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Real-time measurement of the absolute frequency of continuous-wave terahertz (CW-THz) radiation 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 radiation 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.
© 2015 Optical Society of America
1. Introduction
When femtosecond mode-locked laser light is incident onto a photoconductive antenna (PCA) for detecting terahertz (THz) radiation, sub-picosecond photoconductive switching is repeated in the PCA in synchronization with the laser pulses. 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 [4–7]. 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) radiation is mixed with the generated PC-THz comb. Finally, the resultant signal is down-converted 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 radiation [4–7] and even broadband THz combs [1, 8–10]. Similar approaches for CW-THz radiation have been demonstrated in combination with free-space electro-optics sampling [11, 12] or 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 radiation 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 [4–7]. Since it is 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 radiation, 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 radiation. Also, use of a precisely stabilized femtosecond laser often hinders the easy use of this 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 radiation 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 dual THz-comb-referenced spectrum analyzers. 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 radiation.
2. Principle
THz-comb-referenced spectrum analyzer is based on a heterodyne technique involving photoconductive mixing between a PC-THz comb and a CW-THz radiation, 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 CW-THz radiation ( = fTHz) is given by
where m is the order of the comb mode nearest in frequency to the CW-THz radiation, frep is the repetition frequency of the femtosecond laser, and fbeat is the lowest frequency of the beat signals. Since frep and fbeat can be measured directly in the RF region, the value of m and the sign of fbeat have to be determined to obtain fTHz. To this end, one has to measure two fbeat values (fbeat1 and fbeat2) corresponding to two different frep values (frep1 and frep2), because the relation between them is given bySince previous studies have been based on a single PC-THz comb, it is essential to measure the beat frequencies (fbeat1 and fbeat2) before and after shifting the frequency spacing of the PC-THz comb by laser control (frep1 and frep2) [4–6]. However, such temporally serial, two-step measurement with a single PC-THz comb hinders the real-time determination of fTHz. For real-time determination, temporally parallel, that is, simultaneous, measurement of fbeat1, fbeat2, frep1, and frep2 should be performed. To this end, the use of dual PC-THz combs with different frequency spacings will be useful.Figure 1 shows the principle of real-time determination of fTHz based on dual PC-THz combs with different frequency spacings. When two femtosecond laser lights with frep1 and frep2 are incident onto two different PCAs (PCA1 and PCA2), two PC-THz combs (PC-THz-comb1 and PC-THz-comb2) with slightly detuned frequency spacings (frep1 and frep2) are respectively induced in them. Then, a measured CW-THz radiation (frequency fTHz) is incident on both PCA1 and PCA2, resulting in the generation of two beat signals with frequencies fbeat1 and fbeat2. From Eq. (2), the m value can be obtained by
The sign of (fbeat2 - fbeat1)/(frep2 – frep1) equals inverse of that of fbeat in Eq. (1). Finally, fTHz can be determined by measuring frep1, frep2, fbeat1, and fbeat2 becauseIf frep1 and frep2 are stabilized at known values by laser control, we need to measure just fbeat1 and fbeat2 to determine fTHz. If frep1 and frep2 are fluctuated due to the free-running operation, fbeat1, fbeat2, frep1, and frep2 should be measured at the same time.
Fig. 1 Principle of real-time absolute frequency measurement of CW-THz radiation based on simultaneous measurement of fbeat1, fbeat2, frep1, and frep2 with dual PC-THz combs having different frequency spacings.
3. 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 radiation in real time. Figure 2 shows a schematic diagram of the experimental setup. We used dual mode-locked Er-doped fiber lasers (ASOPS TWIN 100 with P100, Menlo Systems; center wavelength = 1550 nm, pulse duration = 50 fs) with slightly mismatched repetition frequencies (frep1 and frep2) to generate dual PC-THz combs with different frequency spacings in the PCAs. In the principle of Fig. 1, we assumed the same m value for frep1 and frep2. To this end, frequency difference between mfrep1 and mfrep2 has to be less than a half of frep1. The maximum frequency difference is given by frep1/(2m). For example, when frep1 = 100 MHz, fTHz = 0.1 THz, and m = 1,000 demonstrated later, the maximum frequency difference is 50 kHz. When the repetition-frequency stabilization systems were activated, frep1 and frep2 were phase-locked to a rubidium (Rb) frequency standard (Stanford Research Systems FS725 with frequency = 10 MHz, accuracy = 5 × 10−11, stability = 2 × 10−11 at 1 s).
