1. Introduction

CW THz spectroscopy systems based on photomixing have been rapidly evolving in the last years. Nowadays, CW THz spectrometers spanning a bandwidth of almost 3 THz, with a step size of < 2 MHz are commercially available [1]. The advancing technology provides the vital base for the venture of THz radiation into various new fields. In particular, the high frequency resolution combined with the coherent detection of THz radiation provides a powerful tool for many applications [2]. The coherent detection of THz radiation in THz frequency-domain (FD) systems is now commonly achieved with fiber coupled photoconductive antennas (PCAs). The measured photocurrent I_ph at the detecting PCA depends on the amplitude of the THz field E THz but also the phase difference Δφ between the incoming THz radiation and the optical beat signal used to trigger the detection:

In order to retrieve the instantaneous amplitude (envelope) and phase of the THz signal from the detected photocurrent I_ph, the phase difference Δφ has to be modulated. Looking at Eq. (1) it is obvious that one can either sweep the THz frequency f or modulate the path difference ΔL between emitter and detector. Due to its simplicity, the common approach in a standard commercially available CW THz spectrometer is to sweep the THz frequency [3,4]. However, traditionally this method extracts the instantaneous amplitude and phase of the THz signal merely from the extrema of the detected photocurrent. In consequence, this method provides an effective frequency resolution given by the fringe spacing of the oscillating photocurrent. Therefore, the achievable effective frequency resolution is usually one or two orders of magnitude lower than the actual step size of < 2 MHz of state of the art CW THz spectrometer. A straightforward approach to decrease the fringe spacing and thus increase the maximum relative frequency resolution achievable by this method is to increase the path difference ΔL between emitter and detector. Latter could be achieved by incorporating a fiber patch cable to one of the two optical arms. However, for measurements requiring high precision, this asymmetric setup is prone to thermal fluctuations of ΔL, and additionally, a large ΔL increases the impact of a possible drift of the THz frequency f [5]. Both these effects add to the uncertainty in the phase difference Δφ. A symmetric setup with a small ΔL, including the THz path, is favorable. Finally, very fine frequency steps are required in order to clearly determine the position of the extrema, which requires a significant data acquisition time.

As mentioned earlier, an alternative method is to modulate the path difference ΔL between emitter and detector. Various approaches like mechanical delay stages or fiber stretchers in the emitter and detector arm have been proposed to efficiently vary ΔL [5]. Those systems allow to retrieve the instantaneous amplitude and phase at each frequency step. However, changing ΔL adds complexity to a standard CW THz spectrometer and increases the costs of commercial systems considerably. Furthermore, it increases the acquisition time compared to a standard setup.

In this work, we present data analysis to extract the instantaneous amplitude (envelope) and phase of the THz signal at each frequency step without complex and expensive extensions to a standard commercially available CW THz spectrometer. In the proposed method, the analytic signal I_a of the detected photocurrent I_ph is calculated, utilizing Hilbert spectral analysis. To demonstrate the effectiveness of the proposed data analysis, we investigate the amplitude and relative phase shift of a WGM THz bubble resonator at critical coupling [6].

2. Data analysis

Hilbert transformation is commonly used in signal processing to calculate the envelope and instantaneous phase from an oscillating signal [7,8]. Here we apply the Hilbert transformation to calculate the complex-valued analytical signal I_a from the detected oscillating photocurrent I_ph as a function of the phase difference Δφ [7,8]:

Finally, the complex spectral properties of a sample can be characterized by comparing a sample scan with an appropriate reference scan. The amplitude transmission t of the sample is given as:

The relative phase shift Φ induced by the sample is:

