Waveguide Bandgap in Crystalline Bandgap Slows Down Surface

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Waveguide Bandgap in Crystalline Bandgap Slows down Surface Plasmon Polariton Hikaru Saito, Naoki Yamamoto, and Takumi Sannomiya ACS Photonics, Just Accepted Manuscript • Publication Date (Web): 01 May 2017 Downloaded from http://pubs.acs.org on May 1, 2017

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Waveguide Bandgap in Crystalline Bandgap Slows down Surface Plasmon Polariton Hikaru Saito*,1, Naoki Yamamoto2, and Takumi Sannomiya*,2 1

Department of Electrical and Materials Science, Kyushu University, 6-1 Kasugakoen, Kasuga, Fukuoka 816-8580, Japan 2 Department of Materials Science and Engineering, Tokyo Institute of Technology, Nagatsuta, Midori-ku, Yokohama, Kanagawa 226-8503, Japan. *corresponding authors

e-mail address HS : [email protected] NY: [email protected] TS : [email protected] Corresponding author’s phone number, fax number Phone: HS 81-92-583-7579, TS 81-45-924-5674 Fax: HS 81-92-583-7580

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Abstract Next generation on-chip optical devices require light manipulation in time and space, i. e. control of group velocity of light in sub-wavelength dimensions. A waveguide in plasmonic crystal fulfills such requirements offering nano-scale light confinement in the dispersion-tunable plasmonic crystal matrix. However, there has been no direct access to the local dispersion of the waveguide mode itself, and the group velocity of light could not be evaluated. Herein for the first time we experimentally clarify the dispersion of the waveguide modes by use of angle-resolved cathodoluminescence scanning transmission electron microscopy. Their group velocity can be extremely slowed down by the existence of a bandgap formed in the waveguide in the energy range of the plasmonic crystal bandgap. Keywords: angle-resolved spectroscopy; cathodoluminescence; plasmonic crystal; plasmonic bandgap; slow light; waveguide dispersion

Manipulation of light propagation in small on-chip devices is one of the important technologies for high speed driving circuit, where information is transferred by light. Enhancement of light– material interaction by e.g. increasing interaction time, or slowing light, is a key issue for optical elements such as light switches, sensors as well as signal converters. Photonic bandgap engineering has enabled strong compression of pulsed light in photonic crystal (PhC) waveguides, which can reduce the group velocity of light to a few hundredths of that in vacuum.1 Such pulsed slow light propagation in a PhC waveguide has been observed in real space using scanning near-field optical microscopy (SNOM).2 Utilization of metal surfaces is an alternative way to enhance light–material interaction because electromagnetic energy can be confined near the metal surface in the form of surface plasmon polaritons (SPPs),3 which can reduce modal volume beyond the diffraction limit, and thus potentially shorten response time and reduce driving power of optical elements. Similar to slow light in dielectrics, slow propagation of a SPP pulse has been probed using SNOM,4 and even lasing using slow SPPs in a metallic Moiré cavity was demonstrated.5 Using periodically


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structured metal surfaces, referred to as plasmonic crystals (PlCs), manipulation of slow SPPs in chip-size devices is also possible.6,7 The bandgap formed in PlCs can be wider than that in PhCs,8 offering more engineering possibilities. Especially, PlCs with a triangular lattice (Tri-PlCs) are attractive due to full bandgap (FBG) formation, where no SPP propagation is allowed.9 In the study by Kitson et al., a Tri-PlC structure was found to have a plasmonic FBG with a measured energy width of around 0.09 eV (22 THz),9 suggesting that Tri-PlC waveguides could support ultra-short pulses with durations of several tens of femtoseconds. Although SPP propagation in Tri-PlC waveguides has been experimentally probed,

6,7

the dispersion

characteristics of the guided modes, which are essential to control SPP pulses, have not been measured or understood. Herein we perform local dispersion measurements of PlC waveguides by using cathodoluminescence (CL) combined with a scanning transmission electron microscopy (STEM). We chose this measurement technique because STEM-CL is most suitable for dispersion measurement and high resolution visualization of local light distribution and propagation. Electron microscope-based spectroscopy techniques such as electron energy-loss spectroscopy (EELS) and CL are powerful characterization tools to observe electromagnetic modes as well as electronic states in materials, owing to their high spatial resolution.10 Even three-dimensional analyses of nanoscopic electromagnetic properties have been demonstrated by tomography combined with EELS11,12 and CL.13 EELS has an advantage in full modal analysis including dark modes14–18 while CL provides emission properties such as polarization characteristics,19 lifetimes,20 and quantum states.21 To measure dispersion relations, momentum-resolved spectroscopy in electron microscopy is required.22–27 Among these techniques, CL combined with an angle-resolved light detection system is the most powerful


