EP4658481A1

EP4658481A1 - High dimensionality volumetric additive manufacturing

Info

Publication number: EP4658481A1
Application number: EP24750866.6A
Authority: EP
Inventors: Robert R. Mcleod
Original assignee: University of Colorado System; University of Colorado Colorado Springs; University of Colorado Denver
Current assignee: University of Colorado System; University of Colorado Colorado Springs; University of Colorado Denver
Priority date: 2023-01-30
Filing date: 2024-01-30
Publication date: 2025-12-10
Also published as: WO2024163474A1; KR20260033470A; MX2025008837A; CN120835826A; JP2026505053A

Abstract

A three-dimensional (3D) printing apparatus includes: a light source comprising a plurality of pixels configured to emit a plurality of beams toward an optical objective; the optical objective configured to diverge the plurality of beams as the plurality of beams exits the optical objective toward a volume of resin; and one or more motors configured to translate the optical objective with respect to the volume of resin. The one or more motors are configured to translate the optical objective relative to the volume of resin to apply that plurality of beams to a plurality of voxels inside the volume of resin. The application of the plurality of beams to the plurality of voxels solidifies the plurality of voxels. The optical objective applies the plurality of beams such that each of the plurality of voxels are exposed by at least two different beams among the plurality of beams.

Description

HIGH DIMENSIONALITY VOLUMETRIC ADDITIVE MANUFACTURING

FIELD OF THE DISCLOSURE

[0001] Embodiments described herein relate generally to an apparatus, systems, and methods for manufacturing a three-dimensional (3D) object. More particularly, embodiments described herein relate to apparatus, systems, and methods for using light exposure to solidify a photosensitive resin in order to form the 3D object.

BACKGROUND

[0002] Conventional volumetric additive manufacturing (VAM) printing uses three or less degrees of freedom (hereinafter, will be referred to as “dimension”) to fabricate an object. Examples of the conventional VAM methods are Computed Axial Lithography (CAL) and Xolography, which both use three dimensions for printing. In CAL, two dimensional images are exposed, typically in a normal angle, onto transparent cylinder containing resin rotating around one axis, for three total dimensions of light exposure. Volume helical additive manufacturing is a variant of CAL which additionally translates the cylindrical vial along the axis of rotation, using this additional translation to extend the effective size of the 2D image projector and not to access an independent degree of freedom and is thus can be classified as using the identical dimensions as CAL. Xolography uses a sheet of one color of light translating linearly in one dimension, sequentially projecting two dimensional images from an orthogonal dimension. However, additional degrees of freedom may increase printing resolution, fidelity and/or enable printing more complex structures.

SUMMARY

[0003] This summary is provided to introduce a selection of concepts that are further described below in the detailed description. This summary is not intended to identify key or essential features of the claimed subject matter, nor is it intended to be used as an aid in limiting the scope of the claimed subject matter.

[0004] One or more embodiments described herein are related to a three-dimensional (3D) printing apparatus. The 3D printing apparatus includes: a light source comprising a plurality of pixels configured to emit a plurality of beams toward an optical objective; the optical objective configured to diverge the plurality of beams as the plurality of beams exits the optical objective toward a volume of resin; and one or more motors configured to translate the optical objective with respect to the volume of resin. The one or more motors are configured to translate the optical objective relative to the volume of resin to apply that plurality of beams to a plurality of voxels inside the volume of resin. The application of the plurality of beams to the plurality of voxels solidifies the plurality of voxels. The optical objective applies the plurality of beams such that each of the plurality of voxels are exposed by at least two different beams among the plurality of beams. The at least two different beams expose the each voxel at two different times such that the at least two different beams spatially overlap only in the each voxel. A total exposure that solidifies the volume of resin in the each voxel equals to at least a sum of exposure of the at least two different beams at the each voxel.

[0005] One or more embodiments described herein are related to a method for operating a 3D printing apparatus. The method includes: controlling a light source comprising a plurality of pixels to emit a plurality of beams toward an optical objective; diverging the plurality of beams, via the optical objective, as the plurality of beams exits the optical objective toward a volume of resin; translating, via one or more motors, the optical objective with respect to the volume of resin; by translating the optical objective relative to the volume of resin, applying the plurality of beams to a plurality of voxels inside the volume of resin; solidifying the plurality of voxels by the application of the plurality of beams to the plurality of voxels; applying the plurality of beams, via the optical objective, such that each of the plurality of voxels are exposed by at least two different beams among the plurality of beams; and exposing, via the at least two different beams, the each voxel at two different times such that the at least two different beams spatially overlap only in the each voxel. A total exposure that solidifies the volume of resin in the each voxel equals to at least a sum of exposure of the at least two different beams at the each voxel.

[0006] One or more embodiments described herein are related to a system for 3D printing. The system includes: a light source comprising a plurality of pixels configured to emit a plurality of beams toward an optical obj ecti ve; the optical obj ective configured to diverge the plurality of beams as the plurality of beams exits the optical objective toward a volume of resin; one or more motors configured to translate the optical objective with respect to the volume of resin; a processor configured to control the one or more motors and thereby control translation of the optical objective over the volume of resin; a memory that stores instructions for translating the optical objective over the volume of resin; wherein the processor controls, based on the instructions, the translation of the optical objective relative to the volume of resin to apply that plurality of beams to a plurality of voxels inside the volume of resin. The application of the plurality of beams to the plurality of voxels solidifies the plurality of voxels. The processor controls the translation of the optical objective to apply the plurality of beams such that each of the plurality of voxels are exposed by at least two different beams among the plurality of beams. The processor controls the translation of the optical objective such that the at least two different beams expose the each voxel at two different times. The at least two different beams at the two different times spatially overlap only in the each voxel. A total exposure that solidifies the volume of resin in the each voxel equals to at least a sum of exposure of the at least two different beams at the each voxel.

[0007] Other aspects and advantages of the claimed subject matter will be apparent from the following description and the appended claims.

BRIEF DESCRIPTION OF DRAWINGS

[0008] Specific embodiments of the disclosed technology will now be described in detail with reference to the accompanying figures. Like elements in the various figures are denoted by like reference numerals for consistency. The sizes and relative positions of elements in the draw ings are not necessarily drawn to scale. For example, the shapes of various elements and angles are not necessarily drawn to scale, and some of these elements may be arbitrarily enlarged and positioned to improve drawing legibility. Further, the particular shapes of the elements as drawn are not necessarily intended to convey any information regarding the actual shape of the particular elements and have been solely selected for ease of recognition in the drawing.

[0009] FIGs. 1A-1C illustrate printing dimensionality (i.e., degrees of freedom) with respect to angle of illumination. [0010] FIG. 2 illustrates six printing dimensions for illumination of a resin volume with the light from a single point on the image display chip, here labeled the Point Spread Function (PSF).

[0011] FIG. 3A illustrates a 3D schematic of a 3D printer, in accordance with one or more embodiments disclosed herein.

[0012] FIG. 3B illustrates a longitudinal cross-section of the schematic show n in FIG. 3A.

[0013] FIGs. 4A and 4B illustrate an operation of a 3D printer, in accordance with one or more embodiments.

[0014] FIGs. 5A-5C illustrate discretizing an object model, in accordance with one or more embodiments.

[0015] FIG. 6 illustrates a system for 3D printing, in accordance with one or more embodiments.

[0016] FIG. 7 shows a flowchart for 3D printing, in accordance with one or more embodiments.

[0017] FIGs. 8A-8C illustrate 3D printing in cylindrical coordinates, in accordance with one or more embodiments.

[0018] FIGs. 9A-9D respectively illustrate a target object, calculated intensity, calculated dose, and error for 3D printing of the target obj ect, in cylindrical coordinates, in accordance with one or more embodiments.

[0019] FIGs. 10A and 10B respectively illustrate a target object and a close-up view of a boundary of the target object, in accordance with one or more embodiments.

[0020] FIG. 11 A illustrates a dose goal and a close-up view of the dose goal for the target object of FIG. 10 A, in accordance with one or more embodiments.

[0021] FIG. 12 shows an image set calculated by forward projection of the dose goal shown in FIG. 11, in accordance with one or more embodiments.

[0022] FIG. 13 shows calculated error between a backward projected object and the dose goal shown in FIG. 11, in accordance with one or more embodiments. [0023] FIGs. 14A-14C show forward projection of the error shown in FIG. 13 respectively for interior of the object, exterior of the object, and the final sum of signed errors, in accordance with one or more embodiments.

[0024] FIG. 15 shows errors with respect to iteration, in accordance with one or more embodiments.

[0025] FIG. 16 shows a final image set corresponding to modifying the image set shown in FIG. 12 based on the errors shown in FIGs. 14A-14C and 15.

[0026] FIG. 17 shows the final object to be printed based on the final image set of FIG. 16, in accordance with one or more embodiments.

[0027] FIGs. 18A and 18B respectively show errors between the final object of FIG. 17 and the dose goal of FIG. 11 and a close-up view of the errors, in accordance with one or more embodiments.

[0028] FIG. 19 shows a flowchart for determining a final image set for 3D printing of an object, in accordance with one or more embodiments.

[0029] FIG. 20 shows a computer system for 3D printing an object, in accordance with one or more embodiments.

DETAILED DESCRIPTION

[0030] In the following detailed description of embodiments of the disclosure, numerous specific details are set forth in order to provide a more thorough understanding of the disclosure. However, it will be apparent to one of ordinary skill in the art that the disclosure may be practiced without these specific details. In other instances, well-known features have not been described in detail to avoid unnecessarily complicating the description.

