Abstract

The forward and backward scattering off linear systems with discrete rotational symmetries Rz(2π/n) with n ≥ 3 are shown to be restricted by symmetry reasons. Along the symmetry axis, forward scattering can only be helicity preserving and backward scattering can only be helicity flipping. These restrictions do not exist for n < 3. If, in addition to the n ≥ 3 discrete rotational symmetry, the system has duality symmetry (obeys the helicity conservation law), it will exhibit zero backscattering. The results pinpoint the underlying symmetry reasons for some notable scattering properties of Rz(2π/4) symmetric systems that have been reported in the metamaterials and radar literature. Applications to planar metamaterials and solar cells are briefly discussed.

© 2013 Optical Society of America

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2013 (7)

I. Fernandez-Corbaton and G. Molina-Terriza, “Role of duality symmetry in transformation optics,” Phys. Rev. B88, 085111 (2013).
[CrossRef]

X. Zambrana-Puyalto, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Dual and anti-dual modes in dielectric spheres,” Opt. Express21, 17520–17530 (2013).
[CrossRef] [PubMed]

I. Fernandez-Corbaton, X. Zambrana-Puyalto, N. Tischler, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Electromagnetic duality symmetry and helicity conservation for the macroscopic maxwells equations,” Phys. Rev. Lett.111, 060401 (2013).
[CrossRef]

X. Zambrana-Puyalto, I. Fernandez-Corbaton, M. L. Juan, X. Vidal, and G. Molina-Terriza, “Duality symmetry and kerker conditions,” Opt. Lett.38, 1857–1859 (2013).
[CrossRef] [PubMed]

I. Fernandez-Corbaton, X. Vidal, N. Tischler, and G. Molina-Terriza, “Necessary symmetry conditions for the rotation of light,” J. Chem. Phys.138, 214311 (2013).
[CrossRef] [PubMed]

B. Hopkins, W. Liu, A. E. Miroshnichenko, and Y. S. Kivshar, “Optically isotropic responses induced by discrete rotational symmetry of nanoparticle clusters,” Nanoscale5, 6395–6403 (2013).
[CrossRef] [PubMed]

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. berg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, and M. T. Borgstrm, “InP nanowire array solar cells achieving 13.8% efficiency by exceeding the ray optics limit,” Science339, 1057–1060 (2013).
[CrossRef] [PubMed]

2012 (3)

I. Fernandez-Corbaton, X. Zambrana-Puyalto, and G. Molina-Terriza, “Helicity and angular momentum: A symmetry-based framework for the study of light-matter interactions,” Phys. Rev. A86, 042103 (2012).
[CrossRef]

J. Kaschke, J. K. Gansel, and M. Wegener, “On metamaterial circular polarizers based on metal n-helices,” Opt. Express20, 26012–26020 (2012).
[CrossRef] [PubMed]

A. O. Karilainen and S. A. Tretyakov, “Isotropic chiral objects with zero backscattering,” IEEE Trans. Antennas Propagat.60, 4449–4452 (2012).
[CrossRef]

2010 (2)

A. Alu and N. Engheta, “How does zero forward-scattering in magnetodielectric nanoparticles comply with the optical theorem?” J. Nanophotonics4, 041590 (2010).
[CrossRef]

M. Decker, R. Zhao, C. M. Soukoulis, S. Linden, and M. Wegener, “Twisted split-ring-resonator photonic metamaterial with huge optical activity,” Opt. Lett.35, 1593–1595 (2010).
[CrossRef] [PubMed]

2009 (1)

I. Lindell, A. Sihvola, P. Yla-Oijala, and H. Wallen, “Zero backscattering from self-dual objects of finite size,” IEEE Trans. Antennas Propag.57, 2725–2731 (2009).
[CrossRef]

1994 (1)

I. Bialynicki-Birula, “On the wave function of the photon,” Acta Phys. Pol.86, 97–116 (1994).

1987 (1)

1983 (1)

1968 (1)

D. Zwanziger, “Quantum field theory of particles with both electric and magnetic charges,” Phys. Rev.176, 1489–1495 (1968).
[CrossRef]

1965 (1)

M. G. Calkin, “An invariance property of the free electromagnetic field,” Am. J. Phys.33, 958 (1965).
[CrossRef]

Alu, A.

