Multiscale Stereolithography Using Shaped Beams

Since stereolithography (SL) was invented in the 1980s [1], various research have been done to improve its fabrication performance including fabrication speed [2,3], part size [4], geometry resolution [5,6], and scalability [4,7,8], etc. However, three main tradeoffs need to be made among the fabrication performances in the SL process, including: (1) fabrication speed versus feature resolution, (2) part size versus feature resolution, and (3) scalability versus fabrication speed. How to address such tradeoffs is the main motivation of our research, which is discussed as follows:

1. (1)

   Fabrication speed versus feature resolution.

   Two types of feature resolutions need to be considered in the SL process, the XY planar resolution and the Z-axis resolution. For the layer-based SL processes such as LSL [9], PμSL [6], and LaPμSL [4], the tradeoff between the Z resolution and the fabrication speed exists as well. In order for the SL processes to achieve a high Z-axis resolution, a thin layer thickness is required, which will lead to an increased number of layers. Because extra time is required in the layer-based fabrication processes to separate the built layers from the platform and to refill liquid resin, the increased number of layers leads to a longer fabrication time. Consequently, the fabrication speed will be reduced. Due to the tradeoff, the commercial SL machines typically use a practical layer thickness such as 50 μm or 100 μm.

   Many approaches have been proposed to reduce the additional time of separating the built layers from the platform and refilling liquid resin. For example, separating built layers by a sliding mechanism may require less time than the separating approach of moving the building platform up and down [3]. The computer numerical control accumulation [10,11] is another method to achieve simultaneous resin curing and recoating by immersing the curing tool inside the photocurable resin; hence, the separation and refilling time can be reduced. However, these approaches can only reduce the part separation and resin refilling time to certain extent. Consequently, using less layer number in the SL process is still preferred to increase the fabrication speed, while it will reduce the Z resolution of the printing process at the same time. Recently, the continuous liquid production method (CLIP) [2] creates a liquid interface between the curing part and the bottom film to facilitate the fast recoating and to eliminate the waiting time for part separation. However, the CLIP process has significant difficulty in fabricating objects with large cross section areas since resin cannot quickly flow into the center of a large area.

   As to the XY resolution, only the laser-based SL process [9] needs to compromise between the XY resolution and the fabrication speed. That is, using a larger laser spot size, one layer can be filled more quickly; however, the feature resolution will be worse at the same time. In comparison, if a smaller laser spot size is used, the quality can be improved; however, the required fabrication time will also increase.

2. (2)

   Part size versus feature resolution

   The projection-based SL processes utilize a digital micromirror device (DMD) to generate a mask image that can be used to cure the whole layer simultaneously. Such capability enables the projection-based SL process to have a faster fabrication speed. However, a DMD has limited number of pixels, e.g., a typical DMD can only have a maximum of 1920 × 1080 pixels. Hence, the part-size-to-feature-size ratio in the projection-based SL processes (e.g., PμSL [6] and CLIP [2]) is limited, which leads to the difficulty in fabricating a large-size part with highly detailed features.

3. (3)

   Scalability versus fabrication speed

   The term “scalability” here refers to the part-size-to-feature-size ratio. As discussed before, in the conventional SL processes, compromise needs to be made between the part-size-to-feature-size ratio and the fabrication speed. It is difficult to directly scale up the micro-SL process to build objects with large sizes. For example, an intuitive method to scale up the micro-SL process is by translating the part in the XY plane so that the overall printing area can be increased [7]. Another scale-up approach is to utilize the scanning mirror to shift the projected image into a larger building area [4]. However, these direct scale-up approaches will take much longer time when fabricating large-size parts.

A multiscale fabrication process is desired to address the aforementioned dilemma between fabrication speed, resolution, and scalability. Several multiscale fabrication approaches have been developed before. For example, by dynamically changing the focus of a laser spot [12], variable laser spot size can be used to cure features at varied sizes. In our previous work [13], we also presented a multiscale fabrication approach using an optical filter with high-contrast gratings. However, only the resolution in the XY plane was considered; the multiscales in the Z-axis have not been considered [13]. In this paper, a complete multiscale fabrication method to achieve high resolutions in all the three axes is presented. The developed fabrication method can tackle the aforementioned tradeoffs based on two mechanisms: multiscale laser beams and multiscale layer thickness.

