Surface-plasmon-polariton waves guided by the uniformly moving planar interface of a metal film and dielectric slab
aa r X i v : . [ phy s i c s . op ti c s ] J un Surface–plasmon–polariton waves guided by theuniformly moving planar interface of a metal film anddielectric slab
Tom G. Mackay School of Mathematics and Maxwell Institute for Mathematical SciencesUniversity of Edinburgh, Edinburgh EH9 3JZ, UK and
NanoMM — Nanoengineered Metamaterials GroupDepartment of Engineering Science and MechanicsPennsylvania State University, University Park, PA 16802–6812, USA
Akhlesh Lakhtakia NanoMM — Nanoengineered Metamaterials GroupDepartment of Engineering Science and MechanicsPennsylvania State University, University Park, PA 16802–6812, USA
Abstract
We explored the effects of relative motion on the excitation of surface–plasmon–polariton (SPP) wavesguided by the planar interface of a metal film and a dielectric slab, both materials being isotropic andhomogeneous. Electromagnetic phasors in moving and non–moving reference frames were related directlyusing the corresponding Lorentz transformations. Our numerical studies revealed that, in the case of auniformly moving dielectric slab, the angle of incidence for SPP-wave excitation is highly sensitive to (i) theratio β of the speed of motion to speed of light in free space and (ii) the direction of motion. When thedirection of motion is parallel to the plane of incidence, the SPP wave is excited by p -polarized (but not s -polarized) incident plane waves for low and moderate values of β , while at higher values of β the totalreflection regime breaks down. When the direction of motion is perpendicular to the plane of incidence, theSPP wave is excited by p -polarized incident plane waves for low values of β , but s -polarized incident planewaves at moderate values of β , while at higher values of β the SPP wave is not excited. In the case of auniformly moving metal film, the sensitivity to β and the direction of motion is less obvious. Keywords: surface plasmon polariton, Lorentz transformation, modified Kretschmann configuration
In quantum mechanical terms, a surface plasmon–polariton (SPP) is a quasiparticle. Created by the in-teraction of photons in a dielectric material and electrons in a metal, a SPP travels along the interface ofthe metal and the dielectric material [1]. In classical terms, a SPP wave propagates guided by the interfacewith an amplitude that decreases exponentially with distance from the interface. Over the past few decades,SPP waves have been widely investigated, in part because of the opportunities that they present for optical E–mail: [email protected]. E–mail: [email protected] ×
