Patent Document

[0001]    This application claims benefits of Japanese Application No. 2009-212134 filed in Japan on Sep. 14, 2009, No. 2009-212135 filed in Japan on Sep. 14, 2009 and No. 2009-212136 filed in Japan on Sep. 14, 2009, the contents of which are hereby incorporated by reference. 
       BACKGROUND OF THE INVENTION 
       [0002]    1. Field of Invention 
         [0003]    This invention relates to a variable power optical system in which aberrations are corrected well while the value of the total length of the optical system relative to image height is being made to become small, and to an image pickup apparatus having the same. 
         [0004]    2. Description of the Related Art 
         [0005]    Digital cameras, which are provided with a solid-state imaging sensor like CCD (Charge Coupled Device) or CMOS (Complementary Metal-Oxide Semiconductor), have become mainstream instead of film-based cameras in recent years. These digital cameras include various kinds of digital cameras which range from high performance-type digital camera for business to compact popular-type digital camera. 
         [0006]    And, in such digital cameras, compact popular-type digital cameras have improved in downsizing because of desires that users easily enjoy photography, so that digital cameras which can be put well in pockets of clothes or bags and are convenient to be carried have appeared. Accordingly, it has become necessary to downsize variable power optical systems for such digital cameras yet more. However, it has been required that variable power optical systems for such digital cameras have not only a small size but also high optical performance (aberrations are corrected well in such variable power optical systems). 
         [0007]    Variable power optical systems which meet such requirements include variable power optical systems which are disclosed in International Publication WO 2006/115107 and Japanese Patent Kokai No. 2008-233611 respectively. International Publication WO 2006/115107 relates to a variable power optical system for correcting chromatic aberration of magnification. When an attempt to achieve a super-small-sized variable power optical system is made, the refractive power of a second lens group becomes strong, so that chromatic aberration of magnification at the telephoto end position becomes a problem. In International Publication WO 2006/115107, this problem is corrected by making the Abbe&#39;s number of a fourth lens group proper. Japanese Patent Kokai No. 2008-233611 relates to a variable power optical system which has high performances and is downsized and the first lens group of which consists of two lens. 
       SUMMARY OF THE INVENTION 
       [0008]    A variable power optical system according to the present first invention is characterized in that: the variable power optical system at least includes, in order from the object side, a first lens group with negative refractive power, a magnification-changing group with positive refractive power, and a last lens group with positive refractive power; the magnification-changing group includes a first lens element with positive refractive power, a second lens element, and a third lens element in that order from the object side; the second lens element has a convex shape on the object side; the last lens group includes a positive lens; and the following conditions (1) and (2) are satisfied: 
         [0000]      10 ≦VdLg≦ 45  (1)
 
         [0000]      −1.0&lt;( R 2 a−R 2 b )/( R 2 a+R 2 b )&lt;1.0  (2)
 
         [0000]    where VdLg denotes the Abbe&#39; Number of the positive lens of the last lens group with respect to the d line, R2a denotes the radius of curvature of the object-side surface of the second lens element, and R2b denotes the radius of curvature of the image-side surface of the third lens element. 
         [0009]    Also, in a variable power optical system according to the present first invention, it is preferred that: the variable power optical system consists of a first lens group with negative refractive power, a second lens group with positive refractive power, a third lens group with negative refractive power, and a fourth lens group with positive refractive power, in that order from the object side; and the magnification-changing group is the second lens group and the last lens group is the fourth lens group. 
         [0010]    Also, in a variable power optical system according to the present first invention, it is preferred that the following condition (3) is satisfied: 
         [0000]      0.45 ≦|f 1|/( FLw×FLt ) 1/2 ≦1.60  (3)
 
         [0000]    where f1 denotes the focal length of the first lens group, FLw denotes the focal length of the whole of the variable power optical system in the wide angle end position, and FLt denotes the focal length of the whole of the variable power optical system in the telephoto end position. 
         [0011]    Also, in a variable power optical system according to the present first invention, it is preferred that the following condition (4) is satisfied: 
         [0000]      0.30 ≦fv /( FLw×FLt ) 1/2 ≦1.10  (4)
 
         [0000]    where fv denotes the focal length of the magnification-changing group, FLw denotes the focal length of the whole of the variable power optical system in the wide angle end position, and FLt denotes the focal length of the whole of the variable power optical system in the telephoto end position. 
         [0012]    Also, in a variable power optical system according to the present first invention, it is preferred that: the positive lens in the last lens group has a concave shape on the object side; and the following condition (5) is satisfied: 
         [0000]      0.1≦( RLa−RLb )/( RLa+RLb )&lt;1.0  (5)
 
         [0000]    where RLa denotes the radius of curvature of the object-side surface of the positive lens in the last lens group, and RLb denotes the radius of curvature of the image-side surface of the positive lens in the last lens group. 
         [0013]    Also, an image pickup apparatus according to the present first invention is characterized in that: the image pickup apparatus includes one of the above-described variable power optical systems according to the present first invention, and an imaging sensor; and the following condition (6) is satisfied: 
         [0000]      1.0 ≦|f 1 |/IH≦ 2.8  (6)
 
         [0000]    where f1 denotes the focal length of the first lens group, and IH denotes the image height of the imaging sensor. 
         [0014]    Also, an image pickup apparatus according to the present first invention is characterized in that the image pickup apparatus includes one of the above described variable power optical systems according to the first present invention, and an imaging sensor; and the following condition (7) is satisfied: 
         [0000]      0.2 ≦|fv|/IH≦ 1.8  (7)
 
         [0000]    where fv denotes the focal length of the magnification-changing group, and IH denotes the image height of the imaging sensor. 
         [0015]    Also, a variable power optical system according to the present second invention is characterized in that: the variable power optical system includes, in order from the object side, a first lens group with negative refractive power, a second lens group with positive refractive power, a third lens group with negative refractive power, and a fourth lens group with positive refractive power; the second lens group includes at least one positive lens and a negative lens; the fourth lens group includes a positive lens; and the following conditions (8), (9), and (10) are satisfied: 
         [0000]      10 ≦Vd 4 g≦ 45  (8)
 
         [0000]      2.2 ≦|α/f 1|+(α/ f 2)−0.026 ×Vd 4 g≦ 5.0  (9)
 
         [0000]      10 ≦Vd max− Vd min≦80  (10)
 
         [0000]    where α=(FLw×FLt) 1/2 , FLw denotes the focal length of the whole of the variable power optical system in the wide angle end position, FLt denotes the focal length of the whole of the variable power optical system in the telephoto end position, f1 denotes the focal length of the first lens group, f2 denotes the focal length of the second lens group, Vd4g denotes the Abbe&#39; Number of the positive lens of the fourth lens group with respect to the d line, Vdmax denotes the Abbe&#39;s Number of a glass material having the lowest dispersion characteristic of those of glass materials which are used for lenses of the second lens group, with respect to the d line, and Vdmin denotes the Abbe&#39;s Number of a glass material having the highest dispersion characteristic of those of glass materials which are used for lenses of the second lens group, with respect to the d line. 
         [0016]    Also, in a variable power optical system according to the present second invention, it is preferred that the following condition (3) is satisfied: 
         [0000]      0.45 ≦|f 1|/( FLw×FLt ) 1/2 ≦1.60  (3)
 
         [0000]    where f1 denotes the focal length of the first lens group, FLw denotes the focal length of the whole of the variable power optical system in the wide angle end position, and FLt denotes the focal length of the whole of the variable power optical system in the telephoto end position. 
         [0017]    Also, in a variable power optical system according to the present second invention, it is preferred that the following condition (11) is satisfied: 
         [0000]      0.30 ≦f 2/( FLw×FLt ) 1/2 ≦1.10  (11)
 
         [0000]    where f2 denotes the focal length of the second lens group, FLw denotes the focal length of the whole of the variable power optical system in the wide angle end position, and FLt denotes the focal length of the whole of the variable power optical system in the telephoto end position. 
         [0018]    Also, in a variable power optical system according to the present second invention, it is preferred that: the second lens group consists of a first lens element with positive refractive power, a second lens element, and a third lens element in that order from the object side; the first lens element has a convex shape on the object side; and the following condition (2) is satisfied: 
         [0000]      −1.0&lt;( R 2 a−R 2 b )/( R 2 a+R 2 b )&lt;1.0  (2)
 
         [0000]    where R2a denotes the radius of curvature of the object-side surface of the second lens element, and R2b denotes the radius of curvature of the image-side surface of the third lens element. 
         [0019]    Also, in a variable power optical system according to the present second invention, it is preferred that: the positive lens in the fourth lens group has a concave shape on the object side; and the following condition (12) is satisfied: 
         [0000]      0&lt;( R 4 a−R 4 b )/( R 4 a+R 4 b )&lt;1.0  (12)
 
         [0000]    where R4a denotes the radius of curvature of the object-side surface of the positive lens in the fourth lens group, and R4b denotes the radius of curvature of the image-side surface of the positive lens in the fourth lens group. 
         [0020]    Also, an image pickup apparatus according to the present second invention is characterized in that: the image pickup apparatus includes one of the above-described variable power optical systems according to the second present invention, and an imaging sensor. 
         [0021]    Also, in an image pickup apparatus according to the present second invention, it is preferred that: the image pickup apparatus includes a variable power optical system forming an optical image of an object, and an imaging sensor; the imaging sensor transforms the optical image formed by the variable power optical system into electrical image signals; the variable power optical system includes, in order from the object side, a first lens group with negative refractive power, a second lens group with positive refractive power, a third lens group with negative refractive power, and a fourth lens group with positive refractive power; the second lens group includes at least one positive lens and a negative lens; the fourth lens group includes a positive lens; and the following conditions (8), (13), and (10) are satisfied: 
         [0000]      10 ≦Vd 4 g≦ 45  (8)
 
         [0000]      −0.27 ≦|IH/f 1|+( IH/f 2)−0.05 ×Vd 4 g≦ 3.0  (13)
 
         [0000]      10 ≦Vd max− Vd min≦80  (10)
 
         [0000]    where IH denotes the image height of the imaging sensor, f1 denotes the focal length of the first lens group, f2 denotes the focal length of the second lens group, Vd4g denotes the Abbe&#39; Number of the positive lens of the fourth lens group with respect to the d line, Vdmax denotes the Abbe&#39;s Number of a glass material having the lowest dispersion characteristic of those of glass materials which are used for lenses of the second lens group, with respect to the d line, and Vdmin denotes the Abbe&#39;s Number of a glass material having the highest dispersion characteristic of those of glass materials which are used for lenses of the second lens group, with respect to the d line. 
         [0022]    Also, in an image pickup apparatus according to the present second invention, it is preferred that the following condition (6) is satisfied: 
         [0000]      1.0 ≦|f 1 |/IH≦ 2.8  (6)
 
         [0000]    where f1 denotes the focal length of the first lens group, and IH denotes the image height of the imaging sensor. 
         [0023]    Also, in an image pickup apparatus according to the present second invention, it is preferred that the following condition (14) is satisfied: 
         [0000]      0.2 ≦|f 2 |/IH≦ 1.8  (14)
 
         [0000]    where f2 denotes the focal length of the second lens group, and IH denotes the image height of the imaging sensor. 
         [0024]    Also, in an image pickup apparatus according to the present second invention, it is preferred that: the second lens group consists of a first lens element with positive refractive power, a second lens element, and a third lens element in that order from the object side; the first lens element has a convex shape on the object side; and the following condition (2) is satisfied: 
         [0000]      −1.0&lt;( R 2 a−R 2 b )/( R 2 a+R 2 b )&lt;1.0  (2)
 
         [0000]    where R2a denotes the radius of curvature of the object-side surface of the second lens element, and R2b denotes the radius of curvature of the image-side surface of the third lens element. 
         [0025]    Also, in an image pickup apparatus according to the present second invention, it is preferred that: the positive lens in the fourth lens group in the variable power optical system has a concave shape on the object side; and the following condition (12) is satisfied: 
         [0000]      0&lt;( R 4 a−R 4 b )/( R 4 a+R 4 b )&lt;1.0  (12)
 
         [0000]    where R4a denotes the radius of curvature of the object-side surface of the positive lens in the fourth lens group, and R4b denotes the radius of curvature of the image-side surface of the positive lens in the fourth lens group. 
         [0026]    Also, a variable power optical system according to the present third invention is characterized in that: the variable power optical system includes, in order from the object side, a first lens group with negative refractive power, a second lens group with positive refractive power, a third lens group with negative refractive power, and a fourth lens group with positive refractive power; the first lens group includes one negative lens and one positive lens in that order from the object side, and an air distance is provided between the negative and positive lenses of the first lens group; and the following conditions (15), (16), and (17) are satisfied: 
         [0000]      1.75 ≦Nd 1 g≦ 2.50  (15)
 
         [0000]      15 ≦Vd 1 g≦ 43  (16)
 
         [0000]      3 ≦VdN−VdP≦ 28  (17)
 
         [0000]    where Nd1g denotes the refractive index of each of lenses constituting the first lens group, with respect to the d line, Vd1g denotes the Abbe&#39;s Number of each of lenses constituting the first lens group, with respect to the d lines, VdN denotes the Abbe&#39;s Number of the negative lens in the first lens group, with respect to the d lines, and VdP denotes the Abbe&#39;s Number of the positive lens in the first lens group, with respect to the d lines. 
         [0027]    Also, in a variable power optical system according to the present third invention, it is preferred that the following condition (18) is satisfied: 
         [0000]      0.03 ≦D /( FLw×FLt ) 1/2 ≦0.26  (18)
 
         [0000]    where D denotes the axial air distance between the negative and positive lenses of the first lens group, FLw denotes the focal length of the whole of the variable power optical system in the wide angle end position, and FLt denotes the focal length of the whole of the variable power optical system in the telephoto end position. 
         [0028]    Also, in a variable power optical system according to the present third invention, it is preferred that: an air lens which has a convex shape on the object side is formed nearer to the image-plane side than the negative lens of the first lens group; and the following condition (19) is satisfied: 
         [0000]      −0.25≦( r 2 −r 3)/( r 2 +r 3)≦−0.07  (19)
 
         [0000]    where r2 denotes the radius of curvature of the image-side surface of the negative lens of the first lens group, and r3 denotes the radius of curvature of the object-side surface of the positive lens of the first lens group. 
         [0029]    Also, in a variable power optical system according to the present third invention, it is preferred that the following condition (3) is satisfied: 
         [0000]      0.45 ≦|f 1|/( FLw×FLt ) 1/2 ≦1.60  (3)
 
         [0000]    where f1 denotes the focal length of the first lens group, FLw denotes the focal length of the whole of the variable power optical system in the wide angle end position, and FLt denotes the focal length of the whole of the variable power optical system in the telephoto end position. 
         [0030]    Also, in a variable power optical system according to the present third invention, it is preferred that the following condition (20) is satisfied: 
         [0000]      −0.5 ≦FLn/FLp≦− 0.3  (20)
 
         [0000]    where FLn denotes the focal length of the negative lens of the first lens group, and FLp denotes the focal length of the positive lens of the first lens group. 
         [0031]    Also, in a variable power optical system according to the present third invention, it is preferred that the following condition (11) is satisfied: 
         [0000]      0.30 ≦f 2/( FLw×FLt ) 1/2 ≦1.10  (11)
 
         [0000]    where f2 denotes the focal length of the second lens group, FLw denotes the focal length of the whole of the variable power optical system in the wide angle end position, and FLt denotes the focal length of the whole of the variable power optical system in the telephoto end position. 
         [0032]    Also, in a variable power optical system according to the present third invention, it is preferred that: the negative lens of the first lens group has a convex shape on the object side; and the following condition (21) is satisfied: 
         [0000]      0.2≦( r 1 −r 2)/( r 1 +r 2)≦1.0  (21)
 
         [0000]    where r1 denotes the radius of curvature of the object-side surface of the negative lens of the first lens group, and r2 denotes the radius of curvature of the image-side surface of the negative lens of the first lens group. 
         [0033]    Also, in a variable power optical system according to the present third invention, it is preferred that: the fourth lens group consists of one lens with positive refractive power; and the following condition (22) is satisfied: 
         [0000]      10 ≦Vd 4 g≦ 40  (22)
 
         [0000]    where Vd4g denotes the Abbe&#39;s Number of the positive lens of the fourth lens group with respect to the d line. 
         [0034]    Also, an image pickup apparatus according to the present third invention is characterized in that: the image pickup apparatus includes one of the above-described variable power optical systems according to the present third invention, and an imaging sensor; and the following condition (6) is satisfied: 
         [0000]      1.0 ≦|f 1 |/IH≦ 2.8  (6)
 
         [0000]    where f1 denotes the focal length of the first lens group, and IH denotes the image height of the imaging sensor. 
         [0035]    Also, an image pickup apparatus according to the present third invention is characterized in that: the image pickup apparatus includes one of the above-described variable power optical systems according to the present third invention, and an imaging sensor; and the following condition (14) is satisfied: 
         [0000]      0.2 ≦|f 2 |/IH≦ 1.8  (14)
 
         [0000]    where f2 denotes the focal length of the second lens group, and IH denotes the image height of the imaging sensor. 
         [0036]    The present invention can offer a variable power optical system in which aberrations are corrected well while the value of the total length of the variable power optical system relative to image height is being made to become small, and an image pickup apparatus having the same. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0037]      FIG. 1  is a sectional view showing a variable power optical systems of an embodiment 1 according to the present invention, taken along the optical axis,  FIG. 1A  shows the configuration of lenses in the wide angle end position,  FIG. 1B  shows the configuration of lenses in the middle position, and  FIG. 1C  shows the configuration of lenses in the telephoto end position. 
           [0038]      FIG. 2  are aberration diagrams in focusing at infinity in the embodiment 1,  FIGS. 2A to 2D  show a state in the wide angle end position,  FIGS. 2E to 2H  show a state in the middle position, and  FIGS. 2I to 2L  show a state in the telephoto end position. 
           [0039]      FIG. 3  are aberration diagrams in focusing at close range in the embodiment 1,  FIGS. 3A to 3D  show a state in the wide angle end position,  FIGS. 3E to 3H  show a state in the middle position, and  FIGS. 3I to 3L  show a state in the telephoto end position. 
           [0040]      FIG. 4  are aberration diagrams showing coma in the embodiment 1,  FIG. 4A  shows a state in focusing at infinity in the wide angle end position,  FIG. 4B  shows a state in focusing at infinity in the middle position,  FIG. 4C  shows a state in focusing at infinity in the telephoto end position,  FIG. 4D  shows a state in focusing at close range in the wide angle end position,  FIG. 4E  shows a state in focusing at close range in the middle position, and  FIG. 4F  shows a state in focusing at close range in the telephoto end position. 
           [0041]      FIG. 5  is a sectional view showing a variable power optical system of an embodiment 2 according to the present invention, taken along the optical axis,  FIG. 5A  shows the configuration of lenses in the wide angle end position,  FIG. 5B  shows the configuration of lenses in the middle position, and  FIG. 5C  shows the configuration of lenses in the telephoto end position. 
           [0042]      FIG. 6  are aberration diagrams in focusing at infinity in the embodiment 2,  FIGS. 6A to 6D  show a state in the wide angle end position,  FIGS. 6E to 6H  show a state in the middle position, and  FIGS. 6I to 6L  show a state in the telephoto end position. 
           [0043]      FIG. 7  are aberration diagrams in focusing at close range in the embodiment 2,  FIGS. 7A to 7D  show a state in the wide angle end position,  FIGS. 7E to 7H  show a state in the middle position, and  FIGS. 7I to 7L  show a state in the telephoto end position. 
           [0044]      FIG. 8  are aberration diagrams showing coma at 70 percent of image height in the embodiment 2,  FIG. 8A  shows a state in focusing at infinity in the wide angle end position,  FIG. 8B  shows a state in focusing at infinity in the middle position,  FIG. 8C  shows a state in focusing at infinity in the telephoto end position,  FIG. 8D  shows a state in focusing at close range in the wide angle end position,  FIG. 8E  shows a state in focusing at close range in the middle position, and  FIG. 8F  shows a state in focusing at close range in the telephoto end position. 
           [0045]      FIG. 9  is a sectional view showing a variable power optical system of an embodiment 3 according to the present invention, taken along the optical axis,  FIG. 9A  shows the configuration of lenses in the wide angle end position,  FIG. 9B  shows the configuration of lenses in the middle position, and  FIG. 9C  shows the configuration of lenses in the telephoto end position. 
           [0046]      FIG. 10  are aberration diagrams in focusing at infinity in the embodiment 3,  FIGS. 10A to 10D  show a state in the wide angle end position,  FIGS. 10E to 10H  show a state in the middle position, and  FIGS. 10I to 10L  show a state in the telephoto end position. 
           [0047]      FIG. 11  are aberration diagrams in focusing at close range in the embodiment 3,  FIGS. 11A to 11D  show a state in the wide angle end position,  FIGS. 11E to 11H  show a state in the middle position, and  FIGS. 11I to 11L  show a state in the telephoto end position. 
           [0048]      FIG. 12  are aberration diagrams showing coma at 70 percent of image height in the embodiment 3,  FIG. 12A  shows a state in focusing at infinity in the wide angle end position,  FIG. 12B  shows a state in focusing at infinity in the middle position,  FIG. 12C  shows a state in focusing at infinity in the telephoto end position,  FIG. 12D  shows a state in focusing at close range in the wide angle end position,  FIG. 12E  shows a state in focusing at close range in the middle position, and  FIG. 12F  shows a state in focusing at close range in the telephoto end position. 
           [0049]      FIG. 13  is a sectional view showing a variable power optical system of an embodiment 4 according to the present invention, taken along the optical axis,  FIG. 13A  shows the configuration of lenses in the wide angle end position,  FIG. 13B  shows the configuration of lenses in the middle position, and  FIG. 13C  shows the configuration of lenses in the telephoto end position. 
           [0050]      FIG. 14  are aberration diagrams in focusing at infinity in the embodiment 4,  FIGS. 14A to 14D  show a state in the wide angle end position,  FIGS. 14E to 14H  show a state in the middle position, and  FIGS. 14I to 14L  show a state in the telephoto end position. 
           [0051]      FIG. 15  are aberration diagrams in focusing at close range in the embodiment 4,  FIGS. 15A to 15D  show a state in the wide angle end position,  FIGS. 15E to 15H  show a state in the middle position, and  FIGS. 15I to 15L  show a state in the telephoto end position. 
           [0052]      FIG. 16  are aberration diagrams showing coma at 70 percent of image height in the embodiment 4,  FIG. 16A  shows a state in focusing at infinity in the wide angle end position,  FIG. 16B  shows a state in focusing at infinity in the middle position, and  FIG. 16C  shows a state in focusing at infinity in the telephoto end position,  FIG. 16D  shows a state in focusing at close range in the wide angle end position,  FIG. 16E  shows a state in focusing at close range in the middle position, and  FIG. 16F  shows a state in focusing at close range in the telephoto end position. 
           [0053]      FIG. 17  is a sectional view showing a variable power optical system of an embodiment 5 according to the present invention, taken along the optical axis,  FIG. 17A  shows the configuration of lenses in the wide angle end position,  FIG. 17B  shows the configuration of lenses in the middle position, and  FIG. 17C  shows the configuration of lenses in the telephoto end position. 
           [0054]      FIG. 18  are aberration diagrams in focusing at infinity in the embodiment 5,  FIGS. 18A  to  18 D show a state in the wide angle end position,  FIGS. 18E to 18H  show a state in the middle position, and  FIGS. 18I to 18L  show a state in the telephoto end position. 
           [0055]      FIG. 19  are aberration diagrams in focusing at close range in the embodiment 5,  FIGS. 19A to 19D  show a state in the wide angle end position,  FIGS. 19E to 19H  show a state in the middle position, and  FIGS. 19I to 19L  show a state in the telephoto end position. 
           [0056]      FIG. 20  are aberration diagrams showing coma at 70 percent of image height in the embodiment 5,  FIG. 20A  shows a state in focusing at infinity in the wide angle end position,  FIG. 20B  shows a state in focusing at infinity in the middle position,  FIG. 20C  shows a state in focusing at infinity in the telephoto end position,  FIG. 20D  shows a state in focusing at close range in the wide angle end position,  FIG. 20E  shows a state in focusing at close range in the middle position, and  FIG. 20F  shows a state in focusing at close range in the telephoto end position. 
           [0057]      FIG. 21  is a sectional view showing a variable power optical system of an embodiment 6 according to the present invention, taken along the optical axis,  FIG. 21A  shows the configuration of lenses in the wide angle end position,  FIG. 21B  shows the configuration of lenses in the middle position, and  FIG. 21C  shows the configuration of lenses in the telephoto end position. 
           [0058]      FIG. 22  are aberration diagrams in focusing at infinity in the embodiment 6,  FIGS. 22A to 22D  show a state in the wide angle end position,  FIGS. 22E to 22H  show a state in the middle position, and  FIGS. 22I to 22L  show a state in the telephoto end position. 
           [0059]      FIG. 23  are aberration diagrams in focusing at close range in the embodiment 6,  FIGS. 23A to 23D  show a state in the wide angle end position,  FIGS. 23E to 23H  show a state in the middle position, and  FIGS. 23I to 23L  show a state in the telephoto end position. 
           [0060]      FIG. 24  are aberration diagrams showing coma at 70 percent of image height in the embodiment 6,  FIG. 24A  shows a state in focusing at infinity in the wide angle end position,  FIG. 24B  shows a state in focusing at infinity in the middle position,  FIG. 24C  shows a state in focusing at infinity in the telephoto end position,  FIG. 24D  shows a state in focusing at close range in the wide angle end position,  FIG. 24E  shows a state in focusing at close range in the middle position, and  FIG. 24F  shows a state in focusing at close range in the telephoto end position. 
           [0061]      FIG. 25  is a sectional view showing a variable power optical system of an embodiment 7 according to the present invention, taken along the optical axis,  FIG. 25A  shows the configuration of lenses in the wide angle end position,  FIG. 25B  shows the configuration of lenses in the middle position, and  FIG. 25C  shows the configuration of lenses in the telephoto end position. 
           [0062]      FIG. 26  are aberration diagrams in focusing at infinity in the embodiment 7,  FIGS. 26A to 26D  show a state in the wide angle end position,  FIGS. 26E to 26H  show a state in the middle position, and  FIGS. 26I to 26L  show a state in the telephoto end position. 
           [0063]      FIG. 27  are aberration diagrams showing coma at 70 percent of image height in the embodiment 7,  FIG. 27A  shows a state in focusing at infinity in the wide angle end position,  FIG. 27B  shows a state in focusing at infinity in the middle position, and  FIG. 27C  shows a state in focusing at infinity in the telephoto end position. 
           [0064]      FIG. 28  is a sectional view showing a variable power optical system of an embodiment 8 according to the present invention, taken along the optical axis,  FIG. 28A  shows the configuration of lenses in the wide angle end position,  FIG. 28B  shows the configuration of lenses in the middle position, and  FIG. 28C  shows the configuration of lenses in the telephoto end position. 
           [0065]      FIG. 29  are aberration diagrams in focusing at infinity in the embodiment 8,  FIGS. 29A to 29D  show a state in the wide angle end position,  FIGS. 29E to 29H  show a state in the middle position, and  FIGS. 29I to 29L  show a state in the telephoto end position. 
           [0066]      FIG. 30  are aberration diagrams showing coma at 70 percent of image height in the embodiment 8,  FIG. 30A  shows a state in focusing at infinity in the wide angle end position,  FIG. 30B  shows a state in focusing at infinity in the middle position, and  FIG. 30C  shows a state in focusing at infinity in the telephoto end position. 
           [0067]      FIG. 31  is a sectional view showing a variable power optical system of an embodiment 9 according to the present invention, taken along the optical axis,  FIG. 31A  shows the configuration of lenses in the wide angle end position,  FIG. 31B  shows the configuration of lenses in the middle position, and  FIG. 31C  shows the configuration of lenses in the telephoto end position. 
           [0068]      FIG. 32  are aberration diagrams in focusing at infinity in the embodiment 9,  FIGS. 32A to 32D  show a state in the wide angle end position,  FIGS. 32E to 32H  show a state in the middle position, and  FIGS. 32I to 32L  shows a state in the telephoto end position. 
           [0069]      FIG. 33  are aberration diagrams showing coma at 70 percent of image height in the embodiment 9,  FIG. 33A  shows a state in focusing at infinity in the wide angle end position,  FIG. 33B  shows a state in focusing at infinity in the middle position, and  FIG. 33C  shows a state in focusing at infinity in the telephoto end position. 
           [0070]      FIG. 34  is a sectional view showing a variable power optical system of an embodiment 10 according to the present invention, taken along the optical axis,  FIG. 34A  shows the configuration of lenses in the wide angle end position,  FIG. 34B  shows the configuration of lenses in the middle position, and  FIG. 34C  shows the configuration of lenses in the telephoto end position. 
           [0071]      FIG. 35  are aberration diagrams in focusing at infinity in the embodiment 10,  FIGS. 35A to 35D  show a state in the wide angle end position,  FIGS. 35E to 35H  show a state in the middle position, and  FIGS. 35I to 35L  show a state in the telephoto end position. 
           [0072]      FIG. 36  are aberration diagrams showing coma at 70 percent of image height in the embodiment 10,  FIG. 36A  shows a state in focusing at infinity in the wide angle end position,  FIG. 36B  shows a state in focusing at infinity in the middle position, and  FIG. 36C  shows a state in focusing at infinity in the telephoto end position. 
           [0073]      FIG. 37  is a sectional view showing a variable power optical system of an embodiment 11 according to the present invention, taken along the optical axis,  FIG. 37A  shows the configuration of lenses in the wide angle end position,  FIG. 37B  shows the configuration of lenses in the middle position, and  FIG. 37C  shows the configuration of lenses in the telephoto end position. 
           [0074]      FIG. 38  are aberration diagrams in focusing at infinity in the embodiment 11,  FIGS. 38A to 38D  show a state in the wide angle end position,  FIGS. 38E to 38H  show a state in the middle position, and  FIGS. 38I to 38L  show a state in the telephoto end position. 
           [0075]      FIG. 39  are aberration diagrams showing coma at 70 percent of image height in the embodiment 11,  FIG. 39A  shows a state in focusing at infinity in the wide angle end position,  FIG. 39B  shows a state in focusing at infinity in the middle position, and  FIG. 39C  shows a state in focusing at infinity in the telephoto end position. 
           [0076]      FIG. 40  is a sectional view showing a variable power optical system of an embodiment 12 according to the present invention, taken along the optical axis,  FIG. 40A  shows the configuration of lenses in the wide angle end position,  FIG. 40B  shows the configuration of lenses in the middle position, and  FIG. 40C  shows the configuration of lenses in the telephoto end position. 
           [0077]      FIG. 41  are aberration diagrams in focusing at infinity in the embodiment 12,  FIGS. 41A to 41D  show a state in the wide angle end position,  FIGS. 41E to 41H  show a state in the middle position, and  FIGS. 41I to 41L  show a state in the telephoto end position. 
           [0078]      FIG. 42  are aberration diagrams showing coma at 70 percent of image height in the embodiment 12,  FIG. 42A  shows a state in focusing at infinity in the wide angle end position,  FIG. 42B  shows a state in focusing at infinity in the middle position, and  FIG. 42C  shows a state in focusing at infinity in the telephoto end position. 
           [0079]      FIG. 43  is a front perspective view showing the appearance of a digital camera into which a variable power optical system of one of the embodiments according to the present invention is incorporated. 
           [0080]      FIG. 44  is a rear perspective view showing the appearance of the digital camera which is shown in  FIG. 43 . 
           [0081]      FIG. 45  is a perspective view schematically showing the constitution of the digital camera which is shown in  FIG. 43 . 
       
