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 & 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. doi: 10.1002/mana.200310026. Google Scholar

[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. doi: 10.1137/S0036139998334160. Google Scholar

[3]

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

[4]

T. Arens, Scattering by biperiodic layered media: The integral equation approach,, Habilitation Thesis, (2010). Google Scholar

[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. doi: 10.1093/imamat/64.3.245. Google Scholar

[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. doi: 10.1515/jiip.2008.001. Google Scholar

[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. Google Scholar

[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. doi: 10.1002/mma.1670170502. Google Scholar

[9]

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

[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. doi: 10.1063/1.1286052. Google Scholar

[11]

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

[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. doi: 10.1142/S0218202507001991. Google Scholar

[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. doi: 10.1016/j.crma.2010.11.019. Google Scholar

[14]

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

[15]

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

[16]

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

[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). doi: 10.1103/PhysRevLett.106.185501. Google Scholar

[18]

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

[19]

S. Heumann, The Factorization Method for Inverse Scattering from Chiral Media,, PhD thesis, (2012). Google Scholar

[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. doi: 10.1109/8.43564. Google Scholar

[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. doi: 10.1088/0266-5611/14/6/009. Google Scholar

[22]

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

[23]

A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems,, Oxford Lecture Series in Mathematics and its Applications 36, (2008). Google Scholar

[24]

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

[25]

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

[26]

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

[27]

A. Lakhtakia, V. V. Varadan and V. K. Varadan, Scattering by periodic achiral-chiral interfaces,, J. Opt. Soc. Am. A, 6 (1989), 1675. doi: 10.1364/JOSAA.6.001675. Google Scholar

[28]

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

[29]

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

[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. doi: 10.1007/s10444-013-9295-2. Google Scholar

[31]

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

[32]

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

[33]

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

[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. doi: 10.1002/mma.2627. Google Scholar

[35]

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

[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. doi: 10.1093/imamat/68.6.621. Google Scholar

[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, (2012). Google Scholar

[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. doi: 10.2528/PIER11112605. Google Scholar

[39]

K. Sandfort, The factorization method for inverse scattering from periodic inhomogeneous media,, PhD thesis, (2010). Google Scholar

[40]

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

[41]

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

show all references

References:
[1]

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

[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. doi: 10.1137/S0036139998334160. Google Scholar

[3]

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

[4]

T. Arens, Scattering by biperiodic layered media: The integral equation approach,, Habilitation Thesis, (2010). Google Scholar

[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. doi: 10.1093/imamat/64.3.245. Google Scholar

[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. doi: 10.1515/jiip.2008.001. Google Scholar

[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. Google Scholar

[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. doi: 10.1002/mma.1670170502. Google Scholar

[9]

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

[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. doi: 10.1063/1.1286052. Google Scholar

[11]

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

[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. doi: 10.1142/S0218202507001991. Google Scholar

[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. doi: 10.1016/j.crma.2010.11.019. Google Scholar

[14]

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

[15]

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

[16]

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

[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). doi: 10.1103/PhysRevLett.106.185501. Google Scholar

[18]

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

[19]

S. Heumann, The Factorization Method for Inverse Scattering from Chiral Media,, PhD thesis, (2012). Google Scholar

[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. doi: 10.1109/8.43564. Google Scholar

[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. doi: 10.1088/0266-5611/14/6/009. Google Scholar

[22]

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

[23]

A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems,, Oxford Lecture Series in Mathematics and its Applications 36, (2008). Google Scholar

[24]

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

[25]

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

[26]

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

[27]

A. Lakhtakia, V. V. Varadan and V. K. Varadan, Scattering by periodic achiral-chiral interfaces,, J. Opt. Soc. Am. A, 6 (1989), 1675. doi: 10.1364/JOSAA.6.001675. Google Scholar

[28]

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

[29]

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

[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. doi: 10.1007/s10444-013-9295-2. Google Scholar

[31]

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

[32]

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

[33]

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

[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. doi: 10.1002/mma.2627. Google Scholar

[35]

