# Optical Diffraction Tomography Using Nearly In-Line Holography with a Broadband LED Source
## 5 ODT Using Wavelength Diversity  
Finally, 3D RI reconstruction based on wavelength diversity is discussed in this section. Since the LED ring supports \"RGB\" colors, we captured images at three different wavelengths: red (623 nm), green (515 nm), and blue (468 nm). Theoretically, this corresponded to mapping at different Ewald's sphere with different radii \(k=\frac{2\pi}{\lambda_{\text{all}}}\) as shown in Figure 9. By following the same procedures discussed in the previous sections at each particular wavelength, the 3D refractive index reconstruction was retrieved. Practically, objective lenses suffer from chromatic aberrations, which results in the image not being in best focus at different wavelengths, as shown in Figure 10. This results in distortions (aberrations) when mapping the projections onto the Ewald's sphere resulting in an inaccurate estimation of the final 3D RI distribution. We can correct for this by acquiring different intensity images at different focal planes and correct for the aberrations by calculating the cross-correlation function between those images and a reference image [32], which effectively corrects for these aberrations. To correct for the chromatic aberrations, a different approach based on the Fresnel propagation was taken to refocus the image digitally, since we have access to the scattered field. This was performed in two steps; first, we reconstructed the 3D RI distribution without calibrating for the aberrations at red and blue (given that green channel is in focus). Then, by monitoring the reconstructions, the fields were backpropagated by the distance where the sample was displaced from the best plane of focus (z = 0). Figure 11 shows an example for the reconstructions along YZ for green and blue. Note how the reconstruction for the blue illumination was displaced from the best plane of focus representing the error due to the chromatic aberrations introduced by the objective lens. Figure 10: Chromatic aberrations of the objective lens. (**a**) Sample is in focus at green illuminations, whereas in (**b**) we see that at blue illumination the sample is out of focus due to the aberrations. Figure 9: Filling of the Ewald’s sphere for 2 projections at different illumination wavelength (i.e., red and blue). (**a**) Transverse spatial frequencies plane at \(K_{z}\) = 0. (**b**) \(K_{x}(K_{y})\),\(K_{z}\) plane at \(K_{y}(K_{x})\) = 0. The second step was to backpropagate the complex field extracted from the intensity image by the distance \(\Delta z\) to refocus it using the Fresnel propagation as follows:  
\[E_{calib}=IFFT\Big{\{}FFT\{FFT\{E_{uncailb}\}e^{iKz\Delta z}\Big{\}} \tag{10}\]  
where \(E_{uncailb}\) is the chromatic aberrated field and \(E_{calib}\) is the calibrated field after removing the chromatic aberrated. Figure 12 shows the effect of refocusing on the displacement along the optical axis on the XY slice at \(z=0\) (best plane of focus) in which the image came into focus after calibrating for the chromatic aberrations. After calibrating each wavelength channel, the 3D RI distribution was retrieved by combing all the calibrated projections into the 3D Fourier-space. However, this was based on the assumption that the sample did not have strong dispersion, and thus, the RI value was almost constant at different wavelengths. Figure 13 shows the 3D frequency support in the Fourier-space and the corresponding XY slices at different depths. While we did not observe enhancement in the optical resolution by going from one wavelength to three wavelengths, imaging at multiple wavelengths can still be advantageous in other aspects. For example, instead of capturing 24 projections where each projection corresponded to each LED, we could simultaneously operate three LED pixels each with a different color and then decouple them in the postprocessing, since we had an RGB camera. This increased the throughput of the system by a factor of three [31,33]. In addition, the incoherent superposition of the three RGB images increased the SNR of the reconstructed 3D object.