Abstract:
An optical element has a light receiving surface covering a light source arranged on a plane and an exit surface covering the light receiving surface. When an axis passing through the center of the light source and is perpendicular to the plane is designated as an optical axis and the point of intersection of the optical axis and the light receiving surface is designated as O1, the light receiving surface is concaved around the optical axis with respect to the periphery. When an angle which a normal to the light receiving surface on a point P thereon forms with the optical axis is designated as φh and distance in the optical axis direction from O1 to P is designated as z, φh has at least one local maximum value and at least one local minimum value with respect to z while P is moved along the light receiving surface.

Description:
BACKGROUND 
     1. Field 
     The present invention relates to an optical element configured to diffuse lights from the light source. 
     2. Description of Related Art 
     Recently LED (light emitting diode) light sources have been widely used. Since a large portion of lights of a LED light source is emitted toward the front, an optical element configured to diffuse lights from the LED light source is commonly used in combination with the LED light source. Particularly, when LED light sources are used as light sources of an illumination unit for illuminating a large area, such as that for backlight, optical elements configured to diffuse lights from the LED light sources over a large angle are used such that a compact illumination unit can be realized with a small number of LED light sources (for example, Patent Document 1). 
     An LED light source for a large amount of light consists of a light emitting chip for emitting shorter-wavelength lights such as blue light and a fluorescent material which emits longer-wavelength fluorescences such as green, yellow or red. In many cases, in such an LED light source, the light emitting chip for emitting shorter-wavelength lights is arranged at the center while the fluorescent material which emits longer-wavelength fluorescences is arranged around the light emitting chip. In such an LED light source, the position of the portion emitting shorter-wavelength lights and the position of the portion emitting longer-wavelength lights are dissimilar from each other. Accordingly, when the optical device is used to diffuse lights from the light source, in some cases there exist directions in which shorter-wavelength lights are stronger and directions in which longer-wavelength lights are stronger. As a result, in some cases the color of light may become bluish in some directions while may become reddish in other directions. That is, the color of light may vary depending on the direction. For the use in illumination units, it is not preferable that color of light varies depending on the direction. However, an optical element configured to diffuse lights from the light source, which can reduce color difference of lights which occurs due to direction, has not been developed so far. 
     Patent Document 1: JP2006-92983A (JP3875247B) 
     Accordingly, there is a need for an optical element configured to diffuse lights from the light source, which can reduce color difference of lights which occurs due to direction. 
     SUMMARY 
     An optical element according to a first aspect of the present invention is an optical element including a light receiving surface which is configured to cover a light source arranged on a plane and an exit surface which covers the light receiving surface, the optical element being configured such that lights from the light source passes through the light receiving surface and the exit surface and goes to the outside for illumination. When an axis which passes through the center of the light source and which is perpendicular to the plane is designated as an optical axis and the point of intersection of the optical axis and the light receiving surface is designated as O1, the light receiving surface is concaved around the optical axis with respect to the periphery. In a cross section of the optical element, the cross section containing the optical axis and being perpendicular to the plane, when an angle which a normal to the light receiving surface on a point P on the light receiving surface forms with the optical axis is designated as φh and distance in the optical axis direction from the point O1 to the point P is designated as z, the light receiving surface is configured such that φh has at least one local maximum value and at least one local minimum value with respect to z while the point P is moved along the light receiving surface from the point O1 to the plane. 
     In the optical element according to the present aspect, the light receiving surface is configured such that φh has at least one local maximum value and at least one local minimum value with respect to z, and therefore when used in combination with a light source, rays from each point on the light source are refracted in various directions depending on location on the light receiving surface which each ray reaches. Accordingly, color difference of lights which occurs due to direction in which light is emitted from the optical element can be reduced. 
     An optical element according to an embodiment of the present invention is an optical element of the first aspect in which the light receiving surface is shaped rotationally symmetric around the optical axis. 
     The optical element according to the present embodiment can be manufactured without great difficulty by injection molding or the like. 
     An optical element according to another embodiment of the present invention is an optical element of the first aspect in which a space around the optical axis is partitioned based on angle around the optical axis into plural zones and the light receiving surface is configured to have different shapes in respective zones. 
     According to the present embodiment, different light distributions can be realized for respective directions corresponding to zones around the optical axis. 
     An optical element according to another embodiment of the present invention is an optical element of the first aspect in which in some of the zones alone, the light receiving surface is configured such that φh has at least one local maximum value and at least one local minimum value with respect to z while the point P is moved along the light receiving surface the from the point O1 to the plane. 
     According to the present embodiment, in some of the zones around the optical axis alone, color difference of lights which occurs due to the direction can be reduced. 
     In an optical element according to another embodiment of the present invention, when the point of intersection of the optical axis and the plane is designated as a point P0 and an angle which a line connecting the point P0 and the point P on the light receiving surface forms with the optical axis is designated as θr, the light receiving surface is configured such that φh has at least one local maximum value and at least one local minimum value with respect to z in the range 30°&lt;θr&lt;90 °. 
     In the optical element according to the present embodiment, in the range 30°&lt;θr&lt;90°, in which inclination of φh data graphed with respect to z would be substantially constant if there were no local maximum value or no local minimum value, the light receiving surface is configured such that φh has at least one local maximum value and at least one local minimum value with respect to z. As a result, when used in combination with a light source, rays from each point on the light source are refracted in more various directions depending on location on the light receiving surface which each ray reaches, compared with the case that there is no local maximum value or no local minimum value. Accordingly, color difference of lights which occurs due to direction in which light is emitted from the optical element can be reduced. 
     An optical element according to another embodiment of the present invention is an optical element of the first aspect in which there exist a local maximum value and a local minimum value which are adjacent to each other and between which a difference in φh is 10 degrees or more. 
     When the optical element of the present embodiment is used in combination with a light source, direction in which a ray from each point on the light source travels after having been refracted on the light receiving surface remarkably varies depending on location on the light receiving surface which the ray reaches. Accordingly, color difference of lights which occurs due to direction in which light is emitted from the optical element can be reduced. 
     An optical element according to another embodiment of the present invention is an optical element of the first aspect in which there exist a local maximum value and a local minimum value which are adjacent to each other and between which a difference in φh is 20 degrees or more. 
     When the optical element of the present embodiment is used in combination with a light source, direction in which a ray from each point on the light source travels after having been refracted on the light receiving surface remarkably varies depending on location on the light receiving surface which the ray reaches. Accordingly, color difference of lights which occurs due to direction in which light is emitted from the optical element can be reduced. 
