1. Introduction

A substance that cannot be superimposed on its mirror image is known as a chiral substance. When two compounds are chiral, they are known as enantiomers and are distinguished as an R-enantiomer (right-handed) and S-enantiomer (left-handed). The separation of chiral materials is a challenging task in various research fields. Specifically, most methods for separating enantiomers, such as high-performance liquid chromatography [1], are based on the chemical characteristics of molecules, and a universal separation method has not yet been established.

The difference in the optical response of chiral materials to circularly polarized light (CPL) is attributed to the microscopic nature of the electronic excited states caused by the geometric structure of the materials. If this difference can be transformed to the mechanical motion of chiral materials, a highly versatile separation technique that does not rely on the chemical characteristics can be obtained. Several theoretical schemes to separate chiral molecules have been proposed. For example, some researchers considered that a linearly polarized plane wave exerts a lateral optical force on chiral particles on a substrate and can thus separate enantiomers in the case of particles with opposite handedness [2]. This study was followed by notable related theoretical research [3–5]. Several other researchers theoretically reported on the chiral selectivity using a circularly polarized laser beam [6,7] and the selective trapping of enantiomers by plasmonic apertures [8–10].

Moreover, experimental studies on the separation of chiral particles [11–13] were reported, including one in which lateral optical force was used to separate chiral particles [14,15]. Notably, in these studies, only micrometer-sized particles were considered. In addition, the dissymmetry factor was enhanced by using superchiral light [16,17], indicating that high enantioselectivity can be achieved by devising ingenious systems to realize such a peculiar light field.

As a potential approach to generate an effective optical force to separate small chiral molecules, optical manipulation based on the electronic resonance effect has been considered [18–20], in which the microscopic information of the electronic system in nanomaterials is transformed to the macroscopic motion of the target material. It has been theoretically demonstrated that the optical force under resonant conditions is remarkably stronger than that under non-resonant conditions, and this principle has been applied to realize the selective optical transport of semiconducting nanoparticles in superfluid helium [21] and selective trapping and transport of single-walled carbon nanotubes at room temperature [22,23]. In particular, the selective manipulation of nano-diamonds with NV-centers exhibiting quantum-mechanical properties has been successfully achieved [24,25]. Thus, the potential of resonant optical manipulation has been demonstrated through various theoretical proposals and experimental applications [26].

Considering these aspects, this study is aimed at formulating chirality-dependent optical forces under resonance conditions. The existing formula to determine the optical force on a chiral substance is based on the dipole approximation [27–29] and phenomenological chiral susceptibility. On the other hand, the chiral susceptibility has been discussed from the first-principle viewpoint, and certain issues regarding consistency with using the phenomenological chiral susceptibility, especially in the resonance region, have been highlighted [30]. Thus, in this study, we derive the formula of the optical force by incorporating the microscopic structure of the chiral molecules in the nonlocal optical response theory. We solve the simultaneous equations of the constitutive equation and Maxwell equation in a self-consistent manner. The constitutive equation is obtained based on the microscopic linear response theory with the transition dipole densities in the chiral configuration without using the phenomenological chiral susceptibility. The results can help clarify the mechanism of the influence of the optical force on the chiral molecules under the resonance condition and the relationship between the optical force spectra and other spectra such as those of the optical activity and circular dichroism of absorptive molecules. Furthermore, we numerically evaluate the exerted force considering the realistic parameter values of chiral molecules and analyze the Brownian dynamics in solvents assuming the irradiation of counter-propagating light waves with opposite circular polarizations. The results can provide guidance for the design of schemes to select chiral molecules by optical forces.

2. Theory

2.1 Optical force on the chiral molecule

This section presents the theoretical framework for evaluating the resonant optical force based on the microscopic nonlocal optical response theory [20,31] to evaluate the possibility of mechanical manipulation of chiral materials by light. According to the liner response theory presented by Kubo [32], the induced polarization caused by the transitions between the ground and excited state of the molecules is

By substituting Eqs. (1)–(3) into the following expression of the time-averaged Lorentz force exerted on a particle with polarization $\boldsymbol {P}_{\mathrm {res}}$ in an electromagnetic field given by [20]

In Eq. (10), the first term on the right-hand side represents the resonant optical force exerted on the target materials by the incident field, and the second term represents the force exerted by the induced field from different excited states.

