NANO LETTERS
Slow Propagation, Anomalous Absorption, and Total External Reflection of Surface Plasmon Polaritons in Nanolayer Systems
2006 Vol. 6, No. 11 2604-2608
Mark I. Stockman* Laboratoire de Photonique Quantique et Mole´ culaire, Ecole Normale Supe´ rieure de Cachan, 61, AVenue du Pre´ sident Wilson, 94235-Cachan, France, and Department of Physics and Astronomy, Georgia State UniVersity, Atlanta, Georgia 30303 Received September 4, 2006; Revised Manuscript Received September 25, 2006
ABSTRACT I predict that a nanoscopic, high-permittivity layer on the surface of a plasmonic metal can cause total external reflection of surface plasmon polaritons (SPPs). Such a layer can be used as a mirror in nanoplasmonics, in particular for resonators of nanolasers and spasers and can also be used in adiabatic nanooptics. I also show that the earlier predicted slow propagating SPP modes, especially those with negative refraction, are highly damped.
Surface plasmon polaritons (SPPs) are electromagnetic waves propagating at the surface of a metal or interface between metal and dielectric, see, for example, refs 1 and 2. The SPPs possess many remarkable optical properties. Their wavelength can be significantly smaller than the wavelength in vacuum, and they can propagate long distances in comparison to their wavelength, which can be used in various plasmonic devices including waveguides3,4 and microscopes.5 I have introduced nanoconcentration of energy due to slowing and stopping of SPPs in tapered nanoplasmonic waveguides.6 This effect allows one to transfer energy and coherence from the far zone to the near zone without major losses. I have also proposed the adiabatic transformation of energy in other types of graded nanostructures: a two-layer system with graded dielectric permittivity and a thin dielectric layer with variable thickness over metal.7 In this system, I have predicted slow propagation and stopping of SPPs. The dielectric layers on the metal systems have been considered earlier as lenses and toy models of black holes.8 Later, systems consisting of a thin dielectric layer over metal were shown to possess slow SPP propagation in a wide spectral band.9 In this Letter, I show, in particular, that in this region of the slow SPPs, the imaginary part of the wavevector becomes on the order of its real part, so the propagation is actually absent. Therefore, even for the best plasmonic metals such as silver, this effect of slow propagation cannot actually be used in most applications envisioned in ref 9. The physical cause of this phenomenon is that in * E-mail:
[email protected]. 10.1021/nl062082g CCC: $33.50 Published on Web 10/20/2006
© 2006 American Chemical Society
the region of slow propagation the electric fields of the SPPs are concentrated in the metal, which leads to the anomalously high losses. Even more important, I predict that besides this disappointing property, there is a new, interesting effect with significant potential applications in such systems, which has been overlooked before: total external reflection of SPPs. A nanolayer of a dielectric can be a highly reflecting mirror for SPPs thus opening the way to create resonators for SPP nanolasers and spasers.10 Consider a system where a plane layer of material with dielectric permittivity 2 and thickness d is sandwiched between two semi-infinite layers with permittivities 1 and 3 as illustrated in Figure 1a. For definiteness, I set that the layers are in the xy plane, and the z direction is normal to the interfaces. I consider the SPP to be a TM wave propagating along the interfaces in the y direction whose magnetic field B is along the x axis, and the electric field possesses both the transverse (z) and longitudinal (y) components. The behavior of the wave in the direction of propagation is described by a complex wavevector k and in the normal (z) direction by the exponential decrements (increments) κi, which depend on the material (i ) 1, 2, 3). All of the fields are assumed to be uniform in the x direction. I will assume that layer 1 is a metal for which Re1(ω) < 0, where ω is the wave frequency. Layers 2 and 3 are assumed to be dielectrics with real permittivities where 2 > 3, that is, the metal is covered with a layer of a material with the permittivity higher than that of the surrounding
Figure 1. (a) Schematic of the system and coordinates. (b) Real part of the right-hand side f(k,ω) of the dispersion relation (2) as a function of wave vector k for fixed frequency ω ) 3.1 eV. Curves from bottom up correspond to n ) 0, 1, and 2. The vertical dashed line denotes the wave vector for the light wave propagation in the dielectric, k ) x2ω/c. The gray rectangle denotes the gap, i.e., the values of dielectric thickness d for which there is no surface plasmon polariton propagation. (c) The lower curve of the dispersion relation describing the fundamental mode is shown magnified and extended, for the same frequency. The horizontal lines correspond to different thicknesses, d, as discussed in the text.
