1. Introduction

Passive enantioseparation of chiral molecules (a pair of non-superposable enantiomers) is one of the most important and challenging problems of chemical research and its possible solution would contribute to huge economic implications in the chiral pharmaceutical industry [1]. It is known that chiral molecules only show their chiral attributes when they interact with a chiral environment and given by its subtle nature it is tricky to differentiate them. Currently, traditional analytic methods, such as Chiral Chromatography (CC) and Capillary Electrophoresis (CE), continue to be the most commons methods for enantioseparation due to its efficacy, high-selectivity and fast evaluation in the identification of enantiomers [2–5]. However, such techniques employing chiral fluid phases are very much complex [6,7] due to the chiral interaction between the enantiomers and chiral selectors carefully chosen in both CC and CE methods making the non-passive processes present many scientific and industrial disadvantages. On the other hand, with the advent of chiral metamaterials and nanophotonic devices, it was possible to improve the chiral sensibility [8–10] and to probe in details the internal structure of biomolecules [11–14] employing superchiral electromagnetic fields (or chiral near-fields), which represented important developments in chiral spectroscopy. Chiral near-fields are generated by the optical excitation of chiral (or achiral) plasmonic (or photonic) nanostructures [15–19] and act as a chiral material environment, being highly sensitive and strongly located in nanoregions of space being able to locally interact with the chiral molecules [20]. Now, if instead of using chiral selectors we use chiral fields, since they are less invasives, we could guarantee the passivity of the enantioselective process. Current researches in this direction showed that the main underlying mechanism of an EOP is that the interaction of chiral specimens with chiral fields leads to two distinct optical forces. On the one hand, the forces appear laterally in opposite directions [21–28] and, on the other hand, one of the forces captures an enantiomer (gradient force) while the other one repels it (pulling force) [29–32], depending in both cases on the handedness of enantiomers. However, such enantioselective optical forces (sub-piconewton scale) correspond to chiral nanoparticles with a strong chiro-optical response [33,34] and not to chiral biomolecules, generating intense debates about the validity of the dipole approximation and back-action effects when considering artificial chiral molecules in EOP [35,36]. Although optomechanical mechanisms for the separation of nanoparticle enantiomers were reported [37–40], optical enantioseparation of chiral biomolecules still represents a challenge. Some theoretical attempts to reach such enantioselectivity were first reported by Chi-Sing Ho et al., [41] who proposed the use of silicon nanospheres as the source of chiral near-fields; intensifying the optical chirality (local density of chirality) and improving, thus, the efficiency of an selective photoexcitation of chiral molecules (photolyzing an enantiomer and leaving the other intact). On the other hand, Tun Cao et al., [42] employing the strong field confinement of the chiral near-fields produced by graphene-coated nanowires [43] managed to extend the EOP to sub-10-nm chiral specimens and more recently, the same group of researchers, designed and fabricated a particular type of asymmetric plasmonic nanostructure (disc-double split ring resonator) which allowed them to demonstrate theoretically the possibility of chiral sorting of artificial nanoparticles [44]. Two important concepts were used to improve a possible EOP of chiral molecules: localized chiral field and optical chirality, which measure, respectively, the local field intensification and the local degree of chiral dissymmetry in electromagnetic fields [45–49] and, then, many plasmonic nanostructures [15–18,50–56], dielectric [9,41,57] and graphene [58,59] were proposed in order to maximize and tune them. In this context, inspired by meta and nanomaterials and their potential application in control and manipulation in chiral electromagnetic fields, we propose an enantioselective all-optical technique of chiral biomolecules employing optimized asymmetric plasmonic nanoapertures with high field intensification and maximum optical chirality.

