Zoom lens

Abstract

A zoom lens having a range of equivalent focal lengths extending from a range below the dimension of the diagonal of the image frame to a dimension substantially above and having a zooming range of up to six to one or greater where the lens has a plus, minus, plus, plus lens unit configuration with a strongly negative second lens unit and a fourth lens unit designed to have a given power and telephoto ratio to compensate for a remote aperture stop located in the third lens unit.

Description

FIELD OF THE INVENTION

This invention relates to zoom lenses and, more particularly, relates to zoom lenses having a range of equivalent focal lengths (EFL) from a dimension below the diagonal of the image frame to a long focal length which may be as great as six times the diagonal of the image plane.

BACKGROUND OF THE INVENTION

Wide angle to long focal length zoom lenses have been available for some time. However, there is a constant effort to design such lenses which are more compact, while having a relatively fast aperture. These wide angle to long focal length zoom lenses require different design considerations than a so-called telephoto zoom lens, which, in the thirty-five millimeter format, may be a 70-210 millimeter zoom lens, as exemplified in U.S. Pat. No. 3,817,600.

In the telephoto zoom lenses, it is customary that a rear stationary lens unit have a low telephoto ratio, generally less than one, and the aperture defining iris is included in this rear stationary lens unit. The telephoto ratio is the ratio of the equivalent focal length of the lens or lens lens unit to the front vertex distance (FVD), where the front vertex distance is the distance from the front apex of the lens to the image plane, generally when the lens if focused to infinity. An example of a widely used wide angle to long focal length zoom lens is disclosed in U.S. Pat. No. 4,299,454.

It is common practice to use a basic triplet or variation thereof as a rear lens unit in a zoom lens. The optical powers of a triplet are in a plus-minus-plus configuration separated by small air spaces. The lens units may be subdivided or compounded, but the overall shape remains biconvex. The positive lens units may be and often are meniscus shaped. A triplet, in its simplest form, can be corrected for all primary aberrations, and is particularly well suited for large aperture requirements; however, it is limited with respect to angular coverage because of inherent astigmatism.

In wide angle to long focal length zoom lenses, where the rear lens unit moves to vary the EFL, the iris defining aperture cannot be placed in the rear lens unit. Otherwise, the diameter of the front of the lens would become excessive.

The fourth lens unit thus has to operate with the handicap of having a removed aperture stop. Generally, in wide angle to long focal length zoom lenses utilizing four lens units, a compromise is made in that the aperture stop is positioned closely adjacent the third lens unit and may move with the third lens unit, as disclosed in U.S. Pat. No. 4,299,454, or may remain stationary with the third lens unit, as disclosed in U.S. Pat. No. 4,256,381. If the aperture stop is positioned too far away from the rear lens grouping, it becomes very difficult to control aberration correction without a complex rear lens unit, sacrificing compactness.

A major design constraint of the wide angle to long focal length zoom lens is the size of the rear lens opening. Typically, the clear aperture must be less than thirty millimeters (24.times.36 mm image size). In the wide angle position, light rays pass through the third lens unit at a large angle to the optical axis normally requiring large clear apertures of the elements of the fourth lens unit. If compactness of the lens is not emphasized, then the optical power of the fourth lens unit need not be large and the size of the opening of the fourth lens unit is not so critical. However, in the use of the thirty-five millimeter camera, lens compactness is a primary concern.

Accordingly, the present invention provides a four lens unit wide angle to long focal length zoom lens of plus, minus, plus, plus configuration, having a new and improved design in the fourth lens unit which compensates for the problem of a remote aperture stop, and diameter of the fourth lens unit, and which is relatively simple in a design that contributes to the compactness of the overall lens.

SUMMARY OF THE INVENTION

The present invention is directed to a wide angle to long focal length zoom lens of the types shown in U.S. Pat. Nos. 4,299,454 and 4,256,381. The aforementioned patents disclose wide angle to long focal length zoom lens comprising a first positive lens unit, a second very strong negative lens unit, a third positive lens unit, and a fourth positive lens unit. In commercial forms of these lenses, generally the first and fourth lens units move in the same direction for zooming, while the second lens unit moves in a direction opposite to the direction of movement of the first and fourth lens units. The third lens unit may move or may remain stationary. Alternatively, the second lens unit may be fixed and the third lens unit move in fixed relation to the aperture defining iris. A characteristic of this form of lens is its very strong negative second lens unit. This contributes to the compactness of the lens over a wide range of equivalent focal lengths (EFL) which may be greater than five to one. Typically, the second negative lens unit may have an absolute power which has a ratio to the geometric mean of the extremes of the EFL's of the lens as great as five to one.

