1. Introduction

Photonic devices made from dielectric materials have fundamental size limitations that are imposed by the diffraction properties of electromagnetic (EM) waves. Typical photonic device features, on the order of a half wavelength or greater, are at least one or two orders of magnitude larger than the nanoscale electronic counterparts. Recently, it was proposed that optical plasmonic structures and devices have advantages for overcoming these limitations, for applications such as on-chip optical integrated circuits, surface or interface technology, data storage [1,2], and bio-sensing [3]. What makes the plasmonic structure unique is its potential for spatial confinement of EM waves over distances that are significantly smaller than the EM wavelength. This is particularly true where the frequencies of the EM wave are close to the plasma frequency of the conductor, and surface plasmon polaritons (SPPs) can be used to confine the EM field within subwavelength dimensions. While most studies of SPPs focus on the visible and infrared frequency range, in recent years there has been increasing interest in studying SPPs for the terahertz (THz) frequency range, on both flat [4–7] and cylindrically shaped [8–12] surfaces.

THz radiation bridges the gap between the microwave and optical regimes, and offers significant scientific and technological potential in many fields [13]. However, waveguiding in this spectral region still remains challenging. Neither conventional metal waveguides for microwave radiation, nor dielectric fibers for visible and near-infrared radiation are able to guide terahertz waves over a long distance, owing to the high loss from the finite conductivity of metals or the high absorption coefficient of dielectric materials in this spectral range. Because of the extremely low loss and low dispersion of SPPs in the THz frequency range, metallic or heavy-doped semiconductor SPPs waveguides might be useful for future highly miniaturized integrated optical devices for THz photonics or optoelectronics [9,14]. However, at THz frequencies the EM energy cannot be confined at the flat metallic surfaces. Consequently, it is difficult to confine the THz wave within subwavelength dimensions on these surfaces. To overcome this problem, certain surface structures were proposed to mimic the SPP behavior such as spoof SPPs over a wide spectral range [15]. The concept of “designer” surface modes opens an opportunity for us to control and to direct EM radiations at surfaces within a subwavelength region, especially in the range from GHz [16] to THz [17]. This could enable compact THz photonic devices. Many schemes for coupling laser light into SPP modes have been demonstrated using prisms, gratings, and nanoscale defects (such as surface protrusions or holes) [18]. But what has been missing in the past is an effective method for ensuring that the generated SPPs only travel in the desired directions from a subwavelength scale launching point [19]. Recently, we numerically demonstrated that two surface waveguide structures placed on the opposite sides of a slit can be introduced to confine and to guide light of different frequencies in the range from visible to infrared [20]. Almost at the same time, it was reported that a device utilizing SPPs could be launched on a chip in a predetermined direction using an asymmetric waveguide placed at the two sides of the slit [21]. In this article, we propose a frequency splitter operating in THz domain based on a bidirectional subwavelength slit simulated using two-dimensional FDTD modeling. We believe that such a splitter can be readily incorporated into THz integrated circuits.

2. Dispersion curves of the metal grating structures

To study the spoof SPP behavior at THz frequencies, we first analyze the dispersion curves of a microstructured metallic surface. The structure, illustrated Fig. 1 (inset), consists of a one-dimensional groove array engraved in the metal plane with depth h, width d, and lattice constant p. For a perfect electrical conductor (PEC), the dispersion relation of TM-polarized (E_x, E_z, and H_y) EM waves in region I propagating along x axis can be obtained from the following equations:

where A_n are constants, β_n=β₀+2πn/p, and β²_n-τ²_n=k²=ω²µ₀ε₀. For region II, the EM fields are zero except in the grooves where each of the field components is a subset of the infinite series consisting of all the eigenfunctions that satisfy the boundary conditions. To simplify our calculation, we consider an approximate solution for a TEM mode propagating in the ±x directions. Hence we have

where B is a constant and the phase shift along z is regarded as a discontinuous function

$e^{-j{\beta }_0\mathrm{mp}}$

. Once the EM fields are obtained, the dispersion equation can be determined by utilizing the boundary conditions, i.e. E_z1(0)=E_z2(0) and ∫_d/2-^d/2 H_y1(0)dz=H_y2(0)d (approximately matched). Following such procedure, the dispersion equation can be obtained as:

where β is the propagation constant, β_n=β+2πn/p.

