Patent Publication Number: US-9838590-B2

Title: Phase-detection auto-focus pixel array and associated imaging system

Description:
BACKGROUND 
     Many digital cameras have autofocus capability. Autofocus may be fully automatic such that the camera identifies objects in the scene and focuses on the objects. In some cases, the camera may even decide which objects are more important than other objects and subsequently focus on the more important objects. Alternatively, autofocus may utilize user input specifying which portion or portions of the scene are of interest. Based thereupon, the autofocus function identifies objects within the portion(s) of the scene, specified by the user, and focuses the camera on such objects. 
     To achieve market adoption, the autofocus function must be reliable and fast such that every time a user captures an image, the camera quickly brings the desired portion, or portions, of the scene into focus. Preferably, the autofocus function is sufficiently fast that the user does not notice any delay between pressing the trigger button and image capture. The autofocus is particularly important for cameras having no means for manual focus, such as compact digital cameras and camera phones. 
     Many digital cameras use contrast autofocus, wherein the autofocus function adjusts the imaging objective to maximize contrast in at least a portion of the scene, thus bringing that portion of the scene into focus. More recently, phase-detection autofocus has gained popularity because it is faster than contrast autofocus. Phase-detection autofocus directly measures the degree of misfocus by comparing light passing through one portion of the imaging objective, e.g., the left portion, with light passing through another portion of the imaging objective, e.g., the right portion. Some digital single-lens reflex cameras include a dedicated phase-detection sensor in addition to the image sensor that captures images. 
     However, this solution is not feasible for more compact and/or less expensive cameras. Therefore, camera manufacturers are developing image sensors with on-chip phase detection, i.e., image sensors with integrated phase detection capability via the inclusion of phase-detection auto-focus (PDAF) pixels in the image sensor&#39;s pixel array. 
       FIG. 1  illustrates one exemplary image sensor  101  with PDAF pixels in an exemplary use scenario  190 . Image sensor  101  is implemented in a digital camera  180  for imaging a scene  150 . Digital camera  180  is, for example, a camera phone or a compact digital camera. Digital camera  180  utilizes the on-chip phase detection capability of image sensor  101  to focus on scene  150 . When focused, digital camera  180  utilizes image sensor  101  to capture a focused image  120 , instead of a defocused image  130 , of scene  150 . 
     Image sensor  101  has a pixel array  200 A that includes at least one dual-diode PDAF pixel  200 .  FIG. 2  is a cross-sectional view of the dual-diode PDAF pixel  200  of pixel array  200 A. Dual-diode PDAF pixel  200  includes photodiodes  211  and  212  having a common color filter  221  and microlens  230 . Microlens  230  has an optical axis  231  centered between photodiodes  211  and  212 . Photodiodes  211  and  212  have respective top surfaces  211 T and  212 T. Dual-diode PDAF pixel  200  may be viewed as including phase-detection pixels  200 L and  200 R, which include photodiode  211  and photodiode  212  respectively. 
       FIGS. 3A-3C  are cross-sectional views of a PDAF imaging system  300  in which a lens  310  forms an image  352  of an off-axis object  350  at an image plane  312  proximate pixel array  200 A. Lens  310  has an optical axis  310 A that intersects pixel array  200 A at a pixel-array center  200 AC. Image  352  is at a radial distance  352 R from optical axis  310 A and pixel-array center  200 AC. Image plane  312  and lens  310  are separated by an image distance  312 Z. 
       FIGS. 3A-3C  illustrate propagation of a chief ray  351 (0), an upper marginal ray  351 (1), and a lower marginal ray  351 (−1). In the cross-sectional view of  FIGS. 3A-3C , pixel array  200 A includes a column of dual-diode PDAF pixels  200  of  FIG. 2 . In  FIG. 3A , pixel array  200 A is in front of image plane  312 . In  FIG. 3B , pixel array  200 A is coplanar with image plane  312 . In  FIG. 3C , pixel array  200 A is behind image plane  312 . 
