As mentioned previously, the passage of the laser beam through the transparent 3D system provides additional locations for determining the direction of the beam: an exit (P2) in addition to an entry (P1) (Fig. 3e). Figure 3f shows the mapping result for a representative case. As indicated in the bar graphs, the photoresponses of devices nearest to P1 are slightly stronger than those at P2, due simply to partial reflections and absorption (the device surface transmits 87% of the incident light). The direction of the incident beam (P1 → P2) follows from the spherical locations of the P1 and P2 vectors with respect to the hemisphere center. The P1 → P2 direction can be defined in terms of the azimuthal angle θ in the x–y plane and the polar angle φ between the x–y plane and z direction (Fig. 3g). This simple example extends naturally to more complex cases involving movement of the laser beam in θ, φ, and both planes. For present purposes, we explore three scenarios: (1) increasing θ from −45.0° to 45.0° while fixing φ at 90.0°, (2) decreasing φ from 90.0° to 67.5° while fixing θ at 0.0°, and (3) increasing θ from 45.0° to 135° and φ from 67.5° to 112.5°. In each, three combinations of θ and φ are tested, as shown in Fig. 4, with the laser beam incident on top of the 3D system. The direction (P1 → P2) can be expressed in spherical coordinates θ and φ. In all combinations in the three scenarios (nine experiments in total), the θ and φ coordinates calculated from the measured photoresponses match the coordinates directly measured with a protractor scale (Supplementary Fig. 17), as summarized in panels (a)–(c) of Fig. 4 along with the spatial mapping of the photoresponse. Corresponding bar graphs are in Supplementary Fig. 18. Note that although the system can map 360° rotations of θ in the x−y plane, the φ movement is limited by the array geometry, which imposes lower and upper circumferences on the hemispherical surface; see Supplementary Fig. 19. Simple modifications allow up to 45° movement of φ (Supplementary Fig. 20). All measurements and analysis can be performed in real time, as shown in the video clip of the supporting materials (Supplementary Movie 3). Further, the degree of accuracy in determining the position and direction improves with decreases in the distances between photoresistors and in the laser spot sizes (Supplementary Figs. 21 and 22). Moreover, the representative 3D photodetector showed good imaging capability in kHz range (Supplementary Fig. 23).Fig. 4Sensing characteristics of a 3D photodetection and imaging system. a Maps of photocurrent measured across the 3D surfaces. The incident angle θ ranges from −45° to 45° with φ fixed at 90°. The measured θ and φ from the device arrays are shown for comparison. b Similar results with φ ranging from 90° to 67.5° and θ = 0°. c Photocurrent maps of the photodetector array on the 3D surfaces randomly penetrated at two points by the incident laser. The incident angles are (from left to right) θ = 45°, φ = 67.5°, θ = 90°, φ = 90°, and θ = 135°, φ = 112.5°