PATENT DOCUMENT

Abstract:
One example of a solar photovoltaic concentrator has a primary mirror with multiple free-form panels, each of which forms a Köhler integrator with a respective panel of a lenticular secondary lens. The Köhler integrators are folded by a common intermediate mirror. The resulting plurality of integrators all concentrate sunlight onto a common photovoltaic cell. Luminaires using a similar geometry are also described.

Full Description:
CROSS REFERENCE TO RELATED APPLICATION 
       [0001]    This application claims benefit of U.S. Provisional Application No. 61/268,129 titled “Kohler Concentrator”, filed Jun. 8, 2009 in the names of Miñano et al., which is incorporated herein by reference in its entirety. 
     
    
       [0002]    Reference is made to commonly-assigned International Patent Applications Nos. WO 2007/016363 to Miñano et al. and WO 2007/103994 to Benítez et al. which are incorporated herein by reference in their entirety. 
         [0003]    Embodiments of the devices described and shown in this application may be within the scope of one or more of the following U.S. patents and patent applications and/or equivalents in other countries: U.S. Pat. Nos. 6,639,733, issued Oct. 28, 2003 in the names of Miñano et al., 6,896,381, issued May 24, 2005 in the names of Benítez et al., 7,152,985 issued Dec. 26, 2006 in the names of Benítez et al., and 7,460,985 issued Dec. 2, 2008 in the names of Benítez et al.; WO 2007/016363 titled “Free-Form Lenticular Optical Elements and Their Application to Condensers and Headlamps” to Miñano et al and US 2008/0316761 of the same title published Dec. 25, 2008 also in the names of Miñano et al; WO 2007/103994 titled “Multi-Junction Solar Cells with a Homogenizer System and Coupled Non-Imaging Light Concentrator” published Sep. 13, 2007 in the names of Benítez et al; US 2008/0223443, titled “Optical Concentrator Especially for Solar Photovoltaic” published Sep. 18, 2008 in the names of Benítez et al.; and US 2009/0071467 titled “Multi-Junction Solar Cells with a Homogenizer System and Coupled Non-Imaging Light Concentrator” published Mar. 19, 2009 in the names of Benítez et al. 
       GLOSSARY 
       [0004]    Concentration-Acceptance Product (CAP)—A parameter associated with any solar concentrating architecture, it is the product of the square root of the concentration ratio times the sine of the acceptance angle. Some optical architectures have a higher CAP than others, enabling higher concentration and/or acceptance angle. For a specific architecture, the CAP is nearly constant when the geometrical concentration is changed, so that increasing the value of one parameter lowers the other. 
         [0005]    Fresnel Facet—Element of a discontinuous-slope concentrator lens that deflects light by refraction. 
         [0006]    TIR Facet—Element of a discontinuous-slope concentrator lens that deflects light by total internal reflection. 
         [0007]    Primary Optical Element (POE)—Optical element that receives the light from the sun or other source and concentrates it towards the Intermediate Optical Element, if any, or to the Secondary Optical Element. 
         [0008]    Intermediate Optical Element (JOE)—Optical element that receives the light from the Primary Optical Element and concentrates it towards the Secondary Optical Element. 
         [0009]    Secondary Optical Element (SOE)—Optical element that receives the light from the Primary Optical Element or from the Intermediate Optical element, if any, and concentrates it towards the solar cell or other target. 
         [0010]    Cartesian Oval—A curve (strictly a family of curves) used in imaging and non-imaging optics to transform a given bundle of rays into another predetermined bundle, if there is no more than one ray crossing each point of the surface generated from the curve. The so-called Generalized Cartesian Oval can be used to transform a non-spherical wavefront into another. See Reference [10], page 185, Reference [16]. 
       BACKGROUND 
       [0011]    Triple-junction photovoltaic solar cells are expensive, making it desirable to operate them with as much concentration of sunlight as practical. The efficiency of currently available multi-junction photovoltaic cells suffers when local concentration of incident radiation surpasses ˜2,000-3,000 suns. Some concentrator designs of the prior art have so much non-uniformity of the flux distribution on the cell that “hot spots” up to 9,000-11,000× concentration happen with 500× average concentration, greatly limiting how high the average concentration can economically be. Kaleidoscopic integrators can reduce the magnitude of such hot spots, but they are more difficult to assemble, and are not suitable for small cells. 
         [0012]    There are two main design problems in Nonimaging Optics, and both are relevant here. The first is called “bundle-coupling” and its objective is to maximize the proportion of rays in a given input bundle that are transformed into a given output bundle. In a solar concentrator, that is effectively to maximize the proportion of the light power emitted by the sun or other source that is delivered to the receiver. The second problem, known as “prescribed irradiance,” has as its objective to produce a particular illuminance pattern on a specified target surface using a given source emission. 
         [0013]    In bundle-coupling, the design problem consists in coupling two ray bundles M i  and M o , called the input and the output bundles respectively. Ideally, this means that any ray entering into the optical system as a ray of the input bundle M i  exits it as a ray of the output bundle M o , and vice versa. Thus the successfully coupled parts of these two bundles M i  and M o  comprise the same rays, and thus are the same bundle M c . This bundle M c  is in general M c =M i ∩M o . In practice, coupling is always imperfect, so that M c ⊂M i  and M c ⊂M o . 
         [0014]    In prescribed-irradiance, however, it is only specified that one bundle must be included in the other, M i  in M o . Any rays of M i  that are not included in M o  are for this problem disregarded, so that M i  is effectively replaced by M c . In this type of solution an additional constraint is imposed that the bundle M c  should produce a prescribed irradiance on a target surface. Since M c  is not fully specified, this design problem is less restrictive than the bundle coupling one, since rays that are inconvenient to a particular design can be deliberately excluded in order to improve the handling of the remaining rays. For example, the periphery of a source may be under-luminous, so that the rays it emits are weaker than average. If the design edge rays are selected inside the periphery, so that the weak peripheral region is omitted, and only the strong rays of the majority of the source area are used, overall performance can be improved. 
         [0015]    Efficient photovoltaic concentrator (CPV) design well exemplifies a design problem comprising both the bundle coupling problem and the prescribed irradiance problem. M i  comprises all rays from the sun that enter the first optical component of the system. M o  comprises those rays from the last optical component that fall onto the actual photovoltaic cell (not just the exterior of its cover glass). Rays that are included in M i  but are not coupled into M o  are lost, along with their power. (Note that in computer ray tracing, rays from a less luminous part of the source will have less flux, if there are a constant number of rays per unit source area.) The irradiance distribution of incoming sunlight must be matched to the prescribed (usually uniform) irradiance on the actual photovoltaic cell, to preclude hot-spots. Optimizing both problems, i.e., to obtain maximum concentration-acceptance product as well a uniform irradiance distribution on the solar cell&#39;s active surface, will maximize efficiency. Of course this is a very difficult task and therefore only partial solutions have been found. 
         [0016]    Good irradiance uniformity on the solar cell can be potentially obtained using a light-pipe homogenizer, which is a well known method in classical optics. See Reference [1]. When a light-pipe homogenizer is used, the solar cell is glued to one end of the light-pipe and the light reaches the cell after some bounces on the light-pipe walls. The light distribution on the cell becomes more uniform with light-pipe length. The use of light-pipes for concentrating photo-voltaic (CPV) devices, however, has some drawbacks. A first drawback is that in the case of high illumination angles the reflecting surfaces of the light-pipe must be metalized, which reduces optical efficiency relative to the near-perfect reflectivity of total internal reflection by a polished surface. A second drawback is that for good homogenization a relatively long light-pipe is necessary, but increasing the length of the light-pipe both increases its absorption and reduces the mechanical stability of the apparatus. A third drawback is that light pipes are unsuitable for relatively thick (small) cells because of lateral light spillage from the edges of the bond holding the cell to the end of the light pipe, typically silicone rubber. Light-pipes have nevertheless been proposed several times in CPV systems, see References [2], [3], [4], [5], [6], and [7], which use a light-pipe length much longer than the cell size, typically 4-5 times. 
         [0017]    Another strategy for achieving good uniformity on the cell is the Köhler illuminator. Köhler integration can solve, or at least mitigate, uniformity issues without compromising the acceptance angle and without increasing the difficulty of assembly. 
         [0018]    Referring to  FIG. 2 , the first photovoltaic concentrator using Kohler integration was proposed (see Reference [8]) by Sandia Labs in the late 1980&#39;s, and subsequently was commercialized by Alpha Solarco. A Fresnel lens  21  was its primary optical element (POE) and an imaging single surface lens  22  (called SILO, for SIngLe Optical surface) that encapsulates the photovoltaic cell  20  was its secondary optical element (SOE). That approach utilizes two imaging optical lenses (the Fresnel lens and the SILO) where the SILO is placed at the focal plane of the Fresnel lens and the SILO images the Fresnel lens (which is uniformly illuminated) onto the photovoltaic cell. Thus, if the cell is square the primary can be square trimmed without losing optical efficiency. That is highly attractive for doing a lossless tessellation of multiple primaries in a module. On the other hand, the primary optical element images the sun onto the secondary surface. That means that the sun image  25  will be formed at the center of the SILO for normal incidence rays  24 , and move towards position  25  on the secondary surface as the sun rays  26  move within the acceptance angle of the concentrator due to tracking perturbations and errors. Thus the concentrator&#39;s acceptance is determined by the size and shape of the secondary optical element. 
         [0019]    Despite the simplicity and high uniformity of illumination on the cell, the practical application of the Sandia system is limited to low concentrations because it has a low concentration-acceptance product of approximately 0.3 (±1° at 300×). The low acceptance angle even at a concentration ratio of 300× is because the imaging secondary cannot achieve high illumination angles on the cell, precluding maximum concentration. 
         [0020]    Another previously proposed approach uses four optical surfaces, to obtain a photovoltaic concentrator for high acceptance angle and relatively uniform irradiance distribution on the solar cell (see Reference [9]). The primary optical element (POE) of this concentrator should be an element, for example a double aspheric imaging lens, that images the sun onto the aperture of a secondary optical element (SOE). Suitable for a secondary optical element is the SMS (Simultaneous Multiple Surface) designed RX concentrator described in References [10], [11], [12]. This is an imaging element that works near the thermodynamic limit of concentration. In this notation, the surfaces of the optical device are listed in the order in which the light beam encounters them: I denotes a totally internally reflective surface, R denotes a refractive surface, and X denotes a reflective surface that may be opaque. If a light beam encounters the same surface twice, it is listed at both encounters with the correct type for each encounter. 
         [0021]    A good strategy for increasing the optical efficiency of the system (which is a critical merit function) is to integrate multiple functions in fewer surfaces of the system, by designing the concentrator optical surfaces to have at least a dual function, e.g., to illuminate the cell with wide angles, at some specified approximation to uniformity. That entails a reduction of the degrees of freedom in the design compared to the ideal four-surface case. Consequently, there is a trade-off between the selected geometry and the homogenization method, in seeking a favorable mix of optical efficiency, acceptance angle, and cell-irradiance uniformity. 
         [0022]    There are two ways to achieve irradiance homogenization. The first is a Köhler integrator, as mentioned before, where the integration process is along both dimensions of the ray bundle, meridional and sagittal. This approach is also known as a 2D Köhler integrator. The other strategy is to integrate in only one of the ray bundle&#39;s dimensions; thus called a 1D Köhler integrator. These integrators will typically provide a lesser homogeneity than is achievable with in 2D, but they are easier to design and manufacture, which makes them suitable for systems where uniformity is not too critical. A design method for calculating fully free-form 1D and 2D Köhler integrators was recently developed (see References [13], [14]), where optical surfaces are used that have the dual function of homogenizing the light and coupling the design&#39;s edge rays bundles. 
         [0023]    In all the embodiments of the present invention, the primary optical element is reflective. The use of reflective primaries is old in solar concentrators, since the parabolic mirror has been in the public domain since centuries. More recently, advanced high-performance free-form asymmetric mirror designs that use a free-form lens with a short kaleidoscope homogenizer protruding from it [14]. designs have been developed. Also recently, the use of two-mirror Cassegrain type concentrators, common in antenna and telescope design, has been extended to solar concentrators with the addition of a kaleidoscope homogenizer [6], and with radial Kohler integration [14] [15]. 
       SUMMARY 
       [0024]    Embodiments of the present invention provide different photovoltaic concentrators that combine high geometric concentration, high acceptance angle, and high irradiance uniformity on the solar cell. In all the embodiments, the primary optical element is reflective in the sense that the light rays exit the primary on the same side that the light rays impinged from. Also in all the embodiments, the primary and secondary optical elements are each lenticulated to form a plurality of segments. In some embodiments, an intermediate optical element, not necessarily segmented, is used in between the primary and the secondary. A segment of the primary optical element and a segment of the secondary optical element combine to form a Köhler integrator. The multiple segments result in a plurality of Köhler integrators that collectively focus their incident sunlight onto a common target, such as a photovoltaic cell. Any hotspots are typically in different places for different individual Köhler integrators, with the plurality further averaging out the multiple hotspots over the target cell. 
         [0025]    In some embodiments, the optical surfaces are modified, typically by lenticulation (i.e., the formation on a single surface of multiple independent lenslets that correspond to the segments mentioned before) to produce Köhler integration. Although the modified optical surfaces behave optically quite differently from the originals, they are macroscopically very similar to the unmodified surface. This means that they can be manufactured with the same techniques (typically plastic injection molding or glass molding) and that their production cost is the same. 
         [0026]    An embodiment of the invention provides an optical device comprising: a primary optical element having a plurality of segments, which in an example are four in number; and a secondary optical element having a plurality of segments, which in an example are four lenticulations of an optical surface of a lens; wherein each segment of the primary optical element, along with a corresponding segment of the secondary optical element, forms one of a plurality of Köhler integrators. The plurality of Köhler integrators are arranged in position and orientation to direct light from a common source onto a common target. The common source, where the device is a light collector, or the common target, where the device is a luminaire, may be external to the device. For example, in the case of a solar photovoltaic concentrator, the source is the sun. Whether it is the common source or the common target, the other may be part of the device or connected to it. For example, in a solar photovoltaic concentrator, the target may be a photovoltaic cell. Further embodiments of the device, however, could be used to concentrate or collimate light between an external common source and an external common target. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0027]    The above and other aspects, features and advantages of the present invention will be apparent from the following more particular description thereof, presented in conjunction with the following drawings wherein: 
           [0028]      FIG. 1  shows design rays used for calculating the desired shape of a radial Köhler refractive lenticulation pair. 
           [0029]      FIG. 2  shows certain principles of the Fresnel-SILO concentrator developed by Sandia Labs. 
           [0030]      FIG. 3  shows a two mirror Cassegrain-type reflective concentrator of sunlight. 
           [0031]      FIG. 4A  shows a perspective view of a quad-lenticular XXR Köhler concentrator that uses azimuthal integration. 
           [0032]      FIG. 4B  shows aside view of the quad-lenticular XXR Köhler concentrator of  FIG. 4A . 
           [0033]      FIG. 5  is a first diagram of a design process for the concentrator shown in  FIG. 4A . 
           [0034]      FIG. 6  is a second diagram of the design process of  FIG. 5 . 
           [0035]      FIG. 7  is a third diagram of the design process of  FIG. 5 . 
           [0036]      FIG. 8  is a fourth diagram of the design process of  FIG. 5 . 
           [0037]      FIG. 9  is a perspective view similar to part of  FIG. 4A , illustrating a second stage of the design process of  FIGS. 5 to 8 . 
           [0038]      FIG. 10A  is an axial sectional view of another embodiment of XXR concentrator, showing ray paths in the plane of section. 
           [0039]      FIG. 10B  is a perspective view of the concentrator of  FIG. 10A , showing ray paths over the whole area of the optical elements. 
           [0040]      FIG. 11  is a graph of the performance of the concentrator of  FIG. 10A . 
           [0041]      FIG. 12A  is an axial sectional view of another form of concentrator. 
           [0042]      FIG. 12B  is a perspective view of the concentrator of  FIG. 12A . 
           [0043]      FIG. 12C  is a perspective view of a further form of concentrator. 
           [0044]      FIG. 13  is a perspective view of another form of concentrator. 
           [0045]      FIG. 14  is an axial sectional view of a further form of concentrator. 
           [0046]      FIG. 15A  is a perspective view of another form of concentrator. 
           [0047]      FIG. 15B  is an enlarged view of one mirror of the concentrator of  FIG. 15A . 
       
