Compound surface-plasmon-polariton waves guided by a thin metal layer sandwiched between a homogeneous isotropic dielectric material and a structurally chiral material
Francesco Chiadini, Vincenzo Fiumara, Antonio Scaglione, Akhlesh Lakhtakia
aa r X i v : . [ phy s i c s . op ti c s ] S e p Compound surface-plasmon-polariton waves guided by a thinmetal layer sandwiched between a homogeneous isotropic dielectricmaterial and a structurally chiral material
Francesco Chiadini , Vincenzo Fiumara , Antonio Scaglione , Akhlesh Lakhtakia D epartment of Industrial Engineering, University of Salerno,via Giovanni Paolo II, 132 - Fisciano (SA), 84084, ITALYe-mail: [email protected], [email protected] S chool of Engineering, University of Basilicata,Viale dell’Ateneo Lucano 10, 85100 Potenza, ITALYemail: vfi[email protected] D epartment of Engineering Science and Mechanics, Pennsylvania StateUniversity,University Park, PA 16802–6812, USAe-mail: [email protected] K eywords: compound surface wave, metal, structurally chiral material, surface-plasmon-polariton wave Abstract
Multiple compound surface plasmon-polariton (SPP) waves can beguided by a structure consisting of a sufficiently thick layer of metal sand-wiched between a homogeneous isotropic dielectric (HID) material and adielectric structurally chiral material (SCM). The compound SPP wavesare strongly bound to both metal/dielectric interfaces when the thick-ness of the metal layer is comparable to the skin depth but just to oneof the two interfaces when the thickness is much larger. The compoundSPP waves differ in phase speed, attenuation rate, and field profile, eventhough all are excitable at the same frequency. Some compound SPPwaves are not greatly affected by the choice of the direction of propaga-tion in the transverse plane but others are, depending on metal thickness.For fixed metal thickness, the number of compound SPP waves dependson the relative permittivity of the HID material, which can be useful forsensing applications.
The propagation of a surface-plasmon-polariton (SPP) wave is guided by a pla-nar metal/dielectric interface, the field strengths decaying exponentially awayfrom the interface in both materials [1]. If the dielectric material is homoge-neous and isotropic, only one SPP wave can propagate parallel to the interfaceat a specific frequency. If the dielectric material is periodically nonhomogeneousin the direction normal to the interface, multiple SPP waves that differ in po-larization state, phase speed, attenuation rate, and field profile, can be guidedsimultaneously by the interface at a specific frequency [2].1his multiplicity is attractive for optical sensing applications, as the sensi-tivity and reliability of sensing can be enhanced thereby [3]. Furthermore, thenumber of the simultaneously detected analytes can than also be greater thanone, the usual number [4]. The same multiplicity will enhance SPP-wave-basedmicroscopy [5] and communications [6] as well.For about three decades, one way to further increase the number of SPPwaves is to interpose a thin metal layer between two suitably chosen homoge-neous dielectric materials [7, 8]. The two SPP waves, each guided separatelyby a metal/dielectric interface when the metal layer is thick, hybridize intocompound SPP waves. Motivated by that possibility, recently we analyzed thepropagation of multiple compound SPP waves guided by an isotropic metal layersandwiched between a homogeneous isotropic dielectric (HID) material and aperiodically multilayered isotropic dielectric (PMLID) material. We demon-strated that compounding occurs even when one of the two metal/dielectricinterfaces is capable of guiding multiple SPP waves by itself [9].In this paper, we extend the scope of the multiple-compound-SPP-wave phe-nomenon to encompass anisotropic dielectric materials by replacing the PM-LID material by a dielectric structurally chiral material (SCM). Exemplified byReusch piles [10, 11], cholesteric liquid crystals [12, 13], and chiral sculpturedthin films [14], a