       FIGS. 3A-3C  also include schematic pixel column responses  303  and  304 , which represent response of, within a column of phase-detection pixels PDAF pixel  200 , (a) left photodiodes  211  and (b) right photodiodes  212 , respectively. 
     In  FIG. 3A , pixel array  200 A is behind image plane  312  such that image  352  is out of focus at pixel array  200 A. Pixel array  200 A is at a distance  311 A from lens  310 , which corresponds to a misfocus distance Δz=Δz A &gt;0 from image plane  312 . Pixel column response  303 A illustrates that a column of left phase-detection pixels detects one intensity peak  303 A′ corresponding to upper marginal ray  351 (1). Pixel column response  304 A illustrates that a column of right phase-detection pixels detects one intensity peak  304 A′ corresponding to lower marginal ray  351 (−1). Intensity peak  304 A′ is closer to optical axis  310 A than intensity peak  303 A′. On pixel array  200 A, intensity peaks  303 A′ and  304 A′ are separated by a distance Δx=Δx A &gt;0. 
     In  FIG. 3B , pixel array  200 A is located at image plane  312  such that image  352  is in focus. Pixel array  200 A is at a distance  311 B from lens  310 , which corresponds to a misfocus distance Δz=Δz B =0 from image plane  312 . Pixel column response  303 B illustrates that a column of left phase-detection pixels detects one intensity peak  303 B′ corresponding rays  351 (−1,0,1) being incident on the same left-phase-detection pixel in the column. Pixel column response  304 B illustrates that a column of right phase-detection pixels detects one intensity peak corresponding to rays  351 (−1,0,1) being incident on the same right-phase-detection pixel in the column. On pixel array  200 A, intensity peaks  303 B and  304 B′ are separated by a distance ΔX=ΔX B , which is illustrated as equal to zero in  FIG. 3B   
     In  FIG. 3C , pixel array  200 A is in front of image plane  312  such that image  352  is out of focus at pixel array  200 A. Pixel array  200 A is at a distance  311 C from lens  310 , which corresponds to a misfocus distance Δz=Δz C &lt;0 from image plane  312 . Pixel column response  303 C illustrates that a column of left phase-detection pixels detects one intensity peak corresponding to upper marginal ray  351 (1). Pixel column response  304 C illustrates that a column of right phase-detection pixels detects one intensity peak corresponding to lower marginal ray  351 (−1). Intensity peak  304 C′ is further from optical axis  310 A than is intensity peak  303 C′. On pixel array  200 A, intensity peaks  303 C′ and  304 C′ are separated by a distance Δx=Δx C &lt;0. 
     One indicator of the accuracy of phase-detection auto-focusing by image sensor  101 , hereinafter “PDAF accuracy,” is how well the magnitude of Δx indicates the magnitude of misfocus Δz. Specifically, with reference to  FIG. 3B , zero misfocus (Δz=0) should correspond to Δx=0. Hence, the smaller the magnitude of Δx is when Δz=0, the higher the PDAF accuracy. 
     SUMMARY OF THE INVENTION 
     In a first embodiment, a PDAF pixel array is disclosed. The PDAF pixel array includes a first pixel and a second pixel. The first pixel is located at a first distance from a center of the PDAF pixel array and includes a first inner photodiode and a first outer photodiode with respect to the center. The first inner photodiode and the first outer photodiode occupy respectively a first inner area and a first outer area. The first inner area divided by the first outer area equals a first ratio. The second pixel is located at a second distance from the center and includes a second inner photodiode and a second outer photodiode with respect to the center. The second inner photodiode and the second outer photodiode occupy respectively a second inner area and a second outer area. The second inner area divided by the second outer area equals a second ratio. The second distance exceeds the first distance and the second ratio exceeds the first ratio. 
     In a second embodiment, a PDAF imaging system is disclosed. The PDAF imaging system includes an imaging lens, and an image sensor, aligned with the imaging lens, having the PDAF pixel array. 
    
