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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 10 — May. 20, 2013
  • pp: 11973–11983
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Size dependence of surface plasmon modes in one-dimensional plasmonic crystal cavities

Masahiro Honda and Naoki Yamamoto  »View Author Affiliations


Optics Express, Vol. 21, Issue 10, pp. 11973-11983 (2013)
http://dx.doi.org/10.1364/OE.21.011973


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Abstract

The characteristics of surface plasmon polaritons (SPPs) confined in a one-dimensional plasmonic crystal (1D-PlC) cavity are investigated using a cathodoluminescence (CL) detection system equipped with a 200 keV scanning transmission electron microscope (STEM). The dispersion curves of SPPs near the Γ point are derived from the angle-resolved CL spectra, and the SPP cavity modes are observed inside the band gap region. The mode number and wavenumber of the cavity modes are determined from the beam scan CL spectral images. The energy of the cavity mode depends on the cavity length and the angular distribution of the emission from the cavity changes with the mode number of the cavity mode. We also reveal that the phase shift due to the reflection at the cavity edge changes significantly with the resonant energy.

© 2013 OSA

1. Introduction

A surface plasmon polariton (SPP) is a transverse magnetic (TM) electromagnetic wave confined to the interface between a metal and a dielectric, and is coupled to the collective oscillations of free electrons at the metal surface [1

1. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).

]. SPPs propagate along a flat interface, and are reflected or scattered into free-space photons by defects, such as ridges and grooves. A plasmonic crystal (PlC) is a sub-wavelength periodic structure on a metal surface. Similar to band structures for electrons in solid substances and photons in photonic crystals [2

2. M. Notomi, “Manipulating light with strongly modulated photonic crystals,” Rep. Prog. Phys. 73(9), 096501 (2010). [CrossRef]

], PlCs can form plasmonic band structures. Because SPPs with energies within the band gap cannot propagate in a PlC, they can be confined in a PlC cavity when sandwiched by PlCs. The characteristics of SPPs in PlCs and PlC cavities have been studied both numerically and experimentally due to their potential in light emitting devices [3

3. T. Okamoto, J. Simonen, and S. Kawata, “Plasmonic band gaps of structured metallic thin films evaluated for a surface plasmon laser using the coupled-wave approach,” Phys. Rev. B 77(11), 115425 (2008). [CrossRef]

], single molecule sensors [4

4. R. M. Gelfand, L. Bruderer, and H. Mohseni, “Nanocavity plasmonic device for ultrabroadband single molecule sensing,” Opt. Lett. 34(7), 1087–1089 (2009). [CrossRef] [PubMed]

], waveguides [5

5. C. Marquart, S. I. Bozhevolnyi, and K. Leosson, “Near-field imaging of surface plasmon-polariton guiding in band gap structures at telecom wavelengths,” Opt. Express 13(9), 3303–3309 (2005). [CrossRef] [PubMed]

], cavity quantum electrodynamics (QED) [6

6. Y. Gong and J. Vučković, “Design of plasmon cavities for solid-state cavity quantum electrodynamics applications,” Appl. Phys. Lett. 90(3), 033113 (2007). [CrossRef]

], and SPP-light couplers [7

7. E. Devaux, T. W. Ebbesen, J.-C. Weeber, and A. Dereux, “Launching and decoupling surface plasmons via micro-gratings,” Appl. Phys. Lett. 83(24), 4936–4938 (2003). [CrossRef]

,8

8. K. Takeuchi and N. Yamamoto, “Visualization of surface plasmon polariton waves in two-dimensional plasmonic crystal by cathodoluminescence,” Opt. Express 19(13), 12365–12374 (2011). [CrossRef] [PubMed]

].

Compared to optical methods, a high spatial resolution (~10 nm) can be easily achieved using experimental methods with fast electron beams, such as electron energy loss spectroscopy (EELS) or a cathodoluminescence (CL) technique. Although EELS is powerful for simple samples such as nanoparticles [14

14. J. Nelayah, M. Kociak, O. Stéphan, F. J. García de Abajo, M. Tencé, L. Henrard, D. Taverna, I. Pastoriza-Santos, L. M. Liz-Marzán, and C. Colliex, “Mapping surface plasmons on a single metallic nanoparticle,” Nat. Phys. 3(5), 348–353 (2007). [CrossRef]

,15

15. V. Myroshnychenko, J. Nelayah, G. Adamo, N. Geuquet, J. Rodríguez-Fernández, I. Pastoriza-Santos, K. F. MacDonald, L. Henrard, L. M. Liz-Marzán, N. I. Zheludev, M. Kociak, and F. J. García de Abajo, “Plasmon Spectroscopy and Imaging of Individual Gold Nanodecahedra: A Combined Optical Microscopy, Cathodoluminescence, and Electron Energy-Loss Spectroscopy Study,” Nano Lett. 12(8), 4172–4180 (2012). [CrossRef] [PubMed]