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method to investigate emission characteristics of locally-modified structures introduced into periodic structures as seen in the previous studies.28–32 In the present study, STEM-CL-based angle-resolved spectroscopy (ARS) is used to analyze the local dispersion characteristics of slow SPPs in PlC waveguides. ANGLE-RESOLVED CL SPECTROSCOPY A scanning transmission electron microscope (JEOL JEM-2100F) was used to excite SPPs. In ARS measurements, SPP modes inside the light cone hold the following relation between the energy E and the emission angle θ:

k=

E sin θ hc

, (1)

where k is the wave number of the SPP mode, ħ is Planck’s constant divided by 2π and c is the velocity of light in vacuum. When the dispersion curve crosses the light cone, θ reaches 90°. Figure 1a depicts the experimental setup for CL measurements. Details of the used technique is described elsewhere.25,26 We define XYZ coordinates fixed on the parabolic mirror and xyz coordinates fixed on the sample. These coordinate systems overlap when the sample is not tilted. For the ARS measurements, the light emitted parallel to the XZ plane was detected by successively changing the pinhole position in the Z direction. PLASMONIC CRYSTAL WITH A TRIANGULAR LATICE The investigated Tri-PlC is composed of a silver dot array on a silver surface. The distance between the lattice points (P) is 300 nm, and the diameter and the height of the dot are about 150 nm and 100 nm, respectively (Fig. 1b). This Tri-PlC specimen was produced on an InP substrate (several hundred micrometer thick) by electron beam lithography and physical vapor deposition of silver (see details in Experimental section).


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Figures 1c and 1d show the dispersion patterns of the defect-less Tri-PlC in the Γ–M and the Γ–K directions taken by the ARS measurement. The sample was set so that the Γ–M or Γ–K direction was parallel to the X axis in the measurements. The FBG could not be measured directly by this method since the band-edges were located outside the light cone (Fig. 1e). However, it can be inferred that the upper edge of the FBG lies approximately at 2.3 eV by extrapolating the dispersion curve towards the M edge in Fig. 1c. DESIGN OF PLASMONIC CRYSTAL WAVEGUIDES Waveguide structures composed of line defects in Tri-PlCs were fabricated in the same process as above (Fig. 2a). The defect width W is defined as the distance between the dot centers as shown in the figure. The SPPs with the energies in the FBG are confined in the flat waveguide area sandwiched by the two Tri-PlCs and are guided parallel to the Γ–K direction (the x direction in Fig. 2a). In order to design the PlC waveguide, we modeled the guided mode as a propagating SPP wave between two interfaces separated by a certain defect width W, as illustrated in Fig. 2b. The wave number k = k of the SPP on a flat silver surface is written as

E ε Ag  k = k x2 + k y2 = Re ,  hc ε Ag + 1 

(2)

where ε Ag is the dielectric function of silver.33 We used Drude’s model for ε Ag with parameters derived from the permittivity data given by Palik.34 Now we introduce a phase shift Φ associated with the reflection at the interface. Then the y wave vector component k y of the guided wave should satisfy the following guiding condition:


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k yW + Φ = n π ,

(3)

where n is an integer.29,35 Honda and Yamamoto have reported that Φ is approximately linear to the resonant energy within the band gap in one-dimensional PlC cavities.29 Considering that the defect width W is defined as the distance between the center of the lattice dots at the waveguide edges, Φ = 0 corresponds to a standing wave with nodes at the waveguide edges, and Φ = π to that with antinodes at the edges, as illustrated in Fig. 2b. In order to quantify Φ, we now assume that Φ is 0 at the lower band-edge energy and π at the upper band-edge energy of the Tri-PlC. This assumption is based on a defect-free crystal where the lower band-edge mode has antinodes on the lattice points (dots), and the upper band-edge mode has nodes on the lattice points (see Supporting Information A). This way, the defect modes, or waveguide modes in this case, lying inside the FBG can continuously bridge the phase of the standing waves at the upper and lower band-edges of the FBG.

