Abstract:
A solid-state imaging device includes a plurality of photo receivers arranged in a two-dimensional array on a semiconductor substrate, the photo receivers receiving exit light from an imaging optical system; and a plurality of condensing lenses arranged over the said plurality of photo receivers, respectively. At least either the plurality of condensing lenses or the plurality of photo receivers are arranged at smaller pitches where an absolute value of an exit pupil position in the imaging optical system is large and at greater pitches where the absolute value of the exit pupil position is small.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a Division of and claims the benefit of priority under 35 U.S.C. §120 from U.S. Ser. No. 11/358,410 filed Feb. 22, 2006, and claims the benefit of priority under 35 U.S.C. §119 from Japanese Patent Application No. 2005-047460 filed Feb. 23, 2005, the entire contents of each of which are incorporated herein by reference. 
    
    
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to a solid-state imaging device and a camera apparatus using the same. More specifically, the invention relates to a miniature camera including an image sensor of charge coupled device (CCD) type or complementary metal oxide semiconductor (CMOS) type and commonly built in a mobile telephone or the like. 
     2. Description of the Related Art 
     Conventionally, CCD type and CMOS type image sensors (solid-state imaging devices) have widely been employed as miniature cameras built in, for example, mobile telephones. Such an image sensor is known including a plurality of photodiodes, each photodiode has a light converging micro-lens provided on the light incident side thereof for increasing the efficiency of photo detection in an image area. 
     An incidence angle of light incident on each photodiode after emitted from a camera lens (imaging optical system) is different between the center and the edge of the imaging area. For this reason, the edge of the imaging area (particularly at each of the four corners in case of an area sensor) is lower in the photosensitivity (the efficiency of photo detection) than the center of the imaging area when the micro-lenses and the photodiodes are arranged at equal pitches, i.e., when the photodiodes are aligned with the corresponding micro-lenses in position. This is because there is employed, as the camera lens, a lens having a lens characteristic according to so-called paraxial ray approximation substantially expressed by z/p=tan θ=θ, where θ is an exit angle of chief-ray from a final plane of an optical system; z is an image height; and e is an exit pupil position, while the exit pupil position hardly depends on the image height z. Meanwhile, the p represents an array pitch of the photodiodes. 
     Upon usage of the camera lens having a lens characteristic according to paraxial ray approximation, the micro-lenses are arranged such that array pitches of the micro-lenses are equally smaller than those of the photodiodes, so that the greater the image height (departing outwardly toward the edge from the center of the imaging area), the greater the dislocation between the photodiode and the micro-lens increases. Namely, the light is made to be obliquely incident on the photodiodes at the edge of the imaging area. Thus, the position of each micro-lens with respect to the position of the corresponding photo diode is gradually offset in the direction of the center as it shifts from the center to the edge of the imaging area. This allows the efficiency of light concentration to be improved over the photodiodes at the edge. Accordingly, the shading at the edge of the imaging area can be corrected, thus ensuring substantially an uniform level of the photosensitivity throughout the imaging area. 
     As camera lenses have recently been thinned and minimized in overall dimensions, many camera lenses having a lens characteristic deviated from the paraxial ray approximation (such as plastic lenses) have been used. Suppose a case of using, for example, a camera lens having a lens characteristic largely deviated from the paraxial ray approximation such that an exit angle of chief-ray increases as the image height becomes greater and then, the exit angle of chief-ray decreases. In this case, the photosensitivity at the edge of the imaging area is lowered in the case where the micro-lenses are arranged so as to be largely displaced from the photodiodes as shifting from the center of the imaging area. 
     With the use of an imaging optical system where the exit angle of chief-ray from the final plane of the optical system is not continuously increased as the image height from the optical axis becomes greater, the array pitch of the micro-lenses in the solid-state imaging device is decreased at least at a region between the center and a predetermined position of the edge of the imaging area and is increased at least at a region of the edge exceeding the predetermined position than the array pitch at the predetermined position. Accordingly, the correction of shading at the edge of the imaging area can be suppressed. Such an attempt has been disclosed in the prior art (see, for example, Jpn. Pat. Appln. KOKAI Publication No. 2004-228645). 
     However, the foregoing attempt permits a plurality of micro-lenses to be arranged at different pitches which are decreased at least at a region between the center and a predetermined position of the edge of the imaging area. When another imaging optical system is used where the exit angle of chief-ray from the final plane of the optical system is increased until the image height rises up to a specific level from the optical axis and then decreased after the specific level of the image height, the correction of shading can hardly be attenuated. 
     In brief, even when the imaging optical system is used where the exit angle of chief-ray from the final plane of the optical system is increased until the image height rises up to a specific level from the optical axis and then decreased after the specific level of the image height, the paraxial ray approximation is substantially established at a region where the image height is close to zero. Accordingly, the efficiency of light concentration can be ensured when the array pitches of the micro-lenses are substantially equally set smaller than the array pitches of the photodiodes. In contrast, when the micro-lenses are arranged at decremental pitches at least at a region between the zero point and the intermediate point of the image height, the photosensitivity is declined. As a result, the efficiency of light concentration can hardly be maintained throughout the imaging area. 
     BRIEF SUMMARY OF THE INVENTION 
     According to a first aspect of the present invention, there is provided a solid-state imaging device comprising: a plurality of photo receivers arranged in a two-dimensional array on a semiconductor substrate, the photo receivers receiving exit light from an imaging optical system; and a plurality of condensing lenses arranged over the said plurality of photo receivers, respectively, wherein at least either said plurality of condensing lenses or said plurality of photo receivers are arranged at smaller pitches where an absolute value of an exit pupil position in the imaging optical system is large and at greater pitches where the absolute value of the exit pupil position is small. 
     According to a second aspect of the present invention, there is provided a camera apparatus comprising: a solid-state imaging device including a plurality of photo receivers arranged in a two-dimensional array on a semiconductor substrate and a plurality of condensing lenses arranged over said plurality of photo receivers, respectively; and an imaging optical system which emits incident light received on said plurality of photo receivers via said plurality of condensing lenses, respectively, wherein at least either said plurality of condensing lenses or said plurality of photo receivers are arranged at smaller pitches where an absolute value of an exit pupil position in the imaging optical system is large and at greater pitches where the absolute value of the exit pupil position is small. 
    
