1. Introduction

Wideband (>10 GHz) instantaneous radio frequency (RF) spectral analysis is increasingly important in the fields of wireless communications, radar systems, electronic warfare, and space exploration [1–3]. However, analog-to-digital converters (ADCs) capable of operating continuously are limited to a speed of ∼1 GHz [4]. While faster ADCs exist, the immense quantity of data collected limits these systems to recording short snapshots of the RF spectrum. Thus, conventional electronic systems can either monitor a wide swath of the RF spectrum for a short period of time, or continuously monitor a relatively narrow ∼1 GHz band. Researchers have proposed a variety of electronic and optical techniques to address this trade-off.

One common approach is to fold the wideband RF information into the available ∼1 GHz ADC bandwidth using a Nyquist folding receiver. This scheme has been investigated both in the electronic [5,6] and optical domain (using optical frequency combs) [7]. While this approach allows a wideband spectrum to be monitored continuously, it has several well-known limitations: RF signals in different bands can overlap in a folding receiver [8]; it is challenging to identify the original carrier frequency of a folded RF signal [9]; and the folding penalty can degrade the measurement signal-to-noise ratio [6]. Attempts to address these trade-offs (e.g. by modulating an optical frequency comb to enable carrier disambiguation [9]) introduce additional challenges and tend to perform best when measuring sparse RF spectra.

Ultimately, RF spectrum analyzers must detect high frequency signals at sub-Nyquist sampling rates. Compressed sensing (CS) [10] is potentially well-suited for this type of application, particularly if the receiver has prior information that the RF spectrum is sparse in a given domain [11,12]. Initially, researchers attempted to recover sparse RF spectra by sampling the spectra with a high-frequency pseudo random bit sequences (PRBS). As with the Nyquist folding receivers, these systems have been implemented in the electronic [11,13–15] and optical domains [16–20]. However, these systems require high bandwidth electronics and are sensitive to timing and amplitude noise in the PRBS, resulting in modest performance with resolution typically limited to ∼100 MHz and restrictions on the sparsity of the signal.

While optical folding receivers and compressive sensing systems followed the same general approach as their electronic counterparts, optics has the potential to enable distinctive solutions without obvious electronic analogs. For example, researchers have used spectral hole burning to optically transfer an RF spectrum to the absorption spectrum of a crystal [21,22] or encode high-resolution gratings for RF analysis [23]. These schemes have enabled broadband operation (>10 GHz) with high resolution (∼MHz), but rely on complex architectures using free-space optics and cryogenically cooled crystals.

A conceptually simpler optical approach is to encode the RF spectrum on an optical carrier and directly measure these sidebands using an optical spectrometer. However, this would require a spectrometer with ∼MHz resolution, ∼10 GHz of bandwidth, and a fast update rate (ideally >100 kHz). These specifications are well beyond the capabilities of traditional grating-based spectrometers, both in terms of resolution and update rate.

One promising approach to achieve higher resolution is to use to a “speckle spectrometer” [24–31]. These spectrometers use a dispersive material (e.g. multimode fiber [24,28], scattering media [26], on-chip multimode waveguides [27,32,33], or integrating spheres [29]) to produce wavelength dependent speckle patterns that are used as fingerprints to identify the optical spectrum. This approach is also well-suited for compressed sensing problems since the dispersive material forms random projections of the optical spectrum [27]. However, simultaneously achieving the resolution, bandwidth, and update rate required for RF spectral analysis remains challenging. Although achieving the MHz resolution required for RF analysis is possible, most speckle spectrometers capable of reaching this resolution are restricted to detecting a single wavelength at a time (i.e. operating as a wavemeter). This single wavelength restriction allows the system to identify changes in wavelength that are much smaller than the spectral correlation width of the speckle pattern [30,31].

As a result, only a few attempts have been made to use speckle spectrometers for RF analysis. A recent work encoded the RF signal on a chirped optical pulse and used a speckle spectrometer to measure the spectral components of the pulse [34–36]. This approach achieved 100 MHz resolution with a fast update rate, but required an array of 16 detectors and ADCs and was limited to measuring ∼240 spectral channels. The same group also used a speckle spectrometer constructed using a 100 m multimode fiber and a high speed (800 kHz) camera to directly measure the RF spectrum encoded on an optical carrier [37,38]. While this approach was able to resolve individual lines with ∼MHz uncertainty, the high-speed camera could increase the system cost and the spectral correlation width (∼100 MHz) of the multimode fiber limited the ability of the system to resolve closely spaced RF lines.

