1. Introduction

Instantaneous frequency measurement (IFM) of an unknown radio frequency (RF) signal is a significant application in wireless communication, sensing, and 5G/6G technology [1–4]. The most straightforward method is using a high-speed oscilloscope or spectrum analyzer, which is known as direct measurement. On the other hand, by mapping the RF frequency into other parameters, the indirect measurement method is introduced. Compared with direct measurement, the MWP-based indirect measurement system may achieve up to THz measurement range, high accuracy, and real-time response [3–6]. Therefore, the MWP approach has emerged as a competitive solution for designing ultra-broad bandwidth measurement systems when compared to conventional electronic systems. There are three common types of MWP based indirect frequency measurement systems. The frequency-to-frequency mapping is usually specially designed for THz signal [5]. By modulating the THz signal into the optical domain, and beating it with the second carrier, the THz frequency can be down-converted to a GHz signal, which can be easily detected by an RF spectrum analyzer. The frequency-to-time mapping also uses an oscilloscope to complete the measurement, and the bandwidth of the oscilloscope can be lower than the input signal bandwidth [7–9]. Although these two approaches can reduce the required bandwidth of the detector, they still require a direct measurement of frequency in the last step. The last approach is mapping the RF frequency to RF power. Therefore, the measurement process can be completed by an RF power meter. To achieve frequency to amplitude (or power) mapping, the frequency response of some devices and systems can be made use of, such as Bragg gratings [6], micro-ring/disk resonators [10,11], and MWP filters [3,12,13]. Nevertheless, most of the frequency to amplitude mapping system is multiband, due to the periodic characteristic of the system frequency response. There are two limitations for multiband systems. First, one measured power may be mapped into more than one possible frequency. Also, the area between two measurement bands cannot be used for measurement because of the unacceptable error, known as blind spots. One possible solution is using multiple MWP filter responses and making the final decision by comparing the results. This method is called the amplitude comparison function (ACF) [2,3]. However, a single MWP filter should be able to do as good as the ACF method, especially if the MWP filter can be programable and reconfigurable. In details, the ideal characteristics of a frequency to amplitude mapping system should include the following:

As a programable system, the MWP filter is an ideal solution to meet all requirements above. Generally, the MWP filter is a digital finite impulse response (FIR) filter implemented in an optical approach. However, the MWP filter is a completely analog system, which suggests that it will never be limited by the speed of analog to digital conversion. To build a MWP filter, a source that can create multi-wavelength carriers (also referred to as taps) with uniform wavelength spacing is required [14]. Usually, the source can be implemented by a laser array or an optical frequency comb (OFC) source. There are three typical on-chip OFC sources. The first one is the micro-ring resonator (MRR), also known as the Kerr OFC source [15–17]. As a passive device, the comb spacing (or free spectrum range (FSR)) is fixed after fabrication. Also, it requires an external laser as the pump. The second OFC source is based on cascaded Mach-Zehnder modulators (MZMs) [18–20]. One advantage of this OFC source is the tunable FSR. Nevertheless, this system still requires an external laser as the carrier, and the comb spectrum is not flat. The last approach is the mode-locked laser (MLL), such as quantum dash (QDash) and quantum dot (QDot) lasers. Compared with the MRR and the cascaded MZMs based OFC sources, one of the significant advantages of the QDash MLL is its quasi-flat comb spectrum. For example, the power difference between the highest and lowest comb lines of the MRR-based OFC source in [15] is ∼40 dB compared to ∼ 5 dB difference for the QDash laser. In MWP filter design, comb shaping is an essential process. For non-flat OFC sources, the total output power is limited to the comb lines with the lowest power. Therefore, the QDash laser can offer more flexibility and efficiency for comb shaping. Since MWP filters are based on optical intensity modulation, the relative intensity noise (RIN) will also be an important parameter for selecting a particular OFC source.