    
     
       BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING 
         FIGS. 1A and 1B  are views each showing an arrangement example of a CMOS image sensor according to a first embodiment of the present invention; 
         FIG. 2  is a view showing the relationship between a camera lens and the image sensor shown in  FIGS. 1A and 1B ; 
         FIG. 3  is a graph for explaining a lens characteristic of the camera lens (between the angle of incidence of chief-ray and the image height); 
         FIG. 4  is a view showing a method for optimizing a scaling amount of a micro-lens; 
         FIGS. 5A and 5B  are graphs illustrating results of the optimum array pitch of the micro-lenses and a displacement amount of the micro-lens from its corresponding photodiode calculated by the optimizing method; 
         FIGS. 6A and 6B  are graphs illustrating comparison between the results shown in  FIGS. 5A and 5B  and calculated by the optimizing method and those of a prior art (Prior art 1); 
         FIGS. 7A and 7B  are graphs illustrating comparison between the results shown in  FIGS. 5A and 5B  and calculated by the optimizing method and those of a prior art (Prior art 2); 
         FIG. 8  is a cross sectional view showing an arrangement example of an image sensor for use in sensitivity calculation; 
         FIG. 9  is a graph illustrating comparison between the results of the sensitivity calculation and those of the prior arts (Prior arts 1 and 2); 
         FIGS. 10A and 10B  are graphs illustrating a lens characteristic of a camera lens (between the angle of incidence of chief-ray and the image height) and the relationship in array pitch between photodiodes and micro-lenses in case of using the camera lens as an imaging optical system, according to a second embodiment of the present invention; 
         FIG. 11  is a cross sectional view of an arrangement example of a CMOS image sensor according to another embodiment of the present invention; and 
         FIG. 12  is a cross sectional view of an arrangement example of a CMOS image sensor according to still another embodiment of the present invention. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Embodiments of the present invention will be described in mode detail with reference to the accompanying drawings. It should be noted that the drawings are illustrative where the dimensions, ratios and the like represent no actual measurements. As a matter of course, there are no relationship in the dimensions and ratios between the drawings. 
     First Embodiment 
       FIGS. 1A and 1B  illustrate a basic configuration of a CMOS image sensor (solid-state imaging device)  10  according to a first embodiment of the present invention.  FIG. 1A  is a plan view of an imaging area provided to the image sensor, and  FIG. 1B  is a cross sectional view taken along the line IB-IB of  FIG. 1A . The embodiment describes, as an example, a CMOS area sensor for used in a miniature camera built in a mobile telephone or the like. The embodiment explains a case of improving (attenuating) the shading characteristic in such a manner that, upon usage of a ca era lens (imaging optical system) having a lens characteristic deviated from the paraxial ray approximation, a scaling amount of micro-lenses (light concentrating lenses) is modified (referred to as array pitch hereinafter) depending on the position of pixels in the imaging area, and the light emitted from a final plane of the camera lens can be incident at higher efficiency on each photodiode (photo-receiver) in every regions of the imaging area, thereby preventing decline in the photosensitivity at, particularly, an intermediate region between the center and the edge of the imaging area. 
     As shown in  FIGS. 1A and 1B , an imaging area  11   a  is formed on the upper surface of a semiconductor substrate  11 . A plurality of photodiodes  13  are arranged in a two-dimensional array on the imaging area  11   a . The plurality of photodiodes  13  in this embodiment are arranged at equal pitches P (P 12 =P 23 =P 34 =P 45 ). 
     A light transmissive film  15  having a uniform thickness is provided on the upper surface of the semiconductor substrate  11 . The light transmissive film  15  has provided thereon a light shielding film  17  for controlling the entrance of light into the photodiodes  13 . The light shielding film  17  has apertures  17   a  in response to the positions of the photodiodes  13 . More specifically, the apertures  17   a  in the light shielding film  17  are substantially identical in the position to the photodiodes  13 . 
     Micro-lenses  19  are provided on the upper surface of the light transmissive film  15  so as to correspond to the respective photodiodes  13 . The micro-lenses  19  are arrayed in the image area  11   a  at different pitches L in this embodiment. For example, the array pitches L of the micro-lenses  19  are smaller than the array pitch P of the photodiodes  13  at the center of the imaging area  11   a , and the array pitches L of the micro-lenses  19  are greater than the array pitch P of the photodiodes  13  at the edge of imaging area  11   a  (L 12 =L 45 &gt;L 23 =L 34 , and L 12 =L 45 &gt;P 12 =P 23 =P 34 =P 45  &gt;L 23 =L 34 ). 
