Abstract:
In a zoom lens system that consists of a plurality of lens units and that performs zooming by varying the distances between the lens units, one of the lens units other than the lens unit disposed at the object side end includes a hand-shake correction lens unit that is decentered in a direction perpendicular to the optical axis for hand-shake correction and a fixed lens unit that is disposed on the image side of the hand-shake correction lens unit and that is kept in a fixed position during hand-shake correction. With respect to the lens element disposed at the image-side end of the hand-shake correction lens unit and the lens element disposed next to the image-side surface of the former lens element and kept in a fixed position during hand-shake correction, the relations between their shape factors and refractive powers are defined.

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to an optical system or zoom lens system having a function for correcting an image blur caused by vibration of the optical system. More particularly, the present invention relates to an optical system, such as a zoom lens system or single-focal-length lens system, that is capable of preventing the blurring of an image due to vibration (hereinafter referred to as hand shakes) of the optical system such as occurs when shooting is performed with a camera held by hand. 
     2. Description of the Prior Art 
     Conventionally, typical causes of unsuccessful photographing have been hand shakes and improper focusing. However, since most cameras are nowadays equipped with an autofocus mechanism, and since the focusing accuracy of such autofocus mechanisms has been steadily improving, unsuccessful photographing is caused by improper focusing far less often now than before. On the other hand, as more and more cameras are equipped with a zoom lens, rather than with a single-focal-length lens, they are designed to have higher magnifications and to be more suitable for telephoto photography, and accordingly they are now more susceptible to hand shakes. As a result, it can safely be said that, today, unsuccessful photographing is caused exclusively by hand shakes. For this reason, a hand-shake correction function is indispensable in photographing optical systems. 
     As optical systems having a hand-shake correction function, such optical systems have been proposed in which part of their lens units are decentered for hand-shake correction. For example, U.S. Pat. No. 5,502,594 proposes a telephoto zoom lens system consisting of, from an object side, a first lens unit having a positive refractive power, a second lens unit having a negative refractive power, a third lens unit having a negative refractive power, a fourth lens unit having a positive refractive power, and a fifth lens unit having a negative refractive power, wherein the third lens unit is decentered in a direction perpendicular to the optical system to achieve hand-shake correction. 
     In an optical system having a hand-shake correction function, it is required that the optical system offer satisfactory optical performance not only in its normal state (hereinafter also referred to as the pre-decentering state), but also in its hand-shake correction state (hereinafter also referred to as the post-decentering state), without causing unduly large aberrations (hereinafter also referred to as the decentering aberrations) as the result of the decentering of the lenses. However, the above-mentioned five-unit zoom lens system according to U.S. Pat. No. 5,502,594 is defective in that it does not offer satisfactory aberration characteristics in the hand-shake correction state (i.e. after decentering) when it corrects hand shakes of large angles. The hand-shake correction performance of this zoom lens system is evaluated, in its publication, with hand shakes of approximately 0.15°. However, while it is often necessary to correct hand shakes of larger angles in actual shooting of night scenes or other with the camera held by hand, this zoom lens system inconveniently exhibits intolerably large aberrations with hand shakes of large angles. 
     SUMMARY OF THE INVENTION 
     An object of the present invention is to provide an optical system that has a hand-shake correction function and that is still capable of correcting various aberrations satisfactorily both in its normal state and in its hand-shake correction state. 
     To achieve the above object, according to one aspect of the present invention, an optical system having a hand-shake correction function is provided with a hand-shake correction lens unit that is decentered in a direction perpendicular to the optical axis for hand-shake correction, and a fixed lens unit that is disposed on the image side of the hand-shake correction lens unit and that is kept in a fixed position during hand-shake correction. 
     Moreover, this optical system satisfies conditions (1), (2), and (4), or conditions (1), (3), and (4) below. Here, the lens element disposed at the image-side end of the above-mentioned hand-shake correction lens unit is represented by PF, and the lens element that is disposed next to the image-side surface of the lens element PF and that is kept in a fixed position during hand-shake correction is represented by PR. 
     
         5&lt;S(PR)/S(PF)&lt;0                                            (1) 
    
     
         1.0&lt;S(PF)                                                  (2) 
    
     
         S(PF)&lt;0                                                    (3) 
    
     
         P(PR)/P(PF)&lt;0                                              (4) 
    
     where 
     S(PF): shape factor of the lens element PF; 
     S(PR): shape factor of the lens element PR; 
     P(PF): refractive power of the lens element PF; 
     P(PR): refractive power of the lens element PR; and the shape factor is defined as follows: 
     
         SF=(CRR+CRF)/(CRR-CRF);                                    (A) 
    
     where 
     SF: shape factor of a lens element; 
     CRF: radius of curvature of the object-side surface of the lens element; 
     CRR: radius of curvature of the image-side surface of the lens element. 
     According to another aspect of the present invention, a zoom lens system having a hand-shake correction function consists of a plurality of zoom lens units and performs zooming by varying the distances between the zoom lens units. Moreover, in this zoom lens system, one zoom lens unit includes a hand-shake correction lens unit that is decentered in a direction perpendicular to the optical axis for hand-shake correction, and a fixed lens unit that is disposed on the image side of the hand-shake correction lens unit and that is kept in a fixed position during hand-shake correction. 
     Furthermore, this zoom lens system satisfies conditions (1), (2), and (4), or conditions (1), (3), and (4) above. Here, the lens element disposed at the image-side end of the above-mentioned hand-shake correction lens unit is represented by PF, and the lens element that is disposed next to the image-side surface of the lens element PF and that is kept in a fixed position during hand-shake correction is represented by PR. 
     As described above, according to the present invention, hand-shake correction is achieved by decentering the hand-shake correction lens unit in a direction perpendicular to the optical axis (that is, by decentering it translationally). As can be seen from the aberration coefficients (described in detail later) that are defined for an optical system having a hand-shake correction function, it is possible, by carefully designing the spherical aberration coefficient of the hand-shake correction lens unit, to correct axial coma aberrations, which is one type of aberrations that occur during hand-shake correction (that is, decentering aberrations). In other words, by disposing a lens element having an appropriate spherical aberration coefficient in the hand-shake correction lens unit, it is possible to correct the axial coma aberrations that occur during hand-shake correction. However, the presence of such a lens element in the hand-shake correction lens unit adversely affects the balance of aberrations in the normal state, increasing in particular spherical aberrations greatly. 
     To avoid this, according to the present invention, the lens element PF that is disposed at the image-side end of the hand-shake correction lens unit is chosen as the one that is designed to have an appropriate spherical aberration coefficient. Moreover, the lens element PR that is kept in a fixed position during hand-shake correction is disposed next to the image-side surface of the lens element PF. Accordingly, the effects that the lens element PF has on aberrations in the normal state are canceled out by the effects that the lens element PR has on them. Moreover, since the lens element PR is disposed on the image side of the hand-shake correction lens unit that is decentered for hand-shake correction, the lens element PR does not affect aberrations during hand-shake correction. As a result, not only the decentering aberrations that occur in the hand-shake correction state can be corrected properly on the one hand, but also the aberrations over the entire system in the normal state can be kept within a tolerable range on the other hand. 
     FIGS. 19 and 20 show the relationship between the shape factor SF and the aberration coefficients I, II, III, P, V in the case where light from infinity is incident on a single lens element. From these figures, it is understood that a positive lens element has a positive spherical aberration coefficient I and a negative lens element has a negative spherical aberration coefficient I, and also that the value of the spherical aberration coefficient varies greatly with the shape factor. It is further understood that the coma aberration coefficient II also varies with the shape factor, and especially that it even changes its sign. It is for these reasons that, in order to enable the lens elements PF and PR to cancel out the spherical and other aberrations originating from themselves, they are designed to have opposite refractive powers and to satisfy a combination of previously noted conditions (1), (2), and (4), or conditions (1), (3), and (4). 
     Condition (1) defines the ratio of the shape factors of the lens elements PF and PR. If the upper or lower limit of condition (1) is exceeded, the differences in the spherical aberration coefficient I and the coma aberration coefficient II between the lens elements PF and PR are so large that it is not possible to cancel out the aberrations between them. This makes it difficult to correct properly the aberrations, in particular spherical and coma aberrations, in the normal state. 
     Alternatively, to obtain better optical performance in the normal state, it is preferable that the lens elements PF and PR satisfy condition (1a) below: 
     
