Abstract:
Provided is a fly eye lens capable of preventing the generation of steps on the boundaries between cells. The fly eye lens includes a plurality of cells arranged on first plane α. The plurality of cells includes a pair of cells adjacent to each other, which have spherical centers on second plane β parallel to first plane a and parts of spherical surfaces whose spherical radius differ, from each other, those parts being surfaces. The pair of cells satisfies the relationship of R I   2 −L I   2 =R O   2 −L O   2 , where R I  is a spherical radius of the surface of one of the cells, R O  is a spherical radius of the surface of the other cell, L I  is a distance between the spherical center of the surface of one of the cells and a boundary surface between the pair of cells, and L O  is a distance between the spherical center of the surface of the other cell and the boundary surface between the pair of cells.

Description:
BACKGROUND ART 
       [0001]    A certain projection display device that projects an image to a screen or the like includes a fly eye lens configured to uniformize the illuminance distribution of light emitted from a light source. The fly eye lens includes a plurality of rectangular lenses (hereinafter, “cells”) arranged in a matrix. Each cell is, for example, a plano-convex lens that has a front surface formed to be part of a spherical surface and a back surface formed to be flat. 
         [0002]    The projection display device generally includes two fly eye lenses, which makes a pair. One of the two fly eye lenses that is close to the light source is referred to as a first fly eye lens, and the other that is far from the light source is referred to as a second fly eye lens. The first fly eye lens and the second fly eye lens have cells corresponding to each other. In other words, the light emitted from the light source is transmitted through each cell of the first fly eye lens, and then enters to the corresponding cell of the second fly eye lens. 
         [0003]      FIG. 1  shows the surface of a general fly eye lens. This fly eye lens has seventy six cells arranged in ten vertical columns and eight rows. In this case, the row direction of each cell is a horizontal direction, while the column direction of each cell is a vertical direction. 
         [0004]    In  FIG. 1 , the position of the center of a spherical surface (hereinafter, “spherical center”) defining the shape of the surface of each cell is indicated by a mark x. In this fly eye lens, the spherical center is located on an axis (hereinafter, “cell center axis”) that passes through the intersection point of the two diagonal lines of the cell and that is vertical to the cell arrangement plane. In other words, the surface of each cell has a vertex on the cell center axis and a plane-symmetrical shape in the horizontal direction and the vertical direction. 
         [0005]    In the projection display device, it is desirable to reduce light loss so that a bright image can be displayed. Thus, in order to ensure that as much light as possible that is transmitted through the first fly eye lens enters the second eye lens, each cell of the second eye lens may be formed larger than the corresponding cell of the first eye lens. In such a projection display device, the light that is transmitted through each cell of the first fly eye lens must accurately enter the corresponding cell of the second fly eye lens. 
         [0006]    This necessitates changing the traveling direction of the light that enters each cell of the first fly eye lens to the direction of the corresponding cell of the second fly eye lens. This configuration can be achieved by making each cell eccentric. In this case, eccentricity means shifting the position of the spherical center of the surface of each cell of the first fly eye lens from the cell center axis of each cell. 
         [0007]      FIGS. 2 and 3  show the surface of the first fly eye lens where each cell is made eccentric. Hereinafter, the amount of shifting of the position of the spherical center of the surface of each cell from the cell center axis of each cell is referred to as an “eccentric amount”. Each cell of the second fly eye lens that makes a pair with the first fly eye lens shown in  FIG. 2  is larger in the horizontal direction than the corresponding cell of the first fly eye lens. Each cell of the second fly eye lens that makes a pair with the first fly eye lens shown in  FIG. 3  is larger in the horizontal direction and the vertical direction than the corresponding cell of the first fly eye lens. 
         [0008]    Each cell of the first fly eye lens shown in  FIG. 2  is made eccentric from the inside to the outside in the horizontal direction. The eccentric amount of each cell of this first fly eye lens becomes gradually larger from the cell closest to the cell center axis to the outside in the horizontal direction. The eccentric amounts of the cells in the same column are equal to each other. The eccentric amount of each cell is determined according to the size or the like of the second fly eye lens. 
         [0009]    In this first fly eye lens, since each cell is made eccentric from the inside to the outside in the horizontal direction, the traveling direction of the light that enters to each cell changes outward in the horizontal direction according to the eccentric amount of each cell. Thus, the light transmitted through each cell of the first fly eye lens is output to the corresponding cell of the second fly eye lens. As a result, light output from each cell of the first fly eye lens accurately enters the corresponding cell of the second fly eye lens. 
         [0010]    Each cell of the first fly eye lens shown in  FIG. 3  is made eccentric from the inside to the outside in the horizontal direction and the vertical direction. The eccentric amount of each cell of this first fly eye lens becomes gradually larger from the cell closest to the cell center axis to the outside. The eccentric amounts in the vertical direction of the cells in the same row are equal to each other, and the eccentric amounts in the horizontal direction of the cells in the same column are equal to each other. The eccentric amount of each cell is determined according to the size or the like of the second fly eye lens. 
         [0011]    In this first fly eye lens, since each cell is made eccentric from the inside to the outside in the horizontal direction and the vertical direction, the traveling direction of the light that enters each cell changes outward in the horizontal direction and the vertical direction according to the eccentric amount of each cell. Thus, light transmitted through each cell of the first fly eye lens is output to the corresponding cell of the second fly eye lens. As a result, light output from each cell of the first fly eye lens accurately enters the corresponding cell of the second fly eye lens. 
         [0012]    Patent Literature 1 and Patent Literature 2 describe technologies of making each cell of the fly eye lens eccentric. 
       CITATION LIST 
     Patent Literature 
       [0013]    Patent Literature 1: JP2000-140261A 
         [0014]    Patent Literature 2: JP10-115870A 
       SUMMARY OF INVENTION 
     Problems to be Solved by Invention 
       [0015]      FIG. 4  is a sectional view cut along the line A-A′ of the fly eye lens shown in  FIG. 1 .  FIG. 5  is a sectional view cut along the line B-B′ of the fly eye lens shown in  FIG. 2 . 
         [0016]    In the fly eye lens shown in  FIG. 1  where each cell is not eccentric, no step is generated on the boundary between the cells as shown in  FIG. 4 . However, in the fly eye lens shown in  FIG. 2  where each cell is eccentric, a step is generated on the boundary between the cells as shown in  FIG. 5 . 
         [0017]    The fly eye lens is generally made of glass. To form the fly eye lens, a dedicated mold based on its shape is prepared. The mold is generally prepared by cutting. 
         [0018]    Thus, in the mold for the fly eye lens, it may be difficult to form a part that corresponds to the step on the boundary between the cells into an accurate shape. In such a case, the boundary between the cells of the fly eye lens is not formed into a shape as designed. This causes shape defects to easily occur on the boundary between the cells of the fly eye lens where the step is generated on the boundary between the cells. 
         [0019]    In the projection display device that uses the first fly eye lens in which each cell has a shape defect, the light emitted from the light source is neither normally transmitted through the defective part nor does the light enter the corresponding cell of the second fly eye lens. As a result, in an image projected by the projection display device, the defective part of the first fly eye lens appears as a shadow. 
         [0020]    Thus, the outer edge of the image projected by the projection display device that uses the first fly eye lens that has the step on the boundary between the cells is likely to become dark. 
         [0021]    In the fly eye lens shown in  FIG. 3 , steps are generated not only on the boundary between the cells adjacent to each other in the horizontal direction but also on the boundary between the cells adjacent to each other in the vertical direction, and the steps on the boundaries between the cells are larger than those in the case of the fly eye lens shown in  FIG. 2 . As the step on the boundary between the cells is larger, the shape defect on the boundary between the cells is larger, thus enlarging an area where the outer edge of the mage projected by the projection display device is dark. 
       Solution to Problems 
       [0022]    The present invention provides a fly eye lens that includes a plurality of cells arranged on a first plane. The plurality of cells includes a pair of cells adjacent to each other, which have spherical centers on a second plane parallel to the first plane and parts of spherical surfaces different from each other in spherical radius as surfaces. The pair of cells satisfies the following relationship, where R I  is a spherical radius of the surface of one of the cells, R O  is a spherical radius of the surface of the other cell, L I  is a distance between the spherical center of the surface of one of the cells and a boundary surface between the pair of cells, and L O  is a distance between the spherical center of the surface of the other cell and the boundary surface. 
         [0000]        R   O =√{square root over ( R   I   2   −L   I   2   +L   O   2 )}  [Expression 1]
 
