1. Introduction

Over the past decades, the extraordinary optical transmission (EOT) through subwavelength metallic structures has been studied extensively to drive the emerging field of nanophotonics [1,2]. Generally, for the periodic aperture arrays there are two different types of mechanisms responsible for the EOT occurrence: tunneling surface plasmon polaritons (SPPs) through up-down interfaces and waveguiding SPPs resonance mode inside the apertures, and which effect dominated depends on the metal thickness [3–6]. Recently, the auxiliary-cavity assisted optical transmission has been also investigated extensively by some researchers. For example, Ebbesen et al. developed a single subwavelength metal slit with periodic corrugations on its illumination surface [7], being so-called bull’s eye structure, and consequently obtained the transmission enhancement through the interference between the incident light and the SPPs generated by corrugations [8–10]. Physically, this type of lateral setting grooves on the sides of the slit are utilized to collect more external light incidence and pull them into the slit. Different from the parallel connection method, Min et al. fabricated a longitudinal cascade structure by placing a compact microcavity close above the single subwavelength slit on a gold film, and the optical transmission through the slit was observed to enhance by a factor of 2 [11]. Numerical calculations revealed that SPPs standing waves in the microcavity could interact with the slit mode to provide the source energy for the transmission enhancement.

More interestingly, through designing a single step within a subwavelength metal slit, Lockyear et al. firstly found that the transmission resonant wavelength could be changed substantially [12]. The subsequent investigations tried to describe characteristics of such internal stubbing structures associated with the single slit and the periodic arrays [13–15]. Up to now, however, most of them emphasize on the wavelength filtering function for the plasmonic devices. In addition, during these studies the spatial arrangement of stubs in the subwavelength slit also was somewhat arbitrary, so that the obtained transmission intensities usually decreased in comparison with the straight structure [13]. Actually, in the cases of EOT through Fabry-Perot (F-P) resonance, the thickness of subwavelength slit structure is critical because of the higher-order modes and the destructive interference. On the other hand, the long-distance waveguiding SPPs is usually provided by storing vast optical energy in electron oscillations, which inevitably leads to high optical loss. Therefore, the efficient controlling of SPPs along a given slit is becoming a challenge.

In this paper, we demonstrate that by spatial periodic breaking a single subwavelength metallic slit into the minimum unit chains with embedded air grooves, a further boost or depression of the optical transmission can be achieved when compared with the corresponding smooth slit. On the basis of the finite-difference time-domain (FDTD) simulation method, we show how such an interesting modification occurs with varying the groove parameters, and an additional mechanism in terms of fundamental plasmon resonances coupling is suggested, leading to the SPPs hopping energy transfer along the slit. This approach naturally extends manipulative capabilities of the slit structure for the subwavelength optoelectronic devices.

2. Simulation results and discussion

A scheme of a spatially quantized single metallic slit is sketched in Fig. 1 , composed of the symmetric distribution of embeddable grooves in inner walls with the air surroundings. The geometric parameters as well as the coordinate axes are employed to define the structure, which can be denoted as follows: w for the width of the central slit, t for the length of metal ridges, L for the total thickness of the slit, N for the number of the groove pairs, h and d for the height and depth of the identical grooves, respectively. Only an incidence of TM polarized plane wave at the wavelength of λ = 1.0 μm is considered here, implying that the magnetic field vector is parallel to the z axis. The metal is chosen to be silver with the dielectric constant at this wavelength as ε = −48.8 + 3.16i.

 

Fig. 1 Schematic plot of a spatially quantized single slit structure in a metallic film, associated with identical air grooves embedded in the interior walls.

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First, we investigate the optical transmission properties of a smooth single slit, which is essentially important to explore the spatially quantized slit structure. For a given smooth slit with width of w = 100 nm, the calculated transmittance as a function of the film thickness is displayed in Fig. 2(a) . In this case the variation of transmittance presents a periodic oscillation tendency due to the F-P interference [16,17], but the peak amplitudes become decreased with the gradual increase of the slit thickness. In particular, the first peak transmission occurring at L = 280 nm has the largest magnitude, which corresponds to the fundamental waveguided resonance in the slit [12,15]. Physically, this well known fact suggests an optimal condition for the extraordinary transmission of the incident light, associated with the best mode quality and lowest loss. Of course, this kind of optimal thickness becomes different depending on the slit width [18,19]. Anyway, if we select this critical thickness as a unit to spatially quantize the smooth slit by embedded grooves, or when some slit units with the critical thickness are cascaded together with a between interval of the air grooves, what will happen for the corresponding optical transmission?

