Optical system having a camera shake compensating function

BACKGROUND OF THE INVENTION

1. Field of the Invention

0001The present invention relates to an optical system having a camera shake compensating function, and more specifically, to an optical system (e.g. zoom lens system, fixed focal length lens system) having a camera shake compensating function capable of preventing image blur due to camera shake (e.g. shake of the camera caused when the user holds the camera to perform photographing).

2. Description of the Prior Art

0002Conventionally, the failure in photographing was mostly attributed to camera shake and being out of focus. In recent years, however, most cameras employ the automatic focusing mechanism and with the improvement of focusing accuracy of the automatic focusing mechanism, the problem of the failure in photographing due to being out of focus has been practically solved. On the other hand, the lens system normally incorporated in the camera has shifted from a fixed focal length lens system to a zoom lens system, and with the shift, the magnification and the longest focal length have been increased. Consequently, camera shake very frequently occurs. As a result, presently, it is no exaggeration to say that the failure in photographing is caused by camera shake. For this reason, a camera shake compensating function is indispensable for the lens system (particularly, zoom lens system).

0003As optical systems having a camera shake compensating function, optical systems have been proposed in which camera shake is compensated for by decentering a part of the lenses (hereinafter, this type of optical system will be referred to as "camera shake compensating optical system"). For example, Japanese Laid-open Patent Application No. <patcit id="pcit0001" dnum="JPH6123836B"><text>H6-123836</text></patcit> discloses a five unit zoom lens system of positive, negative, negative, positive, negative configuration in which camera shake compensation is made by decentering the negative third lens unit. This five unit zoom lens system is based on a four unit telephoto zoom lens system having positive, negative, positive, negative configuration and the negative second lens unit is divided into two negative lens units. The five unit zoom lens system is compact as a whole since it is based on the four unit system of positive, negative, positive, negative configuration.

0004In the camera shake compensating optical system, not only optical performance is necessarily excellent in the normal condition (i.e. "pre-decentering condition") but also the generation of aberrations due to the decentering of the lens is necessarily restrained to maintain desired optical performance in the compensated condition (i.e. "post-decentering condition").

0005However, the five unit zoom lens system of Japanese Laid-open Patent Application No. <patcit id="pcit0002" dnum="JPH6123836B"><text>H6-123836</text></patcit> has a problem that the aberration performance after the camera shake compensation (i.e. after decentering) is inferior. Although Japanese Laid-open Patent Application No. <patcit id="pcit0003" dnum="JPH6123836B"><text>H6-123836</text></patcit> positively appraises the performance of the system at an angle of compensation of approximately 0.15°, when the compensation angle is greater, the aberration degradation is impermissible.

0006In particular, axial coma, which is generated in proportion to the third power of the focal length, increases with an increase in focal length. That is, in a normal telephoto zoom lens system for use in a single-lens reflex camera, the focal length is 120mm or more (corresponding to a half angle of view of 10° or less) on the longer focal length side in 35mm format and the minimum F-number is approximately 6.7 or less. When a camera shake compensating system is incorporated therein, since axial coma is proportional to the third power of the focal length and inversely proportional to the second power of the F-number, if the telephoto zoom lens system is provided with a camera shake compensating function, axial coma is very great, so that it is very difficult to excellently correct aberrations (i.e. aberrations including axial coma) with good balance in the entire zoom range.

0007If a lens unit capable of decentering is added as the first lens unit, aberration performance such as off-axial image point movement error, one-side blur and axial coma after the camera shake compensation can be excellently corrected in the entire zoom range. However, since the first lens unit has a large diameter, a driving mechanism for decentering it should be large. This increases the size of the lens barrel.

SUMMARY OF THE INVENTION

0008An object of the present invention is to provide a compact optical system having a camera shake compensating function, capable of excellently correcting aberrations both in the normal condition and in the compensated condition. The object is solved by the features of claim 1 or of claim 2.

BRIEF DESCRIPTION OF THE DRAWINGS

0009This 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: <ul id="ul0001" list-style="none" compact="compact"><li><figref idref="f0001">Fig. 1</figref> shows the lens arrangement of a first embodiment of the present invention;</li><li><figref idref="f0002">Figs. 2A to 2C</figref> show longitudinal aberrations of the first embodiment before decentering at the shortest focal length condition;</li><li><figref idref="f0002">Figs. 2D to 2F</figref> show longitudinal aberrations of the first embodiment before decentering at the longest focal length condition;</li><li><figref idref="f0003">Figs. 3A and 3B</figref> show meridional lateral aberrations of the first embodiment at the shortest focal length condition before decentering;</li><li><figref idref="f0004">Figs. 4A to 4C</figref> show meridional lateral aberrations of the first embodiment at the shortest focal length condition after decentering;</li><li><figref idref="f0005">Figs. 5A and 5B</figref> show sagittal lateral aberrations of the first embodiment at the shortest focal length condition before decentering;</li><li><figref idref="f0006">Figs.6A to 6C</figref> show sagittal lateral aberrations of the first embodiment at the shortest focal length condition after decentering;</li><li><figref idref="f0007">Figs. 7A to 7B</figref> show meridional lateral aberrations of the first embodiment at the longest focal length condition before decentering;</li><li><figref idref="f0008">Figs. 8A to 8C</figref> show meridional lateral aberrations of the first embodiment at the longest focal length condition after decentering;</li><li><figref idref="f0009">Figs. 9A and 9B</figref> show sagittal lateral aberrations of the first embodiment at the longest focal length condition before decentering;</li><li><figref idref="f0010">Figs. 10A to 10C</figref> show sagittal lateral aberrations of the first embodiment at the longest focal length condition after decentering;</li><li><figref idref="f0011">Fig. 11</figref> shows the lens arrangement of a second embodiment of the present invention;</li><li><figref idref="f0012">Figs. 12A to 12C</figref> show longitudinal aberrations of the second embodiment before decentering at the shortest focal length condition;</li><li><figref idref="f0012">Figs. 12D to 12F</figref> show longitudinal aberrations of the second embodiment before decentering at the longest focal length condition;</li><li><figref idref="f0013">Figs. 13A and 13B</figref> show meridional lateral aberrations of the second embodiment at the shortest focal length condition before decentering;</li><li><figref idref="f0014">Figs. 14A to 14C</figref> show meridional lateral aberrations of the second embodiment at the shortest focal length condition after decentering;</li><li><figref idref="f0015">Figs. 15A and 15B</figref> show sagittal lateral aberrations of the second embodiment at the shortest focal length condition before decentering;</li><li><figref idref="f0016">Figs. 16A to 16C</figref> show sagittal lateral aberrations of the second embodiment at the shortest focal length condition after decentering;</li><li><figref idref="f0017">Figs. 17A and 17B</figref> show meridional lateral aberrations of the second embodiment at the longest focal length condition before decentering;</li><li><figref idref="f0018">Figs. 18A to 18C</figref> show meridional lateral aberrations of the second embodiment at the longest focal length condition after decentering;</li><li><figref idref="f0019">Figs. 19A and 19B</figref> show sagittal lateral aberrations of the second embodiment at the longest focal length condition before decentering;</li><li><figref idref="f0020">Figs. 20A to 20C</figref> show sagittal lateral aberrations of the second embodiment at the longest focal length condition after decentering;</li><li><figref idref="f0021">Fig. 21</figref> shows the lens arrangement of a third embodiment of the present invention;</li><li><figref idref="f0022">Figs. 22A to 22C</figref> show longitudinal aberrations of the third embodiment before decentering;</li><li><figref idref="f0023">Figs. 23A and 23B</figref> show meridional lateral aberrations of the third embodiment before decentering;</li><li><figref idref="f0024">Figs. 24A to 24C</figref> show meridional lateral aberrations of the third embodiment after decentering;</li><li><figref idref="f0025">Figs. 25A and 25B</figref> show sagittal lateral aberrations of the third embodiment before decentering;</li><li><figref idref="f0026">Figs. 26A to 26C</figref> show sagittal lateral aberrations of the third embodiment after decentering;</li><li><figref idref="f0027">Fig. 27</figref> shows the lens arrangement of the fourth embodiment of the present invention;</li><li><figref idref="f0028">Figs. 28A to 28C</figref> show longitudinal aberrations of the fourth embodiment before decentering at the shortest focal length condition;</li><li><figref idref="f0028">Figs. 28D to 28F</figref> show longitudinal aberrations of the fourth embodiment before decentering at the longest focal length condition;</li><li><figref idref="f0029">Figs. 29A and 29B</figref> show meridional lateral aberrations of the fourth embodiment at the shortest focal length condition before decentering;</li><li><figref idref="f0030">Figs. 30A to 30C</figref> show meridional lateral aberrations of the fourth embodiment at the shortest focal length condition after decentering;</li><li><figref idref="f0031">Figs. 31A and 31B</figref> show sagittal lateral aberrations of the fourth embodiment at the shortest focal length condition before decentering;</li><li><figref idref="f0032">Figs. 32A to 32C</figref> show sagittal lateral aberrations of the fourth embodiment at the shortest focal length condition after decentering;</li><li><figref idref="f0033">Figs. 33A and 33B</figref> show meridional lateral aberrations of the fourth embodiment at the longest focal length condition before decentering;</li><li><figref idref="f0034">Figs. 34A to 34C</figref> show meridional lateral aberrations of the fourth embodiment at the longest focal length condition after decentering;</li><li><figref idref="f0035">Figs. 35A and 35B</figref> show sagittal lateral aberrations of the fourth embodiment at the longest focal length condition before decentering;</li><li><figref idref="f0036">Figs. 36A to 36C</figref> show sagittal lateral aberrations of the fourth embodiment at the longest focal length condition after decentering;</li><li><figref idref="f0037">Figs. 37A to 37C</figref> show meridional lateral aberrations of the fourth embodiment at the longest focal length condition after decentering when no aspherical surfaces are provided in the compensating lens unit and the non-decentering lens;</li><li><figref idref="f0038">Figs. 38A to 38C</figref> show sagittal lateral aberrations of the fourth embodiment at the longest focal length condition after decentering when no aspherical surfaces are provided in the compensating lens unit and the non-decentering lens;</li><li><figref idref="f0039">Figs. 39A to 39D</figref> are views of assistance in explaining the factors of image degradation of the camera shake compensating optical system;</li><li><figref idref="f0040">Fig. 40</figref> is a view of assistance in explaining a relationship between the optical system and the coordinates;</li><li><figref idref="f0041">Figs. 41A and 41B</figref> are views of assistance in explaining the difference in light beam passing position due to decentering;</li><li><figref idref="f0041">Fig. 42</figref> is a view of assistance in explaining rotational conversion of the object surface;</li><li><figref idref="f0042">Fig. 43</figref> is a view of assistance in explaining aberration coefficients of reversal and non-reversal optical systems;</li><li><figref idref="f0043">Figs. 44A and 44B</figref> are views of assistance in explaining rotational conversion; and</li><li><figref idref="f0043">Fig. 45</figref> is a view of assistance in explaining conversion to the image surface.</li></ul>

DESCRIPTION OF THE PREFERRED EMBODIMENTS

0010An optical system having a camera shake compensating function according to a first implementation comprises a compensating lens unit capable of decentering for camera shake compensation, a lens located closer to an image than the compensating lens unit and not decentering in camera shake compensation, and an aperture. In the compensating lens unit, an aspherical surface is formed. In the lens not decentering in camera shake compensation, an aspherical surface which tends to counteract an aspherical effect of the aspherical surface of the compensating lens unit is formed. The following conditions (1) and (2) are fulfilled: <maths id="math0001" num="(1)"><math display="block"><mi mathvariant="normal">ω</mi><mo>≦</mo><mn>10</mn><mo>⁢</mo><mi>°</mi></math><img file="EP0707229A2_D0001.tif" /></maths><maths id="math0002" num="(2)"><math display="block"><mfrac><mi>D</mi><mi>f</mi></mfrac><mo>≦</mo><mn>0.3</mn></math><img file="EP0707229A2_D0002.tif" /></maths> where ω is a half angle of view, D is a distance from the aperture stop to the decentering lens unit, and f is a focal length of the entire lens system.

0011An optical system having a camera shake compensating function according to a second implementation comprises a plurality of lens units and performs zooming by varying the distances between the lens units. The following are included: a compensating lens unit located in a lens unit other than the most object side lens unit and capable of decentering for camera shake compensation; and a lens located closer to the image than the compensating lens unit. In the compensating lens unit, an aspherical surface is formed. In the lens not decentering in camera shake compensation, an aspherical surface which tends to counteract an aspherical effect of the aspherical surface of the compensating lens unit is formed.

0012As previously mentioned, in a lens system having a half angle of view of approximately 10° or less (i.e. a telephoto lens system having a focal length f of 120mm or more in 35mm format) and an F-number smaller than 6.7, axial coma is very large as is apparent from subsequently-shown expressions of aberration coefficients of camera shake compensating optical systems. For example, in a four unit zoom lens system of positive, negative, positive, negative configuration where aberrations generated in the normal condition are excellently corrected, if the second lens unit or a part of the second lens unit is used as the decentering lens unit (i.e. compensating lens unit), of the decentering aberrations, axial coma is particularly largely generated in camera shake compensation. It is apparent from the subsequently-shown expressions of aberration coefficients that the largest reason for the generation of axial coma is (the term of) spherical aberration of the decentering lens unit. Therefore, by using an aspherical surface as one surface of the decentering lens unit as described above, spherical aberration of the decentering lens unit can be excellently corrected. As a result, the decentering aberrations in camera shake compensation can be all corrected excellently. However, in the normal condition, the balance of aberrations of the entire lens system is broken by the aspherical surface of the decentering lens unit and spherical aberration is particularly large.

0013In the optical systems having a camera shake compensating function according to the first and second implementations, an aspherical surface which tends to counteract the aspherical effect of the aspherical surface of the decentering lens unit is formed in a lens located closer to the image than the decentering lens unit and not decentering in camera shake compensation so that the effects of the aspherical surfaces (hereinafter referred to as "two aspherical surfaces") counteract each other. Consequently, while the aberrations of the entire lens system is returned to excellent condition and the aberration balance is maintained excellent in the normal condition, axial coma is restrained to be small and the decentering aberrations can be excellently corrected in camera shake compensation.

0014The "aspherical surface which tends to counteract the aspherical effect of the aspherical surface of the decentering lens unit" will be described. For example, when the aspherical surface of the decentering lens unit (i.e. compensating lens unit) is a surface whose object side is glass and whose image side is air and the displacement direction of the aspherical surface relative to a reference spherical surface is a direction such that the aspherical surface displaces toward the object side from the optical axis to the edge along the height, the aspherical surface formed in the lens not decentering in camera shake compensation displaces toward the image side when its object side is glass and its image side is air and displaces toward the object side when its object side is air and its object side is glass. That is, the "aspherical surface which tends to counteract the aspherical effect of the aspherical surface of the decentering lens unit" is an aspherical surface which counteracts the aspherical effect of the other aspherical surface when one aspherical surface is a positive aspherical surface and the other aspherical surface is a negative aspherical surface and when the refractive index, the displacement direction of the aspherical surface and the type of the aspherical surface are decided as shown in Table 1.

0015Since the aspherical surface of the compensating lens unit and the aspherical surface of the lens not decentering in camera shake compensation should be as close to each other as possible in order that their effects excellently counteract each other, a distance d_ASP between the two aspherical surfaces preferably fulfills the following condition (3). In addition, the lenses including the two aspherical surfaces are preferably arranged to adjoin each other. <maths id="math0003" num="(3)"><math display="block"><msub><mi mathvariant="normal">d</mi><mi>ASP</mi></msub><mo>&lt;</mo><mn mathvariant="normal">0.1</mn><mo>×</mo><msub><mi mathvariant="normal">f</mi><mi mathvariant="normal">T</mi></msub></math><img file="EP0707229A2_D0003.tif" /></maths> where d_ASP is the distance between the two aspherical surfaces and f_T is a focal length of the entire lens system at the longest focal length condition.

0016Since it is difficult to excellently correct aberrations on both the longer and shorter focal length sides if the distance d_ASP between the two aspherical surfaces is largely varied during zooming, it is preferable that the distance d_ASP is not varied or that, if it is varied, the variation amount Δd_ASP fulfills the following condition (4): <maths id="math0004" num="(4)"><math display="block"><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msub><mi mathvariant="normal">d</mi><mi>ASP</mi></msub><mo>&lt;</mo><mn mathvariant="normal">0.08</mn><mo>×</mo><msub><mi mathvariant="normal">f</mi><mi mathvariant="normal">T</mi></msub></math><img file="EP0707229A2_D0004.tif" /></maths> where Δd_ASP is a displacement amount of the distance between the two aspherical surfaces.

0017The conditions (1) and (2) will be explained. The condition (1) means that the half angle of view ω of the camera shake compensating optical system is 10° or less. The condition (2) means that the compensating lens unit is arranged within ±0.3×f in the front or the rear of the aperture stop.

0018In a (standard to wide angle) compensating optical system exceeding the limit of the condition (1), as is apparent from the subsequently-shown expressions of aberration coefficients of camera shake compensating optical systems, axial coma which is proportional to the third power of the focal length is not conspicuous and instead, off-axial image point movement error which is inversely proportional to the focal length is large. In such a camera shake compensating optical system exceeding the limit of the condition (1), if the compensating lens unit is arranged close to the aperture stop (i.e. if the condition (2) is fulfilled), off-axial principal ray passes the compensating lens unit at a position very close to the optical axis. Therefore, in this case, even if the curvature of a surface of the compensating lens unit is changed or an aspherical surface is added, the characteristic of the off-axial ray is not changed very much, so that the off-axial image point movement error remains large and is hardly improved.

0019The normal telephoto lens system employs a telephoto type structure in which the entire lens system is roughly divided into a positive front lens unit and a negative rear lens unit. The telephoto lens system of this type is characterized in that the total length is shorter than the focal length f and that when the aperture stop is arranged in the negative rear lens unit, the diameters of the negative rear lens unit and the aperture stop are small. However, since the diameter of the positive front lens unit is very large compared to that of the negative rear lens unit, if the compensating lens unit is arranged in the positive front lens unit, a driving mechanism for decentering a large lens unit is necessary. As a result, the diameter of the lens barrel increases as a whole. Therefore, the compensating lens unit is preferably arranged close to the aperture stop having a small diameter.