    
    
     DETAILED DESCRIPTION OF THE EMBODIMENTS 
       [0048]    A better understanding of various features and advantages of the present invention may be obtained by reference to the following detailed description of embodiments of the invention and accompanying drawings, which set forth illustrative embodiments in which various principles of the invention are utilized. 
         [0049]    Two types of secondary optical elements are described herein: one comprising an array of refractors, the second an array of reflectors. Both exhibit overall N-fold symmetry. In the embodiments taught in this specification the primary reflective elements have the same N-fold symmetry as the secondary optic. In some embodiments the primary is asymmetric so the rest of elements are not located in front of the primary but on the side. Two types of intermediate optical elements are described herein: reflective type, and refractive type. The reflective intermediate optical element folds the ray path, permitting the removal of the secondary optical element and the solar cell (and heat-sink) from in front of the primary. 
         [0050]    As may be seen from  FIGS. 4 and 9  to  10 B, symmetrical XXR configurations allow the photovoltaic cell to be placed close to, at, or even behind the primary mirror. Heat can then be removed to the rear of the primary mirror, greatly reducing the cooling problems of some prior designs, and the mounting for the PV cell can also be provided behind the primary mirror. Suitable heat sinks and mountings, are already known, and in the interests of clarity have been omitted from the drawings. 
         [0051]    Several Köhler integrating solar concentrators are described herein. They are the first to combine a non flat array of Köhler integrators with concentration optics. Although, the embodiments of the invention revealed herein have quadrant symmetry, the invention does not limit embodiments to this symmetry but can be applied, by those skilled in the art, to other configurations (preferably N-fold symmetry, where N can be any number greater than two) once the principles taught herein are fully understood. 
         [0052]      FIG. 1  shows lenticulation  10 , comprising two refractive off-axis surfaces, primary optical element (POE)  11  and secondary optical element (SOE)  12 , through which a light source outside the drawing illuminates cell  13 . The final Radial Köhler concentrator will be the combination of several such lenticulation pairs, with common rotational axis  14  shown as a dot-dashed line. Solid lines  15  define the spatial edge rays and dotted lines  16  define the angular edge rays. They show the behavior of parallel and converging rays, respectively. In an embodiment, each optical element lenticulation  11 ,  12  may be one or more optical surfaces, each of which may be continuous or subdivided. For example, POE  11  may be a Fresnel lens, with one side flat and the other side formed of arcuate prisms. 
         [0053]    Radial Köhler concentrators are 1D Köhler integrators with rotational symmetry. This makes the design process much easier than a 1D free-form Köhler integrator. Furthermore, rotational symmetry makes the manufacturing process as simple for a lenticular form as for any other aspheric rotational symmetry. The design process, however, first designs a 2D optical system, and then applies rotational symmetry. 
         [0054]    Although the irradiance distribution produced by a radial Köhler concentrator has a hotspot, it is much milder than that produced by an imaging system. If α is the system acceptance angle, α s  is the sun&#39;s angular radius, and k is a constant that depends on the shape of the cell&#39;s active area (where k=1 for a round cell and k=4/π for a square cell), it can easily be seen that the hotspot generated by a radial Köhler approach is proportional to k*(α/α s ) times the average optical concentration, while the hotspot generated by an aplanatic device is proportional to k*(α/α s ) 2  times the average optical concentration. For instance, if α=1°, α s =¼° (the angular radius of the sun as seen from Earth), and k=1, the hotspot created by a radial Köhler is around 4 times the average concentration, while the aplanatic design produces a hotspot 16 times the average concentration. For a square cell (k=4/π) the corresponding hotspots are 5 and 20 times the average concentration. 
         [0055]    The radial Köhler concept has been applied in CPV systems to a two-mirror Cassegrain-type reflective concentrator (see Reference [15] and above-referenced WO 2007/103994).  FIG. 3  shows a prior art two-mirror Cassegrain-type reflective concentrator  30 , comprising lenticulated primary mirror  31 , secondary mirror  32 , and encapsulated solar cell  33  mounted on heat sink  34 . Each concave reflector-lenticulation segment  31 L is an annulus, and reflects incoming rays  35  as converging rays  36  focusing onto a corresponding annular lenticulation segment of secondary mirror  32 , which in turn spreads them over cell  33 , a 1 cm 2  cell of the triple junction type. Concentrator  30  is designed to work at C g =650× with ±0.9° of acceptance angle, and has optical efficiency of 78%, with a maximum irradiance peak on the cell of 1200 suns. In the Radial Köhler design of  FIG. 3 , integration takes place only in the radial (meridional) direction, and not in the azimuthal or tangential (sagittal) direction. Also, the Kohler integrators are all different, because they are concentric rings, which both increases complexity and reduces uniformity. It is possible to configure the radial Köhler device to produce uniform irradiation of the photovoltaic cell with the sun on axis, but a hot spot then appears when the sun is off axis. In addition, Kohler integration with circular primary segments produces a circular irradiation on the photovoltaic cell, which is less than optimal because most commercially available PV cells are square. 
         [0056]    In this Radial Köhler design, the average concentration and the peak concentration can be high, so that it is necessary to introduce a further degree of freedom in the radial Köhler design, in order to keep the irradiance peak below 2000 suns. To perform the integration in a second direction, the present application comprises a concentrator with four subsystems (having quad-symmetry), hereinafter referred to as segments, that symmetrically compose a whole that achieves azimuthal integration, while keeping each of the four subsystems rotationally symmetric and thus maintaining ease of manufacture, since each is actually a part of a complete rotationally symmetric radial Köhler system, analogous to those of  FIG. 2  and  FIG. 3 . 
         [0057]    Better homogenization is produced when using a two-directional free-form Köhler integrator instead of a rotational-symmetric one. A possible type of free-form Köhler system is the same XXR, comprising a primary reflector, and intermediate reflector and a secondary refractor, in which the Kohler integration is performed between the primary and secondary elements.  FIG. 4A  and  FIG. 4B  show an embodiment of an XXR Köhler concentrator  40 , comprising four-fold segmented primary mirror  41 , four-fold segmented secondary lens  42 , an intermediate mirror  44  and photovoltaic cell  43 . 
         [0058]    The photovoltaic receiver has preferably a square flat active area, and without loss of generality can be considered as located in a coordinate system in which the receiver plane is z=0 and the sides of the active area are parallel to the x and y axes, and the origin is in the center of the active area. Because of the symmetry, defining the unit in the region x&gt;0, y&gt;0 fully defines the primary optical element. The intermediate optical element will preferably have rotational symmetry around the z axis. The secondary optical element will preferably have the same four-fold symmetry as the primary. In the particular embodiment shown in  FIG. 4A  and  FIG. 4B , the units of the primary and secondary optical elements in regions x&gt;0, y&gt;0 are Köhler pairs, but other correspondences are obviously possible. 
         [0059]    The design process has then three stages. First, the diagonal cross section profiles of the primary and intermediate mirrors are designed as in two dimensions using the SMS2D method (detailed below) with the conditions that the edge rays impinging on the entry aperture tilted +α and −α (α being the design acceptance angle) are focused in two dimensions (i.e., all the rays are contained in a plane) on close to the boundary points A and B of its corresponding lenticulation of the secondary lens, see  FIG. 5 . Second and third stages correspond to the design in three dimensions of the free-form surface of the primary and secondary, respectively. 
         [0060]    The first stage of the design is done with the following process, illustrated by  FIG. 5  to  FIG. 8 , and generates a cross-section through the three optical surfaces in the x=y plane  90  (see  FIG. 9 ). 
         [0061]    1. Choose β, which is the direction of the normal to the optical surface at B. 
         [0062]    2. Choose the x coordinates of R (&amp; R′), which are the corner points of the active area of the PV cell  43 , the x and z coordinates of point B and of point E, which is the outer corner of the selected lenticulation of the primary  41 , and the z coordinate of point D, which is on the rim of the intermediate optical element  44 . 
         [0063]    3. Calculate the x coordinate of D by tracing the reversed ray R′-B-D. 
         [0064]    4. Calculate the optical path length R′-B-D-E. 
         [0065]    5. Choose α. 
         [0066]    6. Calculate the normal vector at E so as to reflect the known reversed ray D-E into the direction −α. 
         [0067]    7. Choose the z coordinate z A  of point A, Calculate the x coordinate of point A using the formula x A =(2 1/2 −1)/(2 1/2 +1)x B . 
         [0068]    8. Calculate the line of the intermediate mirror from D to C as a “distortion-free imaging oval” so that there is a linear mapping between tilt (sin) angles of rays at E in the range +/−α and points along the straight segment A to B. (See  FIG. 6 ). 
         [0069]    9. Calculate the points of the secondary lens, starting from B, so that the rays from E reflected off the intermediate mirror are focused by refraction to R′ (using the optical path length condition, if desired). This is most conveniently done at the same time each point of the intermediate mirror is calculated. 
         [0070]    10. The secondary lens calculated in step 9 will usually not pass through the previously chosen point A. The intersection of the secondary lens with the line x=x A  gives a better estimation of z A . So go back to step 7, substitute the new value of z A , and do an “iteration loop of z A ,” repeating steps 8 and 9, and optionally repeating this step 10. 
         [0071]    11. Calculate the primary and intermediate mirrors with SMS2D to form an image of the incident light from angle −α in B and of the incident light from angle +α in A. (See  FIGS. 7 and 8 .) 
         [0072]    12. When the primary arrives at the z-axis, if the ray from +α at G after refraction at A does not reach R but a different point R″ on the receiver surface, go back to step 5 and choose a better a with value α *|R′R|/|R′R″|. Then repeat the subsequent steps. 
         [0073]    13. If the x-coordinate of the last calculated point of the intermediate mirror (i.e., the closest to the z-axis) is not properly allocated (for instance, is negative), go back to step 2 and choose a different value for the coordinate x B  of point B. Then repeat the subsequent steps. 
         [0074]    14. Generate the three-dimensional intermediate mirror by revolution of the profile with respect to the z-axis. 
         [0075]    In the second stage of the design, illustrated in  FIG. 9 , the section x&gt;0, y&gt;0 of primary optical element  91  is designed in three-dimensions as the free-form mirror that forms an approximate image of the sun on the paired section of the secondary optical element through the rotationally symmetric intermediate mirror  94 . Such a free-form primary mirror can be designed, for instance, as the Generalized reflective Cartesian oval that focuses all the +α rays in three dimensions, which are parallel to direction (−sin α, −sin α, −cos α), onto the point A after reflection on the intermediate mirror. 
         [0076]    In the third step of the design, the secondary free-form lens is designed to form an image of the paired section of the primary optical element, reflected in the intermediate optical element, on the solar cell. Again, such a free-form lens can be designed, for instance, as the Generalized refractive Cartesian oval that receives rays passing through corner point E of the primary and reflected on the rotational intermediate mirror, and focuses them in three dimensions on the corner point R of the cell. 
         [0077]    Note that the calculation in three dimensions of the primary and secondary is consistent with the two dimensional design, which means that the curves  95  and  96  contained in the free-form mirror and lens at the intersection of the diagonal x=y plane  90  in  FIG. 9A  coincide with the profiles calculated in the two-dimensional plane of  FIG. 5  to  FIG. 8 . 
         [0078]    The contour of the primary mirror in three dimensions is given by the image of the photovoltaic cell projected by the secondary lens. A notional cell larger than the real cell can be considered here, to allow for cell placement tolerances. The minimum contour size of the secondary lens units is defined by the image of the three-dimensional acceptance area (that is, the cone of radius α). 
         [0079]    The intermediate mirror designed as described in the first stage differs very significantly from the aplanatic two mirror imaging design used in reference [6]. The aplanatic design produced focusing of the on-axis input rays onto an on-axis point, while the focal region of the on-axis input rays in the intermediate mirror designed according to the present embodiment is approximately centered in the off-axis segment AB. The difference is specially clear if the three-dimensional design is done using the intermediate mirror described in reference [6] and both +α rays and −α rays are traced as in  FIG. 7  and  FIG. 8 , respectively. Even though the primary mirror is redesigned in three dimensions to perfectly focus the +a rays (rays incident parallel to (−sin α, −sin α, −cos α)) onto A, the use of the mirror of reference [6] as the intermediate optical element causes the focal region of the −α rays (parallel to (+sin α, +sin α, −cos α)) to be formed very far from the rim B of the secondary, specifically at a much higher z. 
         [0080]    In another preferred embodiment, the intermediate mirror is also free-form and the primary and intermediate mirrors are designed using the SMS3D method, so four edge rays of the acceptance angle cone are approximately focused on four points at the rim of its corresponding lenticulation of the secondary in 3D geometry. 
         [0081]    Referring to  FIGS. 10A and 10B  (collectively “FIG.  10 ”),  FIG. 10A  shows an XXR system similar to that of  FIG. 4B  with rays contained in a diagonal plane.  FIG. 10B  shows a close-up view of converging rays (in this case traced though the whole aperture) focusing to points  101  on the surface of secondary lens  103  (shown de-emphasized), and then spread out to uniformly cover cell  102 . The irradiance thereupon is the sum of the four images of the primary mirror segments. 
         [0082]    An embodiment of the XXR Köhler in  FIG. 10  achieves a geometric concentration Cg=2090× (ratio of primary projected aperture area to cell area) with an acceptance of ±0.85°, which is a very good result for this concentration level as compared to the prior art. This high concentration level allows reduced cell costs in the system, and the acceptance angle is still high enough to provide the manufacturing tolerances needed for low cost. Shadowing of primary mirror  41  by intermediate mirror  45  is smaller than 5%. 
         [0083]      FIG. 11  shows graph  110  with abscissa  111  plotting off-axis angle and ordinate  112  plotting relative transmission  113  of the XXR Köhler in  FIG. 10 . Vertical dashed line  114  corresponds to 0.85°, and horizontal dashed line  115  corresponds to the 90% threshold at which the acceptance angle is defined. The spectral dependence of the optical performance (optical efficiency, acceptance angle and irradiance distribution) is very small (which is an advantage of using mirrors). 
         [0084]    Tables 1 to 3 (placed at the end of the description) provide an example of a concentrator according to  FIG. 10 . Table 1 contains the X-Y-Z coordinates of points of the free-form primary mirror of said design. The points correspond to the octant X&gt;0, Y&gt;X. Corresponding points in the remaining octants can be generated by interchanging the X and Y coordinates and/or changing the sign of the X and/or Y coordinate. Table 2 contains the p-Z coordinates of the profile points of the intermediate mirror. Since the design is rotationally symmetric, the whole mirror can be generated by rotation of the given coordinates around the Z axis. Finally, Table 3 contains the X-Y-Z coordinates of points of the free-form secondary lens of said design, also in the octant X&gt;0, Y&gt;X. 
         [0085]      FIG. 15A  shows a device  150  which is a modification of the XXR design of  FIG. 10  using grooved reflectors  151  and  152  and the same secondary  153  as in  FIG. 10 . Grooved reflectors are described in U.S. patent application Ser. No. 12/456,406 (Publication Number: US 2010/0002320 A) titled “Reflectors Made of Linear Grooves,” filed 15 Jun. 2009, which is incorporated herein by reference in its entirety, and in which is disclosed how arbitrary rotational aspheric and free-form mirrors can be substituted by dielectric free-form structured equivalents that work by Total Internal Reflection (TIR). TIR is of interest in this XXR device to reduce the reflection losses due to metallic reflection, save the mirror coating cost and avoid the risk of the metal coating corrosion.  FIG. 15B  shows a detail of the intermediate mirror  152 , and the ray  154  coming from the primary is twice totally internal reflected on free-form facets  155  and  156 . In a CPV implementation, the mirrors  150 ,  152  are typically formed as the back surfaces of thin sheets of transparent material. In  FIGS. 15A and 15B  the refractive front surfaces of the dielectric grooved reflectors are not shown for clarity. In other embodiments, the space between the grooved reflectors  150 ,  152  may be a solid block of dielectric material with the grooved reflectors formed on opposite surfaces. 
         [0086]    The present embodiments are a particular realization of the devices described in the above-mentioned patent application WO 2007/016363 to Miñano et al. 
         [0087]    Variations can be obtained by designers skilled in the art. For instance, the number of cells, also called sections or lenslets, on each of the primary and secondary optical elements can be increased, for instance, to nine. Also the cell can be rectangular and not square, and then the four units of the primary mirror will preferably be correspondingly rectangular, so that each unit still images easily onto the photovoltaic cell. Alternatively, or in addition, the number of array units could be reduced to two, or could be another number that is not a square, so that the overall primary is a differently shaped rectangle from the photovoltaic cell. Where each segment is further subdivided into lenslets, the desirable number of lenslets in each primary and secondary lens segment may depend on the actual size of the device, as affecting the resulting size and precision of manufacture of the lens features. 
         [0088]    Examples of such variations are shown in  FIG. 12A  to  FIG. 14 .  FIGS. 12A and 12B  show an embodiment of a two-unit array XR concentrator comprising an asymmetric tilted primary mirror and a refractive secondary to illuminate solar cell  120 , so no intermediate optical element is used in this case. The Kohler pairs are  122   a - 122   b  and  121   a - 121   b . The tilt of the mirror allows the secondary to be placed outside the beam of light incident on the primary, avoiding the shading produced by the secondary and heat sink in conventional centered systems.  FIG. 12C  shows a similar XR configuration with Kohler integration using four units:  123   a  to  136   a  and  123   b  to  126   b.    
         [0089]      FIG. 13  shows a four-unit tilted XR, in which compared to the previous ones the unit is rotated 45 degrees with respect to an axis normal to its surface passing through its center, so the full primary mirror  131  shows the same 45 degree rotation. Each unit has its own secondary lens  130  and PV cell  137  placed at the outer corner of the primary mirror opposite its own primary mirror  131 , in the arrangement shown in  FIG. 13 . Note that the primary  131  receives light from the sun as shown by ray  132  and illuminates the PV cell located behind the secondary  130 . Each primary mirror  131  and each secondary lens  130  is segmented into the Kohler lenticulations, as  133  to  136 . This relative positioning of the primaries and secondaries allows the whole primary to be supported from the secondary positions at the corners, and even the heatsink  137  can be extended along the perimeter to become a supporting frame that eventually can also support a front glass cover. 
         [0090]      FIG. 14  shows an example in which the intermediate optical surface  144  is not a mirror but a lens, while both primary ( 141   a  and  142   a ) and secondary Kohler integrating surfaces ( 141   b  and  142   b ) work by reflection. One secondary reflector  141   b  is metalized (XRX) and the other is a TIR surface (XRI). 
         [0091]    Although various specific embodiments have been shown and described, the skilled reader will understand how features of different embodiments may be combined in a single photovoltaic collector, luminaire, or other device to form other devices within the scope of the present invention. When the photovoltaic cell is replaced by an LED or an LED array, or other light source, the present embodiments provide optical devices that can collimate the light with a quite uniform intensity for the directions of emission, because all points on the source are carried to every direction. This can be used to mix the colors of different LEDs of a source array or to make the intensity of the emission more uniform without the need to bin the chips. 
         [0092]    The preceding description of the presently contemplated best mode of practicing the invention is not to be taken in a limiting sense, but is made merely for the purpose of describing the general principles of the invention. The full scope of the invention should be determined with reference to the Claims. 
         [0000]    
       
