Abstract:
A variable magnification optical system having an image stabilizing function includes, in order from an object side, a first lens unit of negative refractive power, a second lens unit of positive refractive power, a third lens unit of negative refractive power and a fourth lens unit of positive refractive power, wherein the separation between every successive two of the first to fourth lens units is varied to effect variation of magnification, and the third lens unit is so moved as to have directional components perpendicular to an optical axis to compensate for shaking of an image caused by vibration of the variable magnification optical system, the variable magnification optical system satisfying the following conditions: 
     
       
         
           D 
           1 
           W&gt;D 
           1 
           T 
         
       
     
     
       
         
           D 
           2 
           W&lt;D 
           2 
           T 
         
       
     
     
       
         
           D 
           3 
           W&gt;D 
           3 
           T 
         
       
     
     
       
         1.5&lt;| f 3 |/f 2&lt;2.5 
       
     
     
       
         2.1&lt; f 4 /f 2&lt;10 
       
     
     where DiW and DiT are separations between the i-th lens unit and the (i+1)st lens unit at a wide-angle end and at a telephoto end, respectively, and fi is a focal length of the i-th lens unit.

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to variable magnification optical systems and, more particularly, to a variable magnification optical system having an image stabilizing function in which one lens unit, constituting a part of the variable magnification optical system, is so moved as to have directional components perpendicular to an optical axis to optically compensate for the shaking of a photographed image caused by the vibration (tilting) of the variable magnification optical system, thereby obtaining a stabilized image, and which optical system is suited to photographic cameras, electronic cameras, video cameras or the like. 
     2. Description of Related Art 
     When shooting is being performed by using a photographic system on a running car, or flying air plane, or the like moving vehicle, the vibrations propagate to the photographic system, thereby causing the shaking of a photographed image in some cases. Also, when not on the moving vehicle, but when the photographer unstably holds the photographic system by the hand, it often happens that the vibration of the hand causes the shaking of a photographed image. As the focal length of the photographic system increases to the telephoto region, in particular, the frequency of occurrence of the shaking of a photographed image increases rapidly. 
     Heretofore, there have been proposed a variety of optical systems having an image stabilizing function of preventing the shaking of a photographed image from occurring in such situations. 
     For example, in Japanese Laid-Open Patent Application No. Hei 5-232410, there is disclosed a zoom lens of the telephoto type comprising four lens units of positive, negative, positive and positive refractive powers in this order from the object side, wherein the second lens unit is made to move in directions perpendicular to an optical axis, thereby stabilizing the image. 
     In another Japanese Laid-Open Patent Application No. Hei 7-152002, there is disclosed a zoom lens comprising four lens units of negative, positive, negative and positive refractive powers in this order from the object side, wherein the third lens unit is made to move in directions perpendicular to an optical axis, thereby stabilizing the image. 
     Also, in Japanese Laid-Open Patent Application No. Hei 7-199124, there is disclosed a variable magnification optical system comprising four lens units of positive, negative, positive and positive refractive powers in this order from the object side, wherein the entirety of the third lens unit is made to move in directions perpendicular to an optical axis, thereby stabilizing the image. 
     In general, an image stabilizing optical system in which a lens constituting a part of the optical system is made to decenter in parallel with directions perpendicular to the optical axis so as to stabilize the image, has such an advantage that there is no need to use an additional special optical element for stabilizing the image. 
     However, this method requires creation of a space in which to move the moving lens unit. Another problem of this method is that, if an improper one of the lens units is selected to decenter in parallel, the produced amount of decentering aberrations becomes larger. 
     BRIEF SUMMARY OF THE INVENTION 
     The present invention is applied to a variable magnification optical system in which a lens unit constituting a part of the variable magnification optical system is so moved as to have directional components perpendicular to an optical axis to compensate for the shaking of an image caused by the vibration (tilting) of the variable magnification optical system, and sets forth appropriate rules of design for the form and the construction and arrangement of the constituent lenses of the variable magnification optical system. It is, therefore, an object of the invention to provide a variable magnification optical system having an image stabilizing function with the bulk and size of the entire system reduced to a compact form, with the operating system simplified in structure and with the load on the driving means lowered, while, when the lens unit is made to decenter to compensate for the shaking of an image, still permitting the produced amount of decentering aberrations to be suppressed to a minimum. 
     To attain the above object, in accordance with an aspect of the invention, there is provided a variable magnification optical system having an image stabilizing function, which comprises, in order from an object side, a first lens unit of negative refractive power, a second lens unit of positive refractive power, a third lens unit of negative refractive power and a fourth lens unit of positive refractive power, wherein the separation between every successive two of the first to fourth lens units is varied to effect variation of magnification, and the third lens unit is so moved as to have directional components perpendicular to an optical axis to compensate for shaking of an image caused by vibration of the variable magnification optical system, the variable magnification optical system satisfying the following conditions: 
     
       
           D   1   W&gt;D   1   T   (1) 
       
     
     
       
           D   2   W&lt;D   2   T   (2) 
       
     
     
       
           D   3   W&gt;D   3   T   (3) 
       
     
     
       
         1.5 &lt;|f 3|/ f 2&lt;2.5  (4) 
       
     
     
       
         2.1 &lt;f 4 /f 2&lt;10  (5) 
       
     
     where DiW and DiT are separations between the i-th lens unit and the (i+1)st lens unit in a wide-angle end and in a telephoto end, respectively, and fi is the focal length of the i-th lens unit. 
    
