1. Introduction

Guided-wave devices afford a simple and compact means for transmitting electromagnetic radiation from one point to another, as well as across complex geometries. At optical frequencies, waveguide devices have been realized using a wide variety of dielectric structures in planar and fiber geometries. However, it is primarily through the use of planar structures, in the form of integrated optics, that a broad range of novel architectures have been conceived, enabling a wide variety of information technologies and sensing applications. As one moves to longer wavelengths, particularly the terahertz (THz) spectral range, fewer options exist. Although a number of waveguide geometries have been developed [1–6], none have been shown to straightforwardly allow for the development of functional planar THz waveguides capable of leading to the development of complex geometries.

Absorption, scattering, and dispersion play fundamental roles in determining the optimal materials for the fabrication of guided-wave devices. The last parameter is typically only significant in devices designed for broadband or multi-wavelength operation. At optical frequencies, a wide variety of dielectric materials have been shown to exhibit extremely low loss, while the introduction of metals typically increases the loss dramatically. However, at THz frequencies, conventional dielectric materials tend to be lossy, while metals, because of the large conductivities, tend to exhibit low propagation losses. Despite this fact, metallic structures that allow for low loss propagation of surface plasmon-polaritons (SPPs), such as cylindrical metal wires [5], cannot support well-confined surface waves. This is because the imaginary component of the dielectric function for conventional metals in terahertz regime [7] tends to be very large, leading to a loosely bound surface waves [8–10]. The spatial extent of these loosely bound surface waves, which extends over many wavelengths in free space [10], severely limits the viability of this approach for complex guided-wave geometries owing to the possibility of large radiation losses.

In this investigation, we experimentally demonstrate the ability to obtain wavelength-scale spatial confinement and low-loss propagation of THz radiation across complex planar-geometries by using periodically perforated metal films. We accomplish this by engineering the dispersion relation of SPPs by periodically structuring of metal films. We measure both the transmission and propagation properties of the proposed plasmonic guided-wave devices using THz time-domain spectroscopy. This experimental technique is unique in that it allows for direct measurement of the THz electric field, yielding both amplitude and phase information. This information is used to determine the complex dielectric function, ε(ω), for the effective plasmonic medium without resorting to traditional Kramers-Kronig transformations, where somewhat arbitrary assumptions about asymptotic behavior are usually made. Further, this technique can be used to directly measure the complete vector nature of the THz surface electric field as well as map out its spatial distribution [10]. We use these capabilities and the results of the ensuing measurements to design, fabricate and characterize several planar plasmonic THz guided-wave devices.

2. Experimental details

We used a conventional THz time-domain spectroscopy setup to characterize the plasmonic behavior of the one-dimensional array of periodically spaced rectangular apertures using transmission measurements. Single cycle THz pulses were generated using optical rectification in a nonlinear crystal, which were subsequently detected using electro-optic sampling. Two off-axis paraboloidal mirrors were used to collimate and collect the THz radiation. The THz beam was incident on the waveguide array structures at different incident angles θ. Samples were attached to a solid metal plate with a 5 cm×5 cm opening that was placed in the path of the collimated THz beam. Reference transmission spectra were taken with just the metal holder in the THz beam path using the same setup. The metallic foils were completely opaque prior to fabrication of the arrays. Therefore transmission through the perforated metallic films was uniquely due to the apertures in the metal foil. The detected transient photocurrent is then Fourier transformed and normalized to the reference transmission spectra, yielding the electric field transmission spectrum, t(ω), in the range ~0.05 to 0.6 THz. By transforming the time-domain data to the frequency domain, we are able to determine independently both the magnitude and phase of the amplitude transmission coefficient using the relation:

In this expression, E_incident and E_transmitted are the incident and transmitted THz fields, respectively, |t(ω)| and φ(ω) are the magnitude and phase of the amplitude transmission coefficient respectively, and ω/2π is the THz frequency.

We used a modified THz time-domain spectroscopy setup to characterize the surface waveguiding properties of the periodically spaced apertures. The generation and detection of the THz radiation is based on the same principles as in a conventional setup. However, only one off-axis paraboloidal mirror was used to collimate the THz radiation from the emitter to the samples. The collimated THz radiation was normally incident on a semi-circular groove fabricated on the waveguide samples and was subsequently coupled to THz surface plasmons propagating along waveguide direction. On the detection side, we used a second nonlinear optical crystal to measure the out-of-plane component of the THz surface electric field. To measure the guided wave through the row of apertures, we placed the detection ZnTe crystal ~2 mm after the last aperture, so that the field within the apertures could couple back to an electric field propagating only along the metal surface. It should be noted that by moving the optical probe beam and the ZnTe detector simultaneously, we could measure the surface electric field at any position above the sample surface.

