Abstract
Maxwell’s equations are solved when the amplitude and phase of the electromagnetic field are determined at all points in space. Generally, the Stokes parameters can only capture the amplitude and polarization state of the electromagnetic field in the radiation (far) zone. Therefore, the measurement of the Stokes parameters is, in general, insufficient to solve Maxwell’s equations. In this Letter, we solve Maxwell’s equations for a set of objects widely used in Nanophotonics using the Stokes parameters alone. These objects are lossless, axially symmetric, and well described by a single multipolar order. Our method for solving Maxwell’s equations endows the Stokes parameters an even more fundamental role in the electromagnetic scattering theory.
Keywords: Electromagnetism, Nanophotonics, Scattering Theory, Polarization of Light
The determination of the amplitude and phase of the electromagnetic field at all points in space solves Maxwell’s equations.1 Currently, electromagnetic software packages can provide the numerical solution to Maxwell’s equations.2 However, solving Maxwell’s equations in the optical laboratory is nearly infeasible. One needs to measure the components of the scattered field at all points of the radiation zone. On top of that, the internal field induced in the excited object is experimentally inaccessible. In stark contrast, the Stokes parameters can be readily measured using a photodiode and waveplates.3,4 The Stokes parameters can capture the amplitude and polarization state of the electromagnetic field in the radiation (far) zone. Following the notation of ref (4), we can write the Stokes parameters as
 |
1 |
 |
2 |
As eqs 1 and 2 show, the Stokes parameters
depend
on the transversal components of the scattered electromagnetic field
evaluated in the radiation (far) zone, i.e., Eθ and Eφ. (Note that
the longitudinal component of the scattered electromagnetic fields
identically vanishes in far-field, namely, Er = 0.) If the amplitude and phase of Eθ and Eφ are measured, the Stokes parameters can be calculated using eqs 1 and 2.5,6 The converse is not generally true. To demonstrate
this fact, we now write 
and 
, where ξθ and ξφ are real-valued phases. Taking into
account this notation,
we can obtain the following relations from eqs 1 and 2
 |
3 |
 |
4 |
On the one hand, eq 3 reveals that the measurement of s0 and s1 grants access to the amplitudes |Eθ| and |Eφ|. On the other hand, eq 4 shows that measuring s2 and s3 provides the phase difference ξθ – ξφ but falls short of providing the individual phases ξθ and ξφ. The determination of the phase of the scattered field is necessary to solve Maxwell’s equations. Therefore, one could conclude that measuring the Stokes parameters is insufficient to solve Maxwell’s equations.
In this Letter, we demonstrate that a measurement of the Stokes parameters at a single scattering angle is sufficient to solve Maxwell’s equations for a set of objects. These objects share the following features: they are lossless, axially symmetric and their optical response is well-described by a single multipole order. Notably, several works have tackled such objects in different branches of Nanophotonics. Examples include optically resonant nanoantennas,7−11 Kerker conditions,12−15 surface-enhanced Raman scattering,16 surface-enhanced optical chirality,17,18 among many others.19−22 Note that refs (7−22) are experimental studies widely recognized by the Nanophotonics community.
The key to our procedure lies in linking the measurement of the Stokes parameters in the radiation (far) zone with the electric and magnetic scattering coefficients of the multipolar expansion of the scattered field. As we show, the determination of these coefficients solves Maxwell’s equations at all points of the radiation zone, ranging from far-to-near field. Additionally, in the case of spherical objects, we solve Maxwell’s equations at all points in space (also inside the object).