    
     
       BRIEF DESCRIPTION OF THE FIGURES 
         FIG. 1  illustrates a prior-art image sensor that includes a prior-art pixel array having PDAF pixels in a use scenario. 
         FIG. 2  is a cross-sectional view of a prior-art dual-diode PDAF pixel of the pixel array of  FIG. 1 . 
         FIGS. 3A-3C  are cross-sectional views of a PDAF imaging system in which the pixel array is at different positions with respect to the focal plane. 
         FIG. 4  is a schematic angular selectivity plot of the on-axis dual-diode PDAF pixel of  FIG. 2 . 
         FIG. 5  is a cross-sectional view of an off-axis multi-diode PDAF pixel of a PDAF pixel array, in an embodiment. 
         FIG. 6  is a schematic angular selectivity plot of the off-axis multi-diode PDAF pixel of  FIG. 5 . 
         FIG. 7  is a cross-sectional view of an off-axis multi-diode PDAF pixel of a PDAF pixel array, in an embodiment. 
         FIG. 8  is a schematic angular selectivity plot of the off-axis multi-diode PDAF pixel of  FIG. 7 . 
         FIG. 9  is a plan view of a first PDAF pixel array that includes a plurality of off-axis multi-diode PDAF pixels of  FIG. 7 , in an embodiment. 
         FIG. 10  is a plan view of a second PDAF pixel array that includes a plurality of off-axis multi-diode PDAF pixels of  FIG. 7 , in an embodiment. 
         FIG. 11  is a plan view of a third PDAF pixel array that includes a plurality of off-axis multi-diode PDAF pixels of  FIG. 7 , in an embodiment. 
         FIG. 12  is a cross-sectional view of the  FIG. 10  PDAF pixel array at an image plane of an imaging system, in an embodiment. 
         FIG. 13  is a plot of exemplary chief-ray angles as a function of normalized position on the image plane of  FIG. 12 . 
         FIG. 14  shows an exemplary plot of optimal differences of inner and outer photodiode widths, vs. chief-ray angle, of PDAF pixels of the  FIG. 7  PDAF pixel array, in an embodiment. 
         FIG. 15  shows an exemplary plot of optimal ratios of inner and outer photodiode widths of a PDAF pixel, vs. chief-ray angle, of the  FIG. 7  PDAF pixel array, in an embodiment. 
         FIG. 16  shows a plot of exemplary photodiode width ratios as a as a function of normalized distance from the center of the PDAF pixel array if  FIG. 7 , in an embodiment. 
     
    
    
     DETAILED DESCRIPTION OF THE EMBODIMENTS 
     Applicant has determined that PDAF accuracy depends on angular sensitivity of dual-diode PDAF pixels  200 .  FIG. 4  is a schematic angular selectivity plot  400  of an on-axis dual-diode PDAF pixel  200 (0), where on-axis refers to where optical axis  310 A of lens  310  intersects pixel array  200 A. Plot  400  includes photodiode response  411  of left photodiode  211  and photodiode response  412  of right photodiode  212  as a function of incident light angle θ. Since dual-diode PDAF pixel  200 (0) is aligned with optical axis  310 A of lens  310 , a chief ray incident thereon is normal to photodiode top surfaces  212 T and  211 T. Photodiode  212  has a peak response for positive incident light angles θ&gt;0. Photodiode  211  has a peak response for negative incident light angles θ&lt;0. Photodiode responses  411  and  412  are equal at relative angle θ r =θ x =0° and have respective peak regions  411 P and  412 P that are symmetric about θ x . Herein, relative angle θ x  denotes the minimum absolute value of θ r  at which photodiode responses of a multi-diode PDAF pixel are equal. Applicant has determined that PDAF accuracy depends on angular sensitivity of dual-diode PDAF pixels, such as dual-diode PDAF pixels  200 . 
       FIG. 5  is a cross-sectional view of an off-axis multi-diode PDAF pixel  500  of a PDAF pixel array  500 A. PDAF pixel array  500 A is for example part of a CMOS image sensor. PDAF pixel  500  is similar to dual-diode PDAF pixel  200  except that microlens  230  is not aligned with photodiodes  211  and  212 . Optical axis  231  is offset by a distance  510  from an interface  531  between photodiode  211  and  212 . Microlens  230  has a principal plane  232  that intersects optical axis  231  at a focus  232 P. A color filter  521  is between microlens  231  and photodiodes  211 ,  212 . 
     PDAF pixel  500  is at a distance r p  from the center of PDAF pixel array  500 A, where r p  is measured from the pixel array center to a location related to PDAF pixel  500 , such as optical axis  231  or interface  531 . Distance r p  is similar to distance  352 R,  FIG. 3 . In an exemplary use scenario, PDAF pixel array  500 A is at the image plane of lens  310  having an effective focal length f. Lens  310  transmits light at a plurality of chief-ray angles (CRAs) χ with respect to optical axis  310 A, such that a “design” CRA χ p  of a chief ray incident on PDAF pixel  500  depends on distance r p . Design CRA χ p  may be related to distance r p  according to tan 
                 χ   p     =     tan   ⁢           ⁢       r   p       d   pa           ,         
hereinafter referred to as Equation (1). For example, Eq. (1) applies at least for a singlet lens.
 