       FIG. 2  illustrates the relationship between the CMOS image sensor  10  having the above configuration and a camera lens  21 . The light emitted from a final plane of the camera lens  21  is made to be incident to the photodiodes  13  on the image sensor  10  depending on the angles of incidence of chief-ray according to the position of pixel, where images of different image heights corresponding to the angles of incidence of chief-ray are formed (light is received). 
     The term “chief-ray” refers to a center beam of a flux of light to be focused at a certain point, and the term “image height” refers to a distance from the optical axis on an imaging plane. In the embodiment of the present invention, an exit pupil position is defined by the following condition. That is, when the angle of incidence of chief-ray is θ at the position of the image height z, the exit pupil position e(z) at the image height z is defined by tan θ=z/e(z). 
       FIG. 3  shows a lens characteristic of the camera lens  21  (the relationship between the angle of incidence of chief-ray and the image height). As the camera lens  21  is made of a plastic material for down-sizing or thickness-reduction, its exit angle (equal to the angle of incidence on the image sensor) becomes greater along the image height being increased. The exit angle of the camera lens  21  is increased at a uniform rate to a level of the image height (around 3.5 mm in this embodiment) from the optical axis and then declined gradually. 
     As described above, the camera lens  21  has a lens characteristic significantly deviated from the conventional paraxial ray approximation (denoted by the broken line in  FIG. 3 ). When such a camera lens  21  not conforming to the paraxial ray approximation is used as the imaging optical system, its angle of incidence of chief-ray towards the micro-lenses  19  becomes close to zero in the vicinity of the center and the edge of the imaging area  11   a  as shown in  FIG. 1B . However, in the intermediate region between the center and the edge of the imaging area  11   a , the angle of incidence of chief-ray towards the micro-lenses  19  is increased monotonously. 
     Consequently, the array pitches L of the micro-lenses  19  are set greater than the array pitch P of the photodiodes  13  at the edge of the imaging area  11   a , so that the micro-lenses  19  are substantially aligned with the corresponding photodiodes  13 . As a result, even in combination with the camera lens  21  having such a lens characteristic as denoted by the real line in  FIG. 3 , the exit light from the camera lens  21  can remain not interrupted by the shielding film  17  at the edge (particularly at four corners) of the imaging area  11   a , thus ensuring the sufficient photosensitivity (the efficiency of photo detection). 
     In the intermediate region between the center and the edge of the imaging area  11   a , the array pitches L of the micro-lenses  19  are set smaller than the array pitch P of the photodiodes  13 . This allows the micro-lenses  19  to be positioned inwardly of the photodiodes  13  (towards the center). As a result, even in combination with the camera lens  21  with the lens characteristic denoted by the real line in  FIG. 3 , the exit light from the camera lens  21  can remain not interrupted by the shielding film  17  at the intermediate region between the center and the edge of the imaging area  11   a  but received efficiently. 
     In other words, the array pitches L of the micro-lenses  19  are not uniform in the imaging area  11   a . More specifically, the array pitches L of the micro-lenses  19  are smaller than the array pitch P of the photodiodes  13  at the center of the imaging area  11   a  (where the absolute value of the exit pupil position is greater) and greater than the array pitch P of the photodiodes  13  at the edge of the imaging area  11   a  (where the absolute value of the exit pupil position is smaller). Accordingly, the photodiodes  13  can be improved in the photosensitivity throughout the imaging area  11   a.    
     A method for optimizing array pitches of the micro-lenses  19  in relation to the corresponding photodiodes  13  will now be described in more detail in case of using the camera lens  21  having the lens characteristic denoted by the real line in  FIG. 3 . 
       FIG. 4  is a view illustrating the method for optimizing the array pitches of the micro-lenses. Explanation will be made for a case of optimizing the positional relationship between the photodiode and the micro-lens at a pixel spaced from the center of the imaging area (or the camera lens). 
     Assuming as shown in  FIG. 4  that the light emitted at an exit angle θ 1  from the camera lens  21  is incident to the micro-lens  19  (at an angle of incidence θ 2 ) having an index of refraction of n 2 , the following formula 1 is established according to the Snell&#39;s law of refraction: 
                 n   1     ⁢   sin   ⁢           ⁢     θ   1       =       n   2     ⁢   sin   ⁢           ⁢     θ   2                     θ   2     =       sin     -   1       ⁡     (         n   1     ⁢   sin   ⁢           ⁢     θ   1         n   2       )             
where n 1  is the index of refraction of air.
 
