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1.IntroductionAnticipating the discovery of cholesteric liquid crystals by about two decades,1,2 Reusch3 proposed in 1869 that a periodically nonhomogeneous multilayered material reflects normally incident circularly polarized light of one handedness, but not of the opposite handedness, provided that all layers are made of the same homogeneous, uniaxial dielectric material such that the optic axis in each layer is rotated about the thickness direction with respect to the optic axis in the adjacent layer by a fixed angle. Such a periodically nonhomogeneous dielectric material is nowadays called a Reusch pile. Extensive theoretical and experimental work by Joly and colleagues4–7 showed that the circular-polarization-selective reflection of normally incident light by a Reusch pile may occur in several spectral regimes. This selective reflection of circularly polarized light of one handedness, but very little of the other, in a given spectral regime is commonly called the circular Bragg phenomenon.8,9 A classification scheme based on the number of layers in each period of a Reusch pile was developed by Hodgkinson et al.10 If , the Reusch pile is classified as an equichiral material; if , but not very large, it can be called an ambichiral material; and if , it is a finely chiral material. Equichiral materials do not exhibit the circular Bragg phenomenon. Ambichiral materials may exhibit the circular Bragg phenomenon in several spectral regimes, depending on the variations of their constitutive parameters with frequency. A cholesteric liquid crystal11 can be considered as a finely chiral Reusch pile made of uniaxial dielectric layers. Reusch piles can also be made of biaxial dielectric material such as columnar thin films (CTFs).12 A chiral sculptured thin film (STF)8 can be considered to be a finely chiral Reusch pile comprising biaxial CTFs. Chiral STFs were first fabricated by Young and Kowal13 in 1959 and were rediscovered in the 1990s.14 They have been extensively studied since then for optical applications exploiting the circular Bragg phenomenon.8,9 The effect of the number of layers in a period on the circular Bragg phenomenon has been studied.15 Both and the total number of periods have to be substantially large for the circular Bragg phenomenon to fully develop.15 Now, the planar interface of an isotropic homogeneous metal and an ambichiral dielectric material can guide surface-plasmon-polariton (SPP) waves. The planar interface of an isotropic, homogeneous dielectric material and an ambichiral dielectric material can guide Dyakonov–Tamm waves. What is the effect of on both types of surface waves? The results reported in this communication elucidate the evolution of the solution(s) of the dispersion equation for surface-wave propagation with . The plan of this communication is as follows. Section 2 succinctly presents the common formulation of the canonical boundary-value problem for both types of surface waves. Numerical results are presented and discussed in Sec. 3. An dependence on time is implicit, with denoting the angular frequency and . Furthermore, and , respectively, represent the free-space wavenumber and free-space wavelength, where is the permeability and is the permittivity of free space. Vectors are in boldface, dyadics are double-underlined, and the three Cartesian unit vectors are denoted by , , and . 2.Theoretical PreliminariesThe canonical boundary-value problem of surface-wave propagation is depicted schematically in Fig. 1. The half space is occupied by an isotropic and homogeneous material with relative permittivity . The half space is occupied by an ambichiral dielectric material comprising homogeneous layers each of thickness , the ’th layer occupying the region , . The relative permittivity dyadic is given as where the reference permittivity dyadic contains the eigenvalues of . The dyadic depends on the tilt angle with respect to the plane, the dyadic represents a rotation about the axis by an angle , with being the number of layers in each period , is the half period, right-handed rotation is represented by and left-handed rotation by , and is an angular offset with respect to the axis.In the region , the electric field phasor may be written as16–18 where , , is the complex-valued wavenumber of the surface wave, for attenuation as , and and are unknown scalars with the same units as the electric field.For field representation in the region , let us write18 The Cartesian components of the electric and magnetic field phasors tangential to the plane are used to form the column vector which satisfies the matrix differential equation where the matrix depends not only on but also on .The piecewise-uniform approximation technique8 can be used to determine the matrix that appears in the relation for specific values of . Let , , be the eigenvector corresponding to the ’th eigenvalue of . After ensuring that , we set for surface-wave propagation, where and are unknown dimensionless scalars; the other two eigenvalues of pertain to waves that amplify as and cannot therefore contribute to the surface wave. At the same time, can be obtained from Eq. (5) and the corresponding magnetic field phasor .Enforcement of the usual boundary conditions across the plane requires that , which may be rearranged as the matrix equation leading to the dispersion equation3.Numerical Results and DiscussionThe dispersion equation (12) was solved using the Newton–Raphson method,19 with fixed at 633 nm. For all numerical results presented here, the ambichiral dielectric material was taken to comprise CTFs made by directing a collimated evaporant flux of patinal titanium oxide in a low-pressure chamber at a fixed angle with respect to a planar substrate.20 For the chosen CTF, according to Hazel and co-workers.20 We fixed , while varying (so that was simultaneously varied) and . Furthermore, was fixed so that , , and .3.1.Surface-Plasmon-Polariton WavesLet the isotropic homogeneous partnering material be thin-film aluminum with (Ref. 21) at . Then, all solutions of the dispersion equation (12) represent SPP waves.16,18 These solutions are complex valued because is complex valued. Only one solution of the dispersion equation exists for any when the anisotropic partnering material is homogeneous (i.e., ).22 However, when that partnering