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Abstract
A coherent Doppler LIDAR (CDL) with long-duration frequency-modulated pulses was demonstrated and validated by analyzing the data observed by a prototype. In traditional CDL using short-duration single-frequency pulses (PCDL; pulsed CDL), there exists a trade-off relationship between distance and velocity resolution. Meanwhile, in earlier work, a theoretical framework of CDL using long-duration frequency-modulated pulses (FMCDL; frequency-modulated CDL) was put forth to eliminate the trade-off. We developed the prototype to be operated as both a PCDL and FMCDL. Analyses of data observed by the PCDL and FMCDL modes showed that the FMCDL worked in good agreement with the PCDL for wind ranging and velocimetry. Furthermore, the performance of the FMCDL in terms of received power and resolution of distance and velocity was quantitatively consistent with ones theoretically expected.
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1. Introduction
In recent years, coherent Doppler LIDARs (CDLs) have been utilized for various applications such as wind resource assessment [1] and airport safety by detecting windshear, turbulence, and wake vortex [2–6]. They are also used for aviation safety during flights to detect windshear, turbulence, and volcanic ash [7–11]. In addition, they are used to improve efficient power generation by directional controls of wind turbines [12] as well as weather forecasts using spaceborne wind lidars [13]. These real-world applications are crucial for creating a safe and sustainable society.
When measuring wind, having precise specifications for distance and velocity resolution is essential. Each application has its demands on resolution. Wind resource assessment, for example, needs distance resolution which is comparable with the size of wind turbines. For aircraft wake vortex detection, wake vortices whose sizes and rotational velocities are several meters and meters per second respectively, need to be resolved [4]. A significant constraint in CDLs’ specifications is that both distance and velocity resolution are determined by the duration of pulses [14]. Typical pulse duration (hundreds–thousands of nanoseconds) restricts distance and velocity resolution to around one hundred meters and a few meters per second, respectively. Furthermore, there is a trade-off relationship between the resolution where a shorter pulse duration results in higher distance resolution but lower velocity resolution, and vice versa.
Yoshikawa and Ushio proposed a new concept of CDLs, which eliminates the constraint [14]. While typical CDLs radiate short-duration single-frequency pulses, the new-concept CDLs utilize long-duration frequency-modulated pulses. The new concept naturally derives CDLs with frequency-modulated continuous waves. Hereafter, this paper refers to typical pulsed CDLs as PCDLs and new-concept frequency-modulated CDLs as FMCDLs. Theoretical analyses revealed that FMCDLs’ distance and velocity resolution are independently specified by the bandwidth of frequency modulation and pulse duration, respectively. The FMCDLs are not restricted by the trade-off relationship, making them more broadly applicable than PCDL.
In the following work of [14], a prototype of FMCDLs was developed (for the first time) and initial observations were carried out. By analyzing the observed data, the feasibility of FMCDL was demonstrated. This paper reports on the following work. Specifically, details of hardware and software of the prototype, specifications of the initial observation, and analyses and their results for the demonstration are explained. Furthermore, technical caveats found in developing the prototype are discussed.
2. Prototype
A block diagram of the prototype is shown in Fig. 1. The prototype was configured in the headquarters of the National Institute of Information and Communications Technology (NICT). A laser source generates a coherent continuous wave (CW) with a wavelength of 1,550 nanometers, which branches at the fiber coupler (Coupler 1) to a circulator (Circulator 1) and the other coupler (Coupler 2). The CW laser goes through Circulator 1 to the fiber Acousto-Optic Modulator (AOM). The AOM modulates the laser and the modulation is applied twice by the fiber mirror installed immediately after the AOM, which increases modulation accuracy. The modulation wave is generated on a radio frequency (RF) carrier of 40 MHz by an Arbitrary Waveform Generator (AWG). That is, the two-time modulation creates modulated optical pulses whose carrier frequency is shifted by 80 MHz. The modulated optical pulses are then amplified by the Erbium-doped fiber amplifier (EDFA). After passing the other fiber circulator (Circulator 2), the optical pulses are radiated to free space by the collimator, expanded by the telescope, and then arbitrarily directed by the gimbaled mirror. Only at the time of observation, the gimbaled mirror is slid out on a rail mechanism to outdoors. Pulses returned from aerosols are received by the same mirror, telescope, and collimator to insert them into the fiber. The returned pulses are delivered by Circulator 2 to Coupler 2 where they are mixed with the CW laser branched at Coupler 1. Then, the balanced photo-detector down-converts the returned pulses into RF. The optics of the prototype is totally polarization-maintained. The down-converted returned pulses are recorded by the analog-to-digital converter (ADC). Between the photo-detector and the ADC, a high-pass filter is equipped to remove unwanted around-DC components. The ADC begins to record with a specified sampling frequency and the number of acquiring samples by a trigger signal which the AWG produces synchronously to the modulated pulse generation.
