Optical member thermal conditioning method

Abstract

Method to condition an optical member in an initial thermal state to a predetermined final thermal state, the method comprising: - establishing a finite element thermal model of the optical member, - determining a thermal state error representing a deviation of a thermal state vector representing a temperature distribution within the optical member from a desired thermal state at a given timestep, - establishing a cost function comprising the thermal state error, performing an optimisation operation on the cost function, wherein a thermal input of the one or more thermal actuators is optimised to minimise a value of the cost function, wherein the optimisation operation results in an optimal control signal to - control the one or more thermal actuators to achieve an optimal thermal conditioning process, - controlling the one or more thermal actuators according to the optimal control signal to thermally condition the optical member.

Description

OPTICAL MEMBER THERMAL CONDITIONING METHOD CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority of EP application 24190033.1 which was filed on July 22, 2024 and EP application 24194916.3 which was filed on August 16, 2024 and EP application 24210549.2 which was filed on November 4, 2024 and which are incorporated herein in its entirety by reference. FIELD

[0002] The present invention relates to a method for thermally conditioning an optical member. The present invention furthermore relates to a computer program having instructions which when executed by an information processing device causes the information processing device to perform a method for thermally conditioning an optical member. The present invention furthermore relates to a lithographic apparatus which is configured to thermally condition an optical member thereof. BACKGROUND

[0003] Light generated by means of a radiation source can be used by exposure apparatuses for semiconductor manufacturing processes. Examples of such exposure apparatuses are a lithographic apparatus, a metrology, or an inspection apparatus, more specifically a mask inspection apparatus and even more specifically an actinic mask inspection apparatus.

[0004] A lithographic apparatus is a machine constructed to apply a desired pattern onto a substrate. A lithographic apparatus can be used, for example, in the manufacture of integrated circuits (ICs). A lithographic apparatus may, for example, project a pattern at a patterning device (e.g., a mask) onto a layer of radiation-sensitive material (e.g., a photoresist or resist) provided on a substrate. To project a pattern on a substrate a lithographic apparatus may use electromagnetic radiation. The wavelength of this radiation determines the minimum size of features which can be formed on the substrate. A lithographic apparatus, which uses EUV radiation, having a wavelength within the range 4-20 nm, for example 6.7 nm or 13.5 nm, may be used to form smaller features on a substrate than a lithographic apparatus which uses, for example, radiation with a wavelength of 193 nm.

[0005] A lithographic apparatus comprises optical members such as lenses or reflective surfaces (e.g. mirrors) to guide the radiation to e.g. the substrate. Any deviation or variation in the shape of these optical members may affect the interaction of the radiation with the optical members, potentially altering the path of the radiation. This may result in a beam of radiation not being properly focused, yielding lower performance of the lithographic apparatus. Such quality decreases could result in improper patterns projected on the substrate, which may yield substandard or even defective end products.

[0006] In order to control the optical member shape, the temperature of the optical members is controlled during operation of the lithographic device. This is done by monitoring, heating and / or cooling of the optical members.

[0007] The optical members may be made from Ultra-Low Expansion (ULE) material to minimize optical aberrations that result from thermal drift caused by the absorption of EUV light during lot production. ULE exhibits a nonlinear temperature-strain relation, and as a result it has a low thermal sensitivity around the zero-crossing temperature (ZCT, i.e., the temperature at which the gradient of the coefficient of thermal expansion is zero). To achieve minimal aberrations during lot production, thermal actuators are used to keep the mirror temperature close to the zero-crossing temperature when the lithographic apparatus is in a production operational state. Before lot production is possible, thermal conditioning is required to bring the mirrors up to the desired operating temperature. Such thermal conditioning may also be referred to as ‘recovery’ or ‘thermal recovery’, referring to recovering the desired or optimal temperature distribution and / or thermal state of the optical member.

[0008] Typically, when the lithographic device is started after being inoperative, the optical members are at a temperature below their optimal operating temperature. The optical members may also stray from their optimal operating due to other reasons, for example drift during lot production from prolonged exposure to an environment and / or exposure to radiation used to project the pattern on the substrate. Whatever the cause of the optical members not being at their optimal operating temperature, it is desirable to bring the optical members to their optimal operating temperature reliably and quickly. The optical members are thermally conditioned to bring them to a temperature for which the lithographic device is calibrated, where the optical members are not or almost not deformed due to temperature deviations when compared to the preferred operating state.

[0009] To achieve this thermal conditioning, thermal actuators such as heating devices and / or cooling devices may be used to bring the optical members to the desired temperature. This desired temperature may be constant over the whole optical member, but it may also vary. A device that performs such a conditioning method is disclosed in WO2022 / 002519A1.

[0010] Usually, the thermal actuators are heating devices and the desired temperature of the optical members is above the ambient temperature. In these cases, thermal conditioning is achieved by providing a heat input to the optical members from the heating devices. Alternatively, the thermal actuators may be cooling devices, for example when the desired temperature of the optical members is below the ambient temperature. A combination of heating devices and cooling devices may also be used, for example to provide maximum flexibility.

[0011] Thermal conditioning may be achieved in one of two ways. The first is by applying a thermal input from the thermal actuators which corresponds to the steady state thermal input required by the thermal actuators. The temperature distribution within the optical member will then in time converge towards the steady state preferred temperature distribution. This takes a relatively long time, which may result in significant machine downtime. The thermal input may be a heating input if the thermalactuators comprise heating members and / or may be a cooling input, i.e. a removal of heat from the optical member, if the thermal actuators comprise cooling members.

[0012] In order to reduce this machine downtime, a second conditioning method may be used wherein the thermal input from the heating members is controlled by a feedback system, where the temperature is measured on one or more measurement positions on the optical member and the thermal input is controlled to keep the temperature at the measurement positions at the preferred temperature corresponding to the preferred temperature distribution. In practice, this means that more thermal power is used, in particular at the beginning of the conditioning process. This results in a lower time required for the thermal conditioning process, yielding a lower machine downtime.

[0013] In both methods, the temperature distribution within the optical member gradually converges towards the preferred temperature distribution. As a condition to determine when the conditioning process is finished, either a maximum allowable time for the conditioning process is chosen, or a maximum allowable deviation from the preferred temperature distribution is chosen. Once the specified condition is met, the conditioning process is finished and the lithographic apparatus can commence further operations.

[0014] It is desired to reduce the time required for the conditioning process to reduce downtime of the lithographic apparatus. It is furthermore desired to provide better thermal conditioning, with a lower deviation from the preferred temperature distribution at the end of the thermal conditioning process. SUMMARY

[0015] It is an object of the invention to provide an optical member thermal conditioning method. It is a further object of the invention to provide a computer program having instructions which when executed by an information processing device causes the information processing device to perform a method for thermally conditioning an optical member. It is a further object of the invention to provide a lithographic apparatus which is configured to thermally condition an optical member thereof.

[0016] In a first aspect of the invention, the invention provides a method according to claim 1.

[0017] In an embodiment of the first aspect of the invention, the invention provides a method according to claim 2.

[0018] In an embodiment of the first aspect of the invention, the invention provides a method according to claim 3.

[0019] In an embodiment of the first aspect of the invention, the invention provides a method according to claim 4.

[0020] In an embodiment of the first aspect of the invention, the invention provides a method according to claim 5.

