This is an electronic reprint of the original article. This reprint may differ from the original in pagination and typographic detail. Caloz, Christophe; Sihvola, Ari Electromagnetic Chirality, Part 2: The Macroscopic Perspective [Electromagnetic Perspectives] Published in: IEEE Antennas and Propagation Magazine DOI: 10.1109/MAP.2020.2969265 Published: 01/04/2020 Document Version Peer reviewed version Please cite the original version: Caloz, C., & Sihvola, A. (2020). Electromagnetic Chirality, Part 2: The Macroscopic Perspective [Electromagnetic Perspectives]. IEEE Antennas and Propagation Magazine, 62(2), 82-98. [9051770]. https://doi.org/10.1109/MAP.2020.2969265 This material is protected by copyright and other intellectual property rights, and duplication or sale of all or part of any of the repository collections is not permitted, except that material may be duplicated by you for your research use or educational purposes in electronic or print form. You must obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or otherwise to anyone who is not an authorised user. Powered by TCPDF (www.tcpdf.org) 1 Electromagnetic Chirality, Part II: Macroscopic Perspective Christophe Caloz, Fellow, IEEE, and Ari Sihvola, Fellow, IEEE Abstract—This paper is the second part of a two-part paper guidelines to design practical chiral metamaterials. Finally, presenting a bottom-up description of electromagnetic chirality, Sec. IX enumerates the main result of this part, and provide which occurs in materials composed of particles with structural an overall conclusion to the two part-paper formed by this handedness. This part first infers, from the microscopic study of the first part [1], the chiral constitutive relations as a subset of paper with [1]. the most general bianisotropic relations. Then, it establishes the fundamental spatial parity conditions in chiral media upon the II. CONSTITUTIVE RELATIONS basis of space-reversal symmetry considerations. Next, it shows that the eigenstates of chiral media are circularly polarized A. General Bianisotropic Relations waves, with polarization rotation depending on the sign of the The behavior of an electromagnetic medium can gener- chiral parameter, and describes the resulting polarization rota- ally be expressed by the relations [2], [3], [4] tion, or (reciprocal) gyrotropy, in terms of circular birefringence. The following part presents introduces an explicit formulation of D ²0E Pe and B ¹0H Pm, (1) chirality based on spatial frequency dispersion or nonlocality, Æ Å Æ Å analyzes the temporal frequency dispersion or nonlocality of where ²0E and ¹0H respectively represent the electric and chiral media, and finally provides guidelines to design practical magnetic responses of free space (spacings between the chiral metamaterials. molecules or metaparticles), while Pe and Pe represent Index Terms—Chirality, optical activity, chiral media, materi- the response of the particle forming the medium, given als and metamaterials, mirror asymmetry, polarization rotation, by [1].(21). Equations (1) correspond to the most usual magnetoelectric coupling, bianisotropy, Tellegen and Pasteur electromagnetics convention of [D;B] being expressed in media, polarizability and susceptibility dyadic tensors, spatial dispersion or nonlocality, parity conditions, temporal dispersion terms of [E;H], where E and H are considered as the or nonlocality, circular birefringence and circular dichroism, excitations while B and D are considered as the medium reciprocal and nonreciprocal gyrotropy. responses [3], [4]2. We now relate the field responses to the medium suscep- tibilities by inserting [1].(21) into (1), which yields I. INTRODUCTION This paper is the second part of a two-part paper on D ² I  E p² ¹  H, (2a) Æ 0 Å ee ¢ Å 0 0 em ¢ chiral electromagnetics and metamaterials. It completes ³ ´ B p² ¹  E ¹ I  H. (2b) the bottom-up description of this topic by by extending Æ 0 0 me ¢ Å 0 Å mm ¢ ³ ´ its microscopic description of the first part [1] to their Defining macroscopic description. We invite the reader to peruse [1] 1 as the key reference to this paper , starting with the general ² » ²0 I Âee p²0¹0 Âem introduction to chirality provided in Sec. [1].I. Å (3) ó ¹! Æ 0p²³ ¹  ´ ¹ I  1 This part of the paper is organized as follows. Section II 0 0 me 0 Å mm @ ³ ´A infers, from the microscopic study of the first part [1], the transforms (2) into the conventional bianisotropic3 rela- chiral constitutive relations as a subset of the most general tions [2], [6], [7] bianisotropic relations. Section III establishes the funda- mental spatial parity conditions in chiral media upon the D ² E » H and B ³ E ¹ H, (4) Æ ¢ Å ¢ Æ ¢ Å ¢ basis of space-reversal symmetry considerations. Section IV shows that the eigenstates