Abstract:
Methods for maximizing a fraction of light energy absorbed in each of three classes of light concentrators (rectangular parallelepipeds, paraboloids and prisms) by choice of incident angle of radiation and of one or more geometrical or physical parameters (absorber thickness, paraboloid dimensions, location of paraboloid focus, prism angles, concentrator material, cladding, prism angles, etc.). Alternatively, the light energy absorbed plus the light energy that escapes through non-total internal reflection within the light concentrator can be minimized.

Description:
ORIGIN OF THE INVENTION 
     The invention described herein was made in the performance of work under a NASA contract and is subject to the provisions of Public Law 96-517 (35 U.S.C. §202) in which the contractor elected not to retain title. 
    
    
     FIELD OF THE INVENTION 
     This invention relates to conversion of solar energy to other useful forms of energy, using optimized solar concentration and conversion cells. 
     BACKGROUND OF THE INVENTION 
     Polymeric and inorganic semiconductors offer relatively high quantum efficiencies, as high as 80 percent in the near-infrared and ultraviolet regions, and are much less expensive to fabricate than non-amorphous silicon wafers. An optical fiber and cladding can be designed and fabricated to confine light for transport within ultraviolet and near-IT media, using evanescent waves, and to transmit visible wavelength light for direct lighting. By using polymeric and less expensive and easily processable materials for fabrication, and by designing for optimum solar energy absorption for different solar concentrator configurations, the cost effectiveness of a solar energy conversion system can be increased substantially. 
     What is needed is one or more solar energy cell conversion configurations that are optimized with respect to choices of one, two or more important parameters, such as light incidence angle, geometric parameters of the conversion material, and choice(s) of conversion materials. Preferably, the materials should have reduced cost, relative to the costs of conventional systems, and the optimal configurations should be straightforward to implement. In addition to molding and casting processes, three dimensional additive manufacturing techniques can be used to implement the desired configurations. 
     SUMMARY OF THE INVENTION 
     These needs are met by the invention, which provides several solar conversion cells (SCCs) that are individually optimized to provide maximum energy absorption EA in a selected wavelength region, by appropriate choices of light incidence angle, one or more geometric parameters of the concentrator and/or choice(s) of conversion materials. 
     In a first embodiment, a broad or narrow light beam with an incidence angle θ 1  is received and internally reflected many times within a rectangular parallelepiped of thickness h of solar conversion material, EA can be optimized with respect to the incidence angle θ 1 , the thickness h, a reflection coating of the initial light-receiving surface, and the parallelepiped material refractive index ratio n 2 /n 1 . The second reflecting surface of the parallelepiped has substantially constant internal reflection (not necessarily 100 percent) for the incidence angle and wavelength region chosen. 
     In a second embodiment, a broad light beam is received by a truncated paraboloid surface, propagating parallel to the paraboloid axis, and the light is absorbed by a sphere or cylinder (solar concentrator) of solar conversion material of radius d whose center is coincident with the paraboloid focus. The energy absorption EA is optimized with respect to the location of the focus and the concentrator diameter  2   d.    
     In a third embodiment, a light beam is received at a selected incidence angle θ 0  by one surface of a triangular prism of solar conversion material, and undergoes an unlimited number of internal reflections within the prism. The energy absorption EA can be optimized with respect to initial incidence angle θ 0 , prism angles, light reflection coefficient at the initial light-receiving surface, and material chosen for the prism. Each of the surfaces of the prism, other than the initial light-receiving surface, has a percent reflection determined with respect to an internal incidence angle θ 1  and wavelength region chosen. In each of the embodiments, most or all of the energy of the incident light is absorbed, not lost, if the body material is capable of absorbing all energy in the chosen wavelength region. 
     The solar conversion material for any of the first, second and/or third embodiments may at least one inorganic material, drawn from a group comprising bismuth oxides (Bi a O b ), bismuth iodides (Bi c I d ), lead iodide (PbI e ), cadmium sulfide (CdS), cadmium selenide (CdSe), cadmium telluride (CdTe) and lead sulfide (PbS),where a, b, c, d and e are positive numerical values. 
     The solar conversion material for any of the first, second and/or third embodiments may be at least one polymer, drawn from a group comprising poly[2-methoxy-5-(2′-ethylhexyloxy)-p-phenylenevinylene]] (MEH-PPV), olelamine, polythiophene and derivatives thereof, and poly[4,8-bis-substituted-benzo[1,2-b;4,5-b′dithophene-2,6-diyl-alt-4-substitued-thienol[3,4-b[thipphene-2,6-diyl]] (PBBTTT-CF). 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  schematically illustrates effects of passage of solar radiation having different wavelengths through an air mass adjacent to the Earth&#39;s surface. 
         FIGS. 2-5  illustrate optimum energy absorption configurations for three representative types of solar concentrators: a rectangular parallelepiped ( FIG. 2 ); a paraboloid ( FIGS. 3 and 4 ); and a prism ( FIG. 5 ). 
         FIGS. 6 and 7  illustrate a light collector mechanism that accepts and aligns light beams from a variety of directions. 
     
