PATENT DOCUMENT

Abstract:
Provided are a variable power optical system in which the value of the entire length with respect to the image height is reduced and the aberration is well corrected, and an image pickup device comprising the same. Specifically provided is a variable power optical system comprising, in order from the object side, at least a first lens group having negative refractive power, a variable power group having positive refractive power, and a final lens group having positive refractive power, wherein the variable power group is provided with, in order from the object side, a first lens element having positive refractive power, a second lens element, and a third lens element, the second lens element has a convex shape on the object side, and the final lens group comprises a positive lens, the variable power optical system satisfying the following conditional expression:
 
10≦ VdLg ≦45
 
−1.0&lt;( R 2 a−R 2 b )/( R 2 a+R 2 b )&lt;1.0
 
where VdLg is the Abbe number of the positive lens of the final lens group with respect to the d line, R 2   a  is the curvature radius of the object-side surface of the second lens element, and R 2   b  is the curvature radius of the image-side surface of the third lens element.

Full Description:
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 
     1. Field of Invention 
     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. 
     2. Description of the Related Art 
     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. 
     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). 
     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 
     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:
 
10≦VdLg≦45  (1)
 
−1.0&lt;( R 2 a−R 2 b )/( R 2 a+R 2 b )&lt;1.0  (2)
 
where VdLg denotes the Abbe&#39; Number of the positive lens of the last lens group with respect to the d line, R 2   a  denotes the radius of curvature of the object-side surface of the second lens element, and R 2   b  denotes the radius of curvature of the image-side surface of the third lens element.
 
     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. 
     Also, in a variable power optical system according to the present first invention, it is preferred that the following condition (3) is satisfied:
 
0.45≦| f 1|/( FLw×FLt ) 1/2 ≦1.60  (3)
 
where f 1  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.
 
     Also, in a variable power optical system according to the present first invention, it is preferred that the following condition (4) is satisfied:
 
0.30≦ fv /( FLw×FLt ) 1/2 ≦1.10  (4)
 
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.
 
     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:
 
0.1≦( RLa−RLb )/( RLa+RLb )&lt;1.0  (5)
 
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.
 
     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:
 
1.0≦| f 1|/IH≦2.8  (6)
 
where f 1  denotes the focal length of the first lens group, and IH denotes the image height of the imaging sensor.
 
     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:
 
0.2≦| fv| /IH≦1.8  (7)
 
where fv denotes the focal length of the magnification-changing group, and IH denotes the image height of the imaging sensor.
 
     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:
 
10≦Vd4g≦45  (8)
 
2.2≦|α/ f 1|+(α/ f 2)−0.026× Vd 4 g≦ 5.0  (9)
 
10≦ Vd max− Vd min≦80  (10)
 
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, f 1  denotes the focal length of the first lens group, f 2  denotes the focal length of the second lens group, Vd 4 g 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.
 
     Also, in a variable power optical system according to the present second invention, it is preferred that the following condition (3) is satisfied:
 
0.45≦| f 1|/( FLw×FLt ) 1/2 ≦1.60  (3)
 
where f 1  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.
 
     Also, in a variable power optical system according to the present second invention, it is preferred that the following condition (11) is satisfied:
 
0.30&lt; f 2/( FLw×FLt ) 1/2 ≦1.10  (11)
 
where f 2  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.
 
     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:
 
−1.0&lt;( R 2 a−R 2 b )/( R 2 a+R 2 b )&lt;1.0  (2)
 
where R 2   a  denotes the radius of curvature of the object-side surface of the second lens element, and R 2   b  denotes the radius of curvature of the image-side surface of the third lens element.
 
     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:
 
0&lt;( R 4 a−R 4 b )/( R 4 a+R 4 b )&lt;1.0  (12)
 
where R 4   a  denotes the radius of curvature of the object-side surface of the positive lens in the fourth lens group, and R 4   b  denotes the radius of curvature of the image-side surface of the positive lens in the fourth lens group.
 
     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. 
     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:
 
10≦Vd4g≦45  (8)
 
−0.27≦|IH/ f 1|+(IH/ f 2)−0.05× Vd 4 g≦ 3.0  (13)
 
10≦ Vd max− Vd min≦80  (10)
 
where IH denotes the image height of the imaging sensor, f 1  denotes the focal length of the first lens group, f 2  denotes the focal length of the second lens group, Vd 4 g 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.
 
     Also, in an image pickup apparatus according to the present second invention, it is preferred that the following condition (6) is satisfied:
 
1.0≦| f 1|/IH≦2.8  (6)
 
where f 1  denotes the focal length of the first lens group, and IH denotes the image height of the imaging sensor.
 
     Also, in an image pickup apparatus according to the present second invention, it is preferred that the following condition (14) is satisfied:
 
0.2≦| f 2|/IH≦1.8  (14)
 
where f 2  denotes the focal length of the second lens group, and IH denotes the image height of the imaging sensor.
 
     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:
 
−1.0&lt;( R 2 a−R 2 b )/( R 2 a+R 2 b )&lt;1.0  (2)
 
where R 2   a  denotes the radius of curvature of the object-side surface of the second lens element, and R 2   b  denotes the radius of curvature of the image-side surface of the third lens element.
 
     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:
 
0&lt;( R 4 a−R 4 b )/( R 4 a+R 4 b )&lt;1.0  (12)
 
where R 4   a  denotes the radius of curvature of the object-side surface of the positive lens in the fourth lens group, and R 4   b  denotes the radius of curvature of the image-side surface of the positive lens in the fourth lens group.
 