         -1.5&lt;S(PR)/S(PF)&lt;-0.2                                      (1a) 
    
     Conditions (2) and (3) define the shape factor of the lens element PF. If the lower limit of condition (2) or the upper limit of condition (3) is exceeded, the absolute value of the spherical aberration coefficient I of the lens element PF is so small, as is clear from FIGS. 19 and 20, that it is difficult to correct properly the axial coma aberrations that occur during hand-shake correction. 
     Alternatively, to further reduce the aberrations that occur during hand-shake correction, it is preferable that the lens element PF satisfy condition (2a) or (3a) below: 
     
         2.2&lt;S(PF)                                                  (2a) 
    
     
         S(PF)&lt;-0.7                                                 (3a) 
    
     Condition (4) defines the ratio of the refractive powers of the lens elements PF and PR. As with condition (1), if the upper limit of condition (4) is exceeded, the difference in the spherical aberration coefficient I between the lens elements PF and PR is so large that it not possible to cancel out the aberrations between them. 
     Moreover, it is preferable that the lens element PR satisfy conditions (5) and (7), or conditions (6) and (7) below: 
     
         1.0&lt;S(PR)                                                  (5) 
    
     
         S(PR)&lt;0                                                    (6) 
    
     
         0.1&lt;IP(PR)/P&lt;2.5                                           (7) 
    
     where 
     P: refractive power of the entire system. 
     Conditions (5) and (6) define the shape factor of the lens element PR. If the lower limit of condition (5) or the upper limit of condition (6) is exceeded, the spherical aberration coefficient I of the lens element PR is so small that it is difficult to correct properly the axial coma aberrations that occur during hand-shake correction. 
     Alternatively, to further reduce the aberrations that occur during hand-shake correction, it is preferable that the lens element PR satisfy condition (5a) or (6a) below: 
     
         2.2&lt;S(PR)                                                  (5a) 
    
     
         S(PR)&lt;-0.7                                                 (6a) 
    
     Condition (7) defines the refractive power of the lens element PR. If the lower limit of condition (7) is exceeded, the refractive power of the lens element PR is so weak that, in order to obtain a large enough spherical aberration coefficient I, it is necessary to make the shape factor very large. This requires that the radius of curvature of the lens element PR be extremely small, and it is impossible to manufacture such a lens element by any existing method. In contrast, if the upper limit of condition (7) is exceeded, the refractive power of the lens element PR is so strong that it is difficult to correct properly the large aberrations that occur in the lens element PR. 
     Alternatively, to obtain better optical performance, it is preferable that the lens element PR satisfy condition (7a) below: 
     
         0.1&lt;IP(PR)I/P&lt;1.0                                          (7a) 
    
     For the same reasons as stated above in connection with condition (7), it is preferable that the lens element PF, too, satisfy condition (8) below: 
     
         0.1&lt;IP(PF)L/P&lt;2.5                                          (8) 
    
     Alternatively, to obtain better optical performance, it is preferable that the lens element PF satisfy condition (8a) below: 
     
         0.1&lt;IP(PF)L/P&lt;1.0                                          (8a) 
    
     To cancel out the spherical aberrations between the lens elements PF and PR, it is necessary that these two lens elements cause approximately the same degree of spherical aberrations. To achieve this, it is preferable that the lens elements PF and PR satisfy condition (9) below: 
     
         0.003&lt;d(PF, PR)*P&lt;0.1                                      (9) 
    
     where 
     d(PF, PR): axial distance between the first and second lens elements L1 and L2. 
     Condition (9) defines the axial distance between the lens elements PF and PR (that is, the axial distance between the image-side surface of the lens element PF and the object-side surface of the lens element PR). If the upper limit of condition (9) is exceeded, the distance between the lens elements PF and PR is so large that light beams pass through quite different points between when passing through the lens element PF and when passing through the lens element PR. This makes it difficult to cancel out the spherical and coma aberrations simultaneously. If the lower limit of condition (9) is exceeded, the lens elements PF and PR are so close to each other that they collide when the hand-shake correction lens unit is decentered translationally during hand-shake correction. 
     In general, in a zoom photographing optical system for use in a single-lens reflex camera, the first lens unit is the largest of all the lens units, and therefore the lens elements constituting it are considerably heavy. Accordingly, it is not preferable to perform hand-shake correction by moving the lens elements of the first lens unit in a direction perpendicular to the optical axis (that is, by decentering them translationally), because such a construction requires a larger mechanism for driving the hand shake-correction lens unit. For this reason, it is preferable to arrange the hand-shake correction lens unit in a zoom lens unit other than the first lens unit. 
     Since the above-mentioned lens element PF is moved for hand-shake correction, it needs to be light enough to minimize the load to be borne by the hand-shake correction lens unit driving mechanism. Accordingly, it is preferable that the lens element PF be a plastic lens. The use of a plastic lens here not only minimizes, because of its light weight, the load to be borne by the hand-shake correction lens unit driving mechanism, but also offers the advantage of reducing the cost. Moreover, it is preferable that also the above-mentioned lens element PR be a plastic lens, the use of which offers the advantage of reducing the cost. 
     Ideally, the movement amount by which the hand-shake correction lens unit is moved for hand-shake correction (hereinafter referred to as the hand-shake correction movement amount) needs to be approximately the same between at the wide-angle end and at the telephoto end of the zoom range. Accordingly, it is preferable that the optical system according to the present invention further satisfy condition (10) below: 
     
         0.4&lt;MT/MW&lt;2.5                                              (10) 
    
     where 
     MT: movement amount of the hand-shake correction lens units, at the telephoto end; 
     MW: movement amount of the hand-shake correction lens units, at the wide-angle end. 
     If the upper or lower limit of condition (10) is exceeded, the difference between the hand-shake correction movement amounts at the wide-angle and telephoto ends of the zoom range is too large. This makes it impossible to calculate the hand-shake correction amount at an arbitrary focal length without considerable calculation errors. 
     When the hand-shake correction lens unit is translationally decentered for hand-shake correction, there occur axial lateral chromatic aberrations, which are one type of the decentering aberrations. To reduce such aberrations, the hand-shake correction lens unit itself needs to be capable of correcting chromatic aberrations that occur in itself. Accordingly, it is preferable that the hand-shake correction lens unit satisfy condition (11) below: 
     
         νp&gt;νn                                                (11) 
    
     where 
     νp: Abbe number of the positive lens elements in the hand-shake correction lens units; 
     νn: Abbe number of the negative lens elements in the hand-shake correction lens units. 
     When, in the hand-shake correction state (that is, in the post-decentering state), the hand-shake correction lens unit is moved in a direction perpendicular to the optical axis for hand-shake correction, light beams pass through a portion through which they never pass in the normal state (that is, in the pre-decentering state). In such a situation, light beams become stray light beams, and may degenerate the imaging performance of the optical system. To avoid this, and thereby to maintain satisfactory imaging performance even in the hand-shake correction state, it is preferable that a fixed aperture diaphragm be provided on the object side of the hand-shake correction lens unit, or within the hand-shake correction lens unit, or on the image side of the hand-shake correction lens unit. 
     &lt;&lt;Decentering Aberrations and Decentering Aberration Coefficients&gt;&gt; 
     Next, with reference to FIGS. 21A to 21D, descriptions and definitions will be given as to various types of decentering aberrations that occur in an optical system having a hand-shake correction function (hereinafter referred to as a hand-shake correction optical system) such as a zoom lens system according to the present invention. All types of the decentering aberrations shown in FIGS. 21A to 21D (off-axial image-point movement errors, one-side blur, axial coma, and axial lateral chromatic aberrations) degrade imaging performance of a hand-shake correction optical system. 
     [Off-axial image-point movement errors] {FIG. 21A} 
     In a decentered optical system, in addition to normal distortion, additional distortion occurs as the result of the decentering. For this reason, in a hand-shake correction optical system, if a hand shake is corrected in such a way that axial image points (that is, image points at the center of the image area) are brought to a rest, off-axial image points do not stop completely, and thus cause an image blur. In FIG. 21A, reference numeral 1 represents a film surface, reference numeral 2 represents image points in the hand-shake correction state (post-decentering state), reference numeral 3 represents image points in the normal state (pre-decentering state), and reference numeral 4 indicates the direction in which a hand shake is corrected. 
     Here, assume that the optical axis is the X axis, and the direction of a hand shake is the Y axis (thus, the hand shake is corrected in the Y-axis direction). Further, let Y(y&#39;, z&#39;, θ&#39;) be the actual Y coordinate, at a hand-shake correction angle θ, of a light beam with a paraxial image point at (y&#39;, z&#39;) (note that y(0, 0, θ)=0 in any case, since correction is performed such that axial image points are brought to a rest). Then, expression (a) below holds. 
     
