Abstract:
A device for improving resolution capability of an x-ray optical apparatus for an x-ray incident from a direction of incidence includes a mirror element including a mirror edge formed as a cylindrical shell section around an edge axis. The mirror element is spaced apart, in a radial direction, from a focal axis that is parallel to the direction of incidence. The edge axis is oriented at a first non-zero angle relative to the focal axis when viewed along a radial axis. The edge axis is oriented at a second non-zero angle relative to the focal axis.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
   The present application claims priority under 35 U.S.C. § 119 of German Patent Application No. 10 2006 051 912.4, filed Oct. 31, 2006, the disclosure of which is hereby expressly incorporated by reference herein in its entirety. 
   BACKGROUND OF THE INVENTION 
   1. Field of the Invention 
   The invention relates to a device for improving the spatial resolution of a micropore optics system for x-rays. 
   2. Discussion of Background Information 
   In the construction of a telescope for x-rays the problem arises that no suitable lenses exist for x-ray radiation because of the low refraction and the strong absorption in matter. Mirrors in the conventional sense cannot be used either, since the reflectivity for x-rays, unlike for visible light, is too low by far. Adequate reflectivity values result only for very large angles of incidence close to 90 degrees. This effect can be used to build a reflecting telescope for x-rays, provided that suitably designed surfaces are found. The x-rays must thereby strike the reflecting surface at a very small angle (e.g., grazing incidence), since x-rays are reflected by polished surfaces only when the incidence of the rays is almost grazing. One possibility for realizing an x-ray telescope is therefore to use a parabolic reflector. However, the parabolic reflector has very large image errors under the conditions of a grazing incidence. 
   A Wolter Type I telescope is known from the prior art (see, e.g., the publication “Spiegelsysteme streifenden Einfalls als abbildende Optiken für Röntgenstrahlen,” H. Wolter, Analen der Physik, 10, 1952, p. 94-114). A telescope of this type utilizes the reflection of x-ray radiation with grazing incidence on metal surfaces. The basic concept is that a hyperboloid is placed behind the paraboloid as a correcting reflector, on which hyperboloid the x-rays are reflected for a second time. 
   The mirror arrangement of the Wolter Type I (e.g., Wolter-I) is composed of metallic (e.g., often comprising only coated foils) paraboloids of revolution multiply nested within one another, each of which is followed by a hyperboloid of revolution. These mirrors together have similar imaging properties like conventional telescopes in the visible range of light. The rays are first reflected on a small section of a parabolic reflector and subsequently on a section of a hyperbolic reflector. In order to achieve greater intensities, several mirror systems of this type were nested within one another, since, due to the grazing incidence, each pair of mirrors has only a very narrow range in which it can collect x-ray light and focus it in the focal point. For example, in the mirror system of the ROSAT x-ray satellite, four Wolter double mirrors with the same focal length are nested within one another in order to obtain a large collecting area. 
   An approximation of the Wolter-I optics is known from the prior art. The approximation uses several stacks of cylindrical areas with single tilt, which replaces the paraboloids and hyperboloids. This type of approximation can be tolerated if large focal lengths are chosen. 
   Furthermore, an x-ray lens has hitherto been produced by a pore optics system, the reflecting surfaces of which an ideal Wolter-I optics system approximates through two cylindrical areas. A pore optics system of this type is shown in  FIGS. 1A and 1B . An approximation occurs for production-relevant reasons. In the pore optics system, the cylindrical mirror shells  12  are applied layer by layer on a cylindrical base  10 , as shown in  FIG. 1A  (see  FIG. 1B ). A mirror shell is polished on the front face and provided on the back with many webs  14 . The webs  14  of the mirror shell  12  last applied are connected to the mirror surface of the mirror shell  12  lying underneath, so that the last mirror surface is curved exactly like the one underneath it. This production method requires the spaces remaining between the webs  14  and the mirror shells  12 , the pores, to have a rectangular cross section. 
   The advantage of a pore optics system is to be able to produce many mirror shells precisely and to mount them one behind the other. The mirror shells are connected to one another by webs, which leads to the geometry of many small pores. However, one disadvantage of the prior art is that the spatial resolution of the x-ray optics of known solutions no longer meets current requirements. 
   SUMMARY OF THE INVENTION 
   Embodiments of the present invention provide a device for an x-ray optics system that achieves an improved spatial resolution compared to the prior art. 
   According to aspects of the invention, there is a device for improving the resolution capability of an x-ray optical apparatus for an x-ray incident from a direction of incidence. In embodiments, the device comprises a mirror element with a mirror edge (e.g., slope, side, flank, etc.), the first mirror edge being formed by a first cylindrical shell section around an edge axis. The mirror element is arranged spaced apart with respect to a focal axis parallel to the direction of incidence by a focal point of the x-ray optical apparatus in a radial direction. The mirror element is arranged rotated about an axis extending in the radial direction with respect to the direction of incidence such that the edge axis is tilted with respect to the direction of incidence. 
   In implementations of the invention, an approximation of the parabolic and hyperbolic form can be achieved through a rotation of the mirror element about the radial axis an approximation of the parabolic and hyperbolic form can be achieved. According to aspects of the invention, this approximation is closer to the optimal form than is rendered possible by a simple approximation of cylindrical shells. 