     
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         [0023]      FIG. 1  shows the surface of a general fly eye lens. 
           [0024]      FIG. 2  shows the surface of a general fly eye lens. 
           [0025]      FIG. 3  shows the surface of a general fly eye lens. 
           [0026]      FIG. 4  is a sectional view cut along the line A-A′ of the fly eye lens shown in  FIG. 1 . 
           [0027]      FIG. 5  is a sectional view cut along the line B-B′ of the fly eye lens shown in  FIG. 2 . 
           [0028]      FIG. 6  shows the surface of a fly eye lens according to the first embodiment of the present invention. 
           [0029]      FIG. 7  is an enlarged view of a portion surrounded with a dashed line shown in  FIG. 6 . 
           [0030]      FIG. 8A  is a sectional view cut along the line D-D′ shown in  FIG. 7 . 
           [0031]      FIG. 8B  is a sectional view cut along the line E-E′ shown in  FIG. 7 . 
           [0032]      FIG. 9  is a partially enlarged view of  FIG. 8A . 
           [0033]      FIG. 10A  shows a step between cells adjacent to each other in a horizontal direction in the fly eye lens shown in  FIG. 7 . 
           [0034]      FIG. 10B  shows a step between cells adjacent to each other in a vertical direction in the fly eye lens shown in  FIG. 7 . 
           [0035]      FIG. 11  shows the surface of a fly eye lens according to Comparative Example 1. 
           [0036]      FIG. 12A  is a sectional view cut along the line F-F′ shown in  FIG. 11 . 
           [0037]      FIG. 12B  is a sectional view cut along the line G-G′ shown in  FIG. 11 . 
           [0038]      FIG. 13A  shows a step between cells adjacent to each other in a horizontal direction in the fly eye lens shown in  FIG. 11 . 
           [0039]      FIG. 13B  shows a step between cells adjacent to each other in a vertical direction in the fly eye lens shown in  FIG. 11 . 
           [0040]      FIG. 14  shows the surface of a fly eye lens according to the second embodiment of the present invention. 
           [0041]      FIG. 15A  is a sectional view cut along the line H-H′ shown in  FIG. 14 . 
           [0042]      FIG. 15B  is a sectional view cut along the line I-I′ shown in  FIG. 14 . 
           [0043]      FIG. 16  is a partially enlarged view of  FIG. 15A . 
           [0044]      FIG. 17A  shows a step between cells adjacent to each other in a horizontal direction in the fly eye lens shown in  FIG. 14 . 
           [0045]      FIG. 17B  shows a step between cells adjacent to each other in a vertical direction in the fly eye lens shown in  FIG. 14 . 
           [0046]      FIG. 18  shows the surface of a fly eye lens according to Comparative Example 2. 
           [0047]      FIG. 19A  is a sectional view cut along the line J-J′ shown in  FIG. 18 . 
           [0048]      FIG. 19B  is a sectional view cut along the line K-K′ shown in  FIG. 18 . 
           [0049]      FIG. 20A  shows a step between cells adjacent to each other in a horizontal direction in the fly eye lens shown in  FIG. 18 . 
           [0050]      FIG. 20B  shows a step between cells adjacent to each other in a vertical direction in the fly eye lens shown in  FIG. 18 . 
           [0051]      FIG. 21  shows the surface of a fly eye lens according to Comparative Example 3. 
           [0052]      FIG. 22A  shows a step between cells adjacent to each other in a horizontal direction in the fly eye lens shown in  FIG. 21 . 
           [0053]      FIG. 22B  shows a step between cells adjacent to each other in a vertical direction in the fly eye lens shown in  FIG. 21 . 
           [0054]      FIG. 23  schematically shows the configuration of a projection display device according to the third embodiment of the present invention. 
       