Recently, we presented the use of MWP filters with linear responses for IFM [13,21]. In this paper, we summarize the principles of the MWP filter and how DSP FIR filter theory is used to obtain MWP filters with linear responses. We also present additional experimental results that characterize the performance of the IFM system. By applying down-sampling to the MWP filters we just obtained, the maximum frequency range can be expanded from 12.5 GHz to 20 GHz. Please note that 20 GHz is the limit associated with our measurement equipment and not the approach. The negative taps are also used to create four MWP filters in total, whose RMSE for IMF is around 51∼66 MHz. Finally, by letting two MWP filters work together, the RMSE is further improved to 42.2 MHz for linear unit and 30.7 MHz for dB unit. The main novelty is the linear relationship between frequency and amplitude, which allows for using polynomial representations to describe the mapping pattern. In other words, during the measurement, we find the input frequency by solving a polynomial equation based on the measured power which is an algebraic solution. For non-linear frequency to amplitude mapping, numerical methods (e.g., look-up tables) are used which may involve interpolation and thus reduce accuracy.

2. Quantum dash mode-locked laser

We use an InAs/InP-based QDash MLL, whose cross-section is shown in Fig. 1(a), as the OFC source. The core area of the QDash laser is a 350 nm InGaAsP containing a five-layer InAs QD with a period of 108 Å surrounded by a p-type InP cladding. More physical details about this laser are available in [22,23]. The threshold current of this QDash laser is ∼50 mA. Also, the QD laser sustains mode-locked operation with a relatively flat comb spectrum when the pump current is greater than 200 mA. In our experiments, we use an operating current of around 280∼300 mA at 20 °C, where the laser can generate 41 comb lines in the C-band with 25 GHz (∼0.2 nm) comb spacing, as plotted in Fig. 1(b). At this operation point, each comb line exhibits an independent thermal power drift of approximately 0.15 dB. The RIN of this laser is -130.5 dB/Hz in average, and the RF linewidth is 9.98 ± 1.58 kHz. Furthermore, a uniform comb spacing is an essential condition for MWP design. By using the beating between two comb lines, the frequency drift of the comb spacing is ±37.4 kHz, which is negligible compared to the 25 GHz comb spacing. Moreover, the impact of power and frequency drift will be removed by taking averages during the measurement.

 

Fig. 1. (a) Schematic cross-sectional diagram of the QD laser [24]. (b) The optical spectrum of QDash laser with an injection current of 280 mA in 20 °C

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3. Principles, results, and discussions

The principle of the MWP filter system has been illustrated in Fig. 2, where the mathematical model is a discrete-time system that processes a continuous-time signal [25]. Based on the schematic diagram at the top of Fig. 2, the multi-wavelength optical source creates discrete taps (or OFC lines) in the optical domain with a uniform wavelength spacing Δλ. After propagating in a dispersive medium, a uniform delay $\mathrm{\Delta }T = D \cdot \mathrm{\Delta }\lambda $ will be introduced on all taps, where D is the dispersion parameter in ps/nm. At the same time, there is a comb shaping unit to change the power of each tap individually (see the comb spectrum example in Fig. 2). As a discrete spectrum, it could be described by a discrete series p[n] where the power of the lowest wavelength comb line is defined as the first comb line, and it is mapped to p[0]. Thus, the frequency response of the whole system can be described as follows [14,26–28]:

 

Fig. 2. Schematic diagram of a MWP filter (top) and its mathematical model (bottom).

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Please note that the sampling frequency is imaginary because there is no sampling unit in MWP filter, i.e., it is a completely analog system. It is clear that the term 2πf/Fs in brackets is the definition of normalized angular frequency ω. Thus, Eq. (2) can be simplified to

Therefore, the frequency response of the MWP filter is the discrete-time Fourier transform (DTFT) of the series p[n]. Since the number of comb lines is finite, p[n] can be treated as the impulse response of a finite impulse response (FIR) filter, where the total number of taps represents the order of the FIR filter. Nevertheless, the input and output signals of the MWP filter are analog. Hence, the mathematical model for a MWP filter is processing continuous-time signals by a discrete-time system, where the imaginary sampling period is the delay introduced by fiber dispersion.

