Size dependence of band-gaps in a one-dimensional plasmonic crystal

Hiroaki Watanabe; Masahiro Honda; Naoki Yamamoto

doi:10.1364/OE.22.005155

1. Introduction

A surface plasmon polariton (SPP) is a transverse magnetic (TM) mode electromagnetic wave propagating at a metal/dielectric interface, which is evanescently confined in the perpendicular direction [1]. Metal surfaces with periodic structures on the sub-wavelength scale have been used as couplers to convert SPPs to photons and vice versa; they have recently been named “plasmonic crystals (PlCs)” in the developing field of plasmonics [2]. Because of its band gap structure, a PlC acts as a mirror for SPPs with energy in the band gap region [3], and can be used in a wave guide [4] or a cavity [5] for SPPs. Knowledge of the properties of the SPP band gaps and band edge states is key to controlling SPPs in plasmonic devices, such as chemical sensors [6], plasmonic solar cells [7] and plasmonic lasers [8, 9]. Many studies of the band gap properties have been carried out both experimentally [10–13] and theoretically [14–17] thus far. Nevertheless, we do not have sufficient knowledge of the parameters that govern the detailed properties of the plasmonic band gap, because of the lack of systematic experimental studies on the dependence of the band gap on surface shape of the PlC.

Band structures in one-dimensional (1D) and two-dimensional (2D) PlCs have been treated theoretically by the finite-difference time-domain (FDTD) method [18] and the rigorous coupled-wave analysis (RCWA) method [19]. The analytical expression of band edge energy was given by Barnes et al. [14] for a 1D periodic surface with a surface shape function composed of only the first- and second-order Fourier components. They showed that the band gap at Γ depends on the second-order Fourier component, and that the phase difference between the first and second Fourier components causes the degree of coupling efficiency between photon and a standing SPP at the band edge. Their expressions cannot be directly applied to the typically used plasmonic crystals with a rectangular cross-section, because the influence of the higher-order Fourier components becomes large. The RCWA method gives a rigorous solution for periodic structures with a rectangular cross-section. However, applicability of this theory has not yet been sufficiently confirmed through direct comparisons with the experimental data. In the present study, we have prepared 1D PlCs with a rectangular cross-section of different terrace width (D) / period (P) ratios, and we have examined their plasmonic band gaps in order for systematic comparison between the theory and the observed data taken by cathodoluminescence (CL).

We fabricated 1D periodic structures by electron beam lithography. Stripes (30-μm long with a rectangular cross-section) were produced from a resist layer (ZEP520A) on an InP substrate. Forty stripes were aligned in each structure, as shown in Fig. 1(a), and structures with two different periods of 600 and 800 nm were made. The terrace width was changed from 100 to 550 nm, at an increment of 50 nm; terrace heights were 20, 30, 50, 70, and 100 nm. A 200-nm thick silver layer was evaporated onto the structure by thermal evaporation in vacuum (Fig. 1(b)). The terrace width increased by about 30 nm due to the deposition of silver, so the terrace width was measured from a scanning image for each structure. As in the previous experiment [20–22], we employed a scanning transmission electron microscope (STEM) combined with a light detection system, which operates at an acceleration voltage of 200 kV. The electron beam has a diameter of 10 nm with a beam current of 1 nA.

Fig. 1 (a) Structure of a 1D PlC, (b) a cross-section of a 1D PlC, (c) arrangement of the angle-resolved measurement, and (d) dispersion relation of the SPP in a 1D PlC. The red line indicates the light line, and red circles indicate band gaps at the point Γ and X.

Download Full Size | PDF

The experimental setup for the CL measurement in the STEM was described in a previous paper [21]. The sample was set with its y-axis parallel to the Y-axis, and the surface normal direction (the z-axis) was tilted from the incident beam direction by about 15° towards the X-axis (Fig. 1(c)), where the xyz coordinates were fixed at the sample, and the XYZ ones were fixed at the mirror and the X-Y stage. The emission direction was selected by the pinhole in the mask supported by the X-Y stage. Emission spectra were successively recorded with the pinhole moving parallel to the Y-axis, which corresponded to the change of the polar angle (θ_y) towards the y-axis. An angle-resolved spectral (ARS) pattern is created by these set of spectra showing emission intensity distribution in the energy (E)–θ_y plane [21].

