CONTROLLED POLYMERIZATION OF A TARGET ZONE IN A PHOTOPOLYMERIZABLE MEDIUM

Abstract

A method is described for controlled polymerization of a target zone in a photopolymerizable medium wherein the method comprises a processor connectable to an exposure system for illuminating a target zone in a photopolymerizable medium receiving a 3D data representation of a 3D model of an object and using the 3D model to determine a volume of the target zone, the volume being shaped according to the 3D model; the processor determining a target energy field E.sub.0 defining an energy for volume elements in the medium that is needed to achieve polymerization inside the target zone, the determining being based on a model of the polymerization process in the medium; the processor using the target energy field E.sub.0 and a light propagation model M to compute a solution I0 for the equation E.sub.0=t.Math.M[I.sub.0] wherein I.sub.0 is a direction-dependent illumination field needed for achieving energy deposition in the medium according to the target energy field E.sub.0 and t is the exposure time; and, the processor controlling the exposure system based on the direction-dependent illumination field I.sub.0, the exposure including generating a plurality of direction-dependent illumination beams to deposit energy within the target zone according to the target energy field E.sub.0.

Claims

1. A method of controlled polymerization of a target zone in a photopolymerizable medium comprising: a processor connectable to an exposure system for illuminating a target zone in a photopolymerizable medium receiving a data representation of a 3D model of an object and using the 3D model to determine volume elements of the target zone, the photopolymerizable medium including a photo-activation compound for activating polymerization of the medium based on a first illumination field I.sub.+ of a first wavelength and a photo-deactivation compound for deactivating the polymerization in the medium based on a second illumination field I.sub.− of a second wavelength; the processor determining a target energy field E.sub.0, the target energy field E.sub.0 defining an energy to be absorbed by the volume elements of the target zone to achieve polymerization inside the target zone, the determining by the processor being based on a polymerization model of the polymerization process in the medium; the processor computing a direction-dependent illumination field I.sub.o for depositing within an exposure time t the target energy field E.sub.0 in the volume elements of the medium, the computing of I.sub.o including using a light propagation model for an attenuating medium to compute the first illumination field I.sub.+ for depositing within the exposure time t a first deposited energy E.sub.+ in the volume elements for activating and maintaining polymerization and the second illumination field I.sub.− for depositing within the exposure time t a second deposited energy E.sub.− in the volume elements for deactivation of the polymerization; and the processor controlling the exposure system based on the direction-dependent illumination field I.sub.o, the exposure including generating a plurality of direction-dependent illumination beams to deposit energy within the volume elements of the medium according to the target energy field E.sub.0.

2. The method according to claim 1, wherein computing a direction-dependent illumination field I.sub.o further comprises: determining a solution I.sub.o for a system of equations of the type: E.sub.0=t.Math.[I.sub.o].

3. The method according to claim 1, wherein computing a direction-dependent illumination field I.sub.o further comprises: iteratively computing an approximate solution I.sub.o based on the light propagation model , the computing including minimizing a difference between the target energy field E.sub.0 and a deposited energy field E predicted by the light propagation model .

4. The method according to claim 1, wherein computing a direction-dependent illumination field I.sub.o further comprises: iteratively computing an approximate solution I.sub.0 based on the polymerization model [E] and the light propagation model [I.sub.0], the computing including minimizing a difference between the target monomer conversion .sub.0 and a monomer conversion [E] achieved due to the deposited energy field E predicted by the light propagation model [I.sub.0].

5. The method according to claim 1, wherein the polymerization model is a linear approximation of the type: E.sub.0=E.sub.+−β.Math.E.sub.− wherein β is a proportionality constant and wherein E.sub.0>E.sub.crit inside the target zone and E.sub.0<E.sub.crit outside the target zone and E.sub.crit being a critical energy needed to achieve polymerization.

6. The method according to claim 1, wherein E.sub.0 is determined based on a polymerization model [E.sub.0]=.sub.0, wherein .sub.0 is the target monomer conversion in a volume element.

7. The method according to claim 1, wherein generating a plurality of direction-dependent illumination beams comprises: the processor controlling a rotatable and/or movable illumination system, the illumination system including at least one spatial light modulator or a laser galvanometer scanner for generating the plurality of direction-dependent illumination beams.

8. The method according to claim 1, the method further comprising: the processor controlling a detection system to measure the intensity of a part of the plurality of illumination beams that was not absorbed by the medium.

9. The method according to claim 8, further comprising: recomputing the direction-dependent optimal illumination field I.sub.0 based on the target energy field E.sub.0, the light propagation model and the computed absorptivity μ of the medium; and, the processor controlling the exposure system based on the recomputed optimal direction-dependent illumination field I.sub.o.

10. The method according to claim 1, wherein the system for illuminating a target zone comprises a container comprising the photopolymerizable medium and a rotatable and/or movable illumination system, the illumination system including an optical system comprising one or more light sources and optical elements, light modulating apertures and a support structure including motors and/or actuators configured to rotate and/or move the illumination system around the container while locally exposing the medium to light of one or more predetermined wavelengths.

11. The method according to claim 1, wherein the exposure system for illuminating a target zone further comprises a 3D scanning system, and wherein the determination of the volume elements of the target zone comprise: using the 3D scanning system to determine a 3D surface representation of a physical object that is positioned in the medium; aligning an orientation of the 3D model with the orientation of the 3D surface representation; and using the aligned 3D model and the 3D surface representation to determine the volume elements around the physical object, the volume elements defining the target zone.

12. A method of controlled polymerization of a target zone in a photopolymerizable medium comprising: a processor connectable to an exposure system for illuminating a target zone in a photopolymerizable medium in a transparent container receiving a data representation of a 3D model of an object and using the 3D model to determine volume elements of the target zone, the volume defined by the volume elements in the medium being shaped according to the 3D model; the processor determining a target energy field E.sub.0, the target energy field E.sub.0 defining an energy to be absorbed by the volume elements of the target zone to achieve polymerization inside the target zone, the determining by the processor being based on a polymerization model of the polymerization process in the medium; the processor computing a direction-dependent illumination field I.sub.o for depositing within an exposure time t the target energy field E.sub.0 in the volume elements of the medium, the computing being based on a light propagation model configured to predict the deposited energy field E in an attenuating medium, the deposited energy field E being generated upon exposure of the volume elements of the target zone to the direction-dependent illumination field I during the exposure time t; the processor controlling the exposure system based on the direction-dependent illumination field I.sub.0, the exposure including generating a plurality of direction-dependent illumination beams to deposit energy within the volume elements of the medium according to the target energy field E.sub.0; and, the processor controlling an imaging system comprising one or more camera sensors arranged to capture images of the medium and determining changes of the medium, during exposure based on the captured images.

13. An exposure system adapted to photopolymerize a target zone in a photopolymerizable medium comprising: a computer connectable to a rotatable illumination system for exposing the photopolymerizable medium, the computer comprising a computer readable storage medium having at least part of a program embodied therewith; and a processor coupled to the computer readable storage medium, wherein responsive to executing the computer readable program code, the processor is configured to perform executable operations comprising: receiving a data representation of a 3D model of an object and using the 3D model to determine volume elements of the target zone, the photopolymerizable medium including a photo-activation compound for activating polymerization of the medium based on a first illumination field I.sub.+ of a first wavelength and a photo-deactivation compound for deactivating the polymerization in the medium based on a second illumination field I.sub.− of a second wavelength; determining a target energy field E.sub.0, the target energy field E.sub.0 defining an energy to be absorbed by the volume elements of of the target zone to achieve polymerization inside the target zone, the determining by the processor being based on a polymerization model of the polymerization process in the medium; computing a direction-dependent illumination field I.sub.o for depositing within an exposure time t the target energy field E.sub.0 in the volume elements of the medium, the computing of I.sub.o including using a light propagation model for an attenuating medium to compute the first illumination field I.sub.+ for depositing within the exposure time t a first deposited energy E.sub.+ in the volume elements for activating and maintaining polymerization and the second illumination field I.sub.− for depositing within the exposure time t a second deposited energy E.sub.− in the volume elements for deactivation of the polymerization; controlling the exposure system based on the direction-dependent illumination field I.sub.o, the exposure including generating a plurality of direction-dependent illumination beams to deposit energy within the volume elements of the medium according to the target energy field E.sub.0.

