Surface Plasmon Polaritons Sustained at the Interface of a Nonlocal

Home , Fresnel equations

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B(r,ω)= H(r,ω) (2) with material parameters ǫ, γ, and η. For convenience, we assume that the coordinate system of the laboratory is aligned to the coordinate system of the principle axis of the metamaterial. Hence, all material parameters are considered to be diagonal tensors. By a suitable gauge FIG. 1. Schema of surface polaritons sustained at an inter- transformation, γ can be recast to appear as a disper- face. The interface at z = 0 divides two extended half-spaces. sion in the local permeability. For that reason it is called One is vacuum while the other one is made initially from an a weak spatial dispersion. In contrast, η is associated isotropic homogeneous medium with permittivity ǫ and per- to a strong spatial dispersion, it is a nonlocal material meability µ. Later, also a more complicated constitutive re- parameter. These constitutive relations allow for a rigor- lation is considered. The light is not transmitted but excites ous mathematical derivation of interface conditions that surface plasmon polaritons (SPP) on the interface. For this extend the known conditions from basic electrodynam- purpose, it is required to have a wave vector component in ics [19]. Therefore, not just light propagation in bulk the direction of the interface that is larger than the length of material can be described but also functional elements the wave vector in vacuum. thereof. In this contribution, we continue a long standing tra- dispersion relation in the presence of nonlocality where dition in metamaterials research in where basic physical we particular emphasize the question how the onset of a effects are explored once a specific material with given weak nonlocality changes the dispersion relation of the properties has been assumed. On the one hand, such propagating SPP sustained at the interface between vac- research is intellectual appealing, as novel effects that uum and the metamaterial. In a last step, experimental were predicted with those materials constitute a major features are predicted as observable in an attenuated to- driving force to finally identify materials that offer these tal reflection setup. Finally, we conclude on our research. properties. This may have started with the seminal work from Veselago [20]. There, he simply has been assuming the availability of a material with a dispersive permittiv- II. MATERIAL MODELS AND DISPERSION ity and a dispersive permeability and studied afterwards RELATIONS observable optical effects. More contemporary examples would be the field of transformation optics or the sug- In order to model a metamaterial with negative in- gestion for a perfect lens [21–23]. On the other hand, dex behavior, we consider a homogeneous, isotropic, and from such consideration we can predict experimentally nonlocal metamaterial with a Drude permittivity observable features that would be a smoking gun to de- cide whether a particular constitutive relation indeed is ω2 necessary for the effective description of a metamaterial ǫ(ω)=1 − p (3) or not. ω(ω + iΓǫ) Here, we investigate propagating surface plasmon po- and a permeability described by a Lorentz model laritons (SPP) at the interface between an ordinary material and a nonlocal metamaterial that exhibits a negative Fω2 refractive index in some frequency range of interest. A µ(ω)=1 − , (4) ω2 − ω2 + iΓ ω list of metamaterials that have been designed to undergo 0 µ such properties can be found in [24]. Our contribution with parameters ωp, ω0, Γǫ, Γµ and F , which will be is particularly inspired by the work of Ruppin [25] that commented on below. Via a gauge transformation pioneered the study of surface waves sustained at the ′ interface between metamaterials described by local con- D (r,ω)= D(r,ω)+ ∇× Q(r,ω) (5) stitutive relations and ordinary media. Specifically, we investigate the dispersion relation of surface waves and and the reflection to be observed in a frustrated total inter- ′ nal reflection geometry and how these are affected by a H (r,ω)= H − ik0Q(r,ω), (6) strong nonlocality. With that, we offer blueprints for experiments that aim to elucidate nonlocal properties in the parameter γ(ω) that is associated to a weak spatial metamaterials. dispersion can be expressed through µ(ω): Our manuscript is structured such that we introduce in the next section the material models we consider. We µ(ω) − 1 γ(ω)= 2 , (7) study then the interface Fresnel equations, which need µ(ω)k0 to be known in order to derive the dispersion relation ! of the propagating SPPs. In section four, we study the with k0 = ω/c, by choosing Q(r,ω) = −γ∇× E(r,ω). 3

