# US 4,392,724 A · Lens system capable of short distance photography

> Agents: this data is available as JSON from the patentgrep API and MCP server — see https://patentgrep.com/llms.txt

- Number: US 4,392,724 A
- Document ID: US4392724A
- Status: Granted
- Kind: A
- Application number: 06/226,011
- Filed: 1981-01-19
- Granted: 1983-07-12
- Earliest claimed: 1980-01-31
- Assignees: Nippon Kogaku K.K.
- Inventors: Yoshinari Hamanishi
- Web page: https://patentgrep.com/patent/US4392724A

## CPC

- G02B 15/22: with movable lens means specially adapted for focusing at close distances
- G02B: OPTICAL ELEMENTS, SYSTEMS OR APPARATUS

## Abstract

Compact lens systems capable of continuously effecting infinity photography to high magnification proximity photography and having a large aperture ratio and excellent imaging performance comprise, in order from the object side, a first convergent lens group, a second convergent lens group including a diaphragm, and a third divergent lens group. The first and second groups are movable toward the object side relative to the third group, while enlarging their spacing, when focusing is effected from infinity to a short distance.

## Claims

### Claim 1

A lens system capable of continuously effecting infinity photography to high magnification proximity photography, comprising:
a convergent lens group which is a first group positioned most adjacent to the object side;
a convergent lens group which is a second group positioned rearwardly of said first group;
a divergent lens group which is a third group positioned rearwardly of said second group; and
a diaphragm provided between said first group and said third group;
said first group and said second group being movable toward the object side relative to said third group while enlarging the spacing between said first and second groups when focusing is effected from infinity to a short distance, and the diaphragm being movable with the second group.

### Claim 2 (depends on 1)

A lens system according to claim 1, satisfying the following conditions:
1.1<f/f.sub.12 <2.2
1.7<f.sub.1 /f.sub.12 <4.0
1.6<f.sub.1 /f.sub.2 <5.0
0.15<dy/dx<0.6
where f is the focal length of the entire lens system, f.sub.1 and f.sub.2 are the focal lengths of said first and said second groups, respectively, f.sub.12 is the combined focal length of said first and second groups, and dy/dx is the differentiated value of an amount of variation y in the spacing between said first group and said second group to an amount of variation x in the spacing between said second group and said third group, y and x being in a predetermined functional relation.

### Claim 3 (depends on 2)

A lens system according to claim 2, wherein said first group has, in succession from the object side, two positive lens components and a negative lens component, said second group has a meniscus lens component having its convex surface facing the image side and a positive lens component, and said third group has at least one negative lens component.

### Claim 4 (depends on 3)

A lens system according to claim 3, wherein when r.sub.1 and r.sub.2 are, respectively, the radii of curvature of the object side surface and the image side surface of the positive lens component in said first group which is most adjacent to the object side, and the shape factor q.sub.1 is defined as (r.sub.2 +r.sub.1)/(r.sub.2 -r.sub.1), said shape factor q.sub.1 satisfies the condition that 0.6<q.sub.1 <1.3, and wherein the second positive lens component in said first group is a positive meniscus lens having its convex surface facing the object side, and the negative lens component in said first group is a negative meniscus lens having its convex surface facing the object side.

