Abstract

Understanding the interaction of light with a highly scattering material is essential for optical microscopy of optically thick and heterogeneous biological tissues. Ensemble-averaged analytic solutions cannot provide more than general predictions for relatively simple cases. Yet, biological tissues contain chiral organic molecules and many of the cells’ structures are birefringent, a property exploited by polarization microscopy for label-free imaging. Solving Maxwell’s equations in such materials is a notoriously hard problem. Here we present an efficient method to determine the propagation of electro-magnetic waves in arbitrary anisotropic materials. We demonstrate how the algorithm enables large scale calculations of the scattered light field in complex birefringent materials, chiral media, and even materials with a negative refractive index.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. O. G. Ernst and M. J. Gander, “Why it is difficult to solve Helmholtz problems with classical iterative methods,” in Numerical Analysis of Multiscale Problems, I. G. Graham, T. Y. Hou, O. Lakkis, and R. Scheichl, eds. (Springer, 2012).
  2. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light - Second Edition (Princeton University, 2008).
  3. H. C. van de Hulst, Light Scattering by Small Particles (Dover Publications Inc., 2003).
  4. S. Rotter and S. Gigan, “Light fields in complex media: Mesoscopic scattering meets wave control,” Rev. Mod. Phys. 89, 015005 (2017).
    [Crossref]
  5. I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32(16), 2309–2311 (2007).
    [Crossref] [PubMed]
  6. I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322 (2010).
    [Crossref]
  7. A. K. Glaser, Y. Chen, and J. T. Liu, “Fractal propagation method enables realistic optical microscopy simulations in biological tissues,” Optica 3(8), 861–869 (2016).
    [Crossref] [PubMed]
  8. Z. Salamon and G. Tollin, “Optical anisotropy in lipid bilayer membranes: coupled plasmon-waveguide resonance measurements of molecular orientation, polarizability, and shape,” Biophys. J. 80(3), 1557–1567 (2001).
    [Crossref] [PubMed]
  9. M. Everett, K. Schoenenberger, B. Colston, and L. Da Silva, “Birefringence characterization of biological tissue by use of optical coherence tomography,” Opt. Lett. 23(3), 228–230 (1998).
    [Crossref]
  10. W.-C. Kuo, S.-C. Lin, and C.-Y. Chuang, “Birefringence measurement in polarization-sensitive optical coherence tomography using differential-envelope detection method,” Rev. Sci. Instrum. 81, 053705 (2010).
    [Crossref] [PubMed]
  11. G. Jarry, F. Henry, and R. Kaiser, “Anisotropy and multiple scattering in thick mammalian tissues,” J. Opt. Soc. Am. A 17(1), 149–153 (2000).
    [Crossref]
  12. J. F. De Boer, T. E. Milner, M. J. van Gemert, and J. S. Nelson, “Two-dimensional birefringence imaging in biological tissue by polarization-sensitive optical coherence tomography,” Opt. Lett. 22(12), 934–936 (1997).
    [Crossref] [PubMed]
  13. S. Alali, K. J. Aitken, A. Schröder, D. J. Bagli, and I. A. Vitkin, “Optical assessment of tissue anisotropy in ex vivo distended rat bladders,” J. Biomed. Opt. 17(8), 0860101 (2012).
    [Crossref]
  14. A. Pierangelo, A. Nazac, A. Benali, P. Validire, H. Cohen, T. Novikova, B. H. Ibrahim, S. Manhas, C. Fallet, M.-R. Antonelli, and A.-D. Martino, “Polarimetric imaging of uterine cervix: a case study,” Opt. Express 21(12), 14120–14130 (2013).
    [Crossref] [PubMed]
  15. M. Koike-Tani, T. Tani, S. B. Mehta, A. Verma, and R. Oldenbourgh, “Polarized light microscopy in reproductive and developmental biology,” Mol. Reprod. Dev. 82(7–8), 548–562 (2013).
    [Crossref] [PubMed]
  16. D. C. Adams, L. P. Hariri, A. J. Miller, Y. Wang, J. L. Cho, M. Villiger, J. A. Holz, M. V. Szabari, D. L. Hamilos, R. Scott Harris, J. W. Griffith, B. E. Bouma, A. D. Luster, B. D. Medoff, and M. J. Suter, “Birefringence microscopy platform for assessing airway smooth muscle structure and function in vivo,” Sci. Transl. Med. 8(359), 359ra131 (2016).
    [Crossref] [PubMed]
  17. G. Osnabrugge, S. Leedumrongwatthanakun, and I. M. Vellekoop, “A convergent Born series for solving the inhomogeneous Helmholtz equation in arbitrarily large media,” J. Comput. Phys. 322, 113–124 (2016).
    [Crossref]
  18. B. Krüger, T. Brenner, and A. Kienle, “Solution of the inhomogeneous Maxwell’s equations using a Born series,” Opt. Express 25(21), 25165–25182 (2017).
    [Crossref] [PubMed]
  19. I. Koutromanos, Fundamentals of Finite Element Analysis: Linear Finite Element Analysis (John Wiley & Sons, 2018).
  20. M. Born, “Quantenmechanik der stoßvorgänge,” Z. Phys. 38, 803 (1926).
    [Crossref]
  21. E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University, 2007), chap. Coherent backscattering of light.
    [Crossref]
  22. R. G. Newton, Scattering Theory of Waves and Particles (Springer, 2013).
  23. S. A. R. Horsley, M. Artoni, and G. C. L. Rocca, “Spatial kramers-kronig relations and the reflection of waves,” Nat. Photonics 9, 436 (2015).
    [Crossref]
  24. J. S. Walker, Fast Fourier Transforms (CRC Press, 2017).
  25. T. Vettenburg, “Macromax: A python 3 library for solving macroscopic maxwell’s equations for electromagnetic waves in gain-free heterogeneous (bi-)(an)isotropic (non)magnetic materials,” figshare (2019). [Retrieved 17 Jan 2019], https://doi.org/10.6084/m9.figshare.7600100 .
  26. B. D. H. Tellegen, “The gyrator, a new electric network element,” Philips Res. Rep. 3(2), 81–101 (1948).
  27. Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60(14), 1351–1354 (1988).
    [Crossref] [PubMed]
  28. M. V. Berry and P. Shukla, “Typical weak and superweak values,” J. Phys. A 43, 354024 (2010).
    [Crossref]
  29. G. Waldman, Introduction to Light: The Physics of Light, Vision, and Color (Dover, 2002).
  30. J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7(4), 308–313 (1965).
    [Crossref]
  31. A. Priou, A. Sihvola, S. Tretyakov, and A. Vinogradov, Advances in Complex Electromagnetic Materials (Springer, 2012).
  32. D. R. Smith, “Calculation and measurement of bianisotropy in a split ring resonator metamaterial,” J. Appl. Phys. 100, 024507 (2006).
    [Crossref]
  33. A. Lakhtakia, Beltrami Fields in Chiral Media (World Scientific, 1994).
    [Crossref]
  34. L. D. Landau, L. P. Pitaevskii, and E. M. Lifshitz, Electrodynamics of Continuous Media (Butterworth-Heinemann, 2004).
  35. I. S. Gradsheyn and I. M. Ryzhik, Table of Integrals Series and Products (Academic Press, 2000).
  36. R. Merlin, “Metamaterials and the Landau–Lifshitz permeability argument: Large permittivity begets high-frequency magnetism,” Proc. Natl. Acad. Sci. U.S.A. 106(6), 1693–1698 (2009).
    [Crossref] [PubMed]
  37. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science. 312(5781), 1780–1782 (2006).
    [Crossref] [PubMed]
  38. B. Vial, Y. Liu, S. A. R. Horsley, T. G. Philbin, and Y. Hao, “A class of invisible inhomogeneous media and the control of electromagnetic waves,” Phys. Rev. B 94, 245119 (2016).
    [Crossref]
  39. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ∊ and μ,” Sov. Phys. Uspekhi 10(4), 509–514 (1967).
    [Crossref]
  40. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
    [Crossref] [PubMed]
  41. B. H. L. Novotny, Principles of Nano-Optics (Cambridge University, 2012).
    [Crossref]
  42. A. Dutt and V. Rokhlin, “Fast fourier transforms for nonequispaced data,” SIAM J. Sci. Comput. 14(6), 1368–1393 (1993).
    [Crossref]
  43. J. A. Fessler and B. P. Sutton, “Nonuniform fast fourier transforms using min-max interpolation,” IEEE Trans. Signal Process. 51(2), 560–574 (2003).
    [Crossref]
  44. T. E. Oliphant, A guide to NumPy (Trelgol Publishing, 2006).
  45. S. S. Dragomir, Inequalities for the Numerical Radius of Linear Operators in Hilbert Spaces (Springer, 2013).
    [Crossref]

