Exhibition of circular Bragg phenomenon by hyperbolic, dielectric, structurally chiral materials

The relative permittivity dyadic of a dielectric structurally chiral material (SCM) varies helicoidally along a fixed direction; in consequence, the SCM exhibits the circular Bragg phenomenon, which is the circular-polarization-selective reflection of light. The introduction of hyperbolicity in an SCM---by making either one or two but not all three eigenvalues of the relative permittivity dyadic acquire negative real parts---does not eliminate the circular Bragg phenomenon, but significantly alters the regime for its exhibition. Physical vapor deposition techniques appear to be suitable to fabricate hyperbolic SCMs.


Introduction
Chiral liquid crystals 1,2 and chiral sculptured thin films 3,4 are dielectric examples of structurally chiral materials (SCMs)-which are anisotropic and helicoidally nonhomogeneous along a fixed axis. If that fixed axis is parallel to the z-axis of a Cartesian coordinate system with unit vectorŝ u x ,û y , andû z , the frequency-domain constitutive relations of dielectric SCMs may be set down as ; (1) where the superscript T denotes the transpose; μ 0 and ε 0 are the permeability and permittivity of free space; the rotational nonhomogeneity is expressed through the dyadic ̳ S z ðzÞ ¼û zûz þ ðû xûx þû yûy Þ cos hπz Ω þ ðû yûx −û xûy Þ sin hπz Ω ; (2) with 2Ω as the helical pitch and either h ¼ þ1 for structural right-handedness or h ¼ −1 for structural left-handedness; the dyadic ̳ S y ðχÞ ¼ ðû xûx þû zûz Þ cos χ þ ðû zûx −û xûz Þ sin χ þû yûy (3) containing χ ∈ ½0 deg; 90 deg as the angle of rise of the helical morphology; and ε a , ε b , and ε c are the three z-independent eigenvalues of the relative permittivity dyadic ̳ ε r ðzÞ. Typically, dissipation is small enough to be ignored and ε a;b;c > 0; hence, ̳ ε r ðzÞ is positive definite. 5 Figure 1 shows a cross-sectional image of a chiral sculptured thin film. For SCMs of this kind, the adequacy of the continuum constitutive equations (1) at sufficiently large free-space wavelengths has been established experimentally. 6,7 A nanoscopic-to-continuum model based on the Bruggeman homogenization formalism has also been developed. 8 The optical signature of an SCM is a circular-polarization-sensitive stopband. The center wavelength and the width of this stopband depend on the direction of the wavevector of an incident circularly polarized plane wave. Most significantly, the stopband is exhibited when the incident plane wave's handedness is the same as the structural handedness of the SCM, but not otherwise. The stopband is best seen when the thickness of the SCM exceeds a certain number of helical pitches. 6,[9][10][11] When dissipation is small enough to be ignored, ̳ ε r ðzÞ is positive definite, and the variations of ε a;b;c with respect to the free-space wavelength λ 0 are also small enough to be ignored, the circular-polarization-sensitive stopband can be delineated as 4 where θ is the angle of incidence with respect to the z-axis and Provided that Re½ε σ ≫ jIm½ε σ j for all σ ∈ fa; b; cg, the estimates provided by Eq. (4) can be used with ε c replaced by jε c j and ε d by jε d j. The exhibition of the circular-polarization-sensitive stopband is called the circular Bragg phenomenon. During the last 10 years, attention has been paid to dielectric-magnetic materials with indefinite permeability and permittivity dyadics. 12,13 Although the practical realization of such materials remains a matter of conjecture, there is no doubt of the existence in nature of dielectric materials the real parts of whose permittivity dyadics are indefinite. 5 Graphite, 14 triglycine sulfate, 15,16 sapphire, 17 and bismuth 16 are examples. Metal nanowire arrays 18 and periodic metal/ dielectric multilayers 19 provide examples of manufactured anisotropic dielectric materials whose effective permittivity dyadics have indefinite real parts. 