Lens system with image blur correction

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.

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.degree.. 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. 
EQU 5&lt;S(PR)/S(PF)&lt;0 (1) 
EQU 1.0&lt;S(PF) (2) 
EQU S(PF)&lt;0 (3) 
EQU 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: 
EQU 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: 
EQU -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: 
EQU 2.2&lt;S(PF) (2a) 
EQU 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: 
EQU 1.0&lt;S(PR) (5) 
EQU S(PR)&lt;0 (6) 
EQU 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: 
EQU 2.2&lt;S(PR) (5a) 
EQU 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: 
EQU 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: 
EQU 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: 
EQU 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: 
EQU 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: 
EQU 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: 
EQU .nu.p&gt;.nu.n (11) 
where 
.nu.p: Abbe number of the positive lens elements in the hand-shake 
correction lens units; 
.nu.n: 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', z', .theta.') be the actual Y coordinate, 
at a hand-shake correction angle .theta., of a light beam with a paraxial 
image point at (y', z') (note that y(0, 0, .theta.)=0 in any case, since 
correction is performed such that axial image points are brought to a 
rest). Then, expression (a) below holds. 
EQU .DELTA.Y(Y', Z', .theta.)=Y(Y', Z', .theta.)-Y(Y', z', 0) (a) 
Unless otherwise specified, the off-axial image-point movement error 
.DELTA.Y.sub.Y ' of an image point on the Y axis and the off-axial 
image-point movement error .DELTA.Y.sub.z ' 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. 
EQU .DELTA.Y.sub.Y'={.DELTA.Y (0.7field, 0, 0.7.degree.)+.DELTA.Y(-0.7field, 0, 
0.7.degree.)}/2 (b) 
EQU .DELTA.Y.sub.z'=.DELTA.Y (0, 0.7field, 0.7.degree.) (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 
.DELTA.M' and a sagittal one-side blur .DELTA.S' occur, which are 
respectively represented by expressions (d) and (e) below. 
EQU .DELTA.M'={meridional value(y'=0.7field, z=0, 
.theta.=0.7.degree.)-meridional value(y'=-0.7field, z=0, .theta.=0.7+)}/2 
(d) 
EQU .DELTA.S'={sagittal value(y'=0.7field, z=0, .theta.=0.7.degree.)-sagittal 
value(y'=-0.7field, z=0, .theta.=0.7.degree.)}/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. 
EQU AXCM={Y(upper zonal, .theta.=0.7.degree.)+Y(lower zonal, 
.theta.=0.7.degree.)}/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. 
EQU (Axial Lateral Chromatic Aberration)={Y(g-lines, 
.theta.=0.7.degree.)-Y(d-lines, .theta.=0.7.degree.)} (g) 
In connection with the decentering aberration coefficients defined above, 
their applications are discussed in a thesis by Yoshiya Matsui, titled "On 
the Third-degree Aberrations in Optical Systems Involving Decentering" 
(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, .OMEGA. 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, .phi..OMEGA. is its azimuth, R represents the radius of the 
entrance pupil as seen on the object-side principal plane, and .phi.R 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 .DELTA.Y and .DELTA.Z on the image plane 
(image surface) IS are represented by expressions (1A) and (1B) below. 
##EQU2## 
Here, if it is assumed that 
(.DELTA.E).nu.: prismatic effect (lateral deviation of the image), 
(VE1).nu., (VE2).nu.: rotationally asymmetrical distortion, 
(IIIE).nu., (PE).nu.: rotationally asymmetrical astigmatism and image-plane 
inclination, 
(IIE).nu.: 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.sub.1 represents a light 
ray in a normal state without decentering, and L.sub.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.sub.1 represents a light beam before hand-shake correction 
and M.sub.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.sub.1 and the image plane IS are 
in fixed positions. However, in a hand-shake correction optical system, 
the object plane OS.sub.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.sub.1 does not rotate. In FIG. 24, OS.sub.1 represents the object plane 
before hand-shake correction, and OS.sub.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, .sup.R () indicates a 
reversed system, and N represents the refractive index. 
______________________________________ 
.sup.R .alpha. = .sup.R N/.sup.R g$ = -.alpha.' 
(2A) 
.sup.R .alpha.# = .alpha.'# (2B) 
.sup.R .alpha..mu.' = -.alpha..nu. 