 

Fig. 2 (a) Transmittance of a smooth single slit (w = 100 nm) as a function of the film thickness. (b) Transmittance of the N = 1 spatially quantized slit structure with varying the groove depth for several different h. The slit unit is [w, t] = [100, 280] nm. (c) Peak transmittances of the N = 1 spatially quantized slit structure with varying the groove height for several different w.

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Next, we study the spatial quantization effect on the slit transmission properties with FDTD simulations. When the unit length is chosen as t = 280 nm for the slit of w = 100 nm, Fig. 2(b) shows the calculated transmittances of the slit-groove-slit sandwich-like structure (N = 1) as a function of the groove depth for several different h. From this plot, we find that for a given h the enhanced transmission tends to appear periodically with increasing the groove depth, alike the observed profile in Fig. 2(a). At this time, however, the optical transmission seems to be depressed violently, as stated in reference [13], which implies an efficient light blocking function of the composite structure. Another interesting feature is that with increasing the groove height, either the enhancement or the suppression of the optical transmission becomes pronounced, resulting in a high peak-dip ratio. For instance, at h = 200 nm, the peak transmissions can be attained as high as 44%, while the dip transmissions are surprised to drop down to about zero. Such a complete suppression behavior makes a sharp contrast to the smooth slit shown in Fig. 2(a). Moreover, we can find from Fig. 2(b) that the same order of transmission peak is right-shifted for the larger groove height. Conclusively, for the spatially quantized single slit the upgraded transmission can always be achieved just through tuning the transverse groove depth.

The above phenomena might be relevant to the interference of SPPs waves localized into the embedded groove, which results in the depth dependence of the plasmon intensity for the incidence on the second slit unit. Since the real part of the effective index in the groove is reduced with increasing the groove height [6], the longer groove depth is often required to satisfy the condition of F-P resonance, leading to the observed movement of the transmission peaks. Moreover, according to the previous study [20], the output beam from the first unit can be considered as a single point source, which generates a semi-cylindrical wave to propagate radially into space. When the groove height gradually becomes larger, the more SPPs waves will be collected and coupled efficiently into the embedded grooves. As a result, the constructively interfered SPPs waves in the grooves will get the higher magnitudes, giving rise to the pronounced transmission enhancement.

A detailed description of the groove height dependence of the peak transmission through the spatially quantized structure is depicted in Fig. 2(c) for several different slit widths w, wherein only two slit units are cascaded together with variable gaps. Clearly, when h = 0 nm (i.e., the two units are closely contacted), the obtained structures, being equivalent to the smooth slits with different thicknesses, produce the transmittances less than 20%. As the groove height is increased, the calculated transmittances tend to further increase, which makes the maximum transmissions at h = 250 nm to reach about 36.3%, 44% and 66.4% for w = 160 nm, 100 nm and 40 nm, respectively. The narrower the slit width, the greater the transmission enhancement becomes. Notably, such improved transmittances can be almost kept with relatively large varying the groove height. Indeed, this result can be also analyzed qualitatively with the transmission-line theory [21], wherein each slit unit and the embedded groove can be characterized by the impedance. Here the input characteristic impedance is given by the first slit, and the equivalent load impedance is adjusted by varying the height h of the side grooves. Therefore, the lower transmittance can be attributed to the smaller scattering energy into the side grooves because of the high degree of the impedance mismatch. With increasing the groove height, the impedance mismatch tends to be gradually diminished with accompany of the larger transmittance. However, the excessive groove height values will again increase the impedance mismatch to reduce the final transmittance, which is consistent with our observations in Fig. 2(c). In addition, since the total thickness of the spatially quantized slit is expressed by L = (N + 1)*t + N*h, we can also compare its favorable transmission with that of the corresponding smooth slit, as summarized in Table 1 . For example, at L = 600 nm, the transmittance of the smooth slit is about 22.9%, while it reaches 36.8% for the quantized slit, leading to a transmittance increase by 60%. But when the slit length is at L = 880 nm, the optical transmittance for the quantized slit is found to increase by 270%. Since the optical transmittance depends on the slit width, the transmission increase factor tends to grow for the narrower width. Remarkably, in the case of w = 40 nm and L = 700 nm, the obtained transmittance of the smooth slit is only about 6.6%, while it rises up to 67.1% for the quantized slit, resulting in the transmittance increase by as high as about 1000%. These results reveal that the spatial quantization of the slit plays an important role in further manipulating the optical transmissions. In order to gain an insight to the basic physical processes involved in the above phenomena, we calculate the time-averaged density distributions of the electric and magnetic fields for the L = 880 nm slit quantized spatially with one pair of grooves. The simulated near-field results are shown in Fig. 3 , where electromagnetic (EM) distributions of the corresponding smooth slit are also given for comparison. It is clear from this figure that the smooth slit transmission is associated with the second-order waveguided mode to generate two and a half antinodes in EM fields. And their asymmetric distributions along the slit yield the dip transmittance of 11.4%. With respect to the spatially quantized slit, when the embedded air grooves (h = 320 nm) have the depth of d = 520 nm, which corresponds to the peak observation in Fig. 2(b), the whole transmission process is featured by the excitation of the perfect fundamental plasmon resonance in each slit unit [13,14]. It means that the original one second-order waveguided mode is split into two separated fundamental resonance modes respectively within the two slit units, each of which is associated with the integrality and substantiveness. To be exact, there exist two different styles of the mode generation between the smooth and the spatially quantized slit. Most interestingly, the scattering and coupling between the two fundamental plasmon resonances can be evidenced by the local E_x field distribution in the middle of grooves. Within the interaction regime, the coupled EM fields tend to present the relatively low intensities in the smooth slit. On the other hand, as the depth of the embedded grooves is d = 320 nm, which corresponds to the dip observation in Fig. 2(b), the transmission of the quantized slit seems to fade away by only showing the strong plasmon resonance in the first unit. And the disappearance of the plasmonic coupling makes the EM fields in the second slit unit almost null. Consequently, we conclude that the coupling between the fundamental plasmon resonances should upgrade the transmission enhancement through the spatially quantized single slit.