2. Enantioselective optical potential and chiral dissymmetry

A small chiral molecule immersed in an electromagnetic field can be modeled as a magneto- electric dipole [60] with a coupled electric dipole moment, $\textbf {p}$, and magnetic dipole moment, $\textbf {m}$. The molecule response due to an electromagnetic field is completely determined by the polarizability tensor or by the components of such tensor: electric polarizability $\alpha$, magnetic polarizability $\beta$, and chiral polarizability $\gamma$. Such physical quantities are related with $\textbf {p}$ and $\textbf {m}$ through the following equation:

The interaction potential of a randomly oriented enantiomer $X$ immersed in a chiral electromagnetic field is given by [21,57]:

It is known from enantiodetection processes that the optical chirality introduced by Lipkin [62] and after interpreted by Tang and Cohem [45] measure the asymmetry in the rates of excitation between a small chiral molecule and its mirror image, technically known as circular dichroism (CD) based in the differential light absorption. In this manuscript, we conveniently used the normalized optical chirality $\hat {\mathcal {C}}$ defined as:

Aiming to determine the efficiency of the EOP, we define the chiral dissymmetry factor $g$, which is a generalization of the dissymmetry factor for enantiomorphs electromagnetic fields [41] and is defined by the preferential optical potential normalized to $U=(U_{X}^{+} + U_{X}^{-})/2$:

3. Computational method and asymmetric nanoaperture

The asymmetric Plasmonic Crescent Moon (PCM) nanoapertures are periodic nanostrutures with a fixed period ($P$=400 nm) in both the directions of $x$ e $y$ as shown in Fig. 1(a). It is composed by a glass substrate with a refraction index of $\eta _{s}=1.45$ coated by a silver thin film with fixed thickness ($t$=120 nm) where perforations were made with an asymmetric crescent moon format and filled with a dielectric with the same refraction index of the glass. These asymmetric PCM nanoapertures were projected for two symmetry breaking in a plasmonic coaxial aperture with inner radius of $r$=50 nm and outer of $R$=100 nm changing first the parameter $x$ to intensify the electric and magnetic field, and later changing the parameter $y$ to introduce chiral asymmetry in the aperture, which is shown in Fig. 1(c). The chiral electric $\textbf {E}^{\pm }$ and magnetic $\textbf {H}^{\pm }$ near-fields, as well as to their norms $\left \| \textbf {E}^{\pm } \right \|$, $\left \| \textbf {H}^{\pm } \right \|$ were calculated by computational simulation employing the Finite-Difference Time-Domain code (FDTD Solutions 8.5, Lumerical Inc., Canada) in host medium with refractive index $\eta _m=1.59$ and CPL with normal incidence in respect to the device, i.e., along the z-axis was used to excite the nanostructure. To generate CPL in the simulations, two linearly polarized light sources with a phase shift of $\pi /2$ were superimposed. The chiral near-fields were calculated in distinct transverse planes above the aperture from $z$=0 nm to $z$=20 nm employing a 2 nm step. Figure 1(b) shows the unit cell of the structure to calculate the normalized transmission spectra (output power to incident power $P_{out}/P_{in}$) for CPL together with contour periodic conditions in $x$ and $y$ and 12 perfectly matched layers in the $z$-direction. Finally, employing the dipole approximation, we put a pair of chiral molecules (enantiomers R and S) inside the near field region in such a way that the molecule [Fig. 1(b)] does not significantly alter the space distribution of the field. Physically, this approximation is justified by the small size of the considered chiral molecule (effective radius of 5 nm) which is much smaller than the wavelengths of the incident light (near-infrared). Then, once the chiral near-fields are calculated, the enantioselective optical potentials (Eq. (2)) corresponding to the enantiomers can be found, as well as the optical chirality (Eq. (3)) and the chiral dissymmetry factor (Eq. (5)).

 

Fig. 1. a) Schematic representation of the plasmonic device formed by a periodic arrange of asymmetric PCM nanoapertures ($P$=400 nm, $t$=120 nm) on a dielectric substrate with refraction index of $\eta _{s}=1.45$. b) Unit cell formed by an asymmetric nanoaperture with CPL incident from below. Above the nanoaperture, the enantiomers R and S are inserted in a transverse plane. c) Front view of the unit cell with its geometrical parameters: outer radius ($R$), inner radius ($r$) and the asymmetry parameters $x$ and $y$.

