# Nonlinear circular dichroism in Mie-resonant nanoparticle dimers

Kristina Frizyuk<EMAIL_ADDRESS>Department of Physics and
Engineering, ITMO University, St Petersburg 197101, Russia  Elizaveta Melik-
Gaykazyan  Nonlinear Physics Centre, Research School of Physics, Australian
National University, Canberra ACT 2601, Australia  Jae-Hyuck Choi  Department
of Physics, Korea University, Seoul 02841, Republic of Korea  Mihail Petrov
Department of Physics and Engineering, ITMO University, St Petersburg 197101,
Russia  Hong-Gyu Park  Department of Physics, Korea University, Seoul 02841,
Republic of Korea  KU-KIST Graduate School of Converging Science and
Technology, Korea University, Seoul 02841, Republic of Korea  Yuri Kivshar
<EMAIL_ADDRESS>Nonlinear Physics Centre, Research School of Physics,
Australian National University, Canberra ACT 2601, Australia

###### Abstract

We study nonlinear response of a dimer composed of two identical Mie-resonant
dielectric nanoparticles illuminated normally by a circularly polarized light.
We develop a general theory describing hybridization of multipolar modes of
the coupled nanoparticles, and reveal nonvanishing nonlinear circular
dichroism (CD) in the second-harmonic generation (SHG) signal enhanced by the
multipolar resonances in the dimer provided its axis is oriented under an
angle to the crystalline lattice of the dielectric material. We present
experimental results for this SHG-CD effect obtained for the AlGaAs dimers
placed on an engineered substrate which confirm the basic prediction of our
general multipolar hybridization theory.

Many phenomena in nature, including multiple biochemical processes, are
governed by the fundamental property of chirality. An object is called chiral
when its mirror image cannot be superimposed with the original object, and
many examples of chirality can be found at all scales in nature, from
organisms to biomolecules and amino acids, which often occur only in one
handedness. Circular dichroism spectroscopy was suggested as a powerful
optical technique for the study of chiral materials and molecules. It gives
access to an enantioselective signal based on the differential absorption of
right and left circularly polarized light [1, 2, 3, 4].

In natural media, chiral effects are generally weak, and chiral plasmonic
structures [5, 6, 7, 8] and chiral metamaterials [9, 10, 11] were suggested as
a new tool for achieving strong chiroptical responses. Chiroptical activity is
commonly quantified as a linear effect being characterized in terms of the so-
called circular dichroism (CD). Most of the studies of CD effects in
metamaterials have been focused on the scattering properties including
reflection and transmission, and such effects attracted considerable attention
due to their potential applications for biological sensing [12], spin-
sensitive resonant transmission [13], and chiral meta-holograms [14]. Also,
chiroptical effects have been studied in systems of a few reconfigurable
chiral meta-molecules using metallic and dielectric particles as artificial
building blocks [15, 16, 17, 18].

Circularly polarised light interacting with resonances of nanostructures in
the nonlinear regime and nonlinear chiral metasurfaces can become extremely
sensitive to slight asymmetries and thus give rise to a circular dichroic
signal that is order of magnitude higher with respect to the same signal
obtained with the fundamental frequency [19, 20, 21, 22, 23, 24]. One of the
first studies of nonlinear circular dichroism in the second-harmonic
generation (SHG) signal was reported for chiral nanostructures consisting of
$G$-shaped elements made of gold [25, 26], and most recently for the third-
harmonic generation in a planar system of shifted gold bars [27]. It was shown
that in such plasmonic structures the nonlinear signal originates from “hot
spots” being dependant on the handedness of the nanostructures [28].

Although metal nanostructures were explored first for nonlinear chiral
nanophotonics, high-index dielectric nanoparticles can sustain strong magnetic
resonances and higher-order Mie-type modes in the visible and near-infrared
frequency range [29], and they are highly suitable for nonlinear nanophotonics
[30, 31, 32, 33] and nonlinear chiral metasurfaces [34] underpinning a strong
nonlinear multipolar response, in contrast to the metal structures where the
resonant fields are tightly confined to the surface.

Semiconductor compounds and alloys of the III-V group such as AlGaAs often
exhibit a large second-order susceptibility because of a lack of inversion
symmetry of their crystal lattice. Consequently, efficient SHG has been
observed in isolated semiconductor nanoantennas, being enhanced by Mie
resonances, optical anapoles, or bound states in the continuum [35, 36, 37,
38, 39, 40].

Figure 1: (a) Concept of nonlinear circular dichroism in Mie-resonant
nanoparticle dimers. Dimer axis is parallel to the $x$-axis, $\beta$ \- angle
between AlGaAs crystalline axis [100] and the $x$-axis. (b) Asymmetric SHG
driven by a circularly polarized light is generated from a dimer with an
arbitrarily oriented crystalline lattice. For other orientations such as
$[100]||x$ and $[110]||x$, nonlinear circular dichroism is not observed.

In this Letter, we develop the general hybridization theory for multipolar
Mie-resonant modes and then study nonlinear response of a nanoparticle dimer
(shown in Fig. 1 (a)) illuminated normally by a circularly polarized light. We
observe, for the first time to our knowledge, that nonvanishing circular
dichroism of the second-harmonic (SH) signal may occur due to the interplay of
multipolar magnetic and electric Mie resonances provided the dimer axes is
oriented under an angle to the crystalline lattice of the dielectric material
(Fig. 1 (b)). We present a general formalism and modal theory to explain the
origin of this unexpected phenomenon, and also verify our general multipolar
hybridization theory in the experiment for the Mie-resonant SHG from the
AlGaAs dimers placed on an engineered substrate.

##  I Results and Discussion

We consider a dimer structure, which consists of two AlGaAs cylinders with
height $h=635$ nm and radius $r=475$ nm, placed on an engineered substrate,
300-nm ITO on glass with an added SiO2 spacer 350-nm thick, similar to the
structures studied earlier for the observation of quasi-bound states in the
continuum for isolated nanoantennas [40]. Distance between two cylinders
$d=30$ nm. Incident plane wave is circularly polarized propagating in the
negative direction of the z-axis (see Fig. 2 (b)), while the SH power is
collected in the upper media. We show the difference between left and right
circular polarized incident waves (LCP and RCP, respectively) appears in the
pattern of second-harmonic radiation generated by a single particle, while
overall intensity of SHG stays the same. At the same time, for a dimer
structure one can achieve difference in the integral SHG intensity as well,
but only under proper orientation between the crystalline lattice of the
dielectric material and dimer axis.