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
       [0082]    Prior to the description of the embodiments, operation and its effect in variable power optical systems according to the present invention and in image pickup apparatuses having the same will be explained. 
         [0083]    A variable power optical system of the first embodiment is characterized in that: the variable power optical system at least includes, in order from the object side, a first lens group with negative refractive power, a magnification-changing group with positive refractive power, and a last lens group with positive refractive power; the magnification-changing group includes a first lens element with positive refractive power, a second lens element, and a third lens element in that order from the object side; the second lens element has a convex shape on the object side; the last lens group includes a positive lens; and the following conditions (1) and (2) are satisfied: 
         [0000]      10 ≦VdLg≦ 45  (1)
 
         [0000]      −1.0&lt;( R 2 a−R 2 b )/( R 2 a+R 2 b )&lt;1.0  (2)
 
         [0000]    where VdLg denotes the Abbe&#39; Number of the positive lens of the last lens group with respect to the d line, R2a denotes the radius of curvature of the object-side surface of the second lens element, and R2b denotes the radius of curvature of the image-side surface of the third lens element. 
         [0084]    The condition (1) shows the Abbe&#39;s Number of the positive lens of the last lens group. The condition (2) shows the shape factor for the second lens element and the third lens element. When the position of the principal point of a positive group that is the magnification-changing group is moved near the object side in retrofocus-type optical systems in general, it is possible to shorten the total lengths of the retrofocus-type optical systems while the positive group is not physically intercepting with the negative group. Also, the magnification-changing group has a meniscus shape which becomes convex on the object side, by making the optical systems satisfy the condition (2). In this case, the principal point of the magnification-changing group can be moved near the object side. As a result, it is possible to control the variations in various aberrations. 
         [0085]    If VdLg is below the lower limit of the condition (1), there is no actual glass material for the positive lens, so that it is impossible to achieve desired optical systems. On the other hand, if VdLg is beyond the upper limit of the condition (1), it becomes difficult to correct chromatic aberration of magnification well in the telephoto end position. 
         [0086]    If (R2a−R2b)/(R2a+R2b) is below the lower limit of the condition (2), it is impossible to control the variations in spherical aberration and coma in changing magnification. On the other hand, if (R2a−R2b)/(R2a+R2b) is beyond the upper limit of the condition (2), it is impossible to move near the object side the principal point of the magnification-changing group. In addition, it is impossible to control the variations in spherical aberration and coma in changing magnification. 
         [0087]    When the conditions (1) and (2) are satisfied at the same time as described above, it is possible to achieve a variable power optical system in which the value of the total length of the variable power optical system relative to image height is made to become small and in which various aberrations are corrected well. Specifically, it is possible to achieve a variable power optical system in which chromatic aberration of magnification in the telephoto end position and the variations in spherical aberration and coma in changing magnification are particularly corrected well (in particular, the variations in spherical aberration and coma are controlled better). 
         [0088]    Also, it is preferred that the variable power optical system of the first embodiment satisfies the following conditions (1-1) and (2-1) instead of the conditions (1) and (2): 
         [0000]      15 ≦VdLg≦ 40  (1-1)
 
         [0000]      −0.8&lt;( R 2 a−R 2 b )/( R 2 a+R 2 b )&lt;0.88  (2-1)
 
         [0089]    When the conditions (1-1) and (2-1) are satisfied, it is possible to achieve a variable power optical system in which the value of the total length of the variable power optical system relative to image height is small and in which various aberrations are corrected better. Specifically, it is possible to achieve a variable power optical system in which chromatic aberration of magnification in the telephoto end position and the variations in spherical aberration and coma in changing magnification are particularly corrected better. 
         [0090]    Also, in a variable power optical system of the first embodiment, it is preferred that the second lens element in the magnification-changing group has positive refractive power and the third lens element in the magnification-changing group has negative refractive power. 
         [0091]    Also, in a variable power optical system of the first embodiment, it is preferred that: the variable power optical system consists of a first lens group with negative refractive power, a second lens group with positive refractive power, a third lens group with negative refractive power, and a fourth lens group with positive refractive power, in that order from the object side; and the magnification-changing group is the second lens group and the last lens group is the fourth lens group. 
         [0092]    Also, in a variable power optical system of the first embodiment, it is preferred that the first lens group is made to keep still in changing magnification from the wide angle end position to the telephoto end position or in performing shooting by switching from shooting at infinity to shooting in close range. 
         [0093]    Because the total length is fixed in changing magnification, it is possible to easily secure the strength of a lens frame. In addition, because the structure of the lens frame can be simplified, it is possible to downsize the optical system. 
         [0094]    Also, in a variable power optical system of the first embodiment, it is preferred that the fourth lens group is made to keep still in changing magnification from the wide angle end position to the telephoto end position or in performing shooting by switching from shooting at infinity to shooting in close range. 
         [0095]    It is possible to make the movable components with two lens groups the number of which is the minimum number, by fixing the fourth lens group. As a result, the structure of the lens frame can be simplified, so that it is possible to downsize the optical system. In addition, it is possible to control the variations in aberrations, by arranging fixed groups on the object side and the image plane side of the two movable groups. 
         [0096]    Also, in a variable power optical system according to the first embodiment, it is preferred that the following condition (3) is satisfied: 
         [0000]      0.45 ≦|f 1|/( FLw×FLt ) 1/2 ≦1.60  (3)
 
         [0000]    where f1 denotes the focal length of the first lens group, FLw denotes the focal length of the whole of the variable power optical system in the wide angle end position, and FLt denotes the focal length of the whole of the variable power optical system in the telephoto end position. 
         [0097]    Because the refractive power of the first lens group is strong, it is possible to move near the image plane side the point at which a virtual image is formed by the first lens group. As a result, it is possible to shorten the total length of the optical system. However, when the refractive power becomes large, it generally becomes difficult to correct aberrations. When the condition (3) is satisfied, it is possible to achieve a variable power optical system in which the value of the total length of the optical system relative to image height is small and in which various aberrations are corrected well. Specifically, it is possible to achieve a variable power optical system in which the variations in spherical aberration and coma in changing magnification are particularly corrected (controlled) well. 
         [0098]    If |f1|/(FLw×FLt) 1/2  is below the lower limit of the condition (3), it is impossible to control the variations in spherical aberration and coma in changing magnification. On the other hand, if |f1|/(FLw×FLt) 1/2  is beyond the upper limit of the condition (3), it becomes difficult to move near the image plane side the point at which the virtual image is formed by the first lens group, which is undesirable. 
         [0099]    Also, in a variable power optical system according to the first embodiment, it is preferred that the following condition (3-1) is satisfied instead of the condition (3): 
         [0000]      0.50 ≦|f 1|/( FLw×FLt ) 1/2 ≦1.04  (3-1)
 
         [0100]    When the condition (3-1) is satisfied, it is possible to achieve a variable power optical system in which the value of the total length of the optical system relative to image height is small and in which various aberrations are corrected better. Specifically, it is possible to achieve a variable power optical system in which the variations in spherical aberration and coma in changing magnification are particularly corrected (controlled) better. 
         [0101]    Also, in a variable power optical system according to the first embodiment, it is preferred that the first lens group consists of two or less lens elements. 
         [0102]    Also, in a variable power optical system according to the first embodiment, it is preferred that the following condition (4) is satisfied: 
         [0000]      0.30 ≦fv /( FLw×FLt ) 1/2 ≦1.10  (4)
 
         [0000]    where fv denotes the focal length of the magnification-changing group, FLw denotes the focal length of the whole of the variable power optical system in the wide angle end position, and FLt denotes the focal length of the whole of the variable power optical system in the telephoto end position. 
         [0103]    The condition (4) shows the focal length of the magnification-changing group. It generally becomes possible to reduce an amount of movement of the magnification-changing group in changing magnification by making the magnification-changing group have sufficiently strong refractive power. As a result, it is possible to shorten the total length of the optical system. However, when the refractive power becomes large, it generally becomes difficult to correct aberrations. When the condition (4) is satisfied, it is possible to achieve a variable power optical system in which the value of the total length of the variable power optical system relative to image height is small and in which various aberrations are corrected well. Specifically, it is possible to achieve a variable power optical system in which spherical aberration is particularly corrected well. 
         [0104]    If fv/(FLw×FLt) 1/2  is below the lower limit of the condition (4), spherical aberration inevitably becomes worse, which is undesirable. On the other hand, if fv/(FLw×FLt) 1/2  is beyond the upper limit of the condition (4), an amount of movement of the magnification-changing group inevitably increases in changing magnification, which is undesirable. 
         [0105]    Also, in a variable power optical system according to the first embodiment, it is preferred that the following condition (4-1) is satisfied instead of the condition (4): 
         [0000]      0.35 ≦fv /( FLw×FLt ) 1/2 ≦0.62  (4-1)
 
         [0106]    When the condition (4-1) is satisfied, it is possible to achieve a variable power optical system in which the value of the total length of the optical system relative to image height is small and in which various aberrations are corrected better. Specifically, it is possible to achieve a variable power optical system in which spherical aberration is particularly corrected better. 
         [0107]    Also, in a variable power optical system according to the present first invention, it is preferred that the last lens group consists of one lens element having positive refractive power. 
         [0108]    Also, in a variable power optical system according to the first embodiment, it is preferred that: the positive lens in the last lens group has a concave shape on the object side; and the following condition (5) is satisfied: 
         [0000]      0.1≦( RLa−RLb )/( RLa+RLb )&lt;1.0  (5)
 
         [0000]    where RLa denotes the radius of curvature of the object-side surface of the positive lens in the last lens group, and RLb denotes the radius of curvature of the image-side surface of the positive lens in the last lens group. 
         [0109]    The condition (5) shows the shape factor of the positive lens of the last lens group. When the condition (5) is satisfied, the shape of the positive lens becomes a meniscus shape which becomes convex on the object side. As a result, it is possible to achieve a variable power optical system in which various aberrations are corrected well. Specifically, it is possible to achieve a variable power optical system in which the variation in field curvature in changing magnification is particularly corrected (controlled) well. On the other hand, if the condition (5) is not satisfied, it is impossible to control the variation in field curvature in changing magnification. 
         [0110]    Also, in a variable power optical system according to the first embodiment, it is preferred that the following condition (5-1) is satisfied instead of the condition (5): 
         [0000]      0.1≦( RLa−RLb )/( RLa+RLb )&lt;0.9  (5-1)
 
         [0111]    When the condition (5-1) is satisfied, it is possible to achieve a variable power optical system in which the value of the total length of the optical system relative to image height is small and in which various aberrations are corrected better. Specifically, it is possible to achieve a variable power optical system in which the variation in field curvature in changing magnification is particularly corrected (controlled) better. 
         [0112]    Also, an image pickup apparatus of a first embodiment according to the present invention is characterized in that: the image pickup apparatus includes one of the above-described variable power optical systems according to the first embodiment, and an imaging sensor; and the following condition (6) is satisfied: 
         [0000]      1.0 ≦|f 1 |/IH≦ 2.8  (6)
 
         [0000]    where f1 denotes the focal length of the first lens group, and IH denotes the image height of the imaging sensor. 
         [0113]    Because the refractive power of the first lens group is strong, it is possible to move near the image plane side the point at which a virtual image is formed by the first lens group. As a result, it is possible to shorten the total length of the optical system. However, when the refractive power becomes large, it generally becomes difficult to correct aberrations. When the condition (6) is satisfied, it is possible to achieve a variable power optical system in which the value of the total length of the variable power optical system relative to image height is small and in which various aberrations are corrected well. Specifically, it is possible to achieve a variable power optical system in which the variations in spherical aberration and coma in changing magnification are particularly corrected (controlled) well. 
         [0114]    In this case, IH denotes the image height of the imaging sensor. In a more detailed explanation, IH is half as long as the diagonal length of the image plane of the imaging sensor. Besides, the height of an image formed on the imaging sensor may be used as IH (where the height of an image formed on the imaging sensor is the distance between the optical axis and the maximum image height). 
         [0115]    If |f1|/IH is below the lower limit of the condition (6), it is impossible to control the variations in spherical aberration and coma in changing magnification. On the other hand, if |f1|/IH is beyond the upper limit of the condition (6), it is becomes difficult to move near the image plane side the point at which a virtual image is formed by the first lens group, which is undesirable. 
         [0116]    Also, in an image pickup apparatus according to the first embodiment, it is preferred that the following condition (6-1) is satisfied instead of the condition (6): 
         [0000]      1.8 ≦|f 1 |/IH≦ 2.6  (6-1)
 
         [0117]    When the condition (6-1) is satisfied, it is possible to achieve a variable power optical system in which the value of the total length of the optical system relative to image height is small and in which various aberrations are corrected better. Specifically, it is possible to achieve a variable power optical system in which the variations in spherical aberration and coma in changing magnification are particularly corrected (controlled) better. 
         [0118]    Also, an image pickup apparatus according to the first embodiment is characterized in that the image pickup apparatus includes one of the above described variable power optical systems according to the first embodiment, and an imaging sensor; and the following condition (7) is satisfied: 
         [0000]      0.2 ≦|fv|/IH≦ 1.8  (7)
 
         [0000]    where fv denotes the focal length of the magnification-changing group, and IH denotes the image height of the imaging sensor. 
         [0119]    It is generally possible to reduce an amount of movement of the magnification-changing group in changing magnification by making the magnification-changing group have sufficiently strong refractive power. As a result, it is possible to shorten the total length of the optical system. However, when the refractive power becomes large, it generally becomes difficult to correct aberrations. When the condition (7) is satisfied, it is possible to achieve a variable power optical system in which the value of the total length of the variable power optical system relative to image height is small and in which various aberrations are corrected well. Specifically, it is possible to achieve a variable power optical system in which spherical aberration is particularly corrected better. Besides, IH (image height) is as described above. 
         [0120]    If |fv|/IH is below the lower limit of the condition (7), spherical aberration inevitably becomes worse, which is undesirable. On the other hand, if |fv|/IH is beyond the upper limit of the condition (7), an amount of movement of the magnification-changing group inevitably increases in changing magnification, which is undesirable. 
         [0121]    Also, in an image pickup apparatus according to the first embodiment, it is preferred that the following condition (7-1) is satisfied instead of the condition (7): 
         [0000]      1.0 ≦|fv|/IH≦ 1.5  (7-1)
 
         [0122]    When the condition (7-1) is satisfied, it is possible to achieve a variable power optical system in which the value of the total length of the optical system relative to image height is small and in which various aberrations are corrected better. Specifically, it is possible to achieve a variable power optical system in which spherical aberration is particularly corrected better. 
         [0123]    Also, a variable power optical system according to the second embodiment is characterized in that: the variable power optical system includes, in order from the object side, a first lens group with negative refractive power, a second lens group with positive refractive power, a third lens group with negative refractive power, and a fourth lens group with positive refractive power; the second lens group includes at least one positive lens and a negative lens; the fourth lens group includes a positive lens; and the following conditions (8), (9), and (10) are satisfied: 
         [0000]      10 ≦Vd 4 g≦ 45  (8)
 
         [0000]      2.2 ≦|α/f 1|+(α/ f 2)−0.026 ×Vd 4 g≦ 5.0  (9)
 
         [0000]      10 ≦Vd max− Vd min≦80  (10)
 
         [0000]    where α=(FLw×FLt) 1/2 , FLw denotes the focal length of the whole of the variable power optical system in the wide angle end position, FLt denotes the focal length of the whole of the variable power optical system in the telephoto end position, f1 denotes the focal length of the first lens group, f2 denotes the focal length of the second lens group, Vd4g denotes the Abbe&#39; Number of the positive lens of the fourth lens group with respect to the d line, Vdmax denotes the Abbe&#39;s Number of a glass material having the lowest dispersion characteristic of those of glass materials which are used for lenses of the second lens group, with respect to the d line, and Vdmin denotes the Abbe&#39;s Number of a glass material having the highest dispersion characteristic of those of glass materials which are used for lenses of the second lens group, with respect to the d line. 
         [0124]    In order to achieve an optical system in which the value of the total length of the variable power optical system relative to image height is small and in which various aberrations are corrected well, various aberrations have to be corrected in each of the lens groups while the power of each of the lens groups is being strengthened. For example, in the case where the fourth lens groups is composed of one positive lens, the positive lens should be made of a low dispersion material in order to control the occurrence of chromatic aberration by a single lens. However, if the value of the total length of the variable power optical system relative to image height is made to become smaller, the required power of each of the first and second lens groups increases, so that it becomes impossible to balance the power with the corrections of various aberrations (monochromatic aberration/chromatic aberration) in each of the lens groups. 
         [0125]    Specifically, required power in the first lens group increases, so that the variations in monochromatic aberrations (spherical aberration/coma) become large in changing magnification. Accordingly, in order to correct the monochromatic aberrations, the variations in monochromatic aberrations (spherical aberration/coma) are mainly controlled in the first lens group. However, the occurrence of chromatic aberration in the first lens group becomes frequent with the control of the variations in monochromatic aberrations. In addition, required power in the second lens group increases and chromatic aberration frequently occurs. 
         [0126]    Accordingly, in the variable power optical system of the second embodiment, the fourth lens group is made to satisfy the condition (8) in order to correct these aberrations well. The condition (8) shows the Abbe&#39;s Number of the positive lens in the fourth lens group. The condition (9) shows the relation between the Abbe&#39;s Number of the positive lens in the fourth lens group and the powers of the first and second lens groups. The condition (10) shows the difference between a glass material having the lowest dispersion characteristic of those of grass materials for the second lens group and a glass material having the highest dispersion characteristic of those of the grass materials for the second lens group. 
         [0127]    When both of the conditions (8) and (9) are satisfied, it is possible to achieve a variable power optical system in which the value of the total length of the variable power optical system relative to image height is small and in which various aberrations are corrected well. In addition, when the condition (10) is also satisfied, it is possible to achieve a variable power optical system in which various aberrations are corrected better. 
         [0128]    If Vd4g is below the lower limit of the condition (8), there is no grass material, so that it is impossible to achieve a desired optical system. On the other hand, if Vd4g is beyond the upper limit of the condition (8), it becomes difficult to correct chromatic aberration of magnification well in the telephoto end position. 
         [0129]    If |α/f1|+(α/f2)−0.026×Vd4g is below the lower limit of the condition (9), the powers of the first and second lens group are inadequate, so that it is impossible to shorten the total length of the optical system. On the other hand, if |α/f1|+(α/f2)−0.026×Vd4g is beyond the upper limit of the condition (9), the powers of the first and second lens groups become too strong, so that it is impossible to control the variations in spherical aberration and coma in changing magnification. 
         [0130]    If Vdmax−Vdmin is below the lower limit of the condition (10), the correction of chromatic aberration in the second lens group mainly becomes inadequate. On the other hand, if Vdmax−Vdmin is beyond the upper limit of the condition (10), the correction of chromatic aberration in the second lens group mainly becomes surplus. 
         [0131]    As described above, when the conditions (8), (9), and (10) are satisfied at the same time, it is possible to achieve a variable power optical system in which the value of the total length of the variable power optical system relative to image height is small and in which various aberrations are corrected well. Specifically, it is possible to achieve a variable power optical system in which the variations in spherical aberration and coma in changing magnification and chromatic aberration are corrected particularly well. 
         [0132]    Also, in a variable power optical system according to the second embodiment, it is preferred that the following conditions (8-1), (9-1), and (10-1) are satisfied instead of the conditions (8), (9), and (10): 
         [0000]      15 ≦Vd 4 g≦ 40  (8-1)
 
         [0000]      2.25 ≦|α/f 1|+(α/ f 2)−0.026 ×Vd 4 g≦ 4.5  (9-1)
 
         [0000]      15 ≦Vd max− Vd min≦70  (10-1)
 
         [0133]    When the conditions (8-1), (9-1), and (10-1) are satisfied, it is possible to achieve a variable power optical system in which the value of the total length of the optical system relative to image height is small and in which various aberrations are corrected better. Specifically, it is possible to achieve a variable power optical system in which the variations in spherical aberration and coma in changing magnification and chromatic aberration are particularly corrected better. 
         [0134]    Also, in a variable power optical system according to the second embodiment, it is preferred that the first lens group is made to keep still in changing magnification from the wide angle end position to the telephoto end position or in performing shooting by changing shooting at infinity to shooting in close range. 
         [0135]    Because the total length of the variable power optical system is fixed in changing magnification, it is possible to easily secure the strength of a lens frame. In addition, because the structure of the lens frame can be simplified, it is possible to downsize the optical system. 
         [0136]    Also, in a variable power optical system according to the second embodiment, it is preferred that the fourth lens group is made to keep still in changing magnification from the wide angle end position to the telephoto end position or in performing shooting by changing shooting at infinity to shooting in close range. 
         [0137]    Because the fourth lens group is fixed in changing magnification, a minimum of two lens groups can be used as movable components. As a result, the structure of the lens frame can be simplified, so that it is possible to downsize the optical system. In addition, when fixed lens groups are arranged on the object and image-plane sides of the two movable lens groups, it is possible to control the variations in aberrations. 
         [0138]    Also, in a variable power optical system according to the present second embodiment, it is preferred that the following condition (3) is satisfied: 
         [0000]      0.45 ≦|f 1|/( FLw×FLt ) 1/2 ≦1.60  (3)
 
         [0000]    where f1 denotes the focal length of the first lens group, FLw denotes the focal length of the whole of the variable power optical system in the wide angle end position, and FLt denotes the focal length of the whole of the variable power optical system in the telephoto end position. 
         [0139]    Because the refractive power of the first lens group is strong, it is possible to move near the image plane side the point at which a virtual image is formed by the first lens group. As a result, it is possible to shorten the total length of the optical system. However, when the refractive power becomes large, it generally becomes difficult to correct aberrations. When the condition (3) is satisfied, it is possible to achieve a variable power optical system in which the value of the total length of the optical system relative to image height is small and in which various aberrations are corrected well. Specifically, it is possible to achieve a variable power optical system in which the variations in spherical aberration and coma in changing magnification are particularly corrected (controlled) well. 
         [0140]    If |f1|/(FLw×FLt) 1/2  is below the lower limit of the condition (3), it is impossible to control the variations in spherical aberration and coma in changing magnification. On the other hand, if |f1|/(FLw×FLt) 1/2  is beyond the upper limit of the condition (3), it becomes difficult to move near the image plane side the point at which the virtual image is formed by the first lens group, which is undesirable. 
         [0141]    Also, in a variable power optical system according to the second embodiment, it is preferred that the following condition (3-1) is satisfied instead of the condition (3): 
         [0000]      0.50 ≦|f 1|/( FLw×FLt ) 1/2 ≦1.04  (3-1)
 
         [0142]    When the condition (3-1) is satisfied, it is possible to achieve a variable power optical system in which the value of the total length of the optical system relative to image height is small and in which various aberrations are corrected better. Specifically, it is possible to achieve a variable power optical system in which the variations in spherical aberration and coma in changing magnification are particularly corrected (controlled) better. 
         [0143]    Also, in a variable power optical system according to the second embodiment, it is preferred that the first lens group consists of two or less lens elements. 
         [0144]    Also, in a variable power optical system according to the second embodiment, it is preferred that the following condition (11) is satisfied: 
         [0000]      0.30 ≦f 2/( FLw×FLt ) 1/2 ≦1.10  (11)
 