29

The linear relation of Φ to the energy E, which is the first

approximation, can be written as Φ (E ) =

π ∆E

(E − E ), −

(4)

where ∆E is the energy width of the FBG and E − the energy of the lower band-edge. Here we set E − = 1.8 eV and ∆E = 0.5 eV, corresponding to the FBG of a perfect PlC. Although only the upper band-edge energy (~2.3 eV) close to the light cone has been estimated in the previous discussion of PlCs, the lower band-edge far outside the light cone will also be experimentally derived by using grating coupling, which is discussed in the later section. Figure 2c shows W dependence of the waveguide dispersion for n = 2 mode calculated using Eqs. (2)–(4). The dispersion curve shifts to lower energy and becomes steep in the kx direction


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with increase of W. This way the group velocity of the waveguide mode can be slowed down by small W. On the other hand, for Broad-band applications of the waveguide, the threshold energy EM at the M point should be located at lower energy side in the crystalline FBG. This criteria requires nπ /W (≈ k y ) equal to kM = 2π / 3P = 0.012nm−1 , giving W = 520 nm as a proper defect width for the n = 2 mode. In the same way, W of 780 nm (n = 3) and 1040 nm (n = 4) are selected for the higher order modes. The dispersion of the waveguide mode (e.g. W =520 nm in Figure 2c) nicely lies within the FBG although the M point is slightly off the dispersion curve because of the finite phase shift Φ(EM ) . DISPERSION MEASUREMENTS OF WAVEGUIDE MODES Figures 3a and 3b show the dispersion patterns taken from the PlC waveguide with W = 520 nm and W = 780 nm, which are transformed from the measured ARS patterns (see Refs. 25, 26). The dispersion curve of the waveguide mode is outside the light cone owing to the nature of SPPs, and thus the mode might be considered non-radiative. However, radiation in the XZ plane is possible because the streak-like dispersion band of the waveguide mode partially penetrates into the light cone due to the confinement of the wave in the ky direction as illustrated in Fig. 3c. Some fraction of the streak can be detected in the present ARS measurement with the finite sized pinhole moving vertically while keeping ky = 0 (see Fig. 1a). The dispersion curves of the n = 2 waveguide modes actually appear in Figs. 3a and 3b. However, the higher order modes (n= 3, 4) were not observed in the present measurements due to their small contributions because the higher order modes are located further away from the kx axis (Fig. 3d). The n = 1 mode was not detected in this experiment even though it lies closer to the kx axis than the n = 2 mode. The reason is discussed in Supporting Information B.


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For the waveguide with W = 520 nm (Fig. 3a), the group velocity of the waveguide mode in the x direction ( ∂ω / ∂k x ) is 7.5 times slower than the light speed in vacuum by assuming a linear dispersion in the range of 1.95 to 2.10 eV. SPPs in the waveguide are intrinsically slowed down due to the presence of ky components. In addition, waveguide dispersion greatly changes around kx = 0.01 nm-1: The slope of the dispersion curve becomes almost horizontal in Fig. 3a, corresponding to zero group velocity. For W = 780 nm, discontinuity of the dispersion curve is observed, which is indicated as “energy gap” in Fig. 3b. The horizontal dispersion curve of W = 520 nm near the light cone in Fig. 3a is likely originating from this waveguide bandgap (WBG). The upper band-edge of the WBG is located above the crystalline bandgap which is outside the light cone and cannot be detected. The discontinuity of the dispersion at the WBG flattens the dispersion curve around WBG, making the group velocity of the waveguide mode even slower. More detailed analyses and discussions about WBG will be performed in the later section after visualizing the field distribution of the modes. Now, we examine our total internal reflection model expressed by Eqs. (2)–(4) for the continuous region of the dispersion without WBG. Figure 3e shows a comparison between the measured dispersion plot for W = 650 nm and theoretical curves for n = 2. Although the experimental curve follows the theoretical line with constant Φ = π in the higher energy region, it significantly deviates from Φ = π line and approaches the theoretical line with constant Φ = 0 as the energy decreases. The theoretical model with the energy-dependent phase shift

Φ(E ) using Eq. (4) bridges the two theoretical lines with constant Φ and better fits the

experimental curve. For all the defect widths, this model reproduces the experimental results better than those with constant Φ, as shown in Fig. 3f (dispersion patterns for W = 650 nm and