    
    
     DESCRIPTION OF THE INVENTION 
     A light beam, emitted from a solar energy source such as the Sun, will pass through the Earth&#39;s atmosphere at different angular orientations, depending upon the time of day or night in a diurnal cycle and upon the location on the Earth&#39;s surface relative to the source of the incident light beam (e.g., the Sun). At high noon on the Equator (incidence angle θ=0° at edge of atmosphere) in mid-Summer or mid-Winter, the light beam will pass through one “air mass” (AM=1.0), measured by a line integral of the local air density, integrated along the path of the light beam. As the incidence angle θ increases toward 90° at the horizon, the number of equivalent air masses AM increases monotonically 
       FIG. 1  schematically illustrates how components of solar energy received at an edge of the atmosphere and at the Earth&#39;s surface vary with wavelength. The higher energy (lower wavelength) components will tend to decrease faster than the lower energy fractions as air mass increases above 1.0, except for certain wavelength troughs in the atmosphere. 
     Rectangular Parallelepiped. 
       FIG. 2  illustrates a first configuration of a solar energy concentrator, a rectangular parallelepiped with a solar conversion cell (SCC) material M 2 , with associated refractive index n 2  for unpolarized light and thickness h (e.g., h=1 μm−1 mm), with a first surface S 1  that receives an incident light beam LB 0  (with a narrow wavelength band, centered at λ=λ0), propagating in a first material M 1 , having a refractive index n 1  at an initial incidence angle θ 1 . The incident light beam is refracted into the SCC at a first surface S 1  at a refraction angle θ 2 . A fraction, t 1 (θ 1 ) of the incident light beam is transmitted into the SCC with refraction angle θ 2 , propagates at an incident angle θ 2  relative to a second parallel surface S 2 , is internally reflected at the incidence angle θ 2  at the second surface S 2 , propagates at incidence angle θ 2  toward the surface S 1 , is reflected at the first surface S 1  at incidence angle θ 2 , and repeats the reflection cycle (S 1 →S 2 →S 1 ) an unlimited number of times (assuming the length of the SCC is very large or is infinite relative to a light beam cycle length  2   h ·secθ 2 . The angles θ 1  and θ 2  for unpolarized light are related by Snell&#39;s law,
 
 n 1 sin θ1= n 2 sin θ2,  (1)
 
cos θ1={1−( n 2 sin θ2/ n 1) 2 } 1/2 ,  (2)
 
 dθ 1/ dθ 2=cos θ2/{( n 1/ n 2) 2 −sin 2 θ2} 1/2 ,  (3)
 
with appropriate modifications where polarized incident light is used. Light received from the sun is largely unpolarized, and the incident light received here is assumed to be unpolarized.
 
