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We describe the optical concentration properties of periodic arrays of conically tapered metallic apertures measured using terahertz (THz) time-domain spectroscopy. As a first step in this process, we optimize the geometrical properties of individual apertures, keeping the output aperture diameter fixed, and find that the optimal taper angle is 30°. A consequence of increasing the taper angle is that the effective cutoff frequency red shifts, which can be readily explained using conventional waveguide theory. We then fabricate and measure the transmission properties of a periodic (hexagonal) array of optimized tapered apertures. In contrast to periodic arrays of subwavelength apertures in thin metal films, which are characterized by narrowband transmission resonances associated with the periodic spacing, here we observe broadband enhanced transmission above the effective cutoff frequency. Further enhancement in the concentration capabilities of the array can be achieved by tilting the apertures towards the array center, although the optical throughput of individual tapered apertures is reduced with increasing tilt angle. Finally, we discuss possible future directions that utilize cascaded structures, as a means for obtaining further enhancement in the amplitude of the transmitted THz radiation.
©2013 Optical Society of America
1. Introduction
The ability to concentrate optical radiation is a topic of long-standing interest that has utility in a broad range of applications [1–4]. In the case of freely propagating radiation, the diffraction limit constrains the minimum spot size to a dimension that is on the order of the free space wavelength. One approach to circumvent this limitation is through the use of Plasmonics. Surface plasmon-polaritons (SPPs) are surface electromagnetic waves that propagate along the interface between a metal and a dielectric and are characterized by dispersion properties that differ dramatically from those of free space radiation. Using SPPs, there have been numerous theoretical and experimental studies that rely on metallic structures to focus or concentrate electromagnetic radiation to dimensions much smaller than the associated free space wavelength [5]. Common implementations of this have relied on tapered wires [6–8], holes [9–12] and plates [13–15].
As an example, we recently demonstrated that individual conically tapered cylindrical apertures could be used to efficiently concentrate broadband terahertz (THz) radiation [11]. In that work, we kept the input aperture diameter and the taper angle fixed and found that as the diameter of the exit plane aperture was decreased, we obtained an increase in the magnitude of the transmitted THz electric field, which varied inversely with the output aperture diameter. For the smallest aperture that we fabricated, we obtained a ~50 fold increase in the transmitted THz intensity.
While individual tapered structures are useful for applications such as raster scanned near-field imaging [1,2], the sequential acquisition of single pixel data greatly reduces the overall collection efficiency of the system, thereby increasing the total acquisition time. In such cases, the use of multiple identical structures may allow for parallel acquisition. Furthermore, if the radiation from arrays of tapered structures can be focused, it may allow for enhanced nonlinear optical conversion efficiency. In recent years, there have only been a few reports associated with ‘plasmonic lattices’ in which the individual elements are tapered [16,17]. In both cases, since thin metal films with small taper angles were used, the individual transmission resonances were enhanced and the narrowband nature of the enhancement remained unchanged.
In this submission, we demonstrate that periodic arrays of conically tapered cylindrical apertures concentrate broadband THz radiation for frequencies above an effective cutoff frequency. This frequency is determined by both the diameter of the output aperture and the taper angle. As expected, we find that there is an optimal taper angle that maximizes the optical throughput. We then fabricate a hexagonal array containing 19 such apertures and show that the transmitted radiation is also broadband. In contrast to prior work [16–18], the large metal thickness allows for large differences between the input and output aperture diameters, while maintaining taper angles that can be relatively small. As we show, a consequence of this design is that the transmission of each aperture is effectively independent of all of the other apertures. Further concentration of the transmitted radiation can be obtained by tilting the tapered apertures (TAs) toward the array center, thereby decreasing the effective output cross-sectional area, or by cascading the TA structures.
2. Experimental details
We fabricated a series of individual TAs in 3 mm thick stainless steel disks using wire electrical discharge machining (wire EDM). Each TA was fabricated on a separate metal disk and consisted of a circular output aperture with a fixed diameter D2 of 400 µm and a conical taper full angle, α, of 15°, 30°, 45°, 60°, 75°, or 90°. Therefore, the diameter, D1, of the circular input aperture varied as a function of α and took on values of 1.19 mm, 2.00 mm, 2.88 mm, 3.86 mm, 5.00 mm, and 6.40 mm, respectively. For reference purposes, we also fabricated a 400 µm diameter aperture in a 75 µm thick free-standing stainless steel foil and a TA with a taper angle α of 0°, corresponding to a straight 400 µm diameter aperture, in the 3 mm thick metal disk. These are shown schematically in the top and left portion of Fig. 1(a).
