Real-time absolute frequency measurement of continuous-wave terahertz wave based on dual terahertz combs of photocarriers with different frequency spacings

Real-time measurement of the absolute frequency of continuous-wave terahertz (CW-THz) waves is required for characterization and frequency calibration of practical CW-THz sources. We proposed a method for real-time monitoring of the absolute frequency of CW-THz waves involving temporally parallel, i.e., simultaneous, measurement of two pairs of beat frequencies and laser repetition frequencies based on dual THz combs of photocarriers (PC-THz combs) with different frequency spacings. To demonstrate the method, THz-comb-referenced spectrum analyzers were constructed with a dual configuration based on dual femtosecond lasers. Regardless of the presence or absence of frequency control in the PC-THz combs, a frequency precision of 10-11 was achieved at a measurement rate of 100 Hz. Furthermore, large fluctuation of the CW-THz frequencies, crossing several modes of the PC-THz combs, was correctly monitored in real time. The proposed method will be a powerful tool for the research and development of practical CW-THz sources, and other applications.


Introduction
When femtosecond mode-locked laser light is incident onto a photoconductive antenna (PCA) for detecting terahertz (THz) waves, sub-picosecond photoconductive switching is repeated in the PCA in synchronization with the laser pulses. The sequence of switching operations is essentially copies of the same switching operation separated by an interval equal to the laser repetition period. This highly stable, switching pulse train in the time domain can be synthesized by a series of frequency spikes of photocarrier generation regularly separated by the laser repetition frequency in the frequency domain [1]. This structure is referred to as a THz frequency comb of photocarriers, or a PC-THz comb. Since the absolute frequencies of all frequency modes in the PC-THz comb can be phase-locked to a microwave frequency standard by control of the laser repetition frequency, such a frequency-comb structure enables us to use a PC-THz comb as a precise ruler for measuring THz frequency.
Recently, the potential of PC-THz combs in THz frequency metrology has been recognized [2,3], for example, as a THz-comb-referenced spectrum analyzer or frequency counter for absolute frequency measurement [4][5][6][7]. This type of spectrum analyzer is capable of precise frequency measurement within the frequency coverage of the PC-THz comb at room temperature by using the following procedure: First, a PC-THz comb is generated in a PCA. Second, a continuous-wave THz (CW-THz) wave is mixed with the generated PC-THz comb. Finally, the resultant signal is -4-beat-down to the radio-frequency (RF) region by photoconductive mixing. A THz-comb-referenced spectrum analyzer based on this technique has been successively applied to the absolute frequency measurement of narrow-linewidth CW-THz waves [4][5][6][7] and even broadband THz combs [1,[8][9][10]. Similar approaches for CW-THz waves have been demonstrated in combination with free-space electro-optics sampling [11,12] and an interferometric method [13] in place of the photoconductive detection. Also, PC-THz combs have been used in the phase and its slope measurements of tunable CW-THz waves for THz distance measurement of optically rough objects [14]. Furthermore, the generation of a frequency standard signal has been achieved by using a PC-THz comb in combination with frequency control of the CW-THz sources [15,16].
In previous studies on THz-comb-referenced spectrum analyzers, a single PC-THz comb has been used [4][5][6]. In this case, it has been necessary to measure two beat frequencies respectively corresponding to two different frequency spacings of the PC-THz comb in order to determine the comb mode number nearest in frequency to the CW-THz wave. Therefore, two beat frequencies have been measured before and after shifting the frequency spacing of the PC-THz comb by the laser control. This temporally serial, two-step measurement with a single PC-THz comb has been an obstacle in applying this technique to the real-time absolute frequency measurement of frequency-fluctuating CW-THz waves. Also, use of a precisely stabilized femtosecond laser comb often hinders the easy use of this -5-spectrum analyzer. If the real-time absolute frequency measurement of practical CW-THz sources with rapid, large frequency variations could be implemented using unstabilized femtosecond lasers, the scope of applications would be greatly expanded.
In the work described in this article, we determined the absolute frequency of a frequency-fluctuating CW-THz wave in real time based on temporally parallel, i.e. simultaneous, measurement of two pairs of beat frequencies and repetition frequencies of dual PC-THz combs with different frequency spacings, that is to say, by using a dual THz-comb-referenced spectrum analyzer. We also investigated the possibility of using a PC-THz comb without frequency stabilization for the real-time absolute frequency measurement of the frequency-fluctuating CW-THz wave.