] and nanowires [16

16. D. Rossouw, M. Couillard, J. Vickery, E. Kumacheva, and G. A. Botton, “Multipolar Plasmonic Resonances in Silver Nanowire Antennas Imaged with a Subnanometer Electron Probe,” Nano Lett. 11(4), 1499–1504 (2011). [CrossRef] [PubMed]

], it has difficulty with thick complicated samples because the electron beams must transmit or pass through near the sample. In contrast, the CL technique has few sample limitations. It can handle not only nanoparticles [17

17. N. Yamamoto, K. Araya, and F. J. García de Abajo, “Photon emission from silver particles induced by a high energy electron beam,” Phys. Rev. B 64(20), 205419 (2001). [CrossRef]

,18

18. N. Yamamoto, S. Ohtani, and F. J. García de Abajo, “Gap and Mie Plasmons in Individual Silver Nanospheres near a Silver Surface,” Nano Lett. 11(1), 91–95 (2011). [CrossRef] [PubMed]

] and nanorods [19

19. T. Coenen, E. J. R. Vesseur, and A. Polman, “Deep Subwavelength Spatial Characterization of Angular Emission from Single-Crystal Au Plasmonic Ridge Nanoantennas,” ACS Nano 6(2), 1742–1750 (2012). [CrossRef] [PubMed]

] but also complicated samples, such as plasmonic crystals [8

8. K. Takeuchi and N. Yamamoto, “Visualization of surface plasmon polariton waves in two-dimensional plasmonic crystal by cathodoluminescence,” Opt. Express 19(13), 12365–12374 (2011). [CrossRef] [PubMed]

,20

20. M. Kuttge, E. J. R. Vesseur, A. F. Koenderink, H. J. Lezec, H. A. Atwater, F. J. García de Abajo, and A. Polman, “Local density of states, spectrum, and far-field interference of surface plasmon polaritons probed by cathodoluminescence,” Phys. Rev. B 79(11), 113405 (2009). [CrossRef]

,21

21. T. Suzuki and N. Yamamoto, “Cathodoluminescent spectroscopic imaging of surface plasmon polaritons in a 1-dimensional plasmonic crystal,” Opt. Express 17(26), 23664–23671 (2009). [CrossRef] [PubMed]

]. Furthermore, the angular dependence of the emission spectrum from a sample can be investigated using the CL technique with a special experimental setup, which cannot be measured by EELS.

Herein we employ a CL technique to investigate the characteristics of SPPs in a one-dimensional plasmonic crystal (1D-PlC) and 1D-PlC cavities. In particular, we studied the properties of SPPs confined in cavities using a scanning transmission electron microscope (STEM) operated at an acceleration voltage of 200 kV. The angle-resolved spectral (ARS) pattern was measured to determine the 1D-PlC band gap size. The ARS patterns for cavities with various lengths were used to deduce the dependence of cavity energy on cavity length. The wavenumber of the SPP inside the cavity and the phase shift of the SPP at the cavity edge were deduced from the data and compared to hypothetical models.

2. Theory

Figure 1(a)
Fig. 1 (a) Schematic drawing of a one-dimensional plasmonic crystal (1D-PlC) cavity. Cavity edge is defined as the center of the terrace. (b) Schematic diagram of the dispersion curves near the Γ point of a 1D-PlC. Energy of the higher (lower) band gap edge is E+(E). (c) Standing wave patterns of the normal electric field component at each band edge. Surface charge distribution is also shown.
schematically depicts the 1D-PlC cavity considered here. The cavity is sandwiched by two 1D-PlCs with a grating period of Λ, a terrace width of using a filling factor f, and cavity length of L. The resonance condition for a SPP in a cavity is given by [12

12. J.-C. Weeber, A. Bouhelier, G. Colas des Francs, L. Markey, and A. Dereux, “Submicrometer In-Plane Integrated Surface Plasmon Cavities,” Nano Lett. 7(5), 1352–1359 (2007). [CrossRef] [PubMed]

]
kxSPPL+Φ=nπ,
(1)
where kxSPP is the x-component of the SPP wave vector, Φ is the phase shift for the reflection by the 1D-PlC, and n is the mode number (integer).

In this paper, we consider the cavity mode with energy inside the band gap at the Γ point (kSPP=0). Using a single SPP dispersion plane on a flat surface, a set of dispersion planes are formed by shifting it by the reciprocal lattice vectors, which can approximate the dispersion plane of a SPP on a PlC (the empty lattice approximation). In this approximation, the dispersion curves cross at the Γ point (the black dashed lines in Fig. 1(b)). Depending on the surface morphology of the PlC, the plasmonic band gap actually opens up at the Γ point (the red lines in Fig. 1(b)). The SPPs at the upper (E=E+) and lower (E=E) band edges form standing waves.