14. The exposure system according to claim 13, wherein computing a direction-dependent illumination field I.sub.o, further comprises: determining a solution I.sub.o for a system of equations of the type: E.sub.0=t.Math.[I.sub.o].

15. The exposure system according to claim 13, wherein computing a direction-dependent illumination field I.sub.0 further comprises: iteratively determining an approximate solution I.sub.o based on the light propagation model , the computing including minimizing a difference between the target energy field E.sub.0 and a deposited energy field E predicted by the light propagation model .

16. The exposure system according to claim 13, wherein computing a direction-dependent illumination field I.sub.o further comprises: iteratively computing an approximate solution I.sub.0 based on the polymerization model [E] and the light propagation model [I.sub.0], the computing including minimizing a difference between the target monomer conversion .sub.0 and a monomer conversion [E] achieved due to the deposited energy field E predicted by the light propagation model [I.sub.0].

17. The exposure system according to claim 13, wherein the polymerization model is a linear approximation of the type: E.sub.0=E.sub.+−β.Math.E.sub.− wherein β is a proportionality constant and wherein E.sub.0>E.sub.crit inside the target zone and E.sub.0<E.sub.crit, outside the target zone and E.sub.crit being a critical energy needed to achieve polymerization.

18. The exposure system according to claim 13, wherein E.sub.0 is determined based on a polymerization model [E.sub.0]=.sub.0, wherein .sub.0 is the target monomer conversion in each volume element.

19. The exposure system according to claim 13, wherein generating a plurality of direction-dependent illumination beams includes: the processor controlling a rotatable and/or movable illumination system, the illumination system including at least one spatial light modulator or a laser galvanometer scanner for generating the plurality of direction-dependent illumination beams.

20. The exposure system according to claim 13, wherein the executable operations further comprise: controlling a detection system to measure the intensity of a part of the plurality of illumination beams that was not absorbed by the medium.

21. The exposure system according to claim 20, wherein the executable operations further comprise: recomputing the direction-dependent illumination field I.sub.o based on the target energy field E.sub.0, the light propagation model and the computed absorptivity μ of the medium; and, controlling the exposure system based on the recomputed direction-dependent illumination field I.sub.o.

22. A computer program product comprising software code portions configured for, when run in the memory of a computer, executing the method of claim 1.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0058] FIG. 1 depicts a schematic of a system for forming a 3D object using photopolymerization according to an embodiment of the invention;

[0059] FIG. 2 depicts part of a schematic of a system for forming a 3D object using photopolymerization according to an embodiment of the invention;

[0060] FIG. 3A-3E depict computed energy deposition based on a superposition of multiple illumination beams according to various embodiments of the invention;

[0061] FIG. 4A-4D depict computed energy deposition resulting from a superposition of multiple illumination beams that were precomputed and generated according to an analytical model valid for media with low absorptivity;

[0062] FIG. 5 depicts a flow diagram of a method for determining an illumination energy for a target polymerization according to an embodiment of the invention;

[0063] FIG. 6A-6D depict examples of computed energy deposition for a 3D object based on the low absorptivity model according to another embodiment of the invention;

[0064] FIG. 7 depicts a flow diagram of a method for determining an illumination energy for a target polymerization zone based according to an embodiment of the invention;

[0065] FIG. 8A-8D depict examples of computed energy deposition for a 3D object based on the iterative approach according to an embodiment of the invention;

[0066] FIG. 9 depicts a schematic of a system for forming a 3D object using photopolymerization according to an embodiment of the invention;

[0067] FIG. 10 depicts a schematic of a system for forming a 3D object using photopolymerization according to another embodiment of the invention;

[0068] FIG. 11 is a block diagram illustrating an exemplary data computing system that may be used for executing methods and software products described in this disclosure.

DETAILED DESCRIPTION

[0069] FIG. 1 depicts a schematic of an exposure system for illumination of a target zone in a photopolymerizable medium (or, in short, a medium) according to an embodiment of the invention. As shown in FIG. 1, the system 100, a 3D photopolymerization system, may comprise a container 104 enclosing a medium of a certain volume and a rotatable and/or movable illumination system, including an optical system comprising light sources 106 and optical elements 108, light modulating aperture 110 and a support structure including motors and/or actuators 102 configured to rotate and/or move the illumination system around the container at high speed (either around a single axis or multiple axes) while locally exposing the medium to light of at least one predetermined wavelength. This way, the medium is exposed by light beams of predetermined intensities from different angles within a short span of time. In an embodiment, if the properties of the medium allow, the container may be rotated instead or at the same time with the rotatable illumination system.

[0070] The exposure system allows controlled exposure of locations in the medium by light of a certain intensity and wavelength may be achieved for locally triggering a photopolymerization process within a predetermined volume of the medium. The volume in which photopolymerization takes place may be referred to as the target polymerization zone or in short the target zone. Further, the total volume of the medium in the container may be referred to as the build volume. During the exposure of the target zone with an illumination beam of a predetermined intensity and wavelength incident from a single direction, a certain amount of the light energy will be absorbed by volume elements 120 along the propagation path of that beam. Exposure of each volume element by the plurality of illumination beams incident from a large number of directions may cause deposition (by absorption of part of the light) of a certain total amount of energy. This energy may hereafter be referred to as the deposited energy field E defined for volume elements in the medium. Intensity and exposure times, corresponding to the plurality of illumination beams of each wavelength, should be configured such that the deposited energy field E is as close as possible to a target energy field E.sub.0 at which the medium in a volume element will polymerize. Each incident illumination beam will propagate through the medium and a non-absorbed part will leave the medium where it may be detected by a detection system, e.g. an imaging system 112.

[0071] A photoinduced chemical process, such as a photopolymerization process or a monomer cross-linking process, can be induced through controlled application of electromagnetic radiation (typically ultra-violet, visible light or, in some cases, infra-red) to a suitable photopolymerizable medium. Photopolymerizable media may include various types of chemical systems which may be based on e.g. free radical polymerization, cationic polymerization, anionic polymerization or acid catalysed polymerization or any other suitable polymerization process. A photopolymerizable medium may contain a photo-initiator component that, when exposed to light of a certain intensity for a certain period, may generate chemical species that participate in the reaction of polymerization. The rate of the reaction will depend, among other parameters, on the concentration of the photo-initiator, its quantum yield, and the amount of illumination energy absorbed by it. In order to improve photo-initiator sensitivity to illumination of a particular wavelength, photosensitizers can be used.

[0072] The rotatable illumination system may include one or more light sources 106 for producing one or more source beams that can be pre-shaped by the optical elements 108 and modulated by a computer-controlled spatial light modular (SLM) 110. Various readily available types of lasers or light-emitting diode sources may be used as a light source. Depending on the photopolymerization process, different sources of different wavelengths may be used. The optical system may include optical elements positioned before and after the spatial light modulator for collimation, beam shaping and focusing. The optical elements may be selected to shape a wide, sufficiently homogeneous beam for exposing the SLM. The SLM comprises individually controllable light valves (pixels) for spatially modulating the intensity of the source beam, producing what is referred to as the illumination beam. It is assumed that laws of geometrical or ray optics are applicable and that the computer-controlled light valves (the pixels) of the SLM may be used to generate a plurality of substantially parallel light rays forming a single illumination beam, while the combination of illumination beams incident from different directions is referred to as illumination field. The intensity of each illumination ray may be controlled by (at least one) of the light valves of the SLM and, depending on implementation, illumination rays that belong to one source beam can be generated simultaneously or in asynchronous manner.

[0073] An intensity modulating SLM may be used to modulate the intensity of an illumination beam depending on the direction in which the illumination beam propagates through the space in which the medium is located. The rotatable illumination system can be controlled to expose a predetermined target location of the medium by directing one or more parallel collimating intensity-modulated illumination beams from different source locations towards the target location. This way, a large number of illumination beams which intersect at the target location can be produced. The SLM may be implemented using a liquid-crystal display (LCD), digital light processing (DLP), liquid crystal on silicon (LCoS) or any other optical technology that enables modulation of illumination beams. In an embodiment, if the source used for illumination is coherent, the beam may be generated using a holographic technique instead of an intensity modulation technique. In that case, a phase modulating SLM may be used instead of an intensity modulating SLM.