Table 1. Optical Transmittances through the Smooth and the Discretized Single Slit Structures (w = 100 nm)

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Fig. 3 The calculated density distributions of EM fields inside different single slits but with the same parameters of [L, w] = [880, 100] nm. (a) and (b) represent the electric and magnetic fields distribution profiles, respectively. Among them, the left pictures correspond to the smooth slits, the middle pictures for the N = 1 spatially quantized slits with the unit of [h, d] = [320, 520] nm, and the right pictures for the N = 1 spatially quantized slits with the unit of [h, d] = [320, 320] nm. (c) and (d) summarize the electric and magnetic field variations over the center line passing through the slits, respectively. The vertical solid lines mark the beginning and end of the slit structures, whereas the dashed vertical lines stand for the positions of the embedded grooves.

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We now investigate how the interaction of the two fundamental plasmon resonances is varied with the groove parameters. In the spatially quantized structure, the exit of the first slit unit can act as a source of the evanescent fields, which remits evanescent waves along the x direction in the air gap. Subsequently, their propagation and reflection from the groove bottoms are interfered to modify the spatial distribution of the evanescent fields. If the groove depth is set properly to make the constructive interference locate exactly at the entrance of the second slit, in which the fundamental plasmon resonance in the slit unit is provoked strongly, the energy coupling will be optimized towards the transmission enhancement. However, if the selected groove depth produces the destructive interference at the entrance of the second slit, wherein the fundamental plasmon mode and its resonance in the slit unit are no longer excited, the energy coupling and the optical transmission will be suppressed. These analyses can be confirmed by the simulation results in Fig. 3. In addition, different heights of the embedded grooves also alter the coupling strength between the two fundamental plasmon resonances. When h is small enough, the embedded transverse grooves can be considered as only a slight perturbation to the central single slit [12,15], and most of evanescent waves emerging from the first slit unit will flow directly into the second one, so that the residual energy leaking into the air grooves becomes too weak to influence the optical transmission. While h is widened gradually, the direct flow of energy from the first slit into the second one will reduce, and the evanescent waves left for the air-groove interference will become so strong to upgrade the transmission enhancement. This explanation agrees well with our observations in Fig. 2(c). Conclusively, we can recognize that the embedded grooves with different depths can provide the constructive or destructive interference filed, leading to the controllable coupling of fundamental plasmon resonances within the two adjacent slit units.