Figure 1(c) shows the standing wave patterns at the band edge energies, which represent the surface charge distributions, that is, the electric fields of the surface normal component. The filling factor f is less than 0.5, which corresponds to that of our sample. Barnes et al. have theoretically derived these patterns using a surface shape function involving only two Fourier components with a lower order [23

23. W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B Condens. Matter 54(9), 6227–6244 (1996). [CrossRef] [PubMed]

]. The nodes and antinodes have opposite positions in these two standing waves at the upper and lower band edges. This configuration is reversed if the filling factor exceeds 0.5.

The phase shift Φ of the SPP wave for a reflection at the cavity edge should approach + π (0) as the energy of the cavity mode Ecav approaches the energy of the band edge E+ (E) due to the continuity between the standing SPP waves in the cavity and the plasmonic crystal at the cavity edge (see Fig. 1(c)) [12

12. J.-C. Weeber, A. Bouhelier, G. Colas des Francs, L. Markey, and A. Dereux, “Submicrometer In-Plane Integrated Surface Plasmon Cavities,” Nano Lett. 7(5), 1352–1359 (2007). [CrossRef] [PubMed]

]. This suggests that the phase shift should depend on the energy in the energy range within the band gap. A standing wave has a fixed (open) end when Φ equals + π (0). Here, we temporarily assume that Φ is a linear function of Ecav in the band gap energy range, which is expressed as
Φ(Ecav)=πΔE(EcavE),
(2)
where ΔE=E+E represents the band gap width. This equation satisfies the boundary condition for Φ at the band edges.

Figure 2(a)
Fig. 2 (a) Resonant energy calculated using Eq. (3a) (dashed lines) and Eq. (3b) (solid lines) as a function of cavity length. Values of E+ and E are obtained from the experimental result shown in Fig. 4. (b) Standing wave patterns of the normal electric field component at the central energy of the band gap. Symmetric (asymmetric) standing wave arises inside cavities when n is even (odd) number.
shows the relationship between Ecav and cavity length L derived from Eqs. (1)-(3b) under the two aforementioned assumptions. In the calculation, the dielectric function of silver in Eq. (3b) is from the data book by Palik [22

22. E. D. Palik, Handbook of Optical Constants of Solids (Academic, London, 1985).

], and the values of E+ and E are from the present experimental results. As the energy of the cavity modes decreases with L, the difference between the dashed (Eq. (3a)) and solid lines (Eq. (3b)) increases because the difference between the wavenumbers defined by Eqs. (3a) and (3b) increases as the cavity energy inside the energy gap decreases. The energy of the cavity mode should vary according to these theoretical curves as the cavity length changes. Figure 2(b) shows the expected standing wave patterns of the cavity modes when Ecav is equal to the central energy of the band gap (Φ = + π/2). The (n1) number of antinodes appears inside the cavities, and the standing SPP wave with the symmetric mode (anti-symmetric mode) is formed when n is an even (odd) number, as indicated by the blue (red) color. The antinodes at the cavity edges are not counted, because their amplitudes are not as large as those inside the cavity.

3. Experimental setting

Figure 3(a)
Fig. 3 (a) SEM image of a sample with a one-dimensional plasmonic crystal cavity, and a diagram of the cross-section. The grating (period = 600 nm, filling factor = 0.25, height = 100 nm) with a cavity fabricated on an InP substrate is coated with a 200 nm thick silver layer. Scale bar is 1 μm. (b) Geometry of the angle-resolved measurement with a parabolic mirror and a pinhole. Sample is tilted to detect light emitted in the surface normal direction (red line). (c) Angular distribution of the emission from the sample tilted by α = 16° in the y direction. Dashed (solid) lines indicate equi-θ (φ) lines. The pinhole is moved along the red line through the green point (the surface normal direction) to obtain the ARS patterns shown. Angle φ can be regarded as 90° in the range of θ ≤ 10°.
shows the SEM image of the 1D-PlC cavity and a diagram of the cross-section. Electron beam lithography fabricated one-dimensional periodic nanostructures with a period of 600 nm. A 100 nm thick resist layer of ZEP520A was spin-coated onto an InP substrate as a positive resist. Each segment in the structure had a rectangular cross-section with a width of 150 nm, a height of 100 nm, and a length in the x direction of 30 μm. The cavity length L, which is defined by the distance between two PlCs as shown in Fig. 3(a), was between 450 and 1500 nm. A 200 nm thick silver layer was deposited on the substrate by thermal evaporation in a vacuum. In the present study, we used STEM (JEM-2000FX) combined with a light detection system, which operated at an accelerating voltage of 200 kV with an electron beam diameter of ~10 nm.