     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:
 
1.75≦Nd1g≦2.50  (15)
 
15≦Vd1g≦43  (16)
 
3≦ VdN−VdP≦ 28  (17)
 
where Nd 1 g denotes the refractive index of each of lenses constituting the first lens group, with respect to the d line, Vd 1 g 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.
 
     Also, in a variable power optical system according to the present third invention, it is preferred that the following condition (18) is satisfied:
 
0.03≦ D /( FLw×FLt ) 1/2 ≦0.26  (18)
 
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.
 
     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:
 
−0.25≦( r 2− r 3)/( r 2+ r 3)≦−0.07  (19)
 
where r 2  denotes the radius of curvature of the image-side surface of the negative lens of the first lens group, and r 3  denotes the radius of curvature of the object-side surface of the positive lens of the first lens group.
 
     Also, in a variable power optical system according to the present third invention, it is preferred that the following condition (3) is satisfied:
 
0.45 ≦|f 1|/( FLw×FLt ) 1/2 ≦1.60  (3)
 
where f 1  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.
 
     Also, in a variable power optical system according to the present third invention, it is preferred that the following condition (20) is satisfied:
 
−0.5 ≦FLn/FLp ≦−0.3  (20)
 
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.
 
     Also, in a variable power optical system according to the present third invention, it is preferred that the following condition (11) is satisfied:
 
0.30≦ f 2/( FLw×FLt ) 1/2 ≦1.10  (11)
 
where f 2  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.
 
     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:
 
0.2≦( r 1− r 2)/( r 1+ r 2)&lt;1.0  (21)
 
where r 1  denotes the radius of curvature of the object-side surface of the negative lens of the first lens group, and r 2  denotes the radius of curvature of the image-side surface of the negative lens of the first lens group.
 
     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:
 
10≦Vd4g≦40  (22)
 
where Vd 4 g denotes the Abbe&#39;s Number of the positive lens of the fourth lens group with respect to the d line.
 
     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:
 
1.0≦| f 1|/ IH≦ 2.8  (6)
 
where f 1  denotes the focal length of the first lens group, and IH denotes the image height of the imaging sensor.
 
     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:
 
0.2≦| f 2|/IH≦1.8  (14)
 
where f 2  denotes the focal length of the second lens group, and IH denotes the image height of the imaging sensor.
 
     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 
         FIGS. 1A to 1C  are sectional views showing a variable power optical system 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. 
         FIGS. 2A to 2L  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. 21 to 2L  show a state in the telephoto end position. 
         FIGS. 3A to 3L  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. 31 to 3L  show a state in the telephoto end position. 
         FIGS. 4A to 4F  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. 
         FIGS. 5A to 5C  are sectional views 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. 
         FIGS. 6A to 6L  are aberration diagrams in focusing at infinity in the embodiment 2, FIGS.  6 A to  6 D show a state in the wide angle end position,  FIGS. 6E to 6H  show a state in the middle position, and  FIGS. 61 to 6L  show a state in the telephoto end position. 
         FIGS. 7A to 7L  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. 71 to 7L  show a state in the telephoto end position. 
         FIGS. 8A to 8F  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. 
         FIGS. 9A to 9C  are sectional views 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. 
         FIGS. 10A to 10L  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. 
         FIGS. 11A to 11L  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. 
         FIGS. 12A to 12F  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. 
         FIGS. 13A to 13C  are sectional views 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. 
         FIGS. 14A to 14L  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. 
         FIGS. 15A to 15L  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. 
         FIGS. 16A to 16F  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. 
         FIGS. 17A to 17C  are sectional views 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. 
         FIGS. 18A to 18L  are aberration diagrams in focusing at infinity in the embodiment 5, FIGS.  18 A 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. 
         FIGS. 19A to 19L  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. 
         FIGS. 20A to 20F  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. 
         FIGS. 21A to 21C  are sectional views 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. 
         FIGS. 22A to 22L  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. 
         FIGS. 23A to 23L  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. 
         FIGS. 24A to 24F  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. 
         FIGS. 25A to 25C  are sectional views 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. 
         FIGS. 26A to 26L  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. 
         FIGS. 27A to 27C  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. 
         FIGS. 28A to 28C  are sectional views 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. 
         FIGS. 29A to 29L  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. 
         FIGS. 30A to 30C  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. 
         FIGS. 31A to 31C  are sectional views 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. 
         FIGS. 32A to 32L  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. 
         FIGS. 33A to 33C  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. 
         FIGS. 34A to 34C  are sectional views 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. 
         FIGS. 35A to 35L  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. 
         FIGS. 36A to 36C  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. 
         FIGS. 37A to 37C  are sectional views 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. 
         FIGS. 38A to 38L  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. 
         FIGS. 39A to 39C  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. 
         FIGS. 40A to 40C  are sectional views 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. 
         FIGS. 41A to 41L  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. 
         FIGS. 42A to 42C  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. 
         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. 
         FIG. 44  is a rear perspective view showing the appearance of the digital camera which is shown in  FIG. 43 . 
         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 
     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. 
     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:
 
10≦VdLg≦45  (1)
 
−1.0&lt;( R 2 a−R 2 b )/( R 2 a+R 2 b )&lt;1.0  (2)
 
where VdLg denotes the Abbe&#39; Number of the positive lens of the last lens group with respect to the d line, R 2   a  denotes the radius of curvature of the object-side surface of the second lens element, and R 2   b  denotes the radius of curvature of the image-side surface of the third lens element.
 