[0074] In an embodiment, the optical system may comprise a detection system 112, e.g. one or more imaging systems, which may be used to measure the intensity of the light beam that was not absorbed by the medium. The transmitted intensity of the part of the illumination beam that was not absorbed may be used to compute the current absorptivity of the medium during exposure.

[0075] The system may include one rotatable illumination system or multiple illumination systems in order to achieve faster energy deposition within the medium. Different illumination systems may comprise light sources of different wavelengths, for instance, to allow Fourier transform infrared (FTIR) measurements.

[0076] In order to generate the optimal illumination field and, hence, to deposit the target energy at predetermined locations in the target zone in the medium, the rotatable illumination system may be controlled by a computer system 116. In particular, the computer system comprises a processor for executing one or more software programs 118 that are configured to control the system. The computer system may include a data storage connected to the processor. The data storage may comprise a 3D data representation 114 of an object of a certain shape that may be reproduced through photopolymerization. The 3D data representation may have a predetermined data format e.g. a voxel representation, point cloud representation or a 3D surface mesh representation of the 3D object 122.

[0077] Additionally, the data structure may define volume elements 119 in the 3D object in which a certain energy field needs to be deposited by one or more direction-dependent illumination beams. The processor may use the 3D data representation of the 3D object to determine a target polymerization zone 124 in the photopolymerizable medium. The target polymerization zone (or, in short, the target zone) may define a volume within the medium that is shaped according to the 3D model. In particular, the target zone may define a volume within the medium that will be exposed by the rotatable illumination system to light of an intensity and a wavelength such that photopolymerization will take place. This way, a photopolymerization process in the target zone can be triggered (activated), maintained and, optionally, inhibited. The result of the photopolymerization process may be solid polymer copy of the object that was provided as a 3D data representation to the computer system.

[0078] It is submitted that the system depicted in FIG. 1 is only a non-limiting example of an exposure system and many variations are possible without departing from the invention. For example, the imaging system may include one or more camera sensors arranged around the reservoir so that the system can image the polymerization process from different angles. Based on these images the system may locally determine relevant parameters, e.g. absorptivity or refractive index of the medium, during exposure and use these measured parameters to update the model that is used to determine the energy that needs to be deposited within the target zone to realize the creation of a 3D object trough photopolymerization.

[0079] FIG. 2 depicts a part of the system shown in FIG. 1 in more detail. In particular, FIG. 2 illustrates an SLM 202 that is configured to move (rotate) around a transparent container comprising a photopolymerizable medium 204. As described with reference to FIG. 1, the SLM may be used to spatially modulate a source beam through the action of individually addressable light valves, thereby generating intensity modulated illumination beam. Further, various optics can be placed before and after the SLM for focusing and collimation of the illumination beam. Each computer-controlled light valve of the SLM may be used to generate an illumination ray 206 of a predetermined intensity which will contribute to the total deposited energy along its propagation path through the medium. The SLM and the container comprising the medium may be located in a space defined by a coordinate system 208. The coordinate system may be coordinate system that is suitable for describing the target zone 222 and the propagation of illumination beams through the space, e.g. a Cartesian coordinate system based on three orthogonal axes {x, y, z} or any other coordinate system that is suitable for defining positions in a 3D space.

[0080] To simplify the mathematical description of the processes executed by the system, it is assumed that the plane of SLM device is positioned in space parallel to the axis z and can rotate around it. It is further assumed that, at least within the boundaries of the photopolymerizable medium, illumination rays propagate in a parallel fashion. In this case, the mathematical description of the transport of the illumination energy is translation invariant along axis z. This allows to formulate a model of energy transport and deposition based on illuminated beams modulated by any single horizontal row of SLM pixels 210 in any x-y plane 212 independently from the others and omit z axis in the derivations. In practice, a set of illumination beams propagating in a parallel fashion can only be generated for rays significantly wider than the diffraction limit of the optical system. It is assumed that either the target resolution is low enough so that it is not limited by diffraction or that a parallel beam can be emulated through the use of Bessel lenses or rapid adaptive lenses such as TAG lenses or similar devices.

[0081] If the illumination beams generated by the SLM propagate in a parallel fashion, every illumination beam generated by a pixel of the SLM may be defined by: [0082] an illumination angle θ, 214 defining an angle between the direction of the illumination beam and the x-axis, [0083] an offset r, 216 defining the position of the pixel within the horizontal row 210, [0084] a height z, and, [0085] a beam length (or depth) ρ 218 defining a distance between the origin of the ray and a location {x.sub.0, y.sub.0, z.sub.0} inside the medium 220.

[0086] Controlling each row of pixels while rotating the SLM around the photopolymerizable medium allows to generate the illumination field I.sub.0(r,θ,z) pre-computed according to the predetermined target energy E.sub.0(x,y,z) 224.

[0087] While a more detailed model of monomer to polymer conversion can be considered, in an embodiment, a model may be used which assumes that to sustain a photopolymerization process in the target zone, the energy E(x,y,z) absorbed by the photo-initiator at positions in the target zone needs to exceed a certain critical threshold E.sub.crit. In another embodiment, when radiation comprising a plurality (several) wavelengths is used to concurrently control initiation and, optionally, inhibition of the photopolymerization reaction, a conversion model can be introduced where a certain linear combination of the corresponding deposited energies associated with the different wavelengths should exceed a critical value E.sub.crit. Using the intensity-modulated illumination beams, the polymerization process may be locally controlled in a plurality of positions in the target zone, a 3D object can be formed in the medium.

[0088] The formation of a certain 3D object however may require a complex direction-dependent illumination process of the medium by the illumination system wherein parameters that contribute to the attenuation of the light during exposure may be considered. In an embodiment, a computational approach based on an analytical model describing a weakly attenuating medium may be used to determine an exposure time t and illumination intensity I.sub.0(r,θ,z) to sustain and control the polymerization reaction within an accurately defined target polymerization zone. A single or multiple exposures of the target polymerization zone may generate a solid polymer object as defined by the 3D data representation of the object that was provided to the input of the computer. Hereunder, methods and systems are described to control the system initiation and inhibition of the polymerization reaction in a target zone of a photopolymerizable medium. These embodiments are described hereunder in more detail.

[0089] The fraction of the incoming electromagnetic energy that is absorbed by a volume element of any component of the photopolymerizable medium upon illumination may be described by an absorptivity function μ. Absorptivity is related to commonly used quantities, namely a molar extinction ε and a concentration of the photoreactive component C.sub.p, as: μ=ε.Math.C.sub.p. Instead of the standard definition of absorptivity coefficient (as commonly used in chemistry), a modified absorptivity may be used that allows to shorten mathematical notation in the mathematical description of the various embodiments in this application. The modified absorptivity may be defined as μ=ε.Math.C.sub.p.Math.ln(10), where ln(10)≈2.303.

[0090] This function depends on the wavelength of the electromagnetic radiation and can be determined empirically or computed. When a homogeneous photopolymerizable medium is illuminated by radiation of a wavelength λ, the radiation intensity I at depth ρ is described by Beer-Lambert's law:

I(λ,ρ)=I(λ,0).Math.exp(−μ.sub.tot(λ).Math.ρ)  (1)

wherein I(λ, 0) is the intensity upon incidence, μ.sub.tot(λ) is the combined absorptivity of all components of the photopolymerizable medium (photo-initiators, photosensitizers, photo-inhibitors, dyes, other attenuating additives).

