## Abstract

The solution of a dispersion equation indicates the theoretical existence of multiple modes of surface plasmon polariton wave propagation at the planar interface of a metal and a chiral sculptured thin film (STF). One mode appears to occur over a wide range of the structural period of the chiral STF, while all other modes exist only above some minimum value of the structural period, the minimum value being different for each mode. In order to excite the different modes, the interface can be incorporated in the commonplace Kretschmann configuration, for which our calculations show that the efficient excitation of different modes would require different numbers of structural periods of the chiral STF.

## 1. Introduction

Research on electromagnetic surface waves dates back at least to 1907 when Zenneck (1907) proposed a mode of oscillation of the electromagnetic field at radio frequencies along the air–ground interface. The mode propagates parallel to the interface with an amplitude that decreases exponentially with distance from the interface. Basically the same phenomenon at optical frequencies propagating along a metal–dielectric interface, the surface plasmon polariton (SPP) wave has been the object of intense study (Agronovich & Mills 1982; Raether 1983; Kalele *et al*. 2007) from the middle part of the twentieth century. Being highly localized to the interface, the propagation of the SPP wave is highly dependent on interfacial conditions. This has resulted in a flurry of applications of SPP waves to extremely sensitive detection of chemical and biochemical species (Homola *et al*. 1999; Abdulhalim *et al*. 2008).

Initial investigations of SPP waves were focused on the interface of a metal and an isotropic dielectric material (Kretschmann & Raether 1968; Simon *et al*. 1975). The theory was later extended to interfaces of metals and anisotropic dielectric materials (Singh & Thyagarajan 1991; Mihalache *et al*. 1994). With the current surge of interest in application of SPP-wave-based techniques for biosensing (Homola *et al*. 1999; Abdulhalim *et al*. 2008), imaging (Aoki *et al*. 2005; Kanda *et al*. 2005) and information transmission in computer chips (Maier *et al*. 2001), the propagation of SPPs at the interface of a metal and manufactured structures such as photonic crystals is now being explored (Huang & Zhu 2007; Hassani *et al*. 2008).

In the same context, we previously investigated the propagation of SPP waves at the interface of a columnar thin film (CTF) and a metal (Lakhtakia & Polo 2008*a*). CTFs are assemblies of nominally parallel, straight and identical nanorods. At optical wavelengths, a CTF is akin to a biaxial dielectric material (Hodgkinson *et al*. 1998). CTFs are usually fabricated by physical vapour deposition: collimated vapour directed at a substrate in a vacuum at suitable temperature coalesces into nanorods. Generally, the angle *Χ* describing the tilt of the nanorods relative to the substrate plane is greater than the vapour incidence angle *Χ*_{v}, as depicted schematically in figure 1. The vapour incidence angle also controls the eigenvalues of the effective permittivity tensor of the CTF. We found that the selection of a higher value for *Χ*_{v} when growing a CTF leads to SPP wave propagation with a phase velocity of lower magnitude and a shorter propagation range.

As is now known well (Young & Kowal 1959; Robbie *et al*. 1996), nanohelices form instead of nanorods, if during deposition the substrate is slowly rotated about an axis passing normally through it. The resulting assembly is a sculptured thin film (STF) that is structurally chiral (Lakhtakia & Messier 2005), just like a chiral smectic liquid crystal (de Gennes & Prost 1993). In the Kretschmann configuration (Kretschmann & Raether 1968; Simon *et al*. 1975), SPP wave propagation along the planar interface of a metal and a structurally chiral material has been theoretically demonstrated (Lakhtakia 2007); furthermore, we recently showed that the choice of the vapour incidence angle *Χ*_{v} during fabrication must significantly affect the propagation characteristics (Lakhtakia & Polo 2008*b*), if the structurally chiral material is a chiral STF.

In comparison with a CTF, a chiral STF has one more morphological attribute: the pitch of the nanohelices. This is determined by the rotation rate of the substrate during deposition (Messier *et al*. 2000). As the effect of the pitch on SPP wave propagation had not been explored, we decided to solve two boundary-value problems, the first relating to SPP wave propagation at the interface of a semi-infinitely thick metal and a semi-infinitely thick chiral STF, and the second to the plane-wave response of the planar interface of a finitely thick metallic layer and a chiral STF in the Kretschmann configuration.

