Abstract

When considering light scattering from a sphere, the ratios between the expansion coefficients of the scattered and the incident field in a spherical basis are known as the Mie coefficients. Generally, Mie coefficients depend on many degrees of freedom, including the dimensions and electromagnetic properties of the spherical object. However, for fundamental research, it is important to have easy expressions for all possible values of Mie coefficients within the existing physical constraints and which depend on the least number of degrees of freedom. While such expressions are known for spheres made from non-absorbing materials, we present here, for the first time to our knowledge, corresponding expressions for spheres made from absorbing materials. To illustrate the usefulness of these expressions, we investigate the upper bound for the absorption cross section of a trimer made from electric dipolar spheres. Given the results, we have designed a dipolar ITO trimer that offers a maximal absorption cross section. Our approach is not limited to dipolar terms, but indeed, as demonstrated in the manuscript, can be applied to higher order terms as well. Using our model, one can scan the entire accessible parameter space of spheres for specific functionalities in systems made from spherical scatterers.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2020 (1)

J. Olmos-Trigo, D. R. Abujetas, C. Sanz-Fernández, J. A. Sánchez-Gil, and J. J. Sáenz, “Optimal backward light scattering by dipolar particles,” Phys. Rev. Res. 2(1), 013225 (2020).
[Crossref]

2019 (7)

A. Rahimzadegan, D. Arslan, D. Dams, A. Groner, X. Garcia-Santiago, R. Alaee, I. Fernandez-Corbaton, T. Pertsch, I. Staude, and C. Rockstuhl, “Beyond dipolar Huygens’ metasurfaces for full-phase coverage and unity transmittance,” Nanophotonics 9(1), 75–82 (2019).
[Crossref]

H. K. Shamkhi, K. V. Baryshnikova, A. Sayanskiy, P. Kapitanova, P. D. Terekhov, P. Belov, A. Karabchevsky, A. B. Evlyukhin, Y. Kivshar, and A. S. Shalin, “Transverse scattering and generalized kerker effects in all-dielectric Mie-resonant metaoptics,” Phys. Rev. Lett. 122(19), 193905 (2019).
[Crossref]

O. Quevedo-Teruel, H. Chen, A. Díaz-Rubio, G. Gok, A. Grbic, G. Minatti, E. Martini, S. Maci, G. V. Eleftheriades, M. Chen, N. I. Zheludev, N. Papasimakis, S. Choudhury, Z. A. Kudyshev, S. Saha, H. Reddy, A. Boltasseva, V. M. Shalaev, A. V. Kildishev, D. Sievenpiper, C. Caloz, A. Alú, Q. He, L. Zhou, G. Valerio, E. Rajo-Iglesias, Z. Sipus, F. Mesa, R. Rodriguez-Berral, F. Medina, V. Asadchy, S. Tretyakov, and C. Craeye, “Roadmap on metasurfaces,” J. Opt. 21(7), 073002 (2019).
[Crossref]

A. B. Evlyukhin, K. V. Nerkararyan, and S. I. Bozhevolnyi, “Core-shell particles as efficient broadband absorbers in infrared optical range,” Opt. Express 27(13), 17474–17481 (2019).
[Crossref]

R. Dezert, P. Richetti, and A. Baron, “Complete multipolar description of reflection and transmission across a metasurface for perfect absorption of light,” Opt. Express 27(19), 26317–26330 (2019).
[Crossref]

A. Rahimzadegan, D. Arslan, R. Suryadharma, S. Fasold, M. Falkner, T. Pertsch, I. Staude, and C. Rockstuhl, “Disorder-induced phase transitions in the transmission of dielectric metasurfaces,” Phys. Rev. Lett. 122(1), 015702 (2019).
[Crossref]

R. Alaee, C. Rockstuhl, and I. Fernandez-Corbaton, “Exact multipolar decompositions with applications in nanophotonics,” Adv. Opt. Mater. 7(1), 1800783 (2019).
[Crossref]

2018 (4)

