Attenuation of Surface Plasmon Intensity by Transverse and Longitudinal. Slits

Transcription

1 Attenuation of Surface Plasmon Intensity by Transverse and Longitudinal Slits Michael I. Haftel, 1 Brian S. Dennis, 2 Vladimir Aksyuk, 3 Timothy Su, 2 and Girsh Blumberg 2 1 University of Colorado at Colorado Springs, Colorado Springs, CO 80918, USA 2 Rutgers, The State University of New Jersey, Piscataway, NJ 08854, USA 3 Center for Nanoscale Science and Technology, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA Abstract: We investigate the losses in surface plasmon polariton (SPP) intensity as the SPP wave propagates on gold surfaces encountering certain surface features generic to device components: specifically, transverse and longitudinal slits and grooves. We first consider the losses of a SPP jet, of vacuum wavelength 732 nm, as it traverses a single slit of widths 50 nm to one wavelength. Our study includes finite-difference-time-domain (FDTD) simulations, theoretical treatment, and experimental measurements. The theoretical approach is an application of the tight-binding-like technique of Lopez-Tjeira et al. While all three approaches differ slightly quantitatively, they all indicate initially increasing losses, mainly through scattering to the vacuum, but with a leveling off of losses (or perhaps a slight decrease) when the slit width is an integral multiple of half wavelengths. Losses at a half a wavelength slit width are about 50% over the gap. We then consider SPs propagating on parallel gold strips (of 0.75 µm width) separated by gaps of 50 nm to 400 nm. Here FDTD simulations, experiment, and an extension of the theoretical treatment are in essential agreement with the SPP range decreasing from about 33 µm for zero strip separation to about 4 µm for 400 nm separation. We discuss the impacts of these results on device design.

2 I. Introduction The emerging field of nanoplasmonics has shown considerable promise in revolutionizing the field of optical devices. The main virtue of the use of surface plasmons, which we use for short for surface plasmon polaritons (SPPs) electromagnetic (em) fields confined to the surface of metals, usually in the visible and IR is that this confinement to a surface allows lateral focusing to dimensions substantially smaller than the light wavelength at comparable frequencies. 1-4 The confinement to surfaces facilitates the control of the flow of optical energy 5 and wave properties (e.g., phase, interference) by interaction with nanostructures on such surfaces and their ultimate integration with silicon technology. Since SPPs are typically generated by optical means (e.g., lasers at appropriate frequencies) the efficient conversion of optical energy to SPPs is critical in designing practical nanoplasmonic devices, especially in light of the finite range of SPPs, usually measured in tens of microns. Thus ways to efficiently generate SPPs and to minimize losses as they interact with control nanostructures become critical issues. The present authors addressed some aspects of the efficient conversion of optical to SPP conversion in a previous work 6,7. This paper mainly addresses the issue of attenuation and losses as SPPs propagate in the presence of certain device components consisting of patterns of slits. Propagating SPPs in devices may have to jump transverse gaps as well as propagate along longitudinal strips, both of which involve losses due to reflection/transmission from these structures and/or alteration of the propagation vector. This paper assesses the basic question of losses in SPP intensity as the SPP jet propagates in the presence of longitudinal and transverse slits. In this paper we investigate, experimentally and computationally, the losses in SPP intensity due to jumping transverse gaps or propagating along longitudinal strips. We first describe the experimental setup