The remainder of the paper is organized as follows. The constitutive relations of a chiral STF are presented in §2 along with the formulation of both boundary-value problems. Numerical results are provided and discussed in §3, followed by concluding remarks in §4. An exp(−i*ωt*) time dependence is implicit here, with *ω* denoting the angular frequency. The free-space wavenumber, the free-space wavelength and the intrinsic impedance of free space are denoted by , *λ*_{o}=2*π*/*k*_{o} and , respectively, with *μ*_{o} and *ϵ*_{o} being the permeability and permittivity of free space. Vectors are in boldface, dyadics are in italic boldface, and column vectors and matrixes are in italic boldface and enclosed within square brackets. Cartesian unit vectors are identified as *u*_{x}, *u*_{y} and *u*_{z}.

## 2. Boundary-value problems

### (a) First boundary-value problem

Let the half-space *z*≤0 be occupied by a metal of relative permittivity *ϵ*_{met}. The half-space *z*≥0 is occupied by a chiral STF with unidirectionally non-homogeneous relative permittivity dyadic given by (Venugopal & Lakhtakia 2000; Lakhtakia & Messier 2005)(2.1)where the reference relative permittivity dyadic(2.2)indicates local orthorhombic symmetry. The dyadic function(2.3)with(2.4)contains 2*Ω* as the structural period; *h*=±1 as the structural handedness parameter; and *γ* as an angular offset with respect to the *x*-axis in the plane *z*=0. The tilt dyadic(2.5)involves the tilt angle *Χ*. The superscript ^{T} denotes the transpose. Parenthetically, chiral liquid crystals (de Gennes & Prost 1993) can also be accommodated in the foregoing constitutive description of chiral STFs.

In order to investigate SPP wave propagation, we adopted a procedure devised to investigate the propagation of Dyakonov–Tamm waves (Shiyanovskii 1990; Lakhtakia & Polo 2007). Let the SPP wave propagate parallel to the unit vector along the interface *z*=0, and attenuate as *z*→±∞. Therefore, in the region *z*≤0, the wavevector may be written as(2.6)where ; *κ* is complex valued; and Im(*α*_{met})>0 for attenuation as *z*→−∞. Accordingly, the field phasors in the region *z*≤0 may be written as(2.7)and(2.8)where *a*_{1} and *a*_{2} are unknown scalars, and .

For field representation in the half-space *z*≥0, let us write(2.9)and create the column vector(2.10)This column vector satisfies the matrix differential equation (Lakhtakia & Messier 2005)(2.11)where the 4×4 matrix(2.12)and .

Two independent techniques (Lakhtakia & Weiglhofer 1997; Schubert & Herzinger 2001; Polo & Lakhtakia 2004) exist to solve (2.11). Either of the two may be harnessed to determine the matrix [** N**] that appears in the relation(2.13)to characterize the optical response of one period of the chiral STF for specific values of

*κ*and

*ψ*. We used the piecewise uniform approximation technique (Polo & Lakhtakia 2004).

By virtue of the Floquet–Lyapunov theorem (Yakubovich & Starzhinskii 1975), a matrix [** Q**] can be defined such that(2.14)Both [

**] and [**

*N***] share the same eigenvectors, and their eigenvalues are also related. Let [**

*Q***]**

*τ*^{(n)},

*n*∈[1,4], be the eigenvector corresponding to the

*n*th eigenvalue

*σ*

_{n}of [

**]; then, the corresponding eigenvalue**

*N**α*

_{n}of [

**] is given by(2.15)**

*Q*After ensuring that Im(*α*_{1,2})>0, we set (Martorell *et al*. 2006)(2.16)for SPP wave propagation, where *b*_{1} and *b*_{2} are unknown scalars; the other two eigenvalues of [** Q**] pertain to waves that amplify as

*z*→∞ and cannot therefore contribute to the SPP wave. At the same time,(2.17)by virtue of (2.7) and (2.8). Continuity of the tangential components of the electric and magnetic field phasors across the plane

*z*=0 requires that , which may be rearranged as(2.18)For a non-trivial solution, the 4×4 matrix [

**] must be singular, so that(2.19)is the dispersion equation for the SPP wave. This equation has to be solved in order to determine the SPP wavenumber**

*Y**κ*.

