Patent Publication Number: US-2009231739-A1

Title: Matrix formulation of kohler integrating system and coupled non-imaging light concentrator

Description:
CROSS-REFERENCES TO RELATED APPLICATIONS 
     The present application claims priority to and is a non-provisional application of U.S. Provisional Application Ser. No. 60/916,515, filed May 7, 2007, the disclosure of which is hereby incorporated by reference in its entirety for all purposes. 
    
    
     BACKGROUND 
     The present invention relates generally to optical concentrator systems and methods utilizing solar cells for collecting the concentrated light energy, and more particularly to a matrix formulation for designing optical concentrator systems incorporating homogenizer elements. 
     Solar cells for electrical energy production are very well known but have limited utility due to the very high cost of production. For example, although substantial research has been ongoing for many years, the cost per Killowatt-hour (Kwh) still is about ten times that of conventional electric power production. To compete with wind power or other alternative energy sources, the efficiency of production of electricity from solar cells should be drastically improved. 
     One prior system described in US Patent Application publication 2006-0207650-A1, titled “Multi-Junction Solar Cells with an Aplanatic Imaging System and Coupled Non-Imaging Light Concentrator,” which is hereby incorporated by reference in its entirety, considers a two mirror aplanatic system (which produce sharp imaging of normal incidence rays on the cell center and satisfies the Abe sine condition) which may be combined with a non-imaging concentrator.  FIG. 1  shows such a two-mirror aplanatic system without a non-imaging concentrator. Primary mirror  10  concentrates the light onto the secondary mirror  11 , which illuminates the target solar cell  12 . That system has a clear limitation because it produces a highly non-uniform illumination on the solar cell, which reduces the cell efficiency and system reliability. This is because the optics is imaging the plane at infinity onto the plane of the target, where the cell is placed, and thus the sun is imaged on the cell. The angular acceptance of the two mirror aplanatic concentrator is several times (for instance, 3) greater than the angular size of the sun to allow for tolerances. The imaging mapping makes the acceptance angle-to-sun angle ratio the same as the cell diameter-to-sun image diameter ratio. Therefore, the area of the round target would be 3 2  times greater than the sun image area. If the average concentration of the prior art design is 500 suns, the local concentration can reach as much as 3 2 ×500=4,500 suns. This value cannot be tolerated by the present high-efficiency multi-junction cells, which show an abrupt drop in efficiency if they operate above 2,000-3,000 suns. 
     Other related prior art is disclosed in a paper by L. W. James, “Use of Imaging Refractive Secondaries in Photovoltaic Concentrators”, SAND89-7029, Alburquerque, N. Mex., 1989. In that paper, and as shown in  FIG. 2 , a Kohler integrator system is used as a photovoltaic concentrator. The Kohler integrator consists of two imaging optical elements (primary and secondary) with positive focal length (i.e., producing a real image of an object at infinity). The secondary is placed at the focal plane of the primary, and the secondary images the primary on the cell. In James&#39; paper, the photovoltaic Kohler concentrator is composed by a Fresnel lens  20  as primary and a single-surface imaging lens  21  as secondary that encapsulates the cell  22 , as illustrated in  FIG. 2 . The primary images the sun on the secondary aperture  23 . As the primary is uniformly illuminated by the sun, the irradiance distribution on the cell is also uniform, and it will remain unchanged when the sun moves within the acceptance angle (equivalently when the sun image  24  moves within the secondary aperture). The concentration-acceptance angle product that can be attained with this configuration is very limited, because numerical aperture on the cell is small. Additionally, the system is necessarily not compact because the optics used are refractive and includes a single Kohler integration element. 
     Therefore it is desirable to provide optical concentrator systems and methods that overcome the above and other problems. In particular, it is desirable to provide efficient methodologies for designing such optical systems. 
     BRIEF SUMMARY 
     The present invention provides methods for designing optical systems, including homogenizer element(s), that concentrate light from a distant source, such as the sun, onto a target device, such as a solar cell. In certain aspects, devices made according to the methods of the present invention include reflective and/or refractive elements. For example a device might operate such that light impinging from the distant source is focused or imaged by a plurality of primary reflective segments of a primary mirror element onto a plurality of corresponding secondary reflective segments. The primary and secondary reflective segments might be radially symmetric. The secondary mirror segments image the corresponding primary segments onto an exit aperture such that the exit aperture is uniformly illuminated. A target cell may be located proximal to the exit aperture, or an entry aperture of a non-imaging concentrator may be positioned proximal the exit aperture, wherein the concentrator concentrates the reflected light onto the target cell. 
