Scattering from anisotropic geometries of arbitrary shape is relatively difficult to interpret physically, involving the intricate interplay between material and geometric effects. Insights into complex scattering mechanisms are often enabled by modal methods that decompose the response into the well-understood multipolar resonances. Here, we extend the generalized normal mode expansion to lossy and anisotropic scatterers. Unique to the method is that it decomposes the total response of any anisotropic resonator into the modes of the corresponding isotropic resonator. This disentangles the material and geometric contributions to the scattering of any anisotropic resonator. Furthermore, the method can identify absorption and scattering resonances with separate sets of modes. We illustrate our method by considering an infinitely long cylinder with concentric metallic/dielectric layers, targeting the complex case of an effective hyperbolic response. We show that by scanning the material composition of the hyperbolic medium, we can achieve any desired scattering effect, including backscattering cancellation.
REFERENCES
1.
J.
Yao
, X.
Yang
, X.
Yin
, G.
Bartal
, and X.
Zhang
, “Three-dimensional nanometer-scale optical cavities of indefinite medium
,” Proc. Natl. Acad. Sci. U.S.A.
108
, 11327
–11331
(2011
). 2.
X.
Yang
, J.
Yao
, J.
Rho
, X.
Yin
, and X.
Zhang
, “Experimental realization of three-dimensional indefinite cavities at the nanoscale with anomalous scaling laws
,” Nat. Photonics
6
, 450
–454
(2012
). 3.
C.
Wu
, A.
Salandrino
, X.
Ni
, and X.
Zhang
, “Electrodynamical light trapping using whispering-gallery resonances in hyperbolic cavities
,” Phys. Rev. X
4
, 021015
(2014
). 4.
K.
Feng
, G.
Harden
, D. L.
Sivco
, and A. J.
Hoffman
, “Subdiffraction confinement in all-semiconductor hyperbolic metamaterial resonators
,” ACS Photonics
4
, 1621
–1626
(2017
). 5.
M. R.
Shcherbakov
, S.
Liu
, V. V.
Zubyuk
, A.
Vaskin
, P. P.
Vabishchevich
, G.
Keeler
, T.
Pertsch
, T. V.
Dolgova
, I.
Staude
, I.
Brener
, and A. A.
Fedyanin
, “Ultrafast all-optical tuning of direct-gap semiconductor metasurfaces
,” Nat. Commun.
8
, 17
(2017
). 6.
N.
Nookala
, J.
Lee
, M.
Tymchenko
, J.
Sebastian Gomez-Diaz
, F.
Demmerle
, G.
Boehm
, K.
Lai
, G.
Shvets
, M.-C.
Amann
, A.
Alu
, and M.
Belkin
, “Ultrathin gradient nonlinear metasurface with a giant nonlinear response
,” Optica
3
, 283
(2016
). 7.
D.
Bergman
, “The dielectric constant of a simple-cubic array of identical spheres
,” J. Phys. C
12
, 4947
(1979
). 8.
T. C.
Choy
, Effective Medium Theory: Principles and Applications
(Oxford University Press
, 2015
).9.
S. N.
Samaddar
, “Scattering of plane waves from an infinitely long cylinder of anisotropic materials at oblique incidence with an application to an electronic scanning antenna
,” Appl. Sci. Res. Sect. B
10
, 385
–411
(1962
). 10.
J.
Monzon
and N.
Damaskos
, “Two-dimensional scattering by a homogeneous anisotropic rod
,” IEEE Trans. Antennas Propag.
34
, 1243
–1249
(1986
). 11.
J.
Monzon
, “Three-dimensional scattering by an infinite homogeneous anisotropic circular cylinder: A spectral approach
,” IEEE Trans. Antennas Propag.
35
, 670
–682
(1987
). 12.
R.-B.
Wu
and C. H.
Chen
, “Variational reaction formulation of scattering problem for anisotropic dielectric cylinders
,” IEEE Trans. Antennas Propag.
34
, 640
–645
(1986
). 13.
J. C.
Monzon
, “On a surface integral representation for homogeneous anisotropic regions: Two-dimensional case
,” IEEE Trans. Antennas Propag.
36
, 1401
–1406
(1988
). 14.
B.
Beker
and K.
