The system under study is schematically presented in Figure 1 – it is formed by an array of chiral scatterers, such as chiral molecules, positioned inside a planar optical cavity. The chiral scatterers are expected to couple with the vacuum field of the Fabry-Pérot cavity mode, leading to the emergence of new hybrid eigenstates, which we refer to as cavity-polaritons, and giving rise to mode splitting in transmission and reflection spectra [26], [27], [28]. Due to the scatterers’ inherent chirality, the system as a whole has to exhibit CD in the transmitted signal. Owing to the formation of polariton eigenstates, one may expect that not only transmission and reflection spectra, but also the CD spectra of the initial chiral system will also exhibit a similar mode splitting.

Figure 1: Sketch of the system under study: chiral scatterers are placed inside a planar cavity.

As a starting point of the analysis, we have modeled the chiral coupled system as a planar multilayer structure and calculated its optical response with the use of the transfer-matrix method (TMM, see Appendix A). The test structure consists of a bi-isotropic chiral layer placed between two gold mirrors that create a Fabry-Pérot cavity, as sketched in Figure 2A. The constitutive relations for a generic non-magnetic bi-isotropic material in SI units read [29] $\left(\begin{array}{c}D\\ B\end{array}\right)=\left(\begin{array}{cc}\epsilon {\epsilon}_{0}& -i\kappa /c\\ i\kappa /c& {\mu}_{0}\end{array}\right)\left(\begin{array}{c}E\\ H\end{array}\right),$ where *ε*_{0} and *μ*_{0} are the vacuum permittivity and permeability, *c* is the speed of light, *ε* is the relative permittivity, and *κ* is the Pasteur parameter that couples the electric and magnetic fields. We modeled the chiral layer with the Lorentzian permittivity $\epsilon =1+{f}_{0}\frac{{\omega}_{0}^{2}}{{\omega}_{0}^{2}-{\omega}^{2}-i\gamma \omega}$ and the corresponding dispersive Pasteur parameter $\kappa ={\kappa}_{0}\frac{{\omega}_{0}^{2}}{{\omega}_{0}^{2}-{\omega}^{2}-i\gamma \omega},$ where *ω*_{0} is the resonance frequency of the chiral transition, *γ* is its linewidth, *f*_{0} is the oscillator strength, and *κ*_{0} is the amplitude of the Pasteur parameter [29].

Figure 2: Transfer-matrix calculations of a chiral multilayer system.

(A) Sketch of the system studied with the TMM method: a bi-isotropic chiral layer characterized by a resonant permittivity *ε* and Pasteur parameter *κ* placed between two gold mirrors. (B) Transmission spectrum of a 20 nm thick bi-isotropic chiral layer in free space, and transmission spectra of an empty cavity as a function of its thickness. (C) CD spectra of the coupled system with the chiral film placed in the middle of the cavity as the function of the cavity thickness. (D) CD spectra of the 20 nm thick chiral film placed in the middle of the cavity of various thicknesses (solid curves) compared to that of the same film in vacuum (dashed grey). Note logarithmic scale in *y*-axis.

Figure 2B shows transmission spectra of a bare cavity formed by two 20 nm thick gold mirrors as a function of the cavity thickness *L*, and that of a 20 nm thick bi-isotropic chiral layer in free space for *ω*_{0}=1.55 eV, *γ*=50 meV, *f*_{0}=0.5, and *κ*_{0}=0.03. Owing to the Lorentzian dielectric response, the bi-isotropic layer is expected to strongly couple with the cavity mode and exhibit mode splitting in the transmitted signal. We then calculated CD spectra of the same chiral layer placed in the middle of the cavity as a function of the cavity thickness, Figure 2C. The calculated map of CD spectra clearly demonstrates anti-crossing between the two cavity-polariton modes. We note that we calculated CD as the difference in transmission between two different CP incident waves, *T*_{+}–*T*_{−} (where + and−refer to clockwise and counter-clockwise rotating electric field polarization). If a system is isotropic (or at least possesses *C*_{3,}_{z} rotational symmetry), it preserves circular polarization and the difference in reflection is zero due to reciprocity [30], thus the use of differential transmission and absorption is equivalent. If, however, it has at most *C*_{2,}_{z} symmetry, interpretation of both differential transmission and absorption becomes more complicated. To be consistent with further experiments, in which only transmission spectra were measured, we therefore used differential transmission *T*_{+}–*T*_{−} to quantify dichroic response.

Remarkably, the closer the polariton dispersion gets to the uncoupled chiral transition of the medium, the higher the CD magnitude is. This is an expected feature of the dispersion, since the composition of those polariton modes (in terms of the Hopfield coefficients) is dominated by the chiral material resonance, while strongly detuned polaritons have more cavity-like character. At the same time, for all thicknesses in Figure 2C the CD is strongly suppressed exactly at the wavelength of the material resonance, which is another manifestation of strong coupling.