• (a)

  Laser beams with different sizes can solidify geometric features in different size scales. A large laser beam can cure the main body of a computer-aided design (CAD) model while a small laser beam can fabricate sharp and critical features.

• (b)

  A small layer thickness is applied to fabricate the sharp features while a large layer thickness is used to refill resin and cure the main body of a CAD model.

Based on them, the developed multiscale fabrication process can effectively address the aforementioned tradeoffs and fabricate a macroscale part with microscale features using a fast fabrication speed.

The rest of the paper is organized as follows: Section 2 discusses the multiscale fabrication process and system design. Section 3 gives the tool path generation algorithm for the multiscale fabrication process. Section 4 explains a variety of performed experiments to demonstrate the effectiveness of the presented fabrication process. Section 5 gives the experimental results and discussion. Finally, conclusions are drawn with future work in Sec. 6.

The idea of the proposed method is to use laser beams with optimal shape and scale to fabricate features at different size scales. That is, a large laser beam can be used to cure the main portion of a layer while a small laser beam can be used to solidify boundary contours and sharp features. In addition, laser beams with different shapes can be used to fabricate complex patterns that are appropriate for the laser beam shapes. Such a curing strategy enables fast fabrication speed and high feature resolution since the large laser beam can save time in curing a larger area while the small laser beam can fabricate sharp features.

In order to realize the proposed multiscale fabrication process, we first discuss how to create laser beams with different sizes and shapes. Figure 1 shows the optical path to generate shaped beams. Unlike the optics of the conventional laser-based SL process, a special device with designed apertures is added in the optical path as shown in Fig. 1. The collimated laser beam passes through the controlled aperture before passing through the focusing lenses.

The purpose of adding a designed aperture is to modify the shape of the laser beam so that the correspondingly cured features will have desired shapes and sizes. Figure 2 demonstrates four different apertures and their correspondingly cured geometric features. In Fig. 2, the aperture at the leftmost column is a circle (diameter of 0.5 mm), which enables the whole laser beam pass through. Accordingly, the output cured feature is an elliptical shape, which reflects the profile of the input laser. The second aperture to the left is a small hole (diameter of 0.05 mm), which blocks the outer portion of the input laser beam and let only the beam's center portion pass through. The corresponding cured feature is a 50 μm cylinder, which validates that the proposed aperture can modify the input laser beam regarding its shape and size. The right two apertures show modified laser beams with different shapes; accordingly, the cured features are related micropatterns. Such shaped beams are useful in quickly fabricating a functional surface with different features such as micropatterns of pillars.

The shape of the laser beam's cross section can be modified into any geometry in a fashion similar to the shaped-beam electron-beam lithography, which has been used in the semiconductor industry. Due to the small aperture sizes, an added aperture plate can have a library of beam shapes with hundreds of different sizes and shapes. By integrating variable apertures in the optics, our SL process can simultaneously improve the fabrication speed and surface quality. The aperture plate with desired sizes and shapes can be fabricated by common lithography technologies such as photolithography and nanoimprint lithography [14].

The meaning of “multiscale” in this paper is twofold. One is multiscale beams, which is related to the XY resolution; another is multiscale layer thickness, which is related to the Z resolution. Different scales have their unique advantages. For example, a small laser beam and a small layer thickness can cure features with high resolution, while a large laser beam can quickly solidify a large portion of resin, and a large layer thickness will require less resin recoating time.

In this section, we present a multiscale fabrication process based on multiscale laser beam and multiscale layer thickness. The method of selecting appropriate laser beam sizes and layer thicknesses based on the given CAD model's geometry is presented. Figure 3 shows the framework of such a multiscale fabrication process. First, a given three-dimensional (3D) model is sliced into a set of two-dimensional (2D) layers using a large-scale layer thickness. Then, each large layer is further sliced into many small 2D layers using a small-scale layer thickness. We then use the following curing strategy to fabricate the sliced layers.

• Use the small-scale laser beam to fabricate the boundary of each small layer, which contains the shapes of all the detail geometric features.

• After the boundary portions of a small layer have been cured, move the Z linear stage up by the small-layer thickness.

• After all the small-layers' boundaries in this large layer have been fabricated, move the Z stage up and down so that the part can be completely recoated.

• After the fabricated layers were fully recoated with resin, the common interior area of the large layer is cured using the large-scale laser beam.