     An optical element according to a second aspect of the present invention is an optical element including a light receiving surface which is configured to cover a light source arranged on a plane and an exit surface which covers the light receiving surface, the optical element being configured such that lights from the light source passes through the light receiving surface and the exit surface and goes to the outside for illumination. When an axis which passes through the center of the light source and which is perpendicular to the plane is designated as an optical axis, the point of intersection of the optical axis and the light receiving surface is designated as O1, and the point of intersection of the optical axis and the plane is designated as P0, the light receiving surface is concaved around the optical axis with respect to the periphery. In a cross section of the optical element, the cross section containing the optical axis and being perpendicular to the plane, when an angle which a line connecting the point P0 and a point P on the light receiving surface forms with the optical axis is designated as θr, and a direction of light which travels inside the optical element after having traveled from the point P0 to the point P forms with the optical axis is designated as θi, the light receiving surface is configured such that θi has at least one local maximum value and at least one local minimum value with respect to θr while the point P is moved along the light receiving surface from the point O1 to the plane. 
     In the optical element according to the present aspect, the light receiving surface is configured such that θi has at least one local maximum value and at least one local minimum value with respect to θr, and therefore when used in combination with a light source, rays from each point on the light source are refracted in various directions depending on location on the light receiving surface which each ray reaches. Accordingly, color difference of lights which occurs due to direction in which light is emitted from the optical element can be reduced. 
     An optical element according to another embodiment of the present invention is an optical element of the second aspect in which the light receiving surface is shaped rotationally symmetric around the optical axis. 
     The optical element according to the present embodiment can be manufactured without great difficulty by injection molding or the like. 
     An optical element according to another embodiment of the present invention is an optical element of the second aspect in which a space around the optical axis is partitioned based on angle around the optical axis into plural zones and the light receiving surface is configured to have different shapes in respective zones. 
     According to the present embodiment, different light distributions can be realized for respective directions corresponding to zones around the optical axis. 
     An optical element according to another embodiment of the present invention is an optical element of the second aspect in which in some of the zones alone, the light receiving surface is configured such that θi has at least one local maximum value and at least one local minimum value with respect to θr while the point P is moved along the light receiving surface the from the point O1 to the plane. 
     According to the present embodiment, in some of the zones around the optical axis alone, color difference of lights which occurs due to the direction can be reduced. 
     An optical element according to another embodiment of the present invention is an optical element of the second aspect in which the light receiving surface is configured such that θi has at least one local maximum value and at least one local minimum value with respect to θr in the range 30°&lt;θr&lt;90 °. 
     In the optical element according to the present embodiment, in the range 30°&lt;θr&lt;90°, in which inclination of θi data graphed with respect to θr were substantially constant if there had been no local maximum value or no local minimum value, the light receiving surface is configured such that θi has at least one local maximum value and at least one local minimum value with respect to θr. As a result, when used in combination with a light source, rays from each point on the light source are refracted in more various directions depending on location on the light receiving surface which each ray reaches, compared with the case that there is no local maximum value or no local minimum value. Accordingly, color difference of lights which occurs due to direction in which light is emitted from the optical element can be reduced. 
     An optical element according to another embodiment of the present invention is an optical element of the second aspect in which there exist a local maximum value and a local minimum value which are adjacent to each other and between which a difference in θi is 5 degrees or more. 
     When the optical element of the present embodiment is used in combination with a light source, direction in which a ray from each point on the light source travels after having been refracted on the light receiving surface remarkably varies depending on location on the light receiving surface which the ray reaches. Accordingly, color difference of lights which occurs due to direction in which light is emitted from the optical element can be reduced. 
     An optical element according to another embodiment of the present invention is an optical element of the second aspect in which there exist a local maximum value and a local minimum value which are adjacent to each other and between which a difference in θi is 10 degrees or more. 
     When the optical element of the present embodiment is used in combination with a light source, direction in which a ray from each point on the light source travels after having been refracted on the light receiving surface remarkably varies depending on location on the light receiving surface which the ray reaches. Accordingly, color difference of lights which occurs due to direction in which light is emitted from the optical element can be reduced. 
     An illumination unit according to a third aspect of the present invention is an illumination unit including a light source and the optical element according to the first aspect or the second aspect of the present invention. 
     The illumination unit according to the present aspect uses the optical element according to any one of the aspects of the present invention, and therefore color difference of lights which occurs due to direction in which light is emitted from the optical element can be reduced. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         FIGS. 1A and 1B  show an example of a LED light source used with an optical element according to the present invention; 
         FIG. 2  shows a cross section of an optical element used to diffuse lights from the light source according to an embodiment of the present invention, the cross section containing the central axis AX of the optical element; 
         FIG. 3  shows an enlarged view of the portion of the light receiving surface in the cross section of  FIG. 2 ; 
         FIG. 4  shows an example of the configuration of an illumination unit in which plural sets of the light source and the optical element are arranged on a plane; 
         FIG. 5  shows a relationship between z and angle φh which a normal to the light receiving surface forms with the central axis AX in the optical element of Example 1; 
         FIG. 6  shows a relationship between θr and θi in the optical element of Example 1; 
         FIG. 7  shows a relationship between θr and θe in the optical element of Example 1; 
         FIG. 8  shows a light intensity distribution for the case of a combination of the light source shown in  FIGS. 1A and 1B  and the optical element of Example 1; 
         FIG. 9  shows a light intensity distribution for the case of a combination of the light source shown in  FIGS. 1A and 1B  and the optical element of Comparative Example 1; 
         FIG. 10  shows an intensity distribution of rays emitted from a point P0 which is shown in  FIG. 3  in the optical element of Example 1; 
         FIG. 11  shows an intensity distribution of rays emitted from a point P1 which is shown in  FIG. 3  in the optical element of Example 1; 
         FIG. 12  shows an intensity distribution of rays emitted from a point P2 which is shown in  FIG. 3  in the optical element of Example 1; 
         FIG. 13  shows a relationship between z of the light receiving surface and angle φh which a normal to the light receiving surface forms with the central axis AX in the optical element of Example 2; 
         FIG. 14  shows a relationship between θr and θi in the optical element of Example; 
         FIG. 15  shows a relationship between θr and θe in the optical element of Example 2; 
         FIG. 16  shows a light intensity distribution for the case of a combination of the light source shown in  FIGS. 1A and 1B  and the optical element of Example 2; 
         FIG. 17  shows a light intensity distribution for the case of a combination of the light source shown in  FIGS. 1A and 1B  and the optical element of Comparative Example 2; 
         FIG. 18  shows a relationship between z and angle φh which a normal to the light receiving surface forms with the central axis AX in the optical element of Example 3; 
         FIG. 19  shows a relationship between θr and θi in the optical element of Example 3; 
         FIG. 20  shows a relationship between θr and θe in the optical element of Example 3; 
         FIG. 21  shows a light intensity distribution for the case of a combination of the light source shown in  FIGS. 1A and 1B  and the optical element of Example 3; 
         FIG. 22  shows a light intensity distribution for the case of a combination of the light source shown in  FIGS. 1A and 1B  and the optical element of Comparative Example 3; 
         FIGS. 23A and 23B  show the case in which a resin gate is arranged around the center of the exit surface of an optical element; 
         FIGS. 24A and 24B  show the case in which a portion in the form of a truncated cone is provided around the center of the exit surface of an optical element, and a resin gate is arranged on the portion; 
         FIG. 25  shows the case in which a single resin gate is arranged on the bottom face  105  of an optical element; 
         FIG. 26  shows the case in which two resin gates and are arranged on the bottom face of an optical element; 
         FIG. 27  shows a construction of an optical element which is provided with a diffusing structure or a diffusing material on the periphery of the exit surface; and 
         FIG. 28  shows a construction of an optical element which is provided with a diffusing structure or a diffusing material on the bottom face. 
     