[0091] By taking a derivative of the illumination intensity: I(λ, ρ) and multiplying it by the exposure time t, the energy E.sub.tot(λ, ρ) absorbed by an element of volume at depth ρ can be calculated:

[00003]$\begin{array}{cc}{E}_{\mathrm{tot}}\left(\lambda ,\rho \right)=-\frac{\partial }{\partial \rho }I\left(\lambda ,\rho \right).Math.t& \left(2\right)\\ E_{\mathrm{tot}}\left(\lambda ,\rho \right)={\mu }_{\mathrm{tot}}\left(\lambda \right).Math.t.Math.I\left(\lambda ,0\right).Math.\exp \left(-{\mu }_{\mathrm{tot}}\left(\lambda \right).Math.\rho \right)& \left(3\right)\end{array}$

Typically, only a fraction of that energy will be absorbed by the photo-initiator or/and the photosensitizer, so the amount of “useful” energy E.sub.+(λ, ρ) initiating the polymerization reaction may be determined by the following expression:

E.sub.+(λ,ρ)=μ.sub.+(λ).Math.t.Math.I(λ,0).Math.exp(−μ.sub.tot(λ).Math.ρ)  (4)

where μ.sub.+(λ) is the absorptivity of the photo-initiator or the photosensitizer. The degree of polymerization that leads to formation of a solid material is typically associated with a certain critical energy E.sub.crit, such that:

E.sub.+(λ,ρ)>E.sub.crit  (5)

where, E.sub.crit, may be determined empirically or calculated using a reaction kinetics model.

[0092] When multiple illumination beams intersecting at a certain depth are used to initiate the reaction of polymerization, the combined intensity at the intersection point should be at least greater than the intensity at zero depth. Considering the Beer-Lambert's law, a simple condition for the minimum number of illumination beams N.sub.θ that are needed to illuminate a single point at the centre of the volume at depth ρ can be calculated:

N.sub.θ>exp(μ.sub.tot(λ).Math.ρ)  (6)

This condition provides a rough notion of the number of illumination beams needed to create a sufficient contrast to polymerize the deepest point in the volume of a certain size before the less deep parts of the volume are polymerized. For instance, for absorptivity μ.sub.tot=1 cm.sup.−1 (absorptivity considered in One-step volumetric additive manufacturing of complex polymer structures, Science Advances 8 Dec. 2017, vol. 3) at least three illumination beams are needed to create enough contrast at a depth of 1 cm. However, to increase the illumination depth to 5 cm more than 300 intersecting illumination beams are required. Polymerizable media with lower absorptivity will be suitable for illumination at higher depths.

[0093] Even when condition expressed by Eq. 6 is satisfied, the illumination energy will be absorbed by the photo-initiator on its path to the target zone, thereby dramatically limiting the resolution with which the boundary of the target zone can be defined. In order to overcome the problem of limited boundary contrast, the polymerization reaction may be terminated (inhibited) in some of the previously illuminated areas. In that case, a mechanism for polymerization inhibition or polymerization deactivation can be employed. Examples of such mechanisms are known from Resolution Augmentation through Photoinduced Deactivation (RAPID) lithography, PhotoInhibited Super Resolution (PInSR) lithography and multi-colour photolithography. In these approaches, a single source or multiple sources operating at different wavelengths are used to initiate the photopolymerization reaction and subsequently deactivate or substantially inhibit it.

[0094] For instance, in PInSR a photo-initiator camphorquinone sensitive to blue light is used in combination with photo-inhibitor tetraethylthiuram disulfide sensitive to ultra-violet light. In this approach, photoinitiation and photoinhibition reactions are controlled by two light sources: a blue diode-pumped solid-state laser with 473 nm wavelength and ultra-violet argon ion laser with a 365 nm wavelength. The 473 nm wavelength light is absorbed by the photo-initiator which leads to production of free radicals involved in polymerization reaction, while 365 nm light is absorbed by the photo-inhibitor and generates radical traps that terminate polymerization.

[0095] Depending on the type of the photopolymerizable medium and the type of inhibition or deactivation approach, accurate description of the kinetics of the polymerization reaction may require more complex models: typically, rate equations based on partial differential equations). Typically, rate equations based on partial differential equations, random graph or reactive force field approaches. In that case, in an embodiment, the polymerization reaction model may be regarded as an optimization problem where a conversion described by a model [E.sub.+,E.sub.−] has to exceed a critical conversion .sub.crit inside the target zone and stay below .sub.crit outside of the target zone, i.e.: [E.sub.+, E.sub.−]>.sub.crit.

[0096] In an embodiment, a semi-empirical model for describing the activation and inhibition of the polymerization process may be used. In such scheme, polymerization may be achieved when a certain linear combination of the energy E.sub.+ associated with initiation and the energy E.sub.− associated with inhibition (or deactivation) of the polymerization exceeds a critical value:

E.sub.+(λ.sub.+,ρ)−β.Math.E.sub.−(λ.sub.−,ρ)>E.sub.crit,  (7)

wherein the parameter β is a proportionality constant and the wavelengths λ.sub.+ and λ.sub.− are respectively wavelengths at which the photo-initiator and the photo-inhibitor absorb photons respectively. Constant β may be computed using rate equations or measured for a particular photopolymerizable medium. A resulting combined deposited energy E(λ, ρ) may define a linear combination of deposited energies E.sub.+(λ.sub.+,ρ)−β.Math.E.sub.−(λ.sub.−,ρ) for systems with “active” inhibition. In case of “passive” inhibition, the deposited energy can be described by the single term E.sub.+(λ.sub.+,ρ) minus a constant.

[0097] In an embodiment, one or more time dependent terms or non-linear terms of the form E.sub.+(λ.sub.+,ρ).Math.E.sub.−(λ.sub.−,ρ) may be used to yield a better approximation of the model [E.sub.+,E.sub.−]. For instance, time dependent terms may be used to represent effects of depletion of the initiator or inhibitor molecules during the illumination if such effects become apparent in a particular regime. An expression for the deposited energy with a larger number of terms can be considered for systems with a larger number of photosensitive components.

[0098] In an embodiment, a computational approach may be used to compute the illumination intensity for each direction of illumination that is required to facilitate deposition of the correct quantities of initiation energy E.sub.+ and inhibition energy E.sub.− in every location of the polymerization volume. The underlying mathematical model of the deposited energy E(x,y,z) may be defined as follows:

E(x,y,z)=t.Math.[I.sub.0(r,θ,z)],  (8)

wherein is an operator that describes the deposition of energy in the volume of photopolymerizable medium given a direction-dependent illumination field I.sub.0(r,θ,z). An optimal illumination field I.sub.0(r,θ,z) may be computed by searching for a solution for Eq. 8 after substituting the target energy field E.sub.0(x,y,z) in the right-hand side of the equation. If an inverse operator .sup.−1 can be calculated, an analytical solution may be used:

[00004]$\begin{array}{cc}{I}_0\left(r,\theta ,z\right)=\frac{1}{t}.Math.{ℳ}^{-1}\left[E_0\left(x,y,z\right)\right]& \left(9\right)\end{array}$

In a further embodiment, an iterative solver algorithm may be used to find an approximated solution to Eq. 8. For instance, a well-known Least-Squares minimization algorithm may be employed to find I.sub.0(r,θ,z) corresponding to a minimum of an objective function:

[00005]$\begin{array}{cc}\underset{I_0}{\mathrm{argmin}}.Math.t.Math.ℳ\left[I_0\left(r,\theta ,z\right)\right]-E_0\left(x,y,z\right).Math.\begin{array}{c}2\\ 2\end{array}& \left(10a\right)\end{array}$

Wherein the notation |.Math.|.sub.2.sup.2 denotes the Euclidean norm (or L2 norm). A regularization term of some sort can be added to the L2 term to constrain possible solutions for I.sub.0(r,θ,z) to physically more appropriate ones.

[0099] In a further embodiment, the polymerization model may be a non-linear polymerization model [E.sub.+,E.sub.−]. In that case, an iterative solver algorithm may be used to find I.sub.+(r,θ,z) and I.sub.−(r,θ,z) corresponding to a more ‘high-level’ objective function:

[00006]$\begin{array}{cc}\underset{I_{+},I_{-}}{\mathrm{argmin}}.Math.\mathrm{��}\left[E_{+},E_{-}\right]-{\mathrm{��}}_0\left(x,y,z\right).Math.\begin{array}{c}2\\ 2\end{array}& \left(10b\right)\end{array}$

[0100] Here, E.sub.+ and E.sub.− may be computed using the light propagation and absorption model (8). While the optimization problem above is designed to solve equations of the type [E]=.sub.0, different approaches may be used to solve inequalities of the type [E]≥P.sub.0 to allow for over-exposure of already solidified parts of build volume. For instance, model [E] can be expressed via a sigmoid function, allowing for over-exposure.

