Blank made of titanium-doped silica glass and method for the production thereof

Abstract

A blank made of titanium-doped silica glass for a mirror substrate for use in EUV lithography is provided. The blank includes a surface portion to be provided with a reflective film and having an optically used area (CA) over which a coefficient of thermal expansion (CTE) has a two-dimensional inhomogeneity (dCTE) distribution profile averaged over a thickness of the blank. A maximum inhomogeneity (dCTE.sub.max) of less than 5 ppb/K is defined as a difference between a CTE maximum value and a CTE minimum value. The dCTE.sub.max is at least 0.5 ppb/K. The CA forms a non-circular area having a centroid. The dCTE distribution profile is not rotation-symmetrical and is defined over the CA, such that straight profile sections normalized to a unit length and extending through the centroid of the area yield a dCTE family of curves forming a curve band with a bandwidth of less than 0.5dCTE.sub.max.

Claims

1. A blank made of titanium-doped silica glass for a mirror substrate for use in EUV lithography, the blank comprising a surface portion configured to be provided with a reflective film and having an optically used area (CA) over which a coefficient of thermal expansion (CTE) has a two-dimensional inhomogeneity (dCTE) distribution profile averaged over a thickness of the blank, a maximum inhomogeneity (dCTE.sub.max) of less than 5 ppb/K being defined as a difference between a CTE maximum value and a CTE minimum value, wherein the dCTE.sub.max is at least 0.5 ppb/K, wherein the CA forms a non-circular area having a centroid, wherein the dCTE distribution profile is not rotation-symmetrical and is defined over the CA such that straight profile sections normalized to a unit length and extending through the centroid of the non-circular area yield a dCTE family of curves forming a curve band with a bandwidth of less than 0.5dCTE.sub.max, and wherein the dCTE distribution profile can be mathematically described in a biunique manner by stretching a rotation-symmetrical distribution profile in at least one spatial direction, and wherein the stretching factor is at least 1.2.

2. The blank according to claim 1, wherein the bandwidth is less than 0.3dCTE.sub.max.

3. The blank according to claim 1, wherein the dCTE distribution profile comprises a closed isoline with a dCTE value of 0.5 dCTE.sub.max, of which a sub-length of at least 80% of a total length of the isoline extends within the CA.

4. The blank according to claim 3, wherein the isoline fully extends within the CA.

5. The blank according to claim 1, wherein the non-circular area of the CA is defined by a non-circular outline along which a dCTE maximum value and a dCTE minimum value of the dCTE distribution profile are positioned, wherein the difference (PV.sub.CA) between the dCTE maximum value and the dCTE minimum value is not more than 0.5dCTE.sub.max.

6. The blank according to claim 5, wherein the difference PV.sub.CA is not more than 0.3dCTE.sub.max.

7. A blank made of titanium-doped silica glass for a mirror substrate for use in EUV lithography, the blank comprising a surface portion configured to be provided with a reflective film and having an optically used area (CA) over which a coefficient of thermal expansion (CTE) has a two-dimensional inhomogeneity (dCTE) distribution profile averaged over a thickness of the blank, a maximum inhomogeneity dCTE.sub.max) of less than 5 ppb/K being defined as a difference between a CTE maximum value and a CTE minimum value, wherein the dCTE.sub.max is at least 0.5 ppb/K, wherein the CA forms a non-circular area having a centroid, wherein the dCTE distribution profile is not rotation-symmetrical and is defined over the CA such that straight profile sections normalized to a unit length and extending through the centroid of the non-circular area yield a dCTE family of curves forming a curve band with a bandwidth of less than 0.5dCTE.sub.max, and wherein the dCTE distribution profile can be mathematically described in a biunique manner by stretching the round form in plural spatial directions, and wherein the spatial directions extend in a common deformation plane extending in parallel with the optically used area CA.

8. A blank made of titanium-doped silica glass for a mirror substrate for use in EUV lithography, the blank comprising a surface portion configured to be provided with a reflective film and having an optically used area (CA) over which a coefficient of thermal expansion (CTE) has a two-dimensional inhomogeneity (dCTE) distribution profile averaged over a thickness of the blank, a maximum inhomogeneity (dCTE.sub.max) of less than 5 ppb/K being defined as a difference between a CTE maximum value and a CTE minimum value, wherein the dCTE.sub.max is at least 0.5 ppb/K, wherein the CA forms a non-circular area having a centroid, wherein the dCTE distribution profile is not rotation-symmetrical and is defined over the CA such that straight profile sections normalized to a unit length and extending through the centroid of the non-circular area yield a dCTE family of curves forming a curve band with a bandwidth of less than 0.5dCTE=.sub.max, and wherein the dCTE distribution profile can be described by stretching the circular form in three directions extending in the same deformation plane and enclosing an angle of 120 degrees.

9. A method for producing a blank from titanium-doped silica glass for a mirror substrate for use in EUV lithography according to claim 1, the method comprising the following steps: (a) providing a glass cylinder of Ti-doped silica glass with a rotation-symmetrical dCTE distribution profile, and (b) shaping the glass cylinder by softening the cylinder and forming the softened cylinder under an action of a shaping force having a force component acting in a direction perpendicular to a longitudinal axis of the glass cylinder, thereby stretching the rotation-symmetrical dCTE distribution profile in at least one direction so as to obtain a cylindrical blank which has a non-circular cross-section and a non-rotation symmetrical dCTE distribution profile.

10. The method according to claim 9, wherein the shaping according to step (b) comprises a shaping step in which the glass cylinder along a vertically-oriented longitudinal axis thereof is arranged in a melt mold having a non-rotation symmetrical inner geometry, is heated therein to a temperature of at least 1,200 C. and is softened thereby such that softened glass flows out laterally into the melt mold under the action of gravity.

11. The method according to claim 10, wherein the inner geometry of the melt mold, when viewed in cross section in a direction perpendicular to the longitudinal axis of the glass cylinder, has a long axis and a shorter axis in comparison therewith.

12. The method according to claim 11, wherein the inner geometry of the melt mold is oval or rectangular in cross section.

13. The method according to claim 9, wherein the shaping according to step b) comprises a plurality of shaping steps, wherein the glass body obtained after a first shaping step is further deformed in a second and subsequent shaping step.

14. The method according to claim 9, wherein providing the glass cylinder according to step (a) comprises: aa) producing a porous soot body of SiO.sub.2 and TiO.sub.2 by flame hydrolysis of starting substances containing silicon and titanium, bb) drying and sintering the soot body to form an elongated glass pre-product of Ti-doped silica glass, cc) homogenizing the glass pre-product in a homogenization process in which the pre-product is heated to a temperature of more than 1,500 C., is softened therein and is shaped into the glass cylinder.