   A device according to aspects of the invention has the advantage that it can lead to an improvement in the spatial resolution of an x-ray image, which can have a wide field of application with a broad use of x-ray optical devices. In other words, one advantage of a device according to aspects of the invention is that it results in less blurring of the image, which in turn leads to a better image quality. The desired reduction of the image blurring can be dependent on the stack length and the focal length. In implementations of the invention, the improvement in the resolution can be, for example, in the range of a factor of 3. 
   According to embodiments of the invention, a second mirror edge can be provided adjacent to the mirror edge, which second mirror edge is formed about a second edge axis by a second cylindrical shell section. The mirror element can be arranged such that a plane comprising the edge axis and second edge axis is tilted with respect to the direction of incidence. In this manner, the transition between the first and second mirror edge is better approximated, and the mirror edge the second mirror edge can be produced cost-effectively by cylinder approximation. 
   In accordance with aspects of the invention, in order to produce a mirror element that corresponds particularly well to the Wolter-I optics, the mirror edge can correspond to an approximation of a parabolic form and the second mirror edge to an approximation of a hyperbolic form. 
   In accordance with further aspects of the invention, in order for the mirror element to represent a particularly good approximation of the Wolter-I optics, the mirror element can have a width that is smaller than approximately a tenth of the radial distance of the mirror element regarding the focal axis. This can ensure that the approximation range does not become too large, so that the approximation does not become inadmissible. 
   According to embodiments of the invention, the mirror element can have a width that corresponds to an arc length of less than approximately two degrees in the radial direction. This range of the width of the mirror element provides a better approximation of the form of the Wolter-I optics, since the range to be approximated is very small compared to the entire parabolic and hyperbolic form of the Wolter-I optics. In this manner, the approximation does not cause any major errors. 
   According to embodiments of the invention, an incline between the edge axis and the direction of incidence can be in a range between approximately half a degree and approximately five degrees. In accordance with aspects of the invention, this provides a particularly good inclination range for improving the resolution capability of the x-ray optical apparatus. 
   In embodiments, in order to achieve a further improvement of the resolution capability of the x-ray optical apparatus, another mirror element with a third mirror edge and a fourth mirror edge adjacent to the third mirror edge can be provided. The third mirror edge is formed by a third cylindrical shell section around a third edge axis and the other mirror element is arranged spaced apart with respect to the focal axis in another radial direction. The other mirror element is furthermore arranged as being rotated about another axis extending in the other radial direction with respect to the direction of incidence, such that the third edge axis is tilted with respect to the direction of incidence. Through the provision of another mirror element of this type, an improvement of the yield of the incident x-rays can thus be achieved. 
   According to embodiments of the invention, the other mirror element can be adjacent to the mirror element and be arranged at a distance from the focal axis that corresponds to the spacing of the mirror element from the focal axis. Moreover, a lateral transition between the mirror element and the other mirror element can have a stepped offset. Through this tilted arrangement of the mirror elements, the area of the vertical expansion of the border line between the first and second mirror edge or the third and fourth mirror edge can be kept in a very narrow range. In this way, incident x-rays on both mirror elements can be deflected to a very small focal area or focal point. If the arrangement of the mirror elements were chosen such that the border lines between the first and the second mirror edge and the third and the fourth mirror edge touched, an arrangement of this type would not cause an optimal focusing on a joint focal point. 
   According to embodiments of the invention, it can be advantageous if a device according to aspects of the invention comprises a plurality of additional mirror elements that form a ring of mirror elements around the focal axis. This causes x-rays from a plurality of mirror elements to be deflected to a single focal range or focal point, which in turn increases the intensity of the light spot in the focal point. Accordingly, a better detection or evaluation capability of the incident x-rays is possible. 
   A device according to embodiments of the invention can also have an additional mirror element, which is arranged spaced apart from the focal axis in the radial direction. A spacing of the additional mirror element from the focal axis is larger than the spacing of the mirror element from the focal axis. In particular, a device of this type is advantageous when the additional mirror element has two mirror edges that are tilted with respect to one another so that an x-ray incident in the direction of incidence is reflected to an essentially identical focal point, like an x-ray that is deflected on the mirror element. An improvement of the resolution behavior can thus likewise be achieved by a nested arrangement. 
   In accordance with a first aspect of the invention, there is a device for improving resolution capability of an x-ray optical apparatus for an x-ray incident from a direction of incidence comprising a mirror element comprising a mirror edge formed as a cylindrical shell section around an edge axis. The mirror element is spaced apart, in a radial direction, from a focal axis that is parallel to the direction of incidence, the edge axis is oriented at a first non-zero angle relative to the focal axis when viewed along a radial axis, and the edge axis is oriented at a second non-zero angle relative to the focal axis. 