    
    
     DETAILED DESCRIPTION 
       FIGS. 1A and 1B  show an example of a LED light source  200  used with an optical element according to the present invention.  FIG. 1A  shows a cross section perpendicular to the light-emitting surface of the LED light source  200 .  FIG. 1B  shows a plan view of the LED light source  200 . In general, an LED light source for a large amount of light consists of a light emitting chip for emitting shorter-wavelength lights such as blue light and a fluorescent agent which emits longer-wavelength fluorescences such as green, yellow or red. In  FIGS. 1A and 1B , a light emitting chip  201  of blue light is arranged at the center of the LED light source  200  while a fluorescent agent  203  is arranged in an area which is larger than the area occupied by the light emitting chip such that the fluorescent agent  203  covers the light emitting chip  201 . In the plan view of  FIG. 1B , the light emitting chip  201  is a square with sides of 1.0 millimeter while the fluorescent agent  203  is shaped as a circle with a diameter of 3.0 millimeters. A blue ray A is emitted by the light emitting chip  201  located around the center. A ray B of longer wavelength is emitted by the fluorescent agent arranged in an area which includes the periphery of the LED light source. In a LED light source having such a structure as shown in  FIGS. 1A and 1B , the location where blue rays are emitted and the location where rays of longer wavelengths are emitted are dissimilar from each other. 
       FIG. 2  shows a cross section of an optical element  100  used to diffuse lights from the light source  200  according to an embodiment of the present invention. The cross section contains the central axis AX of the optical element  100 . The optical element  100  according to the present embodiment is of a shape having rotational symmetry around the central axis AX. A face  105  which faces the light source  200  has an area recessed relative to the periphery, around the central axis AX. The surface of the recessed area forms a light receiving surface  101 . The face  105  which faces the light source  200  is referred to as a bottom face  105  in the present specification. The surface of the optical element  100  besides the light receiving surface  101  and the bottom face  105  forms an exit surface  103 . 
     The optical element  100  and the light source  200  are arranged such that the central axis AX of the optical element  100  passes through the center of the light source  200 , that is, the center of the circle shown in  FIG. 1B . In this case, the central axis AX forms the optical axis of the optical system including the optical element  100  and the LED light source  200 . 
     Lights emitted by the light source  200  enter the optical element  100  through the light receiving surface  101  and are emitted to the outside through the exit surface  103 . In this case, lights emitted by the light source  200  are refracted at most portions of the light receiving surface  101  and the exit surface  103  such that the lights travel away from the central axis AX. As a result, the lights are diffused. 
     In the present embodiment, the surface of the LED light source  200  is planar. However, the surface of the light source  200  does not necessarily have to be planar. The present invention can be applied to any light sources arranged on a plane, in which the position of the portion emitting shorter-wavelength lights and the position of the portion emitting lights differ from each other. 
       FIG. 3  shows an enlarged view of the portion of the light receiving surface in the cross section of  FIG. 2 . The point of intersection of the light emitting surface  205  of the light source  200  and the central axis AX is designated as a point P0. The angle which a travelling direction of a ray emitted from the point P0 forms with the central axis AX is designated as θr, and the angle which a travelling direction of the ray which travels in the optical element  100  after having been refracted at the light receiving surface  101  forms with the central axis AX is designated as θi. The angle which a travelling direction of the ray which travels after having been refracted at the exit surface forms with the central axis AX is designated as θe (See  FIG. 2 ). In  FIG. 3 , a foot of a perpendicular line from a point representing a side of the emitting chip  201  to a line representing the emitting surface  205  is designated as P1, and a point at an edge of the fluorescent agent, that is, a point on the circumstance of the circle which forms the periphery of the fluorescent agent is designated as P2. 
     The light receiving surface  101  is determined such that θr&lt;θi is satisfied for rays emitted at θr in a certain range. In  FIG. 3 , the certain range is from 0 degree to approximately 20 degrees. In the above-described range, angle θi monotonously increases as angle θr increases. 
     The exit surface  103  is determined such that θr&lt;θe is satisfied for rays emitted at θr which is in the above-described certain range. 
     A shape of the exit surface around the central axis AX is not limited to convex, nor to concave. The shape may be convex, concave or planar. A shape of the exit surface which does not generate total reflection inside the lens is also preferable. In this case, when refractive index of the optical element is designated as n, an angle φ between a ray travelling in the optical element and the normal to the exit surface satisfies the following relationship.
 
φ&lt;sin −1 (1/ n )
 
     Further, in  FIG. 3 , an angle which a normal to the light receiving surface  101  forms with the central axis AX is designated as φh. The angle is measured with reference to the downward direction in  FIG. 3 . That is, the following equation holds at the top of the light receiving surface  101 .
 
φ h= 180 degrees
 
     In the area of the light receiving surface  101  which lights emitted from the point P0 at an angle θr in the range from 0 degree to approximately 20 degrees reach, angle φh monotonously decreases as angle θr increases. In the area of the light receiving surface  101  which lights emitted from the point P0 at an angle θr which is greater than approximately 20 degrees reach, angle φh repeatedly fluctuates as angle θr increases. This area of the light receiving surface  101  is referred to as a diffusing area of the light receiving surface in the present specification. A shape of the diffusing area of the light receiving surface  101  will be described in detail later. 
       FIG. 4  shows an example of the configuration of an illumination unit in which plural sets of the light source  200  and the optical element  100  are arranged on a plane  300 . The illumination unit is further provided with a diffuser  400 . The illumination unit permits uniform illumination on an area ahead of the illumination unit (above the illumination unit in  FIG. 4 ). 
     Examples of the optical elements according to the present invention and their comparative examples will be described below. The material of the optical elements of the examples and the comparative examples is polymethyl methacrylate (PMMA), refractive index of which is 1.492 (d line, 587.56 nm) and Abbe&#39;s number of which is 56.77 (d line, 587.56 nm). Further, in the examples and the comparative examples, unit of length is millimeter unless otherwise designated. 
     Example 1 
     In  FIG. 2 , the coordinates of the point of intersection of the light receiving surface  101  and the central axis AX are represented as O1 while the coordinates of the point of intersection of the exit surface  103  and the central axis AX are represented as O2. 
     In the present example, the distance T between P0 and O2 is given as below. 
     T=5.752 mm 
     The distance h between P0 and O1 is given as below. 
     h=4.400 mm 
     When distance from O1 in the direction of the central axis AX is represented as z, a shape of the light receiving surface  101  can be represented by the following equation in the range where z is between 0 and 1.5 mm inclusive (0≦z≦1.5 mm). 
     
       
         
           
             
               
                 
                   
                     z 
                     = 
                     
                       
                         
                           cr 
                           2 
                         
                         
                           1 
                           + 
                           
                             
                               1 
                               - 
                               
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     k 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   c 
                                   2 
                                 
                                 ⁢ 
                                 
                                   r 
                                   2 
                                 
                               
                             
                           
                         
                       
                       + 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           N 
                         
                         ⁢ 
                         
                           
                             A 
                             i 
                           
                           ⁢ 
                           
                             r 
                             i 
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     c 
                     = 
                     
                       1 
                       / 
                       R 
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     In the equation, r represents distance from the central axis AX, c represents curvature, R represents radius of curvature, k represents conic constant and Ai represents aspheric coefficient. 
     Table 1 shows numerical values of constants in Equation (1) which represents the light receiving surface  101  of Example 1. 
     
       
         
               
               
               
             
           
               
                   
                 TABLE 1 
               
               
                   
                   
               
             
             
               
                   
                 R 
                 −1.201 
               
               
                   
                 K 
                 −0.7990 
               
               
                   
                 A1 
                 0.000 
               
               
                   
                 A2 
                 0.000 
               
               
                   
                 A3 
                 0.000 
               
               
                   
                 A4 
                 0.000 
               
               
                   
                   
               
             
          
         
       
     
     A shape of the area of the light receiving surface  101  which extends from z=1.5 mm to the face  105 , that is, a shape of the diffusing area is represented as a third-order spline curve, a point group of which is given below. A third-order spline curve is a smooth curve which passes through given points, in which each segment between adjacent points is connected by an individual third-order polynomial and the individual polynomials are made continuous at all the points. 
     Table 2 shows the above-described point group. 
     
