1. Introduction

The development of practical terahertz-range systems is dependent upon beamforming techniques, in order to direct the flow of terahertz power and manipulate the shape of propagating wavefronts [1]. In terahertz imaging [2,3] and spectroscopy [4,5], such techniques are deployed to focus terahertz power into a small volume, thereby concentrating the interaction of terahertz waves with a given sample. On the other hand, terahertz communications applications benefit significantly from collimated beams with a large, planar wavefront that is resistant to diffraction, to achieve high antenna gain [6,7]. For these reasons, optics such as convex terahertz lenses and curved reflector dishes, which are capable of both focusing and collimation, are near-ubiquitous in practical demonstrations of terahertz technology.

Integrated terahertz systems are necessary in order to develop compact devices that are mass-producible. This entails a transition from discrete terahertz components that require assembly towards mask-oriented lithographic processes that realize a large number of interconnected components simultaneously. However, currently available terahertz optics are not amenable to this transition, despite their importance to terahertz technology. This is because such optics are typically manufactured by means of subtractive machining [8–10], molding [11], casting [12], and 3D-printing [13]. Such techniques are not mass-producible on a scale comparable to electronic integrated circuits, nor are they amenable to concurrent fabrication of interconnected optical components. As such, the assembly and often-manual alignment of the optical components of a given terahertz system must necessarily take place after the manufacture of those optics. Furthermore, the requirement of a separate mount or holder for each optical component typically engenders a bulky overall system.

We aim to develop integrated lenses for the terahertz range in a micro-structured silicon platform. It is a requirement that the lenses be manufactured with mask-oriented etching processes, and that several lenses are concurrently fabricable in their appropriate relative alignment, from a single high-resistivity silicon wafer. So as to meet this requirement, we eschew the free-space beams that conventional terahertz lenses interact with, and substitute slab-mode beams within a silicon wafer in their place, as illustrated in Fig. 1. In this way, terahertz waves that are confined into a silicon wafer serve as a two-dimensional analog of the three-dimensional beams that propagate in free space.

 

Fig. 1. Concept to support the integration of terahertz lenses, whereby (a) a three-dimensional beam traveling in free-space is replaced with (b) radiation that is confined in the $z$-dimension into a dielectric slab mode in an un-patterned silicon wafer. The radiation is collimated in the $xy$-plane, and in the context of this work, is termed a “slab-mode beam.”

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In this work, we devise an effective and broadband technique to launch a collimated terahertz slab-mode beam. The integrated lens that we employ to develop this beam is a half-Maxwell fisheye lens [14–16]. This optic is derived from a Maxwell fisheye lens, which has previously been realized in the terahertz range using the dispersion of metallic parallel-plate waveguides [17]. The half-Maxwell fisheye lens is a compact, high-numerical-aperture gradient index (GRIN) optic that is amenable to our chosen silicon slab-mode platform, as a given bespoke index distribution is straightforwardly implemented with a through-hole array of subwavelength pitch [18–20]. It is noted that other approaches are viable to create GRIN optics, such as nonuniform-thickness [21] and multi-layer dielectric structures [22], however such techniques may lead to an increase in complexity, and render the structures incompatible with mask-based etch processes. Metamaterials present another option [23], but the resonant nature of conventional metamaterials typically reduces bandwidth and efficiency. In addition to these factors, the use of a through-hole array in a silicon slab also offers versatility as, aside from GRIN optics, terahertz-range photonic crystal waveguides may also be implemented in the same platform [24–28]. We employ waveguides of this sort as an interface to the integrated optic, as they exhibit narrow field confinement, for strong localization of terahertz waves in the vicinity of its focus. Furthermore, terahertz-range photonic crystal waveguides have previously proven suitable for monolithic integration with GRIN optics [20,29], as well as with functional passive components such as frequency-diplexers [30]. Aside from passive components, active terahertz components such resonant tunneling diodes have previously been coupled to terahertz photonic crystal waveguides [31], thereby providing a means to incorporate sources and detectors, for complete terahertz systems.

The functionality of the slab-mode beam launcher that is the main subject of this work is experimentally validated by means of a two-lens device, which couples radiation into- and out-of a slab-mode beam. In this way, the existence of the slab-mode beam is verified over a $\sim$40% terahertz-range bandwidth, with peak efficiency of $\sim$86%. On the other hand, we observe undesired variation in our measured results. This is ascribed to reflections that are exacerbated by the presence of photonic crystal adjacent to the feed point of the integrated optic.

