The refraction of surface plasmon polaritons
Jonathan J. Foley IV, Jeffrey M. McMahon, George C. Schatz, Stephen K. Gray
TThe Refraction of Surface Plasmon Polaritons
Jonathan J. Foley, IV, Jeffrey M. McMahon, George C. Schatz, and Stephen K. Gray Center for Nanoscale Materials, Argonne National Laboratory, Argonne, IL 60439 Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801 Department of Chemistry, Northwestern University, Evanston, IL 60208 (Dated: September 13, 2018)
Abstract
We show how a complex Snell’s law can be used to describe the refraction of surface plasmonpolaritons (SPPs) at an interface between two metals, validating its predictions with 3-D electro-dynamics simulations. Refraction gives rise to peculiar SPP features including inhomogeneitiesin the waveform and dispersion relations that depend on the incident wave and associated mate-rial. These features make it possible to generate SPPs propagating away from the interface withsignificant confinement normal to the propagation direction. We also show that it is possible toencode optical properties of the incident material into the refracted SPP. These consequences ofmetal-metal SPP refraction provide new avenues for the design of plasmonics-based devices. a r X i v : . [ c ond - m a t . m e s - h a ll ] D ec urface plasmon polaritons (SPPs), surface waves created by coupling light into charge-density oscillations at a metal–dielectric interface, continue to be of current interest [1–7].Systems that permit the excitation of SPPs can exhibit interesting and unexpected opticalproperties, including extraordinary optical transmission [8] and super-lensing [9–11]. Suchproperties are also relevant to a wide range of applications including imaging and sensing[12–16] and optoelectronics [17–20]. Therefore, learning how to control and manipulateSPPs as they propagate along a metal surface is a major goal of nanophotonics research. Aparticular goal for optoelectronics applications involves introducing lateral confinement ofthe SPPs without sacrificing the propagation length [19, 21–23]. For sensing applications,sensitivity figures of merit depend not on the features of the SPP wave itself, but on the SPPdispersion relations, where a desirable feature is a strong dependence of the SPP dispersionon the dielectric environment [15, 16].SPPs propagate on a 2-D metal surface, and are exponentially confined both above andbelow this surface, suggesting that one can attempt to describe and manipulate their motionusing ideas from classical optics applied to the 2-D propagation plane. Successful examplesinclude the focusing of SPPs created by an array of holes or slits in metal films [24, 25] andthe generation of Talbot effect intensity patterns [26–28]. SPPs have also been shown ex-perimentally to exhibit refraction behavior when they propagate across an interface betweentwo metal/dielectric interfaces with differing optical properties [29, 30]. Negative refractionof SPP-dominated waveguide modes has also been achieved [31]. In this Letter, we analyzeSPP refraction, presenting and discussing the implications of a complex generalization ofSnell’s law (CSL). The CSL predicts that refracted SPPs will be inhomogeneous and willobey dispersion relations that depend not only on the medium supporting the refracted SPP,but on the incident wave and medium. The inhomogeneous character of the refracted wavecan be exploited to introduce significant confinement of the SPP without sacrificing propa-gation length. The dependence of the dispersion of the refracted SPP on the details of theincident wave introduces the possibility of encoding or imprinting the plasmonic propertiesof one material onto another, including anomalous dispersion phenomena (back-bending)and ‘slow-light’ [32]. Dispersion encoding imparts an unexpected environmental sensitivityto the refracted SPP, which may be useful for sensing applications.