We assume that the constituent molecules in the chiral molecular systems are sufficiently small compared to the wavelength of light, and the transition dipole moment is expressed as

The incident CPL is expressed as follows:

As a simple example, consider a case involving two molecules $i$ and $j$ (or $i$ and $j^{\prime }$), which are the components of a dimer molecule. In this case, Eq. (7) can be solved analytically, as follows:

If the excitation energy and sum of the radiative and nonradiative widths of each molecule are identical, $\bar {E} = \bar {E}_i = \bar {E}_j$ and $\bar {\Gamma } = \bar {\Gamma }_i = \bar {\Gamma }_j$. $\mathbf {e}_z$ is the unit vector in the direction of the $z$-axis, and $A_{\mathrm {R}}$ and $A_{\mathrm {I}}$ represent the real and imaginary parts of $A_{ij} (= A_{ji})$, respectively. In this equation, the pair of dipoles $i$ [$\boldsymbol {r}_{i} = (x_i,y_i,z_i)$, $\boldsymbol {d}_{i} = (d_{ix},d_{iy},d_{iz})$] and $j$ [$\boldsymbol {r}_{j} = (x_j,y_j,z_j)$, $\boldsymbol {d}_{j} = (d_{jx},d_{jy},d_{jz})$] is defined as the R-enantiomer $(l=1)$, and the pair of dipoles $i$ and $j^{\prime }$ [$\boldsymbol {r}_{j^{\prime }} = (x_{j^{\prime }},y_{j^{\prime }},z_{j^{\prime }}) = (x_j,-y_j,z_j)$, $\boldsymbol {d}_{j^{\prime }} = (d_{j^{\prime }x},d_{j^{\prime }y},d_{j^{\prime }z}) = (d_{jx},-d_{jy},d_{jz})$] is defined as the S-enantiomer $(l=2)$. Dipoles $j$ and $j^{\prime }$ are mirror images of the $xz$-plane. Notably, the term associated with the induced field cannot be neglected if strong light scatterers such as metallic structures are present near the molecules.

In Eq. (17), the first term on the right-hand side is the sum of the force on the single dipole, the second term is the force associated with the coaxial dipole interaction, and the third term is used to account for the differences owing to the type of enantiomer $(l=1,2)$ and CPL.

2.2 Relationship with classical theory

We discuss the correspondence between the proposed expression and that derived from the Born–Kuhn model (coupled-oscillator model) [33], which is often used to explain the chiral optical response. According to the Born–Kuhn model, when a chiral material is irradiated with light, two oscillators $i$ and $j$ vibrating in different directions are excited in the material, and they interact with each other. Consider an excited state in which the light is irradiated in the $z$-axis. Oscillator $i$ oscillates in the $x$-axis at coordinates $\boldsymbol {r}_{i} = (x(t), 0, -l_{z}/2)$, and oscillator $j$ oscillates in the $y$-axis at coordinates $\boldsymbol {r}_{j} = (0, y(t), l_{z}/2)$. The potential energy $U$ of the excited state is defined as

Using Eqs. (23) and (24) instead of Eqs. (21) and (22), the following simultaneous equations can be obtained:

Therefore, the polarization can be expressed as

Kirkwood considered the oscillator in the Born–Kuhn model to be a dipole [34]. In this framework, the interaction between the dipoles can be specified by the potential energy $U_{ij}$ associated with the dipole–dipole interaction:

In particular, only the resonant part can be extracted as follows:

According to this equation, in the Kuhn–Kirkwood model, the circular dichroism and optical rotation can be attributed to the interaction between two dipoles that are not on the same wavefront and are oriented in different directions in the chiral material.

