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Robust wide-range chirp rate measurement based on a flexible photonic fractional Fourier transformer

Di Peng, Huaqing Li, Yuwen Qin, Jianping Li, and Songnian Fu

Open Access Open Access

Abstract

We propose a photonic-assisted approach to measure the chirp rate of a linear frequency modulation waveform (LFMW) with a long duration, based on tunable photonic fractional Fourier transform (FrFT). Since the FrFT order can be continuously tuned by varying the frequency shift in an optical frequency-shifting loop (FSL), a specific FrFT order leads the fundamental frequency component arising in the output electrical spectrum to reach its maximum value, after the photonic-to-electrical conversion. Based on the designated FrFT order and the corresponding fundamental frequency in the output electrical spectrum, the chirp rate measurement over a wide range can be accessed, even the signal-to-noise ratio (SNR) of the input LFMW is substantially low. Simulation results indicate that the chirp rate of a 0.16-ms LFMW over a frequency range from 20 GHz to 26 GHz can be precisely characterized, with a relative measurement error of less than 0.13%, under the SNR condition of 0 dB. Moreover, an unambiguous chirp-rate measurement within the range of -5200 MHz/µs to 550 MHz/µs can be achieved. Hence, the proposed chirp rate measurement is featured with broadband operation, robust noise tolerance, low-frequency detection, and long-duration LFMW characterization.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Microwave signal recognition is of critical importance in electronic reconnaissance, including the frequency measurement, the chirp rate identification, and the pulse width characterization. As for modern radar systems, broadband linear frequency modulation waveforms (LFMWs) with long durations are widely used to enhance both the distance detection range and the spatial resolution [1–3]. Therefore, the chirp rate identification of broadband long-duration LFMWs is necessary for electronic warfare. In general, the chirp rate identification can be realized by the time-frequency analysis in the digital domain, which is faced with the following two problems. The first one is that both the sampling rate and the analog bandwidth of commercially available analog-to-digital converters (ADCs) are generally below multi-GS/s and several GHz, respectively, due to the well-known electronic bottleneck [4,5]. Hence, for the broadband LFMW with a long duration, frequency down-conversion or channelization architecture is inevitable before digitization, leading to the enhancement of the system complexity and the reduction of the receiver flexibility [6]. The second problem comes from the large data-storage and the computation complexity for a broadband long-duration LFWM arising in real-time applications.

To conquer those problems, various photonic-assisted chirp rate measurement solutions for the long-duration LFMW have been proposed [7–10]. In general, the key point of all existing photonic-assisted solutions is to transform the LFMW into a Fourier-transform-limited pulse, with the help of a tunable matched filter. Consequently, the chirp rate of the input LFMW can be calculated from the filter parameters. However, the measurement accuracy is constrained by the rough resolution of the employed optical filters or microwave photonic filters [11–14]. Alternatively, photonic fractional Fourier transform (FrFT) is regarded as a potential candidate to identify the LFMW with high accuracy [15]. Photonic FrFT decomposes the input signal into a superposition of linearly chirped signals in the analog domain. Therefore, the chirp rate of the input LFMW can be identified, when the FrFT order is properly set to compress the LFMW into a Fourier-transform-limited pulse. However, the earliest photonic FrFT is realized in the form of bulky free-space optical devices based on Fresnel diffraction, where the FrFT order is tuned by varying the optical propagation length [16–18]. Based on the space-time duality, photonic FrFT can be implemented in the time domain by the optical propagation over the dispersive medium (e.g., either a chirped fiber Bragg grating or a spool of optical fiber) [19]. However, the relatively fixed FrFT order is not qualified for identifying the chirp rate of an unknown LFMW in the electronic warfare application. The FrFT order can be flexibly adjusted by a tunable time lens based on the quadratic phase modulation. Nevertheless, the temporal aperture of the time lens is generally below 1 ns, which is too small for a single radar signal in practice [20]. Recently, an agile photonic fractional Fourier transformer based on an optical frequency-shifting loop (FSL) has been demonstrated to realize pulse compression of a LFMW with a duration beyond several microseconds and even in the order of millisecond [21–24]. As for such scheme, a series of frequency-shifted and time-delayed signal copies from the FSL output is temporally overlapped. Consequently, an extremely large equivalent group-velocity dispersion (GVD) can be obtained to realize the chirp compensation of the input LFMW, with a high resolution of several MHz/ms [21,24]. Moreover, the FrFT order can be finely tuned by changing the ratio between the frequency shift and the free spectral range (FSR) of the loop. Hence, such scheme has the potential to realize high-accuracy chirp rate measurement of broadband long-duration LFMWs. However, since the proper FrFT order is determined by the fact that the input LFMW has been compressed to a Fourier-transform-limited pulse in the time domain, it is challenging to identify the real FrFT order without ambiguity, especially when the received LFMW signal has a low signal-to-noise ratio (SNR) value. In addition, a high-speed photodetector (PD) and a broadband oscilloscope are indispensable for avoiding the pulse broadening and maintaining the measurement resolution. Therefore, the high-accuracy measurement cannot be fully exploited by the judgement of temporal pulse compression. Recently, another approach to estimate the chirp rates of LFWMs and piecewise LFWMs has been demonstrated by the photonic self-fractional Fourier transform [25]. The scheme uses a pass-through double beam interference structure to achieve the de-chirping of the input signals, where the chirp rates are retrieved from the frequencies of the generated single-tone signals. In comparison with the FSL-based method, the most prominent advantage of that scheme is its small latency time to achieve the real-time chirp rate estimation, as it does not need to sweep all the FrFT orders. Nevertheless, it may be invalid for the scenario of worse SNR, due to its implementation of double beam interference, while the FSL scheme with multi-beam interference characteristic has a good tolerance of SNR variation.