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1040 nm are shown in Supporting Information C). The energy-dependent phase Φ (E ) also helps slowing the group velocity over a wide energy range by inclining the dispersion curve. VISUALIZATION OF SPP PROPAGATION IN WAVEGUIDES In order to study the propagation behavior of the guided SPP modes as well as to visualize the modes outside the light cone, the guided wave must be converted to a detectable signal, such as far field radiation, at a certain distance from the input.36,37 To do so, we connected the PlC waveguide (W = 1040 nm) to a one-dimensional (1D) grating structure with a period of 520 nm (Figs. 4a and 4b) (Supporting Information D). By setting a grating with a proper period, the SPP entering into the grating from the PlC waveguide is diffracted, and the dispersion curve enters inside the light cone to make the SPP detected as radiated light. We note that propagation directions of the SPP entering into the grating and the radiation by the grating can be opposite (Fig. 4b). For this measurement, a large pinhole with a diameter of 1.0 mm, which corresponds to the angular range from 0° to 22 ° in the XZ plane, was used to collect the SPP related radiation with a range of kx ≤ 0.024 nm-1. The details about SPP–photon conversion by the 1D grating is written in Supporting Information E. Figure 4c shows spectra taken by irradiating the waveguide area (red curve) and Tri-PlC area (black curve), and collecting light emission from the 1D grating. The beam positions are indicated in the inset of Fig. 4c. The spectrum from the waveguide area (red) shows strong intensities in the range from 1.8 to 2.3 eV, corresponding to the light emission of the guided SPPs entering into the 1D grating. In contrast, the spectrum from the Tri-PlC area exhibits weak signals in the corresponding energy range. The spectral intensity from the pure Tri-PlC area is strong below 1.8 eV, which means that SPP modes exist in the Tri-PlC below this energy and propagate into the 1D grating to emit light. Considering that the upper energy limit of the existent lower band SPP modes towards K point in the Tri-PlC is 1.8V,


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the lower band-edge energy in the band gap of the Tri-PlC at the K point can be determined to be about 1.8 eV, which corresponds to the lower band-edge of FBG (see Fig. 1e). From this value together with the upper band-edge energy estimated before (2.3 eV) at the M point (Fig. 1c and 1e), it is deduced that the Tri-PlC has a FBG energy of about 0.5 eV, ranging from 1.8 to 2.3 eV. This bandgap, formed in the visible region, is much wider than that reported in the previous work for a similar structure (Supporting Information F).9 This confirmation of the FBG energy range also assures that the guided SPPs lie in the FBG. WAVEGUIDE BANDGAP Photon mapping of the guided SPP modes was performed by collecting light emission from the 1D grating, as shown in Figs. 5a–5d. As a representative example, we chose W = 1040 nm. High intensity contrasts appear inside the waveguide, indicating that the guided SPPs propagate to the end and enter into the 1D grating to emit light. The detected CL intensity is directly given by the SPP-photon conversion of the SPP wave entering into the 1D-grating, while the SPP intensity depends on the three factors; first is the excitation probability of the waveguide mode by the incident electron, which is approximately proportional to time-averaged electric field 2

projected along the z direction, E Z ,38–40 and second is the decay of the SPP propagating along the waveguide, and third is the reflection at the boundary between the waveguide and the 1D grating. Since the SPP-photon conversion process in the 1D grating is independent of the incident electron position and the decay process only causes monotonic decrease in the SPP intensity, the main feature of the CL contrast is given by the excitation probability distribution of 2

the waveguide mode, i.e., the pattern of E Z . As for the third factor, the reflection at the grating boundary causes interference between the traveling wave towards the boundary and the reflected wave to form a standing wave along the waveguide. Such an oscillation contrast is


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visible near the boundary in the photon maps at the energy of 1.8 eV in Figure 5d. This reflection fringe period is apparently larger than lattice period, confirming the reflection whereas the oscillation contrasts observed in the photon maps of Figs. 5a and 5b are intrinsic in this waveguide and appear along the waveguide even far from the boundary. This intrinsic oscillation pattern directly corresponds to the excitation probability of the waveguide mode by the electron beam. Since we observed only a single antinode in the y direction, the observed mode is the symmetric mode of the lowest order (n = 2). For comparison, photon maps of the waveguide without a grating are shown in the left side insets, where almost no signal is observed in the waveguide. Here we investigate the intrinsic periodic oscillation features in the waveguide observed at 2.1 eV and 2.0 eV in Figs. 5a and 5b. The oscillation period is 300 nm corresponding to the inter-dot distance for both energies. This oscillation pattern suggests that the dot array facing the waveguide causes Bragg reflection of the guided SPPs in the direction of the waveguide (x axis). A more precise look into the photon maps in left panels of Fig. 5e and 5f clarifies that the positions of the highest intensities with respect to the surrounding lattice dots are different for these two energies. This energy range of 2.0 to 2.1 eV well corresponds to the WBG observed in the dispersion measurement in Fig. 3c. Therefore it is indicated that the WBG is formed depending on the position of the node of the wave. This WBG formation mechanism is also supported by the experimental fact that WBG is always formed at the same kx values corresponding to the inter-dot distance for all the defect widths, as shown in Fig. 3f. Considering that the contrast in the photon map gives information about the Z component of the electric field distribution EZ ,38–40 the bright contrasts in the oscillation pattern should correspond to the antinodes of the surface charge distributions of the guided SPP modes at the band-edges of the


