1.

Introduction

Anticipating 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 colleagues^4–7 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 $N$ of layers in each period of a Reusch pile was developed by Hodgkinson et al.¹⁰ If $N=2$, the Reusch pile is classified as an equichiral material; if $N>2$, but not very large, it can be called an ambichiral material; and if $N\to \infty$, 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 $N$ of layers in a period on the circular Bragg phenomenon has been studied.¹⁵ Both $N$ 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 $N$ 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 $N$.

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 $\exp (-i\omega t)$ dependence on time $t$ is implicit, with $\omega$ denoting the angular frequency and $i=\sqrt{-1}$. Furthermore, $k_0=\omega \sqrt{{\mu }_0{\epsilon }_0}$ and ${\lambda }_0=2\pi /k_0$, respectively, represent the free-space wavenumber and free-space wavelength, where ${\mu }_0$ is the permeability and ${\epsilon }_0$ is the permittivity of free space. Vectors are in boldface, dyadics are double-underlined, and the three Cartesian unit vectors are denoted by ${\widehat{\mathbf{u}}}_x$, ${\widehat{\mathbf{u}}}_y$, and ${\widehat{\mathbf{u}}}_z$.

2.

Theoretical Preliminaries

The canonical boundary-value problem of surface-wave propagation is depicted schematically in Fig. 1. The half space $z<0$ is occupied by an isotropic and homogeneous material with relative permittivity ${\epsilon }_s$. The half space $z>0$ is occupied by an ambichiral dielectric material comprising homogeneous layers each of thickness $D$, the $\ell$’th layer occupying the region $(\ell -1)D<z<\ell D$, $\ell \in [1,\infty )$. The relative permittivity dyadic is given as

Fig. 1

Schematic of the canonical boundary-value problem.

In the region $z<0$, the electric field phasor may be written as^16–18

For field representation in the region $z>0$, let us write¹⁸

The piecewise-uniform approximation technique⁸ can be used to determine the matrix $[\underline{\underline{\tilde{Q}}}]$ that appears in the relation

Enforcement of the usual boundary conditions across the plane $z=0$ requires that $[\mathbf{f}(0-)]=[\mathbf{f}(0+)]$, which may be rearranged as the matrix equation

3.

Numerical Results and Discussion

The dispersion equation (12) was solved using the Newton–Raphson method,¹⁹ with ${\lambda }_0$ 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 ${\chi }_v\in (0\mathrm{deg},90\mathrm{deg}]$ with respect to a planar substrate.²⁰ For the chosen CTF,

3.1.

Surface-Plasmon-Polariton Waves

Let the isotropic homogeneous partnering material be thin-film aluminum with ${\epsilon }_s=-14.65+i5.85$ (Ref. 21) at ${\lambda }_0=633\mathrm{nm}$. Then, all solutions $q$ of the dispersion equation (12) represent SPP waves.^16,18 These solutions are complex valued because ${\epsilon }_s$ is complex valued.

Only one solution of the dispersion equation exists for any $\gamma \in [0\mathrm{deg},90\mathrm{deg}]$ when the anisotropic partnering material is homogeneous (i.e., $N=1$).²² However, when that partnering material is periodically nonhomogeneous (i.e., $N>1$), the solutions can be organized into two branches for $\gamma =0\mathrm{deg}$ and 45 deg but only one for $\gamma =90\mathrm{deg}$, as shown in Fig. 2. Convergence on either branch is monotonic for $N>2$.

Fig. 2

Real and imaginary parts of the solutions $q$ of Eq. (12) as functions of $N\in [1,15]$ and $\gamma \in \{0\mathrm{deg},45\mathrm{deg},90\mathrm{deg}\}$ for SPP waves guided by the planar interface of aluminum (${\epsilon }_s=-14.65+i5.85$) and an ambichiral dielectric material characterized by Eq. (13) with ${\chi }_v=20\mathrm{deg}$, when ${\lambda }_0=633\mathrm{nm}$ and $\mathrm{\Omega }=200\mathrm{nm}$. Top panels: First branch of solutions. Bottom panels: Second branch of solutions.

The most notable feature of the solutions presented in Fig. 2 is their evolution and convergence as $N$ increases. Although not shown in this figure, both the real and imaginary parts of $q$ on the first branch in Fig. 2 lie within 0.1% of the corresponding solutions for the metal/chiral-STF interface¹⁶ when $N\ge 18$, and on the second branch in that figure lie within 0.1% when $N\ge 55$. Also, $\mathrm{Re}(q)$ converges faster with respect to $N$ than $\mathrm{Im}(q)$ does.