The prototype operated as a PCDL and FMCDL. The transmitting signal of the PCDL was designed to produce a short-duration single-frequency pulse as its definition. The signal of the FMCDL is a combination of nonlinear up-chirp and down-chirp signals with long duration, which were placed in the first and second halves of the entire pulse. These chirp signals were generated so that their spectra have a raised-cosine shape. This modulation is different from one proposed in the previous research [14], which was phase shift keying. While modulation bandwidth and pulse duration determine distance and velocity resolution (mainlobe), the modulation method affects sidelobes. However, a relation between modulation method and sidelobes is not clearly understood and we have not thoroughly tested various modulation methods in this study. Therefore, this paper does not focus on sidelobes depending on modulation methods in terms of FMCDL.
Waveforms of the two transmitting signals are presented in Fig. 2. The blue and orange lines are respectively for the PCDL and FMCDL. These waveforms were measured by a loop-back setting, in which the output of the EDFA is connected to Coupler 2 with attenuation. The attenuation level was adjusted so that the transmitting signals were recorded properly in a dynamic range of the ADC. Both the waveforms rise at a time of 1 µs because the AWG was set to produce trigger signals 1 µs before the transmitting signals.
Fig. 2. Waveforms of transmitting signals of the PCDL and FMCDL. The blue and orange lines are respectively for the PCDL and FMCDL.
The waveforms were not ideal shapes due to the EDFA. The EDFA was a product of CW laser which distorted pulsed laser. Since few generic products of EDFA for pulses with hundreds to thousands of nanoseconds are lined up, CW EDFAs are used in many CDLs. The waveforms were analyzed as summarized in Table 1. Duration of the pulses is equivalent to differences of the minimum and maximum timestamps having digital outputs more than 500, which were 1.01 and 4.02 µs in the PCDL and FMCDL, respectively. Energies of the pulses were calculated as 15.1 and 17.2 dB in the PCDL and FMCDL, respectively. Note that these energies are relative values because they are calculated based on the digital output of the ADC. Distance resolution was measured by (absolute-squared) auto-correlation functions of the waveforms which are shown in Fig. 3(a). The functions were normalized by values at a delay time of zero. Their -3-dB widths were 0.62 and 0.67 µs respectively in the PCDL and FMCDL, which correspond to theoretically expected distance resolution of 93 m ($=(c/2)$ 0.62 µs) and 101 m ($=(c/2)$ 0.67 µs), where $c$ is the speed of light. Figure 3(b) shows (power) spectra of envelopes of the waveforms by which velocity resolution can be confirmed. The spectra were also normalized by values at a frequency of zero. Their -3-dB widths were 0.93 and 0.20 MHz respectively in the PCDL and FMCDL, which correspond to theoretically expected velocity resolution of 0.72 m/s ($=(\lambda /2)$ 0.93 MHz) and 0.16 m/s ($=(\lambda /2)$ 0.20 MHz), where $\lambda$ is a wave length of the laser (1550 nm).
Fig. 3. (a) Absolute-squared auto-correlation functions and (b) power spectra of the transmitting waveform envelopes of the PCDL and FMCDL, all of which were normalized by values at zero of their horizontal axes. The blue and orange lines are respectively for the PCDL and the FMCDL.
Table 1. Specifications of observations
Table 1 also shows the other specifications. The PCDL and FMCDL were alternatively operated with a switching frequency of 2 kHz. Namely, a pulse repetition frequency was 1 kHz in either of the modes. The ADC recorded received signals with a sampling frequency, vertical resolution, and the number of time samples of 200 MHz, 14 bits, and 8,192 respectively.
Optical power was measured by installing a power meter after Circulator 2 (before the collimator) as roughly ten milliwatts on average. The gimbaled mirror was set to a tilt angle of 44 degrees, which directed the optical pulses to an elevation angle of 92 degrees. Although the elevation angle could be set to exactly 90 degrees, it was shifted by 2 degrees. This was done to maintain the quality of received signals since the optical path was close to an outer wall of the building.