     The transmission coefficient t 1 (θ 1 ) at the first surface S 1  and the internal reflection coefficient r 2 (θ 2 ) at the first surface S 1  or the second surface S 2  may depend upon wavelength λ(λ≈λ0) and upon θ 1  and θ 2 , and will behave according to the Fresnel reflection-transmission relations set forth in the following. A portion of the propagating light beam is absorbed with absorption coefficient α(λ) per unit length propagated within the SCC material. A fraction β(λ) (assumed≈1 here) of the absorbed or scattered light beam energy (0≦β≦1; β assumed=1 here) may converted by the SCC material into electromagnetic energy with a higher wavelength (e.g., near-infrared, λ=λ1=0.8-2.0 μm). This converted wavelength light (“CWL”) may be emitted preferentially in a transverse, longitudinal or other direction, for example at an angle ψ relative to the local direction of the path of the arriving light beam in the SCC material. The SCC has a small thickness h (e.g., 1 μm or a fraction thereof), measured in a direction perpendicular to a plane in which the light beam propagates (plane of the paper in  FIG. 2 ). 
     The incident light beam enters the SCC at the first surface S 1  with a transmission coefficient t 1 (θ 1 ) (also dependent upon wavelength λ). In a first reflection cycle (S 1 →S 2 →S 1 ), this light beam propagates from the first surface S 1  to the second parallel surface S 2  at the angle θ 2 , is reflected at S 2  with reflection coefficient r 2 (θ 2 ), returns to S 1  at the angle θ 2 , and is reflected at S 1  at the angle θ 2  as shown with reflection coefficient r 2 (θ 2 ). For a first path segment (S 1 →S 2 ): (1) the fraction of initial energy absorbed by or deposited within the SCC material M 2  for a single path segment is t 1 (θ 1 )·(1−e), where e=exp{−α h secθ 2 }); (2) the fraction of energy transmitted (lost) through the surface S 2  is t 1 (θ 1 )(1−r 2 (θ 2 ))·e; and the fraction that is reflected at the surface S 2  and propagates toward the surface S 1  is t 1 (θ 1 )e r 2 (θ 2 ). For a second segment (S 2 →S 1 ): (1) the fraction of initial energy absorbed by the SCC material is t 1 (θ 1 )·(1−e)(e r 2 ); (2) the fraction transmitted (lost) through the surface S 1  is t 1 r 2  e 2 (1−r 2 ),; and (3) the fraction that is reflected at the surface S 1  and propagates toward the surface S 2  is t 1 (θ 1 )·(r 2  e) 2 . 
       FIG. 2  also indicates (i) the fractions of energy absorbed along each path segment, S 1 →S 2  or S 2 →S 1 ; (ii) the fraction of energy that approaches each surface, S 1  or S 2 ; and (iii) the fractions of energy present after the beam is reflected at each surface, S 1  or S 2 . For an infinite number of beam reflections at each of the surfaces, S 1  and S 2 , the fraction of energy deposited within the SCC is 
     
       
         
           
             
               
                 
                   
                     
                       t 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       1 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         EA 
                         ⁡ 
                         
                           ( 
                           
                             
                               θ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                             
                             , 
                             h 
                             , 
                             λ 
                           
                           ) 
                         
                       
                     
                     } 
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           m 
                           = 
                           0 
                         
                         ∞ 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         t 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         1 
                         ⁢ 
                         
                           ( 
                           
                             1 
                             - 
                             e 
                           
                           ) 
                         
                         ⁢ 
                         
                           
                             ( 
                             
                               r 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               e 
                             
                             ) 
                           
                           m 
                         
                       
                     
                     = 
                     
                       t 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       1 
                       ⁢ 
                       
                         
                           ( 
                           
                             1 
                             - 
                             e 
                           
                           ) 
                         
                         / 
                         
                           
                             { 
                             
                               1 
                               - 
                               
                                 ( 
                                 
                                   r 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   e 
                                 
                                 ) 
                               
                             
                             ) 
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   } 
                 
               
             
           
         
       
     
     Because the incidence angle θ 2  is constant for each cycle, the surfaces S 1  and S 2  can be coated with a thin film interference coating that approximately maximizes the reflection coefficient r 2  or the coefficient (r 2  e) for a given choice of the incidence angle θ 2 . This improvement can be incorporated in the SCC and in the reflection value used for r 2  or for (r 2  e). Determination of an optimal anti-reflection coating for a given incidence angle θ 2  is discussed by H. A. MacLeod,  Thin Film Optical Filters , Institute of Physics Publishing, 2001.
 
 t 1(θ1,θ2)={(2  n 1 cos θ1)/( n 2 cos θ1+ n 1 cos θ2)} 2 ,=energy transmission coefficient at  M 1/ M 2 interface,,  (5)
 
 r 2(θ1,θ2)={( n 1 cos θ1− n 2 cos θ2)/( n 2 cos θ1+ n 1 cos θ2)} 2 ,={( n 1{1−( n 2 sin θ2/ n 1) 2 } 1/2   −n 2 cos θ2)/( n 2{1−( n 2 sin θ2/ n 1) 2 } 1/2   −n 1 cos θ2) 2 =internal reflection coefficient at  M 1/ M 2 interface  (6)
 