[0021] In an embodiment of the first aspect of the invention, the invention provides a method according to claim 6.

[0022] In an embodiment of the first aspect of the invention, the invention provides a method according to claim 7.

[0023] In an embodiment of the first aspect of the invention, the invention provides a method according to claim 8.

[0024] In an embodiment of the first aspect of the invention, the invention provides a method according to claim 9.

[0025] In an embodiment of the first aspect of the invention, the invention provides a method according to claim 10.

[0026] In an embodiment of the first aspect of the invention, the invention provides a method according to claim 11.

[0027] In an embodiment of the first aspect of the invention, the invention provides a method according to claim 12.

[0028] In an embodiment of the first aspect of the invention, the invention provides a method according to claim 13.

[0029] In an embodiment of the first aspect of the invention, the invention provides a method according to claim 14.

[0030] In an embodiment of the first aspect of the invention, the invention provides a method according to claim 15.

[0031] In an embodiment of the first aspect of the invention, the invention provides a method according to claim 16.

[0032] In an embodiment of the first aspect of the invention, the invention provides a method according to claim 17.

[0033] In an embodiment of the first aspect of the invention, the invention provides a method according to claim 18.

[0034] In an embodiment of the first aspect of the invention, the invention provides a method according to claim 19.

[0035] In an embodiment of the first aspect of the invention, the invention provides a method according to claim 20.

[0036] In an embodiment of the first aspect of the invention, the invention provides a method according to claim 21.

[0037] In an embodiment of the first aspect of the invention, the invention provides a method according to claim 22.

[0038] In an embodiment of the first aspect of the invention, the invention provides a method according to claim 23.

[0039] In an embodiment of the first aspect of the invention, the invention provides a method according to claim 24.

[0040] In an embodiment of the first aspect of the invention, the invention provides a method according to claim 25.

[0041] In an embodiment of the first aspect of the invention, the invention provides a method according to claim 26.

[0042] In a second aspect of the invention, the invention provides a computer program according to claim 27.

[0043] In a third aspect of the invention, the invention provides a lithographic apparatus according to claim 28. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] Embodiments of the invention will now be described, by way of example only, with reference to the accompanying schematic drawings, in which: - Figure 1 schematically depicts a lithographic system comprising a lithographic apparatus and a radiation source; - Figure 2 schematically depicts an embodiment of a projection optics box according to an embodiment of the invention; and detailed view of a part of the lithographic apparatus of Figure 1; - Figure 3 shows a graph of the thermal state error with respect to time for three methods, including two prior art methods (the first two methods) and the method according to the invention (the third method); - Figure 4 shows a graph of the power of all thermal actuators combined with respect to time for the three methods of figure 3; - Figure 5 shows a graph of the temperature with respect to time of a number of nodes for the third method of figure 3; - Figure 6 shows a graph of the thermal state error with respect to time for three methods, including two prior art methods (the first and second method) and one methods according to an embodiment of the invention (the third method); - Figure 7 shows a graph of the wavefront error with respect to time for the three methods of figure 6; - Figure 8 shows a graph of heater power with respect to time for the thermal actuators, in accordance with Method 3 of figures 6 and 7; and - Figure 9 shows a graph of the temperature with respect to time of a number of nodes for the third method of figures 6 and 7. DETAILED DESCRIPTION

[0045] Figure 1 shows a lithographic system comprising a radiation source SO and a lithographic apparatus LA. The radiation source SO is configured to generate an EUV radiation beam B and tosupply the EUV radiation beam B to the lithographic apparatus LA. The lithographic apparatus LA comprises an illumination system IL, a support structure MT configured to support a patterning device MA (e.g., a mask), a projection system PS and a substrate table WT configured to support a substrate W.

[0046] The illumination system IL is configured to condition the EUV radiation beam B before the EUV radiation beam B is incident upon the patterning device MA. Thereto, the illumination system IL may include a facetted field mirror device 10 and a facetted pupil mirror device 11. The faceted field mirror device 10 and faceted pupil mirror device 11 together provide the EUV radiation beam B with a desired cross-sectional shape and a desired intensity distribution. The illumination system IL may include other mirrors or devices in addition to, or instead of, the faceted field mirror device 10 and faceted pupil mirror device 11.

[0047] After being thus conditioned, the EUV radiation beam B interacts with the patterning device MA. As a result of this interaction, a patterned EUV radiation beam B’ is generated. The projection system PS is configured to project the patterned EUV radiation beam B’ onto the substrate W. For that purpose, the projection system PS may comprise a plurality of mirrors 13,14 which are configured to project the patterned EUV radiation beam B’ onto the substrate W held by the substrate table WT. The projection system PS may apply a reduction factor to the patterned EUV radiation beam B’, thus forming an image with features that are smaller than corresponding features on the patterning device MA. For example, a reduction factor of 4 or 8 may be applied. Although the projection system PS is illustrated as having only two mirrors 13,14 in Figure 1, the projection system PS may include a different number of mirrors (e.g., six or eight mirrors).

[0048] The substrate W may include previously formed patterns. Where this is the case, the lithographic apparatus LA aligns the image, formed by the patterned EUV radiation beam B’, with a pattern previously formed on the substrate W.

[0049] A relative vacuum, i.e. a small amount of gas (e.g. hydrogen) at a pressure well below atmospheric pressure, may be provided in the radiation source SO, in the illumination system IL, and / or in the projection system PS.

[0050] The lithographic apparatus LA and radiation source SO described herein can be used in method for performing a circuit layout patterning process. A circuit layout patterning method comprises receiving a substrate with a photoresist layer. The method further comprises directing EUV radiation from radiation source to the photoresist layer to form a patterned photoresist layer. The method further comprises developing and etching the patterned photoresist layer to form a circuit layout.

[0051] The radiation source SO shown in Figure 1 is, for example, of a type which may be referred to as a laser produced plasma (LPP) source. A laser system 1, which may, for example, include a CO2 laser, is arranged to deposit energy via a laser beam 2 into a fuel (i.e., a target material), such as tin (Sn) which is provided from, e.g., a fuel generator 3. Although tin is referred to in the following description, any suitable fuel may be used. The fuel may, for example, be in liquid form, and may, for example, bea metal or alloy. The fuel generator 3 may comprise a nozzle configured to direct the fuel, e.g. in the form of droplets, along a trajectory towards a plasma formation region 4. The laser beam 2 is incident upon the fuel at the plasma formation region 4. The deposition of laser energy into the tin creates a plasma 7 at the plasma formation region 4. Radiation, including EUV radiation, is emitted from the plasma 7 during de-excitation and recombination of electrons with ions of the plasma 7.

[0052] The EUV radiation from the plasma 7 is collected and focused by a collector 5. Collector 5 comprises, for example, a near-normal incidence radiation collector 5 (sometimes referred to more generally as a normal-incidence radiation collector). The collector 5 may have a multilayer mirror structure which is arranged to reflect EUV radiation (e.g., EUV radiation having a desired wavelength such as 13.5 nm). The collector 5 may have an ellipsoidal configuration, having two focal points. A first one of the focal points may be at the plasma formation region 4, and a second one of the focal points may be at an intermediate focus 6, as discussed below.

[0053] The laser system 1 may be spatially separated from the radiation source SO. Where this is the case, the laser beam 2 may be passed from the laser system 1 to the radiation source SO with the aid of a beam delivery system (not shown) comprising, for example, suitable directing mirrors and / or a beam expander, and / or other optics. The laser system 1, the radiation source SO and the beam delivery system may together be considered to be a radiation system.

