Introduction

In the recent decades, the extraordinary optical transmission (EOT) through a subwavelength metallic nanostructure [1] has drawn significant attentions due to its capability of controlling the electromagnetic (EM) wave in both the spectral [2,3,4] and spatial domains [5,6,7,8,9,10,11]. Many potential applications, such as bio-sensing [12], optical antennas [13], plasmonic lithography [14], and color filters [15, 16], have thus been proposed.

The enhancement of transmission through various configurations was extensively studied to understand the underlying physics, such as a single aperture/slit [17,18,19,20,21,22,23,24,25], aperture/slit array [2, 4, 26,27,28,29], aperture/slit surrounded by grooves [3, 6, 7, 10, 11, 30,31,32,33], or particular composite structures [8, 9, 34, 35]. Besides the excitation of surface plasmons and the Fabry-Pérot resonance [18], the light funneling effect [17, 23,24,25,26, 36] was believed to be the more fundamental mechanism and could be exploited for the extensive light control [37,38,39]. This effect utilizes the charges accumulated at a discontinuous surface [40, 41] when illuminated and draws the nearby wave energy due to the diffraction [42]. Therefore, the effect is intrinsic to the structure and does not require the plasmonic coupling. A single slit in a film of perfect electric conductor (PEC) was shown to have the capability of enhancing the optical transmission [17]. The coupling between a slit and the grooves surrounded [30], a composite aperture and slit array [34], a metallic slit combined with epsilon-near-zero materials [43], etc., could further enhance the funneling capability.

The study of the funneled energy flow should help us gain more insight into the enhanced light transmission; however, it has been rarely investigated [42]. To characterize the funneling and its profile, the interaction of a p-polarized incident wave with a subwavelength slit in a PEC film is studied. With the finite-difference time-domain (FDTD) simulation, the instant EM fields and Poynting vector show the light funneling and its mechanism. A boundary is defined to indicate the area where the energy flows toward the slit. The time-averaged Poynting vector further verifies the existence of the boundary. As the slit width and thickness determine the transmitted energy density, we will show that the size of the funneling area is positively correlated.

Section II introduces the funneling of light into a subwavelength slit from the FDTD simulation. The boundary of the funneling profile defined by the obtained Poynting vector is also presented. The funneling into a slit of various widths and finite thicknesses are proposed in sections III and IV, respectively. Section V is a summary.

Funneling of Light into a Subwavelength Slit

In this article, we consider a subwavelength slit of width w = 80 nm that is placed in the midst of a PEC film, as shown in Fig. 1. A p-polarized plane wave of wavelength λ = 640 nm is incident; the source plane is one λ away from the film.

Fig. 1
figure 1

Schematic of a subwavelength slit of width w in a PEC film. The film is illuminated by a p-polarized wave and its wavelength is λ = 640 nm. In the simulation, the wave source plane is located at one λ away from the film

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In the two-dimensional FDTD simulation, the cell size of the system of both directions is 2.5 nm and the system size is 30 × 3 μm2. The time step of updating the EM field is 0.001 T (T is the period of the incident wave), and the time step of diagnosing is 0.01 T. The PEC film is simulated with the Drude model with the plasma frequency ωp = 1.0 × 1030 Hz and zero damping coefficient. Perfectly matched layers (PMLs) are employed as the boundary conditions. The electric field amplitude of the incident wave is 1 V/m. The impedance of vacuum is normalized to unity so that the magnetic field amplitude of the incident wave is also normalized to unity.

In our simulation, we let the film thickness h to be large enough, i.e., h = 2 μm, and let the diagnostic interval be from t = 4 T to t = 5 T. In this case, the wave reflection from the exit of the slit is avoided. The light transmission through only the entrance of the slit can be obtained.

