Patent Publication Number: US-6668040-B2

Title: Refractive X-ray arrangement

Description:
This application is a continuation of International Application No. PCT/SE00/01502 filed on Jul. 17, 2000 claims the benefit of U.S. Provisional Application No. 60/144,523 filed on Jul. 19, 1999. 
    
    
     TECHNICAL FIELD 
     The present invention relates to x-rays and, more specifically, to X-ray focussing using a refractive X-ray arrangement. The refractive arrangement for X-rays comprises a member of low-Z material, said part of low-Z material having a first end adapted to receive x-rays emitted from an x-ray source and a second end from which emerge said x-rays received at said first end, and first and second surfaces. The invention also concerns a lens and a method for manufacturing the arrangement. 
     BACKGROUND OF THE INVENTION 
     With the advent of 3 rd  generation synchrotron x-ray sources, hard x-ray optics is a field of growing interest with applications in research, material testing, chemical analysis and medical imaging and therapy. Prior art focusing elements in this energy region use the methods of reflection and diffraction, e.g. best crystals, curved mirrors, Fresnel zone plates and capillary optics. These elements are generally expensive and technologically challenging to manufacture, limiting their use in commercial-grade applications. 
     Another shortcoming associated with prior art high-energy x-ray focusing techniques, such prior art attempts are limited to generating a single-peak energy distribution. Hence, such experimental methods are not well suited to applications requiring more than one x-ray energy peak, such as dual-energy x-ray imaging. 
     PRIOR ART 
     It is well known that the refractive index of any material can be expressed by 
     
       
           n =1−δ− iβ   (1) 
       