     Also, the displacement x between the micro-lens  19  and the photodiode  13  is calculated from the following formula 2: 
     
       
         
           
             
               
                 
                   x 
                   = 
                   
                     h 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     tan 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       θ 
                       2 
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     h 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     tan 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           sin 
                           
                             - 
                             1 
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 n 
                                 1 
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 θ 
                                 1 
                               
                             
                             
                               n 
                               2 
                             
                           
                           ) 
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
     When the micro-lens  19  is dislocated by the amount shown in the following formula 3 from the photodiode  13 , it allows the light having passed through the center of the micro-lens  19  to be incident to the center of the photodiode  13 , thereby theoretically permitting no declination in the photosensitivity. 
     
       
         
           
             h 
             ⁢ 
             
                 
             
             ⁢ 
             tan 
             ⁢ 
             
                 
             
             ⁢ 
             
               ( 
               
                 
                   sin 
                   
                     - 
                     1 
                   
                 
                 ⁡ 
                 
                   ( 
                   
                     
                       
                         n 
                         1 
                       
                       ⁢ 
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         θ 
                         1 
                       
                     
                     
                       n 
                       2 
                     
                   
                   ) 
                 
               
               ) 
             
           
         
       
     
     Assuming that the shift amount of the micro-lens  19  from its corresponding photodiode  13  at the nth pixel from the center of the imaging area  11   a  is x(n) (in this case, when the photodiode  13  is displaced to and from the center, its displacement is positive and negative respectively), it has been found that the layout design so as to satisfy the following formula 4 is optimum in optical design. 
               x   ⁡     (   n   )       =     h   ⁢           ⁢     tan   ⁡     (       sin     -   1       ⁡     (         n   1     ⁢   sin   ⁢           ⁢       θ   1     ⁡     (   pn   )           n   2       )       )               
where p is the array pitch (pixel pitch) of the photodiodes  13 , and has a relationship of z=np (z being the image height).
 
     Next, the pixel pitch is determined. First, a value pm(n) of the array pitch of the micro-lens  19  at the nth pixel from the center of the imaging area  11   a  is calculated from the following formula (5): 
     
       
         
           
             
               
                 p 
                 m 
               
               ⁡ 
               
                 ( 
                 n 
                 ) 
               
             
             = 
             
               
                 p 
                 - 
                 
                   ( 
                   
                     
                       x 
                       ⁡ 
                       
                         ( 
                         
                           n 
                           + 
                           1 
                         
                         ) 
                       
                     
                     - 
                     
                       x 
                       ⁡ 
                       
                         ( 
                         n 
                         ) 
                       
                     
                   
                   ) 
                 
               
               ≈ 
               
                 p 
                 - 
                 
                   
                     ∂ 
                     
                       x 
                       ⁡ 
                       
                         ( 
                         n 
                         ) 
                       
                     
                   
                   
                     ∂ 
                     n 
                   
                 
               
             
           
         
       
     
     When x(n) optimized by the above formula 4 is assigned to the formula 5, the value pm(n) of the array pitch of the micro-lens  19  is expressed by the following formula (6): 
     
       
         
           
             
               
                 
                   
                     
                       p 
                       m 
                     
                     ⁡ 
                     
                       ( 
                       n 
                       ) 
                     
                   
                   = 
                   
                     p 
                     - 
                     
                       
                         ∂ 
                         
                           x 
                           ⁡ 
                           
                             ( 
                             n 
                             ) 
                           
                         
                       
                       
                         ∂ 
                         n 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     p 
                     - 
                     
                       h 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           sec 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               sin 
                               
                                 - 
                                 1 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   
                                     n 
                                     1 
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       θ 
                                       1 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       np 
                                       ) 
                                     
                                   
                                 
                                 
                                   n 
                                   2 
                                 
                               
                               ) 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         1 
                         
                           
                             1 
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     
                                       n 
                                       1 
                                     
                                     ⁢ 
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       
                                         θ 
                                         1 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         z 
                                         ) 
                                       
                                     
                                   
                                   
                                     n 
                                     2 
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                       
                       ⁢ 
                       
                         
                           n 
                           1 
                         
                         
                           n 
                           2 
                         
                       
                       ⁢ 
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           θ 
                           1 
                         
                         ⁡ 
                         
                           ( 
                           np 
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           ∂ 
                           
                             
                               θ 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               np 
                               ) 
                             
                           
                         
                         
                           ∂ 
                           n 
                         
                       
                     
                   
                 
               
             
           
         
       
     
     Consequently, it has been found that the arrangement of the micro-lens  19  at the nth pixel from the center of the imaging area  11   a  according to the formula (6) is optimum in optical design. 
     For examining the value pm(n) of the array pitch of the micro-lens  19 , the above formula 6 is differentiated with n. Then, the result is expressed by the following formula (7): 
     
       
         
           
             
               
                 
                   
                     
                       
                         p 
                         m 
                       
                       ⁡ 
                       
                         ( 
                         n 
                         ) 
                       
                     
                     
                       ∂ 
                       n 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       ∂ 
                       
                         ∂ 
                         n 
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           - 
                           h 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             n 
                             1 
                           
                           
                             n 
                             2 
                           
                         
                         ⁢ 
                         
                           
                             sec 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 sin 
                                 
                                   - 
                                   1 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     
                                       n 
                                       1 
                                     
