1. Introduction

It is well known that a metal wedge structure plays an essential role in scattering/diffracting light into free space and in exciting plasmon waves on metal surfaces [1–7]. Possessing a singular point at the corner, a wedge structure offers a broad-range continuum of spatial frequencies. This can help compensate the momentum mismatch between surface-bound plasmons and free-space waves, enabling interactions among the dissimilar wave components. The physical picture of the electromagnetic interaction with a metal wedge, however, is not clearly established, despite evident connection of the subject phenomenon to many related structures. A metal nanoslit, for example, has been commonly viewed as a basic building block of many plasmonic structures [8–11]. The funneling (field concentration) effect that is observable in individual nanoslits is considered one of the crucial factors in extraordinary transmission of nanoslit arrays [12–14]. In this work we consider a semi-infinite 90° metallic wedge as a more elemental structure in understanding various phenomena associated with nanoslits. We show that the essential features of optical interaction with a nanoslit can be derived from the understanding of a metal wedge structure. This investigation focuses on the fundamental interactions that shape the energy flow distribution around the corner.

2. Generation of free-space and surface-bound waves

In this work we have employed a finite-difference time-domain (FDTD) method in analyzing the interaction of light with a metal wedge. The electromagnetic field distributions show rather complicated patterns, and in interpreting the numerical simulation results we take the following approach. The waves produced by a metal wedge in response to an incident wave are categorized into free-space or surface-bound waves (Fig. 1(a)). The free-space waves are further classified into reflection (with planar wavefronts) or diffraction (with non-planar wavefronts) components. Note that in this scheme the total fields are decomposed into planar and non-planar components by the geometry of wavefronts. This geometry-based classification of wave components is intended to help develop a physical picture of the complex electromagnetic phenomena that are analyzed by FDTD simulations.

Figures 1(b), 1(c), 1(d) show a two-dimensional (2D) FDTD simulation result of a Ag wedge with a magnetic polarized planar wave. A planar incident wave was generated at the bottom side of the calculation window, and is expressed as E _in = x̂ E ₀ e ^iky-iωt and H _in = (-)ẑ H ₀ e ^iky-iωt, where k is the propagation constant in free space. A perfectly matched layer (PML) was used for boundary condition. The dielectric constant of Ag was chosen as -17 + i1.15 at 650 nm wavelength [15]. A uniform grid size of 10 nm was used for the entire simulation window. (In some other cases that the dielectric function of metal is chosen such that the surface plasmon penetration depth in metal becomes significantly small, the grid size of simulation needs to be adjusted to proportionally smaller ones. In that case, a sub-grid technique was used, with the primary grid size of 10 nm and the 2nd nested subgrid size of 2.5 nm [16].)

The snapshot images of the distributions of H_z- and E_x-fields (total fields) show the dominance of the incident and reflection wave components (Figs. 1(b), 1(c)). The diffraction or surface plasmon wave components, being of relatively minor intensity, are not clearly resolved in these images. In contrast, the E_y distribution can reveal the diffraction and surface plasmon waves (Fig. 1(d)). This is because the incident and reflection waves, being magnetic polarized, do not contain the E_y-field component. The concentric, circular-shaped wavefronts clearly indicate presence of a boundary (edge) diffraction wave in free space, while surface-bound waves are visible on both perpendicular faces. In order to elucidate the diffraction component of the H_z and E_x fields, the incident and reflection components (planar wavefronts) are subtracted away one by one from the total field. First, upon the elimination of the incident wave, the reflection wave is clearly revealed, being present exclusively on the metal side, that is, in the 4th quadrant of the simulation window (Fig. 1(e)). The fine features of diffraction wave are buried under this strong reflection component, but further subtraction of the planar wavefronts of reflection uncovers circular wavefronts of diffraction (Figs. 1(f), 1(g)).