Principle
THz-comb-referenced spectrum analyzer is based on a heterodyne technique involving photoconductive mixing between a PC-THz comb and a CW-THz wave, which is described in detail elsewhere [4,5]. There are two essential points in this method: First, a PCA is used as a heterodyne receiver having high, broadband spectral sensitivity in the THz region without the need for cryogenic cooling. Second, the PC-THz comb functions as a local oscillator with multiple frequencies, fully covering the THz region.
In the photoconductive mixing, the absolute frequency of the measured -6-CW-THz wave (= f THz ) is given by where m is the order of the comb mode nearest in frequency to the CW-THz wave, f rep is the repetition frequency of the femtosecond laser, and f beat is the lowest frequency of the beat signals. Since f rep and f beat can be measured directly in the RF region, the value of m and the sign of f beat have to be determined to obtain f THz . To this end, one has to measure two f beat values (f beat1 and f beat2 ) corresponding to two different f rep values (f rep1 and f rep2 ), because the relation between them is given by Since previous studies have been based on a single PC-THz comb, it is essential to measure the beat frequencies (f beat1 and f beat2 ) before and after shifting the frequency spacing of the PC-THz comb by laser control (f rep1 and f rep2 ) [4][5][6]. However, such temporally serial, two-step measurement with a single PC-THz comb hinders the real-time determination of f THz . For real-time determination, temporally parallel, that is, simultaneous, measurement of f beat1 , f beat2 , f rep1 , and f rep2 should be performed. To this end, the use of dual PC-THz combs with different frequency spacings will be useful.
Finally, f THz can be determined by measuring f rep1 , f rep2 , f beat1 , and f beat2 because If f rep1 and f rep2 are stabilized at known values by laser control, we need to measure just f beat1 and f beat2 to determine f THz . If f rep1 and f rep2 are fluctuated due to the free-running operation, f beat1 , f beat2 , f rep1 , and f rep2 should be measured at the same time.

Experimental setup
The THz-comb-referenced spectrum analyzer that we developed was composed of femtosecond lasers, a PCA for THz detection, and data acquisition electronics. Two of these THz-comb-referenced spectrum analyzers were constructed with a dual configuration based on dual femtosecond lasers, and these were effectively used to determine the absolute frequency of a CW-THz wave in real time. Figure 2 shows a schematic diagram of the experimental setup. We used dual Therefore, we need to average at least 1,000 signals, corresponding to a measurement rate of 10 kHz, to determine m correctly. We also investigated the frequency precision of f THz and the corresponding frequency error with respect to the signal-to-noise ratio (SNR) when the measurement rate was set to 10 Hz, as shown in Fig. 4(b). For comparison, we also show the relation between them for frequency measurement with an RF frequency counter, which has been used in previous studies [4,5,7]. Although both methods showed the dependence of the frequency precision on the SNR, their dependence characteristics were different from each other. An SNR of only 10 dB was sufficient for achieving a frequency precision of

Use of dual PC-THz combs without stabilization of frequency spacing
In previous studies, the frequency spacing of the PC-THz comb has been precisely stabilized by using laser control [4,5,7]. However, use of a stabilized femtosecond laser has often restricted the use of the THz-comb-referenced spectrum analyzer in various applications, despite its superior performance. If the real-time absolute frequency measurement with dual PC-THz combs could be implemented using free-running, that is, unstabilized, lasers, the scope of application of the spectrum analyzer would be greatly expanded. Recently, temporally serial, two-step frequency measurement has been performed using a single PC-THz comb without stabilization of the frequency spacing [6]; however, there have been no attempts to perform the real-time frequency measurement using dual PC-THz combs without -13-stabilization of the frequency spacing, that is, free-running dual PC-THz combs. We attempted to determine the absolute frequency of a CW-THz wave in real time using free-running dual PC-THz combs. Next, we measured f rep1 , f rep2 , f beat1 , and f beat2 for the frequency-stabilized CW-THz wave, and then determined f THz . In this experiment, the absolute frequency of the CW-THz wave was fixed at 100,001,004,000 Hz, whereas f rep1 (≈ 100,000,007 Hz) and f rep2 (≈ 100,000,217 Hz) were not stabilized.

Conclusions
We demonstrated real-time, precise measurement of the absolute frequency of a CW-THz wave by simultaneous measurement of f rep1 , f rep2 , f beat1 , and f beat2 using dual PC-THz combs with different frequency spacings. Regardless of the presence or absence of frequency control of the PC-THz combs, a frequency precision of 10 -11 was achieved at a measurement rate of 100 Hz. The proposed method was successfully applied to real-time monitoring of f THz with large frequency fluctuations -15-across several PC-THz comb modes, indicating the high potential of our method to practical CW-THz sources with free-running operation or mode hopping. The proposed method will be a practical tool for the characterization and frequency calibration of a variety of CW-THz sources, including THz quantum cascade lasers [17], photomixing sources [18], resonant tunneling diodes [19], and so on.
One may consider that the need for dual femtosecond lasers is still an obstacle for the practical use of this method, even though free-running lasers can be used. Recently, a dual-wavelength mode-locked fiber laser has been realized under certain cavity configurations [20]. Because of dispersion, resulting in different refractive indexes at the two wavelengths in the fiber cavity, the two wavelength lights have different repetition frequencies. This laser will be preferable for the real-time absolute frequency measurement based on dual PC-THz combs with different frequency spacings. Another possible method is to use a single, free-running femtosecond laser with frequency modulation of f rep . Work is in progress to perform real-time monitoring of f THz with a single, f rep -modulated femtosecond laser. The proposed method, in combination with these lasers, will further allow the practical use of THz-comb-referenced spectrum analyzers, and will hence accelerate their adoption in real-world applications. Hz. The measurement rate was set at 100 Hz for both measurements. -25-