Sihvola, Ari Electromagnetic Chirality, Part 2: the Macroscopic Perspective [Electromagnetic Perspectives]

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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

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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

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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 µ0H Pm, (1) chirality based on spatial frequency dispersion or nonlocality, = + = + analyzes the temporal frequency dispersion or nonlocality of where ²0E and µ0H 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 susceptibilities 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 χ E p² µ χ H, (2a) = 0 + ee · + 0 0 em · chiral electromagnetics and metamaterials. It completes ³ ´ B p² µ χ E µ I χ H. (2b) the bottom-up description of this topic by by extending = 0 0 me · + 0 + 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 ξ ²0 I χee p²0µ0 χem introduction to chirality provided in Sec. [1].I. + (3) Ãζ µ! = p²³ µ χ ´ µ I χ  This part of the paper is organized as follows. Section II 0 0 me 0 + mm  ³ ´ 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 fundamental spatial parity conditions in chiral media upon the D ² E ξ H and B ζ E µ H, (4) = · + · = · + · basis of space-reversal symmetry considerations. Section IV shows that the eigenstates of chiral media are circularly where ², µ, ξ and ζ 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); = + × 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 (2π/ω)/4 π/(2ω). − = = 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 π/2 (or T /4) ahead H + 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 ξ and ζ – 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) + H E that the latter is π/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, µ, ξ and ζ in (4) H ( i) ⇒ LH · − reduce to ²I, µI, ξI and ζI, 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 ξH and B ζE µH, (5) E = − = mm ⇒ H = + = + 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, ξ, ζ and µ. 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 = + = − coupling parameters (Fig. [1].9). In terms of the constitutive We have thus, with respect to the references of ² and parameters in (5), this translates into µ, that ξ iχ and ζ iχ. This reformulates the chiral = + = − ζ ξ and ∠ζ,∠ξ ∠² π/2 ∠µ π/2, (6) constitutive relations (5) as = − = § = § where the last equality expresses the fact the electric and D ²E iχH and B iχE µH, (7) magnetic responses of a plane wave in a simple biisotropic = + = − + medium – such as free space – are in phase, as seen by where χ is called the chiral parameter. A chiral medium, 4 inserting [1].(4) into [1].(5) . In order to make sure to select characterized by the relations (7), is also called a Pasteur the correct signs in (6), let us examine the behavior of medium. Inserting (7) into [1].(5) provides then the fol- the RH helix particle described in Fig. [1].9 in some more lowing sourceless explicit form of Maxwell equations for details. a chiral or Pasteur medium: For this purpose, Fig. 1 compares the electric and magnetic responses to the electric and magnetic fields. First, the E iω iχE µH ω χE iµH , (8a) panel at the center of the figure recalls the result established ∇ × = − + = + H iω ²E iχH ω i²E χH . (8b) in conjunction with Fig. [1].9 that the RH helix particle ∇ × = − ¡ + ¢= ¡− + ¢ rotates the field polarization in the transverse plane in the ¡ ¢ ¡ ¢ Note the Greek symbol χ is now associated with both direction corresponding to the left hand with the thumb the susceptibility, from [1].(21), and with the chirality factor, pointing in the propagation direction, or in the y to x from (7). However, the former will always appear with the direction, corresponding to phasor multiplication by the double subscript ee, em, me or mm, whereas the latter is

4The conditions (6) assume losslessness (and passivity), in addition to always on its own. So, this coincidence should not pose any reciprocity. In the case of a bianisotropic medium, with the susceptibility notational ambiguity. ¯T ¯ ¯T ¯ ¯T ¯ tensors in (3), they generalize to χ¯me χ¯em∗ , χ¯ee χ¯ee∗ and χ¯mm χ¯mm∗ . The reciprocity of a chiral medium, which is imposed by These relations also approximately hold= in the= presence of small= loss, but break down at higher loss if the electric and magnetic losses are the assumed absence of an external force or of nonlinearity unbalanced. combined with spatial asymmetry [10], is expressed in the 3 constitutive relations (7) by the fact that ξ iχI ζ which correspond to a chiral of Pasteur medium if τ 0 = = − = satisﬁes the reciprocity condition and χ 0 and to a Tellegen medium if τ 0 and χ 0. 6= 6= = T It is useful to relate the three parameters ², µ and χ in (7) ξ ζ , (9) to the corresponding susceptibilities, which are themselves = − obtainable from the metaparticle polarizabilities via [1].(20). following from the Lorentz nonreciprocity relations [2], [11], These relations may be found by reducing the tensors to which is the macroscopic counterpart of the microscopic scalars in (3) and using (11) with τ 0, i.e., ξ iχ and = = relation of Eq. [1].(12) ζ iχ, which yields As we have seen in Sec. [1].V-D, the metamaterial consti- = − tuted by the triatomic helix metaparticle shown in Fig. [1].10 ² χ ²0 1 χee ip²0µ0 χem is isotropic [Eq. [1].(18)]. Therefore, it indeed corresponds + − , (13) χ µ = ip²0µ0χme µ0 1 χmm to the biisotropic chiral relations (7). Such a medium is µ ¶ µ ¡ ¢ + ¶ ¡ ¢ represented in Fig. 2. implying that χ χ , consistently with the previ- me = − em ously found results for the twisted Omega or helix particle (Sec. [1].V-D).

D. Loss

According to the Poynting theorem, a medium is lossless if the divergence of the Poynting vector in it is purely imaginary [3], [4], i.e., Re{ S} 0, with S E H∗, where the ‘*’ ∇ · = = × superscripts denotes the complex conjugate operation. The divergence of S can be expanded as follows:

1 1 S E H∗ H∗ ( E) E H∗ ∇ · = 2 ∇ · × = 2 · ∇ × − · ∇ × (14) ω ¡ 2¢ £ 2 ¡ ¢¤ i²∗ E iµ H χ χ∗ E H∗ = 2 − | | + | | + − · £ ¡ ¢ ¤ where we have applied the identity (a b) b ( a) ∇ · × = · ∇ × − a ( b) in the second equality, and replaced E and · ∇ × ∇ × H by their Maxwell expressions (8). Nullifying the real ∇ × Fig. 2. Chiral metamaterial made of a 3D array of twisted Omega or helix part of the last expression in (14) yields then the lossless metaparticles [Fig. [1].5(b)] oriented in the x, y and z directions of space conditions (Fig. [1].10).

Re i²∗,iµ 0 or ²,µ R and χ χ∗ 0 or χ R. (15) − = ∈ − = ∈ © ª C. Generalized Biisotropic Media It is easy to verify that in a Tellegen medium τ is also purely The chiral relations (7) do not not represent the only form real in the lossless case and complex in the lossy case. of biisotropy. The magnetoelectric responses can also be in In the presence of loss, the parameters ², µ and χ may phase with their responses. In this case, we can write ξ τ be complex, i.e., = and ζ τ, which particularizes (5) to = ² ²0 i²00, µ µ0 iµ00, χ χ0 iχ00, (16) D ²E τH and B τE µH. (10) = + = + = + = + = + where the primed and unprimed quantities are real num- These equations (10) are the constitutive relations of a bers denoting the real and imaginary parts. Therefore, the Tellegen medium. Such medium is nonreciprocal since ξ RH and LH eigenwavenumbers, which will be derived in T = τI ζ ζ violates the reciprocity condition (9). Therefore, Sec. IV-A as β+ ω(p²µ χ) [Eq. (21a)], generally do not = = § = § it must involve an external force, speciﬁcally permanent or have only different real parts but also different imaginary induced electric and magnetic dipole moments parallel or parts, and then different absorptions. This difference in antiparallel to each other [7]. absorption for the RH and LH waves results in transform- Thus, the coupling parameters ξ and ζ of a biisotropic ing the lossless circularly polarized states into elliptical medium [Eq. (5)] may generally be written as [2], [7] polarized states, as will become clear in Sec. IV-B. This phenomenon is referred to as circular dichroism [12]. ξ τ iχ and ζ τ iχ. (11) = + = − Finally, note that the imaginary parts of (²,χee) and (µ,χmm) are purely positive in a lossy (or dissipative) Inserting (11) into (5) provides the alternative general bi- iωt isotropic constitutive relations medium with the assumed e− harmonic time dependence while the passive lossy condition on (χ,χem,me) is D ²E (τ iχ)H and B (τ iχ)E µH, (12) less restrictive [13] (see Appendix B). = + + = − + 4

