JP2924117B2 - Zoom lens - Google Patents

Description

Description: BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a compact zoom lens used for a lens shutter camera (hereinafter referred to as an LS camera).

2. Description of the Related Art In an LS camera with a built-in zoom lens, there has been a demand for a compact and low-cost photographing lens in order to achieve compactness and low cost. In order to make the lens system compact, including the amount of movement of the lens during zooming, it is necessary to increase the refractive power of each lens group. It can be said that the direction is to increase. On the other hand, it is effective to reduce the number of lenses in order to reduce the cost. As described above, the compactness and the cost reduction of the lens system include many contradictory elements.

By the way, in recent years, technological progress in plastic molding, glass molding and the like has been remarkable, and it has become possible to produce aspherical surfaces at low cost.

SUMMARY OF THE INVENTION In view of such circumstances, it is an object of the present invention to reduce the cost by configuring a compact zoom lens for an LS camera with a small number of lenses by effectively using an aspheric surface.

Means for Solving the Problems In order to reduce the size of the zoom lens,
It is necessary to shorten the overall length and further reduce the amount of movement. Especially
As a type having a short lens back, such as a zoom lens for an LS camera, a lens having two components of a first group having a positive refractive power and a second group having a negative refractive power is generally used.
In the present invention, three components of positive, negative, and positive, which allow higher magnification, are used. In zooming from the wide-angle end to the telephoto end, all the units move to the object side. At this time, the first lens group and the second
The distance between the lens groups increases, and the distance between the second lens group and the third lens group decreases.

By the way, in order to reduce the size of a positive, negative and positive three-component zoom lens as in the present invention and to secure a sufficient back focus, it is necessary to increase the refracting power of each lens unit, which tends to deteriorate the aberration. There is.

In the present invention, at least one lens has an aspherical surface on both sides in order to prevent this tendency. When a double-sided aspherical surface is used in the first lens unit, various aberrations that cannot be suppressed only by the front surface are corrected by the rear surface. For example, when the lens closest to the object side in the first lens group is aspherical on both sides, coma in the peripheral portion of the screen which cannot be suppressed only by the front side when the aspherical surface on one side is corrected on the rear side. Become. At this time, the rear surface also serves to correct spherical aberration. Further, the aspherical surface in the first lens group is also useful for correcting distortion. When a double-sided aspherical surface is used in the second lens group, spherical aberration that has been overcorrected on the front surface is corrected on the rear surface. In addition, the generation of higher-order coma that cannot be suppressed by the first lens group due to these aspheric surfaces is also prevented. Even when a double-sided aspherical surface is used in the third lens unit, aberrations that cannot be suppressed only by the front surface are corrected on the rear surface, as in the case of using a double-sided aspherical surface in the other lens units. For example, if the lens closest to the object in the third lens group is aspherical on both surfaces, coma in the peripheral portion of the screen that cannot be suppressed only by the front surface is corrected on the rear surface. The front surface at this time also corrects distortion near the wide-angle end.

By acting on these aspherical surfaces, the number of components of the lens system can be greatly reduced, and the overall length is 5 times smaller than in the past.
It became possible to shorten it by ~ 10mm.

Further, in the present invention, it is desirable to use at least three aspheric surfaces in the lens system in order to correct aberrations and obtain good performance while reducing the size of the lens system.

Hereinafter, effective aspherical shapes in the present invention will be described.

If the first lens group has an aspheric surface, at least one
It is desirable that the surface satisfies the following conditions.

When the maximum effective diameter of the aspheric surface is Ymax, 0.7Ymax <Y
<For any height Y in the optical axis direction of Ymax, Where φ ₁ : refractive power of the first lens group N: refractive index of the aspherical object side medium N ′: refractive index of the aspherical image side medium X (y): aspherical surface shape X ₀ (y) : Reference spherical shape of aspherical surface Condition (1) is a condition for correcting spherical aberration, coma, and flare. If the upper limit is exceeded, spherical aberration will be undercorrected or overcorrected over the entire zoom range, and inward coma and flare will occur.

More preferably, all aspheric surfaces in the first lens group should satisfy the following condition.

In the case of a shape in which the positive refractive power becomes weaker (the negative refractive power becomes stronger) as the aspherical surface becomes closer to the periphery.

