1. Introduction

Chirality, which refers to structures lacking any mirror symmetry planes, is a very intriguing property of enantiomers. For example, many biologically active molecules are chiral, which plays a pivotal role in biochemistry and the evolution of life itself. When a protein loses its original chirality, it can become toxic to cells [1]. Thus, separating and trapping of enantiomers have gained traction in the biochemical and pharmaceutical industries [2,3]. Since enantiomers are identical in their chemical composition and scalar physical properties, prevailing chemical methods remain inefficient in separation and purification of enantiomer mixtures of chiral biomolecules [4].

Optical manipulation is usually used to separate enantiomers [5–7] where the optical force acted on the particle with different chirality is different due to the opposite sign of their chiral polarizabilities [8–13]. In general, the light force is very weak; such a method is only suitable for the separation of special enantiomers. Plasmonic optical tweezers increasing the optical forces by enhancing the near-field gradients have emerged as a powerful strategy to separate enantiomers [14–25]. Selective optical capture of sub-10 nm chiral particles has been achieved by using an achiral plasmonic tweezer [14]. However, because of the small size of the plasmonic tweezer and the lack of strong chiral light-matter interaction, the amount of the separated chiral particles should be limited. In addition, plasmonic optical tweezers can also induce a giant photo-thermal effect, which leads to denaturation of the chiral particles [26,27].

On the other hand, optical chirality [$C ={-} \frac{{{\varepsilon _0}{\mu _0}\omega }}{2}{\mathop{\rm Im}\nolimits} ({\textbf{E}^ \ast } \cdot \textbf{H})$] has been developed to enhance the interaction between chiral lights and matters [28–41]. Recent investigations have shown that superchiral surface waves exhibit many advantages for all-optical enantiomer separation [42]. Based on such an approach, chiral optical forces can be improved by 2 orders of magnitude than those obtained with circularly polarized lights (CPLs) [42]. To further increase the chiral light-matter interaction, our recent investigations have proved that optical fields with a much larger optical chirality can be achieved using photonic crystal (PhC) slabs that utilized the coupling and hybridizing of the transverse-electric (TE)-like and transverse-magnetic (TM)-like modes [43–47]. A large portion of electromagnetic (EM) energy concentrated in the hole can be used to perform surface enhanced CD and Raman optical activity of chiral molecules [43,44,48].

Motivated by the above investigations, in this work we propose a scheme to separate and trap enantiomers with dielectric PhC slabs. Our designed PhC slabs possess the quasi-fourfold degenerate mode. In such a case, a large gradient of optical chirality can be generated near the PhC slabs. Thus, under the excitation of CPLs, the chiral optical force can push one of the enantiomers toward certain regions of the PhC slab, but let the other one move away. Our theory indicates that achiral PhC slabs, providing a possible route toward all-optical enantiopure syntheses, can mediate the chiral enantiomers.

2. Theory and method

We consider a dielectric PhC slab with a square array of cylindrical holes, as shown in Fig. 1(a). The PhC slab is placed on the hollow SiO₂ substrate with a refractive index of 1.46. Figure 1(b) shows a unit cell of the PhC slab. The lattice constant, the radius of the nanohole and the thickness of the PhC slab are marked by a, r and t, respectively. To ensure the symmetry of the system, the PhC slab with a refraction index n = 2.02 (Si₃N₄) is immersed in a medium with the refraction index being n_b= 1.46. To fulfill the chiral separation, the selected medium is in terms of not only the refractive index but also density, viscosity, biocompatibility. For example, the CCl₄ (Carbon tetrachloride) solution is a good choice. Considering the periodicity of the PhC slab in the XY plane, the resonant properties of the PhC slab can be described by a photonic band structure, as shown in Figs. 1(c) and 1(d). Figures 1(c) and 1(d) show the dispersion relation of the eigenfrequency as a function of Bloch wave vector normalized by 2π/a at t = 280 nm and t = 326.2 nm, respectively. Here, other structural parameters are chosen to be a = 336 nm and r = 80 nm. Because the PhC slab is symmetric with respect to the plane z = 0 nm, the Bloch modes can be classified into TE-like and TM-like modes, which are marked in Figs. 1(c) and 1(d) with blue squares and orange triangles, respectively. At the Brillouin zone center (Γ point), the structures support both double degenerate TE-like and TM-like modes. We enlarge the plot near the double degenerate TE-like and TM-like modes, as illustrated in Figs. 1(c) and 1(d). The blue squares and orange triangles in the insets correspond to the double degenerate TE-like and TM-like modes, respectively. We find that the double degenerate TE-like and TM-like modes are separated at t = 280 nm, and the two different kinds of modes become coincident, giving rise to a quasi-fourfold degenerate mode at t = 326.2 nm.