0020In view of this, by arranging the compensating optical system which fulfills the condition (1) so as to further fulfill the condition (2), the weight and size of the compensating lens unit can be reduced. Then, since it is unnecessary for the driving mechanism to have a great power, the size of the entire lens barrel can be reduced. Conversely, if the limit of the condition (2) is exceeded (i.e. if the compensating lens is away from the aperture stop by 0.3xf or more) in the lens system fulfilling the condition (1), the diameter of the compensating lens unit increases, so that the size of the entire lens barrel increases.

0021Subsequently, a definition of aberration degradation in a camera shake compensating optical system like that of the present invention will be described with reference to <figref idref="f0039">Figs. 39A to 39D</figref>. The decentering aberrations (off-axial image point movement error, one-side blur, axial coma and axial lateral chromatic aberration) shown in the figures cause the image degradation of the camera shake compensating optical system.

[Off-axial image point movement error]

{Fig. 39A}

0022In a decentered optical system, a distortion due to the decentering is generated in addition to the normal distortion. For this reason, in a camera shake compensating optical system, when camera shake is compensated for so that the axial (in the center of the image plane) image point completely stops, the off-axial image point does not completely stop to cause an image blur. In <figref idref="f0039">Fig. 39A</figref>, reference numeral 1 represents a film plane, reference numeral 2 represents an image point in the compensated condition (post-decentering condition), reference numeral 3 represents an image point in the standard condition (pre-decentering condition), and reference numeral 4 represents a direction of camera shake compensation.

0023When the optical axis is along the X-axis, the camera shake direction is along the Y-axis (i.e. a camera shake compensation direction 4 is also along the Y-axis) and Y(y', z', θ') is a Y coordinate (always y(0, 0, θ)=0 since camera shake is compensated for so that the axial image point completely stops) of the actual image point of a light beam with a paraxial image point (y', z') at a compensation angle θ, the following expression (a) holds: <maths id="math0005" num="(a)"><math display="block"><mi>ΔY</mi><mfenced separators=""><mi>yʹ</mi><mo>⁢</mo><mi>zʹ</mi><mo>⁢</mo><mi mathvariant="normal">θ</mi></mfenced><mo>=</mo><mi mathvariant="normal">Y</mi><mfenced separators=""><mi>yʹ</mi><mo>⁢</mo><mi>zʹ</mi><mo>⁢</mo><mi mathvariant="normal">θ</mi></mfenced><mo>-</mo><mi mathvariant="normal">Y</mi><mfenced separators=""><mi>yʹ</mi><mo>⁢</mo><mi>zʹ</mi><mo>⁢</mo><mn>0</mn></mfenced></math><img file="EP0707229A2_D0005.tif" /></maths>

0024Unless specifically indicated, an off-axial image point movement error ΔY_Y' with respect to the image point on the Y-axis and an off-axial image point movement error ΔY_Z' with respect to the image point on the Z-axis are represented by the following expressions (b) and (c): <maths id="math0006" num="(b)"><math display="block"><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msub><mi mathvariant="normal">Y</mi><mi mathvariant="normal">Y</mi></msub><mo>⁢</mo><mi mathvariant="normal">ʹ</mi><mo>=</mo><mfenced open="{" close="}" separators=""><mi mathvariant="normal">ΔY</mi><mfenced separators=""><mn mathvariant="normal">0.7</mn><mo>⁢</mo><mi>field</mi><mo>,</mo><mn mathvariant="normal">0</mn><mo>,</mo><mn mathvariant="normal">0.7</mn><mo>⁢</mo><mi mathvariant="normal">°</mi></mfenced><mo>+</mo><mi mathvariant="normal">ΔY</mi><mo>⁢</mo><mfenced separators=""><mo>-</mo><mn mathvariant="normal">0.7</mn><mo>⁢</mo><mi>field</mi><mo>,</mo><mn mathvariant="normal">0</mn><mo>,</mo><mn mathvariant="normal">0.7</mn><mo>⁢</mo><mi mathvariant="normal">°</mi></mfenced></mfenced><mo>/</mo><mn mathvariant="normal">2</mn></math><img file="EP0707229A2_D0006.tif" /></maths><maths id="math0007" num="(c)"><math display="block"><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msub><mi mathvariant="normal">Y</mi><mi mathvariant="normal">Z</mi></msub><mo>⁢</mo><mi mathvariant="normal">ʹ</mi><mo>=</mo><mi mathvariant="normal">ΔY</mi><mfenced separators=""><mn mathvariant="normal">0</mn><mo>,</mo><mn mathvariant="normal">0.7</mn><mo>⁢</mo><mi>field</mi><mo>,</mo><mn mathvariant="normal">0.7</mn><mo>⁢</mo><mi mathvariant="normal">°</mi></mfenced></math><img file="EP0707229A2_D0007.tif" /></maths> where 0.7field is approximately 15mm in 35mm film.

[One-side blur]

{Fig. 39B}

0025Referring to <figref idref="f0039">Fig. 39B</figref>, reference numeral 5 represents an image plane which is asymmetric with respect to an optical axis AX, and reference numeral 6 represents an image plane which is symmetric with respect to the optical axis. Because of the asymmetry of the optical system, the image plane 5 is asymmetric with respect to the optical axis AX. Consequently, a meridional one-side blur ΔM' and a sagittal one-side blur ΔS' are represented by the following expressions (d) and (e), respectively: <maths id="math0008" num="(d)"><math display="block"><mi mathvariant="normal">ΔMʹ</mi><mo>=</mo><mfenced open="{" close="}" separators=""><mi>meridional value</mi><mspace width="1em" /><mfenced separators=""><mi mathvariant="normal">yʹ</mi><mo>=</mo><mn mathvariant="normal">0.7</mn><mo>⁢</mo><mi>field</mi><mo>,</mo><mi mathvariant="normal">z</mi><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi mathvariant="normal">θ</mi><mo>=</mo><mn mathvariant="normal">0.7</mn><mo>⁢</mo><mi mathvariant="normal">°</mi></mfenced><mo>-</mo><mi>meridional value</mi><mo>⁢</mo><mfenced separators=""><mi mathvariant="normal">yʹ</mi><mo>=</mo><mo>-</mo><mn mathvariant="normal">0.7</mn><mo>⁢</mo><mi>field</mi><mo>,</mo><mi mathvariant="normal">z</mi><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi mathvariant="normal">θ</mi><mo>=</mo><mn mathvariant="normal">0.7</mn><mo>⁢</mo><mi mathvariant="normal">°</mi></mfenced></mfenced><mo>/</mo><mn mathvariant="normal">2</mn></math><img file="EP0707229A2_D0008.tif" /></maths><maths id="math0009" num="(e)"><math display="block"><mi mathvariant="normal">ΔSʹ</mi><mo>=</mo><mfenced open="{" close="}" separators=""><mi>sagittal value</mi><mspace width="1em" /><mfenced separators=""><mi mathvariant="normal">yʹ</mi><mo>=</mo><mn mathvariant="normal">0.7</mn><mo>⁢</mo><mi>field</mi><mo>,</mo><mi mathvariant="normal">z</mi><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi mathvariant="normal">θ</mi><mo>=</mo><mn mathvariant="normal">0.7</mn><mo>⁢</mo><mi mathvariant="normal">°</mi></mfenced><mo>-</mo><mi>sagittal value</mi><mo>⁢</mo><mfenced separators=""><mi mathvariant="normal">yʹ</mi><mo>=</mo><mo>-</mo><mn mathvariant="normal">0.7</mn><mo>⁢</mo><mi>field</mi><mo>,</mo><mi mathvariant="normal">z</mi><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi mathvariant="normal">θ</mi><mo>=</mo><mn mathvariant="normal">0.7</mn><mo>⁢</mo><mi mathvariant="normal">°</mi></mfenced></mfenced><mo>/</mo><mn mathvariant="normal">2</mn></math><img file="EP0707229A2_D0009.tif" /></maths>

[Axial coma]

{Fig. 39C}

0026Referring to <figref idref="f0039">Fig. 39C</figref>, reference numeral 7 represents an axial luminous flux, and reference numeral 8 represents an axial principal light beam. As shown in the figure, the axial luminous flux 7 is not symmetric with respect to the axial principal light beam 8, so that coma is generated. An axial coma AXCM generated at the axial luminous flux 7 is represented by the following expression (f): <maths id="math0010" num="(f)"><math display="block"><mi>AXCM</mi><mo>=</mo><mfenced open="{" close="}" separators=""><mi mathvariant="normal">Y</mi><mfenced separators=""><mi>upper zonal</mi><mo>,</mo><mi mathvariant="normal">θ</mi><mo>=</mo><mn mathvariant="normal">0.7</mn><mo>⁢</mo><mi mathvariant="normal">°</mi></mfenced><mo>+</mo><mi mathvariant="normal">Y</mi><mfenced separators=""><mi>lower zonal</mi><mo>,</mo><mi mathvariant="normal">θ</mi><mo>=</mo><mn mathvariant="normal">0.7</mn><mo>⁢</mo><mi mathvariant="normal">°</mi></mfenced></mfenced><mo>/</mo><mn mathvariant="normal">2</mn></math><img file="EP0707229A2_D0010.tif" /></maths>

[Axial lateral chromatic aberration]

{Fig. 39D}

0027The image point, which shifts according to the difference in wavelength, shifts on the axial light beam when the optical system is asymmetric. The axial lateral chromatic aberration generated in the axial principal light beam is represented by the following expression (g): <maths id="math0011" num="(g)"><math display="block"><mfenced><mi>axial lateral chromatic aberration</mi></mfenced><mo>=</mo><mfenced open="{" close="}" separators=""><mi mathvariant="normal">Y</mi><mo>⁢</mo><mfenced separators=""><mi mathvariant="normal">g</mi><mo>-</mo><mi>line</mi><mo>,</mo><mi mathvariant="normal">θ</mi><mo>=</mo><mn mathvariant="normal">0.7</mn><mo>⁢</mo><mi mathvariant="normal">°</mi></mfenced><mo>-</mo><mi mathvariant="normal">Y</mi><mo>⁢</mo><mfenced separators=""><mi mathvariant="normal">d</mi><mo>-</mo><mi>line</mi><mo>,</mo><mi mathvariant="normal">θ</mi><mo>=</mo><mn mathvariant="normal">0.7</mn><mo>⁢</mo><mi mathvariant="normal">°</mi></mfenced></mfenced></math><img file="EP0707229A2_D0011.tif" /></maths>

0028With respect to the above-described decentering aberrations, an application method is shown in a paper "Theory of Tertiary Aberration of an Optical System Where Decentering Exits" by Mr. Yoshiya Matsui (JOEM, June, 1990). The method is suitable for a case where a normal taking lens is decentered due to an attachment error. However, it cannot be directly applied to a camera shake compensating optical system where a co-axial relationship among the object plane, the taking lens and the image plane is shifted. In order that the method of the paper can be directly applied to the camera shake compensating optical system, the actual aberrations of the camera shake compensating optical system are represented by tertiary aberration coefficients by performing the subsequently-described conversion of expressions.

[Application of decentering aberration coefficients to camera shake compensating optical system]

0029Referring to <figref idref="f0040">Fig. 40</figref> showing a relationship between the optical system and coordinates, how to obtain the decentering aberration coefficients will be described. First, the expressions are defined as follows: <maths id="math0012" num=""><math display="block"><mi mathvariant="normal">tanω</mi><mo>⋅</mo><mi mathvariant="normal">cosφω</mi><mo>=</mo><mfrac><mi>Y</mi><mrow><mi>g</mi><mo>⁢</mo><mi>$</mi></mrow></mfrac></math><img file="EP0707229A2_D0012.tif" /></maths><maths id="math0013" num=""><math display="block"><mi mathvariant="normal">tanω</mi><mo>⋅</mo><mi mathvariant="normal">sinφω</mi><mo>=</mo><mfrac><mi>Z</mi><mrow><mi>g</mi><mo>⁢</mo><mi>$</mi></mrow></mfrac></math><img file="EP0707229A2_D0013.tif" /></maths><maths id="math0014" num=""><math display="block"><mi>R</mi><mo>⋅</mo><mi mathvariant="normal">cosφ</mi><mo>⁢</mo><mi>R</mi><mo>=</mo><mfrac><mrow><mi>g</mi><mo>⁢</mo><mi>$</mi></mrow><mi>g</mi></mfrac><mo>⋅</mo><mi>Y</mi><mo>*</mo></math><img file="EP0707229A2_D0014.tif" /></maths><maths id="math0015" num=""><math display="block"><mi>R</mi><mo>⋅</mo><mi mathvariant="normal">sinφ</mi><mo>⁢</mo><mi>R</mi><mo>=</mo><mfrac><mrow><mi>g</mi><mo>⁢</mo><mi>$</mi></mrow><mi>g</mi></mfrac><mo>⋅</mo><mi>Z</mi><mo>*</mo></math><img file="EP0707229A2_D0015.tif" /></maths> where g and g$ are an entrance pupil surface and the distance from the object side principal plane to an object plane (object surface) OS, respectively, ω is an angle of a straight line between the object point and the object side principal point H, to a reference axis and φω is its azimuth, and R is a radius of entrance pupil converted on the object side principal plane and φR is its azimuth.

0030Image point movement amounts ΔY and ΔZ on an image plane (image surface) IS when a vth surface from the object side is parallely decentered by a slight amount Ev in the Y direction relative to the reference axis are represented by the following expressions (1A) and (1 B): <maths id="math0016" num="(1A)"><math display="block"><mtable><mtr><mtd><mi mathvariant="normal">ΔY</mi><mo>=</mo><mo>-</mo><mfenced separators=""><mi mathvariant="normal">Eν</mi><mo>/</mo><mn mathvariant="normal">2</mn><mo>⁢</mo><msub><mi mathvariant="normal">α</mi><mi mathvariant="normal">k</mi></msub><mo>⁢</mo><mi mathvariant="normal">ʹ</mi></mfenced><mo>⋅</mo><mfenced open="[" close="]" separators=""><mfenced><mi mathvariant="normal">ΔE</mi></mfenced><mo>⁢</mo><mi mathvariant="normal">ν</mi><mo>+</mo><msup><mfenced separators=""><mi mathvariant="normal">N</mi><mo>⋅</mo><mi mathvariant="normal">tanω</mi></mfenced><mn mathvariant="normal">2</mn></msup><mo>⋅</mo><mfenced open="{" close="}" separators=""><mfenced separators=""><mn mathvariant="normal">2</mn><mo>+</mo><mi>cos</mi><mn mathvariant="normal">2</mn><mo>⁢</mo><mi mathvariant="normal">φω</mi></mfenced><mo>⋅</mo><mfenced separators=""><mi>VE</mi><mo>⁢</mo><mn mathvariant="normal">1</mn></mfenced><mo>⁢</mo><mi mathvariant="normal">ν</mi><mo>-</mo><mfenced separators=""><mi>VE</mi><mo>⁢</mo><mn mathvariant="normal">2</mn></mfenced><mo>⁢</mo><mi mathvariant="normal">ν</mi></mfenced><mo>+</mo><mn mathvariant="normal">2</mn><mo>⁢</mo><mi mathvariant="normal">R</mi><mo>⋅</mo><mfenced separators=""><mi mathvariant="normal">N</mi><mo>⋅</mo><mi mathvariant="normal">tanω</mi></mfenced><mo>⋅</mo><mfenced open="{" close="}" separators=""><mfenced separators=""><mn mathvariant="normal">2</mn><mo>⁢</mo><mi>cos</mi><mfenced separators=""><mi mathvariant="normal">φR</mi><mo>-</mo><mi mathvariant="normal">φω</mi></mfenced><mo>+</mo><mi>cos</mi><mfenced separators=""><mi mathvariant="normal">φR</mi><mo>+</mo><mi mathvariant="normal">φω</mi></mfenced></mfenced><mo>⋅</mo><mfenced><mi>IIIE</mi></mfenced><mo>⁢</mo><mi mathvariant="normal">ν</mi><mo>+</mo><mi mathvariant="normal">cosφR</mi><mo>⋅</mo><mi mathvariant="normal">cosφω</mi><mo>⋅</mo><mfenced><mi>PE</mi></mfenced><mo>⁢</mo><mi mathvariant="normal">ν</mi></mfenced><mo>+</mo><msup><mi mathvariant="normal">R</mi><mn mathvariant="normal">2</mn></msup><mo>⋅</mo><mfenced separators=""><mn mathvariant="normal">2</mn><mo>+</mo><mi>cos</mi><mn mathvariant="normal">2</mn><mo>⁢</mo><mi mathvariant="normal">φR</mi></mfenced><mo>⋅</mo><mfenced><mi>IIE</mi></mfenced><mo>⁢</mo><mi mathvariant="normal">ν</mi></mfenced></mtd><mtd><mspace width="1em" /></mtd><mtd><mspace width="1em" /></mtd></mtr></mtable></math><img file="EP0707229A2_D0016.tif" /></maths><maths id="math0017" num="(1B)"><math display="block"><mtable><mtr><mtd><mi mathvariant="normal">ΔZ</mi><mo>=</mo><mo>-</mo><mfenced separators=""><mi mathvariant="normal">Eν</mi><mo>/</mo><mn mathvariant="normal">2</mn><mo>⁢</mo><msub><mi mathvariant="normal">α</mi><mi mathvariant="normal">k</mi></msub><mo>⁢</mo><mi mathvariant="normal">ʹ</mi></mfenced><mo>⋅</mo><mfenced open="[" close="]" separators=""><msup><mfenced separators=""><mi mathvariant="normal">N</mi><mo>⋅</mo><mi mathvariant="normal">tanω</mi></mfenced><mn mathvariant="normal">2</mn></msup><mo>⋅</mo><mi>sin</mi><mn mathvariant="normal">2</mn><mo>⁢</mo><mi mathvariant="normal">φω</mi><mo>⋅</mo><mfenced separators=""><mi>VE</mi><mo>⁢</mo><mn mathvariant="normal">1</mn></mfenced><mo>⁢</mo><mi mathvariant="normal">ν</mi><mo>+</mo><mn mathvariant="normal">2</mn><mo>⁢</mo><mi mathvariant="normal">R</mi><mo>⋅</mo><mfenced separators=""><mi mathvariant="normal">N</mi><mo>⋅</mo><mi mathvariant="normal">tanω</mi></mfenced><mo>⋅</mo><mfenced open="{" close="}" separators=""><mi>sin</mi><mfenced separators=""><mi mathvariant="normal">φR</mi><mo>+</mo><mi mathvariant="normal">φω</mi></mfenced><mo>⋅</mo><mfenced><mi>IIIE</mi></mfenced><mo>⁢</mo><mi mathvariant="normal">ν</mi><mo>+</mo><mi mathvariant="normal">sinφR</mi><mo>⋅</mo><mi mathvariant="normal">sinφω</mi><mo>⋅</mo><mfenced><mi>PE</mi></mfenced><mo>⁢</mo><mi mathvariant="normal">ν</mi></mfenced><mo>+</mo><msup><mi mathvariant="normal">R</mi><mn mathvariant="normal">2</mn></msup><mo>⋅</mo><mi>sin</mi><mn mathvariant="normal">2</mn><mo>⁢</mo><mi mathvariant="normal">φR</mi><mo>⋅</mo><mfenced><mi>IIE</mi></mfenced><mo>⁢</mo><mi mathvariant="normal">ν</mi></mfenced></mtd><mtd><mspace width="1em" /></mtd><mtd><mspace width="1em" /></mtd></mtr></mtable></math><img file="EP0707229A2_D0017.tif" /></maths>