Parenthetically, were the metal to be replaced by an insulator, the SPP wave would convert to a Dyakonov–Tamm wave (Shiyanovskii 1990; Lakhtakia & Polo 2007). If, in addition, the chiral STF were to be replaced by a CTF, the Dyakonov–Tamm wave would convert to a Dyakonov wave (Polo *et al*. 2007; Takayama *et al*. 2008).

### (b) Second boundary-value problem

The dispersion equation obtained in §2*a* and its solutions are theoretically interesting. Whether the solutions are experimentally observable, and thus may have technological applications, requires a somewhat different consideration.

The Kretschmann configuration is a common experimental arrangement for the excitation and detection of SPP waves (Kretschmann & Raether 1968; Simon *et al*. 1975). In this configuration, the region −*L*_{met}≤*z*≤0 is occupied by the metal of relative permittivity *ϵ*_{met}, the region 0≤*z*≤*L*_{stf} by a chiral STF described by (2.1), and without significant loss of generality the half-spaces *z*≤−*L*_{met} and *z*≥*L*_{stf} by a homogeneous isotropic dielectric material of relative permittivity . Dissipation is considered to be negligibly small except in the metal; furthermore, *ϵ*_{l} must exceed the maximum of *ϵ*_{a}, *ϵ*_{b} and *ϵ*_{c}. We also take the chiral STF to contain an integral number of periods, so that *l*_{stf}=*L*_{stf}/2*Ω* is an integer.

A plane wave is supposed to be launched in the half-space *z*≤−*L*_{met} towards the metal layer in order to excite the SPP wave along the interface *z*=0. The wavevector of this exciting plane wave is oriented at an angle *θ*∈[0,*π*/2) to the *z*-axis and at an angle *ψ*∈[0,2*π*) to the *x*-axis in the *xy* plane. The electromagnetic field phasors associated with the exciting plane wave are represented as(2.20)The amplitudes of the s- and the p-polarized components of the exciting plane wave, denoted by *a*_{s} and *a*_{p}, respectively, are assumed given, and(2.21)The reflected electromagnetic field phasors are expressed as(2.22)and the transmitted electromagnetic field phasors as(2.23)

The procedure to determine the amplitudes *r*_{s}, *r*_{p}, *t*_{s} and *t*_{p} in terms of *a*_{s} and *a*_{p} is standard (Lakhtakia & Messier 2005, ch. 10). It suffices to state here that the following set of four algebraic equations emerges (in matrix notation):(2.24)

The matrix(2.25)depends on the refractive index *n*_{l}, whereas the matrix(2.26)captures the response of the metal layer. The matrix is calculated exactly in the same way as the matrix [** N**], except that the real-valued replaces the complex-valued

*κ*.

The solution of (2.24) yields the reflection and transmission coefficients that appear as the elements of the 2×2 matrixes in the following relations:(2.27)Co-polarized 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, *R*_{sp}=|*r*_{sp}|^{2} is the reflectance corresponding to the reflection coefficient *r*_{sp}, and so on.

In general, the power density of the exciting plane wave is partially reflected into the half-space *z*≤−*L*_{met}, partially transmitted into the half-space *z*≥*L*_{stf}, and partially absorbed in the metal layer −*L*_{met}≤*z*≤0. The absorbances(2.28)are obtained as functions of *θ* for fixed *ψ*. A sharp, high peak in the plot of an absorbance versus *θ* indicates the excitation of an SPP wave localized to the interface *z*=0.

## 3. Numerical results and discussion

Although chiral STFs have been fabricated by evaporating a wide variety of materials (Lakhtakia & Messier 2005, ch. 1), measurements of complete sets of constitutive parameters of chiral STFs have not been reported. However, the constitutive parameters of certain CTFs are known. We expect the functional relationships connecting *ϵ*_{a,b,c} and *Χ* to *Χ*_{v} for CTFs to substantially apply for chiral STFs, since the vapour incidence angle *Χ*_{v} remains constant during the deposition of thin films of either kind.

Empirical relationships have been determined for titanium oxide CTFs at *λ*_{o}=633 nm by Hodgkinson *et al*. (1998) as(3.1)(3.2)(3.3)and(3.4)where *Χ*_{v} and *Χ* are in radian. 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 and *γ*=0°; *ϵ*_{met}=−56+21*i*, a typical value for aluminium at *λ*_{o}=633 nm; and *ϵ*_{l}=6.656, which is typical for zinc selenide.

