# Quantitative Study on Circular Dichroism Induction from Achiral
Nanostructure-Chiral Matter Near Field Interactions

Abuduwaili. Abudukelimu<EMAIL_ADDRESS>Ziyan Zhang  School of Physics
and Information Technology, Shaanxi Normal University, Xi’an, 710119, China
Tursunay. Yibibulla  School of Physics and Electronics, Central South
University, Changsha, 410083, P. R. China  Muhammad. Ikram  School of Physics
and Information Technology, Shaanxi Normal University, Xi’an, 710119, China

###### Abstract

Abstract

The circular dichroism (ICD) induced from interaction of chiral molecules with
achiral plasmonic metal nanostructures has improved sensitivity of molecular
chirality detection. Although experimental results have presented several
orders magnitude of ICD enhancement, but the cause of ICD induction has not
arise from previous studies. In this paper, a theory presented for explain
mechanism of ICD induction from near field interaction of chiral molecule with
achiral nanostructure. By introducing dipole approaching of chiral molecule
and a near field approximation of molecule-nanostructure interaction, we
derived an analytical formula for ICD. This theory explains mechanism of ICD
is that the dipole radiation of chiral molecules breaks the enantiomorphic
relationship of the electric field strengths in the nanostructure.
Furthermore, the derived formula exhibits ICD proportional to molecules
chirality parameter and number density, and it exponentially increases with
the inverse cube of distance between molecule and nanostructure. Most
importantly, the formula shows that ICD is proportional to strengths of
electric field in nanostructure and magnetic field in molecular region. To
verify the analytical results, we studied the ICD properties of graphene
nanohole metasurface (GNM) and chiral molecule composed system. Numerical
results were found consistent with analytical results. Our investigation
results are helpful for understanding mechanism of ICD induction, pushing the
limit of ICD enhancement and realizing supersensitive detection of molecular
chirality.

Keywords: Induced circular dichroism, Surface plasmon, Chiral molecules,
dipole approaching, near field approximation.

##  I Introduction

Circular dichroism (CD), absorption difference of right-handed circularly
polarized (RCP) and left-handed circularly polarized (LCP) light, is widely
used in determining handiness and absolute conformations of chiral molecules
Fasman ; Tullius ; YZhao ; Besteiro ; Maoz , and enantioselective separation
of chiral molecules Aitzol2017 ; Aitzol2019 . However, CD of chiral molecules
(MCD) is weak and only observable in the ultraviolet (UV) region, the long
exposure to UV radiation can severely affect chiral molecular samples.
Therefore, it is desired to develop new techniques to increase the limit of
molecular chirality sensitivity or transfer CD signal to long wavelength, so
as to achieve the ultimate goal of single-molecule chirality detection. Thus
the surface plasmon enhancing technique has been used for enhancing the MCD
Hendry . In the interaction between chiral molecules and nanostructure, on one
hand, the optical chiral field enhanced by surface plasmon increases the MCD
in the UV region Schaferling2016 , on the other hand, the molecular chirality
transfers to nanostructure and generates the induced circular dichroism (ICD)
signal YZhao ; Hendry . The ICD provides a way to detect chiral molecules in
the visual or microwave region, which is easily detectable by CD spectroscopy.

In order to further enhance the ICD signal, effects of different factors such
as molecules location BHan ; dawei , molecules density rongyao ; jianjia ,
molecular orientation Govorov2010 ; TLeviBelenkova , separation distance
between molecule and nanostructure Maoz , and shape of nanostructure YZhao ;
ehk ; ZZhu2012  on ICD have been investigated. The studies show that ICD is
proportional to the near field strength in molecular region MLNesterov , thus
ICD is larger when chiral molecule is placed on ends than placed on sides for
nanorod-chiral molecule coupled system SHou2014 ; SHou2015 ; BHan  due to the
stronger near field at end of nanorod. Furthermore, ICD is sensitive to
slightly shape change of nanostructure ehk , becouse shape changes lead to
near field changes. Despite the properties of ICD is gradually being clear,
but mechanism of ICD induction is need to investigate. And the above studies
suggests the cause of ICD induction is strongly associated with near field
characteristics such as electric field strength, magnetic field strength and
near field distribution. Although previous studies have obtained the origin of
CD enhancement in chiral molecule-nanostructure interaction based on chiral
Poynting’s theorem SLee , but cause of ICD induction has not explained. The
molecule-plasmon Coulomb interaction model correctly reproduces ICD through
incident light electric field strength and size parameters of nanostructure
Govorov2010 ; AOGovorov2012 , but it can not also explain the cause of ICD
induction due to ICD is strongly associated with characteristics of near
field. Therefore, quantitative studies on near field interactions between
chiral molecule and nanostructure, and find mechanism of chirality transfer
from chiral molecule to achiral structure is desirable. Specifically,
quantitatively studying the effects of electric field strength, magnetic field
strength, molecular chirality parameters, molecular-nanostructure distance and
molecular density on ICD to discover the mechanism of circular dichroism
induction is helpful to improve the sensitivity of molecular chirality
detection.

In this paper, a theory based on chiral molecular dipole approaching and
molecule-nanostructure near field interaction approximation is presented to
analyze the ICD induction in chiral molecule-achiral nanostructure
interaction, and we use the theory to explain the mechanism of ICD induction
in achiral nanostructure. To prove the theory, a chiral molecule-graphene
nanohole metasurface (GNM) composed system is proposed for simulation. The
simulated results are consistent with the analytical results. These results
would be helpful for realizing supersensitive detection of molecular
chirality.

##  II Theory

Absorption of nanostructure is formally calculated by volume integral
$\frac{\omega}{2}\textrm{Im}(\varepsilon)\oint_{V}|\textrm{{E}}|^{2}dV$ SLee ;
AOGovorov2012 . In presence of chiral molecule, electric field strength in
nanostructure comes from the contributions of the electric field generated by
surface plasmons without chiral molecules and the electric field radiated by
chiral molecular dipole. Thus, absorption of the nanostructure under
circularly polarized light (CPL) is given by BHan ; zwx

${A}_{\pm}^{ns}=\frac{\omega}{2}\mathrm{Im}(\varepsilon_{ns})\int_{V_{ns}}({\mathrm{\textbf{E}}}_{\pm}^{0}+{\mathrm{\textbf{E}}}_{\pm}^{d})^{\ast}\cdot({\mathrm{\textbf{E}}}_{\pm}^{0}+{\mathrm{\textbf{E}}}_{\pm}^{d})dV,$
(1)

where, ${\mathrm{\textbf{E}}}_{\pm}^{0}$ (+ represents RCP light, and -
represents LCP light) is the electric field strength inside the nanostructure
without the chiral molecule, ${\mathrm{\textbf{E}}}_{\pm}^{d}$ is the dipole
radiated electric field strength in nanostructure, ${A}_{\pm}^{ns}$ is the
absorption of nanostructure, $\omega$ is the angular frequency of light,
$\varepsilon_{ns}$, $V_{ns}$ are the permittivity and volume of nanostructure.

The electric fields ${\mathrm{\textbf{E}}}_{\pm}^{d}$ can be written in terms
of dyadic Green’s function ${\mathrm{\textbf{E}}}_{\pm}^{d}=i\omega
n_{mol}{\overline{\overline{G}}_{\pm}s\left(r\right)}\textbf{P}_{\pm}^{mol}$,
where, $n_{mol}$ is the molecules number density, $\textbf{P}_{\pm}^{mol}$ is
dipole strength of chiral molecule, and
$\overline{\overline{G}}_{\pm}\left(r\right)$ is the dyadic Green’s function
which is the electric field corresponding to molecular dipole at arbitrary
location. The Green function $\overline{\overline{G}}\left(r\right)$ of chiral
medium is given by IVLindell

$\displaystyle\overline{\overline{G}}\left(r\right)$ $\displaystyle=$
$\displaystyle\overline{\overline{G}}_{+}\left(r\right)+\overline{\overline{G}}_{-}\left(r\right),$
(2) $\displaystyle\overline{\overline{G}}_{\pm}\left(r\right)$
$\displaystyle=$
$\displaystyle-\frac{i\eta}{2}\left[k_{\pm}\mathrm{G}_{\pm}\left(r\right)\mathds{I}\pm\nabla\mathrm{G}_{\pm}\left(r\right)\times\mathds{I}+\nabla\nabla\left(\frac{\mathrm{G}_{\pm}\left(r\right)}{k_{\pm}}\right)\right],$

where, $\eta=\sqrt{\frac{\mu}{\varepsilon}}$ is the electrical impedance of
chiral medium, $\varepsilon$, $\mu$ are permittivity and magnetic permeability
of chiral medium, $\mathds{I}$ is unit dyadic, $r$ is the distance between
molecule and location of the observed electric field, and the function
$\mathrm{G}_{\pm}\left(r\right)$ is equal to
$\mathrm{G}_{\pm}\left(r\right)=\frac{1}{4\pi r}e^{-ik_{\pm}r}$. Here,
$k_{\pm}=(k\pm\frac{\kappa}{n_{c}})$ are wave vectors of RCP and LCP, $k$ is
wave vector of linear polarized light, $n_{c}$ is refractive index of chiral
molecule. The chiral molecule and nanostructure are interact through near
filed, thus implementing the near-field interaction approximation $k_{\pm}r\ll
1$ to calculate the gradients of function $\mathrm{G}_{\pm}\left(r\right)$, we
derive

$\displaystyle\nabla\mathrm{G}_{\pm}\left(r\right)$ $\displaystyle=$
$\displaystyle\left(ik_{\pm}+\frac{1}{r}\right)\mathrm{G}_{\pm}\left(r\right){\hat{u}_{r}}\approx\frac{1}{r}\mathrm{G}_{\pm}\left(r\right){\hat{u}_{r}},$
$\displaystyle\nabla\nabla\mathrm{G}_{\pm}\left(r\right)$ $\displaystyle=$
$\displaystyle\left[\left(ik_{\pm}+\frac{1}{r}\right)^{2}+\frac{1}{r^{2}}\right]\mathrm{G}_{\pm}\left(r\right){\hat{u}_{r}}{\hat{u}_{r}}-\frac{1}{r^{2}}\mathrm{G}_{\pm}\left(r\right)\left(\mathds{I}-{\hat{u}_{r}}{\hat{u}_{r}}\right)$
$\displaystyle\approx$
$\displaystyle\frac{3}{r^{2}}\mathrm{G}_{\pm}\left(r\right){\hat{u}_{r}}{\hat{u}_{r}}-\frac{1}{r^{2}}\mathrm{G}_{\pm}\left(r\right)\mathds{I},$

where, ${\hat{u}_{r}}{\hat{u}_{r}}$ is a diadic and ${\hat{u}_{r}}$ is a unit
vector along distance r. Replacing $\nabla\mathrm{G}_{\pm}\left(r\right)$ and
$\nabla\nabla\mathrm{G}_{\pm}\left(r\right)$ by their respective relations
from Eq. (II) and implementing near field approximation $k_{\pm}r\ll 1$, Eq.
(2) can be written as