We now consider the scattered Esca(kr) and internal Eint(kr) electromagnetic fields produced by an arbitrary object. In the usual basis of electric and magnetic multipoles,23 we can write the scattered and internal electromagnetic fields as
 |
5 |
 |
6 |
Here 
and 
, 
and 
are spherical Hankel and Bessel functions
of the first kind, respectively, 
(θ,
φ) represents the usual
vector spherical harmonics,23
, where s = {j, h}, and 
and m denote the multipolar
order and the total angular momentum, respectively. Moreover r = {r, θ, φ} is the observational
point, k is the radiation wavenumber, ki = mk, m being the
refractive index of the object, and E0 is the amplitude of the incident wavefield. Importantly, 
and 
denote the electric and magnetic
scattering
coefficients, respectively, and 
and 
are the internal electric and
magnetic
coefficients, respectively.
Equations 5 and 6 show that the
determination of the set {
, 
, 
, 
} grants access to the amplitude
and phase
of both the scattered and internal electromagnetic fields at all points
of space. However, capturing the set {
, 
, 
, 
} is exceptionally demanding due
to the
need to measure the components of the scattered and internal electromagnetic
fields in all directions.23 In fact and
to the best of our knowledge, none of the magnitudes of the set has
been experimentally measured.
As previously mentioned, the Stokes
vector S = {s0, s1, s2, s3} can be measured in
the radiation (far) zone with conventional optical components such
as a photodiode and waveplates. Hereafter, we consider objects well-described
by fixed values of m and 
. In other
words, we deal with axially symmetric
objects whose optical response is described by a single multipolar
order. We recall that such objects have been widely studied in Nanophotonics.7−20,20−22 In this setting
(
is fixed), let us insert the far-field
limit (kr → ∞) of eq 5 into eqs 1 and 2). After some
algebra (see Supporting Information S1),
we get24
 |
7 |
Equation 7 shows that all the quadratic combinations
of {
, 
} can be attained
from a single Stokes vector
measurement in the far-field. As proved in ref (24), all that one needs to
do is compute the 4 × 4 matrix 
and apply it
to the Stokes measurement.
However, we anticipate that even if the object is well-described by
fixed values of m and 
, the phases
of 
and 
cannot be attained
using eq 7. To prove
it, we now write 
and 
, where ϕa and ϕb are real-valued phases.
In this setting, the last two rows of the left side of eq 7 can be manipulated to yield
 |
8 |
Equation 8 provides access to the phase difference ϕa – ϕb but
not to the individual phases ϕa and
ϕb. Consequently, without knowledge
of these individual phases, determining the scattering coefficients 
and 
becomes impossible. Due to this
infeasibility,
we cannot capture the scattered field Esca(kr) using eq 7, and the solution to Maxwell’s equations
is not attained.
In the forthcoming, we show that the phase-indetermination of eq 8 is resolved if the light-scattering system is lossless.
We now consider the extinction and scattering cross sections, denoted by σext = σeext + σmext and σsca = σesca + σmsca, respectively. Note that here e and m denote the electric and magnetic contributions, respectively. The extinction and scattering cross sections can be written as25
 |
9 |
 |
10 |
Here 
and 
denote the beam-shape coefficients
of the
incident wavefield.26 Objects without optical
losses satisfy σext = σsca. According
to eqs 9 and 10, lossless objects fulfill
 |
11 |
Equation 11 shows that the amplitude of the electric (and magnetic) scattering coefficient is a function of the electric (and magnetic) phase.27−29 That noted, by expanding eq 11 and manipulating eq 8, we arrive to
 |
12 |
 |
13 |
 |
14 |
We now reach notable results. The system of eqs 12–14 can be unambiguously solved yielding ϕa and ϕb (in the
correct quadrant). Note that the right-side of eqs 12–14 can be obtained
using eq 7. Moreover,
the beam-shape coefficients 
are usually
known quantities since the
incident electromagnetic field can be controlled. Now, capturing ϕa and ϕb along with the amplitudes |
| and |
| allows us to determine the electric
and
magnetic scattering coefficients 
and 
. As eq 5 shows, the determination of 
and 
grants access to all the components
of
the scattered field evaluated at all points of the radiation zone,
ranging from far-to-near-field. In simple words, the determination
of 
and 
solves Maxwell’s equations
in the
radiation zone.