First, we develop a general theory that demonstrates the origin of nonlinear
CD in nanoparticle dimers. We write the SH polarization as follows [41]:

${\bf{P}}^{2\omega}(\mathbf{r})=\varepsilon_{0}\hat{\chi}^{(2)}\mathbf{E}^{inc}(\mathbf{r})\mathbf{E}^{inc}(\mathbf{r}),$
(1)

where $\mathbf{E}^{inc}(\mathbf{r})$ is the fundamental field inside the
nanoparticle. To obtain SH field, we employ the dyadic Green function
formalism [31, 42]

$\mathbf{E}^{2\omega}({\mathbf{r}})=(2\omega)^{2}\mu\int\limits_{V}dV^{\prime}\hat{{\bf
G}}({\bf
r,r^{\prime}},k)\mathbf{P}^{2\omega}(\mathbf{r}^{\prime})=(2\omega)^{2}\mu\int\limits_{V}dV^{\prime}\sum_{n}\frac{\mathbf{E}_{n}(\mathbf{r})\otimes\mathbf{E}_{n}\left(\mathbf{r}^{\prime}\right)}{2k\left(k-k_{n}\right)}\mathbf{P}^{2\omega}(\mathbf{r}^{\prime}),$
(2)

Figure 2: SHG from an isolated dielectric cylinder. (a) Energy density (red
dashed curve) and SH intensity into upper half-space (blue solid curve). Field
amplitude at the cylinder surface is shown at the resonance. (b) Linear
scattered and nonlinear power flows into upper half-space shown for
$\lambda=1460$nm. Linear power flow has rotational symmetry, while nonlinear
has $C_{4}$ symmetry. The color-bar is presented in linear color-scale.

where we employ the resonant-state expansion of the Green function [43], and
calculate the integral over the particle’s volume,
$\mathbf{E}_{n}\left(\mathbf{r}\right)$ being system’s eigenstate. The key
role in our results is played by the overlap integral[44] that appears in Eq.
(2) as follows,

$D_{n}=\int\limits_{V}dV^{\prime}\mathbf{E}_{n}\left(\mathbf{r}^{\prime}\right)\mathbf{P}^{2\omega}(\mathbf{r}^{\prime})$
(3)

To calculate the overlap integral, we need to know both polarization and mode
content in two cases–a single nanoparticle and a dimer, and below we consider
these two cases subsequently.

###  I.1 SHG for a single nanoparticle

We performed numerical modeling with COMSOL Multiphysics™ simulation software.
The spectral dependence of the SH power flow emitted by a single dielectric
resonator integrated over upper half space is shown in Fig. 2(a) by the blue
solid curve. The spectrum of electromagnetic energy of the fundamental field
inside the nanoparticle is shown by a red dashed curve. The SH intensity is
maximal when the particle has a resonance at both fundamental and SH
wavelengths. The inset of Fig. 2(a) shows the distribution of SH field at the
nanoparticle surface at the resonant wavelength corresponding to excitation of
a Mie mode, which properties will be discussed later. The energy flux of the
linear scattered field in the upper half-space is depicted in Fig. 2(b) along
with the SH power flux. One can notice that the linear scattering pattern has
a cylindrical symmetry whereas the SH radiation pattern has $C_{4}$ symmetry
characterized by four petals. The rotation angle and shape of the petals
depend on the wavelength. If we change the incident polarization from LCP to
RCP, this four-lobe radiation pattern will be reflected in a mirror plane
coinciding with the (100) crystalline plane cut through the center of the
nanoparticle.

The particular shape of the radiation patterns can be explained in terms of
the multipolar decomposition of the eigenmodes [45, 46]. Multipoles
$\mathbf{W}_{pm\ell}$ include both electric multipoles $\mathbf{N}_{pm\ell}$
and magnetic multipoles $\mathbf{M}_{pm\ell}$, which are defined in the
previous work by [45]. The symmetry group of a single cylindrical nanoparticle
is $C_{\infty v}$, and its eigenmodes correspond to irreducible
representations of this group [47, 48, 45, 49], while mode refers to a
particular azimuthal number $m$, see Fig. 3(b).

In order to provide a simpler physical picture, we will operate in terms of
scalar spherical functions here rather than vector functions as they have the
same symmetry, which is important for explaining the observed effects.
However, the more strict analysis in terms of vector functions is provided in
Supporting Information file. We use the cylindrical coordinate system
corresponding to the $z$-axis, and the incident circularly polarized wave has
$e^{\pm i\varphi}$ dependence. Here and below, the upper sign ”$+$”refers to
the LCP fields and lower ”$-$” to the RCP fields. Under the normal
illumination, only modes with $|m|=1$ are excited (marked with the red tick in
Fig. 3(b)). Thus, the fundamental field inside the nanoparticle can be written
in the form, $E^{inc}(r,z,\varphi)\propto E_{0}^{inc}(r,z)e^{\pm i\varphi}$.

Rewriting the tensor $\hat{\chi}^{(2)}$ in the cylindrical coordinates, one
can make sure that it behaves as $\sin(2\varphi)$. Thus, after substituting
the results into Eq. (11), we can write
$P^{2\omega}(r,z,\varphi)\propto\sin(2\varphi)\exp(\pm 2i\varphi)\propto
1-\exp(\pm 4i\varphi).$ If we rotate the crystalline lattice by the angle
$\beta$ as shown in Fig. 1 (b), the polarization is transformed to the form

$P^{2\omega}(r,z,\varphi)\propto\sin[2(\varphi-\beta)]\exp(\pm
2i\varphi)\propto 1-\exp[\pm 4i(\varphi-\beta)].$ (4)

Similar result is obtained in Supporting Information in the framework of more
rigorous vectorial analysis. A nonvanishing overlap integral Eq. (3) involves
only the modes for which $m=0$ and $m=\pm 4$ (Fig. 3(b), blue ticks).

With help of Eq. (2), the SH field then can be decomposed into two terms with
according angular dependence:

${E}^{2\omega}=\sum_{n}g_{n}(\omega)E_{0n}^{2\omega}(r,z)+\sum_{\nu}q_{\nu}(\omega)E_{4\nu}^{2\omega}(r,z)e^{\pm
4i(\varphi-\beta)}\propto a(r,z,\omega)+b(r,z,\omega)e^{i\alpha(r,z,\omega)\pm
4i(\varphi-\beta)}$ (5)

where $n,\nu$ are the mode indices, $E_{0n}^{2\omega}$ and
$E_{4\nu}^{2\omega}$ are the fields of the corresponding eigenmodes being
proportional to the overlap integral. While $g_{n}(\omega)$ and
$q_{\nu}(\omega)$ are complex coefficient, the coefficients $a(r,z,\omega)$
and $b(r,z,\omega)$ are real-valued, and the function $\alpha(r,z,\omega)$
describes the relative phase.

Figure 3:  Hybridisation of the multipolar dimer modes. (a) Hybridisation is
shown for the modes with $m=0$ (non-degenerate) and $m=4$ (doubly degenerate)
in terms of field inside the particles. Scalar yellow-blue functions have the
same symmetry as vector spherical harmonics, and they are shown as a guide for
an eye. Modes excited in SHG are shown by the blue ticks, and in linear
scattering - by the red ticks. (b) Multipolar content of the single
cylindrical nanoparticle’s modes. (c) Multipolar content of the dimer modes
far-field. (d) Both m=0 and m=4 contribute into the same radiation channel,
resulting in SH dichroism.