15. The blank according to claim 7, wherein the bandwidth is less than 0.3dCTE.sub.max.

16. The blank according to claim 7, wherein the dCTE distribution profile comprises a closed isoline with a dCTE value of 0.5 dCTE.sub.max, of which a sub-length of at least 80% of a total length of the isoline extends within the CA.

17. The blank according to claim 8, wherein the bandwidth is less than 0.3dCTE.sub.max.

18. The blank according to claim 8, wherein the dCTE distribution profile comprises a closed isoline with a dCTE value of 0.5 dCTE.sub.max, of which a sub-length of at least 80% of a total length of the isoline extends within the CA.

19. A method for producing a blank from titanium-doped silica glass for a mirror substrate for use in EUV lithography, the blank comprising a surface portion configured to be provided with a reflective film and having an optically used area (CA) over which a coefficient of thermal expansion (CTE) has a two-dimensional inhomogeneity (dCTE) distribution profile averaged over a thickness of the blank, a maximum inhomogeneity (dCTE.sub.max) of less than 5 ppb/K being defined as a difference between a CTE maximum value and a CTE minimum value, wherein the dCTE.sub.max is at least 0.5 ppb/K, wherein the CA forms a non-circular area having a centroid, wherein the dCTE distribution profile is not rotation-symmetrical and is defined over the CA such that straight profile sections normalized to a unit length and extending through the centroid of the non-circular area yield a dCTE family of curves forming a curve band with a bandwidth of less than 0.5dCTE.sub.max, the method comprising the following steps: (a) providing a glass cylinder of Ti-doped silica glass with a rotation-symmetrical dCTE distribution profile, and (b) shaping the glass cylinder by softening the cylinder and forming the softened cylinder under an action of a shaping force having a force component acting in a direction perpendicular to a longitudinal axis of the glass cylinder, thereby stretching the rotation-symmetrical dCTE distribution profile in at least one direction so as to obtain a cylindrical blank which has a non-circular cross-section and a non-rotation symmetrical dCTE distribution profile, wherein the shaping comprises a plurality of shaping steps, wherein the glass body obtained after a first shaping step is further deformed in a second and subsequent shaping step.

20. A method for producing a blank from titanium-doped silica glass for a mirror substrate for use in EUV lithography, the blank comprising a surface portion configured to be provided with a reflective film and having an optically used area (CA) over which a coefficient of thermal expansion (CTE) has a two-dimensional inhomogeneity (dCTE) distribution profile averaged over a thickness of the blank, a maximum inhomogeneity (dCTE.sub.max) of less than 5 ppb/K being defined as a difference between a CTE maximum value and a CTE minimum value, wherein the dCTE.sub.max is at least 0.5 ppb/K, wherein the CA forms a non-circular area having a centroid, wherein the dCTE distribution profile is not rotation-symmetrical and is defined over the CA such that straight profile sections normalized to a unit length and extending through the centroid of the non-circular area yield a dCTE family of curves forming a curve band with a bandwidth of less than 0.5dCTE.sub.max, the method comprising the following steps: (a) providing a glass cylinder of Ti-doped silica glass with a rotation-symmetrical dCTE distribution profile, and (b) shaping the glass cylinder by softening the cylinder and forming the softened cylinder under an action of a shaping force having a force component acting in a direction perpendicular to a longitudinal axis of the glass cylinder, thereby stretching the rotation-symmetrical dCTE distribution profile in at least one direction so as to obtain a cylindrical blank which has a non-circular cross-section and a non-rotation symmetrical dCTE distribution profile, wherein providing the glass cylinder according to step (a) comprises: aa) producing a porous soot body of SiO.sub.2 and TiO.sub.2 by flame hydrolysis of starting substances containing silicon and titanium, bb) drying and sintering the soot body to form an elongated glass pre-product of Ti-doped silica glass, cc) homogenizing the glass pre-product in a homogenization process in which the pre-product is heated to a temperature of more than 1,500 C., is softened therein and is shaped into the glass cylinder.

Description

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

(1) The foregoing summary, as well as the following detailed description of the invention, will be better understood when read in conjunction with the appended drawings. For the purpose of illustrating the invention, there are shown in the drawings embodiments which are presently preferred. It should be understood, however, that the invention is not limited to the precise arrangements and instrumentalities shown.

(2) In the drawings:

(3) FIG. 1 shows a dCTE distribution profile of a SiO.sub.2TiO.sub.2 blank with a round cross-section in a false-color representation (in gray values) together with a gray value scale (in ppb/K);

(4) FIG. 2 shows the dCTE distribution profile of FIG. 1 in a section through the center in a diagram representation;

(5) FIG. 3 shows the dCTE distribution profile of FIG. 1 with plotted hypothetical CA with pentagonal form and six distribution-profile intersection lines S1 to S6;

(6) FIG. 4 shows a dCTE distribution profile according to an embodiment of the present invention which has been produced by lateral deformation, with plotted CA with pentagonal form, isolines and distribution-profile intersection lines S1 to S7;

(7) FIG. 5 is a diagram which shows the dCTE evolution of the profile of FIG. 3 along the normalized intersection lines S1 to S6;

(8) FIG. 6 is a diagram which shows the dCTE evolution of FIG. 4 along the normalized intersection lines S1 to S7;

(9) FIG. 7 is a diagram with two curves that show the dCTE evolution along the CA outline in the distribution profile of FIG. 3 and the dCTE evolution, respectively, along the CA outline in the distribution profile of FIG. 4;

(10) FIG. 8 shows the dCTE distribution profile of FIG. 3 with plotted hypothetical CA with elliptical shape and distribution-profile intersection lines;

(11) FIG. 9 shows a dCTE distribution profile according to an embodiment of the present invention which has been produced by lateral deformation, with plotted CA with elliptical shape, isolines and distribution-profile intersection lines;

(12) FIG. 10 is a diagram which shows the dCTE evolution of the profile of FIG. 8 along the normalized intersection lines;

(13) FIG. 11 is a diagram which shows the dCTE evolution of FIG. 9 along the normalized intersection lines;

(14) FIG. 12 is a diagram with two curves that show the dCTE evolution along the CA outline in the distribution profile of FIG. 8 and the dCTE evolution, respectively, along the CA outline in the distribution profile of FIG. 9;