     Design CRA χ p  may be defined without reference to an imaging lens. For example, PDAF pixel  500  may include an opaque structure  525  that has an aperture  525 A therethrough. Aperture  525 A has a center axis  525 A′. Design CRA χ p  may correspond to the propagation angle of a chief ray transmitted by microlens  230  that passes through a specific position within aperture  525 A, such as through center axis  525 A′. Alternatively, design CRA χ p  may be an angle formed by optical axis  231  and a line connecting point  232 P and a point on center axis  525 A′. 
     Alternatively, design CRA χ p  may be defined with reference to edges of photodiodes  211  and  212 . Photodiode  211  has a left edge  211 L. Photodiode  212  has a right edge  212 R. Design CRA χ p  may be the propagation angle of a chief ray transmitted by microlens  230  that passes through a mid-point between edges  211 L and  212 R. Alternatively, design CRA χ p  may be an angle formed by optical axis  231  and a line connecting point  232 P and a mid-point between edges  211 L and  212 R. 
     In Eq. (1), distance d pa  is a characteristic distance between pixel array  200 A and lens  310  along the z-axis of coordinate system  298 . Herein, distances from lens  310  are referenced to a principal plane of lens  310 , unless noted otherwise. Distances  311 A—C of  FIG. 3  are examples of distance d pa . Distance d pa  is for example within a range of image plane distances  312 Z between focal plane  312  and lens  310  where image plane distances range from f to an integer multiple of f . Alternatively, d pa =f . 
     In  FIG. 5 , pixel array  500 A is at a focal plane of an imaging system, not shown, that transmits a chief-ray  551 (0) and marginal rays  551 (±1) thereto. Chief ray  551 (0) propagates to a focus  531 P and forms an angle χ p  with optical axis  231 . Lens  230  refracts marginal rays  551 (±1), also propagating at angle χ p , to a focus  531 P where they intersect chief ray  551 (0). As focus  531 P is within photodiode  212  and rays  551  propagate a longer distance in photodiode  212  than they do in photodiode  211 , photodiode  212  has a stronger response to rays  551  than does photodiode  211 , which is illustrated in  FIG. 6 . 
       FIG. 6  is a schematic angular selectivity plot  600  of off-axis multi-diode PDAF pixel  500  showing photodiode response as a function of relative CRA θ r , which is the chief-ray angle of incident light offset by design CRA χ p . Plot  600  includes photodiode response  611  of left photodiode  211  and photodiode response  612  of right photodiode  212 . Photodiode responses  611  and  612  are equal when angle θ r  equals a “crossing angle” θ x , which in this example is θ x ≈−9°. Photodiode responses  611  and  612  have respective peak regions  611 P and  612 P that are symmetric about crossing angle θ x . 
     Applicant has determined that PDAF accuracy decreases as crossing angle θ x  deviates from zero degrees. For multi-diode PDAF pixels with photodiodes  211  and  212  having equal width, crossing angle θ x  increases with radial distance r p  (e.g., distance  352 R,  FIG. 3 ) of the PDAF pixel from the imaging lens optical axis (e.g., optical axis  310 A). 
       FIG. 7  is a cross-sectional view of an off-axis multi-diode PDAF pixel  700  included in a PDAF pixel array  700 A. PDAF pixel array  700 A is for example part of a CMOS image sensor, and is compatible for use in image sensor  101  as a replacement for pixel array  200 A. In  FIG. 7 , pixel array  700 A is depicted at a focal plane of an imaging system, not shown, that transmits a chief-ray  551 (0) and marginal rays  551 (±1) thereto. 
     Pixel  700  has photodiodes  711  and  712  that are further and closer respectively to a geometric center of PDAF pixel array  700 A. Photodiodes  711  and  712  have respective widths  711 W and  712 W where width  711 W exceeds  712 W. Interface  731  between photodiodes  711  and  712  is at a distance  710  from optical axis  231 . Distance  710  exceeds distance  510  of PDAF pixel  500 ,  FIG. 5 . Photodiodes  711  and  712  are shown having common color filter  521  thereon. Photodiodes  711  and  712  may have different respective color filters thereon without departing from the scope hereof. 
     Pixel  700 A may include opaque structure  525  and corresponding aperture  525 A. Photodiode  711  has a bottom edge  711 B. Photodiode  712  has a top edge  712 T. Design CRA χ p  may be defined in with respect to aperture  525 A or edges  711 B and  712 A in analogous ways to those discussed with respect to pixel  500 A.  FIG. 8  is a schematic angular selectivity plot  800  of off-axis multi-diode PDAF pixel  700  showing photodiode response as a function of relative CRA θ r . Plot  800  includes photodiode response  812  of right photodiode  712  and photodiode response  811  of left photodiode  711 . Photodiode responses  811  and  812  are equal at a value of θ r  within an angular range 820 that includes θ r =0. As photodiode responses  811  and  812  are respective attributes of photodiodes  711  and  712  of PDAF pixel  700 , crossing angle θ x  is an attribute of PDAF pixel  700 . Angular range 820 is for example ±4° about θ r =0. Angular range 820 may be larger, for example, ±8° or ±18° about θ r =0. 
     In an embodiment, the ratio of width  711 W to width  712 W varies with distance r p  of optical axis  231  to a geometric center of PDAF pixel array  700 A, as illustrated in  FIG. 9 .  FIG. 9  is a plan view of a PDAF pixel array  900 A having a plurality of pixels  900 ( m,n ), where 1&lt;m&lt;M and 1&lt;n&lt;N and m, n, M, and N are each positive integers. Each pixel  900  and pixel array  900 A are examples of pixel  700  and PDAF pixel array  700 A, respectively. Each pixel  900  and pixel array  900 A are examples of pixel  700  and PDAF pixel array  700 A, respectively, and are compatible for use in image sensor  101 . 
     Pixel-array center  900 AC is analogous to pixel-array center  200 AC, such that when pixel array  900 A is part of an imaging system including a lens, the lens optical axis is aligned with pixel-array center  900 AC to within tolerances attainable in the art. 
     In the embodiment of  FIG. 9 , pixel array  900 A includes a center pixel 
             900   ⁢           ⁢     (       ⌈     M   2     ⌉     ,     ⌈     N   2     ⌉       )           
(hereinafter “ 900 (C)”) at pixel-array center  900 AC. Pixel array  900 A includes center pixel  900 (C) when, for example, the total number of pixel rows and the total number of pixel columns of pixel array  900 A are each odd. In a different embodiment, pixel array  900 A does not include a center pixel, such that pixel-array center  900 AC is between adjacent pixels  900 , for example, when the total number of pixel rows and the total number of pixel columns of pixel array  900 A are each even.
 
