E48 · Magnetic electron imaging

Electron Holography: Recover the Hidden Phase

Can electron holography separate the magnetic part of the phase from the electric part? It can, provided you reverse the magnetization and change nothing else. A single phase map mixes the electric and magnetic parts. Take a second map with the magnetization flipped and subtract: the electric part cancels and the magnetic part remains, but only if the electric phase is unchanged and the two maps line up exactly.

Use a biprism to overlap the wave that crossed the specimen with a reference wave that crossed vacuum. Record the specimen in two opposite magnetic states, pick out a Fourier sideband and unwrap its phase. Then separate the magnetic part of the phase before taking its gradient.

Recovered relative projected-potential RMS $\text{—}$

Band-limited projected-charge RMS $\text{—}$

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

Projected-induction relative RMS reference error $\text{—}$

More measurements and target

Object-plane fringe spacing $\text{—}$

Pixels per fringe $\text{—}$

Magnetic-phase RMS reference error $\text{—}$

Mask clearance from centre and Nyquist $\text{—}$

Experiment target $\text{—}$

Simulated examples; each plot identifies its data source.

Peak prescribed induction

B_0

0.5\,\mathrm{T}

Assumed mean electric potential

V_0

10\,\mathrm{V}

Calibrated biprism carrier · fringes across field

c

12

Row phase unwrapping: 0 off, 1 on

u

0

Magnetic reversal separation: 0 off, 1 on

s

0

Advanced settings

Electric inversion bandwidth in Fourier bins

b_e

3

Foil thickness

t

60\,\mathrm{nm}

Applied field · assumed texture shift

B_{\mathrm{app}}

0\,\mathrm{mT}

Intrinsic fringe visibility

\mathcal V

0.5

Combined incident electrons per pixel per state

N_0

1000

Fourier sideband mask radius · bins

r

5

Learning goal and model limits

Make a prediction, then test it

Can electron holography separate the magnetic part of the phase from the electric part?

Target: separate the two opposite magnetization states and unwrap their phase. Keep the Fourier mask clear of the central band and the Nyquist edge, and record at least 1000 counts per pixel per state on average. Recover the projected induction with a relative RMS error of at most 12 percent against the known reference.

Model assumptions

The beam is fixed at 200 keV and travels along positive z, with x to the right and y up. The field of view is periodic, 800 nm across, sampled on 128 by 128 points at 6.25 nm pitch. The in-plane induction is a prescribed smooth sinusoid, not a micromagnetic solution, and the foil has uniform thickness. The optional electric phase comes from a specified potential with a smooth modulation. Absorption, crystal diffraction, stray-field calculation, out-of-plane induction, aberrations, damage and calibrated material identification are not included. The detector records 80 percent of electrons. Fixed seeds generate independent Poisson records, with a normal approximation above 40 counts. Counts stay linear; display colors are mapped separately. In the apparatus view, thickness, electron paths, angular separation and magnification are schematic. Each acquisition stores two independent holograms of opposite magnetic states, assuming the electric potential and exact pixel registration do not change between them. The vacuum reference wave is idealized as flat and field-free, and the object wave has constant amplitude. The carrier frequency is a whole number of fringes across the field. It stands for a calibrated biprism optical setting, not a model of the electrode voltage. Fringe visibility includes a pixel MTF evaluated at the carrier only, assuming the specimen phase is sampled locally; full subpixel integration is not done. Reconstruction uses only the stored counts, the carrier and a hard-edged circular Fourier mask, and the processing sliders reuse those counts. Row-by-row unwrapping assumes the horizontal phase steps are smooth and smaller than half a cycle, and it removes an arbitrary offset from every row. It cannot handle phase singularities and cannot recover the absolute potential or vertical phase gradients. The magnetic phase is half the difference between the two reconstructed states. Its gradient gives the horizontal-derivative component of projected induction, in tesla nanometres; this is not the magnetization or a 3D field. A separate smooth two-dimensional unwrap retains relative vertical offsets, assuming every neighbouring phase step is below half a cycle and there are no defects. Its half-sum yields a mean-zero relative projected potential using the 200 keV interaction constant. A selectable low-pass bandwidth limits noise before a periodic Laplacian gives projected charge in vacuum permittivity. This charge requires matched longitudinal normal derivatives at the integration boundaries; unknown fringing-field boundary terms are not recovered. The constant potential remains unknown, and thickness is not inferred. The known induction and the error readouts are labeled as forward references from the model input. Non-periodic charge inversion, fringing fields in the reference wave, alignment drift and dynamical-diffraction correction are not included.