     For a fixed SCC material M 2 , fixed thickness h, and fixed wavelength range, the total fraction F 1  is maximized by determining an incidence angle θ 2 , satisfying Eq. (1), for which
 
 ∂F 1/∂θ2=0,  (7)
 
 ∂e/∂θ 2=−α  h  {sec θ2 tan θ2}  e,.   (8)
 
where e and r 2  depend explicitly on θ 2  and t 1  depends implicitly upon θ 2  through the Snell&#39;s law relations, (1) and (2) and the Fresnel relations, Eqs. (4) and (6). The quantity ∂r 2 /∂θ 2  is computed using Eq. (6) and the Snell&#39;s law relations.
 
     For a fixed incidence angle θ 2  and fixed wavelength range, the total fraction t 1  EA 1  is maximized with respect to thickness h by determining an SCC material thickness h for which
 
∂( t 1  EA )1/∂ h= 0;  (9A)
 
or equivalently,
 
∂( t 1  EA )/∂ e =0,  (9B)
 
Equation (11B) is satisfied where
 
1− r 2=0,  (9C)
 
which requires that r 2 =1 (unrealistic). A more realistic optimization is to require that energy absorption per unit volume, proportional to (t 1  EA)/h, be maximized. This requirement is equivalent to
 
∂( F 1/ h )/∂ h= 0,  (10)
 
which represents optimization of energy absorbed per unit SCC thickness. The fraction F 1  will increase monotonically with increasing attenuation coefficient α, but α is not a parameter that is controllable by the user.
 
Paraboloid.
 
       FIG. 3  illustrates a second configuration of a solar cell energy concentrator, a two dimensional or three-dimensional parabola, defined by y=x 2 /2k(−D≦x≦D), having a thickness Δh 1  (e.g., ≈1 μm), with a parabolic-shaped inner surface P 1 . An initial light beam LB 0 , moving parallel to a parabola central axis at a distance x from the y-axis (d≦x≦D), and with a narrow wavelength band, centered at λ≈λ0, is received at an initial incidence angle θ 1 =θ 1 (x) relative to a local portion of the inner surface P 1  (depending upon x), and is reflected as a first light beam portion LB 1  toward the parabola focus F, which has coordinates (x=0, y=k/2). This reflected first light beam LB 1  is received by and transmitted at normal incidence into a cylinder-shaped or sphere-shaped SCC of diameter  2   d , having a small thickness Δh 2  (e.g., a fraction of 1 mm) measured in a direction perpendicular to the plane of light beam propagation (plane of the paper in  FIG. 3 ). As in the configuration of  FIG. 2 , a portion or all of the propagating light beam is absorbed by the SCC material. Within the volume enclosed by the parabolic surface P 1  and external to the SCC concentrator, the material is M 1  (which may be a partial or full vacuum or air with a selected density), and the SCC is comprised of an energy absorbing material M 2  with absorption coefficient α(λ) per unit path length in the material M 2  (analogous to Eq. (2) for the rectangular parallelepiped material in the first configuration). 
     The first light beam intersects the parabola inner surface at an angle given by
 
 cotθ 1= x/k,   (11)
 
relative to the x-axis and is reflected with a reflection coefficient r 1 (θ 1 ), which may depend upon the refractive indices, n 1  and n 2 , and upon a surface coating for the surface P 1  This light beam is reflected by the surface P 1  at an angle  2 θ 1 , intersects the SCC surface P 2  at normal incidence, and passes through the parabola focus (x=0,y=k/2).
 
     The surface P 2  of the SCC defines a cylinder/or sphere of SCC material, having a diameter of length  2   d , an absorption coefficient of α(λ0) per unit path length within the interior of the paraboloid for the wavelength λ0, and an associated absorption exp(−2αd) for a light beam that passes along a diameter of the circular surface P 2 . Preferably, 2αd&gt;&gt;1 The reflection coefficient r 1  associated with reflection at normal incidence of the first light beam from the inner surface P 1  is determined as
 
 r 1(θ2=0)={( n 2− n 1)/( n 2+ n 1)} 2 ,  (12)
 
where n 1  and n 2  are the (wavelength-dependent refractive indices of the materials, M 1  and M 2 , respectively, indicated in  FIG. 3 . Optionally, it may be assumed that the material M 1  is substantially a vacuum so that n 1 =1. The reflection coefficients r 1 (θ 1 ) and r 2 (θ 2 ) are determined using the Fresnel relations in Eqs. (5) and (6), The reflection coefficients r 1  and r 2  can be modified, for a given wavelength λ0, by providing a reflective or anti-reflective coating on the corresponding reflecting surface, P 1  or P 2 ,
 