The intensity depends on $\varphi$ as

$|E^{2\omega}|^{2}\propto|a+be^{i\alpha}e^{\pm
4i(\varphi-\beta)}|^{2}$$=a^{2}+b^{2}+2ab\cos[\pm 4(\varphi-\beta)+\alpha)]$,
which results in appearance of four petals in Fig. 2(b). Note, that both $m=0$
and $m=4$ modes play role in forming these petals, however, at the resonance
Fig. 2(a) mode $m=4$ appears to have dominant contribution. Under the flipping
of the polarization from RCP to LCP (corresponding to sign change from $+$ to
$-$) the angular behaviour of the field intensity changes. However, the
overall integral intensity does not depend on the signs $\pm$, and two types
of modes contribute independently and equally for RCP and LCP:
${I}^{2\omega,total}=I_{0}+I_{4}$, somewhat resembling the mechanism of a
hidden chirality discussed previously [50, 51].

###  I.2 SHG for nanoparticle dimers

Now, we consider a dimer composed of two identical cylindrical nanoparticles.
We assume that the polarisation $P^{2\omega}(r,z,\varphi)$ remains the same in
each single nanoparticle being similar to that of an isolated nanoparticle,
and we study the effect of mode hybridisation on the SH fields.

In our approximation, the difference between isolated nanoparticles and
nanoparticle dimers appears at the level of the coupling integral. In case of
a single nanoparticle, the modes proportional to $\sin(4\varphi)$ and
$\cos(4\varphi)$ are degenerate. For a dimer, the hybridisation of these two
degenerate modes leads to four modes with different symmetries and energies
(see Fig. 3(a) )[52, 53, 54, 55, 56, 57, 58]. According to the selection rules
[59], only the modes $A_{1}$ and $A_{2}$ are excited in the SH field, and
comparing Fig. 2(a) and Fig. 4(a) we see how one resonant peak ($E_{4}$)
transformed into two ($A_{1}$ and $A_{2}$). Based on the mode symmetry, we can
calculate the coupling integral over one nanoparticle, while the integral over
the second nanoparticle will be the same. First, we use the approximation that
only single multipole $\propto\cos(4\varphi)$ dominates in each nanoparticle.
However, the far-field radiation pattern has a complex multipole composition
as a sum of fields generated by two nanoparticles as shown in the table of
Fig. 3(c).

The coupling integral with this mode takes the form

Figure 4: Numerically calculated (a) SH intensity for both RCP and LCP
fundamental beams, and (b) resulting nonlinear circular dichroism SH-
CD=$2(I^{2\omega}_{RCP}-I^{2\omega}_{LCP})/(I^{2\omega}_{RCP}+I^{2\omega}_{LCP})$[21,
24] of the SH signal for the nanoparticle dimer.

$D^{dim}_{4}\propto\int_{0}^{2\pi}d\varphi\cos(4\varphi)\left[1-e^{\pm
4i(\varphi-\beta)}\right]\propto e^{\mp 4i\beta}.$ (6)

Now, we calculate the coupling integral for the mode with dominating $m=0$
multipoles,

$D^{dim}_{0}\propto\int_{0}^{2\pi}d\varphi(1-e^{\pm 4i(\varphi-\beta)})\propto
2\pi.$ (7)

Figure 5: Experimental SHG for a nanoparticle dimer. (a) Scanning electron
micrograph of the nanoparticle dimer with diameters 945 $\pm$ 5 nm and height
635 nm. (b) Measured SH spectra of the nanoparticle pumped by circularly
polarized laser beams: RCP (orange) and LCP (blue). (c) Circular dichroism
obtained from the SH signals of (b) generated by the nanoparticle dimer. Thick
solid line is a guide for an eye.

The most important result of this analysis is that these two modes can
contribute to the overall intensity, while they both are transformed by the
$A_{1}$ representation and radiate into the same radiation channel. Being
excited with different relative phases, the mode overlap results in strong
dichroism. Indeed, let us consider a very illustrative case when one of these
two modes is resonant and the second is not. The phase difference between the
mode coefficients equals $\pi/2$; multipolar contributions to $A_{1}$ SH far-
field (Fig. 3(c)) will come with different phases
$c_{\mathbf{W}}=a+\exp(i\pi/2)b\exp(\mp 4i\beta)=a+ib\exp(\mp 4i\beta)$, here
$c_{\mathbf{W}}$ is the amplitude of a particular multipole. Now one can
immediately see that this amplitude starts to have strong polarization
dependence. When $\beta=Z\pi/4$, $Z=0,1,2..$, the difference between LCP and
RCP beam excitations should not be observed in SH signal, and that, indeed,
was confirmed in our numerical simulations. Once $\beta=\pi/8$, for LCP we can
write $a+ibe^{-i\pi/2}=a+b$, and for RCP the same coefficient will become
$a-b$, and CD is maximal in this case. We believe, that by proper engineering
modes’ phases, amplitudes, and $\beta$ one can achieve even higher CD values.
For this particular $\beta$, we obtain a local maximum close to resonance
(Fig. 4(b)), however, the resonance is not required for the existence of CD.
The required phases and amplitudes can appear for nonresonant wavelengths, as
well as the conditions will be different for different values of $\beta$. For
$A_{2}$ modes all considerations are analogous.

In order to verify our general analysis and theoretical findings, we conduct
nonlinear spectroscopy from the dimers of two AlGaAs nanoparticles fabricated
by a consequent transfer of individual nanoparticles on a glass substrate as
described earlier [60]. The dimers are formed (see an SEM picture in Fig.
5(a)) with the random relative orientation of crystalline lattices, which
breaks the symmetry required for observation of CD in SHG. The dimer is
illuminated normally by circularly polarized focused optical pulses of 2 ps
duration in 1410-1450 nm range. The excitation occurs through an objective
with 0.7 NA from above the substrate. The generated SH signal is collected by
an objective lens with 0.9 NA. We observe experimentally the effect of
nonlinear CD in the dimer, as shown in Fig. 5(b). More specifically, the
nonlinear CD is enhanced at the resonant fundamental frequency, and, at the
same time, it takes place at nonresonant frequencies (see Fig. 5(c)) in
accordance with the theoretical predictions. The difference between the
simulated spectra (Fig. 4) and experimental results can be attributed to a
different geometry of the cylindrical nanoparticles being slightly conical and
not absolutely identical in the experiment. We note that the relative angle of
the crystalline axis in two nanoparticles is arbitrary, and it is hard to
determine it exactly in our case. Consequently, these experimental results
reflect well the major predictions of our general multipolar hybridization
theory, and also confirm a dramatic enhancement of all the effects in the
vicinity of Mie resonances.