         
               
               
               
             
               
               
               
             
           
               
                 TABLE 1 
               
               
                   
               
               
                 x 
                 y 
                 z 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 192.287 
                 195.167 
                 58.140 
               
               
                 56.261 
                 163.672 
                 −12.523 
               
               
                 57.652 
                 167.477 
                 −10.307 
               
               
                 1.976 
                 2.092 
                 −57.289 
               
               
                 1.719 
                 2.311 
                 −57.288 
               
               
                 1.168 
                 2.702 
                 −57.285 
               
               
                 0.578 
                 3.028 
                 −57.281 
               
               
                 −0.041 
                 3.290 
                 −57.274 
               
               
                 5.427 
                 5.579 
                 −57.309 
               
               
                 4.679 
                 6.201 
                 −57.308 
               
               
                 3.873 
                 6.744 
                 −57.305 
               
               
                 3.453 
                 6.985 
                 −57.302 
               
               
                 2.581 
                 7.407 
                 −57.296 
               
               
                 2.134 
                 7.587 
                 −57.292 
               
               
                 1.221 
                 7.889 
                 −57.283 
               
               
                 0.758 
                 8.011 
                 −57.278 
               
               
                 0.058 
                 8.159 
                 −57.269 
               
               
                 9.454 
                 9.667 
                 −57.230 
               
               
                 8.171 
                 10.737 
                 −57.228 
               
               
                 7.065 
                 11.481 
                 −57.225 
               
               
                 6.489 
                 11.815 
                 −57.222 
               
               
                 5.300 
                 12.404 
                 −57.215 
               
               
                 4.690 
                 12.660 
                 −57.211 
               
               
                 3.446 
                 13.095 
                 −57.200 
               
               
                 2.814 
                 13.274 
                 −57.194 
               
               
                 2.178 
                 13.428 
                 −57.187 
               
               
                 0.896 
                 13.662 
                 −57.172 
               
               
                 0.253 
                 13.743 
                 −57.164 
               
               
                 12.965 
                 13.232 
                 −57.085 
               
               
                 11.397 
                 14.580 
                 −57.083 
               
               
                 10.052 
                 15.538 
                 −57.079 
               
               
                 9.353 
                 15.975 
                 −57.076 
               
               
                 7.910 
                 16.764 
                 −57.068 
               
               
                 7.168 
                 17.115 
                 −57.063 
               
               
                 5.652 
                 17.732 
                 −57.050 
               
               
                 4.880 
                 17.997 
                 −57.043 
               
               
                 3.313 
                 18.443 
                 −57.026 
               
               
                 2.522 
                 18.623 
                 −57.017 
               
               
                 0.927 
                 18.899 
                 −56.996 
               
               
                 0.125 
                 18.996 
                 −56.984 
               
               
                 19.039 
                 19.394 
                 −56.643 
               
               
                 18.231 
                 20.140 
                 −56.644 
               
               
                 16.530 
                 21.533 
                 −56.642 
               
               
                 14.730 
                 22.789 
                 −56.637 
               
               
                 13.797 
                 23.364 
                 −56.634 
               
               
                 11.874 
                 24.406 
                 −56.624 
               
               
                 10.887 
                 24.873 
                 −56.618 
               
               
                 8.870 
                 25.698 
                 −56.603 
               
               
                 7.843 
                 26.057 
                 −56.594 
               
               
                 5.759 
                 26.666 
                 −56.574 
               
               
                 4.705 
                 26.916 
                 −56.562 
               
               
                 1.508 
                 27.453 
                 −56.521 
               
               
                 −0.102 
                 27.602 
                 −56.497 
               
               
                 22.715 
                 23.123 
                 −56.261 
               
               
                 20.824 
                 24.804 
                 −56.261 
               
               
                 19.832 
                 25.587 
                 −56.260 
               
               
                 17.764 
                 27.035 
                 −56.256 
               
               
                 15.598 
                 28.321 
                 −56.249 
               
               
                 14.483 
                 28.902 
                 −56.244 
               
               
                 12.196 
                 29.939 
                 −56.231 
               
               
                 11.029 
                 30.395 
                 −56.223 
               
               
                 8.654 
                 31.180 
                 −56.205 
               
               
                 7.449 
                 31.510 
                 −56.194 
               
               
                 5.015 
                 32.045 
                 −56.169 
               
               
                 3.788 
                 32.251 
                 −56.155 
               
               
                 1.323 
                 32.539 
                 −56.124 
               
               
                 0.086 
                 32.623 
                 −56.107 
               
               
                 26.721 
                 26.173 
                 −55.821 
               
               
                 25.697 
                 27.162 
                 −55.822 
               
               
                 24.638 
                 28.112 
                 −55.822 
               
               
                 22.420 
                 29.890 
                 −55.819 
               
               
                 20.080 
                 31.497 
                 −55.813 
               
               
                 18.870 
                 32.236 
                 −55.809 
               
               
                 16.378 
                 33.580 
                 −55.796 
               
               
                 15.099 
                 34.185 
                 −55.789 
               
               
                 12.486 
                 35.258 
                 −55.770 
               
               
                 11.155 
                 35.726 
                 −55.758 
               
               
                 8.452 
                 36.524 
                 −55.732 
               
               
                 7.083 
                 36.854 
                 −55.718 
               
               
                 4.318 
                 37.374 
                 −55.685 
               
               
                 1.529 
                 37.710 
                 −55.647 
               
               
                 0.129 
                 37.808 
                 −55.626 
               
               
                 30.068 
                 30.583 
                 −55.222 
               
               
                 28.898 
                 31.669 
                 −55.224 
               
               
                 27.687 
                 32.708 
                 −55.224 
               
               
                 25.800 
                 34.178 
                 −55.224 
               
               
                 23.165 
                 35.966 
                 −55.222 
               
               
                 21.802 
                 36.785 
                 −55.219 
               
               
                 18.996 
                 38.269 
                 −55.211 
               
               
                 16.098 
                 39.549 
                 −55.199 
               
               
                 13.874 
                 40.374 
                 −55.187 
               
               
                 10.854 
                 41.294 
                 −55.167 
               
               
                 9.325 
                 41.676 
                 −55.155 
               
               
                 6.237 
                 42.288 
                 −55.126 
               
               
                 4.681 
                 42.517 
                 −55.110 
               
               
                 1.557 
                 42.821 
                 −55.073 
               
               
                 −0.009 
                 42.897 
                 −55.053 
               
               
                 34.298 
                 34.875 
                 −54.472 
               
               
                 31.643 
                 37.265 
                 −54.473 
               
               
                 30.252 
                 38.385 
                 −54.472 
               
               
                 27.356 
                 40.468 
                 −54.468 
               
               
                 25.092 
                 41.888 
                 −54.462 
               
               
                 23.544 
                 42.766 
                 −54.457 
               
               
                 20.363 
                 44.355 
                 −54.443 
               
               
                 18.734 
                 45.065 
                 −54.435 
               
               
                 16.250 
                 46.022 
                 −54.419 
               
               
                 14.570 
                 46.589 
                 −54.407 
               
               
                 12.017 
                 47.330 
                 −54.387 
               
               
                 10.297 
                 47.752 
                 −54.371 
               
               
                 6.822 
                 48.420 
                 −54.336 
               
               
                 3.314 
                 48.855 
                 −54.296 
               
               
                 1.552 
                 48.985 
                 −54.274 
               
               
                 −0.211 
                 49.056 
                 −54.250 
               
               
                 40.643 
                 41.311 
                 −53.125 
               
               
                 37.574 
                 44.096 
                 −53.126 
               
               
                 34.320 
                 46.656 
                 −53.121 
               
               
                 30.898 
                 48.980 
                 −53.112 
               
               
                 27.329 
                 51.063 
                 −53.097 
               
               
                 23.629 
                 52.897 
                 −53.076 
               
               
                 19.815 
                 54.478 
                 −53.048 
               
               
                 15.907 
                 55.802 
                 −53.014 
               
               
                 11.920 
                 56.865 
                 −52.972 
               
               
                 7.871 
                 57.666 
                 −52.923 
               
               
                 3.777 
                 58.201 
                 −52.867 
               
               
                 1.719 
                 58.369 
                 −52.836 
               
               
                 −0.344 
                 58.469 
                 −52.804 
               
               
                 43.944 
                 44.660 
                 −52.319 
               
               
                 42.325 
                 46.180 
                 −52.321 
               
               
                 40.653 
                 47.642 
                 −52.321 
               
               
                 37.163 
                 50.382 
                 −52.319 
               
               
                 35.350 
                 51.657 
                 −52.315 
               
               
                 32.551 
                 53.449 
                 −52.308 
               
               
                 30.637 
                 54.562 
                 −52.302 
               
               
                 26.702 
                 56.587 
                 −52.285 
               
               
                 24.687 
                 57.498 
                 −52.274 
               
               
                 20.572 
                 59.115 
                 −52.246 
               
               
                 18.477 
                 59.820 
                 −52.230 
               
               
                 14.224 
                 61.020 
                 −52.191 
               
               
                 12.070 
                 61.515 
                 −52.169 
               
               
                 7.719 
                 62.293 
                 −52.120 
               
               
                 5.527 
                 62.575 
                 −52.092 
               
               
                 0.015 
                 62.969 
                 −52.014 
               
               
                 51.111 
                 51.931 
                 −50.333 
               
               
                 47.346 
                 55.353 
                 −50.335 
               
               
                 44.373 
                 57.740 
                 −50.334 
               
               
                 42.325 
                 59.244 
                 −50.331 
               
               
                 39.161 
                 61.364 
                 −50.325 
               
               
                 36.994 
                 62.685 
                 −50.318 
               
               
                 34.784 
                 63.932 
                 −50.310 
               
               
                 32.534 
                 65.103 
                 −50.301 
               
               
                 30.247 
                 66.198 
                 −50.289 
               
               
                 27.925 
                 67.216 
                 −50.276 
               
               
                 25.570 
                 68.156 
                 −50.260 
               
               
                 23.186 
                 69.019 
                 −50.243 
               
               
                 20.775 
                 69.803 
                 −50.223 
               
               
                 18.340 
                 70.507 
                 −50.201 
               
               
                 15.882 
                 71.132 
                 −50.176 
               
               
                 13.405 
                 71.677 
                 −50.150 
               
               
                 10.910 
                 72.141 
                 −50.121 
               
               
                 8.402 
                 72.525 
                 −50.090 
               
               
                 5.881 
                 72.827 
                 −50.056 
               
               
                 3.351 
                 73.047 
                 −50.021 
               
               
                 0.814 
                 73.184 
                 −49.983 
               
               
                 −0.456 
                 73.221 
                 −49.963 
               
               
                 55.323 
                 56.204 
                 −48.984 
               
               
                 51.268 
                 59.873 
                 −48.990 
               
               
                 48.064 
                 62.427 
                 −48.993 
               
               
                 45.855 
                 64.032 
                 −48.994 
               
               
                 42.443 
                 66.291 
                 −48.994 
               
               
                 40.106 
                 67.695 
                 −48.993 
               
               
                 37.724 
                 69.017 
                 −48.990 
               
               
                 35.298 
                 70.256 
                 −48.986 
               
               
                 32.833 
                 71.412 
                 −48.980 
               
               
                 30.332 
                 72.483 
                 −48.973 
               
               
                 27.797 
                 73.469 
                 −48.963 
               
               
                 25.231 
                 74.369 
                 −48.952 
               
               
                 22.637 
                 75.184 
                 −48.939 
               
               
                 20.019 
                 75.913 
                 −48.924 
               
               
                 17.378 
                 76.555 
                 −48.906 
               
               
                 14.718 
                 77.110 
                 −48.887 
               
               
                 12.041 
                 77.578 
                 −48.865 
               
               
                 9.351 
                 77.959 
                 −48.840 
               
               
                 6.649 
                 78.252 
                 −48.814 
               
               
                 3.940 
                 78.457 
                 −48.785 
               
               
                 1.226 
                 78.574 
                 −48.754 
               
               
                 −0.132 
                 78.600 
                 −48.737 
               
               
                 62.828 
                 63.819 
                 −46.338 
               
               
                 58.291 
                 67.964 
                 −46.339 
               
               
                 54.710 
                 70.864 
                 −46.336 
               
               
                 52.244 
                 72.693 
                 −46.332 
               
               
                 45.824 
                 76.892 
                 −46.313 
               
               
                 41.813 
                 79.148 
                 −46.295 
               
               
                 39.079 
                 80.539 
                 −46.281 
               
               
                 36.300 
                 81.838 
                 −46.264 
               
               
                 33.479 
                 83.046 
                 −46.245 
               
               
                 30.620 
                 84.159 
                 −46.224 
               
               
                 27.724 
                 85.179 
                 −46.200 
               
               
                 24.796 
                 86.103 
                 −46.174 
               
               
                 21.838 
                 86.931 
                 −46.145 
               
               
                 18.853 
                 87.662 
                 −46.114 
               
               
                 15.845 
                 88.296 
                 −46.080 
               
               
                 12.815 
                 88.831 
                 −46.043 
               
               
                 9.769 
                 89.268 
                 −46.004 
               
               
                 6.707 
                 89.605 
                 −45.963 
               
               
                 3.635 
                 89.842 
                 −45.918 
               
               
                 0.555 
                 89.979 
                 −45.872 
               
               
                 65.901 
                 66.937 
                 −45.142 
               
               
                 61.161 
                 71.270 
                 −45.144 
               
               
                 57.420 
                 74.301 
                 −45.140 
               
               
                 54.844 
                 76.214 
                 −45.136 
               
               
                 48.138 
                 80.607 
                 −45.117 
               
               
                 45.358 
                 82.204 
                 −45.105 
               
               
                 42.526 
                 83.707 
                 −45.091 
               
               
                 39.646 
                 85.116 
                 −45.075 
               
               
                 36.721 
                 86.429 
                 −45.057 
               
               
                 33.753 
                 87.645 
                 −45.036 
               
               
                 30.746 
                 88.764 
                 −45.013 
               
               
                 27.704 
                 89.783 
                 −44.987 
               
               
                 24.628 
                 90.703 
                 −44.958 
               
               
                 21.522 
                 91.522 
                 −44.927 
               
               
                 18.390 
                 92.239 
                 −44.893 
               
               
                 15.234 
                 92.855 
                 −44.856 
               
               
                 12.058 
                 93.368 
                 −44.816 
               
               
                 8.865 
                 93.777 
                 −44.774 
               
               
                 4.052 
                 94.195 
                 −44.706 
               
               
                 0.831 
                 94.344 
                 −44.657 
               
               
                 75.988 
                 77.170 
                 −40.786 
               
               
                 70.575 
                 82.117 
                 −40.791 
               
               
                 63.362 
                 87.761 
                 −40.787 
               
               
                 54.124 
                 93.697 
                 −40.768 
               
               
                 47.658 
                 97.129 
                 −40.744 
               
               
                 44.344 
                 98.682 
                 −40.729 
               
               
                 37.569 
                 101.458 
                 −40.689 
               
               
                 30.624 
                 103.788 
                 −40.637 
               
               
                 27.095 
                 104.782 
                 −40.607 
               
               
                 23.535 
                 105.661 
                 −40.573 
               
               
                 16.332 
                 107.072 
                 −40.497 
               
               
                 12.697 
                 107.602 
                 −40.453 
               
               
                 5.378 
                 108.309 
                 −40.357 
               
               
                 1.701 
                 108.484 
                 −40.304 
               
               
                 −0.140 
                 108.527 
                 −40.276 
               
               
                 80.478 
                 81.726 
                 −38.607 
               
               
                 74.754 
                 86.936 
                 −38.617 
               
               
                 67.119 
                 92.865 
                 −38.627 
               
               
                 57.340 
                 99.076 
                 −38.632 
               
               
                 50.498 
                 102.648 
                 −38.627 
               
               
                 46.991 
                 104.260 
                 −38.622 
               
               
                 39.829 
                 107.128 
                 −38.604 
               
               
                 32.491 
                 109.517 
                 −38.577 
               
               
                 28.767 
                 110.530 
                 −38.559 
               
               
                 25.011 
                 111.420 
                 −38.538 
               
               
                 17.419 
                 112.831 
                 −38.487 
               
               
                 13.591 
                 113.350 
                 −38.457 
               
               
                 5.890 
                 114.014 
                 −38.387 
               
               
                 2.026 
                 114.158 
                 −38.347 
               
               
                 0.092 
                 114.183 
                 −38.326 
               
               
                 85.008 
                 86.323 
                 −36.318 
               
               
                 78.994 
                 91.826 
                 −36.322 
               
               
                 70.980 
                 98.107 
                 −36.319 
               
               
                 60.715 
                 104.718 
                 −36.299 
               
               
                 53.528 
                 108.538 
                 −36.275 
               
               
                 49.842 
                 110.267 
                 −36.260 
               
               
                 40.394 
                 114.046 
                 −36.210 
               
               
                 34.582 
                 115.936 
                 −36.172 
               
               
                 30.658 
                 117.036 
                 −36.143 
               
               
                 26.697 
                 118.006 
                 −36.112 
               
               
                 18.684 
                 119.553 
                 −36.040 
               
               
                 14.641 
                 120.129 
                 −36.000 
               
               
                 6.502 
                 120.880 
                 −35.911 
               
               
                 2.416 
                 121.054 
                 −35.863 
               
               
                 0.370 
                 121.091 
                 −35.837 
               
               
                 94.197 
                 95.645 
                 −31.214 
               
               
                 87.579 
                 101.716 
                 −31.216 
               
               
                 78.766 
                 108.658 
                 −31.206 
               
               
                 75.085 
                 111.225 
                 −31.197 
               
               
                 65.531 
                 117.096 
                 −31.165 
               
               
                 61.579 
                 119.220 
                 −31.147 
               
               
                 57.558 
                 121.214 
                 −31.126 
               
               
                 53.474 
                 123.075 
                 −31.101 
               
               
                 45.127 
                 126.394 
                 −31.043 
               
               
                 36.573 
                 129.163 
                 −30.970 
               
               
                 32.229 
                 130.338 
                 −30.929 
               
               
                 27.846 
                 131.371 
                 −30.884 
               
               
                 18.980 
                 133.008 
                 −30.783 
               
               
                 14.506 
                 133.609 
                 −30.727 
               
               
                 10.011 
                 134.065 
                 −30.667 
               
               
                 0.978 
                 134.536 
                 −30.537 
               
               
                 −1.286 
                 134.562 
                 −30.502 
               
               
                 97.395 
                 98.890 
                 −29.308 
               
               
                 90.566 
                 105.157 
                 −29.310 
               
               
                 81.470 
                 112.325 
                 −29.299 
               
               
                 77.672 
                 114.975 
                 −29.291 
               
               
                 67.812 
                 121.038 
                 −29.258 
               
               
                 63.733 
                 123.232 
                 −29.240 
               
               
                 59.584 
                 125.291 
                 −29.218 
               
               
                 55.368 
                 127.213 
                 −29.193 
               
               
                 46.753 
                 130.641 
                 −29.134 
               
               
                 37.924 
                 133.500 
                 −29.061 
               
               
                 33.440 
                 134.713 
                 −29.019 
               
               
                 28.915 
                 135.780 
                 −28.974 
               
               
                 19.763 
                 137.469 
                 −28.872 
               
               
                 15.145 
                 138.089 
                 −28.816 
               
               
                 10.506 
                 138.558 
                 −28.756 
               
               
                 1.181 
                 139.042 
                 −28.625 
               
               
                 −1.156 
                 139.067 
                 −28.590 
               
               
                 107.402 
                 109.043 
                 −22.913 
               
               
                 99.909 
                 115.927 
