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Abstract
We propose and experimentally demonstrated an all-optical radio-frequency (RF) spectrum analyzer, based on asynchronous optical sampling (ASOPS) assisted temporal convolution. The RF spectrum is mapped onto the time axis with the help of the temporal convolution system. In combination with the bandwidth compression capability of the ASOPS scheme, up to 28-GHz RF spectrum can be directly read out by an acquisition system with bandwidth as low as 20 MHz. The experimental results demonstrated about 100-MHz resolution and 28-GHz observation bandwidth. The resolution can be improved by increasing the amount of temporal dispersion or optical spectral bandwidth, and the bandwidth can be further extended by compensating the higher-order dispersion, although it is currently mainly limited by that of the electro-optic modulator. The frame rate is flexibly tunable by changing the repetition rate difference between the two mode-locked fiber lasers. Moreover, nearly 25-dB dynamic range indicates this system has a promising application prospect.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
As a fundamental measurement technology, radio frequency (RF) spectrum analysis plays an important role in numerous applications, e.g. the electronic warfare, the radio astronomy, the radio-over-fiber systems and device testing [1–4]. Indeed, electronic solutions, like the widely applied electrical spectrum analyzer, are the most straightforward method to measure the RF spectrum, with high resolution and flexibility. However, the problem is that such electronic solutions show drawbacks in terms of observation bandwidth as well as the frame rate. Microwave photonics has been proved to be a promising way for overcoming these limitations. Furthermore, it has the advantages of low loss and immune to electromagnetic interference [5].
Therefore, a plenty of photonics-assisted approaches for RF spectrum measurement have been proposed and demonstrated in recent years [6]. Most instantaneous frequency measurement systems based on the frequency-to-power mapping can achieve high frequency accuracy; however, unable to perform the measurement of simultaneous multi-frequency components [7–10]. Moreover, there exist a trade-off between the measurement range and accuracy of these schemes. Although the combination of the stimulated Brillouin scattering (SBS) filter and the distributed amplitude comparison function successfully circumvents these problems, large bandwidth photo-detector (PD) and frequency-scanning receiver are required [11]. Spectrum analysis based on photonics-assisted microwave channelization is competent for multi-frequency measurement. In this method, the RF signal is modulated onto an optical carrier, and the modulated optical signal is divided into many parallel channels and detected individually, thus complicated structure and receiver array are required [12–15]. Moreover, the resolution is limited by the optical channel bandwidth, though it can be improved by some nonlinear filtering process, such as the SBS processing unit [16]. Spectrum analysis based on a scanning receiver is another widely studied technology [17–19], but its frame rate is confined by the sweep speed of the optical tunable filter. Other schemes, e.g. compressed sensing [20], and time-stretch based analog-to-digital converter [21], require computer-aided algorithms.
Alternatively, the frequency-to-time mapping (FTM) converts the RF spectrum onto the time axis, thus enables real-time serial detection by a single-pixel PD. Among them, real-time Fourier transform, based on frequency shifted feedback laser [22], spectrally-discrete chromatic dispersion [23], or RF bandwidth magnification [24], achieved high resolution whereas the measurement bandwidth is limited. FTM based on frequency shifting recirculating delay line realized 250-MHz resolution within 20-GHz bandwidth, but the measurement error can be up to 250 MHz [25]. FTM, based on time-domain optical processing in fiber mapping the RF frequency to the time position of an ultrafast pulse, therefore autocorrelation detection technique is required to ensure high resolution, which would greatly hinder the operation speed [26]. Our previous work [27] relaxed the acquisition bandwidth through the temporal magnification technology, thereby achieved over 20-GHz bandwidth at tens of megahertz frame rate; however, the 1-GHz resolution is mainly limited by the imperfect temporal magnification system.
In this letter, we propose and experimentally demonstrated an all-optical RF spectrum analyzer based on the widely used temporal convolution system (TCS) [28] in combination with the asynchronous optical sampling (ASOPS) technique. The TCS helps to convert the RF spectrum onto the time axis, while the ASOPS technique greatly compresses the large optical bandwidth. Therefore, a single-pixel PD and real-time oscilloscope with very low bandwidth can directly capture the RF spectrum, without the need for any post-processing or synchronization.
