Abstract:
A projection lens arranged to project an image formed on a display panel onto a screen includes a plurality of lens units movable along an axis for varying magnification, wherein at least one of lenses constituting the plurality of lens units has at least one lens surface of shape having no symmetry with respect to the axis, so that the trapezoidal deformation which tends to occur when the image formed on the display panel is projected onto the screen is corrected well.

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to projection lenses and image projection apparatuses using the same and, more particularly, to projection lenses suited to, for example, liquid crystal projectors of the type in which an original projection image displayed on a display panel, such as a liquid crystal display element, is projected obliquely onto a screen in an enlarged size. Still more particularly, the present invention relates to such a liquid crystal projector that has a projection lens appropriately designed to correct well the distortion and trapezoidal deformation of the projected image and the declination of the image plane so that a projected image of good quality can be obtained. 
     2. Description of Related Art 
     Heretofore, there have been proposed a variety of projection apparatuses arranged to project an original projection image displayed on a display panel, such as a liquid crystal display element, onto a screen. 
     FIG. 21 is a diagram of geometry of the arrangement of a projection apparatus of the oblique projection type in which an original projection image LCD is obliquely projected onto a screen S by a projection lens LP. 
     In the projection apparatus shown in FIG. 21, for the purpose of preventing the projected image from being distorted when the original projection image LCD is projected on the screen S, the so-called image frame shifting is utilized in which the center LCDa of the original projection image LCD is shifted downward with respect to an optical axis Ax of the projection lens LP. 
     In the conventional liquid crystal projector, when installed in the apparatus, despite a short projection distance as usual, the projected image has to appear more upper. To this purpose, the projection lens is made wider in the image angle. The center of the liquid crystal panel is thus allowed to take an offset (shifted) position relative to the optical axis of the projection lens, so that the center of the projected image appears upper than the optical axis of the projection lens. 
     However, when the image angle of the projection lens is widened, distortion, field curvature, lateral chromatic aberration and other aberrations become larger, causing an increase of the difficulty of correcting these various aberrations. 
     Particularly with the distortion left large, because the projected image is offset from the optical axis, the projected image is distorted asymmetrically in the vertical direction, giving rise to a problem of detracting the quality of the projected image. 
     To solve this problem, among others, a method of introducing an axially-symmetric aspheric surface into the projection lens, too, has been put into practice. However, since the original projection image LCD is offset as shown in FIG. 22, it is difficult correct the distortion throughout an image circle IC including the entirety of the original projection image LCD. 
     BRIEF SUMMARY OF THE INVENTION 
     It is an object of the present invention to provide, by using an aspheric surface having no axial symmetry in a part of an optical system, a projection lens which can correct trapezoidal deformation and distortion as an original projection image is projected obliquely, and an image projection apparatus using such a projection lens. 
     To attain the above object, in accordance with an aspect of the invention, there is provided a projection lens arranged to project an image formed on a display panel onto a screen, the projection lens comprising a plurality of lens units movable along an axis for varying magnification, wherein at least one of lenses constituting the plurality of lens units has at least one lens surface of shape having no symmetry with respect to the axis. 
     Further, in the above projection lens, the display panel and the axis are almost orthogonal with each other, and a center of the display panel is offset from the axis. 
     Further, in the above projection lens, a normal of the display panel makes an angle θ with the axis, wherein the following condition is satisfied: 
     
       
         −5°&lt;θ&lt;5° 
       
     
     Further, in the above projection lens, at least one of lenses constituting the plurality of lens units has at least one aspheric surface of shape having symmetry with respect to the axis. 
     Further, the above projection lens comprises, in order from the screen side, a negative lens unit having a negative refractive power and a positive lens unit having a positive refractive power. 
     Further, in the above projection lens, the negative lens unit has a lens surface of shape having no symmetry with respect to the axis. 
     Further, in the above projection lens, the negative lens unit comprises, in order from the screen side, a lens subunit of meniscus form convex toward the screen side, a lens subunit of bi-concave form, and a lens subunit of bi-convex form. 
     Further, in the above projection lens, the lens subunit of meniscus form has a lens surface, facing the screen side, of shape having no symmetry with respect to the axis. 
     Further, in the above projection lens, the positive lens unit has an aspheric surface having symmetry with respect to the axis. 
     The above and further objects and features of the invention will become apparent from the following detailed description of preferred embodiments thereof taken in conjunction with the accompanying drawings. 
    
