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Nanoscale chiral imaging under complex optical field excitation with controllable oriented chiral dipole moment

Guanghao Rui, Yulin Ji, Bing Gu, Yiping Cui, and Qiwen Zhan

Open Access Open Access

Abstract

Since chirality is a fundamental building block of nature, the identification of the chiral specimen’s structure is of great interest, especially in applications involving the modification and utilization of proteins. In this work, by exploiting photoinduced force exerted on an achiral tip placed in the vicinity of a reciprocal chiral sample, a novel technique is proposed to detect the sample’s chirality in nanoscale spatial resolution. Under separate excitation of focal field carrying chiral dipole moment with opposite handedness, there is a differential optical force ΔF exerted on the tip apex, which is connected to the enantiomer type and quasi-linearly depends on specific component of the sample’s chirality parameter. With the help of time-reversal approach, we prove that the required excitation can be derived by radiation fields from the superposition of parallel electric and magnetic dipoles. Through adjusting the orientation of the chiral dipole moment, all the diagonal components of the sample’s chirality can be exclusively retrieved. In addition, the sensitivity of the proposed technique is demonstrated to enantiospecify nanoscale chiral samples with chirality parameter on the order of 0.001. The proposed technique may open new avenue for wide applications in biomedicine, material science and pharmaceutics.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

As one of the structural characteristics, chirality refers to that an object cannot superimpose with its mirror image via rotation or translation [1]. Chirality is ubiquitous in chemistry and biology, most of biomolecules (such as L-amino acids, D-sugars, proteins, and nucleic acids) are chiral. Chiral substances with opposite handedness, which are called enantiomers, usually exhibit different chemical properties. For example, drug made of chiral molecule with one enantiomer is effective for a kind of disease, however the other type of enantiomer may be ineffective or even toxic. Indeed, discrimination of enantiomers is an extremely critical process in the pharmaceutical and chemical industries [2]. Besides recognizing the type of molecule, the structural determination is another key point in molecular identification techniques. For example, proteins are highly complex macromolecules consisting of one or more long chains of amino acids linked together by peptide bonds. The structure of protein has been described using four basic structural levels of the organization: primary, secondary, tertiary, and quaternary [3]. The first three structural levels can exist in molecules composed of a single polypeptide chain. In contrast, the last structural level involves interactions of polypeptides within a multi chained protein molecule. Unscrambling the structure information of protein is an important and fundamental requirement to modify and utilize proteins in novel applications such as antibody drug conjugate.

Due to the optical activity of chiral materials, the optical response of the enantiomers strongly depends on the polarization of light. The optical activity originates from the modification to the refractive index n± = nm ± κ, where nm is the refractive index, κ is the chirality parameter, and ± indicates left- and right-handed circular polarization (LCP and RCP) [4]. The imaginary component of the differential index leads to circular dichroism (CD), meaning that the absorption of chiral material is dependent on the handedness of light. The real component of the differential index of a material is related to circular birefringence (CB), describing that the speed of light with different handedness are different in chiral medium. Consequently, these chiroptical effects enable optical analytical techniques to determine the structure of chiral samples. For example, CD spectroscopy measures the differential absorption when chiral molecule interacts with LCP and RCP light, making it a powerful tool to identify and characterize secondary and tertiary structures of protein, which exhibit CD in the ultraviolet (UV) spectral region [5–7]. In addition, optical rotatory dispersion (ORD) measures the rotation angle of a linearly polarized light after passing through a chiral medium, which can be understood by CB induced phase difference between LCP and RCP [8]. To improve the sensitivity of chirality sensing, nanophotonic materials are applicable for enhancing chiroptical signals from biomolecules. Chiral plasmonic metamaterials show promise for the detection of chiral analytes with higher sensitivity than commercial UV CD spectroscopy. Since ORD is more sensitive to higher orders of protein structure than CD, Kadodwala et al. made use of the ORD line shapes of chiral plasmonic metamaterials to sense secondary and tertiary protein structures [9]. However, since the structure of chiral molecule is smaller than the wavelength of light it is interacted with, CD signal from molecule is inherently weak, thus it demands a large quantity of chiral molecules to acquire adequate signal-to-noise ratio [10]. Consequently, it is urgent to develop near-field measurement techniques to probe high resolution structural details.

In the past decade, the rapid development of nano-manufacturing technology technology has greatly promoted the research of nanophotonics, it provides a new idea for the chirality detection of micro and nano structures. For example, chiral metasurfaces and asymmetric nanostructures provides a new choice for the detection of chiral particles [11–14]. In addition, the interaction between polarized optical fields and metal or dielectric nanostructures may produce superchiral field whose chirality is stronger than circularly polarized light in the near-field region, chiral molecules in a superchiral near field will generate a stronger circular dichroic signal than when excited by circularly polarized light [15]. Although the plasmonic metamaterials have a strong local electric field enhancement effect, their enhancement of the magnetic field is very limited [16]. Therefore, plasma nanostructures with geometric chirality are preferred in experiments [17–19]. Although the background signal strength introduced by the chiral metamaterials is sometimes comparable to the intrinsic signal of the object to be measured, it can be eliminated by the structural design of the racemic structure [20–22]. Different from plasma materials, dielectric metamaterials can show the simultaneous enhancement of electric field and magnetic field at similar resonance frequencies, and have the advantages of low loss and weak thermal effect, so dielectric metamaterials [23–26] and plasma / dielectric hybrid metamaterials [27–29] bring new research direction.

Recently, optical tweezers has emerged as a powerful tool to identify the type of enantiomers [30–32]. When optical field is scattered or absorbed by a chiral material, the molecular chirality would induce cross-polarization between electric and magnetic field, leading to the generation of optical chiral force, which points to opposite directions for chiral molecules with different handedness. It has found that structured light can exert conservative forces perpendicular to the light’s propagation. Once the contribution from achiral force is suppressed, the emergence of lateral chiral force provides a nondestructive method for sorting nanomaterials by the handedness [33,34]. To create the lateral chiral force, it is necessary to realize the asymmetric distribution of Poynting vector in the near field region of chiral material, which can be achieved by placing samples on top of a surface [33,34], in the interference field [13], or sculpted focal field with transverse spin angular momentum [35]. In addition, plasmonic tweezers (coaxial nanoaperture [36], elliptical nanohole [37], double split ring resonator [38], etc.) has also been reported to enhance the lateral chiral force by large light field gradient, which is attributed to the evanescent characteristics of surface plasmon. Furthermore, photoinduced force microscopy (PiFM) has been utilized to report not only the type of enantiomers but also the chirality structure. By exciting the tip-sample interactive system with external light beam, the exerted force is due to the interaction between the polarized sample and tip. It has been proved that excitation with circular polarization is suitable to detect the transverse component of sample chirality, while the longitudinal component of chirality requires excitation of optical vortex with a longitudinal electric/magnetic field component [39,40].

In this work, a novel technique is presented to completely probe the chirality of reciprocal sample with high resolution on the sub-100 nm scale. By taking advantage of PiFM, we prove that the proper excitation for detecting individual component of sample’s chirality is the parallel electric and magnetic field along specific direction with π/2 phase difference. This complex optical field can be mimicked by a chiral dipole with controllable orientation, which is realized by applying time-reversal approach on radiated field from a combination of electric and magnetic dipoles. By measuring the exerted force on the tip apex in the vicinity of the sample, simultaneously adjusting the excitation beam in terms of the orientation of the chiral dipole moment, different components of the sample handedness can be identified through the magnetoelectric polarizability. We believe this work has the potential to advance studies of chirality of samples and molecular concentrations in nanoscale.

