Algorithms and image formation in orbital tomography

Orbital tomography has recently been established as a technique to reconstruct molecular orbitals directly from photoemission data using iterative phase retrieval algorithms. In this work, we present a detailed description of steps for processing of the photoemission data followed by an improved iterative phase retrieval procedure and the interpretation of reconstructed two-dimensional orbital distributions. We address the issue of background subtraction by suggesting a signal restoration routine based on the maximization of mutual information algorithm and solve the problem of finding the geometrical center in the reconstruction by using a tight-centered object support in a two-step phase retrieval procedure. The proposed image processing and improved phase retrieval procedures are used to reconstruct the highest occupied molecular orbital of pentacene on Ag(110), using photoemission data only. The results of the reconstruction agree well with the density functional theory simulation, modified to comply with the experimental conditions. By comparison with photoelectron holography, we show that the reconstructed two-dimensional orbital distribution can be interpreted as a superposition of the in-focus orbital distribution evaluated at the $z=0$ plane and out-of-focus distributions evaluated at other $z=\mathrm{const}$ planes. Three-dimensional molecular orbital distributions could thus be reconstructed directly from two-dimensional photoemission data, provided the axial resolution of the imaging system is high enough.

Figure 1

(a) PAD recorded from a submonolayer of pentacene molecules at 1.2 eV binding energy. Sum of 50 geometrically aligned data sets. (b) Ag(110) substrate background at 1.2 eV binding energy. (c) The same PAD as in (a), but after the subtraction of the background in (b) registered with the PAD in (a) using the intensity interpolation method, in which only a mutual translation was accounted for. (d) The same PAD as in (a), but after the subtraction of the background in (b) registered with the PAD in (a) using the maximization of mutual information algorithm via iterative application of affine transformation to the PAD in (b). (e) Difference between the PAD in (b) and the same PAD, but registered with the PAD in (a) using the maximization of mutual information algorithm via iterative application of affine transformation to the PAD in (b). (f)–(h) The same PADs as in (a), (c), and (d), but normalized by the $|\mathit{A}·{\mathit{k}}_{\mathrm{f}}{|}^2$ factor.

Figure 2

Reconstruction of the HOMO of pentacene on Ag(110). (a) Final PAD used as an input for the phase retrieval algorithm, obtained after subtraction of the mean intensity value from each pixel in the PAD shown in Fig. 1. (b) Amplitude and (c) phase distributions reconstructed with the shrinkwrap algorithm using the uncentered support constraint. (d) Amplitude and (e) phase distributions reconstructed using the centered tight support obtained from the amplitude distribution in (b). The transparency of the phase images is weighted with the corresponding amplitude values for illustration purposes. The crossing dotted lines mark the geometrical centers of the computational domains. Images (b)–(e) are $70\times 70$ pixel sections cut out from $2000\times 2000$ pixel reconstructed images.

Figure 3

(a),(b) Results of the DFT simulation of the pentacene HOMO. (a) $\psi (x,y,z)$: 3D orbital distribution, represented as isosurface at the value of $50\%$ of maximum of $\left|\psi \right(x,y,z\left)\right|$. Inset images on the sides of the cube: Cross sections through $\psi (x,y,z)$, computed at the planes located at $x\approx 2.61\AA ,y\approx 1.54\AA$,  and $z\approx -0.15\AA$. The phase values of the cross sections are weighted with the corresponding amplitude values of $\psi (x,y,z)$ for illustration purposes. (b) $\mathrm{\Psi }(k_x,k_y,k_z)$: Fourier transform of $\psi (x,y,z)$, represented as an isosurface at the value of $50\%$ of the maximum of $\left|\mathrm{\Psi }\right(k_x,k_y,k_z\left)\right|$. (c)–(j) Simulation of the experimental conditions using the results of the DFT simulation shown in (a) and (b). (c) Transfer function $H(k_x,k_y,k_z)$: $H=1$ on a segment of a hemisphere of radius $k_0=3.0\AA {}^{-1}$ within the field of view of the parallel components of the momenta $k_{\parallel ,\max }=\pm 2\AA {}^{-1},H=0$ elsewhere. (d) Inset images on the sides of the cube: Cross sections through the amplitude of the response function $h(x,y,z)={\mathcal{F}}^{-1}\left\{H(k_x,k_y,k_z)\right\}$, computed at the planes located at $x=0\AA ,y=0\AA$,  and $z=0\AA$. (e) $\mathrm{\Psi }(k_x,k_y,k_z)$ multiplied with the corresponding values of the transfer function $H$. (f) Squared modulus of the parallel projection of (e) onto the $(k_x,k_y)$ plane. To be compared with the experimental PAD shown in Fig. 2. (g),(h) Amplitude and phase distributions of the parallel projection of (e) onto the $(k_x,k_y)$ plane. (i) and (j) Amplitude and phase distributions in real space obtained by computing the inverse Fourier transform of (g) and (h). The phase values are weighted with the corresponding amplitude values for illustration purposes. To be compared with the reconstructed orbital distributions shown in Figs. 2 and 2. In (a) and (d), the gray borders of the cubes mark $110\times 110\times 110$ pixel sections cut out from $512\times 512\times 512$ pixel DFT data. In (b), (c), and (e), the gray borders of the cubes mark $75\times 75\times 75$ pixel sections cut out from $512\times 512\times 512$ pixel DFT data.

Figure 4

(a) Experimental geometry. The 40 eV $p$-polarized light was incident on the sample at a grazing angle of $\alpha ={25}^{\circ }$. The azimuthal angle between the plane of incidence and the $\left[1\overline{1}0\right]$ high-symmetry direction of the Ag(110) crystal was $\varphi =5^{\circ }$. The vector potential and wave vector of light are denoted by $\mathit{A}$ and ${\mathit{k}}_{\mathrm{ph}}$, respectively. The photoelectrons were collected by the PEEM objective lens. ${\mathit{k}}_{\mathrm{f},\parallel }$ and ${\mathit{k}}_{\mathrm{f},\perp }$ denote parallel and normal components of the final state wave vector of the photoelectrons. (b) Secondary photoelectron horizon recorded at 7 eV photoelectron kinetic energy. The full width of the horizon, $N_{\text{h}}$, corresponds to $2|{\mathit{k}}_{\mathrm{f},\parallel }^{\max }|\approx 2.7\AA {}^{-1}$ and was equal to $N_{\mathrm{h}}=365\pm 4$ pixels.