                 −22.915 
               
               
                 89.931 
                 123.805 
                 −22.902 
               
               
                 85.764 
                 126.720 
                 −22.891 
               
               
                 79.341 
                 130.831 
                 −22.870 
               
               
                 74.949 
                 133.393 
                 −22.852 
               
               
                 68.208 
                 136.963 
                 −22.819 
               
               
                 63.618 
                 139.157 
                 −22.792 
               
               
                 58.957 
                 141.201 
                 −22.762 
               
               
                 49.440 
                 144.827 
                 −22.691 
               
               
                 42.150 
                 147.137 
                 −22.628 
               
               
                 37.227 
                 148.478 
                 −22.582 
               
               
                 32.259 
                 149.658 
                 −22.531 
               
               
                 22.208 
                 151.531 
                 −22.419 
               
               
                 17.136 
                 152.221 
                 −22.357 
               
               
                 12.039 
                 152.744 
                 −22.291 
               
               
                 1.795 
                 153.290 
                 −22.149 
               
               
                 −0.773 
                 153.321 
                 −22.111 
               
               
                 111.075 
                 112.769 
                 −20.379 
               
               
                 103.326 
                 119.870 
                 −20.387 
               
               
                 93.005 
                 127.983 
                 −20.389 
               
               
                 88.694 
                 130.980 
                 −20.387 
               
               
                 79.786 
                 136.534 
                 −20.375 
               
               
                 75.200 
                 139.087 
                 −20.366 
               
               
                 70.531 
                 141.486 
                 −20.354 
               
               
                 65.785 
                 143.731 
                 −20.338 
               
               
                 60.966 
                 145.818 
                 −20.320 
               
               
                 51.132 
                 149.516 
                 −20.272 
               
               
                 43.603 
                 151.864 
                 −20.227 
               
               
                 38.520 
                 153.224 
                 −20.192 
               
               
                 33.392 
                 154.418 
                 −20.153 
               
               
                 23.024 
                 156.304 
                 −20.063 
               
               
                 17.793 
                 156.993 
                 −20.011 
               
               
                 12.539 
                 157.511 
                 −19.956 
               
               
                 1.983 
                 158.033 
                 −19.832 
               
               
                 −0.662 
                 158.055 
                 −19.799 
               
               
                 121.459 
                 123.305 
                 −12.789 
               
               
                 113.030 
                 131.058 
                 −12.789 
               
               
                 101.808 
                 139.934 
                 −12.772 
               
               
                 97.123 
                 143.220 
                 −12.759 
               
               
                 87.440 
                 149.322 
                 −12.723 
               
               
                 82.453 
                 152.132 
                 −12.700 
               
               
                 77.375 
                 154.778 
                 −12.673 
               
               
                 72.212 
                 157.257 
                 −12.641 
               
               
                 66.969 
                 159.568 
                 −12.606 
               
               
                 50.807 
                 165.462 
                 −12.474 
               
               
                 39.725 
                 168.507 
                 −12.365 
               
               
                 34.109 
                 169.758 
                 −12.304 
               
               
                 22.753 
                 171.710 
                 −12.167 
               
               
                 17.026 
                 172.408 
                 −12.093 
               
               
                 5.501 
                 173.242 
                 −11.930 
               
               
                 −0.284 
                 173.376 
                 −11.842 
               
               
                 124.320 
                 126.208 
                 −10.568 
               
               
                 115.701 
                 134.134 
                 −10.569 
               
               
                 104.226 
                 143.211 
                 −10.553 
               
               
                 99.435 
                 146.570 
                 −10.540 
               
               
                 89.534 
                 152.809 
                 −10.506 
               
               
                 84.434 
                 155.681 
                 −10.483 
               
               
                 79.242 
                 158.386 
                 −10.456 
               
               
                 73.963 
                 160.921 
                 −10.425 
               
               
                 68.601 
                 163.282 
                 −10.390 
               
               
                 52.075 
                 169.306 
                 −10.260 
               
               
                 40.744 
                 172.418 
                 −10.151 
               
               
                 35.002 
                 173.696 
                 −10.089 
               
               
                 23.391 
                 175.690 
                 −9.954 
               
               
                 17.535 
                 176.402 
                 −9.879 
               
               
                 5.752 
                 177.252 
                 −9.717 
               
               
                 −0.163 
                 177.387 
                 −9.629 
               
               
                 133.343 
                 135.362 
                 −3.215 
               
               
                 124.122 
                 143.840 
                 −3.218 
               
               
                 111.847 
                 153.546 
                 −3.205 
               
               
                 106.720 
                 157.139 
                 −3.194 
               
               
                 98.816 
                 162.206 
                 −3.173 
               
               
                 93.412 
                 165.365 
                 −3.153 
               
               
                 87.906 
                 168.346 
                 −3.131 
               
               
                 82.304 
                 171.145 
                 −3.104 
               
               
                 76.611 
                 173.761 
                 −3.073 
               
               
                 70.833 
                 176.190 
                 −3.038 
               
               
                 64.976 
                 178.430 
                 −2.999 
               
               
                 59.045 
                 180.479 
                 −2.955 
               
               
                 43.934 
                 184.749 
                 −2.827 
               
               
                 37.791 
                 186.112 
                 −2.767 
               
               
                 25.373 
                 188.234 
                 −2.634 
               
               
                 19.111 
                 188.990 
                 −2.560 
               
               
                 6.510 
                 189.888 
                 −2.400 
               
               
                 0.186 
                 190.027 
                 −2.313 
               
               
                 136.402 
                 138.466 
                 −0.604 
               
               
                 126.981 
                 147.136 
                 −0.604 
               
               
                 114.440 
                 157.066 
                 −0.585 
               
               
                 109.203 
                 160.743 
                 −0.572 
               
               
                 101.129 
                 165.931 
                 −0.544 
               
               
                 95.608 
                 169.167 
                 −0.521 
               
               
                 89.982 
                 172.221 
                 −0.493 
               
               
                 84.259 
                 175.091 
                 −0.462 
               
               
                 78.442 
                 177.772 
                 −0.426 
               
               
                 72.537 
                 180.263 
                 −0.386 
               
               
                 66.551 
                 182.562 
                 −0.342 
               
               
                 60.489 
                 184.665 
                 −0.293 
               
               
                 45.042 
                 189.052 
                 −0.150 
               
               
                 38.762 
                 190.454 
                 −0.085 
               
               
                 26.065 
                 192.639 
                 0.058 
               
               
                 19.660 
                 193.420 
                 0.137 
               
               
                 6.773 
                 194.352 
                 0.308 
               
               
                 0.304 
                 194.500 
                 0.399 
               
               
                 146.053 
                 148.259 
                 8.057 
               
               
                 135.987 
                 157.523 
                 8.056 
               
               
                 122.586 
                 168.133 
                 8.073 
               
               
                 116.990 
                 172.062 
                 8.086 
               
               
                 105.427 
                 179.358 
                 8.124 
               
               
                 99.471 
                 182.719 
                 8.149 
               
               
                 87.241 
                 188.850 
                 8.212 
               
               
                 80.978 
                 191.614 
                 8.251 
               
               
                 68.187 
                 196.525 
                 8.341 
               
               
                 61.672 
                 198.668 
                 8.394 
               
               
                 48.434 
                 202.313 
                 8.514 
               
               
                 41.724 
                 203.812 
                 8.582 
               
               
                 28.157 
                 206.151 
                 8.733 
               
               
                 21.314 
                 206.988 
                 8.816 
               
               
                 7.544 
                 207.989 
                 8.997 
               
               
                 0.631 
                 208.151 
                 9.094 
               
               
                 149.453 
                 151.709 
                 11.263 
               
               
                 139.157 
                 161.180 
                 11.260 
               
               
                 125.450 
                 172.025 
                 11.273 
               
               
                 119.726 
                 176.040 
                 11.284 
               
               
                 107.897 
                 183.492 
                 11.315 
               
               
                 101.804 
                 186.924 
                 11.336 
               
               
                 89.292 
                 193.179 
                 11.390 
               
               
                 82.886 
                 195.998 
                 11.423 
               
               
                 69.802 
                 201.001 
                 11.502 
               
               
                 63.139 
                 203.181 
                 11.548 
               
               
                 49.600 
                 206.884 
                 11.653 
               
               
                 42.739 
                 208.402 
                 11.713 
               
               
                 28.868 
                 210.763 
                 11.845 
               
               
                 21.873 
                 211.602 
                 11.918 
               
               
                 7.800 
                 212.590 
                 12.078 
               
               
                 0.737 
                 212.737 
                 12.165 
               
               
                 146.886 
                 170.086 
                 19.368 
               
               
                 132.458 
                 181.527 
                 19.393 
               
               
                 126.433 
                 185.765 
                 19.410 
               
               
                 113.983 
                 193.639 
                 19.458 
               
               
                 107.571 
                 197.268 
                 19.488 
               
               
                 94.400 
                 203.890 
                 19.564 
               
               
                 87.655 
                 206.877 
                 19.609 
               
               
                 73.878 
                 212.188 
                 19.714 
               
               
                 66.859 
                 214.506 
                 19.774 
               
               
                 52.596 
                 218.453 
                 19.911 
               
               
                 45.366 
                 220.077 
                 19.987 
               
               
                 30.744 
                 222.616 
                 20.154 
               
               
                 23.368 
                 223.526 
                 20.246 
               
               
                 8.523 
                 224.621 
                 20.444 
               
               
                 1.070 
                 224.802 
                 20.551 
               
               
                 160.387 
                 162.802 
                 22.070 
               
               
                 149.371 
                 172.949 
                 22.072 
               
               
                 134.709 
                 184.578 
                 22.098 
               
               
                 128.587 
                 188.886 
                 22.115 
               
               
                 115.935 
                 196.890 
                 22.164 
               
               
                 109.418 
                 200.578 
                 22.195 
               
               
                 96.033 
                 207.310 
                 22.272 
               
               
                 89.179 
                 210.347 
                 22.318 
               
               
                 75.177 
                 215.746 
                 22.425 
               
               
                 68.044 
                 218.103 
                 22.486 
               
               
                 53.549 
                 222.117 
                 22.624 
               
               
                 46.200 
                 223.768 
                 22.701 
               
               
                 31.340 
                 226.350 
                 22.870 
               
               
                 23.843 
                 227.275 
                 22.963 
               
               
                 8.756 
                 228.389 
                 23.163 
               
               
                 1.182 
                 228.573 
                 23.271 
               
               
                 165.711 
                 168.203 
                 27.618 
               
               
                 154.342 
                 178.677 
                 27.620 
               
               
                 139.209 
                 190.679 
                 27.645 
               
               
                 132.891 
                 195.126 
                 27.663 
               
               
                 119.833 
                 203.387 
                 27.712 
               
               
                 113.107 
                 207.194 
                 27.743 
               
               
                 99.292 
                 214.143 
                 27.821 
               
               
                 92.218 
                 217.278 
                 27.867 
               
               
                 77.767 
                 222.851 
                 27.974 
               
               
                 70.405 
                 225.283 
                 28.036 
               
               
                 55.444 
                 229.425 
                 28.175 
               
               
                 47.859 
                 231.129 
                 28.253 
               
               
                 32.522 
                 233.793 
                 28.424 
               
               
                 24.784 
                 234.748 
                 28.517 
               
               
                 9.212 
                 235.895 
                 28.720 
               
               
                 1.395 
                 236.085 
                 28.829 
               
               
                 168.389 
                 170.920 
                 30.478 
               
               
                 156.842 
                 181.559 
                 30.481 
               
               
                 141.474 
                 193.751 
                 30.508 
               
               
                 135.057 
                 198.268 
                 30.527 
               
               
                 121.796 
                 206.662 
                 30.577 
               
               
                 114.965 
                 210.530 
                 30.610 
               
               
                 100.936 
                 217.590 
                 30.690 
               
               
                 93.751 
                 220.775 
                 30.737 
               
               
                 79.074 
                 226.439 
                 30.848 
               
               
                 71.597 
                 228.911 
                 30.911 
               
               
                 56.401 
                 233.121 
                 31.053 
               
               
                 48.697 
                 234.853 
                 31.132 
               
               
                 33.118 
                 237.561 
                 31.306 
               
               
                 25.259 
                 238.532 
                 31.401 
               
               
                 173.887 
                 176.499 
                 36.502 
               
               
                 161.975 
                 187.475 
                 36.505 
               
               
                 146.121 
                 200.055 
                 36.533 
               
               
                 139.500 
                 204.715 
                 36.551 
               
               
                 125.819 
                 213.376 
                 36.603 
               
               
                 118.772 
                 217.367 
                 36.636 
               
               
                 104.297 
                 224.651 
                 36.716 
               
               
                 96.884 
                 227.937 
                 36.764 
               
               
                 81.742 
                 233.780 
                 36.875 
               
               
                 74.027 
                 236.330 
                 36.939 
               
               
                 176.578 
                 179.229 
                 39.525 
               
               
                 164.486 
                 190.370 
                 39.527 
               
               
                 148.391 
                 203.137 
                 39.553 
               
               
                 141.670 
                 207.867 
                 39.571 
               
               
                 127.782 
                 216.654 
                 39.619 
               
               
                 120.628 
                 220.704 
                 39.651 
               
               
                 105.934 
                 228.095 
                 39.729 
               
               
                 98.409 
                 231.428 
                 39.775 
               
               
                 83.039 
                 237.355 
                 39.884 
               
               
                 182.202 
                 184.935 
                 45.998 
               
               
                 169.736 
                 196.423 
                 46.001 
               
               
                 153.144 
                 209.591 
                 46.030 
               
               
                 146.216 
                 214.470 
                 46.050 
               
               
                 131.898 
                 223.536 
                 46.103 
               
               
                 124.523 
                 227.714 
                 46.138 
               
               
                 109.375 
                 235.340 
                 46.222 
               
               
                 185.063 
                 187.838 
                 49.375 
               
               
                 172.405 
                 199.502 
                 49.378 
               
               
                 155.557 
                 212.869 
                 49.405 
               
               
                 148.522 
                 217.821 
                 49.423 
               
               
                 133.983 
                 227.022 
                 49.474 
               
               
                 126.494 
                 231.262 
                 49.506 
               
               
                 139.296 
                 235.862 
                 58.265 
               
               
                 194.459 
                 197.371 
                 60.841 
               
               
                 181.184 
                 209.612 
                 60.848 
               
               
                 163.518 
                 223.648 
                 60.886 
               
               
                 156.141 
                 228.850 
                 60.911 
               
               
                 161.667 
                 236.846 
                 69.607 
               
               
                 203.483 
                 206.524 
                 72.398 
               
               
                 189.612 
                 219.316 
                 72.405 
               
               
                 171.152 
                 233.983 
                 72.444 
               
               
                 210.278 
                 213.418 
                 81.456 
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
               
               
             
               
               
               
             
           
               
                   
                 TABLE 2 
               
               
                   
                   
               
               
                   
                 ρ 
                 z 
               
               
                   
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 0.000 
                 78.022 
               
               
                   
                 0.085 
                 78.013 
               
               
                   
                 0.255 
                 78.005 
               
               
                   
                 0.388 
                 77.999 
               
               
                   
                 0.475 
                 77.996 
               
               
                   
                 0.647 
                 77.988 
               
               
                   
                 0.816 
                 77.981 
               
               
                   
                 0.981 
                 77.975 
               
               
                   
                 1.142 
                 77.968 
               
               
                   
                 1.300 
                 77.962 
               
               
                   
                 1.454 
                 77.955 
               
               
                   
                 1.605 
                 77.949 
               
               
                   
                 1.753 
                 77.943 
               
               
                   
                 1.896 
                 77.938 
               
               
                   
                 2.037 
                 77.932 
               
               
                   
                 2.173 
                 77.927 
               
               
                   
                 2.372 
                 77.920 
               
               
                   
                 2.563 
                 77.913 
               
               
                   
                 2.746 
                 77.907 
               
               
                   
                 2.921 
                 77.901 
               
               
                   
                 3.088 
                 77.895 
               
               
                   
                 3.399 
                 77.886 
               
               
                   
                 3.589 
                 77.881 
               
               
                   
                 3.808 
                 77.875 
               
               
                   
                 4.005 
                 77.870 
               
               
                   
                 4.182 
                 77.867 
               
               
                   
                 4.380 
                 77.864 
               
               
                   
                 4.576 
                 77.862 
               
               
                   
                 4.762 
                 77.861 
               
               
                   
                 4.937 
                 77.862 
               
               
                   
                 5.109 
                 77.864 
               
               
                   
                 5.278 
                 77.866 
               
               
                   
                 5.443 
                 77.867 
               
               
                   
                 5.604 
                 77.869 
               
               
                   
                 5.762 
                 77.871 
               
               
                   
                 5.917 
                 77.873 
               
               
                   
                 6.215 
                 77.877 
               
               
                   
                 6.499 
                 77.882 
               
               
                   
                 6.736 
                 77.886 
               
               
                   
                 6.994 
                 77.891 
               
               
                   
                 7.238 
                 77.896 
               
               
                   
                 7.440 
                 77.900 
               
               
                   
                 7.685 
                 77.906 
               
               
                   
                 7.937 
                 77.913 
               
               
                   
                 8.167 
                 77.920 
               
               
                   
                 8.396 
                 77.928 
               
               
                   
                 8.565 
                 77.935 
               
               
                   
                 8.866 
                 77.947 
               
               
                   
                 9.106 
                 77.959 
               
               
                   
                 9.348 
                 77.974 
               
               
                   
                 9.481 
                 77.984 
               
               
                   
                 9.695 
                 77.999 
               
               
                   
                 9.903 
                 78.014 
               
               
                   
                 10.107 
                 78.029 
               
               
                   
                 10.304 
                 78.044 
               
               
                   
                 10.496 
                 78.059 
               
               
                   
                 10.683 
                 78.073 
               
               
                   
                 10.865 
                 78.087 
               
               
                   
                 11.041 
                 78.102 
               
               
                   
                 11.211 
                 78.116 
               
               
                   
                 11.568 
                 78.146 
               
               
                   
                 11.722 
                 78.159 
               
               
                   
                 11.899 
                 78.175 
               
               
                   
                 12.069 
                 78.190 
               
               
                   
                 12.231 
                 78.205 
               
               
                   
                 12.386 
                 78.220 
               
               
                   
                 12.557 
                 78.236 
               
               
                   
                 12.719 
                 78.252 
               
               
                   
                 12.890 
                 78.270 
               
               
                   
                 13.050 
                 78.287 
               
               
                   
                 13.213 
                 78.304 
               
               
                   
