Special Section on Nanostructured Thin Films: Latest Developments in Theory and Practice

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

[+] Author Affiliations
Akhlesh Lakhtakia

National Taipei University of Technology, Department of Electro-Optical Engineering, Taipei 106, Taiwan

Pennsylvania State University, Department of Engineering Science and Mechanics, University Park, Pennsylvania 16802-6812

J. Nanophoton. 8(1), 083998 (Feb 27, 2014). doi:10.1117/1.JNP.8.083998
History: Received December 7, 2013; Revised January 20, 2014; Accepted January 24, 2014
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Abstract.  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. Significantly wider circular-polarization-sensitive stopbands may be exhibited by hyperbolic SCMs in comparison to nonhyperbolic SCMs. Physical vapor deposition techniques appear to be suitable to fabricate hyperbolic SCMs.

Figures in this Article

Chiral liquid crystals1,2 and chiral sculptured thin films3,4 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 vectors u^x, u^y, and u^z, the frequency-domain constitutive relations of dielectric SCMs may be set down as Display Formula

D(r)=ε0ε̳r(z)E(r)=ε0z(z)y(χ)(εau^zu^z+εbu^xu^x+εcu^yu^y)yT(χ)zT(z)E(r)B(r)=μ0H(r),},(1)
where the superscript T denotes the transpose; μ0 and ε0 are the permeability and permittivity of free space; the rotational nonhomogeneity is expressed through the dyadic Display Formula
z(z)=u^zu^z+(u^xu^x+u^yu^y)cos(hπzΩ)+(u^yu^xu^xu^y)sin(hπzΩ),(2)
with 2Ω as the helical pitch and either h=+1 for structural right-handedness or h=1 for structural left-handedness; the dyadic Display Formula
y(χ)=(u^xu^x+u^zu^z)cosχ+(u^zu^xu^xu^z)sinχ+u^yu^y(3)
containing χ[0deg,90deg] as the angle of rise of the helical morphology; and εa, εb, and εc are the three z-independent eigenvalues of the relative permittivity dyadic ε̳r(z). Typically, dissipation is small enough to be ignored and εa,b,c>0; hence, ε̳r(z) 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

Graphic Jump LocationF1 :

Cross-sectional scanning electron micrograph of a chiral sculptured thin film made by thermal evaporation of a chalcogenide glass.

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.6,911 When dissipation is small enough to be ignored, ε̳r(z) is positive definite, and the variations of εa,b,c with respect to the free-space wavelength λ0 are also small enough to be ignored, the circular-polarization-sensitive stopband can be delineated as4Display Formula

λ02Ω{[εc,εd]cos1/2θ,εc<εd,[εd,εc]cos1/2θ,εc>εd,(4)
where θ is the angle of incidence with respect to the z-axis and Display Formula
εd=εaεbεacos2χ+εbsin2χ.(5)
Provided that Re[εσ]|Im[εσ]| for all σ{a,b,c}, the estimates provided by Eq. (4) can be used with εc replaced by |εc| and εd by |εd|. 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 bismuth16 are examples. Metal nanowire arrays18 and periodic metal/dielectric multilayers19 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 materials24,25 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 sputtering26—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 model8 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 Re(εa), Re(εb), and Re(εc) 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=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 exp(iωt) dependence on time t is implicit, with ω denoting the angular frequency and i=1. The free-space wavenumber is denoted by k0=ωε0μ0=2π/λ0.

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=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[0deg,90deg) with respect to the +z-axis. Accordingly, the electric field phasor of the incident plane wave may be written as Display Formula

Einc=(aLiu^yp+2aRiu^y+p+2)exp[ik0(xsinθ+zcosθ)],z0,(6)
where aL and aR are the known amplitudes of the left- and right-circularly polarized components, respectively, and the vectors Display Formula
p±=u^xcosθ+u^zsinθ(7)
are of unit magnitude. The reflected plane wave’s electric field phasor is given by Display Formula
Eref=(rLiu^yp2+rRiu^y+p2)exp[ik0(xsinθzcosθ)],z0,(8)
with unknown amplitudes rL and rR. The procedure to determine rL and rR in terms of aL and aR 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: Display Formula

[rLrR]=[rLLrLRrRLrRR][aLaR].(9)
The copolarized reflectances of the SCM half space are denoted by RLL=|rLL|2 and RRR=|rRR|2, and the cross-polarized ones by RLR=|rLR|2 and RRL=|rRL|2. The principle of conservation of energy requires that RR=RRR+RLR1 and RL=RLL+RRL1.