dielectric 3, similar to the system of ref 7. For such a system, the dispersion equation for the SPP dispersion relation k(ω) can be cast into the form
] }
(1)
1 ω q2 - i, υi ) xi - q2, k ) q i x c
(2)
f(k,ω) )
{
d ) f(k,ω)
[
υ2(u1 + u3) 1 -Arctan + nπ , 2υ2 u1u3 - υ22 n ) 0, 1, 2, ..., where
ui )
There are two characteristic frequencies, ω2 and ω3, for this equation that are the SP frequencies for the metal interfaces with the dielectrics 2 and 3, correspondingly; they are given by the following equations: 1(ω2) + 2 ) 0, 1(ω3) + 3 ) 0
(3)
Assuming that 2 > 3, it is true that ω3 > ω2. In all specific computations and examples, I set silver as the metal and use its experimentally measured complex dielectric function.11 I set 2 ) 5, which is realistic for semiconductors or some polymers, and 3 ) 1 for vacuum as the surrounding medium. This leads to pω2 ) 3.02 eV and pω3 ) 3.68 eV. The most interesting situation is when the frequency of the SPPs is between these two frequencies, ω3 > ω > ω2. In this case, SPPs can propagate for a very thin layer of dielectric 2 (d f 0) when this layer can simply be neglected, and the SPP propagation effectively occurs at the interface Nano Lett., Vol. 6, No. 11, 2006
of the metal with dielectric 3. However, when d increases, the situation eventually changes to resemble an infinite layer of dielectric 2 over the metal where the SPP propagation is impossible. Therefore, a layer of a high-permittivity dielectric on the surface of a metal plays the role of a near-perfect mirror causing the total reflection of SPPs from it. Although this is evident from these qualitative arguments, it is still unknown how thin the layer of the dielectric should be to cause such a total reflection, which can be called the total external reflection because it occurs when SPPs are incident on the area of the high- dielectric covering the metal. To answer this question, I plot in Figure 1b the right-hand side f(k,ω) of eq 1 as a function of Re k for ω ) 3.1 eV that is between ω2 and ω3. Note that f(k,ω) is equal to layer thickness d. From this figure, it follows that there is a gap in the values of thickness d between lower critical thickness d1 ) 12 nm and upper critical thickness du ) 82 nm where there are no solutions. Thus, minimum thickness d1 of the externally reflecting plasmonic mirror is nanoscopic, which may have wide applications in nanooptics. This effect went unnoticed in the earlier related publications. Considering thin (d < du ) 82 nm) dielectric layers, on Figure 1 (c), I illustrate three possible cases depending on d. For thickness d0 > d1 ) 12 nm, there are no propagating SPPs. The thickness d ) d1 ) 12 nm is a bifurcation point at which there is a single solution of eq 1. For a smaller thickness, d2 < d1 ) 12 nm, there are two solutions. The first of them is a long wavelength (small k), fast SPP (a root indicated by the left dot in the figure). The second solution describes a short wavelength (large k), slow SPP indicated by the right dot on the line d2. In contrast, for frequency ω < ω2, there is only one nearly real, small-k root of eq 1 corresponding to a fast SPP. There is also the second root, but it has a very large imaginary part and does not correspond to any propagating SPPs (data not shown). In refs 6 and 7, the idea of adiabatic nanooptics was introduced. A system that carries SPP is characterized by its parameters varying in space. Such parameters can be a geometric size of a metal waveguide,6 chemical composition, or thickness of the dielectric covering the metal.7 As a result, the wavevector of the SPPs changes in space. If a parameter changes very slowly in space, then the SPP wave adiabatically follows, adjusts to these parameters without back reflection, scattering in the plane, or scattering into the three dimensions. This provides a possibility, in particular, to slow down and stop SPP transforming them into standing surface plasmons. Other possibilities include refraction and focusing of SPPs. Adiabatic plasmonic optics has a significant advantage over conventional optics with discreet elements because any sharp edges of such elements can trap surface plasmons and scatter them out of the plane of propagation into the three dimensions. For the adiabatic nanooptics, it is important to know how the wavevector of the propagating SPPs depends on the thickness of the dielectric layer. Such a dependence is displayed in Figure 2a for frequency ω ) 3.1 eV, which is between ω2 and ω3, when the total external reflection is possible. As one can see, as the thickness of the dielectric 2605
Figure 2. (a) For a thin dielectric layer on silver, dependence of the wave vector of the surface plasmon polariton on the thickness of this layer for fixed frequency ω ) 3.1 eV. The solid curve corresponds to Re k; the dashed curve corresponds to Im k. (b) The same for ω ) 2.5 eV; Im k is multiplied by a factor of 10 (actual values of Im k are 10 times smaller than those on the graph). (c) Dependence of critical thickness d1 on frequency for three values of 2: 2, 5, and 10 (as shown for curves from top to bottom). (d) The same for upper critical thickness du. (e) Schematic of the total external reection of a SPP wave from a 20-nm thin dielectric layer on metal (silver).
