CONTROL OF AN OPTICAL IMAGE DEVICE

Abstract

The present invention relates to a control arrangement for an optical imaging device (101) for microlithography, in particular for the use of light in the extreme ultraviolet (EUV) range, comprising a determination device (106.1) for determining at least one thermal state information for at least one optical element (108) of the imaging device (101). The thermal state information is representative of a thermal state of the optical element (108), wherein the thermal state information is configured to be used in determining an imaging error information that is representative of at least one optical imaging error of the optical imaging device (101). The thermal state information is representative of a temperature distribution in at least a first element part (108.4) of the optical element (108), wherein the first element part (108.4) the part of the optical element (108) that, during normal operation, has a non-negligible, thermally induced influence on the optical aberration. The determination device (106.1) is configured to determine the thermal state information in a plurality of successive determination steps using a finite element model of the optical element (108), each with a determination accuracy sufficient for real-time correction of the at least one optical aberration, and to output it at an output frequency AF. The finite element model has a plurality of nodes (109) and a mean node spacing MKA, where the mean node spacing is 0.1 mm to 30 mm. The determination device (106.1) is further configured to apply at least one complexity reduction method to the finite element model such that the output frequency AF of the thermal state information is 0.1 Hz to 1 Hz.000 Hz.

Description

[0001] The present invention relates to a control arrangement and a method for controlling an optical imaging device for microlithography, which is suitable for the use of UV light, in particular light in the extreme ultraviolet (EUV) range. Furthermore, the invention relates to an optical imaging device with such a control arrangement. The invention can be used in conjunction with any optical imaging method. It can be used particularly advantageously in the manufacture or inspection of microelectronic circuits and the optical components used therefor (e.g., optical masks).

[0002] Typically, the optical systems used in connection with the fabrication of such microelectronic circuits comprise a plurality of optical element modules comprising optical elements such as lenses, mirrors, gratings, etc., arranged in the path of the light. These optical elements typically cooperate in an exposure process to illuminate a pattern formed on a mask, reticle, or the like and to transfer an image of this pattern to a substrate such as a wafer. The optical elements are usually grouped into one or more functionally distinct optical element groups, which may be contained within different optical element units.

[0003] The ever-increasing miniaturization of semiconductor devices is leading to an ever-growing demand for increased resolution in the optical systems used to manufacture these semiconductor devices. This demand for increased resolution necessitates the need for increased numerical aperture (NA) and increased imaging accuracy in the optical systems.

[0004] One approach to achieving increased optical resolution is to reduce the wavelength of the light used in the imaging process. In recent years, there has been an increased development of systems that use light in the so-called extreme ultraviolet (EUV) range, typically at wavelengths from 5 nm to 20 nm, in most cases at a wavelength of around 13 nm. In this EUV range, it is no longer possible to use conventional refractive optical systems. This is due to the fact that the materials used for refractive optical systems have an absorption coefficient in this EUV range that is too high to achieve acceptable imaging results with the available light power. Consequently, reflective optical systems must be used for imaging in this EUV range.

[0005] This transition to purely reflective optical systems with high numerical aperture (e.g., NA > 0.4 to 0.5) in the EUV range leads to significant challenges with regard to the design of the imaging device.

[0006] The above factors lead to very strict requirements regarding the position and / or orientation of the optical elements involved in imaging relative to each other, as well as the deformation of the individual optical elements, in order to achieve the desired imaging accuracy. Furthermore, it is necessary to maintain this high imaging accuracy throughout the entire operation, ultimately over the lifetime of the system.

[0007] As a consequence, the components of the optical imaging device (e.g., the optical elements of the illumination device, the mask, the optical elements of the projection device, and the substrate) that interact during imaging must be supported in a well-defined manner to maintain a predetermined, well-defined spatial relationship between these components and minimize unwanted deformation of these components, ultimately achieving the highest possible imaging quality. Accordingly, a particularly high level of effort is required for deformation-decoupled support of the optical elements in order to keep them as free as possible from parasitic stresses that could otherwise lead to unwanted deformations of the optical elements.

[0008] A common challenge here is that the optical elements are exposed to a thermal load (e.g., in the form of heat from absorbed useful light) and deform under this thermal load. This deformation usually results from the thermal expansion of the optical material. This deformation of the optically effective surfaces usually causes a deviation of the optical surface from its target geometry (unless this can already be anticipated during the manufacturing process of the optical element), which is often referred to as a fit error. This fit error impairs the imaging properties of the component, e.g., due to a deviation of the curvature of the optical surface from its target curvature.

[0009] Such thermally induced deformation can be reduced by appropriate cooling of the optical element. In EUV exposure systems, the optical elements are located in a vacuum, which is why heat must be dissipated primarily by thermal conduction to achieve sufficient heat dissipation. This requires a cooler that contacts the optical element. This cooler must be held accordingly (for example, via a support frame or the like). The problem here is that the contact of the cooler with the optical element introduces an additional load path or force flow that runs kinematically parallel to the deformation-decoupling support of the optical element and may at least partially bridge it, thus leading to the introduction of undesirable parasitic stresses in the optical element.In addition, this can lead to a change in the dynamic behavior of the arrangement as well as to the introduction of vibrations (e.g. flow-induced vibrations) into the optical element.

[0010] Another possibility for correcting such aberrations caused by thermally induced deformations is the active deformation of the affected optical element or another optical element in the beam path.

[0011] A prerequisite for a targeted correction of such thermally induced aberrations is often sufficiently precise knowledge of the aberration and / or the thermal state of the optical element in question, for example the temperature distribution in the optical element and, if applicable, its temporal development.

[0012] However, actually measuring this temperature distribution with sufficient spatial resolution is often difficult or even impossible. An alternative to actual measurement is the sufficiently precise simulation of the thermal state of the optical element using appropriate numerical methods, such as so-called finite element methods (FEM). While such a simulation is fundamentally possible using correspondingly high-resolution models, the problem is that these models, even with sufficiently good spatial resolution, usually require too much computing power or are too slow to output the results of the individual simulation loops (e.g., the calculated temperature distribution in the optical element) at an output frequency that is sufficient in the field of microlithography as a basis for controlling the associated compensation device(s) in real time.

[0013] An alternative or complement to this approach is to determine or measure the actual imaging error and use it to control the associated compensation device(s). The problem here is that the imaging error typically cannot be determined during the actual imaging, but rather intermittently between individual imaging processes, which naturally undesirably reduces the throughput of the imaging device. BRIEF SUMMARY OF THE INVENTION

[0014] The invention is therefore based on the object of providing a control arrangement for an imaging device for microlithography, a corresponding optical imaging device, a method for controlling such an imaging device and an optical imaging method which does not have the aforementioned disadvantages or at least has them to a lesser extent and in particular enables improved compensation of thermally induced imaging errors in a simple manner.

[0015] The invention solves this problem with the features of the independent claims.

[0016] The invention is based on the technical teaching that improved compensation of thermally induced aberrations can be achieved in a simple manner by determining, as geometric state information relevant for correcting the aberration, information representative of a deformation in at least one relevant part of an optical element via sufficiently fast numerical modeling, and then using this information to control the associated compensation device(s) in real time. It may be sufficient to model only the part of the optical element that, during normal operation of the optical imaging device, has a non-negligible influence on the relevant optical aberration caused by thermally induced deformation.

[0017] Sufficiently fast numerical modeling can be achieved by first performing a sufficiently high-resolution model to determine the thermal state (or temperature distribution) and / or the material properties and / or the deformation of the relevant part of the optical element using a conventional finite element model. This basic modeling of the temperature distribution and / or the material properties and / or the deformation naturally exhibits such a high dimensionality at the spatial resolution required (for application in microlithography) that, due to the required computing time, the simulation results cannot be used to control the compensation device(s) in real time.In order to enable the use of the simulation results for a real-time correction of the respective optical aberration, thus achieving a sufficiently high output frequency of the simulation results, at least one complexity reduction method is applied to the finite element model, which sufficiently accelerates the simulation.

[0018] It has been shown that, through a suitable design or choice of the complexity reduction method, it is possible to reduce the dimensionality of the (thus obtained) simplified or dimensionally reduced model to such a low value that the required computing time is drastically reduced, thus enabling the simulation results to be used for real-time correction of the relevant optical aberration. The sufficiently precise simulation results of the simplified, dimensionally reduced model or the respective submodel can then be easily and quickly transferred back to the high-dimensional initial model, i.e., the original finite element model, so that the subsequently obtained information about the temperature distribution has a spatial resolution sufficient for application in microlithography.In this way, output frequencies of the thermal state information can be achieved that are in the range of 0.1 Hz to 1,000 Hz, thus in the range of a control bandwidth as required in microlithography for real-time correction of the respective optical imaging error.

[0019] According to one aspect, the invention therefore relates to a control arrangement for an optical imaging device for microlithography, in particular for the use of light in the extreme UV range (EUV), comprising a determination device for determining at least one item of geometric state information for at least one optical element of the imaging device. The at least one item of geometric state information is representative of a geometric state of the at least one optical element. The at least one item of geometric state information is designed to be used in the determination of aberration information that is representative of at least one optical aberration of the optical imaging device.The determination device is designed to determine at least one item of thermal state information that is representative of a temperature distribution in at least a first element part of the optical element, wherein the first element part is the part of the optical element that, during normal operation of the optical imaging device, has a non-negligible influence on the at least one optical imaging error caused by thermally induced deformation.

[0020] The determination device is further configured to determine at least one piece of material state information representative of a distribution of at least one material property in at least the first element part, wherein the at least one material property has a non-negligible influence on the thermally induced deformation of the first element part. The determination device is further configured to determine the at least one piece of geometric state information using the at least one piece of thermal state information and the at least one piece of material state information.The determination device is designed to determine the at least one piece of geometric state information in a plurality of successive determination steps using a finite element model of the at least one optical element, each with a determination accuracy sufficient for real-time correction of the at least one optical aberration, and to output it at a possible output frequency AF. The finite element model has a plurality of nodes and an average node spacing MKA, wherein the average node spacing is 0.1 mm to 30 mm, preferably 0.5 mm to 25 mm, more preferably 2 mm to 10 mm. The determination device is further designed to apply at least one complexity reduction method to the finite element model such that the output frequency AF of the geometric state information is 0.1 Hz to 1.000 Hz, preferably 10 Hz to 200 Hz, more preferably 50 Hz to 100 Hz.

[0021] It can therefore be achieved that the period time or runtime of the simulation (i.e. the time that elapses for each run of the simulation loop, corresponding to the output frequency AF, until a newly determined thermal or geometric state information is output) is in the range of 1 ms to 10 s, for example 5 ms to 5 s, preferably 10 ms to 2 s. It is understood that additional processing time may be required for the actual control of the imaging device, for example for real-time correction of the relevant optical imaging error. Therefore, it can be provided that the geometric state information newly determined in the most recent simulation loop is not actually fed into the control of the imaging device or used therein at the output frequency AF.In particular, in certain variants, the output frequency AF can be a theoretical value at which the at least one piece of geometric state information could be practically output thanks to the high speed of the simulation, while in reality the output is delayed accordingly in order to synchronize it with the control cycle of the imaging device (for example, for real-time correction of the at least one optical imaging error). However, the simulation process can also run without a pause between the simulation loops (with the at least one piece of geometric state information then actually being output at the output frequency AF), and the most recent thermal state information available from the simulation process can then be used to control the imaging device.

[0022] It should be mentioned again here that the invention is particularly suitable for use in the EUV range or for systems with a particularly high numerical aperture (e.g., NA > 0.4 to 0.5). However, it can also be advantageously used in systems that operate with other wavelengths of useful light, particularly in the so-called DUV (deep UV) range. The invention is also suitable for use in systems with a lower numerical aperture (e.g., NA < 0.4).

[0023] In principle, depending on the complexity of the required calculations, it may be sufficient to perform a corresponding complexity reduction for only one or more of the temperature distribution, the material properties, and the deformation. Thus, the determination device can be configured to determine the at least one piece of thermal state information using a thermal submodel of the finite element model, wherein the determination device can be configured, in particular, to apply the at least one complexity reduction method to the thermal submodel.