2017 (2)

S. Rotter and S. Gigan, “Light fields in complex media: Mesoscopic scattering meets wave control,” Rev. Mod. Phys. 89, 015005 (2017).
[Crossref]

B. Krüger, T. Brenner, and A. Kienle, “Solution of the inhomogeneous Maxwell’s equations using a Born series,” Opt. Express 25(21), 25165–25182 (2017).
[Crossref] [PubMed]

2016 (4)

D. C. Adams, L. P. Hariri, A. J. Miller, Y. Wang, J. L. Cho, M. Villiger, J. A. Holz, M. V. Szabari, D. L. Hamilos, R. Scott Harris, J. W. Griffith, B. E. Bouma, A. D. Luster, B. D. Medoff, and M. J. Suter, “Birefringence microscopy platform for assessing airway smooth muscle structure and function in vivo,” Sci. Transl. Med. 8(359), 359ra131 (2016).
[Crossref] [PubMed]

G. Osnabrugge, S. Leedumrongwatthanakun, and I. M. Vellekoop, “A convergent Born series for solving the inhomogeneous Helmholtz equation in arbitrarily large media,” J. Comput. Phys. 322, 113–124 (2016).
[Crossref]

A. K. Glaser, Y. Chen, and J. T. Liu, “Fractal propagation method enables realistic optical microscopy simulations in biological tissues,” Optica 3(8), 861–869 (2016).
[Crossref] [PubMed]

B. Vial, Y. Liu, S. A. R. Horsley, T. G. Philbin, and Y. Hao, “A class of invisible inhomogeneous media and the control of electromagnetic waves,” Phys. Rev. B 94, 245119 (2016).
[Crossref]

2015 (1)

S. A. R. Horsley, M. Artoni, and G. C. L. Rocca, “Spatial kramers-kronig relations and the reflection of waves,” Nat. Photonics 9, 436 (2015).
[Crossref]

2013 (2)

A. Pierangelo, A. Nazac, A. Benali, P. Validire, H. Cohen, T. Novikova, B. H. Ibrahim, S. Manhas, C. Fallet, M.-R. Antonelli, and A.-D. Martino, “Polarimetric imaging of uterine cervix: a case study,” Opt. Express 21(12), 14120–14130 (2013).
[Crossref] [PubMed]

M. Koike-Tani, T. Tani, S. B. Mehta, A. Verma, and R. Oldenbourgh, “Polarized light microscopy in reproductive and developmental biology,” Mol. Reprod. Dev. 82(7–8), 548–562 (2013).
[Crossref] [PubMed]

2012 (1)

S. Alali, K. J. Aitken, A. Schröder, D. J. Bagli, and I. A. Vitkin, “Optical assessment of tissue anisotropy in ex vivo distended rat bladders,” J. Biomed. Opt. 17(8), 0860101 (2012).
[Crossref]

2010 (3)

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322 (2010).
[Crossref]

W.-C. Kuo, S.-C. Lin, and C.-Y. Chuang, “Birefringence measurement in polarization-sensitive optical coherence tomography using differential-envelope detection method,” Rev. Sci. Instrum. 81, 053705 (2010).
[Crossref] [PubMed]

M. V. Berry and P. Shukla, “Typical weak and superweak values,” J. Phys. A 43, 354024 (2010).
[Crossref]

2009 (1)

R. Merlin, “Metamaterials and the Landau–Lifshitz permeability argument: Large permittivity begets high-frequency magnetism,” Proc. Natl. Acad. Sci. U.S.A. 106(6), 1693–1698 (2009).
[Crossref] [PubMed]

2007 (1)

2006 (2)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science. 312(5781), 1780–1782 (2006).
[Crossref] [PubMed]

D. R. Smith, “Calculation and measurement of bianisotropy in a split ring resonator metamaterial,” J. Appl. Phys. 100, 024507 (2006).
[Crossref]

2003 (1)

J. A. Fessler and B. P. Sutton, “Nonuniform fast fourier transforms using min-max interpolation,” IEEE Trans. Signal Process. 51(2), 560–574 (2003).
[Crossref]

2001 (1)

Z. Salamon and G. Tollin, “Optical anisotropy in lipid bilayer membranes: coupled plasmon-waveguide resonance measurements of molecular orientation, polarizability, and shape,” Biophys. J. 80(3), 1557–1567 (2001).
[Crossref] [PubMed]

2000 (2)

1998 (1)

1997 (1)

1993 (1)

A. Dutt and V. Rokhlin, “Fast fourier transforms for nonequispaced data,” SIAM J. Sci. Comput. 14(6), 1368–1393 (1993).
[Crossref]

1988 (1)

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60(14), 1351–1354 (1988).
[Crossref] [PubMed]

1967 (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ∊ and μ,” Sov. Phys. Uspekhi 10(4), 509–514 (1967).
[Crossref]

1965 (1)

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7(4), 308–313 (1965).
[Crossref]

1948 (1)

B. D. H. Tellegen, “The gyrator, a new electric network element,” Philips Res. Rep. 3(2), 81–101 (1948).

1926 (1)

M. Born, “Quantenmechanik der stoßvorgänge,” Z. Phys. 38, 803 (1926).
[Crossref]

Adams, D. C.