2, 4 × × × I = ˆ x ˆ x + ˆ y ˆ y + ˆ z ˆ z and the null dyadic being0. The permittivity and permeability of free space are denoted by ǫ and µ ; the free–space wavenumber atangular frequency ω is k = ω √ ǫ µ ; and c = ω/k . The partition of space into four distinct layers provides the backdrop for our analysis:(i) the half–space z < ǫ i with respectto the inertial reference frame Π, which is the laboratory frame;(ii) a thin metal film of relative permittivity scalar ǫ m with respect to the inertial reference frame Π fillsthe layer 0 < z < L m ;(iii) a dielectric slab of relative permittivity scalar ǫ ′ d with respect to the inertial reference frame Π ′ fillsthe layer L m < z < L Σ ; and(iv) the half–space z > L Σ is occupied by a dielectric material with relative permittivity scalar ǫ t withrespect to the inertial reference frame Π.All four materials are homogeneous and their relative permittivity scalars are frequency-dependent in theirrespective co-moving inertial reference frames. Dissipation is small enough to be ignored in both materialsoccupying the two half–spaces as well as in the dielectric slab. The reference frame Π ′ moves at uniformvelocity v = v ˆ v , in the xy plane, with respect to the laboratory frame Π. This setup, schematically illustratedin Fig. 1 for the case ˆ v = ˆ x , represents a modification [16] of the standard Kretschmann configuration [4],suitable for launching SPP waves guided by the planar interface of the metal film and the dielectric slab;i.e., along z = L m .Suppose that an arbitrarily polarized plane wave in the half–space z < xz plane, making an angle θ i ∈ [0 , π/
2) to the + z axis, in the laboratory frameΠ. In consequence, a reflected plane wave is generated in the half–space z < z > L Σ . In the laboratory frame, the total electric field phasor in the half–space z < E ( r , ω ) = (cid:8) [ a s u y + a p p + ( θ i )] exp ( ik √ ǫ i z cos θ i ) + [ r s u y + r p p − ( θ i )] exp ( − ik √ ǫ i z cos θ i ) (cid:9) × exp ( iκx ) , z < , (1)whereas that in the half–space z > L Σ may be written as E ( r , ω ) = [ t s u y + t p p + ( θ t ) ] exp [ ik √ ǫ t ( z − L Σ ) cos θ t ] exp ( iκx ) , z > L Σ . (2)2ere p ± ( θ ) = ∓ u x cos θ + u z sin θ , κ = k √ ǫ i sin θ i and the angle of transmission θ t in frame Π satisfies √ ǫ i sin θ i = √ ǫ t sin θ t . (3)Our task is to relate the complex–valued reflection and transmission amplitudes—namely r s , r p , t s and t p —to the corresponding amplitudes a s and a p of the s - and p -polarized components of the incident planewave.For later use, we note that the wavevector k i = κ ˆ x + k √ ǫ i cos θ i ˆ z , z < , (4)of the incident plane wave in frame Π is related to its counterpart k ′ i in frame Π ′ by the Lorentz transformation[19, 20] k i = γ (cid:18) k ′ i · ˆ v + ω ′ vc (cid:19) ˆ v + (cid:0) I − ˆ v ˆ v (cid:1) · k ′ i , (5)where the scalar parameters γ = 1 p − β , β = vc . (6)The angular frequencies in the two frames are related per ω = γ ( ω ′ + k ′ i · v ) . (7)In the metal film, the electric and magnetic field phasors with respect to the laboratory frame Π, namely E ( r , ω ) = e ( z, κ, ω ) exp ( iκx ) and H ( r , ω ) = h ( z, κ, ω ) exp ( iκx ), are related via the Maxwell curl postulates;thus, ddz [ f ( z, κ, ω )] = i [ P ( ǫ m , κ, ω )] [ f ( z, κ, ω )] , < z < L m , (8)where the 4–vector [ f ( z, κ, ω )] = e ( z, κ, ω ) · ˆ xe ( z, κ, ω ) · ˆ yh ( z, κ, ω ) · ˆ xh ( z, κ, ω ) · ˆ y (9)and the 4 × P ( ǫ, κ, ω )] = ω µ − κ ω ǫ ǫ − µ κ ω µ − ǫ ǫ ǫ ǫ . (10)The solution to the matrix ordinary differential equation (8) is conveniently stated as[ f ( L m , κ, ω )] = [ M ( ǫ m , L m , κ, ω )] [ f (0 , κ, ω )] , (11)where the transfer matrix [ M ( ǫ m , L m , κ, ω )] = exp { iL m [ P ( ǫ m , κ, ω )] } . (12)In a similar fashion, the electric and magnetic field phasors with respect to frame Π ′ , namely E ′ ( r ′ , ω ′ ) = e ′ ( z, κ ′ , ω ′ ) exp ( iκ ′ x ′ ) and H ′ ( r ′ , ω ′ ) = h ′ ( z, κ ′ , ω ′ ) exp ( iκ ′ x ′ ), satisfy the matrix ordinary differential equa-tion ddz [ f ′ ( z, κ ′ , ω ′ )] = i [ P ( ǫ ′ d , κ ′ , ω ′ )] [ f ′ ( z, κ ′ , ω ′ )] , L m < z < L Σ (13)3n the dielectric slab, where the 4–vector[ f ′ ( z, κ ′ , ω ′ )] = e ′ ( z, κ ′ , ω ′ ) · ˆ xe ′ ( z, κ ′ , ω ′ ) · ˆ yh ′ ( z, κ ′ , ω ′ ) · ˆ xh ′ ( z, κ ′ , ω ′ ) · ˆ y , (14)the angular frequency ω ′ is specified by (7) and κ ′ = k ′ i · ˆ x with k ′ i being specified by (5). The solution tothe matrix ordinary differential equation (13) is conveniently stated as[ f ′ ( L Σ , κ ′ , ω ′ )] = [ M ( ǫ ′ d , L Σ − L m , κ ′ , ω ′ )] [ f ′ ( L m , κ ′ , ω ′ )] . (15)We now seek to express the solution (15) in terms of the field phasors in the laboratory frame Π. Usingthe fact that the z components of e ′ ( z, κ ′ , ω ′ ) and h ′ ( z, κ ′ , ω ′ ) are related via the Maxwell curl postulatesper e ′ ( z, κ ′ , ω ′ ) · ˆ z = − κ ′ ω ′ ǫ ǫ ′ d h ′ ( z, κ ′ , ω ′ ) · ˆ yh ′ ( z, κ ′ , ω ′ ) · ˆ z = κ ′ ω ′ µ e ′ ( z, κ ′ , ω ′ ) · ˆ y , (16)we introduce the 6–vector extension of [ f ′ ( z, κ ′ , ω ′ )], namely h ˜ f ′ ( z, κ ′ , ω ′ ) i = e ′ ( z, κ ′ , ω ′ ) · ˆ xe ′ ( z, κ ′ , ω ′ ) · ˆ ye ′ ( z, κ ′ , ω ′ ) · ˆ zh ′ ( z, κ ′ , ω ′ ) · ˆ xh ′ ( z, κ ′ , ω ′ ) · ˆ yh ′ ( z, κ ′ , ω ′ ) · ˆ z , (17)and the 6 × M ( ǫ ′ d , L Σ − L m , κ ′ , ω ′ )], namely h ˜ M ( ǫ ′ d , L Σ − L m , κ ′ , ω ′ ) i , with components h ˜ M ( ǫ ′ d , L Σ − L m , κ ′ , ω ′ ) i ˜ p ˜ q = [ M ( ǫ ′ d , L Σ − L m , κ ′ , ω ′ )] pq , ˜ p, ˜ q ∈ { , , , } ; p, q ∈ { , , , } ; (18) h ˜ M ( ǫ ′ d , L Σ − L m , κ ′ , ω ′ ) i q = − κ ′ ω ′ ǫ ǫ ′ d [ M ( ǫ ′ d , L Σ − L m , κ ′ , ω ′ )] q , ˜ q ∈ { , , , } ; q ∈ { , , , } ;(19) h ˜ M ( ǫ ′ d , L Σ − L m , κ ′ , ω ′ ) i q = κ ′ ω ′ µ [ M ( ǫ ′ d , L Σ − L m , κ ′ , ω ′ )] q , ˜ q ∈ { , , , } ; q ∈ { , , , } ;(20) h ˜ M ( ǫ ′ d , L Σ − L m , κ ′ , ω ′ ) i ˜ p ˜ q = 0 , ˜ p, ˜ q ∈ { , } . (21)Thus, the solution (15) may be extended as h ˜ f ′ ( L Σ , κ ′ , ω ′ ) i = h ˜ M ( ǫ ′ d , L Σ − L m , κ ′ , ω ′ ) i h ˜ f ′ ( L m , κ ′ , ω ′ ) i . (22)Now in the dielectric slab the electromagnetic phasors in frames Π and Π ′ are related by the Lorentztransformations [20] e ( z, κ, ω ) = A d · e ′ ( z, κ ′ , ω ′ ) h ( z, κ, ω ) = A d · h ′ ( z, κ ′ , ω ′ ) ) , L m < z < L Σ , (23)4here the 3 × A d = (1 − γ ) ˆ v ˆ v + γI − γω ′ (cid:0) v × I (cid:1) · (cid:0) k ′ d × I (cid:1) , (24)with k ′ d = κ ′ ˆ x ′ + q ( ω ′ ) ǫ ǫ ′ d µ − ( κ ′ ) ˆ z , L m < z < L Σ . (25)Thus, (22) leads to