$\overline{\overline{G}}_{\pm}\left(r\right)=\frac{-i\eta}{2r^{2}k_{\pm}}\left[3\mathrm{G}_{\pm}\left(r\right){\hat{u}_{r}}{\hat{u}_{r}}-\mathrm{G}_{\pm}\left(r\right)\mathds{I}\right].$
(4)

Dipole moment of chiral molecule is given by LDBarron ; YCTang

$\textbf{P}_{\pm}^{mol}=\alpha{\mathrm{\textbf{E}}}_{\pm}^{c}-i\frac{\kappa}{c}{\mathrm{\textbf{B}}}_{\pm}^{c},$
(5)

where, $\mathrm{\textbf{E}}_{\pm}^{c}$ and $\mathrm{\textbf{B}}_{\pm}^{c}$ are
respectively electric field strength and magnetic flux density in molecular
region, $\alpha$ and $\kappa$ are respectively the electric polarizability and
chirality parameter of chiral molecules, and $c$ is the speed of light.

Finally, we derive the dipole radiated electric field
${\mathrm{\textbf{E}}}_{\pm}^{d}$

$\displaystyle\mathrm{\textbf{E}}_{\pm}^{d}$ $\displaystyle=$ $\displaystyle
i\omega
n_{m}ol\alpha\overline{\overline{G}}{\mathrm{\textbf{E}}}_{\pm}^{c}+\frac{\omega
n_{m}ol\kappa}{c}\overline{\overline{G}}{\mathrm{\textbf{B}}}_{\pm}^{c}$ (6)
$\displaystyle=$ $\displaystyle\frac{\omega\eta
n_{mol}\alpha}{2r^{2}k}G\left(3\hat{u}_{r}\hat{u}_{r}-\mathds{I}\right)\cdot{\mathrm{\textbf{E}}}_{\pm}^{c}-\frac{i\omega\eta
n_{mol}}{2r^{2}k}\frac{\kappa}{c}G\left(3\hat{u}_{r}\hat{u}_{r}-\mathds{I}\right)\cdot{\mathrm{\textbf{B}}}_{\pm}^{c}.$

where, $G_{+}\left(r\right)+G_{-}\left(r\right)=G$.

In order to derive the ICD, we first calculate the field intensity difference
in nanostrcture

$\displaystyle(\textbf{E}_{+}^{0}+\textbf{E}_{+}^{d})\cdot(\textbf{E}_{+}^{0}+\textbf{E}_{+}^{d})^{*}-(\textbf{E}_{-}^{0}+\textbf{E}_{-}^{d})\cdot(\textbf{E}_{-}^{0}-\textbf{E}_{-}^{d})^{*}=2\textrm{Re}\left[(\textbf{E}_{+}^{0*}\cdot\textbf{E}_{+}^{d})-(\textbf{E}_{-}^{0*}\cdot\textbf{E}_{-}^{d})\right].$
(7)

Replacing $\textbf{E}_{\pm}^{d}$ to calculate the
$(\textbf{E}_{\pm}^{0*}\cdot\textbf{E}_{\pm}^{d})$, we derive

$\displaystyle(\textbf{E}_{\pm}^{0*}\cdot\textbf{E}_{\pm}^{d})=\textbf{E}_{\pm}^{0*}\cdot\left[\frac{\omega\eta
n_{mol}\alpha}{2r^{2}k}G\left(3\hat{u}_{r}\hat{u}_{r}-\mathds{I}\right){\mathrm{\textbf{E}}}_{\pm}^{c}+\frac{-i\omega\eta
n_{mol}}{2r^{2}k}\frac{\kappa}{c}G\left(3\hat{u}_{r}\hat{u}_{r}-\mathds{I}\right){\mathrm{\textbf{B}}}_{\pm}^{c}\right]$
$\displaystyle=\frac{3\omega\eta
n_{mol}\alpha}{2r^{2}k}\frac{1}{r^{2}}G\sum_{\nu}\sum_{\gamma}x_{\nu}x_{\gamma}\textrm{E}_{\pm}^{0\nu*}\textrm{E}_{\pm}^{c\gamma}-\frac{\omega\eta
n_{mol}\alpha}{2r^{2}k}G\sum_{\nu}\textrm{E}_{\pm}^{0\nu*}\textrm{E}_{\pm}^{c\nu}$
$\displaystyle-\frac{3i\omega\eta
n_{mol}\kappa}{2r^{2}kc}\frac{1}{r^{2}}G\sum_{\nu}\sum_{\gamma}x_{\nu}x_{\gamma}\textrm{E}_{\pm}^{0\nu*}\textrm{B}_{\pm}^{c\gamma}+\frac{i\omega\eta
n_{mol\kappa}}{2r^{2}kc}G\sum_{\nu}\textrm{E}_{\pm}^{0\nu*}\textrm{B}_{\pm}^{c\nu},$

Where, $\nu,\gamma=x,y,z$.

Suppose the incident light is propagate along $z$ direction, and the achiral
structure is symmetric along $y$ axis. The near field in any $xy$ plain has
properties $\textrm{E}_{+}^{x}(x,y)=\textrm{E}_{-}^{x}(x,-y)$,
$\textrm{E}_{+}^{y}(x,y)=\textrm{-E}_{-}^{y}(x,-y)$,
$\textrm{E}_{+}^{z}(x,y)=\textrm{E}_{-}^{z}(x,-y)$,
$\textrm{B}_{+}^{x}(x,y)=\textrm{-B}_{-}^{x}(x,-y)$,
$\textrm{B}_{+}^{y}(x,y)=\textrm{B}_{-}^{y}(x,-y)$,
$\textrm{B}_{+}^{z}(x,y)=\textrm{-B}_{-}^{z}(x,-y)$. Implementing these
properties to calculate Eq. (II), the field intensity difference in
nanostrcture becomes

$\displaystyle\textbf{E}_{+}^{0}\cdot\textbf{E}_{+}^{d\ast}-\textbf{E}_{-}^{0}\cdot\textbf{E}_{-}^{d\ast}$
$\displaystyle=$ $\displaystyle\frac{-3i\omega\eta
n_{mol}}{r^{2}k}\frac{\kappa}{c}G\frac{1}{r^{2}}\sum_{\nu}x_{\nu}^{2}\textrm{E}_{+}^{0\nu*}\textrm{B}_{+}^{c\nu}+\frac{i\omega\eta
n_{mol}}{r^{2}k}\frac{\kappa}{c}G\sum_{\nu}\textrm{E}_{+}^{0\nu*}\textrm{B}_{+}^{c\nu}.$
(9)

For periodic structure in $x$ and $y$ directions, Eq. (9) can approximate as

$\displaystyle(\textbf{E}_{+}^{0}+\textbf{E}_{+}^{d})\cdot(\textbf{E}_{+}^{0}+\textbf{E}_{+}^{d})^{*}-(\textbf{E}_{-}^{0}+\textbf{E}_{-}^{d})\cdot(\textbf{E}_{-}^{0}-\textbf{E}_{-}^{d})^{*}$
(10) $\displaystyle=$ $\displaystyle\frac{\eta n_{mol}}{\pi r^{3}}\textrm{
Re}\left[-i\kappa
e^{-ikr}\left(\textrm{E}_{+}^{0x*}\textrm{B}_{+}^{cx}+\textrm{E}_{+}^{0y*}\textrm{B}_{+}^{cy}\right)\right].$

Finally, the ICD generated by a molecular dipole strength P is can be written
as

$\displaystyle\textsc{ICD}=\frac{\eta
n_{mol}}{2\pi}\mathrm{Im}(\varepsilon_{m})\int_{V_{ns}}\frac{1}{r^{3}}\textrm{
Re}\left[-i\kappa
e^{-ikr}\left(\textrm{E}_{+}^{0x*}\textrm{B}_{+}^{cx}+\textrm{E}_{+}^{0y*}\textrm{B}_{+}^{cy}\right)\right]dV.$
(11)

According to Eq. (11), there are linear relationship between the ICD and
number density of chiral molecules and chirality parameter of chiral
molecules. Moreover, the ICD exponentially increases with the inverse cube of
distance between chiral molecules and nanostructure. Most importantly, ICD
proportional to electric field strength in nanostructure and magnetic flux
density in chiral molecular region. These results not only show relationships
between ICD and near field, this theory also can explain the cause of ICD
induction (Section IV).

Figure 1: Schematic drawing of chiral molecule-GNM composed system.

##  III Structure and Computational Method

In order to prove Eq. (11), we propose a chiral molecule and graphene nanohole
metasurface (GNM) composed system. Generally, the SPR wavelength of graphene
is located in microwave region, thus the condition $k_{\pm}r\ll 1$ is
completely satisfied. Figure 1(a) depicts the proposed chiral molecule-GNM
composed system. The GNM was supposed to be immersed in the bottom of
$100\,nm$ thick chiral molecule solution. The periods were fixed as
$P_{x}=P_{y}=600\,nm$, and radius of nanohole was chosen to be $150\,nm$.
Generally, the chiral molecules have absorption and MCD peak located in UV
region Fasman , for example the $\alpha$-helix has an absorption and MCD peak
located at $192\,nm$. Therefore, the chiral molecules are modeled with complex
refractive index and chirality parameter that has absorption and MCD peak at
$192\,nm$.

