Iterative reconstruction of a refractive-index profile from x-ray or neutron reflectivity measurements

Analysis of x-ray and neutron reflectivity is usually performed by modeling the density profile of the sample and performing a least square fit to the measured (phaseless) reflectivity data. Here we address the uniqueness of the reflectivity problem as well as its numerical reconstruction. In particular, we derive conditions for uniqueness, which are applicable in the kinematic limit (Born approximation), and for the most relevant case of box model profiles with Gaussian roughness. At the same time we present an iterative method to reconstruct the profile based on regularization methods. The method is successfully implemented and tested both on simulated and real experimental data.

Figure 1

Inner CGNE iteration in the Newton-CG algorithm.

Figure 2

(Color online) Results from a reconstruction of simulated data assuming a synchrotron radiation beam intensity ($I_0=7\times {10}^9$ photons). (Top left) Several electron density profiles are shown: The original model used for data simulation (dashed line, blue), resembling a phospholipid bilayer on a solid silicon support, furthermore the initial guess for the reconstruction (dashed-dotted line, green). While for one reconstructed profile (solid line, red) the algorithm was stopped according to the discrepancy principle after 9 iterations, for the other reconstruction (dotted line, black) it was continued for illustrative reasons up to iteration 20. Obviously, overfitting leads to unphysical oscillations in the profile here. The shaded area marks the interval $\left[\alpha ,\beta \right]$, in which the initial profile was assumed to be a correct representation of the true one. (Top right) Reflectivity curves corresponding to profiles on the left, normalized by Fresnel reflectivity, i.e., the simulated data points with added noise (circles with error bars, black), the simulated reflectivity based on the same model, but without added noise (dashed line, blue) and the reconstruction after 9 iterations, obtained according to the discrepancy principle (solid line, red). (Center right) Residuals of the reconstructed reflectivity, normalized to the expectation value of the simulated errors. The majority of points lies within the error bars. (Bottom left) Misfit ${\xi }^{\left(k\right)}/{\xi }^{\left(1\right)}$ of the real space profile at iteration $k$ with the original model, normalized for clarity to the misfit of the initial profile. Here ${\xi }^{\left(k\right)}$ is defined as ${\xi }^{\left(k\right)}={\Vert {\varphi }^{\left(k\right)}-{\varphi }_{\text{model}}\Vert }_2$, where ${\varphi }_{\text{model}}$ denotes the original model. (Bottom right) Data misfit ${\Vert F\left({\varphi }^{\left(k\right)}\right)-y^{\delta }\Vert }_Y^2$. The misfit after iteration 9 is about 2.6, representing a good fit to the data.

Figure 3

(Top) Maximum gradient change ${\left(\delta \rho \right)}_0$ in initial profile versus misfit of data and reconstructed reflectivities obtained after stopping of the algorithm by the discrepancy principle. The same simulated data as presented in Fig. 2 were used. The data point corresponding to the true gradient in the initial profile is marked by an arrow and corresponds to the fit result presented in Fig. 2. (Bottom) Gradient change ${\left(\delta \rho \right)}_0$ in initial profile versus misfit of the reconstructed real space profiles corresponding to the reflectivity curves evaluated in the top graph. For comparison, the points are normalized to the misfit ${\xi }_{\text{original}}$ resulting from reconstruction with the true gradient ${\left(\delta \rho \right)}_0$ in the initial profile.

Figure 4

(Color online) Results from a reconstruction of simulated data assuming a laboratory source beam intensity ($I_0=7\times {10}^6$ photons). For nomenclature refer to Fig. 2.

Figure 5

(Color online) Comparison of the regularization method with a reconstruction scheme based on a genetic algorithm (MOTOFIT). (Top left) Several density profiles are shown, more importantly (i) the reconstructed density profile resulting from the regularization scheme with abort due to the discrepancy principle (solid line, red), and (ii) reconstruction by application of a genetic algorithm in MOTOFIT (solid line, cyan). In addition, the profile used as the start profile for both reconstruction schemes is shown (“initial guess,” dashed-dotted line, green). Finally, the profile used for simulation of the data with added noise, i.e., the “true” solution is shown (“original model,” dashed line, blue). (Top right) Reflectivity curves—not normalized to Fresnel reflectivity here—corresponding to profiles on the left, i.e., simulated data with added noise (“experimental data”), reconstructed reflectivity by regularization (solid line, red) and by application of the genetic algorithm (solid line, cyan). (Bottom left) Data misfit versus iteration number for the reconstruction by regularization. (Bottom right) Residuals of reconstructed reflectivity curve by regularization with respect to the theoretical reflectivity curve without added noise.

Figure 6

(Color online) (Top left) reconstructed profile for a DPPC monolayer on OTS-silicon support (solid line). For comparison, a box-model fit to the same data is shown (dashed-dotted line), as well as the initial guess for the reconstruction (dotted line). (Top right) experimental data with errors and reflectivity simulations corresponding to the profiles shown on the left. (Bottom left) Misfit ${\Vert F\left({\varphi }^{\left(k\right)}\right)-y^{\delta }\Vert }_Y^2$. (Bottom right) Residuals for the reconstructed profile.