[0024] Additionally or alternatively, the determination device can be configured to determine the at least one piece of material state information using a material partial model of the finite element model, wherein the determination device can be configured, in particular, to apply the at least one complexity reduction method to the material partial model. This advantageously makes it possible to consider any complex or non-linear material properties of the optical element that have a corresponding non-negligible influence on the deformation of the optical element. These can, for example, be material properties that vary locally and / or over time (for example, depending on the temperature and / or the current stress state and / or the microstructure of the material in question).Such a varying material property can be, for example, the thermal expansion coefficient and / or the thermal conductivity.

[0025] Additionally or alternatively, the determination device can be configured to determine the at least one piece of geometric state information using a geometric submodel of the finite element model, wherein the determination device can be configured, in particular, to apply the at least one complexity reduction method to the geometric submodel. Preferably, at least one complexity reduction method is applied to all three submodels, since this allows for a particularly fast simulation.

[0026] In principle, any suitable complexity reduction method can be used which, in addition to a sufficiently strong reduction of the dimensionality from the initial model to the dimensionally reduced model or partial model then used for the actual simulation calculation (and thus a correspondingly strong reduction of the computing time for the respective determination step), achieve a sufficient accuracy of the simulation or numerical determination of the deformation after the back transformation into the high-dimensional initial model (i.e. the original finite element model) (with regard to the actual temperature distribution and / or material property distribution and / or deformation).

[0027] In particularly favorable variants with a strong dimensionality reduction and high accuracy of the determined temperature distribution, material property distribution, or deformation, the complexity reduction method is a reduced basis method. In the course of the complexity reduction method, a plurality of MBVs of basis vectors of at least one orthogonal basis vector system are applied to a first differential equation system DGS1 of the finite element model in such a way that at least one reduced second differential equation system DGS2 is obtained with a second dimension D2 that is reduced compared to a first dimension D1 of the first differential equation system DGS1 of the finite element model.It can be provided that the basis vectors for creating the at least one reduced second differential equation system DGLS2 are each multiplied from the right by the first differential equation system DGLS1 of the finite element model. This allows a correspondingly reduced differential equation system DGLS2 with significantly lower dimensionality to be achieved in a particularly simple and rapid manner. It is understood that the dimensionalities D2 of the relevant submodels (or differential equation systems DGLS2) can differ for the temperature distribution, the material property distribution, and the deformation, whereby the results of a submodel can then be easily converted into input values ​​for a subsequent submodel via a transformation operation (which changes the dimensionality accordingly).

[0028] In particularly favorable variants, the MBV of basis vectors is 100 to 20,000, preferably 200 to 10,000, and more preferably 1,000 to 4,000, since this allows a corresponding reduction in dimensionality with significantly reduced computing times while maintaining sufficient simulation accuracy. For example, using an orthogonal basis vector system with a number of MBV = 80 to 100 basis vectors can accelerate the required computing time by a factor of 1,500 to 2,500. In general, a smaller number of basis vectors results in a greater reduction in computing time, but this also results in a reduction in simulation accuracy.

[0029] In preferred variants, the first dimension D1 of the first differential equation system DGLS1 of the finite element model can be 10,000 to 2,000,000, preferably 50,000 to 1,000,000, more preferably 200,000 to 500,000. This makes it possible, in particular, to achieve a sufficiently good spatial resolution of the determined temperature distribution or material property distribution or deformation for application in the field of microlithography. In certain variants, the second dimension D2 of the at least one reduced second differential equation system DGLS2 can be from 100 to 20,000, preferably 200 to 10,000, more preferably 1,000 to 4,000. This makes it possible, in particular, to achieve a sufficiently good reduction in computing time with sufficient simulation accuracy for real-time control of aberration correction or compensation in the field of microlithography.

[0030] The discretization of the first element part of the optical element relevant for aberration correction using a finite element model for simulating the temperature distribution can be carried out using the continuous heat conduction equation, according to which in homogeneous media: ∂∂tT(x→,t)−α⋅ΔT(x→,t)=0 in Ω×[tmin,tmax] B(T(x→,t),∇T(x→,t),Q(x→,t))=0 at ∂Ω×[tmin,tmax] T(x→,t)=T0(x→) at t=tmin. where the boundary conditions B(T(x→,t),∇T(x→,t),Q(x→,t)) are not linear due to the Stefan-Boltzmann law, but can be linearized in the preheated state, and where: Temperature field for the first element part; t time; Δ Laplace operator; ∇ Nabla Operator; Heat load for the first element part; Ω geometry of the body to be simulated; ∂Ω surface of the simulated body; α thermal conductivity.

[0031] With the mass matrix M of the first element part and the stiffness matrix S of the first element part, one then obtains a typically high-dimensional first differential equation system DGLS1 via the FEM discretization with: ∂∂tMT→(t)−ST→(t)=MQ→(t) Temperature vector for the first element part; Heat load vector for the first element part; t time; M mass matrix M of the first element part; S Stiffness matrix S of the first element part.

[0032] The dimensionality or first dimension D1 of the first differential equation system DGLS1 of the finite element model can have the values ​​mentioned above.

[0033] In preferred variants, the basis vectors of the orthogonal basis vector system are chosen such that the temperature vector dependent on time t T→(t) for the first element part can be represented by the basis vectors, so that: T→(t)=∑n=1kcn(t)Ψ→n with Ψ→nMΨ→m=δnm where: Temperature vector for the first element part; t time; c n (t) development coefficient n; Basis vectors n, m; M mass matrix of the finite element model for the first element part; δ nm Kronecker Delta.

[0034] In certain variants, the basis vectors of the orthogonal basis vector system are chosen such that the heat load vector dependent on time t Q→(t) for the first element part can be represented by the basis vectors, so that: Q→(t)=∑n=1kdn(t)Ψ→n with Ψ→nMΨ→m=δnm where: Heat load vector for the first element part; t time; d n (t) development coefficient n; Basis vectors n, m; M mass matrix of the finite element model for the first element part; δ nm Kronecker Delta.

[0035] The discretization of the first element part of the optical element relevant for aberration correction using a finite element model for the simulation of the deformation can be carried out using the continuous elasticity equation, according to which in homogeneous media the following applies: ∇⋅σ(u→)=0 in Ω σ(u→)=λtr(ε(u→))I+2με(u→)−(3λ+2μ)αT(T)I=0 ε(u→)=12(∇u→+(∇u→)T) σ(u→)⋅n→=0 in ∂Ω where the problem in the displacement vector u→ is linear and where: displacement vector; σ stress tensor; ε strain tensor; ∇ Nabla Operator; λ first Lamé constant; tr matrix trace operator; I identity matrix; µ second Lamé constant; α Ttemperature-dependent thermal expansion coefficient; T temperature; Ω geometry of the body to be simulated; ∂Ω surface of the simulated body; Displacement vector.

[0036] Using the stiffness matrix S of the first element part, one then obtains a typically high-dimensional first differential equation system DGLS1 via the FEM discretization with: Su→=v→ where: Displacement vector on the FEM nodes; Thermal expansion vector on the FEM nodes; S Stiffness matrix S of the first element part.

[0037] The dimensionality or first dimension D1 of the first differential equation system DGLS1 of the finite element model can have the values ​​mentioned above.

[0038] In certain variants, the basis vectors of the at least one orthogonal basis vector system are selected at least for the geometric partial model such that the displacement vector dependent on the thermal expansion v u→(v) for the first element part can be represented by the basis vectors, so that: u→(v)=∑n=1dcn(v)Ψ→n with Ψ→nSΨ→m=δnm where: Displacement vector for the first element part; v thermal expansion; c n (v) development coefficient n; Basis vectors n, m; S stiffness matrix of the finite element model for the first element part; δ nm Kronecker Delta;

[0039] It is understood that the thermal submodel and the geometric submodel may have different structures. For example, different stiffness matrices may be used.

[0040] In certain variants, the basis vectors of the at least one orthogonal basis vector system are selected at least for the geometric submodel such that the thermal expansion vector dependent on the temperature T v→(T) for the first element part can be represented by the basis vectors, so that: v→(T)=∑n=1ddn(T)Ψ→n with Ψ→nSΨ→m=δnm where: Thermal expansion vector for the first element part; T temperature; d n (T) development coefficient n; Basis vectors n, m; S stiffness matrix of the finite element model for the first element part; δ nm Kronecker Delta.

[0041] In certain variants, the material submodel is a part of the geometric submodel and the basis vectors of the at least one orthogonal basis vector system are selected at least for the material submodel of the geometric submodel such that the material property α(T) dependent on the temperature T for the first element part can be mapped via the basis vectors of the geometric submodel, where: α(T)=α(T(x,y,z);x;y;z) where: α(T) material property, in particular a thermal expansion coefficient, for the first element part; T(x,y,z) Temperature at the point of the first element part with the coordinates (x, y, z).

[0042] The thermal expansions V of the at least one orthogonal basis vector system for the material submodel of the geometric submodel can be determined starting from a complete basis of the thermal expansions V of the finite element model by determining a Fekete basis and the relevant modes of the Fekete basis. A corresponding method is described, for example, in the article "Computing Approximate Fekete Points by QR Factorizations of Vandermonde Matrices" (A. Sommariva, M. Vianello in Computers and Mathematics with Applications 57 (2009), pages 1324 to 1336, Elsevir Ltd.) (the entire disclosure of which is incorporated herein by reference).The basis of thermal expansion V of the geometric submodel was orthogonalized and normalized, and for the complexity reduction procedure, the modes of the basis of thermal expansion V were sorted using a QR decomposition with pivoting to determine at least one reduced second differential equation system of the geometric submodel with the second dimension D2 as the reduced basis of thermal expansion V. This advantageously makes it possible to transform the corresponding values ​​between the first and second differential equation systems.

[0043] In certain variants, the determination device is designed to transfer the results of the solution of the reduced second differential equation system of the thermal submodel into the reduced second differential equation system of the material submodel of the geometric submodel using first operators to determine the at least one piece of geometric state information. Additionally or alternatively, the determination device can be designed to integrate the results of the solution of the reduced second differential equation system of the material submodel of the geometric submodel into the reduced second differential equation system of the geometric submodel to determine the at least one piece of geometric state information using second operators.

[0044] In certain variants, the geometric state information can be determined entirely using the reduced second differential equation system (i.e., with reduced dimensionality) before the final result is then transferred back to the first differential equation system (i.e., into the high dimensionality D1). This allows a significant reduction in computing time. This allows the precalculation of parameters that are consistent throughout the calculation, for example, that are constant. For time-dependent quantities such as temperature, the described operators can thus be formulated in such a way that a mapping from the temperature basis (i.e., the thermal submodel) to the reduced basis with the reduced dimensionality is possible.Likewise, the result of the thermal submodel can be mapped into the material submodel of the geometric submodel (i.e., a correspondingly evaluated material law) via appropriately precalculated operators, and then directly into the reduced basis of the geometric submodel (i.e., the displacement basis) via appropriately precalculated operators. Thus, no further evaluation is required in the first differential equation system, and a significantly faster evaluation can be performed with sufficient accuracy using the reduced second differential equation system.

[0045] The solution of the reduced second differential equation system DGLS2 determined in the respective determination step can, depending on the achieved computing speed, be processed together with the solution of one or more previous and / or subsequent determination steps before the thus processed solution is transferred back to the first differential equation system. For example, an average can be calculated from the jointly processed solutions of the second differential equation system DGLS2, and then this average can be transferred back to the first differential equation system.Such an approach can be chosen if, despite the delay caused by such joint processing of several solutions of the reduced second differential equation system DGLS2, an output frequency AF of the geometric state information in the dimensional space of the first differential equation system DGLS1 that is sufficiently high for real-time control is still achieved.

[0046] In particularly simple variants with a high output frequency AF, the determination device is designed to determine a solution of the reduced second differential equation system in each of the determination steps and to transfer the determined solution of the reduced second differential equation system DGLS2 back into the dimensional space of the first differential equation system DGLS1 in order to obtain the geometric state information.