[15, 16, 18]Refraction of 3-D plane waves from a dielectric medium into an absorbing medium isa well-known problem [33–35] and there are also treatments of refraction that allow for2he medium of the incident wave to be absorbing [36–38]. Here we adapt the particularlytransparent treatment of Chang, Walker and Hopcraft[38] to the case of 2-D SPP refractionat the boundary between different metal surfaces and discuss several consequences. Thesystem of interest involves an SPP, generated by conventional means, propagating on top ofa metal surface 1 that has dielectric material 1 above it; it is reflected and refracted at theinterface with a different metal surface 2 with possibly different dielectric 2 above it (Fig.1). The refracted SPP propagates on the surface of metal 2. The propagation of the SPPson each surface can be described with 2-D waveforms moving in the x − y plane of Fig. 1, E j = E ,j exp ( i k j · r − iωt ) , (1)where k j is a complex SPP wavevector associated with incident ( j = 1) or refracted ( j = 2)waves. (For the present purposes reflection is not of relevance.) A 2-D medium refractiveindex may be defined based on the standard SPP dispersion relation: η j + iκ j = (cid:32) (cid:15) j (cid:15) Dj (cid:15) j + (cid:15) Dj (cid:33) / , (2)where (cid:15) j = (cid:15) j ( ω ) is the frequency-dependent permittivity of metal j and (cid:15) Dj is the permit-tivity of the dielectric material above it. The incident SPP wavevector is k = ωc ( η + iκ ) ˆ e , (3)where ˆ e = sin( θ )ˆ x + cos( θ )ˆ y is a unit vector indicating the direction of propagation. Thelines of constant phase and constant amplitude for E are parallel and ˆ e is the directionnormal to both these types of lines. E is therefore homogeneous. In contrast, the refractedSPP is allowed to be inhomogeneous; its lines of constant phase and amplitude are notnecessarily parallel. The wavevector for the refracted SPP is thus taken to be k = ωc (cid:16) N ˆ a + iK ˆ b (cid:17) (4)where ˆ a = sin( θ )ˆ x + cos( θ )ˆ y is a unit vector normal to the lines of constant phase, ˆ b =sin( φ )ˆ x + cos( φ )ˆ y is a unit vector normal to the lines of constant amplitude, and theeffective indices N and K depend on the medium refractive indices η , κ , η , κ andincident angle, θ . The boundary conditions and wave equation determine N and K interms of the known quantities. The SPP phases k j · r must be continuous at the y = 03 IG. 1. (a) Diagram of SPP refraction at a metal-dielectric/metal-dielectric interface. The incidentSPP propagates on the 2-D interface between metal 1 and dielectric 1, with direction ˆ e beingassociated with both the real and imaginary parts of its wavevector. The wavevector of the refractedSPP propagating in the interface region of metal 2 and dielectric 2 has direction ˆ a associated withits real part and direction ˆ b associated with its imaginary part. (b) 2-D effective medium pictureof the refraction in the x − y plane of (a). metal-metal interface, implying η sin( θ ) = N sin( θ ) (5) κ sin( θ ) = K sin( φ ) . (6)We refer to Eqs. (5) and (6) as the complex Snell’s law (CSL) because they determine theangles of refraction of the real and imaginary parts of the complex refracted SPP wavevector.All the quantities involved in these equations are still real and so the complication of complexangles is avoided. To determine N and K , inserting E of the form Eq. (1) into the usualsecond-order electromagnetic wave equation gives( k · k ) E = ω c ( η + iκ ) E , (7)which is satisfied if N − K = η − κ (8)4nd N K cos( θ − φ ) = η κ . (9)Using Eqs. (5) and (6), Eq. (9) is equivalent to η κ − α β = (cid:113) N − α (cid:113) N − ( η − κ ) − β , (10)where α = η sin( θ ) and β = κ sin( θ ). Squaring both sides of Eq. 10 gives a quarticequation for N , which has the following root of interest N = 1 √ (cid:113) a + √ b, (11)where a = α + β + η − κ ,b = (cid:0) ( κ − β ) + ( η − α ) (cid:1) (cid:0) ( κ + β ) + ( η + α ) (cid:1) Once N is known, K may be determined readily using Eq. (8); θ and φ are then foundfrom form the CSL, Eqs. (5) and (6).In the case of normal incidence ( θ = 0) the CSL leads to N = η , K = κ , and θ = φ ,i.e. a refracted SPP in medium 2 that is an ordinary medium 2 SPP. Otherwise, there aretwo new features: (1) θ (cid:54) = φ , i.e. the waveform is inhomogeneous with its lines of constantphase and constant