Figure 3(b) shows the experimental setup of an angle-resolved measurement. A parabolic mirror placed above the sample collected the emission from the sample. The position of the mirror was adjusted so that the electron beam was incident at the focus of the mirror. When the sample surface was perpendicular to the electron beam, the emission from the sample to the surface normal direction could not be observed because it passed through the hole in the mirror where the electron beam passed. To observe the dispersion near the Γ point, we tilted the sample in the y direction as schematically illustrated in Fig. 3(b) [8

8. K. Takeuchi and N. Yamamoto, “Visualization of surface plasmon polariton waves in two-dimensional plasmonic crystal by cathodoluminescence,” Opt. Express 19(13), 12365–12374 (2011). [CrossRef] [PubMed]

]. The emission collected by the parabolic mirror was transformed into a parallel ray that passed through a polarizer before reaching the pinhole. Here, we define the p-polarization as parallel to the emission plane subtended by the axis of parabola, and s-polarization as parallel to the x-axis as shown in Fig. 3(b). We used the s-polarized emission to observe the spectra shown in the following section.

The pinhole had a diameter of 0.5 mm, and it could selectively detect light emission from a sample in a particular direction. The light was then focused onto a slit of a monochrometer by a lens and detected by a CCD detector. We measured the angular dependence of an emission spectrum by moving the position of the pinhole. Aligning the observed spectra with respect to the emission angle produced the ARS pattern. We could also observe the dependence of an emission spectrum on the electron beam position by linearly scanning the electron beam with the pinhole fixed at the proper position. Aligning the observed spectra with respect to the electron beam position produced a beam-scan spectral (BSS) image. In this study, the typical integration time to acquire each spectrum for one pixel was 5 sec.

Figure 3(c) is the front view of the parabolic mirror used in this work and shows the angular distribution of the emission from the sample tilted by α = 16°. The gray area represents the area to collect light, while the white circle represents the hole where the electron beam passes. We moved the pinhole along the red line in Fig. 3(c). The pinhole should trace the blue line in Fig. 3(c) to measure the dispersion along the Γ-X line where the wave vector component was parallel to the y axis. That is, kyph=Ephcsinθcosφ equaled zero, i.e., φ = 90°. The value of φ gradually deviated from 90° as the pinhole departed from the central position. However, the deviation between the red and blue lines was negligible compared to the diameter of the pinhole (red circle) when θ ≤ 10°. Diameter of the pinhole is 0.5 mm, which corresponded to the half angle of 3.6° at the hole position.

4. Results and discussion

4.1. 1D-plasmonic crystal

The intensity of the E mode emission to the surface normal direction (θ = 0°) is much weaker than that of the E+ mode. This phenomenon can be easily understood; the antinodes of the E+ mode standing wave, which have the opposite sign as the nodes, are located near the terrace edges. Therefore, the surface charges of the SPPs form oscillating electric dipoles on the terrace, which are parallel to the surface and generate a strong emission in the surface normal direction. On the other hand, the antinodes of the E mode standing wave exist at the center of the terraces and grooves (see Fig. 1(c)). Because the oscillating electric dipoles do not form parallel to the surface, the emission in the surface normal direction is weak.

4.2. 1D-PlC with a cavity

Figures 5(a)
Fig. 5 ARS patterns (upper part) and BSS images (lower part) of the four cavities with different cavity lengths: (a) 450 nm, (b) 800 nm, (c) 1200 nm, and (d) 1500 nm. Mode number is deduced from the number of peaks inside each cavity. Emission at θ = 0° occurs when n is an odd number. In contrast, the emission at θ = 5-10° occurs when n is an even number. (e) BSS image over a wider area of the sample (L = 1500 nm). Cross-section of the sample is depicted below each BSS image.
-5(d) show the ARS patterns of 1D-PlCs with cavity lengths of 450, 800, 1200, and 1500 nm, respectively. The corresponding BSS image of the cavity mode with energy inside the band gap is shown below each ARS pattern. For each measurement, the electron beam was scanned over an area of 1.5 × 1.5 μm2, which included the cavity. These ARS patterns qualitatively reveal the dispersion relation near the Γ point. Although the sample without a cavity does not exhibit an emission inside the band gap (Fig. 4), samples with cavity lengths of 450, 800, 1200, and 1500 nm display emissions at energies 1.859, 1.865, 1.805, and 1.85 eV inside the band gap region, respectively (Fig. 5). Consequently, these emissions are attributed to the SPP cavity modes.

The BSS images in Fig. 5 were acquired by scanning the electron beam along the x direction across the cavities with the pinhole fixed at the position of the maximum emission intensity; the BSS images in Figs. 5(b), 5(d), and 5(e) (Figs. 5(a) and 5(c)) were taken with the emission in the surface normal (tilted angle) direction. These BSS images indicate that the standing waves formed in the cavities have 1 to 4 antinodes. From Fig. 2(b), the mode number is expressed as n=2-5 for those cavities. Figure 5(e) shows a BSS image of the cavity with a length of 1500 nm taken from a wider region than those in Figs. 5(a)5(d); the SPP state of the cavity mode is spatially localized at the cavity, and the extension of the field is only about one period of the PlC outside the cavity.