### Claim 5 (depends on 1)

A lens system according to claim 1, wherein numerical data are as follows:
__________________________________________________________________________ Focal length f = 105 F-number F = 2.8 Angle of view 2.omega. = 23.25.degree. d.sub.0 = .infin..about.254.9907 __________________________________________________________________________ r.sub.1 = 74,500 d.sub.1 = 5.0 n.sub.1 = 1.77279 .nu..sub.1 = 49.4 L.sub.11 r.sub.2 = -1042,700 d.sub.2 = 0.15 First r.sub.3 = 29,000 d.sub.3 = 5.0 n.sub.2 = 1.71700 .nu..sub.2 = 48.1 L.sub.12 Group r.sub.4 = 57,000 d.sub.4 = 2.0 G.sub.1 r.sub.5 = 91,280 d.sub.5 = 2.0 n.sub.3 = 1.74950 .nu..sub.3 = 35.2 L.sub.13 r.sub.6 = 24,131 d.sub.6 = 13.9866 .about. 20.6820 r.sub.7 = -31,190 d.sub.7 = 2.0 n.sub.4 = 1.68893 .nu..sub.4 = 31.1 L.sub.21 Second r.sub.8 = 300,000 d.sub.8 = 10.5 n.sub.5 = 1.73350 .nu..sub.5 = 51.1 Group r.sub.9 = -41,983 d.sub.9 = 3.0 G.sub.2 r.sub.10 = 310,300 d.sub.10 = 5.0 n.sub.6 = 1.69350 .nu..sub.6 = 53.8 L.sub.22 r.sub.11 = -70,248 d.sub.11 = 8.1258 .about. 34.3666 r.sub.12 = -191,637 d.sub.12 = 4.0 n.sub.7 = 1.59507 .nu..sub.7 = 35.5 L.sub.31 r.sub.13 = -55,637 d.sub.13 = 2.0 r.sub.14 = -114,216 d.sub.14 = 1.8 n.sub.8 = 1.79668 .nu..sub.8 = 45.5 L.sub.32 Third r.sub.15 = 50,880 d.sub.15 = 1.8 Group r.sub.16 = 45,690 d.sub.16 = 6.0 n.sub.9 = 1.59507 .nu..sub.9 = 35.5 L.sub.33 G.sub.3 r.sub.17 = -175,850 d.sub.17 = 4.6 r.sub.18 = -38,545 d.sub.18 = 2.0 n.sub.10 = 1.80454 .nu..sub.10 = 39.6 L.sub.34 r.sub.19 = -77,213 f.sub.1 = 153,000 f.sub.12 = 75.0 .DELTA.y/.DELTA.x = 0.25515 f.sub.2 = 80,769 q.sub.1 = 0.867 P = 0.00114 f.sub.3 = -118,525 The diaphragm lies 1.5mm ahead of the fifth lens L.sub.22 ; __________________________________________________________________________
where r.sub.1, r.sub.2, r.sub.3, . . . represent the radii of curvature of the successive lens surfaces from the object side, d.sub.1, d.sub.2, d.sub.3, . . . represent the center thicknesses and air spaces of the respective lenses, and n.sub.1, n.sub.2, . . . and .nu..sub.1, .nu..sub.2, . . . represent the refractive indices and Abbe numbers, respectively, of the respective lenses, f.sub.1, f.sub.2 and f.sub.3 are the focal lengths of the first, second and third groups, respectively, f.sub.12 is the combined focal length of the first and second groups, .DELTA.y/.DELTA.x is the average variation rate of an amount of variation y in the spacing between the first and second groups to an amount of variation x in the spacing between the second and third groups, d.sub.0 is the object distance, P is the Petzval sum of the entire system, and q.sub.1 is the shape factor defined as (r.sub.2 +r.sub.1)/(r.sub.2 -r.sub.1).

