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Abstract
Two-dimensional photonic crystal waveguides can support guided modes with low loss. Interfacing such a guided mode with free-space propagation modes is crucial for photonic integrated circuits. Here we propose a dielectric resonator antenna (DRA) fully integrated with a photonic crystal waveguide for endfire radiation. High radiation efficiency can be achieved from the DRA that relies on oscillating displacement currents in a low-loss dielectric material. The antenna is designed to operate at a high-order resonance for high gain. The reflection loss at the interface between the two components is minimized via a matching air hole, the mechanism of which is qualitatively described via temporal coupled-mode theory. As a proof of concept, the all-dielectric integrated structure is realized on a single intrinsic silicon wafer to operate at terahertz frequencies. The antenna footprint is only about one square operational wavelength. The experimental validation confirms the maximum gain of over 10.6 dBi with 3-dB angular beam widths of 29.0 degrees and 45.7 degrees in orthogonal dimensions. The impedance bandwidth obtained from simulation is 6%, spanning 311 to 331 GHz. Given a suitable low-loss dielectric material, this all-dielectric structure holds potential for scaling to infrared and visible light frequencies.
© 2017 Optical Society of America
1. Introduction
Photonic crystals have been conceived and studied extensively by the optics community as a mean for efficient control of light [1–8]. The crystal is typically built from a periodic arrangement of two low-loss dielectric materials with a sufficient contrast in their permittivity values. This arrangement establishes stop-bands known as photonic bandgaps that prevent waves from traversing the crystal. A point defect in the periodicity can lead to the presence of localized discrete photonic modes inside bandgaps of the structure. With the same principle, a one-dimensional defect can create a channel for guided modes with no energy leakage. This concept has led to 2D photonic crystal waveguides, which are compatible with standard planar fabrication technologies, for efficient and compact routing of optical signals.
One important functionality for photonic crystal waveguides is the radiative coupling between guided waves and free-space waves. It enables inter-chip communications in photonic integrated circuits. So far, the coupling was accomplished via leaky modes in photonic crystals themselves [9, 10] or through dedicated diffraction gratings along the waveguide [11, 12]. Nevertheless, in both cases the radiation is associated with beam squint, i.e. strong spatial dispersion of the main radiation lobe due to the phase-matching condition. Alternatively, a single defect with a radiative channel was employed to realize an emitter for photonic crystals [13–15]. However, the matching efficiency was not optimal, and the directivity is presumably low due to strong diffraction from a small aperture. Additionally, in these emitters, the radiative Q factor is high, and therefore, the bandwidth is relatively limited.
On the other side of the spectrum, a large variety of microwave antennas have been developed for over a century, and some of them could be integrated with photonic crystal waveguides to achieve desirable performance. Metal-based resonant antennas are susceptible to ohmic loss at higher frequencies due to increasing sheet resistance, and their structures do not conform with 2D photonic crystal waveguides. While providing a large bandwidth, metallic horn antennas raise an issue on fabrication complexity and compactness at higher frequencies [16]. Non-resonant tapered dielectric rod antennas operate with broad bandwidth and high gain, at the cost of compactness [17]. As a potential candidate, dielectric resonator antennas (DRAs) maintain high efficiency at millimeter-wave frequencies and higher, owing to the absence of ohmic loss [18–20]. This antenna type can achieve appreciable gain over a moderate bandwidth with a compact size, and can be integrated seamlessly with photonic crystal waveguides.
Here, we propose an all-dielectric integration between a DRA and a 2D photonic crystal waveguide. This DRA is placed at the termination of and fed by a photonic crystal waveguide to achieve endfire radiation with high total efficiency. A main challenge lies in the DRA-waveguide interface that must be electrically matched and physically realizable with standard fabrication techniques. As a proof of concept, the design is implemented and experimentally validated at terahertz frequencies. The entire structure is fabricated on a single silicon wafer to operate around 0.32 THz, which lies within an atmospheric transmission window [21]. This realization is of technological relevance, since it offers a much-needed high-gain low-loss antenna platform to overcome free-space path loss in terahertz transmission. Because of scale invariance in electro-magnetics and the availability of low-loss dielectric materials, the design is fully scalable to shorter wavelengths for infrared and visible-light operation.
2. Materials and design
Silicon is chosen as the only constituent material to create an all-dielectric integration between a waveguide and an antenna. Float-zone intrinsic silicon has exceptionally low absorption coefficient, and is nondispersive within the terahertz frequency range [22]. Coincidentally, its moderate relative permittivity of 11.68 (equivalent to the refractive index of 3.418) is excellent for the operation of photonic crystals [23, 24] and dielectric resonator antennas [25, 26]. Specifically, the operation of photonic crystals requires a periodic profile of permittivity to create complete photonic bandgaps that prohibit wave propagation. This periodic profile must have sufficient permittivity contrast, which can be derived from silicon and air. A dielectric resonator antenna, likewise, requires a material with a moderate permittivity to create a resonant cavity with sensible radiative losses and bandwidth. Therefore, silicon is a material of choice for such purposes.