amplitude no longer being parallel and (2) N + iK (cid:54) = η + iκ , i.e.the complex propagation constant (and dispersion relation) is not the same as that for anordinary SPP in medium 2.The propagation length ( L P ) of an SPP is the distance, measured along the propagationdirection, that the SPP propagates when the intensity decays to | E | /e . For an ordinary(homogeneous) SPP in medium 2, L P = 1 / (2 k κ ), where k is the free-space wavevector ofthe exciting light, k = 2 π/λ . For a refracted SPP in medium 2, this distance is measuredalong ˆ a and is given by L P = 1 / (2 k K cos( θ − φ )). Utilizing Eq. (9), the ratio of therefracted SPP propagation length to an ordinary SPP propagation length is simply N /η and so if N > η there will be propagation length enhancement. We define a confinementlength ( L C ) as the distance in the direction perpendicular to propagation over which the SPPintensity decays to | E | /e . The confinement length of an inhomogeneous SPP is given by L C = 1 / (2 k K sin( θ − φ )), where a more strongly confined SPP has a shorter confinement5 IG. 2. Field profiles of incoming SPP (below white dashed line) excited on an Au surface by532 nm light with incident angle 25 ◦ refracting onto an Ag surface (above white dashed line).Air is assumed to be above both surfaces. (a) and (b): Analytical CSL and FDTD electric fieldintensities, respectively. (c) and (d): Analytical and FDTD instantaneous electric field components,respectively. The FDTD results are x − y cuts taken at a z level 120 nm above metal surfaces andare associated with the z component of the electric field. length. In the event of propagation length enhancement, K itself will be larger than κ ;hence propagation length enhancement is also associated with strong confinement of therefracted SPP.We first consider Au for metal 1 and Ag for metal 2 with air as the dielectric aboveeach metal, and an incident SPP on Ag excited with λ = 532 nm light. We take (cid:15) = − .
762 + 2 . i and (cid:15) = − .
825 + 0 . i [39] which gives medium refractive indices η = 1 . κ = 0 . η = 1 . κ = 0 . η and η are similar inmagnitude, κ is significantly larger than κ and so φ rises very rapidly with θ (see Fig.S3). When an incident angle of 25 ◦ is considered, the CSL predicts θ to be 26 ◦ and φ
6o be 112 ◦ (see Fig. S3). ( φ can be larger than 90 ◦ because the arcsine has two uniquevalues, a principal value between 0 and π/ π/ π ; oneor the other of these values is the physically correct one. See the Supporting Material) Theanalytical CSL electric field intensity ( | E z | ) and instantaneous field map, Re( E z ), are shownin Fig. 2 (a) and (c), respectively. To validate these predictions, we use rigorous 3-D finite-difference time-domain (FDTD) calculations [40, 41] (see Supporting Material) to simulatethe refraction phenomena and plot | E z | and Re( E z ) in Fig. 2 (b) and (d), respectively.(Other FDTD field components give similar results.) The FDTD results show a high degreeof similarity in the SPP wavelength, propagation direction, and attenuation behavior withthat predicted by CSL. Values of the simulated electric field intensity are sampled alongthe propagation direction ( ˆa ) and fit an exponential to allow accurate inference of L P (seeFig. S1). The FDTD fields do differ from the CSL ones in that there is a fast decay of theFDTD field in the upper left region of Figs. 2 (b) and (d). This is simply due to the finitesize of the excitation source used in the simulations (see Supplementary Material). One canalso see slight interference fringes in the FDTD results due to interference of incident andreflected waves.We find that the propagation length of the refracted SPP as determined by rigorous 3-Delectrodynamics calculation is approximately 28 µ m, compared to 27 µ m from analyticalpredictions using CSL. The FDTD instantaneous field allows us to infer a propagationdirection of 27 ◦ , compared to the CSL prediction of 26 ◦ . The simulated propagation lengththus closely matches the propagation length predicted for a normal SPP excited on a silversurface, but the refracted SPP has significant