This study is focused on resonance-induced polarization. According to Eq. (14), $\boldsymbol {P}_{\mathrm {res}} = \sum _{\lambda } X_{\lambda }(\omega ) \boldsymbol {\rho }_{\lambda }(\boldsymbol {r})$. If the transition dipole densities at sites $i$ and $j$ have the same magnitude and only have components that are perpendicular to each other, $\boldsymbol {\rho }_{i}(\boldsymbol {r}) = \rho (\boldsymbol {r})(1,0)$ and $\boldsymbol {\rho }_{j}(\boldsymbol {r}) = \rho (\boldsymbol {r})(0,1)$. Therefore,

Because $\boldsymbol {X}^{\mathrm {b}}$ and $\boldsymbol {E}^{\mathrm {b}}$ are related by Eq. (5), the resonant induced polarization corresponds to Eq. (30) derived from the classical theory.

The coefficient matrix of polarization in Eq. (31) derived from the nonlocal response theory is composed of sites $i$ and $j$, whereas the polarization rate tensor in Eq. (30) is composed of the coordinate axes ($x$- and $y$-axes). A comparison of these two formulations indicates that the former has a generalized form that includes the latter in terms of certain approximations and limited conditions. Moreover, in contrast to the latter form, the quantities appearing in Eq. (31) are self-consistently determined from the Schrödinger and Maxwell equations through the Green function propagating both the longitudinal and retarded transverse fields. Consequently, the interaction term includes not only the instantaneous Coulomb interactions but also the radiative interaction. Furthermore, $\boldsymbol {\rho }(\boldsymbol {r})$ is defined in terms of microscopic quantities that yield the quantum-mechanically evaluated oscillator strength. $A_{ij}$ and $X_{i}^{\mathrm {b}}$ represent information regarding the microscopic spatial structure of the induced polarization. These terms enable the inclusion of the quantum-mechanical information of chiral molecules and nanostructured materials with arbitrary sizes, shapes, and internal structures of molecules, which cannot be explicitly treated in the classical model. These features of the present formulation, including nonlocality and self-consistency, are of significance, especially in cases in which the excitonic coherence in molecular aggregates or nanocrystals is prominent [35].

3. Model and calculation method

This section describes the model of chiral molecules considered for the numerical demonstrations. The single molecular model pertains to Tröger’s base porphyrin synthesized by Crossley et al. [36]. The model consists of two porphyrin molecules that are twisted to form a dimeric chiral molecule. We modeled this porphyrin dimer as a bonded transition dipole based on Gouterman’s four-orbital model [37,38]. In reality, transition dipoles also arise in the direction orthogonal to each transition dipole under consideration; however, these dipoles can be ignored because their transition energies are shifted. Although we consider model dipoles for the simplified demonstration in this study, we can configure the corresponding coupled-dipole model by using the first-principle orbital calculation, if necessary.

Figure 1 shows the coordinate system for the porphyrin dimer. The pair of porphyrin $i$ [$\boldsymbol {r}_{i} = (r_p,0,0)$, $\boldsymbol {d}_{i} = (d,0,0)$] and porphyrin $j$ [$\boldsymbol {r}_{j} = (r_p\sin \theta \cos \phi,r_p\sin \theta \sin \phi,r_t+r_p\cos \theta )$, $\boldsymbol {d}_{j} = (d\sin \theta \cos \phi,d\sin \theta \sin \phi,d\cos \theta )$] is defined as the R-enantiomer, and that of porphyrin $i$ and porphyrin $j^{\prime }$ [$\boldsymbol {r}_{j^{\prime }} = (r_p\sin \theta \cos \phi,-r_p\sin \theta \sin \phi,r_t+r_p\cos \theta )$, $\boldsymbol {d}_{j^{\prime }} = (d\sin \theta \cos \phi,-d\sin \theta \sin \phi, d\cos \theta )$] is defined as the S-enantiomer. The molecules are considered to rotate in the solvent owing to thermal fluctuations. Therefore the Euler angle variables $\alpha$, $\beta$, and $\gamma$ are introduced in the coordinate system of the molecule. We irradiate a circularly polarized plane wave along the $z$-axis from the positive side.