The present invention provides zoom lenses which may have a zoom ratio of over six to one, using a four lens unit configuration of plus, minus, plus, plus configuration. The lens is arranged with the iris defining aperture positioned with the third lens unit in fixed relation thereto. The fourth lens unit comprises a first sub lens unit, which is a positive element followed by a doublet comprising a positive element and a negative element. The second sub lens unit may take various forms. However, it is important that the telephoto ratio of the fourth lens unit be within a given range, greater than unity, and the optical power of the lens be within a specified range, as hereinafter disclosed. The disclosed lens form is one in which the first and fourth lens units move in the same direction during zooming, and most often in fixed relation.

An object of this invention is to provide a new and improved zoom lens of the type described.

Another object of this invention is to provide a zoom lens of the type described having a new and improved fourth lens unit which contributes to compactness of the lens.

The features of the invention which are believed to be novel are particularly pointed out and distinctly claimed in the concluding portion of this specification. However, the invention, both as to its organization and operation, together with further objects and advantages thereof, may best be appreciated by reference to the following detailed description taken in conjunction with the drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1-8 are schematic side elevations of lenses embodying the invention.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS OF THE INVENTION

Lenses embodying the invention comprise four lens units, G1, G2, G3, and G4, as exemplified in all figures of the drawings. The lens units are referenced as G1-G4 from the object end to the image end.

Lens elements are identified by the reference L followed by an arabic numeral progressively from the object end to the image end of the lens.

Lens element surfaces are identified by the reference S followed by an arabic numeral progressively from the object end to the image end of the lens.

The focal plane of the lenses shown in the drawings are denoted by the reference FP and the optical axis by A. The prescriptions of all lenses set forth in the Tables II-IX are scaled to an image format of 24.times.36 mm.

In FIGS. 1-6, lens unit G1 comprises a biconvex doublet L1, either air spaced or cemented, L2, and a positive meniscus L3.

In FIGS. 1-6, the second lens unit G2 comprises a negative meniscus L4, a biconcave component L5, and a doublet comprising a positive element L6 concave to the object and a negative element L7. Lens having the configuration of this second negative lens unit are disclosed and claimed in U.S. Pat. No. 4,299,454, assigned to the same assignee as the present invention.

The third lens unit G3 in all cases is positioned in fixed relation to the aperture defining iris I. In FIGS. 1, 2, 4, and 6, lens unit G3 is a doublet comprising a biconvex element L8 and a meniscus L9 concave to the object In FIG. 3, lens unit G3 is a single element L8. In FIG. 4, lens unit G3 comprises a biconvex element L8, followed by a biconvex doublet L9, L10.

Lens unit G4 in all examples consist of first and second sub lens units, hereinafter designated in Table I as G4a and G4b, respectively. The first sub lens unit comprises a positive element L10 in FIGS. 1, 2, 4, and 6, L9 in FIG. 3, and L11 in FIG. 5. This positive component is followed by a doublet L11, L12 (FIGS. 1, 5, and 6), L12, L13 (FIG. 4), L10, L11 (FIGS. 2 and 3), either cemented or air spaced as shown in FIGS. 1 and 3, which comprises a positive element and a negative element.

The second sub lens unit comprises two elements, L13, L14 (FIGS. 1, 2, 4, and 6), L12, L13 (FIG. 3), L15 (FIG. 5).

In FIG. 7, lens unit G1 comprises a biconcave element L1, followed by two biconcave elements L2 and L3. Lens unit G2 consists of a negative meniscus L4, convex to the object and a doublet comprising biconcave element L5 and a positive meniscus L6. Lens unit G3 comprises a positive element L7 convex to the object and a meniscus concave to the object. In lens unit G4 the first sub lens unit comprises a biconvex element L9 followed by a cemented doublet in the form of positive meniscus L11 convex to the object and a negative meniscus L11 also convex to the object. The second sub lens unit consists of a single meniscus L12 having an aspheric surface S21.

The lens of FIG. 8 has a front lens unit G1 comprising elements L1, L2 and L3 in a configuration the same as FIGS. 2 and 3. The second lens unit G2 comprises elements L4, L5 and L6 in the same configuration as FIG. 7. Lens unit G3 comprises elements L7 and L8 in the same configuration as FIG. 7. The first sub lens unit of lens unit G4 comrpises elements L9, L10 and L11 in the same configuration as FIG. 7 while the second sub lens unit consists of a biconcave element L12 and and biconvex element L13. The surface S22 of L12 is aspheric.

The lenses of FIGS. 7 and 8 include an element having at least one aspheric surface defined by the equation ##EQU1## where X is the surface sag at a semi-aperture distance Y from the axis A of the lens, C is the curvature of a lens surface of the optical axis A equal to the reciprocal of the radius of the optical axis, K is a conic constant.

