Abstract

The weak scattering approximation is used when designing optical media that couple fields together, but to account for the interactions of multiple fields in a volume or to achieve the best efficiency, the solution must be consistent with Maxwell’s equations. We describe a method based on the variational formulation of Maxwell’s equations typically employed in the finite element method (FEM) that finds both the fields and the medium that couples incident and scattered fields together, and so can be considered an extension of the FEM when both the field and the medium are allowed to vary. The method iteratively updates estimates of the field and the medium and can be readily implemented. We demonstrate designs of diffractive and refractive elements that couple fields together using an iteratively updated finite-difference-frequency-domain (FDFD) solution. Such methods that are fully consistent with Maxwell’s equations are needed to design metamaterials that fully exploit strongly interacting metamaterial elements.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. J.-M. Jin, The Finite Element Method in Electromagnetics (Wiley, 2014).
  2. J. L. Volakis, A. Chatterjee, and L. C. Kempel, Finite Element Method for Electromagnetics (Wiley, 1998).
    [Crossref]
  3. R.-E. Plessix, “A review of the adjoint-state method for computing the gradient of a functional with geophysical applications,” Geophys. J. Int. 167, 495–503 (2006).
    [Crossref]
  4. M. B. Giles and N. A. Pierce, “An introduction to the adjoint approach to design,” Flow. Turbul. Combust. 65, 393–415 (2000).
    [Crossref]
  5. P. R. McGillivray and D. W. Oldenburg, “Methods for calculating Frechet derivatives and sensitivities for the non-linear inverse problem: a comparative study,” Geophys. Prospect. 38, 499–524 (1990).
    [Crossref]
  6. S. J. Norton, “Iterative inverse scattering algorithms: Methods of computing Frechet derivatives,” J. Acoust. Soc. Am. 106, 2653–2660 (1999).
    [Crossref]
  7. F. Hettlich, “Frechet derivatives in inverse obstacle scattering,” Inverse Probl. 11, 371 (1995).
    [Crossref]
  8. I. T. Rekanos and T. D. Tsiboukis, “A combined finite element-nonlinear conjugate gradient spatial method for the reconstruction of unknown scatterer profiles,” IEEE T. Magn. 34, 2829–2832 (1998).
    [Crossref]
  9. I. T. Rekanos, T. V. Yioultsis, and T. D. Tsiboukis, “Inverse scattering using the finite-element method and a nonlinear optimization technique,” IEEE T. Microw. Theory 47, 336–344 (1999).
    [Crossref]
  10. I. T. Rekanos, “Inverse scattering in the time domain: an iterative method using an FDTD sensitivity analysis scheme,” IEEE T. Magn. 38, 1117–1120 (2002).
    [Crossref]
  11. D. Bertsimas, O. Nohadani, and K. M. Teo, “Robust optimization in electromagnetic scattering problems,” J. Appl. Phys. 101, 074507 (2007).
    [Crossref]
  12. Y. Cao, J. Xie, Y. Liu, and Z. Liu, “Modeling and optimization of photonic crystal devices based on transformation optics method,” Opt. Express 22, 2725–2734 (2014).
    [Crossref] [PubMed]
  13. H. Khalil, S. Bila, M. Aubourg, D. Baillargeat, S. Verdeyme, F. Jouve, C. Delage, and T. Chartier, “Shape optimized design of microwave dielectric resonators by level-set and topology gradient methods,” Int. J. RF Microw. C. E. 20, 33–41 (2010).
  14. C. M. Lalau-Keraly, S. Bhargava, O. D. Miller, and E. Yablonovitch, “Adjoint shape optimization applied to electromagnetic design,” Opt. Express 21, 21693–21701 (2013).
    [Crossref] [PubMed]
  15. A. A. Oberai, N. H. Gokhale, and G. R. Feijoo, “Solution of inverse problems in elasticity imaging using the adjoint method,” Inverse Probl. 19, 297 (2003).
    [Crossref]
  16. G. Bao and P. Li, “Inverse medium scattering for three-dimensional time harmonic Maxwell equations,” Inverse Probl. 20, L1 (2004).
    [Crossref]
  17. G. Bao and P. Li, “Inverse medium scattering problems for electromagnetic waves,” SIAM J. Appl. Math. 65, 2049–2066 (2005).
    [Crossref]
  18. A. Berk, “Variational principles for electromagnetic resonators and waveguides,” IRE T. Antenn. Propag. 4, 104–111 (1956).
    [Crossref]
  19. D. Jones, “A critique of the variational method in scattering problems,” IRE T. Antenn. Propag. 4, 297–301 (1956).
    [Crossref]
  20. T. G. Hazel and A. Wexler, “Variational formulation of the Dirichlet boundary condition,” IEEE T. Microw. Theory 20, 385–390 (1972).
    [Crossref]
  21. B. H. McDonald, M. Friedman, and A. Wexler, “Variational solution of integral equations,” IEEE T. Microw. Theory 22, 237–248 (1974).
    [Crossref]
  22. A. Konrad, “Vector variational formulation of electromagnetic fields in anisotropic media,” IEEE T. Microw. Theory 24, 553–559 (1976).
    [Crossref]
  23. K. Morishita and N. Kumagai, “Unified approach to the derivation of variational expression for electromagnetic fields,” IEEE T. Microw. Theory 25, 34–40 (1977).
    [Crossref]
  24. A. Mohsen, “On the variational solution of electromagnetic problems in lossy anisotropic inhomogeneous media,” J. Phys. A-Math. Gen. 11, 1681 (1978).
    [Crossref]
  25. W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, 1962).
  26. C. H. Chen and C.-D. Lien, “The variational principle for non-self-adjoint electromagnetic problems,” IEEE T. Microw. Theory 28, 878–886 (1980).
    [Crossref]
  27. G. Jeng and A. Wexler, “Self-adjoint variational formulation of problems having non-self-adjoint operators,” IEEE T. Microw. Theory 26, 91–94 (1978).
    [Crossref]
  28. W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, 1995).
  29. R. Kleinman and P. den Berg, “A modified gradient method for two-dimensional problems in tomography,” J. Comput. Appl. Math. 42, 17–35 (1992).
    [Crossref]
  30. K. Kilgore, S. Moskow, and J. C. Schotland, “Convergence of the Born and inverse Born series for electromagnetic scattering,” Appl. Anal. 96, 1737–1748 (2017).
    [Crossref]