3 and measurements, and then present the simulation results and compare the measurements with the simulations as well as to a theoretical approach 8 that utilizes an electromagnetic analogue to tight-binding theory. The simulations utilize the finite-difference-time-domain (FDTD) method using the High Accuracy Scattering and Propagation code (HASP). 9,10 We calculate the reflection/transmission at a transverse gap as a function of its width for widths of up to one wavelength, which is 732 nm in this study. In the case of longitudinal strips the losses can be mainly described by an attenuation constant, and we examine how this attenuation constant depends on the air gap between Au strips (which are taken to be 0.75 µm wide). The experimental, simulation, and theoretical results are in reasonable agreement and we conclude with a discussion of how these resulting trends impact the design of such device features. II. Experiment A 230 nm Au film was sputtered onto a glass substrate. Nanostructures were ion milled using a focused Ga ion beam and included transmission in-coupler gratings (13 slits of width 400 nm with period 665 nm), out-coupler slits (width 400 nm), longitudinal strips and transverse slits (see Fig. 1). 732 nm laser light (spot size 10 µm) from a Kr + pumped tunable dye laser was incident on the in-coupling grating at 10 with p- polarization in the direction of SP propagation. The wavelength was experimentally determined to give the strongest out-coupling. Out-coupled light was imaged with a CCD camera and the intensity was integrated around the out-coupler as in Fig. 1b. All measurements had identical in- and out-couplers and were fabricated side by side with the SP propagation directions parallel (Fig. 1). Comparison of the attenuation of out-coupled intensities due to the intermediate structures (longitudinal strips or transverse slits) was accomplished by moving the incouplers into the stationary laser beam and measuring the out-coupling as desribed above. Reference couplers with no intermediate structures were also used.

4 The two different longitudinal strip both have strip widths 750 nm and length 10 µm with longitudinal gaps 250 nm and 400 nm. The three different tranverse slits are all 8 µm long with widths 255 nm, 360 nm and 490 nm. Fig. 1: a) Schematic showing in and out coupling of surface plasmon after propagation over longitudinal strips; b) general in and out coupling optical image (no intermediate structure) to illustrate out coupled intensity integration area (white dotted box); SEM images with longitudinal slits (c) and transverse slit (d). The coordinate axes employed, relative to the structure, are also shown with the z being the direction of SPP propagation, x the direction across the structure, and y coming out of the page. III. Simulation, Experimental, and Theoretical Analysis The problem that we numerically simulate, by use of the HASP code, is schematically shown in Fig. 1, as well as the definition of the coordinate axes we employ. The details of the HASP code and our handling of the frequency dependence of the gold dielectric constant appear in previous works. 9 In this paper we

5 simulate the propagation of SPPs of a frequency corresponding to that of wavelength 732 nm light in vacuum. The SPP wavelength is 712 nm. The SPPs are launched in a source region, which could be a grating (as in the experiment) or an appropriate current source. We choose a current source for reasons we will discuss shortly. The SPPs propagate away from the source region (in the z direction) encountering a series of longitudinal and/or transverse slits (Fig. 1c,d). An important consideration for both types of slits is to balance the needs of small losses, typical of narrow slits, to those of ease of fabrication and wavefront control, which might favor wider slits. The HASP code 9 calculates, by the FDTD method, 10 the spatial and temporal dependence of the em fields. We assess the SP intensity through the value of S z (y 0), the Poynting vector in the direction of SP propagation near the of the Au-Air interface. The extent of the computational box is about 30 µm in the z direction and 5 µm to 10 µm in the y (vertical) direction. We truncate the computational space by perfectly matched layer (PML) boundary conditions 11,12 in the y and z directions, and periodic boundary conditions in the x direction. Since the HASP code employs a variation of the Yee algorithm 10 that vastly improves the numerical accuracy, we can employ a spatial grid step of 10 nm and a time step of 0.02 fs and obtain numerically stable, accurate results. Two main factors influence the simulation accuracy: reflections at the ends of the computational box, which is a numerical feature handled by imposing absorbing boundary conditions such as PML, and truncation errors introduced by the finite mesh size and time step. The HASP algorithm 9 employs a nonstandard extension of the usual Yee algorithm 10 that drastically reduces the truncation error, both for amplitude and phase. For a an electromagnetic (em) wave propagating in a uniform dielectric region the error for the Yee algorithm with a coarse grid ( 20 points per wavelength) is about 10%, whereas it is about 10-6 for the HASP algorithm. Of course errors are larger in nonuniform regions usually encountered. On the other hand we use about 50 to 100 points per wavelength in the present work. From