The results shown in this section were computed for *Χ*_{v}=20°, which is a realistic value of the vapour incidence angle. For the first boundary-value problem, the results shown here were computed for *ψ*=0°; some calculations were carried out also for *ψ*=45° and 90°, with results qualitatively similar to those for *ψ*=0°. For the second boundary-value problem, we fixed *L*_{met}=15 nm, but the integer *l*_{stf}∈[1,∞) was allowed to vary. The half-pitch *Ω* was normalized with respect to *Ω*_{o}=197 nm; in the limit *Ω*_{o}/*Ω*→0, the chiral STF transforms into a CTF.

### (a) First boundary-value problem

Figure 2 shows the calculated values of *k*_{o}/Re(*κ*) and 1/Im(*κλ*_{o}) versus *Ω*_{o}/*Ω* for *ψ*=0°. The first quantity is the phase speed of the SPP wave relative to *c*_{o} (the speed of light in free space), and the second is the *e*-folding distance along the *x*-axis relative to the free-space wavelength *λ*_{o}. From the figure, we conclude that SPP wave propagation is possible in five different modes labelled 1 to 5.

Let us recall that, in the limit *Ω*→∞, a chiral STF is a CTF. Only one mode of SPP wave propagation is then possible (Lakhtakia & Polo 2008*a*). We found that four other modes arise and then vanish as *Ω*→0. Mode 5 vanishes first, followed successively by modes 4, 3 and 2. Mode 1 alone survives as *Ω* is reduced further. Although the lower bounds of the *Ω* ranges of existence of modes 2–5 were easily ascertained, serious computational difficulties were encountered in determining the corresponding upper bounds (if any). No difficulty was found, however, in calculating the limiting CTF behaviour by eliminating the non-homogeneity of the chiral STF *ab initio* (Lakhtakia & Polo 2008*a*); this result is also shown in figure 2 at *Ω _{o}*/

*Ω*=0.

Figure 2*a* indicates that the phase speed of any of the five modes is a decreasing function of *Ω*. As *Ω*_{o}/*Ω*→0, the phase speeds of all modes decrease towards a limiting value that equals the phase speed of the sole mode that can propagate at the metal–CTF interface.

Figure 2*b* shows the *e*-folding distance relative to *λ*_{o} along the direction of travel for each one of the five modes. The *e*-folding distance is inversely proportional to the attenuation rate. The *e*-folding distance of mode 1 increases slowly at first as *Ω* increases. In the vicinity of *Ω*_{o}/*Ω*=0.3, the *e*-folding distance increases so rapidly as *Ω* increases that two points had to be omitted from the plot so as not to obscure details for the other modes. At *Ω*_{o}/*Ω*=0.13, the relative *e*-folding distance is over 2200, which represents an absolute *e*-folding distance of more than 0.4 mm. The *e*-folding distance of mode 2 initially shows a mild decrease with increasing *Ω* until *Ω*_{o}/*Ω*=0.13, at which point it too shows a sharp upturn with increasing *Ω*. The *e*-folding distances of modes 3, 4 and 5 are all decreasing functions of *Ω*, with that of mode 3 showing a hint of a turnaround for the largest value of *Ω*.

The eigenvalues *α*_{1} and *α*_{2} of the matrix [** Q**] must have positive imaginary parts that may be termed as

*decay constants*, because they indicate the decay of the electromagnetic field phasors as

*z*→+∞. A decay constant for the metallic half-space

*z*≤0 may be defined as

*α*

_{met}=(

*k*

_{o}

*ϵ*

_{met}−

*κ*

^{2})

^{1/2}, with Im(

*α*

_{met})>0 indicating decay as

*z*→−∞.

The two decay constants in the chiral STF, Im(*α*_{1}) and Im(*α*_{2}), are plotted against *Ω*_{o}/*Ω* in figure 3*a–e* for modes 1–5, while the decay constant Im(*α*_{met}) in the metal is shown for all five modes in figure 3*f*. All modes exhibit a sharp peak in the variation of Im(*α*_{1}) against *Ω*_{o}/*Ω*=0. The height of the peak decreases with mode number. Furthermore, as *Ω* approaches the lower bound of the *Ω* range of existence of any of the modes numbered 2–5, Im(*α*_{2}) approaches zero; hence, at the lower bound of the *Ω* range of existence, the particular SPP wave mode inside the chiral STF becomes delocalized from the bimaterial interface *z*=0. In figure 3*b*, Im(*α*_{1}) and Im(*α*_{2}) for mode 2 are very nearly equal and the markers for Im(*α*_{2}) are obliterated by the bolder markers for Im(*α*_{1}) at the lower end of the *Ω* range. In contrast to modes 2–5, Im(*α*_{2}) for mode 1 reaches a plateau at low values of *Ω*.

