Patent Publication Number: US-9897786-B2

Title: Two-surface narrow field-of-view compounds lens

Description:
BACKGROUND 
     Digital camera modules are used in a variety of consumer, industrial and scientific imaging devices to produce still images and/or video. Applications of digital camera modules include image-based recognition applications such as barcode scanning and iris recognition. A camera for such applications may include an imaging lens with relatively large depth of field compared to conventional lenses. Such a large depth of field enables a device using the camera to recognize an object to be relatively insensitive to the object&#39;s distance from the imaging lens. 
     For a fixed imaging lens focal length, the depth of field of the imaging lens is approximately linearly proportional to the lens&#39;s f-number N, where N is the ratio of the lens&#39;s effective focal length to its entrance pupil diameter D. See, for example, The Manual of Photography, 9th ed. by Jacobson et al, Focal Press, 2000. The field of view 2α of a camera with an imaging lens having focal length f and an image sensor with diagonal length d is 
               2   ⁢   α     =     2   ⁢       arctan   ⁡     (     d     2   ⁢   f       )       .             
Expressed in terms of f-number N=f/D,
 
               α   =     2   ⁢     arctan   ⁡     (     d     2   ⁢     D   ·   N         )           ,         
which illustrates that for a constant entrance pupil diameter D, field of view a decreases as f-number N increases. Since depth of field is approximately linearly proportional to the lens&#39;s f-number N, field of view 2α also decreases as depth of field increases.
 
     Image-based recognition devices require a camera module having a lens with a smaller field of view (FOV) than lenses in conventional camera modules, while producing images with line-width resolution minimally reduced compared to images formed by conventional camera modules. 
     Conventional narrow-FOV camera modules achieve a small point of view while maintaining image quality of a larger FOV camera by employing telescope-like compound lenses that include several optical surfaces. A disadvantage of such camera modules is that the manufacturing cost of a compound lens increases with number of optical surfaces. 
     SUMMARY OF THE INVENTION 
     A two-surface narrow field-of-view (FOV) compound lens for producing an image of an object at an image plane of an imaging system is disclosed. In an embodiment, the lens includes a biplanar substrate between a plano-convex lens and a plano-concave lens having a common optical axis. The plano-convex lens has a first planar surface on a first side of the biplanar substrate and is formed of a material having a first Abbe number. The plano-concave lens has a second planar surface on a second side of the biplanar substrate opposite the first side, and is formed of a material having a second Abbe number less than the first Abbe number. 
     In an embodiment, the first Abbe number exceeds 50 and the second Abbe number being less than 35. In an embodiment, the biplanar substrate is formed of a material having a third Abbe number that exceeds the second Abbe number. In an embodiment, the plano-convex lens has a focal length F1, the plano-concave lens has a focal length F2, the ratio F2/F1 satisfying −1.4&lt;F2/F1&lt;−0.9. In an embodiment, the biplanar substrate, the plano-convex lens and the plano-concave lens collectively have an effective focal length F such that the image is formed at the image plane located a distance T from an intersection of the optical axis and an object-side convex surface of the plano-convex lens, and the ratio T/F satisfies 0.88&lt;T/F&lt;0.98. 
    
    
     
       BRIEF DESCRIPTION OF THE FIGURES 
         FIG. 1  illustrates an exemplary narrow field-of-view compound lens in a use scenario, according to an embodiment. 
         FIG. 2  is an exemplary cross-sectional view of an embodiment of the two-surface narrow field-of-view compound lens of  FIG. 1 . 
         FIG. 3  is a cross-sectional view of an imaging system showing an embodiment of the two-surface narrow field-of-view compound lens, of  FIGS. 1 and 2 , in relationship to a coverglass of an imager. 
         FIG. 4  shows a table of exemplary parameters of the compound lens of  FIG. 3 . 
         FIG. 5  is a plot of the longitudinal aberration of the compound lens within the imaging system of  FIG. 3 . 
         FIG. 6  is a plot of the f-theta distortion of the compound lens within the imaging system of  FIG. 3 . 
         FIG. 7  is a plot of the Petzval field curvature of the compound lens within the imaging system of  FIG. 3 . 
         FIG. 8  is a plot of the lateral color error of the compound lens within the imaging system of  FIG. 3 . 
         FIG. 9  is a cross-sectional view of an imaging system showing an embodiment of the two-surface narrow field-of-view compound lens, of  FIGS. 1 and 2 , in relationship to a coverglass of an imager. 
         FIG. 10  shows a table of exemplary parameters of the compound lens of  FIG. 9 . 
         FIG. 11  is a plot of the longitudinal aberration of the compound lens within the imaging system of  FIG. 9 . 
         FIG. 12  is a plot of the f-theta distortion of the compound lens within the imaging system of  FIG. 9 . 
         FIG. 13  is a plot of the Petzval field curvature of the compound lens within the imaging system of  FIG. 9 . 
         FIG. 14  is a plot of the lateral color error of the compound lens within the imaging system of  FIG. 9 . 
     