    
     BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING 
     FIG. 1 is a longitudinal section view of a numerical example 1 of the lens of the invention. 
     FIG. 2 is a longitudinal section view of a numerical example 2 of the lens of the invention. 
     FIG. 3 is a longitudinal section view of a numerical example 3 of the lens of the invention. 
     FIG. 4 is a longitudinal section view of a numerical example 4 of the lens of the invention. 
     FIG. 5 shows graphic representations of the aberrations in the numerical example 1 of the invention in the wide-angle end. 
     FIG. 6 shows graphic representations of the aberrations in the numerical example 1 of the invention in the telephoto end. 
     FIG. 7 is a plan view for explaining the distortion in the numerical example 1 of the invention. 
     FIG. 8 shows graphic representations of the aberrations in the numerical example 2 of the invention in the wide-angle end. 
     FIG. 9 shows graphic representations of the aberrations in the numerical example 2 of the invention in the telephoto end. 
     FIG. 10 is a plan view for explaining the distortion in the numerical example 2 of the invention. 
     FIG. 11 shows graphic representations of the aberrations in the numerical example 3 of the invention in the wide-angle end. 
     FIG. 12 shows graphic representations of the aberrations in the numerical example 3 of the invention in the telephoto end. 
     FIG. 13 is a plan view for explaining the distortion in the numerical example 3 of the invention. 
     FIG. 14 shows graphic representations of the aberrations in the numerical example 4 of the invention in the wide-angle end. 
     FIG. 15 shows graphic representations of the aberrations in the numerical example 4 of the invention in the telephoto end. 
     FIG. 16 is a plan view for explaining the distortion in the numerical example 4 of the invention. 
     FIG. 17 is a diagram for explaining an original projection image in view of the distortion according to the invention. 
     FIG. 18 is a plan view for explaining the distortion according to the invention. 
     FIG. 19 is an enlarged view of FIG.  4 . 
     FIG. 20 is a graph for explaining the coordinates for the lens surface of the projection lens according to the invention. 
     FIG. 21 is a diagram for explaining the conventional liquid crystal projector. 
     FIG. 22 is a plan view for explaining a part of the liquid crystal projector shown in FIG.  21 . 
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Hereinafter, preferred embodiments of the invention will be described in detail with reference to the drawings. 
     FIGS. 1 to  4  in block diagram show numerical examples 1 to 4 of the lens of the invention, whose data will be given later. 
     In FIGS. 1 to  4 , a projection lens (projection optical system) LP, an optical block G, such as an optical filter, a color separation prism, etc., and a projection image LCD formed on a display panel, such as a liquid crystal element, are arranged in this order from a screen S. 
     Further, reference character ZRN denotes an aspheric surface having no axial symmetry, and reference character ASP denotes an axially-symmetric aspheric surface. The projection lens LP has a lens unit L 1  of negative refractive power and a lens unit L 2  of positive refractive power, both of which axially move toward the screen S, while decreasing the separation therebetween, to vary the focal length from the wide-angle end to the telephoto end. Accordingly, the size of the projected image on the screen S is made variable. Further, a fixed lens (r 17 -r 18 ) is disposed immediately before the optical block G. 
     The center LCDa of the projection image LCD is shifted (decentered in parallel) from an optical axis (reference axis) Ax of the projection lens LP. 
     In the numerical examples 1 to 3, as shown in FIGS. 1 to  3 , the screen S and the projection image LCD are parallel to each other, and both the screen S and the projection image LCD are perpendicular to the optical axis Ax of the projection lens LP. The projection lens LP projects the projection image LCD onto the screen S in enlarged scale at various magnifications. 
     In the numerical example 4, as shown in FIG. 4, while the projection image LCD is perpendicular to the optical axis Ax of the projection lens LP, the optical axis Ax of the projection lens LP is made tilted with respect to the screen S, (or the optical axis Ax is made to have a predetermined angle α with the normal Sa of the screen S). 
     Further, in the numerical examples 1 to 4, with the use of the aspheric surfaces ZRN and ASP mentioned before, the distortion produced by the oblique projection is corrected. (Incidentally, the term “reference axis (reference optical axis)” used in the invention means an optical axis specified by at least two lens surfaces in the projection lens LP.) The projection lens LP projects the projection image LCD onto the screen S in enlarged scale at various magnifications. 