Fig. 1 (a) Schematic diagram of the different apertures structures studied. Top: cross-section of the reference aperture with an aperture diameter, Do = 400 μm and metal thickness, do = 75 μm. Middle left and bottom left: cross-section and top view, respectively, of a single tapered aperture with a variable taper full angle, α, and a fixed output aperture diameter, D2 = 400 μm. The taper angle determines the input aperture diameter, D1. The thickness of the metal disk is d = 3 mm. Middle right and bottom right: cross-section and top view, respectively, of the hexagonal lattice of tapered aperture arrays with nineteen apertures, fabricated normal to the disk surface, with a taper full angle, α’ = 30°, an output aperture diameter, D2 = 400 μm, and an input aperture diameter, D1’ = 2 mm. Once again, the thickness of the metal disk is d = 3 mm. (b) Photographs of the stainless steel tapered aperture array structure with 19 apertures, each with α’ = 30°. (c) Schematic diagram of the THz time-domain spectroscopy system. A collimated THz beam was normally incident on the sample. The radiated electromagnetic wave was detected using a photoconductive device for coherent broadband THz detection.
We also fabricated a periodic array of 19 TAs that were placed in a hexagonal lattice, as shown schematically in the lower right portion of Fig. 1(a). Each TA in this array was identical having a 400 µm diameter output aperture and a 30° full angle taper, corresponding to a 2 mm diameter input aperture, with the aperture axis normal to the disk surface. The center-to-center spacing between the TAs was 2 mm. Photographs of the array input and output faces are shown in Fig. 1(b).
To characterize these samples, we used a THz time-domain spectroscopy (THz-TDS) setup as shown schematically in Fig. 1(c). Details of the experimental setup and the advantageous properties of this approach have been discussed previously [19], and thus are presented only briefly here. Photoconductive devices were used for both emission and coherent detection. An off-axis paraboloidal mirror was used to collect and collimate the THz radiation as it propagated from the emitter to the sample. We used two different incident beam sizes for these measurements because the cross-sectional areas of the structures (i.e. individual TAs and TA arrays) were very different: a 1/e THz beam diameter of ~8 mm was used for the individual TAs, while a 1/e THz beam diameter of ~12 mm was used for the TA arrays. In both cases the incident beam size was larger than the cross-sectional area of the input apertures. Coherent THz pulses radiated from the output face of the apertures were focused onto a hyper-hemispherical silicon lens coupled photoconductive dipole detector that was located ~10 mm from the output plane of the apertures. The detected transient photocurrent was then Fourier transformed and normalized to the reference transmission; using this procedure, we obtained an electric field transmission spectrum that spanned the frequency range of ~0.05 - 0.8 THz.
We also performed numerical finite-difference time-domain (FDTD) simulations of the SPP propagation properties, where the metal was modeled as a perfect electrical conductor (PEC), since metal conductivities are typically high at THz frequencies, while the surrounding dielectric medium was assumed to be air. We used a spatial grid size of 50 µm, which was sufficient to ensure convergence of the numerical calculations, and perfectly matched layer absorbing boundary conditions for all boundaries. For the input electric field, we used a plane wave that was modeled temporally as the derivative of a Gaussian pulse. The resulting bipolar THz pulse had the same bandwidth and approximately the same pulse shape as was available in the experimental work. All of the simulated results were measured at specific spatial points in the vicinity of the output plane of the structure or averaged across area of the output aperture(s).
3. Experimental results, simulation and discussion
Figure 2 summarizes the basic concentration properties of individual TAs as a function of the taper full angle, α. In Fig. 2(a), we show the spectral field amplitude concentration factor, fE(ν), for each tapered aperture, which is defined as
where tα(ν) is the measured transmission amplitude spectrum through an individual 3 mm thick TA of angle α, and tref(ν) is the measured transmission amplitude spectrum through an individual cylindrical reference aperture of 400 µm diameter fabricated in a 75 µm thick stainless steel foil. There are several points to note regarding these data. For a TA with α = 0° (i.e. a straight 400 µm diameter aperture in a 3 mm thick metal disk), the transmission enhancement below ~0.5 THz appears to be strongly suppressed and there is essentially no transmission through the TA below ~0.4 THz. This behavior is reasonable for a circular aperture, where we would expect increasingly suppressed transmission below the cut-off frequency, νc ( = 1.841c/(πD2)) [20], where c is the speed of light in vacuum). In the case of the 400 µm diameter reference aperture, the film thickness (75 µm) is sufficiently small that frequencies below νc (~0.44 THz) exhibit only modest transmission suppression. However, in the TA structure with 0° taper angle (i.e. the straight aperture in the 3 mm thick disk), the waveguide cutoff phenomenon is more evident because of the metal thickness. Moreover, above ~0.6 THz, the transmission enhancement is ~1. Above the cutoff frequency the propagation constant is largely real, since the high conductivity of metals allows for low loss propagation through the aperture. Therefore there is relatively little difference between the 75 µm thick reference aperture and the 3 mm thick straight aperture at frequencies well above cutoff.Fig. 2 The experimentally measured spectral transmission properties of an individual tapered aperture structure as a function of the taper full angle, α. (a) Spectra of the field amplitude concentration factor, fE(ν), with respect to α, as noted. (b) Electric field concentration maximum, fE, as a function of the taper angle, α.