         [0000]    where f2 denotes the focal length of the second lens group, FLw denotes the focal length of the whole of the variable power optical system in the wide angle end position, and FLt denotes the focal length of the whole of the variable power optical system in the telephoto end position. 
         [0145]    When the refractive power of the second lens group is strong, it is generally possible to reduce an amount of movement of the lens group in changing magnification. As a result, it is possible to shorten the total length of the optical system. However, when the refractive power becomes large, it generally becomes difficult to correct aberrations. When the condition (11) is satisfied, it is possible to achieve a variable power optical system in which the value of the total length of the optical system relative to image height is small and in which various aberrations are corrected well. Specifically, it is possible to achieve a variable power optical system in which spherical aberration is particularly corrected well. 
         [0146]    If f2/(FLw×FLt) 1/2  is below the lower limit of the condition (11), spherical aberration inevitably becomes worse, which is undesirable. On the other hand, if f2/(FLw×FLt) 1/2  is beyond the upper limit of the condition (11), an amount of movement of the lens group inevitably increases in changing magnification, which is undesirable. 
         [0147]    Also, in a variable power optical system according to the second embodiment, it is preferred that the following condition (11-1) is satisfied instead of the condition (11): 
         [0000]      0.35 ≦f 2/( FLw×FLt ) 1/2 ≦0.62  (11-1)
 
         [0148]    When the condition (11-1) is satisfied, it is possible to achieve a variable power optical system in which the value of the total length of the optical system relative to image height is small and in which various aberrations are corrected better. Specifically, it is possible to achieve a variable power optical system in which spherical aberration is particularly corrected better. 
         [0149]    Also, in a variable power optical system according to the second embodiment, it is preferred that: the second lens group consists of a first lens element (L21) with positive refractive power, a second lens element (L22), and a third lens element (L23) in that order from the object side; the first lens element (L21) has a convex shape on the object side; and the following condition (2) is satisfied: 
         [0000]      −1.0&lt;( R 2 a−R 2 b )/( R 2 a+R 2 b )&lt;1.0  (2)
 
         [0000]    where R2a denotes the radius of curvature of the object-side surface of the second lens element (L22), and R2b denotes the radius of curvature of the image-side surface of the third lens element (L23). 
         [0150]    The condition (2) shows the shape factor for the second lens element (L22) and the third lens element (L23). When the position of the principal point of a positive group that is the main magnification-changing group is moved near the object side in retrofocus-type optical systems in general, it is possible to shorten the total lengths of the retrofocus-type optical systems while the positive group is not physically intercepting with the negative group. 
         [0151]    When the condition (2) is satisfied, the magnification-changing group has a meniscus shape which becomes convex on the object side, so that the principal point of the second lens group can be moved near the object side. As a result, it is possible to achieve a variable power optical system in which various aberrations are corrected well. Specifically, it is possible to achieve a variable power optical system in which the variations in spherical aberration and coma in changing magnification are particularly corrected well. 
         [0152]    If (R2a−R2b)/(R2a+R2b) is below the lower limit of the condition (2), it is impossible to control the variations in spherical aberration and coma in changing magnification. On the other hand, if (R2a−R2b)/(R2a+R2b) is beyond the upper limit of the condition (2), it is impossible to move near the object side the position of the principal point of the second lens group. In addition, it is impossible to control the variations in spherical aberration and coma in changing magnification. 
         [0153]    Also, it is preferred that the variable power optical system of the second embodiment satisfies the following condition (2-1) instead of the condition (2): 
         [0000]      −0.8&lt;( R 2 a−R 2 b )/( R 2 a+R 2 b )&lt;0.72  (2-1)
 
         [0154]    When the condition (2-1) are satisfied, it is possible to achieve a variable power optical system in which the value of the total length of the variable power optical system relative to image height is small and in which various aberrations are corrected better. Specifically, it is possible to achieve a variable power optical system in which the variations in spherical aberration and coma in changing magnification are particularly corrected better. 
         [0155]    Also, in a variable power optical system of the second embodiment, it is preferred that the second lens element (L22) has positive refractive power, and the third lens element (L23) has negative refractive power. 
         [0156]    Also, in a variable power optical system of the second embodiment, it is preferred that the fourth lens group consists of one lens element with positive refractive power. 
         [0157]    Also, in a variable power optical system according to the second embodiment, it is preferred that: the positive lens in the fourth lens group has a concave shape on the object side; and the following condition (12) is satisfied: 
         [0000]      0&lt;( R 4 a−R 4 b )/( R 4 a+R 4 b )&lt;1.0  (12)
 
         [0000]    where R4a denotes the radius of curvature of the object-side surface of the positive lens in the fourth lens group, and R4b denotes the radius of curvature of the image-side surface of the positive lens in the fourth lens group. 
         [0158]    The condition (12) shows the shape factor for the positive lens of the fourth lens group. When the condition (12) is satisfied, the shape of the positive lens becomes a meniscus shape which becomes concave on the object side. As a result, it is possible to achieve a variable power optical system in which various aberrations are corrected well. Specifically, it is possible to achieve a variable power optical system in which the variation in field curvature in changing magnification is particularly corrected (controlled) well. On the other hand, if the condition (12) is not satisfied, it is impossible to control the variation in field curvature in changing magnification. 
         [0159]    Also, it is preferred that the variable power optical system of the second embodiment satisfies the following condition (12-1) instead of the condition (12): 
         [0000]      0.1≦( R 4 a−R 4 b )/( R 4 a+R 4 b )&lt;0.9  (12-1)
 
         [0160]    When the condition (12-1) are satisfied, it is possible to achieve a variable power optical system in which the value of the total length of the variable power optical system relative to image height is small and in which various aberrations are corrected better. Specifically, it is possible to achieve a variable power optical system in which the variation in field curvature in changing magnification is particularly corrected (controlled) better. 
         [0161]    Also, an image pickup apparatus according to the second embodiment includes one of the above-described variable power optical systems according to the second embodiment, and an imaging sensor. 
         [0162]    Also, in an image pickup apparatus according to the second embodiment, it is preferred that: the image pickup apparatus includes a variable power optical system forming an optical image of an object, and an imaging sensor; the imaging sensor transforms the optical image formed by the variable power optical system into electrical image signals; the variable power optical system includes, in order from the object side, a first lens group with negative refractive power, a second lens group with positive refractive power, a third lens group with negative refractive power, and a fourth lens group with positive refractive power; the second lens group includes at least one positive lens and a negative lens; the fourth lens group includes a positive lens; and the following conditions (8), (13), and (10) are satisfied: 
         [0000]      10 ≦Vd 4 g≦ 45  (8)
 
         [0000]      −0.27 ≦|IH/f 1|+( IH/f 2)−0.05 ×Vd 4 g≦ 3.0  (13)
 
         [0000]      10 ≦Vd max− Vd min≦80  (10)
 
         [0000]    where IH denotes the image height of the imaging sensor, f1 denotes the focal length of the first lens group, f2 denotes the focal length of the second lens group, Vd4g denotes the Abbe&#39; Number of the positive lens of the fourth lens group with respect to the d line, Vdmax denotes the Abbe&#39;s Number of a glass material having the lowest dispersion characteristic of those of glass materials which are used for lenses of the second lens group, with respect to the d line, and Vdmin denotes the Abbe&#39;s Number of a glass material having the highest dispersion characteristic of those of glass materials which are used for lenses of the second lens group, with respect to the d line. 
         [0163]    When both of the conditions (8) and (13) are satisfied, it is possible to achieve a variable power optical system in which the value of the total length of the variable power optical system relative to image height is small and in which various aberrations are corrected well. In addition, when the condition (10) is also satisfied, it is possible to achieve a variable power optical system in which various aberrations are also corrected better. In this case, IH denotes the image height of the imaging sensor. In a more detailed explanation, IH is half as long as the diagonal length of the image plane of the imaging sensor. Besides, the height of an image formed on the imaging sensor may be used as IH (where the height of an image formed on the imaging sensor is the distance between the optical axis and the maximum image height). 
         [0164]    The conditions (8) and (10) have been already explained. Also, the condition (13) has the same technical significance and the same operation effects as the condition (9) does. 
         [0165]    As described above, when the conditions (8), (13), and (10) are satisfied at the same time, it is possible to achieve a variable power optical system in which the value of the total length of the variable power optical system relative to image height is small and in which various aberrations are corrected well. Specifically, it is possible to achieve a variable power optical system in which the variations in spherical aberration and coma in changing magnification and chromatic aberration are particularly corrected well. 
         [0166]    Also, in an image pickup apparatus according to the second embodiment, it is preferred that the following conditions (8-1), (13-1), and (10-1) are satisfied instead of the conditions (8), (13), and (10): 
         [0000]      15 ≦Vd 4 g≦ 40  (8-1)
 
         [0000]      0.25 ≦|α/f 1|+(α/ f 2)−0.026 ×Vd 4 g≦ 2.5  (13-1)
 
         [0000]      15 ≦Vd max− Vd min≦70  (10-1)
 
         [0167]    When the conditions (8-1), (13-1), and (10-1) are satisfied, it is possible to achieve a variable power optical system in which the value of the total length of the optical system relative to image height is small and in which various aberrations are corrected better. Specifically, it is possible to achieve a variable power optical system in which the variations in spherical aberration and coma in changing magnification and chromatic aberration are particularly corrected better. 
         [0168]    Also, in an image pickup apparatus according to the second embodiment, it is preferred that the image pickup apparatus includes a variable power optical system in which the first lens group is made to keep still in changing magnification from the wide angle end position to the telephoto end position or in performing shooting by changing shooting at infinity to shooting in close range. Also, in an image pickup apparatus according to the present embodiment, it is preferred that the image pickup apparatus includes a variable power optical system in which the fourth lens group is made to keep still in changing magnification from the wide angle end position to the telephoto end position or in performing shooting by changing shooting at infinity to shooting in close range. The matter of the constitution in which the first and fourth lens groups are fixed has been already explained. 
         [0169]    Also, in an image pickup apparatus according to the second embodiment, it is preferred that the following condition (6) is satisfied: 
         [0000]      1.0 ≦|f 1 |/IH≦ 2.8  (6)
 
         [0000]    where f1 denotes the focal length of the first lens group, and IH denotes the image height of the imaging sensor. 
         [0170]    The condition (6) has the same technical significance and the same operation effects as the condition (3) does. Besides, the explanation about the image height has been described above. 
         [0171]    Also, in an image pickup apparatus according to the second embodiment, it is preferred that the following condition (6-1) is satisfied instead of the condition (6): 
         [0000]      1.8 ≦|f 1 |/IH≦ 2.6  (6-1)
 
         [0172]    Also, in an image pickup apparatus according to the second embodiment, it is preferred that the first lens group in the variable power optical system consists of two or less lens elements. 
         [0173]    Also, in an image pickup apparatus according to the second embodiment, it is preferred that the following condition (14) is satisfied: 
         [0000]      0.2 ≦|f 2 |/IH≦ 1.8  (14)
 
         [0000]    where f2 denotes the focal length of the second lens group, and IH denotes the image height of the imaging sensor. 
         [0174]    The condition (14) has the same technical significance and the same operation effects as the condition (11) does. Besides, the explanation about IH has been described above. 
         [0175]    Also, in an image pickup apparatus according to the second embodiment, it is preferred that the following condition (14-1) is satisfied instead of the condition (14): 
         [0000]      1.0 ≦|f 2 |/IH≦ 1.5  (14-1)
 
         [0176]    Also, in an image pickup apparatus according to the second embodiment, it is preferred that: the second lens group in the variable power optical system consists of a first lens element with positive refractive power, a second lens element, and a third lens element in that order from the object side; the first lens element has a convex shape on the object side; and the following condition (2) is satisfied: 
         [0000]      −1.0&lt;( R 2 a−R 2 b )/( R 2 a+R 2 b )&lt;1.0  (2)
 
         [0000]    where R2a denotes the radius of curvature of the object-side surface of the second lens element, and R2b denotes the radius of curvature of the image-side surface of the third lens element. 
         [0177]    The technical significance and the operation effects of the condition (2) have been already explained. 
         [0178]    Also, in an image pickup apparatus according to the second embodiment, it is preferred that the following condition (2-1) is satisfied instead of the condition (2): 
         [0000]      −0.8&lt;( R 2 a−R 2 b )/( R 2 a+R 2 b )&lt;0.72  (2-1)
 
         [0179]    Also, in an image pickup apparatus according to the second embodiment, it is preferred that the second lens element in the variable power optical system has positive refractive power and the third lens element in the variable power optical system has negative refractive power, respectively. 
         [0180]    Also, in an image pickup apparatus according to the second embodiment, it is preferred that the fourth lens group in the variable power optical system consists of one lens element with positive refractive power. 
         [0181]    Also, in an image pickup apparatus according to the second embodiment, it is preferred that: the positive lens in the fourth lens group in the variable power optical system has a concave shape on the object side; and the following condition (12) is satisfied: 
         [0000]      0&lt;( R 4 a−R 4 b )/( R 4 a+R 4 b )&lt;1.0  (12)
 
         [0000]    where R4a denotes the radius of curvature of the object-side surface of the positive lens in the fourth lens group, and R4b denotes the radius of curvature of the image-side surface of the positive lens in the fourth lens group. 
         [0182]    The technical significance and the operation effects of the condition (12) have been already explained. 
         [0183]    Also, in an image pickup apparatus according to the second embodiment, it is preferred that the following condition (12-1) is satisfied instead of the condition (12): 
         [0000]      0.1≦( R 4 a−R 4 b )/( R 4 a+R 4 b )&lt;0.9  (12-1)
 
         [0184]    Also, a variable power optical system according to the third embodiment is characterized in that: the variable power optical system includes, in order from the object side, a first lens group with negative refractive power, a second lens group with positive refractive power, a third lens group with negative refractive power, and a fourth lens group with positive refractive power; the first lens group includes one negative lens and one positive lens in that order from the object side, and an air distance is provided between the negative and positive lenses of the first lens group; and the following conditions (15), (16), and (17) are satisfied: 
         [0000]      1.75 ≦Nd 1 g≦ 2.50  (15)
 
         [0000]      15 ≦Vd 1 g≦ 43  (16)
 
         [0000]      3 ≦VdN−VdP≦ 28  (17)
 
         [0000]    where Nd1g denotes the refractive index of each of lenses constituting the first lens group, with respect to the d line, Vd1g denotes the Abbe&#39;s Number of each of lenses constituting the first lens group, with respect to the d lines, VdN denotes the Abbe&#39;s Numbers of the negative lens in the first lens group, with respect to the d lines, and VdP denotes the Abbe&#39;s Numbers of the positive lens in the first lens group, with respect to the d lines. 
         [0185]    The variable power optical system according to the third embodiment is characterized in that both of the negative and positive lenses constituting the first lens group have high refractive index and high dispersion characteristic. The condition (15) shows the refractive index of each of the lenses constituting the first lens group. The condition (16) shows the Abbe&#39;s Number of each of the lenses constituting the first lens group. The condition (17) shows the difference between the Abbe&#39;s Numbers of the negative and positive lenses constituting the first lens group. 
         [0186]    When the condition (15) is satisfied, it is possible to strengthen the refractive power while the radius of curvature of each of the lenses constituting the first lens group is being made to become large. Small radius of curvature generally makes the variations in various aberrations large. That is to say, it is possible to control the variations in various aberrations and it is possible to achieve desired refractive power, by making the radius of curvature large. In addition, when both of the conditions (16) and (17) are satisfied, it is possible to correct various aberrations in the first lens group well while desired refractive power is being achieved in the first lens group. 
         [0187]    If Nd1g is below the lower limit of the condition (15), it is impossible to achieve desired refractive power with the variations in various aberrations in each of the lenses being controlled. On the other hand, if Nd1g is beyond the upper limit of the condition (15), there is no glass material for the lenses constituting first lens group, so that it is impossible to achieve a desired optical system. 
         [0188]    If Vd1g is below the lower limit of the condition (16), there is no glass material for the lenses constituting the first lens group, so that it is impossible to achieve a desired optical system. On the other hand, if Vd1g is beyond the upper limit of the condition (16), actual glass materials cause a decline in the refractive power, so that it is impossible to achieve a desired refractive index of the first lens group. 
         [0189]    If VdN−VdP is below the lower limit of the condition (17), the correction of chromatic aberration inevitably becomes inadequate. On the other hand, if VdN−VdP is beyond the upper limit of the condition (17), the correction of chromatic aberration inevitably becomes surplus. 
         [0190]    As described above, when the conditions (15), (16), and (17) are satisfied at the same time, it is possible to achieve a variable power optical system in which the value of the total length of the variable power optical system relative to image height is small and in which various aberrations are corrected well. Specifically, it is possible to achieve a variable power optical system in which chromatic aberration is particularly corrected well. 
         [0191]    Also, in a variable power optical system according to the third embodiment, it is preferred that the following conditions (15-1), (16-1), and (17-1) are satisfied instead of the conditions (15), (16), and (17): 
         [0000]      1.82 ≦nd 1 g≦ 2.40  (15-1)
 
         [0000]      16 ≦vd 1 g≦ 38.7  (16-1)
 
         [0000]      9 ≦vdN−vdP≦ 22.7  (17-1)
 
         [0192]    When the conditions (15-1), (16-1), and (17-1) are satisfied, it is possible to achieve a variable power optical system in which the value of the total length of the optical system relative to image height is small and in which various aberrations are corrected better. Specifically, it is possible to achieve a variable power optical system in which chromatic aberration is particularly corrected better. 
         [0193]    In a variable power optical system according to the third embodiment, it is preferred that the following condition (18) is satisfied: 
         [0000]      0.03 ≦D /( FLw×FLt ) 1/2 ≦0.26  (18)
 
         [0000]    where D denotes the axial air distance between the negative and positive lenses of the first lens group, FLw denotes the focal length of the whole of the variable power optical system in the wide angle end position, and FLt denotes the focal length of the whole of the variable power optical system in the telephoto end position. 
         [0194]    When the condition (18) is satisfied, it is possible to achieve a variable power optical system in which various aberrations are corrected well while the thickness of the first lens group is being thinned. Specifically, it is possible to achieve a variable power optical system in which the variations in spherical aberration and coma in changing magnification are particularly corrected (controlled) well. 
         [0195]    If D/(FLw×FLt) 1/2  is below the lower limit of the condition (18), it is impossible to control the variations in spherical aberration and coma in changing magnification. On the other hand, if D/(FLw×FLt) 1/2  is beyond the upper limit of the condition (18), the thickness of the first lens group increases, so that it is impossible to achieve a desired optical system. 
         [0196]    Also, in a variable power optical system according to the third embodiment, it is preferred that the following condition (18-1) is satisfied instead of the condition (18): 
         [0000]      0.05 ≦D /( FLw×FLt ) 1/2 ≦0.20  (18-1)
 
         [0197]    When the condition (18-1) is satisfied, it is possible to achieve a variable power optical system in which various aberrations are corrected better while the thickness of the first lens group is being thinned. Specifically, it is possible to achieve a variable power optical system in which the variations in spherical aberration and coma in changing magnification are particularly corrected (controlled) better. 
         [0198]    Also, in a variable power optical system according to the third embodiment, it is preferred that: an air lens which has a convex shape on the object side is formed nearer to the image-plane side than the negative lens of the first lens group; and the following condition (19) is satisfied: 
         [0000]      −0.25≦( r 2 −r 3)/( r 2 +r 3)≦−0.07  (19)
 
         [0000]    where r2 denotes the radius of curvature of the image-side surface of the negative lens of the first lens group, and r3 denotes the radius of curvature of the object-side surface of the positive lens of the first lens group. 
         [0199]    The condition (19) shows the shape factor of the air lens of the first lens group. When the condition (19) is satisfied, it is shown that the air lens becomes a meniscus lens having a convex shape on the object side and having positive refractive power. As a result, it is possible to achieve a variable power optical system in which the variations in spherical aberration and coma in changing magnification are corrected (controlled) well. 
         [0200]    If (r2−r3)/(r2+r3) is below the lower limit of the condition (19), the refractive power of the air lens is reduced, so that it is impossible to control the variations in spherical aberration and coma in changing magnification. On the other hand, if (r2−r3)/(r2+r3) is beyond the upper limit of the condition (19), the refractive index of the air lens becomes high. That is to say, the radius of curvature of the object-side surface of the positive lens in the first lens group becomes large, so that it is impossible to correct various aberrations well while desired refractive power of the first lens group is being achieved. 
         [0201]    Also, in a variable power optical system according to the third embodiment, it is preferred that the following condition (19-1) is satisfied instead of the condition (19): 
         [0000]      −0.20≦( r 2 −r 3)/( r 2 +r 3)≦−0.10  (19-1)
 
         [0202]    When the condition (19-1) is satisfied, it is possible to achieve a variable power optical system in which the value of the total length of the optical system relative to image height is small and in which various aberrations are corrected better. Specifically, it is possible to achieve a variable power optical system in which the variations in spherical aberration and coma in changing magnification are particularly corrected (controlled) better. 
         [0203]    Also, in a variable power optical system of the third embodiment, it is preferred that the first lens group is made to keep still in changing magnification from the wide angle end position to the telephoto end position or in performing shooting by switching from shooting at infinity to shooting in close range. 
         [0204]    Because the total length is fixed in changing magnification, it is possible to easily secure the strength of a lens frame. In addition, because the structure of the lens frame can be simplified, it is possible to downsize the optical system. 
         [0205]    Also, in a variable power optical system of the third embodiment, it is preferred that the fourth lens group is made to keep still in changing magnification from the wide angle end position to the telephoto end position or in performing shooting by switching from shooting at infinity to shooting in close range. 
         [0206]    It is possible to use a minimum of two lens groups as movable components by fixing the fourth lens group. As a result, the structure of the lens frame can be simplified, so that it is possible to downsize the optical system. In addition, it is possible to control the variations in aberrations, by arranging fixed groups on the object side and the image-plane side of the two movable groups. 
         [0207]    Also, in a variable power optical system according to the present third embodiment, it is preferred that the following condition (3) is satisfied: 
         [0000]      0.45 ≦|f 1|/( FLw×FLt ) 1/2 ≦1.60  (3)
 
         [0000]    where f1 denotes the focal length of the first lens group, FLw denotes the focal length of the whole of the variable power optical system in the wide angle end position, and FLt denotes the focal length of the whole of the variable power optical system in the telephoto end position. 
         [0208]    The first lens group has high refractive power, so that it is possible to move near the image plane side a point at which a virtual image is formed by the first lens group. As a result, it is possible to shorten the total length of the optical system. However, when refractive power becomes high, it generally becomes difficult to correct aberrations. When the condition (3) is satisfied, it is possible to achieve a variable power optical system in which the value of the total length of the optical system relative to image height is small and in which various aberrations are corrected well. Specifically, it is possible to achieve an optical system in which variations in spherical aberration and coma in changing magnification are particularly corrected (controlled) well. 
         [0209]    If |f1|/(FLw×FLt) 1/2  is below the lower limit value of the condition (3), it becomes impossible to control variations in spherical aberration and coma in changing magnification. On the other hand, if |f1|/(FLw×FLt) 1/2  is beyond the upper limit value of the condition (3), it becomes difficult to move near the image-plane side a point at which a virtual image is formed by the first lens group, which is undesirable. 
         [0210]    Also, in a variable power optical system according to the present third invention, it is preferred that the following condition (3-2) is satisfied instead of the condition (3): 
         [0000]      0.70 ≦|f 1|/( FLw×FLt ) 1/2 ≦1.20  (3-2)
 
         [0211]    When the condition (3-2) is satisfied, it is possible to achieve a variable power optical system in which the value of the total length of the optical system relative to image height is small and in which various aberrations are corrected better. Specifically, it is possible to achieve an optical system in which the variations in spherical aberration and coma in changing magnification are particularly corrected (controlled) better. 
         [0212]    Also, in a variable power optical system according to the present third embodiment, it is preferred that the following condition (20) is satisfied: 
         [0000]      −0.5 ≦FLn/FLp≦− 0.3  (20)
 
         [0000]    where FLn denotes the focal length of the negative lens of the first lens group and FLp denotes the focal length of the positive lens of the first lens group. 
         [0213]    The condition (20) shows the ratio of power of the negative lens to power of the positive lens in the first lens group. When the condition (20) is satisfied, it is possible to achieve a variable power optical system in which the value of the total length of the optical system relative to image height is small and in which various aberrations are corrected well. Specifically, it is possible to achieve a variable power optical system in which chromatic aberration is particularly corrected well. 
         [0214]    If FLn/FLp is below the lower limit value of the condition (20), a correction of chromatic aberration inevitably becomes surplus one. On the other hand, if FLn/FLp is beyond the upper limit value of the condition (20), a correction of chromatic aberration inevitably becomes insufficient one. 
         [0215]    Also, in a variable power optical system according to the third embodiment, it is preferred that the following condition (11) is satisfied: 
         [0000]      0.30 ≦f 2/( FLw×FLt ) 1/2 ≦1.10  (11)
 
         [0000]    where f2 denotes the focal length of the second lens group, FLw denotes the focal length of the whole of the variable power optical system in the wide angle end position, and FLt denotes the focal length of the whole of the variable power optical system in the telephoto end position. 
         [0216]    When the refractive power of the second lens group is sufficiently strong, it is generally possible to reduce an amount of movement of the lens group in changing magnification. As a result, it is possible to shorten the total length of the optical system. However, when the refractive power becomes high, it generally becomes difficult to correct aberrations. When the condition (11) is satisfied, it is possible to achieve a variable power optical system in which the value of the total length of the optical system relative to image height is small and in which various aberrations are corrected well. Specifically, it is possible to achieve a variable power optical system in which spherical aberration is particularly corrected well. 
         [0217]    If f2/(FLw×FLt) 1/2  is below the lower limit value of the condition (11), spherical aberration inevitably becomes worse, which is undesirable. On the other hand, if f2/(FLw×FLt) 1/2  is beyond the upper limit value of the condition (11), an amount of movement of the lens group inevitably increases in changing magnification, which is undesirable. 
         [0218]    Also, in a variable power optical system according to the third embodiment, it is preferred that the following condition (11-2) is satisfied instead of the condition (11): 
         [0000]      0.45 ≦f 2/( FLw×FLt ) 1/2 ≦0.70  (11-2)
 
         [0219]    When the condition (11-2) is satisfied, it is possible to achieve a variable power optical system in which the value of the total length of the optical system relative to image height is small and in which various aberrations are corrected better. Specifically, it is possible to achieve a variable power optical system in which spherical aberration is particularly corrected better. 
         [0220]    Also, in a variable power optical system according to the third embodiment, it is preferred that: the negative lens in the first lens group has a convex shape on the object side; and the following condition (21) is satisfied: 
         [0000]      0.2≦( r 1 −r 2)/( r 1 +r 2)≦1.0  (21)
 
         [0000]    where r1 denotes the radius of curvature of the object-side surface of the negative lens of the first lens group and r2 denotes the radius of curvature of the image-side surface of the negative lens of the first lens group. 
         [0221]    The condition (21) shows the shape factor of the negative lens of the first lens group. When the condition (21) is satisfied, it is possible to achieve a variable power optical system in which the value of the total length of the optical system relative to image height is small and in which various aberrations are corrected well. Specifically, it is possible to achieve a variable power optical system in which the variations in spherical aberration and coma in changing magnification are particularly corrected (controlled) well. 
         [0222]    If (r1−r2)/(r1+r2) is below the lower limit of the condition (21), the power of the negative lens of the first lens group is reduced. In this case, it becomes difficult to move near the image-plane side the point at which a virtual image is formed by the first lens group, which is undesirable. On the other hand, if (r1−r2)/(r1+r2) is beyond the upper limit of the condition (21), the negative lens of the first lens group inevitably has a biconcave shape, so that it is impossible to control the variations in spherical aberration and coma in changing magnification. 
         [0223]    Also, in a variable power optical system according to the third embodiment, it is preferred that: the fourth lens group consists of one lens with positive refractive power; and the following condition (22) is satisfied: 
         [0000]      10 ≦Vd 4 g≦ 40  (22)
 