Physics tutorial

Electron Holography: Recover the Hidden Phase

BackgroundAn electron hologram records interference fringes between a wave that crossed the specimen and a reference wave that crossed vacuum. The specimen's magnetic and electric fields add phase to the object wave, and the fringes shift with it. You recover that phase from the recorded counts and separate out the magnetic part.

Why it mattersThe goal: use reversal separation and phase unwrapping, keep the Fourier mask clear of both the central band and the Nyquist edge, and record at least 1000 counts per pixel per state on average. The recovered projected induction must then match the known reference within 12 percent relative RMS error.

Start with the essentials

Focus question: Can electron holography separate the magnetic part of the phase from the electric part?
One-sentence intuition: A single phase map mixes the electric and magnetic contributions together.

Core mathematical model

Expected counts in each hologram

\begin{aligned}I_\pm&=N_d[1+\mathcal V\cos\Phi_\pm]\\\Phi_\pm&=2\pi q_cx+\phi_e\pm\phi_m\end{aligned}

The reference wave is flat, and the two opposite magnetic states carry independent counting noise.

Separating the phase by reversal

\phi_m=\frac{\phi_+-\phi_-}{2},\qquad\phi_e=\frac{\phi_++\phi_-}{2}

This works only if the electric phase is unchanged and the two images are exactly registered. The magnetic derivative uses row-centering; the separate electric path preserves relative row offsets.

Projected induction

\int B_y\,dz=\frac{\hbar}{e}\frac{\partial\phi_m}{\partial x}

This gives one component of the induction integrated along the beam path, not the magnetization.

Phase from the electric potential

\phi_e=C_E\int V\,dz,\qquad C_E=\frac{e}{\hbar v}

The electric path uses a smooth two-dimensional unwrap to preserve vertical differences. Its global mean is removed, so the potential zero remains unknown. A periodic Laplacian estimates band-limited projected charge only when the two longitudinal boundary derivatives match.

Common difficulties

The reference map is not data

Typical misconceptionThe known reference map is the field the microscope recorded or recovered.

Better mental modelThe reference map is the prescribed specimen input. The holograms are counted and processed without it; the reference is used only to score the error.

Run the experiment

01
Recover electric contrast and charge
Use Separate and recover, then change the electric inversion bandwidth while keeping the recorded holograms fixed.
What to observe: The relative projected potential preserves vertical structure. Differentiation amplifies noise; bandwidth trades detail for stability. A constant potential remains unobservable.
02
Record the two states
Compare the two holograms and the Fourier spectrum, then move the Calibrated biprism carrier slider.
What to observe: The sideband has to fit between the central component and the sampling edge.
03
Process the stored counts
Change the Fourier sideband mask radius and turn on Row phase unwrapping. Nothing is reacquired.
What to observe: A narrow mask wipes out fine detail, and phase left wrapped produces false gradients.
04
Separate before differentiating
Compare the Single state preset with Magnetic reversal separation turned on, then reach the target.
What to observe: The electric phase is the same in both states, so it cancels in the half-difference.

Core mathematical model

Expected counts in each hologram

Separating the phase by reversal

Projected induction

Phase from the electric potential

Common difficulties

The reference map is not data

Run the experiment

Recover electric contrast and charge

Record the two states

Process the stored counts

Separate before differentiating