Fig. 2 Experimental setup. PPLN: periodically-poled-lithium-niobate crystal; BS: beam splitter; L: objective lens; PCA1 and PCA2: photoconductive antennas; PC-THz-comb1 and PC-THz-comb2: THz comb of photocarriers with different frequency spacings; AMP: current preamplifier.
The two laser-beams were individually focused on bowtie-shaped, low-temperature-grown GaAs photoconductive antennas (PCA1 and PCA2) after wavelength conversion by second harmonic generation (SHG) with periodically-poled-lithium-niobate (PPLN) crystals. Dual PC-THz combs with frequency spacings of frep1 and frep2 (PC-THz-comb1 and PC-THz-comb2) were respectively induced in PCA1 and PCA2. To test the spectrum analyzers, we measured a CW-THz radiation (frequency fTHz) from an active frequency multiplier chain (Millitech AMC-10-R0000, multiplication factor = 6, tuning range = 75–110 GHz, linewidth <0.6 Hz, and average power = 2.5 mW), which multiplied the output frequency of a microwave frequency synthesizer (Agilent E8257D, frequency = 12.5–18.33 GHz, and linewidth <0.1 Hz) by six. The frequency synthesizer is phase-locked to the Rb frequency standard. When this CW-THz radiation was incident on both PCA1 and PCA2 together with the two laser beams, groups of current beat signals between the CW-THz radiation and PC-THz-comb1/PC-THz-comb2 were generated in the RF region as a result of photoconductive mixing. The beat signals at the lowest frequency (frequencies fbeat1 and fbeat2) were extracted by amplifying and low-pass-filtering with current preamplifiers (AMP; bandwidth = 40 MHz and sensitivity = 100,000 V/A). The signals with frequencies frep1 and frep2 were measured by detecting the laser beams with fast photodetectors. The four temporal waveforms for fbeat1, fbeat2, frep1, and frep2 were acquired simultaneously by a fast digitizer (resolution = 14 bit, sampling rate = 100 MHz) using the Rb frequency standard’s ouput for an external clock. We applied a software-based, instantaneous-frequency-calculation algorithm to the acquired temporal waveforms [6]. This algorithm is consisting of Fourier transform, digital frequency filtering, inverse Fourier transform, Hilbert transform, and time differential of instantaneous phase. Finally, we determined fTHz by substituting the instantaneous values of fbeat1, fbeat2, frep1, and frep2 into Eq. (4).
4. Results
4.1 Use of dual PC-THz combs with stabilization of frequency spacing
First, we demonstrated real-time monitoring of fTHz for the frequency-fixed CW-THz radiation based on dual PC-THz combs with stabilization of frep1 and frep2, that is to say, using stabilized dual PC-THz combs. Figures 3(a) and 3(b) show temporal changes of instantaneous values of fbeat1 and fbeat2 when frep1 and frep2 were fixed at 100,000,000 Hz and 100,000,050 Hz by laser stabilization control. From these values, we obtained the temporal change of fTHz as shown in Fig. 3(c). The mean and standard deviation of fTHz were 100,001,000,040 Hz and 60 Hz, respectively. Since fTHz was stabilized at 100,001,000,040 Hz within a frequency range of 5 Hz by providing the Rb frequency standard’s output to the frequency synthesizer as an external reference, the variation of fTHz in Fig. 3(c) was mainly due to the error in the instantaneous-frequency-calculation algorithm of fbeat1 and fbeat2.
Fig. 3 Temporal changes of (a) fbeat1, (b) fbeat2, and (c) fTHz when frep1 and frep2 were fixed at 100,000,000 Hz and 100,000,050 Hz by laser stabilization control. Sampling rate was 10 MHz.