III.SPATIAL PARITY CONDITIONS IV. CHIRAL EIGENSTATES The operation of mirror reflection, which substantiates A. Modal Solutions the definition of chirality (Sec. [1].I), is equivalent to the Consider a z-propagating plane-wave excitation. Such + operation of space reversal [3]. Space reversal is defined a wave has the phasor form [1].(4), corresponding in a via the operator S (called the parity operator – P – in Cartesian coordinate system to the fields quantum mechanics) as iβ z iβ z E+(z) (xÊ yÊ )e+ + , H+(z) (xˆH yˆH )e+ + , (20) = x + y = x + y S {r} r0 r or S : r r0 r (17a) = = − 7→ = − where the superscript has been introduced to denote z + + when trivially applied to the space variable, r, and generally, propagation. Substituting (20) into the coupled system of when applied to any other physical quantity Ψ(r), as equations (8) yields the modal wavenumbers and modal fields (see Appendix C) S {Ψ(r)} Ψ0(r0) Ψ( r) Ψ( r), (17b) = = − = § − β+ ω p²µ χ , (21a) § = § where positive and negative signs correspond to even parity ¡ ¢iβ+z E+(z) E0 xˆ iyˆ e+ § , (21b) and odd parity, respectively. This section recalls the fun- § = § damental rules of electromagnetic space-reversal symme- zˆ E+(z) E ¡ ¢ i × 0 iβ+z tries [3], and deduces from them useful parity properties H+(z) § yˆ ixˆ e+ § E+(z), (21c) § = η = η ∓ = ∓η § for the electromagnetic constitutive parameters introduced ¡ ¢ in Sec. II. Note that the space reversal operation (17) where E0 is assumed (choice of phase origin) to be real, and η µ/² is the medium intrinsic impedance. In these corresponds to mirror reflection if it is applied once or an = odd number of times, since S m(r) ( 1)mr only if m is an relations, the signs correspond to RH/LH handedness, = − p § odd integer5. respectively, according to the observations in Sec. [1].V-D. According to (17), r is odd under space reversal, or space- reversal odd or antisymmetric, by definition. It follows that B. Circular Polarization the vectorial spatial derivative operator , involving ∇ = ∇r The physical waves corresponding to the chiral only spatial quantities of single multiplicity, is also space- modes (21) are found from [1].(3) as reversal odd. In contrast, the charge and charge density, q iωt i(ωt β z) and ρ, are space-reversal even, since reversing the coordi- +(z,t) Re E+(z)e− Re E xˆ iyˆ e− − + = = 0 § § nate system leaves charges unaffected. The space-reversal E§ § E0Ren xˆ iyˆ cos(o ωtn β¡+z) i¢sin(ωt β+oz) symmetry properties of and ρ imply then, via Gauss law = § − § − − § ∇ ( D ρ) that D is space-reversal odd, which also implies E0 xˆ cos(©¡ ωt ¢£β+z) yˆ sin(ωt β+z) . (22)¤ª ∇ · = = − § § − § space-reversal odd parity for E, considering for instance Figures 3(a)£ and (b) show the temporal evolution¤ of the the simple case of vacuum. Combining this last result with electric field vector +(z,t) in a given plane transverse to Maxwell-Faraday law ( E iωB) indicates then that B is E+ ∇ × = the propagation direction (here z 0). The tip of +(0,t) = E+ space-reversal even, and so is then also H. We have thus traces a circle following the fingers of the right hand with the thumb pointing in the direction of wave propagation, [D,E]( r) [D,E](r) and [B,H]( r) [B,H](r). (18) − = − − = + as shown in Fig. 3(a); it is thus right-handed (RH) circularly 7 Applying these field space-reversal parities to the consti- polarized . In contrast, the tip of the vector +(z,t) traces E− tutive relations in Sec. II provides then that the parity a circle following the fingers of the left hand with the properties of the different constitutive parameters: ²,² and thumb pointing in the direction of wave propagation, as shown in Fig. 3(b); it is thus left-handed (LH) circularly µ,µ are even, while ξ,ξ, ζ,ζ and τ,χ or odd. Specifically, polarized. Thus, the modes of a chiral medium are RH and enforcing parity compatibility in (7) reveals that ² and µ LH circularly polarized waves. are even functions of space while χ is an odd function of It should be emphasized that the notion of circular space (mirror antisymmetry), or6 polarization refers here to the temporal variation of the ²( r) ²(r), µ( r) µ(r) and χ( r) χ(r). (19) field, and not to its spatial evolution, which is opposite. − = + − = + − = − This easily seen by setting t 0 (instead of z 0) in (22). = = These relations may serve as a precious sanity check in The handedness of the spatial spiral formed by the tip the modeling of chiral materials. Incidentally, note that the 7 Tellegen parameter [Eq. (10)] is odd, as χ, i.e., τ( r) τ(r). This corresponds to the polarization handedness convention gener- − = − ally used in the electrical engineering community (e.g. [2], [14], [15], [16]), iωt for which all the results following the time-harmonic convention e− 5We have S 1(r) S (r) r by definition [Eq. (17)] (mirror symmetry), (used in this paper) apply upon substituting i j for transformation to 2 = = − 3 − → S (r) S [S (r)] S ( r) r (return to original spatial frame), S (r) the time-harmonic e jωt convention (e.g., Eq. (22): the resulting opposite = = − = = + S [S 2(r)] S (r) r (mirror symmetry), etc., which is rather trivial. The signs in both the complex vector and the complex phase eventually cancel = = − situation is more subtle when rotations in different directions are involved, out to yield the same real result.). The physics and optics community gen- as in Sec. [1].V-A. erally follows a convention leading to the opposite handedness, referring 6Note the last of these relations seem to imply that χ 0, which would to clock direction rather than handedness, and considering that when the absurdly imply that chiral effects cannot occur! However, this= is only a local field rotation is counterclockwise (resp. clockwise) with the observer facing condition, whereas a chiral medium is fundamentally nonlocal, as will be the oncoming wave, the wave is left-handed (resp. right-handed) (e.g. [3], discussed in Sec. VI-A. [17]). This is all quite confusing, but important to note! 5

y y with right-handedness in positive propagation [Fig. 3(a)], (0,t) (0,t) but left-handedness in backward propagation [Fig. 3(d)]. E E β+ ω pǫµ χ β+ ω pǫµ χ + = + ω − = − ω E + E0(xˆ iyˆ) E + E0(xˆ iyˆ) + = ¡ + ¢ − = ¡ − ¢ x x D. Eigenstate and Modal Isotropy Perspective Substituting the chiral modes (21) and (23) into the RH LH constitutive relations (7) respectively yields for the forward propagation propagation and backward propagation directions z (a) z (b) χ propagation propagation y y D+ ²E+ iχH+ ² E+ ²+E+, (24a) § = § + § = § η § = § § (0,t) (0,t) µ ¶ E E B+ iχE+ µH+ µ χη H+ µ+H+, (24b) ω ω § = − § + § = § § = § § ¡ ¢ x x and z z β− ω pǫµ χ β− ω pǫµ χ χ − = − − = − + E E (xˆ iyˆ) +E E (xˆ iyˆ) D− ²E− iχH− ² E− ²−E−, (25a) RH − ¡ 0 ¢ LH − ¡ 0 ¢ § = § + § = ∓ η § = § § (c) − = − (d)+ = + µ ¶ B− iχE− µH− µ χη H− µ−H−. (25b) § = − § + § = ∓ § = § § Fig. 3. Circular polarization (CP) of the chiral modes in the plane z 0 ¡ ¢ = Equations (24) and (25) reveal that D§ E§ with amplitude [Eqs. (21) and (23)]. (a) FWD-prop. ( ) and (xˆ iyˆ) phasor ( ), leading §k § to RH-CP. (b) FWD-prop. ( ) and (xˆ +iyˆ) phasor+ ( ), leading+ to LH-CP. ratio ²§ and B§ H§ with amplitude ratio µ§ , i.e., that + − − § §k § § (c) BWD-prop. ( ) and (xˆ iyˆ)( ) phasor, leading to RH-CP. (d) BWD- the CP waves (E§,H§) and parameters (²§,µ§) are the prop. ( ) and (xˆ− iyˆ)( ) phasor,− − leading to LH-CP. § § § § − + + eigenvectors and eigenvalues of the chiral medium. One may alternatively say that, although generally biisotropic [Eq. (7)], a chiral medium is seen as monoisotropic (χ§ of the field in space is always opposite to the temporal § = ζ§ 0) to CP waves, and hence preserves the polarization handedness, as properly illustrated in [16]. = and§ handedness of these waves, altering only their phase The fact that the medium presents different wavenum- and magnitude. bers (or refractive indices) to RH-CP waves (β+) and LH-CP + waves (β+) is referred to as circular birefringence, and is the fundamental− cause of polarization rotation, as will be seen V. POLARIZATION ROTATION OR GYROTROPY9 in Sec. V. How does such a chiral medium affect linearly polarized (LP) waves? The most natural way to address this question C. Opposite Propagation Direction is to decompose a test LP wave into the medium (RH- Reversing the direction of wave propagation from z to CP and LH-CP) eigenstates, since these states form a sim- + z implies reversing the sign of the argument of the spatial ple, isotropic basis with well-defined eigenwavenumbers. − Consider for instance the xˆ-polarized z-propagating plane exponentials – and corresponding superscripts – in (20), + without altering the complex vector xˆ iyˆ . This operation wave in the plane z 0, which reads, according to [1].(4), § = transforms (21) to8 E+ (z 0) xÊ0. The chiral medium resolves such a wave ¡ ¢ LP = = into its RH-CP and a LH-CP components as E+ (z 0) LP = = β− ω p²µ χ ω p²µ χ , (23a) E0 xˆ iyˆ E0 xˆ iyˆ . The propagation along the medium § = − ∓ = − § 2 + + 2 − ¡ ¢ ¡ iβ−z ¢ is then found by assigning to the RH-CP and LH-CP waves E−(z) E0 xˆ iyˆ e+ § , (23b) ¡ ¢ ¡ ¢ § = § composing the LP wave their respective eigenwavenumbers, zˆ E¡ −(z) ¢ i × § β+ and β+, given by (21a), which results in H−(z) E−(z), (23c) + − § = − η = §η § E E 0 iβ+z 0 iβ+z which results in reversing the circular-polarization handed- E+ (z) xˆ iyˆ e + xˆ iyˆ e − . (26) LP = 2 + + 2 − · ¸ ness, as shown in Figs. 3(c) and 3(d), since the thumb is ¡ ¢ ¡ ¢ iωt now pointing in the opposite direction whereas the signs where multiplication by e− shows that the two CP-wave of the complex vector have not changed. vectors composing ELP+ (z) rotate at different spatial rates, One must be very careful to avoid confusion in deter- since β+ β+, resulting in a net rotation of the (linear) 6= mining the (temporal) wave handedness. In (21) and (23), polarization,+ − or gyrotropy, as illustrated in Fig. 4. This is the superscripts unambiguously indicate the direction of the circular birefringence explanation of Fresnel [18], where § wave propagation, while the subscripts unambiguously different medium properties are seen by the two CP waves. § indicate the sign of the complex vector xˆ iyˆ. But neither § 9 the superscripts nor the subscripts indicate handedness. For The terminology “gyrotropy” has been traditionally restricted to magnetized plasmas and ferrites, which are respectively characterized by instance, as shown in Fig. 3, the vector xˆ iyˆ is associated + permittivity and permeability tensors that involve antisymmetric terms [4]. We take here the liberty to prefer ethymological logics (‘gyro-’ (γυ˜%oς) 8Here, we chose to reverse the sign of β while¡ leaving¢ the sign of z meaning ‘circle’ or ‘spinning’ in Geek) to tradition, and hence apply the unchanged. Doing the opposite would transform (23a) to the equivalent term “gyrotropy” also to chiral media, which rotate the polarization just result β− ω p²µ χ in the reversed coordinate system, where the as magnetized plasmas and ferrites, with the only difference that they do reversal§ of= the sign∓ of χ compared to (21a) is a manifestation of this it in a reciprocal rather than a nonreciprocal fashion, as explained in this ¡ ¢ space-reversal odd symmetry of chirality [Eq. (21a)]. section. 6