[0101] Operator may be based on a model of photon propagation and absorption inside the photopolymerizable medium and its particular form may depend on one or more parameters, including e.g. parameters of the illumination system, type of the medium and the level of acceptable approximations. This operator may hereafter be referred to as the propagation operator .

[0102] Below a more detailed embodiment of an implementation of the propagation operator based on the Radon transform is provided. In this embodiment, it is assumed that photons propagate along straight paths in a parallel fashion. More accurate models such as a model based on the so-called Radiative Transport Equations or a Monte Carlo approach may be used when effects such as refraction, diffraction and/or scattering are considered.

[0103] FIG. 3A-3E depict a simulated energy deposition resulting from a superposition of multiple illumination beams, wherein beams of uniform intensity are used. In this example, the target shape is a solid cylinder in a photopolymerizable medium with absorptivity of 1 cm.sup.−1 inside a container of dimensions 1×1 cm (in x-y plane). FIG. 3A depicts the relation between the {x,y} coordinates and the {r,ρ,θ} coordinates (as described in detail with reference to FIG. 2. FIG. 3B depicts a normalized intensity profile of a 0.4 cm wide illumination beam of uniform intensity. FIG. 3C depicts a simulated energy field deposited in a photopolymerizable medium when two orthogonal illumination beams of uniform intensity are used. By extending the illumination process to a higher beam number, it is possible to generate a deposited energy field that is closer to a circular target energy field. FIGS. 3D and 3E depict examples of the deposited energy field resulting from 16 and 314 illumination beams respectively. In FIG. 3E a circular distribution of the energy is achieved wherein each point of the medium that absorbs energy larger than a critical energy E.sub.crit will be polymerized. However, as shown in these figures, the polymerization zone does not exhibit a sharp boundary in this example since energy of every incoming illumination beam is absorbed along the whole propagation path. This will lead to a substantial loss of resolution due to fluctuations in the properties of the photopolymerizable medium and errors in the illumination.

[0104] As described with reference to FIG. 3A, any position (defined by the coordinates {x,y}) on a 2D plane can be illuminated by a plurality of illumination rays characterized by illumination angle θ, offset r and depth ρ. This two-dimensional description is sufficient for the case of parallel beam illumination with rotation around axis z (due to translational invariance along the rotation axis). In more complex cases, the scheme may be easily generalized to three-dimensions based on a suitable coordinate system.

[0105] The intensity of the illumination may be described by a direction depended illumination field I.sub.0(r,θ). A location {x, y} can be illuminated only by sources located on straight lines intersecting with that location, so that the offset r.sub.0 of these illumination sources may be described by the following expression:

r.sub.0(θ)=x.Math.cos(θ)+y.Math.sin(θ)  (11)

and the total intensity that is directed to a location {x,y} can be described by an integral:

I(x,y)=∫.sub.0.sup.2π∫.sub.−∞.sup.∞I.sub.0(r,θ).Math.δ(x.Math.cos(θ)+y.Math.sin(θ)−r)drdθ  (12)

[0106] Here δ(.Math.) represents a Dirac function. To account for attenuation of light along its propagation path, an integral absorptivity factor is introduced in accordance with Beer-Lambert's law. To do that, an absorptivity has to be defined on the x-y grid parametrized by the depth ρ and offset r.sub.0(θ):

x=r.sub.0(θ).Math.cos(θ)+ρ.Math.sin(θ)

y=r.sub.0(θ).Math.sin(θ)−ρ.Math.cos(θ)  (13)

[0107] Combining Eq. 11 and Eq. 13, one finds an expression for Beer-Lambert's law of attenuation based on the total absorptivity of the polymerization medium μ.sub.tot(x,y) for the direction θ. This attenuation term T (x,y,θ) looks as follows:

T(x,y,θ)=exp(−∫.sub.−∞.sup.ρ.sup.oμ.sub.tot(ρ.Math.sin(θ)+(x.Math.cos(θ)+y.Math.sin(θ)).Math.cos(θ),−ρ.Math.cos(θ)+(x.Math.cos(θ)+y.Math.sin(θ)).Math.sin(θ))dρ  (14)

where depth ρ=−∞ corresponds to the outwards direction and depth ρ=ρ.sub.0 corresponds to the location {x, y} inside the medium. Value of ρ.sub.0 can be expressed through x, y and θ as:

ρ.sub.0=x.Math.cos(θ)−y.Math.sin(θ)  (15)

[0108] The attenuated total illumination intensity Î at location {x, y} can be derived by combining the expression for the total illumination intensity from Eq. 12 with the attenuation term from Eq. 14:

Î(x,y)=∫.sub.0.sup.2π∫.sub.−∞.sup.∞I.sub.0(r,θ).Math.δ(x.Math.cos(θ)+y.Math.sin(θ)−r)dr.Math.T(x,y,θ)dθ   (16)

Finally, assuming that the absorptivity is constant within the boundaries of the photopolymerizable medium, the energy E(x,y) associated with photo-initiation at any point {x, y} can be computed similarly to Eq. 4:

E(x,y)=μ.sub.+.Math.t.Math.∫.sub.0.sup.2π∫.sub.−∞.sup.∞I.sub.0(r,θ).Math.δ(x.Math.cos(θ)+y.Math.sin(θ)−r)dr.Math.T(x,y,θ)dθ   (17)

Now, the direction-dependent illumination intensity I.sub.0(r,θ) can be computed by solving Eq. 17 after substitution of the target energy E.sub.0(x,y) into the right-hand side. In a system with active photo-initiation and photo-inhibition, two terms should be considered, such that E(x,y)=E.sub.+(x,y)−β.Math.E.sub.−(x,y) and:

E(x,y)>E.sub.crit, for {x,y} inside the target polymerization zone,

E(x,y)<E.sub.crit, for {x,y} outside of the target polymerization zone  (18)

[0109] To find an analytical solution for Eq. 18 a weakly attenuating medium may be assumed, neglecting the attenuation term T(x,y,θ) in the Eq. 17. In that case, the expression for the deposited illumination energy E(x,y) may be simplified:

E(x,y)=μ.sub.+.Math.t.Math.∫.sub.0.sup.2π∫.sub.−∞.sup.∞I.sub.0(r,θ).Math.δ(x.Math.cos(θ)+y.Math.sin(θ)−r)drdθ  (19)

In fact, the integral in Eq. 19 is a form of the so-called adjoint Radon transform and allows to compute an optimal illumination intensity analytically. A definition for the Radon transform and its adjoint form is provided hereunder. The Radon transform is a linear transformation that defines a series of line integrals applied to a function ƒ(x,y) defined on a 2D plane. It can be written in the following form:

p(r,θ)=∫.sub.−∞.sup.∞∫.sub.−∞.sup.∞ƒ(x,y).Math.δ(x.Math.cos(θ)+y.Math.sin(θ)−r)dxdy  (20)

this transformation may be referred to as the forward Radon transform. Here, the direction-dependent function p(r,θ) may be regarded as a collection of projections of an image defined by ƒ(x,y), which is often used in Computed Tomography applications. The adjoint Radon transform can be defined by the following integral:

ƒ(x,y)=∫.sub.−∞.sup.∞∫.sub.0.sup.2π{circumflex over (p)}(r,θ).Math.δ(x.Math.cos(θ)+y.Math.sin(θ)−r)dθdr  (21)

wherein the function {circumflex over (p)}(r,θ), to which the adjoint Radon transform is applied, can be related to the result of the forward Radon transform p(r,θ) (Eq. 20) through convolution or Fourier filtering (a so-called ramp filter or similar):

{circumflex over (p)}(r,θ)=∫.sub.−∞.sup.∞|ω|(∫.sub.−∞.sup.∞p(r,θ).Math.exp(−j2πrω)dr).Math.exp(j2πrω)dω  (22)

Integrals in Eq. 20-22 can be written in the form of linear operators leading to the following expressions:

p(r,θ)=ƒ(x,y),

ƒ(x,y)=*{circumflex over (p)}(r,θ),

{circumflex over (p)}(r,θ)=p(r,θ),  (23)

wherein and * are the Radon transform and its adjoint version respectively, and is the Fourier filtering operator. Using this notation, a function ƒ(x,y) can be expressed through the adjoint Radon transform of its own filtered projection:

ƒ(x,y)=*ƒ(x,y)  (24)

[0110] Hence, according to Eq. 19, in an embodiment, the deposited energy can be modelled as an adjoint Radon transform of the direction-dependent illumination intensity I.sub.0(r,θ):

E(x,y)=μ.sub.+.Math.t.Math.*l.sub.0(r,θ)  (25)

[0111] Comparing Eq. 25 with Eq. 24, it can be seen that the illumination intensity I.sub.0(r,θ) that corresponds to a certain deposited energy E(x,y), can be computed using a filtered Radon transform:

[00007]$\begin{array}{cc}{I}_0\left(r,\theta \right)=\frac{1}{{\mu }_{+}.Math.t}\mathrm{ℱℛ}\left[E\left(x,y\right)\right]& \left(26\right)\end{array}$

Hence, this equation is a special case of Eq. 9 and may be used to calculate the direction-dependent intensity of illumination analytically in case of a weakly attenuating polymerization medium.