     Aspects of the present invention are directed to methods for designing optical devices and systems that provide extremely high solar flux onto a multi-junction solar cell, or other target cell, to produce efficient electrical output. An aplanatic optical imaging system, according to certain aspects, includes a Kohler homogenizer primary and secondary mirror subsystem that directs and concentrates illumination to a solar cell positioned proximal an exit aperture such that uniform irradiance conditions are achieved for high intensity light concentration onto the solar cell. As used herein, “aplanatic” generally refers to the condition of freedom from spherical aberration and coma. Thus, as used herein, “aplanatic optics” or “aplanatic optical system” or similar phrases generally refer to optical elements or systems that correct for, or are substantially free from, spherical aberration or coma. In certain aspects, a non-imaging light concentrator, or flux booster, is efficiently coupled to the primary and secondary mirrors. 
     Aspects of the present invention provide a methodology that enable designing a variety of Kohler homogenizer and planar optical systems. The optical systems, in certain aspects, are formed by two mirrors and can provide the necessary components to deliver light to a multi-junction solar cell or other solar cell. In one embodiment, for example, symmetric mirror segments on both primary and secondary mirrors are pair-wise correlated so that a segment on the primary images the field of view onto the corresponding secondary segment, while the secondary segment in turn, images the primary segment on the target. In one embodiment, a secondary mirror is co-planar with the entrance aperture, and the exit aperture is co-planar with the vertex of the primary mirror. In another embodiment, the inter-mirror space is filled with a dielectric with index of refraction, n, such that the numerical aperture (“NA”) is increased by a factor of n. In another embodiment, a non-imaging light concentrator is disposed at the exit aperture of the primary mirror. In one aspect, the non-imaging concentrator is a θ 1 /θ 2  concentrator with θ 1  chosen to match the NA of the two-mirror system (sin θ 1 =NA/n) while θ 2  is chosen to satisfy a subsidiary condition, such as maintaining total internal reflection (“TIR”) or limiting the angle of irradiance on the multi-junction solar cell. In another aspect, the non-imaging concentrator is a flow line concentrator or a tailored non-imaging concentrator. 
     According to one aspect of the present invention, a method is provided for designing an optical device that typically comprises an aplanatic optical imaging system including a segmented primary reflective element defining an entrance aperture and having a plurality of aspherical primary reflective segments, and a segmented secondary reflective element having a plurality of aspherical secondary reflective segments. The primary and secondary reflective segments are pair-wise correlated so that a primary reflective segment images a field of view onto a corresponding secondary reflective segment, and the secondary reflective segment images the corresponding primary segment onto an exit aperture. In certain aspects, the device typically includes a target element positioned proximal to a vertex of the primary reflective element. One type of target element is a solar cell. In certain aspects, the primary reflective segments have a substantially parabolic reflecting surface and the secondary reflective segments have a substantially elliptical reflecting surface. In certain aspects, the primary reflective segments forming the primary reflective element include slope discontinuities where they meet such that the second derivative of the primary reflective element surface is discontinuous, and the secondary reflective segments forming the secondary reflective element include slope discontinuities where they meet such that the second derivative of the secondary reflective element surface is discontinuous. 
     According to another aspect of the present invention, a method is provided for designing an optical device that typically includes an aplanatic optical imaging system that includes a segmented primary reflective element defining an entrance aperture and having a plurality of primary reflective segments, and a segmented secondary reflective element having a plurality of secondary reflective segments. The device also typically includes a target element positioned proximal to an exit aperture. The primary and secondary reflective segments are pair-wise correlated so that light impinging on a primary reflective segment is focused onto a corresponding secondary reflective segment, and the secondary reflective segment images the corresponding primary segment onto the exit aperture such that the target element is substantially uniformly illuminated. In certain aspects, the primary reflective segments and the secondary reflective segments are aspherical. In certain aspects, the target element includes a solar cell. In certain aspects, the device further includes a non-imaging concentrator having an entrance aperture disposed proximal to the exit aperture. In certain aspects, the primary reflective segments forming the primary reflective element include slope discontinuities where they meet such that the second derivative of the primary reflective element surface is discontinuous, and the secondary reflective segments forming the secondary reflective element include slope discontinuities where they meet such that the second derivative of the secondary reflective element surface is discontinuous. 
     In certain aspects, the methods include using a matrix representation of the optical elements of the form 
     