Recently, a novel speckle spectrometer architecture was proposed which combines exceptionally high-resolution with a fast measurement speed. This scheme relies on the speckle formed by Rayleigh backscattered light in single-mode optical fiber, which can be recorded on a single, high-speed photodetector [39–41]. The spectral resolution is dictated by the fiber length and pulse duration while the measurement speed is dictated by the round-trip time in the fiber. This scheme has enabled a spectral correlation width of ∼1 MHz with an update rate of 40 kHz [39]. Although the number of spectral channels that could be measured simultaneously was limited, this combination of native resolution and measurement speed has enormous potential for RF spectral analysis.

In this work, we constructed an RF spectrum analyzer using a Rayleigh-backscattering speckle spectrometer. In order to monitor a broad bandwidth with high spectral resolution, we introduce a dual-resolution scheme to aid the spectral recovery, enabling the system to monitor 15,000 spectral channels simultaneously. This approach is uniquely suited to a Rayleigh backscattering configuration since the spectral correlation width can be adjusted by controlling the pulse duration of light coupled into the fiber. We also introduce a simple scheme to mitigate the environmental sensitivity of the Rayleigh-backscattering spectrometer, reducing the need for recalibration. This approach enables an RF spectrum analyzer capable of continuously monitoring a 15 GHz band with an update rate of 385 kHz. Our approach is able to resolve RF tones separated by 25 MHz and identify the frequency of individual tones with an accuracy of 1 MHz. The system is constructed using off-the-shelf fiber coupled components and low-bandwidth electronics, providing an accessible scheme for wide-band RF spectral analysis.

2. Spectrum analyzer design and operation

The RF spectrum analyzer operates by encoding an RF spectrum onto an optical carrier (e.g. using an electro-optic modulator) and using a Rayleigh-backscattering (RBS) based speckle spectrometer to record the resulting optical spectrum. The challenge is in designing the spectrometer to meet the requirements for wideband RF analysis. Ideally, the spectrometer would provide ∼MHz resolution over a bandwidth of at least 10 GHz while maintaining a fast update rate (>100 kHz) [3].

In the RBS speckle spectrometer, the optical signal under test is modulated into a short pulse and injected into a single mode fiber. The Rayleigh backscattered light is then measured as a function of time, providing the speckle pattern that will be used to recover the input spectrum [39]. As in any speckle spectrometer, a calibration step is required to construct a transfer matrix, ${\boldsymbol T}$. The transfer matrix is an ${M_t} \times {N_f}$ matrix that converts the input spectrum, S (described as a vector with ${N_f}$ spectral channels), to the measured speckle pattern, I (described as a vector containing ${M_t}$ temporal samples). Calibration is performed by recording the speckle pattern formed by each spectral channel in S to construct ${\boldsymbol T}$ one column at a time. In this work, we used a compressed sensing approach to recover the input spectrum, S, from the measured speckle pattern, I, by solving the following minimization problem [28,42]:

The resolution, bandwidth, and measurement rate of the RBS spectrometer can be adjusted by selecting the pulse duration and the fiber length. The pulse duration, $\tau $, dictates the spectral resolution of the spectrometer since the spectral correlation width of the speckle pattern, $\mathrm{ \delta }f,$ scales as $\mathrm{\delta }f\textrm{ }\sim 1/\tau $ [43]. The length of the fiber, L, determines the measurement rate: ${f_{meas}} = ({c/n} )/({2L} ),$ as well as the number of measured temporal channels, or speckle grains, which can be estimated as ${M_t} \cong 2nL\textrm{ / }({c\tau } )$, where c is the speed of light and n is the effective index in the fiber. The measurement bandwidth, $\mathrm{\Delta }F$, depends on the acceptable compression ratio and is limited to $\mathrm{\Delta }F = {N_f}\mathrm{\delta }f \cong ({{N_f}/{M_t}} )[{2nL\textrm{ / }({c{\tau^2}} )} ]$.

Based on these expressions, we can attempt to design a spectrometer with the desired 1 MHz resolution, 10 GHz bandwidth, and 100 kHz update rate. This resolution will require a pulse duration of $\tau$ ~ 1 μs while the update rate will limit the fiber length to $L\sim $ km. Unfortunately, this would result in a speckle pattern containing ${M_t}\sim 10$ speckle grains and require an unrealistic compression ratio of ∼1000 to recover the ${N_f}\sim {10^4}$ spectral channels needed to support a bandwidth of 10 GHz.

