Electromagnetic surface waves guided by the interface of a metal and a tightly interlaced matched ambidextrous bilayer

The existence and characteristics of electromagnetic surface waves (ESWs) whose propagation is guided by the planar interface of metal and a tightly interlaced matched ambidextrous bilayer (TIMAB) were theoretically investigated, a TIMAB being a periodic unidirectionally nonhomogeneous material whose unit cell consists of one period each of two structurally chiral materials that are identical except in structural handedness. Thus, the structural handedness flips in the center of the unit cell. A canonical boundary-value problem was formulated and a dispersion equation was solved, the ESWs being classified as surface-plasmon-polariton (SPP) waves. Flipping the structural handedness once in the unit cell can greatly enhance the number of possible SPP waves, one or more of which may be superluminal.


Introduction
Electromagnetic Surface Waves (ESWs) propagate guided by the interface of two distinct mediums with different constitutive relations [1][2][3].The distinguishing characteristic of an ESW is that its amplitude generally decays in the direction orthogonal to the interface on both sides.Depending on the constitutive properties of both partnering mediums, the ESW is given a different name [3][4][5].
ESWs have attracted a lot of attention because their fields have appreciable magnitudes only in a stripe of finite width, making them useful when the dimensions of a guiding structure are small and strong field confinement is needed [6][7][8][9].ESWs can theoretically propagate infinitely along the interface when the partnering mediums are non-dissipative [10,11].ESWs can propagate for several wavelengths when the mediums are moderately dissipative [12][13][14][15].
Typically, only one ESW can propagate in a given direction in the interface plane at a specified frequency, if both partnering mediums are homogeneous [1][2][3], but there are exceptions [16].However, if one of both of the partnering mediums are periodically nonhomogeneous in the direction normal to the interface, it is possible to trigger at the same time the propagation of more than one ESW, each with its own propagation characteristics [17][18][19][20][21][22][23].In some instances it has been observed that two ESWs propagate with complex-valued wavenumbers that are very close to each other giving rise to a sort of a doublet [24,25].The simultaneous triggering of several ESWs is interesting to enhance reliability and detection accuracy in measurements but also for the possibility of sensing multiple analytes with the same device [26].In addition, superluminal ESWs can exist [16,27], which is attractive for the excitation of Čerenkov radiation [28][29][30][31].
Experiments have been conducted with structurally chiral materials with a helicoidal variation of the relative permittivity direction along a fixed axis (identified as the z axis of a Cartesian coordinate system in this paper) that is oriented normal to the planar interface, in the present context [19,32,33].Fabricated using well-known methods of physical vapor deposition [34,35], these solid-state structurally chiral materials are called chiral sculptured thin films (CSTFs) [36].A CSTF is either structurally right-handed or structurally left-handed, but not both.The planar interface of a metal and a CSTF can guide multiple ESWs called surfaceplasmon-polariton (SPP) waves in a given direction at a specified frequency [19,26,32,33,37].Since chiral smectic liquid crystals (CSLCs) and cholesteric liquid crystals (CLCs) [38,39] are structurally chiral materials with a similar helicoidal variation of the relative permittivity direction along a fixed axis, experimental and theoretical findings about SPPwave propagation guided by a planar metal/CSTF interface are also applicable to metal/CSLC and metal/CLC interfaces.
CLCs and CSLCs have a 135-year-old history going back to 1888, when Reinitzer reported the discovery of temperaturedependent color effects in cholesteryl benzoate (C 34 H 56 O 2 ), an ester of cholesterol and benzoic acid [40,41].The history of CSTFs began 65 years ago with the fabrication of fluorite CSTF by Young and Kowal in 1959 [42,43], though research on these materials was dormant until revival in 1995 [36,44].A new type of structurally chiral material was proposed about 15 years ago [45] and first fabricated using physical vapor deposition [36] in 2022.Called a tightly interlaced matched ambidextrous bilayer (TIMAB), this material differs from CSTFs, CSLCs, and CLCs in one attribute: structural handedness.Whereas the structural handedness of a CSTF/CSLC/CLC is invariant along the z axis, the structural handedness flips in the center of the unit cell of a TIMAB because each unit cell consists of one period each of two structurally chiral materials that are identical except in structural handedness, as depicted in figure 1.As a consequence, a TIMAB exhibits a bandgap for both left-and right-circularly polarized incident plane waves [46], whereas a CSTF/CSLC/CLC exhibits a bandgap for either left-or right-circularly polarized incident plane waves [47][48][49][50][51][52][53][54][55].This difference has been shown both theoretically and experimentally [46].
In our quest to distinguish TIMABs from CSTFs, CSLCs, and CLCs in their optical response characteristics, here we theoretically examine the propagation of SPP waves guided by a planar metal/TIMAB interface, and contrast it with the propagation of SPP waves guided by the planar interface of a metal and a CSTF [19,26,32,33,37].This paper is organized as follows.First, the canonical boundary-value problem is formulated to yield the dispersion equation for SPPwave propagation.Next, this dispersion equation is numerically solved to find the complex-valued wavenumbers of SPP waves.Then, the wavenumbers are analyzed to extract the phase speed and propagation distance of every SPP wave guided by the metal/TIMAB interface.The paper concludes with some remarks on the significance of the obtained results.
An exp (−iωt) dependence on time t is used, with ω as the angular frequency and i = √ −1 as the imaginary unity; k 0 = ω √ ε 0 µ 0 is the free-space wavenumber and λ 0 = 2π/k 0 is the free-space wavelength, with µ 0 as the free-space permeability and ε 0 as the free-space permittivity; c 0 = 1/ √ ε 0 µ 0 is the speed of light in free space; and η 0 = √ µ 0 /ε 0 is the intrinsic impedance of free space.Vectors are represented in boldface; dyadics are doubly underlined; and Cartesian unit vectors are denoted by u x , u y , and u z .Column vectors are in boldface and bracketed, and matrices are double underlined and bracketed.Finally, the asterisk denotes the complex conjugate, and the superscript T the matrix transpose.