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

[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. doi: 10.1093/imamat/68.6.621. Google Scholar

[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, (2012). Google Scholar

[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. doi: 10.2528/PIER11112605. Google Scholar

[39]

K. Sandfort, The factorization method for inverse scattering from periodic inhomogeneous media,, PhD thesis, (2010). Google Scholar

[40]

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

[41]

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

[1]

Qinghua Wu, Guozheng Yan. The factorization method for a partially coated cavity in inverse scattering. Inverse Problems & Imaging, 2016, 10 (1) : 263-279. doi: 10.3934/ipi.2016.10.263

[2]

Andreas Kirsch, Albert Ruiz. The Factorization Method for an inverse fluid-solid interaction scattering problem. Inverse Problems & Imaging, 2012, 6 (4) : 681-695. doi: 10.3934/ipi.2012.6.681

[3]

Jun Guo, Qinghua Wu, Guozheng Yan. The factorization method for cracks in elastic scattering. Inverse Problems & Imaging, 2018, 12 (2) : 349-371. doi: 10.3934/ipi.2018016

[4]

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

[5]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271

[6]

Guanghui Hu, Andreas Kirsch, Tao Yin. Factorization method in inverse interaction problems with bi-periodic interfaces between acoustic and elastic waves. Inverse Problems & Imaging, 2016, 10 (1) : 103-129. doi: 10.3934/ipi.2016.10.103

[7]

Kaitlyn (Voccola) Muller. A reproducing kernel Hilbert space framework for inverse scattering problems within the Born approximation. Inverse Problems & Imaging, 2019, 13 (6) : 1327-1348. doi: 10.3934/ipi.2019058

[8]

Bastian Gebauer, Nuutti Hyvönen. Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem. Inverse Problems & Imaging, 2008, 2 (3) : 355-372. doi: 10.3934/ipi.2008.2.355

[9]

Simopekka Vänskä. Stationary waves method for inverse scattering problems. Inverse Problems & Imaging, 2008, 2 (4) : 577-586. doi: 10.3934/ipi.2008.2.577

[10]

Fang Zeng, Pablo Suarez, Jiguang Sun. A decomposition method for an interior inverse scattering problem. Inverse Problems & Imaging, 2013, 7 (1) : 291-303. doi: 10.3934/ipi.2013.7.291

[11]

Beatrice Bugert, Gunther Schmidt. Analytical investigation of an integral equation method for electromagnetic scattering by biperiodic structures. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 435-473. doi: 10.3934/dcdss.2015.8.435

[12]

Jingzhi Li, Jun Zou. A direct sampling method for inverse scattering using far-field data. Inverse Problems & Imaging, 2013, 7 (3) : 757-775. doi: 10.3934/ipi.2013.7.757

[13]

Armin Lechleiter. The factorization method is independent of transmission eigenvalues. Inverse Problems & Imaging, 2009, 3 (1) : 123-138. doi: 10.3934/ipi.2009.3.123

[14]

Nuutti Hyvönen, Harri Hakula, Sampsa Pursiainen. Numerical implementation of the factorization method within the complete electrode model of electrical impedance tomography. Inverse Problems & Imaging, 2007, 1 (2) : 299-317. doi: 10.3934/ipi.2007.1.299

[15]

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

[16]

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

[17]

Pedro Serranho. A hybrid method for inverse scattering for Sound-soft obstacles in R3. Inverse Problems & Imaging, 2007, 1 (4) : 691-712. doi: 10.3934/ipi.2007.1.691

[18]

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

[19]

Masaya Maeda, Hironobu Sasaki, Etsuo Segawa, Akito Suzuki, Kanako Suzuki. Scattering and inverse scattering for nonlinear quantum walks. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3687-3703. doi: 10.3934/dcds.2018159

[20]

Francesco Demontis, Cornelis Van der Mee. Novel formulation of inverse scattering and characterization of scattering data. Conference Publications, 2011, 2011 (Special) : 343-350. doi: 10.3934/proc.2011.2011.343

2018 Impact Factor: 1.469

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]