Finite element method (FEM) was used to perform the simulations. To simulate
for chiral medium, Maxwell equations were modulated by implementing a chiral
constitutive relation given by AHSihvola

D $\displaystyle=$
$\displaystyle\varepsilon_{0}\varepsilon_{c}{{\textbf{E}}}-\frac{i\kappa}{c}{\mathrm{\textbf{H}}},$
B $\displaystyle=$
$\displaystyle\mu_{0}\mu_{c}{{\textbf{H}}}+\frac{i\kappa}{c}{\mathrm{\textbf{E}}},$
(12)

where, $\varepsilon_{c}$, $\mu_{c}$ are the relative permittivity and magnetic
permeability of the chiral molecule, $\kappa$ is the molecular chirality
parameter, $\varepsilon_{0}$ and $\mu_{0}$ are the vacuum permittivity and
magnetic permeability, respectively. The permittivity $\varepsilon_{c}$ and
molecular chirality parameter $\kappa$ are given by LDBarron

$\displaystyle\varepsilon_{c}$ $\displaystyle=$
$\displaystyle\varepsilon_{sol}+n_{mol}\omega_{0}d\frac{\omega_{0}^{2}-\omega^{2}+i\omega\Gamma}{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+\omega^{2}\Gamma^{2}},$
$\displaystyle\kappa$ $\displaystyle=$ $\displaystyle n_{mol}\omega
R_{rot}\frac{\omega_{0}^{2}-\omega^{2}+i\omega\Gamma}{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+\omega^{2}\Gamma^{2}},$
(13)

where, $\varepsilon_{sol}$ is the relative permittivity of solvent, $d$
represents the electric dipole strength, $R_{rot}$ is the rotatory strength,
$\omega_{0}$ is molecular absorption band frequency. The parameters were set
as $\varepsilon_{sol}=1.77$, $d=1.6\times 10^{-9}m^{3}s^{-1}$,
$R_{rot}=1.6\times 10^{-11}m^{3}s^{-1}$, $n_{mol}=5\times 10^{22}m^{-3}$,
$\Gamma=3.77\times 10^{14}s^{-1}$, $\omega_{0}=9.8\times 10^{15}\mathrm{Hz}$.
The chirality parameter of a chiral molecule is represented by a matrix. The
initial molecular orientation is defined as parallel to $z$-axis, and the
unitary transformation $\kappa_{f}=R(\varphi)\kappa_{i}R^{-1}(\varphi)$ is
used for changing of molecular orientation, where, $\kappa_{i}$ is the initial
orientation and $\kappa_{f}$ is the final orientation, $R(\varphi)$ is unit
rotation matrix for angle $\varphi$ which is different for rotations about
$x$, $y$ and $z$-axis AAbudukelimu .

The conductivity of graphene was computed within the local random phase
approximation, which is a function of the frequency of incident light EHHwang
; GWHanson

$\displaystyle\sigma(\omega)$ $\displaystyle=$
$\displaystyle\frac{2ik_{B}T}{\pi{\hbar}^{2}\left(\omega+i\tau^{-1}\right)}\ln\left[2\cosh\left(\frac{E_{f}}{2k_{B}T}\right)\right]$
$\displaystyle+$
$\displaystyle\frac{e^{2}}{4\pi\hbar^{2}}\left\\{0.5+\frac{1}{\pi}\arctan\left(\frac{-\hbar\omega+2E_{f}}{2k_{B}T}\right)-\frac{i}{2\pi}\ln\left[\frac{\left(\hbar\omega+2E_{f}\right)^{2}}{\left(\hbar\omega-2E_{f}\right)^{2}+(2k_{B}T)^{2}}\right]\right\\},$

where, $T$ is temperature, $\tau$ is carrier relaxation time, $k_{B}$ is
Boltzmann constant, $v_{F}$ is Fermi velocity, $k_{B}T$ and $E_{f}$ represent
temperature energy and Fermi energy, respectively. The parameters were set as
$T=300\,K$, $E_{f}=0.6\,eV$, $v_{F}=c/300$ ($c$ is speed of light),
$\tau=\frac{{\rho}_{DC}E_{f}}{e{v}_{F}^{2}}$ and ${\rho}_{DC}=10^{4}\,cm/Vs$.

##  IV Results and Discussion

Mechanism of circular dichroism induction. According to Eq. (1) and Eq. (7),
ICD comes from different electric field strengths in nanostructure under RCP
and LCP illumination. Here, we use our theory to explain the mechanism of
generating the different electric field strengths.

The racemic mixture of chiral molecule does not possess chiral properties and
it cannot induce ICD. The radiated electric field by the racemic mixture
molecular dipole is

$\mathrm{\textbf{E}}_{\pm}^{rd}=\frac{\omega\eta
n_{mol}\alpha}{2r^{2}k}G\left(3\hat{u}_{r}\hat{u}_{r}-\mathds{I}\right)\cdot{\mathrm{\textbf{E}}}_{\pm}^{c}.$
(15)

The near field has properties $\textrm{E}_{+}^{x}=\textrm{E}_{-}^{x}$,
$\textrm{E}_{+}^{y}=\textrm{-E}_{-}^{y}$, and
$\textrm{E}_{+}^{z}=\textrm{E}_{-}^{z}$. According to Eq. (15),
$\mathrm{\textbf{E}}_{+}^{rd}$ and $\mathrm{\textbf{E}}_{-}^{rd}$ has same
enantiomorphic relationship with $\mathrm{\textbf{E}}_{+}^{0}$ and
$\mathrm{\textbf{E}}_{-}^{0}$. Thereby, the electric field in achiral
nanostructure still has properties
$\textrm{E}_{+}^{0x}+\textrm{E}_{+}^{rdx}=\textrm{E}_{-}^{0x}+\textrm{E}_{-}^{rdx}$,
$\textrm{E}_{+}^{0y}+\textrm{E}_{+}^{rdy}=-\textrm{E}_{-}^{0y}-\textrm{E}_{-}^{rdy}$,
and
$\textrm{E}_{+}^{0z}+\textrm{E}_{+}^{rdz}=\textrm{E}_{-}^{z}+\textrm{E}_{-}^{rdz}$.
This means field intensities in achiral nanostructure for RCP and LCP
illumination is equal. Therefore, racemic mixture of chiral molecule cannot
induce ICD in nanostructure.

The radiated electric field by pure chiral molecule is given in Eq. (6).
Calculate the increments of electric field strength
$\triangle\mathrm{\textbf{E}}_{\pm}^{d}=\mathrm{\textbf{E}}_{\pm}^{d}-\mathrm{\textbf{E}}_{\pm}^{rd}$
which is generated by molecular chirality, we derive (Supporting information,
Section II. B)

$\displaystyle\triangle\mathrm{\textbf{E}}_{\pm}^{d}=\mathrm{\textbf{E}}_{\pm}^{d}-\mathrm{\textbf{E}}_{\pm}^{rd}=\frac{-i\eta
n_{mol}\kappa}{2r^{2}}G\left(3\hat{u}_{r}\hat{u}_{r}-\mathds{I}\right)\cdot{\mathrm{\textbf{B}}}_{\pm}^{c}.$
(16)

If calculate the $x$ component of electric field increments
$\triangle\mathrm{{E}}_{\pm}^{dx}$ which is generated by dipole moment along
$x$ direction, we derive

$\triangle\mathrm{{E}}_{\pm}^{dx}=\frac{-i\eta n_{mol}\kappa}{16\pi
r^{3}}{\mathrm{\text{B}}}_{\pm}^{cx}e^{-ikr}.$ (17)

In a similar manner, the $y$ component of electric field increments
$\triangle\mathrm{{E}}_{\pm}^{dy}$ generated by dipole moment along $y$
direction is

$\triangle\mathrm{{E}}_{\pm}^{dy}=\frac{-i\eta n_{mol}\kappa}{16\pi
r^{3}}{\mathrm{\text{B}}}_{\pm}^{cy}e^{-ikr}.$ (18)

The $\triangle\mathrm{{E}}_{\pm}^{dx}$ has anti-enantiomorphic relation
$\triangle\mathrm{E}_{+}^{dx}=-\triangle\mathrm{E}_{-}^{dx}$ while
$\mathrm{{E}}_{\pm}^{dx}$ has enantiomorphic relation
$\mathrm{E}_{+}^{0x}=\mathrm{E}_{-}^{0x}$, and
$\triangle\mathrm{{E}}_{\pm}^{dy}$ has enantiomorphic relation
$\triangle\mathrm{{E}}_{+}^{dy}=\triangle\mathrm{{E}}_{-}^{dy}$ while
$\mathrm{{E}}_{\pm}^{dy}$ has anti-enantiomorphic relation
$\mathrm{E}_{+}^{0y}=\mathrm{-E}_{-}^{0y}$. Since the enantiomorphic relation
of $\textbf{E}^{0}$ and $\textbf{E}^{d}$ is different, electric field strength
in nanostructure generated by RCP and LCP incident light is not equal anymore,
that is
$\textrm{E}_{+}^{0x}+\textrm{E}_{+}^{rdx}\neq\textrm{E}_{-}^{0x}+\textrm{E}_{-}^{rdx}$,
$\textrm{E}_{+}^{0y}+\textrm{E}_{+}^{rdy}\neq-\textrm{E}_{-}^{0y}-\textrm{E}_{-}^{rdy}$,
and
$\textrm{E}_{+}^{0z}+\textrm{E}_{+}^{rdz}\neq\textrm{E}_{-}^{z}+\textrm{E}_{-}^{rdz}$.
The nonequal electric field strengths leading to different absorption for RCP
and LCP. This is the mechanism of ICD induction in chiral molecule-achiral
nanostructure near field interaction.

Figure 2: (a) Absorption and ICD spectrum of GNM, (b) Charge distributions on
surface of GNM.

Comparison of the analytical results to the simulation results. To prove the
theory, we proposed the chiral molecule-GNM composed system, and investigated
the ICD properties. Figure 2(a) depicts the absorption spectra of GNM. Three
SPR modes (marked as I, II, and III) in the absorption spectrum are located at
wavelengths $13.6\,\mu m$, $10.8\,\mu m$ and $9.2\,\mu m$, respectively.
Figure 2(b) depicts the surface charge distribution on GNM. The red color
corresponds to positive charge, the blue color corresponds to negative charge,
and the charge distributions for RCP and LCP has enantiomorphic relation. Mode
I is a dipolar mode because the positive and negative charges are clearly
separated in two regions. Mode II is a hexapolar mode, the positive and
negative charges are located in the six apexes of hexagon. Mode III is an
octupolar mode composed of a hexapolar and dipolar mode. Figure. 2(c) shows
the ICD spectra of GNM. Three ICD dips are appear at three modes. Associated
whit Eq. (11), we can conclude that the three ICD dips arise from electric
field strength $\textbf{E}^{0}$ and magnetic flux density $\textbf{B}^{c}$
enhancement at three modes. Since the field enhancement is largest at mode I
while smallest at mode III, the absorption and ICD are also largest at mode I
while smallest at mode III. The ICD induced by oriented molecules is larger
than that in the case of the isotropic molecules and it obtains the maximum
when molecular orientation is parallel to $x$-axis. Therefore, molecular
orientation is set as parallel to $x$\- axis in the following.

