2.1.

High-Speed Operation

To demonstrate high-speed data acquisition, the standard laboratory camera was replaced with a high-speed camera (Photron FASTCAM Nova S6, monochrome). A $128\times 16\text{pixel}$ window was selected, which resulted in operation at a rate of 800,000 frames per second and an approximate integration time of $1\mu \mathrm{s}$. With respect to the previous setup, the focus of the lens system was changed, so more speckle features were imaged onto the smaller active region of the camera. In this mode of operation, the laser power at the output of the modulator was measured to be $22\mu \mathrm{W}$, which corresponded to operation safely below the pixel saturation threshold. Meanwhile, a microwave frequency generator (HP 8340B) sweeper was repeatedly swept from 2 to 7 GHz at a rate of $555\mathrm{GHz}/\text{second}$. A complete scan consisted of 7711 images, and each scan was recorded in $\sim 9\mathrm{ms}$ of lab time.

3.1.

Data Collection

During the calibration process, a sequence of equally spaced RF tones is played over the operation range of interest, in which the frequency spacing of the tones is typically set to the resolution of interest. While each tone is being played, a two-dimensional array from the camera—the speckle image—is recorded, converted to a one-dimensional array, and saved. The full set of these arrays is the calibration matrix and serves as the dictionary for recovery. Finally, one or more signals of interest are applied to the same setup, and the data array is stored.

3.2.

Spectrum Recovery

After both the calibration dictionary and the signal of interest are acquired and stored, the spectrum of interest must be computationally reconstructed. We have primarily explored regression analyses to determine the spectrum of the unknown signal.

Regression analysis seeks to estimate the relationship between dependent and independent variables. In this case, the dependent variables are the spectral coefficients scaling the recorded pixel intensities of each frequency component of the dictionary to the measured pixel intensities of the SUT. The data coefficient matrix (dictionary), $X$, is composed of $N$ measurements (columns, frequencies), consisting of $M$ elements (rows, pixel intensities) each. In most cases presented here, $M>N$, and the system is overdetermined. Although this is not true in some of the cases discussed in Sec. 4, such as that of the high frame rate camera, the same data processing procedure was used for consistency.

Before regression analysis, each calibration and SUT frequency dataset is set to have zero mean. Ordinary least squares is found to perform poorly in the presence of experimental noise and returns too many nonzero spectral coefficients. Thus, we initially begin our analysis by focusing on the lasso (penalized $l^1$-norm) algorithm⁴¹ because it promotes sparsity in the number of returned nonzero spectral coefficients with an appropriate choice of the tuning parameter, $\alpha$, that controls the size of the regularization term in lasso, relative to the measurement constraint. However, lasso performs slowly when the system is overdetermined, so to reduce the computational time, the SVD of the data coefficient matrix $X$ is used to reduce the size of the problem. $X$ is decomposed into three matrices as

Fig. 2

(a) SVD of the calibration matrix. (b) Projection of the calibration data onto the space spanned by the calibration measurements.^39,40

After performing the SVD, it is instructive to look at the Pearson autocorrelation coefficients of the dictionary to determine the correlation width of the dictionary. See Fig. 3 for an example. The cross-correlation coefficients between the dictionary and SUT data set(s) can also be computed to judge the stability of recovery. If the SUT is composed of a single frequency, such as the typical scenario in which an IFM receiver is used, the maximum of the correlation function serves as the recovered frequency. Computing the correlation coefficient maximum is considerably faster than a lasso computation and provides the same result.

Fig. 3

(a) Spatially averaged spectral correlation function.²² (b) Auto correlation coefficient of a dictionary element at 5500 MHz.^39,40

The spectrum can be more finely recovered using lasso or elastic-net (a regression that incorporates L1 and L2 norm penalties) regressions,⁴² selectively tailoring $\alpha$ to promote sparsity in the number of positive spectral coefficients relative to ordinary least squares ($\alpha =0$). However, this process can be slow and requires user input or automation to correctly predict the spectrum. Lasso-based recovery becomes problematic when the SUT has multiple frequency components, particularly when they are not of the same intensity. The frequency accuracy of recovered spectral components is also affected when multiple components are spaced more closely than the correlation width of the spectrometer. For these reasons, we employed orthogonal matching pursuit (OMP)^42,43 when signals are composed of multiple frequency components. See Sec. 4 for further details.

4.1.

Recovery Speed Improvements

In the interest of increasing the data processing and recovery speed, a smaller subset of pixels is randomly chosen. As expected, the time to compute the SVD, the most computationally expensive task in recovery, drops rapidly as the number of pixels is reduced. Although a full SVD is only required whenever a new dictionary is measured, the time for a matrix multiplication, required every time a spectrum is determined, also scales similarly with the number of pixels, but the time required is small and poorly determined due to large relative variations in computation time among tests (Fig. 5).