When the maximum effective diameter of the aspherical surface is Ymax, 0 <Y <0.7Y
max for any height Y in the vertical direction of the optical axis When the value exceeds the upper limit of the condition (2), the spherical aberration of the orbicular zone has a large negative value, and a shift of the focus position due to the stop-down becomes a problem. On the other hand, if the lower limit is exceeded, the spherical aberration correcting effect on the annular light flux becomes excessive, and it becomes difficult to correct other aberrations and spherical aberration in a well-balanced manner. (In this case, the spherical aberration tends to have a wavy shape.) In the case of a shape in which the negative refractive power becomes weaker (positive refractive power becomes stronger) as the aspherical surface becomes closer to the periphery.

When the maximum effective form of the aspheric surface is Ymax, 0 <Y <0.07
For any height Y in the vertical direction of the optical axis of Ymax When the value goes below the lower limit of the condition (3), the orbicular spherical aberration has a large value of loss, and there is a problem that the focus position shifts due to the aperture stop. On the other hand, if the upper limit is exceeded, the spherical aberration correction effect on the annular luminous flux becomes excessive, and it becomes difficult to correct other aberrations and spherical aberration in a well-balanced manner. (In this case, the spherical aberration tends to have a wavy shape.) Further, when a double-sided aspherical lens is used in the first lens group, the front surface thereof satisfies the following condition (4) or (6). It is desirable that the rear surface satisfies the following condition (5) or (7).

In the case of a shape in which the positive refractive power becomes weaker (the negative refractive power becomes stronger) as the front aspherical surface becomes closer to the periphery.

When the maximum effective diameter of the front aspheric surface is Ymax, 0.7Yma
x <Y <Ymax for any vertical height Y in the optical axis direction When the maximum effective diameter of the rear aspheric surface is Ymax, 0.7Yma
x <Y <Ymax for any vertical height Y in the optical axis direction In the first lens group, the conditional expression (4) means that the positive refractive power becomes weaker (negative refractive power becomes stronger) near the aspherical surface. If the lower limit is exceeded, the fall of spherical aberration to under cannot be corrected to the over side in the range of the third-order aberration region. By the way, the on-axis luminous flux passing through a place far from the optical axis of the lens has excessively corrected spherical aberration and falls to the over side. Therefore, in order to return this light beam to the under side, an aspherical surface having a negative refractive power weaker (positive refractive power stronger) as the periphery satisfies conditional expression (5) may be introduced to the rear surface.

These aspheric surfaces also prevent the occurrence of coma aberration. For example, when the lower limit of the condition (4) is exceeded, the lower light portion of the off-axis lateral aberration hangs down, and the inward chromatic aberration occurs. Frames occur. Desirably, the deviation amount of the aspherical surface on the side satisfying the conditional expression (4) from the reference spherical surface is larger than that on the side satisfying the conditional expression (5).

In the case of a shape in which the positive refractive power becomes stronger (the negative refractive power becomes weaker) as the front aspheric surface becomes closer to the periphery.

When the maximum effective diameter of the front aspheric surface is Ymax, 0.7Yma
x <Y <Ymax for any vertical height Y in the optical axis direction When the maximum effective diameter of the rear aspheric surface is Ymax, 0.7Yma
x <Y <Ymax for any vertical height Y in the optical axis direction In the first lens group, it means that the negative refractive power becomes weaker (positive refractive power becomes stronger) as the aspherical surface satisfies the conditional expression (6) becomes closer to the periphery. If the upper limit is exceeded, it is not possible to correct the over-fall of spherical aberration to the under side in the range of the third-order aberration region. By the way, with respect to an on-axis light beam passing through a place far from the optical axis of the lens, the spherical aberration is excessively corrected and falls to the under side. Therefore, in order to return this light beam to the over side, an aspherical surface having a negative refractive power stronger (positive refractive power weaker) as the periphery thereof satisfies conditional expression (7) may be introduced to the rear surface.

In addition, these aspheric surfaces also prevent the occurrence of flare. For example, when the upper limit of conditional expression (6) is exceeded, the lower light portion of off-axis lateral aberration jumps upward, and flare occurs. I will.

Desirably, the deviation amount of the aspherical surface on the side satisfying the conditional expression (7) from the reference spherical surface is larger than that on the side satisfying the conditional expression (6).

Further, when a lens having both aspheric surfaces is used in the first lens group, it is desirable that the lens satisfy the following conditions.