    
     The above and further objects and features of the invention will become apparent from the following detailed description of preferred embodiments thereof taken in conjunction with the accompanying drawings. 
     BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING 
     FIG. 1 is a longitudinal section view of a numerical example 1 of the variable magnification optical system at the wide-angle end. 
     FIG. 2 is a longitudinal section view of a numerical example 2 of the variable magnification optical system at the wide-angle end. 
     FIG. 3 is a longitudinal section view of a numerical example 3 of the variable magnification optical system at the wide-angle end. 
     FIG. 4 is a longitudinal section view of a numerical example 4 of the variable magnification optical system at the wide-angle end. 
     FIGS.  5 A 1  and  5 A 2  are graphic representations of the aberrations for the normal state of the variable magnification optical system of the numerical example 1 at the wide-angle end, and FIGS.  5 B 1  to  5 B 3  are similar graphs obtained during the image stabilizing operation for the tilting of 0.5° of the variable magnification optical system. 
     FIGS.  6 A 1  and  6 A 2  are graphic representations of the aberrations for the normal state of the variable magnification optical system of the numerical example 1 at the telephoto end, and FIGS.  6 B 1  to  6 B 3  are similar graphs obtained during the image stabilizing operation for the tilting of 0.5° of the variable magnification optical system. 
     FIGS.  7 A 1  and  7 A 2  are graphic representations of the aberrations for the normal state of the variable magnification optical system of the numerical example 2 at the wide-angle end, and FIGS.  7 B 1  to  7 B 3  are similar graphs obtained during the image stabilizing operation for the tilting of 0.5° of the variable magnification optical system. 
     FIGS.  8 A 1  and  8 A 2  are graphic representations of the aberrations for the normal state of the variable magnification optical system of the numerical example 2 at the telephoto end, and FIGS.  8 B 1  to  8 B 3  are similar graphs obtained during the image stabilizing operation for the tilting of 0.5° of the variable magnification optical system. 
     FIGS.  9 A 1  and  9 A 2  are graphic representations of the aberrations for the normal state of the variable magnification optical system of the numerical example 3 in the wide-angle end, and FIGS.  9 B 1  to  9 B 3  are similar graphs obtained during the image stabilizing operation for the tilting of 0.5° of the variable magnification optical system. 
     FIGS.  10 A 1  and  10 A 2  are graphic representations of the aberrations for the normal state of the variable magnification optical system of the numerical example 3 at the telephoto end, and FIGS.  10 B 1  to  10 B 3  are similar graphs obtained during the image stabilizing operation for the tilting of 0.5° of the variable magnification optical system. 
     FIGS.  11 A 1  and  11 A 2  are graphic representations of the aberrations for the normal state of the variable magnification optical system of the numerical example 4 at the wide-angle end, and FIGS.  11 B 1  to  11 B 3  are similar graphs obtained during the image stabilizing operation for the tilting of 0.5° of the variable magnification optical system. 
     FIGS.  12 A 1  and  12 A 2  are graphic representations of the aberrations for the normal state of the variable magnification optical system of the numerical example 4 at the telephoto end, and FIGS.  12 B 1  to  12 B 3  are similar graphs obtained during the image stabilizing operation for the tilting of 0.5° of the variable magnification optical system. 
     FIG. 13 is a schematic diagram of the main parts of a camera with the variable magnification optical system of any one of the numerical examples 1 to 4 applied to the photographic lens thereof. 
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Hereinafter, preferred embodiments of the invention will be described in detail with reference to the drawings. 
     FIGS. 1 to  4  are sectional views respectively showing first to fourth numerical examples 1 to 4 of the variable magnification optical system in the wide-angle end, whose data will be given later. 
     In FIGS. 1 to  4 , the variable magnification optical system comprises, in order from an object side, a first lens unit L 1  of negative refractive power, a second lens unit L 2  of positive refractive power, a third lens unit L 3  of negative refractive power and a fourth lens unit L 4  of positive refractive power. Reference symbol SP denotes a stop, and reference symbol P denotes a flare cutter. 
     In the numerical examples 1, 2 and 4 shown in FIGS. 1,  2  and  4 , respectively, during zooming from the wide-angle end to the telephoto end, the first lens unit L 1  axially moves in a locus convex toward the image side, and the second lens unit L 2  to the fourth lens unit L 4  axially move toward the object side, as shown by the arrows. 
     In the numerical example  3  shown in FIG. 3, during zooming from the wide-angle end to the telephoto end, the first lens unit L 1  axially moves in a locus convex toward the image side, and the second lens unit L 2  and the fourth lens unit L 4  axially move toward the object side, as shown by the arrows. In all the numerical examples 1 to 4, the second lens unit L 2  and the fourth lens unit L 4  move integrally with each other. 
     During zooming, the stop SP moves integrally with the third lens unit L 3 . 
     In the present embodiments, the third lens unit L 3  is made to move in directions perpendicular to an optical axis to compensate for the shaking of an image caused by the vibration of the variable magnification optical system. Incidentally, the stop SP remains stationary during the image stabilizing operation. 
     In the present embodiments, the stop SP is disposed just before the third lens unit L 3  and is moved integrally with the third lens unit L 3  during zooming, thereby lessening the variation of the aberrations attributable to the movable lens units. Also, the separation between the lens units that precede the stop SP, i.e., the first lens unit L 1  and the second lens unit L 2 , is made shorter as zooming goes to the telephoto end, thereby making it easy to achieve reduction of the diameter of the front lens members. Focusing is performed by moving the first lens unit L 1 , but one of the other lens units may be used instead. 
     A variable magnification optical system having an image stabilizing function according to the invention is attained with all the lens units arranged in such a manner as described above, during zooming and during the image stabilizing operation. Moreover, the above-described conditions (1) to (5) are satisfied to thereby simultaneously fulfill the requirements of advantageously securing the desired zoom ratio and of minimizing the decentering aberrations during the image stabilizing operation. The difference in image quality between in the normal state (i.e., when the image stabilizing operation is not performed) and during the image stabilizing operation is thus reduced to a minimum for a good optical performance. 
     The technical significance of each of the above-described conditions (1) to (5) is explained below. 
     The inequalities of conditions (1) to (3) specify the relation in which all the lens units of certain refractive powers axially move during zooming from the wide-angle end to the telephoto end. By this relation, the zoom ratio is secured at a predetermined value with a high efficiency, while still assuring minimization of the size of the entire optical system. 
     The inequalities of conditions (4) and (5) give ranges for the focal lengths of the third lens unit L 3  and the fourth lens unit L 4  relative to the focal length of the second lens unit L 2  and have an aim chiefly to maintain good stability of optical performance during the image stabilizing operation. When the lower limits of the conditions (4) and (5) are exceeded, as this means that the refractive powers of the third lens unit L 3  and the fourth lens unit L 4  are too strong, large decentering aberrations are objectionably produced during the image stabilizing operation. When the refractive powers of the third lens unit L 3  and the fourth lens unit L 4  become too weak beyond the upper limits of the conditions (4) and (5), the total length of the entire optical system becomes undesirably longer. 
     Next, the way of finding out a paraxial refractive power arrangement for satisfying the above-described conditions is described. Letting the powers at the wide-angle end and the telephoto end of the entire optical system be denoted by φw and φt, respectively, letting the power of the first lens unit L 1  be denoted by φ 1 , letting the power of the second lens unit L 2  be denoted by φ 2 , and letting the principal point intervals at the wide-angle end and the telephoto end between the i-th lens unit and the (i+1)st lens unit be denoted by Eiw and Eit, respectively, for given values of these parameters, the power φ 3  of the third lens unit L 3  and the power φ 4  of the fourth lens unit L 4  as unknown quantities are determined to obtain Gaussian brackets Aijw, Cijw and Aijt, Cijt of the i-th lens unit to the (i+1)st lens unit at the wide-angle end and the telephoto end. Then, the following expressions are obtained: 
     