For the parameter expressing the strong spatial disper- part. In the lossless case, for the existence of SPPs, it is sion we make a model assumption and set its frequency usually required that the wave vector components away dependency to from the interface are purely imaginary, so that the wave is evanescent in that direction. However, in the more GF ω2 realistic case of an absorptive medium as considered here, η(ω)= , (8) 2 2 this condition is not feasible [26]. Instead, we only require ω0 − ω − iΓηω the radiation away from the interface to be very strongly where G is a constant parameter scaling the strength of dampened, so the nonlocality. With that we tight the nonlocality to the (i) (i) magnetic response. This seems to be meaningful as in |ℑ(kz )| > |ℜ(kz )|. (12) the resonance an enhanced light-matter-interaction can be observed that will cause the nonlocal response. The Using this condition, SPP with small radiative losses can resonance frequency of the material is chosen to be ω =4 also be discussed. 0 III. FRESNEL COEFFICIENTS GHz, the plasma frequency to be ωp = 10 GHz. Further, we set F = 0.56 and the damping terms shall be Γǫ = 0.03 ωp and Γµ = Γη = 0.03 ω0. For the local material As mentioned above, interface conditions have been de- parameters, the same values were used by Ruppin in [25]. rived for the model of nonlocality discussed here. Going The exact values are of no importance and only serve the on from these, the corresponding Fresnel equations have purpose to make everything definite. also been found [19]. They read We restrict ourselves to a plane geometry where the ER I half-space below the x-axis is filled with the metamate- z kz  (1) I   rial and the half-space above it with vacuum, as illus- FTM Ez = −Ez 1 (13) (2) trated in Fig. 1. Because the metamaterial is modeled Ez   0  as a homogeneous material and the vacuum is trivial, the eigenmodes and hence the fields can be described by for TM polarization with the TM Fresnel matrix plane waves R (1) (2) kz −kz −kz (i) (i)   E (r,ω) ∝ exp[i(hx + kz z)]. (9) FTM ≡ 1 −ǫ −ǫ , (14) (1) (1) 2 (2) (2) 2  0 ηkz k ηkz k  (0) E (r,ω) shall be the field in the vacuum with 2 where k(i) = (k(i))2 + h2, and (0) 2 2 2 z (kz ) = k0 − h (10) R Ex 1 being the coresponding dispersion relation. For the meta- (1) F E  = −EI kI  (15) material with a strong spatial dispersion, characterized TE x x z (2) 0 by the constitutive relation given in Eq. (1), the disper- Ex    sion relation reads as for TE polarization, where we introduced the TE Fresnel matrix (k(i))2 = −h2 + p ± p2 − q (11) z p 1 −1 −1 − 2 1 (1) (2) with q = ǫ/η and p = (2k0ηµ) [19].  R  FTE ≡ kz kz A1 kz A2 (16) This relation has four mathematical solutions in total, (1) (2)  0 ηk ηk  two of which are physical. We denote these two with the z z indexes i = 1 and i = 2. Note, that they only differ in the 2 with A = −µ−1 + ηk2 k(i) . Using these equations, sign in front of the square root. The others result in waves i h 0 i diverging towards infinity due to their positive imaginary the reflection coefficients

I 2 (1) 2 (2) 2 (1) (2) (1) (2) (1) (2) ǫkz hh + (kz ) + (kz ) + kz kz i − kz kz (kz + kz ) rTM = (17) R 2 (1) 2 (2) 2 (1) (2) (1) (2) (1) (2) ǫkz hh + (kz ) + (kz ) + kz kz i − kz kz (kz + kz ) for TM polarization and

I (1) (2) 2 (1) (2) 2 (1) 2 (2) 2 TE µkz(kz + kz )+ h − kz kz − ηµk0 k k r = 2 2 (18) µkR(k(1) + k(2))+ h2 − k(1)k(2) − ηµk2 k(1) k(2) z z z z z 0