     As illustrated in  FIG. 9 , both M and N are odd, though at least one of M and N may be even without departing from the scope hereof.  FIG. 9  illustrates selected pixels  900  in three regions  901 ,  905 , and  909 . Each pixel  900  has a respective photodiode pair: an inner photodiode  911  and an outer photodiode  912 , where for a given pixel  900 , at least part of inner photodiode  911  is between outer photodiode  912  and pixel-array center  900 AC. For example, pixel  900 (1,1) includes photodiodes  911 (1,1) and  912 (1,1). In  FIG. 9 , inner photodiodes are white and outer photodiodes are hatched. 
     Pixel  900 (C) is centered on the origin (x,y)=(0,0) of coordinate system  298 . For any pixel  900 ( m, n ) centered at coordinates (x m ,y n ), the area ratio R of its inner photodiode  911 ( m, n ) to its outer photodiode  912 ( m, n ) is a function of (x m ,y n ) such that ratio R(x m ,y n ) increases according to the distance between pixel  900 ( m,n ) and pixel-array center  900 AC. In  FIG. 9  for example, ratio R in region  909  exceeds ratio R in region  905 , which exceeds ratio R in region  901 . For example, ratio R(x m ,y n ) is a polynomial in at least one of x m  and y n : R(x m ,y n )=Σ k a k |x m | k +b k |y n | k +c 0 , hereinafter referred to as Equation (2). In Eq. (2), k is a non-negative integer and R(0,0) =c 0 , where for example c 0 =1 and (x m ,y n ) are related to r p  as r p   2 =r m,n   2 =x m   2 +y n   2 . Alternatively, R(x m ,y n ) may be expressed as a function of design CRA χ p  (m, n). For example, R(x m ,y n ) increases linearly as a function of design CRA χ p  (m, n). 
     Pixels along an image sensor diagonal, such as pixel  900 (1,1), are shown to have photodiodes  901  that are triangular. Such diagonal pixels may have photodiodes  901  that are rectangular, for example, similar to those of pixel 
             900   ⁢           ⁢     (     1   ,     ⌈     N   2     ⌉       )           
or pixel
 
     
       
         
           
             900 
             ⁢ 
             
                 
             
             ⁢ 
             
               
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                     ⌈ 
                     
                       M 
                       2 
                     
                     ⌉ 
                   
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                   1 
                 
                 ) 
               
               . 
             