A. Rahimzadegan, C. Rockstuhl, and I. Fernandez-Corbaton, “Core-shell particles as building blocks for systems with high duality symmetry,” Phys. Rev. Appl. 9(5), 054051 (2018).
[Crossref]

J. Peurifoy, Y. Shen, L. Jing, Y. Yang, F. Cano-Renteria, B. G. DeLacy, J. D. Joannopoulos, M. Tegmark, and M. Soljačić, “Nanophotonic particle simulation and inverse design using artificial neural networks,” Sci. Adv. 4(6), eaar4206 (2018).
[Crossref]

S. M. Kamali, E. Arbabi, A. Arbabi, and A. Faraon, “A review of dielectric optical metasurfaces for wavefront control,” Nanophotonics 7(6), 1041–1068 (2018).
[Crossref]

D. Tzarouchis and A. Sihvola, “Light scattering by a dielectric sphere: perspectives on the Mie resonances,” Appl. Sci. 8(2), 184 (2018).
[Crossref]

2017 (9)

R. Dezert, P. Richetti, and A. Baron, “Isotropic Huygens dipoles and multipoles with colloidal particles,” Phys. Rev. B 96(18), 180201 (2017).
[Crossref]

P. Lalanne and P. Chavel, “Metalenses at visible wavelengths: past, present, perspectives,” Laser Photonics Rev. 11(3), 1600295 (2017).
[Crossref]

I. Staude and J. Schilling, “Metamaterial-inspired silicon nanophotonics,” Nat. Photonics 11(5), 274–284 (2017).
[Crossref]

A. Rahimzadegan, R. Alaee, I. Fernandez-Corbaton, and C. Rockstuhl, “Fundamental limits of optical force and torque,” Phys. Rev. B 95(3), 035106 (2017).
[Crossref]

M. Rahmani, L. Xu, A. E. Miroshnichenko, A. Komar, R. Camacho-Morales, H. Chen, Y. Zárate, S. Kruk, G. Zhang, D. N. Neshev, and Y. S. Kivshar, “Reversible thermal tuning of all-dielectric metasurfaces,” Adv. Funct. Mater. 27(31), 1700580 (2017).
[Crossref]

X. Zhang, H. Liu, Z. Zhang, Q. Wang, and S. Zhu, “Controlling thermal emission of phonon by magnetic metasurfaces,” Sci. Rep. 7(1), 41858 (2017).
[Crossref]

R. Alaee, M. Albooyeh, and C. Rockstuhl, “Theory of metasurface based perfect absorbers,” J. Phys. D: Appl. Phys. 50(50), 503002 (2017).
[Crossref]

I. Fernandez-Corbaton and C. Rockstuhl, “Unified theory to describe and engineer conservation laws in light-matter interactions,” Phys. Rev. A 95(5), 053829 (2017).
[Crossref]

M. I. Abdelrahman, C. Rockstuhl, and I. Fernandez-Corbaton, “Broadband suppression of backscattering at optical frequencies using low permittivity dielectric spheres,” Sci. Rep. 7(1), 14762 (2017).
[Crossref]

2016 (6)

M. Z. Alam, I. De Leon, and R. W. Boyd, “Large optical nonlinearity of indium tin oxide in its epsilon-near-zero region,” Science 352(6287), 795–797 (2016).
[Crossref]

A. Jouanin, J. P. Hugonin, and P. Lalanne, “Designer colloidal layers of disordered plasmonic nanoparticles for light extraction,” Adv. Funct. Mater. 26(34), 6215–6223 (2016).
[Crossref]

A. Ghanekar, L. Lin, and Y. Zheng, “Novel and efficient Mie-metamaterial thermal emitter for thermophotovoltaic systems,” Opt. Express 24(10), A868–A877 (2016).
[Crossref]

M. M. Hossain and M. Gu, “Radiative cooling: Principles, progress, and potentials,” Adv. Sci. 3(7), 1500360 (2016).
[Crossref]

J. Proust, F. Bedu, B. Gallas, I. Ozerov, and N. Bonod, “All-dielectric colored metasurfaces with silicon Mie resonators,” ACS Nano 10(8), 7761–7767 (2016).
[Crossref]