In this paper, a high-precision chirp rate measurement method based on the tunable photonic FrFT is proposed, with the help of an FSL. The real FrFT order for an input LFMW under test can be acquired by finely tuning the frequency shift of the FSL, when the power of fundamental frequency component arising in the output electrical spectrum reaches its maximum. Based on the measured FrFT order and the fundamental frequency, the chirp rate of the input LFMW even with a substantially low SNR can be precisely obtained over a broadband range. By evaluating the power of the fundamental frequency component arising in the electrical spectrum, rather than the temporal pulse compression, our proposed method is featured with the low-frequency detection, robust noise tolerance, wide range measurement and high accuracy, which can be used to achieve chirp rate identification of broadband long-duration LFMWs.

2. Operation principle

Figure 1 shows the schematic diagram of the proposed chirp rate measurement method, based on a flexible photonic fractional Fourier transformer. A narrow-linewidth laser diode (LD) with a center frequency of f0 is used to generate the continuous-wave (CW) carrier with high temporal coherence. The LFMW under test is modulated onto the CW light through the carrier-suppressed single-sideband (CS-SSB) modulation, with the help of a dual-parallel Mach-Zehnder modulator (DPMZM) and a 90° electrical hybrid coupler. Then, the generated optical linear frequency sweep signal enters an FSL via a 2×2 optical coupler (OC), where the FSL includes an acousto-optic frequency shifter (AOFS), an erbium-doped optical fiber amplifier (EDFA), and an optical bandpass filter (OBPF). Within the FSL, the optical signal experiences a frequency shift of fs and a time delay of τc per round trip. Thereinto, the EDFA is necessary to compensate the loss of fiber loop propagation. The OBPF with a 3-dB spectral width of $\Delta \upsilon$ is utilized to manage the effective round-trip numbers N and eliminate the out-of-band amplified spontaneous emission (ASE) noise. At the output port of the FSL, the frequency-shifted and time-delayed optical signal copies overlap with each other, and are coherently superimposed, leading to the generation of a series of optical combs with a frequency interval of fs. For any input single-tone signal with a frequency of fin, the relative phase shift of each comb tone is $2\pi n{f_{in}}{\tau _c} + \pi n({n + 1} ){f_s}{\tau _c}$, where $n = 0,\textrm{ }1,\textrm{ }2, \cdots ,\textrm{ }N$ is the round-trip number. Based on the fact that the phase shift corresponding to each frequency-shifted comb increases with the square of the round-trip number n, an equivalent GVD is realized. Therefore, the optical output from the FSL is the FrFT of the input signal, and the FrFT order can be finely tuned through varying the frequency shift fs over a broadband range. When the FrFT order is properly set, the chirp of the optical linear frequency sweep signal is perfectly compensated. In such condition, the FSL output is a series of Fourier-transform-limited optical pulses with a repetition rate of xfc, where fc is the FSR of the FSL, and x is a positive integer no more than Nfs/fc. Simultaneously, the power of the fundamental frequency component at xfc arising in the electrical spectrum reaches its maximum value, after the photonic-to-electrical conversion. Hence, the chirp rate of the input LFMW under test can be obtained, according to the matched frequency shift fs and the corresponding fundamental frequency xfc. The distinct advantage of our proposed measurement method is to determine the FrFT order with the low-frequency detection and analysis, because xfc is normally below several hundred MHz. Actually, the output optical signal from the FSL can be directly sent to a low-speed PD for measuring the fundamental frequency component via the spectrum analysis. Nevertheless, in order to compare with the temporal waveform characterization method [21], the output optical signal from the FSL is divided into two branches, where one part is used for the low-frequency detection, and the other part is sent to a high-speed PD for measuring the temporal waveform via an ultra-high-speed real-time oscilloscope.

 figure: Fig. 1.

Fig. 1. Schematic diagram of the proposed chirp rate measurement based on a flexible photonic fractional Fourier transformer. LD: laser diode, HC: hybrid coupler, DPMZM: dual-parallel Mach-Zehnder modulator, OC: optical coupler, OBPF: optical bandpass filter, EDFA: erbium-doped optical fiber amplifier, AOFS: acousto-optic frequency shifter, EA: electrical amplifier, DDS: direct digital synthesizer, PD: photodetector, EC: electrical coupler, ESA: electrical spectrum analyzer, OSC: oscilloscope. Solid line: optical path; dashed line: electrical path.

