Superluminal Motion-Assisted Four-Dimensional Light-in-Flight Imaging

Advances in high-speed imaging techniques have opened new possibilities for capturing ultrafast phenomena such as light propagation in air or through media. Capturing light in flight in three-dimensional $xyt$ space has been reported based on various types of imaging systems, whereas reconstruction of light-in-flight information in the fourth dimension $z$ has been a challenge. We demonstrate the four-dimensional light-in-flight imaging based on the observation of a superluminal motion captured by a new time-gated megapixel single-photon avalanche diode camera. A high-resolution light-in-flight video is generated without laser scanning, camera translation, interpolation, or dark noise subtraction. An unsupervised machine-learning technique is applied to analyze the measured spatiotemporal data set. A theoretical formula is introduced to perform least-square regression for numerically solving a nonlinear inverse problem, and extra-dimensional information is recovered without prior knowledge. The algorithm relies on the mathematical formulation equivalent to the superluminal motion in astrophysics, which is scaled by a factor of a quadrillionth. The reconstructed light-in-flight trajectory shows good agreement with the actual geometry of the light path. Applicability of the reconstruction approach to more complex scenes with multiple overlapped light trajectories is verified based on a data set generated by Monte Carlo simulations. Our approach could potentially provide novel functionalities to high-speed imaging applications such as non-line-of-sight imaging and time-resolved optical tomography.

Popular Summary

Recent progress in high-speed imaging techniques enables researchers to optically observe phenomena that occur in mere picoseconds or nanoseconds. However, conventional camera systems with picosecond time resolution have had relatively low spatial resolution, which severely limits the image quality. Using a newly developed high-speed megapixel single-photon camera, we successfully capture the propagation of a laser pulse reflected by multiple mirrors in unprecedented spatiotemporal resolution.

The recorded light-in-flight video exhibits a counterintuitive behavior of light propagation: The velocity of light appears to exceed the speed of light in vacuum when the light propagates toward the camera. We discover that this pseudorelativistic phenomenon can be interpreted based on an astrophysical phenomenon called “superluminal motion.”

In-depth modeling and machine-learning-based analysis of this exotic phenomenon enables us to reconstruct the complete 4D dynamics of the laser propagation based on the measured 3D data set. The recorded apparent speed of light is as high as 3.57 times the speed of light in vacuum. The result indicates that our high-speed camera enables us to reproduce the astrophysical phenomenon in the laboratory, scaled by a factor of a quadrillionth.

Our methods can potentially be extended for new scientific observations based on optics and photonics while paving the way to new applications in industry. Applications of interest include unconventional imaging techniques called “imaging around the corner” and “imaging behind objects” for automotive safety and security, and “imaging inside objects” for next-generation optical-computed tomography.

Figure 1

Principle and experimental setup for four-dimensional light-in-flight imaging. (a) Schematic illustration of light propagation and scattering in Minkowski space at $y=0$. Propagating light is depicted as a purple dashed arrow, and scattered light is shown as a light cone. (b) Schematic plot of $t^{\prime }$ dependence of $x$. The slope of the dashed line indicates the apparent velocity. (c) Experimental setup for light-in-flight imaging. The picosecond laser and SPAD camera are both controlled by a pulse generator. When performing the light-in-flight measurements, all the mirrors are confined by a large transparent acrylic box (not shown), and a fog generator forms a small amount of mist in the box to enhance the scattering of the laser in air. The origin of the Cartesian coordinate is set at the optical center of the lens of the SPAD camera.

Figure 2

Video generation of three-dimensional light-in-flight imaging. (a) Data processing flow for generation of light-in-flight video. (b) Laser pulse propagating over time, shown along the dashed line in the top-left image. Superluminal motion of the laser pulse is observed in the latter frames. The light-in-flight video is taken under dark conditions, and each frame is superimposed with a 12-bit two-dimensional intensity image independently taken with the same SPAD camera under room light. The image resolution is $1024\times 500\mathrm{pixels}$.

Figure 3

Analysis of extracted spatiotemporal data set. (a) Color-coded plot of observation time for the light path. Red segments are observed earlier, and blue segments are observed later. Dark regions have no timing data. (b) Light path data subdivided into five data clusters of straight paths using 2D GMM fitting. (c) Schematic view of the coordinate system for light-in-flight reconstruction. The laser beam image is projected onto the focal plane by an objective lens in a flipped geometry. (d) Observation time $t^{\prime }$ as a function of $x_p$, the $x$ position on the focal plane. Each data cluster is individually fitted using a theoretical formula. Dashed lines are the fitting curves. (e) $t^{\prime }$ as a function of $y_p$.

Figure 4

Reconstructed four-dimensional light-in-flight observation. (a) Four-dimensional point cloud reproduced from the measured three-dimensional spatiotemporal data. The origin of the Cartesian coordinate system is defined at the optical center of the lens for the SPAD camera. The ($x$, $y$, $z$, $t$) coordinates of each data point are reproduced using the obtained fitting parameters for the corresponding data cluster. (b,c) Comparison of fitted light propagation angles $\theta$ and $\varphi$, respectively, with the actual angles for five data clusters. The fitted results show good agreement with the actual angles.

Figure 5

Reconstructed apparent transverse velocity distribution in $xyz$ space. The color of each dot corresponds to the locally estimated apparent transverse velocity, normalized by the speed of light in vacuum.

Figure 6

Virtual setup and data set generated by Monte Carlo simulation of four-dimensional light-in-flight imaging for two separate light trajectories. (a) Schematic illustration for top view of the virtual setup. (b) Schematic illustration for side view of the virtual setup. (c) Time stamps for primary events detected at each pixel, generated by Monte Carlo simulation. The black region indicates “no data.” (d) Generated time stamps for secondary events.

Figure 7

Analysis of generated spatiotemporal data set. (a) Data points classified as the first data cluster based on 3D GMM. (b) Data points classified as the second data cluster. (c) Top view of reconstructed four-dimensional point cloud. (d) Side view of reconstructed four-dimensional point cloud.