To reduce the errors in fTHz, we performed signal averaging of fbeat1 and fbeat2 after calculating their instantaneous values. Figure 4(a) shows the frequency error with respect to the number of averaged signals in fbeat1 and fbeat2, and the upper horizontal axis gives the corresponding measurement rate. It is clear that the frequency error depends on the number of averaged signals or the measurement rate. To avoid the incorrect determination of m, we have to determine fbeat1 and fbeat2 within a frequency error of ± 25 Hz because |frep2 - frep1| in Eq. (3) was set to 50 Hz in this experiment. 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 fTHz 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 measurement precision for frequency measurement with an RF frequency counter, which has been used in previous studies [4, 5, 7]. The minimum detectable power of CW-THz radiation is given elsewhere [5]. 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 10−11 in the instantaneous-frequency-calculation algorithm, whereas the frequency counter method required an SNR of at least 30 dB to perform the absolute frequency determination due to the measurement principle of the RF frequency counter. Thus, a combination of dual PC-THz combs with the instantaneous-frequency-calculation algorithm enables precise measurement of fTHz in real time and/or at low SNR. We achieved the frequency precision of 8.2 × 10−12 at a measurement rate of 100 Hz. It should be note that this value indicates the relative precision of frequency measurement because the output signal of the Rb frequency standard is used as a common time-base signal in this setup. The absolute precision of frequency measurement is determined by the accuracy of the Rb frequency standard or the relative precision whichever is better. In this experiment, the final precision was limited by the accuracy of the Rb frequency standard. If we use more accurate frequency standard, the present system has further precision gains that can be implemented.
Fig. 4 (a) Frequency error in fbeat1 with respect to number of averaged signals or measurement rate. Red dotted line indicates the acceptance frequency error in fbeat1 and fbeat2 to correctly determine m. (b) Comparison of frequency precision and the corresponding frequency error in fTHz with respect to signal-to-noise ratio between Hilbert transformation method and frequency counter method. The measurement rate was set at 10 Hz.
Next, we tracked the temporal changes in fTHz to demonstrate the real-time capability of this method. Figure 5(a) and Media 1 show the temporal changes in fTHz when fTHz was manually changed within a frequency range of 0.1 THz ± 100 Hz (measurement rate = 100 Hz). A slight change of several Hz to a few tens of Hz in fTHz was sensitively reflected in the chart, indicating the potential of this method for precise measurement of absolute frequency in real time.
Fig. 5 (a) Temporal change of fTHz when the fTHz was manually changed within a frequency range of 0.1 THz ± 100 Hz (Media 1). (b) Temporal change of fTHz when fTHz was stepwise tuned at intervals of 200 MHz from 99,801,000,000 Hz to 100,440,000,000 Hz. The measurement rate was set at 100 Hz for both measurements.
On the other hand, practical CW-THz sources often exhibit large fluctuations of fTHz due to free-running operation or mode hopping. In particular, frequency jumps due to mode hopping are sometimes as high as several hundreds MHz to several GHz. In this case, the CW-THz radiation may suddenly cross over many frequency modes of the PC-THz comb because the frequency modes of PC-THz comb are too densely distributed along the frequency axis compared with the frequency change. Such large and/or instantaneous fluctuations of fTHz cannot be observed by the conventional method with a single PC-THz comb due to the non-real-time two-step measurement principle used in that method. To evaluate the applicability of our method to such CW-THz radiation, we stepwise tuned fTHz at intervals of 200 MHz from 99,801,000,000 Hz to 100,440,000,000 Hz. Figure 5(b) shows the result of real-time monitoring of fTHz (measurement rate = 100 Hz), indicating that the CW-THz radiation crossed two PC-THz comb modes in every stepwise tuning of fTHz. This demonstration reveals the potential of our method for measuring large and rapid fluctuations of fTHz.
4.2 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 stabilization of the frequency spacing, that is, free-running dual PC-THz combs. We attempted to determine the absolute frequency of a CW-THz radiation in real time using free-running dual PC-THz combs.
Consider Eq. (3) when both frep1 and frep2 fluctuate. This relation must always be true every moment regardless of the fluctuations of frep1 and frep2 because a free-running PC-THz comb can also be used as a frequency ruler with a linear scale at every moment. This means that precise stabilization of frep1 and frep2 in dual lasers is not essential if frep1, frep2, fbeat1, and fbeat2 are measured simultaneously. In other words, free-running dual PC-THz combs can be used in the proposed method.