2017 (1)

K. Kilgore, S. Moskow, and J. C. Schotland, “Convergence of the Born and inverse Born series for electromagnetic scattering,” Appl. Anal. 96, 1737–1748 (2017).
[Crossref]

2014 (1)

2013 (1)

2010 (1)

H. Khalil, S. Bila, M. Aubourg, D. Baillargeat, S. Verdeyme, F. Jouve, C. Delage, and T. Chartier, “Shape optimized design of microwave dielectric resonators by level-set and topology gradient methods,” Int. J. RF Microw. C. E. 20, 33–41 (2010).

2007 (1)

D. Bertsimas, O. Nohadani, and K. M. Teo, “Robust optimization in electromagnetic scattering problems,” J. Appl. Phys. 101, 074507 (2007).
[Crossref]

2006 (1)

R.-E. Plessix, “A review of the adjoint-state method for computing the gradient of a functional with geophysical applications,” Geophys. J. Int. 167, 495–503 (2006).
[Crossref]

2005 (1)

G. Bao and P. Li, “Inverse medium scattering problems for electromagnetic waves,” SIAM J. Appl. Math. 65, 2049–2066 (2005).
[Crossref]

2004 (1)

G. Bao and P. Li, “Inverse medium scattering for three-dimensional time harmonic Maxwell equations,” Inverse Probl. 20, L1 (2004).
[Crossref]

2003 (1)

A. A. Oberai, N. H. Gokhale, and G. R. Feijoo, “Solution of inverse problems in elasticity imaging using the adjoint method,” Inverse Probl. 19, 297 (2003).
[Crossref]

2002 (1)

I. T. Rekanos, “Inverse scattering in the time domain: an iterative method using an FDTD sensitivity analysis scheme,” IEEE T. Magn. 38, 1117–1120 (2002).
[Crossref]

2000 (1)

M. B. Giles and N. A. Pierce, “An introduction to the adjoint approach to design,” Flow. Turbul. Combust. 65, 393–415 (2000).
[Crossref]

1999 (2)

S. J. Norton, “Iterative inverse scattering algorithms: Methods of computing Frechet derivatives,” J. Acoust. Soc. Am. 106, 2653–2660 (1999).
[Crossref]

I. T. Rekanos, T. V. Yioultsis, and T. D. Tsiboukis, “Inverse scattering using the finite-element method and a nonlinear optimization technique,” IEEE T. Microw. Theory 47, 336–344 (1999).
[Crossref]

1998 (1)

I. T. Rekanos and T. D. Tsiboukis, “A combined finite element-nonlinear conjugate gradient spatial method for the reconstruction of unknown scatterer profiles,” IEEE T. Magn. 34, 2829–2832 (1998).
[Crossref]

1995 (1)

F. Hettlich, “Frechet derivatives in inverse obstacle scattering,” Inverse Probl. 11, 371 (1995).
[Crossref]

1992 (1)

R. Kleinman and P. den Berg, “A modified gradient method for two-dimensional problems in tomography,” J. Comput. Appl. Math. 42, 17–35 (1992).
[Crossref]

1990 (1)

P. R. McGillivray and D. W. Oldenburg, “Methods for calculating Frechet derivatives and sensitivities for the non-linear inverse problem: a comparative study,” Geophys. Prospect. 38, 499–524 (1990).
[Crossref]

1980 (1)

C. H. Chen and C.-D. Lien, “The variational principle for non-self-adjoint electromagnetic problems,” IEEE T. Microw. Theory 28, 878–886 (1980).
[Crossref]

1978 (2)