[0047] The basis vectors of the orthogonal basis vector system can, in principle, be determined in any suitable manner. Preferably, the basis vectors of the orthogonal basis vector system are or were determined by determining the associated thermal state information as a simulation result for the thermal or geometric submodel with the first differential equation system DGLS1 in a plurality of simulation steps, each with a heat load Q. From the plurality of simulation results, the basis vectors of the respective orthogonal basis vector system were determined using a singular value decomposition method (so-called "single-value decomposition method" or "SVD method") or a so-called proper orthogonal decomposition method ("POD method"). Such SVD methods or POD methods are sufficiently well known, so they will not be discussed in detail here.

[0048] It can be particularly advantageous if the number VS of simulation steps is 100 to 100,000, preferably 500 to 10,000, more preferably 1,000 to 5,000.

[0049] In certain variants, a heat load Q that is as realistic as possible is used, which is assumed in the successive simulation steps in order to achieve a reliable determination of the basis vectors of the orthogonal basis vector system. The heat load Q can vary between successive simulation steps or successive groups of simulation steps. It can also be provided that, in the simulation steps, training takes place with a large number of representative individual heat load scenarios of appropriate resolution, from which real heat loads can be simulated by superimposing them. In this case, the results from the relevant reduced second differential equation system DGLS2 can also be superimposed or combined, resulting in a particularly flexible system with which almost any real heat load can be simulated or represented.The number of simulation steps VS can then be, in particular, 50 to 10,000, preferably 500 to 2,000. However, a particularly favorable or realistic simulation can be achieved if the heat load Q is assumed or specified to be at least substantially constant over at least some of the VS simulation steps, in particular over all VS simulation steps. It can be advantageous if the heat load Q in the respective simulation step is 0.2 W to 2 W, preferably 0.5 W to 1.5 W, more preferably 0.8 W to 1.0 W.

[0050] In certain variants, the basis vectors of the orthogonal basis vector system are determined by determining the corresponding geometric state information as a simulation result for the geometric submodel with the first differential equation system in a plurality VS of simulation steps at a temperature T, and by determining the basis vectors of the orthogonal basis vector system from the plurality VS of simulation results using a singular value decomposition method or a proper orthogonal decomposition method. The plurality VS of simulation steps can be 100 to 100,000, preferably 500 to 10,000, more preferably 1,000 to 5,000.

[0051] In certain variants, the basis vectors of the orthogonal basis vector system of the geometric submodel are determined by simulating the first differential equation system for the geometric submodel in a large number of simulation steps, each with a given (theoretical or synthetic) thermal expansion vector v→(T) In each case, the corresponding geometric state information is determined as a simulation result, and from the multitude VS of simulation results, the basis vectors of the orthogonal basis vector system are determined by a singular value decomposition method or by a proper orthogonal decomposition method. Here, too, a large number of representative thermal expansion vectors can be used in the simulation steps. v→(T) with appropriate resolution, from which real thermal expansions can be superimposed. In this case, the results from the corresponding reduced second differential equation system DGLS2 can also be superimposed or combined, resulting in a particularly flexible system with which almost any real thermal expansion can be simulated or represented. The number VS of simulation steps can be 100 to 100,000, preferably 500 to 10,000, more preferably 1,000 to 5,000.

[0052] The basis vectors of the orthogonal basis vector system, in particular their number, can in principle be chosen in such a way that they achieve a good compromise between the reduction of the computing time and the reduction of the simulation accuracy for the respective application, which, adapted to the respective application for microlithography, still allows the control of the aberration compensation in real time.

[0053] In certain advantageous variants, the basis vectors of the orthogonal basis vector system are selected in such a way, in particular the number of basis vectors for the thermal submodel is selected in such a way that an absolute error of the thermal state information determined in the respective determination step compared to an actual value of the thermal state information is at most 0.0005 K to 0.5 K, preferably 0.001 K to 0.01 K, more preferably 0.001 K to 0.005 K. This allows very good results to be achieved in the field of microlithography with regard to the control of the imaging error compensation in real time.The same applies to variants in which the basis vectors of the orthogonal basis vector system for the thermal submodel are selected in such a way, in particular the number of basis vectors is selected such that a relative error of the thermal state information determined in the respective determination step is at most 0.01% to 5%, preferably 0.05% to 1%, more preferably 0.1% to 0.5%, of an actual value of the thermal state information. In the field of microlithography, it is typically particularly advantageous if the relative error is approximately 0.1% or less.

[0054] The actual value of the thermal state information can generally be determined in any suitable manner. For example, it can be determined based on actual measurements on the optical element. The actual value of the thermal state information can be determined particularly easily using the finite element model (FEM), i.e., the first differential equation system (DGLS1). In other words, with sufficiently precise modeling, the value of the thermal state information (TZI) resulting from the time-consuming conventional solution of the first differential equation system (DGLS1) can be used as the actual value of the thermal state information (TZI).

[0055] In certain advantageous variants, the basis vectors of the orthogonal basis vector system for the geometric partial model are selected in such a way, in particular the number of basis vectors is selected in such a way that an absolute error of the geometric state information determined in the respective determination step compared to an actual value of the geometric state information is at most 2 pm to 40 pm, preferably 4 pm to 30 pm, more preferably 8 pm to 20 pm. Additionally or alternatively, the basis vectors of the orthogonal basis vector system can be selected for the geometric partial model in such a way, in particular the number of basis vectors is selected in such a way that a relative error of the geometric state information determined in the respective determination step is at most 0.01% to 5%, preferably 0.05% to 1%, more preferably 0.1% to 0.5%, of an actual value of the geometric state information.It is understood that the above-mentioned accuracy specifications preferably apply to the entire system or the entire modeling and are adhered to.

[0056] The present invention further relates to an optical imaging device, in particular for microlithography, comprising an illumination device with a first optical element group, an object device for accommodating an object, a projection device with a second optical element group, and an image device, wherein the illumination device is designed to illuminate the object and the projection device is designed to project an image of the object onto the image device. During normal operation, the illumination device and / or the projection device are controlled by at least one control device according to the invention in order to correct at least one imaging error of the imaging device using the determined geometric state information.This allows the variants and advantages described above to be realized to the same extent, so that reference is made to the above statements in this regard.

[0057] The present invention further relates to a method for controlling an optical imaging device for microlithography, in particular for the use of light in the extreme UV range (EUV), in which at least one piece of geometric state information is determined for at least one optical element of the imaging device. The at least one piece of geometric state information is representative of a geometric state of the at least one optical element, wherein the at least one piece of geometric state information is designed to be used in determining aberration information that is representative of at least one optical aberration of the optical imaging device.At least one piece of thermal state information is determined, which is representative of a temperature distribution in at least a first element part of the optical element, wherein the first element part is the part of the optical element that, during normal operation of the optical imaging device, has a non-negligible influence on the at least one optical imaging error caused by thermally induced deformation. Furthermore, at least one piece of material state information is determined, which is representative of a distribution of at least one material property in at least the first element part, wherein the at least one material property has a non-negligible influence on the thermally induced deformation of the first element part. Using the at least one piece of thermal state information and the at least one piece of material state information, the at least one piece of geometric state information is then determined.The at least one piece of geometric state information is determined in a plurality of successive determination steps using a finite element model of the at least one optical element, each with a determination accuracy sufficient for real-time correction of the at least one optical aberration, and is output at an output frequency AF. The finite element model has a plurality of nodes and an average node spacing MKA, wherein the average node spacing MKA is from 0.1 mm to 30 mm, preferably 0.5 mm to 25 mm, more preferably 2 mm to 10 mm. Furthermore, at least one complexity reduction method is applied to the finite element model such that the (theoretically possible) output frequency AF of the geometric state information is 0.1 Hz to 1,000 Hz, preferably 10 Hz to 200 Hz, more preferably 50 Hz to 100 Hz.The variants and advantages described above can also be realized to the same extent, so that reference is made to the above explanations in this regard.

[0058] Finally, the present invention relates to an optical imaging method, in particular for microlithography, in which an illumination device having a first optical element group illuminates an object, and a projection device having a second optical element group projects an image of the object onto an image device. In this case, the illumination device and / or the projection device is controlled during normal operation by at least one method according to the invention in order to correct at least one imaging error of the imaging device using the determined thermal state information. The variants and advantages described above can also be realized to the same extent in this way, so that reference is made to the above explanations in this regard.

[0059] Further aspects and embodiments of the invention emerge from the dependent claims and the following description of preferred embodiments, which refers to the accompanying figures. All combinations of the disclosed features, regardless of whether they are the subject of a claim or not, are within the scope of the invention. BRIEF DESCRIPTION OF THE DRAWINGS Fig. 1 is a schematic representation of a preferred embodiment of a projection exposure system according to the invention, in which a preferred embodiment of the control arrangement according to the invention is used. Fig. 2 is a schematic plan view of an optical element of Fig. 1. Fig. 3 is a flow chart of a preferred embodiment of a method according to the invention for controlling an optical imaging device, which is carried out during a preferred embodiment of the optical imaging method according to the invention in the projection exposure apparatus of Fig. 1 can be used. Fig. 4 is a flowchart of a method for determining an orthogonal basis vector system used in the method of Fig. 3 can be used. DETAILED DESCRIPTION OF THE INVENTION

[0060] In the following, with reference to the Fig. 1 to 4, a preferred embodiment of a projection exposure apparatus 101 according to the invention for microlithography is described, which comprises a preferred embodiment of a control arrangement according to the invention. To simplify the following explanations, an x, y, z coordinate system is indicated in the drawings, wherein the z direction runs parallel to the direction of the gravitational force. The x direction and the y direction accordingly run horizontally, wherein the x direction in the representation of the Fig. 1 runs perpendicularly into the drawing plane. Of course, in further embodiments, it is possible to choose any orientations deviating from these of an x, y, z coordinate system.

[0061] In the following, with reference to the Fig. 1 describes the essential components of a projection exposure system 101 by way of example. The description of the basic structure of the projection exposure system 101 and its components is not intended to be limiting.

[0062] An illumination device or illumination system 102 of the projection exposure system 101 comprises, in addition to a radiation source 102.1, an optical element group in the form of an illumination optics 102.2 for illuminating an object field 103.1 (shown schematically). The object field 103.1 lies in an object plane 103.2 of an object device 103. A reticle 103.3 (also referred to as a mask) arranged in the object field 103.1 is illuminated. The reticle 103.3 is held by a reticle holder 103.4. The reticle holder 103.4 can be displaced, in particular in one or more scanning directions, via a reticle displacement drive 103.5. In the present example, such a scanning direction runs parallel to the y-axis.

[0063] The projection exposure system 101 further comprises a projection device 104 with a further optical element group in the form of a projection optics system 104.1. The projection optics system 104.1 serves to image the object field 103.1 into a (schematically shown) image field 105.1, which lies in an image plane 105.2 of an image device 105. The image plane 105.2 runs parallel to the object plane 103.2. Alternatively, an angle other than 0° between the object plane 103.2 and the image plane 105.2 is also possible.

[0064] During exposure, a structure of the reticle 103.3 is imaged onto a light-sensitive layer of a substrate in the form of a wafer 105.3, wherein the light-sensitive layer is arranged in the image plane 105.2 in the region of the image field 105.1. The wafer 105.3 is held by a substrate holder or wafer holder 105.4. The wafer holder 105.4 can be displaced via a wafer displacement drive 105.5, in particular along the y-direction. The displacement of the reticle 103.3 via the reticle displacement drive 103.5, on the one hand, and of the wafer 105.3 via the wafer displacement drive 105.5, on the other hand, can be synchronized with each other. This synchronization can, for example, be achieved via a common (in Fig. 1 only shown very schematically and without control paths) control device 106.