As the taper angle increases, the spectral concentration factor changes both in shape and amplitude. The shape of the concentration spectrum is related to the cutoff characteristics of the TA and is discussed in greater detail below. For the concentration amplitude, we can define a concentration factor maximum, fE (shown in Fig. 2(b)), that corresponds to the maximum value of fE(ν); we plot fE(ν) as a function of α in Fig. 2(a). It is clearly seen that fE(ν) is maximized for a taper angle α = 30°. Below 30° the small taper angle corresponds to an input aperture diameter that is only slightly larger than the output aperture diameter, thus limiting the maximum concentration factor. Above 30°, although the input aperture diameter becomes larger, there is increased reflectivity from the aperture sidewalls, which reduces the measured concentration factor. While the data presented correspond to a specific metal thickness, we have performed additional numerical simulations with different taper lengths between 3 and 10 mm and found that 30° appears to be the optimal taper angle in each case.
In order to confirm the experimental observations, we used analytical calculations as well as 3D FDTD simulations. In both cases we based our calculation on geometrical parameters that were used in the experiments. As we noted above, the effective cutoff frequencies for the TAs appear to red shift with increasing taper angle, in apparent contradiction with the more usual view that the cutoff frequency is associated only with the diameter of the output aperture. This red shift can be readily explained by the fact that the TA diameter varies along the length of the aperture. Consequently, for a straight 400 µm diameter aperture (i.e. α = 0°), the cutoff frequency is νc = 1.841c/(πD) = 0.44 THz along the entire length of the aperture. However, for α = 90°, the TA input aperture diameter is 6.4 mm, corresponding to νc = 0.03 THz, while the output aperture diameter is 400 µm, corresponding to νc = 0.44 THz. Therefore, the incident radiation experiences a continuously varying diameter as it propagates through the aperture. In order to model this geometry analytically, we analyzed the propagation properties of the lowest order mode of a cylindrical waveguide [21] as the waveguide diameter varied continuously, using the geometrical parameters of the TA. The resulting spectral concentration factors are shown in Fig. 3. We note that the mathematical details of this analysis are beyond the scope of this manuscript (and therefore are not given here). In order to compare the calculated and experimental spectral properties, each curve in Fig. 3 was scaled to match the maximum amplitude shown in Fig. 2(a). From the good agreement between the analytical calculations and experimental data, we conclude that the red shift in the effective cutoff frequency can be attributed to the aperture variation of the TA with α.
Fig. 3 Calculated fE(ν) spectra of single tapered apertures for various full taper angles, α, as given. The calculation is based on cylindrical waveguide theory as described in the text. The result for each α was scaled to match the experimental concentration amplitude shown in Fig. 2(a).
While the analytical model describes well the shift in the spectral transmission properties with α, it does not properly predict the concentration factor amplitude as a function of α. In order to model this property, we used 3D FDTD simulations for each of the TAs studied. As an example of the comparison between experiment and simulation, we show in Fig. 4(a) the spectral concentration factor for a TA with α = 30°. The model calculation agrees very well with the experiment, both in terms of spectral shape and amplitude. In Fig. 4(b) we show fE as a function of α obtained using these simulations, compared with the experimental values from Fig. 2(b). The agreement is good for all taper angles and demonstrates again that α = 30° is indeed the optimal TA angle.
Fig. 4 Numerical simulations of the field properties for individual tapered aperture structures. (a) The enhancement spectrum fE(ν) for a TA with α = 30° from simulation (red trace) and experiment (blue trace). (b) Field concentration factor maximum, fE, of the TAs with various α obtained from simulation (red dots); the experimental fE results from Fig. 2(b) are shown for comparison (blue dots).