D. C. Adams, L. P. Hariri, A. J. Miller, Y. Wang, J. L. Cho, M. Villiger, J. A. Holz, M. V. Szabari, D. L. Hamilos, R. Scott Harris, J. W. Griffith, B. E. Bouma, A. D. Luster, B. D. Medoff, and M. J. Suter, “Birefringence microscopy platform for assessing airway smooth muscle structure and function in vivo,” Sci. Transl. Med. 8(359), 359ra131 (2016).
[Crossref] [PubMed]

Aharonov, Y.

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60(14), 1351–1354 (1988).
[Crossref] [PubMed]

Aitken, K. J.

S. Alali, K. J. Aitken, A. Schröder, D. J. Bagli, and I. A. Vitkin, “Optical assessment of tissue anisotropy in ex vivo distended rat bladders,” J. Biomed. Opt. 17(8), 0860101 (2012).
[Crossref]

Akkermans, E.

E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University, 2007), chap. Coherent backscattering of light.
[Crossref]

Alali, S.

S. Alali, K. J. Aitken, A. Schröder, D. J. Bagli, and I. A. Vitkin, “Optical assessment of tissue anisotropy in ex vivo distended rat bladders,” J. Biomed. Opt. 17(8), 0860101 (2012).
[Crossref]

Albert, D. Z.

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60(14), 1351–1354 (1988).
[Crossref] [PubMed]

Antonelli, M.-R.

Artoni, M.

S. A. R. Horsley, M. Artoni, and G. C. L. Rocca, “Spatial kramers-kronig relations and the reflection of waves,” Nat. Photonics 9, 436 (2015).
[Crossref]

Bagli, D. J.

S. Alali, K. J. Aitken, A. Schröder, D. J. Bagli, and I. A. Vitkin, “Optical assessment of tissue anisotropy in ex vivo distended rat bladders,” J. Biomed. Opt. 17(8), 0860101 (2012).
[Crossref]

Benali, A.

Berry, M. V.

M. V. Berry and P. Shukla, “Typical weak and superweak values,” J. Phys. A 43, 354024 (2010).
[Crossref]

Born, M.

M. Born, “Quantenmechanik der stoßvorgänge,” Z. Phys. 38, 803 (1926).
[Crossref]

Bouma, B. E.

D. C. Adams, L. P. Hariri, A. J. Miller, Y. Wang, J. L. Cho, M. Villiger, J. A. Holz, M. V. Szabari, D. L. Hamilos, R. Scott Harris, J. W. Griffith, B. E. Bouma, A. D. Luster, B. D. Medoff, and M. J. Suter, “Birefringence microscopy platform for assessing airway smooth muscle structure and function in vivo,” Sci. Transl. Med. 8(359), 359ra131 (2016).
[Crossref] [PubMed]

Brenner, T.

Chen, Y.

Cho, J. L.

D. C. Adams, L. P. Hariri, A. J. Miller, Y. Wang, J. L. Cho, M. Villiger, J. A. Holz, M. V. Szabari, D. L. Hamilos, R. Scott Harris, J. W. Griffith, B. E. Bouma, A. D. Luster, B. D. Medoff, and M. J. Suter, “Birefringence microscopy platform for assessing airway smooth muscle structure and function in vivo,” Sci. Transl. Med. 8(359), 359ra131 (2016).
[Crossref] [PubMed]

Chuang, C.-Y.

W.-C. Kuo, S.-C. Lin, and C.-Y. Chuang, “Birefringence measurement in polarization-sensitive optical coherence tomography using differential-envelope detection method,” Rev. Sci. Instrum. 81, 053705 (2010).
[Crossref] [PubMed]

Cohen, H.

Colston, B.

De Boer, J. F.

Dragomir, S. S.

S. S. Dragomir, Inequalities for the Numerical Radius of Linear Operators in Hilbert Spaces (Springer, 2013).
[Crossref]

Dutt, A.

A. Dutt and V. Rokhlin, “Fast fourier transforms for nonequispaced data,” SIAM J. Sci. Comput. 14(6), 1368–1393 (1993).
[Crossref]

Ernst, O. G.

O. G. Ernst and M. J. Gander, “Why it is difficult to solve Helmholtz problems with classical iterative methods,” in Numerical Analysis of Multiscale Problems, I. G. Graham, T. Y. Hou, O. Lakkis, and R. Scheichl, eds. (Springer, 2012).

Everett, M.

Fallet, C.

Fessler, J. A.

J. A. Fessler and B. P. Sutton, “Nonuniform fast fourier transforms using min-max interpolation,” IEEE Trans. Signal Process. 51(2), 560–574 (2003).
[Crossref]

Gander, M. J.

O. G. Ernst and M. J. Gander, “Why it is difficult to solve Helmholtz problems with classical iterative methods,” in Numerical Analysis of Multiscale Problems, I. G. Graham, T. Y. Hou, O. Lakkis, and R. Scheichl, eds. (Springer, 2012).

Gigan, S.

S. Rotter and S. Gigan, “Light fields in complex media: Mesoscopic scattering meets wave control,” Rev. Mod. Phys. 89, 015005 (2017).
[Crossref]

Glaser, A. K.

Gradsheyn, I. S.

I. S. Gradsheyn and I. M. Ryzhik, Table of Integrals Series and Products (Academic Press, 2000).

Griffith, J. W.

D. C. Adams, L. P. Hariri, A. J. Miller, Y. Wang, J. L. Cho, M. Villiger, J. A. Holz, M. V. Szabari, D. L. Hamilos, R. Scott Harris, J. W. Griffith, B. E. Bouma, A. D. Luster, B. D. Medoff, and M. J. Suter, “Birefringence microscopy platform for assessing airway smooth muscle structure and function in vivo,” Sci. Transl. Med. 8(359), 359ra131 (2016).
[Crossref] [PubMed]

Hamilos, D. L.

D. C. Adams, L. P. Hariri, A. J. Miller, Y. Wang, J. L. Cho, M. Villiger, J. A. Holz, M. V. Szabari, D. L. Hamilos, R. Scott Harris, J. W. Griffith, B. E. Bouma, A. D. Luster, B. D. Medoff, and M. J. Suter, “Birefringence microscopy platform for assessing airway smooth muscle structure and function in vivo,” Sci. Transl. Med. 8(359), 359ra131 (2016).
[Crossref] [PubMed]

Hao, Y.

B. Vial, Y. Liu, S. A. R. Horsley, T. G. Philbin, and Y. Hao, “A class of invisible inhomogeneous media and the control of electromagnetic waves,” Phys. Rev. B 94, 245119 (2016).
[Crossref]

Hariri, L. P.

D. C. Adams, L. P. Hariri, A. J. Miller, Y. Wang, J. L. Cho, M. Villiger, J. A. Holz, M. V. Szabari, D. L. Hamilos, R. Scott Harris, J. W. Griffith, B. E. Bouma, A. D. Luster, B. D. Medoff, and M. J. Suter, “Birefringence microscopy platform for assessing airway smooth muscle structure and function in vivo,” Sci. Transl. Med. 8(359), 359ra131 (2016).
[Crossref] [PubMed]

Henry, F.