θ+ LP+ C P P E ELP+ 2 2 θ θ + C C + + E− θ+ θ+ E− E+ z + θ+ z ψ(t) ψ( t) + + − + E+ ELP ELP + − x x θ+ y y − 0 0 time-reversal + + + + symmetry E+ E− χ 0 E+ E− χ 0 = 6= maintained x x P1 (a) P1

(a) (b) P2 P2 θ θ Fig. 4. Chiral rotation of an LP wave (figure inspired by [19]). F F (a) Monoisotropic medium (χ 0), where the RH/LH-CP phasors rotate ψ(t) ψ( t) = − by the same amounts θ+ ωp²µz/2, resulting in no LP rotation. (b) Chiral medium (χ 0),§ where= § the RH/LH-CP phasors rotate by the 6= 0 0 different amounts θ ω(p²µ χ)z/2, resulting in a chiral rotation angle 2θ + = § B0 F B0 of θ θ θ §ωχz. Note that this graph also holds for Faraday time-reversal C+ = + − + = rotation [19]+ upon− replacing the electric phasor and quantities by their symmetry magnetic counterparts, and χ by the magnetization saturation Ms [14], broken from P1 (b) P1 [20]. non-reversed B0 Fig. 5. Comparison of chirality gyrotropy and Faraday gyrotropy using direct-time (left) and time-reversed (right) experiments. (a) Chiral medium This may be mathematically shown by grouping the (reciprocal), where θC(z) ωχz. (b) Faraday medium (nonreciprocal), terms of same polarization in (26), and next factoring out where µ and κ form the= permeability tensor as θ (z) ω(p²(ν κ) F = + − p²(ν κ))z/2 with µ [ν, iκ,0;iκ,ν,0;0,0,µ ] [14], [20]. the resulting exponential sums and difference to form sines − = − 0 and cosines, i.e., eiβ+z eiβ+z eiβ+z eiβ+z + − + − The reciprocity of chiral rotation is readily apparent ELP+ (z) E0 xˆ + iyˆ − = " 2 + 2 # (27a) in (28). Since reversing the direction of wave propagation iβe+z is equivalent to reversing the sign of the argument of the e E0 xˆ cos βo+z yˆ sin βo+z , = − spatial phase, i.e., of either β or z, consider reversing the β+ β©+ ¡ ¢ ¡β+ ¢ªβ+ sign of βo while maintaining the coordinate system fixed where βe+ + + − ωp²µ, βo+ + − − ωχ. (27b) = 2 = = 2 = in (28). This results in The last expression of (27a) reveals that the chiral medium rotates the polarization of the LP wave in space as it θC(z; βo+) ( βo+)z βo+z θC(z;βo+), (29) propagates a distance z by the angle − = − − = = − which shows that that if the LP wave has accumulated a 1 Ey 1 sin(βo+z) θC(z;β+) tan− tan− − rotation angle of θ as propagating in the z direction, this o = E = cos(β z) µ x ¶ · o+ ¸ (28a) + 1 angle is undone as the wave returns, propagating in the z tan− tan(β+z) β+z ωχz − = − o = − o = − direction, so that the initial polarization state is retrieved, as and with the phase shift£ ¤ shown in Fig. 5(a). Such a medium is therefore time-reversal symmetric, and hence reciprocal, as it should be due to the φ (z;β+) β+z ωp²µz. (28b) C e = e = absence of external force and asymmetric nonlinearity [11]. So a chiral medium preserves the LP nature of an LP wave, At the microscopic scale of the helix chiral particle but rotates its polarization as it propagates, by an amount [Fig. [1].5(b)], reciprocity may be understood by reversing that is proportional to the difference of the RH-CP and LH- the direction of propagation from z to z in Fig. [1].9, + − CP wavenumbers, or refractive indices n§ β§/k cβ§/ω, which corresponds to reversing the direction of either = 0 = and with a phase shift that is proportional§ to§ the sum§ of E or H so as to preserve the handedness of the triad the same RH-CP and LH-CP wavenumbers. (E,H,k). First, consider the top of Fig. [1].9, with E still This spatial rotation effect for an LP wave seems essen- pointing towards x, and hence H pointing now towards + tially identical to Faraday rotation [21] in magnetized plas- y; since the induced dipolar moments are unchanged, H − mas [22] and magnetized ferrites [20] or transistor-loaded rotates from the y direction to the x direction. Similarly, − + magnetless metaferrites [23], despite the fact that none of consider now the bottom of Fig. [1].9, with H still pointing these media involve chiral particles (proof of in Fig. [1].3) towards x, and hence E pointing now towards y; given + + and are purely monoanisotropic (non magnetelectric cou- the again unchanged dipolar moments, E rotates from the pling), with anisotropy ² for plasmas and anisotropy µ y direction to the x direction. The electromagnetic field + − for ferrites or metaferrites (proof of in Fig. [1].3). has therefore rotated about the z axis in the direction → However, there is a fundamental difference between chiral corresponding to the right-hand with the thumb pointing rotation and Faraday rotation [21]: the former is reciprocal, in the propagation direction ( z), which is the opposite − as shown from the medium reciprocity condition in Sec. II, rotation direction than that found for z propagation. Note + whereas the latter breaks that reciprocity condition, and is that with the chosen convention for rotation handedness, hence nonreciprocal [11]. The two types of gyrotropies are depicted in Fig. 3, a RH helices (as in Fig. [1].9) induce LH illustrated in Fig. 5. CP (and RH spatial spirals), and LH helices induce RH CP 7

