On the surface plasmon polariton wave at the planar interface of a metal and a chiral sculptured thin film

John A Polo, Akhlesh Lakhtakia

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.

Keywords:

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 2008a). 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 Χv, 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 Χv when growing a CTF leads to SPP wave propagation with a phase velocity of lower magnitude and a shorter propagation range.

Figure 1

Schematic of the nanowire morphology of a CTF and its deposition.

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 Χv during fabrication must significantly affect the propagation characteristics (Lakhtakia & Polo 2008b), 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(−iωt) 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 Embedded Image, λo=2π/ko and Embedded Image, respectively, with μo and ϵo 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 ux, uy and uz.

2. Boundary-value problems

(a) First boundary-value problem

Let the half-space z≤0 be occupied by a metal of relative permittivity ϵmet. 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)Embedded Image(2.1)where the reference relative permittivity dyadicEmbedded Image(2.2)indicates local orthorhombic symmetry. The dyadic functionEmbedded Image(2.3)withEmbedded Image(2.4)contains 2Ω 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 dyadicEmbedded Image(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 Embedded Image along the interface z=0, and attenuate as z→±∞. Therefore, in the region z≤0, the wavevector may be written asEmbedded Image(2.6)where Embedded Image; κ is complex valued; and Im(αmet)>0 for attenuation as z→−∞. Accordingly, the field phasors in the region z≤0 may be written asEmbedded Image(2.7)andEmbedded Image(2.8)where a1 and a2 are unknown scalars, Embedded Image and Embedded Image.

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

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 relationEmbedded Image(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 thatEmbedded Image(2.14)Both [N] and [Q] share the same eigenvectors, and their eigenvalues are also related. Let [τ](n), n∈[1,4], be the eigenvector corresponding to the nth eigenvalue σn of [N]; then, the corresponding eigenvalue αn of [Q] is given byEmbedded Image(2.15)

After ensuring that Im(α1,2)>0, we set (Martorell et al. 2006)Embedded Image(2.16)for SPP wave propagation, where b1 and b2 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,Embedded Image(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 Embedded Image, which may be rearranged asEmbedded Image(2.18)For a non-trivial solution, the 4×4 matrix [Y] must be singular, so thatEmbedded Image(2.19)is the dispersion equation for the SPP wave. This equation has to be solved in order to determine the SPP wavenumber 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 §2a 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 −Lmetz≤0 is occupied by the metal of relative permittivity ϵmet, the region 0≤zLstf by a chiral STF described by (2.1), and without significant loss of generality the half-spaces z≤−Lmet and zLstf by a homogeneous isotropic dielectric material of relative permittivity Embedded Image. Dissipation is considered to be negligibly small except in the metal; furthermore, ϵl must exceed the maximum of ϵa, ϵb and ϵc. We also take the chiral STF to contain an integral number of periods, so that lstf=Lstf/2Ω is an integer.

A plane wave is supposed to be launched in the half-space z≤−Lmet 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∈[0,2π) to the x-axis in the xy plane. The electromagnetic field phasors associated with the exciting plane wave are represented asEmbedded Image(2.20)The amplitudes of the s- and the p-polarized components of the exciting plane wave, denoted by as and ap, respectively, are assumed given, andEmbedded Image(2.21)The reflected electromagnetic field phasors are expressed asEmbedded Image(2.22)and the transmitted electromagnetic field phasors asEmbedded Image(2.23)

The procedure to determine the amplitudes rs, rp, ts and tp in terms of as and ap is standard (Lakhtakia & Messier 2005, ch. 10). It suffices to state here that the following set of four algebraic equations emerges (in matrix notation):Embedded Image(2.24)

The matrixEmbedded Image(2.25)depends on the refractive index nl, whereas the matrixEmbedded Image(2.26)captures the response of the metal layer. The matrix Embedded Image is calculated exactly in the same way as the matrix [N], except that the real-valued Embedded Image 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:Embedded Image(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, Rsp=|rsp|2 is the reflectance corresponding to the reflection coefficient rsp, and so on.