dielectric SCM is anisotropic and helically nonhomogeneousalong a fixed axis. Whereas a planar metal/HID interface by itself can guide asingle SPP wave, the planar interface of a metal and a dielectric SCM by itselfcan guide multiple SPP waves [15].This paper is organized as follows. A description of the boundary-value prob-lem is provided in Section 2, but we have elected not to describe the procedure toobtain the dispersion equation for compound SPP waves, as the methodology isavailable in detail elsewhere [2, Chap. 3]. In Section 3 numerical results showingthe compounding of the SPP waves guided by the metal/HID and metal/SCMinterfaces are presented in relation to both the thickness of the metal layer,the relative permittivity of the HID, and the direction of propagation in thetransverse plane. Concluding remarks are given in Section 4.An exp ( − iωt ) dependence on time t is implicit, with ω denoting the an-gular frequency and i = √−
1. Furthermore, k = ω √ ε µ and λ = 2 π/k ,respectively, represent the wavenumber and the wavelength in free space, with µ as the permeability and ε as the permittivity of free space. Vectors are inboldface; dyadics are double underlined; Cartesian unit vectors are denoted by u x , u y , and u z ; and the asterisk denotes the complex conjugate. The geometry of the boundary-value problem for the propagation of compoundSPP waves is schematically illustrated in Figure 1. The half space z < − L isoccupied by a HID material with real-valued relative permittivity ε d >
0. Anisotropic metal layer of thickness L and complex-valued relative permittivity ε m separates the HID material from a dielectric SCM which occupies the half space2 IDmetal z = - L dielectricSCM z = 0 xzy Figure 1: Schematic of the boundary-value problem. A metal layer of thickness L separates a half space occupied by a HID material and another half spaceoccupied by a dielectric SCM. z >
0. The chosen SCM is periodically nonhomogeneous along the z axis withperiod 2Ω and can be characterized by the relative-permittivity dyadic [2] ε SCM ( z ) = S z ( hπz/ Ω) • S y ( χ ) • ε oref • S − y ( χ ) • S − z ( hπz/ Ω) , (1)where h = 1 for a right-handed SCM or h = − ε oref = ε a u z u z + ε b u x u x + ε c u y u y (2)contains the principal relative permittivity scalars ε a , ε b , and ε c ; and the rota-tion dyadics S z ( ζ ) = u z u z + ( u x u x + u y u y ) cos ζ + ( u y u x − u x u y ) sin ζ (3)and S y ( χ ) = u y u y + ( u x u x + u z u z ) cos χ + ( u z u x − u x u z ) sin χ (4)capture the helical nonhomogeneity. Whereas ε a = ε c = ε b and χ = 0 forcholesteric liquid crystals, ε a = ε b = ε c and χ ∈ (0 , π/
2] for chiral smecticliquid crystals [16] and chiral sculptured thin films [17]. All three materials areassumed to be nonmagnetic. 3ithout loss of generality, we consider a compound SPP wave propagatingparallel to the unit vector u x cos ψ + u y sin ψ , ψ ∈ [0 ◦ , ◦ ), in the transverse(i.e., xy ) plane and decaying far away from the metal layer. The electric andmagnetic field phasors can be written everywhere as E ( x, y, z ) = [ e x ( z ) u x + e y ( z ) u y + e z ( z ) u z ] ·· exp[ iq ( x cos ψ + y sin ψ )] H ( x, y, z ) = [ h x ( z ) u x + h y ( z ) u y + h z ( z ) u z ] ·· exp[ iq ( x cos ψ + y sin ψ )] ,z ∈ ( −∞ , ∞ ) , (5)where q is the complex-valued wavenumber. The procedure to obtain a disper-sion equation for the wavenumber q is provided in detail elsewhere [2]. Once q has been numerically determined from that dispersion equation, the piecewisedetermination of the functions e x,y,z ( z ) and h x,y,z ( z ) is also possible. In this section, we present the normalized solutions ˜ q = q/k of the dispersionequation for compound SPP waves. The equation was solved numerically usingMathematica (Version 10) on a Windows XP laptop computer. For all calcu-lations, we fixed λ = 633 nm. The HID material was taken to be SF11 glass( ε d = 3 . ε d ∈ [1 ,
4] for the results presented inSec. 3.2. The metal was taken to be silver ( ε m = − . i . δ m = 1 / Im (cid:0) k √ ε m (cid:1) = 26 .
