Patent Number: 
Section: description

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(xcex3+xcex1)=n sin(xcex3+xcex1+xcex94xcex1)xe2x80x83xe2x80x83(i) Since xcex94xcex1 is very small and xcex1 greater than  greater than xcex3, this can be written sin(xcex3+xcex1)=n sin(xcex3+xcex1)+n cos(xcex3+xcex1)xcex94xcex1xe2x80x83xe2x80x83(ii)                     Δα        =                                                                              (                                      1                    -                    n                                    )                                ⁢                                  sin                  ⁡                                      (                                          γ                      +                      α                                        )                                                                              n                ⁢                                  xe2x80x83                                ⁢                                  cos                  ⁡                                      (                                          γ                      +                      α                                        )                                                                        ≈                          δ              ⁢                              xe2x80x83                            ⁢                              tan                ⁡                                  (                  γ                  )                                                              =                      δ                          tan              ⁡                              (                β                )                                                                        (        iii        )             where n=1xe2x88x92xcex4 and xcex2+xcex3=xcfx80/2. After passage of N sawteeth the total deflection angle will be xcex94xcex1tot=2N xcex4/tan(xcex2)xe2x80x83xe2x80x83(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          ⁢                      xe2x80x83                    ⁢                                    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              ⁢                              xe2x80x83                            ⁢                              α                tot                                                    2              ⁢              δ                                =                                    y              ⁢                              xe2x80x83                            ⁢                              tan                ⁡                                  (                  β                  )                                                                    2              ⁢              δ              ⁢                              xe2x80x83                            ⁢              f                                                          (        vi        )             The distance a ray has to travel before seeing an additional tooth can be calculated from                               y          ⁡                      (            N            )                          =                                                            2                ⁢                N                ⁢                                  xe2x80x83                                ⁢                δ                ⁢                                  xe2x80x83                                ⁢                f                                            tan                ⁡                                  (                  β                  )                                                      ⇒                                          y                ⁡                                  (                  i                  )                                            -                              y                ⁡                                  (                                      i                    -                    1                                    )                                                              =                                    2              ⁢              δ              ⁢                              xe2x80x83                            ⁢              f                                      tan              ⁡                              (                β                )                                                                        (        vii        )             and an additional path length is obtained in the material                               x          ⁡                      (            y            )                          =                                                            2                ⁢                y                                            tan                ⁢                                  xe2x80x83                                ⁢                                  (                  β                  )                                                      ⇒                          Δ              ⁢                              xe2x80x83                            ⁢              x                                =                                                    x                ⁡                                  (                  i                  )                                            -                              x                ⁡                                  (                                      i                    -                    1                                    )                                                      =                                          4                ⁢                δ                ⁢                                  xe2x80x83                                ⁢                f                                                              tan                  2                                ⁡                                  (                  β                  )                                                                                        (        viii        )             The total path-length follows from summation of all contributions:                                                                         X                ⁡                                  (                  y                  )                                            =                              Δ                ⁢                                  xe2x80x83                                ⁢                                  x                  ⁡                                      (                                          1                      +                      2                      +                      …                      +                                              N                        ⁡                                                  (                          y                          )                                                                                      )                                                                                                                          =                              Δ                ⁢                                  xe2x80x83                                ⁢                x                ⁢                                                                            1                      2                                        ⁡                                          [                                              N                        ⁡                                                  (                          y                          )                                                                    ]                                                        2                                                                                                        =                                                                    4                    ⁢                    δ                    ⁢                                          xe2x80x83                                        ⁢                    f                                                                              tan                      2                                        ⁡                                          (                      β                      )                                                                      ⁢                                  1                  2                                ⁢                                                      (                                                                  y                        ⁢                                                  xe2x80x83                                                ⁢                        tan                        ⁢                                                  xe2x80x83                                                ⁢                        β                                                                    2                        ⁢                        δ                        ⁢                                                  xe2x80x83                                                ⁢                        f                                                              )                                    2                                                                                                        =                                                y                  2                                                  2                  ⁢                  δ                  ⁢                                      xe2x80x83                                    ⁢                  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=xcex4f . 