The arrangement of the lens elements in lens units G1-G4 is tabulated and summarized in Table I. The subunits of lens unit G4 are denoted as G4a and G4b.

TABLE I______________________________________G1 G2 G3 G4a G4b______________________________________FIG. 1 L1-L3 L4-L7 L8,L9 L10-L12 L13,L14FIG. 2 L1-L3 L4-L7 L8,L9 L10-L12 L13-L15FIG. 3 L1-L3 L4-L7 L8 L9-L11 L12,L13FIG. 4 L1-L3 L4-L7 L8,L9 L10-L12 L13,L14FIG. 5 L1-L3 L4-L7 L8-L10 L11-L13 L14,L15FIG. 6 L1-L3 L4-L7 L8,L9 L10-L12 L13,L14FIG. 7 L1-L3 L4-L6 L7,L8 L9-L11 L12FIG. 8 L1-L3 L4-L6 L7,L8 L9-L11 L12,L13______________________________________ where G4a and G4b are the front and rear subunits of lens unit G4 respectively.

Lens unit G4 has a telephoto ratio TR, where

1.5>TR>1.0

The telephoto ratio TR is the ratio of the front vertex distance (FVD) of lens unit G4 to its equivalent focal length. The FVD is the dimension from the front apex of lens unit G4 to the focal plane FP when focused to infinity.

The ratio of the optical power K.sub.4 of lens unit G4 to the geometric mean of the power of the overall lens K.sub.M is

1.7>K.sub.4 /K.sub.M >1.1

Additionally, the optical power K.sub.2 of the second lens unit is very strong, and

5.0>.vertline.K.sub.2 /KM.vertline.>1.9

The geometric mean of the power of a zoom lens (K.sub.M) is

.sqroot.K.sub.S K.sub.L where K.sub.S is the optical power of the lens expressed as the reciprocal of the shortest EFL in millimeters and K.sub.L is the optical power of the lens expressed as the reciprocal of the largest EFL in millimeters.

The various lenses disclosed herein may have different movements of the various lens units as hereinafter described. In the tables hereinafter set forth to describe the various lenses, the back focal length (BFL) is the distance from the rear apex of lens unit G4 at the optical axis to the film plane FP. The spacings between the various lens units Z1, Z2, and Z3 are as noted on the drawings and corresponding tables for various focal lengths.

A lens as shown in FIG. 1 is substantially described in Table II. In this lens lens unit G1 and G4 move in fixed relation. Lens unit G2 moves in a direction opposite to lens units G1 and G4, while lens unit G3 has reciprocating motion.

A lens as shown in FIG. 2 is substantially described in Table III. In this lens lens units G1 and G4 move in fixed relation. Group G2 is stationary, and lens unit G3 moves as noted in Table III.

The lens of FIG. 3 is substantially described in Table IV. Lens units G1 and G4 move in fixed relation. Lens unit G3 and iris I are stationary while lens unit G2 moves.

A lens as shown in FIG. 4 is substantially described in Table V. Here the first lens unit G1 and the fourth lens unit G4 move in fixed relation, lens unit G2 moves in the opposite direction and lens unit G3 moves as indicated.

FIG. 5 shows a lens as substantially described in Table VI. Here the first and fourth lens units, G1 and G4 move in the same direction but not in fixed relation. Both of lens units G2 and G3 also move. The iris I moves in fixed relation to lens unit G3.

FIG. 6 shows a lens as substantially described in Table VII. Here the first lens unit G1 and fourth lens unit G4 move in fixed relation, while both lens units G2 and G3 also move as noted in Table VII.

FIG. 7 shows a lens as substantially described in Table VIII. Here the first lens unit G1 and fourth lens unit G4 move in fixed relation. The second lens unit G2 and third lens unit G3 also move as noted in Table VIII. Here surfaces S5 and S22 are aspheric.

FIG. 8 shows a lens as substantially described in Table IX. Here the first lens unit G1 and fourth lens unit G4 move in fixed relation and lens units G2 and G3 move as noted in Table IX. Here surface S22 is aspheric. The aspheric surfaces are provided for purposes of optical correction.

FIG. 1 also exemplifies a lens as described in Table X with minor differences. In Table X, elements L1 and L2 are slightly air spaced and element L13 is of negative power. The lens of Table X has an aspheric surface S24.