Fig. 5

Time to compute the SVD versus the number of pixels included in the recovery, for a 1000-frequency dataset with a maximum of 65,356 pixels. The computation time is not standardized and is subject to some amount of variation.⁴⁰

The instrument speed increases as the number of pixels is reduced, whereas the dynamic range of the instrument and the recovery accuracy, to a lesser extent, suffer due to lower cross-correlation contrast as a result of using fewer measurements. See Fig. 6.

Fig. 6

Cross-correlation plots used for recovery with (a) 66,820, (b) 2048, and (c) 64 randomly selected pixels.⁴⁰

Not all pixels are equally well suited for use in recovery. As shown in Fig. 7, some pixels have significantly higher spectral correlation maxima than the mean. The use of pixels above the correlation maximum mean provides better recovery than the use of all pixels and is much better compared with those below the mean. See Fig. 8 for a comparison.

Fig. 7

Spectral correlation maxima along a randomly selected row of pixels with a dashed mean.⁴⁰ The dataset is the same as Fig. 3(a).

Fig. 8

Recovery of a tone at 5500 MHz using only pixels with spectral correlation function maxima above (blue) and below (orange) the mean.

If the “good” pixels can be expeditiously identified, then the downsides of using a lower number of pixels can be offset. The locations of the good pixels do not depend on the RF signal but may be changed by variable environmental conditions. The pixel-by-pixel correlation can be calculated as described in Ref. 2, but this is computationally expensive and slow; however, a pixel’s maximum of the spectral correlation function was found to have a positive association with the variance of the values that the pixel takes as frequency is varied during a calibration scan. Unfortunately, high variance does not guarantee a strong correlation; thus, selection by variance does not considerably increase the recovery accuracy. However, a selection of pixels with variance above the mean decreased the computation time by 70%, similar to selection by correlation maximum, and had nearly the same recovery accuracy relative to a computation using the full dataset. Selection by correlation maximum increased recovery by 10% relative to the full dataset.

4.2.

Recovery of Multitone Spectra

Figure 9 shows the recovery in the presence of multiple tones. Lasso regression struggles to recover the correct frequencies when two or more tones are positioned within the correlation width of the spectrometer and incorrectly recovers the frequency lower tone (2250 MHz). The situation is more promising when the tones are spaced farther than the correlation width of the spectrometer; however, an algorithm that varies $\alpha$ until a specified number of positive coefficients are reached requires more positive coefficients than tones present in the SUT. This results in uncertainty in the recovered tones and incorrect spectral amplitudes. OMP finds the correct tone frequencies in both cases, the correct number of tones, and the appropriate relative amplitudes. Although OMP requires specification of the number of nonzero coefficients or some other stopping criterion, this value can be increased above the number present in the SUT, and extras can be filtered out on the basis of coefficient intensity or sign.

Fig. 9

(a) Comparison of the recovery of multitone (2250 and 2750 MHz) spectra near or below the correlation limit of the spectrometer. (b) Comparison of the recovery of multitone (2500 and 4000 MHz) spectra above the correlation limit of the spectrometer.³⁹

4.3.

Experimental Results with a High Frame Rate Camera

The mean recovery error for the data acquired with the high frame rate camera is 1.32 MHz (1.66 MHz, root mean square error) for identical scans in succession, which excludes points at the beginning of the scan when the sweeper is increasing power and has low contrast. This is worse than typical results obtained with the experimental setup using the lower-rate camera ($<1\mathrm{MHz}$ mean recovery error). Also, the dynamic range, related to the correlation floor, is degraded. See Figs. 10 and 7 for illustrative recoveries with the fast camera and the previous camera, respectively. The differences are attributed to the reduced number of pixels in the images (66,820 versus 2048, here), lower bit depth of the saved Photron camera images (could be improved), microwave sweeper settling time and instability, and significantly lower camera integration time.

Fig. 10

Recovered frequency offset histogram (a). Correlation plot for a recovered tone at 4500 MHz (b).⁴⁰

4.4.