     
     Refractive lenses can easily be fabricated for use in the visible light region, since materials having a refractive index n far from unity and a small absorption in this region are readily available. In contrast, optical elements utilizing refraction are intrinsically difficult to fabricate for use in the x-ray region, since in this energy region, all materials have an index of refraction n near unity and exhibit a large absorption. Consider a concave piece of material having a circular revolution with the radius of curvature R. Such a piece of material will focus a plane-wave entering parallel to the axis at a focal distance of f. The focal length is given by              f   =     R   δ             (   2   )                         
     A lens fabricated according to eq. 2 would have a very large focal length, since d is typically 10 −5  or 10 −6  in the hard x-ray region. Examples of such lenses were given by Suehiro et al (Nature 352 (1991), pp. 385-386). In a correspondence, this approach was ruled out for any practical application by Michette (Nature 353 (1991), p. 510). The extent to which the focal length can be shortened by reducing R has limitations in terms of fabrication technology and practical use. 
     A significant improvement was achieved when Snigirev et at (Nature 384 (1996), pp. 49-51) cascaded N drilled holes in a piece of aluminum. This corresponds to 2N concave surfaces, thereby reducing the focal length by the same factor. The total focal length of the compound lens is given by              F   =       f     2      N       =     R     2      δ                 N                 (   3   )                         
     This lens still suffered from spherical aberration and high absorption and focusing was only achieved in one dimension. These shortcomings have been addressed by several authors. Similar solutions are also known through U.S. Pat. No. 5,594,773 and U.S. Pat. No. 5,684,852. 
     Low-Z materials have been used for decreased absorption and two-dimensional focusing has been achieved by, e.g. Elleaume Nucl. Instr. and Meth. A 412 (1998), pp. 483-506) by means of crossing two linear arrays. 
     Another lens is described in a U.S.A Patent Application entitled “A COMPOUND REFRACTIVE X-RAY LENS”, now U.S. Pat. No. 6,091,798, which discloses a novel manufacturing technique to make parabolic profiles by splitting the lens in two halves at the symmetry axis, thereby reducing spherical aberration and absorption. 
     However, aberration free compound reflective x-ray lenses still rely on elaborate and expensive manufacturing techniques. Hence, such refractive lenses are not well suited to commercial-grade applications. Furthermore, such prior art refractive x-ray lenses are limited to generating a single-peak energy distribution. As yet another disadvantage, prior art refractive x-ray lenses have, for a given energy, a fixed focal length, which cannot be varied. 
     SUMMARY OF THE INVENTION 
     Thus, a need exists for a refractive x-ray lens, which is well suited for commercial applications and which does not suffer from the disadvantageous inherited by the known lense. Still another need exists for a refractive x-ray lens, which is able to generate a dual energy distribution from an x-ray source. Yet another need exists for a refractive x-ray lens for which the focal length for a given energy can easily be varied. Still, another need exists for a high-energy x-ray leas able to generate a dual energy distribution from a broadband x-ray source. 
     A further need exists for a method readily to form a refractive x-ray lens at a low cost, e.g. so that high-energy x-ray optics should find its way from specialized research facilities into general applications in industry and commercial R&amp;D. 
     The present invention provides an x-ray lens which is well suited for commercial applications. The present invention further provides a method readily to form a compound refractive x-ray lens. The present invention also provides a refractive x-ray lens able to generate a dual-energy distribution from a broad energy x-ray source. Furthermore, the present invention provides an x-ray lens for which the focal length for a given energy can easily be varied. The present invention achieves the above accomplishments with a novel x-ray focusing apparatus, novel x-ray lens formation methods and novel methods for focusing of x-rays. 
     Moreover, the present invention has as an objective to increase the flux on a scanned slit. For these reasons, the initially mentioned refractive arrangement for X-rays further comprises a plurality of substantially sawtooth formed grooves disposed between said first and second ends on at least one of said first or second surfaces. Said plurality of grooves oriented such that said x-rays which are received at said first end, pass through said member of low-Z material and said plurality of grooves, and emerge from said second end, are refracted to a focal point. 
     Preferably, said member of low-Z material consists of a plastic material, specially one of from the group comprising polymethylmethacrylate, vinyl and PVC. It may also consist of beryllium. 
     Preferably, said grooves have the form of sawteeth with substantially straight cuts. 
     In an advantageous embodiment said pluralities of grooves have varying sizes, decreasing or increasing continuously from said first end towards said second end. 
     The refractive X-ray lens according to the invention comprises a volume of low-Z material, said volume having a first end adapted to receive x-rays emitted from an x-ray source and a second end from which emerge said x-rays received at said first end and first and second surfaces. The volume further comprises a plurality of substantially sawtooth formed grooves disposed between said first and second ends on at least one of said at least two surface, said plurality of grooves oriented such that said x-rays which are received at said first end, pass through said volume of low-Z material and said plurality of grooves, and emerge from said second end, are refracted to a focal point. 
     In one advantageous embodiment the lens comprises two volumes arranged such that the surfaces with the plurality of grooves are facing each other. Preferably, said two volumes each have a tilt angle to an optical axis of said X-ray. Said volumes have non-coincident focal points. 
     Preferably, a focal length of each of the two volumes of the lens is varied by separately varying each tilt angle. 
     Said volume of low-Z material consists of a plastic material, specially one from the group comprising polymethylmethacrylate, vinyl and PVC or said volume of low-Z material consists of beryllium. 
     Moreover, the invention concerns an X-ray system and a method for two-dimensional focusing of X-rays and including at least two leases according to above. The focusing is obtained by arranging said at least two lenses, such that each x-ray traverses both of lenses in sequence and that one of said at least two lenses are rotated around an optical axis with respect to the other lens. 
     In one preferred application said refractive lens is coupled to at least one second commercial-grade compound refractive x-ray lens such that an array of compound refractive x-ray lenses is formed. 
     The method of fabricating the saw-tooth profile refractive x-ray lens is characterized by: transferring shapes of grooves onto a carrier by means of an engraving arrangement producing a master, and using said master to pressing grooves on a suitable material. 
    