    
    
     DESCRIPTION OF EMBODIMENTS 
     First Embodiment 
       [0055]      FIG. 6  shows the surface of a fly eye lens according to the first embodiment of the present invention. This fly eye lens is formed to be symmetrical in a horizontal direction and a vertical direction. Thus, in this embodiment, an upper right portion surrounded with a dashed line corresponding to ¼ of the fly eye lens shown in  FIG. 6  is described. 
         [0056]      FIG. 7  is an enlarged view of the portion surrounded with the dashed line in the fly eye lens of this embodiment shown in  FIG. 6 . As shown in  FIG. 7 , numerals of  1  to  5  are allocated to the cell rows of the fly eye lens in order from the inside to the outside in the vertical direction. Numerals of  1  to  4  are allocated to the cell columns of the fly eye lens in order from the inside to the outside in the horizontal direction. The cell of an on row and an n column is represented by Cmn. 
         [0057]    Specifically, a center cell in the horizontal direction and the vertical direction in the fly eye lens is C 11 . Cells in the same first row as that of cell C 11  are C 12 , C 13 , and C 14  in order from the C 11  to the outside of the horizontal direction. Second to fifth rows are similar to the first row. Cells in the same first column as that of cell C 11  are C 21 , C 31 , C 41 , and C 51  in order from the C 11  to the outside of the vertical direction. Second to fourth columns are similar to the first row. 
         [0058]    Each cell of the fly eye lens is made eccentric outside in the horizontal direction. The eccentric amounts of the cells in the same column are equal to one another, and the eccentric amounts are larger in the cells of the outside column. Each cell is not made eccentric in the vertical direction. 
         [0059]      FIG. 8A  is a sectional view cut along the line D-D′ shown in  FIG. 7 .  FIG. 8B  is a sectional view cut along the line E-E′ shown in  FIG. 7 . The spherical centers O of the surfaces of the cells of both cases are located on plane β parallel to cell arrangement plane a. 
         [0060]    As shown in  FIG. 8A , in the fly eye lens, the radius R of a spherical surface (hereinafter, “spherical radius R”) defining the shape of the surface of the cell is larger in the cell of the outside column. As shown in  FIG. 8B , the spherical radiuses of cells in the same column are equal to each other. 
         [0061]    In the fly eye lens according to this embodiment, generation of steps on the boundaries between the cells is prevented by determining the spherical radius R of the surface of each cell by a method described below. 
         [0062]    According to this embodiment, first, the spherical radius R of the surface of cell C 11  and the eccentric amounts of the cells in each column are determined according to the size or the like of a second fly eye lens that makes a pair with this fly eye lens. Specifically, the spherical radius R of the surface of cell C 11  is determined according to the focal length of each cell, and the eccentric amounts of the cells in each column are determined according to the changing amount of the traveling direction of light entered to each cell. 
         [0063]    The spherical radius of the surface of cell Cmn is represented by R mn , and the spherical center of the surface of cell Cmn is represented by O mn . A distance from the spherical center O mn  of the surface of cell Cmn to a plane that becomes a boundary between cell Cmn and cell Cm(n+1) is represented by L mn . A distance from the spherical center O mn  of the surface of cell Cmn to a plane that becomes a boundary between cell Cmn and cell Cm(n−1) is represented by L mn ′. 
         [0064]    Referring to  FIG. 9 , a method for calculating the spherical radius R 12  of the surface of cell C 12  is described.  FIG. 9  is a partially enlarged view of  FIG. 8A . A surface that becomes a boundary surface between cells C 11  and C 12  is set as a boundary surface y. An intersection point on this section between the boundary surface y and the surface of the fly eye lens is set as a point a, and an intersection point between the boundary surface y and line segment O 11  O 12  is set as a point b. A line segment ab has a length x. 
         [0065]    Thus, L 11  is a length of the line segment O 11 b, and L 12 ′ is a length of the line segment bO 12 . Since the positions of O 11  and O 12  are determined based on the eccentric amounts of cells C 11  and C 12 , values of L 1 , and L 12 ′ are calculated from the eccentric amounts of cells C II and C 12 . 
         [0066]      FIG. 9  shows two right triangles abO 11  and abO 12 . When the Pythagorean theorem is applied to these right triangles, the following two expressions are established. 
         [0000]    
       