h ˜ f ( L Σ , κ, ω ) i = h ˜ N ( ǫ ′ d , L Σ − L m , κ, ω ) i h ˜ f ( L m , κ, ω ) i (26)in the laboratory frame Π, with the 6 × h ˜ N ( ǫ ′ d , L Σ − L m , κ, ω ) i = " A d A d − h ˜ M ( ǫ ′ d , L Σ − L m , κ ′ , ω ′ ) i " A d A d (27)and the 6–vector h ˜ f ( z, κ, ω ) i = e ( z, κ, ω ) · ˆ xe ( z, κ, ω ) · ˆ ye ( z, κ, ω ) · ˆ zh ( z, κ, ω ) · ˆ xh ( z, κ, ω ) · ˆ yh ( z, κ, ω ) · ˆ z . (28)Finally, the counterpart of (15) in the laboratory frame Π emerges as[ f ( L Σ , κ, ω )] = [ N ( ǫ ′ d , L Σ − L m , κ, ω )] [ f ( L m , κ, ω )] , (29)wherein the components of the 4 × N ( ǫ ′ d , L Σ − L m , κ, ω )] are given by[ N ( ǫ ′ d , L Σ − L m , κ, ω )] pq = h ˜ N ( ǫ ′ d , L Σ − L m , κ, ω ) i ˜ p ˜ q , p, q ∈ { , , , } ; ˜ p, ˜ q ∈ { , , , } . (30)For later use, we introduce the 4 × Q ( ǫ ′ d , κ, ω )] via[ N ( ǫ ′ d , L Σ − L m , κ, ω )] = exp { i ( L Σ − L m ) [ Q ( ǫ ′ d , κ, ω )] } . (31)The matrixes [ N ( ǫ ′ d , L Σ − L m , κ, ω )] and [ Q ( ǫ ′ d , κ, ω )] have the same eigenvectors. The eigenvalues σ ℓ , ℓ ∈{ , , , } , of [ N ( ǫ ′ d , L Σ − L m , κ, ω )] are related to the eigenvalues α ℓ of [ Q ( ǫ ′ d , κ, ω )] as follows: α ℓ = ln σ ℓ i ( L Σ − L m ) , ℓ ∈ { , , , } . (32)Upon combining the solutions (11) and (29), and invoking the boundary conditions on the tangentialcomponents of the electric and magnetic phasors at the planar interfaces, we arrive at the algebraic relation t s t p = [ K ( ǫ t , θ t ) ] − [ N ( ǫ ′ d , L Σ − L m , κ, ω )] [ M ( ǫ m , L m , κ, ω )] [ K ( ǫ i , θ i ) ] a s a p r s r p , (33)where the 4 × K ( ǫ, θ ) ] = − cos θ θ − √ ǫ cos θη √ ǫ cos θη − √ ǫη − √ ǫη . (34)5 straightforward manipulation of (33) yields " r s r p = " r ss r sp r ps r pp a s a p (35)and " t s t p = " t ss t sp t ps t pp a s a p (36)wherein the reflection coefficients r ss,sp,ps,pp and transmission coefficients t ss,sp,ps,pp are introduced. Thesquare magnitude of a reflection coefficient provides the corresponding reflectance; i.e., R mn = | r mn | , m, n ∈ { s, p } , (37)while the four transmittances are specified by T mn = √ ǫ t Re [ cos θ t ] √ ǫ i cos θ i | t mn | , m, n ∈ { s, p } . (38)The absorbances for incident plane waves of the p - and s -polarization states are defined as A p = 1 − ( R pp + R sp + T pp + T sp ) A s = 1 − ( R ss + R ps + T ss + T ps ) ) . (39)These absorbances are used to identify SPP waves in the modified Kretschmann configuration. A sharp highpeak in the graph of an absorbance versus θ i , occurring at θ i = θ ♯i say, is a distinctive characteristic of SPPexcitation at the z = L m interface provided that(i) θ ♯i does not vary when the thickness of the dielectric slab (i.e., L Σ − L m ) changes beyond a minimum,and(ii) all four eigenvalues of the matrix [ Q ( ǫ ′ d , κ ′ , ω ′ )] evaluated at θ ♯i have non–zero imaginary parts. Let us explore by numerical means the effects of relative motion on the propagation of SPP waves guidedby the planar interface z = L m . We fix the free-space wavelength in the laboratory frame Π at 633 nm.Next, we choose ǫ i = 6 .