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WBG. Such field distribution can be reproduced by numerically simulated E Z

2

as shown in

the right panels of Figs. 5e and 5f. Details of the finite difference time domain (FDTD) simulation are described in the Supporting Information G. These two modes found in simulation, corresponding to the lower and upper “band-edge modes”, respectively have the energy of 2.03 eV and 2.07 eV, which agrees with the experimentally observed band-edge energies shown in Fig. 3f. The field of the lower band-edge mode spreads into the lattice between the dots (Fig. 5f) while the field of the upper band-edge mode is more tightly confined within the flat waveguide area (Fig. 5e). Thus these two modes are considered to have different field extension in the y direction, generating the energy difference between the band-edge modes, i.e., WBG. Such energy increase due to reduction of the waveguide width has been reported for metal stripe waveguides.41 Now we try to explain how the WBG energy relates to the defect width based on the waveguide model. Since the guided mode satisfies the Eq. (3), changes of the field extension in the y direction can be translated into changes of the wave vector in the y direction ky and the phase shift Φ. kx is identical for both modes, which equals to π/P (inter-dot distance P = 300 nm, see Fig. 6a). The lower band-edge mode should have a smaller ky value than that of the upper band-edge mode. This ky difference at the WBG, ∆ky,WBG, is related to the phase shift difference ∆ΦWBG from Eq. (3) as

∆ k y ,WBG

− + Φ WBG − Φ WBG ∆Φ WBG = = , W W

(5)


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± are the phase shifts at the band-edge energies of the WBG. The small change in ky , where Φ WBG

i.e., ∆ky,WBG, is also related to the wave number difference ∆kWBG by differentiating Eq. (2) and using k yW = nπ − Φ as

∆ k WBG ≈

ky

k

∆ k y ,WBG =

nπ − Φ ∆ k y ,WBG , kW

(6)

where variables with upper bars correspond to the values at the center of the WBG energy. In the present visible light energy range, the dispersion is very close to the light cone and almost linear. Therefore the WBG energy ∆EWBG is approximately proportional to ∆kWBG, i.e., ∆EWBG ∝ ∆kWBG . Thus, Eqs. (5) and (6) suggest the following relation:

∆ E WBG ∝

nπ − Φ nπ − Φ ∆Φ WBG ∝ ∆Φ WBG . 2 kW EW 2

(7)

In order to examine Eq. (7), the measured E , Φ , and ∆ΦWBG are plotted as a function of W + in Figs. 6b and 6c, respectively. Φ was calculated by Eqs. (2), (3), and the measured EWBG − and EWBG . It should be noted that Φ and ∆ΦWBG hardly depend on W in the present

measurements range of W. In addition, when the gentle reduction of E as a function of W is not considered, Eq. (7) is roughly approximated as ∆ EWBG ∝ W −2 .

(8)

By the fitting the power of W in the experimental results as shown in Fig. 6d, we obtained ∆ EWBG ∝ W −1.928 ± 0.005 , which is well consistent with the model equation (8) and supports the

model.


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PROPAGATION LOSS The SPP modes in the PlC waveguide should obey Bloch’s theorem since they are affected by the periodicity in the direction of the x axis as described above. The Z component of the electric field distribution EZ can be written as a sum of Bloch harmonics:42

EZ (x, y) = ∑ EZ ,mum ( y) exp[i (kx x + 2π mx P + φm )] ,

(9)

m

where m is an integer, um ( y) the wave function in the y direction, and φm the phase. The z- and time-dependent terms are omitted here. In the two wave approximation involving only m = -1, 0 terms, the excitation probability of the SPP mode, ηex (x, y ) , is written as follows,

ηex ( x, y) ∝ EZ2,−1u−21 ( y) + EZ2,0u02 ( y) + 2EZ ,−1EZ ,0u−1 ( y)u0 ( y) cos(2π x P − φ−1 + φ0 ) ,

(10)

where the cross term makes the oscillation pattern with the period of P, which corresponds to the observed oscillation patterns in Figs. 5e and 5f. The contrast decay in the waveguide from the grating position in Fig. 5a–5d can be considered as the decay of the guided mode. Thus the −x λ contrast of the photon map can be expressed as I ∝ ηex e with a decay constant λ. This decay

originates from radiation loss, ohmic loss and leakage to the Tri-PlCs. Radiation loss can exist because the SPP modes, observed via the 1D grating, partially include modes inside the light cone due to the streaking. The intensity profiles of the guided SPP mode along the waveguide are shown in Figs. 7a–7d (black circle). The background intensities were measured from a waveguide without a 1D grating and were subtracted from the profiles. The intensity profiles are compared with the fitted curves (colored lines) which have the following form,

f ( x ) = { a12 + a22 + 2a1a2 cos(2π x a3 + a4 )}e−x λ .