Umashankar
, “Analysis of electromagnetic scattering by arbitrarily shaped two-dimensional anisotropic objects: Combined field surface integral equation formulation
,” Electromagnetics
9
, 215
–229
(1989
). 15.
N.
Uzunoglu
, P.
Cottis
, and J.
Fikioris
, “Excitation of electromagnetic waves in a gyroelectric cylinder
,” IEEE Trans. Antennas Propag.
33
, 90
–99
(1985
). 16.
V. V.
Varadan
, A.
Lakhtakia
, and V. K.
Varadan
, “Scattering by three-dimensional anisotropic scatterers
,” IEEE Trans. Antennas Propag.
37
, 800
–802
(1989
). 17.
R. D.
Graglia
and P.
Uslenghi
, “Electromagnetic scattering from anisotropic materials, part I: General theory
,” IEEE Trans. Antennas Propag.
32
, 867
–869
(1984
). 18.
R. D.
Graglia
, P. L. E.
Uslenghi
, and R. S.
Zich
, “Moment method with isoparametric elements for three-dimensional anisotropic scatterers
,” Proc. IEEE
77
, 750
–760
(1989
). 19.
C. M.
Rappaport
and E. B.
Smith
, “Anisotropic FDFD computed on conformal meshes
,” IEEE Trans. Magn.
27
, 3848
–3851
(1991
). 20.
W.
Sun
and C. A.
Balanis
, “Edge-based FEM solution of scattering from inhomogeneous and anisotropic objects
,” IEEE Trans. Antennas Propag.
42
, 627
–632
(1994
). 21.
X. B.
Wu
and K.
Yasumoto
, “Three-dimensional scattering by an infinite homogeneous anisotropic circular cylinder: An analytical solution
,” J. Appl. Phys.
82
, 1996
–2003
(1997
). 22.
W.
Ren
, “Contributions to the electromagnetic wave theory of bounded homogeneous anisotropic media
,” Phys. Rev. E
47
, 664
–673
(1993
). 23.
E. A.
Muljarov
, W.
Langbein
, and R.
Zimmermann
, “Brillouin–Wigner perturbation theory in open electromagnetic systems
,” Europhys. Lett.
92
, 50010
(2010
). 24.
P. T.
Leung
, S. Y.
Liu
, S. S.
Tong
, and K.
Young
, “Time-independent perturbation theory for quasinormal modes in leaky optical cavities
,” Phys. Rev. A
49
, 3068
–3073
(1994
). 25.
P.
Lalanne
, W.
Yan
, K.
Vynck
, C.
Sauvan
, and J.-P.
Hugonin
, “Light interaction with photonic and plasmonic resonances
,” Laser Photon. Rev.
12
, 1700113
(2018
). 26.
R.
Sammut
and A. W.
Snyder
, “Leaky modes on a dielectric waveguide: Orthogonality and excitation
,” Appl. Opt.
15
, 1040
–1044
(1976
). 27.
E. A.
Muljarov
and W.
Langbein
, “Resonant-state expansion of dispersive open optical systems: Creating gold from sand
,” Phys. Rev. B
93
, 075417
(2016
). 28.
C.
Sauvan
, J. P.
Hugonin
, I. S.
Maksymov
, and P.
Lalanne
, “Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators
,” Phys. Rev. Lett.
110
, 237401
(2013
). 29.
E. A.
Muljarov
and T.
Weiss
, “Resonant-state expansion for open optical systems: Generalization to magnetic, chiral, and bi-anisotropic materials
,” Opt. Lett.
43
, 1978
–1981
(2018
). 30.
R.
Colom
, R.
McPhedran
, B.
Stout
, and N.
Bonod
, “Modal expansion of the scattered field: Causality, nondivergence, and nonresonant contribution
,” Phys. Rev. B
98
, 085418
(2018
). 31.
M. I.
Abdelrahman
and B.
Gralak
, “Completeness and divergence-free behavior of the quasi-normal modes using causality principle
,” OSA Continuum
1
, 340
(2018
). 32.
P. Y.
Chen
, D. J.
Bergman
, and Y.
Sivan
, “Generalizing normal mode expansion of electromagnetic Green’s tensor to open systems
,” Phys. Rev. Appl.
11
, 044018
(2019
). 33.
D. J.
Bergman
and D.
