Objects of Maximum Electromagnetic Chirality

We introduce a definition of the electromagnetic chirality of an object and show that it has an upper bound. Reciprocal objects attain the upper bound if and only if they are transparent for all the fields of one polarization handedness (helicity). Additionally, electromagnetic duality symmetry, i.e., helicity preservation upon interaction, turns out to be a necessary condition for reciprocal objects to attain the upper bound. We use these results to provide requirements for the design of such extremal objects. The requirements can be formulated as constraints on the polarizability tensors for dipolar objects or on the material constitutive relations for continuous media. We also outline two applications for objects of maximum electromagnetic chirality: a twofold resonantly enhanced and background-free circular dichroism measurement setup, and angle-independent helicity filtering glasses. Finally, we use the theoretically obtained requirements to guide the design of a specific structure, which we then analyze numerically and discuss its performance with respect to maximal electromagnetic chirality.

Popular Summary

From the weak interactions between elementary particles to the chiral shape of most building blocks of living organisms, chirality is entrenched in nature.

Whether an object is chiral or not can be easily answered by placing it in front of a mirror: If the image in the mirror cannot be superimposed onto the original object, the object is chiral. Chirality is very common and has a very clear definition, yet it still poses important challenges. For example, given two chiral objects, it is impossible to state which object is more chiral; the current definition of chirality does not allow for its quantification. This situation prevents the systematic design of chiral structures. In the context of light-matter interactions, we solve this problem by abandoning the geometrical definition of chirality and instead adopting a definition that is based on the interaction of the object with electromagnetic fields. Here, we rank objects according to their electromagnetic chirality.

We theoretically show that an electromagnetically chiral object is one for which the information obtained from experiments using a fixed incident polarization handedness cannot be obtained using the opposite polarization handedness. In other words, it is possible to quantify electromagnetic chirality on the basis of how an object interacts with fields of different helicities. Contrary to the geometrical definition of chirality, electromagnetic chirality has an upper bound. We show that the objects that achieve the upper bound are invisible to one of the polarization handedness and do not change the handedness of the incident field upon interaction. Using numerical analyses, we focus on a double-turn silver helix and show that maximum electromagnetic chirality is nearly achievable.

We expect that our work will pave the way for promising applications of such extremely chiral objects, in particular, if they can be fabricated to function at optical frequencies.

Figure 1

Interaction in the Hilbert space of transverse Maxwell fields. An incident field $|{\mathrm{\Phi }}_{\mathrm{in}}⟩$ interacts with an object, characterized by its interaction operator $S$, and produces a scattered field $|{\mathrm{\Phi }}_{\mathrm{out}}⟩=S|{\mathrm{\Phi }}_{\mathrm{in}}⟩$. A detector like the one in the bottom right corner of the figure obtains information about the field scattered through the solid angle $\mathrm{\Omega }$: $f(⟨\mathrm{\Omega }|{\mathrm{\Phi }}_{\mathrm{out}}⟩)=f(⟨\mathrm{\Omega }|S|{\mathrm{\Phi }}_{\mathrm{in}}⟩)$, where $⟨\mathrm{\Psi }|\mathrm{\Gamma }⟩$ is the scalar product of the two vectors $|\mathrm{\Psi }⟩$ and $|\mathrm{\Gamma }⟩$. The interaction operator $S$ also describes absorption and near-field illumination and/or measurement.

Figure 2

Interaction of a general object (a) and a reciprocal maximally em-chiral object (b) with fields of pure helicity $\pm 1$. Fields of helicity $+1$ are blue and marked with a “$+$.” Fields of helicity $-1$ are red and marked with a “$-$.” Incoming fields are drawn as bulletlike shapes and scattered fields as clouds surrounding the scatterers. A general object interacts with and mixes both helicities (a). For reciprocal objects (b), maximal electromagnetic chirality occurs if and only if the object is transparent to one helicity. It also implies that the object preserves helicity upon interaction: The scattered field has the same helicity as the incident field. The object hence must have electromagnetic duality symmetry.

Figure 3

Double resonant enhancement for circular dichroism measurements. Panel (a): Two resonant maximally electromagnetically chiral and reciprocal objects are placed close to each other. A chiral molecule is in their vicinity. The external illumination excites only one of the objects, whose resonance illuminates the molecule with a strong field of the same helicity as the incident beam. Panel (b): Upon illumination, the molecule produces a weak scattered field containing both helicities, which excite the two resonant objects. Panel (c): The scattered field of helicity opposite to the initial one is measured. This field exists because of the presence of the molecule. The other half of the circular dichroism measurement is obtained by interchanging the input and measured helicities. The final difference features the two resonant enhancements of opposite helicity: one in illumination and one in the amplification of the field scattered by the molecule.

Figure 4

Two slabs containing lossy maximally electromagnetically chiral and reciprocal objects of opposite handedness. For large enough slab thickness, particle density, or losses, each slab filters out one of the helicities by absorption. The other one passes right through. This behavior is independent of the angle of incidence. The slabs can be used to design glasses for viewing 3D projections where the images destined for each eye are encoded in the two circular polarizations. The glasses would allow us to see the 3D effect even at large angles from the perpendicular of the projector.

Figure 5

Normalized em-chirality $\chi /\sqrt{C_{\mathrm{int}}}$, measure of duality breaking $\cancel{D}$, and contrast of helicity cross sections $\mathrm{\Delta }$ for the two-turn silver helix shown in the inset. Large values of $\chi /\sqrt{C_{\mathrm{int}}}$ coincide with simultaneously large values of $\mathrm{\Delta }$ and small values of $\cancel{D}$. The dimensions of the helix are as follows: major radius $a=6.48\mu \mathrm{m}$, height $b=8.52\mu \mathrm{m}$, and wire radius $c=0.8\mu \mathrm{m}$.

Figure 6

Interaction cross sections and the ratio of absorption to interaction cross sections for each helicity $\pm$. At $200\mu \mathrm{m}$, the maximum of em-chirality in Fig. 5, there is non-negligible absorption of the helicity with the dominant interaction cross section.