an arrangement for the thermal actuation of a mirror is explained below initially with reference to fig1 and fig2 a - b on the basis of a first embodiment of the invention . according to the perspective schematic representation of fig1 , the mirror 100 has a plurality of access channels 110 , 111 , 112 , . . . , which extend from the rear side of the mirror ( i . e . the side of the mirror 100 that is opposite from the optically effective surface 101 ) through the mirror substrate denoted by “ 102 ” into the vicinity of the optically effective surface 101 , the access channels 110 , 111 , 112 , . . . according to fig1 being in a matrix - like arrangement . in fig2 a - b , only one access channel 110 is shown in section , the mirror 100 and the other access channels 111 , 112 , . . . being configured in an analogous way . on the optically effective surface 101 of the mirror 100 or the mirror substrate 102 there is a reflective coating ( not represented ). fig1 and 2 , and similarly the further figures too , respectively represent schematic illustrations that are not to scale , it being possible for dimensions that are given merely by way of example of the access channel 110 ( without restricting the invention to them ) to comprise a diameter of the respective bores in the range of 5 to 30 mm and also a distance of the respective end faces 110 a from the optically effective surface 101 in the range of likewise 5 to 30 mm . typical thicknesses of the mirror 100 itself ( likewise merely by way of example and without restricting the invention ) lie , for example , in the range of approximately 50 to 120 mm . as can be seen best from fig2 a , a cooling element 120 of a tubular geometry in the exemplary embodiment extends into the access channel 110 , this cooling element 120 being able to be set variably in its cooling power ( by connection to a controllable cooler that is not shown ). as likewise indicated in fig2 a - b , the arrangement also has a heat source in the form of a heat emitter 130 , which in the exemplary embodiment is arranged in the end portion of the access channel 110 that is facing the optically effective surface 101 and has a heating power that can likewise be set variably . in further embodiments , the heat emitter 130 may also be arranged in a different region of the access channel 110 , in particular also in the region of the rear side of the mirror , in order in this case to couple in the heating radiation by way of the access channel 110 or reflection on the side walls of the access channel 110 , preferably sufficiently shallow reflection angles with what is known as “ grazing incidence ”. furthermore , as explained in still more detail below , the heat source may also be configured in the form of at least one heating wire . the electromagnetic radiation that is radiated — once again with reference to fig1 and 2 a , b — by way of the heat emitter 130 and coupled into the mirror substrate material may , for example , have a wavelength of at least 2 . 5 μm , more particularly a wavelength of at least 5 μm , as may be realized for instance by way of what is known as a low - temperature emitter at temperatures of up to 400 ° c ., in particular in the range up to 200 ° c . in other exemplary embodiments , a high - temperature emitter , such as for example a spiral - wound filament ( with an operating temperature of typically up to 3000 ° c .) may also be used . the arrangement chosen in fig2 a - b of the cooling element 120 , on the one hand , and the heat source or heat emitter 130 , on the other hand , achieves the effect in particular that the heat transfer from the mirror substrate material to the cooling element 120 ( set to a temperature of t =− 20 ° c .) takes place substantially along the entire surface of the access channel 110 , whereas the coupling of heating radiation ( for example infrared radiation ) by way of the heat emitter 130 into the mirror substrate material takes place substantially only in the end portion of the access channel 110 containing the end face 110 a . the combined use of the cooling element 120 , on the one hand , and the heat emitter 130 , on the other hand , allows the temperature of the mirror substrate material in the region of the optically effective surface 101 to be kept constantly at a value that corresponds substantially to the zero - crossing temperature of the mirror substrate material concerned ( for example when there is an increasing optical load in the region of the optically effective surface 101 as a result of the impinging electromagnetic ( euv ) radiation during the operation of the projection exposure apparatus ). conversely , independently of the thermal balance maintained in the region of the optically effective surface 101 , a temperature that deviates from the zero - crossing temperature and makes sufficient sensitivity of the mirror substrate material to thermal loads possible , and consequently specific thermal actuation or deformation of the mirror 100 , can be set in