1. Introduction
Chirality describes the asymmetric feature of a three-dimensional object, whose mirror image is different from itself [
1]. Objects with chirality are quite common in nature. In the human body, most of the important molecules have chirality, such as DNA, enzyme, and protein. As a wave phenomenon, the light field also has chirality [
2]. A well-known example is circularly polarized light (CPL). Left circularly polarized (LCP) and right circularly polarized (RCP) light beams are a pair of enantiomers that have opposite spins. The interaction between the light field and chiral objects has attracted great interest from researchers [
3]. The chiral properties of the objects can be evaluated by the different responses under illumination of LCP and RCP lights. This chiral signal is useful to distinguish and measure the chirality of a sample. In natural materials, the chiral signal is very weak (
–
) [
1]. The main reason is the mismatch between the size of the chiral molecules and the light wavelength. In previous studies, it has been shown that plasmonic structures can increase chiral signals by orders of magnitude owing to the strong light confinement near the metallic surface [
4]. The plasmonic chiral structure is used to improve the sensing performance of enantiomers [
5]. In a nonchiral structure, optical chirality can be largely increased [
6]. It has been shown that the local chirality of the light field can be tailored near the plasmonic structure. The optical antenna theory can be used to design the chiral light [
7]. Symmetric metal–dielectric–metal (MDM) metamaterial structures have also been proposed to enhance the chiral light–matter interaction [
8]. However, metals have intrinsic loss, which fundamentally limits the practical performance of the designed plasmonic structures. Recently, dielectric nanoparticles with high refractive indices have also been used to manipulate the light field at the nanoscale [
9]. The supercavity with high q-factor can be designed by subwavelength high-index particles [
10]. The electric and magnetic resonances in silicon nanoparticles can be utilized to enhance the quantum efficiency of silicon nanoparticles [
11]. The anapole mode in such kind of particles has been intensively studied recently [
12]. The simplest case of an anapole is caused by the destructive interference between the electric dipole and the toroidal moment. In 2015, the radiationless anapole mode in visible was first observed [
13]. By tuning the geometry of the scattering particle, an obvious dip can be found in the far-field scattering spectrum, which is caused by the excitation of the anapole mode. Moreover, by tailoring the incident field, the anapole mode can also be excited in a dielectric sphere [
14]. The electric and magnetic resonances can be strongly excited inside the high-index particles with low loss, which may further enable the chirality enhancement. Besides the enhancement mechanism arising from near-field structures, the enhancement of a chiral signal can also be realized by tailoring the light field. Traditionally, circularly polarized light is believed to have the highest chirality. However, Cohen proposed the superchiral field, which has larger chirality than circular polarized light [
15,
16]. Recently, it was demonstrated by us that a localized superchiral hotspot can be generated by tightly focusing a vectorial light beam with orbital angular momentum [
17].
In this study, a method based on the T-matrix is employed to efficiently analyze the scattering process by a chiral sphere. It is demonstrated that two enhancement mechanisms can be employed to enhance the scattering circular dichroism (CD) signal of a chiral sphere simultaneously.
2. Materials and Methods
To consider the chiral property of the sphere in our theoretical model, the following constitutive relations are used [
18]:
In Equations (1) and (2),
,
, and
are the permittivity, permeability, and chirality parameters of the material in the sphere. Following the procedure proposed by Bohren [
19], the electromagnetic field in the chiral sphere (
,
) can be expressed by the linear transformation relation:
Because of the transformed fields,
and
should satisfy the wave equation,
. They can be expanded by the linear combination of vector spherical harmonics (VSHs):
where
, and
m and
n are both integers. In this study,
and
are regular VSHs whose values at origin are finite, and
and
are outgoing VSHs with singularity points at origin. The definitions of these VSH functions can be found in [
20]. Outside the sphere, the incident field (
,
and scattering field (
,
) can be expressed as the combinations of VSHs:
In Equations (6) and (7), the expansion coefficients
and
can be determined by the incident light field. The multipolar decomposition of various types of focused beams was considered in previous studies [
21]. The expansion coefficients of
and
for the scattering field can be calculated by the T-matrix method. According to the boundary condition on the surface of the sphere (
r =
), the T-matrix for a chiral sphere can be obtained from:
In Equation (10), the elements of the matrices
and
are given below:
In Equations (11)–(18), the Riccati–Bessel functions are defined as
and
.