                                     ⁢ 
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       
                                         θ 
                                         1 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         np 
                                         ) 
                                       
                                     
                                   
                                   
                                     n 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           1 
                           
                             
                               1 
                               - 
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         n 
                                         1 
                                       
                                       ⁢ 
                                       sin 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         
                                           θ 
                                           1 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           np 
                                           ) 
                                         
                                       
                                     
                                     
                                       n 
                                       2 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                             
                           
                         
                         ⁢ 
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             θ 
                             1 
                           
                           ⁡ 
                           
                             ( 
                             np 
                             ) 
                           
                         
                         ⁢ 
                         
                           
                             ∂ 
                             
                               
                                 θ 
                                 1 
                               
                               ⁡ 
                               
                                 ( 
                                 np 
                                 ) 
                               
                             
                           
                           
                             ∂ 
                             n 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       2 
                       ⁢ 
                       h 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           n 
                           1 
                         
                         
                           n 
                           2 
                         
                       
                       ⁢ 
                       
                         
                           sec 
                           3 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               sin 
                               
                                 - 
                                 1 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   
                                     n 
                                     1 
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       θ 
                                       1 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       np 
                                       ) 
                                     
                                   
                                 
                                 
                                   n 
                                   2 
                                 
                               
                               ) 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           
                             n 
                             1 
                           
                           ⁢ 
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               θ 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               np 
                               ) 
                             
                           
                         
                         
                           n 
                           2 
                         
                       
                       ⁢ 
                       
                         1 
                         
                           
                             1 
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     
                                       n 
                                       1 
                                     
                                     ⁢ 
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       
                                         θ 
                                         1 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         np 
                                         ) 
                                       
                                     
                                   
                                   
                                     n 
                                     2 
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                       
                       ⁢ 
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           θ 
                           1 
                         
                         ⁡ 
                         
                           ( 
                           np 
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           ∂ 
                           
                             
                               θ 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               np 
                               ) 
                             
                           
                         
                         
                           ∂ 
                           n 
                         
                       
                     
                     + 
                   
                 
               
             
             
               
                 
                     
                   ⁢ 
                   
                     
                       
                         h 
                         2 
                       
                       ⁢ 
                       
                         
                           n 
                           1 
                         
                         
                           n 
                           2 
                         
                       
                       ⁢ 
                       
                         
                           sec 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               sin 
                               
                                 - 
                                 1 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   
                                     n 
                                     1 
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       θ 
                                       1 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       np 
                                       ) 
                                     
                                   
                                 
                                 
                                   n 
                                   2 
                                 
                               
                               ) 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           ( 
                           
                             1 
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     
                                       n 
                                       1 
                                     
                                     ⁢ 
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       
                                         θ 
                                         1 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         np 
                                         ) 
                                       
                                     
                                   
                                   
                                     n 
                                     2 
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                           ) 
                         
                         
                           - 
                           
                             3 
                             2 
                           
                         
                       
                       ⁢ 
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           θ 
                           1 
                         
                         ⁡ 
                         
                           ( 
                           np 
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           ∂ 
                           
                             
                               θ 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               np 
                               ) 
                             
                           
                         
                         
                           ∂ 
                           n 
                         
                       
                     
                     + 
                   
                 
               
             
             
               
                 
                     
                   ⁢ 
                   
                     
                       h 
                       ⁢ 
                       
                         
                           n 
                           1 
                         
                         
                           n 
                           2 
                         
                       
                       ⁢ 
                       
                         
                           sec 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               sin 
                               
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   
                                     n 
                                     1 
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       θ 
                                       1 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       np 
                                       ) 
                                     
                                   
                                 
                                 
                                   n 
                                   2 
                                 
                               
                               ) 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         1 
                         
                           
                             1 
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     
                                       n 
                                       1 
                                     
                                     ⁢ 
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       
                                         θ 
                                         1 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         np 
                                         ) 
                                       
                                     
                                   
                                   
                                     n 
                                     2 
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                       
                       ⁢ 
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           θ 
                           1 
                         
                         ⁡ 
                         
                           ( 
                           np 
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           ∂ 
                           
                             
                               θ 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               np 
                               ) 
                             
                           
                         
                         
                           ∂ 
                           n 
                         
                       
                     
                     - 
                   
                 
               
             
             
               
                 
                     
                   ⁢ 
                   
                     h 
                     ⁢ 
                     
                       
                         n 
                         1 
                       
                       
                         n 
                         2 
                       
                     
                     ⁢ 
                     
                       
                         sec 
                         2 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             sin 
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   n 
                                   1 
                                 
                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     θ 
                                     1 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     np 
                                     ) 
                                   
                                 
                               
                               
                                 n 
                                 2 
                               
                             
                             ) 
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       1 
                       
                         
                           1 
                           - 
                           
                             
                               ( 
                               
                                 
                                   
                                     n 
                                     1 
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       θ 
                                       1 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       np 
                                       ) 
                                     
                                   
                                 
                                 
                                   n 
                                   2 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                     ⁢ 
                     cos 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         θ 
                         1 
                       
                       ⁡ 
                       
                         ( 
                         np 
                         ) 
                       
                     
                     ⁢ 
                     
                       
                         
                           ∂ 
                           2 
                         
                         ⁢ 
                         
                           
                             θ 
                             1 
                           
                           ⁡ 
                           
                             ( 
                             np 
                             ) 
                           
                         
                       
                       
                         ∂ 
                         
                           n 
                           2 
                         
                       
                     
                   
                 
               
             
           
         
       
     
     When the image height is about 0 mm, the above formula 7 approximates to the following formula 8.
 