         ΔY(Y&#39;, Z&#39;, θ)=Y(Y&#39;, Z&#39;, θ)-Y(Y&#39;, z&#39;, 0)  (a) 
    
     Unless otherwise specified, the off-axial image-point movement error ΔY Y  &#39; of an image point on the Y axis and the off-axial image-point movement error ΔY z  &#39; of an image point on the Z axis are respectively represented by expressions (b) and (c) below. Here, 0.7field is approximately 12 mm for new-generation 24 mm film. 
     
         ΔY.sub.Y&#39;={ΔY (0.7field, 0, 0.7°)+ΔY(-0.7field, 0, 0.7°)}/2                                           (b) 
    
     
         ΔY.sub.z&#39;=ΔY (0, 0.7field, 0.7°)        (c) 
    
     [One-side blur] {FIG. 21B} 
     In FIG. 21B, reference numeral 5 represents an image plane that is asymmetrical with respect to the optical axis AX, and 6 represents an image plane that is symmetrical with respect to the optical axis AX. When an optical system is asymmetrical, the image plane 5 is asymmetrical with respect to the optical axis AX. In such a case, a meridional one-side blur ΔM&#39; and a sagittal one-side blur ΔS&#39; occur, which are respectively represented by expressions (d) and (e) below. 
     
         ΔM&#39;={meridional value(y&#39;=0.7field, z=0, θ=0.7°)-meridional value(y&#39;=-0.7field, z=0, θ=0.7+)}/2 (d) 
    
     
         ΔS&#39;={sagittal value(y&#39;=0.7field, z=0, θ=0.7°)-sagittal value(y&#39;=-0.7field, z=0, θ=0.7°)}/2          (e) 
    
     [Axial coma] {FIG. 21C} 
     In FIG. 21C, reference numeral 7 represents an axial light beam, and 8 represents an axial principal light ray. As shown in the figure, when the axial light beam 7 is not symmetrical with respect to the axial principal light ray 8, coma occurs. The axial coma AXCM developed in the axial light beam 7 is represented by expression (f) below. 
     
         AXCM={Y(upper zonal, θ=0.7°)+Y(lower zonal, θ=0.7°)}/2                                   (f) 
    
     [Axial lateral chromatic aberration] {FIG. 21D} 
     Since the position in which a light ray forms its image point varies with its wavelength, even an axial light ray exhibits deviation of the image point in an asymmetrical optical system. For an axial principal light ray, the axial lateral chromatic aberration is represented by expression (g) below. 
     
         (Axial Lateral Chromatic Aberration)={Y(g-lines, θ=0.7°)-Y(d-lines, θ=0.7°)}     (g) 
    
     In connection with the decentering aberration coefficients defined above, their applications are discussed in a thesis by Yoshiya Matsui, titled &#34;On the Third-degree Aberrations in Optical Systems Involving Decentering&#34; (JOEM, June, 1990). The method presented there is however intended to be applied to such a situation where an ordinary taking lens is decentered as a result of improper assembly, and therefore it cannot be applied directly to a hand-shake correction optical system, where the co-axial relationship among the object plane, taking lens, and image plane is intentionally broken. To make it possible to apply the method presented in the above-mentioned thesis to a hand-shake correction optical system, it is necessary to express the aberrations that actually occur in the hand-shake correction optical system as aberration coefficients of the third degree, with the help of the transformation formulae and techniques described hereinafter 
     [Application of the decentering aberration coefficients to a hand-shake correction optical system] 
     With reference to FIG. 22, which defines the relationship between the optical system and a coordinate system, the methods for calculating the decentering aberration coefficients will be described. First of all, the following expressions hold: ##EQU1## where g represents the distance from the entrance-pupil plane to the object plane (object surface) OS, g$ represents the distance from the object-side principal plane to the object plane OS, Ω represents the angle of the straight line from the object point to the object-side principal point H with respect to the reference axis of the optical system, φΩ is its azimuth, R represents the radius of the entrance pupil as seen on the object-side principal plane, and φR is its azimuth. 
     When the v-th surface from the object side is decentered translationally a slight distance Ev in the Y direction with respect to the reference axis, the image-point movement amounts ΔY and ΔZ on the image plane (image surface) IS are represented by expressions (1A) and (1B) below. ##EQU2## 
     Here, if it is assumed that 
     (ΔE)ν: prismatic effect (lateral deviation of the image), 
     (VE1)ν, (VE2)ν: rotationally asymmetrical distortion, 
     (IIIE)ν, (PE)ν: rotationally asymmetrical astigmatism and image-plane inclination, 
     (IIE)ν: rotationally asymmetrical coma that occurs even with axial light rays, 
     then the decentering aberration coefficients that represent the effects of the decentering are represented, on the basis of the aberration coefficients of the lens surfaces from the vth surface to the image plane, by expressions (1C) to (1H) (here, items followed by # are ones related to the object plane). Note that expressions (1A) to (1H) can be used also in the case of rotational decentering. ##EQU3## 
     However, in applying the decentering aberration coefficients to a hand-shake correction optical system, it is necessary, by reversing the optical system, to replace the image plane IS with the object plane OS in order to obtain aberration coefficients as seen from the image plane IS. That is, the image-point movement amounts need to be converted into those as seen on the object plane OS. The reasons are as follows. 
     First, there is a difference in how the paths of light rays are affected by the decentering. As shown in FIG. 23A (here, L 1  represents a light ray in a normal state without decentering, and L 2  represents the same light ray in a decentered state), the method described in the above-mentioned thesis by Y. Matsui deals only with such cases in which it is between the decentered lens LS and the image plane IS that the paths of light rays are affected by the decentered lens LS. In such cases, the decentering aberration coefficients depend on the aberration coefficients of the decentered lens LS and of the lenses disposed between the decentered lens LS and the image plane IS. In contrast, as shown in FIG. 23B (here, M 1  represents a light beam before hand-shake correction and M 2  represents the same light beam after hand-shake correction), in a hand-shake correction optical system, it is (ideally) on the downstream side of the decentered lens LS that light rays take different paths before and after hand-shake correction. In this case, the decentering aberration coefficients depend on the aberration coefficients of the decentered lens LS and of the lenses disposed on the downstream side of the decentered lens LS. 
     Second, rotational conversion of the object plane may cause additional aberrations. The method described in the above-mentioned thesis by Y. Matsui assumes that the object plane OS 1  and the image plane IS are in fixed positions. However, in a hand-shake correction optical system, the object plane OS 1  rotates, as shown in FIG. 24. As a result, the off-axial image-point movement errors and the one-side blur occur in a considerably different manner from in cases where the object plane OS 1  does not rotate. In FIG. 24, OS 1  represents the object plane before hand-shake correction, and OS 2  represents the object plane after hand-shake correction. 
     [Aberration coefficients of a reversed optical system and aberration coefficients of a non-reversed optical system] 
     For the reasons stated above, it is necessary to convert the image-point movement amounts into those as seen on the object plane. Specifically, the coefficients defined by expressions (1A) to (1H) above are converted according to expressions (2A) to (2J) below, which hold in a reversed optical system as shown in FIG. 25. Note that, here,  R  () indicates a reversed system, and N represents the refractive index. 
     