3. Results

To experimentally demonstrate the precision of the proposed data analysis, we investigate the spectral characteristics of a WGM THz bubble resonator at critical coupling. The WGMs of the THz bubble resonator are THz waves confined in the wall with sub-wavelength thickness of the spherical resonator by index guiding. If the wavelength of the traveling THz wave is an integral multiple of the circumference of the bubble, the THz wave interferes constructively and is forming a resonant mode of the bubble - the WGM. The WGMs of the bubble resonator can be excited by coupling into the evanescent field of the mode. The latter is commonly achieved with the evanescent field of a sub-wavelength waveguide. Thereby the coupling efficiency is highly dependent on the waveguide-resonator distance. The case when the coupling losses compensate the intrinsic loss of the WGM is referred to as critical coupling. In this experiment, we use a single mode silica fiber with a diameter of 200 μm to evanescently couple into the THz bubble resonator. The waveguide-resonator distance can be precisely optimized with two computer-controlled translation stages. A microscope image of the fiber and the THz bubble resonator can be seen in the inset of Fig. 1. For the coherent generation and detection of the THz field propagating in the THz fiber, we are using the TeraScan 1550nm system from Toptica with two fiber coupled InGaAs PCAs (see Fig. 1) [1]. The Toptica system has a specified absolute frequency resolution of <2 GHz and a step size of <2 MHz. The linewidth of the in-built DFB lasers is stated as 960 kHz [1].

 

Fig. 1 Schematic diagram of the experimental setup with the Toptica TeraScan 1550nm system. The THz path length is about 0.8 m. The inset shows a microscope picture of the silica fiber and the THz bubble resonator. The sub-wavelength fiber has a diameter of 200 μm. The bubble resonator made of quartz glass has a diameter of 6.3 mm and a wall thickness of 78 μm.

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To characterize the WGM’s spectral properties, we analyze the THz field transmitted through the fiber with and without the THz bubble resonator in close proximity of the fiber. The photocurrents of the reference scan (without resonator) and sample scan (with resonator) are shown in Fig. 2 (a) in black and red dots respectively. The scans shown cover the frequency range from 0.4642 THz to 0.4672 THz in 1 MHz frequency steps and are averaged over five measurements. With a long integration time of 300 ms, the acquisition time of a single scan is about 25 min. A shorter integration time for the lock-in detection significantly decreases the measurement time, but reduces the signal-to-noise ratio. Also, the very low scanning speed of 3.3 MHz/s allows sufficient time for the temperature control loop of the lasers to adjust to the swept frequencies. Please note, that the experimental setup is placed in a standard temperature controlled laboratory environment, and no further means to control the temperature are taken.

 

Fig. 2 (a) Detected photocurrent in the frequency range from 0.4642 THz to 0.4672 THz for the sample (red dots) and reference (black dots) scans, respectively. The frequency step size is 1 MHz and the data is averaged over 5 scans. The red and black solid lines show the corresponding envelopes retrieved from the analytical signal and (b) shows the phase of both sample and reference scan obtained from the data processing and (c) shows the amplitude ratio of the sample and reference envelope shown in (a). The absorption due to the resonance of the THz bubble is clearly visible and (d) presents the relative phase shift of the WGM. A very steep transition of π at the resonance frequency of 0.46574 THz is clearly visible.

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The envelopes of the sample and reference scan calculated with Eq. (1) are shown in Fig. 2(a) with red and black solid lines, respectively. The strong dip in the sample envelope at 0.46574 THz is due to the resonant coupling from the fiber to the WGM of the THz bubble resonator. The small oscillations in the envelope of both sample and reference scan are caused by standing waves in the THz path of the experimental setup [1], but as expected these are not visible in the ratio as shown in Fig. 2(c). In Fig. 2(b) the corresponding instantaneous phase of the sample and reference scan calculated with eq. 2 are plotted with red and black solid lines, respectively. Please note that the unwrap function of the python package ’numpy’ was applied to remove artificial jumps of 2π in the calculated phase [10]. Figures 2(c) and 2(d) show the amplitude transmission and relative phase shift calculated with Eq. (4) and Eq. (5), respectively. The amplitude ratio shows a significant absorption of > 99 % at the resonance frequency of the WGM, clearly indicating that the mode is very close to critical coupling. At the same time, the relative phase shift shows a very interesting profile as can be seen in Fig. 2(d). The phase undergoes an overall shift of 2π with a very steep transition of π centered at the resonance frequency of the WGM of 0.46574 THz. Intriguingly, the analytical theory for a 2D ring resonator predicts a jump of π with a step function-like behavior exclusively for the case of critical coupling [11]. The very steep transition in the phase profile observed in Fig. 2(d) at the resonance frequency indicates that the WGM is indeed very close to critical coupling. In the following, we focus on a much narrower frequency range around the resonance frequency of the WGM and optimize the waveguide-resonator distance to achieve critical coupling.