TABLE II______________________________________ AXIAL DISTANCE SURFACE BETWEENLENS RADII(mm) SURFACES(mm) N.sub.d V.sub.d______________________________________ S1 160.690L1 2.500 1.805 25.5 S2 56.190L2 10.270 1.517 64.2 S3 -144.312 0.200 S4 40.250L3 4.100 1.806 40.7 S5 69.987 Z1 S6 61.000L4 1.500 1.806 40.7 S7 15.180 5.180 S8 -40.000L5 1.400 1.773 49.6 S9 49.500 0.200 S10 28.000L6 7.200 1.673 32.2 S11 -15.415L7 1.200 1.773 49.6 S12 -619.000 Z2Aperture 1.500 S13 63.588L8 3.530 1.697 48.5 S14 -42.087 0.480 S15 -30.682L9 1.650 1.806 40.7 S16 -64.500 Z3 S17 34.420L10 4.630 1.581 40.9 S18 -156.800 0.200 S19 114.940L11 9.390 1.517 52.2 S20 -31.690L12 1.220 1.805 25.5 S21 34.420 2.480 S22 -88.000L13 2.400 1.806 40.7 S23 -38.900 0.200 S24 104.500L14 2.860 1.806 40.7 S25 -140.000______________________________________ZOOM SPACING DATAEFL Z1 Z2 Z3 BFLFVD______________________________________ 28.83 mm 1.00 mm 19.33 mm 16.61 mm 56.93 mm 158.16 mm 50.01 14.92 11.77 10.25 68.19169.41 5.01 26.31 6.24 4.38 78.01179.24131.00 34.52 1.51 0.91 83.26184.48______________________________________ TABLE III______________________________________ AXIAL DISTANCE SURFACE BETWEENLENS RADII(mm) SURFACES(mm) N.sub.d V.sub.d______________________________________ S1 102.879L1 2.300 1.741 27.8 S2 48.694 0.500 S3 48.183L2 10.400 1.514 64.2 S4 -189.534 0.100 S5 43.872L3 4.000 1.541 47.2 S6 75.339 Z1 S7 84.254L4 1.660 1.834 37.3 S8 20.443 5.908 S9 -32.836L5 1.440 1.773 49.6 S10 57.306 0.220 S11 43.367L6 7.200 1.717 29.5 S12 -16.903L7 1.330 1.773 49.6 S13 -333.557 Z2 S14 64.932L8 5.000 1.604 60.2 S15 -48.454 1.126 S16 -29.934L9 2.000 1.673 32.2 S17 -47.916 0.500Aperture Z3 S18 125.580L10 3.670 1.648 33.8 S19 -68.714 0.180 S20 88.886L11 6.000 1.517 64.2 S21 -32.051L12 5.500 1.785 25.7 S22 -83.481 8.017 S23 118.446L13 6.000 1.581 40.9 S24 -42.226L14 2.000 1.717 29.5 S25 37.606 1.900 S26 341.397L15 4.800 1.805 25.5 S27 -95.200______________________________________ZOOM SPACING DATAEFL Z1 Z2 Z3 BFLFVD______________________________________ 38.50 mm 7.00 mm 19.19 mm 23.74 mm 48.36 mm 186.24 mm 73.00 22.36 10.63 13.68 66.89201.50136.00 36.30 4.64 5.02 81.83215.74194.06 42.86 0.59 1.06 89.52221.97______________________________________ TABLE IV______________________________________ AXIAL DISTANCE SURFACE BETWEENLENS RADII(mm) SURFACES(mm) N.sub.d V.sub.D______________________________________ S1 97.471L1 2.000 1.805 25.5 S2 46.414 0.961 S3 44.857L2 7.500 1.517 64.2 S4 -204.325 0.100 S5 33.332L3 5.000 1.658 50.9 S6 67.053 Z1 S7 48.154L4 1.400 1.834 37.3 S8 16.552 9.047 S9 -29.301L5 1.200 1.773 49.6 S10 37.735 0.200 S11 30.244L6 5.800 1.673 32.2 S12 -13.650L7 1.200 1.773 49.6 S13 -101.590 Z2Aperture 1.040 S14 651.451L8 2.500 1.834 37.3 S15 -75.022 Z3 S16 148.051L9 2.664 1.834 37.3 S17 -59.655 0.100 S18 17.272L10 3.600 1.517 64.2 S19 80.238 0.570 S20 -2295.605L11 6.060 1.785 26.1 S21 17.700 6.307 S22 47.296L12 2.000 1.805 25.5 S23 28.126 0.931 S24 29.714L13 6.000 1.517 64.2 S25 -40.167______________________________________ZOOM SPACING DATAEFL Z1 Z2 Z3 BFL FVD______________________________________ 35.90 mm 0.50 mm 10.79 mm 10.77 mm 49.46 mm 141.60 54.93 9.65 7.07 5.34 53.89 147.03 85.00 17.79 2.84 1.43 57.80 150.94101.42 20.76 0.60 0.70 58.53 151.67______________________________________ TABLE V______________________________________ AXIAL DISTANCE SURFACE BETWEENLENS RADII(mm) SURFACES(mm) N.sub.D V.sub.D______________________________________ S1 