### Claim 6 (depends on 1)

A lens system according to claim 1, wherein numerical data are as follows:
__________________________________________________________________________ Focal length f = 105.187 F-number F = 2.8 Angle of view 2.omega. = 23.21.degree. d.sub.0 = .infin..about.265.1865 __________________________________________________________________________ r.sub.1 = 107.669 d.sub.1 = 4.15 n.sub.1 = 1.76684 .nu..sub.1 = 46.8 L.sub.11 r.sub.2 = -533.176 d.sub.2 = 0.7 First r.sub.3 = 29.357 d.sub.3 = 6.7 n.sub.3 = 1.79668 .nu..sub.3 = 45.5 L.sub.12 Group r.sub.4 = 84.686 d.sub.4 = 2.05 G.sub.1 r.sub.5 = 204.175 d.sub.5 = 3.3 n.sub.3 = 1.75520 .nu..sub.3 = 27.6 L.sub.13 r.sub.6 = 23.623 d.sub.6 = 14.4023.about.21.0977 r.sub.7 = -82.105 d.sub.7 = 2.3 n.sub.4 = 1.75520 .nu..sub.4 = 27.6 L.sub.21 Second r.sub.8 = 26.484 d.sub.8 = 9.1 n.sub.5 = 1.76684 .nu..sub.5 = 46.8 Group r.sub.9 = -64.715 d.sub.9 = 3.0 G.sub.2 r.sub.10 = -7971.249 d.sub.10 = 4.3 n.sub.6 = 1.71300 .nu..sub.6 = 54 L.sub.22 r.sub.11 = -84.941 d.sub.11 = 4.8129.about.31.0537 r.sub.12 = -822.692 d.sub.12 = 4.0 n.sub.7 = 1.59507 .nu..sub.7 = 35.5 L.sub.31 r.sub.13 = -35.108 d.sub.13 = 2.0 Third r.sub.14 = -32.776 d.sub.14 = 1.8 n.sub.8 = 1.79668 .nu..sub.8 = 45.5 L.sub.32 Group r.sub.15 = 51.278 d.sub.15 = 1.8 G.sub.3 r.sub.16 = 43.213 d.sub.16 = 5.0 n.sub.9 = 1.59507 .nu..sub.9 = 35.5 L.sub.33 r.sub.17 = 168888.708 f.sub.1 = 153.00 f.sub.12 = 75.0 .DELTA.y/.DELTA.x = 0.25515 f.sub.2 = 80.769 q.sub.1 = 0.664 P = 0.00169 f.sub.3 = -118.00 The diaphragm lies 1.5mm ahead of the fifth lens L.sub.22 ; __________________________________________________________________________
where r.sub.1, r.sub.2, r.sub.3, . . . represent the radii of curvature of the successive lens surfaces from the object side, d.sub.1, d.sub.2, d.sub.3, . . . represent the center thicknesses and air spaces of the respective lenses, and n.sub.1, n.sub.2, . . . and .nu..sub.1, .nu..sub.2, . . . represent the refractive indices and Abbe numbers, respectively, of the respective lenses, f.sub.1, f.sub.2 and f.sub.3 are the focal lengths of the first, second and third groups, respectively, f.sub.12 is the combined focal length of the first and second groups, .DELTA.y/.DELTA.x is the average variation rate of an amount of variation y in the spacing between the first and second groups to an amount of variations x in the spacing between the second and third groups, d.sub.0 is the object distance, P is the Petzval sum of the entire system, and q.sub.1 is the shape factor defined as (r.sub.2 +r.sub.1)/(r.sub.2 -r.sub.1).

### Claim 7 (depends on 1)

A lens system according to claim 1, wherein numerical data are as follows:
__________________________________________________________________________ Focal length f = 105.187 F-number F = 2.8 Angle of view 2.omega. = 23.21.degree. d.sub.0 = .infin. __________________________________________________________________________ .about.268.6618 r.sub.1 = 78.231 d.sub.1 = 4.15 n.sub.1 = 1.76684 .nu..sub.1 = 46.8 L.sub.11 First r.sub.2 = 663.375 d.sub.2 = 0.7 Group r.sub.3 = 31.990 d.sub.3 = 6.7 n.sub.2 = 1.76684 .nu..sub.2 = 46.8 L.sub.12 G.sub.1 r.sub.4 = 79.989 d.sub.4 = 4.0 n.sub.3 = 1.58267 .nu..sub.3 = 46.5 L'.sub.12 r.sub.5 = -177.053 d.sub.5 = 3.3 n.sub.4 = 1.75520 .nu..sub.4 = 27.6 L.sub.13 r.sub.6 = 23.920 d.sub.6 = 9.8444.about.16.5398 r.sub.7 = -43.896 d.sub.7 = 8.0 n.sub.5 = 1.67003 .nu..sub.5 = 47.1 L.sub.21 Second r.sub.8 = -58.056 d.sub.8 = 3.0 Group r.sub.9 = 309.188 d.sub.9 = 2.0 n.sub.6 = 1.71736 .nu..sub.6 = 29.5 G.sub.2 L.sub.22 r.sub.10 = 63.787 d.sub.10 = 4.0 n.sub.7 = 1.73350 .nu..sub.7 = 51.1 r.sub.11 = -63.636 d.sub.11 = 2.5291.about.28.7699 r.sub.12 = -111.194 d.sub.12 = 4.0 n.sub.8 = 1.66755 .nu..sub.8 = 42.0 L.sub.31 r.sub.13 = -53.353 d.sub.13 = 2.0 Third r.sub.14 = -113.983 d.sub.14 = 1.8 n.sub.9 = 1.80218 .nu..sub.9 = 44.7 L.sub.32 Group r.sub.15 = 45.948 d.sub.15 = 1.8 G.sub.3 r.sub.16 = 41.777 d.sub.16 = 6.0 n.sub.10 = 1.66446 .nu..sub.10 = 35.9 L.sub.33 r.sub.17 = -130.887 d.sub.17 = 4.6 r.sub.18 = -40.915 d.sub.18 = 2.0 n.sub.11 = 1.79631 .nu..sub.11 = 41 L.sub.34 r.sub.19 = -109.967 f.sub.1 = 153.0 f.sub.12 = 75.0 .DELTA.y/.DELTA.x = 0.25515 f.sub.2 = 80.769 q.sub.1 = 1.267 P = 0.00127 f.sub.3 = -118.00 The diaphragm lies 1.5mm ahead of the sixth lens L.sub.22 ; __________________________________________________________________________
where r.sub.1, r.sub.2, r.sub.3, . . . represent the radii of curvature of the successive lens surfaces from the object side, d.sub.1, d.sub.2, d.sub.3, . . . represent the center thicknesses and air spaces of the respective lenses, and n.sub.1, n.sub.2, . . . and .nu..sub.1, .nu..sub.2, . . . represent the refractive indices and Abbe numbers, respectively, of the respective lenses, f.sub.1, f.sub.2 and f.sub.3 are the focal lengths of the first, second and third groups, respectively, f.sub.12 is the combined focal length of the first and second groups, .DELTA.y/.DELTA.x is the average variation rate of an amount of variation y in the spacing between the first and second groups to an amount of variation x in the spacing between the second and third groups, d.sub.0 is the object distance, P is the Petzval sum of the entire system, and q.sub.1 is the shape factor defined as (r.sub.2 +r.sub.1)/(r.sub.2 -r.sub.1).