 

Fig. 1. Electromagnetic interaction of a metal wedge. (a) Schematic illustration of free-space and surface-bound waves generated by a metal wedge for a normally incident planar wave: (from the bottom layer) incident; reflection; boundary diffraction; reflection of boundary diffraction; and surface plasmon waves. (b) to (g), Snapshot images of field distributions around a Ag corner calculated by FDTD for a 650 nm wavelength, magnetic polarized incident light: H_z,total (b); E_x,total (c); E_y (d); H_z,total - H_z,in (e); H_z,total - H_z,in - H_z,refl (f); E_x,total - E_x,in - E_x,refl (g).

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(Here, for the subtraction of semi-infinite planar reflection wavefronts, first the infinite planar reflection wavefronts were calculated assuming an infinite metal surface with a planar wave normally incident to the surface. The right-half part (x > 0) of the planar reflection was taken and used as semi-infinite, uniform planar reflection wavefronts in the subtraction process. This simple geometry-based division/interpretation of the FDTD results implies that the non-uniform distribution of fields originating from the boundary diffraction is taken care of by the remaining, non-planar wavefront part of the total fields.)

The diffraction wave shows an antiphase distribution across the boundary between the planar reflection and circular diffraction regimes, that is, on the border of the 3rd and 4th quadrants. Note that the sharp discontinuity of diffraction wavefronts at the boundary stems from the step function profile assumed for the semi-infinite planar reflection wave. The amplitudes of the antiphase diffraction fields are found to be equal at the boundary and are 0.5 times the planar reflection field (Fig. 2). Therefore the total field (reflection plus diffraction) becomes continuous at the border, satisfying the requisite boundary conditions. It is interesting to note that the field amplitude of the diffraction component is unattenuating along the vertical (-y) direction although the field distribution broadens in the transverse direction. The anisotropic nature of the radiation pattern suggests that the sources of the diffraction wave are not necessarily confined to the corner point itself, because otherwise a circularly symmetric pattern would be expected from a line (point) source. The simulation result rather suggests that an extended distribution of source elements is responsible for the observed diffraction pattern. In order to elucidate the nature of this boundary diffraction wave and its connection to the reflection wave, the electromagnetic response of the horizontal surface is modelled as follows.

 

Fig. 2. Anti-symmetric phase distribution of magnetic field (H_z) of a boundary diffraction wave around a Ag wedge. (a) H_z field profiles scanned along the y-direction at x = 0- (blue curve) or at x = 0 + (red curve). H_z field of planar reflection (x ≥ 0, green curve) is also shown for comparison with boundary diffraction components in their phase and amplitude relations. (b) Snapshot image of H_z field distribution of a boundary diffraction wave. Note the opposite phase distribution across x = 0. A magnetic polarized light (650 nm wavelength) is incident from the bottom side of simulation window.

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3. A model for boundary diffraction

In order to develop a better understanding of how the boundary diffraction wave (with non-planar wavefronts) is generated in conjunction with reflection of a planar incident wave by a metal edge, a model H_z-source of uniform reflection field (H_z,refl) was assumed to be placed on the bottom face of a perfect electric conductor (PEC) wedge, and the resulting emanating waves were analyzed by FDTD (Fig. 3(a)). The wavefronts produced by this semi-infinite uniform H_z source (i.e., a half-plane H_z,refl source) well match the FDTD simulation result of a metal wedge in most regions (See the 3rd and 4th quadrants of Fig. 3(a), and compare with those of Fig. 1(e)). In the 2nd quadrant, a discrepancy appears: the circular wavefronts partially remain strong around the vertical surface (Fig. 1(e)). This local, strong field around the sidewall is caused by the interaction of the diffraction wave with the sidewall as will be discussed below.