0031Here, when (ΔE)ν is a prismatic effect (lateral shift of the image), (VE1)ν and (VE2)v are rotationally asymmetric distortions, (IIIE)ν and (PE)ν are a rotationally asymmetric astigmatism and an inclination of the image surface, respectively, and (IIE)ν is a rotationally asymmetric coma which is present also on the axis, the decentering aberration coefficients representing the effects of the decentering are represented by the following expressions (1 C) to (1 H) based on the aberration coefficients of from the vth surface to the image surface (# is a suffix representative of "on the object surface"). In the case of rotational decentering, the decentering aberration coefficients are represented by expressions similar to the expressions (1 A) to (1 H): <maths id="math0018" num="(1C)"><math display="block"><mfenced><mi mathvariant="normal">ΔE</mi></mfenced><mo>⁢</mo><mi mathvariant="normal">ν</mi><mo>=</mo><mo>-</mo><mn mathvariant="normal">2</mn><mo>⁢</mo><mfenced separators=""><mi mathvariant="normal">ανʹ</mi><mo>-</mo><mi mathvariant="normal">αν</mi></mfenced></math><img file="EP0707229A2_D0018.tif" /></maths><maths id="math0019" num="(1D)"><math display="block"><mfenced separators=""><mi>VE</mi><mo>⁢</mo><mn mathvariant="normal">1</mn></mfenced><mo>⁢</mo><mi mathvariant="normal">ν</mi><mo>=</mo><mfenced open="[" close="]" separators=""><mfenced open="{" close="}" separators=""><mi mathvariant="normal">ανʹ</mi><mo>⋅</mo><mfenced separators=""><mi mathvariant="normal">μ</mi><mo>=</mo><mi mathvariant="normal">ν</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo>→</mo><mi mathvariant="normal">k</mi></mfenced><mo>⁢</mo><mi mathvariant="normal">ΣVμ</mi></mfenced><mo>-</mo><mfenced open="{" close="}" separators=""><mi mathvariant="normal">αν</mi><mo>⋅</mo><mfenced separators=""><mi mathvariant="normal">μ</mi><mo>=</mo><mi mathvariant="normal">ν</mi><mo>→</mo><mi mathvariant="normal">k</mi></mfenced><mo>⁢</mo><mi mathvariant="normal">ΣVμ</mi></mfenced></mfenced><mo>-</mo><mfenced open="[" close="]" separators=""><mfenced open="{" close="}" separators=""><mi mathvariant="normal">ανʹ#</mi><mo>⋅</mo><mfenced separators=""><mi mathvariant="normal">μ</mi><mo>=</mo><mi mathvariant="normal">ν</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo>→</mo><mi mathvariant="normal">k</mi></mfenced><mo>⁢</mo><mi mathvariant="normal">ΣIIIμ</mi></mfenced><mo>-</mo><mfenced open="{" close="}" separators=""><mi mathvariant="normal">αν#</mi><mo>⋅</mo><mfenced separators=""><mi mathvariant="normal">μ</mi><mo>=</mo><mi mathvariant="normal">ν</mi><mo>→</mo><mi mathvariant="normal">k</mi></mfenced><mo>⁢</mo><mi mathvariant="normal">ΣIIIμ</mi></mfenced></mfenced></math><img file="EP0707229A2_D0019.tif" /></maths><maths id="math0020" num="(1E)"><math display="block"><mfenced separators=""><mi>VE</mi><mo>⁢</mo><mn mathvariant="normal">2</mn></mfenced><mo>⁢</mo><mi mathvariant="normal">ν</mi><mo>=</mo><mfenced open="{" close="}" separators=""><mi mathvariant="normal">ανʹ#</mi><mo>⋅</mo><mfenced separators=""><mi mathvariant="normal">μ</mi><mo>=</mo><mi mathvariant="normal">ν</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo>→</mo><mi mathvariant="normal">k</mi></mfenced><mo>⁢</mo><mi mathvariant="normal">ΣPμ</mi></mfenced><mo>-</mo><mfenced open="{" close="}" separators=""><mi mathvariant="normal">αν#</mi><mo>⋅</mo><mfenced separators=""><mi mathvariant="normal">μ</mi><mo>=</mo><mi mathvariant="normal">ν</mi><mo>→</mo><mi mathvariant="normal">k</mi></mfenced><mo>⁢</mo><mi mathvariant="normal">ΣPμ</mi></mfenced></math><img file="EP0707229A2_D0020.tif" /></maths><maths id="math0021" num="(1F)"><math display="block"><mfenced><mi>IIIE</mi></mfenced><mo>⁢</mo><mi mathvariant="normal">ν</mi><mo>=</mo><mfenced open="[" close="]" separators=""><mfenced open="{" close="}" separators=""><mi mathvariant="normal">ανʹ</mi><mo>⋅</mo><mfenced separators=""><mi mathvariant="normal">μ</mi><mo>=</mo><mi mathvariant="normal">ν</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo>→</mo><mi mathvariant="normal">k</mi></mfenced><mo>⁢</mo><mi mathvariant="normal">ΣIIIμ</mi></mfenced><mo>-</mo><mfenced open="{" close="}" separators=""><mi mathvariant="normal">αν</mi><mo>⋅</mo><mfenced separators=""><mi mathvariant="normal">μ</mi><mo>=</mo><mi mathvariant="normal">ν</mi><mo>→</mo><mi mathvariant="normal">k</mi></mfenced><mo>⁢</mo><mi mathvariant="normal">ΣIIIμ</mi></mfenced></mfenced><mo>-</mo><mfenced open="[" close="]" separators=""><mfenced open="{" close="}" separators=""><mi mathvariant="normal">ανʹ#</mi><mo>⋅</mo><mfenced separators=""><mi mathvariant="normal">μ</mi><mo>=</mo><mi mathvariant="normal">ν</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo>→</mo><mi mathvariant="normal">k</mi></mfenced><mo>⁢</mo><mi mathvariant="normal">ΣIIμ</mi></mfenced><mo>-</mo><mfenced open="{" close="}" separators=""><mi mathvariant="normal">αν#</mi><mo>⋅</mo><mfenced separators=""><mi mathvariant="normal">μ</mi><mo>=</mo><mi mathvariant="normal">ν</mi><mo>→</mo><mi mathvariant="normal">k</mi></mfenced><mo>⁢</mo><mi mathvariant="normal">ΣIIμ</mi></mfenced></mfenced></math><img file="EP0707229A2_D0021.tif" /></maths><maths id="math0022" num="(1G)"><math display="block"><mfenced><mi>PE</mi></mfenced><mo>⁢</mo><mi mathvariant="normal">ν</mi><mo>=</mo><mfenced open="{" close="}" separators=""><mi mathvariant="normal">ανʹ</mi><mo>⋅</mo><mfenced separators=""><mi mathvariant="normal">μ</mi><mo>=</mo><mi mathvariant="normal">ν</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo>→</mo><mi mathvariant="normal">k</mi></mfenced><mo>⁢</mo><mi mathvariant="normal">ΣPμ</mi></mfenced><mo>-</mo><mfenced open="{" close="}" separators=""><mi mathvariant="normal">αν</mi><mo>⋅</mo><mfenced separators=""><mi mathvariant="normal">μ</mi><mo>=</mo><mi mathvariant="normal">ν</mi><mo>→</mo><mi mathvariant="normal">k</mi></mfenced><mo>⁢</mo><mi mathvariant="normal">ΣPμ</mi></mfenced></math><img file="EP0707229A2_D0022.tif" /></maths><maths id="math0023" num="(1H)"><math display="block"><mfenced><mi>IIE</mi></mfenced><mo>⁢</mo><mi mathvariant="normal">ν</mi><mo>=</mo><mfenced open="[" close="]" separators=""><mfenced open="{" close="}" separators=""><mi mathvariant="normal">ανʹ</mi><mo>⋅</mo><mfenced separators=""><mi mathvariant="normal">μ</mi><mo>=</mo><mi mathvariant="normal">ν</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo>→</mo><mi mathvariant="normal">k</mi></mfenced><mo>⁢</mo><mi mathvariant="normal">ΣIIμ</mi></mfenced><mo>-</mo><mfenced open="{" close="}" separators=""><mi mathvariant="normal">αν</mi><mo>⋅</mo><mfenced separators=""><mi mathvariant="normal">μ</mi><mo>=</mo><mi mathvariant="normal">ν</mi><mo>→</mo><mi mathvariant="normal">k</mi></mfenced><mo>⁢</mo><mi mathvariant="normal">ΣIIμ</mi></mfenced></mfenced><mo>-</mo><mfenced open="[" close="]" separators=""><mfenced open="{" close="}" separators=""><mi mathvariant="normal">ανʹ#</mi><mo>⋅</mo><mfenced separators=""><mi mathvariant="normal">μ</mi><mo>=</mo><mi mathvariant="normal">ν</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo>→</mo><mi mathvariant="normal">k</mi></mfenced><mo>⁢</mo><mi mathvariant="normal">ΣIμ</mi></mfenced><mo>-</mo><mfenced open="{" close="}" separators=""><mi mathvariant="normal">αν#</mi><mo>⋅</mo><mfenced separators=""><mi mathvariant="normal">μ</mi><mo>=</mo><mi mathvariant="normal">ν</mi><mo>→</mo><mi mathvariant="normal">k</mi></mfenced><mo>⁢</mo><mi mathvariant="normal">ΣIμ</mi></mfenced></mfenced></math><img file="EP0707229A2_D0023.tif" /></maths>

0032However, in order to apply the decentering aberration coefficients to the camera shake compensating optical system, it is necessary to replace the image surface IS with the object surface OS by a reversal of the optical system to use aberration coefficients from the image surface IS. That is, the image point movement amounts must be converted to those of the object surface OS. The reasons therefor will be described.

0033The first reason is that the light beam passing position shifts due to decentering. Referring to <figref idref="f0041">Fig. 41A</figref> (L₁ represents a light beam before decentering and L₂ represents a light beam after decentering), according to the above-described method of Mr. Matsui's paper, the light beam passing position on the image surface IS side of a decentering lens LS is shifted by the decentering lens LS. Consequently, the aberration coefficients of the decentering lens LS and of the decentering lens LS to the image surface IS relate to the decentering aberration coefficients. On the contrary, referring to <figref idref="f0041">Fig. 41B</figref> (M₁ represents a light beam before camera shake compensation and M₂ represents a light beam after camera shake compensation), in the camera shake compensating optical system (ideally), the light beam passing position on the object side of the decentering lens LS differs between before camera shake compensation and after camera shake compensation. Consequently, the aberration coefficients of the decentering lens LS and the lenses located on the object side of the decentering lens LS relate to the decentering aberration coefficients.

0034The second reason is that the aberrations sometimes degrade due to a rotational conversion of the object surface. According to the above-described method of Mr. Matsui's paper, neither the object surface OS₁ nor the image surface IS moves, whereas in the camera shake compensating optical system, the object surface OS₁ rotates as shown in <figref idref="f0041">Fig. 42</figref>. For this reason, the off-axial image point movement error and the one-side blur are great compared to the case where the object surface does not rotate. In <figref idref="f0041">Fig. 42</figref>, OS₁ represents an object surface before camera shake compensation and OS₂ represents an object surface after camera shake compensation.

[Aberration coefficients of reversal optical system and aberration coefficients of non-reversal optical system]

0035Since the image point movement amounts must be converted to those of the object surface for the above-described reasons, the coefficients of the expressions (1A) to (1H) are converted according to the following expressions (2A) to (2J) represented based on <figref idref="f0042">Fig. 43</figref> (non-reversal optical system): <maths id="math0024" num="(2A)"><math display="block"><mmultiscripts><mi mathvariant="normal">α</mi><mprescripts /><none /><mi mathvariant="normal">R</mi></mmultiscripts><mo>=</mo><mmultiscripts><mi mathvariant="normal">N</mi><mprescripts /><none /><mi mathvariant="normal">R</mi></mmultiscripts><mo>/</mo><mmultiscripts><mi>g$</mi><mprescripts /><none /><mi mathvariant="normal">R</mi></mmultiscripts><mo>=</mo><mo>-</mo><mi mathvariant="normal">αʹ</mi></math><img file="EP0707229A2_D0024.tif" /></maths><maths id="math0025" num="(2B)"><math display="block"><mmultiscripts><mi mathvariant="normal">α</mi><mprescripts /><none /><mi mathvariant="normal">R</mi></mmultiscripts><mi>#</mi><mo>=</mo><mi mathvariant="normal">αʹ#</mi></math><img file="EP0707229A2_D0025.tif" /></maths><maths id="math0026" num="(2C)"><math display="block"><mmultiscripts><mi mathvariant="normal">α</mi><mprescripts /><none /><mi mathvariant="normal">R</mi></mmultiscripts><mi mathvariant="normal">μʹ</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">αν</mi></math><img file="EP0707229A2_D0026.tif" /></maths><maths id="math0027" num="(2D)"><math display="block"><mmultiscripts><mi mathvariant="normal">α</mi><mprescripts /><none /><mi mathvariant="normal">R</mi></mmultiscripts><mi mathvariant="normal">μʹ#</mi><mo>=</mo><mi mathvariant="normal">αν#</mi></math><img file="EP0707229A2_D0027.tif" /></maths><maths id="math0028" num="(2E) same"><math display="block"><mmultiscripts><mi mathvariant="normal">P</mi><mprescripts /><none /><mi mathvariant="normal">R</mi></mmultiscripts><mi mathvariant="normal">μ</mi><mo>=</mo><mi mathvariant="normal">Pν</mi></math><img file="EP0707229A2_D0028.tif" /></maths><maths id="math0029" num="(2F) same"><math display="block"><mmultiscripts><mi mathvariant="normal">φ</mi><mprescripts /><none /><mi mathvariant="normal">R</mi></mmultiscripts><mi mathvariant="normal">μ</mi><mo>=</mo><mi mathvariant="normal">φν</mi></math><img file="EP0707229A2_D0029.tif" /></maths><maths id="math0030" num="(2G) same"><math display="block"><mmultiscripts><mi mathvariant="normal">I</mi><mprescripts /><none /><mi mathvariant="normal">R</mi></mmultiscripts><mi mathvariant="normal">μ</mi><mo>=</mo><mi mathvariant="normal">Iν</mi></math><img file="EP0707229A2_D0030.tif" /></maths><maths id="math0031" num="(2H) reverse"><math display="block"><mmultiscripts><mi>II</mi><mprescripts /><none /><mi mathvariant="normal">R</mi></mmultiscripts><mi mathvariant="normal">μ</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">IIν</mi></math><img file="EP0707229A2_D0031.tif" /></maths><maths id="math0032" num="(2I) same"><math display="block"><mmultiscripts><mi>III</mi><mprescripts /><none /><mi mathvariant="normal">R</mi></mmultiscripts><mi mathvariant="normal">μ</mi><mo>=</mo><mi mathvariant="normal">IIIν</mi></math><img file="EP0707229A2_D0032.tif" /></maths><maths id="math0033" num="(2J) reverse"><math display="block"><mmultiscripts><mi mathvariant="normal">V</mi><mprescripts /><none /><mi mathvariant="normal">R</mi></mmultiscripts><mi mathvariant="normal">μ</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">Vν</mi></math><img file="EP0707229A2_D0033.tif" /></maths> where ^R( ) represents a reversal optical system and N is a refractive index.