6 running the HASP code at different mesh lengths we estimate the truncation error at about 3 %. We also tested the effect of artificial reflections by varying the size of the computational box. Results were stable to about 1 %. From these considerations we conclude that the field intensities (and Poynting vectors) are accurate to 3 % to 5 %. We find that an efficient way to launch SPs computationally is to introduce a current source that mimics the time and vertical (y) spatial dependence of the SP field. In this case we omit the input slits in Fig. 1 and introduce a source current J S (r,t) into the FDTD solution of Maxwell s equations, where J S (r,t) = y J 0 (z) exp(-κ i y ) exp(-iωt), (1) where y is the unit vector in the y direction, ω is the angular frequency, J 0 (z) is a constant in a finite region of z (which we take to be a 50 nm interval at the beginning of what would be the input slit region), zero elsewhere, and κ i is the SPP decay constant in the direction normal to the air-au interface for material i (with i being Au or air), with y = 0 being the Au-air interface. This SPP decay constants, which are essentially the imaginary parts of the SPP wave vectors in the y direction, are given by κ i = (k 2 SP - ω 2 ε i / c 2 ) 1/2, (2) with ε i being the dielectric constant in region i and k SP being the well - known SP wave number 13 k sp = ω (ε air ε Au / (ε air + ε Au )) 1/2 / c. (3) We call this method of launching SPPs current launched. The current in Eq. (1) should be regarded as merely a device to launch SPPs as we are mainly interested in how the SPPs, once launched, lose intensity as they traverse the transverse and longitudinal slits on the right-hand-sides of Figs. 1c and 1d. We confirm numerically that the current source in Eq. (1) produces SPPs with the correct wavelength and vertical decay outside of the current source region. Another way to launch SPPs is to impinge incident light on a grating of the appropriate spacing as shown in Fig. 1a. The advantage of grating launched

7 SPPs is that it is close to the experimental arrangement; the advantage of current launched is that it can be used to examine the propagation of SPs without the extraneous influences of scattering and diffraction from the input slits. In this work the simulations employ the current launching of SPPs. Figure 2: Surface plasmon intensity, for various transverse gap widths, as measured by the averaged Poynting vector component S z from zero to 600 nm above the Au surface, as a function of distance from the current source and gap width (w). The relative value of S z is normalized to a value of 1.0 for the averaged Poynting vector 3.1 µm from the current source when there is no gap. The diagram at the top illustrates the gold film (filled in blue) relative to the z coordinate on the abscissa and the position of the gap at 13.6 µm from the current source. The Au film is 200 nm thick. The numbers labeling the curves are the widths of the gap in nanometers.

8 Figure 2 shows the loss of SP intensity, as measured by the Poynting vector, S z averaged over 600 nm (roughly the vertical decay length of the SPP field in the vacuum) above the gold film, normalized to the Poynting vector 3.1 µm from the current source when there is no gap. The current-launched SPP propagates on the Au film over a single transverse gap, of various widths, where the beginning of the gap is positioned 13.6 µm from the current source, as shown on the diagram at the top of the figure. Note that losses exist even when the gap is absent, with the SP having a range of about 27 µm. Losses occur here because the dielectric constant of Au at this frequency has an imaginary part, i.e., there is absorption. Figure 3: Surface plasmon transmission across a transverse gap as a function of the gap width and experimental results. The transmission is relative value of S z, averaged vertically as described in Fig. 2, but normalized to a value of 1.0 for the averaged Poynting vector 16.6 µm from the current source when there is no gap (see text). Results are also shown using the tight binding like theory of ref. 8 for the total SPP transmission, losses due to scattering of the electromagnetic energy from the SPPs into the vacuum at the gap, and the back reflection of SPPs from the gap. Experimental uncertainty at 95 % confidence level (2 standard deviations of the mean) is indicated by vertical error bars and was determined by statistical analysis of the reproducibility of the outcoupled intensity.