The dispersion relation for the EM waves propagating along the x axis on the surface of the PEC with a 1D groove array can be generated numerically from Eq. (3) [22]. The solutions can be made to converge well by including at least four orders of the infinite series in Eq. (3), i.e., n≥4 (a simpler expression neglecting higher orders was derived in [23]). Using this approximate dispersion relation, we can design the structures characterized by various parameters (h, d, p) for the surface modes at frequencies from GHz to THz. For instance, Fig. 1 shows the dispersion relation calculated for d=20µm and p=50µm, and different groove depths. One can see that for h=40 to 60µm the cutoff frequency of the dispersion curve is larger than 1THz. Therefore, the corresponding structures are capable of spatially confining the EM wave as surface modes at the grating surface at 1THz, which has been verified by us using the FDTD simulation. For groove heights in the range of 70–110µm, the cutoff frequency for the grating structure is larger than 0.5THz. Therefore the EM wave at 0.5THz could be coupled into the grating structure and then be confined at the surface. It should be noted that with decreasing frequency, the dispersion curves are increasingly close to the light line (dotted line c in Fig. 1). Strong confinement of the surface modes will result from a large mismatch between the wave vectors of the guided surface waves and any free-space waves. When the wave frequency is close to the cutoff frequency, the EM wave should be strongly confined at the surface. As a result, it should be feasible to realize a bidirectional subwavelength slit splitter operating at THz frequencies. For instance, for h=50µm (line 1 in Fig. 1), the dispersion curve is close to the light line at 0.5THz, and the EM wave could not be confined well at the surface. For h = 110µm (line 2 in Fig. 1), the cutoff frequency is close to 0.5THz, and the wave could be strongly confined at the surface. On the other hand, for a frequency of 1THz, the EM wave should be confined well by the structure with a depth of 50 µm, but cannot be coupled into the structure with a depth of 110µm since its frequency is larger than the cutoff frequency of the structure (line 2 in Fig. 1). Consequently, the splitter mechanisms on the two sides of the slit are different. If we place the two structures on opposite sides of a single slit, it is feasible for the EM wave at these two frequencies to be split and transmitted in two opposite directions. In this way, spoof SPPs at different THz frequencies are launched on a chip in the two predetermined opposite directions.

 

Fig. 1. Dispersion curves calculated for d=20µm and p=50µm with different groove depths. Inset: Schematic illustration of the envisioned structure.

Download Full Size | PDF

3. FDTD model of the bidirectional subwavelength slit splitter

Next, we introduce a two-dimensional FDTD model to investigate the expression given in Eq. (3). The proposed structure is shown in Fig. 2(a); the thickness of the metal is 400µm. The wavelength of the incident wave is set to 300µm (a frequency of 1THz). According to Ref. [24, 25], a tapered waveguide can be used to achieve an intensity-enhanced spot at the tip of the waveguide for a TM mode. Similarly, we propose to use a tapered slit to achieve an intensity-enhanced spot in the tip of the slit. The top and bottom widths of the tapered slit are 20µm and 300µm, respectively. The intensity at the top slit can be enhanced by 30 to 40 times the incident intensity according to our simulation. The dimension of the simulated region is 3600µm×800µm, with a uniform cell of Δx=1µm and Δz=2µm. This region is surrounded by a perfectly matched layer absorber. Figure 2(a) also describes the one-dimensional steady-state THz intensity at a location of 2µm above the metal surface [lower part of Fig. 2(a)]. We have found that the THz intensity at the metal/air interface is relatively weak. A similar result is expected for the incident wavelength of 600µm (a frequency of 0.5THz). The THz wave cannot be confined or guided at the metal surface; rather it is confined at the tip of the tapered slit. It is well known that when the EM wave is transmitted through a subwavelength slit or aperture, evanescent waves with relatively low phase and group velocities are excited around the slit or aperture. If the phase velocity of the transmitted evanescent wave matches that for the surface modes supported by the out-coupling surfaces, it can be coupled and guided along the structured surfaces. The coupling between the incident wave and SPPs is subwavelength in dimension. Moreover, the long wavelength of the incident THz wave does not limit the dimension of a waveguide or a THz circuit placed at the out-coupling surface. This mechanism is related to the SP wave excited by a single defect at the metallic surface [26], which is different from the conventional coupling approaches using grating or attenuated internal reflection coupling [18].

In recent reports, a single slit was employed as a wave source to excite SPPs [21, 27, 28]. In Figs. 2(b) and (c), two surface structures on the different sides of the slit are proposed. The period, width, and depth of the grating on the left-hand side are 50µm, 20µm, and 50µm, respectively. On the other hand, on the right-hand side the period and width are the same with those on the left, but the depth is now 110 µm. For the configuration shown in Fig 2(b), most of the EM wave at 0.5THz is coupled into the surface grating and guided towards the right-hand side. As shown in the lower part of Fig. 2(b), the peak intensity within the region of 1800 µm is 7–9 times the peak intensity of the incident wave. We would like to note that the steady-state intensity at the left-hand side surface is relatively weak. On the other hand, as shown in Fig. 2(c), when the THz frequency is 1 THz, most of the wave is coupled and guided towards the left-hand side. The peak intensity within the region of 1800µm is 10–13 times the peak intensity of the incident wave [the lower part of Fig. 2(c)].

