High-resolution, Nyquist rate sampling of gigahertz-band RF signals rapidly generates huge amounts of data. Compressive sensing (CS) has been developed to address this general issue for sparse signals and images [1–3]. In CS systems, a sparse input signal $x$ (dimension $N$) is recovered from a measurement vector $y$ (dimension $M$) with $M\ll N$. The vector $y$ is obtained from $x$ after multiplication by a measurement matrix $\mathrm{\Phi }$, as in Eq. (1):

Both electronic [5,6] and microwave photonic CS systems [4,7–16] have been demonstrated that recover sparse RF signals. In the gigahertz band, electronic CS systems suffer from the same sources of error as Nyquist rate analog-to-digital converters (ADCs), timing jitter and amplitude noise. Photonic CS systems have the equivalent of timing jitter and amplitude noise, but in many cases, the distortions are static or lower in frequency and amenable to calibration. Previous photonic systems have used pseudo-random bit sequences (PRBS) for the MM and modulated them on optical carriers with light valves [7–9] or optical modulators [10–16]. Here we demonstrate that the propagation of an optically chirped signal through a multimode optical fiber or planar waveguide performs the function of a CS MM.

The CS system proposed here in Fig. 1 uses a multimode waveguide to replace the two-dimensional spatial light modulator in earlier work [8, Fig. 1] and performs the function of the PRBS in the modulated wideband converter [6, Fig. 3]. Pulses from a femtosecond mode-locked laser (MLL) pass though a dispersion-compensating fiber (DCF) or other dispersive device with dispersion chosen to stretch the pulse to the interpulse time, pass through a Mach–Zehnder modulator (MZM) that impresses the RF signal on the optical intensity, enter a multimode waveguide, and finally are split spatially at the output of the guide and directed to an array of $M$ photodiodes. The integration times of the photodiodes are matched to the MLL period, and the electrical signals from the photodiode array are digitized by an array of ADCs clocked to the MLL pulse repetition frequency (PRF). Optical pulse compression, by placing after the MZM a dispersive element of opposite sign to the DCF, can be used to facilitate signal integration [14]. The components on the left and the right of the multimode waveguide are similar to earlier works [7–9], and again we exploit time-wavelength mapping as depicted by the rainbow-colored pulse icons. At the output of the multimode waveguide are formed speckle patterns that vary with the wavelength (and hence the time via the time-wavelength mapping of the MLL plus DCF combination), and small changes in the wavelength can give completely different patterns after modest propagation distances [17]. For example, Fig. 2 shows speckle patterns at the end of a 1 m-long, 105 μm-core diameter, 0.22 NA step-index fiber observed at 1539.44 and 1539.52 nm. An optional fiber mode scrambler (Newport Corporation model FM-1) is used near the input end of the fiber to fully excite the fiber modes. Without the mode scrambler, $>10\mathrm{m}$ of multimode fiber is required to realize speckle patterns of similar spatial-frequency content. As seen in Fig. 3, wavelength scans at 4 different locations within the output image of the fiber appear random and uncorrelated. Each of these wavelength scans would correspond to a row of the MM $\mathrm{\Phi }$, with the sampling in time set by the time-wavelength mapping property. (In generating Fig. 3, speckle pattern images were recorded while sweeping the wavelength of a single-frequency tunable laser. The combination of the camera frame rate and laser sweep rate provided a wavelength resolution of 0.02 nm.)

 

Fig. 1. Compressive sensing system for measuring sparse RF signals using a multimode waveguide to implement the measurement matrix.

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Fig. 2. Speckle patterns at end of 1 m long, 105 μm diameter, 0.22 NA multimode for $\lambda =1539.44\mathrm{nm}$ (left) and 1539.52 nm (right).

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Fig. 3. Measured intensity as a function of wavelength at 4 locations within the output plane of a 1 m, 105 μm, 0.22 NA step-index fiber.