G. Jeng and A. Wexler, “Self-adjoint variational formulation of problems having non-self-adjoint operators,” IEEE T. Microw. Theory 26, 91–94 (1978).
[Crossref]

A. Mohsen, “On the variational solution of electromagnetic problems in lossy anisotropic inhomogeneous media,” J. Phys. A-Math. Gen. 11, 1681 (1978).
[Crossref]

1977 (1)

K. Morishita and N. Kumagai, “Unified approach to the derivation of variational expression for electromagnetic fields,” IEEE T. Microw. Theory 25, 34–40 (1977).
[Crossref]

1976 (1)

A. Konrad, “Vector variational formulation of electromagnetic fields in anisotropic media,” IEEE T. Microw. Theory 24, 553–559 (1976).
[Crossref]

1974 (1)

B. H. McDonald, M. Friedman, and A. Wexler, “Variational solution of integral equations,” IEEE T. Microw. Theory 22, 237–248 (1974).
[Crossref]

1972 (1)

T. G. Hazel and A. Wexler, “Variational formulation of the Dirichlet boundary condition,” IEEE T. Microw. Theory 20, 385–390 (1972).
[Crossref]

1956 (2)

A. Berk, “Variational principles for electromagnetic resonators and waveguides,” IRE T. Antenn. Propag. 4, 104–111 (1956).
[Crossref]

D. Jones, “A critique of the variational method in scattering problems,” IRE T. Antenn. Propag. 4, 297–301 (1956).
[Crossref]

Aubourg, M.

H. Khalil, S. Bila, M. Aubourg, D. Baillargeat, S. Verdeyme, F. Jouve, C. Delage, and T. Chartier, “Shape optimized design of microwave dielectric resonators by level-set and topology gradient methods,” Int. J. RF Microw. C. E. 20, 33–41 (2010).

Baillargeat, D.

H. Khalil, S. Bila, M. Aubourg, D. Baillargeat, S. Verdeyme, F. Jouve, C. Delage, and T. Chartier, “Shape optimized design of microwave dielectric resonators by level-set and topology gradient methods,” Int. J. RF Microw. C. E. 20, 33–41 (2010).

Bao, G.

G. Bao and P. Li, “Inverse medium scattering problems for electromagnetic waves,” SIAM J. Appl. Math. 65, 2049–2066 (2005).
[Crossref]

G. Bao and P. Li, “Inverse medium scattering for three-dimensional time harmonic Maxwell equations,” Inverse Probl. 20, L1 (2004).
[Crossref]

Berk, A.

A. Berk, “Variational principles for electromagnetic resonators and waveguides,” IRE T. Antenn. Propag. 4, 104–111 (1956).
[Crossref]

Bertsimas, D.

D. Bertsimas, O. Nohadani, and K. M. Teo, “Robust optimization in electromagnetic scattering problems,” J. Appl. Phys. 101, 074507 (2007).
[Crossref]

Bhargava, S.

Bila, S.

H. Khalil, S. Bila, M. Aubourg, D. Baillargeat, S. Verdeyme, F. Jouve, C. Delage, and T. Chartier, “Shape optimized design of microwave dielectric resonators by level-set and topology gradient methods,” Int. J. RF Microw. C. E. 20, 33–41 (2010).

Cao, Y.

Chartier, T.

H. Khalil, S. Bila, M. Aubourg, D. Baillargeat, S. Verdeyme, F. Jouve, C. Delage, and T. Chartier, “Shape optimized design of microwave dielectric resonators by level-set and topology gradient methods,” Int. J. RF Microw. C. E. 20, 33–41 (2010).

Chatterjee, A.

J. L. Volakis, A. Chatterjee, and L. C. Kempel, Finite Element Method for Electromagnetics (Wiley, 1998).
[Crossref]

Chen, C. H.

C. H. Chen and C.-D. Lien, “The variational principle for non-self-adjoint electromagnetic problems,” IEEE T. Microw. Theory 28, 878–886 (1980).
[Crossref]

Chew, W. C.

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, 1995).

Delage, C.

H. Khalil, S. Bila, M. Aubourg, D. Baillargeat, S. Verdeyme, F. Jouve, C. Delage, and T. Chartier, “Shape optimized design of microwave dielectric resonators by level-set and topology gradient methods,” Int. J. RF Microw. C. E. 20, 33–41 (2010).

den Berg, P.

R. Kleinman and P. den Berg, “A modified gradient method for two-dimensional problems in tomography,” J. Comput. Appl. Math. 42, 17–35 (1992).
[Crossref]

Feijoo, G. R.

A. A. Oberai, N. H. Gokhale, and G. R. Feijoo, “Solution of inverse problems in elasticity imaging using the adjoint method,” Inverse Probl. 19, 297 (2003).
[Crossref]

Friedman, M.

B. H. McDonald, M. Friedman, and A. Wexler, “Variational solution of integral equations,” IEEE T. Microw. Theory 22, 237–248 (1974).
[Crossref]

Giles, M. B.