       
         φ w=C   14   w=−A   13   w·E   3   w ·φ 3 ·φ 4 − C   12   w·E 3 w ·φ4 +A   13   w ·φ 4   +A   13   w ·φ 3 + C   12   w   (a) 
       
     
     
       
         φ t=C   14   t=−A   13   t·E   3   t·φ   3 ·φ 4   −C   12   t·E   3   t·φ   4   +A   13   t·φ   4 + A   13   t·φ   3 + C   12   t   (b) 
       
     
     Vw≡C 12 w·E 3 w−A 13 w 
     Ww≡φw−C 12 w 
     Xw≡−E 3 w·A 13 w 
     Yw≡A 13 w 
     Vt≡C 12 t·E 3 t−A 13 t 
     wt≡φt−C 12 t 
     Xt≡−E 3 t·A 13 t 
     Yt≡A 13 t 
     From the equations (a) and (b) described above, the following equations are obtained: 
     
       
         φ 3 =( Vw·φ   4 + Ww )/( Xw·φ   4 + Yw )  (c) 
       
     
     
       
         φ 3 =( Vt·φ   4 + Wt )/( Xt·φ   4 + Yt )  (d) 
       
     
     From the equations (c) and (d), φ 4  is found as two solutions of the following equation: 
     
       
         ( VwXt−VtXw )φ 42 +( VwYt+XtWw−VtYw−XwWt )φ 4 +( WwYt−WtYw )=0  (e) 
       
     
     For φ 3 , too, from the equation (c), two values are found in correspondence to the respective values of φ 4  described above. 
     In the present embodiments, among the two pairs of solutions, the weaker pair of solutions in refractive power is selected to be given to the third lens unit L 3  and the fourth lens unit L 4 , thereby making it possible to correct aberrations well even during the image stabilizing operation. 
     The above-described conditions suffice for realizing a variable magnification optical system having an image stabilizing function according to the invention. However, to further improve the optical performance with the limitation of the total length of the entire optical system to a minimum, it is preferable to satisfy at least one of the following features or conditions. 
     (a-1) The third lens unit L 3  consists of a positive lens and a negative lens, and the fourth lens unit L 4  includes a negative lens and a positive lens. 
     By this arrangement, on the premise of selecting such a paraxial refractive power arrangement as described above, the aberrations are corrected well. 
     (a-2) The distance OK 4  (from the most image side surface of the fourth lens unit) of the rear principal point of the fourth lens unit L 4  lies within the following range: 
     