Placing a uniform H_z source on a metal edge is equivalent to assuming a uniform surface current on the metal surface. The implication of employing this model source can be understood as follows. For a planar wave normally incident to a metal surface, the metal responds by inducing a surface current, J = n × H. Here H refers to the total magnetic field vector on metal surface and n is the unit normal vector to that face. The induction of a surface current results in generation of a reflection wave, whose electric and magnetic fields are set such that they satisfy the boundary conditions on the metal surface. The reflection wave generated at the metal surface (y = 0) is related to the incident wave as follows: E_x,refl = rE_x,in and H_z,refl = -rH_z,in. Here r is the reflection coefficient of air/metal interface for normal incidence. For most metals |r| ~ 1, and the phase angle of r is close to 180 degrees, e.g., 180 or 153 degrees for perfect conductor or Ag at 650 nm, respectively. The surface current that corresponds to the planar reflection wavefronts can then be expressed as J = (1-r)H_z,in≅2H_z,in (The electromagnetic interaction with a metal wedge produces a complex distribution of fields. The induced surface current also takes a non-uniform distribution around the corner [17 , 18]. For the exact fields, the edge diffraction can be evaluated by taking the physical optics approach with a modified surface current [18].)

According to the geometry-based classification scheme taken in this work, the total magnetic field can be decomposed into two parts, one corresponding to the planar reflection wave and the other one to non-planar wavefronts (diffraction and surface-bound waves). The uniform surface current component by the planar reflection field is separated out from the total surface current, and is used as a model source in studying the response of the metal edge to an incident wave. As shown in Fig. 3(a), it is found that the half-plane uniform H_z source placed on the front surface of a metal wedge produces the non-planar diffraction wavefronts as well as the planar reflection wavefronts.

When a planar incident wave hits the bottom surface of a metal wedge, it produces planar reflection and non-planar diffraction (and surface plasmons) as discussed above. Note that these reflection/diffraction wave components propagate mostly in the 3rd and 4th quadrants of the simulation window. The 2nd quadrant region is affected by the presence of another diffraction wave component.

 

Fig. 3. A model for boundary diffraction. (a) H_z,total field distribution calculated by FDTD for a half-plane H_z source placed on the bottom face of a perfect electric conductor wedge. (b) H_z,total. (c) E_y. (d) (H_z,total - H_z,pl) field distributions calculated for a half-plane H_z source placed on the border of the 2nd and 3rd quadrants (x < 0, y = 0) in free space.

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When an infinite-planar incident wave enters into the 2nd quadrant (x < 0 and y > 0), it suddenly becomes of semi-infinite extent confined to the air side of the space. Similar to the case of an semi-infinite uniform H_z source placed on the bottom surface of the metal wedge for reflection, this propagating semi-infinite wave is expected to generate/bring non-planar diffraction components along with a planar one and to interact with the vertical sidewall of metal.

The nature of this boundary diffraction wave generation (by the propagating planar wave at the air side) and its interaction with the vertical sidewall of metal are further studied as follows. A half-plane uniform H_z soft (transparent) source is placed on the border of the 2nd and 3rd quadrants (x < 0, y = 0) in free space, and FDTD analysis is carried out for the emanating waves [Figs. 3(b), 3(c), 3(d) for the total H_z, E_y, and (H_z,total - H_z,pl) field distributions, respectively]. Comparing with the corresponding fields in Figs. 1(e), 1(d), 1(g), it is clear that the circular-shaped, anti-symmetric diffraction wavefronts originate from a semi-infinite uniform H_z source. It is important to note that unlike the case of reflection by the bottom surface, there is no surface current involved in describing the diffraction wave generation by a propagating semi-infinite planar wave in free space.

A semi-infinite planar source (both cases of Figs. 3(a) and 3(b)) is shown to produce composite, mixed-symmetry wavefronts, and this characteristic distribution of fields can be understood from the reasoning based on the Huygens-Fresnel principle: each point on a wavefront serves as a source of wavelets, and superposition of all the wavelets from the primary wavefront results in formation of next wavefronts [19]. Imagine an infinite plane source of H_z in free space. The wavefronts emanating from the source would be perfectly planar without any circular fringe component. This is because the normal component of electric field (E_y) at any point would be cancelled out by anti-symmetric contributions originating from the opposing half-sections of the infinite plane source. As such the resulting field has only the horizontal component (E_x). When the H_z source becomes of semi-infinite extent, cancellation of the normal component would not be perfect, especially around the edge region. The fringe (diffraction) field on the source-free side originates from the source side, that is, where the semi-infinite H_z is placed. By contrast, the fields on the source side lack contributions from the opposite side (source-free side).