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In theory, the output optical field of the FSL with a frequency shift ${f_s}$ and a time delay ${\tau _c}$ per round trip is [21]

$${E_{out}}(t )= {A_\alpha }\sum\limits_{n = 0}^N {{f_{\alpha \textrm{ - }{{\tilde{E}}_{in}}}}(z ){e^{\frac{j}{2}\cot \alpha \left( {{z^2} - \frac{{2t}}{{\tau \cos \alpha }}z + \frac{{{t^2}}}{{{\tau^2}}}} \right)}}}$$
where ${A_\alpha }$ is the complex amplitude, and $\tau$ is a dimensionless variable for the time scale variation. ${f_{\alpha \textrm{ - }{{\tilde{E}}_{in}}}}(z )$ is the FrFT of the input optical field ${E_{in}}(t )$ with a coefficient of $\alpha$, and the FrFT coefficient $\alpha$ ($0 \le \alpha \le {\pi / 2}$) satisfies $\cot \alpha \textrm{ = } - {{2\pi {\tau ^2}\Delta {f_s}} / {{\tau _c}}}$. $\varDelta {f_s}$ is the frequency-shift deviation defined as $\varDelta {f_s}\textrm{ = }{f_s} - k{f_c}$, where k is a positive integer, and ${f_c}$ is the FSR of the FSL. In addition, z can be calculated as
$$z = \frac{\tau }{{{\tau _c}\csc \alpha }}({{\omega_s}t^{\prime} - \Delta {\omega_s}t + 2\pi n} )\textrm{ = }\frac{\tau }{{{\tau _c}}}\sin \alpha \left( {2\pi k\frac{t}{{{\tau_c}}} - \frac{{\psi^{\prime}}}{2} + 2\pi n} \right)$$
where $\psi ^{\prime} = 2\pi \Delta {f_s}{\tau _c}$ is the phase shift induced by the frequency-shift deviation. As shown in Eq. (1) and Eq. (2), the output waveform from the FSL is composed of multiple delayed FrFT samples of the input signal. The FrFT coefficient $\alpha$ can be tuned by the variation of ${f_s}$. The time interval between two neighboring samples is equal to ${{{\tau _c}} / k}$ only under the condition of $\sin \alpha = 1$, and varies with the frequency shift. The LFMW under test ${V_0}\cos ({{\omega_{\textrm{RF}}}t\textrm{ + }\pi a{t^2}} )$ is loaded onto the CW carrier ${E_0}(t )$ via the CS-SSB modulation, where ${V_0}$, ${\omega _{\textrm{RF}}}$ and a are the voltage amplitude, the initial angular frequency, and the chirp rate of the input LFMW, respectively. The CS-SSB modulation is realized by using a DPMZM and a 90° hybrid coupler, where the two sub-MZMs and the main-MZM in the DPMZM are biased at the minimum transmission point and the quadrature point, respectively. Under such condition, the input optical field of the FSL is
$$\begin{array}{c} {E_{in}}(t )= \frac{{{E_0}(t )}}{2}\left[ { - \textrm{sin}\left( {\frac{m}{2}\cos ({{\omega_{RF}}t + \pi a{t^2}} )} \right){e^{j\frac{3}{4}\pi }} - \textrm{sin}\left( {\frac{m}{2}\sin ({{\omega_{RF}}t + \pi a{t^2}} )} \right){e^{ - j\frac{3}{4}\pi }}} \right]\\ \textrm{ = }{E_0}(t )\left[ {{J_1}\left( {\frac{m}{2}} \right){e^{j({{\omega_{RF}}t + \pi a{t^2}} )}} - {J_3}\left( {\frac{m}{2}} \right){e^{ - j({3{\omega_{RF}}t + 3\pi a{t^2}} )}} +{\cdot}{\cdot} \cdot } \right]{e^{ - j\frac{\pi }{4}}} \end{array}$$
where $m = \pi {{{V_0}} / {{V_\pi }}}$ is the modulation index, and ${V_\pi }$ is the half-wave voltage of the two sub-MZMs within the DPMZM. ${J_n}\left( {\frac{m}{2}} \right)$ is the nth-order Bessel function of the first kind. By neglecting the high-order terms of Eq. (3), the input optical filed of the FSL can be simplified as
$${E_{in}}(t )\textrm{ = }{J_1}\left( {\frac{m}{2}} \right)E_{in}^0(t ){e^{j\pi a{t^2}}}$$
where $E_{in}^0(t )= {E_0}(t ){e^{ - {{j\pi } / 4}}}{e^{j{\omega _{RF}}t}}$. Hence, the output optical field of the FSL is
$${E_{out}}(t )= {A_\alpha }{J_1}\left( {\frac{m}{2}} \right){e^{j\pi a{t^2}}}\sum\limits_{n = 0}^N {{f_{\alpha \textrm{ - }\tilde{E}{{_{in}^0}_{}}}}(\upsilon )} {e^{j\frac{{\cot \alpha }}{2}\left( {{\upsilon^2} - \frac{{2t}}{{\tau \cos \alpha }}\upsilon + \frac{{{t^2}}}{{{\tau^2}}}} \right)}}$$
where $\upsilon = {\tau / {{\tau _c}}}\sin \alpha [{2\pi ({{k / {{\tau_c}}} + a{\tau_c}} )t - {{\psi^{\prime}} / 2} + 2\pi n} ]$ is satisfied, and the FrFT coefficient $\alpha$ is calculated as
$$\cot \alpha \textrm{ = } - 2\pi \left( {\frac{{\Delta {f_s}}}{{{\tau_c}}}\textrm{ + }a} \right){\tau ^2}$$
As shown in Eq. (5), the output optical signal from the wo FSL is a series of delayed pulses, and each pulse is the FrFT of the input optical field $E_{in}^0(t )$ multiplied by a time-dependent quadratic phase term. The time interval between two neighboring pulses also varies with $\alpha$, i.e., the frequency shift. Those optical pulses are coherently superimposed to generate the optical output from the FSL. After photodetection, the output current can be calculated as
$$I \propto {E_{out}}(t )E_{out}^\ast (t )= A_\alpha ^2J_1^2(m )\left[ {\sum\limits_{n = 0}^N {{f_{\alpha \textrm{ - }\tilde{E}{{_{in}^0}_{}}}}(\upsilon )} {e^{j\frac{{\cot \alpha }}{2}\left( {{\upsilon^2} - \frac{{2t}}{{\tau \cos \alpha }}\upsilon + \frac{{{t^2}}}{{{\tau^2}}}} \right)}}} \right]\left[ {\sum\limits_{n = 0}^N {{f_{\alpha \textrm{ - }\tilde{E}{{_{in}^0}_{}}}}(\upsilon )} {e^{ - j\frac{{\cot \alpha }}{2}\left( {{\upsilon^2} - \frac{{2t}}{{\tau \cos \alpha }}\upsilon + \frac{{{t^2}}}{{{\tau^2}}}} \right)}}} \right]$$
The average output current after photodetection will reach its maximum values only when the time-dependent quadratic phase term disappears, i.e., $\cot \alpha \textrm{ = }0$. As shown in Eq. (6), through the finely adjusting ${f_s}$ within the range of ${k_0}{f_\textrm{c}}$ to $({{k_0}\textrm{ + }1} ){f_\textrm{c}}$ (${k_0}$ is a positive integer), the FrFT coefficient can be set to $\alpha \textrm{ = }{\pi / 2}$, i.e., $\cot \alpha \textrm{ = }0$. In such condition, a series of Fourier-transform-limited pulses with maximum power and a repetition rate of $x{f_\textrm{c}}$ is observed, and the corresponding fundamental frequency component at $x{f_\textrm{c}}$ in the electrical spectrum reaches its maximum power. Thus, the chirp rate of the input LFMW satisfies
$$a\textrm{ = } - \frac{{\Delta {f_s}}}{{{\tau _c}}} ={-} \Delta {f_s}{f_c}$$
Therefore, according to the matched frequency shift ${f_s}$ and the measured fundamental frequency $x{f_\textrm{c}}$, the chirp rate of the input LFMW can be calculated as
$$a\textrm{ = }({{f_s} - x{f_c}} ){f_c}$$
where $x = 1,\textrm{ }2,\textrm{ }3, \cdots ,\textrm{ }{k_0}$ for the case that the input LFWM is with a positive chirp (i.e., $a > 0$), and $x = {k_0} + 1,\textrm{ }{k_0} + 2, \cdots ,\textrm{ }{{{\Delta \upsilon } / f}_\textrm{c}}$ for the case that the input LFWM is with a negative chirp (i.e., $a < 0$).