A. Alu and N. Engheta, “How does zero forward-scattering in magnetodielectric nanoparticles comply with the optical theorem?” J. Nanophotonics4, 041590 (2010).
[CrossRef]

Anttu, N.

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. berg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, and M. T. Borgstrm, “InP nanowire array solar cells achieving 13.8% efficiency by exceeding the ray optics limit,” Science339, 1057–1060 (2013).
[CrossRef] [PubMed]

Asoli, D.

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. berg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, and M. T. Borgstrm, “InP nanowire array solar cells achieving 13.8% efficiency by exceeding the ray optics limit,” Science339, 1057–1060 (2013).
[CrossRef] [PubMed]

berg, I.

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. berg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, and M. T. Borgstrm, “InP nanowire array solar cells achieving 13.8% efficiency by exceeding the ray optics limit,” Science339, 1057–1060 (2013).
[CrossRef] [PubMed]

Bialynicki-Birula, I.

I. Bialynicki-Birula, “On the wave function of the photon,” Acta Phys. Pol.86, 97–116 (1994).

I. Bialynicki-Birula, “Photon wave function,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1996), Vol. 36, pp. 245–294.
[CrossRef]

Borgstrm, M. T.

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. berg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, and M. T. Borgstrm, “InP nanowire array solar cells achieving 13.8% efficiency by exceeding the ray optics limit,” Science339, 1057–1060 (2013).
[CrossRef] [PubMed]

Calkin, M. G.

M. G. Calkin, “An invariance property of the free electromagnetic field,” Am. J. Phys.33, 958 (1965).
[CrossRef]

Decker, M.

Deppert, K.

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. berg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, and M. T. Borgstrm, “InP nanowire array solar cells achieving 13.8% efficiency by exceeding the ray optics limit,” Science339, 1057–1060 (2013).
[CrossRef] [PubMed]

Dimroth, F.

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. berg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, and M. T. Borgstrm, “InP nanowire array solar cells achieving 13.8% efficiency by exceeding the ray optics limit,” Science339, 1057–1060 (2013).
[CrossRef] [PubMed]

Engheta, N.

A. Alu and N. Engheta, “How does zero forward-scattering in magnetodielectric nanoparticles comply with the optical theorem?” J. Nanophotonics4, 041590 (2010).
[CrossRef]

Fernandez-Corbaton, I.

X. Zambrana-Puyalto, I. Fernandez-Corbaton, M. L. Juan, X. Vidal, and G. Molina-Terriza, “Duality symmetry and kerker conditions,” Opt. Lett.38, 1857–1859 (2013).
[CrossRef] [PubMed]

I. Fernandez-Corbaton, X. Vidal, N. Tischler, and G. Molina-Terriza, “Necessary symmetry conditions for the rotation of light,” J. Chem. Phys.138, 214311 (2013).
[CrossRef] [PubMed]

I. Fernandez-Corbaton and G. Molina-Terriza, “Role of duality symmetry in transformation optics,” Phys. Rev. B88, 085111 (2013).
[CrossRef]

I. Fernandez-Corbaton, X. Zambrana-Puyalto, N. Tischler, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Electromagnetic duality symmetry and helicity conservation for the macroscopic maxwells equations,” Phys. Rev. Lett.111, 060401 (2013).
[CrossRef]

I. Fernandez-Corbaton, X. Zambrana-Puyalto, and G. Molina-Terriza, “Helicity and angular momentum: A symmetry-based framework for the study of light-matter interactions,” Phys. Rev. A86, 042103 (2012).
[CrossRef]

Fuss-Kailuweit, P.

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. berg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, and M. T. Borgstrm, “InP nanowire array solar cells achieving 13.8% efficiency by exceeding the ray optics limit,” Science339, 1057–1060 (2013).
[CrossRef] [PubMed]

Gansel, J. K.

Giles, C. L.

Herb, P.

Hopkins, B.

