Limitations of a superchiral field

Recently, Tang and Cohen [Y. Tang and A. E. Cohen, Science 332, 333 (2011)] proposed and demonstrated the use of “superchiral” electromagnetic fields to enhance optical enantioselectivity. Their work generated much excitement as enantioselective signals are typically quite small, and it appeared that the enhancement factor could be extremely large. In this paper we explicitly show the limitations of such fields by including the magnetic susceptibility term. This term is small and is ignored in most cases compared to the electric polarizability term. However, for the fields used, the enhancement was obtained at the electric field energy node. Due to conservation of field energy, the magnetic field energy is then maximum, and the magnetic susceptibility contribution can no longer be ignored. This then is what limits the enhancement of the optical enantioselectivity. For a counterpropagating left- and right-circularly polarized light field, as used in the aforementioned experiment, we show that this fundamentally limits the enhancement to one or two orders of magnitude in general, determined by the ratio of the magnetic susceptibility to the electric polarizability of the material used. We also generalize the dissymmetry factor to include optical rotation effects present in chiral media, as opposed to fields being in vacuum. In the process, we generalize Lipkin's “$Z^{000}$ zilch” (or “optical chirality”) to that for a linear medium. This generalization shows that chirality of the material cannot be completely separated from chirality of the field and that opposite enantiomers are symmetric in terms of the dissymmetry factor enhancement. Finally, an analogy between ellipsometric chiroptical signal enhancement and enhanced optical enantioselectivity using a standing wave chiral field is discussed. Our analysis and generalization can be used as a guide for future searches of locally enhanced chiral fields.

Figure 1

Dissymmetry factor enhancement $(g/g_{\mathrm{CPL}})$ for the standing wave (SWCF) arrangement, as a function of $(k_{\mathrm{ave}}z/\pi )$. $R=0.72$, $\Delta n/n_{\mathrm{ave}}^0=0.00818$, and ${\gamma }_{\mathrm{ave}}^{\mathrm{CPL}}=0.000781$. This plot shows that the maximum enhancement occurs when $k_{\mathrm{ave}}z=l\pi$, where $l\in \mathbb{Z}$.

Figure 2

Reflectivity $R_{\max }^0={[(1-\sqrt{\gamma })/(1+\sqrt{\gamma })]}^2$ and $R_{\max }(\Delta n/n_{\mathrm{ave}}^0=0.00818)$ for maximum enhancement $g/g_{\mathrm{CPL}}$, for ${\gamma }_{\mathrm{ave}}^{\mathrm{CPL}}\in [{10}^{-6},{10}^{-3}]$. $\left(k_{\mathrm{ave}}z\right)$ set for $U_e$ node. For $R_{\max }$, $g_{\mathrm{CPL}}=1.42\times {10}^{-3}$. Both are indistinguishable here. Solid line, $R_{\max }$; dot-dashed line, $R_{\max }^0\left(\gamma \to {\gamma }_{\mathrm{ave}}^{\mathrm{CPL}}\right)$.

Figure 3

Maximum enhancement $g_0/g_{\mathrm{CPL}}=1/\left(2\sqrt{\gamma }\right)$ and $g/g_{\mathrm{CPL}}(\Delta n/n_{\mathrm{ave}}^0=0.00818)$, for ${\gamma }_{\mathrm{ave}}^{\mathrm{CPL}}\in [{10}^{-6},{10}^{-3}]$. $\left(k_{\mathrm{ave}}z\right)$ set for $U_e$ node. For $g/g_{\mathrm{CPL}}$, $g_{\mathrm{CPL}}=1.42\times {10}^{-3}$. Both are indistinguishable here. Solid line, $g/g_{\mathrm{CPL}}$; dot-dashed line, $g_0/g_{\mathrm{CPL}}\left(\gamma \to {\gamma }_{\mathrm{ave}}^{\mathrm{CPL}}\right)$.