           
         
       
     
     For a given pixel  900 , the interface between its inner and outer photodiodes may be at an angle with respect to a line between the center of the pixel and pixel-array center  900 AC. For example, pixel  900 ( m   1   , n   1 ) has an interface  922  that is orthogonal to a line  922 N in the direction of pixel-array center  900 AC such that its photodiodes are trapezoidal. 
       FIG. 9  illustrates pixel array  900 A with every pixel thereof being a PDAF pixel  700 , PDAF pixel  900  in this instance. In an embodiment, pixel array  900 A includes pixels that differ from PDAF pixel  700 , such pixels with single photodiodes, or dual-photodiode pixels with equal-sized photodiodes.  FIG. 9  includes lines  950  superimposed on pixel array  900 A that intersect the region occupied by center pixel  900 (C), for example, at pixel-array center  900 AC. Lines  950 (1-3) are shown as examples and are not meant to limit orientation of a line  950  through the region occupied by center pixel  900 (C). In an embodiment, pixel array  900 A includes at least one of (a) a plurality of PDAF pixels  700  located beneath line  950 (1), (b) a plurality of PDAF pixels  700  located beneath line  950 (2), and (c) a plurality of PDAF pixels  700  located beneath line  950 (3). 
     In an embodiment, a pixel  900  in region  909 , a pixel  900  in region  905 , and a pixel in region  901  are collinear. Pixels in region  901  are closer to pixel-array center  900 AC than a percentage of the total number of pixels constituting pixel array  900 A. This percentage for example exceeds ninety-five percent, or alternatively is between ninety percent and ninety-five percent. 
       FIG. 10  is a plan view of a PDAF pixel array  1000 A having a plurality of PDAF pixels  1000 ( m,n ), where, as in  FIG. 9 , 1&lt;m&lt;M and 1&lt;n&lt;N and m, n, M, and N are each positive integers. PDAF pixels  1000  and PDAF pixel array  1000 A are examples of PDAF pixel  700  and PDAF pixel array  700 A, respectively. PDAF pixel array  1000 A is similar to PDAF pixel array  900 A, except that for each pixel  1000 , the division between photodiodes is either along the x direction or y direction denoted by coordinate system  298 . 
       FIG. 11  is a plan view of a PDAF pixel array  1100 A having a plurality of PDAF pixels  1100 ( m,n ), where, as in  FIG. 10 , 1&lt;m&lt;M and 1&lt;n&lt;Nand m, n, M, and N are each positive integers. PDAF pixels  1100  and PDAF pixel array  1100 A are examples of PDAF pixel  700  and PDAF pixel array  700 A, respectively. PDAF pixel array  1100 A is similar to PDAF pixel array  1000 A, except that for each pixel  1100 , the division between photodiodes is along the x direction denoted by coordinate system  298 . 
     Each PDAF pixel  700 ,  900 ,  1000 , and  1100  are shown as having two photodiodes. A PDAF pixel, such as PDAF pixels  700 ,  900 ,  1000 , and  1100 , may have more than two photodiodes without departing from the scope hereof. 
     When a pixel array is a the focal plane of an imaging system, the incident angle of a chief ray incident on the pixel array at distance r p  from the pixel array center, that is, CRA χ p , depends on distance r p . In  FIG. 10  illustrates normalized distances  1060 , that is, r p  normalized to the half-width of pixel array  1000 A in the x direction of coordinate system  298 . 
       FIG. 12  is a cross-sectional view of an imaging system  1200  that includes PDAF pixel array  1000 A aligned with optical axis  1200 A of a lens  1210 . Pixel array  1000 A is at image plane  1212  plane of imaging system  1200 . Inclusion of PDAF pixel array  1000 A instead of other examples of PDAF pixel array  700 A is for illustrative purposes only. Other examples of PDAF pixel array  700 A may replace PDAF pixel array  1000 A in  FIG. 12 . Lens  1210  is for example a compound lens that includes a thick lens and a surface designed to reduce chief-ray angles near pixel-array edges. 
       FIG. 12  illustrates a plurality of chief-rays  1250 ( x ) imaged onto PDAF pixel array  1000 A by lens  1210 . Each chief ray  1250 ( x ) is incident on pixel array  1000 A at a respective distance x from optical axis  1200 A, indicated by normalized distances  1060  and at a respective angle  1251 ( x ) with respect to optical axis  1210 A. For example, chief ray  1250 (0.0) propagates parallel to optical axis  1210 A, such that angle  1251 (0.0) is zero degrees, and chief ray  1250 (1.0) propagates at angle  1251 (1.0) 
       FIG. 13  is a plot  1300  of chief-ray angles  1351  as a function of normalized distances  1060  on image plane  1212 . Chief-ray angles  1351  are examples of chief-ray angles  1251  of chief rays  1250 . In particular, chief-ray angles  1351 (0.2, 0.4, 0.6, 0.8, 1.0) represent chief-ray angles of chief rays  1250 (0.2, 0.4, 0.6, 0.8, 1.0), respectively. Plot  1300  may be generated by optical design software used in the art, and provides a mapping between image sensor position and chief-ray angle incident thereon. 