a region remote therefrom ( that is in the “ lower ” region in fig2 a - b ). merely by way of example ( and without restricting the invention to this ), the temperature set in the region of the optically effective surface 101 may be 30 ° c . and in the region remote from the optically effective surface 101 may be − 15 ° c ., in which case there is already a significant change in the coefficient of linear expansion depending on the mirror substrate material . the inventive concept in particular involves creating , using the independently realizable adjustability of the cooling power of a cooling element on the one hand and the heating power of at least one heat source on the other hand , a thermal flux between the optically effective surface and the surface of the mirror not corresponding to the optically effective surface ( e . g . the backside of the mirror ). this thermal flux results in a temperature gradient and a related local variation of a value of the coefficient of thermal expansion in the mirror substrate , which leads to a behaviour of the mirror substrate being comparable , in terms of thermal sensitivity or the varying coefficient of thermal expansion ( cte ), to a bimetal . this concept is also schematically illustrated in fig7 ( where components or parts being analogous to fig1 have same reference numbers ). as shown in fig7 , the thermal flow along e . g . the access channel 110 and the created temperature gradient between the optically effective surface 101 and the backside of the mirror 100 makes it possible to maintain the optically effective surface 101 of the mirror 100 at a temperature for which the mirror material is substantially insensitive against thermal loads occurring during operation of the optical system ( namely the so - called zero crossing temperature ), while at the same time enabling a region in the mirror substrate remote from the optically effective surface to be more sensitive or thermally actuable due to the different cte - value in order to create a desired specific deformation of the mirror . according to an aspect of the invention , a temperature variation is created in a region of the mirror substrate facing away from the optically effective surface ( e . g . at the backside of the mirror ) such that , due to the thermal sensitivity achieved by the value of the coefficient of thermal expansion in that region of the mirror substrate , a deformation in the mirror substrate is effected , the deformation being mechanically transferred to the optically effective surface ( having a substantially constant temperature close to the zero crossing temperature ). by superposing all of the basic functions of the cooling elements 120 , desired deformation profiles can be set . this is a non - linear optimization problem ( because of the proximity to the “ zero - crossing point ” of the coefficient of linear expansion ), the resolution of which can be obtained numerically by iteratively carrying out the following steps : a ) generating a pattern of holes ; b ) calculating the impact matrix for the surface temperature and surface deformation for the temperature operating point ; c ) determining the individual heating and cooling power , so that , according to the linear calculation , the desired deformation profile is obtained under the secondary condition that the temperature distribution is kept close to the zero - crossing temperature over the optically effective surface ; d ) determining the actual deformation profile via non - linear calculation ; and e ) changing the temperature profile to correct the deformation profile deviation according to the linear calculation . in fig2 c - f , the result of the calculation of the impact matrices is shown merely by way of example in the form of contour diagrams for a cooling element and the associated access channel , fig2 c showing the profile dt / dq f , fig2 d showing the profile dw / dq f , fig2 e showing the profile dt / dt f and fig2 f showing the profile dw / dt f , where w denotes the z deformation (= deformation in the z direction ), where q denotes the thermal flow and t denotes the temperature . as described above , the corresponding impact matrices ( and the resultant temperature base functions and deformation base functions ) can be suitably combined to set a desired ( as homogeneous as possible ) temperature profile and also a desired deformation profile . by repeatedly or iteratively carrying out steps d ) and e ), the achieved deviation in the deformation becomes smaller . in this respect , for example , deformations of the order of magnitude of ± 0 . 