Letters

Left/right asymmetry in reflection and transmission by a planar anisotropic dielectric slab with topologically insulating surface states

[+] Author Affiliations
Akhlesh Lakhtakia

Pennsylvania State University, Department of Engineering Science and Mechanics, University Park, Pennsylvania 16802, United States

Tom G. Mackay

Pennsylvania State University, Department of Engineering Science and Mechanics, University Park, Pennsylvania 16802, United States

University of Edinburgh, School of Mathematics and Maxwell Institute for Mathematical Sciences, Edinburgh EH9 3FD, Scotland, United Kingdom

J. Nanophoton. 10(2), 020501 (Jun 22, 2016). doi:10.1117/1.JNP.10.020501
History: Received April 11, 2016; Accepted May 31, 2016
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Abstract.  The reflection and transmission of plane waves by a homogeneous anisotropic dielectric slab—as represented by a columnar thin film—with topologically insulating surface states was theoretically investigated. Copolarized and cross-polarized reflectances and transmittances were calculated by solving the associated boundary-value problem. Numerical calculations revealed that all four reflectances and all four transmittances were asymmetric with respect to reversal of projection of the propagation direction of the incident plane wave on the illuminated surface of the slab. This left/right reflection and transmission asymmetry arises due to the combined effects of the slab’s anisotropy and surface states.

Figures in this Article

The modest aim of this communication is to theoretically show that a homogeneous anisotropic dielectric slab with topologically insulating surface states (TISS)1,2 reflects and transmits light in such a way as to exhibit asymmetry with respect to reversal of projection of the propagation direction of the incident plane wave on the illuminated surface of the slab. We refer to this phenomenon as left/right asymmetry.

A homogeneous anisotropic material is characterized by a frequency-dependent relative permittivity dyadic ϵ__. Suppose that this material occupies the region Vin bounded by the surface S, which separates Vin from the vacuous region Vout. If the anisotropic material possesses TISS, then the boundary conditions3Display Formula

n^(rS)×[Eout(rS)Ein(rS)]=0n^(rS)×[Hout(rS)Hin(rS)]=γ˜n^(rS)×Ein(rS)},rSS,(1)
hold, with the unit normal vector n^(rS) at rSS pointing into Vout and the admittance γ˜ describing the TISS. Although optical scattering by isotropic dielectric materials, i.e., ϵ__=ϵ(u^xu^x+u^yu^y+u^zu^z), with TISS has been investigated theoretically47 as well as experimentally,2,8 this communication is possibly the first report of optical scattering by an anisotropic dielectric material with TISS. The existence of such materials is deemed possible because the isotropic materials with TISS are chalcogenides,1,2 columnar thin films (CTFs) of other chalcogenides have been fabricated,9 and CTFs function as anisotropic dielectric materials at sufficiently low frequencies.10 Furthermore, periodically multilayered composite materials11,12 comprising laminas of an isotropic topological insulator and some other material should function as effectively anisotropic continuums at sufficiently low frequencies.13

We have found that the TISS induce the exhibition of left/right asymmetry in reflection and transmission by a homogeneous anisotropic dielectric slab. This asymmetry could be exploited for one-way optical devices. The boundary-value problem of reflection and transmission of an obliquely incident plane wave by a homogeneous anisotropic dielectric slab with TISS is described and solved in Sec. 2. Illustrative numerical results are presented and discussed in Sec. 3.

The free-space wavenumber, the free-space wavelength, and the intrinsic impedance of free space are denoted by k0=ωϵ0μ0, λ0=2π/k0, and η0=μ0/ϵ0, respectively, with μ0 and ϵ0 being the permeability and permittivity of free space. We denote the fine structure constant by α˜=(qe2η0)/2h˜, where qe is the quantum of charge and h˜ is the Planck constant. Vectors are in boldface, dyadics are underlined twice, column vectors are in boldface and enclosed within square brackets, while matrixes are underlined twice and similarly bracketed. Cartesian unit vectors are identified as u^x, u^y, and u^z.