2. Theoretical model of probing the handedness of chiral sample with photoinduced force

When excited by an optical field, the optical property of a chiral sample can be modelled with the photoinduced electric and magnetic dipole moment ps and ms [41]:

$$\left[ {\begin{array}{c} {{{\boldsymbol p}_s}}\\ {{{\boldsymbol m}_s}} \end{array}} \right] = \left[ {\begin{array}{cc} {\bar{{\boldsymbol \alpha }}_s^{ee}}&{i\bar{{\boldsymbol \alpha }}_s^{em}}\\ {i{{(\bar{{\boldsymbol \alpha }}_s^{me})}^T}}&{\bar{{\boldsymbol \alpha }}_s^{mm}} \end{array}} \right]\left[ {\begin{array}{c} {{{\boldsymbol E}^{loc}}({{\boldsymbol r}_s})}\\ {{{\boldsymbol H}^{loc}}({{\boldsymbol r}_s})} \end{array}} \right], $$
where $\bar{{\boldsymbol \alpha }}_s^{ee}\; $, $\bar{{\boldsymbol \alpha }}_s^{mm}$, $\bar{{\boldsymbol \alpha }}_s^{em}$ and $\bar{{\boldsymbol \alpha }}_s^{me}$ are second-rank tensors describing electric, magnetic, magnetoelectric and electromagnetic polarizabilities of the sample, and T denotes the tensor transpose. Eloc(rs) and Hloc(rs) are the local electric and magnetic field vector acting on the sample at location rs. For reciprocal chiral sample, the electromagnetic polarizability tensor can be transposed into magnetoelectric one via $\bar{{\boldsymbol \alpha }}_s^{em} ={-} {({\bar{{\boldsymbol \alpha }}_s^{me}} )^T}$ [42,43], thus Eq. (1) is rewritten as:
$$\left[ {\begin{array}{c} {{{\boldsymbol p}_s}}\\ {{{\boldsymbol m}_s}} \end{array}} \right] = \left[ {\begin{array}{cc} {\bar{{\boldsymbol \alpha }}_s^{ee}}&{i\bar{{\boldsymbol \alpha }}_s^{em}}\\ { - i\bar{{\boldsymbol \alpha }}_s^{em}}&{\bar{{\boldsymbol \alpha }}_s^{mm}} \end{array}} \right]\left[ {\begin{array}{c} {{{\boldsymbol E}^{loc}}({{\boldsymbol r}_s})}\\ {{{\boldsymbol H}^{loc}}({{\boldsymbol r}_s})} \end{array}} \right], $$
In addition, only the diagonal components of the magnetoelectric polarizability tensor of a reciprocal chiral sample are nonvanishing, which can be expressed in Cartesian coordinates as:
$$\bar{{\boldsymbol \alpha }}_s^{em} = \left[ {\begin{array}{ccc} {\alpha_{s,xx}^{em}}&0&0\\ 0&{\alpha_{s,yy}^{em}}&0\\ 0&0&{\alpha_{s,zz}^{em}} \end{array}} \right], $$
It is worth nothing that the sample’s magnetoelectric polarization is highly correlated with the chirality parameter κ. Assuming a chiral particle with radius as much smaller than the wavelength of incident light, its magnetoelectric polarizability can be described under the dipolar approximation [44]:
$$\left\{ \begin{array}{l} \alpha_{s,xx}^{em} = {[\frac{{12i\pi a_s^3}}{c}\frac{{{\kappa_x}}}{{({\varepsilon_s} + 2)({\mu_s} + 2) - \kappa_x^2}}]^{\ast }}\\ \alpha_{s,yy}^{em} = {[\frac{{12i\pi a_s^3}}{c}\frac{{{\kappa_y}}}{{({\varepsilon_s} + 2)({\mu_s} + 2) - \kappa_y^2}}]^{\ast }}\\ \alpha_{s,zz}^{em} = {[\frac{{12i\pi a_s^3}}{c}\frac{{{\kappa_z}}}{{({\varepsilon_s} + 2)({\mu_s} + 2) - \kappa_z^2}}]^{\ast }} \end{array} \right., $$
where εs and µs are the relative permittivity and permeability of the sample, c is the speed of light in vacuum, the asterisk represents the complex conjugate, and κi (i = x, y, z) are the diagonal components of the sample’s chirality parameter in Cartesian coordinate. For simplicity, both the electric and magnetic polarizability of the sample are assumed to be isotropic ($\alpha _{s,xx}^{ee} = \alpha _{s,yy}^{ee} = \alpha _{s,zz}^{ee}$, $\alpha _{s,xx}^{mm} = \alpha _{s,yy}^{mm} = \alpha _{s,zz}^{mm}$) [44]:
$$\left\{ \begin{array}{l} \alpha_s^{ee} = \frac{{4\pi a_s^3{\varepsilon_0}}}{{{\varepsilon_s} + 2}}[{\varepsilon_s} - 1 + \frac{{3{\kappa^2}}}{{{\kappa^2} - ({\varepsilon_s} + 2)({\mu_s} + 2)}}]\\ \alpha_s^{mm} = \frac{{4\pi a_s^3{\mu_0}}}{{{\mu_s} + 2}}[{\mu_s} - 1 + \frac{{3{\kappa^2}}}{{{\kappa^2} - ({\varepsilon_s} + 2)({\mu_s} + 2)}}] \end{array} \right., $$
where ε0 and µ0 are the relative permittivity and permeability in vacuum.

In this work, the goal is to characterize all three components of the sample’s chirality parameter separately, which is crucial to understand the structure information of a chiral molecule. According to Eq. (4), the chirality parameter can be exclusively determined by measuring the corresponding diagonal component of the magnetoelectric polarizability, which would be reflected in the exerted optical force. To precisely measure the force in the vicinity of the sample, PiFM composed of a nanoscale microscope tip is considered, which is placed close enough to the sample. Although circularly polarized waves are adequate to detect the transverse polarizability components of the sample, they are unable to distinguish the contributions from x- and y-component of the magnetoelectric polarizability. To overcome this inadequacy, we adopt a kind of complex optical beam carrying chiral dipole moment as the illumination, which is the combination of purely parallel electric and magnetic field components with π/2 phase difference. In this section, we will demonstrate that all the diagonal components of the sample’s chirality can be exclusively retrieved through measuring the optical force exerted on the tip apex, which is induced by incident chiral dipole moment with controllable orientation.

As the PiFM setup shown in Fig. 1(a), the interactive system composed of a nanoscale chiral sample and a nonchiral tip is illuminated by two counter-propagating focused light, which is crucial to generate chiral dipole and will be discussed in Sect. 3. Assuming both the tip apex and the sample are small enough compared to the wavelength of the incident light, they can be approximated as two nanospheres (shown in Fig. 1(b)). The optical response of the tip apex can be modelled as:

$$\left\{ \begin{array}{l} {{\boldsymbol p}_t} = \alpha_t^{ee} \cdot {{\boldsymbol E}^{loc}}({{\boldsymbol r}_t})\\ {{\boldsymbol m}_t} = \alpha_t^{mm} \cdot {{\boldsymbol H}^{loc}}({{\boldsymbol r}_t}) \end{array} \right., $$
where pt and mt are the photoinduced electric and magnetic dipole moments of the tip apex respectively, $\alpha_t^{e e}$ and $\alpha_t^{mm}$ are the electric and magnetic polarizability of the tip, which can also be calculated with Eq. (5) by setting κ to 0 and replacing the relative permittivity, permeability, and radius of the sample to that of the tip. Eloc(rt) and Hloc(rt) are the local electric and magnetic field vector acting on the tip apex at location rt, which can be expressed as:
$$\left\{ \begin{array}{l} {{\boldsymbol E}^{loc}}({{\boldsymbol r}_t}) = {{\boldsymbol E}^{inc}}({{\boldsymbol r}_t}) + {{\boldsymbol E}_{scat}}{|_{s \to t}}\textrm{ = }{{\boldsymbol E}^{inc}}({{\boldsymbol r}_t}) + {{\boldsymbol G}^{EE}}({{\boldsymbol r}_t},{{\boldsymbol r}_s}) \cdot {{\boldsymbol p}_s} + {{\boldsymbol G}^{EM}}({{\boldsymbol r}_t},{{\boldsymbol r}_s}) \cdot {{\boldsymbol m}_s}\\ {{\boldsymbol H}^{loc}}({{\boldsymbol r}_t}) = {{\boldsymbol H}^{inc}}({{\boldsymbol r}_t}) + {{\boldsymbol H}_{scat}}{|_{s \to t}}\textrm{ = }{{\boldsymbol H}^{inc}}({{\boldsymbol r}_t}) + {{\boldsymbol G}^{ME}}({{\boldsymbol r}_t},{{\boldsymbol r}_s}) \cdot {{\boldsymbol p}_s} + {{\boldsymbol G}^{MM}}({{\boldsymbol r}_t},{{\boldsymbol r}_s}) \cdot {{\boldsymbol m}_s} \end{array} \right., $$
where Escat|s→t and Hscat|s→t are the scattering electric and magnetic fields from the sample to the tip, Einc(rt) and Hinc(rt) represent the incident field at location rt. GEE(GEM) and GME(GMM) are the dyadic Green’s functions that provide the electric and magnetic fields due to the electric (magnetic) dipole ps (ms) respectively, which can be expressed in terms of free-space scalar Green’s function G = exp[ik0(r − r0)/4π(r − r0)], as $({1/{\varepsilon_0}} )\left( {k_0^2\overline{\overline I} + \nabla \nabla } \right){\boldsymbol G}$ and $- i\omega \left( {\nabla {\boldsymbol G} \times \overline{\overline I} } \right)$ respectively, where k0 is the free-space wavenumber, $\overline{\overline I} $ is the unit dyad, r refer to the tip location (sample location) and r0 represents sample location (tip location), respectively [45–47]. Under the illumination, electric and magnetic polarization currents would be induced on both the tip apex and the sample, afterwards they can be thought of as secondary electromagnetic wave source that reradiate into free space. Since the tip apex is located to contact with the near field of the sample, the local fields given in Eq. (7) include the contributions from both the incident field and the scattering field. Accordingly, a time-averaged optical force would be exerted on the tip apex due to the momentum transfer from the scattered photons [48–50]:
$${\boldsymbol F} = \frac{1}{2}Re [(\nabla {{\boldsymbol E}^{loc}}{({{\boldsymbol r}_t})^\ast }) \cdot {{\boldsymbol p}_t} + (\nabla {{\boldsymbol H}^{loc}}{({{\boldsymbol r}_t})^ \ast }) \cdot {{\boldsymbol m}_t} - \frac{{c{k_1}^4}}{{6\pi }}\frac{1}{{\sqrt {\varepsilon \mu } }}({{\boldsymbol p}_t} \times {{\boldsymbol m}_t}^ \ast )], $$

 figure: Fig. 1.

Fig. 1. (a) Schematic of PiFM capable of probing different components of the handedness of a chiral sample. (b) Simplified model of the tip-sample interactive system. (c) Differential force exerted on the tip apex versus κx of the sample for excitations of left- and right-hand chiral dipole respectively.

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Firstly, we assume the tip apex and the sample are aligned along x axis with center-to-center distance of d, and this dimer structure is illuminated by an optical field carrying x-oriented chiral dipole moment. Since GEM = −µ0GME and GMM = ε0GEE, Eq. (7) can be expanded in matrix form:

$$\begin{array}{l} \left[ {\begin{array}{c} {E_x^{loc}({r_t})}\\ {E_y^{loc}({r_t})}\\ {E_z^{loc}({r_t})} \end{array}} \right] = {E^{inc}}({r_t}) + \frac{{k_0^2}}{{{\varepsilon _0}}}\frac{{exp (i{k_0}d)}}{{4\pi d}}\left( {\begin{array}{ccc} {\frac{2}{{i{k_0}d}} + \frac{2}{{k_0^2{d^2}}}}&0&0\\ 0&{1 - \frac{1}{{i{k_0}d}} - \frac{1}{{k_0^2{d^2}}}}&0\\ 0&0&{1 - \frac{1}{{i{k_0}d}} - \frac{1}{{k_0^2{d^2}}}} \end{array}} \right)\left( {\begin{array}{c} {{p_{s,x}}}\\ {{p_{s,y}}}\\ {{p_{s,z}}} \end{array}} \right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; - {\eta _0}k_0^2\frac{{exp (i{k_0}d)}}{{4\pi d}}(1 - \frac{1}{{i{k_0}d}})\left( {\begin{array}{ccc} 0&0&0\\ 0&0&{ - 1}\\ 0&1&0 \end{array}} \right)\left( {\begin{array}{c} {{m_{s,x}}}\\ {{m_{s,y}}}\\ {{m_{s,z}}} \end{array}} \right) \end{array}, $$
$$\begin{array}{l} \left[ {\begin{array}{c} {H_x^{loc}({{\boldsymbol r}_t})}\\ {H_y^{loc}({{\boldsymbol r}_t})}\\ {H_z^{loc}({{\boldsymbol r}_t})} \end{array}} \right] = {E^{inc}}({{\boldsymbol r}_t}) + {c_0}k_0^2\frac{{exp (i{k_0}d)}}{{4\pi d}}(1 - \frac{1}{{i{k_0}d}})\left( {\begin{array}{ccc} 0&0&0\\ 0&0&{ - 1}\\ 0&1&0 \end{array}} \right)\left( {\begin{array}{c} {{p_{s,x}}}\\ {{p_{s,y}}}\\ {{p_{s,z}}} \end{array}} \right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; + k_0^2\frac{{exp (i{k_0}d)}}{{4\pi d}}\left( {\begin{array}{ccc} {\frac{2}{{i{k_0}d}} + \frac{2}{{k_0^2{d^2}}}}&0&0\\ 0&{1 - \frac{1}{{i{k_0}d}} - \frac{1}{{k_0^2{d^2}}}}&0\\ 0&0&{1 - \frac{1}{{i{k_0}d}} - \frac{1}{{k_0^2{d^2}}}} \end{array}} \right)\left( {\begin{array}{c} {{m_{s,x}}}\\ {{m_{s,y}}}\\ {{m_{s,z}}} \end{array}} \right) \end{array}, $$
If retaining only the stronger terms in Eq. (9) and (10), the scattering field in the near-field region can be approximated by an electric ps and a magnetic ms dipole [46]:
$$\left\{ {\begin{array}{c} {{{\boldsymbol E}_{scat}}{|_{s \to t}} \approx \frac{{{e^{i{k_0}d}}}}{{4\pi {\varepsilon_0}{d^3}}}[3\hat{{\boldsymbol x}}\textrm{(}\hat{{\boldsymbol x}} \cdot {{\boldsymbol p}_s}\textrm{)} - {{\boldsymbol p}_s}]}\\ {{{\boldsymbol H}_{scat}}{|_{s \to t}} \approx \frac{{{e^{i{k_0}d}}}}{{4\pi {d^3}}}[3\hat{{\boldsymbol x}}\textrm{(}\hat{{\boldsymbol x}} \cdot {{\boldsymbol m}_s}\textrm{)} - {{\boldsymbol m}_s}]} \end{array}} \right., $$
where $\hat{{\boldsymbol x}}$ is the unit vector of x-direction in Cartesian coordinates centered at the source location. Eq. (11) can be expanded as [47]:
$$\left\{ {\begin{array}{c} {{{\boldsymbol E}_{scat}}{|_{s \to t}} ={-} \frac{{{G_0}}}{{{\varepsilon_0}}}( - 2{p_{s,x}}\hat{x} + {p_{s,y}}\hat{y} + {p_{s,z}}\hat{z})}\\ {{{\boldsymbol H}_{scat}}{|_{s \to t}} ={-} {G_0}( - 2{m_{s,x}}\hat{x} + {m_{s,y}}\hat{y} + {m_{s,z}}\hat{z})} \end{array}} \right., $$
where G0 ≈ exp[ik0|xt−xs|/4π|xt−xs|3], the positions of the tip apex and the sample are denoted by xt and xs, respectively. Substituting Eq. (2) and Eq. (12) into Eq. (7), the local electric and magnetic field acting on the tip apex are expressed as:
$$\left[ {\begin{array}{c} {E_x^{loc}({x_t})}\\ {E_y^{loc}({x_t})}\\ {E_z^{loc}({x_t})} \end{array}} \right] = \left[ {\begin{array}{c} {E_x^{inc}({x_t})}\\ {E_y^{inc}({x_t})}\\ {E_z^{inc}({x_t})} \end{array}} \right] - \frac{{{G_0}}}{{{\varepsilon _0}}}\left[ {\begin{array}{c} { - 2\alpha_s^{ee}E_x^{loc}({x_s}) - 2i\alpha_{s,xx}^{em}H_x^{loc}({x_s})}\\ {\alpha_s^{ee}E_y^{loc}({x_s}) + i\alpha_{s,yy}^{em}H_y^{loc}({x_s})}\\ {\alpha_s^{ee}E_z^{loc}({x_s}) + i\alpha_{s,zz}^{em}H_z^{loc}({x_s})} \end{array}} \right], $$
$$\left[ {\begin{array}{c} {H_x^{loc}({x_t})}\\ {H_y^{loc}({x_t})}\\ {H_z^{loc}({x_t})} \end{array}} \right] = \left[ {\begin{array}{c} {H_x^{inc}({x_t})}\\ {H_y^{inc}({x_t})}\\ {H_z^{inc}({x_t})} \end{array}} \right] - {G_0}\left[ {\begin{array}{c} {2i\alpha_{s,xx}^{em}E_x^{loc}({z_s}) - 2\alpha_s^{mm}H_x^{loc}({z_s})}\\ { - i\alpha_{s,yy}^{em}E_y^{loc}({z_s}) + \alpha_s^{mm}H_y^{loc}({z_s})}\\ { - i\alpha_{s,zz}^{em}E_z^{loc}({z_s}) + \alpha_s^{mm}H_z^{loc}({z_s})} \end{array}} \right], $$
By neglecting the contribution from mt and considering $E_x^{inc}({{x_t}} )\approx E_x^{inc}({{x_s}} )$ (quasistatic limit under k0d→0), the local field is dominated by its x component since both the electromagnetic field of the excitation and the dimer axis are along x axis:
$$E_x^{loc}({x_t}) = \frac{{ - {\varepsilon _0}}}{{4G_0^2\alpha _s^{ee}\alpha _t^{ee} - \varepsilon _0^2}}[({\varepsilon _0} + 2{G_0}\alpha _s^{ee})E_x^{inc}({x_t}) + 2i{G_0}\alpha _{s,xx}^{em}H_x^{inc}({x_t})], $$
$$\begin{array}{l} H_x^{loc}({x_t}) = \frac{{(4i\alpha _t^{ee}\alpha _{s,xx}^{em}{\varepsilon _0}{G_0}^2 + 2i\varepsilon _0^2\alpha _{s,xx}^{em}{G_0})E_x^{inc}({x_t})}}{{4{G_0}^2\alpha _s^{ee}\alpha _t^{ee} - \varepsilon _0^2}} + \\ \frac{{( - 8\alpha _t^{ee}{{(\alpha _{s,xx}^{em})}^2}G_0^3 + 8\alpha _t^{ee}\alpha _s^{ee}{\alpha _{mm}} + 4\alpha _t^{ee}\alpha _s^{ee}G_0^2 - 2\varepsilon _0^2\alpha _s^{mm}{G_0} - \varepsilon _0^2)H_x^{inc}({x_t})}}{{4{G_0}^2\alpha _s^{ee}\alpha _t^{ee} - \varepsilon _0^2}} \end{array}, $$
In this case, the time-averaged optical force exerted on the tip apex is also dominated by its x component:
$${F_x} \approx \frac{1}{2}R\textrm{e[}{(2{p_{s,x}}\frac{\partial }{{\partial x}}\frac{{{G_0}}}{{{\varepsilon _0}}})^\ast }{p_{t,x}}\textrm{]}, $$
By substituting Eq. (15) and (16) into Eq. (17), and neglecting the high-order terms, Eq. (17) can be simplified as:
$${F_x} \approx \frac{{ - 3}}{{4\pi {\varepsilon _0}{d^4}}}R\textrm{e[}\alpha _t^{ee}E_x^{inc}{(\alpha _s^{ee}E_x^{inc} + i\alpha _{s,xx}^{em}H_x^{inc})^ \ast }\textrm{]}, $$
We approximate the excitation field (x-oriented chiral dipole) as |Ex| = |E0| and |Hx| = |E0|e ± iπ/2/η0, where η0 is the impedance in vacuum, consequently the force difference induced by chiral dipole moments with opposite handedness can be derived as:
$$\begin{array}{l} \Delta {F_x} \approx \frac{{ - 3{{|{{E_0}} |}^2}}}{{4\pi \sqrt {{\varepsilon _0}{\mu _0}} {d^4}}}[Re (i\alpha _t^{ee}{(\alpha _{s,xx}^{em})^ \ast }{e^{i\frac{\pi }{2}}}) - Re (i\alpha _t^{ee}{(\alpha _{s,xx}^{em})^ \ast }{e^{ - i\frac{\pi }{2}}})]\;\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \frac{{ - 3{{|{{E_0}} |}^2}}}{{4\pi \sqrt {{\varepsilon _0}{\mu _0}} {d^4}}}[2{\mathop{\rm Im}\nolimits} (i\alpha _t^{ee}{(\alpha _{s,xx}^{em})^ \ast })], \end{array}$$
Equation (19) clearly demonstrates that the differential force depends linearly on the x-component of the sample’s magnetoelectric polarizability, which is also linearly proportional to κx according to Eq. (4).

To verify it, we consider an example when the tip-sample system interacts with a x-oriented chiral dipole with wavelength λ = 532 nm, both the sample and the tip apex are assumed to have equal radii as = at = 50 nm with center-to-center distance d = 110 nm. Furthermore, the relative permittivity of the sample and the gold tip apex are εs = 2.5 and εt = -4.6810 + i2.4266, respectively. When the chirality of the dipole is respectively left- and right-hand, the time-averaged differential force exerted on the tip apex versus chirality parameter of the sample have been depicted in Fig. 1(c). As expected, the exerted differential force on the tip apex is zero for an achiral sample. As the magnitude of κx of the sample grows, the differential force exerted on the tip apex also increases. In addition, the handedness of the sample can also be identified by the sign of this differential force.

So far, we have demonstrated the possibility of detecting x-component of the sample’s chirality with the excitation of x-oriented chiral dipole moment. Similarly, with the use of chiral dipole moment with opposite handedness oriented along y (z) direction, simultaneously align the tip apex with the sample along y (z) direction, κy (κz) of the sample can also be detected accordingly by measuring the differential optical force exerted on the tip apex. Next, we will discuss the strategy to generate the required chiral dipole moment with controllable orientation.