For an individual TA having α = 30° and an output aperture diameter of 400 µm in a 3 mm thick metal slab, we find that the electric field concentration factor is ~4. Since an input aperture of an individual TA is typically much smaller than the incident THz beam, the overall optical throughput is limited. In such cases, it may be advantageous to use multiple apertures simultaneously and thereby more fully utilizing the incident THz beam. With this in mind, we fabricated a periodic array of TAs placed in a close packed hexagonal geometry, where the aperture axes are all normal to the metallic disk surface (Fig. 1(b)). Since the cross-sectional area of the output surface of this structure is significantly larger than the single TA, the detection geometry needed some slight modification. Specifically for all of the relevant samples, we now inserted a 50 mm focal length TPX lens between the sample and detector. In measuring the field amplitude concentration factor fE(ν) for the TA array and single TA, both sets of data were normalized using the spectrum of the same single reference aperture.
In Fig. 5, we show the spectral concentration factor for a single TA taken from Fig. 2(a), and TA array containing 19 TAs in a hexagonal pattern. In both cases, the apertures had a taper angle of α = 30°. Similar to the single TA that exhibits a relatively flat spectrum well above a cutoff frequency, the spectrum associated with the TA array also appears to be rather broad. However there is a decrease in the enhancement factor at high frequencies. Furthermore the concentration enhancement for the TA array at ~0.5 THz is only ~10 times larger than that observed for a single TA, rather than the factor of 19 expected from the number of apertures in the array. Both of these observations can be readily explained by the fact that the incident THz beam has a frequency dependent Gaussian spatial profile. We noted that the 1/e THz beam diameter used here was ~12 mm, but this only corresponds to the beam diameter at ~0.3 THz. For higher frequencies, the beam diameter is correspondingly smaller. Thus, the factor of 19 enhancement could only be achieved if the incident beam was uniform over the array area. The fact that the beam becomes increasingly smaller with increasing THz frequency may also explain why the enhancement is not flat above ~0.5 THz. We therefore conclude that, while a larger concentration enhancement can be achieved using a larger TA array, it is also necessary that the incident beam be larger than the input cross-sectional area of the array at all relevant THz frequencies.
Fig. 5 The spectral amplitude concentration factors, fE(ν) measured for the tapered aperture array and a single TA having α = 30° full angle. Both spectra were normalized by the transmitted field of the same reference obtained from a single cylindrical aperture.
Given that the apertures in the array are placed in a periodic geometry, it is reasonable to determine whether or not any transmission resonances should be present based on the excitation of surface plasmon-polaritons (SPPs). These resonances arise from constructive interference between SPPs scattered from discontinuities (apertures, in this case) that are present on both the input and output faces of the structure. In a conventional hexagonal array of subwavelength apertures, we expect well-defined transmission resonances at wavelengths given by [18]
where P is the periodic spacing between apertures, i and j are integers, and nSPP is the effective refractive index for a propagating SPP (for a metal-air interface, nSPP ~1 at THz frequencies). Thus the lowest order resonance (i = ± 1, j = 0) is expected to occur at , which for P = 2 mm corresponds to 0.17 THz. This frequency, however lies in the low transmission regime of the TA array and, therefore, has minimal impact on the resulting spectrum. The next two higher order resonances are expected to occur at ν = 0.3 THz (i = 1, j = 1) and ν = 0.34 THz (i = 2, j = 0 or i = 0, j = 2). However, these resonances are also suppressed and are not clearly present in the data. This suppression arises primarily from the fact that the output aperture diameter (400 µm) is significantly smaller than the periodic spacing (2 mm), resulting in weak resonances below the cutoff frequency associated with the output aperture (0.44 THz) [22].Based on the available enhancement with a TA array, it is reasonable to consider modifications to the geometry that would allow for tighter focusing. Specifically, for the TA array discussed above, the input cross-sectional area and the output cross-sectional area are approximately the same. Thus, there is minimal spatial focusing caused by such structures. We now consider the effect of tilting the axes of the outer TAs in the array towards the array center, thereby reducing the output span of the apertures, which allows for radiation concentration to a smaller area on the output plane. By maintaining the hexagonal geometry and tilting the axes of the outer TAs (inclination angle, θ with regard to the normal axis), we maintain a six-fold rotationally symmetric pattern on the output surface of the array. In this case, the input surface has a periodic spacing p1 = 2 mm, while the output surface has a periodic spacing p2, which is determined by the tilt angle θ. This is shown schematically in Fig. 6(a). Figure 6(b) displays a snapshot of the electric field pattern near the exit plane of a TA array structure with p2 = 1.0 mm. It can be seen that the field strength of the six outer TAs is weaker than that of the central TA. Since these are time-domain simulations, it is also worth noting that the transit time through the tilted apertures is slightly longer than the apertures that are not tilted. Therefore, the field maxima are not present in the output plane at the same time for the central aperture and the outer apertures. Nevertheless, as we show below, tilting the apertures reduces the optical throughput.