    
     These and other advantages of the present invention will no doubt become obvious to those of ordinary skill in the art after having read the preferred embodiments which are illustrated in the various drawing figures. 
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The present invention will become more fully apparent from the appended claims and the description as it proceeds in connection with the drawings illustrating some preferred embodiments of the invention. In the drawings: 
     FIG. 1 is a schematic perspective view of a refractive x-ray lens in accordance with one embodiment of the present invention, 
     FIG. 2 is a schematic perspective view of a section of a sawtooth profile refractive x-ray lens in accordance with a second embodiment of the present invention, 
     FIG. 3 a  is a schematic side view of the sawtooth profile refractive x-ray lens comprising the sections according to FIG. 2, 
     FIG. 3 b  is an imaginary projection showing the parabolic lenticular shape achieved with the sawtooth shape, 
     FIG. 4 is a schematic side view of a refractive x-ray lens in accordance with a second embodiment, 
     FIG. 5 is a side view of the one-dimensional focusing geometry of the sawtooth profile refractive x-ray lens in accordance with the embodiment shown in FIG. 4, 
     FIGS. 6 a  and  6   b  show the side and the top views, receptively, of another embodiment, 
     FIGS. 7 to  9  show representations of a sawtooth given for theocratical explanations, 
     FIG. 10 is a schematic illustration of an arrangement for crystallographical application comprising a lens according to the present invention, and 
     FIGS. 11 and 12 a  schematic illustration of an microscope involving a lens according to the present invention. 
    
    
     BASIC THEORY 
     In the following well-known ray-optics is applied to a sawtooth geometry. The thin lens approximation is made. The definitions are illustrated in FIG. 7 illustrating a substantially triangular sawtooth. 
     The law of refraction yields 
     
       
         sin(γ+α)= n  sin(γ+α+Δα)  (i) 
       
     
     Since Δα is very small and α&gt;&gt;γ, this can be written 
     
       
         sin(γ+α)= n  sin(γ+α)+ n  cos(γ+α)Δα  (ii) 
       
     
     
       
         
           
             
               
                 
                   Δα 
                   = 
                   
                     
                       
                         
                           
                             ( 
                             
                               1 
                               - 
                               n 
                             
                             ) 
                           
                            
                           
                             sin 
                              
                             
                               ( 
                               
                                 γ 
                                 + 
                                 α 
                               
                               ) 
                             
                           
                         
                         
                           n 
                            
                           
                               
                           
                            
                           
                             cos 
                              
                             
                               ( 
                               
                                 γ 
                                 + 
                                 α 
                               
                               ) 
                             
                           
                         
                       
                       ≈ 
                       
                         δ 
                          
                         
                             
                         
                          
                         
                           tan 
                            
                           
                             ( 
                             γ 
                             ) 
                           
                         
                       
                     
                     = 
                     
                       δ 
                       
                         tan 
                          
                         
                           ( 
                           β 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   iii 
                   ) 
                 
               
             
           
         
         
         
             
         
       
     
     where n=1−δ and β+γ=π/2. 
     After passage of N sawteeth the total deflection angle will be 
     
       
         Δα tot =2N δ/tan(β)  (iv) (see also FIG.  8 ) 
       
     
     This angle is so small that it will be assumed that the ray will traverse the lens in a straight line parallel to the axis. The geometry above shows that                Δ                     α   tot          (   y   )         =         y     s   o       +     y     s   i         ≡     y   f               (   v   )                         
     where f is the focal length of the compound lens. 
     Combination of (iv) and (v) gives the number of teeth seen by a ray at a distance y from the axis,                N        (   y   )       =           tan        (   β   )          Δ                   α   tot         2      δ       =       y                   tan        (   β   )           2      δ                 f                 (   vi   )                         
     The distance a ray has to travel before seeing an additional tooth can be calculated from                y        (   N   )       =           2      N                 δ                 f       tan        (   β   )         ⇒       y        (   i   )       -     y        (     i   -   1     )           =       2      δ                 f       tan        (   β   )                   (   vii   )                         
     and an additional path length is obtained in the material                x        (   y   )       =           2      y       tan                   (   β   )         ⇒     Δ                 x       =         x        (   i   )       -     x        (     i   -   1     )         =       4      δ                 f         tan   2          (   β   )                     (   viii   )                         
     The total path-length follows from summation of all contributions:                      X        (   y   )       =     Δ                   x        (     1   +   2   +   …   +     N        (   y   )         )                     =     Δ                 x            1   2          [     N        (   y   )       ]       2                   =         4      δ                 f         tan   2          (   β   )              1   2            (       y                 tan                 β       2      δ                 f       )     2                   =       y   2       2      δ                 f                     (   ix   )                         
     Thus, it is shown that the path-length as a function of y will be parabolic. If y is the height of the first and largest tooth, the radius of curvature is R=δf . In reality, it is not a continuous function since a finite number of sawteeth exist, and the parabola will be approximated by a few hundred straight lines. This could give a perceptible aberration effects in some imaging applications, However, the effect should be small and neglectable. 
     Considering the case of a finite source perfectly projected onto a slit with size d s . The attenuation length is denoted λ. A ray that has lateral displacement y is attenuated by a factor.                exp        (       -     X        (   y   )         λ     )       =     exp        (     -       y   2       2      δ                 f                 λ         )               (   x   )                         
     Thus, the rms beam spread is 
     