     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. 
     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. 
     If (R 2   a −R 2   b )/(R 2   a +R 2   b ) 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 (R 2   a −R 2   b )/(R 2   a +R 2   b ) 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. 
     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). 
     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):
 
15≦VdLg≦40  (1-1)
 
−0.8&lt;( R 2 a−R 2 b )/( R 2 a+R 2 b )&lt;0.88  (2-1)
 
     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. 
     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. 
     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. 
     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. 
     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. 
     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. 
     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. 
     Also, in a variable power optical system according to the first embodiment, it is preferred that the following condition (3) is satisfied:
 
0.45≦| f 1/( FLw×FLt ) 1/2 ≦1.60  (3)
 
where f 1  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.
 
     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. 
     If |f 1 |/(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 |f 1 |/(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. 
     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):
 
0.50≦| f 1|/( FLw×FLt ) 1/2 ≦1.04  (3-1)
 
     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. 
     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. 
     Also, in a variable power optical system according to the first embodiment, it is preferred that the following condition (4) is satisfied:
 
0.30≦ fv /( FLw×FLt ) 1/2 ≦1.10  (4)
 
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.
 
     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. 
     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. 
     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):
 
0.35≦ fv /( FLw×FLt ) 1/2 ≦0.62  (4-1)
 
     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. 
     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. 
     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:
 
0.1≦( RLa−RLb )/( RLa+RLb )&lt;1.0  (5)
 
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.
 
     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. 
     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):
 
0.1≦( RLa−RLb )/( RLa+RLb )&lt;0.9  (5-1)
 
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.
 
     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:
 
1.0≦| f 1|/IH≦2.8  (6)
 
where f 1  denotes the focal length of the first lens group, and IH denotes the image height of the imaging sensor.
 
     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. 
     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). 
     If |f 1 |/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 |f 1 |/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. 
     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):
 
1.8≦| f 1|/IH≦2.6  (6-1)
 
     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. 
     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:
 
0.2≦| fv| /IH≦1.8  (7)
 
where fv denotes the focal length of the magnification-changing group, and IH denotes the image height of the imaging sensor.
 
     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. 
     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. 
     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):
 
1.0≦| fv| /IH≦1.5  (7-1)
 
     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. 
     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:
 
10≦Vd4g≦45  (8)
 
2.2≦|α/ f 1|+(α/ f 2)−0.026× Vd 4 g≦ 5.0  (9)
 
10≦ Vd max− Vd min≦80  (10)
 
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, f 1  denotes the focal length of the first lens group, f 2  denotes the focal length of the second lens group, Vd 4 g 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.
 
     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. 
     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. 
     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. 
     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. 
     If Vd 4 g 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 Vd 4 g is beyond the upper limit of the condition (8), it becomes difficult to correct chromatic aberration of magnification well in the telephoto end position. 
     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/f 1 |+(α/f 2 )−0.026×Vd 4 g 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. 
     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. 
     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. 
     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):
 
15≦Vd4g≦40  (8-1)
 
2.25≦|α/ f 1|+(α/ f 2)−0.026× Vd 4 g≦ 4.5  (9-1)
 
15≦ Vd max− Vd min≦70  (10-1)
 
     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. 
     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. 
     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. 
     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. 
     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. 
     Also, in a variable power optical system according to the present second embodiment, it is preferred that the following condition (3) is satisfied:
 
0.45≦| f 1|/( FLw×FLt ) 1/2 ≦1.60  (3)
 
where f 1  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.
 
     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. 
     If |f 1 |/(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 |f 1 |/(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. 
     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):
 
0.50≦| f 1|/( FLw×FLt ) 1/2 ≦1.04  (3-1)
 
     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. 
     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. 
     Also, in a variable power optical system according to the second embodiment, it is preferred that the following condition (11) is satisfied:
 
0.30≦ f 2/( FLw×FLt ) 1/2 ≦1.10  (11)
 
where f 2  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.
 
     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. 
     If f 2 /(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 f 2 /(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. 
     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):
 
0.35≦ f 2/( FLw×FLt ) 1/2 ≦0.62  (11-1)
 
     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. 
     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 (L 21 ) with positive refractive power, a second lens element (L 22 ), and a third lens element (L 23 ) in that order from the object side; the first lens element (L 21 ) has a convex shape on the object side; and the following condition (2) is satisfied:
 
−1.0&lt;( R 2 a−R 2 b )/( R 2 a+R 2 b )&lt;1.0  (2)
 
where R 2   a  denotes the radius of curvature of the object-side surface of the second lens element (L 22 ), and R 2   b  denotes the radius of curvature of the image-side surface of the third lens element (L 23 ).
 
     The condition (2) shows the shape factor for the second lens element (L 22 ) and the third lens element (L 23 ). 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. 
     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. 
     If (R 2   a −R 2   b )/(R 2   a +R 2   b ) 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 (R 2   a −R 2   b )/(R 2   a +R 2   b ) 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. 
     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):
 
−0.8&lt;( R 2 a−R 2 b )/( R 2 a+R 2 b )&lt;0.72  (2-1)
 
     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. 
     Also, in a variable power optical system of the second embodiment, it is preferred that the second lens element (L 22 ) has positive refractive power, and the third lens element (L 23 ) has negative refractive power. 
     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. 
     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:
 
0&lt;( R 4 a−R 4 b )/( R 4 a+R 4 b )&lt;1.0  (12)
 
where R 4   a  denotes the radius of curvature of the object-side surface of the positive lens in the fourth lens group, and R 4   b  denotes the radius of curvature of the image-side surface of the positive lens in the fourth lens group.
 
