1. Introduction

For many years, the spectral region between 0.1 - 10 THz was seen as inaccessible. Instruments used for generating or detecting radiation on either the lower frequency side (electronic) or the high frequency side (optical) were insufficient at THz frequencies, resulting in this being called the "THz gap". However, there has recently been a significant focus on the development of efficient, cost-effective and compact generation and detection of THz radiation [1–5]. This has been propelled by the applications in communication, imaging and sensing ranging from medical to security and defense [6–9]. A lot of the discussion and investigation into designing and exploring materials and structures for manipulating THz radiation is fueled by the idea to use platforms and fabrication schemes that are regularly used in electronics and optics, particularly in implementing silicon (Si) photonic structures for THz [10–12]. There is also notable interest in developing integrated platforms for efficient nonlinear processes combined with a small footprint for THz sources and sensors [13–16].

Here, we contribute to the field by presenting the fabrication and characterisation of a THz whispering gallery mode resonator (WGMR) made out of gallium arsenide (GaAs). We present a GaAs WGMR that is designed for use in the 150 GHz to 380 GHz spectral range. This spectral region has been of interest for applications ranging from identifying signatures of aromatic chiral molecules for interstellar searches [17] to demonstrating wireless communication at a high data rate [18–20]. To the best of our knowledge, GaAs has not been explored in the form of a WGMR in the THz domain before. WGMRs are useful as compact resonators in applications such as tunable filters [21] and isolators [22], but they are also implemented as sensitive detectors and low-threshold sources [23,24], as well as for sensitive material characterisation [25,26] and for fundamental studies [27,28]. High-Q(uality) WGMRs provide a platform of tight confinement of the resonant fields in a compact volume for an extended period of time. This increase in the interaction between the fields and the material can lower the threshold of nonlinear optical processes, as has been demonstrated at optical frequencies [29] as well as in the THz domain [30–34]. As a crystal with second-order optical nonlinearity, WGMRs fabricated out of GaAs could be used for nonlinear optical mixing processes such as difference frequency generation to implement THz sources [35,36], as well as frequency doubling and up-conversion to optical frequencies for THz detection [24,37,38]. In addition, as GaAs is a material that is used in the semiconductor industry, one could take advantage of existing fabrication techniques for mass manufacturing of integrated chip-scale THz hybrid systems.

2. Experimental methods

2.1 Design and fabrication of the GaAs WGMR

The WGMR to be fabricated was designed with the help of finite element method (FEM) modelling of an axis-symmetric 2D cross-section of the WGMR disk using COMSOL Multiphysics, using the refractive index to be 3.6 and the absorption loss to be below 0.5 cm⁻¹ [39]. The eigenmodes of the GaAs WGMR in different polarisations for the corresponding azimuthal mode number $m$ (i.e., the number of wavelengths that fit in the circumference of the WGMR) are calculated. From the array of vertically-polarized eigenmodes, the free spectral range (FSR) of the resonator is defined by the spectral spacing between consecutive modes of the same mode family. The radius of the resonator was chosen to be 2 mm such that the FSR is 6.77 GHz at around 300 GHz. The cross-section of the normalized electric field strength of the vertically-polarized mode at 304.97 GHz and $m=39$ is shown in Fig. 1(a).

 

Fig. 1. Normalized electric field strength at 304.97 GHz in the (a) 2-mm-radius GaAs WGMR, and (b) 640 $\mathrm{\mu}\textrm {m} \times {350} \mathrm{\mu}\textrm {m}$ HRFZ-Si waveguide cross-section. (c) Microscope image of the WGMR and waveguide in the experimental set-up. (d) THz spectrometric system from Toptica Photonics with photoconductive antennae (PCA) for the THz transmitter (TX) and detected at the receiver (RX); the radiation is coupled into and from a HRFZ-Si waveguide using UHMWPE lenses and coupled between the waveguide and GaAS WGMR by evanescent field coupling.

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The GaAs WGMR was fabricated from a (500 $\pm$ 20) $\mathrm{\mu}\textrm {m}$-thick wafer with orientation (001) from American Elements. The wafer was characterized by the crystal manufacturer to have a resistivity of up to 4.4×10⁸ Ω cm. The resistivity goes inversely with the absorption of the material in the THz domain, and therefore high resistivity is necessary to fabricate high quality microresonators [25]. A disk was drilled out of the wafer, and was then mounted on a brass rod and spun on a lathe, while a single-point diamond tool was used to cut the edge of the spinning disk. The angle of the diamond tool with respect to the spinning wafer edge (i.e. the rake angle) was roughly 40° [40]. The disk was then polished by hand using 1 µm diamond slurry on a tissue until the scratches on the surface due to the cutting process are smoothed out. The resulting GaAs disk WGMR, $(1.997 \pm 0.010)\;\textrm {mm}$ in radius and $(508 \pm 9)\;\mathrm{\mu}\textrm {m}$ in thickness, was mounted on an aluminium rod with a narrow neck of 0.5 mm in radius to be held in the experimental set-up.