     A second light beam portion LB 2  moves parallel to the parabola axis, is displaced from the parabola axis by a distance x (−d≦x≦d, with d&lt;D), and is not intercepted by or reflected by the parabola inner surface P 1 . This second light beam LB 2  encounters the circular body at an incidence angle
 
3=sin −1 (| x|/d )  (13)
 
which is generally non-perpendicular, as illustrated in  FIG. 4 , which illustrates LB 1  and LB 2  relative to the SCC. The corresponding refraction angle θ 3  within the SCC (across surface P 2 ) is determined from Snell&#39;s law
 
 n 1 sin θ2= n 2 sin θ3,  (14)
 
for the M 1 /M 2  interface. The path length  2   d ′ within the SCC for the refracted portion of the second component of the light beam LB 2  is
 
2 d′= 2 d  cos θ3,  (15)
 
and the attenuated fraction of the refracted component of the second light beam LB 2 t is
 
 f ( d′ )−1−exp{−2α d′}= 1−exp{−2α d′  cos θ}.  (16)
 
     Assuming that the parabolic surface P 1  has a lateral extent of 2D (−D≦x≦D), the first components of the light beam, with −D≦x&lt;−d and/or d&lt;x≦D, will be received by the circular surface P 2  at normal incidence, for maximum transmission into the SCC material; and the second components of the light beam, with −d≦x≦d, will be received directly by the circular surface P 2  at an incidence angle θ 2 =sin −1 (|x|/d) with reduced transmission into the SCC material). 
     For first light beam components, received at P 2  at normal incidence, (θ 2 =0°), an uncoated surface will provide a transmission coefficient
 
 t 1(θ2=0)={(2  n 1  n 2)/( n 2+ n 1)} 2 .  (17)
 
The refraction angle for the first light beam components is θ 2 =0; and each refracted first light beam component moves radially across the SCC concentrator to an antipodal point on the surface, P 2  with path distance  2   d . For a second light beam component received at non-normal incidence (θ 2 =sin −1 (|x|/d)&gt;0), the refracted second light beam component moves non-radially across the cylinder. However, because the relevant refractive indices n 1  and n 2  satisfy n 1  (=1)&lt;n 2 , the refracted angle θ 4  will be much less than the incidence angle θ 3 , and the refracted light beam path will be closer to a radial path within the SCC. A reasonable approximation here is that a second light beam component, corresponding to −d≦x≦d, also moves across the SCC with total length  2   d . With this perspective adopted, the fraction of refracted light beam energy deposited within the SCC by any first or second light beam component is
 
 f ( d )=1−exp{−2 α  d},   (18)
 
A first light beam component (reflected from the parabola surface) has a total fraction of light beam energy deposited within the SCC of
 
 EA 1(θ1; d,k )= r 1(θ1) r 2(θ2=0){1−exp{−2 α  d }}( d≦x≦D )  (19-1)
 
θ1=sin −1 (| x|/k ),  (19-2)
 
A second light beam component (received directly by the cylinder P 2 ) has a total fraction of light beam energy deposited within the SCC of
 
 EA 2(θ3, d )= r 3(θ3){1−exp{−2 α  d }}(0≦ x≦d ),  (20-1)
 
θ3=tan −1 ( x/d ).  (20-2)
 
     Each of the fractions EA 1  and EA 2  should be doubled to take account of the reflections and refractions that occur for the region −D≦x≦0, but this will have no effect on the optimization procedure. The fraction EA=EA 1 +EA 2  for the paraboloid configuration shown in  FIGS. 3 and 4  is optimized with respect to a choice of the radius d, and/or the focus parameter k,:
 
(∂/∂ d ){∫ 0   d   F 22(θ3; d ) dx+∫   d   D   F 21(θ1; d ) dx}= 0,  (21-1)
 
(∂/∂ k ){∫ 0   d   F 22(θ3; d ) dx+∫   d   D   F 21(θ1; d ) dx}= 0,  (21-2)
 
where Eqs. (1) and (2) are to be used for sin θ 1  and cos θ 1 , respectively. The energy absorbed will increase monotonically with increasing attenuation coefficient α, but α is not a parameter that is controllable by the user.
 