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4. Numerical results and discussion

In Fig. 2(a), we show a graph of the optical potential using formula 2 for left-handed chiral light (L-CL) as function of the local electric field intensification $I_{\textbf {E}}$ for a chiral spherical molecule (enantiomer S and R) with effective radius of 5 nm and several angles between the fields $i\textbf {E}$ and $\textbf {H}$ given by the $\cos (\theta _{i\textbf {E},\textbf {H}})$. In this work, we have considered non-magnetic enantiomers with dual-symmetry conditions $(\epsilon _{r} \approx 1, \mu _{r} \approx 1)$ in such a way that the polarizabilities $\alpha '$, $\beta '$ and $\gamma '$ of the enantiomers only dependent on chirality parameter $\kappa$ or normalized chirality parameter, $\hat \kappa =\kappa /\eta$, where $\eta$ is the index of refraction of the biomolecule. We chose $\alpha '= \frac {3.33}{\epsilon _{o}} \times 10^{-27}m^{3}$, $\beta '=-\frac {6.83}{\mu _{o}} \times 10^{-30}m^{3}$ and $\gamma '=\pm \frac { 5.23}{c} \times 10^{-27}m^{3}$, which correspond to the polarizabilities of macromolecules (proteins, carbohydrates or lipids) [64] with chirality parameter $\kappa = \pm 0.01$ (or $\hat \kappa = \pm 0.00625$) and $\eta =1.6$. It is important to consider that these polarizabilities are three orders of magnitude lower than the correspondents polarizabilities of artificial chiral nanoparticles [31]. The spacial relationship ${\textbf {H}}=f({\textbf {E}})$ was fit and extrapolated using an optimized plasmonic split-ring nanoaperture with opening angle of $60^{o}$ [65]. As can be seen in Fig. 2(a), the optical potentials for the enantiomer S (dotted lines) reveals that these biomolecules are weakly attracted and unstable (curves above the reference line: $U^{+}=-10$kT$/100$mW) strickly due to the thermal fluctuations for all the considered angles ($0\le \theta _{i\textbf {E},\textbf {H}} \le \pi /2,$) and $I_{\textbf {E}}< 100$; showing a slightly capture for $I_{\textbf {E}}=100$ and $\theta _{i\textbf {E},\textbf {H}}=0$. Once defined $\cos (\theta _{i\textbf {E},\textbf {H}})$ as the intrinsic optical chirality, we have that for $\theta _{i\textbf {E},\textbf {H}} = 0$ the fields $i\textbf {E}$ and $\textbf {H}$ are parallel leading thus to a maximum chirality and for $\theta _{i\textbf {E},\textbf {H}}=\pi /2$ the same fields are orthogonal and lead to a minimum chirality. On the other hand, although the R enantiomer [solid lines in Fig. 2(a)] are also weakly attracted for low values of intensification $I_{\textbf {E}}<70$, such enantiomer starts to be captured appreciably (curves below the reference line) when the local electric near-field and the intrinsic optical chirality are intensified until their maximum value. It is important to mention also that there is a value of $I_{\textbf {E}} \approx 90$ where the enantiomer S is repelled and the enantiomer R is captured for all values of $\theta _{i\textbf {E},\textbf {H}}$, being the most efficient one with maximum chirality ($\theta _{i\textbf {E},\textbf {H}} = 0$). In other words, an intensification in the optical chirality leads to an increase in the trap depth and therefore a better efficiency in the capture (pulling) of the enantiomer R (S).

