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Angle-resolved measurements on a cathodoluminescence (CL) detection system equipped with a 200 keV transmission electron microscope were performed on a 1-dimensional silver grating. The dispersion curves of surface plasmon polaritons (SPPs) on the grating were derived from the angle-resolved CL spectra, and patterns of SPP standing waves at the plasmonic band edge were directly observed.
©2009 Optical Society of America
1. Introduction
A surface plasmon polariton (SPP) is a transverse magnetic (TM) mode electromagnetic wave, coupled with a surface charge density wave propagating at a metal/dielectric interface [1]. The electric field of an SPP can be confined into the nanoscale region, beyond the diffraction limit of light, therefore, SPPs have significant potential as information carriers in future nanophotonic integrated devices. Fundamental science and technology of utilizing and controlling SPPs with nanostructures created a new field of “plasmonics” [2, 3]. Control of the conversion between propagating light and an SPP, mediated by metallic nanostructures is a critical issue in plasmonics [4–6]. A grating is a well-known tool for such a purpose as it is a 1-dimensional plasmonic crystal [7, 8]. However, at present available experimental methods to investigate the conversion process are very limited in number, due to the high spatial resolution needed, which is smaller than the wavelength of light.
An SPP can be excited by the incidence of electrons onto metal surfaces [8–10]. An electron beam can converge to a small nanometer-sized probe to produce a point source of SPPs at any position on the surface. Recently, several groups have developed light detection components equipped with a scanning electron microscope (SEM) to detect SPP-induced light using a grating [11, 12], which is called the “cathodoluminescence (CL) technique.” High resolution images were obtained, although the light was collected over a wide solid angle of the collection mirror, which makes it difficult to visualize individual SPP modes formed on the grating. We have developed a CL detection system equipped with a 200 keV transmission electron microscope (TEM) [13], and recently added an angular resolution tool. The advantage of using higher voltage is the resulting increases both in the excitation probability of an SPP and in the light emission intensity.
Subsequently, the dispersion relation of an SPP on a flat Ag surface was derived from the angle-resolved measurement [14]. In this paper we present the dispersion curve of an SPP on an Ag-coated grating derived from the angle-resolved measurement by TEM-CL. A grating is a simple 1-dimensional plasmonic crystal with a band gap. We also demonstrate standing SPP waves at the plasmonic band edge energies in the spectrum image.
2. Experimental setting
Figure 1(a) shows a diagram of a 1-dimensional periodic structure with a period of 800 nm on an InP substrate created by electron-beam lithography. Silver layer 200 nm thick was evaporated onto the substrate by thermal evaporation in vacuum. In cross-section each segment in the structure is rectangular, with a width of 320 nm and a height of 50 nm. In the present experiment, we use a TEM combined with a light detection system operated at an accelerating voltage of 200 kV. The electron beam has a diameter of 10 nm with a beam current of 1 nA.
The experimental setup of an angle-resolved measurement is shown in Fig. 1(b). A parabolic mirror is set up above the sample, and its position adjusted so that the electron beam is incident at the focus of the mirror. The incident electron beam excites a surface plasmon polariton, and the SPP propagates into the surroundings as a spherical wave on the sample surface. The SPP induces light emission when it propagates on the periodic structure. The emitted light is then collected by the parabolic mirror and transformed to a parallel ray to be detected by a CCD camera. The polarization direction of light is selected by a polarizer placed between the parabolic mirror and the camera. Here we take the x axis parallel to the axis of parabola and the z axis normal to the sample surface. We define p-polarization to be parallel to the z axis and s-polarization - perpendicular to the x and z axes.
Fig. 1. (a) A schematic drawing of a grating with a rectangular cross section. The period is 800 nm, terrace width is 320 nm and height is 50 nm. (b) Arrangement of a parabolic mirror and sample for an angle-resolved measurement.
For an angle-resolved measurement, a mask with a small hole is inserted between the polarizer and the CCD detector. Solid angle subtended by the hole is 0.1% of the total solid angle of hemisphere at θ=0° and 0.4% at θ=90°. The mask can be moved in a plane perpendicular to the x axis by an X-Y stage. Only a part of the parallel ray can go through a small hole in the mask, and this means that the emission direction of the detected light specified by angles θ and ϕ is selected by the mask position. We move the mask vertically under the condition of ϕ=0° and measure emission spectra successively to find the θ -dependence. Then the direction of the detected light lies on the x-z plane, i.e., the azimuth angle ϕ=0°. The mixing of the polarization by the reflection at the mirror can be neglected for the light emitted in the direction parallel to the x-z plane. This measurement gives an emission intensity pattern of an angle-resolved spectrum (ARS) which shows variation with emission angle θ. The variation in the observed spectrum due to detection efficiency variability is corrected after the measurement, and the θ -dependence due to the solid angle subtended by the hole is also taken into account.