4 for TE polarization have been calculated. The reflection that even for divergent wave numbers, as they occur in coefficients of a local medium can be reproduced from the local limit, the SPP conditions proposed here remain (i) this by performing the limit η → 0. One of the kz is meaningful. Finally, due to causality, only points outside (2) the light cone are relevant, i.e. requiring h > k0. The divergent in this limit, without restriction let it be kz . (i) (2) 2 2 −1 resulting dispersion curves are displayed in Fig. 2 for TM Now, using limη→0 kz η = 0, limη→0(kz ) η = (k0µ) (1) polarization and in Fig. 3 for TE polarization. and lim → (k )2η = 0 leads to the correct local limit η 0 z Turning to the TM polarization first, the quasilocal for both reflection coefficients, case shown in Fig. 2 (a) exhibits a dispersion relation (1) similar to the one obtained by Ruppin in [25] but for a ǫ(ω)kI − lim → k lim rTM = z η 0 z (19) lossy medium. Quasilocal means that this curve can ei- η→0 R (1) ǫ(ω)kz − limη→0 kz ther be obtained by a very low nonlocality or the classical derivation going from the purely local wave equation and and interface conditions. Instead of two divergent branches I (1) in the lossless case, it is now one connected curve that ex- k − µ(ω) lim → kz lim rTE = z η 0 . (20) hibits back-bending in place of the divergence. Opposed η→0 R (1) kz − µ(ω) limη→0 kz to the lossless case, not all solutions outside the light cone fulfill the SPP conditions. Scaling up to a low nonlocality with G = 10−8 m4 [see Fig. 2 (b)], the dispersion rela- IV. SPP DISPERSION WITH NONLOCALITY tion stays similar to the quasilocal case, although a slight broadening with a decrease in height can be observed for The SPP dispersion relation can be found from the the peaks in the dispersion relation. Again, SPP exist on poles of the reflection coefficients in Eqs. (18) and (17) the upward part of the peaks. Going to a moderate mag- R I (0) −6 4 [27] with kz = −kz = −kz . Making use of the simplifi- nitude of the nonlocality with G = 10 m [see Fig. 2 cations (c)], the first peak in the area of the resonance frequency flattens out much further. The higher frequency peak 2 (1) (2) 2 4 ǫ h roughly stays the same. The frequency range in which (kz kz ) = h + − 2 (21) η k0ηµ the SPP conditions are fulfilled also doesn’t change much for the higher frequencies while it shrinks for lower fre- and quencies. Finally, the highest nonlocality amplitude con- − − 4 4 (k(1))2 + (k(2))2 = −2h2 + (k2ηµ) 1, (22) sidered is G = 10 m due to the effect being of fourth z z 0 order. In fact, even higher nonlocalities don’t alter the these equations are solved. The solution formulas are shape of the dispersion curve any further. As can be seen very long and not displayed here for readability. Instead, in Fig. 2 (d), the lower peak has entirely vanished and numerical values as given above have been used to study now follows the light line. The upper peak didn’t change the effect of the nonlocality on the SPP dispersion in its shape much again. SPPs can now exist in an even Figs. 2 and 3. smaller frequency range for lower frequencies or on the Inequality (12) gives an existence condition for SPP. upward slope of the peak. All in all, the nonlocality has This however, is not entirely sufficient. Calculating the led to a vanishing lower frequency peak while it didn’t al- ratio of the energy transmitted by each of the waves from ter the higher frequency peak much. Looking at the cho- the Fresnel equations (15) and (13), sen form of the nonlocality, we can conclude that in the region where the nonlocality has its maximum, for high 2 2 (1) 2 (1) 2 nonlocalities the medium becomes non-dispersive along (2) kz h + (kz ) tTM h i the surface and SPP cannot exist there anymore. τTM = = (23) (1) (2) 2 (2) 2 tTM kz h + (kz ) For TE polarization, the dispersion curves are dis- h i played in Fig. 3. There are only valid solutions to the for TM polarization and dispersion relation in the frequency range where the permeability takes on negative values. Similar to the TM 2 2 (2) 2 (1) 2 dispersion relation, a comparison with the lossless case tTE h + (kz ) τTE = (1) = (2) (24) discussed by Ruppin [25] shows that instead of the sharp t h2 + (k )2 TE z divergence of the dispersion curve there is a back bend-

(i) ing into the peak as shown in Fig. 3 (a). SPPs can exist for TE polarization, we see that if one of the kz is very on the downward slope of that peak. Figure 3 (b) shows large compared to the other one, the associated wave the change when taking into account a small nonlocality − carries a lot less energy. We require 0.01 < τTE/TM < 100 with G = 10 8 m4. There is a slight change in the shape as an additional condition, such that both waves carry at of the dispersion curve, but the part that lies outside the least 1% of the energy. If this additional condition is not light cone stays almost the same, just as the frequency met, the original condition in Eq. (12) is waived for the range in that the SPP conditions are fulﬁlled. Going to (i) −6 4 very large kz . This needs to be done in order to ensure a moderate nonlocality of G = 10 m , the shape again 5

60 60 a quasilocal, G<10-8 m4 b G=10-8 m4 50 50 -1 40 -1 40

30 30 in m in in m in h 20 h 20

10 10

0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 k -1 k -1 0 in m 0 in m 60 60 c G=10-6 m4 d G=10-4 m4 50 50 -1 40 -1 40