2. Concept: Half-Maxwell fisheye lens as slab-mode beam launcher

A silicon-slab platform is innately amenable to GRIN lenses, as an effective medium of engineerable refractive index is readily readily patterned into the dielectric slab in the form of a periodic through-hole lattice [20,29,32]. The effective index is mediated by the filling factor of air that is introduced to the silicon by the through-holes, which are necessarily of subwavelength pitch.

As indicated in Fig. 2(a), a Maxwell fisheye lens is a radially-symmetrical optic that maps a point source to a diametrically opposed focal point—both of which being situated at the circumference. The required refractive index distribution to achieve this beamforming behavior is given by [33],

 

Fig. 2. Maxwell fisheye lens, showing (a) conceptual schematic of its beamforming functionality, i.e. mapping a circumferential point source to a diametrically opposed focus, (b) a half-Maxwell fisheye lens, implemented with a GRIN effective medium through-hole lattice, employed as a slab-mode beam coupler, (c) the refractive index profile chosen in the present work, for both the bulk and modal index, and (d) the corresponding through-hole diameter, $D$, as a function of radial position in the lens, in units of lattice constant $a$.

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If a two-dimensional Maxwell fisheye lens is bisected to form a semicircle, as illustrated in Fig. 2(b), then a half-Maxwell fisheye lens is realized. The beamforming functionality of the Maxwell fisheye lens is essentially interrupted, and it will collimate radiation from a point source situated at the apex of its $\pi$-radian arc. If the flat portion of the semicircle’s perimeter is coupled to a dielectric slab, then this half-Maxwell fisheye lens is in-principle capable of launching a slab-mode beam. Although there is non-negligible index mismatch at this interface to the high-index dielectric slab, it is noted that a Maxwell fisheye lens’ index is highest in the center, as aforementioned. Thus, the value of $n_{\mathrm {max}}$ can be set as close as possible to that of the slab-mode, so as to reduce reflections due to index mismatch at the center of the beam, where intensity is likely highest. It is noted that equivalent collimating functionality is provided by a Luneburg lens [20], but the use of such an optic would produce higher index mismatch to the slab, as it would make contact with the slab at its edge, where the refractive index is at minimum.

The aperture width of the half-Maxwell slab-mode beam launcher is twice the distance from the focus to the un-patterned slab, corresponding to high numerical aperture with f-number 1/2. This is because the Maxwell fisheye lens essentially planarizes the outgoing wavefront whilst simultaneously allowing it to spread out, rather than correcting the wavefront after diffraction has caused it to diverge and cover a large area, as in [29]. In the cited work, a micro-scale GRIN silicon lens was integrated directly with a terahertz photonic crystal waveguide. However, the lens was of width 8 mm and focal length 10.2 mm, corresponding to larger f-number 1.3. As such, a half-Maxwell fisheye lens facilitates the development of a wider-aperture slab-mode beam in a comparatively short propagation distance, engendering a more-compact device. This compactness is a major factor that motivates the specific choice of the half-Maxwell fisheye lens as slab-mode beam launcher. Additional to that, the use of slab-mode beams in general also engenders compactness; as the refractive index of the slab is higher than that of free space, an in-medium wavelength is smaller. Thus, less absolute physical size is needed to achieve an aperture of several wavelengths across, which is a requirement to avoid excessive diffraction of the collimated slab-mode beam.

3. Design

3.1 Half-Maxwell fisheye lens

As stated in Section 2, the GRIN lens is implemented using an effective medium composed of an array of holes in a silicon slab. Specifically, this array is an equilateral triangular lattice of cylindrical through-holes, which is a structure that we have previously employed to realize effective media [20,32,34]. In order to deploy this structure to realize an integrated lens, a predictive model of the effective index of the hole lattice, $n_{\mathrm {eff}}$, is required. It is noted that, in the present work, the orientation of the electric field is always perpendicular to that of the cylindrical holes. Thus, we may employ a two-dimensional version of the Maxwell Garnett effective medium approximation [35],