(2C) 
.sup.R .alpha..mu.'# = .alpha..nu.# 
(2D) 
.sup.R P.mu. = P.nu. 
&lt;non-reversed&gt; 
(2E) 
.sup.R .phi..mu. = .phi..nu. 
&lt;non-reversed&gt; 
(2F) 
.sup.R I.mu. = I.nu. 
&lt;non-reversed&gt; 
(2G) 
.sup.R II.mu. = -II.nu. 
&lt;reversed&gt; (2H) 
.sup.R III.mu. = III.nu. 
&lt;non-reversed&gt; 
(2I) 
.sup.R V.mu. = -V.nu. 
&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 .nu. 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.about.j are decentered. When a hand-shake correction lens unit 
is decentered translationally, the decentering amounts Ei.about.Ej of all 
the decentered surfaces i.about.j are equal. This means that, in handling 
the aberration coefficients, it is only necessary to consider their 
respective sum totals. For example, 
EQU (.DELTA.E)i.about.j=(.nu.=i.fwdarw.j).SIGMA.{-2.multidot.(.alpha..nu.'-.alp 
ha..nu.)} 
Further, from .alpha..nu.'=.alpha..nu.+1, the following expression is 
obtained: 
EQU (.DELTA.E)i.about.j=-2.multidot.(.alpha.j'-.alpha.i) 
In similar manners, the intermediate terms of .SIGMA. in the other 
aberration coefficients can be eliminated. For example, 
EQU (PE)i.about.j=(.mu.=i.fwdarw.j).SIGMA.{.alpha..nu.'.multidot.(.mu.=.nu.+1.f 
wdarw.k).SIGMA.P.mu.-.alpha..nu..multidot.(.mu.=.nu..fwdarw.k).SIGMA.P.mu.} 
=.alpha.j'.multidot.(.mu.=j+1.fwdarw.k).SIGMA.P.mu.-.alpha.i.multidot.(.mu. 
=i.fwdarw.k).SIGMA.P.mu. 
This is further transformed into 
EQU (PE)i.about.j=(.alpha.j'-.alpha.i).multidot.(.mu.=j+1.fwdarw.k).SIGMA.P.mu. 
-.alpha.i.multidot.(.mu.=i.fwdarw.j).SIGMA.P.mu. 
where 
(u=j+1.fwdarw.k).SIGMA.P.mu.: sum of P's (Petzval sum) of the lenses 
disposed on the downstream side of the hand-shake correction lens unit; 
(.mu.=i.fwdarw.j).SIGMA.P.mu.: sum of P's (Petzval sum) of the lenses 
constituting the hand-shake correction lens unit. 
Eventually, the following expression is obtained: 
EQU (PE)i.about.j=(.alpha.j'-.alpha.i)P.sub.R -.alpha.i.multidot.P.sub.D 
where 
().sub.R : the sum of the aberration coefficients of the lenses disposed on 
the downstream side of the hand-shake correction lens unit; 
().sub.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.about.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 .DELTA.E, 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). 
EQU .DELTA.Y#=-(E/2.alpha.'.sub.k).multidot.[.DELTA.E+(N.multidot.tan 
.OMEGA.).sup.2 .multidot.{(2+cos.sup.2 .phi..OMEGA.)VE1-VE2}](4A) 
EQU .DELTA.Z#=-(E/2.alpha.').multidot.{(N.multidot.tan .OMEGA.).sup.2 
.multidot.sin 2.phi..OMEGA.).multidot.VE1} (4B) 
From expressions (4A) and (4B), expressions (4C) and (4D) below are 
obtained (for an axial light ray, tan .OMEGA.=0). 
EQU .DELTA.Y.sub.0 #=-(E/2.alpha.'.sub.k).multidot..DELTA.E (4C) 
EQU .DELTA.Z.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: 
EQU Y#=g$.sub.k .multidot.tan .OMEGA. 