Figure 3*f* shows that Im(*α*_{met}) is virtually independent of *Ω* for every SPP wave mode. Also, a comparison of figure 3*f* with figure 3*a–e* shows that the decay constant in the metal is about an order of magnitude greater than the two decay constants in the chiral STF, indicating thereby that the SPP waves are more closely constrained to the interface on the metal side of the bimaterial interface than on the chiral STF side.

At the planar interface between a metal and an isotropic dielectric material, an SPP wave has to be p-polarized, i.e. its electric field does not have a *y*-directed component and its magnetic field is wholly parallel to the *y*-axis (Kretschmann & Raether 1968; Simon *et al*. 1975; Abdulhalim *et al*. 2008). By contrast, the magnetic field of an s-polarized wave does not have a *y*-directed component and its electric field is wholly parallel to the *y*-axis.

In order to examine the polarization states of the SPP wave modes at the planar interface of a metal and a chiral STF, the polarization parameterwas calculated at *z*=0− for every mode. Figure 4*a–e* shows the results for modes 1–5 in that order. Mode 1 is predominantly p-polarized, and becomes more so at the upper and lower bounds of the *Ω* range over which the calculations were performed. Each of the modes 2–5 shows the opposite behaviour, with maximum p-polarization content somewhere in the middle of its *Ω* range of calculation. Modes 3 and 5 are, similar to mode 1, predominantly p-polarized over their entire *Ω* range of calculation, but modes 2 and 4 acquire roughly equal amounts of p- and s-polarization contents as *Ω* becomes small. The s-polarization content of the electric field in the metal is due to the structural handedness of the chiral STF.

### (b) Second boundary-value problem

Plots of the absorbances *A*_{p} and *A*_{s} versus the angle of incidence *θ* were drawn for a large number of situations covered in figure 2. The angle of incidence for every sharp peak in the absorbance plots was identified and used to calculate the corresponding real-valued quantity , which was then compared with the set of complex-valued quantities *κ* in figure 2.

Figure 5*a* exemplifies the dependences of *A*_{p} and *A*_{s} on *θ*, when the chiral STF in the Kretschmann configuration has two structural periods (i.e. *l*_{stf}=2), *γ*=*ψ*=0° and *Ω*_{o}/*Ω*=0.6. Two sharp peaks are present in the plot of *A*_{p} in this figure, one at *θ*=40.62° corresponding to mode 2 with an absorbance of 0.92 and the other at *θ*=48.21° corresponding to mode 1 with an absorbance of 0.97. As with homogeneous isotropic dielectric materials, the SPP wave is excited only by a p-polarized plane wave for the chosen parameters; the curve for *A*_{s} is extremely smooth with no peaks.

Figure 5*b* was also drawn for *l*_{stf}=2 and *γ*=*ψ*=0°, but when *Ω*_{o}/*Ω*=1.5. Only a portion of the absorbance curves covering the *θ* range of [20°, 40°] where mode 2 appears is shown. Mode 2 is near the lower bound of its *Ω* range, and only a hint of its existence is indicated by a small bump with a maximum *A*_{p}=0.31 at *θ*=30.37°. A slight rise in *A*_{s} also exists at the same angle of incidence. When the number *l*_{stf} of structural periods in the chiral STF is increased, however, the peak in *A*_{p} corresponding to mode 2 becomes much more sharply defined. This is clearly indicated in figure 5*c*, which shows the two absorbances when *l*_{stf}=20. The peak in *A*_{p} corresponding to mode 2 is now very sharp with a magnitude of 0.72. Also, the location of the peak in *A*_{p} has shifted to *θ*=28.25°. A peak in *A*_{s} at the same value of *θ* but of lesser magnitude can also be seen. The peak magnitude in *A*_{p} decreases for every mode, as *Ω* approaches the lower bound of the *Ω* range for that mode.