[0031] Throughout the application, ordinal numbers (e.g. , first, second, third, etc.) may be used as an adjective for an element (i.e., any noun in the application). The use of ordinal numbers is not to imply or create any particular ordering of the elements nor to limit any element to being only a single element unless expressly disclosed, such as using the terms "before," "after," "single," and other such terminology. Rather, the use of ordinal numbers is to distinguish between the elements. By way of an example, a first element is distinct from a second element, and the first element may encompass more than one element and succeed (or precede) the second element in an ordering of elements.

[0032] In the following description of FIGs. 1-20, any component described regarding a figure, in various embodiments disclosed herein, may be equivalent to one or more like-named components described with regard to any other figure. For brevity, descriptions of these components will not be repeated regarding each figure. Thus, each and every embodiment of the components of each figure is incorporated by reference and assumed to be optionally present within every other figure having one or more like-named components. Additionally, in accordance with various embodiments disclosed herein, any description of the components of a figure is to be interpreted as an optional embodiment which may be implemented in addition to, in conjunction with, or in place of the embodiments described with regard to a corresponding like-named component in any other figure.

[0033] It is to be understood that the singular forms “a,” “an,” and “the” include plural referents unless the context clearly dictates otherwise. Thus, for example, reference to “a wellbore” includes reference to one or more of such wellbores.

[0034] Terms such as “approximately,” “substantially,” etc., mean that the recited characteristic, parameter, or value need not be achieved exactly, but that deviations or variations, including for example, tolerances, measurement error, measurement accuracy limitations and other factors known to those of skill in the art, may occur in amounts that do not preclude the effect the characteristic was intended to provide.

[0035] It is to be understood that one or more of the steps shown in the flowcharts may be omitted, repeated, and/or performed in a different order than the order shown. Accordingly, the scope disclosed herein should not be considered limited to the specific arrangement of steps shown in the flowcharts.

[0036] Embodiments disclosed herein are regarding a three-dimensional (3D) printing of an object by shining light on a resin. Throughout this disclosure, the 3D printing is also referred to as volumetric additive manufactunng (VAM). To this end, an image set is created using an object model. Then, the image set is illuminated through an optical objective (hereinafter, “objective”) into the resin. The illumination of the set of images on the resin solidifies the resin in areas that correspond to the object model. Then the solidified resin, which will be the printed object, will be extracted from the resin. To this end, the solidified resin may be put in a bath to wash away the nonsolidified resin.

[0037] Embodiments disclosed here in describe a new class of architectures for VAM. The classic VAM method, Computed Axial Lithography (CAL), projects 2D images into a volume that is rotated around a single axis, providing 3D spatial projection of light onto the resin (hereinafter, “spatial addressing”). This arrangement is shown in FIG. 1A. Specifically, FIG. 1 A shows a Digital Micromirror Device (DMD) (104) that proj ects 2D images onto a resin (102). Note that the lenses implementing the proj ection are not shown but are well known to a practitioner in the art. The resin (102), which may be in a liquid form, is disposed in a cylindrical container. By rotating the cylindrical container around its rotation axis, which is its central axis, the DMD (104) can illuminate the images onto the resin in three dimensions, two dimensions for the two dimensional array of pixels arranged on the DMD projection and one rotational dimension of the resin (102). For example, a single voxel of the resin (102) in FIG. 1 A can be uniquely addressed by intersecting rays at the voxel by choosing appropriate pixels on the DMD (104) to turn on in each image in an image set wherein one image from each image in the set illuminates the resin at one rotation angle of the resin.

[0038] Tilting the rotation axis with respect to the projection direction of the DMD (104) can be viewed as reducing the dimensionality of the spatial addressing to less than 3 with a consequent reduction in printing capability. The reduction of the spatial addressing is shown in FIGs. IB and 1C. Specifically, if the images are projected parallel to the rotation axis of the resin (102), as shown in FIG. 1C, the 3D addressing is lost. This is because the rotation of the resin (102) does not provide any additional addressing of the resin (102) with respect to the 2D illumination of the DMD (104). Thus, the configuration of FIG. 1C is reduced to a 2D printer, similar to traditional lithography. FIG. IB shows a transition between FIGs. 1A and 1C. In FIG. IB, the angle between the optical projection of the DMD (104) and the rotation axis of the resin (102) is between 0 and 90 degrees. Accordingly, the control over printed structures, in FIG. IB, varies between 3D corresponding to FIG. 1A and 2D corresponding to FIG. 1C. In other words, the configuration of FIG. IB can be considered to have fractional dimensionality addressing of the resin (102) between 2 and 3 independent dimensions. Thus, one can characterize the configurations illustrated in FIGs. 1 A-1C as providing a variable dimensional addressing of the resin (102) where the dimensionality ranges from 3 to 2.

[0039] Various 3D printing metrics improve as the dimensionality increases from 2 (where 3D printing is impossible) to 3 (like traditional CAL). These metrics include the minimum size of the 3D printed voxel, the complexity of objects that can be printed, and the contrast of the optical dose deposited by the printer between in-part and out-of- part regions of the object being printed. According to one or more embodiments, increasing the dimensionality of the writing images may increase such printing quality metrics.

[0040] One or more embodiments describe new printing architectures with effective printing dimensionality greater than 3. As shown in FIG. 2, the intensity projected from a single pixel of the display device has a 3D shape, which is called the point spread function (PSF). A common PSF is a Gaussian beam. Like a solid body, a PSF has six degrees of freedom, commonly referred to as “rigid body modes/’ Among the six degrees of freedom, three are displaced along Cartesian axes (202) and the remaining three are rotations (204) around the Cartesian axes. Commonly, PSFs are rotationally symmetric around the optical axis of propagation. In the case of symmetry, rotation around that axis would not change the light distribution in the resin, reducing the useful number of modes to five. Further, conventional CAL uses a large depth-of-focus “pencil beam” PSF that is invariant to translation along the optical axis, for example along the vertical axis among the Cartesian axes shown in FIG. 2. This configuration reduces the number of effective modes to four. However, Gaussian beams with depth- of-focus smaller than the resin are well known in imaging tomography and have been explored for CAL.

[0041] As shown in FIG. 1 A, conventional CAL uses a DMD (104) to translate the PSF through two transverse dimensions corresponding to the two-dimensional grid of pixels. Additionally, the resin (102) is rotated on an axis orthogonal to the optical propagation direction. Accordingly, the conventional CAL corresponds to three of the possible six degrees of freedom illustrated in FIG. 2 - two translations along two Cartesian axes of the projected DMD pixels (202) and one independent rotation of the resin relative to the projected image. FIG. 2, however, illustrates that it is possible to rotate and translate projected light through as many as six dimensions (i.e. , degrees of freedom).

[0042] According to one or more embodiments, in VAM printing, initially a set of optical images must be determined. Illumination of the optical images from an exterior of the resin volume causes the resin to solidify in a shape approximating the desired 3D object. This is an inverse problem to be solved. In the field of tomographic imaging, this inverse problem may be easier to solve as “useful” additional images are added. “Useful” images mean those images that add significant information not available in the remainder of the image set. For example, increasing the density of images presented during rotation may be useful but only up to a limit known in the tomographic imaging. Conversely, according to one or more embodiments, the quality of the inverse solution may be improved by adding images that are incident at positions or angles quite distinct from those in the original set. This can be understood by considering the desired dose delivered to the resin volume as a set of constraints and the intensity of every pixel in the set of images as variables available to meet the constraints. The intensity of every pixel, sampled at every voxel in the resin volume forms a set of linear equations that relate variables (pixel intensities) to constraints (voxel doses). The greater the number of independent variables, the better the possible solution. “Independent” in this context is essentially the same concept as “useful” as introduced before. That is, adding image sets whose PSFs are distinct may improve the solution.

[0043] FIGs. 3A-3B and 4A-4B illustrate a 3D printer (i.e., a 3D printing apparatus), in accordance with one or more embodiments. In one or more embodiments, the 3D printer is configured such that multiple independent rays (i.e., beams of light) may pass through a voxel in the resin, at different angles. To this end, a set of images that provide the a set of angles may be determined. A light source (402) including a plurality of pixels illuminates a plurality of beams of light (302) through an objective (306). When the plurality of pixels are on, a cone of light (408) is formed. The objective (306) diverges the exiting beams (exiting from the objective) (304) traveling in the resin (310). Upon moving the objective (306) with respect to the resin (310) and specifically choosing an image in the image set, the 3D printer can control the angles of illumination of diverged beams (304) as well as the voxels, in the resin (310), to which the beams are illuminated. Controlled application of the diverged beams (304) to the voxels solidifies the voxels and create the object.

[0044] In one or more embodiments, the divergence of the beams may be between a half angle from the optical axis to 30 degrees or more, measured in air. According to one or more embodiments, the higher divergence of the beams may increase the resolution of the final printed object. This is because the higher divergence enables greater rotational angle of the individual beams illuminating individual voxels, more fully exploiting the associated rotational degree of freedom. Therefore, the higher divergence may increase printing quality metrics as previously described.