[Decentering aberration coefficients and camera shake aberration coefficients when compensating lens unit parallely decenters]

0036Since the previously-mentioned expressions (1 A) to (1 H) show a case where only one surface ν is decentered, they are converted to expressions showing a case where a plurality of surfaces i to j are decentered. When the compensating lens unit is parallely decentered, since the decentering amounts Ei to Ej of the decentered surfaces i to j are the same, the aberration coefficients can be treated as a sum as shown in the following expression: <maths id="math0034" num=""><math display="block"><mfenced><mi mathvariant="normal">ΔE</mi></mfenced><mo>⁢</mo><mi>i to j</mi><mo>=</mo><mfenced separators=""><mi mathvariant="normal">ν</mi><mo>=</mo><mi mathvariant="normal">i</mi><mo>→</mo><mi mathvariant="normal">j</mi></mfenced><mo>⁢</mo><mi mathvariant="normal">Σ</mi><mfenced open="{" close="}" separators=""><mo>-</mo><mn mathvariant="normal">2</mn><mo>⋅</mo><mfenced separators=""><mi mathvariant="normal">ανʹ</mi><mo>-</mo><mi mathvariant="normal">αν</mi></mfenced></mfenced></math><img file="EP0707229A2_D0034.tif" /></maths> From αν' = αν+1 , the following expression is obtained: <maths id="math0035" num=""><math display="block"><mfenced><mi mathvariant="normal">ΔE</mi></mfenced><mo>⁢</mo><mi>i to j</mi><mo>=</mo><mo>-</mo><mn>2</mn><mo>⋅</mo><mfenced separators=""><mi mathvariant="normal">αjʹ</mi><mo>-</mo><mi mathvariant="normal">αi</mi></mfenced></math><img file="EP0707229A2_D0035.tif" /></maths>

0037Likewise, with respect to other aberration coefficients, the terms between Σs are deleted. For example, <maths id="math0036" num=""><math display="block"><mtable><mtr><mtd><mfenced><mi>PE</mi></mfenced><mo>⁢</mo><mi>i to j</mi><mo>=</mo><mfenced separators=""><mi mathvariant="normal">μ</mi><mo>=</mo><mi mathvariant="normal">i</mi><mo>→</mo><mi mathvariant="normal">j</mi></mfenced><mo>⁢</mo><mi mathvariant="normal">Σ</mi><mfenced open="{" close="}" separators=""><mi mathvariant="normal">ανʹ</mi><mo>⋅</mo><mfenced separators=""><mi mathvariant="normal">μ</mi><mo>=</mo><mi mathvariant="normal">ν</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo>→</mo><mi mathvariant="normal">k</mi></mfenced><mo>⁢</mo><mi mathvariant="normal">ΣPμ</mi><mo>-</mo><mi mathvariant="normal">αν</mi><mo>⋅</mo><mfenced separators=""><mi mathvariant="normal">μ</mi><mo>=</mo><mi mathvariant="normal">ν</mi><mo>→</mo><mi mathvariant="normal">k</mi></mfenced><mo>⁢</mo><mi mathvariant="normal">ΣPμ</mi></mfenced></mtd></mtr><mtr><mtd><mo>=</mo><mi mathvariant="normal">αjʹ</mi><mo>⋅</mo><mfenced separators=""><mi mathvariant="normal">μ</mi><mo>=</mo><mi mathvariant="normal">j</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo>→</mo><mi mathvariant="normal">k</mi></mfenced><mo>⁢</mo><mi mathvariant="normal">ΣPμ</mi><mo>⋅</mo><mi mathvariant="normal">αi</mi><mo>⋅</mo><mfenced separators=""><mi mathvariant="normal">μ</mi><mo>=</mo><mi mathvariant="normal">i</mi><mo>→</mo><mi mathvariant="normal">k</mi></mfenced><mo>⁢</mo><mi mathvariant="normal">ΣPμ</mi></mtd></mtr></mtable></math><img file="EP0707229A2_D0036.tif" /></maths>

0038This is further converted to <maths id="math0037" num=""><math display="block"><mfenced><mi>PE</mi></mfenced><mo>⁢</mo><mi>i to j</mi><mo>=</mo><mfenced separators=""><mi mathvariant="normal">αjʹ</mi><mo>-</mo><mi mathvariant="normal">αi</mi></mfenced><mo>⋅</mo><mfenced separators=""><mi mathvariant="normal">μ</mi><mo>=</mo><mi mathvariant="normal">j</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo>→</mo><mi mathvariant="normal">k</mi></mfenced><mo>⁢</mo><mi mathvariant="normal">ΣPμ</mi><mo>-</mo><mi mathvariant="normal">αi</mi><mo>⋅</mo><mfenced separators=""><mi mathvariant="normal">μ</mi><mo>=</mo><mi>i to j</mi></mfenced><mo>⁢</mo><mi mathvariant="normal">ΣPμ</mi></math><img file="EP0707229A2_D0037.tif" /></maths> where (µ=j+1 →k)ΣPµ is the sum of P (Petzval) of the lenses arranged behind the compensating lens unit, and (µ=i to j)ΣPµ is the sum of P of the compensating lens unit. <maths id="math0038" num=""><math display="block"><mfenced><mi>PE</mi></mfenced><mo>⁢</mo><mi>i to j</mi><mo>=</mo><mfenced separators=""><mi mathvariant="normal">αjʹ</mi><mo>-</mo><mi mathvariant="normal">αi</mi></mfenced><mo>⁢</mo><msub><mi mathvariant="normal">P</mi><mi mathvariant="normal">R</mi></msub><mo>-</mo><mi mathvariant="normal">αi</mi><mo>⋅</mo><msub><mi mathvariant="normal">P</mi><mi mathvariant="normal">D</mi></msub></math><img file="EP0707229A2_D0038.tif" /></maths> where ( )_R is the sum of the aberration coefficients of the lenses arranged behind the compensating lens and ( )_D is the sum of the aberration coefficients of the compensating lens unit.

0039As described above, by the conversion of the image point movement amounts to those of the object surface and the conversion of the expressions to the expressions showing the case where a plurality of surfaces i to j are decentered, the decentering aberration coefficients represented by the following expressions (3A) to (3F) are obtained. By redefining the decentering aberration coefficients according to the expressions (3A) to (3F), the expressions (1 A) to (1 H) can be used as they are as expressions representing the image point movement amounts on the object surface. <maths id="math0039" num="(3A)"><math display="block"><mfenced><mi mathvariant="normal">ΔE</mi></mfenced><mo>⁢</mo><mi>i to j</mi><mo>=</mo><mo>-</mo><mn>2</mn><mo>⋅</mo><mfenced separators=""><mi mathvariant="normal">αjʹ</mi><mo>-</mo><mi mathvariant="normal">αi</mi></mfenced></math><img file="EP0707229A2_D0039.tif" /></maths><maths id="math0040" num="(3B)"><math display="block"><mfenced separators=""><mi>VE</mi><mo>⁢</mo><mn mathvariant="normal">1</mn></mfenced><mo>⁢</mo><mi>i to j</mi><mo>=</mo><mfenced separators=""><mi mathvariant="normal">αjʹ</mi><mo>-</mo><mi mathvariant="normal">αi</mi></mfenced><mo>⋅</mo><msub><mi mathvariant="normal">V</mi><mi mathvariant="normal">R</mi></msub><mo>-</mo><mfenced separators=""><mi mathvariant="normal">αjʹ#</mi><mo>-</mo><mi mathvariant="normal">αi#</mi></mfenced><mo>⋅</mo><msub><mi>III</mi><mi mathvariant="normal">R</mi></msub><mo>-</mo><mfenced separators=""><mi mathvariant="normal">αi</mi><mo>⋅</mo><msub><mi mathvariant="normal">V</mi><mi mathvariant="normal">D</mi></msub><mo>-</mo><mi mathvariant="normal">αi#</mi><mo>⋅</mo><msub><mi>III</mi><mi mathvariant="normal">D</mi></msub></mfenced></math><img file="EP0707229A2_D0040.tif" /></maths><maths id="math0041" num="(3C)"><math display="block"><mfenced separators=""><mi>VE</mi><mo>⁢</mo><mn>2</mn></mfenced><mo>⁢</mo><mi>i to j</mi><mo>=</mo><mfenced separators=""><mi mathvariant="normal">αj#</mi><mo>-</mo><mi mathvariant="normal">αi#</mi></mfenced><mo>⋅</mo><msub><mi mathvariant="normal">P</mi><mi mathvariant="normal">R</mi></msub><mo>-</mo><mi mathvariant="normal">αi#</mi><mo>⋅</mo><msub><mi mathvariant="normal">P</mi><mi mathvariant="normal">D</mi></msub></math><img file="EP0707229A2_D0041.tif" /></maths><maths id="math0042" num="(3D)"><math display="block"><mfenced><mi>IIIE</mi></mfenced><mo>⁢</mo><mi>i to j</mi><mo>=</mo><mfenced separators=""><mi mathvariant="normal">αjʹ</mi><mo>-</mo><mi mathvariant="normal">αi</mi></mfenced><mo>⋅</mo><msub><mi>III</mi><mi mathvariant="normal">R</mi></msub><mo>-</mo><mfenced separators=""><mi mathvariant="normal">αj#</mi><mo>-</mo><mi mathvariant="normal">αi#</mi></mfenced><mo>⋅</mo><msub><mi>II</mi><mi mathvariant="normal">R</mi></msub><mo>-</mo><mfenced separators=""><mi mathvariant="normal">αi</mi><mo>⋅</mo><msub><mi>III</mi><mi mathvariant="normal">D</mi></msub><mo>-</mo><mi mathvariant="normal">αi#</mi><mo>⋅</mo><msub><mi>II</mi><mi mathvariant="normal">D</mi></msub></mfenced></math><img file="EP0707229A2_D0042.tif" /></maths><maths id="math0043" num="(3E)"><math display="block"><mfenced><mi>PE</mi></mfenced><mo>⁢</mo><mi>i to j</mi><mo>=</mo><mfenced separators=""><mi mathvariant="normal">αjʹ</mi><mo>-</mo><mi mathvariant="normal">αi</mi></mfenced><mo>⋅</mo><msub><mi mathvariant="normal">P</mi><mi mathvariant="normal">R</mi></msub><mo>-</mo><mi mathvariant="normal">αi</mi><mo>⋅</mo><msub><mi mathvariant="normal">P</mi><mi mathvariant="normal">D</mi></msub></math><img file="EP0707229A2_D0043.tif" /></maths><maths id="math0044" num="(3F)"><math display="block"><mfenced><mi>IIE</mi></mfenced><mo>⁢</mo><mi>i to j</mi><mo>=</mo><mfenced separators=""><mi mathvariant="normal">αjʹ</mi><mo>-</mo><mi mathvariant="normal">αi</mi></mfenced><mo>⋅</mo><msub><mi>II</mi><mi mathvariant="normal">R</mi></msub><mo>-</mo><mfenced separators=""><mi mathvariant="normal">αjʹ#</mi><mo>-</mo><mi mathvariant="normal">αi#</mi></mfenced><mo>⋅</mo><msub><mi mathvariant="normal">I</mi><mi mathvariant="normal">R</mi></msub><mo>-</mo><mfenced separators=""><mi mathvariant="normal">αi</mi><mo>⋅</mo><msub><mi>II</mi><mi mathvariant="normal">D</mi></msub><mo>-</mo><mi mathvariant="normal">αi#</mi><mo>⋅</mo><msub><mi mathvariant="normal">I</mi><mi mathvariant="normal">D</mi></msub></mfenced></math><img file="EP0707229A2_D0044.tif" /></maths>

[Off-axial image point movement error]

0040Subsequently, the off-axial image point movement error will be described. The decentering aberration coefficients (of the reversal optical systems) are represented by ΔE, VE1, VE2, IIIE, PE and IIE. The movements of the image point (before rotational conversion on the object surface) due to decentering on the object surface are represented by the following expressions (4A) and (4B) (in the principal ray (R=0)). The expressions (4A) and (4B) are the expressions (1 A) and (1B) where R=0. <maths id="math0045" num="(4A)"><math display="block"><mi mathvariant="normal">ΔY#</mi><mo>=</mo><mo>-</mo><mfenced separators=""><mi mathvariant="normal">E</mi><mo>/</mo><mn mathvariant="normal">2</mn><mo>⁢</mo><msub><mi mathvariant="normal">αʹ</mi><mi mathvariant="normal">k</mi></msub></mfenced><mo>⋅</mo><mfenced open="[" close="]" separators=""><mi mathvariant="normal">ΔE</mi><mo>+</mo><msup><mfenced separators=""><mi mathvariant="normal">N</mi><mo>⋅</mo><mi mathvariant="normal">tanω</mi></mfenced><mn mathvariant="normal">2</mn></msup><mo>⋅</mo><mfenced open="{" close="}" separators=""><mfenced separators=""><mn mathvariant="normal">2</mn><mo>+</mo><msup><mi>cos</mi><mn mathvariant="normal">2</mn></msup><mo>⁢</mo><mi mathvariant="normal">φω</mi></mfenced><mo>⁢</mo><mi>VE</mi><mo>⁢</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>VE</mi><mo>⁢</mo><mn mathvariant="normal">2</mn></mfenced></mfenced></math><img file="EP0707229A2_D0045.tif" /></maths><maths id="math0046" num="(4B)"><math display="block"><mi mathvariant="normal">ΔZ#</mi><mo>=</mo><mo>-</mo><mfenced separators=""><mi mathvariant="normal">E</mi><mo>/</mo><mn mathvariant="normal">2</mn><mo>⁢</mo><msub><mi mathvariant="normal">αʹ</mi><mi mathvariant="normal">k</mi></msub></mfenced><mo>⋅</mo><mfenced open="{" close="}" separators=""><msup><mfenced separators=""><mi mathvariant="normal">N</mi><mo>⋅</mo><mi mathvariant="normal">tanω</mi></mfenced><mn mathvariant="normal">2</mn></msup><mo>⋅</mo><mi>sin</mi><mn>2</mn><mo>⁢</mo><mi mathvariant="normal">φω</mi><mo>⋅</mo><mi>VE</mi><mo>⁢</mo><mn mathvariant="normal">1</mn></mfenced></math><img file="EP0707229A2_D0046.tif" /></maths>

0041Based on the expressions (4A) and (4B), the following expressions (4C) and (4D) are obtained (axial ray, tanω=0): <maths id="math0047" num="(4C)"><math display="block"><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msub><mi mathvariant="normal">Y</mi><mn mathvariant="normal">0</mn></msub><mo>⁢</mo><mi mathvariant="normal">#</mi><mo>=</mo><mo>-</mo><mfenced separators=""><mi mathvariant="normal">E</mi><mo>/</mo><mn mathvariant="normal">2</mn><mo>⁢</mo><msub><mi mathvariant="normal">αʹ</mi><mi mathvariant="normal">k</mi></msub></mfenced><mo>⋅</mo><mi mathvariant="normal">ΔE</mi></math><img file="EP0707229A2_D0047.tif" /></maths><maths id="math0048" num="(4D)"><math display="block"><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msub><mi mathvariant="normal">Z</mi><mn mathvariant="normal">0</mn></msub><mo>⁢</mo><mi>#</mi><mo>=</mo><mn>0</mn></math><img file="EP0707229A2_D0048.tif" /></maths>

0042Subsequently, the rotational conversion will be described with reference to <figref idref="f0043">Figs. 44A and 44B</figref>. From <figref idref="f0043">Fig. 44A</figref>, the following expression holds: <maths id="math0049" num=""><math display="block"><mi>Y#</mi><mo>=</mo><msub><mi>g$</mi><mi mathvariant="normal">k</mi></msub><mo>⋅</mo><mi mathvariant="normal">tanω</mi></math><img file="EP0707229A2_D0049.tif" /></maths> From the sine theorem, <maths id="math0050" num=""><math display="block"><mi mathvariant="normal">Yʹ#</mi><mo>/</mo><mfenced open="{" close="}" separators=""><mi>sin</mi><mfenced separators=""><mi mathvariant="normal">π</mi><mo>/</mo><mn mathvariant="normal">2</mn><mo>-</mo><mi mathvariant="normal">ωʹ</mi></mfenced></mfenced><mo>=</mo><mfenced separators=""><mi>Y#</mi><mo>+</mo><mi mathvariant="normal">ΔY#</mi><mo>-</mo><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msub><mi mathvariant="normal">Y</mi><mn mathvariant="normal">0</mn></msub><mo>⁢</mo><mi mathvariant="normal">#</mi></mfenced><mo>/</mo><mfenced open="{" close="}" separators=""><mi>sin</mi><mfenced separators=""><mi mathvariant="normal">π</mi><mo>/</mo><mn mathvariant="normal">2</mn><mo>+</mo><mi mathvariant="normal">ωʹ</mi><mo>-</mo><mi mathvariant="normal">θ</mi></mfenced></mfenced></math><img file="EP0707229A2_D0050.tif" /></maths> ΔY'# after the rotational conversion is represented by the following expression: <maths id="math0051" num=""><math display="block"><mtable><mtr><mtd><mi mathvariant="normal">ΔYʹ#</mi><mo>=</mo><mfenced><mi mathvariant="normal">Yʹ#</mi></mfenced><mo>-</mo><mfenced><mi>Y#</mi></mfenced></mtd></mtr><mtr><mtd><mo>=</mo><mfenced open="[" close="]" separators=""><mi>Y#</mi><mo>⋅</mo><mi mathvariant="normal">cosωʹ</mi><mo>+</mo><mfenced open="{" close="}" separators=""><mfenced><mi mathvariant="normal">ΔY#</mi></mfenced><mo>-</mo><mfenced separators=""><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msub><mi mathvariant="normal">Y</mi><mn mathvariant="normal">0</mn></msub><mo>⁢</mo><mi mathvariant="normal">#</mi></mfenced></mfenced><mo>⋅</mo><mi mathvariant="normal">cosωʹ</mi><mo>-</mo><mi>Y#</mi><mo>⋅</mo><mi>cos</mi><mfenced separators=""><mi mathvariant="normal">ωʹ</mi><mo>-</mo><mi mathvariant="normal">θ</mi></mfenced></mfenced><mo>/</mo><mi>cos</mi><mfenced separators=""><mi mathvariant="normal">ωʹ</mi><mo>-</mo><mi mathvariant="normal">θ</mi></mfenced></mtd></mtr></mtable></math><img file="EP0707229A2_D0051.tif" /></maths> Only the numerators of this expression are converted. <maths id="math0052" num=""><math display="block"><mtable><mtr><mtd><mfenced open="[" close="]" separators=""><mi>Y#</mi><mo>⋅</mo><mi mathvariant="normal">cosωʹ</mi><mo>+</mo><mfenced open="{" close="}" separators=""><mfenced><mi mathvariant="normal">ΔY#</mi></mfenced><mo>-</mo><mfenced separators=""><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msub><mi mathvariant="normal">Y</mi><mn mathvariant="normal">0</mn></msub><mo>⁢</mo><mi mathvariant="normal">#</mi></mfenced></mfenced><mo>⋅</mo><mi mathvariant="normal">cosωʹ</mi><mo>-</mo><mi>Y#</mi><mo>⋅</mo><mi>cos</mi><mfenced separators=""><mi mathvariant="normal">ωʹ</mi><mo>-</mo><mi mathvariant="normal">θ</mi></mfenced></mfenced></mtd></mtr><mtr><mtd><mo>=</mo><mi>Y#</mi><mo>⋅</mo><mi mathvariant="normal">cosωʹ</mi><mo>+</mo><mfenced open="{" close="}" separators=""><mfenced><mi mathvariant="normal">ΔY#</mi></mfenced><mo>-</mo><mfenced separators=""><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msub><mi mathvariant="normal">Y</mi><mn mathvariant="normal">0</mn></msub><mo>⁢</mo><mi mathvariant="normal">#</mi></mfenced></mfenced><mo>⋅</mo><mi mathvariant="normal">cosωʹ</mi><mo>-</mo><mi>Y#</mi><mo>⋅</mo><mi mathvariant="normal">cosθ</mi><mo>⋅</mo><mi mathvariant="normal">cosωʹ</mi><mo>-</mo><mi>Y#</mi><mo>⋅</mo><mi mathvariant="normal">sinθ</mi><mo>⋅</mo><mi mathvariant="normal">sinωʹ</mi></mtd></mtr><mtr><mtd><mo>=</mo><mfenced separators=""><mn mathvariant="normal">1</mn><mo>-</mo><mi mathvariant="normal">cosθ</mi></mfenced><mo>⋅</mo><mi>Y#</mi><mo>⋅</mo><mi mathvariant="normal">cosωʹ</mi><mo>+</mo><mfenced open="{" close="}" separators=""><mfenced><mi mathvariant="normal">ΔY#</mi></mfenced><mo>-</mo><mfenced separators=""><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msub><mi mathvariant="normal">Y</mi><mn mathvariant="normal">0</mn></msub><mo>⁢</mo><mi mathvariant="normal">#</mi></mfenced></mfenced><mo>⋅</mo><mi mathvariant="normal">cosωʹ</mi><mo>-</mo><mi>Y#</mi><mo>⋅</mo><mi mathvariant="normal">sinθ</mi><mo>⋅</mo><mi mathvariant="normal">sinωʹ</mi></mtd></mtr></mtable></math><img file="EP0707229A2_D0052.tif" /></maths>

0043Here, since θ is small and ignorable compared to the other values, (1-cosθ) ≒ θ²/2, sinθ ≒ θ, and cosω'/{cos(ω'-θ)} ≒ 1 , sinω'/{cos(ω'-θ)} ≒ tanω.