The cavity mode emission is pointed in the direction of θ = 0° in Figs. 5(b) and 5(d), whereas the emission is pointed in the direction tilted by θ = 5-10° from the surface normal in Figs. 5(a) and 5(c). The cavity mode emissions that are observed inside the band gap region clearly depend on the mode number. The strong emission at θ = 0° appears for an odd mode number. In contrast, the emission at θ = 0° disappears for an even mode number, and a strong emission appears at θ = 5-10°. This property can be explained by interference between the two oscillating dipoles near both edges of the cavity. The oscillating dipoles are mainly responsible for light emission from the cavity because SPPs propagating on a flat surface do not contribute to light emission [19

19. T. Coenen, E. J. R. Vesseur, and A. Polman, “Deep Subwavelength Spatial Characterization of Angular Emission from Single-Crystal Au Plasmonic Ridge Nanoantennas,” ACS Nano 6(2), 1742–1750 (2012). [CrossRef] [PubMed]

,26

26. T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Optical nanorod antennas modeled as cavities for dipolar emitters: evolution of sub- and super-radiant modes,” Nano Lett. 11(3), 1020–1024 (2011). [CrossRef] [PubMed]

]. The standing wave pattern of the odd cavity mode is asymmetric with respect to the center of the cavity, and the two dipoles always turn toward the same direction. Therefore, their oscillations are in phase, resulting in constructive interference for the emission toward the surface normal (θ = 0°). On the other hand, the standing wave pattern of the even cavity mode is symmetric, and consequently, the phase difference between the two dipole oscillations is π. Therefore, destructive interference occurs for the emission toward θ = 0°.

Figures 6(a)
Fig. 6 BSS images of three cavities with the same order (n = 5) but different cavity lengths of (a) 1400 nm, (b) 1450 nm, and (c) 1500 nm. Red shift occurs as cavity length increases (black arrows). Distance between two antinodes equals the half-wavelength of the cavity mode. (d) Experimental energy of the cavity modes versus the cavity length. Theoretical curves (Fig. 2(a)) are also shown.
-6(c) show the BSS images of cavities with lengths of 1400, 1450, and 1500 nm, respectively. These images were acquired by scanning the electron beam across the cavities with the pinhole fixed at θ = 0° (the Γ point). The cavity modes of n = 5 order are clearly visualized. The energy of the cavity mode redshifts with increasing cavity length, which is denoted by the arrows in Fig. 6. Figure 6(d) plots the energy of the cavity mode as a function of cavity length. Red shifts also occur for cavity modes of n = 3 and 4.

The emission contrast appears inside the cavity at an energy of E=E+ in the BSS images of Figs. 6(a)-6(c), indicating that part of the standing SPP in the PlC leaks into the cavity region from the PlC. This SPP leakage from the two-bounded PlC’s interference with each other forms the standing wave in the cavity. The superposition of this standing wave and LSP at the terrace edges is also seen at E=E+ in Figs. 6(a)-6(c). The formation mechanism remains unclear, but further observations using a sample with a larger cavity length should provide additional evidence to propose a reliable model.

When the cavity length is large, the experimental results support the second approximation expressed by Eq. (3b) for the n = 4 and 5 modes as seen in Fig. 6(d). But from the same result of Fig. 6(d), we cannot conclude which approximation is suitable for the n = 2 and 3 modes. Therefore, we directly deduced the wavenumber of the SPP cavity mode from the experimental results. Because the BSS image mimics the SPP standing wave pattern, the wavenumber of SPP cavity mode should be derived from the BSS image. BSS images of the cavity mode near the cavity edges are somehow affected by the LSP located at the steps of the terrace edge. Therefore, the half-wavelength of the SPP, λxSPP/2=π/kxSPP, is taken as the distance between the two antinodes (nodes) in the cavity center when n is an odd (even) number (see Fig. 6(c)). Figure 7(a)
Fig. 7 (a) Relation between energy and wavenumber of the cavity modes derived from the BSS images. Solid line indicates the dispersion curve of SPPs at the vacuum/silver interface. (b) Phase shift versus energy of the cavity mode. Solid line indicates the relation in Eq. (2). Red, green, and blue circles indicate n = 3, 4 and 5, respectively.
compares the deduced values of kxSPP with the dispersion curve (solid line) of SPPs at a vacuum/silver interface calculated by the index data of Ref [22

22. E. D. Palik, Handbook of Optical Constants of Solids (Academic, London, 1985).

]. The color of the plotted circle indicates the mode number n. From the result shown in Fig. 7(a), we conclude that the wavenumber of the cavity modes coincides with that of the SPP propagating on a flat silver surface. This result clearly shows the validity of the second approximation that the SPP in the cavity obeys the dispersion relation of SPP on the flat surface.

