Abstract

With the conservation of power, a phase diagram defined by amplitude square and phase of scattering coefficients for each spherical harmonic channel is introduced as a universal map for any passive electromagnetic scatterers. Physically allowable solutions for scattering coefficients in this diagram clearly show power competitions among scattering and absorption. It also illustrates a variety of exotic scattering or absorption phenomena, from resonant scattering, invisible cloaking, to coherent perfect absorber. With electrically small core-shell scatterers as an example, we demonstrate a systematic method to design field-controllable structures based on the allowed trajectories in this diagram. The proposed phase diagram and inverse design can provide tools to design functional electromagnetic devices.

© 2016 Optical Society of America

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References

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  5. A. Alú and N. Engheta, “Polarizabilities and effective parameters for collections of spherical nanoparticles formed by pairs of concentric double-negative, single-negative, and/or double-positive metamaterial layers,” J. Appl. Phys. 97, 094310 (2005).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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2015 (2)

V. Grigoriev, N. Bonod, J. Wenger, and B. Stout, “Optimizing nanoparticle designs for ideal absorption of light,” ACS Photonics 2, 263–270 (2015).
[Crossref]

J.-Y. Lee, M.-C. Tsai, P.-C. Chen, T.-T. Chen, K.-L. Chan, C.-Y. Lee, and R.-K. Lee, “Thickness effects on light absorption and scattering for nano-particles in shape of hollow-spheres,” J. Phys. Chem. C 119, 25754–25760 (2015).
[Crossref]

2014 (5)

A. Mirzaei, A. E. Miroshnichenko, I. V. Shadrivov, and Y. S. Kivshar, “Superscattering of light optimized by a genetic algorithm,” Appl. Phys. Lett. 105, 011109 (2014).
[Crossref]

N. M. Estakhri and A. Alú, “Minimum-scattering superabsorbers,” Phys. Rev. B. 89, 121416 (2014).
[Crossref]

S.-W. Chu, T.-Y. Su, R. Oketani, Y.-T. Huang, H.-Y. Wu, Y. Yonemaru, M. Yamanaka, H. Lee, G.-Y. Zhuo, M.-Y. Lee, S. Kawata, and K. Fujita, “Measurement of a saturated emission of optical radiation from gold nanoparticles: Application to an ultrahigh resolution microscope,” Phys. Rev. Lett. 112, 017402 (2014).
[Crossref] [PubMed]

R. Fleury, J. Soric, and A. Alú, “Physical bounds on absorption and scattering for cloaked sensors,” Phys. Rev. B. 89, 045122 (2014).
[Crossref]

J. Shi, F. Monticone, S. Elias, Y. Wu, D. Ratchford, X. Li, and A. Alú, “Modular assembly of optical nanocircuits,” Nat. Commun. 5, 3896 (2014).
[Crossref] [PubMed]

2013 (5)

F. Monticone, C. Argyropoulos, and A. Alú, “Multi-layered plasmonic covers for comblike scattering response and optical tagging,” Phys. Rev. Lett. 110, 113901 (2013).
[Crossref]

H. Noh, S.M. Popoff, and H. Cao, “Broadband subwavelength focusing of light using a passive sink,” Opt. Express 21, 17435–17446 (2013).
[Crossref] [PubMed]

S. Muhlig, A. Cunningham, J. Dintinger, M. Farhat, S. B. Hasan, T. Scharf, T. Burgi, F. Lederer, and C. Rockstuhl, “A self-assembled three-dimensional cloak in the visible,” Sci. Rep. 3, 2328 (2013).
[Crossref] [PubMed]

X. Zhang, Y. L. Chen, R. S. Liu, and D. P. Tsai, “Plasmonic photocatalysis,” Rep. Prog. Phys. 76, 046401 (2013).
[Crossref] [PubMed]

A. Mirzaei, I. V. Shadrivov, A. E. Miroshnichenko, and Y. S. Kivshar, “Cloaking and enhanced scattering of core-shell plasmonic nanowires,” Opt. Express 21, 10454–10459 (2013).
[Crossref] [PubMed]

2012 (2)

M. Farhat, S. Mhlig, C. Rockstuhl, and F. Lederer, “Scattering cancellation of the magnetic dipole field from macroscopic spheres,” Opt. Express 20, 13896–13906 (2012).
[Crossref] [PubMed]

H. Noh, Y. Chong, A.D. Stone, and H. Cao, “Perfect coupling of light to surface plasmons by coherent absorption,” Phys. Rev. Lett. 108, 186805 (2012).
[Crossref] [PubMed]

2011 (4)

M. I. Tribelsky, “Anomalous light absorption by small particles,” Europhys. Lett. 94, 14004 (2011).
[Crossref]

Z. Ruan and S. Fan, “Design of subwavelength superscattering nanospheres,” Appl. Phys. Lett. 98, 043101 (2011).
[Crossref]

S. Muhlig, M. Farhat, C. Rockstuhl, and F. Lederer, “Cloaking dielectric spherical objects by a shell of metallic nanoparticles,” Phys. Rev. B 83, 195116 (2011).
[Crossref]

M. I. Tribelsky, A. E. Miroshnichenko, Y. S. Kivshar, B. S. Lukyanchuk, and A. R. Khokhlov, “Laser pulse heating of spherical metal particles,” Phys. Rev. X 1, 021024 (2011).

2010 (3)

Y. Pu, R. Grange, C. L. Hsieh, and D. Psaltis, “Nonlinear optical properties of core-shell nanocavities for enhanced second-harmonic generation,” Phys. Rev. Lett. 104, 207402 (2010).
[Crossref] [PubMed]

Z. Ruan and S. Fan, “Temporal coupled-mode theory for Fano resonance in light scattering by a single obstacle,” J. Phys. Chem. C 114, 7324 (2010).
[Crossref]

Z. Ruan and S. Fan, “Superscattering of light from subwavelength nanostructures,” Phys. Rev. Lett. 105, 013901 (2010)
[Crossref] [PubMed]

2009 (1)

A. Alú and N. Engheta, “Cloaking a sensor,” Phys. Rev. Lett. 102, 233901 (2009).
[Crossref] [PubMed]

2008 (1)

2007 (3)

A. Alú and N. Engheta, “Plasmonic materials in transparency and cloaking problems: mechanism, robustness, and physical insights,” Opt. Express 15, 3318–3332 (2007).
[Crossref] [PubMed]

R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljacic, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A 75, 053801 (2007).
[Crossref]

N. Engheta, “Circuits with light at nanoscales: optical nanocircuits inspired by metamaterials,” Science 317, 1698–1702 (2007).
[Crossref] [PubMed]