Prism.
 
       FIG. 5  illustrates a triangular prism Pr that receives an incident light beam LB 0  at a location (x,y)=(x0,0) on a prism surface S 1  with incidence angle θ 0 . A portion of the light beam t(θ 1 ) is transmitted and refracted into the prism material with refraction angle θ 1 , and the refracted light beam is internally reflected without end from the three prism surfaces, S 1 , S 2  and S 3 , as indicated in  FIG. 5 . The triangular prism is defined, in part, by the three prism angles, ψ 1 , ψ 2  and ψ 3  (assumed to be fixed), which satisfy
 
Ψ1+ψ2+ψ3=π.  (22)
 
       FIG. 5  also illustrates an ordered sequence of reflection angles, θ 1 , θ 1 , θ 3 , θ 4 , θ 5 , θ 6 , θ 7 , etc. of internal reflections from the prism surfaces. The incident light beam encounters the first prism surface S 1  at initial incident angle θ 0  and is transmitted into the prism and refracted at an initial refraction angle θ 1 . The once-refracted light beam LB 1  approaches the second prism surface S 2  at an incident angle θ 2 , where
 
ψ1+(π/2−θ1)+(π/2−θ2}=π,  (23-1)
 
θ2=ψ1−θ1,  (23-2)
 
The once-refracted, once-internally reflected light beam LB 2  approaches the third prism surface S 3  at an incident angle θ 3 , where
 
(π/2−θ2)+ψ2+(π/2−θ3)=π.  (24-1)
 
θ3=ψ2 θ2=ψ2 ψ1+θ1,  (24-2)
 
The twice-internally reflected light beam LB 3  approaches the first prism surface S 1  at an incident angle θ 4 , where
 
(π/2−θ3)+ψ3+(π/2+θ4)=π.  (25-1)
 
θ4=−ψ3−θ3=−ψ3−ψ2+ψ1−θ1,  (25-2)
 
     The next three angles of reflection, illustrated in  FIG. 5 , are determined by the relations
 
θ5=−ψ3−θ4=−ψ2−ψ1+ψ1+θ1,  (26)
 
θ6=−ψ3−θ5=−ψ2−ψ1+θ1,  (27)
 
θ7=−ψ3−θ6=θ1,  (28)
 
From this point on, it appears that the sequence of reflection angles repeats for general prisms.
 
     A shorter pattern of reflection angles may also develop. The incidence angle (−)θ 4  in Eq. (25-2) is equal to the initial refracted angle θ 1  if and only if
 
Ψ1−ψ2+ψ3=0,  (29)
 
θ4=(−)θ1.  (30)
 
Equations (22) and (2-25) require that
 
ψ2=π/2,  (31)
 
ψ1+ψ3=ψ2=π/2.  (32)
 
in order that the light beam execute cycles within the prism with a sequence of repeating incidence angle triples, {θ 1 , θ 2 , θ 3 , θ 1 , θ 2 , θ 3 , etc.} Note that the locations on the prism surfaces where the incident light beams are reflected will normally not repeat. However, the reflection angle triples {θ 1 , θ 2 , θ 3 } do repeat and are therefore predictable and determinable from θ 1 , or from θ 2  or from θ 3 . A surface coating can therefore be determined for each of the prism surfaces, based on the respective reflection angle θ 1 , θ 2  or θ 3  for that surface, that maximizes the reflection coefficient and minimizes transmission coefficient at an M 1 /M 2  interface. The choice of prism angles Ψ 1 , ψ 2  and ψ 3  in Eq. (32) may lead to some simplifications in the subsequent analysis. However, in order to keep the scope as general as possible, general prism angles, Ψ 1 , ψ 2  and ψ 3  will be assumed here, including but not limited to the special constraint set forth in Eq. (32).
 