     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. 
     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):
 
0.1≦( R 4 a−R 4 b )/( R 4 a+R 4 b )&lt;0.9  (12-1)
 
     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. 
     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. 
     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:
 
10≦Vd4g≦45  (8)
 
−0.27≦| IH/f 1|+( IH/f 2)−0.05× Vd 4 g≦ 3.0  (13)
 
10≦ Vd max− Vd min≦80  (10)
 
where IH denotes the image height of the imaging sensor, f 1  denotes the focal length of the first lens group, f 2  denotes the focal length of the second lens group, Vd 4 g 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.
 
     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). 
     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. 
     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. 
     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):
 
15≦Vd4g≦40  (8-1)
 
−0.25≦|α/ f 1|+(α/ f 2)−0.026× Vd 4 g≦ 2.5  (13-1)
 
15≦ Vd max− Vd min≦70  (10-1)
 
     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. 
     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. 
     Also, in an image pickup apparatus according to the second embodiment, it is preferred that the following condition (6) is satisfied:
 
1.0≦| f 1|/ IH≦ 2.8  (6)
 
where f1 denotes the focal length of the first lens group, and IH denotes the image height of the imaging sensor.
 
     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. 
     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):
 
1.8≦| f 1/IH≦2.6  (6-1)
 
     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. 
     Also, in an image pickup apparatus according to the second embodiment, it is preferred that the following condition (14) is satisfied:
 
0.2≦| f 2|/IH≦1.8  (14)
 
is where f 2  denotes the focal length of the second lens group, and IH denotes the image height of the imaging sensor.
 
     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. 
     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):
 
1.0≦| f 2|/IH≦1.5  (14-1)
 
     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:
 
−1.0&lt;( R 2 a−R 2 b )/( R 2 a+R 2 b )&lt;1.0  (2)
 
where R 2   a  denotes the radius of curvature of the object-side surface of the second lens element, and R 2   b  denotes the radius of curvature of the image-side surface of the third lens element.
 
     The technical significance and the operation effects of the condition (2) have been already explained. 
     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):
 
−0.8&lt;( R 2 a−R 2 b )/( R 2 a+R 2 b )&lt;0.72  (2-1)
 
     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. 
     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. 
     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:
 
0&lt;( R 4 a−R 4 b )/( R 4 a+R 4 b )&lt;1.0  (12)
 
where R 4   a  denotes the radius of curvature of the object-side surface of the positive lens in the fourth lens group, and R 4   b  denotes the radius of curvature of the image-side surface of the positive lens in the fourth lens group.
 
     The technical significance and the operation effects of the condition (12) have been already explained. 
     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):
 
0.1≦( R 4 a−R 4 b )/( R 4 a+R 4 b )&lt;0.9  (12-1)
 
     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:
 
1.75≦Nd1g≦2.50  (15)
 
15≦Vd1g≦43  (16)
 
3≦ VdN−VdP≦ 28  (17)
 
where Nd 1 g denotes the refractive index of each of lenses constituting the first lens group, with respect to the d line, Vd 1 g 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.
 
     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. 
     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. 
     If Nd 1 g 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 Nd 1 g 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. 
     If Vd 1 g 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 Vd 1 g 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. 
     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. 
     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. 
     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):
 
1.82≦nd1g≦2.40  (15-1)
 
16≦vd1g≦38.7  (16-1)
 
9≦ vdN−vdP≦ 22.7  (17-1)
 
     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. 
     In a variable power optical system according to the third embodiment, it is preferred that the following condition (18) is satisfied:
 
0.03≦ D /( FLw×FLt ) 1/2 ≦0.26  (18)
 
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.
 
     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. 
     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. 
     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):
 
0.05≦ D /( FLw×FLt ) 1/2 ≦0.20  (18-1)
 
     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. 
     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:
 
−0.25≦( r 2− r 3)/( r 2+ r 3)≦−0.07  (19)
 
where r 2  denotes the radius of curvature of the image-side surface of the negative lens of the first lens group, and r 3  denotes the radius of curvature of the object-side surface of the positive lens of the first lens group.
 
     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. 
     If (r 2 −r 3 )/(r 2 +r 3 ) 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 (r 2 −r 3 )/(r 2 +r 3 ) 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. 
     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):
 
−0.20≦( r 2− r 3)/( r 2+ r 3)≦−0.10  (19-1)
 
     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. 
     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. 
     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. 
     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. 
     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. 
     Also, in a variable power optical system according to the present third embodiment, it is preferred that the following condition (3) is satisfied:
 
0.45≦| f 1|/( FLw×FLt ) 1/2 ≦1.60  (3)
 
where f 1  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.
 
     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. 
     If |f 1 |/(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 |f 1 |/(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. 
     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):
 
0.70≦| f 1|/( FLw×FLt ) 1/2 ≦1.20  (3-2)
 
     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. 
     Also, in a variable power optical system according to the present third embodiment, it is preferred that the following condition (20) is satisfied:
 
−0.5≦ FLn/FLp≦− 0.3  (20)
 
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.
 
     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. 
     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. 
     Also, in a variable power optical system according to the third embodiment, it is preferred that the following condition (11) is satisfied:
 
0.30≦ f 2/( FLw×FLt ) 1/2 ≦1.10  (11)
 
where f 2  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.
 
     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. 
     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. 
     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):
 
0.45≦ f 2/( FLw×FLt ) 1/2 ≦0.70  (11-2)
 
     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. 
     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:
 
0.2≦( r 1− r 2)/( r 1+ r 2)≦1.0  (21)
 
where r 1  denotes the radius of curvature of the object-side surface of the negative lens of the first lens group and r 2  denotes the radius of curvature of the image-side surface of the negative lens of the first lens group.
 