      
       R 
       11 
       2 
       −L 
       11 
       2 
       =x 
       2  
      
     
         [0000]        R   12   2   −L   12 ′ 2   =x   2  
 
         [0067]    The right-hand sides of these two expressions are both x 2 , while the left-hand sides of the two expressions are equal. The following expression can accordingly be acquired. 
         [0000]        R   11   2   −L   11   2   =R   12   2   −L   12 ′ 2  
 
         [0068]    When this expression is modified, R 12  is represented by the following expression. 
         [0000]        R   12 =√{square root over ( R   11   2   −L   11   2   +L   12 ′ 2 )}  [Expression 2]
 
         [0069]    Similarly, the spherical surface R 13  of cell C 13  and the spherical surface R 14  of cell C 14  are sequentially calculated. 
         [0000]        R   13 =√{square root over ( R   12   2   −L   12   2   +L   13 ′ 2 )}
 
         [0000]        R   14 =√{square root over ( R   13   2   −L   13   2   +L   14 ′ 2 )}  [Expression 3]
 
         [0070]    The spherical radiuses R 21  to R 51  of the surfaces of cells C 21  to C 51  in the same first column as that of cell C 11  are equal to the spherical surface R 11  of the surface of cell C 11 , and the spherical radiuses R 22  to R 52  of the surfaces of cells C 22  to C 52  in the same second column as that of cell C 12  are equal to the spherical surface R 12  of the surface of cell C 12 . The spherical radiuses R 23  to R 53  of the surfaces of cells C 23  to C 53  in the same third column as that of cell C 13  are equal to the spherical surface R 13  of the surface of cell C 13 , and the spherical radiuses R 24  to R 44  of the surfaces of cells C 24  to C 44  in the same fourth column as that of cell C 14  are equal to the spherical surface R 14  of the surface of cell C 14 . 
         [0071]    Thus, when the spherical radius R 11  of the surface of cell C 11  and the eccentric amount of each cell are determined, the spherical radiuses R of the surfaces of all the cells are determined. 
         [0072]    In short, in the fly eye lens according to this embodiment, a given pair of cells adjacent to each other in the horizontal direction satisfies the relationship of the following formula (1). 
         [0000]      [Expression 4] 
         [0000]        R   O1 =√{square root over ( R   I1   2   −L   I1   2   +L   O1 ′ 2 )}  (1)
 
         [0073]    R I1  is a spherical radius of the surface of an inner cell of the pair of cells, and L I1  is a distance between the spherical center O of the surface of the inner cell and the boundary surface between the pair of cells. R O1  is a spherical radius of the surface of an outer cell of the pair of cells, and L O1  is a distance between the spherical center O of the surface of the outer cell and the boundary surface between the pair of cells. 
         [0074]    The formula (1) can be applied to a fly eye lens that includes the cells of all row and column numbers. Further, the formula (1) can be applied not only to the fly eye lens of this embodiment that includes the cells made eccentric outside in the horizontal direction but also to a fly eye lens that includes cells made eccentric inside in the horizontal direction. 
         [0075]    Different from the case of the fly eye lens according to this embodiment, it is not essential for a given pair of cells adjacent to each other in the horizontal direction to satisfy the relationship of the formula (1). Even when only one of the two cells adjacent to each other in the horizontal direction satisfies the relationship of the formula (1), the influence of steps in the boundaries between the cells in the entire fly eye lens can be reduced. 
         [0076]    Concerning the thickness T of each cell of the fly eye lens, when the thickness T 11  of cell C 11  is determined, the thicknesses T 12  to T 14  of cells C 12  to C 14  in the same first row as that of cell C 11  are sequentially calculated. 
         [0000]    
       
      
       T 
       12 
       =T 
       11 
       +R 
       12 
       −R 
       11  
      
     
         [0000]    
       
      
       T 
       13 
       =T 
       12 
       +R 
       13 
       −R 
       12  
      
     
         [0000]    
       