[0054] Radiation that is reflected by the collector 5 forms the EUV radiation beam B. The EUV radiation beam B is focused at intermediate focus 6 to form an image at the intermediate focus 6 of the plasma present at the plasma formation region 4. The image at the intermediate focus 6 acts as a virtual radiation source for the illumination system IL. The radiation source SO is arranged such that the intermediate focus 6 is located at or near to an opening 8 in an enclosing structure 9 of the radiation source SO.

[0055] Although Figure 1 depicts the radiation source SO as a laser produced plasma (LPP) source, any suitable source such as a free electron laser (FEL) or a discharge produced plasma (DPP) source may be used to generate EUV radiation.

[0056] Figure 2 depicts schematically a projection optics box POB which is part of a projection system PS of a lithographic apparatus LA as depicted in Figure 1. The projection optics box POB comprises a number of optical elements to guide the radiation beam B from the patterning device to the substrate W. Such optical elements may be mirrors or lenses. In the imaged embodiment, two mirrors are shown in the projection optics box POB.

[0057] The radiation beam B entering the projection optics box POB is reflected by a first mirror 15A, and impinges on a second mirror 15B, in particular a reflective surface thereof on which the radiation beam B is directed. The radiation beam is reflected by the second mirror 15B and leaves the projection optics box POB. In practice the projection optics box POB may comprise more than two mirrors to guide the radiation beam B.

[0058] The lithographic apparatus LA comprises one or more thermal actuators 16 to provide a heat flux to or from the second mirror 15B. In the embodiment shown in figure 2, multiple thermal actuators 16 are held by a thermal assembly 17. In the imaged embodiment, the thermal actuators 16 comprise heating devices which can heat the second mirror 15B. Figures 3 – 5 also relate to embodiments wherein the thermal actuators 16 comprise heating devices which can heat the optical member 15. The first mirror 15A and the second mirror 15B shown in figure 2 are an example of an optical member 15. Two mirrors 15A, 15B are shown in figure 2, but other mirrors or other optical members 15 may also be used. These optical members 15 can be heated or cooled by the thermal actuators 16 which are controlled by a control unit 19. Furthermore, the temperature of the optical member 15 can be measured in one or more positions by one or more temperature sensors. In the embodiments discussed here, the thermal actuators 16 are heating devices that can heat the optical members 15. Alternatively or additionally, cooling elements may be used to thermally condition the optical member 15.

[0059] After a period of inactivity of the lithographic apparatus LA or another circumstance such as after lot production or in between production of two different lots, an optical member 15 may have strayed from its steady state thermal state. The thermal state comprises the temperature distribution within the optical member 15. In order to bring the optical member 15 back to the operating state, the thermal actuators can be used to thermally condition the optical member 15 by supplying heat to the optical member 15 or by removing heat from the optical member 15.

[0060] In the method according to the invention, a finite element thermal model is established of the optical member 15. The thermal model is discretised in space by defining a mesh comprising member nodes. The mesh divides the optical member 15 in a plurality of member elements. Each member element comprises a section of the optical member 15. The member elements are defined by the member nodes, the member nodes are points in space that surround a particular member element. The one or more temperature sensors monitor the temperature of the optical member 15 at one or more nodes. The thermal model is discretised in time by defining a timestep length. The timestep length may be constant, but may also be variable. For example, the timestep length may be decreased when the thermal state error decreases and / or when constraints such as maximum node temperatures are nearly reached.

[0061] The thermal model is configured to describe an evolution over time of a temperature distribution of the optical member 15. Factors that influence this evolution may include internal thermal energy transfer within the optical member 15, thermal interaction between the optical member 15 and an environment 18, and thermal input from the one or more thermal actuators 16. The thermal model includes transfer of heat within the optical member 15 and may additionally include transfer of heat from the optical member 15 to an environment 18 or from the environment 18 to the optical member 15 by e.g. conduction, radiative heat transfer and / or convection from e.g. liquid cooling of the optical member 15.

[0062] The thermal model comprises the following equation:^^^^^ = ^^^ + ^^^ (1)Herein, x represents a thermal state vector, E represents a thermal capacity matrix, A represents a thermal conductance matrix, B represents an irradiance matrix of the thermal actuators 16 and u represents a thermal forcing vector of the one or more thermal actuators 16. The subscripts with the vectors represent the timestep at which they are taken. Thus, as on the right hand side of the equation only subscript k is shown while on the left hand side only subscript k+1 is shown, it is clear that the model uses the situation on timestep k to determine the thermal state on step k+1 (i.e. the second timestep directly following the first timestep k). The input used by the model to determine the thermal state vector xk+1 at the second timestep k+1 is the thermal state vector xk at the first timestep k and the thermal input to the optical member 15 from the one or more thermal actuators 16 during the interval between the first timestep k and the second timestep k+1, incorporated via the thermal forcing vector u.

[0063] The thermal state vector x comprises Nxcomponents, wherein Nxis the number of member nodes of the optical member 15 in the thermal model. The thermal state vector represents the thermal state of the optical member. The thermal state relates to the temperature distribution throughout the optical member 15, specifying the temperature at each of the member nodes. The thermal capacity matrix E has dimensions Nx by Nx. The thermal capacity matrix comprises capacity components with a first capacity index and a second capacity index, wherein the first capacity index and the second capacity index define the position of the associated capacity component in the thermal capacity matrix. The multiplication of the thermal capacity matrix and the thermal state vector x produces a vector having Nx components. This vector is a thermal energy content vector which represents a thermal energy content of the optical member 15 at the member nodes.

[0064] The thermal transfer matrix A represents transfer coefficients of internal thermal energy transfer during one timestep within the optical member 15 at the member nodes and / or thermal interaction during one timestep between the optical member 15 and an environment 18 at the member nodes. The thermal transfer matrix has dimensions Nx by Nx. The thermal transfer matrix comprises transfer components with a first transfer index and a second transfer index, wherein the first transfer index and the second transfer index define the position of the associated transfer component in the thermal transfer matrix. Each transfer component represents a heat transfer coefficient which is representative for heat transfer to a member node with an element index in the thermal state vector which corresponds to the first transfer index from a member node with an element index in the thermal state vector which corresponds to the second transfer index and / or which is representative for thermal interaction between the optical member 15 and the environment 18 at a member node with an element index in the thermal state vector which corresponds to the first transfer index. Additionally, each transfer component may represent transfer of heat from the optical member 15 to an environment 18 or from the environment 18to the optical member 15 by e.g. conduction, radiative heat transfer and / or convection from e.g. liquid cooling of the optical member 15. The thermal transfer matrix A may thus represent evolution of the temperature distribution or thermal state of the optical member 15 without the influence of the one or more thermal actuators 16.

[0065] The multiplication of the thermal transfer matrix A with the thermal state vector x provides a thermal transfer vector which represents an energy transfer related change in the thermal energy content of the optical member 15 at the member nodes due internal thermal energy transfer within the optical member 15 and / or thermal interaction between the optical member 15 and the environment 18 within one timestep. The thermal transfer vector has Nx components.