9 Figure 3 gives the SP transmission across the gap, as a function of gap width. The losses shown in Fig. 3 are those we attribute just to the gap itself, which includes the effects of reflection of SPPs and scattering of electromagnetic energy into the vacuum. We extract the transmission across the gap by comparing the Poynting vector S z, averaged as in Fig. 2, at a point 2.9 µm after the beginning of the transverse gap (corresponding to the nm mark in Fig. 2) with that of the zero-gap result at the same point, which is thus normalized to unity. While the Poynting vector at this observation point contains the effects of further losses just from the propagation on the gold surface from the end of the gap to this observation point, the renormalization removes this effect that is not related to the gap itself. The simulations use a spatial mesh of (6 nm) 3 and a time step of 1/200 of the wave period. When compared with results of a (10 nm) 3 and time step of 1/120 of a period, the transmission increases by 3-5%. Thus we estimate the numerical error in the transmission as 3-5%. Fig. 3 also gives our corresponding experimental measurement of losses due to the gap. The setups of the simulation and experiment differ slightly e.g., the gap in the simulation is 13.6 µm past the current source, whereas it is about 18 µm past the end of the grating in the experiment. Also the experiment measures intensity by measuring the light coming from the output slit (30 µm past the grating) whereas the simulation directly measures the SP intensity by calculating the Poynting vector. To make a meaningful comparison we normalize the experimental intensities to the simulation result for the zero-width gap. This isolates the effect of the transmission across the gap and eliminates the effects of further losses because of the propagation distance between the gap and the output slit and of the output slit itself (assuming that the output light intensity is proportional to the SP intensity). The simulation results in Figs. 2 and 3 exhibit two trends. As might be expected from scattering or diffraction arguments, initially the transmission rapidly decreases with gap size. However, as the gap approaches λ SP /2 (356 nm) the decrease quenches somewhat. From gap widths λ SP /2 to λ SP the trend

10 repeats: a decrease in SP intensity with w followed by a leveling off as the gap approaches λ SP. In fact, there is actually a local maximum near λ SP. Overall there is a general increase in losses with increasing w, but superimposed on this is a reflective interference effect of period λ SP /2 similar to that of a Fabry-Perot (FP) resonator. The experimental measurements, while generally indicating more transmission than the simulations, exhibit a similar alternating minima maxima pattern. The minima are close to the minima in the simulations, but the maxima appear closer together than the simulations - at about 450 nm and 620 nm gaps, as opposed to about 355 nm (here actually a leveling in the decrease) and 720 nm in the simulations. The general decrease of transmission with gap along with local maxima at ~nλ SP /2 is argued qualitatively as follows. The SP field has a longitudinal electric field and the finite conductivity at the relevant frequency yields a current flow. The coupling between two Au sides would then be capacitive (via Coulomb fields) and we would expect strong capacitive coupling for tiny gaps (typically λ/8 imperfections do little damage to the wave) and weakening for larger gaps. The SP field also has transverse E and H components, and this part resembles an ordinary propagating plane wave in which case the gap would act like a Fabry-Perot resonator giving an alternating maximum minimum structure with gap width.. Reference 8 studies the interaction of SPs with slits or grooves using an analogy of the tight-binding method used in solid-state physics. (We will refer to this theory by the shorthand TB). Here the fields in the grooves are expanded in modes similar to those enclosed in perfect conductors, and the fields are matched at the top and bottom boundaries of the metal to calculate the fields in all space. The dotted curves in Fig. 3 give the results of this approach for SPs propagating on top of a 200 nm gold film traversing a single gap of various widths. The losses in this approach due to reflection and scattering to the vacuum also appear in the figure. While there are some differences between the TB results and