30 30 in m in in m in h 20 h 20

10 10

0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 k -1 k -1 0 in m 0 in m

Re(h) (SPP conditions fulfilled) Re(h) Im(h) Light line

FIG. 2. SPP dispersion curve for TM polarized light. stays almost the same apart from a slight decrease in the ometry, commonly referred to as the Otto configuration, peak maximum. Just as previously, SPPs exist on the is illustrated in Fig. 4. Using a transfer matrix formalism downward slope of the peak outside the light cone. Fi- [29] and the previously derived reflection coefficients in nally, the highest nonlocality of G = 10−4 m4 decreases Eq. (17) and Eq. (18), the reflectivity of such a setup the peak height further and leads to a very slightly nar- has been calculated for both polarizations. For the meta- rower peak but only significantly changes the shape of material, the same parameters as above have been used. the dispersion curve below the light line. In summary, it The permittivity of the prism is chosen to be nondisper- can be said that this type of nonlocality does not affect sive with ǫP = 3. the TE mode SPP a lot and opposed to the TM mode, where the changes are quite significant, only causes slight Turning to TM polarization first, with an angle of in- changes in the dispersion relation. cidence of θ = π/4 and thickness d = 3cm, the spectra in Fig. 5 have been obtained. The quasilocal curve has two SPP peaks, one for each SPP branch, and is identi- V. ATR SPECTRA cal to the one predicted by Ruppin [25]. For all different magnitudes of the nonlocality, the higher frequency peak stays almost unchanged. Belonging to the higher fre- Surface plasmon polariton excitation can be observed quency peak in the SPP dispersion relation (Fig. 2), this using the method of attenuated total reflection (ATR) observation confirms the prediction that there is no sig- spectroscopy. A geometry for that was proposed by Otto nificant change to that peak. The lower frequency peak [28]. in the spectrum however, gets less pronounced and drifts It consists of a prism with permittivity ǫP, an air layer off to lower frequencies. This is associated with shrinking of thickness d, and the medium to be analyzed, so in of the frequency range that the SPP conditions are ful- this case the metamaterial. Light is sent in at an angle filled for with increasing nonlocality. Correspondingly, θ that it is totally reflected at the prism-air interface. the frequencies that SPP can be excited are lower and The evanescent waves penetrating the air layer can then lower. Additionally, the decreasing depth of the peak excite SPPs at the air-metamaterial interface. This ge- implies that the coupling strength to the SPP mode gets 6

60 60 a quasilocal, G<10-8 m4 b G=10-8 m4 50 50 -1 40 -1 40

30 30 in m in in m in h 20 h 20

10 10

0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 k -1 k -1 0 in m 0 in m 60 60 c G=10-6 m4 d G=10-4 m4 50 50 -1 40 -1 40

30 30 in m in in m in h 20 h 20

10 10

0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 k -1 k -1 0 in m 0 in m

Re(h) (SPP conditions fulfilled) Re(h) Im(h) Light line

FIG. 3. SPP dispersion curve for TE polarized light.

while the one at lower frequencies is due to the excitation of bulk polaritons. An increase of the nonlocality leads to the bulk polariton peak moving to lower frequencies and becoming less deep. The SPP peak keeps its form and position independent of the nonlocality. This is in compliance with the dispersion relation (Fig. 3), which predicted no signiﬁcant eﬀect despite the increasing nonlocality.

FIG. 4. Schema of the Otto configuration for attenuated total reflection (ATR) spectroscopy. The angle of incidence θ is VI. CONCLUSIONS chosen to be the angle of total internal reflection resulting in an evanescent wave in the air layer, that couples to the SPPs in the medium at the air-metamaterial interface, where z = 0. Concluding, we have discussed the Fresnel equations for an interface between a nonlocal, homogeneous and isotropic metamaterial and vacuum and derived expres- less with an increase in nonlocality. The very small peak sions for the reflection coefficients for both TE and TM seen inbetween the two SPP peaks is due to the excitation polarizations. Further, we proposed appropriate exis- of bulk polaritons [25]. With an increase in nonlocality, tence conditions for SPP in lossy, nonlocal media. Us- this peak also becomes less pronounced. ing these results, we obtained the dispersion relation for The spectrum for TE polarization is shown in Fig. 6. surface plasmon polaritons and discussed the effect of a The parameters used here are θ = π/3 for the angle gradually increasing nonlocality on it. We observed that of incidence and d = 1 cm for the distance between the the nonlocality has no significant effect in the case of TE prism and the metamaterial. Again, the quasilocal curve polarized light. For TM polarized light however, it leads is identical to Ruppin’s work. The minimum at higher to the collapse of the lower frequency dispersion peak to frequencies of that curve is due to the excitation of SPP a non-dispersive form while the higher frequency peak 7

1.0 1.0

0.8 0.8

0.6 0.6 R R quasilocal, 0.4 0.4 quasilocal, G G <10-8 m4 G <10-8 m4 G = 10-8 m4 G = 10-8 m4 0.2 0.2 G = 10-6 m4 G = 10-6 m4 G = 10-4 m4 G = 10-4 m4 0.0 0.0 5 10 15 20 25 5 10 15 20 25 -1 -1 k0 in m k0 in m

FIG. 5. Reflection coefficient of TM polarized light calculated FIG. 6. Reflection coefficient of TE polarized light calculated for Otto configuration with d =3 cm and θ = π/4. for Otto configuration with d =1 cm and θ = π/3. does not change its shape significantly. These observa- 1173. K.M. also acknowledges support from the Karl- tions were backed up by calculating and discussing the sruhe School of Optics and Photonics (KSOP). C.S. ac- ATR spectrum in an Otto configuration. knowledges the support of the Klaus Tschira Stiftung.

ACKNOWLEDGEMENTS

We gratefully acknowledge ﬁnancial support by the Deutsche Forschungsgemeinschaft (DFG) through CRC

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