Z0 is the vacuum impedance.
k0 = ω/c is the wave vector in vacuum. According to the Mie theory, the scattering power can be calculated by:
The radius of the sphere is
= 75 nm, and the parameters of the chiral material are
= 25,
= 1, and
= 0.01. The scattering spectrum under the illumination of linearly polarized light in free space (
= 1) is shown in
Figure 1a. In the frequency range of 200–700 THz, the scattering power is mainly caused by terms related to an electric dipole (ED), magnetic dipole (MD), and magnetic quadrupole (MQ) [
22]. Their contributions to the scattering power can be analyzed by the following three terms from the total scattering power:
In
Figure 1b, the spectral curves for the three scattering terms are helpful to analyze the mechanisms of the Mie resonances. It can be clearly demonstrated that the three peaks of
in
Figure 1a are caused by the MD, ED, and MQ resonances at frequencies of 388.5, 538.5, and 563.5 THz, respectively. Linearly polarized light can be represented as the superposition of left- and right-hand circularly polarized light. The circularly polarized light is useful to explore the properties of materials [
23]. To evaluate the chiral response of the sphere, the scattering CD parameter
is calculated, which is defined as
. In this definition,
and
are the scattering energies by the chiral sphere under the illuminations of LCP and RCP light. There are two regions in the spectrum in
Figure 1c, where the scattering CD single is enhanced. One region is near the MD resonance wavelength, and the other is near the ED resonance wavelength. However, the signs of CD values in the two ranges are opposite. The inversion of the optical chiral response can be attributed to the main absorption mechanism, which is converted from MD to ED when the light frequency is increased. In the region where ED absorption is dominant, there exists a deep valley at a frequency of 563.5 THz, which is exactly the MQ resonance frequency.
As discussed above, the light field can be tailored to gain larger chirality than CPL. To evaluate the chirality of the light field, the parameter of the CD enhancement factor is defined as
g/
, where
g and
are the CD values of a chiral dipole excited by the tailored light and CPL light. When
g/
> 1, the tailored light can produce a larger CD signal than CPL light. This kind of light is called superchiral field. Recently, it was reported by us a highly localized superchiral hotspot by tightly focusing a radially polarized beam with orbital angular momentum near the dielectric interface [
17]. In this study, the capability of the superchiral field to increase the scattering CD signal can be verified, when the particle is at the Mie resonance. It has been proved that when the incident angle of a focused beam is slightly smaller than the critical angle of totally internal reflection, the chirality of the light field (
g/
) can be largely enhanced. Therefore, the configuration in
Figure 2 is considered. The chiral sphere is on the substrate with a refractive index of
, which is larger than
. To analyze the interaction between the chiral sphere on the substrate and the superchiral field, the T-matrix in Equation (10) should be modified to incorporate the substrate into the theoretical model [
22,
24]. The scattering process is illustrated in
Figure 2. The light propagates from the substrate (
) to the air (
), and the chiral sphere is illuminated by the transmitted light (
,
). The scattering field is expressed as [
,
]
T =
T[
,
]
T, where the T-matrix is given in Equations (11)–(18). As shown by the red arrow in
Figure 2, a part of the scattering field can be reflected by the interface, and then the reflected field can also be scattered by the sphere again. One can see that there is a round trip of light between the sphere and the interface. Therefore, the multiple reflections between the sphere and the interface should be considered to build the accurate scattering model. In this study, the matrix of
is the reflection matrix, and the reflected field of the scattering field can be expressed as
[
,
]T. The final scattering field of the multiscattering process can be calculated by Equation (23) [
24].