0044Therefore, the following expression is obtained: <maths id="math0053" num=""><math display="block"><mi mathvariant="normal">ΔYʹ#</mi><mo>≒</mo><mfenced separators=""><mi mathvariant="normal">ΔY#</mi><mo>-</mo><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msub><mi mathvariant="normal">Y</mi><mn mathvariant="normal">0</mn></msub><mo>⁢</mo><mi mathvariant="normal">#</mi></mfenced><mo>-</mo><msub><mi mathvariant="normal">Y</mi><mi mathvariant="normal">#</mi></msub><mo>⋅</mo><mi mathvariant="normal">θ</mi><mo>⋅</mo><mi mathvariant="normal">tanω</mi></math><img file="EP0707229A2_D0053.tif" /></maths> (ΔY#-ΔY₀#) represents the off-axial image point movement error of parallel decentering and Y# · θtanω is an additional term (irrelevant to the aberration coefficients) due to rotation. Since ω at this time is on an X-Y cross section, <maths id="math0054" num="(5A)"><math display="block"><mi mathvariant="normal">ΔYʹ#</mi><mo>≒</mo><mfenced separators=""><mi mathvariant="normal">ΔY#</mi><mo>-</mo><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msub><mi mathvariant="normal">Y</mi><mn mathvariant="normal">0</mn></msub><mo>⁢</mo><mi mathvariant="normal">#</mi></mfenced><mo>-</mo><mi>Y#</mi><mo>⋅</mo><mi mathvariant="normal">θ</mi><mo>⋅</mo><mi mathvariant="normal">tanω</mi><mo>⋅</mo><mi mathvariant="normal">cosφω</mi></math><img file="EP0707229A2_D0054.tif" /></maths>

0045Subsequently, the conversion to the image surface IS will be described with reference to <figref idref="f0043">Fig. 45</figref>. A magnification β is represented by the following expression: <maths id="math0055" num=""><math display="block"><mi mathvariant="normal">β</mi><mo>=</mo><msub><mi>g$</mi><mn mathvariant="normal">1</mn></msub><mo>/</mo><msub><mi>g$</mi><mi mathvariant="normal">k</mi></msub><mo>=</mo><msub><mi mathvariant="normal">α</mi><mi mathvariant="normal">k</mi></msub><mo>⁢</mo><mi mathvariant="normal">ʹ</mi><mo>/</mo><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></math><img file="EP0707229A2_D0055.tif" /></maths> where α₁ = 1/g$₁. The relationship between the image surface IS and the object surface OS is represented by the following expression: <maths id="math0056" num=""><math display="block"><mi mathvariant="normal">Y</mi><mo>=</mo><mi mathvariant="normal">β</mi><mo>⋅</mo><mi>Y#</mi></math><img file="EP0707229A2_D0056.tif" /></maths> Y# and ΔY# which take the form of 1/α_k'×( ) are converted as follows: <maths id="math0057" num=""><math display="block"><mtable><mtr><mtd><mi mathvariant="normal">Y</mi><mo>=</mo><mi mathvariant="normal">β</mi><mo>⋅</mo><mi>Y#</mi></mtd></mtr><mtr><mtd><mo>=</mo><mfenced separators=""><msub><mi mathvariant="normal">α</mi><mi mathvariant="normal">k</mi></msub><mo>⁢</mo><mi mathvariant="normal">ʹ</mi><mo>/</mo><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced><mo>⋅</mo><mfenced separators=""><mn mathvariant="normal">1</mn><mo>/</mo><msub><mi mathvariant="normal">α</mi><mi mathvariant="normal">k</mi></msub><mo>⁢</mo><mi mathvariant="normal">ʹ</mi></mfenced><mo>×</mo><mrow><mo>(</mo></mrow><mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mo>=</mo><msub><mi>g$</mi><mn mathvariant="normal">1</mn></msub><mo>×</mo><mrow><mo>(</mo></mrow><mrow><mo>)</mo></mrow></mtd></mtr></mtable></math><img file="EP0707229A2_D0057.tif" /></maths> Here, if g$_k'→∞, g$₁=-F1. Therefore, the following expression holds: <maths id="math0058" num=""><math display="block"><mtable><mtr><mtd><mi mathvariant="normal">Y</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">F</mi><mo>⁢</mo><mn mathvariant="normal">1</mn><mo>×</mo><mrow><mo>(</mo></mrow><mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mo>=</mo><mo>-</mo><mi mathvariant="normal">F</mi><mo>⁢</mo><mn mathvariant="normal">1</mn><mo>×</mo><msub><mi mathvariant="normal">α</mi><mi mathvariant="normal">k</mi></msub><mo>⁢</mo><mi mathvariant="normal">ʹ</mi><mo>×</mo><mi>Y#</mi></mtd></mtr></mtable></math><img file="EP0707229A2_D0058.tif" /></maths>

0046Subsequently, the off-axial image point movement error on the image surface will be described. From the expression (4C) and α_k'=1/g_k'$, a decentering amount E is represented by the following expressions: <maths id="math0059" num=""><math display="block"><mi mathvariant="normal">θ</mi><mo>=</mo><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msub><mi mathvariant="normal">Y</mi><mn mathvariant="normal">0</mn></msub><mo>⁢</mo><mi mathvariant="normal">#</mi><mo>/</mo><msub><mi>g$</mi><mi mathvariant="normal">k</mi></msub><mo>⁢</mo><mi mathvariant="normal">ʹ</mi><mo>=</mo><mi mathvariant="normal">E</mi><mo>⋅</mo><mi mathvariant="normal">ΔE</mi><mo>/</mo><mn mathvariant="normal">2</mn></math><img file="EP0707229A2_D0059.tif" /></maths><maths id="math0060" num=""><math display="block"><mi mathvariant="normal">E</mi><mo>=</mo><mn mathvariant="normal">2</mn><mo>⋅</mo><mi mathvariant="normal">θ</mi><mo>/</mo><mi mathvariant="normal">ΔE</mi></math><img file="EP0707229A2_D0060.tif" /></maths> Normalization is performed so that the camera shake compensation angle θ is constant (0.7deg = 0.0122173rad).

0047By performing parallel decentering (rotational decentering is not performed) to image-surface-convert ΔY=(ΔY#-ΔY₀#) (here, N · tanω=Φ/F1, Φ²=Y²+Z²), the following expressions (6A) to (6D) are obtained: <maths id="math0061" num="(6A)"><math display="block"><mi mathvariant="normal">ΔY</mi><mo>=</mo><mfenced separators=""><mi mathvariant="normal">θ</mi><mo>⋅</mo><msup><mi mathvariant="normal">Φ</mi><mn mathvariant="normal">2</mn></msup><mo>/</mo><mi mathvariant="normal">F</mi><mo>⁢</mo><mn mathvariant="normal">1</mn></mfenced><mo>⋅</mo><mfenced open="[" close="]" separators=""><mfenced open="{" close="}" separators=""><mrow><mn mathvariant="normal">2</mn><mo>+</mo><mi>cos</mi><mn mathvariant="normal">2</mn><mo>⋅</mo><mi mathvariant="normal">φω</mi><mo>)</mo></mrow><mo>⋅</mo><mi>VE</mi><mo>⁢</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>VE</mi><mo>⁢</mo><mn mathvariant="normal">2</mn></mfenced><mo>/</mo><mi mathvariant="normal">ΔE</mi></mfenced></math><img file="EP0707229A2_D0061.tif" /></maths><maths id="math0062" num="(6B)"><math display="block"><mi mathvariant="normal">ΔZ</mi><mo>=</mo><mfenced separators=""><mi mathvariant="normal">θ</mi><mo>⋅</mo><msup><mi mathvariant="normal">Φ</mi><mn mathvariant="normal">2</mn></msup><mo>/</mo><mi mathvariant="normal">F</mi><mo>⁢</mo><mn mathvariant="normal">1</mn></mfenced><mo>⋅</mo><mfenced open="[" close="]" separators=""><mfenced open="{" close="}" separators=""><mfenced separators=""><mi>sin</mi><mn mathvariant="normal">2</mn><mo>⋅</mo><mi mathvariant="normal">φω</mi></mfenced><mo>⋅</mo><mi>VE</mi><mo>⁢</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>VE</mi><mo>⁢</mo><mn mathvariant="normal">2</mn></mfenced><mo>/</mo><mi mathvariant="normal">ΔE</mi></mfenced></math><img file="EP0707229A2_D0062.tif" /></maths>

0048Y₊ image point, Y_ image point {φω=0, π of the expressions (6A) and (6B)}: <maths id="math0063" num="(6C)"><math display="block"><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msub><mi mathvariant="normal">Y</mi><mi mathvariant="normal">Y</mi></msub><mo>=</mo><mfenced separators=""><mi mathvariant="normal">θ</mi><mo>⋅</mo><msup><mi mathvariant="normal">Y</mi><mn mathvariant="normal">2</mn></msup><mo>/</mo><mi mathvariant="normal">F</mi><mo>⁢</mo><mn mathvariant="normal">1</mn></mfenced><mo>⋅</mo><mfenced open="{" close="}" separators=""><mfenced separators=""><mn>3</mn><mo>⋅</mo><mi>VE</mi><mo>⁢</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>VE</mi><mo>⁢</mo><mn mathvariant="normal">2</mn></mfenced><mo>/</mo><mi mathvariant="normal">ΔE</mi></mfenced></math><img file="EP0707229A2_D0063.tif" /></maths>

0049Z image point {φω=π/2 of the expressions (6A) and (6B)} <maths id="math0064" num="(6D)"><math display="block"><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msub><mi mathvariant="normal">Y</mi><mi mathvariant="normal">Z</mi></msub><mo>=</mo><mfenced separators=""><mi mathvariant="normal">θ</mi><mo>⋅</mo><msup><mi mathvariant="normal">Z</mi><mn mathvariant="normal">2</mn></msup><mo>/</mo><mi mathvariant="normal">F</mi><mo>⁢</mo><mn mathvariant="normal">1</mn></mfenced><mo>⋅</mo><mfenced open="{" close="}" separators=""><mfenced separators=""><mi>VE</mi><mo>⁢</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>VE</mi><mo>⁢</mo><mn mathvariant="normal">2</mn></mfenced><mo>/</mo><mi mathvariant="normal">ΔE</mi></mfenced></math><img file="EP0707229A2_D0064.tif" /></maths>

0050Then, rotational conversion is performed. Since Y#=- Y/(F1×α_k'), with respect to -Y# · θ · tanω · cosφω of the expression (5A), the following expression holds: <maths id="math0065" num=""><math display="block"><mo>-</mo><mi>Y#</mi><mo>⋅</mo><mi mathvariant="normal">θ</mi><mo>⋅</mo><mi mathvariant="normal">tanω</mi><mo>⋅</mo><mi mathvariant="normal">cosφω</mi><mo>=</mo><mi mathvariant="normal">Y</mi><mo>/</mo><mfenced separators=""><mi mathvariant="normal">F</mi><mo>⁢</mo><mn mathvariant="normal">1</mn><mo>×</mo><msub><mi mathvariant="normal">α</mi><mi mathvariant="normal">k</mi></msub><mo>⁢</mo><mi mathvariant="normal">ʹ</mi></mfenced><mo>⋅</mo><mi mathvariant="normal">θ</mi><mo>⋅</mo><mi mathvariant="normal">tanω</mi><mo>⋅</mo><mi mathvariant="normal">cosφω</mi></math><img file="EP0707229A2_D0065.tif" /></maths> At the Y, image point and the Y, image point, since φω0, π and tanω/α_k'=Y, -Y# · θ · tanω · cosφω on the image surface is -Y# · θ · tanω · cosφω=Y² · θ/F1. By adding this to the expression (6C), the following expression (6E) is obtained. At the Z image point, since φω=π/2, -Y# · θ · tanω · cosφω on the image surface is -Y# · θ · tanω · cosφω=0. By adding this to the expression (6D), the following expression (6F) is obtained: <maths id="math0066" num="(6E)"><math display="block"><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msub><mi mathvariant="normal">Y</mi><mi mathvariant="normal">Y</mi></msub><mo>⁢</mo><mi>ʹ</mi><mo>=</mo><mfenced separators=""><mi mathvariant="normal">θ</mi><mo>⋅</mo><msup><mi mathvariant="normal">Y</mi><mn mathvariant="normal">2</mn></msup><mo>/</mo><mi mathvariant="normal">F</mi><mo>⁢</mo><mn mathvariant="normal">1</mn></mfenced><mo>⋅</mo><mfenced open="{" close="}" separators=""><mfenced separators=""><mn>3</mn><mo>⋅</mo><mi>VE</mi><mo>⁢</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>VE</mi><mo>⁢</mo><mn mathvariant="normal">2</mn><mo>-</mo><mi mathvariant="normal">ΔE</mi></mfenced><mo>/</mo><mi mathvariant="normal">ΔE</mi></mfenced></math><img file="EP0707229A2_D0066.tif" /></maths><maths id="math0067" num="(6F)"><math display="block"><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msub><mi mathvariant="normal">Y</mi><mi mathvariant="normal">Z</mi></msub><mo>⁢</mo><mi>ʹ</mi><mo>=</mo><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msub><mi mathvariant="normal">Y</mi><mi mathvariant="normal">Z</mi></msub></math><img file="EP0707229A2_D0067.tif" /></maths>

[One-side blur]

0051Subsequently, the one-side blur will be described. From the expressions (1 A) and (1 B), ΔM is {(primary term of R) of ΔY φR=0}×g$_k' and ΔS is {(primary term of R) of ΔZ φ = π/2}×g$_k'. Before rotation, the following expression holds on the object surface OS (here, α_k'=N_k'/g$_k' and E/2=θ/ΔE are used): <maths id="math0068" num=""><math display="block"><mi mathvariant="normal">ΔM#</mi><mo>=</mo><mfenced separators=""><mo>-</mo><msup><msub><mi>g$</mi><mi mathvariant="normal">kʹ</mi></msub><mn mathvariant="normal">2</mn></msup><mo>⋅</mo><mi mathvariant="normal">θ</mi><mo>/</mo><msub><mi mathvariant="normal">N</mi><mi mathvariant="normal">kʹ</mi></msub></mfenced><mo>×</mo><mn mathvariant="normal">2</mn><mo>⋅</mo><mi mathvariant="normal">R</mi><mo>⋅</mo><mfenced separators=""><mi mathvariant="normal">N</mi><mo>⋅</mo><mi mathvariant="normal">tanω</mi></mfenced><mo>⋅</mo><mi mathvariant="normal">cosφω</mi><mo>⋅</mo><mfenced open="{" close="}" separators=""><mfenced separators=""><mn mathvariant="normal">3</mn><mo>⋅</mo><mi>IIIE</mi><mo>+</mo><mi>PE</mi></mfenced><mo>/</mo><mi mathvariant="normal">ΔE</mi></mfenced></math><img file="EP0707229A2_D0068.tif" /></maths> After the rotation, the following expression holds: <maths id="math0069" num=""><math display="block"><mi mathvariant="normal">ΔMʹ#</mi><mo>≒</mo><mi mathvariant="normal">ΔM#</mi><mo>+</mo><mi mathvariant="normal">θY#</mi></math><img file="EP0707229A2_D0069.tif" /></maths>