2.2 Design and fabrication of the silicon waveguide

In order to excite the THz modes in the GaAs WGMR, one needs to design and fabricate a waveguide that supports modes that are phase-matched to the WGMR modes. From the azimtuhal mode number $m$ and the eigenfrequency $f$, the effective modal index of the different modes in the WGMR with radius $R$ can be calculated as: $n_{eff}=m c/(2\pi R f)$, where $c$ is the speed of light. Since the effective index of the WGMR modes (for $m=38$, $f={304.97}\;\textrm{GHz}$, $n_{eff}=2.974$) is close to the refractive index of high resistivity float zone (HRFZ) Si (3.42 [41]), we identified HRFZ-Si (resistivity larger than 20 kΩ cm as characterized by the manufacturer) as a good material candidate for the waveguide. Furthermore, the absorption losses of HRFZ-Si (0.002 cm⁻¹ [41]) are significantly lower than that of GaAs (0.5 cm⁻¹ [39]) which makes it attractive for a coupling waveguide. 2D FEM modelling of the cross-section of a cuboidal HRFZ-Si waveguide was done using COMSOL Multiphysics, which calculates the effective index in the waveguide. The dimensions of the waveguide were optimized to match the effective indices in both structures (i.e., phase-matched).

The required HRFZ-Si rod waveguide was found to have dimensions of 640 µm in height and 350 µm in width for exciting vertically-polarized modes in the WGMR. It was fabricated by cutting a HRFZ-Si wafer of thickness 350 µm using a laser micromachining system, using the system mentioned in [42]. The micromachining system used 20 W 120 fs pulses centred at 1030 nm at a repetition rate of 20 kHz with the stage moving at a speed of 20 mm s⁻¹. Figure 1(b) shows the cross-section of the normalized electric field strength at 304.97 GHz in the sub-wavelength waveguide. Figure 1(c) shows a microscope image of the GaAs WGMR and HRFZ-Si waveguide.

2.3 Characterization of the GaAs WGMR

The set-up for characterizing the Q factor of the resonator modes by coupling the radiation into the WGMR and exciting modes in it is shown in Fig. 1(d). The commercially-available frequency domain THz spectrometric system from Toptica Photonics [43] was used. The THz radiation is generated using a pair of lasers at telecom wavelengths whose frequency separation gives the frequency of the THz radiation being emitted from the photoconductive antenna (PCA) on the left shown as TX (transmitter), and detected coherently as a photocurrent at the antenna on the right labelled RX (receiver). Teflon lenses are used to collimate and focus the beam from and to the antennae. A pair of ultra-high-molecular-weight polyethylene (UHMWPE) lenses [44] are used to couple the radiation to and from the sub-wavelength HRFZ-Si rod waveguide. The UHMWPE lenses were mounted in X-Y lens mounts to centre the transmission axis and mounted on Z-axis translation stages to adjust their focal position. A pinhole in aluminium foil was used to mark the focal point of the two UHMWPE lenses, which served as a starting point to optimize the position of one end of the HRFZ-Si waveguide. The 6-cm long waveguide was mounted on a 5-axis stage to control the X-Y position on both ends and to bring each end into the focal plane of the corresponding UHMWPE lens. Once the waveguide was aligned to show a sinusoidally oscillating photocurrent across a range of frequencies, the WGMR was mounted on a translational stage to bring it closer to the waveguide. The radiation in the waveguide then excites modes in the WGMR by evanescent field coupling. The separation between the waveguide and the WGMR, when coupled, ranged from $\sim$200 µm to $\sim$300 µm.

The coherent frequency-domain spectroscopy followed by data analysis using the Hilbert transform [45] involves a two-step measurement. First, a reference measurement was taken while the WGMR was far away and not coupled to the waveguide. This smoothly oscillating photocurrent is shown in blue in Fig. 2(a). Then, a sample measurement was taken with the WGMR closer and coupled to the waveguide. As can be seen in the orange curve in Fig. 2(a), a small kink appears at 169.8 GHz due to the coupling of the resonant radiation into the WGMR. A Hilbert transform of the photocurrent signal gives an instantaneous amplitude and phase corresponding to the reference and sample measurements. The ratio of the amplitudes (labelled the "Relative amplitude" of the WGMR) shows a dip at the resonant frequency [Fig. 2(b)]. The difference in the phases (labelled the "Relative phase" of the WGMR) shows a sharp change in phase at the same frequency [Fig. 2(c)]. This coherent detection thus provides us additional information with the relative phase profile: when the system is undercoupled, the shape of the phase shift is as shown in Fig. 2(c); and when overcoupled, the phase shift is a jump of 2$\pi$ [46].

 

Fig. 2. (a) Photocurrent signal from 168.5GHz to 171.2GHz with (sample:orange) and without (reference:blue) the WGMR coupled to the waveguide. Relative (b) amplitude and (c) phase from Hilbert transform analysis of the sample and reference profiles.

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3. Results and discussion

The THz spectroscopy system was scanned across different spectral ranges within 150GHz to 380GHz. Reference and sample measurements are acquired as described before, and from this section onward, we show only the relative amplitude and phase profiles of the WGMR extracted from the Hilbert transform analysis.