       
         
               
               
               
             
           
               
                   
                 TABLE 2 
               
               
                   
                   
               
               
                   
                 z 
                 r 
               
               
                   
                   
               
             
             
               
                   
                 1.500 
                 1.70 
               
               
                   
                 1.661 
                 1.80 
               
               
                   
                 1.822 
                 1.82 
               
               
                   
                 1.983 
                 1.85 
               
               
                   
                 2.144 
                 1.95 
               
               
                   
                 2.306 
                 1.97 
               
               
                   
                 2.467 
                 2.00 
               
               
                   
                 2.628 
                 2.10 
               
               
                   
                 2.789 
                 2.12 
               
               
                   
                 2.950 
                 2.15 
               
               
                   
                 3.111 
                 2.25 
               
               
                   
                 3.272 
                 2.27 
               
               
                   
                 3.433 
                 2.30 
               
               
                   
                 3.594 
                 2.40 
               
               
                   
                 3.756 
                 2.42 
               
               
                   
                 3.917 
                 2.45 
               
               
                   
                 4.078 
                 2.55 
               
               
                   
                 4.239 
                 2.57 
               
               
                   
                 4.400 
                 2.60 
               
               
                   
                   
               
             
          
         
       
     
       FIG. 5  shows a relationship between z of the light receiving surface  101  and angle φh which a normal to the light receiving surface  101  forms with the central axis AX in the optical element of Example 1. The horizontal axis of  FIG. 5  represents z while the vertical axis represents φh. According to  FIG. 5 , in the range where z is 1.5 mm or less, φh monotonously decreases as z increases. In the range where z is greater than 1.5 mm, φh repeatedly fluctuates as z increases. In other words, in the range where z is greater than 1.5 mm, φh which is a function of z has local maximum values and local minimum values. 
     Specifically, in  FIG. 5 , φh has 6 local maximum values and 6 local minimum values. Minor fluctuations of φh around the local minimum values have been ignored. Difference in φh between a local maximum value and a local minimum value which are adjacent to each other is approximately 30 degrees. 
     When distance from O2 in the direction of the central axis AX is represented as z, a shape of the exit surface  103  around the central axis AX is what does not cause total reflection of rays from the light source on the exit surface and can be represented by the following equation. 
     
       
         
           
             
               
                 
                   
                     z 
                     = 
                     
                       
                         
                           cr 
                           2 
                         
                         
                           1 
                           + 
                           
                             
                               1 
                               - 
                               
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     k 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   c 
                                   2 
                                 
                                 ⁢ 
                                 
                                   r 
                                   2 
                                 
                               
                             
                           
                         
                       
                       + 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           N 
                         
                         ⁢ 
                         
                           
                             A 
                             i 
                           
                           ⁢ 
                           
                             r 
                             i 
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     c 
                     = 
                     
                       1 
                       / 
                       R 
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     In the equation, r represents distance from the central axis AX, c represents curvature, R represents radius of curvature, k represents conic constant and Ai represents aspheric coefficient. 
     Table 3 shows numerical values of constants in Equation (2) which represents the exit surface of Example 1. 
     
       
         
               
               
               
             
           
               
                   
                 TABLE 3 
               
               
                   
                   
               
             
             
               
                   
                 R 
                 −2.222  
               
               
                   
                 K 
                 −7.513  
               
               
                   
                 A1 
                 0.000 
               
               
                   
                 A2 
                 −3.75E−02 
               
               
                   
                 A3 
                 −4.78E−04 
               
               
                   
                 A4 
                 −2.51E−04 
               
               
                   
                 A5 
                 0.000 
               
               
                   
                 A6 
                 0.000 
               
               
                   
                 A7 
                 0.000 
               
               
                   
                 A8 
                 0.000 
               
               
                   
                 A9 
                 0.000 
               
               
                   
                 A10 
                 0.000 
               
               
                   
                   
               
             
          
         
       