0052By converting the aberration coefficients to those on the image surface and substituting N_k'=1 and N=1, the following expression is obtained: <maths id="math0070" num=""><math display="block"><mtable><mtr><mtd><mi mathvariant="normal">ΔMʹ</mi><mo>=</mo><msup><mi mathvariant="normal">β</mi><mn mathvariant="normal">2</mn></msup><mo>⋅</mo><mi mathvariant="normal">ΔMʹ#</mi></mtd></mtr><mtr><mtd><mo>=</mo><mo>-</mo><msup><msub><mi>g$</mi><mn mathvariant="normal">1</mn></msub><mn mathvariant="normal">2</mn></msup><mo>⋅</mo><mi mathvariant="normal">θ</mi><mo>×</mo><mn mathvariant="normal">2</mn><mo>⋅</mo><mi mathvariant="normal">R</mi><mo>⋅</mo><mi mathvariant="normal">tanω</mi><mo>⋅</mo><mi mathvariant="normal">cosφω</mi><mo>⋅</mo><mfenced open="{" close="}" separators=""><mfenced separators=""><mn mathvariant="normal">3</mn><mo>⋅</mo><mi>IIIE</mi><mo>+</mo><mi>PE</mi></mfenced><mo>/</mo><mi mathvariant="normal">ΔE</mi></mfenced><mo>+</mo><mi mathvariant="normal">β</mi><mo>⋅</mo><mi mathvariant="normal">Y</mi><mo>⋅</mo><mi mathvariant="normal">θ</mi></mtd></mtr></mtable></math><img file="EP0707229A2_D0070.tif" /></maths> If the object surface OS is ∞ (here, g$₁ =-F₁, β→0, tanω=Y/F1 and φω=0), the following expression (7A) representative of a meridional one-side blur ΔM' is obtained. Likewise, an expression (7B) representative of a sagittal one-side blur is obtained. <maths id="math0071" num="(7A)"><math display="block"><mi mathvariant="normal">ΔMʹ</mi><mo>=</mo><mo>-</mo><mn mathvariant="normal">2</mn><mo>⋅</mo><mi mathvariant="normal">F</mi><mo>⁢</mo><mn mathvariant="normal">1</mn><mo>⋅</mo><mi mathvariant="normal">Y</mi><mo>⋅</mo><mi mathvariant="normal">θ</mi><mo>⋅</mo><mi mathvariant="normal">R</mi><mo>⋅</mo><mfenced open="{" close="}" separators=""><mfenced separators=""><mn mathvariant="normal">3</mn><mo>⋅</mo><mi>IIID</mi><mo>+</mo><mi>PE</mi></mfenced><mo>/</mo><mi mathvariant="normal">ΔE</mi></mfenced></math><img file="EP0707229A2_D0071.tif" /></maths><maths id="math0072" num="(7B)"><math display="block"><mi mathvariant="normal">ΔSʹ</mi><mo>=</mo><mo>-</mo><mn mathvariant="normal">2</mn><mo>⋅</mo><mi mathvariant="normal">F</mi><mo>⁢</mo><mn mathvariant="normal">1</mn><mo>⋅</mo><mi mathvariant="normal">Y</mi><mo>⋅</mo><mi mathvariant="normal">θ</mi><mo>⋅</mo><mi mathvariant="normal">R</mi><mo>⋅</mo><mfenced open="{" close="}" separators=""><mfenced separators=""><mi>IIID</mi><mo>+</mo><mi>PE</mi></mfenced><mo>/</mo><mi mathvariant="normal">ΔE</mi></mfenced></math><img file="EP0707229A2_D0072.tif" /></maths>

[Axial coma]

0053Subsequently, axial coma will be described. Based on the expression (1 A), coma of upper ray by ω=0 and decentering is represented by the following expression: <maths id="math0073" num=""><math display="block"><mtable><mtr><mtd><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msub><mi mathvariant="normal">Y</mi><mi>Upper</mi></msub><mo>⁢</mo><mi mathvariant="normal">#</mi><mo>=</mo><mi mathvariant="normal">ΔY#</mi><mfenced separators=""><mi mathvariant="normal">ω</mi><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mi mathvariant="normal">φ</mi><mi mathvariant="normal">R</mi></msub><mo>=</mo><mn mathvariant="normal">0</mn></mfenced><mo>-</mo><mi mathvariant="normal">ΔY#</mi><mfenced separators=""><mi mathvariant="normal">ω</mi><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi mathvariant="normal">R</mi><mo>=</mo><mn mathvariant="normal">0</mn></mfenced></mtd></mtr><mtr><mtd><mo>=</mo><mo>-</mo><mi mathvariant="normal">E</mi><mo>/</mo><mfenced separators=""><mn mathvariant="normal">2</mn><mo>⋅</mo><mi mathvariant="normal">αʹ</mi></mfenced><mo>×</mo><msup><mi mathvariant="normal">R</mi><mn mathvariant="normal">2</mn></msup><mo>×</mo><mn mathvariant="normal">3</mn><mo>⋅</mo><mi>IIE</mi></mtd></mtr></mtable></math><img file="EP0707229A2_D0073.tif" /></maths> Coma of lower ray by ω=0 and decentering (the same as ΔY_Upper # including the sign) is represented by the following expression: <maths id="math0074" num=""><math display="block"><mtable><mtr><mtd><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msub><mi mathvariant="normal">Y</mi><mi>Lower</mi></msub><mo>⁢</mo><mi mathvariant="normal">#</mi><mo>=</mo><msub><mi mathvariant="normal">ΔY</mi><mi>#</mi></msub><mfenced separators=""><mi mathvariant="normal">ω</mi><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mi mathvariant="normal">φ</mi><mi mathvariant="normal">R</mi></msub><mo>=</mo><mi mathvariant="normal">π</mi></mfenced><mo>-</mo><mi mathvariant="normal">ΔY#</mi><mfenced separators=""><mi mathvariant="normal">ω</mi><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi mathvariant="normal">R</mi><mo>=</mo><mn mathvariant="normal">0</mn></mfenced></mtd></mtr><mtr><mtd><mo>=</mo><mo>-</mo><mi mathvariant="normal">E</mi><mo>/</mo><mfenced separators=""><mn mathvariant="normal">2</mn><mo>⋅</mo><mi mathvariant="normal">αʹ</mi></mfenced><mo>×</mo><msup><mi mathvariant="normal">R</mi><mn mathvariant="normal">2</mn></msup><mo>×</mo><mn mathvariant="normal">3</mn><mo>⋅</mo><mi>IIE</mi></mtd></mtr></mtable></math><img file="EP0707229A2_D0074.tif" /></maths>

0054Since ω=0, axial coma is hardly varied by the rotational conversion. By the conversion from the object surface OS to the image surface IS (ΔY=β · ΔY#, E/2=θ/ΔE), the following expression is obtained <maths id="math0075" num=""><math display="block"><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msub><mi mathvariant="normal">Y</mi><mi>Upper</mi></msub><mo>=</mo><mi mathvariant="normal">F</mi><mo>⁢</mo><mn mathvariant="normal">1</mn><mo>×</mo><mi mathvariant="normal">θ</mi><mo>×</mo><msup><mi mathvariant="normal">R</mi><mn mathvariant="normal">2</mn></msup><mo>×</mo><mfenced separators=""><mn mathvariant="normal">3</mn><mo>⋅</mo><mi>IIE</mi><mo>/</mo><mi mathvariant="normal">ΔE</mi></mfenced><mo>=</mo><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msub><mi mathvariant="normal">Y</mi><mi>Lower</mi></msub></math><img file="EP0707229A2_D0075.tif" /></maths> The axial coma AXCM is represented by the following expression (8A): <maths id="math0076" num=""><math display="block"><mtable><mtr><mtd><mi>AXCM</mi><mo>=</mo><mfenced separators=""><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msub><mi mathvariant="normal">Y</mi><mi>Upper</mi></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msub><mi mathvariant="normal">Y</mi><mi>Lower</mi></msub></mfenced><mo>/</mo><mn mathvariant="normal">2</mn></mtd></mtr><mtr><mtd><mo>=</mo><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msub><mi mathvariant="normal">Y</mi><mi>Upper</mi></msub></mtd></mtr></mtable></math><img file="EP0707229A2_D0076.tif" /></maths>

0055A part of each of the expressions (6E), (6F), (7A), (7B) and (8A) is newly defined as an aberration coefficient represented by the following expressions (9A) to (9E):

0056Off-axial image point movement error of the image point on the Y-axis <maths id="math0077" num="(9A)"><math display="block"><msub><mi>VE</mi><mi mathvariant="normal">Y</mi></msub><mo>=</mo><mfenced open="{" close="}" separators=""><mfenced separators=""><mn mathvariant="normal">3</mn><mo>⋅</mo><mi>VE</mi><mo>⁢</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>VE</mi><mo>⁢</mo><mn mathvariant="normal">2</mn><mo>-</mo><mi mathvariant="normal">ΔE</mi></mfenced><mo>/</mo><mi mathvariant="normal">ΔE</mi></mfenced></math><img file="EP0707229A2_D0077.tif" /></maths>

0057Off-axial image point movement error of the image point on the Z-axis <maths id="math0078" num="(9B)"><math display="block"><msub><mi>VE</mi><mi mathvariant="normal">Z</mi></msub><mo>=</mo><mfenced open="{" close="}" separators=""><mfenced separators=""><mi>VE</mi><mo>⁢</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>VE</mi><mo>⁢</mo><mn mathvariant="normal">2</mn></mfenced><mo>/</mo><mi mathvariant="normal">ΔE</mi></mfenced></math><img file="EP0707229A2_D0078.tif" /></maths><maths id="math0079" num="(9C)"><math display="block"><mi>Marginal one</mi><mo>-</mo><mi>side blur</mi><mspace width="1em" /><msub><mi>IIIE</mi><mi mathvariant="normal">M</mi></msub><mo>=</mo><mfenced open="{" close="}" separators=""><mfenced separators=""><mn mathvariant="normal">3</mn><mo>⋅</mo><mi>IIIE</mi><mo>+</mo><mi>PE</mi></mfenced><mo>/</mo><mi mathvariant="normal">ΔE</mi></mfenced></math><img file="EP0707229A2_D0079.tif" /></maths><maths id="math0080" num="(9D)"><math display="block"><mi>Sagittal one</mi><mo>-</mo><mi>side blur</mi><mspace width="1em" /><msub><mi>IIIE</mi><mi mathvariant="normal">S</mi></msub><mo>=</mo><mfenced open="{" close="}" separators=""><mfenced separators=""><mi>IIIE</mi><mo>+</mo><mi>PE</mi></mfenced><mo>/</mo><mi mathvariant="normal">ΔE</mi></mfenced></math><img file="EP0707229A2_D0080.tif" /></maths><maths id="math0081" num="(9E)"><math display="block"><mi>Axial coma</mi><mspace width="1em" /><msub><mi>IIE</mi><mi mathvariant="normal">A</mi></msub><mo>=</mo><mfenced open="{" close="}" separators=""><mfenced separators=""><mn mathvariant="normal">3</mn><mo>⋅</mo><mi>IIE</mi></mfenced><mo>/</mo><mi mathvariant="normal">ΔE</mi></mfenced></math><img file="EP0707229A2_D0081.tif" /></maths>

0058By substituting the expressions (3A) to (3F) in the expressions (9A) to (9B) representative of the camera shake aberration coefficients, the following expressions (10A) to (10E) representative of camera shake aberration coefficients are obtained: <maths id="math0082" num="(10A)"><math display="block"><msub><mi>VE</mi><mi mathvariant="normal">Y</mi></msub><mo>=</mo><mo>-</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn><mo>⋅</mo><mfenced open="{" close="}" separators=""><mn mathvariant="normal">3</mn><mo>⁢</mo><msub><mi mathvariant="normal">V</mi><mi mathvariant="normal">R</mi></msub><mo>-</mo><mn mathvariant="normal">3</mn><mo>⁢</mo><msub><mi mathvariant="normal">V</mi><mi mathvariant="normal">D</mi></msub><mo>⋅</mo><mi mathvariant="normal">A</mi><mo>+</mo><mn mathvariant="normal">2</mn><mo>-</mo><mfenced separators=""><mn mathvariant="normal">3</mn><mo>⋅</mo><msub><mi>III</mi><mi mathvariant="normal">R</mi></msub><mo>+</mo><msub><mi mathvariant="normal">P</mi><mi mathvariant="normal">R</mi></msub></mfenced><mo>⋅</mo><mi>H#</mi><mo>+</mo><mfenced separators=""><mn mathvariant="normal">3</mn><mo>⋅</mo><msub><mi>III</mi><mi mathvariant="normal">D</mi></msub><mo>+</mo><msub><mi mathvariant="normal">P</mi><mi mathvariant="normal">D</mi></msub></mfenced><mo>⋅</mo><mi>A#</mi></mfenced></math><img file="EP0707229A2_D0082.tif" /></maths><maths id="math0083" num="(10B)"><math display="block"><msub><mi>VE</mi><mi mathvariant="normal">Z</mi></msub><mo>=</mo><mo>-</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn><mo>⋅</mo><mfenced open="{" close="}" separators=""><msub><mi mathvariant="normal">V</mi><mi mathvariant="normal">R</mi></msub><mo>-</mo><msub><mi mathvariant="normal">V</mi><mi mathvariant="normal">D</mi></msub><mo>⋅</mo><mi mathvariant="normal">A</mi><mo>-</mo><mfenced separators=""><msub><mi>III</mi><mi mathvariant="normal">R</mi></msub><mo>+</mo><msub><mi mathvariant="normal">P</mi><mi mathvariant="normal">R</mi></msub></mfenced><mo>⋅</mo><mi>H#</mi><mo>+</mo><mfenced separators=""><msub><mi>III</mi><mi mathvariant="normal">D</mi></msub><mo>+</mo><msub><mi mathvariant="normal">P</mi><mi mathvariant="normal">D</mi></msub></mfenced><mo>⋅</mo><mi>A#</mi></mfenced></math><img file="EP0707229A2_D0083.tif" /></maths><maths id="math0084" num="(10C)"><math display="block"><msub><mi>IIIE</mi><mi mathvariant="normal">M</mi></msub><mo>=</mo><mo>-</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn><mo>⋅</mo><mfenced open="{" close="}" separators=""><mfenced separators=""><mn mathvariant="normal">3</mn><mo>⋅</mo><msub><mi>III</mi><mi mathvariant="normal">R</mi></msub><mo>+</mo><msub><mi mathvariant="normal">P</mi><mi mathvariant="normal">R</mi></msub></mfenced><mo>-</mo><mfenced separators=""><mn mathvariant="normal">3</mn><mo>⋅</mo><msub><mi>III</mi><mi mathvariant="normal">D</mi></msub><mo>+</mo><msub><mi mathvariant="normal">P</mi><mi mathvariant="normal">D</mi></msub></mfenced><mo>⋅</mo><mi mathvariant="normal">A</mi><mo>-</mo><mn mathvariant="normal">3</mn><mo>⋅</mo><msub><mi>II</mi><mi mathvariant="normal">R</mi></msub><mo>⋅</mo><mi>H#</mi><mo>+</mo><mn mathvariant="normal">3</mn><mo>⋅</mo><msub><mi>II</mi><mi mathvariant="normal">D</mi></msub><mo>⋅</mo><mi>A#</mi></mfenced></math><img file="EP0707229A2_D0084.tif" /></maths><maths id="math0085" num="(10D)"><math display="block"><msub><mi>IIIE</mi><mi mathvariant="normal">S</mi></msub><mo>=</mo><mo>-</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn><mo>⋅</mo><mfenced open="{" close="}" separators=""><mfenced separators=""><msub><mi>III</mi><mi mathvariant="normal">R</mi></msub><mo>+</mo><msub><mi mathvariant="normal">P</mi><mi mathvariant="normal">R</mi></msub></mfenced><mo>-</mo><mfenced separators=""><msub><mi>III</mi><mi mathvariant="normal">D</mi></msub><mo>+</mo><msub><mi mathvariant="normal">P</mi><mi mathvariant="normal">D</mi></msub></mfenced><mo>⋅</mo><mi mathvariant="normal">A</mi><mo>-</mo><msub><mi>II</mi><mi mathvariant="normal">R</mi></msub><mo>⋅</mo><mi>H#</mi><mo>+</mo><msub><mi>II</mi><mi mathvariant="normal">D</mi></msub><mo>⋅</mo><mi>A#</mi></mfenced></math><img file="EP0707229A2_D0085.tif" /></maths><maths id="math0086" num="(10E)"><math display="block"><msub><mi>IIE</mi><mi mathvariant="normal">A</mi></msub><mo>=</mo><mo>-</mo><mn mathvariant="normal">3</mn><mo>/</mo><mn mathvariant="normal">2</mn><mo>⋅</mo><mfenced separators=""><msub><mi>II</mi><mi mathvariant="normal">R</mi></msub><mo>+</mo><msub><mi>II</mi><mi mathvariant="normal">D</mi></msub><mo>⋅</mo><mi mathvariant="normal">A</mi><mo>-</mo><msub><mi mathvariant="normal">I</mi><mi mathvariant="normal">R</mi></msub><mo>⋅</mo><mi>H#</mi><mo>+</mo><msub><mi mathvariant="normal">I</mi><mi mathvariant="normal">D</mi></msub><mo>⋅</mo><mi>A#</mi></mfenced></math><img file="EP0707229A2_D0086.tif" /></maths> where <ul id="ul0002" list-style="none" compact="compact"><li>( )_D is the sum of the aberration coefficients of the compensating lens unit;</li><li>( )_R is the sum of the aberration coefficients of the lens units arranged behind (on the object side of) the compensating lens unit;</li><li>A=αi/(αj'-αi) (here, the compensating lens units are i to j), A#=αi#/(αj'-αi) ; and</li><li>H#=(αi'#-αi#)/(αj'-αi).</li><li>ΔE=-2 · (αj'-αi) (here, (αj'-αi) is ±0.0122173 when 0.7°/mm) which is the coefficient of (camera shake compensation angle)/(decentering amount) aims substantially at a predetermined value (however, the sign differs according to whether the compensating lens units are positive or negative). Therefore, A is an incident angle of a marginal ray to the compensating lens units (viewed from the image side) and A# is proportional to the incident angle of the principal ray. When h# and h do not vary so much in the compensating lens units, H# represents a ratio between h# of the principal ray and h of the marginal ray.</li></ul>