The above curing strategy is repeated for every large layer until all the 2D layers have been fabricated. The algorithm to divide a sliced layer into boundary and interior portions, and accordingly, to plan the scanning paths for different size-scale laser beams will be discussed in Sec. 3.

Such a fabrication process significantly improves the printing speed in two aspects. First, only the boundary is built for each small layer. This can save a significant amount of scanning time since the inner portion is not filled by the small laser beam. Second, the separation time and recoating time for each small layer are significantly less. In each small layer, only the boundary portion is cured, which is bounded with the constrained window; such boundary portion can be easily separated and recoated due to the small area. The details of how these three aspects help to speed up the proposed process will be further discussed in Sec. 4.

Figure 4 shows the schematic diagram of our prototype system. Figure 5 demonstrates the benefits of the presented multiscale SL process. The pyramids (a) and (d) were fabricated using a similar fabrication time. However, pyramid (a), which was fabricated by the multiscale process, has a much smoother slope surface with a sharp corner. In comparison, pyramid (d) was fabricated using the traditional layer-based SL process, which shows typically stair-stepping defects. More results can be found in Sec. 5.

An algorithm to divide a sliced 2D layer into boundary and interior and to plan the scanning paths for different size-scale laser beams is presented in this section. The algorithm is inspired by the infeasible features identification method [15], in which a computation method is presented to recognize the minimum feature size enabled by a 3D printing machine to ensure the small features of an input CAD model can be fabricated. Similarly, in the multiscale SL process, we need to identify which regions can be cured by a large-size laser beam and which features require a small-size laser beam instead. Accordingly, different tool paths for both small and large-size laser beams will be generated. Our method is mainly based on two geometric operations: the grow- and shrink-based offsetting operations with a distance r on a given solid S [16]. Suppose a ball is defined as $br$ with radius r, the two offsetting operations can be defined as:

1. (1)

   S grown by r as $S↑r=S⊕br$⁠, and

2. (2)

   S shrunk by r as $S↓r=S⊗br$⁠.

We now introduce our multiscale toolpath generation approach for the multiscale SL process. The algorithm structure is first presented followed by the algorithm implementation.

A toolpath p(r, h) is defined as a function of radius r and thickness h, which is evaluated on every slicing at the desired resolution. Within each slice, the generation process can be broken into two steps: (1) computing p_A(r_small, h_thin) and p_B(r_large, h_thin) on the XY plane with thickness h_thin, and (2) computing p_C(r_large, h_thick) along the Z-axis with thickness h_thick—this is the case when the interior of a large region is filled. Note that in this paper, we only use three types of beam (p_A, p_B, p_C) to illustrate the multiscale fabrication idea. Our method is general and can be used for more beam size and shape variations.

Toolpaths on the XY plane: In the algorithm, the first step is to determine the offset region of the large beam $pB$⁠. Given a two-dimensional loop defined in E² as shown in Fig. 6 (bottom row), the interior region D will be fabricated by toolpaths $pA$ and $pB$⁠. The region is first offsetted by the large laser beam's radius $↓r$⁠, and we can obtain the start point at the top-left corner of $D↓r$ for the toolpath. The tool tip (at center of beam) then sweeps from left to right and top to bottom until it reaches the bottom-right corner in $D↓r$⁠, producing scanline pattern for p_B. Via subtracting the offset region of p_B, D − $pB↑r$ is the rest of the region that is separated for the small beam p_A (outlined at the bottom-right of Fig. 6). Toolpath p_A is built upon the same process. The overall procedure is illustrated in Fig. 6. For toolpath planning with N scales of beam, N − 1 times of subtraction are performed to produce N number of toolpath regions. The process will be terminated whenever the region is empty since there will be no starting point for current tool tip.

We next discuss how to generate the toolpath for different Z thicknesses.

Toolpaths in the Z-axis: Integrating toolpath p_C with large radius r_large and thickness h_thick in the process can effectively reduce the fabrication time. Given m number of slices W, which represent the cross section of an object with thickness h_thin, and the thickness h_thick of p_C should equal to m × h_thin. We first search for any common interior regions that can be fabricated in all of the m slices. By doing so, intersection operations between these m slices are performed and region E for p_C is produced (as shown in the top row of Fig. 6). After that, we shrink the region E with r_large to generate p_C and grow the path p_C with r_large to acquire the offset region of p_C. Once the offset region O_c is determined, we can produce toolpaths p_A and p_B from the subtracted region (D − O_c) for every slice $∈W$⁠. This process will be repeated in every m number of slices of the object. Figure 7 (left) illustrates the main idea. Those m layers are fabricated without the common region, and this region will be filled right after the mth layer is produced. Figure 3 illustrates the overall process in the XY plane and along the Z direction.

