Our research aims to optimize the self-interference digital holography (SIDH) system by using the Collins formula. We focus on simplifying the calculation of the reconstruction distance and improving the phase accuracy in holographic image reconstruction. SIDH systems, unlike conventional holography systems that rely on coherent light sources, can use incoherent light sources such as natural light, LED lamps, and flashlights. This flexibility eliminates problems related to speckle noise, which often occur in laser-based holography, and thus improves the quality of holographic imaging. However, traditional diffraction-based models in SIDH systems cause great computational complexity, especially when dealing with complex optical configurations. This complexity not only reduces the imaging performance but also restricts the system’s adaptability to various practical applications. In this study, we address these challenges by integrating optical transformation matrices with the Collins formula to optimize the SIDH system. A key aspect of the research is to derive a simplified reconstruction distance formula based on the system’s optical parameters. This formula depends only on four optical transfer matrix elements—B1, B2, D1, and D2. Additionally, we introduce the parameter k to measure the phase variations on the reconstruction plane. These variations are affected by the system’s geometric parameters, interference region limitations, and diffraction distance. By optimizing this parameter, we improve the system’s image resolution and its adaptability to different experimental setups.

Our methodology focuses on combining the Collins diffraction formula with optical transformation matrices to model and optimize the light propagation in the SIDH system. The Collins formula provides a mathematical framework for calculating the diffraction patterns generated when light passes through optical elements. This enables accurate prediction of the light’s phase and amplitude variations. In the SIDH system, geometric phase lens (GPL) is employed to split incident light into right-handed circularly polarized (RCP) and left-handed circularly polarized (LCP) components. The RCP component acts as if it has passed through a converging lens, while the LCP component acts as if it has passed through a diverging lens. This facilitates self-interference and enables hologram capture. The system’s key geometric parameters include the distance between the object and the GPL (z0?) and the distance between the GPL and the imaging sensor (zh?) (Fig. 1). We describe the propagation of RCP and LCP light fields using optical transfer matrices M1 and M2?, respectively. These matrices account for the system’s geometric effects on the light waves. We simplify the computation of the reconstruction distance zrec? by applying the Collins formula together with the optical transfer matrices. This approach allows us to derive the simplified formula for the reconstruction distance, which greatly reduces the computational complexity compared to traditional diffraction models. Additionally, the parameter k is introduced to represent the phase variation due to geometric asymmetries and diffraction effects in the system. By optimizing this parameter, we can improve the system’s phase accuracy and image resolution. This allows us to fine-tune the SIDH system’s configuration to obtain high-quality holographic images.

The simplified reconstruction distance formula derived in our study reduces the computational burden of diffraction calculations in SIDH systems. To verify the proposed method, we conduct three sets of experiments using a compact SIDH system equipped with a GPL of focal length 40mm and a monochromatic polarized imaging sensor. We use a USAF1951 resolution target to evaluate the system’s performance. The experimental setup captures holograms of the target, and then we process them to reconstruct amplitude and phase information (Fig. 6). In experiment 1, with z0=14mm and zh=30mm, we obtain a reconstructed image resolution of 25.398lp/mm at an optimal reconstruction distance of 189.3mm (Fig. 7). This value closely matches the theoretical reconstruction distance of 186.3mm, demonstrating the accuracy of the simplified formula. In experiment 2, with z0=14mm and zh=80mm, the reconstruction distance is calculated to be 821.6mm. At a slightly adjusted distance of 826.0mm, the system achieves a resolution of 40.318lp/mm (Fig. 9). Experiment 3 tests at z0=15mm and zh=48mm, resulting in a calculated reconstruction distance of 324.0mm. The experimental results show the highest resolution of 50.797lp/mm at a reconstruction distance of 328.0mm, further validating the simplified formula (Fig. 9). These experimental results confirm that the derived reconstruction distance formula provides accurate predictions, with discrepancies between theoretical and experimental values of less than 2%. Furthermore, optimizing the parameter k improves the phase accuracy. Smaller values of this parameter correspond to reduced geometric asymmetry, enhancing the image resolution.

In this study, we successfully apply the Collins formula and optical transformation matrices to optimize the SIDH system. By deriving a simplified reconstruction distance formula that depends only on four optical transfer matrix elements, we significantly reduce the computational complexity of holographic image reconstruction. The introduction of the parameter k enables effective optimization of the phase accuracy, ensuring high-quality image reconstruction under different experimental conditions. The experimental results, with resolutions of up to 50.797lp/mm, demonstrate the robustness and accuracy of the proposed approach. This research provides a practical method for enhancing the performance of SIDH systems, bringing significant improvements in computational efficiency and image quality for future applications in incoherent digital holography.