H.-T. Chen, A. J. Taylor, and N. Yu, “A review of metasurfaces: physics and applications,” Rep. Prog. Phys. 79(7), 076401 (2016).
[Crossref]

2015 (6)

Y. Ra’Di, C. Simovski, and S. Tretyakov, “Thin perfect absorbers for electromagnetic waves: theory, design, and realizations,” Phys. Rev. Appl. 3(3), 037001 (2015).
[Crossref]

M. Albooyeh, S. Kruk, C. Menzel, C. Helgert, M. Kroll, A. Krysinski, M. Decker, D. N. Neshev, T. Pertsch, C. Etrich, and R. Carsten, “Resonant metasurfaces at oblique incidence: interplay of order and disorder,” Sci. Rep. 4(1), 4484 (2015).
[Crossref]

P. Genevet and F. Capasso, “Holographic optical metasurfaces: a review of current progress,” Rep. Prog. Phys. 78(2), 024401 (2015).
[Crossref]

T. Geng, S. Zhuang, J. Gao, and X. Yang, “Nonlocal effective medium approximation for metallic nanorod metamaterials,” Phys. Rev. B 91(24), 245128 (2015).
[Crossref]

G. Zheng, H. Mühlenbernd, M. Kenney, G. Li, T. Zentgraf, and S. Zhang, “Metasurface holograms reaching 80% efficiency,” Nat. Nanotechnol. 10(4), 308–312 (2015).
[Crossref]

C. Guclu, V. A. Tamma, H. K. Wickramasinghe, and F. Capolino, “Photoinduced magnetic force between nanostructures,” Phys. Rev. B 92(23), 235111 (2015).
[Crossref]

2014 (2)

S. Tretyakov, “Maximizing absorption and scattering by dipole particles,” Plasmonics 9(4), 935–944 (2014).
[Crossref]

E. Karimi, S. A. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light: Sci. Appl. 3(5), e167 (2014).
[Crossref]

2013 (4)

X. Ni, A. V. Kildishev, and V. M. Shalaev, “Metasurface holograms for visible light,” Nat. Commun. 4(1), 2807 (2013).
[Crossref]

L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen, B. Bai, Q. Tan, G. Jin, K.-W. Cheah, C.-W. Qiu, J. Li, T. Zentgraf, and S. Zhang, “Three-dimensional optical holography using a plasmonic metasurface,” Nat. Commun. 4(1), 2808 (2013).
[Crossref]

B. Hopkins, W. Liu, A. E. Miroshnichenko, and Y. S. Kivshar, “Optically isotropic responses induced by discrete rotational symmetry of nanoparticle clusters,” Nanoscale 5(14), 6395–6403 (2013).
[Crossref]

X. Zambrana-Puyalto, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Dual and anti-dual modes in dielectric spheres,” Opt. Express 21(15), 17520–17530 (2013).
[Crossref]

2011 (3)

2010 (1)

2009 (2)

Q. Zhao, J. Zhou, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today 12(12), 60–69 (2009).
[Crossref]

M. Silveirinha, J. Baena, L. Jelinek, and R. Marques, “Nonlocal homogenization of an array of cubic particles made of resonant rings,” Metamaterials 3(3-4), 115–128 (2009).
[Crossref]

2008 (1)

M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophotonics 2(1), 021875 (2008).
[Crossref]

2006 (1)

M. Alam and Y. Massoud, “A closed-form analytical model for single nanoshells,” IEEE Trans. Nanotechnol. 5(3), 265–272 (2006).
[Crossref]

2005 (2)

J. Shen, “Algorithm of numerical calculation on lorentz Mie theory,” PIERS Online 1(6), 691–694 (2005).
[Crossref]

A. Moroz, “A recursive transfer-matrix solution for a dipole radiating inside and outside a stratified sphere,” Ann. Phys. 315(2), 352–418 (2005).
[Crossref]

2003 (1)