θ 1 ( np )≅0, sin θ 1 ( np )≅0, cos θ 1 ( np )≅sec θ 1 ( np )≅1
 
     Then, the above formula 7 is expressed by the following formula (9): 
     
       
         
           
             
               
                 
                   p 
                   m 
                 
                 ⁡ 
                 
                   ( 
                   n 
                   ) 
                 
               
               
                 ∂ 
                 n 
               
             
             = 
             
               
                 - 
                 h 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                 
                   n 
                   1 
                 
                 
                   n 
                   2 
                 
               
               ⁢ 
               
                 
                   
                     ∂ 
                     2 
                   
                   ⁢ 
                   
                     
                       θ 
                       1 
                     
                     ⁡ 
                     
                       ( 
                       np 
                       ) 
                     
                   
                 
                 
                   ∂ 
                   
                     n 
                     2 
                   
                 
               
             
           
         
       
     
     In case of the camera lens satisfying the following formula (10), the value pm(n) of the array pitch of the micro-lenses  19  is a monotonously increasing function for the image height of about 0 mm. 
     
       
         
           
             
               
                 
                   ∂ 
                   2 
                 
                 ⁢ 
                 
                   
                     θ 
                     1 
                   
                   ⁡ 
                   
                     ( 
                     np 
                     ) 
                   
                 
               
               
                 ∂ 
                 
                   n 
                   2 
                 
               
             
             &lt; 
             0 
           
         
       
     
     It has been then found that this embodiment is more advantageous in optical design of pitch arrangement than the prior art (disclosed in Jpn. Pat. Appln. KOKAI Publication No. 2004-228645) where the array pitches of micro-lenses are decrementally varied about 0 mm of the image height. 
     In an actual camera lens, the above formula (10) is often established. For example, the known report about such a camera lens, “Basic and development of photographic lenses” by Toshinori Ogura, Asahi Sonorama, 1995, describes that flux of light to fall on a film is projected as if departing from the exit pupil, which is an aperture image viewed from the rear side of the lens, and focused on an imaging point. 
     In case of using the camera lens (where the exit pupil position remains constant regardless of the image height) having a lens characteristic conforming to the above report, the relationship among the angle of incidence θ 1 , the exit pupil position e, and the image height z is expressed by the following formula (11): 
     
       
         
           
             z 
             = 
             
               e 
               ⁢ 
               
                   
               
               ⁢ 
               tan 
               ⁢ 
               
                   
               
               ⁢ 
               θ 
             
           
         
       
       
         
           
             θ 
             = 
             
               
                 
                   tan 
                   
                     - 
                     1 
                   
                 
                 ⁡ 
                 
                   ( 
                   
                     z 
                     e 
                   
                   ) 
                 
               
               = 
               
                 
                   tan 
                   
                     - 
                     1 
                   
                 
                 ⁡ 
                 
                   ( 
                   
                     np 
                     e 
                   
                   ) 
                 
               
             
           
         
       
     
     Then, the following formula (12) is established. 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∂ 
                         2 
                       
                       ⁢ 
                       
                         
                           θ 
                           1 
                         
                         ⁡ 
                         
                           ( 
                           np 
                           ) 
                         
                       
                     
                     
                       ∂ 
                       
                         n 
                         2 
                       
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       ∂ 
                       
                         ∂ 
                         n 
                       
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           
                             θ 
                             1 
                           
                           ⁡ 
                           
                             ( 
                             np 
                             ) 
                           
                         
                       
                       
                         ∂ 
                         n 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       p 
                       e 
                     
                     ⁢ 
                     
                       ∂ 
                       
                         ∂ 
                         n 
                       
                     
                     ⁢ 
                     
                       1 
                       
                         1 
                         + 
                         
                           
                             ( 
                             
                               np 
                               e 
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       p 
                       e 
                     
                     ⁢ 
                     
                       ∂ 
                       
                         ∂ 
                         n 
                       
                     
                     ⁢ 
                     
                       1 
                       
                         1 
                         + 
                         
                           
                             ( 
                             
                               np 
                               e 
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       - 
                       2 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         n 
                         ⁡ 
                         
                           ( 
                           
                             p 
                             e 
                           
                           ) 
                         
                       
                       3 
                     
                     ⁢ 
                     
                       
                         ( 
                         
                           1 
                           
                             1 
                             + 
                             
                               
                                 ( 
                                 
                                   np 
                                   e 
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                         ) 
                       
                       2 
                     
                   
                 
               
             
           
         
       
     