In order to prove the linear relation between molecular density and ICD, the
ICD for different molecular densities are calculated. For the sake of
simplicity, the density of molecule is normalized by
$\hat{n}=\frac{n_{mol}}{6.25\times 10^{21}\,m^{-3}}$. Figure 3(a) presents the
ICD spectra for different molecule number densities $\hat{n}$. As expected,
the ICD peak value increases with the increase of $\hat{n}$. In order to
compare with the analytical result, the linear fitting was made at wavelengths
of three modes. Figure 3(b) shows linear fitting of $\hat{n}-\textsc{ICD}$
relation at mode I (black line), mode II (red line) and mode III (blue line).
It can be seen that there are indeed linear relations between $\hat{n}$ and
ICD, and the goodness of fitting is $0.99$. This is consistent with analytical
result of Eq. (11).

Figure 3: (a) The ICD spectra for different molecule densities, (b) Linear
fitting (solid line) of ICD and molecule densities at the three modes.

Similarly, to prove the linear relation between molecular parameter $\kappa$
and ICD, the ICD for different molecular chirality parameters are calculated.
For the sake of simplicity, the chirality parameter is normalized by
$\hat{\kappa}=\frac{\kappa_{mol}}{\kappa_{mol}^{0}}$. Here, $\kappa_{mol}^{0}$
is the initial molecular chirality parameter which corresponds to
$R_{rot}=1.6\times 10^{-11}m^{3}s^{-1}$ and $n_{mol}=5\times 10^{22}m^{-3}$.
The ICD spectra with the change of $\hat{\kappa}$ is given by Fig. 4(a). The
ICD increases with increasing of $\hat{\kappa}$ at all three modes. Figure
4(b) shows the linear fitting at mode I (yellow line), mode II (olive line)
and mode III (orange line). The fitting shows linear relationships between
$\hat{\kappa}$ and ICD. Same as fitting of $\hat{n}-\textsc{ICD}$ , goodness
of fitting for $\hat{\kappa}-\textsc{ICD}$ relation is 0.99. These results
further prove proposed theory.

Figure 4: (a) The ICD spectra for different chirality parameter, (b) Linear
fitting (sold line) of ICD (Square) and chirality parameters at the three
modes.

In an effort to determine the relation between ICD and molecule-nanostructure
distance $r$, the distance $h_{r}$ from the GNM to bottom of molecular region
is changed by a step of $10\,nm$. As the GNM is moved from the bottom to the
center of the chiral molecules layer, the ICD at mode I increases and
approaches the maximum when GNM reaches the center of chiral molecular region.
And the ICD at mode II and III are decreased with increase of $h_{r}$.
According to Eq. (11), the relationship between $e^{ikr}\textsc{ICD}$ and
$\frac{1}{r^{3}}$ is linear, but electric field strength $\textbf{E}^{0}$ and
magnetic flux density $\textbf{B}^{c}$ change with $h_{r}$ thereby affecting
the ICD. Through comparing the $\textbf{E}^{0*}\cdot\textbf{B}^{c}$ and
$\frac{\textbf{E}^{0*}\cdot\textbf{B}^{c}}{r^{3}}$ spectra (Fig. S5), we
concluded that the influence of distance r is obvious, and the
$\frac{\textbf{E}^{0*}\cdot\textbf{B}^{c}}{r^{3}}$ spectra are more close to
ICD spectra. To eliminate the effect of $\textbf{E}^{0}$ and $\textbf{B}^{c}$
changes and inconvenience for fitting due to different peak positions for
different $h_{r}$, three off-resonance wavelengths $18\,{\mu}m$ , $19\,\mu m$
and $20\,{\mu}m$ are chosen to get liner fitting of
$e^{ikr}\textsc{ICD}-\frac{1}{r^{3}}$ relation. Because effects of mode shape
changes on $\textbf{E}^{0}$ and $\textbf{B}^{c}$ for different $h_{r}$ are
negligible at the three wavelengths. To fitting liner relation of
$e^{ikr}\textsc{ICD}$ and $\frac{1}{r^{3}}$ , the averaged distance between
molecule layer to GNM is calculated by $r=\frac{\sum_{i}(z_{i}-z_{0})}{i}$,
here, $z_{0}$ is the $z$ coordinate of graphene and the $z_{i}$ is the $z$
coordinate of chiral molecule layers. Figure. 5(b) shows the linear fitting at
the three wavelengths. The fitting shows linear relations between
$e^{ikr}\textsc{ICD}$ and$\frac{1}{r^{3}}$, and the
$e^{ikr}\textsc{ICD}-\frac{1}{r^{3}}$ relations are well consistent with the
analytical results.

Figure 5: (a) The ICD spectra for different $h_{r}$, (b) The linear fitting of
$\textsc{ICD}e^{ikr}-\frac{1}{r^{3}}$ relation.

For the purpose of investigating the effects of $\textbf{E}^{0}$ and
$\textbf{B}^{c}$ on the ICD, we change the structural parameters to regulate
$\textbf{E}^{0}$ and $\textbf{B}^{c}$ (Supporting information, Section II. D).
A small additional hole is set at the center of every four adjacent original
holes. And the radius of additional hole is $R_{\alpha}$ is changed from
$20\,nm$ to $100\,nm$ by a step of $20\,nm$. Figure 6(a) shows the ICD spectra
for different $R_{\alpha}$. After adding the hole, the ICD decreases at mode
I, increases at mode II and mode III. Figure 6(b) and (c) shows the averaged
$\textbf{E}^{0}$ and $\textbf{B}^{c}$ spectra for different $R_{\alpha}$. From
figure 6 it can be observed that the change of the $\textbf{E}^{0}$,
$\textbf{B}^{c}$ and ICD are similar, the reason is that ICD spectra is
regulated by the electric field $\textbf{E}^{0}$ and magnetic field
$\textbf{B}^{c}$. At mode I, ICD is increased by by $\textbf{E}^{0}$ and
$\textbf{B}^{c}$, whereas at mode II and III the ICD gets decreased by
$\textbf{E}^{0}$ and $\textbf{B}^{c}$ which agrees with Eq. (11).

Figure 6: The (a) ICD, (b) $\textsc{E}^{0}$ and (c) $\textsc{B}^{c}$ spectra
for different $R_{\alpha}$.

##  V Conclusion

Based on chiral molecule dipole approaching and chiral molecule-achiral
nanostructure near field approximation, we present a theory for quantitative
analysis of ICD from chiral molecule-achiral structure near field interaction.
The theory clarifies the mechanism of ICD is that the dipole radiation of
chiral molecules breaks the enantiomorphic relationship of the electric field
strengths in the nanostructure. Furthermore, the ICD properties of graphene
nanohole metasurface and chiral molecule coupled system are simulated to prove
the theory. The simulated results are consistent with the theoretical ones.
There are linear relations between the ICD and molecular density, molecular
chirality parameter, and the ICD exponentially increases with the inverse cube
of molecule-nanostructure distance. Finally, a smaller hole is added at the
center of four original holes to regulate the near field, and investigate the
effect of electric field strength $\textbf{E}^{0}$ in nanostructure and
magnetic flux density $\textbf{B}^{c}$ in chiral molecule region on the ICD.
As expected, the ICD is proportional to $\textbf{E}^{0}$ and $\textbf{B}^{c}$.
Both, the analytic and simulation results show two ways of enhancing the ICD.
First, decreasing the molecule-nanostructure distance, and second, enhancing
magnetic field in molecular region or enhancing electric field inside
nanostructure. These results are useful for analyzing experimental or
calculated data, and beneficial for realizing supersensitive detection of
molecular chirality by plasmon enhancing technique.

##  VI Acknowledgment

We would like to thank Zhongyue Zhang for useful discussion and valuable
comments. This work was supported by Fundamental Research Funds for the
Central Universities (Grant No. 2018TS066).

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Supporting Informations for

Quantitative Study on Circular Dichroism Induction from Achiral Nanostructure-
Chiral Matter Near Field Interactions

Abuduwaili. Abudukelimu,1 ***Email<EMAIL_ADDRESS>Zhang,1
Tursunay. Yibibulla2 and Muhammad. Ikram1

1\. School of Physics and Information Technology, Shaanxi Normal University,
Xi’an, 710119, China
2\. School of Physics and Electronics, Central South University, Changsha,
410083, P. R. China

##  I Circular dichroism induction by dipole radiated electric field

Circular dichroism (CD) is absorption difference of right circularly polarized
(RCP) and left circularly polarized (LCP) light, and it is a characteristic
property of chiral media. The essential difference between chiral and achiral
media lies in the constitutive relations. The constitutive relation for a
chiral medium is given by SAHSihvola

$\begin{array}[]{l}\textbf{D}=\varepsilon_{0}\varepsilon_{c}{{\textbf{E}}}-\frac{i\kappa}{c}{\mathrm{\textbf{H}}}\\\
\textbf{B}=\mu_{0}\mu_{c}{{\textbf{H}}}+\frac{i\kappa}{c}{\mathrm{\textbf{E}}}\end{array}$
(S1)

where, $\varepsilon_{c}$, $\mu_{c}$ are respectively the relative permittivity
and magnetic permeability of the chiral molecule, $\kappa$ is the molecular
chirality parameter, $\varepsilon_{0}$ and $\mu_{0}$ are the vacuum
permittivity and magnetic permeability, respectively.