Holz, J. A.

D. C. Adams, L. P. Hariri, A. J. Miller, Y. Wang, J. L. Cho, M. Villiger, J. A. Holz, M. V. Szabari, D. L. Hamilos, R. Scott Harris, J. W. Griffith, B. E. Bouma, A. D. Luster, B. D. Medoff, and M. J. Suter, “Birefringence microscopy platform for assessing airway smooth muscle structure and function in vivo,” Sci. Transl. Med. 8(359), 359ra131 (2016).
[Crossref] [PubMed]

Horsley, S. A. R.

B. Vial, Y. Liu, S. A. R. Horsley, T. G. Philbin, and Y. Hao, “A class of invisible inhomogeneous media and the control of electromagnetic waves,” Phys. Rev. B 94, 245119 (2016).
[Crossref]

S. A. R. Horsley, M. Artoni, and G. C. L. Rocca, “Spatial kramers-kronig relations and the reflection of waves,” Nat. Photonics 9, 436 (2015).
[Crossref]

Ibrahim, B. H.

Jarry, G.

Joannopoulos, J. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light - Second Edition (Princeton University, 2008).

Johnson, S. G.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light - Second Edition (Princeton University, 2008).

Kaiser, R.

Kienle, A.

Koike-Tani, M.

M. Koike-Tani, T. Tani, S. B. Mehta, A. Verma, and R. Oldenbourgh, “Polarized light microscopy in reproductive and developmental biology,” Mol. Reprod. Dev. 82(7–8), 548–562 (2013).
[Crossref] [PubMed]

Koutromanos, I.

I. Koutromanos, Fundamentals of Finite Element Analysis: Linear Finite Element Analysis (John Wiley & Sons, 2018).

Krüger, B.

Kuo, W.-C.

W.-C. Kuo, S.-C. Lin, and C.-Y. Chuang, “Birefringence measurement in polarization-sensitive optical coherence tomography using differential-envelope detection method,” Rev. Sci. Instrum. 81, 053705 (2010).
[Crossref] [PubMed]

Lagendijk, A.

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322 (2010).
[Crossref]

Lakhtakia, A.

A. Lakhtakia, Beltrami Fields in Chiral Media (World Scientific, 1994).
[Crossref]

Landau, L. D.

L. D. Landau, L. P. Pitaevskii, and E. M. Lifshitz, Electrodynamics of Continuous Media (Butterworth-Heinemann, 2004).

Leedumrongwatthanakun, S.

G. Osnabrugge, S. Leedumrongwatthanakun, and I. M. Vellekoop, “A convergent Born series for solving the inhomogeneous Helmholtz equation in arbitrarily large media,” J. Comput. Phys. 322, 113–124 (2016).
[Crossref]

Lifshitz, E. M.

L. D. Landau, L. P. Pitaevskii, and E. M. Lifshitz, Electrodynamics of Continuous Media (Butterworth-Heinemann, 2004).

Lin, S.-C.

W.-C. Kuo, S.-C. Lin, and C.-Y. Chuang, “Birefringence measurement in polarization-sensitive optical coherence tomography using differential-envelope detection method,” Rev. Sci. Instrum. 81, 053705 (2010).
[Crossref] [PubMed]

Liu, J. T.

Liu, Y.

B. Vial, Y. Liu, S. A. R. Horsley, T. G. Philbin, and Y. Hao, “A class of invisible inhomogeneous media and the control of electromagnetic waves,” Phys. Rev. B 94, 245119 (2016).
[Crossref]

Luster, A. D.

D. C. Adams, L. P. Hariri, A. J. Miller, Y. Wang, J. L. Cho, M. Villiger, J. A. Holz, M. V. Szabari, D. L. Hamilos, R. Scott Harris, J. W. Griffith, B. E. Bouma, A. D. Luster, B. D. Medoff, and M. J. Suter, “Birefringence microscopy platform for assessing airway smooth muscle structure and function in vivo,” Sci. Transl. Med. 8(359), 359ra131 (2016).
[Crossref] [PubMed]

Manhas, S.

Martino, A.-D.

Mead, R.

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7(4), 308–313 (1965).
[Crossref]

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light - Second Edition (Princeton University, 2008).

Medoff, B. D.

D. C. Adams, L. P. Hariri, A. J. Miller, Y. Wang, J. L. Cho, M. Villiger, J. A. Holz, M. V. Szabari, D. L. Hamilos, R. Scott Harris, J. W. Griffith, B. E. Bouma, A. D. Luster, B. D. Medoff, and M. J. Suter, “Birefringence microscopy platform for assessing airway smooth muscle structure and function in vivo,” Sci. Transl. Med. 8(359), 359ra131 (2016).
[Crossref] [PubMed]

Mehta, S. B.

M. Koike-Tani, T. Tani, S. B. Mehta, A. Verma, and R. Oldenbourgh, “Polarized light microscopy in reproductive and developmental biology,” Mol. Reprod. Dev. 82(7–8), 548–562 (2013).
[Crossref] [PubMed]

Merlin, R.

R. Merlin, “Metamaterials and the Landau–Lifshitz permeability argument: Large permittivity begets high-frequency magnetism,” Proc. Natl. Acad. Sci. U.S.A. 106(6), 1693–1698 (2009).
[Crossref] [PubMed]

Miller, A. J.

D. C. Adams, L. P. Hariri, A. J. Miller, Y. Wang, J. L. Cho, M. Villiger, J. A. Holz, M. V. Szabari, D. L. Hamilos, R. Scott Harris, J. W. Griffith, B. E. Bouma, A. D. Luster, B. D. Medoff, and M. J. Suter, “Birefringence microscopy platform for assessing airway smooth muscle structure and function in vivo,” Sci. Transl. Med. 8(359), 359ra131 (2016).
[Crossref] [PubMed]

Milner, T. E.

Montambaux, G.

E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University, 2007), chap. Coherent backscattering of light.
[Crossref]

Mosk, A. P.

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322 (2010).
[Crossref]

I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32(16), 2309–2311 (2007).
[Crossref] [PubMed]

Nazac, A.

Nelder, J. A.

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7(4), 308–313 (1965).
[Crossref]

Nelson, J. S.

Newton, R. G.

R. G. Newton, Scattering Theory of Waves and Particles (Springer, 2013).

Novikova, T.

Novotny, B. H. L.

B. H. L. Novotny, Principles of Nano-Optics (Cambridge University, 2012).
[Crossref]

Oldenbourgh, R.

M. Koike-Tani, T. Tani, S. B. Mehta, A. Verma, and R. Oldenbourgh, “Polarized light microscopy in reproductive and developmental biology,” Mol. Reprod. Dev. 82(7–8), 548–562 (2013).
[Crossref] [PubMed]

Oliphant, T. E.

T. E. Oliphant, A guide to NumPy (Trelgol Publishing, 2006).

Osnabrugge, G.