      
       T 
       14 
       =T 
       13 
       +R 
       14 
       −R 
       13  
      
     
         [0077]    The thicknesses T 21  to T 51  of cells C 21  to C 51  in the same first row as that of cell C 11  are equal to the thickness T 11  of cell C 11 , and the thicknesses T 22  to T 52  of cells C 22  to C 52  in the same second row as that of cell C 12  are equal to the thickness T 12  of cell C 12 . The thicknesses T 23  to T 53  of cells C 23  to C 53  in the same third row as that of cell C 13  are equal to the thickness T 13  of cell C 13 , and the thicknesses T 24  to T 54  of cells C 24  to C 44  in the same fourth row as that of cell C 14  are equal to the thickness T 14  of cell C 14 . 
         [0078]    The steps in the boundaries between the cells in the fly eye lens according to this embodiment were measured.  FIG. 10A  shows the step in the boundary between the cells adjacent to each other in the horizontal direction, and  FIG. 10B  shows the step in the boundary between the cells adjacent to each other in the vertical direction. 
         [0079]    In  FIGS. 10A and 10B , a vertical axis indicates the steps in the boundaries between the cells, and a horizontal axis indicates positions corresponding to the data of the steps. The position “0.0 mm” of the horizontal axis is the center of each boundary, and the absolute values of the horizontal axis indicate distances from the center of each boundary. For example, a position “1.5 mm” is away by 1.5 millimeters from the center of the boundary, and a position “−1.0 mm” is away by 1.0 millimeter from the center of the boundary in a direction opposite that of the position “1.5 mm”. 
         [0080]    For example, the data of the boundary between cells C 11  and C 12  is represented by C 11 -C 12 , and the data of the boundary between cells C 11  and C 21  is represented by C 11 -C 21 . 
         [0081]    It can be understood from  FIGS. 10A and 10B  that no step is generated in any boundary between the cells in the fly eye lens of this embodiment. 
         [0082]      FIG. 11  shows the surface of a fly eye lens according to Comparative Example 1. The size and the eccentric amount of each cell of this fly eye lens are equal to those of the fly eye lens of the embodiment shown in  FIG. 7 . The spherical radiuses of the surfaces of all the cells of this fly eye lens are equal to one another. 
         [0083]      FIG. 12A  is a sectional view cut along the line F-F′ shown in  FIG. 11 .  FIG. 12B  is a sectional view cut along the line G-G′ shown in  FIG. 11 . In this fly eye lens, the thickness of each cell is adjusted no that the maximum value of steps in the boundaries between the cells can be smallest. 
         [0084]    The steps in the boundaries between the cells in the fly eye lens according to Comparative Example 1 were measured.  FIG. 13A  shows a step in the boundary between the cells adjacent to each other in a horizontal direction, and  FIG. 13B  shows a step in the boundary between the cells adjacent to each other in a vertical direction. 
         [0085]    In this fly eye lens, each cell is not made eccentric in the vertical direction, and hence no step is generated in the boundary between the cells adjacent to each other in the vertical direction as shown in  FIG. 3B . However, as shown in  FIG. 13A , a step up to about 30 micrometers is generated in the boundary between the cells adjacent to each other in the horizontal direction. 
       Second Embodiment 
       [0086]      FIG. 14  shows the surface of a fly eye lens according to the second embodiment of the present invention. The cells of the fly eye lens according to this embodiment are made eccentric outside not only in a horizontal direction but also a vertical direction. The eccentric amounts of the cells in the same column in the horizontal direction are equal to one another, and eccentric amounts in the horizontal direction are larger in the cells in the outside column. The eccentric amounts of the cells in the same row in the vertical direction are equal to one another, and eccentric amounts in the vertical direction are larger in the cells in the outside row. 
         [0087]      FIG. 15A  is a sectional view cut along the line H-H′ shown in  FIG. 14 , and  FIG. 15B  is a sectional view cut along the line I-I′ shown in  FIG. 14 . The spherical centers O of the surfaces of all the cells are located on plane β parallel to cell arrangement plane α. 
         [0088]    As shown in  FIG. 15A , the spherical radiuses R of the surfaces of given cells in the same row are larger in the cells in the outside column. As shown in  FIG. 15B , the spherical radiuses R of the surfaces of given cells in the same column are larger in the cells in the outside row. 
         [0089]    In the fly eye lens according to this embodiment, generation of steps in the boundaries between the cells can be prevented by determining the radius R of the surface of each cell by a method described below. 
         [0090]    According to this embodiment, first, the spherical radius R 11  of the surface of cell C 11 , the eccentric amounts of the cells in each column in the horizontal direction, and the eccentric amounts of the cells in each row in the vertical direction are determined according to the size or the like of a second fly eye lens that makes a pair with this fly eye lens. Specifically, the spherical radius R of the surface of cell C 11  is determined according to the focal length of each cell, and the eccentric amounts of the cells in each column and the eccentric amounts of the cells in each row in the vertical direction are determined according to the changing amount of the traveling direction of light entered to each cell. 
         [0091]    Based on the spherical radius R 11  of the surface of cell C 11 , the eccentric amounts of the cells in each column in the horizontal direction, and the eccentric amounts of the cells in each row in the vertical direction, the radiuses R of the surfaces of the cells other than cell C 11  are calculated by a method described below. 
         [0092]    First, the spherical radiuses R 12  to R 14  of the surfaces of cells C 12  to C 14  in the first row are calculated by the same method as that of the first embodiment. 
         [0000]        R   12 =√{square root over ( R   11   2   −L   11   2   +L   12 ′ 2 )}
 
