Dyakonov-Tamm wave at the planar interface of a chiral sculptured thin film and an isotropic dielectric material

Surface waves, named here as Dyakonov-Tamm waves, can exist at the planar interface of an isotropic dielectric material and a chiral sculptured thin film (STF). Due to the periodic nonhomogeneity of a chiral STF, the range of the refractive index of the isotropic material is smaller but the range of the propagation direction in the interface plane is much larger, in comparison to those for the existence of Dyakonov waves at the planar interface of an isotropic dielectric material and a columnar thin film.


INTRODUCTION
Less than two decades ago, Dyakonov [1] theoretically predicted the propagation of a surface wave at the planar interface of an isotropic dielectric material and a positively uniaxial dielectric material with its optic axis wholly parallel to the interface plane. If ψ indicates the angle between the optic axis and the direction of surface-wave propagation, and n s is the refractive index of the isotropic dielectric material, then the Dyakonov wave exists for rather narrow ranges of ψ and n s . The consequent significance of Dyakonov waves for optical sensing and waveguiding was recognized thereafter [2,3]. Since then, the concept of the Dyakonov wave has been extended to the planar interfaces of isotropic and biaxial dielectric materials [4]. The possibility of the anisotropic material being artificially engineered, either as a photonic crystal with a short period in comparison to the wavelength [5] or as a columnar thin film (CTF) [6], has also emerged. Let us note here that the Dyakonov wave still remains to be experimentally observed, in part due to the narrow range of ψ for its existence [5].
The anisotropic material is taken to be homogeneous in all of the foregoing and other reports on the Dyakonov wave. What if the anisotropic material were to be chosen as periodically nonhomogeneous in a direction normal to the bimaterial interface? This question initiated a research project, the first results of which are being communicated here. Being a natural extension of a CTF, a chiral sculptured thin film (STF) was chosen as the periodically nonhomogeneous anisotropic material [7].
A chiral STF is made by directing a vapor flux in vacuum at an oblique angle onto a rotating substrate. Under suitable conditions, an assembly of parallel nanohelixes of the evaporated species forms, with the helical axes perpendicular to the substrate. An example of a single nanohelix is illustrated in Figure 1. By adjusting deposition parameters, both the pitch 2Ω and the angle of inclination χ ∈ (0, π/2] can be controlled. Each nanohelix, composed of multimolecular clusters with ∼ 3 nm diameter, is effectively a continuously bent column of ∼ 100-nm cross-sectional diameter. Therefore, at visible frequencies and lower, a chiral STF may be regarded as a linear, locally orthorhombic, unidirectionally nonhomogeneous continuum whose relative permittivity dyadic is akin to that of chiral smectic liquid crystals [8].
In formulating the surface-wave-propagation problem on the planar interface of an isotropic, homogeneous, dielectric material and a chiral STF, we adopted a methodology originally developed by Tamm in 1932 for a realistic Kronig-Penney model. Instead of assuming the solid to occupy the entire space, as is commonplace in solid-state physics [9], Tamm assumed the solid to occupy only a half-space. The incorporation of the oft-neglected surface led to the emergence of electronic states localized to the surface. Tamm states were experimentally observed in 1990 on the surfaces of superlattices [10], and their optical analogs for superlattices of isotropic materials are being investigated these days [11,12].
Given the braiding of Dyakonov waves and Tamm states in this communication, we decided to name the surface wave at the planar interface of an isotropic, homogeneous, dielectric material and a chiral STF as the Dyakonov-Tamm wave. Section 2 presents the boundary-value problem and the dispersion equation for the Dyakonov-Tamm wave. Section 3 contains numerical results when the chiral STF is chosen to be made of titanium oxide [6,13]. An exp(−iωt) timedependence is implicit, with ω denoting the angular frequency. The free-space wavenumber, the free-space wavelength, and the intrinsic impedance of free space are denoted by

Geometry and permittivity
Let the half-space z ≤ 0 be occupied by an isotropic, homogeneous, nondissipative, dielectric material of refractive index n s . The region z ≥ 0 is occupied by a chiral STF with unidirectionally nonhomogeneous permittivity dyadic given by [7] where the reference relative permittivity dyadic indicates the locally orthorhombic symmetry of the chiral STF. The dyadic function contains 2Ω as the structural period, ψ as an angular offset, and h = ±1 as the handedness parameter. The tilt dyadic involves the angle of inclination χ. The superscript T denotes the transpose.
Without loss of generality, we take the Dyakonov-Tamm wave to propagate parallel to the x axis in the plane z = 0. There is no dependence on the y coordinate, whereas the Dyakonov-Tamm wave must attenuate as z → ±∞.

