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A cathodoluminescence technique using a 200-keV transmission electron microscope revealed the dispersion patterns of surface plasmon polaritons (SPPs) in a two- dimensional plasmonic crystal with cylindrical hole arrays. The dispersion curves of the SPP modes involving the Γ point were derived from the angle-resolved spectrum patterns. The contrast along the dispersion curves changed with the polarization direction of the emitted light due to the property of the SPP modes. The SPP modes at the Γ point were identified from the photon maps, which mimicked standing SPP waves in a real space. The beam-scan spectral images across the plasmonic crystal edge clearly demonstrated the dependence of the SPP to light conversion efficiency on the emission angle and polarization of light.
©2011 Optical Society of America
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]. SPPs can mutually be converted with light, and metal surfaces with periodic arrangements on a sub-wavelength scale have been utilized as couplers. Similar to electrons in a crystal potential, the dispersion of SPPs on a flat metal surface folds at the Brillouin zone boundaries to form a band structure. An SPP whose energy is within the band gap cannot propagate in the periodic structure. In contrast, the conversion efficiency of an SPP whose energy is at the band edge is enlarged due to the increased density of states. These properties allow devices that manipulate SPPs to be produced, including mirrors [2], wave guides [3], cavities [4], and bio-sensors [5,6]. These types of metal surfaces with periodic structures are called plasmonic crystals, and are recognized as valuable elements in the field of plasmonics.
To design and optimize an SPP−light coupler and other SPP devices using plasmonic crystals, details about the properties of the SPP modes and spatial distribution of the energies of the SPP waves within the band gaps or at the band edges are important. Although optical methods lack the spatial resolution to provide sufficient information about SPPs in plasmonic crystals, high-energy electrons can excite SPPs on metal surfaces, and the dispersion relation has been derived from electron excitation experiments [1,7–9]. An electron beam can converge into a small nanometer-sized probe to produce a point source of SPPs at any positions on the surface. Recently, several groups have developed light detection components equipped with a scanning electron microscope (SEM) [10,11] or a transmission electron microscope (TEM) [9,12–14] to detect SPP-induced light using a grating. This setup is called the cathodoluminescence (CL) technique, and its high spatial resolution power has been applied to studies on the localized surface plasmon around a metal particle [12,14] and a sub-wavelength scale hole in metal films [15] as well as propagation of SPPs [9] and dispersion of SPPs in metal gratings [13].
In this letter, we employ a CL technique to measure SPPs in a two-dimensional plasmonic crystal (2D-PLC) with a periodic array of cylindrical holes. This technique is implemented using a 200-keV transmission electron microscope (TEM) to measure the angle-resolved and polarization-selected spectrum patterns. These patterns are subsequently transformed into the SPP dispersion patterns of the 2D-PLCs because the SPP dispersion patterns show the property of the SPP to light conversion by the 2D-PLCs. The SPP modes at the Γ point are identified from the photon maps, which mimic standing SPP waves at the resonant energies.
2. Experimental setting
Figure 1(a) shows the SEM image of an array of cylindrical holes on an InP substrate created by electron-beam lithography. A square lattice in the array has a periodicity of 600 nm, and the holes have a diameter and depth of 300 nm and 100 nm, respectively. A 200-nm thick silver layer is evaporated onto the substrate by thermal evaporation in a vacuum. In the present experiment, we employ a TEM combined with a light detection system, which operates at an acceleration voltage of 200 kV. The electron beam has a diameter of 10 nm and a beam current of 1 nA. The incident electron beam excites an SPP, which then propagates into the surroundings as a spherical wave on the specimen surface (Fig. 1(b)).
Fig. 1 (a) SEM image of a specimen with a periodic array of cylindrical holes and a period of 600 nm, and a diagram of the cross-section. Hole diameter and depth are 300 nm and 100 nm, respectively. InP substrate is coated by a 200-nm thick silver layer. (b) SPP to photon conversion process on the 2D-plasmonic crystal after excitation by an electron beam. (c) Geometry of the angle resolved measurement with a parabolic mirror. (d) Relation among the wave vectors of an emitting photon and associated SPPs.