Planar Waveguide Speckle Correlation as a Function of Frequency

Multimode planar waveguides have been used by many groups to obtain speckle patterns for optical and RF spectrometers,^24,33,35,44 and they offer the opportunity to make far more compact devices than would be possible with a spool of multimode fiber. It may also be simpler to perform the environmental conditioning (temperature, vibration) needed for a high-resolution spectrometer in a small waveguide. Here, we compare calculations of silicon and silicon nitride waveguide speckle with the speckle observed in our multimode fiber. The best basis for this comparison is the correlation as a function of RF for the pixels of a camera or photodiode array, as shown in Fig. 11. We perform the calculations along the lines described by Ashner et al.,⁴⁵ which is essentially an updated version of the standard waveguide solutions from the 1970s. The calculated results assume an optical wavelength of 1550 nm and a $50\text{-}\mu \mathrm{m}$ wide, 40-cm long, single vertical mode planar waveguide in Si or ${\mathrm{Si}}_3{\mathrm{N}}_4$ with indices of 3.4774 and 1.9963, respectively, and clad by a material with refractive index = 1.44. The waveguides are straight, but clearly for applications, one would spiral or bend the waveguides as in Refs. 24, 33, 35, and 45, which is expected to enhance the speckle mixing. Figure 11(a) shows measured results for the multimode fiber, and Figs. 11(b) and 11(c) show simulations for the Si and ${\mathrm{Si}}_3{\mathrm{N}}_4$ planar waveguides, respectively. Note that the three families of correlation functions are very similar. This suggests that for the same number of pixels, we should be able to obtain a similar performance for planar waveguides as we obtained for the single mode fiber.

Fig. 11

Correlation as a function of frequency difference. (a) Experimental results for 100 m long multimode fiber at a wavelength of 780 nm. (b) Calculated results for $50\mu \mathrm{m}$ wide, 40 cm long Si waveguide at 1550 nm. (c) Calculated results for $50\mu \mathrm{m}$ wide, 40 cm long ${\mathrm{Si}}_3{\mathrm{N}}_4$ waveguide at 1550 nm.⁴⁰

5.1.

Differences between Optical and RF Speckle Spectrometers—Electro-Optic Modulator

The RF speckle spectrometer differs from optical speckle-based implementations in several ways because of the use of an electro-optic modulator (EOM). First, the intensity modulator is a double sideband, and thus, the laser spectrum associated with an RF frequency has multiple spectral components, whereas in most optical spectrometers, a single- or shifted-frequency laser is used to form the calibration set. The optical implementations cannot afford a second sideband unless it is also present in the spectrum of the signal of interest. As a result of the sidebands, the observed speckle patterns in the RF spectrometer may have reduced contrast relative to a single sideband speckle pattern, although the effect should be relatively insignificant for narrowband signals.

Not only is the observed speckle pattern composed of some combination of the upper and lower sidebands, but the residual unextinguished or leaked fundamental laser spectral component contributes to the observed speckle pattern as well. To reduce this effect, a background image was acquired before the calibration step and subtracted from each image to reduce the influence of the leakage power on the image and ultimately to the dynamic range of the spectrometer; however, the residual light still adds to the noise. The experiments reported here employed an EOM with a relatively modest 18 dB extinction ratio. An EOM with a better extinction ratio (optical null depth) could be employed to reduce the intensity of the unmodulated laser fundamental, and EOMs with extinction ratios exceeding 50 dB are commercially available.⁴⁶

The stability of the spectrometer is limited by the time-dependent variation of the null bias point of the Mach-Zehnder modulator. This bias point has a strong influence on the intensity and speckle pattern of the leaked light. When closely examining our longer scans, which had durations of up to 20 min in some cases, the dynamic range, as measured by the ratio of the spectral correlation maximum to correlation minimum, was lower at the end of the scan than at the beginning. Bias drift is a well-known effect in ${\mathrm{LiNbO}}_3$ optical modulators.⁴⁷ The voltage required to maintain the optimal bias point tends to drift over time, and this limits the performance of the spectrometer over time. This effect may be mitigated through the use of a modulator bias controller to maintain a high extinction ratio throughout the experiment. This would reduce the time-dependent extinction ratio. The same effect might instead be mitigated through periodic one-null-frame recalibrations. Finally, it is likely possible to minimize the time-dependent optical carrier speckle pattern contribution in post-processing through fitting and subtraction of the slowly varying component.

The RF speckle spectrometer reported here is limited in dynamic range, in part due to the dynamic range of the modulator, as previously discussed, and in part due to the noise of the camera. Even for very short integration or shutter times, there is enough laser power to easily bring the camera near saturation, and thus, the pixel full well depth, read noise, and digitizer bit depth determine the effective dynamic range of the camera. Improvements to any of these aspects improve the performance of the spectrometer. In this light, we tested several cameras throughout our experiments. In fact, our RF speckle spectrometer was first implemented in the optical C-band,^35,38 but the camera performance was found to be a limiting factor, and this prompted the move to a silicon-compatible wavelength.