layer over the silver increases from 0 to 12 nm, the real part of the wavevector increases by a factor of ∼2 while its imaginary part stays small, suggesting a good propagation with small losses. Such a dependence k(d) substituted in the eikonal equation for SPP phase, (∇φ)2 ) k2(d), defines the adiabatic refraction of the SPPs. When thickness d riches the critical value, which for this frequency is 12 nm, the propagation terminates: the imaginary part of k starts to dominate, which implies the evanescent, exponentially decaying tail of the fields in the direction of the propagation. Note that this large Im k is a property of dispersion eqs 1 and 2 and not of the imaginary part of the metal’s dielectric permittivity that is not very large in the spectral region under consideration (Im1 , |Re1|). For layer thickness d between d1 and du (for ω ) 3.1 eV, this interval is 82 nm > d > 12 nm), the SPP propagation is impossible, and such a dielectric layer causes the total external reflection. One needs, however, to caution that the finite losses in the metal bring about gradual rise of Im k and decline of Re k (Figure 2a). Consequently, Im k dominates over Re k, which would be characteristic of the total internal reflection in dielectrics, only for a layer thickness significantly exceeding the lower critical value d1 2606
) 12 nm; actually, for d J 20 nm that is still nanoscale. With this difference in mind, this is a unique case of the total external reflection at optical frequencies. This effect is illustrated schematically in Figure 2e where a SPP wave (shown by the red wavy arrows) is totally externally reflected from a 20-nm layer of dielectric on massive silver. Note that this reflection is not related to the effective index mismatch but is due to the absence of the propagating SPP modes. The effect that a nanoscopic layer of dielectric with thickness d ≈ 20 nm can cause the total external reflection and thus serve as high-efficiency two-dimensional (2d) mirror for SPPs can find applications in nanotechnology to create 2d resonant cavities for SPPs. Among possible applications will be optical nanofilters, nanospectrometers, nanolasers, and spasers.10 Given the strong dependence on the parameters near the critical thickness of d ) d1 ≈ 12 nm, such a layer can also be used as an efficient electrooptical or all-optical modulator in SPP nanooptics. For relatively thick layers (d > du ) 82 nm), as one can understand from Figure 1b, there will be oscillations of the propagation properties of the SPP because of the higher bands involved corresponding to modes in the z direction. This class of phenomena is of lesser interest to nanooptics (because of the relatively thick layers) and will not be considered in this Letter. Now let me turn to the lower-frequency range ω < ω2 where the SPP propagation is possible for arbitrary d. For this case, the dependence of the SPP wavevector on d is displayed in Figure 2b for a frequency of ω ) 2.5 eV. In contrast to panel a, there is no critical thickness; Re k increases monotonically with d, interpolating between the value of k ) (ω/c)[13/(1 + 3)](1/2) for the metal/dielectric 3 interface for d f 0 and the magnitude of k ) (ω/c)[12/ (1 + 2)]1/2 that one predicts for the metal/dielectric 2 interface for d f ∞. There is no total external reflection: Im k , |Re k| everywhere. The large interval of change of the effective index neff ) k(c/ω) can be used efficiently in adiabatic nanooptics to control SPP propagation. Dependence of the critical thickness d1 (for which the total external reflection sets on) on frequency is shown in Figure 2c for three values of 2. In a wide range of parameters, this thickness is nanoscopic, d1 ≈ 10 nm. Panel d displays similar dependencies for the upper critical thickness, du, for which the total external reflection disappears and a continuous band of propagation begins. This thickness is much greater, du ≈ 100 nm, which corresponds to the formation of modal structure in the Fabri-Perot resonator formed by the layer. Earlier, the idea of the control of SPP propagation by a dielectric lens over metal has been developed on the basis of understanding that a thin dielectric layer acts as an infinite layer with an effective dielectric constant, neff, that smoothly depends on its thickness.5,8 This approach misses the existence of the forbidden zone of thickness d1 < d < du described above and an oscillating structure of modes in d > du films. It also lacks the effects of the SPP slow propagation9 and the associated negative refraction and anomalous losses (see below). Nano Lett., Vol. 6, No. 11, 2006
Figure 3. Dispersion relation, i.e., the dependence of frequency of surafce plasmon polaritons on the real part of wave vector Re k (panel a). Frequency dependence of the imaginary part of the wave vector (panel b); the upper and lower branches of this dependence correspond to the positive and negative group velocities (i.e., righthanded and left-handed propagation, respectively). The discontinuity in Im k is shown by the vertical line. Dielectric layer with 2 ) 5 and thickness of 5 nm on silver.