[0112] The optimal direction-dependent illumination intensity I.sub.0(r,θ) according to Eq. 26 may be solved for a finite number of illumination angles and associated illumination beams, thus integrals in linear operators are replaced by finite sums. To avoid sampling artefacts, the number of offsets (i.e. illumination rays) N.sub.r in each illumination beam may correspond to the desired spatial resolution of E(x,y). Further, to avoid aliasing artefacts, the number of illumination angles (i.e. illumination beams) N.sub.θ may be related to the number of offsets N.sub.r by using the expression: N.sub.θ≥πN.sub.r. For instance, when an image of 1024×1024 pixels needs to be formed by multi-directional illumination of a target zone, approximately 3200 illumination beams with 1024 rays each may be generated in 0 to 2π angular range.

[0113] FIG. 4A-4D depict a simulated energy deposition resulting from a superposition of multiple illumination beams, wherein beams were precomputed and generated according to an analytical model valid for media with weak absorptivity. The multiple illumination beams were pre-computed using an analytical model described above. FIG. 4A depicts a Radon transform (projected image) of a target energy [E(x,y)]. It is the same in every direction, as the target polymerization zone is circular in this example. FIG. 4B depicts an illumination field I.sub.0(r,θ) computed by applying a Fourier filter (e.g. ramp filter) to the Radon transform of the target energy field as defined by Eq. 26. It is apparent that the computed illumination intensity has negative values which do not have any physical meaning in a read-world system. The negative values however can be associated with the intensity of the light interacting with the inhibition/deactivation agent. In an embodiment, in case of passive (ambient) inhibition, a constant offset can be added to the illumination intensity that matches the effect of the inhibitor. This way the illumination intensity will always be above zero. Alternatively, in another embodiment, in case of active inhibition, I.sub.0(r,θ) can be separated into two terms related to the initiation and inhibition: I.sub.0(r,θ)=I.sub.+(r,θ)−β.Math.I.sub.−(r,θ). In that case, the terms can be defined as:

[00008]$\begin{array}{cc}{I}_{+}\left(r,\theta \right)=\{\begin{array}{cc}.Math.I_0\left(r,\theta \right).Math.,& \mathrm{if}{I}_0\left(r,\theta \right)>0\\ 0,& \mathrm{otherwise}\end{array}{I}_{-}\left(r,\theta \right)=\{\begin{array}{cc}-.Math.I_0\left(r,\theta \right).Math./\beta ,& \mathrm{if}{I}_0\left(r,\theta \right)<0\\ 0,& \mathrm{otherwise}\end{array}& \left(27\right)\end{array}$

FIG. 4C depicts the intensity of the photo-initiation illumination field I.sub.+(r,θ) (402 continuous line) and the intensity of the photo-inhibition illumination field L(r,θ) (404 dotted line). The associated (combined) deposited energy field E(x,y) resulting from illumination by 314 beams is depicted in FIG. 4D. The wavelength of the initiation illumination differs from the wavelength of the inhibition illumination. As the linear absorptivity may be wavelength dependent, absorptivity associated with the initiation illumination may be different from the absorptivity associated with the inhibition illumination. In contrast to the example with the uniform intensity beam illumination (FIG. 3E), an approach described above leads to a sharply defined polymerization zone allowing for high resolution of the printed object boundary.

[0114] The computation described above is based on the assumption of weak absorption, where the direction-dependent absorption term is neglected. Reviewing Eq. 17, where the energy is expressed including the absorption term T(x,y,θ), a Radon-based computation method can be derived that incorporates self-absorption by the medium. To simplify the description, it may be assumed that the total absorptivity at the wavelength that corresponds to I.sub.+(r,θ) term and to I.sub.−(r,θ) is the same: μ.sub.+=μ.sub.−=μ. To this end, Eq. 17 can be rewritten for the photo-initiation and photo-inhibition energy terms using a short operator notation:

E.sub.+(x,y)=μ.Math.t.Math.*I.sub.+(r,θ)

E.sub.−(x,y)=μ.Math.t.Math.*I.sub.−(r,θ)  (28)

where * is a modified adjoint Radon transform incorporating an attenuation term T(x,y,θ). Now, Eq. 28 can be condensed into the following form:

E(x,y)=μ.Math.t.Math.*I.sub.0(r,θ)  (29)

Combining this result with the polymerization condition expressed in Eq. 18, the optimal illumination intensity functions I.sub.+(r,θ) and I.sub.−(r,θ) can be computed for a given target energy E.sub.0(x,y) using a Least-Squares approach described by Eq. 10 where the propagation operator equals to μ.Math.*:

[00009]$\begin{array}{cc}\underset{I_0\left(r,\theta \right)}{\mathrm{argmin}}.Math.\mu .Math.t.Math.{ℛ}^{*}{I}_0\left(r,\theta \right)-E_0\left(x,y\right).Math.\begin{array}{c}2\\ 2\end{array}& \left(30\right)\end{array}$

[0115] FIG. 5 depicts a flow diagram describing a method of controlling an illumination system for illuminating a target zone in a photopolymerizable medium according to an embodiment of the invention. The method may include determining direction-dependent illumination field based on target energy field determined for a target polymerization zone based and a low absorptivity light propagation model. The method may be implemented on a computer in the form of program code stored in the memory of the computer. The program code may be executed by a processor of the computer thereby enabling the computer to determine direction-dependent illumination intensities which can be used to control an exposure system, e.g. a system as described with reference to FIG. 1. The computed illumination intensity may be used to control and generate a plurality of direction-dependent exposures leading to the desired photopolymerization of the target zone.

[0116] In a first step, a 3D data representation representing a 3D model of an object may be provided to a computer that is connected to an exposure system for illumination of the target zone, e.g. an exposure system described with reference to FIGS. 1 and 2. The computer may use the 3D model to determine a volume of a target polymerization zone in the photopolymerizable medium 502. The volume may be shaped according to the 3D model. In a next step 504, the computer may determine a target energy field E.sub.0 defining an energy for volume elements in the medium that is needed to achieve polymerization inside the target zone. Here, the computer may determine a target energy field E.sub.0 based on a model [E] of the polymerization process in the medium. The target energy field may define the energy that is needed to achieve polymerization inside the target polymerization zone or in short, the target zone. The determination of the target energy field may be based on a model [E] of the photopolymerization process.

[0117] Here the polymerization model may be based on an empirical model of the photopolymerization process or a model based on the polymerization kinetics. In an embodiment, target energy E.sub.0 may be a function of E.sub.+ and E.sub.− and polymerization is achieved in case E.sub.0 exceeds a certain critical value E.sub.crit. For example, a linear polymerization model of the type E.sub.0=E.sub.+−β.Math.E.sub.− may be used to determine the target energy wherein polymerization is achieved when a weighted difference between the energy absorbed by the photo-initiator E.sub.+ and the energy absorbed by the inhibitor (or involved in deactivation of polymerization on another way) E.sub.− exceeds E.sub.crit.