[0045] According to one or more embodiments, the objective that diverges the beams may be a single objective or may be one or more optical components that diverge the beams. For example, the objective may include any of one or more lenses, one or more mirrors, or one or more customized or commercial objectives, which in combination can bend the beams. Thus, the phrase “objective” as used here refers to a set of optical components arranged to transform the array of pixels with modulated intensity into an array of point spread functions with desirable size, shape and direction of propagation within the resin.

[0046] Throughout this disclosure, solidifying the voxels means that the viscosity of the resin in the voxel is increased, via light exposure, to an extent that the printed object can be extracted from the remaining resin. For example, in one or more embodiments, the solidified voxels may be in a form of gel that can keep its shape. One of average skill in the art of photochemistry will realize that many materials and reactions are available that meet this description including photopolymerization and photoisomenzation.

[0047] Specifically, the 3D printer in accordance with one or more embodiments includes a light source (402) that includes a plurality⁷ of pixels configured to emit a plurality of beams (302) toward an objective (406-1, 406-2). The objective (406-1, 406-2) is configured to diverge the plurality of beams (404) as the plurality⁷ of beams (404) exits the objective (406-1, 406-2) toward a volume of resin (410). The resin (410) is contained in a container (412). The container (412) is configured such that the diverged beams (404) can pass through the container (412) in to the resin (410). For example, the container (412) may include a transparent top side such that the diverged beams (404) can pass through to reach the resin (410). In one or more embodiments, the container (412) may not include a top face, and the resin (410) can be directly exposed to the diverged beams (404). The objective may be separated from the top side or the resin by air or, as is known in the art, the intermediate space may be filled with a material of higher refractive index than air (i. e. , higher than 1) in order to enable greater angles of beam propagation within the resin. This coupling material may include oil, water, various elastomers or could be the resin itself.

[0048] In one or more embodiments, the 3D printer includes one or more motors configured to translate the objective (406-1, 406-2) with respect to the resin (410). The motor(s) are configured to translate the objective (406-1, 406-2) with respect to the resin (410) to apply the beams (404) to a plurality of voxels inside the resin (410). To this end, the motor(s) are configured to translate the objective (406-1, 406-2) with respect to the resin (410). For example, the motor(s) may be coupled to and move the objective (406-1, 406-2). Alternatively, the motor(s) may be coupled to and move the container (412), or to both the objective (406-1, 406-2) and the container (412), to translate the objective (406-1, 406-2) and resin (410) with respect to one another. In one or more embodiments, the motor(s) may be stepper motors, voice coil motors, piezoelectric motors, or a combination thereof. However, the embodiments of the invention are not limited to the type of the motor(s).

[0049] In the embodiments disclosed herein, moving the objective with respect to the resin is not limited to moving only the objective, unless specifically stated. Moving the objective with respect to the resin may be any of moving the objective, moving the resin, or moving both. Additionally, these motors may move only a portion of the elements comprising the objective, moving or modifying the light within the resin without moving the entirety of the objective with respect to the resin. For example, a steering mirror may be rotated to redirect or reposition light directed to the resin. Other actuators known in the art include deformable mirrors and liquid lenses that may be used to change the focal depth or shape of the PSF. These actuators need not be mechanical but can use optoelectronic processes such as electro-optic, liquid crystal or other w ell-known physics to modify the function of the objective such that the PSFs are modulated in a “useful” manner, as defined previously.

[0050] In one or more embodiments, the motor(s) move the objective (406-1, 406-2) to apply the diverged beams (404) such that each of the plurality of voxels are exposed by at least two different beams (404) among the plurality of beams. To this end, the two different beams (404) expose a voxel at two different times such that the two different beams spatially overlap only in the voxel. Specifically, the motor(s) move the objective (406-1) to position 1. In position 1, the light source (402) illuminates image 1 , where a pixel of the light source (402) illuminates a beam (404) such that the diverged beam (404) exiting the objective (406-1) passes through the voxel. Then, the motor(s) move the same objective, which was in position 1, to position 2. The objective at position 2 is represented by reference character “(406-2).” In position 2, the light source (402) illuminates image 2, where a pixel of the light source (402) illuminates a beam (404) such that the diverged beam (404) exiting the obj ective (406- 2) passes through the voxel at a different angle from the diverged beam (404) corresponding to position 1. Accordingly, the voxel receives the exposure of the beam (404) corresponding to position 1 and of the beam (404) corresponding to position 2. In a similar way, the 3D printer exposes the plurality of voxels by beams at different angles in order to solidify the resin at those voxels to create the object.

[0051] According to one or more embodiments, the 3D printer may solidify the voxel by exposing the voxel with at least two beams at different angles. An example of exposing the voxel with two beams at different angles is shown in FIG. 4A. The total exposure that solidifies the resin (410) in the voxel equals to at least a sum of exposure of the at least two different beams (404) at the voxel.

[0052] In one or more embodiments, the image set or the movement of the objective (406-1, 406-2) may be selected such that more than two beams at respectively different angles expose the voxel to desirably solidify the resin at the voxel. For example, the objective shown in FIG. 4 A may then move to position 3 and illuminate the voxel at a third angle different from the angles corresponding to positions 1 and 2. According to one or more embodiments, the number of illumination, the images in the image set, and the angles of illumination are calculated for each voxel in the printing area such that: the voxels inside object, which form the object, are exposed enough to be solidified; and the voxels outside the object are exposed sufficiently low to maintain their liquid form, to be separable from the object.

[0053] Calculating the necessary illumination at each voxel is demonstrated in FIGs. 5A-5C. Specifically, FIG. 5 A shows that an object model (502) is discretized onto a 3D grid (506) of voxels. Then, the voxels are printed using a beam (504). Assume that in order to print a voxel, the beam (504) must be illuminated on point (xo, yo, zo), as shown in FIG. 5B. As shown in FIG. 5C, when the beam (504) is illuminated on voxels (508), the beam (504) also passes through other voxels (510, 512) in the grid. This illumination on the other voxels (510, 512) can be thought of as residual exposure. According to one or more embodiments, in order to determine the image set and the translation of the objective with respect to the resin, the collective (total) exposure at each of the voxels, including the residual exposure, may be calculated. In other words, the image set and the translation of the objective with respect to the resin are determined such that the collective exposure at each voxel inside the object model is sufficiently high to solidify the resin at the voxel. Conversely, the image set and the translation of the objective with respect to the resin are determined such that the collective exposure at each voxel outside the object model is sufficiently low to not solidify the resin at the voxel.

[0054] Further, FIG. 4A shows that in each of positions 1 and 2, only one pixel is turned on. This illustration is for simplicity of demonstrating the operation of the 3D printer in accordance with one or more embodiments. However, at each position, the image corresponding to the position may be chosen such that multiple pixels are turned on. This way, multiple voxels can be addressed at each position of the objective with respect to the resin. For example, FIG. 4B shows a prototype, in accordance with one or more embodiments, where the beams (414) are illuminated from the objective at position 1 (406-1) and at position 2 (406-2) into the resin that is disposed in the container (412). According to one or more embodiments, the container (412) may be configured to contain the resin (410) such that a surface of the resin (410) from where the beams (404) enter is flat and normal to the axial direction (i.e., optical axis) of the objective (406-1, 406-2).

[0055] According to one or more embodiments, the light source (402) may include a DMD configured to reflect the plurality of beams toward the resin (410). [0056] According to one or more embodiments, the light source (402) may include one or more lasers that generate the plurality of beams. The light source (402) may include one or more minors or prisms that reflect the beams toward the objective (406-1, 406- 2).

[0057] According to one or more embodiments, the 3D printer may collimate the plurality of beams before entering the objective. For example, the 3D printer may include a collimator, which includes one or more lenses, one or more prisms, or one or more mirrors, before the objective. An example is shown in FIG. 3B, where the beams (302) are collimated before entering the objective (306).

[0058] According to one or more embodiments, the objective may be an oil-immersion objective. The output of an oil-immersion objective is immersed in, for example, a droplet or thin film of oil, the resin itself, or an elastomeric solid. In one or more embodiments, immersing the objective in a medium with index greater than one enables larger angular divergence of the beams within the resin of refractive index greater than one. This technique is used in various optical technologies including lithography and microscopy and is known to include a variety of coupling materials including water or elastomers as well as evanescent coupling across a thin air gap referred to as “solid immersion.”

[0059] According to one or more embodiments, the motor(s) are configured to move the objective, with respect to the resin, along at least two axes that are orthogonal to an axial direction of the objective. For example, in FIGs. 3A and 4A, the motor(s) may move the objective with respect to the resin along two in-plane axes that are orthogonal to the axial direction of the objective (306, 406-1, 406-2). The axial direction of the objective (306, 406-1, 406-2) in FIGs. 3A and 4A is a vertical axis along the illumination of the collimated beams (302). According to one or more embodiments the two in-plane axes may be orthogonal with one another.

[0060] In one or more embodiments, the motor(s) may move the objective with respect to the resin along the axial direction of the objective. This additional degree of movement enables shaping the size of the beam at the voxel.