       
           δ={square root over (δfλ)}   (xi) 
       
     
     The gain will be a product of the geometrical gain and the transmission through the lens.                G        (     y   d     )       =                    2        y   d         d   s                s   0     +     s   i         s   0            1     y   d              ∫   0     y   d              exp        (     -       y   2       2        σ   2           )               y                       =                    2        (     1   +     M   y       )         d   s            2     π              π     2            ∫   0       y   d         2      π                  exp        (     -     ξ   2       )            2        σ           ξ                       =                    1   +     M   y         d   s              2      π            σ   ·     erf        (       y   d         2        σ       )                                 
     M y  is the lateral magnification and the error function is used:                erf        (   z   )       =       2     π              ∫   0   z            exp        (     -     x   2       )               x                   (   xii   )                         
     The error function will approach unity when the height is increased, and in the limiting y d →∝,                G   max     =         2      π            (     1   +     M   y       )          σ     d   s                 (   xiii   )                         
     This is evidently an unphysical limit. However, the error-function approaches unity quickly. The growth of the length of the lens quadratically with y d  will not contribute much for a fixed focal length. Since the length should be kept down for practical and economical reasons. 
     Once the geometry and lens parameters are fixed, the system will be optimized for one single energy. Calculating the gain in this case is less straightforward. Assuming that the beam from a point source on the optical axis is focused at s 1 +Δ, it follows that (referring to FIG. 9)                  1     s   0       +     1       s   i     +   Δ         =     1   f             (   xiv   )                     d   s     /   2     Δ     =     h       s   l     +   Δ               (     x                 v     )                         
     The maximal angle a ray can make horizontally and still encounter the slit is                θ   =       h     s   0       =           d   s     /   2         s   0          s   1              1   ɛ                
        where           (   xvi   )               ɛ   =     |       1     s   0       +     1     s   1       -     1   f       |             (     x                 v                 i                 i     )                         
     The absolute value makes the relation valid even if the focus lies in front of the slit, However, h must not be greater than the height of the lens, y d , in which case the ray would miss the lens entirely. In the absence of the lens, the fraction of the x-rays emitted by the source that would encounter the slit would be (the normalization factor I/2π is omitted)                I   0     =     d       s   0     +     s   i                 (   xviii   )                         
     With the lens present, but with no absorption of the x-rays, this would be increased to 
       I   lens =θ  (ixx) 
     Including absorption, the flux falling on the slit is given by an integral over the angle α of the ray from the source;                I   lens     a                 b                 s       =       ∫     -     min        (     θ   ,       y   d     /     s   0         )           min        (     θ   ,       y   d     /     s   0         )                exp   (         -     s   0   2            α   2         2        σ   2         )             α                 (   xx   )                         
     Here a simplification is made. The aperture is limited either by θ or by y d =s 0 . However, even in the last case integration is made to θ. This is a good approximation, since rays that far from the optical axis will be strongly absorbed and only have a small contribution to the flux.                I   lens     a                 b                 s       =           2      π          σ        1     s   0            erf        (         θs   0        1         2        σ       )         =         2      π          σ        1     s   0            erf        (       d   s       2      σ                   s   i        ɛ        2         )                   (   xxi   )                         
     The gain will be                G        (   0   )       =         I   lens     a                 b                 s       /     I   0       =         2      π                s   0     +     s   i         s   l            σ     d   s            erf        (       d   s       2        2        σɛ                     s                i         )                   (   xxii   )                         
     Now assuming that the point source is located at y s  from the optical axis and a similar geometrical exercise gives (omitting the algebraic details)                G        (     y   s     )       =         π   2                s   0     +     s   i         s   0              σ     d   s            [       erf        (         d   s       2        2        σɛ                     s                l         +       y   s       2        2        σɛ                     s                0           )       -     erf        (       -       d   s       2        2        σɛ                     s                i           +       y   s       2        2        σɛ                     s                0           )         ]                 (   xxiii   )                         
     It is interesting to study how the maximal gain depends on the material properties of the lens. From Eqs. xi and xiii is obtained 
     