Generally, the absorption of a nanostructure is given bySAOGovorov2012

${A}_{\pm}^{ns}=\frac{\omega}{2}\int_{V_{ns}}\left[\mathrm{Im}\left(\varepsilon_{ns}\right)\left({\mathrm{\textbf{E}}}_{\pm}^{0}\cdot{\mathrm{\textbf{E}}}_{\pm}^{0*}\right)+\mathrm{Im}\left(\mu_{ns}\right)\left({\mathrm{\textbf{H}}}_{\pm}^{0}\cdot{\mathrm{\textbf{H}}}_{\pm}^{0*}\right)\right]dV,$
(S2)

where, $\omega$ is angular frequency of CPL, $\varepsilon_{ns}$ and $\mu_{ns}$
are relative permittivity and permeability of nanostructure,
${\mathrm{\textbf{E}}}_{\pm}^{0}$ and ${\mathrm{\textbf{H}}}_{\pm}^{0}$ are
electric and magnetic field strengths. The magnetic susceptibility of
plasmonic nanostructure is very small, therefore magnetic loss can be
neglected. In presence of chiral molecule, ${\mathrm{\textbf{E}}}_{\pm}^{0}$
will be affected by the electric field radiated by dipole moment of chiral
molecule. Then Eq. (S2) can be written SBHan

${A}_{\pm}^{ns}=\frac{\omega}{2}\mathrm{Im}\left(\varepsilon_{ns}\right)\int_{V_{ns}}\left({\mathrm{\textbf{E}}}_{\pm}^{0}+{\mathrm{\textbf{E}}}_{\pm}^{d}\right)^{\ast}\cdot\left({\mathrm{\textbf{E}}}_{\pm}^{0}+{\mathrm{\textbf{E}}}_{\pm}^{d}\right)dV,$
(S3)

where, ${\mathrm{\textbf{E}}}_{\pm}^{d}$ is dipole radiated electric field
which can be written in terms of Green’s function. For time harmonic
electromagnetic fields, the Green functions are dyadic functions, because the
source is vector. Thus, the dipole radiated electric field can be written in
terms of Green dyadic integral form

$\mathrm{\textbf{E}}^{d}\left(r\right)=\int_{V^{\prime}}\left[\overline{\overline{G}}_{ee}\left(\textbf{r}-\textbf{r}^{\prime}\right)\qquad\overline{\overline{G}}_{em}\left(\textbf{r}-\textbf{r}^{\prime}\right)\right]\left(\begin{array}[]{ccc}\textbf{J}\\\
\textbf{M}\\\ \end{array}\right)$ (S4)

where the J and M electric and magnetic current. The Green functions satisfy
the Maxwell equations. Using chiral constitutive relation to solve the Maxwell
equations we derive Green functions for chiral mediumSIVLindell

$\displaystyle\overline{\overline{G}}_{ee}\left(r\right)$ $\displaystyle=$
$\displaystyle\overline{\overline{G}}_{ee+}\left(r\right)+\overline{\overline{G}}_{ee-}\left(r\right)$
$\displaystyle\overline{\overline{G}}_{ee\pm}\left(r\right)$ $\displaystyle=$
$\displaystyle-\frac{i\eta}{2}\left[k_{\pm}\mathrm{G}_{\pm}\left(r\right)\mathds{I}\pm\nabla\mathrm{G}_{\pm}\left(r\right)\times\mathds{I}+\nabla\nabla\left(\frac{\mathrm{G}_{\pm}\left(r\right)}{k_{\pm}}\right)\right],$
(S5) $\displaystyle\overline{\overline{G}}_{em}\left(r\right)$
$\displaystyle=$
$\displaystyle\overline{\overline{G}}_{em+}\left(r\right)+\overline{\overline{G}}_{em-}\left(r\right)$
$\displaystyle\overline{\overline{G}}_{em\pm}\left(r\right)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left[\pm
k_{\pm}\mathrm{G}_{\pm}\left(r\right)\mathds{I}+\nabla\mathrm{G}_{\pm}\left(r\right)\times\mathds{I}\pm\nabla\nabla\left(\frac{\mathrm{G}_{\pm}\left(r\right)}{k_{\pm}}\right)\right]$

where $\mathrm{G}_{\pm}\left(r\right)=\frac{e^{-ik_{\pm}r}}{4\pi r}$,
$k_{\pm}=k\left(1\pm\frac{\kappa}{n_{c}}\right)$ and $n_{c}$ is refractive
index of chiral medium. Using the approximation $k_{\pm}r\ll 1$ for near field
calculation, and calculating the gradient terms in Eq. (I) becomes

$\displaystyle\nabla\mathrm{G}_{\pm}\left(r\right)$ $\displaystyle=$
$\displaystyle\left(-ik_{\pm}+\frac{1}{r}\right)\mathrm{G}_{\pm}\left(r\right)\hat{u}_{r}\approx\frac{1}{r}\mathrm{G}_{\pm}\left(r\right)\hat{u}_{r},$
$\displaystyle\nabla\nabla\mathrm{G}_{\pm}\left(r\right)$ $\displaystyle=$
$\displaystyle\left[\left(ik_{\pm}+\frac{1}{r}\right)^{2}+\frac{1}{r^{2}}\right]\mathrm{G}_{\pm}\left(r\right)\hat{u}_{r}\hat{u}_{r}-\left(ik_{\pm}+\frac{1}{r}\right)\frac{1}{r}\left(\mathds{I}-\hat{u}_{r}\hat{u}_{r}\right)\mathrm{G}_{\pm}\left(r\right)$
$\displaystyle\approx$
$\displaystyle\frac{3}{r^{2}}\frac{\mathrm{G}_{\pm}\left(r\right)}{k_{\pm}}\hat{u}_{r}\hat{u}_{r}-\frac{1}{r^{2}}\frac{\mathrm{G}_{\pm}\left(r\right)}{k_{\pm}}\mathds{I}.$

where $\hat{u}_{r}$ is unit vector in r direction, $\hat{u}_{r}\hat{u}_{r}$ is
dyadic, and can be written as,

$\displaystyle\hat{u}_{r}=\frac{1}{r}\left(\begin{array}[]{ccc}x\\\ y\\\ z\\\
\end{array}\right),\qquad\hat{u}_{r}\hat{u}_{r}=\frac{1}{r^{2}}\left(\begin{array}[]{ccc}x^{2}&xy&xz\\\
yx&y^{2}&yz\\\ zx&zy&z^{2}\\\ \end{array}\right).$ (S13)

Thus the dyadic Green functions becomes

$\displaystyle\overline{\overline{G}}_{ee+}\left(r\right)$ $\displaystyle=$
$\displaystyle\frac{-i\eta}{2}\left[k_{+}G_{+}\left(r\right)-\frac{1}{r}G_{+}\left(r\right)\hat{u}_{r}\times\mathds{I}+\frac{3}{r^{2}}\frac{G_{+}\left(r\right)}{k_{+}}\hat{u}_{r}\hat{u}_{r}-\frac{1}{r^{2}}\frac{G_{+}\left(r\right)}{k_{+}}\mathds{I}\right],$
$\displaystyle\overline{\overline{G}}_{ee-}\left(r\right)$ $\displaystyle=$
$\displaystyle\frac{-i\eta}{2}\left[k_{-}G_{-}\left(r\right)+\frac{1}{r}G_{-}\left(r\right)\hat{u}_{r}\times\mathds{I}+\frac{3}{r^{2}}\frac{G_{-}\left(r\right)}{k_{-}}\hat{u}_{r}\hat{u}_{r}-\frac{1}{r^{2}}\frac{G_{-}\left(r\right)}{k_{-}}\mathds{I}\right].$
(S14)

Implementing $k_{\pm}r\ll 1$, we derive

$\displaystyle\overline{\overline{G}}_{ee+}\left(r\right)$ $\displaystyle=$
$\displaystyle\frac{-i\eta}{2r^{2}k}\left[{3}G_{+}\left(r\right)\hat{u}_{r}\hat{u}_{r}-G_{+}\left(r\right)\mathds{I}\right],$
$\displaystyle\overline{\overline{G}}_{ee-}\left(r\right)$ $\displaystyle=$
$\displaystyle\frac{-i\eta}{2r^{2}k}\left[{3}G_{-}\left(r\right)\hat{u}_{r}\hat{u}_{r}-G_{-}\left(r\right)\mathds{I}\right],$
(S15)

where $k_{\pm}=k(1\pm\frac{\kappa}{1.77})$. Since $\kappa$ is of the order of
$10^{-5}$, we can approximate $k_{\pm}=k$. Finally, we derive the dyadic Green
functions for near field

$\displaystyle\overline{\overline{G}}_{ee}\left(r\right)$ $\displaystyle=$
$\displaystyle\frac{-i\eta}{2r^{2}k}\left[G_{+}\left(r\right)+G_{-}\left(r\right)\right]\left({3}\hat{u}_{r}\hat{u}_{r}-\mathds{I}\right),$
$\displaystyle=$ $\displaystyle\frac{-i\eta}{2r^{2}k}\frac{1}{4\pi
r}e^{-ikr}\left(e^{-ik\frac{\kappa}{n_{c}}}+e^{ik\frac{\kappa}{n_{c}}}\right)\left({3}\hat{u}_{r}\hat{u}_{r}-\mathds{I}\right),$
$\displaystyle=$ $\displaystyle\frac{-i\eta}{2r^{2}k}\frac{1}{2\pi
r}e^{-ikr}cosh\left(-ikr\frac{\kappa}{n_{c}}\right)\left({3}\hat{u}_{r}\hat{u}_{r}-\mathds{I}\right),$

In the same method we derive

$\displaystyle\overline{\overline{G}}_{em}\left(r\right)$ $\displaystyle=$
$\displaystyle\frac{-1}{2r^{2}k}\left[G_{-}\left(r\right)+G_{-}\left(r\right)\right]\left({3}\hat{u}_{r}\hat{u}_{r}-\mathds{I}\right),$
$\displaystyle=$ $\displaystyle\frac{-1}{2r^{2}k}\frac{1}{4\pi
r}e^{-ikr}\left(e^{-ik\frac{\kappa}{n_{c}}}-e^{ik\frac{\kappa}{n_{c}}}\right)\left({3}\hat{u}_{r}\hat{u}_{r}-\mathds{I}\right),$
$\displaystyle=$ $\displaystyle\frac{-1}{2r^{2}k}\frac{1}{2\pi
r}e^{-ikr}sinh\left(-ikr\frac{\kappa}{n_{c}}\right)\left({3}\hat{u}_{r}\hat{u}_{r}-\mathds{I}\right)\approx
0.$

From Eqs. (S4), (I) and (I), we derive

$\textbf{E}_{\pm}^{d}=\int_{V^{\prime}}\overline{\overline{G}}_{ee}\cdot\textbf{J}dV^{\prime}.$
(S18)

The electric current can be calculated by
$\textbf{J}_{\pm}=\frac{\partial}{\partial
t}\textbf{P}_{\pm}=i\omega\textbf{P}_{\pm}$. And the polarization strength of
chiral molecule is given by SYCTang

$\textbf{P}_{\pm}=n_{mol}\alpha\textbf{E}_{\pm}^{c}-n_{mol}\frac{i\kappa}{c}\textbf{B}_{\pm}^{c},$
(S19)

where $n_{mol}$ is the number density of molecule. Thus, the radiated electric
field by chiral molecular dipole is