    
    
     DETAILED DESCRIPTION 
       FIG. 1  illustrates the imaging of a barcode  190  with a narrow field-of-view compound lens  100  within a camera module  150  of a mobile device  160 . Camera module  150  and compound lens  100  are depicted with dashed lines because they are visible on a side of mobile device  160  facing barcode  190 . An image  191  of barcode  190  is shown on output display  161  of mobile device  160 . It should be appreciated that narrow field-of-view compound lens  100  may be included in alternate locations on the mobile device  160 , such as on the front, back, top, bottom or sides of the device  160 . Furthermore, narrow field-of-view compound lens  100  may be included on other devices including, but not limited to, laptop computers, tablet computers, barcode scanners, and image-based recognition devices without departing from the scope hereof. 
       FIG. 2  is a cross-sectional view of a two-surface narrow field-of-view compound lens  200 , which is an embodiment of two-surface narrow field-of-view compound lens  100  of  FIG. 1 . Compound lens  200  includes a substrate  220  between a plano-convex lens  210  and a plano-concave lens  230 . Plano-convex lens  210  and plano-concave lens  230  share a common optical axis  279 , and are hence coaxial. Substrate  220  may be either monolithic, or alternatively be formed of more than one optical element. 
     Plano-convex lens  210  includes a convex surface  211  and a planar surface  212 . Planar surface  212  adjoins a planar surface  221  of substrate  220 , which also includes a planar surface  222  opposing the planar surface  221 . Planar surface  222  adjoins a planar surface  231  of plano-concave lens  230 , which also includes a concave surface  232  opposing the planar surface  231 . At least one of surfaces  211  and  232  may be aspheric. At least one of surfaces  211  and  232  may be spherical without departing from the scope hereof. Surfaces  212  and  221  are shown being in direct contact with each other, but may be indirectly adjoined, via an additional optical element, without departing from the scope hereof. Likewise, surfaces  222  and  231  are shown being in direct contact with each other, but may be indirectly adjoined without departing from the scope hereof. 
     Plano-convex lens  210  and plano-concave lens  230  may each be singlet lenses. In an embodiment of compound lens  200 , at least one of plano-convex lens  210  and plano-concave lens  230  may be non-singlet lenses without departing from the scope hereof. 
     Surface  211  of plano-convex lens  210  may be optimized to collect incident rays, control the propagation direction of those rays into compound lens  200 , such as through lenses  220  and  230 , and guide the incident rays passing through an aperture stop  225 . Surface  232  of plano-concave lens  230  may be optimized to correct chromatic aberration and spherical aberration of an image formed by compound lens  200 . In  FIG. 2 , aperture stop  225  is illustrated as a physical aperture for illustrative purposes only. Without departing from the scope hereof, aperture stop  225  may be a non-physical aperture, that is, formed at least in-part by an element other than lenses  210 ,  230 , and substrate  220 . 
     Used in an imaging system, compound lens  200  may have aperture stop  225  between substrate  220  and plano-concave lens  230 , which corresponds to a plane between adjoining surfaces  222  and  231 . Locating aperture stop  225  within compound lens  200  helps to maintain symmetry of ray cones from field coordinates, which decreases selected aberrations and contributes to the imaging system having an optimal modulation transfer function (“MTF”). 
     Plano-convex lens  210  has a focal length F1 and plano-concave lens  230  has a focal length F2. Embodiments of compound lens  200  may have a quotient F2/F1 between −1.4 and −0.9. Limiting the quotient F2/F1 to this range allows for limiting chromatic and spherical aberration in an image formed by compound lens  200  to values that may be adequately corrected for effective image-based recognition. Images formed by a lens with quotient F2/F1 outside of this range have chromatic and spherical aberrations that exceed a threshold beyond which the aberrations cannot be adequately corrected for effective image-based recognition. 
       FIG. 2  shows compound lens  200  focusing parallel rays  250  onto an image plane  278 . Converging rays  254  exit compound lens  200  and converge at image plane  278 . Extensions of rays  250  and  254  into compound lens  200  intersect at a principal plane  274 .  FIG. 2  shows principal plane  274  is intersecting compound lens  200 . Without departing from the scope here of, embodiments of compound lens  200  may have principal plane  274  located such that it does not intersect compound lens  200 . 