     FIGS. 5 to  7  show the lateral aberrations of the numerical example 1 in the wide-angle end and in the telephoto end and a lattice used for explaining the state of distortion. 
     FIGS. 8 to  10  show the lateral aberrations of the numerical example 2 in the wide-angle end and in the telephoto end and a lattice used for explaining the state of distortion. 
     FIGS. 11 to  13  show the lateral aberrations of the numerical example 3 in the wide-angle end and in the telephoto end and a lattice used for explaining the state of distortion. 
     FIGS. 14 to  16  show the lateral aberrations of the numerical example 4 in the wide-angle end and in the telephoto end and a lattice used for explaining the state of distortion. 
     The lateral aberrations are concerned with a central point P 1  and two corners P 2  and P 3  in the projection image LCD, as shown in FIG. 17, and calculated by tracing rays of light from the screen S to the projection image LCD. The distortion represents the projection image itself when rays of light are traced from the projection image LCD to the screen S. 
     The first lens unit L 1  of the projection lens LP is composed of, in order from the screen side (longer conjugate point), a lens (r 1 -r 2 ) of meniscus form convex toward the screen side, a negative lens (r 3 -r 4 ) of bi-concave form and a positive lens (r 5 -r 6 ) of bi-convex form. 
     The second lens unit L 2  of the projection lens LP is composed of, in order from the screen side (longer conjugate point), a positive lens (r 7 -r 8 ) having a convex surface facing the projection image LCD (shorter conjugate point), a positive lens (r 9 -r 10 ) having a convex surface facing the screen side, a negative lens (r 11 -r 12 ) of bi-concave form, a positive lens (r 13 -r 14 ) having a convex surface facing the screen side, and a positive lens (r 15 -r 16 ) of bi-convex form. 
     Then, in the numerical example 1, a lens surface r 1 , facing the screen side, of the lens L 11  is made to be an aspheric surface having no axial symmetry. 
     Also, in the numerical example 2, a lens surface r 2 , facing the projection image LCD, of the lens L 11  is made to be an aspheric surface having no axial symmetry, and, in the numerical example 3, a lens surface r 17 , facing the screen side, of the positive lens L 26  is made to be an aspheric surface having no axial symmetry. 
     Also, in the numerical example 4, both lens surfaces r 1  and r 2  of the lens L 11  and both lens surfaces r 17  and r 18  of the positive lens L 26  each are made to be an aspheric surface having no axial symmetry. 
     In the present invention, at least one of the surfaces constituting the projection lens LP is provided with an aspheric surface having no axial symmetry, so that it is possible to choose, for the design purposes, only an area corresponding to the liquid crystal panel LCD which is offset. 
     By this arrangement, despite a few lens surfaces, the distortion can be corrected well. Particularly, for the lens members on the enlarge projection side of the projection lens (that is, the ones nearest to the screen), and for the lens members nearest to the liquid crystal panel, the light beam utilizes only the upper or lower parts of the lens members. On this account, such places are selected to use the aspheric surface or surfaces having no axial symmetry, thus effectively correcting various aberrations (especially, distortion). 
     In the numerical examples 1 to 3, the projection lens LP (with exclusion of the aspheric surface having no axial symmetry) has its optical axis Ax set perpendicular to the enlarge projection plane (screen) S and the liquid crystal panel LCD. 
     In order to project the image even more upward onto the screen S, the liquid crystal projector is raised at its end nearest to the screen, being tilted upward. In this situation, as shown in FIG. 18, in addition to the corrected distortion, the projection lens LP produces a trapezoidal deformation due to the tilting. 
     The numerical example 4 is, as shown in FIG. 19 in the extended form, an example of design for the projection lens LP when used with its optical axis Ax tilted with respect to the enlarge projection plane (screen) S, while still permitting the resultant distortion to be corrected. 
     In FIG. 19, the reference axis (optical axis) Ax makes an angle of α=8° with a normal Sa of the screen S at the cross point S 1  of the screen surface and the reference axis Ax, and the liquid crystal panel LCD is put below the normal Sa with the cross point LCDa of the liquid crystal panel and the reference axis Ax set at a distance of ΔY=534.5 mm. 
     In this connection, it should be noted that, in the projection lens according to the invention, if an angle e which the reference axis (reference optical axis) of the projection lens makes with the normal of the display panel LCD lies within the following range: 
     
