## 5 Discussion and Conclusions  
Holographic interferometry is a mature optical tool of interest in multiple imaging applications. In-line holographic configurations are especially interesting when some error sources, as those related to mechanical vibrations or refractive index fluctuations, need to be minimized. However, those configurations have the disadvantage of presenting the conjugated images superimposed next to the reconstructed ones, which results in a loss of efficiency in the final reconstructed images. Under this scenario, some authors have presented a combination of the in-line interferometer with the DSB filter, providing the usefulness of the technique with removed conjugated images. However, no analysis of the magnification associated with such systems was presented in the literature, with this information being crucial for efficiently applying the method in real applications. Figure 13: (**a**) Experimental M\({}_{\mathrm{L}}\) as a function of the microsphere displacement \(\Delta z_{o}\) and (**b**) experimental M\({}_{\mathrm{T}}\) as a function of the microsphere displacement \(\Delta z_{o}\). The error bars are associated with micrometer resolution of the linear translation stage and the resolution of the precision ruler used for measuring the image distance, respectively. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{6}{c}{**USAF 1951**} & \multicolumn{6}{c}{**Microsphere**} \\ \hline  & & **Ent** & **TC** & **GI** & **Ent** & **TC** & **GI** \\ \hline M\({}_{\mathrm{L}}\) & Ave. & \(-9.4867\) & \(-9.4490\) & \(-9.4543\) & \(-9.4695\) & \(-9.4742\) & \(-9.4857\) \\ S.D. & S.D. & \(0.4799\) & \(0.2184\) & \(0.2471\) & \(0.4454\) & \(0.1191\) & \(0.2718\) \\ M\({}_{\mathrm{T}}\) & Ave. & \(-3.0834\) & \(-3.0787\) & \(-3.0776\) & \(-3.1198\) & \(-3.0931\) & \(-3.6058\) \\ S.D. & \(0.0286\) & \(0.0324\) & \(0.0323\) & \(0.2085\) & \(0.1586\) & \(0.2772\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Summary of average (Ave.) and standard deviation (S.D.) values of the experimental M\({}_{\mathrm{L}}\) and M\({}_{\mathrm{T}}\) magnifications for the tested microscopic objects: the USAF 1951 resolution test and the glass microsphere. In this work, we provide a thorough study of the magnification relations (the transverse and longitudinal magnifications) associated with an in-line interferometer system based on a DSB filter. This analysis is combined with methods for hologram propagation and optimal focused image identification. Results provided in this work highlight the suitability of the approach for object position determination. In particular, the presented simulations include computing the diffracted wavefront by microscopic objects, applying the Rayleigh-Sommerfeld diffraction equation to model the propagation of the diffracted wavefront. The numerical Fourier and the inverse Fourier transforms were used to block half of the spatial-frequency spectrum of the diffracted wavefront. Additional computing processes were used for simulating the transverse magnification produced by the optical system. Finally, the magnified focused image was reconstructed using the Rayleigh-Sommerfeld equation to propagate the digital hologram to the reconstruction plane. The reconstruction distance, related to longitudinal magnification, was estimated using three focusing criteria: the Tamura coefficient, the Gini Index, and the entropy. The Tamura coefficient and the entropy demonstrated the lowest standard deviation, suggesting that they are the most effective evaluation criteria for achieving the best focused images. The simulation results have confirmed the validity of our theoretical analysis. The fluctuations of the magnification values are not greater than 1%. The presented methods were experimentally validated by analyzing the images obtained of three different micro-objects: glass microspheres, a micrometric reticle, and a resolution test chart USAF 1951. Experimental average values of the magnifications fall within the expected range, considering the experimental errors.