[0061] FIG. 6 shows a schematic of a system (600) in accordance with one or more embodiments. The system (600) includes, for example, a buffer (602) and a 3D printer (606), an example of which was discussed above with reference to FIGs. 3A-3B and 4A-4B. The buffer (602) may be implemented in hardware (i.e., circuitry), software, or any combination thereof. The buffer (602) is configured to store a printing object file (604). The printing object file (604) may include an object model and instructions for printing the object model using the printer (606). Based on the printing object file (604), the printer (606) can print the object corresponding to the object model. To do the printing function, the printer (606) includes one or more motors (608), a light source (610), an objective (612), and a container (614) that contains the resin. Examples of the motor(s) (608), light source (610), objective (612), and container (614) are described above with reference to FIGs. 3A-3B and 4A-4B. As described further below with reference to FIG. 20, the system (600) may include a processor that controls the operation of the one or more motors (608) and light source (610) for performing the printing functions. For example, based on printing object file (604), the processor may control the light source (610) for projecting the set of images and control the translation of the objective (612) with respect to the resin, as discussed above with reference to FIGs. 4A-4B.

[0062] FIG. 7 illustrates a flowchart of a method for operating a 3D printer, in accordance with one or more embodiments. In one or more embodiments, one or more of the steps shown in FIG. 7 may be omitted, repeated, and/or performed in a different order than the order shown in FIG. 7. Accordingly, the scope of the invention should not be considered limited to the specific arrangement of steps shown in FIG. 7. Steps 700 to 735 shown in FIG. 7 are explained below.

[0063] In Step 700, a light source that includes a plurality of pixels is controlled to emit a plurality of beams toward an objective. Examples of this step were shown above with reference to FIGs. 3A-3B, 4A-4B, and 6. For example, a processor may control the light source (402) shown in FIG. 4A to project the image set.

[0064] In Step 705, the plurality of beams are diverged, via the objective, as the plurality of beams exits the objective toward a volume of resin. Examples of this step were shown above with reference to FIGs. 3A-3B and 4A-4B. For example, the objective (306) shown in FIG. 3B diverges the beams (304) before the beams (304) enter the resin (310). [0065] In Step 710, the one or more motors translate the objective with respect to the resin.

[0066] In Step 715, by translating the optical objective, the plurality of beams is applied to a plurality of voxels inside the volume of resin.

[0067] In Step 720, by the application of the plurality of beams to the plurality of voxels, the plurality of voxels is solidified.

[0068] In Step 725, the objective applies the plurality of beams such that each of the plurality of voxels are exposed by at least two different beams among the plurality of beams.

[0069] In Step 730, the at least two different beams expose the each voxel at two different times such that the at least two different beams spatially overlap only in the each voxel.

[0070] In Step 735, a total exposure that solidifies the volume of resin in the each voxel is determined to be equal to at least a sum of exposure of the at least two different beams at the each voxel.

[0071] Examples of Step 710 to Step 735 are also shown above with reference to FIGs. 4A-4B and 6.

[0072] Based on the embodiments disclosed herein, the method may comprise one or more steps. For example, according to one or more embodiments, the application of the at least two beams may be such the at least two different beams have respectively different angles from one another. According to one or more embodiments, the method may comprise reflecting the plurality of beams toward the volume of resin. According to one or more embodiments, the method may comprise collimating the plurality of beams before the plurality of beams enters the optical objective. According to one or more embodiments, the method may comprise translating the objective along at least two axes that are orthogonal to the axial direction of the optical objective.

[0073] The 3D printer in accordance with one or more embodiments may be advantageous compared to the conventional VAM. For example, unlike CAL where the sample of resin is rotated, for example in a cylindrical container, in front of projected light, one or more embodiments disclosed herein may provide more angular diversity. In addition, the 3D printer according to one or more embodiments frees the resin to be packaged in virtually any geometry such as, for example, a flat package as shown in FIGs. 3A-3B and 4A-4B. Because of the higher degrees of freedom, for example four or more in accordance with one or more embodiments, more complex patterns with higher resolution may potentially be printed. To elaborate, in accordance with one or more embodiments, a single voxel can experience light (at different times) that is tilted in two orthogonal axes, not just one. According to one or more embodiments, the translation of the beams with respect to the resin is not limited, unlike the conventional VAM; the motor(s) can potentially translate the objective to any location that is scaled by a commercialized unit.

[0074] In conventional VAM, beam bundles corresponding to each pixel must be approximately collimated through the thickness of the resin, enforcing a minimum beam bundle diameter due to the physics of diffraction. On the other hand, in the 3D printer according to one or more embodiments, the thickness of the resin and the transverse dimensions of translation of the objective are not coupled.

[0075] One or more embodiments disclosed herein are directed toward a method and a system for printing a 3D object based on an object model. Specifically, one or more embodiments are directed toward a method and a system for generating the set of images that will be projected by the light source. An example of the set of images projected by a light source (402) was illustrated above with reference to FIG. 4 A. The set of images must be chosen/ calculated as a function of the translation of the obj ective to realize a desired 3D object. To this end, intensities for each pixel at each angle must be chosen in order to fabricate a 2D slice of the 3D part. According to one or more embodiments, the 3D object model is addressed by images with four effective dimensions (degrees of freedom); pixel row + pixel column = 2D, but then translated through two dimensions = two more dimensions. These translations are independent dimensions because the plurality of beams emerging from the objective is diverging into two angles. If the plurality of beams were parallel or nearly parallel, translation would not apply a “useful” change of the illumination, as explained previously and as illustrated in Fig. 2. [0076] To determine the image set and its translation to the object, backward (image to object) and forward (object to image) projections must be determined. Backward projection is also referred to as “back projection” in this disclosure. Backward projection may be defined as a matrix (B) that translates the intensity of images in the image space (i.e., a space where the image set exists) into the dose deposited in the object in the object space (i.e., a space where the object and object model exist). Conversely, forward projection may be defined as a matrix (F) that translates a quantity defined across the object or object model (e.g. dose or an error in the desired dose relative to a goal) in the object space into the images in the image space. The B and F matrices may be determined by projecting light through the optical system of the 3D printer and subsequently determining the light refraction through the resin and/or via simulations or calculations corresponding to the physics of refraction, diffraction, lens aberrations, scatter or other relevant optical and material physics.

[0077] Turning to an explanation of the method of determining the image set and translating the image set to the object space, the method is explained in cylindrical coordinates for simplicity, such as shown in FIGs. 8A-8C in accordance with one or more embodiments. However, these embodiments are not limited to describing the object in cylindrical coordinates and can use other coordinate systems such as Cartesian coordinates, an example of which was described above with reference to FIG. 4A.

[0078] FIG. 8A shows 3D printing of an object described in cylindrical coordinates, in accordance with one or more embodiments. Specifically, in FIG. 8A, a 2D pixel array (S) (802) illuminates beams into a cylinder containing a resin. Similar to a conventional VAM, by rotating the cylinder and projecting a set of images, by the pixel array, linked to the rotation of the cylinder, a 3D object can be printed. FIG. 8B shows a slice of the printing area only for angle 0 = 0. FIG. 8C shows a close-up view of the slice shown on FIG. 8B as a horizontal cut. In FIGs. 8A-8C, “x” and “z” coordinates are along the plane of the pixel array (S) and are perpendicular to “y.” The intensity levels on the right side of FIGs. 8B and 8C show the dose intensity in the spatial area shown in the figures. Dose intensity is also referred to as object intensity, dose intensity, or object dose throughout this disclosure. [0079] According to one or more embodiments, an advantage of the cylindrical coordinate system may be that for a rotationally symmetric object, the calculation may need to be done only for a single angle. All other angles can be computed through a circular shift (i.e., incrementing 0), because the motion trajectory and the representation of the object are in the same coordinate system. For example, for a typical VAM system using one degree angular spacing, the rotational symmetry may save a factor of 360 in computing time and possibly memory. Similarly, for a system in which the motion of the objective relative to the resin is most easily described as translations in Cartesian coordinates, it may be advantageous to describe the object in Cartesian coordinates. In general, the choice of coordinate system to describe the object may exploit symmetry of the printer to reduce memory or other computational resources.

[0080] According to one or more embodiments, the relation of object dose to image intensity is considered linear. That is, the back projection operation can be represented as a matrix vector multiplication B I=O where I is a vector of all image pixels (at all positions s and all angles) and O is vector of all object voxels (here at all radius “r” and all angles 0). B is a matrix with rows and columns equal to the number of all image pixels and the number of all object voxels. As mentioned before, this may be impractically large to generate or store. However, according to one or more embodiments, the computation may be reduced by calculating a sub-matrix of B over just the pixel array S at one position of the motion system, for example 0 in a rotational printer like CAL and all object voxels. The sub-matrix of B may be quite sparse, based on the number of zero values as illustrated in FIG. 8B where most of the slice is black (i.e., no dose intensity). Accordingly, the sub-matrix of B may be represented numerically with a “sparse matrix” that only stores the non-zero values. Storage and math on such “sparse matrices” is standard in modem computational languages such as MATLAB. For typical problems, the sparse matrix may be less than 1 % of the full matrix size. In addition, computation at only one motion position such as one angle and matrix sparsity may reduce memory usage and time requirements by 4 to 5 orders of magnitude, typically. While illustrated for the cylindrical coordinate system typical of CAL in which a cylindrical resin vial is rotated on its axis, the discussion applies broadly to any volumetric printer in which the motion of the objective relative to the resin can be chosen as the axis of a coordinate system. Thus, the translational motion illustrated in FIGs. 4A and 4B may be most efficiently represented in a Cartesian coordinate system that, if used to represent the object, may result in memory savings for the projection matrices as described above.