Suppose that the regions Vin={(x,y,z):z(0,L)} and Vout={(x,y,z):z[0,L]} are separated by the surface S={(x,y,z):z{0,L}}.

A plane wave, propagating in the half-space z<0 at an angle θ[0,π/2) to the z-axis and at an angle ψ[0,2π) to the x-axis in the xy plane, is incident on the slab, as shown in Fig. 1. The electromagnetic field phasors associated with the incident plane wave are represented as Display Formula

Einc(r)=(ass+app+)exp[iκ(xcosψ+ysinψ)+ik0zcosθ]Hinc(r)=1η0(asp+aps)exp[iκ(xcosψ+ysinψ)+ik0zcosθ]},z<0.(2)
The amplitudes of the s- and the p-polarized components of the incident plane wave, denoted by as and ap, respectively, are assumed given, whereas Display Formula
κ=k0sinθ,s=u^xsinψ+u^ycosψp±=(u^xcosψ+u^ysinψ)cosθ+u^zsinθ}.(3)

Graphic Jump Location
Fig. 1
F1 :

A plane wave is incident on the slab Vin={(x,y,z):z(0,L)}; the wave vector of the incident plane wave inclined at an angle θ with respect to the z-axis and at an angle ψ with respect to the x-axis in the xy plane. Also shown is the angle χ.

The reflected electromagnetic field phasors are expressed as Display Formula

Eref(r)=(rss+rpp)exp[iκ(xcosψ+ysinψ)ik0zcosθ]Href(r)=1η0(rsprps)exp[iκ(xcosψ+ysinψ)ik0zcosθ]},z<0,(4)
and the transmitted electromagnetic field phasors as Display Formula
Etr(r)=(tss+tpp+)exp[iκ(xcosψ+ysinψ)+ik0(zL)cosθ]Htr(r)=1η0(tsp+tps)exp[iκ(xcosψ+ysinψ)+ik0(zL)cosθ]},z>L.(5)
The reflection amplitudes rs and rp, as well as the transmission amplitudes ts and tp, have to be determined by the solution of a boundary-value problem.

The frequency-domain electromagnetic constitutive relations of the homogeneous anisotropic dielectric material in Vin can be written as10Display Formula

D(r)=ϵ0ϵ__·E(r),B(r)=μ0H(r),z(0,L),(6)
where the dyadics Display Formula
ϵ__=S__y·(ϵau^zu^z+ϵbu^xu^x+ϵcu^yu^y)·S__yT,(7)
Display Formula
S__y=(u^xu^x+u^zu^z)cosχ+(u^zu^xu^xu^z)sinχ+u^yu^y(8)
involve the angle χ[0,π/2]. The superscript T denotes the transpose. The principal relative permittivity scalars ϵa, ϵb, and ϵc, as well as the angle χ, can be chosen for application to either natural crystals14 or the manufactured CTFs.10

In Vin, the electric and magnetic field phasors can be represented as15Display Formula

E(r)=e(z)exp[iκ(xcosψ+ysinψ)]H(r)=h(z)exp[iκ(xcosψ+ysinψ)]},(9)
where the vector functions e(z) and h(z) are unknown. Substitution of Eqs. (6) and (9) in the Maxwell curl postulates followed by certain algebraic manipulations leads to the 4×4-matrix ordinary differential equations Display Formula
ddz[f(z)]=i[P__]·[f(z)],zVin,(10)
where the column vector Display Formula
[f(z)]=[ex(z),ey(z),hx(z),hy(z)]T,(11)
the 4×4 matrix Display Formula
[P__]=ω[000μ000μ000ϵ0ϵc00ϵ0ϵd000]+κϵd(ϵaϵb)2ϵaϵbsin(2χ)[cosψ000sinψ000000000sinψcosψ]+κ2ωϵ0ϵdϵaϵb[00cosψsinψcos2ψ00sin2ψcosψsinψ00000000]+κ2ωμ0[00000000cosψsinψcos2ψ00sin2ψcosψsinψ00],(12)
and the scalarDisplay Formula
ϵd=ϵaϵbϵacos2χ+ϵbsin2χ.(13)