3. Results and discussion
Angle-resolved measurements were performed with the sample in two different experimental arrangements shown in Fig. 2. In the first arrangement, shown in Fig. 2(a), the sample was placed such that the grating direction [the X direction in Fig.1(a)] was parallel to the x axis. Figure 2(b) and 2(2c) show ARS patterns of p- and s-polarized emissions, respectively. During the measurement the electron beam was scanned over an area of 5×5µm 2. In the p-polarized ARS pattern of Fig. 2(b), a bright curved shape was observed, in which anti-crossing occurred at the crossing points of the curves. However, in the s-polarized ARS pattern of Fig. 2(c), the emission vanished and no contrast appeared. These results indicate that the detected emission is purely p-polarized in the setup of Fig. 2(a). Figure 2(e) and 2(f) were the ARS pattern measured in the setup shown in Fig. 2(d), in which the grating direction was perpendicular to the x axis. In the p-polarized ARS pattern of Fig. 2(e), spectra with a broad emission peak were observed over a large angular range. This emission is mainly due to the transition radiation (TR) emitted when an electron passes through a metal surface [15]. It is well known that the TR is polarized in the direction parallel to the emission plane subtended by the emission direction and the surface normal, i.e. parallel to the p-polarization direction. The TR is also seen in Fig. 2(b) as a weak background. In the s-polarized ARS pattern of Fig. 2(f), several curved bright spots appear, which differ in shape from those in Fig. 2(b). The emission in the lower energy region near 1 eV is due to the cathodoluminescence excited in the InP substrate by the deeply entered electrons.
Fig. 2. (a) The arrangement for the angle-resolved measurements with the grating direction (the x direction) parallel to the parabolic axis of the mirror. (b, c) ARS patterns taken with the p- and s-polarized light, respectively, in the (a) arrangement. (d) The arrangement with a grating direction perpendicular to the parabolic axis. (e, f) ARS patterns taken with the p- and s-polarized light, respectively, arranged as in (d).
A surface plasmon polariton on a periodic structure is similar to a Bloch wave. When the wave vector is far from the Brillouin zone boundary, the SPP is plane wave-like, whereas it becomes a standing wave when the wave vector is at the Brillouin zone boundary. The SPP can be converted to a photon to emit light when it propagates along a periodic structure. The condition for the light converted at each element of the periodic structure to interfere is
where kp is the wave vector of the SPP, k ‖ is the surface parallel component of the wave vector of light, and G is a reciprocal lattice vector of the surface structure. In Eq. (2), ESPP and Eph are energies of the SPP and light, respectively. If a basic lattice vector of the periodic structure in the x direction is a, a reciprocal lattice vector can then be expressed as G=na*, where a* is a basic reciprocal lattice vector and n is an integer. The observed ARS pattern can be transformed to a dispersion pattern by changing the emission angle to the wave vector component kx using the relation
where k is a wave vector of the emitted light, h̄ Plank constant divided by 2π and c the velocity of light. After the conversion of the abscissa, the ARS patterns in Fig. 2(b) and 2(f) are transformed into the dispersion patterns in Fig. 3(a) and 3(b), respectively. Then the entire ARS pattern is confined inside the light line.
In the experimental configuration in Fig. 2(a), the reciprocal lattice vector G is parallel to the x axis. Equation (1) imposes a strict condition for the SPP wave vector kp and the surface parallel component of the wave vector of light k ‖. Since the wave vector k of the detected photon lies in the x-z plane (ky=0), only the SPPs propagating in the x direction can couple with the detected p-polarized photon, according to Eq. (1). In other words, k ‖ ‖G‖ x and thus kp ‖ x [see Fig. 3(c)]. Consequently, the SPPs which contribute to the detected light are restricted to those having wave vectors parallel to the x axis.