     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. 
     If (r 1 −r 2 )/(r 1 +r 2 ) 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 (r 1 −r 2 )/(r 1 +r 2 ) 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. 
     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:
 
10≦Vd4g≦40  (22)
 
where Vd 4 g denotes Abbe&#39;s Number of the positive lens of the fourth lens group with respect to the d line.
 
     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. 
     If Vd 4 g 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 Vd 4 g is beyond the upper limit of the condition (22), it becomes difficult to correct chromatic aberration of magnification well in the telephoto end position. 
     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:
 
1.0≦| f 1|/IH≦2.8  (6)
 
where f 1  denotes the focal length of the first lens group, and IH denotes the image height of the imaging sensor.
 
     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. 
     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):
 
1.8≦| f 1|/IH≦2.6  (6-1)
 
     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:
 
0.2 ≦|f 2|/IH≦1.8  (14)
 
where f 2  denotes the focal length of the second lens group, and IH denotes the image height of the imaging sensor.
 
     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. 
     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):
 
1.0≦|f2|/IH≦1.5  (14-1)
 
[Embodiment]
 
     Embodiments for variable power optical systems according to the present invention and image pickup apparatuses having the same are explained using the drawings, below. 
     First, the embodiments 1 to 12 for variable power optical systems according to the present invention will be explained. 
     The sectional view of the variable power optical system of the embodiment 1 is shown in  FIGS. 1A to 1C , the sectional view of the variable power optical system of the embodiment 2 is shown in  FIGS. 5A to 5C , the sectional view of the variable power optical system of the embodiment 3 is shown in  FIGS. 9A to 9C , the sectional view of the variable power optical system of the embodiment 4 is shown in  FIGS. 13A to 13C , the sectional view of the variable power optical system of the embodiment 5 is shown in  FIGS. 17A to 17C , the sectional view of the variable power optical system of the embodiment 6 is shown in  FIGS. 21A to 21C , the sectional view of the variable power optical system of the embodiment 7 is shown in  FIGS. 25A to 25C , the sectional view of the variable power optical system of the embodiment 8 is shown in  FIGS. 28A to 28C , the sectional view of the variable power optical system of the embodiment 9 is shown in  FIGS. 31A to 31C , the sectional view of the variable power optical system of the embodiment 10 is shown in  FIGS. 34A to 34C , the sectional view of the variable power optical system of the embodiment 11 is shown in  FIGS. 37A to 37C , and the sectional view of the variable power optical system of the embodiment 12 is shown in  FIGS. 40A to 40C . 
     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 
     The optical constitution of the variable power optical system of the present embodiment is explained using  FIGS. 1A to 1C . 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 G 1  with negative refractive power, a second lens group G 2  with positive refractive power, a third lens group G 3  with negative refractive power, and a fourth lens group G 4  with positive refractive power, these lens groups being located on the optical axis Lc. 
     The first lens group G 1  is composed of a negative meniscus lens L 11  the convex surface of which faces toward the object side and a positive meniscus lens L 12  the convex surface of which faces toward the object side, in that order from the object side. And, the first lens group G 1  as a whole has negative refractive power. 
     The second lens group G 2  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 L 21  which becomes the first lens element, an aperture stop S, a biconvex positive lens L 22  which becomes the second lens element, and a biconcave negative lens L 23  which becomes the third lens element and is joined to the biconvex positive lens L 22 , in that order from the object side. And, the second lens group G 2  has positive refractive power as a whole and has a main magnification change function. 
     The third lens group G 3  is composed of one biconcave negative lens L 3 . Besides, the biconcave negative lens L 3  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. 
     The fourth lens group G 4  is composed of one positive meniscus lens L 4  the convex surface of which faces toward the image side. 
     In changing magnification from the wide angle end position to the telephoto end position, the first lens group G 1  and the fourth lens group G 4  keep still, and the second lens group G 2  and the third lens group G 3  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 G 2  and G 3  may be moved. 
     Embodiment 2 
     The optical constitution of the variable power optical system of the present embodiment is explained using  FIGS. 5A to 5C . 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 G 1  with negative refractive power, a second lens group G 2  with positive refractive power, a third lens group G 3  with negative refractive power, and a fourth lens group G 4  with positive refractive power, these lens group being located on the optical axis Lc. 
     The first lens group G 1  is composed of a negative meniscus lens L 11  the convex surface of which faces toward the object side and a positive meniscus lens L 12  the convex surface of which faces toward the object side, in that order from the object side. And, the first lens group G 1  as a whole has negative refractive power. 
     The second lens group G 2  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 G 2  is composed of a biconvex positive lens L 21  which becomes the first lens element, an aperture stop S, a biconvex positive lens L 22  which becomes the second lens element, and a biconcave negative lens L 23  which becomes the third lens element and is joined to the biconvex positive lens L 22 , in that order from the object side. And, the second lens group G 2  has positive refractive power as a whole and has a main magnification change function. 
     The third lens group G 3  is composed of one negative meniscus lens L 3  the concave surface of which faces toward the object side. Besides, the negative meniscus lens element L 3  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. 
     The fourth lens group G 4  is composed of one positive meniscus lens L 4  the convex surface of which faces toward the image side. 
     In changing magnification from the wide angle end position to the telephoto end position, the first lens group G 1  and the fourth lens group G 4  keep still, and the second lens group G 2  and the third lens group G 3  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 G 2  and G 3  may be moved. 
     It is possible to correct variation in field curvature well in focusing on an object, by moving the both lens groups. 
     Embodiment 3 
     The optical constitution of the variable power optical system of the present embodiment is explained using  FIGS. 9A to 9C . The total length of the variable power optical system of the present embodiment is about 16 mm. 
     The variable power optical system of the present embodiment includes, in order from the object side, a first lens group G 1  with negative refractive power, a second lens group G 2  with positive refractive power, a third lens group G 3  with positive refractive power, and a fourth lens group G 4  with positive refractive power, these lens groups being located on the optical axis Lc. 
     The first lens group G 1  is composed of one biconcave negative lens L 1 . 
     The second lens group G 2  is composed of one positive meniscus lens L 2  the convex surface of which faces toward the object side. 