Constitutive parameters
The geometry of the canonical boundary-value problem is schematically illustrated in figure 2. The half space z < 0 is occupied by a metal with complex-valued relative permittivity ε m , whereas the TIMAB occupies the half-space z > 0. The relative permittivity dyadic ε TIMAB (z) of the TIMAB is periodic along the z axis with so that the TIMAB period is 4Ω.In the first unit cell 0 ⩽ z ⩽ 4Ω we have ( In this equation, the reference relative permittivity dyadic is where ε a , ε b , and ε c are the three principal relative permittivity scalars.The rotation dyadic is a product of two dyadics, with denoting a rotation by an angle πυζ about the z axis and denoting a rotation by angle χ about the y axis.In equations ( 4) and ( 5), the parameter υ = 1 for structural right-handedness whereas υ = −1 for structural left-handedness.
The TIMAB is structurally right-handed (resp.left-handed) in the region 0 ⩽ z ⩽ 2Ω if h = +1 (resp.h = −1) in equation ( 2).Also, s = −1 so that the TIMAB has the opposite structural handedness in the region 2Ω ⩽ z ⩽ 4Ω relative to that in the region 0 ⩽ z ⩽ 2Ω.In contrast, s = 1 for CSTFs, CSLCs, and CLCs so that their period is 2Ω along the z axis.

Electromagnetic fields
Let us consider an SPP wave propagating in the xy plane along the generic direction identified by the unit vector u prop = u x cos ψ + u y sin ψ.After defining u s = −u x sin ψ + u y cos ψ and the wavevector k m = qu prop − α m u z , where k 2 0 ε m = q 2 + α 2 m and q is the complex-valued SPP wavenumber, the electric and magnetic field phasors in the metallic half-space are expressed [3] and where A 1 and A 2 are unknown coefficients to be determined.Note that Im (α m ) > 0, in compliance with the condition of attenuation as z → −∞.
In the half-space z ⩾ 0 occupied by the TIMAB, the electric and magnetic field phasors can be written as and The Cartesian components e x,y (z) and h x,y (z) of the functions e(z) and h(z), respectively, in accordance with the source-free Maxwell equations, can be arranged in a 4×4 matrix ordinary differential equation as [3,36] where is a column vector.A detailed expression for the 4 × 4 matrix is provided in the appendix.Finally, [f (4Ω)] can be obtained after solving numerically equation ( 9) using the piecewise uniform approximation technique [36] as where is the characteristic matrix of the unit cell of the TIMAB.

Dispersion equation
Boundary conditions on the interface z = 0 require where from equations (7a) and (7b).Substitution of equations ( 12) and ( 14) in equation ( 13) leads to the matrix equation where This is the dispersion equation that must be solved to evaluate the SPP wavenumber q.Not only do its solutions depend on the constitutive parameters of the metal and the TIMAB as well as on ω and Ω, but also on the direction of propagation in the xy plane as delineated by the angle ψ.A convenient way to solve the dispersion equation is to use the search method [56].