3. Signal processing
(Digital) signal processing was performed to estimate distance-velocity profiles from received signals, which can be majorly divided into preprocessing and linear inversion.
The preprocessing sequentially performs a mode classification, the Hilbert transform, a band-pass filter, a down-conversion, and a sampling rate reduction. The mode classification determines whether the received signals were on the PCDL or FMCDL by measuring the duration of replicas of transmitted pulses. The replicas were found to appear by direct input of the transmitted pulses to Coupler 2, which is caused by the cross-talk on Circulator 2 or reflections on the collimator or telescope. This classification was necessary for the prototype because the ADC missed some triggers. The Hilbert transform converts each received signal from real to complex. This conversion preserves all information in the received signals and simplifies linear inversion implementation. The band-pass filter passes frequencies between 67.5–92.5 MHz. The pass frequency band was calculated considering the carrier frequency of transmission (80 MHz) and possible Doppler shifts (several megahertz) to include most of the information of received signals. The down-conversion lowers the frequencies by 67.5 MHz. This resulted in the frequency band of each received signal in the 0–25 MHz range. Finally, the sampling rate reduction is performed by an integer factor of 8, by which the number of time samples and the sampling frequency of each received signal are reduced from 8,192 to 1,024 and from 200 to 25 MHz, respectively. The reduction omits frequency components in 25-200 MHz which have become (approximately) zero by the preceding processes and allows the processing of the received signals using a small memory size.
Each distance-velocity profile can be estimated from a bunch of preprocessed received signals by any linear inversion methods. This study used a method based on Capon’s proposal, details of which are explained in the Appendix. It is well known that, for calculating a single spectrum from a bunch of incoherent time-series signals, the Capon method allows us to estimate the correct spectrum with very low sidelobes. This study used it for calculating distance-velocity profiles that can be considered as a bunch of spectra. The Capon method was analogously expected to give correct estimation with low distance-velocity sidelobes.
Specifications of the inversion were designated as shown in Table 2. The number of received signals was around 10,000. Its fluctuation is caused by the ADC missing triggers. To acquire (around) 10,000 received signals, 1 kHz PRF demands roughly 10 s which is the temporal resolution of the estimated distance-velocity profiles. A time difference between a pair of the PCDL’s and FMCDL’s distance-velocity profiles is around 1 ms (a reciprocal of 1 kHz) which is neglectably small compared with the temporal resolution. Calculation resolution for distance is derived from the sampling frequency of the preprocessed signal, which is 6 m ($=(c/2)/$ 25 MHz). The calculation resolution of Doppler shift is arbitrary and was determined as 50 kHz in this study, which resulted in 0.04-m/s calculation resolution for velocity ($=(\lambda /2)$ 50 kHz).
Table 2. Specifications of linear inversion method
The inversion method requires a replica of the transmitting signal (also known as a reference signal). We therefore obtained about 500,000 received signals of each mode by the loop-back setting. A replica of each mode was created by coherently averaging the loop-backed signals, by which system noises of the loop-back setting were mitigated.
4. Observation
An observation exemplified in this paper was carried out in two temporal ranges of 15:52–16:03 and 16:41–16:52 on June 30, 2022 (JST). In each of the two temporal ranges, 1,000,000 received signals (about 500,000 for each mode) were obtained. These were separated into fifty bunches of received signals. A single bunch contains 20,000 consecutive received signals (about 10,000 for each mode). Each bunch of the received signals produced a single distance-velocity profile for each of the two modes. Therefore, 100 distance-velocity profiles were calculated in each mode.
