HRTEM Image Formation¶

The HRTEM image is formed from the exit-surface wavefunction — the transmission coefficients \(T_{\mathbf g}\) obtained from the dynamical core — by passing it through the objective lens. ReciPro offers two models: the fast quasi-coherent approximation and the more rigorous transmission cross coefficient (TCC) model. See also the HRTEM simulator GUI page.

Symbols¶

Symbol	Meaning
\(\mathbf R\)	X–Y component in real space (image plane)
\(\mathbf K\)	X–Y component of the incident wavevector
\(\mathbf G, \mathbf H\)	X–Y components of reciprocal-lattice vectors
\(\mathbf u\)	spatial frequency (e.g. \(\mathbf K+\mathbf G\))
\(\chi(\mathbf u)\)	lens aberration function
\(A(\mathbf u)\)	objective-aperture function
\(\Delta f\)	defocus value
\(C_s\)	spherical aberration coefficient
\(C_c\)	chromatic aberration coefficient
\(\beta\)	illumination semi-angle (finite source size)
\(\Delta E\)	\(1/e\) width of the electron energy fluctuations
\(\Delta_0\)	\(1/e\) width of the defocus spread (Gaussian), \(\Delta_0 = C_c\,\Delta E / E\)

Lens aberration function and aperture¶

\[\chi(\mathbf u) = \pi\lambda\Delta f\, u^2 + \tfrac{1}{2}\pi\lambda^3 C_s\, u^4 = \pi\lambda u^2\!\left(\Delta f + \tfrac{1}{2}\lambda^2 C_s u^2\right)\]

\[A(\mathbf u) = \begin{cases} 1 & (\mathbf u\ \text{inside the objective aperture})\\[2pt] 0 & (\mathbf u\ \text{outside the objective aperture})\end{cases}\]

Quasi-coherent model¶

A fast approximation: each diffracted beam is modulated by the lens transfer and damped by coherence envelopes, then summed coherently.

\[I(\mathbf R) = |\psi(\mathbf R)|^2\]

\[\psi(\mathbf R) = \sum_{\mathbf g} T_{\mathbf g}\,\exp\!\left[2\pi i(\mathbf K+\mathbf G)\cdot\mathbf R\right]\exp\!\left[-i\chi(\mathbf K+\mathbf G)\right]A(\mathbf K+\mathbf G)\,E_c(\mathbf K+\mathbf G)\,E_s(\mathbf K+\mathbf G)\]

with the temporal and spatial coherence envelopes

\[E_c(\mathbf u) = \exp\!\left[-\tfrac{1}{2}\left(\pi\lambda\Delta_0\, u^2\right)^2\right], \qquad E_s(\mathbf u) = \exp\!\left[-\pi^2\beta^2 u^2\!\left(\Delta f + \lambda^2 C_s u^2\right)^2\right]\]

Transmission cross coefficient (TCC) model¶

The rigorous treatment of partial coherence: every pair of beams \((\mathbf g, \mathbf h)\) interferes through the transmission cross coefficient.

\[I(\mathbf R) = \sum_{\mathbf g}\sum_{\mathbf h} T_{\mathbf g}\,T_{\mathbf h}^{*}\,\exp\!\left[2\pi i(\mathbf G-\mathbf H)\cdot\mathbf R\right]\mathrm{TCC}(\mathbf K+\mathbf G,\ \mathbf K+\mathbf H)\]

\[\mathrm{TCC}(\mathbf u, \mathbf u') = A(\mathbf u)\,A(\mathbf u')\,\exp\!\left[-i\{\chi(\mathbf u)-\chi(\mathbf u')\}\right]E_c(\mathbf u, \mathbf u')\,E_s(\mathbf u, \mathbf u')\]

with the mixed coherence envelopes

\[E_c(\mathbf u, \mathbf u') = \exp\!\left[-\tfrac{1}{2}\left(\pi\lambda\Delta_0\right)^2\!\left(u^2 - u'^2\right)^2\right]\]

\[E_s(\mathbf u, \mathbf u') = \exp\!\left[-\pi^2\beta^2\left\{\Delta f(\mathbf u-\mathbf u') + \lambda^2 C_s\!\left(u^2\mathbf u - u'^2\mathbf u'\right)\right\}^2\right]\]

In the limit \(\mathbf u' \to \mathbf u\) the TCC reduces to the quasi-coherent envelopes above.

Reducing the cost of the TCC model¶

The double sum of the TCC model evaluates \(\mathrm{TCC}\) once per pair of beams, so it is expensive. Because the image intensity \(I(\mathbf R)\) is real, the cost can be roughly halved.

First, beams outside the objective aperture (\(A(\mathbf K+\mathbf G)=0\)) do not contribute, so it is sufficient to sum only over the beams inside the aperture (\(A=1\)).

Next, the TCC is Hermitian,

\[\mathrm{TCC}(\mathbf u', \mathbf u) = \mathrm{TCC}(\mathbf u, \mathbf u')^{*}\]

(\(A\) is real; \(E_c, E_s\) are real functions invariant under \(\mathbf u\leftrightarrow\mathbf u'\); the phase term \(\exp[-i\{\chi(\mathbf u)-\chi(\mathbf u')\}]\) is complex-conjugated). Together with \(\exp[2\pi i(\mathbf H-\mathbf G)\cdot\mathbf R]=\bigl(\exp[2\pi i(\mathbf G-\mathbf H)\cdot\mathbf R]\bigr)^{*}\) and \(T_{\mathbf h}T_{\mathbf g}^{*}=\bigl(T_{\mathbf g}T_{\mathbf h}^{*}\bigr)^{*}\), the \((\mathbf g,\mathbf h)\) and \((\mathbf h,\mathbf g)\) terms are complex conjugates of each other, so their sum equals twice the real part:

\[F(\mathbf g,\mathbf h) + F(\mathbf h,\mathbf g) = 2\,\mathrm{Re}\{F(\mathbf g,\mathbf h)\}, \qquad F(\mathbf g,\mathbf h) \equiv T_{\mathbf g}T_{\mathbf h}^{*}\,\exp[2\pi i(\mathbf G-\mathbf H)\cdot\mathbf R]\,\mathrm{TCC}(\mathbf K+\mathbf G,\ \mathbf K+\mathbf H)\]

The double sum therefore reduces to the diagonal plus the upper triangle (one side, once an arbitrary ordering is assigned to the beams), halving the number of \(\mathrm{TCC}\) evaluations:

\[I(\mathbf R) = \sum_{\mathbf g} |T_{\mathbf g}|^2\,A(\mathbf K+\mathbf G)^2 \;+\; 2\sum_{\mathbf g}\sum_{\mathbf h > \mathbf g} \mathrm{Re}\!\left\{ T_{\mathbf g}T_{\mathbf h}^{*}\,\exp[2\pi i(\mathbf G-\mathbf H)\cdot\mathbf R]\,\mathrm{TCC}(\mathbf K+\mathbf G,\ \mathbf K+\mathbf H)\right\}\]

For the diagonal term \(\mathrm{TCC}(\mathbf u,\mathbf u)=A(\mathbf u)^2\), i.e. \(|T_{\mathbf g}|^2\) inside the aperture.

Furthermore, the phase factor \(\exp[2\pi i(\mathbf G-\mathbf H)\cdot\mathbf R]\) takes the same value many times within this sum. Storing and reusing these values accelerates the computation further.