       
         
           
             
               
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               = 
               
                 
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     where n is the refractive index of material surrounding the target cell, x-θ and x′-θ′ are the space angle coordinates at the target cell and input aperture, respectively, and μ is a free parameter and C is the concentration ratio. 
     In certain aspects, the methods include using a matrix representation of the optical elements of the form 
     
       
         
           
             
               
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     where n, n′ are the refractive indices at the input aperture and target cell respectively, P 1  is the power of the first Kohler element and the re-scaling parameter is redefined to resemble concentration C. 
     Reference to the remaining portions of the specification, including the drawings and claims, will realize other features and advantages of the present invention. Further features and advantages of the present invention, as well as the structure and operation of various embodiments of the present invention, are described in detail below with respect to the accompanying drawings. In the drawings, like reference numbers indicate identical or functionally similar elements. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  illustrates a solid-dielectric aplanatic two mirror design disclosed in prior art; 
         FIG. 2  illustrates a photovoltaic Kohler concentrator; 
         FIG. 3  illustrates a solid dielectric two-mirror Kohler radial homogenizer optical system according to one embodiment; 
         FIG. 4  illustrates a 3-dimensional (3D) view of the two-mirror Kohler radial homogenizer optical system of  FIG. 3 ; 
         FIG. 5  illustrates the operation of a two-mirror Kohler radial homogenizer optical system when the sun is off-center but still within the design acceptance angle according to one embodiment; 
         FIG. 6  illustrates a non-imaging concentrator added to the two-mirror Kohler radial homogenizer optical system to increase the concentration ratio of the acceptance angle according to one embodiment; 
         FIG. 7  illustrates a two-mirror Kohler radial homogenizer optical system that includes volumes of different refractive indices according to one embodiment; 
         FIG. 8  illustrates a Kohler homogenizer optical system that includes volumes of different refractive indices, and where Kohler homogenization is performed between the two surfaces of the front top dielectric cover according to one embodiment; 
         FIG. 9  illustrates a Kohler radial homogenizer optical system that includes volumes of different refractive indices, and where Kohler radial homogenization is performed between the entry surface and the primary mirror according to one embodiment; 
         FIG. 10  illustrates a Kohler radial homogenizer optical system that includes volumes of different refractive indices, and where Kohler radial homogenization is performed between the entry surface and the secondary mirror according to one embodiment; 
         FIG. 11  illustrates a Kohler radial homogenizer optical system that includes volumes of different refractive indices, and where Kohler radial homogenization is performed between the entry surface and the exit surface according to one embodiment; 
         FIG. 12  illustrates a Kohler radial homogenizer optical system that includes volumes of different refractive indices, and where Kohler radial homogenization is performed between the inner surface of the top cover and the primary mirror according to one embodiment; 
         FIG. 13  illustrates a Kohler radial homogenizer optical system that includes volumes of different refractive indices, and where Kohler radial homogenization is performed between the inner surface of the top cover and the secondary mirror according to one embodiment; 
         FIG. 14  illustrates a Kohler radial homogenizer optical system that includes volumes of different refractive indices, and where Kohler radial homogenization is performed between the inner surface of the top cover and the exit surface according to one embodiment; 
         FIG. 15  illustrates a Kohler radial homogenizer optical system that includes volumes of different refractive indices, and where Kohler radial homogenization is performed between the primary mirror and the exit surface according to one embodiment; 
         FIG. 16  illustrates a Kohler radial homogenizer optical system that includes volumes of different refractive indices, and where Kohler radial homogenization is performed between the secondary mirror and the exit surface according to one embodiment; and 
         FIGS. 17A and 17B  illustrate the shaping of the exit surface to improve the uniformity of the system when the sun is on axis according to one embodiment. 
         FIG. 18   a  illustrates a perspective view of a specific optical system having segmented primary and secondary mirror segments.  FIG. 18   b  shows an example of relative angular transmission as a function of incidence angle for the system of  FIG. 18   a .  FIG. 18   c  illustrates a plurality of systems mounted on multiple heat sinks. 
     