(and LH spatial spirals), which corresponds to (29) and the namely at the straight sections and at the looped section, rotation rewinding in Fig. 5. respectively; they are thus nonlocal. The medium is thus In contrast to chiral rotation, in Faraday rotation, the globally nonlocal. The nonlocality of the twisted Omega or angle acquired by the wave propagating along the z direc- helix particle may be understood from the same current + tion, θ, keeps accumulating in the same absolute direction, continuity thought experiment, with local polarizabilities xx xx xx xx dictated by the fixed bias field B0, when propagating back αee and αmm, and nonlocal polarizabilities αme and αem. in the z direction. This leads to a total rotation angle of 2θ − for the round trip, as shown in Fig. 5(b). So, the initial state C. Plane-Wave Description of the system is not retrieved upon time reversal, and the It is instructive to consider the particular case of plane- system is nonreciprocal [11], which allows the realization wave propagation in the chiral medium. In this case, the of devices such as isolators, circulators and nonreciprocal source may be considered to be at infinity and the current phase shifters. at any point of the medium may hence be set to zero, which leads to insightful relations between the response VI. SPATIAL-FREQUENCY PROPERTIES and excitation fields. A. Spatial Dispersion (or Spatial Nonlocality) We first solve the sourceless relations (8) for E and for H in terms of E and H, which yields At frequencies where the size of the particles and the ∇ × ∇ × distance between them are not both much smaller than 1 1 E iµ H χ E and H i² E χ H , the wavelength (say not smaller then λ/4), the polarization = ωγ ∇ × − ∇ × = −ωγ ∇ × + ∇ × response of the medium at a given point depends on the ¡ ¢ ¡ (31)¢ with γ ²µ χ2. Next, we substitute these relations into field scattered at surrounding points, i.e., = − ˆ the expressions of (7) that are associated with chirality, i.e., E P e (r) χ¯ab(r,r0) (r0)dr0, (30) having the coefficient χ. Specifically, inserting the second m = · H Ω ½ ¾ equation of (31) into the first equation of (7) and the first © ª where ab ee, em, me or mm, and Ω is the volume of the equation of (31) into the second equation of (7) provides = medium over which this neighboring effect extends10. This the alternative constitutive relations effect is called spatial dispersion (or spatial nonlocality) [3], χ D ²E ² E iχ H , (32a) [24], [25], [26], and the expansion of (30) in Fourier integral = + ωγ ∇ × − ∇ × χ ¡ ¢ with respect to space (r), for easier treatment in terms of B µH µ H iχ E , (32b) a set of plane waves, leads to k-dependent susceptibilities, = + ωγ ∇ × + ∇ × namely, χ¯ χ¯˜ (k) [24], [25], [26], [27]. Spatial dispersion ¡ ¢ ab → ab which seem to be presented here for the first time. is very common in metamaterials, where the condition a ≤ Given the assumed linearity, one may then Fourier- p λ [Eq. [1].(6)] is only mildly satisfied at the edge of the ¿ transform (32), which leads for each plane wave of the ik r homogeneization range [25], [27]. Chirality can be shown corresponding angular spectrum expansion [28], Ψ˜ (k)e · to be an effect of first-order spatial dispersion, where the (Ψ D,B,E,H), the relations = field response is associated with the first spatial derivative χ D˜ ²E˜ i²k E˜ χk H˜ , (33a) of the field excitation (see Appendix D). = + ωγ × + × χ ¡ ¢ B˜ µH˜ iµk H˜ χk E˜ . (33b) B. Microscopic Origin of Spatial Dispersion = + ωγ × − × As mentioned in Sec. VI-A, spatial dispersion macroscop- Projecting these equations¡ onto the x, y and¢ z directions ically involves a dependence of the medium properties on finally yields the wave direction (k). However, this effect has naturally ˜ ˜ a microscopic origin. In the case of chiral metamaterials, D˜ ²(k) ξ(k) E˜ , (34a) this origin can be easily understood from the microscopic B˜ = ˜ ˜  · H˜ µ ¶ ζ(k) µ(k) µ ¶ analysis of the Omega and helix particles in [1], where   the local and nonlocal responses have been indicated by ˜ ˜ ²(k) ξ(k) ²I a k I a k I the respective labels ‘L’ and ‘NL’ in Figs. [1].8 and 9. First, where + ² × χ × , (34b) ˜ ˜  = consider the planar Omega particle. The direct responses ζ(k) µ(k) Ã aχk I µI aµk I! y − × + × px , corresponding to αxx , and p− , corresponding to   e,Ex ee m,Hy x y which involves the skewon (or antisymmetric) dyadic k I α are produced exactly at the points of excitation, × mm that characterizes gyrotropic media [29]. The relations rep- namely at the straight sections and at the looped section, resent the plane-wave description of the chiral medium respectively; they are thus local. In contrast, the cross re- y x yx In the general bianisotropic case, the relations (32) sponses p and p− , respectively corresponding to αme m,Ex e,Hy and (33) respectively generalize to (see Appendix E) x y and αem, are produced away from the points of excitation, i 1 1 1 D ² E ξ µ ζ ²− ξ − E ζ ²− H , (35a) 10If the medium is operated in its homogeneous regime, then its = · − ω · − · · · ∇ × + · · ∇ × ³ 1 ´ ³ ´ response depends only on the distance between the source (r0) and i 1 − 1 observation (r) points, and not on their absolute positions. Therefore, B ζ ² ξ µ− ζ H ξ µ− E µ H, ¯ ¯ =ω · − · · · ∇ × + · · ∇ × + · we can substitute χ¯ab (r,r0) χ¯ab (r r0) in (30), which becomes then a ³ ´ ³ ´ convolution integral. → − (35b) 8

(Sec. [1].V-D) and since its eigenstates are consequently CP 1 modes (Sec. IV), the medium resolves LP incident waves D˜ ² T k I E˜ T ζ ²− k I H˜ , (36a) = + D · × · + D · · · × · into their two CP states (RH and LH) (Sec. V), given by (21) 1 ³ ´ 1 ³ 1 − ´ and (23). This means that the k in (36) corresponds to the with T D ξ µ ζ ²− ξ (36b) = ω · − · · eigenstates k§ in reference to (21) and (23). Let us consider, 1 ³ ´ as in Fig. [1].9,§ a forward LH-CP wave propagating along the B˜ T ξ µ− k I E˜ µ T k I H˜ (36c) = B · · · × · + + B · × · z direction, which corresponds to 1 + ³ ´1 ³ 1 −´ with T B ζ ² ξ µ− ζ , (36d) = −ω · − · · k+ β+zˆ ω p²µ χ zˆ, (40a) ³ ´ − = − = − which leads to more complex tensorial relations [33]. ¡ ¢ E+(z) 1 iβ+z − E0 xˆ iyˆ e − . (40b) H+(z) = i/η − ½ ¾ ½ ¾ D. Application to the Omega Particles − ¡ ¢ For this wave, we have to set k k+ β+zˆ (kx ky 0, This section applies the plane-wave formulation of chi- = − = − = = k k ), E˜ E˜ + E xˆ iyˆ and H˜ H˜ + i(E /η) xˆ iyˆ rality presented in Sec. VI-C to better relate the micro- z = 0 = = 0 − = = 0 − in (36). The result− is − scopic behavior of Omega-type particles (Sec. [1].V) to their ¡ ¢ ¡ ¢ xx macroscopic response (Secs. II). xx χem xx xx D˜ x ²0 ² i ² χ E0, (41a) Let us start with the planar Omega particle. Apply- = r + ϕ r + me · ¸ ing (36) to the tensorial structure [1].(13) yields, after xx ¡ ¢ xx χme xx xx E0 B˜x µ0 µ i χ µ , (41b) translating [1].(13) into its macroscopic form with (3) and = r − ϕ em + r η ˜ ˜ ˜ ˜ · ¸ 0 setting Ey Ez Hx Hz kx ky 0 (kz k0) for the ¡ µ0 ¢ ======D˜ i² E , B˜ i E , D˜ B˜ 0, (41c) (E˜x , H˜ y ,kz ) polarization in Fig. [1].8, y 0 0 y 0 z z = − = − η0 = = x y x y 2 χ p²0µ0 χem xx xx 2 xx xx xx D˜ ² ²xx 1 em E˜ H˜ , (37a) with ϕ 1 χ χ χ χ χ . (41d) x 0 r x y = + ee + em + mm + ee mm = + ϕ − ϕ¡ ¢ µ ¶ 2 x y 2 x y where χxx χxx χxx¡ following¢ from [1].(16). These p²0µ0 χem y y χem em = − em me B˜y E˜x µ0µ 1 H˜ y , (37b) relations are harder to decipher than their planar-Omega = − ϕ + r − ϕ ¡ ¢ ¡ ¢ µ ¶ counterparts, but certainly include complementary in- D˜ D˜ B˜ B˜ 0, (37c) y = z = x = z = formation on the bianisotropic properties of the chiral x y 2 y y y y medium. If the monoatomic helical particle corresponding with ϕ 1 χxx χ χ χxx χ (37d) = + ee + em + mm + ee mm this medium would be transformed into the triatomic parti- x y 2 ¡ x y¢ yx where the term χem χemχme from [1].(12), and cle in Fig. [1].10 to form the biisotropic chiral metamaterial = − shown in Fig. 2, the complicated bianisotropic relations (41) xx ¡ xx ¢ y y y y µ0 ² 1 χ , µ 1 χ , η0 . (38) would naturally reduce to more fundamental and practical r = + ee r = + mm = ² r 0 relations (34). The relations (37) are in agreement with the observations made in Sec. [1].V-C, and provide in addition the exact VII. TEMPORAL-FREQUENCY PROPERTIES quantitative response of the related medium. Note that, applying (36) to the perpendicular polarization (E˜ , H˜ ,k ) A. Temporal Dispersion (or Temporal Nonlocality) y − x z where the wave does not interact with the particle, properly In general, due to the inertia of matter to elastic and retrieves the free-space solution D˜ ² E˜ and B˜ µ H˜ y = 0 y x = − 0 x damping forces, the response of a medium at a given time (with D˜ D˜ B˜ B˜ 0). In the case of the oblique x = z = y = z = depends not only on the excitation at that time but also at incident polarization [(E˜ ,E˜ ),(H˜ , H˜ ),k ], applying (36) x y y − x z previous times, i.e., naturally adds, by superposition, D˜ ² E˜ and B˜ µ H˜ ˆ y = 0 y x = − 0 x to the response in (37), leading to the polarization angle E P e (t) χ¯ab(t,t 0) (t 0)dt 0, (42) m = · H ˜ ˜ T ½ ¾ 1 D y 1 ²0Ey © ª θ tan− tan− , (39a) D xx x y where ab ee, em, me or mm, and T is the duration of this = Ã D˜ x ! = Ã ² E˜x ξ H˜ y ! = + memory effect. This effect is called temporal dispersion (or x y x y 2 temporal dispersion) – which is naturally the temporal coun- k χ kz χ xx xx z em x y em 11 with ² ² ²0 , ξ , (39b) terpart of spatial dispersion [Eq. (30)] –, and the expansion = r + η ωϕ = − ωϕ µ 0 ¶ ¡ ¢ of (42) in Fourier integral with respect to time (t), for easier consistently with the waveplate effects already noted in the treatment in terms of a set of harmonic waves, leads to ω- microscopic analysis of Sec. [1].V-C. dependent susceptibilities, namely, χ¯ χ¯˜ (ω) [2], [3], ab → ab Let us now analyze the medium composed of twisted [4]. Omega or helix particle. Analogously to the case of the straight Omega particle, we shall apply (36) to the tensorial 11Here, if the medium is operated in a time-invariant regime, then structure [1].(17), after translating [1].(17) into its macro- its response depends only on the difference between the excitation (t0) and response (t) times, and not on their absolute positions. Therefore, scopic form with (3) and for the appropriate polarization. we can substitute χ¯ab (t,t0) χ¯ab (t t0) in (42), which becomes then a However, since this structure induces polarization rotation convolution integral. → − 9