 

Fig. 1. (a) Schematic diagram of the PhC slab placed on the SiO₂ substrate. There is a gap between the PhC slab and the substrate. (b) Geometry and coordinate of a unit cell for the PhC slab. The lattice constant, the radius of the nanohole and the thickness of the PhC slab are marked by a, r and t, respectively. (c) Dispersion relation in ΓΧ and ΓΜ directions as a function of |k|a/2π. Where k is the Bloch wave vector. The structural parameters are chosen as a = 395 nm, r = 60 nm, t = 280 nm. Moreover, c is the vacuum speed of light. (d) Dispersion relation in ΓΧ and ΓΜ directions as a function of Bloch wave vector normalized by 2π/a. The structural parameters are chosen as a = 395 nm, r = 60 nm, t = 326.2 nm. (e)-(f) The distributions of electric ($\widetilde {\textbf E}$) and magnetic ($\widetilde {\textbf H}$) fields for the double degenerate TM-like and TE-like modes in the YZ plane of the unit cell, respectively. The blue arrows denote the real part of the electric and magnetic fields. In addition, the black solid line frames the Si₃N₄ of the unit cell. The parameters of the PhC slab are chosen as a = 395 nm, r = 60 nm, and t = 280 nm. (g)-(i) The averaged enhancements of electric, magnetic and optical chirality in the cylindrical nanohole of the PhC slab with the cases of t = 280 nm and 326.2 nm.

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In order to explain the quasi-fourfold degenerate mode more clearly and show its advantages, in Figs. 1(e) and 1(f), we plot the distributions of electric and magnetic fields for the TM-like and TE-like modes in a unit cell, respectively. The blue arrows in Figs. 1(e) and 1(f) denote the real part of the electric and magnetic fields. We note that the electric (magnetic) fields for the TE-like (TM-like) mode are mainly concentrated at the center region of the nanohole while the electric (magnetic) fields for the TM-like (TE-like) mode circulate around the dielectric region. According to the definition of optical chirality, $C ={-} \frac{{{\varepsilon _0}{\mu _0}\omega }}{2}{\mathop{\rm Im}\nolimits} ({\textbf{E}^ \ast } \cdot \textbf{H})$, it should reach maximum when the electric field and magnetic field are both significantly enhanced at the same region, and have parallel components with the phase difference being π/2. From the distributions of $\widetilde {\textbf E}$ and $\widetilde {\textbf H}$ fields in Figs. 1(e) and 1(f), it is expected that the PhC slab would not give rise to extremely large optical chirality if only the double degenerate TM-like or TE-like mode is excited alone. This is because, firstly, the region where the electric fields have large values does not overlap with that of the magnetic fields, and, more importantly, the electric fields are nearly perpendicular with the magnetic fields. These issues can be solved by the degenerate state of TM-like and TE-like modes (the quasi-fourfold degenerate mode), which can be simultaneously excited, making the optical chirality of near-fields be extremely enhanced.

