.1286052.

[11]

V. M. Churikov, V. I. Kopp and A. Z. Genack, Chiral diffraction gratings in twisted microstructured fibers, Opt. Lett., 35 (2010), p342. doi: 10.1364/OL.35.000342.

[12]

P. Ciarlet Jr. and G. Legendre, Well-posedness of the Drude-Born-Fedorov model for chiral media, Math. Models Methods Appl. Sci., 17 (2007), 461-484. doi: 10.1142/S0218202507001991.

[13]

P.-H. Cocquet, P.-A. Mazet and V. Mouysset, On well-posedness of some homogenized Drude-Born-Fedorov systems on a bounded domain and applications to metamaterials, C. R. Math., 349 (2011), 99-103. doi: 10.1016/j.crma.2010.11.019.

[14]

D. Colton, J. Coyle and P. Monk, Recent developments in inverse acoustic scattering theory, SIAM Review, 42 (2000), 369-414. doi: 10.1137/S0036144500367337.

[15]

K. M. Flood and D. L. Jaggard, Band-gap structure for periodic chiral media, J. Opt. Soc. Am. A, 13 (1996), p1395. doi: 10.1364/JOSAA.13.001395.

[16]

T. Gerlach, The two-dimensional electromagnetic inverse scattering problem for chiral media, Inverse Problems, 15 (1999), 1663-1675. doi: 10.1088/0266-5611/15/6/315.

[17]

B. Gompf, J. Braun, T. Weiss, H. Giessen, M. Dressel and U. Hübner, Periodic nanostructures: Spatial dispersion mimics chirality, Phys. Rev. Lett., 106 (2011), 185501. doi: 10.1103/PhysRevLett.106.185501.

[18]

S. Guenneau and F. Zolla, Homogenization of 3D finite chiral photonic crystals, Physica B: Condensed Matter, 394 (2007), 145-147. doi: 10.1016/j.physb.2006.12.021.

[19]

S. Heumann, The Factorization Method for Inverse Scattering from Chiral Media, PhD thesis, Karlsruher Institut für Technologie, 2012.

[20]

D. L. Jaggard, N. Engheta, M. W. Kowarz, P. Pelet, J. C. Liu and Y. Kim, Periodic chiral structures, IEEE Trans. Antennas Propagat., 37 (1989), 1447-1452. doi: 10.1109/8.43564.

[21]

A. Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator, Inverse Problems, 14 (1998), 1489-1512. doi: 10.1088/0266-5611/14/6/009.

[22]

A. Kirsch, The factorization method for Maxwell's equations, Inverse Problems, 20 (2004), S117-S134. doi: 10.1088/0266-5611/20/6/S08.

[23]

A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford Lecture Series in Mathematics and its Applications 36, Oxford University Press, 2008.

[24]

V. I. Kopp and A. Z. Genack, Twist defect in chiral photonic structures, Phys. Rev. Lett., 89 (2002), 033901. doi: 10.1103/PhysRevLett.89.033901.

[25]

V. I. Kopp and A. Z. Genack, Chiral fibres: Adding twist, Nature Photonics, 5 (2011), 470-472. doi: 10.1038/nphoton.2011.158.

[26]

V. V. Kravchenko and H. Oviedo, Time-dependent electromagnetic fields in chiral media, J. Phys. A: Math. Theor., 43 (2010), 455213, 9pp. doi: 10.1088/1751-8113/43/45/455213.

[27]

A. Lakhtakia, V. V. Varadan and V. K. Varadan, Scattering by periodic achiral-chiral interfaces, J. Opt. Soc. Am. A, 6 (1989), 1675-1681; erratum 7, 951 (1990). doi: 10.1364/JOSAA.6.001675.

[28]

A. Lechleiter, The Factorization method is independent of transmission eigenvalues, Inverse Probl. Imag., 3 (2009), 123-138. doi: 10.3934/ipi.2009.3.123.

[29]

A. Lechleiter and D.-L. Nguyen, Factorization method for electromagnetic inverse scattering from biperiodic structures, SIAM J. Imaging Sci., 6 (2013), 1111-1139. doi: 10.1137/120903968.

[30]

A. Lechleiter and D.-L. Nguyen, A trigonometric Galerkin method for volume integral equations arising in TM grating scattering, Adv. Comput. Math., 40 (2014), 1-25. doi: 10.1007/s10444-013-9295-2.

[31]

S. Li, An inverse problem for Maxwell's equations in bi-isotropic media, SIAM J. Math. Anal., 37 (2005), 1027-1043. doi: 10.1137/S003614100444366X.

[32]

S. R. McDowall, Boundary determination of material parameters from electromagnetic boundary information, Inverse Problems, 13 (1997), 153-163. doi: 10.1088/0266-5611/13/1/012.

[33]

S. R. McDowall, An electromagnetic inverse problem in chiral media, T. Am. Math. Soc., 352 (2000), 2993-3013. doi: 10.1090/S0002-9947-00-02518-6.

[34]

S. Nicaise, Time domain study of the Drude-Born-Fedorov model for a class of heterogeneous chiral materials, Math. Meth. Appl. Sci., 36 (2013), 794-813. doi: 10.1002/mma.2627.

[35]

P. Ola, Boundary integral equations for the scattering of electromagnetic waves by a homogeneous chiral obstacle, J. Math. Phys., 35 (1994), 3969-3980. doi: 10.1063/1.530836.

[36]

R. Potthast and I. G. Stratis, On the domain derivative for scattering by impenetrable obstacles in chiral media, IMA J. Appl. Math., 68 (2003), 621-635. doi: 10.1093/imamat/68.6.621.

[37]

G. F. Roach, I. G. Stratis and A. N. Yannacopoulos, Mathematical Analysis of Deterministic and Stochastic Problems in Complex Media Electromagnetics, Princeton Series in Applied Mathematics, Princeton University Press, 2012.

[38]

C. Sabah and H. G. Roskos, Design of a terahertz polarization rotator based on a periodic sequence of chiral-metamaterial and dielectric slabs, Prog. Electromagn. Res., 124 (2012), 301-314. doi: 10.2528/PIER11112605.

[39]

K. Sandfort, The factorization method for inverse scattering from periodic inhomogeneous media, PhD thesis, Karlsruher Institut für Technologie, 2010, URL http://digbib.ubka.uni-karlsruhe.de/volltexte/1000019400.

[40]

G. Schmidt, On the diffraction by biperiodic anisotropic structures, Appl. Anal., 82 (2003), 75-92. doi: 10.1080/0003681031000068275.

[41]

I. G. Stratis, Electromagnetic scattering problems in chiral media: A review, Electromagnetics, 19 (1999), 547-562. doi: 10.1080/02726349908908673.

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