(15) FIG. 13 shows a second dCTE distribution profile with plotted hypothetical CA with rectangular shape and three distribution-profile intersection lines S1, S2, S3;

(16) FIG. 14 shows a dCTE distribution profile according to an embodiment of the present invention which has been produced by lateral deformation of a round initial plate, with plotted CA with rectangular shape, isolines and three distribution-profile intersection lines S1, S2, S3;

(17) FIG. 15 is a diagram with three curves that show the dCTE evolution of the profile of FIG. 13 along the normalized intersection lines S1, S2, S3;

(18) FIG. 16 is diagram with three curves that show the dCTE evolution of FIG. 14 along the normalized intersection lines S1, S2, S3;

(19) FIG. 17 shows a third rotation-symmetrical dCTE distribution profile with plotted hypothetical CA with elliptical shape and distribution-profile intersection lines S1 to S6;

(20) FIG. 18 shows a dCTE distribution profile according to an embodiment of the present invention which has been produced by lateral deformation, with plotted CA with an elliptical shape, isolines and distribution-profile intersection lines;

(21) FIG. 19 is a diagram which shows the dCTE evolution of the profile of FIG. 17 along the normalized intersection lines;

(22) FIG. 20 is a diagram which shows the dCTE evolution of FIG. 18 along the normalized intersection lines; and

(23) FIG. 21 shows an embodiment of the present invention including a mirror substrate blank with triangular shape, embedded in the melt mold used for production, in a top view.

DETAILED DESCRIPTION OF THE INVENTION

First Embodiment of a Method for Producing a Cylindrical Pre-Product for a Mirror Substrate Blank

(24) A soot body doped with about 8% by wt. of TiO.sub.2 is produced preferably with the help of the OVD method by flame hydrolysis of octamethylcyclotetrasiloxane (OMCTS) and titanium isopropoxide [Ti(O.sup.iPr).sub.4] as starting substances for the formation of SiO.sub.2TiO.sub.2 particles.

(25) The soot body is dehydrated at a temperature of 1150 C. in a heating furnace with a heating element of graphite under vacuum. The dehydration treatment preferably ends after 2 hours.

(26) The soot body dried in this way is subsequently vitrified in a sintering furnace at a temperature of about 1500 C. under reduced pressure (10.sup.2 mbar) into a transparent blank consisting of TiO.sub.2SiO.sub.2 glass. The mean OH content of the glass is about 170 wt. ppm.

(27) The glass is then homogenized by thermo-mechanical homogenization (twisting) and formation of a cylinder of TiO.sub.2SiO.sub.2 glass. To this end, a rod-like start body is clamped into a glass lathe equipped with an oxyhydrogen burner and is homogenized on the basis of a shaping process, as described in EP 673 888 A1, for the complete removal of layers. In this process, the starting body is heated by means of the oxyhydrogen burner locally to more than 2,000 C. and thereby softened. In this process, the oxyhydrogen burner is fed with 1.8 mole hydrogen per 1 mole oxygen, and an oxyhydrogen flame with an oxidizing effect is thereby produced.

(28) The starting body is twisted about its longitudinal axis by relative movement of the two holders relative to each other, the softened glass mass being thoroughly mixed under formation of a twist body in radial direction over the whole length of the starting body. An elongated twisted body with a diameter of about 90 mm and a length of about 960 mm is thereby obtained.

(29) A round starting plate of TiO.sub.2SiO.sub.2 glass with a diameter of 200 mm and a thickness of 195 mm is formed from the twisted body. FIG. 1 shows a false-color representation (shades of gray) of the dCTE distribution profile measured over the thickness of the starting plate 1, in a top view on the top side of the plate. According to the gray scale shown at the right side of FIG. 1 the relative dCTE values are between zero and 5 ppb/K. This gray scale is also the basis for the false-color representation (shades of gray) of FIGS. 3, 4, 8, 9, 13, 14, 18 and 19.

(30) The distribution profile is substantially rotation-symmetrical with respect to the center axis 2 of the plate, with the dCTE values decreasing from the outside to the inside. The relative zero value (the minimum CTE value of the profile) is located on the center axis 2.

(31) This is also demonstrated in the diagram of FIG. 2, in which the dCTE distribution profile of the starting plate 1 is shown in a section through the center axis 2. The dCTE value is here plotted in ppb/K against the position P (diameter) in mm. Hence, dCTE is evenly decreasing inwardly from a value of about 5 ppb/K to zero. The total homogeneity dCTE.sub.max as the difference between the maximum inhomogeneity value on the edge and the minimum inhomogeneity value (zero) in the center is thus 5 ppb/K.

(32) The maximum difference of the CTE values is relatively small at 5 ppb/K. The differences are primarily due to variations of the TiO.sub.2 concentration and to fluctuations of the fictive temperature. At this relatively low dCTE level, slight variations in the manufacturing process, e.g. in the deposition or homogenization step, may lead to fundamentally different dCTE distribution profiles. In particular, a distribution profile which is horizontally mirrored in comparison with FIG. 1 is often obtained with a CTE maximum value in the region of the center axis and the dCTE zero value on the edge.

(33) It is, however, important for the present invention that the dCTE distribution profile of the respective starting plate is substantially rotation-symmetrical. This is ensured in the embodiment by the homogenization process.

(34) Such round starting plates are the starting point for the manufacture of different molded bodies, as shall be explained hereinafter with reference to examples.

Example 1

Manufacture of the Mirror Substrate Blank from the Pre-Product

(35) The starting plate 1 of TiO.sub.2SiO.sub.2 glass with a diameter of 200 mm and a thickness of 195 mm is shaped in a furnace by lateral shaping into a polygonal plate 4 with five corners, as is schematically shown in FIG. 4. To this end, the starting plate 1 is centrally inserted into a melt mold of graphite which has a pentagonal inner cross-section, similarly as explained further below in more detail for a shaping into a plate having a triangular cross-section. The melt mold is evacuated and heated to 1350 C. and subsequently with a ramp of 9 C./min to 1700 C., and thereafter with a ramp of 2 C./min to a temperature of 1780 C. At this temperature, the quartz glass mass softens and the softened quartz glass deforms under its own weight and the weight of a graphite plate additionally placed thereon for the purpose of acceleration and flows out in lateral direction, thereby filling the bottom of the melt mold completely.