       
         0&lt; OK   4   (6) 
       
     
     The term “distance OK 4 ” used herein is taken as positive when measured from the fourth lens unit L 4  to the image side. The inequality of condition (6) is for constructing the fourth lens unit L 4  in the form of the retro type. With this arrangement, while the bulk and size of the entire optical system are minimized, all aberrations are corrected well. 
     (a-3) The fourth lens unit L 4  has at least one aspheric surface. 
     In this case, the aspheric surface is applied to a convex surface on the image side of the positive lens of the fourth lens unit L 4  and is formed to such a shape that the negative refractive power becomes progressively stronger as the distance increases from the center of the lens to the margin, thereby correcting well the sagittal field curvature of higher orders that is produced in the wide-angle region. 
     (a-4) The stop SP is provided as disposed on the object side or image side of the third lens unit L 3  to remain stationary (i.e., not to decenter in parallel) during the image stabilizing operation. 
     This enables the outer diameters of the first lens unit L 1  and the fourth lens unit L 4  to be minimized and assures good balance of the corrected aberrations. 
     (a-5) The second lens unit L 2  and the fourth lens unit L 4  move integrally with each other during zooming. 
     This assists in simplifying the structure of construction of the zooming mechanism, while securing the predetermined zoom ratio. 
     (a-6) In order from the object side, the first lens unit L 1  consists of a negative lens of meniscus form concave toward the image side, a negative lens having a concave surface facing the image side and a positive lens of meniscus form convex toward the object side, the second lens unit L 2  consists of a negative lens of meniscus form concave toward the image side, a positive lens of bi-convex form and a positive lens having a convex surface facing the object side, the third lens unit L 3  consists of a positive lens of meniscus form convex toward the image side and a negative lens of bi-concave form, and the fourth lens unit L 4  consists of a negative lens of meniscus form concave toward the image side and a positive lens of bi-convex form. 
     This arrangement improves the compact form of the entire optical system, simplifies the structure of the mounting mechanism, and reduces the load on the operating mechanism, while, when the lens unit L 3  is made to decenter for stabilizing an image, still permitting the produced amount of decentering aberrations to be suppressed to a minimum. 
     (a-7) Letting the sensitivity to parallel decentering of the third lens unit L 3  at the telephoto end be denoted by TS 3 t, letting the focal lengths in the wide-angle end and the telephoto end of the entire optical system be denoted by fW and fT, respectively, letting the F-number in the telephoto end of the entire optical system be denoted by FNot, and letting the amounts of variation of the separations between the first lens unit L 1  and the second lens unit L 2  and between the second lens unit L 2  and the third lens unit L 3  during zooming from the wide-angle end to the telephoto end be denoted by Δ 12  and Δ 23 , respectively, at least one of the following conditions is satisfied: 
     
       
         1.1&lt;| TS 3 t|   (7) 
       
     
     
       
         0.6&lt;| f 1 |/{square root over (fW·ft )}&lt; 1.3  (8) 
       
     
     
       
         1.3&lt; f 2· FNot/fT&lt; 3.5  (9) 
       
     
     
       
         0.2&lt;|Δ23/Δ12|&lt;0.6  (10) 
       
     
     In order to limit the consumption of energy to a minimum in the driving motion of the lens unit for stabilizing an image, it is necessary to decrease the size of the moving mechanism and the weight of the image-stabilizing lens unit. Nonetheless, it is also necessary to increase the sensitivity to parallel decentering of the image-stabilizing lens unit. 
     The term “sensitivity to parallel decentering” herein used means the ratio of the amount of movement of the image point on the focal plane to the amount of movement of the lens unit perpendicular to the optical axis. 
     In general, the sensitivity TSi to parallel decentering of the i-th lens unit is calculated by the following formula: 
     
       
           TSi=( 1 −βi )βi+1·  ·β n   
       
     
     where βi is the magnification of the i-th lens unit. 
     In the present embodiments, such a lens arrangement as to satisfy the condition (7) is adopted. When the lower limit of the condition (7) is exceeded, as this means that the sensitivity to parallel decentering of the image-stabilizing lens unit is too small, and the moving mechanism for stabilizing an image increases in size objectionably. To lighten the weight of the image-stabilizing lens unit, the aperture stop is fixed in the direction perpendicular to the optical axis during the image stabilizing operation. 
     The conditions (8), (9) and (10) in application to the 4-unit zoom lens of minus-plus-minus-plus refractive power arrangement, have an aim to achieve the desired specification, while still permitting the image quality to be improved to a satisfactory level with the bulk and size of the entire optical system reduced to a compact form. 
     When the lower limit of the condition (8) is exceeded, as this means that the negative refractive power of the first lens unit L 1  is too strong, it becomes difficult to correct aberrations well. When the negative refractive power of the first lens unit L 1  is too weak beyond the upper limit, the size of the optical system increases objectionably. When the lower limit of the condition (9) is exceeded, as this means that the positive refractive power of the second lens unit L 2  is too strong, it favors the shortening of the total length of the entire optical system, but all aberrations the second lens unit L 2  produces increase, which become difficult to correct well in good balance. When the upper limit is exceeded, the total length of the entire optical system increases objectionably. When the lower limit of the condition (10) is exceeded, as this means that the amount of variation of the separation between the second lens unit L 2  and the third lens unit L 3  during zooming from the wide-angle end to the telephoto end is shorter than that of the separation between the first lens unit L 1  and the second lens unit L 2 , the configuration becomes close to the 2-unit zoom lens of minus-plus refractive power arrangement, so that the advantage of improving the compact form by adopting the multi-unit technique diminishes. When the upper limit is exceeded, the outer diameter of the fourth lens unit L 4  increases objectionably. 
     Next, four numerical examples 1 to 4 of the invention are shown. In the numerical data for the examples 1 to 4, ri is the radius of curvature of the i-th surface when counted from the object side, di is the i-th lens thickness or air separation when counted from the object side, and ni and vi are respectively the refractive index and Abbe number of the glass of the i-th lens element when counted from the object side. 
     The values of the factors in the above-described conditions (4) to (10) for the numerical examples 1 to 4 are listed in Table-1. 
     The shape of the aspheric surface is expressed in the coordinates with an X axis in the axial direction and an H axis in the direction perpendicular to the optical axis, the direction in which light advances being taken as positive, by the following equation:        X   =           (     1   /   R     )          H   2         1   +       1   -       (     H   /   R     )     2             +     b                   H   4       +     c                   H   6       +     d                   H   8       +     e                   H   10                                
     where R is the radius of the osculating sphere, and b, c, d and e are the aspheric coefficients. The values of the aspheric coefficients are also tabulated where the notation “e-0X” means “10 −X ”. 
     Numerical Example 1: 
     