The instant EM fields from the simulation are shown in Fig. 2; simulated Ex field at t = 4.22 T is shown in Fig. 2a. At this instant, the tangential electric field of the incident wave at the metal surface (y = 0) is in the positive phase so that it drives the free electrons on the surface to move in the −x direction as the +Jx current and to let Ex = 0 at the surface. The current prevents the incident wave from penetration into the metal and then reflects the light in the direction opposite to that of the incidence. However, the current will not be continuous along the x direction near the slit; positive and negative charges are therefore accumulated at the left and right sides of the slit entrance, respectively, due to the conservation of the charges.

Fig. 2
figure 2

Instant electric field and magnetic field from the simulation. The dashed lines indicate the boundary of the slit, and the dotted curves in c, e, g, i, k, and m indicate the instant funneling boundaries

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These accumulated charges form an electrical dipole [40] to diffract the light into the incident region (y > 0) and into the slit region (y < 0, |x| < w/2) as well. That is, the presence of the slit changes the direction of the incident energy flow in various angles. Since the slit width is much smaller than the wavelength, the diffracted wave is cylindrical in the incident region [44]. Besides the incident and reflected wave, the simulated Ex field above the slit, shown in Fig. 2a, is superposed by that of the diffracted cylindrical wave. In the slit region, the field amplitude measured at y = − 200 nm is 1.367 in the case studied; the enhanced field indicates that the energy is concentrated toward the slit.

We show the Hz field at the same instant in Fig. 2b. The field above the slit is depressed because the magnetic field of the diffracted wave is destructive to those of the incident and reflected wave. Similar to the case of the Ex field, the field amplitude measured at y = − 200 nm is also enhanced to 1.366. The Ey field is shown in Fig. 2c. The accumulated charges at the left and right sides of the slit entrance result in the positive and negative phase of the field close to the entrance, respectively. The field oscillates along the ±x direction, respectively, and quickly decays in the +y direction.

The instant Poynting vector \( \mathbf{S}=\widehat{x}{S}_x+\widehat{y}{S}_y \) is shown in Fig. 2c. Since S = E × H, we have Sy = −ExHz and Sx = EyHz. The phases of the Ex and Hz fields above the slit are positive so that Sy is negative; it indicates the energy flows in the −y direction. The magnitude of Sy decreases in the +y direction since Hz is changing from the positive to the zero phase. The phases of the Ey field at the left and right entrance cause Sx to be positive and negative, respectively. That is, the energy around the slit flows into the slit as the funneling effect. Therefore, the transmission through the entrance is enhanced. Away from the entrance, while Hz in the x direction remains the same positive phase, the oscillation of the Ey field in the x direction changes the sign of Sx so that the energy becomes to flow outward. We find the locations around the slit where Sx = 0 along the x direction at every y to define the boundary of the funneling profile, as the dotted curve shown in Fig. 2c.

When the time increases to t = 4.31 T, the Ex field is shown in Fig. 2d. The field of the positive phase shrinks and moves toward the slit while that of the negative phase is approaching. Figure 2e shows that the Ey field propagates in the ±x direction to have the larger area distribution. While the Hz field has the similar phase distribution to that in Fig. 2b (not shown), the Poynting vector as shown in Fig. 2e is therefore in the +y direction, except that adjacent to the entrance. The boundary of the funneling profile as shown in Fig. 2e becomes wider but lower. The transmitted energy in the slit continues to propagate. At t = 4.47 T, the zero phase of the Ex field is about to arrive at the entrance, as shown in Fig. 2f. The Ey field in Fig. 2g continues to propagate in the ±x direction. The Poynting vector is close to zero above the slit, and the funneling boundary becomes the widest. The energy transmitted at the entrance also becomes smaller.

The Hz field is shown in Fig. 2h as t continues to 4.53 T. The phase near the entrance is zero while that away from it is negative. The Ey field as shown in Fig. 2i indicates the field propagates further. But, no charges are accumulated. The Poynting vector in Fig. 2i is therefore deflected from the slit. The funneling boundary coincides with the entrance and the energy transmitted is about zero.