At this stage in the design of the half-Maxwell fisheye lens, it is necessary to select a value for $n_{\mathrm {max}}$. As stated in Section 2, it is desirable to maximize this value so as to minimize reflections at the interface to the un-patterned dielectric slab. However, there is a practical constraint that precludes the use of the modal index of the un-patterned slab itself for this purpose. This is because there is a minimum viable hole diameter that can be reliably fabricated, namely 20 µm for our foundry, and hence effective index would not be continuously variable over the range that demands a smaller hole radius. This would engender deviation from the refractive index distribution that is given in Eq. (1), thereby producing aberration. For this reason, $n_{\mathrm {max}}$ is chosen such that it corresponds to a fabricable hole diameter. The slab-mode index found in Eq. (5) is mildly dispersive, and hence its effective modal index varies with frequency. A consequence is that $n_{\mathrm {max}}$ and the ensuing function $n(r)$ must correspond to a specific design frequency. We choose 330 GHz for this purpose, as it is readily accessible to our available experimental equipment.

The thickness of our silicon slab is 200 µm, and hence the slab-mode index is $\sim$3.04 at the design frequency, according to Eqs. (4) and (5). The value of $n_{\mathrm {max}}=2.9$ is chosen, as it is lower-than, but close-to the un-patterned slab index, and Eq. (1) yields the refractive index distribution shown in Fig. 2(c). Thereafter, Eqs. (4) and (5) are employed to convert these slab-modal index values into bulk indices, which is also plotted in Fig. 2(c). Finally, Eqs. (2) and (3) are employed to convert the desired distribution of bulk index into a distribution of hole diameters as a function of radial position, and this is given in Fig. 2(d). This information is presented in relative terms, and hence it can in-principle scale to any choice of $r_{\mathrm {max}}$ and $a$, although in practice the former must be several wavelengths long, and the latter must be subwavelength. We select 4 mm for the lens radius, as it corresponds to a diameter of over 26.5$\lambda$ of the TE$_0$ slab mode at the design frequency. For the lattice constant, it is noted that the minimum value of hole diameter is $D(r=0) \sim 0.233 a$. It is stated above that the minimum viable hole diameter is 20 µm, and hence the lattice constant must be at least $a \geq 20$ µm$/0.233 \sim 86$ µm in order to abide this constraint. Aside from this requirement, $a$ is left variable at this stage.

3.2 Broadband photonic crystal waveguide

In order to employ the half-Maxwell fisheye lens for its intended purpose as a slab-mode beam launcher, a feed structure is required to couple terahertz waves to the focus at the apex of its $\pi$-radian arc. There are several constraints upon this design. Firstly, an all-silicon structure that can be fabricated concurrently with the GRIN lens is necessary, as we aim to realize a monolithically integrated device. This feed structure must also exhibit strong field confinement at the point of contact with the half-Maxwell fisheye lens, so as to resemble a point source as closely as possible. Finally, reflections at the feed point due to mismatch with the low-index slab-mode should be minimized. In the interest of brevity, the latter two constraints are termed the “confinement constraint” and the “index-matching constraint,” respectively. Both of these constraints must hold in broadband.

Previously, we have employed rectangular cross-section dielectric waveguides to feed a closely-related integrated lens [20]. In that work, the waveguide was narrowed at the point of contact with the lens. In a manner similar to a fibre edge-coupler, narrowing a dielectric waveguide in this way essentially de-localizes the fields of its mode into the lower-index surrounding material, thereby reducing the effective modal index [37]. Thus, by narrowing the waveguide cross-section at the point of contact with the lens, the matching constraint is satisfied. However, this improved matching presents an unfortunate trade-off with the confinement constraint; due to the aforementioned de-localization phenomenon, the modal field distribution becomes large, and hence it is a poor approximation of a point source, leading to low aperture efficiency. Aside from narrowing the dielectric waveguide, it is also possible to reduce the index of the medium in the waveguide volume, e.g. with effective medium techniques [32], but this lowers index-contrast with the surrounding medium, which also engenders a large modal field distribution. For these reasons, it appears that a dielectric waveguide of lower modal index will also exhibit weaker field confinement. Thus, a dielectric waveguide that satisfies the index-matching constraint will in-general be less-well-suited to satisfy the confinement constraint. We therefore deem such dielectric waveguides to be unsuitable as a feed structure for GRIN lenses of this sort.