11 simulations (e.g., the simulations show less transmission at for small gaps) the trends are in good agreement, especially for the muted FP effect mentioned above. In the usual FP effect minimum reflection and maximum transmission occurs at half wavelengths. In our situation the reflection is evidently always small (Fig. 3), and losses are almost entirely due to scattering of the SSPs into the upper vacuum, which is minimized at gaps of multiples of half wavelengths. The simulations, theory, and experiments all resemble the usual FP case in that transmission minima occur near odd multiples of quarter wavelengths. Figure 4 shows the SP intensity as it propagates along a series of longitudinal gold strips separated by air gaps as shown in the diagram at the top of the figure. In our experimental arrangement we usually employ about 8 or 9 strips of width 0.75 µm (as opposed to three shown in the figure). The simulations employ periodic boundary conditions in the x direction, which corresponds to an infinite series of such strips (and gaps). The SPs in the simulation are current-launched, and the spatial grid used is (10 nm) 3 and the time step 1/120 of a period. By comparison to simulations with a (15 nm) 3 grid, we estimate that the error in the ranges in the inset of Fig. 4 at 10-15%. The strip structure starts 3.6 µm from the end of the launching current region and extends 11 µm, as in the experiment, at which point an ungrooved Au film resumes. Extending the grooved film further to the end of the 30 µm box has negligible effect on the curves of Fig. 4. To a good degree all the curves in Fig. 4 have exponential decay, even when there is no gap, as previously discussed. The inset of Fig. 4 gives the range, as a function of gap size, based on a least squares fit, for the seven gap sizes considered. The losses here may be described in terms of scattering from the strip structure and modifications of the SP propagation vector due to the periodic strip structure, as illustrated by the agreement with the theoretical curve (labeled TB ) in the inset, as will be described below. The inset to Fig. 4 also shows results of experimental measurements for a 9 strip SP waveguide. The

12 experimental range is extracted from the transmission data at the output slit taking into account that the strip region is 11 µm long. As with Fig. 3 the experimental measurements have been normalized to simulation results for the zero gap case, and thus the ranges in the inset correctly shows the difference in ranges between the simulations and experiment (and also difference between the finite gap cases and zero gap). Generally there is good agreement between experiment, theory, and simulation for the overall decrease of range with gap size. Simulations give a fairly uniform decrease in range with gap width, while the theory and experiment indicate some maximum minimum structure (inset, Fig. 4). However the positions of the maxima and minima in theory and experiment differ. For device applications we would like ranges > 15 µm, which indicates that gaps up to about 150 nm could be tolerated. One reason for the disagreement between simulations and experiment could be due to the simulation employing an infinite number of strips while the experiment employs a finite number (8 or 9). The finite number of strips used in the experiment would necessarily lead to a somewhat shorter SP range than in the simulation. The simulations also indicate a range of about 28 µm for the ungrooved case compared to a range of 33 µm gotten from the imaginary part of the SP wave number at 732 nm for a flat Au film. Ebbesen et al. 14 have discussed the usefulness of metal strips as SP waveguides. Previous treatments of SP propagation on strips have mainly concentrated on a single metal strip, 15,16 whereas our case employs multiple (e.g., 8) strips. The tight binding theory of ref. 8 does not directly address longitudinal gaps but restricts itself to SPs normally incident on transverse gaps. We extend this approach as follows: We use the methods of ref. 8 to calculate the transmission and reflection of SPs normally incident on a series of n transverse gaps (usually n = 8) with the same width and spacing as the longitudinal gaps we are considering. We then approximate the problem in a 2D way where an electromagnetic (em) wave of wave number k SP and frequency ω in the metallic regions encounters a dielectric region with a complex dielectric constant and width ( n- 1) d + a, where a is the gap width of the slits and d is the periodicity of the slits. The effective dielectric constant in the metallic regions of width d a is determined by k SP and

13 ω, and we calculate what effective complex dielectric constant in the region of width ( n- 1) d + a would account for the transmission and reflection results of the TB theory of ref. 8. (This has a significant imaginary part since most of the losses are due to scattering, not reflection, and this is treated as an effective absorption). We then estimate a typical change in the SP propagation vector due to interaction Figure 4. Surface plasmon intensity, for current launched SPs, as measured by the summed Poynting vector component S z from zero to 600 nm above the Au surface, as a function of distance from the beginning of a striped region and gap width. The number associated with each curve gives the gap width in nm, whereas the Au strip width is always 0.75 µm. The position of the gap is illustrated at the top of