The repetition rate of the output pulse train from the PD, i.e., $x{f_\textrm{c}}$, is relevant to the chirp rate under test, whose maximum and minimum values are equal to the optical comb spectrum width $\Delta \upsilon$ and the FSR ${f_\textrm{c}}$, respectively. Based on the proposed recovery algorithm in Eq. (8), the unambiguous measurement range of the chirp rate is $[{({{f_s} - \Delta \upsilon } ){f_c},({{f_s} - {f_c}} ){f_c}} ]$. For a LFMW with a positive chirp rate, i.e., $a > 0$, the matched frequency shift is

$${f_s}\textrm{ = }\left\{ \begin{array}{ll} {k_0}{f_c} + \varDelta {f_s}^\prime({0 \le \varDelta {f_s}^\prime \le {f_c}} ), &0 \le a \le {f_c}^2\\ x{f_c} + \varDelta {f_s}^\prime ({\varDelta {f_s}^\prime > {f_c},x < {k_0}} ), &{f_c}^2 \le a \le ({{f_s} - {f_c}} ){f_c} \end{array} \right. $$
where the item on the left side of the plus sign is the fundamental frequency component measured in the output electrical spectrum, and the item on the right side of the plus sign is the frequency deviation. Thus, the frequency deviation is always positive. For a chirp rate within the range of 0 to ${f_c}^2$, the frequency deviation is smaller than ${f_c}$, and the fundamental frequency of the output pulse train from the PD is equal to the lowest value of the sweeping frequency shift ${k_0}{f_\textrm{c}}$. As the chirp rate value increases, the measured fundamental frequency decreases, and the corresponding frequency deviation increases. The fundamental frequency ultimately declines to ${f_\textrm{c}}$. Hence, the maximum chirp rate value to be measured is ${a_{\max }}\textrm{ = }({{f_s} - {f_c}} ){f_c}$, based on the recovery algorithm in Eq. (8). Similarly, for a LFMW with a negative chirp rate, i.e., $a < 0$, the matched frequency shift is written as
$${f_s}\textrm{ = }\left\{ \begin{array}{ll} ({{k_0}\textrm{ + }1} ){f_c} + \varDelta {f_s}^\prime ({|{\varDelta {f_s}^\prime } |\le {f_c},\varDelta {f_s}^\prime < 0} ), &- {f_c}^2 \le a < 0\\ x{f_c} + \varDelta {f_s}^\prime ({|{\varDelta {f_s}^\prime } |> {f_c},\varDelta {f_s}^\prime < 0,({{k_0}\textrm{ + }1} ){f_c} < x{f_c} < \Delta \upsilon } ), &({{f_s} - \Delta \upsilon } ){f_c} \le a \le - {f_c}^2 \end{array} \right. $$
where the frequency deviation is always negative in this situation. For a chirp rate within the range of $- {f_c}^2$ to 0, the modulus of the frequency deviation $\varDelta {f_s}^\prime$ is smaller than ${f_c}$, and the fundamental frequency of the output pulse train from the PD is equal to $({{k_0} + 1} ){f_\textrm{c}}$. As the chirp rate value decreases, the measured fundamental frequency and the corresponding modulus of the frequency deviation increase simultaneously. Since the maximum fundamental frequency of the output pulse train is limited to $\Delta \upsilon$, the minimum chirp rate value to be measured is ${a_{\max }}\textrm{ = }({{f_s} - \Delta \upsilon } ){f_c}$.