         [0000]    where Vd4g denotes Abbe&#39;s Number of the positive lens of the fourth lens group with respect to the d line. 
         [0224]    The condition (22) shows the Abbe&#39;s Number of the positive lens of the fourth lens group. The achievement of the condition (22) makes it possible to achieve a variable power optical system in which the value of the total length of the optical system relative to image height is small and in which various aberrations are corrected well. Specifically, it is possible to achieve a variable power optical system in which chromatic aberration of magnification is particularly corrected well in the telephoto end position. 
         [0225]    If Vd4g is below the lower limit of the condition (22), there is no actual glass material, so that it is impossible to achieve a desired optical system. On the other hand, if Vd4g is beyond the upper limit of the condition (22), it becomes difficult to correct chromatic aberration of magnification well in the telephoto end position. 
         [0226]    Also, in an image pickup apparatus according to the third embodiment, it is preferred that: the image pickup apparatus includes one of the above-described variable power optical systems according to the third embodiment, and an imaging sensor; and the following condition (6) is satisfied: 
         [0000]      1.0 ≦|f 1 |/IH≦ 2.8  (6)
 
         [0000]    where f1 denotes the focal length of the first lens group, and IH denotes the image height of the imaging sensor. 
         [0227]    The condition (6) has the same technical significance and the same operation effect as the condition (3) does. In this case, IH denotes the image height of the imaging sensor. In a more detailed explanation, IH is half as long as the diagonal length of the image plane of the imaging sensor. Besides, the height of an image formed on the imaging sensor (the distance between the optical axis and the maximum image height) may be used as IH. 
         [0228]    Also, in an image pickup apparatus according to the third embodiment, it is preferred that the following condition (6-1) is satisfied instead of the condition (6): 
         [0000]      1.8 ≦|f 1 |/IH≦ 2.6  (6-1)
 
         [0229]    Also, in an image pickup apparatus according to the third embodiment, it is preferred that: the image pickup apparatus includes one of the above-described variable power optical systems according to the third embodiment, and an imaging sensor; and the following condition (14) is satisfied: 
         [0000]      0.2 ≦|f 2 |/IH≦ 1.8  (14)
 
         [0000]    where f2 denotes the focal length of the second lens group, and IH denotes the image height of the imaging sensor. 
         [0230]    The condition (14) has the same technical significance and the same operation effects as the condition (11) does. Besides, the explanation about IH has been described above. 
         [0231]    Also, in an image pickup apparatus according to the third embodiment, it is preferred that the following condition (14-1) is satisfied instead of the condition (14): 
         [0000]      1.0 ≦|f 2 |/IH≦ 1.5  (14-1)
 
       Embodiment 
       [0232]    Embodiments for variable power optical systems according to the present invention and image pickup apparatuses having the same are explained using the drawings, below. 
         [0233]    First, the embodiments 1 to 12 for variable power optical systems according to the present invention will be explained. 
         [0234]    The sectional view of the variable power optical system of the embodiment 1 is shown in  FIG. 1 , the sectional view of the variable power optical system of the embodiment 2 is shown in  FIG. 5 , the sectional view of the variable power optical system of the embodiment 3 is shown in  FIG. 9 , the sectional view of the variable power optical system of the embodiment 4 is shown in  FIG. 13 , the sectional view of the variable power optical system of the embodiment 5 is shown in  FIG. 17 , the sectional view of the variable power optical system of the embodiment 6 is shown in  FIG. 21 , the sectional view of the variable power optical system of the embodiment 7 is shown in  FIG. 25 , the sectional view of the variable power optical system of the embodiment 8 is shown in  FIG. 28 , the sectional view of the variable power optical system of the embodiment 9 is shown in  FIG. 31 , the sectional view of the variable power optical system of the embodiment 10 is shown in  FIG. 34 , the sectional view of the variable power optical system of the embodiment 11 is shown in  FIG. 37 , and the sectional view of the variable power optical system of the embodiment 12 is shown in  FIG. 40 . 
         [0235]    In the embodiment 1, the image height (IH) is 2.9 mm, and the pixel pitch of the imaging sensor is 1.4 μm. In the embodiment 10, the image height (IH) is 2.25 mm, and the pixel pitch of the imaging sensor is 1.1 μm, in the below explanation. However, the image height and the pixel pitch in each of the below-described embodiments are not limited to these numerical values. For example, the pixel pitch of the imaging sensor may be 2.00 μm, 1.75 μm, 1.40 μm, or 1.1 μm. The diameter of the aperture stop in the telephoto end position is larger than that of the aperture stop in the wide angle end position. As a result, it is possible to prevent the deterioration of the performance due to the diffraction limit, in the telephoto end position. However, the aperture diameter may be unchangeable if there is no practical problem. 
       Embodiment 1 
       [0236]    The optical constitution of the variable power optical system of the present embodiment is explained using  FIG. 1 . The total length of the variable power optical system of the present embodiment is about 13 mm. The variable power optical system of the present embodiment includes, in order from the object side, a first lens group G1 with negative refractive power, a second lens group G2 with positive refractive power, a third lens group G3 with negative refractive power, and a fourth lens group G4 with positive refractive power, these lens groups being located on the optical axis Lc. 
         [0237]    The first lens group G1 is composed of a negative meniscus lens L11 the convex surface of which faces toward the object side and a positive meniscus lens L12 the convex surface of which faces toward the object side, in that order from the object side. And, the first lens group G1 as a whole has negative refractive power. 
         [0238]    The second lens group G2 includes a first lens element with positive refractive power, a second lens element, and a third lens element, in that order from the object side. The first and second lens elements have convex shapes on the object side, respectively. Specifically, the second lens group is composed of a biconvex positive lens L21 which becomes the first lens element, an aperture stop S, a biconvex positive lens L22 which becomes the second lens element, and a biconcave negative lens L23 which becomes the third lens element and is joined to the biconvex positive lens L22, in that order from the object side. And, the second lens group G2 has positive refractive power as a whole and has a main magnification change function. 
         [0239]    The third lens group G3 is composed of one biconcave negative lens L3. Besides, the biconcave negative lens L3 may be replaced with a negative meniscus lens the convex surface of which faces toward the object side or with a negative meniscus lens the concave surface of which faces toward the object side. 
         [0240]    The fourth lens group G4 is composed of one positive meniscus lens L4 the convex surface of which faces toward the image side. 
         [0241]    In changing magnification from the wide angle end position to the telephoto end position, the first lens group G1 and the fourth lens group G4 keep still, and the second lens group G2 and the third lens group G3 move toward the object side. In changing from an object point at infinity to a close object point to focus the optical system on the close object point, both of the second and third lens groups G2 and G3 may be moved. 
       Embodiment 2 
       [0242]    The optical constitution of the variable power optical system of the present embodiment is explained using  FIG. 5 . The total length of the variable power optical system of the present embodiment is about 13 mm. 
         [0243]    The variable power optical system of the present embodiment includes, in order from the object side, a first lens group G1 with negative refractive power, a second lens group G2 with positive refractive power, a third lens group G3 with negative refractive power, and a fourth lens group G4 with positive refractive power, these lens group being located on the optical axis Lc. 
         [0244]    The first lens group G1 is composed of a negative meniscus lens L11 the convex surface of which faces toward the object side and a positive meniscus lens L12 the convex surface of which faces toward the object side, in that order from the object side. And, the first lens group G1 as a whole has negative refractive power. 
         [0245]    The second lens group G2 includes a first lens element with positive refractive power, a second lens element, and a third lens element, in that order from the object side. The first and second lens elements have convex shapes on the object side, respectively. Specifically, the second lens group G2 is composed of a biconvex positive lens L21 which becomes the first lens element, an aperture stop S, a biconvex positive lens L22 which becomes the second lens element, and a biconcave negative lens L23 which becomes the third lens element and is joined to the biconvex positive lens L22, in that order from the object side. And, the second lens group G2 has positive refractive power as a whole and has a main magnification change function. 
         [0246]    The third lens group G3 is composed of one negative meniscus lens L3 the concave surface of which faces toward the object side. Besides, the negative meniscus lens element L3 the concave surface of which faces toward the object side may be replaced with a negative meniscus lens the convex surface of which faces toward the object side or with a biconcave negative lens. 
         [0247]    The fourth lens group G4 is composed of one positive meniscus lens L4 the convex surface of which faces toward the image side. 
         [0248]    In changing magnification from the wide angle end position to the telephoto end position, the first lens group G1 and the fourth lens group G4 keep still, and the second lens group G2 and the third lens group G3 move toward the object side. In changing from an object point at infinity to a close object point to focus the optical system on the close object point, both of the second and third lens groups G2 and G3 may be moved. 
         [0249]    It is possible to correct variation in field curvature well in focusing on an object, by moving the both lens groups. 
       Embodiment 3 
       [0250]    The optical constitution of the variable power optical system of the present embodiment is explained using  FIG. 9 . The total length of the variable power optical system of the present embodiment is about 16 mm. 
         [0251]    The variable power optical system of the present embodiment includes, in order from the object side, a first lens group G1 with negative refractive power, a second lens group G2 with positive refractive power, a third lens group G3 with positive refractive power, and a fourth lens group G4 with positive refractive power, these lens groups being located on the optical axis Lc. 
         [0252]    The first lens group G1 is composed of one biconcave negative lens L1. 
         [0253]    The second lens group G2 is composed of one positive meniscus lens L2 the convex surface of which faces toward the object side. 
         [0254]    The third lens group G3 includes a first lens element with positive refractive power, a second lens element, and a third lens element, in that order from the object side. The second lens element has a convex shape on the object side. Specifically, the second lens group G3 is composed of an aperture stop S, a biconvex positive lens L31 which becomes the first lens element, a negative meniscus lens L32 which becomes the second lens element and the convex surface of which faces toward the object side, and a negative meniscus lens L33 which becomes the third lens element and the convex surface of which faces toward the object side, in that order from the object side. And, the third lens group G3 as a whole has positive refractive power. 
         [0255]    The fourth lens group G4 is composed of one biconvex positive lens L4. 
         [0256]    In changing magnification from the wide angle end position to the telephoto end position, the first lens group G1 and the fourth lens group G4 keep still, and the second lens group G2 and the third lens group G3 move toward the object side. 
         [0257]    In changing from an object point at infinity to a close object point to focus the optical system on the close object point, both of the second and third lens groups G2 and G3 may be moved. 
         [0258]    It is possible to correct variation in field curvature well in focusing on an object, by moving the both lens groups. 
       Embodiment 4 
       [0259]    The optical constitution of the variable power optical system of the present embodiment is explained using  FIG. 13 . The total length of the variable power optical system of the present embodiment is about 13 mm. 
         [0260]    The variable power optical system of the present embodiment includes, in order from the object side, a first lens group G1 with negative refractive power, a second lens group G2 with positive refractive power, a third lens group G3 with negative refractive power, and a fourth lens group G4 with positive refractive power, these lens groups being located on the optical axis Lc. 
         [0261]    The first lens group G1 is composed of a negative meniscus lens L11 the convex surface of which faces toward the object side and a positive meniscus lens L12 the convex surface of which faces toward the object side, in that order from the object side. And, the first lens group G1 as a whole has negative refractive power. 
         [0262]    The second lens group G2 includes a first lens element with positive refractive power, a second lens element, and a third lens element, in that order from the object side. The first and second lens elements have convex shapes on the object side, respectively. Specifically, the second lens group G2 is composed of a biconvex positive lens L21 which becomes the first lens element, an aperture stop S, a biconvex positive lens L22 which becomes the second lens element, and a biconcave negative lens L23 which becomes the third lens element and is joined to the biconvex positive lens L22, in that order from the object side. And, the second lens group G2 has positive refractive power as a whole and has a main magnification change function. 
         [0263]    The third lens group G3 is composed of one biconcave negative lens L3. Besides, the biconcave negative lens L3 may be replaced with a negative meniscus lens the convex surface of which faces toward the object side or with a negative meniscus lens the concave surface of which faces toward the object side. 
         [0264]    The fourth lens group G4 is composed of one positive meniscus lens L4 the convex surface of which faces toward the image side. 
         [0265]    In changing magnification from the wide angle end position to the telephoto end position, the first lens group G1 and the fourth lens group G4 keep still, and the second lens group G2 and the third lens group G3 move toward the object side. 
         [0266]    In changing from an object point at infinity to a close object point to focus the optical system on the close object point, both of the second and third lens groups G2 and G3 may be moved. 
       Embodiment 5 
       [0267]    The optical constitution of the variable power optical system of the present embodiment is explained using  FIG. 17 . The total length of the variable power optical system of the present embodiment is about 13 mm. 
         [0268]    The variable power optical system of the present embodiment includes, in order from the object side, a first lens group G1 with negative refractive power, a second lens group G2 with positive refractive power, a third lens group G3 with negative refractive power, and a fourth lens group G4 with positive refractive power, these lens groups being located on the optical axis Lc. 
         [0269]    The first lens group G1 is composed of a negative meniscus lens L11 the convex surface of which faces toward the object side and a positive meniscus lens L12 the convex surface of which faces toward the object side, in that order from the object side. And, the first lens group G1 as a whole has negative refractive power. 
         [0270]    The second lens group G2 includes a first lens element with positive refractive power, a second lens element, and a third lens element, in that order from the object side. The first and second lens elements have convex shapes on the object side, respectively. Specifically, the second lens group is composed of a biconvex positive lens L21 which becomes the first lens element, an aperture stop S, a biconvex positive lens L22 which becomes the second lens element, and a biconcave negative lens L23 which becomes the third lens element and is joined to the biconvex positive lens L22, in that order from the object side. And, the second lens group G2 has positive refractive power as a whole and has a main magnification change function. 
         [0271]    The third lens group G3 is composed of one biconcave negative lens L3. Besides, the biconcave negative lens L3 may be replaced with a negative meniscus lens the convex surface of which faces toward the object side or with a negative meniscus lens the concave surface of which faces toward the object side. 
         [0272]    The fourth lens group G4 is composed of one positive meniscus lens L4 the convex surface of which faces toward the image side. 
         [0273]    In changing magnification from the wide angle end position to the telephoto end position, the first lens group G1 and the fourth lens group G4 keep still, and the second lens group G2 and the third lens group G3 move toward the object side. In changing from an object point at infinity to a close object point to focus the optical system on the close object point, both of the second and third lens groups G2 and G3 may be moved. 
       Embodiment 6 
       [0274]    The optical constitution of the variable power optical system of the present embodiment is explained using  FIG. 21 . The total length of the variable power optical system of the present embodiment is about 13.5 mm. 
         [0275]    The variable power optical system of the present embodiment includes, in order from the object side, a first lens group G1 with negative refractive power, a second lens group G2 with positive refractive power, a third lens group G3 with negative refractive power, and a fourth lens group G4 with positive refractive power, these lens group being located on the optical axis Lc. 
         [0276]    The first lens group G1 is composed of a biconcave negative lens L11 and a positive meniscus lens L12 which is jointed to the biconcave negative lens L11 and the convex surface of which faces toward the object side, in that order from the object side. And, the first lens group G1 as a whole has negative refractive power. When the L12 is made of energy curable resin, it is possible to make the first lens group G1 thin, so that it is possible to sufficiently secure an amount of movement of the second lens group G2 in changing magnification. As a result, it is possible to shorten the total length of the optical system while good performance of the optical system is being maintained. 
         [0277]    The second lens group G2 includes a first lens element with positive refractive power, a second lens element, and a third lens element, in that order from the object side. The first and second lens elements have convex shapes on the object side, respectively. Specifically, the second lens group G2 is composed of a biconvex positive lens L21 which becomes the first lens element, an aperture stop S, a biconvex positive lens L22 which becomes the second lens element, and a biconcave negative lens L23 which becomes the third lens element and is joined to the biconvex positive lens L22, in that order from the object side. And, the second lens group G2 as a whole has positive refractive power. 
         [0278]    The third lens group G3 is composed of one biconcave negative lens L3. Besides, the biconcave negative lens element L3 may be replaced with a negative meniscus lens the convex surface of which faces toward the object side or with a negative meniscus lens the concave surface of which faces toward the object side. 
         [0279]    The fourth lens group G4 is composed of one positive meniscus lens L4 the convex surface of which faces toward the image side. 
         [0280]    In changing magnification from the wide angle end position to the telephoto end position, the first lens group G1 and the fourth lens group G4 keep still, the second lens group G2 moves toward the object side, and the third lens group G3 moves to the position nearest to the object side in the middle of the optical system. 
         [0281]    In changing from an object point at infinity to a close object point to focus the optical system on the close object point, both of the second and third lens groups G2 and G3 may be moved. 
       Embodiment 7 
       [0282]    The optical constitution of the variable power optical system of the present embodiment is explained using  FIG. 25 . The total length of the variable power optical system of the present embodiment is about 13 mm. 
         [0283]    The variable power optical system of the present embodiment includes, in order from the object side, a first lens group G1 with negative refractive power, a second lens group G2 with positive refractive power, a third lens group G3 with negative refractive power, and a fourth lens group G4 with positive refractive power, these lens groups being located on the optical axis Lc. 
         [0284]    The first lens group G1 is composed of a biconcave negative lens L11 and a positive meniscus lens L12 which is jointed to the biconcave negative lens L11 and the convex surface of which faces toward the object side, in that order from the object side. And, the first lens group G1 as a whole has negative refractive power. 
         [0285]    When the L12 is made of energy curable resin, it is possible to make the first lens group G1 thin, so that it is possible to sufficiently secure an amount of movement of the second lens group G2 in changing magnification. As a result, it is possible to shorten the total length of the optical system while good performance of the optical system is being maintained. 
         [0286]    The second lens group G2 includes a first lens element with positive refractive power, a second lens element, and a third lens element, in that order from the object side. The first and second lens elements have convex shapes on the object side, respectively. Specifically, the second lens group G2 is composed of a biconvex positive lens L2 which becomes the first lens element, an aperture stop S, the biconvex positive lens L22 which becomes the second lens element, a biconcave negative lens L23 which becomes the third lens element and is jointed to the biconvex positive lens L22, in that order from the object side. And, the second lens group G2 has positive refractive power as a whole and has a main magnification change function. 
         [0287]    The third lens group G3 is composed of one biconcave negative lens L3. Besides, the biconcave negative lens L3 may be replaced with a negative meniscus lens the convex surface of which faces toward the object side or with a negative meniscus lens the concave surface of which faces toward the object side. 
         [0288]    The fourth lens group G4 is composed of one positive meniscus lens L4 the convex surface of which faces toward the image side. 
         [0289]    In changing magnification from the wide angle end position to the telephoto end position, the first lens group G1 and the fourth lens group G4 keep still, and the second lens group G2 moves toward the object side, and the third lens group G3 moves to the position nearest to the object side in the middle of the optical system. 
         [0290]    In changing from an object point at infinity to a close object point to focus the optical system on the close object point, both of the second and third lens groups G2 and G3 may be moved. 
       Embodiment 8 
       [0291]    The optical constitution of the variable power optical system of the present embodiment is explained using  FIG. 28 . The total length of the variable power optical system of the present embodiment is about 13 mm. 
         [0292]    The variable power optical system of the present embodiment includes, in order from the object side, a first lens group G1 with negative refractive power, a second lens group G2 with positive refractive power, a third lens group G3 with negative refractive power, and a fourth lens group G4 with positive refractive power, these lens groups being located on the optical axis Lc. 
         [0293]    The first lens group G1 is composed of a biconcave negative lens L11 and a positive meniscus lens L12 which is jointed to the biconcave negative lens L11 and the convex surface of which faces toward the object side, in that order from the object side. And, the first lens group G1 as a whole has negative refractive power. 
         [0294]    When the L12 is made of energy curable resin, it is possible to make the first lens group G1 thin, so that it is possible to sufficiently secure an amount of movement of the second lens group G2 in changing magnification. As a result, it is possible to shorten the total length of the optical system while good performance of the optical system is being maintained. 
         [0295]    The second lens group G2 includes a first lens element with positive refractive power, a second lens element, and a third lens element, in that order from the object side. The first and second lens elements have convex shapes on the object side, respectively. Specifically, the second lens group G2 is composed of a biconvex positive lens L21 which becomes the first lens element, an aperture stop S, a biconvex positive lens L22 which becomes the second lens element, and a biconcave negative lens L23 which becomes the third lens element and is joined to the biconvex positive lens L22, in that order from the object side. And, the second lens group G2 has positive refractive power as a whole and has a main magnification change function. 
         [0296]    The third lens group G3 is composed of one negative meniscus lens L3 the convex surface of which faces toward the object sides. Besides, the negative meniscus lens L3 the convex surface of which faces toward the object side may be replaced with a negative meniscus lens the concave surface of which faces toward the object side or with a biconcave negative lens. 
         [0297]    The fourth lens group G4 is composed of one positive meniscus lens L4 the convex surface of which faces toward the image side. 
         [0298]    In changing magnification from the wide angle end position to the telephoto end position, the first lens group G1 and the fourth lens group G4 keep still, and the second lens group G2 and the third lens group G3 move toward the object side. 
       Embodiment 9 
       [0299]    The optical constitution of the variable power optical system of the present embodiment is explained using  FIG. 31 . The total length of the variable power optical system of the present embodiment is about 13.5 mm. 
         [0300]    The variable power optical system of the present embodiment includes, in order from the object side, a first lens group G1 with negative refractive power, a second lens group G2 with positive refractive power, a third lens group G3 with negative refractive power, and a fourth lens group G4 with positive refractive power, these lens groups being located on the optical axis Lc. 
         [0301]    The first lens group G1 is composed of a biconcave negative lens L11 and a positive meniscus lens L12 which is jointed to the biconcave negative lens L11 and the convex surface of which faces toward the object side, in that order from the object side. And, the first lens group G1 as a whole has negative refractive power. 
         [0302]    When the L12 is made of energy curable resin, it is possible to make the first lens group G1 thin, so that it is possible to sufficiently secure an amount of movement of the second lens group G2 in changing magnification. As a result, it is possible to shorten the total length of the optical system while good performance of the optical system is being maintained. 
         [0303]    The second lens group G2 includes a first lens element with positive refractive power, a second lens element, and a third lens element, in that order from the object side. The first and second lens elements have convex shapes on the object side, respectively. Specifically, the second lens group is composed of a biconvex positive lens L21 which becomes the first lens element, an aperture stop S, a biconvex positive lens L22 which becomes the second lens element, and a biconcave negative lens L23 which becomes the third lens element and is joined to the biconvex positive lens L22, in that order from the object side. And, the second lens group G2 has positive refractive power as a whole and has a main magnification change function. 
         [0304]    The third lens group G3 is composed of one biconcave negative lens L3. Besides, the biconcave negative lens L3 may be replaced with a negative meniscus lens the convex surface of which faces toward the object side or with a negative meniscus lens the concave surface of which faces toward the object side. 
         [0305]    The fourth lens group G4 is composed of one positive meniscus lens L4 the convex surface of which faces toward the image side. 
         [0306]    In changing magnification from the wide angle end position to the telephoto end position, the first lens group G1 keeps still, the second lens group G2 moves toward the object side, the third lens group G3 moves to the position nearest to the object side in the middle of the optical system, and the fourth lens group G4 moves toward the image side. 
       Embodiment 10 
       [0307]    The optical constitution of the variable power optical system of the present embodiment is explained using  FIG. 34 . The total length of the variable power optical system of the present embodiment is about 10 mm. 
         [0308]    The variable power optical system of the present embodiment includes, in order from the object side, a first lens group G1 with negative refractive power, a second lens group G2 with positive refractive power, a third lens group G3 with negative refractive power, and a fourth lens group G4 with positive refractive power, these lens group being located on the optical axis Lc. 
         [0309]    The first lens group G1 is composed of a biconcave negative lens L11 and a positive meniscus lens L12 the convex surface of which faces toward the object side, in that order from the object side. And, the first lens group G1 as a whole has negative refractive power. 
         [0310]    The second lens group G2 includes a first lens element with positive refractive power, a second lens element, and a third lens element, in that order from the object side. The first and second lens elements have convex shapes on the object side, respectively. Specifically, the second lens group G2 is composed of a biconvex positive lens L21 which becomes the first lens element, an aperture stop S, a biconvex positive lens L22 which becomes the second lens element, and a biconcave negative lens L23 which becomes the third lens element and is joined to the biconvex positive lens L22, in that order from the object side. And, the second lens group G2 has positive refractive power as a whole and has a main magnification change function. 
         [0311]    The third lens group G3 is composed of one biconcave negative lens L3. Besides, the biconcave negative lens element L3 may be replaced with a negative meniscus lens the convex surface of which faces toward the object side or with a negative meniscus lens the concave surface of which faces toward the object side. 
         [0312]    The fourth lens group G4 is composed of one positive meniscus lens L4 the convex surface of which faces toward the image side. 
         [0313]    In changing magnification from the wide angle end position to the telephoto end position, the first lens group G1 and the fourth lens group G4 keep still, and the second lens group G2 and the third lens group G3 move toward the object side. In changing from an object point at infinity to a close object point to focus the optical system on the close object point, both of the second and third lens groups G2 and G3 may be moved. 
       Embodiment 11 
       [0314]    The optical constitution of the variable power optical system of the present embodiment is explained using  FIG. 37  The total length of the variable power optical system of the present embodiment is about 10 mm. 
         [0315]    The variable power optical system of the present embodiment includes, in order from the object side, a first lens group G1 with negative refractive power, a second lens group G2 with positive refractive power, a third lens group G3 with negative refractive power, and a fourth lens group G4 with positive refractive power, these lens groups being located on the optical axis Lc. 
         [0316]    The first lens group G1 is composed of a negative meniscus lens L11 the convex surface of which faces toward the object side and a positive meniscus lens L12 the convex surface of which faces toward the object side, in that order from the object side. And, the first lens group G1 as a whole has negative refractive power. 
         [0317]    The second lens group G2 includes a first lens element with positive refractive power, a second lens element, and a third lens element, in that order from the object side. The first and second lens elements have convex shapes on the object side, respectively. Specifically, the second lens group G2 is composed of a biconvex positive lens L21 which becomes the first lens element, an aperture stop S, a biconvex positive lens L22 which becomes the second lens element, and a biconcave negative lens L23 which becomes the third lens element and is jointed to the biconvex positive lens L22, in that order from the object side. And, the third lens group G2 has positive refractive power as a whole and has a main magnification change function. 
         [0318]    The third lens group G3 is composed of one negative meniscus lens L3 the concave surface of which faces toward the object side. Besides, the negative meniscus lens L3 the concave surface of which faces toward the object side may be replaced with a negative meniscus lens the convex surface of which faces toward the object side or with a biconcave negative lens. 
         [0319]    The fourth lens group G4 is composed of one positive meniscus lens L4 the convex surface of which faces toward the image side. 
         [0320]    In changing magnification from the wide angle end position to the telephoto end position, the first lens group G1 and the fourth lens group G4 keep still, and the second lens group G2 and the third lens group G3 move toward the object side. In changing from an object point at infinity to a close object point to focus the optical system on the close object point, both of the second and third lens groups G2 and G3 may be moved. 
       Embodiment 12 
       [0321]    The optical constitution of the variable power optical system of the present embodiment is explained using  FIG. 40 . The total length of the variable power optical system of the present embodiment is about 14 mm. 
         [0322]    The variable power optical system of the present embodiment includes, in order from the object side, a first lens group G1 with negative refractive power, a second lens group G2 with positive refractive power, a third lens group G3 with negative refractive power, and a fourth lens group G4 with positive refractive power, these lens groups being located on the optical axis Lc. 
         [0323]    The first lens group G1 is composed of one biconcave negative lens L1. 
         [0324]    The second lens group G2 is composed of a first lens element with positive refractive power, a second lens element, and a third lens element, in that order from the object side. The first lens element has a convex shape on the object side. Specifically, the second lens group G2 is composed of a biconvex positive lens L21 which becomes the first lens element, a biconcave negative lens L22 which becomes the second lens element and is joined to the biconvex positive lens L21, an aperture stop S, and a biconvex positive lens L23 which becomes the third lens element, in that order from the object side. And, the second lens group G2 has positive refractive power as a whole and has a main magnification change function. 
         [0325]    The third lens group G3 is composed of one negative meniscus lens L3 the convex surface of which faces toward the object side. Besides, the negative meniscus lens L3 the convex surface of which faces toward the object side may be replaced with a negative meniscus lens the concave surface of which faces toward the object side or with a biconcave negative lens. 
         [0326]    The fourth lens group G4 is composed of one positive meniscus lens L4 the convex surface of which faces toward the image side. 
         [0327]    In changing magnification from the wide angle end position to the telephoto end position, the first lens group G1 and the fourth lens group G4 keep still, and the second lens group G2 and the third lens group G3 move toward the object side. 
         [0328]    Next, in each of the embodiments 1 to 12, the numerical data of the optical members constituting each of the variable power optical systems will be given. The embodiment 1 corresponds to a numerical embodiment 1. The embodiment 2 corresponds to a numerical embodiment 2. The embodiment 3 corresponds to a numerical embodiment 3. The embodiment 4 corresponds to a numerical embodiment 4. The embodiment 5 corresponds to a numerical embodiment 5. The embodiment 6 corresponds to a numerical embodiment 6. The embodiment 7 corresponds to a numerical embodiment 7. The embodiment 8 corresponds to a numerical embodiment 8. The embodiment 9 corresponds to a numerical embodiment 9. The embodiment 10 corresponds to a numerical embodiment 10. The embodiment 11 corresponds to a numerical embodiment 11. The embodiment 12 corresponds to a numerical embodiment 12. 
         [0329]    Besides, in the numerical data and the drawings, r denotes the radius of curvature of each of lens surfaces, d denotes the thickness of each of lenses or air spacing between lenses, nd denotes the refractive index of each of lenses with respect to the d line (587.56 nm), vd denotes the Abbe&#39;s number of each of lenses with respect to the d line (587.56 nm), and * (asterisk) expresses aspherical surface. A unit of length is mm in the numerical data. 
         [0330]    Also, when z is taken as a coordinate in the direction along the optical axis, y is taken as a coordinate in the direction perpendicular to the optical axis, K denotes a conic constant, and A4, A6, A8, and A10 denote an aspherical coefficient, the shapes of aspherical surfaces are expressed by the following formula (I): 
         [0000]        z =( y   2   /r )/[1+{1−(1 +K )( y/r ) 2 } 1/2   ]+A 4 y   4   +A 6 y   6   +A 8 y   8   +A 10 y   10   (I)
 