SPPs that propagate on a periodic structure induce light emissions. This emitted light is collected by a parabolic mirror, and transformed into a parallel ray, which is then focused onto a slit of a monochrometer by a lens and detected by a CCD detector (Fig. 1(c)). The mirror is 10 mm in width and 5 mm in height. The distance p between the focal point and mirror surface along the x direction in Fig. 1(c) is 2 mm. A single lens is used for transforming an emission image on the mirror into a mask plane, though not depicted in Fig. 1(c) for simplicity. The polarization direction of light is determined by a polarizer, which is placed between the parabolic mirror and detector. Here we define p-polarization as parallel to the emission plane subtended by the axis of parabola and normal to the specimen surface, and s-polarization is normal to the emission plane. For an angle-resolved measurement, a mask with a small hole is inserted between the polarizer and the CCD detector. The mask position selects the emission direction of the detected light, which is specified by angles θ and φ. The mask is moved vertically when the azimuth angle φ = 0° and the emission spectra are successively measured to determine the θ-dependence. This procedure provides an emission intensity distribution of an angle-resolved spectrum (ARS), which shows variations with angle θ and forms an ARS pattern, I (θ,E) [13]. After the measurement, the variations in the observed spectrum due to the variability in the detection efficiency are corrected and the θ-dependence due to the solid angle subtended by the hole is taken into account. Typical integration time for acquiring each spectrum for one pixel is 10 s in the ARS image, and is 1 s in the photon map.
The parabolic curve of the mirror is expressed as using the coordinates shown in Fig. 1(c), where the origin is taken at the focal point of parabola. The emission is acquired with moving the hole along the z axis, so the observed intensity is expressed as a function of the photon energy and position of the hole. The data array at fixed energy is transformed to those at the same energy but with different width in ARS pattern and dispersion pattern as shown in Figs. 2(a) –2(c). The intensity along the z axis is multiplied by a proper factor when converted into the angular space, as
because should be satisfied. The factor in the right side expresses the dependence of the solid angle subtended by the hole on the position of the hole z, and indicates that the solid angle increases by 4 times from z = 2p (θ = 0°) to z = 0 (θ = 90°). Diameter of the hole is 0.5 mm, so the solid angle is 1.2 × 10−2 sr at θ = 0°and 5 × 10−2 sr at θ = 90°.Fig. 2 Conversion of the intensity distribution from (a) the original data acquired by changing the hole position, to (b) an ARS pattern and (c) dispersion pattern. Each square indicates an interval between neighboring data points corresponding to the original ones in (a).
3. Results and discussion
We observed the ARS patterns in arrangements where the [1,0] and [1,1] directions of the specimen are parallel to the parabolic mirror axis. During the measurement, the electron beam is scanned over a 5 × 5 μm2 area. The observed ARS patterns are then transformed into the dispersion patterns, I (k //,E), by changing the emission angle to the wave vector component parallel to the surface, k //, using the relation
where k is the wave vector for the emitted light, ℏ is Plank constant divided by 2π, and c is the velocity of light. In the conversion from the ARS pattern to the dispersion pattern, the intensity should be multiplied by a factor asHowever, this factor amplifies the noise in the angular range near θ = 0°and 90°, so we neglect this factor for convenience. The intensity data array is redistributed inside the light line as shown in Fig. 2(c).Figures 3(a) –3(d) show the dispersion patterns along the Γ−X direction for (a) p-polarized and (b) s-polarized emissions, and along the Γ−M direction for (c) p-polarized and (d) s-polarized emissions. The emission near the Γ point disappears because the hole in the mirror where the electron beam passes is incident to the specimen surface. These patterns were made by folding a right-half pattern to the left.
Fig. 3 Dispersion patterns along the Γ−X direction derived from the ARS patterns for (a) p-polarized and (b) s-polarized emissions, and along the Γ−M direction for (c) p-polarized and (d) s-polarized emissions. Green lines indicate the SPP dispersion curves of the plasmonic crystal in the empty lattice approximation.