2. Principle of operation
The schematic diagram of the proposed spectrum analyzer is shown in Fig. 1. Two pulse sources with slightly different repetition rate (Δf) serve as the probe and the local oscillator (LO), respectively. The probe, with repetition rate f1 and pulse shape u1(t), interacts with the RF signal via the TCS. The TCS is composed of two dispersive elements with opposite dispersion value and an electro-optic modulator (EOM) between them. Assuming that the EOM implements carrier-suppressed double sidebands modulation and the small-signal approximation is satisfied, the TCS response waveform can be expressed as [27]
where Φ is the group-delay dispersion (GDD) of the dispersive element, ω0 is the RF frequency, and the sign ⊗ denotes convolution. The first convolution u'1(t) = u1(t) ⊗ δ(t ± Φω0) realized the RF frequency-to-time mapping, while the second convolution means that u'1(t) periodically appears with repetition rate of f1. The four-wave mixing (FWM) based ASOPS is adopted here to down convert u'1(t) onto a low frequency band [29]. The LO pulses, with repetition rate f2 = f1 + ∆f and pulse shape u2(t), are served as the FWM pump, and the intensity profile of the generated idler can be expressed as:where U2 and U' 1 are the Fourier transforms of u2 2(t) and u' 1(t), respectively. Since U' 1 is an even function, we have ∑ +∞ n= - ∞ U' 1(2πnf1)exp(–2πnf1t) = ∑ +∞ n= - ∞U' 1(2πnf1)exp(2πnf1t), thus Eq. (2) can be rewritten as [30]:where p = n – m. An approximation f1 + ∆f ≈f1 is made in the derivations as ∆f is much small than f1. The exponential term means that the spectrum of g(t) is periodically copied in the frequency domain, with frequency interval of f2. The fundamental band replica (p = 0), namely I' o(t) = ∑ +∞ n= -∞U2(2πnf1)U' 1(2πnf1)exp(–j2πn∆ft), can be selected out by a low-pass filter (LPF), and its Fourier transform can be derived aswhere M = f1/∆f is the bandwidth compression factor. The temporal intensity, namely the inverse Fourier transform of Eq. (4), is given bywhere M denotes the temporal magnification factor, and the approximation is reasonable as long as the pulsewidth of u2(t) is short enough. It's clear that the RF spectrum is mapping onto the time axis with a frequency-to-time relation of t = 2πMΦfRF. The convolution in Eq. (5) means that the temporal mapped RF spectrum periodically appears on the time axis with interval of 1/∆f. The frequency resolution is the ratio of the output pulsewidth to the frequency-to-time mapping factor, that is δfRF = δτ1/(2πΦ) ∝ 1/Tw, where δτ1 and Tw represent the probe pulsewidth before and after the first dispersion element. Tw can be also defined as the RF observation window, which mainly decides the resolution.
Fig. 1 Schematic diagram of the proposed RF spectrum analyzer. TCS: temporal convolution system, ASOPS: asynchronous optical sampling.
This scheme can realize accurate spectrum measurement since the spectrum is mapped onto the time axis and the ultrafast pulses provided a super fine time resolution. Thanks to the enormous bandwidth compression or temporal magnification factor provided by the ASOPS scheme, low bandwidth acquisition system is capable of receiving the signal directly, thus the system cost can be effectively reduced. Moreover, the measurement bandwidth has potential to be extended to hundreds of gigahertz [31], although temporarily limited by the bandwidth of the state-of-the-art EOM.