3. Experimental results and discussion

In Fig. 1(a), we show the schematic diagram of an effective plasmonic medium fabricated by periodically perforating a row of subwavelength rectangular apertures in a stainless steel metal foil. The apertures were fabricated using conventional laser micromachining techniques. A photograph of the one-dimensional array of apertures is shown in Fig. 1(b). Since the effective medium exhibits unique transmission and dispersion properties for TM polarized radiation, the THz field is incident such that the H-field lies parallel to the y-axis [11–16]. The rectangular apertures have dimensions of 500 µm×50 µm, with the long axis of the apertures parallel to the y-axis, and a periodicity of 250 µm. Compared to the wavelength (λ) of interest, discussed below, the aperture width is ~λ/8 and the periodicity is ~λ/4, indicating that we are operating in the long wavelength limit.

 

Fig. 1. Transmission properties of a linear row of periodically spaced rectangular apertures for a TM-polarized incident THz field. (a) Schematic diagram showing the experimental configuration to measure the dielectric properties of the effective plasmonic media. The TM-polarized THz radiation is normally incident on the structured metal film. The dimensions of the apertures are: length s=500 µm, width a=50 µm, periodicity d=250 µm and thickness l=635 µm. The total length of this structure is 8 cm, corresponding to approximately 320 apertures. (b) A top-view photograph of the actual row of apertures. (c) The amplitude transmittance spectra, t(ω)measured using THz time-domain spectroscopy (red trace). The black trace corresponds to the calculated transmittance through an equivalent dispersive dielectric slab using the FDTD technique. (d) The corresponding measured phase spectra φ(ω). (e) Change in the resonance frequencies ω_mnp as a function of incident angle θ. The three traces correspond to the different resonances in the perforated medium: TM₁₀₀ (red), TM₁₀₁ (blue), and TM₁₀₂ (black).

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Using terahertz time-domain spectroscopy, we measured the amplitude, t(ω), and phase, φ(ω), of the transmittance spectra for this effective medium. The measured t(ω) (red curve) and the simulated t(ω) (black curve) along with the measured φ(ω) (Inset) are shown in Fig. 1(c). The simulation results are derived from numerical finite-difference time-domain (FDTD) calculations based on the same geometry and plane-wave excitation source. We modeled the stainless steel metal foil as a perfect electrical conductor in the FDTD simulations, since the frequencies examined here are much smaller than the bulk plasma frequency of metals.

It is apparent that there are three distinct peaks in the t(ω) spectra lying at 0.30 THz, 0.35 THz and 0.50 THz. The locations of these frequencies can be explained if each individual aperture is considered as a rectangular metallic cavity with both interfaces open to free space. The eigen-frequencies of the cavity modes in the rectangular cavity are given by [17]:

where m, n, and p are integers, l is the metal thickness, and c is the speed of light in vacuum. Indeed, the three resonant frequencies correspond to the TM₁₀₀, TM₁₀₁ and TM₁₀₂ modes in the cavity. Moreover, these shape-dependent resonances are independent of the incident angle of the incident TM radiation, as confirmed by the angle dependent transmission measurements, shown in Fig. 1(e). It is worth noting that the angular independence of the fundamental resonance (TM₁₀₀) is due to cut-off properties of the individual apertures, while the angular independence of the higher-order resonances (TM₁₀₁ and TM₁₀₂) is analogous to what would be expected from Fabry-Perot resonances in a planar dielectric slab. Therefore, from a macroscopic point of view, a one-dimensional row of apertures can be modeled as a homogeneous, albeit anisotropic, plasmonic slab for TM polarized excitation. Before moving on, it should be pointed out that the lowest frequency resonance associated with the enhanced optical transmission phenomenon [18] would occur at 1.2 THz, which is well outside the spectral window of our measurements.

Using the amplitude and phase of the transmittance spectra, we directly calculated the real and imaginary components of ε(ω) for the effective medium, as shown in Fig. 2 (red curves) [19]. ε(ω) exhibits the dispersion properties of a plasma, similar to that of a metallic medium, but with an effective plasma frequency approximately equal to the cutoff frequency, f_c of the fundamental cavity mode for a rectangular aperture. Higher-order resonances corresponding to TM₁₀₁ and TM₁₀₂ mode in Fig. 1(c) are due to the Fabry-Perot effects, and are taken into account while calculating the effective dielectric properties of the aperture array medium [19]. To demonstrate this, we modeled ε(ω) for the effective medium with the equation of a ‘lossy plasma’:

where $\tilde{\omega }$ _P is the effective plasma frequency, ε _∞ is the high frequency dielectric constant, and γ is the plasma relaxation rate. Using Eq. (3) with $\tilde{\omega }$ _P/2π=0.30 THz, ε_∞=32, and γ=0.025 rad/ps, we obtained an excellent fit to the real and imaginary components of ε(ω), as shown in Fig. 2 (black curves).