An SPP can be converted into a photon to emit light when it propagates on a periodic structure. The condition where the converted light interferes with each element in the periodic structure is expressed as
where is the wave vector of the SPP, and G is the reciprocal lattice vector of the surface structure. In Eq. (4b), andare the energies of the SPP and light, respectively. The relation among these vectors is schematically illustrated in Fig. 1(d). If a and b are the basic lattice vectors of the periodic structure, then the reciprocal lattice vector can be expressed as G = n a* + mb* where a* and b* are the basic reciprocal lattice vectors and n and m are integers. The dispersion plane of an SPP on a plasmonic crystal can be approximated by a set of dispersion planes obtained from a single dispersion plane of an SPP on a flat surface by shifting the SPP dispersion plane by the reciprocal lattice vectors (the empty lattice approximation). The dispersion plane of an SPP on a flat silver surface has a cone-like shape in the E −space, which opens around the E axis [13]. The green lines in Fig. 3 indicate SPP dispersion curves along the Γ−X and Γ−M directions. According to Eqs. (4a) and (4b), an SPP is converted into a photon at the crossing point between the dispersion lines of the SPP and light in the free space. Consequently, light emission occurs along the SPP dispersion curves inside the light line.In the dispersion patterns, bright curved contrasts, which appear along the SPP dispersion curves, depend on the polarization direction of the emitted light. In the dispersion pattern in the Γ−X direction, split line contrasts along the horizontal dispersion curve appear in the s-polarization (Fig. 3(b)), but disappear in the p-polarization (Fig. 3(a)). However, the other line contrasts are elongated in tilted directions from nearly the same energy position (~1.9 eV) at the Γ point. These contrasts appear in the p-polarization (Fig. 3(a)), but disappear in the s-polarization. The dispersion patterns in the Γ−M direction exhibit a similar dependence on the polarization direction; a gap occurs at the closing point of the dispersion curves in the p-polarization (Fig. 3(c)), but not in the s-polarization (Fig. 3(d)).
Here we treat the SPP states at the Γ point on the basis of group theory [16]. An SPP in a plasmonic crystal can be expressed by the Bloch wave function, and can be solved from Maxwell equation under the boundary condition of the periodic surface structure. The wave function of the SPP field is generally expressed as
where n indicates the branch of the Bloch mode. Similar to electrons in a crystal, energy band gaps widen at the Brillouin zone boundary in the SPP band structure due to the Bragg reflection by a periodic lattice. In the case of a surface structure with a square lattice, four dispersion planes emerge from the <1,0> reciprocal lattice points, which intersect at the Γ point (k p = 0) to form the first order branch of the Bloch states. The Bloch wave functions at the Γ point are mainly composed of the four terms, which are associated with the same type of the reciprocal lattice vectors G = a*, b*,-a*,-b* (Fig. 4(a) ), and can be approximated asThe lattice has a four-fold rotational symmetry, C 4, around the origin of the lattice, causing these wave functions to be invariant to a four-fold rotational operation. Group theory indicates four eigenvectors and eigenvalues, although two of them are degenerate. The field distributions of the four eigenstates are shown in Figs. 4(b)–4(e) by calculating where the red and blue colors denote opposite signs. These patterns can also be interpreted as the surface charge distributions because the induced surface charge is proportional to the electric field component normal to the surface. Figure 4(b) corresponds to the total symmetry mode (A mode), which has a maximum at the lattice point and the center of the unit cell. In contrast, the B mode in Fig. 4(c) has minimum at these points. The E1 and E2 modes in Figs. 4(d) and 4(e), respectively, have inversion symmetry around the lattice point and are energetically degenerate.Fig. 4 (a) Four reciprocal lattice points associated with the first-order Bloch states at the Γ point. (b)−(e) represent the Bloch wave patterns of the four SPP modes at the Γ point. Solid circles indicate holes and squares indicate a unit cell.
Figures 5(a) and 5(b) show the ARS patterns of (a) p-polarized and (b) s-polarized emissions near the surface normal direction acquired by tilting the specimen in the [1,0] direction with respect to the incident beam direction. These patterns were made by folding a right-half pattern to the left. These reveal the dispersion pattern of the SPP bands around the Γ point. The contrast disappears in the larger k 10 due to the hole in the mirror; the dispersion line contrasts are actually related to those in Figs. 3(a) and 3(b). The four dispersion lines gather at the Γ point, as illustrated in Fig. 5(c). The line contrast of the highest energy mode appears only in p-polarization, although it disappears at the Γ point (Fig. 3(a)). Therefore, this mode should correspond to the A mode, which does not emit light, due to the symmetrical charge distribution around the hole. The dispersion line contrast of the lowest energy mode in p-polarization intersects with the energy axis at the same position as the one of the line contrasts that extends in the horizontal direction in s-polarization. This suggests that these lines are connected via the E mode at the Γ point.