 

Fig. 1. Coordinate system for Tröger’s base porphyrin dimer. The image of the molecule is reprinted from Ref. [36]. The images of the red and blue arrows have been added by the authors of the present study. The pair of porphyrin $i$ [$\boldsymbol {r}_{i} = (r_p,0,0)$, $\boldsymbol {d}_{i} = (d,0,0)$] and porphyrin $j$ [$\boldsymbol {r}_{j} = (r_p\sin \theta \cos \phi,r_p\sin \theta \sin \phi,r_t+r_p\cos \theta )$, $\boldsymbol {d}_{j} = (d\sin \theta \cos \phi,d\sin \theta \sin \phi,d\cos \theta )$] is defined as the R-enantiomer, and the pair of porphyrin $i$ and porphyrin $j^{\prime }$ [$\boldsymbol {r}_{j^{\prime }} = (r_p\sin \theta \cos \phi,-r_p\sin \theta \sin \phi,r_t+r_p\cos \theta )$, $\boldsymbol {d}_{j^{\prime }} = (d\sin \theta \cos \phi,-d\sin \theta \sin \phi,d\cos \theta )$] is defined as the S-enantiomer.

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In addition, to examine the dependence of the optical force on the number of dipoles, we consider a model in which the dipoles are stacked in a spiral. As a real substance close to this model, a molecule consisting of porphyrins twisted and bound together between planes (porphyrin chains) has been reported [39]. Figure 2 shows the helical nanostructures corresponding to a single rotation by four dipoles. The framework is established such that the distance between adjacent dipoles along the $z$-axis is $l_z$, the deviation from the central axis ($z$-axis) is $l_0$, and the rotation angle of the spiral is $\theta = \pi /2$. In this case, the coordinates of the $i$-th transition dipole in the right-handed helix are $\boldsymbol {r}^{\mathrm {R}}_{i} = (l_0 \sin (i\theta ),-l_0\cos (i\theta ),(i-1)l_z)$, and the associated dipole moment is $\boldsymbol {d}^{\mathrm {R}}_{i} = (d_i \sin (i\theta ),-d_i\cos (i\theta ),0)$. Similarly, the coordinates of the $i$-th transition dipole in the left-handed helix are $\boldsymbol {r}^{\mathrm {L}}_{i} = (l_0 \sin (i\theta ),l_0\cos (i\theta ),(i-1)l_z)$, and the corresponding dipole moment is $\boldsymbol {d}^{\mathrm {L}}_{i} = (d_i \sin (i\theta ),d_i\cos (i\theta ),0)$.

 

Fig. 2. Model in which the dipoles are stacked in a spiral. The figure shows helical nanostructures corresponding to a single rotation by four dipoles. The right(left)-handed spiral is defined as the right(left)-handed helix, respectively. The distance between adjacent dipoles along the $z$-axis is $l_z$, the deviation from the central axis ($z$-axis) is $l_0$, and the rotation angle of the spiral is $\theta = \pi /2$. In this case, the coordinates of the $i$-th transition dipole in the right-handed helix are $\boldsymbol {r}^{\mathrm {R}}_{i} = (l_0 \sin (i\theta ),-l_0\cos (i\theta ),(i-1)l_z)$, and the corresponding dipole moment is $\boldsymbol {d}^{\mathrm {R}}_{i} = (d_i \sin (i\theta ),-d_i\cos (i\theta ),0)$. Similarly, the coordinates of the $i$-th transition dipole in the left-handed helix are $\boldsymbol {r}^{\mathrm {L}}_{i} = (l_0 \sin (i\theta ),l_0\cos (i\theta ),(i-1)l_z)$, and the corresponding dipole moment is $\boldsymbol {d}^{\mathrm {L}}_{i} = (d_i \sin (i\theta ),d_i\cos (i\theta ),0)$.