First, we evaluated the frequency fluctuation of mfrep1 (m = 1,000 and frep1 = 100 MHz) for free-running and stabilized PC-THz combs with respect to various gate times, as shown in Fig. 6. Although the frequency fluctuation of the free-running PC-THz comb was larger than that of the stabilized PC-THz comb, the former was still smaller than the frequency spacing of the PC-THz combs. This means that we could determine the absolute frequency of fTHz using the same m value of the PC-THz combs during the measurement.
Fig. 6 Comparison of frequency fluctuation in mfrep1 with respect to gate time between stabilized and free-running PC-THz combs.
Next, we measured frep1, frep2, fbeat1, and fbeat2 for the frequency-stabilized CW-THz radiation, and then determined fTHz. In this experiment, the absolute frequency of the CW-THz radiation was fixed at 100,001,004,000 Hz, whereas frep1 (≈100,000,007 Hz) and frep2 (≈100,000,217 Hz) were not stabilized. Figure 7 shows the results for the real-time monitoring of (a) frep1, (b) frep2, (c) fbeat1, (d) fbeat2, and (e) the corresponding fTHz (sampling rate = 10 MHz). Temporal behavior of these values was somewhat different from that of them in Fig. 3. In the instantaneous-frequency-calculation algorithm for the free-running dual PC-THz combs, we used the digital band-pass filter having a pass-band width of 100 kHz to reduce the error by noise components in unstable fbeat1, fbeat2, frep1, frep2 values. On the other hand, in the case of the stabilized dual PC-THz combs, another digital frequency filtering with the shaper band-pass characteristic (bandwidth = 10 kHz) was applied because of stable fbeat1 and fbeat2 values. Difference of the temporal behavior between them is mainly due to different band-pass characteristics in the digital frequency filtering rather than the fluctuation of these values themselves. However, we achieved the frequency precision of 4.0 × 10−11 at a measurement rate of 100 Hz. This precision was comparable to that in the stabilized dual PC-THz combs, indicating that the frequency precision achieved in the proposed method is independent of the frequency stability of frep1 and frep2 (see Fig. 6) if they are precisely measured at the same time.
Fig. 7 Temporal changes of (a) fbeat1, (b) fbeat2, (c) frep1, (d) frep2, and (e) fTHz when frep1 and frep2 were not stabilized. Sampling rate was 10 MHz.
Finally, we performed real-time monitoring of fTHz, which was stepwise tuned at intervals of 200 MHz from 99,801,000,000 Hz to 100,401,000,000 Hz to evaluate the applicability of our method for measuring fTHz with large fluctuations. Figure 8 shows the result of real-time monitoring of fTHz (measurement rate = 100 Hz), indicating that the CW-THz radiation crossed two PC-THz comb modes in every stepwise tuning of fTHz. Thus, we successfully demonstrated the real-time and precise monitoring of fTHz with free-running dual PC-THz combs.
Fig. 8 Temporal change of fTHz when fTHz was stepwise tuned at intervals of 200 MHz from 99,801,000,000 Hz to 100,401,000,000 Hz. The measurement rate was set at 100 Hz.
5. Conclusions
We demonstrated real-time, precise measurement of the absolute frequency of a CW-THz radiation by simultaneous measurement of frep1, frep2, fbeat1, and fbeat2 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 fTHz with large frequency fluctuations 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. Although such effectiveness was demonstrated using the 0.1-THz CW-THz radiation in this article, upper limit of the frequency measurement in the proposed method depends on the spectral sensitivity in PCA and the fluctuation of frep1 and frep2. The frequency precision limits the frequency range of this method when fluctuation of frep1 and frep2 are too large. However, frep1 and frep2 are still stable even in the free-running operation as shown in Fig. 6. In this case, the spectral sensitivity in PCA determines the upper frequency limit. For example, frequency measurement of 2.5-THz CW-THz radiation was achieved using the stabilized PC-THz comb [15]. Therefore, 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 frep. Work is in progress to perform real-time monitoring of fTHz with a single, frep-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.
Acknowledgment
This work was supported by Collaborative Research Based on Industrial Demand from Japan Science and Technology Agency, and Grants-in-Aid for Scientific Research No. 26246031 from the Ministry of Education, Culture, Sports, Science, and Technology of Japan. The authors wish to acknowledge Yumi Sumino for her help in preparation of the manuscript.
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