14 the figure with the filled region indicating the Au surface. The inset gives the e folding range as a function of gap width (see text) from simulations and the theory of ref. 8 labeled TB theory. Error bars of ± 3% (too small to be seen) are assigned to the experimental points in the inset based on an experimental uncertainty at 95 % confidence level (2 standard deviations of the mean) and was determined by statistical analysis of the reproducibility of the out coupled intensity. with the series of slits represented by a corresponding series of alternating dielectric regions, which is in the x direction, i.e., normal to the slits. These alternating regions consist of a metallic region d a = 0.75µm wide followed by a region of width a region ( n- 1) d + a of its dielectric constant. The change in the wave number can have both real and imaginary parts where the latter will correspond to the attenuation the diverted SP suffers as it propagates in the direction perpendicular to the slits (as well as parallel). There are a number of ways one could estimate the change in the SP wave number (Δk SP ) due to the longitudinal grating. One could employ diffraction theory to obtain a distribution F(Δk x ) of x-component wave number changes calculated, for example, from a suitable summation of by the sum of Fresnel like diffraction integrals, and use some kind of average Δk x. Another way, appropriate for a periodic collection of alternating (complex) dielectric regions, is to solve for Bloch states. In this case one obtains both real and imaginary parts of the Bloch q vector (which here only has an x component). We employ the latter method. The real part of q, which we look for in the first Brillouin zone, indicates the typical displacement of the wave vector in the x direction, and the imaginary part gives the decay constant of the SP wave amplitude in the x direction. The overall decay constant κ for the SP intensity I as measured along the principal propagation direction along the z axis is then given by!" κ =! =!!!"!!"!"!"!!"!"!!" = 2Im q!"!!"(!!" ) + 2Im k!", 4

15 where the factor two appears because of converting decay constants for the amplitude to that for the intensity. The e-folding range is simply 1/κ. With this ansatz the Re(q) ranges from nm -1 for 50 nm air gaps to.0026 nm -1 for 400 nm gaps, and Im(q) ranges from nm -1 to nm -1- over the same interval of gap widths, compared to k SP = i nm -1. The ranges obtained from Eq. (4) appear in curve labeled TB theory in the inset of Fig. 4. The theoretical results are in good agreement at the experimental points (which have about ± 3% error bars) and with results obtained from simulations, except the simulation results appear to be slightly too low for the narrowest gaps. We lastly describe the losses due to a structure similar to the device shown in Fig. 1c. Fig. 5 gives the losses for SPs propagating 12 µm along 0.75 µm wide gold longitudinal strips with 150 nm between strips terminated by a transverse slit of various widths. This determines the losses one would encounter for a typical design configuration as in Fig. 1. In Fig. 5 the SSP e-folding range is about 12 µm in the region of the longitudinal slits, consistent with Fig. 4. Overall the losses are 81% to 88% over this typical device configuration terminated at the transverse slit. This may be a (barely) tolerable loss in applications. The results shown in Figs. 3-5 give a realistic assessment of how such design features would affect device performance from losses and would greatly impact device design. Working at longer wavelengths than

16 732 nm would help in that losses even for a flat Au surface would be considerably less. Fig. 5: FDTD simulation results for the surface plasmon intensity, defined as in Figs. 2 4, as it propagates along a series of gold strips followed by a transverse gap leading to a gold film (without strips). This is a typical configuration for the bottom part of the device shown in Fig. 1. The gold strips are 750 nm wide separated by 150 nm gaps, followed by a transverse gap of various widths. IV Summary and Conclusions We have assessed how transverse and longitudinal slits may affect the performance of SP devices, such as in Fig. 1, mainly in terms of losses. We have concentrated on subwavelength gaps. A single subwavelength transverse slit can lead to losses of 50 % to 60 % for gaps greater than 100 nm. The losses