3. Generation of chiral dipole moment with controllable orientation

In the helicity basis, chiral dipole consists of parallel electric and magnetic dipole moments of equal amplitude and a relative phase of π/2, corresponding to a helicity of ±1. The chiral dipole is related to the chiral geometry and observed in the fundamental mode of chiral nanostructures [51,52]. With the rapid development of the optical engineering, tailoring the spatial distribution of the optical field at will becomes possible [53–55]. For example, z-oriented chiral dipole has been experimentally demonstrated by exciting an achiral nanoparticle with an in-phase superposition of tightly focused radially and azimuthally polarized beams. In this case, the unique focusing properties of cylindrical vector beams lead to the purely electric and magnetic field along the optical axis, and the phase shift between the fields and the resulting dipole moment can be realized due to the wavelength-dependent electric and magnetic polarizabilities of the particle.

In this section, our intention is the tailored excitation and verification of chiral dipole moments with controllable orientation in free space. The orientation of the dipole moment is difficult to adjust in both paraxial beam and focusing beam since the transverse electric and magnetic fields are perpendicular to each other. Besides, the phase-delayed electric and magnetic dipole moments is difficult to achieve without the introduction of nanostructure. One possible solution is the inverse design of the chiral dipole moment, which can be mimicked by the superposition of electric and magnetic dipoles with collinear oscillating direction. By locating the dipoles at the focal point of a high numerical aperture (NA) lens and reversing their radiation fields, the required spatial distribution of the illumination in the pupil plane of the lens can be obtained [56–58]. In addition, the phase retardation between electric and magnetic dipole moments would result in the light emission with predominantly positive or negative helicity.

As the schematic diagram illustrated in Fig. 2(a), a x-oriented chiral dipole with helicity of +1 is synthesized by the superposition of one electric dipole and one magnetic dipole oscillating along x axis. These dipoles are placed at the focal point of a 4Pi focusing system, which is consists of two identical sine-type aplanatic lens with the same NA and aimed to correct the asymmetric radiation patterns of the dipoles.

 figure: Fig. 2.

Fig. 2. (a) Diagram of the proposed setup to generate chiral dipole with controllable orientation. (b) Generation of x-oriented chiral dipole. The first row show both the intensity and SOP distributions of the incident left- and right-propagation optical fields in the left and right pupil plane of the 4Pi focusing system. The second and third rows show the calculated distributions of focal electric and magnetic fields intensities and relative phases of tightly focused beam. (c) The calculated distributions of focal electric and magnetic fields intensities and relative phases of tightly focused beam, demonstrating the generation of (c) y-oriented and (d) z-oriented chiral dipole.

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Through time-reversal approach, the corresponding radiation fields in the left pupil plane from the dipoles can be expressed as:

$$\begin{array}{l} {{\boldsymbol E}_{li}}(r,\phi ) = \{ [ - \textrm{cos}\theta {\cos ^2}\phi - {\sin ^2}\phi + (\textrm{cos}\theta \sin \phi \cos \phi - \sin \phi \cos \phi ){e^{i\pi /2}}]{{\vec{e}}_x} + \\ {[\sin \phi \cos \phi - \textrm{cos}\theta \sin \phi \cos \phi + ( - \textrm{cos}\theta {\cos ^2}\phi - {\sin ^2}\phi ){e^{i\pi /2}}]}{{\vec{e}}_y}\} /\sqrt {\cos \theta } \end{array}, $$
where r = f·sinθ (f is the focal length of the lens), θ and ϕ are the refraction angle and azimuthal angle respectively in the image space. Similarly, the expression of the incident field in the pupil plane of the right lens can be derived:
$$\begin{array}{l} {{\boldsymbol E}_{ri}}(r,\phi ) = \{ [\textrm{cos}\theta {\cos ^2}\phi + {\sin ^2}\phi + (\textrm{cos}\theta \sin \phi \cos \phi - \sin \phi \cos \phi ){e^{j\pi /2}}]{{\vec{e}}_x} + \\ {[\sin \phi \cos \phi - \textrm{cos}\theta \sin \phi \cos \phi + ( - \textrm{cos}\theta {\cos ^2}\phi - {\sin ^2}\phi ){e^{j\pi /2}}]}{{\vec{e}}_y}\} /\sqrt {\cos \theta } \end{array}, $$
The exponential terms provide the π/2 phase difference to create the required helicity. With Richard-Wolf vectorial diffraction theory, the electric field in the vicinity of the focus in the case of the left-propagating incident field can be described as:
$${{\boldsymbol E}_{lf}}({r_f},{\varphi _f},{z_f}) = \frac{i}{\lambda }\int_0^{\theta \max } {\int_o^{2\pi } {({{X_{lf}}\hat{x} + {Y_{lf}}\hat{y} + {Z_{lf}}\hat{z}} )} } \times {e^{j{k_0}{r_f}\sin \theta \cos (\phi - {\varphi _f}) + j{k_0}{z_f}\cos \theta }}\sin \theta d\theta d\phi, $$
$${X_{lf}} ={-} {\cos ^2}\theta {\cos ^2}\phi - {\sin ^2}\phi , $$
$${Y_{lf}} = ( - {\cos ^2}\theta \sin \phi \cos \phi + \sin \phi \cos \phi ) - \textrm{cos}\theta {e^{j\pi /2}}, $$
$${Z_{lf}} = \sin \theta \cos \theta \cos \phi + \textrm{sin}\theta \sin \phi {e^{j\pi /2}}, $$
where rf = (x2 + y2)1/2 and φf = tan−1(y/x) are the polar coordinates in the focal volume. θmax is the maximum focusing angle determined by NA (close to 1) of the lens. In the same way, the electric field in the vicinity of the focus in the case of the right-propagating incident field is given:
$${{\boldsymbol E}_{rf}}({r_f},{\varphi _f},{z_f}) = \frac{i}{\lambda }\int_0^{\theta \max } {\int_o^{2\pi } {({{X_{rf}}\hat{x} + {Y_{rf}}\hat{y} + {Z_{rf}}\hat{z}} )} } \times {e^{j{k_0}{r_f}\sin \theta \cos (\phi - {\varphi _f}) + j{k_0}{z_f}\cos \theta }}\sin \theta d\theta d\phi, $$
$${X_{rf}} = {\cos ^2}\theta {\cos ^2}\phi + {\sin ^2}\phi , $$
$${Y_{rf}} = ( - {\cos ^2}\theta \sin \phi \cos \phi + \sin \phi \cos \phi ) - \textrm{cos}\theta {e^{j\pi /2}}, $$
$${Z_{rf}} ={-} \sin \theta \cos \theta \cos \phi + \textrm{sin}\theta \sin \phi {e^{j\pi /2}}, $$
To demonstrate the proposed method for generating x-oriented chiral dipole, we assume the wavelength and power of the illumination are 532 nm and 100 mW respectively, and numerically calculate the intensity patterns of the required incident optical field. As shown in the first row of Fig. 2(b), in which the state of polarization (SOP) at the beam cross-section is also indicated by the polarization ellipses (green and blue colors represent left- and right-handedness respectively), both the amplitude and the SOP of the illumination are not homogeneously distributed. Generally, the left- and right-propagating incident light possess opposite handedness, and very low energy accumulates in the left and right sides of the beam. Besides, the SOP would change its polarization from circular to elliptical with increased radius, and the elevation angle would also be varied.

Next, the intensity distributions of the focal field in the focal plane are calculated and shown in the second and third row of Fig. 2(b). One can find out that the patterns of |Ex| and |Hx| are solid spot with homogenous phase distribution, while the other components of the electromagnetic field would split into two lobes with π phase difference. As expected, both E- and H-field are x-polarized and parallel along x axis, and the phase difference between Ex and Hx is found to be π/2. In contrast, y- and z-field components are antisymmetric with respect to the x axis. Consequently, a chiral dipole moment with helicity of +1 is synthesized at the focal point, with its orientation along x axis. By changing the phase term of the illumination from +π/2 to –π/2, the helicity of the chiral dipole can be reversed accordingly.