(36) The pentagonal plate 4 (FIG. 4) obtained thereby consists of homogenized glass having a high silicic-acid content that contains 8% by wt. of titanium oxide and has a mean hydroxyl group content of around 170 wt. ppm. FIG. 4 shows a false-color representation of the dCTE distribution profile (as shade of gray with the gray scale of FIG. 1), measured over the plate thickness of the pentagonal plate 4 in a top view on the top side thereof. The dCTE distribution profile is now no longer rotation-symmetrical as in the starting plate 1, but substantially polygonal with mirror symmetry with respect to the center axis 41. Based on the total cross-section of the pentagonal plate, the dCTE extreme values with 5 ppb/k and 0 ppb/K are the same as in the starting plate 1.

(37) By comparison, FIG. 3 shows a hypothetical dCTE distribution profile of a round plate 3 which can be produced from the starting plate 1 by way of outflowing. This is not the subject of the present invention, but just serves comparative purposes. The starting plate 1 is here shaped by way of outflowing into a melt mold with a round cross-section. The starting plate 1 is here centrally mounted within the melt mold with vertically oriented longitudinal axis. The softened quartz glass flows out in lateral direction, thereby filling the bottom of the melt mold completely. The round plate 3 formed in this way has a diameter of 280 mm and a thickness of 100 mm.

(38) During the hypothetical deformation into the round plate 3, the viscous quartz-glass mass of the starting plate 1 moves in the direction of the surrounding melt mold edge and reaches the same theoretically at all points at the same time. The dCTE distribution profile of the round plate 3 is thus identicalin all sections through the center 2with the profile shown in FIG. 2.

(39) By contrast, the softened quartz-glass mass upon deformation of the starting plate 1 into the pentagonal plate 4 hits against an obstacle in at least one direction relatively early and accumulates on the obstacle, whereas otherwise it can still freely flow out in other directions. In these directions the dCTE distribution profile is thus more strongly stretched in comparison with the other direction. Therefore, the dCTE distribution profile of the pentagonal plate 4 is also almost identical with the profile shown in FIG. 2 in a section through the center axis M of the plate along the relatively short mirror axis 41, but it is more strongly stretched in the relatively long section through the center axis M of the plate along the axis 42 extending perpendicular thereto.

(40) Irrespective of this, there is similarity between the dCTE distribution profiles of the original starting plate 1, the round plate 3 and the pentagonal plate 4, in the sense that the essential features of the dCTE distribution profile in the pentagonal form, namely the number of the relative and absolute extreme values of the distribution and also their relative position to one another, are the same as in the starting plate 1 and in the round plate 3. In this respect the dCTE distribution profile of FIG. 4 already fulfills part of the above-explained basic conditions that form the basis for the design principle for a mirror substrate blank according to this invention. It is not rotation-symmetrical, but by way of geometric transformation and in consideration of simple stretching deformations it can be transformed mathematically uniquely into the originally rotation-symmetrical distribution of the round form of the starting plate 1.

(41) Moreover, by comparison with the original, rotation-symmetrical distribution profile, the dCTE distribution profile of FIG. 4 shows an interaction with a comparatively insignificant impairment of the imaging quality of the mirror in cooperation with a non-circular, optically used area CA. This shall be explained in more detail hereinafter:

(42) (a) Bandwidth of Intersection Lines

(43) As a supplement to FIG. 1, the outline L(CA) of an optically used area CA with pentagonal form and rounded edges is plotted in FIG. 3. Within CA a plurality of intersection lines S1 to S6 extending through the center 2 are plotted. The center 2 is the centroid of the area of CA at the same time.

(44) When dCTE values are mentioned hereinafter, these refer to the region within CA. The dCTE values are thus calculated as the amount of the local deviation from an absolute minimum value CTE.sub.min of the CTE distribution profile within CA (dCTE=CTECTE.sub.min).

(45) The diagram of FIG. 5 shows the dCTE distribution profiles along the intersection lines S1 to S6 (within CA; only the most prominent intersection lines of relevance to the assessment are provided with reference numerals). On the ordinate of the diagram, the relevant dCTE value is plotted (in ppb/K) and, on the abscissa, a position value P normalized to the respective intersection line length (in relative unit), which thereby covers the range of 0 to 1. The maximum inhomogeneity dCTE.sub.max within CA is thus about 3.4 ppb/K. The minimum value dCTE.sub.min within CA is zero by definition. Distinctly different dCTE values are found in the region of the edge positions 0 and 1 between the section profiles K(S3) and K(S5) on the one hand and the section profiles K(S1) and K(S6) on the other hand. In the maximum (marked with block arrow 51) the difference is about 2.8 ppb/K. The maximum 51 represents the maximum bandwidth of the family of curves formed by the curves K(S1) to K(S6); it is about 82% of dCTE.sub.max (3.4 ppb/K).

(46) The same outline L(CA) of the optically used area CA with pentagonal form and rounded-off edges is schematically plotted also in the distribution profile of FIG. 4. The centroid of the area of CA is in point M. The diagram of FIG. 6 shows the dCTE distribution profiles along some prominent intersection lines S1 to S6 (within CA) in the case of the dCTE distribution profile of FIG. 4. In this case, too, the corresponding dCTE value is plotted (in ppb/K) on the ordinate of the diagram, and the position value normalized to the respective intersection line length on the abscissa. It is evident from this that all section profiles K(S1) to K(S6) are relatively similar and form a small curve band. The maximum inhomogeneity dCTE.sub.max within CA is thus about 1.6 ppb/K. The maximum difference of the dCTE values (marked with block arrow 61) is only about 0.75 ppb/K. Consequently, the maximum bandwidth of the family of curves formed by the curves K(S1) to K(S6) (maximum 61) is here about 47% of dCTE.sub.max. Moreover, it has been possible through the use of the same start material to reduce dCTE.sub.max within the CA area from 3.4 ppb/K (see FIGS. 3 and 5) to 1.6 ppb/K (see FIGS. 4 and 6), which is accompanied by an improved imaging quality in the application as a mirror for EUV lithography.

(47) (b) Height Difference on the CA Outline L(CA)

(48) The diagram of FIG. 7 shows a comparison of the evolutions of the dCTE distribution profile along the CA outlines L(CA) in the blanks of FIGS. 3 and 4, each starting with a clockwise circulation and ending at position P0 (position P0 marked in FIGS. 3 and 4). On the y-axis, the dCTE value is plotted in ppb/K against the position P normalized to the respective length of the outline L(CA) (in relative unit) (the lengths of the outlines L(CA) are identical in this case).