 

Fig. 4. (a) |H_z| distributions scanned along the x-direction at y = -3λ (λ = 650 nm): for a half-plane H_z source placed on the bottom face of a perfect electric conductor wedge (red); for a perfect electric conductor half-plane sheet placed at x < 0 and y = 0 (blue) with a magnetic polarized light normally incident to the PEC sheet. These FDTD calculated simulation results (red and blue) are compared with the analytical result (green) (after adjusting phases) calculated from the Sommerfeld half-plane problem, which concerns the same geometry as the PEC sheet case (blue) [1]. A good agreement is observed among the three results (especially between blue and green curves). (b) Snapshot image of H_z distribution for the case of a half-plane H_z source placed on the bottom face of a perfect electric conductor corner.

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Therefore the total field on the source side will be less the amount of the missing contribution than the otherwise perfect planar wavefronts. This symmetry-based reasoning provides an explanation on how a semi-infinite plane source produces such characteristic anti-symmetric circular wavefronts for H_z (and E_x) in conjunction with the semi-infinite planar wavefronts. We note that the field distribution of the waves (planar + circular) emanating from a half-plane source in free space takes the form of complementary error function, or equivalently of the Fresnel integral, an analytical expression well established in association with the knife-edge diffraction problem [20–23] (Fig. 4). [It is interesting to note in Fig. 3 that the total E_y field (Fig. 3(c)) shows an isotropic distribution whereas the total H_z field (Fig. 3(b)) is anisotropic. An analytical expression can be derived for the total fields originating from a half-plane uniform H_z source placed in free space. The half-plane source is considered as a semi-infinite array of densely-spaced line sources of infinite length (point sources when viewed on the x-y domain). The total fields are then calculated by integrating the contributions from all source elements. The expression for the E_y field is found to take the form of the complex complementary error function (or the complex Fresnel integral) and the field strength is a function of the distance from the edge (corner point) of the semi-infinite source. This functional dependence explains the isotropic nature (circular symmetry) of the E_y field.]

4. Interference of boundary diffraction

Now we move on to the region around the vertical sidewall (the 2nd quadrant). The planar wavefronts of the total H_z field shown in Fig. 1(b) significantly bend in the region close to the sidewall. The (H_z,total - H_z,in - H_z,refl) and (E_x,total - E_x,in - E_x,refl) field distributions shown in Figs. 1(f), 1(g) also confirm strong presence of circular wavefronts localized in the same region. These seemingly peculiar distributions require a proper understanding. The incident wave entering into the 2nd quadrant can be modeled with a half-plane H_z source that is placed on the border with the 3rd quadrant (x < 0, y = 0). This H_z source is expected to develop fringe (diffraction) fields of the same nature as those of a half-plane H_z source placed in free space (Fig. 3(d)) discussed above. The boundary diffraction fields that are associated with the propagating semi-infinite planar wavefronts (Fig. 5(a)) and that fall onto the metal side (the 1st quadrant) are glancing incident in the region near/along the vertical sidewall and reflect back to the air side (the 2nd quadrant) (Fig. 5(b)).

 

Fig. 5. Schematics of boundary diffraction wavefront generation by a propagating semi-infinite planar H_z source in free space and its interaction with the vertical sidewall of a metal wedge. (a) A half-plane H_z source placed in free space (an array of brown dots with crosses inside: placed on the border of the 2nd and 3rd quadrants, x < 0, y = 0) generates anitiphase non-planar boundary diffraction wavefronts (red and blue solid curves) besides the planar wavefronts (brown solid lines). (b) When a metal wedge is introduced in the 1st quadrant (green block: x > 0, y > 0), the non-planar diffraction wave components in the 1st quadrant (red dotted curves) reflect back to the 2nd quadrant (red solid curves) and are superposed to the diffraction components propagating in that region (blue solid curves). The ‘r’ indicates the reflection coefficient at the vertical sidewall. The red arrows denote the propagation direction of wavefronts.

