# Optical Diffraction Tomography Using Nearly In-Line Holography with a Broadband LED Source
## 2 Theory
This is followed by the spatial filtering of the \"Principal\" image with a circular filter whose size is determined by the \(NA\) of the objective lens as shown in Figure 2.  
From Figure 2, it can be seen that only when the illumination angle is at the edge of the imaging \(NA\) can we extract the complex scattered field from the intensity image. This can be experimentally demonstrated by illumination along a circular cone whose center is perfectly aligned with the imaging objective lens. This is demonstrated in the experimental setup described in the following section. Figure 1: Intensity images and their 2D Fourier transforms for on-axis and off-axis configurations with different illumination angles. As can be seen from the figures, as the incident illumination vector \(k_{in}\) approaches the numerical aperture of the objective lens, the 2 cross terms can be decoupled, and the principal term can be retrieved. Scale bar = 8 \(\upmu\)m.  
By multiplying the intensity image with the incident plane-wave to shift the spectrum in the Fourier domain, the scattered field spectrum becomes centered around the origin. To filter out the complex scattered field in the Fourier domain, we apply a low pass filter given by the following equation:  
\[\widehat{U}_{s}(k_{x},k_{y})=LPF\Big{\{}FFT_{2D}\Big{\{}I_{te}^{j\varphi_{i}} \Big{\}}\Big{\}} \tag{5}\]  
where \(\widetilde{U}_{s}(k_{x},k_{y})\) is the 2D Fourier transform of \(U_{s}(x,y)\), and \(LPF\{.\}\) represents a circular low pass filter whose radius is given by \(k_{0}NA\), where \(k_{0}\) is the wave number in free-space. By extracting the complex field along the direction \(k=(k_{x},k_{y},k_{z})\) for each illumination k-vector \(k_{in}=(k_{x}^{in},k_{y}^{in},k_{z}^{in})\), one Fourier component of the 3D spectrum of the scattering potential \(\widetilde{F}(\overrightarrow{\kappa})\) (which is directly related to the index distribution) can be retrieved [1]:  
\[\widetilde{F}(\overrightarrow{\kappa})=\frac{k_{z}}{2\pi j}\widetilde{U}_{s}( k_{x},k_{y}) \tag{6}\]  
where  
\[\overrightarrow{\kappa}=\overrightarrow{k}-\overrightarrow{k}_{i}=\left[ \begin{array}{c}k_{x}-k_{x}^{in}\\ k_{y}-k_{z}^{in}\\ \sqrt{\left(k_{0}n_{0}\right)^{2}-k_{x}^{2}-k_{y}^{2}}-k_{z}^{in}\end{array} \right]\]  
where \(\overrightarrow{\kappa}\) is the 3D spatial frequency of the scattering potential, \(k_{0}=\frac{2\pi}{\lambda}\), \(\lambda\) is the wavelength of the illumination beam, and \(n_{0}\) is the refractive index of the surrounding medium. We refer to Equation (6) as the Wolf transform [1,29]. By applying Equation (6) for different 2D projections and accumulating the various spectral components in the 3D Fourier domain, the 3D spectrum \(\widetilde{F}(\overrightarrow{\kappa})\) can be measured. Figure 2: Processing of the 2D intensity images before mapping into the 3D Fourier space. The left-most panel shows the intensity measurements and the corresponding Fourier transform. The middle panel shows the effect of multiplying by the incident plane wave, which results in centering the scattered field highlighted by the white circles. The final step is the filtering of the scattered field with a circular filter whose size matches the size of the numerical aperture in the Fourier space declared by the red circle. Scale bar = 8 μm. Subsequently, \(F(r)\) can be spatially reconstructed using an inverse 3D Fourier transform.