Parametric calculations were made with the SCM assumed to be structurally right-handed (h=1), with all three of εa,b,c chosen to have very small and positive imaginary parts (that are indicative of weak dissipation). The reflectances RLL, RRL, RRR, and RLR were computed as functions of the angle of incidence θ and either (i) the normalized wavelength λ0/2Ω for fixed angle of rise χ or (ii) χ for fixed λ0/2Ω.

In order to set a baseline for discussion, Fig. 2 displays all four reflectances as functions of the normalized wavelength λ0/2Ω and the angle of incidence θ, when χ=60deg and the SCM is of the regular (i.e., nonhyperbolic) type: εa=3.26(1+0.001i), εb=4.46(1+0.001i), and εc=3.78(1+0.001i). A sigmoid ridge of high values of RRR is evident in this figure. The limits provided in Eq. (4) with εσ replaced by |εσ|, σ{a,b,c}, are satisfied by this ridge. For θ<70deg, RLL is negligible in that portion of the (λ0/2Ω)θ plane which is occupied by the high-RRR ridge; additionally, both cross-polarized reflectances are very small. The huge excess of RRR over RLL accompanied by very small values of the two other reflectances is the chief manifestation of the circular Bragg phenomenon.

Graphic Jump LocationF2 :

Reflectances RLL, RRL, RRR, and RLR as functions of λ0/2Ω and θ, when εa=3.26(1+0.001i), εb=4.46(1+0.001i), εc=3.78(1+0.001i), χ=60deg, and h=1. The color coding employs the spectrum of the rainbow with the deepest violet denoting 0 and the deepest red denoting 1.0.

When the sign of Re(εb) was altered from positive to negative, the SCM of Fig. 2 became hyperbolic and the sigmoid high-RRR ridge of that figure disappeared. However, a search with somewhat higher values of λ0/2Ω soon revealed a portion of the (λ0/2Ω)θ plane in which (i) RRR exceeds RLL by a large margin and (ii) the excess of RR over RL is even greater, with RLL very close to 0. In Fig. 3, circular-polarization-selective reflection is clearly evident for λ0/2Ω[1.95,2.4] and θ[0deg,20deg] as well as in the upper left neighborhood of that rectangular zone, for the hyperbolic SCM.

Graphic Jump LocationF3 :

Same as Fig. 2, except that εb=4.46(1+0.001i).

Next, for the computation of the reflectances presented in Fig. 4 as functions of λ0/2Ω and θ, the following parameters were used: εa=3.26(1+0.001i), εb=4.46(1+0.001i), εc=3.78(1+0.001i), and χ=15deg. Thus, two of the three eigenvalues of ε̳r(z) now have negative real parts. Circular-polarization-selective reflection with RLL almost equal to 0 is evident in Fig. 4 for λ0/2Ω[2,2.2] and θ[0deg,70deg] as well as on the outskirts of this rectangular zone in the (λ0/2Ω)θ plane.

Graphic Jump LocationF4 :

Same as Fig. 2, except that εa=3.26(1+0.001i), εc=3.78(1+0.001i), and χ=15deg.

The circular-polarization-sensitive stopband for normal incidence can be identified as 1.87<(λ0/2Ω)<1.95 in Fig. 2, 1.95<(λ0/2Ω)<2.4 in Fig. 3, and 1.8<(λ0/2Ω)<2.3 in Fig. 4. 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 λ0-dependences of εa,b,c is taken into account.