     Compound lens  200  has an effective focal length  276  (herein also denoted by f eff ), between principal plane  274  and image plane  278 . A plane  272  is tangent to surface  211  at optical axis  279  and perpendicular to optical axis  279 . Total track length  275  defines a distance T between plane  272  and image plane  278 . Embodiments of compound lens  200  may have a quotient T/f eff  between 0.88 and 0.98. Limiting the quotient T/f eff  to this range limits total track length  275  and the length of an imaging system that includes compound lens  200 . 
     In compound lens  200 , plano-convex lens  210  has an Abbe number V d &gt;50 and plano-concave lens  230  has an Abbe number V d &lt;35. These constraints on Abbe numbers allow for limiting chromatic aberration in imaging systems that include compound lens  200 , such as imaging systems  301  and  901  discussed herein, to values that may be adequately corrected for effective image-based recognition. Images formed by a lenses Abbe numbers outside of this range have chromatic aberration that exceeds a threshold beyond which the aberration cannot be adequately corrected for effective image-based recognition. Herein, all refractive index values and Abbe numbers correspond to λ d =587.6 nm unless otherwise specified. 
     Transparent optical materials with V d &gt;50 include polymethyl methacrylate (PMMA), alicyclic acrylate (e.g., Optrez OZ1330®), and cycloolefin polymers (e.g., APEL™ 5014 DP, TOPAS® 5013, and ZEONEX® 480R). The lens material with V d &gt;50 may be plastic, glass, or any non-plastic optical material without departing from the scope hereof. 
     Transparent optical materials with V d &lt;35 include PANLITE®, a brand-name polycarbonate, Udel® P-1700, a brand-name polysulfone, and OKP-4, a brand-name optical polyester. The lens material with V d &lt;35 may be plastic, glass, or any non-plastic optical material without departing from the scope hereof. 
     Lenses  210  and  230  may be formed of a solder-reflow compatible material via a wafer-level optics replication process. Lenses  210  and  230  may also be formed via injection molding or other methods known in the art. Alternatively, lenses  210  and  230  may be formed of glass via precision glass molding (also known as ultra-precision glass pressing) or other methods known in the art. 
     Two-surface Narrow Field-of-view Compound Lens, Example 1 
       FIG. 3  is a cross-sectional view of a two-surface narrow field-of-view compound lens  300  within an imaging system  301 . Compound lens  300  is an embodiment of two-surface narrow field-of-view compound lens  200 . Compound lens  300  includes a substrate  320  between a plano-convex lens  310  and a plano-concave lens  330 . Substrate  320 , plano-convex lens  310 , and plano-concave lens  330  are embodiments of substrate  220 , plano-convex lens  210 , and plano-concave lens  230 , respectively, of compound lens  200 . Plano-convex lens  310  and plano-concave lens  330  have a common optical axis  379  and are thus coaxial. Compound lens  300  has an aperture stop  325  resulting from diameter  339  of lens  330 . Aperture stop  325  and diameter  339  are similar to aperture stop  225  and diameter  239  of compound lens  200 . Diameter  339  equals 0.936 mm. 
     Plano-convex lens  310  includes a convex surface  311  and a planar surface  312 . Planar surface  312  adjoins a planar surface  321  of substrate  320 , which also includes a planar surface planar surface  322 . Planar surface  322  adjoins a planar surface  331  of plano-concave lens  330 , which also includes a concave surface  332 . Surfaces  311 ,  312 ,  321 ,  322 ,  331 , and  332  are embodiments of surfaces  211 ,  212 ,  221 ,  222 ,  231 , and  232 , respectively. 
     In addition to including compound lens  300 , imaging system  301  may also include a cover glass  350 . Cover glass  350  includes surfaces  351  and  352  and covers a pixel array of an image sensor, not shown, located at image plane  378 . The specific type of pixel array and image sensor may vary and is thus not discussed in detail herein. 
       FIG. 4  shows a table  400  of exemplary parameters of each surface of compound lens  300 . Table  400  includes columns  404 ,  406 ,  408 ,  410 ,  412 , and  421 - 426 . Surface column  421  denotes surfaces  311 ,  312 ,  321 ,  322 ,  331 ,  332 ,  351 ,  352 , and image plane  378  shown in  FIG. 3 . Column  423  includes on-axis thickness values, in millimeters, between adjacent surfaces of imaging system  301 . Column  423  includes center thicknesses of lens  310 , substrate  320 , lens  330 , and cover glass  350 . Specifically, lens  310  has a center thickness  313  equal to 0.33 mm, substrate  320  has a center thickness  323  equal to 0.30 mm, lens  330  has a center thickness  333 , and cover glass  350  has a center thickness  353  equal to 0.40 mm. Surface  332  of plano-concave lens  330  and surface  351  of cover glass  350  are separated by a distance  343  equal to 4.120 mm. 
     It should be appreciated that imaging system  301  need not include cover glass  350 , in which case parameters of compound lens  300  may be reoptimized to form an image at image plane  378  absent cover glass  350 . Surface  352  and image plane  363  are separated by a distance  363  equal to 0.040 mm. 
     Surfaces  311  and  332  are defined by surface sag z sag , Eqn. 1. 
     