         ______________________________________.sup.R α = .sup.R N/.sup.R g$ = -α&#39;                         (2A).sup.R α# = α&#39;#   (2B).sup.R αμ&#39; = -αν                         (2C).sup.R αμ&#39;# = αν#                         (2D).sup.R Pμ = Pν              &lt;non-reversed&gt;                         (2E).sup.R φμ = φν              &lt;non-reversed&gt;                         (2F).sup.R Iμ = Iν              &lt;non-reversed&gt;                         (2G).sup.R IIμ = -IIν              &lt;reversed&gt; (2H).sup.R IIIμ = IIIν              &lt;non-reversed&gt;                         (2I).sup.R Vμ = -Vν              &lt;reversed&gt; (2J)______________________________________ 
    
     [Decentering aberration coefficients and hand-shake aberration coefficients when a hand-shake correcting lens unit is decentered translationally] 
     Expressions (1A) to (1H) above assume that only one surface ν is decentered. Accordingly, next, expressions (1A) to (1H) need to be further transformed into expressions that can deal with cases where two or more surfaces i˜j are decentered. When a hand-shake correction lens unit is decentered translationally, the decentering amounts Ei˜Ej of all the decentered surfaces i˜j are equal. This means that, in handling the aberration coefficients, it is only necessary to consider their respective sum totals. For example, 
     
         (ΔE)i˜j=(ν=i→j)Σ{-2·(αν&#39;-.alpha.ν)} 
    
     Further, from αν&#39;=αν+1, the following expression is obtained: 
     
         (ΔE)i˜j=-2·(αj&#39;-αi) 
    
     In similar manners, the intermediate terms of Σ in the other aberration coefficients can be eliminated. For example, 
     
         (PE)i˜j=(μ=i→j)Σ{αν&#39;·(μ=ν+1.fwdarw.k)ΣPμ-αν·(μ=ν→k)ΣPμ}=αj&#39;·(μ=j+1→k)ΣPμ-αi·(μ=i→k)ΣPμ 
    
     This is further transformed into 
     
         (PE)i˜j=(αj&#39;-αi)·(μ=j+1→k)ΣPμ-αi·(μ=i→j)ΣPμ 
    
     where 
     (u=j+1→k)ΣPμ: sum of P&#39;s (Petzval sum) of the lenses disposed on the downstream side of the hand-shake correction lens unit; 
     (μ=i→j)ΣPμ: sum of P&#39;s (Petzval sum) of the lenses constituting the hand-shake correction lens unit. 
     Eventually, the following expression is obtained: 
     
         (PE)i˜j=(αj&#39;-αi)P.sub.R -αi·P.sub.D 
    
     where 
     () R  : the sum of the aberration coefficients of the lenses disposed on the downstream side of the hand-shake correction lens unit; 
     () D  : the sum of the aberration coefficients of the lenses constituting the hand-shake correction lens unit. 
     After necessary conversions as described above, which are performed to obtain the image-point movement amounts as seen on the object plane and to cope with cases where two or more surfaces i˜j are decentered, the decentering aberration coefficients are reduced to expressions (3A) to (3F) below. Now that the decentering aberration coefficients are redefined by expressions (3A) to (3F), it is possible to use expressions (1A) to (1H), as they are, as definitions of the image-point movement amounts on the object plane. ##EQU4## [Off-axial image-point movement errors] 
     Next, the off-axial image-point movement errors will be described. Assume that the decentering aberration coefficients (of a reversed optical system) are ΔE, VE1, VE2, IIIE, PE, and IIE. For a principal light ray (R=0), the amounts of the image-point movements caused on the object plane as the result of the decentering (before performing rotational conversion on the object plane) are represented by expressions (4A) and (4B) below. Expressions (4A) and (4B) are obtained by substituting R=0 in expressions (1A) and (1B). 
     
         ΔY#=-(E/2α&#39;.sub.k)·[ΔE+(N·tan Ω).sup.2 ·{(2+cos.sup.2 φΩ)VE1-VE2}](4A) 
    
     
         ΔZ#=-(E/2α&#39;)·{(N·tan Ω).sup.2 ·sin 2φΩ)·VE1}                (4B) 
    
     From expressions (4A) and (4B), expressions (4C) and (4D) below are obtained (for an axial light ray, tan Ω=0). 
     
         ΔY.sub.0 #=-(E/2α&#39;.sub.k)·ΔE    (4C) 
    
     
         ΔZ.sub.0 #=0                                         (4D) 
    
     Next, the rotational conversion will be described, with reference to FIGS. 26A and 26B. From FIG. 26A, the following expression is obtained: 
     
         Y#=g$.sub.k ·tan Ω 
    
     This is transformed, using the sine theorem, into 
     
         Y&#39;#/{sin(π/2-Ω&#39;)}=(Y#+ΔY#-ΔY.sub.0 #)/{sin(π/2+Ω&#39;-θ)} 
    
     then, the ΔY&#39;# after the rotational conversion is represented by 
     
         ΔY&#39;#=(Y&#39;#)-(Y#)=[Y#·cos Ω&#39;+{(ΔY#)-(ΔY.sub.0 #)}·cos Ω&#39;-Y#·cos(Ω&#39;-θ)]/cos(Ω&#39;-θ) 
    
     The numerator of this expression is transformed into ##EQU5## Here, since e is small and negligible compared with the other two terms, (1-cos θ)≈θ 2  /2, sin θ≈=θ. Moreover, cos θ&#39;/{cos (θ&#39;-θ)}≈1, sin Ω&#39;/{cos(Ω&#39;-θ)≈tan Ω. 
     Thus, the following expression is obtained. 
     
         ΔY&#39;#≈(ΔY#-ΔY.sub.0 #)-Y#·θ·tan Ω 
    
     where (ΔY#-ΔY 0  #) represents the off-axial image-point movement errors resulting from the translational decentering, and Y#·θ·tan Ω is an additional term related to the rotation (but not related to the aberration coefficients). Note that, since Ω here is on the X-Y cross section, 
     
         ΔY&#39;#≈(ΔY#-ΔY.sub.0 #)-Y#·θ·tan Ω·cos φΩ(5A) 
    
     Next, the conversion to the image plane IS will be described, with reference to FIG. 27. The magnification β is defined by 
     
         β=g$.sub.1 /g$.sub.k =α.sub.k /α.sub.1 
    
     Here, α 1  =1/g$ 1 . On the other hand, the image plane IS and the object plane OS have a relation 
     
         Y=β·Y# 
    
     Further, since Y# and ΔY# retain the form of 1/α k  &#39;×(), the above expression is further transformed, as 
     
         Y=β·Y# 
    
     
         =(α.sub.k &#39;/α.sub.1)·(1/α.sub.k &#39;)×() 
    
     
         =g$.sub.1 ×() 
    
     Here, if it is assumed that g$ 1  &#39;→∞, then g$ 1  =-F1. Hence, 
     
         Y=-F1×() 
    
     
         =-F1×α.sub.k &#39;×Y.sub.# 
    
     Next, the off-axial image-point movement errors on the image plane will be described. From expression (4C) and α k  &#39;=1/g k  &#39;$, the decentering amount E is obtained as 
     
         θ=ΔY.sub.0 #/g$.sub.k &#39;=E·ΔE/2 
    
     
         E=2·θ/ΔE 
    
     Then, normalization is performed to make the hand-shake correction angle θ constant (0.7 deg=0.0122173 rad). 
     As the result of translational decentering (involving no rotational decentering), ΔY=(ΔY#-ΔY 0  #) is subjected to image-plane conversion (here, N·tanΩ=Φ/F1, Φ 2  =Y 2  +Z 2 ). Thus, expressions (6A) to (6D) below are obtained. 
     
         ΔY=(θ·Φ.sup.2 /F1)·[{(2+cos 2·φΩ)·VE1-VE2}/ΔE]      (6A) 
    
     
         ΔZ=(θ·Φ.sup.2 /F1)·[{(sin 2·φΩ)·VE1-VE2}/ΔE]      (6B) 
    
     Y +   Image Point, Y -   Image Point {φΩ=0, π of expressions (6A) and (6B)}: 
     
         ΔY.sub.Y =(θ·Y.sup.2 /F1)·{(3·VE1-VE2)/ΔE}             (6C) 
    
     Z Image Point{φΩ=π/2 of expressions (6A) and (6B)}: 
     
         ΔY.sub.Z =(θ·Z.sup.2 /F1)·{(VE1-VE2)/ΔE}(6D) 
    
     These expressions are then subjected to rotational conversion. Since Y#=-Y/(F1×a k  &#39;), the term -Y#·θ·tan Ω·cos φΩ of the expression (5A) can be expressed as 
     
         -Y#·θ·tan Ω·cos φΩ=Y/(F1×α.sub.k &#39;)·θ·tan Ω·cos φΩ 
    
     At the Y +   and Y -   image points, φΩ=0, π, and tan Ω/α k  &#39;=Y. Hence, on the image plane, -Y#·θ·tan Ω·cos φΩ=Y 2  ·θ/F1. By adding this to expression (6C), expression (6E) below is obtained. On the other hand, at the Z image point, φΩ=π/2. Hence, on the image plane, -Y#·θ·tan Ω·cos φΩ=0. By adding this to expression (6D), expression (6F) below is obtained. 
     
