Generation of Optical Chirality Patterns with Plane Waves, Evanescent Waves and Surface Plasmon Waves

We systematically investigate the generation of optical chirality patterns by applying the superposition of two waves in three scenarios, namely plane waves in free space, evanescent waves of totally reflected light at dielectric interface and propagating surface plasmon waves on a metallic surface. In each scenario, the general analytical solution of the optical chirality pattern is derived for different polarization states and propagating directions of the two waves. The analytical solutions are verified by numerical simulations. Spatially structured optical chirality patterns can be generated in all scenarios if the incident polarization states and propagation directions are correctly chosen. Optical chirality enhancement can be obtained from the constructive interference of free-space circularly polarized light or enhanced evanescent waves of totally reflected light. Surface plasmon waves do not provide enhanced optical chirality unless the near-field intensity enhancement is sufficiently high. The structured optical chirality patterns may find applications in chirality sorting, chiral imaging and circular dichroism spectroscopy.


TOC Graphic Introduction
A chiral object can exist in two different forms that are non-superposable mirror images of each other.
This asymmetry, called chirality, can be found at scales ranging from a tiny molecule to a giant galaxy.
Similarly, an optical field can also be chiral. For example, circularly polarized light (CPL) and elliptically polarized light (EPL) are chiral and possess different handedness. The chirality of an optical field can be quantified by a conservative quantity called "optical chirality (OC)" 1,2 where and are the permittivity and permeability of free space, respectively, and is the angular frequency of light. and are the real parts of the complex local electric ̃ and magnetic field ̃. This quantity has been linked to the characterization of dissymmetry of chiral molecules. 2 This has led to great research interest in OC engineering for the enhancement of chiral light-matter interaction. For instance, engineered "superchiral" fields [3][4][5] with larger OC than that of CPL can enhance the chiroptical response of chiral molecules. In this context, various OC engineering approaches using plasmonic nanostructures [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] , dielectric nanostructures [24][25][26][27][28] and one-dimensional photonic crystals 29,30 have been proposed to obtain enhanced chiral optical fields.
Apart from enhancing OC, spatially structuring OC has been shown promising for chirality sorting [31][32][33][34][35][36][37][38][39][40][41] , circular dichroism (CD) measurement 42 and super-resolution chiral imaging 43 . For example, an optical field can induce a time-averaged optical force on chiral objects with a reactive and a dissipative component, where the reactive one is proportional to the spatial gradient of the OC patterns. 31 Spatially modulated OC patterns with uniform intensity have also been applied to acquire CD spectrum within a single camera snapshot. 42 Another example is the recent theoretical demonstration of chiral structured illumination microscopy, where spatially structured OC patterns are employed to form moiré patterns with chiral samples and thereby obtain images of chiral domains at sub-wavelength resolution. 43 In addition, inhomogeneous OC can be used to imprint chiral patterns in materials, e.g., liquid crystal polymers, which may find applications in imaging the helicity of light. 44 According to eq (1), a chiral optical field can be obtained only when its electric field has components which are parallel to and not in phase with its magnetic field. In the far-field region, linearly polarized light (LPL) has no OC, whereas CPL has non-zero and spatially uniform OC. Similar to the generation of light intensity patterns by the superposition of coherent waves, it's also possible to generate structured OC patterns using wave superposition. 34,44,45 In this work, we comprehensively investigate the generation of OC patterns by the superposition of (i) two plane waves in free space, (ii) the near field of two total internal reflection (TIR)-based evanescent waves (EWs) and (iii) the near field of two propagating surface plasmon waves (SPWs). Different polarization states and incident directions of the two waves are considered in the three scenarios. Analytical solutions of the produced OC patterns for each scenario have been derived and verified by numerical simulations. The results show that spatially structured OC patterns can be generated in both far field and near field. OC enhancement can be obtained in three conditions, namely the constructive interference of free-space CPL, enhanced near field of TIR-based EWs, and SPWs with sufficiently high near-field intensity enhancement. This work provides a guideline for generating OC patterns by the superposition of two waves.