In general, the power density of the exciting plane wave is partially reflected into the half-space z≤−Lmet, partially transmitted into the half-space zLstf, and partially absorbed in the metal layer −Lmetz≤0. The absorbancesEmbedded Image(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 ϵa,b,c and Χ to Χv for CTFs to substantially apply for chiral STFs, since the vapour incidence angle Χv remains constant during the deposition of thin films of either kind.

Empirical relationships have been determined for titanium oxide CTFs at λo=633 nm by Hodgkinson et al. (1998) asEmbedded Image(3.1)Embedded Image(3.2)Embedded Image(3.3)andEmbedded Image(3.4)where Χv 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°; ϵmet=−56+21i, a typical value for aluminium at λo=633 nm; and ϵl=6.656, which is typical for zinc selenide.

The results shown in this section were computed for Χv=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 Lmet=15 nm, but the integer lstf∈[1,∞) was allowed to vary. The half-pitch Ω was normalized with respect to Ωo=197 nm; in the limit Ωo/Ω→0, the chiral STF transforms into a CTF.

(a) First boundary-value problem

Figure 2 shows the calculated values of ko/Re(κ) and 1/Im(κλo) versus Ωo/Ω for ψ=0°. The first quantity is the phase speed of the SPP wave relative to co (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 λo. From the figure, we conclude that SPP wave propagation is possible in five different modes labelled 1 to 5.

Figure 2

Characteristics of SPP wave propagation with respect to the inverse-periodicity parameter Ωo/Ω for the first boundary-value problem, when Χv=20°, h=1, γ=0°, λo=633 nm and ψ=0°. The metal is aluminium (ϵmet=−56+21i), whereas the constitutive parameters of the titanium oxide chiral STF are given by (3.3)–(3.4). (a) Phase speed relative to co and (b) e-folding distance relative to λo along the direction of travel. Cross, CTF; diamonds, mode 1; squares, mode 2; triangles, mode 3; circles, mode 4; rectangles, mode 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 2008a). 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 2008a); this result is also shown in figure 2 at Ωo/Ω=0.

Figure 2a indicates that the phase speed of any of the five modes is a decreasing function of Ω. As Ωo/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 2b shows the e-folding distance relative to λo 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 Ωo/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 Ωo/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 Ωo/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 α1 and α2 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 αmet=(koϵmetκ2)1/2, with Im(αmet)>0 indicating decay as z→−∞.

The two decay constants in the chiral STF, Im(α1) and Im(α2), are plotted against Ωo/Ω in figure 3a–e for modes 1–5, while the decay constant Im(αmet) in the metal is shown for all five modes in figure 3f. All modes exhibit a sharp peak in the variation of Im(α1) against Ωo/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(α2) 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 3b, Im(α1) and Im(α2) for mode 2 are very nearly equal and the markers for Im(α2) are obliterated by the bolder markers for Im(α1) at the lower end of the Ω range. In contrast to modes 2–5, Im(α2) for mode 1 reaches a plateau at low values of Ω.

Figure 3

Decay constants versus the inverse-periodicity parameter Ωo/Ω for the first boundary-value problem. See the caption of figure 2 for details. Decay constants Im(α1) and Im(α2) in the chiral STF for (a) mode 1 (filled diamonds, 2Ω Im(α1); open diamonds, 2Ω Im(α2)), (b) mode 2 (filled squares, 2Ω Im(α1); open squares, 2Ω Im(α2)), (c) mode 3 (filled triangles, 2Ω Im(α1); open triangle, 2Ω Im(α2)), (d) mode 4 (filled circles, 2Ω Im(α1); open circles, 2Ω Im(α2)) and (e) mode 5 (filled rectangles, 2Ω Im(α1); open rectangles, 2Ω Im(α2)); and (f) Im(αmet) in the metal for all modes. Diamonds, mode 1; squares, mode 2; triangles, mode 3; circles, mode 4; rectangles, mode 5.

Figure 3f shows that Im(αmet) is virtually independent of Ω for every SPP wave mode. Also, a comparison of figure 3f with figure 3a–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 parameterEmbedded Imagewas calculated at z=0− for every mode. Figure 4a–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.