[0065] The radiation source 102.1 is an EUV radiation source (extreme ultraviolet radiation). The radiation source 102.1 emits, in particular, EUV radiation 107, which is also referred to below as useful radiation or illumination radiation. The useful radiation has, in particular, a wavelength in the range between 5 nm and 30 nm, in particular a wavelength of approximately 13 nm. The radiation source 102.1 can be a plasma source, for example, an LPP source (laser produced plasma, i.e., plasma generated using a laser) or a DPP source (gas discharged produced plasma, i.e., plasma generated by gas discharge). It can also be a synchrotron-based radiation source. However, the radiation source 102.1 can also be a free-electron laser (FEL).

[0066] Since the projection exposure system 101 operates with useful light in the EUV range, the optical elements used are exclusively reflective optical elements. In further embodiments of the invention, it is of course also possible (particularly depending on the wavelength of the illumination light) to use any type of optical element (refractive, reflective, diffractive) alone or in any combination.

[0067] The illumination radiation 107 emanating from the radiation source 102.1 is focused by a collector 102.3. The collector 102.3 can be a collector with one or more ellipsoidal and / or hyperboloidal reflection surfaces. The at least one reflection surface of the collector 102.3 can be exposed to the illumination radiation 107 at grazing incidence (Gl), i.e., at angles of incidence greater than 45°, or at normal incidence (NI), i.e., at angles of incidence less than 45°. The collector 11 can be structured and / or coated, on the one hand, to optimize its reflectivity for the useful radiation and, on the other hand, to suppress stray light.

[0068] After the collector 102.3, the illumination radiation 107 propagates through an intermediate focus in an intermediate focal plane 107.1. In certain variants, the intermediate focal plane 107.1 can represent a separation between the illumination optics 102.2 and a radiation source module 102.4, which comprises the radiation source 102.1 and the collector 102.3.

[0069] The illumination optics 102.2 comprise a deflecting mirror 102.5 and a downstream first facet mirror 102.6 along the beam path. The deflecting mirror 102.5 can be a flat deflecting mirror or, alternatively, a mirror with a beam-influencing effect beyond the pure deflection effect. Alternatively or additionally, the deflecting mirror 102.5 can be designed as a spectral filter that at least partially separates so-called stray light from the illumination radiation 107, the wavelength of which deviates from the useful light wavelength. If the optically effective surfaces of the first facet mirror 102.6 are arranged in the region of a plane of the illumination optics 102.2 that is optically conjugate to the object plane 103.2 as the field plane, the first facet mirror 102.6 is also referred to as a field facet mirror. The first facet mirror 102.6 comprises a plurality of individual first facets 102.7, which are also referred to as field facets. These first facets and their optical surfaces are shown in the . Fig. 1 is only indicated schematically by the dashed contour 102.7.

[0070] The first facets 102.7 can be designed as macroscopic facets, in particular as rectangular facets or as facets with an arcuate or partially circular edge contour. The first facets 102.7 can be designed as facets with a planar or, alternatively, with a convex or concave curved optical surface.

[0071] As is known, for example, from DE 10 2008 009 600 A1 (the entire disclosure of which is incorporated herein by reference), the first facets 102.7 themselves can each also be composed of a plurality of individual mirrors, in particular a plurality of micromirrors. The first facet mirror 102.6 can, in particular, be designed as a microelectromechanical system (MEMS system), as described in detail, for example, in DE 10 2008 009 600 A1.

[0072] In the present example, the illumination radiation 107 runs horizontally between the collector 102.3 and the deflection mirror 102.5, i.e., along the y-direction. However, it is understood that a different orientation may be chosen for other variants.

[0073] In the beam path of the illumination optics 102.2, a second facet mirror 102.8 is arranged downstream of the first facet mirror 102.6. If the optically effective surfaces of the second facet mirror 102.8 are arranged in the region of a pupil plane of the illumination optics 102.2, the second facet mirror 102.8 is also referred to as a pupil facet mirror. The second facet mirror 102.8 can also be arranged at a distance from a pupil plane of the illumination optics 102.2. In this case, the combination of the first facet mirror 102.6 and the second facet mirror 102.8 is also referred to as a specular reflector. Such specular reflectors are known, for example, from US 2006 / 0132747 A1, EP 1 614 008 B1 or US 6,573,978 (the entire disclosure of which is incorporated herein by reference).

[0074] The second facet mirror 102.8 in turn comprises a plurality of second facets which are arranged in the Fig. 1 are only indicated very schematically by the dashed contour 102.9. In the case of a pupil facet mirror, the second facets 102.9 are also referred to as pupil facets. The second facets 102.9 can basically be designed like the first facets 102.7. In particular, the second facets 102.9 can also be macroscopic facets, which can, for example, be round, rectangular, or hexagonal in shape. Alternatively, the second facets 102.9 can be facets composed of micromirrors. The second facets 102.9 can in turn have planar or, alternatively, convex or concave curved reflection surfaces. In this regard, reference is again made to DE 10 2008 009 600 A1.

[0075] In the present example, the illumination optics 102.2 thus form a double-faceted system. This basic principle is also referred to as a fly's-eye integrator. In certain variants, it may also be advantageous not to arrange the optical surfaces of the second facet mirror 102.8 exactly in a plane that is optically conjugate to a pupil plane of the projection optics 104.1.

[0076] In a further, not-shown embodiment of the illumination optics 102.2, a transmission optics 102.10 (shown only very schematically) can be arranged in the beam path between the second facet mirror 102.8 and the object field 103.1, which transmission optics contributes in particular to the imaging of the first facets 102.7 into the object field 103.1. The transmission optics 102.10 can have exactly one mirror, but alternatively also two or more mirrors, which are arranged one behind the other in the beam path of the illumination optics 102.2. The transmission optics 102.10 can in particular comprise one or two mirrors for normal incidence (NI mirrors, normal incidence mirrors) and / or one or two mirrors for grazing incidence (GL mirrors, grazing incidence mirrors).

[0077] The illumination optics 102.2 has, in the design as shown in the Fig. 1, exactly three mirrors are arranged downstream of the collector 102.3, namely the deflecting mirror 102.5, the first facet mirror 102.6 (e.g., a field facet mirror), and the second facet mirror 102.8 (e.g., a pupil facet mirror). In a further embodiment of the illumination optics 102.2, the deflecting mirror 102.5 can also be omitted, so that the illumination optics 102.2 can then have exactly two mirrors downstream of the collector 102.3, namely the first facet mirror 102.6 and the second facet mirror 102.8.

[0078] With the help of the second facet mirror 102.8, the individual first facets 102.7 are imaged into the object field 103.1. The second facet mirror 102.8 is the last beam-forming mirror, or actually the last mirror, for the illumination radiation 107 in the beam path before the object field 103.1. The image of the first facets 102.7 by means of the second facets 102.9 or with the second facets 102.9 and a transmission optics 102.10 into the object plane 103.2 is usually only an approximate image.

[0079] The projection optics 104.1 comprises a plurality of mirrors Mi, which are numbered according to their arrangement along the beam path of the projection exposure system 101. In the Fig. In the example shown in Figure 1, the projection optics 104.1 comprises six mirrors M1 to M6. Alternatives with four, eight, ten, twelve or a different number of mirrors Mi are also possible. The penultimate mirror M5 and the last mirror M6 can each have a passage opening (not shown in detail) for the illumination radiation 107. In the present example, the projection optics 104.1 is a doubly obscured optic. The projection optics 104.1 has an image-side numerical aperture NA that is greater than 0.5. In particular, the image-side numerical aperture NA can also be greater than 0.6. For example, the image-side numerical aperture NA can be 0.7 or 0.75.

[0080] The reflection surfaces of the mirrors Mi can be designed as freeform surfaces without a rotational symmetry axis. Alternatively, the reflection surfaces of the mirrors Mi can be designed as aspherical surfaces with exactly one rotational symmetry axis of the reflection surface shape. The mirrors Mi, like the mirrors of the illumination optics 102.2, can have highly reflective coatings for the illumination radiation 107. These coatings can be constructed from several layers (multilayer coatings); in particular, they can be designed with alternating layers of molybdenum and silicon.

[0081] In the present example, the projection optics 104.1 has a large object-image offset in the y-direction between a y-coordinate of a center of the object field 103.1 and a y-coordinate of the center of the image field 105.1. This object-image offset in the y-direction can be approximately as large as a distance between the object plane 103.2 and the image plane 105.2 in the z-direction.

[0082] The projection optics 104.1 can, in particular, be anamorphic. It has, in particular, different imaging scales βx, βy in the x- and y-directions. The two imaging scales βx, βy of the projection optics 104.1 are preferably (βx; βy) = (+ / - 0.25; + / - 0.125). A positive imaging scale β means imaging without image inversion. A negative sign for the imaging scale β means imaging with image inversion. In the present example, the projection optics 104.1 thus leads to a reduction in the ratio of 4:1 in the x-direction, i.e., in the direction perpendicular to the scanning direction. In contrast, the projection optics 104.1 leads to a reduction in the ratio of 8:1 in the y-direction, i.e., in the scanning direction. Other imaging scales are also possible. Image scales with the same sign and absolutely equal in the x and y directions are also possible, for example with absolute values ​​of 0.125 or 0.25.

[0083] The number of intermediate image planes in the x- and y-directions in the beam path between the object field 103.1 and the image field 105.1 can be the same. Likewise, the number of intermediate image planes can vary depending on the design of the projection optics 104.1. Examples of projection optics with different numbers of such intermediate images in the x- and y-directions are known, for example, from US 2018 / 0074303 A1 (the entire disclosure of which is incorporated herein by reference).

[0084] In the present example, each of the pupil facets 102.9 is assigned to exactly one of the field facets 102.7 to form a respective illumination channel for illuminating the object field 103.1. This can, in particular, result in illumination according to the Köhler principle. The far field is divided into a plurality of object fields 103.1 using the field facets 102.7. The field facets 102.7 generate a plurality of images of the intermediate focus on the pupil facets 102.9 assigned to them.

[0085] The field facets 102.7 are each imaged onto the reticle 103.3 by an associated pupil facet 102.9, with the images overlapping, resulting in an overlapping illumination of the object field 103.1. The illumination of the object field 103.1 is preferably as homogeneous as possible. It preferably has a uniformity error of less than 2%. Field uniformity can be achieved by superimposing different illumination channels.

[0086] By arranging the pupil facets 102.9, the illumination of the entrance pupil of the projection optics 104.1 can be geometrically defined. By selecting the illumination channels, in particular the subset of the pupil facets 102.9 that guide light, the intensity distribution in the entrance pupil of the projection optics 104.1 can be adjusted. This intensity distribution is also referred to as the illumination setting of the illumination system 102. A likewise preferred pupil uniformity in the area of ​​defined illuminated sections of an illumination pupil of the illumination optics 102.2 can be achieved by redistributing the illumination channels. The aforementioned settings can be made for actively adjustable facets by appropriate control via the control device 106.

[0087] Further aspects and details of the illumination of the object field 103.1 and in particular of the entrance pupil of the projection optics 104.1 are described below.

[0088] The projection optics 104.1 can, in particular, have a homocentric entrance pupil. This can be accessible or inaccessible. The entrance pupil of the projection optics 104.1 can often not be precisely illuminated with the pupil facet mirror 102.8. When imaging the projection optics 104.1, which telecentrically images the center of the pupil facet mirror 102.8 onto the wafer 105.3, the aperture rays often do not intersect at a single point. However, a surface can be found in which the pairwise determined spacing of the aperture rays is minimized. This surface represents the entrance pupil or a surface conjugate to it in spatial space. In particular, this surface exhibits a finite curvature.

[0089] In certain variants, the projection optics 104.1 may have different entrance pupil positions for the tangential and sagittal beam paths. In this case, it is preferred if an imaging optical element, in particular an optical component of the transmission optics 102.10, is provided between the second facet mirror 102.8 and the reticle 103.3. With the help of this imaging optical element, the different positions of the tangential entrance pupil and the sagittal entrance pupil can be taken into account.