$\textbf{E}_{\pm}^{d}=i\omega
n_{mol}\alpha\overline{\overline{G}}\cdot\textbf{E}_{\pm}^{c}+\omega
n_{mol}\frac{\kappa}{c}\overline{\overline{G}}\cdot\textbf{B}_{\pm}^{c}.$
(S20)

The induced ICD in the nanostucture is

ICD $\displaystyle=$
$\displaystyle\frac{\omega}{2}\mathrm{Im}(\varepsilon_{ns})\int_{V_{ns}}({\mathrm{\textbf{E}}}_{+}^{0}+{\mathrm{\textbf{E}}}_{+}^{d})^{\ast}\cdot({\mathrm{\textbf{E}}}_{+}^{0}+{\mathrm{\textbf{E}}}_{+}^{d})dV-\frac{\omega}{2}\mathrm{Im}(\varepsilon_{ns})\int_{V_{ns}}({\mathrm{\textbf{E}}}_{-}^{0}+{\mathrm{\textbf{E}}}_{-}^{d})^{\ast}\cdot({\mathrm{\textbf{E}}}_{-}^{0}+{\mathrm{\textbf{E}}}_{-}^{d})dV$
(S21) $\displaystyle=$
$\displaystyle\frac{\omega}{2}\mathrm{Im}(\varepsilon_{ns})\int_{V_{ns}}\left[\left({\mathrm{\textbf{E}}}_{+}^{0}+{\mathrm{\textbf{E}}}_{+}^{d}\right)^{\ast}\cdot\left({\mathrm{\textbf{E}}}_{+}^{0}+{\mathrm{\textbf{E}}}_{+}^{d}\right)-\left({\mathrm{\textbf{E}}}_{-}^{0}+{\mathrm{\textbf{E}}}_{-}^{d}\right)^{\ast}\cdot({\mathrm{\textbf{E}}}_{-}^{0}+{\mathrm{\textbf{E}}}_{-}^{d})\right]dV.$

The dipole radiated electric field $\textbf{E}^{d}$ is much smaller than
original near field in nanostructure. Therefore, after neglecting the second
order term $\textbf{E}^{d}\cdot\textbf{E}^{d\ast}$ we derive

${\textrm{ICD}}=\omega\mathrm{Im}(\varepsilon_{ns})\int_{V_{ns}}\textrm{Re}\left(\textbf{E}_{+}^{0}\cdot\textbf{E}_{+}^{d\ast}-\textbf{E}_{-}^{0}\cdot\textbf{E}_{-}^{d\ast}\right)dV.$
(S22)

Replace $\textbf{E}^{d}$ to calculate
$\textbf{E}_{+}^{0}\cdot\textbf{E}_{-}^{d\ast}$

$\displaystyle\left(\textbf{E}_{\pm}^{0*}\cdot\textbf{E}_{\pm}^{d}\right)=\textbf{E}_{\pm}^{0*}\cdot\left[\frac{\omega\eta
n_{mol}\alpha}{2r^{2}k}G\left(3\hat{u}_{r}\hat{u}_{r}-\mathds{I}\right){\mathrm{\textbf{E}}}_{\pm}^{c}+\frac{-i\omega\eta
n_{mol}}{2r^{2}k}\frac{\kappa}{c}G\left(3\hat{u}_{r}\hat{u}_{r}-\mathds{I}\right){\mathrm{\textbf{B}}}_{\pm}^{c}\right]$
$\displaystyle=\frac{3\omega\eta
n_{mol}\alpha}{2r^{2}k}\frac{1}{r^{2}}G\textrm{E}_{\pm}^{0x*}\left(x^{2}\textrm{E}_{\pm}^{cx}+xy\textrm{E}_{\pm}^{cy}+xz\textrm{E}_{\pm}^{cz}\right)+\frac{3\omega\eta
n_{mol}\alpha}{2r^{2}k}\frac{1}{r^{2}}G\textrm{E}_{\pm}^{0y*}\left(yx\textrm{E}_{\pm}^{cx}+y^{2}\textrm{E}_{\pm}^{cy}+yz\textrm{E}_{\pm}^{cz}\right)$
$\displaystyle+\frac{3\omega\eta
n_{mol}\alpha}{2r^{2}k}\frac{1}{r^{2}}G\textrm{E}_{\pm}^{0z*}\left(zx\textrm{E}_{\pm}^{cx}+zy\textrm{E}_{\pm}^{cy}+z^{2}\textrm{E}_{\pm}^{cz}\right)-\frac{\omega\eta
n_{mol}\alpha}{2r^{2}k}G\left(\textrm{E}_{\pm}^{0z*}\textrm{E}_{\pm}^{cx}+\textrm{E}_{\pm}^{0z*}\textrm{E}_{\pm}^{cy}+\textrm{E}_{\pm}^{0z*}\textrm{E}_{\pm}^{cz}\right)$
$\displaystyle-\frac{3i\omega\eta
n_{mol}\kappa}{2r^{2}kc}\frac{1}{r^{2}}G\textrm{E}_{\pm}^{0x*}\left(x^{2}\textrm{B}_{\pm}^{cx}+xy\textrm{B}_{\pm}^{cy}+xz\textrm{B}_{\pm}^{cz}\right)-\frac{3i\omega\eta
n_{mol}\kappa}{2r^{2}kc}\frac{1}{r^{2}}G\textrm{E}_{\pm}^{0y*}\left(yx\textrm{B}_{\pm}^{cx}+y^{2}\textrm{B}_{\pm}^{cy}+zy\textrm{B}_{\pm}^{cz}\right)$
$\displaystyle-\frac{3i\omega\eta
n_{mol}\kappa}{2r^{2}kc}\frac{1}{r^{2}}G\textrm{E}_{\pm}^{0z*}\left(zx\textrm{B}_{\pm}^{cx}+zy\textrm{B}_{\pm}^{cy}+z^{2}\textrm{B}_{\pm}^{cz}\right)+\frac{i\omega\eta
n_{mol\kappa}}{2r^{2}kc}G\left(\textrm{E}_{\pm}^{0x*}\textrm{B}_{\pm}^{cx}+\textrm{E}_{\pm}^{0x*}\textrm{B}_{\pm}^{cy}+\textrm{E}_{\pm}^{0x*}\textrm{B}_{\pm}^{cz}\right),$

where $G=G_{+}\left(r\right)+G_{-}\left(r\right)$.

If we suppose that the incident light propagates along $z$ direction and the
achiral naonostructure is symmetric to $y$ or $x$-axis (any achiral structure
has a symmetry axis), then near field in the $xy$-plane has properties
$\textrm{E}_{+}^{x}(x,y)=\textrm{E}_{-}^{x}(x,-y)$,
$\textrm{E}_{+}^{y}(x,y)=\textrm{-E}_{-}^{y}(x,-y)$,
$\textrm{E}_{+}^{z}(x,y)=\textrm{E}_{-}^{z}(x,-y)$,
$\textrm{B}_{+}^{x}(x,y)=\textrm{-B}_{-}^{x}(x,-y)$,
$\textrm{B}_{+}^{y}(x,y)=\textrm{B}_{-}^{y}(x,-y)$,
$\textrm{B}_{+}^{z}(x,y)=\textrm{-B}_{-}^{z}(x,-y)$. Applying coordinate
transformations $y\rightarrow-y$ or $x\rightarrow-x$ (depending on the axis of
symmetry) to
$\frac{1}{r^{2}}(x-x^{\prime})^{2}\textrm{E}_{+}^{0x^{\prime}\ast}(x^{\prime},y^{\prime})\textrm{E}_{+}^{cx}(x,y)$
, we derive

$\frac{1}{r^{2}}(x-x^{\prime})^{2}\textrm{E}_{+}^{0x^{\prime}\ast}(x^{\prime},-y^{\prime})\textrm{E}_{+}^{cx}(x,-y)=\frac{1}{r^{2}}(x-x^{\prime})^{2}\textrm{E}_{-}^{0x^{\prime}\ast}(x^{\prime},y^{\prime})\textrm{E}_{-}^{cx}(x,y).$
(S24)

Eq. (S24) means the term
$\frac{1}{r^{2}}(x-x^{\prime})^{2}\textrm{E}^{0x^{\prime}\ast}(x^{\prime},y^{\prime})\textrm{E}^{cx}(x,y)$
is equal for RCP and LCP illumination. Using the same coordinate
transformations to other terms , we finally derive

$\displaystyle\textbf{E}_{+}^{0}\cdot\textbf{E}_{+}^{d\ast}-\textbf{E}_{-}^{0}\cdot\textbf{E}_{-}^{d\ast}$
$\displaystyle=$ $\displaystyle\frac{-6i\omega\eta
n_{mol}}{r^{2}k}\frac{\kappa}{c}G\frac{1}{r^{2}}\left(x^{2}\textrm{E}_{+}^{0x\ast}\textrm{B}_{+}^{cx}+y^{2}\textrm{E}_{+}^{0y\ast}\textrm{B}_{+}^{cy}+z^{2}\textrm{E}_{+}^{0z\ast}\textrm{B}_{\pm}^{cz}\right)$
$\displaystyle+$ $\displaystyle\frac{2i\omega\eta
n_{mol}}{r^{2}k}\frac{\kappa}{c}G\left(\textrm{E}_{+}^{0x\ast}\textrm{B}_{+}^{cx}+\textrm{E}_{+}^{0y\ast}\textrm{B}_{+}^{cy}+\textrm{E}_{+}^{0z\ast}\textrm{B}_{+}^{cz}\right).$

Eq. (I) is general form of field intensity difference generated by chiral
molecular dipole. The $z$ component of near field is much smaller than $x$ and
$y$ components, neglecting the $z$ component we derive

$\displaystyle\textbf{E}_{+}^{0}\cdot\textbf{E}_{+}^{d\ast}-\textbf{E}_{-}^{0}\cdot\textbf{E}_{-}^{d\ast}$
$\displaystyle=$ $\displaystyle\frac{-6i\omega\eta
n_{mol}}{r^{2}k}\frac{\kappa}{c}G\frac{1}{r^{2}}\left(x^{2}\textrm{E}_{+}^{0x\ast}\textrm{B}_{+}^{cx}+y^{2}\textrm{E}_{+}^{0y\ast}\textrm{B}_{+}^{cy}\right)$
$\displaystyle+$ $\displaystyle\frac{2i\omega\eta
n_{mol}}{r^{2}k}\frac{\kappa}{c}G\left(\textrm{E}_{+}^{0x\ast}\textrm{B}_{+}^{cx}+\textrm{E}_{+}^{0y\ast}\textrm{B}_{+}^{cy}\right).$