         [0000]        R   13 =√{square root over ( R   12   2   −L   12   2   +L   13 ′ 2 )}
 
         [0000]        R   14 =√{square root over ( R   13   2   −L   13   2   +L   14 ′ 2 )}  [Expression 5]
 
         [0093]    Referring to  FIG. 16 , a method for calculating the spherical radius R 21  of the surface of cell C 21  is described.  FIG. 16  is a partially enlarged view of  FIG. 15B . A surface that becomes a boundary between cells C 11  and C 21  is set as a boundary surface y. An intersection point on this section between the boundary surface y and the surface of the fly eye lens is set as a point a, and an intersection point between the boundary surface y and line segment O 11  O 12  is set as a point b. A line segment ab has a length x. 
         [0094]      FIG. 16  shows two right triangles abO 11  and abO 21 . When the Pythagorean theorem is applied to these right triangles as in the case of the first embodiment, the following expression is established. 
         [0000]        R   11   2   =L   11   2   =R   21   2   −L   21 ′ 2  
 
         [0095]    When this expression is modified, R 21  is represented by the following expression. 
         [0000]        R   21 =√{square root over ( R   11   2   −L   11   2   +L   21 ′ 2 )}  [Expression 6]
 
         [0096]    Similarly, the spherical surfaces R 31  to R 51  of the surfaces of cells C 31  to C 51  are sequentially calculated. 
         [0000]        R   31 =√{square root over ( R   21   2   −L   21   2   +L   31 ′ 2 )}
 
         [0000]        R   41 =√{square root over ( R   31   2   −L   31   2   +L   41 ′ 2 )}
 
         [0000]        R   51 =√{square root over ( R   41   2   −L   41   2   +L   51 ′ 2 )}  [Expression 7]
 
         [0097]    As in the case of the cells in the first column, the radiuses R of the surfaces of the cells in the second row and after of the second to fourth columns are calculated as follows. 
         [0000]        R   22 =√{square root over ( R   12   2   −L   12   2   +L   22 ′ 2 )}
 
         [0000]        R   32 =√{square root over ( R   22   2   −L   22   2   +L   32 ′ 2 )}
 
         [0000]        R   42 =√{square root over ( R   32   2   −L   32   2   +L   42 ′ 2 )}
 
         [0000]        R   52 =√{square root over ( R   42   2   −L   42   2   +L   52 ′ 2 )}
 
         [0000]        R   23 =√{square root over ( R   13   2   −L   13   2   +L   23 ′ 2 )}
 
         [0000]        R   33 =√{square root over ( R   23   2   −L   23   2   +L   33 ′ 2 )}
 
         [0000]        R   43 =√{square root over ( R   33   2   −L   33   2   +L   43 ′ 2 )}
 
         [0000]        R   53 =√{square root over ( R   43   2   −L   43   2   +L   53 ′ 2 )}
 
         [0000]        R   24 =√{square root over ( R   14   2   −L   14   2   +L   24 ′ 2 )}
 
         [0000]        R   34 =√{square root over ( R   24   2   −L   24   2   +L   34 ′ 2 )}
 
         [0000]        R   44 =√{square root over ( R   34   2   −L   34   2   +L   44 ′ 2 )}  [Expression 8]
 
         [0098]    Thus, when the spherical radius R 11  of the surface of cell C 11  and the eccentric amount of each cell are determined, the spherical radiuses R of the surfaces of all the cells are determined. 
         [0099]    In short, in the fly eye lens according to this embodiment, as in the case of the fly eye lens of the first embodiment, a given pair of cells adjacent to each other in the horizontal direction satisfies the relationship of the following formula (2). 
         [0000]      [Expression 9] 
         [0000]        R   O1 =√{square root over ( R   I1   2   −L   I1   2   +L   O1   2 )}  (2)
 
         [0100]    Further, in the fly eye lens according to this embodiment, a given pair of cells adjacent to each other in the vertical direction satisfies the relationship of the following formula (3). 
         [0000]      [Expression 10] 
         [0000]        R   O2 =√{square root over ( R   I2   2   −L   I2   2   +L   O2   2 )}  (3)
 