       
         
           
             
               
                 
                   
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                                 c 
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                         N 
                       
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     In Eqn. 1, z sag  is a function of radial coordinate r, where directions z and r are shown in coordinate axes  398 ,  FIG. 3 . Quantity i is a positive integer and N=6. In Eqn. 1, the parameter c is the reciprocal of the surface radius of curvature 
                 r   c     ⁢     :     ⁢           ⁢   c     =       1     r   c       .           
Column  422  of  FIG. 4  lists r c  values for surfaces  311  and  332 . Parameter k denotes the conic constant, shown in column  426 . Columns  404 ,  406 ,  408 ,  410 , and  412  contain values of aspheric coefficients α 4 , α 6 , α 8 , α 10 , and α 12 , respectively. The units of quantities in  FIG. 3  are consistent with z sag  in Eqn. 1 being expressed in millimeters.
 
     Column  424  lists the material&#39;s refractive index n d  at free-space wavelength λ=587.6 nm, and column  425  lists the corresponding Abbe numbers V d . Plano-convex lens  310  has refractive index n d =1.511, Abbe number V d =57, and includes object-side surface  311  and image-side surface  312 . Substrate  320  has refractive index n d =1.517, Abbe number V d =63, and includes object-side surface  321  and image-side surface  322 . Plano-concave lens  330  has refractive index n d =1.595, Abbe number V d =30, and includes object-side surface  331  and image-side surface  332 . 
     Compound lens  300  has a total track length  375  equal to 5.217 mm, which is the sum of thickness values in column  423  of table  400 . Referring to  FIG. 3 , total track length  375  is between plane  372  and image plane  378 , where plane  372  is tangent to surface  311  at optical axis  379 . At free-space wavelength λ=587.6 nm, compound lens  300  has an effective focal length  376  (f eff ) equal to 5.706 mm between a principal plane  374  and image plane  378 . Principal plane  374 , total track length  375  and effective focal length  376  are similar to principal plane  274 , total track length  275  and effective focal length  276 , respectively. The ratio of total track length  375  to effective focal length  376  equals 0.914. Compound lens  300  has a working f-number equal to 4.8 and, for an image sensor with a 1.7-mm diagonal length, a 16-degree field of view. 
     Plano-convex lens  310  and plano-concave lens  330  have focal lengths F1 and F2 respectively, which may be approximated using the lensmaker&#39;s equation. Referring to plano-convex lens  310 , object-side surface  311  has a 0.888-mm radius of curvature, and image-side surface  312  is planar hence has an infinite radius of curvature. Using these radii of curvature, center thickness  313 , and n d =1.511, the lensmaker&#39;s equation yields F1≈1.738 mm. Referring to plano-concave lens  330 , object-side surface  331  is planar and hence has an infinite radius of curvature, and image-side surface  332  has a radius of curvature R 4 =1.105. Using these radii of curvature, center thickness  333 , and n d =1.595, the lensmaker&#39;s equation yields F2≈−1.857 mm. Ratio F2/F1≈−1.069. 
       FIGS. 5-8  are plots of longitudinal aberration, f-theta distortion, field curvature, and lateral color, respectively, of compound lens  300  within imaging system  301  as computed by Zemax®. 
       FIG. 5  is a plot of the longitudinal aberration of compound lens  300  within imaging system  301 . In  FIG. 5 , longitudinal aberration is plotted in units of millimeters as a function of normalized radial coordinate r/r p , where r p =0.6334 mm is the maximum entrance pupil radius. Longitudinal aberration curves  548 ,  558 , and  565  are computed at the blue, green, and red Fraunhofer F-, d- and C-spectral lines: λ F =486.1 nm, λ d =587.6 nm, and λ C =656.3 nm respectively. 
       FIG. 6  is a plot of the f-theta distortion, versus field angle, of compound lens  300  within imaging system  301 . The maximum field angle plotted in  FIG. 6  is θ max =8.149°. Distortion curve  658  is computed at wavelength λ d . For clarity, distortion curves corresponding to wavelength λ F  and λ C  are not shown, as they overlap distortion curve  658  to within its line thickness as plotted in  FIG. 6 . 