         ΔY.sub.Y &#39;=(θ·Y.sup.2 /F1)·{(3·VE1-VE2-ΔE)/ΔE}    (6E) 
    
     
         ΔY.sub.Z &#39;=ΔY.sub.Z                            (6F) 
    
     [One-side blur] 
     Next, the one-side blur will be described. From the expressions (1A) and (1B), it is known that ΔM equals {ΔY with φR=0 in the first-degree terms with respect to R}×g$ k  &#39; and ΔS equals {ΔZ with φR=π/2 in the first-degree terms with respect to R}×g$ k  &#39;. On the object plane OS before rotation, the following expression holds (here, it is assumed that a k  &#39;=N k  &#39;/g$ k  &#39; and E/2=/ΔE): 
     
         ΔM#=(-g$.sub.k &#39;.sup.2 ·θ/N.sub.k &#39;)×2·R·(N·tan Ω)·cos φΩ·{(3·IIIE+PE)/ΔE} 
    
     After rotation, the following expression holds: 
     
         ΔM&#39;#≈ΔM#+θY# 
    
     By converting the aberration coefficients to those as seen on the image plane and substituting N k  &#39;=1 and N=1, the following expression is obtained: 
     
         ΔM&#39;=β.sup.2 ·ΔM&#39;#=-g$.sub.1.sup.2 ·θ×2·R·tan Ω·cos φΩ·{(3·IIIE+PE)/ΔE}+β·Y.multidot.θ 
    
     Assume that the object plane OS is at ∞ (that is, g$ 1  =-F1, β→0, tan Ω=Y/F1, and φΩ=0). Then, the meridional one-side blur ΔM&#39; is represented by expression (7A) below; likewise, the sagittal one-side blur is represented by expression (7B) below. 
     
         ΔM&#39;=-2-F1·Y·θ·R·{(3.multidot.IIIE+PE)/ΔE}                                       (7A) 
    
     
         ΔS&#39;=-2·F1·Y·θ·R·(IIIE+PE)/ΔE}                                          (7B) 
    
     [Axial coma] 
     Next, the axial coma will be described. From expression (1A), it is known that the coma resulting from upward (upper) decentering of Ω=0 is represented by the following expression: 
     
         ΔY.sub.upper #=ΔY#(Ω=0, φ.sub.R =0)-ΔY#(Ω=0, R=0) 
    
     
         =-E/(2·α&#39;)×R.sup.2 ×3·IIE 
    
     On the other hand, the coma resulting from downward (lower) decentering by Ω=0 is represented by the following expression (having the same value and sign as ΔY upper  #): 
     
         ΔY.sub.Lower #=ΔY#(Ω=0, φ.sub.R =π)-ΔY#(Ω=0, R=0) 
    
     
         =-E/(2·α&#39;)×R.sup.2 ×3·IIE 
    
     Since Ω=0, the axial coma is little affected by the rotational conversion. As the result of the conversion from the object plane OS to the image plane IS (ΔY=β·ΔY#, E/2=θ/ΔE), the following expression is obtained: 
     
         ΔY.sub.Upper =F1×θ×R.sup.2 ×(3·IIE/ΔE)=ΔY.sub.Lower 
    
     Hence, the axial coma AXCM is represented by expression (8A) below: 
     AXCM=(ΔY Upper  +ΔY Lower ) /2 
     
         =ΔY.sub.upper                                        (8A) 
    
     By using relevant portions of thus obtained expressions (6E), (6F), (7A), (7B), and (8A), the hand-shake aberration coefficients are now defined by expressions (9A) to (9E) below: 
     Off-axial image-point movement error of an image point on the Y axis: 
     
         VE.sub.Y ={(3·VE1-VE2-ΔE)/ΔE}         (9A) 
    
     Off-axial image-point movement error of an image point on the Z axis: 
     
         VE.sub.Z ={(VE1-VE2)/ΔE}                             (9B) 
    
     Meridional single-sided blur: 
     
         IIIE.sub.M ={(3·IIIE+PE)/ΔE}                (9C) 
    
     Sagittal single-sided blur: 
     
         IIIE.sub.S ={(IIIE+PE)/ΔE}                           (9D) 
    
     Off-axial coma: 
     
         IIE.sub.A ={(3·IIE)/ΔE}                     (9E) 
    