The dielectric properties of this effective medium can be explained in terms of recent theoretical work proposing that surfaces of perfect electric conductors perforated by periodic subwavelength aperture arrays are capable of supporting surface electromagnetic modes [11–13]. An important aspect of these theories is that the effective plasma frequency of this fictitious, imperfect flat metal surface is given by the cut-off frequency (f_c) of the waveguide mode in the aperture. Therefore, the effective plasma frequency can be designed to occur at almost any frequency below the natural plasma frequency of the metal film, providing great flexibility in designing materials that exhibit unique electromagnetic properties. In terms of the measurements discussed above, the value of $\tilde{\omega }$ _P/2π agrees well with the calculated f_c=0.30 THz and is consistent with the predictions of the “spoof plasmon” model [11]. It is important to note that in contrast to the aforementioned theories, the perfect conductor approximation is not strictly valid in the THz frequency range, since the conductivity is finite. Thus, the complete complex dielectric properties of the medium, which includes attenuation, must be taken into account.

 

Fig. 2. Dielectric properties of a linear row of periodically spaced rectangular apertures (shown in Fig. 1) for a TM-polarized incident THz field. The real and imaginary components of the effective dielectric constant, ε_real(ω) and ε_imag(ω) calculated from the measured t(ω) and φ(ω) spectra shown in Fig. 1. The black traces shows the fit to ε_real(ω) and ε_imag(ω) using the Drude model of Eq. (3).

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Before discussing practical device implementations, it is of value to compare the properties of conventional unperforated metal films versus the effective plasmonic medium discussed here. At THz frequencies, conventional planar metals are characterized by real and imaginary components of the dielectric constant that are very large. This leads to a low-loss propagating surface waves that are loosely bound to the metal-dielectric interface. The 1/e field decay length into free-space for these waves at THz frequencies has been measured to be ~5–7 mm [9,10] and the propagation losses along the metal surface are typically ~0.023 cm^-1 [9]. Although the low value for the propagation loss allows for the possibility of large-scale terahertz devices, the fact that the waves are loosely bound creates difficulties in realizing useful devices, because of the potential for high radiation losses on waveguide bends and the lack of field confinement in the lateral (in-plane) direction.

We now experimentally consider these issues for the effective medium discussed above. In Fig. 3(a), we show a linear waveguide that is essentially identical to the structure shown in Fig. 1(a), although for this structure we chemically etched a semi-circular groove at one end to couple free-space terahertz radiation to a propagating THz SPP wave. The semi-circular groove has a radius of ~1 cm with a groove width of ~300 µm and depth of ~100 µm. The groove is oriented such that its origin is overlapped with the center of the first aperture. We have previously shown that a single semi-circular groove can couple a free-space THz pulse to a focused single-cycle surface wave, with the focal point occurring at the origin of the circle [9,20]. The dimensions noted above correspond to the optimal groove geometry [21] and acts as an efficient means of coupling incident radiation into the waveguide. We then use an electro-optic crystal to measure the time-domain properties of the guided-wave radiation at different points along the waveguide.

 

Fig. 3. Linear waveguiding properties of the planar plasmonic waveguide fabricated using periodically spaced rectangular apertures. (a) Schematic diagram showing the experimental configuration. A semi-circular groove is used to couple a single-cycle THz pulse to a surface wave pulse, which is subsequently focused at the input of the waveguide. The guided wave is then detected by electro-optic sampling using a (110) ZnTe crystal placed at the output of the waveguide. (b) The guided-wave transmission spectrum, t_WG(ω) measured after propagation along the entire waveguide (red trace). The black trace corresponds to the guided-wave transmission spectrum calculated using the FDTD technique. (c)–(e) shows the total electric field distributions in the yz-plane in the rectangular aperture, corresponding to the resonance modes of (c) TM₁₀₀, (d) TM₁₀₁ and (e) TM₁₀₂.