656 which is the relative permittivity of zinc selenide, and for simplicity we take ǫ t = ǫ i . For the relative permittivity of the dielectric slab we choose ǫ ′ d = 2, while for the metal film wechoose ǫ m = −
56 + 21 i which is the relative permittivity of aluminum. The dielectric slab is taken to have athickness of 1000 nm, while the metal film has a thickness of 15 nm; i.e., L Σ = 1015 nm and L m = 15 nm.Before considering the effects of relative motion, we must first establish the baseline for our studies, whichis represented by the scenario wherein there is no relative motion. In Fig. 2, the absorbances A p and A s are plotted versus angle of incidence for the case β = 0. Also plotted on these graphs are the quantities R p = R pp + R sp and R s = R ss + R ps , calculated when the metal film is absent. A sharp high peak in the graphof A p at θ i = 34 . ◦ — which lies in the angular regime R p for total reflection in the absence of the metalfilm — is the signature of SPP excitation by a p -polarized incident plane wave; for the chosen parameters, R p = { θ i | θ i > . ◦ } . This identification is further confirmed by the facts that (i) the A p -peak remains at θ i = 34 . ◦ when the thickness of the dielectric slab is increased beyond 1000 nm and (ii) the eigenvalues of[ Q ( ǫ ′ d , κ ′ , ω ′ )] evaluated at θ i = 34 . ◦ all have non–zero imaginary parts. There is no corresponding sharphigh peak in the graph of A s in Fig. 2, in accordance with standard results [3, 4], although an angular regime R s for total reflection in the absence of the metal film does exist; note that R p ≡ R s when β = 0.6 .1 Moving dielectric slab Now we investigate the scenario schematically depicted in Fig. 1, where the dielectric slab moves at constantvelocity parallel to the x axis; i.e., ˆ v = ˆ x . The absorbance A p is plotted versus θ i for the relative speeds β ∈ { . , . , . , . , . , . } in Fig. 3. To confirm whether any possible SPP peaks occur in the angularregime R p , the quantity R p = R pp + R sp , calculated when the metal film is absent, is also plotted.The SPP peak in the plots of A p is found to arise at lower values of θ i when β increases from zero.Furthermore, the SPP peak in the plot of A p observed for β ≤ .
85 is joined by another prominent peakwhen β > .
85. For example, at β = 0 .
9, there are prominent A p -peaks at θ i = θ ♯ i = 24 . ◦ and at θ i = θ ♯ i = 45 . ◦ . These values of θ ♯ i and θ ♯ i do not vary when the thickness of the dielectric slab isincreased beyond 1000 nm; and all four eigenvalues of the matrix [ Q ( ǫ ′ d , κ ′ , ω ′ )] evaluated at θ ♯ i and θ ♯ i havenon–zero imaginary parts.The plots of R p for L m = 0 in Fig. 3 indicate that the total reflection is not possible at β > .