2006 (1)

M. I. Tribelsky and B. S. Lukyanchuk, “Anomalous light scattering by small particles,” Phys. Rev. Lett. 97, 263902 (2006).
[Crossref]

2005 (2)

A. Alú and N. Engheta, “Polarizabilities and effective parameters for collections of spherical nanoparticles formed by pairs of concentric double-negative, single-negative, and/or double-positive metamaterial layers,” J. Appl. Phys. 97, 094310 (2005).
[Crossref]

A. Alú and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 72, 016623 (2005).
[Crossref]

2003 (1)

L. R. Hirsch, R. J. Stafford, J. A. Bankson, S. R. Sershen, B. Rivera, R. E. Price, J. D. Hazle, N. J. Halas, and J. L. West, “Nanoshell-mediated near-infrared thermal therapy of tumors under magnetic resonance guidance,” Proc. Natl. Acad. Sci. U. S. A. 100, 13549–13554 (2003).
[Crossref] [PubMed]

Alú, A.

J. Shi, F. Monticone, S. Elias, Y. Wu, D. Ratchford, X. Li, and A. Alú, “Modular assembly of optical nanocircuits,” Nat. Commun. 5, 3896 (2014).
[Crossref] [PubMed]

N. M. Estakhri and A. Alú, “Minimum-scattering superabsorbers,” Phys. Rev. B. 89, 121416 (2014).
[Crossref]

R. Fleury, J. Soric, and A. Alú, “Physical bounds on absorption and scattering for cloaked sensors,” Phys. Rev. B. 89, 045122 (2014).
[Crossref]

F. Monticone, C. Argyropoulos, and A. Alú, “Multi-layered plasmonic covers for comblike scattering response and optical tagging,” Phys. Rev. Lett. 110, 113901 (2013).
[Crossref]

A. Alú and N. Engheta, “Cloaking a sensor,” Phys. Rev. Lett. 102, 233901 (2009).
[Crossref] [PubMed]

A. Alú and N. Engheta, “Plasmonic materials in transparency and cloaking problems: mechanism, robustness, and physical insights,” Opt. Express 15, 3318–3332 (2007).
[Crossref] [PubMed]

A. Alú and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 72, 016623 (2005).
[Crossref]

A. Alú and N. Engheta, “Polarizabilities and effective parameters for collections of spherical nanoparticles formed by pairs of concentric double-negative, single-negative, and/or double-positive metamaterial layers,” J. Appl. Phys. 97, 094310 (2005).
[Crossref]

Argyropoulos, C.

F. Monticone, C. Argyropoulos, and A. Alú, “Multi-layered plasmonic covers for comblike scattering response and optical tagging,” Phys. Rev. Lett. 110, 113901 (2013).
[Crossref]

Bankson, J. A.

L. R. Hirsch, R. J. Stafford, J. A. Bankson, S. R. Sershen, B. Rivera, R. E. Price, J. D. Hazle, N. J. Halas, and J. L. West, “Nanoshell-mediated near-infrared thermal therapy of tumors under magnetic resonance guidance,” Proc. Natl. Acad. Sci. U. S. A. 100, 13549–13554 (2003).
[Crossref] [PubMed]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Bonod, N.

V. Grigoriev, N. Bonod, J. Wenger, and B. Stout, “Optimizing nanoparticle designs for ideal absorption of light,” ACS Photonics 2, 263–270 (2015).
[Crossref]

Born, M.

M. Born and E. Wolf, Principle of Optics, 7th ed. (Cambridge University, 1999).
[Crossref]

Burgi, T.

S. Muhlig, A. Cunningham, J. Dintinger, M. Farhat, S. B. Hasan, T. Scharf, T. Burgi, F. Lederer, and C. Rockstuhl, “A self-assembled three-dimensional cloak in the visible,” Sci. Rep. 3, 2328 (2013).
[Crossref] [PubMed]

Cao, H.

H. Noh, S.M. Popoff, and H. Cao, “Broadband subwavelength focusing of light using a passive sink,” Opt. Express 21, 17435–17446 (2013).
[Crossref] [PubMed]

H. Noh, Y. Chong, A.D. Stone, and H. Cao, “Perfect coupling of light to surface plasmons by coherent absorption,” Phys. Rev. Lett. 108, 186805 (2012).
[Crossref] [PubMed]

Catchpole, K. R.

Chan, K.-L.

J.-Y. Lee, M.-C. Tsai, P.-C. Chen, T.-T. Chen, K.-L. Chan, C.-Y. Lee, and R.-K. Lee, “Thickness effects on light absorption and scattering for nano-particles in shape of hollow-spheres,” J. Phys. Chem. C 119, 25754–25760 (2015).
[Crossref]

Chen, P.-C.

J.-Y. Lee, M.-C. Tsai, P.-C. Chen, T.-T. Chen, K.-L. Chan, C.-Y. Lee, and R.-K. Lee, “Thickness effects on light absorption and scattering for nano-particles in shape of hollow-spheres,” J. Phys. Chem. C 119, 25754–25760 (2015).
[Crossref]

Chen, T.-T.

J.-Y. Lee, M.-C. Tsai, P.-C. Chen, T.-T. Chen, K.-L. Chan, C.-Y. Lee, and R.-K. Lee, “Thickness effects on light absorption and scattering for nano-particles in shape of hollow-spheres,” J. Phys. Chem. C 119, 25754–25760 (2015).
[Crossref]

Chen, Y. L.

X. Zhang, Y. L. Chen, R. S. Liu, and D. P. Tsai, “Plasmonic photocatalysis,” Rep. Prog. Phys. 76, 046401 (2013).
[Crossref] [PubMed]

Chong, Y.

H. Noh, Y. Chong, A.D. Stone, and H. Cao, “Perfect coupling of light to surface plasmons by coherent absorption,” Phys. Rev. Lett. 108, 186805 (2012).
[Crossref] [PubMed]

Chu, S.-W.

S.-W. Chu, T.-Y. Su, R. Oketani, Y.-T. Huang, H.-Y. Wu, Y. Yonemaru, M. Yamanaka, H. Lee, G.-Y. Zhuo, M.-Y. Lee, S. Kawata, and K. Fujita, “Measurement of a saturated emission of optical radiation from gold nanoparticles: Application to an ultrahigh resolution microscope,” Phys. Rev. Lett. 112, 017402 (2014).
[Crossref] [PubMed]

Cunningham, A.