Figures 1(g)–1(i) show the averaged enhancements of electric field intensity ($\int_V {{{|\textbf{E} |}^2}} /{|{{\textbf{E}_0}} |^2}dV$), magnetic field intensity ($\int_V {{{|\textbf{H} |}^2}} /{|{{\textbf{H}_0}} |^2}dV$) and optical chirality ($\int_V {C/{C_0}} dV$) in the hole of the PhC slab under the CPL excitation along the z-axis. Here, V denotes the volume of the nanoholes of the PhC slab; E and H are the electric and magnetic fields with the existence of PhC slabs. E₀, H₀ and C₀ represent the electric field, magnetic field and optical chirality of the CPL in the background, respectively. The red (circle) and blue (square) lines correspond to the cases with t = 280 nm and 326.2 nm. It is worthy to note that the two peaks of the case with t = 280 nm (red circle lines) are attributed to the excitation of the double degenerate TM-like modes (at λ = 705.2 nm) and TE-like modes (at λ = 710.3 nm). To further understand the PhC slab with different Bloch modes, we show the distributions of electric and magnetic fields in the unit cell of the PhC slab under the excitation of CPL in Supplement 1. Comparing the enhancement factors of optical chirality with t = 280 nm and 326.2 nm in Fig. 1(i), it is clearly shown that the enhancement factor of the system sustaining quasi-fourfold degenerate mode is much larger than that with separated double degenerate TE-like and TM-like modes. All numerical results are calculated by COMSOL. The eigenfrequency simulation is used for the dispersion relation calculating, and the wavelength domain simulation is used for the optical fields and the gradient of optical chirality calculating.

Now, we put a chiral particle on the PhC slab. We focus on the chiral nanoparticles with the radius of 10 nm. Actually, there are numerous synthetic chiral nanoparticles that are around 10 nm, such as carbon nanotubes, chiral nanocrystals, chiral quantum dots, and DNA-assembled nanoparticles [49–52]. This approach assumes that the chiral specimens do not perturb the fields, which is typical for specimens that are much smaller than the aperture and within the Rayleigh regime [14]. Here, the size of the chiral particle considered in our work is much smaller than the incident wavelength (λ = 719.8 nm), where the polarizability of the chiral particle of 10 nm is considered for all numerical results. The chiral particle can be modeled as a pair of interacting electric and magnetic dipoles. When the EM waves act on the chiral particle, the induced dipole moments of such a chiral particle can be expressed as [13,53]:

It is clearly shown that the sign of electric polarizability α_ee or magnetic polarizability α_mm is independent of κ. In general, the optical force acting on a dipolar chiral particle can be written as [55–57]:

Substitute Eq. (1) into Eq. (5), we can obtain the expression of optical force as [13]:

The first term in Eq. (6) corresponds to the gradient force, which plays a major role in the design process. We denote the gradient force as F_g, F_g can be expressed as:

According to the optical chirality ($C ={-} \frac{{{\varepsilon _0}{\mu _0}\omega }}{2}{\mathop{\rm Im}\nolimits} ({\textbf{E}^ \ast } \cdot \textbf{H})$), the gradient of optical chirality $\nabla C$ can be written as:

The subscripts x, y, and z are the coordinate components. It can be seen from Eq. (9), the chiral gradient force is directly related to the gradient of optical chirality $\nabla C$. Because the signs of chiral gradient forces are different for enantiomers, hence, when the chiral gradient force is larger than the dielectric gradient, the separation of enantiomers can be realized.

3. Numerical results and discussion

As shown by Eq. (6), the optical force is not only related to the gradients of electric and magnetic fields, but also depends on the Poynting vector and spin angular momentum of the fields. Firstly, we focus on the contribution of each term of the optical force on the chiral particle. The red solid and black dash lines in Fig. 2(a) show the total optical force F in Eq. (6) and the gradient force F_g in Eq. (7) under the excitation by the left-handed circularly polarized (LCP) light, respectively. The target chiral compounds are modeled as spherical nanoparticles with the refractive index n_r = 1.45. The chiral particle is placed at the position as shown in the inset of Fig. 2(a). The radius of the chiral particle is 10 nm and the particle chirality is taken as κ = +0.5, such a chirality value is achievable with synthetic chiral structures, including the chiral nanoparticles [8,9,14,28,42,52]. It is seen clearly that the magnitude of the gradient force F_g is nearly identical to the magnitude of the total optical force F. Such a phenomenon is similar to many other optical tweezers [15,58–61]. Hence, the expression of optical force near the PhC slab can be simplified as [13,14]:

Based on the discussion above, we know that the PhC slab with quasi-fourfold degenerate mode can produce strong optical chirality under the excitation of CPL. While the PhC slab with a single double degenerate TM-like or TE-like mode can only produce strong electric or magnetic fields. In this case, the large and uneven distribution of the optical chirality near the PhC slab with quasi-fourfold degenerate mode can make the gradient of optical chirality be extremely enhanced. To clearly show the difference between the structures with and without the degenerate modes, the PhC slab with different thicknesses, that is t = 326.2 nm (t = 280 nm) corresponds to the system with (without) the quasi-fourfold degenerate mode, are investigated. In Figs. 2(b) and 2(c), we calculate the magnitudes of F_d and F_k for cases with t = 326.2 nm and 280 nm, respectively. As shown by Eqs. (8) and (9), the dielectric gradient force F_d is related to the gradients of electric field and magnetic field, and the chiral gradient force F_k is related to the gradient of optical chirality. The orange solid and blue dash lines in Figs. 2(b) and 2(c) correspond to the magnitude of F_k and F_d. We find that the magnitude of F_k is around 4 times larger than F_d at the resonance wavelength of the quasi-fourfold degenerate mode (λ = 719.8 nm) due to the large gradient of optical chirality. In contrast, the magnitude of F_d is always larger than F_k at the resonance wavelength of separated TM-like (λ = 705.2 nm) or TE-like (λ = 710.3 nm) mode. Hence, the PhC slab sustaining the quasi-fourfold degenerate modes possesses a much better role to realize chiral separations.

 

Fig. 2. (a) Numerically calculated the total optical force F and the gradient force F_g acting on the chiral particles with the excitation of LCP light. The radius of the chiral particle is 10 nm. Moreover, the chiral particle is placed 20 nm above the PhC slab. (b) The magnitude of dielectric gradient force F_d and chiral gradient force F_k with the thickness of the PhC slab t = 326.2 nm. (c) The magnitude of dielectric gradient force F_d and chiral gradient force F_k with the thickness of the PhC slab t = 280 nm. The radius of the chiral particle is 10 nm with the refractive index n_r = 1.45 and the particle chirality is κ = +0.5. The other structural parameters are chosen as a = 336 nm, r = 80 nm.

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To further illustrate the chiral separation mechanism, Fig. 3(a) presents the intensity distribution and direction (marked by the white arrows) of optical forces in the YZ plane for cases with κ = +0.5 and κ = −0.5, respectively. Here, the thickness of the PhC slab is t = 326.2 nm which has a quasi-fourfold degenerate mode. When the PhC slab is illuminated by the LCP light along the z-axis, the optical force can push the chiral particles with the chirality parameter κ = +0.5 toward the nanohole, but let the chiral particles with κ = −0.5 move away from the nanohole. Thus, the chiral particles with the chirality parameter κ = +0.5 are trapped inside the nanohole of the PhC slab. Moreover, the chirality of the trapped particles is opposite when the incident LCP light is changed to the right-handed circularly polarized (RCP) light. For comparison, in Fig. 3(b), we plot the distributions of optical forces in the unit cell for the case of t = 280 nm. Here, the optical force acted on particles with different chirality have the same directions, and cannot be used for chiral separations. Consequently, the PhC slab with the quasi-fourfold degenerate mode can separate a pair of enantiomers while capturing one of the enantiomers with specific chirality, which cannot be achieved by the PhC slab with separated TM-like and TE-like modes.