       FIG. 14  shows an exemplary plot  1400  of optimal photodiode width differences  1402 (1-4) as a function of chief-ray angle for an embodiment of PDAF pixel array  700 A.  FIG. 15  shows a plot  1500  of the same data as  FIG. 14 , but with optimum photodiode sizes expressed as photodiode width ratios  1502 (1-4). Photodiode width differences  1402  and photodiode width ratios  1502  result from a finite-difference time-domain simulations of pixel optical response. 
     Photodiode widths refer to, for example, photodiode width  711 W of inner photodiode  711  and photodiode width  712 W of outer photodiode  712 . When inner photodiode  711  and outer photodiode  712  have equal height along the y-axis of coordinate system  298 , each photodiode width ratio  1502  also represents an area ratio of inner photodiode  711  to outer photodiode  712  in a same PDAF pixel  700 . Optimum width differences and optimum width ratios correspond to when photodiode responses  811  and  812  are equal at a value of θ r  within angular range 820, as shown in  FIG. 8 . 
     Width difference  1402 (1) and width ratio  1502 (1) for example correspond to pixel  1000 ( m   1   , n   1 ) ( FIG. 10 ) having inner photodiode  1011 ( m   1   , n   1 ) and outer photodiode  1012 ( m   1   , n   1 ). Width difference  1402 (4) and width ratio  1502 (4) for example correspond to pixel  1000 ( m   4   , n   4 ) having inner photodiode  1011 ( m   4   , n   4 ) and outer photodiode  1012 ( m   4   , n   4 ). Pixels  1000  each of a have a total photodiode height  1023 , as illustrated for pixel  1000 ( m   4   , n   4 ). In the simulations that generated width differences  1402  and width ratios  1502 , height  1023  equals 1.3 μm. 
     Width differences  1402  are fit by a line  1404  with slope m 12 =0.0111 μm/degree and a coefficient of determination R 2 =0.9991. Width ratios  1502  are fit by a line  1504  having a slope m 13 =0.0228/degree and a coefficient of determination R 2 =0.9908. Lines  1404  and  1504  intersect the origin of plots  1400  and  1500 , respectively, which correspond to a chief-ray angle equal to zero and, in a PDAF pixel  700 , photodiode width  711  being equal to photodiode with  712 . 
       FIG. 16  shows a plot  1600  of exemplary photodiode width ratios  1610  (solid curve, r(x n )) as a as a function of normalized distance  1060  ( x   n ) from the center of a PDAF pixel array  700 A. Ratios  1610  correspond to ratios of line  1504  of plot  1500  plotted against normalized distances  1060  instead of chief-ray angle. Normalized distances  1060  are obtained from chief-ray angles of plot  1500  from plot  1300  of  FIG. 13 . 
     Plot  1600  also includes a parabolic fit  1610 F to photodiode width ratios  1610 . Parabolic fit  1610 F satisfies r(x n )=ax n   2 +bx n +1.0, where fit values of a and b are a=−1.16, b=1.91. The root-mean-square error between width ratios  1610  and parabolic fit  1610 F is Δ RMS =1.011×10 −2 . 
     Combinations of Features 
     Features described above as well as those claimed below may be combined in various ways without departing from the scope hereof. The following examples illustrate some possible, non-limiting combinations: 
     (A1) A PDAF pixel array includes a first pixel and a second pixel. The first pixel is located at a first distance from a center of the PDAF pixel array and includes a first inner photodiode and a first outer photodiode with respect to the center. The first inner photodiode and the first outer photodiode occupy respectively a first inner area and a first outer area. The first inner area divided by the first outer area equals a first ratio. The second pixel is located at a second distance from the center and includes a second inner photodiode and a second outer photodiode with respect to the center. The second inner photodiode and the second outer photodiode occupy respectively a second inner area and a second outer area. The second inner area divided by the second outer area equals a second ratio. The second distance exceeds the first distance and the second ratio exceeds the first ratio. 
     (A2) In the PDAF pixel array denoted by (A1), a response of the first inner photodiode and the first outer photodiode may be equal in response to light incident within five degrees of a design chief-ray angle associated with the first pixel. 
     (A3) A PDAF pixel array denoted by one of (A1) and (A2) may further include a third pixel located at a third distance from the center and includes a third inner photodiode and a third outer photodiode with respect to the center. The third inner photodiode and the third outer photodiode occupy respectively a third inner area and a third outer area. The third inner area divided by the third outer area equals a third ratio. The third distance exceeds the second distance and the third ratio exceeds the second ratio. 
     (A4) In the PDAF pixel array denoted by (A3), the first pixel may at least one of (1) overlap the pixel-array center, (2) be closer to the pixel array center than all other pixels of the PDAF pixel array, and (3) be closer to the pixel-array center than ninety percent of all other pixels in the PDAF pixel array. 