Figures 4(a) and (b) show distance-velocity profiles of signal-to-noise ratios (SNRs) of the PCDL and FMCDL, respectively. Positive velocities are toward the prototype (almost downward). The first of 20,000 consecutive received signals used to produce the profiles was acquired at 15:58:32. Note that velocity profiles at every distance were normalized by their peaks and these velocities are indicated by the solid lines. The two dashed lines are velocities at which SNRs are -3 dB from the peaks. Figure 4(a) (PCDL) indicated that distances in the first 75 m were affected by the direct input of transmitted signals. In Fig. 4(b) (FMCDL), the same effect appeared in the first 300-m distances. This effect majorly consists of sidelobes of the direct input, which was anticipated in [14]. Thus, the direct input affected first distances with spatial ranges of transmitting pulses. Beyond the first distances to 2,000 m, where signals from aerosols appeared, both a and b indicated consistent velocities with each other. At a distance just beyond 2,000 m, a very sharp velocity profile was seen in b. Since very high power was received at this distance compared with other distances around (as shown later), they are considered as signals from a cloud bottom. Figure 5 shows exemplified velocity profiles of (non-normalized) SNRs at a distance of 1,974 m, which are considered to be from aerosols. The dashed and solid lines correspond to the PCDL and FMCDL, respectively. It is seen that the FMCDL produced a detailed velocity profile which is, in the PCDL, blurred to lower resolution.
Fig. 4. Distance-velocity profiles of peak-normalized SNRs of (a) PCDL and (b) FMCDL at 15:58:32 (JST). Velocity profiles at every distance were normalized by their peak values. The solid and dashed lines respectively indicate velocities at which 0 and -3 dB of peak-normalized SNRs.
Fig. 5. Velocity profiles of SNRs of the PCDL and FMCDL acquired at 1,974 m and 15:58:32 (JST), which are shown in the dashed and solid lines, respectively.
Signals from cloud bottoms can appear with very narrow profiles, especially in the distance axis since most of the energy of transmitted pulses is scattered and lost there. The cloud bottom at 15:58:32 was identified at a distance-velocity of 2,070 m–0.00 m/s. Figure 6 shows distance profiles of SNRs at the cloud bottom’s velocity (0.00 m/s). -3-dB widths with respect to the cloud bottom were measured as 90 and 108 m in the PCDL and FMCDL, respectively. Both the measured widths are comparable to the theoretically expected distance resolution shown in Table 1 (93 and 101 m). Figure 7 shows velocity profiles at a distance of the cloud bottom (2,070 m). It is seen that the FMCDL produced a narrower velocity profile than the PCDL. -3-dB widths of the PCDL and FMCDL were measured as 0.81 and 0.27 m/s, respectively. The measured widths were also comparable to the theoretically expected velocity resolution (0.72 and 0.16 m/s). Differences between the widths and the velocity resolution in both modes were almost 0.1 m/s, which might be a narrow velocity dispersion of the cloud bottom. It was thus confirmed that the distance resolution of the FMCDL is not determined by its pulse duration but bandwidth of its frequency modulation (strictly, auto-correlation of transmitting waveform). Moreover, the velocity resolution of the FMCDL was confirmed to be determined by pulse duration (namely, spectra of transmitting waveform envelope). These properties are as theoretically indicated by [14].
Fig. 6. Distance profiles of SNRs of the PCDL and FMCDL acquired at 0.00 m/s and 15:58:32 (JST), which are shown in the dashed and solid lines, respectively.
Fig. 7. Velocity profiles of SNRs of PCDL and FMCDL acquired at 2,070 m distance and 15:58:32 (JST), which are shown in the dashed and solid lines, respectively.
Figures 8(a), (b), and (c) show distance profiles of peak powers, peak velocities, and -3-dB widths of velocity profiles at 15:58:32, respectively. In peak powers (Fig. 8(a)), large differences are seen in distances below 300 m, which can be considered as effects of the direct input (and its sidelobes) as also seen in Fig. 4. Similarly, it is seen that, immediately beyond the cloud bottom (2,070 m), the FMCDL’s peak powers more gently drop due to sidelobes of the cloud bottom. In distances of 300–2,000 m, where signals from aerosols appeared, the FMCDL outputs higher SNRs than the PCDL. Their differences were comparable to the difference in transmitting pulse energy between the two modes (2.1 dB ($=17.2-15.1$ dB)). Variation of the FMCDL in this range of distances appeared to be smaller than that of the PCDL, which is also considered to be caused by sidelobes. In peak velocities (Fig. 8(b)), the PCDL and FMCDL were consistent with each other in distances from 300 m to the cloud bottom. In -3-dB widths (Fig. 8(c)), also, the PCDL and FMCDL were similar despite that they are different in velocity resolution. This would be because the velocity dispersions of aerosols were much larger than the difference of the velocity resolution of the two modes. Figure 5 also shows that the FMCDL produced a more detailed shape with its higher velocity resolution than the PCDL but their -3-dB widths were dominated by aerosols’ velocity dispersion.