S. Muhlig, A. Cunningham, J. Dintinger, M. Farhat, S. B. Hasan, T. Scharf, T. Burgi, F. Lederer, and C. Rockstuhl, “A self-assembled three-dimensional cloak in the visible,” Sci. Rep. 3, 2328 (2013).
[Crossref] [PubMed]

Dintinger, J.

S. Muhlig, A. Cunningham, J. Dintinger, M. Farhat, S. B. Hasan, T. Scharf, T. Burgi, F. Lederer, and C. Rockstuhl, “A self-assembled three-dimensional cloak in the visible,” Sci. Rep. 3, 2328 (2013).
[Crossref] [PubMed]

Elias, S.

J. Shi, F. Monticone, S. Elias, Y. Wu, D. Ratchford, X. Li, and A. Alú, “Modular assembly of optical nanocircuits,” Nat. Commun. 5, 3896 (2014).
[Crossref] [PubMed]

Engheta, N.

A. Alú and N. Engheta, “Cloaking a sensor,” Phys. Rev. Lett. 102, 233901 (2009).
[Crossref] [PubMed]

A. Alú and N. Engheta, “Plasmonic materials in transparency and cloaking problems: mechanism, robustness, and physical insights,” Opt. Express 15, 3318–3332 (2007).
[Crossref] [PubMed]

N. Engheta, “Circuits with light at nanoscales: optical nanocircuits inspired by metamaterials,” Science 317, 1698–1702 (2007).
[Crossref] [PubMed]

A. Alú and N. Engheta, “Polarizabilities and effective parameters for collections of spherical nanoparticles formed by pairs of concentric double-negative, single-negative, and/or double-positive metamaterial layers,” J. Appl. Phys. 97, 094310 (2005).
[Crossref]

A. Alú and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 72, 016623 (2005).
[Crossref]

Estakhri, N. M.

N. M. Estakhri and A. Alú, “Minimum-scattering superabsorbers,” Phys. Rev. B. 89, 121416 (2014).
[Crossref]

Fan, S.

Z. Ruan and S. Fan, “Design of subwavelength superscattering nanospheres,” Appl. Phys. Lett. 98, 043101 (2011).
[Crossref]

Z. Ruan and S. Fan, “Superscattering of light from subwavelength nanostructures,” Phys. Rev. Lett. 105, 013901 (2010)
[Crossref] [PubMed]

Z. Ruan and S. Fan, “Temporal coupled-mode theory for Fano resonance in light scattering by a single obstacle,” J. Phys. Chem. C 114, 7324 (2010).
[Crossref]

Farhat, M.

S. Muhlig, A. Cunningham, J. Dintinger, M. Farhat, S. B. Hasan, T. Scharf, T. Burgi, F. Lederer, and C. Rockstuhl, “A self-assembled three-dimensional cloak in the visible,” Sci. Rep. 3, 2328 (2013).
[Crossref] [PubMed]

M. Farhat, S. Mhlig, C. Rockstuhl, and F. Lederer, “Scattering cancellation of the magnetic dipole field from macroscopic spheres,” Opt. Express 20, 13896–13906 (2012).
[Crossref] [PubMed]

S. Muhlig, M. Farhat, C. Rockstuhl, and F. Lederer, “Cloaking dielectric spherical objects by a shell of metallic nanoparticles,” Phys. Rev. B 83, 195116 (2011).
[Crossref]

Fleury, R.

R. Fleury, J. Soric, and A. Alú, “Physical bounds on absorption and scattering for cloaked sensors,” Phys. Rev. B. 89, 045122 (2014).
[Crossref]

Fujita, K.

S.-W. Chu, T.-Y. Su, R. Oketani, Y.-T. Huang, H.-Y. Wu, Y. Yonemaru, M. Yamanaka, H. Lee, G.-Y. Zhuo, M.-Y. Lee, S. Kawata, and K. Fujita, “Measurement of a saturated emission of optical radiation from gold nanoparticles: Application to an ultrahigh resolution microscope,” Phys. Rev. Lett. 112, 017402 (2014).
[Crossref] [PubMed]

Grange, R.

Y. Pu, R. Grange, C. L. Hsieh, and D. Psaltis, “Nonlinear optical properties of core-shell nanocavities for enhanced second-harmonic generation,” Phys. Rev. Lett. 104, 207402 (2010).
[Crossref] [PubMed]

Grigoriev, V.

V. Grigoriev, N. Bonod, J. Wenger, and B. Stout, “Optimizing nanoparticle designs for ideal absorption of light,” ACS Photonics 2, 263–270 (2015).
[Crossref]

Halas, N. J.

L. R. Hirsch, R. J. Stafford, J. A. Bankson, S. R. Sershen, B. Rivera, R. E. Price, J. D. Hazle, N. J. Halas, and J. L. West, “Nanoshell-mediated near-infrared thermal therapy of tumors under magnetic resonance guidance,” Proc. Natl. Acad. Sci. U. S. A. 100, 13549–13554 (2003).
[Crossref] [PubMed]

Hamam, R. E.

R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljacic, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A 75, 053801 (2007).
[Crossref]

Hasan, S. B.

S. Muhlig, A. Cunningham, J. Dintinger, M. Farhat, S. B. Hasan, T. Scharf, T. Burgi, F. Lederer, and C. Rockstuhl, “A self-assembled three-dimensional cloak in the visible,” Sci. Rep. 3, 2328 (2013).
[Crossref] [PubMed]

Hazle, J. D.

L. R. Hirsch, R. J. Stafford, J. A. Bankson, S. R. Sershen, B. Rivera, R. E. Price, J. D. Hazle, N. J. Halas, and J. L. West, “Nanoshell-mediated near-infrared thermal therapy of tumors under magnetic resonance guidance,” Proc. Natl. Acad. Sci. U. S. A. 100, 13549–13554 (2003).
[Crossref] [PubMed]

Hirsch, L. R.

L. R. Hirsch, R. J. Stafford, J. A. Bankson, S. R. Sershen, B. Rivera, R. E. Price, J. D. Hazle, N. J. Halas, and J. L. West, “Nanoshell-mediated near-infrared thermal therapy of tumors under magnetic resonance guidance,” Proc. Natl. Acad. Sci. U. S. A. 100, 13549–13554 (2003).
[Crossref] [PubMed]

Hsieh, C. L.

Y. Pu, R. Grange, C. L. Hsieh, and D. Psaltis, “Nonlinear optical properties of core-shell nanocavities for enhanced second-harmonic generation,” Phys. Rev. Lett. 104, 207402 (2010).
[Crossref] [PubMed]

Huang, Y.-T.