B. Lorentz Response C. Metamaterial Resonance and Plasma Frequencies The medium susceptibilities must exhibit temporal fre- A metamaterial maximally interacts with electromagnetic quency (ω) dependences that obey causality or, mathe- energy near its lowest resonance (Sec. [1].IV-B), which matically, that follows the Kramers-Kronig relations [3]. For occurs at the frequency where the incoming wave best composites made of relatively low-loss particles, such as the matches the boundary conditions of the metaparticles straight and looped sections of the Omega-type particles forming the metamaterial, while fulfilling the subwave- considered in this paper (Fig. [1].5), dispersion follows the length dimensional constraint [1].(6). In the case of single- Lorentz model [2], [3], [4] (also see Appendix F) block particles, such as the Omega-type particles, this corresponds to the situation where the size of the unfolded ωp,ab ωx ω if ab ee, mm, structure, `, is about half the wavelength at resonance χ (ω) ,ω p,ab = ab 2 2 x (` λres/2) if it is highly conductive, or somewhat less = −ω ω iωνab = ( iω if ab em, me, ≈ − 0,ab + − = (λ /10 ` λ /2) if it is dielectric or plasmonic, due (43) res < < res to field penetration, and not to the separate sizes of the where ω0,ab, ωp,ab and νab denote the resonance frequencies, plasma frequencies and damping factors, respec- straight-pair section and looped section. There is thus a tively12. unique and common resonance frequency for the three parameters in (44), i.e., According to (13), the dispersive susceptibility relations (43) correspond to the relative chiral parameters are ω ω ω ω . (46) 0 = 0,² = 0,µ = 0,χ Contrarily to the resonance frequency, the plasma fre- 2 2 2 quencies may differ from each other. Indeed, the plasma ²(ω) ω ωp,² ω0,² iων² ² (ω) 1 χ (ω) − − + , (44a) r ee 2 2 frequencies in (44) are proportional to the density of the = ²0 = + = ω ω iων − 0,² + ² corresponding dipole moments13, which are generally different for the electric (p) and magnetic (m) responses. For 2 2 2 µ(ω) ω ωp,µ ω0,µ iωνµ instance, in the case of the Omega particles, we have seen µ (ω) 1 χ (ω) − − + , (44b) r = = + mm = 2 2 in Sec. [1].IV that most of the p response is related to the µ0 ω ω0,µ iωνµ − + straight section while most of the m response is related to the looped section, which leads to particularizing [1].(9) as χ(ω) ωp,χ ω ˆ ˆ χr(ω) iχem(ω) , (44c) = ² µ = − = 2 2 µ0 p 0 0 ω ω0,χ iωνχ p r0ρ(r0)dr0, p r0 J(r0)dr0, (47) − + e = m = × Vstraight 2 Vlooped where we have replaced wherever appropriate the sub- where the relevant integration volumes have been ex- scripts ee, mm and em by the subscripts ², µ and χ, plicited. Omega particles with a much larger (resp. respectively, for notational convenience. In these relations, smaller) straight section and a much smaller (resp. larger) ω , ω and ω are usually interdependent, as will be 0,² 0,µ 0,χ looped section, following the antagonistic dimensional con- shown in Sec. VII-C. straint [1].(10), will then be characterized by larger densities Using low-loss materials, we typically have ων ω2, ²,µ,χ ¿ of electric (resp. magnetic) dipole moments, and hence by and the damping terms ων²,µ,χ can then be neglected larger electric (resp. magnetic) plasma frequency, while the to the first order. Moreover, the resonance frequency is cross-coupling frequency, involving the same geometrical often considerably lower than the plasma and operating parts, will be intermediate, i.e., frequencies, i.e., ω ω ,ω. Under such conditions, 0 ¿ p,²,µ,χ ≷ ≷ Eqs. (44) can be approximated by the Drude lossless forms 2d Às ωp,² ωp,χ ωp,µ. (48) ¿ ⇐⇒ ω2 D. Parametric Study 2 p,² ω 2 Figure 6 plots the dispersive responses (44) for a chiral ² (ω) χ (ω) − (ω ) r ee p,µ medium with equal permittivity, permeability and chiral 1 2 , (45a) (µr(ω)) = + (χmm(ω)) ≈ ω factor plasma frequencies, ω ω ω ω , occurring p,² = p,µ = p,χ = p when the straight and looped sections of the helix particle ωp,χ have comparable dimensions. As predicted by (45), the real χr(ω) . (45b) ≈ ω parts of the permittivity and permeability are positive (positive refractive index) and negative (negative refractive index) These relations provide precious insight into the dispersion above and below that frequency, respectively [10], while χ response of a low-loss chiral medium. They most impor- is positive in that frequency range. Moreover, the imaginary tantly reveal that in such media ²(ω) and µ(ω) tend to be parts of ², µ and χ are always positive, consistently with the respectively positive and negative above and below their implicit assumption of passivity (Sec. II-D and Appendix B). plasma frequency while χ(ω) follow the opposite polarities. 13This is well-known to be the case in regular plasmas, where 12 The odd factor ω in χem,me ensures that these susceptibilities have the plasma frequency of the conduction electrons is given by ωp 2 = the opposite parity of χee,mm, as shown in Appendix F to be required, Ne /(m∗²0), where N, e and m∗ are the density, charge and effective while the factor i ensures that χ and χem have the same polarity, as mass of these electron [3], but this is also true in metamaterials, as − p required from considering that χ ip²0µ0χem according to (13) while shown in [30] for split-ring resonators upon safely defining the effective χ R in the lossless case according= to− (15). permeability as the ratio of the B and D fields. ∈ 10

ǫ,µ,χ ǫ,µ,χ ǫ,µ,χ ǫ,µ,χ 1 3 3 ǫ ,µ ǫ′′,µ′′ ǫ′′,µ′′ 10 ′r′ r′′ χr′ 2 2 χ′ χ′ ǫ stopband µ stopband ǫ′ ,µ′ r r 0.5 ω ω 5 ǫr′′,µr′′ 1 p,∗ χ µ′ 1 p∗,χ ǫ′ ωp,∗ µ ωp∗,ǫ χr′ ωp,∗ ǫ ωp∗,µ 0 ω/ω0 ω/ω0 ω∗ ω∗ ω∗ 0,χ 0ǫ,µ p,ǫµ ωp∗,χ 1 ω 2 3 4 5 1 ω 342 5 0 0 0∗ 0∗ 21 34 21 34 ω/ω0 ω/ω0 -1 -1 µ χ µ µ χ′′ χ′′ -5 χ′r′ ǫ′ ,µ′ -2 -2 -0.5 r r ǫ′ χ µ′ χ χr′′ ǫ ǫ ǫ -3 -3 -10

-1 Fig. 7. Dispersion responses (44) for ωp,χ 3ω0 and ν² νµ νχ 0.1 (ω /ω ω /ω ω /ω 1.00). (a) Helix= with straight= section= much= 0,∗ ² 0 ≈ 0,∗ µ 0 ≈ 0,∗ χ 0 ≈ Fig. 6. Dispersion responses (44) for the parameters ωp,² ωp,µ ωp,χ larger than looped section (2d s), and ωp,² 4.5ω0 and ωp,µ 2ω0 ω 3ω , corresponding to helix particle with similar straight-section= = and= À = = p 0 (ωp,∗ ²/ω0 4.609, ωp,∗ µ/ω0 2.333). (b) Helix with looped section much looped-section= dimensions, ν ν 0.8ω and ν 0.3ω . (a) Full scale. = = ² µ 0 χ 0 larger than straight section (s 2d), and ωp,² 2ω0 and ωp,µ 4.5ω0 (b) Zoom about the abscissa.= The= exact poles are=ω /ω 1.000 and À = = 0∗χ 0 ≈ (ωp,∗ ²/ω0 2.333, ωp,∗ µ/ω0 4.609). ω /ω ω /ω 1.038. = = 0∗² 0 = 0∗µ 0 ≈