[0118] The computer may then use the target energy field E.sub.0(x,y,z) and a light propagation model for a attenuating medium to compute a direction-dependent illumination field I.sub.0(r,θ,z) (step 506). In a weakly attenuating medium a projection of the target energy field E.sub.0(x,y,z) can be computed for every direction of illumination. A projection p(r,θ,z) of the target energy E.sub.0(x,y,z) can be computed by applying a forward projection operator based on the Radon transform (or similar transforms computing projections of a 3D image) to the target energy field. Subsequently a linear convolutional filter (e.g. ramp or Ram-Lak filter) may be applied to the projection of the energy thereby defining a direction-dependent illumination field I.sub.0(r,θ,z). Hence, the illumination field may be determined on the basis of a filtered Radon transform of the target energy field as defined by eq. (26) above.

[0119] In a next step 508, the computed illumination field I.sub.0(r,θ,z) may be used by the computer to control the exposure system based on the illumination field, wherein the exposure may include generating a plurality of direction-dependent illumination beams for depositing energy at positions within the target zone according to the target energy E.sub.0(x,y,z). Here, it is assumed that the power of the illumination source and the exposure time are kept constant, while the intensity I.sub.0(r,θ,z) is controllably varied, e.g. by controlling the pixels of the SLM and the illumination angle. The illumination angles and associated illumination intensities may be used to control the rotation of the rotatable illumination system and to control the SLM in the rotatable illumination system depicted in FIG. 1. It is submitted that the target energy field and the intensity field as mentioned above may be described based on coordinate systems that are suitable for a particular case. Selection of such coordinate systems, as well as the type of the Fourier filter and the projection transformation may depend on the symmetry of the setup (e.g. cylindrical or spherical symmetry)

[0120] FIG. 6A-6D depict examples of computed energy deposition for a 3D object (x-y slice displayed) based on the weak absorptivity model according to an embodiment of the invention. In particular, FIG. 6A-6D depict examples of computed energy deposition for a fairly complex object, in this case a 1×1 cm image of a Yin-Yang shaped object. FIG. 6A-B and FIG. 6C-D depict computed energy deposition for the 3D object in a moderate absorptivity medium (μ=1 cm.sup.−1) and a high absorptivity medium (μ=5 cm.sup.−1) medium respectively. FIGS. 6B and 6D represent cross-sections through the mid-line 602.sub.1,2 of FIGS. 6A and 6C respectively, wherein the solid line represents the profile of the target energy and wherein the dashed line represents the actual deposited energy. The images depicted in these figures are formed by 314 beams of 100 rays each. FIGS. 6A and 6B show that in a moderately attenuating medium, an accurate image of the object can be formed by multi-directional illumination with intensity calculated using weak absorptivity model. FIGS. 6C and 6D show that when the absorbance of the medium is high (or the size of the volume is too large), such approach may provide poor contrast. In that case, an iterative approach may be used to determine the illumination intensity. This approach is described hereunder in more detail with reference to FIG. 7 and FIG. 8.

[0121] The photopolymerizable medium and the illumination wavelength may be selected to minimize these effects. Eventually, a more precise model relating the illumination field to the deposited energy field (for the photo-initiator and the inhibitor) can be used for a particular implementation of the illumination system.

[0122] A further assumption that was made in order to derive Eq. 29 is that the absorptivity is constant in time and does not change as a result of illumination. Concentrations of the photo-initiator and the inhibitor however may locally change during the illumination of the medium by the illumination beams leading to the change in absorptivity. Similarly, as a result of polymerization dynamics, the proportionality constant β in the polymerization condition as expressed by Eq. 7 may vary during the illumination.

[0123] To account for unknown changes in the parameters of the photopolymerizable medium, the computation of energy deposition and the associated illumination may be separated into a number of iterations, where during each iteration the medium is exposed to a relatively low illumination energy wherein parameters of the medium can be considered constant. One way to monitor parameters of the medium is to measure the intensity of light transmitted through the medium during the illumination process. If such measurement is available, the linear absorptivity of the medium can be computed using known Computed Tomography methods as for example described by Herman, G. T., Fundamentals of computerized tomography: Image re-construction from projection, 2nd edition, Springer, 2009, using e.g. a filtered back projection method:

[00010]$\begin{array}{cc}\mu \left(x,y\right)={ℛ}^{*}ℱ\left[-\log \left(\frac{I_{\mathrm{out}}\left(r,\theta \right)}{I_0\left(r,\theta \right)}\right)\right]& \left(31\right)\end{array}$

wherein, I.sub.out(r,θ) is the intensity of the light transmitted through the medium.

[0124] Propagation-based Phase-Contrast Tomography can be used similarly to the conventional tomography, in order to detect local changes in the refractive index of the medium due to polymerization in real-time. This method could allow to monitor the degree of polymerization indirectly even when changes in absorptivity of the medium are too small to detect. The calculated refractive index can be incorporated in the light propagation model in addition to the linear absorptivity. This will, however, require a different model rather than a Radon-based one.

[0125] A direct real-time measurement of the degree of polymerization can be obtained using more elaborate spectral measurements of transmitted light using Fourier-transform infrared spectroscopy (FTIR) tomography methods as described in 3D spectral imaging with synchrotron Fourier transform infrared spectro-microtomography Nature Methods volume 10, pages 861864 (2013).

[0126] The embodiments described in this application may not only be used for creating a 3D object in the photopolymerizable medium from scratch. In a further embodiment, a 3D object may be printed around or on an existing 3D object. Such object can be introduced into the reservoir prior to the printing process. The presence of the 3D object will alter the total absorptivity of the volume. Hence, in that case, the exposure system of FIG. 1 may include a 3D scanning mode, wherein the system selects one or more scanning beams of a wavelength that does not affect the photopolymerizable medium. During the scanning process, the surface of the 3D object is scanned and either the reflected or transmitted light is detected by the imaging system of the system. The imaging system may include one or more camera sensors arranged around the reservoir so that the system is able to detect reflections of the scanning beams from different viewing angles. The captured images may be used to reconstruct the surface of the 3D object in the reservoir.

[0127] The computed reconstructed surface of the 3D object may be matched with a known 3D model to determine its orientation. To that end, a known registration method, e.g. a rigid surface registration method, may be used to register the reconstructed surface with the 3D model. This way, the orientation of the target zone can be aligned with orientation of the 3D object in the target zone. The orientation of the target zone and the computed absorptivity μ(x,y,z) may be used to calculate the optimal illumination intensity I.sub.0(r,θ,z) to deposit around an external object using the methods described in this application.

[0128] FIG. 7 depicts a flow diagram of a method for determining an illumination field I.sub.0(r,θ,z) fora target polymerization zone according to another embodiment. In particular, FIG. 7 depicts a flow diagram of a method for determining an illumination energy based on an iterative approach for media with a moderate absorptivity according to an embodiment of the invention. In such iterative approach, an approximated solution for an illumination intensity may be computed for each direction through an iterative minimization of an objective function that describes the discrepancy between a target energy field E.sub.0(x,y,z) and a predicted deposited energy field E(x,y,z)=t.Math.[I.sub.0 (r,θ,z)].

[0129] A propagation operator may be used for predicting the energy deposition. For example, a propagation operator based on the adjoint Radon transform as described with reference to FIG. 5 may be used. Such adjoint Radon transform uses a propagation model with an absorptivity term and assumes that photons propagate along straight lines in a parallel fashion. It may be replaced by more accurate models (for instance Radiative Transport Equations) in situations when such assumption fails. A wide range of algorithms may be used to solve the Least Squares-type minimization problem. In FIG. 7 a basic approach is described that is applicable to cases with linear propagation model .

[0130] The iterative minimization method may be used once a target energy field is determined (as in step 504 of FIG. 5) and the output of the method may be used to control and generate direction-dependent illumination field defining a plurality of exposures of a target zone which leads to photopolymerization of the target zone (as described in set 510 of FIG. 5). This analytically determined illumination intensity function may be a first estimate of the illumination field that is used in the iterative minimization method (step 702).

[0131] In a next step 704 the deposited energy field E(x,y,z) is predicted. In an embodiment, the predicted energy may be computed using a modified adjoint Radon transform with an absorptivity term represented by a linear operator . This modified adjoint Radon transform * is described with reference to eq. (28)-(30) above.