The resultant relative phase shift in a frequency range of 40 MHz on either side of the resonance frequency of the WGM measured in 1 MHz steps is shown in Fig. 3. The waveguide-resonator distance is highly optimized with a precision of about 0.2 μm, corresponding to ∼ λ₀/3200 at 0.466 THz.

 

Fig. 3 Relative phase shift (black dots) of a WGM of a THz bubble resonator at critical coupling. The measurements are averaged over seven scans and the error bars show the standard deviation of the calculated relative phase shift. The measurements are taken with a 1 MHz step size and are analyzed with Hilbert spectral analysis. The frequency axis is centered around the resonance frequency of the WGM. The experimental results are fitted (cyan solid line) with the analytical equation for a 2D ring resonator for critical coupling, convoluted with a Gaussian beam. The fitted FWHM of the Gaussian beam is ∼ 4 MHz. For comparison, the red solid line shows the phase profile for critical coupling of the analytical model without the convolution.

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The presented data is averaged over seven scans with 300 ms integration time. The acquisition time for one scan is about four minutes. The error bars show the standard deviation of the calculated relative phase shift. The averaged data resembles the step function-like behavior predicted by the analytical model for critical coupling. The phase jump of π occurs over a frequency range of just 3–4 MHz. More importantly, individual scans show that the jump of π is indeed occurring within the 1 MHz step size of the frequency sweep. The latter strongly indicates that the WGM is genuinely at critical coupling. However, effects such as jitter of the swept frequency, the beat laser linewidth and potentially a minor thermal drift in the fiber-resonator position lead to an averaged phase profile as observed in Fig. 3. To take all the mentioned effects into account, we fit the experimentally obtained data with a convolution of a Gaussian function and the analytical model for critical coupling. The resultant curve is shown in Fig. 3 as the cyan solid line. For comparison, the red solid line shows the analytical model with the same parameters from the fit, but without the Gaussian convolution. The fit shows an exceptional good agreement with the relative phase shift extracted from the experimental data using Hilbert spectral analysis. The fitted full-width half-maximum (FWHM) of the Gaussian is ∼ 4 MHz. The implication is that Fig. 3 clearly shows that the experimental data extracted via Hilbert spectral analysis resolve the step function in the relative phase shift of the WGM with a frequency resolution of ∼ 4 MHz. Considering a measured jitter of up to 3–4 MHz of the swept frequency and the specified laser linewidth of ∼ 1 MHz, the Gaussian broadening of ∼ 4 MHz is very close to the smallest frequency resolution of this spectrometer. This result is also a strong indication that the resonator is indeed at critical coupling, and that a steady thermal drift in the waveguide-resonator position is negligible. Please note that in the sophisticated experimental setup, we are constraint to a relatively long THz path of about 0.8 m, potentially deteriorating the phase stability of the system compared to an ideal configuration with a very small ΔL [1]. The effective frequency resolution with the Hilbert spectral analysis of 4 MHz obtained experimentally is an exceptional result. As a comparison, Fig. 2(a) displays the frequency spacing between extrema in the photocurrent to be about 0.13 GHz, which is equivalent to the effective frequency resolution achievable with the traditional method. Therefore, the effective frequency resolution of 4 MHz demonstrated with the proposed data analysis is about 33 times better, compared to the effective resolution achievable with the traditional analysis for an identical experimental setup.

4. Conclusion

We have shown that Hilbert spectral analysis is a powerful tool for analyzing CW THz spectroscopy measurements. Experimental characterization of the relative phase shift of a WGM in a THz bubble resonator at critical coupling highlights the capability of the proposed method. The measurements are taken with a standard commercially available CW THz spectrometer without complex extensions. The effective frequency resolution obtained from the measurements of ∼ 4 MHz is limited by the small fluctuations within the system, and its resolution is about 33 times better compared to the traditional method. Moreover, since less frequency steps are needed to obtain the same effective frequency resolution, the acquisition time of scans is significantly reduced. Ultimately, for a single scan, Hilbert spectral analysis clearly showed the expected step function within the frequency resolution given by the step size (1 MHz) of the laser system.