Absorbance curves indicating the excitation of modes 3, 4 and 5 in the Kretschmann configuration are shown in figure 6. Two peaks in *A*_{p} corresponding to modes 3 and 4 are present in figure 6*a* at *θ*=43.93° and 41.02°, respectively, when *Ω*_{o}/*Ω*=0.2 and *l*_{stf}=2. The peak value of *A*_{p} for mode 3 is 0.96, while that for mode 4 is 0.91. As with modes 1 and 2 in figure 5*a*, the *A*_{s} curve is nearly featureless over the same range. Similar behaviour can be seen in figure 5*b* for mode 5, also for *l*_{stf}=2, but when *Ω*_{o}/*Ω*=0.13. The maximum value of *A*_{p} (=0.94) occurs at *θ*=42.09°.

Figures 5 and 6 indicate that not all modes of SPP wave propagation *may* be advantageously excited in the Kretschmann configuration, if the thickness of the metal layer and the number of structural periods of the chiral STF are fixed; instead, different modes or different sets of modes are going to be excited.

Calculations of for SPP waves in the Kretschmann configuration were performed for many of the values of the inverse-periodicity parameter *Ω*_{o}/*Ω* that appear in figures 2 and 3, as well as several other values of the parameter. The values of were determined by the angles of incidence corresponding to peaks in the *A*_{p} versus *θ* curves. Identical values of *θ* were found from peaks in the *A*_{s} versus *θ* curves when the peaks were well defined as in figure 5*c*; however, such situations were limited and appeared to occur when the value of the polarization parameter describing the first boundary-value problem, displayed in figure 4, is large at the same value of *Ω*_{o}/*Ω*. For all calculations of , we fixed *L*_{met}=15 nm, *h*=1 and *γ*=*ψ*=0°. The thickness of the STF was set to *L*_{stf}=4*Ω* for all calculations, except for the three lowest values of *Ω* for mode 2 for which we set *L*_{stf}=40*Ω* in order to clearly observe a peak in the *A*_{p} versus *θ* curve.

A plot of the normalized phase speed against *Ω*_{o}/*Ω* is shown in figure 7, wherein five SPP wave modes are identifiable. Visual comparison of values in figure 7 with the values of *k*_{o}/Re(*κ*) in figure 2*a* for the five modes indicates that the solutions of both boundary-value problems yield nearly the same values for the relative phase speed. Indeed, a detailed numerical comparison revealed a maximum difference in relative phase speed of less than 6 per cent, with most differences less than 2 per cent. We must caution that, near the lower bound of the *Ω* range for each mode, the peak magnitude of *A*_{p} peak is not very large, so that the excitation of the particular SPP wave mode may be considered to be inefficient then. Additionally, the somewhat abrupt change in slope of versus *Ω*_{o}/*Ω* for the last three data points may be due to the previously mentioned change in the number of structural periods of the chiral STF for these three points.

To demonstrate that the energy of the exciting plane wave is absorbed via conversion to the energy of an SPP wave if the right conditions prevail, the time-averaged Poynting vector(3.5)was calculated within the metal layer, for many instances at values of *θ* for which a sharp peak in *A*_{p} indicates the excitation of a SPP wave mode, with *a*_{p}=1 V m^{−1} and *a*_{s}=0 V m^{−1}. The *x*- and *z*-directed components of the time-averaged Poynting vector as functions of *z*/*L*_{met}∈(−1,0) are shown in figure 8 for modes 1 and 2 for the same conditions as figure 5*a*. In figure 8*a*, the magnitude of *P*_{x} for mode 1 rises from 0.25 W m^{−2} upon entering the metal from the zinc selenide side to 1.08 W m^{−2} at the interface with the chiral STF, when *θ*=48.21°; correspondingly, the magnitude of *P*_{z} plummets from 1.66 to 0.000068 W m^{−2} in figure 8*b*. The rise in the magnitude of *P*_{x} accompanied by reduction in the magnitude of *P*_{z} is consistent with a transfer of energy from the exciting plane wave travelling at an angle *θ* to the *z*-axis to the SPP wave travelling along the *x*-axis.

The time-averaged Poynting vector in the metal layer for mode 2 is shown in figure 8*c*,*d* for the same conditions as figure 5*a*, when *θ*=40.62°. The magnitude of *P*_{x} rises from 0.24 W m^{−2} on the zinc selenide side to 0.97 W m^{−2} to the chiral STF side, as shown in figure 8*c*. The magnitude of *P*_{z} in figure 8*d*, however, drops from 1.85 to 0.046 W m^{−2}. Again, the rise in |*P*_{x}| accompanied by reduction in |*P*_{z}| is consistent with a transfer of energy from the exciting plane wave to an SPP wave.