As mentioned above, p[n] is not only the power of all comb lines but also the impulse response of the FIR filter. In DSP, the value of the impulse response can be positive and negative, but the negative power is meaningless in physics. So, although negative taps are applied, all taps in the optical domain are always positive. At the same time, as the designer, we know which taps should be negative. Thus, we may separate the positive and negative taps, and detect them with a balanced photodiode (BPD). Mathematically:

The design methodology and the experimental setup is illustrated in Fig. 3. The objective is to design a MWP filter with linear frequency response. Based on the principles above, a FIR filter with linear frequency response is required. To achieve this requirement, we use MATLAB filter builder to implement our designs [29]. For example, the amplitude response of the expected filter is linear decreasing from 0 to 20 GHz. After importing the expected filter response into the MATLAB filter builder, the toolbox will calculate and optimize the impulse response, i.e., p[n] in Eq. (3). The p[n] represents the power of each tap (or comb line) in experiments. Therefore, a real-time feedback loop (dash line path in Fig. 3) is designed to implement the comb shaping. The comb spectrum is detected by an optical spectrum analyzer (OSA), and the power of each comb line is controlled by two waveshapers (Finisar Waveshaper 1000S and 4000S). The waveshaper is a programable optical filter that can apply attenuation on each tap individually. It is important to note that a single waveshaper (4000s) can be used for both comb shaping and de-multiplexing of the positive and negative taps, and the first one can, in fact, be removed. We also use a BPD to implement negative taps. By converting all taps into an electrical domain, and letting positive taps substrate negative taps in BPD, the negative p[n] has been implemented, where the linearity of DTFT also supports this operation in theory. Note that the primary objective of the feedback loop is to minimize the error between the measured and designed comb shaping, rather than serving as a stabilizer to counteract comb power drift or frequency drifts in the comb spacing. In our experimental setup, the power drift of the QDash laser is exclusively managed by the thermoelectric cooler (TEC). Instead, the feedback loop can also apply pre-distortion to deal with the non-uneven frequency responses, such as the non-flat gain of EDFA.

 

Fig. 3. Experimental setup. TEC: thermoelectric cooler, PC: polarization controller, EDFA: Erbium-doped fiber amplifier, MZM: Mach-Zehnder modulator, PA: (electrical) power amplifier

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We first use all 41 comb lines. In this case, the maximum operation frequency is half of the comb spacing or 12.5 GHz. The dispersive medium here is a 5 km standard single-mode fiber (SSMF, D≈17 ps/km·nm). For example, we want to design a MWP filter whose amplitude response is linear decrease when the frequency changes from 0 to 12.5 GHz. By plotting the negative frequency part, it is clear that the frequency response is a triangle function in the range of -12.5∼12.5 GHz. Therefore, the impulse response is the inverse DTFT of a triangle function, i.e., a sinc square function. After optimizing that sinc square function with MATLAB filter builder toolbox, the coefficient of comb shaping p[n] is obtained. All shaping of comb power in this paper is normalized under linear unit scale. The comb shaping is following implemented by the feedback loop in Fig. 3, and the experiment result is plotted in Fig. 4(a). Next, the frequency response is measured by a vector network analyzer (VNA, Rohde and Schwarz ZNB20), and the results are shown by the orange line in Fig. 4(c), where the back-to-back (B2B) system response (modulator and fiber has been accounted. The green dash-dot line is the simulation results based on the designed filter coefficients (black dots in Fig. 4(a)). The measurement is repeated 100 times and the results are organized as the colormap to evaluate the measurement error. By comparing the data in colormap with its polynomial fitting, a 99.9% of goodness (R²) is achieved, and their RMSE is 48.5 MHz. Please note that this value is the worst case. We simulate the impact of power fluctuations in the comb lines and frequency drifting in the comb spacing. Under ideal conditions, where only a frequency drift of ±37.4 kHz is present, the error between the designed and expected filter response amounts to $1.2 \times {10^{ - 6}}$. However, in the presence of solely a power drift of 0.15 dB, the error escalates to $2.5 \times {10^{ - 3}}$. These results suggests that the predominant source of error stems from the thermal power drift exhibited by the QDash laser. During measurements, the average operation can also reduce the impact of power variations of the QDash laser. Therefore, by using the CW mode of VNA to repeat the measurement from 1 GHz to 12.5 GHz with 0.5 GHz spacing, averaging the measured power, and obtaining the measured RF frequency. The results are shown in Fig. 4(d), and the overall RMSE is 46.3 MHz. Finally, we want the frequency response to keep the linear pattern, but the unit of the y-axis is dB. To achieve this requirement, the frequency response of MWP should be exponential decay under the linear unit. Hence, the impulse response and p[n] should be an inverse proportion function, as illustrated in Fig. 4(b). Although the error of simulation (green dash-dot line) and experiment (black line) in Fig. 4(c) is larger than the linear unit case, they still have >99% coherence coefficient. The RMSE of the dB unit case is 55.6 MHz, as plotted in Fig. 4(e).