Equation (10) has the following straightforward solution: Display Formula

[f(L)]=exp{i[P__]L}·[f(0+)],(14)
where the notation [f(a±)] stands for limδ0[f(a±δ)] with δ0. Application of the boundary conditions [Eq. (1)] to the planes z=0 and z=L leads to Display Formula
[f(0)]=[V__]·[f(0+)],(15)
Display Formula
[f(L+)]=[V__]·[f(L)],(16)
respectively, where the matrix Display Formula
[V__]=[10000100γ˜0100γ˜01].(17)
Combining Eqs. (14)–(16), we get Display Formula
[f(L+)]=[V__]·exp{i[P__]L}·[V__]1·[f(0)].(18)

But the elements of [f(0)] are known by virtue of Eqs. (2) and (4), and those of [f(L+)] by virtue of Eq. (5). Accordingly, Eq. (18) may be written as Display Formula

[tstp00]=[K__]1·[V__]·exp{i[P__]L}·[V__]1·[K__]·[asaprsrp],(19)
where Display Formula
[K__]=[sinψcosψcosθsinψcosψcosθcosψsinψcosθcosψsinψcosθ(1η0)cosψcosθ(1η0)sinψ(1η0)cosψcosθ(1η0)sinψ(1η0)sinψcosθ(1η0)cosψ(1η0)sinψcosθ(1η0)cosψ].(20)

The solution of Eq. (19) yields the reflection and transmission coefficients that appear as the elements of the 2×2 matrixes in the following relations: Display Formula

[rsrp]=[rssrsprpsrpp][asap],[tstp]=[tsstsptpstpp][asap].(21)
Copolarized coefficients have both subscripts identical, but cross-polarized coefficients do not. The square of the magnitude of a reflection or transmission coefficient is the corresponding reflectance or transmittance; thus, Rsp=|rsp|2 is the reflectance corresponding to the reflection coefficient rsp, and so on. The principle of conservation of energy mandates the constraints Rss+Rps+Tss+Tps1 and Rpp+Rsp+Tpp+Tsp1. Let us note here that a real-valued γ˜ does not cause dissipation.

Let the left side of the xy plane be specified by ψ[0,π] and the right side by ψ[π,2π]. In order to delineate the characteristics of and the factors responsible for left/right asymmetry of reflection and transmission, we need to consider four distinct cases as follows.

Case I: Suppose that ϵa=ϵb=ϵc and γ˜=0. Then the material in Vin is a homogeneous isotropic dielectric material and the TISS are absent. The boundary-value problem then turns into a textbook reflection/transmission problem.16 None of the four reflectances (Rss, Rps, Rpp, and Rss) and the four transmittances (Tss, Tps, Tpp, and Tss) then depend on ψ. Furthermore, the cross-polarized remittances are null valued. In other words, the following relationships hold: Display Formula

Rss(θ,ψ)=Rss(θ,0),Rpp(θ,ψ)=Rpp(θ,0)Tss(θ,ψ)=Tss(θ,0),Tpp(θ,ψ)=Tpp(θ,0)Rps(θ,ψ)=Rsp(θ,ψ)0,Tps(θ,ψ)=Tsp(θ,ψ)0}.(22)
In particular, all reflectances and transmittances are unchanged upon replacing ψ with ψ+π; i.e., all are left/right symmetric.

Case II: Suppose next that ϵa=ϵb=ϵc but γ˜0. Then the material in Vin is a homogeneous isotropic dielectric material with TISS. None of the eight remittances (Rss, and so on, and Tss, and so on) then depend on ψ, and the cross-polarized remittances are not identically zero. Analysis of numerical results reveals that the following relationships hold: Display Formula

Rss(θ,ψ)=Rss(θ,0),Rpp(θ,ψ)=Rpp(θ,0)Tss(θ,ψ)=Tss(θ,0),Tpp(θ,ψ)=Tpp(θ,0)Rps(θ,ψ)=Rsp(θ,ψ)0,Tps(θ,ψ)=Tsp(θ,ψ)0}.(23)
A comparison of Eqs. (22) and (23) indicates that the TISS are responsible for de-polarization on both reflection and transmission. As in case I, all eight remittances are left/right symmetric.