The modified ARS pattern in Fig. 3(a) reveals the dispersion relation of the SPP propagating in the G direction. Here the dispersion curves for an SPP propagating on a grating with a period of 800 nm are shown by solid lines. The observed bright curved lines in the right half of Fig. 3(a) coincide with the SPP dispersion curves, indicating that the emission originates from an SPP propagating in the G direction. We also note that the band gap opens up at the crossing point of the dispersion curves. This is because standing waves of SPPs with different energies are formed at the zone boundaries, where the dispersion curves cross, as in the band structure of electrons in crystals. The observed bright line deviates from the SPP dispersion curves of the empty lattice approximation near the zone boundaries and reveals a band structure characteristic of a plasmonic crystal. The deviation of energies from the crossing points of the SPP dispersion curves at the zone boundaries were calculated from theory [16], resulting in a good correspondence between observed and calculated values.
In the arrangement in Fig. 2(d), G is perpendicular to the x axis. Thus the SPPs coupled with the detected photons are restricted to wave vectors kp=k ‖+G, where k ‖ is perpendicular to G [Fig. 3(d)]. The SPP with a wave vector parallel to the x axis [the b* axis in Fig. 3(d)] does not couple with light, because its |kp| is always larger than |k| when Eq. (3) holds, while Eq. (1) cannot be satisfied for G=0. Therefore, SPPs propagating in directions inclined to the x axis contribute to light emission as illustrated in Fig. 3(d). A set of dispersion curves for such SPPs along the b* axis are given by the intersection (in the ω-b* plane) of two dispersion cones shifted from the origin by G and-G, as shown in Fig. 3(e).
A splitting of the bright lines along each dispersion curves (white solid lines) appears in the modified ARS pattern in Fig. 3(b). According to the analogy with the band theory of electrons in crystals, two standing wave functions are expected for the SPP modes on each dispersion curve. They can be expressed as follows:
where G 1 is taken to be a* for the n=1 band. These wave functions have a standing wave form in the a* direction and behave as plane waves in the b* direction. The spatial distributions of the two SPP waves are different with respect to the grating structure, so they have different energies. This causes the splitting of the dispersion curve in Fig. 3(b). The split curves at low energies have minima at the point Γ(k=0), i.e. at band edge energies E +=1.67eV and E-=1.44 eV, with a gap of 0.23 eV. There are five bright lines in Fig. 3(b). The middle line does not correspond to an SPP dispersion curve, and the origin of it is not yet clear. One possible explanation is that the line reflects the dispersion of the wave-guide mode of an SPP propagating along the grooves or terrace edges.
Fig. 3. (a, b) Dispersion patterns transformed from Fig. 2(b) and 2(f), respectively. (c, d) The relation between the wave vectors of an SPP and a photon in the emission processes of (a) and (b), respectively. (e) The dispersion plane of an SPP in the 1-dimensional periodic structure. A set of cones are drawn so as that the dispersion cone of the SPP on a flat silver surface is shifted by G (the empty lattice approximation). Solid lines in (a) and (b) are the dispersion curves along the a* and b* directions in (e) for a period of 800 nm.
Emission spectra were successively measured by scanning the electron beam along the x direction in the experimental arrangement of Fig. 2(b), with the mask fixed at the position indicated by an arrow (θ=20°) in Fig. 4(a). Figure 4(b) shows a composite spectrum image created by aligning the observed spectra with respect to the electron beam position, where the vertical axis indicates photon energy. The scanning length corresponds to five periods of the grating. Bright spots align horizontally at the band edge energies of E +=h̄ω+=2.28eV and E-=h̄ω-=2.09 eV, 1/3 of the period apart. The intensity at the higher energy has a maximum at the center of the terrace, whereas at the lower energy there is a minimum at the center. The two intensities are exactly out of phase with each other. These intensities reflect the patterns of the SPP standing waves at the band edges of the plasmonic crystal. These are the SPP modes at kX=a*/2, and the conditions Eqs. (1) and (2) require that the standing waves be composed of two plane wave SPPs with wave vectors kp=kX+G=±3a*/2 (G=+a*,-2a*). There is a difference in emission intensity between the two modes in Fig. 4(b): the lower energy mode shows a stronger intensity contrast than the higher energy mode. This difference can be qualitatively explained from the electric field distribution of the two modes as illustrated in Fig. 4(c). From comparison between Fig. 4(b) and 4(c), the modes indicated by ω +and ω- correspond to the higher and lower energy modes, respectively. In the ω- mode, antinodes of the SPP standing wave are at the step edges, whereas in the ω+ mode the anti-nodes reside on the flat surfaces of the terraces and grooves. Since the SPP is converted to a photon at step edges, it is likely that the emission intensity of the ω- mode is stronger than that of the ω+ mode.
Fig. 4. (a) The ARS pattern for Fig. 2(b). (b) A spectrum image taken with a scanning electron beam across the grating. (c) Illustration of charge distribution and the electric field of the two SPP standing waves.
