General framework for quantitative three-dimensional reconstruction from arbitrary detection geometries in TEM

In Phys. Rev. Lett. 109, 245502 (2012), a method for retrieving the object's three-dimensional potential distribution by inverting the dynamical scattering was presented and validated by a reconstruction from simulated atomic resolution transmission electron microscopy (TEM) images. In this paper, an extension to ptychography and scanning confocal electron microscopy is demonstrated and validated with simulations. Ultimately, this will make it possible to operate the microscope in the mode that yields the best reconstruction instead of accommodating the microscope settings to the linear approximation to the specimen-electron interaction used in most reconstruction algorithms. Furthermore, simultaneous estimation of the object and the unknown defocus from simulated atomic resolution TEM images is attained. This is an important step towards experimental reconstructions.

Figure 1

Simplified ray diagram of the HRTEM geometry. The projector lens transforms the spherical wave emanating from the electron gun to a plane wave that illuminates the specimen. The objective lens produces an image on the CCD.

Figure 2

The ANN and the BP algorithm. (a) The ANN that describes the image formation in HRTEM. The incoming wave ${\psi }_1$ is multiplied with the transmission function $t_1$. The next layer of edges and nodes encodes a real-space convolution with the Fresnel propagator $p$, resulting in ${\psi }_2$. This is repeated until the exit wave ${\psi }_{N+1}$ is produced. Then, a real space convolution with $\mathrm{LF}$ produces ${\psi }_{N+2}$, and the intensity $I$ is computed. The last layers compute $E$ from the measurements $J$. (b) The BP algorithm. Each node contains the derivative with respect to the node input. $E$ is replaced with 1 and the network is run backwards with the same weights, incoming information to each node is added and multiplied with the stored derivatives, resulting in the derivative of $E$ with respect to that node's input.

Figure 3

Error as a function of defocus. The correct defocus resides in a valley of about $6.5$ nm wide.

Figure 4

Simplified ray diagrams. (a) Ptychography. The objective lens illuminates the specimen with a defocused electron probe and the post-specimen projector lens images a diffraction pattern on the CCD. (b) Conventional SCEM. The first objective lens illuminates the specimen with a condensed electron probe and the second objective lens images the beam crossover. The pinhole detector rejects most of the intensity from above and below the crossover, thus creating depth sensitivity.

Figure 5

Five typical simulated HRTEM images. From left to right, the $\alpha$ tilt is $-{10}^{\circ }$, $-5^{\circ }$, $0^{\circ }$, $+5^{\circ }$, and $+{10}^{\circ }$ and the $\beta$ tilt equals $0^{\circ }$.

Figure 6

The defoci at the start of the reconstruction process (circles) and at the end (squares).

Figure 7

Reconstruction error with (full line) and without (dashed line) defocus correction. After 4096 iterations, only little improvement is to be expected.

Figure 8

Reconstruction with defocus correction. Upper two rows: Slices with the original potential $V$. Middle two rows: Reconstructed potential $V$. Lower two rows: Reconstructed absorptive potential $W$.

Figure 9

Reconstruction without defocus correction, compare to the upper two rows of Fig. 8.

Figure 10

Five typical diffraction patterns for ptychography, displayed on a logarithmic gray scale. From left to right, the $\alpha$ tilt is $-{10}^{\circ }$, $-5^{\circ }$, $0^{\circ }$, $+5^{\circ }$, and $+{10}^{\circ }$ and the $\beta$ tilt is $0^{\circ }$. Note the noise in the higher frequencies.

Figure 11

The error of the ptychographic reconstruction as a function of iteration number. After 1024 iterations, the error has leveled off.

Figure 12

Ptychographic reconstruction. Upper two rows: Slices with the original potential $V$. Middle two rows: Reconstructed potential $V$. Lower two rows: Reconstructed absorptive potential $W$.

Figure 13

Five typical SCEM measurements, displayed on a linear gray scale. From left to right, the $\beta$ tilt equals $-{10}^{\circ }$, $-5^{\circ }$, $0^{\circ }$, $+5^{\circ }$, and $+{10}^{\circ }$ and the $\alpha$ tilt equals $0^{\circ }$.

Figure 14

The error of the SCEM reconstruction as a function of iteration number. After 1024 iterations, the error has leveled off.

Figure 15

SCEM reconstruction. Upper two rows: Slices with the original potential $V$. Middle two rows: Reconstructed potential $V$. Lower two rows: Reconstructed absorptive potential $W$.

Figure 16

Detail of the ANN, the weights $u_{k\ell }$ encode a convolution of the object exit wave with the lens function $\mathrm{LF}$. The elements of the upper layer are indexed with $k$, those of the lower layer with $\ell$. The weights $u_{k\ell }$ equal ${\mathrm{LF}}_{k-\ell }$.