5 nm are achievable . fig3 shows a diagram in which the coefficient of linear expansion for different mirror substrate materials is plotted in dependence on the temperature . the zero crossing of the corresponding curves respectively corresponds to the zero - crossing temperature , which , for example , for ule ™ lies at approximately t = 30 ° c . if , as a result of a suitably chosen combination of the heating power coupled in by way of the heat emitter 130 and the cooling power set by way of the cooling element 120 , the region of the mirror substrate 102 in the vicinity of the optically effective surface 101 is at this zero - crossing temperature , optical loads caused by the electromagnetic radiation impinging on the optically effective surface 101 during the operation of the projection exposure apparatus do not result in any deformation , since the coefficient of linear expansion of the mirror substrate material is equal to zero . conversely , according to fig3 , when there is a deviation from this temperature , there is an increasing sensitivity of the respective mirror substrate material to temperature changes , which can be used for the thermal actuation or deformation of the mirror 100 that is described above . a further embodiment of the invention is explained below with reference to fig4 a - 4 b , components that are analogous or have substantially the same function as in fig2 being designated by corresponding reference numerals increased by “ 100 ”. the embodiment according to fig4 a - 4 b differs from that according to fig2 in that the mirror substrate 202 is produced from two different mirror substrate materials , the mirror 200 is therefore formed as a “ composite mirror ”. in fact , the mirror substrate 202 has a first mirror substrate region 202 a and a second mirror substrate region 202 b , which is more remote in relation thereto from the optically effective surface 201 of the mirror 200 , the mirror substrate regions 202 a , 202 b being produced from different materials . in the actual exemplary embodiment , the mirror substrate material of the first mirror substrate region is ule ™ and the mirror substrate material of the second mirror substrate region 202 b is quartz glass ( sio 2 ). however , the invention is not restricted to these materials , so that in further embodiments the mirror substrate regions 202 a , 202 b may also be produced from different mirror substrate materials , the mirror substrate material in the second mirror substrate region 202 b in each case having a coefficient of linear expansion that is greater in terms of the absolute amount at a prescribed temperature than the mirror substrate material in the first mirror substrate region . as a result of this configuration in the exemplary embodiment of fig4 a - 4 b , the desired greater sensitivity of the mirror substrate material that is described above and is desired for the purpose of specific deformation is obtained in the region more remote from the optically effective surface 201 not only because of the temperature gradient forming within the mirror substrate material but also because of the transition between different mirror substrate materials , so that the embodiment according to fig4 a - 4 b are suitable in particular in situations in which the temperature gradient alone is not yet sufficient to achieve the sensitivity to thermal loads desired for the deformation while at the same time maintaining the zero - crossing temperature at the optically effective surface . fig5 a and 5 b serve for explaining further embodiments of the invention . according to fig5 a , an absorbent layer 530 for the at least partial absorption of the heating radiation that is coupled into the access channel 110 is arranged between the mirror substrate 102 and the reflective coating 550 . according to fig5 b , instead of the heat emitter 130 that is present in the embodiments described above , an arrangement 540 of heating wires is provided as the heat source , by which the region between the mirror substrate 102 and the reflective coating 550 can be heated directly with heating power that can be set variably . an arrangement according to the invention for the thermal actuation of a mirror may be used both in conjunction with a mirror in the illumination device and in conjunction with a mirror in the projection lens of a microlithographic projection exposure apparatus . fig6 shows merely as an example of an application the path of rays of a projection lens 600 , which is disclosed in us 2008 / 0170310 a1 ( see fig2 there ). the projection lens 500 is designed for operation in euv and has six mirrors m 1 to m 6 . in this case , for example , the mirrors m 2 and m 6 that are respectively close to the pupil or else another of the mirrors according to one of the embodiments described above may be configured as thermally actuable . even though the invention has been described on the basis of specific embodiments , numerous variations and alternative embodiments are evident to a person skilled in the art , for example by combining and / or exchanging features of individual embodiments . accordingly , it goes without saying for a person skilled in the art that such variations and alternative embodiments are also covered by the present invention , and the scope of the invention is only restricted by the constraints of the accompanying patent claims and the equivalents thereof .