                 13.406 
                 78.326 
               
               
                   
                 13.608 
                 78.350 
               
               
                   
                 13.803 
                 78.374 
               
               
                   
                 13.963 
                 78.396 
               
               
                   
                 14.141 
                 78.421 
               
               
                   
                 14.316 
                 78.446 
               
               
                   
                 14.487 
                 78.470 
               
               
                   
                 14.655 
                 78.494 
               
               
                   
                 14.819 
                 78.518 
               
               
                   
                 14.980 
                 78.542 
               
               
                   
                 15.138 
                 78.565 
               
               
                   
                 15.292 
                 78.588 
               
               
                   
                 15.443 
                 78.611 
               
               
                   
                 15.591 
                 78.633 
               
               
                   
                 15.735 
                 78.656 
               
               
                   
                 15.876 
                 78.677 
               
               
                   
                 16.014 
                 78.699 
               
               
                   
                 16.181 
                 78.725 
               
               
                   
                 16.344 
                 78.751 
               
               
                   
                 16.501 
                 78.777 
               
               
                   
                 16.653 
                 78.801 
               
               
                   
                 16.800 
                 78.826 
               
               
                   
                 16.942 
                 78.849 
               
               
                   
                 17.079 
                 78.872 
               
               
                   
                 17.211 
                 78.895 
               
               
                   
                 17.363 
                 78.921 
               
               
                   
                 17.508 
                 78.947 
               
               
                   
                 17.645 
                 78.971 
               
               
                   
                 17.797 
                 78.998 
               
               
                   
                 17.940 
                 79.024 
               
               
                   
                 18.091 
                 79.053 
               
               
                   
                 18.327 
                 79.098 
               
               
                   
                 18.648 
                 79.161 
               
               
                   
                 18.951 
                 79.225 
               
               
                   
                 19.229 
                 79.286 
               
               
                   
                 19.499 
                 79.345 
               
               
                   
                 19.762 
                 79.403 
               
               
                   
                 20.307 
                 79.526 
               
               
                   
                 20.547 
                 79.581 
               
               
                   
                 20.780 
                 79.635 
               
               
                   
                 20.969 
                 79.679 
               
               
                   
                 21.153 
                 79.722 
               
               
                   
                 21.367 
                 79.773 
               
               
                   
                 21.574 
                 79.823 
               
               
                   
                 21.806 
                 79.879 
               
               
                   
                 21.998 
                 79.926 
               
               
                   
                 22.183 
                 79.972 
               
               
                   
                 22.361 
                 80.016 
               
               
                   
                 22.533 
                 80.059 
               
               
                   
                 22.751 
                 80.115 
               
               
                   
                 22.907 
                 80.155 
               
               
                   
                 23.080 
                 80.200 
               
               
                   
                 23.245 
                 80.243 
               
               
                   
                 23.401 
                 80.285 
               
               
                   
                 23.647 
                 80.352 
               
               
                   
                 23.903 
                 80.422 
               
               
                   
                 24.034 
                 80.459 
               
               
                   
                 24.251 
                 80.521 
               
               
                   
                 24.363 
                 80.553 
               
               
                   
                 24.573 
                 80.615 
               
               
                   
                 24.725 
                 80.660 
               
               
                   
                 24.875 
                 80.705 
               
               
                   
                 25.023 
                 80.750 
               
               
                   
                 25.170 
                 80.794 
               
               
                   
                 25.314 
                 80.837 
               
               
                   
                 25.457 
                 80.881 
               
               
                   
                 25.599 
                 80.924 
               
               
                   
                 25.739 
                 80.967 
               
               
                   
                 25.876 
                 81.009 
               
               
                   
                 26.013 
                 81.051 
               
               
                   
                 26.147 
                 81.093 
               
               
                   
                 26.280 
                 81.134 
               
               
                   
                 26.411 
                 81.175 
               
               
                   
                 26.541 
                 81.215 
               
               
                   
                 26.669 
                 81.255 
               
               
                   
                 26.795 
                 81.295 
               
               
                   
                 26.919 
                 81.334 
               
               
                   
                 27.042 
                 81.373 
               
               
                   
                 27.164 
                 81.412 
               
               
                   
                 27.283 
                 81.450 
               
               
                   
                 27.401 
                 81.488 
               
               
                   
                 27.517 
                 81.526 
               
               
                   
                 27.632 
                 81.563 
               
               
                   
                 27.857 
                 81.636 
               
               
                   
                 28.075 
                 81.707 
               
               
                   
                 28.287 
                 81.777 
               
               
                   
                 28.492 
                 81.845 
               
               
                   
                 28.691 
                 81.912 
               
               
                   
                 28.884 
                 81.977 
               
               
                   
                 29.071 
                 82.041 
               
               
                   
                 29.252 
                 82.103 
               
               
                   
                 29.426 
                 82.163 
               
               
                   
                 29.595 
                 82.221 
               
               
                   
                 29.758 
                 82.278 
               
               
                   
                 29.914 
                 82.333 
               
               
                   
                 30.065 
                 82.387 
               
               
                   
                 30.210 
                 82.439 
               
               
                   
                 30.349 
                 82.489 
               
               
                   
                 30.483 
                 82.537 
               
               
                   
                 30.610 
                 82.583 
               
               
                   
                 30.733 
                 82.628 
               
               
                   
                 30.849 
                 82.671 
               
               
                   
                 30.960 
                 82.713 
               
               
                   
                 31.066 
                 82.752 
               
               
                   
                 31.174 
                 82.793 
               
               
                   
                 31.349 
                 82.859 
               
               
                   
                 31.464 
                 82.902 
               
               
                   
                 31.579 
                 82.946 
               
               
                   
                 31.749 
                 83.011 
               
               
                   
                 31.862 
                 83.054 
               
               
                   
                 31.974 
                 83.096 
               
               
                   
                 32.086 
                 83.139 
               
               
                   
                 32.197 
                 83.182 
               
               
                   
                 32.471 
                 83.287 
               
               
                   
                 32.687 
                 83.370 
               
               
                   
                 32.900 
                 83.453 
               
               
                   
                 33.111 
                 83.535 
               
               
                   
                 33.318 
                 83.616 
               
               
                   
                 33.524 
                 83.697 
               
               
                   
                 33.876 
                 83.836 
               
               
                   
                 34.074 
                 83.914 
               
               
                   
                 34.269 
                 83.992 
               
               
                   
                 34.462 
                 84.069 
               
               
                   
                 34.651 
                 84.145 
               
               
                   
                 34.839 
                 84.221 
               
               
                   
                 34.978 
                 84.277 
               
               
                   
                 35.206 
                 84.369 
               
               
                   
                 35.385 
                 84.442 
               
               
                   
                 35.562 
                 84.515 
               
               
                   
                 35.737 
                 84.586 
               
               
                   
                 35.909 
                 84.657 
               
               
                   
                 36.078 
                 84.727 
               
               
                   
                 36.245 
                 84.796 
               
               
                   
                 36.410 
                 84.865 
               
               
                   
                 36.572 
                 84.933 
               
               
                   
                 36.731 
                 85.000 
               
               
                   
                 36.888 
                 85.066 
               
               
                   
                 37.043 
                 85.131 
               
               
                   
                 37.195 
                 85.196 
               
               
                   
                 37.345 
                 85.259 
               
               
                   
                 37.493 
                 85.322 
               
               
                   
                 37.638 
                 85.385 
               
               
                   
                 37.781 
                 85.446 
               
               
                   
                 37.990 
                 85.537 
               
               
                   
                 38.195 
                 85.625 
               
               
                   
                 38.394 
                 85.712 
               
               
                   
                 38.588 
                 85.798 
               
               
                   
                 38.778 
                 85.881 
               
               
                   
                 38.962 
                 85.962 
               
               
                   
                 39.141 
                 86.042 
               
               
                   
                 39.315 
                 86.120 
               
               
                   
                 39.484 
                 86.196 
               
               
                   
                 39.637 
                 86.265 
               
               
                   
                 39.778 
                 86.329 
               
               
                   
                 39.919 
                 86.393 
               
               
                   
                 40.060 
                 86.457 
               
               
                   
                 40.199 
                 86.520 
               
               
                   
                 40.338 
                 86.584 
               
               
                   
                 40.477 
                 86.647 
               
               
                   
                 40.615 
                 86.710 
               
               
                   
                 40.752 
                 86.773 
               
               
                   
                 40.889 
                 86.835 
               
               
                   
                 41.025 
                 86.898 
               
               
                   
                 41.161 
                 86.960 
               
               
                   
                 41.296 
                 87.022 
               
               
                   
                 41.430 
                 87.084 
               
               
                   
                 41.564 
                 87.146 
               
               
                   
                 41.697 
                 87.207 
               
               
                   
                 41.830 
                 87.269 
               
               
                   
                 41.962 
                 87.330 
               
               
                   
                 42.093 
                 87.391 
               
               
                   
                 42.289 
                 87.482 
               
               
                   
                 42.484 
                 87.573 
               
               
                   
                 42.678 
                 87.663 
               
               
                   
                 42.870 
                 87.753 
               
               
                   
                 43.061 
                 87.843 
               
               
                   
                 43.251 
                 87.932 
               
               
                   
                 43.439 
                 88.021 
               
               
                   
                 43.626 
                 88.109 
               
               
                   
                 43.812 
                 88.197 
               
               
                   
                 43.997 
                 88.285 
               
               
                   
                 44.181 
                 88.372 
               
               
                   
                 44.363 
                 88.459 
               
               
                   
                 44.544 
                 88.545 
               
               
                   
                 44.724 
                 88.631 
               
               
                   
                 44.903 
                 88.717 
               
               
                   
                 45.080 
                 88.802 
               
               
                   
                 45.256 
                 88.887 
               
               
                   
                 45.432 
                 88.971 
               
               
                   
                 45.605 
                 89.055 
               
               
                   
                 45.778 
                 89.138 
               
               
                   
                 45.950 
                 89.222 
               
               
                   
                 46.120 
                 89.304 
               
               
                   
                 46.289 
                 89.387 
               
               
                   
                 46.457 
                 89.469 
               
               
                   
                 46.624 
                 89.550 
               
               
                   
                 46.790 
                 89.631 
               
               
                   
                 46.955 
                 89.712 
               
               
                   
                 47.118 
                 89.793 
               
               
                   
                 47.280 
                 89.873 
               
               
                   
                 47.442 
                 89.952 
               
               
                   
                 47.602 
                 90.031 
               
               
                   
                 47.761 
                 90.110 
               
               
                   
                 47.919 
                 90.188 
               
               
                   
                 48.075 
                 90.266 
               
               
                   
                 48.231 
                 90.344 
               
               
                   
                 48.386 
                 90.421 
               
               
                   
                 48.539 
                 90.498 
               
               
                   
                 48.691 
                 90.574 
               
               
                   
                 48.843 
                 90.650 
               
               
                   
                 48.993 
                 90.726 
               
               
                   
                 49.142 
                 90.801 
               
               
                   
                 49.290 
                 90.875 
               
               
                   
                 49.437 
                 90.950 
               
               
                   
                 49.583 
                 91.024 
               
               
                   
                 49.728 
                 91.097 
               
               
                   
                 49.872 
                 91.170 
               
               
                   
                 50.015 
                 91.243 
               
               
                   
                 50.157 
                 91.316 
               
               
                   
                 50.344 
                 91.411 
               
               
                   
                 50.483 
                 91.483 
               
               
                   
                 50.622 
                 91.554 
               
               
                   
                 50.759 
                 91.624 
               
               
                   
                 50.895 
                 91.695 
               
               
                   
                 51.122 
                 91.812 
               
               
                   
                 51.306 
                 91.907 
               
               
                   
                 51.490 
                 92.002 
               
               
                   
                 51.673 
                 92.097 
               
               
                   
                 51.947 
                 92.239 
               
               
                   
                 52.128 
                 92.333 
               
               
                   
                 52.310 
                 92.428 
               
               
                   
                 52.491 
                 92.522 
               
               
                   
                 52.671 
                 92.616 
               
               
                   
                 52.852 
                 92.710 
               
               
                   
                 53.031 
                 92.804 
               
               
                   
                 53.210 
                 92.897 
               
               
                   
                 53.389 
                 92.991 
               
               
                   
                 53.568 
                 93.084 
               
               
                   
                 53.746 
                 93.178 
               
               
                   
                 53.923 
                 93.271 
               
               
                   
                 54.100 
                 93.364 
               
               
                   
                 54.277 
                 93.457 
               
               
                   
                 54.453 
                 93.549 
               
               
                   
                 54.629 
                 93.642 
               
               
                   
                 54.804 
                 93.734 
               
               
                   
                 54.979 
                 93.827 
               
               
                   
                 55.154 
                 93.919 
               
               
                   
                 55.328 
                 94.011 
               
               
                   
                 55.502 
                 94.103 
               
               
                   
                 55.675 
                 94.195 
               
               
                   
                 55.848 
                 94.287 
               
               
                   
                 56.020 
                 94.378 
               
               
                   
                 56.193 
                 94.470 
               
               
                   
                 56.364 
                 94.561 
               
               
                   
                 56.536 
                 94.652 
               
               
                   
                 56.707 
                 94.743 
               
               
                   
                 56.877 
                 94.834 
               
               
                   
                 57.047 
                 94.925 
               
               
                   
                 57.217 
                 95.016 
               
               
                   
                 57.386 
                 95.106 
               
               
                   
                 57.555 
                 95.197 
               
               
                   
                 57.724 
                 95.287 
               
               
                   
                 58.060 
                 95.468 
               
               
                   
                 58.228 
                 95.558 
               
               
                   
                 58.395 
                 95.647 
               
               
                   
                 58.561 
                 95.737 
               
               
                   
                 58.728 
                 95.827 
               
               
                   
                 58.894 
                 95.916 
               
               
                   
                 59.059 
                 96.006 
               
               
                   
                 59.225 
                 96.095 
               
               
                   
                 59.389 
                 96.184 
               
               
                   
                 59.554 
                 96.273 
               
               
                   
                 59.718 
                 96.362 
               
               
                   
                 59.882 
                 96.451 
               
               
                   
                 60.045 
                 96.540 
               
               
                   
                 60.208 
                 96.628 
               
               
                   
                 60.371 
                 96.717 
               
               
                   
                 60.533 
                 96.805 
               
               
                   
                 60.695 
                 96.893 
               
               
                   
                 60.857 
                 96.981 
               
               
                   
                 61.018 
                 97.069 
               
               
                   
                 61.179 
                 97.157 
               
               
                   
                 61.340 
                 97.245 
               
               
                   
                 61.500 
                 97.333 
               
               
                   
                 61.660 
                 97.420 
               
               
                   
                 61.820 
                 97.508 
               
               
                   
                 61.979 
                 97.595 
               
               
                   
                 62.138 
                 97.682 
               
               
                   