Now I consider the SPP dispersion relation obtained as a solution of eq 1 for d ) 5 nm. The real part of it is displayed in Figure 3a as a dependence ω(k). In accord with our discussion of Figure 1c, for ω < ω2 ) 3.02 eV, there is only one root, and it does correspond to a fast SPP whose phase velocity is close to that of a light wave. The SPP dispersion curve dramatically deviates from this behavior close to its maximum at a frequency ωm ) 3.40 eV. Note that ωm is indeed smaller than the upper limit ω3 ) 3.68 eV. It is actually ωm that determines the highest allowed frequency of the SPPs redefining (narrowing) the range of the existence of the negative refraction as ωm > ω > ω2 narowing it from the side of the high frequencies compared with what was discussed above in conjunction with eq 3. Note that the specific value of ωm is a function of d. For ωm > ω > ω2, there are two branches of SPPs that correspond to the two solutions illustrated by Figure 1c. The small-k root corresponds to fast-propagating SPP with normal dispersion (positive refraction), whereas the high-k root corresponds to slow SPPs with negative refraction. Note that the positive or negative refraction is described by the sign of the group velocity Vg ) ∂ω/∂k. In the negative refraction region and close to it, Vg is anomalously low, which is displayed by a very flat dispersion curve. This appears to be in a perfect agreement with the main result of ref 9. However, I have not yet considered the imaginary part of the dispersion relation, which appears not to have been taken into consideration in ref 9. This imaginary part is shown in Figure 3b. In the region of fast propagation (ω < ω2 ) 3.02 eV), Im k is very small comparable to Re k, which corresponds to weakly absorbed, well-propagating, long-range SPPs. In contrast for ω J ω2, Im k is comparable to Re k, which implies that there are no well-defined, propagating SPPs. There is a discontinuity pronounced in Figure 3b, where Im k changes its sign. Such a discontinuity should generally take place at the maximum ωm of the dispersion curve ω(k) where the positive refraction that corresponds to Im k > 0 changes to the negative one where Im k < 0. Thus, complex function k(ω) has an essential singularity at a point ω ) ωm. In the negative refraction region, the negative value of Im k is in accord with the propagation of energy opposite to the direction of the phase propagation. For the negative refraction branch, Nano Lett., Vol. 6, No. 11, 2006
Figure 4. For a thin (5 nm) dielectric layer (2 ) 5) between silver and vacuum, components of the local fields of surface plasmon polaritons are shown in the plane of propagation normal to the interfaces. The direction of propagation is y; the normal to the interfaces is z. Fields are shown in arbitrary units consistent within any single panel. The upper and lower panels display the two branches of the surface plasmon polaritons: the low-k, positiverefraction branch (upper panels) and high-k, negative-refraction branch (lower panels), as indicated in the graphs. All of the panels correspond to the same frequency pω ) 3.2 eV. Silver occupies the lower half-space (z < 0); the dielectric nanolayer is at 5 nm g z g 0; vacuum is in the upper half-space (z > 5 nm).