(49) Curve U1 represents the evolution of the dCTE distribution profile along the CA outline L(CA) in the blank of FIG. 3. This yields a difference PV.sub.CA between the dCTE extreme values (maximum value (3.4 ppb/K) and minimum value (0.7 ppb/K) of 2.7 ppb/K along the CA outline L(CA). PV.sub.CA is thus about 0.79dCTE.sub.max. Curve U2 represents the evolution of the dCTE distribution profile along the CA outline L(CA) in the blank of FIG. 4. This yields a difference PV.sub.CA between the dCTE extreme values (maximum value (1.6 ppb/K) and minimum value (1.2 ppb/K) of 0.4 ppb/K along the CA outline L(CA). PV.sub.CA is thus about 0.25dCTE.sub.max.

(50) By comparison with the dCTE distribution profile of FIG. 3, the mirror substrate blank 4 shows a better imaging behavior upon optical exposure with the non-circular CA. This may be due to the fact that the dCTE distribution profile is well fitted to the shape of CA and designed such that the CA outline L(CA) is intersected by a number of isolines of the distribution profile that is as small as possible.

(51) (c) Extension of the Isoline with the Level 0.5dCTE.sub.max

(52) An isoline H1 for the dCTE value of 0.5dCTE.sub.max is schematically plotted in FIG. 4. It is discernible that the isoline H1 extends with its whole isoline length within the outline L(CA) of the optically used area CA. This means that the shape of the dCTE inhomogeneity profile in the mirror substrate blank 4 of FIG. 4 is matched to the outline L(CA) to the extent that the CA outline L(CA) is intersected by a number of isolines of the distribution profile that is as small as possible.

(53) Thus blank 4 in combination with its specific, optically used area CA in pentagonal form meets all conditions of the general design principle according to the invention, namely the demand made on the dCTE distribution profile of the blank and the interaction between the dCTE distribution profile and the optically exposed area CA with non-circular outline L(CA).

Example 2

(54) In a further example of the invention, the starting plate 1 of TiO.sub.2SiO.sub.2 glass with a diameter of 200 mm and a thickness of 195 mm is shaped in a furnace by lateral shaping into a plate 9 with oval cross-section, as is schematically shown in FIG. 9. To this end the starting plate 1 is centrally inserted into a melt mold of graphite which has an oval inner cross-section. Otherwise, the lateral deformation by softening and outflowing into the melt mold is carried out in the way as has already been explained with reference to Example 1. In comparison with the starting plate 1 of FIG. 1, the ratio of the long oval half-axis 91 to the short axis 92 is about 1.45.

(55) The oval plate 9 obtained thereby consists of homogenized glass having a high-silicic acid content, which contains 8% by wt. of titanium oxide and has a mean hydroxyl group content of around 170 wt. ppm.

(56) The false-color representation of FIG. 9 (as shade of gray with the gray scale of FIG. 1) almost shows mirror symmetry with respect to the main axes 91 and 92 of the oval. The dCTE extreme values (5 ppb/K) and (0 ppb/K) within the whole area of the blank 9 are the same as in the starting plate 1. There is similarity between the dCTE distribution profiles of the starting plate 1 and the plate 9 with oval cross-section in the sense that the essential features of the dCTE distribution profile in the oval form, namely the number of the relative and absolute extreme values of the distribution and also their relative mutual position, are the same with respect to one another. In this respect, the dCTE distribution profile of FIG. 9 already fulfills part of the above-explained basic conditions that form the basis for the design principle for a mirror substrate blank according to this invention. It is not rotation-symmetrical, but by way of geometric transformation and in consideration of simple stretching deformations it can be transformed mathematically uniquely into the originally rotation-symmetrical distribution.

(57) Moreover, by comparison with a rotation-symmetrical distribution profile, the dCTE distribution profile of FIG. 9 shows an interaction with a comparatively insignificant impairment of the imaging quality of the mirror in cooperation with a non-circular, optically used area CA. This shall be explained in more detail hereinafter:

(58) (a) Bandwidth of Intersection Lines

(59) By comparison, FIG. 8 shows the round plate 3 again (as explained with respect to FIG. 3). The outline L(CA) of an optically used area CA is schematically plotted therein, which outline could be called ellipsoidal. Within CA, a plurality of intersection lines S1 to S5 that extend through the center 2 are plotted. The center 2 simultaneously forms the centroid of the area of CA.

(60) The diagram of FIG. 10 shows the dCTE distribution profiles along the intersection lines S1 to S5 (within CA; only the most prominent lines that are of importance to the determination of the width of the family of curves are provided with reference numerals). On the ordinate of the diagram, the relevant dCTE value is plotted (in ppb/K) and, on the abscissa, a position value P normalized to the respective intersection line length (in relative unit), which thereby covers the range of 0 to 1. Distinctly different dCTE values in the region of the edge positions 0 and 1 are found between the section profiles K(S1) to K(S5). In the bandwidth maximum (marked with block arrow 101), the difference between minimum value and maximum value is about 3.1 ppb/K. The maximum 101 represents the maximal bandwidth of the family of curves formed by the curves K(S1) to K(S5); it is about 79% of dCTE.sub.max.

(61) The same outline L(CA) of the optically used area CA with elliptical form is schematically plotted also in the distribution profile of FIG. 9. The centroid of the area of CA lies in point M. The diagram of FIG. 11 shows the dCTE distribution profiles along intersection lines S1 to S5 (within CA) in the case of the dCTE distribution profile of FIG. 9. In this case, too, the corresponding dCTE value is plotted (in ppb/K) on the ordinate of the diagram, and the position value P normalized to the respective intersection line length (in relative units) on the abscissa. It is evident from this that all section profiles K(S1) to K(S6) are relatively similar and form a small curve band. The maximum difference of the dCTE values (marked with block arrow 111) is only about 0.6 ppb/K. The maximum inhomogeneity dCTE.sub.max within CA is thus about 1.55 ppb/K. Consequently, the maximum bandwidth of the family of curves formed by the curves K(S1) to K(S5) (maximum 111) is here about 39% of dCTE.sub.max.

(62) (b) Height Difference on the CA Outline L(CA)

(63) The diagram of FIG. 12 shows a comparison of the evolutions of the dCTE distribution profile along the CA outlines L(CA) in the blanks of FIGS. 8 and 9, each starting with a clockwise circulation and ending at position P0. On the y-axis, the dCTE value is plotted (in ppb/K) against the position P normalized to the respective length of the outline L(CA) (the lengths of the outlines L(CA) are here also identical).