B. Hopkins, W. Liu, A. E. Miroshnichenko, and Y. S. Kivshar, “Optically isotropic responses induced by discrete rotational symmetry of nanoparticle clusters,” Nanoscale5, 6395–6403 (2013).
[CrossRef] [PubMed]

Hu, C.-R.

Huffman, M.

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. berg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, and M. T. Borgstrm, “InP nanowire array solar cells achieving 13.8% efficiency by exceeding the ray optics limit,” Science339, 1057–1060 (2013).
[CrossRef] [PubMed]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1998).

Juan, M. L.

Karilainen, A. O.

A. O. Karilainen and S. A. Tretyakov, “Isotropic chiral objects with zero backscattering,” IEEE Trans. Antennas Propagat.60, 4449–4452 (2012).
[CrossRef]

Kaschke, J.

Kattawar, G. W.

Kerker, M.

Kivshar, Y. S.

B. Hopkins, W. Liu, A. E. Miroshnichenko, and Y. S. Kivshar, “Optically isotropic responses induced by discrete rotational symmetry of nanoparticle clusters,” Nanoscale5, 6395–6403 (2013).
[CrossRef] [PubMed]

Lindell, I.

I. Lindell, A. Sihvola, P. Yla-Oijala, and H. Wallen, “Zero backscattering from self-dual objects of finite size,” IEEE Trans. Antennas Propag.57, 2725–2731 (2009).
[CrossRef]

Linden, S.

Liu, W.

B. Hopkins, W. Liu, A. E. Miroshnichenko, and Y. S. Kivshar, “Optically isotropic responses induced by discrete rotational symmetry of nanoparticle clusters,” Nanoscale5, 6395–6403 (2013).
[CrossRef] [PubMed]

Magnusson, M. H.

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. berg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, and M. T. Borgstrm, “InP nanowire array solar cells achieving 13.8% efficiency by exceeding the ray optics limit,” Science339, 1057–1060 (2013).
[CrossRef] [PubMed]

Miroshnichenko, A. E.

B. Hopkins, W. Liu, A. E. Miroshnichenko, and Y. S. Kivshar, “Optically isotropic responses induced by discrete rotational symmetry of nanoparticle clusters,” Nanoscale5, 6395–6403 (2013).
[CrossRef] [PubMed]

Molina-Terriza, G.

X. Zambrana-Puyalto, I. Fernandez-Corbaton, M. L. Juan, X. Vidal, and G. Molina-Terriza, “Duality symmetry and kerker conditions,” Opt. Lett.38, 1857–1859 (2013).
[CrossRef] [PubMed]

I. Fernandez-Corbaton, X. Vidal, N. Tischler, and G. Molina-Terriza, “Necessary symmetry conditions for the rotation of light,” J. Chem. Phys.138, 214311 (2013).
[CrossRef] [PubMed]

X. Zambrana-Puyalto, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Dual and anti-dual modes in dielectric spheres,” Opt. Express21, 17520–17530 (2013).
[CrossRef] [PubMed]

I. Fernandez-Corbaton and G. Molina-Terriza, “Role of duality symmetry in transformation optics,” Phys. Rev. B88, 085111 (2013).
[CrossRef]

I. Fernandez-Corbaton, X. Zambrana-Puyalto, N. Tischler, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Electromagnetic duality symmetry and helicity conservation for the macroscopic maxwells equations,” Phys. Rev. Lett.111, 060401 (2013).
[CrossRef]

I. Fernandez-Corbaton, X. Zambrana-Puyalto, and G. Molina-Terriza, “Helicity and angular momentum: A symmetry-based framework for the study of light-matter interactions,” Phys. Rev. A86, 042103 (2012).
[CrossRef]

Parkin, M. E.

Samuelson, L.

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. berg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, and M. T. Borgstrm, “InP nanowire array solar cells achieving 13.8% efficiency by exceeding the ray optics limit,” Science339, 1057–1060 (2013).
[CrossRef] [PubMed]

Siefer, G.

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. berg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, and M. T. Borgstrm, “InP nanowire array solar cells achieving 13.8% efficiency by exceeding the ray optics limit,” Science339, 1057–1060 (2013).
[CrossRef] [PubMed]

Sihvola, A.