    
    
     DETAILED DESCRIPTION 
     The present invention provides methods for designing optical imaging systems using homogenizers to concentrate and uniformly irradiate a target cell. 
     Optical Design 
     Kohler illumination techniques are well known in optics for producing uniform illuminance on a target. One advantageous way to implement the technique is to use a matrix representation of paraxial optics. A general on-axis Kohler concentrator in two dimensions has the form: 
     
       
         
           
             
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                       ′ 
                     
                   
                 
                 
                   
                     
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             = 
             
               
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     where n is the refractive index of material surrounding the target cell, x-θ and x′-θ′ are the space angle coordinates at the target cell and input aperture, respectively, and μ is a free parameter. One important property of this configuration is that M 2,1 =0. It then follows that the spatial image distribution of an object at infinity is simply a re-scaling of the spatial distribution on the first lens, which is uniform, with a scaling factor equal to the concentration ratio (in two dimensions in this example). From this, the simplest Kohler system can be made with two positive lenses: one first lens placed at the input aperture and a second lens located at the first lens focal plane, with the target cell placed at the image plane of the second lens when considering the first lens as an object. 
     In certain aspects, it is advantageous to rewrite this in a somewhat revised form: 
     
       
         
           
             
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     Here, n, n′ are the retractive indices at the input aperture and target cell respectively (n′ is typically 1), P 1  is the power of the first Kohler element (e.g., 1/f 1  for a positive lens) and the re-scaling parameter is redefined to resemble concentration C. 
     Two important features to this matrix include 1) M 2,1 =0 implies uniform illumination of the target, and 2) M 1,2 =P 1  implies that the second Kohler element is at the focal plane of the first. The overall minus sign is just the reversal of the image and does not change the conclusion that illumination is uniform. 
     A more general approach to the matrix is given in Appendix A of  The Optics of Nonimaging Concentrators , by W. T. Welford and R. Winston, ACADEMIC PRESS, 1978, which is hereby incorporated by reference. Here the discussion is general, in full two dimensions. The small angle (paraxial) approximation is not required. The variables are x, y, p, q where p=nL, q=nM, L, M are direction cosines. For small angles, p=nq x  and so on. The key condition M 2,1 =0 becomes 
     
       
         
           
             
               
                 ∂ 
                 
                   x 
                   ′ 
                 
               
               
                 ∂ 
                 p 
               
             
             = 
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     and its companion in the y,q variables, 
     
       
         
           
             
               
                 ∂ 
                 
                   y 
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                 ∂ 
                 q 
               
             
             = 
             0. 
           
         
       