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Subsequently, we describe the method of separating the chiral molecules in solvents. As shown in Fig. 3, we irradiate molecules with counter-propagating light waves [11,24,25] with different CPL characteristics. The separation can be realized because the optical force on the chiral material is different under irradiation by L- or R-CPL. In general, larger particles are transported for a larger period if irradiated from only one side; however, in this approach, because of the counter-propagating light, the size dependency is negated, and separation can be realized through only the difference in chirality. Notably, successful separation has been realized in the case of micrometer-sized chiral liquid crystals [11]. However, it remains unclear whether particles that are considerably smaller than the wavelength of light can be separated using this method. In the following analyses, the abovementioned method is considered to quantitatively evaluate the incident light intensity and exposure time.

 

Fig. 3. Schematic of enantiomeric separation methods. We irradiate molecules with counter-propagating light waves [11,24,25] with different CPL. The separation can be realized because the optical force on the chiral material is different under irradiation by L- or R-CPL.

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4. Results and discussion

This section describes the numerical demonstrations of the chiral selective transport of molecules. First, we discuss the criterion for enantio-sorting. We assume that only a few particles are present in the medium, and they move almost independently. In this case, the particle motion obeys the Langevin equation: [40],

In this study, the particles are transported by the optical force. However, particles may also be transported by diffusion. Therefore, we examine the condition in which the transport distance $X_h$ exceeds the diffusion distance $X_d$. This condition can be defined through the following equation [24]:

By substituting Eqs. (37) and (38) into Eq. (36), we obtain

Equation (39) can be used to quantitatively determine the intensity of the incident light and irradiation time required for isolating the target molecules. Similar discussion of the influence of Brownian motion on the sorting time and efficiency has been made and generally the same results have been obtained in Ref. [42]. Furthermore, it should be noted that experimental separations of racemic mixtures using optical force have been reported, where the light exposure time was evaluated from the configuration of each enantiomer [43].

Next, we present the results of the numerical calculation for the single molecule case. According to the X-ray crystal structure, the parameters for the structure of Tröger’s base porphyrin can be extracted from Ref. [36] as follows: The angle between porphyrin $i$ and porphyrin $j$ is $\Phi =81^{\circ }$, the slope of porphyrin $j$ with respect to porphyrin $i$ is $\theta = 30.8^{\circ }$, and the distance between the central metals (zinc) in the porphyrins is $L=8.38$ Å. According to the results of the X-ray analysis of porphyrins with zinc as the central metal, presented in Ref. [44], the distance between zinc and nitrogen (pyrrole ring) is 2.063 Å, distance between nitrogen (pyrrole ring) and carbon is 1.392 Å, and distance between carbon units is 1.467 Å. However, these elements do not lie in a straight line, and thus, the distances cannot be simply added. For simplicity, we estimate the distance between the porphyrin central metal zinc and Tröger’s base as $r_p = 4$ Å through an analogy with the abovementioned values. Although the actual X-ray structure analysis was conducted for the case of porphyrins with palladium as the center metal, the same parameters are used for porphyrins with zinc as the center metal because the conformation of the molecule is considered to be strongly dependent on Tröger’s base. The parameters for the transition dipole of a simple substance porphyrin are determined from the quantum chemical calculations presented in Ref. [45]: The excitation energy including the self-interaction is $\bar {E}_{i,j}=2.89\ \mathrm {eV}$, and the magnitude of the transition dipole moment is $|\boldsymbol {d}_{i,j}|$=8.06 Debye.