For planner nanostructure (in our GNM-chiral molecule composed system, $x$ and
$y$ are infinite while $z$ is up to $\textrm{100}\,nm$ ) the the distance $r$
can approximated as $r=(x^{2}+y^{2})^{1/2}$, then Eq. (I) becomes

$\displaystyle\textbf{E}_{+}^{0}\cdot\textbf{E}_{+}^{d\ast}-\textbf{E}_{-}^{0}\cdot\textbf{E}_{-}^{d\ast}=$
(S27) $\displaystyle\frac{-6i\omega\eta
n_{mol}\kappa}{r^{2}kc}G\left(cos^{2}{\alpha}\textrm{E}_{+}^{0x\ast}\textrm{B}_{+}^{cx}+sin^{2}{\alpha}\textrm{E}_{+}^{0y\ast}\textrm{B}_{+}^{cy}\right)+\frac{2i\omega\eta
n_{mol}\kappa}{r^{2}kc}G\left(\textrm{E}_{\pm}^{0x\ast}\textrm{B}_{+}^{cx}+\textrm{E}_{+}^{0y\ast}\textrm{B}_{+}^{cy}\right).$

where $\alpha$ is angle between $r$ and $x$-axis. If the chiral molecule is
uniformly arranged around the achiral structure, taking average values of
$sin^{2}(\alpha)$ and $cos^{2}(\alpha)$, Eq. (S27) becomes

$\displaystyle\textbf{E}_{+}^{0}\cdot\textbf{E}_{+}^{d\ast}-\textbf{E}_{-}^{0}\cdot\textbf{E}_{-}^{d\ast}=-\frac{i\omega\eta
n_{mol}}{r^{2}k}\frac{\kappa}{c}G\left(\textrm{E}_{+}^{0x\ast}\textrm{B}_{+}^{cx}+\textrm{E}_{+}^{0y\ast}\textrm{B}_{+}^{cy}\right)$
(S28) $\displaystyle=$ $\displaystyle-\frac{i\omega\eta
n_{mol}}{r^{2}k}\frac{\kappa}{c}\frac{1}{4\pi
r}e^{-ikr}\left(e^{-ik\frac{\kappa}{n_{c}}}+e^{ik\frac{\kappa}{n_{c}}}\right)\left(\textrm{E}_{+}^{0x\ast}\textrm{B}_{+}^{cx}+\textrm{E}_{+}^{0y\ast}\textrm{B}_{+}^{cy}\right)$
$\displaystyle=$ $\displaystyle-\frac{i\omega\eta
n_{mol}}{r^{2}k}\frac{\kappa}{c}\frac{1}{2\pi
r}e^{-ikr}cosh\left(-ikr\frac{\kappa}{n_{c}}\right)\left(\textrm{E}_{+}^{0x\ast}\textrm{B}_{+}^{cx}+\textrm{E}_{+}^{0y\ast}\textrm{B}_{+}^{cy}\right)$
$\displaystyle=$ $\displaystyle-\frac{i\eta n_{mol}}{2\pi r^{3}}\kappa
e^{-ikr}\left(\textrm{E}_{+}^{0x\ast}\textrm{B}_{+}^{cx}+\textrm{E}_{+}^{0y\ast}\textrm{B}_{+}^{cy}\right),$

where $cosh\left(-ikr\frac{\kappa}{n_{c}}\right)\approx 1$.

Finally, we derive the ICD generated by a molecular dipole strength P in the
nanostructure

$\textsc{ICD}=\frac{\eta
n_{mol}}{2\pi}\mathrm{Im}(\varepsilon_{ns})\int_{V_{ns}}\frac{1}{r^{3}}\textrm{
Re}\left[-i\kappa
e^{-ikr}\left(\textrm{E}_{+}^{0x\ast}\textrm{B}_{+}^{cx}+\textrm{E}_{+}^{0y\ast}\textrm{B}_{+}^{cy}\right)\right]dV.$
(S29)

##  II Analyzing and approximations

###  II.1 Enantiomorphic properties of near field around achiral structure

Suppose if the incident light propagates along $z$ direction, the near field
in $xy$-plain has enantiomorphic relations
$\textrm{E}_{+}^{x}(x,y)=\textrm{E}_{-}^{x}(x,-y)$,
$\textrm{E}_{+}^{z}(x,y)=\textrm{E}_{-}^{z}(x,-y)$,
$\textrm{B}_{+}^{y}(x,y)=\textrm{B}_{-}^{y}(x,-y)$ and anti-elastomeric
relation $\textrm{E}_{+}^{y}(x,y)=\textrm{-E}_{-}^{y}(x,-y)$,
$\textrm{B}_{+}^{x}(x,y)=\textrm{-B}_{-}^{x}(x,-y)$,
$\textrm{B}_{+}^{z}(x,y)=\textrm{-B}_{-}^{z}(x,-y)$. Figure S1 shows the
electric field and magnetic flux distribution of GNM at mode II. The average
electric and magnetic field values satisfy
$\textrm{E}_{+}^{x}=\textrm{E}_{-}^{x}$,
$\textrm{E}_{+}^{y}=\textrm{-E}_{-}^{y}$,
$\textrm{E}_{+}^{z}=\textrm{E}_{-}^{z}$,
$\textrm{B}_{+}^{x}=\textrm{-B}_{-}^{x}$,
$\textrm{B}_{+}^{y}=\textrm{B}_{-}^{y}$,
$\textrm{B}_{+}^{z}=\textrm{-B}_{-}^{z}$,
${\textrm{E}_{+}^{z}}\ll{\textrm{E}_{+}^{x}}\,,\,{\textrm{E}_{+}^{y}}$ and
${\textrm{B}_{+}^{z}}\ll{\textrm{B}_{+}^{x}}\,,\,{\textrm{B}_{+}^{y}}$ .
Figure S2 shows the average electric field and magnetic flux spectra of GNM.

Figure S1:  Electric and magnetic field distribution at surface of GNM Figure
S2: Average $x$, $y$ and $z$ components of (a-c) electric field
$\textbf{E}^{c}$, (d-f) magnetic field $\textbf{B}^{c}$ and (g-i) electric
field $\textbf{E}^{0}$.

###  II.2 Enetiomeric relation breaking of electric fields in achiral
nanostructure

The racemic mixture of chiral molecule has no chirality and it does not induce
ICD. The pure chiral molecule breaks the electric field symmetry in achiral
structure and induces the ICD signal. The averaged electric field in achiral
nanostrcture and molecular region has properties
$\textrm{E}_{+}^{0x}=\textrm{E}_{-}^{0x}$,
$\textrm{E}_{+}^{0y}=\textrm{-E}_{-}^{0y}$,
$\textrm{E}_{+}^{0z}=\textrm{E}_{-}^{0z}$, and
${\mathrm{\textbf{E}}}_{\pm}^{c}$ satisfies the constitutive relations

$\displaystyle\textbf{D}^{rc}$ $\displaystyle=$
$\displaystyle\varepsilon_{0}\varepsilon_{c}\textbf{E}^{c},$
$\displaystyle\textbf{B}^{rc}$ $\displaystyle=$
$\displaystyle\mu_{0}\mu_{c}\textbf{H}^{c}.$

The polarization strength of racemic mixture is given by

$\displaystyle\textbf{P}_{\pm}^{r}=n_{mol}\alpha\textbf{E}^{c},$

The electric field radiated by the racemic mixture molecular dipole is

$\mathrm{\textbf{E}}_{\pm}^{rd}=\frac{\omega\eta
n_{mol}\alpha}{2r^{2}k}G\left(3\hat{u}_{r}\hat{u}_{r}-\mathds{I}\right)\cdot{\mathrm{\textbf{E}}}_{\pm}^{c}.$
(S30)

According to Eq. (S30) $\mathrm{\textbf{E}}_{+}^{rd}$ and
$\mathrm{\textbf{E}}_{-}^{rd}$ has same enantiomorphic relationship with
$\mathrm{\textbf{E}}_{+}^{0}$ and $\mathrm{\textbf{E}}_{-}^{0}$. The electric
field in achiral nanostructure still hass properties
$\textrm{E}_{+}^{0x}+\textrm{E}_{+}^{rdx}=\textrm{E}_{-}^{0x}+\textrm{E}_{-}^{rdx}$,
$\textrm{E}_{+}^{0y}+\textrm{E}_{+}^{rdy}=-\textrm{E}_{-}^{0y}-\textrm{E}_{-}^{rdy}$,
and
$\textrm{E}_{+}^{0z}+\textrm{E}_{+}^{rdz}=\textrm{E}_{-}^{z}+\textrm{E}_{-}^{rdz}$.
With the properties, field intensities in achiral nanostructure for RCP and
LCP illumination is euqal. Therefore, racemic mixture of chiral molecule can
not induce ICD in nanostructure.

If we calculate the difference of $\mathrm{\textbf{E}}_{\pm}^{rd}$ and
$\mathrm{\textbf{E}}_{\pm}^{d}$, we derive

$\triangle\mathrm{\textbf{E}}_{\pm}^{d}=\mathrm{\textbf{E}}_{\pm}^{d}-\mathrm{\textbf{E}}_{\pm}^{rd}=\frac{-i\eta
n_{mol}\kappa}{2r^{2}}G\left(3\hat{u}_{r}\hat{u}_{r}-\mathds{I}\right)\cdot{\mathrm{\textbf{B}}}_{\pm}^{c}.$
(S31)

Calculating the electric field difference $\triangle\mathrm{{E}}_{\pm}^{dx}$
which is generated by dipole moment $n_{mol}\textrm{P}^{x}\hat{u}_{x}$ along
$x$ axis, we derive

$\displaystyle\triangle\mathrm{{E}}_{\pm}^{dx}$ $\displaystyle=$
$\displaystyle\mathrm{{E}}_{\pm}^{d}-\mathrm{{E}}_{\pm}^{rd}=\frac{-i\eta
n_{mol}\kappa}{2r^{2}}G\left(3\hat{u}_{r}\hat{u}_{r}-\mathds{I}\right)\cdot{\mathrm{{B}}}_{\pm}^{cx}$
(S32) $\displaystyle=$ $\displaystyle\frac{-i\eta
n_{mol}\kappa}{2r^{2}}\frac{1}{4\pi
r}e^{-ikr}e^{-ikr\frac{\kappa}{n_{c}}}\left(\frac{3x^{2}}{r^{2}}{{\text{B}}}_{\pm}^{cx}-{\mathrm{{B}}}_{\pm}^{cx}\right)$
$\displaystyle=$ $\displaystyle\frac{-i\eta n_{mol}\kappa}{8\pi
r^{3}}e^{-ikr}\left(3{cos}^{2}\alpha{\mathrm{\text{B}}}_{\pm}^{cx}-{\mathrm{\text{B}}}_{\pm}^{cx}\right)=\frac{-i\eta
n_{mol}\kappa}{16\pi r^{3}}{\mathrm{\text{B}}}_{\pm}^{cx}e^{-ikr},$

where $e^{-ikr\frac{\kappa}{n_{c}}}\approx 1$, and average value of
${cos}^{2}$ is 0.5.

