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Vector Tomography​

  • Accelerated electron beams interacting with a thin sample acquire phase shifts:
ϕ(r⊥)=ϕe(r⊥)+ϕm(r⊥)=eℏv∫LV(r⊥,z)dz−eℏ∫LA(r⊥,z)⋅dr\begin{aligned} \phi\left(\boldsymbol{r}_{\perp}\right) &= \phi_e\left(\boldsymbol{r}_{\perp}\right) + \phi_m \left(\boldsymbol{r}_{\perp}\right) \\ &= \frac{e}{\hbar v} \int_L V(\boldsymbol{r}_{\perp},z) dz - \frac{e}{\hbar} \int_L \boldsymbol{A}(\boldsymbol{r}_{\perp},z) \cdot d \boldsymbol{r} \end{aligned}
  • Can we reconstruct the 3D vector potential using tilt-series magnetic projections?
    • Using Maxwell's equations, we can convert this to two scalar Radon transforms for the magnetic field
Vector field tomography
D Wolf et al., Nat. Nano. 17 250-255 (2022)
  • How about using the tilt-series of 4D-STEM data directly?
    • Yes, we can use vector ptychographic tomography