17 generally increase with gap size, but not uniformly. The increase of losses is quenched or perhaps even reversed at gap sizes near a half or a single SP wavelength. This trend is evident, in varying degrees, in the simulations, experiments, and in the theoretical treatment based on ref. 8. The losses occurring with longitudinal strips, used in Fig. 1c for wavefront control, are of a somewhat different nature since they lead to exponential decay of the SP intensity characterized by a decay constant or corresponding range. For series of 0.75 µm gold strips the SP ranges decrease steadily with gap size up to about 150 nm (for our case of 732 nm vacuum wavelength), and then less rapidly with ranges 4 µm to 8 µm for gap sizes of 400 nm. Again this trend is evident in all our forms of analysis simulation, experiment, and an extension of the theory of ref. 8. How do these results impact device design? Transverse gaps are used for outcoupling and for separation of the longitudinal strips in Fig. 1 from the subsequent gold film where the SPs propagate. For outcoupling a larger gap is favorable to maximize the outcoupled light; however, this leads to greater losses. The results of this work, for instance, show that one can get around this difficulty somewhat by having half-wavelength transverse gaps. In the case of the longitudinal strips in the device in Fig. 1, the region of the strips is also the region where the SP wave number is changing and hence also the phase emerging from the strips. One could use this property for wavefront control, e.g., in designing an SP lens. Larger gaps produce larger changes in wave number and phase, but at the same time larger gaps mean larger losses. The important consideration is whether the losses are tolerable over a long enough strip length to get sufficient phase changes. This length is of the order of 10 µm, which is also the range of the SP for gaps of 150 nm between 0.75 µm metal strips. We conclude that devices employing the strip design of Fig. 1 have tolerable losses with gap widths of less than 200 nm.

18 Of course, in different device designs the considerations could be somewhat different than the device of Fig. 1. However, the loss mechanisms we have studied should be useful in general for any SP devices employing transverse and horizontal slits and grooves. The methods proposed in ref. 8 should be useful in the investigation of SP loss mechanisms. We have used an approximate extension of this theory in our assessment of longitudinal strips. A more precise extension of this technique would be useful in future studies to complement simulation and experimental methods. Acknowledgements This research was supported by AFOSR under Grant No. FA Computations were supported in part by the Dept. of Defense High Performance Computation Modernization Project. Nanodevices were fabricated at the NIST Center for Nanoscale Science and Technology. References 1. I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, Science 315, 1699 (2007). 2. D. K. Gramotnev and S. I. Bozhevolnyi, Nat. Photon. 4, 83 (2010). 3. W. L. Barnes, A. Dereux, and T. W. Ebbesen, Nature 424, 824 (2003). 4. A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, Phys. Rep. 408, 131 (2005). 5. D. Egorov, B. S. Dennis, and G. Blumberg, and M.I. Haftel, Phys. Rev. B 70, (2004). 6. B.S. Dennis, V. Aksyuk, M.I. Haftel, S.T. Koev, G. Blumberg, Journ. Appl. Phys. 110, (2011). 7. S.T. Koev, A. Agrawal, H.J. Lezec, and V. Aksyuk, online reference: Plasmonics: DOI: /S F. López-Tejeira, F.J. García-Vidal, and L. Martín-Moreno, Appl. Phys. A89, 251 (2007). 9. M.I. Haftel, C. Schlockermann, and G. Blumberg, Phys. Rev. B74, (2006). 10. K. Yee, IEEE Trans. Ant. and Prop. 14, 302 (1966).

19 11. J.-P. Berenger, J. Comp.. Phys.,110, 185 ( 1994). 12. S.D. Gedney, IEEE Trans. Ant. and Prop. 44, 1630 (1996). We use this version of PML. 13. H. Raether, Surface Plasmons (Springer, Berlin, 1988). 14. T.W. Ebbesen, C. Genet, and S.I. Bozhevolnyi, Physics Today, May 2008, pp B. Lamprecht et al., Appl. Phys. Lett. 79, 51 (2001). 16. P. Berini, Phys. Rev. B61, (2000); Phys. Rev. B63, (2001).