At this point, let us highlight the main features of our method for solving Maxwell’s equations in the radiation zone:
Simplicity in the measurement: Our method relies on a measurement of the Stokes parameters at a single angle. From an experimental standpoint, we must avoid any propagation direction where the incident wavefield has a component. Otherwise, the total field measured in the optical laboratory will be the sum of the scattered and incident wavefields, thus invalidating our method.
Generality of the material and shape of the object: Our method works for axially symmetrical objects such as disks, pillars, spheres, and spheroids. Note that these objects can be composed of a single material or not (coated objects). The only requirement is that the absorption cross-section of such objects must be zero.
Wide range of illumination conditions: Our method can accommodate plane waves, Gaussian beams,36 or even vortex wavefields.26,30 The only requirement is that the total angular momentum m of the light-scattering system must be well-defined.
At this point, let us illustrate the relevance of eqs 12–14 with one of the most canonical examples used in Nanophotonics: all-dielectric spherical nanoparticles excited by a plane wave. For details on the beam-shape coefficients of a plane wave, check Supporting Information S2. Additionally, check Supporting Information S3 to learn how eqs 12–14 simplify for spherical and lossless particles.
First,
we illustrate the accuracy of our method to capture a1 and b1. In Figure 1, we depict 
, 
, 
, and 
calculated from a Stokes measurement at
the scattering angle θ = 90° and utilizing Mie theory (exact
solution). Note that φ does not play a role due to the symmetries
of the light-scattering system. Additionally, notice that other scattering
angles θ could have been selected. The calculation of the Mie
coefficients obtained from the Stokes measurement shows an excellent
agreement with the exact solution in the broadband interval of refractive
index contrasts 2 < m < 4 and optical sizes 0.6 < x < 1. Here x = ka = 2πa/λ,
λ and a being the radiation wavelength and the radius of the
object, respectively. To get a deeper insight, the percentage relative
error between the exact solution and our novel procedure, as summarized
in eqs 12–14, is shown Figure 1 i-l). As previously stated, the overall agreement
is remarkable. Note that the significant error in Figure 1) arises since our novel procedure
captures 
at slightly different spectral
points compared
to the exact solution.
Figure 1.
Real and imaginary parts of the dipolar electric and magnetic Mie coefficients obtained from both a Stokes measurement at θ = 90° (see a–d) and using exact Mie theory (see e–h). The excitation wavefield is a circularly polarized plane wave in both cases. The real and imaginary parts of the Mie coefficients are depicted vs the refractive index contrast m and the optical size x = ka = (2πa)/λ, λ and a being the radiation wavelength and the radius of the spherical nanoparticle. The intense red colors indicate the Mie resonances. The percentage relative error (i–l) calculated from using Mie theory and the Stokes measurement at θ = 90° is also shown to gain further insights.
Note that for 0 < x < 0.6,
our approach
works as it holds for objects described by an electric and/or magnetic
response. We stress that the scattering Mie coefficients 
and 
do not depend on the incident illumination.
Therefore, once we determine 
and 
using eqs 12–14 for a specific illumination,
such as a plane wave, we can subsequently explore the scattering features
of the spherical object under general illumination conditions.
Interestingly, the dipolar Mie coefficients are biunequivocally determined by the electric and magnetic polarizabilities, often denoted as αE and αM, respectively. For the sake of clarity, let us write the correspondence between Mie coefficients and polarizabilities31
 |
15 |
Equation 15 can be calculated from eqs 12–14, and thus, our Stokes method can be employed to retrieve the electric and magnetic polarizabilities when dealing with dipolar Mie objects.
At this point, we show the accuracy of our method to solve the Maxwell equations in the radiation zone with a realistic material. Particularly, we consider a Gallium Phosphide (GaP) nanoparticle of radius a = 75 nm excited by a circularly polarized plane wave.11 We select GaP as it is a material with high potential for metasurface-based devices operating across the visible, as it presents a high-refractive index (m > 3.3) and negligible losses.32
Figure 2a-d
shows 
, 
, 
, and 
calculated from a Stokes measurement at
θ = 90° and θ = 60° and employing Mie theory.