 

Fig. 2. a) Trapping optical potential for L-CL as a function of $I_{\textbf {E}}$ for the non-magnetic enantiomers R and S and several angles $\theta _{i\textbf {E},\textbf {H}}$. Trapping optical potential for fixed $I_{\textbf {E}}=100$ as a function of chiral parameter $\kappa$ and several angles $\theta _{i\textbf {E},\textbf {H}}$ for b) L-CL and c) R-CL. The non-magnetic enantiomers satisfies the dual-symmetric conditions: $\epsilon _{r} \approx 1, \mu _{r} \approx 1$ with refractive index of the host medium of $\eta _m=1.59$.

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Now let’s consider the case of $I_{\textbf {E}} = 100$ in such a way that for this value the incident laser power exactly matches the trapping threshold. Currently, giant electric near-field can be reached employing metallic nanoantennas in a classic [66,67] or quantum [68] regime. Figures 2(b) and 2(c) show graph of the trapping optical potentials using formula 2 for L-CL and right-handed chiral light (R-CL), respectively, as a function of the chirality parameter $\kappa$ of the non-magnetic enantiomers and several angles $\theta _{i\textbf {E},\textbf {H}}$. Considering as a reference the optical potential $U^{\pm }$ for $\cos \theta _{i\textbf {E},\textbf {H}} = 0$, it’s observed that for L-CL and $\kappa < 0$ the depth of the optical trap increases as the intrinsic optical chirality intensifies $(\cos \theta _{i\textbf {E},\textbf {H}} > 0)$ makes it a stable trap for the enantiomer R. On the other hand, when the R-CL is employed [Fig. 2(c)] now the depth of the trap increases for positive chirality parameters ($\kappa > 0$) as the intrinsic optical chirality increases and, thus, leading to a stable trap now for enantiomer S. This symmetry property that occurs when we exchange L-CL to R-CL and $\kappa$ with $-\kappa$ not altering the optical potentials is only valid when it is considered enantiomorphic chiral fields descendant from achiral nanostructures [57]. In the case of more complex nanostructures, for instance, chiral metamaterials, it will exist a slight difference between the optical potentials triggered by L-CL and R-CL not depending on the chirality of the biomolecule. It is also important to note from Figs. 2(b) and 2(c) that it is possible efficiently capture chiral molecules with small chirality parameters ($|\kappa | < 0.01$) considering maximum optical chirality that were not yet reported in the literature. For instance, at first it is possible to sort enantiomers with $\kappa =\pm 0.001$ considering high values of field intensification ($I_{\textbf {E}}>100$) and intrinsic optical chirality close to 1 where the enantioselective optical potentials present a relation approximately linear with $\kappa$ and not parabolic (Eq. (1)) and, moreover, the L-CL e R-CL act as turn on and off keys to the optical trap.

Figure 3(a) shows a graph of chiral discrimination (absolute value of the differential optical potential) using the formula 4 for four chiral molecules with different $\kappa$ as a function of $\Delta \hat {\mathcal {C}}$ for enantiomorph fields (solid lines) and non-enantiomorph fields (dotted lines). It is noted that a referential displacement of the lines $|\Delta U|=m_{\kappa }\Delta \hat {\mathcal {C}}$ (where $m_{\kappa }$ is the slope) exists upon considering non-enantiomorph fields with $\Delta I_{\textbf {E}} = 50$. Such displacement of $U_a=9kT/100$mW produced and introduced in the lines’ equation $|\Delta U|=m_{\kappa }\Delta \hat {\mathcal {C}}+U_a$ is an indicator of the increase in the efficiency of the enantioselective process due to the intensification of the chiral discrimination or the metric increase between the enantioselective optical potentials. For instance, upon considering enantiomorph fields, it is possible to obtain an appreciable dissymmetry and, thus, an efficient enantiomeric separation for the chirality parameters between 0.01 and 0.05 and for all values of $\Delta \hat {\mathcal {C}}$. Already, upon considering small values of $\kappa$, for instance, $\kappa <0.01$, the inclination of the lines decreases reducing the chiral discrimination and making the enantioselective process inefficient. However, when we consider non-enantiomorphic fields, the asymmetry increases $\Delta U$ and this increment can be controlled by $\Delta I_{\textbf {E}}$ and therefore it possible to obtain very efficient enantioselective processes including for small values of $\kappa$, this is shown in line $|\Delta U|=m_{0.005}\Delta \hat {\mathcal {C}}+9kT/100$mW of Fig. 3(a). The quantification and the chiral limits of the discrimination can be better visualized and understood employing the chiral dissymmetry factor $g_c$ (effect of the asymmetry introduced in the achiral dissymmetry factor due to enantiomorph fields).