The exhibition of the circular Bragg phenomenon by a regular SCM for fixed values of λ0/2Ω and θ is delineated by Eq. (4). One just has to ensure the appropriate choices of εc and εd, the correct choice of the latter parameter being determined by the correct choices of εa, εb, and χ. If all three of the eigenvalues of ε̳r(z) are fixed as well, then an appropriate value of sinχ 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 χθ plane for fixed values of εa, εb, εc, and λ0/2Ω.

Figure 5 shows all four reflectances computed as functions of χ[0deg,90deg] and θ[0deg,90deg), when εa=3.26(1+0.001i), εb=4.46(1+0.001i), εc=3.78(1+0.001i), and λ0/2Ω=1.918. Thus, only one of the three eigenvalues of ε̳r(z) has a negative real part. At least four distinct zones of high values of RRR accompanied by almost zero values of RLL and very low values of both cross-polarized reflectances can be identified in this figure.

Graphic Jump LocationF5 :

Reflectances RLL, RRL, RRR, and RLR as functions of χ and θ, when εa=3.26(1+0.001i), εb=4.46(1+0.001i), εc=3.78(1+0.001i), λ0/2Ω=1.918, and h=1. The color coding employs the spectrum of the rainbow with the deepest violet denoting 0 and the deepest red denoting 1.0. The thin white strips for θ<20deg and χ40deg indicate a failure of the computational algorithm (30).

Similar data computed for εa=3.26(1+0.001i), εb=4.46(1+0.001i), and εc=3.78(1+0.001i) are displayed in Fig. 6. Now, two of the three eigenvalues of ε̳r(z) have negative real parts. Circular-polarization-selective reflection with RLL0 and very low values of RLR and RRR is evident for χ[0deg,15deg] and θ[0deg,70deg].

Graphic Jump LocationF6 :

Same as Fig. 5, except εa=3.26(1+0.001i), εb=4.46(1+0.001i), and εc=3.78(1+0.001i).

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 crystals9 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-polarization-sensitive stopbands may be exhibited by hyperbolic SCMs in comparison to their nonhyperbolic analogs.

The National Taipei University of Technology is thanked by the author for an honorary international chair professorship. The Charles Godfrey Binder Endowment at Penn State is thanked for partial support of the author’s research activities.