       
         
               
               
               
             
               
               
               
               
             
           
               
                   
               
             
             
               
                 f = 29.6˜81.5 
                 Fno = 4.1˜5.9 
                 2ω = 72.3°˜29.7° 
               
             
          
           
               
                  r 1 =  43.437 
                 d 1 = 1.80 
                 n 1 = 1.69680 
                 ν 1 = 55.5 
               
               
                  r 2 =  21.458 
                 d 2 = 6.56 
               
               
                  r 3 = 229.741 
                 d 3 = 1.50 
                 n 2 = 1.72916 
                 ν 2 = 54.7 
               
               
                  r 4 =  30.449 
                 d 4 = 1.90 
               
               
                  r 5 =  25.886 
                 d 5 = 2.73 
                 n 3 = 1.84666 
                 ν 3 = 23.8 
               
               
                  r 6 =  37.968 
                 d 6 = Variable 
               
               
                  r 7 =  24.231 
                 d 7 = 1.30 
                 n 4 = 1.80518 
                 ν 4 = 25.4 
               
               
                  r 8 =  14.955 
                 d 8 = 5.00 
                 n 5 = 1.55963 
                 ν 5 = 61.2 
               
               
                  r 9 = −88.239 
                 d 9 = 0.15 
               
               
                  r10 =  31.026 
                 d10 = 2.00 
                 n 6 = 1.69680 
                 ν 6 = 55.5 
               
               
                  r11 = 224.789 
                 d11 = Variable 
               
               
                  r12 = ∞ (stop) 
                 d12 = 1.75 
               
               
                  r13 = −33.283 
                 d13 = 1.70 
                 n 7 = 1.84666 
                 ν 7 = 23.8 
               
               
                  r14 = −16.122 
                 d14 = 1.10 
                 n 8 = 1.60562 
                 ν 8 = 43.7 
               
               
                  r15 =  39.023 
                 d15 = Variable 
               
               
                  r16 =  77.037 
                 d16 = 1.30 
                 n 9 = 1.84666 
                 ν 9 = 23.8 
               
               
                  r17 =  30.801 
                 d17 = 1.60 
               
               
                  r18 =  35.956 
                 d18 = 4.00 
                 n10 = 1.51633 
                 ν10 = 64.1 
               
               
                 *r19 = −39.173 
                 d19 = Variable 
               
               
                  r20 = ∞ 
               
               
                   
               
               
                 *) Aspheric Surface  
               
             
          
         
       
     
     
       
         
               
               
               
             
               
               
               
               
               
             
           
               
                   
                   
               
               
                   
                 Variable 
                 Focal Length 
               
             
          
           
               
                   
                 Separation 
                 29.62 
                 54.22 
                 81.49 
               
               
                   
                   
               
               
                   
                 d 6 
                 34.28 
                 10.47 
                  1.13 
               
               
                   
                 d11 
                  2.14 
                  5.93 
                  9.72 
               
               
                   
                 d15 
                  8.62 
                  4.83 
                  1.04 
               
               
                   
                 d19 
                  0.00 
                 14.49 
                 28.98 
               
               
                   
                   
               
             
          
         
       
     
     Aspheric Coefficients: 
     
       
         
               
               
               
             
           
               
                   
               
             
             
               
                 r19: 
                 b = 1.416455e-05 
                 c = 2.139864e-09 
               
               
                   
                 d = 9.652477e-10 
                 e = −5.607167e-12 
               
               
                   
               
             
          
         
       
     
     Numerical Example 2: 
     
       
         
               
               
               
             
               
               
               
               
             
           
               
                   
               
             
             
               
                 f = 29.0˜81.9 
                 Fno = 4.1˜5.9 
                 2ω = 73.4°˜29.6° 
               
             
          
           
               
                  r 1 =  61.487 
                 d 1 = 1.80 
                 n 1 = 1.69680 
                 ν 1 = 55.5 
               