Field representations
In the region z ≤ 0, the wave vector may be written as where κ is positive and real-valued for unattenuated propagation along the x axis, and Im [α s ] > 0 for attenuation as z → −∞. Accordingly, the field phasors in the region z ≤ 0 may be written as and where A 1 and A 2 are unknown scalars.
The field representation in the region z ≥ 0 is more complicated. It is best to write and create the column vector This column vector satisfies the matrix differential equation [7] d dz where the 4×4 matrix and Two independent techniques [14,15] exist to solve (11), which may be harnessed to determine the matrix [N ] that appears in the relation to characterize the optical response of one period of the chiral STF. By virtue of the Floquet-Lyapunov theorem [16], we can define a matrix [Q] such that Both [N ] and [Q] share the same eigenvectors, and their eigenvalues are also related. Let [t] (n) , (n = 1, 2, 3, 4), be the eigenvector corresponding to the nth eigenvalue σ n of [N ]; then, the corresponding eigenvalue α n of [Q] is given by

Dispersion equation for Dyakonov-Tamm wave
For the Dyakonov-Tamm wave to propagate along the x axis, we must ensure that Im[α 1,2 ] > 0, and set where B 1 and B 2 are unknown scalars; the other two eigenvalues of [Q] describe waves that amplify as z → ∞ and cannot therefore contribute to the Dyakonov-Tamm wave. At the same time, by virtue of (7) and (8). Continuity of the tangential components of the electric and magnetic field phasors across the plane z = 0 requires that which may be rearranged as For a nontrivial solution, the 4×4 matrix [M ] must be singular, so that is the dispersion equation for the Dyakonov-Tamm wave.