Following this strategy, y- and z-oriented chiral dipole moments can also be generated by rotating the oscillating direction of the electric and magnetic dipoles. The expressions of the required illuminations are summarized in Table 1, and the corresponding focal fields are calculated and shown in Fig. 2(c) and 2(d). It can be clearly seen that both the electric field and magnetic field are parallel with phase difference of +π/2 along y- and z-axis respectively, demonstrating the feasibility of creating chiral dipole with controllable orientation.

Tables Icon

Table 1. The expressions of the left- and right- propagating incident fields for generating y- and z-oriented chiral dipole.

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4. Probing the chirality of sample with the use of chiral dipole moment

The simplified model in Sect. 2 provides valuable insights into the force characteristics of chiral sample excited by chiral dipole moment. To take all the effects into consideration and validate the analytical derivations, here rigorous numerical calculation is performed to investigate the photoinduced force in the tip-sample interactive system, which is excited by focal field carrying chiral dipole moment.

To probe the x- component of the chirality, the tip-sample interactive system should be excited by the light field carrying x-oriented chiral dipole moment. Moreover, the sample and tip should be aligned along x-axis, since the location of the tip is crucial to measure the required force component. Accordingly, the local electric and magnetic field in Eq. (13) and Eq. (14) can be solved:

$$\left\{ \begin{array}{l} E_x^{loc}({r_t}) = \frac{{ - 8{\varepsilon_0}({N_{1x}} + \frac{1}{2}{N_{2x}} - \frac{1}{4}{N_{3x}} - \frac{1}{8}E_x^{inc}{\varepsilon_0})}}{{16{D_1} - 4{D_2} + \varepsilon_0^2}}\\ E_y^{loc}({r_t}) = \frac{{{\varepsilon_0}({N_{1y}} - {N_{2y}} - {N_{3y}} + E_y^{inc}{\varepsilon_0})}}{{{D_1} - {D_2} + \varepsilon_0^2}}\\ E_z^{loc}({r_t}) = \frac{{{\varepsilon_0}({N_{1z}} - {N_{2z}} - {N_{3z}} + E_z^{inc}{\varepsilon_0})}}{{{D_1} - {D_2} + \varepsilon_0^2}} \end{array} \right., $$
$$\left\{ \begin{array}{l} H_x^{loc}({r_t}) = \frac{{ - 8({N_{4x}} + \frac{1}{2}{N_{5x}} + \frac{1}{4}{N_{6x}} - \frac{1}{8}H_x^{inc}\varepsilon_0^2)}}{{16{D_1} - 4{D_2} + \varepsilon_0^2}}\\ H_y^{loc}({r_t}) = \frac{{({N_{4y}} - {N_{5y}} + {N_{6y}} + H_y^{inc}\varepsilon_0^2)}}{{{D_1} - {D_2} + \varepsilon_0^2}}\\ H_z^{loc}({r_t}) = \frac{{({N_{4z}} - {N_{5z}} + {N_{6z}} + H_z^{inc}\varepsilon_0^2)}}{{{D_1} - {D_2} + \varepsilon_0^2}} \end{array} \right., $$
where the expression of the field components can be found in Tab. 2. During the derivation, G0 is approximated by:
$${G_0} \approx \frac{{{e^{i{k_0}|{{x_t} - {x_s}} |}}}}{{4\pi {{|{{x_t} - {x_s}} |}^3}}}, $$
where xt and xs are the spatial position of the tip and the sample, respectively. Considering the same parameters used in Fig. 1(c), the optical force measured by the PiFM can be calculated accordingly with Eq. (8). For a chiral sample with varied κx, Fig. 3(a)–3(c) compares the optical force components induced by x-oriented chiral dipole with opposite handedness. Firstly, since the photoinduced electric and magnetic dipole moments of the sample are related to its chirality, the exerted force would also change with chirality parameter. Secondly, since the sample and tip are aligned along x axis, the measured force is dominated by Fx. Lastly, owing to the optical activity of chiral materials, the sample would react differently to chiral dipole moments with opposite handedness, leading to the differential force ΔFx exerted on the tip apex, which is able to separate the effect of chirality of the sample from its electric and magnetic properties. Furthermore, Fig. 3(d) shows the dependence of ΔFx on different components of the chirality parameter. As the theoretical derivation predicted in Sect. 2, the differential force is zero for achiral sample, while it would quasi-linearly decrease with the increase of κx (red curve in Fig. 3(d)). Since the sign of the differential force depends on the sign of the chirality parameter, this method can identify not only κx but also the type of the enantiomers.

 figure: Fig. 3.

Fig. 3. (a) x-component, (b) y-component, and (c) z-component of the exerted force on the tip apex for local excitation of left- and right-hand x-oriented chiral dipole respectively. Differential force exerted on the tip apex versus different components of the chirality parameter for local excitation of (d) x-oriented, (e) y-oriented, and (f) z-oriented chiral dipole. The configurations of the tip-sample interactive system are also shown in the insets.

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Tables Icon

Table 2. Expressions of the local field near the tip apex when illuminated by x-oriented chiral dipole.

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However, since the local excitation lacks both y- and z-component of the electromagnetic field, and the exerted force on the tip apex depends on the induced electric and magnetic dipole moments of the sample, both the y- and z-component of the sample’s chirality parameter do not contribute to induced electric and magnetic dipole moments of the sample, giving rise to a constant differential force for varied κy and κz (blue and green curves in Fig. 3(d)). Following similar strategy, both κy and κz of the sample can be individually determined by adopting the focal field carrying chiral dipole moment oriented along y and z axis, respectively. Besides, the tip apex is required to align with the sample along y and z axis, respectively. By measuring the optical force exerted on the tip apex under the excitation of chiral dipole moment with opposite handedness, the differential force can be approximated as ΔFy and ΔFz, which also depends quasi-linear on κy and κz, respectively (shown in Fig. 3(e) and 3(f)).

5. Discussions

So far, we have demonstrated that it is feasible to detect the sample chirality on nanoscale with the use of photoinduced force under the excitation of chiral dipole moment. Here, we will discuss the resolution of our proposed method. A color map in Fig. 4(a) and 4(b) illustrates the relation among ΔF, chirality parameter and the radius of sample, in which the 0.1 pN force boundary is marked with a black solid line since it is the sensitivity of general instruments. With the decreased sample radius from 50 nm to 37 nm, the smallest detectable transverse (κx and κy) and longitudinal (κz) component of chirality parameter gradually increases from 0.007 to 0.014 and from 0.011 to 0.02, respectively. On the other hand, the smallest detectable radius for a sample with chirality parameter |κx (κy)| = 0.015 and |κz| = 0.015 are 36 nm and 43 nm, respectively. It is worthy of noting that the advantage of our technique is the capability of detecting chiral sample on the order of sub-100 nm. By replacing the current tip with nanotips with stronger electric polarizability or magnetic response, detection of chiral samples with smaller chirality parameter can be realized.

 figure: Fig. 4.

Fig. 4. ΔF is shown versus the (a) transverse and (b) longitudinal component of the chirality parameter and radius of the sample, illustrating the sensitivity of our proposed technique. ΔF versus lateral displacement of the tip when probing (c) transverse and (d) longitudinal component of the chirality parameter. Effect of (e, g) transverse and (f, h) lateral displacement of the tip-sample system with respect to the excitation beam on differential force ΔF for probing (e, f) transverse and (g, h) longitudinal component of the chirality parameter.