               
                  r 2 =  27.511 
                 d 2 = 5.06 
               
               
                  r 3 = 307.265 
                 d 3 = 1.50 
                 n 2 = 1.69680 
                 ν 2 = 55.5 
               
               
                  r 4 =  26.277 
                 d 4 = 1.90 
               
               
                  r 5 =  24.934 
                 d 5 = 2.73 
                 n 3 = 1.84666 
                 ν 3 = 23.8 
               
               
                  r 6 =  34.627 
                 d 6 = Variable 
               
               
                  r 7 =  25.731 
                 d 7 = 1.30 
                 n 4 = 1.80518 
                 ν 4 = 25.4 
               
               
                  r 8 =  15.576 
                 d 8 = 5.00 
                 n 5 = 1.55963 
                 ν 5 = 61.2 
               
               
                  r 9 = −55.177 
                 d 9 = 0.15 
               
               
                  r10 =  26.449 
                 d10 = 2.00 
                 n 6 = 1.69680 
                 ν 6 = 55.5 
               
               
                  r11 =  89.109 
                 d11 = Variable 
               
               
                  r12 = ∞ (Stop) 
                 d12 = 1.75 
               
               
                  r13 = −31.195 
                 d13 = 1.70 
                 n 7 = 1.84666 
                 ν 7 = 23.8 
               
               
                  r14 = −15.143 
                 d14 = 1.10 
                 n 8 = 1.60562 
                 ν 8 = 43.7 
               
               
                  r15 =  38.686 
                 d15 = Variable 
               
               
                  r16 =  93.992 
                 d16 = 1.30 
                 n 9 = 1.84666 
                 ν 9 = 23.8 
               
               
                  r17 =  32.153 
                 d17 = 1.60 
               
               
                  r18 =  33.217 
                 d18 = 4.00 
                 n10 = 1.51633 
                 ν10 = 64.1 
               
               
                 *r19 = −39.152 
                 d19 = Variable 
               
               
                  r20 = ∞ 
               
               
                   
               
               
                 *) Aspheric Surface  
               
             
          
         
       
     
     
       
         
               
               
               
             
               
               
               
               
               
             
           
               
                   
                   
               
               
                   
                 Variable 
                 Focal Length 
               
             
          
           
               
                   
                 Separation 
                 29.04 
                 54.00 
                 81.87 
               
               
                   
                   
               
               
                   
                 d 6 
                 32.92 
                  9.94 
                  1.20 
               
               
                   
                 d11 
                  2.14 
                  7.22 
                 12.29 
               
               
                   
                 d15 
                 11.38 
                  6.30 
                  1.23 
               
               
                   
                 d19 
                  0.00 
                 12.96 
                 25.93 
               
               
                   
                   
               
             
          
         
       
     
     Aspheric Coefficients: 
     
       
         
               
               
               
             
           
               
                   
               
             
             
               
                 r19: 
                 b = 1.495083e-05 
                 c = 7.974152e-08 
               
               
                   
                 d = −1.602771e-10 
                 e = 3.818354e-13 
               
               
                   
               
             
          
         
       
     
     Numerical Example 3 
     
       
         
               
               
               
             
               
               
               
               
             
           
               
                   
               
             
             
               
                 f = 29.6˜74.8 
                 Fno = 3.5˜4.5 
                 2ω = 72.2°˜32.2° 
               
             
          
           
               
                  r 1 =  59.321 
                 d 1 = 1.80 
                 n 1 = 1.77250 
                 ν 1 = 49.6 
               
               
                  r 2 =  23.928 
                 d 2 = 6.90 
               
               
                  r 3 = −281.686 
                 d 3 = 1.60 
                 n 2 = 1.77250 
                 ν 2 = 49.6 
               
               
                  r 4 =  46.196 
                 d 4 = 0.10 
               
               
                  r 5 =  33.427 
                 d 5 = 3.90 
                 n 3 = 1.80518 
                 ν 3 = 25.4 
               
               
                  r 6 =  83.983 
                 d 6 = Variable 
               
               
                  r 7 =  58.116 
                 d 7 = 1.45 
                 n 4 = 1.84666 
                 ν 4 = 23.8 
               
               
                  r 8 =  26.317 
                 d 3 = 0.12 
               
               
                  r 9 =  27.522 
                 d 9 = 4.00 
                 n 5 = 1.69680 
                 ν 5 = 55.5 
               
               
                  r10 = −134.358 
                 d10 = 0.10 
               
               
                  r11 =  36.762 
                 d11 = 3.15 
                 n 6 = 1.69680 
                 ν 6 = 55.5 
               
               
                  r12 = −228.351 
                 d12 = Variable 
               
               
                  r13 = ∞ (Stop) 
                 d13 = 1.75 
               
               
                  r14 =  −47.294 
                 d14 = 2.10 
                 n 7 = 1.85026 
                 ν 7 = 32.3 
               
               
                  r15 =  −23.767 
                 d15 = 1.30 
                 n 8 = 1.60311 
                 ν 8 = 60.6 
               
               
                  r16 =  50.584 
                 d16 = Variable 
               
               
                  r17 =  168.851 
                 d17 = 1.00 
                 n 9 = 1.71736 
                 ν 9 = 29.5 
               