D. G. Grier, “A revolution in optical manipulation.,” Nature 424(6950), 810–816 (2003).
[Crossref]

2001 (1)

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Figures (7)

Fig. 1.
Fig. 1. General idea of our proposal: A minimal model to describe the electromagnetic response of spheres. The proposed Mie angles (right side of the figure) reduce the many degrees of freedom of an actual homogeneous or core-multishell sphere (left side of the figure) to only four Mie angles per multipolar order. These Mie angles express every possible Mie coefficient the scattering object may offer.
Fig. 2.
Fig. 2. Potential and actual Mie coefficient values: a) Here, all physically possible values for Mie coefficients in the complex plane based on the optical theorem (using Eq. (10)) are illustrated. For a non-absorbing sphere, the physically possible values are on the red circle. For an absorbing sphere, the physically possible values are within the blue area. The yellow dot in the center shows the point of critical coupling. b) Electric and c) magnetic dipolar Mie coefficient of homogeneous spheres embedded in the free space with discrete radius $R_s\in [0.45:0.005:0.75]\lambda$, discrete permittivity $\Re (\epsilon _r)\in [0:0.5:20]$, $\Im (\epsilon _r)\in [0:0.025:1]$ and permeability $\mu _r=1$ using Eq. (19). Each marker point shows an individual realization. The purpose is to show that indeed with homogeneous spheres in this parameter regime, every physically possible electric/magnetic dipolar Mie coefficient can be achieved. The remaining empty spaces in these figures remain because of a finite sampling of our data plots, we plot here only a discrete number of samples. The entire disk will be covered if finer data points are plotted.
Fig. 3.
Fig. 3. Optical cross sections as a function of the Mie angles: The normalized a) scattering, b) extinction, and c) absorption cross sections as a function of the Mie angles (i.e. $\theta _{\mathrm {E}j}$ and $\theta _{\mathrm {E}j}^{\prime }$) using Eqs. (11)–(13).
Fig. 4.
Fig. 4. Non-absorbing particle: The dipolar electric (a) and magnetic (b) Mie angles (i.e. $\theta _{\mathrm {E1}}$ and $\theta _{\mathrm {M1}}$) for a non-absorbing and non-magnetic sphere embedded in the free space as a function of sphere radius $R_s$ and $\Re (\epsilon _r)$. Derived from the calculated Mie coefficients using Eq. (19) and Eq. (10).
Fig. 5.
Fig. 5. Absorbing particle: The dipolar electric and magnetic Mie angles (i.e. $\theta _{\mathrm {E1}}$ and $\theta _{\mathrm {M1}}$) for an absorbing sphere with a normalized radius of $R_s/\lambda =0.25$ as a function of the real and imaginary part of the sphere’s material. Derived from the calculated Mie coefficients using Eqs. (19) and Eq. (10).
Fig. 6.
Fig. 6. Maximum absorption cross section of a trimer: a) The full map of the normalized absorption cross section of an equilateral trimer made from isotropic electric dipoles with a side length of $d=0.5\lambda$ as a function of the Mie angles of the individual particle. The polarization and direction of the excitation are shown in the inset of part (b). b) The maximum absorption cross section of the trimer for any given distance. The global maximal absorption cross section occurs at a distance of $d=0.725\lambda$ and the required Mie angles are: $\theta _{\mathrm {E1}}=4^o,\,\theta _{\mathrm {E}1}^{\prime }=34.5^o$ (i.e. $a_1=0.59+0.02\mathrm {i}$). The inset figure shows the trimer with the assumed plane wave excitation polarization. c) The maximal absorption cross section for an optimized ITO trimer at a wavelength of $\lambda =$1740 nm designed based on the theoretical results derived from Fig. 6(b). The radius of the ITO spheres in the trimer is 179 nm.
Fig. 7.
Fig. 7. Maximum possible absorption cross section of three multipolar trimers: a) The maximum absorption cross section of three different equilateral trimers made from isotropic electric dipoles, shown with legend $\mathrm {E}_{1}$, isotropic electric quadrupoles, shown with legend $\mathrm {E}_{2}$, and isotropic electric dipoles and quadrupoles, shown with the legend $\mathrm {E}_{1,2}$ for any given distance. The plane wave excitation polarization is the same as in Fig. 6. The dotted lines of the same color show the maximum theoretical absorption cross section of the trimer, if no coupling is considered, i.e. at a distance of infinity. For $\mathrm {E}_{1}$: $3\times \frac {3}{4}\times \frac {\lambda ^2}{2\pi }$. For $\mathrm {E}_{2}$: $3\times \frac {5}{4}\times \frac {\lambda ^2}{2\pi }$. For $\mathrm {E}_{1,2}$: $3\times (\frac {3}{4}+\frac {5}{4})\times \frac {\lambda ^2}{2\pi }$. b) The required Mie angles to maximize the absorption cross section of the trimer made out of three electric dipoles, c) three electric quadrupole, and, d) three electric dipole - electric quadrupoles. Note that, whenever not mentioned, the other multipoles are assumed to be zero.