then the medium should be operated at the common Interestingly, at the common – or balanced – plasma plasma frequency, ωp; so, set the plasma frequency frequency, we have ² µ 0 – and hence zero refractive equal to the operation frequency, ω , i.e., ω ω . = = op p = op index [10] – and χ 0. Equations (7) reduce then to Otherwise, set ω r ω with r 0 for negative- 6= p = op > index response or r 0 for positive-index response; D(ωp) iχ(ωp)H(ωp) and B(ωp) iχ(ωp)E(ωp), (49) < = = − r might initially be in the order of 1.1 or 0.9 for | | which correspond to a purely chiral medium, also referred operation at 10% off the plasma frequency. to as chiral nihility medium [31]. In this case, an LP wave 3) Select a particle with a 90◦ twist with between its is still rotated by the angle θC(z) ωχz [Eq. (28a), where straight and looped parts, e.g., a helix. = ∓ θC(z) does not depend on (²,µ)], but it does not undergo 4) Since the operation frequency must be larger than any phase shift, since φC(z) ωp²µz 0 [Eq. (77b)]. the resonance frequency, ω ω , and since the = § = op > 0 At the resonance frequency, the real parts of all the unfolded length of the particle is about half the wave- parameters change sign, which specifically means for χ(ω) length at resonance (` λ /2), that length much ≈ res a change of the polarization-rotation direction, according be somewhat greater at the operation frequency, i.e., to (28a), but this region of the spectrum is highly lossy and ` c/(2f ) πc/ω . Set an initial guess for it, > op = op should therefore be avoided unless really needed. possibly using some analytical formulas [32]. Figure 7 compares chiral media with different plasma 5) Simulate the scattering parameters of the particle frequencies, with Fig. 7(a) and Fig. 7(b) respectively cor- as indicated in Fig. 8, and extract the (frequency- responding to a helix particle with a straight section that is dependent) parameters χ, ² and µ from the formulas much longer and much shorter than the looped section, (see Appendix G) where the plasma frequencies follow the two sequences 1 yx xx tan− S / S in (48). Here, the difference in the plasma frequencies leads χ 21 21 , (50a) to the opening of forbidden bands, or stopbands, corre- = − ¡¯ ωp¯ ¯ ¯¢ ¯ ¯ ¯ ¯ sponding imaginary phase shifts in (77b). These stopbands yx 1 ∠S 1 Sxx correspond to the frequency range extending between the 21 − 11 ² xx , (50b) double-negative (or negative refractive index) and double- = Zport ωp 1 S Ã !µ + 11 ¶ positive (or positive refractive index) bands. The same ∠ yx xx comment as above applies for the common resonance S21 1 S11 µ Zport + , (50c) frequency. = ωp 1 Sxx Ã !µ − 11 ¶ where p is the period of the metastructure. Note that VIII.DESIGN GUIDELINES this technique automatically accounts for interpar- A chiral metamaterial may be designed as follows: ticle coupling in case the dilute-medium condition 1) Choose the balanced design (Fig. 6) rather than an is not satisfied, thanks to the utilization of periodic unbalanced design (Fig. 7) to avoid permittivity or boundary conditions (infinite periodicity) in the x permeability stopbands. We have then ω ω and y directions and to the utilization of a sufficient p,² = p,µ = ω ω , corresponding to a particle with balanced number of periods in the z direction for simulation p,χ = p straight-section and looped-section dimensions. convergence to the periodic regime. 2) If one targets a purely chiral response (polarization 6) Fine-tune the structure parameters using a full-wave rotation without phase shift), corresponding to (49), optimization tool until satisfaction. 11

5) In contrast to Faraday rotation, which is nonreciprocal x (and monoanisotropic), chiral or Pasteur polarization Ports 2 periodic rotation (reciprocally) unwinds to its original state boundary y upon time reversal. conditions 6) Different right-handed and left-handed eigenstate absorptions in the presence of loss distort circular polarization into elliptic polarization, a phenomenon called circular dichroism. x N p 7) A chiral medium is spatially dispersive (or spatially Ports 1 z nonlocal), i.e., its response at a given point of space p depends also on the excitation in the vicinity of that y point. Spatial dispersion leads to an alternative of p the standard bianisotropic or biisotropic constitutive relations where the component of the response fields Fig. 8. Full-wave simulation set up of the chiral particle for synthesis of the can be explicitly written in terms of the components corresponding chiral metamaterial by iterative analysis. The computational box is a z-oriented cylinder with rectangular cross section of dimension p of the excitation fields. This formulation of chirality is p and length N p (i.e., N periods, N [5,10] – here N 7 – for convergence× explicit but tensorial, whereas the standard formula- ∈ = to a periodic response upon washing out the port-edge aperiodic effects), tion is scalar but implicit. whose two sides perpendicular to zˆ are defined as input and output ports while the four parallel sides parallel to zˆ are defined as periodic boundary 8) A chiral medium is also temporally dispersive (or x y x y temporally nonlocal), due to causality, typically with conditions. The ports 1 are P1 and P1 while the ports 2 are P2 and P2 yx x y (e.g. S21 represents the transmission from the port P1 to the port P2 ). Lorentz dispersion in its three constitutive parameters (permittivity, permeability and chirality factor). In the lossless case, the three parameters are purely real, and IX.CONCLUSIONS all are complex in the general, lossy case. The three parameters share the same resonance frequency, cor- In the second part of this two-part paper, we have responding to the unfolded length of the particle, but presented a macroscopic description of electromagnetic have different and antagonistic electric and magnetic chirality and materials. The main conclusions and results plasma frequencies, depending on the densities of of this part may be summarized as follows: the straight and folded parts of the chiral particle. In 1) A biisotropic medium is not necessarily chiral. It is the balanced regime, where the electric and magnetic chiral if its electric and magnetic responses are re- plasma frequencies are merged, the medium is purely spectively in quadrature with their magnetic and elec- chiral at the common plasma frequency, with dou- tric counterpart excitations. In this case, the medium ble negative (or negative index) and double positive is indeed chiral, and is also called a Pasteur medium, (or positive index) below and above that frequency, and such a medium is reciprocal. If these responses respectively. In the unbalanced regime, a stopband and excitations are in phase, the biisotropic medium exists between the two plasma frequencies. is called a Tellegen medium; such a medium is non- 9) A chiral metamaterial can be designed following a reciprocal, and therefore requires magnetization. given procedure that leverages the concepts devel- 2) Whereas the permittivity and permeability functions oped in the paper and uses full-wave simulation are even under space reversal, the chirality parameter extraction in a setup involving periodic boundary function is odd under space reversal. conditions and orthogonal ports. 3) The fundamental modes, or eigenstates, of a chiral Although discovered more then 200 years ago, chirality medium are circularly polarized left-handed or right- has been relatively little applied in electromagnetics (mi- handed waves, whose handedness changes with the crowave, terahertz and optical) to date, probably because direction of propagation. The chiral eigenstates ‘see’ of implementation difficulties and theoretical complexity. the chiral medium as monoisotropic. We expect that current developments of nanotechnologies 4) The polarization rotation in a chiral medium is a and modern metamaterial/metasurface concepts will spur direct consequence of its circularly polarized eigen- a diversity of developments in this area, and we hope that states: the chiral medium resolves an incident linearly this paper will contribute to such developments. polarized wave in its right-handed and left-handed components and assigns them the corresponding ACKNOWLEDGMENTS different wavenumbers (or refractive indices), which As in [1], the three-dimensional drawings of the particles results in a net polarization rotation, with rotation in the figures have been realized by Amar Al-Bassam. direction depending on the handedness of the chiral Moreover, the authors wish to thank Mário G. Silveirinha particles forming the medium. This is called circular for illuminating discussions related to spatial dispersion. birefringence, not to be confused with phase birefringence (e.g. case of the planar Omega particle or REFERENCES standard birefringent crystals) leading to waveplate- [1] C. Caloz and A. Sihvola, “Electromagnetic chirality, Part I: Microscopic type polarization transformation. perspective,” IEEE Antennas Propag. Mag., 2019. to be published. 12