     The third lens group G 3  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 G 3  is composed of an aperture stop S, a biconvex positive lens L 31  which becomes the first lens element, a negative meniscus lens L 32  which becomes the second lens element and the convex surface of which faces toward the object side, and a negative meniscus lens L 33  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 G 3  as a whole has positive refractive power. 
     The fourth lens group G 4  is composed of one biconvex positive lens L 4 . 
     In changing magnification from the wide angle end position to the telephoto end position, the first lens group G 1  and the fourth lens group G 4  keep still, and the second lens group G 2  and the third lens group G 3  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 G 2  and G 3  may be moved. 
     It is possible to correct variation in field curvature well in focusing on an object, by moving the both lens groups. 
     Embodiment 4 
     The optical constitution of the variable power optical system of the present embodiment is explained using  FIGS. 13A to 13C . 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 G 1  with negative refractive power, a second lens group G 2  with positive refractive power, a third lens group G 3  with negative refractive power, and a fourth lens group G 4  with positive refractive power, these lens groups being located on the optical axis Lc. 
     The first lens group G 1  is composed of a negative meniscus lens L 11  the convex surface of which faces toward the object side and a positive meniscus lens L 12  the convex surface of which faces toward the object side, in that order from the object side. And, the first lens group G 1  as a whole has negative refractive power. 
     The second lens group G 2  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 G 2  is composed of a biconvex positive lens L 21  which becomes the first lens element, an aperture stop S, a biconvex positive lens L 22  which becomes the second lens element, and a biconcave negative lens L 23  which becomes the third lens element and is joined to the biconvex positive lens L 22 , in that order from the object side. And, the second lens group G 2  has positive refractive power as a whole and has a main magnification change function. 
     The third lens group G 3  is composed of one biconcave negative lens L 3 . Besides, the biconcave negative lens L 3  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. 
     The fourth lens group G 4  is composed of one positive meniscus lens L 4  the convex surface of which faces toward the image side. 
     In changing magnification from the wide angle end position to the telephoto end position, the first lens group G 1  and the fourth lens group G 4  keep still, and the second lens group G 2  and the third lens group G 3  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 G 2  and G 3  may be moved. 
     Embodiment 5 
     The optical constitution of the variable power optical system of the present embodiment is explained using  FIGS. 17A to 17C . 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 G 1  with negative refractive power, a second lens group G 2  with positive refractive power, a third lens group G 3  with negative refractive power, and a fourth lens group G 4  with positive refractive power, these lens groups being located on the optical axis Lc. 
     The first lens group G 1  is composed of a negative meniscus lens L 11  the convex surface of which faces toward the object side and a positive meniscus lens L 12  the convex surface of which faces toward the object side, in that order from the object side. And, the first lens group G 1  as a whole has negative refractive power. 
     The second lens group G 2  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 L 21  which becomes the first lens element, an aperture stop S, a biconvex positive lens L 22  which becomes the second lens element, and a biconcave negative lens L 23  which becomes the third lens element and is joined to the biconvex positive lens L 22 , in that order from the object side. And, the second lens group G 2  has positive refractive power as a whole and has a main magnification change function. 
     The third lens group G 3  is composed of one biconcave negative lens L 3 . Besides, the biconcave negative lens L 3  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. 
     The fourth lens group G 4  is composed of one positive meniscus lens L 4  the convex surface of which faces toward the image side. 
     In changing magnification from the wide angle end position to the telephoto end position, the first lens group G 1  and the fourth lens group G 4  keep still, and the second lens group G 2  and the third lens group G 3  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 G 2  and G 3  may be moved. 
     Embodiment 6 
     The optical constitution of the variable power optical system of the present embodiment is explained using  FIGS. 21A to 21C . The total length of the variable power optical system of the present embodiment is about 13.5 mm. 
     The variable power optical system of the present embodiment includes, in order from the object side, a first lens group G 1  with negative refractive power, a second lens group G 2  with positive refractive power, a third lens group G 3  with negative refractive power, and a fourth lens group G 4  with positive refractive power, these lens group being located on the optical axis Lc. 
     The first lens group G 1  is composed of a biconcave negative lens L 11  and a positive meniscus lens L 12  which is jointed to the biconcave negative lens L 11  and the convex surface of which faces toward the object side, in that order from the object side. And, the first lens group G 1  as a whole has negative refractive power. When the L 12  is made of energy curable resin, it is possible to make the first lens group G 1  thin, so that it is possible to sufficiently secure an amount of movement of the second lens group G 2  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. 
     The second lens group G 2  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 G 2  is composed of a biconvex positive lens L 21  which becomes the first lens element, an aperture stop S, a biconvex positive lens L 22  which becomes the second lens element, and a biconcave negative lens L 23  which becomes the third lens element and is joined to the biconvex positive lens L 22 , in that order from the object side. And, the second lens group G 2  as a whole has positive refractive power. 
     The third lens group G 3  is composed of one biconcave negative lens L 3 . Besides, the biconcave negative lens element L 3  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. 
     The fourth lens group G 4  is composed of one positive meniscus lens L 4  the convex surface of which faces toward the image side. 
     In changing magnification from the wide angle end position to the telephoto end position, the first lens group G 1  and the fourth lens group G 4  keep still, the second lens group G 2  moves toward the object side, and the third lens group G 3  moves to the position nearest to the object side in the middle of the optical system. 
     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 G 2  and G 3  may be moved. 
     Embodiment 7 
     The optical constitution of the variable power optical system of the present embodiment is explained using  FIGS. 25A to 25C . 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 G 1  with negative refractive power, a second lens group G 2  with positive refractive power, a third lens group G 3  with negative refractive power, and a fourth lens group G 4  with positive refractive power, these lens groups being located on the optical axis Lc. 