       
         −5°&lt;θ&lt;5°, 
       
     
     an image displayed on the display panel LCD can be projected well onto the screen S. 
     Next, four numerical examples 1 to 4 of the invention are shown. In the numerical data for the examples 1 to 4, ri is the radius of curvature of the i-th lens surface, when counted from the enlarge projection side, di is the i-th axial lens thickness or air separation, when counted from the enlarge projection side, and ndi and vdi are respectively the refractive index and Abbe number of the i-th lens element. 
     The shape of an aspheric surface having axial symmetry is expressed in the coordinates with a z axis in the optical axis direction and a height h in the direction perpendicular to the optical axis, the direction from the screen to the display panel being taken as positive, by the following equation:        z   =         Ch   2       1   +       1   -       (     1   +   K     )          C   2          h   2               +     Ah   4     +     Bh   6     +     Ch   8     +     Dh   10     +     Eh   12                              
     where C is the curvature of the osculating sphere, and K, A, B, C, D and E are the aspheric coefficients. 
     The shape of an aspheric surface having no axial symmetry is expressed in the coordinates with a z axis in the optical axis direction, a y axis in the direction perpendicular to the optical axis, and an x axis in the direction perpendicular to the z and y axes, the direction from the screen to the display panel being taken as positive, by the following equation:        z   =         c        (       x   2     +     y   2       )         1   +       1   -       (     1   +   K     )            c   2          (       x   2     +     y   2       )                 +       ∑   j          zj   ·     p   j                                  
     where c is the curvature of the osculating sphere, K and zj are the aspheric coefficients, and 
     p 3 =y 
     p 4 =x 2 −y 2    
     p 5 =2x 2 +2y 2 −1 
     p 9 =3x 2 y+3y 3 −2y 
     p 10 =3x 2 y−y 3    
     p 11 =x 4 −6x 2 y 2 +y 4    
     p 12 =4x 4 −4y 4 −3x 2 +3y 2    
     p 13 =6x 4 +12x 2 y 2 +6y 4 −6x 2 −6y 2 +1 
     p 23 =6x 6 −30x 4 y 2 −30x 2 y 4 +6y 6 −5x 4 +30x 2 y−5y 4    
     p 24 =15x 6 +15x 4 y 2 −15x 2 y 4 −15y 6 −20x 4 +20y 4 +6x 2 −6y 2    
     p 25 =20x 6 +60x 4 y 2 +60x 2 y 4 +20y 6 −30x 4 −60x 2 y 4 −30y 4 +12x 2 +12y 2 +1. 
     In each numerical example, taking a common axis of symmetry of all the spherical surfaces as the reference axis, the aspheric surface having no axial symmetry has its original point of the equation for the shape lying not on this reference axis, or moved therefrom in coordinates as shown in FIG.  20 . Therefore, data of the original point of the equation for the shape of the aspheric surface having no axial symmetry are described as decentering data together with the lens data. 
     NUMERICAL EXAMPLE 1 
     
       
         
               
             
               
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
             
               
               
               
               
             
           
               
                   
               
               
                 f:47.2˜73.2 Fno:2.51˜3.69 ωx:15.63˜10.22 ωy:19.72˜13.01 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 r 
                 d 
                 nd 
                 νd 
               
               
                   
                   
               
             
          
           
               
                 1 
                 Asphere 
                 4.20 
                 1.492 
                 57.4 
               
             
          
           
               
                 2 
                 26.141 
                 24.99 
                   
                   
               
               
                 3 
                 −24.377 
                 2.00 
                 1.581 
                 40.8 
               
               
                 4 
                 63.361 
                 0.18 
               
               
                 5 
                 66.940 
                 4.28 
                 1.805 
                 25.4 
               
             
          
           
               
                 6 
                 −84.774 
                 Variable 
                   
               
             
          
           
               
                 7 
                 894.781 
                 3.58 
                 1.603 
                 60.7 
               
               
                 8 
                 −91.106 
                 0.20 
               
               
                 9 
                 42.480 
                 5.15 
                 1.639 
                 55.4 
               
               
                 10 
                 367.733 
                 20.58 
               
               
                 11 
                 −56.665 
                 2.50 
                 1.741 
                 27.8 
               
               
                 12 
                 45.375 
                 1.02 
               
               
                 13 
                 69.625 
                 4.31 
                 1.492 
                 57.4 
               
             
          
           
               
                 14 
                 Asphere 
                 2.98 
                   
               
             
          
           
               
                 15 
                 84.679 
                 9.45 
                 1.516 
                 64.1 
               
             
          
           
               
                 16 
                 −40.811 
                 Variable 
                   
               
             
          
           
               
                 17 
                 111.360 
                 6.00 
                 1.516 
                 64.1 
               
               
                 18 
                 −216.146 
                 9.98 
               
             
          
           
               
                 19 
                 inf. 
                 40.00 
                 1.516 
                 64.1 
               
               
                 20 
                 inf. 
               