In order to mitigate this trade-off between spectral resolution and bandwidth, we introduce a dual-resolution scheme in which we probe the fiber with a short pulse followed by a longer pulse. This scheme is similar to the multi-scale or progressive compressive sensing techniques that have been proposed for high-resolution, large area compressive imaging applications [44]. In our system, the speckle pattern produced by the short pulse is used to obtain a broadband measurement with modest resolution while the speckle pattern produced by the long pulse is used to improve the resolution of the spectrum obtained with the short pulse. Provided the RF spectrum is relatively sparse, the short-pulse can be used to screen the overall spectrum for regions where RF signals are present so that we can reduce the search bandwidth (and compression ratio) when we use the long pulse data to improve the resolution.

3. Experimental setup

The RF spectrum analyzer design is shown in Fig. 1(a). Light from a narrow linewidth laser (<10 Hz, OEwaves 4028) was directed to an electro-optic modulator (EOM). The EOM was driven by the RF signal under test in dual-sideband suppressed-carrier mode, creating two copies of the RF signal. A tunable bandpass filter was used to select a band from 5 to 20 GHz above the original laser frequency. The optical signal carrying the RF information was then coupled into the Rayleigh backscattering speckle spectrometer, shown in Fig. 1(b).

 

Fig. 1. (a) The RF spectrum analyzer components are shown in the blue box. The RF signal is encoded on an optical carrier and measured using a RBS speckle spectrometer. (b) The RBS speckle spectrometer operates by modulating the optical signal into a pair of pulses which are coupled into a 100 m fiber. The Rayleigh backscattering pattern is used to reconstruct the original RF spectrum. (c) An example speckle pattern recorded on one of the photodetectors showing the RBS pattern generated by the 10 ns pulse followed by the pattern generated by the 100 ns pulse. Note that in the experiments reported in this work, an acousto-optic modulator (not shown) was included after the EOM within the RBS spectrometer to increase the pulse extinction and 100 GHz bandpass filters were included after each EDFA to suppress amplified spontaneous emission.

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Within the spectrometer, the optical signal was modulated into alternating 10 and 100 ns pulses using an EOM. The pulses were separated by 1.3 $\mu s$, as shown in the inset in Fig. 1(b). This pulse separation was slightly longer than required based on the 100 m fiber length in order to clearly separate the speckle patterns formed by the two pulses. The modulator transmission was adjusted so that the 100 ns pulse had ∼30% of the peak power of the 10 ns pulse. This ensured that the RBS speckle pattern from each pulse would have similar power levels, since the RBS power is dictated by the input peak power and pulse duration [45]. The interrogation pulses were amplified to an average power of ∼2 mW using an Er-doped fiber amplifier (EDFA) and directed into 100 m of standard single mode fiber (SMF-28e+) via a circulator. The fiber was placed in an insulated metal box to reduce the environmental fluctuations, although the box was not temperature controlled for these experiments. The backscattered speckle pattern was amplified using a second EDFA and directed through a polarizing beam splitter (PBS) onto a pair of 125 MHz photodetectors and recorded at 1 GS/s. A typical speckle pattern recorded on one of the photodetectors is shown in Fig. 1(c). The first ∼1 μs contains the speckle pattern produced by the 10 ns pulse, revealing a dense speckle pattern, while the speckle pattern produced by the 100 ns pulse contains far fewer speckle grains.

To operate the spectrometer, we first calibrated the transfer matrices associated with the 10 ns and 100 ns pulses. This was accomplished by recording the speckle patterns formed while driving the EOM at frequencies from 5 GHz to 20 GHz in steps of 1 MHz using an RF signal generator. Portions of the two transfer matrices are shown in Fig. 2(a,b). As expected, the 10 ns pulse produced speckle patterns with more temporal features, but a larger spectral correlation width than the 100 ns pulse. We calculated the spectral correlation width associated with both transfer matrices, as shown in Fig. 2(c). The 10 ns pulse produced speckle patterns with a half-width at half-maximum (HWHM) spectral correlation width of ∼30 MHz while the 100 ns pulse produced speckle patterns with a ∼3 MHz correlation width, following the expected $1/\tau $ dependence [43]. We also calculated the temporal correlation width associated we both transfer matrices, as shown in Fig. 2(d). The 10 ns pulse produced a pattern with a temporal HWHM of ∼7 ns, while the 100 ns pulse produced a pattern with temporal HWHM of ∼52 ns, following the expected $\tau $ dependence.