Superposition of two plane waves
We first investigate the superposition of two plane waves propagating in free space. The wave vectors can be described as where is the wavenumber of light in vacuum, is the incident angle, and is the orientation angle respect to the +x axis, as depicted in Fig. 1. Here, the numbers "1" and "2" denote the first and the second wave, respectively. The superposition of the two plane waves creates an overlapping volume. In this work, we focus on the OC patterns on the xy plane with the largest cross section. In principle, the polarization state of a plane wave can be described by any arbitrary set of two orthogonal polarization components. For simplicity, we have chosen to use the s-polarized (s-pol.) and p-polarized (p-pol.) components relative to the incident plane to define the polarization state of each wave. The s-pol. component has a phase difference with respect to the p-pol. one. When 0 or ± , the plane wave represents LPL or CPL. Other arbitrary corresponds to EPL. The complex electric field of the s-pol. component for each plane wave is where ̂ ̂ ̂ is the spatial coordinate, is the amplitude and the initial phase.
Accordingly, the complex magnetic field of the s-pol. component can be expressed as For the p-pol. component, the complex magnetic and electric field are given by where is the magnetic field amplitude. Upon superposition, the total electric and magnetic field are ̃ ∑ ̃ and ̃ ∑ ̃ ( ) , respectively. Taking and , the OC distribution of the superimposed plane waves calculated by eq (1) is where . From eq (6), one can get the analytical solution of OC for a specific case by setting the corresponding quantities , , , and . In this work, we focus on two representative configurations of the two propagating plane waves, i.e., the cross-propagation ( 0, ) and the counter-propagation configuration ( 0, ). To characterize the OC enhancement, we set the intensity of the superimposed field to be the same as that of free-space CPL.  Table 1. Analytical solutions of the normalized OC generated by the superposition of two free-space plane waves cross-propagating (second column) and counter-propagating (third column) at incident angles of .
Finite-difference time-domain method (FDTD Solutions, Lumerical) has been applied to numerically simulate the analytically predicted OC patterns in Table 1. The OC distribution of a left-handed CPL beam propagating in free space is also simulated in order to provide the value of for normalization.

Superposition of two evanescent waves
Previously, it has been demonstrated that two cross-propagating EWs with transverse electric modes in TIR can generate patterns of local circular polarization in the optical near field. 46 In this section, we derive the general solution of OC for the superposition of two EWs in TIR. As depicted in Here,  (11). Again, for a fair comparison, we set the total intensity of the incident light beams to be the same as that of free-space CPL and the OC values are normalized to . The results of several cases for the cross-propagating ( 0, ) and counter-propagating ( 0, ) EWs are listed in Table 2.  Table 2. Analytical solutions of the normalized OC generated by the superposition of two TIR-based EWs excited by two incident plane waves cross-propagating (second column) and counter-propagating (third column) at incident angles of .
To verify the validity of the analytical solutions, we performed FDTD simulations on several configurations of the incident beams. The theoretically predicted OC patterns are all well reproduced by FDTD simulations (Figs. 4(a)-(j)). Compared to the superposition of two plane waves in free space, the OC patterns generated here are in the near field region bounded to the prism surface. The amplitude of the OC patterns is influenced by the refractive index of the prism ( > ) and the value of ( ), which is about 6 when is very close to the critical angle (see Supporting Information, Fig. S.1).
Therefore, the near-field OC can be enhanced by applying the TIR-based EWs. For example, the superposition of the EWs of two cross-propagating totally reflected beams with orthogonal polarization state, e.g., s-pol. and p-pol., yields a pattern with enhanced OC (Fig. 4(a)) compared to the pattern obtained with two far-field cross-polarized beams ( Fig. 2(a)). However, no OC can be generated by using other configurations of two incident LPL beams (Figs. 4(b), (f) and (g)). Even higher enhancement in the OC with spatially invariant handedness (Fig. 4(c)) can be generated by the interference of the EWs of two totally reflected CPL beams with the same handedness. Interestingly, the superposition of two cross-propagating EWs excited by two incident CPL beams with the opposite handedness can lead to a non-zero structured OC pattern (Fig. 4(d)). This is very different from the superposition of two farfield CPL beams with the opposite handedness, which results in zero OC (Fig. 2(d)). By analyzing the corresponding solution in Table 2, we found that the non-zero OC comes from the term ( ), which is related to the transmission phase shift of the sand p-component induced in TIR. By applying two totally reflected CPL beams with the opposite handedness in counter-propagation configuration, the OC of the superimposed EWs remains zero (Fig. 4(i)). This is similar to the case of two counter-propagating far-field CPL beams with the opposite handedness ( Fig. 2(i)). Finally, the superposition of the EW of a LPL beam and that of a CPL beam in cross-propagation configuration (Fig. 4(e)) leads to a structured OC pattern with relatively lower OC enhancement compared to Fig. 4(c). If enhanced OC with uniform spatial distribution is needed, one may employ two counter-propagating CPL beams with the same handedness ( Fig. 4(h)) or counter-propagating CPL and LPL beam (Fig. 4(j)).
Another important feature of the OC pattern generated by the TIR-based EWs is that, with the same orientation angle difference | |, the spatial frequency is times larger than that in the scenario of far-field plane waves. This can be easily confirmed by identifying the finer OC stripes in Figs of EWs compared to that of the plane waves in free space. For trapping and sorting applications, enhanced OC with higher spatial frequency provides larger spatial gradient of the OC and thus a larger reactive component of the chiral optical force. 31 For high resolution chiral domain imaging 43 , higher spatial frequency of the OC pattern promises higher spatial resolution.