3. Experimental results and discussions
Figure 2 gives the detailed experimental setup of the proposed RF spectrum analyzer. The probe and LO are two mode-locked fiber lasers (MLFLs) based on nonlinear polarization rotation, with repetition rates of 91.862 MHz and 91.863 MHz, respectively. The repetition rates are stabilized by the active phase locking loops to reduce the impact of the timing jitter of the mode-locked lasers on the frequency-to-time mapping accuracy [32]. Two optical band-pass filters (OBPFs) centered at 1572 nm (7.0-nm bandwidth) and 1559.5 nm (7.0-nm bandwidth) are used to filter the two lasers' spectra, respectively. The probe pulse is stretched by the dispersion-compensating fiber (DCF) to provide a long time window for RF signal multiplexing at the electro-optic intensity modulator (FTM7938EZ, 25-GHz analog bandwidth, Vπ = 3.5 V). Figure 3(a) gives the waveforms before and after modulated by a 2-GHz RF signal. After the probe pulse is recompressed by the 80-km single mode fiber (SMF, with dispersion of –1.45 ns/nm), the RF spectrum is mapped onto the time axis. The average power of the LO laser is –5.3 dBm before coupling into the 100-m highly nonlinear fiber (HNLF) (zero dispersion wavelength λ0 = 1561.5 nm, nonlinear parameter γ = 10 w−1 km−1), and serve as the pump of the FWM process. Low bandwidth acquisition system, including a 400-MHz PD and 4-GHz oscilloscope, is used for receiving the generated idler.
Fig. 2 Detailed experimental setup. EDF: Erbium doped fiber, PC: polarization controller, PZT: piezoelectric ceramic transducer; PID: proportional-integral-differential controller; SG: signal generator.
Fig. 3 (a) Stretched probe pulses before (blue) and after (red) the EOM. (b) Spectra of the FWM process. (c) The output signal of the system (black), its envelope (red) is extracted through 20-MHz filtering in the post-processing. (d) Results of the dispersion optimization. (e) Measurement results of an RF signal with three components at 1-kHz frame rate. (f) Zoom-in observation of a single frame.
The FWM spectra are shown in Fig. 3(b), where the pump broadening is caused by the self-phase modulation (SPM). The black trace in Fig. 3(c) is the idler waveform recorded by the oscilloscope, with the inset as the zoom-in observation. Its envelope (red line) is the magnified u1(t), and can be extracted through 20-MHz filtering in the post-processing. This filtering process is not required if a 20-MHz PD is used. By fine tuning the dispersions, the duration of the envelope is optimized to 112 ns (the red line in Fig. 3(d)), which is even shorter than the ASOPS result of the probe pulse before the TCS (the black dashed line in Fig. 3(d)). It indicates that the probe pulses are slightly pre-chirped by the intra-cavity dispersion of the MLFL. An RF signal with three components of 5 GHz, 10 GHz and 12 GHz from an arbitrary waveform generator (AWG, Keysight M8195A) is first tested by the system. The result of three continuous frames in Fig. 3(e) verified a frame rate of 1-kHz, namely the repetition rate offset between the two lasers. The zoom-in observation of a single frame is given in Fig. 3(f), and the multi-frequency measurement ability of the system is demonstrated. The two sidebands contain identical RF spectrum information, while the right side one has finer resolution due to the impact of the third order dispersion (TOD). This can be explained as follows. The two sideband pulses have different optical carrier frequencies due to the modulation induced frequency shift [33]. Therefore, the two sidebands undergo different dispersions in the SMF since the dispersion is frequency-dependent. The dispersion of the right side pulses is better matched, so that finer resolution are achieved. Actually, the pulses corresponding to different RF frequencies also undergo different dispersions, which leads to the result that the resolution varies with frequency.
Scanning the RF frequency from 0.5 GHz to 25 GHz with 500 MHz step, the measurement results are depicted in Fig. 4(a). It can be seen that over 20-GHz observation bandwidth is achieved. Theoretically, the observation bandwidth is the ratio of the non-overlapping time 1/2∆f to the frequency-to-time mapping factor 2πMΦ, i.e., B = 1/4πΦf1. About 468-GHz non-overlapped bandwidth is achievable under the parameters in this paper, and it can be even extended to over 900 GHz when single sideband modulation is applied. In practical, however, the fibers' TOD will greatly decrease the bandwidth. The practical frequency-to-time mapping factor inferred from the results in Fig. 4(a) is 1.088 μs/GHz. According to this mapping factor, resolution at different frequency can be calculated out by measuring the output pulsewidth. The resolution versus frequency in Fig. 4(c) indicates that the best resolution is about 100 MHz at around 12.5 GHz, where we called the optimum dispersion matching point (ODMP). By fine tuning the fiber length, the ODMP can be moved to a higher frequency as Figs. 4(b) and (d) shown. The bandwidth is simultaneously extended to 28 GHz, which is mainly limited by the bandwidth of the EOM. The results of distinguishing two frequencies are given in Fig. 4(e), where 150-MHz multi-tone resolution is demonstrated. The dynamic range of the system is also investigated in Fig. 4(f), where the power of the RF signal is attenuated from 22 dBm to –2 dBm with 1-dB step. The output signal is gradually submerged by noise when the RF power is below –2 dBm, while higher power than 22 dBm will break the small signal approximation and harmonics appear. The linear fitting result (the black line) indicated that about 25-dB dynamic range is achieved.
