1. Introduction

High-Q devices with very narrow spectral features operating in the THz frequency range are essential to fully exploit this unique frequency range. Potential applications are for example highly sensitive sensors and very narrow-band filters for THz communication. Despite an on-going effort, Q-factors higher than 10³ are hardly achieved [1–18]. The breakthrough of high-Q THz devices is mainly hampered by the strong material absorption present in the THz frequency range.

Our group recently demonstrated THz WGMs in a low-loss HRFZ-Si sphere with a Q-factor of 1.5 × 10⁴ at 0.62 THz [19]. The results manifest a significant advancement towards the development of high-Q THz devices. We excited the THz WGMs via evanescent coupling from a sub-wavelength single-mode air-silica step-index waveguide. However, maintaining single-mode operation of and efficient coupling to the waveguide over a broad frequency range is not straight forward [20]. Moreover, the excitation of the fundamental WGM in the HRFZ-Si sphere is not possible due to the large refractive index mismatch between the silica waveguide and the HRFZ-Si sphere. Designing a broadband single-mode waveguide that is phase-matched to the fundamental mode in a HRFZ-Si sphere is very challenging.

Prism coupling could provide a viable approach to overcome the above mentioned aspects of waveguide coupling of THz WGMs. Prism coupling is traditionally used in the optical domain, as it provides a very robust excitation scheme that also allows coupling to the fundamental mode of high refractive index WGM resonators (WGMRs) [21]. Coupling can be achieved by using the phase-matched evanescent field of a beam that experiences total internal reflection at the base of the prism. Using a prism avoids the challenge of coupling to a sub-wavelength waveguide and maintaining its single-mode behavior over the desired bandwidth. However, prism coupling imposes the prerequisite of mode-matching between the WGMs and the totally reflected beam. Furthermore, it is essential to detect the entire beam coupled to the WGMR in order to avoid distortions of the measured WGMs [22]. Both, mode-matching and detection, are particularly challenging in the THz frequency range due to technical limitations.

In this work, we investigate prism coupling of THz WGMs using a HRFZ-Si prism and spherical HRFZ-Si WGMRs. We experimentally demonstrate the excitation of high-Q THz WGMs over more than two octaves from 250 GHz to 1060 GHz in the same device. The results provide exciting opportunities for high-Q devices operating over an extremely broad bandwidth in the THz frequency range.

2. Methods and experiment

A schematic of the experimental setup is shown in Fig. 1. Generation and detection of THz radiation is realized using a standard coherent continuous wave (CW) THz spectroscopy setup based on heterodyne detection (Toptica TeraScan 1550 nm) [23]. The THz radiation is focused onto the base of the prism using specially developed symmetric-pass (s-p) polymer lenses [24]. The polarization of the THz radiation is parallel to the base of the prism (s-polarization) or out of the plane of Fig. 1. In order to provide phase- and mode-matching with the desired THz WGMs, the apex angle and material of the prism as well as the focal-spot size of the s-p lenses have to be chosen very carefully. Here, we are using a HRFZ-Si prism with an apex angle of about 77 ^◦ as this angle assures that the output beam is collinear with the input beam. With a material refractive index of HRFZ-Si of 3.416 [25], this corresponds to a phase refractive index of the parallel component of the incoming beam of 3.0 [26]. The latter provides phase-matching to the fundamental mode of an 8 mm diameter HRFZ-Si WGMR at around 250 GHz, as calculated with Mie theory (see below). To maximize mode-matching with the tightly confined THz WGMs [see e.g. the mode cross-section in Figs. 2(d) and 2(e)], the deployed s-p lenses have a very small free-space focal spot of about 0.7λ and a numerical aperture (NA) of 1. The large NA supports detection of the entire beam transmitted through the prism.

 

Fig. 1 (a) Schematic of the CW THz spectroscopy system with fiber coupled photo-conductive antennas (PCAs) using two-inch diameter s-p THz lenses to focus the THz radiation onto the base of the HRFZ-Si prism. The length of the base of the prism is about 13 mm. The spherical HRFZ-Si WGMR is mounted on a 3D manual translation stage, and the position is observed using two microscope cameras. (b) and (c) show the corresponding top and side view, respectively, of the 8 mm HRFZ-Si sphere next to the HRFZ-Si prism. Please note that (c) is focused on the prism surface.