In addition, we are also interested to explore the dependence of the transmission enhancement on the number of groove pairs. Figure 4(a) compares the calculated transmittances with varying the groove depth in cases of N = 1, 2, 3 and 4. For these different structures, the minimum slit units are all given by [w, t] = [100, 280] nm, and the groove heights are h = 240 nm. The total thickness of the single slit corresponds to L = 800 nm, 1320 nm, 1840 nm and 2360 nm, respectively. It is seen that with increasing the number of slit units, the groove-depth dependent transmittances appear to have the similar oscillating profiles associated with the maxima generation at the same positions, but the magnitudes tend to reduce gradually with accompany of narrowing the line-width. This observation indicates that the optical transmission through the spatially multi-quantized slit structure becomes more sensitive to the embedded groove depth, which may originate from the multiple scattering and coupling of the fundamental plasmon resonance processes. Moreover, compared with the transmissions of the corresponding smooth slits, the peak transmittances of the multiple quantized slits still have further enhancements, as shown in Table 1.

 

Fig. 4 (a) Dependence of the transmittance on the groove depth for multiple pairs of embedded grooves. The slit unit is [w, t] = [100, 280] nm and the groove height is h = 240 nm. (b) Variation of |S_y| at the center line passing through the N = 4 spatially quantized slit with the total length L = 2500 nm. The dashed vertical lines represent the entrance and exit openings of the whole structures.

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Figure 5 depicts the simulated distributions of H_z and S_y, which is the y-component of Poynting vector of $S=1/2\mathrm{Re}\text{al}(E\times H^{*})$, around the N = 4 spatially quantized slit with the total thickness of L = 2500 nm. Here the situations of the smooth slit with the equivalent film thickness are also employed for comparison. First, in the case of the smooth slit, the magnetic field shows six and a half antinodes formation in the slit, whose spatial arrangement is spanned by about 400 nm. Once the single slit is quantized spatially by the multi-pair of grooves, only five antinodes of H_z field are observed and exactly localized in slit units, which is associated with the spatial period of as large as 560 nm. Correspondingly, the original one higher-order waveguided mode in the slit is split into many separated fundamental plasmon resonances within the slit units, among which the coupling is mediated through the near-field interference in the embedded grooves. As concerning the Poynting vector, the smooth single slit reveals that the propagation of SPPs energy experiences a continuous decaying process to get the dip transmittance of 10.8%, while the spatially quantized slit shows that the SPPs energy tends to have multiple hoppings with the gradual intensity attenuation [22], in which the less energy is stored in the embedded gaps and the transmission can grow up to 38.9%. The more quantitative analysis of their difference is shown in Fig. 4(b), wherein the modeled S_y are on the center line passing through the slit. Notably, in the case of the spatially quantized structure, the variation of S_y presents a step-function of the propagation distance. Physically, the role of each embedded groove can be considered as an energy collector of SPPs waves to give out its processed (interference) signal for the incidence on the subsequent slit unit, so that the enhanced optical transmission can be extended into a very long distance.

 

Fig. 5 The simulated magnetic field distributions (a) and y-component of Poynting vectors (b) for the smooth (left) and N = 4 spatially quantized (right) single slit structures. Geometrical parameters for the both situations are [w, t] = [100, 280] nm, h = 275 nm, d = 480 nm and L = 2500 nm.

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The last but not the lest, our observed EM energy transfer associated with the SPPs hopping processes along the compound subwavelength slit is much the same way as the plasmon waveguide made from the metal nanoparticle chains [23]. The individual quantized slit unit is the counterpart of the isolated nanoparticle, and the fundamental plasmon resonance in the slit units corresponds to the SPPs field bound to the nanoparticles. However, in contrast to the nanoparticles waveguiding behaviors, the strength of the plasmonic coupling between our quantized slit units can be tuned conveniently through altering the embedded transverse groove.

3. Conclusions

In conclusion, we have presented a new approach to spatially quantize a single smooth slit with embedded air gaps. The observed optical transmittance has been possible to further increase up by 1000%. Multiple scattering and coupling of fundamental plasmon resonances among the slit units are visualized through FDTD simulation results. Correspondingly, the propagation of SPPs energy along the compound slit is found to change into the hopping style. Moreover, this effect could be controlled by the embedded groove parameters. We can also understand that such an interiorly roughened metallic slit might benefit its transmission rather than increasing its losses, which is contrary to our conversional idea. This slit spatial quantization method improves the operational flexibility of subwavelength structures and could therefore have important applications such as long-range channeling SPPs, enhanced SPPs sensors, as well as optically integrated circuits.