lateral confinement compared to an ordinarySPP. We follow a similar procedure to extract L C , this time sampling the electric fieldintensity along the direction perpendicular to ˆa , and nearly parallel to ˆb . We find L C ofthe refracted SPP is 1.1 µ m as determined by FDTD calculation, which agrees closely tothe CSL prediction of 1.7 µ m (see Fig. S1). Note that an ordinary SPP in this mediumwould propagate with no such confinement lateral to the propagation direction. Analysis ofthe FDTD lines of constant amplitude allows us to infer an attenuation direction of 115 ◦ ,similar to the CSL prediction of 112 ◦ . In this first example, φ is relatively large relativerelative to θ , i.e., the direction of the amplitude decay is nearly normal to the propagationdirection, a dramatic change relative to the incident wave that had amplitude decay in thesame direction as the propagation direction. Regarding the effective indices in medium 2,7 IG. 3. Field profiles of the refracted SPP on an Al surface arising from refraction of an incidentSPP on a Au surface. An (cid:15) D = 4 material is assumed to be above the Au surface and glass, (cid:15) D =2.25, is assumed to be above the Al surface. The incident SPP was excited with 780 nm light andhad incident angle 50 ◦ . (a) and (b): Analytical CSL and FDTD electric field intensities, respec-tively. (c) and (d): Analytical and FDTD instantaneous electric field components, respectiely. (e)Dispersion relation for the SPP generated on Al (solid red) showing how refraction imparts strongfeatures of Au’s SPP dispersion (dashed blue) onto the refracted wave. it turns out that K = 0.0252, which is significantly larger that the medium value of κ =0.0015, but that N ≈ η = 1.045. The next example will involve more significant changesin N , which is proportionate to the real part of the propagation vector and determines thedispersion relation.As a second example consider λ = 780 nm excitation of SPPs on Au as metal 1 ( (cid:15) = − .
660 + 1 . i [39]), with a high refractive index material ( (cid:15) D = 4, e.g. TiO ) above.8hese SPPs refract at an interface with aluminum as metal 2 ( (cid:15) = − .
263 + 45 . i [42])with glass ( (cid:15) D = 2 .
25) above. These result in medium refractive indices of η = 2 . κ = 0 . η = 1 . κ = 0 . θ = 50 ◦ θ ≈ ◦ while φ ≈ ◦ , making this akin to total internal reflection (TIR) (see Fig. S4). FDTD resultsagain show a high degree of similarity in propagation direction and attenuation behavior(see Fig. 3 (a)-(d)) and a strong quantitative agreement in the predicted propagation andconfinement lengths (see Fig. S2). We focus only on the fields in medium 2, but againinterference patterns can be seen in the FDTD fields (Figs. 4 (c) and (e)) resulting frominterference of the reflected and incident SPP in medium 1. Analysis of the lines of constantphase and amplitude from FDTD simulations gives a propagation direction of 90 ◦ and anattenuation direction of approximately 0.5 ◦ , respectively, in excellent agreement with CSLpredictions. Interestingly, some experimental evidence for this particular consequence ofSnell’s law (TIR plasmons) has already been reported [30].The FDTD propagation length of the refracted SPP is approximately 5.6 µ m, and CSLalso predicts 5.6 µ m (see Fig. S2). The corresponding ordinary SPP on an Al/glass surfacewould have a propagation length of 5.0 µ m, so this provides an example of modest propaga-tion length enhancement. The refraction also produces an extremely confined mode: FDTD L C ≈ . µ m; CSL L C = 0.08 µ m (see Fig. S2). L P may be enhanced proportionally to N /η , so it is expected the refracted SPP will have a shorter wavelength than an ordinarySPP propagating in medium 2. Indeed, we find the refracted FDTD λ SP P to be 486 nmcompared to a wavelength of 462 nm predicted by CSL. An ordinary SPP propagating inmedium 2 would have a wavelength of 514 nm. In this example both the real and imaginaryparts of the effective index of the refracted SPP are different from the medium 2 values: N = 1.688 compared to η = 1.517 and K = 0.738 compared to κ = 0.012.It is also interesting for this