Download Full Size | PDF

 

Fig. 2 2D simulation of a Gaussian beam (focal spot size: 0.7λ) propagating through the HRFZ-Si prism at 500 GHz (a) without WGMR, (b) with 2 mm diameter HRFZ-Si WGMR, and (c) with 8 mm diameter HRFZ-Si WGR. The solid black lines show the outline of the prism. The mode cross-sections (|E|²) of the fundamental THz WGMs at around 500 GHz in a 2 mm and 8 mm diameter HRFZ-Si WGMR are shown in (d) and (e), respectively. The solid black lines indicates the surfaces of the WGMRs. (f) shows the THz beam profile cross-sections extracted at the lines indicated in (c) with the blue and green solid lines, before and after the prism, respectively.

Download Full Size | PDF

In the following we present 2D simulations of the prism coupling between the prism and the THz WGMRs using the deployed s-p lenses. The frequency-domain finite-element method simulations are performed with COMSOL multiphysics. Second-order scattering boundary conditions are applied at the boundaries of the computational cell. The polarization of the THz radiation is out of the plane of Figs. 2(a)–2(c). While the 2D simulations are insufficient for a quantitative analysis of the experimental results, nonetheless, they provide valuable qualitative insights and explanations for the effects observed in the experiment. Figure 2(a) shows |E|² of a Gaussian beam with a free-space focal spot size of 0.7 λ at 500 GHz propagating from left to right, focused onto the base of the prism. Please notice the strong secondary reflection at the output face of the prism due to the large refractive index contrast of HRFZ-Si and air. Only about 60% of the s-polarized THz radiation is transmitted. However, while the HRFZ-Si prism experiences high reflection losses, it has negligible losses due to material absorption (0.015 cm⁻¹ at 0.6 THz [19]). Experimentally, we observed a much higher overall transmission with the HRFZ-Si prism compared to the sub-wavelength air-silica step-index waveguide used in previous work [25]. The higher losses for the waveguide are caused by stronger material absorption as well the coupling to a sub-wavelength structure. The higher throughput with the prism offers a significant ease in the experimental implementation compared to the waveguide coupling.

To minimize unwanted interference with the main beam, in the experiment a second prism was added perpendicular to the main prism at the position where the secondary reflection hits the input face of the main prism [see black colored bar in Fig. 2(a)]. Great care is taken not to effect the input beam. For clarity, in the simulation a layer of highly doped Si with a strong material absorption was placed at the position where the prism is in the experiment, to suppress any further secondary reflections.

Please also note, because of the relatively large extent of the highly converging beam on the input face of the prism, the output beam only resembles a quasi-Gaussian profile, and deviations from the input beam are observed in the simulations [e.g. by comparison of the beam width before and after the prism in Fig. 2(a)]. The deviations can be minimized by using a more collimated beam. However, the consequently larger focal spot size at the base of the prism would be undesirable for the mode-matching criteria, as discussed below.

Figure 2(b) visualizes the impact of insufficient mode-matching, when a THz WGM in a 2 mm diameter HRFZ-Si resonator is coupled (close to critical coupling) to the prism shown in Fig. 2(a). Please note that critical coupling is found by iteratively changing the WGMR-prism distance. The focal spot at the base of the prism is too large compared to the small mode size of the WGM as shown in Fig. 2(d). As a consequence, the totally reflected beam is significantly distorted [26], and a detection of the entire beam is extremely difficult with the available THz optics and small area detectors like photo-conductive antennas. The beam distortion can be significantly reduced by increasing the mode-matching, i.e. choosing a larger diameter resonator or decreasing the focal spot size. However, an even smaller focal spot leads to further distortions from the prism itself. For comparison Fig. 2(c) shows the same coupling scenario but with an 8 mm diameter WGMR. Due to better mode-matching with the larger mode profile of the WGM [as shown in Fig. 2(e)], the distortions of the reflected beam are significantly reduced. However, the WGMR still alters the beam profile of the totally reflected beam, as can be seen from the beam cross-sections in Fig. 2(f) extracted along the corresponding blue and green solid lines in Fig. 2(c). In particular additional reflection losses at the output face of the prism are introduced. The latter can be nicely observed in Fig. 2(c). Consequently, the overall transmission including the additional reflection losses is lower than expected for the ideal scenario. Please note that the deviations of the reflected beam are only observed for strong coupling to the WGMR. However, despite the distorted output beam, if one could detect the entire beam, the measured WGM would not deviate from the theoretical Lorentzian lineshape [22].