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Using the grid of red dots in Fig. 2 as locations of the output photodiode array, we derive an MM for the multimode fiber from the speckle pattern images as a function of the wavelength. For $0.4\mathrm{\mu m}\times 0.4\mathrm{\mu m}$ apertures, this yields 112 measurements of the optical intensity as a function of the wavelength, of which 4 are shown in Fig. 3, and an MM with dimensions $112\times 2048$.

We performed several tests to test the measured speckle MM for application to CS. First, we tried to recover several different types of sparse signals: sparse in time (identity transform, $\mathrm{\Psi }=\mathrm{I}$), sparse in frequency (discrete cosine transform), and sparse after the Harr wavelet transform. Figure 4 shows the probabilities that the RF signals are recovered as a function of the small dimension $M$ of the MM. A signal is classified as recovered if all $K$ of its unknown frequencies, pulse locations, or Haar coefficients are recovered and the amplitudes are recovered to better than 1 part in 10,000. For each basis, the sparse vector $s$ consists of $K$ equal-amplitude numbers randomly placed on a 2048-point grid, and each curve is for 100 realizations of the sparse vector $s$. We used a standard LASSO code [18] to recover the vector $x_{\text{rec}}$ and varied the small dimension $M$ of the MM by stripping rows off the $112\times 2048\mathrm{MM}$. The results are consistent with the well-known formula for the minimum dimension of the MM, $M_{\text{min}}\sim K\log (N/K)$ [2,3], and the measured speckle MM works well for sparse signals in all three bases. Note that the pulses with the identity matrix for $\mathrm{\Psi }$ are recovered somewhat better than the sinusoids or Haars. Apparently, the speckle MM is more correlated with the Haar and cosine basis vectors than with the identity basis.

 

Fig. 4. Probability of signal recovery as a function of small dimension of the measurement matrix for 100 trials for $K=2$, 4, 8, and 16 and sparse signals in the identity, discrete cosine, and Haar wavelet transforms.

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A second test involves the coherence between the rows of the MM $\mathrm{\Phi }$, which ideally should be uncorrelated with each other so that each component of $y$ is an independent measurement of the input $x$ [19]. This can be quantified with the normalized mutual coherence $C_{ij}$:

 

Fig. 5. Mutual coherence between rows for speckle MM (left) and Gaussian random-number MM (right).

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A third well-known test for a CS MM is the restricted isometry property (RIP) [1–3]. Unfortunately, proving RIP appears to be computationally intractable. A surrogate to proving RIP suggested by Ref. [20] is performing numerical experiments to determine if the MM produces a recovery “phase transition.” The “phase transition” is seen as a sharp boundary between regions of high probability of recovery and regions of low probability of recovery in a three-dimensional plot of probability as a function of $M$ and $K$. Figure 6 shows the phase transitions for the multimode-fiber speckle MM and the random MM for sparse signals in time, and the difference is minimal. CS MMs must be equally effective against all possible bases in which an input signal may be sparse, and this occurs when the rows of the MM are uncorrelated with the bases of interest. It is not possible to test all possible bases, but we have also performed CS recovery calculations with sparse signals in cosines and Haars with the results shown in Fig. 7. It can be seen that the measured fiber MM yields very similar sharp phase transitions when used with all three bases.

 

Fig. 6. Probability of recovery as a function of sparsity $K$ and number of measurements $M$ for the speckle MM (left) and the Gaussian random-number MM (right).

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Fig. 7. Probability of recovery as a function of sparsity $K$ and the number of measurements $M$ for the measured speckle MM with a signal composed of $K$ sinusoids (left) and a signal that is $K$ sparse under the Haar wavelet transform (right).