Using Eq. (1) and the experimental results shown in Figs. 6(d) and 7(a), we deduce the phase shift of the SPP wave that suffers from the reflection at the boundary between the cavity and 1D-PlC. Figure 7(b) shows the dependence of the phase shift on the energy inside the band gap. The solid line indicates the relation of Eq. (2) as a function of the cavity energy. The data plots are distributed near the solid line. However, the solid curve and the data plots deviate; the slope of the observed Φ tends to becomes gentler in the middle of the band gap, but steeper in the region close to the band edges. This observation corresponds to the deviation between the solid curves and data plots in the graph of energy vs. cavity length in Fig. 6(d).

5. Conclusion

We investigated the characteristics of the standing SPP modes in the 1D-PlC and the cavities sandwiched by them when the energy is located inside the band gap at the Γ point. The PlC with the filling factor f ~0.25 used in this work has a wide band gap. The emission from the cavities is observed with the energy inside the band gap. The angular distribution of the emission from the cavity and the symmetry of the standing SPP mode change with the order of the cavity mode. The resonant energy of the cavity shifts according to the condition kxSPPL+Φ=nπ. Upon examining how the wavenumber kxSPP and the phase shift Φ change with the cavity length L, we found the following facts: (1) kxSPP changes according to the dispersion relation of a SPP on a flat metal surface and (2) Φ changes from 0 to π, and is nearly proportional to the cavity energy, but the change tends to decrease around the middle of the band gap. These results should provide useful information for practical applications of a 1D-PlC cavity to plasmonic devices.

Acknowledgments

This work was supported by Japan-Spain Research Cooperative Program of JSPS, Grants-in-Aid for Scientific Research (Nos. 19101004 and 21340080) from the MEXT of Japan, and MEXT Nanotechnology platform 12025014 (F-12-IT-0010).

References and links

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H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).

2.

M. Notomi, “Manipulating light with strongly modulated photonic crystals,” Rep. Prog. Phys. 73(9), 096501 (2010). [CrossRef]

3.

T. Okamoto, J. Simonen, and S. Kawata, “Plasmonic band gaps of structured metallic thin films evaluated for a surface plasmon laser using the coupled-wave approach,” Phys. Rev. B 77(11), 115425 (2008). [CrossRef]

4.

R. M. Gelfand, L. Bruderer, and H. Mohseni, “Nanocavity plasmonic device for ultrabroadband single molecule sensing,” Opt. Lett. 34(7), 1087–1089 (2009). [CrossRef] [PubMed]

5.

C. Marquart, S. I. Bozhevolnyi, and K. Leosson, “Near-field imaging of surface plasmon-polariton guiding in band gap structures at telecom wavelengths,” Opt. Express 13(9), 3303–3309 (2005). [CrossRef] [PubMed]

6.

Y. Gong and J. Vučković, “Design of plasmon cavities for solid-state cavity quantum electrodynamics applications,” Appl. Phys. Lett. 90(3), 033113 (2007). [CrossRef]

7.

E. Devaux, T. W. Ebbesen, J.-C. Weeber, and A. Dereux, “Launching and decoupling surface plasmons via micro-gratings,” Appl. Phys. Lett. 83(24), 4936–4938 (2003). [CrossRef]

8.

K. Takeuchi and N. Yamamoto, “Visualization of surface plasmon polariton waves in two-dimensional plasmonic crystal by cathodoluminescence,” Opt. Express 19(13), 12365–12374 (2011). [CrossRef] [PubMed]

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12.

J.-C. Weeber, A. Bouhelier, G. Colas des Francs, L. Markey, and A. Dereux, “Submicrometer In-Plane Integrated Surface Plasmon Cavities,” Nano Lett. 7(5), 1352–1359 (2007). [CrossRef] [PubMed]

13.

S. Balci, E. Karademir, C. Kocabas, and A. Aydinli, “Direct imaging of localized surface plasmon polaritons,” Opt. Lett. 36(17), 3401–3403 (2011). [CrossRef] [PubMed]

14.

J. Nelayah, M. Kociak, O. Stéphan, F. J. García de Abajo, M. Tencé, L. Henrard, D. Taverna, I. Pastoriza-Santos, L. M. Liz-Marzán, and C. Colliex, “Mapping surface plasmons on a single metallic nanoparticle,” Nat. Phys. 3(5), 348–353 (2007). [CrossRef]

15.