       
         Max gain ασ=sqrt{ fδλ}   (xxiv) 
       
     
     and thus δλ should be maximized. The attenuation length is a strong function of the atomic number and it is obvious a material with the lowest possible Z is interested. In this energy region it is a good approximation to take δ∝E −2  and a parameterization of the X-ray cross-section in barns (∝½) is (from fitting totabulated values) 
     
       
         24.15 Z   4.2   E   −9 +0.56 Z   (xxv) 
       
     
     where the two terms Z and E are photo and Compton effect, respectively (E in keV). Then the optimum energy may be calculated using: 
     
       
           d/dE (δ, λ)=0 =E   opt =2.78 Z   1.07  keV  (xxvi) 
       
     
     For example for Beryllium and PMMA, the optimal energies are 12 keV and 19 keV, respectively. PVC with a higher effective Z and thus lower contribution from Compton scattering has a much higher optimum around 48 keV. While PMMA is 3 times better than vinyl at 18 keV, it is only 84% better at 40 keV. This is due to the high Compton scattering at high energies for the very low-Z materials. 
     DETAILED DESCRIPTION OF THE EMBODIMENTS 
     A refracting arrangement, which can be used as a lens in x-ray applications is schematically illustrated in FIG.  1 . The arrangement  100 , hereinafter referred to as lens, comprises a volume having a first end  105 , a second end  106  opposite said first end  105 , and longitudinal surfaces  107 - 110 . Within the volume are arranged cavities  102  extending substantially from said first end  105  to said second end  106 . The cavities are so arranged that the longitudinal axis of each cavity is substantially parallel to the said first and second ends. 
     Each cavity  102  comprises a first (e.g. upper) and a second (e.g. lower) ridge shaped groove  103  and  104 , which consecutively form a sawtooth formed first (e.g. upper) and a second (e.g. lower) lens sections  101 . The theory behind the design of the cavities is described above. 
     During the operation, the lens  100  is arranged to receive X-rays, e.g. through the first end  105 , and the X-rays after being refracted are emerged from the second end  106 . 
     Preferably, the volume material should have an atomic number as low as possible, i.e. a low Z-material; good candidates are, e.g. beryllium and plastics such as polymethylmethacrylate (PMMA). 
     In FIG. 2, a section  201  (e.g. lower part) of another sawtooth profiled refractive x-ray lens according to the present invention is illustrated. Sawtooth shaped grooves are arranged on one surface  207  of the section while the opposite surface  208  is plane. According to this embodiment, the size of the grooves  203  vary by decreasing the depth of the grooves is linearly from a first end  205  towards a the second end  206  of the volume. In a preferred embodiment the section contains, e.g. approximately 300 straight cut grooves with depth 211 decreasing linearly from about 100 to 0 microns and a bottom angle  212  of approximately 90°. This will give a total length of 30 mm. However, the bottom angle is a free parameter and can be optimized with respect to practical and manufacturing issues. The width  213  of the section can be varied according to the requirements, ranging from mm to dm. 
     In one embodiment, the invention is a split saw-tooth profile refractive x-ray lens. FIG. 3 a  shows a cut through an embodiment of the lens  300  consisting of two sections  201  according to FIG.  2 . The sawtooth profile refractive x-ray lens includes two volumes  201  of low-Z material, placed on opposite sides of the optical axis. The volumes  201  of low-Z material form a first end  305  that receives x-rays, preferably of commercially-applicable power emitted from a commercial-grade x-ray source. From the opposite, second end  306  the x-rays emerge. The plurality of grooves are oriented such that the x-rays which are received at the first surface, pass through the volume of low-Z material and through the plurality of grooves. In so doing, the x-rays of a single energy that emerge are refracted to a single focal point. If the x-ray source emits x-rays of variable energy, the spectrum of x-rays received at a single focal point will be enhanced near a unique energy. 