[0066] The irradiance matrix B represents a thermal input of each of the one or more thermal actuators 16 to the optical member 15 at the member nodes for a unit thermal power amount of each of the one or more thermal actuators 16. The thermal forcing vector represents for each thermal actuator the thermal power. The irradiance matrix has dimensions Nxby Nu, wherein Nuis the number of thermal actuators 16. The thermal forcing vector has Nu components.

[0067] The irradiance matrix comprises irradiance components with a first irradiance index and a second irradiance index, wherein the first irradiance index and the second irradiance index define the position of the associated irradiance component in the irradiance matrix.

[0068] Each irradiance component represents a thermal input for a unit thermal power amount to the optical member 15 at a member node with an element index in the thermal state vector which corresponds to the first irradiance index of said irradiance component of a thermal actuator with a heating index in the thermal forcing vector which corresponds to the second irradiance index of said irradiance component.

[0069] The multiplication of the irradiance matrix with the thermal forcing vector provides a thermal input vector which represents a thermal input to the optical member 15 at the member nodes within one timestep. The thermal input vector has Nxcomponents.

[0070] The multiplications of the thermal capacity matrix E with the thermal state vector x, of the thermal transfer matrix A with the thermal state vector x and of the irradiance matrix B with the thermal forcing vector u are matrix multiplications, each resulting in a vector. These matrix multiplications adhere to the known rules for matrix multiplications, thus if E is a matrix of dimensions m by n and F is a matrix of dimensions n by p: ^^^ ^^^ … ^^^ ^^^ ^^^ … ^^^ ^^^ ^^^ … ^^^^ = ^^^^ ^^^ … ^^^^ , ^ = ^^^^ ^^^ … ^^^^ , ^ = ^^^^ ^^^ … ^^^^ (2)^ = ^^ → ^ = ^ ^ + ^ ^ + ⋯ + ^^ ^^ ∑^^^ ^^ ^^ ^^ ^^ ^ ^ = ^^^ ^^^^^^ (3)

[0071] letter and their indices, i.e. dijis a component of matrix D with its first index equal to i and its second index equal to j. Thus, if a matrix of dimensions Nxby Nxis multiplied with a vector of dimensions Nxby 1, the result is a vector of dimensions Nx by 1, as is the case for the multiplications of the thermal capacity matrix E with the thermal state vector x and of the thermal transfer matrix A with the thermal state vector x. The multiplication of the irradiance matrix B with the thermal forcing vector u also results in a vector of dimensions Nxby 1, as the dimensions of B are Nxby Nuand the dimensions of u are Nuby 1.The equation (1) shown above describes how the thermal state, represented by the thermal state vector x, evolves over time, from one timestep to the next. In the optical member 15 thermal conditioning method according to the invention, a thermal state error is defined on each timestep. This thermal state error represents a deviation of the thermal state vector from a desired thermal state. A cost function is defined which comprises the thermal state error. To find the control input needed to be fed to the thermal actuators in order to optimally thermally condition the optical member 15, a minimisation operation is performed on the cost function.

[0072] The cost function is shown below: ^^^',( ^ ^!, ^"#$ , %& = ^^^',( ^) − ^ +"#$& , ^) − ^"#$& (4)^^ = −Herein, J is the cost function, x0 is the initial thermal state vector of the optical member 15 at the start of the thermal conditioning method, xref is the target thermal state vector which represents an optimally thermally conditioned optical member 15, xNis the thermal state vector at timestep N, capital X is the collection of thermal state vectors from timestep 1 to N, capital U is the collection of thermal forcing vectors from timestep 0 to N-1, and P is a weighing matrix. xN-xrefis an error vector which represents the error of the thermal state with respect to the target thermal state at timestep N. (xN-xref)Tis the transposed error vector. Timestep N is the final timestep of the thermal conditioning process. The cost function thus only depends on the thermal state error at the end of the thermal conditioning, not on the error throughout the thermal conditioning process. The thermal model as summarised in equation (1) computes xNand may compute xkfor k<N as well.

[0073] Capital . and capital 3 define constraints for the thermal state vector x and the thermal forcing vector u. These constraints may optionally be different per time step. Capital . defines a maximum and / or minimum thermal power of the one or more thermal actuators 16. This may result from a physicallimitation, as for example the thermal actuators 16 may have a maximum power setting and may not be able to cool (i.e. provide negative heat).

[0074] Capital 3 defines the maximum temperatures of the optical member 15 at one or more nodes of the mesh, i.e. the maximum values one or more the elements of the thermal state vector x. This maximum temperature is above the desired temperature. This maximum temperature may be chosen to prevent damage to the optical member 15 from overheating. Capital 3 may define the maximum value of every element of the thermal state vector x, but may instead also define the maximum value of a subset of the elements of the thermal state vector x, representing the most critical nodes of the mesh of the optical member 15. This may considerably reduce the computational resources required for the optimisation of the cost function. The maximum temperature does not need to be specified at nodes where the temperature is monitored by a temperature sensor, because the temperature distribution throughout the optical member 15 at each timestep is calculated by the thermal model in advance, before starting the actual heating of the optical member 15 via the one or more thermal actuators 16.

[0075] The weighing matrix P is optional. The weighing matrix may for example be used to provide a higher contribution to the thermal state error of temperature deviations from the reference temperature at some nodes of the mesh of the optical member 15 as compared to other nodes. This may be beneficial if temperature deviations at some nodes have a bigger influence on the way electromagnetic radiation of the radiation beam B interacts with the optical member 15 than temperature deviations at other nodes.

[0076] In the optimisation operation, optimal values are determined for the thermal forcing vector for each of the timesteps. As a further constraint, a timestep amount of allowable timesteps or an error threshold may be fixed. These further constraints may for example be fixed by a user, an operator, or by an algorithm. In the former case, the optimisation operation comprises optimising the thermal input of the one or more thermal actuators 16 to achieve the smallest possible thermal state error after the timestep amount of timesteps has passed. In the latter case, the optimisation operation comprises optimising the thermal input of the one or more thermal actuators 16 to achieve the lowest possible number of timesteps required until the thermal state error falls below the error threshold.

[0077] The method may be a computer implemented method. A computer program may have instructions which when executed by an information processing device 22 causes the information processing device 22 to perform the method. The optimisation operation may be formulated as a compact quadratic programming problem that can efficiently be solved, for example, by using quadprog in MATLAB. Other programs such as Python may also be used. Other formulations for the optimisation problem may also be possible.

[0078] The method may be applied in a lithographic apparatus comprising one or more thermal actuators 16, one or more temperature sensors for sensing a temperature of the optical member 15, a control unit 19 to control the one or more thermal actuators 16 based on a control signal, and an information processing device 22 comprising a processor 23 and a memory 24. The informationprocessing device 22 is configured to determine the optimal control signal based on the optimisation and to communicate the optimal control signal to the control unit. Then, the control unit 19 can control the one or more thermal actuators 16 to provide a time dependent thermal input to the optical member 15 to optimally condition the optical member 15. In figure 2, the control unit 19 and the information processing device 22 comprising the processor 23 and the memory 24 are integrated into a single unit. In other embodiments, these components may be split up. The control unit 19 and the information processing device 22 communicate with the lithographic apparatus via the dotted lines representing communication lines 20, 21, these communication lines 20, 21 comprising a temperature communication line 20 for receiving the temperature measurements from the temperature sensors 57 and a control communication line 21 for sending control instructions to the thermal actuators 16. The temperature communication line 20 and the control communication line 21 may be physically wired connections, or may alternatively be wireless connections. The control unit 19 and / or the information processing device 22 may be positioned at or near the lithographic apparatus LA, but one or both may alternatively be placed remote from the lithographic apparatus LA. The control unit 19 and the information processing device 22 do not need to be placed near each other. The information processing device 22 may comprise a computer such as a personal computer or a supercomputer. The information processing device 22 may additionally or alternatively comprise a cloud computing environment.