When we compared these CD spectra to that of the bi-isotropic layer in vacuum, we found that CD from the layer in vacuum at all wavelengths exceeds that of the layer inside the cavity, Figure 2D. Noteworthy, such suppression of CD occurs despite enhanced electromagnetic field inside the cavity at the resonance, Figure 3A, and strong interaction between the cavity mode and the chiral layer manifested in the anti-crossing. The reason for this limitation becomes apparent when we look at the chirality density $C=-\frac{{\epsilon}_{0}\omega}{2}\text{Im}\mathrm{(}{E}^{\text{*}}\cdot B\mathrm{)}$ induced by an incident CP wave inside the cavity. As shown in Figure 3B, chirality density normalized by that of the incident CP wave at all wavelengths is less than unity. Since the asymmetry of light-matter interaction and CD magnitude is linked to the chirality density [22], CD produced by a chiral layer inside a planar Fabry-Pérot cavity, correspondingly, is also smaller than the free space value. In fact, this limitation of chirality density is fundamental to any isotropic passive planar cavity (i.e. without gain), and we discuss the origin of this limitation in the next paragraph.

Figure 3: Limitation of optical chirality in planar isotropic cavities.

(A) Electric field enhancement inside an empty 300 nm thick cavity induced by a normally incident plane wave at a wavelength of 750 nm corresponding to the first order Fabry-Pérot resonance. (B) Spectrum of normalized chirality density *C*/*C*_{inc} induced by a normally incident CP wave inside a 300 nm thick Fabry-Pérot cavity formed by 20 nm thick Au mirrors. (C) Illustration of chirality density limitation inside isotropic planar cavities: an incident CP plane wave experiences a series of reflections inside the cavity, reversing its handedness each time.

Figure 3C illustrates the nature of this limitation: let us consider a CP wave of a certain handedness and carrying the corresponding chirality density ${C}_{0}=-\frac{{\epsilon}_{0}\omega}{2}\text{Im}\mathrm{(}{E}_{0}^{\text{*}}\cdot {B}_{0}\mathrm{)}$ and intensity ${I}_{0}=\frac{1}{2}\text{Re}\mathrm{(}{E}_{0}\times {H}_{0}\mathrm{)}$ incident onto a planar cavity; let us assume for concreteness *C*_{0}>0. The cavity mirrors may have arbitrary composition; we will be interested in the empty cavity region between the mirrors where a chiral layer can be positioned later. As the wave experiences a series of reflections inside the cavity, the intensity *I*_{i} of each consecutive wave can only decrease. At the same time, the handedness of the travelling wave, which determines the sign of the chirality density, is reversed upon each reflection [29]. Furthermore, since *C* is a bi-linear function of fields, the chirality density of each consecutive wave in the empty region between the mirrors is *C*_{i}=(−1)^{i}^{−1}(*I*_{i}/*I*_{0})*C*_{0}. The total chirality density in this field distribution is $C\text{\hspace{0.05em}}=-\frac{{\epsilon}_{0}\omega}{2}\text{I}\text{\hspace{0.05em}}\text{m}\mathrm{(}{\mathrm{(}{E}_{1}+{E}_{2}+\dots +{E}_{i}+\text{\hspace{0.17em}}\text{\hspace{0.17em}}\dots \mathrm{)}}^{*}\text{}\cdot \mathrm{(}{B}_{1}+{B}_{2}+\text{\hspace{0.17em}}\text{\hspace{0.17em}}\dots \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{B}_{i}+\text{\hspace{0.17em}}\text{\hspace{0.17em}}\dots \mathrm{)}\mathrm{)},$ where **E**_{i} and **B**_{i} are the electric and magnetic field amplitudes of *i*-th CP wave. By direct calculations, one can easily verify that terms $\text{Im}\mathrm{(}{E}_{i}^{*}\cdot {B}_{j}+{E}_{j}^{*}\cdot {B}_{i}\mathrm{)}$ vanish for *i*≠*j*, so the total chirality density reduces to the sum of chirality densities of individual waves:

$C={C}_{1}+{C}_{2}+\dots +{C}_{i}+\dots .$

This is a sign-alternating sum, such that |*C*_{i}_{+1}|<|*C*_{i}|; therefore, it converges, and clearly |*C*|<*C*_{0}. This proves that the chirality density inside any planar passive isotropic cavity is smaller than that of the incident CP wave, which imposes bounds on the CD produced by a chiral layer placed inside a planar Fabry-Pérot cavity. Of course, this bound can be lifted by incorporating a gain medium, which would allow transmitted waves to have larger intensity and chirality densities. However, from the energy point of view it is equivalent to increase the incident CP wave intensity in a passive system.

Preserving the helicity of reflected waves is crucial to overcoming this fundamental limitation with planar structures [31]. A possible way to overcome the free space limit is to employ anisotropic mirrors with at most *C*_{2,}_{z} symmetry. Broken isotropy of the mirrors will allow to convert the polarization rotation direction upon reflection (clockwise to counter-clockwise and vice versa), thus preserving the sign of the chirality density of the travelling wave and enhancing the chirality density in the middle of the cavity [32]. Another recently suggested approach to bypass this limitation is based on oblique-propagating modes, which do not flip chirality upon reflection from specially designed metasurface mirrors [33]. Potentially, both of these approaches could allow obtaining enhanced CD and chiral mode splitting at the same time.

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.