[0081] According to one or more embodiments, generating the complete B matrix may be accomplished by indexing the sub-matrix of B (which is at one position of the objective, for example 9) appropriately. This may save memory because a sparse matrix may make it practical to compute and store B. It is mentioned above that where rotational symmetry exists for the object in cylindrical coordinates, the calculations may need to be done only once for only one angle. Similar logic applies to a matrix to execute the forward projection (F), which depends on the same calculation and can be written with the same sparse matrix indexing as 0><F=I.

[0082] According to one or more embodiments, by using the sparse matrices, forward and back projections can be performed faster than conventional approaches. According to one or more embodiments, a single sub-matrix may be calculated, then matrix multiples of this sub matrix with the object (O) or intensity (I) vectors may be executed, and then a sum over the motion (e.g., rotation angle) with appropriate indexing may be executed to express the motion. This method may replace one dimension of the operation done by the full B and F matrices with an iterative sum (e.g., a “for” loop) and thus, may save memory.

[0083] Accordingly, by choosing a coordinate system for discretizing the obj ect that is invariant to the motion of the object relative to the image, one realizes dramatic memory or speed improvements to the mathematical operations. According to one or more embodiments, for translational motion, a grid that is invariant to one (for ID translation) or two (for 2D translation) dimensional translation may be selected. This may include a Cartesian grid, but other choices may be possible. For example, a 2D translation layout may not need to translate as a 2D Cartesian raster, but instead may move in a Cylindrical coordinate system.

[0084] There are three broad classes of methods ( 1 )-(3) that may be used in calculation of the image set for projection. (1) Filtered back projection (FBP) (known from the art of computed tomography) - this method generates negative intensities that are unphysical. The negative intensities are truncated with a minimum value of zero. This method may result in poor print fidelity.

(2) Gradient descent (GD) in the image space - This method uses a traditional optimization method to improve the object. It is computationally expensive and requires knowledge of the derivative of a merit function representing object quality with every pixel. These gradients are expensive to calculate and not always accurate, leading to poor convergence.

(3) Object space model optimization (OSMO) - This method applies the FBP method to an “object model,” in the object space, that is allowed to be nonphysical (e.g., allowed to use negative values). But back projection of the model results in an optimal object. The method attempts to force the interior of the object (to be solidified) to be above an upper critical dose intensity D_upPer and the exterior of the object (that must stay liquid) to receive below a lower critical dose intensity Diower.

[0085] The method described below in accordance with one or more embodiments may provide at least the following advantages. One advantage of the method may be deriving feedback from a simple comparison of the object to two target dose thresholds (Du_PPer and Diower). Conversely, gradient descent in image space requires estimation of derivates of some error metric with every pixel intensity. Another advantage may be deriving an error metric from the object. The error metric, which will be described further below in accordance with one or more embodiments, may make it simple and natural to include material physics that occur within the object. The material physics may include inhibition, nonlinear response, or diffusion. These physics may be more difficult to include using gradient descent of the images.

[0086] New methods of 1) direct matrix solution and 2) forward projection of object errors are described below, in accordance with one or more embodiments.

1) Direct Matrix Solution

[0087] According to one or more embodiments, when the entire object is known (that is, dose values, not inequalities, are given), the back projection matrix (B) described above can be used to find a least-squares optimal solution. That is, the linear system BxI=O can be solved for I if B is of sufficient rank. It is well known in the field of linear algebra that this solution does not require inversion of the matrix B but instead the system can be solved by matrix decomposition such as QR decomposition. These solutions minimize the error between the realized object OB I and the goal object. An example is shown in FIG. 9, in accordance with one or more embodiments. In FIG. 9. the photo of the woman’s face is chosen as an example object where the dose throughout the part is known.

[0088] This approach does not guarantee satisfaction of the non-negativity constraint. That is, negative intensities are allowed. However, reducing the contrast and smoothing the object can be used to satisfy this constraint. FIG. 9 A shows the target object, which is the goal for printing. In FIG. 9A, the minimum dose is set to 0.9, and additionally may be smoothed using a low-pass filter, to reduce the contrast and smooth the image. As the contrast is reduced and/or the object is smoothed, the minimum intensity increases and the range of intensities decreases. In FIG. 9A, the range of intensities is between 0.9 and 1. Accordingly, the target object may be modified to meet minimum and maximum intensity constraints.

[0089] FIG. 9B shows image intensity (I) (in the image space) with respect to the pixel array (S) and the rotational angle (0), in accordance with one or more embodiments. The image intensity, which represents the set of images with respect to various angles, is calculated by forward projection of the target object shown in FIG. 9 A using the forward projection matric (F). Specifically, OxF=I.

[0090] FIG. 9C shows the dose intensity (in the object space) with respect to Cartesian coordinates of a slice in the object space, in accordance with one or more embodiments. The dose intensity represents the dose in the object space that corresponds to the image intensity shown in FIG. 9B. The dose intensity is calculated by backward projection of the image intensity (I) using the backward projection matrix (B). Specifically, as mentioned above, BxI=O.

[0091] FIG. 9D shows the error, which is the difference between the target object shown in FIG. 9A and the dose intensity shown in FIG. 9C. In other words, FIG. 9D shows the deviation of the dose intensity shown in FIG. 9C from the target object shown in FIG. 9A. One or more embodiments described below provide a method for reducing the error.

2) Forward Projection of Object Errors

[0092] According to one or more embodiments, this method may apply to more typical VAM processing for printing a 3D object that is solid within a target region where the outside region of the target remains liquid. Because intensity constraints in the image space are coupled by back projection to dose constraints in the object space, iteration between these two spaces may be performed. The OSMO method uses forward projection (object to image) to create an “intermediate” model that responds to errors (unmet constraints) in the object in order to find improved image sets. Here, in one or more embodiments, the model is eliminated by calculating errors in the object space and directly forward propagating the errors to the image space to calculate a new image set.

[0093] Mult pie steps corresponding to the Forward Projection of Object Errors are described below in an alphabetical order. The alphabetical order is set forth only to facilitate understanding the embodiments disclosed herein and are not meant to limit the invention to only this particular ordering or steps.

[0094] A) According to one or more embodiments, a target object that one wishes to print is first discretized onto a voxel grid. An example of this step is shown in FIGs. 10A-10B. FIG. 10A shows a target object (1006) inside a printing region (1008) in a Cylindrical coordinate system. The white area, which is the target object (1006), is intended to be solidified (printed) while the darker area (1008), which is outside (exterior) the target object, is intended to remain liquid. FIG. 10B shows an edge of the object (1004) when the target object (1006) is discretized on the voxel grid. The difference between the boarder of the target object (1002) and the edge of the discretized object (1004) is shown in FIG. 10B. According to one or more embodiments, the fractional area of each voxel that overlaps the target object may be calculated, or a binary representation in which voxels are categorized as fully interior (e.g., 1) or fully exterior (e.g., 0) to the object may be used. [0095] B) According to one or more embodiments, in addition to the target object to be printed (step A), three quantities may be specified. These three quantities are described below.

[0096] The first quantity is Dupper. Dupper is the minimum desired object dose interior to the target. In other words, Dupper is the minimum object dose for the voxels that are intended to be solidified to form the object. This quantity may be chosen from photorheology' tests, which determine the minimum dose that sufficiently hardens the material for post processing including removal from the printer and solvent washing.

[0097] The second quantity is Diower. Diower is the maximum desired object dose exterior to the target. In other words, Diower is the maximum object dose for the voxels that are intended to remain liquid or with low viscosity such that the resin of the exterior can be separated from the printed/solidified object. This quantity may be chosen from photo-rheology tests, which determine the maximum dose for which the resin stays liquid. The maximum dose is often referred to as the gelation dose.

[0098] The third quantity' is A. A is the desired print resolution. This quantity is determined from printer properties, such as the diffraction limit of the optical projections, and also resin properties, such as the characteristic scale of species diffusion in the print time. These limits may enforce a minimum feature size that can be physically printed. The set of feasible solutions expands as the target resolution is relaxed (the minimum feature size is decreased). Thus the optimal design may be one that matches but does not exceed the printer resolution. That is, the design resolution and printer/material resolution may be chosen to be the same.

[0099] According to one or more embodiments, the above three quantities may be used to create a goal dose distribution with Dupper in the interior of the target and Diower in the exterior of the target. Then the goal dose may be spatially filtered by a low-pass filter, such as Gaussian or moving average, to set the resolution based on A. An example of this filtering is shown in FIG. 11, in accordance with one or more embodiments. Specifically, FIG. 11 shows the interior of the object (1106) that is intended to be printed and the exterior of the object (1108) that is not intended to be printed. The boundary' (1110) between the interior (1106) and exterior (1108) of the object is smoothed based on A. The close-up view in FIG. 11 shows the smoothed area at the boundary (1110).

[00100] Without the smoothing step, finding a solution that maximally separates Du_PPer and Diower for a binary target may be impossible or difficult because the dose distribution is continuous, even with ideal binary image projection and no chemical diffusion. Conventional reported solutions that report optimal results with a finite gap (called the “process window”) may be incorrect. The reported gap may be an artifact of the discrete sampling of the object. If the sampling is made sufficiently fine, the gap vanishes. Thus, the smoothing step with a finite resolution, A, described in one or more embodiments may improve calculations.