     (A5) A PDAF pixel array denoted by one of (A3) and (A4) may further include a fourth pixel located at a fourth distance from the center. The fourth pixel includes a fourth inner photodiode and a fourth outer photodiode with respect to the center. The fourth inner photodiode and the fourth outer photodiode occupy respectively a fourth inner area and a fourth outer area. The fourth inner area divided by the fourth outer area equal a fourth ratio. The fourth distance exceeds the third distance and the fourth ratio exceeds the third ratio. The first, second, third, and fourth ratios are representable as a parabolic function of the respective first, second, third, and fourth distances. 
     (A6) In the PDAF pixel array denoted by (A5), the parabolic function may be a parabolic curve fit to at least four coordinates {(x 1 ,y 1 ), (x 2 ,y 2 ), (x 3 ,y 3 ), (x 4 ,y 4 )} and may have a root-mean-square error relative to the at least four coordinates less than 0.02, where {x 1 ,x 2 ,x 3 ,x 4 } correspond to the first, second, third, and fourth distances respectively and {y 1 ,y 2 ,y 3 ,y 4 } correspond to the first, second, third, and fourth ratios respectively. 
     (A7) A PDAF pixel array denoted by one of (A1) through (A6) may further include a third pixel located at a third distance from the center. The third pixel includes a third inner photodiode and a third outer photodiode with respect to the center. The third inner photodiode and the third outer photodiode occupy respectively a third inner area and a third outer area. The third inner area divided by the third outer area equals a third ratio. The third distance may be less than the second distance, and the third ratio equals the first ratio. Alternatively, the third distance exceeds the second distance, and the third ratio equals the second ratio. 
     (B1) A PDAF imaging system includes an imaging lens and an image sensor aligned with the imaging lens. The image sensor has the PDAF pixel array denoted by (A1). 
     (B2) In the PDAF imaging system denoted by (B1), the first pixel (a) may be located on the PDAF pixel array associated with a first chief-ray angle transmitted by the imaging lens aligned with the PDAF pixel array. A response of the first inner photodiode and the first outer photodiode may be equal in response to light incident within five degrees of the first chief-ray angle. 
     (B3) In a PDAF imaging system denoted by one of (B1) and (B2), the PDAF pixel array may further include a third pixel located at a third distance from the center. The third pixel includes a third inner photodiode and a third outer photodiode. The third inner photodiode and the third outer photodiode occupy respectively a third inner area and a third outer area. The third inner area divided by the third outer area equals a third ratio. The third distance exceeds the second distance and the third ratio exceeds the second ratio. 
     (B4) In a PDAF imaging system denoted by (B3), the first pixel may be at least one of (1) overlap the pixel-array center, (2) be closer to the pixel array center than all other pixels of the PDAF pixel array, and (3) be closer to the pixel-array center than ninety percent of all other pixels in the PDAF pixel array. 
     (B5) In a PDAF imaging system denoted by one of (B3) and (B4), the first, second, and third ratios may be representable a linear function of respective first, second, and third distances. 
     (B6) In a PDAF imaging system denoted by (B5), the linear function may be a least-squares line of best fit to three coordinates {(x 1 ,y 1 ), (x 2 ,y 2 ), (x 3 ,y 3 )}, and may have a coefficient of determination exceeding 0.98, where {x 1 ,x 2 ,x 3 } correspond to the first, second, and third distances respectively and {y 1 ,y 2 ,y 3 } correspond to the first, second, and third ratios respectively. 
     (B7) A PDAF pixel array denoted by one of (B1) through (B6) may further include a third pixel located at a third distance from the center. The third pixel includes a third inner photodiode and a third outer photodiode with respect to the center. The third inner photodiode and the third outer photodiode occupy respectively a third inner area and a third outer area. The third inner area divided by the third outer area equals a third ratio. The third distance may be less than the second distance, and the third ratio equals the first ratio. Alternatively, the third distance exceeds the second distance, and the third ratio equals the second ratio. 
     Changes may be made in the above methods and systems without departing from the scope hereof. It should thus be noted that the matter contained in the above description or shown in the accompanying drawings should be interpreted as illustrative and not in a limiting sense. The following claims are intended to cover all generic and specific features described herein, as well as all statements of the scope of the present method and system, which, as a matter of language, might be said to fall therebetween.