0059Since the decentering aberration coefficients in the expressions (10A) to (10E) are defined as those of the reversal optical system, it is necessary to return them to the coefficients of the non-reversal optical system. Returning the coefficients in the expressions (10A) to (10E) by using the expressions (2A) to (2J), the following expressions (11 A) to (11E) are obtained: <maths id="math0087" num="(11A)"><math display="block"><msub><mi>VE</mi><mi mathvariant="normal">Y</mi></msub><mo>=</mo><mo>+</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn><mo>⋅</mo><mfenced open="{" close="}" separators=""><mn mathvariant="normal">3</mn><mo>⁢</mo><msub><mi mathvariant="normal">V</mi><mi mathvariant="normal">F</mi></msub><mo>-</mo><mn mathvariant="normal">3</mn><mo>⁢</mo><msub><mi mathvariant="normal">V</mi><mi mathvariant="normal">D</mi></msub><mo>⋅</mo><mi mathvariant="normal">A</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo>+</mo><mfenced separators=""><mn mathvariant="normal">3</mn><mo>⋅</mo><msub><mi>III</mi><mi mathvariant="normal">F</mi></msub><mo>+</mo><msub><mi mathvariant="normal">P</mi><mi mathvariant="normal">F</mi></msub></mfenced><mo>⁢</mo><mi>H#</mi><mo>-</mo><mfenced separators=""><mn mathvariant="normal">3</mn><mo>⋅</mo><msub><mi>III</mi><mi mathvariant="normal">D</mi></msub><mo>+</mo><msub><mi mathvariant="normal">P</mi><mi mathvariant="normal">D</mi></msub></mfenced><mo>⋅</mo><mi>A#</mi></mfenced></math><img file="EP0707229A2_D0087.tif" /></maths><maths id="math0088" num="(11B)"><math display="block"><msub><mi>VE</mi><mi mathvariant="normal">Z</mi></msub><mo>=</mo><mo>+</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn><mo>⋅</mo><mfenced open="{" close="}" separators=""><msub><mi mathvariant="normal">V</mi><mi mathvariant="normal">F</mi></msub><mo>-</mo><msub><mi mathvariant="normal">V</mi><mi mathvariant="normal">D</mi></msub><mo>⋅</mo><mi mathvariant="normal">A</mi><mo>+</mo><mfenced separators=""><msub><mi>III</mi><mi mathvariant="normal">F</mi></msub><mo>+</mo><msub><mi mathvariant="normal">P</mi><mi mathvariant="normal">F</mi></msub></mfenced><mo>⋅</mo><mi>H#</mi><mo>-</mo><mfenced separators=""><msub><mi>III</mi><mi mathvariant="normal">D</mi></msub><mo>+</mo><msub><mi mathvariant="normal">P</mi><mi mathvariant="normal">D</mi></msub></mfenced><mo>⋅</mo><msub><mi mathvariant="normal">A</mi><mi>#</mi></msub></mfenced></math><img file="EP0707229A2_D0088.tif" /></maths><maths id="math0089" num="(11C)"><math display="block"><msub><mi>IIIE</mi><mi mathvariant="normal">M</mi></msub><mo>=</mo><mo>-</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn><mo>⋅</mo><mfenced open="{" close="}" separators=""><mfenced separators=""><mn mathvariant="normal">3</mn><mo>⋅</mo><msub><mi>III</mi><mi mathvariant="normal">F</mi></msub><mo>+</mo><msub><mi mathvariant="normal">P</mi><mi mathvariant="normal">F</mi></msub></mfenced><mo>-</mo><mfenced separators=""><mn mathvariant="normal">3</mn><mo>⋅</mo><msub><mi>III</mi><mi mathvariant="normal">D</mi></msub><mo>+</mo><msub><mi mathvariant="normal">P</mi><mi mathvariant="normal">D</mi></msub></mfenced><mo>⋅</mo><mi mathvariant="normal">A</mi><mo>+</mo><mn mathvariant="normal">3</mn><mo>⋅</mo><msub><mi>II</mi><mi mathvariant="normal">F</mi></msub><mo>⋅</mo><mi>H#</mi><mo>-</mo><mn mathvariant="normal">3</mn><mo>⋅</mo><msub><mi>II</mi><mi mathvariant="normal">D</mi></msub><mo>⋅</mo><mi>A#</mi></mfenced></math><img file="EP0707229A2_D0089.tif" /></maths><maths id="math0090" num="(11D)"><math display="block"><msub><mi>IIIE</mi><mi mathvariant="normal">S</mi></msub><mo>=</mo><mo>-</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn><mo>⋅</mo><mfenced open="{" close="}" separators=""><mfenced separators=""><msub><mi>III</mi><mi mathvariant="normal">F</mi></msub><mo>+</mo><msub><mi mathvariant="normal">P</mi><mi mathvariant="normal">F</mi></msub></mfenced><mo>-</mo><mfenced separators=""><msub><mi>III</mi><mi mathvariant="normal">D</mi></msub><mo>+</mo><msub><mi mathvariant="normal">P</mi><mi mathvariant="normal">D</mi></msub></mfenced><mo>⋅</mo><mi mathvariant="normal">A</mi><mo>+</mo><msub><mi>II</mi><mi mathvariant="normal">F</mi></msub><mo>⋅</mo><mi>H#</mi><mo>-</mo><msub><mi>II</mi><mi mathvariant="normal">D</mi></msub><mo>⋅</mo><mi>A#</mi></mfenced></math><img file="EP0707229A2_D0090.tif" /></maths><maths id="math0091" num="(11E)"><math display="block"><msub><mi>IIE</mi><mi mathvariant="normal">A</mi></msub><mo>=</mo><mo>+</mo><mn mathvariant="normal">3</mn><mo>/</mo><mn mathvariant="normal">2</mn><mo>⋅</mo><mfenced separators=""><msub><mi>II</mi><mi mathvariant="normal">F</mi></msub><mo>-</mo><msub><mi>II</mi><mi mathvariant="normal">D</mi></msub><mo>⋅</mo><mi mathvariant="normal">A</mi><mo>+</mo><msub><mi mathvariant="normal">I</mi><mi mathvariant="normal">F</mi></msub><mo>⋅</mo><mi>H#</mi><mo>-</mo><msub><mi mathvariant="normal">I</mi><mi mathvariant="normal">D</mi></msub><mo>⋅</mo><msub><mi mathvariant="normal">A</mi><mi>#</mi></msub></mfenced></math><img file="EP0707229A2_D0091.tif" /></maths> where <ul id="ul0003" list-style="none" compact="compact"><li>( )_D is the sum of the aberration coefficients of the compensating lens units and the non-reversal optical system;</li><li>( )_F is the sum of the aberration coefficients of the lens units arranged in front of the compensating lens units;</li></ul><maths id="math0092" num=""><math display="block"><mi mathvariant="normal">A</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">αnʹ</mi><mo>/</mo><mfenced separators=""><mi mathvariant="normal">αnʹ</mi><mo>-</mo><mi mathvariant="normal">αm</mi></mfenced><mo>;</mo></math><img file="EP0707229A2_D0092.tif" /></maths><maths id="math0093" num=""><math display="block"><mi>A#</mi><mo>=</mo><mi mathvariant="normal">αnʹ#</mi><mo>/</mo><mfenced separators=""><mi mathvariant="normal">αnʹ</mi><mo>-</mo><mi mathvariant="normal">αm</mi></mfenced><mo>;</mo></math><img file="EP0707229A2_D0093.tif" /></maths><maths id="math0094" num=""><math display="block"><mi mathvariant="normal">H</mi><mo>=</mo><mo>-</mo><mfenced separators=""><mi mathvariant="normal">αnʹ#</mi><mo>-</mo><mi mathvariant="normal">αm#</mi></mfenced><mo>/</mo><mfenced separators=""><mi mathvariant="normal">αnʹ</mi><mo>-</mo><mi mathvariant="normal">αm</mi></mfenced><mo>=</mo><mo>-</mo><mfenced separators=""><mi mathvariant="normal">Σhμ#</mi><mo>⋅</mo><mi mathvariant="normal">φμ</mi></mfenced><mo>/</mo><mfenced separators=""><mi mathvariant="normal">Σhμ</mi><mo>⋅</mo><mi mathvariant="normal">φμ</mi></mfenced><mo>;</mo></math><img file="EP0707229A2_D0094.tif" /></maths> and <maths id="math0095" num=""><math display="block"><mi mathvariant="normal">ΔE</mi><mo>=</mo><mo>-</mo><mn mathvariant="normal">2</mn><mo>⁢</mo><mfenced separators=""><mi mathvariant="normal">αnʹ</mi><mo>-</mo><mi mathvariant="normal">αm</mi></mfenced><mo>⁢</mo><mfenced separators=""><mi>compensating lens units</mi><mo>:</mo><mspace width="1em" /><mi mathvariant="normal">m</mi><mo>→</mo><mi mathvariant="normal">n</mi><mo>,</mo><mspace width="1em" /><mi>non</mi><mo>-</mo><mi>reversal lens system</mi><mo>:</mo><mspace width="1em" /><mi mathvariant="normal">j</mi><mo>←</mo><mi mathvariant="normal">i</mi></mfenced><mn mathvariant="normal">.</mn></math><img file="EP0707229A2_D0095.tif" /></maths>

0060From the expressions (11A) to (11E), the following are understood: First, as described previously, while the decentered lens unit and the lens units arranged therebehind relate to the performance according to the method of Mr. Matsui's paper, the decentered lens unit and the lens units arranged therebefore relate to the performance in the expressions (11A) to (11E). Secondly, the off-axial image point movement error tends to increase in wide angle optical systems (the focal length F1 of the compensating lens unit is the denominator) and one-side blur and axial coma tend to increase in telephoto optical systems.

0061Thirdly, although the degradation of aberrations due to decentering decreases by reducing the aberration coefficients of the decentered lens unit and the lens units arranged there-before, a constant (-2 in {} of the expression (11A)) remains in the coefficient VE_Y of the off-axial image point movement error ΔY_Y'. This is a term generated since the object surface OS and the image surface IS incline relative to each other due to rotational camera shake. The off-axial image point movement error due to the constant term (-2) is remarkably great in wide angle optical systems. For example, the off-axial image point movement error ΔY_Y'=-72µm at a focal length F1 of 38mm, which is not ignorable. The off-axial image point movement error due to the constant term (-2) remains even if the aberration coefficients are 0. Therefore, it is preferable to set the aberration coefficients so that the constant term (-2) is canceled. The condition (3) is a condition therefor.

0062Fourthly, in order to reduce the aberration degradation due to decentering, it is necessary to reduce the aberration coefficients and the coefficients such as A, A# and H# multiplied by the aberration coefficients. In order to reduce A and A#, it is necessary to increase the denominator α_n'-α_m. However, since this is directly connected to ΔE=-2(α_n'-α_m), if it is too great, the camera shake compensation sensitivity (how many degrees the luminous flux is bent at when the lens is decentered by predetermined millimeters) is too high, so that a mechanical driving accuracy is necessary. With respect to H#, the closer the compensating lens unit is to the aperture stop, the smaller h# of each surface is, so that H# is also small.

0063Numerical data of first to fourth embodiments of the present invention are shown in Tables 2 to 5. In each table, <i>ri</i> (<i>i</i>=1, 2, 3, ...) is a radius of curvature of an ith surface counted from the object side, d<i>i</i> (<i>i</i>=1, 2, 3, ...) is an ith axial distance counted from the object side, and N<i>i</i> (<i>i</i>=1, 2, 3, ...) and ν<i>i</i> (<i>i</i>=1, 2, 3, ...) are a refractive index and an Abbe number to the d-line of an <i>i</i>th lens counted from the object surface (in the fourth embodiment, the refractive indices and Abbe numbers of lenses L11F and L11R are represented by N11F, N11R, ν11F and ν11R). f is a focal length of the entire lens system. FNO is an F-number (in the first, second and fourth embodiments, focal lengths f at the shortest focal length condition (W) and at the longest focal length condition (T) and corresponding F-numbers are shown).

0064In each table, the surfaces marked with asterisks are aspherical and defined by the following expression representative of the surface configuration of an aspherical surface: <maths id="math0096" num=""><math display="block"><mi>X</mi><mo>=</mo><mfrac><mrow><mi>C</mi><mo>⋅</mo><msup><mi>Y</mi><mn>2</mn></msup></mrow><mrow><mn>1</mn><mo>+</mo><msup><mfenced separators=""><mn>1</mn><mo>-</mo><mi mathvariant="normal">ε</mi><mo>⋅</mo><msup><mi>Y</mi><mn>2</mn></msup><mo>⋅</mo><msup><mi>C</mi><mn>2</mn></msup></mfenced><mfrac><mi mathvariant="normal">l</mi><mn>2</mn></mfrac></msup></mrow></mfrac><mo>+</mo><mstyle displaystyle="false"><mstyle displaystyle="true"><munder><mo>∑</mo><mi>i</mi></munder></mstyle><mi mathvariant="italic">Ai</mi><mo>⁢</mo><msup><mi>Y</mi><mi>i</mi></msup></mstyle></math><img file="EP0707229A2_D0096.tif" /></maths> where X is an amount of displacement from the reference surface along the optical axis, Y is a height in a direction vertical to the optical axis, C is a paraxial curvature, ε is a conic constant, and A<i>i</i> is an <i>i</i>th aspherical coefficient.

0065<figref idref="f0001">Figs. 1</figref>, <figref idref="f0011">11</figref>, <figref idref="f0021">21</figref> and <figref idref="f0027">27</figref> show the lens arrangements of the first to fourth embodiments. The first embodiment (<figref idref="f0001">Fig. 1</figref>), the second embodiment (<figref idref="f0011">Fig. 11</figref>) and the fourth embodiment (<figref idref="f0027">Fig. 27</figref>) are zoom lens systems, and the third embodiment (<figref idref="f0021">Fig. 21</figref>) is a fixed focal length telephoto lens system (in each figure, AX represent the optical axis). With respect to the zoom lens systems, <figref idref="f0001">Figs. 1</figref>, <figref idref="f0011">11</figref> and <figref idref="f0027">27</figref> show the lens arrangements at the shortest focal length condition (W). Arrows m1, m2, m3 and m4 in the figures schematically show the movements of a first lens unit Gr1, a second lens unit Gr2, a third lens unit Gr3 and a fourth lens unit Gr4 from the shortest focal length condition (W) to the longest focal length condition (T), respectively.

0066The first to third embodiments are embodiments according to the first implementation and are optical systems including a compensating lens unit GrA capable of decentering for camera shake compensation and a lens (hereinafter, referred to as "non-decentering lens") N located closer to the image than the compensating lens unit GrA and not decentering in camera shake compensation, and an aperture stop S. In the compensating lens unit GrA, an aspherical surface is formed. In the non-decentering lens N not decentering in camera shake compensation, an aspherical surface is formed which tends to counteract the aspherical effect of the aspherical surface formed in the compensating lens unit GrA. In addition, the previously-described conditions (1) and (2) are fulfilled as shown in Table 6.

0067The first, second and fourth embodiments are embodiments according to the second implementation and are zoom lens systems including four lens units Gr1 to Gr4 and performing zooming by varying the distances between the lens units. Further, the following are included: a compensating lens unit GrA located in a lens unit (i.e. the second lens unit Gr2 or the third lens unit Gr3 or the fourth lens unit Gr4) other than the most object side lens unit (i.e. the first lens unit Gr1) and capable of decentering for camera shake compensation; and a lens N located closer to the image than the compensating lens unit GrA. In the compensating lens unit GrA, an aspherical surface is formed. In the lens N not decentering in camera shake compensation, an aspherical surface is formed which tends to counteract the aspherical effect of the aspherical surface formed in the compensating lens unit GrA.

[Description of the first embodiment]

0068The first embodiment comprises from the object side: <ul id="ul0004" list-style="none" compact="compact"><li>a first lens unit Gr1 including a negative meniscus lens element concave to the image side, a positive bi-convex lens element, and a positive meniscus lens element convex to the object side;</li><li>a second lens unit Gr2 including a doublet lens element (hatched portion; this constitutes the compensating lens unit GrA) consisting of a positive meniscus lens element convex to the image side and a negative bi-concave lens element (whose image side surface is aspherical), and a positive bi-convex lens element (this constitutes the non-decentering lens N and its object side surface is aspherical);</li><li>a third lens unit Gr3 including an aperture stop S, a negative meniscus lens element concave to the image side, and two positive bi-convex lens elements; and</li><li>a fourth lens unit Gr4 including a negative meniscus lens element concave to the image side, and a doublet lens element consisting of a positive meniscus lens element convex to the image side and a negative meniscus lens element concave to the object side.</li></ul>

0069The first and second embodiments are four-unit zoom lens systems of positive, negative, positive, negative configuration which are new and very compact as telephoto type lens systems. In the first embodiment, in the four-unit zoom lens system of positive, negative, positive, negative configuration, the negative doublet lens element in the negative second lens unit Gr2 is used as the compensating lens unit GrA and its image side surface is aspherical. The displacement direction of the aspherical surface relative to the reference surface is a direction such that the surface displaces toward the object side from the optical axis to the edge along the height. Therefore, the aspherical surface of the compensating lens unit GrA is a positive aspherical surface where the negative power decreases (compared to spherical surfaces) from the center to the edge along the height.

0070The aspherical surface for counteracting the effect of the aspherical surface of the compensating lens unit GrA is formed on the object side of the positive bi-convex lens element of the second lens unit Gr2 which is located right on the image side of the negative doublet lens element. The aspherical surface is a negative aspherical surface having a displacement in a direction to counteract the effect of the aspherical surface of the compensating lens unit GrA. Since the bi-convex lens element used as the non-decentering lens N is arranged in the same lens unit as the lens unit in which the doublet lens element used as the compensating lens unit GrA is arranged, the distance between the compensating lens unit GrA and the nondecentering lens N does not vary during zooming. Further, the compensating lens unit GrA and the non-decentering lens N adjoin each other and their aspherical surfaces face each other and are in a closest positional relationship. Therefore, the decentering aberrations in camera shake compensation can be excellently corrected, and by the aspherical surface of the bi-convex lens element, the aberrations in the normal condition are returned to excellent condition in the entire zoom range.