Superposition of two surface plasmon waves
To further increase the spatial frequency of the OC pattern, one may exploit surface plasmon waves. In this section, we derive the general analytical solution of OC generated by the superposition of two SPWs propagating in arbitrary in-plane (xy plane) directions respect to the +x axis ( Fig. 5(a)). Considering the transverse magnetic nature of the SPWs, the magnetic field of each SPW is described by where is the magnetic field amplitude and is the initial phase of each SPW. and are the in-plane and out-of-plane (along z-axis) complex wave vectors satisfying the relationship ( ) . Accordingly, the electric field of each SPW can be expressed as (13) and the OC distribution of the superimposed SPWs is where is the electric field amplitude of each SPW, and the two exponential terms denote the out-of-plane and in-plane damping factors, respectively. Equation (14) indicates that the OC of the superimposed SPWs is proportional to , the real part of the out-of-plane wave vector of the SPWs. It is worth noting that the out-of-plane wave vector of SPWs possesses a small but non-zero real part (Table S.1, Supporting Information), whereas that of the EWs in TIR is purely imaginary. This is an important difference between these two near fields and is the reason why the superposition of the EWs excited by two totally reflected p-pol. light beams is achiral (Figs. 4(b) and (g)). Equation (14) also suggests that a near-field OC pattern can be generated as long as the propagating directions of the two SPWs are not parallel to each other, i.e., To verify the analytical prediction, the near-field OC patterns are numerically simulated using the FDTD method. Mode sources have been used to launch the SPWs. The orientation angles of the SPWs are set to 0 , for cross-propagation and 0 , for counter-propagation configuration (see Methods for details). The superposition of two propagating SPWs results in non-zero OC ( Fig. 5(b)), except for the case of counter-propagation configuration (Fig. 5(c)). The magnitude of the OC pattern in Fig. 5(b) decreases along the propagating directions of the SPWs because of the plasmon damping. To quantify the OC enhancement relative to free-space CPL, a factor which describes the field intensity enhancement between the excitation light and the excited SPWs should be considered. This factor depends on the methods used for photon-to-plasmon coupling, e.g., Kretschmann configuration at the best coupling angle yields 0 . 48 According to the simulated OC values normalized to ( Fig. 5(b)), an OC enhancement larger than 1 requires to be at least 50. From eq (14), the amplitude of the OC on the metal surface ( 0) depends on the real part of the out-of-plane wave vector and the near-field intensity. For SPWs, the quantity is about two orders of magnitude smaller than the wavenumber of the light in vacuum (Table S.1, Supporting Information). Therefore, within the interference area, the enhancement of OC is always smaller than that of the near-field intensity. Nevertheless, the spatial frequency of the structured OC pattern generated here is higher than the corresponding cases in the previous two scenarios owing to the relatively large wave vector of the SPWs.

Conclusion
We have theoretically investigated the generation of OC patterns by the superposition of two waves in three scenarios, namely free-space plane waves, TIR-based EWs and propagating SPWs. In each scenario, the general analytical solution of OC is derived and several practically realizable configurations are discussed. The analytically predicted OC patterns are all well reproduced by FDTD numerical simulations. The results show that spatially structured OC patterns can be generated with farfield waves as well as near-field EWs and SPWs. Enhancement in OC can be obtained from the constructive interference of free-space CPL, the enhanced EWs in TIR, and the SPWs with sufficiently high intensity enhancement. This work may serve as a guideline for generating OC patterns by twowave-superposition for applications in chirality sorting, chiral imaging and snapshot CD measurement.

Supporting Information
Supporting Information Available: <Transmittance of the incident light in total internal reflection, Calculation parameters for the surface plasmon waves.> This material is available free of charge via the Internet at http://pubs.acs.org

S.1 Transmittance of the incident light in total internal reflection
The confined evanescent wave in total internal reflection possesses an enhanced electric field compared to the incident light beam. The enhancement is determined by the transmission coefficient calculated by Fresnel equations. The relationship between the transmittance and the incident angle is plotted in Fig. S.1. The calculation parameters are consistent with those in the numerical simulations.