     
     From the general discussion, we find that the matrix has unit determinant. Therefore there are three independent matrix elements. Two of these are C and 1/C, the concentration and its reciprocal. The Kohler condition, M 2,1 =0, has been shown. The only other matrix element to be determined is M 1,2 . The Kohler condition (for an object at infinity) places the second element at the focal plane of the first. Taking the special case q′ x =0, the incident plane wave becomes a spherical wave of radius 1/P 1 . It follows that M 1,2 =−P 1 . 
     To elaborate on this point, as an example, it may be useful to use a pinhole as a toy model of a lens. A pinhole is a lens of any focal length (infinite depth of field) and it&#39;s easy to analyze—just think of a pinhole camera. In this case, the Kohler system has a positive lens of power P 1  followed by a pinhole. What then, are the matrix elements? We have already seen that the diagonals are 1/C and C. Uniform illumination requires M 2,1 =0. That leaves M 1,2  to be determined. Clearly, to have any throughput the pinhole is placed at the focus of the lens, hence M 1,2 =−P 1 . 
     In certain aspects, three dimensional systems may be formed by designing a two dimensional system and making the system rotationally symmetric about a defined axis (e.g., sweeping the space curve rotationally about a central axis), or by translating the space curve along a straight line or other curve, for example, to make a trough-shaped optical device. 
     As one example, the matrix methods provide for two-surface design Kohler integrator arrays on aspheric or free-form surfaces. Examples shown below include two mirrors in an array of radial Kohler elements, allowing for ultra compact configurations with enhanced concentration-acceptance angle product and achromacity. 
     Systems and Devices 
     An optical system constructed in accordance with one embodiment of the invention is shown in  FIG. 3 . As shown, in one embodiment, a segmented secondary mirror  32  is substantially co-planar with the entrance aperture  30  of a primary mirror  31 . The entrance aperture  30  and the exit aperture  36  are substantially flat. The segments on the primary  1   a ,  2   a , etc. are essentially parabolic, with the focus of each at the associated mirror segment  1   b ,  2   b , etc. on the secondary and vertical axis. The secondary mirror segments  1   b ,  2   b , etc. are essentially elliptic with foci at the locations of the associated primary mirror segment and the target  33 . There is continuity in profile but discontinuity in slope in both the secondary and primary mirrors. The target plane of the combination of the primary mirror  31  and the secondary mirror  32  resides at the target  33 . In one aspect, the target  33  includes a solar cell. The optical system as shown in  FIG. 3  operates to receive and radially integrate incoming light from a distant source, e.g., the sun, and concentrate the light onto the target  33 . However, a target  22  may include a light source or an illumination element, in which case the optical system performs high collimation and radial integration of outgoing light. This reversibility applies to all embodiments disclosed herein. 
     In this embodiment, the edges of a given segment of the secondary mirror are designed so that their images at infinity through the associated primary segment match the design acceptance angle. On the other hand, the edges of a given segment of the primary are designed so that their images on the target through the associated secondary segment match the target size, e.g., cell size. Because the segments on the primary are uniformly illuminated, the illumination on the cell is also uniform in two dimensions. 
     The mirror segments can also be configured to optimize the global performance. For instance, the focus position of the parabola or ellipses, and the parabola axis could be considered a parameter, and a multi-parameter optimization program can be used to optimize the acceptance angle of the entire system. Alternatively, the parabola axis could be chosen to coincide with one of the edges of the acceptance angle and its focus placed at one of the edges of the associated secondary mirror, and also the ellipses can have their focus coinciding with the edges of the cell and of the associated primary mirrors. 
     An example of a three-dimensional (3D) device is rotationally symmetric, as shown in  FIG. 4 , so the segments of primary and secondary form rings or annulus structures. Because the Kohler integration is performed in the meridian cross-section, no uniformity is gained in the saggital direction. This means that if the acceptance angle is 3 times the solar disk, the local concentration is only 3 times the average. Therefore, for a 500 suns average concentration, the maximum local concentration on the cell is 1,500 (which is acceptable). 
     Regarding the local concentration produced on the secondary mirror, when the sun is centered on axis, the irradiance pattern on each segment annulus of the secondary mirror is a thin ring centered on the segment with about 100-150 suns concentration, which is also acceptable for the mirror durability. 
     If the sun is off-center, but still within the design acceptance angle, the irradiance thin ring on each segment annulus of the secondary is displaced; but it is still inside the segment. In the meridian cross-section, the sun images are thus displaced as shown in  FIG. 5 . The maximum irradiance levels on the secondary and on the cell remain unchanged. 
     Dispersion due to the variation of the refractive index of the dielectric material (i.e. glass or acrylic) used significantly limits the solar flux concentration with reasonable acceptance angle (&gt;15 mrad half angle) by a well-designed flat Fresnel lens to ≈150 suns. The angular dispersion due to refraction is: 
     
       
         
           
             
               
                 
                   
                     
                       δ 
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                       θ 
                     
                     
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                        
                       
                           
                       
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                   = 
                   
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                         δ 
                          
                         
                             
                         
                          
                         n 
                       
                       n 
                     
                   
                 
               
               