87.132L1 2.500 1.805 25.5 S2 50.967L2 10.500 1.517 64.2 S3 -218.779 0.100 S4 38.314L3 4.000 1.618 55.2 S5 56.924 Z1 S6 117.702L4 1.400 1.834 37.3 S7 17.798 4.702 S8 -50.677L5 1.300 1.589 61.3 S9 49.929 0.100 S10 27.135L6 6.500 1.699 30.1 S11 -18.768L7 1.200 1.773 49.6 S12 65.374 Z2Aperture 1.700 S13 46.163L8 4.600 1.697 48.5 S14 -61.782 0.950 S15 -31.839L9 1.600 1.847 37.3 S16 -87.982 Z3 S17 2658.419L10 3.500 1.806 40.7 S18 -45.955 0.100 S19 55.327L11 5.400 1.487 70.4 S20 -30.009L12 1.300 1.805 25.5 S21 -166.729 10.527 S22 93.945L13 1.600 1.834 37.3 S23 37.521 1.956 S24 157.252L14 3.500 1.786 43.9 S25 -74.928______________________________________ZOOM SPACING DATAEFL Z1 Z2 Z3 BFLFVD______________________________________ 28.88 mm 1.75 mm 20.26 mm 16.96 mm 50.66 mm 174.73 mm 50.00 14.87 12.47 11.62 63.63177.70105.00 29.25 5.24 4.47 83.17197.24154.41 36.29 1.87 0.80 90.30204.37______________________________________ TABLE VI______________________________________ AXIAL DISTANCE SURFACE BETWEENLENS RADII(mm) SURFACES(mm) N.sub.d V.sub.d______________________________________ S1 155.350L1 2.200 1.805 25.5 S2 62.701L2 9.300 1.517 64.2 S3 -159.115 0.100 S4 47.500L3 3.500 1.723 38.0 S5 73.500 Z1 S6 151.230L4 1.504 1.834 37.3 S7 20.622 5.415 S8 -38.920L5 1.400 1.598 61.3 S9 94.000 0.100 S10 37.400L6 7.700 1.699 30.1 S11 -19.236L7 1.100 1.773 49.6 S12 215.750 Z2Aperture 1.500 S13 109.780L8 3.300 1.743 49.2 S14 -144.100 0.100 S15 71.820L9 5.800 1.589 61.3 S16 -37.400L10 1.500 1.834 37.3 S17 -2090.000 Z3 S18 102.599L11 3.800 1.744 44.9 S19 -110.900 0.100 S20 57.000L12 5.500 1.487 70.4 S21 -49.500L13 1.500 1.805 25.5 S22 -350.000 10.306 S23 288.000L14 1.800 1.834 37.3 S24 35.570 1.663 S25 129.810L15 3.200 1.834 37.3 S26 -95.800______________________________________ZOOM SPACING DATAEFL Z1 Z2 Z3 BFLFVD______________________________________ 39.50 mm 8.97 mm 23.72 mm 24.87 mm 55.41 mm 187.60 mm 70.08 26.06 14.23 14.13 70.87199.93137.21 42.17 5.56 4.60 90.13217.10193.26 49.82 0.95 0.94 97.09223.94______________________________________ TABLE VII______________________________________ AXIAL DISTANCE SURFACE BETWEENLENS RADII(mm) SURFACES(mm) N.sub.d V.sub.d______________________________________ S1 243.578L1 2.200 1.805 25.5 S2 58.863L2 8.500 1.517 64.2 S3 -121.610 0.100 S4 44.309L3 4.600 1.664 35.9 S5 99.626 Z1 S6 92.673L4 1.400 1.834 37.3 S7 17.808 5.358 S8 -36.818L5 1.400 1.517 64.2 S9 56.813 0.100 S10 27.885L6 6.800 1.699 30.1 S11 -17.481L7 1.100 1.773 49.6 S12 61.149 Z2Aperture 1.500 S13 46.648L8 4.100 1.603 60.7 S14 -56.897 0.900 S15 -31.060L9 1.500 1.834 37.3 S16 -56.434 Z3 S17 184.516L10 3.800 1.664 35.9 S18 -52.262 0.100 S19 52.773L11 5.500 1.487 70.4 S20 -32.229L12 1.500 1.805 25.5 S21 -157.768 9.691 S22 200.032L13 1.800 1.834 37.3 S23 37.027 1.692 S24 141.797L14 3.200 1.743 49.2 S25 -67.446______________________________________ZOOM SPACING DATAEFLZ1 Z2Z3 BFLFVD______________________________________ 28.88 mm 1.00 mm 22.71 mm 19.79 mm 55.01 mm 167.60 mm 66.5221.03 11.5710.90 79.41192.00135.0034.22 5.164.12 104.19216.77193.0040.61 1.950.94 115.22227.80______________________________________ TABLE VIII______________________________________ AXIAL DISTANCE SURFACE BETWEENLENS RADII(mm) SURFACES(mm) N.sub.d V.sub.d______________________________________ S1 -781.242L1 2.500 1.805 25.5 S2 156.226 0.105 S3 75.329L2 7.473 1.487 70.4 S4 -728.237 0.110 S5 58.221L3 8.716 1.491 57.2 S6 -1188.187 Z1 S7 64.040L4 1.600 1.834 37.3 S8 18.666 5.066 S9 -63.869L5 2.000 1.815 65.2 S10 18.672L6 3.537 1.805 25.5 S11 56.132 Z2Aperture 0.800 S12 40.929L7 2.500 1.564 60.8 S13 -356.168 1.575 S14 -22.864L8 1.500 1.517 52.2 S15 -47.470 Z3 S16 36.349L9 7.011 1.640 60.2 S17 -106.713 0.100 S18 24.204L10 6.572 1.678 55.5 S19 165.775L11 1.600 1.805 25.5 S20 20.436 3.220 S21 48.161L12 2.924 1.491 57.2 S22 -528.121______________________________________Surface S5 and S22 are asphericS5 S22______________________________________K = -1.000 D .2551 .times. 