However, d _DSASP1 : the core thickness of a double-sided aspherical lens H _DSASP1 : the effective optical path diameter of a double-sided aspherical lens This is a condition that defines the core thickness of a lens when a double-sided aspherical surface is used in the first lens group. If the lower limit is exceeded, the position (length) through which light passes between the front and rear surfaces of the lens is almost the same, so that the aberration correction effect on the rear surface, especially for off-axis light, is almost eliminated. (It is meaningless to use two-sided aspheric surfaces.) If the upper limit is exceeded, the core thickness of the lens becomes too large, and it becomes difficult to manufacture the lens.

If the second lens group has an aspheric surface, at least one
It is desirable that the surface satisfies the following conditions.

When the maximum effective diameter of the aspheric surface is Ymax, 0.7Ymax <Y
<For any height Y in the vertical direction of the optical axis of Ymax Here, φ ₂ : refractive power of the second lens group Condition (9) is a condition for correcting spherical aberration. If the upper limit is exceeded, spherical aberration will be undercorrected or overcorrected over the entire zoom range.

More preferably, all aspheric surfaces in the second lens group should satisfy the following condition.

In the case of a shape in which the positive refractive power becomes weaker (the negative refractive power becomes stronger) as the aspherical surface becomes closer to the periphery.

When the maximum effective diameter of the aspherical surface is Ymax, 0 <Y <0.7Y
max for any height Y in the vertical direction of the optical axis When the value exceeds the upper limit of the condition (10), the orbicular spherical aberration has a large negative value, and there is a problem of a shift of the focus position due to the aperture stop. On the other hand, if the lower limit is exceeded, the spherical aberration correction effect on the annular luminous flux becomes excessive, and it becomes difficult to correct other various aberrations and the spherical aberration in a well-balanced manner. (In this case, the spherical aberration tends to have a wavy shape.) In the case of a shape in which the negative refractive power becomes weaker (positive correction power becomes stronger) as the aspherical surface becomes closer.

When the maximum effective diameter of the aspherical surface is Ymax, 0 <Y <0.7Y
max for any height Y in the vertical direction of the optical axis When the value goes below the lower limit of the condition (11), the orbicular spherical aberration has a large negative value, and there is a problem of a shift of the focus position due to the aperture stop. On the other hand, if the upper limit is exceeded, the spherical aberration correction effect on the annular luminous flux becomes excessive, and it becomes difficult to correct other aberrations and spherical aberration in a well-balanced manner. (In this case, the spherical aberration tends to have a wavy shape.) When a double-sided aspherical lens is used in the second lens group, the front surface satisfies the following condition (12) or (14). It is desirable that the rear surface satisfies the following condition (13) or (15).

In the case of a shape in which the positive refractive power becomes weaker (the negative refractive power becomes stronger) as the front aspherical surface becomes closer to the periphery.

When the maximum effective diameter of the front aspheric surface is Ymax, 0.7Yma
x <Y <Ymax for any vertical height Y in the optical axis direction When the maximum effective diameter of the rear aspheric surface is Ymax, 0.7Yma
x <Y <Ymax for any vertical height Y in the optical axis direction In the second lens group, the conditional expression (12) means that the positive refractive power becomes weaker (the negative refractive power becomes stronger) near the aspherical surface. If the lower limit is exceeded, the fall of spherical aberration to under cannot be corrected to the over side in the range of the third-order aberration region. By the way, the on-axis luminous flux passing through a place far from the optical axis of the lens has excessively corrected spherical aberration and falls to the over side. Therefore, in order to return this light beam to the under side, a conditional expression (1
It is sufficient to introduce an aspherical surface such that the negative refractive power becomes weaker (positive refractive power becomes stronger) as the periphery satisfies 3).

These aspheric surfaces also prevent the occurrence of higher-order coma aberration that cannot be suppressed by the first lens unit. For example, if the lower limit of the condition (12) is exceeded, off-axis peripheral coma and annular zone coma Becomes large, and the lateral aberration tends to be wavy.

In the case of a shape in which the positive refractive power becomes stronger (the negative refractive power becomes weaker) as the front aspheric surface becomes closer to the periphery.