G. Osnabrugge, S. Leedumrongwatthanakun, and I. M. Vellekoop, “A convergent Born series for solving the inhomogeneous Helmholtz equation in arbitrarily large media,” J. Comput. Phys. 322, 113–124 (2016).
[Crossref]

Pendry, J. B.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science. 312(5781), 1780–1782 (2006).
[Crossref] [PubMed]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[Crossref] [PubMed]

Philbin, T. G.

B. Vial, Y. Liu, S. A. R. Horsley, T. G. Philbin, and Y. Hao, “A class of invisible inhomogeneous media and the control of electromagnetic waves,” Phys. Rev. B 94, 245119 (2016).
[Crossref]

Pierangelo, A.

Pitaevskii, L. P.

L. D. Landau, L. P. Pitaevskii, and E. M. Lifshitz, Electrodynamics of Continuous Media (Butterworth-Heinemann, 2004).

Priou, A.

A. Priou, A. Sihvola, S. Tretyakov, and A. Vinogradov, Advances in Complex Electromagnetic Materials (Springer, 2012).

Rocca, G. C. L.

S. A. R. Horsley, M. Artoni, and G. C. L. Rocca, “Spatial kramers-kronig relations and the reflection of waves,” Nat. Photonics 9, 436 (2015).
[Crossref]

Rokhlin, V.

A. Dutt and V. Rokhlin, “Fast fourier transforms for nonequispaced data,” SIAM J. Sci. Comput. 14(6), 1368–1393 (1993).
[Crossref]

Rotter, S.

S. Rotter and S. Gigan, “Light fields in complex media: Mesoscopic scattering meets wave control,” Rev. Mod. Phys. 89, 015005 (2017).
[Crossref]

Ryzhik, I. M.

I. S. Gradsheyn and I. M. Ryzhik, Table of Integrals Series and Products (Academic Press, 2000).

Salamon, Z.

Z. Salamon and G. Tollin, “Optical anisotropy in lipid bilayer membranes: coupled plasmon-waveguide resonance measurements of molecular orientation, polarizability, and shape,” Biophys. J. 80(3), 1557–1567 (2001).
[Crossref] [PubMed]

Schoenenberger, K.

Schröder, A.

S. Alali, K. J. Aitken, A. Schröder, D. J. Bagli, and I. A. Vitkin, “Optical assessment of tissue anisotropy in ex vivo distended rat bladders,” J. Biomed. Opt. 17(8), 0860101 (2012).
[Crossref]

Schurig, D.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science. 312(5781), 1780–1782 (2006).
[Crossref] [PubMed]

Scott Harris, R.

D. C. Adams, L. P. Hariri, A. J. Miller, Y. Wang, J. L. Cho, M. Villiger, J. A. Holz, M. V. Szabari, D. L. Hamilos, R. Scott Harris, J. W. Griffith, B. E. Bouma, A. D. Luster, B. D. Medoff, and M. J. Suter, “Birefringence microscopy platform for assessing airway smooth muscle structure and function in vivo,” Sci. Transl. Med. 8(359), 359ra131 (2016).
[Crossref] [PubMed]

Shukla, P.

M. V. Berry and P. Shukla, “Typical weak and superweak values,” J. Phys. A 43, 354024 (2010).
[Crossref]

Sihvola, A.

A. Priou, A. Sihvola, S. Tretyakov, and A. Vinogradov, Advances in Complex Electromagnetic Materials (Springer, 2012).

Silva, L. Da

Smith, D. R.

D. R. Smith, “Calculation and measurement of bianisotropy in a split ring resonator metamaterial,” J. Appl. Phys. 100, 024507 (2006).
[Crossref]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science. 312(5781), 1780–1782 (2006).
[Crossref] [PubMed]

Suter, M. J.

D. C. Adams, L. P. Hariri, A. J. Miller, Y. Wang, J. L. Cho, M. Villiger, J. A. Holz, M. V. Szabari, D. L. Hamilos, R. Scott Harris, J. W. Griffith, B. E. Bouma, A. D. Luster, B. D. Medoff, and M. J. Suter, “Birefringence microscopy platform for assessing airway smooth muscle structure and function in vivo,” Sci. Transl. Med. 8(359), 359ra131 (2016).
[Crossref] [PubMed]

Sutton, B. P.

J. A. Fessler and B. P. Sutton, “Nonuniform fast fourier transforms using min-max interpolation,” IEEE Trans. Signal Process. 51(2), 560–574 (2003).
[Crossref]

Szabari, M. V.

D. C. Adams, L. P. Hariri, A. J. Miller, Y. Wang, J. L. Cho, M. Villiger, J. A. Holz, M. V. Szabari, D. L. Hamilos, R. Scott Harris, J. W. Griffith, B. E. Bouma, A. D. Luster, B. D. Medoff, and M. J. Suter, “Birefringence microscopy platform for assessing airway smooth muscle structure and function in vivo,” Sci. Transl. Med. 8(359), 359ra131 (2016).
[Crossref] [PubMed]

Tani, T.

M. Koike-Tani, T. Tani, S. B. Mehta, A. Verma, and R. Oldenbourgh, “Polarized light microscopy in reproductive and developmental biology,” Mol. Reprod. Dev. 82(7–8), 548–562 (2013).
[Crossref] [PubMed]

Tellegen, B. D. H.

B. D. H. Tellegen, “The gyrator, a new electric network element,” Philips Res. Rep. 3(2), 81–101 (1948).

Tollin, G.

Z. Salamon and G. Tollin, “Optical anisotropy in lipid bilayer membranes: coupled plasmon-waveguide resonance measurements of molecular orientation, polarizability, and shape,” Biophys. J. 80(3), 1557–1567 (2001).
[Crossref] [PubMed]

Tretyakov, S.

A. Priou, A. Sihvola, S. Tretyakov, and A. Vinogradov, Advances in Complex Electromagnetic Materials (Springer, 2012).

Vaidman, L.

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60(14), 1351–1354 (1988).
[Crossref] [PubMed]

Validire, P.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover Publications Inc., 2003).

van Gemert, M. J.

Vellekoop, I. M.

G. Osnabrugge, S. Leedumrongwatthanakun, and I. M. Vellekoop, “A convergent Born series for solving the inhomogeneous Helmholtz equation in arbitrarily large media,” J. Comput. Phys. 322, 113–124 (2016).
[Crossref]

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322 (2010).
[Crossref]

I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32(16), 2309–2311 (2007).
[Crossref] [PubMed]

Verma, A.

M. Koike-Tani, T. Tani, S. B. Mehta, A. Verma, and R. Oldenbourgh, “Polarized light microscopy in reproductive and developmental biology,” Mol. Reprod. Dev. 82(7–8), 548–562 (2013).
[Crossref] [PubMed]

Veselago, V. G.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ∊ and μ,” Sov. Phys. Uspekhi 10(4), 509–514 (1967).
[Crossref]

Vial, B.

B. Vial, Y. Liu, S. A. R. Horsley, T. G. Philbin, and Y. Hao, “A class of invisible inhomogeneous media and the control of electromagnetic waves,” Phys. Rev. B 94, 245119 (2016).
[Crossref]

Villiger, M.