Stroud
, “Theory of resonances in the electromagnetic scattering by macroscopic bodies
,” Phys. Rev. B
22
, 3527
–3539
(1980
). 34.
C.
Forestiere
, G.
Gravina
, G.
Miano
, M.
Pascale
, and R.
Tricarico
, “Electromagnetic modes and resonances of two-dimensional bodies
,” Phys. Rev. B
99
, 155423
(2019
). 35.
M. S.
Agranovich
, B. Z.
Katsenelenbaum
, A. N.
Sivov
, and N. N.
Voitovich
, Generalized Method of Eigenoscillations in Diffraction Theory
(Wiley-VCH
, 1999
).36.
A.
Farhi
and D. J.
Bergman
, “Analysis of a Veselago lens in the quasistatic regime
,” Phys. Rev. A
90
, 013806
(2014
). 37.
D. J.
Bergman
and K.-J.
Dunn
, “Bulk effective dielectric constant of a composite with a periodic microgeometry
,” Phys. Rev. B
45
, 13262
–13271
(1992
). 38.
M. I.
Stockman
, “Spaser action, loss compensation, and stability in plasmonic systems with gain
,” Phys. Rev. Lett.
106
, 156802
(2011
). 39.
O.
Schnitzer
, V.
Giannini
, S. A.
Maier
, and R. V.
Craster
, “Surface plasmon resonances of arbitrarily shaped nanometallic structures in the small-screening length limit
,” Proc. R. Soc. A
472
, 20160258
(2016
). 40.
K. N.
Reddy
, P. Y.
Chen
, A. I.
Fernández-Domínguez
, and Y.
Sivan
, “Revisiting the boundary conditions for second-harmonic generation at metal-dielectric interfaces
,” J. Opt. Soc. Am. B
34
, 1824
–1832
(2017
). 41.
C.
Forestiere
, G.
Miano
, G.
Rubinacci
, A.
Tamburrino
, R.
Tricarico
, and S.
Ventre
, “Volume integral formulation for the calculation of material independent modes of dielectric scatterers
,” IEEE Trans. Antennas Propag.
66
, 2505
–2514
(2018
). 42.
P. Y.
Chen
, Y.
Sivan
, and E. A.
Muljarov
, “An efficient solver for the generalized normal modes of non-uniform open optical resonators
,” J. Comput. Phys.
422
, 109754
(2020
). 43.
P. Y.
Chen
and Y.
Sivan
, “Robust location of optical fiber modes via the argument principle method
,” Comput. Phys. Commun.
214
, 105
–116
(2017
). 44.
P. Y.
Chen
, D. J.
Bergman
, and Y.
Sivan
, “Spectral decomposition of the Lippmann–Schwinger equation applied to cylinders,” arXiv:1705.01747[physics] (2017).45.
G.
Rosolen
, B.
Maes
, P. Y.
Chen
, and Y.
Sivan
, “Overcoming the bottleneck for quantum computations of complex nanophotonic structures: Purcell and Förster resonant energy transfer calculations using a rigorous mode-hybridization method
,” Phys. Rev. B
101
, 155401
(2020
). 46.
C.
Forestiere
and G.
Miano
, “Material-independent modes for electromagnetic scattering
,” Phys. Rev. B
94
, 201406
(2016
). 47.
Y.
Kantor
and D. J.
Bergman
, “Elastostatic resonances—A new approach to the calculation of the effective elastic constants of composites
,” J. Mech. Phys. Solids
30
, 355
–376
(1982
). 48.
M.
Pascale
, G.
Miano
, R.
Tricarico
, and C.
Forestiere
, “Full-wave electromagnetic modes and hybridization in nanoparticle dimers
,” Sci. Rep.
9
, 1
–21
(2019
). 49.
H.
Kettunen
, H.
Wallén
, and A.
Sihvola
, “Tailoring effective media by Mie resonances of radially-anisotropic cylinders
,” Photonics
2
, 509
–526
(2015
). 50.
H. L.
Chen
and L.
Gao
, “Anomalous electromagnetic scattering from radially anisotropic nanowires
,” Phys. Rev. A
86
, 033825
(2012
). 51.
P. Y.
Chen
and Y.
Sivan
, “Resolving the Gibbs phenomenon via a discontinuous basis in a mode solver for open optical systems
,” J. Comput. Phys.