V. Myroshnychenko, J. Nelayah, G. Adamo, N. Geuquet, J. Rodríguez-Fernández, I. Pastoriza-Santos, K. F. MacDonald, L. Henrard, L. M. Liz-Marzán, N. I. Zheludev, M. Kociak, and F. J. García de Abajo, “Plasmon Spectroscopy and Imaging of Individual Gold Nanodecahedra: A Combined Optical Microscopy, Cathodoluminescence, and Electron Energy-Loss Spectroscopy Study,” Nano Lett. 12(8), 4172–4180 (2012). [CrossRef] [PubMed]

16.

D. Rossouw, M. Couillard, J. Vickery, E. Kumacheva, and G. A. Botton, “Multipolar Plasmonic Resonances in Silver Nanowire Antennas Imaged with a Subnanometer Electron Probe,” Nano Lett. 11(4), 1499–1504 (2011). [CrossRef] [PubMed]

17.

N. Yamamoto, K. Araya, and F. J. García de Abajo, “Photon emission from silver particles induced by a high energy electron beam,” Phys. Rev. B 64(20), 205419 (2001). [CrossRef]

18.

N. Yamamoto, S. Ohtani, and F. J. García de Abajo, “Gap and Mie Plasmons in Individual Silver Nanospheres near a Silver Surface,” Nano Lett. 11(1), 91–95 (2011). [CrossRef] [PubMed]

19.

T. Coenen, E. J. R. Vesseur, and A. Polman, “Deep Subwavelength Spatial Characterization of Angular Emission from Single-Crystal Au Plasmonic Ridge Nanoantennas,” ACS Nano 6(2), 1742–1750 (2012). [CrossRef] [PubMed]

20.

M. Kuttge, E. J. R. Vesseur, A. F. Koenderink, H. J. Lezec, H. A. Atwater, F. J. García de Abajo, and A. Polman, “Local density of states, spectrum, and far-field interference of surface plasmon polaritons probed by cathodoluminescence,” Phys. Rev. B 79(11), 113405 (2009). [CrossRef]

21.

T. Suzuki and N. Yamamoto, “Cathodoluminescent spectroscopic imaging of surface plasmon polaritons in a 1-dimensional plasmonic crystal,” Opt. Express 17(26), 23664–23671 (2009). [CrossRef] [PubMed]

22.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, London, 1985).

23.

W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B Condens. Matter 54(9), 6227–6244 (1996). [CrossRef] [PubMed]

24.

M. Kociak and F. J. García de Abajo, “Nanoscale mapping of plasmons, photons, and excitons,” MRS Bull. 37(01), 39–46 (2012). [CrossRef]

25.

N. Yamamoto and T. Suzuki, “Conversion of surface plasmon polaritons to light by a surface step,” Appl. Phys. Lett. 93(9), 093114 (2008). [CrossRef]

26.

T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Optical nanorod antennas modeled as cavities for dipolar emitters: evolution of sub- and super-radiant modes,” Nano Lett. 11(3), 1020–1024 (2011). [CrossRef] [PubMed]

OCIS Codes
(240.6680) Optics at surfaces : Surface plasmons
(250.1500) Optoelectronics : Cathodoluminescence
(250.5403) Optoelectronics : Plasmonics

ToC Category:
Optics at Surfaces

History
Original Manuscript: January 25, 2013
Revised Manuscript: April 24, 2013
Manuscript Accepted: May 2, 2013
Published: May 8, 2013

Citation
Masahiro Honda and Naoki Yamamoto, "Size dependence of surface plasmon modes in one-dimensional plasmonic crystal cavities," Opt. Express 21, 11973-11983 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-10-11973