S.-W. Chu, T.-Y. Su, R. Oketani, Y.-T. Huang, H.-Y. Wu, Y. Yonemaru, M. Yamanaka, H. Lee, G.-Y. Zhuo, M.-Y. Lee, S. Kawata, and K. Fujita, “Measurement of a saturated emission of optical radiation from gold nanoparticles: Application to an ultrahigh resolution microscope,” Phys. Rev. Lett. 112, 017402 (2014).
[Crossref] [PubMed]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Joannopoulos, J. D.

R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljacic, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A 75, 053801 (2007).
[Crossref]

Karalis, A.

R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljacic, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A 75, 053801 (2007).
[Crossref]

Kawata, S.

S.-W. Chu, T.-Y. Su, R. Oketani, Y.-T. Huang, H.-Y. Wu, Y. Yonemaru, M. Yamanaka, H. Lee, G.-Y. Zhuo, M.-Y. Lee, S. Kawata, and K. Fujita, “Measurement of a saturated emission of optical radiation from gold nanoparticles: Application to an ultrahigh resolution microscope,” Phys. Rev. Lett. 112, 017402 (2014).
[Crossref] [PubMed]

Khokhlov, A. R.

M. I. Tribelsky, A. E. Miroshnichenko, Y. S. Kivshar, B. S. Lukyanchuk, and A. R. Khokhlov, “Laser pulse heating of spherical metal particles,” Phys. Rev. X 1, 021024 (2011).

Kivshar, Y. S.

A. Mirzaei, A. E. Miroshnichenko, I. V. Shadrivov, and Y. S. Kivshar, “Superscattering of light optimized by a genetic algorithm,” Appl. Phys. Lett. 105, 011109 (2014).
[Crossref]

A. Mirzaei, I. V. Shadrivov, A. E. Miroshnichenko, and Y. S. Kivshar, “Cloaking and enhanced scattering of core-shell plasmonic nanowires,” Opt. Express 21, 10454–10459 (2013).
[Crossref] [PubMed]

M. I. Tribelsky, A. E. Miroshnichenko, Y. S. Kivshar, B. S. Lukyanchuk, and A. R. Khokhlov, “Laser pulse heating of spherical metal particles,” Phys. Rev. X 1, 021024 (2011).

Lederer, F.

S. Muhlig, A. Cunningham, J. Dintinger, M. Farhat, S. B. Hasan, T. Scharf, T. Burgi, F. Lederer, and C. Rockstuhl, “A self-assembled three-dimensional cloak in the visible,” Sci. Rep. 3, 2328 (2013).
[Crossref] [PubMed]

M. Farhat, S. Mhlig, C. Rockstuhl, and F. Lederer, “Scattering cancellation of the magnetic dipole field from macroscopic spheres,” Opt. Express 20, 13896–13906 (2012).
[Crossref] [PubMed]

S. Muhlig, M. Farhat, C. Rockstuhl, and F. Lederer, “Cloaking dielectric spherical objects by a shell of metallic nanoparticles,” Phys. Rev. B 83, 195116 (2011).
[Crossref]

Lee, C.-Y.

J.-Y. Lee, M.-C. Tsai, P.-C. Chen, T.-T. Chen, K.-L. Chan, C.-Y. Lee, and R.-K. Lee, “Thickness effects on light absorption and scattering for nano-particles in shape of hollow-spheres,” J. Phys. Chem. C 119, 25754–25760 (2015).
[Crossref]

Lee, H.

S.-W. Chu, T.-Y. Su, R. Oketani, Y.-T. Huang, H.-Y. Wu, Y. Yonemaru, M. Yamanaka, H. Lee, G.-Y. Zhuo, M.-Y. Lee, S. Kawata, and K. Fujita, “Measurement of a saturated emission of optical radiation from gold nanoparticles: Application to an ultrahigh resolution microscope,” Phys. Rev. Lett. 112, 017402 (2014).
[Crossref] [PubMed]

Lee, J.-Y.

J.-Y. Lee, M.-C. Tsai, P.-C. Chen, T.-T. Chen, K.-L. Chan, C.-Y. Lee, and R.-K. Lee, “Thickness effects on light absorption and scattering for nano-particles in shape of hollow-spheres,” J. Phys. Chem. C 119, 25754–25760 (2015).
[Crossref]

Lee, M.-Y.

S.-W. Chu, T.-Y. Su, R. Oketani, Y.-T. Huang, H.-Y. Wu, Y. Yonemaru, M. Yamanaka, H. Lee, G.-Y. Zhuo, M.-Y. Lee, S. Kawata, and K. Fujita, “Measurement of a saturated emission of optical radiation from gold nanoparticles: Application to an ultrahigh resolution microscope,” Phys. Rev. Lett. 112, 017402 (2014).
[Crossref] [PubMed]

Lee, R.-K.

J.-Y. Lee, M.-C. Tsai, P.-C. Chen, T.-T. Chen, K.-L. Chan, C.-Y. Lee, and R.-K. Lee, “Thickness effects on light absorption and scattering for nano-particles in shape of hollow-spheres,” J. Phys. Chem. C 119, 25754–25760 (2015).
[Crossref]

Li, X.

J. Shi, F. Monticone, S. Elias, Y. Wu, D. Ratchford, X. Li, and A. Alú, “Modular assembly of optical nanocircuits,” Nat. Commun. 5, 3896 (2014).
[Crossref] [PubMed]

Liu, R. S.

X. Zhang, Y. L. Chen, R. S. Liu, and D. P. Tsai, “Plasmonic photocatalysis,” Rep. Prog. Phys. 76, 046401 (2013).
[Crossref] [PubMed]

Lukyanchuk, B. S.

M. I. Tribelsky, A. E. Miroshnichenko, Y. S. Kivshar, B. S. Lukyanchuk, and A. R. Khokhlov, “Laser pulse heating of spherical metal particles,” Phys. Rev. X 1, 021024 (2011).

M. I. Tribelsky and B. S. Lukyanchuk, “Anomalous light scattering by small particles,” Phys. Rev. Lett. 97, 263902 (2006).
[Crossref]

Maier, S. A.

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

Mhlig, S.

Miroshnichenko, A. E.

A. Mirzaei, A. E. Miroshnichenko, I. V. Shadrivov, and Y. S. Kivshar, “Superscattering of light optimized by a genetic algorithm,” Appl. Phys. Lett. 105, 011109 (2014).
[Crossref]

A. Mirzaei, I. V. Shadrivov, A. E. Miroshnichenko, and Y. S. Kivshar, “Cloaking and enhanced scattering of core-shell plasmonic nanowires,” Opt. Express 21, 10454–10459 (2013).
[Crossref] [PubMed]

M. I. Tribelsky, A. E. Miroshnichenko, Y. S. Kivshar, B. S. Lukyanchuk, and A. R. Khokhlov, “Laser pulse heating of spherical metal particles,” Phys. Rev. X 1, 021024 (2011).