M. B. Giles and N. A. Pierce, “An introduction to the adjoint approach to design,” Flow. Turbul. Combust. 65, 393–415 (2000).
[Crossref]

Gokhale, N. H.

A. A. Oberai, N. H. Gokhale, and G. R. Feijoo, “Solution of inverse problems in elasticity imaging using the adjoint method,” Inverse Probl. 19, 297 (2003).
[Crossref]

Hazel, T. G.

T. G. Hazel and A. Wexler, “Variational formulation of the Dirichlet boundary condition,” IEEE T. Microw. Theory 20, 385–390 (1972).
[Crossref]

Hettlich, F.

F. Hettlich, “Frechet derivatives in inverse obstacle scattering,” Inverse Probl. 11, 371 (1995).
[Crossref]

Jeng, G.

G. Jeng and A. Wexler, “Self-adjoint variational formulation of problems having non-self-adjoint operators,” IEEE T. Microw. Theory 26, 91–94 (1978).
[Crossref]

Jin, J.-M.

J.-M. Jin, The Finite Element Method in Electromagnetics (Wiley, 2014).

Jones, D.

D. Jones, “A critique of the variational method in scattering problems,” IRE T. Antenn. Propag. 4, 297–301 (1956).
[Crossref]

Jouve, F.

H. Khalil, S. Bila, M. Aubourg, D. Baillargeat, S. Verdeyme, F. Jouve, C. Delage, and T. Chartier, “Shape optimized design of microwave dielectric resonators by level-set and topology gradient methods,” Int. J. RF Microw. C. E. 20, 33–41 (2010).

Kempel, L. C.

J. L. Volakis, A. Chatterjee, and L. C. Kempel, Finite Element Method for Electromagnetics (Wiley, 1998).
[Crossref]

Khalil, H.

H. Khalil, S. Bila, M. Aubourg, D. Baillargeat, S. Verdeyme, F. Jouve, C. Delage, and T. Chartier, “Shape optimized design of microwave dielectric resonators by level-set and topology gradient methods,” Int. J. RF Microw. C. E. 20, 33–41 (2010).

Kilgore, K.

K. Kilgore, S. Moskow, and J. C. Schotland, “Convergence of the Born and inverse Born series for electromagnetic scattering,” Appl. Anal. 96, 1737–1748 (2017).
[Crossref]

Kleinman, R.

R. Kleinman and P. den Berg, “A modified gradient method for two-dimensional problems in tomography,” J. Comput. Appl. Math. 42, 17–35 (1992).
[Crossref]

Konrad, A.

A. Konrad, “Vector variational formulation of electromagnetic fields in anisotropic media,” IEEE T. Microw. Theory 24, 553–559 (1976).
[Crossref]

Kumagai, N.

K. Morishita and N. Kumagai, “Unified approach to the derivation of variational expression for electromagnetic fields,” IEEE T. Microw. Theory 25, 34–40 (1977).
[Crossref]

Lalau-Keraly, C. M.

Li, P.

G. Bao and P. Li, “Inverse medium scattering problems for electromagnetic waves,” SIAM J. Appl. Math. 65, 2049–2066 (2005).
[Crossref]

G. Bao and P. Li, “Inverse medium scattering for three-dimensional time harmonic Maxwell equations,” Inverse Probl. 20, L1 (2004).
[Crossref]

Lien, C.-D.

C. H. Chen and C.-D. Lien, “The variational principle for non-self-adjoint electromagnetic problems,” IEEE T. Microw. Theory 28, 878–886 (1980).
[Crossref]

Liu, Y.

Liu, Z.

McDonald, B. H.

B. H. McDonald, M. Friedman, and A. Wexler, “Variational solution of integral equations,” IEEE T. Microw. Theory 22, 237–248 (1974).
[Crossref]

McGillivray, P. R.

P. R. McGillivray and D. W. Oldenburg, “Methods for calculating Frechet derivatives and sensitivities for the non-linear inverse problem: a comparative study,” Geophys. Prospect. 38, 499–524 (1990).
[Crossref]

Miller, O. D.

Mohsen, A.

A. Mohsen, “On the variational solution of electromagnetic problems in lossy anisotropic inhomogeneous media,” J. Phys. A-Math. Gen. 11, 1681 (1978).
[Crossref]

Morishita, K.

K. Morishita and N. Kumagai, “Unified approach to the derivation of variational expression for electromagnetic fields,” IEEE T. Microw. Theory 25, 34–40 (1977).
[Crossref]

Moskow, S.

K. Kilgore, S. Moskow, and J. C. Schotland, “Convergence of the Born and inverse Born series for electromagnetic scattering,” Appl. Anal. 96, 1737–1748 (2017).
[Crossref]

Nohadani, O.

D. Bertsimas, O. Nohadani, and K. M. Teo, “Robust optimization in electromagnetic scattering problems,” J. Appl. Phys. 101, 074507 (2007).
[Crossref]

Norton, S. J.

S. J. Norton, “Iterative inverse scattering algorithms: Methods of computing Frechet derivatives,” J. Acoust. Soc. Am. 106, 2653–2660 (1999).
[Crossref]

Oberai, A. A.