     The first lens group G 1  is composed of a biconcave negative lens L 11  and a positive meniscus lens L 12  which is jointed to the biconcave negative lens L 11  and the convex surface of which faces toward the object side, in that order from the object side. And, the first lens group G 1  as a whole has negative refractive power. 
     When the L 12  is made of energy curable resin, it is possible to make the first lens group G 1  thin, so that it is possible to sufficiently secure an amount of movement of the second lens group G 2  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. 
     The second lens group G 2  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 G 2  is composed of a biconvex positive lens L 2  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 L 23  which becomes the third lens element and is jointed to the biconvex positive lens L 22 , in that order from the object side. And, the second lens group G 2  has positive refractive power as a whole and has a main magnification change function. 
     The third lens group G 3  is composed of one biconcave negative lens L 3 . Besides, the biconcave negative lens L 3  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. 
     The fourth lens group G 4  is composed of one positive meniscus lens L 4  the convex surface of which faces toward the image side. 
     In changing magnification from the wide angle end position to the telephoto end position, the first lens group G 1  and the fourth lens group G 4  keep still, and the second lens group G 2  moves toward the object side, and the third lens group G 3  moves to the position nearest to the object side in the middle of the optical system. 
     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 G 2  and G 3  may be moved. 
     Embodiment 8 
     The optical constitution of the variable power optical system of the present embodiment is explained using  FIGS. 28A to 28C . 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 G 1  with negative refractive power, a second lens group G 2  with positive refractive power, a third lens group G 3  with negative refractive power, and a fourth lens group G 4  with positive refractive power, these lens groups being located on the optical axis Lc. 
     The first lens group G 1  is composed of a biconcave negative lens L 11  and a positive meniscus lens L 12  which is jointed to the biconcave negative lens L 11  and the convex surface of which faces toward the object side, in that order from the object side. And, the first lens group G 1  as a whole has negative refractive power. 
     When the L 12  is made of energy curable resin, it is possible to make the first lens group G 1  thin, so that it is possible to sufficiently secure an amount of movement of the second lens group G 2  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. 
     The second lens group G 2  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 G 2  is composed of a biconvex positive lens L 21  which becomes the first lens element, an aperture stop S, a biconvex positive lens L 22  which becomes the second lens element, and a biconcave negative lens L 23  which becomes the third lens element and is joined to the biconvex positive lens L 22 , in that order from the object side. And, the second lens group G 2  has positive refractive power as a whole and has a main magnification change function. 
     The third lens group G 3  is composed of one negative meniscus lens L 3  the convex surface of which faces toward the object sides. Besides, the negative meniscus lens L 3  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. 
     The fourth lens group G 4  is composed of one positive meniscus lens L 4  the convex surface of which faces toward the image side. 
     In changing magnification from the wide angle end position to the telephoto end position, the first lens group G 1  and the fourth lens group G 4  keep still, and the second lens group G 2  and the third lens group G 3  move toward the object side. 
     Embodiment 9 
     The optical constitution of the variable power optical system of the present embodiment is explained using  FIGS. 31A to 31C . The total length of the variable power optical system of the present embodiment is about 13.5 mm. 
     The variable power optical system of the present embodiment includes, in order from the object side, a first lens group G 1  with negative refractive power, a second lens group G 2  with positive refractive power, a third lens group G 3  with negative refractive power, and a fourth lens group G 4  with positive refractive power, these lens groups being located on the optical axis Lc. 
     The first lens group G 1  is composed of a biconcave negative lens L 11  and a positive meniscus lens L 12  which is jointed to the biconcave negative lens L 11  and the convex surface of which faces toward the object side, in that order from the object side. And, the first lens group G 1  as a whole has negative refractive power. 
     When the L 12  is made of energy curable resin, it is possible to make the first lens group G 1  thin, so that it is possible to sufficiently secure an amount of movement of the second lens group G 2  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. 
     The second lens group G 2  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 L 21  which becomes the first lens element, an aperture stop S, a biconvex positive lens L 22  which becomes the second lens element, and a biconcave negative lens L 23  which becomes the third lens element and is joined to the biconvex positive lens L 22 , in that order from the object side. And, the second lens group G 2  has positive refractive power as a whole and has a main magnification change function. 
     The third lens group G 3  is composed of one biconcave negative lens L 3 . Besides, the biconcave negative lens L 3  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. 
     The fourth lens group G 4  is composed of one positive meniscus lens L 4  the convex surface of which faces toward the image side. 
     In changing magnification from the wide angle end position to the telephoto end position, the first lens group G 1  keeps still, the second lens group G 2  moves toward the object side, the third lens group G 3  moves to the position nearest to the object side in the middle of the optical system, and the fourth lens group G 4  moves toward the image side. 
     Embodiment 10 
     The optical constitution of the variable power optical system of the present embodiment is explained using  FIGS. 34A to 34C . The total length of the variable power optical system of the present embodiment is about 10 mm. 
     The variable power optical system of the present embodiment includes, in order from the object side, a first lens group G 1  with negative refractive power, a second lens group G 2  with positive refractive power, a third lens group G 3  with negative refractive power, and a fourth lens group G 4  with positive refractive power, these lens group being located on the optical axis Lc. 
     The first lens group G 1  is composed of a biconcave negative lens L 11  and a positive meniscus lens L 12  the convex surface of which faces toward the object side, in that order from the object side. And, the first lens group G 1  as a whole has negative refractive power. 
     The second lens group G 2  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 G 2  is composed of a biconvex positive lens L 21  which becomes the first lens element, an aperture stop S, a biconvex positive lens L 22  which becomes the second lens element, and a biconcave negative lens L 23  which becomes the third lens element and is joined to the biconvex positive lens L 22 , in that order from the object side. And, the second lens group G 2  has positive refractive power as a whole and has a main magnification change function. 
     The third lens group G 3  is composed of one biconcave negative lens L 3 . Besides, the biconcave negative lens element L 3  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. 
     The fourth lens group G 4  is composed of one positive meniscus lens L 4  the convex surface of which faces toward the image side. 