The ARS pattern approximately represents the dispersion pattern of the SPP near Γ, as shown in Fig. 1(d). A beam-scan spectral (BSS) image can be obtained by successively recording emission spectra with scanning the electron beam across the PlC with the pinhole fixed at the θ_y = 0° position (the point Γ); this reveals the spatial distribution of the standing SPP wave in the band edge state [13, 20–22]. On the other hand, in order to observe the dispersion pattern near the X point, a different setting was used, where the y-axis of the sample was parallel to the parabolic axis (the X-axis). The ARS patterns were taken by moving the pinhole along the vertical direction (the Z-direction) [13, 22].

2. Theory

An SPP in a plasmonic crystal can be described as a Bloch wave. The conversion between a photon and an SPP mediated by the plasmonic crystal must satisfy the following conditions in wave vector and energy:

k_{/ /} - k_{p} = G,

E_{S P P} = E_{p h},

where

k_{p}

is the wave vector of the SPP, _{$k_{/ /}$} is the surface parallel component of the wave vector of the photon, and G is a reciprocal lattice vector of the surface structure. In Eq. (1b), E_SPP and E_ph are energies of the SPP and the photon, respectively. If a is the basic lattice vector of the periodic structure, then the reciprocal lattice vector can be expressed as G = na*, where a* is the basic reciprocal lattice vector and n is an integer. For the emission in the surface normal direction, i.e., _{$k_{/ /} = 0$} (the point Γ), the condition (1a) requires that _{$k_{p} = G$}. In the 1D PlC with a period of P, an SPP wave with a wave vector _{$k_{p}$} at the Brillouin zone boundary is Bragg reflected by the periodic structure to create a reflected wave with a wave vector

{k^{'}}_{p}

, which satisfies

{k^{'}}_{p} - k_{p} = G^{'}

, and consequently forms a standing wave. A standing wave at the lowest band gap edges at Γ is mainly composed of the two waves with wave vectors of _{$k_{p}, {k^{'}}_{p} = \pm a^{*}$}, which have the same magnitude of G = 2π/P, and is expressed as

≅ C_{1} \exp (i 2 π \frac{x}{P}) + C_{- 1} \exp (- i 2 π \frac{x}{P}),

where ψ represents the surface charge density of the SPP and thus the surface normal component of the electric field. If the origin is placed at the center of the terrace in the 1D PlC, the surface structure has a two-fold symmetry of rotation about the surface normal axis at the origin (the z-axis). The transformation of the wave function by the two-fold rotational operation C₂ is written as C₂ψ = κψ, and then κ² = 1, because twice the operation returns the initial state. Therefore, the eigenvalues of C₂ are κ = ±1. Eigenfunctions for them are deduced as follows:

ψ^{S} (x) ≅ \sqrt{2} \cos (2 π \frac{x}{P}), ψ^{A} (x) ≅ \sqrt{2} \sin (2 π \frac{x}{P}) .

Here ψ^S and ψ^A indicate symmetric and anti-symmetric standing waves with respect to the center of the terrace. Energies of the electromagnetic fields are different for the two standing waves, which causes the band gap at Γ. In a similar way, the standing wave functions at the point X are obtained as

ψ^{S} (x) ≅ \sqrt{2} \cos (3 π \frac{x}{P}), ψ^{A} (x) ≅ \sqrt{2} \sin (3 π \frac{x}{P}),

where the wave vectors of the related photon and SPP are _{$k_{/ /} = \frac{a *}{2}$} and _{$k_{p} = \frac{3 a *}{2}$}.

Barnes et al. have shown that the band edge energy at the Brillouin zone boundary can be deduced from the corresponding Fourier component of the surface shape function [14]. They treated band edge energies of SPPs on a 1D surface structure composed of the first- and second-order Fourier components, and derived those energies analytically. The mode frequencies of the band edges at Γ are expressed as

(\frac{ω_{}^{A, S}}{c}) = {[{(\frac{ω_{0}}{c})}^{2} {1 - {(k_{p} h_{2})}^{2}} \pm 2 (k_{p} h_{2}) \frac{k_{p}^{2}}{\sqrt{- ε_{0} ε}} {1 - \frac{7}{2} {(k_{p} h_{2})}^{2}}]}^{\frac{1}{2}}

where ε₀ and ε are the relative dielectric constants of vacuum and the metal, respectively, and k_p is the wavenumber of the SPP at Γ, with magnitude of G = 2π/P; ω₀ is the mode frequency in the case of no corrugation. The dispersion relation of an SPP on a flat surface is as follows:

k_{p} = \frac{2 π}{P} = \frac{ω_{0}}{c} \sqrt{\frac{ε_{0} ε (ω_{0})}{ε (ω_{0}) + ε_{0}}}