0071The reason why a doublet lens element consisting of two lens elements is used as the compensating lens unit GrA is that if the compensating lens unit GrA includes only one negative lens element, axial lateral chromatic aberration is largely generated due to the decentering in camera shake compensation. Therefore, in order to restrain the axial lateral chromatic aberration, the camera shake compensating lens unit GrA necessarily includes a plurality of lens elements. However, if the number of lens elements of the compensating lens unit GrA is too large, its weight increases, so that a large driving mechanism is necessary to quickly decenter the compensating lens unit GrA for camera shake compensation. Therefore, it is the most desirable that the compensating lens unit GrA should include two lens elements.

0072When the compensating lens unit GrA includes two lens elements (including a doublet lens element) and has a negative refractive power like in the first embodiment, it is preferable that the compensating lens unit GrA includes a positive lens element with an Abbe number of ν_d<35 and a negative lens element with an Abbe number of ν_d>40.

[Description of the second embodiment]

0073The second embodiment comprises from the object side: <ul id="ul0005" list-style="none" compact="compact"><li>a first lens unit Gr1 including a negative meniscus lens element concave to the image side, a positive bi-convex lens element, and a positive meniscus lens element convex to the object side;</li><li>a second lens unit Gr2 including a negative bi-concave lens element, a positive bi-convex lens element (hatched portion; this constitutes the compensating lens unit GrA and its image side surface is aspherical), and a positive meniscus lens element convex to the object side (this constitutes the non-decentering lens N and its object side surface is aspherical);</li><li>a third lens unit Gr3 including an aperture stop S, a negative meniscus lens element concave to the image side, and two positive bi-convex lens elements; and</li><li>a fourth lens unit Gr4 including a negative bi-concave lens element, and a doublet lens element consisting of a positive meniscus lens element convex to the image side and a negative meniscus lens element concave to the object side.</li></ul>

0074In the second embodiment, the second lens unit (from the object side, this lens unit includes a negative lens element and a positive lens element) of the basic four-unit zoom lens system of positive, negative, positive, negative configuration is used as the compensating lens unit GrA and the image side surface of the positive lens element in the negative second lens unit is aspherical, so that the generation of axial coma in camera shake compensation is restrained to be small.

0075For example, if the aspherical surface of the non-de-centering lens N is formed in the positive third lens unit Gr3, the aspherical surface is largely away from the aspherical surface of the second lens unit Gr2 used as the compensating lens unit GrA and the distance d_ASP between the two aspherical surfaces relatively varies during zooming, so that it is impossible to return the aberrations in the normal condition to excellent condition in the entire zoom range. In this embodiment, in order to return the aberrations in the normal condition to excellent condition, a non-decentering lens N which hardly has any power is arranged closer to the image than the compensating lens unit GrA of the negative second lens unit Gr2 and an aspherical surface is formed therein.

0076Since the non-decentering lens N adjoin the compensating lens unit GrA and moves simultaneously therewith, in the second embodiment, the optimum power arrangement is changed if the non-decentering lens N has a high power. Therefore, in the lens arrangement like that of the second embodiment, the following condition (5) is preferably fulfilled: <maths id="math0097" num="(5)"><math display="block"><mfenced open="|" close="|"><mfrac><msub><mi mathvariant="normal">φ</mi><mi>N</mi></msub><msub><mi mathvariant="normal">φ</mi><mi>A</mi></msub></mfrac></mfenced><mo>&lt;</mo><mn>0.35</mn></math><img file="EP0707229A2_D0097.tif" /></maths> where φ_N is the power of the non-decentering lens N and φ_A is the power of the compensating lens unit GrA.

[Description common to the first and second embodiments]

0077In both the first and second embodiments, the power φ_A of the compensating lens unit GrA relates to how much the compensating lens unit GrA is decentered to perform a camera shake compensation of a predetermined angle. If the power φ_A of the compensating lens unit GrA is high, camera shake compensation is made with a large compensation angle θ only by slightly de-centering the compensating lens unit GrA. In that case, however, since it is necessary to decenter the compensating lens unit GrA with an extremely high accuracy, the cost of the mechanism for driving the compensating lens unit GrA increases. Conversely, if the power φ_A of the compensating lens unit GrA is weak, although less accuracy is required for the decentering, the decentering amount of the compensating lens unit GrA necessary for a compensation of a predetermined angle increases, so that the sizes of the compensating lens unit and its driving mechanism both increase. Therefore, with respect to the power φ_A of the compensating lens unit GrA, the following condition (6) is preferably fulfilled: <maths id="math0098" num="(6)"><math display="block"><mn>1.5</mn><mo>&lt;</mo><mfenced open="|" close="|"><mfrac><msub><mi mathvariant="normal">φ</mi><mi>A</mi></msub><msub><mi mathvariant="normal">φ</mi><mi>T</mi></msub></mfrac></mfenced><mo>&lt;</mo><mn>12.5</mn></math><img file="EP0707229A2_D0098.tif" /></maths> where φ_T is the power of the entire lens system at the longest focal length condition.

0078Here, a case will be examined where the camera shake compensating system is employed for the four-unit telephoto zoom lens system of positive, negative, positive, negative configuration disclosed in Japanese Laid-open Patent Application No. H1-197713. The negative second lens unit of this zoom lens system comprises from the object side a negative lens element and a positive lens element. When this lens unit is used as the compensating lens unit GrA, the decentering aberrations other than axial coma generated in camera shake compensation are excellently corrected (1) when only the negative lens element is used and (2) when both the negative and positive lens elements are used. However, in a camera shake compensation of a compensation angle θ of 0.7°, axial coma is 240µm in the case of (1) and 320µm in the case of (2), which are ten or more times as large as the permissible aberration amount.

[Description of the third embodiment]

0079The third embodiment comprises from the object side two positive bi-convex lens element, a negative bi-concave lens element, a negative meniscus lens element concave to the object side, a positive bi-convex lens element, a doublet lens element consisting of a positive meniscus lens element convex to the image side and a negative bi-concave lens element, a doublet lens element (hatched portion; this constitutes the compensating lens unit GrA) consisting of a positive meniscus lens element convex to the image side and a negative bi-concave lens element (whose image side surface is aspherical), a negative meniscus lens element concave to the image side (this constitutes the non-decentering lens N and its object side surface is aspherical), an aperture stop S, and a doublet lens element consisting of a negative meniscus lens element concave to the image side and a positive bi-convex lens element.

0080Although the third embodiment is a fixed focal length lens system, the workings, associated with camera shake compensation, of the compensating lens unit GrA, the non-decentering lens N and their aspherical surfaces are similar to those of the first and second embodiments. Specifically, when the image side doublet lens element is used as the compensating lens unit GrA as described above, of the decentering aberrations generated in camera shake compensation, axial coma is very largely generated. This axial coma is restrained by using an aspherical surface as the image side surface of the doublet lens element, arranging in close vicinity thereto the non-decentering lens N which hardly has any power to counteract the effect of the aspherical surface of the doublet lens element and forming an aspherical surface in the non-decentering lens N. Further, in this embodiment, by using a doublet lens element consisting of two lens elements as the compensating lens unit GrA, the generation of axial lateral chromatic aberration is restrained in camera shake compensation.

[Description of the fourth embodiment]

0081The fourth embodiment comprises from the object side: <ul id="ul0006" list-style="none" compact="compact"><li>a first lens unit Gr1 including a doublet lens element consisting of a negative meniscus lens element concave to the image side and a positive bi-convex lens element, and a positive meniscus lens element convex to the object side;</li><li>a second lens unit Gr2 including a negative meniscus lens element concave to the image side (whose object side surface is aspherical), a doublet lens element consisting of a negative bi-concave lens element and a positive lens element convex to the object side, and a positive meniscus lens element convex to the object side;</li><li>a third lens unit Gr3 including an aperture stop S, a positive bi-convex lens element, a doublet lens element consisting of a positive bi-convex lens element and a negative bi-concave lens element, and a light restricting plate S'; and</li><li>a fourth lens unit Gr4 including a piano-convex lens element L11F convex to the object side (hatched portion; this constitutes the compensating lens unit GrA and its image side surface is aspherical), a plano-convex lens element L11R convex to the image side (this constitutes the non-decentering lens N and its object side surface is aspherical), and a negative meniscus lens element L12 concave to the object side (whose object side surface is aspherical).</li></ul>

0082The fourth embodiment is a four-unit high-magnification standard zoom lens system of positive, negative, positive, positive configuration. The aperture S is included in the third lens unit Gr3 of a positive power. The lens diameters of the lens units Gr2 to Gr4 other than the first lens unit Gr1 are all small. The working of the fourth embodiment associated with camera shake compensation is also similar to the first to third embodiments. Specifically, by dividing the positive element of the positive fourth lens unit Gr4 into the lens element L11F and the lens element L11R from the object side to use the lens L11F as the compensating lens unit GrA and the lens L11R as the non-decentering lens N and by using aspherical surfaces which counteract each other as the image side surface of the lens L11F and as the object side surface of the lens L11R, axial coma on the longer focal length side is reduced to excellently correct the decentering aberrations in camera shake compensation.

0083Since, simply by dividing the positive lens element of the fourth lens unit Gr4 into the lens elements L11F and L11R to use the lens L11F as the compensating lens unit GrA, axial coma is very large, the decentering aberrations generated in camera shake compensation cannot be excellently corrected. <figref idref="f0037">Figs. 37</figref> and <figref idref="f0038">38</figref> show lateral aberrations generated in camera shake compensation when no aspherical surfaces are formed on the lens elements L11F and L11R.

0084In the fourth embodiment, the compensating lens unit GrA includes only one lens element unlike the first and second embodiments. This is because the lateral chromatic aberration generated in camera shake compensation causes no problem since the fourth embodiment is a standard zoom lens system whose longest focal length is at most 103mm. Still, it is necessary to reduce the lateral chromatic aberration to some degree. To do so, it is preferable that the Abbe number ν_d of the compensating lens unit GrA is 50 or more.

0085<figref idref="f0002">Figs. 2A to 2F</figref>, <figref idref="f0012">12A to 12F</figref>, <figref idref="f0022">22A to 22C</figref> and <figref idref="f0028">28A to 28F</figref> show longitudinal aberrations of the first to fourth embodiments in the normal condition (pre-decentering condition). <figref idref="f0002">Figs. 2A to 2C</figref>, <figref idref="f0012">12A to 12C</figref> and <figref idref="f0028">28A to 28C</figref> show aberrations at the shortest focal length condition. <figref idref="f0002">Figs. 2D to 2F</figref>, <figref idref="f0012">12D to 12F</figref> and <figref idref="f0028">28D to 28F</figref> show aberrations at the longest focal length condition. The solid line d represents aberration to the d-line, the broken line SC represents sine condition. The broken line DM and the solid line DS show astigmatisms to the d-line on the meridional and sagittal image planes, respectively.

0086<figref idref="f0003">Figs. 3A to 3B</figref>, <figref idref="f0004">4A to 4C</figref>, <figref idref="f0005">5A to 5B</figref>, <figref idref="f0006">6A to 6C</figref>, <figref idref="f0007">7A to 7B</figref>, <figref idref="f0008">8A to 8C</figref>, <figref idref="f0009">9A to 9B</figref> and <figref idref="f0010">10A to 10C</figref> show lateral aberrations of the compensating lens unit GrA of the first embodiment before and after decentering. <figref idref="f0013">Figs. 13A to 13B</figref>, <figref idref="f0014">14A to 14C</figref>, <figref idref="f0015">15A to 15B</figref>, <figref idref="f0016">16A to 16C</figref>, <figref idref="f0017">17A to 17B</figref>, <figref idref="f0018">18A to 18C</figref>, <figref idref="f0019">19A to 19B</figref> and <figref idref="f0020">20A to 20C</figref> show lateral aberrations of the compensating lens unit GrA of the second embodiment before and after decentering. <figref idref="f0023">Figs. 23A to 23B</figref>, <figref idref="f0024">24A to 24C</figref>, <figref idref="f0025">25A to 25B</figref> and 26A to 26C show lateral aberrations of the compensating lens unit GrA of the third embodiment before and after decentering. <figref idref="f0029">Figs. 29A to 29B</figref>, <figref idref="f0030">30A to 30C</figref>, <figref idref="f0031">31A to 31B</figref>, <figref idref="f0032">32A to 32C</figref>, <figref idref="f0033">33A to 33B</figref>, <figref idref="f0034">34A to 34C</figref>, <figref idref="f0035">35A to 35B</figref> and <figref idref="f0036">36A to 36C</figref> show lateral aberrations of the compensating lens unit GrA of the fourth embodiment before and after de-centering. The aberrations of the compensating lens unit GrA after decentering are aberrations under a compensated condition where the compensating lens unit GrA is decentered at a camera shake compensation angle θ of 0.7°.

0087The values corresponding to the conditions (1) to (6) are shown in Table 6 with respect to the first to fourth embodiments.

0088As described above, according to the first to the fourth embodiments, since an aspherical surface is formed in the compensating lens unit GrA, the decentering aberrations such as axial coma are corrected, and since an aspherical surface which tends to counteract the aspherical effect of the aspherical surface of the compensating lens unit GrA is formed in the non-decentering lens N, aberrations generated in the normal condition are maintained excellent. Thus, aberrations are excellently corrected both in the normal condition and in the compensated condition. Further, according to the first to fourth embodiments, compared to the zoom lens system proposed by the previously-mentioned Japanese Laid-open Patent Application No. <patcit id="pcit0004" dnum="JPH6123836B"><text>H6-123836</text></patcit> (assuming that the camera shake compensating angles θ are the same), axial coma is restrained to a fraction, and when a large camera shake occurs, image degradation in compensation is very small.

0089Further, according to the first to third embodiments, since off-axial image point movement error is restrained to be small by fulfilling the condition (1) and the condition (2) is further fulfilled, the weight and size of the compensating lens unit GrA are reduced to reduce the size of the entire optical system. Since it is unnecessary for the driving mechanism to have a great power because of the size reduction of the optical system, the size of the entire lens barrel can be reduced.

0090According to the first, second and fourth embodiments, although aberrations are excellently corrected both in the normal and compensated conditions in the entire zoom range from the shortest focal length condition (W) to the longest focal length condition (T), since the compensating lens unit GrA is located in the lens unit Gr2 or Gr3 or Gr4 other than the first lens unit Gr1, not only the size of the compensating lens unit GrA but also the size of the driving mechanism necessary for its decentering can be reduced, so that the entire optical system having the camera shake compensating function can be reduced.

0091As described above, according to the optical systems having a camera shake compensating function of the first and second implementations, since an aspherical surface is formed in the compensating lens unit capable of decentering for camera shake compensation, decentering aberrations such as axial coma is corrected, and since an aspherical surface which tends to counteract the aspherical effect of the aspherical surface formed in the compensating lens unit is formed in a lens located closer to the image than the compensating lens unit and not decentering in camera shake compensation, aberrations in the normal condition are maintained excellent. Thus, aberrations are excellently corrected both in the normal condition and in the compensated condition.

0092Further, according to the optical system having a camera shake compensating function of the first implementation, since off-axial image point error is restrained to be small by fulfilling the condition (1) and the condition (2) is further fulfilled, the weight and size of the compensating lens unit can be reduced to reduce the size of the entire optical system. Since it is unnecessary for the driving mechanism to have a great power because of the reduction in size of the compensating lens unit, the size of the entire lens barrel can be reduced.

0093Moreover, according to the optical system having a camera shake compensating function of the second implementation, since the compensating lens unit is located in a lens unit other than the most object side lens unit, not only the size of the compensating lens unit but also the size of the driving mechanism necessary for the decentering can be small, so that the size of the entire optical system having a camera shake compensating function can be reduced.

0094Obviously, many modifications and variations of the present invention are possible in light of the above teachings. It is therefore to be understood that within the scope of the appended claims, the invention may be practiced other than as specifically described. <tables id="tabl0001" num="0001"><table frame="all"><title>TABLE 1</title><tgroup cols="3"><colspec colnum="1" colname="col1" colwidth="39mm" /><colspec colnum="2" colname="col2" colwidth="67mm" /><colspec colnum="3" colname="col3" colwidth="45mm" /><thead><row><entry align="center" valign="top">Refractive Index</entry><entry align="center" valign="top">Displacement Direction of Aspherical Surface</entry><entry align="center" valign="top">Type of Aspherical Surface</entry></row></thead><tbody><row><entry align="center">Object side > Image side</entry><entry align="center">Toward object side</entry><entry>Positive aspherical surface</entry></row><row><entry align="center">Object side < Image side</entry><entry align="center">Toward object side</entry><entry>Negative aspherical surface</entry></row><row><entry align="center">Object side > Image side</entry><entry align="center">Toward image side</entry><entry>Negative aspherical surface</entry></row><row><entry align="center">Object side < Image side</entry><entry align="center">Toward image side</entry><entry>Positive aspherical surface</entry></row></tbody></tgroup></table></tables><tables id="tabl0002" num="0002"><img file="EP0707229A2_D0099.tif" /></tables><tables id="tabl0003" num="0003"><img file="EP0707229A2_D0100.tif" /></tables><tables id="tabl0004" num="0004"><img file="EP0707229A2_D0101.tif" /></tables><tables id="tabl0005" num="0005"><img file="EP0707229A2_D0102.tif" /></tables><tables id="tabl0006" num="0006"><img file="EP0707229A2_D0103.tif" /></tables><tables id="tabl0007" num="0007"><img file="EP0707229A2_D0104.tif" /></tables><tables id="tabl0008" num="0008"><img file="EP0707229A2_D0105.tif" /></tables><tables id="tabl0009" num="0009"><img file="EP0707229A2_D0106.tif" /></tables><tables id="tabl0010" num="0010"><img file="EP0707229A2_D0107.tif" /></tables>