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In Fig. 3(b), we show the experimental and numerically simulated spectra, t_WG(ω), of the guided-wave THz radiation measured at the end the linear waveguide. In the FDTD simulations, the same geometry was used as in the transmission simulations shown in Fig. 1(b). However, instead of plane-wave excitation, a z-directed dipole was placed on the metal surface at one end of the rectangular array and the output E_z field was sampled on the surface at the other end. While it is apparent that the frequencies corresponding to the first and third resonance peaks agree quite well, there is a noticeable discrepancy between the experiment and simulation for the second resonance peak. We do not fully understand the source of this discrepancy; however, it is worth noting that the anti-resonance frequencies on the high frequency side of each resonance do agree rather well. This has previously been observed in the context of enhanced THz transmission through subwavelength apertures [22]. Since the metal is modeled as a perfect conductor, the absence of absorption in the metal film corresponds to narrower linewidths. The total electric field distributions in the yz-plane in the rectangular aperture for each resonance mode are shown in Figs. 3(c)–3(e). Based on theories that consider one-dimensionally perforated metal films [14, 15], we attribute the low frequency resonance to a plasmonic mode, where the guided-wave energy is highly confined near the surface. The higher frequency resonances, however, are dielectric slab modes, where energy can flow into the apertures and bounce back and forth between two interfaces. In comparison to the free-space transmission measurements described in Fig. 1, there are two points that are of particular interest. First, there is in good agreement between the transmission and waveguide resonance properties as shown in Figs. 1 and 3, respectively. Second, the experimentally observed lineshapes of the guided-wave resonances are quite different from those observed in the free-space transmission data. We attribute this to the longer interaction lengths of the propagating THz radiation with the effective plasmonic medium in the waveguide geometry.

We now examine the propagation properties of the low-frequency plasmonic mode, which peaks at 0.28 THz, as it propagates along the waveguide. The electro-optic detection scheme used to measure the propagating surface electric field utilized a (110) ZnTe crystal. Based on the orientation of the crystal and the polarization of the THz and optical probe beams, we were only sensitive to the out-of-plane (E_z) component of the propagating surface electric field [10,20], rather than the in-plane (E_x) component, which dominates within the apertures. In Fig. 4(a), we show the magnitude of the E_z field component measured along the length (x-axis) of the waveguide. From these measurements, we find that the waveguide loss is ~0.013 mm^-1. This loss is comparable to that observed in other recently reported waveguide geometries (0.018 mm^-1) [6], although as noted earlier, there have been no reports of planar waveguide embodiments of these other geometries. In Fig. 4(b), we show the magnitude of the lateral (along y-axis) field distribution measured at two different cross sections along x-axis: 5 cm and 7 cm from the waveguide input. The lateral field distribution at the two cross-sections exhibits a Gaussian shape mode profile with the full-width at half-maximum (FWHM) mode size of ~2.2 mm, indicating tight confinement in the lateral direction for the plasmonic mode as it propagates along the waveguide. We attribute the slight difference in magnitude between these two spatial distributions to waveguide losses. In Fig. 4(c), we show the magnitude of the E_z field component as a function of distance (along z-axis) above the waveguide surface. As expected, the field decays exponentially away from the metal-dielectric interface. However, the 1/e decay length is ~1.69 mm, which is ~4 times smaller than that on a planar metal surface [10]. Similar wavelength scale confinement of THz surface electric field in the out-of-plane direction has recently been demonstrated by texturing the metal surface with a two-dimensional array of dimples [23].

 

Fig. 4. Propagation properties of the TM₁₀₀ mode in the planar plasmonic waveguide shown in Fig. 3. (a) The field amplitude |E_z| measured along the x-axis at different positions along the length of the waveguide. (b) The |E_z| component of the electric field measured along the y-axis at two different positions along the waveguide: 5 cm and 7 cm from the waveguide input. (c) The |E_z| component of the electric field measured along the z-axis at different heights above the sample surface. It is apparent from the figures that the plasmonic mode propagates with very low-loss and remains tightly confined to the waveguide.

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Therefore, the plasmonic waveguide mode, although not subwavelength (only the E_x field component is subwavelength), exhibits tight confinement and low propagation loss along the perforated metallic waveguide. These results are generally consistent with theoretical work concerning the properties of periodically spaced rectangular apertures, which have been shown theoretically to allow for both in-plane and out-of-plane confinement of the propagating surface fields [15]. It is worth considering whether or not the electromagnetic field can be more tightly confined to the surface. The out-of-plane 1/e decay length can be expressed as

where k_z is the propagation constants along the z-axis. The propagation constant along the x-axis, k_x, is limited only by the Brillouin zone boundary π/d [14,15], at any given frequency. Thus, the minimum out-of-plane 1/e decay length would be

Therefore, it should be possible to achieve a more tightly confined electric field reducing the periodicity between the rectangular apertures.