               
                  r18 =  29.281 
                 d18 = 0.40 
               
               
                  r19 =  32.347 
                 d19 = 6.00 
                 n10 = 1.60311 
                 ν10 = 60.6 
               
               
                 *r20 =  −39.707 
                 d20 = Variable 
               
               
                  r21 = ∞ 
               
               
                   
               
               
                 *) Aspheric Surface  
               
             
          
         
       
     
     
       
         
               
               
               
             
               
               
               
               
               
             
           
               
                   
                   
               
               
                   
                 Variable 
                 Focal Length 
               
             
          
           
               
                   
                 Separation 
                 29.65 
                 51.68 
                 74.83 
               
               
                   
                   
               
               
                   
                 d 6 
                 37.09 
                 11.08 
                  0.91 
               
               
                   
                 d12 
                  2.34 
                 10.72 
                 19.09 
               
               
                   
                 d16 
                 18.45 
                 10.07 
                  1.70 
               
               
                   
                 d20 
                  0.00 
                  8.37 
                 16.75 
               
               
                   
                   
               
             
          
         
       
     
     Aspheric Coefficients: 
     
       
         
               
               
               
             
           
               
                   
               
             
             
               
                 r20: 
                 b = 5.259191e-06 
                 c = −8.361761e-09 
               
               
                   
                 d = 1.127100e-10 
                 e = −3.658179e-13 
               
               
                   
               
             
          
         
       
     
     Numerical Example 4: 
     
       
         
               
               
               
             
               
               
               
               
             
           
               
                   
               
             
             
               
                 f = 29.5˜76.1 
                 Fno = 3.5˜4.5 
                 2ω = 72.5°˜31.7° 
               
             
          
           
               
                  r 1 =  54.782 
                 d 1 = 1.80 
                 n 1 = 1.77250 
                 ν 1 = 49.6 
               
               
                  r 2 =  23.790 
                 d 2 = 6.90 
               
               
                  r 3 = −626.792 
                 d 3 = 1.60 
                 n 2 = 1.77250 
                 ν 2 = 49.6 
               
               
                  r 4 =  38.638 
                 d 4 = 0.50 
               
               
                  r 5 =  31.839 
                 d 5 = 3.90 
                 n 3 = 1.80518 
                 ν 3 = 25.4 
               
               
                  r 6 =  79.657 
                 d 6 = Variable 
               
               
                  r 7 =  54.334 
                 d 7 = 1.45 
                 n 4 = 1.84666 
                 ν 4 = 23.8 
               
               
                  r 8 =  24.872 
                 d 8 = 0.12 
               
               
                  r 9 =  25.990 
                 d 9 = 4.00 
                 n 5 = 1.69680 
                 ν 5 = 55.5 
               
               
                  r10 = −193.042 
                 d10 = 0.10 
               
               
                  r11 =  33.276 
                 d11 = 3.15 
                 n 6 = 1.69680 
                 ν 6 = 56.5 
               
               
                  r12 = −203.247 
                 d12 = Variable 
               
               
                  r13 = ∞ (Stop) 
                 d13 = 1.75 
               
               
                  r14 =  −45.434 
                 d14 = 2.10 
                 n 7 = 1.85026 
                 ν 7 = 32.3 
               
               
                  r15 =  −22.412 
                 d15 = 1.30 
                 n 8 = 1.60311 
                 ν 8 = 60.6 
               
               
                  r16 =  51.298 
                 d16 = Variable 
               
               
                  r17 =  151.346 
                 d17 = 1.00 
                 n 9 = 1.71736 
                 ν 9 = 29.5 
               
               
                  r18 =  26.620 
                 d18 = 0.40 
               
               
                  r19 =  29.218 
                 d19 = 6.00 
                 n10 = 1.58313 
                 ν10 = 59.4 
               
               
                 *r20 =  −42.587 
                 d20 = Variable 
               
               
                  r21 = ∞ 
               
               
                   
               
               
                 *) Aspheric Surface  
               
             
          
         
       
     
     
       
         
               
               
               
             
               
               
               
               
               
             
           
               
                   
                   
               
               
                   
                 Variable 
                 Focal Length 
               
             
          
           
               
                   
                 Separation 
                 29.50 
                 52.01 
                 76.08 
               
               
                   
                   
               
               
                   
                 d 6 
                 37.70 
                 11.30 
                  0.86 
               
               
                   
                 d12 
                  2.34 
                  8.93 
                 15.51 
               
               
                   
                 d16 
                 15.00 
                  8.42 
                  1.84 
               
               
                   
                 d20 
                  0.00 
                 10.49 
                 20.98 
               
               
                   
                   
               
             
          
         
       
     
     Aspheric Coefficients: 
     
       
         
               
               
               
             
           
               
                   
               
             
             
               
                 r20: 
                 b = 7.335475e-06 
                 c = −2.472863e-08 
               
               
                   
                 d = 4.115772e-10 
                 e = −1.490244e-12 
               
               
                   
               
             
          
         
       
     
     
       
         
               
               
               
             
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 Condition 
                 Limit 
                 Numerical Example 
               
             
          
           
               
                 No. &amp; Factor 
                 Lower 
                 Upper 
                 1 
                 2 
                 3 
                 4 
               
               
                   
               
             
          
           