A. A. Oberai, N. H. Gokhale, and G. R. Feijoo, “Solution of inverse problems in elasticity imaging using the adjoint method,” Inverse Probl. 19, 297 (2003).
[Crossref]

Oldenburg, D. W.

P. R. McGillivray and D. W. Oldenburg, “Methods for calculating Frechet derivatives and sensitivities for the non-linear inverse problem: a comparative study,” Geophys. Prospect. 38, 499–524 (1990).
[Crossref]

Panofsky, W. K. H.

W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, 1962).

Phillips, M.

W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, 1962).

Pierce, N. A.

M. B. Giles and N. A. Pierce, “An introduction to the adjoint approach to design,” Flow. Turbul. Combust. 65, 393–415 (2000).
[Crossref]

Plessix, R.-E.

R.-E. Plessix, “A review of the adjoint-state method for computing the gradient of a functional with geophysical applications,” Geophys. J. Int. 167, 495–503 (2006).
[Crossref]

Rekanos, I. T.

I. T. Rekanos, “Inverse scattering in the time domain: an iterative method using an FDTD sensitivity analysis scheme,” IEEE T. Magn. 38, 1117–1120 (2002).
[Crossref]

I. T. Rekanos, T. V. Yioultsis, and T. D. Tsiboukis, “Inverse scattering using the finite-element method and a nonlinear optimization technique,” IEEE T. Microw. Theory 47, 336–344 (1999).
[Crossref]

I. T. Rekanos and T. D. Tsiboukis, “A combined finite element-nonlinear conjugate gradient spatial method for the reconstruction of unknown scatterer profiles,” IEEE T. Magn. 34, 2829–2832 (1998).
[Crossref]

Schotland, J. C.

K. Kilgore, S. Moskow, and J. C. Schotland, “Convergence of the Born and inverse Born series for electromagnetic scattering,” Appl. Anal. 96, 1737–1748 (2017).
[Crossref]

Teo, K. M.

D. Bertsimas, O. Nohadani, and K. M. Teo, “Robust optimization in electromagnetic scattering problems,” J. Appl. Phys. 101, 074507 (2007).
[Crossref]

Tsiboukis, T. D.

I. T. Rekanos, T. V. Yioultsis, and T. D. Tsiboukis, “Inverse scattering using the finite-element method and a nonlinear optimization technique,” IEEE T. Microw. Theory 47, 336–344 (1999).
[Crossref]

I. T. Rekanos and T. D. Tsiboukis, “A combined finite element-nonlinear conjugate gradient spatial method for the reconstruction of unknown scatterer profiles,” IEEE T. Magn. 34, 2829–2832 (1998).
[Crossref]

Verdeyme, S.

H. Khalil, S. Bila, M. Aubourg, D. Baillargeat, S. Verdeyme, F. Jouve, C. Delage, and T. Chartier, “Shape optimized design of microwave dielectric resonators by level-set and topology gradient methods,” Int. J. RF Microw. C. E. 20, 33–41 (2010).

Volakis, J. L.

J. L. Volakis, A. Chatterjee, and L. C. Kempel, Finite Element Method for Electromagnetics (Wiley, 1998).
[Crossref]

Wexler, A.

G. Jeng and A. Wexler, “Self-adjoint variational formulation of problems having non-self-adjoint operators,” IEEE T. Microw. Theory 26, 91–94 (1978).
[Crossref]

B. H. McDonald, M. Friedman, and A. Wexler, “Variational solution of integral equations,” IEEE T. Microw. Theory 22, 237–248 (1974).
[Crossref]

T. G. Hazel and A. Wexler, “Variational formulation of the Dirichlet boundary condition,” IEEE T. Microw. Theory 20, 385–390 (1972).
[Crossref]

Xie, J.

Yablonovitch, E.

Yioultsis, T. V.

I. T. Rekanos, T. V. Yioultsis, and T. D. Tsiboukis, “Inverse scattering using the finite-element method and a nonlinear optimization technique,” IEEE T. Microw. Theory 47, 336–344 (1999).
[Crossref]

Appl. Anal. (1)

K. Kilgore, S. Moskow, and J. C. Schotland, “Convergence of the Born and inverse Born series for electromagnetic scattering,” Appl. Anal. 96, 1737–1748 (2017).
[Crossref]

Flow. Turbul. Combust. (1)

M. B. Giles and N. A. Pierce, “An introduction to the adjoint approach to design,” Flow. Turbul. Combust. 65, 393–415 (2000).
[Crossref]

Geophys. J. Int. (1)

R.-E. Plessix, “A review of the adjoint-state method for computing the gradient of a functional with geophysical applications,” Geophys. J. Int. 167, 495–503 (2006).
[Crossref]

Geophys. Prospect. (1)

P. R. McGillivray and D. W. Oldenburg, “Methods for calculating Frechet derivatives and sensitivities for the non-linear inverse problem: a comparative study,” Geophys. Prospect. 38, 499–524 (1990).
[Crossref]