         [0331]    Also, e denotes a power of ten. Besides, these symbols for these various values are also common to the following numerical data of the embodiments. 
         [0332]    Besides, BF denotes the distance from the last surface in lenses to a paraxial image plane in the form of air equivalent amount, and lens total length denotes a value obtained by adding the distance between the first surface and the last surface in lenses to BF. On the other hand, BF# denotes the distance from the last surface in lenses to an image plane in the form of air equivalent amount, and lens total length# denotes a value obtained by adding the distance between the first surface and the last surface in lenses to BF#. 
         [0000]    
       
         
               
             
               
               
               
               
               
               
             
               
             
               
               
             
               
             
               
               
               
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
               
               
               
             
               
               
               
               
             
               
             
               
               
               
               
               
               
             
               
             
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
               
               
               
             
           
               
                   
               
               
                 Numerical Embodiment 1 
               
               
                   
               
             
             
               
                 Surface data 
               
               
                 Unit: mm 
               
             
          
           
               
                 Surface No. 
                 r 
                 d 
                 nd 
                 vd 
                 Effective diameter 
               
               
                   
               
               
                 Object plane 
                 ∞ 
                 ∞ 
               
               
                  1* 
                 29.7496 
                 0.5421 
                 1.90270 
                 31.00 
                 2.690 
               
               
                  2* 
                 2.6180 
                 0.5723 
                   
                   
                 2.165 
               
               
                  3* 
                 3.5600 
                 0.7172 
                 2.10223 
                 16.77 
                 2.185 
               
               
                  4* 
                 5.6420 
                 D4 
                   
                   
                 2.097 
               
               
                  5* 
                 2.1763 
                 0.9187 
                 1.59201 
                 67.02 
                 1.427 
               
               
                  6* 
                 −44.5379 
                 0.2000 
                   
                   
                 1.318 
               
               
                  7 (Stop) 
                 ∞ 
                 0.0000 
                   
                   
                 (Variable) 
               
               
                  8* 
                 5.9838 
                 0.6894 
                 1.85135 
                 40.10 
                 1.207 
               
               
                  9 
                 −6.5431 
                 0.5215 
                 1.82114 
                 24.06 
                 1.137 
               
               
                 10* 
                 5.0933 
                 D10 
                   
                   
                 1.011 
               
               
                 11* 
                 −4.9399 
                 0.4461 
                 1.77377 
                 47.17 
                 1.198 
               
               
                 12* 
                 53.6530 
                 D12 
                   
                   
                 1.427 
               
               
                 13* 
                 −14.4249 
                 0.8801 
                 1.82114 
                 24.06 
                 2.808 
               
               
                 14* 
                 −5.0670 
                 0.2353 
                   
                   
                 2.868 
               
               
                 15 
                 ∞ 
                 0.3000 
                 1.51633 
                 64.14 
                 2.929 
               
               
                 16 
                 ∞ 
                 0.4000 
                   
                   
                 2.947 
               
               
                 Image plane 
                 ∞ 
               
               
                   
               
             
          
           
               
                 Aspherical surface data 
               
               
                   
               
             
          
           
               
                   
                 The first surface 
               
               
                   
                 K = −224.489, A4 = 6.10008e−03, A6 = −1.22957e−03, A8 = 5.75218e−05 
               
               
                   
                 The second surface 
               
               
                   
                 K = 0.086, A4 = −6.58139e−03, A6 = 3.39688e−03, A8 = −7.32903e−04 
               
               
                   
                 The third surface 
               
               
                   
                 K = −6.605, A4 = −3.67877e−03, A6 = 1.95938e−03, A8 = −8.93682e−05 
               
               
                   
                 The fourth surface 
               
               
                   
                 K = −21.342, A4 = −4.83678e−03, A6 = 6.39538e−04, A8 = 5.70942e−05 
               
               
                   
                 The fifth surface 
               
               
                   
                 K = −0.989, A4 = 2.50732e−03, A6 = 7.21049e−04, A8 = 3.61962e−04 
               
               
                   
                 The sixth surface 
               
               
                   
                 K = −791.631, A4 = −1.12792e−02, A6 = 8.93540e−03, A8 = −1.87646e−03 
               
               
                   
                 The eighth surface 
               
               
                   
                 K = 4.845, A4 = −9.49342e−04, A6 = 9.01956e−03, A8 = −2.80204e−03 
               
               
                   
                 The tenth surface 
               
               
                   
                 K = −32.788, A4 = 6.96684e−02, A6 = −4.68501e−03, A8 = 9.73012e−03 
               
               
                   
                 The eleventh surface 
               
               
                   
                 K = −3.112, A4 = 3.30281e−03, A6 = −6.05974e−03, A8 = −4.24602e−03 
               
               
                   
                 The twelfth surface 
               
               
                   
                 K = −40363.004, A4 = 2.29435e−02, A6 = −1.24625e−02, A8 = 9.19254e−04 
               
               
                   
                 The thirteenth surface 
               
               
                   
                 K = −1.394, A4 = 4.97914e−03, A6 = −1.14929e−03, A8 = 9.38867e−05 
               
               
                   
                 The fourteenth surface 
               
               
                   
                 K = −2.722, A4 = 1.16889e−02, A6 = −2.73333e−03, A8 = 1.79573e−04 
               
               
                   
                   
               
             
          
           
               
                 Various data 
               
               
                 Zoom ratio: 2.850 
               
             
          
           
               
                   
                   
                   
                   
                   
                   
                 Telephoto 
               
               
                   
                 Wide 
                   
                 Telephoto 
                 Wide angle end 
                 Middle 
                 end 
               
             
          
           
               
                   
                 angle end 
                 Middle 
                 end 
                 (focusing on a close range) 
               
               
                   
                   
               
             
          
           
               
                 Focal length 
                 3.754 
                 6.025 
                 10.699 
                   
                   
                   
               
               
                 F No. 
                 3.200 
                 4.369 
                 5.200 
               
               
                 Angle of view 2ω 
                 82.828 
                 51.090 
                 29.527 
               
               
                 Image height 
                 2.900 
                 2.900 
                 2.900 
               
               
                 BF 
                 0.833 
                 0.833 
                 0.833 
               
               
                 Lens total length 
                 12.898 
                 12.898 
                 12.898 
               
               
                 (Distance from an 
                 ∞ 
                 ∞ 
                 ∞ 
                 100.00 
                 500.00 
                 800.00 
               
               
                 object point) 
               
               
                 D4 
                 3.941 
                 2.207 
                 0.200 
                 3.225 
                 2.257 
                 0.483 
               
               
                 D10 
                 1.025 
                 0.863 
                 1.714 
                 0.969 
                 0.883 
                 1.519 
               
               
                 D12 
                 1.611 
                 3.507 
                 4.664 
                 2.384 
                 3.438 
                 4.575 
               
               
                 Stop diameter 
                 0.987 
                 0.987 
                 1.215 
               
               
                 (Entrance pupil 
                 3.159 
                 2.655 
                 1.791 
               
               
                 position) 
               
               
                 (Exit pupil 
                 −6.368 
                 −14.458 
                 −35.576 
               
               
                 position) 
               
               
                 (The position of 
                 4.966 
                 6.304 
                 9.349 
               
               
                 the front side 
               
               
                 principle point) 
               
               
                 (The position of 
                 −3.317 
                 −5.641 
                 −10.271 
               
               
                 the rear side 
               
               
                 principle point) 
               
               
                   
               
             
          
           
               
                   
                 Lens 
                 The first surface of lens 
                 Focal length of lens 
               
               
                   
                   
               
               
                   
                 L11 
                 1 
                 −3.210 
               
               
                   
                 L12 
                 3 
                 7.413 
               
               
                   
                 L21 
                 5 
                 3.531 
               
               
                   
                 L22 
                 8 
                 3.767 
               
               
                   
                 L23 
                 9 
                 −3.419 
               
               
                   
                 L3 
                 11 
                 −5.827 
               
               
                   
                 L4 
                 13 
                 9.125 
               
               
                   
                   
               
             
          
           
               
                 Zoom lens group data 
               
             
          
           
               
                   
                 The first 
                   
                 Composition 
                 The position 
                 The position 
               
               
                   
                 surface 
                 Focal length of 
                 length 
                 of the front side 
                 of the rear side 
               
               
                 Group 
                 of lens group 
                 lens group 
                 of lens group 
                 principle point 
                 principle point 
               
               
                   
               
               
                 G1 
                 1 
                 −5.731 
                 1.832 
                 0.276 
                 −0.873 
               
               
                 G2 
                 5 
                 3.191 
                 2.330 
                 −0.304 
                 −1.515 
               
               
                 G3 
                 11 
                 −5.827 
                 0.446 
                 0.021 
                 −0.230 
               
               
                 G4 
                 13 
                 9.125 
                 0.880 
                 0.715 
                 0.251 
               
               
                   
               
             
          
           
               
                 Magnification of lens group 
               
             
          
           
               
                   
                 Wide angle end 
                 Middle 
                 Telephoto end 
               
             
          
           
               
                   
                 Wide angle end 
                 Middle 
                 Telephoto end 
                 (focusing on a close range) 
               
               
                   
                   
               
             
          
           
               
                 G1 
                 0.000 
                 0.000 
                 0.000 
                 0.054 
                 0.011 
                 0.007 
               
               
                 G2 
                 −0.453 
                 −0.600 
                 −0.964 
                 −0.530 
                 −0.602 
                 −0.898 
               
               
                 G3 
                 1.553 
                 1.868 
                 2.075 
                 1.689 
                 1.859 
                 2.063 
               
               
                 G4 
                 0.932 
                 0.938 
                 0.933 
                 0.930 
                 0.936 
                 0.931 
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
             
               
             
               
             
               
             
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
               
               
               
             
               
               
               
               
             
               
             
               
               
               
               
               
               
             
               
             
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
               
               
               
             
           
               
                   
               
               
                 Numerical embodiment 2 
               
               
                   
               
             
             
               
                 Surface data 
               
               
                 Unit: mm 
               
             
          
           
               
                   
                 Surface No. 
                 r 
                 d 
                 nd 
                 vd 
                 Effective diameter 
               
               
                   
                   
               
               
                   
                 Object plane 
                 ∞ 
                 ∞ 
               
               
                   
                  1* 
                 31.1191 
                 0.5500 
                 1.85135 
                 40.10 
                 2.825 
               
               
                   
                  2* 
                 2.5330 
                 0.5132 
                   
                   
                 2.187 
               
               
                   
                  3* 
                 3.1051 
                 0.8424 
                 1.82114 
                 24.06 
                 2.203 
               
               
                   
                  4* 
                 5.5712 
                 D4 
                   
                   
                 2.110 
               
               
                   
                  5* 
                 2.0868 
                 1.0720 
                 1.59201 
                 67.02 
                 1.438 
               
               
                   
                  6* 
                 −181.2845 
                 0.2000 
                   
                   
                 1.296 
               
               
                   
                  7 (Stop) 
                 ∞ 
                 0.0000 
                   
                   
                 (Variable) 
               
               
                   
                  8* 
                 5.2934 
                 0.6793 
                 1.85135 
                 40.10 
                 1.161 
               
               
                   
                  9 
                 −9.4008 
                 0.4687 
                 1.82114 
                 24.06 
                 1.086 
               
               
                   
                 10* 
                 4.3096 
                 D10 
                   
                   
                 0.980 
               
               
                   
                 11* 
                 −4.5361 
                 0.4000 
                 1.77377 
                 47.17 
                 1.185 
               
               
                   
                 12* 
                 −4863.2721 
                 D12 
                   
                   
                 1.405 
               
               
                   
                 13* 
                 −15.0779 
                 0.8960 
                 1.82114 
                 24.06 
                 2.787 
               
               
                   
                 14* 
                 −4.9623 
                 0.2027 
                   
                   
                 2.854 
               
               
                   
                 15 
                 ∞ 
                 0.3000 
                 1.51633 
                 64.14 
                 2.920 
               
               
                   
                 16 
                 ∞ 
                 0.4000 
                   
                   
                 2.953 
               
               
                   
                 Image plane 
                 ∞ 
               
               
                   
                   
               
             
          
           
               
                 Aspherical surface data 
               
               
                   
               
             
          
           
               
                 The first surface 
               
               
                 K = 9.975, A4 = 5.45942e−03, A6 = −1.17913e−03, A8 = 6.14112e−05 
               
               
                 The second surface 
               
               
                 K = 0.028, A4 = −5.92450e−03, A6 = 3.54360e−03, A8 = −7.18809e−04 
               
               
                 The third surface 
               
               
                 K = −4.944, A4 = −2.93256e−03, A6 = 2.56724e−03, A8 = −1.52620e−04 
               
               
                 The fourth surface 
               
               
                 K = −16.419, A4 = −5.87501e−03, A6 = 9.32666e−04, A8 = 4.10944e−05 
               
               
                 The fifth surface 
               
               
                 K = −0.869, A4 = 3.97321e−03, A6 = 4.46620e−04, A8 = 3.23707e−04 
               
               
                 The sixth surface 
               
               
                 K = −9645.985, A4 = −2.36383e−02, A6 = 1.14055e−02, A8 = −2.04346e−03 
               
               
                 The eighth surface 
               
               
                 K = −2.017, A4 = −1.25319e−02, A6 = 1.04830e−02, A8 = −2.75134e−03 
               
               
                 The tenth surface 
               
               
                 K = −37.283, A4 = 8.97816e−02, A6 = −2.64567e−02, A8 = 2.13844e−02 
               
               
                 The eleventh surface 
               
               
                 K = −3.217, A4 = 3.81585e−03, A6 = −1.49213e−02, A8 = 1.03508e−03 
               
               
                 The twelfth surface 
               
               
                 K = −14369395.106, A4 = 1.43170e−02, A6 = −1.10677e−02, A8 = 1.12467e−03 
               
               
                 The thirteenth surface 
               
               
                 K = −1.990, A4 = 5.00050e−03, A6 = −1.15848e−03, A8 = 9.16097e−05 
               
               
                 The fourteenth surface 
               
               
                 K = −2.578, A4 = 1.16150e−02, A6 = −2.73641e−03, A8 = 1.80048e−04 
               
               
                   
               
             
          
           
               
                 Various data 
               
               
                 Zoom ratio: 2.850 
               
             
          
           
               
                   
                 Wide angle end 
                 Middle 
                 Telephoto end 
               
             
          
           
               
                   
                 Wide angle end 
                 Middle 
                 Telephoto end 
                 (focusing on a close range) 
               
               
                   
                   
               
             
          
           
               
                 Focal length 
                 3.754 
                 6.021 
                 10.699 
                   
                   
                   
               
               
                 F No. 
                 3.200 
                 4.384 
                 5.200 
               
               
                 Angle of view 2ω 
                 82.691 
                 50.978 
                 29.424 
               
               
                 Image height 
                 2.900 
                 2.900 
                 2.900 
               
               
                 BF 
                 0.801 
                 0.801 
                 0.801 
               
               
                 Lens total length 
                 12.898 
                 12.898 
                 12.898 
               
               
                 (Distance from an 
                 ∞ 
                 ∞ 
                 ∞ 
                 100.00 
                 500.00 
                 800.00 
               
               
                 object point) 
               
               
                 D4 
                 3.931 
                 2.208 
                 0.200 
                 3.736 
                 2.222 
                 0.266 
               
               
                 D10 
                 0.964 
                 0.788 
                 1.612 
                 0.981 
                 0.811 
                 1.598 
               
               
                 D12 
                 1.581 
                 3.480 
                 4.663 
                 1.759 
                 3.443 
                 4.612 
               
               
                 Stop diameter 
                 0.944 
                 0.944 
                 1.173 
               
               
                 (Entrance pupil 
                 3.288 
                 2.795 
                 1.956 
               
               
                 position) 
               
               
                 (Exit pupil 
                 −6.289 
                 −14.879 
                 −41.689 
               
               
                 position) 
               
               
                 (The position of 
                 5.061 
                 6.495 
                 9.958 
               
               
                 the front side 
               
               
                 principle point) 
               
               
                 (The position of 
                 −3.330 
                 −5.679 
                 −10.344 
               
               
                 the rear side 
               
               
                 principle point) 
               
               
                   
               
             
          
           
               
                   
                 Lens 
                 The first surface of lens 
                 Focal length of lens 
               
               
                   
                   
               
               
                   
                 L11 
                 1 
                 −3.268 
               
               
                   
                 L12 
                 3 
                 7.403 
               
               
                   
                 L21 
                 5 
                 3.492 
               
               
                   
                 L22 
                 8 
                 4.064 
               
               
                   
                 L23 
                 9 
                 −3.544 
               
               
                   
                 L3 
                 11 
                 −5.868 
               
               
                   
                 L4 
                 13 
                 8.662 
               
               
                   
                   
               
             
          
           
               
                 Zoom lens group data 
               
             
          
           
               
                   
                   
                   
                   
                 The position 
                 The position 
               
               
                   
                 The first surface 
                 Focal length 
                 Composition length 
                 of the front side 
                 of the rear side 
               
               
                 Group 
                 of lens group 
                 of lens group 
                 of lens group 
                 principle point 
                 principle point 
               
               
                   
               
               
                 G1 
                 1 
                 −5.825 
                 1.906 
                 0.340 
                 −0.873 
               
               
                 G2 
                 5 
                 3.180 
                 2.420 
                 −0.387 
                 −1.600 
               
               
                 G3 
                 11 
                 −5.868 
                 0.400 
                 −0.000 
                 −0.226 
               
               
                 G4 
                 13 
                 8.662 
                 0.896 
                 0.705 
                 0.232 
               
               
                   
               
             
          
           
               
                 Magnification of lens group 
               
             
          
           
               
                   
                 Wide angle end 
                 Middle 
                 Telephoto end 
               
             
          
           
               
                   
                 Wide angle end 
                 Middle 
                 Telephoto end 
                 (focusing on a close range) 
               
               
                   
                   
               
             
          
           
               
                 G1 
                 0.000 
                 0.000 
                 0.000 
                 0.055 
                 0.012 
                 0.007 
               
               
                 G2 
                 −0.450 
                 −0.596 
                 −0.954 
                 −0.486 
                 −0.601 
                 −0.948 
               
               
                 G3 
                 1.536 
                 1.844 
                 2.048 
                 1.576 
                 1.842 
                 2.043 
               
               
                 G4 
                 0.932 
                 0.941 
                 0.940 
                 0.926 
                 0.939 
                 0.938 
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
             
               
             
               
             
               
             
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
               
               
               
             
               
               
               
               
             
               
             
               
               
               
               
               
               
             
               
             
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
               
               
               
             
           
               
                   
               
               
                 Numerical embodiment 3 
               
               
                   
               
             
             
               
                 Surface data 
               
               
                 Unit: mm 
               
             
          
           
               
                   
                 Surface No. 
                 r 
                 d 
                 nd 
                 vd 
                 Effective diameter 
               
               
                   
                   
               
               
                   
                 Object plane 
                 ∞ 
                 ∞ 
               
               
                   
                  1* 
                 −51.6608 
                 0.5500 
                 1.53071 
                 55.67 
                 3.911 
               
               
                   
                  2* 
                 3.8768 
                 D2 
                   
                   
                 2.973 
               
               
                   
                  3* 
                 4.5535 
                 0.7634 
                 1.63493 
                 23.89 
                 2.471 
               
               
                   
                  4* 
                 8.1682 
                 D4 
                   
                   
                 2.332 
               
               
                   
                  5 (Stop) 
                 ∞ 
                 0.0000 
                   
                   
                 (Variable) 
               
               
                   
                  6* 
                 2.2847 
                 1.2110 
                 1.53071 
                 55.67 
                 1.442 
               
               
                   
                  7* 
                 −4.8589 
                 0.2000 
                   
                   
                 1.407 
               
               
                   
                  8* 
                 27.5864 
                 0.5000 
                 1.63493 
                 23.89 
                 1.317 
               
               
                   
                  9* 
                 1.9602 
                 0.5251 
                   
                   
                 1.202 
               
               
                   
                 10* 
                 1.9235 
                 0.7100 
                 1.53071 
                 55.67 
                 1.500 
               
               
                   
                 11* 
                 1.9019 
                 D11 
                   
                   
                 1.610 
               
               
                   
                 12* 
                 13.5849 
                 1.5632 
                 1.63493 
                 23.89 
                 2.944 
               
               
                   
                 13* 
                 −23.9264 
                 1.2468 
                   
                   
                 2.814 
               
               
                   
                 14 
                 ∞ 
                 0.4000 
                 1.51633 
                 64.14 
                 2.946 
               
               
                   
                 15 
                 ∞ 
                 0.4000 
                   
                   
                 2.973 
               
               
                   
                 Image plane 
                 ∞ 
               
               
                   
                   
               
             
          
           
               
                 Aspherical surface data 
               
               
                   
               
             
          
           
               
                 The first surface 
               
               
                 K = 5.000, A4 = −2.36892e−03, A6 = 2.28612e−04, A8 = −5.01395e−06 
               
               
                 The second surface 
               
               
                 K = −5.000, A4 = 6.35893e−03, A6 = −6.43852e−04, A8 = 6.13916e−05 
               
               
                 The third surface 
               
               
                 K = −4.992, A4 = 2.91448e−03, A6 = −7.05027e−04, A8 = 8.65670e−05 
               
               
                 The fourth surface 
               
               
                 K = −1.084, A4 = −3.57126e−03, A6 = −4.87021e−05, A8 = 6.46356e−05 
               
               
                 The sixth surface 
               
               
                 K = −1.312, A4 = 5.44552e−03, A6 = 3.11419e−03, A8 = −8.98067e−04 
               
               
                 The seventh surface 
               
               
                 K = −4.146, A4 = 2.83431e−02, A6 = −1.00090e−02, A8 = 6.33877e−04 
               
               
                 The eighth surface 
               
               
                 K = 0.000, A4 = 2.55526e−02, A6 = −8.57471e−03 
               
               
                 The ninth surface 
               
               
                 K = −3.832, A4 = 3.21058e−02, A6 = 7.16734e−03, A8 = 7.08642e−04 
               
               
                 The tenth surface 
               
               
                 K = −3.755, A4 = −1.67129e−02, A6 = −5.64958e−03, A8 = 3.28290e−03 
               
               
                 The eleventh surface 
               
               
                 K = −1.122, A4 = −4.68131e−02, A6 = 2.86083e−03, A8 = 8.24444e−04 
               
               
                 The twelfth surface 
               
               
                 K = 0.756, A4 = −3.58875e−03, A6 = 7.72697e−04, A8 = −1.41169e−05 
               
               
                 The thirteenth surface 
               
               
                 K = −5.000, A4 = −5.33300e−03, A6 = 7.14335e−04, A8 = 2.85829e−05 
               
               
                   
               
             
          
           
               
                 Various data 
               
               
                 Zoom ratio: 2.862 
               
             
          
           
               
                   
                 Wide angle end 
                 Middle 
                 Telephoto end 
               
             
          
           
               
                   
                 Wide angle end 
                 Middle 
                 Telephoto end 
                 (focusing on a close range) 
               
               
                   
                   
               
             
          
           
               
                 Focal length 
                 4.156 
                 6.693 
                 11.893 
                   
                   
                   
               
               
                 F No. 
                 3.200 
                 4.396 
                 5.200 
               
               
                 Angle of view 2ω 
                 77.136 
                 47.218 
                 26.480 
               
               
                 Image height 
                 2.900 
                 2.900 
                 2.900 
               
               
                 BF 
                 1.911 
                 1.911 
                 1.911 
               
               
                 Lens total length 
                 15.864 
                 15.864 
                 15.864 
               
               
                 (Distance from an 
                 ∞ 
                 ∞ 
                 ∞ 
                 100.00 
                 500.00 
                 800.00 
               
               
                 object point) 
               
               
                 D2 
                 3.202 
                 0.703 
                 0.327 
                 2.875 
                 0.774 
                 0.401 
               
               
                 D4 
                 3.820 
                 3.980 
                 0.909 
                 3.675 
                 3.824 
                 0.787 
               
               
                 D11 
                 0.909 
                 3.247 
                 6.694 
                 1.381 
                 3.333 
                 6.742 
               
               
                 Stop diameter 
                 1.080 
                 1.080 
                 1.292 
               
               
                 (Entrance pupil 
                 4.327 
                 3.912 
                 1.855 
               
               
                 position) 
               
               
                 (Exit pupil 
                 −4.270 
                 −9.011 
                 −23.780 
               
               
                 position) 
               
               
                 (The position of 
                 5.704 
                 6.491 
                 8.219 
               
               
                 the front side 
               
               
                 principle point) 
               
               
                 (The position of 
                 −3.721 
                 −6.324 
                 −11.604 
               
               
                 the rear side 
               
               
                 principle point) 
               
               
                   
               
             
          
           
               
                   
                 Lens 
                 The first surface of lens 
                 Focal length of lens 
               
               
                   
                   
               
               
                   