85 forsufficiently large angles of incidence and therefore the significance of the two A p -peaks at θ ♯ i and θ ♯ i isunclear. However, the A p -peak at θ ♯ i corresponds to the SPP peak observed in the plots of A p for β ≤ . β is continuously varied. The plots of A s (not shown here)corresponding to those of Fig. 3 for β ∈ (0 ,
1) do not exhibit SPP peaks.Next we consider the scenario where the dielectric slab moves at constant velocity parallel to the y axis;i.e., ˆ v = ˆ y . The absorbance A p , and the quantity R p calculated when the metal film is absent, are plottedversus θ i for the relative speeds β ∈ { . , . , . , . } in Fig. 4. Unlike for the case of ˆ v = ˆ x in Fig. 3,now the value of θ i at which the A p -peak indicating the excitation of an SPP wave occurs increases as β increases. Furthermore, this peak vanishes for β & . s -polarized incident plane wave, provided in Fig. 5, exhibita sharp peak for mid–range values of β , at angles of incidence in the angular regime R s . For example, this A s -peak arises at θ i = 47 . ◦ for β = 0 .
6. Further calculations have revealed that this peak is insensitive tochanges in the thickness of the dielectric slab and all eigenvalues of [ Q ( ǫ ′ d , κ ′ , ω ′ )] evaluated at θ i = 47 . ◦ have non–zero imaginary parts.We therefore infer that, in the case of ˆ v = ˆ y , the SPP wave is excited by an incident p -polarized planewave for low values of β , but excited by an incident s –polarized plane wave at higher values of β . As β approaches unity, no absorbance peak indicating the excitation of an SPP wave is observed, for incidentplane waves of either linear polarization state.Clearly, from Figs. 3–5, the direction of motion has a major bearing on the excitation of SPP waves.This sensitivity to direction of motion reflects the fact that κ = κ ′ when motion is parallel to the plane ofincidence but κ = κ ′ when motion is perpendicular to the plane of incidence, as may be inferred from (5).In order to further illuminate this issue, let us investigate the case where the direction of motion is neitherparallel nor perpendicular to the plane of incidence; i.e., ˆ v = ˆ x cos ψ + ˆ y sin ψ where, for example, we choosethe angle ψ = 45 ◦ . In Fig. 6, the absorbance A p , and the quantity R p calculated when the metal film isabsent, are plotted versus θ i for the relative speeds β ∈ { . , . , . , . , . , . } . The plots in Fig. 6display some features of the plots presented in both Figs. 3 and 4. As is the case when motion is parallel tothe plane of incidence, the solitary A p -peak observed at low and moderate values of β is joined by a secondpeak at higher values of β . A second A p -peak begins to emerge in Fig. 6 at β ≈ .
88; it is fully developedat β = 0 .
9; and as β increases to approximately 0.95, the second A p -peak coalesces with the first A p -peak.For example, at β = 0 .
9, there are A p -peaks at θ i = θ ♯ i = 32 . ◦ and at θ i = θ ♯ i = 64 . ◦ . The peak at θ ♯ i corresponds to the solitary SPP peak which is observed at low and moderate values of β . As is thecase for the two high– β A p -peaks in Fig. 3, the values of θ ♯ i and θ ♯ i do not vary when the thickness of thedielectric slab is increased beyond 1000 nm; and all four eigenvalues of the matrix [ Q ( ǫ ′ d , κ ′ , ω ′ )] evaluatedat θ ♯ i and θ ♯ i have non–zero imaginary parts. We note that the A p -peak at θ ♯ i is considerably sharper thanthe A p -peak at θ ♯ i .The manifestation of an A s -peak at moderate values of β , as may be observed in Fig. 5 when the motion isdirected perpendicular to the plane of incidence, also occurs when motion is directed along ˆ v = (ˆ x + ˆ y ) / √ A s plots in Fig. 7 exhibit two A s -peaks7t high values of β , which coalesce as β approaches unity, in a similar manner to the two A p -peaks observedin Fig. 6. Next, we turn to a rather different situation. The dielectric slab is now held fixed relative to the half–spaces z < z > L Σ , whereas the metal film moves with velocity v = βc ˆ v in the xy plane. The correspondingformulation is isomorphic to that presented in Sec. 2, but with the treatments for the metal film and thedielectric slab interchanged. For the case ˆ v = ˆ x , plots of A p versus θ i (not shown here) are largely insensitiveto β . That is, the A p -peak indicating the excitation of an SPP wave occurs at approximately the same valueof θ i and has approximately the same amplitude, regardless of β . Furthermore, there are no notable peaksin the corresponding plots of A s .However, plots of A p in the case of ˆ v = ˆ y are much more sensitive to β . As we can see in Fig. 8, the A p -peak indicating the excitation of an SPP wave occurs at slightly higher values of θ i as β increases. Also,the amplitude of this peak gradually diminishes as β increases, to such an extent that the peak is barelydiscernible at β = 0 .