In summary, we have systematically studied the SHG from the dimer composed of
two identical III-V semiconductor nanoparticles. We have illuminated the dimer
by a circularly polarized light and observed, both theoretically and
experimentally, that the nonlinear signal possesses circular dichroism. We
have revealed that this effect can be attributed to the Mie-resonant
multipolar response of the nanoparticles, and demonstrated that the result is
nonzero when the dimer axes is oriented under an angle to the crystalline
lattice of the dielectric material.

##  II Acknowledgement

The authors thank S. Kruk for a technical assistance, as well as P. Banzer, A.
Krasnok, and V. Valev for useful comments and suggestions. K.F. thanks A.A.
Bogdanov, K. Koshelev and A.A. Nikolaeva for fruitful discussions. This work
was supported by the Australian Research Council (grant DP210101292), the
National Research Foundation (NRF) of Korea (grant no. 2021R1A2C3006781 funded
by the Korean Government, MSIT) and the Russian Science Foundation (grant no.
18-72-10140). E.M. acknowledges a support of the ACT Node of the Australian
National Fabrication Facility. H.-G.P. acknowledges a support from the Samsung
Research Funding & Incubation Center of Samsung Electronics (SRFC-MA2001-01).
K.F. acknowledges support from the Foundation for the Advancement of
Theoretical Physics and Mathematics ”BASIS” (Russia).