                 L1 
                 1 
                 −6.772 
               
               
                   
                 L2 
                 3 
                 14.977 
               
               
                   
                 L31 
                 6 
                 3.111 
               
               
                   
                 L32 
                 8 
                 −3.349 
               
               
                   
                 L33 
                 10 
                 30.688 
               
               
                   
                 L4 
                 12 
                 13.872 
               
               
                   
                   
               
             
          
           
               
                 Zoom lens group data 
               
             
          
           
               
                   
                   
                   
                   
                 The position 
                 The position 
               
               
                   
                 The first surface 
                 Focal length 
                 Composition length 
                 of the front side 
                 of the rear side 
               
               
                 Group 
                 of lens group 
                 of lens group 
                 of lens group 
                 principle point 
                 principle point 
               
               
                   
               
               
                 G1 
                 1 
                 −6.772 
                 0.550 
                 0.333 
                 −0.025 
               
               
                 G2 
                 3 
                 14.977 
                 0.763 
                 −0.544 
                 −0.975 
               
               
                 G3 
                 5 
                 6.085 
                 3.146 
                 −2.433 
                 −3.367 
               
               
                 G4 
                 12 
                 13.872 
                 1.563 
                 0.352 
                 −0.620 
               
               
                   
               
             
          
           
               
                 Magnification of lens group 
               
             
          
           
               
                   
                 Wide angle end 
                 Middle 
                 Telephoto end 
               
             
          
           
               
                   
                 Wide angle end 
                 Middle 
                 Telephoto end 
                 (focusing on a close range) 
               
               
                   
                   
               
             
          
           
               
                 G1 
                 0.000 
                 0.000 
                 0.000 
                 0.063 
                 0.013 
                 0.008 
               
               
                 G2 
                 2.712 
                 1.867 
                 1.784 
                 2.386 
                 1.863 
                 1.787 
               
               
                 G3 
                 −0.278 
                 −0.646 
                 −1.193 
                 −0.360 
                 −0.661 
                 −1.208 
               
               
                 G4 
                 0.815 
                 0.820 
                 0.826 
                 0.814 
                 0.819 
                 0.823 
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
             
               
             
               
             
               
             
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
               
               
               
             
               
               
               
               
             
               
             
               
               
               
               
               
               
             
               
             
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
               
               
               
             
           
               
                   
               
               
                 Numerical embodiment 4 
               
               
                   
               
             
             
               
                 Surface data 
               
               
                 Unit: mm 
               
             
          
           
               
                   
                 Surface No. 
                 r 
                 d 
                 nd 
                 vd 
                 Effective diameter 
               
               
                   
                   
               
               
                   
                 Object plane 
                 ∞ 
                 ∞ 
               
               
                   
                  1* 
                 24.9824 
                 0.4982 
                 1.90270 
                 31.00 
                 2.680 
               
               
                   
                  2* 
                 2.6593 
                 0.5867 
                   
                   
                 2.159 
               
               
                   
                  3* 
                 3.7965 
                 0.6883 
                 2.10223 
                 16.77 
                 2.159 
               
               
                   
                  4* 
                 5.9430 
                 D4 
                   
                   
                 2.120 
               
               
                   
                  5* 
                 2.1596 
                 0.9525 
                 1.59201 
                 67.02 
                 1.435 
               
               
                   
                  6* 
                 −48.6138 
                 0.2000 
                   
                   
                 1.322 
               
               
                   
                  7 (Stop) 
                 ∞ 
                 0.0000 
                   
                   
                 (Variable) 
               
               
                   
                  8* 
                 6.5103 
                 0.6801 
                 1.85135 
                 40.10 
                 1.201 
               
               
                   
                  9 
                 −7.3491 
                 0.5708 
                 1.82114 
                 24.06 
                 1.134 
               
               
                   
                 10* 
                 5.5872 
                 D10 
                   
                   
                 1.015 
               
               
                   
                 11* 
                 −4.6844 
                 0.4504 
                 1.77377 
                 47.17 
                 1.212 
               
               
                   
                 12* 
                 100.7637 
                 D12 
                   
                   
                 1.437 
               
               
                   
                 13* 
                 −13.7627 
                 0.8839 
                 1.90270 
                 31.00 
                 2.801 
               
               
                   
                 14* 
                 −5.1236 
                 0.2065 
                   
                   
                 2.866 
               
               
                   
                 15 
                 ∞ 
                 0.3000 
                 1.51633 
                 64.14 
                 2.928 
               
               
                   
                 16 
                 ∞ 
                 0.4000 
                   
                   
                 2.946 
               
               
                   
                 Image plane 
                 ∞ 
               
               
                   
                   
               
             
          
           
               
                 Aspherical surface data 
               
               
                   
               
             
          
           
               
                 The first surface 
               
               
                 K = −397.963, A4 = 6.59381e−03, A6 = −1.17236e−03, A8 = 5.02427e−05 
               
               
                 The second surface 
               
               
                 K = 0.088, A4 = −6.56207e−03, A6 = 3.84046e−03, A8 = −7.40701e−04 
               
               
                 The third surface 
               
               
                 K = −7.634, A4 = −3.69106e−03, A6 = 1.79132e−03, A8 = −1.86618e−04 
               
               
                 The fourth surface 
               
               
                 K = −23.212, A4 = −5.62260e−03, A6 = 5.95203e−04, A8 = −6.89594e−05 
               
               
                 The fifth surface 
               
               
                 K = −0.956, A4 = 2.77532e−03, A6 = 7.11842e−04, A8 = 9.68824e−05 
               
               
                 The sixth surface 
               
               
                 K = 218.608, A4 = −1.28903e−02, A6 = 7.79247e−03, A8 = −1.56315e−03 
               
               
                 The eighth surface 
               
               
                 K = 0.801, A4 = −2.76409e−03, A6 = 7.97038e−03, A8 = −2.01303e−03 
               
               
                 The tenth surface 
               
               
                 K = −38.926, A4 = 6.44034e−02, A6 = −6.22281e−03, A8 = 1.13101e−02 
               
               
                 The eleventh surface 
               
               
                 K = −3.245, A4 = 4.36822e−03, A6 = −5.62552e−03, A8 = −2.76127e−03 
               
               
                 The twelfth surface 
               
               
                 K = −575054.125, A4 = 1.94775e−02, A6 = −8.94047e−03, A8 = 3.15805e−04 
               
               
                 The thirteenth surface 
               
               
                 K = −0.780, A4 = 4.95196e−03, A6 = −1.14942e−03, A8 = 9.43150e−05 
               
               
                 The fourteenth surface 
               
               
                 K = −2.819, A4 = 1.17394e−02, A6 = −2.72948e−03, A8 = 1.79738e−04 
               
               
                   
               
             
          
           
               
                 Various data 
               
               
                 Zoom ratio: 2.849 
               
             
          
           
               
                   
                 Wide angle end 
                 Middle 
                 Telephoto end 
               
             
          
           
               
                   
                 Wide angle end 
                 Middle 
                 Telephoto end 
                 (focusing on a close range) 
               
               
                   
                   
               
             
          
           
               
                 Focal length 
                 3.754 
                 6.019 
                 10.695 
                   
                   
                   
               
               
                 F No. 
                 3.200 
                 4.367 
                 5.200 
               
               
                 Angle of view 2ω 
                 82.905 
                 51.227 
                 29.493 
               
               
                 Image height 
                 2.900 
                 2.900 
                 2.900 
               
               
                 BF 
                 0.804 
                 0.804 
                 0.804 
               
               
                 Lens total length 
                 12.898 
                 12.898 
                 12.898 
               
               
                 (Distance from an 
                 ∞ 
                 ∞ 
                 ∞ 
                 100.00 
                 500.00 
                 800.00 
               
               
                 object point) 
               
               
                 D4 
                 3.931 
                 2.207 
                 0.200 
                 3.137 
                 2.256 
                 0.534 
               
               
                 D10 
                 1.026 
                 0.876 
                 1.749 
                 0.968 
                 0.895 
                 1.511 
               
               
                 D12 
                 1.626 
                 3.500 
                 4.633 
                 2.478 
                 3.432 
                 4.537 
               
               
                 Stop diameter 
                 0.983 
                 0.983 
                 1.210 
               
               
                 (Entrance pupil 
                 3.137 
                 2.642 
                 1.790 
               
               
                 position) 
               
               
                 (Exit pupil 
                 −6.729 
                 −16.061 
                 −45.914 
               
               
                 position) 
               
               
                 (The position of 
                 5.032 
                 6.511 
                 10.036 
               
               
                 the front side 
               
               
                 principle point) 
               
               
                 (The position of 
                 −3.308 
                 −5.634 
                 −10.298 
               
               
                 the rear side 
               
               
                 principle point) 
               
               
                   
               
             
          
           
               
                   
                 Lens 
                 The first surface of lens 
                 Focal length of lens 
               
               
                   
                   
               
               
                   
                 L11 
                 1 
                 −3.332 
               
               
                   
                 L12 
                 3 
                 8.164 
               
               
                   
                 L21 
                 5 
                 3.517 
               
               
                   
                 L22 
                 8 
                 4.149 
               
               
                   
                 L23 
                 9 
                 −3.790 
               
               
                   
                 L3 
                 11 
                 −5.774 
               
               
                   
                 L4 
                 13 
                 8.624 
               
               
                   
                   
               
             
          
           
               
                 Zoom lens group data 
               
             
          
           
               
                   
                   
                   
                   
                 The position 
                 The position 
               
               
                   
                 The first surface 
                 Focal length 
                 Composition length 
                 of the front side 
                 of the rear side 
               
               
                 Group 
                 of lens group 
                 of lens group 
                 of lens group 
                 principle point 
                 principle point 
               
               
                   
               
               
                 G1 
                 1 
                 −5.700 
                 1.773 
                 0.255 
                 −0.878 
               
               
                 G2 
                 5 
                 3.194 
                 2.403 
                 −0.286 
                 −1.548 
               
               
                 G3 
                 11 
                 −5.774 
                 0.450 
                 0.011 
                 −0.242 
               
               
                 G4 
                 13 
                 8.624 
                 0.884 
                 0.706 
                 0.263 
               
               
                   
               
             
          
           
               
                 Magnification of lens group 
               
             
          
           
               
                   
                 Wide angle end 
                 Middle 
                 Telephoto end 
               
             
          
           
               
                   
                 Wide angle end 
                 Middle 
                 Telephoto end 
                 (focusing on a close range) 
               
               
                   
                   
               
             
          
           
               
                 G1 
                 0.000 
                 0.000 
                 0.000 
                 0.054 
                 0.011 
                 0.007 
               
               
                 G2 
                 −0.454 
                 −0.602 
                 −0.969 
                 −0.539 
                 −0.604 
                 −0.889 
               
               
                 G3 
                 1.555 
                 1.867 
                 2.066 
                 1.705 
                 1.859 
                 2.055 
               
               
                 G4 
                 0.932 
                 0.939 
                 0.937 
                 0.930 
                 0.937 
                 0.934 
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
             
               
             
               
             
               
             
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
               
               
               
             
               
               
               
               
             
               
             
               
               
               
               
               
               
             
               
             
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
               
               
               
             
           
               
                   
               
               
                 Numerical embodiment 5 
               
               
                   
               
             
             
               
                 Surface data 
               
               
                 Unit: mm 
               
             
          
           
               
                   
                 Surface No. 
                 r 
                 d 
                 nd 
                 vd 
                 Effective diameter 
               
               
                   
                   
               
               
                   
                 Object plane 
                 ∞ 
                 ∞ 
               
               
                   
                  1* 
                 32.5181 
                 0.4000 
                 1.90270 
                 31.00 
                 2.662 
               
               
                   
                  2* 
                 2.6859 
                 0.5841 
                   
                   
                 2.171 
               
               
                   
                  3* 
                 3.9263 
                 0.7023 
                 2.10223 
                 16.77 
                 2.172 
               
               
                   
                  4* 
                 6.3251 
                 D4 
                   
                   
                 2.123 
               
               
                   
                  5* 
                 2.3656 
                 0.8877 
                 1.59201 
                 67.02 
                 1.442 
               
               
                   
                  6* 
                 −19.9615 
                 0.2000 
                   
                   
                 1.349 
               
               
                   
                  7 (Stop) 
                 ∞ 
                 0.0000 
                   
                   
                 (Variable) 
               
               
                   
                  8* 
                 5.6753 
                 0.6799 
                 1.85135 
                 40.10 
                 1.257 
               
               
                   
                  9 
                 −8.0095 
                 0.5194 
                 1.82114 
                 24.06 
                 1.184 
               
               
                   
                 10* 
                 5.2122 
                 D10 
                   
                   
                 1.039 
               
               
                   
                 11* 
                 −4.7550 
                 0.4439 
                 1.77377 
                 47.17 
                 1.186 
               
               
                   
                 12* 
                 241.8954 
                 D12 
                   
                   
                 1.373 
               
               
                   
                 13* 
                 −17.7526 
                 1.0020 
                 1.58347 
                 30.25 
                 2.836 
               
               
                   
                 14* 
                 −5.0153 
                 0.2387 
                   
                   
                 2.891 
               
               
                   
                 15 
                 ∞ 
                 0.3000 
                 1.51633 
                 64.14 
                 2.940 
               
               
                   
                 16 
                 ∞ 
                 0.4000 
                   
                   
                 2.952 
               
               
                   
                 Image plane 
                 ∞ 
               
               
                   
                   
               
             
          
           
               
                 Aspherical surface data 
               
               
                   
               
             
          
           
               
                 The first surface 
               
               
                 K = −204.602, A4 = 6.38593e−03, A6 = −1.31082e−03, A8 = 6.42265e−05 
               
               
                 The second surface 
               
               
                 K = 0.140, A4 = −4.48908e−03, A6 = 3.35561e−03, A8 = −7.34479e−04 
               
               
                 The third surface 
               
               
                 K = −8.010, A4 = −3.58577e−03, A6 = 2.02086e−03, A8 = −1.85601e−04 
               
               
                 The fourth surface 
               
               
                 K = −27.131, A4 = −5.26129e−03, A6 = 7.06800e−04, A8 = −5.83212e−05 
               
               
                 The fifth surface 
               
               
                 K = −1.224, A4 = −2.34964e−04, A6 = 1.88499e−08, A8 = 3.10798e−04 
               
               
                 The sixth surface 
               
               
                 K = −178.856, A4 = −8.86247e−03, A6 = 7.81377e−03, A8 = −1.53400e−03 
               
               
                 The eighth surface 
               
               
                 K = 6.120, A4 = 6.42116e−03, A6 = 8.89900e−03, A8 = −2.09052e−03 
               
               
                 The tenth surface 
               
               
                 K = −38.050, A4 = 7.18749e−02, A6 = −5.04696e−03, A8 = 1.13516e−02 
               
               
                 The eleventh surface 
               
               
                 K = −3.616, A4 = 4.31319e−03, A6 = −1.04263e−03, A8 = −3.53360e−03 
               
               
                 The twelfth surface 
               
               
                 K = −1640587.993, A4 = 1.96904e−02, A6 = −4.81382e−03, A8 = −6.24215e−04 
               
               
                 The thirteenth surface 
               
               
                 K = −5.864, A4 = 5.12076e−03, A6 = −1.13861e−03, A8 = 9.49549e−05 
               
               
                 The fourteenth surface 
               
               
                 K = −2.353, A4 = 1.15022e−02, A6 = −2.74683e−03, A8 = 1.78418e−04 
               
               
                   
               
             
          
           
               
                 Various data 
               
               
                 Zoom ratio: 2.850 
               
             
          
           
               
                   
                 Wide angle end 
                 Middle 
                 Telephoto end 
               
             
          
           
               
                   
                 Wide angle end 
                 Middle 
                 Telephoto end 
                 (focusing on a close range) 
               
               
                   
                   
               
             
          
           
               
                 Focal length 
                 3.754 
                 6.038 
                 10.699 
                   
                   
                   
               
               
                 F No. 
                 3.200 
                 4.349 
                 5.200 
               
               
                 Angle of view 2ω 
                 82.899 
                 51.642 
                 29.898 
               
               
                 Image height 
                 2.900 
                 2.900 
                 2.900 
               
               
                 BF 
                 0.837 
                 0.837 
                 0.837 
               
               
                 Lens total length 
                 12.898 
                 12.898 
                 12.898 
               
               
                 (Distance from an 
                 ∞ 
                 ∞ 
                 ∞ 
                 100.00 
                 500.00 
                 800.00 
               
               
                 object point) 
               
               
                 D4 
                 3.975 
                 2.194 
                 0.200 
                 3.198 
                 2.281 
                 0.556 
               
               
                 D10 
                 1.006 
                 0.888 
                 1.793 
                 0.960 
                 0.900 
                 1.528 
               
               
                 D12 
                 1.661 
                 3.560 
                 4.649 
                 2.484 
                 3.461 
                 4.557 
               
               
                 Stop diameter 
                 1.033 
                 1.033 
                 1.255 
               
               
                 (Entrance pupil 
                 3.012 
                 2.510 
                 1.668 
               
               
                 position) 
               
               
                 (Exit pupil 
                 −5.919 
                 −11.530 
                 −20.540 
               
               
                 position) 
               
               
                 (The position of 
                 4.693 
                 5.589 
                 7.011 
               
               
                 the front side 
               
               
                 principle point) 
               
               
                 (The position of 
                 −3.312 
                 −5.686 
                 −10.305 
               
               
                 the rear side 
               
               
                 principle point) 
               
               
                   
               
             
          
           
               
                   
                 Lens 
                 The first surface of lens 
                 Focal length of lens 
               
               
                   
                   
               
               
                   
                 L11 
                 1 
                 −3.264 
               
               
                   
                 L12 
                 3 
                 8.143 
               
               
                   
                 L21 
                 5 
                 3.626 
               
               
                   
                 L22 
                 8 
                 3.993 
               
               
                   
                 L23 
                 9 
                 −3.778 
               
               
                   
                 L3 
                 11 
                 −6.022 
               
               
                   
                 L4 
                 13 
                 11.643 
               
               
                   
                   
               
             
          
           
               
                 Zoom lens group data 
               
             
          
           
               
                   
                   
                   
                   
                 The position 
                 The position 
               
               
                   
                 The first surface 
                 Focal length 
                 Composition length 
                 of the front side 
                 of the rear side 
               
               
                 Group 
                 of lens group 
                 of lens group 
                 of lens group 
                 principle point 
                 principle point 
               
               
                   
               
               
                 G1 
                 1 
                 −5.552 
                 1.686 
                 0.169 
                 −0.919 
               
               
                 G2 
                 5 
                 3.194 
                 2.287 
                 −0.238 
                 −1.451 
               
               
                 G3 
                 11 
                 −6.022 
                 0.444 
                 0.005 
                 −0.245 
               
               
                 G4 
                 13 
                 11.643 
                 1.002 
                 0.857 
                 0.242 
               
               
                   
               
             
          
           
               
                 Magnification of lens group 
               
             
          
           
               
                   
                 Wide angle end 
                 Middle 
                 Telephoto end 
               
             
          
           
               
                   
                 Wide angle end 
                 Middle 
                 Telephoto end 
                 (focusing on a lose range) 
               
               
                   
                   
               
             
          
           
               
                 G1 
                 0.000 
                 0.000 
                 0.000 
                 0.053 
                 0.011 
                 0.007 
               
               
                 G2 
                 −0.455 
                 −0.610 
                 −0.986 
                 −0.537 
                 −0.607 
                 −0.898 
               
               
                 G3 
                 1.571 
                 1.870 
                 2.058 
                 1.707 
                 1.857 
                 2.046 
               
               
                 G4 
                 0.945 
                 0.953 
                 0.949 
                 0.945 
                 0.951 
                 0.948 
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
             
               
             
               
             
               
             
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
               
               
               
             
               
               
               
               
             
               
             
               
               
               
               
               
               
             
               
             
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
               
               
               
             
           
               
                   
               
               
                 Numerical embodiment 6 
               
               
                   
               
             
             
               
                 Surface data 
               
               
                 Unit: mm 
               
             
          
           
               
                   
                 Surface No. 
                 r 
                 d 
                 nd 
                 vd 
                 Effective diameter 
               
               
                   
                   
               
               
                   
                 Object plane 
                 ∞ 
                 ∞ 
               
               
                   
                  1* 
                 −20.7452 
                 0.4261 
                 1.76802 
                 49.24 
                 2.660 
               
               
                   
                  2* 
                 3.1798 
                 0.5199 
                 1.63387 
                 23.38 
                 2.348 
               
               
                   
                  3* 
                 11.5413 
                 D3 
                   
                   
                 2.338 
               
               
                   
                  4* 
                 2.5653 
                 0.7836 
                 1.69350 
                 53.21 
                 1.425 
               
               
                   
                  5* 
                 −9.5581 
                 0.1627 
                   
                   
                 1.304 
               
               
                   
                  6 (Stop) 
                 ∞ 
                 0.1056 
                   
                   
                 (Variable) 
               
               
                   
                  7* 
                 6.3829 
                 0.6523 
                 1.77377 
                 47.17 
                 1.173 
               
               
                   
                  8 
                 −16.9084 
                 0.3267 
                 1.84666 
                 23.78 
                 1.048 
               
               
                   
                  9* 
                 2.6017 
                 D9 
                   
                   
                 0.900 
               
               
                   
                 10* 
                 −1663.5639 
                 0.3688 
                 1.58913 
                 61.14 
                 1.798 
               
               
                   
                 11* 
                 9.0733 
                 D11 
                   
                   
                 1.796 
               
               
                   
                 12* 
                 −6.3464 
                 0.7403 
                 1.82114 
                 24.06 
                 2.400 
               
               
                   
                 13* 
                 −4.0025 
                 2.2296 
                   
                   
                 2.510 
               
               
                   
                 14 
                 ∞ 
                 0.3796 
                 1.51633 
                 64.14 
                 2.909 
               
               
                   
                 15 
                 ∞ 
                 0.4000 
                   
                   
                 2.942 
               
               
                   
                 Image plane 
                 ∞ 
               
               
                   
                   
               
             
          
           
               
                 Aspherical surface data 
               
               
                   
               
             
          
           
               
                 The first surface 
               
               
                 K = −375.930, A4 = −1.15480e−02, A6 = 2.04009e−03, A8 = −2.03463e−04, A10 = 9.13064e−06 
               
               
                 The second surface 
               
               
                 K = −4.767 
               
               
                 The third surface 
               
               
                 K = 19.716, A4 = −8.39462e−03, A6 = 8.74090e−04, A8 = −3.16644e−05, A10 = −9.08607e−06 
               
               
                 The fourth surface 
               
               
                 K = −5.335, A4 = 3.95944e−02, A6 = −2.26682e−04, A8 = 1.21738e−03, A10 = −2.20735e−05 
               
               
                 The fifth surface 
               
               
                 K = −1.433, A4 = 5.93078e−02, A6 = −5.72481e−03, A8 = −1.00605e−05, A10 = −4.23420e−04 
               
               
                 The seventh surface 
               
               
                 K = 0.827, A4 = 7.27634e−02, A6 = −2.04547e−02, A8 = 2.74787e−03, A10 = −2.76359e−03 
               
               
                 The ninth surface 
               
               
                 K = 3.493, A4 = 2.51814e−02, A6 = −9.44244e−04, A8 = −9.17287e−03, A10 = −9.81182e−03 
               
               
                 The tenth surface 
               
               
                 K = 0.000, A4 = 2.01378e−02, A6 = 2.42834e−03, A8 = −6.61110e−04, A10 = −3.37042e−05 
               
               
                 The eleventh surface 
               
               
                 K = 1.229, A4 = 2.32836e−02, A6 = 1.19434e−03, A8 = −5.32506e−05, A10 = −1.30308e−04 
               
               
                 The twelfth surface 
               
               
                 K = −0.983, A4 = 3.30959e−03, A6 = 1.23354e−04 
               
               
                 The thirteenth surface 
               
               
                 K = −1.651, A4 = −8.10707e−05, A6 = −7.42911e−05, A8 = 1.46926e−05 
               
               
                   
               
             
          
           
               
                 Various data 
               
               
                 Zoom ratio: 2.849 
               
             
          
           
               
                   
                 Wide angle end 
                 Middle 
                 Telephoto end 
               
             
          
           
               
                   
                 Wide angle end 
                 Middle 
                 Telephoto end 
                 (focusing on a close range) 
               
               
                   
                   
               
             
          
           
               
                 Focal length 
                 4.521 
                 7.640 
                 12.880 
                   
                   
                   
               
               
                 F No. 
                 3.757 
                 4.702 
                 5.248 
               
               
                 Angle of view 2ω 
                 70.567 
                 43.949 
                 24.572 
               
               
                 Image height 
                 2.900 
                 2.900 
                 2.900 
               
               
                 BF 
                 2.880 
                 2.880 
                 2.880 
               
               
                 Lens total length 
                 13.374 
                 13.371 
                 13.371 
               
               
                 (Distance from an 
                 ∞ 
                 ∞ 
                 ∞ 
                 100.00 
                 500.00 
                 800.00 
               
               
                 object point) 
               
               
                 D3 
                 4.604 
                 2.224 
                 0.151 
                 3.959 
                 2.507 
                 0.265 
               
               
                 D9 
                 0.557 
                 0.550 
                 4.157 
                 0.781 
                 0.549 
                 4.037 
               
               
                 D11 
                 1.247 
                 3.631 
                 2.097 
                 1.665 
                 3.348 
                 2.102 
               
               
                 Stop diameter 
                 0.900 
                 1.000 
                 1.250 
               
               
                 (Entrance pupil 
                 3.479 
                 2.544 
                 1.301 
               
               
                 position) 
               
               
                 (Exit pupil 
                 −4.272 
                 −11.074 
                 −15.850 
               
               
                 position) 
               
               
                 (The position of 
                 5.143 
                 6.001 
                 5.324 
               
               
                 the front side 
               
               
                 principle point) 
               
               
                 (The position of 
                 −4.121 
                 −7.240 
                 −12.480 
               
               
                 the rear side 
               
               
                 principle point) 
               
               
                   
               
             
          
           
               
                   
                 Lens 
                 The first surface of lens 
                 Focal length of lens 
               
               
                   
                   
               
               
                   
                 L11 
                 1 
                 −3.562 
               
               
                   
                 L12 
                 2 
                 6.761 
               
               
                   
                 L21 
                 4 
                 2.996 
               
               
                   
                 L22 
                 7 
                 6.063 
               
               
                   
                 L23 
                 8 
                 −2.643 
               
               
                   
                 L3 
                 10 
                 −15.316 
               
               
                   
                 L4 
                 12 
                 11.553 
               
               
                   
                   
               
             
          
           
               
                 Zoom lens group data 
               
             
          
           
               
                   
                   
                   
                   
                 The position 
                 The position 
               
               
                   
                 The first surface 
                 Focal length 
                 Composition length 
                 of the front side 
                 of the rear side 
               
               
                 Group 
                 of lens group 
                 of lens group 
                 of lens group 
                 principle point 
                 principle point 
               
               
                   
               
               
                 G1 
                 1 
                 −7.332 
                 0.946 
                 0.301 
                 −0.251 
               
               
                 G2 
                 4 
                 4.054 
                 2.031 
                 −1.113 
                 −1.816 
               
               
                 G3 
                 10 
                 −15.316 
                 0.369 
                 0.231 
                 −0.001 
               
               
                 G4 
                 12 
                 11.553 
                 3.349 
                 0.963 
                 −1.872 
               
               
                   
               
             
          
           
               
                 Magnification of lens group 
               
             
          
           
               
                   
                 Wide angle end 
                 Middle 
                 Telephoto end 
               
             
          
           
               