The presented algorithm is simple, easy to implement, and able to generate scanning paths for laser beams with different sizes and thicknesses. One implementation of the algorithm is discussed as follows.

We implement our slicing using the layered depth normal image (LDNI) model [17] since a computed LDNI model can be easily converted into high resolution images by classifying the in/out state of a given point. We slice at a resolution between 20 μm to 80 μm per pixel.

Contour: To ensure small features can be made in the process, we reserve a boundary region for contour generation (see Fig. 6). Contour toolpath is defined by a chain of small segments that describe the features along the surfaces tangent. We use the small beam (r_small, h_thin) to fabricate the boundary, and the toolpath construction is based on the work [18]. We reserve the boundary region by shrinking the cross section O with radius equal to $d=2×rsmall$ (diameter of small beam $B0$⁠), and the computed offset $O↓d$ will then become the input for toolpath planning of $pC.$

Distance between scanline pattern: For toolpaths $pA, pB$⁠, and $pC$⁠, one of the parameters that can be adjusted by users is the distance between each scanline. The tool paths need certain overlapping since the laser beam may not be exactly of the same radius as expected. This could be caused by the noise and vibration from the machine. In order to ensure the tool paths are blended together, we set the overlapping rate $α$ as 0.7 such that the distance between two scanlines is equal to $2×r×α.$

Buffer region for toolpath$pC$⁠: Another concern considered is that the extracted common region E could lead to empty region (or nearly empty) in some slices from W. An example is illustrated in Fig. 7. Even though we have boundary region to prevent E from completely empty, it is difficult for the (i + 1)th layer to support the rest of the layer from W. Therefore, a buffer region, with radius $β$⁠, is also added to prevent potential failure. The pseudo-code of our whole toolpath generation is shown in Algorithm 1.

The theoretical analysis of the main characteristics of the developed multiscale fabrication process is presented in the section. We first present how to control the beam shape for different sizes. The process of recoating resin is then explained. Finally, the fabrication speed based on the analytic results is discussed.

In order to control the focused spot shape, we analyze how the laser beam changes after passing through apertures that are a set of pinholes. Essentially small pinholes are a low-pass filter, which can filter out the high-frequency laser mode, resulting in a clean Gaussian beam. Therefore, a pinhole aperture can be used to regulate the laser beam and to significantly reduce the size of the laser spot.

Specifically, due to the Fraunhofer diffraction when light beam passes through a pinhole, the output beam follows a complex airy pattern, which is derived from Fraunhofer diffraction theory [19]. As shown in Fig. 8, the airy pattern has a series of sharp dark rings alternating with broader bright rings. The core of the airy pattern contains 86% of the total light in the image, and this core ring can be approximated by a Gaussian beam with the same peak and full width at half maximum diameter [20]. This Gaussian beam approximation works nicely in our fabrication process since only the core peak region can reach the critical resin curing energy $Ec$⁠.

Equation (1) governs the size of a beam shape, which shows that simultaneously increasing the exposure time and decreasing the aperture diameter can improve the feature resolution. However, increasing the scanning speed will sacrifice the XY scanning position resolution, which provokes a large scanning distortion. Also, the shape feature cannot be changed only by tuning the exposure time. In comparison, decreasing the aperture diameter can improve feature resolution without affecting the scanning position accuracy. This is the major benefit of using the dynamic apertures to change the beam shape and size as discussed in our method. In addition, we conducted dosage tests (i.e., finding the appropriate scanning speed) for each aperture to compensate for the laser power change due to different aperture diameters.

The main reason for fast recoating of small layers is shown in Fig. 9. The resin flow can be modeled as Hele–Shaw flow [11]. Denote the small layer thickness as $h$ (20 μm), the boundary width as $b$ (0.5 mm) and the boundary perimeter as $L$⁠. According to Hele–Shaw flow modeling, the refilling liquid resin flow is

This equation indicates that the recoating time has a second-order relationship with the boundary width. In our implementation, the viscosity of the resin is around $μ=0.1 Pa⋅s$⁠, the air pressure is $P=101325 pa$⁠, $h=0.02 mm,$ and $b=0.5 m$⁠. Hence, the recoating time is $t=12×0.1×(0.52/(101325×0.022))=7.4 ms$⁠, which is sufficiently fast for the 3D printing process. In contrast, the conventional SL process recoats the whole layer instead of the boundary, and the recoating width of each layer can be as large as the whole platform, e.g., 50 mm. Hence, the recoating time for the whole layer can reach to $t=12×0.1×(502/(101325×0.022))=74 s$⁠, which is significantly longer. This explains why the up-and-down motion is required for the conventional stereolithography apparatus (SLA) process in order to speed up the resin recoating process.