The scattering Mie coefficients calculated from the Stokes measurements
at the specified angles exhibit excellent agreement with the exact
calculations (and with each other) in the range 475 nm < λ
< 700 nm.
Figure 2.
Real and imaginary parts of the scattering (see a–d) and internal (see e–h) Mie coefficients of a GaP spherical object with radii a = 75 nm obtained from both a Stokes measurement at θ = 90° (dashed) and θ = 60° (dotted). The excitation wavefield is a circularly polarized plane wave in all cases. The scattering and internal Mie coefficients are depicted vs the incident wavelength λ.
It is worth noting that the results obtained from
the Stokes vector
measurement exhibit slight deviations from each other (and from the
exact result) at shorter wavelengths, specifically in the range 450
nm < λ < 475 nm. This deviation arises because, in this
wavelength range, the scattering cannot be fully described by 
= m = 1 due to the presence
of the magnetic quadrupole. In addition to this, GaP presents non-negligible
optical losses for λ < 475 nm.33 Due to these facts, our method, as summarized in eqs 12–14, cannot be applied for shorter wavelengths λ < 475 nm since
the setting we assumed—a lossless dipolar object—is
not fulfilled.
Importantly, our Stokes method robustly detects that the setting we assumed does not fully hold as the curves for θ = 90° and θ = 60° deviate in the range 450 nm < λ < 475 nm.
We now summarized the scope of the method. Our Stokes
method is
reliable if the calculated coefficients remain identical regardless
of the scattering angle θ of the collection. In this scenario,
one can experimentally retrieve the full solution of Maxwell equations.
If such coefficients differ, then the scattering cannot be fully described
by the selected values of 
and m in the 
matrix or/and
the light-scattering system
is not lossless. In other words, the Stokes method is robust and can
be trusted upon measuring the Stokes parameters at different scattering
angles of collection.
In Supporting Information S5, we discuss the role of optical losses and higher multipolar orders in the accuracy of the Stokes method. In particular, we consider a different GaP nanosphere at shorter wavelengths, where losses are high.
Next, we show that the determination of the scattering Mie coefficients grants access to the internal Mie coefficients.
In 1908, Gustav
Mie solved the scattering of a plane wave by a
spherical object.34 Specifically, Mie determined
the scattering {
, 
} and internal coefficients {
, 
}. The relation between {
, 
} {
, 
} and can be compactly written as4
 |
16 |
Equation 16 shows that the
internal Mie coefficients {
, 
} can be determined from the scattering
coefficients {
, 
}. Thus, the internal Mie coefficients can
be captured using eqs 12–14 particularized for spherical particles.
In Figure 2, we
plot 
, 
, 
, and 
using eq 16. For this
calculation, we have employed a1 and b1, which, in turn,
have been previously obtained using eqs 12–14 at θ
= 90° and θ = 60°. As could be expected, the calculation
of the internal Mie coefficients shows an excellent agreement with
the exact calculation in the wavelength interval 475 nm < λ
< 700 nm.
As mentioned in the introduction, the determination of internal and scattering coefficients gives rise to the exact solution to Maxwell’s equations. Since every electromagnetic physical magnitude originates from the electromagnetic field, we can also access, for instance, the exact internal dipolar moments, denoted as p and m in ref (35).
In conclusion, we have presented a method that solves Maxwell’s equations at all points in the radiation zone based on a single measurement of the Stokes parameters in the far-field. We have illustrated the accuracy of our method with one of the most studied systems in Nanophotonics: a spherical nanoparticle excited by a plane wave. In this setting, we have also determined the internal Mie coefficients, solving Maxwell’s equations at all points in the space.