I. Lindell, A. Sihvola, P. Yla-Oijala, and H. Wallen, “Zero backscattering from self-dual objects of finite size,” IEEE Trans. Antennas Propag.57, 2725–2731 (2009).
[CrossRef]

Soukoulis, C. M.

Tischler, N.

I. Fernandez-Corbaton, X. Zambrana-Puyalto, N. Tischler, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Electromagnetic duality symmetry and helicity conservation for the macroscopic maxwells equations,” Phys. Rev. Lett.111, 060401 (2013).
[CrossRef]

I. Fernandez-Corbaton, X. Vidal, N. Tischler, and G. Molina-Terriza, “Necessary symmetry conditions for the rotation of light,” J. Chem. Phys.138, 214311 (2013).
[CrossRef] [PubMed]

Tretyakov, S. A.

A. O. Karilainen and S. A. Tretyakov, “Isotropic chiral objects with zero backscattering,” IEEE Trans. Antennas Propagat.60, 4449–4452 (2012).
[CrossRef]

Vidal, X.

I. Fernandez-Corbaton, X. Zambrana-Puyalto, N. Tischler, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Electromagnetic duality symmetry and helicity conservation for the macroscopic maxwells equations,” Phys. Rev. Lett.111, 060401 (2013).
[CrossRef]

I. Fernandez-Corbaton, X. Vidal, N. Tischler, and G. Molina-Terriza, “Necessary symmetry conditions for the rotation of light,” J. Chem. Phys.138, 214311 (2013).
[CrossRef] [PubMed]

X. Zambrana-Puyalto, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Dual and anti-dual modes in dielectric spheres,” Opt. Express21, 17520–17530 (2013).
[CrossRef] [PubMed]

X. Zambrana-Puyalto, I. Fernandez-Corbaton, M. L. Juan, X. Vidal, and G. Molina-Terriza, “Duality symmetry and kerker conditions,” Opt. Lett.38, 1857–1859 (2013).
[CrossRef] [PubMed]

Wallen, H.

I. Lindell, A. Sihvola, P. Yla-Oijala, and H. Wallen, “Zero backscattering from self-dual objects of finite size,” IEEE Trans. Antennas Propag.57, 2725–2731 (2009).
[CrossRef]

Wallentin, J.

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. berg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, and M. T. Borgstrm, “InP nanowire array solar cells achieving 13.8% efficiency by exceeding the ray optics limit,” Science339, 1057–1060 (2013).
[CrossRef] [PubMed]

Wang, D. S.

Wegener, M.

Witzigmann, B.

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. berg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, and M. T. Borgstrm, “InP nanowire array solar cells achieving 13.8% efficiency by exceeding the ray optics limit,” Science339, 1057–1060 (2013).
[CrossRef] [PubMed]

Xu, H. Q.

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. berg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, and M. T. Borgstrm, “InP nanowire array solar cells achieving 13.8% efficiency by exceeding the ray optics limit,” Science339, 1057–1060 (2013).
[CrossRef] [PubMed]

Yla-Oijala, P.

I. Lindell, A. Sihvola, P. Yla-Oijala, and H. Wallen, “Zero backscattering from self-dual objects of finite size,” IEEE Trans. Antennas Propag.57, 2725–2731 (2009).
[CrossRef]

Zambrana-Puyalto, X.

X. Zambrana-Puyalto, I. Fernandez-Corbaton, M. L. Juan, X. Vidal, and G. Molina-Terriza, “Duality symmetry and kerker conditions,” Opt. Lett.38, 1857–1859 (2013).
[CrossRef] [PubMed]

X. Zambrana-Puyalto, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Dual and anti-dual modes in dielectric spheres,” Opt. Express21, 17520–17530 (2013).
[CrossRef] [PubMed]

I. Fernandez-Corbaton, X. Zambrana-Puyalto, N. Tischler, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Electromagnetic duality symmetry and helicity conservation for the macroscopic maxwells equations,” Phys. Rev. Lett.111, 060401 (2013).
[CrossRef]

I. Fernandez-Corbaton, X. Zambrana-Puyalto, and G. Molina-Terriza, “Helicity and angular momentum: A symmetry-based framework for the study of light-matter interactions,” Phys. Rev. A86, 042103 (2012).
[CrossRef]

Zhao, R.