NUMERICAL RESULTS AND DISCUSSION
Although chiral STFs may be made by evaporating a wide variety of materials [7, Chap. 1], the constitutive parameters of chiral STFs have not been extensively measured. However, the constitutive parameters of certain columnar thin films (CTFs) are known. CTFs are assemblies of nanorods oriented at an angle χ to the substrate and are produced by directing the vapor at an angle χ v onto a stationary substrate, as shown in Fig. 2; the vapor incidence angle χ v (in addition to the evaporant species) determines the constitutive parameters [13]. When the substrate is rotated about a normal passing through it at a constant angular velocity of reasonable magnitude, parallel nanohelixes grow instead of parallel nanorods, and a chiral STF is deposited instead of a CTF [7,17]. Although the substrate is nonstationary, the functional relationships connecting ǫ a,b,c and χ to χ v for CTFs would substantially apply for chiral STFs, since the vapor incidence angle χ v remains constant during the deposition of thin films of either kind.
Among the CTFs which have been characterized are those made of titanium oxide, a material important in many practical applications [6]. Empirical relationships have been determined for titanium-oxide CTFs at λ o = 633 nm by Hodgkinson et al. [13] as ǫ a = 1.0443 + 2.7394 and tan χ = 2.8818 tan where χ v and χ are in radians. We must caution that the foregoing expressions are applicable to CTFs produced by one particular experimental apparatus, but may have to be modified for CTFs produced by others on different apparatuses; hence, we used these expressions for the numerical results presented in this section for chiral STFs simply for illustration. Furthermore, we set h = 1, Ω = 197 nm, and χ v = 7.2 • . Following Walker et al. [4] and Polo et al. [6], we left ψ and n s as variable parameters. All numerical results presented in this section were computed for λ o = 633 nm.
As mentioned in Section 2, the matrix [N ] can be calculated using two numerical techniques [15]: approximation technique was selected for calculations reported here. Basically, the technique consists of subdividing the chiral STF into a series of electrically thin sublayers parallel to the interface, and assuming the dielectric properties to be spatially uniform in each sublayer. The accuracy of this technique depends on the thickness of the sublayers, with thinner ones yielding more accurate results. Based on experience [7,15], a sublayer thickness of 2 nm gives reasonable results.
The magnitude of the phase velocity of the Dyakonov-Tamm wave was compared with that of the phase velocity of the electromagnetic wave in the bulk isotropic material. For this purpose, we defined the relative phase speed where v DT = ω/κ is the phase speed of the Dyakonov-Tamm wave and v s = 1/n s √ ǫ o µ o is the phase speed of the electromagnetic wave in the bulk isotropic material. Figure 3 shows v as a function of ψ for several values of n s . The phase velocity of the Dyakonov-Tamm wave, like several other surface waves [1]- [6], was found to be lower in magnitude than the phase velocity of the electromagnetic wave in the bulk isotropic material.
The minimum and maximum values of n s (1.631 and 1.65, respectively) in Figure 3 represent the approximate limits of the n s -range for which the determinantal equation (21) representing the boundary conditions between the two material could be solved. Outside this n s -range, the Dyakonov-Tamm wave can not exist for the chosen parameters.
Each of the curves in Figure 3 was drawn over the continuous ψ-range for which the Dyakonov-Tamm wave was found to exist. The plot in Figure 3  The mid-point of the ψ-range is ψ m = 37 • . As n s approaches either end of the n s -range, ∆ψ diminishes. At the low end of the n s -range, ∆ψ = 51 • when n s = 1.635, but ∆ψ = 20 • when n s = 1.631. At the high end of the n s range, ∆ψ = 54 • when n s = There seems to be a slight increase in ψ m as n s increases. The values of ψ m , however, are only approximate since the end-points of the ψ-range were only determined by the last whole degree lying inside the range. This may account for why ψ m at n s = 1.65 is slightly lower than that at n s = 1.645.
Every curve in Figure 3 is smooth with a broad minimum in the vicinity of 35 • to 40 • which levels off at both ends of the ψ-range. The minimum is deepest for curves representing midrange values of n s , while curves at extreme values of n s are nearly flat. As n s decreases, the v vs. ψ curve shifts downward.
The confinement of the Dyakonov-Tamm wave to the interface is described by the decay constants which are given by the imaginary part of the two eigenvalues in the chiral STF (α 1 and α 2 ) and the single eigenvalue α s in the isotropic dielectric material. We found all three eigenvalues to be purely imaginary, which are shown in Figure 4 as functions of ψ for several values of n s .
In Figure 4, Similar results are obtained at other values of χ v . As an example, Figure 5 displays v when χ v = 25 • . As in Figure 3, the maximum and minimum values of n s displayed in Figure 5 represent the approximate limits of the range of n s over which surface-wave propagation is possible. The width of the n s -range for χ v = 25 • is roughly half of that obtained for χ v = 7.2 • . Additional calculations, not shown here, were made for an identical chiral STF except with structural left-handedness (h = −1). The exact same results were obtained with the lefthanded chiral STF as with the right-handed chiral STF presented in Figures 2-4. It must borne in mind that, by changing h from ±1 to ∓1 in (3), we invert not only the structural handedness of the chiral STF but also the sense of rotation brought about by ψ = 0.

CONCLUDING REMARKS
To conclude, we examined the phenomenon of surface-wave propagation at the planar interface of an isotropic dielectric material and a chiral sculptured thin film. The boundary-value problem was formulated by marrying the usual formalism for the Dyakonov wave at the planar interface of an isotropic dielectric material and a columnar thin film with the methodology for Tamm states in solid-state physics. The solution of the boundary-value problem let us deduce the existence of Dyakonov-Tamm waves.
In constitutive terms, the major difference between a CTF and a chiral STF is the periodic nonhomogeneity enshrined in S z (z); in the limit Ω → ∞, a nanohelix uncurls into a nanorod, and a chiral STF transmutes into a CTF. However, the distinction between chiral STFs and CTFs is nontrivial, as may be deduced from the Floquet-Lyapunov theorem [16,18]. In comparison to the Dyakonov wave localized to the planar interface of an isotropic dielectric material and a CTF [6], we found that the n s -range for the existence of a Dyakonov-Tamm wave at the planar interface of an isotropic dielectric material and a chiral STF is smaller. However, the ψ-range is much larger in width: in comparison to ∆ψ < 1 • with CTFs [6] -and ∆ψ < 5 • with effectively uniaxial, short-period photonic crystals [5] -the width of the ψ-range is as high as 98 • in Figure 3 with chiral STFs. This implies that Dyakonov-Tamm waves could be detected much more easily than Dyakonov waves.