Figure 4 shows the optical force applied to the R-enantiomer and S-enantiomer and the difference in the optical forces for the R-enantiomer and S-enantiomer when irradiated with CPL in the positive direction of the $z$-axis. We assume a cryogenic temperature corresponding to liquid helium, which has been used in the optical transport of quantum dots [21]. The relaxation constant is set as $\bar {\Gamma }_{i,j}=0.2\ \mathrm {meV}$. Although the reported relaxation constants in cryogenic conditions are widely scattered because they depend strongly on the environment [46], we choose a value that is much smaller than that at room temperature, but is not smaller than the larger group of values reported in cryogenic conditions. The optical force is plotted in the unit [fN/($\mathrm {MW/cm^{2}}$)] assuming the linear response of molecules. (If laser intensity is strong, the nonlinear response might occur [47]. However, we leave this issue to address in future work and limit ourselves to the linear response. The possible effect of the nonlinear response is mentioned later.) We introduce $\alpha$, $\beta$, and $\gamma$ in the coordinate system of the molecules and consider the molecular rotation to ensure that the effect of the molecular orientation is averaged out. Specifically, the orientation averages are determined by rotating $\alpha$, $\beta$, and $\gamma$ by $10^{\circ }$ each. Depending on the orientation, the optical force difference may be zero or a large value. The optical force and optical force difference post averaging are smaller than those before averaging. However, a significant difference can be noted between the R-enantiomer and S-enantiomer in terms of the optical force. As shown in Fig. 4, two strong peaks of the optical force occur owing to the interaction of the molecules composed of the dimer, and the total optical force for the R-enantiomer and S-enantiomer exhibits a difference of approximately $0.8\ \%$. (This difference is reasonable if considering the ratio of the light wavelength and the size of molecules, and also the observed circular dichroism in absorption in the literature [48].) Moreover, the spectra of the optical force difference for R-CPL and L-CPL exhibit opposite signs. These results indicate that the optical force associated with the R- and S-enantiomer differs according to the combination of their three-dimensional structures and L- and R-CPL.

 

Fig. 4. Dependence of optical force and optical force difference of a porphyrin dimer on the incident light energy when irradiated with L- and R-CPL from the positive side along the $z$-axis. Assuming a cryogenic temperature, the relaxation constants for the two dipoles are set as $\bar {\Gamma } = 0.2\ \mathrm {meV}$. Optical force is plotted in units of [fN/($\mathrm {MW/cm^{2}}$)], assuming that the light intensity is in the linear response regime. Optical force for R-enantiomer for (a) R-CPL, (b) L-CPL, and for S-enantiomer for (c) R-CPL, (d) L-CPL. Difference in the optical forces for the R-enantiomer and S-enantiomer for (e) R-CPL, (f) L-CPL. The values are obtained by subtracting the optical force on S-enantiomer from that on R-enantiomer.

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Although the manipulation efficiency is extremely high in a liquid helium environment [21], special techniques are necessary to introduce materials in liquid helium. Thus, we perform the same examination at room temperature. The corresponding results are presented in Fig. 5. Regarding the relaxation constants, the values at room temperature are several orders of magnitude larger than that in a cryogenic condition and are typically around 10 meV [49]. However, this value possibly changes depending on environment, although the temperature dependence was reported to be weak [50]. Thus, we assume the typical, larger and smaller relaxation constants, namely, $\bar {\Gamma }_{i,j}=10\ \mathrm {meV}, 30\ \mathrm {meV}$ and 2$\ \mathrm {meV}$ to see the dependence on the relaxation constant. In this calculation, the effect of the molecular orientation is averaged out. Compared with those shown in Fig. 4, the shape of the spectra is broadened, and the magnitude of the optical force is reduced. However, a significant difference exists between the R-enantiomer and S-enantiomer in terms of the optical force.

 

Fig. 5. Dependence of optical force and optical force difference for porphyrin dimer on the incident light energy when irradiated with L- and R-CPL in the positive direction of $z$-axis. Assuming room temperature, the relaxation constants for the two dipoles are set as $\bar {\Gamma } = 2, 10$ and $30\ \mathrm {meV}$. Optical force is plotted in unit of [fN/($\mathrm {MW/cm^{2}}$)] assuming the light intensity is in the linear response regime. Optical force for R-enantiomer for (a) R-CPL, (b) L-CPL, and for S-enantiomer for (c) R-CPL, (d) L-CPL. Difference in the optical forces for the R-enantiomer and S-enantiomer for (e) R-CPL, (f) L-CPL. The values are obtained by subtracting the optical force of S-enantiomer from that of R-enantiomer.

