## 2 Short Theory of Polarized Neutron Imaging  
It has been shown that polarized neutrons are particularly suitable for visualizing magnetic fields [19; 20; 21; 22; 23; 24; 26]. Using tomographic methods, shape, size, amount of magnetic fields in superconducting matter in large (\(\sim\)cm\({}^{3}\)) samples could be determined [22; 27].  
Based on absorption, radiography and tomography with (thermal) neutrons are quite similar to well-known X-ray techniques, but due to their magnetic moment \(\mu_{n}\) (\(\mu_{n}=-0.96623651\times 10^{-26}[JT^{-1}]\)) imaging with polarized neutrons involves in addition the interaction of the neutron spin with a magnetic field. If a neutron enters a magnetic field, its spin starts rotations around B with the so-called Larmor frequency \(\omega_{L}=\gamma_{n}B\), \(\gamma_{n}=1.83247172\times 10^{8}[T^{-1}s^{-1}]\)[28; 29] (all values see [30; 31; 32]). The rotation angle \(\phi\)depends on the velocity vs. of the neutron and can be used to determine the amount of \(B\), calculating the path integral along \(B\),  
\[\phi=\omega_{L}t=\gamma_{n}B\cdot t=\frac{\gamma_{n}}{\mathrm{v}}\int_{path} \vec{B}\cdot d\vec{s} \tag{1}\]  
If's' is the path length in the sample \(\phi\) becomes  
\[\phi=\frac{\gamma_{n}\cdot\lambda\cdot m}{h}\cdot B\cdot s \tag{2}\]  
Assuming the sample placed in a {xyz} coordinate system, and considering only two projections (0 and 90\({}^{\circ}\)), the path integral (Equation (1)) then becomes [22]  
\[\begin{array}{l}R_{0,\alpha}\{B(x,y,z)\}=\\ \int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}B(x,z)\cdot\delta(p- x\cdot\cos(\alpha)-z\cdot\sin(\alpha))\cdot dx\cdot dz\\ R_{0,\alpha}\{B(x,y,z)\}=\\ \int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}B(y,z)\cdot\delta( p-y\cdot\cos(\alpha)-z\cdot\sin(\alpha))\cdot dy\cdot dz\end{array} \tag{3}\]  
\(R_{0,\alpha}\{B(x,y,z)\}\) is the 2D-Radon transform of the sample when its rod axis is perpendicular to the {xz} plane and \(R_{90,\alpha}\{B(x,y,z)\}\) the 2D-Radon transform of the sample when its rod axis is perpendicular to the {yz} plane. For \(R_{0,\alpha}\{B(x,y,z)\}\)\(\alpha\) is constant in this calculation, it is usually the (rotation) angle of the sample with respect either to the x or z axis, and for \(R_{90,\alpha}\{B(x,y,z)\}\) parallel either to the y or z axis. p is the scanning parameter and describes the parallel beam scanning across a slice. The intensity registered by a 2D neutron detector is then given as  
\[\begin{array}{l}I(x,z)=\underbrace{I_{0}\cdot T\cdot\exp(-\int\limits_{path }\Sigma(s)ds)}_{I_{int}(x,z)}\cdot\underbrace{\frac{1}{2}(1+\cos\phi(x,z))}_{ I_{spin}(x,z)}\\ I(y,z)=\underbrace{I_{0}\cdot T\cdot\exp(-\int\limits_{path}\Sigma(s)ds)}_{ I_{int}(y,z)}\cdot\underbrace{\frac{1}{2}(1+\cos\phi(y,z))}_{I_{spin}(y,z)} \end{array} \tag{4}\]  
\(I_{o}\) is the incident intensity, \(\Sigma(s)\) the linear attenuation coefficient of the sample, \(I_{spin}\) is the'magnetic' part of the transmitted intensity \(I(x,z)\) and \(I(y,z)\), respectively. \(I(x,z)\) and \(I(y,z)\) are spatially resolved two-dimensional depolarization images. Each trapped magnetic field causes a characteristic depolarization image, which can be calculated from first principles using Equations (2)-(4) and compared with experimental neutron images.