Case III: Suppose that ϵa, ϵb, and ϵc are all dissimilar, but γ˜=0. Then the material in Vin is a homogeneous anisotropic dielectric material and the TISS are absent. Calculations then show the following symmetries: Display Formula

Rss(θ,ψ)=Rss(θ,ψ+π),Rpp(θ,ψ)=Rpp(θ,ψ+π)Rps(θ,ψ)=Rsp(θ,ψ+π)0,Tps(θ,ψ)=Tsp(θ,ψ)0}.(24)
While both Rss and Rpp are left/right symmetric, both Tss and Tpp are not. Furthermore, as Rps(θ,ψ)Rsp(θ,ψ) and Tps(θ,ψ)Tsp(θ,ψ+π), it follows that all cross-polarized remittances are left/right asymmetric. In summary, the following inequalities are entirely due to anisotropy: Display Formula
Rps(θ,ψ)Rps(θ,ψ+π),Rsp(θ,ψ)Rsp(θ,ψ+π)Tss(θ,ψ)Tss(θ,ψ+π),Tpp(θ,ψ)Tpp(θ,ψ+π)Tps(θ,ψ)Tps(θ,ψ+π),Tsp(θ,ψ)Tsp(θ,ψ+π)}.(25)

Case IV: Finally, ϵa, ϵb, and ϵc are all dissimilar and γ˜0, so that the material in Vin is a homogeneous anisotropic dielectric material with TISS. All eight remittances then depend on ψ, the cross-polarized remittances are not identically zero, and only one relationship can be found. Display Formula

Tps(θ,ψ)=Tsp(θ,ψ)0.(26)
All eight remittances exhibit left/right asymmetry, i.e., Display Formula
Rss(θ,ψ)Rss(θ,ψ+π),Rpp(θ,ψ)Rpp(θ,ψ+π)Rps(θ,ψ)Rps(θ,ψ+π),Rsp(θ,ψ)Rsp(θ,ψ+π)Tss(θ,ψ)Tss(θ,ψ+π),Tpp(θ,ψ)Tpp(θ,ψ+π)Tps(θ,ψ)Tps(θ,ψ+π),Tsp(θ,ψ)Tsp(θ,ψ+π)}.(27)
Furthermore, by comparison with cases I to III, we deduce that these inequalities arise due to the combined effects of the slab’s anisotropy and the presence of TISS. Without anisotropy, cases I and II show that the TISS are responsible for the cross-polarized reflectances and transmittances, but do not give rise to left/right asymmetry. Without TISS, case III shows that only the two copolarized transmittances (out of the eight remittances) are left/right asymmetric when the slab is made of an anisotropic material.

The complete left/right asymmetry that arises for case IV is illustrated in Figs. 2 and 3, wherein all reflectances and transmittances, respectively, are plotted as functions of the incidence angles θ[0,π/2) and ψ[0,2π). For these representative calculations, we chose ϵa=2.14, ϵb=3.67, ϵc=2.83, χ=38  deg, γ˜=100α˜/η0, and L=1.4λ0. The chosen values of ϵa,b,c emerged from a homogenization model for dielectric CTFs,15 whereas γ˜1719 was chosen to clearly highlight left/right asymmetry, in the absence of any experimental data for anisotropic topological insulators.

Graphic Jump Location
Fig. 2
F2 :

Reflectances Rss, Rps, Rpp, and Rsp as functions of the incidence angles θ[0,π/2) and ψ[0,2π) when ϵa=2.14, ϵb=3.67, ϵc=2.83, χ=38  deg, γ˜=100α˜/η0, and L=1.4λ0. The color coding employs the spectrum of the rainbow with the deepest violet denoting 0 and the deepest red denoting 1.0.