               
                   
               
             
          
           
               
                   
                 Variable 
                 Zooming Position 
                   
               
             
          
           
               
                   
                 Separation 
                 W 
                 T 
               
               
                   
                   
               
               
                   
                 d6 
                 15.56 
                  3.00 
               
               
                   
                 d16 
                 39.48 
                 90.96 
               
               
                   
                   
               
             
          
         
       
     
     Aspheric Coefficients of aspheric surface having axial symmetry: 
     For r 14 , c(1/r): −2.641e-03 K: −1.345e+01 A: 4.626e-06 B: 1.900e-10 C: 7.502e-13 D: −8.734e-15 E: 0.000e+00 
     Aspheric Coefficients of aspheric surface having no axial symmetry: 
     For r  1 , c(1/r): 1.763e-02 K: 3.886e+00 z  4 : −9.177e-07 z  5 : 2.131e-05 z 11 : 3.457e-09 z 12 : 1.826e-09 z 13 : 3.898e-07 z 23 : −3.750e-12 z 24 : −1.587e-12 z 25 : −2.517e-12 
     NUMERICAL EXAMPLE 2 
     
       
         
               
             
               
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
             
               
               
               
               
             
           
               
                   
               
               
                 f:47.2˜73.2 Fno:2.51˜3.69 ωx:15.63˜10.22 ωy:19.72˜13.01 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 r 
                 d 
                 nd 
                 νd 
               
               
                   
                   
               
             
          
           
               
                 1 
                 Asphere 
                 4.20 
                 1.49 
                 57.4 
               
               
                 2 
                 Asphere 
                 24.89 
               
             
          
           
               
                 3 
                 −24.294 
                 2.00 
                 1.581 
                 40.8 
               
               
                 4 
                 62.607 
                 0.17 
               
               
                 5 
                 66.026 
                 4.29 
                 1.805 
                 25.4 
               
             
          
           
               
                 6 
                 −85.178 
                 Variable 
                   
               
             
          
           
               
                 7 
                 1075.082 
                 3.60 
                 1.603 
                 60.7 
               
               
                 8 
                 −92.697 
                 0.20 
               
               
                 9 
                 42.537 
                 5.23 
                 1.639 
                 55.4 
               
               
                 10 
                 455.826 
                 20.43 
               
               
                 11 
                 −56.178 
                 2.50 
                 1.741 
                 27.8 
               
               
                 12 
                 45.490 
                 0.98 
               
               
                 13 
                 68.240 
                 4.33 
                 1.492 
                 57.4 
               
             
          
           
               
                 14 
                 Asphere 
                 3.09 
                   
               
             
          
           
               
                 15 
                 85.674 
                 9.44 
                 1.516 
                 64.1 
               
             
          
           
               
                 16 
                 −40.755 
                 Variable 
                   
               
             
          
           
               
                 17 
                 111.796 
                 5.99 
                 1.516 
                 64.1 
               
               
                 18 
                 −216.795 
                 9.98 
               
             
          
           
               
                 19 
                 inf. 
                 40.00 
                 1.516 
                 64.1 
               
               
                 20 
                 inf. 
               
               
                   
               
             
          
           
               
                   
                 Variable 
                 Zooming Position 
                   
               
             
          
           
               
                   
                 Separation 
                 W 
                 T 
               
               
                   
                   
               
               
                   
                 d6 
                 15.49 
                  3.00 
               
               
                   
                 d16 
                 39.66 
                 91.18 
               
               
                   
                   
               
             
          
         
       
     
     Aspheric Coefficients of aspheric surface having axial symmetry: 
     For r  1 , c(1/r): 1.815e-02 K: 3.645e+00 A: 2.058e-06 B:−2.203e-10 C: 7.966e-13 D: 0.000e+00 E: 0.000e+00 r 14 , c(1/r):−2.616e-03 K:−7.274e+01 A: 4.638e-06 B: 4.770e-10 C:−3.106e-14 D:−7.398e-15 E: 0.000e+00 
     Aspheric Coefficients of aspheric surface having no axial symmetry: 
     For r  2 , c(1/r): 3.900e-02 K: 9.493e-02 z  3 : −2.911e-05 z  4 : −1.614e-07 z  5 : −2.198e-05 z 13 : −1.782e-07 z 25 : −9.723e-11 
     NUMERICAL EXAMPLE 3 
     