     These expressions (9A) to (9E), which represent the hand-shake aberration coefficients, are then rearranged by substituting expressions (3A) to (3F) into them, and are eventually transformed into expressions (10A) to (10E) below. ##EQU6## where () D  : sum of the aberration coefficients of the lenses constituting the hand-shake correction lens unit; 
     () R  : sum of the aberration coefficients of the lenses disposed on the downstream side of the hand-shake correction lens unit; 
     A=αi/(αj&#39;-αi) (here, the hand-shake correction lens unit is assumed to include surfaces i˜j); 
     A#=αi#/(αj&#39;-αi); 
     H#=(αi&#39;#-αi#)/(αj&#39;-αi). 
     Since ΔE=-2·(αj&#39;-αi) (here, (αj&#39;-αi)=±0.0122173 if 0.7°/mm) is a coefficient representing (hand-shake correction angle)/(decentering amount), it converges approximately on a predetermined value (though the sign depends on whether the hand-shake correction lens unit has a positive or negative power). Therefore, A represents an incident angle of a marginal light ray to the hand-shake correction lens units (as seen from the image side), and A# varies in proportion to the incident angle of a principal light ray. In cases where h# and h vary only slightly in the hand-shake correction lens unit, then H# represents the ratio of h# of the principal light ray to h of the marginal light ray. 
     The decentering aberration coefficients defined by expressions (10A) to (10E) are based on a reversed optical system. Accordingly, they now need to be converted back into coefficients based on a non-reversed optical system. To achieve this, expressions (10A) to (10E) are converted, by using expressions (2A) to (2J) noted previously, into expressions (11A) to (11E) below, which represents the aberration coefficients based on a non-reversed optical system. ##EQU7## where () D  : sum of the aberration coefficients of the lenses constituting the hand-shake correction lens unit, as seen in a non-reversed optical system; 
     () F  : sum of the aberration coefficients of the lenses disposed on the upstream side of the hand-shake correction lens unit; 
     A=-αn&#39;/(αn&#39;-αm); 
     A#=αn&#39;#/(αn&#39;-αm); 
     H=-(αn&#39;#-αm#)/(αn&#39;-αm)=-(Σhμ#·.phi.μ)/(Σhμ-μ); 
     ΔE=-2(αn&#39;-αm), 
     (Here, it is assumed that the hand-shake correction lens unit includes surfaces m→n, the non-reversed optical system j←i). 
     From expressions (11A) to (11E) above, the following conclusions are drawn. 
     First, as noted previously, whereas the method described in Y. Matsui&#39;s thesis is directed to cases where the hand-shake correction lens unit (i.e. the decentered lens LS) and the lenses disposed on the downstream side thereof affect the optical performance, expressions (11A) to (11E) are directed to cases where the hand-shake correction lens unit and the lenses disposed on the upstream side thereof affect the optical performance. 
     Second, whereas the off-axial image-point movement errors tend to be larger in wide-angle optical systems (since the focal length F1 of the hand-shake correction lens unit is in the denominator), the one-side blur and axial coma tend to be larger in telephoto optical systems. 
     Third, although it is possible to reduce the aberrations resulting from the decentering by reducing the aberration coefficients of the hand-shake correction lens unit and the lenses disposed on the upstream side thereof, there still remains a constant term (-2 in { } of expression (11A)) in the coefficient VE Y  representing the off-axial image-point movement errors ΔY Y  &#39;. This term indicates that the object plane OS and the image plane IS become inclined with respect to each other as a result of a rotational hand shake. And this term (-2) contributes to a considerable increase in off-axial image-point movement errors in wide-angle optical systems. For example, at a focal length F1 of 38 mm, the off-axial image-point movement errors are as large as ΔY Y  &#39;=-72 μm, and are therefore not negligible. Moreover, the off-axial image-point movement errors due to the constant term (-2) remain even when all the aberration coefficients are set to 0. Accordingly, it is preferable to set the aberration coefficients such that the constant term (-2) is canceled out. 
     Fourth, to reduce the aberrations resulting from the decentering, it is necessary to reduce the aberration coefficients as well as the factors such as A, A#, and H# included therein. As for A and A#, this can be achieved by increasing their denominator α n  &#39;-αm. However, since such an operation directly affects ΔE=-2(α n  &#39;-αm), an excessive increase in α n  &#39;-αm leads to an excessively high hand-shake correction sensitivity (how much (°) a light beam is inclined by a unit amount (mm) of decentering), which necessitates high accuracy in the driving mechanism. As for H#, as the hand-shake correction lens unit is disposed closer to aperture diaphragm, h# of each surface becomes smaller, and thus H# also becomes smaller. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     This and other objects and features of this invention will become clear from the following description, taken in conjunction with the preferred embodiments with reference to the accompanied drawings in which: 
     FIG. 1 is a lens construction diagram of the first embodiment of the present invention; 
     FIGS. 2A to 2I are aberration diagrams showing the longitudinal aberrations in the first embodiment before decentering; 
     FIGS. 3A to 3F are aberration diagrams showing the meridional lateral aberrations in the first embodiment before and after decentering, at the wide-angle end; 
     FIGS. 4A to 4F are aberration diagrams showing the meridional lateral aberrations in the first embodiment before and after decentering, at the telephoto end; 
     FIG. 5 is a lens construction diagram of the second embodiment of the present invention; 
     FIGS. 6A to 6C are aberration diagrams showing the longitudinal aberrations in the first embodiment before decentering; 
     FIGS. 7A to 7F are aberration diagrams showing the meridional lateral aberrations in the second embodiment before and after decentering; 
     FIG. 8 is a lens construction diagram of the third embodiment of the present invention; 
     FIGS. 9A to 9C are aberration diagrams showing the longitudinal aberrations in the third embodiment before decentering; 
     FIGS. 10A to 10F are aberration diagrams showing the meridional lateral aberrations in the third embodiment before and after decentering; 
     FIG. 11 is a lens construction diagram of the fourth embodiment of the present invention; 
     FIGS. 12A to 12I are aberration diagrams showing the longitudinal aberrations in the fourth embodiment before decentering; 
     FIGS. 13A to 13F are aberration diagrams showing the meridional lateral aberrations in the fourth embodiment before and after decentering, at the wide-angle end; 
     FIGS. 14A to 14F are aberration diagrams showing the meridional lateral aberrations in the fourth embodiment before and after decentering, at the telephoto end; 
     FIG. 15 is a lens construction diagram of the fifth embodiment of the present invention; 
     FIGS. 16A to 16I are aberration diagrams showing the longitudinal aberrations in the fifth embodiment before decentering; 
     FIGS. 17A to 17F are aberration diagrams showing the meridional lateral aberrations in the fifth embodiment before and after decentering, at the wide-angle end; 
     FIGS. 18A to 18F are aberration diagrams showing the meridional lateral aberrations in the fifth embodiment before and after decentering, at the telephoto end; 
     FIG. 19 is a graph showing the relations between the shape factor of a negative lens and the aberration coefficients; 
     FIG. 20 is a graph showing the relations between the shape factor of a positive lens and the aberration coefficients; 
     FIGS. 21A to 21D are diagrams explaining the factors causing image degradation in a hand-shake correction optical system; 
     FIG. 22 is a diagram explaining the relationship between an optical system and a coordinate system; 
     FIGS. 23A and 23B are diagrams explaining light-ray path shifts resulting from decentering; 
     FIG. 24 is a diagram explaining the rotational conversion of the object plane; 
     FIG. 25 is a diagram explaining the aberration coefficients in a reversed and a non-reversed optical system; 
     FIGS. 26A and 26B are diagrams explaining the rotational conversion; and 
     FIG. 27 is a diagram explaining the use of the image plane in place of the object plane. 
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     Hereinafter, optical systems having a hand-shake correction function according to the present invention will be described with reference to the drawings. FIGS. 1, 5, 8, 11, and 15 are diagrams showing respectively the lens construction of a first to a fifth embodiment of the present invention, each illustrating the lens construction in the normal (pre-decentering) state at the wide-angle end [W]. In each figure, ri (i=1, 2, 3, . . . ) represents the radius of curvature of the i-th surface from the object side, and di (i=1, 2, 3, . . . ) represents the i-th axial distance from the object side. In FIGS. 1, 11, and 15, arrows ml to m4 schematically show the movement directions of the first lens unit Gr1, the second lens unit Gr2, the aperture diaphragm S together with the third lens unit Gr3, and the fourth lens unit Gr4, respectively, during zooming from the wide-angle end [W] to the telephoto end [T]. In FIGS. 1, 5, 8, 11, and 15, arrow C schematically show the movement direction of a hand-shake correction lens unit. 
     The zoom lens system of the first embodiment consists of, from the object side, a first lens unit Gr1 having a positive refractive power, a second lens unit Gr2 having a negative refractive power, a third lens unit having a positive refractive power, and a fourth lens unit having a positive refractive power, and zooming is performed by varying the distances between these lens units. In the first embodiment, the third lens unit Gr3 is divided into, from the object side, a front lens unit GrA and a rear lens unit GrB. Here, the front lens unit GrA serves as a hand-shake correction lens unit, and hand-shake correction is achieved by decentering it translationally (that is, by decentering it in a direction perpendicular to the optical axis AX). 
     The single-focal-length lens systems of the second and third embodiments consist of, from the object side, a first lens unit Gr1 having a positive refractive power, a second lens unit Gr2 having a negative refractive power, and a third lens unit Gr3 having a positive refractive power, and focusing is performed by moving the second lens unit Gr2. In the second and third embodiments, the second lens unit Gr2 is divided into, from the object side, a fixed lens unit GrL, a hand-shake correction lens unit GrM, and another fixed lens unit GrN, and hand-shake correction is achieved by decentering the hand-shake correction lens unit GrM translationally. Moreover, a protective glass is provided at the image-side end of these single-focal-length lens systems. 
     The zoom lens systems of the fourth and fifth embodiments consist of, from the object side, a first lens unit Gr1 having a positive refractive power, a second lens unit Gr2 having a negative refractive power, a third lens unit Gr3 having a positive refractive power, and a fourth lens unit Gr4 having a negative refractive power, and zooming is performed by varying the distances between these lens units. In the fourth and fifth embodiments, the second lens unit Gr2 is divided into, from the object side, a front lens unit GrA and a rear lens unit GrB. Here, the front lens unit GrA serves as a hand-shake correction lens unit, and hand-shake correction is achieved by decentering it translationally. 
     In all the embodiments of the present invention, the lens element PF disposed at the image-side end of the hand-shake correction lens unit and the lens element PR disposed next to the image-side surface of the lens element PF and kept in a fixed position during hand-shake correction satisfy conditions (1), (2), and (4), or conditions (1), (3), and (4). This makes it possible to correct various aberrations properly both in the normal state and in the hand-shake correction state. 
     Tables 1 to 5 list the construction data, aberration characteristics, and other data of the optical systems of the above described first to fifth embodiments (FIGS. 1, 5, 8, 11, and 15), respectively. In these tables, ri (i=1, 2, 3, . . . ) represents the radius of curvature of the i-th surface from the object side, di (i=1, 2, 3, . . . ) represents the i-th axial distance from the object side (as observed before decentering), and Ni (i=1, 2, 3, . . . ) and vi (i=1, 2, 3, . . . ) represent the refractive index (Nd) and the Abbe number (νd), for d-lines, of the i-th lens element from the object side, respectively. Moreover, for each of such axial distances that vary with zooming, three values are given which are the actual axial distances between the lens units involved when the optical system is set at the wide-angle end [W], at the middle focal length [M], and at the telephoto end [T], respectively. Likewise, for the focal length f and the f-number FNO of the entire system, three values are listed which respectively correspond to the above three states of the optical system. 
     In the first embodiment, the surface having the radius of curvature r28* is an aspherical surface. It is assumed that an aspherical surface is defined by formula (AS) below. 
     