Zwanziger, D.

D. Zwanziger, “Quantum field theory of particles with both electric and magnetic charges,” Phys. Rev.176, 1489–1495 (1968).
[CrossRef]

Acta Phys. Pol. (1)

I. Bialynicki-Birula, “On the wave function of the photon,” Acta Phys. Pol.86, 97–116 (1994).

Am. J. Phys. (1)

M. G. Calkin, “An invariance property of the free electromagnetic field,” Am. J. Phys.33, 958 (1965).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Antennas Propag. (1)

I. Lindell, A. Sihvola, P. Yla-Oijala, and H. Wallen, “Zero backscattering from self-dual objects of finite size,” IEEE Trans. Antennas Propag.57, 2725–2731 (2009).
[CrossRef]

IEEE Trans. Antennas Propagat. (1)

A. O. Karilainen and S. A. Tretyakov, “Isotropic chiral objects with zero backscattering,” IEEE Trans. Antennas Propagat.60, 4449–4452 (2012).
[CrossRef]

J. Chem. Phys. (1)

I. Fernandez-Corbaton, X. Vidal, N. Tischler, and G. Molina-Terriza, “Necessary symmetry conditions for the rotation of light,” J. Chem. Phys.138, 214311 (2013).
[CrossRef] [PubMed]

J. Nanophotonics (1)

A. Alu and N. Engheta, “How does zero forward-scattering in magnetodielectric nanoparticles comply with the optical theorem?” J. Nanophotonics4, 041590 (2010).
[CrossRef]

J. Opt. Soc. Am. (1)

Nanoscale (1)

B. Hopkins, W. Liu, A. E. Miroshnichenko, and Y. S. Kivshar, “Optically isotropic responses induced by discrete rotational symmetry of nanoparticle clusters,” Nanoscale5, 6395–6403 (2013).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. (1)

D. Zwanziger, “Quantum field theory of particles with both electric and magnetic charges,” Phys. Rev.176, 1489–1495 (1968).
[CrossRef]

Phys. Rev. A (1)

I. Fernandez-Corbaton, X. Zambrana-Puyalto, and G. Molina-Terriza, “Helicity and angular momentum: A symmetry-based framework for the study of light-matter interactions,” Phys. Rev. A86, 042103 (2012).
[CrossRef]

Phys. Rev. B (1)

I. Fernandez-Corbaton and G. Molina-Terriza, “Role of duality symmetry in transformation optics,” Phys. Rev. B88, 085111 (2013).
[CrossRef]

Phys. Rev. Lett. (1)

I. Fernandez-Corbaton, X. Zambrana-Puyalto, N. Tischler, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Electromagnetic duality symmetry and helicity conservation for the macroscopic maxwells equations,” Phys. Rev. Lett.111, 060401 (2013).
[CrossRef]

Science (1)

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. berg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, and M. T. Borgstrm, “InP nanowire array solar cells achieving 13.8% efficiency by exceeding the ray optics limit,” Science339, 1057–1060 (2013).
[CrossRef] [PubMed]

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I. Bialynicki-Birula, “Photon wave function,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1996), Vol. 36, pp. 245–294.
[CrossRef]

J. D. Jackson, Classical Electrodynamics (Wiley, 1998).

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Figures (3)

Fig. 1
Fig. 1

A field composed by the superposition of five plane waves has a well defined helicity equal to one if all the plane waves have left handed polarization (left panel), equal to minus one if they all have right handed polarization (central panel) and does not have a well defined helicity, i.e. it is not an eigenstate of the helicity operator, if all the plane waves do not have the same handedness (right panel).