Mirzaei, A.

A. Mirzaei, A. E. Miroshnichenko, I. V. Shadrivov, and Y. S. Kivshar, “Superscattering of light optimized by a genetic algorithm,” Appl. Phys. Lett. 105, 011109 (2014).
[Crossref]

A. Mirzaei, I. V. Shadrivov, A. E. Miroshnichenko, and Y. S. Kivshar, “Cloaking and enhanced scattering of core-shell plasmonic nanowires,” Opt. Express 21, 10454–10459 (2013).
[Crossref] [PubMed]

Monticone, F.

J. Shi, F. Monticone, S. Elias, Y. Wu, D. Ratchford, X. Li, and A. Alú, “Modular assembly of optical nanocircuits,” Nat. Commun. 5, 3896 (2014).
[Crossref] [PubMed]

F. Monticone, C. Argyropoulos, and A. Alú, “Multi-layered plasmonic covers for comblike scattering response and optical tagging,” Phys. Rev. Lett. 110, 113901 (2013).
[Crossref]

Muhlig, S.

S. Muhlig, A. Cunningham, J. Dintinger, M. Farhat, S. B. Hasan, T. Scharf, T. Burgi, F. Lederer, and C. Rockstuhl, “A self-assembled three-dimensional cloak in the visible,” Sci. Rep. 3, 2328 (2013).
[Crossref] [PubMed]

S. Muhlig, M. Farhat, C. Rockstuhl, and F. Lederer, “Cloaking dielectric spherical objects by a shell of metallic nanoparticles,” Phys. Rev. B 83, 195116 (2011).
[Crossref]

Noh, H.

H. Noh, S.M. Popoff, and H. Cao, “Broadband subwavelength focusing of light using a passive sink,” Opt. Express 21, 17435–17446 (2013).
[Crossref] [PubMed]

H. Noh, Y. Chong, A.D. Stone, and H. Cao, “Perfect coupling of light to surface plasmons by coherent absorption,” Phys. Rev. Lett. 108, 186805 (2012).
[Crossref] [PubMed]

Oketani, R.

S.-W. Chu, T.-Y. Su, R. Oketani, Y.-T. Huang, H.-Y. Wu, Y. Yonemaru, M. Yamanaka, H. Lee, G.-Y. Zhuo, M.-Y. Lee, S. Kawata, and K. Fujita, “Measurement of a saturated emission of optical radiation from gold nanoparticles: Application to an ultrahigh resolution microscope,” Phys. Rev. Lett. 112, 017402 (2014).
[Crossref] [PubMed]

Polman, A.

Popoff, S.M.

Price, R. E.

L. R. Hirsch, R. J. Stafford, J. A. Bankson, S. R. Sershen, B. Rivera, R. E. Price, J. D. Hazle, N. J. Halas, and J. L. West, “Nanoshell-mediated near-infrared thermal therapy of tumors under magnetic resonance guidance,” Proc. Natl. Acad. Sci. U. S. A. 100, 13549–13554 (2003).
[Crossref] [PubMed]

Psaltis, D.

Y. Pu, R. Grange, C. L. Hsieh, and D. Psaltis, “Nonlinear optical properties of core-shell nanocavities for enhanced second-harmonic generation,” Phys. Rev. Lett. 104, 207402 (2010).
[Crossref] [PubMed]

Pu, Y.

Y. Pu, R. Grange, C. L. Hsieh, and D. Psaltis, “Nonlinear optical properties of core-shell nanocavities for enhanced second-harmonic generation,” Phys. Rev. Lett. 104, 207402 (2010).
[Crossref] [PubMed]

Ratchford, D.

J. Shi, F. Monticone, S. Elias, Y. Wu, D. Ratchford, X. Li, and A. Alú, “Modular assembly of optical nanocircuits,” Nat. Commun. 5, 3896 (2014).
[Crossref] [PubMed]

Rivera, B.

L. R. Hirsch, R. J. Stafford, J. A. Bankson, S. R. Sershen, B. Rivera, R. E. Price, J. D. Hazle, N. J. Halas, and J. L. West, “Nanoshell-mediated near-infrared thermal therapy of tumors under magnetic resonance guidance,” Proc. Natl. Acad. Sci. U. S. A. 100, 13549–13554 (2003).
[Crossref] [PubMed]

Rockstuhl, C.

S. Muhlig, A. Cunningham, J. Dintinger, M. Farhat, S. B. Hasan, T. Scharf, T. Burgi, F. Lederer, and C. Rockstuhl, “A self-assembled three-dimensional cloak in the visible,” Sci. Rep. 3, 2328 (2013).
[Crossref] [PubMed]

M. Farhat, S. Mhlig, C. Rockstuhl, and F. Lederer, “Scattering cancellation of the magnetic dipole field from macroscopic spheres,” Opt. Express 20, 13896–13906 (2012).
[Crossref] [PubMed]

S. Muhlig, M. Farhat, C. Rockstuhl, and F. Lederer, “Cloaking dielectric spherical objects by a shell of metallic nanoparticles,” Phys. Rev. B 83, 195116 (2011).
[Crossref]

Ruan, Z.

Z. Ruan and S. Fan, “Design of subwavelength superscattering nanospheres,” Appl. Phys. Lett. 98, 043101 (2011).
[Crossref]

Z. Ruan and S. Fan, “Superscattering of light from subwavelength nanostructures,” Phys. Rev. Lett. 105, 013901 (2010)
[Crossref] [PubMed]

Z. Ruan and S. Fan, “Temporal coupled-mode theory for Fano resonance in light scattering by a single obstacle,” J. Phys. Chem. C 114, 7324 (2010).
[Crossref]

Scharf, T.

S. Muhlig, A. Cunningham, J. Dintinger, M. Farhat, S. B. Hasan, T. Scharf, T. Burgi, F. Lederer, and C. Rockstuhl, “A self-assembled three-dimensional cloak in the visible,” Sci. Rep. 3, 2328 (2013).
[Crossref] [PubMed]

Sershen, S. R.

L. R. Hirsch, R. J. Stafford, J. A. Bankson, S. R. Sershen, B. Rivera, R. E. Price, J. D. Hazle, N. J. Halas, and J. L. West, “Nanoshell-mediated near-infrared thermal therapy of tumors under magnetic resonance guidance,” Proc. Natl. Acad. Sci. U. S. A. 100, 13549–13554 (2003).
[Crossref] [PubMed]

Shadrivov, I. V.