material is periodically nonhomogeneous (i.e., ), the solutions can be organized into two branches for and 45 deg but only one for , as shown in Fig. 2. Convergence on either branch is monotonic for . The most notable feature of the solutions presented in Fig. 2 is their evolution and convergence as increases. Although not shown in this figure, both the real and imaginary parts of on the first branch in Fig. 2 lie within 0.1% of the corresponding solutions for the metal/chiral-STF interface16 when , and on the second branch in that figure lie within 0.1% when . Also, converges faster with respect to than does. In order to delineate the effect of on the localization of SPP waves to the interface , the penetration depth into aluminum is presented in Fig. 3 in relation to for all solutions of Eq. (12). The penetration depth converges to for the first branch and to for the second branch as increases. Since the ambichiral material is periodically nonhomogeneous and anisotropic, two decay constants are defined18 to quantify localization in the anisotropic partnering material. The smaller that is, the higher is that localization of the SPP wave to the interface. The decay constants for all SPP waves found are presented in Fig. 4. The SPP waves on the first branch are highly localized to the interface within the ambichiral material as both decay constants converge to values with increasing . The SPP waves on the second branch are loosely bound to the interface, and the degree of localization strongly depends on the direction of propagation. Both decay constants converge to for , but both converge to for . Furthermore, data for (not shown) indicated that , , and converge to within 0.1% of their respective values when on the first branch and on the second branch. 3.2.Dyakonov–Tamm WavesNext, let the isotropic homogeneous partnering material be magnesium fluoride—a dielectric material with at —instead of a metal. Every solution of the dispersion equation (12) represents a Dyakonov–Tamm wave17—named thus because this wave has the attributes of both the Dyakonov wave23,24 and the Tamm wave18,25—since both partnering materials are dielectric materials and one of the two is anisotropic and periodically nonhomogeneous normal to the waveguiding interface for .17,18 For every , only one solution of Eq. (12) was found. Dissipation being absent in both partnering materials, is real valued. The dependence of on is shown in Fig. 5. Also provided in the same figure are and in relation to . For , the anisotropic partner is homogeneous and the solution for represents a Dyakonov wave;23,24 no solution was found for . For , only one solution was found regardless of the value of , and that solution represents a Dyakonov–Tamm wave. Figure 5 indicates the typical difference between Dyakonov and Dyakonov–Tamm waves:17 The range of is much larger for surface waves of the latter type than for the surface waves of the former type. Just like the solutions presented in Fig. 2 for SPP waves, those presented in Fig. 5 for Dyakonov–Tamm waves also evolve as increases. Specifically, the solutions in Fig. 5 converge monotonically to within 0.1% of the corresponding solution for the isotropic-dielectric/chiral-STF interface17 when . Figure 5 shows that the penetration depth of the Dyakonov–Tamm wave into the homogeneous partnering material depends upon and, hence, the direction of propagation, but is about four times greater than for SPP waves. The plots of the decay constants show that the degree of localization of the Dyakonov–Tamm wave in the ambichiral material increases as increases. The values of converge to for and to for and 90 deg. Furthermore, , , and converge to within 0.1% when . 3.3.Sufficient Value of NIn order to fabricate a structurally chiral material with continuous variation of with as an ambichiral material, a sufficient value of must be chosen based upon the tolerances in the values of the wavenumber , penetration depth , and the smaller decay constant . If all tolerances are chosen to be 0.1%, as in the previous subsections, is sufficient for the SPP wave, and for the Dyakonov–Tamm wave. However, if all tolerances are chosen to be 1%, suffices for the SPP wave and for the Dyakonov–Tamm wave. Let us note that the thickness of each CTF in the ambichiral material is for and for . If the thickness of each CTF either equals or is less than one-tenth of the smallest wavelength inside the ambichiral material, then each CTF is electrically thin.26 For the chosen parameters, one-tenth of the smallest wavelength inside the ambichiral material is since . Therefore, if , the ambichiral material has a sufficiently smooth variation of with and all important quantities characterizing a surface wave will converge to within . 4.Concluding RemarksThe canonical boundary-value problem of surface-wave propagation guided by the planar interface of an isotropic homogeneous material and an ambichiral material was set up and solved. Both SPP and Dyakonov–Tamm waves were investigated. As the number of layers per period in the ambichiral partnering material was increased, the solutions of the dispersion equation for surface-wave propagation were found to converge to those for the ambichiral material replaced by the corresponding finely chiral material. The convergence is faster when the homogeneous partnering material is dielectric than when it is metallic. The real part of the wavenumber of an SPP wave converges faster than the imaginary part with respect to . 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BiographyMuhammad Faryad received his MSc and MPhil degrees in electronics from Quaid-i-Azam University in 2006 and 2008, respectively, and his PhD (2012) degree in engineering science and mechanics from the Pennsylvania State University, where he is presently a postdoctoral research scholar. His research interests include modeling of thin-film solar cells, electromagnetic surface waves, chemical sensing, circular Bragg phenomenon, and sculptured thin films. Akhlesh Lakhtakia is 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 (United Kingdom). He was the sole recipient of the 2010 SPIE Technical Achievement Award. His current research interests include nanophotonics, surface multiplasmonics, complex materials including mimumes and sculptured thin films, bone refacing, bioreplication, and forensic science. |