### Claim 8 (depends on 1)

A lens system according to claim 1, wherein numerical data are as follows:
__________________________________________________________________________ Focal length f = 105.0 F-number F = 2.8 Angle of view 2.omega. = 23.25.degree. d.sub.0 = .infin. __________________________________________________________________________ .about.149.1795 r.sub.1 = 99.370 d.sub.1 = 4.5 n.sub.1 = 1.77279 .nu..sub.1 = 49.4 L.sub.11 r.sub.2 = -682.579 d.sub.2 = 0.1 r.sub.3 = 37.643 d.sub.3 = 4.5 n.sub.2 = 1.77279 .nu..sub.2 = 49.4 L.sub.12 First r.sub.4 d.sub.4 = 4.0 Group r.sub.5 = -2018.698 d.sub.5 = 2.0 n.sub.3 = 1.69895 .nu..sub.3 = 80.1 G.sub.1 L'.sub.12 r.sub.6 = 39.556 d.sub.6 = 5.0 n.sub.4 = 1.79631 .nu..sub.4 = 39.6 r.sub.7 = 159.067 d.sub.7 = 0.1 r.sub.8 = 51.905 d.sub.8 = 2.0 n.sub.5 = 1.59507 .nu..sub.5 = 35.5 L.sub.13 r.sub.9 = 24.944 d.sub.9 = 12.347.about.26.347 Second r.sub.10 = -30.955 d.sub.10 = 1.5 n.sub.6 = 1.64831 .nu..sub.6 = 33.8 L.sub.21 Group r.sub.11 = 251.790 d.sub.12 = 9.1 n.sub.7 = 1.80411 .nu..sub.7 = 46.4 G.sub.2 r.sub.12 = -39.137 d.sub.12 = 0.1 r.sub.13 = 5315.299 d.sub.13 = 4.3 n.sub.8 = 1.80411 .nu..sub.8 = 46.4 L.sub.22 r.sub.14 = -94.504 d.sub.14 = 1.4438.about.56.3132 r.sub.15 = 121.0 d.sub.15 = 4.0 n.sub.9 = 1.59507 .nu..sub.9 = 35.5 L.sub.31 r.sub.16 = -100.350 d.sub.16 = 2.0 r.sub.17 = -118.300 d.sub.17 = 1.8 n.sub.10 = 1.79668 .nu..sub.10 = 45.5 L.sub.32 Third r.sub.18 = 35.5 d.sub.18 = 1.8 Group r.sub.19 = 36.77 d.sub.19 = 6.0 n.sub.11 = 1.59507 .nu..sub.11 = 35.5 L.sub.33 G.sub.3 r.sub.20 = -5164.00 d.sub.20 = 4.6 r.sub.21 = -100.00 d.sub.21 = 1.5 n.sub.12 = 1.79668 .sub.12 = 45.5 L.sub.34 r.sub.22 = 70.000 d.sub.22 = 0.2 r.sub.23 = 54.000 d.sub.23 = 9 n.sub.13 = 1.51009 .nu..sub.13 = 63.4 L.sub.35 r.sub.24 = -179.445 f.sub. 1 = 153.0 f.sub.12 = 5.0 .DELTA.y/.DELTA.x = 0.25515 f.sub.2 = 80.769 q.sub.1 = 0.746 P = 0.00125 f.sub.3 = -125.525 The diaphragm lies 4mm ahead of the fifth lens L.sub.21 ; __________________________________________________________________________
where r.sub.1, r.sub.2, r.sub.3, . . . represent the radii of curvature of the successive lens surfaces from the object side, d.sub.1, d.sub.2, d.sub.3, . . . represent the center thicknesses and air spaces of the respective lenses, and n.sub.1, n.sub.2, . . . and .nu..sub.1, .nu..sub.2, . . . represent the refractive indices and Abbe numbers, respectively, of the respective lenses, f.sub.1, f.sub.2 and f.sub.3 are the focal lengths of the first, second and third groups, respectively, f.sub.12 is the combined focal length of the first and second groups, .DELTA.y/.DELTA.x is the average variation rate of an amount of variation y in the spacing between the first and second groups to an amount of variation x in the spacing between the second and third groups, d.sub.0 is the object distance, P is the Petzval sum of the entire system, and q.sub.1 is the shape factor defined as (r.sub.2 +r.sub.1)/(r.sub.2 -r.sub.1).