Graphic Jump Location
Fig. 3
F3 :

Same as Fig. 2, except that the transmittances Tss, Tps, Tpp, and Tsp are displayed as functions of θ and ψ.

The inequalities in Eq. (27) are readily observed in the two figures. The left/right asymmetry is most easily discernible in the plots of Rss (Fig. 2) and Tpp (Fig. 3), but can be identified in the plots of the remaining six remittances too for mid-range values of θ.

In further numerical calculations (not presented here), the left/right asymmetry was found to be even more conspicuous for various remittances, when the magnitude of γ˜ was increased. Intrinsic topological insulators are characterized by γ˜=±α˜/η0,1 but a very thin coating of a magnetic material can be used to realize γ˜=(2q+1)α˜/η0, q{0,±1,±2,±3,}.17,18 Values of q other than 1 and 0 can also be obtained by immersing a topological insulator in a magnetostatic field.19

Practically oriented research on topological insulators is embryonic though steady progress is being made in the identification of several relevant materials.2,20 As stated in Sec. 1, attention is chiefly being given to isotropic topological insulators, although the fabrication of anisotropic topological insulators appears possible. The exploitation of left/right asymmetry theoretically shown here to be possible with anisotropic topological insulators is promising for one-way optical devices, which could reduce backscattering noise21 in optical communication networks, microscopy, and tomography, for example. But high magnitudes of η0γ˜/α˜ would be needed for practical implementation.

A.L. is grateful to the Charles Godfrey Binder Endowment at Penn State for the ongoing support of his research. T.G.M. acknowledges the support of EPSRC Grant EP/M018075/1.

Hasan  M. Z., and Kane  C. L., “Topological insulators,” Rev. Modern Phys.. 82, (4 ), 3045 –3067 (2010). 0034-6861 CrossRef
Di Pietro  P., Optical Properties of Bismuth-Based Topological Insulators. ,  Springer ,  Cham ,  Switzerland  (2014).
Lakhtakia  A., and Mackay  T. G., “Classical electromagnetic model of surface states in topological insulators,” J. Nanophoton.. 10, (3 ), 033004  (2016). 1934-2608 CrossRef
Chang  M.-C., and Yang  M.-F., “Optical signature of topological insulators,” Phys. Rev. B. 80, (11 ), 113304  (2009).CrossRef
Liu  F.  et al., “Goos–Hänchen and Imbert–Fedorov shifts at the interface of ordinary dielectric and topological insulator,” J. Opt. Soc. Am. B. 30, (5 ), 735 –741 (2013). 0740-3224 CrossRef
Liu  F., , Xu  J., and Yang  Y., “Polarization conversion of reflected electromagnetic wave from topological insulator,” J. Opt. Soc. Am. B. 31, (4 ), 735 –741 (2014). 0740-3224 CrossRef
Lakhtakia  A., and Mackay  T. G., “Electromagnetic scattering by homogeneous, isotropic, dielectric-magnetic sphere with topologically insulating surface states,” J. Opt. Soc. Am. B. 33, (4 ), 603 –609 (2016). 0740-3224 CrossRef
LaForge  A. D.  et al., “Optical characterization of Bi2Se3 in a magnetic field: infrared evidence for magnetoelectric coupling in a topological insulator material,” Phys. Rev. B. 81, (12 ), 125120  (2010).CrossRef
Martín-Palma  R. J.  et al., “Retardance of chalcogenide thin films grown by the oblique-angle-deposition technique,” Thin Solid Films. 517, (10 ), 5553 –5556 (2009). 0040-6090 CrossRef
Hodgkinson  I. J., and Wu  Q. H., Birefringent Thin Films and Polarizing Elements. ,  World Scientific ,  Singapore  (1997).
Abelès  F., “Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés. Application aux couches minces (partie),” Ann. Phys. (Paris). 5, , 596 –640 (1950). 0003-4169 
Baumeister  P. W., Optical Coating Technology. ,  SPIE ,  Bellingham ,  Washington  (2004).
Lakhtakia  A., “Constraints on effective constitutive parameters of certain bianisotropic laminated composite materials,” Electromagnetics. 29, (6 ), 508 –514 (2009). 0272-6343 CrossRef
Gribble  C. D., and Hall  A. J., Optical Mineralogy: Principles and Practice. ,  University College Press ,  London ,  United Kingdom  (1993).
Lakhtakia  A., and Messier  R., Sculptured Thin Films: Nanoengineered Morphology and Optics. ,  SPIE Press ,  Bellingham ,  Washington  (2005).
Born  M., and Wolf  E., Principles of Optics. , 7th (expanded) ed.,  Cambridge University Press ,  Cambridge ,  United Kingdom  (1999).
Qi  X.-L., , Hughes  T. L., and Zhang  S.-C., “Topological field theory of time-reversal invariant insulators,” Phys. Rev. B. 78, (19 ), 195424  (2008).CrossRef
Qi  X.-L., , Hughes  T. L., and Zhang  S.-C., “Erratum: topological field theory of time-reversal invariant insulators [Phys. Rev. B 78, 195424 (2008)],” Phys. Rev. B. 81, (15 ), 155901  (2010).CrossRef
Maciejko  J.  et al., “Topological quantization in units of the fine structure constant,” Phys. Rev. Lett.. 105, (16 ), 166803  (2010). 0031-9007 CrossRef
Amaricci  A.  et al., “First-order character and observable signatures of topological quantum phase transitions,” Phys. Rev. Lett.. 114, (18 ), 185701  (2015). 0031-9007 CrossRef
Sayrin  C.  et al., “Nanophotonic optical isolator controlled by the internal state of cold atoms,” Phys. Rev. X. 5, (4 ), 041036  (2015). 2160-3308 CrossRef