20 Periodic graphene/dielectric multilayers have also been proffered as candidates. 21 Although dissipation due to conduction in metals and graphene has been predicted to be offsetable by using dielectric materials with optical gain, 22 the effective-medium approximations underlying such predictions must be handled with some care. 23 Experience with ambichiral materials 24,25 indicates that hyperbolic, dielectric SCMs ought to be practically realizable as nanoengineered periodic multilayers. A variety of physical deposition techniques-such as thermal evaporation, electron-beam evaporation, and sputtering 26 -can be used to deposit alternating layers of a metal and a dielectric material on a suitably rotating planar substrate. 4 In these fabrication techniques, collimated vapor fluxes of both materials must be directed very obliquely toward the substrate in order to engender biaxiality. Furthermore, the nominal thickness of each metal layer must be a small fraction of the nominal thickness of each dielectric layer, 23 and all layers must be electrically thin. 27,28 Equation (1) would still apply, although the use of the nanoscopic-to-continuum model 8 developed for chiral sculptured thin films may be perilous if the metal volume fraction is not sufficiently small. 29 Anyhow, either one or two of Reðε a Þ, Reðε b Þ, and Reðε c Þ could be negative, with the remainder being positive. Then, with dissipation assumed to be sufficiently small, the estimates provided by Eq. (4) would become dubious. Indeed, the question arises: will a hyperbolic SCM exhibit the circular Bragg phenomenon?
In order to answer this question, a one-point boundary-value problem was formulated and solved. In this problem, the half space z < 0 is vacuous while the half space z > 0 is occupied by the hyperbolic SCM, and a circularly polarized plane wave is obliquely incident on the interface z ¼ 0 from its vacuous side. As it is known that the circular Bragg phenomenon develops as the thickness of an SCM increases, 4,10 an SCM half space should conceptually deliver the best developed circular Bragg phenomenon. The underlying boundary-value problem is introduced briefly in Sec. 2, the detailed procedure to solve it being available elsewhere. 30 Numerical results are provided and discussed in Sec. 3. An expð−iωtÞ dependence on time t is implicit, with . The free-space wavenumber is denoted by

Boundary-Value Problem
Let the half space z < 0 be vacuous, while the half space z > 0 be occupied by an SCM described by Eqs. (1)-(3). An arbitrarily polarized plane wave is obliquely incident on the interface z ¼ 0 from the vacuous half space. Without significant loss of generality, let the wave vector of this plane wave lie wholly in the xz-plane and make an angle θ ∈ ½0 deg; 90 degÞ with respect to the þz-axis. Accordingly, the electric field phasor of the incident plane wave may be written as where a L and a R are the known amplitudes of the left-and right-circularly polarized components, respectively, and the vectors are of unit magnitude. The reflected plane wave's electric field phasor is given by Fig. 3 Same as Fig. 2, except that ε with unknown amplitudes r L and r R . The procedure to determine r L and r R in terms of a L and a R is described in detail elsewhere. 30 The reflection amplitudes are related to the incidence amplitudes by the four reflection coefficients entering the 2 × 2 matrix in the following relation: The copolarized reflectances of the SCM half space are denoted by R LL ¼ jr LL j 2 and R RR ¼ jr RR j 2 , and the cross-polarized ones by R LR ¼ jr LR j 2 and R RL ¼ jr RL j 2 . The principle of conservation of energy requires that

Numerical Results and Discussion