In the experiment, the studied THz WGMs are spectrally characterized by detecting the transmitted THz radiation from the HRFZ-Si prism coupled to the WGMR (sample scan) normalized to the transmission without WGMR (reference scan). The instantaneous amplitude (envelope) and instantaneous phase of the measured photocurrents are analyzed using Hilbert transformation [27]. The intensity and phase profile of the THz WGMs are retrieved by calculating the ratio of the envelopes and difference of the instantaneous phases of the sample and reference scans, respectively. Finally, the data is analyzed using the intensity and phase profile of the complex amplitude transmission coefficient t describing evanescent coupling of a spherical WGMR [21]:

The coupling rate δ_c can be conveniently controlled in the experiment by varying the distance between the WGMR and the prism. The position of strongest coupling is initially found by locating the focal spot at the base of the prism by applying a small water droplet. The water droplet causes a significant absorption in the measured photocurrent and clearly marks the position where best coupling is expected. Consequently, coupling is maximized by measuring the maximum coupling rate under slight variations of the WGMR’s position at a constant WGMR-prism distance. Finally, the observed THz WGMs are unambiguously identified by using the measured phase profile and calculations based on Mie theory, as described in our previous publications [19,28].

3. Results and discussion

The measured intensity profiles of an 8 mm diameter HRFZ-Si WGMR coupled to the HRFZ-Si prism in three representative scans covering more than two full octaves of the spectrum are shown in Fig. 3. Particularly the frequency regions from 250 GHz to 260 GHz, 500 GHz to 510 GHz, and 1050 GHz to 1060 GHz have been chosen [see Figs. 3(a)–3(c), respectively]. Please note that the setup was not changed or re-aligned but for the WGMR-prism distance to ensure close-to critical coupling of the THz WGMs in the corresponding frequency ranges. The typical WGMR-prism distances range from 10 µm to about 90 µm in the frequency range from 1060 GHz to 250 GHz, respectively. The measurements impressively reveal the excitation of high-Q THz WGMs over more than two octaves. The minor deviations of the intensity profile of the THz WGMs from the expected Lorentzian line shape evident from Fig. 3 are discussed in detail below.

 

Fig. 3 Measured intensity profiles of the THz WGMs in an 8 mm diameter HRFZ-Si spherical WGMR in the frequency ranges from (a) 250 GHz to 260 GHz, (b) 500 GHz to 510 GHz, and (c) 1050 GHz to 1060 GHz. The arrows in (a) indicate the FSRs of the fundamental mode (3.62 GHz) and first higher order radial mode (3.74 GHz).

Download Full Size | PDF

Using the mode-identification procedure [19,28], the two dominant WGM families present in Fig. 3(a) can be clearly identified. The measured free-spectral range (FSR) of 3.62 GHz and 3.74 GHz are in excellent agreement with the calculated FSRs of 3.62 GHz, and 3.73 GHz of the fundamental and first higher order radial mode. This is expected since the HRFZ-Si prism was designed to be phase-matched to the phase-refractive index of about 3.08 at the surface of the WGMR for the fundamental mode at around 250 GHz. The calculated phase-refractive index of the first higher order radial mode is about 2.82. Interestingly, the phase refractive index at the surface is much lower than the material refractive index of the HRFZ-Si WGMR, despite that most of the WGM is inside the bulk of the resonator [see Figs. 2(d)–2(e)]. The phase refractive index ${\mathrm{n}}_{\text{ph}}{|}_{{\mathrm{r}}_0}$ at the surface of the resonator has been calculated using the simple equation: ${\mathrm{n}}_{\text{ph}}{|}_{{\mathrm{r}}_0}=\left(mc\right)/\left(2\pi {\mathrm{r}}_{0}f\right)$. Here, c is the speed of light, r₀ the WGMR radius of 4 mm, m is the number of wavelengths in the circumference of the WGMR, and f the WGM resonance frequency. m and f are obtained from the calculations based on Mie theory [29].

Furthermore, with increasing frequency [see Figs. 3(b) and 3(c)], more THz WGMs are phase-matched and are observed in the measurements. This observation is in accordance with the calculations based on Mie theory. For example, in the 500 GHz range scan shown in Fig. 3(b) the fundamental mode and at least four higher order radial modes are excited. However, the fundamental mode can only be weakly excited due to its large phase refractive index of about 3.20 in this frequency range. The strongest coupling is observed for the first and second higher order radial mode with phase refractive indices of 3.04 and 2.90, respectively. Overall, the experimentally observed coupling behavior of the THz WGMs clearly confirms the approach of calculating the phase refractive index at the surface of the resonator, despite the bulk of the WGM localized inside the HRFZ-Si sphere.