(64) Curve U1 represents the evolution of the dCTE distribution profile along the CA outline L(CA) in the blank of FIG. 8. This yields a difference PV.sub.CA between the dCTE extreme values (maximum value (4 ppb/K) and minimum value (0.8 ppb/K) of 3.2 ppb/K along the CA outline L(CA). PV.sub.CA is thus about 0.80dCTE.sub.max.

(65) Curve U2 represents the evolution of the dCTE distribution profile along the CA outline L(CA) in the blank of FIG. 9. This yields a difference PV.sub.CA between the dCTE extreme values (maximum value (1.5 ppb/K) and minimum value (1 ppb/K) of 0.5 ppb/K along the CA outline L(CA). PV.sub.CA is thus about 0.33dCTE.sub.max.

(66) (c) Extension of the Isoline with the Level 0.5dCTE.sub.max

(67) An isoline H1 for the dCTE value of 0.5dCTE.sub.max is schematically plotted in FIG. 9. It is discernible that the isoline H1 extends with almost its whole isoline length (with more than 80% of its length) within the outline L(CA) of the optically used area CA. This means that the shape of the dCTE inhomogeneity profile in the mirror substrate blank 9 of FIG. 9 is matched to the outline L(CA) to the extent that the CA outline L(CA) is intersected by a number of isolines of the distribution profile that is as small as possible.

(68) Thus, blank 9 in combination with its specific, optically used area CA also satisfies all conditions of the general design principle according to the invention, namely the demand made on the dCTE distribution profile of the blank and the interaction between the dCTE distribution profile and the optically exposed area CA with non-circular outline L(CA).

Second Embodiment of Method for Producing a Cylindrical Pre-Product for a Mirror Substrate Blank

(69) A soot body which is doped with about 8% by wt. of TiO.sub.2 is produced with the help of the OVD method by flame hydrolysis of octamethylcyclotetrasiloxane (OMCTS) and titanium isopropoxide [Ti(O.sup.iPr).sub.4] as start substances for the formation of SiO.sub.2TiO.sub.2 particles. By comparison with the above-explained first procedure, the surface temperature of the soot body is kept slightly higher during the whole deposition process. This small difference leads to a different distribution of the TiO.sub.2 concentration.

Example 3

(70) The starting plate of TiO.sub.2SiO.sub.2 glass with a diameter of 200 mm and a thickness of 195 mm is shaped in a furnace by lateral shaping into a rectangular plate with the lateral dimensions 400 mm250 mm and a thickness of 60 mm. To this end the starting plate is centrally inserted into a melt mold of graphite which has a rectangular inner cross-section with a short side b=250 mm and a long side a=400 mm. Otherwise, the lateral deformation by softening and outflowing into the melt mold is carried out as has already been explained with reference to Example 1.

(71) The rectangular plate 14 obtained thereby consists of homogenized glass having a high-silicic acid content, which contains 8% by wt. of titanium oxide and has a mean hydroxyl group content of around 170 wt. ppm.

(72) The false-color representation of FIG. 14 (as shade of gray with the gray scale of FIG. 1) shows the dCTE distribution profile measured over the plate thickness of the rectangular plate 14 with the plate sides a and b in a top view on the top side thereof. The dCTE distribution profile is now no longer rotation-symmetrical as in the round plate 13 made of the same starting plate, as is schematically shown for comparison in FIG. 13, but it has an approximately 2-axis symmetry with the two main axes a and b of the rectangular plate 14 as the mirror axes. In this profile the dCTE values are increasing from the outside to the inside up to the center axis M. The relative extreme value dCTE.sub.max, viewed over the whole blank, is 5 ppb/K.

(73) The deformation of the starting plate in relation to the rectangular plate 14 differs from the deformation in relation to the round plate 13 (as shown in FIG. 13; the dCTE distribution profile thereof corresponds to that of the starting plate) in that in the direction of the long plate side a stronger stretching (stretching factor in relation to the starting plate=2) has taken place than in the direction of the short plate side b (stretching factor=1.25). Therefore, the dCTE distribution profile in a section through the center axis M of the plate along the long plate side a in comparison with the profile of the round plate 13 of FIG. 13 is stretched, and in the direction of the short plate side b it is almost identical with the profile shown in FIG. 13.

(74) Similarity exists between the dCTE distribution profiles of the original starting plate and the rectangular plate 14 in the sense that the essential features of the dCTE distribution profile in the rectangular form, namely the number of the relative and absolute extreme values of the distribution and also the mutual relative position thereof are the same as in the starting plate. In this respect, the dCTE distribution profile of FIG. 14 already fulfills part of the above-explained basic conditions that form the basis for the design principle for a mirror substrate blank according to this invention. It is not rotation-symmetrical, but by way of geometric transformation and in consideration of simple stretching deformations it can be transformed mathematically uniquely into the originally rotation-symmetrical distribution of the round form of the starting plate.

(75) Moreover, by comparison with this rotation-symmetrical distribution profile, the dCTE distribution profile of FIG. 14 shows an interaction with a comparatively slight impairment of the imaging quality of the mirror in cooperation with a non-circular, optically used area CA. This shall be explained in more detail hereinafter:

(76) (a) Bandwidth of Intersection Lines

(77) In FIG. 13, the outline L(CA) of an optically used area CA with rectangular form and rounded-off edges is schematically plotted. Within CA, three intersection lines S1, S2 and S3 that extend through the center 2 are plotted. S1 extends along the short axis of the rectangular form, S2 along the long axis, and S3 along the diagonal.

(78) The diagram of FIG. 15 shows the dCTE distribution profiles along the intersection lines S1 to S3 (within CA). On the ordinate of the diagram, the relevant dCTE value is plotted (in ppb/K) and, on the abscissa, a position value P normalized to the respective intersection line length (in relative units), which thereby covers the range of 0 to 1. It follows that the section profiles K(S2) and K(S3) along the long rectangle axis S2 and along the diagonal S3, respectively, are similar and almost come to lie one upon the other, but both differ considerably from the section profile K(S1) along the short rectangle axis S1. In the respective center region the dCTE values of all sections show a common maximum value with dCTE.sub.max=4.5 ppb/K (calculated as the difference between maximum value (5 ppb/K) and minimum value (0.5 ppb/K) within CA). At the edge positions 0 and 1, however, there is a great difference that is about 3.3 ppb/K in the maximum (marked with block arrow 151). The maximum 151 represents the maximum bandwidth of the family of curves formed by curves K(S1), K(S2), and K(S3), which is about 73% of dCTE.sub.max.