   The mirror element may further comprise a second mirror edge adjacent the mirror edge. The second mirror edge is formed as a second cylindrical shell section around a second edge axis, and a plane comprising the edge axis and the second edge axis is tilted with respect to the direction of incidence. Moreover, the mirror edge may correspond to an approximation of a hyperbolic form, while the second mirror edge corresponds to an approximation of a parabolic form. 
   A width of the mirror element may be smaller than approximately one tenth of a radial distance between the mirror element and the focal axis. Additionally, the width of the mirror element corresponds to an arc length of less than approximately two degrees in the radial direction. In embodiments, a magnitude of tilt between the edge axis and the direction of incidence is in a range of approximately one half a degree to approximately five degrees. 
   The device may also comprise another mirror element having a third mirror edge and a fourth mirror edge adjacent the third mirror edge. In embodiments, the third mirror edge is formed as a third cylindrical shell section around a third edge axis, and the other mirror element is spaced apart, in another radial direction, from the focal axis. Also, the third edge axis is oriented at the first non-zero angle relative to the focal axis when viewed along another radial axis extending in the other radial direction, and the third edge axis is oriented at the second non-zero angle relative to the focal axis. 
   The other mirror element may be adjacent the mirror element, while the mirror element and the other mirror element are spaced apart from the focal axis by a same distance. Furthermore, a transition between the mirror element and the other mirror element comprises a stepped offset. Even further, the device may include a plurality of additional mirror elements forming a ring around the focal axis. 
   The device may comprise an additional mirror element spaced apart from the focal axis in the radial direction. A spacing of the additional mirror element from the focal axis is larger than a spacing of the mirror element from the focal axis. 
   In embodiments, the mirror element is spaced apart, in the radial direction, from the focal axis that is parallel to the direction of incidence by a focal point of the x-ray optical apparatus. 
   According to another aspect of the invention, there is a device for improving resolution of an x-ray optical apparatus associated with an x-ray incident from a direction of incidence, the device comprising a mirror element having a first portion and a second portion adjacent the first portion. The first portion comprises a first cylindrical shell section formed around a first edge axis tilted at a first non-zero angle relative to a focal axis that is parallel to the direction of incidence. The second portion comprises a second cylindrical shell section formed around a second edge axis tilted at a second non-zero angle relative to the focal axis, the second non-zero angle being different from the first non-zero angle. The first edge axis and the second edge axis are oriented at a third non-zero angle relative to the focal axis when viewed along a radial axis in a radial direction. 
   In embodiments, a plane containing the first edge axis and the second edge axis is tilted at an offset angle relative to the focal axis. Moreover, the offset angle is in a range of approximately one half a degree to approximately five degrees, and a width of the mirror element is smaller than approximately one tenth of a radial distance between the mirror element and the focal axis. 
   The first portion may comprise an approximation of a hyperbolic form. The second portion may comprises an approximation of a parabolic form. 
   The device may also include another mirror element having a third portion and a fourth portion adjacent the third portion. The third portion comprises a third cylindrical shell section. The fourth portion comprises a fourth cylindrical shell section tilted relative to the third portion. The other mirror element is arranged in a stepped offset relative to the mirror element. The other mirror element is rotated about another radial axis that is perpendicular to the focal axis. The device may also include a plurality of additional mirror elements, wherein the mirror element, the other mirror element, and the plurality of additional mirror elements form a ring around the focal axis. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     Further advantages and application possibilities of the present invention are shown by the following description in conjunction with the exemplary embodiments shown in the drawings, in which: 
       FIGS. 1A and 1B  show representations of the structure of a pore optics system; 
       FIG. 2  shows a diagrammatic representation of two mirror shells one after the other according to the Wolter-I arrangement; 
       FIG. 3  shows a representation of cylinder segments of a mirror shell; 
       FIG. 4  shows a representation of an exemplary embodiment of the rotation of cylinder segments around the radial axis of the telescope arrangement; 
       FIGS. 5A and 5B  show representations of an exemplary embodiment of the present invention in two different sectional views; 
       FIG. 6  shows a representation of the deflection of light rays at the telescope; 
       FIG. 7  shows a spot diagram in the focal plane, conical mirror shells being used to generate the spot diagram; 
       FIG. 8  shows a diagram of the deviation when using a mirror element of two cylinder surfaces; 
       FIGS. 9A and 9B  show diagrams of the deviation of the cylinder approximation of a conical surface for a paraboloid ( FIG. 9A ) or a hyperboloid ( FIG. 9B ); 
       FIG. 10  shows a representation of the illumination of the mirror element of two cylindrical shells; 
       FIG. 11  shows a spot diagram in the focal plane, which is generated by a mirror element with cylinder surfaces; 
       FIGS. 12A and 12B  show diagrams of the deviation of an exemplary embodiment of the present invention from an unmodified cylinder approximation for the first mirror edge ( FIG. 12A ) and the second mirror edge ( FIG. 12B ); and 
       FIG. 13  shows a spot diagram in the focal plane that is obtained through a mirror element according to an exemplary embodiment of the present invention. 
   