 

Fig. 3. (a) and (b) Calculated optical forces for both enantiomers (κ = ±0.5) in the unit cell with the excitation of LCP light. The gray areas represent Si₃N₄ material. Moreover, the black solid line frames the Si₃N₄ of the unit cell. The non-dielectric regions include the nanohole and the range of 360 nm above and below the unit cell. The color map shows the magnitude of the optical force, and the white arrows show the directions of the optical force. The optical force along the z-axis (-z-axis) is marked by the upward (downward) arrows. The excitation wavelengths with the thickness of the PhC slab (a) t = 326.2 nm and (b) t = 280 nm are 719.8 nm and 705.2 nm, respectively. (c) The trapping potential along the z-axis for both enantiomers when the thickness of the PhC slab is 326.2 nm under the excitation of LCP light. Here, the excitation wavelength is 719.8 nm. (d) The trapping potential along the z-axis for both enantiomers when the thickness of the PhC slab is 280 nm under the excitation of LCP light. Here, the excitation wavelength is 705.2 nm. The red solid and blue dashed lines represent the trapping potential of the chiral particle with the chirality being 0.5 and −0.5, respectively. The trapping potential is calculated when the power of incident light is 100 mW in the unit cell.

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Indeed, it is important to ensure that the particles are stably trapped. To trap a particle stably, the trapping potential must be larger than 10 k_BT, which is enough to oppose the Brownian motion of the nanoparticles suspended in solution [14,58]. We have calculated the one-dimensional trapping potential of the chiral particle along the z-axis. Figure 3(c) shows the trapping potential for both enantiomers when the thickness of the PhC slab is 326.2 nm under the excitation of LCP light. Here, the trapping potential is calculated when the power of incident light is 100 mW in the unit cell. The red solid and blue dashed lines represent the trapping potential of the chiral particle with the chirality being 0.5 and −0.5, respectively. As we can see, the PhC slab with quasi-fourfold degenerate mode provides a positive trapping potential for the chiral particle with the chirality being −0.5, while for the chiral particle with the chirality being 0.5, the trapping potential is negative at the same location. This distinct trapping potential stably captures only the chiral particle with the chirality being 0.5 at the range of 185 nm above the PhC slab, keeping the chiral particles with the chirality being −0.5 away from the nanohole. To trap the other handedness, one may simply reverse the incidence polarization to RCP light. According to the previous work [14,25], the chiral particles are generally placed at the position of 20 nm above the structure. Here, the chiral particles placed at the position of 20 nm above the PhC slab can be trapped by a trapping potential of more than 10 k_BT using 5 mW (33 mW/µm²) of the illumination power. Figure 3(d) shows the trapping potential for both enantiomers when the thickness of the PhC slab is 280 nm. Without quasi-fourfold degenerate mode, the PhC slab provides both positive trapping potentials for chiral particles with different chirality. This is the reason that the chiral enantiomers cannot be separated by using the separated TM-like and TE-like modes.

To clarify the origin of the above phenomena, in Figs. 4(a) and 4(b), we plot the distributions of the near-field gradient of optical chirality with the thicknesses of PhC slabs being t = 326.2 nm and 280 nm, respectively. Here, the red arrows denote the directions of $\nabla C$. This shows clearly that the chiral gradients are in opposite directions on two sides of the PhC slab, and both directions are towards the nanohole. We note that the gradient of optical chirality $\nabla C$ for the system sustaining quasi-fourfold degenerate mode is much larger than that with separated double degenerate TE-like and TM-like modes. Based on such a giant gradient of optical chirality, the particles with positive (negative) κ can be trapped under the excitation of LCP (RCP) light.

 

Fig. 4. (a) Near-field distribution of the gradient of optical chirality with the thickness of the PhC slab t = 326.2 nm at the excitation wavelength λ = 719.8 nm. (b) Near-field distribution of the gradient of optical chirality with the thickness of the PhC slab t = 280 nm at the excitation wavelength λ = 705.2 nm. The color map shows the magnitude of the gradient of optical chirality $\nabla C$ and the red arrows denote the directions of $\nabla C$.

