Simultaneous Reconstruction

• Accelerated electron beams interacting with a thin sample acquire phase shifts:

$$\begin{aligned} \phi\left(\boldsymbol{r}_{\perp}\right) &= \phi_e\left(\boldsymbol{r}_{\perp}\right) + \phi_m \left(\boldsymbol{r}_{\perp}\right) \\ &= \frac{e}{\hbar v} \int_L V(\boldsymbol{r}_{\perp},z) dz - \frac{e}{\hbar} \int_L \boldsymbol{A}(\boldsymbol{r}_{\perp},z) \cdot d \boldsymbol{r} \end{aligned}$$

'Naive' Subtraction

• Electrostatic contribution → invariant under time-reversal symmetry
• Magnetic contribution → changes sign under time-reversal symmetry
  • Inversion-symmetry: "flip" sample and record two measurements:

$$\phi_e = \frac{1}{2}\left(\phi^{\rightarrow} {\color{#b880ff} +} \phi^{\leftarrow} \right) \qquad \text{and} \qquad \phi_m = \frac{1}{2}\left(\phi^{\rightarrow} {\color{#b880ff} -} \phi^{\leftarrow} \right)$$

Simultaneous Adjoint Operator

• Forward-model for our measurement as we "flip" the sample

$$\begin{aligned} \psi^{\rightarrow} &= \mathcal{P} \times \exp\left(i \left(V_p {\color{#b880ff} +} i A_z\right)\right) \\ \psi^{\leftarrow} &= \mathcal{P} \times \exp\left(i \left(V_p {\color{#b880ff} -} i A_z\right)\right) \end{aligned}$$

• Simultaneous electrostatic and magnetic adjoint operator:

$$\begin{aligned} \Delta V_p &\propto \overbrace{\Re\left\{ -i \exp(-i (V_p{\color{#b880ff} +}A_z)) \times \mathcal{P}^* \times \Delta \psi^{\rightarrow} \right\}}^{{\color{#b880ff} \text{Forward Measurement Contribution}}} {\color{#b880ff} +} \overbrace{\Re\left\{ -i \exp(-i (V_p{\color{#b880ff} -}A_z)) \times \mathcal{P}^* \times \Delta \psi^{\leftarrow} \right\}}^{{\color{#b880ff} \text{Reverse Measurement Contribution}}} \\ \Delta A_z & \propto \Re\left\{ -i \exp(-i (V_p{\color{#b880ff} +}A_z)) \times \mathcal{P}^* \times \Delta \psi^{\rightarrow} \right\} {\color{#b880ff} -} \Re\left\{-i \exp(-i (V_p{\color{#b880ff} -}A_z)) \times \mathcal{P}^* \times \Delta \psi^{\leftarrow} \right\} \end{aligned}$$