                 
                   ( 
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     where n is the relative refractive index at the interface and θ is the refracted angle. 
     For the concentrator shown in  FIG. 3 , there are two refractive surfaces: the surface defining the entry aperture  30  and the exit surface  36 . In the first refractive surface, if the refractive entrance aperture  30  is flat, the incidence angle is limited to the acceptance angle and equation (1) states (approximating the tangent function by its argument) that relative dispersion of the refraction angle is equal (in absolute value) to relative dispersion of the refractive index, which is below 1% in optical dielectrics. In the second refractive surface, the effect is even smaller since the angular acceptance at that surface is very wide (close to ±90°) and since the dispersion of the relative refractive index δn is much smaller (assuming a limited cell illumination angle (up to about ±45°) since the concentrator dielectric material and the cell encapsulant have a more close variation with wavelength than that of the dielectric and air of the first refraction). 
     Therefore, the dielectric optical system in  FIG. 3  is for practical purposes achromatic. In fact, Equation (1) indicates some flexibility in design. For instance, the dielectric/air interface (the entrance aperture  30 ) does not have to be strictly normal to the beam. A modest inclination is allowable, just as long as chromatic effects, as determined by Equation (1) are kept reasonably bounded. If the entrance surface is not flat, the rays should be traced in the design. For instance, if the segments of the primary mirror element  31  are parabolas when the entrance surface is flat, when it is non-flat, its shape should be calculated to enable the impinging parallel rays, after the refraction on the non-flat aperture, to be focused on the associated secondary. This calculation is called generalized Cartesian Oval, which in general solves the inverse problem of calculating the optical surface (reflective or refractive) that couples the rays normal to two given wavefronts. The same considerations apply to the change of the exit surface from flat. A hemispherical shape, for instance, could also be used. 
     In cases where an increase in concentration or acceptance angle is desired, it is desirable to add a final stage non-imaging concentrator as shown in  FIG. 6 . This concentrator  60  could be a θ 1 /θ 2  non-imaging concentrator where θ 1  is chosen to match the numerical aperture (NA 1 ) at the exit of the two mirror system and where sin θ 1 =NA 1 /n. The θ 2  is chosen to satisfy a subsidiary condition, such as maintaining total internal reflection (TIR) on the non-imaging concentrator sides or limiting angles of irradiance onto a multi-junction cell. The concentration or flux boost of the terminal stage approaches the fundamental limit of (sin θ 2 /sin θ 1 ) 2 . The overall concentration can approach the etendue limit of (n/sin θ 0 ) 2  where sin θ 0 =n sin θ 1 . In alternative embodiments, the non-imaging concentrator can be a known tailored non-imaging concentrator or a flow line concentrator. 
     The planar all-dielectric optical system presented here embodies inexpensive high-performance forms that are capable of (a) concentrating the solar radiation with acceptable non-uniform irradiance levels, (b) incurring negligible chromatic aberration even at ultra-high concentration, (c) passive cooling of the cell, (d) accommodating liberal optical tolerances, (e) mass production with existing glass and polymeric molding techniques, and (f) high compactness. 
       FIG. 7  illustrates a two-mirror Kohler radial homogenizer optical system that may be composed of volumes  70 ,  71 ,  72  and  73  of different refractive indices. If volume  72  is air and volumes  70  and  71  are the same dielectric material (so interface  702  does not exist), the device in  FIG. 7  reduces to that in  FIG. 3 . The optical design of the device in  FIG. 7  is done in substantially the same way as described for the design of  FIG. 3 . 
     Another embodiment considers that volume  71  is air, in which the optical losses due to absorption in that medium are eliminated. In this case, a flat cover  70  prevents dust from accumulation in the system and protects mirror elements from the external environment. If medium  73  is also air, the cell is not encapsulated, which increases the system optical efficiency since the Fresnel reflection on the interface  705  is eliminated. However, to prevent cell degradation by moisture, it is desirable to encapsulate the cell with a dielectric dense medium  73  such as silicone rubber. The prescribed surface  705  of the encapsulating dielectric material will be considered in the design as a prescribed surface through which the rays are traced. Additional surfaces may be included as desired in the system, and the application of the same design procedure of the invention is straightforward for one of skill in the art. For example, the encapsulating lens medium  73  can be made of glass or transparent plastic, and the cell coupled to it with a gel or a silicon rubber. In this case, an additional interface (without a specific optical function) will appear. 
     Instead of establishing a design of the two mirrors, any other two surfaces of the five surfaces  701 ,  702 ,  703 ,  704  and  705  in  FIG. 7  could be designed. Therefore, nine alternative families of devices are illustrated in  FIG. 8  to  FIG. 16 . One skilled in the art will realize that other like configurations can be established using combinations and variations by employing the principals of the invention. For all these cases, the design can be performed in 4 steps according to one aspect:
         (1) Prescribe three of the five surfaces,   (2) Calculate the remaining two surfaces in 2D to make the coupling of two parallel input wavefronts defined by the acceptance angle into the two spherical exit wavefronts defined by the target edges. Alternatively, the target and acceptance angle can be scaled down to converge to zero, and then the resulting two surfaces will be aplanatic (i.e., stigmatic and fulfilling the Abbe sine condition). If the calculated surfaces are not manufacturable, a new selection of the three prescribed surfaces in step (1) is performed,   (3) Select two of the five surfaces,   (4) Recalculate the two selected surfaces in (3) containing the Kohler integrator segments as disclosed above, ray tracing through the prescribed or precalculated surfaces.       

