Frequency-modulated continuous-wave LiDAR compressive depth-mapping

We present an inexpensive architecture for converting a frequency-modulated continuous-wave LiDAR system into a compressive-sensing based depth-mapping camera. Instead of raster scanning to obtain depth-maps, compressive sensing is used to significantly reduce the number of measurements. Ideally, our approach requires two difference detectors. % but can operate with only one at the cost of doubling the number of measurments. Due to the large flux entering the detectors, the signal amplification from heterodyne detection, and the effects of background subtraction from compressive sensing, the system can obtain higher signal-to-noise ratios over detector-array based schemes while scanning a scene faster than is possible through raster-scanning. %Moreover, we show how a single total-variation minimization and two fast least-squares minimizations, instead of a single complex nonlinear minimization, can efficiently recover high-resolution depth-maps with minimal computational overhead. Moreover, by efficiently storing only $2m$ data points from $m<n$ measurements of an $n$ pixel scene, we can easily extract depths by solving only two linear equations with efficient convex-optimization methods

Compressive Wavefront Sensing with Weak Values

We demonstrate a wavefront sensor based on the compressive sensing, single-pixel camera. Using a high-resolution spatial light modulator (SLM) as a variable waveplate, we weakly couple an optical field's transverse-position and polarization degrees of freedom. By placing random, binary patterns on the SLM, polarization serves as a meter for directly measuring random projections of the real and imaginary components of the wavefront. Compressive sensing techniques can then recover the wavefront. We acquire high quality, 256x256 pixel images of the wavefront from only 10,000 projections. Photon-counting detectors give sub-picowatt sensitivity

Fast Hadamard transforms for compressive sensing of joint systems: measurement of a 3.2 million-dimensional bi-photon probability distribution

We demonstrate how to efficiently implement extremely high-dimensional compressive imaging of a bi-photon probability distribution. Our method uses fast-Hadamard-transform Kronecker-based compressive sensing to acquire the joint space distribution. We list, in detail, the operations necessary to enable fast-transform-based matrix-vector operations in the joint space to reconstruct a 16.8 million-dimensional image in less than 10 minutes. Within a subspace of that image exists a 3.2 million-dimensional bi-photon probability distribution. In addition, we demonstrate how the marginal distributions can aid in the accuracy of joint space distribution reconstructions

Compressive Direct Imaging of a Billion-Dimensional Optical Phase-Space

Optical phase-spaces represent fields of any spatial coherence, and are typically measured through phase-retrieval methods involving a computational inversion, interference, or a resolution-limiting lenslet array. Recently, a weak-values technique demonstrated that a beam's Dirac phase-space is proportional to the measurable complex weak-value, regardless of coherence. These direct measurements require scanning through all possible position-polarization couplings, limiting their dimensionality to less than 100,000. We circumvent these limitations using compressive sensing, a numerical protocol that allows us to undersample, yet efficiently measure high-dimensional phase-spaces. We also propose an improved technique that allows us to directly measure phase-spaces with high spatial resolution and scalable frequency resolution. With this method, we are able to easily measure a 1.07-billion-dimensional phase-space. The distributions are numerically propagated to an object placed in the beam path, with excellent agreement. This protocol has broad implications in signal processing and imaging, including recovery of Fourier amplitudes in any dimension with linear algorithmic solutions and ultra-high dimensional phase-space imaging.Comment: 7 pages, 5 figures. Added new larger dataset and fixed typo

Position-Momentum Bell-Nonlocality with Entangled Photon Pairs

Witnessing continuous-variable Bell nonlocality is a challenging endeavor, but Bell himself showed how one might demonstrate this nonlocality. Though Bell nearly showed a violation using the CHSH inequality with sign-binned position-momentum statistics of entangled pairs of particles measured at different times, his demonstration is subject to approximations not realizable in a laboratory setting. Moreover, he doesn't give a quantitative estimation of the maximum achievable violation for the wavefunction he considers. In this article, we show how his strategy can be reimagined using the transverse positions and momenta of entangled photon pairs measured at different propagation distances, and we find that the maximum achievable violation for the state he considers is actually very small relative to the upper limit of $2\sqrt{2}$. Although Bell's wavefunction does not produce a large violation of the CHSH inequality, other states may yet do so.Comment: 6 pages, 3 figure

Photon counting compressive depth mapping

We demonstrate a compressed sensing, photon counting lidar system based on the single-pixel camera. Our technique recovers both depth and intensity maps from a single under-sampled set of incoherent, linear projections of a scene of interest at ultra-low light levels around 0.5 picowatts. Only two-dimensional reconstructions are required to image a three-dimensional scene. We demonstrate intensity imaging and depth mapping at 256 x 256 pixel transverse resolution with acquisition times as short as 3 seconds. We also show novelty filtering, reconstructing only the difference between two instances of a scene. Finally, we acquire 32 x 32 pixel real-time video for three-dimensional object tracking at 14 frames-per-second.Comment: 16 pages, 8 figure

Compressively characterizing high-dimensional entangled states with complementary, random filtering

The resources needed to conventionally characterize a quantum system are overwhelmingly large for high- dimensional systems. This obstacle may be overcome by abandoning traditional cornerstones of quantum measurement, such as general quantum states, strong projective measurement, and assumption-free characterization. Following this reasoning, we demonstrate an efficient technique for characterizing high-dimensional, spatial entanglement with one set of measurements. We recover sharp distributions with local, random filtering of the same ensemble in momentum followed by position---something the uncertainty principle forbids for projective measurements. Exploiting the expectation that entangled signals are highly correlated, we use fewer than 5,000 measurements to characterize a 65, 536-dimensional state. Finally, we use entropic inequalities to witness entanglement without a density matrix. Our method represents the sea change unfolding in quantum measurement where methods influenced by the information theory and signal-processing communities replace unscalable, brute-force techniques---a progression previously followed by classical sensing.Comment: 13 pages, 7 figure

Microprocessor-based digital correlator

We describe the design, construction, and operation of a low-cost, microprocessor-based digital correlator. The device has 128 channels, operates in either the single clipping or single scaling mode, and allows selection of the sample interval with 2-digit precision over the range 100 ns to 9.9 s. The device can be operated in the autocorrelate or cross-correlate mode and may easily be expanded to more than 128 correlation channels