10.sup.-04 E .3653 .times. 10.sup.-07 F .2158 .times. 10.sup.-11 G .5188 .times. 10.sup.-12______________________________________ZOOM SPACING DATAEFL Z1 Z2 Z3 BFL FVD______________________________________ 28.50 mm 0.71 mm 14.95 mm 18.93 mm 40.12 mm 136.62 50.00 16.49 7.08 11.02 48.58 145.08102.00 33.46 0.83 0.30 60.60 157.10______________________________________ TABLE IX______________________________________ AXIAL DISTANCE SURFACE BETWEENLENS RADII(mm) SURFACES(mm) N.sub.d V.sub.d______________________________________ S1 195.772L1 2.500 1.805 25.5 S2 67.344 0.500 S3 66.244L2 7.000 1.613 54.6 S4 -411.314 0.100 S5 46.824L3 4.000 1.834 42.4 S6 66.472 Z1 S7 100.775L4 1.600 1.834 37.3 S8 21.211 5.851 S9 -153.496L5 2.000 1.481 69.6 S10 19.263L6 3.537 1.824 24.6 S11 45.706 Z2Aperture 0.800 S12 42.808L7 2.500 1.581 59.3 S13 -8183.436 1.575 S14 -24.474L8 1.500 1.776 40.0 S15 -37.379 Z3 S16 35.468L9 5.400 1.663 57.9 S17 -104.707 0.100 S18 26.104L10 4.800 1.692 54.4 S19 238.734L11 1.600 1.805 25.5 S20 23.704 5.602 S21 -402.429L12 2.924 1.491 57.2 S22 -402.429 0.200 S23 107.084L13 3.000 1.482 55.9 S24 -405.409______________________________________Aspheric Surface S22______________________________________ D .2455 .times. 10.sup.-04 E .5226 .times. 10.sup.-07 F -.6774 .times. 10.sup.-10 G .3605 .times. 10.sup.-12______________________________________ZOOM SPACING DATAEFL Z1 Z2 Z3 BFLFVD______________________________________ 28.51 mm 0.71 mm 20.48 mm 18.70 mm 38.51 mm 139.51 mm 50.00 18.89 9.95 11.04 46.17147.17 75.03 30.18 4.82 4.89 52.33153.33102.04 38.16 1.22 0.50 56.72157.72______________________________________ TABLE X______________________________________ AXIAL DISTANCE SURFACE BETWEENLENS RADII(mm) SURFACES(mm) N.sub.d V.sub.d______________________________________ S1 122.608L1 2.500 1.805 25.5 S2 55.746 0.500 S3 54.539L2 10.600 1.517 64.2 S4 -249.144 0.100 S5 41.949L3 4.328 1.702 41.2 S6 75.883 Z1 S7 84.733L4 1.500 1.834 37.3 S8 17.252 5.757 S9 -52.546L5 1.300 1.773 49.6 S10 45.963 0.200 S11 29.582L6 6.738 1.673 32.2 S12 -17.554L7 1.200 1.773 49.6 S13 -532.786 Z2Aperture 1.000 S14 77.815L8 2.452 1.834 37.3 S15 -58.953 0.550 S16 -34.685L9 1.300 1.717 29.5 S17 -70.992 Z3 S18 33.486L10 4.446 1.720 50.3 S19 -174.911 0.100 S20 37.651L11 5.299 1.717 48.0 S21 -55.577L12 2.000 1.785 26.1 S22 23.790 5.147 S23 -32.331L13 2.300 1.491 57.2 S24 -110.442 0.200 S25 150.096L14 6.350 1.487 70.4 S26 -31.538______________________________________Aspheric Surface S24______________________________________ D .1995 .times. 10.sup.-04 E .5118 .times. 10.sup.-07 F -.1442 .times. 10.sup.-09 G .2947 .times. 10.sup.-12______________________________________ZOOM SPACING DATAEFL Z1 Z2 Z3 BLFFVD______________________________________ 28.60 mm 1.50 mm 15.87 mm 17.87 mm 41.80 mm 144.94 mm 50.00 17.14 8.62 9.48 50.91153.87 75.00 26.79 4.36 4.09 56.92159.89102.00 33.36 1.00 0.88 60.76163.76______________________________________