     As understood, the above formula (12 ) always represents a negative value. It is thus apparent from the formula (12) that the optimizing method of this embodiment is highly effective 
     Although the foregoing description concerns the specific configuration for ease of the understanding, its actual optimizing process incorporates the following steps. 
     &lt;1&gt; Check the specifications (the angle of incidence θ 1  from the cameral lens  21 , the thickness of the light transmissive film  15  (the height h), the index of refraction n 2 , and so on). 
     &lt;2&gt; Calculate the displacement x between the position of incident light received by the micro-lens  19  and the position of incident light projected on the imaging plane of the corresponding photodiode  13  by optical calculation. 
     &lt;3&gt; Shift the position of the micro-lens  19  by the displacement x calculated in the step &lt;2&gt;. 
     &lt;4&gt; Subject the shift amount calculated by the step &lt;3&gt; to a differential operation to calculate the value pm(n) of the array pitch of the micro-lens  19 . 
     The array pitches L of the micro-lenses  19  are not strictly set to the value pm(n) obtained by the above optimizing method. The array pitches L may be set in consideration of economical effects without departing from the scope of the present invention. In this case, it is necessary that the photosensitivity is determined with an optical simulator by using the value pm(n) of the array pitches L of the micro-lenses  19 , thereby confirming that no significant drop in the photosensitivity is measured. 
     Particularly when the overall configuration becomes complex, for example, a plurality of light transmissive films exist which are different in the index of refraction, the optical calculation based on the teaching of the present invention is actually conducted. Using the displacement x calculated by the optical calculation, the array pitches L of the micro-lenses  19  may finally be determined. 
     The specific design and photosensitivity calculation of the array pitches of the micro-lenses will be described in more detail. It is now assumed that the array pitch of the photodiodes  13  is fixed to 5 μm with the camera lens  21  having such a lens characteristic as denoted by the real line in  FIG. 3 . 
       FIGS. 5A and 5B  illustrates results (optimum design value) of actual calculation of the optimum array pitch L of the micro-lens  13  and the displacement x of the micro-lens  13  from the corresponding photodiode (PD)  13  by the optimizing method described above. The profile is based on the configuration shown in  FIG. 4  where the indices of refraction n 1  and n 2  are 1.0 and 1.5 respectively while the height h is 10 μm. The profile shown in  FIG. 5A  represents the relationship between the image height and the array pitch L of the micro-lens  19 . The profile shown in  FIG. 5B  represents the relationship between the image height and the displacement x of the micro-lens  19 . 
     As apparent from  FIG. 5A , the array pitch P of the photo-diode  13  is uniform (5 μm in this embodiment) regardless of the image height. The array pitch L of the micro-lens  19  is approximately 4.9964 μm where the image height is 0 mm and monotonously increased to about 5.005 μm at the image height of 5 mm. 
     As shown in  FIG. 5B , the displacement x of the micro-lens  19  from the corresponding photodiode  13  is 0 μm where the image height is 0 mm. The displacement x is then increased to the maximum (about 1.85 μm) where the image height is about 3.5 mm (70% in percentage calculated from 3.5/5.0). As the image height further increases, the displacement x starts being declined and is finally minimized to about 1.2 μm where the image height is 5 mm. 
     In this embodiment, the image sensor  10  shown in  FIGS. 1A and 1B  is fabricated with the obtained optimum design value. More specifically, when the image height is 0 mm or the absolute value of the exit pupil position is greater, the array pitch L of the micro-lens  19  stays at approximately 4.9964 μm as denoted by the broken line in  FIG. 5A , which is smaller than the array pitch P of the photodiode  13 . When the image height is 5 mm or the absolute value of the exit pupil position is smaller, the array pitch L of the micro-lens  19  is increased to approximately 5.005 μm which is greater than the array pitch P of the photodiode  13 . 
       FIGS. 6A and 6B  and  FIGS. 7A and 7B  illustrate comparisons in both the optimum array pitch L of the micro-lens  19  and the displacement x of the micro-lens  19  from its corresponding photodiode  13  between the settings of this embodiment and the settings of the prior art where the array pitch L of the micro-lens  19  is decreased when the image height is low and increased when the same is high. 
     The array pitch in Prior art 1 shown in  FIG. 6A  is equal to the optimum design value of this embodiment where the image height is approximately 0 mm. As the image height increases, the array pitch becomes smaller. For example, as shown in  FIG. 6B , the displacement x of the micro-lens from the photodiode in Prior art 1 is close to the optimum design value of this embodiment where the image height is approximately 0 mm. As the image height increases, the discrepancy in the displacement becomes great. When the image height is about 3.5 mm, the displacement in Prior art 1 is deviated by approximately 1 μm from the optimum design value of this embodiment. Finally, when the image height increases to about 5 mm, the displacement becomes similar to the optimum design value. 
     Alternatively, the array pitch in Prior art 2 shown in  FIG. 7A  is as greater (as 4.998 μm) than the optimum design value of this embodiment where the image height is approximately 0 mm. As the image height increases, the array pitch becomes smaller. As shown in  FIG. 7B , the displacement x of the micro-lens from the photodiode in Prior art 2 is smaller by 0.4 μm than the optimum design value of this embodiment where the image height is approximately 2 mm. As the image height increases to about 4 mm, the displacement in Prior art 2 becomes greater by about 0.5 μm than the optimum design value. Finally, when the image height reaches approximately 5 mm, the displacement becomes decreased as close to the optimum design value. 