       FIG. 8  illustrates a Kohler homogenizer optical system that includes volumes  80 ,  81 ,  82  and  83  of different refractive indices. The Kohler homogenization is done between the two surfaces  801  and  802  of the front top dielectric cover  80 . In one embodiment, the medium  82  is air and the lenses in cover  80  are top-down symmetric. The Kohler homogenization can be only in the radial direction. Alternatively, the Kohler homogenization can be done in both the radial and saggital directions using with rotational symmetric lens units in the lens array displaced either in a rectangular or hexagonal pattern. This embodiment would increase further the illumination homogeneity of the lens. 
       FIG. 9  illustrates a Kohler radial homogenizer optical system that includes volumes  90 ,  91 ,  92  and  93  of different refractive indices. Kohler radial homogenization is done between the entry surface  901  and the primary mirror  903 . In this case, the focal length and pitch of the lens and mirror segment decrease from the optical axis to the rim, due to the progressively smaller separation of the Kohler integrator pairs. 
       FIG. 10  illustrates a Kohler radial homogenizer optical system that includes volumes  100 ,  101 ,  102  and  103  of different refractive indices; the Kohler radial homogenization is done between the entry surface  1001  and the secondary mirror  1004 . In one embodiment, material  101  is air and dielectric material materials  100  and  102  are equal (so interface  1002  does not exist), so that the optical system can be manufactured as a single piece. 
       FIG. 11  illustrates a Kohler radial homogenizer optical system that includes volumes  110 ,  111 ,  112  and  113  of different refractive indices. The Kohler radial homogenization is done between the entry surface  1101  and the exit surface  1105 . In one embodiment, where the material  112  is air, the lens segments of exit surface  1105  are concave. In another embodiment, where the material  112  is a dielectric and volume  113  is air, the lens segments of exit surface  1105  are convex. 
       FIG. 12  illustrates a Kohler radial homogenizer optical system that includes volumes  120 ,  121 ,  122  and  123  of different refractive indices; the Kohler radial homogenization is done between the inner surface  1202  of the top cover and the primary mirror  1203 . Also in this case, the focal length and pitch of the lens and mirror segment will decrease from the optical axis to the rim, due to the progressively smaller separation of the Kohler integrator pairs (i.e., pair-wise correlated primary and secondary mirror segments). 
       FIG. 13  illustrates a Kohler radial homogenizer optical system that includes volumes  130 ,  131 ,  132  and  133  of different refractive indices; the Kohler radial homogenization is done between the inner surface of the top cover  1302  and the secondary mirror  1304 . In one embodiment, material  1301  is air and dielectric material materials  1300  and  1302  are equal (so interface  1002  does not exist). In this embodiment, the optical system can advantageously be manufactured as a single piece. 
       FIG. 14  illustrates a Kohler radial homogenizer optical system that includes volumes  140 ,  141 ,  142  and  143  of different refractive indices; the Kohler radial homogenization is done between the inner surface of the top cover  1402  and the exit surface  1405 . In one embodiment, material  141  is air and dielectric material materials  140  and  142  are equal (so interface  1402  does not exist). In this embodiment, the optical system can advantageously be manufactured as a single piece. In one embodiment, where the material  112  is air, the lens segments of exit surface  1405  are concave. In another embodiment, where the material  142  is a dielectric and volume  143  is air, the lens segments of exit surface  1405  are convex. 
     The use of the inner surface instead of outer surface of the cover (for instance, surface  1202  in  FIG. 12  instead of surface  901  in  FIG. 9 ) to allocate the Kohler integrator segments is beneficial in cases where the system is used outdoors (as it is usually the case in photovoltaic applications), because the slope discontinuities between segments in the top surface will tend to accumulate dust and thus will need a higher cleaning maintenance. 