         [0101]    R 12  is the spherical radius of the surface of the inner cell of the pair of cells adjacent to each other in the vertical direction, and L 12  is the distance between spherical center O of the surface of the inner cell and the boundary surface between the pair of cells. R O2  is the spherical radius of the surface of the outer cell of the pair of cells adjacent to each other in the vertical direction, and L O2  is the distance between the spherical center O of the surface of the outer cell and the boundary surface between the pair of cells. 
         [0102]    The formulas (2) and (3) can be applied to a fly eye lens that includes the cells of all row and column numbers. Further, the formulas (2) and (3) can be applied not only to the fly eye lens of this embodiment that includes the cells made eccentric from the inside to the outside in the horizontal direction and the vertical direction but also to a fly eye lens that includes cells made eccentric from the outside to the inside in the horizontal direction and the vertical direction. 
         [0103]    Different from the case of the fly eye lens according to this embodiment, it is not essential for a given pair of cells adjacent to each other in the horizontal direction to satisfy the relationship of formula (2) and for a given pair of cells adjacent to each other in the vertical direction to satisfy the relationship of formula (3). Even when only one of the two cells adjacent to each other in the horizontal direction satisfies the relationship of formula (2) and only one of the two cells adjacent to each other in the vertical direction satisfies the relationship of formula (3), the influence of steps in the boundaries between the cells in the entire fly eye lens can be reduced. 
         [0104]    Concerning thickness T of each cell of the fly eye lens, when thickness T 11  of cell C 11  is determined, thicknesses T 12  to T 14  of cells C 12  to C 14  in the same first row as that of cell C 11  are sequentially calculated. 
         [0000]    
       
      
       T 
       12 
       =T 
       11 
       +R 
       12 
       −R 
       11  
      
     
         [0000]    
       
      
       T 
       13 
       =T 
       12 
       +R 
       13 
       −R 
       12  
      
     
         [0000]    
       
      
       T 
       14 
       =T 
       13 
       +R 
       14 
       −R 
       13  
      
     
         [0105]    Similarly, thicknesses T 21  to T 51  of cells C 21  to C 51  in the same first column as that of cell C 11  are sequentially calculated. 
         [0000]    
       
      
       T 
       21 
       =T 
       11 
       +R 
       21 
       −R 
       11  
      
     
         [0000]    
       
      
       T 
       31 
       =T 
       21 
       +R 
       31 
       −R 
       21  
      
     
         [0000]    
       
      
       T 
       41 
       =T 
       31 
       +R 
       41 
       −R 
       31  
      
     
         [0000]    
       
      
       T 
       51 
       =T 
       41 
       +R 
       51 
       −R 
       41  
      
     
         [0106]    Further, as in the case of the cells in the first column, thicknesses T of the cells of the second row and after of the second to fourth columns are sequentially calculated. 
         [0000]    
       
      
       T 
       22 
       =T 
       12 
       +R 
       22 
       −R 
       12  
      
     
         [0000]    
       
      
       T 
       32 
       =T 
       22 
       +R 
       32 
       −R 
       22  
      
     
         [0000]    
       
      
       T 
       42 
       =T 
       32 
       +R 
       42 
       −R 
       32  
      
     
         [0000]    
       
      
       T 
       52 
       =T 
       42 
       +R 
       42 
       −R 
       42  
      
     
         [0000]    
       
      
       T 
       23 
       =T 
       13 
       +R 
       23 
       −R 
       13  
      
     
         [0000]    
       
      
       T 
       33 
       =T 
       23 
       +R 
       33 
       −R 
       23  
      
     
         [0000]    
       
      
       T 
       43 
       =T 
       33 
       +R 
       43 
       −R 
       33  
      
     
         [0000]    
       
      
       T 
       53 
       =T 
       43 
       +R 
       53 
       −R 
       43  
      
     
         [0000]    
       
      
       T 
       24 
       =T 
       14 
       +R 
       24 
       −R 
       14  
      
     
         [0000]    
       
      
       T 
       34 
       =T 
       24 
       +R 
       34 
       −R 
       24  
      
     
         [0000]    
       