A. Mirzaei, A. E. Miroshnichenko, I. V. Shadrivov, and Y. S. Kivshar, “Superscattering of light optimized by a genetic algorithm,” Appl. Phys. Lett. 105, 011109 (2014).
[Crossref]

A. Mirzaei, I. V. Shadrivov, A. E. Miroshnichenko, and Y. S. Kivshar, “Cloaking and enhanced scattering of core-shell plasmonic nanowires,” Opt. Express 21, 10454–10459 (2013).
[Crossref] [PubMed]

Shi, J.

J. Shi, F. Monticone, S. Elias, Y. Wu, D. Ratchford, X. Li, and A. Alú, “Modular assembly of optical nanocircuits,” Nat. Commun. 5, 3896 (2014).
[Crossref] [PubMed]

Soljacic, M.

R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljacic, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A 75, 053801 (2007).
[Crossref]

Soric, J.

R. Fleury, J. Soric, and A. Alú, “Physical bounds on absorption and scattering for cloaked sensors,” Phys. Rev. B. 89, 045122 (2014).
[Crossref]

Stafford, R. J.

L. R. Hirsch, R. J. Stafford, J. A. Bankson, S. R. Sershen, B. Rivera, R. E. Price, J. D. Hazle, N. J. Halas, and J. L. West, “Nanoshell-mediated near-infrared thermal therapy of tumors under magnetic resonance guidance,” Proc. Natl. Acad. Sci. U. S. A. 100, 13549–13554 (2003).
[Crossref] [PubMed]

Stone, A.D.

H. Noh, Y. Chong, A.D. Stone, and H. Cao, “Perfect coupling of light to surface plasmons by coherent absorption,” Phys. Rev. Lett. 108, 186805 (2012).
[Crossref] [PubMed]

Stout, B.

V. Grigoriev, N. Bonod, J. Wenger, and B. Stout, “Optimizing nanoparticle designs for ideal absorption of light,” ACS Photonics 2, 263–270 (2015).
[Crossref]

Su, T.-Y.

S.-W. Chu, T.-Y. Su, R. Oketani, Y.-T. Huang, H.-Y. Wu, Y. Yonemaru, M. Yamanaka, H. Lee, G.-Y. Zhuo, M.-Y. Lee, S. Kawata, and K. Fujita, “Measurement of a saturated emission of optical radiation from gold nanoparticles: Application to an ultrahigh resolution microscope,” Phys. Rev. Lett. 112, 017402 (2014).
[Crossref] [PubMed]

Tribelsky, M. I.

M. I. Tribelsky, A. E. Miroshnichenko, Y. S. Kivshar, B. S. Lukyanchuk, and A. R. Khokhlov, “Laser pulse heating of spherical metal particles,” Phys. Rev. X 1, 021024 (2011).

M. I. Tribelsky, “Anomalous light absorption by small particles,” Europhys. Lett. 94, 14004 (2011).
[Crossref]

M. I. Tribelsky and B. S. Lukyanchuk, “Anomalous light scattering by small particles,” Phys. Rev. Lett. 97, 263902 (2006).
[Crossref]

Tsai, D. P.

X. Zhang, Y. L. Chen, R. S. Liu, and D. P. Tsai, “Plasmonic photocatalysis,” Rep. Prog. Phys. 76, 046401 (2013).
[Crossref] [PubMed]

Tsai, M.-C.

J.-Y. Lee, M.-C. Tsai, P.-C. Chen, T.-T. Chen, K.-L. Chan, C.-Y. Lee, and R.-K. Lee, “Thickness effects on light absorption and scattering for nano-particles in shape of hollow-spheres,” J. Phys. Chem. C 119, 25754–25760 (2015).
[Crossref]

van de Hulst, H.C.

H.C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

Wenger, J.

V. Grigoriev, N. Bonod, J. Wenger, and B. Stout, “Optimizing nanoparticle designs for ideal absorption of light,” ACS Photonics 2, 263–270 (2015).
[Crossref]

West, J. L.

L. R. Hirsch, R. J. Stafford, J. A. Bankson, S. R. Sershen, B. Rivera, R. E. Price, J. D. Hazle, N. J. Halas, and J. L. West, “Nanoshell-mediated near-infrared thermal therapy of tumors under magnetic resonance guidance,” Proc. Natl. Acad. Sci. U. S. A. 100, 13549–13554 (2003).
[Crossref] [PubMed]

Wolf, E.

M. Born and E. Wolf, Principle of Optics, 7th ed. (Cambridge University, 1999).
[Crossref]

Wu, H.-Y.

S.-W. Chu, T.-Y. Su, R. Oketani, Y.-T. Huang, H.-Y. Wu, Y. Yonemaru, M. Yamanaka, H. Lee, G.-Y. Zhuo, M.-Y. Lee, S. Kawata, and K. Fujita, “Measurement of a saturated emission of optical radiation from gold nanoparticles: Application to an ultrahigh resolution microscope,” Phys. Rev. Lett. 112, 017402 (2014).
[Crossref] [PubMed]

Wu, Y.

J. Shi, F. Monticone, S. Elias, Y. Wu, D. Ratchford, X. Li, and A. Alú, “Modular assembly of optical nanocircuits,” Nat. Commun. 5, 3896 (2014).
[Crossref] [PubMed]

Yamanaka, M.

S.-W. Chu, T.-Y. Su, R. Oketani, Y.-T. Huang, H.-Y. Wu, Y. Yonemaru, M. Yamanaka, H. Lee, G.-Y. Zhuo, M.-Y. Lee, S. Kawata, and K. Fujita, “Measurement of a saturated emission of optical radiation from gold nanoparticles: Application to an ultrahigh resolution microscope,” Phys. Rev. Lett. 112, 017402 (2014).
[Crossref] [PubMed]

Yonemaru, Y.

S.-W. Chu, T.-Y. Su, R. Oketani, Y.-T. Huang, H.-Y. Wu, Y. Yonemaru, M. Yamanaka, H. Lee, G.-Y. Zhuo, M.-Y. Lee, S. Kawata, and K. Fujita, “Measurement of a saturated emission of optical radiation from gold nanoparticles: Application to an ultrahigh resolution microscope,” Phys. Rev. Lett. 112, 017402 (2014).
[Crossref] [PubMed]

Zhang, X.