In case of silver and P = 600nm, E₀ = ħω₀ = 2.0 eV. The amplitude of the surface corrugation with the wavenumber of 2k_p is written as h₂, which is the second-order Fourier component of the surface shape function. In general, the nth order Fourier component of the rectangular cross-section is written as

h_{n} = \frac{h}{n π} \sin (n π \frac{D}{P})

In Eq. (5), magnitudes of the frequencies change with the sign of h₂. The + and − in Eq. (5) correspond to the A and S modes, respectively, as will be confirmed from the observed images.

3. Results and discussion

Figures 2(a)–2(e) show ARS patterns taken from 1D PlCs with a period of 600 nm and a height of 70 nm, and with terrace widths of (a) 100 nm, (b) 200 nm, (c) 300 nm, (d) 400 nm, and (e) 500 nm. During the measurement, the incident electron beam was scanned over a 3 × 3 μm² area. The horizontal axis indicates the emission angle measured from the surface normal direction towards the y-direction. Because of the relation of $k_{y} = k_{p h} \sin θ_{y}$ between the emission angle and wave vector k_y, the central position of θ_y = 0° corresponds to Γ, and these ARS patterns reveal approximately the SPP dispersion relation around the lowest energy band gap at Γ. The dispersion curves are seen to cross at D/P ~1/2, and open up at the other D/P values.

Fig. 2 ARS patterns taken from 1D PlCs of a period of 600 nm and a height of 70 nm with terrace widths of (a) 100 nm, (b) 200 nm, (c) 300 nm, (d) 400 nm, and (e) 500 nm. (f) Emission spectra from the PlCs with various D/Ps taken in the surface normal direction.

Download Full Size | PDF

Emission spectra from the PlCs with a fixed period of 600 nm and with various D/P taken in the surface normal direction are shown in Fig. 2(f). These are the spectra taken at θ_y = 0° in Figs. 2(a)–2(e). There are two distinct peaks in each spectrum, which appear at the band edge energies at the Γ. Integrated intensity becomes maximum near D/P = 1/2, and tends to decrease as D/P approaches 0 or 1. In the range where D/P is smaller than 1/2, the emission intensity is much stronger at the higher band edge than at the lower one, whereas in the range where D/P is larger than 1/2, the situation is reversed. The band gap width becomes maximal when D/P is close to 1/4 or 3/4. Additionally, the full width at half maximum (FWHM) of the lower energy peak reaches a maximum at D/P = 3/4 (ΔE = 0.15 eV). We note that near D/P = 1/2, the spectral peak is split into three closely spaced peaks with a large FWHM (Fig. 2(f)).

The observed variation of the band gap structure with terrace height is shown in Fig. 3. The ARS patterns taken from the PlCs of 30, 70, and 100 nm heights are aligned from top to bottom in Fig. 3. The three patterns from left to right were taken from the PlCs with different D/P values, 1/4, 1/2, and 3/4, respectively. The band gap is seen to increase with terrace height from top to bottom. The intensity distributions along the dispersion curve are similar in the patterns with the same D/P, regardless of the terrace height. Figures 3(j)–3(l) show simulated images of the absorption spectrum as a function of the incident angle, which are calculated for the 70-nm height PlC using the RCWA method. These images reproduce well the contrast and width of the dispersion curves in the ARS patterns in Figs. 3(d)–3(f).

Fig. 3 ARS patterns taken from the PlCs with (a)–(c) 30 nm, (d)–(f) 70 nm, and (g)–(i) 100 nm height, and D/P values of 1/4, 1/2, and 3/4, from left to right. (j)–(l) Simulated images composed of the absorption spectra as a function of incident angle, calculated for the 70-nm tall PlC using the RCWA method.

