## 1 Introduction  
A varied-line-spacing (VLS) plane grating is a straight groove periodic structure with a desired groove density function. VLS plane grating can self-focus and reduce aberration. The efficiency and resolving power of instruments can be improved. Compared with the concave grating, the substrate is easy to process. VLS plane gratings are widely used as key components in extreme ultraviolet spectrum analysis, monochromators, spectral imaging, and position sensors [1; 2; 3; 4; 5; 6].  
The most popular methods to pattern VLS plane gratings are diamond ruling, holographic imaging, and electron or laser beam lithography. Traditional direct-write methods such as diamond ruling and electron-beam or laser beam lithography can be extremely slow and expensive, and result in gratings with undesired phase errors [1; 2; 7]. Holographic gratings written by two-beam interference introduced a cost-effective path to more rapidly produce gratings with smoother phases. However, holographic methods require spherical or aspheric wavefront recording optics. The desired groove density frequently has difficulty satisfying geometrical constraints in spherical wavefront recording systems, and groove bending decreases the spectral resolution of the grating. Aspheric wavefront recording optics are complicated to design, manufacture, and align. The deviation and error from the recording optics introduce errors into the groove density distribution [8; 9].  
A variable-period scanning beam interference lithography (VP-SBIL) system is an effective tool to pattern high-fidelity general periodic structures in one or two dimensions within a short period. The VP-SBIL system uses small Gaussian beams with diameters of several micrometers to millimeters to form fringes in a small interference image. The image has a very high phase fidelity. The fringe period and orientation can be progressively changed in a desirable fashion. The fringe phase can be locked using a high-bandwidth fringe locking system. The grating substrate is scanned under the image using a high-performance stage. A uniform exposure dose is achieved by overlapping the subsequent scans. The patterning process is fast and cost-effective because the interference image contains several hundred or thousands of fringes, and expensive aspheric optics are not required in VP-SBIL [10; 11; 12].  
Fabricating a VLS grating with VP-SBIL is not as straightforward as the direct-write method. The groove density of the grating represents the grating phase distribution, which is determined by the exposure phase modulation. To fabricate a straight groove VLS grating using VP-SBIL, three questions must be answered. First, how to express the total exposure dose for patterning a VLS grating with VP-SBIL? Second, which parameters are chosen for the exposure phase modulation? Third, how to cooperatively regulate these parameters to acquire the desired phase distribution? In the published articles about VP-SBIL, the optical topology and basic manufacturing principle for variable-period structures are illustrated. However, details about the phase modulation are not mentioned [10; 11; 12]. We attempt to provide possible and simple answers to these questions. This is a key issue for practical applications of VP-SBIL [13; 14; 15].  
This paper establishes a model of the total exposure dose for a VLS plane grating patterned with VP-SBIL. The exposure phase distribution and exposure contrast in an intuitive manner are verified, and the phase modulation effects of the parameters are given. The parameter selection for the phase modulation and the optimal coordinate for phase compensation to pattern the VLS grating are proposed. The results for an actual VLS plane grating are given. Finally, the reasons for the differences in the phase modulation performance and phase compensation are discussed.