Figure 4

The polarization parameter Embedded Image at z=0− versus the inverse-periodicity parameter Ωo/Ω for the first boundary-value problem. See the caption of figure 2 for details. (a) Mode 1, (b) mode 2, (c) mode 3, (d) mode 4 and (e) mode 5.

(b) Second boundary-value problem

Plots of the absorbances Ap and As 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 Embedded Image, which was then compared with the set of complex-valued quantities κ in figure 2.

Figure 5a exemplifies the dependences of Ap and As on θ, when the chiral STF in the Kretschmann configuration has two structural periods (i.e. lstf=2), γ=ψ=0° and Ωo/Ω=0.6. Two sharp peaks are present in the plot of Ap 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 As is extremely smooth with no peaks.

Figure 5

Absorbances Ap and As in the Kretschmann configuration as functions of the angle of incidence θ. The metal is aluminium (ϵmet=−56+21i, Lmet=15 nm), and the material occupying the two half-spaces z≤−Lmet and zLstf is zinc selenide (ϵl=6.656). The constitutive parameters of the titanium oxide chiral STF are given by (3.3)–(3.4) along with Χv=20°, γ=0° and h=1. The free-space wavelength λo=633 nm and ψ=0°. (a) lstf=2 and Ωo/Ω=0.6, (b) lstf=2 and Ωo/Ω=1.5 and (c) lstf=20 and Ωo/Ω=1.5.

Figure 5b was also drawn for lstf=2 and γ=ψ=0°, but when Ωo/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 Ap=0.31 at θ=30.37°. A slight rise in As also exists at the same angle of incidence. When the number lstf of structural periods in the chiral STF is increased, however, the peak in Ap corresponding to mode 2 becomes much more sharply defined. This is clearly indicated in figure 5c, which shows the two absorbances when lstf=20. The peak in Ap corresponding to mode 2 is now very sharp with a magnitude of 0.72. Also, the location of the peak in Ap has shifted to θ=28.25°. A peak in As at the same value of θ but of lesser magnitude can also be seen. The peak magnitude in Ap 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 Ap corresponding to modes 3 and 4 are present in figure 6a at θ=43.93° and 41.02°, respectively, when Ωo/Ω=0.2 and lstf=2. The peak value of Ap for mode 3 is 0.96, while that for mode 4 is 0.91. As with modes 1 and 2 in figure 5a, the As curve is nearly featureless over the same range. Similar behaviour can be seen in figure 5b for mode 5, also for lstf=2, but when Ωo/Ω=0.13. The maximum value of Ap (=0.94) occurs at θ=42.09°.

Figure 6

Absorbances as in figure 5 with lstf=2 for (a) Ωo/Ω=0.2 showing modes 3 and 4, and (b) Ωo/Ω=0.13 showing mode 5.

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 Embedded Image for SPP waves in the Kretschmann configuration were performed for many of the values of the inverse-periodicity parameter Ωo/Ω that appear in figures 2 and 3, as well as several other values of the parameter. The values of Embedded Image were determined by the angles of incidence corresponding to peaks in the Ap versus θ curves. Identical values of θ were found from peaks in the As versus θ curves when the peaks were well defined as in figure 5c; 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 Ωo/Ω. For all calculations of Embedded Image, we fixed Lmet=15 nm, h=1 and γ=ψ=0°. The thickness of the STF was set to Lstf=4Ω for all calculations, except for the three lowest values of Ω for mode 2 for which we set Lstf=40Ω in order to clearly observe a peak in the Ap versus θ curve.

A plot of the normalized phase speed Embedded Image against Ωo/Ω is shown in figure 7, wherein five SPP wave modes are identifiable. Visual comparison of Embedded Image values in figure 7 with the values of ko/Re(κ) in figure 2a 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 Ap 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 Embedded Image versus Ωo/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.

Figure 7

Relative phase speed Embedded Image of SPP wave modes in the Kretschmann configuration for the same parameter values as in figure 5. See the text for values of Lstf/(2Ω) used for these calculations. Diamonds, mode 1; squares, mode 2; triangles, mode 3; circles, mode 4; rectangles, mode 5.