The photonic crystal waveguide design is based on the earlier work of some of the authors [23]. The silicon wafer has a resistivity of >10 kΩ-cm and a thickness of 200 μm. As shown in Fig. 1, a triangular lattice of through-holes is formed in this wafer with a lattice constant and a hole diameter of a = 240 μm and dw = 0.6a = 144 μm, respectively. By removing a row of holes, a waveguide is created in this hole lattice. In effect, a two-dimensional photonic bandgap results in a single non-leaky mode within a band of 315–336 GHz. Within this band, transverse-electric waves (electric field in plane; magnetic field out of plane) can be confined in-plane via the photonic-bandgap effect. The out-of-plane confinement is achieved via total-internal reflection at the silicon-air interface. Between 326 and 331 GHz, the propagation loss is less than 0.1 dB/cm.
Fig. 1 DRA fed via a photonic crystal waveguide. (a) Perspective view. (b) Dimensions of the waveguide and DRA: a = 240 μm, dw = 0.6a = 144 μm, dr = 642 μm, da = 1016 μm, dm = 100 μm, sr = 50 μm, and sm = 405 μm. The silicon thickness is 200 μm. The DRA and surrounding air ring share the same center. The photonic crystal on each side of this waveguide extends to 5.6 mm, but is not shown here for clarity.
For endfire radiation (+z) of guided waves, a cylindrical DRA is incorporated into one end of this photonic crystal waveguide, as shown in Fig. 1. An air ring encircles the DRA to create a sharp reflection boundary, and thus a cavity for the displacement current. A limited antenna gain would be attained if a fundamental resonance mode is used for this DRA because of the limited aperture size and strong diffraction. Therefore, in order to access a high-order resonance mode, the diameter of this DRA is designed to be about 2.5 times the operation wavelength inside silicon. As a result, the electrical size and the gain of this antenna can be enhanced [27]. Additionally, a large DRA facilitates the integration with the photonic crystal waveguide that has a relatively large physical cross section. Nevertheless, the entire antenna footprint measures merely 1.1λ0 × 1.1λ0, where λ0 denotes the free-space wavelength at 325 GHz. An air hole with diameter dm = 100 μm is inserted between the DRA and the waveguide feed to match the coupling and thus maximize the transmission. The mechanism of this air hole can be perceived in simple terms — on resonance, the wave decaying from the DRA into the waveguide destructively interferes with the wave directly reflected by the matching air hole. A rigorous analysis of this air hole is discussed in the next paragraph.
The necessity of the matching air hole can be understood in the framework of coupled-mode theory. Under the assumption of weak coupling, temporal coupled-mode theory can express the transmission (radiation) and reflection of a coupled resonant system in terms of its decay rates 1/τ = ω0/(2Q) [7, 28]. Here, the DRA together with the photonic crystal waveguide can be simplified into Fig. 2, and the DRA couples to the waveguide and free space with decay rates 1/τw and 1/τr, respectively. The transmission (radiation) power spectrum is given as [7]
Based on this equation, in order to attain 100% transmission on resonance, the lifetimes τw and τr must be equal. Without the matching air hole, the DRA-waveguide coupling is too strong, or equivalently, τw is smaller than τr. In effect, the decay from the DRA into the waveguide is much faster than the decay into free space. The air hole located between the DRA and the waveguide can limit their coupling strength to meet the condition τw = τr. The size and location of this air hole must be optimized numerically to achieve the minimum reflection [7].
Fig. 2 Conceptual diagram for the coupling between the waveguide, DRA, and free space. Here, ω0 corresponds to a resonance frequency of the DRA, E represent propagating fields in isolated incident (+) and reflected (-) components, and τw = 2Qw/ω0 and τr = 2Qr/ω0 are the lifetimes of the resonator coupling with the waveguide and free space, respectively. A shorter lifetime implies stronger coupling and lower Q factor. A single resonance mode and a single waveguide mode are considered, and absorption losses are neglected.