IEEE T. Magn. (2)

I. T. Rekanos and T. D. Tsiboukis, “A combined finite element-nonlinear conjugate gradient spatial method for the reconstruction of unknown scatterer profiles,” IEEE T. Magn. 34, 2829–2832 (1998).
[Crossref]

I. T. Rekanos, “Inverse scattering in the time domain: an iterative method using an FDTD sensitivity analysis scheme,” IEEE T. Magn. 38, 1117–1120 (2002).
[Crossref]

IEEE T. Microw. Theory (7)

I. T. Rekanos, T. V. Yioultsis, and T. D. Tsiboukis, “Inverse scattering using the finite-element method and a nonlinear optimization technique,” IEEE T. Microw. Theory 47, 336–344 (1999).
[Crossref]

C. H. Chen and C.-D. Lien, “The variational principle for non-self-adjoint electromagnetic problems,” IEEE T. Microw. Theory 28, 878–886 (1980).
[Crossref]

G. Jeng and A. Wexler, “Self-adjoint variational formulation of problems having non-self-adjoint operators,” IEEE T. Microw. Theory 26, 91–94 (1978).
[Crossref]

T. G. Hazel and A. Wexler, “Variational formulation of the Dirichlet boundary condition,” IEEE T. Microw. Theory 20, 385–390 (1972).
[Crossref]

B. H. McDonald, M. Friedman, and A. Wexler, “Variational solution of integral equations,” IEEE T. Microw. Theory 22, 237–248 (1974).
[Crossref]

A. Konrad, “Vector variational formulation of electromagnetic fields in anisotropic media,” IEEE T. Microw. Theory 24, 553–559 (1976).
[Crossref]

K. Morishita and N. Kumagai, “Unified approach to the derivation of variational expression for electromagnetic fields,” IEEE T. Microw. Theory 25, 34–40 (1977).
[Crossref]

Int. J. RF Microw. C. E. (1)

H. Khalil, S. Bila, M. Aubourg, D. Baillargeat, S. Verdeyme, F. Jouve, C. Delage, and T. Chartier, “Shape optimized design of microwave dielectric resonators by level-set and topology gradient methods,” Int. J. RF Microw. C. E. 20, 33–41 (2010).

Inverse Probl. (3)

A. A. Oberai, N. H. Gokhale, and G. R. Feijoo, “Solution of inverse problems in elasticity imaging using the adjoint method,” Inverse Probl. 19, 297 (2003).
[Crossref]

G. Bao and P. Li, “Inverse medium scattering for three-dimensional time harmonic Maxwell equations,” Inverse Probl. 20, L1 (2004).
[Crossref]

F. Hettlich, “Frechet derivatives in inverse obstacle scattering,” Inverse Probl. 11, 371 (1995).
[Crossref]

IRE T. Antenn. Propag. (2)

A. Berk, “Variational principles for electromagnetic resonators and waveguides,” IRE T. Antenn. Propag. 4, 104–111 (1956).
[Crossref]

D. Jones, “A critique of the variational method in scattering problems,” IRE T. Antenn. Propag. 4, 297–301 (1956).
[Crossref]

J. Acoust. Soc. Am. (1)

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Supplementary Material (5)

NameDescription
» Visualization 1       Diffractive solution of a single incident field coupled to a single scattered field.
» Visualization 2       Refractive solution without smoothing of a single incident field coupled to a single scattered field.
» Visualization 3       Refractive solution with smoothing of a single incident field coupled to a single scattered field.
» Visualization 4       Diffractive solution of two incident fields coupled to their respective scattered fields.
» Visualization 5       Refractive solution with smoothing of two incident fields coupled to their respective scattered fields.

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Figures (7)