                   
                 Wide angle end 
                 Middle 
                 Telephoto end 
                 (focusing on a close range) 
               
               
                   
                   
               
             
          
           
               
                 G1 
                 0.000 
                 0.000 
                 0.000 
                 0.068 
                 0.014 
                 0.009 
               
               
                 G2 
                 −0.578 
                 −0.874 
                 −1.579 
                 −0.690 
                 −0.842 
                 −1.551 
               
               
                 G3 
                 1.329 
                 1.485 
                 1.385 
                 1.356 
                 1.466 
                 1.385 
               
               
                 G4 
                 0.803 
                 0.803 
                 0.803 
                 0.803 
                 0.803 
                 0.803 
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
               
               
             
               
             
               
               
             
               
             
               
               
               
               
             
               
               
               
               
             
               
             
               
               
               
               
               
               
             
               
             
               
               
               
               
               
             
           
               
                   
               
               
                 Numerical embodiment 7 
               
               
                   
               
             
             
               
                 Surface data 
               
               
                 Unit: mm 
               
             
          
           
               
                 Surface No. 
                 r 
                 d 
                 nd 
                 vd 
                 Effective diameter 
               
               
                   
               
               
                 Object plane 
                 ∞ 
                 ∞ 
               
               
                  1* 
                 −11.1356 
                 0.4975 
                 1.76802 
                 49.24 
                 2.436 
               
               
                  2* 
                 4.3635 
                 0.2782 
                 1.68086 
                 20.03 
                 2.161 
               
               
                  3* 
                 13.0403 
                 D3 
                   
                   
                 2.143 
               
               
                  4* 
                 2.4354 
                 1.1497 
                 1.69350 
                 53.21 
                 1.500 
               
               
                  5* 
                 −10.2972 
                 0.0965 
                   
                   
                 1.338 
               
               
                  6 (Stop) 
                 ∞ 
                 0.1122 
                   
                   
                 (Variable) 
               
               
                  7* 
                 6.9467 
                 0.6344 
                 1.76802 
                 49.24 
                 1.209 
               
               
                  8 
                 −10.0264 
                 0.3871 
                 1.84666 
                 23.78 
                 1.142 
               
               
                  9* 
                 2.7369 
                 D9 
                   
                   
                 1.053 
               
               
                 10* 
                 −1593.9529 
                 0.4096 
                 1.58913 
                 61.14 
                 1.938 
               
               
                 11* 
                 7.2355 
                 D11 
                   
                   
                 1.949 
               
               
                 12* 
                 −7.0449 
                 1.0575 
                 1.92286 
                 20.88 
                 2.686 
               
               
                 13* 
                 −3.9192 
                 0.8750 
                   
                   
                 2.866 
               
               
                 14 
                 ∞ 
                 0.3796 
                 1.51633 
                 64.14 
                 2.965 
               
               
                 15 
                 ∞ 
                 0.4000 
                   
                   
                 2.977 
               
               
                 Image plane 
                 ∞ 
               
               
                   
               
             
          
           
               
                 Aspherical surface data 
               
               
                   
               
             
          
           
               
                   
                 The first surface 
               
               
                   
                 K = −43.263, A4 = −1.24712e−02, A6 = 1.73364e−03, A8 = −6.78483e−05 
               
               
                   
                 The second surface 
               
               
                   
                 K = −13.143 
               
               
                   
                 The third surface 
               
               
                   
                 K = 11.728, A4 = −8.61588e−03, A6 = 1.31833e−03, A8 = 3.44649e−05 
               
               
                   
                 The fourth surface 
               
               
                   
                 K = −1.810, A4 = 1.47470e−02, A6 = 1.63827e−03, A8 = 1.52183e−04 
               
               
                   
                 The fifth surface 
               
               
                   
                 K = 10.732, A4 = 1.76981e−02, A6 = −3.95829e−03, A8 = 3.18646e−04 
               
               
                   
                 The seventh surface 
               
               
                   
                 K = −0.652, A4 = 6.62402e−03, A6 = −8.64488e−03, A8 = −8.05063e−04 
               
               
                   
                 The ninth surface 
               
               
                   
                 K = 3.404, A4 = −4.48263e−03, A6 = −5.81944e−03, A8 = −5.86170e−03 
               
               
                   
                 The tenth surface 
               
               
                   
                 K = 0.000, A4 = 6.85162e−03, A6 = 5.94218e−04 
               
               
                   
                 The eleventh surface 
               
               
                   
                 K = 1.002, A4 = 8.32696e−03, A6 = 6.31067e−04, A8 = −4.05088e−05 
               
               
                   
                 The twelfth surface 
               
               
                   
                 K = −4.931 
               
               
                   
                 The thirteenth surface 
               
               
                   
                 K = −3.651, A4 = −2.91002e−03, A6 = −8.23036e−05, A8 = −3.97711e−07 
               
               
                   
                   
               
             
          
           
               
                 Various data 
               
               
                 Zoom ratio: 2.846 
               
             
          
           
               
                   
                 Wide angle end 
                 Middle 
                 Telephoto end 
               
               
                   
               
               
                 Focal length 
                 4.526 
                 7.640 
                 12.878 
               
               
                 F No. 
                 3.590 
                 5.050 
                 5.131 
               
               
                 Angle of view 2ω 
                 72.782 
                 43.815 
                 24.659 
               
               
                 Image height 
                 2.900 
                 2.900 
                 2.900 
               
               
                 BF 
                 1.525 
                 1.525 
                 1.525 
               
               
                 Lens total length 
                 12.867 
                 12.873 
                 12.875 
               
               
                 Distance from an object point 
                 ∞ 
                 ∞ 
                 ∞ 
               
               
                 D3 
                 4.309 
                 2.213 
                 0.196 
               
               
                 D9 
                 1.296 
                 1.055 
                 4.311 
               
               
                 D11 
                 1.114 
                 3.457 
                 2.220 
               
               
                 Stop diameter 
                 0.900 
                 0.900 
                 1.250 
               
               
                 Entrance pupil position 
                 3.292 
                 2.510 
                 1.389 
               
               
                 Exit pupil position 
                 −7.123 
                 −23.689 
                 −42.013 
               
               
                 The position of the front side principle point 
                 5.450 
                 7.835 
                 10.458 
               
               
                 The position of the rear side principle point 
                 −4.122 
                 −7.242 
                 −12.481 
               
               
                   
               
             
          
           
               
                   
                 Lens 
                 The first surface of lens 
                 Focal length of lens 
               
               
                   
                   
               
               
                   
                 L11 
                 1 
                 −4.026 
               
               
                   
                 L12 
                 2 
                 9.508 
               
               
                   
                 L21 
                 4 
                 2.949 
               
               
                   
                 L22 
                 7 
                 5.431 
               
               
                   
                 L23 
                 8 
                 −2.505 
               
               
                   
                 L3 
                 10 
                 −12.225 
               
               
                   
                 L4 
                 12 
                 8.235 
               
               
                   
                   
               
             
          
           
               
                 Zoom lens group data 
               
             
          
           
               
                   
                 The first 
                   
                 Composition 
                 The position 
                 The position 
               
               
                   
                 surface 
                 Focal length 
                 length 
                 of the front side 
                 of the rear side 
               
               
                 Group 
                 of lens group 
                 of lens group 
                 of lens group 
                 principle point 
                 principle point 
               
               
                   
               
               
                 G1 
                 1 
                 −6.977 
                 0.776 
                 0.202 
                 −0.239 
               
               
                 G2 
                 4 
                 3.819 
                 2.380 
                 −1.106 
                 −1.918 
               
               
                 G3 
                 10 
                 −12.225 
                 0.410 
                 0.257 
                 −0.001 
               
               
                 G4 
                 12 
                 8.235 
                 2.312 
                 1.066 
                 −0.532 
               
               
                   
               
             
          
           
               
                 Magnification of lens group 
               
             
          
           
               
                   
                   
                 Wide angle end 
                 Middle 
                 Telephoto end 
               
               
                   
                   
               
               
                   
                 G1 
                 0.000 
                 0.000 
                 0.000 
               
               
                   
                 G2 
                 −0.579 
                 −0.848 
                 −1.536 
               
               
                   
                 G3 
                 1.265 
                 1.456 
                 1.355 
               
               
                   
                 G4 
                 0.886 
                 0.887 
                 0.887 
               
               
                   
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
               
               
             
               
             
               
               
             
               
             
               
               
               
               
             
               
               
               
               
             
               
             
               
               
               
               
               
               
             
               
             
               
               
               
               
               
             
           
               
                   
               
               
                 Numerical embodiment 8 
               
               
                   
               
             
             
               
                 Surface data 
               
               
                 Unit: mm 
               
             
          
           
               
                 Surface No. 
                 r 
                 d 
                 nd 
                 vd 
                 Effective diameter 
               
               
                   
               
               
                 Object plane 
                 ∞ 
                 ∞ 
               
               
                  1* 
                 −5.6531 
                 0.5000 
                 1.58913 
                 61.14 
                 2.376 
               
               
                  2 
                 9.8000 
                 0.2500 
                 1.63494 
                 23.22 
                 2.159 
               
               
                  3* 
                 16.8078 
                 D3 
                   
                   
                 2.137 
               
               
                  4* 
                 2.9908 
                 0.9110 
                 1.85135 
                 40.10 
                 1.500 
               
               
                  5* 
                 −31.7144 
                 0.1000 
                   
                   
                 1.355 
               
               
                  6 (Stop) 
                 ∞ 
                 0.1518 
                   
                   
                 (Variable) 
               
               
                  7* 
                 6.6515 
                 0.7500 
                 1.76802 
                 49.24 
                 1.268 
               
               
                  8 
                 −4.5000 
                 0.4000 
                 1.84666 
                 23.78 
                 1.185 
               
               
                  9* 
                 4.8113 
                 D9 
                   
                   
                 1.058 
               
               
                 10 
                 44.8931 
                 0.5000 
                 1.81474 
                 37.03 
                 1.181 
               
               
                 11* 
                 3.8464 
                 D11 
                   
                   
                 1.268 
               
               
                 12 
                 −32.1035 
                 1.3000 
                 1.82114 
                 24.06 
                 2.886 
               
               
                 13* 
                 −4.0000 
                 0.4217 
                   
                   
                 3.000 
               
               
                 14 
                 ∞ 
                 0.3796 
                 1.51633 
                 64.14 
                 2.961 
               
               
                 15 
                 ∞ 
                 0.2900 
                   
                   
                 2.957 
               
               
                 Image plane 
                 ∞ 
               
               
                   
               
             
          
           
               
                 Aspherical surface data 
               
               
                   
               
             
          
           
               
                   
                 The first surface 
               
               
                   
                 K = −1.000, A4 = −3.19351e−03, A6 = 2.63420e−04, A8 = 1.16363e−05 
               
               
                   
                 The third surface 
               
               
                   
                 K = −1.000, A4 = −2.77149e−03, A6 = 2.14794e−04, A8 = 5.18982e−05 
               
               
                   
                 The fourth surface 
               
               
                   
                 K = −1.000, A4 = 1.88539e−03, A6 = 3.55638e−04 
               
               
                   
                 The fifth surface 
               
               
                   
                 K = −1.000, A4 = 5.68662e−03, A6 = −7.24378e−04 
               
               
                   
                 The seventh surface 
               
               
                   
                 K = 0.000, A4 = 1.78188e−02, A6 = −8.88828e−04 
               
               
                   
                 The ninth surface 
               
               
                   
                 K = 11.588, A4 = 2.35797e−02, A6 = −1.00377e−04, A8 = 1.61425e−04 
               
               
                   
                 The eleventh surface 
               
               
                   
                 K = 0.000, A4 = −1.32598e−03, A6 = 2.94203e−04, A8 = −3.40185e−04 
               
               
                   
                 The thirteenth surface 
               
               
                   
                 K = −1.000, A4 = 5.47362e−03, A6 = −6.78501e−04, A8 = 2.75097e−05 
               
               
                   
                   
               
             
          
           
               
                 Various data 
               
               
                 Zoom ratio: 2.821 
               
             
          
           
               
                   
                 Wide angle end 
                 Middle 
                 Telephoto end 
               
               
                   
               
               
                 Focal length 
                 4.570 
                 7.640 
                 12.890 
               
               
                 F No. 
                 3.137 
                 4.383 
                 5.181 
               
               
                 Angle of view 2ω 
                 72.832 
                 41.710 
                 24.978 
               
               
                 Image height 
                 2.900 
                 2.900 
                 2.900 
               
               
                 BF 
                 0.962 
                 0.962 
                 0.962 
               
               
                 Lens total length 
                 12.871 
                 12.871 
                 12.871 
               
               
                 Distance from an object point 
                 ∞ 
                 ∞ 
                 ∞ 
               
               
                 D3 
                 4.231 
                 2.167 
                 0.172 
               
               
                 D9 
                 0.758 
                 0.721 
                 1.542 
               
               
                 D11 
                 2.057 
                 4.157 
                 5.332 
               
               
                 Stop diameter 
                 1.100 
                 1.100 
                 1.300 
               
               
                 Entrance pupil position 
                 3.165 
                 2.343 
                 1.167 
               
               
                 Exit pupil position 
                 −17.126 
                 44.144 
                 18.147 
               
               
                 The position of the front side principle point 
                 6.580 
                 11.335 
                 23.725 
               
               
                 The position of the rear side principle point 
                 −4.280 
                 −7.350 
                 −12.600 
               
               
                   
               
             
          
           
               
                   
                 Lens 
                 The first surface of lens 
                 Focal length of lens 
               
               
                   
                   
               
               
                   
                 L11 
                 1 
                 −6.013 
               
               
                   
                 L12 
                 2 
                 36.513 
               
               
                   
                 L21 
                 4 
                 3.249 
               
               
                   
                 L22 
                 7 
                 3.600 
               
               
                   
                 L23 
                 8 
                 −2.693 
               
               
                   
                 L3 
                 10 
                 −5.192 
               
               
                   
                 L4 
                 12 
                 5.451 
               
               
                   
                   
               
             
          
           
               
                 Zoom lens group data 
               
             
          
           
               
                   
                 The first 
                   
                 Composition 
                 The position 
                 The position 
               
               
                   
                 surface 
                 Focal length 
                 length 
                 of the front side 
                 of the rear side 
               
               
                 Group 
                 of lens group 
                 of lens group 
                 of lens group 
                 principle point 
                 principle point 
               
               
                   
               
               
                 G1 
                 1 
                 −7.196 
                 0.750 
                 0.116 
                 −0.345 
               
               
                 G2 
                 4 
                 3.400 
                 2.313 
                 −0.522 
                 −1.599 
               
               
                 G3 
                 10 
                 −5.192 
                 0.500 
                 0.303 
                 0.026 
               
               
                 G4 
                 12 
                 5.451 
                 2.101 
                 0.799 
                 −0.573 
               
               
                   
               
             
          
           
               
                 Magnification of lens group 
               
             
          
           
               
                   
                   
                 Wide angle end 
                 Middle 
                 Telephoto end 
               
               
                   
                   
               
               
                   
                 G1 
                 0.000 
                 0.000 
                 0.000 
               
               
                   
                 G2 
                 −0.433 
                 −0.588 
                 −0.897 
               
               
                   
                 G3 
                 1.742 
                 2.147 
                 2.373 
               
               
                   
                 G4 
                 0.842 
                 0.842 
                 0.842 
               
               
                   
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
               
               
             
               
             
               
               
             
               
             
               
               
               
               
             
               
               
               
               
             
               
             
               
               
               
               
               
               
             
               
             
               
               
               
               
               
             
           
               
                   
               
               
                 Numerical embodiment 9 
               
               
                   
               
             
             
               
                 Surface data 
               
               
                 Unit: mm 
               
             
          
           
               
                 Surface No. 
                 r 
                 d 
                 nd 
                 vd 
                 Effective diameter 
               
               
                   
               
               
                 Object plane 
                 ∞ 
                 ∞ 
               
               
                  1* 
                 −19.3192 
                 0.4988 
                 1.76802 
                 49.24 
                 2.583 
               
               
                  2* 
                 3.7522 
                 0.4683 
                 1.63387 
                 23.38 
                 2.307 
               
               
                  3* 
                 11.3663 
                 D3 
                   
                   
                 2.287 
               
               
                  4* 
                 2.6413 
                 0.8593 
                 1.69350 
                 53.21 
                 1.473 
               
               
                  5* 
                 −9.2555 
                 0.1569 
                   
                   
                 1.381 
               
               
                  6 (Stop) 
                 ∞ 
                 0.1202 
                   
                   
                 (Variable) 
               
               
                  7* 
                 6.3775 
                 0.6959 
                 1.76802 
                 49.24 
                 1.175 
               
               
                  8 
                 −17.7460 
                 0.4046 
                 1.84666 
                 23.78 
                 1.035 
               
               
                  9* 
                 2.5760 
                 D9 
                   
                   
                 0.900 
               
               
                 10* 
                 −2150.7307 
                 0.4011 
                 1.58913 
                 61.14 
                 1.778 
               
               
                 11* 
                 8.3413 
                 D11 
                   
                   
                 1.857 
               
               
                 12* 
                 −7.5498 
                 0.9821 
                 1.82114 
                 24.06 
                 2.555 
               
               
                 13* 
                 −4.0253 
                 D13 
                   
                   
                 2.724 
               
               
                 14 
                 ∞ 
                 0.3796 
                 1.51633 
                 64.14 
                 2.944 
               
               
                 15 
                 ∞ 
                 0.4000 
                   
                   
                 2.966 
               
               
                 Image plane 
                 ∞ 
               
               
                   
               
             
          
           
               
                 Aspherical surface data 
               
               
                   
               
             
          
           
               
                   
                 The first surface 
               
               
                   
                 K = −55.536, A4 = −9.05980e−03, A6 = 1.14637e−03, A8 = −5.51750e−05 
               
               
                   
                 The second surface 
               
               
                   
                 K = −5.943 
               
               
                   
                 The third surface 
               
               
                   
                 K = 3.375, A4 = −8.51272e−03, A6 = 1.43185e−03, A8 = −6.70214e−05 
               
               
                   
                 The fourth surface 
               
               
                   
                 K = −2.992, A4 = 1.71186e−02, A6 = −1.00737e−03, A8 = 4.64173e−04 
               
               
                   
                 The fifth surface 
               
               
                   
                 K = −0.989, A4 = 8.39214e−03, A6 = −8.19252e−04, A8 = 2.03175e−04 
               
               
                   
                 The seventh surface 
               
               
                   
                 K = 0.788, A4 = 5.33856e−03, A6 = −1.80989e−03, A8 = −4.20069e−04 
               
               
                   
                 The ninth surface 
               
               
                   
                 K = 1.499, A4 = 6.20768e−03, A6 = −5.64490e−04, A8 = −1.23795e−04 
               
               
                   
                 The tenth surface 
               
               
                   
                 K = 958794.047, A4 = 1.75804e−03, A6 = 1.99109e−03, A8 = −4.76253e−04 
               
               
                   
                 The eleventh surface 
               
               
                   
                 K = −0.095, A4 = 3.52958e−03, A6 = 1.43329e−03, A8 = −3.48716e−04 
               
               
                   
                 The twelfth surface 
               
               
                   
                 K = −0.216 
               
               
                   
                 The thirteenth surface 
               
               
                   
                 K = −0.637, A4 = 4.61435e−04, A6 = −1.26973e−04, A8 = −1.09864e−06 
               
               
                   
                   
               
             
          
           
               
                 Various data 
               
               
                 Zoom ratio: 2.838 
               
             
          
           
               
                   
                 Wide angle end 
                 Middle 
                 Telephoto end 
               
               
                   
               
               
                 Focal length 
                 4.538 
                 7.640 
                 12.880 
               
               
                 F No. 
                 3.339 
                 4.651 
                 5.209 
               
               
                 Angle of view 2ω 
                 72.624 
                 42.356 
                 24.646 
               
               
                 Image height 
                 2.900 
                 2.900 
                 2.900 
               
               
                 BF 
                 2.364 
                 2.306 
                 2.176 
               
               
                 Lens total length 
                 13.369 
                 13.374 
                 13.368 
               
               
                 Distance from an object point 
                 ∞ 
                 ∞ 
                 ∞ 
               
               
                 D3 
                 4.526 
                 2.251 
                 0.195 
               
               
                 D9 
                 0.863 
                 0.766 
                 4.061 
               
               
                 D11 
                 1.028 
                 3.465 
                 2.347 
               
               
                 D13 
                 1.714 
                 1.655 
                 1.526 
               
               
                 Stop diameter 
                 1.000 
                 1.000 
                 1.250 
               
               
                 Entrance pupil position 
                 3.512 
                 2.608 
                 1.385 
               
               
                 Exit pupil position 
                 −5.255 
                 −15.770 
                 −27.703 
               
               
                 The position of the front side principle point 
                 5.348 
                 7.018 
                 8.713 
               
               
                 The position of the rear side principle point 
                 −4.136 
                 −7.241 
                 −12.480 
               
               
                   
               
             
          
           
               
                   
                 Lens 
                 The first surface of lens 
                 Focal length of lens 
               
               
                   
                   
               
               
                   
                 L11 
                 1 
                 −4.053 
               
               
                   
                 L12 
                 2 
                 8.631 
               
               
                   
                 L21 
                 4 
                 3.053 
               
               
                   
                 L22 
                 7 
                 6.186 
               
               
                   
                 L23 
                 8 
                 −2.633 
               
               
                   
                 L3 
                 10 
                 −14.103 
               
               
                   
                 L4 
                 12 
                 9.329 
               
               
                   
                   
               
             
          
           
               
                 Zoom lens group data 
               
             
          
           
               
                   
                 The first 
                   
                 Composition 
                 The position 
                 The position 
               
               
                   
                 surface 
                 Focal length 
                 length 
                 of the front side 
                 of the rear side 
               
               
                 Group 
                 of lens group 
                 of lens group 
                 of lens group 
                 principle point 
                 principle point 
               
               
                   
               
               
                 G1 
                 1 
                 −7.490 
                 0.967 
                 0.314 
                 −0.247 
               
               
                 G2 
                 4 
                 4.036 
                 2.237 
                 −1.225 
                 −1.943 
               
               
                 G3 
                 10 
                 −14.103 
                 0.401 
                 0.251 
                 −0.001 
               
               
                 G4 
                 12 
                 9.329 
                 0.982 
                 1.026 
                 0.547 
               
               
                   
               
             
          
           
               
                 Magnification of lens group 
               
             
          
           
               
                   
                   
                 Wide angle end 
                 Middle 
                 Telephoto end 
               
               
                   
                   
               
               
                   
                 G1 
                 0.000 
                 0.000 
                 0.000 
               
               
                   
                 G2 
                 −0.576 
                 −0.854 
                 −1.511 
               
               
                   
                 G3 
                 1.306 
                 1.472 
                 1.379 
               
               
                   
                 G4 
                 0.805 
                 0.812 
                 0.825 
               
               
                   
                 G5 
                 1.000 
                 1.000 
                 1.000 
               
               
                   
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
               
               
             
               
             
               
               
             
               
             
               
               
               
               
             
               
               
               
               
             
               
             
               
               
               
               
               
               
             
               
             
               
               
               
               
               
             
           
               
                   
               
               
                 Numerical embodiment 10 
               
               
                   
               
             
             
               
                 Surface data 
               
               
                 Unit: mm 
               
             
          
           
               
                 Surface No. 
                 r 
                 d 
                 nd 
                 vd 
                 Effective diameter 
               
               
                   
               
               
                 Object plane 
                 ∞ 
                 ∞ 
               
               
                  1* 
                 −23.8081 
                 0.4000 
                 1.90270 
                 31.00 
                 1.764 
               
               
                  2* 
                 2.2994 
                 0.3419 
                   
                   
                 1.467 
               
               
                  3* 
                 2.8941 
                 0.4964 
                 2.10223 
                 16.77 
                 1.475 
               
               
                  4* 
                 5.3026 
                 D4 
                   
                   
                 1.420 
               
               
                  5* 
                 2.1625 
                 0.8676 
                 1.76802 
                 49.24 
                 1.201 
               
               
                  6* 
                 −6.2445 
                 0.1994 
                   
                   
                 1.091 
               
               
                  7 (Stop) 
                 ∞ 
                 0.0000 
                   
                   
                 (Variable) 
               
               
                  8* 
                 6.5304 
                 0.5563 
                 1.49700 
                 81.54 
                 0.943 
               
               
                  9 
                 −11.0440 
                 0.4000 
                 2.10223 
                 16.77 
                 0.864 
               
               
                 10* 
                 6.7381 
                 D10 
                   
                   
                 0.803 
               
               
                 11* 
                 −4.9986 
                 0.4000 
                 1.53071 
                 55.67 
                 0.915 
               
               
                 12* 
                 6.1075 
                 D12 
                   
                   
                 1.016 
               
               
                 13* 
                 −4.3703 
                 0.6790 
                 2.10223 
                 16.77 
                 2.225 
               
               
                 14* 
                 −2.6586 
                 0.3758 
                   
                   
                 2.242 
               
               
                 15 
                 ∞ 
                 0.3000 
                 1.51633 
                 64.14 
                 2.270 
               
               
                 16 
                 ∞ 
                 0.3500 
                   
                   
                 2.276 
               
               
                 Image plane 
                 ∞ 
               
               
                   
               
             
          
           
               
                 Aspherical surface data 
               
               
                   
               
             
          
           
               
                   
                 The first surface 
               
               
                   
                 K = −10.000, A4 = 4.48967e−03, A6 = −2.58325e−03, A8 = 2.70892e−04 
               
               
                   
                 The second surface 
               
               
                   
                 K = 0.528, A4 = −1.10216e−02, A6 = 6.00177e−03, A8 = −3.60751e−03 
               
               
                   
                 The third surface 
               
               
                   
                 K = −1.683, A4 = −1.62024e−02, A6 = 1.06888e−02, A8 = −2.57278e−03 
               
               
                   
                 The fourth surface 
               
               
                   
                 K = −10.000, A4 = −1.24411e−02, A6 = 6.03283e−03, A8 = −1.54408e−03 
               
               
                   
                 The fifth surface 
               
               
                   
                 K = −0.330, A4 = −6.66268e−03, A6 = −7.36472e−04, A8 = 2.13320e−03 
               
               
                   
                 The sixth surface 
               
               
                   
                 K = 3.642, A4 = 2.02813e−02, A6 = 5.86325e−03, A8 = −1.69631e−03 
               
               
                   
                 The eighth surface 
               
               
                   
                 K = −7.584, A4 = 6.35851e−02, A6 = 3.93734e−02, A8 = −1.23472e−02 
               
               
                   
                 The tenth surface 
               
               
                   
                 K = −10.000, A4 = 5.93583e−02, A6 = 2.28767e−02, A8 = 3.08419e−02 
               
               
                   
                 The eleventh surface 
               
               
                   
                 K = −9.807, A4 = 2.12775e−02, A6 = −4.53840e−02, A8 = −2.43365e−03 
               
               
                   
                 The twelfth surface 
               
               
                   
                 K = −0.858, A4 = 5.63611e−02, A6 = −5.02019e−02, A8 = 8.82711e−03 
               
               
                   
                 The thirteenth surface 
               
               
                   
                 K = 1.905, A4 = 1.75077e−02, A6 = 1.42222e−03 
               
               
                   
                 The fourteenth surface 
               
               
                   
                 K = 0.102, A4 = 2.50940e−02, A6 = −3.51630e−04, A8 = 3.23344e−04 
               
               
                   
                   
               
             
          
           
               
                 Various data 
               
               
                 Zoom ratio: 2.800 
               
             
          
           
               
                   
                 Wide angle end 
                 Middle 
                 Telephoto end 
               
               
                   
               
               