Fig. 5 ARS patterns near the Γ point taken by (a) p-polarized and (b) s-polarized emissions. (c) Schematic diagram of the dispersion curves along the Γ–X direction. Photon maps taken by the p-polarized emission at peak energies of (d) 2.067 eV, (e) 2.002 eV, (f) 1.944 eV, and by the s-polarized emission at (g) 2.002 eV and (h) 1.944 eV. (i)−(k) show patterns of the time-averaged square modulus of the field amplitude associated with the SPP modes at the Γ point. Red circles indicate holes.
Figures 5(d)–5(h) are photon maps taken with light emitted near the surface normal direction at the peak energies of the Γ point for the p- and s-polarizations where the white circles denote hole positions. It has been reported that a photon map acquired via the cathodoluminescence technique under the proper conditions mimics the field distribution of standing SPP waves or the local electromagnetic density of states (EMLDOS) [13,17–19]. Similar mapping experiment was performed by electron energy loss spectroscopy to reveal the photonic density of states in Si photonic structures [20]. Figures 5(i)–5(k) show patterns of the time-averaged field strength associated with the SPP modes at the Γ point, i.e., the square modulus of the field amplitude in Fig. 4. There is a clear correlation between the photon maps and calculated patterns; (d) corresponds to (i), (e) and (g) correspond to (j), and (f) and (h) correspond to (k). Thus, we can identify the modes at the Γ point, as indicated in Fig. 5(c).
The SPP to photon conversion mechanism in the plasmonic crystal can be explained by the idea that a collective oscillation of an induced surface charge of the SPP forms an oscillating electric dipole around the edge of the hole to emit light. In first order states at the Γ point, the symmetric charge distribution of the A mode does not form an electric dipole around the hole, and consequently, light is not emitted. In contrast, the B mode produces a quadrupole-like charge distribution, and light can be emitted. In the E mode, the charge distribution with inversion symmetry around the hole produces a distinct electric dipole, which emits strong light compared to the other modes. Here we ignore the influence of localized surface plasmons inside the hole because their contribution to the emission is weak when the electron beam is incident normal to the specimen surface.
The SPP contributing to the emission in the Γ−X direction forms a plane wave instead of a standing wave. This occurs when the emission direction is tilted from the surface normal to the [1,0] direction, and the surface parallel component of the wave vector of photon is k // = εa*. In the region of ε>0, the A mode changes to a forward plane wave mode with a positively increasing dispersion, while the E mode changes to a backward plane wave mode with a negatively decreasing dispersion. The dispersion of the former mode is derived from the dispersion cone that emerges from the reciprocal lattice point of G = -a*. According to Eq. (4a), the relation between the wave vectors of the SPP and the emitting photon can be written as k p = k // + a* = (ε + 1)a*. The SPP mode contributing to this emission is the only one to propagate in the [1,0] direction. Hence, the emitted light is purely polarized in the p-direction. Similarly, the dispersion of the latter mode is derived from the dispersion cone emerging from the reciprocal lattice point of G = a*, and the relation of the wave vectors can be expressed as k p = k //-a* = (ε-1)a*. Only the SPP mode propagating in the [−1,0] direction contributes to the emission, and consequently, the emission is purely polarized in the p-direction.
In the split modes with dispersion lines extending in the horizontal direction in Fig. 3(b), the higher energy mode corresponds to the B mode at the Γ point, while the lower energy mode corresponds to one of the E mode. The SPP of these modes contributing to the emission in the Γ−X direction has a wave vector of k p = εa* ± b*. In the region of ε>0, the SPPs of these modes have a plane wave form with a long wavelength, although they still form a standing wave in the [0,1] direction. The standing SPP wave associated with the B mode has maximum at the center and the middle of the hole arrays along the [1,0] direction, but the opposite is associated with the E mode. Therefore, these modes can emit s-polarized light. However, the SPP with a long wavelength produces only a small electric dipole around the hole. Consequently, the p-polarized component of the emission is weak except at the Γ point (Fig. 3(a)).