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Analytical solutions for the modes of multimode waveguides [21–23] can be used to calculate speckle patterns and MMs. For a cylindrical guide like our fiber (105-μm core, 1-m long, 0.22 NA) with equal power in each mode to approximate the mode-scrambler, rows of the calculated MM look qualitatively the same as for the measured MM in Fig. 3. For planar silicon-on-insulator (SOI) guides with equal power per mode at the entrances of the guides, we similarly calculate output speckle patterns as functions of wavelength. Figure 8(a) shows output 1D speckle patterns for two closely spaced wavelengths for several 10-cm long guides, and Fig. 8(b) shows speckle patterns for a band of wavelengths for several 25.4-μm wide guides. Based on Fig. 8, we select a $25.4\mathrm{\mu m}\times 5\mathrm{cm}$ guide and calculate its MM. MMs for both the planar and cylindrical waveguides are used in CS simulations to recover signals sparse in time. Figure 9 (left) shows phase transition plots for the calculated fiber MM that are nearly the same as those for the measured MM in Fig. 6, and Fig. 9 (right) shows similar results for the calculated planar waveguide MM.

 

Fig. 8. Speckle patterns for planar guides. (a) 10 cm long by width 5 to 30 μm at $\lambda =1.537$ and 1.53701 mm. (b) 25.4 μm wide by 1 mm to 1 m long for 50 wavelengths ($\mathrm{\Delta \lambda }=0.01\mathrm{nm}$).

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Fig. 9. Probability of recovery as a function of sparsity $K$ and the small dimension of the MM $M$ for the calculated fiber MM (left) and the calculated planar waveguide MM (right).

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The practical use of speckle in a multimode waveguide for CS requires that the speckle MM, the MLL pulse, and any other dispersion in the system be stable and amenable to calibration. Previous work [8,9,14,15] has found that commercial mode-locked lasers were sufficiently stable for CS. The stability of speckle from a multimode waveguide depends on control of the temperature and mechanical stresses (e.g., bending of the fiber). In our laboratory measurements with the multimode fiber, basic precautions were taken to ensure stable speckle patterns, namely securing the fiber from perturbations. Figure 10 shows the match between two wavelength calibrations taken more than 1 h apart (using a tunable single-frequency laser) for a single row of an MM obtained with a single photodiode placed in the image plane of the multimode fiber output. Other work using speckle in a similar 1 m, 105 μm, 0.22 NA fiber for spectroscopic applications discusses the stability of the speckle pattern in detail [17], Sections 8 and 9, and states that “for a 1 m long fiber, the temperature would need to change by $\sim 8°\mathrm{C}$ to decorrelate the speckle pattern.”

 

Fig. 10. Two independent calibrations of the multimode fiber separated in time by more than 1 h.

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A potential factor limiting the RF bandwidth of a speckle-based CS system is that the frequency content of the RF signal itself may modify the speckle pattern. At an optical wavelength of 1550 nm, should the speckle pattern vary on a 0.05 nm wavelength scale, the MM will be frequency dependent for RF signals with frequency content above 6.24 GHz. However, the speckle MM can still be used for CS if the system is calibrated by measuring the response for all basis vectors in which the RF signal is sparse. Referring to Eq. (1), the calibration consists of measuring $y_i$ for each basis vector $i$. This $M\times N$ matrix, in which $N$ is now the number of possible basis vectors in the RF signal, can be used as the dictionary to recover the signal using an orthogonal matching pursuit recovery algorithm [9]. The variation of the speckle MM with RF frequency suggests that it may be possible to measure RF signals modulated on a stable single-frequency laser directly from the change in speckle pattern, a subject for future investigation.

To conclude, we show that optical speckle in multimode fibers and planar waveguides satisfies 3 tests for a compressive sensing measurement matrix: (1) CS simulations show the expected recovery as a function of the number of measurements, (2) rows of the speckle MMs show coherence properties similar to an MM formed from Gaussian random numbers, and (3) recovery plotted in the sparsity/measurement plane ($K-M$) shows sharp phase transitions for all measured and calculated speckle MMs and for 3 classes of sparse signals. The next step is to couple an array of photodiodes and ADCs to the output of the multimode guide and demonstrate a full CS system.