Im k is very high, whereas for the positive refraction branch it becomes smaller in the vicinity of ω2. This interesting behavior notwithstanding, these effects may not be readily observable because of the large Im k, that is, strong absorption of the SPPs in this spectral region, which is missed in ref 9. To gain more insight into this anomalous behavior of the SPPs in the frequency interval ωm > ω > ω2, I consider the distribution of the SPP local fields in the nanovicinty of the dielectric layer. It has been conventionally obtained as a solution of the Maxwell equations evanescent in the outer layers 1 and 3 satisfying the continuity conditions for the tangential fields at the interfaces. I choose frequency ω ) 3.2 eV that is close enough to ω2, so the SPPs with the positive refraction possess not very high Im k and can propagate, cf. Figure 3b. In contrast, the negative refraction SPPs (with high k) always have very large values of |Im k| ∼ Re k and, therefore, cannot propagate. The corresponding distributions of the local electric field in the plane of propagation normal to the interface (yz plane) are shown in Figure 4. Panels a and b show the local electric fields for the positive-refraction SPPs. The SPPs propagate with wavelengths that are indeed smaller than those in 2607
vacuum, and their fields are delocalized into the vacuum to distances ∼50 nm. There is appreciable decay of the SPPs along the direction of propagation (y) at distances of ∼1 µm, as is expected from the values of Im k shown in Figure 3b. In a sharp contrast, the negative-refraction (high-k) SPPs whose local electric fields are displayed in Figure 4c and d do dramatically decay on a distance of ∼50 nm. (Note the propagation trend from right to left, i.e., counter to the wave vector.) Hence in practical terms, the negative-refraction SPPs do not propagate. The physical reason for that is evident from Figure 4c and d: the local fields are significantly localized in the metal where there is a strong absorption due to the Ohmic losses. To conclude briefly, I have considered propagation of SPPs at a metal surface covered with a dielectric layer with nanoscopic thickness whose permittivity is higher than that for the surrounding space. There are two main results that I establish. (i) There is a range of thickness (typically between ∼10 nm and ∼100 nm) for which propagation of SPPs is forbidden. A layer with thickness in this forbidden range causes total external reflection of SPPs that propagate at the interface toward it, see Figure 2e. This opens up a possibility with a single dielectric layer to fabricate high-quality mirrors and resonators for plasmonic nanooptics. (ii) The slow propagation of SPPs predicted earlier9 is intrinsically related to strong losses due to the localization of fields in the metal. In the region of negative refraction, these losses cause dissipation of SPPs at distances on order of their wavelength. I trust that the much better way to achieve small group velocity and, in addition, small phase velocity (slowing down and stopping) of SPPs with acceptable losses is using the adiabatically tapered plasmonic waveguides.6
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Acknowledgment. This work was supported by grants from the Chemical Sciences, Biosciences and Geosciences Division of the Office of Basic Energy Sciences, Office of Science, U.S. Department of Energy, a NIRT grant CHE0507147 from the National Science Foundation, and a grant from the U.S.-Israel Binational Science Foundation. I appreciate the hospitality of and useful discussions with J. Zyss during my Invited Professorship at ENS Cachan (France). References (1) Barnes, W. L.; Dereux, A.; Ebbesen, T. W. Nature 2003, 424, 824830. (2) Zayats, A. V.; Smolyaninov, I. I. J. Opt. A 2003, 5, S16-S50. (3) Bozhevolnyi, S. I.; Volkov, V. S.; Devaux, E.; Laluet, J.-Y.; Ebbesen, T. W. Nature 2006, 440, 508-511. (4) Bozhevolnyi, S. I.; Erland, J.; Leosson, K.; Skovgaard, P. M. W.; Hvam, J. M. Phys. ReV. Lett. 2001, 86, 3008-3011. (5) Smolyaninov, I. I.; Elliott, J.; Zayats, A. V.; Davis, C. C. Phys. ReV. Lett. 2005, 94, 057401-1-4. (6) Stockman, M. I. Phys. ReV. Lett. 2004, 93, 137404-1-4. (7) Stockman, M. I. Delivering Energy to Nanoscale: Rapid Adiabatic Transformation, Concentration, and Stopping of Radiation in NanoOptics. In Plasmonics: Metallic Nanostructures and Their Optical Properties II; Halas, N. J.; Huser, T. R., Eds.; SPIE: Denver, CO, 2004; Vol. 5512. (8) Smolyaninov, I. I. New J. Phys. 2003, 5, 147.1-147.8. (9) Karalis, A.; Lidorikis, E.; Ibanescu, M.; Joannopoulos, J. D.; Soljacic, M. Phys. ReV. Lett. 2005, 95, 063901-1-4. (10) Bergman, D. J.; Stockman, M. I. Phys. ReV. Lett. 2003, 90, 0274021-4. (11) Johnson, P. B.; Christy, R. W. Phys. ReV. B 1972, 6, 4370-4379.
NL062082G
Nano Lett., Vol. 6, No. 11, 2006