In view of the above-stated limitations upon dielectric waveguides, we clad the feed structure with photonic crystal in the present work, thereby realizing a photonic crystal waveguide [25,38]. In brief, photonic crystal is a form of periodically-structured medium that can exhibit a photonic bandgap, which is essentially a range of frequencies over which no propagating modes exist. As with the effective medium that comprises the integrated optic, a two-dimensional photonic crystal may be implemented with a through-hole array in a silicon slab, albeit the pitch and diameter of the holes is usually larger in the case of photonic crystal. Introducing a line defect into this periodic structure can produce a track along which terahertz waves may propagate, and this is termed a photonic crystal waveguide. A waveguide of this sort is not subject to the same undesirable trade-off between index-matching and confinement as the conventional dielectric waveguide. The reason for this is that the presence of photonic crystal rapidly attenuates the evanescent field that surrounds the waveguide, thereby producing narrow in-plane confinement, and this remains true even in cases of lower modal index.

Although beneficial for confinement and matching, the use of photonic crystal waveguides poses a potential disadvantage, as such waveguides typically exhibit narrower relative bandwidth than conventional dielectric waveguides due to the Bragg-mirror effects that arise from periodic modulation of the waveguide’s dimensions by the photonic crystal structure [39]. This limitation upon bandwidth is undesirable for practical applications of terahertz waves, and hence we have previously developed a photonic crystal waveguide that suppresses these Bragg-mirror effects [28]. In this way, a broadband terahertz photonic crystal waveguide was realized, and the structure of this waveguide is illustrated in Fig. 3(a). Only a superficial description of the waveguide design is warranted here, as it was reported in detail in the cited work. The photonic crystal medium is implemented with an array of through-holes in a 200 µm-thick silicon slab. There is no supporting substrate; the silicon slab is surrounded on all sides by air, and in experiment, the sample was held in place using ordinary tweezers. The holes are of radius 137.8 µm, arranged in an equilateral lattice of pitch 336 µm. The waveguide track is 459.7 µm-wide, and is clad on both sides by semi-cylindrical through-holes. This feature reduces the periodic modulation of waveguide dimensions that is commonly observed in photonic crystal waveguides, thereby suppressing Bragg mirror effects, and resulting in a dominant mode with broad usable bandwidth, as shown in Fig. 3(b). The waveguide is also observed to exhibit numerous undesired higher-order propagating modes over this bandwidth, but it is possible to avoid exciting them in practice.

 

Fig. 3. The broadband photonic crystal waveguide that serves as an interface to the integrated optic, showing (a) a schematic diagram, with annotated dimensions, and (b) simulated transmission of the dominant mode through a 1 cm length of this waveguide.

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3.3 Slab-mode beam launcher

We employ the broadband terahertz photonic crystal waveguide that was presented in [28] as an interface to the half-Maxwell fisheye lens-based slab-mode beam launcher that is detailed in Section 3.1. It is intended to provide adequate index matching whilst simultaneously exhibiting narrow field confinement over a broad usable bandwidth. Due to the nonuniform nature of photonic crystal waveguides, the specific point at which the periodic photonic crystal structure is interrupted in order to transition to the lens is relevant to the confinement constraint. In the cited work, modal field distributions are given that show that field confinement is stronger in the space in-between two of the consecutive semi-cylindrical through-holes that clad the waveguide track, and hence the waveguide is terminated at this point.

The periodic lattice structure of the effective medium continues beyond the circumference of the integrated lens, up to the point at which it makes contact with the photonic crystal medium that clads the waveguide. In this lens-external region, the holes are at their maximum size, corresponding to the outermost edge of the lens itself. This is included as, if un-patterned silicon were to surround the lens body, then this would likely attract terahertz waves into this high-index region, and detract from the intended guided beam. Another option is to simply remove all of the silicon that surrounds the lens, and essentially fill this lens-external region with air. This approach is likely to perform well electromagnetically, but it is eschewed for mechanical stability reasons. Without any form of cladding, the only point of contact between the lens and the photonic crystal waveguide would be at the feed point itself, and this would be extremely fragile. Thus, surrounding the lens with low-index effective medium is chosen in view of of the trade-off between electromagnetic performance and mechanical stability.