At t = 4.64 T, the Hz and Ey fields are shown in Fig. 2j, k, respectively. The phase of the Hz field at the entrance becomes negative. At this instant, charges are accumulated at both sides of the entrance. But at this time, the signs of the charges are interchanged in contrast to those shown in Fig. 2c. The Poynting vector shown in Fig. 2k indicates that the energy flows into the slit again, and the funneling boundary becomes larger. At t = 4.72 T, the Ex, Hz, and Ey fields are shown in Fig. 2i–n, respectively. Since it is 0.5 T later than that for the fields shown in Fig. 2a–c, the field distributions are similar but the phases become opposite. The Poynting vector and the funneling boundary are therefore the same as those shown in Fig. 2c. The flow of the energy and the variation of the funneling boundary repeat as discussed when t increases.

The normalized time-averaged Poynting vector in the x and y directions are obtained from

$$ \left\langle {S}_x\left(x,y\right)\right\rangle =\frac{1}{S_0}\frac{1}{T}{\int}_{t_a}^{t_b}{S}_x\left(x,y,z\right)\mathrm{dt}, $$
(1a)
$$ \left\langle {S}_y\left(x,y\right)\right\rangle =\frac{1}{S_0}\frac{1}{T}{\int}_{t_a}^{t_b}{S}_y\left(x,y,z\right)\mathrm{dt}, $$
(1b)

where ta = 4 T, tb = ta + T = 5 T, and S0 = 1/2 is the magnitude of the Poynting vector of the incident wave. The Poynting vector \( \left\langle \mathbf{S}\right\rangle =\widehat{x}\left\langle {S}_x\right\rangle +\widehat{y}\left\langle {S}_y\right\rangle \) and its magnitude contour |〈S〉| = (〈Sx〉2 + 〈Sy〉2)1/2 are shown in Fig. 3. We obtain that the energy above and near the slit is funneled into the slit, while that away from the slit is deflected. Similar to the instant Poynting vector, we find the locations where 〈Sx〉 = 0 along the x direction at every y to define the funneling boundary and show it in Fig. 3 as the dotted curve. Inside the boundary, the energy flows toward the slit. That is, the slit collects the nearby energy so that |〈S〉| in the slit is larger than unity. In our case, we obtain that |〈S〉| is uniform when y < < 0 and is about 1.864.

Fig. 3
figure 3

Time-averaged Poynting vector and its magnitude contour. The dotted curve indicates the funneling boundary

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Funneling Boundary Related to the Slit Width

The normalized energy flux of the wave transmitted through the slit is defined as

$$ P=\frac{1}{w}{\int}_{\frac{-w}{2}}^{\frac{w}{2}}\left\langle {S}_y\left(x,y\right)\right\rangle \mathrm{dx}, $$
(2)

for y < 0. For w < λ/2, 〈Sy〉 in the slit is constant in the x direction since there only exists the fundamental mode [17]. Therefore, we obtain P = 〈Sy〉.

We change the slit width w in the simulation to obtain the normalized energy flux P and show the results in Fig. 4a. It is obtained that, when w decreases, P increases. When w = 80, 160, and 240 nm, P is 1.864, 1.259, and 1.042, respectively. An exponent fitting function indicates that, when w approaches to zero, P approaches to a constant of four. Our simulation results are consistent to the previous studies of Refs. [23,24,25].

Fig. 4
figure 4

a Transmitted energy flux normalized to the slit width (blue circles) and exponential fitting function P(w) = 2.957exp(−0.014w) + 0.961 (red curve). b Funneling boundary obtained when w = 80 nm (blue circles), 160 nm (red triangles), and 240 nm (black diamonds). In b, the lines indicate the boundary of the various slit widths; blue lines: w = 80 nm, red lines: w = 160 nm, black lines: w = 240 nm

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The funneling boundaries for the cases of w = 80, 160, and 240 nm are further shown in Fig. 4b. The width of the boundary remains about 340, 343, and 377 nm, respectively. However, the height of the boundary for the smallest w = 80 nm case is the largest 121 nm while those for the other two cases are 99 and 76 nm, respectively. The estimated funneling areas for the three widths are 3.388 × 104, 2.760 × 104, and 2.462 × 104 nm2. This indicates that, while the narrower slit enhances P, it also enlarges the funneling area.