As can be seen in Fig. 3, there are multiple dips in the relative amplitude spectrum which are spaced from each other by 6.75 GHz within the range from 310GHz to 350GHz. This corresponds to the FSR expected from the FEM in this frequency range, if the refractive index of the material in the FEM simulations is modified from 3.6 [39] to 3.65. These modes are identified by comparison to the FEM simulations to be the vertically-polarized fundamental modes, i.e., single-lobed in radial and polar directions.

 

Fig. 3. Relative (a) amplitude and (b) phase profiles showing the modes in the GaAs WGMR in a range from 310 GHz to 350 GHz with an FSR of 6.75 GHz.

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Adjusting the coupling such that a mode at 300.48 GHz was undercoupled, a scan with a smaller frequency step size of 5 MHz was taken to plot and fit the 300.48 GHz mode. A linewidth of 213 MHz was obtained by the simultaneous fitting of both the relative amplitude [Fig. 4(a)] and relative phase [Fig. 4(b)] profiles of the mode. This corresponds to a $Q$ of $1.41\text {k}\pm 0.02\text {k}$ at 300.48 GHz.

 

Fig. 4. Simultaneous fitting of the relative (a) amplitude and (b) phase profiles at 300.48 GHz and 151.16 GHz to obtain to a $Q$ of $\sim 1.41\text {k}$ and $\sim 2.21\text {k}$ respectively.

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In order to compare the $Q$ at lower frequencies, the system was adjusted to undercouple modes at around 150 GHz (which would be overcoupled when measuring undercoupled modes at 300 GHz). On simultaneous fitting of the relative amplitude and phase profiles at 151.16 GHz, a linewidth of 68 MHz was obtained as shown in Fig. 4(c)-(d), which gives a $Q$ of $2.21\text {k}\pm 0.06\text {k}$ at 151.16 GHz.

As can be seen from the relative amplitude in both cases, there was significant absorption (the relative amplitude at resonance is $<25{\%}$). This indicates that the $Q$ was a loaded $Q$, i.e., the waveguide has an effect on the absorption losses in the system.

A useful feature of such THz microresonators is that the frequency position of the WGMR modes can be adjusted by introducing a piece of metal into the vicinity of the THz mode [47], as shown in Fig. 5. Here, we used a block of aluminium and brought it about 0.2 mm away from the edge of the GaAs disk. The coupling of the system was adjusted such that we see one overcoupled mode at $\sim {287.5}\;\textrm{GHz}$ between two undercoupled modes at 281 GHz and at 294 GHz. As we moved the aluminium piece closer by a few micrometers, we saw the THz modes shifting from 280.81GHz to 281.07GHz, 287.49GHz to 287.77GHz and 294.26GHz to 294.56GHz corresponding to the blue to the green curves. The loaded Q factor improves slightly with the blue-shifting, eg. it goes from $975\pm 18$ at 280.81 GHz to $1032\pm 21$ at 281.07 GHz, and from $1341\pm 24$ at 287.5 GHz to $1750\pm 78$ at 287.77 GHz. This can be attributed to a slight change in coupling strength.

 

Fig. 5. Relative (a) amplitude and (b) phase profiles showing the blue-shifting of the modes as an aluminium piece moves closer to the WGMR (blue to orange to green).

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Such tunability of around 0.3 GHz can be very useful to relax the stringent requirements during fabrication of the WGMR, since the positions of the THz modes in a WGMR depend strongly on its physical dimensions. This is particularly important for the implementation of nonlinear optical processes in such microresonators since the dimensions of the WGMRs play an important role in ensuring phase-matching between THz modes and optical modes as discussed, eg., in [48,49]. For instance, a 10 µm change in radius of the disk WGMR results in a shift in the FSR of 0.04 GHz, and a 20 µm change in thickness of the disk WGMR causes a 0.02 GHz change in the FSR.

4. Conclusion

We presented a disk-shaped GaAs WGMR of radius 2 mm which was fabricated by single-point diamond-turning on a lathe followed by polishing with 1 µm sized diamond slurry. The resonator was characterized using a THz spectroscopy scheme and found to have quality factors of 2.21k at $\sim {150}\;\textrm{GHz}$ and 1.41k at $\sim {300}\;\textrm{GHz}$. We showed that the frequency positions of the modes can be tuned slightly by introducing a metal in their vicinity. This additional degree of freedom could eliminate the need for iterative fabrication steps to achieve the exact dimensions required for phase-matching of THz modes involved in nonlinear optical processes in the resonator.

Our evaluation of the GaAs microdisk WGMR shows that it can be a major component in the rapidly developing field of nonlinear THz photonics, especially as part of on-chip platforms for nonlinear generation and detection schemes. THz-photonics sources based on optical-to-THz wavelength converters via difference frequency generation [50], can be realized at low threshold powers in resonance cavities, as has been demonstrated in resonant designs before [51–53], including WGMRs [24]. Similarly, a detector based on the reverse process, i.e., by nonlinear optical up-conversion of a THz signal to the optical domain, would benefit from such a WGMR-based implementation. For instance, as shown in [48], the photon conversion efficiency of the up-conversion process in such a WGMR would go linearly with the Q factor of the mode that the THz signal occupies.