                 62.297 
                 97.769 
               
               
                   
                 62.455 
                 97.856 
               
               
                   
                 62.613 
                 97.943 
               
               
                   
                 62.771 
                 98.030 
               
               
                   
                 62.928 
                 98.117 
               
               
                   
                 63.085 
                 98.203 
               
               
                   
                 63.241 
                 98.290 
               
               
                   
                 63.398 
                 98.376 
               
               
                   
                 63.554 
                 98.462 
               
               
                   
                 63.709 
                 98.549 
               
               
                   
                 63.865 
                 98.635 
               
               
                   
                 64.020 
                 98.721 
               
               
                   
                 64.174 
                 98.806 
               
               
                   
                 64.329 
                 98.892 
               
               
                   
                 64.483 
                 98.978 
               
               
                   
                 64.636 
                 99.063 
               
               
                   
                 64.790 
                 99.149 
               
               
                   
                 64.943 
                 99.234 
               
               
                   
                 65.096 
                 99.319 
               
               
                   
                 65.248 
                 99.404 
               
               
                   
                 65.401 
                 99.489 
               
               
                   
                 65.552 
                 99.574 
               
               
                   
                 65.704 
                 99.659 
               
               
                   
                 65.855 
                 99.744 
               
               
                   
                 66.006 
                 99.828 
               
               
                   
                 66.157 
                 99.913 
               
               
                   
                 66.307 
                 99.997 
               
               
                   
                 66.458 
                 100.081 
               
               
                   
                 66.607 
                 100.165 
               
               
                   
                 66.757 
                 100.250 
               
               
                   
                 66.906 
                 100.333 
               
               
                   
                 67.055 
                 100.417 
               
               
                   
                 67.204 
                 100.501 
               
               
                   
                 67.352 
                 100.585 
               
               
                   
                 67.500 
                 100.668 
               
               
                   
                   
               
             
          
         
       
     
         [0000]    
       
         
               
               
               
             
               
               
               
             
           
               
                 TABLE 3 
               
               
                   
               
               
                 x 
                 y 
                 z 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 0.900 
                 15.153 
                 16.072 
               