[0132] In the following step, 708 a gradient should be computed for an objective function based on the difference between the predicted and the target energy. If the objective function is of an L2-type (Euclidian distance) the gradient as computed by applying * i.e. an adjoint version of the linear operator , to the difference E(x,y,z)−E.sub.0(x,y,z). An update of the current guess of the illumination field I.sub.0(r,θ,z) is computed by subtracting the calculated gradient of the objective function scaled by a constant factor α. The result of this operation is the new guess of the illumination intensity function I.sub.0(r,θ,z) 710.

[0133] Different stopping conditions 712 can be implemented. For instance, to stop when the sum of squares of the discrepancy E(x,y,z)−E.sub.0(x,y,z) is lower than some small constant ϵ. If the condition is not satisfied a new iteration of the method is started by substitution of the updated illumination intensity function I.sub.0(r,θ,z) 710 into step 704. If it is satisfied, the current guess of the illumination intensity may be used to control the illumination system 714. Also in this case, the target energy field and the intensity field may be described based on coordinate systems that are suitable for a particular case.

[0134] FIG. 8A-8D depict examples of computed energy deposition for a 3D object based on the iterative approach for media with moderate absorptivity according to an embodiment of the invention. FIG. 8A depicts a computed deposited energy field in a high absorptivity medium (μ=5 cm.sup.−1) for a Yin Yang shaped object. FIG. 8B depicts the target energy field. FIG. 8C represents a cross-section through the mid-line 802 of FIG. 8A, wherein the solid line represents the profile of the target energy and wherein the dashed line represents the actual deposited energy. The images depicted in these figures are formed by 314 beams of 100 rays each. FIG. 8D shows the illumination field intensity profile for one of the directions of illumination, wherein the solid line represents the intensity profile precomputed using the analytical solution for the weak absorptivity model and the dashed line represents the intensity profile precomputed using an iterative approach applied to the model with significant absorptivity. When the absorptivity of the medium is high (or the size of the volume is too large), the approach based on the model for a weak absorptivity medium delivers poorer contrast and eventually fails. In that case, an iterative approach may be used to determine the illumination intensity. This approach is described above with a reference to FIG. 7.

[0135] FIG. 9 depicts a schematic of a system for forming a 3D object using photopolymerization according to an embodiment of the invention. FIG. 9 shows a schematic of an illumination system wherein the illumination sources 902 and cameras 906 remain stationary, while moving mirrors 903.sub.1-903.sub.6 ensure positioning of the illumination beam during exposure. Different variations of this concept can be realized using mirrors or other light guiding systems. If the viscosity of the medium allows, the container itself 904 can be rotated, while the illumination system remains stationary.

[0136] In further embodiments, the illumination system may deviate from the standard tomographic geometry as depicted in FIG. 1 wherein light beams are projected at a 90 degree angle to the rotation axis, e.g. the z-axis. A possible implementation of such illumination system is depicted in FIG. 10. As shown in this figure, an illumination system may be rotatable mounted above a reservoir 1002 comprising a photopolymerizable medium. In this example, the dimensions of the reservoir in the x,y direction (perpendicular to the rotation axis 1006 of the illumination system) are substantially larger than the dimensions of the reservoir in the z direction (parallel to the rotation axis). This way, the reservoir may define one or more shallow planar-shaped target zones similar to an illumination system as used in laminography, which is a subtype of tomography.

[0137] The illumination system may comprise one or more optical elements 1008 for projecting one or more light beams from the top (or bottom) at a grazing angle θ to the flat surface of the reservoir (wherein the angle between the one or more light beams and the rotation axis is smaller than 90 degrees). While rotating, the illumination system exposes the photopolymerizable medium in a target zone 1012 in the reservoir.

[0138] As shown in FIG. 10B, the illumination center of the rotation can be moved in a direction perpendicular to the rotation axis. Hence, each time the illumination system has exposed an area, the rotational axis of the illumination system is moved relative to the reservoir so that an area next to the exposed area can be exposed. This process can be repeated many times so that multiple target zones 1014.sub.1-4 may define a large exposed area. Hence, by moving the illumination system relative to the reservoir in the x-y plane, the illumination system effectively exposes a large area of the planar reservoir. The same computational approach as for a standard geometry (e.g. as depicted in FIG. 1) may be used for this method as long as the light propagation direction is accounted for. While this embodiment may yield lower vertical resolution, it can be advantageous when printing has to be realized across the surface of a planar reservoir.

[0139] FIG. 11 is a block diagram illustrating an exemplary data processing system that may be used in as described in this disclosure. Data processing system 1100 may include at least one processor 1102 coupled to memory elements 1104 through a system bus 1106. As such, the data processing system may store program code within memory elements 1104. Further, processor 1102 may execute the program code accessed from memory elements 1104 via system bus 1106. In one aspect, data processing system may be implemented as a computer that is suitable for storing and/or executing program code. It should be appreciated, however, that data processing system 1100 may be implemented in the form of any system including a processor and memory that is capable of performing the functions described within this specification.

[0140] Memory elements 1104 may include one or more physical memory devices such as, for example, local memory 1108 and one or more bulk storage devices 1110. Local memory may refer to random access memory or other non-persistent memory device(s) generally used during actual execution of the program code. A bulk storage device may be implemented as a hard drive or other persistent data storage device. The processing system 1100 may also include one or more cache memories (not shown) that provide temporary storage of at least some program code in order to reduce the number of times program code must be retrieved from bulk storage device 1110 during execution.

[0141] Input/output (I/O) devices depicted as input device 1012 and output device 1114 optionally can be coupled to the data processing system. Examples of input device may include, but are not limited to, for example, a keyboard, a pointing device such as a mouse, or the like. Examples of output device may include, but are not limited to, for example, a monitor or display, speakers, or the like. Input device and/or output device may be coupled to data processing system either directly or through intervening I/O controllers. A network adapter 1116 may also be coupled to data processing system to enable it to become coupled to other systems, computer systems, remote network devices, and/or remote storage devices through intervening private or public networks. The network adapter may comprise a data receiver for receiving data that is transmitted by said systems, devices and/or networks to said data and a data transmitter for transmitting data to said systems, devices and/or networks. Modems, cable modems, and Ethernet cards are examples of different types of network adapter that may be used with data processing system 1150.

[0142] As pictured in FIG. 11, memory elements 1104 may store an application 1118. It should be appreciated that data processing system 1100 may further execute an operating system (not shown) that can facilitate execution of the application. Application, being implemented in the form of executable program code, can be executed by data processing system 1100, e.g., by processor 1102. Responsive to executing application, data processing system may be configured to perform one or more operations to be described herein in further detail.

[0143] In one aspect, for example, data processing system 1100 may represent a client data processing system. In that case, application 1118 may represent a client application that, when executed, configures data processing system 1100 to perform the various functions described herein with reference to a “client”. Examples of a client can include, but are not limited to, a personal computer, a portable computer, a mobile phone, or the like. In another aspect, data processing system may represent a server. For example, data processing system may represent an (HTTP) server in which case application 1118, when executed, may configure data processing system to perform (HTTP) server operations. In another aspect, data processing system may represent a module, unit or function as referred to in this specification.

[0144] The terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of the invention. As used herein, the singular forms “a,” “an,” and “the” are intended to include the plural forms as well, unless the context clearly indicates otherwise. It will be further understood that the terms “comprises” and/or “comprising,” when used in this specification, specify the presence of stated features, integers, steps, operations, elements, and/or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and/or groups thereof.

[0145] The corresponding structures, materials, acts, and equivalents of all means or step plus function elements in the claims below are intended to include any structure, material, or act for performing the function in combination with other claimed elements as specifically claimed. The description of the present invention has been presented for purposes of illustration and description, but is not intended to be exhaustive or limited to the invention in the form disclosed. Many modifications and variations will be apparent to those of ordinary skill in the art without departing from the scope and spirit of the invention. The embodiment was chosen and described in order to best explain the principles of the invention and the practical application, and to enable others of ordinary skill in the art to understand the invention for various embodiments with various modifications as are suited to the particular use contemplated.