Noninvasive Imaging of Three-Dimensional Micro and Nanostructures by Topological Methods

Abstract

We present topological derivative and energy based procedures for the imaging of micro and nanostructures using one beam of visible light of a single wavelength. Objects with diameters as small as 10 nm can be located, and their position tracked with nanometer precision. Multiple objects distributed either on planes perpendicular to the incidence direction or along axial lines in the incidence direction are distinguishable. More precisely, the shape and size of plane sections perpendicular to the incidence direction can be clearly determined, even for asymmetric and non-convex scatterers. Axial resolution improves as the size of the objects decreases. Initial reconstructions may proceed by glueing together 2D horizontal slices between axial peaks or by locating objects at 3D peaks of topological energies, depending on the effective wavenumber. Below a threshold size, topological derivative based iterative schemes improve initial predictions of the location, size and shape of objects by postprocessing fixed measured data. For larger sizes, tracking the peaks of topological energy fields that average information from additional incident light beams seems to be more effective.

When topological derivatives met regularized Gauss-Newton iterations in holographic 3D imaging

“…Then, the objects are more clearly located tracking the peaks of a companion field [16], the topological energy E T (x, R 3 ) = |E| 2 |P| 2 , where E and P are the forward and adjoint fields governed by (11) and (12). This issue is further discussed in [5,7,26]. Here, we focus on improving the reconstructions of objects whose sizes are similar to or smaller than the light wavelength to avoid this transition.…”

“…Once we have an initial reconstruction of the objects Ω ap we may improve it by a topological derivative based iteration. We construct a new approximation Ω new from Ω ap as follows [5,8]:…”

“…The procedure described in (14) allows for the creation of new components forming the objects (if missed in the previous iteration), for merging existing components (if they are close and intermediate points are indentified), for the destruction of proposed components (if they turn out to be spurious), and for the creation of holes inside them [8]. It does not rely on any object parametrization and can be used to reconstruct non convex and multiple objects [5,6].…”

“…The idea is to optimize the error functionals with respect to arbitrary shapes and arbitrary functions representing the imaged objects and their optical properties. When illuminating simple object configurations with time harmonic and polarized light, the problem somewhat simplifies, since we can use scalar Helmholtz approximations to simulate the polarized component of light [5,6]. Within this simplified framework, Ref.…”

“…[6] succeeds in producing first guesses of objects and of their optical properties from the holograms they generate by combining topological derivative and gradient based optimization procedures, provided the size of the objects is of the same order or smaller than the employed light wavelength. Helmholtz equations for the forward problems are solved in [6] by coupled boundary element (BEM)/finite element (FEM) methods [48,51], whereas shapes are constructed by means of blobby molecule coverings and signed distance functions as in [5], without imposing any specific parametrization.…”