### Claim 9 (depends on 1)

A lens system according to claim 1, wherein numerical data are as follows:
__________________________________________________________________________ focal length f = 105.0 F-number F = 2.6 Angle of view 2.omega. = 23.25.degree. d.sub.0 = .infin. .about. __________________________________________________________________________ 261.408 r.sub.1 = 122.568 d.sub.1 = 6.08 n.sub.1 = 1.77279 .nu..sub.1 = 49.4 L.sub.11 First r.sub.2 = -2677.350 d.sub.2 = 0.19 Group r.sub.3 = 40.437 d.sub.3 = 10.118 n.sub.2 = 1.84042 .nu..sub.2 = 43.3 L.sub.12 G.sub.1 r.sub.4 = 93.003 d.sub.4 = 1.9 r.sub.5 = 194.456 d.sub.5 = 2.850 n.sub.3 = 1.75692 .nu..sub.3 = 31.7 L.sub.13 r.sub.6 = 31.060 d.sub.6 = 14.501.about. 29.796 r.sub.7 = -35.650 d.sub.7 = 1.853 n.sub.4 = 1.71736 .nu..sub.4 = 29.5 Second L.sub.21 Group r.sub.8 = .infin. d.sub.8 = 9.738 n.sub.5 = 1.78797 .nu..sub.5 = 47.5 G.sub.2 r.sub.9 = -43.010 d.sub.9 = 0.19 r.sub.10 = 1009.388 d.sub.10 = 7.790 n.sub.6 = 1.84042 .nu..sub.6 = 43.3 L.sub.22 r.sub.11 = -105.762 d.sub.11 = 3.043.about. 48.715 Third r.sub.12 = 244.557 d.sub.12 = 2.0 n.sub.7 = 1.46450 .nu..sub.7 = 65.8 L.sub.31 Group G.sub.3 r.sub.13 = 137.036 f.sub.1 = 246.031 f.sub.12 = 95.0 .DELTA.y/.DELTA.x = 0.33489 f.sub.2 = 95.399 q.sub.1 = 0.912 P = 0.00073 f.sub.3 = -675.00 The diaphragm lies 4.75mm ahead of the fourth lens L.sub.21 ; __________________________________________________________________________
where r.sub.1, r.sub.2, r.sub.3, . . . represent the radii of curvature of the successive lens surfaces from the object side, d.sub.1, d.sub.2, d.sub.3, . . . represent the center thicknesses and air spaces of the respective lenses, and n.sub.1, n.sub.2, . . . and .nu..sub.1, .nu..sub.2, . . . represent the refractive indices and Abbe numbers, respectively, of the respective lenses, f.sub.1, f.sub.2 and f.sub.3 are the focal lengths of the first, second and third groups, respectively, f.sub.12 is the combined focal length of the first and second groups, .DELTA.y/.DELTA.x is the average variation rate of an amount of variation y in the spacing between the first and second groups to an amount of variation x in the spacing between the second and third groups, d.sub.0 is the object distance, P is the Petzval sum of the entire system, and q.sub.1 is the shape factor defined as (r.sub.2 +r.sub.1)/(r.sub.2 -r.sub.1).