[0079] Figure 3 shows a comparison of the error in the thermal state over time of an optical member 15 conditioned via three different conditioning methods. The graph shown in figure 3 is generated from a simulation study of a thermal conditioning process of an optical member. The same stimulation study has been used to generate the graphs shown in figures 4 and 5, such that they can be properly compared. In this simulation study, in the conditioning method according to the invention the maximum allowable time for the thermal conditioning process was set at 80 minutes.

[0080] In the first method, the one or more thermal actuators 16 are controlled to continuously apply or remove the same heat per unit time as they would when the system is in steady state. The error then slowly approaches the error threshold. In the second method, the thermal actuators 16 are controlled by monitoring the temperature on one or more positions on the optical member 15 and controlling the one or more thermal actuators 16 to maintain the monitored temperatures at the desired temperature. The graph shows that the error approaches the error threshold at a faster rate than in the first method, especially at the beginning. This is because the one or more thermal actuators 16 can be controlled to provide more thermal input to the optical member 15 than in the first method, resulting in an overall faster heating and thus thermal conditioning of the optical member 15. However, this method still limits thermal input to the optical member 15 as this method aims to maintain the monitored temperatures at or below the desired temperature at any point during thermal conditioning. The monitored temperatures may overshoot the desired temperature during thermal conditioning, but if this happens the power of the thermal actuators 16 will be reduced to bring the monitored temperature back to the desired temperature. Both the first method and the second method are known from prior art.

[0081] Figure 3 also shows the evolution of the thermal state error over time of the conditioning method according to the invention. What is shown, is that the thermal state error initially increases, only to fall rapidly afterwards, resulting in a much lower time required to reach the error threshold and / or a much lower thermal state error after the allowed time has passed. This is achieved by exploiting more freedom during the thermal conditioning process by allowing the thermal state error to increase during the thermal recovery process, which is not possible in the first and second methods. While the second method takes about 190 minutes to reach the error threshold, the third method takes 80 minutes. The first method does not even reach the error threshold in the timeframe shown here.

[0082] An intuitive explanation for the fact that the first conditioning method performs worst while the conditioning method according to the invention performs best, can be found when considering that in order to thermally condition the optical member 15, heat energy must be put into or removed from the optical member 15, as its temperature must be raised. If the heat energy can be put into or removed from the optical member 15 at a faster rate, less time is required. Thus, the first method performs worst, because it uses the lowest average thermal input rate as the power of the one or more thermal actuators 16 is kept at the power needed in the steady state situation. The second method performs better, because that method allows the thermal actuators 16 to use more power when compared to the first method. The third method, i.e. the method according to the invention, performs best, because it allows the largest thermal input rate, i.e. the largest power of the one or more thermal actuators 16. While the second method restricts the temperature on one or more positions on the optical member 15 to the desired temperature, the method according to the invention allows the temperature to go beyond the desired temperature. In the imaged embodiment, the optical member is initially at a lower temperature than desired and the thermal actuators are heating devices, such that the temperature is allowed to rise above the desired temperature, up to a maximum temperature to prevent damage to the optical member 15. Therefore, the method according to the invention can utilise more power of the one or more thermal actuators 16, as it allows higher temperatures of the optical member 15. If the thermal conditioning were to comprise cooling the optical member instead of heating the optical member because the optical member is at a temperature above the desired temperature at the start of the thermal conditioning, temperatures of the optical member below the desired temperature could be allowed, potentially with a minimum temperature constraint to prevent damage to the optical member 15.

[0083] As discussed above, the method according to the invention not only takes into account the current temperature of the optical member 15 like the second method, but also takes into account how thermal energy flows through the optical member 15 and / or between the optical member 15 and the environment 18. Thus, even though the temperatures of a number of nodes of the mesh, i.e. in a number of positions on the optical member 15, may reach values above the desired temperature, the method according to the invention ensures that thermal energy flow ensures that such temperatures fall to near the desired temperature by the time the thermal conditioning process is finished.

[0084] Figure 4 shows the total power of the thermal actuators 16 for the three methods discussed above. It is shown that the total power of the thermal actuators 16 is constant in the first method, while the power starts higher for the second method and gradually decreases towards the power value used in the first method over time. The first method as discussed continuously uses the power level of the thermal actuators 16 which is also used during steady state operation, after thermal conditioning is completed. This explains why the power level of the first method remains constant. This also explains why the power level of the second method approaches the power level of the first method, as the second method converges towards a steady state situation as well.

[0085] From the total power of the thermal actuators for the third method shown in figure 4, i.e. the conditioning method according to the invention, it is clear that, compared to the prior art methods 1 and 2, higher power levels are retained for a longer time. For the third method, the maximum total thermal power is 120 Watts. For the first and second methods, the maximum total thermal power is 60 Watts. The total thermal power spends some time at the maximum power level in the third method, showing that better thermal conditioning performance could be achieved if the maximum total thermal power would be increased. Thermal conditioning performance can be measured as the time required to meet a certain thermal state error threshold or the magnitude of the thermal state error after the conditioning method has been performed for a given amount of time. In the former case, a lower time indicates a better performance whereas in the latter case, a lower thermal state error indicates a better performance.

[0086] The first method does not utilise the full power potential of the prethermal actuators while the second method only utilises the maximum power potential of the thermal actuators for a short time at the beginning of the conditioning operation. Thus, increasing maximum power will not help when using the first method and may help a little when using the second method, while it may significantly decrease the required thermal conditioning time when using the third method, because the third method allows higher temperatures to be reached by the optical member 15 such that a higher maximum total thermal power can be used.

[0087] While figure 4 only shows the total thermal power used by all thermal actuators 16, it is to be understood that, in all three methods, individual thermal actuators 16 can behave differently. In method 1, the power of each individual thermal actuator 16 is specified such that the temperature of the optical member converges to the desired state temperature. In method 2, the power of individual thermal actuators 16 may be individually controlled to maintain the monitored temperature values at the desired temperature. In method 3, the power of individual thermal actuators 16 may be individually controlled independently from other thermal actuators 16 to obtain an optimal thermal conditioning performance.

[0088] Figure 5 shows the temperature of multiple nodes of the mesh for which the temperature is monitored and kept subject to a constraint of a maximum temperature. The nodes are shown to spend some time at the maximum allowable value, in this example about 45 degrees Celsius, before falling to about 30 degrees Celsius (the desired temperature) at the end of the conditioning process. In the first, the temperatures of these nodes would never exceed the desired temperature of 30 degrees, while in thesecond method, the objective is to maintain the desired temperature of 30 degrees throughout the thermal conditioning process. This limits the rate at which heat energy can be fed into or removed from the optical member 15, for both the first method and the second method.