[2] J. A. Kong, Electromagnetic Wave Theory. EMW Publishing, 2008. APPENDIX A [3] J. D. Jackson, Classical Electrodynamics. Wiley, third ed., 1998. UNITSOFTHE CONSTITUTIVE PARAMETERS [4] A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scat- tering: From Fundamentals to Applications. Wiley - IEEE Press, The units of the medium parameters in the bianisotropic second ed., 2017. constitutive relations (4) may be found by first isolating the [5] D. K. Cheng and J.-A. Kong, “Covariant descriptions of bianisotropic media,” Proc. IEEE, vol. 56, pp. 248–251, Mar. 1968. parameter of interest in the same relations, next replacing D [6] J. A. Kong, “Theorems of bianisotropic media,” Proc. IEEE, vol. 60, and B by their respective expressions in Maxwell equations pp. 1036–1046, Sep. 1972. [Eqs. [1].(5)], and finally remembering that the units of , ∇ [7] I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, E, H and ω are rad/m14, V/m, A/m and rad/s, respectively. Electromagnetic Waves in Chiral and Bi-Isotropic Media. Artech, 1994. [8] R. W. Boyd and D. Prato, Nonlinear Optics. Academic Press, third ed., This yields 2008. H (rad/m)(A/m) [9] C. Caloz and Z.-L. Deck-Léger, “Spacetime metamaterials, Part I: D ∇×iω rad/s As General concepts,” IEEE Trans. Antennas Propag. to be published. ² − , (51a) = E = " E # = V/m = Vm [10] C. Caloz and T. Itoh, Electromagnetic Metamaterials: Transmission h i · ¸ Line Theory and Microwave Applications. Wiley and IEEE Press, 2005. E (rad/m)(V/m) [11] C. Caloz, A. Alù, S. Tretyakov, D. Sounas, K. Achouri, and Z.-L. Deck- B ∇×iω rad/s Vs Léger, “Electromagnetic nonreciprocity,” Phys. Rev. Appl., vol. 10, µ , (51b) = H = " H # = A/m = Am pp. 047001:1–26, Oct. 2018. h i · ¸ [12] G. D. Fasman, Circular Dichroism and the Conformational Analysis of H (rad/m)(A/m) Biomolecules. Springer Science & Business Media, seventh ed., 2013. D ∇×iω rad/s s ξ − , (51c) [13] I. V. Lindell, Methods for Electromagnetic Field Analysis. Wiley-IEEE = H = " H # = A/m = m Press, 1992. h i · ¸ [14] D. M. Pozar, Microwave Engineering. Wiley, fourth ed., 2011. or E (rad/m)(V/m) [15] C. A. Balanis, Advanced Engineering Electromagnetics. Wiley, sec- B ∇×iω rad/s s ond ed., 2012. ζ . (51d) = E = " E # = V/m = m [16] F. T. Ulaby and U. Ravaioli, Fundamentals of Applied Electromagnetics. h i · ¸ Pearson, seventh ed., 2014. [17] B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics. Wiley- APPENDIX B Interscience, second ed., 2007. SIGNSOFTHE IMAGINARY PARTSOFTHE CHIRAL [18] A. Fresnel, Oeuvres Complètes d’Augustin Fresnel. Imprimerie impéri- ale, 1868. Vol. 1:731-51; vol. 2:479-596. CONSTITUTIVE PARAMETERSINTHE LOSSY CASE [19] G. F. Dionne, G. A. Allen, P. R. Haddad, C. A. Ross, and B. Lax, Let us first consider the case of ². According to (13), “Circular polarization and nonreciprocal propagation in magnetic media,” Lincoln Lab. J., vol. 15, pp. 323–340, Nov. 2005. ² ²r ²0 i²00 1 χee 1 χ0 iχ00 , (52a) [20] B. Lax and K. J. Button, Microwave Ferrites and Ferrimagnetics. = ² = r + r = + = + ee + ee McGraw-Hill, 1962. 0 [21] M. Faraday, Faraday’s Diary. George Bell and Sons, 1933. Vol. IV, Nov. i.e., ¡ ¢ 12, 1839 - June 26, 1847. ²0 1 χ0 and ²00 χ00 , (52b) [22] T. H. Stix, Waves in Plasmas. American Institute of Physics, 1992. r = + ee r = ee [23] T. Kodera and C. Caloz, “Unidirectional Loop Metamaterial (ULMs) as where the primed and double-primed quantities denote the as magnetless ferrimagnetic materials: principles and applications,” IEEE Antennas Wirel. Propag. Lett., vol. 17, pp. 1943–1947, Nov. 2018. real and imaginary parts, respectively. [24] L. D. Landau, L. P. Pitaevskii, and E. M. Lifshitz, Electrodynamics of Given the assumed plane-wave forward ( r -direction Continuous Media, vol. 8 of Course of theoretical physics. Butterworth- ikr + i(k ik )r propagating) spacetime dependence ψ e e 0+ 00 Heinemann, second ed., 1984. Chap. XII. ik r k r ∝ = = e 0 e− 00 [Eq. [1].(4)], one must have k00 0 for exponential [25] F. Capolino, ed., Metamaterials Handbook. CRC Press, 2009. 2 > volumes. decay in the r direction if the medium is lossy. Assuming [26] A. Serdyukov, I. Semchenko, S. Tretyakov, and A. Sihvola, Electromag- + first a nonmagnetic (µr 1) and nonchiral (χ 0) medium, netics of Bi-anisotropic Materials, Theory and Applications. Routledge, = = 2001. we have [27] A. D. Yaghjian, A. Alù, and M. G. Silveirinha, “Homogenization of ² spatially dispersive metamaterial arrays in terms of generalized elec- k k p² k ² i² k ² 1 i 00 tric and magnetic polarizations,” Photonics Nanostructures: Fundam. 0 r 0 r0 r00 0 r0 = = + = s + ²0 Appl., vol. 11, pp. 374–396, Nov. 2013. q q (53) ²00 ²0 ²00 [28] P. C. Clemmow, The Plane Wave Spectrum Representation of Elec- ¿ k ² 1 i , tromagnetic Fields. The IEEE/OUP Series on Electromagnetic Wave 0 r0 ≈ + 2²0 Theory, IEEE Press, 1996. classical reissue. q µ ¶ [29] F. W. Hehl and Y. N. Obukhov, Foundations of Classical Electrodynam- where we have made the low-loss assumption ²00 ²0 in ics. Birkhäuser Boston, 2003. ¿ the last approximation. We have thus found that k00 [30] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism = k ²00/(2 ² ) and, assuming ²0 0, the k00 positivity required from conductors and enhanced nonlinear phenomena,” IEEE Trans. 0 r r0 > Microw. Theory Tech., vol. 47, pp. 2075–2084, Apr. 1999. for lossy absorption translates into ²00 positivity, i.e., ²00 0. p > [31] S. A. Tretyakov, I. Nefedov, A. Sihvola, S. Maslovski, and C. Simovski, An analogous argument leads to the conclusion that we “Waves and energy in chiral nihility,” J. Electromag. Waves Appl., have also µ00 0. vol. 17, pp. 695–706, Jan. 2003. > [32] S. A. Tretyakov, F. Mariotte, C. R. Simovski, T. G. Kharina, and J.- The case of χ is more subtle. The eigenstates, whose P. Heliot, “Analytical antenna model for chiral scatterers: comparison wavenumbers are given by (21a) as β+ ω(p²µ χ), must with numerical and experimental data,” IEEE Tran. Antennas Propag., § = § satisfy the same condition as k above, i.e., Im β+ 0. At vol. 44, pp. 1006–1014, Jul. 1996. § > the resonance frequency [see (46)], we have ²0 µ0 χ0 0, [33] M. G. Silveirinha, “Design of linear-to-circular polarization trans- =© ª= = formers made of long densely packed metallic helices,” IEEE Trans. Antennas Propag., vol. 56, pp. 390–401, Feb. 2008. 14In Maxwell equations, the differential operator refers to space, i.e., ∇ [34] A. Papoulis, The Fourier Integral and Its Applications. Dover Publica- r, and applies to electromagnetic waves, which, assuming linearity, ∇ = ∇ ik r tions, 2019. decompose into plane waves, Ψ˜ (k)e [28]. We have then that r ik · ∇ × = × [35] Private communication with D. L. Sounas. or, more generally, r ik, whose unit is clearly rad/m. ∇ = 13

so that this condition reduces to χ00(ω ) ² (ω )µ (ω ). APPENDIX D 0 < 00 0 00 0 The absolute value in this relation indicates that there SPATIAL DISPERSION ¯ ¯ p is no general constraint on the¯ sign of¯ χ00, except that, Spatial dispersion is a complex effect and its mathemat- mathematically, it must change sign together with χ0, i.e., ical description is quite involved. We only enumerate here with the medium handedness. the main steps of the related developments that concern chirality, following [24], [25], [26]. APPENDIX C The derivations traditionally [24] treat the medium as a DERIVATION OF THE CHIRAL EIGENSTATES purely dielectric one, embedding the magnetic effects in the permittivity. In the model of (1), this reverts to setting Inserting the z-forward plane-wave fields (20) into Pm 0 and replacing Pe by a generalized polarization the chiral-explicit Maxwell equations (8), using the z- = P Jpol/(iω), where Jpol is the global polarization current propagating plane-wave conditions ∂/∂x ∂/∂y 0, and = − = = density. Equations (1) reduce then to projecting onto the x, y,z axes, yields D ²0E Jpol/(iω) and B µ0H, (60) iβEy ωχEx iωµHx , (54a) = − = − = + where J is computed, similarly to (30), as pol ˆ iβEx ωχEy iωµHy , (54b) = + ¯ Jpol(r) K¯(r,r0) E(r0)dr0, (61) = Ω · iβHy iω²Ex ωχHx , (54c) − = − + ¯ with the kernel K¯(r,r0) accounting, according to the defini- iβHx iω²Ey ωχHy , (54d) = − + tion of Jpol, for both electric and magnetic effects. which form a 4 4 homogeneous linear system of equations Next, substituting E(r0) in this relation by its three- × dimensional Taylor expansion, performing the integral, in- in Ex , Ey , Hx and Hz . Solving now (54a) and (54b) for Hx serting the result into (60), and assuming isotropy, leads to and Hy , respectively, yields a relation of the form χ β Hx i Ex Ey , (55a) D ²E α E β E γ E, (62) = µ − ωµ = + ∇ × + ∇∇ · + ∇ × ∇ × which explicitly shows the first orders – 0 (²), 1 (α) and 2 β χ (β and γ) – of spatial dispersion. Combining this equation Hy Ex i Ey , (55b) = ωµ + µ with Maxwell equations eventually leads to and inserting these results into (54c) and (54d) reduces (54) ωα 1 ωα D ²E i B β E, H ω2γ B i E. (63) to the 2 2 system = + 2 + ∇∇ · = µ − + 2 × µ 0 ¶ β2 ω2 ²µ χ2 E 2iβχωE 0, (56a) − − x + y = Comparing (63) with (5) indicates that the chiral terms £ ¡ ¢¤ are related to the α E term in (62), which reveals that 2iβχωE β2 ω2 ²µ χ2 E 0. (56b) ∇ × − x + − − y = chirality is a first-order spatial dispersion effect. Finally, nullifying the£ determinant¡ of¢¤ this system for a nontrivial solution leads to the quartic equation APPENDIX E BIANISOTROPIC PLANE-WAVE RELATIONS 2 β2 ω2 χ2 ²µ 4β2ω2χ2 0. (57) + − − = Inserting (4) into [1].(4) yields £ ¡ ¢¤ The resolution of this equation provides the four modal E iωζ E iωµ H, (64a) wavenumbers ∇ × = · + · β§ ω p²µ χ , (58) H iω² E iωξ H. (64b) § = § § ∇ × = − · − · ¡ ¢ This system of equations can be solved for E in terms of while inserting these wavenumbers into (56a) or (56b), 1 E and H by pre-dotmultiplying (64a) by ξ µ− and and subsequently comparing the coefficients of Ex and Ey , ∇ × ∇ × · provides the four modal electric fields summing the resulting equation with (64b) so as to eliminate H, and finally isolating E. Similarly, it can be solved for iβ+z E+ E0 xˆ iyˆ e + , (59a) H in terms of E and H by pre-dotmultiplying (64b) + = + 1 ∇ × ∇ × iβ+z − E+ E0 ¡xˆ iyˆ¢e − , (59b) by ζ ² and summing the resulting equation with (64a) − = − · iβ− so as to eliminate E, and finally isolating H, where we E− E0 ¡xˆ iyˆ¢e + , (59c) + = + have avoided to involve the inverses of ξ and ζ because, iβ−z E− E0 ¡xˆ iyˆ¢e − . (59d) − = − in contrast to ² and µ, these tensors may be non invertible. The corresponding four¡ modal¢ magnetic field are then The result is 1 easily found by substituting (59) into (55), or by consid- i 1 − 1 E ² ξ µ− ζ H ξ µ− E , (65a) ering that, for plane-wave excitation, the simple relation = ω − · · · ∇ × + · · ∇ × ³ ´ 1³ ´ H zˆ E/η (η µ/²) holds. i 1 − 1 = × = H µ ζ ²− ξ E ζ ²− H . (65b) = − ω − · · · ∇ × + · · ∇ × p ³ ´ ³ ´ 14