   DETAILED DESCRIPTION OF THE INVENTION 
   To explain the present invention more precisely, first the fundamental concepts are explained in more detail, which lead to a description of devices according to aspects of the invention. Absolute size data in the following description and the drawings are only exemplary data, and do not restrict the invention. 
     FIG. 2  depicts an approximation to the Wolter-I optics for x-ray astronomy. An x-ray telescope can comprise mirror shells  20 ,  22 , which represent a so-called Wolter-I optics. Then the mirror shell  20  facing the object is a section of a paraboloid, and the mirror shell  22  facing the image plane is a section of a hyperboloid. Accordingly, the first mirror shell  20  would be the paraboloid section and the second mirror shell  22 , the hyperboloid section, as shown in  FIG. 2 . 
   In order to always work in the range of the grazing incidence of x-rays  24 , the sections of the paraboloid and of the hyperboloid are narrow mirror shells. They are usually arranged in a staggered manner, in order to image a greater quantity of light on the focal plane  23  at a distance  24   a  from the mirror shells  20 ,  22 . It is customary to approximate the narrow shell-shaped sections of the paraboloid and of the hyperboloid through conical elements. In this case, the mirrors  20  and  22  represent ring-shaped sections of conical surfaces with a radius  26 . The two cones forming the basis have a cone axis that is identical to the telescope symmetry axis (or focal axis  25 ). The included angles are selected such that the conical surfaces at the location of the mirror shells  20  and  22  fit against one another tangentially. In the exemplary embodiment, a conical approximation of a Wolter-I optics system is described by way of introduction. 
   One criterion for assessing the quality of the optical image is the diameter of the light spot  27  in the focal plane  23 . A small spot  27  means that the resolution capability of the telescope is large, while with a large light spot  27  no distinction can be made between two objects lying close together. It is therefore the object of every optical telescope to generate the smallest possible light spot  27  in the focal plane  23 . 
     FIG. 3  depicts a cylinder approximation. Since the production of whole mirror shells  20 ,  22  is complex, it is expedient to undertake an azimuthal segmentation  30 , as shown in  FIG. 3 . A mirror segment  32  of this type or a mirror edge can be described by approximation by a sectional surface of a cylinder. This greatly facilitates the production of the mirror shell segments. However, this approximation also leads to the diameter of the light spot in the focal plane being larger. 
   The cylinder approximation lies in adapting a cylinder surface to the conical surface that represents the paraboloid section. This is very successful, as long as the azimuthal segment size  30  is small compared to the radius of the shells  34 , i.e., b segment &lt;&lt;R shell  applies. The consequence of this approximation is that the light spot becomes larger in the focal plane. 
     FIG. 4  shows a cylinder approximation with rotated cylinder surfaces according to aspects of the invention. Embodiments of the invention employ a modification of the cylinder approximation, which makes it possible to substantially reduce the size of the light spot diameter with respect to the cylinder approximation. It is thus possible to achieve a light spot of the size of the conical approximation to the Wolter-I optics with cylinder shell sections. In this manner, the advantages of the easier production of cylinder shell segments have been effected without any significant loss of the resolution capability of the telescope. 
   According to exemplary embodiments, the modification lies in the cylinder segments  40  being rotated around the radial axis  42  of the mirror shell arrangement of the telescope, which runs through the center of the mirror shell segment. An arrangement of this type having rotation of the cylinder segments around the radial axis of the telescope arrangement is shown in  FIG. 4 . According to aspects of the invention, the diameter of the light spot in the focal plane can be reduced by a factor of three. In embodiments, the improvement is dependent on the distance of the mirror segment from the symmetry axis: it increases with smaller distance. Since exemplary embodiments relate to a tandem mirror of the periphery of the telescope, much smaller light spot diameters are achieved for the inner tandem mirror. 