Fig. 1
Fig. 1 The diffractive solution to the coupling an incident to a scattered beam. The beam is incident from the lower left at an angle of 45° and scattered to the lower right at an angle of 60°. The upper left panel is the real part of the incident/non-adjoint electric field, the upper right panel is the real part of the scattered/adjoint field, the lower left is the final permittivity of the medium. the lower right panel shows both field magnitudes, the red channel being the non-adjoint field and the green channel being the adjoint field. The animation of these iterations is shown in Visualization 1.
Fig. 2
Fig. 2 The refractive solution, without smoothing, to the coupling an incident to a scattered beam. The beam is incident from the lower left at an angle of 45° and scattered to the lower right at an angle of 60°. The upper left panel is the real part of the incident/non-adjoint electric field, the upper right panel is the real part of the scattered/adjoint field, the lower left is the final permittivity of the medium. the lower right panel shows both field magnitudes, the red channel being the non-adjoint field and the green channel being the adjoint field. The animation of these iterations is shown in Visualization 2.
Fig. 3
Fig. 3 The refractive solution, with smoothing, to the coupling an incident to a scattered beam. The beam is incident from the lower left at an angle of 45° and scattered to the lower right at an angle of 60°. The upper left panel is the real part of the incident/non-adjoint electric field, the upper right panel is the real part of the scattered/adjoint field, the lower left is the final permittivity of the medium. the lower right panel shows both field magnitudes, the red channel being the non-adjoint field and the green channel being the adjoint field. The animation of these iterations is shown in Visualization 3.
Fig. 4
Fig. 4 The coupled power between the incident and the scattered fields at each iteration, for (a) the diffractive solution, (b) the refractive solution without smoothing, and (c) the refractive solution with smoothing. While the power is specified in relative units, these may be compared between the three solutions because the same fields are being coupled in all three simulations. The blue curves are the reaction of the incident field source on the scattered field, the red curves are the reaction of the scattered field source on the incident field, and the black curve is the sum of the two, or the reward function P. The blue and red curves follow closely, showing that Lorentz reciprocity is broadly satisfied. Furthermore, the method does optimize P, despite the fact it is not specifically designed to do so.
Fig. 5
Fig. 5 The diffractive solution to the coupling two incident beams to two scattered beams. The left column is the real part of the incident/non-adjoint electric fields, the second column is the real part of the scattered/adjoint fields. The third column is the final permittivity of the medium. The right column shows both field magnitudes, the red channel being the non-adjoint field and the green channel being the adjoint field. The animation of these iterations is shown in Visualization 4.
Fig. 6
Fig. 6 The refractive solution to the coupling two incident beams to two scattered beams. The left column is the real part of the incident/non-adjoint electric fields, the second column is the real part of the scattered/adjoint fields. The third column is the final permittivity of the medium. The right column shows both field magnitudes, the red channel being the non-adjoint field and the green channel being the adjoint field. The animation of these iterations is shown in Visualization 5.
Fig. 7
Fig. 7 The coupled power between the two pairs of incident and the scattered fields at each iteration, for (a) the diffractive solution and (b) the refractive solution with smoothing. While the power is specified in relative units, these may be compared between the solutions because the same fields are being coupled in both simulations. The blue curve is the total coupled power P for the first set of incident and scattered waves, and the red curve is the total coupled power for the second set of waves. While the diffractive solution monotonically approaches the solution, the efficiency oscillates between the two wave pairs for the refractive solution as a solution is found which is consistent with both being coupled efficiently.

Equations (39)