 

Fig. 2. Part of the transfer matrices recorded using the (a) 10 ns and (b) 100 ns pulses. (c) The spectral correlation HWHM for the 10 ns pulse was ∼30 MHz while the HWHM for the 100 ns pulse was ∼3 MHz. (d) The temporal correlation HWHM for the 10 ns pulse was ∼7 ns while the HWHM for the 100 ns pulse was ∼52 ns.

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The measured transfer matrices were oversampled along both the frequency and time axes. To reduce the amount of correlated data, we first applied a low pass filter in the time domain with a passband set at $1/\tau $. We then sub-sampled the 10 ns transfer matrix every 4 ns and the 100 ns transfer matrix every 25 ns. We also sub-sampled the 10 ns transfer matrix in the frequency domain at a step size of 20 MHz (just below the spectral correlation width). Finally, we discarded the specular reflections from the beginning and end of the fiber and stitched together the speckle patterns recorded in the two polarization states. This resulted in a 10 ns transfer matrix with dimensions ${N_f} = 750$ spectral channels and ${M_t} = 180$ uncorrelated temporal channels, enabling a coarse reconstruction of the 15 GHz measurement bandwidth with a modest compression ratio of ∼4. The 100 ns transfer matrix had dimensions ${N_f} = 1.5 \times {10^4}$ and ${M_t} = 15$. In practice, the number of spectral channels used in the 100 ns transfer matrix will be reduced depending on the number of bands that contain RF signals, determined using the initial coarse reconstruction. As an example, if the coarse reconstruction identified RF signals in three of the 20 MHz bands, the fine reconstruction would only be required to search 60 MHz of bandwidth, resulting in a compression ratio of 4. On the other hand, if the entire 15 GHz bandwidth was reconstructed with the fine resolution transfer matrix the compression ratio would be 1000.

After calibration, we tested the ability of the RF spectrum analyzer to recover an unknown RF signal. As with the calibration measurement, the recorded speckle patterns were low pass filtered at $1/\tau $ for each pulse duration. This step served two important functions. First, the filter removed highly correlated information, accelerating the spectral reconstruction. Second, the filter removed the effect of interference between nearby tones in the RF spectrum. Most speckle spectrometers are able to ignore this type of interference effect since the detector bandwidth is well below the spectral resolution of the spectrometer. However, in the RBS spectrometer, interference signals between RF tones separated by less than the 125 MHz detector bandwidth will introduce oscillations in the measured speckle pattern. Since, the compressed sensing formulation in Eq. (1) assumes the speckle patterns produced by distinct frequencies will add incoherently, a low-pass filter is required to remove the interference signal. As we will discuss below, this has implications for the minimum resolvable separation between two RF tones.

As an initial test, we measured an RF spectrum consisting of two tones separated by 100 MHz at 19.45 GHz and 19.55 GHz. We first used the 10 ns speckle pattern to obtain a coarse measurement of the RF spectrum. In this work, we solved Eq. (1) using a lasso algorithm that constrains the elements of S to be nonnegative [46]. Figure 3(a) shows the recovered spectrum revealing two tones near 19.5 GHz. This coarse measurement does not have the resolution to accurately identify the two underlying frequencies; however, it was able to successfully identify the spectral regions containing RF signals. We then used this coarse spectrum to select the spectral bands to search using the 100 ns speckle pattern. To do this, we selected any spectral channel within 20 MHz of an RF signal that was above a normalized threshold value of 0.01 (this threshold was set by the dynamic range of the spectrometer, which will be discussed below). The orange lines in the inset of Fig. 3(a) show the spectral bands that were used to obtain the fine-resolution spectrum.

 

Fig. 3. (a) Recovered spectrum after “Coarse reconstruction” using data obtained with the 10 ns interrogation pulse is shown in blue. The orange regions shown in the inset indicate the frequency bands with RF signals above a threshold (indicated by the black dashed line at an intensity of 0.01). The fine reconstruction was restricted to the frequency bands indicated in orange. (b) Recovered spectrum after the “Fine Reconstruction” revealing 2 lines separated by 100 MHz.