The time-averaged Poynting vector in the metal layer for mode 2—for the same conditions as figure 5*c* and with *a*_{p}=1 V m^{−1}, *a*_{s}=0 V m^{−1} and *θ*=28.25°—is plotted in figure 9. Although the magnitude of *P*_{x} increases only slightly from 0.16 to 0.18 W m^{−2} from the zinc selenide side to the chiral STF side, the magnitude of *P*_{z} concurrently decreases significantly from 1.72 to 0.029 W m^{−2} as it does for modes 1 and 2 in figure 8.

The profile of the time-averaged Poynting vector in the metal layer is shown for mode 3 in figure 10*a*,*b*, when *θ*=43.93° and the other parameters are the same as for figure 6*a*. The magnitude of *P*_{x} rises from 0.25 W m^{−2} on the zinc selenide side to 1.14 W m^{−2} on the chiral STF side, whereas |*P*_{z}| drops from 1.79 to 0.0086 W m^{−2}. The profile of the time-averaged Poynting vector in the metal layer is shown for mode 4 in figure 10*c*,*d*, when *θ*=41.02°. Whereas |*P*_{x}| rises from 0.23 W m^{−2} on the zinc selenide side to 0.89 W m^{−2} on the chiral STF side, |*P*_{z}| drops from 1.81 to 0.04 W m^{−2}. The excitation of an SPP wave by an illuminating plane wave is thereby confirmed in both instances. A similar confirmation is provided by figure 11 for the excitation of mode 5 in figure 6*b*.

Although the results presented heretofore in this paper are for *ψ*=0°, similar results were obtained when *ψ*≠0°. Figure 12 shows the absorbances *A*_{p} and *A*_{s} for the Kretschmann configuration, with all parameters the same as for figure 5 except *ψ*=30°. Two sharp peaks in *A*_{p} are present in figure 12*a* when *Ω _{o}*/

*Ω*=0.6, similar to those in figure 5

*a*when

*ψ*=0°. For

*ψ*=30°, the peaks in

*A*

_{p}are closer together at

*θ*=42.06° and

*θ*=47.8°, with magnitudes of 0.93 and 0.97, respectively. Figure 12

*b*shows a dramatic reduction of the peak magnitude of

*A*

_{p}peak when

*Ω*/

_{o}*Ω*=1.5, as was observed when

*ψ*=0°. Following the trend seen for

*ψ*=0°, the peak in

*A*

_{p}is accentuated for

*Ω*/

_{o}*Ω*=1.5 when

*l*

_{stf}is increased to 20. The peak

*A*

_{p}of 0.76 is now located at

*θ*=29.15° and is also accompanied by a substantial peak in

*A*

_{s}.

## 4. Concluding remarks

We have theoretically demonstrated the existence of multiple modes of SPP wave propagation at the planar interface of a metal and a chiral STF. Except for one mode, all modes (modes 2–5) exist when the structural period of the chiral STF exceeds a certain mode-dependent lower bound, but no minimum value of the structural period for the remaining mode (mode 1) was found. The phase speed and the decay constants appear to approach the values found for a CTF with the same tilt angle as the structural period increases. However, owing to computational difficulties, we could not ascertain whether a (possibly, mode dependent) upper bound on the structural period exists for each SPP wave mode. Calculations of absorbances when the interface is incorporated in the Kretschmann configuration indicate that these modes should be experimentally observable. In many cases, observation appears possible when the chiral STF is as little as two periods thick. However, the excitation of different modes may require different numbers of structural periods of the chiral STF in the Kretschmann configuration. Mode excitation by s-polarized exciting plane waves may be possible, in addition to the usual excitation by p-polarized plane waves. The possibility of multiple SPP wave modes at the interface of a chiral STF and a metal may result in new technological applications, particularly for error-free optical sensing of chemical and biochemical species present in trace amounts.

## Acknowledgments

A.L. is grateful to the Charles Godfrey Binder Endowment at the Pennsylvania State University for financial support.

## Footnotes

- Received May 23, 2008.
- Accepted August 8, 2008.

- © 2008 The Royal Society