Figure S3:  Schematic of electric field symmetry breaking in achiral
nanostructure.

In a similar manner, the $y$ component of electric field increments
$\triangle\mathrm{{E}}_{\pm}^{dy}$ generated by dipole moment along $y$
direction is

$\triangle\mathrm{{E}}_{\pm}^{dy}=\frac{-i\eta n_{mol}\kappa}{16\pi
r^{3}}{\mathrm{\text{B}}}_{\pm}^{cy}e^{-ikr}.$ (S33)

The $\triangle\mathrm{{E}}_{\pm}^{dx}$ has anti-enantiomorphic relation
$\triangle\mathrm{E}_{+}^{dx}=-\triangle\mathrm{E}_{-}^{dx}$ while
$\mathrm{{E}}_{\pm}^{dx}$ has enantiomorphic relation
$\mathrm{E}_{+}^{0x}=\mathrm{E}_{-}^{0x}$, and
$\triangle\mathrm{{E}}_{\pm}^{dy}$ has enantiomorphic relation
$\triangle\mathrm{{E}}_{+}^{dy}=\triangle\mathrm{{E}}_{-}^{dy}$ while
$\mathrm{{E}}_{\pm}^{dy}$ has anti-enantiomorphic relation
$\mathrm{E}_{+}^{0y}=\mathrm{-E}_{-}^{0y}$. Since the enantiomeric relation of
$\textbf{E}^{0}$ and $\textbf{E}^{d}$ is different, electric field strength in
nanostructure generated by RCP and LCP incident light is not equal anymore,
that is
$\textrm{E}_{+}^{0x}+\textrm{E}_{+}^{rdx}\neq\textrm{E}_{-}^{0x}+\textrm{E}_{-}^{rdx}$,
$\textrm{E}_{+}^{0y}+\textrm{E}_{+}^{rdy}\neq-\textrm{E}_{-}^{0y}-\textrm{E}_{-}^{rdy}$,
and
$\textrm{E}_{+}^{0z}+\textrm{E}_{+}^{rdz}\neq\textrm{E}_{-}^{z}+\textrm{E}_{-}^{rdz}$.
The nonequal electric field strengths leading to different absorption for RCP
and LCP. This is the mechanism of ICD induction in chiral molecule-achiral
nanostructure near field interaction. Figure S3 is schematic illustration the
origin of electric field strength difference in achiral nanostructure.
Generally, the increments in electric field by chiral molecule radiation are
opposite to each other for RCP and LCP incident, if the electric field
increments for RCP is positive, the electric field increments for LCP will be
negative. Figure S4 shows the electric field increments in GNM, the results
are same as analytical result.

Figure S4: The electric field strength increments in GNM.

###  II.3 Influence of distance r to the ICD

According to Eq. (S29), there is a linear relationship between ICD and
$\frac{e^{-ikr}}{r^{3}}$. But the electric fields $\textrm{E}^{0}$ and
magnetic fields $\textrm{B}^{c}$ change with $h_{r}$ thereby affecting the
ICD. Thus we analyzed the $\textbf{E}^{0*}\cdot\textbf{B}^{c}$,
$\frac{\textbf{E}^{0*}\cdot\textbf{B}^{c}}{r^{3}}$ and ICD. Figure S5 shows
the $\textbf{E}^{0*}\cdot\textbf{B}^{c}$,
$\frac{\textbf{E}^{0*}\cdot\textbf{B}^{c}}{r^{3}}$ and ICD spectra. From
$\textbf{E}^{0*}\cdot\textbf{B}^{c}$ and
$\frac{\textbf{E}^{0*}\cdot\textbf{B}^{c}}{r^{3}}$ spectra it can be observed
that the influence of distance r is obvious, and the
$\frac{\textbf{E}^{0*}\cdot\textbf{B}^{c}}{r^{3}}$ spectra are more close to
ICD spectra. Another inconvenience to do the linear fitting is that the peak
wavelengths are different for different $h_{r}$. Therefore, for excluding the
effect of $\textrm{E}^{0}$ and $\textrm{B}^{c}$, we chose the $18\,nm$,
$19\,nm$ and $20\,nm$ for linear fitting. Due to these are off-resonance
wavelength, electric and magnetic field strengths are almost same for
different $h_{r}$.

Figure S5: The spectra of (a) $\textbf{E}^{0*}\cdot\textbf{B}^{c}$, (b)
$\textbf{E}^{0*}\cdot\textbf{B}^{c}/{r^{3}}$ and (c) ICD for different
$h_{r}$.

###  II.4 Effect of electric field in nanostructure and magnetic field in
molecular region on ICD

The electric field strength ${{\mathbf{E}}^{0}}$ and magnetic field density
${{\mathbf{B}}^{c}}$ also affect the ICD. For the purpose of investigating the
effects of ${{\mathbf{E}}^{0}}$ and ${{\mathbf{B}}^{c}}$ on the ICD, we change
the structure to regulate these factors. One additional hole is set at the
center of every four adjacent original holes.

The radius of additional holes is changed from 20 $nm$ to 100 $nm$ by step of
20 $nm$. Figure S6 shows the charge distribution at three modes with changing
of ${R_{\alpha}}$. From figure S6 it can be seen that SPR strength gradually
decreases at mode I, and increases at mode II and III. Thereby, the electric
field strength ${{\mathbf{E}}^{0}}$ and magnetic flux density
${{\mathbf{B}}^{c}}$ decreases at mode I, and increases at mode II and I .

Figure S6: The charge distribution of GNM with different radius
${R_{\alpha}}$.

##  III  Calculation of chiral medium by FEM

We used COMSOL software for the simulation. For calculation of chiral medium
we changed part of equations through implementing chiral constitutive relation
SCKelly  . To confirm our calculation results, we applied these modifications
to solve for the transmission coefficients for a chiral slab and compared the
simulation results of the co-and cross-components of the transmission
coefficients at normal incidence from a slab with thickness $L$. The results
are:

$\displaystyle\textrm{T}_{co}$ $\displaystyle=$
$\displaystyle\frac{2\eta\eta_{0}cos(kL)cos(\kappa
kL)}{2\eta\eta_{0}cos(kL)+i(\eta_{0}^{2}+\eta^{2})sin(kL)}$ (S34)
$\displaystyle\textrm{T}_{cr}$ $\displaystyle=$
$\displaystyle\frac{2\eta\eta_{0}cos(kL)sin(\kappa
kL)}{2\eta\eta_{0}cos(kL)+i(\eta_{0}^{2}+\eta^{2})sin(kL)}$

Figure S7 depicts infinite chiral slab in air and calculation result. The
simulated and analytical results are in a good agreement.

Figure S7: (a) Chiral slab (b) Analytical (dashed line) and simulated (solid
line) co-component of transmission (c) Analytical (dashed line) and simulated
(solid line) cross-component of transmission

For anisotropic chiral medium, the constitutive relations are expressed by
tensorial chirality parameter. The unitary transformations are used for
rotation of orientation, for example, for rotation about $y$ axis the
constitutive relations are modulated as

$\displaystyle\left(\begin{array}[]{ccc}\textrm{D}^{x}\\\ \textrm{D}^{y}\\\
\textrm{D}^{z}\\\ \end{array}\right)$ $\displaystyle=$
$\displaystyle\varepsilon_{0}\varepsilon_{c}\left(\begin{array}[]{ccc}\textrm{E}^{x}\\\
\textrm{E}^{y}\\\ \textrm{E}^{z}\\\ \end{array}\right)$ (S41) $\displaystyle-$
$\displaystyle\frac{i}{c}\left(\begin{array}[]{ccc}cos(\alpha)&0&-sin(\alpha)\\\
0&1&0\\\ sin(\alpha)&0&cos(\alpha)\\\
\end{array}\right)\left(\begin{array}[]{ccc}\kappa_{xx}&\kappa_{xy}&0\\\
\kappa_{yx}&\kappa_{yy}&0\\\ 0&0&\kappa_{zz}\\\
\end{array}\right)\left(\begin{array}[]{ccc}cos(\alpha)&0&sin(\alpha)\\\
0&1&0\\\ -sin(\alpha)&0&cos(\alpha)\\\
\end{array}\right)\left(\begin{array}[]{ccc}\textrm{B}^{x}\\\
\textrm{B}^{y}\\\ \textrm{B}^{z}\\\ \end{array}\right)$ (S54)

$\displaystyle\left(\begin{array}[]{ccc}\textrm{B}^{x}\\\ \textrm{B}^{y}\\\
\textrm{B}^{z}\\\ \end{array}\right)$ $\displaystyle=$
$\displaystyle\mu_{0}\mu_{c}\left(\begin{array}[]{ccc}\textrm{H}^{x}\\\
\textrm{H}^{y}\\\ \textrm{H}^{z}\\\ \end{array}\right)$ (S61) $\displaystyle+$
$\displaystyle\frac{i}{c}\left(\begin{array}[]{ccc}cos(\alpha)&0&-sin(\alpha)\\\
0&1&0\\\ sin(\alpha)&0&cos(\alpha)\\\
\end{array}\right)\left(\begin{array}[]{ccc}\kappa_{xx}&\kappa_{xy}&0\\\
\kappa_{yx}&\kappa_{yy}&0\\\ 0&0&\kappa_{zz}\\\
\end{array}\right)\left(\begin{array}[]{ccc}cos(\alpha)&0&sin(\alpha)\\\
0&1&0\\\ -sin(\alpha)&0&cos(\alpha)\\\
\end{array}\right)\left(\begin{array}[]{ccc}\textrm{E}^{x}\\\
\textrm{E}^{y}\\\ \textrm{E}^{z}\\\ \end{array}\right)$ (S74)

## References

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