[0090] In the arrangement of the components of the illumination optics 102.2, as shown in the Fig. 1, the optical surfaces of the pupil facet mirror 102.8 are arranged in a surface conjugate to the entrance pupil of the projection optics 104.1. The first facet mirror 102.6 (field facet mirror) defines a first main extension plane of its optical surfaces, which in the present example is tilted relative to the object plane 5. This first main extension plane of the first facet mirror 102.6 is tilted relative to a second main extension plane defined by the optical surface of the deflection mirror 102.5 in the present example. The first main extension plane of the first facet mirror 102.6 is further tilted relative to a third main extension plane defined by the optical surfaces of the second facet mirror 102.8 in the present example.

[0091] As shown below using the mirror M3 or 108 and the Fig. 2 to 4, the optical elements of the projection exposure system 101, thus also the optical element 108, are exposed to a (in many cases changing) thermal load (e.g., in the form of heat from absorbed useful light or absorbed useful radiation). Thus, the mirror 108 (i.e., the optical element) has a mirror body 108.1, which is constructed from a corresponding mirror body material. An optical surface 108.2 is formed on the mirror body 108.1, typically by a single- or multi-layer coating of a correspondingly high-precision machined surface of the mirror body 108.1. The mirror is connected to its support structure via (in the present example: three) interface units 108.3, wherein the mirror 108 can be supported on the support structure via active support elements (as is shown in Fig. 1 is indicated by the contour 104.2).

[0092] In a number of typical imaging processes, the optical surface 108.2 is exposed to different intensity distributions (so-called settings) of the useful radiation during normal operation of the projection exposure system 101, so that the mirror 108 is subjected to a varying thermal load. Due to the thermal expansion of the materials used, the mirror 108, especially its optical surface 108.2, deforms, resulting in a specific imaging error IE, which is often referred to as a registration error.

[0093] In order to correct or at least reduce this imaging error IE, it is possible to implement one or more correction concepts, some of which were already mentioned as examples. For this purpose, in the present example, before or during operation of the projection exposure system 101, in a step 110.1, a variant of the method for controlling the imaging device 101 is started (see Fig. 3). Next, in step 110.2, a check is made to determine whether a correction of an aberration is to be performed. If not, the program returns to step 110.2; otherwise, the correction of the aberration IE is initiated in a determination step 110.3.

[0094] One of the prerequisites for a targeted correction of such thermally induced aberrations IE in numerous embodiments is sufficiently precise knowledge of the thermal state of the relevant optical element 108, for example, the temperature distribution in the optical element 108 and, if applicable, also its temporal development. A changing thermal state of the optical element 108 (for example, possibly locally varying heating) causes, due to the material properties of the optical element 108, a thermally induced expansion and thus a deformation, i.e., a geometric state that deviates to a greater or lesser extent from a geometric initial state or a geometric target state.

[0095] In this case, the material of the optical element 108 (which may also consist of a combination of different materials) may have arbitrarily complex or non-linear material properties that have a correspondingly non-negligible influence on the deformation of the optical element. These may, for example, be locally and / or temporally varying material properties (for example, depending on the temperature and / or the current stress state and / or the structural state of the material in question). Such a varying material property may, for example, be the thermal expansion coefficient and / or the thermal conductivity.

[0096] As will be explained in more detail below, in the present example, information TVI1, which is representative of a temperature distribution TV1 in at least one first element part 108.4 of the optical element 108 that is relevant to the aberration IE, is determined via sufficiently fast numerical modeling as thermal state information TZI relevant for the correction of the aberration IE. This information TVI1 is then used by the control device 106 to control the associated compensation device(s) (not described in more detail) in real time. Likewise, in the present example, material state information MZI, which is representative of a distribution of at least one material property in the first element part 108, is determined via sufficiently fast numerical modeling as material state information relevant for the correction of the aberration IE.4 is representative and has a non-negligible influence on the thermally induced deformation of the first element part 108.4.

[0097] Using the thermal state information TZI and the material state information MZI, a geometric state information GZI is then determined via a sufficiently fast numerical modeling, which is representative of a geometric state of the first element part 108.4 and is used in the determination of an aberration information IEI, which is representative of the optical aberration IE

[0098] In this case, it may be sufficient to model only the part 108.4 (i.e., the first element part 108.4) of the optical element 108, which, during normal operation of the optical imaging device 101, has a non-negligible influence on the relevant optical aberration IE caused by thermally induced deformation. However, the entire optical element 108 can also be modeled accordingly. Thus, the relevant part 108.4 can extend over the entire optical element 108, which is assumed for the sake of simplicity in the present example.

[0099] Sufficiently fast numerical modeling can be achieved by first performing a sufficiently spatially high-resolution modeling to determine the thermal state or temperature distribution TV1 of the relevant part 108.4 of the optical element 108 using a conventional finite element model (FEM), which results in a thermal subsystem TDGLS1 of a first differential equation system DGLS1. This basic modeling can, in principle, be performed during a step 110.4. Typically, however, it is performed before the start of normal operation of the imaging device 101.

[0100] Analogously, in the present example, a sufficiently spatially high-resolution model is finally carried out to determine the geometric state or deformation DV1 of the relevant part 108.4 of the optical element 108 using a conventional finite element model (FEM), which leads to a geometric subsystem GDGLS1 of a first differential equation system DGLS1. In this case, the material state or a material property distribution MV1 of at least one material property of the relevant part 108.4 of the optical element 108 is also taken into account at the appropriate resolution using a corresponding material law.

[0101] This basic modeling of the temperature distribution TV1, the material property distribution MV1 and the deformation DV1 naturally has such a high first dimensionality D1 at the spatial resolution required (for application in microlithography) that the simulation results are not available sufficiently quickly and thus a direct use of the simulation results from this first differential equation system DGLS1 in a control of the compensation device(s) in real time by the control device 106 is not possible.

[0102] To enable the use of the simulation results for a real-time correction of the relevant optical aberration IE, thus achieving a sufficiently high output frequency AF of the simulation results, in the present example, at least one complexity reduction method KRV is applied to the finite element model FEM in such a way that the simulation is sufficiently accelerated, thus enabling a real-time correction using these simulation results by the control device 106. This application of the complexity reduction method KRV to the finite element model FEM can also, in principle, take place during a step 110.4. Typically, however, it takes place before the start of normal operation of the imaging device 101.

[0103] By appropriately designing or selecting the complexity reduction method (CRV), it is possible to reduce the dimensionality D2 of the (thus obtained) simplified or dimensionally reduced model DRM, for example, a second differential equation system DGLS2, to a significantly lower value (than the dimensionality D1 of the first differential equation system DGLS1). The reduction can be so significant that the required computing time for simulating the temperature distribution is drastically reduced, thus enabling the use of the simulation results for real-time correction of the relevant optical aberration IE by the control device 106.

[0104] The sufficiently precise simulation results of the simplified, dimensionally reduced model DRM could then be easily and quickly transferred back to the high-dimensional initial model, i.e., the finite element model FEM, in a step 110.5, so that the subsequently obtained information DVI about the TV1 deformation DV1 of the optical element 108 has a spatial resolution sufficient for application in microlithography. Preferably, however, the simulation results of the simplified, dimensionally reduced model DRM are used directly (i.e., without back-transformation to the high-dimensional initial model, i.e., the finite element model FEM) to correct the relevant optical aberration IE, whereby a particularly high output frequency AF can be achieved.In this case, output frequencies AF of the thermal state information TZI (here, the information TVI about the temperature distribution TV1), the material state information MZI (here, the information MVI about the material property distribution MV1), and the geometric state information GZI (here, the information GVI about the deformation GV1) can be achieved, which are in the range of 0.1 Hz to 1,000 Hz. These output frequencies AF are therefore within the range of a control bandwidth, as required in microlithography for real-time correction of the relevant optical imaging error IE by means of the control device 106.

[0105] In the present example, the control device 106 comprises a determination device 106.1, which is designed, on the one hand, to determine at least the thermal state information TZI for the optical element 108, wherein the thermal state information TZI is representative of a thermal state of the optical element 108 and, for example, in a step 110.6, is used in the determination of an image aberration information IEI that is representative of at least one optical image aberration IE of the optical imaging device 101. The image aberration information IEI can then be used to correct the image aberration IE in real time in step 110.6. More precisely, the determination device 106.1 is further designed to determine the material state information MZI using the thermal state information TZI.It is also designed to determine the geometric state information GZI using the material state information MZI. In the present example, the geometric state information GZI is then used in step 110.6 to determine an aberration information IEI.

[0106] In step 110.7, a check is then made to determine whether the process flow should end. If so, the process flow is terminated in step 110.8; otherwise, the program returns to step 110.2.

[0107] In the present example, the at least one piece of thermal state information TZI is representative of a temperature distribution TV1 in at least a first element part 108.4 of the optical element 108, wherein the first element part 108.4 is the part of the optical element 108 that, during normal operation of the optical imaging device 101, has a non-negligible influence on the at least one optical aberration IE caused by thermally induced deformation. In the present example, the first element part 108.4 extends, as mentioned, over the entire optical element 108. However, it is understood that in other variants, the first element part 108.4 may not extend over the entire optical element 108.

[0108] The determination device 106.1 determines the at least one thermal state information item TZI, the at least one material state information item MZI and the at least one geometric state information item GZI in a plurality of successive determination steps 110.3 (see Fig. 3). It uses the finite element model FEM of the optical element 108, achieving a determination accuracy and output frequency AF sufficient for real-time correction of the optical aberration IE.

[0109] For this purpose, the finite element model FEM has a large number of nodes 109, as shown in Fig. 2 is shown in a highly schematic manner. The finite element model (FEM) is designed such that a mean nodal distance MKA exists between its neighboring nodes 109 (averaged over the modeled first element part 108.4), with the mean nodal distance MKA in the present example being 2 mm to 10 mm. In certain preferred variants, the mean nodal distance MKA can also be 0.1 mm to 30 mm, preferably 0.5 mm to 25 mm.

[0110] In the present example, the determination device 106.1 is further designed to apply at least one complexity reduction method KRV to the finite element model FEM such that the theoretically possible output frequency AF of the thermal state information is 0.1 Hz to 1,000 Hz, preferably 10 Hz to 200 Hz, more preferably 50 Hz to 100 Hz,

[0111] It can therefore be achieved that the period time or runtime of the simulation (i.e. the time that elapses, according to the output frequency AF, for each run through the simulation loop or simulation step 110.3 until a newly determined thermal state information item TZI, the at least one material state information item MZI, and the at least one geometric state information item GZI of the respective simulation step 110.3 is output) lies in the range of 1 ms to 10 s, for example 5 ms to 5 s, preferably 10 ms to 2 s. It is understood that additional processing time may be required for the actual control of the imaging device 101, for example for real-time correction of the relevant optical imaging error. Therefore, it can be provided that the respective data in the most recent simulation loop or simulation step 110.3 newly determined thermal state information TZI, material state information MZI and geometric state information GZI is not actually fed into the control by the control device 106 of the imaging device 101 or used therein at the output frequency AF. In particular, in certain variants, the output frequency AF can be a theoretical value with which the at least one piece of thermal state information TZI, material state information MZI and geometric state information GZI could theoretically be output thanks to the high speed of the simulation, while in reality the output is delayed accordingly in order to synchronize it with the control cycle of the control device 106 of the imaging device 101 (for example, for real-time correction of the at least one optical imaging error). Likewise, the simulation process can, however, be carried out without a break between the simulation loops orSimulation steps 110.3 run and the most recent thermal state information TZI available from the simulation process is then used to control the imaging device 101 in the control device 106.

[0112] In principle, any suitable complexity reduction method (CRV) (individually or in combination) can be used, which, in addition to a sufficiently strong reduction in dimensionality, achieves sufficient simulation accuracy. Thus, the reduction in dimensionality from the initial FEM model to the dimensionally reduced DRM model then used for the actual simulation calculation should, on the one hand, be so extensive that a correspondingly significant reduction in the computing time for the respective determination step 110.3 is achieved. On the other hand, a sufficient accuracy of the simulation or numerical determination of the deformation GV1 (with respect to the actual temperature distribution TVT, material state distribution MVT, and deformation or geometric state distribution GVT present in the optical element 108) should be achieved even after the back transformation into the high-dimensional initial FEM model (i.e., the original finite element model).