     For a first half cycle of the light beam, involving the reflection angle θ 1 , θ 2 , and θ 3 , (S 1 →S 2 →S 3 ), the fraction of energy EA of initial light beam energy deposited in the material M 2  is a sum of energy increments deposited along the individual path segments and can be expressed as a fraction of the light beam energy originally incident on the first surface S 1 :
 
 EA (1,2,3)= t 1{(1− e 12)+ e 12 r 2(1− e 23)}+ e 12  r 2  e 23  r 3(1− e 31),  (33)
 
 e 12=exp(−β(λ)Δ s 12),  (34-1)
 
 e 23=exp(−β(λ)Δ s 23),  (34-2)
 
 e 31=exp(−β(λ)Δ s 31),  (34-3)
 
where Δsij is the path length between the path intersections with the prism surface Si and the prism surface Sj for this particular light beam path. For each cycle, the path length increments Δsij will vary. However, the sum of the cycle path length increments, Δs 12 +Δs 23 +Δs 31 , may be approximated by a path length value L(avg)=(Δs 12 +Δs 23 +Δs 31 )(avg) that may not vary strongly from one cycle to the next cycle.
 
     For the combined first half and second half of the general path cycle, involving the reflection angle θ 1 , θ 2 , θ 3 , θ 4 , θ 5 , θ 6  and θ 7 , (S 1 →S 2 →S 3 →S 2 →S 1 ), the fraction of energy deposited along the combined first and second halves of this cycle is
 
 t 1  EA (1,2,3,4,5,6)= t 1{(1− e 12)+ e 12  r 2(1− e 23)}+ e 12  r 2  e 23  r 3(1− e 34)+ e 12  r 2  e 23  r 3  e 34  r 4(1− e 45)+ e 12  r 2  e 23  r 3  e 34  r 4  e 45  r 5(1− e 56)++ e 12  r 2  e 23  r 3  e 34  r 4  e 45  r 5  e 56  r 6(1− e 67)}  (35)
 
 e 34=exp(−β(λ)Δ s 34),  (36-1)
 
 e 45=exp(−β(λ)Δ s 45),  (36-2)
 
 e 56=exp(−β(λ)Δ s 56),  (36-3)
 
 e 67=exp(−β(λ)Δ s 67),  (36-4)
 
The fraction of the energy present at the beginning of a second path cycle (first half plus second half) is
 
 t 1  E 0= t 1{ e 12  r 2  e 23  r 3  e 34  r 4  e 45  r 5  e 56 6  e 67  r 7,  (37)
 
and this factor is multiplied by the factor EA(1,2,3,4,5,6) to determine the fraction of energy deposited in the material M 2  by the multiply-reflected light beam during its second full cycle. More generally, for the kth full cycle of the light beam (k≧2), the fraction of energy deposited is the factor t 1 E 0 , multiplied by {EA(1,2,3,4,5,6)} k−1 . Collecting the fraction of energy deposited in the material M 2  for all cycles, this fraction can be expressed as
 
Δ E−t 1  E 0/{1− EA (1,2,3,4,5,6)}  (38)
 
For fixed prism angles, Ψ 1 , ψ 2  and ψ 3  the quantity ΔE is maximized by choice of the initial refraction angle q 1 :
 
 dΔE )( dθ 1)={ d ( t 1)/ dθ 1)= t 1  E 0/{1− EA (1,2,3,4,5,6)}+ t 1( d/dθ 1){ E 0/{1− EA (1,2,3,4,5,6)}}=0,  (39)
 
where Eq. (3) is used to convert d/dθ 0  to d/dθ 1  for the factor t 1 .
 