     The projection of the amount of traversed material for an X-ray entering parallel to the optical axis will be a parabolic profile, as illustrated in FIG. 3 b.  Thus, in principal, the described geometry could be replaced by a single parabolic surface, given by the equation              x   =       y   2       2      R               (   4   )                         
     where R is the radius of curvature and x and y are defined in FIG. 3 a.  This, however, would be extremely difficult to manufacture. One can look at the present invention as a redistribution of the low-Z material to simplify fabrication. With the geometry described above, R=0.167 m. Assume that the low-Z material is beryllium, for which d=8.5×10 −7  at 20 keV. This will, according to Eq. 2, give a focal length F=195 mm for 20 keV X-rays. Consequently, unlike the meter-level focal lengths associated with prior art experimental high-energy X-ray focusing devices, the sawtooth profile refractive X-ray lens  300  of the present embodiment attains a focal length on the order of decimeters. 
     In the embodiment outlined in FIG. 4, the lens  400  comprises to sections  401 , in which the jags (teeth)  416  all have the same size. By slightly tilting the parts  401  with respect to the optical axis  415 , the similar focusing behaviour as in FIG. 3 is achieved. The depth of the grooves is, e.g. about 100 mm. To achieve the same focusing properties as in the previous embodiment, still  300  sawteeth are needed, but the total length of the sawtooth profile refractive lens will be doubled to 60 mm. The separation  413  should be twice the depth of the grooves, i.e. 200 mm. This will give a tilt angle  414  of about 0.1°. These volumes of low-Z material will be substantially easier to manufacture than other geometries. In this embodiment the lens is a tunable sawtooth profile refractive x-ray lens. The volumes  401  of low-Z material including the plurality of straight-cut grooves, through which the x-rays pass, each has thus a small angle to the optical axis. The focal length will be a function of this angle. By varying the angle  414 , the focal point for a given energy will be translated. Alternatively, by varying the angle, at a fixed point, the energy at which the spectrum is enhanced will consequently be varied. 
     FIG. 5 is a side view of a one-dimensional focusing geometry of the sawtooth profile refractive x-ray lens  500  in accordance with the embodiment shown in FIG. 4. A divergent beam from a source S is focussed to a line at the focal point P. The lense according to this embodiment comprises two halves of refractive arrangements which are designed with sawteeth on both faces of the volume instead of only one face. This design may further improve the focusing properties of the lens. 
     FIGS. 6 a  and  6   b  show the side and the top view, respectively, of an embodiment in which two sawtooth profile refractive lenses  600   a  and  600   b  are used to achieve two-dimensional focusing. The second sawtooth profile refractive lens  600   b  is rotated 90° around the optical axis with respect to the first one  600   a.  A divergent beam from the source S is focussed to a point at the focal point P. 
     In still another embodiment (not shown), the present invention recites a method for providing a dual energy distribution from an x-ray source using a sawtooth profile refractive leas. In such an embodiment, the sawtooth profile refractive x-ray lens includes two volumes of low-Z material, placed on opposite sides of the optical axis. The volumes of low-Z material include a plurality of straight-cut grooves through which the x-rays will pass. Each of the volumes has a small unique angle to the optical axis. By having different angles for the two halves, each half will have a separate focal point. At a given point on the optical axis, the x-ray spectrum will he enhanced at two separate energies and thus yield a bimodal energy distribution. 
     According to one preferred method for manufacturing a lens of the invention, the shape of the grooves are transferred onto a (e.g. plastic) carrier by means of an engraving machine, comprising a hot engraving pointer which is controlled by a controlling arrangement transferring the shape of the grooves on to the carrier. Then a (metallic) master is formed using the carrier. The master may be used directly or through intermediate steps to make pressing moulds for pressing the grooves on suitable material. 