To the best of our knowledge, this study represents the first method capable of solving Maxwell’s equations experimentally and from a measurement of the Stokes parameters. This feature endows the Stokes parameters, mostly used to get insight into the polarization state of the electromagnetic radiation, an even more fundamental role in the electromagnetic scattering theory. Consequently, our findings, supported by analytical theory and exact numerical simulations, can find applications in all branches of Nanophotonics and Optics.
Acknowledgments
J.O-T. acknowledges Adrian Juan-Delgado and Dr. Cristina Sanz-Fernández for useful comments.
Supporting Information Available
The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.nanolett.4c01976.
S1: The Stokes method used to derive eq 7. S2: The beam shape coefficients of a plane wave. S3: The Stokes method applied to a magnetodielectric spherical nanoparticle. S4: The role of optical losses and higher multipolar orders in the accuracy of solving Maxwell’s equations from the Stokes method (PDF)
J.O.-T. acknowledges support from the Juan de la Cierva fellowship No. FJC2021–047090-I of MCIN/AEI/10.13039/501100011033 and NextGenerationEU/PRTR and acknowledges financial support from the Spanish Ministry of Science and Innovation (MCIN), AEI and FEDER (UE) through projects PID2022–137569NB-C43 and PID2022–143268NB-I00.
The author declares no competing financial interest.
Supplementary Material
References
- Maxwell J. C. VIII A dynamical theory of the electromagnetic field. Philosophical transactions of the Royal Society of London 1865, 459–512. 10.5479/sil.423156.39088007130693. [DOI] [Google Scholar]
- Multiphysics C.Introduction to COMSOL multiphysics®; COMSOL Multiphysics: Burlington, MA, 1998; Vol. 9, p 32. [Google Scholar]
- Stokes G. G. On the composition and resolution of streams of polarized light from different sources. Transactions of the Cambridge Philosophical Society 1851, 9, 399. [Google Scholar]
- Bohren C. F.; Huffman D. R.. Absorption and scattering of light by small particles; John Wiley & Sons, 2008. [Google Scholar]
- Crichton J. H.; Marston P. L. The measurable distinction between the spin and orbital angular momenta of electromagnetic radiation. Electronic Journal of Differential Equations 2000, 4, 37–50. [Google Scholar]
- Marston P. L. Humblet’s angular momentum decomposition applied to radiation torque on metallic spheres using the Hagen–Rubens approximation. J. Quant. Spectrosc. Radiat. Transfer 2018, 220, 97–105. 10.1016/j.jqsrt.2018.08.019. [DOI] [Google Scholar]
- Kuznetsov A. I.; Miroshnichenko A. E.; Fu Y. H.; Zhang J.; Luk’Yanchuk B. Magnetic light. Sci. Rep. 2012, 2, 492. 10.1038/srep00492. [DOI] [PMC free article] [PubMed] [Google Scholar]
- Evlyukhin A. B.; Novikov S. M.; Zywietz U.; Eriksen R. L.; Reinhardt C.; Bozhevolnyi S. I.; Chichkov B. N. Demonstration of magnetic dipole resonances of dielectric nanospheres in the visible region. Nano Lett. 2012, 12, 3749–3755. 10.1021/nl301594s. [DOI] [PubMed] [Google Scholar]
- Sheikholeslami S. N.; Alaeian H.; Koh A. L.; Dionne J. A. A metafluid exhibiting strong optical magnetism. Nano Lett. 2013, 13, 4137–4141. 10.1021/nl401642z. [DOI] [PubMed] [Google Scholar]