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 ds. The attenuation length is denoted xcex. A ray that has lateral displacement y is attenuated by a factor.                               exp          ⁡                      (                                          -                                  X                  ⁡                                      (                    y                    )                                                              λ                        )                          =                  exp          ⁡                      (                          -                                                y                  2                                                  2                  ⁢                  δ                  ⁢                                      xe2x80x83                                    ⁢                  f                  ⁢                                      xe2x80x83                                    ⁢                  λ                                                      )                                              (        x        )             Thus, the rms beam spread is xcex4={square root over (xcex4fxcex)}xe2x80x83xe2x80x83(xi) The gain will be a product of the geometrical gain and the transmission through the lens.                               G          ⁡                      (                          y              d                        )                          =                  xe2x80x83                ⁢                                            2              ⁢                              y                d                                                    d              s                                ⁢                                                    s                0                            +                              s                i                                                    s              0                                ⁢                      1                          y              d                                ⁢                                    ∫              0                              y                d                                      ⁢                                          exp                ⁡                                  (                                      -                                                                  y                        2                                                                    2                        ⁢                                                  σ                          2                                                                                                      )                                            ⁢                              ⅆ                y                                                                            =                  xe2x80x83                ⁢                                            2              ⁢                              (                                  1                  +                                      M                    y                                                  )                                                    d              s                                ⁢                      2                          π                                ⁢                                    π                        2                    ⁢                                    ∫              0                                                y                  d                                                                      2                    ⁢                    π                                                                        ⁢                                          exp                ⁡                                  (                                      -                                          ξ                      2                                                        )                                            ⁢                              2                            ⁢              σ              ⁢                              ⅆ                ξ                                                                            =                  xe2x80x83                ⁢                                            1              +                              M                y                                                    d              s                                ⁢                                    2              ⁢              π                                ⁢                      σ            ·                          erf              ⁡                              (                                                      y                    d                                                                              2                                        ⁢                    σ                                                  )                                                         My is the lateral magnification and the error function is used:                               erf          ⁡                      (            z            )                          =                              2                          π                                ⁢                                    ∫              0              z                        ⁢                                          exp                ⁡                                  (                                      -                                          x                      2                                                        )                                            ⁢                              ⅆ                x                                                                        (        xii        )             The error function will approach unity when the height is increased, and in the limiting ydxe2x86x92xe2x88x9d,                               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 yd 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 s1+xcex94, 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          ⁢                      xe2x80x83                    ⁢          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                ϵ                                                    ⁢                  
                ⁢        where                            (        xvi        )                                ϵ        =                  |                                    1                              s                0                                      +                          1                              s                1                                      -                          1              f                                |                                    (                  x          ⁢                      xe2x80x83                    ⁢          v          ⁢                      xe2x80x83                    ⁢          i          ⁢                      xe2x80x83                    ⁢          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, yd, 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/2xcfx80 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 xe2x80x83Ilens=xcex8xe2x80x83xe2x80x83(ixx) Including absorption, the flux falling on the slit is given by an integral over the angle xcex1 of the ray from the source;                               I          lens                      a            ⁢                          xe2x80x83                        ⁢            b            ⁢                          xe2x80x83                        ⁢            s                          =                              ∫                          -                              min                ⁡                                  (                                      θ                    ,                                                                  y                        d                                            /                                              s                        0                                                                              )                                                                    min              ⁡                              (                                  θ                  ,                                                            y                      d                                        /                                          s                      0                                                                      )                                              ⁢                                    exp              (                                                                    -                                          s                      0                      2                                                        ⁢                                      α                    2                                                                    2                  ⁢                                      σ                    2                                                              )                        ⁢                          ⅆ              α                                                          (        xx        )             Here a simplification is made. The aperture is limited either by xcex8 or by yd=s0. However, even in the last case integration is made to xcex8. 