## References

  * Berova  _et al._ [2000] N. Berova, K. Nakanishi, and R. W. Woody,  _Circular Dichroism: Principles and Applications, 2nd Edition_ (Wiley-VCH, Hoboken, NJ, USA, 2000).
  * Greenfield [2006] N. J. Greenfield, Using circular dichroism spectra to estimate protein secondary structure, Nature Protocols 1, 2876 (2006).
  * Hopkins  _et al._ [2016] B. Hopkins, A. N. Poddubny, A. E. Miroshnichenko, and Y. S. Kivshar, Circular dichroism induced by Fano resonances in planar chiral oligomers, Laser Photonics Rev. 10, 137 (2016).
  * De Silva  _et al._ [2021] A. H. N. C. De Silva, D. Atri-Schuller, S. Dubey, B. P. Acharya, K. L. Romans, K. Foster, O. Russ, K. Compton, C. Rischbieter, N. Douguet, K. Bartschat, and D. Fischer, Using Circular Dichroism to Control Energy Transfer in Multiphoton Ionization, Phys. Rev. Lett. 126, 023201 (2021).
  * Schäferling  _et al._ [2012] M. Schäferling, D. Dregely, M. Hentschel, and H. Giessen, Tailoring enhanced optical chirality: Design principles for chiral plasmonic anostructures, Phys. Rev. X 2, 031010 (2012).
  * Hentschel  _et al._ [2017] M. Hentschel, M. Schäferling, X. Duan, H. Giessen, and N. Liu, Chiral plasmonics, Science Advances 3, e1602737 (2017).
  * Collins  _et al._ [2017] J. Collins, C. Kuppe, D. Hooper, C. Sibilia, M. Centini, and V. Valev, Chirality and chiroptical effects in metal nanostructures: fundamentals and current trends, Advanced Optical Materials 5, 1700182 (2017).
  * Li  _et al._ [2021] J. Li, M. Wang, Z. Wu, H. Li, G. Hu, T. Jiang, J. Guo, Y. Liu, K. Yao, Z. Chen, J. Fang, D. Fan, B. A. Korgel, A. Alù, and Y. Zheng, Tunable Chiral Optics in All-Solid-Phase Reconfigurable Dielectric Nanostructures, Nano Lett. 21, 973 (2021).
  * Decker  _et al._ [2007] M. Decker, M. W. Klein, M. Wegener, and S. Linden, Circular dichroism of planar chiral magnetic metamaterials, Optics Letters 32, 856 (2007).
  * Wang  _et al._ [2016] Z. Wang, F. Cheng, T. Winsor, and Y. Liu, Optical chiral metamaterials: a review of the fundamentals, fabrication methods and applications, Nanotechnology 27, 412001 (2016).
  * Zhao  _et al._ [2012] Y. Zhao, M. A. Belkin, and A. Alù, Twisted optical metamaterials for planarized ultrathin broadband circular polarizers, Nat. Commun. 3, 1 (2012).
  * Hu  _et al._ [2020] J. Hu, M. Lawrence, and J. A. Dionne, High Quality Factor Dielectric Metasurfaces for Ultraviolet Circular Dichroism Spectroscopy, ACS Photonics 7, 36 (2020).
  * Gorkunov  _et al._ [2020] M. V. Gorkunov, A. A. Antonov, and Y. S. Kivshar, Metasurfaces with Maximum Chirality Empowered by Bound States in the Continuum, Phys. Rev. Lett. 125, 093903 (2020).
  * Chen  _et al._ [2021] C. Chen, S. Gao, W. Song, H. Li, S.-N. Zhu, and T. Li, Metasurfaces with planar chiral meta-atoms for spin light manipulation, Nano Letters 21, XXXX (2021).
  * Lin  _et al._ [2019] L. Lin, S. Lepeshov, A. Krasnok, T. Jiang, X. Peng, B. A. Korgel, A. Alù, and Y. Zheng, All-optical reconfigurable chiral meta-molecules, Mater. Today 25, 10 (2019).
  * Nechayev  _et al._ [2019] S. Nechayev, R. Barczyk, U. Mick, and P. Banzer, Substrate-induced chirality in an individual nanostructure, ACS Photonics 6, 1876 (2019).
  * Nechayev  _et al._ [2018] S. Nechayev, P. Woźniak, M. Neugebauer, R. Barczyk, and P. Banzer, Chirality of Symmetric Resonant Heterostructures, Laser Photonics Rev. 12, 1800109 (2018).
  * Bautista  _et al._ [2012] G. Bautista, M. J. Huttunen, J. Mäkitalo, J. M. Kontio, J. Simonen, and M. Kauranen, Second-Harmonic Generation Imaging of Metal Nano-Objects with Cylindrical Vector Beams, Nano Lett. 12, 3207 (2012).
  * Belardini  _et al._ [2011] A. Belardini, M. C. Larciprete, M. Centini, E. Fazio, C. Sibilia, D. Chiappe, C. Martella, A. Toma, M. Giordano, and F. Buatier de Mongeot, Circular dichroism in the optical second-harmonic emission of curved gold metal nanowires, Phys. Rev. Lett. 107, 257401 (2011).
  * Hooper  _et al._ [2017] D. C. Hooper, A. G. Mark, C. Kuppe, J. T. Collins, P. Fischer, and V. K. Valev, Strong rotational anisotropies affect nonlinear chiral metamaterials, Advanced Materials 29, 1605110 (2017).
  * Belardini  _et al._ [2020] A. Belardini, G. Leahu, E. Petronijevic, T. Hakkarainen, E. Koivusalo, M. Rizzo Piton, S. Talmila, M. Guina, and C. Sibilia, Circular Dichroism in the Second Harmonic Field Evidenced by Asymmetric Au Coated GaAs Nanowires, Micromachines 11, 225 (2020).
  * Rodrigues  _et al._ [2014] S. P. Rodrigues, S. Lan, L. Kang, Y. Cui, and W. Cai, Nonlinear Imaging and Spectroscopy of Chiral Metamaterials, Adv. Mater. 26, 6157 (2014).
  * Bertolotti  _et al._ [2015] M. Bertolotti, A. Belardini, A. Benedetti, and C. Sibilia, Second harmonic circular dichroism by self-assembled metasurfaces [Invited], J. Opt. Soc. Am. B, JOSAB 32, 1287 (2015).
  * Schmeltz  _et al._ [2020] M. Schmeltz, C. Teulon, M. Pinsard, U. Hansen, M. Alnawaiseh, D. Ghoubay, V. Borderie, G. Mosser, C. Aimé, C. Aimé, F. Légaré, G. Latour, G. Latour, and M.-C. Schanne-Klein, Circular dichroism second-harmonic generation microscopy probes the polarity distribution of collagen fibrils, Optica 7, 1469 (2020).
  * Valev  _et al._ [2009] V. Valev, N. Smisdom, A. Silhanek, B. De Clercq, W. Gillijns, M. Ameloot, V. Moshchalkov, and T. Verbiest, Plasmonic ratchet wheels: Switching circular dichroism by arranging chiral nanostructures, Nano Letters 9, 3945 (2009).
  * Valev  _et al._ [2010] V. K. Valev, A. V. Silhanek, N. Verellen, W. Gillijns, P. Van Dorpe, O. A. Aktsipetrov, G. A. E. Vandenbosch, V. V. Moshchalkov, and T. Verbiest, Asymmetric optical second-harmonic generation from chiral $g$-shaped gold nanostructures, Phys. Rev. Lett. 104, 127401 (2010).
  * Wang and Harutyunyan [2019] F. Wang and H. Harutyunyan, Observation of a giant nonlinear chiro-optical response in planar plasmon-photonic metasurfaces, Advanced Optical Materials 7, 1900744 (2019).
  * Tymchenko  _et al._ [2015] M. Tymchenko, J. S. Gomez-Diaz, J. Lee, N. Nookala, M. A. Belkin, and A. Alù, Gradient Nonlinear Pancharatnam-Berry Metasurfaces, Phys. Rev. Lett. 115, 207403 (2015).
  * Koshelev and Kivshar [2021] K. Koshelev and Y. Kivshar, Dielectric resonant metaphotonics, ACS Photonics 8, 102 (2021).
  * Smirnova and Kivshar [2016] D. Smirnova and Y. S. Kivshar, Multipolar nonlinear nanophotonics, Optica 3, 1241 (2016).
  * Frizyuk  _et al._ [2019a] K. Frizyuk, I. Volkovskaya, D. Smirnova, A. Poddubny, and M. Petrov, Second-harmonic generation in mie-resonant dielectric nanoparticles made of noncentrosymmetric materials, Physical Review B 99, 075425 (2019a).
  * Gigli  _et al._ [2019] C. Gigli, G. Marino, A. Borne, P. Lalanne, and G. Leo, All-Dielectric Nanoresonators for $\chi$(2) Nonlinear Optics, Front. Phys. 7, 10.3389/fphy.2019.00221 (2019).
  * Renaut  _et al._ [2019] C. Renaut, L. Lang, K. Frizyuk, M. Timofeeva, F. E. Komissarenko, I. S. Mukhin, D. Smirnova, F. Timpu, M. Petrov, Y. Kivshar, and R. Grange, Reshaping the Second-Order Polar Response of Hybrid Metal–Dielectric Nanodimers, Nano Lett. 19, 877 (2019).