- Zywietz U.; Evlyukhin A. B.; Reinhardt C.; Chichkov B. N. Laser printing of silicon nanoparticles with resonant optical electric and magnetic responses. Nat. Commun. 2014, 5, 3402. 10.1038/ncomms4402. [DOI] [PubMed] [Google Scholar]
- Shima D.; Sugimoto H.; Assadillayev A.; Raza S.; Fujii M. Gallium Phosphide Nanoparticles for Low-Loss Nanoantennas in Visible Range. Advanced Optical Materials 2023, 11, 2203107. 10.1002/adom.202203107. [DOI] [Google Scholar]
- Geffrin J.-M.; García-Cámara B.; Gómez-Medina R.; Albella P.; Froufe-Pérez L. S.; Eyraud C.; Litman A.; Vaillon R.; González F.; Nieto-Vesperinas M.; Sáenz J. J.; Moreno F. Magnetic and electric coherence in forward-and back-scattered electromagnetic waves by a single dielectric subwavelength sphere. Nat. Commun. 2012, 3, 1171. 10.1038/ncomms2167. [DOI] [PubMed] [Google Scholar]
- Fu Y. H.; Kuznetsov A. I.; Miroshnichenko A. E.; Yu Y. F.; Luk’yanchuk B. Directional visible light scattering by silicon nanoparticles. Nat. Commun. 2013, 4, 1527. 10.1038/ncomms2538. [DOI] [PubMed] [Google Scholar]
- Staude I.; Miroshnichenko A. E.; Decker M.; Fofang N. T.; Liu S.; Gonzales E.; Dominguez J.; Luk T. S.; Neshev D. N.; Brener I.; et al. Tailoring directional scattering through magnetic and electric resonances in subwavelength silicon nanodisks. ACS Nano 2013, 7, 7824–7832. 10.1021/nn402736f. [DOI] [PubMed] [Google Scholar]
- Person S.; Jain M.; Lapin Z.; Sáenz J. J.; Wicks G.; Novotny L. Demonstration of zero optical backscattering from single nanoparticles. Nano Lett. 2013, 13, 1806–1809. 10.1021/nl4005018. [DOI] [PubMed] [Google Scholar]
- Dmitriev P. A.; Baranov D. G.; Milichko V. A.; Makarov S. V.; Mukhin I. S.; Samusev A. K.; Krasnok A. E.; Belov P. A.; Kivshar Y. S. Resonant Raman scattering from silicon nanoparticles enhanced by magnetic response. Nanoscale 2016, 8, 9721–9726. 10.1039/C5NR07965A. [DOI] [PubMed] [Google Scholar]
- Negoro H.; Sugimoto H.; Fujii M. Helicity-Preserving Optical Metafluids. Nano Lett. 2023, 23, 5101–5107. 10.1021/acs.nanolett.3c01026. [DOI] [PubMed] [Google Scholar]
- Olmos-Trigo J.; Sugimoto H.; Fujii M. Far-Field Detection of Near-Field Circular Dichroism Enhancements Induced by a Nanoantenna. Laser & Photonics Reviews 2024, 18, 2300948. 10.1002/lpor.202300948. [DOI] [Google Scholar]
- Chaabani W.; Proust J.; Movsesyan A.; Béal J.; Baudrion A.-L.; Adam P.-M.; Chehaidar A.; Plain J. Large-scale and low-cost fabrication of silicon mie resonators. ACS Nano 2019, 13, 4199–4208. 10.1021/acsnano.8b09198. [DOI] [PubMed] [Google Scholar]
- Ishii S.; Chen K.; Okuyama H.; Nagao T. Resonant optical absorption and photothermal process in high refractive index germanium nanoparticles. Advanced Optical Materials 2017, 5, 1600902. 10.1002/adom.201600902. [DOI] [Google Scholar]
- Sugimoto H.; Fujii M. Colloidal Dispersion of Subquarter Micrometer Silicon Spheres for Low-Loss Antenna in Visible Regime. Advanced Optical Materials 2017, 5, 1700332. 10.1002/adom.201700332. [DOI] [Google Scholar]