     The quantities e 12 , e 23 , e 34 , e 45 , e 56  and e 67  would normally vary individually from one cycle to the next, although the reflection coefficients r, 2  r 3 , r 4 , r 5 , r 6  and r 7  will not vary from one cycle to the next cycle, For purposes of the estimate of cumulative light beam energy deposited in the material M 2  over all cycles, the quantities e 12 , e 23 , e 34 , e 45 , e 56  and e 67  are replaced by representative values e 12 (avg), e 23 (avg), e 34 (rep), e 45 (avg), e 56 (avg) and e 67 (avg), respectively. 
     Light Collector. 
     The solar cell concentrators discussed and illustrated in  FIGS. 2-5  implicitly assume that the incident angle θ for the incident light beam is constant, or nearly so. This can be difficult to achieve where the light beam source moves substantially in the ambient medium. However, most useful light from a solar source is received within a cone having a cone half angle of about 35°. 
     One method of partially controlling the incident angle is illustrated in one embodiment of a light collector mechanism (LCM) module  60  in  FIGS. 6 and 7  The LCM module  60  comprises a relatively transparent optical module  61  that is shaped as half a circle or as half a sphere of diameter  2   a  and has a central axis  62 . The optical module  61  has a planar surface  63  that is contiguous to a cylindrically shaped optical fiber  64  having a planar end  65 . The optical module  61  and the optical fiber  64  have respective refractive indices n 1 (λ) and n 2 (λ), where n 1 (λ) at a representative wavelength λ0 is at least as large as, n 2 (λ0). 
     An incident light beam LB 0  is received at a location  66  on the surface of the optical module  61 , and the light beam incident path defines an incidence angle θ 0  with respect to a vector N( 67 ) normal to the surface of the optical module  60  at the location  66 , which has location coordinates (x( 66 ),y( 66 )). The normal vector N( 67 ) defines an angle, Φ=Φ(max), relative to the planes  63  and  65  so that the location coordinates (x( 66 ),y( 66 ))=(a cos Φ,a sin Φ) relative to the origin O at which the central axis  62  intersects the planes  63  and  65 . 
     A portion of the incident light beam LB 0  is refracted into the interior of the optical module  61  at a refraction angle θ 1  relative to the normal vector N( 67 ). The once-refracted light beam LBR 1  propagates within the optical module  61  and the associated light beam path intersects the planes  63  and  65  at a location  68  having location coordinates (x( 68 ),y( 68 ))=(a cos Φ−Δs cos(Φ+θ 1 ), 0), where Δs sin(Φ+θ 1 )=a sin Φ. 
     The once-refracted light beam LBR 1  approaches the planes  63  and  65  at an incident angle θ 2 , where θ 1 +θ 2 +Φ=π, and is refracted at the planes  63  and  65  to a refraction angle θ 3  as a twice-refracted light beam LBR 2 , relative to a vector N( 69 ) that is normal to these planes. The light beam LBR 2  propagates within the optical fiber  64  and intersects a surface of the fiber at an incident angle θ 4 =π/2−θ 3  at a location  70 . 
     Total internal reflection (TIR) occurs at the location  70  if n 2  sin θ 4 ≧n 0 . Total internal reflection is desirable here in order to deliver, to the SCC, shown in  FIG. 7 , the energy contained in the twice-refracted light beam LBR 2  received at the location  70 . By using TIR and modest bending of the optical fiber  64 , the resulting light beam can be delivered to an SCC (rectangular parallelepiped, paraboloid, prism or any other SCC with optimized incidence angle θ) with the appropriate incidence angle. From the preceding discussion, the following constraints are applicable to the light beam path(s) in  FIG. 6 , in order to achieve TIR at the surface location  80 .
 
 n 0 sin θ0= n 1 sin θ1,  (40)
 
( x (76), y (76))=( a  cos Φ, a  sin Φ),  (41)
 
( x (78), y (78))=( a  cos Φ−Δ s  cos(Φ+θ1), 0)=(( a  cos Φ− a  sin Φ cot(Φ−θ1), 0)  (42)
 
Δ s  sin(Φ−θ1)= a  sin Φ,  (43)
 
θ1+θ2+π/2+Φ=π,  (44)
 
 n 1 sin θ2 =n 2 sin θ3,  (45)
 
θ4=π/2−θ3,  (46)
 
 a  cos Φ− a  sin Φ cot(Φ−θ1)≧− a,   (47)
 
 n 2 sin θ4≧ n 0,  (48)
 
     The optical module  71  need not be a full half circle or a full hemisphere but should have a planar lower surface that is contiguous to a planar upper surface of the optical fiber  64 . Providing an optical module  71  that is less than half a circle or less than half a hemisphere will effectively mask a portion of the light source (e.g., solar). This is acceptable if the mask, located at the horizon, has an angular value no more than about 10°-20°. 
     Where an optical module parameters satisfy Eqs. (40)-(48), a remainder of the received solar light beam will be received at or close to a selected (optimum) incident angle. Variation of the actual incident angle from the optimum incident angle will vary with the angular difference θ 4  at  80 . 
     Ideally, the refractive indices n 1  and n 2  are the same (θ 2 =θ 3 ) and are relatively high, and the materials are reasonably transparent for the wavelength λ0 of interest. Impure flint glass has a refractive index in a range n=1.52-1.92. Titanium dioxide, diamond and strontium titanate have refractive indices in a range n=2.41-2.49. Sapphire has a refractive index in a range 1.76-1.78. Most other materials of interest have refractive indices no higher than about 1.6.