When the maximum effective diameter of the front aspheric surface is Ymax, 0.7Yma
x <Y <Ymax for any vertical height Y in the optical axis direction When the maximum effective diameter of the rear aspheric surface is Ymax, 0.7Yma
x <Y <Ymax for any vertical height Y in the optical axis direction In the second lens group, the conditional expression (14) means that the positive refractive power becomes stronger (the negative refractive power becomes weaker) near the aspherical surface. If the upper limit is exceeded, it is not possible to make a correction such that the overshoot of the spherical aberration falls to the underside in the range of the third-order aberration region. By the way, an on-axis luminous flux passing through a place far from the optical axis of the lens is overcorrected and falls down to the underside. Therefore,
In order to return this light beam to the over side, an aspherical surface may be introduced into the rear surface such that the negative refractive power becomes stronger (positive refractive power becomes weaker) as the periphery satisfies conditional expression (15).

These aspheric surfaces also prevent the occurrence of higher-order coma aberration that cannot be suppressed by the first lens unit. For example, if the lower limit of the condition (12) is exceeded, off-axis peripheral coma and annular zone coma And the lateral aberration tends to be wavy.

Further, when a lens having both aspheric surfaces is used in the second lens group, it is desirable that the lens satisfies the following conditions.

However, d _DSASP2 : Core thickness of double-sided aspherical lens H _DSASP2 : Effective optical path diameter of double-sided aspherical lens This is a condition that defines the core thickness of the lens when a double-sided aspherical surface is used in the second lens group. If the lower limit is exceeded, the positions (heights) through which the light passes on the front and rear surfaces of the lens are almost the same, so that the aberration correction effect on the rear surface, especially for off-axis light, is almost negligible. (It is meaningless to use two-sided aspheric surfaces.) If the upper limit is exceeded, the core thickness of the lens becomes too large, and it becomes difficult to manufacture the lens.

If the third lens group has an aspheric surface, at least one
It is desirable that the surface satisfies the following conditions.

When the maximum effective diameter of the aspheric surface is Ymax, 0.8Ymax <Y
<For any height Y in the vertical direction of the optical axis of Ymax Here, φ ₃ : refractive power of the third lens unit Condition (17) is a condition for correcting distortion and field curvature in a well-balanced manner. Beyond this upper limit, the distortion at the wide-angle end takes a large positive value,
The tendency of the image surface to curve in the negative direction over the entire zoom range becomes significant.

More preferably, all aspheric surfaces in the third lens group should satisfy the following condition.

In the case of a shape in which negative refractive power becomes weaker (positive refractive power becomes stronger) as the aspherical surface becomes closer to the periphery.

When the maximum effective diameter of the aspherical surface is Ymax, 0 <Y <0.8Y
max for any height Y in the vertical direction of the optical axis If the condition (18) is exceeded, the positive distortion and the positive shift of the field curvature become large in the intermediate angle of view band from the wide-angle end to the intermediate focal length region. On the other hand, if the lower limit is exceeded, the negative distortion becomes large from the intermediate focal point to the telephoto end. In addition, the negative shift of the field curvature becomes remarkable in the entire zoom range.

In the case of a shape in which the positive refractive power becomes weaker (the negative refractive power becomes stronger) as the aspherical surface becomes closer.

When the maximum effective diameter of the aspherical surface is Ymax, 0 <Y <0.8Y
max for any height Y in the vertical direction of the optical axis When the value goes below the lower limit of the condition (19), the positive distortion and the positive shift of the field curvature become large in the intermediate angle of view band from the wide-angle end to the intermediate focal length region. If the upper limit is exceeded, the negative distortion becomes large in the range from the intermediate focal point to the telephoto end, and the negative shift of the field curvature becomes remarkable in the entire zoom range.

Further, when a double-sided aspheric lens is used in the third lens group, the front surface satisfies the lower condition (1) or (3), and the rear surface satisfies the lower condition (2) or (4). desirable.

In the case of a shape in which the negative refractive power becomes weaker (positive refractive power becomes stronger) as the front aspherical surface becomes closer to the periphery.

When the maximum effective diameter of the front aspheric surface is Ymax, 0.8Yma
x <Y <Ymax for any vertical height Y in the optical axis direction When the maximum effective diameter of the rear aspheric surface is Ymax, 0.8Yma
x <Y <Ymax for any vertical height Y in the optical axis direction In the third lens group, the conditional expression (20) means that the negative refractive power becomes weaker (positive refractive power becomes stronger) near the aspherical surface. If the lower limit is exceeded, distortion near the wide-angle end will increase, and the field curvature will fall to the under side. In addition, by using an aspheric surface that satisfies the condition (21) for the rear surface, the field curvature that cannot be suppressed only by the front surface is favorably corrected.