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Subsequently, we discuss the light exposure time and incident light intensity when separating chiral molecules according to Eq. (39). If an experiment is conducted in a superfluid helium environment [21], the temperature and viscosity of the superfluid helium are $T = 2\ \mathrm {K}$ and $\eta = 1.486 \times 10^{-6}\ \mathrm {Pa \cdot s}$, respectively [51]. When the incident light intensity and energy are set as $50\ \mathrm {kW/cm^{2}}$ and $2.8130\ \mathrm {eV}$, respectively, the optical force difference is $1.3004 \times 10^{-19}\ \mathrm {N}$. For simplicity, assuming that Tröger’s base porphyrin is a sphere of radius $a = 1\ \mathrm {nm}$, according to Eq. (39), the light exposure time can be determined to be 6 min. If an experiment is conducted under realistic conditions, for example, at room temperature in a water environment, $T = 298\ \mathrm {K}$ and $\eta = 8.903 \times 10^{-4}\ \mathrm {Pa \cdot s}$ [52]. When the incident light intensity and energy are set as $100\ \mathrm {MW/cm^{2}}$ and $2.8130\ \mathrm {eV}$, respectively, the optical force difference is $5.187 \times 10^{-18}\ \mathrm {N}$, and the light exposure time is 6 h.

Then, for providing a visual image of the enantioselective manipulation, we perform simple kinetic simulations of porphyrin dimers in solvents. Specifically, we evaluate the trajectories of each enantiomer. As mentioned previously, the porphyrin dimer is assumed to be a sphere of radius 1 nm, and the incident light intensity and energy are $100\ \mathrm {MW/cm^{2}}$ and $2.8130\ \mathrm {eV}$, respectively, in water at room temperature ($T = 298\ \mathrm {K}$). The relaxation constants are set as $\bar {\Gamma }_{i,j}=10\ \mathrm {meV}$. Fig. 6 shows the trajectories of the R-enantiomer (orange curves) and S-enantiomer (blue curves) irradiated with L- and R-CPL along the x-axis direction for 6 h. For reference, the trajectory of an achiral particle (green curves), for which the optical force difference is zero, is also shown. The achiral particle is assumed to have the same radius of 1 nm as the porphyrin dimer. Figure 6 shows that the R-enantiomer (S-enantiomer) with a positive (negative) optical force difference clearly moves toward the positive (negative) direction of the $x$-axis. The resultant transported distance is consistent with the result obtained by Eq. (37) (0.67 cm). In contrast, the achiral particle with a zero optical force difference drifts near the origin.

 

Fig. 6. (a) Trajectories of the R-enantiomer (orange curves) and S-enantiomer (blue curves) irradiated with L- and R-CPL on opposite directions along the x-axis. (Fig. 3.) For reference, the trajectories of achiral particles (green curves), for which the optical force difference is zero, are also shown. (b) The final destinations of several simulations (50 times) for individual particles are shown.

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These demonstrations exhibit the potential enantioselective optical manipulation. However, a high light intensity and long exposure time are required to effectively sort enantiomers. (In the presence of the optical nonlinearity causing a saturation effect, the longer exposure time might be necessary.) For maintaining the high intensity of the present level over a distance longer than the light wavelength, kinds of micro- or tapered nano-capillaries are necessary and useful as in Ref. [23] and Ref. [53], respectively. As another issue, some methods to avoid possible damage of molecules must be developed though an intensity with similar orders of magnitude is often used for the optical trapping of single molecules (about several tens of $\mathrm {MW/cm^{2}}$). A promising scenario is to consider aggregates or nanocrystals (with racemic mixtures) of chiral molecules. The optical force is expected to increase with the number of induced dipoles, thereby enhancing the efficiency of enantioselective optical manipulation. As a simple example of chiral molecular aggregates, we consider helical nanostructures, as shown in Fig. 2, and evaluate the optical force exerted on these structures to examine the potential of enantioselective sorting.