Download Full Size | PDF

Figures 4(a)–4(i) show BSS images taken from the 1D PlCs with heights of (a)–(c) 30nm, (d)–(f) 70nm, and (g)–(i) 100nm, and with D/P values of nearly 1/4, 1/2, and 3/4 for each height, which corresponds to the ARS patterns in Fig. 3. These images were observed with the pinhole in the surface normal direction (θ_y = 0), and reveal the field intensity distribution of the standing SPPs in the PlCs [13,20–22]. The cross-sections of the samples are illustrated in the bottom of the figure, and the white dashed lines indicate the central position of the terrace. We note that in the case of D/P ~1/4, the standing wave image of the higher energy band edge has a node at the center of the terrace, so it should represent the anti-symmetric standing wave (the A mode) in Eq. (3). The standing wave image of the lower energy band edge has an anti-node at the center of the terrace, and should belong to the symmetric standing wave (the S mode). The emission intensity of the A mode is much stronger than that of the S mode, whereas in the case of D/P ~3/4, this situation is reversed. The peak energy broadening of the A mode is seen to be remarkable at D/P ~3/4. In the case of D/P ~1/2, characteristic images appear, as seen in Figs. 4(e) and 4(h), in which the A mode SPP wave is vertically split into two and sandwiches the S mode SPP wave in the spatial and energy spaces.

Fig. 4 (a)–(i) BSS images taken from the 1D PlCs with terrace heights of (a)–(c) 30 nm, (d)–(f) 70 nm, and (g)–(i) 100 nm, and with D/P values of nearly 1/4, 1/2, and 3/4 from left to right. The images correspond to the ARS patterns in Figs. 3(a)–3(i).

Download Full Size | PDF

In terms of emission intensity, the S and the A modes are comparable with each other in the range of D/P near 1/2, though elsewhere the A mode always has a higher intensity than the S. The reason for this was explained by Barnes et al. [14] for a surface structure with two Fourier components. The second-order Fourier component causes the band gap at Γ (see Eq. (5)), but does not emit light without the first-order component. Since the emission corresponding to Γ is in the surface normal direction, this emission is caused by the electric dipole oscillation parallel to the surface. The electric dipoles formed by the surface charge distribution on the surface associated with only the second-order Fourier component cancel out, though coexistence with the first-order component of the surface shape function produces the surface parallel component of the electric dipole to cause light emission. In the case of the PlC with a rectangular cross-section, the charge distribution of the S mode has anti-nodes located at the center of the terraces and grooves, which generate electric fields mainly normal to the surface. Therefore, the S mode cannot couple with light whose electric field is parallel to the surface. On the other hand, in the A mode, charges of the opposite signs are located at the edges of the terraces and grooves to produce the electric dipole parallel to the surface. Then the A mode can couple with light emitting in the normal direction. The sign of the second order Fourier component is reversed between D/P = 1/4 and 3/4, because the position of the ridge and groove in the second order periodic structure is exchanged. Therefore, the energies of the S and A modes are reversed between D/P = 1/4 and 3/4, though the emission property of the two modes does not change.

The band edge energies of the PlC at Γ depend on the D/P ratio, as shown in Fig. 5(a).The data were measured from the samples with terrace height of 30 nm (black lines), 70 nm (red lines), and 100 nm (blue lines). Solid symbols indicate the S mode and open symbols indicates the A mode. The data plots for the 70-nm height sample are shown in Fig. 5(b) for comparison with the theory. Solid and dotted white lines are calculated by Eq. (5), indicating the D/P dependence of the band edge energies of the S and A modes, respectively. These energies are well fitted to the observed data for higher energies, except for the region near D/P = 1/2, but not for lower energies. This is because the higher-order Fourier components are not included in the theory. Nevertheless, we note that the rough behavior of the band edge energy with the D/P ratio is well-represented by this theory.

Fig. 5 (a) A D/P dependence of the band edge energies of the PlC at Γ. Black, red, and blue lines show data from the PlCs with a terrace height h of 30 nm, 70 nm, and 100 nm, respectively. Solid and open circles indicate the S and A modes. (b) The background absorption map calculated by the RCWA method for the 70-nm tall PlC, showing the D/P and (c) that showing h dependence of the band-edge energies of the PlC with D/P = 3/4. Red and white plots indicate the S and A mode data, respectively. A solid and dotted white lines are calculated by Eq. (5), indicating the band edge energies of the S and A modes, respectively.