example to consider the behavior of the refracted SPP inAl/glass across a spectrum of incident frequencies, i.e. its dispersion. Unlike an ordinarySPP dispersion in in an Al/glass system, the refracted SPP dispersion also has informationabout the refractive index, η + iκ and the angle of incidence, θ , in the Au/ (cid:15) D = 4 medium.Mapping out the real part of the propagation vector, ( ω/c ) N , with θ = 50 ◦ shows a startlingdeparture from the ordinary Al/glass result (Fig. 3 (e)). The ordinary Al/glass dispersionis relatively featureless, the wavevector increases uniformly with ω so that the SPP groupvelocity is approximately independent of frequency. The dispersion of the refracted SPP9hows markedly different features, including a dramatic slowing of group velocity in thefrequency range between 1.5 and 2 eV and a back-bending region between 2 and 3 eV.These features mirror dispersion characteristics of SPPs on the gold/ (cid:15) D = 4 medium (seeFig. 3(e)). To see the relationship between the details of the incident SPP and the dispersionof the refracted SPP more clearly, we note that η can be factored out of b Eq. 11 so theroot can be written N = (cid:112) a + η √ f . We can then expand √ f to first order in f , yielding the following approximation of N , N ≈ η + β − η α β κ + 14 η (cid:0) α + β + κ + 2 (cid:0) κ α − κ β + β α (cid:1)(cid:1) . (12) η makes a strong contribution to N , which is to be expected as the dispersion of therefracted SPP should depend on the material properties of the metal/dielectric supportingits propagation. What is interesting to note is the fact the terms involving α ( β ) canmake strong contributions to N when θ is large and η ( κ ) is large. Both η and κ tendto be large in the vicinity of the surface plasmon resonance (SPR) of material 1, which interms of the SPP dispersion, is associated with the back-bending or anomalous dispersionregion [32, 43–45]. Back-bending features can be encoded from material 1 to material 2 ingeneral, but they will often be associated with high loss. Recalling the definitions of L P and L C , we observe that the any additional loss imparted by refraction leads to greaterconfinement of the SPP and not additional attenuation in the propagation direction.In this Letter, we presented a complex generalization of Snell’s law (CSL) to be appliedto the refraction of SPPs propagating across a metal-metal interface. This theory predictsseveral surprising features of the refracted SPPs which were validated using 3-D electrody-namics simulations. In particular, we demonstrated that refraction can generate SPPs thatare inhomogeneous in the plane of propagation. It is possible to introduce significant con-finement and, in certain cases, propagation length enhancement to the refracted modes. Afurther consequence is that the refracted SPPs obey unique dispersion relations that dependon the supporting medium as well as the incident medium. The theory should also be ap-plicable to more complicated structures such as the layered wave guide modes described byAtwater and co-workers for the measurement of negative refraction in the visible spectrum[31], and indeed such structures might provide experimentally realizable systems for thepeculiar predictions that result from the CSL discussed here. The theoretical and numericalresults presented suggest that simple geometric principles such as the CSL discussed here can10ffer novel and powerful strategies for engineering SPPs, which could be particularly usefulfor optoelectronics and sensing applications. For example, although we do not discuss thisapplication in detail, FDTD simulations demonstrate SPP focusing can be achieved usinga metal region acting as a plasmonic lens (see Fig. S6). The focusing through refraction ofsuch a lens could allow the generation of tightly confined and controlled SPP modes with-out loss of propagation length. Optical switching devices could also be constructed usingincident angle or superstrate dielectric constant as nobs to induce a dramatic change in thepropagation behavior of the refracted SPP. 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