(11)


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The fitting parameters a1, a2, a3, a4, and λ are listed in Table 1. The fitted curves well reproduce the observed oscillation patterns, and the oscillation period a3 accords with P in the energy range from 1.9 to 2.1 eV. These results support the validity of the two wave approximation written as Eq. (10). For 1.8 eV, the guided mode no longer behaves as a Bloch wave as indicated by negligibly-small a1/ a2 in Table 1. The deduced decay constant λ decreases with decreasing SPP mode energy. For 1.8 eV, which is below the lower band edge of FBG, strong intensity distribution in the Tri-PlC regions outside the waveguide is observed (Fig. 5d). The shorter propagation with larger damping of the guided mode at this energy is related to this leakage due to coupling to the Tri-PlCs. In order to analyze spatial frequencies of the oscillation patterns in Figs. 7a–7d more precisely, fast Fourier transform (FFT) was performed as shown in Fig. 7e. The results indicate that 1/P is the dominant spatial frequency in the photon maps for 1.9 – 2.1 eV as expected. However, a sub-peak appears in the lower spatial frequency side of the 1/P peak for 1.9 eV. This sub-peak suggests a small contribution of the reflected SPP wave from the 1D grating. By taking account of the reflected wave, another cross term with the wave number of 2kx should be added to Eq. (10), which generates more complicated profiles with beats. We also noticed that the matrix Tri-PlC itself slightly gives CL intensities at the silver dot positions even in the energy range of FBG, where no Bloch mode is expected to exist (Fig. 5a–5c, 1.9 - 2.1 eV range). This suggests light emission from a localized mode at the silver dot, which can be directly detected without using the 1D grating, and may result in propagation loss of the waveguide mode. On the other hand, it is noted that the decay length increases at the band edge energies of the WGB (Figs. 7a and 7b) compared to those at the lower energies. This is because


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the waveguide SPPs at the band edges locate on a flat area between the dots, which decreases the radiation loss for these modes. CONCLUSION We have experimentally determined the dispersion of the SPPs in waveguides created in the Tri-PlCs by the STEM-CL technique. We designed the waveguide so that the guided SPP modes lie in the FBG of the Tri-PlCs. Two essential features are found: i) energy dependent phase shift

Φ(E ) of SPPs reflected at the side of the waveguide and ii) WBG formation at kx =π/P in the propagation direction. Both contribute to slowing down the group velocity of the guided SPP mode. At the WBG, the group velocity seems reaching infinitesimal. The proposed model including energy-dependent phase shift Φ(E ) for the SPP mode in the PlC waveguide qualitatively explains the small ramp in the dispersion curve. The mechanism of the WBG formation has been elucidated by visualizing the difference of the charge distribution at the upper and lower band-edge modes, which have different effective widths of the waveguide. In addition, from the experimental data and modeling, a simple relation between the WBG energy ∆EWBG and the defect width W, i.e. ∆EWBG ∝ W −2 was found. The above results indicate that PlC waveguides are fully engineerable with potential advantages to manipulate ultrashort pulses over a wide energy range. Although the waveguide mode dispersion streaking inside the light cone was measured in this fundamental study, it could be pushed totally outside the light cone for practical use in optical circuits by e.g. adding a dielectric strip on a metal surface.8,43,44

EXPERIMENTAL SECTION Sample fabrication


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The Tri-PlC specimen was produced as follows. A 100 nm thick positive resist layer (ZEP520A) on an InP substrate (several hundred micrometer thick) was fabricated by electron beam lithography. On the patterned triangular lattice of cylindrical pillars, a 10 nm thick chromium layer was deposited as an adhesion layer by direct current sputtering. Finally, a 200 nm thick silver layer was deposited by thermal evaporation in vacuum. Waveguide structures composed of Tri-PlCs with the above structural parameters were fabricated in the same process. Cathodoluminescence measurements A scanning transmission electron microscope (JEOL JEM-2100F) was used to excite SPPs at an acceleration voltage of 80 kV. This electron microscope has a Schottky electron source and a spherical aberration corrector for the probe-forming lens system. The beam current was set to be 4–20 nA. The beam diameter is estimated to be about 2 nm from the lens configuration. Beam spreading due to the specimen thickness does not largely influence the spatial resolution of photon mapping for SPP modes because the excitation point of SPPs is localized at the silver surface. The excitation of SPPs by secondary electrons is also not a problem because such slow electrons hardly excite SPPs.45 The convergence semi-angle was set to be 40 mrad. This finite convergence angle might cause that a contrast related to XY components of the electric fields appears in the photon maps. However, it did not affect the interpretations by the Z component of the electric field distribution in this experiment. An Andor Newton EMCCD spectroscopy detector was used for light detections. A parabolic mirror located around the sample collects the light emitted from the sample under irradiation of the electron beam and the pinhole selects the emission angle. The parabolic curve of the mirror is expressed as 4 p( p − X ) = Z , where p (= 2