               
                 (4) 
                 |f3|/f2 
                 1.5 
                 2.5 
                 1.6 
                 1.6 
                 1.7 
                 1.8 
               
               
                 (5) 
                 f4/f2 
                 2.1 
                 10 
                 3.5 
                 3.5 
                 2.4 
                 3.0 
               
               
                 (6) 
                 OK4 
                 0 
                 — 
                 1.9 
                 2.0 
                 0.8 
                 1.2 
               
               
                 (7) 
                 |TS3t| 
                 1.1 
                 — 
                 1.9 
                 1.8 
                 1.2 
                 1.3 
               
               
                 (8)* 
                   
                 0.6 
                 1.3 
                 0.8 
                 0.7 
                 0.9 
                 0.9 
               
               
                 (9)* 
                   
                 1.3 
                 3.5 
                 1.8 
                 1.7 
                 1.8 
                 1.7 
               
               
                 (10) 
                 |Δ23/Δ12| 
                 0.2 
                 0.6 
                 0.23 
                 0.32 
                 0.46 
                 0.36 
               
               
                   
               
               
                 
                   
                     
                       
                         * 
                         
                           ( 
                           8 
                           ) 
                         
                          
                         
                           : 
                         
                          
                         
                             
                         
                          
                         
                           
                              
                             f1 
                              
                           
                           / 
                           
                             
                               fW 
                               · 
                               fT 
                             
                           
                         
                       
                     
                             
                     
                         
                     
                   
                 
               
               
                 (9):  f2 · FNot/fT  
               
             
          
         
       
     
     FIGS.  5 A 1  and  5 A 2 , FIGS.  7 A 1  and  7 A 2 , FIGS.  9 A 1  and  9 A 2  and FIGS.  11 A 1  and  11 A 2  are graphic representations of the aberrations of the variable magnification optical systems of the numerical examples 1 to 4, respectively, at image heights of y=0 and y=15 at the wide-angle end for the normal state. FIGS.  5 B 1  to  5 B 3 , FIGS.  7 B 1  to  7 B 3 , FIGS.  9 B 1  to  9 B 3  and FIGS.  11 B 1  to  11 B 3  are graphic representations of the aberrations of the variable magnification optical systems of the numerical examples 1 to 4, respectively, at image heights of y=0 and y=±15 at the wide-angle end during the image stabilizing operation for the tilting of 0.5° of the variable magnification optical system. 
     FIGS.  6 A 1  and  6 A 2 , FIGS.  8 A 1  and  8 A 2 , FIGS.  10 A 1  and  10 A 2  and FIGS.  12 A 1  and  12 A 2  are graphic representations of the aberrations of the variable magnification optical systems of the numerical examples 1 to 4, respectively, at image heights of y=0 and y=15 at the telephoto end for the normal state. FIGS.  6 B 1  to  6 B 3 , FIGS.  8 B 1  to  8 B 3 , FIGS.  10 B 1  to  10 B 3  and FIGS.  12 B 1  to  12 B 3  are graphic representations of the aberrations of the variable magnification optical systems of the numerical examples 1 to 4, respectively, at image heights of y=0 and y=±15 at the telephoto end during the image stabilizing operation for the tilting of 0.5° of the variable magnification optical system. 
     Incidentally, in the normal state of the variable magnification optical system, the graphic representation of the aberrations of the variable magnification optical system obtained at an image height of −15 is symmetrical with the graphic representation of the aberrations of the variable magnification optical system obtained at an image height of +15. Therefore, only the graphic representations of the aberrations obtained at an image height of +15 are shown. In addition, the object distance is assumed to be the infinite distance. 
     As described above, a lens unit which constitutes a part of the variable magnification optical system and which is relatively small in size and light in weight is selected to be used for compensating for the shaking of an image by moving the in the directions perpendicular to the optical axis when the variable magnification optical system vibrates (or tilts), and the proper rules of design are set forth for the construction and arrangement of the constituent lenses of the variable magnification optical system. This produces great advantages of improving the compact form of the entire optical system, simplifying the structure of the operating mechanism, and reducing the load on the driving means, while, when the image-stabilizing lens unit is made to decenter, still permitting the produced amount of decentering aberrations to be suppressed to a minimum. Accordingly, a variable magnification optical system whose range is about 3 with a field angle of about 73° at the wide-angle end and which is corrected well for the decentering aberrations is thus achieved. 
     Next, a practical example of application of the variable magnification optical system shown in each of the numerical examples 1 to 4 described above to a camera is described with reference to FIG.  13 . 
     In FIG. 13, the variable magnification optical system  11  of any one of the numerical examples 1 to 4 is housed in lens barrel  10 . A camera body  20  contains a quick-return mirror  21  in the viewing position where a light beam entering through the variable magnification optical system  11  is reflected upward to a focusing screen  22  on which an image of an object being photographed is formed. The inverted image on the focusing screen  22  is erected and laterally reversed by a pentagonal roof prism  23 . The object image formed on the focusing screen  22  is viewed through an eyepiece lens  24 . During shooting, the mirror  21  is retracted from the optical path in response to the operation of a release button (not shown) by the photographer, so that the object image is cast on a silver-halide film  25 . 
     With the variable magnification optical system of each of the numerical examples 1 to 4 applied to the above camera, a photographed image can be prevented as far as possible from being deteriorated by the vibration of the variable magnification optical system during shooting.