     
       FIG. 6  shows a relationship between θr and θi on the light receiving surface in the optical element of Example 1. The horizontal axis of  FIG. 6  represents θr while the vertical axis represents θi. In the range where θr is approximately 30 degrees or less, θi monotonously increases as θr increases. In the range where θr is greater than approximately 30 degrees, θi increases while repeatedly fluctuating as θr increases. In other words, θi which is a function of θr has local maximum values and local minimum values. 
     Specifically, in  FIG. 6 , θi has 6 local maximum values and 6 local minimum values in the range where θr is from approximately 30 degrees to 90 degrees. Minor fluctuations of around the local maximum values of θi have been ignored. Difference in θi between a local maximum value and a local minimum value which are adjacent to each other is approximately 15 degrees. 
       FIG. 7  shows a relationship between θr and θe on the exit surface of the optical element of Example 1. The horizontal axis of  FIG. 7  represents θr while the vertical axis represents θe. In the range where θr is approximately 30 degrees or less, θe monotonously increases as θr increases. In the range where θr is greater than approximately 30 degrees, θe increases while repeatedly fluctuating with a peak-to-peak amplitude of approximately 10 degrees as θr increases. In other words, θe which is a function of θr has local maximum values and local minimum values in the range where θr is greater than approximately 30 degrees. 
     Comparative Example 1 
     In the present comparative example, the distance T between P0 and O2 is given as below. 
     T=5.752 mm 
     The distance h between P0 and O1 is given as below. 
     h=4.400 mm 
     When distance from O1 in the direction of the central axis AX is represented as z, a shape of the light receiving surface can be represented by Equation (1). Further, values of constants in Equation (1) are those shown in Table 1. That is, a shape of the light receiving surface of Comparative Example 1 is identical with that of Example 1 in the range where z is 1.5 mm or less, and in the range where z is greater than 1.5 mm, φh which is a function of z does not have a local maximum value or a local minimum value and monotonously decreases as z increases. In other words, the light receiving surface of the optical element of Comparative Example 1 differs from the light receiving surface of Example 1 in that it does not have a diffusing area of the light receiving surface. 
     When distance from O2 in the direction of the central axis AX is represented as z, a shape of the exit surface around the central axis AX is what does not cause total reflection of rays from the light source on the exit surface and can be represented by Equation (2). Further, values of constants in Equation (2) are those shown in Table 3. That is, a shape of the exit surface of Comparative Example 1 is identical with that of Example 1. 
     Performance Comparison Between Example 1 and Comparative Example 1 
     Performance comparison between Example 1 and Comparative Example 1 will be made by comparing light intensity distribution between the case of a combination of the light source shown in  FIGS. 1A and 1B  and the optical element of Example 1 and the case of a combination of the light source shown in  FIGS. 1A and 1B  and the optical element of Comparative Example 1. 
       FIG. 8  shows a light intensity distribution for the case of a combination of the light source shown in  FIGS. 1A and 1B  and the optical element of Example 1. The horizontal axis of  FIG. 8  represents direction which forms angle θ with the central axis AX. The vertical axis of  FIG. 8  represents relative value of intensity of light which is emitted in the direction which forms angle θ with the central axis AX. The solid line in  FIG. 8  represents relative value of intensity of lights which have wavelengths of less than 500 nanometers (lights in a shorter wavelength range). The relative value of intensity is scaled such that the maximum value is 100%. The dashed line in  FIG. 8  represents relative value of intensity of lights which have wavelengths of 500 nanometers or more (lights in a longer wavelength range). The relative value of intensity is scaled such that the maximum value is 100%. 
       FIG. 9  shows a light intensity distribution for the case of a combination of the light source shown in  FIGS. 1A and 1B  and the optical element of Comparative Example 1. The horizontal axis of  FIG. 9  represents direction which forms angle θ with the central axis AX. The vertical axis of  FIG. 9  represents relative value of intensity of light which is emitted in the direction which forms angle θ with the central axis AX. The solid line in  FIG. 9  represents relative value of intensity of lights which have wavelengths of less than 500 nanometers (lights in a shorter wavelength range). The relative value of intensity is scaled such that the maximum value is 100%. The dashed line in  FIG. 9  represents relative value of intensity of lights which have wavelengths of 500 nanometers or more (lights in a longer wavelength range). The relative value of intensity is r scaled such that the maximum value is 100%. 
     When  FIG. 8  and  FIG. 9  are compared with each other, difference in intensity of light between the shorter wavelength range and the longer wavelength range is greater in  FIG. 9  which relates to Comparative Example 1. The difference between the both is particularly great when 0 is around 60 degrees. When the difference between the both is great, a difference in color is generated. For example, in the case that intensity of the longer wavelength range becomes greater in an area where θ is around 60 degrees as shown in  FIG. 9 , light becomes reddish in the area where θ is around 60 degrees. 
     Thus, the optical element of Example 1 is superior to that of Comparative Example 1 in preventing a difference in color from being generated. 
       FIG. 10  shows an intensity distribution of rays emitted from a point P0 which is shown in  FIG. 3  in the optical element of Example 1. The point P0 is the point of intersection of the emitting surface  205  of the light source  200  and the central axis AX. The horizontal axis of  FIG. 10  represents direction which forms angle θ with the central axis AX. The vertical axis of  FIG. 10  represents relative value of intensity of light which is emitted in the direction which forms angle θ with the central axis AX. The solid line represents an intensity distribution of Example 1 while the dashed line represents an intensity distribution of Comparative Example 1. The relative value of intensity for Example 1 and that for Comparative Example 1 are scaled such that the maximum value is 100%. 
       FIG. 11  shows an intensity distribution of rays emitted from a point P1 which is shown in  FIG. 3  in the optical element of Example 1. The point P1 is a foot of a perpendicular line from a point representing a side of the emitting chip  201  to a line representing the emitting surface  205 . The horizontal axis of  FIG. 11  represents direction which forms angle θ with the central axis AX. The vertical axis of  FIG. 11  represents relative value of intensity of light which is emitted in the direction which forms angle θ with the central axis AX. The solid line in  FIG. 11  represents an intensity distribution of Example 1 while the dashed line represents an intensity distribution of Comparative Example 1. The relative value of intensity for Example 1 and that for Comparative Example 1 are scaled such that the maximum value is 100%. 
       FIG. 12  shows an intensity distribution of rays emitted from a point P2 which is shown in  FIG. 3  in the optical element of Example 1. The point P2 is a point on the circumstance forming the periphery of the fluorescent agent. The horizontal axis of  FIG. 12  represents direction which forms angle θ with the central axis AX. The vertical axis of  FIG. 12  represents relative value of intensity of light which is emitted in the direction which forms angle θ with the central axis AX. The solid line in  FIG. 12  represents an intensity distribution of Example 1 while the dashed line represents an intensity distribution of Comparative Example 1. The relative value of intensity for Example 1 and that for Comparative Example 1 are scaled such that the maximum value is 100%. 
     According to a comparative inspection of rays emitted from P0, P1 and P2 in  FIGS. 10 to 12 , rays of Example 1 are distributed in a wider area than rays of Comparative Example 1. Lights shown in  FIG. 8  and  FIG. 9  are a combination of rays emitted from various points on the surface of the light source. Thus, in the case of Example 1 where rays emitted from each of the various points are distributed in a wider area, difference in position on the surface of the light source has a smaller influence on difference in color. 
     Example 2 
     In  FIG. 2 , the coordinates of the point of intersection of the light receiving surface  101  and the central axis AX are represented as O1 while the coordinates of the point of intersection of the exit surface  103  and the central axis AX are represented as O2. 
     In the present example, the distance T between P0 and O2 is given as below. 
     T=5.513 mm 
     The distance h between P0 and O1 is given as below. 
     h=3.569 mm 
     When distance from O1 in the direction of the central axis AX is represented as z, a shape of the light receiving surface  101  can be represented by the following equation in the range where z is between 0 and 2.689 mm inclusive (0≦z≦2.689 mm). 
     
       
         
           
             
               
                 
                   
                     z 
                     = 
                     
                       
                         
                           cr 
                           2 
                         
                         
                           1 
                           + 
                           
                             
                               1 
                               - 
                               
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     k 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   c 
                                   2 
                                 
                                 ⁢ 
                                 
                                   r 
                                   2 
                                 
                               
                             
                           
                         
                       
                       + 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           N 
                         
                         ⁢ 
                         
                           
                             A 
                             i 
                           
                           ⁢ 
                           
                             r 
                             i 
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     c 
                     = 
                     
                       1 
                       / 
                       R 
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     In the equation, r represents distance from the central axis AX, c represents curvature, R represents radius of curvature, k represents conic constant and Ai represents aspheric coefficient. 
     Table 4 shows numerical values of constants in Equation (1) which represents the light receiving surface  101  of Example 2. 
     
       
         
               
               
               
             
           
               
                   
                 TABLE 4 
               
               
                   
                   
               
             
             
               
                   
                 R 
                 −0.9793 
               
               
                   
                 k 
                 −0.7528 
               
               
                   
                 A1 
                 0.000 
               
               
                   
                 A2 
                 0.000 
               
               
                   
                 A3 
                 0.000 
               
               
                   
                 A4 
                 0.000 
               
               
                   
                   
               
             
          
         
       
     
     A shape of the area of the light receiving surface  101  which extends from z=2.689 mm to the face  105 , that is, a shape of the diffusing area is represented as a third-order spline curve, a point group of which is given below. A third-order spline curve is a smooth curve which passes through given points, in which each segment between adjacent points is connected by an individual third-order polynomial and the individual polynomials are made continuous at all the points. 
     Table 5 shows the above-described point group. 
     
       
         
               
               
               
             
           
               
                   
                 TABLE 5 
               
               
                   
                   
               
               
                   
                 z 
                 r 
               
               
                   
                   
               
             
             
               
                   
                 2.689 
                 1.82 
               
               
                   
                 2.789 
                 1.90 
               
               
                   
                 2.889 
                 1.90 
               
               
                   
                 2.989 
                 1.90 
               
               
                   
                 3.089 
                 1.95 
               
               
                   
                 3.189 
                 1.95 
               
               
                   
                 3.289 
                 1.95 
               
               
                   
                 3.389 
                 2.00 
               
               
                   
                 3.489 
                 2.00 
               
               
                   
                 3.589 
                 2.00 
               
               
                   
                   
               
             
          
         
       
     
       FIG. 13  shows a relationship between z of the light receiving surface  101  and angle φh which a normal to the light receiving surface  101  forms with the central axis AX in the optical element of Example 2. The horizontal axis of  FIG. 13  represents z while the vertical axis represents φh. According to  FIG. 13 , in the range where z is 2.689 mm or less, φh monotonously decreases as z increases. In the range where z is greater than 2.689 mm, φh repeatedly fluctuates as z increases. In other words, in the range where z is greater than 2.689 mm, φh which is a function of z has local maximum values and local minimum values. 
     Specifically, in  FIG. 13 , φh has 3 local maximum values and 3 local minimum values. Minor fluctuations of φh around the local minimum values have been ignored. Difference in φh between a local maximum value and a local minimum value which are adjacent to each other is approximately 30 degrees. 
     When distance from O2 in the direction of the central axis AX is represented as z, a shape of the exit surface  103  around the central axis AX is what does not cause total reflection of rays from the light source on the exit surface and can be represented by the following equation. 
     