In particular, high-precision wide range chirp rate measurement can be realized for a long-duration LFMW, even when the input signal has a relatively low SNR. This can be attributed to the following two facts. Firstly, the agile photonic FrFT generates a proper rotation of the time-frequency distribution, for the ease of extracting the signal out of noise, although they are overlapped in both the frequency and the time domain. In addition, the fundamental frequency detection at the low-frequency band improves the measurement accuracy, and releases the bandwidth constraints of the optoelectronic devices and the low noise floor requirement of measurement equipment. Hence, the chirp rate of a LFMW even with a relatively low SNR can be accessed. Secondly, since the photonic FrFT is realized by coherent superposition of multiple frequency-shifted and time-delayed input LFWM samples, the chirp rate is obtained from the matched frequency shift and the corresponding fundamental frequency power measurement by the electrical spectrum analyzer (ESA). Therefore, the chirp rate of a long-duration LFWM can be precisely identified as long as ${f_s}$ can be finely tuned and ${f_c}$ can be accurately determined.

3. Results and discussions

We carry out numerical simulation to verify the feasibility of the proposed measurement method, based on parameters of commercially available devices and instruments. The linewidth of the optical carrier is 100 Hz (NKT Photonics BASIK E15), which corresponds to a coherence time of 10 ms. The input LFMW is modulated onto the optical carrier by a DPMZM (EOspace IQ-0DVS-35-PFU-PFU-LB) with a 3-dB bandwidth of more than 30 GHz and a 90° electrical hybrid coupler (Marki QH-0440) with an operation frequency from 4 GHz to 40 GHz. The round-trip time delay of the FSL is 200 ns, which corresponds to a FSR of 5 MHz. An AOFS (AA Opto-Electronic MT110-IIR20-Fio-PM) with an operation frequency from 110 MHz to 115 MHz, corresponding to 22-23 times of the FSR, is utilized to secure a flexible FrFT order. Thereinto, the driven signal of the AOFS is obtained through the power amplification of the single-tone electrical signal from a direct digital synthesizer (DDS), having a frequency tuning resolution of below 1 Hz. The OBPF (Finisar WaveShaper) has a Gaussian-shape frequency response, whose 3-dB bandwidth is properly set to determine the effective round-trip amounts N. The gain spectrum of the EDFA has a Gaussian shape and a 3-dB bandwidth of 1 THz. The average propagation loss of the FSL is 0.097 dB/m. Please note that, for the ease of performance comparison with the temporal waveform characterization method, a PD (Thorlabs RX40AF) with a 3-dB bandwidth of 45 GHz is employed to detect the optical pulse trains from the FSL. As for our proposed method, a PD with a 3-dB bandwidth of 2 GHz is sufficient for the fundamental frequency detection. The PD output is measured by an ESA with a bandwidth of 2 GHz. Meanwhile, the output waveform from the high-speed PD is recorded by employing a real-time oscilloscope (LeCroy WaveMaster 816Zi-B) with a sampling rate of 80 GS/s, an analog bandwidth of 16 GHz, and a quantization level of 8 bits. In order to obtain a stable output signal from the FSL, the coherence of the multiple frequency-shifted and time-delayed optical replicas must be carefully considered. Firstly, the coherence time of the optical source must be larger than the time delay between the first replica and the last one. Secondly, all devices within the FSL are recommended to be polarization-maintaining and connected by polarization-maintaining fibers.