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The effect of this reflection at the vertical sidewall would be most significant along the glancing angle direction, because it corresponds to a shortest possible path that a wavelet can take in order to reach an observation point located near the vertical surface. For the case of an interface of a real metal with air, the reflection coefficient becomes -1 for glancing angle incidence (See the analysis below).

For a magnetic polarized light, the reflection coefficient of air/metal interface for oblique incidence is given as [2]:

For air/Ag interface at glancing angle incidence: ε_M = -17 + i1.15; θ_i → 90°, $r\to \frac{-\sqrt{{\epsilon }_M-1}}{+\sqrt{{\epsilon }_M-1}}=-1$

For air/PEC interface at glancing angle incidence: ε_M → i∞; θ_i → 90°,

Since the reflection angles of boundary diffraction for glancing incidence are close to (but not equal to) 90° in reality, $\mid \sqrt{{\epsilon }_M}\cos {\theta }_i\mid \to \infty$ and $r\to \frac{\sqrt{{\epsilon }_M}\cos {\theta }_i}{\sqrt{{\epsilon }_M}\cos {\theta }_i}=1$.

The phase-inverted reflection component is then added to the direct propagation component on the air side (the 2nd quadrant). Since the original diffraction fields existing on the air/metal sides are antiphase to each other with equal amplitudes (blue and red circular curves in Fig. 5(a)), the reflection/superposition would result in (-1) times the original field of the planar component on the air side. (Recall that the diffraction field amplitude at the boundary is 0.5 times the original planar component.) When these total diffraction fields (dark blue circular curve in Fig. 5(b)) are added to the planar wavefronts on the air side (dark red straight line in Fig. 5(b)), a field depletion region develops near the sidewall. Note that the surface plasmons excited at the corner continue to propagate upward unaffected by the presence of the diffraction fields. For the particular case simulated in Fig. 1, that is, Ag at 650 nm wavelength, the surface plasmon propagation constant is similar to that of free-space wave.

 

Fig. 6. Snapshot images of magnetic field (H_z) distributions around a metal wedge. Different dielectric constant values are assumed for FDTD calculation: (a) ε_M = -5 + i1.15; (b) ε_M = -1.5 + i1.15. A magnetic polarized light (650 nm wavelength) is normally incident from the bottom side of the simulation window. Presence of field-depletion region is clearly observed near the vertical sidewall of metal, although hindered by coexistence of surface plasmon fields.

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The field depletion region that is expected to develop near the sidewall is not clearly visible in this simulation, because of the strong presence of surface plasmon fields in the adjacent region. When a different dielectric constant is assumed for metal such that the surface plasmon propagation constant significantly differs from the diffraction wave’s, a field depletion region is clearly observed (Fig. 6). In the case of perfect electric conductor, the reflection coefficient at glancing angle incidence becomes + 1, and reflection/superposition results in perfect cancellation of the antiphase diffraction fields on the air side. This implies the total fields near the sidewall would have only planar wavefronts without any local depletion of fields. The physical model of boundary diffraction discussed above provides a quantitative understanding of mirror imaging on metal surface [2].

We note that the field-depletion mechanism discussed above in conjunction with a metal corner can also explain the phenomenon commonly reported with metal nanoslit structures. It is well known that the radiation pattern of a nanoslit formed in plasmonic metal has a field depletion region in far fields along the glancing angle directions [10,20–22]. In contrast, for the case of a nanoslit formed in perfect electric conductor (or real metal operating in the longer wavelength range, in which the metal behaves like perfect conductor), the radiation pattern shows a uniform angular distribution over the entire span (π radian) without any depletion. The field-depletion in the case of a plamonic nanoslit can be understood as follows. As a light from a nanoslit radiates away (that is, as the radius of curvature of circular wavefronts increases), the wavefronts would look locally more planar, normal to metal surface. The interaction of the quasi-planar wavefronts with the metal surface in the glancing angle direction then becomes similar to that of semi-infinite planar wavefronts propagating along the sidewall of a metal corner discussed above: incident fields are cancelled out near the metal surface via phase-inverted reflection of boundary diffraction waves.