(79) The outline L(CA) of the optically used area CA with rectangular form and rounded-off edges is schematically plotted also in the distribution profile of FIG. 14. The centroid of the area of CA lies in the center axis M. The diagram of FIG. 16 shows the dCTE distribution profiles along intersection lines S1 to S3 (within CA) in the case of the dCTE distribution profile of FIG. 14 through the centroid of the area (center M).

(80) In this case, too, the corresponding dCTE value is plotted (in ppb/K) on the ordinate of the diagram, and the position value P normalized to the respective intersection line length (in relative unit) on the abscissa. It is evident from this that the section profiles K(S2) and K(S3) are similar, but differ from the section profile K(S1) along the short rectangle axis S1 less clearly than in the diagram of FIG. 15. The common maximum value dCTE.sub.max within the CA area of the sections lies again in the center region, but it is only 2 ppb/K (calculated as the difference between maximum value (5 ppb/K) and minimum value (3 ppb/K) within CA); the maximum difference of the dCTE values of the family of curves at a position (marked with block arrow 161) is only about 0.7 ppb/K. Consequently, the maximum bandwidth of the family of curves formed by the curves K(S1), K(S2), and K(S3) is here about 35% of dCTE.sub.max. Due to the smaller variations in dCTE and the more uniform profiles of the sections S1 to S3 this distribution profile is distinguished in comparison with FIG. 13/FIG. 15 by a better adaptation to CA and better imaging properties.

(81) (b) Extension of the Isoline with the Level 0.5dCTE.sub.max

(82) Isolines H1, H2, H3 are schematically plotted in FIG. 14; these indicate dCTE values at the same level. Here,

(83) isoline H3 represents a dCTE value of 1.2dCTE.sub.max,

(84) isoline H2 represents a dCTE value of 0.8dCTE.sub.max, and

(85) isoline H1 represents a dCTE value of 0.5dCTE.sub.max.

(86) It is evident that isoline H1 extends with its whole isoline length within the outline L(CA) of the optically used area CA. Isoline H2 extends with about 15% of its total isoline length outside of CA, whereas isoline H3 extends fully outside of CA, but it does also not belong to the optically exposed area CA.

(87) This means that the contour of the dCTE inhomogeneity profile in the case of the mirror substrate blank of FIG. 14 is matched to the outline L(CA) to the extent that the CA outline L(CA) is intersected by a number of isolines of the distribution profile that is as small as possible.

(88) Thus the blank 14 in combination with its specific, optically used area CA satisfies all conditions of the general design principle according to the invention, namely the demand made on the dCTE distribution profile of the blank and the interaction between the dCTE distribution profile and the optically exposed area CA with non-circular outline L(CA).

Third Embodiment of Method for Producing a Cylindrical Pre-Product for a Mirror Substrate Blank

(89) A soot body which is doped with about 8% by wt. of TiO.sub.2 is produced with the help of the OVD method by flame hydrolysis of octamethylcyclotetrasiloxane (OMCTS) and titanium isopropoxide [Ti(O.sup.iPr).sub.4] as start substances for the formation of SiO.sub.2TiO.sub.2 particles.

(90) In contrast to the above-explained procedures, the surface temperature of the soot body is here slightly varied during the whole deposition process. This small difference leads to a different distribution of the TiO.sub.2 concentration.

Example 4

(91) The starting plate of TiO.sub.2SiO.sub.2 glass with a diameter of 200 mm and a thickness of 195 mm is shaped in a furnace by lateral shaping into an oval plate 18, as schematically shown in FIG. 18. To this end the starting plate is centrally inserted into a melt mold of graphite which has an oval inner cross-section, similarly as already explained for Example 1. The lateral deformation by softening and outflowing into the melt mold is carried out as has been explained with reference to Example 1. The stretching factor in the direction of the long oval half-axis 181 as compared with the round plate 17 of the same starting plate, which is schematically shown in FIG. 17, is about 1.5; the stretching factor in the direction of the short axis 182 is virtually equal to 1.

(92) The oval plate 9 obtained thereby consists of homogenized glass having a high-silicic acid content, which contains 8% by wt. of titanium oxide and has a mean hydroxyl group content of around 170 wt. ppm.

(93) The false-color representation in FIG. 18 (as shade of gray with the gray scale of FIG. 1) shows approximately a mirror symmetry with respect to the main axes 181 and 182 of the oval. The dCTE extreme values (5 ppb/K) and (0 ppb/K) are the same as in the original starting plate and in the previously described Examples 1 to 3. Similarity exists between the dCTE distribution profiles of the original starting plate and the plate 18 with oval cross-section in the sense that the essential features of the dCTE distribution profile in the oval form, namely the number of the relative and absolute extreme values of the distribution and also the mutual relative position thereof are the same. In this respect the dCTE distribution profile of FIG. 18 already fulfills part of the above-explained basic conditions that form the basis for the design principle for a mirror substrate blank according to this invention. It is not rotation-symmetrical, but by way of geometric transformation and in consideration of simple stretching deformations it can be transformed mathematically uniquely into the originally rotation-symmetrical distribution of the round starting plate.

(94) Moreover, by comparison with this rotation-symmetrical distribution profile (which is also manifested by the round plate 17 of FIG. 17), the dCTE distribution profile of FIG. 18 shows an interaction with a comparatively slight impairment of the imaging quality of the mirror in cooperation with a non-circular, optically used area CA. This shall be explained in more detail hereinafter:

(95) (a) Bandwidth of Intersection Lines

(96) In FIG. 17, the outline L(CA) of an optically used area CA is schematically plotted; this outline can also be called elliptical. Within CA, a plurality of intersection lines that extend through the center 2 are plotted, of which the specifically characteristic intersection lines are designated with the reference numerals S1 to S4. The center 2 simultaneously forms the centroid of the area of CA.

(97) The diagram of FIG. 19 shows the dCTE distribution profiles along a representative selection of the intersection lines S1 to S4 (within CA). On the ordinate of the diagram, the relevant dCTE value is plotted (in ppb/K) and, on the abscissa, a position value P normalized to the respective intersection line length (in relative units), which thereby covers the range of 0 to 1. The maximum inhomogeneity dCTE.sub.max within CA is 3.8 ppb/K. Clearly different dCTE values in the region of the edge positions 0 and 1 are found between the section profiles K(S1) and K(S3)/K(S4). In the maximum (marked with block arrow 191), the difference is about 2.8 ppb/K. The maximum 191 represents the maximum bandwidth of the family of curves formed by curves K(S1) to K(S4); it is about 74% of dCTE.sub.max.