According to Equation (23), the T-matrix for the sphere in free space should be modified as the effective one, which is
Teff = [
I −
T]
−1T.
I represents the unit matrix. The elements of
are the reflection coefficients of VSHs by the interface. To calculate these coefficients, the VSH is first expanded by the plane waves. The directions of the plane waves are represented by
, where
is the polar angle and
is the azimuthal angle. The reflection coefficient of each plane wave can be calculated independently by the Fresnel equation. The reflected field of the VSH can be obtained by recombining all the reflected plane wave components. After these operations, the reflected field of each VSH field scattered by the sphere can be calculated. To calculate the elements of
LR, the reflected fields have to be expended by the VSH basis, which has been used to express the incident field of [
,
]
T in Equations (6) and (7). The expansions of the reflected VSHs,
and
can be expressed as:
In Equations (24) and (25), the expansion coefficients can be determined by the following integrals with the integration path of
being
:
In Equations (26)–(29),
and
are the Fresnel reflection coefficients for p-polarized and s-polarized light,
and
are the angle-dependent functions in the Mie theory for convenience [
20]. The expression of
is in Equations (26)–(29), which is only related to the order numbers (
n,
m and
ν):
After obtaining the coefficients in Equations (26)–(29), the reflected matrix
can be expressed as:
Based on the theoretical model, light scattering by the sphere in the superchiral field can be analyzed efficiently. To further simplify the calculation, let us consider the vectorial Bessel beam as the incident light, whose field in a cylindrical coordinate can be expressed as [
25]:
In Equations (32) and (33),
is the topological charge of the incident field.
=
is the wave vector in the substrate. The refractive indexes of the substrate and air are
= 1.518 and
= 1. As shown in
Figure 3a, this vectorial light beam can be regarded as the superposition of the p-polarized plane waves, whose wave vector
k lives in a cone with the polar angle of
and the azimuthal angle
in k-space. The chirality enhancement factor at the focus point (
r = 0) is calculated when the polar angle is changed from zero to the critical angle of the total internal reflection, as shown in
Figure 3b. When
is slightly smaller than the critical angle of
, the hotspot region with high optical chirality can be realized near the focus point. The sign of optical chirality is the same with the topological charge
m0. Therefore, the scattering energies of
and
in the definition of
are calculated when,
= 1 and,
= −1 in Equations (32) and (33).
In
Figure 4a, the scattering CD spectrum for the sphere in the superchiral field with
is calculated as shown by the black curve. For comparison, the red curve represents the scattering CD spectrum for the same sphere in free space when illuminated by CPL light. As discussed above, the chiral response can be largely increased at the Mie resonance wavelength compared with that in the Rayleigh region, where the light wavelength is much larger than the sphere diameter. The peak values of the two CD spectra are 0.0445 and 0.0303 at frequencies of 546 and 579 THz, respectively. It means that, by introducing the superchiral field, the peak value of the scattering CD can be enhanced by 46.8%. The relationship between this enhancement factor and the incident angle has been calculated, as shown by the black curve in
Figure 4b. The red dash line represents the level of the peak CD value for CPL light. When the incident angle approaches the critical angle, the CD enhancement is increased at first, and reaches the maximum value at
. Then the enhancement factor drops dramatically, which is quite different from the prediction by the chiral dipole model in
Figure 3b. To explain the difference, one should notice that the CD value in
Figure 3b is for a chiral dipole, which is an individual point. However, the curve in
Figure 4b is for the chiral sphere with a diameter of
= 75 nm. As it has been reported previously [
17], the area of the superchiral hotspot is decreased when
is close to
, which may make it mismatch the size of the sphere. When
, the superchiral hotspot collapses to a point with infinite optical chirality. Around the singularity point, the light field becomes achiral. Therefore, there exists an optimized incident angle for a chiral sphere with finite size.