Fig. 6 Numerical simulations of structures designed for tighter field focusing. (a) Schematic diagram of the hexagonal tapered aperture array with seven apertures, where the axes of the outer apertures were tilted with an inclination angle, θ. The reduced spacing of the output apertures, p2, is shown in the top view on left top, cross-section on left bottom and isometric view on the right. For all the apertures, the input diameter D1’ is 2 mm and the output diameter D2 is 400 μm. (b) A snapshot of the planar electric field distribution near the output surface. (c) Maximum value of fE calculated for an individual TA with tilted axes of different inclination angles, θ, for the two orthogonal THz field polarizations. (d) Spectra of field concentration intensity, σ(ν), of TA arrays for various θ as given. The inset shows the maximum σ value extracted from σ(ν) at various values of θ.
Clearly, there is a need to consider the optimal tilt angle and what effect an aperture tilt has on the polarization properties of an individual TA. In Fig. 6(c), we show the concentration enhancement, fE, for a tapered aperture with input diameter D1’ = 2 mm and output diameter D2 = 400 μm that is tilted by an angle θ along the x-axis for incident THz radiation that is polarized along the x- and y-axes. Here we present an angle range from 0° to 33.7°. For an aperture with no tilt, the transmission properties are independent of the polarization and the two orthogonal polarizations exhibit identical characteristics, as expected. As the incline angle increases, fE becomes increasingly smaller. Interestingly, y-polarized radiation exhibits higher throughput for low tilt angles and there is a crossover between 20° and 25°. This behavior is reversed for TAs tilted along the y-axis, as expected by symmetry. We expect that this crossover is related to the reflection properties associated with the two polarizations.
We now consider the concentration properties of a TA array that consists of 7 TAs placed in a hexagonal geometry with variable tilt angles. The issue now is not simply calculating the concentration enhancement, fE, of the array, but rather if that enhanced transmission can be focused to a smaller output area. Therefore, we define a spectral field concentration density, given by
where λ is the longest distance in the array between centers of the output apertures. For a hexagonal lattice that contains only 7 TAs, λ = 2p2. In Fig. 6(d), we show the spectra of the field concentration density, as a function of the tilt angle, θ. If we consider the maximum value of σ(ν), which we define as σ, it increases with increasing tilt angle, as shown in the inset of Fig. 6(d), despite the fact that the propagation throughput becomes progressively reduced.The notion that an array based on tilted TA can be used to concentrate and focus radiation suggests that cascaded structures may be useful in increasing the optical throughput further. In Fig. 7, we sketch one possible example of such a combination. More generally, through appropriate design, the TA array in the top layer could be used to focus the field into the input aperture in the lower layer. Such an approach may circumvent practical limitations and allow for greater flexibility in the design of structures for concentration of radiation.
Fig. 7 Schematic diagram for a proposed cascaded taper structure. (a) Cross-section of the structure. The diameter of the input aperture (2.4 mm) on the bottom layer is equal to the longest distance between the edges of the output apertures on the top layer. (b) Decomposition of the structure into two sections: Structure 1 consists of seven tapered apertures forming a hexagonal array with input aperture diameters of 2 mm, output aperture diameters of 400 μm and axis inclination angle of 18.4°; Structure 2 consists of a single tapered aperture with input aperture diameter of 2.4 mm, output aperture diameter of 400 μm and taper full angle of 36.9°.
4. Conclusions
In summary, we have demonstrated experimentally that periodic arrays of tapered apertures are able to concentrate broadband THz radiation. The individual aperture parameters were based on experimental and numerical determination of the optimal taper angle. Interestingly, the effective cutoff frequency red shifted with increasing taper angle, which was a consequence of the rate of change in a continuously varying aperture diameter. With the optimal taper angle in hand, we constructed a hexagonal lattice of nineteen TAs that led to a factor of ~10 increase in the broadband THz throughput for frequencies above the effective cutoff frequency. Further concentration of the transmitted radiation with higher field density was shown to be possible by tilting the apertures in an array toward the array center. Such a demonstration suggests that cascaded TA structures may allow for an even greater level of field concentration than any single layer structure is capable of providing.
Acknowledgments
This work was supported by the NSF MRSEC program at the University of Utah under grant # DMR 1121252.
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