Akhlesh Lakhtakia received degrees from the Banaras Hindu University and the University of Utah. He is the Charles Godfrey Binder Professor of Engineering Science and Mechanics at the Pennsylvania State University. His research interests include surface multiplasmonics, biorepli-cation, forensic science, solar energy, sculptured thin films, and mimumes. He is a fellow of OSA, SPIE, IoP, AAAS, APS, and IEEE. He received the 2010 SPIE Technical Achievement Award and the 2016 Walston Chubb Award for Innovation.

Tom G. Mackay is a reader in the School of Mathematics at the University of Edinburgh and an adjunct professor in the Department of Engineering Science and Mechanics at the Pennsylvania State University. He is a graduate of the Universities of Edinburgh, Glasgow, and Strathclyde, and a fellow of the Institute of Physics (UK) and SPIE. His research interests include the electromagnetic theory of novel and complex materials, including homogenized composite materials.

© 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 and Tom G. Mackay
"Left/right asymmetry in reflection and transmission by a planar anisotropic dielectric slab with topologically insulating surface states", J. Nanophoton. 10(2), 020501 (Jun 22, 2016). ; http://dx.doi.org/10.1117/1.JNP.10.020501


Figures

Graphic Jump Location
Fig. 1
F1 :

A plane wave is incident on the slab Vin={(x,y,z):z(0,L)}; the wave vector of the incident plane wave inclined at an angle θ with respect to the z-axis and at an angle ψ with respect to the x-axis in the xy plane. Also shown is the angle χ.

Graphic Jump Location
Fig. 3
F3 :

Same as Fig. 2, except that the transmittances Tss, Tps, Tpp, and Tsp are displayed as functions of θ and ψ.

Graphic Jump Location
Fig. 2
F2 :

Reflectances Rss, Rps, Rpp, and Rsp as functions of the incidence angles θ[0,π/2) and ψ[0,2π) when ϵa=2.14, ϵb=3.67, ϵc=2.83, χ=38  deg, γ˜=100α˜/η0, and L=1.4λ0. The color coding employs the spectrum of the rainbow with the deepest violet denoting 0 and the deepest red denoting 1.0.