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× E = i ω μ 0 μ r H
× H = J i ω 0 r E
n ^ × ( n ^ × E ) η 0 ( n ^ × H ) = E ( + )
n ^ × ( n ^ × E ) + η 0 ( n ^ × H ) = E ( )
L ^ E = i k 1 η 0 J for L ^ E = r E k 2 × ( μ r 1 × E ) n ^ × ( n ^ × E ) + i k 1 [ n ^ × ( × E ) ] = E ( + )
L ^ a E a = i k 1 η 0 J a for L ^ a E a = r a E a k 2 × ( μ r 1 * × E a ) n ^ × ( n ^ × E a ) i k 1 [ n ^ × ( × E a ) ] = E ( ) *
L ^ E a * = i k 1 η 0 J a for L ^ E a = r E a * k 2 × ( μ r 1 × E a * ) n ^ × ( n ^ × E a * ) + i k 1 [ n ^ × ( × E a * ) ] = E ( )
δ F = V δ E a * ( L ^ E + i k 1 η 0 J ) d V + V δ E ( L ^ a E a + i k 1 η 0 J a ) * d V
δ F = V r δ E a E + i k 1 η 0 δ E a * J k 1 δ E a × [ μ r 1 × E ] d V + V r δ E E a * i k 1 η 0 δ E J a * k 1 δ E × [ μ r 1 × E a * ] d V
V a ( × b ) d V = V ( × a ) b d V S ( a × b ) n ^ d S
δ F = V r δ E a * E + i k 1 η 0 δ E a * J μ r 1 k 2 ( × δ E a ) * ( × E ) d V + V r δ E E a * i k 1 η 0 δ E J a * μ r 1 k 2 ( × δ E ) ( × E a ) * d V + S k 2 [ δ E a * × ( × E ) ] n ^ d S + S k 2 [ δ E × ( × E a ) * ] n ^ d S
δ F = V r δ E a * E + i k 1 η 0 δ E a * J μ r 1 k 2 ( × δ E a ) * ( × E ) d V + V r δ E E a * i k 1 η 0 δ E J a * μ r 1 k 2 ( × δ E ) ( × E a ) * d V S k 2 [ n ^ × ( × E ) ] δ E a * d S S k 2 [ n ^ × ( × E a ) * ] δ E d S
δ F = V r δ E a * E + i k 1 η 0 δ E a * J μ r 1 k 2 ( × δ E a ) * ( × E ) d V + V r δ E E a * i k 1 η 0 δ E J a * μ r 1 k 2 ( × δ E ) ( × E a ) * d V + S i k 1 [ n ^ × ( n ^ × E ) + E ( + ) ] n ^ × ( n ^ × δ E a ) * d S + S i k 1 [ n ^ × ( n ^ × E ( a ) ) + E ( ) * ] * n ^ × ( n ^ × δ E ) d S
F = V r E a * E + i k 1 η 0 ( E a * J E J a * ) μ r 1 k 2 ( × E a ) * ( × E ) d V + S i k 1 [ n ^ × ( n ^ × E ) + E ( + ) ] [ n ^ × ( n ^ × E a ) + E ( ) * ] * d S
F T = n = 1 N w n V r E n a * E n + i k n 1 η 0 ( E n a * J n E n J n a * ) μ r 1 k n 1 ( × E n a ) * ( × E n ) d V + w n S i k n 1 [ n ^ × ( n ^ × E n ) + E n ( + ) ] [ n ^ × ( n ^ × E n a ) + E n ( ) * ] * d S
E n ( p + 1 ) = E n ( p ) β ( L ^ n E n ( p ) + i k 1 η 0 J n )
E n a ( p + 1 ) = E n a ( p ) β ( L ^ n a E n a ( p + 1 ) + i k 1 η 0 J n a )
E n ( p + 1 ) = E n ( p ) β B ^ n a L ^ n a ( L ^ n B ^ n E n ( p ) + i k 1 η 0 J n )
E n a ( p + 1 ) = E n a ( p ) β B ^ n L ^ n ( L ^ n a B ^ n a E n a ( p + 1 ) + i k 1 η 0 J n a )
δ F = V r δ E a * E μ r 1 k 1 ( × δ E a ) * ( × E ) d V + V r δ E a * μ r 1 k 2 ( × δ E ) ( × E a ) * d V + S i k 1 [ n ^ × ( n ^ × E ) + E ( + ) + η 0 J Δ ] n ^ × ( n ^ × δ E a ) * d S + S i k 1 [ n ^ × ( n ^ × E a ) + E ( ) * + η 0 J a Δ ] * n ^ × ( n ^ × δ E ) d S
F T = V r [ n = 1 N w n E n a * E n ] μ r 1 [ n = 1 N w n k n 1 ( × E n a ) * ( × E n ) ] K d V
δ F T = V δ r [ n = 1 N w n E n a * E n ] + δ μ r μ r 2 [ n = 1 N w n k n 2 ( × E n a ) * ( × E n ) ] d V
μ r ( p + 1 ) = μ r ( p ) γ μ F T μ r
r ( p + 1 ) = r ( p ) γ F T r
F T μ r = μ r 2 n = 1 N w n k n 2 ( × E n a ) * ( × E n )
F T r = n = 1 N w n E n a * E n
L ^ E = i k 1 η 0 J and L ^ E a = i k 1 η 0 J a n ^ × ( n ^ × E ) + i k 1 [ n ^ × ( × E ) ] = E ( + ) n ^ × ( n ^ × E a ) i k 1 [ n ^ × ( × E a ) ] = E ( )
μ r ( p + 1 ) = μ r ( p ) γ μ μ r 2 n = 1 N w n k n 2 ( × E n a ) ( × E n )
r ( p + 1 ) = r ( p ) γ n = 1 N w n E n a E n
μ r r = K μ r + n = 1 N w n k n 1 ( × E n a ) * ( × E n ) n = 1 N w n E n a * E n
μ r r = K μ r + n = 1 N w n E m 4 + n = 1 N w n k n 2 ( × E n a ) * ( × E n ) ( E n a E n ) * n = 1 N w n E m 4 + n = 1 N w n | E n a * E n | 2
μ r r = K μ r n = 1 N w n E m 4 + n = 1 N w n k n 2 ( × E n B ) * ( × E n A ) ( E n B * E n A ) * n = 1 N w n E m 4 + n = 1 N w n | E n B * E n A | 2
μ r ( p + 1 ) = μ r ( p ) γ μ μ r 2 n = 1 N w n k n 1 ( × E n B ) ( × E n A )
r ( p + 1 ) = r ( p ) γ n = 1 N w n E n B E n A
V J E a * d V = V J a * E d V
P = | n = 1 N V J n E n a * + J n a * E n d V |
L ^ n E = r E + k n 1 [ 2 E x 2 + 2 E y 2 ] L ^ n a E a = r * E a + k n 2 [ 2 E a x 2 + 2 E a y 2 ] E n ( p + 1 ) = E n ( p ) β L ^ n a ( L ^ n E n ( p ) + i ( k Δ ) 1 E n ( + ) ) E n a ( p + 1 ) = E n a ( p ) β L ^ n ( L ^ n a E n a ( p ) + i ( k Δ ) 1 E n ( ) * )
r ( p + 1 ) = r ( p ) γ E m 2 n = 1 N w n E n B ( p ) E n A ( p )
r ( p + 1 ) = ( 1 γ ) r ( p ) + γ = r ( 0 ) n = 1 N w n E m 4 + n = 1 N w n k n 2 [ E n B ( p ) * x E n A ( p ) x + E n B ( p ) * y E n A ( p ) y ] ( E n B ( p ) * E n A ( p ) * ) n = 1 N w n E m 4 + n = 1 N w n | E n B * ( p ) E n A ( p ) | 2

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