Chirality notions and electromagnetic scattering: a mini review

Abstract

We review mathematical results on the interaction of time-harmonic electromagnetic waves with ‘geometric’ and ‘electromagnetic’ chiral objects and discuss the relation between these two notions.

Keywords:

• Maxwell's equations
• nonconductive homogeneous chiral media
• geometric (material) chirality
• electromagnetic chirality
• scattering

AMS Subject Classifications:

• 78-02
• 78A45
• 35Q61
• 78A25

Acknowledgments

I thank the referees for their useful comments and bibliographical suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 For more details, see [57], on which the following discussion of the notions of handedness is based.

2 Which refers to the direction and behaviour of the electric field vector.

3 These notions have to be accurately defined: according to the U.S. Federal Standard 1037C, the polarisation is defined as right-handed if the temporal rotation is clockwise when viewed from the transmitter (in the propagation direction) and left-handed if the rotation is counterclockwise.

4 In a much more general setting, ‘mirror symmetry’ is an example of a phenomenon known as duality, which occurs when two supposably different physical systems are non-trivially isomorphic. The nontriviality of this isomorphism requires quantum corrections. In Mathematics, an analogy is the Fourier transform: a local concept as the multiplication of two functions is equivalent to a convolution product, requiring integration over the whole space. Finding such dualities leads to solving complicated physical questions in terms of simple ones in the dual framework. A profound understanding of the archetypal processes of duality symmetries is, in general, not yet feasible, with one exception: mirror symmetry. A mathematical framework to rigourise physical statements is already in an advanced stage of development. An excellent source unfolding aspects of this theory for physicists and mathematicians is [58].

5 ‘I call any geometrical figure, or group of points, chiral, and say that it has chirality, if its image in a plane mirror, ideally realised, cannot be brought to coincide with itself’ ([59], p. 619).

6 With Sir J. W. Cornforth.

7 See Chiral Polyhedra, by E. W. Weisstein, in the framework of ‘The Wolfram Demonstrations Project’ (http://demonstrations.wolfram.com/ChiralPolyhedra/).

8 We use this generic expression in order to avoid stating the exact regularity assumptions for the boundary; they may differ for different results.

9 By Green's formulae, it can be shown that this definition is independent of the choice of the particular extension.

10 We use the widetilde diacritic mark over the entries of the above block matrices only for notational purposes, in particular so as the block matrix $\left(\begin{array}{cc}ϵ\left(x\right)& \xi \left(x\right)\\ \zeta \left(x\right)& \mu \left(x\right)\end{array}\right)$ defined below is ‘free’ of any notational marks.

11 Here, and throughout Section 3 we drop the subscript ‘$\mathrm{D}\mathrm{B}\mathrm{F}$’ from ${ϵ}_{\mathrm{D}\mathrm{B}\mathrm{F}}$, ${\mu }_{\mathrm{D}\mathrm{B}\mathrm{F}}$ and ${\text{β}}_{\mathrm{D}\mathrm{B}\mathrm{F}}$ in the Drude–Born–Fedorov constitutive relations (6(6) $D={ϵ}_{\mathrm{D}\mathrm{B}\mathrm{F}}(E+{\text{β}}_{\mathrm{D}\mathrm{B}\mathrm{F}}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}E),B={\mu }_{\mathrm{D}\mathrm{B}\mathrm{F}}(H+{\text{β}}_{\mathrm{D}\mathrm{B}\mathrm{F}}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}H),$(6) ).

12 Note that k is just an abbreviation for $\omega \sqrt{ϵ\mu }$ and (contrary to the achiral case ($\text{β}=0$)) not a wave number. As we will see below, $k(1-\text{β}k{)}^{-1}$ and $k(1+\text{β}k{)}^{-1}$ are the two wave numbers inherent in this problem.

13 Time-dependent Beltrami fields (where λ is a function varying not only spatially, but temporally as well) are also used in the framework of the time domain analysis of electromagnetic fields.

14 This helicity-based decomposition corresponds to the ‘Riemann-Silberstein vector’, see, e.g.[60].

15 With respect to the inner product of $H(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l},\mathcal{O})$.

16 To indicate the dependence of the far-field patterns on the incident direction $\hat{p}$ and the polarisation vector q, we write $U^{\mathrm{\infty }}(\hat{x};\hat{p},q(\hat{p}\left)\right)$, where $U^{\mathrm{\infty }}$ stands for $E^{\mathrm{\infty }}$ or for $H^{\mathrm{\infty }}$.

17 Where ‘pv’ denotes the Cauchy principal value integral.

18 i.e. one for which the Reciprocity Principle (Theorem 3.6) holds.

19 On the contrary, numerical results in [38] suggest that χ is not differentiable.