Fig. 8. Distance profiles of (a) peak power, (b) peak velocity, and (c) -3-dB width at 15:58:32 (JST). The dashed and solid lines correspond to the PCDL and FMCDL, respectively.
In Figs. 9(a), (b), and (c), the two modes were compared in the three parameters. In this comparison, all the distance-velocity profiles acquired in the two temporal ranges were included. In each profile, a distance range of 500–1,500 m was considered, that is, signals only from aerosols were considered, and direct inputs and cloud bottoms were excluded. The number of evaluated values for each parameter was 16,600. Three statistical criteria, bias, standard deviation, and correlation coefficient, were summarized in Table 3. In peak power, it was confirmed that the FMCDL had a positive bias of 2.6 dB compared with the PCDL, which resulted from the difference in energies of transmitted pulses between the two modes (2.1 dB ($=17.2-15.1$ dB)). Peak power’s standard deviation and correlation coefficient indicated that the PCDL and FMCDL agreed with each other. In peak velocity, the two modes agreed very well in all the criteria. The -3-dB velocity widths of the two modes agreed in bias because they were considered to be dominated by the velocity dispersion of aerosols. The standard deviation and correlation coefficient of -3-dB velocity widths showed a low consistency between the two modes, which is possible because detailed (non-smoothed) velocity profiles of the FMCDL resulted in more fluctuated -3-dB widths.
Fig. 9. Histograms comparing the PCDL and the FMCDL in (a) peak power, (b) peak velocity, and (c) -3-dB width of velocity profiles. All the observation time (15:52–16:03 and 16:41–16:52 (JST)) and distances from 500 to 1,500 m were contained.
Table 3. Statistical differences of the FMCDL from the PCDL
5. Discussion
This section discusses three technical issues that were learned through the development and observation of the prototype.
The first one is related to sidelobes. In this study, the effects of sidelobes appeared as blind ranges due to strong signal returns of the direct input and the cloud bottom and as the FMCDL’s smaller variation of the distance profile of aerosols. We found that the transmitted pulses were not very stable probably due to the AOM or EDFA, which produced a mismatch between the received signals and the reference signal. The mismatch can increase sidelobe levels. It is, therefore, desired to create transmitting pulses more stably. Also, it may be effective to use other modulation methods.
The next one is related to the direct input from the transmitter to the receiver side. High direct input levels may cause saturation in the ADC, which forces FMCDLs to be blind in their long first distances. Although the prototype avoided such saturation by using a circulator with a very low cross-talk level, the powers of the direct input signals were much larger than the ones from the aerosols and cloud bottoms. A distance range that is affected by the direct input expands with a larger pulse duration. In order to design an FMCDL with CW transmission (by defining the same temporal length for both pulse duration and pulse repetition time), solutions to extinguish the direct input are essential. The direct input power of normal fiber circulators fluctuates on the order of seconds, which makes it difficult to remove.
The final one is related to missing triggers in the ADC. This issue occurred because of the specifications of the ADC used in the prototype and should be solved by using other ADCs that can rapidly transfer recorded signal data to storage devices.
6. Conclusion
This paper presents the first experimental demonstration of the FMCDL, which was previously only a theoretical concept. [14]. PCDLs, which are traditional, use short-duration single-frequency pulses, where both distance and velocity resolution are specified by pulse duration. Since a shorter pulse duration derives higher (narrower) distance resolution and lower (broader) velocity resolution, the two resolutions of PCDL are in a trade-off relation. Meanwhile, in FMCDL, which uses long-duration frequency-modulated pulses or frequency-modulated continuous waves, distance and velocity resolution depends on the bandwidth of frequency modulation and pulse duration (strictly speaking, auto-correlation function of the transmitting signal and spectrum of transmitting signal envelope), respectively. That is, the FMCDL dissolves the trade-off relation of PCDLs, resulting in CDLs’ flexible design.
In this study, a prototype working as both a PCDL and FMCDL was developed. Specifically, the prototype can alternatively transmit short-duration single-frequency and long-duration frequency-modulated pulses, which are called PCDL and FMCDL modes, respectively. The waveform of the two modes was confirmed by directly connecting the EDFA output to a receiver system of the prototype. Analysis of transmitting signal waveform turned out theoretically expected performances of the PCDL and FMCDL. Theoretically expected distance resolution of the PCDL and FMCDL modes were 93 and 101 m, respectively. Thus, the FMCDL was expected to have a distance resolution that is much higher compared with the spatial length of its pulse (603 m $=(c/2)/$ 4.02 µs). Theoretically expected velocity resolution of the FMCDL was 0.16 m/s which is higher than the PDCL’s one, 0.72 m/s.
