# Optical Diffraction Tomography Using Nearly In-Line Holography with a Broadband LED Source
## 2 Theory  
The intensity pattern captured by the detector of the ODT system is denoted by \(I_{t}(x,y)\) with \(x\) and \(y\) being the horizontal and the vertical dimensions of the 2D intensity pattern. The detected intensity is given by:  
\[I_{t}=\left|{{U_{i}}}\right|^{2}+\left|{{U_{s}}}\right|^{2}+2\left|{{U_{i}}} \right|\left|{{U_{s}}}\right|\cos(\varphi_{s}-\varphi_{i}) \tag{1}\]  
where \(\left|{{U_{i}}}\right|\) is the amplitude of the incident field, \(\left|{{U_{s}}}\right|\) is the amplitude of the scattered field, and \(\varphi_{s}-\varphi_{i}\) is the difference between the phases of the complex scattered and the incident field, carrying the phase information of the sample. For weakly scattering samples (i.e., Born approximation), we can assume that (\(\left|{{U_{s}}}\right|\)\(<\)\(<\)\(\left|{{U_{i}}}\right|\)), and defining \(\left. {{U_{i}}}\right|=e^{j\varphi_{i}}\), \(\left|{{U_{i}}}\right|=1\), Equation (1) can be simplified as follows:  
\[I_{t}\approx 1+2\left|{{U_{s}}}\right|\cos(\Delta\varphi) \tag{2}\]  
where \(\Delta\varphi=\varphi_{s}-\varphi_{i}\). Equation (2) can be rewritten as:  
\[ \begin{array}{l}I_{t}=1+2\left|{{U_{s}}}\right|\cos(\Delta\varphi)\\ \Rightarrow\left.{{I_{t}}}\right.=1+\left|{{U_{s}}}\right|e^{j\Delta\varphi} +\left|{{U_{s}}}\right|e^{-j\Delta\varphi}\end{array}  \tag{3}\]  
Multiplying both sides of Equation (3) by \(E_{i}=e^{j\varphi_{i}}\), we obtain:  
\[ \begin{array}{l}I_{t}e^{j\varphi_{i}} =\left|{{U_{s}}}\right|e^{j\varphi_{s}}+e^{j\varphi_{i}}+\left|{{ U_{s}}}\right|e^{-j\varphi_{s}}e^{2j\varphi_{i}}\\ =\left.{{U_{s}}}\right.={{U_{s}}}+e^{j\varphi_{i}}+{{U_{s}}^{*}}e^{2j\varphi_{ i}}\\ =\left.{{U_{s}}}\right.={{U_{s}}}+e^{j\varphi_{i}}(1+{{U_{s}}^{*}}e^{j\varphi_{ i}})\end{array}  \tag{4}\]  
Equation (4) includes the effect of the scattered field (i.e.,\({{U_{s}}}\)), which we refer to as the \"Principal\" image and its complex conjugate (i.e.,\({{U_{s}}^{*}}\)), which we refer to as the twin image. As has been shown previously [28], at illumination angles less than the numerical aperture, the two terms tend to cancel each other, which results in a low contrast 3D reconstruction while maintaining the high frequency features of the sample. Figure 1 shows the effect of changing the illumination angle on the 2D intensity image of a simulated digital phantom where different illumination angles were assumed. Note the 2D Fourier transform of the corresponding intensity images includes two circles in the Fourier domain. Each circle is the result of the spectral filtering applied by the limited numerical aperture of the objective lens. From Equation (3), we see that we have three terms: a Zero-order term (the first term on the right-hand side) and the two cross terms. The shift of the two circles from the Zero-order term depends on the illumination angle of the incident plane wave. In other words, for normal incidence, the two circles completely overlap with each other. However, as we increase the illumination angle, the shift between the two circles increases until we reach the limit of the numerical aperture, at which point we see that the two circles are tangent to each other. Only when the illumination is at the maximum angle permitted by the \(NA\) of the objective lens can the complex field be retrieved from the intensity image as would be the case for an off-axis interferometric setup with a separate reference arm for holographic detection. Figure 1 shows that for an accurate extraction of the scattered field, the sample should be illuminated with the maximum angle permitted by the \(NA\) of the objective lens [21]. The scattered field can be extracted by simply multiplying the intensity measurement with the incident plane wave, which results in shifting the spectrum in the Fourier domain.