 

Fig. 2. Results Obtained by using two dimensional FDTD simulations: (a) corresponds to the two-dimensional field distribution and one-dimensional optical intensity (|E|²) distribution 2µm above the perfect metal surface, respectively. In this simulation, the frequency of the incident light is 1THz. The simulation result is similar when the incident light wavelength is 600µm and frequency is 0.5THz. (b) and (c) represents the two-dimensional field distribution and one-dimensional optical intensity (|E|²) distribution 2µm above the structured metal surface, respectively. The frequencies of the incident light are 0.5THz in (b) and 1THz in (c), respectively. In these simulations, the thickness of the metal is 400µm, the width (d) and period (p) of the gratings are 20 µm and 50 µm, respectively. The depths (h) of the left-hand side and right-hand side gratings are 50 and 110µm, the top and the bottom width of the slit are 20µm and 300µm. The simulation time T=6500µm/c, here c is the light velocity in vacuum.

Download Full Size | PDF

As illustrated by the dispersion relations in Fig. 1, the phase velocity of the SPP mode is given by the slope of the line drawn from the origin to a particular point in the dispersion curve. The group velocity of the SPP mode is given by the slope of the tangent at this point in the dispersion curve. It is obvious that the group velocity of the SPPs at 0.5THz (h=110µm) is lower than that of the SPPs at 1THz (h=50µm). Consequently, the propagation distance at the surface for SPPs at 0.5THz should be shorter than that for SPPs at 1THz in a given time interval. For the simulation time for Fig. 2 given by T=6500µm/c where c is the wave velocity in vacuum, the envelope of the SPPs at 1 THz can be guided for a longer period than 1800µm, but this is not the case at 0.5THz. Such a difference indicates that the group velocity of the SPPs at 0.5THz is lower than that for the SPPs at 1THz. These simulation results demonstrate an operational principle for a bidirectional subwavelength slit splitter for SPPs working for the dual frequencies of 0.5THz and 1THz.

From the dispersion curves in Fig. 1, we can further analyze the effect of field confinement for different surface structures. It is worth noting that the larger the momentum mismatch between the propagating light and the SPPs modes, the better the confinement of the SPPs modes at the surface. Comparing line 1 with line 2 in Fig. 1, the momentum mismatch at 1THz is slightly larger than that at 0.5 THz, and therefore, the SPPs modes at 1 THz should be better confined, as shown in Fig. 3. Lines 1 and 2 in Fig. 3 show that the confinement depth of the EM energy, which is defined by the distance from the metal surface at which the E filed is reduced to the half, is only 5.5–6.5µm. This result demonstrates that the surface grating structures can channel the EM energy into sub-wavelength structures (5.5µm for 1THz, and 6.5µm for 0.5THz), which are significantly smaller than the incident wavelength (i.e. 300 and 600µm). In addition, since the dispersion curve for the depth of 50µm is close to the light line at 0.5THz, the momentum mismatch is small. Consequently, the spatial extent of the mode is wider, as shown by line 3 in Fig. 3. Finally, 1THz is beyond the cutoff frequency of the dispersion curve for the depth of 110µm (line 2 in Fig. 1), and the wave cannot be coupled into the structure and confined at the surface; rather it is scattered to the far-field. This is the reason why the amplitude of the electric field in the far-field is larger than that in the near-field with respect to the surface (line 4 in Fig. 3).

 

Fig. 3. Field concentration above the two surface structures with different depths (50µm in the left-hand side and 110µm in the right-hand side). Distribution of the E field, evaluated at f=0.5 THz and 1THz, along the vertical direction to the surface structures were calculated from the FDTD results shown in Fig. 2(b) and (c).

Download Full Size | PDF

4. Conclusion

In conclusion, we have proposed a THz frequency splitter based on a bidirectional subwavelength slit resonating with SPPs. This device employs a single narrow slit or aperture as a high-intensity subwavelength-sized excitation source to reduce the dimension of the wave-SPP coupling interface. Two different grating structures are placed on the opposite sides of the slit in order to confine and to guide the EM waves at 0.5THz and 1THz in the two predetermined opposite directions. The peak intensity of the guided THz wave is higher than that for the incident wave by factor of 7–15 over a distance of 1800 µm. In addition, the field confinement by different surface structures is discussed in terms of the dispersion curves. We believe that such a frequency splitter would be useful to THz integrated optical circuits and chemical and biological sensing.