         X=C·Y.sup.2 /{1+(1-.di-elect cons.·C.sup.2 ·Y.sup.2).sup.1/2 }+A4·Y.sup.4 +A6·Y.sup.6 +A8·Y.sup.8 +A10·Y.sup.10 +A12·Y.sup.12) (AS) 
    
     where 
     X: displacement from the reference surface of the optical axis direction; 
     Y: height in a direction perpendicular to the optical axis; 
     C: paraxial curvature; 
     .di-elect cons.: quadric surface parameter; and 
     A4, A6, A8, A10, and A12: aspherical coefficients of the fourth, sixth, eighth, tenth, and twelfth degree. 
     Tables 6 to 9 list the actual values corresponding to the previously noted conditions (1) to (11) as observed in each embodiment, together with related data. 
     FIGS. 2A to 2I, 6A to 6C, 9A to 9C, 12A to 12I, and 16A to 16I are aberration diagrams showing the longitudinal aberrations observed in the first to fifth embodiments, respectively, when the optical system is in the normal (pre-decentering) state. Among these figures, FIGS. 2A to 2C, 12A to 12C, and 16A to 16C show the longitudinal aberrations observed when the optical system is set to the wide-angle end [W], FIG. 2D to 2F, 12D to 12F, and 16D to 16F show those when the optical system is set to the middle focal length [M], and FIG. 2G to 2I, 12G to 12I, and 16G to 16I show those when the optical system is set to the telephoto end [T]. In each aberration diagram, the solid line (d) represents the aberration for d-lines, and the broken line (SC) represents the sine condition. Moreover, the broken line (DM) and the solid line (DS) represent the astigmatism on the meridional and sagittal planes, respectively. 
     FIGS. 3A to 3E and 4A to 4E, 7A to 7E, 10A to 10E, 13A to 13E and 14A to 14E, and 17A to 17E and 18A to 18E are aberration diagrams showing the lateral aberrations observed with respect to a light beam on the meridional plane in the first to fifth embodiments, respectively, with FIGS. 3A to 3E, 13A to 13E, and 17A to 17E showing those when the optical system is set to the wide-angle end [W], and FIGS. 4A to 4E, 14A to 14E, and 18A to 18E showing those when the optical system is set to the telephoto end [T]. FIGS. 3A to 3B and 4A to 4B, 7A to 7B, 10A to 10B, 13A to 13B and 14A to 14B, and 17A to 17B and 18A to 18B show such aberrations observed before the decentering of the hand-shake correction lens unit, whereas FIGS. 3C to 3E and 4C to 4E, 7C to 7E, 10C to 10E, 13C to 13E and 14C to 14E, and 17C to 17E and 18C to 18E show such aberrations after the decentering. Here, it is assumed that, after the decentering, that is, in the hand-shake correction state, the hand-shake correction lens unit is inclined at a hand-shake correction angle of θ=0.7° (=0.0122173 rad). 
     As described heretofore, according to the present invention, it is possible to correct various aberrations properly both in the normal state and in the hand-shake correction state. 
     
                       TABLE 1______________________________________&lt;&lt; Embodiment 1 &gt;&gt;f = 22.6˜50.5˜78.0FNO = 4.24˜6.22˜7.28Radius of Axial      RefractiveCurvature Distance   Index      Abbe Number______________________________________&lt;First Lens Unit Gr1 - positive&gt;r1   971.931         d1     1.300 N1   1.83350                                 ν1                                      21.00r2   88.101         d2     6.550 N2   1.58913                                 ν2                                      61.11r3   -137.987         d3     0.100r4   36.312         d4     4.250 N3   1.71300                                 ν3                                      53.93r5   99.372   d5   1.845˜12.505˜19.997&lt;Second Lens Unit Gr2 - negative&gt;r6   39.377         d6     1.100 N4   1.80420                                 ν4                                      46.50r7   10.701         d7     4.400r8   -32.341         d8     0.950 N5   1.75450                                 ν5                                      51.57r9   21.282         d9     0.300r10  17.036         d10    3.700 N6   1.75000                                 ν6                                      25.14r11  -40.855         d11    0.940r12  -16.652         d12    1.300 N7   1.69680                                 ν7                                      56.47r13  -66.585   d13  11.379˜4.400˜2.000&lt;Aperture Diaphragm S, Third Lens Unit Gr3 - positive&gt;r14  ∞ (Aperture Diaphragm S)   d14  0.500{Front Lens Unit GrA - Hand-shake Correction Lens Unit}r15  75.184         d15    1.500 N8   1.62041                                 ν8                                      60.29r16  -61.919         d16    0.500r17  33.676         d17    1.215 N9   1.51728                                 ν9                                      69.43 -- PFr18  26.682         d18    1.000{Rear Lens Unit GrB}r19  16.915         d19    1.215 N10  1.51728                                 ν10                                      69.43 -- PRr20  24.253         d20    1.500r21  -61.919         d21    1.310 N11  1.62041                                 ν11                                      60.29r22  -32.276         d22    0.110r23  18.287         d23    4.710 N12  1.51742                                 ν12                                      52.15r24  -14.950         d24    1.360 N13  1.80741                                 ν13                                      31.59r25  126.060   d25  5.300˜1.623˜1.000&lt;Fourth Lens Unit Gr4 - positive&gt;r26  34.239         d26    4.820 N14  1.51823                                 ν14                                      58.96r27  -19.452         d27    1.470r28  -106.937         d28    0.100 N15  1.51790                                 ν15                                      52.31r29  -45.739         d29    1.400 N16  1.80500                                 ν16                                      40.97r30  42.176   Σd        66.125˜66.128˜70.598______________________________________     [Aspherical Coefficient]     r28: ε = 1.0000     A4 = -0.10470 × 10.sup.-3     A6 = -0.34147 × 10.sup.-6     A8 = -0.51713 × 10.sup.-9     A10 = -0.14464 × 10.sup.-10     A12 = -0.10659 × 10.sup.-16______________________________________ 
    
     
                       TABLE 2______________________________________&lt;&lt; Embodiment 2 &gt;&gt;f = 470.0FNO = 4.10Radius of Axial      RefractiveCurvature Distance   Index      Abbe Number______________________________________&lt;First Lens Unit Gr1 - positive&gt;r1   252.985         d1     12.855                      N1   1.49520                                 ν1                                      79.74r2   -412.809         d2     0.402r3   145.653         d3     14.863                      N2   1.49520                                 ν2                                      79.74r4   -2674.154         d4     2.410r5   -903.130         d5     5.463 N3   1.65100                                 ν3                                      39.55r6   216.707         d6     134.974&lt;Second Lens Unit Gr2 - negative&gt;{Fixed Lens Unit GrL}r7   -145.734         d7     2.812 N4   1.65446                                 ν4                                      33.86r8   -424.067         d8     1.607r9   160.143         d9     4.178 N5   1.69680                                 ν5                                      56.47r10  1028.521         d10    3.214r11  -874.378         d11    4.820 N6   1.67339                                 ν6                                      29.25r12  -88.133         d12    2.491 N7   1.58913                                 ν7                                      61.11r13  ∞         d13    2.812{Hand-shake Correction Lens Unit GrM}r14  -1716.385r14  -1716.385         d14    2.410 N8   1.78100                                 ν8                                      44.55r15  84.092         d15    1.500r16  56.872         d16    5.000 N9   1.49140                                 ν9                                      57.82 -- PFr17  71.858         d17    3.500{Fixed Lens Unit GrN}r18  93.140r18  93.140         d18    2.000 N10  1.49140                                 ν10                                      57.82 -- PRr19  70.119         d19    11.085&lt;Third Lens Unit Gr3 - positive, Aperture Diaphragm S, Protective Glass&gt;r20  -856.898         d20    4.017 N11  1.61800                                 ν11                                      63.39r21  -101.065         d21    19.282r22  ∞ (Aperture Diaphragm S)         d22    44.212r23  ∞         d23    1.446 N12  1.51680                                 ν12                                      64.20r24  ∞         Σd                287.353______________________________________ 
    