       
         
               
             
               
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
             
               
               
               
               
             
           
               
                   
               
               
                 f:47.2˜73.1 Fno:2.57˜3.73 ωx:15.64˜10.23 ωy:19.73˜13.02 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 r 
                 d 
                 nd 
                 νd 
               
               
                   
                   
               
             
          
           
               
                 1 
                 Asphere 
                 4.20 
                 1.492 
                 57.4 
               
             
          
           
               
                 2 
                 28.832 
                 22.20 
                   
                   
               
               
                 3 
                 −23.880 
                 2.00 
                 1.581 
                 40.8 
               
               
                 4 
                 68.612 
                 0.22 
               
               
                 5 
                 73.835 
                 4.62 
                 1.805 
                 25.4 
               
             
          
           
               
                 6 
                 −73.326 
                 Variable 
                   
               
             
          
           
               
                 7 
                 9046.483 
                 4.31 
                 1.603 
                 60.7 
               
               
                 8 
                 −98.245 
                 0.20 
               
               
                 9 
                 41.994 
                 6.51 
                 1.639 
                 55.4 
               
               
                 10 
                 1023.504 
                 18.90 
               
               
                 11 
                 −57.529 
                 2.50 
                 1.741 
                 27.8 
               
               
                 12 
                 43.285 
                 0.73 
               
               
                 13 
                 55.926 
                 4.45 
                 1.492 
                 57.4 
               
             
          
           
               
                 14 
                 Asphere 
                 3.05 
                   
               
             
          
           
               
                 15 
                 79.980 
                 9.27 
                 1.516 
                 64.1 
               
             
          
           
               
                 16 
                 −42.281 
                 Variable 
                   
               
             
          
           
               
                 17 
                 Asphere 
                 5.78 
                 1.492 
                 57.4 
               
             
          
           
               
                 18 
                 −102.863 
                 9.98 
                   
               
             
          
           
               
                 19 
                 inf. 
                 40.00 
                 1.516 
                 64.1 
               
               
                 20 
                 inf. 
               
               
                   
               
             
          
           
               
                   
                 Variable 
                 Zooming Position 
                   
               
             
          
           
               
                   
                 Separation 
                 W 
                 T 
               
               
                   
                   
               
               
                   
                 d6 
                 15.57 
                  3.00 
               
               
                   
                 d16 
                 41.86 
                 92.49 
               
               
                   
                   
               
             
          
         
       
     
     Aspheric Coefficients of aspheric surface having axial symmetry: 
     For r  1 , c(1/r): 1.166e-02 K: 1.048e+01 A: 2.887e-06 B: −3.749e-10 C: 9.292e-13 D: 0.000e+00 E: 0.000e+00 For r 14 , c(1/r):−1.952e-04 K:−5.410e+04 A: 5.199e-06 B: 1.646e-09 C:−4.933e-12 D:−6.981e-16 E: 0.000e+00 
     Aspheric Coefficients of aspheric surface having no axial symmetry: 
     For r 17 , c(1/r): 4.518e-03 K: −5.813e+01 z  2 : −2.594e-05 z  4 : −2.346e-08 z  5 : −8.619e-05 z 11 :−7.525e-10 z 12 : 9.041e-11 z 13 : 2.958e-08 z 23 : −1.454e-13 z 24 : −1.311e-14 z 25 : −1.094e-11 
     NUMERICAL EXAMPLE 4 
     
       
         
               
             
               
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
             
               
               
               
               
             
           
               
                   
               
               
                 f:47.2˜73.2 Fno:2.51˜3.60 ωx:15.63˜10.23 ωy:19.72˜13.01 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 r 
                 d 
                 nd 
                 νd 
               
               
                   
                   
               
             
          
           
               
                 1 
                 Asphere 
                 4.20 
                 1.492 
                 57.4 
               
               
                 2 
                 Asphere 
                 22.46 
               
             
          
           
               
                 3 
                 −23.833 
                 2.00 
                 1.581 
                 40.8 
               