47 nm at the chosen wavelength. The dielectricSCM was supposed to be a chiral sculptured thin film made of patinal titaniumoxide with Ω = 135 nm, h = 1, ε a = 2 . ε b = 3 . ε c = 2 . χ = 37 . ◦ [18]. The angle ψ was set equal to 0 ◦ for the results presented inSecs. 3.1 and 3.2. Results for ψ = 0 ◦ are presented in the Sec. 3.3. The searchfor solutions was restricted to 0 < Re(˜ q ) ≤ . q of the dispersion equation was found, we determined thefield phasors E ( r ) and H ( r ), and then computed the Cartesian components ofthe time-averaged Poynting vector P ( x, y, z ) = 12 Re [ E ( x, y, z ) × H ∗ ( x, y, z )] . (6)Profiles of the Cartesian components of P ( x, y, z ) are presented for representa-tive solutions in this section. Figures 2 and 3, respectively, show the real part of the normalized wavenumber˜ q and the propagation distance ∆ prop = 1 / Im( q ) in the xy plane found for allsolutions as the thickness L of the metal layer was varied from 10 nm to 60 nm,4 L (nm) R e ( q ) ~ Figure 2: Variation of Re(˜ q ) with the thickness L of the metal layer when ε d = 3 . ψ = 0 ◦ .with ψ = 0 ◦ fixed. As many as three different compound SPP waves can beguided by the chosen HID/metal/SCM structure for L ∈ [10 ,
60] nm. Thesesolutions are organized in three branches labeled 1 to 3 in Figs. 2 and 3. Eachbranch spans the range [ L th ,
60 nm], the branch being absent for
L < L th . Wedetermined that L th ≃
44 nm for branch 1, L th ≃
24 nm for branch 2, and L th ≃
12 nm for branch 3.As L increases from L th , Re (˜ q ) and, therefore, the phase speed v ph = c / Re (˜ q ), are practically constant on branches 1 and 2; however, the propaga-tion distance ∆ prop decreases. Branch 3 has quite different characteristics: both v ph and ∆ prop increase as L increases. However, for a fixed value L >
44 nm,the solution belonging to branch 1 exhibits the highest value of ∆ prop .Table 1 shows the relative wavenumbers ˜ q of the compound SPP wavescomputed for two different thicknesses of the metal layer, L = 60 nm ( > δ m )and L ∼ L th , while ψ = 0 ◦ is fixed. The table also lists the values of ˜ q for (a)the SPP waves guided by the metal/SCM interface by itself [15] and (b) the sole p -polarized SPP wave guided by the metal/HID interface by itself [4]. When L = 60 nm, the relative wavenumbers of the compound SPP waves belonging tothe branches 1 and 2 are very close to the ones of the SPP wave guided by themetal/SCM interface by itself, while the relative wavenumber of the compoundSPP wave belonging to the branch 3 is very close to the one of the SPP wave5 −1 L (nm) Δ p r op ( μ m ) Figure 3: Variation of the propagation distance ∆ prop with the thickness L ofthe metal layer when ε d = 3 . ψ = 0 ◦ .6able 1: Values of ˜ q computed for compound SPP waves belonging tothe three solution branches when ψ = 0 ◦ and the metal layer has eithera thickness L = 60 nm or L ≃ L th , where L th is the minimum thick-ness for which a solution on a specific branch was found. Solutionsobtained for SPP waves guided by either the metal/SCM interfacealone or the metal/HID interface alone are also provided. Branch → L ≃ L th . i . . i . . i . L = 60 nm 1 . i . . i . . i . q met/SCM . i . . i . q met/HID - - 2 . i . L , the compound SPP waves on branches 1 and 2 propagate boundpredominantly to the metal/SCM interface while the compound SPP wave onbranch 3 propagates bound predominantly to the metal/HID interface. Bothsituations are indicative of a weak coupling between the two metal/dielectricinterfaces z = − L and z = 0. As L decreases, solutions on branch 1 remainstrongly bound to the metal/SCM interface, but solutions on branches 2 and 3show stronger coupling between the two metal/dielectric interfaces.The foregoing conclusions are confirmed by the plots of the spatial profilesof the Cartesian components of the time-averaged Poynting vector P ( x, y, z ).Figure 4 shows the spatial variations of P x (0 , , z ), P y (0 , , z ) and P z (0 , , z ) forrepresentative compound SPP waves on the three solution branches, when when ψ = 0 ◦ . From the figure it is evident that the compound SPP wave belonging tobranch 1 propagates bound predominantly to the