       FIG. 7  is a plot of the Petzval field curvature, as a function of field angle, of compound lens  300  within imaging system  301 . The field curvature is plotted for field angles between zero and θ max =8.149°. Field curvature  748 -S and field curvature  748 -T (solid lines) are computed at wavelength λ F  in the sagittal and tangential planes, respectively. Field curvature  758 -S and field curvature  758 -T (short-dashed lines) are computed at wavelength λ d  in the sagittal and tangential planes, respectively. Field curvature  765 -S and field curvature  765 -T (long-dashed lines) correspond to field curvature at wavelength λ C  in the sagittal and tangential planes, respectively. 
       FIG. 8  is a plot of the lateral color error, also known as transverse chromatic aberration, versus field height of compound lens  300  within imaging system  301 . Field height ranges from h min =0 (on-axis) to h max =0.850 mm in image plane  378  Lateral color is referenced to the yellow d Fraunhofer line of helium, λ d =587.6 nm such that lateral color for λ d  is zero for all field heights. Lateral color  848  is computed at wavelength  4 . Lateral color  865  is computed at wavelength λ C . The lateral color error is less than the Airy disk radius for the range of field heights evaluated. 
     Two-surface Narrow Field-of-view Compound Lens, Example 2 
       FIG. 9  is a cross-sectional view of a two-surface narrow field-of-view compound lens  900  within an imaging system  901 . Compound lens  900  is an embodiment of two-surface narrow field-of-view compound lens  200 . Compound lens  900  includes a substrate  920  between a plano-convex lens  910  and a plano-concave lens  930 . Substrate  920 , plano-convex lens  910 , and plano-concave lens  930  are embodiments of substrate  220 , plano-convex lens  210 , and plano-concave lens  230 , respectively, of compound lens  200 . Plano-convex lens  910  and plano-concave lens  930  have a common optical axis  979 . Compound lens  900  has an aperture stop  925  resulting from diameter  939  of lens  930 . Aperture stop  925  and diameter  939  are similar to aperture stop  225  and diameter  239  of compound lens  200 . Diameter  939  equals 0.964 mm. 
     Plano-convex lens  910  includes a convex surface  911  and a planar surface  912 . Planar surface  912  adjoins a planar surface  921  of substrate  920 , which also includes a planar surface planar surface  922 . Planar surface  922  adjoins a planar surface  931  of plano-concave lens  930 , which also includes a concave surface  932 . Surfaces  911 ,  912 ,  921 ,  922 ,  931 , and  932  are embodiments of surfaces  211 ,  212 ,  221 ,  222 ,  231 , and  232 , respectively. In addition to including compound lens  900 , imaging system  901  may also include cover glass  350 . 
       FIG. 10  shows a table  1000  of exemplary parameters of each surface of compound lens  900 . Table  1000  includes columns  1004 ,  1006 ,  1008 ,  1010 ,  1012 , and  1021 - 1026 . Surface column  1021  denotes surfaces  911 ,  912 ,  921 ,  922 ,  931 ,  932 ,  351 ,  352 , and image plane  978  shown in  FIG. 9 . Column  1023  includes on-axis thickness values, in millimeters, between adjacent surfaces of imaging system  901 . Column  1023  includes center thicknesses of lens  910 , substrate  920 , lens  930 , and cover glass  350 . Specifically, lens  910  has a center thickness  913  equal to 0.33 mm, substrate  920  has a center thickness  923  equal to 0.30 mm, lens  930  has a center thickness  933 , and cover glass  350  has center thickness  353 . Surface  932  of plano-convex lens  930  and surface  351  of cover glass  350  are separated by a distance  943  equal to 4.120 mm. Surface  352  and image plane  948  are separated by a distance  963  equal to 0.040 mm. 
     It should be appreciated that imaging system  901  need not include cover glass  350 , in which case parameters of compound lens  900  may be reoptimized to form an image at image plane  978  absent cover glass  350 . 
     Surfaces  911 , and  932  are defined by surface sag z sag , Eqn. 1. Directions z and r are shown in coordinate axes  998 ,  FIG. 9 . In Eqn. 1, the parameter c is the reciprocal of the surface radius of curvature 
                 r   c     ⁢     :     ⁢           ⁢   c     =       1     r   c       .           
Column  1022  of  FIG. 10  lists r c  values for surfaces  911  and  932 . Parameter k denotes the conic constant, shown in column  1026 . Columns  1004 ,  1006 ,  1008 ,  1010 , and  1012  contain values of aspheric coefficients α 4 , α 6 , α 8 , α 10 , and α 12 , respectively. The units of quantities in  FIG. 9  are consistent with z sag  in Eqn. 1 being expressed in millimeters.
 