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To reconstruct the fine resolution spectrum, we used a subset of the original 100 ns transfer matrix that only contained the frequency bands identified in the coarse reconstruction. The 100 ns speckle pattern was then provided to the same lasso algorithm to recover a high-resolution spectrum in the selected frequency bands. The spectral components outside these regions were set to 0. The recovered spectrum over the entire 15 GHz range is shown in Fig. 3(b). This dual resolution approach correctly identified the two input tones, enabling high resolution spectral analysis across the entire 15 GHz range. Note that even though the coarse reconstruction identified two possible RF signals near 13 GHz (which were above the threshold of 0.01), the fine reconstruction correctly identified these signals as spurious and did not include RF content in these bands in the final spectrum.

4. Advantages of dual resolution approach

To illustrate the advantage of the dual resolution approach, we attempted to recover an RF spectrum using the 10 ns speckle pattern on its own, the 100 ns speckle pattern on its own, or the combined approach. In this test, the spectrum contained a single frequency tone at 12.5 GHz. Figure 4(a) shows the recovered spectrum using the coarse reconstruction. In order to perform a direct comparison with the other methods, this coarse reconstruction was performed using a transfer matrix with 1 MHz frequency steps. The inset in Fig. 4(a) shows that this method does not have the spectral resolution to resolve the frequency down to 1 MHz and instead shows a series of lines near 12.5 GHz. Additionally, there are several spurious lines across the spectrum. Figure 4(b) shows the recovered spectrum using only the fine resolution (100 ns) reconstruction. Although there is sufficient resolution to recover the single line, as shown in the inset, the recovered spectrum also shows a series of spurious lines due to the high compression ratio. Finally, Fig. 4(c) shows the recovered spectrum using the dual resolution approach. This approach correctly recovers the single line at 12.5 GHz without spurs, since the compression ratio was relatively modest in both reconstructions.

 

Fig. 4. (a) Coarse resolution reconstruction only using the $10\; ns$ speckle pattern. (b) Fine resolution reconstruction only using the $100\; ns\; $ speckle pattern. (c) Dual resolution reconstruction using both the 10 and 100 $ns\; $ speckle patterns. (d) Processing time per spectral reconstruction for the dual resolution reconstruction scheme compared to only using the 100 ns data (“Fine Only”) to reconstruct the entire spectrum.

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The dual resolution approach also has a significant advantage in computing speed. To demonstrate this advantage, we calculated the time required to recover a spectrum with varying bandwidth using the dual-resolution scheme or performing a single reconstruction using only the fine-resolution ($\tau = 100\; ns)$ data. As shown in Fig. 4(d), the dual resolution scheme enabled a faster spectral reconstruction for all but the smallest bandwidth (specified by the number of spectral channels, where the channel spacing was set to 1 MHz). As mentioned above, reconstructing the entire 15 GHz bandwidth using only the fine resolution measurement requires a compression ratio of 1000, dramatically higher than the compression ratio of 4 or 5 in each stage of the dual resolution approach. This higher compression ratio resulted in slower reconstruction with more errors. For broadband spectra with >10⁴ spectral channels, the dual-resolution scheme provides a ∼10${\times} $ speed-up which is similar to the acceleration demonstrated using progressive compressive sensing in an imaging context [44]. Although we used a publicly available compressed sensing algorithm written in MATLAB running on a desktop computer (Intel Core i7-6820HQ, 2.7 GHz), we expect that a similar speed-up would be maintained in an optimized implementation.

5. Results

To demonstrate the frequency range and resolution of the spectrometer, we measured RF signals ranging from 5 to 20 GHz in steps of 1 MHz. Figure 5(a) shows the recovered spectra over the entire frequency range of the spectrometer and Fig. 5(b) shows a magnified view of the spectra over a 50 MHz range starting at 12.5 GHz. This measurement confirmed that the Rayleigh spectrometer is able to correctly recover narrowband RF signals over the entire 15 GHz range with 1 MHz resolution. Here, the resolution was slightly better than the correlation width due to the high signal-to-noise ratio of the measurement and the sparsity of the recovered spectrum.

 

Fig. 5. (a) Recovered spectra of SUT with a single frequency component over the entire frequency range of the spectrometer. (b) Magnified view of the frequency region from 12.5 to 12.55 GHz showing discrete 1 MHz steps in the recovered spectra.