 

Fig. 3. a) Differential optical potential $\Delta U_{X}$ as a function of $\Delta \hat {\mathcal {C}}$ for enantiomorphic (solid lines) and non-enantiomorphic (dotted lines) fields of an enantiomer with chirality parameter $|\kappa |=0.005; 0.01; 0.03; 0.05$ and differential field intensification $\Delta I_{\textbf {E}} = 50$. b) Achiral dissymmetry factor $g_a$ as a function of $\kappa$ for $\theta _{i\textbf {E},\textbf {H}}=0;0.5;1$ and c) Chiral dissymmetry factor $g_c$ for $\theta _{i\textbf {E},\textbf {H}}=1$ (maximum optical chirality) as a function of $\kappa$ for $\Delta I_{\textbf {E}} = 10, 20, 40$.

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Figures 3(b) and 3(c) show graphs of the achiral and chiral dissymmetry factor (formula 5), respectively, as a function of the chirality parameter of enantiomers for three values of $\cos \theta _{i\textbf {E},\textbf {H}}=0, 0.5, 1$. For the non-chiral dissymmetry factor [Fig. 3(b), enantiomorphic fields] the curve has odd symmetry, i.e., $\hat {g}=g_a(\kappa )+g_a(-\kappa )=0$ and $\hat {f}=g_a(-\kappa )-g_a(\kappa )=2g_a(-\kappa )$, where $\hat {g}$ measures the enantiomorfity effect in the efficiency of an EOP and the factor $\hat {f}$ the metric of this process. When non-enantiomorph fields are considered, as shown in Fig. 3(c), the factors $\hat {g}=g_c(\kappa )+g_c(-\kappa )\ne 0$ and $\hat {f}=g_c(-\kappa )-g_c(\kappa )>2g_c(-\kappa )$ indicate a break of symmetry in the system, and the greater are the values of $\Delta I_{\textbf {E}}$ [Fig. 3(c)] the bigger the asymmetry introduced in the $g$ factor will be. Later we will show that thanks to this break of symmetry we can improve the efficacy and stability of an EOP of chiral molecules with $\kappa <0.01$ using asymmetric plasmonic nanoapertures.

When a chiral plasmonic nanostructure interacts with CPL, this nanostructure generates very complex localized chiro-plasmonic oscillations on the metal-dielectric interface and depending on the handedness of the incident light, the chiral plasmonic near-fields are also L-CL or R-CL. In particular, plasmonic nanoapertures [31,65] and asymmetric nanocrescent [69] have been recently proposed as the source of chiral fields with strong optical chirality and highly localized fields, respectively, but until now, no chiral plasmonic nanostructure with simultaneously high optical chirality and high field intensification was reported.