Chandrasekhar  S., Liquid Crystals. , 2nd ed.,  Cambridge University ,  Cambridge  (1992).
de Gennes  P. G., Prost  J. A., The Physics of Liquid Crystals. , 2nd ed.,  Clarendon ,  Oxford  (1993).
Young  N. O., Kowal  J., “Optically active fluorite films,” Nature. 183, (4654 ), 104 –105 (1959), CrossRef. 0028-0836 
Lakhtakia  A., Messier  R., Chapter 9 in Sculptured Thin Films: Nanoengineered Morphology and Optics. ,  SPIE Press ,  Bellingham, WA  (2005).
Lütkepohl  H., Chapter 9 in Handbook of Matrices. ,  Wiley ,  Chichester  (1996).
Wu  Q., Hodgkinson  I. J., Lakhtakia  A., “Circular polarization filters made of chiral sculptured thin films: experimental and simulation results,” Opt. Eng.. 39, (7 ), 1863 –1868 (2000), CrossRef. 0091-3286 
Sherwin  J. A., Lakhtakia  A., Hodgkinson  I. J., “On calibration of a nominal structure-property relationship model for chiral sculptured thin films by axial transmittance measurements,” Opt. Commun.. 209, (4–6 ), 369 –375 (2002), CrossRef. 0030-4018 
Mackay  T. G., Lakhtakia  A., “Modeling chiral sculptured thin films as platforms for surface-plasmonic-polaritonic optical sensing,” IEEE Sensors J.. 12, (2 ), 273 –280 (2012), CrossRef. 1530-437X 
Fergason  J. L., “Cholesteric structure—I Optical properties,” Mol. Cryst.. 1, (2 ), 293 –307 (1966), CrossRef. 0369-1152 
John  W. D. St. et al., “Bragg reflection from cholesteric liquid crystals,” Phys. Rev. E. 51, (2 ), 1191 –1198 (1995), CrossRef. 1063-651X 
Hodgkinson  I. J. et al., “Spacerless circular-polarization spectral-hole filters using chiral sculptured thin films: theory and experiment,” Opt. Commun.. 184, (1–4 ), 57 –66 (2000), CrossRef. 0030-4018 
Smith  D. R., Kolinko  P., Schurig  D., “Negative refraction in indefinite media,” J. Opt. Soc. Am. B. 21, (5 ), 1032 –1043 (2004), CrossRef. 0740-3224 
Depine  R. A., Inchaussandague  M. E., Lakhtakia  A., “Classification of dispersion equations for homogeneous, dielectric-magnetic, uniaxial materials,” J. Opt. Soc. Am. A. 23, (4 ), 949 –955 (2006), CrossRef. 0740-3232 
Sun  J. et al., “Indefinite permittivity and negative refraction in natural material: graphite,” Appl. Phys. Lett.. 98, (10 ), 101901  (2011), CrossRef. 0003-6951 
Gerbaux  X., Tazawa  M., Hadni  A., “Far IR transmission measurements on triglycine sulphate (TGS), at 5 K,” Ferroelectrics. 215, (1 ), 47 –63 (1998), CrossRef. 0015-0193 
Alekseyev  L. V., Podolskiy  V. A., Narimanov  E. E., “Homogeneous hyperbolic systems for terahertz and far-infrared frequencies,” Adv. Optoelectron.. 2012, , 267564  (2012), CrossRef. 1687-563X 
Schubert  M., Tiwald  T. E., Herzinger  C. M., “Infrared dielectric anisotropy and phonon modes of sapphire,” Phys. Rev. B. 61, (12 ), 8187 –8201 (2000), CrossRef. 1098-0121 
Kanungo  J., Schilling  J., “Experimental determination of the principal dielectric functions in silver nanowire metamaterials,” Appl. Phys. Lett.. 97, (2 ), 021903  (2010), CrossRef. 0003-6951 
Kidwai  O., Zhukovsky  S. V., Sipe  J. E., “Effective-medium approach to planar multilayer hyperbolic metamaterials: strengths and limitations,” Phys. Rev. A. 85, (5 ), 053842  (2012), CrossRef. 1050-2947 
Cortes  C. L. et al., “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt.. 14, (6 ), 063001  (2012), CrossRef. 0150-536X 
Othman  M. A. K., Guclu  C., Capolino  F., “Graphene-dielectric composite metamaterials: evolution from elliptic to hyperbolic wavevector dispersion and the transverse epsilon-near-zero condition,” J. Nanophotonics. 7, (1 ), 073089  (2013), CrossRef. 1934-2608 
Ni  X. et al., “Loss-compensated and active hyperbolic metamaterials,” Opt. Express. 19, (25 ), 25242 –25254 (2011), CrossRef. 1094-4087 
Mackay  T. G., Lakhtakia  A., “On the application of homogenization formalisms to active dielectric composite materials,” Opt. Commun.. 282, (13 ), 2470 –2475 (2009), CrossRef. 0030-4018 
Gerritsen  H. J., Yamaguchi  R. T., “A microwave analog of optical rotation in cholesteric liquid crystals,” Am. J. Phys.. 39, (8 ), 920 –923 (1971), CrossRef. 0002-9505 
Hodgkinson  I. J. et al., “Ambichiral, equichiral and finely chiral layered structures,” Opt. Commun.. 239, (4–6 ), 353 –358 (2004), CrossRef. 0030-4018 
Martín-Palma  R. J., Lakhtakia  A., Chapter 4 in Nanotechnology: A Crash Course. ,  SPIE Press ,  Bellingham, WA  (2010).
King  R. W. P., Sandler  S. S., “The electromagnetic field of a vertical electric dipole in the presence of a three-layered region,” Radio Sci.. 29, (1 ), 97 –113 (1994), CrossRef. 0048-6604 
Bohren  C. F., Xiao  X., Lakhtakia  A., “The missing ingredient in effective-medium theories: standard deviations,” J. Mod. Opt.. 59, (15 ), 1312 –1315 (2012), CrossRef. 0950-0340 
Mackay  T. G., “On the effective permittivity of silverinsulator nanocomposites,” J. Nanophotonics. 1, (1 ), 019501  (2007), CrossRef. 1934-2608 
Lakhtakia  A., “Reflection of an obliquely incident plane wave by a half space filled by a helicoidal bianisotropic medium,” Phys. Lett. A. 374, (37 ), 3887 –3894 (2010), CrossRef. 0375-9601 