 

Fig. 5. Passive plasmonic guided-wave Y-splitter fabricated using the effective medium concept. Upper panel - A schematic diagram of the Y-splitter structure; a semi-circular groove is used to couple a free-space THz radiation into the guiding structure. For clarity, the splitter section of the Y-splitter is expanded in the figure panel. Lower panel: The field amplitude |E_z| measured along y-axis at the end of the Y-splitter (red dots), marked as a black dashed line in the upper panel. The solid red trace is a fit to the experimental data using a sum of two spatially offset Gaussian functions.

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Based on the mode properties ascertained from a linear waveguide, we are now able to extend this approach to develop several simple two-dimensional waveguide devices: a Y-splitter and a 3-dB coupler. In Figs. 5 and 6 we summarize the guided-wave properties of these two waveguide devices fabricated using the effective medium concept. In Fig. 5, we show the schematic diagram of a Y-splitter that is composed of the same periodically spaced rectangular apertures used in the structures described above. The Y-splitter consists of a 32 mm long linear input, two 32 mm long slanted arms, each rotated by 11.2° angle from the input, and two 32 mm long output sections. The total length of this Y-splitter is ~95.4 mm and the center-to-center separation distance between two arms is ~11.6 mm. Free space THz radiation was coupled to the waveguide device via an etched semi-circular groove. We measured the magnitude of the E_z electric field along a cross-section (shown by the dotted line in Fig. 5) at the output of this device. We then fit the experimentally measured data with the sum of two spatially offset Gaussian functions (solid red line). The center positions of the two Gaussian functions occur at y=-5.8 mm and y=5.8 mm, which overlaps with the center of the two output arms of the Y-splitter. Using the fit, the FWHM widths of the two Gaussian functions was found to be ~2.31 mm and ~2.34 mm respectively, which is in good agreement with the mode size shown in Fig. 4(b), indicating guided mode propagation along the Y-splitter. Finally, the resulting distribution exhibits a ~2.5:1 contrast ratio for the peak amplitude associated with each guided mode and the background signal present between the two arms of the splitter.

 

Fig. 6. Passive plasmonic guided-wave 3-dB coupler fabricated using the effective medium concept. Upper panel: A schematic diagram of the 3-dB coupler structure; again, a semicircular groove is used to couple a free-space THz radiation into the guiding structure. For clarity the coupling section of the 3-dB coupler is expanded in the figure panel. Lower panel: The field amplitude |E_z| measured along y-axis at the end of the 3dB-coupler (red dots), marked as a black dashed line in the upper panel. The solid red trace is a fit to the experimental data using a sum of two spatially offset Gaussian functions.

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We also designed, fabricated and tested a 3-dB coupler, as shown schematically in Fig. 6. The device was designed using the same basic approach as that used for the Y-splitter. The central coupler section has a center-to-center distance between the two arms of ~1.55 mm, with a length of 3 cm. The total size of the 3-dB coupler is ~169 mm×12.2 mm. The parameters used to design this device were based on scaling the mode size and operating wavelength of well developed dielectric 3-dB couplers designed for optical frequencies [24]. Once again, we measured the magnitude of the E_z electric field component along a cross-section at the output of this device (shown by the dotted line in Fig. 6). The field distribution measured at the end of the waveguide was again fit using two Gaussian functions. The center positions of the two Gaussian functions occur at the waveguide centers, and have FWHM widths of ~2.12 mm and ~2.27 mm, demonstrating that the mode is indeed guided along the 3-dB coupler. Also, the contrast ratio for the peak amplitude associated with each guided mode and the background signal present between the two arms of the splitter is ~2.3:1.

4. Conclusion

In summary, we have designed, fabricated and characterized planar plasmonic THz guided wave devices using periodically perforated metal films as effective plasmonic media. The spatial distribution of the plasmonic guided-wave mode shows that the electromagnetic field is tightly confined in the waveguide along both the in-plane and out-of-plane axes with a 1/e propagation length of ~8 cm. In measuring the complex dielectric function, ε(ω) at THz frequencies for these structures, we demonstrated that the effective media could be modeled as having an effective ‘plasma-like’ response. More generally, the dispersion properties can be arbitrarily engineered just by changing the geometry of the structure. This is important for designing media with unique electromagnetic properties and is expected to create new and exciting opportunities in the development of useful passive and active plasmonic waveguide architectures for the THz spectral range.