In the following we focus on the three THz WGMs at 258.90 GHz, 503.74 GHz, and 1054.43 GHz present in Fig. 3. The measured intensity profiles of each THz WGM are plotted in Figs. 4(a), 4(c), and 4(e), respectively, while the corresponding phase profiles are shown in Figs. 4(b), 4(d), and 4(f) with black dots. All WGMs are measured at varying WGMR-prism distances, when the phase profile exhibits the distinguished step-function like transition of π in the phase profile associated with critical coupling [30]. Each measurement is averaged over three subsequent scans with a very long integration time of 300 ms. The frequency step size is 4 MHz, corresponding to the effective frequency resolution of the CW-THz system [27]. The errorbars show the corresponding standard deviation of the extracted intensity and phase profiles. The extremely narrow spectral features of the shown WGMs resemble very closely the expectations from the analytical model according to Eq. (1). For example, the WGM at 503.74 GHz has a Q-factor at critical coupling of about 1.4 × 10⁴ in accordance with our previous results with the waveguide coupling. The WGM at 1054.43 GHz has a Q-factor at critical coupling of 2.2 × 10⁴. Please note, that both Q-factors are extracted from the fits to the measured phase-profiles, and are related to Eq. (1) via Q-factor = (2δ_oω₀⁻¹ + 2δ_cω₀⁻¹)⁻¹ (see discussion below).

 

Fig. 4 Measured intensity and phase profile (black dots) of the THz WGM at 258.90 GHz [(a) and (b)], 503.74 GHz [(c) and (d)], and 1054.43 GHz [(e) and (f)]. The solid red and blue lines show the fit of Eq. (1) to the intensity and phase profiles, respectively. The corresponding calculated counterpart (intensity or phase) is shown with red and blue dashed lines.

Download Full Size | PDF

However, a close analysis of the WGM spectral features reveals slight systematic deviations from the analytical model. The deviations are best observed by comparison of the intensity and phase profiles based on the analytical model. The red solid lines in Figs. 4(a), 4(c), and 4(e) show the fit of the intensity of Eq. (1) to the measured intensity profiles. Overall, a very good agreement of the fits is observed. However, the corresponding calculated phase profiles shown with dashed red lines in Figs. 4(d), and 4(f) reveal a clear discrepancy between the calculated phase profiles based on the fit and measurements. The same observations can be made by fitting the phase profile (solid blue lines) and calculating the intensity profile (dashed blue lines). The discrepancy is particularly pronounced for strong coupling at a small WGMR-prism distance. This can be clearly seen from the intensity and phase profiles of the WGM at 1054.43 GHz shown in Figs. 5(a) and 5(b). In contrary, for a large WGMR-prism distance, or weak coupling, there is only a diminutive difference between the fits to the intensity and phase profiles [see Figs. 5(c) and 5(d)]. The above observation clearly indicates that the mismatch between the intensity and phase profile is caused by an incomplete detection of the THz radiation strongly interacting with the WGMR. Since the effect is most evident for strong coupling, but to a smaller extent for weak coupling, the incomplete detection is a consequence of the distortion of the THz beam due to a strong interaction with a mode-mismatched THz WGM as demonstrated above with the 2D simulations (see e.g. Fig. 2). As expected, further experiments (not shown here) with a smaller 4 mm diameter HRFZ-Si THz WGMR clearly reveal even stronger pronounced discrepancy between the intensity and phase profiles, as observed with the 8 mm diameter WGMR. Moreover, the incomplete detection leads to an apparent significant broadening of the observed THz WGMs. Please note, distortions as described above have not been observed in previous work with evanescent coupling from a single-mode waveguide. The mode matching problems discussed above are not applicable for a single-mode waveguide. Signal distortions with the prism can however be minimized by thoroughly aligning the system and carefully designing the experimental setup, such as focal spot size of the lenses, prism and WGMR geometry [31–33].

 

Fig. 5 Measured intensity and phase profiles of the THz WGM at 1054.66 GHz at strong coupling [(a) and (b)], and weak coupling [(c) and (d)]. The red and blue lines show again the fitted and calculated intensity and phase profiles based on Eq. (1).

Download Full Size | PDF

4. Conclusion

Despite the technological challenges in the THz frequency range, the presented results clearly demonstrate the feasibility of using a prism coupling scheme to excite high-Q THz WGMs. Nearly perfect detection of the THz WGMs with only minor deviations from an analytical model based on a complex valued Lorentzian lineshape has been achieved with a spherical WGMR and without any extensive optimization of the coupling. We furthermore show that potential detection issues in the experimental setup can be readily identified comparing the intensity and phase profiles of the observed THz WGMs. Also, the prism coupling of THz WGMs inherently provides stronger signals in the experiment compared to waveguide coupling due to significant waveguide losses. The THz WGMs in the HRFZ-Si spherical resonator excited via prism coupling show very high Q-factors over two octaves reaching up to 2.2 × 10⁴ at 1.1 THz. Finally, the excitation of high-Q THz WGMs via prism coupling from 0.2 THz to 1.1 THz provides manifold opportunities for future developments and applications like ultra-broadband highly-sensitive sensors.