5. Energy flow distribution

Next we elucidate how the various wave components interplay in the near- to far-field regions, shaping the energy flow distribution around the corner. Figures 7(a), 7(b), 7(c), 7(d) show the Poynting vector distributions (|〈S〉|, 〈S_x〉, and 〈S_y〉) calculated for a metal (Ag or perfect electric conductor) wedge. Note that the energy flow in most regions is dominantly along the y-direction, that is, the <S_y> component is an order of magnitude stronger than the <S_x> component. The 2nd quadrant corresponds to the region where primarily the planar incident wave and the circular diffraction wave interplay in far fields. Note the contrasting difference between PEC and Ag cases (Figs. 6(a), 6(b)): the Ag wedge shows virtually-zero energy flow in the region along the sidewall whereas the PEC wedge does not reveal such depletion. This is due to the fact that in Ag case the phase-inverting reflection of a boundary diffraction wave into its anti-symmetric counterpart in the 2nd quadrant results in cancellation of incident fields along the side wall. In PEC case, the sidewall reflection does not induce any phase change and the anti-symmetric diffraction wavefronts cancel each other upon reflection. Therefore the incident wavefronts remain unaffected by reflection of boundary diffraction components. The energy flow carried by surface plasmons is clearly visible in the near field region along the sidewall of Ag. The quasi-parabolic fringe pattern observed in the intermediate- to far-field region indicates interference of two propagating waves, one with planar and the other with circular wavefronts [See Fig. 5 and reference 23]. In the 3rd quadrant, the planar incident and circular diffraction waves are counter-propagating, therefore interference of the two waves produces a fringe pattern, distinctly different from that in the 2nd quadrant. Because of the contra directionality, the fringe spacing is significantly smaller than the co-propagating case. The 4th quadrant reveals much complex patterns. This is because two extra wave components are present in this region, the reflection wave and the surface plasmons besides the incident and diffraction waves.

 

Fig. 7. Energy flow distributions calculated by FDTD. (a) |<S>| for a perfect electric conductor wedge. (b) |<S>| for a Ag corner. (c) |<S_x>| (10 times magnified) for a Ag corner. (d) |<S_y>| for a Ag corner. (e) |<S_x>| scanned along the negative y-direction at x = 0. (f) |<S_y>| scanned along the negative x-direction at y = 0.

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6. Funneling effect

Now we consider how a metal wedge structure would possibly induce a funneling effect on a planar wave incident to the wedge. Figure 7(e) shows the Poynting vector (〈S_x〉) distribution scanned along the border of the 3rd and 4th quadrants (y < 0; x = 0). The energy flow in the horizontal direction shows an oscillatory behavior: it starts with a relatively large negative value at the surface (y = 0) and then alternates between the positive and negative regimes with an oscillation period of λ/2. The amplitudes of the negative peaks are found to be persistently greater than the neighboring positive peaks. This indicates a net energy flow into the negative x-direction (∫_-∞ ⁰〈S_x〉dy <0), that is, a transfer of energy from the 4th quadrant to the 3rd quadrant crossing the border (x = 0; y < 0). Figure 7(f) shows the Poynting vector (〈S_y〉) distribution scanned along the border of the 2nd and 3rd quadrants (x < 0; y = 0). The energy flow along the positive y-direction shows relatively symmetric undulation around the incident flux level, except for the region very near to the corner (-0.1λ < x < 0). The upward energy flow crossing the entire border (x < 0; y = 0) is found to be slightly smaller than the incident energy flow, that is, ∫_-∞ ⁰〈S_y〉dx < ∫_-∞ ⁰〈S_y〉_in dx. In a local view, however, the region near the corner exhibits a net positive flow of energy beyond the level of the planar incident flux (directed to the positive y-direction).

In reviewing the two energy flow distributions (the horizontal and vertical scans around the corner) it becomes evident that a funneling phenomenon does occur, but only in the near-field region around the corner. Note that in Fig. 7 the funneling region is not clearly resolved, but when magnified, the corner region reveals funneling (Fig. 8). The diffraction wave component that is being produced in conjunction with reflection from the front surface causes undulation of energy flow. The near-field region around the corner, where the funneling phenomenon is confined to, is estimated to be approximately 50-nm and 80-nm wide on the front and side wall surfaces, respectively.