     FIGS. 5A and 5B  show an exemplary embodiment of the mirror element according to aspects of the invention in different sectional representations.  FIG. 5A  shows a cross-sectional representation of the exemplary embodiment of the mirror element according to aspects of the invention. In embodiments, the mirror element comprises a first mirror edge  52  and a second mirror edge  54 , both of which are adjacent to one another. As used herein, the term “mirror edge” can also refer to a slope side, flank, etc. The first mirror edge  52  is arranged at a radial spacing  56  around a first edge axis  58 . The first mirror edge  52  is formed from a cylinder segment or a cylinder surface section. The first edge axis  58  is tilted with respect to the focal axis  25 . Furthermore, the second mirror edge  54  also comprises a cylinder surface section, which is arranged at a second radial spacing  60  around a second edge axis  62 . The second edge axis  62  is tilted with respect to the first edge axis  58  so that the play element comprising the first mirror edge  52  and the second mirror edge  54  has a bent form. In embodiments, it can be ensured through this arrangement that x-rays that strike the mirror element at a direction of incidence parallel to the focal axis  25  are focused on a focal point (not shown in  FIG. 5A ). 
     FIG. 5B  show a representation of the mirror element shown in  FIG. 5A  looking along axis  42 , along with other mirror elements arranged adjacent thereto.  FIG. 5B  also shows the focal axis  25  as well as the first edge axis  58  and the second edge axis  62 . The first and second mirror edge  52  or  54  of the individual mirror elements are cylinder surface sections, as described in connection with  FIG. 5A . Furthermore,  FIG. 5B  shows, according to aspects of the invention, a rotation of the mirror element in which the mirror element is turned around a radial axis (not shown here) arranged at right angles to the focal axis  25 . In this manner, an offset angle  64  is formed between the focal axis  25  under the first or second edge axis  58  or  62 . According to aspects of the invention, improvement in the optical resolution can be achieved through this offset angle  64 . 
   Furthermore, adjacent mirror elements, such as those shown by reference numbers  66  and  68  in  FIG. 5B , can also be arranged in a stepped offset  70  so that the structure shown in  FIG. 5B  results. In embodiments, a limit between the first mirror edge  52  and the second mirror edge  54  lies as far as possible in a narrow lateral range so that a focusing of light rays or x-rays from different mirror elements are all focused as far as possible on a small focal point. Also, through utilizing the structure shown in  FIG. 5B , a ring shape can be formed around the focal axis  25 , as shown, e.g., in  FIG. 2 . A shape of this type is indicated to some extent in  FIG. 4 . 
   According to aspects of the invention, an improvement in the focusing of an x-ray is achieved in that a better approximation of the Wolter-I optics is achieved through the offset angle  64  compared to when the boundary line between one of the first mirror edge  52  and the second mirror edge  54  is horizontal (i.e., at right angles to the focal axis  25 ). 
   A specific exemplary embodiment of the present invention compared to a conical as well as a simple cylinder approximation is described in more detail below with respect to  FIGS. 6-13 . 
   A model of the conical approximation of the Wolter-I optics and the unmodified and modified cylinder approximation (i.e., of an exemplary embodiment of the present invention) of the conical approximation of the Wolter-I optics was produced with the aid of the “ASAP” optics program. With the aid of geometrical-optical ray-tracing calculations, the light spot was calculated in the focal plane of the arrangement (spot diagram). The geometry parameters are as follows: a distance between mirror shells and focal plane is given by f=50000 mm, and a radius of the mirror shell boundary is given by R=3500 mm. 
     FIGS. 6 and 7  depict an example of conical approximation of the Wolter-I optics. For the light source, light rays  24  (especially x-rays) fall parallel to the symmetry axis  25  of the telescope on the annular arrangement of a tandem mirror  40 , which is composed of the mirror shell that represents the conical approximation of the paraboloid and the second mirror shell that represents the conical approximation of the hyperboloid. The grid of the light rays is indicated in  FIG. 6 , the left part of the image showing a top plan representation and the right part of the image showing a front view representation of a telescope of this type. 
   Regarding the light spot in the focal plane, a rotationally symmetrical light spot results in the center of the focal plane, the diameter of which light spot is approximately 0.6 mm. This is shown by the dimensions of the diagram shown in  FIG. 7 , which represents the impact points of the rays in the focal plane. As can be expected, the symmetry is maintained and the image points of the individual rays lie on circles. The consequence of a shift of the focal plane along the telescope axis is that the light spot becomes larger, irrespective of the shift direction. This shows that in fact the focal plane is present.  FIG. 7  thereby shows a spot diagram in the focal plane. The two conical mirror shells are illuminated with axial light rays. The spot diameter is 0.42 mm. 