A gradual transition between the waveguide and the lens is desirable so as to avoid a strong discontinuity, and suppress any reflections at this interface. To this end, a hole lattice medium is included in the throat of the waveguide, with lattice constant $a\sim 106.2$ µm, so as to fit an integer number of periods into the waveguide track width. This choice also satisfies the constraint upon minimum lattice constant stated in Section 3.1, where $a \geq 86$ µm in order to ensure its smallest holes may be fabricated accurately. The periodic lattice arrangement of the effective medium is common to both the matching structure and to the lens body itself. That is to say, the lens body is a continuation of the effective medium in the matching structure, so as to avoid disturbing its periodicity. In the matching structure, the holes progressively widen from $\sim$21.1 µm to $\sim$82.2 µm in diameter, over a distance of $6a\sim 636.9$ µm, as measured from hole centers. This design is the result of a careful process of manual optimization, where the ensuing slab-mode beam quality and matching performance are subjectively appraised after each individual simulation run. The final design of the slab-mode beam launcher is shown in Fig. 4(a).

 

Fig. 4. The design of the slab-mode beam launcher, showing (a) an illustration of the overall structure, with optical micrographs of specific features of the fabricated device given as insets. The scale bars are 0.5 mm. (b) Simulated reflection coefficient, and (c)-(h) Simulated field distributions in logarithmic scale, where each is normalized to its own maximum. Subfigures (c), (e), and (g) show the xy-plane, and (d), (f), and (h) show the xz-plane at corresponding frequencies.

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Full-wave simulations are conducted with the commercial finite-element analysis software package CST Microwave Studio. This software provides two solver engines, namely a time-domain and a frequency-domain solver, and the latter is chosen for its greater accuracy for problems of this sort, as its tetrahedral mesh is better-suited to approximate cylindrical holes than the hexahedral mesh of the time-domain solver. In simulation, the dielectric loss is neglected, as the absorption of high-resistivity intrinsic silicon is essentially negligible in the 300-GHz band [40]. Excitation is provided to the integrated lens via a hollow metallic waveguide of inner rectangular cross-section 864 µm$\times$432 µm, which is coupled to a 5.2 mm-long tapered spike at the photonic crystal waveguide’s termination (not shown in Fig. 4(a)) by simply inserting the spike into the hollow waveguide. From simulations of the waveguide in isolation, we extract the loss due to this coupling structure by comparison of overall simulated transmission with propagation loss. In this way, we determine the loss of this coupling structure to be lower than $0.05$ dB for frequencies in the range from 277 to 430 GHz. That said, this loss may be higher in practice due to imperfect alignment in experiment.

It is noted that the un-patterned slab-mode region is necessarily finite in simulation, but reflections from the edge of this slab would likely confound the simulated results. For this reason, waves that are incident upon the boundary of the finitely-sized slab-mode region are simply absorbed. The simulated matching performance is given in Fig. 4(b), and it can be seen that low reflection is achieved over a broad bandwidth. Peak reflection occurs at $\sim$293 GHz, with less than 3% of stimulated power reflected back to the exciting hollow waveguide port. It is of interest to determine the impact of the progressive index matching structure on device performance. For this reason, a second simulation is performed where the through-holes that comprise this structure are removed, and hence the photonic crystal waveguide is connected directly to the effective medium region. The results of this simulation are also plotted in Fig. 4(b), and it is clear that reflected power is indeed significantly larger than in the case with the hole-lattice matching structure. Thus, we may conclude that the hole lattice matching structure reduces reflection at the interface between the photonic crystal waveguide and the integrated optic, as intended. It is noted that it may be possible to optimize the cut position of the photonic crystal waveguide in order to increase coupling efficiency without the need for the hole lattice matching structure [41]. That said, this optimization may negatively impact the field distribution at the feed point. Aside from this feed point, there is a second interface that is common to both simulations, namely the boundary between the half-Maxwell fisheye lens and the un-patterned silicon slab. Some reflection is expected here due to the finite step in effective index. However, as the reflection magnitude is below 10% across the entire bandwidth for both simulations, we may conclude that reflection at this interface is not significant.