      
       T 
       44 
       =T 
       34 
       +R 
       44 
       −R 
       34  
      
     
         [0107]    The steps in the boundaries between the cells in the fly eye lens according to this embodiment were measured.  FIG. 17A  shows the step in the boundary between the cells adjacent to each other in the horizontal direction, and  FIG. 17B  shows the step in the boundary between the cells adjacent to each other in the vertical direction. 
         [0108]    It can be understood from  FIGS. 17A and 17B  that no step is generated in any boundary between the cells in the fly eye lens of this embodiment. 
         [0109]      FIG. 18  shows the surface of a fly eye lens according to Comparative Example 2. The size and the eccentric amount of each cell of this fly eye lens are equal to those of the fly eye lens of the embodiment shown in  FIG. 14 . The spherical radiuses of the surfaces of all the cells of this fly eye lens are equal to one another. 
         [0110]      FIG. 19A  is a sectional view cut along line J-J′ shown in  FIG. 18 .  FIG. 19B  is a sectional view cut along line K-K′ shown in  FIG. 18 . In this fly eye lens, the thickness of each cell is adjusted so that the maximum value of steps in the boundaries between the cells can be smallest. 
         [0111]    The steps in the boundaries between the cells in the fly eye lens according to Comparative Example 2 were measured.  FIG. 20A  shows a step in the boundary between the cells adjacent to each other in the horizontal direction, and  FIG. 20B  shows a step in the boundary between the cells adjacent to each other in the vertical direction. 
         [0112]    In this fly eye lens, as shown in  FIGS. 20A and 20B , steps are generated both in the boundary between the cells adjacent to each other in the horizontal direction and in the boundary between the cells adjacent to each other in the vertical direction. A step up to about 20 micrometers is generated in the boundary between the cells adjacent to each other in the horizontal direction, and a step up to about 30 micrometers is generated in the boundary between the cells adjacent to each other in the vertical direction. 
         [0113]      FIG. 21  shows the surface of a fly eye lens according to Comparative Example 3. This fly eye lens is configured by improving the fly eye lens of Comparative Example 2 to reduce the steps in the boundaries between the cells. 
         [0114]      FIG. 21  shows grids indicated by dashed lines to define the positions of the spherical centers of the surfaces of the cells of the fly eye lens according to Comparative Example 2 shown in  FIG. 18 . The spherical centers of the surfaces of the cells of the fly eye lens according to Comparative Example 3 are slightly shifted from the spherical centers of the surfaces of the cells of the fly eye lens according to Comparative Example 2 so that the steps in the boundaries between the cells can be symmetrical. Thus, the steps in the boundaries between the cells can be reduced. 
         [0115]    The steps in the boundaries between the cells in the fly eye lens according to Comparative Example 3 were measured.  FIG. 22A  shows the step in the boundary between the cells adjacent to each other in the horizontal direction, and  FIG. 22B  shows the step in the boundary between the cells adjacent to each other in the vertical direction. 
         [0116]    As shown in  FIGS. 22A and 22B , the steps in the boundaries between the cells are symmetrical. Still, however, a step up to about 5 micrometers is generated in the boundary between the cells adjacent to each other in the horizontal direction, and a step up to about 5 micrometers is generated in the boundary between the cells adjacent to each other in the vertical direction. 
       Third Embodiment 
       [0117]      FIG. 23  schematically shows the configuration of projection display device  1  according to the third embodiment of the present invention. Projection display device  1  includes illumination optical unit  10  that emits light, image forming unit  20  that modulates the light emitted from the illumination optical unit based on an image signal, and projection lens  30  that magnifies and projects the light modulated by image forming unit  20  to a screen or the like. 
         [0118]    Illumination optical unit  10  includes first fly eye lens  13  according to the first embodiment. First fly eye lens  13  is configured such that the surfaces of the cells are directed to light source  11  side. First fly eye lens  13  constitutes, together with second fly eye lens  14 , a uniformizing optical unit that uniformizes the illuminance of the light emitted from light source  11 . 
         [0119]    Second fly eye lens  14  includes cells corresponding to the cells of first fly eye lens  13 . The cells of second fly eye lens  14  are formed slightly larger than the corresponding cells of first fly eye lens  13 . 
         [0120]    Thus, spherical radius R 11  of the surface of cell C 11  and the eccentric amounts of the cells of first fly eye lens  13  are determined as described above in the first embodiment so that the light transmitted through each cell can enter the corresponding cells of second fly eye lens  14 . 
         [0121]    The light emitted from light source  11  and transmitted through concave lens  12  is transmitted through each cell of first fly eye lens  13  to be divided into a plurality of very small light fluxes, and then enters each cell of second fly eye lens  14 . The light transmitted through each cell of second fly eye lens  14  is transmitted through polarization conversion element  15  to be converted into polarized light, and then transmitted through condenser lens  16  to enter image forming unit  20 . 
         [0122]    The light that has entered image forming unit  20  is separated into three primary colors of R, G, and B sequentially by dichroic mirrors  21   a  and  21   b . Specifically, the light of a blue wavelength included in white light is reflected by dichroic mirror  21   a , the light of a green wavelength is transmitted through dichroic mirror  21   a  and then reflected by dichroic mirror  21   b , and the light of a red wavelength is transmitted through both dichroic mirrors  21   a  and  21   b.    
         [0123]    The blue light reflected by dichroic mirror  21   a , which is included in the light applied to illumination optical unit  10 , is reflected by reflection mirror  22   a , and then sequentially transmitted through field lens  24 B, entrance side polarization plate  25 B, and liquid crystal light bulb  26 B to enter exit side polarization plate  27 B. The light transmitted through exit side polarization plate  27 B enters cross dichroic mirror  28 . 
         [0124]    The green light reflected by dichroic mirror  21   b  is sequentially transmitted through field lens  24 G, entrance side polarization plate  25 G, and liquid crystal light bulb  26 G to enter exit side polarization plate  27 G. The light transmitted through exit side polarization plate  27 G enters cross dichroic mirror  28 . 
         [0125]    The red light reflected by dichroic mirror  21   b  enters field lens  24 R via relay lens  23   a , reflection mirror  22   b , relay lens  23   b , and reflection mirror  21   c . The light transmitted through field lens  24 R is sequentially transmitted through entrance side polarization plate  25 R and liquid crystal light bulb  26 R to enter exit side polarization plate  27 R. The light transmitted through exit side polarization plate  27 R enters cross dichroic mirror  28 . 
         [0126]    Each color light that has entered cross dichroic mirror  28  enters projection lens  30 . Specifically, the red light and the blue light are reflected by cross dichroic mirror  28  to enter projection lens  30 , and the green light is transmitted through cross dichroic mirror  28  to enter projection lens  30 . The light that has entered projection lens  30  is magnified and projected to the screen or the like by projection lens  30 . 
         [0127]    The embodiments of the present invention have been described. However, the embodiments are in no way limitative of the invention. Various changes understandable to those skilled in the art can be made of the configuration of the present invention within the spirit and the scope of the invention.