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One of the major challenges for all speckle spectrometers is environmental sensitivity and the need to frequently recalibrate the transfer matrix. In this system, recalibrating all 15,000 spectral channels would require only 40 ms (in practice, our calibration required ∼8 minutes due to the limited scan speed of the RF signal generator used in our experiments). While this calibration can be quite fast compared to most camera-based speckle spectrometers, the high-resolution also makes the system especially sensitive to changes in temperature. Fortunately, the single-mode fiber platform offers the potential for a much simpler form of re-calibration since a change in temperature and a change in optical frequency have the same effect on the RBS speckle pattern [47]. In other words, if the temperature of the fiber changes, the entire calibrated transfer matrix will still be valid, it will merely be shifted in frequency by ${\sim} 1$GHz/°C. Here, we use this equivalence to extend the time between full re-calibrations.

Rather than re-calibrating all 15,000 spectral channels, we can monitor frequency-shifts in the transfer matrix by periodically measuring the speckle pattern at a single (known) optical frequency (which only requires 2.6 µs). We can then perform a cross-correlation between the measured speckle pattern and the speckle patterns stored in the transfer matrix to find any temperature-induced frequency shifts in the transfer matrix. If we find that the transfer matrix shifted by 100 MHz due to a change in temperature, we can simply shift the reconstructed spectrum by -100 MHz to compensate.

The primary limitation of this simple re-calibration scheme is that it assumes any environmental changes (in temperature or strain) are uniform across the fiber. If the temperature changes non-uniformly, a single frequency shift will not be sufficient to update the transfer matrix. While a more sophisticated algorithm could potentially compensate for some non-uniformity, temperature non-uniformity on a length scale below the spatial extent of the pulse (i.e. $\tau c/({2n} )$, or 10 m for a 100 ns pulse) will produce an uncorrelated speckle pattern that is not stored at other frequencies in the transfer matrix. At this point, a full recalibration is required.

To evaluate this approach, an RF signal with a single frequency component was measured every 0.5 seconds for ∼40 minutes after the initial full calibration. Before each measurement, we performed the single-frequency re-calibration described above, which required 2.6 $\mu s$. Figure 6(a) shows the recovered spectra as a function of time without using the single-frequency re-calibration measurement. The recovered frequency drifts almost immediately (>5 MHz drift after ∼60 s) and deviates by up to 80 MHz during the course of the measurement (corresponding to a temperature drift of ∼0.08 °C). Figure 6(b) shows the recovered spectra using the single frequency re-calibration measurement. In this case, the spectrum analyzer accurately identified the RF signal for ∼20 minutes with an error below 2 MHz. After ∼30 minutes, the recovered spectrum shows significant errors due to non-uniform changes in temperature across the fiber and a full-recalibration is required. Nevertheless, the single-frequency re-calibration enables the system to operate for ∼20 minutes without significant errors. To enable continuous operation during this period, we could employ a second laser to continuously perform the single-frequency re-calibration and separate it from the laser carrying the RF signal using wavelength-division multiplexing filters. Finally, better environmental isolation [48] and/or the use of machine learning algorithms [31] could further reduce the need to perform a full recalibration.

 

Fig. 6. (a) Recovered spectra of SUT over 40 minutes without any re-calibration. (b) Recovered spectra of SUT over 40 minutes with the simple single frequency re-calibration completed prior to each measurement.

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While speckle spectrometers are able to identify single frequencies with accuracy much better than the spectral correlation width [30,31], multi-line measurements are more challenging. In order to measure the minimum resolvable separation between two RF tones, we used an EOM to create a pair of input tones with varying frequency separation. The EOM was positioned immediately before the input to the RBS spectrometer shown in Fig. 1 and operated in dual-sideband suppressed-carrier mode. The frequency separation of the two tones was scanned from 1 to 100 MHz in steps of 0.5 MHz. A spectrogram containing the measured spectra as a function of frequency separation is shown in Fig. 7(a) and several cross-sections are shown in Fig. 7(b). The spectrometer can successfully distinguish between two tones separated by as little as 25 MHz. However, as the tone separation approaches the 10 MHz passband of the low-pass filter, interference effects are no longer fully suppressed. This introduces a dependence on the relative phase between the two tones which is not captured in Eq. (1) and leads to spurious signals in the recovered spectra (see $\mathrm{\Delta }f $= 10 MHz in Fig. 7(b)).

 

Fig. 7. (a) Spectrogram recorded while varying the separation between two RF tones. The dashed line at 10 MHz corresponds to the inverse of the pulse duration, and sets an effective limit on the resolution of the spectrometer. (b) Cross-sections of (a) measured using RF tones with a frequency separation of 100 MHz, 25 MHz, and 10 MHz.