1.5 mm) represents the distance between the focal point (specimen position) and the bottom of the parabola. The XYZ coordinates are fixed on the parabolic mirror position as shown in Fig. 1d.


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For the ARS measurements, the light emitted parallel to the XZ plane was detected by successively changing the pinhole position. The original ARS data array was converted to an ARS pattern or a dispersion pattern according to the previous report.25 The diameter of the −3

pinhole was 0.3 mm for the ARS measurements, corresponding to a solid angle of 7.9 ×10 −2

at θ = 0 ° and 3.1×10

sr

sr at θ = 90° . The typical acquisition time for the ARS

measurements was 3 s for each spectrum. The beam scan area on the sample during the ARS measurements was about 1×1 µm2. Photon map imaging was performed by recording the emission spectra while scanning the electron beam with a fixed pinhole position. For the photon map imaging, the acquisition time was 1 s per pixel.

ASSOCIATED CONTENT The Supporting Information is available free of charge on the ACS Publications website.

AUTHOR INFORMATION Corresponding Authors HS : [email protected] TS : [email protected]

CONFLICT OF INTEREST The authors declare no competing financial interest

ACKNOWLEDGEMENTS


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This work was supported by Kazato Research Foundation, and a part of this work was supported by Tokyo Institute of Technology in “Nanotechnology Platform Project” sponsored by the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan.

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30 Ma, X.; Grüßer, M.; Schuster, R. Angular Dependence of Cathodoluminescence of Linear and Circular Au Gratings: Imaging the Coupling Angles between Surface Plasmon Polaritons and Light. J. Phys. Chem. C 2014, 118, 23247–23255. 31 Saito, H.; Yamamoto, N. Control of light emission by a plasmonic crystal cavity. Nano. Lett. 2015, 15, 5764–5769. 32 Saito, H.; Mizuma, S.; Yamamoto, N. Confinement of surface plasmon polaritons by heterostructures of plasmonic crystals. Nano. Lett. 2015, 15, 6789–6793. 33 Maier, S. A. Plasmonics: Fundamentals and Applications; Springer: New York, 2007. 34 Weeber, J. C.; Bouhelier, A.; Colas des Francs, G.; Markey, L.; Dereux, A. Submicrometer in-plane integrated surface plasmon cavities. Nano. Lett. 2007, 7, 1352–1359. 35 Palik, E. D. Handbook of Optical Constants of Solids; Academic Press: Boston, 1985. 36 Devaux, E.; Ebbesen, T. W.; Weeber, J. C.; Dereux, A. Launching and decoupling surface plasmons via micro-gratings. Appl. Phys. Lett. 2003, 83, 4936(1)–4936(3). 37 Wulf, M.; de Hoogh, A.; Rotenberg, N.; Kuipers, L. Ultrafast Plasmonics on Gold Nanowires: Confinement, Dispersion, and Pulse Propagation. ACS Photonics 2014, 1, 1173– 1180. 38 García de Abajo, F. J.; Kociak, M. Probing the photonic local density of states with electron energy loss spectroscopy. Phys. Rev. Lett. 2008, 100, 106804(1) –106804(4). 39 Kociak, M.; Stéphan, O. Mapping plasmons at the nanometer scale in an electron microscope. Chem. Soc. Rev. 2014, 43, 3865–3883.


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Table 1. Fitting parameters for the decay profiles in Figs. 7a–7d. Energy (eV) 2.1 2.0 1.9 1.8

a1/ a2 0.160 0.132 0.0440 0.00884

a3 (nm) 301 303 303 281

a4 3.77 7.77 7.64 -1.94

λ (µm) 7.94 6.64 4.17 3.12

Figure captions


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Figure 1. (a) A schematic diagram of the experimental setup for angle-resolved CL spectroscopy. The X and Z axes are fixed on the parabolic mirror and light emission in the XZ plane is detected with a certain pinhole position selecting the emission angle. (b) (left) A backscattered electron image of the Tri-PlC with a lattice period of 300 nm and dot size of 150 nm. (right) A reciprocal lattice of the Tri-PlC. (c and d) Dispersion patterns taken from the perfect Tri-PlC without waveguides in the (c) Γ–M and (d) Γ–K directions. (e) A schematic diagram of a band structure of a Tri-PlC (blue). The blue dashed lines represent a band structure in the empty lattice approximation. The light cone is shown in red.