Fig. 4 (a) (b) Bandwidth test results. (c) (d) Resolution degeneration at different frequencies. (e) Multi-tone resolution of the system. (f) Dynamic range of the system.
Figure 5 explored the timing jitter which has an impact on the frequency-to-time mapping accuracy. The long fibers are put into a thermostat, the temperature of which is firstly stabilized to a specific value. Within two minutes the measurement result of 10-GHz frequency is recorded 50 times. As shown in Fig. 5(a), the absolute time positions of the two sideband pulses have considerable fluctuation, which can be mainly attributed to the instability of the trigger. Actually, the repetition rates of the two lasers have small fluctuations, although stabilized by the phase locking loops. Therefore, the period of the beat signal between the direct outputs of the two lasers, which we used as the trigger signal, is unstable. Fortunately, this fluctuation of the right side pulse can be calibrated by using the left side one or the DC component as a reference, since in our method the RF frequency is actually mapped to the relative time delay between them. The calibrated time position of the right side pulse is depicted with the black line in Fig. 5(a), with the inset as the zoom-in observation. The residual timing jitter is caused by the repetition rate instability of the two lasers which leads to the fluctuation of the magnification factor M = f1/∆f, thereby brings fluctuation to the frequency-to-time mapping factor. The root mean square (RMS) of the residual timing jitter is calculated to be 17.8 ns, corresponding to about 16.4 MHz measurement error. Figure 5(c) is an overlay drawing of the 50 measurements after calibration, with different colors represent different time fluctuation range. To investigate the timing jitter caused by the temperature-induced fiber length variation, the thermostat is heating up during the measurement. The results in Figs. 5(b) and 5(d) show that the time drift caused by the temperature variation can also be calibrated using the reference method.
Fig. 5 Time position fluctuation of the two sidebands pulse (blue and red line) and the calibrated right side pulse (black line) in 50 measurements at constant (a) and varying (b) temperature. Overlay drawing of the 50 measurement results after calibration at constant (c) and varying (d) temperature.
Finally, the feasibility for faster measurement is verified in Fig. 6. The frame rate is respectively raised from 1 kHz to 4 kHz, 8 kHz, and 13 kHz by tuning the repetition rate difference between the two lasers. The resolution retains almost the same, although the signal-to-noise ratio (SNR) is degraded with the increased frame rate as the effective sampling points of the ASOPS is reduced.
4. Conclusion
In conclusion, we have demonstrated an all-optical RF spectrum analyzer, which mapped the frequency onto the time axis. Acquisition bandwidth as low as 20 MHz is capable of capturing the time mapped RF spectrum, thanks to the enormous bandwidth compression capability of the ASOPS scheme. Relies on the super fine time resolution provided by the ultrafast pulses of the MLFLs, about 100-MHz frequency resolution is achieved. The resolution is limited by the 8.6-ns duration of the dispersive stretched probe pulse, thus can be further improved by using larger dispersion or shorter probe pulse. 28-GHz bandwidth is demonstrated and can be theoretically extended to sub-terahertz, although limited by that of the EOM. Moreover, the 1-kHz frame rate can be easily raised up to over 10 kHz by tuning the repetition rate offset between the two lasers.
Funding
National Natural Science Foundation of China (Grants No. 61125501, 61320106016, 61505060, 61631166003, 61675081, and 61735006); Wuhan National Laboratory for Optoelectronics (WNLO).
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