Figure 4

Maximum enhancement $g_0/g_{\mathrm{CPL}}=1/\left(2\sqrt{\gamma }\right)$ and $g/g_{\mathrm{CPL}}(\Delta n\ne 0)$, for various ${\gamma }_{\mathrm{ave}}^{\mathrm{CPL}}$ values. $\left(k_{\mathrm{ave}}z\right)$ set for $U_e$ node. The plots provide the full range for $g/g_{\mathrm{CPL}}$ and show that both agree well. Curved blue plot, $g/g_{\mathrm{CPL}}$; constant red plot, $g_0/g_{\mathrm{CPL}}\left(\gamma \to {\gamma }_{\mathrm{ave}}^{\mathrm{CPL}}\right)$.

Figure 5

$\Delta n/n_{\mathrm{ave}}^0=(n^L-n^R)/(n^L+n^R)$ dependence of $g/g_{\mathrm{CPL}}$ for counterpropagating CPLs forming a standing wave (SWCF), for constant $g_{\mathrm{CPL}}=\pm 1.42\times {10}^{-3}$. ($R=0.72$, ${\gamma }_{\mathrm{ave}}^{\mathrm{CPL}}=7.81\times {10}^{-4}$, $\left(k_{\mathrm{ave}}z\right)$ set for $U_e$ node.) Dashed line, positive $g_{\mathrm{CPL}}=+1.42\times {10}^{-3}$; dot-dashed line, negative $g_{\mathrm{CPL}}=-1.42\times {10}^{-3}$. For opposite enantiomers, both $\Delta n/n_{\mathrm{ave}}^0$ and $g_{\mathrm{CPL}}$ change signs. So only the solid lines A and B are the physically valid curves, and only one is valid for a given frequency. Curve A corresponds to the experiment in Ref. [2].

Figure 6

Comparison of $g\left(R\right)/g_{\mathrm{CPL}}$ for our theory (dashed and dot-dashed lines) and that in Ref. [2] (dotted line) for the counterpropagating CPLs forming a SWCF. $g_{\mathrm{CPL}}=\pm 1.42\times {10}^{-3}$, ${\gamma }_{\mathrm{ave}}^{\mathrm{CPL}}=7.81\times {10}^{-4}$, $\Delta n/n_{\mathrm{ave}}^0=\pm 8.18\times {10}^{-3}$, $\left(k_{\mathrm{ave}}z\right)$ set for $U_e$ node. Positive $g_{\mathrm{CPL}}$ and $\Delta n/n_{\mathrm{ave}}^0$ are used for p-enantiomer and negative values for m-enantiomer; the former is dashed line and the latter is dot-dashed line, but both are identical since both enantiomers should have same enhancement factor. The measured values and error bars for p (lower value) and m (higher value) from Ref. [2] are shown for $R=0.72$.

Figure 7

Log plot of the denominator of $g_0$ at the nodal $U_e$ position for the SWCF. A decrease in ${\gamma }_{\mathrm{ave}}^{\mathrm{CPL}}$ corresponds to a decrease in the fluorescent signal by the same amount, as seen by values where the maximum enhancement occurs ($R\approx 0.85–1.0$). Solid line, ${\gamma }_{\mathrm{ave}}^{\mathrm{CPL}}={10}^{-3}$; dot-dashed line, ${\gamma }_{\mathrm{ave}}^{\mathrm{CPL}}={10}^{-4}$; dashed line, ${\gamma }_{\mathrm{ave}}^{\mathrm{CPL}}={10}^{-5}$; dotted line, ${\gamma }_{\mathrm{ave}}^{\mathrm{CPL}}={10}^{-6}$.

Figure 8

Left-handed $(+)$ SWCF arrangement. Incident electric field is LCPL (with amplitude $E_1$), traveling from right to left, consistent with Ref. [3]. $E_1>E_2$ because reflectivity $R<1$. Reflection occurs at $z=0$.

Figure 9

Right-handed $(-)$ SWCF arrangement. Incident electric field is RCPL (with amplitude $E_1^{\prime }$), traveling from right to left. $E_1^{\prime }>E_2^{\prime }$ because reflectivity $R<1$. Reflection occurs at $z=0$.

Figure 10

CD and Kramers-Kronig transformed ORD spectrum (in milliradians) for the m-enantiomer and p-enantiomer used in Ref. [2]. CD data are as given in the supporting online material for Ref. [2]. ORD spectrum was calculated via Kramers-Kronig transformation of the CD data.