Fig. 2
Fig. 2

(a) The helicity of an electromagnetic field is not preserved after interaction with a non-dual symmetric object. An incoming field with well defined helicity, in this case a single plane wave of definite polarization handedness (blue), produces a scattered field that contains components of the opposite helicity (red). The helicity of the scattered field in panel (a) is not well defined because it contains plane waves of different helicities. (b) Helicity preservation after interaction with a dual symmetric object. The helicity of the scattered field is well defined an equal to the helicity of the input field.

Fig. 3
Fig. 3

Forward and backward scattering produced by input plane waves of well defined helicity impinging on structures with discrete rotational symmetries Rz(2π/n): Gray rectangular prism (n = 2), green equilateral triangular prism (n = 3) and green square prism (n = 4). Plane waves of positive helicity (left handed polarization) are blue. Plane waves of negative helicity (right handed polarization) are red. The input plane waves, labeled as “in”, have positive helicity and momentum aligned with the positive z axis. The text shows that for n ≥ 3, the forward scattering can only contain components with the same helicity as the input and the backward scattering can only contain components with helicity opposite to the input one. In the figure, this is reflected in the restricted helicity components drawn in forward and backward scattering for the square (n = 4) and triangular (n = 3) prisms. No such restrictions apply to the rectangular prism (n = 2): Any helicity is allowed both in forward and in backward scattering. If, besides the discrete rotational symmetry, the scatterers had also electromagnetic duality symmetry, all the red right handed plane waves will disappear from the picture because duality enforces helicity preservation in all scattering directions. Therefore, dual objects with discrete rotational symmetries with n ≥ 3 will exhibit zero backscattering.

Tables (1)

Tables Icon

Table 1 Expressions for monochromatic plane waves of well defined helicity (λ = ±1) in the real space representation of electromagnetic fields. The momentum of the plane waves is either parallel or anti-parallel to the z axis ±p. The real space polarization vectors in the expressions are l ^ = ( x ^ + i y ^ ) / 2 and r ^ = ( x ^ i y ^ ) / 2. To have the same handedness with respect to its momentum vector, the real space polarization vector must change when the momentum changes sign.

Equations (12)

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E E θ = E cos θ H sin θ , H H θ = E sin θ + H cos θ .
τ f λ λ ¯ = λ ¯ p z ^ | S | p z ^ λ , τ b λ λ ¯ = λ ¯ p z ^ | S | p z ^ λ .
Λ | ± p z ^ λ = λ | ± p z ^ λ .
Λ | ± p z ^ λ = J P | P | | ± p z ^ λ = i = 1 3 J i P i i = 1 3 P i 2 | ± p z ^ λ = ± J z | ± p z ^ λ ,
J z | ± p z ^ λ = ± λ | ± p z ^ λ .
R z ( α ) | ± p z ^ λ = exp ( i α λ ) | ± p z ^ λ .
λ ± p z ^ | R z ( α ) 1 = ( R z ( α ) | ± p z ^ λ ) = ( R z ( α ) | ± p z ^ λ ) = exp ( ± i α λ ) λ ± p z ^ | ,
R z 1 ( 2 π / n ) S R z ( 2 π / n ) = S .
τ f λ λ ¯ = λ ¯ p z ^ | S | p z ^ λ = 8 λ ¯ p z ^ | R z 1 ( 2 π / n ) S R z ( 2 π / n ) | p z ^ λ = 6 , 7 exp ( i ( λ λ ¯ ) 2 π n ) λ ¯ p z ^ | S | p z ^ λ = exp ( i ( λ λ ¯ ) 2 π n ) τ f λ λ ¯ .
4 π n = 2 π k 2 n = k ,
τ b λ λ ¯ = λ ¯ p z ^ | S | p z ^ λ = 8 λ ¯ p z ^ | R z 1 ( 2 π / n ) S R z ( 2 π / n ) | p z ^ λ = 6 , 7 exp ( i ( λ + λ ¯ ) 2 π n ) λ ¯ p z ^ | S | p z ^ λ = exp ( i ( λ + λ ¯ ) 2 π n ) τ b λ λ ¯ .
τ f λ λ = λ p z ^ | S | p z ^ λ = λ p z ^ | M S M | p z ^ λ = λ p z ^ | S | p z ^ λ = τ f λ λ ,

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