The model for deep ultraviolet lithography illumination systems is typically simplified. Studies have assumed that the illumination beam propagates as a set of plane waves within a given angular range, with each plane wave considered to be incoherent. Here, “incoherent” means that when calculating intensity by summing the squares of the complex amplitudes of the plane waves, the cross terms related to these complex amplitudes are disregarded, and only the squares of the complex amplitude modes are retained. This method is known as intensity superposition. Although any instantaneous light field superposition should theoretically be a superposition of complex amplitudes, the key factor for lithography exposure is the time-averaged intensity. Under incoherent conditions, the cross terms cancel each other out over time, thus they are omitted during incoherent superposition. However, this approximate assumption relies on the source being completely spatially incoherent. Due to the limited exposure time in lithography and the spatial coherence of the lithography source, this assumption is not entirely accurate. The intensity distribution on the image plane results from the coherent superposition of diffracted light from a finite size of a microlens unit and diffracted light from other microlens units illuminating the target surface. When calculating light intensity, it is important to consider the influence of certain cross terms. Gregg et al. have published numerous academic papers on laser speckle and coherence effects on mask surfaces. However, due to the use of KrF or ArF excimer lasers in deep ultraviolet lithography machines, which have multiple transverse modes and low spatial coherence, the complex coherence mode between adjacent microlenses in the illumination pupil is no greater than 2.43%. This results in a negligible decrease in lithographic performance, a matter that has not garnered sufficient attention or discussion in the industry. In recent years, as research on lithography sources has deepened, the demands on illumination systems have increased continuously. Other error factors affecting the illumination field, such as long-term laser irradiation and materials and film systems during actual manufacturing, have been thoroughly analyzed, and corresponding compensation mechanisms have been introduced for correction. However, previously overlooked issues of spatial coherence in sources have become increasingly prominent, and their effects on uniformity degradation need to be accurately reflected in lithographic models. Furthermore, speckle effects caused by them also lead to problems such as the narrowing of process windows, which require further research.

Based on the mutual intensity propagation theory, we build a simplified illumination model for partially coherent systems. The simulation employs parameters relevant to 193 nm lithography, but this model can also be applied to various other types of light sources. Different light sources require varying time coherence lengths and numbers of spatial coherence units, while the selection of spatial coherence units remains consistent. In other words, different light sources can be simulated using the same method by adjusting simulation parameters. After passing through a beam shaping unit, the wide beam emitted by the light source is divided into multiple thin beams, each corresponding to a plane wave illuminated in a specific direction. Traditional lithography models are built on Abbe’s theory, which assumes complete incoherence of the light source and simple superposition of intensity between light source points. However, in practice, due to factors such as finite transverse modes, finite spectral width, and finite exposure time of the light source, there is spatial coherence between light source points, and the complex amplitude cross terms cannot be eliminated. The cross terms generate speckles on both the mask surface and the silicon wafer surface, affecting lithography performance such as illumination uniformity and line width roughness. Abbe’s imaging theory assumption cannot be met due to the spatial coherence of actual light sources, rendering it incapable of accurately describing the spatial coherence’s influence on lithography performance. This research employs a simplified analysis method to partition the light source into a sequence of temporal coherence units and spatial coherence units. We investigate the propagation and superposition process of these coherence units within the illumination system and utilize mutual intensity to characterize the statistical properties of light sources across different spaces. Subsequently, a lithography model grounded on mutual intensity propagation is established.

The results indicate that the perpendicular illumination integration uniformity, concerning the scanning direction, diminishes with increasing coherence across three distinct illumination modes: annular, dipole, and quadrupole. When the spatial coherence falls below 2.43%, the uniformity under annular, dipole, and quadrupole illumination remains at 100%. To meet the illumination requirements, the spatial coherence of annular, dipole, and quadrupole illumination needs to be less than 3.37%, 2.96%, and 3.11%, respectively (Fig. 6). Further analysis of the partial coherence factors’ influence on illumination uniformity reveals that, with a constant spatial coherence function, smaller annular widths correspond to lower illumination uniformity. For dipole and annular illumination, when the annular width is not less than 0.13 and 0.14 (Fig. 7), the mask surface illumination uniformity meets the illumination requirements. Regarding speckle effects, the findings demonstrate a direct correlation between the obviousness of the speckle effect on the mask surface and the coherence size. Speckle contrast increases as spatial coherence does. Moreover, as the number of temporal coherence units grows, the speckle effect induced by spatial coherence becomes less pronounced. The speckle contrasts of annular, dipole, and quadrupole illumination that meet the illumination requirements are less than 2.81%, 2.13%, and 2.33%, respectively (Fig. 11).

Increasing the spatial coherence of lithographic light sources can lead to a reduction in the uniformity of mask surface illumination and cause speckle effects. The current lithographic simulation model is designed for incoherent light sources and cannot adequately account for the impact of spatial coherence on lithographic illumination. This study introduces a lithographic simulation model based on mutual intensity propagation theory and establishes an illumination module using this theory. We examine how the spatial coherence of lithographic light sources affects the uniformity of mask surface illumination and simulate the resulting uniformity under different illumination modes. The findings indicate that, across three different illumination modes-annular, dipole, and quadrupole-the uniformity of illumination perpendicular to the scanning direction decreases as coherence increases. Regarding speckle effects, the results suggest that the visibility of speckle effects on the mask surface is directly correlated with the degree of coherence. Speckle contrast increases with higher spatial coherence. Moreover, as the number of temporal coherence units increases, the speckle effect caused by spatial coherence becomes less prominent.