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Next, the effects of misalignment of the tip on the resolution of the proposed technique is investigated. Ideally, the tip is assumed to be aligned with the sample along the orientation of the chiral dipole moment of the excitation beam. However, a lateral misalignment (normal to the orientation of the chiral dipole) of the tip would decrease the measured differential force. Assuming a chiral sample with purely transverse chirality parameter κx(y) = 0.03, Fig. 4(c) depicts the differential force versus the lateral distance between the tip and sample (as = at = 50 nm, d = 110 nm). As expected, a decreased differential force with full-width-at-half-maximum (FWHM) of 150 nm is observed due to the lateral shift of the tip. As for a chiral sample with purely longitudinal chirality parameter κz = 0.03, the FWHM of the differential force is 100 nm (shown in Fig. 4(d)). These examples show that the radius of the working distance of tip is about 75 nm and 50 nm, within which the tip can be to be able to make an enantiospecific detection of chiral nanosamples.

Lastly, the robustness of the proposed method for detecting chirality parameter is investigated by observing the dependence of the photoinduced force on the tip-sample relative displacement with respect to the excitation beam. Here, transverse and longitudinal displacement are considered, which indicate the direction normal to and along the orientation of the chiral dipole moment of the excitation beam, respectively. Figure 4(e)–4(h) show the effects of the displacement dt and dl of the tip-sample interactive system with respect to the excitation beam on the differential force. It can be seen that differential force is maximized when dt = 0 and dl = 0. With the increase of displacement, the field strength of the chiral dipole moment would decrease accordingly, leading to the decreased differential force. However, even with a displacement as large as dt = 110 nm and dl = 30 nm, the samples with a transverse chirality parameter as small as κx(κy) = 0.015 can still be detectable. As for sample with longitudinal chirality parameter κz = 0.015, the tolerable displacements are found to dt = 80 nm and dl = 18 nm.

6. Conclusions

In this work, based on the concept of photoinduced force in PiFM, a novel method has been proposed to characterize the chirality of reciprocal material in nanoscale, with the use of optical excitation carrying chiral dipole moment with controllable orientation. Through exploring the optical response and the photoinduced force of the tip apex interactive system, we demonstrate a feasible force-detection approach not only to determine the enantiomer type of a chiral sample but also to distinguish among three diagonal components of a sample’s chirality parameter. Particularly, we show that the required complex optical field can be synthesized by the superposition of parallel electric and magnetic dipoles with π/2 phase difference, and the orientation of the chiral dipole moment is determined by the oscillation direction of the dipoles. Consequently, the required illumination is analytically derived through reversing the radiation patterns from the dipoles located at the focal point of a 4Pi microscopy. The idea of using such beams in photoinduced force microscopy enables a method for detection of reciprocal chiral samples down to nanoscale resolution and sample size. A chiral particle (with a radius as small as 50 nm) with a chirality parameter as small as κx (κy) = 0.007 and κz = 0.011 with nanometric resolution can be specified. We believe this work has the potential to advance studies of chirality of samples and molecular concentrations in nanoscale.

Funding

National Natural Science Foundation of China (12274074, 12134013, 12074066, 92050202).

Acknowledgment

G. R. acknowledged the support by the Zhishan Young Scholar Program of Southeast University.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (4)
Fig. 1.
Fig. 1. (a) Schematic of PiFM capable of probing different components of the handedness of a chiral sample. (b) Simplified model of the tip-sample interactive system. (c) Differential force exerted on the tip apex versus κx of the sample for excitations of left- and right-hand chiral dipole respectively.

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Fig. 2.
Fig. 2. (a) Diagram of the proposed setup to generate chiral dipole with controllable orientation. (b) Generation of x-oriented chiral dipole. The first row show both the intensity and SOP distributions of the incident left- and right-propagation optical fields in the left and right pupil plane of the 4Pi focusing system. The second and third rows show the calculated distributions of focal electric and magnetic fields intensities and relative phases of tightly focused beam. (c) The calculated distributions of focal electric and magnetic fields intensities and relative phases of tightly focused beam, demonstrating the generation of (c) y-oriented and (d) z-oriented chiral dipole.

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Fig. 3.
Fig. 3. (a) x-component, (b) y-component, and (c) z-component of the exerted force on the tip apex for local excitation of left- and right-hand x-oriented chiral dipole respectively. Differential force exerted on the tip apex versus different components of the chirality parameter for local excitation of (d) x-oriented, (e) y-oriented, and (f) z-oriented chiral dipole. The configurations of the tip-sample interactive system are also shown in the insets.

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Fig. 4.
Fig. 4. ΔF is shown versus the (a) transverse and (b) longitudinal component of the chirality parameter and radius of the sample, illustrating the sensitivity of our proposed technique. ΔF versus lateral displacement of the tip when probing (c) transverse and (d) longitudinal component of the chirality parameter. Effect of (e, g) transverse and (f, h) lateral displacement of the tip-sample system with respect to the excitation beam on differential force ΔF for probing (e, f) transverse and (g, h) longitudinal component of the chirality parameter.

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Tables (2)

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Table 1. The expressions of the left- and right- propagating incident fields for generating y- and z-oriented chiral dipole.

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Table 2. Expressions of the local field near the tip apex when illuminated by x-oriented chiral dipole.

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Equations (32)