Figures 6(a) –6(f) show the ARS patterns and beam-scan spectral images taken under the arrangement illustrated in Fig. 1(b). Figures 6(a)–6(c) show images with p-polarized light, while 6(d)–6(f) show s-polarized light. The ARS patterns in Figs. 6(a) and 6(d) were acquired with an electron beam fixed 2 μm outside the crystal edge. The beam-scan spectral images were taken by scanning the electron beam across the plasmonic crystal edge from 6 μm outside to 3 μm inside the crystal. In this arrangement, SPP waves propagating in the [1,0] direction are dominant inside the plasmonic crystal, and consequently, the positive increase in light emission by the forward plane wave modes is stronger than that by the backward plane wave modes. The beam-scan spectral images in Figs. 6(b) and 6(e) were acquired using the emission in the surface normal direction (the Γ point), while 6(c) and 6(f) were acquired for the emission in the direction slightly tilted from the surface normal direction to the [1,0], as indicated by arrows 1 and 2 in Fig. 6(a), respectively.
Fig. 6 (a) ARS pattern taken with a fixed electron beam outside the plasmonic crystal under the arrangement illustrated in Fig. 1(b). (b) and (c) are beam-scan spectral images across the plasmonic crystal edge taken by p-polarized emission in the surface normal direction and a tilted direction indicated by arrows 1 and 2 in (a), respectively. (d)–(f) are the ARS pattern and beam-scan spectral images for the s-polarized emission corresponding to (a)–(c).
In the spectra of Figs. 6(b) and 6(e), which were taken at the Γ point, strong contrasts appear at the peak energies inside the plasmonic crystal. These energies correspond to the photon maps in Fig. 5. An SPP excited by the electron beam outside the plasmonic crystal propagates into the crystal region and is transformed into light. However, an electron beam located on a flat silver surface sufficiently far from the plasmonic crystal generates only transition radiation, which emits p-polarized light with a broad spectrum. Although the contribution of the transition radiation is involved in the p-polarized emission, the emission intensity outside the plasmonic crystal indicates the SPP to light conversion efficiency by the crystal.
The images in Figs. 6(b) and 6(e) clearly indicate that the SPP−light conversion efficiency is highest for the E mode for the emission in the surface normal direction. The periodic intensity varies outside the plasmonic crystal in Fig. 6(b), whereas such a variation is not observed in Fig. 6(e). This periodic variation originates from the interference between the SPP induced emission and the transition radiation [9]. The emission intensity due to the standing SPP wave mode at the Γ point should be equal for p- and s-polarized light. Hence, comparing Figs. 6(b) and 6(e) indicates that p-polarized light is stronger than the s-polarized one due to the contribution of the transition radiation.
Similarly the emission intensities in Figs. 6(c) and 6(f) show the SPP−light conversion efficiency for the emission in the direction tilted from the surface normal direction to the [1,0] direction by about 5°. In this direction, the conversion efficiency for the p-polarized emission is highest for the SPP mode connected to the A mode at the Γ point. On the other hand for the s-polarized emission, the efficiency is highest for the SPP mode connected to the B mode. The difference in photon energy between these modes increases with the tilt angle. Thus, switching the polarization direction changes the photon energy of the emission in this direction. For the emission tilted in the opposite [−1,0] direction, the p-polarized emission due to the SPP mode connected to the E mode becomes dominant, and photon energy of the emission decreases.
4. Conclusion
The dispersion relations of SPPs in 2D-PLC with arrays of cylindrical holes are investigated using a 200-keV TEM combined with the angle-resolved CL system. In this technique, a focused electron beam provides a point source of an SPP at any location on the surface, and photon maps can be measured with a high spatial resolution on the nanometer scale. The photon map acquired at the peak energy at the Γ point mimics a standing SPP wave, which allows the SPP mode to be identified. The dispersion patterns of SPPs propagating in special directions are derived from the ARS pattern measured along select emission directions. After considering the excitation probability of an SPP by a 200-keV electron, the dispersion pattern can yield the efficiency of the SPP to photon conversion. These results provide useful information for designing SPP−photon coupling devices using a plasmonic crystal. The resonant energy and bang gap depend on shape factors of the surface structure, such as the diameter and depth of the hole, but further details will be reported elsewhere.
Acknowledgments
This work has been supported from MEXT of Japan by a Grant-in-Aid for Scientific Research (Nos. 19101004 and 21340080) and from MEXT-Nanotech NP.
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