[00101] Therefore, according to one or more embodiments, two dose limits are set (D_upPer and Diower). Then, an image set that meets the constraints on image intensity (e.g. non-negativity) and associated object dose that meets the object dose constraints (e.g. D_upPer and Diower) are found/determined. In regions like edges that have been smoothed by the filtering operation (e.g., the smoothed boundary (1110) in FIG. 11), the dose constraint within the boundary of the target is between D_upPer and Diower. The goal is to reduce the dose in the exterior region, for example far outside the object, to below Diower, between D_upPer and Diower in the smoothed boundary of the object, and above D_upper in the interior of the object.

[00102] According to one or more embodiments, the constraint on image intensity may be explicitly met, while the object dose constraints may be attempted to be met, but an optimal solution may not always reach those metrics.

[00103] According to one or more embodiments, the smoothing of the boundaries may facilitate finding solutions for the image set and object dose with some error. Error in this paragraph means that a portion of the object (e.g., near the edges) does not meet the dose constraints.

[00104] According to one or more embodiments, iteration improves an initial guess to find an optimal image set based on the dose goals.

[00105] C) According to one or more embodiments, an initial guess for an image set is calculated. This can be all ones, all zeros, or random. A guess that may often work well is the forward projection of the target object (e.g., using matrix F), as shown in FIG. 12. In the language of computed tomographic imaging, this transformation may be called Radon transform or “sinogram” of the object. FIG. 12 shows forward projection of the smoothed dose goal shown in FIG. 11.

[00106] D) According to one or more embodiments, the image set calculated in step C is back (backward) projected (e.g., using matrix B) to the object space to compute the expected dose distribution across the object.

[00107] E) According to one or more embodiments, the expected dose calculated in step D is used to compute an error function for every voxel. The error function is calculated by comparing the expected dose calculated in step D with the target object determined in step B. To calculate the error, for every voxel inside the object, the difference between the expected dose and the upper limit Dupper is calculated. If the expected dose in the interior region calculated in step D is greater than Dupper, the error is zero. If not, then the error is computed for example as the square of the difference of the expected dose in the interior region calculated in step D and Dupper in the interior region. Similarly, for every exterior voxel, the expected dose calculated in step D is compared to the lower limit Diower. If the expected dose in the exterior region calculated in Step D is lower than Diower, the error is zero. Otherwise, the error is calculated for example as the square of the difference of the expected dose in the exterior region calculated in Step D and Diower.

[00108] FIG. 13 illustrates an example of the calculations of the error performed in step E, in accordance with one or more embodiments. Specifically, in FIG. 13, the shading in the interior rejoin of the object (1306) shows the root mean square (RMS) of the error that is the difference between the expected dose in interior voxels that have expected dose less than D_upPer and Dupper. Similarly, the shading in the exterior region of the object (1308) shows the RMS of the error that is the difference between the expected dose in exterior voxels that have expected dose higher than Diower and Diower. The close-up view on the right-hand side of FIG. 13 shows a magnified view of the interior and exterior shadings. According to one or more embodiments, the error calculation in step E may ignore the boundaries of the object (1310), as shown in the close-up view in FIG. 13 where there is no shading in the boundaries (1310). Instead, the exposure on the boundaries (1310) may be considered as a smooth curve between Dupper and Di ower.

[00109] F) According to one or more embodiments, the error distribution across the object may be forward projected to the image space. Specifically, the interior error and exterior error may be separately forward projected. This process finds two RMS errors for every voxel illuminated by every pixel. One error expresses insufficient dose in the interior region of the object and the other expresses too much dose in the exterior region of the object. Each of these has been computed for every pixel in the image set.

[00110] FIGs. 14A-14C show an example of step F. As shown in FIG. 14A, the error corresponding to the interior region of the object (final signed RMS internal error) is forward projected to the image space. The error shown in FIG. 14A has a negative sign because the internal dose is in error if and only if it is too low. Similarly, as shown in FIG. 14B, the error corresponding to the exterior region of the object (final signed RMS external error) is forward projected to the image space. The error shown in FIG. 14B has a positive sign because the external dose is in error if and only if it is too high. The forward projection may be performed by using the forward projection matrix F. FIG. 14C shows a final sum of the signed RMS errors, which is the sum of the forward projected errors presented in FIGs. 14A and 14B. Because some of the forward projected errors presented in FIGs. 14A and 14B cancel each other, the sum shown in FIG. 14C includes less intensity compared to the forward projected errors presented in FIGs. 14A and 14B. In other words, the sum may be near zero for most image pixels because the intensity is adjusted to balance the interior and exterior errors.

[00111] G) According to one or more embodiments, the sum of the two signed forward projected errors (for example as shown in FIG. 14C) at every pixel may provide a signed error function that determines how the pixel intensity should be updated in the image space. For example, by taking the square root of the two mean square errors to produce root mean square (RMS) errors, the sum of the two signed forward projected errors can be added to the pixel intensity to update its value. For example, the sum error shown in FIG. 14C may be applied to the pixel intensity of the pixel arrays S to modify the image set. According to one or more embodiments, acceleration techniques may be used to take larger step sizes, leading to convergence in a smaller number of iterations.

[00112] H) According to one or more embodiments, after every update, the image set is bounded to be positive and (optionally) lower than some desired maximum. If intensity bounds are met, the error function is not used to further update the image. In other words, the iteration may be stopped if the solution is not feasible because error will no longer decrease for this image pixel. For example, the iterative process may continue to improve the total error, producing an optimal design for the given inputs, but finite errors will remain and feasibility has been rejected.

[00113] I) According to one or more embodiments, additionally, the image set may be discretized (e.g., to 8 bits) to match the resolution of the light source (e.g. , of an image projection chip such as a DMD) used in the printer. This may reduce errors in the physical print introduced by a continuous numerical solution being discretized to a projector with finite discrete resolution.

[00114] J) According to one or more embodiments, steps D-G may be repeated until an exit condition is reached. The exit condition may be, for example, as maximum number of iterations or slow improvement in reducing the errors.

[00115] K) According to one or more embodiments, if the intensity constraints have not been reached, the magnitude and final slope of the error function may be evaluated to determine feasibility to within some error tolerance. Possible intensity constraints are described above and include non-negativity, an upper bound on intensity, and discretization to a finite number of values such as 256. For example, in the FIG. 15, the total RMS error (1506) has decreased from the initial guess by a factor of 1000 and is continuing to reduce with further iterations. This solution could be defined to be feasible, to within given tolerances on total or maximum object error. In FIG. 15, the RMS of the internal error (1504) and the RMS of the external error (1502) continuously converge and the total RMS error (1506) continuously decreases.

[00116] According to one or more embodiments, at the end of the image correction process, a final image set is found, as shown in FIG. 16. The intensity of the final image set is bounded to be between 0 and 1 and has been discretized at every step to the number of bits that matches the resolution of for example a typical DMD proj ector chip (e.g., 8 bits).

[00117] According to one or more embodiments, the resulting object dose is closer to Dupper in the interior region of the object and closer to Diower in the exterior region of the object. For example, in FIG. 17, Dupper and Diower are respectively 0.4 and 0.35 and the final object dose is close to these numbers respectively in the interior and exterior regions of the object.

[00118] FIGs. 18A and 18B show that the final errors, which are doses that do not meet Dupper and Diower, are concentrated around the edges of the object. The final errors are the errors calculated by comparing backward projection of the final image set and Dupper (for interior of the obj ect) and Diower (for exterior of the obj ect). FIG. 18B shows a magnified view of FIG. 18 A.

[00119] According to one or more embodiments, the method described with reference to steps A-K and FIGs. 8 A to 18B, may provide one or more of the following advantages. Using the sparse matrix multiplication in the optimal (e.g., cylindrical for CAL) coordinate system described above, may provide less iteration than conventional methods. Using proper choice of error function, the errors may continuously and rapidly converge. By accessing both the object space and image space, it may be possible to include object-space physics and image-space bounds to prevent divergence. There may be flexibility in choosing error functions in object space that directly generate gradients in image space. The simplicity, memory efficiency, and speed of the method illustrated here for the 2D telecentric geometry may make it possible to extend the method to more challenging multi-dimensional architectures.

[00120] While the above methods described with reference to steps A-K and FIGs. 8A to 18B are described with reference to Cylindrical coordinates, these methods can be extended to Cartesian coordinates.

[00121] FIG. 19 illustrates a flowchart of a method for determining an image set, in accordance with one or more embodiments. In one or more embodiments, one or more of the steps shown in FIG. 19 may be omitted, repeated, and/or performed in a different order than the order shown in FIG. 19. Accordingly, the scope of the invention should not be considered limited to the specific arrangement of steps shown in FIG. 19. Step 1900 to Step 1940 shown in FIG. 19 are explained below.

[00122] In Step 1900, an initial object model (0) in an object space is determined by discretizing the object onto a voxel grid. One or more examples of this step were described above with reference to step A and FIGs. 10A and 10B. According to one or more embodiments, an upper limit dose (Du_PPer) that is the minimum dose for interior of the object may be defined. According to one or more embodiments, a lower limit dose (Diower) that is the maximum dose for exterior of the object may be defined. According to one or more embodiments, a print resolution (A) that is a minimum printable feature size may be defined. According to one or more embodiments, a low- pass spatial filter with cutoff of A may be applied to smooth boundaries of the object. The dose inside the smoothed boundaries of the object may be smoothed between Du_PPer and Di ower.

[00123] In Step 1905, an image set (I) in an image space is determined by forward projecting (0) using a forward projection matrix (F). One or more examples of this step were described above with reference to step C and FIG. 12.