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 vectorEmbedded Image(3.5)was calculated within the metal layer, for many instances at values of θ for which a sharp peak in Ap indicates the excitation of a SPP wave mode, with ap=1 V m−1 and as=0 V m−1. The x- and z-directed components of the time-averaged Poynting vector as functions of z/Lmet∈(−1,0) are shown in figure 8 for modes 1 and 2 for the same conditions as figure 5a. In figure 8a, the magnitude of Px 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 Pz plummets from 1.66 to 0.000068 W m−2 in figure 8b. The rise in the magnitude of Px accompanied by reduction in the magnitude of Pz 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.

Figure 8

The x- and z-directed components Px and Pz (in W m−2) of the time-averaged Poynting vector in the metal layer for the same parameters as in figure 5a, when ap=1 V m−1 and as=0 V m−1. (a,b) Mode 1 (θ=48.21°) and (c,d) mode 2 (θ=40.62°).

The time-averaged Poynting vector in the metal layer for mode 2 is shown in figure 8c,d for the same conditions as figure 5a, when θ=40.62°. The magnitude of Px 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 8c. The magnitude of Pz in figure 8d, however, drops from 1.85 to 0.046 W m−2. Again, the rise in |Px| accompanied by reduction in |Pz| 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 5c and with ap=1 V m−1, as=0 V m−1 and θ=28.25°—is plotted in figure 9. Although the magnitude of Px 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 Pz concurrently decreases significantly from 1.72 to 0.029 W m−2 as it does for modes 1 and 2 in figure 8.

Figure 9

The x- and z-directed components Px and Pz (in W m−2) of the time-averaged Poynting vector in the metal layer for the same parameters as in figure 5c, when ap=1 V m−1, as=0 V m−1 and θ=28.25°. The mode of the SPP wave excited is labelled 2.

The profile of the time-averaged Poynting vector in the metal layer is shown for mode 3 in figure 10a,b, when θ=43.93° and the other parameters are the same as for figure 6a. The magnitude of Px rises from 0.25 W m−2 on the zinc selenide side to 1.14 W m−2 on the chiral STF side, whereas |Pz| 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 10c,d, when θ=41.02°. Whereas |Px| rises from 0.23 W m−2 on the zinc selenide side to 0.89 W m−2 on the chiral STF side, |Pz| 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 6b.

Figure 10

The x- and z-directed components Px and Pz (in W m−2) of the time-averaged Poynting vector in the metal layer for the same parameters as in figure 6a, when ap=1 V m−1 and as=0 V m−1. (a,b) Mode 3 (θ=43.93°) and (c,d) mode 4 (θ=41.02°).

Figure 11

The x- and z-directed components Px and Pz (in W m−2) of the time-averaged Poynting vector in the metal layer for the same parameters as in figure 6b, when ap=1 V m−1, as=0 V m−1 and θ=42.09°. The mode of the SPP wave excited is labelled 5.

Although the results presented heretofore in this paper are for ψ=0°, similar results were obtained when ψ≠0°. Figure 12 shows the absorbances Ap and As for the Kretschmann configuration, with all parameters the same as for figure 5 except ψ=30°. Two sharp peaks in Ap are present in figure 12a when Ωo/Ω=0.6, similar to those in figure 5a when ψ=0°. For ψ=30°, the peaks in Ap are closer together at θ=42.06° and θ=47.8°, with magnitudes of 0.93 and 0.97, respectively. Figure 12b shows a dramatic reduction of the peak magnitude of Ap peak when Ωo/Ω=1.5, as was observed when ψ=0°. Following the trend seen for ψ=0°, the peak in Ap is accentuated for Ωo/Ω=1.5 when lstf is increased to 20. The peak Ap of 0.76 is now located at θ=29.15° and is also accompanied by a substantial peak in As.

Figure 12

Absorbances in the Kretschmann configuration as in figure 5, but with ψ=30°. (a) lstf=2 and Ωo/Ω=0.6, (b) lstf=2 and Ωo/Ω=1.5 and (c) lstf=20 and Ωo/Ω=1.5.

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.

References

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