D. C. Adams, L. P. Hariri, A. J. Miller, Y. Wang, J. L. Cho, M. Villiger, J. A. Holz, M. V. Szabari, D. L. Hamilos, R. Scott Harris, J. W. Griffith, B. E. Bouma, A. D. Luster, B. D. Medoff, and M. J. Suter, “Birefringence microscopy platform for assessing airway smooth muscle structure and function in vivo,” Sci. Transl. Med. 8(359), 359ra131 (2016).
[Crossref] [PubMed]

Vinogradov, A.

A. Priou, A. Sihvola, S. Tretyakov, and A. Vinogradov, Advances in Complex Electromagnetic Materials (Springer, 2012).

Vitkin, I. A.

S. Alali, K. J. Aitken, A. Schröder, D. J. Bagli, and I. A. Vitkin, “Optical assessment of tissue anisotropy in ex vivo distended rat bladders,” J. Biomed. Opt. 17(8), 0860101 (2012).
[Crossref]

Waldman, G.

G. Waldman, Introduction to Light: The Physics of Light, Vision, and Color (Dover, 2002).

Walker, J. S.

J. S. Walker, Fast Fourier Transforms (CRC Press, 2017).

Wang, Y.

D. C. Adams, L. P. Hariri, A. J. Miller, Y. Wang, J. L. Cho, M. Villiger, J. A. Holz, M. V. Szabari, D. L. Hamilos, R. Scott Harris, J. W. Griffith, B. E. Bouma, A. D. Luster, B. D. Medoff, and M. J. Suter, “Birefringence microscopy platform for assessing airway smooth muscle structure and function in vivo,” Sci. Transl. Med. 8(359), 359ra131 (2016).
[Crossref] [PubMed]

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light - Second Edition (Princeton University, 2008).

Biophys. J. (1)

Z. Salamon and G. Tollin, “Optical anisotropy in lipid bilayer membranes: coupled plasmon-waveguide resonance measurements of molecular orientation, polarizability, and shape,” Biophys. J. 80(3), 1557–1567 (2001).
[Crossref] [PubMed]

Comput. J. (1)

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7(4), 308–313 (1965).
[Crossref]

IEEE Trans. Signal Process. (1)

J. A. Fessler and B. P. Sutton, “Nonuniform fast fourier transforms using min-max interpolation,” IEEE Trans. Signal Process. 51(2), 560–574 (2003).
[Crossref]

J. Appl. Phys. (1)

D. R. Smith, “Calculation and measurement of bianisotropy in a split ring resonator metamaterial,” J. Appl. Phys. 100, 024507 (2006).
[Crossref]

J. Biomed. Opt. (1)

S. Alali, K. J. Aitken, A. Schröder, D. J. Bagli, and I. A. Vitkin, “Optical assessment of tissue anisotropy in ex vivo distended rat bladders,” J. Biomed. Opt. 17(8), 0860101 (2012).
[Crossref]

J. Comput. Phys. (1)

G. Osnabrugge, S. Leedumrongwatthanakun, and I. M. Vellekoop, “A convergent Born series for solving the inhomogeneous Helmholtz equation in arbitrarily large media,” J. Comput. Phys. 322, 113–124 (2016).
[Crossref]

J. Opt. Soc. Am. A (1)

J. Phys. A (1)

M. V. Berry and P. Shukla, “Typical weak and superweak values,” J. Phys. A 43, 354024 (2010).
[Crossref]

Mol. Reprod. Dev. (1)

M. Koike-Tani, T. Tani, S. B. Mehta, A. Verma, and R. Oldenbourgh, “Polarized light microscopy in reproductive and developmental biology,” Mol. Reprod. Dev. 82(7–8), 548–562 (2013).
[Crossref] [PubMed]

Nat. Photonics (2)

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322 (2010).
[Crossref]

S. A. R. Horsley, M. Artoni, and G. C. L. Rocca, “Spatial kramers-kronig relations and the reflection of waves,” Nat. Photonics 9, 436 (2015).
[Crossref]

Opt. Express (2)

Opt. Lett. (3)

Optica (1)

Philips Res. Rep. (1)

B. D. H. Tellegen, “The gyrator, a new electric network element,” Philips Res. Rep. 3(2), 81–101 (1948).

Phys. Rev. B (1)

B. Vial, Y. Liu, S. A. R. Horsley, T. G. Philbin, and Y. Hao, “A class of invisible inhomogeneous media and the control of electromagnetic waves,” Phys. Rev. B 94, 245119 (2016).
[Crossref]

Phys. Rev. Lett. (2)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[Crossref] [PubMed]

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60(14), 1351–1354 (1988).
[Crossref] [PubMed]

Proc. Natl. Acad. Sci. U.S.A. (1)

R. Merlin, “Metamaterials and the Landau–Lifshitz permeability argument: Large permittivity begets high-frequency magnetism,” Proc. Natl. Acad. Sci. U.S.A. 106(6), 1693–1698 (2009).
[Crossref] [PubMed]

Rev. Mod. Phys. (1)

S. Rotter and S. Gigan, “Light fields in complex media: Mesoscopic scattering meets wave control,” Rev. Mod. Phys. 89, 015005 (2017).
[Crossref]

Rev. Sci. Instrum. (1)

W.-C. Kuo, S.-C. Lin, and C.-Y. Chuang, “Birefringence measurement in polarization-sensitive optical coherence tomography using differential-envelope detection method,” Rev. Sci. Instrum. 81, 053705 (2010).
[Crossref] [PubMed]

Sci. Transl. Med. (1)

D. C. Adams, L. P. Hariri, A. J. Miller, Y. Wang, J. L. Cho, M. Villiger, J. A. Holz, M. V. Szabari, D. L. Hamilos, R. Scott Harris, J. W. Griffith, B. E. Bouma, A. D. Luster, B. D. Medoff, and M. J. Suter, “Birefringence microscopy platform for assessing airway smooth muscle structure and function in vivo,” Sci. Transl. Med. 8(359), 359ra131 (2016).
[Crossref] [PubMed]

Science. (1)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science. 312(5781), 1780–1782 (2006).
[Crossref] [PubMed]

SIAM J. Sci. Comput. (1)

A. Dutt and V. Rokhlin, “Fast fourier transforms for nonequispaced data,” SIAM J. Sci. Comput. 14(6), 1368–1393 (1993).
[Crossref]

Sov. Phys. Uspekhi (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ∊ and μ,” Sov. Phys. Uspekhi 10(4), 509–514 (1967).
[Crossref]

Z. Phys. (1)

M. Born, “Quantenmechanik der stoßvorgänge,” Z. Phys. 38, 803 (1926).
[Crossref]

Other (16)

E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University, 2007), chap. Coherent backscattering of light.
[Crossref]

R. G. Newton, Scattering Theory of Waves and Particles (Springer, 2013).

J. S. Walker, Fast Fourier Transforms (CRC Press, 2017).

T. Vettenburg, “Macromax: A python 3 library for solving macroscopic maxwell’s equations for electromagnetic waves in gain-free heterogeneous (bi-)(an)isotropic (non)magnetic materials,” figshare (2019). [Retrieved 17 Jan 2019], https://doi.org/10.6084/m9.figshare.7600100 .