 

Fig. 4. Experiment results of (a) comb shaping of a sinc square function, (b) comb shaping of an inverse proportion function. The normalized power in (a) and (b) is in linear units. (c) frequency responses of the MWP filters after subtracting the back-to-back (B2B) response, where the green dash-dot lines are the simulation results, and (d&e) frequency measurement error.

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After implementing those two linear response MWP filters, it is necessary to evaluate the effect of the total number of comb lines, which represents the order of the MWP filter. If the filter response is perfectly linear, increasing the total numbers of comb lines will not affect the accuracy. However, when we use an FIR to implement a linear response, the filter order will affect the error between the designed and expected filter responses. On the one hand, using a higher filter order can enhance the linearity of the MWP filter. This improvement in linearity substantially enhances the accuracy of polynomial fitting, consequently leading to reduced measurement error. Nevertheless, controlling such a large number of comb lines becomes more challenging. To evaluate this trade-off, we keep the comb shaping the same as what we did in Fig. 4(a and b), but we decrease the number of comb lines from 41 to 21 with a step of 2. (Note: the total number of comb lines must be odd.). Figure 5 presents the experimental results, where the RMSE is calculated from 100 times measurements in statistic (i.e., the colormap in Fig. 4(c)). For the dB unit, the system achieves the best RMSE 50.7 MHz with 35 comb lines. However, the case of the linear unit does not follow this rule because of its sinc square shaping. When changing the total numbers of comb lines, we directly remove the comb lines on the left and right edges (shortest and longest wavelengths) instead of redesigning the comb shaping. This explains why the pattern increases then decreases, just like the sidelobes of the sinc square function. To verify, we simulate this process using MATLAB based on Eq. (1). As shown in Fig. 5(b). Even in the ideal case (all comb lines are perfectly shaped), the RMSE as a function of the number of comb lines (filter order) exhibits variations as it decreases.

 

Fig. 5. (a) RMSE vs. total number of comb lines, where the RMSE is calculated based on 100 times experimental results, and (b) simulation of error between the designed (des.) and expected (exp) filter response vs filter order.

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The maximum RF frequency of results in Fig. 4 is limited at half of the comb spacing 25/2 = 12.5 GHz. There are two possible solutions to overcome this limitation. The first solution is using optical single side-band modulation. The second solution is that we select one comb line from every two comb lines. In that case, a new set of 21 comb lines with 50 GHz is obtained. In digital signal processing, this operation is the representation of down-sampling for the impulse response p[n]. At the same time, the length of SSMF is 3 km here to obtain a similar delay between two neighboring taps. The experimental results are shown in Fig. 6, where the simulation results (blue dash-dot line in (a-d)) are obtained based on the designed comb shaping patterns (black dot in (e-h)). For example, by shaping all 21 OFC lines into a sinc square pattern (Fig. 6(e)), a MWP filter with linear frequency response under linear unit has been achieved in the range of 2-20 GHz, see Fig. 6(a). However, we also notice that the frequency responses of the MWF filters are not perfectly linear in the higher frequency range (such as the region close to 20 GHz in Fig. 6(a)), and the frequency responses may become noisy in the higher frequency range, as shown in Fig. 6(c), in the range close to 20 GHz. The reason is that only positive taps are used, and the slope of linear frequency response is negative. Therefore, higher frequency components will be mapped into lower power, which is more seriously affected by the noise. To solve this problem, MWP filters with negative taps can be used. It is an important note that the power of all comb lines should be positive and negative power is physically meaningless. The negative power in Fig. 6(f and h) is an equivalent value after splitting positive and negative taps and detecting them with a BPD. By applying negative taps, two MWP filters with linear frequency response and positive slope have been implemented, as shown in Fig. 6(b and d).