Firstly, a noise-free LFWM over the frequency range from 20 GHz to 26 GHz and with a duration of 0.16 ms is modulated onto the CW optical carrier, which corresponds to a chirp rate of 37.5 MHz/µs. The modulation index is set to be 0.8. The effective overlap number of the FSL is 10 by properly setting the OBPF bandwidth. Meanwhile, a Gaussian white noise with a power spectral density of -78 dBm/Hz is loaded during each round-trip transmission. Figure 2(a) and (b) show the output pulse train from the PD and its corresponding electrical spectrum, respectively, when the frequency shift is tuned to be a matched value of 112.5 MHz. For the ease of comparison, Fig. 2 (c)-(f) exhibit the output pulse trains and the corresponding electrical spectra when the frequency shift slightly deviates from the matched value. Thereinto, Fig. 2(c) and (d) present the results with a frequency shift of 112.4 MHz, and Fig. 2(e) and (f) show the results with a frequency shift of 112.6 MHz. As shown in Fig. 2(a), (c) and (e), it is difficult to determine the matched frequency shift value based on the temporal waveforms under those three cases. However, there is an obvious difference in the power of the fundamental frequency component, as shown in Fig. 2(b), (d) and (f). Figure 3 shows the measured fundamental frequency and its power under different frequency shifts. The fundamental frequency is 105 MHz under the matched frequency shift of 112.5 MHz. As a result, the chirp rate of the input LFMW is calculated as $({112.5\textrm{MHz} - 105\textrm{MHz}} )\times 5\textrm{MHz = 37}\mathrm{.5MHz/\mu s}$, which agrees well with the setting value. In practice, there is an inevitable power fluctuation of the fundamental frequency, due to the measurement uncertainty of the ESA (<0.3dB), the noise, and the interference instability. Hence, the precise characterization of actual matched frequency shift is constrained by the power fluctuation. Furthermore, the deviation from the matched frequency shift leads to a measurement error of the fundamental frequency. Therefore, the power fluctuation of the fundamental frequency component limits the chirp rate measurement accuracy. If we assume that the power fluctuation is less than 0.5 dB, the corresponding maximum deviation from the matched frequency shift is ±0.051 MHz. Figure 4 exhibits the measured fundamental frequencies and the corresponding errors of the recovered chirp rates under different frequency shift deviations, when the input LFMW is with a chirp rate of 37.5 MHz/µs. The maximum error of the recovered chirp rate is ±0.05 MHz/µs, corresponding to a relative measurement error less than 0.13%. Therefore, the proposed method based on judging the power of the fundamental frequency component arising in the electrical spectrum is capable for the high-precision chirp rate measurement. In addition, please note that, high-precision chirp rate measurement can also be achieved for an LFWM with a longer duration (e.g., >1 ms), since the output signal can be stably constructed via the coherent superimposition of ten frequency-shifted and time-delayed copies with a total time delay of 2 µs (i.e., 200 ns×10). In fact, a frequency shift around 5 MHz is enough to realize the pulse compression for a FSL with a round-trip time delay of 200 ns. In such condition, the fundamental frequency is located at around 5 MHz, which can further reduce the requirement of low-frequency detection. Nevertheless, commercially available AOFS generally works within the frequency range of 40 MHz to 200 MHz. In order to achieve such a small frequency shift, two AOFSs with reversed frequency shifts and slightly different frequencies should be used [21]. The key point is that two independent phase-locked microwave sources are used to drive two AOFSs.

 figure: Fig. 2.

Fig. 2. Calculated results for an input LFMW over the frequency range of 20 GHz to 26 GHz with a duration of 0.16 ms. (a) Output pulse train from the PD, and (b) the corresponding electrical spectrum when the frequency shift is tuned to be a matched value of 112.5 MHz; (c) Output pulse train from the PD, and (d) the corresponding electrical spectrum when the frequency shift is 112.4 MHz; (e) Output pulse train from the PD, and (f) the corresponding electrical spectrum when the frequency shift is 112.6 MHz.

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 figure: Fig. 3.

Fig. 3. Fundamental frequency and its power with respect to the frequency shift

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 figure: Fig. 4.

Fig. 4. Measured fundamental frequencies and the corresponding errors of the recovered chirp rate under the variation of frequency shift deviations.

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Based on the parameters of the FSL, the unambiguous chirp-rate measurement range is from -5200 MHz/µs to 550 MHz/µs. Figure 5 exhibits the matched frequency shifts of the FSL and the corresponding fundamental frequencies under different input chirp rates. As shown in Fig. 5, the matched frequency shift increases periodically, and the fundamental frequency monotonically decreases as the chirp rate increases. A chirp rate within the range of -5200 MHz/µs to 550 MHz/µs can be recovered without ambiguity, because a pair of matched frequency shift and fundamental frequency is unique within such chirp rate range. These results indicate that, the proposed method can be used to realize a wide-range chirp rate measurement.

 figure: Fig. 5.

Fig. 5. Matched frequency shifts and the corresponding fundamental frequencies under the variation of input chirp rates.

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To verify the feasibility of the proposed chirp-rate measurement under the condition of low SNR and small input signal power, an LFMW with a SNR of 0 dB and a chirp rate of 37.5 MHz/µs is modulated onto the CW carrier under a small modulation index of 0.1. Figure 6(a) and (b) show the output pulse train from the PD without the average and its electrical spectrum, respectively, where the frequency shift is tuned to the matched value of 112.5 MHz, and the acquisition time is 1 µs. Figure 6(c) and (d) show the temporal waveform with 50 times average and the corresponding electrical spectrum, respectively, where the acquisition time is 50 µs. As shown in Fig. 6(a), it is unable to recognize the output pulse train due to the serious noise and the low-power signal. However, the frequency comb presented in Fig. 6(b) is still dramatically higher than the noise floor. Especially, the power of the fundamental frequency component is 20 dB higher than the noise floor. In addition, the spectrum shape and the frequency spacing are identical to the measurement case with a regular SNR in Fig. 2(b). Those results indicate that, enabled by the fundamental frequency at its maximum power, the matched frequency shift value can be precisely determined, even under a hostile condition of low SNR and small input signal power. As shown in Fig. 6(c), although the output pulses are much more regular, and the pulse compression becomes obvious after the average, the relatively large noise on the temporal waveform still makes it difficult to judge whether the input LFWM has been compressed to a Fourier-transform-limited pulse or not. Therefore, the proposed method has a better noise tolerance, in comparison with the method based on the measurement of the pulse compression.

 figure: Fig. 6.