       
         
           
             
               
                 
                   
                     z 
                     = 
                     
                       
                         
                           cr 
                           2 
                         
                         
                           1 
                           + 
                           
                             
                               1 
                               - 
                               
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     k 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   c 
                                   2 
                                 
                                 ⁢ 
                                 
                                   r 
                                   2 
                                 
                               
                             
                           
                         
                       
                       + 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           N 
                         
                         ⁢ 
                         
                           
                             A 
                             i 
                           
                           ⁢ 
                           
                             r 
                             i 
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     c 
                     = 
                     
                       1 
                       / 
                       R 
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     In the equation, r represents distance from the central axis AX, c represents curvature, R represents radius of curvature, k represents conic constant and Ai represents aspheric coefficient. 
     Table 6 shows numerical values of constants in Equation (2) which represents the exit surface of Example 2. 
     
       
         
               
               
               
             
           
               
                   
                 TABLE 6 
               
               
                   
                   
               
             
             
               
                   
                 R 
                 −1.000  
               
               
                   
                 K 
                 −8.9797 
               
               
                   
                 A1 
                 0.000 
               
               
                   
                 A2 
                 −5.28E−02 
               
               
                   
                 A3 
                 −4.04E−04 
               
               
                   
                 A4 
                 −6.21E−05 
               
               
                   
                 A5 
                 0.000 
               
               
                   
                 A6 
                  1.72E−06 
               
               
                   
                 A7 
                 0.000 
               
               
                   
                 A8 
                 −3.63E−08 
               
               
                   
                 A9 
                 0.000 
               
               
                   
                 A10 
                 −1.42E−10 
               
               
                   
                   
               
             
          
         
       
     
       FIG. 14  shows a relationship between θr and θi of the optical element of Example 2. The horizontal axis of  FIG. 14  represents θr while the vertical axis represents θi. In the range where θr is approximately 55 degrees or less, θi monotonously increases as θr increases. In the range where θr is greater than approximately 55 degrees, θi increases while repeatedly fluctuating as θr increases. In other words, in the range where θr is greater than approximately 55 degrees, θi which is a function of θr has local maximum values and local minimum values. 
     Specifically, in  FIG. 14 , θi has 3 local maximum values and 3 local minimum values in the range where θr is from approximately 55 degrees to 90 degrees. Minor fluctuations of θi around the local maximum values have been ignored. Difference in θi between a local maximum value and a local minimum value which are adjacent to each other is approximately 15 degrees. 
       FIG. 15  shows a relationship between θr and θe of the optical element of Example 2. The horizontal axis of  FIG. 15  represents θr while the vertical axis represents θe. In the range where θr is approximately 55 degrees or less, θe monotonously increases as θr increases. In the range where θr is greater than approximately 55 degrees, θe increases while repeatedly fluctuating with a peak-to-peak amplitude of approximately 15 degrees as θr increases. In other words, θe which is a function of θr has local maximum values and local minimum values in the range where θr is greater than approximately 55 degrees. 
     Comparative Example 2 
     In the present comparative example, the distance T between P0 and O2 is given as below. 
     T=5.513 mm 
     The distance h between P0 and O1 is given as below. 
     h=3.569 mm 
     When distance from O1 in the direction of the central axis AX is represented as z, a shape of the light receiving surface can be represented by Equation (1). Further, values of constants in Equation (1) are those shown in Table 4. That is, a shape of the light receiving surface of Comparative Example 2 is identical with that of Example 2 in the range where z is 2.689 mm or less, and in the range where z is greater than 2.689 mm, φh which is a function of z does not have a local maximum value or a local minimum value and monotonously decreases as z increases. In other words, the light receiving surface of the optical element of Comparative Example 2 differs from the light receiving surface of Example 2 in that it does not have a diffusing area of the light receiving surface. 
     When distance from O2 in the direction of the central axis AX is represented as z, a shape of the exit surface around the central axis AX is what does not cause total reflection of rays from the light source on the exit surface and can be represented by Equation (2). Further, values of constants in Equation (2) are those shown in Table 6. That is, a shape of the exit surface of Comparative Example 2 is identical with that of Example 2. 
     Performance Comparison Between Example 2 and Comparative Example 2 
     Performance comparison between Example 2 and Comparative Example 2 will be made by comparing light intensity distribution between the case of a combination of the light source shown in  FIGS. 1A and 1B  and the optical element of Example 2 and the case of a combination of the light source shown in  FIGS. 1A and 1B  and the optical element of Comparative Example 2. 
       FIG. 16  shows a light intensity distribution for the case of a combination of the light source shown in  FIGS. 1A and 1B  and the optical element of Example 2. The horizontal axis of  FIG. 16  represents direction which forms angle θ with the central axis AX. The vertical axis of  FIG. 16  represents relative value of intensity of light which is emitted in the direction which forms angle θ with the central axis AX. The solid line in  FIG. 16  represents relative value of intensity of lights which have wavelengths of less than 500 nanometers (lights in a shorter wavelength range). The relative value of intensity is scaled such that the maximum value is 100%. The dashed line in  FIG. 16  represents relative value of intensity of lights which have wavelengths of 500 nanometers or more (lights in a longer wavelength range). The relative value of intensity is scaled such that the maximum value is 100%. 
       FIG. 17  shows a light intensity distribution for the case of a combination of the light source shown in  FIGS. 1A and 1B  and the optical element of Comparative Example 2. The horizontal axis of  FIG. 17  represents direction which forms angle θ with the central axis AX. The vertical axis of  FIG. 17  represents relative value of intensity of light which is emitted in the direction which forms angle θ with the central axis AX. The solid line in  FIG. 17  represents relative value of intensity of lights which have wavelengths of less than 500 nanometers (lights in a shorter wavelength range). The relative value of intensity is scaled such that the maximum value is 100%. The dashed line in  FIG. 17  represents relative value of intensity of lights which have wavelengths of 500 nanometers or more (lights in a longer wavelength range). The relative value of intensity is scaled such that the maximum value is 100%. 
     When  FIG. 16  and  FIG. 17  are compared with each other, difference in intensity of light between the shorter wavelength range and the longer wavelength range is greater in  FIG. 17  which relates to Comparative Example 2. The difference between the both is particularly great in an area where θ is around 60 degrees. When the difference between the both is great, a difference in color is generated. For example, in the case that intensity of the longer wavelength range becomes greater in an area where θ is around 60 degrees as shown in  FIG. 17 , light becomes reddish in the area where θ is around 60 degrees. 
     Thus, the optical element of Example 2 is superior to that of Comparative Example 2 in preventing a difference in color from being generated. 
     Example 3 
     In  FIG. 2 , the coordinates of the point of intersection of the light receiving surface  101  and the central axis AX are represented as O1 while the coordinates of the point of intersection of the exit surface  103  and the central axis AX are represented as O2. 
     In the present example, the distance T between P0 and O2 is given as below. 
     T=5.385 mm 
     The distance h between P0 and O1 is given as below. 
     h=3.829 mm 
     When distance from O1 in the direction of the central axis AX is represented as z, a shape of the light receiving surface  101  can be represented by the following equation in the range where z is between 0 and 1.322 mm inclusive (0≦z≦1.322 mm). 
     