[0089] In an embodiment, the thermal model of the optical member further comprises a deformation of the optical member and a wavefront aberration resulting from the deformation of the optical member, wherein the cost function comprises the deformation and the wavefront aberration. By including the deformation of the optical member and the wavefront aberration resulting from the deformation into the thermal model, the deformation and wavefront aberration resulting from thermal effects may be taken into account in the model. As the deformation and the wavefront aberration resulting from the deformation may be included in the cost function, the optimisation may take into account the deformation and the wavefront aberration resulting from the deformation. Accordingly, the optimisation of the cost function may provide an optimization of the temperature distribution, the deformation and the wavefront aberration resulting from the deformation.

[0090] For example, the method may comprise deriving from the deformation of the optical member a deformation of an optical surface of the optical member. Modelling, in the thermal model, the deformation and the resulting wavefront aberration may enhance an amount of data associated with the optimisation. The aberration resulting from the deformation may, to a large extent, be associated with the deformation of the optical surface of the optical member. Accordingly, deriving the deformation of the optical surface of the optical member may enable to perform the optimization on the basis of a reduced dataset, which may potentially reduce a complexity and computation time of the optimization. A data reduction in the data set associated with the thermal model may be achieved in that the wavefront aberration may be determined from the deformation of the optical surface of the optical member.

[0091] The deformation of the optical member may be determined from at least one of a mechanical stiffness, a thermal stiffness and a strain of the optical member. The mechanical stiffness may be a mechanical stiffness matrix, the thermal stiffness may be a thermal stiffness matrix. For example, the deformation of the optical member may be obtained by solving the equation mechanical stiffness matrix times deformation plus thermal stiffness matrix times strain equals zero. The deformation at time k may accordingly be determined from the mechanical stiffness matrix, the thermal stiffness matrix at time k and the strain at time k.

[0092] The strain of the optical member may be determined from at least one of a coefficient of thermal expansion, the temperature of the optical member, a zero crossing temperature of the optical member and a manufacturing temperature of the optical member. For example, the strain of the optical member may be determined from the coefficient of thermal expansion, the temperature of the optical member, the zero crossing temperature of the optical member and the manufacturing temperature of the optical member. For example, a vertical deformation normal to the optical surface of the optical member may be selected as being relevant, e.g. using a selection matrix, to reduce the data set and enhance a computation efficiency of the modelling and the optimisation.

[0093] In order to combine the different parameters temperature, deformation and aberration in the cost function, the cost function may include at least one of a temperature weighting matrix, a deformation weighting matrix and an aberration weighting matrix.

[0094] The method may further comprise measuring at least one of the temperature, the deformation and the aberration and inputting the measured at least one of the temperature, the deformation and the aberration into the optimization operation. For example, measured values of said parameters at a time during the recovery may be input into the optimization to repeat the optimization procedure for the remaining recovery time using the measured values of said parameters. As another example, the thermal model may be refined on the basis of the measured values of said parameters, e.g. adjusting heater gains as used in the thermal model.

[0095] Thus, by including the deformation and wavefront aberration from the deformation, a model- based control strategy may be provided that yields an enhanced control sequence for thermal actuators to recover the mirrors to a desired final state within a given recovery time. An optimization objective of the approach may be temperature-, deformation- and / or wavefront-based.

[0096] Starting from the linear time invariant thermal dynamics as expressed by equation (1) described above, and applying limitation for allowable optical surface temperature to prevent damaging the coating of the mirror, and applying constant or time varying limits to infrared heater powers to applying heat up to a maximum power per individual fiber and / or per heater head (containing multiple fibers), the deformations ^^induced by the thermal loads may be obtained by solving equation (5):

[0097] 5^^ + 5+6^ = 0 , (5)

[0098] where 5 is the mechanical stiffness matrix, 5+the thermal stiffness matrix, and 6^is the strain, which is a nonlinear function of the temperature distribution in the mirror for ULE material.