This is transformed, using the sine theorem, into 
EQU Y'#/{sin(.pi./2-.OMEGA.')}=(Y#+.DELTA.Y#-.DELTA.Y.sub.0 
#)/{sin(.pi./2+.OMEGA.'-.theta.)} 
then, the .DELTA.Y'# after the rotational conversion is represented by 
EQU .DELTA.Y'#=(Y'#)-(Y#)=[Y#.multidot.cos 
.OMEGA.'+{(.DELTA.Y#)-(.DELTA.Y.sub.0 #)}.multidot.cos 
.OMEGA.'-Y#.multidot.cos(.OMEGA.'-.theta.)]/cos(.OMEGA.'-.theta.) 
The numerator of this expression is transformed into 
##EQU5## 
Here, since e is small and negligible compared with the other two terms, 
(1-cos .theta.).apprxeq..theta..sup.2 /2, sin .theta..apprxeq.=.theta.. 
Moreover, cos .theta.'/{cos (.theta.'-.theta.)}.apprxeq.1, sin 
.OMEGA.'/{cos(.OMEGA.'-.theta.).apprxeq.tan .OMEGA.. 
Thus, the following expression is obtained. 
EQU .DELTA.Y'#.apprxeq.(.DELTA.Y#-.DELTA.Y.sub.0 
#)-Y#.multidot..theta..multidot.tan .OMEGA. 
where (.DELTA.Y#-.DELTA.Y.sub.0 #) represents the off-axial image-point 
movement errors resulting from the translational decentering, and 
Y#.multidot..theta..multidot.tan .OMEGA. is an additional term related to 
the rotation (but not related to the aberration coefficients). Note that, 
since .OMEGA. here is on the X-Y cross section, 
EQU .DELTA.Y'#.apprxeq.(.DELTA.Y#-.DELTA.Y.sub.0 
#)-Y#.multidot..theta..multidot.tan .OMEGA..multidot.cos .phi..OMEGA.(5A) 
Next, the conversion to the image plane IS will be described, with 
reference to FIG. 27. The magnification .beta. is defined by 
EQU .beta.=g$.sub.1 /g$.sub.k =.alpha..sub.k /.alpha..sub.1 
Here, .alpha..sub.1 =1/g$.sub.1. On the other hand, the image plane IS and 
the object plane OS have a relation 
EQU Y=.beta..multidot.Y# 
Further, since Y# and .DELTA.Y# retain the form of 1/.alpha..sub.k 
'.times.(), the above expression is further transformed, as 
EQU Y=.beta..multidot.Y# 
EQU =(.alpha..sub.k '/.alpha..sub.1).multidot.(1/.alpha..sub.k ').times.() 
EQU =g$.sub.1 .times.() 
Here, if it is assumed that g$.sub.1 '.fwdarw..infin., then g$.sub.1 =-F1. 
Hence, 
EQU Y=-F1.times.() 
EQU =-F1.times..alpha..sub.k '.times.Y.sub.# 
Next, the off-axial image-point movement errors on the image plane will be 
described. From expression (4C) and .alpha..sub.k '=1/g.sub.k '$, the 
decentering amount E is obtained as 
EQU .theta.=.DELTA.Y.sub.0 #/g$.sub.k '=E.multidot..DELTA.E/2 
EQU E=2.multidot..theta./.DELTA.E 
Then, normalization is performed to make the hand-shake correction angle 
.theta. constant (0.7 deg=0.0122173 rad). 
As the result of translational decentering (involving no rotational 
decentering), .DELTA.Y=(.DELTA.Y#-.DELTA.Y.sub.0 #) is subjected to 
image-plane conversion (here, N.multidot.tan.OMEGA.=.PHI./F1, .PHI..sup.2 
=Y.sup.2 +Z.sup.2). Thus, expressions (6A) to (6D) below are obtained. 