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[ p s m s ] = [ α ¯ s e e i α ¯ s e m i ( α ¯ s m e ) T α ¯ s m m ] [ E l o c ( r s ) H l o c ( r s ) ] ,
[ p s m s ] = [ α ¯ s e e i α ¯ s e m − i α ¯ s e m α ¯ s m m ] [ E l o c ( r s ) H l o c ( r s ) ] ,
α ¯ s e m = [ α s , x x e m 0 0 0 α s , y y e m 0 0 0 α s , z z e m ] ,
{ α s , x x e m = [ 12 i π a s 3 c κ x ( ε s + 2 ) ( μ s + 2 ) − κ x 2 ] ∗ α s , y y e m = [ 12 i π a s 3 c κ y ( ε s + 2 ) ( μ s + 2 ) − κ y 2 ] ∗ α s , z z e m = [ 12 i π a s 3 c κ z ( ε s + 2 ) ( μ s + 2 ) − κ z 2 ] ∗ ,
{ α s e e = 4 π a s 3 ε 0 ε s + 2 [ ε s − 1 + 3 κ 2 κ 2 − ( ε s + 2 ) ( μ s + 2 ) ] α s m m = 4 π a s 3 μ 0 μ s + 2 [ μ s − 1 + 3 κ 2 κ 2 − ( ε s + 2 ) ( μ s + 2 ) ] ,
{ p t = α t e e ⋅ E l o c ( r t ) m t = α t m m ⋅ H l o c ( r t ) ,
{ E l o c ( r t ) = E i n c ( r t ) + E s c a t | s → t  =  E i n c ( r t ) + G E E ( r t , r s ) ⋅ p s + G E M ( r t , r s ) ⋅ m s H l o c ( r t ) = H i n c ( r t ) + H s c a t | s → t  =  H i n c ( r t ) + G M E ( r t , r s ) ⋅ p s + G M M ( r t , r s ) ⋅ m s ,
F = 1 2 R e [ ( ∇ E l o c ( r t ) ∗ ) ⋅ p t + ( ∇ H l o c ( r t ) ∗ ) ⋅ m t − c k 1 4 6 π 1 ε μ ( p t × m t ∗ ) ] ,
[ E x l o c ( r t ) E y l o c ( r t ) E z l o c ( r t ) ] = E i n c ( r t ) + k 0 2 ε 0 e x p ( i k 0 d ) 4 π d ( 2 i k 0 d + 2 k 0 2 d 2 0 0 0 1 − 1 i k 0 d − 1 k 0 2 d 2 0 0 0 1 − 1 i k 0 d − 1 k 0 2 d 2 ) ( p s , x p s , y p s , z ) − η 0 k 0 2 e x p ( i k 0 d ) 4 π d ( 1 − 1 i k 0 d ) ( 0 0 0 0 0 − 1 0 1 0 ) ( m s , x m s , y m s , z ) ,
[ H x l o c ( r t ) H y l o c ( r t ) H z l o c ( r t ) ] = E i n c ( r t ) + c 0 k 0 2 e x p ( i k 0 d ) 4 π d ( 1 − 1 i k 0 d ) ( 0 0 0 0 0 − 1 0 1 0 ) ( p s , x p s , y p s , z ) + k 0 2 e x p ( i k 0 d ) 4 π d ( 2 i k 0 d + 2 k 0 2 d 2 0 0 0 1 − 1 i k 0 d − 1 k 0 2 d 2 0 0 0 1 − 1 i k 0 d − 1 k 0 2 d 2 ) ( m s , x m s , y m s , z ) ,
{ E s c a t | s → t ≈ e i k 0 d 4 π ε 0 d 3 [ 3 x ^ ( x ^ ⋅ p s ) − p s ] H s c a t | s → t ≈ e i k 0 d 4 π d 3 [ 3 x ^ ( x ^ ⋅ m s ) − m s ] ,
{ E s c a t | s → t = − G 0 ε 0 ( − 2 p s , x x ^ + p s , y y ^ + p s , z z ^ ) H s c a t | s → t = − G 0 ( − 2 m s , x x ^ + m s , y y ^ + m s , z z ^ ) ,
[ E x l o c ( x t ) E y l o c ( x t ) E z l o c ( x t ) ] = [ E x i n c ( x t ) E y i n c ( x t ) E z i n c ( x t ) ] − G 0 ε 0 [ − 2 α s e e E x l o c ( x s ) − 2 i α s , x x e m H x l o c ( x s ) α s e e E y l o c ( x s ) + i α s , y y e m H y l o c ( x s ) α s e e E z l o c ( x s ) + i α s , z z e m H z l o c ( x s ) ] ,
[ H x l o c ( x t ) H y l o c ( x t ) H z l o c ( x t ) ] = [ H x i n c ( x t ) H y i n c ( x t ) H z i n c ( x t ) ] − G 0 [ 2 i α s , x x e m E x l o c ( z s ) − 2 α s m m H x l o c ( z s ) − i α s , y y e m E y l o c ( z s ) + α s m m H y l o c ( z s ) − i α s , z z e m E z l o c ( z s ) + α s m m H z l o c ( z s ) ] ,
E x l o c ( x t ) = − ε 0 4 G 0 2 α s e e α t e e − ε 0 2 [ ( ε 0 + 2 G 0 α s e e ) E x i n c ( x t ) + 2 i G 0 α s , x x e m H x i n c ( x t ) ] ,
H x l o c ( x t ) = ( 4 i α t e e α s , x x e m ε 0 G 0 2 + 2 i ε 0 2 α s , x x e m G 0 ) E x i n c ( x t ) 4 G 0 2 α s e e α t e e − ε 0 2 + ( − 8 α t e e ( α s , x x e m ) 2 G 0 3 + 8 α t e e α s e e α m m + 4 α t e e α s e e G 0 2 − 2 ε 0 2 α s m m G 0 − ε 0 2 ) H x i n c ( x t ) 4 G 0 2 α s e e α t e e − ε 0 2 ,
F x ≈ 1 2 R e[ ( 2 p s , x ∂ ∂ x G 0 ε 0 ) ∗ p t , x ] ,
F x ≈ − 3 4 π ε 0 d 4 R e[ α t e e E x i n c ( α s e e E x i n c + i α s , x x e m H x i n c ) ∗ ] ,
Δ F x ≈ − 3 | E 0 | 2 4 π ε 0 μ 0 d 4 [ R e ( i α t e e ( α s , x x e m ) ∗ e i π 2 ) − R e ( i α t e e ( α s , x x e m ) ∗ e − i π 2 ) ] = − 3 | E 0 | 2 4 π ε 0 μ 0 d 4 [ 2 Im ( i α t e e ( α s , x x e m ) ∗ ) ] ,
E l i ( r , ϕ ) = { [ − cos θ cos 2 ϕ − sin 2 ϕ + ( cos θ sin ⁡ ϕ cos ⁡ ϕ − sin ⁡ ϕ cos ⁡ ϕ ) e i π / 2 ] e → x + [ sin ⁡ ϕ cos ⁡ ϕ − cos θ sin ⁡ ϕ cos ⁡ ϕ + ( − cos θ cos 2 ϕ − sin 2 ϕ ) e i π / 2 ] e → y } / cos ⁡ θ ,
E r i ( r , ϕ ) = { [ cos θ cos 2 ϕ + sin 2 ϕ + ( cos θ sin ⁡ ϕ cos ⁡ ϕ − sin ⁡ ϕ cos ⁡ ϕ ) e j π / 2 ] e → x + [ sin ⁡ ϕ cos ⁡ ϕ − cos θ sin ⁡ ϕ cos ⁡ ϕ + ( − cos θ cos 2 ϕ − sin 2 ϕ ) e j π / 2 ] e → y } / cos ⁡ θ ,
E l f ( r f , φ f , z f ) = i λ ∫ 0 θ max ∫ o 2 π ( X l f x ^ + Y l f y ^ + Z l f z ^ ) × e j k 0 r f sin ⁡ θ cos ⁡ ( ϕ − φ f ) + j k 0 z f cos ⁡ θ sin ⁡ θ d θ d ϕ ,
X l f = − cos 2 θ cos 2 ϕ − sin 2 ϕ ,
Y l f = ( − cos 2 θ sin ⁡ ϕ cos ⁡ ϕ + sin ⁡ ϕ cos ⁡ ϕ ) − cos θ e j π / 2 ,
Z l f = sin ⁡ θ cos ⁡ θ cos ⁡ ϕ + sin θ sin ⁡ ϕ e j π / 2 ,
E r f ( r f , φ f , z f ) = i λ ∫ 0 θ max ∫ o 2 π ( X r f x ^ + Y r f y ^ + Z r f z ^ ) × e j k 0 r f sin ⁡ θ cos ⁡ ( ϕ − φ f ) + j k 0 z f cos ⁡ θ sin ⁡ θ d θ d ϕ ,
X r f = cos 2 θ cos 2 ϕ + sin 2 ϕ ,
Y r f = ( − cos 2 θ sin ⁡ ϕ cos ⁡ ϕ + sin ⁡ ϕ cos ⁡ ϕ ) − cos θ e j π / 2 ,
Z r f = − sin ⁡ θ cos ⁡ θ cos ⁡ ϕ + sin θ sin ⁡ ϕ e j π / 2 ,
{ E x l o c ( r t ) = − 8 ε 0 ( N 1 x + 1 2 N 2 x − 1 4 N 3 x − 1 8 E x i n c ε 0 ) 16 D 1 − 4 D 2 + ε 0 2 E y l o c ( r t ) = ε 0 ( N 1 y − N 2 y − N 3 y + E y i n c ε 0 ) D 1 − D 2 + ε 0 2 E z l o c ( r t ) = ε 0 ( N 1 z − N 2 z − N 3 z + E z i n c ε 0 ) D 1 − D 2 + ε 0 2 ,
{ H x l o c ( r t ) = − 8 ( N 4 x + 1 2 N 5 x + 1 4 N 6 x − 1 8 H x i n c ε 0 2 ) 16 D 1 − 4 D 2 + ε 0 2 H y l o c ( r t ) = ( N 4 y − N 5 y + N 6 y + H y i n c ε 0 2 ) D 1 − D 2 + ε 0 2 H z l o c ( r t ) = ( N 4 z − N 5 z + N 6 z + H z i n c ε 0 2 ) D 1 − D 2 + ε 0 2 ,
G 0 ≈ e i k 0 | x t − x s | 4 π | x t − x s | 3 ,
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