  * Kang  _et al._ [2020] L. Kang, C. Wang, X. Guo, X. Ni, Z. Liu, and D. Werner, Nonlinear chiral meta-mirrors: Enabling technology for ultrafast switching of light polarisation, Nano Letters 20, 2047 (2020).
  * Sanatinia  _et al._ [2014] R. Sanatinia, S. Anand, and M. Swillo, Modal engineering of second-harmonic generation in single gap nanopillars, Nano Letters 14, 5376 (2014).
  * Carletti  _et al._ [2016] L. Carletti, A. Locatelli, D. Neshev, and C. De Angelis, Shaping the Radiation Pattern of Second-Harmonic Generation from AlGaAs Dielectric Nanoantennas, ACS Photonics 3, 1500 (2016).
  * Kruk  _et al._ [2017] S. S. Kruk, R. Camacho-Morales, L. Xu, M. Rahmani, D. A. Smirnova, L. Wang, H. H. Tan, C. Jagadish, D. N. Neshev, and Y. S. Kivshar, Nonlinear optical magnetism revealed by second-harmonic generation in nanoantennas, Nano Letters 17, 3914 (2017).
  * Timofeeva  _et al._ [2018] M. Timofeeva, L. Lang, F. Timpu, C. Renaut, A. Bouravleuv, I. Shtrom, G. Cirlin, and R. Grange, Anapoles in free-standing iii-v nanodisks enhancing second-harmonic generation, Nano Letters 18, 3695 (2018).
  * Sautter  _et al._ [2019] J. Sautter, L. Xu, A. E. Miroshnichenko, M. Lysevych, I. Volkovskaya, D. A. Smirnova, R. Camacho-Morales, K. Zangeneh Kamali, F. Karouta, K. Vora, H. H. Tan, M. Kauranen, I. Staude, C. Jagadish, D. N. Neshev, and M. Rahmani, Tailoring Second-Harmonic Emission from (111)-GaAs Nanoantennas, Nano Lett. 19, 3905 (2019).
  * Koshelev  _et al._ [2020] K. Koshelev, S. Kruk, E. Melik-Gaykazyan, J.-H. Choi, A. Bogdanov, H.-G. Park, and Y. Kivshar, Subwavelength dielectric resonators for nonlinear nanophotonics, Science 367, 288 (2020).
  * Boyd [2003] R. W. Boyd,  _Nonlinear optics_ (Academic press, 2003).
  * Novotny and Hecht [2006] L. Novotny and B. Hecht,  _Principles of Nano-Optics_ (Cambridge University Press, USA, 2006).
  * Doost  _et al._ [2014] M. B. Doost, W. Langbein, and E. A. Muljarov, Resonant-state expansion applied to three-dimensional open optical systems, Phys. Rev. A 90, 013834 (2014).
  * Gigli  _et al._ [2020] C. Gigli, T. Wu, G. Marino, A. Borne, G. Leo, and P. Lalanne, Quasinormal-Mode Non-Hermitian Modeling and Design in Nonlinear Nano-Optics, ACS Photonics 7, 1197 (2020).
  * Gladyshev  _et al._ [2020] S. Gladyshev, K. Frizyuk, and A. Bogdanov, Symmetry analysis and multipole classification of eigenmodes in electromagnetic resonators for engineering their optical properties, Physical Review B 102, 075103 (2020).
  * Brandl  _et al._ [2006] D. W. Brandl, N. A. Mirin, and P. Nordlander, Plasmon modes of nanosphere trimers and quadrumers, The Journal of Physical Chemistry B 110, 12302 (2006).
  * Zheng  _et al._ [2015] X. Zheng, N. Verellen, D. Vercruysse, V. Volskiy, P. Van Dorpe, G. A. E. Vandenbosch, and V. Moshchalkov, On the Use of Group Theory in Understanding the Optical Response of a Nanoantenna, IEEE Trans. Antennas Propag. 63, 1589 (2015).
  * Chikkaraddy  _et al._ [2017] R. Chikkaraddy, X. Zheng, F. Benz, L. J. Brooks, B. de Nijs, C. Carnegie, M.-E. Kleemann, J. Mertens, R. W. Bowman, G. A. E. Vandenbosch, V. V. Moshchalkov, and J. J. Baumberg, How Ultranarrow Gap Symmetries Control Plasmonic Nanocavity Modes: From Cubes to Spheres in the Nanoparticle-on-Mirror, ACS Photonics 4, 469 (2017).
  * Xiong  _et al._ [2020] Z. Xiong, Z. Xiong, Q. Yang, Q. Yang, W. Chen, Z. Wang, J. Xu, J. Xu, J. Xu, W. Liu, and et al., On the constraints of electromagnetic multipoles for symmetric scatterers: eigenmode analysis, Optics Express 28, 3073 (2020).
  * Horrer  _et al._ [2020] A. Horrer, Y. Zhang, D. Gérard, J. Béal, M. Kociak, J. Plain, and R. Bachelot, Local Optical Chirality Induced by Near-Field Mode Interference in Achiral Plasmonic Metamolecules, Nano Lett. 20, 509 (2020).
  * Zu  _et al._ [2018] S. Zu, T. Han, M. Jiang, F. Lin, X. Zhu, and Z. Fang, Deep-Subwavelength Resolving and Manipulating of Hidden Chirality in Achiral Nanostructures, ACS Nano 12, 3908 (2018).
  * Nordlander  _et al._ [2004] P. Nordlander, C. Oubre, E. Prodan, K. Li, and M. I. Stockman, Plasmon Hybridization in Nanoparticle Dimers, Nano Lett. 4, 899 (2004).
  * Gao  _et al._ [2017] Z. Gao, F. Gao, Y. Zhang, H. Xu, Y. Luo, and B. Zhang, Forward/Backward Switching of Plasmonic Wave Propagation Using Sign-Reversal Coupling, Adv. Mater. 29, 1700018 (2017).
  * Deng  _et al._ [2018] T.-S. Deng, J. Parker, Y. Yifat, N. Shepherd, and N. F. Scherer, Dark Plasmon Modes in Symmetric Gold Nanoparticle Dimers Illuminated by Focused Cylindrical Vector Beams, J. Phys. Chem. C 122, 27662 (2018).
  * Pascale  _et al._ [2019] M. Pascale, G. Miano, R. Tricarico, and C. Forestiere, Full-wave electromagnetic modes and hybridization in nanoparticle dimers, Sci. Rep. 9, 1 (2019).
  * Hopkins  _et al._ [2015] B. Hopkins, D. S. Filonov, S. B. Glybovski, and A. E. Miroshnichenko, Hybridization and the origin of fano resonances in symmetric nanoparticle trimers, Phys. Rev. B 92, 045433 (2015).
  * Dmitriev  _et al._ [2021] A. A. Dmitriev, M. V. Rybin, and M. V. Rybin, Optical coupling of overlapping nanopillars, Opt. Lett. 46, 1221 (2021).
  * Song  _et al._ [2021] J.-H. Song, S. Raza, J. van de Groep, J.-H. Kang, Q. Li, P. G. Kik, and M. L. Brongersma, Nanoelectromechanical modulation of a strongly-coupled plasmonic dimer, Nat. Commun. 12, 1 (2021).
  * Frizyuk [2019] K. Frizyuk, Second-harmonic generation in dielectric nanoparticles with different symmetries, JOSA B 36, F32 (2019).
  * Melik-Gaykazyan  _et al._ [2021] E. Melik-Gaykazyan, K. Koshelev, J.-H. Choi, S. S. Kruk, A. Bogdanov, H.-G. Park, and Y. Kivshar, From fano to quasi-bic resonances in individual dielectric nanoantennas, Nano Letters 21, 1765 (2021).
  * STEIN [1961] S. STEIN, Addition theorems for spherical wave functions, Quarterly of Applied Mathematics 19, 15 (1961).
  * Contributors to Wikimedia projects [2021] Contributors to Wikimedia projects, Vector spherical harmonics (2021), [Online; accessed 9. Mar. 2021].
  * Zhang and Han [2008] H. Zhang and Y. Han, Addition theorem for the spherical vector wave functions and its application to the beam shape coefficients, J. Opt. Soc. Am. B, JOSAB 25, 255 (2008).
  * Venkatesh and Schurig [2016] S. Venkatesh and D. Schurig, Computationally fast em field propagation through axi-symmetric media using cylindrical harmonic decomposition, Optics Express 24, 29246 (2016).
  * Bohren and Huffman [1983] C. F. Bohren and D. R. Huffman,  _Absorption and scattering of light by small particles._ (New York Wiley Interscience, 1983) pp. xiv, 530 p.
  * Dresselhaus  _et al._ [2008] M. S. Dresselhaus, G. Dresselhaus, and A. Jorio,  _Group Theory_ (Springer-Verlag, Berlin, Germany, 2008).
  * Ivchenko  _et al._ [2012] E. L. Ivchenko, G. P. Skrebtsov, and G. Pikus,  _Superlattices and Other Heterostructures_ (Springer, Berlin, Germany, 2012).
  * Frizyuk  _et al._ [2019b] K. Frizyuk, I. Volkovskaya, D. Smirnova, A. Poddubny, and M. Petrov, Second-harmonic generation in mie-resonant dielectric nanoparticles made of noncentrosymmetric materials, Phys. Rev. B 99, 075425 (2019b).