       
         
           
             
               
                 
                   
                     z 
                     = 
                     
                       
                         
                           cr 
                           2 
                         
                         
                           1 
                           + 
                           
                             
                               1 
                               - 
                               
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     k 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   c 
                                   2 
                                 
                                 ⁢ 
                                 
                                   r 
                                   2 
                                 
                               
                             
                           
                         
                       
                       + 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           N 
                         
                         ⁢ 
                         
                           
                             A 
                             i 
                           
                           ⁢ 
                           
                             r 
                             i 
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     c 
                     = 
                     
                       1 
                       / 
                       R 
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     In the equation, r represents distance from the central axis AX, c represents curvature, R represents radius of curvature, k represents conic constant and Ai represents aspheric coefficient. 
     Table 7 shows numerical values of constants in Equation (1) which represents the light receiving surface  101  of Example 3. 
     
       
         
               
               
               
             
           
               
                   
                 TABLE 7 
               
               
                   
                   
               
             
             
               
                   
                 R 
                 −0.8668 
               
               
                   
                 k 
                 −0.7490 
               
               
                   
                 A1 
                 0.000 
               
               
                   
                 A2 
                 0.000 
               
               
                   
                 A3 
                 0.000 
               
               
                   
                 A4 
                 0.000 
               
               
                   
                   
               
             
          
         
       
     
     A shape of the area of the light receiving surface  101  which extends from z=1.322 mm to the face  105 , that is, a shape of the diffusing area is represented by the following equation. 
     
       
         
           
             
               
                 
                   
                     z 
                     = 
                     
                       
                         
                           cr 
                           2 
                         
                         
                           1 
                           + 
                           
                             
                               1 
                               - 
                               
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     k 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   c 
                                   2 
                                 
                                 ⁢ 
                                 
                                   r 
                                   2 
                                 
                               
                             
                           
                         
                       
                       + 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           N 
                         
                         ⁢ 
                         
                           
                             A 
                             i 
                           
                           ⁢ 
                           
                             r 
                             i 
                           
                         
                       
                       + 
                       
                         B 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         Kr 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     c 
                     = 
                     
                       1 
                       / 
                       R 
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     In the equation, r represents distance from the central axis AX, c represents curvature, R represents radius of curvature, k represents conic constant and Ai represents aspheric coefficient. Further, K is a constant. The unit of K is 1/mm. 
     Table 8 shows numerical values of constants in Equation (3) which represents the light receiving surface of Example 3. 
     
       
         
               
               
               
             
           
               
                   
                 TABLE 8 
               
               
                   
                   
               
             
             
               
                   
                 R 
                 −0.8668 
               
               
                   
                 k 
                 −0.7490 
               
               
                   
                 A1 
                 0.000 
               
               
                   
                 A2 
                 0.000 
               
               
                   
                 A3 
                 0.000 
               
               
                   
                 A4 
                 0.000 
               
               
                   
                 B 
                 0.050 
               
               
                   
                 K 
                 52.5 
               
               
                   
                   
               
             
          
         
       
     
       FIG. 18  shows a relationship between z of the light receiving surface  101  and angle φh which a normal to the light receiving surface  101  forms with the central axis AX in the optical element of Example 3. The horizontal axis of  FIG. 18  represents z while the vertical axis represents φh. According to  FIG. 18 , in the range where z is 1.322 mm or less, φh monotonously decreases as z increases. In the range where z is greater than 1.322 mm, φh repeatedly fluctuates as z increases. In other words, in the range where z is greater than 1.322 mm, φh which is a function of z has local maximum values and local minimum values. 
     Specifically, in  FIG. 18 , φh has 4 local maximum values and 3 local minimum values. Minor fluctuations of φh around the local minimum values have been ignored. Difference in φh between a local maximum value and a local minimum value which are adjacent to each other is approximately 30 degrees. 
     When distance from O2 in the direction of the central axis AX is represented as z, a shape of the exit surface  103  around the central axis AX is what does not cause total reflection of rays from the light source on the exit surface and can be represented by the following equation. 
     
       
         
           
             
               
                 
                   
                     z 
                     = 
                     
                       
                         
                           cr 
                           2 
                         
                         
                           1 
                           + 
                           
                             
                               1 
                               - 
                               
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     k 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   c 
                                   2 
                                 
                                 ⁢ 
                                 
                                   r 
                                   2 
                                 
                               
                             
                           
                         
                       
                       + 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           N 
                         
                         ⁢ 
                         
                           
                             A 
                             i 
                           
                           ⁢ 
                           
                             r 
                             i 
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     c 
                     = 
                     
                       1 
                       / 
                       R 
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     In the equation, r represents distance from the central axis AX, c represents curvature, R represents radius of curvature, k represents conic constant and Ai represents aspheric coefficient. 
     Table 9 shows numerical values of constants in Equation (2) which represents the exit surface of Example 3. 
     
       
         
               
               
               
             
           
               
                   
                 TABLE 9 
               
               
                   
                   
               
             
             
               
                   
                 R 
                 −0.6625 
               
               
                   
                 k 
                 −8.5998 
               
               
                   
                 A1 
                 0.000 
               
               
                   
                 A2 
                 −5.35E−02 
               
               
                   
                 A3 
                 −5.32E−04 
               
               
                   
                 A4 
                 −8.50E−04 
               
               
                   
                 A5 
                 0.000 
               
               
                   
                 A6 
                  3.31E−06 
               
               
                   
                 A7 
                 0.000 
               
               
                   
                 A8 
                 −3.94E−08 
               
               
                   
                 A9 
                 0.000 
               
               
                   
                 A10 
                 −3.22E−10 
               
               
                   
                   
               
             
          
         
       