Download Full Size | PDF

We performed a rigorous calculation involving the higher-order components using the RCWA method for the 1D PlC with a rectangular cross-section. A background absorption map in Fig. 5(b) is produced from spectra of the optical absorption as a function of D/P calculated for the 70-nm height PlC. The incident direction of light used in the calculation is slightly tilted from the surface normal by 0.5°, because the reflectivity of the S mode vanishes for normal incidence. The observed data were obtained from the integrated intensity of the emission passing through the pinhole, i.e., integrated over a finite solid angle around Γ. Because of the small curvature of the SPP band near Γ, the calculation should be adequate for comparison with the observed data. The loci of intensity maxima in the calculated map are in good agreement with those of the observed data, though the observed plots are located at slightly higher energy positions from those at lower energies. This deviation could be attributed to the imperfection in the rectangular shape of the PlC, such as the rounding of the terrace edge.

The dependence of the band-edge energy on terrace height is indicated in Fig. 5(c). The background absorption map is calculated by the RCWA method as in Fig. 5(b). The higher energy plots are for the S mode and the lower one is for the A mode. Both of the band edge energies approach $E_{0} = ℏ ω_{0}$ as h decreases to 0. The higher band-edge energy (E_S) increases a little with h and approaches the photon energy of the same wave number as the SPP. The lower band-edge energy (E_A) decreases monotonically with h. Consequently, the band gap energy (E_S – E_A) increases almost linearly with h up to h = 150 nm. The calculation reproduces well the behavior of the band-edge energies, though the deviation in E_A between the observed plots and the intensity map is also seen, as in Fig. 5(b). The emission intensity in the ARS pattern becomes weak when terrace height exceeds 100 nm, and was negligible for the sample with h = 200 nm. This suggests that the propagation of the SPP across the terrace tends to be suppressed when the terrace height exceeds 100 nm. This tendency is also suggested from the theoretical work on the SPP scattering by surface defects [17].

The two bands of the S and A modes crossed near D/P = 1/2, and the emission intensity reaches a maximum. The intersection is slightly below 1/2, at D/P = 0.46, both in the experimental data and in the calculated map in Fig. 5(b). In the calculated pattern in Fig. 3(k), the dispersion curves calculated for D/P = 1/2 do not cross at Γ, but split in the k direction. This split is an artifact, which appears due to the destructive interference between the SPP radiation and the specular reflection of the incident light, as suggested by Barnes et al. (Fig. 8 in [14]). Such splitting was not observed in CL, even with a smaller pinhole. The BSS image showed that the A mode peak is split into two near D/P = 1/2, though this does not appear in the calculations. This behavior resembles the plasmonic, electromagnetically induced transparency (EIT), where a high-Q resonator of the bright mode is coupled with a low-Q resonator of the dark mode [23]. However, in the present case, the two modes coexist in the same resonator, and the mechanism of the splitting is not yet clear.

The D/P dependence of the SPP band gap at the point X (k_p = G/2) is shown in Fig. 6 for the sample with a period of 800 nm and a terrace height of 50 nm. In this case, the sample was set to be rotated by 90° from the arrangement shown in Fig. 1(c). Then the grating direction (the y-axis) was parallel to the parabolic axis of the mirror (the X-axis), and the surface normal (the z-axis) was parallel to the Z-axis. The ARS patterns were taken along the emission angle θ_y in the y-z plane. According to the procedure used in previous studies [13, 20, 22], the ARS patterns taken from the samples with various terrace widths were transformed into the dispersion patterns as shown in Fig. 6(a). The band gap was observed to close near D/P = 1/3 and 2/3, and to open up around D/P = 1/6, 1/2, and 5/6. The dispersion patterns in Fig. 6(b) were calculated by the RCWA method corresponding to the patterns in Fig. 6(a). The calculated patterns reproduce well the intensity distribution and the broadening of the width along the dispersion curves seen in the experiments.

Fig. 6 (a) Dispersion patterns transformed from the ARS patterns of the PlCs with various terrace widths. The PlC has a period of 800 nm and height of 50 nm. The right edge in each pattern corresponds to the light line. (b) Dispersion patterns calculated by the RCWA method corresponding to those in (a). (c) Dependence of the band edge energies at X on D/P. A solid white line indicates the energy of a SPP on a flat surface at $k_{p} = 3 π / 2 P$ , and a dotted white line shows the light line limit.

