Most samples of interest in the fields of optical measurements, adaptive optics, materials physics and biomedical imaging are phase object. Such samples change very little in the amplitude part of the transmitted light, but the different refractive index or thickness distribution of the samples results in a very large change in the phase part of the transmitted light. The human eye or detector can only record the change in intensity caused by the object but cannot determine its phase change. Due to its non-invasive nature, label-free capability, and ability to provide quantitative measurements, quantitative phase imaging technology has emerged as a preferred tool for obtaining phase information in these areas. Techniques in quantitative phase imaging, particularly those based on the Transport of Intensity Equation (TIE), offer advantages such as non-interferometric operation, non-iterative processing, and compatibility with modern bright-field microscopes, has already found application in various domains. However, TIE was derived under the assumption of fully coherent light. In the field of optical microimaging, the use of partially coherent light illumination generated by modern optical microscopes is important for improving imaging quality and suppressing coherent noise. Given that the illumination source in the microscope Kohler illumination system is usually in the form of a combination of halogen lamps and band-pass filters, the resulting illumination light is polychromatic with a certain bandwidth. Under the conditions of lower spatial and temporal coherence of illumination, TIE is still applicable if the sample to be measured does not have a strong dispersion of the illumination wavelength. However, since the definition of mean phase is inherently tied to the choice of mean wavelength, a challenge in the phase reconstruction emerges due to the complexity in fixing a specific mean wavelength. The mean illumination wavelength is often approximated by the center wavelength of the bandpass filter. This approximation overlooks the polychromatic light transmission characteristics of different bandpass filters. If the spectral distribution curve of the light after the halogen lamp passes through the bandpass filter has central symmetry, the peak spectral wavelength can be determined as the central wavelength. However, if the spectral distribution curve does not have central symmetry, directly determining the peak spectral wavelength as the mean wavelength may increase the reconstruction error of the mean phase. To address the above problems, this paper proposes an optimal mean wavelength method, we carried out numerical simulation and experiments using broadband polychromatic radiation with known spectral distributions. The polychromatic intensity is obtained by integrating the monochromatic intensity across wavelengths through numerical simulation, and then the axial differentiation of the intensity is calculated. The mean phase of the polychromatic beam is determined based on the fast Fourier transform based TIE solution algorithm, and the reconstructed mean phases at different mean wavelengths are quantitatively analyzed to determine the optimal mean wavelength for different bandpass filters. By adding Gaussian noise to the polychromatic intensity to simulate actual measurement noise and quantization effects, we further investigate the influence of different defocus distances on the selection of optimal mean wavelength and the recovered phase accuracy. Experimental results on microlens arrays with known height information show that when the spectral distribution curve has good central symmetry, using the spectral peak wavelength as the center wavelength performs comparable to the optimal mean wavelength method. When the spectral distribution curve does not have good central symmetry, the optimal mean wavelength method can reduce the reconstruction error of the sagittal height of the microlens array compared with the peak center wavelength method at the same defocus distance.