     
       FIG. 19  shows a relationship between θr and θi of the optical element of Example 3. The horizontal axis of  FIG. 19  represents θr while the vertical axis represents θi. In the range where θr is approximately 32 degrees or less, θi monotonously increases as θr increases. In the range where θr is greater than approximately 32 degrees, θi increases while repeatedly fluctuating as θr increases. In other words, in the range where θr is greater than approximately 32 degrees, θi which is a function of θr has local maximum values and local minimum values. 
     Specifically, in  FIG. 19 , θi has 3 local maximum values and 4 local minimum values in the range where θr is from approximately 32 degrees to 90 degrees. Minor fluctuations of θi around the local maximum values have been ignored. Difference in θi between a local maximum value and a local minimum value which are adjacent to each other ranges from 15 degrees to 20 degrees. 
       FIG. 20  shows a relationship between θr and θe of the optical element of Example 3. The horizontal axis of  FIG. 20  represents θr while the vertical axis represents θe. In the range where θr is approximately 32 degrees or less, θe monotonously increases as θr increases. In the range where θr is greater than approximately 32 degrees, θe increases while repeatedly fluctuating with a peak-to-peak amplitude of approximately 15 degrees as θr increases. In other words, in the range where θr is greater than approximately 32 degrees, θe which is a function of θr has local maximum values and local minimum values. 
     Comparative Example 3 
     In the present comparative example, the distance T between P0 and O2 is given as below. 
     T=5.385 mm 
     The distance h between P0 and O1 is given as below. 
     h=3.829 mm 
     When distance from O1 in the direction of the central axis AX is represented as z, a shape of the light receiving surface can be represented by Equation (1). Further, values of constants in Equation (1) are those shown in Table 7. That is, a shape of the light receiving surface of Comparative Example 2 is identical with that of Example 3 in the range where z is 1.322 mm or less, and in the range where z is greater than 1.322 mm, φh which is a function of z does not have a local maximum value or a local minimum value and monotonously decreases as z increases. In other words, the light receiving surface of the optical element of Comparative Example 3 differs from the light receiving surface of Example 3 in that it does not have a diffusing area of the light receiving surface. 
     When distance from O2 in the direction of the central axis AX is represented as z, a shape of the exit surface around the central axis AX is what does not cause total reflection of rays from the light source on the exit surface and can be represented by Equation (2). Further, values of constants in Equation (2) are those shown in Table 9. That is, a shape of the exit surface of Comparative Example 3 is identical with that of Example 3. 
     Performance Comparison Between Example 3 and Comparative Example 3 
     Performance comparison between Example 3 and Comparative Example 3 will be made by comparing light intensity distribution between the case of a combination of the light source shown in  FIGS. 1A and 1B  and the optical element of Example 3 and the case of a combination of the light source shown in  FIGS. 1A and 1B  and the optical element of Comparative Example 3. 
       FIG. 21  shows a light intensity distribution for the case of a combination of the light source shown in  FIGS. 1A and 1B  and the optical element of Example 3. The horizontal axis of  FIG. 21  represents direction which forms angle θ with the central axis AX. The vertical axis of  FIG. 21  represents relative value of intensity of light which is emitted in the direction which forms angle θ with the central axis AX. The solid line in  FIG. 21  represents relative value of intensity of lights which have wavelengths of less than 500 nanometers (lights in a shorter wavelength range). The relative value of intensity is scaled such that the maximum value is 100%. The dashed line in  FIG. 21  represents relative value of intensity of lights which have wavelengths of 500 nanometers or more (lights in a longer wavelength range). The relative value of intensity is scaled such that the maximum value is 100%. 
       FIG. 22  shows a light intensity distribution for the case of a combination of the light source shown in  FIGS. 1A and 1B  and the optical element of Comparative Example 3. The horizontal axis of  FIG. 22  represents direction which forms angle θ with the central axis AX. The vertical axis of  FIG. 22  represents relative value of intensity of light which is emitted in the direction which forms angle θ with the central axis AX. The solid line in  FIG. 22  represents relative value of intensity of lights which have wavelengths of less than 500 nanometers (lights in a shorter wavelength range). The relative value of intensity is scaled such that the maximum value is 100%. The dashed line in  FIG. 22  represents relative value of intensity of lights which have wavelengths of 500 nanometers or more (lights in a longer wavelength range). The relative value of intensity is scaled such that the maximum value is 100%. 
     When  FIG. 21  and  FIG. 22  are compared with each other, difference in intensity of light between the shorter wavelength range and the longer wavelength range is greater in  FIG. 22  which relates to Comparative Example 3. The difference between the both is particularly great in an area where θ is around 65 degrees. When the difference between the both is great, a difference in color is generated. For example, in the case that intensity of the longer wavelength range becomes greater in an area where θ is around 65 degrees as shown in  FIG. 22 , light becomes reddish in the area where θ is around 65 degrees. 
     Thus, the optical element of Example 3 is superior to that of Comparative Example 3 in preventing a difference in color from being generated. 
     Other Preferred Embodiments 
     Optical elements according to the present invention are preferably manufactured by injection molding in which molds are used. In the process, the position of a resin gate through which resin (plastic) is injected to the mold will affect the product. 
       FIGS. 23A and 23B  show the case in which a resin gate  1031  is arranged around the center of the exit surface  103  of an optical element.  FIG. 23A  shows the state in which the resin gate  1031  is arranged.  FIG. 23B  shows a shape of the optical element which has been manufactured using the resin gate  1031  arranged as shown in  FIG. 23A . A resin gate mark  1033  has a scattering surface, which diffuses high-intensity lights around the center. Further, it is preferable particularly when a plane to be illuminated is located nearby, that the scattering surface helps high-intensity rays around the center of the light source diffuse. 
       FIGS. 24A and 24B  show the case in which a portion  1035  in the form of a truncated cone is provided around the center of the exit surface  103  of an optical element, and a resin gate  1037  is arranged on the portion.  FIG. 24A  shows the state in which the resin gate  1037  is arranged.  FIG. 24B  shows a shape of the optical element which has been manufactured using the resin gate  1037  arranged as shown in  FIG. 24A . The portion  1035  in the form of a truncated cone diffuses high-intensity lights around the center, and a resin gate mark has a scattering surface, which diffuses high-intensity lights around the center. Further, it is preferable particularly when a plane to be illuminated is located nearby, that the scattering surface helps high-intensity rays around the center of the light source diffuse. 
       FIG. 25  shows the case in which a single resin gate  1051  is arranged on the bottom face  105  of an optical element. In this embodiment, a resin gate mark does not affect the optical surfaces. 
       FIG. 26  shows the case in which two resin gates  1051 A and  1051 B are arranged on the bottom face  105  of an optical element. In this embodiment, resin gate marks do not affect the optical surfaces. 
     It is preferable that a portion of the exit surface or the bottom face of an optical element is provided with a diffusing structure or a diffusing material. The diffusing structure can be microscopic depressions or projections in a spherical or an aspherical shape on a surface, each of the depressions or each of the projections being included in a circle of diameter of less than 1 mm on the surface. Alternatively, the diffusing structure can be microscopic depressions or projections in a conical, a triangular pyramid, a quadrangular pyramid shape on a surface, each of the depressions or each of the projections being included in a circle of diameter of less than 1 mm on the surface. Alternatively, the diffusing structure can be a grained surface by roughening, a refracting structure including microscopic curved surfaces or prisms such as a microlens array, or a total-reflecting structure including prisms. The diffusing material can be scattering materials such as acrylic powder, polystyrene particles, silicon powder, silver powder, titanium oxide powder, aluminium powder, white carbon, magnesia oxide and zinc oxide. 
       FIG. 27  shows a construction of an optical element which is provided with a diffusing structure or a diffusing material  1039  on the periphery of the exit surface. Portions marked with circles in  FIG. 27  represent the diffusing structure or the diffusing material. According to the optical element of the present embodiment, lights emitted from the periphery of the exit surface are further diffused. 
       FIG. 28  shows a construction of an optical element which is provided with a diffusing structure or a diffusing material  1053  on the bottom face. According to the optical element of the present embodiment, rays which reach a plane to be illuminated via the bottom face of the optical element can be prevented from generating brightness irregularities on the plane to be illuminated. The rays which reach the plane to be illuminated via the bottom face of the optical element may include rays which have undergone total reflection inside the optical element, rays which have been reflected on the plane to be illuminated, and rays from adjacent optical elements. 
     Further, as the structure of the diffusing area of the light receiving surface, the above-described diffusing structure or diffusing material may be provided in place of the above-described shape of the optical surface. 
     Shapes of the light receiving surface and the exit surface of an optical element are not limited to those which are rotationally symmetric around the axis AX. For example, a space around the axis AX may be partitioned based on angle around the optical axis into plural zones and different shapes may be provided in respective zones. The zones may or may not be those with the same angle, such as four zones with 90 degrees or six zones with 60 degrees. 
     Further, in some of the zones alone, a diffusing area may be provided on the light receiving surface. 
     According to the above-described embodiments, different light distributions can be realized for respective directions corresponding to zones around the axis AX. For example, particularly in a specific direction around the axis AX, color difference can be reduced.