Download Full Size | PDF

The observed band-edge energies were plotted as a function of D/P in Fig. 6(c). The symmetry of the band-edge mode was determined from the BSS images. Red spots indicate the S mode and white spots indicate the A mode, respectively. The behavior of the band edge energy with D/P is qualitatively expressed by Eq. (5) using the third-order Fourier component in Eq. (5), though the energies deviate from the data, especially for the lower-energy mode. The background intensity map is calculated by the RCWA method and fitted quantitatively with the observed data, showing that the loci of the intensity maximum are asymmetric with respect to D/P = 1/2, i.e., the band gap at D/P = 5 /6 is larger than at 1/6.

Figure 7(a) shows a BSS image taken from the 1D PlC with P = 800nm, D = 413 nm and h = 50 nm in the direction of the point X. The change in emission intensity of the two band-edge modes (E_S = 2.26 eV and E_A = 2.05 eV) with the electron beam position is revealed in the image. Figure 7(b) indicates a spatial distribution of the standing SPP wave in the 1D PlC, where the z-component of the electric field is calculated by the RCWA method using an incident light from the direction corresponding to the point X. Figures 7(c) and 7(d) indicate the z-component of the electric field strength at the higher and lower band-edge energies, respectively, showing the spatial distributions in the xz plane. The PlC appears dark at the bottom of each calculated image because the field strength inside the metal is relatively small. The intensity in Fig. 7(b) is obtained by integrating the z-component electric field along the z-axis at each x position. The observed image in Fig. 7(a) is in good agreement with the calculated field strength, from which we can conclude that the BSS image reveals the standing SPP wave in the 1D PlC. It has been reported that a photon map acquired via cathodoluminescence under certain conditions mimics the field distribution of standing SPP waves or the photonic local density of states (LDOS) [13, 20, 23–25].

Fig. 7 (a) A BSS image taken from the 1D PlC with P = 800nm, D = 413 nm, and h = 50 nm at X. (b) The spatial distribution of the electric field strength of the standing SPP wave in the 1D PlC of D/P = 1/2 calculated by the RCWA method under the same condition as in (a). (c),(d) The z-component of the electric field strength at the higher and lower band edge energies in the xz plane.

Download Full Size | PDF

The two standing SPP waves at the higher and lower energy band edges at X should be excited with a similar magnitude by an incident electron when the electron hits an anti-node position of them, because their energies are close to each other. Nevertheless, their emission intensities have a large difference, as seen in Fig. 7(a), with the emission intensity of the lower-energy mode being much higher. The detected emission intensity is given by the product of the excitation rate of the SPP wave by the incident electron and the SPP to photon conversion efficiency. In Fig. 7(b), the SPP wave field is excited due to absorption of the incident plane wave of light from the direction corresponding to the point X, so this is a reversed process of the CL emission. Since both processes should have the same SPP-photon conversion efficiency, the BSS image in Fig. 7(a) and the distribution of the field strength in Fig. 7(b) are expected to show the same intensity distribution. However, the higher-energy mode has rather larger intensity in Fig. 7(b) in contrast to the observation in Fig. 7(a). This discrepancy is due to the fact that the observed intensity was measured by integrating the emission intensity over the finite solid angle subtended by the pin-hole area, whereas the calculated one in Fig. 7(b) is the intensity only at the fixed angle corresponding to the point X. The emission intensity of the higher-energy mode is strong but localized at X, whereas the emission intensity of the lower-energy mode is dispersed around X. Consequently the integrated intensity of the higher-energy mode becomes lower than that of the lower-energy mode. This is also seen in the intensity distributions in Figs. 6(a) and 6(b) at D/P = 1/2.

In summary, we have investigated the dependence of the band-edge energies of the 1D PlC on the D/P ratio of rectangular cross-section by CL technique using a 200 keV STEM. The band-edge energies at Γ and X were observed to change with terrace width and height, in rough correspondence with the Fourier component of the surface shape function. Calculation of the band edge energies by the RCWA method well reproduced the observed data and their dependence on terrace width and height, though some deviation in magnitude remained. The symmetry of the standing SPP wave of the band-edge mode was determined from the BSS image. There exist two band-edge modes, the radiative A mode and non-radiative S mode, which exchange their energy positions in correspondence with the sign of the relevant Fourier component. From these results, once the structural parameters of the rectangular cross-section are given, we can find the energy and symmetry of the band edge mode in the 1D PlC. Thus the present results give us useful information for designing plasmonic devises using a 1D PlC.

