Objective   Multilayer films are critical optical elements for beam energy control in optical systems, and their quality directly influences optical system performance. Ideal monochromatic light serves as the foundation for the design of multilayer optical elements. But there is spectral linewidth in the actual beam. The optical characteristics of the element will differ from the theoretical value and may even result in total failure when operating in non-monochromatic light. In the established convolution model, the calculation technique utilizes monochromatic light conditions to determine the interference superposition of beams in the film, hence disregarding the quasi-monochromatic light interference effect. Based on this, the author proposes a calculation method for the optical properties of multilayer films under quasi-monochromatic light conditions and uses partial coherence theory to calculate the interference superposition of quasi-monochromatic beams.

  Methods   A technique for estimating the optical characteristics of multilayer films in quasi-monochromatic lighting is proposed as a solution to this issue. To quantitatively quantify the linewidth and spectrum distribution of quasi-monochromatic light beams, the normalized power spectral density function is developed. Additionally, partial coherence theory is used to determine the irradiance of quasi-monochromatic light fields at the film interface. This study presents the design of a narrow-band filter with a passband ripple without collapse, a bandwidth of 4.29 nm, a center wavelength of 1 064 nm, and a Rectangle degree of 0.66 (Fig.2). Numerical simulation experiments are used to discuss how substrate thickness, spectral line-shape profile, and beam linewidth affect the optical characteristics of narrow-band filters.

  Results and Discussions   As illustrated in Figure 4, for Gaussian, Lorentz, and rectangular line-shape, the Full Width at Half Maximum (FWHM) drops initially and subsequently increases as linewidth increases. The corresponding lowest values are 3.85 nm, 4.08 nm, and 3.74 nm. And these minimal values correspond to linewidth of 4.5 nm, 2.5 nm, and 5.5 nm, respectively. There is a similar shift trend for the three line-shape conditions for the transmission line's Rectangle Degree (RD). When the linewidth is less than 4 nm, the RD decreases rapidly with the increase of the linewidth. RD steadily diminishes when the linewidth is larger than 4 nm. Under different line-shape conditions, the relationship between the T_max of the transmission spectrum and the linewidth has a significant difference. The beam linewidth and spectra line-shape profile have an important influence on the shape of the spectral line. The increase of the linewidth will lead to the decrease of the transmittance and the variation of the RD. The spectral line-shape profile establishes the precise link between the transmittance, FWHM, and RD with the linewidth. With the right choice of line width value, the FWHM can obtain the smallest value. Figure 7(a) illustrates how, for a given set of four beam linewidths, T_max progressively drops as substrate thickness increases. This decline is limited to 0.8% and is dependent on both the substrate's thickness and extinction coefficient. The smaller the extinction coefficient of the substrate, the smaller the decrease. T_max dramatically drops as linewidth increases when substrate thickness remains constant. The examination of Figure 7(b) demonstrates that, in the case of a constant line width, the FWHM essentially stays constant as the substrate's thickness increases. The FWHM first rises and then falls as the linewidth increases, while the thickness stays constant. It is evident that the transmission spectrum's FWHM, passband shape, and T_max of the narrow-band filter are all significantly influenced by the beam linewidth, while the substrate's thickness primarily determines the transmission spectrum's passband shape.

  Conclusions   This research offers a calculating technique for the optical properties of multilayer films under the condition of quasi-monochromatic light incidence based on partial coherence theory, which can be used to analyze the response characteristics of multilayer optical elements to beam linewidth. Numerical experiments are used to investigate how substrate thickness and beam linewidth affect narrow-band filter performance. Numerical results demonstrate that the narrowband filter's response characteristics are significantly influenced by the beam linewidth and the power spectral density function's line-shape. The incident beam must satisfy the following requirements in order to guarantee the narrow-band filter's passband form: the beam linewidth must be less than half of the theoretical FWHM of the filter, and the spectral line-shape profile must have a tendency toward a rectangular distribution. This study is informative for the design and application of multilayer optical elements in coherent optical systems.