9. In the corresponding plots for a s –polarized incident plane wave (not shown here),there is a mere hint of an A s -peak indicating the excitation of an SPP wave at mid–range values of β , butnothing as obvious as appears in the s -polarization scenario represented in Fig. 5. Uniform motion can have a major effect on the excitation of SPP waves guided by the interface of a metalfilm and a dielectric slab, both isotropic and homogeneous in their respective co-moving inertial referenceframes. In the case of a uniformly moving dielectric slab, the angle of incidence for SPP-wave excitation ishighly sensitive to the relative speed β and the direction of motion. For the specific example considered inSec. 3,(a) when the direction of motion is parallel to the plane of incidence, the SPP wave is excited by p -polarized(but not s -polarized) incident plane waves for low and moderate values of β ; and(b) when the direction of motion is perpendicular to the plane of incidence, the SPP wave is excited by p -polarized incident plane waves for low values of β , but s -polarized incident plane waves at moderatevalues of β , while at higher values of β the SPP wave is not excited.Some insight into this sensitivity to relative motion can be gained by considering the Minkowski constitu-tive relations for the uniformly moving dielectric slab. That is, the electromagnetic response of the uniformlymoving dielectric slab may be represented in the laboratory frame by the bianisotropic constitutive relations[19, Chap. 8] D ( r , ω ) = ǫ ǫ ′ d α · E ( r , ω ) + 1 c (cid:0) m × I (cid:1) · H ( r , ω ) B ( r , ω ) = − c (cid:0) m × I (cid:1) · E ( r , ω ) + µ α · H ( r , ω ) , (40)where α = α I + (1 − α ) ˆ v ˆ v , α = 1 − β − ǫ ′ d β (41)and m = m ˆ v , m = β ( ǫ ′ d − − ǫ ′ d β . (42)Thus, when the direction of motion is parallel to the plane of incidence, for example, the Minkowski consti-tutive dyadics take the form α = ˆ x ˆ x + α (ˆ y ˆ y + ˆ z ˆ z ) m × I = m (ˆ z ˆ y − ˆ y ˆ z ) ) . (43)8he Minkowski constitutive scalars α and m for this case are plotted versus β in Fig. 9. Clearly, theconstitutive scalars are highly sensitive to β . Indeed, both α and m become unbounded as β → / √ . A p and A s reported in Sec. 3 at high values of β .In the case of a uniformly moving metal film, the sensitivity to β is less obvious. Since the metalconsidered in Sec. 3 is dissipative — and quite highly, too — the Minkowski constitutive relations cannot beused here to gain an insight into the sensitivity to β [18].To conclude, our study further extends our understanding of electrodynamic processes at planar interfaces,especially relating to SPP waves. Acknowledgments:
TGM is supported by a Royal Academy of Engineering/Leverhulme Trust SeniorResearch Fellowship. AL thanks the Binder Endowment at Penn State for partial financial support of hisresearch activities.