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There are several approaches to improve the spectral resolution which help illustrate the trade-offs in the RBS spectrometer design. One could simply reduce the low-pass filter cutoff below $1/\tau $ to suppress these interference effects, but this reduces the number of uncorrelated measured speckle grains, ${M_t}$, increasing the compression ratio and degrading the reconstructed spectra. In practice, we found that reducing the low-pass filter in this way did not significantly improve the resolution. Alternately, we could increase the pulse duration and fiber length to improve the resolution without compromising ${M_t}$, although this would reduce the measurement rate. This trade-off could be offset through spatial multiplexing if the RBS patterns from multiple fibers (or multiple modes in a multimode fiber) were measured to increase the speckle diversity at the cost of increased system complexity. Finally, a coherent calibration process similar to the technique described in [35] could also be considered in order to compensate for interference effects, although this would likely increase the complexity of the calibration process and reconstruction algorithm.

The dynamic range is another important metric for RF spectral analysis. Here, dynamic range is defined as the ratio between the largest and smallest signal that the instrument can simultaneously detect. The dynamic range of the spectrometer was evaluated by measuring two tones separated by 200 MHz with varying amplitude. For this measurement, an optical carrier at an offset frequency of 19.35 GHz was divided into two paths and coupled through separate acousto-optic modulators (AOMs) which introduced shifts of ${\pm} 100\; MHz$. Light from the two AOMs was then recombined and coupled into the RBS spectrometer. We then adjusted the transmission through one of the AOMs to vary the relative power between the two tones.

The measured spectra recorded at each power level are shown in Fig. 8. Each spectrum has been shifted in frequency by 5 MHz for clarity. The expected spectral amplitude of the lower sidebands are shown as a series of black circles. The spectrometer successfully measured the two frequencies until their relative power level was separated by more than ∼15 dB. Based on this measurement, we set the threshold level used to identify spectral regions with RF signals in the initial coarse reconstruction to 0.01 (-20 dB).

 

Fig. 8. Measured RF spectrum consisting of two tones separated by 200 MHz with varying relative power. The recovered spectrum are offset in steps of 5 MHz for clarity and the black circles indicate the expected power of the lower frequency sideband. The spectrometer correctly measured two tones with a dynamic range of ∼15 dB.

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A final test of the RF spectrum analyzer was conducted to highlight the fast measurement rate (385 kHz). In this test, we generated a chirped RADAR-like signal and a frequency-hopping RF signal using an EOM. The EOM was positioned between the bandpass filter and the RBS spectrometer in Fig. 1, and was driven in dual-sideband suppressed-carrier mode, creating a pair of sidebands on a carrier centered at 19.5 GHz. The frequency chirp ramped from 100 MHz to 200 MHz in 500 µs. This chirp was followed by a frequency agile carrier that oscillated rapidly between 3 frequencies that were each held for 50 $\mu s$. The entire 15 GHz spectrum was recorded every 2.6 µs. The spectrogram in Fig. 9 shows a magnified view of a 1 GHz region centered around 19.5 GHz as a function of time. This experiment demonstrates the ability of the RF analyzer to continuously monitor a rapidly changing spectrum with broad-bandwidth and high resolution.

 

Fig. 9. Spectrogram recorded of a chirped pulse followed by a frequency agile RF signal that oscillates rapidly between 3 frequencies.

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6. Conclusion

We introduced a wideband RF spectrum analyzer that uses a high-resolution speckle spectrometer to monitor RF sidebands encoded on an optical carrier. The spectrometer relies on a compressed sensing algorithm to recover the RF spectrum from the speckle pattern formed by Rayleigh backscattered light in a standard single mode fiber. This RBS scheme has tremendous potential for RF analysis due to its combination of high resolution and a fast update rate. However, high-resolution speckle spectrometers are extremely sensitive to environmental changes and require a trade-off between resolution and bandwidth. In this work, we introduced a dual-resolution scheme to mitigate the trade-off between resolution, bandwidth, and measurement speed. We also demonstrated a simple technique to compensate for environmental drift, reducing the need for frequent re-calibration. Using this optimized spectrometer design, we were able to construct an RF spectrum analyzer capable of continuously monitoring a 15 GHz wide spectrum with MHz-level resolution and a 385 kHz update rate. The system was constructed using fiber-coupled off-the-shelf components, providing a simple and cost-effective approach to wideband RF detection.