X. Zhang, Y. L. Chen, R. S. Liu, and D. P. Tsai, “Plasmonic photocatalysis,” Rep. Prog. Phys. 76, 046401 (2013).
[Crossref] [PubMed]

Zhuo, G.-Y.

S.-W. Chu, T.-Y. Su, R. Oketani, Y.-T. Huang, H.-Y. Wu, Y. Yonemaru, M. Yamanaka, H. Lee, G.-Y. Zhuo, M.-Y. Lee, S. Kawata, and K. Fujita, “Measurement of a saturated emission of optical radiation from gold nanoparticles: Application to an ultrahigh resolution microscope,” Phys. Rev. Lett. 112, 017402 (2014).
[Crossref] [PubMed]

ACS Photonics (1)

V. Grigoriev, N. Bonod, J. Wenger, and B. Stout, “Optimizing nanoparticle designs for ideal absorption of light,” ACS Photonics 2, 263–270 (2015).
[Crossref]

Appl. Phys. Lett. (2)

Z. Ruan and S. Fan, “Design of subwavelength superscattering nanospheres,” Appl. Phys. Lett. 98, 043101 (2011).
[Crossref]

A. Mirzaei, A. E. Miroshnichenko, I. V. Shadrivov, and Y. S. Kivshar, “Superscattering of light optimized by a genetic algorithm,” Appl. Phys. Lett. 105, 011109 (2014).
[Crossref]

Europhys. Lett. (1)

M. I. Tribelsky, “Anomalous light absorption by small particles,” Europhys. Lett. 94, 14004 (2011).
[Crossref]

J. Appl. Phys. (1)

A. Alú and N. Engheta, “Polarizabilities and effective parameters for collections of spherical nanoparticles formed by pairs of concentric double-negative, single-negative, and/or double-positive metamaterial layers,” J. Appl. Phys. 97, 094310 (2005).
[Crossref]

J. Phys. Chem. C (2)

Z. Ruan and S. Fan, “Temporal coupled-mode theory for Fano resonance in light scattering by a single obstacle,” J. Phys. Chem. C 114, 7324 (2010).
[Crossref]

J.-Y. Lee, M.-C. Tsai, P.-C. Chen, T.-T. Chen, K.-L. Chan, C.-Y. Lee, and R.-K. Lee, “Thickness effects on light absorption and scattering for nano-particles in shape of hollow-spheres,” J. Phys. Chem. C 119, 25754–25760 (2015).
[Crossref]

Nat. Commun. (1)

J. Shi, F. Monticone, S. Elias, Y. Wu, D. Ratchford, X. Li, and A. Alú, “Modular assembly of optical nanocircuits,” Nat. Commun. 5, 3896 (2014).
[Crossref] [PubMed]

Opt. Express (5)

Phys. Rev. A (1)

R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljacic, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A 75, 053801 (2007).
[Crossref]

Phys. Rev. B (1)

S. Muhlig, M. Farhat, C. Rockstuhl, and F. Lederer, “Cloaking dielectric spherical objects by a shell of metallic nanoparticles,” Phys. Rev. B 83, 195116 (2011).
[Crossref]

Phys. Rev. B. (2)

R. Fleury, J. Soric, and A. Alú, “Physical bounds on absorption and scattering for cloaked sensors,” Phys. Rev. B. 89, 045122 (2014).
[Crossref]

N. M. Estakhri and A. Alú, “Minimum-scattering superabsorbers,” Phys. Rev. B. 89, 121416 (2014).
[Crossref]

Phys. Rev. E (1)

A. Alú and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 72, 016623 (2005).
[Crossref]

Phys. Rev. Lett. (7)

A. Alú and N. Engheta, “Cloaking a sensor,” Phys. Rev. Lett. 102, 233901 (2009).
[Crossref] [PubMed]

Z. Ruan and S. Fan, “Superscattering of light from subwavelength nanostructures,” Phys. Rev. Lett. 105, 013901 (2010)
[Crossref] [PubMed]

M. I. Tribelsky and B. S. Lukyanchuk, “Anomalous light scattering by small particles,” Phys. Rev. Lett. 97, 263902 (2006).
[Crossref]

F. Monticone, C. Argyropoulos, and A. Alú, “Multi-layered plasmonic covers for comblike scattering response and optical tagging,” Phys. Rev. Lett. 110, 113901 (2013).
[Crossref]

H. Noh, Y. Chong, A.D. Stone, and H. Cao, “Perfect coupling of light to surface plasmons by coherent absorption,” Phys. Rev. Lett. 108, 186805 (2012).
[Crossref] [PubMed]

Y. Pu, R. Grange, C. L. Hsieh, and D. Psaltis, “Nonlinear optical properties of core-shell nanocavities for enhanced second-harmonic generation,” Phys. Rev. Lett. 104, 207402 (2010).
[Crossref] [PubMed]

S.-W. Chu, T.-Y. Su, R. Oketani, Y.-T. Huang, H.-Y. Wu, Y. Yonemaru, M. Yamanaka, H. Lee, G.-Y. Zhuo, M.-Y. Lee, S. Kawata, and K. Fujita, “Measurement of a saturated emission of optical radiation from gold nanoparticles: Application to an ultrahigh resolution microscope,” Phys. Rev. Lett. 112, 017402 (2014).
[Crossref] [PubMed]

Phys. Rev. X (1)

M. I. Tribelsky, A. E. Miroshnichenko, Y. S. Kivshar, B. S. Lukyanchuk, and A. R. Khokhlov, “Laser pulse heating of spherical metal particles,” Phys. Rev. X 1, 021024 (2011).

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

L. R. Hirsch, R. J. Stafford, J. A. Bankson, S. R. Sershen, B. Rivera, R. E. Price, J. D. Hazle, N. J. Halas, and J. L. West, “Nanoshell-mediated near-infrared thermal therapy of tumors under magnetic resonance guidance,” Proc. Natl. Acad. Sci. U. S. A. 100, 13549–13554 (2003).
[Crossref] [PubMed]

Rep. Prog. Phys. (1)

X. Zhang, Y. L. Chen, R. S. Liu, and D. P. Tsai, “Plasmonic photocatalysis,” Rep. Prog. Phys. 76, 046401 (2013).
[Crossref] [PubMed]

Sci. Rep. (1)

S. Muhlig, A. Cunningham, J. Dintinger, M. Farhat, S. B. Hasan, T. Scharf, T. Burgi, F. Lederer, and C. Rockstuhl, “A self-assembled three-dimensional cloak in the visible,” Sci. Rep. 3, 2328 (2013).
[Crossref] [PubMed]

Science (1)