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-13-IT-0010).

References and links

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

2. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

3. M. U. González, J.-C. Weeber, A.-L. Baudrion, A. Dereux, A. L. Stepanov, J. R. Krenn, E. Devaux, and T. W. Ebbesen, “Design, near-field characterization, and modeling of 45° surface-plasmon Bragg mirrors,” Phys. Rev. B 73(15), 155416 (2006). [CrossRef]

4. S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. 86(14), 3008–3011 (2001). [CrossRef] [PubMed]

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

6. C. Valsecchi and A. G. Brolo, “Periodic metallic nanostructures as plasmonic chemical sensors,” Langmuir 29(19), 5638–5649 (2013). [CrossRef] [PubMed]

7. H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213 (2010). [CrossRef] [PubMed]

8. 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]

9. A. M. Lakhani, M. K. Kim, E. K. Lau, and M. C. Wu, “Plasmonic crystal defect nanolaser,” Opt. Express 19(19), 18237–18245 (2011). [CrossRef] [PubMed]

10. J. C. Weeber, Y. Lacroute, A. Dereux, E. Devaux, T. Ebbesen, C. Girard, M. U. González, and A.-L. Baudrion, “Near-field characterization of Bragg mirrors engraved in surface plasmon waveguides,” Phys. Rev. B 70(23), 235406 (2004). [CrossRef]

11. M. V. Bashevoy, F. Jonsson, A. V. Krasavin, N. I. Zheludev, Y. Chen, and M. I. Stockman, “Generation of traveling surface plasmon waves by free-electron impact,” Nano Lett. 6(6), 1113–1115 (2006). [CrossRef] [PubMed]

12. J. E. Kihm, Y. C. Yoon, D. J. Park, Y. H. Ahn, C. Ropers, C. Lienau, J. Kim, Q. H. Park, and D. S. Kim, “Fabry-Perot tuning of the band-gap polarity in plasmonic crystals,” Phys. Rev. B 75(3), 035414 (2007). [CrossRef]

13. 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]

14. 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]

15. A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408(3–4), 131–314 (2005). [CrossRef]

16. G. Lévêque and O. J. F. Martin, “Optimization of finite diffraction gratings for the excitation of surface plasmons,” J. Appl. Phys. 100, 124301 (2006). [CrossRef]

17. C. Brucoli and L. Martín-Moreno, “Effect of defect depth on surface plasmon scattering by subwavelength surface defects,” Phys. Rev. B 83(7), 075433 (2011). [CrossRef]

18. H. Iwase, D. Englund, and J. Vucković, “Analysis of the Purcell effect in photonic and plasmonic crystals with losses,” Opt. Express 18(16), 16546–16560 (2010). [CrossRef] [PubMed]

19. T. Okamoto and S. Kawata, “Dispersion relation and radiation properties of plasmonic crystals with triangular lattices,” Opt. Express 20(5), 5168–5177 (2012). [CrossRef] [PubMed]

20. 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]

21. M. Honda and N. Yamamoto, “Size dependence of surface plasmon modes in one-dimensional plasmonic crystal cavities,” Opt. Express 21(10), 11973–11983 (2013). [CrossRef] [PubMed]

22. N. Yamamoto, The Transmission Electron Microscope (InTech, 2011), Chap. 15.

23. F. J. García de Abajo and M. Kociak, “Probing the photonic local density of states with electron energy loss spectroscopy,” Phys. Rev. Lett. 100(10), 106804 (2008). [CrossRef] [PubMed]

24. M. Kuttge, E. J. R. Vesseur, A. F. Koenderink, H. J. Lezec, H. Atwater, F. 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]

25. G. Boudarham, N. Feth, V. Myroshnychenko, S. Linden, J. García de Abajo, M. Wegener, and M. Kociak, “Spectral imaging of individual split-ring resonators,” Phys. Rev. Lett. 105(25), 255501 (2010). [CrossRef] [PubMed]

Size dependence of band-gaps in a one-dimensional plasmonic crystal

Abstract

1. Introduction

2. Theory

3. Results and discussion

Acknowledgments

References and links

Cited By

Figures (7)

Equations (8)

Optics Express