TBA
, TBA
(2020
). 52.
A.
Poddubny
, I.
Iorsh
, P.
Belov
, and Y. S.
Kivshar
, “Hyperbolic metamaterials
,” Nat. Photonics
7
, 948
–957
(2013
). 53.
P.
Huo
, S.
Zhang
, Y.
Liang
, Y.
Lu
, and T.
Xu
, “Hyperbolic metamaterials and metasurfaces: Fundamentals and applications
,” Adv. Opt. Mater.
7
, 1801616
(2019
). 54.
A. D.
Rakić
, A. B.
Djurišić
, J. M.
Elazar
, and M. L.
Majewski
, “Optical properties of metallic films for vertical-cavity optoelectronic devices
,” Appl. Opt.
37
, 5271
–5283
(1998
). 55.
T.
Siefke
, S.
Kroker
, K.
Pfeiffer
, O.
Puffky
, K.
Dietrich
, D.
Franta
, I.
Ohlídal
, A.
Szeghalmi
, E.-B.
Kley
, and A.
Tünnermann
, “Materials pushing the application limits of wire grid polarizers further into the deep ultraviolet spectral range
,” Adv. Opt. Mater.
4
, 1780
–1786
(2016
). 56.
P.
Shekhar
, J.
Atkinson
, and Z.
Jacob
, “Hyperbolic metamaterials: Fundamentals and applications
,” Nano Convergence
1
, 14
(2014
). 57.
Special attention is given in the eigensolver routine of Eq. (5). Depending on the case that is being treated, pre-arrangement of the matrix of overlap integrals ease the calculation. In our case, since the permittivity tensor (9) is diagonal, there is no mode mixing between modes of different azimuthal orders (see Appendix C). Thus, we can re-arrange the matrix of overlap integrals to be block diagonal and solve Eq. (5) for each azimuthal order independently.
58.
In other words, higher order longitudinal modes continue to contribute significantly to the expansion. This is because the permittivity is singular at the center of the cylinder, so the electric field tends to infinity, which is difficult to represent via a linear combination of modes that are everywhere finite. Thus, we only achieve up to accuracy at this point, even when including longitudinal modes up to th radial order.
59.
R.
Kumar
and K.
Kajikawa
, “Superscattering from cylindrical hyperbolic metamaterials in the visible region
,” Opt. Express
28
, 1507
–1517
(2020
). 60.
C.
Qian
, X.
Lin
, Y.
Yang
, F.
Gao
, Y.
Shen
, J.
Lopez
, I.
Kaminer
, B.
Zhang
, E.
Li
, M.
Soljačić
, and H.
Chen
, “Multifrequency superscattering from subwavelength hyperbolic structures
,” ACS Photonics
5
, 1506
–1511
(2018
). 61.
N.
Maccaferri
, Y.
Zhao
, T.
Isoniemi
, M.
Iarossi
, A.
Parracino
, G.
Strangi
, and F.
De Angelis
, “Hyperbolic meta-antennas enable full control of scattering and absorption of light
,” Nano Lett.
19
, 1851
–1859
(2019
). 62.
M.
Kerker
, D.-S.
Wang
, and C. L.
Giles
, “Electromagnetic scattering by magnetic spheres
,” J. Opt. Soc. Am.
73
, 765
(1983
). 63.
W.
Liu
and Y. S.
Kivshar
, “Generalized Kerker effects in nanophotonics and meta-optics (invited)
,” Opt. Express
26
, 13085
(2018
). 64.
M.
Born
and E.
Wolf
, Principle of Optics
, 7th ed. (Cambridge
, 1999
).65.
P.-H.
Tsao
, “Derivation and implications of the symmetry property of the permittivity tensor
,” Am. J. Phys.
61
, 823
–825
(1993
). 66.
J. A.
Kong
, “Theorems of bianisotropic media
,” Proc. IEEE
60
, 1036
–1046
(1972
). 67.
A.
Villeneuve
and R.
Harrington
, “Reciprocity relationships for gyrotropic media
,” IRE Trans. Microwave Theor. Tech.
6
, 308
–310
(1958
). 68.
P. S.
Pershan
, “Magneto-optical effects
,” J. Appl. Phys.
38
, 1482
–1490
(1967
). © 2021 Author(s).
2021
Author(s)
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