[0113] In the present example, a strong dimensionality reduction with good accuracy of the determined temperature distribution TV1 is achieved by using a reduced basis method RBV as the complexity reduction method KRV. In the course of the complexity reduction method KRV, a plurality of MBV of basis vectors BV of an orthogonal basis vector system BVS are applied to the thermal subsystem TDGLS1 of the first differential equation system DGLS1 of the finite element model FEM in such a way that a reduced second thermal subsystem TDGLS2 of a second differential equation system DGLS2 is obtained with a second dimension D2 that is reduced compared to a first dimension D1 of the first differential equation system DGLS1 of the finite element model FEM.

[0114] As in the present example, the basis vectors BV for creating the reduced second differential equation system DGLS2 can be multiplied from the right by the first differential equation system DGLS1 of the finite element model FEM. This allows a particularly simple and rapid creation of a reduced differential equation system DGLS2 with a significantly lower dimensionality D2 (i.e., D2 << D1).

[0115] In the present example, the MBV of basis vectors is 100 to 20,000, as this allows for particularly favorable results. In certain advantageous variants, the MBV of basis vectors can also be 200 to 10,000, preferably 1,000 to 4,000, as this also allows for a corresponding reduction in dimensionality with significantly reduced computing times while maintaining sufficient simulation accuracy. For example, using an orthogonal basis vector system (BVS) with a number of MBVs = 80 to 100 basis vectors (BV), the required computing time can be accelerated by a factor of 1,500 to 2,500. For example, with a number of MBV = 91 basis vectors BV, an acceleration of the required computing time by a factor of about 2,000 could be achieved for a first differential equation system DGLS1 of the finite element model FEM with a first dimension D1 of 500,000.Likewise, for example, when using an orthogonal basis vector system BVS with a number of MBV = 1,000 to 10,000 basis vectors BV (compared to a first differential equation system DGLS1 of the finite element model FEM with a first dimension D1 of 100,000), the required computing time can be accelerated by a factor of more than 100. In general, a smaller number of basis vectors BV results in a greater reduction in computing time, but this also results in a reduction in simulation accuracy.

[0116] In the present example, the first dimension D1 of the first differential equation system DGLS1 of the finite element model (FEM) is from 200,000 to 500,000. In certain variants, the first dimension D1 of the first differential equation system DGLS1 can also be from 10,000 to 2,000,000, preferably from 50,000 to 1,000,000. This makes it possible to achieve a sufficiently good spatial resolution of the determined temperature distribution TV1, particularly for applications in the field of microlithography. In the present example, the second dimension D2 of the reduced second differential equation system DGLS2 is still from 1,000 to 4,000. In certain variants, the second dimension D2 can also be from 100 to 20,000, preferably from 200 to 10,000. This allows a reduction in computing time sufficient for real-time control of aberration correction or compensation with sufficient simulation accuracy to be achieved.

[0117] The discretization of the first element part 108.4 of the optical element 108, which is relevant for the aberration correction, by means of the finite element model FEM for the simulation of the temperature distribution TV1 can be carried out using the continuous heat conduction equation, according to which, as already explained above, in homogeneous media, the following applies: ∂∂tT(x→,t)−α⋅ΔT(x→,t)=0 in Ω×[tmin,tmax] B(T(x→,t),∇T(x→,t),Q(x→,t))=0 at ∂Ω×[tmin,tmax] T(x→,t)=T→0 at t=tmin. where the boundary conditions B(T(x→,t),∇T(x→,t),Q(x→,t)) are not linear due to the Stefan-Boltzmann law, but can be linearized in the preheated state, and where: Temperature field for the first element part 108.4; t time; Δ Laplace operator; ∇ Nabla Operator; Heat load for the first element part 108.; Ω geometry of the body to be simulated; ∂Ω surface of the simulated body; α thermal conductivity.

[0118] With the mass matrix M of the first element part 108.4 and the stiffness matrix S of this first element part 108.4, the first differential equation system DGLS1 is obtained via the FEM discretization, which is typically high-dimensional and for which the following applies: ∂∂tMT→(t)−ST→(t)=MQ→(t) where: Temperature vector for the first element part 108.4; Heat load vector for the first element part 108.4; t time; M mass matrix M of the first element part 108.4; S Stiffness matrix S of the first element part 108.4.

[0119] The dimensionality or first dimension D1 of the first differential equation system DGLS1 of the finite element model FEM can have the values ​​mentioned above.

[0120] In the present example, the basis vectors BV of the orthogonal basis vector system BVS are chosen such that the temperature vector T(t) for the first element part 108.4, which depends on the time t, can be represented by the basis vectors BV, so that: T→(t)=∑n=1kcn(t)Ψ→n with Ψ→nMΨ→m=δnm where: Temperature vector for the first element part; t time; c n (t) development coefficient n; Basis vectors n, m (BV); M mass matrix of the finite element model FEM for the first element part 108.4; δ nm Kronecker Delta.

[0121] In the present example, the basis vectors BV of the orthogonal basis vector system BVS are further selected such that the heat load vector dependent on time t Q→(t) for the first element part 108.4 can be represented by the basis vectors BV, so that: Q→(t)=∑n=1kdn(t)Ψ→n with Ψ→nMΨ→m=δnm where: Heat load vector for the first element part 108.4; t time; d n (t) development coefficient n; Basis vectors n, m (BV); M mass matrix of the finite element model FEM for the first element part 108.4; δ nm Kronecker Delta.

[0122] The discretization of the first element part 108.4 of the optical element 108, which is relevant for the aberration correction, by means of a finite element model FEM for the simulation of the deformation is carried out in the present example using the continuous elasticity equation, according to which the following applies in homogeneous media: ∇⋅σ(u→)=0 in Ω σ(u→)=λtr(ε(u→))I+2με(u→)−(3λ+2μ)αT(T)I=0 ε(u→)=12(∇u→+(∇u→)T) σ(u→)⋅n→=0 in ∂Ω where the problem in the displacement vector u→ linear i is linear and where: displacement vector; σ stress tensor; ε strain tensor; ∇ Nabla Operator; λ first Lamé constant; tr matrix trace operator; I identity matrix; µ second Lamé constant; α T temperature-dependent thermal expansion coefficient; T temperature; Ω geometry of the body to be simulated; ∂Ω surface of the simulated body; Displacement vector.

[0123] Using the stiffness matrix S of the first element part 108.4, one then obtains a typically high-dimensional geometric subsystem GDGLS1 of the first differential equation system DGLS1 via the FEM discretization with: Su→=v→ where: Displacement vector on the FEM nodes; Thermal expansion vector on the FEM nodes; S Stiffness matrix S of the first element part.

[0124] In the present example, the basis vectors of the at least one orthogonal basis vector system are selected at least for the geometric submodel GDGLS1 such that the displacement vector dependent on the thermal expansion v u→(v) for the first element part 108.4 can be represented by the basis vectors, so that: u→(v)=∑n=1dcn(v)Ψ→n with Ψ→nSΨ→m=δnm where: Displacement vector for the first element part (108.4); v thermal expansion; c n (v) development coefficient n; Basis vectors n, m; S stiffness matrix of the finite element model for the first element part (108.4); δ nm Kronecker Delta.

[0125] It is understood that the thermal submodel TDGLS1 and the geometric submodel GDGLS1 may have different structures. For example, different stiffness matrices S may be used.

[0126] In the present example, the basis vectors of the at least one orthogonal basis vector system are selected at least for the geometric submodel GDGLS1 such that the thermal expansion vector dependent on the temperature T v→(T) for the first element part 108.4 can be represented by the basis vectors, so that: v→(T)=∑n=1ddn(T)Ψ→n with Ψ→nSΨ→m=δnm where: Thermal expansion vector for the first element part (108.4); T temperature; d n (T) development coefficient n; Basis vectors n, m; S stiffness matrix of the finite element model for the first element part; δ nmKronecker Delta.

[0127] The dimensionality or first dimension D1 of the geometric submodel GDGLS1 of the first differential equation system DGLS1 of the finite element model can have the values ​​mentioned above.

[0128] In the present example, the basis vectors of the at least one orthogonal basis vector system are further selected at least for the material submodel MDGLS1 of the geometric submodel GDGLS1 such that the material property α(T) dependent on the temperature T for the first element part 108.4 can be represented via the basis vectors of the geometric submodel GDGLS1, so that: α(T)=α(T(x,y,z);x;y;z) where: α(T) material property, in particular a thermal expansion coefficient, for the first element part 108.4; T(x,y,z) Temperature at the point of the first element part 108.4 with the coordinates (x, y, z).

[0129] The thermal expansions V of the at least one orthogonal basis vector system for the material submodel MDGLS1 of the geometric submodel GDGLS1 can be determined starting from a complete basis of the thermal expansions V of the finite element model FEM by determining a Fekete basis and the relevant modes of the Fekete basis. A corresponding method is described, for example, in the article "Computing Approximate Fekete Points by QR Factorizations of Vandermonde Matrices" (A. Sommariva, M. Vianello in Computers and Mathematics with Applications 57 (2009), pages 1324 to 1336, Elsevir Ltd.) (the entire disclosure of which is incorporated herein by reference).The basis of thermal expansion V of the geometric submodel GDGLS1 was orthogonalized and normalized, and for the complexity reduction procedure, the modes of the basis of thermal expansion V were sorted using a QR decomposition with pivoting to determine at least one reduced second differential equation system of the geometric submodel GDGLS1 with the second dimension D2 as the reduced basis of thermal expansion V. This advantageously makes it possible to transform the corresponding values ​​between the first differential equation system DGLS1 and the second differential equation system DGLS2.

[0130] In the present example, the determination device 106.1 is configured to transfer the results of the solution of the reduced second differential equation system DGLS2 of the thermal submodel TDGLS2 into the reduced second differential equation system DGLS2 of the material submodel MDGLS2 of the geometric submodel GDGLS1 using first operators OP1 to determine the at least one piece of geometric state information GZI. The determination device is further configured to integrate the results of the solution of the reduced second differential equation system DGLS2 of the material submodel MDGLS2 into the reduced second differential equation system DGLS2 of the geometric submodel GDGLS2 using second operators OP2 to determine the at least one piece of geometric state information GZI.

[0131] In the present example, the geometric state information GZI is determined entirely using the reduced subsystems of the second differential equation system DGLS2 (i.e., with reduced dimensionality D2), before the final result is then transferred back to the first differential equation system DGLS1 (i.e., with high dimensionality D1). This allows a significant reduction in computing time. This allows the precalculation of parameters that are consistent throughout the calculation. For time-dependent quantities such as temperature, the described operators OP1 can thus be formulated in such a way that a mapping from the temperature basis (i.e., the thermal submodel) to the reduced basis with the reduced dimensionality is possible.Likewise, the result of the thermal submodel can be mapped into the material submodel of the geometric submodel (i.e., a correspondingly evaluated material law) via appropriately precalculated operators, and then directly into the reduced basis of the geometric submodel (i.e., the displacement basis) via appropriately precalculated operators OP2. Thus, no further evaluation is required in the first differential equation system DGSL1, and a significantly faster evaluation can be performed with sufficient accuracy in the reduced second differential equation system DGLS2.

[0132] The solution of the reduced second differential equation system DGLS2 determined in the respective determination step 110.3 can, depending on the achieved computing speed, be processed together with the solution of one or more previous and / or subsequent determination steps 110.3 in step 110.4, before the thus processed solution is transferred back to the first differential equation system DGLS1 in step 110.5. For example, an average can be calculated from the jointly processed solutions of the second differential equation system DGLS2, and this average can then be transferred back to the first differential equation system DGLS1.Such an approach can be chosen if, despite the delay caused by such joint processing of several solutions of the reduced second differential equation system DGLS2, an output frequency AF of the geometric state information in the dimensional space of the first differential equation system DGLS1 that is sufficiently high for real-time control is still achieved.