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20 40 60 800.00.20.40.60.81.0 Θ i A p Β = v ` = x ` R p H L m = L Θ i A p Β = v ` = x ` R p H L m = L Θ i A p Β = v ` = x ` R p H L m = L Θ i A p Β = v ` = x ` R p H L m = L Θ i A p Β = v ` = x ` R p H L m = L Θ i A p Β = v ` = x ` R p H L m = L Figure 3: Absorbance A p = 1 − ( R pp + R sp + T pp + T sp ) (red, solid curve) plotted versus angle of incidence θ i (in degree) for the scenario where the dielectric slab moves in the direction ˆ v = ˆ x at relative speeds β ∈ { . , . , . , . , . , . } . Also plotted is the quantity R p = R pp + R sp (blue, dashed curve), calculatedwhen L m = 0. 12
20 40 60 800.00.20.40.60.81.0 Θ i A p Β = v ` = y ` R p H L m = L Θ i A p Β = v ` = y ` R p H L m = L Θ i A p Β = v ` = y ` R p H L m = L Θ i A p Β = v ` = y ` R p H L m = L Figure 4: As Fig. 3 except that the dielectric slab moves in the direction ˆ v = ˆ y and β ∈ { . , . , . , . } .13
20 40 60 800.00.20.40.60.81.0 Θ i A s Β = v ` = y ` R s H L m = L Θ i A s Β = v ` = y ` R s H L m = L Θ i A s Β = v ` = y ` R s H L m = L Θ i A s Β = v ` = y ` R s H L m = L Figure 5: As Fig. 4 except that quantities plotted are A s = 1 − ( R ss + R ps + T ss + T ps ) (red, solid curve)versus angle of incidence θ i (in degree), and the quantity R s = R ss + R ps (blue, dashed curve), calculatedwhen L m = 0. 14
20 40 60 800.00.20.40.60.81.0 Θ i A p Β = v ` = H x ` + y ` L(cid:144) R p H L m = L Θ i A p Β = v ` = H x ` + y ` L(cid:144) R p H L m = L Θ i A p Β = v ` = H x ` + y ` L(cid:144) R p H L m = L Θ i A p Β = v ` = H x ` + y ` L(cid:144) R p H L m = L Θ i A p Β = v ` = H x ` + y ` L(cid:144) R p H L m = L Θ i A p Β = v ` = H x ` + y ` L(cid:144) R p H L m = L Figure 6: Absorbance A p = 1 − ( R pp + R sp + T pp + T sp ) (red, solid curve) plotted versus angle of incidence θ i (in degree) for the scenario where the dielectric slab moves in the direction ˆ v = (ˆ x + ˆ y ) / √ β ∈ { . , . , . , . , . , . } . Also plotted is the quantity R p = R pp + R sp (blue, dashed curve),calculated when L m = 0. 15
10 20 30 40 50 600.00.20.40.60.81.0 Θ i A s Β = v ` = H x ` + y ` L(cid:144) R s H L m = L Θ i A s Β = v ` = H x ` + y ` L(cid:144) R s H L m = L Θ i A s Β = v ` = H x ` + y ` L(cid:144) R s H L m = L Θ i A s Β = v ` = H x ` + y ` L(cid:144) R s H L m = L Θ i A s Β = v ` = H x ` + y ` L(cid:144) R s H L m = L Θ i A s Β = v ` = H x ` + y ` L(cid:144) R s H L m = L Figure 7: As Fig. 6 except that quantities plotted are A s = 1 − ( R ss + R ps + T ss + T ps ) (red, solid curve)versus angle of incidence θ i (in degree), and the quantity R s = R ss + R ps (blue, dashed curve), calculatedwhen L m = 0. 16
20 40 60 800.00.20.40.60.81.0 Θ i A p Β = v ` = y ` R p H L m = L Θ i A p Β = v ` = y ` R p H L m = L Θ i A p Β = v ` = y ` R p H L m = L Θ i A p Β = v ` = y ` R p H L m = L Figure 8: Absorbance A p = 1 − ( R pp + R sp + T pp + T sp ) (red, solid curve) plotted versus angle of incidence θ i (in degree) for the scenario where the metal film moves in the direction ˆ v = ˆ y at relative speeds β ∈{ . , . , . , . } . Also plotted is the quantity R p = R pp + R sp (blue, dashed curve), calculated when L m = 0.17 .0 0.2 0.4 0.6 0.8 1.0 - - - Β Α v ` = x ` m Figure 9: The Minkowski constitutive scalars α (red, solid curve) and m (blue, dashed curve) of the dielectricslab plotted versus relative speed ββ