N. Engheta, “Circuits with light at nanoscales: optical nanocircuits inspired by metamaterials,” Science 317, 1698–1702 (2007).
[Crossref] [PubMed]

Other (4)

M. Born and E. Wolf, Principle of Optics, 7th ed. (Cambridge University, 1999).
[Crossref]

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

H.C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

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Figures (4)

Fig. 1
Fig. 1 A phase diagram for each spherical harmonic channel, labeled by n, is generated by imposing the power conservation on the partial absorption cross section, for TE or TM mode separately. Marked numbers shown in the contour lines correspond to the values of normalized absorption cross section in the individual channel: 2 π ( 2 n + 1 ) λ 2 σ n a b s ( T E , T M ). Colored regions are physically allowed solutions; while uncolored regions represent forbidden solutions. It is noted that the amplitude square is bounded within the range [0, 1]; while the allowed phase is within [π/2, 3π/2]. The Green cross-marker, localed at ( θ n ( T E , T M ) = π , | C n ( T E , T M ) | 2 = 0.25 ), indicates the maximum value, 0.25, in the normalized absorption cross-section.
Fig. 2
Fig. 2 Supported trajectories in the phase diagram are shown for different sets of the parameters: α n ( T E , T M ) and β n ( T E , T M ) defined in Eq. (3). Here, trajectories with a constant β n ( T E , T M ) are shown in Blue dotted-dashed-curves; while trajectories with a constant α n ( T E , T M ) are shown in Red dotted-dashed-curves. Two contours for a constant absorption power are also depicted in the Black color.
Fig. 3
Fig. 3 Absorption and scattering cross sections correspond to the contour shown in Fig. 2, which are depicted in terms of the parametric variable t defined in Eqs. (4)(5). Here, a constant absorption power is requested by setting q 1 T M = 0.2; while there is a degree of freedom in the scattering power. The insect illustrates the core-shell scatterer used as an example to design a passive electromagnetic devices with the constant absorption power.
Fig. 4
Fig. 4 The permittivities to support a constant absorption power are shown as a function of the parametric variable t. For a given material in the shell region, εs = 3.12, found solutions for the real and imaginary parts of the permittivity in the core region are shown in (a) and (b), respectively. For a given material in the core region, εc = 5, two families of found solutions shown in Eq. (8) are denoted as ε s + and ε s for the shell region, with the corresponding real and imaginary parts of the permittivity shown in (c, e) and (d, f), respectively. Results obtained from analytical formulas are depicted in solid-curves; while exact solutions from scattering theory are depicted in dashed-curves. In all cases, the core-shell geometries are fixed with a = 1/24λ and γ = 0.9.

Equations (15)

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σ abs n = 1 σ n abs ( TE ) + σ n abs ( TM ) = n = 1 ( 2 n + 1 ) λ 2 2 π ( Re { C n TM } + | C n TM | 2 + Re { C n TE } + | C n TE | 2 ) ,
σ scat = n = 1 n = ( 2 n + 1 ) λ 2 2 π ( | C n TM | 2 + | C n TE | 2 ) ,
C n ( T E , T M ) = 1 1 + i V n ( T E , T M ) U n ( T E , T M ) 1 1 + i [ α n ( T E , T M ) + i β n ( T E , T M ) ] ,
α n ( T E , T M ) ( t ) = 1 4 [ q n ( T E , T M ) ] 2 1 q n ( T E , T M ) sin ( t ) ,
β n ( T E , T M ) ( t ) = [ 1 1 2 q n ( T E , T M ) ] + 1 4 [ q n ( T E , T M ) ] 2 1 q n ( T E , T M ) cos ( t ) .
V 1 T M U 1 T M = 3 λ 3 2 ( 2 π a ) 3 2 γ 3 ( 1 ε s ) ( ε c ε s ) ( 2 + ε s ) ( ε c + 2 ε s ) γ 3 ( ε s ε c ) ( 2 ε s + 1 ) + ( 1 ε s ) ( ε c + 2 ε s ) ,
ε c = ε s 3 ( 2 ε s 4 2 γ 3 + 2 γ 3 ε s ) 2 ( α 1 T M + i β 1 T M ) ( 2 π a / λ ) 3 ( 2 2 ε s + 2 ε s γ 3 + γ 3 ) 3 ( ε s + 2 + 2 γ 3 ε s 2 γ 3 ) + 2 ( α 1 T M + i β 1 T M ) ( 2 π a / λ ) 3 ( 1 ε s 2 ε s γ 3 γ 3 ) .
ε s ± = g ± g 2 4 f h 2 f ,
f = 2 ( 1 γ 3 ) [ 3 2 ( α 1 T M + i β 1 T M ) ( 2 π a / λ ) 3 ] ,
g = 2 ( α 1 T M + i β 1 T M ) ( 2 π a / λ ) 3 [ γ 3 ( 1 2 ε c ) + 2 ε c ] + 3 ( 2 γ 3 + 2 γ 3 ε c + ε c + 4 ) ,
h = ε c ( 1 γ 3 ) [ 6 + 2 ( α 1 T M + i β 1 T M ) ( 2 π a / λ ) 3 ] .
V 1 T M U 1 T M = Z 1 T M = 3 2 ( λ 2 π a ) 3 2 γ 3 ( 1 ε s ) ( ε c ε s ) ( 2 + ε s ) ( ε c + 2 ε s ) γ 3 ( ε s ε c ) ( 2 ε s + 1 ) + ( 1 ε s ) ( ε c + 2 ε s ) .
ε c { 2 ( 2 π a λ ) 3 Z 1 T M [ 2 ε s γ 3 γ 3 + 1 ε s ] 3 [ 2 γ 3 2 γ 3 ε s 2 ε s ] } = 3 [ 4 ε s 2 ε s 2 2 γ 3 ε s + 2 γ 3 ε s 2 ] 2 ( 2 π a λ ) 3 Z 1 T M [ γ 3 ( 2 ε s 2 + ε s ) + ( 2 ε s 2 ε s 2 ) ] ,
ε s 2 ( 2 γ 3 2 ) [ 2 ( 2 π a λ ) 3 Z 1 T M 3 ] + [ 2 ( 2 π a λ ) 3 Z 1 T M ε c ( 1 γ 3 ) 6 ε c ( γ 3 1 ) ] + ε s [ 2 ( 2 π a λ ) 3 Z 1 T M ( γ 3 2 ε c γ 3 + 2 ε c ) + 3 ( 2 γ 3 + 2 γ 3 ε c + 4 + ε c ) ] = 0.
f ε s 2 + g ε s + h = 0.

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