Akhlesh Lakhtakia is presently the Charles Godfrey Binder (Endowed) Professor of Engineering Science and Mechanics at the Pennsylvania State University. He is a fellow of SPIE, Optical Society of America, American Association for the Advancement of Science, American Physical Society, and Institute of Physics (UK). He was the sole recipient of the 2010 SPIE Technical Achievement Award. He served as the founding editor-in-chief of the Journal of Nanophotonics from 2006 to 2013. His current research interests include nanophotonics, surface multiplasmonics, complex materials including multifunctional materials and sculptured thin films, bone refacing, bioreplication, and forensic science.

© The Authors. Published by SPIE under a Creative Commons Attribution 3.0 Unported License. Distribution or reproduction of this work in whole or in part requires full attribution of the original publication, including its DOI.

Citation

Akhlesh Lakhtakia
"Exhibition of circular Bragg phenomenon by hyperbolic, dielectric, structurally chiral materials", J. Nanophoton. 8(1), 083998 (Feb 27, 2014). ; http://dx.doi.org/10.1117/1.JNP.8.083998


Figures

Graphic Jump LocationF1 :

Cross-sectional scanning electron micrograph of a chiral sculptured thin film made by thermal evaporation of a chalcogenide glass.

Graphic Jump LocationF2 :

Reflectances RLL, RRL, RRR, and RLR as functions of λ0/2Ω and θ, when εa=3.26(1+0.001i), εb=4.46(1+0.001i), εc=3.78(1+0.001i), χ=60deg, and h=1. The color coding employs the spectrum of the rainbow with the deepest violet denoting 0 and the deepest red denoting 1.0.

Graphic Jump LocationF6 :

Same as Fig. 5, except εa=3.26(1+0.001i), εb=4.46(1+0.001i), and εc=3.78(1+0.001i).

Graphic Jump LocationF5 :

Reflectances RLL, RRL, RRR, and RLR as functions of χ and θ, when εa=3.26(1+0.001i), εb=4.46(1+0.001i), εc=3.78(1+0.001i), λ0/2Ω=1.918, and h=1. The color coding employs the spectrum of the rainbow with the deepest violet denoting 0 and the deepest red denoting 1.0. The thin white strips for θ<20deg and χ40deg indicate a failure of the computational algorithm (30).

Graphic Jump LocationF4 :

Same as Fig. 2, except that εa=3.26(1+0.001i), εc=3.78(1+0.001i), and χ=15deg.

Graphic Jump LocationF3 :

Same as Fig. 2, except that εb=4.46(1+0.001i).