In order to better understand the funneling process around the corner, the Poynting vector 〈S_x〉 is decomposed into two parts for the case of PEC: one contributed by boundary diffraction wave itself and the other by cross-coupling of boundary diffraction wave and planar incident and reflection waves.

Here we note that the planar incident and reflection waves do not contain E_y component. The E_y fields are contributed by boundary diffraction waves and they converge to surface plasmon fields on metal surface in the case of a Ag corner. The amplitude and phase plots of the field components (E_y,diff, H_z,diff, and H_z,pl) reveal that the cross-coupling term 〈S_x〉_cross is dominant over the self-diffraction term 〈S_x〉_diff (Figs. 8(e), 8(f), 8(g), 8(h)). This is mainly due to the fact that the H_z field on the metal side is supplemented by an in-phase reflection wave and the total planar wave component becomes much stronger than the diffraction component of H_z.

 

Fig. 8. Close-up view of energy flow around a metal wedge. (a) <S_x> (10 times magnified) around a Ag corner. (b) <S_y> around a Ag corner. (c), (d), Energy flow vectors: distribution of Poyning vectors around a Ag corner (c) or a PEC corner (d). Note that for both Ag and PEC corner cases the energy flow in the near field region of front surface (x < ~60 nm; y = 0 to 20 nm) is directed left, being funneled around the corner. (e), (f), Poynting vector <S_x>, and its components <S_x>_diff and <S_x>_cross around a PEC corner: scanned along the x-direction at y = 0 (e) or along the y-direction at x = 0 (f). Amplitudes (g) and phases (h) of E_y,diff, H_z,diff and H_z,pl fields scanned along the x-direction at y = 0.

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Considering the phase distributions of diffraction fields around the corner (symmetric E_y and anti-symmetric H_z), a diffraction wave itself would radiate the energy away from the corner. Interplay of boundary diffraction fields (E_y with symmetric, linearly rising phases) and planar wavefront fields (H_z with symmetric constant phase), however, enables funneling around the corner. Note that the diffraction field E_y at x = 0 is almost phase-inverted compared to the planar wave field H_z. Due to this phase relationship and the linearly rising phase profile of E_y, cross-coupling of the two fields yields forward energy flow (S_x < 0 and S_y > 0) around the corner in the narrow region of front metal surface (Fig. 8: x < ~100 nm). Note that inside the Ag metal corner Poynting vectors of significant strength appear with orientations opposite to that in the air side (Fig. 8(c): on the horizontal surface of the wedge). The tangential components of Poynting vectors associated with surface plasmons are opposite to each other inside and outside the metal surface, and this is due to the fact the dielectric constant of metal is largely negative real [24–27]. Note that a PEC corner (Fig. 7(e), (f), Fig. 8(d)) shows the same or even slightly stronger funneling than the plasmonic metal case. This indicates that surface plasmons are not essential to induce a funneling phenomenon [28,29]. Overall this analysis elucidates that a metal corner possesses an intrinsic capability of funneling light, which is most commonly observed in a metal nanoslit structure. The energy flow distributions around a corner further suggest that when two corners are brought together to form a slit, the funneling effect of the slit would show characteristic dependence on slit width (Fig. 7(e)). In order to maximize the funneling effect (the transmission normalized by slit width), slit width needs to be smaller than ~160 nm. For larger widths, the amount of light coupled into a slit will show an oscillatory behavior (with a period equal to λ) as a function of slit width [30–32].

7. Field depletion

A silver wedge structure was formed by depositing a thick Ag layer on a cleaved edge structure of a GaAs wafer. A (001)-oriented GaAs wafer (356-μm-thick, single-side polished) was cleaved along the (110) cleavage plane by a razor blade. This process produces a 90-degree wedge structure defined by two crystal planes (001) and (110). The cleaved wafer (1 cm × 2 cm size) was introduced into a vacuum deposition chamber (base pressure of 10^-6 Torr), and a 200-nm-thick Ag layer was deposited on the edge facets by thermally evaporating Ag (4N8 purity: Alfa Aesar). The sample face, the (001) plane, was 45-degree tilted to the evaporation flux so that there be no shadow effect on the corner area and both facets receive the same amount of Ag flux during deposition.