     Column  1024  lists the material&#39;s refractive index n d  at free-space wavelength λ=587.6 nm, and column  1025  lists the corresponding Abbe numbers V d . Plano-convex lens  910  has refractive index n d =1.511, Abbe number V d =57, and includes object-side surface  911  and image-side surface  912 . Substrate  920  has refractive index n d =1.517, Abbe number V d =63, and includes object-side surface  921  and image-side surface  922 . Plano-concave lens  930  has refractive index n d =1.595, Abbe number V d =30, and includes object-side surface  931  and image-side surface  932 . 
     Compound lens  900  has a total track length  975  equal to 4.369 mm, which is the sum of thickness values in column  1023  of table  1000 . Referring to  FIG. 9 , total track length  975  is between a plane  972  and image plane  978 , where plane  972  is tangent to surface  911  at optical axis  979 . At free-space wavelength λ=587.6 nm, compound lens  900  has an effective focal length  976  (f eff ) equal to 4.644 mm. Principal plane  974 , total track length  975  and effective focal length  976  are similar to principal plane  274 , total track length  275  and effective focal length  276 , respectively. The ratio of total track length  975  to effective focal length  976  equals 0.941. Compound lens  900  has a working f-number equal to 3.8 and, for an image sensor with a 1.7-mm diagonal length, a 20-degree field of view. 
     Plano-convex lens  910  and plano-concave lens  930  have focal lengths F1 and F2 respectively, which may be approximated using the lensmaker&#39;s equation. Referring to plano-convex lens  910 , object-side surface  911  has a 0.928-mm radius of curvature, and image-side surface  312  is planar hence has an infinite radius of curvature R 2 . Using these radii of curvature, center thickness  913 , and n d =1.511, the lensmaker&#39;s equation yields F1≈1.816 mm. Referring to plano-concave lens  930 , object-side surface  931  is planar and hence has an infinite radius of curvature, and image-side surface  932  has a 1.358-mm radius of curvature. Using these radii of curvature, center thickness  933 , and n d =1.595, lensmaker&#39;s equation yields F2≈−2.282 mm. Ratio F2/F1≈−1.257. 
       FIGS. 11-14  are plots of longitudinal aberration, f-theta distortion, field curvature, and lateral color, respectively, of compound lens  900  within imaging system  901  as computed by Zemax®. 
       FIG. 11  is a plot of the longitudinal aberration of compound lens  900  within imaging system  901 . In  FIG. 11 , longitudinal aberration is plotted in units of millimeters as a function of normalized radial coordinate r/r p , where r p =0.6288 mm is the maximum entrance pupil radius. Longitudinal aberration curves  1148 ,  1158 , and  1165  are computed at the blue, green, and red Fraunhofer F-, d- and C-spectral lines. 
       FIG. 12  is a plot of the f-theta distortion, versus field angle, of compound lens  900  within imaging system  901 . The maximum field angle plotted in  FIG. 12  is θ max  =10.200°. Distortion curve  1258  is computed at wavelength λ d . For clarity, distortion curves corresponding to wavelength λ F  and λ c  are not shown, as they overlap distortion curve  658  to within its line thickness as plotted in  FIG. 12 . 
       FIG. 13  is a plot of the Petzval field curvature, as a function of field angle, of compound lens  900  within imaging system  901 . The field curvature is plotted for field angles between zero and θ max =10.200°. Field curvature  1348 -S and field curvature  1348 -T (solid lines) are computed at wavelength λ F  in the sagittal and tangential planes, respectively. Field curvature  1358 -S and field curvature  1358 -T (short-dashed lines) are computed at wavelength λ d  in the sagittal and tangential planes, respectively. Field curvature  1365 -S and field curvature  1365 -T (long-dashed lines) correspond to field curvature at wavelength λ C  in the sagittal and tangential planes, respectively. 