[0099] The strain in turn may be calculated according to equation (6) below:[000100] 6^ = 7^ ∗ 9^^ − :;<=∘^ − 7^ ∗ 9:! − :;<=∘^ , (6):!is the manufacturing temperature of the mirror, :;<is the zero-crossing temperature distribution (the zero crossing temperature may possibly be non-constant over the spatial domain), and the symbol ∘ denotes the element-wise power also known as Hadamard power. For imaging, only the vertical deformation ^^;@ABnormal to the optical surface of the mirror may be selected using a selection matrix, as expressed by (7) below:[000102] ^;@AB^ = C;@AB^^ , (7)[000103] where C;@ABis a selection matrix that collects the deformations at the optical surface from the full deformation vector of the mirror. The optical surface deformation normal to the surface impacts the raw wavefront aberrations in terms of Zernikes D^"EF:[000104] D"EF = C ;@AB^ ;^^ , (8)[000105] where C;is the matrix that maps the deformation of the optical surface to raw wavefront distortions at wafer level in terms of Zernikes. These raw aberrations may be partially corrected by rigid-body mirror- and stage-adjustments:[000106] D"#G "EF^ = CHID^ , (9)[000107] where CHIdescribes the behavior of the optical correction mechanism using rigid-body mirror- and stage-adjustments and D^"#Gare the residual wavefront aberrations after correction, expressed in Zernikes. Similar rigid-body mirror-adjustments may be used to correct for mirror deformation:[000108] ^"#G^ = CJI^^ , (10)[000109] where CJIdescribes the behavior of the deformation correction mechanism using rigid- body mirror- adjustments. [000110] Reverting to the optimization, employing the discretized system dynamics in the above equations (1) and (5)-(10), the proposed optimal control strategy in equation (11) may find a controlinput sequence ^K^L,^ ∈ . for / = 0,1, … , 2 − 1, given the available actuation range ., the admissibleset of temperature states of the optical member 3, an initial temperature distribution ^! ∈ 3, a desired(steady state) temperature distribution (and corresponding deformation and wavefront) after recovery^"#$ ∈ 3, the cost weight matrices PN ∈ ℝ^P×^P, PR ∈ ℝ^S×^S and PT ∈ ℝ^U×^U, and the fixedrecovery time set by the user expressed as the number of time samples 2 ∈ ℕ, by solving theoptimization problem in accordance with equation (11):[000111] WXY ^9^ , ^ , Z [',( ! "#$ , %= = W',X(Y ^) − ^"#$& ,N ^) − ^"#$& + (11)),R ) ) ,T )[000113] subject to the constraints in line with the above equations (1) and (5) – (10): ^^^^^ = ^^^ + ^^^ ,5^^ + 5+6^ = 0 ,?^ =∘^ ?= ∗ ^ − − ∗ − ,[000114] where ^)is the achieved thermal state at timestep 2, i.e., the end of recovery. Furthermore, ^) and D"#G) are the deformation and wavefront that result from the final thermal state^), respectively.[000115] The proposed control strategy may enhance the control sequence for the thermal actuators by: [000116] aiming for the full thermal state at the end of the recovery to be as close as possible to a desired (steady state) thermal state, resulting in little or no thermal drift after recovery (this is enforced by ^ − ^ [) "#$& , ^) − ^"#$& ),the mirror deformation at the end of the recovery to be as close asdeformation, resulting in little / no change in mirror deformation after recovery (this is enforced by ^"#G [ "#G) − ^"#$& ,R ^) − ^"#$& ),[000118] and / or the end of the recovery to be as close as possible to a desired (steadywavefront drift after recovery (this is enforced by D"#G) − D"#$&[,T D"#G) − D"#$& ),[000119] model to make predictions of the full temperatureover time. [000120] The including of temperature, deformation and wavefront aberration resulting from the deformation may enable short recovery times, [000121] considering limitations on the operation ranges of the (thermal) actuators to enable using their full actuator potential for as fast as possible recovery within the limitations of the actuators (specified by ^^ ∈ .^, for / = 0,1, … , 2 − 1 ),[000122] considering state constraints, e.g., a maximum allowable mirror surface temperature during recovery, to enable as fast as possible recovery while at all time safeguarding the mirrortemperature range (specified by ^^ ∈ 3^, for / = 1,2, … , 2),[000123] making use of a computationally efficient formulation that allows longer prediction horizons leading to better performance. [000124] By tuning the weight matrices, a tradeoff may be made between the temperature, deformation and wavefront objectives. The temperature domain objective (in combination with deformation and / or wavefront objectives) may be used to provide that a solution is found close to the thermal steady state solution, since due to nonlinearity, multiple solutions may be found optimizing the deformation or wavefront objective, while being far from the desired thermal steady state. As an alternative, one could include in the objective function also a finite time horizon after recovery hasended (when assuming a control roll-out strategy for ^^ for / > 2) to make sure the deformation and / orwavefront performance is optimized after recovery has ended. Computation time may be kept small when choosing ^Nsuch that only selected, e.g. most critical, optical footprint node temperatures are subjected to the state constraints, as this may imply that the rest of the temperatures are also within bounds. Due to the nonlinear temperature-strain relation of the ULE material, the optimization may become non-linear. [000125] In addition to the above embodiment describing the optimization on the basis of temperature, deformation and wavefront objectives, temperature-, deformation- and / or wavefront setpoints may be used, the setpoints may be derived by steady-state wavefront optimization [000126] The determined heater powers may be used as a feedforward signal to the heaters, and the temperature-, deformation- and / or wavefront setpoints may be used as reference signal for a sensor- based feedback control loop, [000127] Temperature, deformation and / or wavefront sensor feedback may be used during recovery by using an online state-estimation scheme such as described in US2024077380A1. This feedback can be used in various ways: - input for the optimization as shown in (11), i.e., repeating the optimization procedure for the remaining recovery time starting from the latest estimated state, - input to refine the model in (1) and (5) - (10) (in particular the heater-gains are not known in advance), [000128] A model reduction may be utilized to improve efficiency of the optimizer. [000129] A moving-horizon Model Predictive Control strategy may be employed as opposed to the fixed-horizon optimal control. [000130] Figures 6 - 9 illustrate a behavior of the above described embodiment wherein the temperature, the deformation and the aberration resulting from the deformation are modelled and employed wherein the temperature and the aberration resulting from the deformation are employed in the cost function. Figure 6 depicts wavefront aberration error on the vertical axis versus time on thehorizontal axis. Three control strategies are compared, namely (1) steady state heater power, indicated in the Figure as Method 1 and represented by the top uninterrupted line, (2) using temperature feedback based on temperature sensors, indicated in the Figure as Method 2 and represented by the bottom uninterrupted line, well as (3) using temperature, deformation and aberration resulting from the deformation in the modelling and using temperature and aberration resulting from the deformation in the cost function according to an embodiment of the invention, indicated in the Figure as Method 3 represented by the dot dash line. Figure 7 depicts residual wavefront error for each one of the control strategies as identified with reference to Figure 6, likewise indicated by Method 1, 2 and 3 respectively. As follows from Figures 6 and 7, a recovery time may be shortened in case the temperature, the deformation and the aberration resulting from the deformation are modelled and whereby the temperature and the aberration resulting from the deformation is employed in the cost function. [000131] Figure 8 and 9 further illustrate the embodiment in accordance with Method 3. Figure 8 depicts heater powers of heaters identified as H1 – H18 on the vertical axis versus time on the horizontal axis. [000132] Figure 9 depicts a temperature (vertical) over time (horizontal) of the optical footprint nodes using the constraint of the maximum temperature Tmax. [000133] Although specific reference may be made in this text to the use of lithographic apparatus in the manufacture of ICs, it should be understood that the lithographic apparatus described herein may have other applications. Possible other applications include the manufacture of integrated optical systems, guidance and detection patterns for magnetic domain memories, flat-panel displays, liquid-crystal displays (LCDs), thin-film magnetic heads, etc. [000134] Although this text refers to thermal actuators only, cooling devices may be used alternatively or in addition to thermal actuators. [000135] Although specific reference may be made in this text to embodiments of the invention in the context of a lithographic apparatus, embodiments of the invention may be used in other apparatus. Embodiments of the invention may form part of a mask inspection apparatus, a metrology apparatus, or any apparatus that measures or processes an object such as a wafer (or other substrates) or mask (or other patterning devices). These apparatus may be generally referred to as lithographic tools. Such a lithographic tool may use vacuum conditions or ambient (non-vacuum) conditions. [000136] Where the context allows, embodiments of the invention may be implemented in hardware, firmware, software, or any combination thereof. Embodiments of the invention may also be implemented as instructions stored on a machine-readable medium, which may be read and executed by one or more processors. A machine-readable medium may include any mechanism for storing or transmitting information in a form readable by a machine (e.g., a computing device). For example, a machine-readable medium may include read only memory (ROM); random access memory (RAM); magnetic storage media; optical storage media; flash memory devices; electrical, optical, acoustical, and others. Further, firmware, software, routines, instructions may be described herein as performingcertain actions. However, it should be appreciated that such descriptions are merely for convenience and that such actions in fact result from computing devices, processors, controllers, or other devices executing the firmware, software, routines, instructions, etc. and in doing that may cause actuators or other devices to interact with the physical world. [000137] While specific embodiments of the invention have been described above, it will be appreciated that the invention may be practiced otherwise than as described. The descriptions above are intended to be illustrative, not limiting. Thus it will be apparent to one skilled in the art that modifications may be made to the invention as described without departing from the scope of the claims set out below.

Claims

CLAIMS 1. Optical member thermal conditioning method configured to condition an optical member in an initial thermal state to a predetermined final thermal state, the method comprising: - providing one or more thermal actuators for influencing a thermal state of the optical member by providing a thermal input to the optical member, - establishing a finite element thermal model of the optical member, wherein the thermal model is discretised in space by defining a mesh comprising member nodes, the mesh dividing the optical member in a plurality of member elements defined by the member nodes, wherein the member nodes are points in space that surround a particular member element, wherein each member element comprises a section of the optical member, wherein the thermal model is discretised in time by defining a timestep length, wherein the thermal model is configured to describe an evolution over time of a temperature distribution of the optical member under the influence of internal thermal energy transfer within the optical member and / or thermal interaction between the optical member and an environment and thermal input from the one or more thermal actuators, wherein the thermal model comprises a thermal state vector representing a thermal state of the optical member, the thermal state comprising a temperature distribution within the optical member, - determining a thermal state error representing a deviation of the thermal state vector from a desired thermal state at a given timestep, - establishing a cost function comprising the thermal state error, - performing an optimisation operation on the cost function, wherein a thermal input of the one or more thermal actuators is optimised to minimise a value of the cost function, wherein the optimisation operation results in an optimal control signal to control the one or more thermal actuators to achieve an optimal thermal conditioning process, - controlling the one or more thermal actuators according to the optimal control signal to thermally condition the optical member.