In the case of a shape in which negative refractive power becomes stronger (positive refractive power becomes weaker) near the front aspherical surface.

When the maximum effective diameter of the front aspheric surface is Ymax, 0.8Yma
x <Y <Ymax for any vertical height Y in the optical axis direction When the maximum effective diameter of the rear aspheric surface is Ymax, 0.8Yma
x <Y <Ymax for any vertical height Y in the optical axis direction In the third lens group, the conditional expression (22) means that the negative refractive power becomes stronger (the positive refractive power becomes weaker) near the aspherical surface. If the value exceeds the upper limit, the field curvature cannot be prevented from falling to the over side. In addition, by providing an aspherical surface that satisfies the condition (23) on the rear surface, the field curvature that cannot be suppressed only by the front surface is favorably corrected.

Further, when a lens having both aspheric surfaces is used in the third lens group, it is desirable that the lens satisfies the following conditions.

However, d _DSASP3 : Core thickness of a double-sided aspherical lens H _DSASP3 : Effective optical path diameter of a double-sided aspherical lens This is a condition that defines the core thickness of a lens when a double-sided aspherical surface is used in the third lens group. If the lower limit is exceeded, the positions (heights) through which light passes between the front and rear surfaces of the lens are almost the same, so that the aberration correction effect on the rear surface, especially for off-axis light, is almost eliminated. (It is meaningless to use two-sided aspheric surfaces.) If the upper limit is exceeded, the core thickness of the lens becomes too large, and it becomes difficult to manufacture the lens.

Although the aspherical shape has been described above, it is desirable that the first lens group and the third lens group are configured to satisfy the following conditions.

Where φ _W : refractive power of the whole system at the wide-angle end φ _T : refractive power of the whole system at the telephoto end β: zoom ratio (β = φ _T / φ _W ) Conditions (25) and (26) are the total length of the lens system, This is a condition for maintaining a good balance between the amount of movement for zooming, the back focus, and the state of correction of various aberrations.

When the value goes below the lower limit of the condition (25), the refractive power of the first lens unit becomes too strong, and it becomes difficult to maintain the back focus at an appropriate value (15% of the focal length at the wide-angle end) at the wide-angle end.
This will increase the diameter of the subsequent lens group. If the upper limit is exceeded, the amount of movement of each group due to zooming becomes excessive, which is disadvantageous in the lens barrel configuration.

When the value exceeds the lower limit of the condition (26), the Petzval sum takes a large value, the image plane remarkably falls in the positive direction, and the distortion at the wide-angle end takes a large positive value. On the other hand, if the upper limit is exceeded, it is necessary to increase the change in the distance between the second and third lens groups due to zooming, so that the second and third lens groups are far apart at the wide-angle end, thereby increasing the total lens length.

 It is also effective to satisfy the following conditions.

The condition (27) defines the ratio between the refractive power of the entire system and the refractive power of the first lens unit at the wide-angle end. Even if a spherical surface is used, it is difficult to correct various aberrations generated there, particularly spherical aberration. On the other hand, if the lower limit is exceeded, the tendency for inward coma to occur around the screen becomes remarkable.

The condition (28) defines the ratio between the refractive power of the entire system at the wide-angle end and the refractive power of the third lens group. If the upper limit of this condition is exceeded, the refractive power of the third lens group becomes excessive, and Even if an aspherical surface is used, it is difficult to correct various aberrations generated there, particularly curvature of field and distortion. On the other hand, if the lower limit is exceeded, an inward coma tends to occur around the screen, and it becomes difficult to secure a sufficient back focus. The aspherical shape X (y) and the reference spherical shape X ₀ (y) are defined by the following equations, respectively.

(Examples) Hereinafter, Examples 1 to 11 of the present invention are shown in Tables 1 to 11, respectively.

Here, f: focal length of the entire system, F: open F number, ri (i = 1, 2, 3,...): Radius of curvature of the i-th lens surface from the object side, di (i = 1, 2, 3,...): I-th axial top surface distance from the object side, Ni (i = 1, 2, 3,...): D-line of the i-th lens from the object side Refractive index, νi (i = 1, 2, 3,...): Abbe number of the i-th lens from the object side.

