E47 · Magnetic electron imaging

Lorentz TEM: Where the Walls Appear

Defocus a magnetic foil, reverse the sign and select one angular half-plane. Compare counted images with the known induction input, then test whether electric phase can imitate a magnetic signal.

Mean recorded counts per pixel $\text{—}$

Peak model magnetic deflection $\text{—}$

Model relative RMS contrast $\text{—}$

Opposite-defocus model correlation $\text{—}$

Model aperture throughput $\text{—}$

Assumed field-induced displacement $\text{—}$

Experiment target $\text{—}$

Predict · record · explain

Why does a magnetic wall change from bright to dark when focus is reversed?

Foil thickness

t

60\,\mathrm{nm}

Peak prescribed induction

B_0

0.8\,\mathrm{T}

Assumed mean electric potential

V_0

0\,\mathrm{V}

Applied field · assumed texture shift

B_{\mathrm{app}}

0\,\mathrm{mT}

Signed defocus · Fresnel only

\Delta f

0\,\mathrm{\mu m}

Detector Gaussian RMS blur

\sigma_d

5\,\mathrm{nm}

Image mode: 0 Fresnel, 1 Foucault

m

0

Accepted angular sign

s

1

Foucault angular cutoff

\theta_c

0\,\mathrm{\mu rad}

Mean incident electrons per pixel

N_0

2000

Target: use positive Fresnel defocus with zero electric potential. Obtain at least 1000 recorded counts per pixel on average, model RMS contrast at least 0.12, and opposite-defocus contrast correlation at most -0.85.

Fixed 200 keV, 800 nm periodic field, 128 by 128 samples, 6.25 nm pitch. The beam travels along positive z; x points right and y up. A smooth sinusoidal in-plane induction is prescribed, not solved by micromagnetics. Uniform thickness is assumed. The optional electric phase comes from a specified potential with smooth modulation. No absorption, crystal diffraction, stray-field solution, out-of-plane induction, aberrations, damage or calibrated material identification. Detection efficiency is 80 percent; fixed seeds generate independent Poisson records, with a normal approximation above 40 counts. Counts are linear; display colors are separate. Thickness, electron paths, angular separation and magnification in the apparatus are schematic. Fresnel images use full complex-wave paraxial propagation, followed by a Gaussian detector blur. Positive defocus is downstream propagation. Foucault mode is an in-focus coherent half-plane aperture; its cutoff is measured in microradians, the direct zero-order beam is blocked and defocus is inactive. The conventional immersion objective is off; a separate field-free Lorentz lens is used. Applied field causes only an assumed reversible rigid displacement of 2 nm per mT, with no hysteresis or domain nucleation. The opposite-defocus image is a noiseless forward reference, not a second acquired record. Bright and dark reversal is approximate at finite defocus, quantified by a correlation. No height or magnetization reconstruction is claimed.

Start with the essentials

Focus question: Why does a magnetic wall change from bright to dark when focus is reversed?
One-sentence intuition: A focused pure-phase foil can look uniform. Defocus turns phase curvature into intensity, while a Foucault aperture selects angular components and sacrifices transmitted counts. Electric phase can also generate Fresnel contrast.

Core mathematical model

Phase gradient and deflection

\begin{aligned}\theta_x&=\frac{\lambda}{2\pi}\frac{\partial\phi_m}{\partial x}\\&=\frac{e\lambda}{h}\int B_y\,dz\end{aligned}

Sign follows a negative electron travelling along positive z.

Fresnel transfer

\begin{aligned}H&=\exp(-i\chi)\\\chi&=\pi\lambda\Delta f(q_x^2+q_y^2)\end{aligned}

Propagation conserves total expected intensity before the aperture.

Small-defocus limit

I(\Delta f)\simeq1-\frac{\lambda\Delta f}{2\pi}\nabla_\perp^2\phi

This limit explains reversal; the actual image uses full complex propagation.

Common difficulties

Distinguish data from reference

Typical misconceptionA known reference map is the same as an acquired or recovered field.

Better mental modelThe reference is a declared specimen input. Counted records and their processing are separate; error comparisons use the reference only for assessment.

Run the experiment

01
Predict the focused foil
Begin in focus, then apply positive and negative defocus.
What to observe: Phase becomes contrast; finite-defocus reversal is approximate.
02
Select an angular half-plane
Switch to Foucault, flip the accepted sign and raise its cutoff.
What to observe: Domain contrast changes while recorded throughput falls.
03
Test an alternative cause
Compare the electric preset, move the assumed field response, then reach the target.
What to observe: An intensity wall alone does not uniquely identify a magnetic contribution.