Figure 2. (a) Backscattered electron image of a PlC waveguide with a defect width of 1040 nm. (b) A model for SPP propagation in the PlC waveguide. The lines in the lower insets correspond to the wave fronts and the filled circles to the antinodes of the standing wave with opposite signs indicated by red and blue colors. (c) Dispersion curves of the n=2 waveguide modes projected on the E–kx plane at ky = 0 (upper panel) and on the ky–kx plane (lower panel) for various defect widths.

Figure 3. Dispersion analysis of the waveguide modes inside the light cone. (a and b) Dispersion patterns taken from the PlC waveguides with (a) W = 520 nm and (b) W = 780 nm. The thick white arrows indicate the dispersion of the guided modes. An energy gap is also indicated by an arrow in panel b. (c) Schematic three-dimensional model of streaked waveguide dispersion in the FBG. The dispersion curve of the waveguide mode lies on the free-SPP dispersion cone (green) accompanying with the streaking in the ky direction (blue). The light cone is in brown color. (d) Dispersion curves of the n=2 and 3 waveguide modes for W = 780 nm calculated from Eqs. (2)–


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(4), which are projected on the E–kx at ky = 0 (upper panel) and on the ky–kx plane (lower panel). (e) Dispersion plots extracted from the ARS pattern of the PlC waveguide with W = 650 nm (green solid curve). Dashed curves represents three theoretical curves calculated for n = 2 with constant phase ( Φ = 0 , Φ = π ) and energy-dependent phase ( Φ(E ) ). (f) Dispersion plots for the PlC waveguides with W = 520 nm (red), W = 650 nm (green), W = 780 nm (blue), and W = 1040 nm (black). The theoretical dispersion with n = 2 and energy dependent phase Φ(E ) are shown as dashed curves with the corresponding color. The experimental mode energy in each dispersion plot is determined by measuring an energy position of the maximum intensity in each fixed kx spectrum in the dispersion pattern.

Figure 4. (a) Schematic drawing of the PlC waveguide (W = 1040 nm) connected to a one-dimensional grating (1D grating) with a period of 520 nm. The SPPs excited by electron beam (yellow) propagate along the waveguide and are converted to photons by the 1D grating. (b) Relation between wave vectors in the SPP–photon conversion by the 1D grating. G is the reciprocal lattice vector of the 1D grating. (c) Spectra taken with the electron beam in the waveguide (red) and in the matrix Tri-PlC (black), which are schematically depicted in the inset.

Figure 5. (a–d) Monochromatic photon maps of a PlC waveguide with W = 1040 nm connected to the 1D grating taken at (a) 2.1 eV, (b) 2.0 eV, (c) 1.9 eV, and (d) 1.8 eV. Photon maps of the waveguide without the grating are shown on the left side. (e and f) Magnified images at (e) 2.1 eV and (f) 2.0 eV, compared with the field distribution by FDTD simulation on the right hand side. For the simulations, z component of the electric field (Ez on the left and E z

2

on the right)


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25 nm above the dots are displayed. The simulated upper and lower band-edge energies are 2.07 eV and 2.03 eV.

Figure 6. (a) Schematic drawing of surface charge distributions at the band-edge energies of the ± WBG. (b–d) Defect width (W) dependence of (b) the band-edge energies of the WBG EWBG ± and their average E , (c) phase shifts at the band-edge energies ΦWBG and their average Φ

and difference ∆ΦWBG, and (d) the WBG energy ∆EWBG . In panel d, the data points are fitted by a power function ∆EWBG ∝ W −α , resulted in α = −1.928 ± 0.005 .

Figure 7. (a–d) The intensity profiles along the waveguide extracted from Figs. 5a–5d, respectively (black circle), averaged inside the flat area in the y direction. The fitted curves using Eq. (11) are shown in red, green, blue, and purple, respectively. The decay lengths of the fitted curves are indicated at the right, respectively. The fitting was performed from position x = 2.36 µm up to 8.00 µm. The waveguide is connected to the 1D grating at x = 1.16 µm as indicated by a red arrow. (e) Fast Fourier transform analyses for the intensity profiles in Figs. 7a–7d.


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Waveguide Bandgap in Crystalline Bandgap Slows Down Surface

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