Parametric calculations were made with the SCM assumed to be structurally right-handed (h ¼ 1), with all three of ε a;b;c chosen to have very small and positive imaginary parts (that are indicative of weak dissipation). The reflectances R LL , R RL , R RR , and R LR were computed as functions of the angle of incidence θ and either (i) the normalized wavelength λ 0 ∕2Ω for fixed angle of rise χ or (ii) χ for fixed λ 0 ∕2Ω. In order to set a baseline for discussion, Fig. 2 displays all four reflectances as functions of the normalized wavelength λ 0 ∕2Ω and the angle of incidence θ, when χ ¼ 60 deg and the SCM is of the regular (i.e., nonhyperbolic) type: ε a ¼ 3.26ð1 þ 0.001iÞ, ε b ¼ 4.46ð1 þ 0.001iÞ, and Fig. 4 Same as Fig. 2, except that ε a ¼ 3.26ð−1 þ 0.001iÞ, ε c ¼ 3.78ð−1 þ 0.001iÞ, and χ ¼ 15 deg. ε c ¼ 3.78ð1 þ 0.001iÞ. A sigmoid ridge of high values of R RR is evident in this figure. The limits provided in Eq. (4) with ε σ replaced by jε σ j, σ ∈ fa; b; cg, are satisfied by this ridge. For θ∼ < 70 deg, R LL is negligible in that portion of the ðλ 0 ∕2ΩÞ − θ plane which is occupied by the high-R RR ridge; additionally, both cross-polarized reflectances are very small. The huge excess of R RR over R LL accompanied by very small values of the two other reflectances is the chief manifestation of the circular Bragg phenomenon. When the sign of Reðε b Þ was altered from positive to negative, the SCM of Fig. 2 became hyperbolic and the sigmoid high-R RR ridge of that figure disappeared. However, a search with somewhat higher values of λ 0 ∕2Ω soon revealed a portion of the ðλ 0 ∕2ΩÞ − θ plane in which (i) R RR exceeds R LL by a large margin and (ii) the excess of R R over R L is even greater, with R LL very close to 0. In Fig. 3, circular-polarization-selective reflection is clearly evident for λ 0 ∕2Ω ∈ ½1.95; 2.4 and θ ∈ ½0 deg; 20 deg as well as in the upper left neighborhood of that rectangular zone, for the hyperbolic SCM.
Thus, a significant conclusion from these three figures is that hyperbolicity tends to widen the stopband. However, this conclusion may have to be modified when material dispersion manifested through the λ 0 -dependences of ε a;b;c is taken into account. The exhibition of the circular Bragg phenomenon by a regular SCM for fixed values of λ 0 ∕2Ω and θ is delineated by Eq. (4). One just has to ensure the appropriate choices of ε c and ε d , the correct choice of the latter parameter being determined by the correct choices of ε a , ε b , and χ. If all three of the eigenvalues of ̳ ε r ðzÞ are fixed as well, then an appropriate value of sin χ must be found. But no physical value of χ may emerge. Therefore, the exhibition of circular-polarization-selective reflection by hyperbolic SCMs was investigated in the χ − θ plane for fixed values of ε a , ε b , ε c , and λ 0 ∕2Ω. Figure 5 shows all four reflectances computed as functions of χ ∈ ½0 deg; 90 deg and θ ∈ ½0 deg; 90 degÞ, when ε a ¼ 3.26ð1 þ 0.001iÞ, ε b ¼ 4.46ð−1 þ 0.001iÞ, ε c ¼ 3.78ð1 þ 0.001iÞ, and λ 0 ∕2Ω ¼ 1.918. Thus, only one of the three eigenvalues of ̳ ε r ðzÞ has a negative real part. At least four distinct zones of high values of R RR accompanied by almost zero values of R LL and very low values of both cross-polarized reflectances can be identified in this figure.
Similar data computed for ε a ¼ 3.26ð−1 þ 0.001iÞ, ε b ¼ 4.46ð1 þ 0.001iÞ, and ε c ¼ 3.78ð−1 þ 0.001iÞ are displayed in Fig. 6. Now, two of the three eigenvalues of ̳ ε r ðzÞ have negative real parts. Circular-polarization-selective reflection with R LL ≈ 0 and very low values of R LR and R RR is evident for χ ∈ ½0 deg; 15 deg and θ ∈ ½0 deg; 70 deg. To conclude, the concept of hyperbolic SCMs was introduced in this communication. The hyperbolicity was found to significantly alter-but not eliminate-the exhibition of the circular Bragg phenomenon, which has long been known to be the distinctive signature of nonhyperbolic SCMs such as cholesteric liquid crystals 9 and chiral sculptured thin films. 6 Although practical realization of hyperbolic SCMs has yet to occur, physical vapor deposition offers suitable techniques to fabricate these materials. Significantly wider circular-polarizationsensitive stopbands may be exhibited by hyperbolic SCMs in comparison to their nonhyperbolic analogs.