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References

  1. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).
  2. M. Notomi, “Manipulating light with strongly modulated photonic crystals,” Rep. Prog. Phys.73(9), 096501 (2010). [CrossRef]
  3. T. Okamoto, J. Simonen, and S. Kawata, “Plasmonic band gaps of structured metallic thin films evaluated for a surface plasmon laser using the coupled-wave approach,” Phys. Rev. B77(11), 115425 (2008). [CrossRef]
  4. R. M. Gelfand, L. Bruderer, and H. Mohseni, “Nanocavity plasmonic device for ultrabroadband single molecule sensing,” Opt. Lett.34(7), 1087–1089 (2009). [CrossRef] [PubMed]
  5. C. Marquart, S. I. Bozhevolnyi, and K. Leosson, “Near-field imaging of surface plasmon-polariton guiding in band gap structures at telecom wavelengths,” Opt. Express13(9), 3303–3309 (2005). [CrossRef] [PubMed]
  6. Y. Gong and J. Vučković, “Design of plasmon cavities for solid-state cavity quantum electrodynamics applications,” Appl. Phys. Lett.90(3), 033113 (2007). [CrossRef]
  7. E. Devaux, T. W. Ebbesen, J.-C. Weeber, and A. Dereux, “Launching and decoupling surface plasmons via micro-gratings,” Appl. Phys. Lett.83(24), 4936–4938 (2003). [CrossRef]
  8. K. Takeuchi and N. Yamamoto, “Visualization of surface plasmon polariton waves in two-dimensional plasmonic crystal by cathodoluminescence,” Opt. Express19(13), 12365–12374 (2011). [CrossRef] [PubMed]
  9. A. Kocabas, S. S. Senlik, and A. Aydinli, “Plasmonic band gap cavities on biharmonic gratings,” Phys. Rev. B77(19), 195130 (2008). [CrossRef]
  10. A. Kocabas, S. S. Senlik, and A. Aydinli, “Slowing down surface plasmons on a Moiré surface,” Phys. Rev. Lett.102(6), 063901 (2009). [CrossRef] [PubMed]
  11. S. Balci, A. Kocabas, C. Kocabas, and A. Aydinli, “Localization of surface plasmon polaritons in hexagonal arrays of Moiré cavities,” Appl. Phys. Lett.98(3), 031101 (2011). [CrossRef]
  12. J.-C. Weeber, A. Bouhelier, G. Colas des Francs, L. Markey, and A. Dereux, “Submicrometer In-Plane Integrated Surface Plasmon Cavities,” Nano Lett.7(5), 1352–1359 (2007). [CrossRef] [PubMed]
  13. S. Balci, E. Karademir, C. Kocabas, and A. Aydinli, “Direct imaging of localized surface plasmon polaritons,” Opt. Lett.36(17), 3401–3403 (2011). [CrossRef] [PubMed]
  14. J. Nelayah, M. Kociak, O. Stéphan, F. J. García de Abajo, M. Tencé, L. Henrard, D. Taverna, I. Pastoriza-Santos, L. M. Liz-Marzán, and C. Colliex, “Mapping surface plasmons on a single metallic nanoparticle,” Nat. Phys.3(5), 348–353 (2007). [CrossRef]
  15. V. Myroshnychenko, J. Nelayah, G. Adamo, N. Geuquet, J. Rodríguez-Fernández, I. Pastoriza-Santos, K. F. MacDonald, L. Henrard, L. M. Liz-Marzán, N. I. Zheludev, M. Kociak, and F. J. García de Abajo, “Plasmon Spectroscopy and Imaging of Individual Gold Nanodecahedra: A Combined Optical Microscopy, Cathodoluminescence, and Electron Energy-Loss Spectroscopy Study,” Nano Lett.12(8), 4172–4180 (2012). [CrossRef] [PubMed]
  16. D. Rossouw, M. Couillard, J. Vickery, E. Kumacheva, and G. A. Botton, “Multipolar Plasmonic Resonances in Silver Nanowire Antennas Imaged with a Subnanometer Electron Probe,” Nano Lett.11(4), 1499–1504 (2011). [CrossRef] [PubMed]
  17. N. Yamamoto, K. Araya, and F. J. García de Abajo, “Photon emission from silver particles induced by a high energy electron beam,” Phys. Rev. B64(20), 205419 (2001). [CrossRef]
  18. N. Yamamoto, S. Ohtani, and F. J. García de Abajo, “Gap and Mie Plasmons in Individual Silver Nanospheres near a Silver Surface,” Nano Lett.11(1), 91–95 (2011). [CrossRef] [PubMed]
  19. T. Coenen, E. J. R. Vesseur, and A. Polman, “Deep Subwavelength Spatial Characterization of Angular Emission from Single-Crystal Au Plasmonic Ridge Nanoantennas,” ACS Nano6(2), 1742–1750 (2012). [CrossRef] [PubMed]
  20. M. Kuttge, E. J. R. Vesseur, A. F. Koenderink, H. J. Lezec, H. A. Atwater, F. J. García de Abajo, and A. Polman, “Local density of states, spectrum, and far-field interference of surface plasmon polaritons probed by cathodoluminescence,” Phys. Rev. B79(11), 113405 (2009). [CrossRef]
  21. T. Suzuki and N. Yamamoto, “Cathodoluminescent spectroscopic imaging of surface plasmon polaritons in a 1-dimensional plasmonic crystal,” Opt. Express17(26), 23664–23671 (2009). [CrossRef] [PubMed]
  22. E. D. Palik, Handbook of Optical Constants of Solids (Academic, London, 1985).
  23. W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B Condens. Matter54(9), 6227–6244 (1996). [CrossRef] [PubMed]
  24. M. Kociak and F. J. García de Abajo, “Nanoscale mapping of plasmons, photons, and excitons,” MRS Bull.37(01), 39–46 (2012). [CrossRef]
  25. N. Yamamoto and T. Suzuki, “Conversion of surface plasmon polaritons to light by a surface step,” Appl. Phys. Lett.93(9), 093114 (2008). [CrossRef]
  26. T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Optical nanorod antennas modeled as cavities for dipolar emitters: evolution of sub- and super-radiant modes,” Nano Lett.11(3), 1020–1024 (2011). [CrossRef] [PubMed]

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