3. Experiment
The antenna-waveguide structure is fabricated by photolithography and deep reactive-ion etching to form straight air holes and gaps through this wafer. Figure 3 shows the fabricated structure and microscopic features. It can be seen that the waveguide and DRA are precisely developed. The length of the photonic crystal waveguide extends to 18.7 mm (78a) for handling purposes. The tapered dielectric waveguide of 3.0 mm long on the opposite side of the antenna can be inserted into a WR-3 hollow rectangular waveguide. Through this tapered waveguide, the TE10 mode [29] in the rectangular waveguide can couple with the fundamental mode inside the photonic crystal waveguide with less than 0.2 dB insertion loss across the spectral band of interest [23].
Fig. 3 Fabricated structure. (a) Complete structure fabricated from a single silicon wafer. Visible in this image from left to right are the tapered dielectric feed to a rectangular waveguide, photonic crystal waveguide, and DRA. (b) Optical microscope image around the antenna part. The matching air hole can be seen mediating the waveguide and antenna.
The antenna is subject to two series of measurements, one to characterize the frequency-dependent gain, and the other to obtain the radiation patterns in two planes. The measurement setup is illustrated in Fig. 4, and can be described as follows. The continuous wave from the signal generator tuned around 40 GHz (LO) is modulated with a 100 kHz sine wave (IF). The modulated signal is fed into a millimeter-wave amplifier and 9× multipliers. The resulting terahertz signal sweeping 290–375 GHz excites the antenna under test. A standard WR-3 conical horn antenna is on the receiving side. Both antennas are aligned for vertical polarization, and are separated by 200 mm for the gain measurement. The separation reduces to 100 mm for radiation pattern measurement to increase the received power off the main lobe. Connected to the receiving horn antenna, a SBD functions as a square-law demodulator to extract the IF signal of 100 kHz. This signal is fed into a low-frequency amplifier followed by a spectrum analyzer. An absorber is placed around the receiving horn antenna to mitigate standing waves. The gain measurement is calibrated with another reference standard conical horn antenna. The radiation pattern measurement is carried out at 325 GHz with an angular step size of 1 degree.
Fig. 4 Continuous-wave electronic transceiver chain. This configuration is used for both the measurements of gain and radiation patterns. SG: signal generator, Amp: amplifier, Tx: transmitter, Rx: receiver, SBD: Schottky barrier diode, SA: spectrum analyzer, IF: intermediate frequency, LO: local oscillator, and RF: radio frequency.
4. Results
Figure 5 compares the reflection coefficients of the waveguide-fed DRA with and without the matching air hole. As can be seen, the DRA with the matching hole has its reflection coefficient lower than −10 dB between 311 and 331 GHz, corresponding to the impedance bandwidth of 6%. This band overlaps with the non-leaky mode band of the waveguide between 315 and 336 GHz. The main resonance takes place around 323 GHz. Another resonance mode at 312 GHz is weak because of the mismatch and the leakage into propagation modes inside the photonic crystal. At frequencies from 331 GHz up to 340 GHz, the strong reflection for the DRA is due to the absence of resonance modes. Without the matching air hole, the reflection level increases significantly in accordance with the description of temporal coupled-mode theory.
Fig. 5 Comparison of simulated amplitude reflection coefficient. Each antenna is fed by the photonic crystal waveguide with a length of 1.92 mm (8a). The simulation results in all figures are carried out with the frequency-domain solver in CST Microwave Studio 2016.
We focus on the operation of the matched DRA and the feeding waveguide in the near-field region at 325 GHz, close to where the reflection coefficient is minimum. From Fig. 6(a) and 6(b), it is evident that this DRA operates in its high-order mode with the E-field oscillating in plane and the H-field oscillating out of plane, and the matching air hole contributes to the wave confinement. Figure 6(c) reveals that the waves are laterally bound inside the waveguide and antenna by the photonic crystal. Strong radiation can be observed towards the endfire direction with small diffraction in the ±x directions. Another cross-sectional view in Fig. 6(d) shows that the main radiation in this plane has a relatively wide angular width, since the electrical size of the DRA in the y dimension is limited by the planar layout. Additionally, small radiation from the DRA and the waveguide-DRA interface can be seen towards the ±y directions.
Fig. 6 Simulated mode field distributions at 325 GHz. (a–b) Vector plots for the E and H fields, respectively. (c–d) Tangential and normal E-field components in plane and out of plane, respectively. The scales for all plots are linear and capped, and the plots are on the symmetry planes.
The radiating near-field features of the matched DRA correspond to the far-field radiation patterns, shown in Fig. 7. Here, the simulation and measurement results show good agreement. We can observe the weak angular ripples in the measured results. This is likely caused by standing waves between the DRA and the receiving horn antenna. The main lobe in the E- and H-planes has 3-dB angular widths of 29.0 degrees and 45.7 degrees, respectively. The wider main lobe in the H-plane is due to the limited aperture size in the y direction, as discussed earlier. In the E-plane, the sidelobe level is as low as −12 dB, while it increases to −5.4 dB in the H-plane. The latter is caused by the radiation from the DRA towards the ±y directions.