               
                 0.971 
                 15.042 
                 16.990 
               
               
                 1.005 
                 14.908 
                 17.770 
               
               
                 0.979 
                 14.562 
                 19.181 
               
               
                 0.924 
                 14.353 
                 19.822 
               
               
                 0.773 
                 13.916 
                 20.901 
               
               
                 0.672 
                 13.678 
                 21.386 
               
               
                 0.543 
                 13.412 
                 21.864 
               
               
                 0.412 
                 13.151 
                 22.283 
               
               
                 0.267 
                 12.879 
                 22.674 
               
               
                 0.110 
                 12.597 
                 23.037 
               
               
                 −0.060 
                 12.305 
                 23.373 
               
               
                 1.812 
                 15.270 
                 15.168 
               
               
                 1.983 
                 15.100 
                 17.054 
               
               
                 1.969 
                 14.626 
                 19.302 
               
               
                 1.840 
                 14.219 
                 20.516 
               
               
                 1.754 
                 14.001 
                 21.044 
               
               
                 1.651 
                 13.771 
                 21.538 
               
               
                 1.535 
                 13.530 
                 21.999 
               
               
                 1.391 
                 13.260 
                 22.457 
               
               
                 1.247 
                 12.997 
                 22.858 
               
               
                 1.090 
                 12.725 
                 23.233 
               
               
                 0.922 
                 12.443 
                 23.583 
               
               
                 0.742 
                 12.151 
                 23.907 
               
               
                 0.533 
                 11.830 
                 24.226 
               
               
                 0.330 
                 11.520 
                 24.502 
               
               
                 0.115 
                 11.200 
                 24.756 
               
               
                 −0.111 
                 10.871 
                 24.987 
               
               
                 2.788 
                 15.254 
                 15.172 
               
               
                 2.887 
                 15.195 
                 16.140 
               
               
                 2.923 
                 14.653 
                 19.296 
               
               
                 2.592 
                 13.819 
                 21.613 
               
               
                 2.033 
                 12.809 
                 23.357 
               
               
                 1.867 
                 12.536 
                 23.720 
               
               
                 1.689 
                 12.254 
                 24.058 
               
               
                 1.500 
                 11.964 
                 24.373 
               
               
                 1.283 
                 11.644 
                 24.684 
               
               
                 1.188 
                 11.502 
                 24.812 
               
               
                 1.072 
                 11.336 
                 24.953 
               
               
                 0.973 
                 11.189 
                 25.071 
               
               
                 0.851 
                 11.019 
                 25.200 
               
               
                 0.747 
                 10.868 
                 25.309 
               
               
                 0.619 
                 10.693 
                 25.426 
               
               
                 0.510 
                 10.538 
                 25.525 
               
               
                 0.377 
                 10.358 
                 25.630 
               
               
                 0.262 
                 10.199 
                 25.719 
               
               
                 0.123 
                 10.014 
                 25.813 
               
               
                 0.003 
                 9.851 
                 25.892 
               
               
                 −0.142 
                 9.660 
                 25.975 
               
               
                 3.763 
                 15.160 
                 15.533 
               
               
                 3.701 
                 14.252 
                 20.515 
               
               
                 3.613 
                 14.051 
                 21.069 
               
               
                 3.500 
                 13.825 
                 21.620 
               
               
                 3.111 
                 13.122 
                 22.989 
               
               
                 2.943 
                 12.852 
                 23.416 
               
               
                 2.416 
                 12.036 
                 24.473 
               
               
                 1.995 
                 11.430 
                 25.084 
               
               
                 1.778 
                 11.124 
                 25.347 
               
               
                 1.675 
                 10.979 
                 25.463 
               
               
                 1.313 
                 10.487 
                 25.811 
               
               
                 1.201 
                 10.333 
                 25.907 
               
               
                 1.065 
                 10.155 
                 26.012 
               
               
                 0.948 
                 9.998 
                 26.099 
               
               
                 0.807 
                 9.814 
                 26.192 
               
               
                 0.685 
                 9.653 
                 26.270 
               
               
                 0.538 
                 9.465 
                 26.352 
               
               
                 0.411 
                 9.299 
                 26.419 
               
               
                 0.126 
                 8.935 
                 26.548 
               
               
                 −0.172 
                 8.561 
                 26.655 
               
               
                 4.714 
                 14.978 
                 16.279 
               
               
                 4.756 
                 14.771 
                 17.988 
               
               
                 4.540 
                 14.110 
                 20.696 
               
               
                 3.751 
                 12.738 
                 23.599 
               
               
                 3.302 
                 12.070 
                 24.514 
               
               
                 3.108 
                 11.791 
                 24.835 
               
               
                 2.790 
                 11.348 
                 25.287 
               
               
                 2.572 
                 11.049 
                 25.556 
               
               
                 2.106 
                 10.427 
                 26.033 
               
               
                 1.973 
                 10.255 
                 26.147 
               
               
                 1.721 
                 9.926 
                 26.345 
               
               
                 1.458 
                 9.589 
                 26.523 
               
               
                 1.185 
                 9.243 
                 26.681 
               
               
                 0.901 
                 8.888 
                 26.819 
               
               
                 0.300 
                 8.148 
                 27.031 
               
               
                 0.009 
                 7.791 
                 27.102 
               
               
                 5.515 
                 14.839 
                 15.487 
               
               
                 5.560 
                 14.439 
                 18.890 
               
               
                 5.433 
                 14.113 
                 20.231 
               
               
                 5.122 
                 13.525 
                 21.921 
               
               
                 4.692 
                 12.838 
                 23.327 
               
               
                 4.059 
                 11.925 
                 24.680 
               
               
                 3.970 
                 11.798 
                 24.834 
               
               
                 3.863 
                 11.652 
                 25.005 
               
               
                 3.769 
                 11.522 
                 25.150 
               
               
                 3.656 
                 11.371 
                 25.309 
               
               
                 3.559 
                 11.237 
                 25.445 
               
               
                 3.441 
                 11.082 
                 25.593 
               
               
                 3.321 
                 10.925 
                 25.737 
               
               
                 3.216 
                 10.786 
                 25.858 
               
               
                 3.091 
                 10.625 
                 25.991 
               
               
                 2.983 
                 10.482 
                 26.102 
               
               
                 2.852 
                 10.316 
                 26.225 
               
               
                 2.740 
                 10.169 
                 26.327 
               
               
                 2.487 
                 9.849 
                 26.533 
               
               
                 2.225 
                 9.520 
                 26.720 
               
               
                 1.953 
                 9.182 
                 26.887 
               
               
                 1.672 
                 8.836 
                 27.034 
               
               
                 1.515 
                 8.647 
                 27.105 
               
               
                 1.217 
                 8.287 
                 27.222 
               
               
                 0.908 
                 7.916 
                 27.318 
               
               
                 0.614 
                 7.563 
                 27.388 
               
               
                 0.282 
                 7.172 
                 27.443 
               
               
                 −0.063 
                 6.770 
                 27.473 
               
               
                 6.346 
                 14.599 
                 15.560 
               
               
                 6.410 
                 14.458 
                 17.399 
               
               
                 6.216 
                 13.900 
                 20.295 
               
               
                 6.116 
                 13.714 
                 20.922 
               
               
                 6.010 
                 13.527 
                 21.473 
               
               
                 5.821 
                 13.219 
                 22.258 
               
               
                 5.671 
                 12.992 
                 22.764 
               
               
                 5.522 
                 12.768 
                 23.211 
               
               
                 5.362 
                 12.535 
                 23.632 
               
               
                 5.192 
                 12.294 
                 24.030 
               
               
                 5.096 
                 12.162 
                 24.233 
               
               
                 4.997 
                 12.028 
                 24.429 
               
               
                 4.911 
                 11.908 
                 24.596 
               
               
                 4.498 
                 11.358 
                 25.279 
               
               
                 4.285 
                 11.080 
                 25.578 
               
               
                 4.063 
                 10.794 
                 25.858 
               
               
                 3.833 
                 10.500 
                 26.118 
               
               
                 3.345 
                 9.890 
                 26.581 
               
               
                 3.088 
                 9.573 
                 26.785 
               
               
                 2.822 
                 9.247 
                 26.969 
               
               
                 2.260 
                 8.571 
                 27.281 
               
               
                 1.965 
                 8.220 
                 27.409 
               
               
                 1.659 
                 7.859 
                 27.516 
               
               
                 1.342 
                 7.488 
                 27.602 
               
               
                 1.013 
                 7.107 
                 27.668 
               
               
                 0.672 
                 6.715 
                 27.712 
               
               
                 0.348 
                 6.341 
                 27.732 
               
               
                 −0.019 
                 5.925 
                 27.730 
               
               
                 −0.223 
                 5.697 
                 27.718 
               
               
                 7.164 
                 14.294 
                 16.059 
               
               
                 7.187 
                 14.230 
                 16.950 
               
               
                 7.164 
                 14.078 
                 18.228 
               
               
                 7.143 
                 14.016 
                 18.621 
               
               
                 7.121 
                 13.957 
                 18.954 
               
               
                 7.089 
                 13.885 
                 19.320 
               
               
                 7.052 
                 13.809 
                 19.673 
               
               
                 7.010 
                 13.728 
                 20.015 
               
               
                 6.971 
                 13.654 
                 20.305 
               
               
                 6.921 
                 13.567 
                 20.626 
               
               
                 6.867 
                 13.475 
                 20.936 
               
               
                 6.757 
                 13.295 
                 21.492 
               
               
                 6.487 
                 12.888 
                 22.543 
               
               
                 6.328 
                 12.661 
                 23.039 
               
               
                 6.002 
                 12.208 
                 23.892 
               
               
                 5.824 
                 11.969 
                 24.282 
               
               
                 5.425 
                 11.449 
                 25.021 
               
               
                 5.006 
                 10.915 
                 25.652 
               
               
                 4.783 
                 10.636 
                 25.938 
               
               
                 4.293 
                 10.035 
                 26.470 
               
               
                 3.786 
                 9.423 
                 26.908 
               
               
                 3.243 
                 8.778 
                 27.273 
               
               
                 2.958 
                 8.444 
                 27.427 
               
               
                 2.824 
                 8.285 
                 27.492 
               
               
                 2.664 
                 8.100 
                 27.562 
               
               
                 2.524 
                 7.938 
                 27.619 
               
               
                 2.359 
                 7.748 
                 27.679 
               
               
                 2.215 
                 7.581 
                 27.726 
               
               
                 2.044 
                 7.386 
                 27.775 
               
               
                 1.895 
                 7.215 
                 27.813 
               
               
                 1.744 
                 7.041 
                 27.847 
               
               
                 1.564 
                 6.838 
                 27.880 
               
               
                 1.408 
                 6.660 
                 27.904 
               
               
                 1.221 
                 6.450 
                 27.925 
               
               
                 1.059 
                 6.267 
                 27.939 
               
               
                 0.895 
                 6.081 
                 27.947 
               
               
                 0.698 
                 5.862 
                 27.950 
               
               
                 0.527 
                 5.671 
                 27.948 
               
               
                 0.322 
                 5.445 
                 27.937 
               
               
                 0.145 
                 5.247 
                 27.923 
               
               
                 −0.036 
                 5.046 
                 27.903 
               
               
                 7.892 
                 13.989 
                 15.461 
               
               
                 7.932 
                 13.878 
                 17.322 
               
               
                 7.852 
                 13.656 
                 18.943 
               
               
                 7.693 
                 13.367 
                 20.302 
               
               
                 7.469 
                 13.021 
                 21.499 
               
               
                 7.342 
                 12.835 
                 22.027 
               
               
                 7.203 
                 12.639 
                 22.527 
               
               
                 7.041 
                 12.420 
                 23.029 
               
               
                 6.881 
                 12.205 
                 23.474 
               
               
                 6.697 
                 11.966 
                 23.921 
               
               
                 6.516 
                 11.734 
                 24.317 
               
               
                 6.326 
                 11.493 
                 24.691 
               
               
                 6.128 
                 11.244 
                 25.043 
               
               
                 5.905 
                 10.970 
                 25.397 
               
               
                 5.689 
                 10.705 
                 25.708 
               
               
                 5.464 
                 10.433 
                 26.000 
               
               
                 5.231 
                 10.153 
                 26.273 
               
               
                 5.103 
                 10.001 
                 26.411 
               
               
                 4.990 
                 9.866 
                 26.527 
               
               
                 4.857 
                 9.710 
                 26.655 
               
               
                 4.741 
                 9.571 
                 26.763 
               
               
                 4.603 
                 9.411 
                 26.881 
               
               
                 4.483 
                 9.269 
                 26.980 
               
               
                 4.340 
                 9.104 
                 27.089 
               
               
                 4.216 
                 8.959 
                 27.180 
               
               
                 4.069 
                 8.789 
                 27.279 
               
               
                 3.941 
                 8.640 
                 27.362 
               
               
                 3.789 
                 8.467 
                 27.451 
               
               
                 3.657 
                 8.314 
                 27.525 
               
               
                 3.500 