Equations (23)

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C s c a = λ 2 2 π j = 1 ( 2 j + 1 ) ( | a j | 2 + | b j | 2 ) ,
C e x t = λ 2 2 π j = 1 ( 2 j + 1 ) ( a j + b j ) ,
C a b s = λ 2 2 π j = 1 ( 2 j + 1 ) [ ( a j + b j ) ( | a j | 2 + | b j | 2 ) ] ,
C s c a , E j = ( 2 j + 1 ) λ 2 2 π | a j | 2 ,
C e x t , E j = ( 2 j + 1 ) λ 2 2 π ( a j ) ,
C a b s , E j = ( 2 j + 1 ) λ 2 2 π [ ( a j ) | a j | 2 ] .
( a j ) = | a j | 2 .
a j = cos ( θ E j ) exp ( i θ E j ) ,
C e x t , E j C s c a , E j ( a j ) | a j | 2 cos θ E j | a j | .
a j = 1 1 cos θ E j exp i θ E j + tan θ E j = 1 1 i tan θ E j + tan θ E j ,
C s c a , E j = ( 2 j + 1 ) λ 2 2 π 1 ( 1 + tan θ E j ) 2 + tan 2 θ E j ,
C e x t , E j = ( 2 j + 1 ) λ 2 2 π 1 + tan θ E j ( 1 + tan θ E j ) 2 + tan 2 θ E j ,
C a b s , E j = ( 2 j + 1 ) λ 2 2 π tan θ E j ( 1 + tan θ E j ) 2 + tan 2 θ E j .
α = 6 π k 0 3 γ r / 2 δ + ( γ n r + γ r ) i / 2 ,
1 a 1 = 2 δ γ r i + γ n r γ r + 1.
tan θ E 1 = 2 δ γ r , tan θ E 1 = γ n r γ r .
a n = i sin α n exp ( i α n ) , b n = i sin β n exp ( i β n ) ,
α n = π 2 θ E j , β n = π 2 θ M j ,
a n = μ e η 2 j n ( η x ) [ x j n ( x ) ] μ s j n ( x ) [ η x j n ( η x ) ] μ e η 2 j n ( η x ) [ x h n ( 1 ) ( x ) ] μ s h n ( 1 ) ( x ) [ η x j n ( η x ) ] , b n = μ s j n ( η x ) [ x j n ( x ) ] μ e j n ( x ) [ η x j n ( η x ) ] μ s j n ( η x ) [ x h n ( 1 ) ( x ) ] μ e h n ( 1 ) ( x ) [ η x j n ( η x ) ] ,
x = ω c ϵ e ( ω ) μ e ( ω ) R s , η = ϵ s ( ω ) μ s ( ω ) ϵ e ( ω ) μ e ( ω ) ,
a j = P j e P j e + i Q j e , b j = P j m P j m + i Q j m ,
1 a j = 1 + i Q j e P j e = 1 + i ( Q j e P j e ) ( Q j e P j e ) .
( Q j e P j e ) = tan θ E j , ( Q j e P j e ) = tan θ E j .