2. Method according to claim 1, wherein the one or more thermal actuators comprise heating devices for heating the optical member wherein the thermal input comprises a heat input and / or cooling devices for cooling the optical member wherein the thermal input comprises a cold input by heat extraction.

3. Method according to claim 1 or 2, wherein the thermal state vector represents the temperatures of all member nodes of the optical member at a given timestep, wherein thethermal state vector comprises Nxcomponents, wherein Nxis the number of member nodes of the optical member.

4. Method according to any of claims 1 – 3, wherein the method comprises monitoring one or more member node temperatures of one or more member nodes.

5. Method according to any of claims 1 – 4, wherein the internal thermal energy transfer within the optical member and / or thermal interaction between the optical member and an environment that is incorporated into the model includes: - transfer of heat within the optical member, and / or - transfer of heat from the optical member to an environment or from the environment to the optical member by e.g. conduction, radiative heat transfer and / or convection from e.g. liquid cooling of the optical member.

6. Method according to any of claims 1 – 5, wherein a timestep amount is determined, wherein the timestep amount is the maximum number of usable timesteps, wherein the optimisation operation comprises optimising the thermal input of the one or more thermal actuators to achieve the smallest possible thermal state error after the timestep amount of timesteps has passed.

7. Method according to any of claims 1 – 5, wherein an error threshold is determined, wherein the error threshold is the maximum allowed value of the thermal state error, wherein the optimisation operation comprises optimising the thermal input of the one or more thermal actuators to achieve the lowest possible number of timesteps required until the thermal state error falls below the error threshold.

8. Method according to any of claims 1 – 7, wherein the thermal model is configured to determine a second temperature distribution of the optical member at a second timestep based on at least: - a first temperature distribution of the optical member at a first timestep preceding the second timestep, and - a thermal input to the optical member from the one or more thermal actuators during the interval between the first timestep and the second timestep.

9. Method according to any of claims 1 – 8, the thermal model comprising a thermal capacity matrix, wherein the thermal capacity matrix represents the thermal capacity of the optical member at the member nodes, wherein a multiplication of the thermal capacity matrix withthe thermal state vector provides a thermal energy content vector which represents a thermal energy content of the optical member at the member nodes.

10. Method according to claim 9, wherein the thermal capacity matrix has dimensions Nx by Nx, wherein the thermal energy content vector has Nxcomponents, wherein the thermal capacity matrix comprises capacity components with a first capacity index and a second capacity index, wherein the first capacity index and the second capacity index define the position of the associated capacity component in the thermal capacity matrix.

11. Method according to claim 9 or 10, the thermal model comprising a thermal transfer matrix, wherein the thermal transfer matrix represents transfer coefficients of internal thermal energy transfer during one timestep within the optical member at the member nodes and / or thermal interaction during one timestep between the optical member and an environment at the member nodes, wherein a multiplication of the thermal transfer matrix with the thermal state vector provides a thermal transfer vector which represents an energy transfer related change in the thermal energy content of the optical member at the member nodes due internal thermal energy transfer within the optical member and / or thermal interaction between the optical member and the environment within one timestep.

12. Method according to claim 11, wherein: - the thermal transfer matrix has dimensions Nx by Nx, - the thermal transfer vector has Nx components, - the thermal transfer matrix comprises transfer components with a first transfer index and a second transfer index, wherein the first transfer index and the second transfer index define the position of the associated transfer component in the thermal transfer matrix, - each transfer component represents a heat transfer coefficient which is representative for heat transfer to a member node with an element index in the thermal state vector which corresponds to the first transfer index from a member node with an element index in the thermal state vector which corresponds to the second transfer index and / or which is representative for thermal interaction between the optical member and the environment at a member node with an element index in the thermal state vector which corresponds to the first transfer index.

13. Method according to any of claims 9 - 12, the thermal model comprising an irradiance matrix of the one or more thermal actuators and a thermal forcing vector of the one or more thermal actuators, wherein the irradiance matrix represents a thermal input of each of the one or morethermal actuators to the optical member at the member nodes for a unit thermal power amount of each of the one or more thermal actuators, wherein the thermal forcing vector represents for each thermal actuator the thermal power, wherein a multiplication of the irradiance matrix with the thermal forcing vector provides a thermal input vector which represents a thermal input to the optical member at the member nodes within one timestep.

14. Method according to claim 13, wherein: - the irradiance matrix has dimensions Nxby Nu, wherein Nuis the number of thermal actuators, - the thermal forcing vector has Nu components, - the irradiance matrix comprises irradiance components with a first irradiance index and a second irradiance index, wherein the first irradiance index and the second irradiance index define the position of the associated irradiance component in the irradiance matrix, - each irradiance component represents a thermal input for a unit thermal power amount to the optical member at a member node with an element index in the thermal state vector which corresponds to the first irradiance index of said irradiance component of a thermal actuator with a thermal index in the thermal forcing vector which corresponds to the second irradiance index of said irradiance component.

15. Method according to any of claims 1 – 14, wherein the thermal model is configured to include heat flow between the optical member and an environment to determine an evolution over time of the temperature distribution of the optical member.

16. Method according to any of claims 1 – 15, wherein the cost function includes a weighing matrix.

17. Method according to any of claims 1 – 16, wherein a maximum temperature is specified for at least one member node as a constraint to the optimisation operation.

18. Method according to any of claims 1 – 17, wherein a maximum thermal power and / or a minimum thermal power is specified for at least one of the one or more thermal actuators as a constraint to the optimisation operation.

19. Method according to any of claims 1 – 18, wherein the thermal model of the optical member further comprises a deformation of the optical member and a wavefront aberration resultingfrom the deformation of the optical member, wherein the cost function comprises the deformation and the wavefront aberration.

20. Method according to claim 19, comprising deriving from the deformation of the optical member a deformation of an optical surface of the optical member.

21. Method according to claim 20, wherein the wavefront aberration is determined from the deformation of the optical surface of the optical member.

22. Method according to any of claims 19 - 21, wherein the deformation of the optical member is determined from at least one of a mechanical stiffness, a thermal stiffness and a strain of the optical member.

23. Method according to claim 22, wherein the strain of the optical member is determined from at least one of a coefficient of thermal expansion, the temperature of the optical member, a zero crossing temperature of the optical member and a manufacturing temperature of the optical member.

24. Method according to any of claims 19 – 23, wherein the cost function includes at least one of a temperature weighting matrix, a deformation weighting matrix and an aberration weighting matrix.

25. Method according to any of claims 19 – 24, further comprising measuring at least one of the temperature, the deformation and the aberration and inputting the measured at least one of the temperature, the deformation and the aberration into the optimization operation. I 26. Method according to any of claims 1 – 25, wherein the method is a computer-implemented method.

27. A computer program having instructions which when executed by an information processing device causes the information processing device to perform the method according to any of claims 1 – 26.

28. Lithographic apparatus comprising: - an optical member, - one or more thermal actuators, - one or more temperature sensors for sensing a temperature of the optical member,- a control unit to control the one or more thermal actuators based on a control signal, - an information processing device comprising a processor and a memory, wherein the information processing device is configured to perform the method of any of claims 1 – 26 and wherein the information processing device is configured to determine the optimal control signal based on the optimisation and to communicate the optimal control signal to the control unit.

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