### Claim 10 (depends on 1)

A lens system according to claim 1, wherein numerical data are as follows:
__________________________________________________________________________ Focal length F = 210.0 F-number F = 4.5 Angle of view 2.omega. = 11.75.degree. d.sub.0 = .infin..about.479.6208 __________________________________________________________________________ r.sub.1 = 56.301 d.sub.1 = 9.5 n.sub.1 = 1.69680 .nu..sub.1 = 55.6 L.sub.11 r.sub.2 = 9668.693 d.sub.2 = 3.1 First r.sub.3 = 42.374 d.sub.3 = 11.5 n.sub.2 = 1.62041 .nu..sub.2 = 60.3 L.sub.12 Group r.sub.4 = 68.455 d.sub.4 = 2.5 G.sub.1 r.sub.5 = 3944.716 d.sub.5 = 5.6 n.sub.3 = 1.72825 .nu..sub.3 = 28.3 L.sub.13 r.sub.6 = 28.393 d.sub.6 = 25.716.about.33.716 Second r.sub.7 = -41.143 d.sub.7 = 3.9 n.sub.4 = 1.59551 .nu. .sub.4 = 39.2 L.sub.21 Group r.sub.8 = -48.910 d.sub.8 = 4.1 G.sub.2 r.sub.9 = 165.842 d.sub.9 = 5.0 n.sub.5 = 1.73350 .nu..sub.5 = 51.1 L.sub.22 r.sub.10 = -76.597 d.sub.10 = 13.830.about.39.948 r.sub.11 = 35.545 d.sub.11 = 1.0 n.sub.6 = 1.79668 .nu..sub.6 = 45.5 L.sub.31 r.sub.12 = 17.780 d.sub.12 = 1.8 r.sub.13 = 32.293 d.sub.13 = 4.8 n.sub.7 = 1.59507 .nu..sub.7 = 35.6 L.sub.32 r.sub.14 = -25.322 d.sub.14 = 1.0 r.sub.15 = -25.590 d.sub.15 = 1.0 n.sub.8 = 1.77279 .nu..sub.8 = 49.4 L.sub.33 Third r.sub.16 = 40.631 d.sub.16 = 2.75 n.sub.9 = 1.59507 .nu..sub.9 = 35.5 Group r.sub.17 = 136.476 d.sub.17 = 14.290 G.sub.3 r.sub.18 = 109.029 d.sub.18 = 1.5 n.sub.10 = 1.79631 .nu..sub.10 = 41.0 L.sub.34 r.sub.19 = 37.535 d.sub.19 = 10.0 n.sub.11 = 1.51823 .nu..sub.11 = 59.0 r.sub.20 = -23.340 d.sub.20 = 0.1 r.sub.21 = -87.100 d.sub.21 = 1 n.sub.12 = 1.78797 .nu..sub.12 = 47.5 L.sub.35 r.sub.22 = 83.340 f.sub.1 = 280.444 f.sub.12 = 105.00 .DELTA.y/.DELTA.x = 0.30629 f.sub.2 = 79.0025 q.sub. 1 = 1.012 P = 0.00073 f.sub.3 = -76.488 The diaphragm lies 7.5mm ahead of the fourth lens L.sub.21 ; __________________________________________________________________________
where r.sub.1, r.sub.2, r.sub.3, . . . represent the radii of curvature of the successive lens surfaces from the object side, d.sub.1, d.sub.2, d.sub.3, . . . represent the center thicknesses and air spaces of the respective lenses, and n.sub.1, n.sub.2, . . . and .nu..sub.1, .nu..sub.2, . . . represent the refractive indices and Abbe numbers, respectively, of the respective lenses, f.sub.1, f.sub.2 and f.sub.3 are the focal lengths of the first, second and third groups, respectively, f.sub.12 is the combined focal length of the first and second groups, .DELTA.y/.DELTA.x is the average variation rate of an amount of variation y in the spacing between the first and second groups to an amount of variation x in the spacing between the second and third groups, d.sub.0 is the object distance, P is the Petzval sum of the entire system, and q.sub.1 is the shape factor defined as (r.sub.2 +r.sub.1)/(r.sub.2 -r.sub.1).