Light Funneling into a Subwavelength Slit of Finite Thickness

In this section, we consider a subwavelength slit in a PEC film of finite thickness h as shown in Fig. 5. When the incident wave is transmitted though the slit entrance, the wave propagates to the slit exit. It is reflected at the exit. Then, the reflected wave propagates back to entrance and is reflected again. The process repeats as the Fabry-Pérot resonance Refs. [17, 18, 36].

Fig. 5
figure 5

Schematic of a subwavelength slit of width w and thickness h in a PEC film

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The film thickness h determines the phase difference between those of the incident and the reflected wave. The normalized energy flux P that is considered here is the energy of the total wave field. We show P as functions of the film thickness h in Fig. 6a for the cases of w = 80 nm, where h is varied from 200 to 800 nm in a step of 5 nm. The function is periodic, and the period is λ/2 = 320 nm, featured as the Fabry-Pérot resonance.

Fig. 6
figure 6

a Transmitted energy flux normalized to the slit width w = 80 nm as a function of slit thickness. b Funneling boundary obtained when h = 545 nm (gold diamonds) and 385 nm (green triangles). In b, the blue lines indicate the boundary of the slit

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When h = 545 nm, P = 2.575 is at the second peak. The Fabry-Pérot resonance therefore enhances P as compared to the w = 80 nm case discussed in the previous section that only considers the light transmission through the slit entrance. According to Fig. 4a, the increasing ratio of P is 2.575/1.864 − 1 = 38.1%.

The corresponding light funneling boundary for the h = 545 nm case is shown in Fig. 6b. The width and the height become 386 and 114 nm, respectively, and the estimated funneling area is 3.686 × 104 nm2. Although the height is slightly decreased when compared with the w = 80 nm case as what we showed in Fig. 4b, we obtain that the width of the boundary is notably increased; consequently, the area is 3.686 × 104/(3.388 × 104) − 1 = 8.8% increased.

We further choose the case of h = 385 nm to study, where P = 0.829 in this case is at the first bottom of Fig. 6a. The energy flux P in this film thickness is 1 − 0.829/1.864 = 55.5% decreased. The corresponding funneling boundary is shown in Fig. 6b. While the width and the height of the boundary are both significantly decreased to 221 and 89 nm, the funneling area is decreased to 1.541 × 104 nm2; that is, the area has the 1 − 1.541 × 104/(3.388 × 104) = 54.5% decrease.

We thus obtain the correspondence that the change of the funneling area is related to the change of the energy flux P. With the Fabry-Pérot resonance in the cases of the finite film thickness, the funneling area is enlarged with the increase of P, and vice versa. The correspondence suggests that the Fabry-Pérot resonance can enhance or reduce the slit capability of collecting and intensifying the light wave energy in the slit.

Summary

In summary, we have shown the funneling mechanism of light transmission through the entrance of a subwavelength slit by the various instant EM fields from the FDTD simulation. The accumulated charge at the edges of the slit drew the light energy adjacent to the slit so that the transmitted energy was enhanced. We defined the boundary of the funneling profile according to the time-averaged Poynting vector in the x direction to explicitly indicate the area where the energy could be collected by the slit. As the normalized energy flux increased with the decrease of the slit width, we obtained that the funneling area was correspondingly increased. The finite slit thickness was considered to study the Fabry-Pérot resonance. At the thickness where the transmitted energy is enhanced, the funneling area was enlarged, and vice versa. Therefore, the slit capability of collecting the light was regulated by the resonance condition so that the funneling area was altered.