Also, ri with an asterisk indicates that the i-th surface from the object side is formed of an aspheric surface.

 Tables 12 and 13 show the values of each example for each condition.

 Table 14 shows the number of aspheric surfaces used in each example.

[Brief description of the drawings]

1 to 11 are lens configuration diagrams corresponding to Examples 1 to 11 of the present invention, respectively. 12 to 22 are aberration diagrams respectively corresponding to Examples 1 to 12 of the present invention.

────────────────────────────────────────────────── ─── Continuing from the front page Examiner Masaaki Moriuchi (56) References JP-A-1-230013 (JP, A) JP-A 1-225216 (JP, A) JP-A 1-225217 (JP, A) JP-A-3-45916 (JP, A) JP-A-2-16515 (JP, A) JP-A-2-16516 (JP, A) JP-A-2-135312 (JP, A) (58) Int.Cl. ⁶ , DB name) G02B 9/00-17/08 G02B 21/02-21/04 G02B 25/00-25/04

Claims (4)

(57) [Claims] 1. A first lens having a positive refractive power in order from the object side.
It consists of three components, a lens group, a second lens group having a positive refractive power, and a third lens group having a negative refractive power. The focal length of the entire system is changed by changing the air spacing between the lens groups. A zoom lens having a double-sided aspheric lens in the first lens group, wherein the aspheric surface in the first lens group satisfies the following conditional expression; When the diameter is y _max , 0.7y _max <y <
For any height y in the vertical direction of the optical axis of 1.0y _max , Where φ ₁ : refractive power of the first lens group, N: refractive index of the aspherical object side medium, N ′: refractive index of the aspherical image side medium, X (y): surface shape of the aspherical surface, X ₀ (y): Reference spherical shape of aspherical surface. 2. A first lens having a positive refractive power in order from the object side.
It consists of three components, a lens group, a second lens group having a positive refractive power, and a third lens group having a negative refractive power. The focal length of the entire system is changed by changing the air spacing between the lens groups. A zoom lens having a double-sided aspheric lens in the second lens group, wherein the aspheric surface in the second lens group satisfies the following conditional expression; When the diameter is y _max , 0.7y _max <y <
For any height y in the vertical direction of the optical axis of 1.0y _max , Where φ ₂ : refractive power of the second lens group, N: refractive index of the aspherical object side medium, N ′: refractive index of the aspherical image side medium, X (y): surface shape of the aspherical surface, X ₀ (y): Reference spherical shape of aspherical surface. 3. A first lens having a positive refractive power in order from the object side.
It consists of three components, a lens group, a second lens group having a positive refractive power, and a third lens group having a negative refractive power. The focal length of the entire system is changed by changing the air spacing between the lens groups. In the zoom lens, the third lens group is composed of two lenses, the third lens group has a double-sided aspheric lens, and the aspheric surface in the third lens group satisfies the following conditional expression. satisfaction zoom characterized by a lens; when the maximum effective diameter of the aspherical surface and y _max, 0.8y _max <y <
For any height y in the vertical direction of the optical axis of 1.0y _max , Here, φ ₃ : refractive power of the third lens group, N: refractive index of the aspherical object side medium, N ′: refractive index of the aspherical image side medium, X (y): aspherical surface shape, X ₀ (y): Reference spherical shape of aspherical surface. 4. A first lens having a positive refractive power in order from the object side.
It consists of three components, a lens group, a second lens group having a positive refractive power, and a third lens group having a negative refractive power. The focal length of the entire system is changed by changing the air spacing between the lens groups. A zoom lens having an aperture on the image side of the second lens group, a double-sided aspheric lens in the third lens group, and an aspheric surface in the third lens group satisfying the following conditional expression: the zoom lens is characterized in that; when a y _max the maximum effective diameter of the aspherical surface, 0.8y _max <y <
For any height y in the vertical direction of the optical axis of 1.0y _max , Here, φ ₃ : refractive power of the third lens group, N: refractive index of the aspherical object side medium, N ′: refractive index of the aspherical image side medium, X (y): aspherical surface shape, X ₀ (y): Reference spherical shape of aspherical surface.