                 Focal length 
                 2.912 
                 4.629 
                 8.154 
               
               
                 F No. 
                 3.500 
                 4.729 
                 4.800 
               
               
                 Angle of view 2ω 
                 84.169 
                 52.256 
                 29.967 
               
               
                 Image height 
                 2.250 
                 2.250 
                 2.250 
               
               
                 BF# 
                 0.924 
                 0.924 
                 0.924 
               
               
                 Lens total length# 
                 9.998 
                 9.998 
                 9.998 
               
               
                 Distance from an object point 
                 ∞ 
                 ∞ 
                 ∞ 
               
               
                 D4 
                 2.912 
                 1.582 
                 0.100 
               
               
                 D10 
                 0.297 
                 0.260 
                 1.135 
               
               
                 D12 
                 1.525 
                 2.891 
                 3.499 
               
               
                 Stop diameter 
                 0.651 
                 0.651 
                 0.954 
               
               
                 Entrance pupil position 
                 2.300 
                 1.927 
                 1.315 
               
               
                 Exit pupil position 
                 −7.694 
                 −35.857 
                 54.178 
               
               
                 The position of the front side principle point 
                 4.231 
                 5.973 
                 10.717 
               
               
                 The position of the rear side principle point 
                 −2.532 
                 −4.291 
                 −7.820 
               
               
                   
               
             
          
           
               
                   
                 Lens 
                 The first surface of lens 
                 Focal length of lens 
               
               
                   
                   
               
               
                   
                 L11 
                 1 
                 −2.306 
               
               
                   
                 L12 
                 3 
                 5.217 
               
               
                   
                 L21 
                 5 
                 2.190 
               
               
                   
                 L22 
                 8 
                 8.345 
               
               
                   
                 L23 
                 9 
                 −3.752 
               
               
                   
                 L3 
                 11 
                 −5.116 
               
               
                   
                 L4 
                 13 
                 5.098 
               
               
                   
                   
               
             
          
           
               
                 Zoom lens group data 
               
             
          
           
               
                   
                 The first 
                   
                 Composition 
                 The position 
                 The position 
               
               
                   
                 surface 
                 Focal length 
                 length 
                 of the front side 
                 of the rear side 
               
               
                 Group 
                 of lens group 
                 of lens group 
                 of lens group 
                 principle point 
                 principle point 
               
               
                   
               
               
                 G1 
                 1 
                 −4.287 
                 1.238 
                 0.105 
                 −0.663 
               
               
                 G2 
                 5 
                 2.446 
                 2.023 
                 −0.336 
                 −1.340 
               
               
                 G3 
                 11 
                 −5.116 
                 0.400 
                 0.116 
                 −0.142 
               
               
                 G4 
                 13 
                 5.098 
                 0.679 
                 0.683 
                 0.415 
               
               
                   
               
             
          
           
               
                 Magnification of lens group 
               
             
          
           
               
                   
                   
                 Wide angle end 
                 Middle 
                 Telephoto end 
               
               
                   
                   
               
               
                   
                 G1 
                 0.000 
                 0.000 
                 0.000 
               
               
                   
                 G2 
                 −0.482 
                 −0.652 
                 −1.079 
               
               
                   
                 G3 
                 1.577 
                 1.834 
                 1.952 
               
               
                   
                 G4 
                 0.894 
                 0.903 
                 0.903 
               
               
                   
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
               
               
             
               
             
               
               
             
               
             
               
               
               
               
             
               
               
               
               
             
               
             
               
               
               
               
               
               
             
               
             
               
               
               
               
               
             
           
               
                   
               
               
                 Numerical embodiment 11 
               
               
                   
               
             
             
               
                 Surface data 
               
               
                 Unit: mm 
               
             
          
           
               
                 Surface No. 
                 r 
                 d 
                 nd 
                 vd 
                 Effective diameter 
               
               
                   
               
               
                 Object plane 
                 ∞ 
                 ∞ 
               
               
                  1* 
                 10.1679 
                 0.4000 
                 1.77377 
                 47.17 
                 1.804 
               
               
                  2* 
                 1.8406 
                 0.3689 
                   
                   
                 1.449 
               
               
                  3* 
                 2.4091 
                 0.5290 
                 1.70000 
                 15.00 
                 1.449 
               
               
                  4* 
                 3.3582 
                 D4 
                   
                   
                 1.400 
               
               
                  5* 
                 2.0501 
                 0.7343 
                 1.76802 
                 49.24 
                 1.149 
               
               
                  6* 
                 −27.4479 
                 0.1994 
                   
                   
                 1.049 
               
               
                  7 (Stop) 
                 ∞ 
                 0.0000 
                   
                   
                 (Variable) 
               
               
                  8* 
                 4.4431 
                 0.5434 
                 1.49700 
                 81.54 
                 0.931 
               
               
                  9 
                 −4.9922 
                 0.4000 
                 1.70000 
                 15.00 
                 0.888 
               
               
                 10* 
                 27.3397 
                 D10 
                   
                   
                 0.846 
               
               
                 11* 
                 −2.0238 
                 0.4000 
                 1.53071 
                 55.67 
                 0.868 
               
               
                 12* 
                 −24.7934 
                 D12 
                   
                   
                 0.946 
               
               
                 13* 
                 −4.1717 
                 0.8192 
                 1.70000 
                 15.00 
                 2.139 
               
               
                 14* 
                 −2.5605 
                 0.5342 
                   
                   
                 2.223 
               
               
                 15 
                 ∞ 
                 0.3000 
                 1.51633 
                 64.14 
                 2.258 
               
               
                 16 
                 ∞ 
                 0.3500 
                   
                   
                 2.263 
               
               
                 Image plane 
                 ∞ 
               
               
                   
               
             
          
           
               
                 Aspherical surface data 
               
               
                   
               
             
          
           
               
                   
                 The first surface 
               
               
                   
                 K = −2.968, A4 = 1.54293e−02, A6 = −8.65844e−03, A8 = 8.28284e−04 
               
               
                   
                 The second surface 
               
               
                   
                 K = 0.002, A4 = 8.96390e−03, A6 = −1.08004e−03, A8 = −5.17338e−03 
               
               
                   
                 The third surface 
               
               
                   
                 K = −0.982, A4 = −4.16429e−02, A6 = 1.42120e−03 
               
               
                   
                 The fourth surface 
               
               
                   
                 K = −4.286, A4 = −4.16742e−02, A6 = −2.66315e−03, A8 = 1.91253e−03 
               
               
                   
                 The fifth surface 
               
               
                   
                 K = −0.227, A4 = 1.02517e−03, A6 = 8.62020e−04, A8 = 2.01632e−03 
               
               
                   
                 The sixth surface 
               
               
                   
                 K = −9.886, A4 = −7.92533e−03, A6 = 1.68283e−02, A8 = −4.56115e−03 
               
               
                   
                 The eighth surface 
               
               
                   
                 K = −3.826, A4 = −2.05536e−02, A6 = 3.91404e−02, A8 = −1.54510e−02 
               
               
                   
                 The tenth surface 
               
               
                   
                 K = −9.474, A4 = 4.13517e−02, A6 = 1.51235e−02, A8 = 2.94491e−02 
               
               
                   
                 The eleventh surface 
               
               
                   
                 K = −9.859, A4 = 6.25595e−02, A6 = −1.76258e−02, A8 = −5.16373e−02 
               
               
                   
                 The twelfth surface 
               
               
                   
                 K = −10.000, A4 = 2.13359e−01, A6 = −9.44745e−02, A8 = 9.69920e−04 
               
               
                   
                 The thirteenth surface 
               
               
                   
                 K = 2.062, A4 = 1.88493e−02, A6 = 1.42105e−03 
               
               
                   
                 The fourteenth surface 
               
               
                   
                 K = 0.080, A4 = 2.24501e−02, A6 = −5.53882e−04, A8 = 3.18452e−04 
               
               
                   
                   
               
             
          
           
               
                 Various data 
               
               
                 Zoom ratio: 2.795 
               
             
          
           
               
                   
                 Wide angle end 
                 Middle 
                 Telephoto end 
               
               
                   
               
               
                 Focal length 
                 2.917 
                 4.633 
                 8.154 
               
               
                 F No. 
                 3.500 
                 4.662 
                 4.800 
               
               
                 Angle of view 2ω 
                 84.108 
                 52.711 
                 30.750 
               
               
                 Image height 
                 2.250 
                 2.250 
                 2.250 
               
               
                 BF# 
                 1.082 
                 1.082 
                 1.082 
               
               
                 Lens total length# 
                 9.998 
                 9.998 
                 9.998 
               
               
                 Distance from an object point 
                 ∞ 
                 ∞ 
                 ∞ 
               
               
                 D4 
                 2.873 
                 1.548 
                 0.100 
               
               
                 D10 
                 0.240 
                 0.227 
                 0.840 
               
               
                 D12 
                 1.408 
                 2.746 
                 3.581 
               
               
                 Stop diameter 
                 0.665 
                 0.665 
                 0.937 
               
               
                 Entrance pupil position 
                 2.457 
                 2.069 
                 1.428 
               
               
                 Exit pupil position 
                 −5.371 
                 −11.293 
                 −24.106 
               
               
                 The position of the front side principle point 
                 4.062 
                 4.966 
                 6.940 
               
               
                 The position of the rear side principle point 
                 −2.533 
                 −4.299 
                 −7.825 
               
               
                   
               
             
          
           
               
                   
                 Lens 
                 The first surface of lens 
                 Focal length of lens 
               
               
                   
                   
               
               
                   
                 L11 
                 1 
                 −2.967 
               
               
                   
                 L12 
                 3 
                 9.905 
               
               
                   
                 L21 
                 5 
                 2.511 
               
               
                   
                 L22 
                 8 
                 4.822 
               
               
                   
                 L23 
                 9 
                 −6.000 
               
               
                   
                 L3 
                 11 
                 −4.178 
               
               
                   
                 L4 
                 13 
                 7.831 
               
               
                   
                   
               
             
          
           
               
                 Zoom lens group data 
               
             
          
           
               
                   
                 The first 
                   
                 Composition 
                 The position 
                 The position 
               
               
                   
                 surface 
                 Focal length 
                 length 
                 of the front side 
                 of the rear side 
               
               
                 Group 
                 of lens group 
                 of lens group 
                 of lens group 
                 principle point 
                 principle point 
               
               
                   
               
               
                 G1 
                 1 
                 −4.046 
                 1.298 
                 0.414 
                 −0.453 
               
               
                 G2 
                 5 
                 2.234 
                 1.877 
                 0.026 
                 −1.159 
               
               
                 G3 
                 11 
                 −4.178 
                 0.400 
                 −0.023 
                 −0.286 
               
               
                 G4 
                 13 
                 7.831 
                 0.819 
                 1.032 
                 0.633 
               
               
                   
               
             
          
           
               
                 Magnification of lens group 
               
             
          
           
               
                   
                   
                 Wide angle end 
                 Middle 
                 Telephoto end 
               
               
                   
                   
               
               
                   
                 G1 
                 0.000 
                 0.000 
                 0.000 
               
               
                   
                 G2 
                 −0.433 
                 −0.582 
                 −0.935 
               
               
                   
                 G3 
                 1.776 
                 2.083 
                 2.281 
               
               
                   
                 G4 
                 0.938 
                 0.945 
                 0.945 
               
               
                   
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
               
               
             
               
             
               
               
             
               
             
               
               
               
               
             
               
               
               
               
             
               
             
               
               
               
               
               
               
             
               
             
               
               
               
               
               
             
           
               
                   
               
               
                 Numerical embodiment 12 
               
               
                   
               
             
             
               
                 Surface data 
               
               
                 Unit: mm 
               
             
          
           
               
                 Surface No. 
                 r 
                 d 
                 nd 
                 vd 
                 Effective diameter 
               
               
                   
               
               
                 Object plane 
                 ∞ 
                 ∞ 
               
               
                  1* 
                 −20.6604 
                 0.6681 
                 1.69350 
                 53.21 
                 2.556 
               
               
                  2* 
                 6.2554 
                 D2 
                   
                   
                 2.275 
               
               
                  3* 
                 2.8735 
                 1.2776 
                 1.77377 
                 47.17 
                 1.550 
               
               
                  4 
                 −41.3684 
                 0.5180 
                 1.79491 
                 25.63 
                 1.357 
               
               
                  5* 
                 10.6571 
                 0.5000 
                   
                   
                 1.208 
               
               
                  6 (Stop) 
                 ∞ 
                 0.1965 
                   
                   
                 (Variable) 
               
               
                  7* 
                 7.5874 
                 0.8263 
                 1.76802 
                 49.24 
                 1.136 
               
               
                  8* 
                 −21.6754 
                 D8 
                   
                   
                 1.217 
               
               
                  9* 
                 279.5342 
                 0.3790 
                 1.82114 
                 24.06 
                 1.225 
               
               
                 10* 
                 3.4792 
                 D10 
                   
                   
                 1.229 
               
               
                 11* 
                 −7.1348 
                 0.9360 
                 1.79491 
                 25.63 
                 2.746 
               
               
                 12* 
                 −3.7927 
                 1.3721 
                   
                   
                 2.868 
               
               
                 13 
                 ∞ 
                 0.4000 
                 1.51633 
                 64.14 
                 2.972 
               
               
                 14 
                 ∞ 
                 0.4000 
                   
                   
                 2.983 
               
               
                 Image plane 
                 ∞ 
               
               
                   
               
             
          
           
               
                 Aspherical surface data 
               
               
                   
               
             
          
           
               
                   
                 The first surface 
               
               
                   
                 K = −28.194, A4 = −9.53942e−03, A6 = 1.14164e−03, A8 = −5.01129e−05 
               
               
                   
                 The second surface 
               
               
                   
                 K = −0.224, A4 = −9.75476e−03, A6 = 1.35826e−03, A8 = −4.78131e−05 
               
               
                   
                 The third surface 
               
               
                   
                 K = 0.624, A4 = −1.58057e−03, A6 = −1.23052e−04, A8 = 7.62687e−06 
               
               
                   
                 The fifth surface 
               
               
                   
                 K = 44.632, A4 = 5.76135e−04, A6 = 1.26800e−03, A8 = −7.09871e−05 
               
               
                   
                 The seventh surface 
               
               
                   
                 K = −1.283, A4 = −2.54498e−02, A6 = −2.93083e−03, A8 = −1.21006e−03 
               
               
                   
                 The eighth surface 
               
               
                   
                 K = 44.767, A4 = −2.16942e−02, A6 = −2.05558e−03, A8 = −1.25759e−05 
               
               
                   
                 The ninth surface 
               
               
                   
                 K = −560206.977 
               
               
                   
                 The tenth surface 
               
               
                   
                 K = −0.582, A4 = 7.46329e−03, A6 = 1.55863e−03, A8 = −8.04564e−04 
               
               
                   
                 The eleventh surface 
               
               
                   
                 K = −4.214 
               
               
                   
                 The twelfth surface 
               
               
                   
                 K = −0.006, A4 = 4.15000e−03, A6 = −2.49039e−05, A8 = −2.17406e−05, 
               
               
                   
                 A10 = 2.21954e−06 
               
               
                   
                   
               
             
          
           
               
                 Various data 
               
               
                 Zoom ratio: 2.816 
               
             
          
           
               
                   
                 Wide angle end 
                 Middle 
                 Telephoto end 
               
               
                   
               
               
                 Focal length 
                 4.574 
                 7.639 
                 12.879 
               
               
                 F No. 
                 3.228 
                 3.726 
                 5.123 
               
               
                 Angle of view 2ω 
                 70.712 
                 41.233 
                 24.715 
               
               
                 Image height 
                 2.900 
                 2.900 
                 2.900 
               
               
                 BF 
                 2.036 
                 2.036 
                 2.036 
               
               
                 Lens total length 
                 13.864 
                 13.864 
                 13.864 
               
               
                 Distance from an object point 
                 ∞ 
                 ∞ 
                 ∞ 
               
               
                 D2 
                 4.476 
                 2.312 
                 0.201 
               
               
                 D8 
                 0.611 
                 0.650 
                 1.363 
               
               
                 D10 
                 1.439 
                 3.564 
                 4.963 
               
               
                 Stop diameter 
                 0.900 
                 1.100 
                 1.100 
               
               
                 Entrance pupil position 
                 3.777 
                 3.155 
                 2.278 
               
               
                 Exit pupil position 
                 −5.287 
                 −14.748 
                 −40.735 
               
               
                 The position of the front side principle point 
                 5.494 
                 7.317 
                 11.279 
               
               
                 The position of the rear side principle point 
                 −4.174 
                 −7.239 
                 −12.479 
               
               
                   
               
             
          
           
               
                   
                 Lens 
                 The first surface of lens 
                 Focal length of lens 
               
               
                   
                   
               
               
                   
                 L1 
                 1 
                 −6.854 
               
               
                   
                 L21 
                 3 
                 3.517 
               
               
                   
                 L22 
                 4 
                 −10.614 
               
               
                   
                 L23 
                 7 
                 7.409 
               
               
                   
                 L3 
                 9 
                 −4.293 
               
               
                   
                 L4 
                 11 
                 9.062 
               
               
                   
                   
               
             
          
           
               
                 Zoom lens group data 
               
             
          
           
               
                   
                 The first 
                   
                 Composition 
                 The position 
                 The position 
               
               
                   
                 surface 
                 Focal length 
                 length 
                 of the front side 
                 of the rear side 
               
               
                 Group 
                 of lens group 
                 of lens group 
                 of lens group 
                 principle point 
                 principle point 
               
               
                   
               
               
                 G1 
                 1 
                 −6.854 
                 0.668 
                 0.300 
                 −0.091 
               
               
                 G2 
                 3 
                 3.459 
                 3.318 
                 0.621 
                 −1.898 
               
               
                 G3 
                 9 
                 −4.293 
                 0.379 
                 0.211 
                 0.003 
               
               
                 G4 
                 11 
                 9.062 
                 2.708 
                 0.990 
                 −1.109 
               
               
                   
               
             
          
           
               
                 Magnification of lens group 
               
             
          
           
               
                   
                   
                 Wide angle end 
                 Middle 
                 Telephoto end 
               
               
                   
                   
               
               
                   
                 G1 
                 0.000 
                 0.000 
                 0.000 
               
               
                   
                 G2 
                 −0.403 
                 −0.539 
                 −0.803 
               
               
                   
                 G3 
                 1.987 
                 2.482 
                 2.808 
               
               
                   
                 G4 
                 0.833 
                 0.833 
                 0.833 
               
               
                   
                   
               
             
          
         
       
     
         [0333]    Next, parameter values which the embodiment 1 (the numeral embodiment 1) to the embodiment 12 (the numeral embodiment 12) have in the conditions (1) to (22) are given. 
         [0334]    Parameter values which the embodiments 1 to 12 have in the respective conditions 
         [0000]    
       
         
               
               
               
               
               
               
               
             
               
               
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
             
               
               
               
               
               
               
               
               
             
               
               
               
               
               
             
           
               
                   
               
             
             
               
                   
                 Condition (1) 
                 Condition (2) 
                 Condition (3) 
                 Condition (4) 
                 Condition (5) 
                 Condition (6) 
               
               
                   
               
               
                 Embodiment 1 
                 24.06 
                 0.08 
                 0.90 
                 0.50 
                 0.48 
                 1.98 
               
               
                 Embodiment 2 
                 24.06 
                 0.10 
                 0.92 
                 0.50 
                 0.50 
                 2.01 
               
               
                 Embodiment 3 
                 23.89 
                 0.87 
                 0.96 
                 0.87 
                 −3.63 
                 2.34 
               
               
                 Embodiment 4 
                 31.00 
                 0.08 
                 0.90 
                 0.50 
                 0.46 
                 1.97 
               
               
                 Embodiment 5 
                 30.25 
                 0.04 
                 0.88 
                 0.50 
                 0.56 
                 1.91 
               
               
                 Embodiment 6 
                 24.06 
                 0.42 
                 0.96 
                 0.53 
                 0.23 
                 2.53 
               
               
                 Embodiment 7 
                 20.88 
                 0.43 
                 0.91 
                 0.50 
                 0.29 
                 2.41 
               
               
                 Embodiment 8 
                 24.06 
                 0.16 
                 0.94 
                 0.44 
                 0.78 
                 2.48 
               
               
                 Embodiment 9 
                 24.06 
                 0.42 
                 0.98 
                 0.53 
                 0.30 
                 2.58 
               
               
                 Embodiment 10 
                 16.77 
                 −0.02 
                 0.88 
                 0.50 
                 0.24 
                 1.91 
               
               
                 Embodiment 11 
                 15.00 
                 −0.72 
                 0.83 
                 0.46 
                 0.24 
                 1.80 
               
               
                 Embodiment 12 
                 25.63 
                 0.31 
                 0.89 
                 0.45 
                 0.31 
                 2.36 
               
               
                   
               
             
          
           
               
                   
                 Condition (7) 
                 Condition (8) 
                 Condition (9) 
                 Condition (10) 
                 Condition (11) 
               
               
                   
               
               
                 Embodiment 1 
                 1.10 
                 24.06 
                 2.47 
                 42.96 
                 0.50 
               
               
                 Embodiment 2 
                 1.10 
                 24.06 
                 2.46 
                 42.96 
                 0.50 
               
               
                 Embodiment 3 
                 2.10 
                 23.89 
                 — 
                 — 
                 0.87 
               
               
                 Embodiment 4 
                 1.10 
                 31.00 
                 2.29 
                 42.96 
                 0.50 
               
               
                 Embodiment 5 
                 1.10 
                 30.25 
                 2.34 
                 42.96 
                 0.50 
               
               
                 Embodiment 6 
                 1.40 
                 24.06 
                 2.30 
                 29.43 
                 0.53 
               
               
                 Embodiment 7 
                 1.32 
                 20.88 
                 2.55 
                 29.43 
                 0.50 
               
               
                 Embodiment 8 
                 1.17 
                 24.06 
                 2.70 
                 25.46 
                 0.44 
               
               
                 Embodiment 9 
                 1.39 
                 24.06 
                 2.29 
                 29.43 
                 0.53 
               
               
                 Embodiment 10 
                 1.09 
                 16.77 
                 2.69 
                 64.77 
                 0.50 
               
               
                 Embodiment 11 
                 0.99 
                 15.00 
                 3.00 
                 66.54 
                 0.46 
               
               
                 Embodiment 12 
                 1.19 
                 25.63 
                 2.67 
                 23.61 
                 0.45 
               
               
                   
               
             
          
           
               
                   
                   
                 Condition (12) 
                 Condition (13) 
                 Condition (14) 
               
               
                   
                   
               
               
                   
                 Embodiment 1 
                 0.48 
                 0.21 
                 1.10 
               
               
                   
                 Embodiment 2 
                 0.50 
                 0.21 
                 1.10 
               
               
                   
                 Embodiment 3 
                 −3.63 
                 — 
                 2.10 
               
               
                   
                 Embodiment 4 
                 0.46 
                 −0.13 
                 1.10 
               
               
                   
                 Embodiment 5 
                 0.56 
                 −0.08 
                 1.10 
               
               
                   
                 Embodiment 6 
                 0.23 
                 −0.09 
                 1.40 
               
               
                   
                 Embodiment 7 
                 0.29 
                 0.13 
                 1.32 
               
               
                   
                 Embodiment 8 
                 0.78 
                 0.05 
                 1.17 
               
               
                   
                 Embodiment 9 
                 0.30 
                 −0.10 
                 1.39 
               
               
                   
                 Embodiment 10 
                 0.24 
                 0.61 
                 1.09 
               
               
                   
                 Embodiment 11 
                 0.24 
                 0.81 
                 0.99 
               
               
                   
                 Embodiment 12 
                 0.31 
                 −0.02 
                 1.19 
               
               
                   
                   
               
             
          
           
               
                   
                 Condition (15) 
                 Condition (16) 
                   
               
             
          
           
               
                   
                 L11 
                 L12 
                 L11 
                 L12 
                 Condition (17) 
                 Condition (18) 
                 Condition (19) 
               
               
                   
               
               
                 Embodiment 1 
                 1.901 
                 2.102 
                 31.00 
                 16.77 
                 14.23 
                 0.09 
                 −0.152 
               
               
                 Embodiment 2 
                 1.851 
                 1.821 
                 40.10 
                 24.06 
                 16.04 
                 0.08 
                 −0.101 
               
               
                 Embodiment 10 
                 1.903 
                 2.102 
                 31.00 
                 16.77 
                 14.23 
                 0.07 
                 −0.115 
               
               
                   
               
             
          
           
               
                   
                   
                 Condition (20) 
                 Condition (21) 
                 Condition (22) 
               
               
                   
                   
               
               
                   
                 Embodiment 1 
                 −0.43 
                 0.84 
                 24.06 
               
               
                   
                 Embodiment 2 
                 −0.44 
                 0.85 
                 24.06 
               
               
                   
                 Embodiment 10 
                 −0.44 
                 1.21 
                 16.77 
               
               
                   
                   
               
             
          
         
       
     
         [0335]    The variable power optical systems according to the embodiments of the present invention as described above can be used for image pickup apparatuses, such as digital camera and video camera, in which shooting is performed by forming on an imaging sensor like CCD an object image that is formed by the variable power optical systems. A concrete example for the image pickup apparatuses is given below. 
         [0336]      FIGS. 43 ,  44 , and  45  are conceptual views showing the constitution of a digital camera including a variable power optical system of one of the embodiments of the present invention,  FIG. 43  is a front perspective view showing the appearance of a digital camera,  FIG. 44  is a rear perspective view showing the appearance of the digital camera which is shown in  FIG. 43 , and  FIG. 45  is a perspective view schematically showing the constitution of the digital camera. 
         [0337]    The digital camera is provided with an opening section  1  for shooting, a finder opening section  2 , and a flash-firing section  3  on the front side of the digital camera. Also, the digital camera is provided with a shutter button  4  on the top of the digital camera. Also, the digital camera is provided with a liquid crystal display monitor  5  and an information input section  6  on the rear side of the digital camera. In addition, the digital camera is provided with a variable power optical system  7  that is formed in the same manner as in the embodiment 1, 12, or 23 for example, a processing means  8 , a recording means  9 , and a finder optical system  10  inside the digital camera. Also, cover members  12  are arranged in the finder opening section  2  and in an opening section  11 , the opening section  11  being located on the exit side of the finder optical system  10  and being provided on the rear side of the digital camera. In addition, a cover member  13  is also arranged in the opening section  1  for shooting. 
         [0338]    When the shutter button  4  which is arranged on the top of the digital camera is pressed, shooting is performed through the variable power optical system  7  in response to the pressing of the shutter button  4 . An object image is formed on the image forming plane of a CCD  7   a  that is a solid-state imaging sensor, through the variable power optical system  7  and the cover glass CG. The image information on the object image which is formed on the image forming plane of the CCD  7   a  is recorded on the recording means  9  through the processing means  8 . Also, recorded image information can be taken through the processing means  8 , and the image information can be also displayed as an electronic image on the liquid crystal display monitor  5  which is provided on the rear side of the camera. 
         [0339]    Also, the finder optical system  10  is composed of a finder objective optical system  10   a , an erecting prism  10   b , and an eyepiece optical system  10   c . Light from an object which enters through the finder opening section  2  is led to the erecting prism  10   b  that is a member for erecting an image, by the finder objective optical system  10   a , and an object image is formed as an erect image in the view finder frame  10   b   1 , and, afterward, the object image is led to an eye E of an observer by the eyepiece optical system  10   c.    
         [0340]    Digital cameras which are formed in such a manner secure good performances while it is possible to achieve downsizing of the digital cameras, because the variable power optical system  7  has a high magnification ratio and is small.

Technology Category: 3