Tables

References

Hasan  M. Z., and Kane  C. L., “Topological insulators,” Rev. Modern Phys.. 82, (4 ), 3045 –3067 (2010). 0034-6861 CrossRef
Di Pietro  P., Optical Properties of Bismuth-Based Topological Insulators. ,  Springer ,  Cham ,  Switzerland  (2014).
Lakhtakia  A., and Mackay  T. G., “Classical electromagnetic model of surface states in topological insulators,” J. Nanophoton.. 10, (3 ), 033004  (2016). 1934-2608 CrossRef
Chang  M.-C., and Yang  M.-F., “Optical signature of topological insulators,” Phys. Rev. B. 80, (11 ), 113304  (2009).CrossRef
Liu  F.  et al., “Goos–Hänchen and Imbert–Fedorov shifts at the interface of ordinary dielectric and topological insulator,” J. Opt. Soc. Am. B. 30, (5 ), 735 –741 (2013). 0740-3224 CrossRef
Liu  F., , Xu  J., and Yang  Y., “Polarization conversion of reflected electromagnetic wave from topological insulator,” J. Opt. Soc. Am. B. 31, (4 ), 735 –741 (2014). 0740-3224 CrossRef
Lakhtakia  A., and Mackay  T. G., “Electromagnetic scattering by homogeneous, isotropic, dielectric-magnetic sphere with topologically insulating surface states,” J. Opt. Soc. Am. B. 33, (4 ), 603 –609 (2016). 0740-3224 CrossRef
LaForge  A. D.  et al., “Optical characterization of Bi2Se3 in a magnetic field: infrared evidence for magnetoelectric coupling in a topological insulator material,” Phys. Rev. B. 81, (12 ), 125120  (2010).CrossRef
Martín-Palma  R. J.  et al., “Retardance of chalcogenide thin films grown by the oblique-angle-deposition technique,” Thin Solid Films. 517, (10 ), 5553 –5556 (2009). 0040-6090 CrossRef
Hodgkinson  I. J., and Wu  Q. H., Birefringent Thin Films and Polarizing Elements. ,  World Scientific ,  Singapore  (1997).
Abelès  F., “Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés. Application aux couches minces (partie),” Ann. Phys. (Paris). 5, , 596 –640 (1950). 0003-4169 
Baumeister  P. W., Optical Coating Technology. ,  SPIE ,  Bellingham ,  Washington  (2004).
Lakhtakia  A., “Constraints on effective constitutive parameters of certain bianisotropic laminated composite materials,” Electromagnetics. 29, (6 ), 508 –514 (2009). 0272-6343 CrossRef
Gribble  C. D., and Hall  A. J., Optical Mineralogy: Principles and Practice. ,  University College Press ,  London ,  United Kingdom  (1993).
Lakhtakia  A., and Messier  R., Sculptured Thin Films: Nanoengineered Morphology and Optics. ,  SPIE Press ,  Bellingham ,  Washington  (2005).
Born  M., and Wolf  E., Principles of Optics. , 7th (expanded) ed.,  Cambridge University Press ,  Cambridge ,  United Kingdom  (1999).
Qi  X.-L., , Hughes  T. L., and Zhang  S.-C., “Topological field theory of time-reversal invariant insulators,” Phys. Rev. B. 78, (19 ), 195424  (2008).CrossRef
Qi  X.-L., , Hughes  T. L., and Zhang  S.-C., “Erratum: topological field theory of time-reversal invariant insulators [Phys. Rev. B 78, 195424 (2008)],” Phys. Rev. B. 81, (15 ), 155901  (2010).CrossRef
Maciejko  J.  et al., “Topological quantization in units of the fine structure constant,” Phys. Rev. Lett.. 105, (16 ), 166803  (2010). 0031-9007 CrossRef
Amaricci  A.  et al., “First-order character and observable signatures of topological quantum phase transitions,” Phys. Rev. Lett.. 114, (18 ), 185701  (2015). 0031-9007 CrossRef
Sayrin  C.  et al., “Nanophotonic optical isolator controlled by the internal state of cold atoms,” Phys. Rev. X. 5, (4 ), 041036  (2015). 2160-3308 CrossRef

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