We previously discussed the optical leakage effect of the EOM but not the RF dynamic range—RF signals that are too weak do not impart a detectable change to the speckle pattern, and RF signals that are too strong can reduce linearity or cause phase wrap when $V_{\pi }$ is exceeded. Increasing the dynamic range of the modulator should help in this regard as well, but other options also exist. This effect might be improved by the addition of an RF limiting amplifier before the EOM or some other form of automated gain control. However, another deleterious effect that limits the dynamic range is the even-order harmonics⁴⁸ of the null-biased modulator. These become significant when the applied RF signal power exceeds $\sim 8\mathrm{dBm}$ and are clearly seen in our data. They are still observable at lower powers but are on the order of the noise in our setup.

5.2.

High-Speed Operation

Using a high-speed camera, we were able to increase the data acquisition rate relative to the initial experiments by a factor of 16,000 while only slightly degrading the recovery error. Although the demonstrated 800 kHz acquisition rate is, to the authors’ knowledge, the highest rate yet demonstrated for a speckle-based spectrometer, the acquisition rate of the spectrometer can be increased to higher rates using other commercially available cameras, capable of maximum frame rates at MHz rates and with active regions composed of a greater number of pixels.⁴⁹ Aside from the greater spectral rate, a more subtle advantage of high-speed operation is that the device might be made more tolerant to environmental and laser frequency drifts over time by rapid reacquisition of the dictionary or tracking of the changes to the speckle pattern over time.⁵⁰ This would likely permit the use of a smaller laser, not locked to an atomic reference, further reducing the size, weight, and power of the instrument. In addition, it might remove some of the strict environmental requirements, especially in combination with the smaller, integrated, multimode waveguides discussed in Sec. 4.4, which would be easier to temperature control and would be less susceptible to air currents. Moreover, a photonically integrated device²⁴ could replace the camera with an integrated high-speed photodiode array to reduce size and improve stability.

The demonstrated increase in data collection speed makes speckle spectrometers competitive with the data rates of real-time spectrum analyzers; however, we were unable to recover spectra at the rate of acquisition. As discussed in Sec. 4.1, reducing the number of pixels used can reduce both recovery and camera frame time; however, this has the effect of increasing the uncertainty in recovery accuracy and lowering the effective dynamic range of the instrument. Improved spectrum recovery algorithms, a real-time operating system, or hardware acceleration might be required for real-time data processing, handling, and storage. A dedicated processor or application-specific integrated circuit would likely help in this regard.

Recently, Murray et al.⁵⁰ demonstrated a high-speed time domain spectrometer based on Rayleigh backscattering that is capable of operating at update rates of 385 kHz. Although the measurement speed is limited by the round-trip time in the fiber in these experiments, it could likely support higher rates with a tradeoff in spectral resolution. The disadvantage of this approach is the need for multiple optical amplifiers, which would increase the size, weight, and power of the spectrometer.

5.3.

Conclusions

Optically multiplexed RF spectrometers offer many attractive features relative to their electronic counterparts, including high-resolution multiline operation over broad and reconfigurable bandwidths. For example, we used the same speckle-based RF spectrometer over the frequency range of 1 to 18 GHz with calibrated ranges spanning 1 to 17 GHz and resolutions commensurate with the calibration frequency step size, typically between 1 to 5 MHz.³⁹ These parameters can likely be improved in future iterations, e.g., the operational band could be shifted to millimeter-wave frequencies by frequency mixing or use of a higher bandwidth EOM.⁵¹ The precision and accuracy of recovery can be increased via improvements to (1) the laser’s frequency stability, (2) the environment of the multimode fiber, and (3) the null bias point of the EOM, in order of importance. The resolution of the spectrometer can likely reach 10 to 100 s of kHz by laser linewidth reduction, improved stabilization, or dictionary shifting via speckle pattern tracking.⁴⁹ The results of the waveguide simulations in Sec. 4.4 suggest that the spectrometer could be considerably miniaturized by photonic integration and replacement of the multimode fiber with a more dispersive material. Real-time data collection speed comparable to that of the best commercially available microwave real-time spectrum analyzers was demonstrated with the high-speed camera implementation, and this could also be enhanced by integrating the spectrometer.

However, the RF speckle-based spectrometer has some disadvantages with respect to the alternatives. Because this is a computational spectrometer, the spectral information is not directly sensed in the time or frequency domains, and thus, our spectrometer is slower, at least in the current implementation. The RF spectrometer has, and will likely continue to have, a lower dynamic range than alternatives, due in part to both the modulator and the camera. With respect to a conventional spectrometer, a disadvantage of a speckle spectrometer is the difficulty in handling broadband signals. A sufficiently broadband or spectrally dense input signal will remove the contrast of the observed speckle pattern, eventually presenting as a homogenous field. Some progress has been made in measuring broadband signals by employing wavelength division multiplexers and multimode bundles to increase the addressable spectral range for dense signals.²⁶