                 8.136 
                 27.605 
               
               
                 3.364 
                 7.979 
                 27.670 
               
               
                 3.202 
                 7.796 
                 27.739 
               
               
                 3.061 
                 7.636 
                 27.796 
               
               
                 2.894 
                 7.448 
                 27.855 
               
               
                 2.748 
                 7.283 
                 27.903 
               
               
                 2.575 
                 7.090 
                 27.952 
               
               
                 2.425 
                 6.921 
                 27.990 
               
               
                 2.091 
                 6.548 
                 28.058 
               
               
                 1.746 
                 6.165 
                 28.104 
               
               
                 1.417 
                 5.800 
                 28.127 
               
               
                 1.048 
                 5.394 
                 28.130 
               
               
                 0.664 
                 4.975 
                 28.108 
               
               
                 0.265 
                 4.542 
                 28.060 
               
               
                 −0.116 
                 4.128 
                 27.991 
               
               
                 8.633 
                 13.611 
                 15.815 
               
               
                 8.591 
                 13.389 
                 18.441 
               
               
                 8.343 
                 12.979 
                 20.535 
               
               
                 8.233 
                 12.819 
                 21.122 
               
               
                 8.100 
                 12.636 
                 21.711 
               
               
                 7.963 
                 12.452 
                 22.232 
               
               
                 7.816 
                 12.259 
                 22.724 
               
               
                 7.647 
                 12.042 
                 23.220 
               
               
                 7.478 
                 11.830 
                 23.659 
               
               
                 7.384 
                 11.713 
                 23.883 
               
               
                 7.287 
                 11.594 
                 24.101 
               
               
                 7.201 
                 11.487 
                 24.286 
               
               
                 7.100 
                 11.364 
                 24.492 
               
               
                 7.010 
                 11.254 
                 24.667 
               
               
                 6.904 
                 11.126 
                 24.862 
               
               
                 6.795 
                 10.996 
                 25.050 
               
               
                 6.699 
                 10.880 
                 25.210 
               
               
                 6.143 
                 10.224 
                 26.005 
               
               
                 6.018 
                 10.079 
                 26.157 
               
               
                 5.780 
                 9.803 
                 26.428 
               
               
                 5.649 
                 9.652 
                 26.565 
               
               
                 5.534 
                 9.519 
                 26.680 
               
               
                 5.139 
                 9.069 
                 27.032 
               
               
                 4.313 
                 8.136 
                 27.597 
               
               
                 3.881 
                 7.654 
                 27.815 
               
               
                 2.933 
                 6.607 
                 28.135 
               
               
                 2.437 
                 6.064 
                 28.228 
               
               
                 1.916 
                 5.499 
                 28.275 
               
               
                 1.545 
                 5.097 
                 28.279 
               
               
                 1.160 
                 4.683 
                 28.259 
               
               
                 0.760 
                 4.255 
                 28.214 
               
               
                 0.377 
                 3.845 
                 28.147 
               
               
                 −0.134 
                 3.307 
                 28.024 
               
               
                 9.329 
                 13.195 
                 16.088 
               
               
                 9.328 
                 13.144 
                 17.034 
               
               
                 9.299 
                 13.071 
                 17.850 
               
               
                 9.207 
                 12.912 
                 19.037 
               
               
                 9.126 
                 12.789 
                 19.716 
               
               
                 9.021 
                 12.643 
                 20.395 
               
               
                 8.909 
                 12.492 
                 20.992 
               
               
                 8.774 
                 12.318 
                 21.592 
               
               
                 8.636 
                 12.143 
                 22.122 
               
               
                 8.558 
                 12.046 
                 22.393 
               
               
                 8.476 
                 11.945 
                 22.656 
               
               
                 8.316 
                 11.750 
                 23.129 
               
               
                 8.226 
                 11.642 
                 23.370 
               
               
                 8.133 
                 11.531 
                 23.605 
               
               
                 8.051 
                 11.432 
                 23.805 
               
               
                 7.953 
                 11.317 
                 24.027 
               
               
                 7.853 
                 11.200 
                 24.243 
               
               
                 7.764 
                 11.095 
                 24.426 
               
               
                 7.660 
                 10.973 
                 24.630 
               
               
                 7.567 
                 10.864 
                 24.803 
               
               
                 7.346 
                 10.610 
                 25.183 
               
               
                 7.012 
                 10.229 
                 25.689 
               
               
                 6.677 
                 9.848 
                 26.131 
               
               
                 6.307 
                 9.433 
                 26.550 
               
               
                 5.917 
                 9.000 
                 26.927 
               
               
                 5.529 
                 8.570 
                 27.248 
               
               
                 5.103 
                 8.101 
                 27.543 
               
               
                 4.678 
                 7.637 
                 27.786 
               
               
                 4.378 
                 7.309 
                 27.930 
               
               
                 4.067 
                 6.974 
                 28.055 
               
               
                 3.748 
                 6.629 
                 28.162 
               
               
                 3.419 
                 6.274 
                 28.250 
               
               
                 3.079 
                 5.910 
                 28.318 
               
               
                 2.728 
                 5.535 
                 28.365 
               
               
                 2.020 
                 4.781 
                 28.396 
               
               
                 1.633 
                 4.371 
                 28.378 
               
               
                 1.449 
                 4.177 
                 28.361 
               
               
                 1.231 
                 3.947 
                 28.335 
               
               
                 1.041 
                 3.746 
                 28.306 
               
               
                 0.847 
                 3.542 
                 28.270 
               
               
                 0.615 
                 3.300 
                 28.221 
               
               
                 0.335 
                 3.009 
                 28.151 
               
               
                 0.037 
                 2.701 
                 28.065 
               
               
                 9.967 
                 12.760 
                 15.331 
               
               
                 9.863 
                 12.522 
                 18.814 
               
               
                 9.821 
                 12.465 
                 19.190 
               
               
                 9.555 
                 12.125 
                 20.852 
               
               
                 9.127 
                 11.616 
                 22.512 
               
               
                 8.875 
                 11.325 
                 23.242 
               
               
                 8.600 
                 11.013 
                 23.912 
               
               
                 8.304 
                 10.682 
                 24.529 
               
               
                 7.989 
                 10.331 
                 25.096 
               
               
                 7.300 
                 9.575 
                 26.089 
               
               
                 6.946 
                 9.189 
                 26.500 
               
               
                 6.556 
                 8.767 
                 26.890 
               
               
                 6.148 
                 8.328 
                 27.238 
               
               
                 5.742 
                 7.893 
                 27.531 
               
               
                 5.296 
                 7.418 
                 27.799 
               
               
                 4.854 
                 6.947 
                 28.015 
               
               
                 4.540 
                 6.614 
                 28.140 
               
               
                 4.217 
                 6.273 
                 28.247 
               
               
                 3.884 
                 5.923 
                 28.335 
               
               
                 3.727 
                 5.757 
                 28.369 
               
               
                 3.541 
                 5.562 
                 28.403 
               
               
                 3.379 
                 5.391 
                 28.428 
               
               
                 3.187 
                 5.191 
                 28.451 
               
               
                 3.020 
                 5.015 
                 28.466 
               
               
                 2.850 
                 4.838 
                 28.477 
               
               
                 2.649 
                 4.628 
                 28.483 
               
               
                 2.474 
                 4.445 
                 28.483 
               
               
                 2.266 
                 4.228 
                 28.477 
               
               
                 2.084 
                 4.039 
                 28.466 
               
               
                 1.680 
                 3.619 
                 28.424 
               
               
                 1.489 
                 3.420 
                 28.396 
               
               
                 1.294 
                 3.218 
                 28.362 
               
               
                 1.062 
                 2.978 
                 28.314 
               
               
                 0.781 
                 2.690 
                 28.245 
               
               
                 0.483 
                 2.385 
                 28.161 
               
               
                 0.134 
                 2.028 
                 28.047 
               
               
                 10.606 
                 12.270 
                 15.924 
               
               
                 10.528 
                 12.142 
                 18.131 
               
               
                 10.249 
                 11.807 
                 20.296 
               
               
                 9.829 
                 11.337 
                 22.084 
               
               
                 9.583 
                 11.067 
                 22.850 
               
               
                 9.022 
                 10.462 
                 24.203 
               
               
                 8.381 
                 9.779 
                 25.347 
               
               
                 7.923 
                 9.294 
                 25.991 
               
               
                 7.682 
                 9.040 
                 26.285 
               
               
                 7.434 
                 8.779 
                 26.561 
               
               
                 7.159 
                 8.491 
                 26.836 
               
               
                 6.895 
                 8.214 
                 27.075 
               
               
                 6.623 
                 7.930 
                 27.296 
               
               
                 6.343 
                 7.638 
                 27.500 
               
               
                 6.055 
                 7.338 
                 27.686 
               
               
                 5.918 
                 7.197 
                 27.767 
               
               
                 5.454 
                 6.715 
                 28.007 
               
               
                 4.487 
                 5.714 
                 28.352 
               
               
                 3.983 
                 5.196 
                 28.459 
               
               
                 3.458 
                 4.655 
                 28.523 
               
               
                 2.907 
                 4.089 
                 28.541 
               
               
                 2.329 
                 3.497 
                 28.509 
               
               
                 1.720 
                 2.874 
                 28.422 
               
               
                 0.907 
                 2.047 
                 28.225 
               
               
                 11.096 
                 11.633 
                 18.186 
               
               
                 11.012 
                 11.543 
                 18.970 
               
               
                 10.908 
                 11.433 
                 19.700 
               
               
                 10.794 
                 11.314 
                 20.342 
               
               
                 10.658 
                 11.172 
                 20.985 
               
               
                 10.518 
                 11.026 
                 21.553 
               
               
                 10.355 
                 10.858 
                 22.124 
               
               
                 10.192 
                 10.689 
                 22.630 
               
               
                 9.524 
                 10.004 
                 24.238 
               
               
                 9.205 
                 9.678 
                 24.833 
               
               
                 8.867 
                 9.333 
                 25.380 
               
               
                 8.511 
                 8.970 
                 25.880 
               
               
                 8.154 
                 8.607 
                 26.316 
               
               
                 8.020 
                 8.470 
                 26.465 
               
               
                 7.622 
                 8.066 
                 26.866 
               
               
                 7.226 
                 7.664 
                 27.209 
               
               
                 6.793 
                 7.225 
                 27.529 
               
               
                 6.363 
                 6.789 
                 27.795 
               
               
                 5.893 
                 6.313 
                 28.035 
               
               
                 5.747 
                 6.165 
                 28.099 
               
               
                 5.576 
                 5.992 
                 28.168 
               
               
                 5.426 
                 5.841 
                 28.224 
               
               
                 5.250 
                 5.663 
                 28.284 
               
               
                 5.096 
                 5.507 
                 28.331 
               
               
                 4.915 
                 5.324 
                 28.380 
               
               
                 4.241 
                 4.643 
                 28.512 
               
               
                 3.703 
                 4.100 
                 28.562 
               
               
                 3.499 
                 3.894 
                 28.569 
               
               
                 3.321 
                 3.715 
                 28.569 
               
               
                 3.110 
                 3.502 
                 28.564 
               
               
                 2.925 
                 3.316 
                 28.554 
               
               
                 2.322 
                 2.708 
                 28.485 
               
               
                 1.608 
                 1.989 
                 28.339 
               
               
                 1.308 
                 1.688 
                 28.256 
               
               
                 0.957 
                 1.336 
                 28.144 
               
               
                 0.593 
                 0.970 
                 28.011 
               
               
                 0.259 
                 0.635 
                 27.873 
               
               
                   
               
             
          
         
       
     
       REFERENCES 
       [0000]    
       
         [1] W. Cassarly, “Nonimaging Optics: Concentration and illumination”, in the Handbook of Optics, 2nd ed., pp. 2.23-2.42, (McGraw-Hill, New York, 2001) 
         [2]H. Ries, J. M. Gordon, M. Laxen, “High-flux photovoltaic solar concentrators with Kaleidoscope based optical designs”, Solar Energy, Vol. 60, No. 1, pp. 11-16, (1997) 
         [3] J. J. O&#39;Ghallagher, R. Winston, “Nonimaging solar concentrator with near-uniform irradiance for photovoltaic arrays” in Nonimaging Optics: Maximum Efficiency Light Transfer VI, Roland Winston, Ed., Proc. SPIE 4446, pp. 60-64, (2001) 
         [4] D. G. Jenkings, “High-uniformity solar concentrators for photovoltaic systems” in Nonimaging Optics Maximum Efficiency Light Transfer VI, Roland Winston, Ed., Proc. SPIE 4446, pp. 52-59, (2001) 
         [5] http://www.daido.co.jp/english/rd/7503.pdf 
         [6] http://www.solfocus.com/, also U.S. 2006/0207650, 2006/0274439 
         [7] www.sol3g.com 
         [8] L. W. James, “Use of imaging refractive secondaries in photovoltaic concentrators”, SAND 89-7029, Albuquerque, N. Mex., (1989) 
         [9] Chapter 13, “Next Generation Photovoltaics”, (Taylor &amp; Francis, CRC Press, 2003) 
         [10] R. Winston, J. C. Miñano, P. Benítez, “Nonimaging Optics”, (Elsevier-Academic Press, New York, 2005) 
         [11] J. C. Miñano, P. Benítez, J. C. González, “RX: a nonimaging concentrator”, Appl. Opt. 34, pp. 2226-2235, (1995) 
         [12] P. Benítez, J. C. Miñano, “Ultrahigh-numerical-aperture imaging concentrator”, J. Opt. Soc. Am. A, 14, pp. 1988-1997, (1997) 
         [13] J. C. Miñano, M. Hernandez, P. Benítez, J. Blen, O. Dross, R. Mohedano, A. Santamaria, “Free-form integrator array optics”, in Nonimaging Optics and Efficient Illumination Systems II, SPIE Proc., R. Winston &amp; T. J. Koshel ed. (2005) 
         [14] US and other patents and patent applications: U.S. Pat. No. 6,639,733; U.S. Pat. No. 6,896,381; U.S. Pat. No. 7,152,985; U.S. Pat. No. 7,460,985; WO 2007/016363; US 2009/0071467; US 2008/0223443; US 2008/0316761. 
         [15] P. Benítez, A. Cvetkovic, R. Winston, G. Diaz, L. Reed, J. Cisneros, A. Tovar, A. Ritschel, J. Wright, “High-Concentration Mirror-Based Köhler Integrating System for Tandem Solar Cells”, 4th World Conference on Photovoltaic Energy Conversion, Hawaii, (2006). 
         [16] J. Chaves, “Introduction to Nonimaging Optics”, CRC Press, 2008, Chapter 17.