E48 · Magnetic electron imaging

Electron Holography: Recover the Hidden Phase

Overlap object and vacuum reference waves with a biprism. Record opposite magnetic states, select a Fourier sideband, unwrap its phase and separate the magnetic contribution before taking its gradient.

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

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

Pixels per fringe $\text{—}$

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

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

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

Experiment target $\text{—}$

Predict · record · explain

Can counted fringes separate magnetic induction from electric phase?

Foil thickness

t

60\,\mathrm{nm}

Peak prescribed induction

B_0

0.5\,\mathrm{T}

Assumed mean electric potential

V_0

10\,\mathrm{V}

Applied field · assumed texture shift

B_{\mathrm{app}}

0\,\mathrm{mT}

Calibrated biprism carrier · fringes across field

c

12

Intrinsic fringe visibility

\mathcal V

0.5

Combined incident electrons per pixel per state

N_0

1000

Fourier sideband mask radius · bins

r

5

Row phase unwrapping: 0 off, 1 on

u

0

Magnetic reversal separation: 0 off, 1 on

s

0

Target: separate opposite states and unwrap their phase. Keep the Fourier mask clear of the central band and Nyquist edge, average at least 1000 recorded counts per pixel per state, and recover projected induction within 12 percent relative RMS error against the known reference.

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. Each acquisition stores two independent opposite-magnetic-state holograms; the electric potential and exact pixel registration are assumed unchanged. A flat field-free vacuum reference and constant object amplitude are idealized. Carrier frequency is an integer number of fringes across the field, a calibrated biprism optical setting rather than an electrode-voltage model. Visibility includes a carrier-only pixel MTF, assuming specimen phase is locally sampled; no full subpixel integration. Reconstruction takes only stored counts, carrier and a circular hard Fourier mask. Processing sliders reuse those counts. Row-wise unwrapping assumes smooth horizontal phase differences below pi and removes an arbitrary offset from each row; it cannot solve phase singularities or recover absolute potential or vertical phase gradients. Magnetic phase is half the difference between reconstructed opposite states; the gradient gives the horizontal derivative component of projected induction in tesla nanometres, not magnetization or a 3D field. The electric diagnostic is only the row-centred half-sum. Known induction and error readouts are explicitly forward references. No charge-density inversion, reference-wave fringing field, alignment drift or dynamical diffraction correction.

Physics tutorial

Electron Holography: Recover the Hidden Phase

BackgroundOverlap object and vacuum reference waves with a biprism. Record opposite magnetic states, select a Fourier sideband, unwrap its phase and separate the magnetic contribution before taking its gradient.

Why it mattersTarget: separate opposite states and unwrap their phase. Keep the Fourier mask clear of the central band and Nyquist edge, average at least 1000 recorded counts per pixel per state, and recover projected induction within 12 percent relative RMS error against the known reference.

Start with the essentials

Focus question: Can counted fringes separate magnetic induction from electric phase?
One-sentence intuition: A single phase map mixes electric and magnetic contributions. Reversal isolates magnetism only when electric phase and registration remain unchanged. Wider Fourier masks preserve detail but admit noise and overlapping bands.

Core mathematical model

Recorded hologram expectation

\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 is flat; opposite magnetic states have independent count noise.

Reversal separation

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

Requires unchanged electric phase and exact registration; row offsets remain unknown.

Projected induction

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

This is a projected induction component, not magnetization.

Electric interaction

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

The absolute potential cannot be recovered after removing row offsets.

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
Record the two states
Compare the two holograms and Fourier spectrum; move the carrier.
What to observe: The sideband must fit between the central component and the sampling edge.
02
Process the stored counts
Vary mask width and enable phase unwrapping without reacquiring.
What to observe: A narrow mask erases detail; wrapping creates false phase gradients.
03
Separate before differentiating
Compare the single-state preset with reversal separation, then reach the target.
What to observe: Unchanged electric phase cancels from the half-difference.