For the simulations that include the progressive matching structure, field plots in the $xy$-plane are included in Figs. 4(c), 4(e), and 4(g). It can be seen from these results that the integrated lens produces a collimated slab-mode beam of high quality over a broad bandwidth of 120 GHz, as intended. This is a larger bandwidth than previous photonic crystal waveguide-coupled all-silicon integrated terahertz lenses [20,29], owing to our choice of a broadband photonic crystal waveguide [28] for the feeding structure. The outgoing wavefronts of our demonstrated device are highly planar, and occupy a large proportion of the half-Maxwell fisheye lens’ available aperture. In the ideal case, the phase of the $y$-component of the electric field is constant across the aperture, along a line that is parallel to the $y$-axis in the $xy$-plane. Thus, beam quality may be quantified with absolute deviation in phase from its value at $y=0$; high deviation is indicative of a poorly collimated beam. We calculate an average of this deviation that is weighted by power density. We subsequently convert the resultant phase value into distance via normalization by the in-medium wavenumber, in order to allow for fair comparison across different frequencies. For all three frequencies shown, this average deviation is 6–8 µm, which is less than 4% of an in-medium wavelength. Visually, the beam quality is highest at the design frequency of 330 GHz, which is to be expected, as deviation from this frequency essentially introduces error into the bespoke index distribution of the Maxwell fisheye lens. At 270 GHz, some undesired field leakage into the photonic crystal slab is evident, which is likely due to the fact that this is close to the waveguide’s cutoff, according to Fig. 3(b). Nevertheless, a clear slab-mode beam remains evident in this case. Field plots in the $xz$-plane are included in Figs. 4(d), 4(f), and 4(h), and they are indicative of minor radiation loss in the vicinity of the point at which the waveguide makes contact with the lens. Analysis of power flow in the full-wave simulation results confirm that such radiation accounts for $\sim -15$ dB of stimulated power at the design frequency, and $<\;-10$ dB across the device’s operation bandwidth.

4. Experiment

We aim to experimentally verify the formation of the slab-mode beam that is observed in the simulations presented in Figs. 4(c)–4(e). However, it is not possible to directly observe the field distribution within the silicon slab without significantly disturbing it. For this reason, we devise a two-port strategy, whereby a slab-mode beam is first launched, and then converted back into a waveguide mode that may be experimentally probed. Due to reciprocity, we reason that the reception of the slab-mode beam may also be performed by the slab-mode beam launcher itself. Thus, our experimental test-case device consists of two identical slab-mode beam launchers that are directed toward each other, and a 2 cm-long un-patterned slab region that separates them. Any transmission through this device is evidence of a slab-mode beam in the un-patterned region, as intended. The test-case sample is fabricated from a 20 k$\Omega {}$-cm silicon wafer using a process involving photolithography and deep-reactive ion etching, thereby producing vertical-walled through-holes of high quality [25]. Optical micrographs of this structure are given as inset to Fig. 4(a), and it is also shown in a photograph in Fig. 5(a). It is noted that the width of the slab in the fabricated sample is enlarged with respect to Fig. 4(a) to allow space to hold the sample with tweezers.

 

Fig. 5. Experimental characterization of Maxwell fisheye lenses, showing (a) a photograph of the experimental characterization of a sample with a 2-cm slab length, (b) measured transmission, compared to the results of full-wave simulations, and (c) smoothed measured transmission, compared to the results of time-gated full-wave simulations. Insets to this plot indicate the structure of the two simulations that were performed; first with the design as-is, and then with all GRIN portions removed by substituting un-patterned silicon for effective medium.

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An all-electronic experimental setup is employed to characterize the fabricated sample. Terahertz-range signals are generated by up-conversion of a continuous-wave (CW) millimeter-wave signal, which is tunable from 28.9 to 43.3 GHz, by means of a $\times$9 multiplier. Terahertz waves are output from the source via a hollow rectangular metallic waveguide of inner dimensions 864 µm$\times$432 µm. This terahertz power is coupled to the slab-mode beam launcher via a tapered spike at the termination of the photonic crystal waveguide, which is simply inserted into the hollow rectangular waveguide, in order to provide broadband index matching. Thereafter, the slab-mode beam is launched by the integrated optic, and after propagating through the 2 cm-long un-patterned region, it is received by the second half-Maxwell fisheye slab-mode beam launcher. The terahertz power is subsequently coupled to a second hollow metallic waveguide, and is down-converted to the millimeter-wave range prior to analysis with a spectrum analyzer. This measurement is repeated in stepped frequency, until a transmission spectrum spanning from 260 to 390 GHz is acquired. Finally, the measured power is normalized by the results of a back-to-back measurement, i.e. with the source and detector coupled directly together.