(98) The outline L(CA) of the optically used area CA with elliptical form is schematically plotted also in the distribution profile of FIG. 18. The centroid of the area of CA lies in point M. The diagram of FIG. 20 shows the dCTE distribution profiles along the intersection lines S1 to S4 (within CA) in the case of the dCTE distribution profile of FIG. 18. In this case, too, the corresponding dCTE value is plotted (in ppb/K) on the ordinate of the diagram, and the position value P normalized to the respective intersection line length (in relative units) on the abscissa. It is evident from this that all of the section profiles K(S1) to K(S4) are relatively similar and form a small curve band. The maximum difference of the dCTE values (marked with block arrow 201) is about 1.3 ppb/K; dCTE.sub.max is 3.8 ppb/K. The maximum bandwidth of the family of curves formed by the curves K(S1) to K(S4) (maximum 201) is thus about 34% of dCTE.sub.max.

(99) The differences in the widths of the bands of the families of curves of FIGS. 19 and 20 are in this case not as striking as in the previous Examples 1 to 3. The amplitudes between dCTE maximum value (about 3.8 ppb/K) and minimum value (0) are also almost identical. It is nevertheless evident that the profile sections K(S1) to K(S4) in the case of the dCTE distribution profile of FIG. 20 within CA show a greater similarity than in the profile of FIG. 19. The greater similarity of the profile sections produces a more uniform distortion of the mirror substrate blank of the invention upon heating and thus accomplishes an easier correction of the imaging errors created thereby.

(100) (b) Extension of the Isoline with the Level 0.5dCTE.sub.max

(101) The dCTE distribution profile of FIG. 18 has two isolines at the level 0.5dCTE.sub.max; a further inwardly extending isoline H2, and an outer isoline fully surrounding the inner isoline. In such profiles the isoline that extends closest to the outline of CA is of relevance to the quality of the adaptation to the non-rotation symmetrical CA. In the profile of FIG. 18, this is the inner isoline H2. In other cases in which it cannot unambiguously be determined which isoline with the level 0.5dCTE.sub.max extends closer to CA or in the case of which such an isoline does not exist, this criterion for adjusting and assessing the adaptation quality of the dCTE distribution profile to CA is not applicable.

(102) In the case of the distribution profile of FIG. 18, it is discernible that the inner isoline H1 with its total isoline length extends within the outline L(CA) of the optically used area CA. This means that the form of the dCTE inhomogeneity profile in the mirror substrate blank 18 of FIG. 18 is also adapted to the outline L(CA) to the extent that the CA outline L(CA) is intersected by a number of isolines of the distribution profile that is as small as possible.

(103) Thus, blank 18 in combination with its specific, optically used area CA also satisfies all conditions of the general design principle according to the invention, namely the demand made on the dCTE distribution profile of the blank and the interaction between the dCTE distribution profile and the optically used area CA with non-circular outline L(CA).

(104) For the manufacture of a mirror substrate, the top side of the mirror substrate blank is subjected to a mechanical treatment, which includes grinding and polishing. A convexly curved surface area is e.g. produced, of which for instance a pentagonal sub-area, as shown in FIG. 4, is specified as a highly exposed surface area CA with particularly high demands made on the quality of the surface and on the homogeneity of the quartz glass. The curved surface of the mirror substrate is provided with a mirror layer and the mirror element obtained is used in a projection system for EUV lithography.

(105) In all examples, the mirror substrate blank according to the invention has been produced by lateral deformation (stretching) of a round plate. This procedure for a mirror substrate blank 210 with triangular form shall be explained in more detail with reference to FIG. 21. The dCTE distribution profile 100 produced in the blank 210 has a threefold symmetry, wherein the corners of the triangle are widened due to the manufacturing process and removed later.

(106) For the manufacture of the blank 201, a graphite mold 211 is used with an inner geometry that is triangular, but with bulges 212 on the triangle tips. A round plate (outlined by way of the dotted circle 213) is inserted into the graphite mold such that the center axes 214 of round plate 213 and graphite mold 211 extend concentrically. In the case of a non-concentric arrangement, which would alternatively be suited for the production of a mirror substrate blank, this would lead to a deviation from the threefold symmetry of the dCTE distribution profile as is here desired.

(107) The inner geometry of the graphite mold 211 is filled by heating, softening and outflowing of the round plate 213 (as has been explained above with reference to the example of FIG. 4). Due to the bulges 212 of the graphite mold 211 there are flow processes on the corners of the mirror substrate blank 210, which flow processes lead to a stronger shaping of the corners of the resulting dCTE distribution profile, as shown by the contour line 219. In the top view on the optically used area, the contour line 219 outlines the form of a Wankel engine piston or a plectrum. The contour line 219 just symbolizes the form of the dCTE distribution profile (as e.g. defined by a dCTE isoline value which is still within CA). The dCTE distribution profile is also continued outwards outside of the contour line 219.

(108) General Considerations for a Special Case of CA

(109) Special cases of a mirror substrate blank of the invention in which the optically used area CA is substantially defined by two axes a, b that are perpendicular to each other and have different lengths (a>b) have been explained with reference to FIGS. 9 and 14. In such special cases, the CTE inhomogeneity evolution dCTE.sub.a or dCTE.sub.b, respectively, in the direction of the two axes can be generally described in conformity with the above-mentioned DE 10 2004 024 808 A1 by the following formulae (2) and (3):
dCTEaC.sub.0+C.sub.1(x/a)+C.sub.2(2(x/a).sup.21)+C.sub.3(6(x/a).sup.46(x/a).sup.2+1)(2)
dCTEb=C.sub.0+C.sub.2(2(y/b).sup.21)+C.sub.3(6(y/b).sup.46(y/b).sup.2+1)(3)
wherein after deduction of dCTE.sub.a from the really existing CTE inhomogeneity distribution over CA a minimal residual inhomogeneity of not more than 0.5 ppb/K remains. The parameters in formulae (2) and (3) mean:
a=long axis, b=short half-axis and
x=distance along axis a,
y=distance along axis b,
C.sub.0, C.sub.1, C.sub.2, C.sub.3=adaptation parameters of the spherical Zernike terms.

(110) However, this general description of non-rotation symmetrical distribution profiles does not replace the above-explained additional requirements regarding the similarity of profile sections through the center of mass of the respective, optically used, non-round CA.

(111) It will be appreciated by those skilled in the art that changes could be made to the embodiments described above without departing from the broad inventive concept thereof. It is understood, therefore, that this invention is not limited to the particular embodiments disclosed, but it is intended to cover modifications within the spirit and scope of the present invention as defined by the appended claims.