Figure 7 shows the dependence of the optical force and optical force difference on the incident light energy of the helical nanostructure. The distance between adjacent dipoles along the $z$-axis is $l_{z} = 1\ \mathrm {nm}$, the deviation from the central axis ($z$-axis) is $l_{0} = 4$ Å, and the parameters of each dipole are $|\boldsymbol {d}| = 8.06\ \mathrm {Debye}$ and $\bar {E} = 2.89\ \mathrm {eV}$. Calculations for $n=2,4,8$, and $12$ are performed using Eq. (14). The relaxation constants for all dipoles are set as $\bar {\Gamma } = 10\ \mathrm {meV}$ assuming room temperature, and the incident light intensity is set as $100\ \mathrm {MW/cm^{2}}$. The spectra of optical force exhibit a broadened structure because of damping; however, the optical force and optical force difference increase in proportion to the number of dipoles. These results suggest that by considering the molecular aggregation and nano-crystallization involving a significant number of molecules, effective enantioselectivity based on optical forces can be realized because the optical force difference increases and diffusion distance decreases with the increase in the number of molecules in an aggregate or a crystal.

 

Fig. 7. Dependence of optical force and optical force difference for the helical nanostructure model on the incident photon energy when irradiated with L- and R-CPL in the positive direction of the $z$-axis. Assuming room temperature, the relaxation constants for all dipoles are set as $\bar {\Gamma } = 10\ \mathrm {meV}$ and the incident light intensity is set as $100\ \mathrm {MW/cm^{2}}$. Calculations for $n=2,4,8$, and $12$ are performed using Eq. (14). Optical force on right-handed helix for (a) R-CPL, (b) L-CPL, and optical force on left-handed helix for (c) R-CPL, (d) L-CPL. Difference in the optical forces for the right-handed helix and left-handed helix for (e) R-CPL, (f) L-CPL. The values are obtained by subtracting the optical force on left-handed helix from that on the right-handed helix.

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5. Concluding remarks

Although several researchers have focused on the use of optical force for separating chiral molecules owing to its versatility, the resonant optical force has not been considered. In this study, by considering the microscopic structure of chiral molecules, we formulated the resonant optical force without using the phenomenological chiral susceptibility. Because the resonant effect enhances the optical force [18], the chiral selectivity can likely be considerably enhanced using this effect. By comparing the proposed formula for the induced polarization with that based on the Born–Kuhn model, we demonstrate that the proposed formula is the extended and generalized version of the latter. Specifically, the quantum-mechanical properties of chiral molecules and nanostructured materials having arbitrary sizes, shapes, and internal structures can be considered in evaluating the chirality-dependent optical force.

The proposed formula is used to examine the method of using counter-propagating CPL waves [11,24,25] to separate chiral molecules. In general, large particles are transported for a larger period when irradiated from one side. However, this effect is canceled out by counter-propagating light waves with opposite circular polarizations. Parameters such as the incident light intensity and irradiation time are quantitatively evaluated by considering the condition in which the transport distance by light exceeds the diffusion distance [24]. Furthermore, to discuss the system size dependence of the optical force difference, we consider a model of helical molecules and perform numerical calculations. The results indicate that the optical force difference is almost proportional to the number of dipoles. Moreover, by considering the molecular aggregation and nano-crystallization involving a significant number of molecules, effective enantioselectivity can be realized through the optical forces because the optical force difference increases and diffusion distance decreases with the number of molecules in the aggregates or crystals. In such cases, the proposed nonlocal and self-consistent formulation can effectively address the large coherent length of coupled dipoles. Notably, to perform kinetic simulations and obtain more accurate results, a more sophisticated method is required to treat small molecules in a solvent. Nevertheless, the present study demonstrates the possibility of the derived formulation in examining the chiral selective optical manipulation and facilitating the development of enantio-separation schemes based on the optical force. Future work can be focused on considering more realistic aggregates or nanocrystals based on the microscopic nonlocal optical response and quantum chemical calculations to derive a coupled transition dipole model.