Tables

References

Chandrasekhar  S., Liquid Crystals. , 2nd ed.,  Cambridge University ,  Cambridge  (1992).
de Gennes  P. G., Prost  J. A., The Physics of Liquid Crystals. , 2nd ed.,  Clarendon ,  Oxford  (1993).
Young  N. O., Kowal  J., “Optically active fluorite films,” Nature. 183, (4654 ), 104 –105 (1959), CrossRef. 0028-0836 
Lakhtakia  A., Messier  R., Chapter 9 in Sculptured Thin Films: Nanoengineered Morphology and Optics. ,  SPIE Press ,  Bellingham, WA  (2005).
Lütkepohl  H., Chapter 9 in Handbook of Matrices. ,  Wiley ,  Chichester  (1996).
Wu  Q., Hodgkinson  I. J., Lakhtakia  A., “Circular polarization filters made of chiral sculptured thin films: experimental and simulation results,” Opt. Eng.. 39, (7 ), 1863 –1868 (2000), CrossRef. 0091-3286 
Sherwin  J. A., Lakhtakia  A., Hodgkinson  I. J., “On calibration of a nominal structure-property relationship model for chiral sculptured thin films by axial transmittance measurements,” Opt. Commun.. 209, (4–6 ), 369 –375 (2002), CrossRef. 0030-4018 
Mackay  T. G., Lakhtakia  A., “Modeling chiral sculptured thin films as platforms for surface-plasmonic-polaritonic optical sensing,” IEEE Sensors J.. 12, (2 ), 273 –280 (2012), CrossRef. 1530-437X 
Fergason  J. L., “Cholesteric structure—I Optical properties,” Mol. Cryst.. 1, (2 ), 293 –307 (1966), CrossRef. 0369-1152 
John  W. D. St. et al., “Bragg reflection from cholesteric liquid crystals,” Phys. Rev. E. 51, (2 ), 1191 –1198 (1995), CrossRef. 1063-651X 
Hodgkinson  I. J. et al., “Spacerless circular-polarization spectral-hole filters using chiral sculptured thin films: theory and experiment,” Opt. Commun.. 184, (1–4 ), 57 –66 (2000), CrossRef. 0030-4018 
Smith  D. R., Kolinko  P., Schurig  D., “Negative refraction in indefinite media,” J. Opt. Soc. Am. B. 21, (5 ), 1032 –1043 (2004), CrossRef. 0740-3224 
Depine  R. A., Inchaussandague  M. E., Lakhtakia  A., “Classification of dispersion equations for homogeneous, dielectric-magnetic, uniaxial materials,” J. Opt. Soc. Am. A. 23, (4 ), 949 –955 (2006), CrossRef. 0740-3232 
Sun  J. et al., “Indefinite permittivity and negative refraction in natural material: graphite,” Appl. Phys. Lett.. 98, (10 ), 101901  (2011), CrossRef. 0003-6951 
Gerbaux  X., Tazawa  M., Hadni  A., “Far IR transmission measurements on triglycine sulphate (TGS), at 5 K,” Ferroelectrics. 215, (1 ), 47 –63 (1998), CrossRef. 0015-0193 
Alekseyev  L. V., Podolskiy  V. A., Narimanov  E. E., “Homogeneous hyperbolic systems for terahertz and far-infrared frequencies,” Adv. Optoelectron.. 2012, , 267564  (2012), CrossRef. 1687-563X 
Schubert  M., Tiwald  T. E., Herzinger  C. M., “Infrared dielectric anisotropy and phonon modes of sapphire,” Phys. Rev. B. 61, (12 ), 8187 –8201 (2000), CrossRef. 1098-0121 
Kanungo  J., Schilling  J., “Experimental determination of the principal dielectric functions in silver nanowire metamaterials,” Appl. Phys. Lett.. 97, (2 ), 021903  (2010), CrossRef. 0003-6951 
Kidwai  O., Zhukovsky  S. V., Sipe  J. E., “Effective-medium approach to planar multilayer hyperbolic metamaterials: strengths and limitations,” Phys. Rev. A. 85, (5 ), 053842  (2012), CrossRef. 1050-2947 
Cortes  C. L. et al., “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt.. 14, (6 ), 063001  (2012), CrossRef. 0150-536X 
Othman  M. A. K., Guclu  C., Capolino  F., “Graphene-dielectric composite metamaterials: evolution from elliptic to hyperbolic wavevector dispersion and the transverse epsilon-near-zero condition,” J. Nanophotonics. 7, (1 ), 073089  (2013), CrossRef. 1934-2608 
Ni  X. et al., “Loss-compensated and active hyperbolic metamaterials,” Opt. Express. 19, (25 ), 25242 –25254 (2011), CrossRef. 1094-4087 
Mackay  T. G., Lakhtakia  A., “On the application of homogenization formalisms to active dielectric composite materials,” Opt. Commun.. 282, (13 ), 2470 –2475 (2009), CrossRef. 0030-4018 
Gerritsen  H. J., Yamaguchi  R. T., “A microwave analog of optical rotation in cholesteric liquid crystals,” Am. J. Phys.. 39, (8 ), 920 –923 (1971), CrossRef. 0002-9505 
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