In order to delineate the effect of $N$ on the localization of SPP waves to the interface $z=0$, the penetration depth ${\delta }_s=1/\mathrm{Im}({\alpha }_s)$ into aluminum is presented in Fig. 3 in relation to $N$ for all solutions of Eq. (12). The penetration depth converges to $\simeq 23\mathrm{nm}$ for the first branch and to $\simeq 24\mathrm{nm}$ for the second branch as $N$ increases.

Fig. 3

The penetration depth ${\delta }_s$ of SPP waves into aluminum for the (top) first and (bottom) second branches of solutions provided in Fig. 2.

Since the ambichiral material is periodically nonhomogeneous and anisotropic, two decay constants ${\beta }_{1,2}=\exp [-2\mathrm{\Omega }\mathrm{Im}({\alpha }_{1,2})]\in (0,1)$ are defined¹⁸ to quantify localization in the anisotropic partnering material. The smaller that $\min \{{\beta }_1,{\beta }_2\}$ 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.

Fig. 4

The decay constants ${\beta }_{1,2}$ of SPP waves for the (top) first and (bottom) second branches of solutions provided in Fig. 2.

The SPP waves on the first branch are highly localized to the interface within the ambichiral material as both decay constants converge to values $<0.05$ with increasing $N$. 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 $\simeq 0.6$ for $\gamma =0\mathrm{deg}$, but both converge to $\simeq 0.9$ for $\gamma =45\mathrm{deg}$. Furthermore, data for $N>15$ (not shown) indicated that ${\delta }_s$, ${\beta }_1$, and ${\beta }_2$ converge to within 0.1% of their respective values when $N\ge 29$ on the first branch and $N\ge 30$ on the second branch.

3.2.

Dyakonov–Tamm Waves

Next, let the isotropic homogeneous partnering material be magnesium fluoride—a dielectric material with ${\epsilon }_s=1.896$ at ${\lambda }_0=633\mathrm{nm}$—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 wave^23,24 and the Tamm wave^18,25—since both partnering materials are dielectric materials and one of the two is anisotropic and periodically nonhomogeneous normal to the waveguiding interface for $N\ge 2$.^17,18

For every $\gamma \in \{0\mathrm{deg},45\mathrm{deg},90\mathrm{deg}\}$, only one solution of Eq. (12) was found. Dissipation being absent in both partnering materials, $q$ is real valued. The dependence of $q$ on $N$ is shown in Fig. 5. Also provided in the same figure are ${\delta }_s$ and ${\beta }_{1,2}$ in relation to $N$.

Fig. 5

Solution $q$, penetration depth ${\delta }_s$ into magnesium fluoride (${\epsilon }_s=1.896$), and the decay constants ${\beta }_{1,2}$ in the ambichiral dielectric material characterized by Eqs. (13) with ${\chi }_v=20\mathrm{deg}$, as functions of $N\in [1,15]$ and $\gamma \in \{0\mathrm{deg},45\mathrm{deg},90\mathrm{deg}\}$, for Dyakonov–Tamm waves guided by the interface of magnesium fluoride and the ambichiral dielectric material, when ${\lambda }_0=633\mathrm{nm}$ and $\mathrm{\Omega }=200\mathrm{nm}$.

For $N=1$, the anisotropic partner is homogeneous and the solution $q$ for $\gamma =0\mathrm{deg}$ represents a Dyakonov wave;^23,24 no solution was found for $\gamma \in \{45\mathrm{deg},90\mathrm{deg}\}$. For $N>1$, only one solution was found regardless of the value of $\gamma$, and that solution represents a Dyakonov–Tamm wave. Figure 5 indicates the typical difference between Dyakonov and Dyakonov–Tamm waves:¹⁷ The range of $\gamma$ 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 $N$ 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 $N\ge 9$.

Figure 5 shows that the penetration depth of the Dyakonov–Tamm wave into the homogeneous partnering material depends upon $\gamma$ 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 $\gamma$ increases. The values of ${\beta }_{1,2}$ converge to $\simeq 0.8$ for $\gamma =0\mathrm{deg}$ and to $\simeq 0.5$ for $\gamma =45\mathrm{deg}$ and 90 deg. Furthermore, ${\delta }_s$, ${\beta }_1$, and ${\beta }_2$ converge to within 0.1% when $N\ge 32$.

4.

Concluding Remarks

The 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 $N$ 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 $q$ of an SPP wave converges faster than the imaginary part with respect to $N$. The real and imaginary parts of the surface wavenumber, the penetration depth into the isotropic partner, and the decay constants in the ambichiral dielectric material converge to within $\sim 1\%$ if the layers in the ambichiral material are electrically thin.