Figure 4(a) shows the normalized transmission spectra for distinct asymmetric plasmonic coaxial nanoapertures (labeled by the symmetry breaking parameter $x$) in 120 nm silver slab, with the inner and outer radius of 50 and 100 nm, respectively, and circularly polarized incident light. It is worth mentioning that all the transmission spectra both for L-CPL and R-CPL were coincident because the asymmetric nanoapertures are achiral. When $x=0$ (coaxial cylindrical aperture) the main peak takes place at $\lambda =965$ nm and corresponds to a non-propagating mode in the coaxial structure [70]. As we broke the nanoaperture symmetry ($x>0$) this non-propagating mode is divided into two guided modes near to the infrared. In the case of $x=50$ nm (symmetric PCM nanoaperture) both resonant modes appear well-defined with resonant wavelengths $\lambda _1=845$ nm and $\lambda _2=988$ nm. Similar resonant peaks were also observed in plasmonic slanted split-ring apertures [62]. On the other hand, in Fig. 4(b) (lower part) are shown the transmission spectra for an asymmetric PCM nanoaperture with chiral asymmetry parameter $y=15$ nm when L-CPL and R-CPL are incidents on the nanoaperture. Clearly, we can observe the appearance of a dissymmetry in the transmission spectra around the high-energy resonant mode ($\lambda =$798 nm). This phenomenon, consequence of the geometric chirality of the nanoaperture is revealed in the circular dichroism transmission CDT=$(T_{L}-T_{R})/(T_{L}+T_{R})$ reaching a maximum value of 0.25 at $\lambda =798$ nm [top part of Fig. 4(b)]. The origin of this chiral dissymmetry in asymmetric PCM nanoapertures is in the asymmetric redistribution of the electromagnetic energy due to the breaking of geometric symmetry [transition from a nonchiral geometry (symmetric PCM nanoaperture) in a chiral geometry (asymmetric PCM nanoaperture)] controlled by the parameter $y$. Figure 4(c) shows this asymmetric distribution of the electric $I_{\textbf {E}^{\pm }}$, magnetic $I_{\textbf {H}^{\pm }}$ and chiral $\cos (\theta _{i\textbf {E}^{\pm },\textbf {H}^{\pm }})$ field intensities in a transverse plane 5 nm above the nanoaperture at $\lambda =798$nm (maximum CDT). The maximum values reached in the intensification of electric $I_{\textbf {E}^{-}}$ and magnetic $I_{\textbf {H}^{-}}$ fields were 144 and 16, respectively, with a reduction of $37.5\%$ for $I_{\textbf {E}^{+}}$ and $I_{\textbf {H}^{+}}$. It is noticed that for L-CPL and R-CPL the bigger localization and intensification of energy are in the complementary probes of the crescent moon-shaped nanostructures. It is also noticed that the values of intrinsic optical chirality oscillate between −0.9 and 0.9, and the chiral field is distributed all over the structure. With maximized values of the chiral field intensification, the asymmetric plasmonic nanoaperture can reach huge values of optical chirality of up to $\hat {\mathcal {C}}\approx \pm 2000$, four times higher than the maximum optical chirality employing graphene nanostructures [59] and chiral dissymmetry factor of up to $|\hat {g}|\approx 1$ and $|\hat {f}|\approx 1.3$ for chirality parameters $\kappa$ between −0.01 and 0.01 with $\kappa \ne 0$.

 

Fig. 4. a) Transmission spectrum of the asymmetric coaxial plasmonic nanoapertures ($R=100$ nm and $r=50$ nm) in 120 nm silver slab for various values of the asymmetry parameter $x$ and CPL. b) Transmission spectra of an asymmetric PCM nanoaperture with $x=50$ nm e $y=15$ nm when the incident light is circularly polarized to the left (L-CPL) and to the right (R-CPL) [below]; its spectra of CD transmission is shown above. c) Intensification of electric ($I_{\textbf {E}^{\pm }}$), magnetic ($I_{\textbf {H}^{\pm }}$) and chiral [$\cos (\theta _{i\textbf {E}^{\pm },\textbf {H}^{\pm }})$] near-field 5 nm away from the asymmetric PCM nanoaperture at $\lambda =798$ nm. The maximum intensification of the electric and magnetic fields for L-CPL in this plane are 144 and 16, respectively, and the maximum/minimum intrinsic optical chirality is $\pm 0.9$.