The experimental results are plotted in Fig. 5(b), and they confirm that non-negligible power is indeed transferred from the transmitter to the detector, which is strong evidence of the existence of the slab-mode beam. However, a degree of undesired variation is observed in this measurement. We aim to determine whether this variation is a real, physical effect that is intrinsic to the device, or if it is simply due to experimental error and noise. To this end, full-wave simulations of the test-case structure are performed. Due to the large size of the device in comparison to a single slab-mode beam launcher (i.e. for the simulations presented in Section 3.3), the frequency-domain solver provided by CST Microwave Studio is eschewed in favor of the time-domain solver that is included in the same software package. Although the latter is of lower accuracy for problems of this sort, it is better-suited to larger simulation domains. This is because it advances a time-domain pulse stepwise through the mesh, whereas the frequency-domain solver must calculate inversions of a large matrix. The results of these simulations are shown in Fig. 5(b), showing a similar trend to the measured results, and an equivalent degree of variation. We may conclude from this result that the variation is not due to experimental error and noise, but rather, it is a genuine feature of the fabricated test-case device.

The existence of undesired reflections is a feasible explanation for the observed variation. Essentially, not all waves that are incident upon the receiving lens will be accepted by the associated photonic crystal waveguide. This is because, for a half-Maxwell fisheye lens, only perfectly planar wavefronts that are incident at the appropriate angle will be directed to its focus. In practice, there is always a finite amount of deviation from this idealized case, either due to diffraction of the guided mode within the slab, or to irregularity in the initially-launched beam, and this reduces overall transmission efficiency. The small proportion of terahertz power that is not perfectly matched to the lens aperture will be directed to locations adjacent to the waveguide mouth, where it encounters the photonic crystal medium itself. The photonic crystal exhibits a photonic bandgap, and hence it cannot accept the incident energy in the form of a propagating slab mode. Thus, the mismatched waves are strongly reflected back into the integrated optic, and subsequently re-enter the silicon slab region with an unspecified field distribution. Thereafter, these waves undergo several reflections at the boundaries of the device before ultimately being absorbed by either the source or detector, or by escaping to free space. As such, the variation may be explained by the existence of waves that are essentially trapped within the device, owing to reflection that is enhanced by the photonic crystal’s bandgap, which incidentally is engineered specifically to reject incident terahertz waves.

In order to confirm the above explanation of the variation that is observed in experiment, a time-gating procedure that is detailed in the appendix is performed upon the simulated results. This facilitates isolation of the first pulse of terahertz power that is incident upon the receiving port, and in doing so, eliminates any contributions arising from multiple reflections. The results of this procedure are plotted in Fig. 5(c), and it is clear that the variation is significantly reduced. This time-gating procedure also facilitates quantitative comparison of the desired and undesired portions of the signal. By integration of signal power in the time-domain, it is found that 4.5% of total received energy is contributed by undesired reflections. The experimental results are derived from scalar CW measurements, and hence it is not possible to subject them to the same time-gating procedure. As such, data smoothing [42] is performed on the frequency-domain trace directly, so as to extract the overall trend, and the results are also shown in Fig. 5(c). It can be seen that the simulation and measurement are in reasonable agreement, despite having been processed in very different ways. We consider this to be adequate validation of the functionality of the integrated lens-based slab-mode beam launcher that is the main subject of this work. Furthermore, the smoothed measured results show a transmission-efficiency of $\sim$75% at a frequency of 321 GHz, which is close to the design frequency of 330 GHz, and the edges of the 3 dB bandwidth are encountered at the limits of the measured frequency range—approximately 40% relative bandwidth—and hence the experimental results indicate that a slab-mode beam is indeed launched and detected in broadband. That said, such values are approximate due to the variation observed in experiment.

We aim to verify that the broadband transfer of terahertz power is indeed due to the existence of the half-Maxwell fisheye lens. To this end, additional simulations are performed with the lens and matching structure removed, for comparison. This entails replacing all of the through-holes that comprise the effective medium with an un-patterned silicon slab. The simulation is repeated with this modified structure, and results are given in Fig. 5(b). There is significant variation, as before, and hence the time-gating procedure is also repeated, with results shown in Fig. 5(c). In this case, the smoothed results indicate peak power transfer of $\sim$6% in the first pulse (i.e. prior to reflections), which is less than a tenth of the power that is observed with the lens in place. This provides further confirmation of the functionality of the half-Maxwell slab-mode beam launcher that is the main subject of this work.