EQU .DELTA.Y=(.theta..multidot..PHI..sup.2 /F1).multidot.[{(2+cos 
2.multidot..phi..OMEGA.).multidot.VE1-VE2}/.DELTA.E] (6A) 
EQU .DELTA.Z=(.theta..multidot..PHI..sup.2 /F1).multidot.[{(sin 
2.multidot..phi..OMEGA.).multidot.VE1-VE2}/.DELTA.E] (6B) 
Y.sub.+ Image Point, Y.sub.- Image Point {.phi..OMEGA.=0, .pi. of 
expressions (6A) and (6B)}: 
EQU .DELTA.Y.sub.Y =(.theta..multidot.Y.sup.2 
/F1).multidot.{(3.multidot.VE1-VE2)/.DELTA.E} (6C) 
Z Image Point{.phi..OMEGA.=.pi./2 of expressions (6A) and (6B)}: 
EQU .DELTA.Y.sub.Z =(.theta..multidot.Z.sup.2 
/F1).multidot.{(VE1-VE2)/.DELTA.E}(6D) 
These expressions are then subjected to rotational conversion. Since 
Y#=-Y/(F1.times.a.sub.k '), the term -Y#.multidot..theta..multidot.tan 
.OMEGA..multidot.cos .phi..OMEGA. of the expression (5A) can be expressed 
as 
EQU -Y#.multidot..theta..multidot.tan .OMEGA..multidot.cos 
.phi..OMEGA.=Y/(F1.times..alpha..sub.k ').multidot..theta..multidot.tan 
.OMEGA..multidot.cos .phi..OMEGA. 
At the Y.sub.+ and Y.sub.- image points, .phi..OMEGA.=0, .pi., and tan 
.OMEGA./.alpha..sub.k '=Y. Hence, on the image plane, 
-Y#.multidot..theta..multidot.tan .OMEGA..multidot.cos 
.phi..OMEGA.=Y.sup.2 .multidot..theta./F1. By adding this to expression 
(6C), expression (6E) below is obtained. On the other hand, at the Z image 
point, .phi..OMEGA.=.pi./2. Hence, on the image plane, 
-Y#.multidot..theta..multidot.tan .OMEGA..multidot.cos .phi..OMEGA.=0. By 
adding this to expression (6D), expression (6F) below is obtained. 
EQU .DELTA.Y.sub.Y '=(.theta..multidot.Y.sup.2 
/F1).multidot.{(3.multidot.VE1-VE2-.DELTA.E)/.DELTA.E} (6E) 
EQU .DELTA.Y.sub.Z '=.DELTA.Y.sub.Z (6F) 
[One-side blur] 
Next, the one-side blur will be described. From the expressions (1A) and 
(1B), it is known that .DELTA.M equals {.DELTA.Y with .phi.R=0 in the 
first-degree terms with respect to R}.times.g$.sub.k ' and .DELTA.S equals 
{.DELTA.Z with .phi.R=.pi./2 in the first-degree terms with respect to 
R}.times.g$.sub.k '. On the object plane OS before rotation, the following 
expression holds (here, it is assumed that a.sub.k '=N.sub.k '/g$.sub.k ' 
and E/2=/.DELTA.E): 
EQU .DELTA.M#=(-g$.sub.k '.sup.2 .multidot..theta./N.sub.k 
').times.2.multidot.R.multidot.(N.multidot.tan .OMEGA.).multidot.cos 
.phi..OMEGA..multidot.{(3.multidot.IIIE+PE)/.DELTA.E} 
After rotation, the following expression holds: 
EQU .DELTA.M'#.apprxeq..DELTA.M#+.theta.Y# 
By converting the aberration coefficients to those as seen on the image 
plane and substituting N.sub.k '=1 and N=1, the following expression is 
obtained: 
EQU .DELTA.M'=.beta..sup.2 .multidot..DELTA.M'#=-g$.sub.1.sup.2 
.multidot..theta..times.2.multidot.R.multidot.tan .OMEGA..multidot.cos 
.phi..OMEGA..multidot.{(3.multidot.IIIE+PE)/.DELTA.E}+.beta..multidot.Y.mu 
ltidot..theta. 
Assume that the object plane OS is at .infin. (that is, g$.sub.1 =-F1, 
.beta..fwdarw.0, tan .OMEGA.=Y/F1, and .phi..OMEGA.=0). Then, the 
meridional one-side blur .DELTA.M' is represented by expression (7A) 
below; likewise, the sagittal one-side blur is represented by expression 
(7B) below. 
