Mirror Optical Activity: Nanophotonic Chiral Sensing from Parity Indefiniteness

Mirror symmetry is among the most fundamental concepts of physics, and its spontaneous breaking at the molecular level allows chiral molecules to exist in two enantiomers that are mirror images of each other. The majority of the chiro-optical effects routinely used to detect enantiomers in mixtures, such as circular dichroism, rely on chiral sensitivity to photon circular polarization, and thus do not harness the full potential of mirror-symmetry breaking, which also involves the spatial profile of the radiation. Here we show that the parity indefiniteness of an electromagnetic field interacting with chiral matter supports mirror optical activity, a chiro-optical effect where a chiral film, when it is probed by the mirror-symmetric field of a nanoemitter, produces a near field whose spatial profile has broken mirror symmetry. The detection of near-field dissymmetry can provide an efficient chiral sensing technique. We specialize the discussion to nanofilms with infrared chirality by using a swift electron in an aloof configuration as the nanoemitter and an off-axis transparent conducting nanoparticle as the near-field probe; the spatial dissymmetry factor of the nanoparticle cathodoluminescence is 1 order of magnitude larger than the dimensionless circular dichroism, and is further enhanced to 2 orders of magnitude if an additional graphene sheet is deposited on the film interface.

Figure 1

Electromagnetic parity indefiniteness in chiral media and MOA. (a) If a field ${\mathbf{E}}_{\omega },{\mathbf{H}}_{\omega }$ exists in a homogeneous chiral medium, its mirror image ${\mathbf{E}}_{\omega }^{\prime },{\mathbf{H}}_{\omega }^{\prime }$ is a physically admissible field in the opposite enantiomeric medium. (b) A mirror-symmetric field (MSF) cannot exist in a chiral medium, since it should exist in the opposite enantiomeric medium as well. (c) MOA produced by a chiral slab sandwiched between two achiral dielectrics, with the potential inclusion of a graphene sheet. The incident ($i$) MSF triggers the excitation of an indefinite-parity field (IPF) in the slab ($s$), in turn producing reflected ($r$) and transmitted ($t$) IPFs. (d) Parameters of the setup used for numerical evaluations. (e) Reflection coefficients of a 50-nm-thick chiral slab at the molecular resonance $\lambda =6.25\mu \mathrm{m}$ as functions of the (normalized) transverse photon momentum, with and without graphene inclusion.

Figure 2

MOA triggered by fast electrons. (a) A fast electron traveling at velocity $v$ through a substrate in an aloof configuration generates a MSF triggering MOA, while an off-axis nanoparticle (NP) at $y=h$ locally probes the near field. Because of the asymmetric near field (yellow), the cathodoluminescence (CL) emission probabilities $\mathrm{\Gamma }(h)$ and $\mathrm{\Gamma }(-h)$ of the nanoparticle when located at two mirror-symmetric positions are unequal, their difference providing MOA detection and chiral sensing. (b,c). Normalized modulus of the total field ${\mathbf{E}}_{\omega }$ and its dissymmetry factor $\mathrm{{\rm Y}}$ in two setups (without nanoparticle): (b) without graphene and (c) with inclusion of graphene. $\mathrm{{\rm Y}}$ provides a local estimation of MOA efficiency.

Figure 3

Graphene-free chiral sensing. (a) Setup and nanoparticle effective polarizability $\alpha$. (b) Cathodoluminescence emission probability $\mathrm{\Gamma }$ and its dissymmetry factor $\delta \mathrm{\Gamma }$, as functions of the wavelength, for a nanoparticle positioned at $h=200\mathrm{nm}$ for various electron energies $K$. The dimensionless circular-dichroism (CD) spectrum $\mathrm{Im}(\kappa )k_{0}L$ of the slab is plotted in the inset for comparison purposes. (c) Dependence of $\mathrm{\Gamma }$ and $\delta \mathrm{\Gamma }$ on the nanoparticle position $h$ for an electron energy $K=395\mathrm{keV}$ at various wavelengths close to the molecular resonance. The detailed dependence of $\delta \mathrm{\Gamma }$ on $h$ and $\lambda$ is plotted in the inset.

Figure 4

Graphene-enhanced chiral sensing. (a) Chiral-sensing setup with inclusion of graphene and electron energy $K=395\mathrm{keV}$. (b) Cathodoluminescence emission probability $\mathrm{\Gamma }$ and its dissymmetry factor $\delta \mathrm{\Gamma }$, as functions of the wavelength, for a nanoparticle positioned at $h=180\mathrm{nm}$. (c) Detailed dependence of $\delta \mathrm{\Gamma }$ on both the nanoparticle position (horizontal axis) and the wavelength (vertical axis). The bottom and side subplots display several slices of $\delta \mathrm{\Gamma }$ for selected wavelengths and nanoparticle positions, respectively. (d) Field dissymmetry factors $\mathrm{{\rm Y}}(h)$ at the three wavelengths selected in the bottom plot of Fig. 4.

Figure 5

Geometry of a chiral slab surrounded by an achiral environment made up of a substrate, a graphene sheet, and a dielectric. Red arrows sketch the field configuration.

Figure 6

Reflection and transmission coefficients of a chiral slab as functions of the normalized parallel wave vector $k_{\parallel }/k_0$ and of the wavelength $\lambda =2\pi c/\omega$, without (a) and with (b) a graphene sheet.

Figure 7

(a) MOA: generation of reflected and transmitted IPFs by a chiral slab in the presence of an incident MSF. (b) The reflected and transmitted fields generated by the opposite enantiomeric slab (with chirality parameter $-\kappa$) are the mirror images of the corresponding fields in (a).

Figure 8

MOA triggered by a fast electron interacting with a chiral slab in an aloof configuration. The electron provides an incident MSF triggering the reflection and transmission of IPFs by the slab.

Figure 9

Symmetric and antisymmetric parts of the $S$ and $P$ amplitudes of the IPF reflected by a chiral slab interacting with an aloof electron. The normalized amplitudes are plotted as functions of the normalized wave number $k_y/k_0$ (for $k_y>0$) and the wavelength $\lambda$, for different parameters $d,\beta$ of the electron interacting with the same substrate, slab, and superstrate (with and without graphene) considered in Fig. 6.

Figure 10

Cathodoluminescence radiation (CLR) produced by a nanoparticle in the substrate.