       FIG. 14  is a plot of the lateral color error, also known as transverse chromatic aberration, versus field height of compound lens  900  within imaging system  901 . Field height ranges from h min =0 (on-axis) to h max =0.850 mm in image plane  978 . Lateral color is referenced to λ d =587.6 nm such that lateral color for λ d  is zero for all field heights. Lateral color  1448  is computed at wavelength λ F . Lateral color  1465  is computed at wavelength λ C . The lateral color error is less than the Airy disk radius for the range of field heights evaluated. 
     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 two-surface narrow field-of-view (FOV) compound lens for producing an image of an object at an image plane of an imaging system may include a biplanar substrate between a plano-convex lens and a plano-concave lens having a common optical axis. The plano-convex lens has a first planar surface on a first side of the biplanar substrate and is formed of a material having a first Abbe number. The plano-concave lens has a second planar surface on a second side of the biplanar substrate opposite the first side, and is formed of a material having a second Abbe number less than the first Abbe number. 
     (A2) In the compound lens denoted as (A1), the first Abbe number may exceed 50 and the second Abbe number may be less than 35. 
     (A3) In either of the compound lenses denoted as (A1) and (A2), the biplanar substrate may be formed of a material having a third Abbe number that exceeds the second Abbe number. 
     (A4) In the compound lenses denoted as (A3), the third Abbe number may exceed the first Abbe number. 
     (A5) In any of the compound lenses denoted as (A1) through (A4), the plano-convex lens may have a focal length F1, the plano-concave lens may have a focal length F2, wherein the ratio F2/F1 satisfies −1.4&lt;F2/F1&lt;−0.9. 
     (A6) In any of the compound lenses denoted as (A1) through (A5), the biplanar substrate, the plano-convex lens and the plano-concave lens may collectively have an effective focal length f eff  such that the image is formed at the image plane located a distance T from an intersection of the optical axis and an object-side convex surface of the plano-convex lens, and the ratio T/f eff  satisfying 0.88&lt;T/f eff &lt;0.98. 
     (A7) In any of the compound lenses denoted as (A1) through (A6), second planar surface may function as an aperture stop. 
     (A8) In any of the compound lenses denoted as (A1) through (A7), the biplanar substrate may have a width exceeding a diameter of the second planar surface. 
     (A9) In any of the compound lenses denoted as (A1) through (A8), at least one of the plano-convex lens and the plano-concave lens may be singlet lens. 
     (A10) In any of the compound lenses denoted as (A1) through (A9), the biplanar substrate, the plano-convex lens and the plano-concave lens may collectively have an effective focal length between four millimeters and six millimeters. 
     (A11) Any of the compound lenses denoted as (A1) through (A10) may have an f-number between 3.5 and 5.5 for increasing a depth of field of the imaging system. 
     (A12) Any of the compound lenses denoted as (A1) through (A11) may further include a cover glass between the plano-concave lens and the image plane. 
     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 there between.