EQU .DELTA.M'=-2-F1.multidot.Y.multidot..theta..multidot.R.multidot.{(3.multido 
t.IIIE+PE)/.DELTA.E} (7A) 
EQU .DELTA.S'=-2.multidot.F1.multidot.Y.multidot..theta..multidot.R.multidot.(I 
IIE+PE)/.DELTA.E} (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 .OMEGA.=0 is 
represented by the following expression: 
EQU .DELTA.Y.sub.upper #=.DELTA.Y#(.OMEGA.=0, .phi..sub.R 
=0)-.DELTA.Y#(.OMEGA.=0, R=0) 
EQU =-E/(2.multidot..alpha.').times.R.sup.2 .times.3.multidot.IIE 
On the other hand, the coma resulting from downward (lower) decentering by 
.OMEGA.=0 is represented by the following expression (having the same 
value and sign as .DELTA.Y.sub.upper #): 
EQU .DELTA.Y.sub.Lower #=.DELTA.Y#(.OMEGA.=0, .phi..sub.R 
=.pi.)-.DELTA.Y#(.OMEGA.=0, R=0) 
EQU =-E/(2.multidot..alpha.').times.R.sup.2 .times.3.multidot.IIE 
Since .OMEGA.=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 (.DELTA.Y=.beta..multidot..DELTA.Y#, 
E/2=.theta./.DELTA.E), the following expression is obtained: 
EQU .DELTA.Y.sub.Upper =F1.times..theta..times.R.sup.2 
.times.(3.multidot.IIE/.DELTA.E)=.DELTA.Y.sub.Lower 
Hence, the axial coma AXCM is represented by expression (8A) below: 
AXCM=(.DELTA.Y.sub.Upper +.DELTA.Y.sub.Lower) /2 
EQU =.DELTA.Y.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: 
EQU VE.sub.Y ={(3.multidot.VE1-VE2-.DELTA.E)/.DELTA.E} (9A) 
Off-axial image-point movement error of an image point on the Z axis: 
EQU VE.sub.Z ={(VE1-VE2)/.DELTA.E} (9B) 
Meridional single-sided blur: 
EQU IIIE.sub.M ={(3.multidot.IIIE+PE)/.DELTA.E} (9C) 
Sagittal single-sided blur: 
EQU IIIE.sub.S ={(IIIE+PE)/.DELTA.E} (9D) 
Off-axial coma: 
EQU IIE.sub.A ={(3.multidot.IIE)/.DELTA.E} (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 ().sub.D : sum of the aberration coefficients of the lenses 
constituting the hand-shake correction lens unit; 
().sub.R : sum of the aberration coefficients of the lenses disposed on the 
downstream side of the hand-shake correction lens unit; 
A=.alpha.i/(.alpha.j'-.alpha.i) (here, the hand-shake correction lens unit 
is assumed to include surfaces i.about.j); 
A#=.alpha.i#/(.alpha.j'-.alpha.i); 
H#=(.alpha.i'#-.alpha.i#)/(.alpha.j'-.alpha.i). 
Since .DELTA.E=-2.multidot.(.alpha.j'-.alpha.i) (here, 
(.alpha.j'-.alpha.i)=.+-.0.0122173 if 0.7.degree./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 ().sub.D : sum of the aberration coefficients of the lenses 
constituting the hand-shake correction lens unit, as seen in a 
non-reversed optical system; 
().sub.F : sum of the aberration coefficients of the lenses disposed on the 
upstream side of the hand-shake correction lens unit; 
A=-.alpha.n'/(.alpha.n'-.alpha.m); 
A#=.alpha.n'#/(.alpha.n'-.alpha.m); 
H=-(.alpha.n'#-.alpha.m#)/(.alpha.n'-.alpha.m)=-(.SIGMA.h.mu.#.multidot..ph 
i..mu.)/(.SIGMA.h.mu.-.mu.); 
.DELTA.E=-2(.alpha.n'-.alpha.m), 
(Here, it is assumed that the hand-shake correction lens unit includes 
surfaces m.fwdarw.n, the non-reversed optical system j.rarw.i). 
From expressions (11A) to (11E) above, the following conclusions are drawn. 
First, as noted previously, whereas the method described in Y. Matsui'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.sub.Y representing the off-axial image-point movement 
errors .DELTA.Y.sub.Y '. 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 .DELTA.Y.sub.Y 
'=-72 .mu.m, 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 .alpha..sub.n '-.alpha.m. 
However, since such an operation directly affects 
.DELTA.E=-2(.alpha..sub.n '-.alpha.m), an excessive increase in 
.alpha..sub.n '-.alpha.m leads to an excessively high hand-shake 
correction sensitivity (how much (.degree.) 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.