We next substitute these relations into the expressions Since it represents a physical quantity, (t) is purely real, C of (4) that are associated with bianisotropy, i.e., having the while C˜ (ω) is complex. Expanding (70b) into its real and coefficients ξ and ζ. Specifically, inserting (65b) into the imaginary parts yields then first equation of (4) and (65a) into the first equation of (4) ˆ +∞ provides the alternative constitutive relations C˜ 0(ω) iC˜ 00(ω) (t)[cos(ωt) i sin(ωt)] dt. + = ˆ C + i 1 1 1 −∞ − − − +∞ D ² E ξ µ ζ ² ξ E ζ ² H , (66a) (t)cos(ωt)dt (71) = · − ω · − · · · ∇ × + · · ∇ × = C ³ 1 ´ ³ ´ ˆ i 1 − 1 −∞ B ζ ² ξ µ− ζ H ξ µ− E µ H, +∞ =ω · − · · · ∇ × + · · ∇ × + · i (t)sin(ωt)dt, ³ ´ ³ ´ (66b) + C −∞ which upon angular spectrum decomposition reduce to which separates as ˆ 1 1 1 1 − − − +∞ D˜ ² E˜ ξ µ ζ ² ξ k E˜ ζ ² k H˜ , (67a) C˜ 0(ω) (t)cos(ωt)dt C˜ 0( ω), (72a) = · + ω · − · · · × + · · × = ˆ C = − 1 ³ 1 1 ´ ³ 1 ´ −∞ ˜ − − ˜ − ˜ ˜ +∞ B ζ ² ξ µ ζ k H ξ µ k E µ H, C˜ (ω) (t)sin(ωt)dt C˜ ( ω), (72b) = − ω · − · · · × + · · × + · I = C = − I − ³ ´ ³ ´ (67b) −∞ where the last equalities, revealing the parity of the real and or, using the identity k a k I a, × = × · imaginary Fourier-transform functions, straightforwardly ³ ´ follow from the parities of the cosine and sine functions. 1 D˜ ² T k I E˜ T ζ ²− k I H˜ , (68a) Let us now decompose the Fourier-transform version of = + D · × · + D · · · × · 1 Eqs. (7), i.e., ³ ´ 1 ³ 1 − ´ with T D ξ µ ζ ²− ξ (68b) = ω · − · · ˜ ˜ ˜ ˜ ˜ ˜ 1 ³ ´ D ²E iχH and B iχE µH, (73) B˜ T ξ µ− k I E˜ µ T k I H˜ (68c) = + = − + = B · · · × · + + B · × · 1 into their real and imaginary parts, while remembering the ³ ´1 ³ 1 −´ with T B ζ ² ξ µ− ζ . (68d) = −ω · − · · parities of ²(ω) and µ(ω) from their Lorentzian form [2], [3] ³ ´ to be15 For a biisotropic medium (Sec. II-C), transforming the ² (ω),µ (ω) are even (e), (74a) tensors into scalars reduces (35) to 0 0 i ξ ζ H D ²E E ∇ × ²00(ω),µ00(ω) are odd (o), (74b) = − ω µ ζξ/² ∇ × + ² µ − ¶µ ¶ i ξ ²E (² E ζ H), (69a) and analyze the parities of the resulting equations. We get = − ω γ ∇ × + ∇ × i ζ ξ E (e) (e)(e) (o)(o) o (o) e (e) B H ∇ × µH D˜ 0 ²0 E˜ 0 ²00E˜ 00 χ0H˜ 00 χ00H˜ 0, (75a) = ω ² ξζ/µ ∇ × + µ + = − − − µ − ¶µ ¶ (o) (e)(o) (o)(e) o (e) e (o) i ζ D˜ 00 ²0 E˜ 00 ²00E˜ 0 χ0H˜ 0 χ00H˜ 00, (75b) ξ E µ H µH, (69b) = + + − = ω γ ∇ × + ∇ × + (e) (e)(e) (o) (o) o (o) e (e) ¡ ¢ B˜ 0 µ0H˜ 0 µ00H˜ 00 χ0E˜ 00 χ00E˜ 0, (75c) where = − + + (o) (e) (o) (o) (e) o (e) e (o) γ ²µ ξζ, (69c) B˜ 00 µ0H˜ 00 µ00H˜ 0 χ0E˜ 0 χ00E˜ 00, (75d) = − = + − + which further reduce to (32) in the chiral case (ξ iχ and = where the known parities have been indicated in brackets ζ iχ). = − and the parities of χ have been deduced. We have thus found, with consistency within and redundancy across the APPENDIX F four equations, that TEMPORAL FREQUENCY PARITYOFTHE CHIRAL PARAMETER χ0(ω) is odd, (76a) Let us refer, for conciseness, to the electromagnetic field functions (t), (t), (t) and (t) by the generic function E H D B χ00(ω) is even. (76b) (t). This function is related to its Fourier transform, C˜ (ω), C by the Fourier transform pair [34] ˆ So, the direct and cross constitutive parameters have 1 +∞ iωt 1 opposite temporal frequency parities. (t) C˜ (ω)e− dω F − C˜ (ω) , (70a) C = 2π = ˆ −∞ © ª 15For consistency with the rest of the paper, we drop here the tildes +∞ iωt C˜ (ω) (t)e+ dt F { (t)}. (70b) corresponding to the Fourier transforms of the permittivity and perme- = C = C ability. −∞ 15

APPENDIX G PARAMETER EXTRACTION FORMULAS We consider a linearly polarized (LP) incident wave. Solving Eqs. (28) respectively for χ and p²µ, with z p = representing the lattice period (Fig. [1].4), yields 1 1 yx xx θC tan− Ey /Ex tan− S / S χ 21 21 , (77a) = −ωp = − ω¡ p ¢ = − ¡¯ ωp¯ ¯ ¯¢ ¯ ¯ ¯ ¯ ∠ xx ∠ yx φC S S p²µ 21 21 , (77b) = ωp = ωp = ωp where the (frequency-dependent) scattering transmission parameters Suv (u,v x, y) correspond to the simulation 21 = setup in Fig. 8. The chiral parameter, χ, is readily provided by (77a), but more information is required to discriminate ² and µ in (77b). Such information may be accessed via the uv (frequency-dependent) scattering reflection parameters S11 (u,v x, y). Due to reciprocity, where any rotation incurred = to the wave in one direction must be undone in the opposite direction, the polarization of the wave reflected by the particle must be identical to the incident wave16. Therefore, one needs to consider only the incident polarization, where the reflection parameter is simply given by the normal- incidence Fresnel coefficient Sxx (η Z )/(η Z ), with 11 = − port + port η µ/². Solving this relation for η yields = xx p 1 S11 µ η Zport + . (78) = 1 Sxx = ² − 11 r Finally, taking successfully the product and the ratio of Eqs. (77b) and (78) provides the missing parameters: ∠ yx xx S21 1 S11 µ Zport + , (79a) = ωp 1 Sxx Ã !µ − 11 ¶ yx 1 ∠S 1 Sxx ² 21 − 11 . (79b) = Z ωp 1 Sxx port Ã !µ + 11 ¶

16This may be demonstrated ad absurdum as follows [35]. If we had yx S 0 (reflection rotation), then we would also have, from isotropy, 11x y 6= x y yx S11 0 and S11 S11 . The latter result following from the fact that the wave6= would rotate= − in the same direction for the two experiments yx refl,a inc,a x y (given the same particle chirality), so that S11 Ey /Ex and S11 refl,b inc,b refl,b inc,b yx = = Ex /Ey ( Ex )/Ey S11 . But reciprocity (or time-reversal = −| x y| yx = − yx x y symmetry) requires S11 S11 . Therefore, one must have S11 S11 0, i.e., no reflection rotation.= = =