of chiral media are circularly where ², ¹, » and ³ are the permittivity, permeability, polarized waves, with polarization rotation depending on magnetic-to-electric coupling and electric-to-magnetic cou- the sign of the chiral parameter. Section V describes the pling dyadic tensors, respectively, which are measured in resulting polarization rotation, or (reciprocal) gyrotropy, in 2The book [2] also includes the alternative formulations [D;H] versus terms of circular birefringence. Finally, Secs. VI and VII [E;B], based on the motivation that E and B are the fields that are respectively discuss the spatial dispersion (or spatial nonlo- directly measurable experimentally via the Lorentz force (F q (E v B); Æ Å £ cality) and the temporal frequency dispersion (or temporal q: charge, v: charge velocity), and [E;H] versus [D;B], based on the motivation that D and B form with the spectral spatial wavevector k nonlocality) aspects of chiral media, while Sec. VIII provides an electromagnetic basis that does not dependent on the nature of the medium. C. Caloz is with Polytechnique Montréal, Montréal, Canada. 3In the term ‘bianisotropic’, coined by Kong in [5], ‘bi’ refers to the fact A. Sihvola is with Aalto University, Espoo, Finland. that each (electric and magnetic) response depends on both the electric Manuscript received Month February, 2019; revised Month xx, 2019. excitation and the magnetic excitation, while ‘anisotropic’ refers to the 1This paper uses the conventions [1].N, [1].n and [1].(n) to respectively fact the responses are not parallel to their excitations or, equivalently, are denote section N, figure n and equation (n) in [1] [e.g. Sec. [1].V-D, characterized by tensorial parameters. From [1].(20), it is clear that this Fig. [1].9 and Eq. [1].(18)]. term applies to the polarizabilities as well as to the susceptibilities. 2 As/Vm, Vs/Am and s/m (see Appendix A). The four medium factor ( i), or yet to the delay T /4 (2¼/!)/4 ¼/(2!). ¡ Æ Æ tensors are generally of dimension 3 3, and involve thus Comparing the excitations associated to the responses D £ E 36 complex parameters overall. and D shows then that the latter are ¼/2 (or T /4) ahead H Å The coupled equations [1].(20) – where D and B are inter- of the former; it must therefore be rotated backward in time dependent through the coupling tensors » and ³ – represent or in the positive (x to y) direction, for synchronization the most general explicit constitutive relations for an LTI with the electric reference, i.e., multiplied by the factor i. Similarly comparing the responses B and B shows (linear time-invariant) medium. The LTI [Assumption 1) Å H E that the latter is ¼/2 (or T /4) behind the former; it must in Sec. [1].III] is indeed a necessary condition for such ¡ relations to hold, since nonlinearity would involve powers therefore be rotated forward in time, in the negative (y to x) of E and H [8], while time variance would require a more direction, for synchronization with the magnetic reference, or multiplied by the factor i. Opposite signs would of complex treatment [9]. ¡ As we have seen in Sec. [1].V-C, the straight Omega par- course be found in the case of a LH helix particle. ticle [Fig. [1].5(a) or [1].8(a)], whether unique or combined with copies of itself in different directions of space, is al- E H ways characterized by non-reducible bianisotropic relations [Eqs. [1].(13), (14) and (15)]. Therefore, it corresponds to the E H t 0 t T /4 most general relations (4), whose tensorial susceptibilities = = + p ( i) are obtained from [1].(13), (14) or (15) via [1].(20). ee E ref. · + E D ⇓ k k DE B. Biisotropic Chiral Media x pem DH In the particular case where the tensors ², ¹, » and ³ in (4) H ( i) ⇒ LH · − reduce to ²I, ¹I, »I and ³I, and hence ultimately to scalars, z E H the medium is called biisotropic. This is for instance the RH y case of the medium formed by the triatomic helix particle H ref. ( i) E shown in Fig. [1].10 and characterized by the polarizability B · − k k tensor [1].(18). Equations (4) reduce then to pme ⇓ B t T /4 t 0 p B D ²E »H and B ³E ¹H, (5) E = − = mm ⇒ H Æ Å Æ Å H ( i) and a biisotropic medium is thus characterized only by the · − Fig. 1. Electric responses (D) and magnetic responses (B) of the RH helix 4 complex parameters ², », ³ and ¹. particle in Fig. [1].5(b) to the electric excitation (E) and magnetic excitation We have established in Sec. [1].V-D that the cross cou- (H). The direct responses DE and BH are synchronized with E and H, pling parameters of a chiral structure were opposite to respectively, whereas the cross responses DH and BE are respectively advanced (t T /4) and delayed (t T /4) with respect to them. each other [Eq. [1].(16)] and in quadrature with the direct ÆÅ Æ¡ coupling parameters (Fig.