[0133] In particularly simple variants with a high output frequency AF, the determination device 106.1 is designed to determine a solution of the reduced second differential equation system DGLS2 in the respective step 110.4 in each of the determination steps 110.3 and to transfer the determined solution of the reduced second differential equation system DGLS2 back into the dimensional space of the first differential equation system DGLS1 in the step 110.5 in order to obtain the geometric state information TZI.

[0134] The basis vectors BV of the orthogonal basis vector system BVS can generally be determined in any suitable manner. This can be done, for example, when performing step 110.4 for the first time or before. In the present example, the basis vectors BV of the orthogonal basis vector system BVS for the thermal submodel TDGLS1 and the geometric submodel GDGLS1 are or were determined in advance, as shown in the example in Fig. 4. The determination is initially started in a step 111.1. Then, the first differential equation system DGLS1 is used in a plurality of simulation steps 111.2, each with different heat loads Q i the corresponding thermal state information TZI i determined as a simulation result.

[0135] For this purpose, in a first run of simulation step 111.2 with the first differential equation system DGLS1 at a first heat load Q1, the associated thermal state information TZI1 is determined as the simulation result. In step 111.3, it is then checked whether a further simulation step 111.2 should be performed. If so, simulation step 111.2 is performed with a modified heat load Q2. The loop of steps 111.2 and 111.3 is executed until it is determined in one of steps 111.3 that no further simulation step 111.2 should be performed.

[0136] From the multitude of VS simulation results TZI iIn a step 111.4, the basis vectors BV of the respective orthogonal basis vector system BVS are then determined using a singular value decomposition method (SVD) and / or a proper orthogonal decomposition method (POD). Such SVD methods or POD methods are well known, so they will not be discussed in detail here.

[0137] It can be particularly advantageous if the multiplicity VS of simulation steps is 100 to 100,000, preferably 500 to 10,000, more preferably 1,000 to 5,000. In certain variants, the heat load Q can vary between successive simulation steps 111.2 or successive groups of simulation steps 111.2. However, a particularly favorable or realistic simulation can be achieved if the heat load Q specified in the respective simulation step 111.2 is assumed or specified to be at least substantially constant over at least some of the VS simulation steps 111.2, in particular over all VS simulation steps 111.2, in order to achieve a reliable and realistic determination of the basis vectors BV of the orthogonal basis vector system BVS.

[0138] Likewise, in certain variants, it may be provided that, in simulation steps 111.2, training is carried out with a large number of representative individual heat load scenarios of appropriate resolution, from which real heat loads can be simulated by superimposing them. In this case, the results from the relevant reduced second differential equation system DGLS2 can also be superimposed or combined, resulting in a particularly flexible system with which almost any real heat load Q can be simulated or represented.

[0139] In this case, it may be advantageous if the heat load Q in the respective simulation step 111.2 is from 0.2 W to 2 W, preferably 0.5 W to 1.5 W, more preferably 0.8 W to 1.0 W.

[0140] Analogously, in the present example, the basis vectors BV of the orthogonal basis vector system BVS are determined by determining the corresponding geometric state information as a simulation result for the geometric submodel GDGLS1 with the first differential equation system DGSL1 in a plurality VS of simulation steps at a temperature T, and by determining the basis vectors of the orthogonal basis vector system from the plurality VS of simulation results using a singular value decomposition method or a proper orthogonal decomposition method. The plurality VS of simulation steps can be 100 to 100,000, preferably 500 to 10,000, more preferably 1,000 to 5,000.

[0141] In certain variants, the basis vectors BV of the orthogonal basis vector system BVS of the geometric submodel are determined by determining the corresponding geometric state information GZI as a simulation result for the geometric submodel GDGLS1 with the first differential equation system DGSL1 in a plurality VS of simulation steps, each for a given (theoretical or synthetic) thermal expansion vector v(T), and from the plurality VS of simulation results, the basis vectors BV of the orthogonal basis vector system BVS are determined by a singular value decomposition method or by a proper orthogonal decomposition method. Here, too, in the simulation steps with a plurality of representative thermal expansion vectors v→(T) with appropriate resolution, from which real thermal expansions can be simulated by superposition. In this case, the results from the relevant reduced second differential equation system DGLS2 can also be superimposed or combined, resulting in a particularly flexible system with which almost any real thermal expansion v can be simulated or mapped. The number VS of simulation steps can be 100 to 100,000, preferably 500 to 10,000, more preferably 1,000 to 5,000.

[0142] The basis vectors BV of the orthogonal basis vector system, in particular their number, can in principle be chosen such that they achieve a good compromise between the reduction of the computing time and the reduction of the simulation accuracy for the respective application, which, adapted to the respective application for microlithography, still allows a control of the aberration compensation in real time.

[0143] In the present example, the basis vectors BV of the orthogonal basis vector system BVS for the thermal submodel TDGLS1 are selected in such a way, in particular the number of basis vectors BV is selected in such a way that an absolute error EA of the in the respective determination step 110.3 (see Fig. 3) the thermal state information TZI determined compared to an actual value of the thermal state information TZIT is at most 0.0005 K to 0.5 K, preferably 0.001 K to 0.01 K, more preferably 0.001 K to 0.005 K. This allows very good results to be achieved in the field of microlithography with regard to the control of the imaging error compensation in real time. Furthermore, the basis vectors BV of the orthogonal basis vector system BVS for the thermal submodel TDGLS1 are selected in such a way, in particular the number of basis vectors is selected such that a relative error ER of the thermal state information TZI determined in the respective determination step is at most 0.01% to 5%, preferably 0.05% to 1%, more preferably 0.1% to 0.5%, of an actual value of the thermal state information TZIT.

[0144] In the present example, the basis vectors BV of the orthogonal basis vector system BVS for the geometric submodel GDGLS are selected in such a way, in particular the number of basis vectors BV is selected such that an absolute error of the geometric state information GZI determined in the respective determination step compared to an actual value of the geometric state information GZIT is at most 2 pm to 40 pm, preferably 4 pm to 30 pm, more preferably 8 pm to 20 pm. Additionally or alternatively, the basis vectors BV of the orthogonal basis vector system BVS for the geometric submodel can be selected in such a way, in particular the number of basis vectors can be selected such that a relative error of the geometric state information GZI determined in the respective determination step is at most 0.01% to 5%, preferably 0.05% to 1%, more preferably 0.1% to 0.5%, of an actual value of the geometric state information GZIT.

[0145] As already explained above, the actual value of the thermal state information TZIT can generally be determined in any suitable manner. For example, it can be determined based on actual measurements on the optical element 108. The actual value of the thermal state information TZIT can be determined particularly easily if the finite element model (FEM) is used, i.e., the first differential equation system DGLS1. In other words, with sufficiently precise modeling of the optical element 108, the value of the geometric state information TZI, which results from the time-consuming conventional solution of the first differential equation system DGLS1, can be used as the actual value of the geometric state information TZIT.

[0146] Depending on the number of basis vectors BV of the orthogonal basis vector system BVS, and thus also the dimension D2 of the second differential equation system DGLS2, significant reductions in the required computing time can be achieved, which are associated with different relative errors ER. For example, for a first differential equation system DGLS1 of the finite element model FEM with a first dimension D1 of 500,000, an orthogonal basis vector system BVS with MBV = 10 basis vectors can achieve a calculation acceleration by a factor of approximately 18,000 with a relative error of approximately ER = 0.2%. With the same basic modeling, an orthogonal basis vector system BVS with MBV = 91 basis vectors can achieve a calculation acceleration by a factor of 100 to approximately 2.000 with a relative error ER = 0.05%, while with an orthogonal basis vector system BVS with MBV = 278 basis vectors BV a calculation acceleration by a factor of more than 100, typically about 200, with a relative error ER = 0.01% can be achieved.

[0147] In particularly favorable variants, the MBV of basis vectors is 100 to 20,000, preferably 200 to 10,000, and more preferably 1,000 to 4,000, since this allows a corresponding reduction in dimensionality with significantly reduced computing times while maintaining sufficient simulation accuracy. For example, using an orthogonal basis vector system with a number of MBV = 80 to 100 basis vectors can accelerate the required computing time by a factor of 1,500 to 2,500. In general, a smaller number of basis vectors results in a greater reduction in computing time, but this also results in a reduction in simulation accuracy.

[0148] In a concrete example, the system described above, compared to a conventional high-dimensional FEM model (i.e., with a dimension D1), achieved an error of less than 20 pm for the low Zernicke polynomials (Z2 to Z15) and an error of less than 8 pm for the high Zernicke polynomials (Z16 to Z49) for an ensemble of transient calculations from different illumination settings and different variations of the coefficients of the nonlinear material law. According to current knowledge, this is sufficient for the use of the described system or method for a feedforward strategy in microlithography.

[0149] The present invention has been described above exclusively using the example of one optical element (mirror 108 or M3). However, it is understood that the invention can be applied in conjunction with multiple optical elements, in particular with all optical elements of the imaging device 101. In this case, the modeling can be limited not only to the optical elements, but also extend to adjacent components up to the entire imaging device 101, which have a non-negligible effect on thermally induced aberrations.

[0150] The present invention has been described above exclusively using examples from the field of microlithography, particularly in the EUV range. However, it is understood that the invention can also be used in connection with any other optical applications, in particular imaging processes at other wavelengths, for example, in the DUV (deep UV) range, where similar problems arise regarding the compensation of thermally induced aberrations.

[0151] Furthermore, the invention can be used in connection with the inspection of objects, such as the so-called mask inspection, in which the masks used for microlithography are examined for their integrity, etc. Instead of the wafer 105.1, Fig.1, for example, a sensor unit that captures the image of the projection pattern of reticle 104.1 (for further processing). This mask inspection can then be performed at essentially the same wavelength used in the subsequent microlithography process. However, any deviating wavelengths can also be used for the inspection.

[0152] The present invention has been described above with reference to specific embodiments, which show specific combinations of the features defined in the following patent claims. It should be expressly noted at this point that the subject matter of the present invention is not limited to these combinations of features, but rather all other combinations of features as they emerge from the following patent claims also belong to the subject matter of the present invention.* * * * * QUOTES CONTAINED IN THE DESCRIPTION

[0000] This list of documents submitted by the applicant was generated automatically and is included solely for the convenience of the reader. This list is not part of the German patent or utility model application. The DPMA assumes no liability for any errors or omissions. Cited patent literature

[0000] DE 10 2008 009 600 A1 [0071, 0074] US 2006 / 0132747 A1

[0073] EP 1 614 008 B1

[0073] US 6,573,978

[0073] US 2018 / 0074303 A1

[0083] Cited non-patent literature

[0000] Computing Approximate Fekete Points by QR Factorizations of Vandermonde Matrices” (A. Sommariva, M. Vianello in Computers and Mathematics with Applications 57 (2009), pages 1324 to 1336

[0042] Computing Approximate Fekete Points by QR Factorizations of Vandermonde Matrices“ (A. Sommariva, M. Vianello in Computers and Mathematics with Applications 57 (2009), Seite 1324 bis 1336, Elsevir Ltd

[0129]