[00124] In Step 1910, an expected dose distribution (D) in the object space is determined by backward projecting I using a backward projection matrix (B). One or more examples of this step were described above with reference to step D.

[00125] In Step 1915, an error set (E) corresponding to D is determined. One or more examples of this step were described above with reference to step E and FIG. 13. According to one or more embodiments, the difference between D and D_upPer for interior voxels of the object may be determined. If an expected dose, in D, of an interior voxel is not less than Du_PPer, an error corresponding to the interior voxel is determined to be zero. If the expected dose of the interior voxel is less than Du_PPer, the error corresponding to the interior voxel is determined based on the difference between the expected dose of the interior voxel and D_upper. According to one or more embodiments, the difference between D and Diower for exterior voxels of the obj ect may be determined. If an expected dose, in D, of an exterior voxel is not greater than Diower, determining an error corresponding to the exterior voxel to be zero. If the expected dose of the exterior voxel is greater than Di_0Wer, the error corresponding to the exterior voxel is determined based on the difference between the expected dose of the exterior voxel and Diower.

[00126] In Step 1920, E is forward projected using F. One or more examples of this step were described above with reference to step F and FIGs. 14A-14C and 15.

[00127] In Step 1925, 1 is modified based on the forward projection of E. One or more examples of this step were described above with reference to steps G-H and FIG. 16. According to one or more embodiments, the modified I is compared to predetermined intensity bounds to determine whether the modified I is within the intensity bounds.

[00128] In Step 1930, the modified I is forward projected using F to determine D. One or more examples of this step were described above with reference to steps H-K and FIG. 17.

[00129] In Step 1935, modification of I is successively performed until a predetermined condition occurs. One or more examples of this step were described above with reference to steps J-K and FIGs. 15, 17, and 18A-18B. According to one or more embodiments, the predetermined condition may be that E becomes less than a first threshold. According to one or more embodiments, the predetermined condition may be reaching a maximum number of iterations. According to one or more embodiments, the predetermined condition may be that the difference between E of one iteration step and E of the immediately previous iteration step being less than a second threshold.

[00130] In Step 1940, a final I that corresponds to the predetermined condition is used to print the object. One or more examples of this step were described above with reference to steps J-K and FIGs. 15, 16, and 18A-18B.

[00131] According to one or more embodiments, the error determination is not performed for boundaries of the object.

[00132] According to one or more embodiments, the error corresponding to the interior voxel and the error corresponding to the exterior voxel are forward projected separately.

[00133] One or more embodiments disclosed herein for the operations of the 3D printer, for example with reference to FIGs. 3A-19, may be implemented on virtually any type of computer system, regardless of the platform being used. The computer system may have programs or algorithms to control the functions/operations of the measurement described in the above embodiments. For example, the computer system may be one or more mobile devices (e.g., laptop computer, smart phone, personal digital assistant, tablet computer, or other mobile device), desktop computers, servers, blades in a server chassis, or any other type of computer system that includes at least the minimum processing power, memory, and input and output device(s) to perform one or more embodiments of the invention.

[00134] An example of the computer system is described with reference to FIG. 20, in accordance with one or more embodiments. FIG. 20 is a block diagram of a computer system used to provide computational functionalities associated with described algorithms, methods, functions, processes, flows, and procedures as described in the instant disclosure, according to an implementation. The illustrated computer (2002) in the computer system is intended to encompass any computing device such as a server, desktop computer, laptop/notebook computer, wireless data port, smart phone, personal data assistant (PDA), tablet computing device, one or more processors within these devices, or any other suitable processing device, including both physical or virtual instances (or both) of the computing device. Additionally, the computer (2002) may include an input device, such as a keypad, keyboard, touch screen, or other device that can accept user information, and an output device that conveys information associated with the operation of the computer (2002), including digital data, visual, or audio information (or a combination of information), or a GUI.

[00135] In one or more embodiments, the 3D printer described in the above embodiments may include the computer (2002), may be in the form of the computer (2002), or may be implemented on the computer (2002) such that the computer (2002) performs the processing and calculations described above with reference to FIGs. SA- 19. For example, the computer (2002) with which the 3D printer is implemented includes tools for performing the processing related to 3D printing and determining the final image set described in the above embodiments.

[00136] The computer (2002) can serve in a role as a client, network component, a server, a database or other persistency, or any other component (or a combination of roles) of a computer system for performing the subject matter described in the instant disclosure. The illustrated computer (2002) is communicably coupled with a network (2030). In some implementations, one or more components of the computer (2002) may be configured to operate within environments, including cloudcomputing-based, local, global, or other environment (or a combination of environments).

[00137] At a high level, the computer (2002) is an electronic computing device operable to receive, transmit, process, store, or manage data and information associated with the described subject matter. According to some implementations, the computer (2002) may also include or be communicably coupled with an application server, e- mail server, web server, caching server, streaming data server, business intelligence (BI) server, or other server (or a combination of servers).

[00138] The computer (2002) can receive requests over network (2030) from a client application (for example, executing on another computer (2002)) and responding to the received requests by processing the said requests in an appropriate software application. In addition, requests may also be sent to the computer (2002) from internal users (for example, from a command console or by other appropriate access method), external or third-parties, other automated applications, as well as any other appropriate entities, individuals, systems, or computers.

[00139] Each of the components of the computer (2002) can communicate using a system bus (2003). In some implementations, any or all of the components of the computer (2002), both hardware or software (or a combination of hardware and software), may interface with each other or the interface (2004) (or a combination of both) over the system bus (2003) using an application programming interface (API) (2012) or a service layer (2013) (or a combination of the API (2012) and service layer (2013)). The API (2012) may include specifications for routines, data structures, and object classes. The API (2012) may be either computer-language independent or dependent and refer to a complete interface, a single function, or even a set of APIs. The service layer (2013) provides software services to the computer (2002) or other components (whether or not illustrated) that are communicably coupled to the computer (2002). The functionality of the computer (2002) may be accessible for all service consumers using this service layer (2013). Software services, such as those provided by the service layer (2013), provide reusable, defined business functionalities through a defined interface. For example, the interface may be software writen in JAVA, C++, Python, or other suitable language providing data in extensible markup language (XML) format or another suitable format. While illustrated as an integrated component of the computer (2002), alternative implementations may illustrate the API (2012) or the service layer (2013) as standalone components in relation to other components of the computer (2002) or other components (whether or not illustrated) that are communicably coupled to the computer (2002). Moreover, any or all parts of the API (2012) or the service layer (2013) may be implemented as child or sub-modules of another software module, enterprise application, or hardware module without departing from the scope of this disclosure.

[00140] The computer (2002) includes an interface (2004). Although illustrated as a single interface (2004) in FIG. 20, two or more interfaces (2004) may be used according to particular needs, desires, or particular implementations of the computer (2002). The interface (2004) is used by the computer (2002) for communicating with other systems in a distributed environment that are connected to the network (2030). Generally, the interface (2004) includes logic encoded in software or hardware (or a combination of software and hardware) and operable to communicate with the network (2030). More specifically, the interface (2004) may include software supporting one or more communication protocols associated with communications such that the network (2030) or interface’s hardware is operable to communicate physical signals within and outside of the illustrated computer (2002).

[00141] The computer (2002) includes at least one computer processor (2005). Although illustrated as a single computer processor (2005) in FIG. 20, two or more processors may be used according to particular needs, desires, or particular implementations of the computer (2002). Generally, the computer processor (2005) executes instructions and manipulates data to perform the operations of the computer (2002) and any algorithms, methods, functions, processes, flows, and procedures as described in the instant disclosure.

[00142] The computer (2002) also includes a memory (2006) that holds data for the computer (2002) or other components (or a combination of both) that can be connected to the network (2030). For example, memory (2006) can be a database storing data consistent with this disclosure. In one example, memory (2006) may store programs or algorithms for controlling the processes directed to 3D printing and determining the final image set described in the above embodiments. Although illustrated as a single memory (2006) in FIG. 20, two or more memories may be used according to particular needs, desires, or particular implementations of the computer (2002) and the described functionality. While memory (2006) is illustrated as an integral component of the computer (2002), in alternative implementations, memory (2006) can be external to the computer (2002).

[00143] The application (2007) is an algorithmic software engine providing functionality according to particular needs, desires, or particular implementations of the computer (2002), particularly with respect to functionality described in this disclosure. For example, the application (2007) can serve as one or more components, modules, applications, etc. In one example, the application (2007) may include programs or algorithms for controlling operation of 3D printer and determining the final image set that are described in the above embodiments. More specifically, in this example, the programs or algorithms may control the 3D printing and determining the final image set described in the above embodiments with reference to FIGs. 3A- 19. Further, although illustrated as a single application (2007), the application (2007) may be implemented as multiple applications (2007) on the computer (2002). In addition, although illustrated as integral to the computer (2002), in alternative implementations, the application (2007) can be external to the computer (2002). In one example, the method described with reference to FIGs. 7 and 19 may be implemented by the application (2007).

[00144] There may be any number of computers (2002) associated with, or external to, a computer system containing computer (2002), each computer (2002) communicating over network (2030). Further, the term “client,” “user,” and other appropriate terminology may be used interchangeably as appropriate without departing from the scope of this disclosure. Moreover, this disclosure contemplates that many users may use one computer (2002), or that one user may use multiple computers (2002). Furthermore, in one or more embodiments, the computer (2002) is a non-transitory computer readable medium (CRM).