     
                       TABLE 3______________________________________&lt;&lt; Embodiment 3 &gt;&gt;f = 236.0FNO = 2.89Radius of Axial      RefractiveCurvature Distance   Index      Abbe Number______________________________________&lt;First Lens Unit Gr1 - positive&gt;r1   118.597         d1     12.800                      N1   1.49520                                 ν1                                      79.74r2   -317.987         d2     0.336r3   91.844         d3     11.600                      N2   1.49520                                 ν2                                      79.74r4   -637.093         d4     2.016r5   -391.734         d5     2.960 N3   1.68150                                 ν3                                      36.64r6   132.684         d6     62.400&lt;Second Lens Unit Gr2 - negative&gt;{Fixed Lens Unit GrL}r7   -104.212         d7     2.000 N4   1.65446                                 ν4                                      33.86r8   -163.145         d8     1.080r9   85.118         d9     3.200 N5   1.60311                                 ν5                                      60.74r10  474.044         d10    2.400{Hand-shake Correction Lens Unit GrM}r11  -1363.419         d11    5.200 N6   1.71736                                 ν6                                      29.42r12  -53.358         d12    1.480 N7   1.60311                                 ν7                                      60.74r13  338.866         d13    2.520r14  -173.400         d14    1.360 N8   1.67000                                 ν8                                      57.07r15  52.435         d15    2.000r16  40.163         d16    1.500 N9   1.58340                                 ν9                                      30.23 -- PFr17  50.774         d17    2.000{Fixed Lens Unit GrN}r18  70.431         d18    2.000 N10  1.58340                                 ν10                                      30.23 -- PRr19  47.223         d19    9.000&lt;Aperture Diaphragm S, Third Lens Unit Gr3 - positive, Protective Glass&gt;r20  ∞ (Aperture Diaphragm S)         d20    1.200r21  521.110         d21    5.600 N11  1.60311                                 ν11                                      60.74r22  -33.634         d22    1.280 N12  1.65446                                 ν12                                      33.86r23  -63.807         d23    24.640r24  ∞         d24    1.440 N13  1.51680                                 ν13                                      64.20r25  ∞         Σd                162.012______________________________________ 
    
     
                       TABLE 4______________________________________&lt;&lt; Embodiment 4 &gt;&gt;f = 82.2˜160.0˜233.6FNO = 4.60˜5.81˜6.19Radius of Axial      RefractiveCurvature Distance   Index      Abbe Number______________________________________&lt;First Lens Unit Gr1 - positive&gt;r1   971.931         d1     1.700 N1   1.61293                                 ν1                                      36.96r2   49.221         d2     6.460 N2   1.49310                                 ν2                                      83.58r3   -1678.106         d3     0.100r4   56.111         d4     3.820 N3   1.49310                                 ν3                                      83.58r5   859.262   d5   3.300˜27.890˜41.425&lt;Second Lens Unit Gr2 - negative&gt;{Front Lens Unit GrA - Hand-shake Correction Lens Unit}r6   -69.399         d6     1.830 N4   1.71300                                 ν4                                      53.93r7   34.412         d7     3.000r8   38.193         d8     2.750 N5   1.67339                                 ν5                                      29.25r9   1893.115         d9     2.000r10  -35.714         d10    1.215 N6   1.51728                                 ν6                                      69.43 -- PFr11  -29.097         d11    2.000{Rear Lens Unit GrB}r12  -24.999         d12    1.215 N7   1.51728                                 ν7                                      69.43 -- PRr13  -30.588   d13  20.004˜4.713˜1.036&lt;Aperture Diaphragrn S, Third Lens Unit Gr3 - positive&gt;r14  ∞ (Aperture Diaphragrn S)         d14    1.380r15  60.855         d15    1.300 N8   1.84666                                 ν8                                      23.82r16  26.095         d16    2.460r17  41.450         d17    2.840 N9   1.51680                                 ν9                                      64.20r18  -111.975         d18    0.400r19  35.623         d19    4.550 N10  1.51680                                 ν10                                      64.20r20  -42.960   d20  20.260˜9.024˜0.874&lt;Fourth Lens Unit Gr4 - negative&gt;r21  206.481         d21    1.080 N11  1.71300                                 ν11                                      53.93r22  24.106         d22    1.540r23  -195.003         d23    3.480 N12  1.67339                                 ν12                                      29.25r24  -18.789         d24    1.130 N13  1.75450                                 ν13                                      51.57r25  ∞   Σd        89.815˜87.878˜89.586______________________________________ 
    
     
                       TABLE 5______________________________________&lt;&lt; Embodiment 5 &gt;&gt;f = 82.2˜160.0˜233.6FNO = 4.60˜5.81˜6.60Radius of Axial      RefractiveCurvature Distance   Index      Abbe Number______________________________________&lt;First Lens Unit Gr1 - positive&gt;r1   103.105         d1     1.700 N1   1.61293                                 ν1                                      36.96r2   47.562         d2     6.460 N2   1.49310                                 ν2                                      83.58r3   -214.862         d3     0.100r4   50.735         d4     3.820 N3   1.49310                                 ν3                                      83.58r5   247.066   d5   3.300˜25.679˜34.769&lt;Second Lens Unit Gr2 - negative&gt;{Front Lens Unit GrA - Hand-shake Correction Lens Unit}r6   -70.232         d6     1.830 N4   1.71300                                 ν4                                      53.93r7   33.675         d7     1.000r8   20.095         d8     2.000 N5   1.51728                                 ν5                                      69.43 -- PFr9   24.008         d9     2.000{Rear Lens Unit GrB}r10  30.117         d10    1.215 N6   1.51728                                 ν6                                      69.43 -- PRr11  19.468         d11    1.000r12  27.326         d12    2.750 N7   1.67339                                 ν7                                      29.25r13  157.462   d13  22.913˜7.665˜1.036&lt;Aperture Diaphragin S, Third Lens Unit Gr3 - positive&gt;r14  ∞ (Aperture Diaphragm S)         d14    1.380r15  86.768         d15    1.300 N8   1.84666                                 ν8                                      23.82r16  28.577         d16    2.460r17  46.617         d17    2.840 N9   1.51680                                 ν9                                      64.20r18  -58.745         d18    0.400r19  34.744         d19    4.550 N10  1.51680                                 ν10                                      64.20r20  -42.470   d20  18.566˜7.464˜0.874&lt;Fourth Lens Unit Gr4 - negative&gt;r21  488.716         d21    1.080 N11  1.71300                                 ν11                                      53.93r22  24.791         d22    1.540r23  -153.247         d23    3.480 N12  1.67339                                 ν12                                      29.25r24  -17.314         d24    1.130 N13  1.75450                                 ν13                                      51.57r25  ∞   Σd        88.815˜84.843˜80.715______________________________________ 
    
     
                       TABLE 6______________________________________    S(PF)     S(PR)   S(PR)/S(PF)______________________________________Embodiment 1      -8.630      5.610   -0.65Embodiment 2      8.593       -7.199  -0.84Embodiment 3      8.570       -5.069  -0.59Embodiment 4      -9.795      9.947   -1.02Embodiment 5      11.270      -4.656  -0.41______________________________________ 
    
     
                       TABLE 7______________________________________    P(PF)     P(PR)   P(PR)/P(PF)______________________________________Embodiment 1      -0.0038     0.0098  -2.58Embodiment 2      0.002       -0.0017 -0.85Embodiment 3      0.0032      -0.0039 -1.22Embodiment 4      0.0035      -0.0035 -1.00Embodiment 5      0.0049      -0.0090 -1.84______________________________________ 
    
     
                       TABLE 8______________________________________      |P(PF)|/P                      |P(PR)|/P      [W]  [T]        [W]    [T]______________________________________Embodiment 1 0.09   0.30       0.22 0.76Embodiment 2 0.94              0.80Embodiment 3 0.76              0.92Embodiment 4 0.29   0.82       0.29 0.82Embodiment 5 0.40   1.14       0.74 2.10______________________________________ 
    
     
                       TABLE 9______________________________________     d(PF, PR) · P    [W]    [T]     MT/MW______________________________________Embodiment 1      0.044    0.013   2.17Embodiment 2    0.0074      (Single Focal Length)Embodiment 3    0.0085      (Single Focal Length)Embodiment 4      0.024    0.0086  1.78Embodiment 5      0.024    0.0086  1.92______________________________________