Fig. 6. Calculated results for an input LFMW with an SNR of 0 dB and a chirp rate of 37.5 MHz/µs under a small modulation index of 0.1, when the frequency shift is tuned to the matched value of 112.5 MHz. (a) Output pulse train from the high-speed PD without the average, and (b) the corresponding electrical spectrum with an acquisition time of 1 µs. (c) Output pulse train from the high-speed PD with 50 times average, and (d) the corresponding electrical spectrum with an acquisition time of 50 µs. The insets are enlarged view of the waveforms.

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Figure 7 presents the dynamic range (DR) under different modulation indices, where the input LFMW is with a chirp rate of 37.5 MHz/µs and SNRs of 0 dB, 20 dB and 30 dB, respectively. In Fig. 7, the acquisition time is 50 µs. The DR is defined as the power ratio between the fundamental frequency component and the highest spurious component in its electrical spectrum. As for the case of small-signal modulation, the highest spurious component is the noise floor. However, when it comes to large-signal modulation, the highest spurious component is the modulation nonlinearity products. It can be seen from Fig. 7 that, in order to obtain the highest DR, the modulation index is optimized to be around 2.1.

 figure: Fig. 7.

Fig. 7. DR variation with respect to the modulation index.

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The vestigial sideband and carrier arising in the CS-SSB modulation are inevitable in practice, due to the imperfection of the hybrid coupler and the DPMZM. These vestigial sideband and carrier will introduce spurious components to the electrical spectrum, leading to the DR reduction. For a commercially available 90° RF hybrid coupler (Marki QH-0440) with a typical phase offset of 8° and a typical amplitude imbalance of 0.6 dB, Fig. 8(a) and (b) show the output pulse train from the PD and the corresponding electrical spectrum, where the input LFMW is with a chirp rate of 37.5 MHz/µs and a SNR of 20 dB, the frequency shift is tuned to the matched value of 112.5 MHz and the modulation index is 2.1. It can be seen from Fig. 8(b) that, the vestigial optical sideband introduced by the RF hybrid coupler brings a series of spurs with an equal spacing in the output electrical spectrum. However, the DR is still 41.41 dB, which is large enough to discriminate the fundamental frequency component from the spurious component. Figure 9(a) and (b) show the contour maps of the DR under various parameter deviations of the RF hybrid coupler and the DPMZM, respectively. It can be seen from Fig. 9 that, the DR is larger than 40 dB, even when the devices employed to achieve CS-SSB modulation have a serious deviation from the ideal parameters. Therefore, the proposed measurement method has a high tolerance to the device imperfections.

 figure: Fig. 8.

Fig. 8. Calculated results for an input LFMW with a chirp rate of 37.5 MHz/µs and a SNR of 20 dB by employing a RF hybrid coupler with a phase offset of 8° and an amplitude imbalance of 0.6 dB, where the frequency shift is tuned to be the matched value of 112.5 MHz and the modulation index is set to be 2.1. (a) Output pulse train from the PD, and (b) the corresponding electrical spectrum.

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 figure: Fig. 9.

Fig. 9. Contour maps of the DR (a) under various power-splitting ratios and phase offsets of the RF hybrid coupler, and (b) under various optical power-splitting ratios and bias-voltage offsets of the DPMZM.

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Finally, the advantages of our proposed method, in comparison with the traditional temporal waveform characterization method, can be summarized as follows. For the temporal waveform characterization method, a high-speed PD and OSC are compulsory [21]. The criterion to determine the analog bandwidths of the PD and the OSC is that their bandwidth must be larger than the bandwidth of the Fourier-transform-limited pulse after the compression. Consequently, we can measure the Fourier-transform-limited pulse without any filtering-induced pulse broadening. If the measured pulses are broadened due to the bandwidth constraint of the PD and the OSC, we cannot accurately evaluate whether the input LFMW has been compressed to a Fourier-transform-limited pulse. As for our proposed method, whether the input LFMW has been compressed to a Fourier-transform-limited pulse is determined by whether the power of the fundamental frequency in the output spectrum reaches its maximum. The most prominent advantage of this method is to avoid the use of costly PD and OSC. In addition, the filtering-induced pulse broadening has no influence on the fundamental frequency power evaluation. Therefore, our method can simplify the measurement setup and improve the measurement accuracy.

4. Conclusion

We have proposed a measurement method of the chirp rate for the broadband long-duration LFMW, based on tunable photonic FrFT by employing an FSL. Through finely varying the frequency shift of the FSL, the specific FrFT order ensures the fundamental frequency component arising in the output electrical spectrum reach its maximum power. Hence, the chirp rate of the input LFMW can be unambiguously evaluated over a wide range, based on the matched frequency shift and the corresponding fundamental frequency. Both theoretical analysis and simulation results verify the feasibility of the proposed chirp rate method, with the capabilities of broadband operation, robust noise tolerance, low-frequency detection, and long-duration LFMW characterization.

Funding

Research and Development Plan in Key Areas of Guangdong Province (2018B010114002); National Natural Science Foundation of China (62175038, 61805046); Guangdong Introducing Innovative and Entrepreneurial Teams of “The Pearl River Talent Recruitment Program” (2019ZT08X340).

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.