5. Conclusion

Owing to the ubiquity of lenses in terahertz technology, we aim to develop integrated terahertz lenses that can be fabricated with a single-mask etch process, so as facilitate miniaturization and mass-production of terahertz optical devices, and avoid the necessity of manual alignment and assembly. We concentrate our efforts upon slab-mode beams in un-patterned silicon of uniform thickness, which serves as a two-dimensional analog of three-dimensional free-space beams.

In this work, we have designed and experimentally validated a half-Maxwell fisheye lens-based slab-mode beam launcher. In experiment, a two-port test case was devised to confirm the existence of the slab-mode beam. This sufficed to demonstrate that terahertz power was indeed transferred into- and out-of the slab-mode beam in broadband, with $\sim$75% peak efficiency over a 40% relative bandwidth. It is noted that, as these performance metrics are derived from a measurement with two slab-mode beam launchers, the peak efficiency of a single slab-mode beam launcher in isolation is likely closer to $\sim$86% (i.e. $\sqrt {0.75}$), assuming losses are shared equally across the emitter and detector, and that the intrinsic silicon slab waveguide exhibits negligible propagation loss.

Non-negligible variation was observed in both simulation and experiment. We ascribe this to waves that undergo multiple reflections whilst trapped in the silicon slab. This is enhanced by the presence of photonic crystal that is deployed adjacent to the feed point in order to provide field confinement. According to simulation, this reflection phenomenon accounts for less than 5% of overall received signal power. Nevertheless, its impact is discernible in transmission, and hence future work in this field should target the inhibition of such reflections. This may be achieved by modifying the photonic crystal structure adjacent to the feed point such that it scatters incident radiation to free-space, rather than reflecting it back into the slab. Additional to modification of its structure, the introduction of free carriers into the dielectric material that comprises the photonic crystal medium can realize a terahertz absorber [43], for the further suppression of reflection. Care must be taken to ensure that these alterations do not disturb the field distribution at the feed point or adversely impact efficiency or bandwidth.

Appendix A. Time-domain gating of simulation results

As stated in Section 4., CST Microwave Studio’s time-domain solver is employed to obtain the results shown in Fig. 5(b). A brief outline of the functionality of this solver is as follows. Firstly, a model of a physical structure is defined by the user, and specific locations on this structure are designated as ports. In simulation, the structure is discretized in a process known as meshing, and one port is excited by a broadband pulse. The energy from this port is propagated through the structure by means of a finite-difference time-domain approximation of Maxwell’s equations, until it is ultimately absorbed in the materials of the model, radiated (i.e. the power escapes the bounding box of the simulation), or accepted by one of the ports in the simulation. A pulse waveform is obtained in the latter case, as power is accepted progressively as the simulation steps through the time-domain. The solver is terminated when the total remaining energy in the mesh reaches some minimum value, which is a user-defined fraction of its peak value. At this point, the time-domain waveforms are converted to the frequency domain via Fourier transform, and the spectra of pulses accepted by ports are normalized by that of the exciting pulse in order to obtain scattering parameters. The pulses that are associated with the results in Fig. 5(b) are shown in Fig. 6.

 

Fig. 6. Pulses derived from time-domain simulations using CST Microwave Studio. In order to show the pulse-shape in more detail, all three pulses are shown as inset with the time-domain expanded $\times 20$. The pulses are also shifted in time, for visibility.

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Time-gating is required to obtain the results that are given in Fig. 5(c). To this end, the transmitted waveforms are inspected visually, so as to determine the time-range that is associated with the initial pulse of terahertz power that is incident upon the receiving port. In this way, the time-domain is restricted to the range from 0 ns to $\sim$0.83 ns, as indicated in Fig. 6. The signal amplitude of the pulse waveform is set to zero outside this time-gate span, which corresponds to a rectangular window function, and this is appropriate as the signal magnitude is essentially negligible at the boundaries of the window. After the gated pulse is obtained, conversion to the frequency domain and normalization by the exciting pulse are repeated, and the results given in Fig. 5(c) are obtained.

Funding

Core Research for Evolutional Science and Technology (JPMJCR1534); Ministry of Education, Culture, Sports, Science and Technology (17H01064).

Acknowledgments

The authors wish to acknowledge the assistance of Mr Xiongbin Yu in conducting experiments.

Disclosures

The authors declare no conflicts of interest related to this article.

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