- Sugimoto H.; Okazaki T.; Fujii M. Mie resonator color inks of monodispersed and perfectly spherical crystalline silicon nanoparticles. Advanced Optical Materials 2020, 8, 2000033. 10.1002/adom.202000033. [DOI] [Google Scholar]
- Jackson J. D.Classical Electrodynamics; John Wiley & Sons: New York, 1999. [Google Scholar]
- Olmos-Trigo J.The Stokes Vector Measurement: A Paradigm Shift in Electric-Magnetic Light Distinction. arXiv preprint arXiv:2310.17946 2023. [Google Scholar]
- Mishchenko M. I.; Travis L. D.; Lacis A. A.. Scattering, absorption, and emission of light by small particles; Cambridge university press, 2002. [Google Scholar]
- Zambrana-Puyalto X.; Vidal X.; Molina-Terriza G. Excitation of single multipolar modes with engineered cylindrically symmetric fields. Opt. Express 2012, 20, 24536–24544. 10.1364/OE.20.024536. [DOI] [PubMed] [Google Scholar]
- Hulst H. C.; van de Hulst H. C.. Light scattering by small particles; Courier Corporation, 1957. [Google Scholar]
- Olmos-Trigo J.; Abujetas D. R.; Sanz-Fernández C.; Zambrana-Puyalto X.; de Sousa N.; Sánchez-Gil J. A.; Sáenz J. J. Unveiling dipolar spectral regimes of large dielectric Mie spheres from helicity conservation. Physical Review Research 2020, 2, 043021. 10.1103/PhysRevResearch.2.043021. [DOI] [Google Scholar]
- Olmos-Trigo J.; Zambrana-Puyalto X. Helicity Conservation for Mie Optical Cavities. Physical Review Applied 2022, 18, 044007. 10.1103/PhysRevApplied.18.044007. [DOI] [Google Scholar]
- Sanz-Fernández C.; Molezuelas-Ferreras M.; Lasa-Alonso J.; de Sousa N.; Zambrana-Puyalto X.; Olmos-Trigo J. Multiple Kerker anapoles in dielectric microspheres. Laser Photonics Rev. 2021, 15, 2100035. 10.1002/lpor.202100035. [DOI] [Google Scholar]
- Zambrana-Puyalto X.; Vidal X.; Juan M. L.; Molina-Terriza G. Dual and anti-dual modes in dielectric spheres. Opt. Express 2013, 21, 17520–17530. 10.1364/OE.21.017520. [DOI] [PubMed] [Google Scholar]
- García-Etxarri A.; Gómez-Medina R.; Froufe-Pérez L. S.; López C.; Chantada L.; Scheffold F.; Aizpurua J.; Nieto-Vesperinas M.; Sáenz J. J. Strong magnetic response of submicron silicon particles in the infrared. Opt. Express 2011, 19, 4815–4826. 10.1364/OE.19.004815. [DOI] [PubMed] [Google Scholar]
- Baranikov A. V.; Khaidarov E.; Lassalle E.; Eschimese D.; Yeo J.; Loh N. D.; Paniagua-Dominguez R.; Kuznetsov A. I. Large Field-of-View and Multi-Color Imaging with GaP Quadratic Metalenses. Laser & Photonics Reviews 2024, 18, 2300553. 10.1002/lpor.202300553. [DOI] [Google Scholar]
- Khmelevskaia D.; Markina D.; Fedorov V.; Ermolaev G.; Arsenin A.; Volkov V.; Goltaev A.; Zadiranov Y. M.; Tzibizov I.; Pushkarev A. Directly grown crystalline gallium phosphide on sapphire for nonlinear all-dielectric nanophotonics. Appl. Phys. Lett. 2021, 10.1063/5.0048969. [DOI] [Google Scholar]
- Mie G. Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen. Annalen der physik 1908, 330, 377–445. 10.1002/andp.19083300302. [DOI] [Google Scholar]
- Alaee R.; Rockstuhl C.; Fernandez-Corbaton I. An electromagnetic multipole expansion beyond the long-wavelength approximation. Opt. Commun. 2018, 407, 17–21. 10.1016/j.optcom.2017.08.064. [DOI] [Google Scholar]
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