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In Figs. 5(a) and 5(b) it is shown the effect of the $x$ parameter (symmetry breaking parameter) and the distance $z$ (transverse plane above the nanoaperture) on the maximum chirality that can be reached with asymmetric nanoapertures. In the case of Fig. 5(a) (symmetric PCM nanoapertures) which corresponds to the resonant mode of higher energy, the maximum optical chirality takes place at $x$=50 nm and $\lambda =845$ nm with $\hat {\mathcal {C}}_{max,min}^{\pm }\approx \pm 1100$, $I_{\textbf {E}^{\pm }}\approx 133$ and keeping $I_{\textbf {H}^{\pm }}$ almost constant. An increase also occurred in the intrinsic optical chirality by a factor of 2 in comparison with coaxial nanostructures [31]. Breaking the symmetry of the PCM nanoaperture again by means of the parameter $y=15$ nm, Fig. 5(b) shows the differential optical chirality for $\lambda =798$ nm (maximum value of CDT) in various transverse planes above the asymmetric nanoaperture. As $z$ increases it was observed that the field intensification and the optical chirality rapidly decrease as well as its differential quantities. The maximum differential optical chirality is reached on the surface of the structure ($z=0$ nm) and vanished to zero for $z>20$ nm. As we are going to show further, the conditions of $z=5$ nm, $\Delta I_{\textbf {E}} \approx 50$ and $\hat {\mathcal {C}}=550$ are more than enough to ensure the functionality of an EOP. Finally, Fig. 5(c) shows the distribution of the differential chiral near-field $\Delta \hat {\mathcal {C}}$ created by the asymmetric PCM nanoaperture when it is illuminated with L-CPL. This transverse distribution was simulated in a $z=5$ nm plane on a nanoaperture and as expected for $y=15$ nm the corresponding energy distribution to the chiral field is asymmetric for differential values $\Delta \hat {\mathcal {C}}_{min}=-1200$ and $\Delta \hat {\mathcal {C}}_{max}=600$. The most important, these high-values with a significant difference quantify the location and the dissymmetry around the complementary tip of the asymmetric nanoaperture. Similar results were observed in chiral gammadion nanostructures [14] where the asymmetric in the arms of the gammadion structures intensified the local field distribution and the optical chirality.

 

Fig. 5. a) Effect of the parameter $x$ on the maximum optical chirality of symmetric PCM nanoapertures for the higher resonance energy [$\lambda = 845$nm in Fig. 4]. b) Differential optical chirality as a function of the distance $z$ above the nanoaperture at $\lambda =798$ nm. c) Differential optical chirality created by an asymmetric PCM nanoaperture 5 nm above the nanoaperture with values $\Delta \hat {\mathcal {C}}_{min}=-1200$ and $\Delta \hat {\mathcal {C}}_{max}=+600$.