       FIG. 15  illustrates a Kohler radial homogenizer optical system that includes volumes  150 ,  151 ,  152  and  153  of different refractive indices; the Kohler radial homogenization is done between the primary mirror  1503  and the exit surface  1505 . In one embodiment, where the material  152  is air, the lens segments of exit surface  1505  are concave. In another embodiment, where the material  152  is a dielectric and  153  is air, the lens segments of exit surface  1505  are convex. 
       FIG. 16  illustrates a Kohler radial homogenizer optical system that includes volumes  160 ,  161 ,  162  and  163  of different refractive indices; the Kohler radial homogenization is done between the secondary mirror  1604  and the exit surface  1605 . In one embodiment, where the material  162  is air, the lens segments of exit surface  1605  are concave. In another embodiment, where the material  162  is a dielectric and  163  is air, the lens segments of exit surface  1605  are convex. 
     When the exit surface is to be selected, it is possible to shape it to improve the uniformity provided by the Kohler homogenizing device when the sun is on-axis, provided that there is a sufficient refractive index difference at both sides of the exit surface.  FIGS. 17A and 17B  illustrate the uniformity improvement of the exit surface profile by having a concavity  171  or a convexity  172  at the center when the cell side  173  has a higher or lower refractive index, respectively. This profiles cause the rays  174  going to the cell center to deflect when closer to the cell rim. 
     According to one embodiment, a cover made of, for example, glass or PMMA or other suitable material, is positioned to cover and protect the optical elements from the environment, e.g., dust or debris.  FIG. 18   a  illustrates an embodiment including a cover. It will be appreciated that any of the various embodiments discussed herein could benefit from the use of a cover, especially where the internal volume is air. 
     According to one embodiment, a heat sink is provided on which to mount one or more optical systems. For example,  FIG. 18   a  illustrates a perspective view of a specific optical system having segmented primary and secondary mirror segments as described in the various embodiments above. The specific system shown includes a primary segmented mirror element with silver coated reflecting surfaces and a secondary segmented mirror element with silver coated reflective surfaces. A glass molded dome covers a multi-junction solar cell. As shown, the optical system is mounted or attached to a heat sink. The heat sink as shown is a U-beam structure or comb structure as is well known, however other structures may be used as desired. The heat sink may also provide a platform on which to mount multiple systems. The target cell may be attached directly to the heat sink, or a heat spreader (e.g., Aluminum Nitride) may be provided to couple the heat sink with the target and enhance heat dissipation from the cell to the heat sink. 
     In certain aspects, a tracking system is provided to reposition the system(s) as needed to track the motion of the sun and maintain the light impinging on the system within a desirable acceptance angle.  FIG. 18   b  shows an example of relative angular transmission as a function of incidence angle for the system of  FIG. 18   a .  FIG. 18   c  illustrates a plurality of systems mounted on multiple heat sinks. A tracking system coupled to servos and motors allows the array of optical systems shown in  FIG. 18   c  to track the motion of the sun. 
     Any algorithms, process or functions described herein may be implemented as software code to be executed by a processor using any suitable computer language such as, for example, Java, C++ or Perl using, for example, conventional or object-oriented techniques. The software code may be stored as a series of instructions, or commands on a computer readable medium, such as a random access memory (RAM), a read only memory (ROM), a magnetic medium such as a hard-drive or a floppy disk, or an optical medium such as a CD-ROM. Any such computer readable medium may reside on or within a single computational apparatus, and may be present on or within different computational apparatuses within a system or network. 
     While the invention has been described by way of example and in terms of the specific embodiments, it is to be understood that the invention is not limited to the disclosed embodiments. To the contrary, it is intended to cover various modifications and similar arrangements as would be apparent to those skilled in the art. Therefore, the scope of the appended claims should be accorded the broadest interpretation so as to encompass all such modifications and similar arrangements.