Simulating Probe Intensity and EELS Zero-Loss Peak from Aberration Functions

Part 1 — STEM Probe

The aberration function $\chi$ assigns a phase (in radians) to every point $(q, \varphi)$ in reciprocal space, where $q$ is the radial coordinate (Å$^{-1}$) and $\varphi$ is the azimuthal angle:

$$\chi(q,\varphi) = \frac{2\pi}{\lambda}\left[ \frac{(\lambda q)^2}{2}\,C_{10} + \sum_{n,m} \frac{(\lambda q)^{n+1}}{n+1}\,C_{nm}\cos\!\bigl(m(\varphi - \theta_{nm})\bigr) \right]$$

Each aberration $C_{nm}$ (e.g. defocus $C_{10}$, spherical $C_{30}$, astigmatism $C_{12}$) distorts the phase by a different power of $q$ and angular symmetry $m$. Setting all amplitudes to zero gives $\chi = 0$ everywhere — the ideal, unaberrated case.

The contrast transfer function (CTF) in reciprocal space is:

$$\text{CTF}(\mathbf{q}) = \begin{cases} e^{-i\,\chi(q,\varphi)} & q \leq q_{\text{aperture}} \\ 0 & \text{otherwise} \end{cases}$$

The aperture is a hard top-hat mask. Inside it, the CTF has unit magnitude — aberrations only twist the phase, not the amplitude. Outside the aperture, the CTF is zero.

The real-space probe wavefunction is the inverse Fourier transform:

$$\psi(\mathbf{r}) = \mathcal{F}^{-1}\!\bigl[\,\text{CTF}(\mathbf{q})\bigr]$$

And the probe intensity is:

$$I(\mathbf{r}) = |\psi(\mathbf{r})|^2$$

With no aberrations, CTF = 1 inside a disk, whose Fourier transform is an Airy function — a tight, diffraction-limited spot. Aberrations (e.g. defocus adds $\sim q^2$ phase, $C_{30}$ adds $\sim q^4$) cause different parts of the aperture to arrive out of phase, broadening the probe.

Part 2 — EELS

For EELS, the energy-loss signal is dispersed along one axis (call it $x$). It is more natural to use a Cartesian aberration function $\chi_2$ that separates the dispersive ($x$) and non-dispersive ($y$) directions:

$$\chi_2(\theta_x, \theta_y) = \frac{2\pi}{\lambda} \sum_{n=0}^{5}\sum_{m=0}^{5} \frac{E_{nm}}{n+1}\,\theta_x^{\,n+1}\,\theta_y^{\,m}$$

where $\theta_x = q_x\lambda$, $\theta_y = q_y\lambda$ are the scattering angles, and $E_{nm}$ is the 6×6 matrix of Cartesian aberration coefficients set via Enm.

The CTF is built identically to before — exp(-i χ₂) inside the aperture — but now a 1D Fourier transform along the dispersive axis replaces the 2D transform:

$$\Psi_{\text{EELS}}(x, q_y) = \mathcal{F}_x\!\bigl[\,\text{CTF}(q_x, q_y)\bigr]$$

By the projection-slice theorem, this is equivalent to projecting the 2D probe intensity onto the $x$-axis. The resulting EELS intensity pattern is:

$$I_{\text{EELS}}(x, q_y) = |\Psi_{\text{EELS}}|^2$$

Aberration terms that couple $x$ and $y$ (e.g. $E_{01}$ — a tilt/dispersion term) shear or broaden the EELS pattern along the dispersive direction, degrading energy resolution and spatial resolution simultaneously.