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In order to examine the enhanced chiro-optical properties of the asymmetric PCM nanoapertures in the passive processes of optical enantioseparation of chiral biomolecules, full-wave electromagnetic simulations were performed. Figures 6(a) and 6(b) show the optical potentials $U^{-}$ and $U^{+}$ in 3D, respectively, for a non magnetic chiral macromolecule with $\kappa =0.005$ (enantiomer S) and dual-symmetric conditions $\epsilon _r\approx 1$ and $\mu _r\approx 1$. These potentials were calculated at 5 nm away from the nanoaperture and it is possible to notice the differences on its topography for both handedness of an incident light. The chiral discrimination results in a difference between their optical potentials $|\Delta U|=15$ kT/100mW, while the potential $U^{-}$ acts as a stable trap for the enantiomer S, the potential $U^{+}$ acts as a barrier by not overcoming the intrinsic thermal fluctuations of the molecule. The cross sections (one-dimensional) of the transverse optical potentials $U_{xy}$ along the axes $y=75$ nm and $x=106$ nm are shown in Figs. 6(c) and 6(d), respectively, and provided a trapping potential (black curves) as deep as $-17.4$kT per 100mW of transmitted power, enough to capture the enantiomer S employing L-CPL and slightly repel it when using R-CPL. The full-width at half-maximum (FWHM) of the trapping potential in the $x$-direction and $y$-direction is about 6 nm illustrating the stronger isotropic trapping in both the directions and with a maximum chiral dissymmetry factor of $|g|$=1.52. These results demonstrate that an asymmetric PCM nanoaperture excited with L-CPL (R-CPL) can trap (repel) chiral biomolecules as small as 5 nm while keeping the required power below 100 mW. Figures 6(e) and 6(f) show optical enantioselective stack bar diagrams for a set of various chiral molecules with $\kappa$ between −0.05 and 0.05 ($\kappa \ne 0$) for both L-CPL and R-CPL, respectively, in a transverse plane located at $z=6\pm 1$ nm. These stack bar diagrams represent the main focus of the manuscript and summarize all the considered results so far. The color bars indicate the maximum values of the optical potentials (potential barriers) located around the complementary tips of the asymmetric plasmonic nanoaperture, while the black bars (below the reference line) indicate the minimal values of the depth correspondent to a stable trap located for a particular enantiomer. For instance, when both enantiomers R and S are immersed in a left-handed chiral near-field [Fig. 6(e)], it is clearly observed that the enantiomer R is repelled, while the enantiomer S is stably trapping because it is below the reference energy, and located because of the FWHM$\sim$ 8nm for all $\kappa$ considered. Additionally, we can use a right-handed chiral near-field [Fig. 6(f)] to reverse the process, i.e., to capture enantiomer R, while the enantiomer S is repelled. Thus, the incident circularly polarized light acts as a switch enantioselective turning on and off the enantioselective preferential process.

 

Fig. 6. Three-dimensional (3D) optical potential for (a) L-CPL and (b) R-CPL of an enantiomer S ($\kappa =0.005$) at 5nm above the asymmetric PCM nanoaperture. One-dimensional (1D) transverse optical potentials $U_{xy}$, where the cross sections are taken along (c) $y=75$ nm and (d) $x=106$ nm for L-CPL (black lines) and R-CPL (red lines) with equal FWHM = 6 nm of the trapping potential along the $x$ and $y$ direction. Enantioselective optical diagrams for chiral macromolecules with values of $\kappa$ between −0.05 and 0.05 for both (e) L-CPL and (f) R-CPL. In both cases, differences in the optical potentials belonging to R ($\kappa <0$) and S ($\kappa >0$) enantiomers provide passive enantioseparation.

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Although the proposed enantioselective optical method demonstrates that it is possible to separate non-magnetic chiral macromolecules (proteins or carbohydrates) under dual-symmetric conditions up to $\kappa =\pm 0.005$, it is possible to decrease it to $\kappa =\pm 0.001$ if the transverse capture plane is located at 2 nm (close to the surface of the asymmetric plasmonic nanoapertures). Under such conditions, the potential depth increases due to increased field intensification (stable capture of one of the enantiomers) with strong localization of the chiral near-field (FWHM $\approx$ 3 nm). Therefore, our plasmon-enhanced chiral-field approach could go toward an efficient passive-optical enantioseparation process of sub-5-nm of single-chiral biomolecules.

5. Conclusions

In summary, we have proposed the use of non-enantiomorphic chiral near-fields created by asymmetric plasmonic nanoapertures for an EOP of chiral macromolecules. By intensifying the chiral dissymmetry factor due mainly to the asymmetry of the superchiral near-fields and maximum optical chirality of the nanoaperture, we managed to separate on a dual-symmetric condition single-chiral enantiomers with small chirality parameter up to $\pm 0.005$ previously inaccessible. Our contribution will allow us to efficiently design new enantioselective nanoplasmonic devices with the discriminatory capability of sub-5nm chiral molecules and ultra-small chirality parameter.