Three-Dimensional Structure from Single Two-Dimensional Diffraction Intensity Measurement

Conventional three-dimensional (3D) imaging methods require multiple measurements of the sample in different orientation or scanning. When the sample is probed with coherent waves, a single two-dimensional (2D) intensity measurement is sufficient as it contains all the information of the 3D sample distribution. We show a method that allows reconstruction of 3D sample distribution from a single 2D intensity measurement, at the $z$ resolution exceeding the classical limit. The method can be practical for radiation-sensitive materials, or where the experimental setup allows only one intensity measurement.

Figure 1

Illustration of the wave front propagation calculation and the used symbols.

Figure 2

Simulated examples of 3D reconstructions from a single 2D intensity measurement. Three-dimensional samples consist of individual objects located in four different planes. (a) Hologram of a sample consisting of opaque objects in the form of letters. Reconstruction obtained by (b) conventional wave front backward propagation, and (c) the MIPR. (d) Hologram of a sample consisting of phase spheres. Phase distributions reconstructed by (e) conventional wave front propagation and (f) the MIPR. The phase range in (e) is from $-0.43$ to 0.49 rad, and (f) from 0 to 0.7 rad. The plots in (f) show phase profiles of the spheres: original distributions (red), reconstructed by conventional reconstruction (gray) and by the MIPR (blue). (g) Diffraction pattern of 3D sample made of opaque objects in the form of letters (shown in logarithmic scale) and (h) its 3D reconstructions. (i) Diffraction pattern of a 3D sample consisting of phase spheres (shown in logarithmic scale) and (j) the reconstructed phase distributions. All distributions are sampled with $400\times 400\mathrm{pixels}$, and only central regions of $200\times 200\mathrm{pixels}$ are shown. The diffraction patterns in (g) and (i) are shown in full.

Figure 3

Simulated 2D electron diffraction pattern of bilayer twisted graphene shown in inverted logarithmic scale (a). (b) Modeled twisted bilayer graphene distribution with defects in one layer. (c),(d) The phase distributions of the reconstructed transmission functions of the individual layers, values range from 0 to 0.24 rad. Scale bar in (b)–(d) is 1 nm.

Figure 4

Experimental results. (a) Experimental optical setup. (b) Hologram. Scale bar is $50\mathrm{\mu }\mathrm{m}$. (c)–(f) Reconstructions obtained by conventional wave front propagation. (c),(d) Amplitude and (e),(f) phase distributions at two different $z$ distances. Amplitude ranges from 0.116 to 1.190 a.u., and phase ranges from $-0.561$ to 1.204 rad. (g)–(j) Reconstructions obtained by the MIPR. (g),(h) Amplitude and (i),(j) phase distributions (inverted contrast) at the two $z$ distances. Amplitude ranges from 0.119 to 1.000 a.u., and phase ranges from 0.000 to 1.371 rad. (c)–(j) Reconstructions of the region marked by the yellow rectangle in (b) are shown. Scale bar is $20\mathrm{\mu }\mathrm{m}$.