     FIGS. 8-11  depict an example of a cylinder approximation without rotated surfaces. An azimuthal segment corresponds, for example, to a 360th of an arc, that is, one degree. For a circular radius of 3500 mm, this result in an arc length of b=(2π/360) 1 deg 3500 mm=61 mm. One cylinder surface respectively has been adapted to the conical surfaces  52 ′ and  54 ′, which correspond to the mirror surfaces  52  and  54 . This is possible, because the arc length is much smaller than the circular radius.  FIG. 8  shows an image of a tandem of this type of two cylinder surfaces. 
   In this example, the deviation of the cylinder surfaces from the conical surfaces is always less than one micrometer.  FIG. 9A  shows that the difference in the case of the mirror surface  52 ′ (corresponding to the mirror edge  52 ) is less than 10 nm.  FIG. 9B  shows the case of the mirror surface  54 ′ (corresponding to mirror edge  54 ) in which the difference is less than 200 nm. Thus, in  FIG. 9A  the deviation from the conical approximation is shown as the difference between the cylinder approximation of the conical surface that describes the paraboloid of the Wolter-I optics. In  FIG. 9B , the deviation from the conical approximation is shown as the difference between the cylinder approximation of the conical surface that describes the hyperboloids of the Wolter-I optics. The y-axis gives the deviation in micrometers. 
   For the light source, the tandem of cylinder mirrors  52 ′,  54 ′ is illuminated with light rays  24 . The light rays  24  run parallel to the telescope axis  25 , their spatial arrangement being shown in  FIG. 10 , in which the illumination of the tandem is shown from two cylinder shells  52 ′ and  54 ′. The arc section is shown in an exaggerated manner; the azimuth angle being in actuality approximately 1 degree. 
   Regarding the light spot in the focal plane, a cylinder tandem mirror generates a light spot in the center of the focal plane, which light spot is unsymmetrical. Its maximum extension lies in the direction perpendicular to the tandem mirror and is approximately 0.82 mm, as can be seen from the spot diagram from  FIG. 11 . In this example, an arrangement of several cylinder tandem mirrors such that a complete ring of mirror shells was produced, resulting in a round light spot in the center of the focal plane, the diameter of which light spot is approximately 0.82 mm.  FIG. 11  shows a spot diagram in the focal plane which generates a tandem from two described cylinder surfaces from the axially incident light rays. The spot diameter is 0.82 mm. 
     FIGS. 12A ,  12 B, and  13  depict an example of a cylinder approximation with rotated cylinder surfaces according to aspects of the invention. In the example, the surfaces of the modified (i.e., rotated according to the invention) and unmodified cylinder approximation of the mirror surface (i.e.,  52  or  52 ′) differ by less than 40 micrometers; while the surfaces of the modified and unmodified cylinder approximation of the mirror surface (i.e.,  54  and  54 ′) differ by less than 60 micrometers. Although these are small numbers compared to the lateral dimensions of the mirror surfaces, they still represent significant deviations, if one considers that the difference between cylinder and conical approximation is smaller by three orders of magnitude.  FIG. 12A  shows the deviation of the modified (rotated) and unmodified cylinder approximation for the mirror surface  52  or  52 ′, while  FIG. 12B  shows the deviation of the modified (rotated) and unmodified cylinder approximation for the mirror surface  2  of the tandem mirror. 
   In the example of  FIGS. 12A ,  12 B, and  13 , the same light source is used as was used in the example of  FIGS. 8-11  in order to have a direct comparison. 
   In the example of  FIGS. 12A ,  12 B, and  13 , the diameter of the light spot in the focal plane is substantially reduced by rotating the tandem of the two cylinder mirrors  52  and  54  around the radial axis of the telescope arrangement by 1.00713 degrees. As shown in  FIG. 13 , the maximum extension of the light spot is approximately 0.25 mm. In this example, an arrangement of several cylinder tandem mirrors forming a complete ring of mirror shells is produced, resulting in a round light spot in the center of the focal plane, the diameter of which light spot is approximately 0.25 mm. That is smaller by approximately a factor of 3.3 compared the case of the unmodified cylinder approximation (e.g.,  FIGS. 8-11 ), and smaller by approximately a factor of 2.4 than in the case of the conical approximation of the Wolter-I optics (e.g.,  FIGS. 6-7 ). 
     FIG. 13  shows a spot diagram in the focal plane that generates a tandem of two rotated cylinder surfaces from the axially incident light rays. The spot diameter is 0.25 mm. 
   