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In the above analysis, we only chose a fixed value for the chirality of manipulated particles. In the following, we discuss the influence of the magnitude for the particle chirality on the separation of enantiomers. In Figs. 5(a) and 5(b), we plot the total optical force F (red solid lines), the dielectric gradient component F_d (orange long dash lines) and the chiral gradient component F_k (blue dash lines) of the PhC slab with t = 326.2 nm and 280 nm as functions of chirality κ under the excitation of LCP light, respectively. Here, the sign of the force represents its direction. The positive (negative) sign means that the direction of the optical force is along the z-axis (–z-axis). We find that the magnitude of F_d is larger than F_k when κ ranges from 0 to ±0.01 for the PhC slab possessing quasi-fourfold degenerate modes, as shown in Fig. 5(a). This means that the chiral particles cannot be trapped by such a structure within this range. Additionally, as for the PhC slab with separated double degenerate TE-like and TM-like modes, the magnitude of ${{\textbf F}_\textrm{d}}$ is always larger than ${{\textbf F}_\textrm{k}}$ with κ =[−0.2, 0.2], as shown in Fig. 5(b). In such a case, the PhC slab with separated double degenerate TE-like and TM-like modes cannot trap the chiral particles. It is worthy to note that, although the PhC slab with the quasi-fourfold degenerate mode can trap the chiral particles with the chirality greater than 0.01 (LCP light excitations) of less than −0.01 (RCP light excitations), such value of chirality is still much larger than that of natural chiral molecules whose chirality is extremely small (about 10⁻⁵) [54]. Therefore, in order to achieve the capture of the natural chiral material, we need to design the structure that can further increase the gradient of the optical chirality while reducing the gradient of the energy density for both electric and magnetic fields.

 

Fig. 5. (a) Optical force F and its components F_d and F_k acting on the chiral particle as a function of chirality κ under the excitation of LCP light at λ = 719.8 nm. The thickness of the PhC slab is 326.2 nm. (b) Optical force F and its components F_d and F_k acting on the chiral particle as a function of chirality κ under the excitation of LCP light at λ = 705.2 nm. The thickness of the PhC slab is 280 nm. The red solid, orange long dash and blue dash lines correspond to the forces F, F_d and F_k respectively. Other parameters in Fig. 5 are the same as Fig. 2. Here, the sign of the force represents its direction. The positive (negative) sign means that the direction of the force is along the + z-axis (–z-axis).

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4. Summary

We have demonstrated that a large gradient of optical chirality can be generated due to the large and uneven distribution of the optical chirality on the PhC slab with a quasi-fourfold degenerate mode. Because of the large gradient of optical chirality, the chiral gradient force can be several times larger than the dielectric gradient force and plays a key role in chiral separating and trapping. As indicated above, the optical forces push one kind of enantiomers toward the PhC slab and trap it, but let the other away from the sample. To trap chiral particles with the other handedness, one can simply change the handedness of the incidence CPLs. Although our study may provide a strategy for realizing all-optical enantiopure syntheses, the current method can only separate chiral particles with κ being around 10⁻², and the method for separating natural chiral molecules, whose chirality is extremely small (about 10⁻⁵), should be further investigated. Actually, it is worthy to note that the sensitivity of chiral separation depends on the ratio of the chiral gradient force F_k and the dielectric gradient force F_d. In this case, we proposed two promising pathways for chiral molecules separations. Refractive index matching can improve the enantioselectivity because it reduces the contribution of the dielectric gradient force F_d while maintaining the differences between the chiral gradient force F_k. Hence, we can engineer the structure to make the refractive index of background media match with that of detected chiral molecules. On the other hand, we can design the PhC slab to sustain the much stronger resonance to further enhance the sensitivity of separation for chiral molecules. For example, the bound states in the continuum, which possesses an infinite Q-factor and can generate the extremely enhanced gradient of chiral near-fields, should improve the sensitivity for chiral separations.