As shown in the earlier discussion, the recoating time of the small layers' boundary is ignorable compared with the large layers' recoating time. Hence, in our multiscale fabrication process, only the boundary portion is cured for each small layer. Hence, only the boundary portions require recoating. Due to the ignorable recoating time for these boundaries, our process can achieve nearly continuous polymerization for the small layers. By making full advantage of it, the proposed method not only maintains the same fabrication speed for the large layer thickness but also preserves the detailed features that can be fabricated using the small layer thickness.

Figure 10 illustrates the fabrication time of the conventional SL process and the proposed multiscale SL process. The meaning of the notations can be found in the nomenclature. In Fig. 10, the bottom row shows the time distribution of the proposed process, i.e., curing five small layer boundary, and then curing one large layer with one up-and-down motion. We omit the boundary recoating time and the aperture changing time since they are much less than others. In the layer fabrication process, the aperture changes twice, from the large beam to the small beam and vice versa. The total changing time in our prototype system is less than 1 s. This changing time can be further shortened if a faster switch mechanism is applied. In comparison, as shown in the top row of Fig. 10, the conventional process to achieve similar feature resolution requires the recoating time $tud$ in order to recoat the whole layer, and $ti0$ by using the small laser beam to cure the whole layer.

Note $ti0≫tud$ happens when the target part has a large solid area. For such cases, our multiscale fabrication approach makes full advantages of the multiscale beam diameters and multiscale layer thicknesses, which is $SZSXY$ times faster than conventional layer-based SL process. For example, in our prototype system, S_Z = 5 and S_XY = 4. Hence, our current implementation is $5×4=20$ times faster than the traditional SL process in order to achieve the similar XYZ resolutions.

When $ti0$ is comparable with $tud$⁠, the time for the layer separation and resin recoating is longer than the resin curing time. This happens when the part is small. In such cases, our process needs only one up-and-down transition for every $SZ$ layers. Compared with the up-and-down transition for every layer in the conventional SL process, our fabrication speed is $SZ=5$ times faster.

Table 1 shows the fabrication time of the tested models. All parts are fabricated using 20 μm layer thickness. A is the bounding box area (unit $mm2$⁠), H is the height of the model (unit $mm$⁠), $tmulti$ and $tsingle$ are the fabrication time of our method and the conventional SL process, respectively (unit minute), and Vm/Vs means the fabrication speed ratio.

The proposed SL process offers two main benefits: (1) providing a cost-effective and robust shaped-beam method to fabricate 3D objects with high-resolution features and (2) presenting a multiscale tool path planning framework to efficiently fabricate 2D layers using shaped beams. We used CAD models with various geometries to verify the presented multiscale SL process.

In general, we can use a single aperture based on a small pinhole to generate a small laser spot, and use it to cure any shape of a given part. However, it requires significantly long scanning time in order to build lattice structures with a large number of beams. In order to speed up the fabrication speed, an aperture with 3 × 3 pinholes can be used to shape the laser beam. The correspondingly fabricated pattern is a 3 × 3 dots array as shown in Figs. 11(a) and 11(b). By using such a shaped beam with multipinholes, the resin curing speed is nine times faster than that of using a single pinhole.

A porous lattice structure has a large elasticity, which does not exist in a solid object using the same material. Figure 11(f) shows the compression test. A 40% elasticity strain is observed. The purposes of using shaped beams, like the one with multiple holes, are mainly to address two issues: (1) making the curing simultaneously so that the fabrication speed can be increased and (2) fabricating specific micropatterns that can modify the surface texture of an object surface. The second issue will be demonstrated in our future work.

In addition to the lattice structures, three more complex parts have been fabricated to verify the proposed multiscale tool path planning framework. The fabrication results based on the accordingly planned tool path are shown in Fig. 12. The successful fabrication of these parts validates the proposed multiscale fabrication process. Especially, the part “Lion” has features ranging from the centimeter's main body to tens of micrometers' hairs.