G. Waldman, Introduction to Light: The Physics of Light, Vision, and Color (Dover, 2002).

A. Priou, A. Sihvola, S. Tretyakov, and A. Vinogradov, Advances in Complex Electromagnetic Materials (Springer, 2012).

A. Lakhtakia, Beltrami Fields in Chiral Media (World Scientific, 1994).
[Crossref]

L. D. Landau, L. P. Pitaevskii, and E. M. Lifshitz, Electrodynamics of Continuous Media (Butterworth-Heinemann, 2004).

I. S. Gradsheyn and I. M. Ryzhik, Table of Integrals Series and Products (Academic Press, 2000).

I. Koutromanos, Fundamentals of Finite Element Analysis: Linear Finite Element Analysis (John Wiley & Sons, 2018).

O. G. Ernst and M. J. Gander, “Why it is difficult to solve Helmholtz problems with classical iterative methods,” in Numerical Analysis of Multiscale Problems, I. G. Graham, T. Y. Hou, O. Lakkis, and R. Scheichl, eds. (Springer, 2012).

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light - Second Edition (Princeton University, 2008).

H. C. van de Hulst, Light Scattering by Small Particles (Dover Publications Inc., 2003).

B. H. L. Novotny, Principles of Nano-Optics (Cambridge University, 2012).
[Crossref]

T. E. Oliphant, A guide to NumPy (Trelgol Publishing, 2006).

S. S. Dragomir, Inequalities for the Numerical Radius of Linear Operators in Hilbert Spaces (Springer, 2013).
[Crossref]

Supplementary Material (1)

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» Code 1       Source Code

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Figures (5)

Fig. 1
Fig. 1 (a) Demonstration of anisotropic permittivity. Diagonally polarized light propagates from left to right through a calcite crystal (light gray box) cut at 45° with respect to its optic axis (indicated by the arrow). It can be seen that, as expected, the in-plane polarized extraordinary ray (e, magenta) is displaced from the ray that is polarized perpendicular to the plane (o, cyan). Some interference can be noticed between the incoming wave and its back-reflection at both the entrance and exit surface of the crystal. (b–e) A circularly polarized Gaussian beam incident from the left on a birefringent vaterite (CaCO3) microrod with a diameter of 20 μm forms a complex scattering pattern instead of a single focus. Although the volume is homogeneous CaCO3, complex, seemingly random, scattering occurs due to its subdivision in crystals of approximately 1 μm in cross-section for which the fast axis is oriented randomly with angles θ shown as hue in panel (e). Panels (b–d) show the field components Ex, Ey, and Ez, respectively. The darkness and hue indicate the field amplitude and phase, respectively, as indicated by the legend in panel (e). An overlaid gray grid outlines the crystal areas for reference, and the inset shows a 4× magnified detail of the field at the exit surface.
Fig. 2
Fig. 2 Demonstration of non-Hermitian anisotropy. An Ex-polarized wave traverses two (a,c,e), or three (b,d,f), polarizers from the left to the right. The first and the last polarizer are in cross-diagonal-orientation, preventing transmission through the first system shown in (a). The intensity (black line) and the real part of the electric field (green line) are shown for the vertical and horizontal components in (c,e) and (d,f), respectively. Arbitrary units are used so that the maximum value of the intensity and field match for clarity of display.
Fig. 3
Fig. 3 Demonstration of impedance matching (a–f), and propagation in a chiral medium (g,h). A plane wave in free space ( = μ = 1) with a wavelength λ0 = 500 nm enters at x = 10 μm from the left into dielectric slabs of thickness 10 μm, with μ = 1 (a,c) and μ = 1.5 (b,d). In both cases the permittivity is 1.5 (green line, panels a and b). The interference between the incoming and reflected wave is clearly visible as oscillations in intensity (|E|2, black line, c). It can be seen that a fraction of the wave is reflected from the slab without impedance matching (μ(z) ∝̸(z) in panel a). In contrast, a constant intensity is seen in panel (d), indicative of the absence of back-reflection for the impedance matched slab (μ in panel b). To facilitate comparison, both the intensity and field are normalized to their respective maximum value in panels (c) and (d). Panels (e,f) show the (truncated) electric field amplitude for a dipole with absorbing (e) and impedance matched (f) boundary layers. The interference with the back reflected wave, visible as beating in panel (e), is suppressed by the impedance matched layers as seen in panel (f). (g,h) Linear polarization rotates upon propagation in a chiral medium with a high chirality that is 100 times of that of saturated glucose (n = 1.45, specific rotation [ α ] 500 nm T 52.7 ° mL g 1 dm 1, at 909 g/L). The constitutive relations can thus be seen to be = 1.45, μ = 1, and ξ = ζ = 52.7 909 λ 0 360 i = 66.53 × 10 6 i. The intensity transfer between the Ex-polarization (solid green) and Ey-polarization (dashed red) can be seen to occur several times over a propagation distance of 10 mm (g). The local angle, θ, of the linear polarization is shown in panel (h). Note the significantly larger length scale for panels (g) and (h).
Fig. 4
Fig. 4 Light-wave propagation through a material with negative refractive index ( = −1.5, μ = −1, gray). The field amplitude (brightness) and phase (hue) are shown for a Gaussian beam that enters the surface at 30° to the normal. The beam can be seen to refract at −19.5°, backwards, into the metamaterial at an angle opposite to that for normal glass.
Fig. 5
Fig. 5 Numerical check that condition αi > max {A1, A2} ensures that the numerical radius of M, given by Eq. (21), is less than unity. To produce this figure we used the condition on the numerical radius, |〈n|M|n〉| < 〈n|n〉, with 40 × 40 matrices. We generated 1 × 106 random matrices Δ (positive definite Δi) and unitary matrices U, along with the corresponding values of αi calculated as indicated in each panel (using Python’s numpy library for random matrix generation [44]). For each matrix we calculated values for |〈n|M|n〉| for a set of 40 random but orthogonal complex vectors, n. The largest magnitude of these values is one of the 106 points plotted in each panel. In (b) (largest magnitude 0.999997) we used the condition (58), which we know analytically to be sufficient to move the eigenvalues of M within the unit circle. In (a) (largest magnitude 0.999999) we show that αi > ‖Δ‖ appears to guarantee also that the eigenvalues are within the unit circle

Tables (3)

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Algorithm 1 Calculation of the electric field in materials with anisotropic permittivity, .

Tables Icon

Algorithm 2 A function that implements the general algorithm for arbitrary materials.

Tables Icon

Table 1 Storage requirements for the algorithm.

Equations (59)

Equations on this page are rendered with MathJax. Learn more.