References

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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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Figures (9)
Fig. 1.
Fig. 1. Schematic diagram of the proposed chirp rate measurement based on a flexible photonic fractional Fourier transformer. LD: laser diode, HC: hybrid coupler, DPMZM: dual-parallel Mach-Zehnder modulator, OC: optical coupler, OBPF: optical bandpass filter, EDFA: erbium-doped optical fiber amplifier, AOFS: acousto-optic frequency shifter, EA: electrical amplifier, DDS: direct digital synthesizer, PD: photodetector, EC: electrical coupler, ESA: electrical spectrum analyzer, OSC: oscilloscope. Solid line: optical path; dashed line: electrical path.

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Fig. 2.
Fig. 2. Calculated results for an input LFMW over the frequency range of 20 GHz to 26 GHz with a duration of 0.16 ms. (a) Output pulse train from the PD, and (b) the corresponding electrical spectrum when the frequency shift is tuned to be a matched value of 112.5 MHz; (c) Output pulse train from the PD, and (d) the corresponding electrical spectrum when the frequency shift is 112.4 MHz; (e) Output pulse train from the PD, and (f) the corresponding electrical spectrum when the frequency shift is 112.6 MHz.

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Fig. 3.
Fig. 3. Fundamental frequency and its power with respect to the frequency shift

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Fig. 4.
Fig. 4. Measured fundamental frequencies and the corresponding errors of the recovered chirp rate under the variation of frequency shift deviations.

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Fig. 5.
Fig. 5. Matched frequency shifts and the corresponding fundamental frequencies under the variation of input chirp rates.

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Fig. 6.
Fig. 6. Calculated results for an input LFMW with an SNR of 0 dB and a chirp rate of 37.5 MHz/µs under a small modulation index of 0.1, when the frequency shift is tuned to the matched value of 112.5 MHz. (a) Output pulse train from the high-speed PD without the average, and (b) the corresponding electrical spectrum with an acquisition time of 1 µs. (c) Output pulse train from the high-speed PD with 50 times average, and (d) the corresponding electrical spectrum with an acquisition time of 50 µs. The insets are enlarged view of the waveforms.

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Fig. 7.
Fig. 7. DR variation with respect to the modulation index.

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Fig. 8.
Fig. 8. Calculated results for an input LFMW with a chirp rate of 37.5 MHz/µs and a SNR of 20 dB by employing a RF hybrid coupler with a phase offset of 8° and an amplitude imbalance of 0.6 dB, where the frequency shift is tuned to be the matched value of 112.5 MHz and the modulation index is set to be 2.1. (a) Output pulse train from the PD, and (b) the corresponding electrical spectrum.

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Fig. 9.
Fig. 9. Contour maps of the DR (a) under various power-splitting ratios and phase offsets of the RF hybrid coupler, and (b) under various optical power-splitting ratios and bias-voltage offsets of the DPMZM.

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Equations (11)

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E o u t ( t ) = A α ∑ n = 0 N f α  -  E ~ i n ( z ) e j 2 cot ⁡ α ( z 2 − 2 t τ cos ⁡ α z + t 2 τ 2 )
z = τ τ c csc ⁡ α ( ω s t ′ − Δ ω s t + 2 π n )  =  τ τ c sin ⁡ α ( 2 π k t τ c − ψ ′ 2 + 2 π n )
E i n ( t ) = E 0 ( t ) 2 [ − sin ( m 2 cos ⁡ ( ω R F t + π a t 2 ) ) e j 3 4 π − sin ( m 2 sin ⁡ ( ω R F t + π a t 2 ) ) e − j 3 4 π ]  =  E 0 ( t ) [ J 1 ( m 2 ) e j ( ω R F t + π a t 2 ) − J 3 ( m 2 ) e − j ( 3 ω R F t + 3 π a t 2 ) + ⋅ ⋅ ⋅ ] e − j π 4
E i n ( t )  =  J 1 ( m 2 ) E i n 0 ( t ) e j π a t 2
E o u t ( t ) = A α J 1 ( m 2 ) e j π a t 2 ∑ n = 0 N f α  -  E ~ i n 0 ( υ ) e j cot ⁡ α 2 ( υ 2 − 2 t τ cos ⁡ α υ + t 2 τ 2 )
cot ⁡ α  =  − 2 π ( Δ f s τ c  +  a ) τ 2
I ∝ E o u t ( t ) E o u t ∗ ( t ) = A α 2 J 1 2 ( m ) [ ∑ n = 0 N f α  -  E ~ i n 0 ( υ ) e j cot ⁡ α 2 ( υ 2 − 2 t τ cos ⁡ α υ + t 2 τ 2 ) ] [ ∑ n = 0 N f α  -  E ~ i n 0 ( υ ) e − j cot ⁡ α 2 ( υ 2 − 2 t τ cos ⁡ α υ + t 2 τ 2 ) ]
a  =  − Δ f s τ c = − Δ f s f c
a  =  ( f s − x f c ) f c
f s  =  { k 0 f c + Δ f s ′ ( 0 ≤ Δ f s ′ ≤ f c ) , 0 ≤ a ≤ f c 2 x f c + Δ f s ′ ( Δ f s ′ > f c , x < k 0 ) , f c 2 ≤ a ≤ ( f s − f c ) f c
f s  =  { ( k 0  +  1 ) f c + Δ f s ′ ( | Δ f s ′ | ≤ f c , Δ f s ′ < 0 ) , − f c 2 ≤ a < 0 x f c + Δ f s ′ ( | Δ f s ′ | > f c , Δ f s ′ < 0 , ( k 0  +  1 ) f c < x f c < Δ υ ) , ( f s − Δ υ ) f c ≤ a ≤ − f c 2
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