Table XI sets forth the powers K1-K4 of each lens lens unit G1-G4, respectively, as a function of the geometric mean K.sub.M of the extremes of the equivalent focal length of each lens where

K.sub.M =.sqroot.K.sub.S K.sub.L, and K.sub.S =the optical power of the lens at its shortest equivalent focal length, and K.sub.L =the optical power of the lens at its longest equivalent focal length.

The optical powers are the reciprocal of the equivalent focal lengths in millimeters.

TABLE XI______________________________________TABLE K.sub.1 /K.sub.M K.sub.2 /K.sub.M K.sub.3 /K.sub.M K.sub.4 /K.sub.M K.sub.M______________________________________II .778 -3.660 .858 1.173 .0162III .940 -4.817 1.278 1.417 .0115IV .093 -3.754 .833 1.513 .0167V .773 -3.860 .687 1.780 .0150VI .833 -4.193 1.263 1.289 .0114VII .851 -4.320 .880 1.433 .0134VIII .530 -2.156 .227 1.584 .0185IX .975 -4.000 .895 1.657 .0162X .627 -2.941 .778 1.162 .0185______________________________________

Table XII sets forth the equivalent focal lengths F4 of the fourth lens units G4 of the lenses, the front vertex distance FVD.sub.4 of the fourth lens units and the telephoto ratios TR.sub.4 of the fourth lens units.

TABLE XII______________________________________TABLE F.sub.4 FVD.sub.4 TR.sub.4______________________________________II 44.35 mm 54.07 mm 1.219III 61.39 71.47 1.164IV 44.03 51.67 1.174V 44.98 54.09 1.202VI 68.21 72.36 1.061VII 51.95 59.34 1.142VIII 34.14 40.45 1.185IX 36.75 42.48 1.156X 46.58 46.58 1.162______________________________________

From the foregoing tables it may be seen that

5.0>.vertline.K.sub.2 /K.sub.m .vertline.>2.0 1.8>K.sub.4 /K.sub.m >1.1 1.5>TR.sub.4 >1.0

These parameters provide well corrected lenses of various ranges of equivalent focal lengths having a fourth lens unit which contributes to the compactness of the overall lens.

Claims

1. In a lens having a range of equivalent focal lengths from a dimension below the diagonal of the image frame of the lens to a dimension at least substantially three times the shortest equivalent focal length where said lens comprises a first positive lens unit, a second negative lens unit, a third positive lens unit, and a fourth positive lens unit, said lens having an aperture defining means in fixed relation to said third lens unit, said first and fourth lens units being axially movable in the same direction and at least one of said second and third lens units being axially movable to vary the equivalent focal length of said lens, said second lens unit having an absolute optical power at least 2.0 as great as the geometrical mean of the extremes of the equivalent focal lengths of said lens; and

1.8>K.sub.4 /K.sub.M >1.0

1.5>TR.sub.4 >1.0

where K.sub.4 is the optical power of said fourth lens unit, K.sub.M is the geometrical mean of the extremes of the equivalent focal lengths of said lens, and TR.sub.4 is the telephoto ratio of said fourth lens unit, said fourth lens unit comprising first and second sub lens units, said first sub lens unit comprising from the object end a first positive element and a doublet formed of a positive element and a following negative element, said second sub lens unit comprising a positive element most closely adjacent the image plane.

2. The lens of claim 1 where said first and fourth lens units move in fixed relation to vary the equivalent focal length of said lens.