Optical intensity profiles around the Ag corner structure were measured by scanning a nanoapertured probe (Veeco Aurora NSOM probe 1720-00: 100-nm-thick Al coated; 80-nm aperture diameter). A magnetic polarized He-Ne laser beam (633 nm wavelength, 1-mm beam diameter) was incident to the corner, normal to the (001) surface. The scanning probe was tilted 30 degree off from the vertical sidewall, the (110) plane, and was scanned from the corner (0 μm) to 6 μm up in the vertical (y) direction, and from the sidewall (0 μm) to -6 μm in the horizontal (x) direction. The step size of scan was 50 nm and 125 nm in the horizontal and vertical directions, respectively.

Figure 9 shows an experimental data of the energy flow distribution around the Ag corner measured by a scanning probe technique. The probe axis is tilted 30 degree off from the vertical direction (Figs. 9(a), 9(b)), and therefore the measured intensity more closely represents the vertical component of Poynting vector (<S_y>) than the horizontal component (<S_x>). The measured fringe pattern (Figs. 9(c), 9(d)) well matches the simulation result (Figs. 7(b), 7(d)).

A well-defined ‘dark’ (virtually-zero energy flow) region is observed near and along the sidewall. The sharply-rising intensity profile near the metal surface corresponds to the surface plasmon power propagating along the positive y-direction. The energy flux incident to this ‘dark’ region parts into the surface plasmon and free-space fluxes. The transverse components of Poynting vectors in this region take opposite signs, splitting the incident flux and diverting them away from the region, one toward the sidewall (with a positive value of <S_x>) and the other into free space (with a negative value of <S_x>) (Figs. 7(c), 7(d)). The fringe locations are determined by interference of planar incident and circular diffraction wavefronts. The <S_y> profile scanned at y = 0 along the negative x-direction shows a fringe spacing well-matching the free-space wavelength of light (633 nm) (Figs. 9(e), 9(f), 9(g)).

 

Fig. 9. Experimentally measured energy flow distribution around a Ag wedge. (a) Schematic of scanning probe measurement. (b) Optical micrograph of a scanning probe aligned to the edge of a Ag wedge. (c) Color map of measured energy flow distribution (primarily <S_y> component). (d) Measured scan profiles. The coordinate (0,0) corresponds to the corner point. Note the depleted-energy-flow region [dark blue in (c)] along the glancing angle direction near the sidewall (x = 0). (e), (f), (g), Energy flow distributions scanned along the x-direction in the 2nd quadrant: comparison between the experimentally measured data (blue) and a FDTD simulation result calculated at 633 nm wavelength (red). The coordinate (x = 0, y = 0) corresponds to the corner point.

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8. Conclusion

The physical nature of electromagnetic interactions of a planar incident wave with a metal wedge is elucidated in this theoretical and experimental study. The boundary diffraction wave is found to play a crucial role in shaping the energy flow distribution around the corner. It is found that a funneling effect is intrinsic to a metal (perfect conductor or real metal) corner structure and is confined to the near field region of a corner. This study confirms that surface plasmons do not play an essential role in funneling at slit edges. A metal wedge’s intrinsic capability to redirect the incident energy flow provides a fundamental basis for the commonly observed phenomenon of concentrating an incident energy flow into a narrow channel region. Scanning probe measurement of energy flow distribution confirms presence of a field depletion region along the glancing angle direction of a sidewall. The zero energy-flow region is caused by destructive interference of boundary diffraction waves near a plasmonic metal surface. The same mechanism provides an explanation of the characteristic radiation pattern (field depletion along the glancing angle direction) of a metal nanoslit structure. A physical understanding of various electromagnetic phenomena associated with a metal wedge structure confirms rich potential of the simple structure as an elemental building block of complex metal nanostructures