The prototype was operated to observe (almost) vertical profiles as the PCDL and FMCDL. This paper presents distance-velocity profiles of PCDL and FMCDL observed at 15:58:32. In regions of aerosols and cloud bottoms, the distance-velocity profiles of the two modes were qualitatively consistent with each other. It was observed that the FMCDL produced a detailed velocity profile for a certain distance, while the PCDL had a blurred profile due to its lower velocity resolution. By using distance and velocity widths with respect to the cloud bottom, accomplished distance and velocity resolution were evaluated as 90 m and 0.81 m/s in the PCDL and 108 m and 0.27 m/s in the FMCDL, which are comparable to the theoretically expected resolutions. It was thus confirmed the two accomplished resolutions of the FMCDL, as well as the PCDL, were as theoretically expected.
By using 100 distance-velocity profiles obtained in each mode, velocity profiles of the two modes were statistically compared. The FMCDL was higher in peak power due to a difference in pulse energies between the two modes. In peak velocity, the two modes were in good agreement in bias, standard deviation, and correlation coefficient. -3-dB widths of the two modes agreed with each other despite that they were different in velocity resolution. This could be because -3-dB widths were not dominated by velocity resolution but by velocity dispersion of aerosols.
Appendix: signal processing
It is supposed that aerosols exist in a channel of laser emitted from a CDL. A received signal of the CDL is expressed as
where $\boldsymbol {\Omega }$ is a set of aerosols on the channel. $y^{(i)} (t)$ is a received signal from the $i$-th aerosol in the set, which is a function of time $t$. $n (t)$ is an additive noise. $y^{(i)} (t)$ is expressed as where $s (t)$ is a transmitting complex waveform. $x^{(i)}$ is a complex amplitude due to the $i$-th aerosols. Strictly, it contains scattering amplitude of the $i$-th aerosols, attenuation on the propagation channel, and a gain of the CDL’s optical antenna. $\tau ^{(i)}$ and $f_d^{(i)}$ are a round-trip time (delay time) of light between the CDL and the $i$-th aerosols and a Doppler frequency shift due to CDL-radial movement of the $i$-th aerosols, respectively.When discretizing delay time and Doppler shift by small portions of them, the received signal is approximately expressed as
whereHere, two approximations as below are applied,
This approximation is effective when the sizes of time-frequency gates are sufficiently small and the numbers of aerosols contained in $\{ \boldsymbol {\Omega }_{l,m} \}$ are sufficiently large.
Equation (3) can be expressed by a matrix formulation as
where $\mathbf {y} = \begin {bmatrix} y (t_1) & y (t_2) & \ldots & y (t_n) & \ldots & y (t_N) \end {bmatrix}^{\rm T}$, and $\mathbf {n} = \left[ n (t_1)\;\;\; n (t_2)\quad \ldots\quad n (t_n)\quad \ldots\quad\right. $ $\left.n (t_N) \right]^{\rm T}$. $\mathbf {S}$ and $\mathbf {x}$ are based on Eq. (4) as below.Meanwhile,
CDLs normally acquire a number of received signals which are incoherent to each other. Here assumes that probabilistic properties of $\mathbf {x}$ do not change while measuring the bunch of $\mathbf {y}$. Now, signal processing is demanded to estimate $\mathbf {x}$ from the bunch of $\mathbf {y}$ via the linear equation of Eq. (8). This study used a signal processing method based on Capon’s proposal [15], by which a power estimate of $x_{l,m}$ is calculated as
Funding
Japan Society for the Promotion of Science (JP17K06484).
Acknowledgment
The authors thank Dr. Hamaki Inokuchi with the Japan Aerospace Exploration Agency for his support in the development of the prototype. The authors also thank Dr. Tomohiro Akiyama and Dr. Nobuki Koatake for fruitful discussions, who were with Japan Aerospace Exploration Agency and are currently with Mitsubishi Electric Corporation. Finally, the authors thank Mr. Biswas Sounak for English correction.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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