               
                 4 
                 68.512 
                 0.82 
               
               
                 5 
                 81.776 
                 4.63 
                 1.805 
                 25.4 
               
             
          
           
               
                 6 
                 −75.185 
                 Variable 
                   
               
             
          
           
               
                 7 
                 −6266.770 
                 3.27 
                 1.603 
                 60.7 
               
               
                 8 
                 −107.530 
                 0.20 
               
               
                 9 
                 42.619 
                 7.11 
                 1.639 
                 55.4 
               
               
                 10 
                 −550.693 
                 17.56 
               
               
                 11 
                 −52.483 
                 2.50 
                 1.741 
                 27.8 
               
               
                 12 
                 45.852 
                 0.80 
               
               
                 13 
                 64.287 
                 4.47 
                 1.492 
                 57.4 
               
             
          
           
               
                 14 
                 Asphere 
                 4.28 
                   
               
             
          
           
               
                 15 
                 99.741 
                 9.43 
                 1.516 
                 64.1 
               
             
          
           
               
                 16 
                 −42.096 
                 Variable 
                   
               
             
          
           
               
                 17 
                 Asphere 
                 5.83 
                 1.516 
                 64.1 
               
               
                 18 
                 Asphere 
                 9.98 
               
               
                 19 
                 inf. 
                 40.00 
                 1.516 
                 64.1 
               
               
                 20 
                 inf. 
               
               
                   
               
             
          
           
               
                   
                 Variable 
                 Zooming Position 
                   
               
             
          
           
               
                   
                 Separation 
                 W 
                 T 
               
               
                   
                   
               
               
                   
                 d6 
                 15.99 
                  3.00 
               
               
                   
                 d16 
                 40.59 
                 90.11 
               
               
                   
                   
               
             
          
         
       
     
     Aspheric Coefficients of aspheric surface having axial symmetry: 
     For r 14 , c(1/r):−3.465e-03 K:−1.157e+01 A: 4.676e-06 B: 1.461e-09 C:−4.218e-12 D:−2.289e-16 E: 0.000e+00 
     Aspheric Coefficients of aspheric surface having no axial symmetry: 
     For r  1 , c(1/r): 1.550e-02 K: 5.961e+00 z  3 : 6.129e-03 z  4 : −3.581e-06 z  5 : 2.009e-04 z  9 : −3.693e-07 z 10 : 7.064e-07 z 11 : 4.792e-07 z 12 : 2.189e-08 z 13 : 2.960e-07 z 23 : 1.450e-11 z 24 : −6.346e-12 z 25 : −6.194e-11 
     For r  2 , c(1/r): 3.615e-02 K: 0.000e+00 z  5 : 1.658e-04 z  9 : −2.011e-06 z 10 : 7.746e-07 z 11 : 5.342e-07 z 12 : 1.973e-08 z 13 : −1.911e-07 z 23 : 1.228e-10 z 24 : −1.137e-14 z 25 : −1.873e-10 
     For r 17 , c(1/r): 6.412e-04 K: 0.000e+00 z  5 : 3.949e-04 z  9 : −3.517e-06 z 10 : 1.393e-06 z 11 : 3.622e-07 z 12 : 1.017e-08 213: 1.276e-07 z 23 : 1.745e-10 z 24 : −2.981e-11 z 25 : 4.313e-11 
     For r 18 , c(1/r): −1.274e-02 K: 0.000e+00 z  5 : 5.047e-04 z  9 : −3.272e-06 z 10 : 4.507e-07 z 11 : 4.402e-07 z 12 : 1.430e-08 z 13 : 3.482e-07 z 23 : 1.704e-10 z 24 : −2.876e-11 z 25 : 3.458e-11 
     Decentering Data 
     
       
         
               
               
               
             
               
               
               
             
           
               
                   
               
               
                 r 
                 dy (mm) 
                 β (degree) 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 1 
                 −0.11 
                 0.06 
               
               
                 2 
                 −0.06 
                 −0.50 
               
               
                 17 
                 1.36 
                 −1.32 
               
               
                 18 
                 1.64 
                 0.02 
               
               
                   
               
             
          
         
       
     
     It will be appreciated from the foregoing that, according to the invention, an aspheric surface having no axial symmetry is employed in part of the optical system. 
     It is, therefore, made possible to achieve a projection lens which, when projecting the image obliquely, corrects well the trapezoidal deformation and distortion, and an image projection apparatus using the same.