 

Fig. 6. Experiment results of linear frequency responses of the MWP filters with (a) negative slope under linear unit, (b) positive slope under linear unit, (c) negative slope under dB unit, (d) positive slope under dB unit. And (e-h) the designed and measured comb shaping patterns where (e) for (a), (f) for (b), (g) for (c), (h) for (d). The normalized power is in linear units.

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Figure 7 illustrates the pattern of the measurement error while doing IFM with all MWP filters in Fig. 6. All results here are measured by the continuous wave (CW) mode of the VNA with average, where 2 GHz,10 GHz, and 20 GHz are reserved for reference. In Fig. 7(a), the case of linear unit, the overall root-mean-square error (RMSE) of 51.1 MHz for the negative slope MWP filter, and the RMSE is 58.1 MHz for the positive slope MWP filter. For the dB unit case, the overall RMSE is 66.2 MHz for the negative slope MWP filter and 63.1 MHz for the positive slope case. Nevertheless, the dB unit systems show worse performances (RMSE) than the linear unit case because it suffers more effect of noise in the low power range. To overcome this limitation, the method of the AFC in [3] can be applied in our system. As a reconfigurable system, our MWP filter based IFM system can reprogram the comb shaping without calibration under the same unit scale. This characteristic allows the MWP filter in Fig. 6(a and b), or (c and h) to work together. The results of the ACF method are shown by the blue triangle symbol in Fig. 7(a and b). The overall RMSEs of the ACF method are 42.2 MHz for the linear unit case and 30.7 MHz for the dB unit case. There is a great improvement in the RMSE of the dB unit case after applying ACF since the ACF avoids the noisy region of the dB unit case.

 

Fig. 7. Measurement errors of IFM using MWP filters in Fig. 6 for the case of (a) linear unit, and (b) dB unit. (c) measurement errors with different input RF power.

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The last group of experimental results is the effect of the input RF power. Please note that all results are obtained based on the statistic of 100 times measurement results in Fig. 6(a, b, c, and h) without average (instead of the CW mode). In other words, the RMSE here is the worst case. This time, the comb shaping is fixed as Fig. 6(d-g), and the output RF power (i.e., the RF power goes into the MZM) is swept from -5 dBm to -20 dBm. The results are plotted in Fig. 7(c). It is not a surprise that the system performance gets worse when the RF power is reduced. Also, we notice that the MWP filters with negative taps are more sensitive to power change. The main reasons are the difficulty of controlling negative taps as well as the limit of the responsivity of the BPD.

Table 1 lists some recent work on MWP-based IFM. First, the photonic devices based IFM systems offer an integrable on-chip solution, but it is difficult to achieve a board and single band because of the periodic frequency response of the photonic devices. Also, the frequency-to-frequency approach suggests the maximum achievable frequency range. Nevertheless, the MWP filter approach is still the best solution because of its high accuracy and reconfigurability. The measurement errors of Ref. [3] and ours are very close. However, they need 4 AFCs and we only need one MWP filter (Note: one ACF requires at least 2 MWP filters). Also, our work is the only one that has linear frequency responses. Finally, because of linear frequency response, our system is described by the polynomial coefficients while Ref. [3] uses a lookup table (numerical analysis). This suggests that our system uses the algebraic method to identify the measured frequency instead of the numerical method.

Table 1. Comparison between some previous IFM achievements

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4. Conclusion

In this paper, we report a reconfigurable IFM system using MWP filters based on a QDash MLL. All MWP filters have linear frequency responses under either linear unit or dB unit. Thanks to the flat comb spectrum and enough numbers of OFC lines, the QDash laser offers a set of ideal OFC lines for comb shaping. After combining the DSP FIR filter principles and the MWP filter system, a group of MWP filters with linear frequency response are designed, simulated, and measured. By applying down-sampling, the maximum frequency range of the MWP based IFM system can achieve 20 GHz, which is the maximum achievable frequency of the VNA. Moreover, as a reconfigurable system, two MWP filters can work together as an AFC to continuously improve the RMSE of measurement. The best results we achieve are 42.2 MHz for linear unit and 30.7 MHz for dB unit.