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            ⁢                          xe2x80x83                        ⁢            b            ⁢                          xe2x80x83                        ⁢            s                          =                                                            2                ⁢                π                                      ⁢            σ            ⁢                          1                              s                0                                      ⁢                          erf              ⁡                              (                                                                            θs                      0                                        ⁢                    1                                                                              2                                        ⁢                    σ                                                  )                                              =                                                    2                ⁢                π                                      ⁢            σ            ⁢                          1                              s                0                                      ⁢                          erf              ⁡                              (                                                      d                    s                                                        2                    ⁢                    σ                    ⁢                                          xe2x80x83                                        ⁢                                          s                      i                                        ⁢                    ϵ                    ⁢                                          2                                                                      )                                                                        (        xxi        )             The gain will be                               G          ⁡                      (            0            )                          =                                            I              lens                              a                ⁢                                  xe2x80x83                                ⁢                b                ⁢                                  xe2x80x83                                ⁢                s                                      /                          I              0                                =                                                    2                ⁢                π                                      ⁢                                                            s                  0                                +                                  s                  i                                                            s                l                                      ⁢                          σ                              d                s                                      ⁢                          erf              ⁡                              (                                                      d                    s                                                        2                    ⁢                                          2                                        ⁢                    σϵ                    ⁢                                          xe2x80x83                                        ⁢                                                                  s                        ⁢                                                  xe2x80x83                                                                    i                                                                      )                                                                        (        xxii        )             Now assuming that the point source is located at ys 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                                                    ⁢                          σϵ                          ⁢                                                      xe2x80x83                                                    ⁢                                                                                    s                              ⁢                                                              xe2x80x83                                                                                      l                                                                                              +                                                                        y                          s                                                                          2                          ⁢                                                      2                                                    ⁢                          σϵ                          ⁢                                                      xe2x80x83                                                    ⁢                                                                                    s                              ⁢                                                              xe2x80x83                                                                                      0                                                                                                                )                                                  -                                  erf                  ⁡                                      (                                                                  -                                                                              d                            s                                                                                2                            ⁢                                                          2                                                        ⁢                            σϵ                            ⁢                                                          xe2x80x83                                                        ⁢                                                                                          s                                ⁢                                                                  xe2x80x83                                                                                            i                                                                                                                          +                                                                        y                          s                                                                          2                          ⁢                                                      2                                                    ⁢                          σϵ                          ⁢                                                      xe2x80x83                                                    ⁢                                                                                    s                              ⁢                                                              xe2x80x83                                                                                      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 xcex1"sgr"=sqrt{fxcex4xcex}xe2x80x83xe2x80x83(xxiv) and thus xcex4xcex 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 xcex4xe2x88x9dExe2x88x922 and a parameterization of the X-ray cross-section in barns (xe2x88x9dxc2xd) is (from fitting totabulated values) 24.15Z4.2Exe2x88x929+0.56Zxe2x80x83xe2x80x83(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(xcex4, xcex)=0=Eopt=2.78Z1.07 keVxe2x80x83xe2x80x83(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. 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 90xc2x0. 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. 3a 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. 3b. 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. 3a. 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=8.5xc3x9710xe2x88x927 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 0.1xc2x0. 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. 6a and 6b show the side and the top view, respectively, of an embodiment in which two sawtooth profile refractive lenses 600a and 600b are used to achieve two-dimensional focusing. The second sawtooth profile refractive lens 600b is rotated 90xc2x0 around the optical axis with respect to the first one 600a. 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 xcexcm (10 cm at 33 rpm in 25 min). In order to have the dimensions of vibration decoupled, the bottom angle must be 90xc2x0 in stereo mode, i.e. xcex2 as defined in the xe2x80x9cBASIC THEORYxe2x80x9d section is 45xc2x0. Thus, if there were no inter-spacing between the grooves, the depth would be 60 xcexcm. 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 xcexcm. However, the cutting is a flexible process with many free parameters. The restriction is the 100 xcexcm lacquer layer on the master that limits the depth to about 90 xcexcm and consequently the width to 180 xcexcm. A master was cut with a depth of 90xc2x0 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 xcexcm separation at the end, the radius of curvature is R=(90 xcexcm) 2=(2/Delta 300 mm)=0:135 xcexcm. 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.