# Method of Multi-Angle Transmission Radiowave Tomography of Dielectric Objects
## 2 Method of Forward Task Solution and Numerical Modeling  
It is proposed to consider the scheme of transmission multi-angle sounding by means of radio waves (Figure 1). The investigated dielectric objects are located on a rotating platform. A broadband plane wave probes objects at each angle of rotation. The amplitude-phase distribution of the field in the plane behind the objects across the set of frequencies is measured. The receiving array is paced at a distance, R, from the center of rotation of the objects under study. Elements in the array are located in steps of less than half the shortest wavelength, that is, the grid is filled. Figure 1: Measurement principle. A direct problem is proposed to be solved using the algorithm described in [4]. The algorithm is based on splitting the medium into thin layers and calculating the propagation field in the plane wave spectrum for homogeneous layers of different materials (Figure 2). Then, solutions for different homogeneous layers would be combined to get the field distribution in heterogeneous media. A plane wave falls on the structure under consideration (Figure 2) and is described by the expression \(U(x,y,z)=exp(ikz)\), where \(z>z_{0}\), \(z_{0}=-R\)--the boundary beginning of the probing area at the \(z\)-axis; \(k=n_{1}\frac{\omega}{c}\)--the wave number of the background medium; and \(\omega\) is the frequency of probing signal. The calculation of the field in the medium is carried out iteratively, when the value of the field \(U(x,y,z+\Delta z)\) is calculated on the basis of the field values \(U(x,y,z)\). The field in an inhomogeneous medium on the basis of the cross-linking of solutions for a finite number, M, of homogeneous media with refractive indices, \(n_{m}\), is calculated as:  
\[U_{m}(x,y,z+\Delta z)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\widetilde {U}\left(k_{x},k_{y},z\right)exp\left(ik_{x}x+ik_{y}y+ik_{mz}\Delta z\right) dk_{x}dk_{y} \tag{1}\]  
where \(k_{mz}=\sqrt{\left(kn_{m}\right)^{2}-k_{x}{}^{2}-k_{y}{}^{2}}\) is the z component of the wave vector in a medium with a refractive index, \(n_{m}\); \(m\) takes values from 1 to \(M\); \(\widetilde{U}\left(k_{x},k_{y},z\right)=\frac{1}{\left(2\pi\right)^{2}}\int_{ -\infty}^{\infty}\int_{-\infty}^{\infty}U(x,y,z)exp\left(-ik_{x}x-ik_{y}y\right)dxdy\) is the spectrum of the plane waves of the field at the output of the previous layer; and \(U_{m}(x,y,z+\Delta z)\) is the output field of the current equivalent uniform layer with a homogeneous refractive index, \(n_{m}\). The solution can be combined by means of the expression:  
\[U(x,y,z+\Delta z)=U_{m}(x,y,z+\Delta z),\ if\ n(x,y,z)=n_{m} \tag{2}\]  
Figure 2: Splitting the medium into thin layers. Figure 3b shows the result of numerical modeling of diffraction on dielectric rectangular prisms with a refractive indices of 1.5 and 2 (Figure 3a). We considered 256 measurement points over the x axis. Similar calculations were made for various angles of rotation of the test objects. As a result, a field distribution was obtained for the given two rectangular objects (Figure 3) in the measurement area, as shown in Figure 4. We modeled additive noise with a uniform statistical distribution in the in-phase and quadrature components of the measured signal at all frequencies to further verify the stability of the method for solving the inverse problem to noise. The in-phase component image (Figure 4a) with the addition of uniform noise with a spread of 30% of the maximum signal amplitude is shown in Figure 4b, and that with the addition of noise with 100% spread, in Figure 4c. We denote the result of forward task solution by the function:  
\[A(x,y,\alpha,\omega)=U(x,y,R)\big{|}_{\alpha,\omega}, \tag{3}\]  
where \(\alpha=\) the angle of the test object rotation in the XOZ plane, \(\omega=\) the cyclic frequency of the probing radiation, and \(R=\) the position of the measurement system in the z axis.