##  III Supporting Information for ”Nonlinear circular dichroism in Mie-
resonant nanoparticle dimers”

Here, we provide more rigorous theoretical considerations to explain the
circular dichroism, which appears in second harmonic signal of AlGaAs dimer
with arbitrarily oriented crystalline lattice. All functions are written in a
cylindrical coordinate system. Note, that when we speak about symmetry
behavior of vector functions and tensors, we should take into account the
behavior of basis vectors
$\mathbf{e}_{r},\mathbf{e}_{z},\mathbf{e}_{\varphi}$. This makes
considerations more sophisticated than in the scalar case. However, when
considering vector functions as a whole (not each component separately), as it
is usually done, for example, for vector spherical functions [61, 62, 63], one
can see, that everything is very similar to a scalar case. Namely, the
$\mathbf{e}_{\varphi}$ component is always written separately from others in
our vectorial considerations, while it behaves diffetently under
$\varphi\rightarrow-\varphi$ symmetry transformation, but vector fields as a
whole have exactly the same symmetry behavior as a scalar functions given in
the main text.

###  III.1 Single dielectric resonator

For the further convenience, let us use the cylindrical coordinate system for
a single particle case, and assume that all fields are decomposed into the
following series of cylindrical waves [64]:

$\mathbf{E}(r,\varphi,z)=\sum_{m=-\infty}^{m=\infty}\mathbf{E}_{m}(r,z)e^{im\varphi}.$
(8)

Normally incident circularly polarized plane wave with contains only one term
with $m=1$ (LCP) or $m=-1$ (RCP):

$\mathbf{E}^{wave}=E_{x}(\mathbf{e}_{x}\pm
i\mathbf{e}_{y})=E_{x}(\mathbf{e}_{r}\pm i\mathbf{e}_{\varphi})e^{\pm
i\varphi}.$ (9)

Thus, only modes with $m=\pm 1$ are excited in a single cylinder, and the
field inside the cylinder:

$\displaystyle\mathbf{E}^{inc}=(E_{r}(r,z)\mathbf{e}_{r}+E_{z}(r,z)\mathbf{e}_{z}\pm
iE_{\varphi}(r,z)\mathbf{e}_{\varphi})e^{\pm i\varphi}.$ (10)

This can be derived from the exact form of vector spherical harmonics [65].
The second-harmonic polarization is written as following:

$\mathbf{P}^{2\omega}(r,\varphi,z)=\varepsilon_{0}\hat{\chi}^{(2)}\mathbf{E}^{inc}(r,\varphi,z)\mathbf{E}^{inc}(r,\varphi,z)$
(11)

Then, let us consider the susceptibility tensor.

For the AlGaAs ($[100]||x,\ [101]||y,\ [001]||z$), the nonlinearity tensor has
only one independent component,
${\chi}_{xyz}={\chi}_{yxz}={\chi}_{xzy}={\chi}_{yzx}={\chi}_{zxy}={\chi}_{zyx}=\chi$.
We can write it in the form

$\hat{\chi}_{ijk}={\chi}_{ijk}\mathbf{e}_{i}\mathbf{e}_{j}\mathbf{e}_{k}$ (12)

where Einstein summation convention is used. Then, using the relations for the
cylindrical coordinate system

$\begin{cases}\mathbf{e}_{x}=\cos\varphi\mathbf{e}_{r}-\sin\varphi\mathbf{e}_{\varphi},\\\
\mathbf{e}_{y}=\sin\varphi\mathbf{e}_{r}+\cos\varphi\mathbf{e}_{\varphi},\\\
\mathbf{e}_{z}=\mathbf{e}_{z},\end{cases}$ (13)

we write the $\hat{\chi}^{(2)}$ in cylindrical coordinates, for example, the
following sum is

$\displaystyle\mathbf{e}_{z}\mathbf{e}_{y}\mathbf{e}_{x}+\mathbf{e}_{z}\mathbf{e}_{x}\mathbf{e}_{y}=$
(14)
$\displaystyle={e^{2i\varphi}}\left(\frac{1}{2i}\mathbf{e}_{z}\mathbf{e}_{r}\mathbf{e}_{r}-\frac{1}{2i}\mathbf{e}_{z}\mathbf{e}_{\varphi}\mathbf{e}_{\varphi}+\frac{1}{2}\mathbf{e}_{z}\mathbf{e}_{\varphi}\mathbf{e}_{r}+\frac{1}{2}\mathbf{e}_{z}\mathbf{e}_{r}\mathbf{e}_{\varphi}\right)+$
$\displaystyle+{e^{-2i\varphi}}\left(-\frac{1}{2i}\mathbf{e}_{z}\mathbf{e}_{r}\mathbf{e}_{r}+\frac{1}{2i}\mathbf{e}_{z}\mathbf{e}_{\varphi}\mathbf{e}_{\varphi}+\frac{1}{2}\mathbf{e}_{z}\mathbf{e}_{\varphi}\mathbf{e}_{r}+\frac{1}{2}\mathbf{e}_{z}\mathbf{e}_{r}\mathbf{e}_{\varphi}\right),$

the whole tensor for AlGaAs has four additional terms,
$\mathbf{e}_{y}\mathbf{e}_{x}\mathbf{e}_{z}+\mathbf{e}_{x}\mathbf{e}_{y}\mathbf{e}_{z}+\mathbf{e}_{y}\mathbf{e}_{z}\mathbf{e}_{x}+\mathbf{e}_{x}\mathbf{e}_{z}\mathbf{e}_{y}$
which look similarly. Note, that nonlinear susceptibility tensor depends on
$\varphi$, but only has two components - $m=2$ and $m=-2$. We are only
interested in a tensor’s symmetry behavior. [66, 67, 68]

First, we consider $[100]||x$\- oriented lattice ($\beta=0$).

Substituting incident field inside the nanoparticle with $m=\pm 1$ and
polarizability tensor into Eq. (11), we obtain, that polarization can be
written as

$\displaystyle\mathbf{P}^{2\omega}(r,\varphi,z)=\varepsilon_{0}\hat{\chi}^{(2)}\mathbf{E}_{1}^{inc}\mathbf{E}_{1}^{inc}\propto\mathbf{P}_{e0}^{2\omega}(r,z)\pm
i\mathbf{P}_{o0}^{2\omega}(r,z)+\left(\mathbf{P}_{e4}^{2\omega}(r,z)\pm
i\mathbf{P}_{o4}^{2\omega}(r,z)\right)e^{\pm 4i\varphi}$ (15)

where the tensor gives its $m=2$ and $m=-2$ contribution and
$\mathbf{P}_{{}^{e}_{o}0}^{2\omega}(r,z)$ and
$\mathbf{P}_{{}^{e}_{o}4}^{2\omega}(r,z)$ can be assumed real, and $\pm$ in
$e^{\pm 4i\varphi}$ refers to two different polarizations. Indexes $e$ and $o$
refer to $\varphi\rightarrow-\varphi$ parity, i.e.
$\mathbf{P}_{o0}\sim\mathbf{e}_{\varphi}$,
$\mathbf{P}_{e0}\sim\mathbf{e}_{z},\mathbf{e}_{r}$.