     Accordingly, the sawtooth lens resembles a vinyl phonograph record. A rough calculation gives that the groove pitch of such a record should be around 120 μm (10 cm at 33 rpm in 25 min). In order to have the dimensions of vibration decoupled, the bottom angle must be 90° in stereo mode, i.e. β as defined in the “BASIC THEORY” section is 45°. Thus, if there were no inter-spacing between the grooves, the depth would be 60 μm. Measurements of the profile of a vinyl record indicated that inter-spacing takes up half of the surface, which gives a depth of only 30 μm. However, the cutting is a flexible process with many free parameters. The restriction is the 100 μm lacquer layer on the master that limits the depth to about 90 μm and consequently the width to 180 μm. A master was cut with a depth of 90° without inter-spacing and a vinyl (PVC) was record-pressed, from which 60 mm long sections were cut out. The surface of the cuts seems to be of rather bad quality and the gain should be expected to be non-optimal. The lens halves were attached to aluminum supports that were adjusted with micrometer screws under a microscope to give the right tilt angle. With, 180 μm separation at the end, the radius of curvature is R=(90 μm) 2=(2/Delta 300 mm)=0:135 μm. This gives a focal length of 218 mm for 23 keV. 
     Above-mentioned methods are given merely as examples and other methods may also be used such as diamond turning techniques, laser cutting etc. 
     The lenses according to the invention may be used in all x-ray applications, such as mammography, bone-density analysis, dental applications, x-ray microscopy or crystallography etc. 
     In an x-ray crystallography arrangement  100 , as shown in FIG. 10, the crystal structure of a sample  101  is determined by detecting the spatial pattern of a diffracted x-ray beam  102  incident on the sample  101 . The divergent beam from a small x-ray source  104  is projected onto the crystal sample by the lens  103 . It is important that the incident beam has a low divergence (cross-fire), more precisely lower or equal to the mosaic spread of the crystal  101 . Thus, the saw-tooth refractive x-ray lens  103  can be applied to x-ray crystallography. Due to the geometry, the beam incident on the sample has a very small divergence. By this, a gain of flux on the sample is obtained and thus image acquisition time is decreased. The minimum distance from source to sample is determined by the constraint on beam divergence. Typical parameters would be: 
     Source size: 20 microns 
     Sample size: 100 microns 
     Source-to-lens distance: 15 cm 
     Lens-sample distance: 75 cm 
     Since the lens is chromatic, a narrow energy peak can be selected from a broad x-ray spectrum from the source. This will enhance the image quality and signal-to-noise ratio. This versatility can be used to choose the optimal energy for every sample. 
     Ideally, two lenses arranged in series could be used to obtain two-dimensional focusing and squared gain. 
     Another application is an x-ray microscope, as shown FIGS. 11 and 12. The lens can be used to form the lens of the x-ray microscope  110  and  120 . In both cases two lenses  111 ,  112 ,  121  and  122  are used to focus the x-ray beam to a very small spot, typically smaller than a few microns. In the arrangement of FIG. 11 the sample  113  is placed in the focal plane. The transmitted beam is incident upon a single x-ray detector  114 . To obtain a full two-dimensional image the object must be scanned point-by-point by a translational stage. The first lens  111  focuses the beam in y direction and the second lens  112  focuses the beam in x direction 
     In the arrangement according to FIG. 12, the sample  123  is stationary and positioned below (or above) the focal point of the lens. A magnified image of the object is seen by a pixelated area detector  124  and no scanning is needed. 
     While the invention is described in conjunction with the preferred embodiments, it is appreciated that there is no intend to limit the invention to these embodiments. On the contrary, the invention is intended to cover alternatives, modifications and equivalents, which may be included within the scope of the invention as defined by the appended claims.