metal/SCM interface with itsenergy confined mostly to several periods of the SCM close to the metal. Thecompound SPP wave on branch 2 is bound to the metal/SCM interface onlywhen L is large enough, and most of its energy is contained in the first periodof the SCM. As L decreases, the wave becomes increasingly bound also to themetal/HID interface, thereby indicating a significant coupling between the twometal/dielectric interfaces. The compound SPP wave belonging to branch 3propagates bound predominantly to the metal/HID interface when L = 60 nmwith a power density confined almost totally in the HID material. The couplingbetween the two interfaces is clearly evident when L = 12 nm, the wave thenpropagating bound to both metal/dielectric interfaces and its energy distributedalmost equally in both the HID material and the dielectric SCM.7a) −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.20−0.0050.0050.0100.0150.0200.0250.030 z ( μ m) ( , , z ) ( W m − ) P x , y , z (b) −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2−0.00500.0050.0100.0150.0200.0250.030 z ( μ m) ( , , z ) ( W m − ) P x , y , z (c) −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50 z ( μ m)−0.0050.0050.0100.0150.0200.0250.030 ( , , z ) ( W m − ) P x , y , z (d) −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50 z ( μ m)−0.0050.0050.0100.0150.0200.0250.030 ( , , z ) ( W m − ) P x , y , z (e) −0.2−0.15−0.1−0.05 0 0.05 0.1 0.15 0.2 0.250 z ( μ m)−0.0050.0050.0100.0150.0200.0250.030 ( , , z ) ( W m − ) P x , y , z (f) −0.4 −0.3 −0.2 −0.1 0 0.1 0.2−0.0200.020.040.060.080.100.12 z ( μ m) ( , , z ) ( W m − ) P x , y , z Figure 4: Variations of P x (0 , , z ) (blue solid lines), P y (0 , , z ) (black dotted-dashed lines), and P z (0 , , z ) (red dashed lines) with respect to z for the com-pound SPP waves when ψ = 0 ◦ and (a) L = 48 nm or (b) L = 60 nm forbranch 1; (c) L = 24 nm or (d) L = 60 nm for branch 2; and (e) L = 12 nmor (f) L = 60 nm for branch 3. All field phasors have been normalized so thatnumerical comparisons are possible. 8 R e ( q ) ~ ε d Figure 5: Variation of Re(˜ q ) with the relative permittivity ε d of the HID materialwhen L = 25 nm and ψ = 0 ◦ . In Sec. 3.1 we showed that the number of the compound SPP waves guidedjointly by the two metal/dielectric interfaces reduces (and even vanishes) as thethickness L of the metal layer decreases; moreover, no compound SPP waveexists when L < δ m / .
2. In order to investigate the influence of the relativepermittivity of the HID material on the number of compound SPP waves, wefixed ψ = 0 ◦ and L = 25 nm (i.e., L is slightly smaller than δ m ), and varied ε d from 1 to 4.Figures 5 and 6, respectively, show plots of Re(˜ q ) and ∆ prop as functions of ε d . The solutions are organized in three branches labeled 1 to 3. Two solutionsexist in two distinct ε d -intervals: (i) 1 ≤ ε d . . . . ε d . .
1. Inthe remaining parts of the range 1 ≤ ε d ≤
4, only one compound SPP waveexists. As ε d increases, the phase speed v ph increases on all three branches,the increase being almost linear on branches 1 and 2. As ε d increases, thepropagation distance ∆ prop first decreases and then increases on branches 1 and2, but decreases almost linearly on branch 3.Figure 7 shows the spatial profiles of the Cartesian components of the time-averaged Poynting vector P (0 , , z ) for compound SPP waves belonging to thebranches 1 and 3 when ε d = 1 .
5. Likewise, Fig. 7 shows the spatial profiles of9 −1 ε d Δ p r op ( μ m ) Figure 6: Variation of the propagation distance ∆ prop with the relative permit-tivity ε d of the HID material when L = 25 nm and ψ = 0 ◦ .10he Cartesian components P (0 , , z ) for compound SPP waves belonging to thebranches 2 and 3 when ε d = 2 .
8. Due to the significant coupling between the twodielectric/metal interfaces, significant fractions of the energy of the compoundSPP waves reside in the both the HID material and the dielectric SCM. When ε d = 1 .
5, most of the energy resides in the HID material for the wave on branch1 and in the dielectric SCM for the wave on branch 3. When ε d = 2 .
8, theenergy of the wave is distributed almost equally in both dielectric materials.