D ( x ) = 0 ( x ) E ( x ) , and B ( x ) = μ 0 H ( x )
× × E ( x ) k 0 2 ( x ) E ( x ) = i ω μ 0 j ( x ) = S ( x ) ,
𝒪 1 = ( 𝒪 h + 𝒪 i ) 1 = ( 𝟙 3 + 𝒪 h 1 𝒪 i ) 1 𝒪 h 1 = [ 𝟙 3 ( 𝒪 h 1 𝒪 i ) + ( 𝒪 h 1 𝒪 i ) 2 + ] 𝒪 h 1 = [ p = 0 ( 𝒪 h 1 𝒪 i ) p ] 𝒪 h 1 ,
× × G ( x , x ) k 0 2 α G ( x , x ) = 𝟙 3 δ ( 3 ) ( x x ) .
E = 𝒪 1 S = [ p = 0 ( k 0 2 G χ ) p ] G S .
G = 1 ( Π T k 2 α k 0 2 Π L k 0 2 α )
E = [ p = 0 M p ] Γ G S , where M k 0 2 Γ G χ Γ + 𝟙 3 ,
D ( x ) = 0 ( x ) E ( x ) ,
× × E k 0 2 α E ( x ) k 0 2 χ ( x ) E ( x ) = S ( x ) .
M i k 0 2 α i χ G χ i α i χ + 𝟙 3 .
max n | n | i k 0 2 α i χ G χ i α i χ + 𝟙 3 | n | < 1
max n [ | n | Δ 2 α i 2 𝟙 3 | n | + | n | χ U χ | n | ] < 2 α i 2
| n | Δ 2 α i 2 𝟙 3 | n | + n | 1 2 ( Δ Δ + Δ Δ ) | n 2 α i n | Δ i | n + α i 2 < 2 α i 2 n : n = 1 ,
| n | 1 2 ( Δ Δ + Δ Δ ) | n α i 2 | + | n | Δ Δ i + Δ i Δ | n | + n | 1 2 ( Δ Δ + Δ Δ ) | n 2 α i n | Δ i | n < α i 2 n : n = 1 .
A 1 = max n 1 2 n | Δ Δ + Δ Δ | n ; A 2 = max n | n | Δ Δ i + Δ i Δ | n | 2 n | Δ i | n .
D ( x ) = 0 ( x ) E ( x ) + 1 c ξ ( x ) H ( x )
B ( x ) = μ 0 μ ( x ) H ( x ) + 1 c ζ ( x ) E ( x ) .
S i ω μ 0 β 1 j
χ β 1 β ξ μ 1 ζ 𝟙 3 α i β ξ μ 1 𝒟 + i β 𝒟 μ 1 ζ + 𝒟 ( 𝟙 3 μ 1 β ) 𝒟 ,
E = [ p = 0 M p ] Γ G S , where
M k 0 2 Γ G χ Γ + 𝟙 3 , and
Γ i α i χ .
× × E ( x ) k 0 2 ( x ) E ( x ) = i ω μ 0 j ( x )
( × × α k 0 2 ) E k 0 2 ( α ) E = S
( 𝒪 h + 𝒪 i ) E = S
𝒪 h G = × × G ( x , x ) k 0 2 α G ( x , x ) = 𝟙 3 δ ( 3 ) ( x x )
G ( x , x ) = d 3 k ( 2 π ) 3 G ˜ ( k ) e i k ( x x )
G = 1 G ˜ ,
k × k × G ˜ ( k ) + k 0 2 α G ˜ ( k ) = 𝟙 3 .
Π L = k k k 2 and Π T 𝟙 3 Π L ,
k 0 2 α g L ( k ) = 1
( k 2 α k 0 2 ) g T ( k ) = 1 ,
G ( x , x ) = d 3 k ( 2 π ) 3 [ Π T k 2 α k 0 2 Π L α k 0 2 ] e i k ( x x )
G = 1 ( Π T k 2 α k 0 2 Π L α k 0 2 )
1 z = 1 2 i Im [ z ] ( 1 z * z ) = 1 2 i Im [ z ] ( 1 e 2 i z ) ,
G = 1 2 i α i k 0 2 ( 𝟙 3 U )
U = 1 ( k 2 α * k 0 2 k 2 α k 0 2 Π T + α * α Π L ) ,
U U = 1 ( k 2 α * k 0 2 k 2 α k 0 2 Π T + α * α Π L ) 1 ( k 2 α k 0 2 k 2 α * k 0 2 Π T + α α * Π L ) = 𝟙 3 ,
Δ E ˜ = i α i χ ˜ * [ G ˜ ( χ ˜ * E ˜ + S ˜ ) E ˜ ] .
E N = Γ G * k 0 2 χ E N 1 Γ E N 1 + E N 1 + Γ G * S
Δ E N = E N E N 1 = Γ [ G * ( k 0 2 χ E N 1 + S ) E N 1 ] .
E N = Γ [ G * ( k 0 2 χ E N 1 + S ) E N 1 ] + E N 1
= M E N 1 + Γ G * S
= M N E 0 + [ p = 0 N 1 M p ] Γ G * S
[ p = 0 M p ] Γ G * S M N E 0 [ p = 0 N 1 M p ] Γ G * S = M N ( E E 0 ) .
M Γ [ 1 2 ( 𝟙 3 U ) i α i χ 𝟙 3 ] + 𝟙 3 .
M Γ [ 1 2 ( 𝟙 3 U ) Γ 𝟙 3 ] + 𝟙 3 = 1 2 ( 𝟙 3 + P 2 ) 1 2 Γ U Γ .
| n Mn | 1 2 | 1 + n P 2 n | + 1 2 Γ n Γ n ,
| n Mn | i = 1 N | c i | 2 | e i M e i | n : n i = 1 N c i e i n = 1 .
| e i Me i | | 1 + λ 2 | 2 + | 1 λ * | | 1 λ | 2 < 1 .
| 1 + λ r 2 λ i 2 | 2 + | 2 λ r λ i | 2 2 + 1 + λ r 2 + λ i 2 2 λ r 2 < 1 .
| 1 + λ r 2 λ i 2 | 2 + | 2 λ r λ i | 2 < ( 1 λ r 2 λ i 2 + 2 λ r ) 2 .
| 1 λ i 2 | λ r 2 < ( 1 λ i 2 ) ( 2 λ r λ r 2 ) + 2 λ r 2 ( 1 λ i 2 ) 2 λ r 3 .
0 < ( 1 λ i 2 ) λ r λ r 3 .
| n | Δ Δ i + Δ i Δ | n | 2 n | Δ i | n = | 2 λ n 0 | Δ | n 0 + η ( n | Δ i Δ | n 0 + n 0 | Δ Δ i | n ) + η λ ( n | Δ | n 0 + n 0 | Δ | n ) + η 2 n | Δ Δ i + Δ i Δ | n | | 2 ( λ + η 2 n | Δ i | n )
A 2 = max n | n | a 1 Δ a + a Δ a 1 | n | 2 n | n
m = 1 2 ( a 1 Δ a + a Δ a 1 ) .
A 2 = 1 2 [ a 1 Δ a + a Δ a 1 ]
A 1 1 2 Δ Δ + 1 2 Δ Δ = | λ max | A 2