In the composite spectrum image of Fig. 4(b), strong extended contrast is also observed at the terrace edges of the grating for higher energies (>2.3eV). We have previously observed this emission in the composite spectrum image of a single step [14]. There are two steps in each terrace; and the step on the side facing the parabolic mirror shows strong contrast compared the step on the other side. The emission is distributed over a large angular range and gives the background intensity in the ARS pattern of Fig. 4(a). It does not give a distinct dispersion in the propagation direction of the SPP. These observations suggest that the emission originates from a localized surface plasmon mode at the step edge - an edge plasmon.
It is worth considering whether or not the emission intensity distribution observed in the spectrum image taken at the band edge energy directly represents the spatial distribution of SPP standing wave fields. The emission intensity of light induced by an SPP depends on two processes, 1) the excitation of an SPP and 2) the conversion of an SPP to a photon during the propagation. In contrast with the near-field optical microscope measurements made using an optical fiber, position in the CL spectrum image indicates an excitation point of an SPP by the electron beam. If this excitation probability is large when the electron beam illuminates the anti-nodes of an SPP standing wave, the emission intensity distribution in the spectrum image reveals directly the SPP standing wave. The excitation of SPP is closely related to the electron energy loss process of incident electrons. Recently it was shown by the EELS theory [17] that the CL emission intensity induced by high energy electrons is proportional to the photonic local density of states (LDOS) projected on the electron trajectory. In the SEM-CL experiment [18] the interference fringe contrast was interpreted on the basis of this idea. However, another process which we must consider is the conversion of an SPP to a photon. An incident electron may excite a 2-dimensional spherical wave of an SPP on a terrace. After the excitation, the emission of light is thought to be generated at step edges of terraces when the 2-dimensional wave of the excited SPP propagates along the grating. Since electron illumination also generates transition radiation, we must consider the interference between the SPP-induced emission and the transition radiation [14]. When the wave vector of an SPP is deviated from the Brillouin zone boundary, the SPP wave function is plane wave-like. In this case, the excitation probability of SPP is nearly constant regardless of the electron beam position, so the second process becomes important. However, the wave vector is at the Brillouin zone boundary, the SPP forms a standing wave, and the first process becomes significant.
We propose a model for the formation mechanism of the standing wave pattern in the spectral imaging. When an electron passes through a metal surface, it forms an electric dipole moment together with an image charge in the metal. Oscillation of the electric dipole will produce transition radiation and at the same time act as a point source of an SPP. The SPP with a wave vector at the Brillouin zone boundary is Bragg-reflected by the lattice when propagating along the grating, and then forms a standing wave. If the incident point locates at the anti-nodes of the SPP standing wave, the dipole oscillation excited by the incident electron is in-phase with the reflected SPP wave at the incident point, and thus they synchronize. On the other hand, if the incident point locates at the nodes of the SPP standing wave, the dipole oscillation is out of phase with the reflected SPP wave, and decays quickly. Thus, the emission intensity due to the SPP standing wave becomes strong when the electron beam illuminates at the antinodes, whereas it becomes weak when the electron beam illuminates at the nodes. The same situation also happens in the transition radiation, because it has a similar origin. Consequently the emission intensity distribution directly reflects the SPP standing wave pattern observed at the band edge energy in the spectrum image. This idea is similar to the explanation based on the photonic LDOS [17, 18], in that the excitation of SPP is significant in the formation of the standing wave pattern.
4. Conclusion
The characteristic properties of CL in SPP excitation are as follows: (1) it can generate an SPP point source at any position on the sample surface, regardless of the shape of the surface, (2) all the SPP modes with continuous energies can be excited, and (3) the excited SPP propagates in all directions from the electron incident point as a 2-dimensional spherical wave. The electron beam can be focused to a nanometer-sized spot, thus high spatial resolution can be achieved for CL imaging. By selecting the direction of the SPP-induced light in the angle-resolved measurement, we can investigate an individual SPP mode with a specific energy and propagation direction. Then the dispersion relation of the SPP in the specific direction was derived from the angle-resolved spectrum. The emission intensity distribution in the spectrum image taken with scanning electron beam at the selected detection angle revealed the SPP standing wave-like patterns at the band edge energies. These results indicate that CL is a useful technique for investigating the properties of SPP in plasmonic crystals
Acknowledgements
The present study was supported by Grant-in-Aid for Scientific Research (No.19101004 and 21340080) from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan, and MEXT-Nanotech NP.
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