[00145] Although only a few example embodiments have been described in detail above, those skilled in the art will readily appreciate that many modifications are possible in the example embodiments without materially departing from this invention. Accordingly, all such modifications are intended to be included within the scope of this disclosure as defined in the following claims.

[00146] Further, one or more embodiments with respect to determining the image set are described below.

[00147] According to one or more embodiments, a method of 3D printing of an object comprises: determining an initial object model (0) in an object space by discretizing the object onto a voxel grid; determining an image set (I), in an image space, by forward projecting (0) using a forward projection matrix (F); determining an expected dose distribution (D), in the object space, by backward projecting I using a backward projection matrix (B); determining an error set (E) corresponding to D; forward projecting E using F; modifying I based on the forward projection of E; forward projecting the modified I using F to determine D; successively performing modification of I until a predetermined condition occurs; and using a final I that corresponds to the predetermined condition, to print the object.

[00148] According to one or more embodiments, determining 0 comprises: defining an upper limit dose (Dupper) that is the minimum dose for interior of the object; defining a lower limit dose (Diower) that is the maximum dose for exterior of the object; defining a print resolution (A) that is a minimum printable feature size; and applying a low- pass spatial filter with cutoff of A to smooth boundaries of the object, wherein the dose inside the smoothed boundaries of the object is smoothed between Du_pper and Diower.

[00149] According to one or more embodiments, determining E comprises: determining the difference between D and Dupper for interior voxels of the object; if an expected dose, in D, of an interior voxel is not less than Dupper, determining an error corresponding to the interior voxel to be zero; if the expected dose of the interior voxel is less than D_upPer, determining the error corresponding to the interior voxel based on the difference between the expected dose of the interior voxel and D_upPer; determining the difference between D and Diower for exterior voxels of the object; if an expected dose, in D, of an exterior voxel is not greater than Diower, determining an error corresponding to the exterior voxel to be zero; and if the expected dose of the exterior voxel is greater than Diower, determining the error corresponding to the exterior voxel based on the difference between the expected dose of the exterior voxel and Diower.

[00150] According to one or more embodiments, the method further comprises comparing the modified I to predetermined intensity bounds so as the modified I be within the intensity bounds.

[00151] According to one or more embodiments, in the method, the predetermined condition is that E becomes less than a first threshold.

[00152] According to one or more embodiments, the predetermined condition is reaching a maximum number of iterations.

[00153] According to one or more embodiments, the predetermined condition is the difference between E of one iteration step and E of the immediately previous iteration step being less than a second threshold.

[00154] According to one or more embodiments, the method further comprises discretizing I to match a resolution of a projection chip that is used to project the final I.

[00155] According to one or more embodiments, in the method, the error determination is not performed for boundaries of the object.

[00156] According to one or more embodiments, in the method, the error corresponding to the interior voxel and the error corresponding to the exterior voxel are forward projected separately.

[00157] According to one or more embodiments, a system for 3D printing of an object comprises: an optical objective; one or more motors that move the optical objective with respect to a resin; a light source coupled to the optical objective; and a processor. The processor is configured to: determine an initial object model (O) in an object space by discretizing the object onto a voxel grid; determine an image set (I), in an image space, by forward projecting (0) using a forward projection matrix (F); determine an expected dose distribution (D), in the object space, by backward projecting I using a backward projection matrix (B); determine an error set (E) corresponding to D; forward project E using F; modify I based on the forward projection of E; forward project the modified I using F to determine D; successively

31 perform modification of I until a predetermined condition occurs; use a final I that corresponds to the predetermined condition for projection of light from the light source; and instruct the one or more motors to move the optical objective with respect to the resin.

[00158] According to one or more embodiments, a non-transitory computer readable medium (CRM) stores instructions for performing an operation for 3D printing. The operation comprises: determining an initial object model (0) in an object space by discretizing the object onto a voxel grid; determining an image set (I), in an image space, by forward projecting (0) using a forward projection matrix (F); determining an expected dose distribution (D), in the object space, by backward projecting I using a backward projection matrix (B); determining an error set (E) corresponding to D; forward projecting E using F; modifying I based on the forward projection of E; forward projecting the modified I using F to determine D; successively performing modification of I until a predetermined condition occurs; and using a final I that corresponds to the predetermined condition, to print the object.

Claims

CLAIMS What is claimed is:

1. A three-dimensional (3D) printing apparatus comprising: a light source comprising a plurality of pixels configured to emit a plurality of beams toward an optical objective; the optical objective configured to diverge the plurality of beams as the plurality of beams exits the optical objective toward a volume of resin; and one or more motors configured to translate the optical objective with respect to the volume of resin; wherein the one or more motors are configured to translate the optical objective relative to the volume of resin to apply that plurality of beams to a plurality of voxels inside the volume of resin, wherein the application of the plurality of beams to the plurality of voxels solidifies the plurality of voxels, wherein the optical objective applies the plurality of beams such that each of the plurality of voxels are exposed by at least two different beams among the plurality of beams, wherein the at least two different beams expose the each voxel at two different times such that the at least two different beams spatially overlap only in the each voxel, and wherein a total exposure that solidifies the volume of resin in the each voxel equals to at least a sum of exposure of the at least two different beams at the each voxel.

2. The 3D printing apparatus according to claim 1, wherein the at least two different beams propagate at respectively different angles to one another.

3. The 3D printing apparatus according to claim 1, wherein the light source comprises a Digital Micromirror Device (DMD) configured to reflect the plurality of beams toward the volume of resin.

4. The 3D printing apparatus according to claim 1, wherein the light source comprises one or more lasers configured to generate the plurality of beams.

5. The 3D printing apparatus according to claim 1, wherein the one or more motors translate the optical objective relative to the volume of resin along an axial direction of the optical objective.

6. The 3D printing apparatus according to claim 1, wherein the optical objective is coupled to the resin with an index-matching material whose refractive index is greater than one.

7. The 3D printing apparatus according to claim 1, wherein the one or more motors are configured to translate the optical objective along at least two axes that are orthogonal to an axial direction of the optical objective.

8. The 3D printing apparatus according to claim 1, wherein, to translate the objective relative to the resin, the one or more motors move both the optical objective and the resin.

9. The 3D printing apparatus according to claim 1, wherein the apparatus comprises a container that contains the volume of resin.

10. The 3D printing apparatus according to claim 9, wherein the container is configured to contain the volume of resin such that a surface of the volume of resin from where the plurality of beams enters is flat and normal to the axial direction of the optical objective.

11. A method for operating a three-dimensional (3D) printing apparatus, the method comprising: controlling a light source comprising a plurality of pixels to emit a plurality of beams toward an optical objective; diverging the plurality of beams, via the optical objective, as the plurality of beams exits the optical objective toward a volume of resin; translating, via one or more motors, the optical objective with respect to the volume of resin; by translating the optical objective relative to the volume of resin, applying the plurality of beams to a plurality of voxels inside the volume of resin; solidifying the plurality of voxels by the application of the plurality of beams to the plurality of voxels; applying the plurality of beams, via the optical objective, such that each of the plurality of voxels are exposed by at least two different beams among the plurality of beams; and exposing, via the at least two different beams, the each voxel at two different times such that the at least two different beams spatially overlap only in the each voxel, wherein a total exposure that solidifies the volume of resin in the each voxel equals to at least a sum of exposure of the at least two different beams at the each voxel.

12. The method according to claim 11, wherein the application of the at least two beams is such the at least two different beams propagate at respectively different angles from one another.

13. The method according to claim 11, further comprising translating the optical objective relative to the volume of resin along an axial direction of the optical objective.

14. The method according to claim 11, further comprising coupling the optical objective to the resin with an index-matching material whose refractive index is greater than one.

15. The method according to claim 11, wherein the translation of the optical objective is along at least two axes that are orthogonal to an axial direction of the optical objective.

16. A system for three-dimensional (3D) printing comprising: a light source comprising a plurality of pixels configured to emit a plurality of beams toward an optical objective; the optical objective configured to diverge the plurality of beams as the plurality of beams exits the optical objective toward a volume of resin; one or more motors configured to translate the optical objective with respect to the volume of resin; a processor configured to control the one or more motors and thereby control translation of the optical objective over the volume of resin; a memory that stores instructions for translating the optical objective over the volume of resin; wherein the processor controls, based on the instructions, the translation of the optical objective relative to the volume of resin to apply that plurality of beams to a plurality of voxels inside the volume of resin, wherein the application of the plurality of beams to the plurality of voxels solidifies the plurality of voxels, wherein the processor controls the translation of the optical objective to apply the plurality of beams such that each of the plurality of voxels are exposed by at least two different beams among the plurality of beams, wherein the processor controls the translation of the optical objective such that the at least two different beams expose the each voxel at two different times, wherein the at least two different beams at the two different times spatially overlap only in the each voxel, and wherein a total exposure that solidifies the volume of resin in the each voxel equals to at least a sum of exposure of the at least two different beams at the each voxel.

17. The system according to claim 16, wherein the at least two different beams propagate at respectively different angles from one another.

18. The system according to claim 16, wherein the light source comprises a Digital Microminor Device (DMD) configured to reflect the plurality of beams toward the volume of resin.

19. The system according to claim 16, wherein the light source comprises one or more lasers configured to generate the plurality of beams.

20. The system according to claim 16, wherein the one or more motors translate the optical objective relative to the volume of resin along an axial direction of the optical objective.