Fig. 7 Normalized radiation patterns in dB at 325 GHz. (a) E-plane and (b) H-plane. The measurement step size is 1 degree. The angular ranges for the measurement are limited by the dynamic range of the setup.
Figure 8 shows the measured and simulated realized gain towards the endfire direction for the matched DRA and two additional dielectric rod antennas. These two rod antennas have their shape similar to the tapered dielectric feed in Fig. 3(a), but with different lengths of 3 mm and 829 μm. The latter dimension is equal to the DRA size measured from the matching hole to the tip of the DRA, and is chosen as such to compare at similar level of integrability. All the gain profiles account for the frequency-dependent loss in the actual 18.7-mm waveguide, which amounts to ∼0.07 dB at 325 GHz [23]. For all antennas, the measured gain is with some fluctuation and frequency-dependent distortion. This distortion is ascribed to the alignment between the tapered dielectric feed and the hollow rectangular waveguide. The non-ideal alignment results in the mode conversion in the WR-3 feed waveguide from the fundamental TE10 mode to high-order modes, which start appearing at a cut-off frequency lower than 347 GHz owing to silicon loading. Additionally, unavoidable reflections at both the transmitting and receiving antennas [10, 30] create standing waves in the reference and sample measurements. The effect is enhanced for an antenna with higher gain, and it can complicate the antenna alignment particularly when only a single frequency is being observed. Nevertheless, for both the measurement and simulation of the DRA, the maximum gain at 325 GHz is around 10.6 dBi. The simulation suggests the 3-dB gain bandwidth of about 6% spanning 315–334 GHz, in a good overlap with the impedance bandwidth and the non-leaky mode in the waveguide. A numerical comparison between lossless silicon and 10-kΩ-cm Drude silicon shows a negligible dissipation loss in this DRA. The comparison with the short tapered dielectric rod antenna shows that the gain of the DRA is significantly higher at a comparable size. By increasing the rod length to 3 mm, the gain of the rod antenna becomes comparable to the DRA gain, at the cost of integrability, i.e. compactness and fragility.
Fig. 8 Realized gain for the matched DRA and two tapered rod antennas. (a) Measurement, and (b) simulation. The shaded regions represent the systematic error due to standing waves. Owing to the noise floor below 315 GHz in the measured gains, the waveguide losses cannot be properly compensated. Instead, both the measured and simulated gains account for the losses incurred in the 18.7-mm waveguide. The gain below 0 dBi observable in the simulation around 310 GHz is because of the cut-off of the photonic crystal waveguide.
It is noteworthy that the operation of this DRA is fundamentally different from the operation of lenses because of the following reasons: (i) The DRA requires wave trapping to establish the leaky resonance mode for directional radiation. On the other hand, the operation of a lens is to regulate the phase front with varying propagation distance, and thus the lens must minimize wave reflection and confinement. (ii) From the numerical results of the DRA in Fig. 5, the maximum reflection within the bandwidth is −10 dB or 10% power equivalence. For a silicon lens, the index mismatch at the silicon-air interface would result in a reflection as high as around −5 dB or 30% of power.
5. Conclusion
We have proposed and experimentally validated a DRA design that operates with a photonic crystal waveguide for directional radiation. The all-dielectric structure is fabricated on a single silicon wafer with standard planar fabrication technology, and measured with a continuous-wave terahertz electronic setup. We have achieved a maximum measured gain of about 10.6 dBi at 325 GHz and an operation bandwidth from the simulation of about 6%. The dissipation loss is negligible in this type of antenna that operates with displacement current. Nevertheless, the DRA bandwidth is limited by its resonant nature. Future work includes enhancement of gain and bandwidth via higher-order multi-resonance modes. Parametric optimization to reduce the radiative Q factor could further increase the bandwidth. Rigorous analysis based on temporal coupled mode theory can be used to quantitatively characterize the coupling of the resonator with the waveguide and free space [31]. The antenna configuration is amendable to broadside radiation. This compact low-loss antenna platform can potentially serve high-capacity inter-chip communications.
Funding
Grant-in-Aid for Scientific Research, the Ministry of Education, Culture, Sports, Science and Technology of Japan; Core Research for Evolutional Science and Technology (CREST) program, Japan Science and Technology Agency (JST); Australian Research Council (ARC DP170101922).
Acknowledgments
The authors thank K. Okamoto, Y. Nishida, K. Nishio, and Y. Inubushi for their assistance in experiments.
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