If we rotate the crystalline lattice around the z-axis by the angle $\beta$,
the induced polarization will be also rotated by the same angle and also will
get an additional phase, which we are not interested in.

Thus, for the rotated lattice $\varphi\rightarrow\varphi-\beta$:

$\mathbf{P}^{2\omega}(r,\varphi,z)\propto\mathbf{P}_{{}^{e}_{o}0}^{2\omega}(r,z)+\mathbf{P}_{{}^{e}_{o}4}^{2\omega}(r,z)e^{\pm
4i(\varphi-\beta)},$ (16)

where $\mathbf{P}_{{}^{e}_{o}}=\mathbf{P}_{e}\pm i\mathbf{P}_{o}$.

To obtain the important conclusions, we don’t need to know how exactly the
eigenmodes and polarization look like, we only need to know, how they depend
on $\varphi$.

#### Cylinder modes.

Each mode corresponds to particular $|m|$. Modes with different $|m|$ have
different eigenfrequency, thus the resonant-state expansion[43] will look as
following:

$\displaystyle\hat{{\bf G}}({\bf
r,r^{\prime}},k)=\sum_{m,n}\frac{\mathbf{E}_{emn}(\mathbf{r})\otimes\mathbf{E}_{emn}\left(\mathbf{r}^{\prime}\right)+\mathbf{E}_{omn}(\mathbf{r})\otimes\mathbf{E}_{omn}\left(\mathbf{r}^{\prime}\right)}{2k\left(k-k_{mn}\right)},$
(17)

where $n$ is number of mode. Importantly, even and odd modes are degenerated.
Eigenvectors $k_{mn}$ are complex. Under circularly polarized plane wave only
modes with $m=0$ and $m=4$ are excited in SH, because all the other overlap
integrals are zero. Modes with $m=0$ and $m=4$ will be excited with different
phases. After all, the SH field will be written as following:

$\mathbf{E}^{2\omega}({\mathbf{r}})=\sum_{n,\nu}\left(g_{n}(\omega)\mathbf{E}_{{}^{e}_{o}0n}^{2\omega}+q_{\nu}(\omega)\mathbf{E}_{{}^{e}_{o}4\nu}^{2\omega}e^{\pm
4i(\varphi-\beta)}\right)$ (18)

where $n$ and $\nu$ is number of a mode, and $g_{n}(\omega)$ and
$q_{\nu}(\omega)$ are complex coefficients.

###  III.2 Nanoparticle dimer

Modes of a dimer can be considered with help of multipole decomposition[45] in
two different ways:

1) Every particle has its own multipole decomposition, as shown in Fig. 3(a)
of the main text. This approach is useful for mode hybridization
consideration. However, in Fig. 3(a), only the main contribution is shown,
actually, there will be an infinite row of multipoles.

2) Consideration of the multipolar content of the far-field. In this approach,
only symmetry of the dimer matters, so the multipolar content will be the same
as for a single elliptical along the $x$-axis particle, for example. This
approach helps to understand, that all modes, which are transformed by the
same irreducible representation, have the same multipole content, and can add
up in the overall intensity.

How the dimerization affect our fields and intensity?

Now, we will use the first approach to describe the modes and fields in each
particle. Thus, after dimerization, the modes of sinle particle are perturbed
as following:

$\displaystyle\mathbf{E}_{0n}^{2\omega}\rightarrow\mathbf{E}_{e0n}^{2\omega}+\mathbf{E}_{e0n4}^{2\omega}\cos(4\varphi)+\mathbf{E}_{o0n4}^{2\omega}\sin(4\varphi)+\dots=E_{0r}\mathbf{e}_{r}+E_{0z}\mathbf{e}_{z}+\dots$
(19)
$\displaystyle\mathbf{E}_{4n}^{2\omega}\rightarrow\mathbf{E}_{e4n4}^{2\omega}\cos(4\varphi)+\mathbf{E}_{o4n4}^{2\omega}\sin(4\varphi)+\mathbf{E}_{e4n0}^{2\omega}+\dots=(E_{4r}\mathbf{e}_{r}+E_{4z}\mathbf{e}_{z})\cos(4\varphi)+E_{4\varphi}\mathbf{e}_{\varphi}\sin(4\varphi)+\dots$
(20)

for even modes ($A_{1}$), and

$\displaystyle\mathbf{E}_{0n}^{2\omega}\rightarrow\mathbf{E}_{o0n0}^{2\omega}+\mathbf{E}_{e0n4}^{2\omega}\sin(4\varphi)+\mathbf{E}_{o0n4}^{2\omega}\cos(4\varphi)+\dots=E_{0\varphi}\mathbf{e}_{\varphi}+\dots$
(21)
$\displaystyle\mathbf{E}_{4n}^{2\omega}\rightarrow\mathbf{E}_{e4n4}^{2\omega}\sin(4\varphi)+\mathbf{E}_{o4n4}^{2\omega}\cos(4\varphi)+\mathbf{E}_{o4n0}^{2\omega}+\dots=(E_{4r}\mathbf{e}_{r}+E_{4z}\mathbf{e}_{z})\sin(4\varphi)+E_{4\varphi}\mathbf{e}_{\varphi}\cos(4\varphi)+\dots$
(22)

for odd modes ($A_{2}$). Infinite sums containing all possible multipoles,
even (Eq. (19), (20)) or odd (Eq. (21), (22)) under
$\varphi\rightarrow-\varphi$ transformation, but below, we only consider
dominating terms.

Let us write the overlap integral between dimer modes and polarization. Let us
consider two $A_{1}$ modes (19), and (20). We write the integral only over one
cylinder (coordinate system located in center of the cylinder), because over
the second will be the same:

$\displaystyle
D_{0}=\int\limits_{V}(E_{0r}\mathbf{e}_{r}+E_{0z}\mathbf{e}_{z})\mathbf{P}_{{}^{e}_{o}0}^{2\omega}(r,z)\propto
2\pi,$ (23) $\displaystyle
D_{4}=\int\limits_{V}dV\left((E_{4r}\mathbf{e}_{r}+E_{4z}\mathbf{e}_{z})\cos(4\varphi)+E_{4\varphi}\mathbf{e}_{\varphi}\sin(4\varphi)\right)\cdot\left(\mathbf{P}_{{}^{e}_{o}4}^{2\omega}(r,z)e^{\pm
4i(\varphi-\beta)}\right)\propto e^{\mp 4i\beta}$ (24)

which coincides with formulae (5) and (6) in the main text. Second approach
(Fig.3 (b), (c)) helps us to understand, why the sum of $m=0$ and $m=4$ modes
lead to CD appearance. Both of them belong to the $A_{1}$ symmetry, hence,
they have the same multipolar content in the far-field. Further considerations
do not require vector functions, and provided in the main text.