Numerical results
Let us now present the solutions of equation ( 16) in terms of the normalized wavenumber q = q/k 0 as functions of the normalized angle ψ/π and limiting the range of data presented to 0 ⩽ ψ/π ⩽ 1.Note that if q is a solution for a certain ψ, then −q is the solution for ψ + π in view of the dependences of the matrix [ P ] on ψ and q; also u prop . Once q is known, the phase speed v ph = ω/Re (q) and the propagation distance ∆ prop = 1/Im (q) of that SPP wave can be calculated.It is worth pointing out that the solutions reported here are the ones that we were able to find, but this does not mean that other solutions cannot exist.In this connection, the search for the solutions was restricted to a reasonable limited range: 0.8 ⩽ Re (q) ⩽ 3.
For all calculations, we fixed λ 0 = 633 nm.The metal was taken to be silver (ε m = −14.461+ i 1.1936) [57] corresponding to a skin depth .5 nm at the chosen wavelength.The TIMAB was supposed to have h = 1, Ω = 162 nm, ε a = 6.65313 + i 0.0429696, ε b = 7.35561 + i 0.050978, ε c = 6.53285 + i 0.042055 and χ = 50 • [54].For calculating the transfer matrix [Q (4Ω)], the period 4Ω of the TIMAB was subdivided in 324 homogeneous layers each of thickness 2 nm.We verified that this value is adequate for ensuring the convergence of the piecewise uniform approximation technique [36].
We also comment on the spatial profiles along the z axis of the components of the time-averaged Poynting vector for representative solutions.With the Cartesian coordinate system defined by the unit vectors u prop , u s , and u z , the components P prop (0, 0, z) = P (0, 0, z) • u prop , P s (0, 0, z) = P (0, 0, z) • u s , and P z (0, 0, z) = P (0, 0, z) • u z were computed by solving equation ( 15) after choosing B 2 = 1.Branches 3 and 5 possess both Re (q) and ∆ prop very close to each other for ψ/π ∈ [0.333, 0.422], suggesting the existence of a doublet as already found in previous research [24,25].The same conclusion follows from examining branches 6 and 7 in the angular range ψ/π ∈ [0.389, 0.489].Even more interestingly, branches 3 and 5 intersect in figure 3 (right) at ψ/π = 0.4, so that the solutions on the two branches have identical propagation distances but slightly different phase speeds.Likewise, branches 6 and 7 intersect in figure 3 (right) at ψ/π = 0.446.

Metal/TIMAB interface
Further insight is provided by an examination of the spatial profile of the time-averaged Poynting vector for two SPP waves with the same ∆ prop .In figure 4 the Cartesian components P prop,s,z (0, 0, z) are plotted against z for the SPP waves on branches labeled 6 and 7 in figure 3 at ψ/π = 0.446.Both SPP waves have the same ∆ prop but differ in Re (q); i.e. both decay at the same rate along the direction of propagation but have different phase speeds.According to figure 4, the time-averaged Poynting vector decays along the z axis differently for the two SPP waves.Of the two SPP waves, the one with the lower phase speed i.e. with higher Re (q), has a much slower decay along the z axis in the TIMAB half-space.Indeed, the spatial profiles of the Cartesian components are presented for 50 structural periods (z ∈ [0, 200Ω]).And the SPP wave on branch 6 has substantial power density at z = 200Ω in figure 4 (left), in comparison to the SPP wave on branch 7 in figure 4 (right).
In addition to the 13 branches in figure 3, there is a 14th branch.As is clear from figure 5, every solution on branch 14 and 14 ′ is a superluminal solution because Re (q) < 1 =⇒ v ph > c 0 .However, the propagation distance ∆ prop on branch 14 and 14 ′ stays in the range of a few micrometers.This type of solution exists also in the ψ-range where six solutions have been identified in figure 3, thus bringing to seven the multiplicity of solutions in that ψ-range.Variations of the components Pprop(0, 0, z) (black solid lines), Ps(0, 0, z) (blue dashed-dotted lines), and Pz(0, 0, z) (red dashed lines) of P(0, 0, z) with z in the TIMAB half-space, for the SPP wave on branch 14 guided by the silver/TIMAB interface when ψ/π = 0.733.
Figure 6 provides the spatial profiles of the Cartesian components P prop,s,z (0, 0, z) for the superluminal SPP wave on branch 14 at ψ/π = 0.733.The phase speed is very high (actually, greater than c 0 ) and the decay on the z axis is very slow.A comparison of figures 4 and 6 seems to indicate that the higher the phase speed the higher the decay rate.
However, that is not a general conclusion.As a counter-example, we examined the fields for the silver/-TIMAB interface on branch 11 for (a) ψ/π = 2/9 and (b) ψ/π = 19/45.We have q = 1.721501 + i 1.3946 × 10 −2 for case (a) but q = 1.721501 + i1.3236 × 10 −2 for case (b).We found that the fields decay on the z axis four times faster for case (a) than for case (b), as is evident from figure 7. So, even with the same phase speed, we have two distinct decay rates on the z axis in the TIMAB half-space.