Claims

[1] Control arrangement for an optical imaging device for microlithography, in particular for the use of light in the extreme UV range (EUV), with - a determination device (106.1) for determining at least one geometric state information item for at least one optical element (108) of the imaging device (101), wherein - the at least one geometric state information is representative of a geometric state of the at least one optical element (108) and - the at least one geometric state information item is designed to be used in determining aberration information that is representative of at least one optical aberration of the optical imaging device (101), characterized by , that - the determination device (106.1) is designed to determine at least one thermal state information item that is representative of a temperature distribution in at least a first element part (108.4) of the optical element (108), wherein the first element part (108.4) is the part of the optical element (108) that has a non-negligible influence on the at least one optical imaging error caused by thermally induced deformation during normal operation of the optical imaging device (101), - the determining device (106.1) is designed to determine at least one piece of material state information that is representative of a distribution of at least one material property in at least the first element part (108.4), wherein the at least one material property has a non-negligible influence on the thermally induced deformation of the first element part (108.4), - the determination device (106.1) is designed to determine the at least one piece of geometric state information using the at least one piece of thermal state information and the at least one piece of material state information, - the determination device (106.1) is designed to determine the at least one geometric state information item in a plurality of successive determination steps using a finite element model of the at least one optical element (108) in each case with a determination accuracy sufficient for a real-time correction of the at least one optical aberration and to output it with a possible output frequency AF, - the finite element model has a plurality of nodes (109) and a mean node spacing MKA, wherein the mean node spacing is 0.1 mm to 30 mm, preferably 0.5 mm to 25 mm, more preferably 2 mm to 10 mm, and - the determination device (106.1) is designed to apply at least one complexity reduction method to the finite element model such that the output frequency AF of the geometric state information is 0.1 Hz to 1,000 Hz, preferably 10 Hz to 200 Hz, more preferably 50 Hz to 100 Hz. [2] Control arrangement according to claim 1, wherein at least one of the following applies: - the determination device (106.1) is designed to determine the at least one thermal state information item using a thermal submodel of the finite element model, wherein the determination device (106.1) is designed in particular to apply the at least one complexity reduction method to the thermal submodel; - the determination device (106.1) is designed to determine the at least one piece of material state information using a material partial model of the finite element model, wherein the determination device (106.1) is designed in particular to apply the at least one complexity reduction method to the material partial model; - the determination device (106.1) is designed to determine the at least one item of geometric state information using a geometric partial model of the finite element model, wherein the determination device (106.1) is designed in particular to apply the at least one complexity reduction method to the geometric partial model. [3] Control arrangement according to claim 1 or 2, wherein - the complexity reduction method is a reduced basis method, - in the complexity reduction method, a plurality of MBVs of basis vectors of at least one orthogonal basis vector system are applied to a first differential equation system of the finite element model in such a way that at least one reduced second differential equation system is obtained with a second dimension D2 which is reduced compared to a first dimension D1 of the first differential equation system of the finite element model, in particular at least one of the following applies: - the basis vectors are multiplied from the right with the first differential equation system of the finite element model to create the at least reduced second differential equation system; - the plurality MBV of basic vectors is 100 to 20,000, preferably 200 to 10,000, more preferably 1,000 to 4,000. [4] Control arrangement according to claim 3, wherein at least one of the following applies: - the first dimension D1 of the first differential equation system of the finite element model is 10,000 to 2,000,000, preferably 50,000 to 1,000,000, more preferably 200,000 to 500,000; - the second dimension D2 of the at least one reduced second differential equation system is 100 to 20,000, preferably 200 to 10,000, more preferably 1,000 to 4,000. [5] Control arrangement according to claim 3 or 4, wherein at least one of the following applies: - the basis vectors of the at least one orthogonal basis vector system are selected at least for the thermal submodel such that the temperature vector dependent on time t T→(t) for the first element part (108.4) can be represented by the basis vectors, so that: T→(t)=∑n=1kcn(t)Ψ→n with Ψ→nMΨ→m=δnm where: Temperature vector for the first element part (108.4); t time; c n (t) development coefficient n; Basis vectors n, m; M mass matrix of the finite element model for the first element part (108.4); δ nm Kronecker Delta; - the basis vectors of the at least one orthogonal basis vector system are selected at least for the thermal submodel such that the heat load vector dependent on time t Q→(t) for the first element part (108.4) can be represented by the basis vectors, so that: Q→(t)=∑n=1kdn(t)Ψ→n with Ψ→nMΨ→m=δnm where: Heat load vector for the first element part (108.4); t time; d n (t) development coefficient n; Basis vectors n, m; M mass matrix of the finite element model for the first element part (108.4); δ nm Kronecker Delta. [6] Control arrangement according to one of claims 3 to 5, wherein at least one of the following applies: - the basis vectors of the at least one orthogonal basis vector system are selected at least for the geometric partial model such that the displacement vector dependent on the thermal expansion v u→(v) for the first element part (108.4) can be represented by the basis vectors, so that: u→(v)=∑n=1dcn(v)Ψ→n with Ψ→nSΨ→m=δnm where: Displacement vector for the first element part (108.4); v thermal expansion; c n (t) development coefficient n; Basis vectors n, m; S stiffness matrix of the finite element model for the first element part (108.4); δ nm Kronecker Delta; - the basis vectors of the at least one orthogonal basis vector system are selected at least for the geometric submodel such that the thermal expansion vector dependent on the temperature T v→(T) for the first element part (108.4) can be represented by the basis vectors, so that: v→(T)=∑n=1ddn(T)Ψ→n with Ψ→nSΨ→m=δnm where: Thermal expansion vector for the first element part (108.4); T temperature; d n (T) development coefficient n; Basis vectors n, m; S stiffness matrix of the finite element model for the first element part (108.4); δ nm Kronecker Delta. [7] Control arrangement according to one of claims 3 to 6, wherein the material submodel is a part of the geometric submodel and at least one of the following applies: - the basis vectors of the at least one orthogonal basis vector system are selected at least for the material submodel such that the material property α(T) dependent on the temperature T for the first element part (108.4) can be mapped via the basis vectors of the geometric submodel, so that: α(T)=α(T(x,y,z);x;y;z) where: α(T) material property, in particular a thermal expansion coefficient, for the first element part (108.4); T(x,y,z) temperature at the point of the first element part (108.4); with the coordinates (x, y, z); - the thermal expansions V of the at least one orthogonal basis vector system for the material part model are determined starting from a complete basis of the thermal expansions V of the finite element model by determining a Fekete basis and the relevant modes of the Fekete basis, wherein the basis of the thermal expansions V was orthogonalized and normalized, wherein for the complexity reduction method the modes of the basis of the thermal expansions V were sorted by a QR decomposition with pivoting in order to determine the at least one reduced second differential equation system with the second dimension D2 as the reduced basis of the thermal expansions V. [8] Control arrangement according to claim 7, wherein at least one of the following applies - the determination device (106.1) is designed to transfer the results of the solution of the reduced second differential equation system of the thermal submodel into the reduced second differential equation system of the material submodel in order to determine the at least one item of geometric state information using first operators; - the determination device (106.1) is designed to integrate the results of the solution of the reduced second differential equation system of the material submodel into the reduced second differential equation system of the geometric submodel in order to determine the at least one item of geometric state information using second operators. [9] Control arrangement according to one of claims 3 to 8, wherein the determining device (106.1) is designed to, in each of the determining steps: - to determine a solution of the reduced second system of differential equations and - to transfer the determined solution of the reduced second differential equation system back into the dimensional space of the first differential equation system in order to obtain the geometric state information. [10] Control arrangement according to one of claims 3 to 9, wherein the basis vectors of the orthogonal basis vector system were determined by - for the thermal submodel with the first differential equation system in a plurality VS of simulation steps at a heat load Q, the corresponding thermal state information is determined as a simulation result, and - the basis vectors of the orthogonal basis vector system were determined from the plurality VS of simulation results by a singular value decomposition method or by a proper orthogonal decomposition method, wherein in particular at least one of the following applies: - the plurality VS of simulation steps is 100 to 100,000, preferably 500 to 10,000, more preferably 1,000 to 5,000; - training is carried out in the plurality VS of simulation steps with different individual heat load scenarios from which real heat loads can be simulated by superposition, wherein the plurality VS of simulation steps is in particular 50 to 10,000, preferably 500 to 2,000; - the heat load Q in the respective simulation step is 0.2 W to 2 W, preferably 0.5 W to 1.5 W, more preferably 0.8 W to 1.0 W. [11] Control arrangement according to one of claims 3 to 10, wherein the basis vectors of the orthogonal basis vector system were determined by - for the geometric submodel with the first differential equation system, the associated geometric state information is determined as a simulation result in a plurality VS of simulation steps at a temperature T, and the basis vectors of the orthogonal basis vector system were determined from the plurality VS of simulation results by a singular value decomposition method or by a proper orthogonal decomposition method; in particular - the plurality VS of simulation steps is 100 to 100,000, preferably 500 to 10,000, more preferably 1,000 to 5,000. [12] Control arrangement according to one of claims 1 to 11, wherein at least one of the following applies: - the basis vectors of the orthogonal basis vector system are selected for the thermal submodel in such a way, in particular the number of basis vectors is selected in such a way that an absolute error of the thermal state information determined in the respective determination step compared to an actual value of the thermal state information is at most 0.0005 K to 0.5 K, preferably 0.001 K to 0.01 K, more preferably 0.001 K to 0.005 K; - the basis vectors of the orthogonal basis vector system are selected for the thermal submodel in such a way, in particular the number of basis vectors is selected in such a way that a relative error of the thermal state information determined in the respective determination step is at most 0.01% to 5%, preferably 0.05% to 1%, more preferably 0.1% to 0.5%, of an actual value of the thermal state information. [13] Control arrangement according to one of claims 1 to 12, wherein at least one of the following applies: - the basis vectors of the orthogonal basis vector system are selected for the geometric submodel in such a way, in particular the number of basis vectors is selected in such a way that an absolute error of the geometric state information determined in the respective determination step compared to an actual value of the geometric state information is at most 2 pm to 40 pm, preferably 4 pm to 30 pm, more preferably 8 pm to 20 pm; - the basis vectors of the orthogonal basis vector system are selected for the geometric partial model in such a way, in particular the number of basis vectors is selected in such a way that a relative error of the geometric state information determined in the respective determination step is at most 0.01% to 5%, preferably 0.05% to 1%, more preferably 0.1% to 0.5%, of an actual value of the geometric state information. [14] Optical imaging device, in particular for microlithography, with - a lighting device (102) with a first optical element group (102.2), - an object device (103) for receiving an object (103.3), - a projection device (104) with a second optical element group (104.1) and - an image device (105), wherein - the illumination device (102) is designed to illuminate the object (103.3) and - the projection device (104) is designed to project an image of the object (103.3) onto the image device (105), characterized by , that - the illumination device (102) and / or the projection device (104) is controlled in normal operation by at least one control device (106) according to one of claims 1 to 13 in order to correct at least one imaging error of the imaging device (101) using the determined thermal state information. [15] Method for controlling an optical imaging device for microlithography, in particular for the use of light in the extreme UV range (EUV), in which - at least one geometric state information item is determined for at least one optical element (108) of the imaging device (101), wherein - the at least one geometric state information is representative of a geometric state of the at least one optical element (108) and - the at least one geometric state information item is designed to be used in determining aberration information that is representative of at least one optical aberration of the optical imaging device (101), characterized by , that - at least one thermal state information item is determined which is representative of a temperature distribution in at least a first element part (108.4) of the optical element (108), wherein the first element part (108.4) is the part of the optical element (108) which, during normal operation of the optical imaging device (101), has a non-negligible influence on the at least one optical imaging error caused by thermally induced deformation, - at least one piece of material state information is determined which is representative of a distribution of at least one material property in at least the first element part (108.4), wherein the at least one material property has a non-negligible influence on the thermally induced deformation of the first element part (108.4), - using the at least one piece of thermal state information and the at least one piece of material state information, the at least one piece of geometric state information is determined, wherein - the at least one geometric state information item is determined in a plurality of successive determination steps using a finite element model of the at least one optical element (108), each with a determination accuracy sufficient for real-time correction of the at least one optical aberration, and is output at an output frequency AF, - the finite element model has a plurality of nodes and a mean node spacing MKA, wherein the mean node spacing is 0.1 mm to 30 mm, preferably 0.5 mm to 25 mm, more preferably 2 mm to 10 mm, and - at least one complexity reduction method is applied to the finite element model such that the output frequency AF of the geometric state information is 0.1 Hz to 1,000 Hz, preferably 10 Hz to 200 Hz, more preferably 50 Hz to 100 Hz. [16] Optical imaging method, especially for microlithography, in which - an illumination device (102) having a first optical element group (102.2), illuminating an object (103.3) and - a projection device (104) having a second optical element group (104.1) projects an image of the object (103.3) onto an image device (105), characterized by , that - the illumination device (102) and / or the projection device (104) is controlled in normal operation by at least one method according to claim 15 in order to correct at least one imaging error of the imaging device (101) using the determined thermal state information.

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