     In changing magnification from the wide angle end position to the telephoto end position, the first lens group G 1  and the fourth lens group G 4  keep still, and the second lens group G 2  and the third lens group G 3  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 G 2  and G 3  may be moved. 
     Embodiment 11 
     The optical constitution of the variable power optical system of the present embodiment is explained using  FIGS. 37A to 37C . The total length of the variable power optical system of the present embodiment is about 10 mm. 
     The variable power optical system of the present embodiment includes, in order from the object side, a first lens group G 1  with negative refractive power, a second lens group G 2  with positive refractive power, a third lens group G 3  with negative refractive power, and a fourth lens group G 4  with positive refractive power, these lens groups being located on the optical axis Lc. 
     The first lens group G 1  is composed of a negative meniscus lens L 11  the convex surface of which faces toward the object side and a positive meniscus lens L 12  the convex surface of which faces toward the object side, in that order from the object side. And, the first lens group G 1  as a whole has negative refractive power. 
     The second lens group G 2  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 G 2  is composed of a biconvex positive lens L 21  which becomes the first lens element, an aperture stop S, a biconvex positive lens L 22  which becomes the second lens element, and a biconcave negative lens L 23  which becomes the third lens element and is jointed to the biconvex positive lens L 22 , in that order from the object side. And, the third lens group G 2  has positive refractive power as a whole and has a main magnification change function. 
     The third lens group G 3  is composed of one negative meniscus lens L 3  the concave surface of which faces toward the object side. Besides, the negative meniscus lens L 3  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. 
     The fourth lens group G 4  is composed of one positive meniscus lens L 4  the convex surface of which faces toward the image side. 
     In changing magnification from the wide angle end position to the telephoto end position, the first lens group G 1  and the fourth lens group G 4  keep still, and the second lens group G 2  and the third lens group G 3  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 G 2  and G 3  may be moved. 
     Embodiment 12 
     The optical constitution of the variable power optical system of the present embodiment is explained using  FIGS. 40A to 40C . The total length of the variable power optical system of the present embodiment is about 14 mm. 
     The variable power optical system of the present embodiment includes, in order from the object side, a first lens group G 1  with negative refractive power, a second lens group G 2  with positive refractive power, a third lens group G 3  with negative refractive power, and a fourth lens group G 4  with positive refractive power, these lens groups being located on the optical axis Lc. 
     The first lens group G 1  is composed of one biconcave negative lens L 1 . 
     The second lens group G 2  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 G 2  is composed of a biconvex positive lens L 21  which becomes the first lens element, a biconcave negative lens L 22  which becomes the second lens element and is joined to the biconvex positive lens L 21 , an aperture stop S, and a biconvex positive lens L 23  which becomes the third lens element, in that order from the object side. And, the second lens group G 2  has positive refractive power as a whole and has a main magnification change function. 
     The third lens group G 3  is composed of one negative meniscus lens L 3  the convex surface of which faces toward the object side. Besides, the negative meniscus lens L 3  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. 
     The fourth lens group G 4  is composed of one positive meniscus lens L 4  the convex surface of which faces toward the image side. 
     In changing magnification from the wide angle end position to the telephoto end position, the first lens group G 1  and the fourth lens group G 4  keep still, and the second lens group G 2  and the third lens group G 3  move toward the object side. 
     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. 
     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. 
     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 A 4 , A 6 , A 8 , and A 10  denote an aspherical coefficient, the shapes of aspherical surfaces are expressed by the following formula (I):
 
 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)
 
     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. 
     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#. 
     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                Wide angle end   Middle   Telephoto end                Wide angle end   Middle   Telephoto 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 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   −5.731   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                    
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                    
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                    
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                    
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                    
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                    
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 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   −6.977   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                        
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 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   −7.196   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                        
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 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   −7.490   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                        
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 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   −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                        
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 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   −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                        
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 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   −6.854   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                        
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.
 
Parameter values which the embodiments 1 to 12 have in the respective conditions
 
     
       
         
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
             
           
               
                   
               
             
             
               
                   
                 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 
                   
                 Condition 
                   
               
               
                   
                 (15) 
                   
                 (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 
               
               
                   
                   
               
             
          
         
       
     
     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. 
       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. 
     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. 
     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. 
     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.    
     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.