Comparison with metal/CSTF interface
To better understand the role of interlacing in the existence of SPP waves, we compare here the results obtained for the silver/TIMAB interface with those for a silver/CSTF interface.The CSTF is identical to the TIMAB except that s = 1 in equation ( 2), so that the structural period of the CSTF is half of that of the TIMAB.
Figure 8 presents Re (q) and ∆ prop as functions of ψ/π for the metal/CSTF interface.There are four major differences between figures 3 and 8 as follows: 1.The number of solution branches shrivels to three for the metal/CSTF interface compared to 14 for the metal/TIMAB interface.2. For a specified ψ, the maximum number of SPP waves possible is just two for the metal/CSTF interface compared to seven for the metal/TIMAB interface.3.For a specified ψ, the metal/TIMAB interface can guide at least one SPP wave, but the metal/CSTF interface may not guide any SPP wave.4. The metal/TIMAB interface can guide a superluminal SPP wave, but evidently the very closely related metal/CSTF cannot.Parenthetically, metal/CSTF interfaces can guide superluminal waves for other constitutive parameters [37].
To clarify the effect of flipping of the TIMAB, we compared the Poynting-vector profiles for a SPP wave guided by the metal/TIMAB interface and one by the metal/CSTF interface, with solutions having Re (q) extremely close.Branch 11 in figure 3 when ψ/π = 0.176, has a solution q = 1.726105 + i 1.391177 × 10 −2 whereas branch 2 in figure 8, when ψ/π = 0.177, has q = 1.726105 + i1.330370 × 10 −2 .
In figures 9(a) and (c) the Cartesian components P prop,s,z (0, 0, z) are plotted against z for the SPP waves on branch labeled 11 in figure 3 at ψ/π = 0.176.In figures 9(b) and (d) for the SPP wave on branch labeled 2 in figure 8, the same is done for ψ/π = 0.177.The SPP waves have the same Re (q) but differ in ∆ prop thus having different decay rates along the direction of propagation but same phase speed.
According to figure 9, the time-averaged Poynting vector decays along the z axis differently for the two SPP waves.Among the two SPP waves, the one with longer propagation distance ∆ prop , has a much slower decay along the z axis in the CSTF half-space.Indeed, the spatial profiles of the Cartesian components are presented for 50 structural periods (z ∈ [0, 200Ω]) and the SPP wave on branch 2 for the metal/CSTF interface has substantial power density at z = 200Ω in figure 9(d), in comparison to the SPP wave on branch 11 for the metal/TIMAB interface in figure 9(c).

Conclusions
In this paper we have formulated the canonical boundaryvalue problem to obtain the dispersion equation of SPP waves guided by the planar interface of silver and a TIMAB.The dispersion equation has been numerically solved to find the complex wavenumbers q of all the possible SPP waves.The solutions for the silver/TIMAB interface have been compared with those for the silver/CSTF interface, in order to highlight the differences brought by the interlacing of the partnering chiral material.Analysis of numerical results shows that interlacing in the TIMAB significantly increases the number of possible SPP waves possibly guided in a specific direction by the silver/TIMAB interface compared to the silver/CSTF interface.In addition, a metal/TIMAB interface can guide a superluminal SPP wave, but evidently the very closely related metal/CSTF cannot.

Figure 2 .
Figure 2. Schematic of the canonical boundary-value problem.The half-space z < 0 is occupied by a metal and the half space z > 0 by a TIMAB.

Figure 3 .
Figure 3. Variations of (left) Re (q) and (right) ∆prop with ψ/π of SPP waves guided by the planar silver/TIMAB interface.The solutions of the dispersion equation are organized in 13 numbered branches.Ranges of ψ/π with the highest number of SPP waves are shaded gray.

Figure 8 .
Figure 8. Variations of (left) Re (q) and (right) ∆prop with ψ/π of SPP waves guided by the planar silver/CSTF interface.The solutions of the dispersion equation are organized in three numbered branches.