       FIG. 8  illustrates an arrangement example of the image sensor  10  used for calculating the photosensitivity. In this case, the photosensitivity of light at 400 nm of the wavelength is measured when the pixel pitch is 5 μm, the height h of the micro-lens  19  including the light transmissive film  15  is 10 μm, the thickness of the micro-lens  19  is 0.9 μm, the width of each aperture  17   a  in the light shielding film  17  is 2 μm, the height of the light shielding film  17  from the photodiode 13 is 0.3 μm, the index of refraction θ 2  of the micro-lens  19  is 1.5, and the index of refraction θ 1  of air is 1.0. The calculation of the photosensitivity is conduced by using a commercial optical simulator for image sensors (e.g., made of Link Research). 
       FIG. 9  illustrates comparisons in the measurement of the photosensitivity between this embodiment and the prior art (Prior arts 1 and 2). As apparent, the image sensor  10  of this embodiment in which the positional relationship between each micro-lens  19  and its corresponding photodiode  13  is optimized exhibits favorably a constant level of the photosensitivity regardless of the image height. 
     On the other hand, the measurement of the photosensitivity in Prior art 1 is declined in a range of 3 to 5 mm of the image height. The measurement of the photosensitivity in Prior art 2 is also as low as a half about 4.5 mm of the image height. 
     As described, this embodiment allows the positional relationship between the micro-lens and its corresponding photodiode to be optimized in the center, the intermediate, and the edge of the imaging area. Accordingly, each photodiode in the imaging area can receive the incident light at a desirable efficiency. More specifically, even if a camera lens is used of which the exit angle of chief-ray from the final plane is increased as the image height increases and then decreased after a particular level of the image height, the photosensitivity on the photodiodes can remain unchanged and uniform regardless of the image height while the correction of shading is properly performed throughout the imaging area. 
     Second Embodiment 
       FIGS. 10A and 10B  illustrate a lens characteristic of a camera lens and the positional relationship between each micro-lens and its corresponding photodiode in an optical imaging system equipped with the camera lens, according to a second embodiment of the present invention. The camera lens will be described assuming that its angle of incidence of chief-ray is non-linearly increased (at a positive gradation) as the image height changes. 
     As shown in  FIG. 10A , the camera lens according to the second embodiment is differentiated from that of the first embodiment (see, for example,  FIG. 3 ) in that the angle of incidence of chief-ray is non-linearly increased as the image height changes and its gradation is not negative. 
     The array pitch of the photodiode remains constant at 3.3 μm regardless of the image height, as shown in  FIG. 10B . However, the array pitch of the micro-lens is varied depending on the image height. For example, the array pitch is about 3.2 μm, which is smaller than the array pitch of the photodiode, when the image height is 0 mm. Also, the array pitch is about 3.29 μm, which is very close to but bit smaller than the array pitch of the photodiode, when the image height is 10 mm. 
     Similar to the first embodiment, the second embodiment allows the array pitch of the micro-lens to be modified throughout the imaging area in response to the characteristic of the camera lens (between the angle of incidence of chief-ray and the image height). More specifically, when the camera lens is used of which the exit angle of chief-ray from the final plane is gradually increased before the image height reaches a specific level, the photosensitivity remain unchanged, thus successfully attenuating the undesired effect of shading throughout the imaging area. As each of the photodiodes in the imaging area receives the incident light at higher efficiency, its pixel sensitivity can be improved thus suppressing the effect of shading. 
     In both the first and second embodiments, the photodiodes are arranged at equal pitches while the micro-lenses are arranged at varying pitches. The arrangement is not limited to the foregoing arrangement but may be constructed in which the micro-lenses  19  are arranged at equal pitches while the photodiodes  13  are arranged at varying pitches as shown in  FIG. 11  (L 12 =L 23 =L 34 =L 45  and P 12 =P 45 &gt;L 12 =L 23 =L 34 =L 45 &gt;P 23 =P 34 ). 
     Alternatively, all the micro-lenses and the photodiodes are arranged at varying pitches as shown in  FIG. 12  (L 12 =L 45 &gt;L 23 =L 34 , P 12 =P 45 &gt;P 23 =P 34 , and L 12 =L 45 =P 12  =P 45 &gt;L 23 =L 34 =P 23 =P 34 ). 
     This is not limited to the area sensor but may be a linear sensor, in which the effect of shading can be attenuated with equal success. 
     Although the micro-lenses are employed for concentration of incident light from a cameral lens, they may be replaced by so-called intra-layer lenses for providing the same advantages. Moreover, a combination of the micro-lenses and the intra-layer lenses can be provided with equal success. 
     The present invention is applicable to any CCD image sensor as well as the CMOS image sensors. The CMOS image sensors are however greater in the efficacy of advantage than the CCD image sensors since it is wider in the distance between the photodiodes and the micro-lenses. 
     Additional advantages and modifications will readily occur to those skilled in the art. Therefore, the invention in its broader aspects is not limited to the specific details and representative embodiments shown and described herein. Accordingly, various modifications may be made without departing from the spirit or scope of the general inventive concept as defined by the appended claims and their equivalents.