Phase retrieval via propagation-based interferometry

We present an extension of the Gerchberg-Saxton algorithm to allow for phase retrieval from interferometric data recorded at multiple sample-detector distances. A system of coupled waves is introduced where the information exchange can be controlled by use of relaxation parameters. Optimal parameters are investigated by numerical simulation and demonstrated to work with experimental data. We demonstrate that systematic errors such as pointing instabilities of the interfering waves involved and position uncertainties of the detector can be overcome by dimensional extension of the search space. Further it is shown that the proposed approach offers superior reconstruction quality as compared to conventional Gerchberg-Saxton type algorithms. We expect that the method described here will open up numerous possibilities for the correction of systematic errors in optical phase retrieval and lensless microscopy.

Figure 1

The experimental setup (top) consists of a spatial filter (L-P-L) generating a collimated beam that traverses a Mach-Zehnder interferometer. The mirrors (M) of the Mach-Zehnder interferometer are aligned to produce beams with different pointing vectors that are used to produce interferometric data (bottom: (a) $|{\psi }_{1,z}{|}^2$, (b) $|{\psi }_{2,z}{|}^2$, (c) $|{\psi }_{1,z}+{\psi }_{2,z}{|}^2$) at multiple object (O)–detector (D) distances ($z$).

Figure 2

Reconstruction results for simulated test data. Hue and brightness encode phase and magnitude, respectively, of the reconstructed wave field. The bottom of each subfigure indicates ($\alpha \left|\beta \right|\gamma |m$), where $m$ indicates the number of basis functions used for reconstruction. Subfigure (a) shows one of the two original exit-surface waves. The coupled iterative scheme proposed, shown in panels (c)–(e), $\alpha =\beta =\gamma <1$, performs better than the uncoupled Gerchberg-Saxton type iteration shown in panel (b), $\alpha =\beta =1,\gamma =0.5$. Comparing panels (c) and (f) indicates that increasing the number of basis functions can help to compensate systematic position errors.

Figure 3

Comparison of error metric (${\chi }^2$) for various parameters vs iteration ($n$) for simulated data in the presence of systematic position errors and Poisson noise. Various coupling parameters and number of basis functions ($\alpha \left|\beta \right|\gamma |m$) where tested for 2 (solid) and $\ge 3$ (dashed) basis functions. Best results for $m=2$ are obtained for $\alpha =\beta =\gamma =0.4$. For $m\ge 3$ the reconstruction quality is less sensitive to the choice of coupling paramters.

Figure 4

Reconstruction of a stained mouse brain from 60 diffraction patterns and interferograms ($z=40,...,135$ mm). Hue represents phase shift, brightness relative modulus of the exit waves. Panels (a) and (b) show the constituting ESWs, and panel (c) is the interferometric ESW. The second ESW (b) exhibits a phase ramp that is robustly reconstructed due to the incorporation of the interferometric constraint. Panel (d) shows a magnified view of the white, dashed region in panel (a). (e) Optical micrograph of the same region as panel (d).