Figure 13 shows the features at different scales that are fabricated by our multiscale SL process. The Lion part has been sliced into 500 layers using 20 μm layer thickness. If the slices are fabricated using the conventional SL process, the small laser beam needs to scan the whole 500 layers, and after curing each layer, a Z stage linear translation will then be followed. On average, it will take ∼90 s to cure the whole layer using a small laser beam, and ∼30 s for the Z stage to move up and down for a large distance so that liquid resin can be recoated to build next layers. In comparison, our multiscale fabrication process saves time both in the XY plane and along the Z direction. In the XY plane, only the boundary area will be cured by the small laser beam while the interior area is quickly cured by the large laser beam. Along the Z direction, for each small layer with 0.02 mm thickness, only the boundary will be cured, and the Z stage only moves up by a small distance of 0.02 mm. The interior area will be cured in every five layers using the large laser beam, and the Z stage will move up and down once. Thus four out of five times of curing the interior area and moving the Z stage up and down would be saved. Combining all these benefits, it only takes <20 s for each layer on average, which is at least six times faster than the conventional SL process.

As discussed in Sec. 1, many novel SL processes have been developed. We compare our multiscale SL process with some representative SL processes in literature, including the laser-based SLA (LSL) [9], the projection-based micro-SLA (PuSL) [6], the two photon polymerization (TPP) [23], the continuous interface liquid production [2], and the large area projection-based micro-SLA (LaPuSL) [4]. Table 2 lists the details of the comparison. We compared these processes in five major fabrication metrics including part size, feature resolution, part-size-to-feature-size ratio, fabrication speed, and cost.

The data in Table 2 are estimated from the referred website or the published papers. The fabrication speed is measured as the time to fabricate the Lion model in Fig. 13. The ratio is calculated as the ratio part size to feature size. As shown in Table 2, the conventional SL processes face tradeoffs among fabrication speed, resolution, scalability, and cost. On the contrary, the proposed method partially solves these tradeoffs by using the idea of the multiscale laser beams in the XY plane and multiscale layer thicknesses in the Z direction. The main components of our setup are off-the-shelf and the total cost of them is less than $2000. In comparison, the other SL processes are more expensive since they use either a more expensive digital micromirror device or a femtosecond laser, as well as some specially designed components such as an oxygen-permeable membrane and a customized F-theta focusing lens. To the best of our knowledge, no other fabrication processes can provide such a combination of large part size, microscale features, fast fabrication speed, and low cost.

The paper presents a novel multiscale fabrication process and the related tool path planning framework for fabricating three-dimensional macroscale objects with microscale features. A major contribution of our work is to present two multiscale fabrication mechanisms to optimize the fabrication speed, feature resolution, and scalability simultaneously. The first multiscale fabrication mechanism is to use shaped laser beams with different sizes and shapes. A simple approach to achieve such shaped laser beams is discussed based on switching the apertures on a plate. A small laser beam can be used to build shape details while a large laser beam can be used to cure the interior region of the object. The second multiscale fabrication mechanism is to use multiscale layer thicknesses in the Z-axis. This novel multiscale curing strategy enables our fabrication process to be not only fast but also capable of achieving a high XYZ resolution. The fabrication results based on the developed process have verified the benefits of the multiscale curing strategy.

One future work is to build 3D objects with microscale surface textures with some unique properties. In addition, we are investigating how to apply the presented multiscale curing strategy in the projection-based SL process as well as the fused deposition modeling process.

• Directorate for Engineering (Grant No. CMMI 1151191).

• University of Southern California (Technology Advancement Grant).

$B0$ =

the size of small beam

$B1$ =

the size of large beam

$h0$ =

the layer thickness of small layers

$h1$ =

the layer thickness of large layers

$I$ =

light intensity

$sXY$ =

scale ratio of beam area in the XY plane

$sZ$ =

scale ratio of layer thickness along the Z direction. $sZ=h1/h0$

$SXY$ =

$B1/B0$

$tb0$ =

time for curing small layers' boundary

$tb1$ =

time for curing large layers' boundary

$ti0$ =

time for curing small layers' inner portion

$ti1$ =

time for curing large layers' inner portion

$tud$ =

time for up-and-down (resin recoating)

$μ$ =

viscosity of resins