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Chirality in Nature and Science

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Chiral Nanophotonics

Part of the book series: Springer Series in Optical Sciences ((SSOS,volume 205))

Abstract

Geometrical chirality in molecules or plasmonic nanostructures can lead to unique optical responses such as circular dichroism or optical rotatory dispersion. It is important to distinguish between such chiral responses and the underlying chiral geometry. In this chapter, we first discuss the geometrical properties of chiral objects from a mathematical point of view including planar chirality, the quantification of chirality, and different handedness definitions. After a short introduction to localized plasmons, we thoroughly derive the electromagnetic properties of geometrically chiral objects. Starting from the Born-Kuhn model for chiral media, we derive the chiral constitutive equations and, subsequently, the chiral wave equation. This wave equation provides the basis for a theoretical discussion of the resulting chiral far-field responses. Exemplary, we analyze the circular dichroism response of sugars and simple plasmonic nanostructures. Additionally, a short review of modern techniques for the fabrication of chiral plasmonic nanostructures is given.

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Notes

  1. 1.

    We will adopt this nomenclature and also speak of enantiomers in case of chiral plasmonic structures. This is common usage in recent literature and also fits to the concept of “plasmonic molecules” [2].

  2. 2.

    Note that it is possible to observe chiral responses for structures that are achiral from a geometrical point of view. This can, for example, occur when the combination of structure and incident wave vector form a chiral object. In some literature, such an arrangement is referred to as possessing “extrinsic chirality” [5].

  3. 3.

    One could also argue that, to resort this problem, both helices in the example must have the same handedness. However, it is straightforward to extend the transformation scheme to transform the structure back into the enantiomorph of the initial helix. Those two must have opposite sign in any useful handedness measure.

  4. 4.

    Additionally, tensors can change their sign in a continuous transformation without crossing zero.

  5. 5.

    Note that the shapes of the keys are not the same as the shapes of the keyholes. This is not necessary to obtain a different response depending on the handedness of the key-hole, only the geometrical chirality is important.

  6. 6.

    Key and keyhole in this example are only planar geometrically chiral. However, it is straight-forward to extend this example to three-dimensional geometrically chiral objects. The planar example has been chosen because it is easier to visualize.

  7. 7.

    Strictly speaking, this is not true. Theoretical quantum chemistry predicts a small energy difference between two enantiomers because of the parity violation of the weak interaction [13, 14]. However, this difference is so small that it could not be measured by now [15, 16].

  8. 8.

    This can be intuitively understood as follows: The rotation in a plane is a planar geometrically chiral system . Depending on the convention, we look from opposite sides on this system, which changes its handedness.

  9. 9.

    In principle, this additionally depends on the handedness of the coordinate system . Usually, a right-handed coordinate system is used.

  10. 10.

    It has been shown that the material properties of metal clusters are similar to those of bulk metals down to cluster sizes of a few nanometers [28]. Therefore, the same Drude parameters as for bulk metals can be used.

  11. 11.

    In general, the term “bi-isotropic medium” comprises a wider class of materials with more general constitutive equations (cf. [47]). They are all characterized by two orthogonal eigenpolarizations with opposite handedness.

  12. 12.

    In chemistry, the differential molar extinction coefficient is often used instead of the differential absorbance. This quantity describes the chiroptical properties of a chiral molecule independent of external influences such as path length or concentration (cf. Appendix A).

  13. 13.

    The following parameters have been used: \({\omega _0 = {500}\,\text {THz}}\), \({\omega _\text {p} = {50}\,\mathrm{{THz}}}\), \({\gamma ={5}\,\mathrm{{THz}}}\), \({\omega _\text {c} = {2.5}\,\mathrm{{THz}}}\) and \({d={1}\,\text {nm}}\). The path length \({l={2}\,{\upmu \text {m}}}\) has been chosen such that strong absorption on resonance is obtained. Therefore, the absorption per unit length is much stronger than for most natural chiral materials.

  14. 14.

    Strictly speaking, only l-glucose and d-galactose are diastereomers. d-glucose and d-galactose, who differ in exactly one chiral center , are called epimers .

  15. 15.

    In this figure, \(\Delta \epsilon \) is the molar differential extinction coefficient. Please refer to Appendix A for an explanation of the different units used in CD spectroscopy.

  16. 16.

    For most plasmonic systems, we show the transmittance difference \(\Delta T\) instead of \(\Delta A\). Both responses contain equivalent information as long as no differential reflectance occurs (cf. Appendix A).

  17. 17.

    In the detector’s view convention , LCP rotates to the left while RCP rotates to the right. Therefore, such analysis can be performed rather intuitively in the chosen convention.

  18. 18.

    Note that the fabrication method does not allow for \(C_3\) or \(C_4\) symmetry. However, the structure has been measured from both sides to eliminate the influence of circular conversion dichroism, which exhibits opposite sign for backward illumination.

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    Martin Schäferling

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Schäferling, M. (2017). Chirality in Nature and Science. In: Chiral Nanophotonics. Springer Series in Optical Sciences, vol 205. Springer, Cham. https://doi.org/10.1007/978-3-319-42264-0_2

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