     
       
             
           
             
             
           
         
             
                 
             
             
               Reference Numbers 
             
             
                 
             
           
           
             
                 
             
           
        
         
             
               10 
               Base 
             
             
               12 
               Cylindrical mirror shells 
             
             
               14 
               Webs 
             
             
               20, 22 
               Mirror shells 
             
             
               23 
               Focal plane 
             
             
               24 
               Light rays, x-rays 
             
             
               24a 
               Spacing between the focal plane and the mirror shells 
             
             
               25 
               Focal axis, telescope axis 
             
             
               26 
               Radius of the mirror shells 
             
             
               27 
               Light spot 
             
             
               30 
               Azimuthal segmentation 
             
             
               32 
               Mirror segment 
             
             
               34 
               Radius of the shells of the cylinder approximation 
             
             
               40 
               Cylinder segment 
             
             
               42 
               Radial axis 
             
             
               52 
               First mirror edge 
             
             
               54 
               Second mirror edge 
             
             
               56 
               Radial spacing from the first mirror edge to the first edge axis 58 
             
             
               58 
               First edge axis 
             
             
               60 
               Radial spacing from the second mirror edge to the second edge 
             
             
                 
               axis 62 
             
             
               62 
               Second edge axis 
             
             
               64 
               Offset angle 
             
             
               66, 68 
               Further mirror elements 
             
             
               70 
               Stepped offset