3. The lens of claim 1 where said doublet is in the form of a meniscus.

4. The lens of claim 1 where the elements of said doublet are air spaced.

5. The lens of claim 1 where said doublet is biconvex.

6. A zoom lens as described in claim 1 scaled to an image frame of 24.times.36 mm described substantially as follows:

______________________________________ AXIAL DISTANCE SURFACE BETWEENLENS RADII(mm) SURFACES(mm) N.sub.d V.sub.d______________________________________ S1 160.690L1 2.500 1.805 25.5 S2 56.190L2 10.270 1.517 64.2 S3 -144.312 0.200 S4 40.250L3 4.100 1.806 40.7 S5 69.987 Z1 S6 61.000L4 1.500 1.806 40.7 S7 15.180 5.180 S8 -40.000L5 1.400 1.773 49.6 S9 49.500 0.200 S10 28.000L6 7.200 1.673 32.2 S11 -15.415L7 1.200 1.773 49.6 S12 -619.000 Z2Aperture 1.500 S13 63.588L8 3.530 1.697 48.5 S14 -42.087 0.480 S15 -30.682L9 1.650 1.806 40.7 S16 -64.500 Z3 S17 34.420L10 4.630 1.581 40.9 S18 -156.800 0.200 S19 114.940L11 9.390 1.517 52.2 S20 -31.690L12 1.220 1.805 25.5 S21 34.420 2.480 S22 -88.000L13 2.400 1.806 40.7 S23 -38.900 0.200 S24 104.500L14 2.860 1.806 40.7 S25 -140.000______________________________________ZOOM SPACING DATAEFL Z1 Z2 Z3 BFLFVD______________________________________ 28.83 mm 1.00 mm 19.33 mm 16.61 mm 56.93 mm 158.16 mm 50.01 14.92 11.77 10.25 68.19169.41 5.01 26.31 6.24 4.38 78.01179.24131.00 34.52 1.51 0.91 83.26184.48______________________________________

where L1-L14 are lens elements progressively from the object to the image end, S1-S25 are the lens surfaces from the object to image end, N.sub.D is the index of refraction of each lens element, and V.sub.D is the dispersion of each lens element as measured by its Abbe number.

7. A zoom lens as described in claim 1 scaled to an image frame of 24.times.36 mm described substantially as follows:

______________________________________ AXIAL DISTANCE SURFACE BETWEENLENS RADII(mm) SURFACES(mm) N.sub.d V.sub.d______________________________________ S1 102.879L1 2.300 1.741 27.8 S2 48.694 0.500 S3 48.183L2 10.400 1.514 64.2 S4 -189.534 0.100 S5 43.872L3 4.000 1.541 47.2 S6 75.339 Z1 S7 84.254L4 1.660 1.834 37.3 S8 20.443 5.908 S9 -32.836L5 1.440 1.773 49.6 S10 57.306 0.220 S11 43.367L6 7.200 1.717 29.5 S12 -16.903L7 1.330 1.773 49.6 S13 -333.557 Z2 S14 64.932L8 5.000 1.604 60.2 S15 -48.454 1.126 S16 -29.934L9 2.000 1.673 32.2 S17 -47.916 0.500Aperture Z3 S18 125.580L10 3.670 1.648 33.8 S19 -68.714 0.180 S20 88.886L11 6.000 1.517 64.2 S21 -32.051L12 5.500 1.785 25.7 S22 -83.481 8.017 S23 118.446L13 6.000 1.581 40.9 S24 -42.226L14 2.000 1.717 29.5 S25 37.606 1.900 S26 341.397L15 4.800 1.805 25.5 S27 -95.200______________________________________ZOOM SPACING DATAEFL Z1 Z2 Z3 BFLFVD______________________________________ 38.50 mm 7.00 mm 19.19 mm 23.74 mm 48.36 mm 186.24 mm 73.00 22.36 10.63 13.68 66.89201.50136.00 36.30 4.64 5.02 81.83215.74194.06 42.86 0.59 1.06 89.52221.97______________________________________

where L1-L15 are lens elements progressively from the object to the image end, S1-S27 are the lens surfaces from the object to image end, N.sub.D is the index of refraction of each lens element, and V.sub.D is the dispersion of each lens element as measured by its Abbe number.

8. A zoom lens as described in claim 1 scaled to an image frame of 24.times.36 mm described substantially as follows:

______________________________________ AXIAL DISTANCE SURFACE BETWEENLENS RADII(mm) SURFACES(mm) N.sub.d V.sub.D______________________________________ S1 97.471L1 2.000 1.805 25.5 S2 46.414 0.961 S3 44.857L2 7.500 1.517 64.2 S4 -204.325 0.100 S5 33.332L3 5.000 1.658 50.9 S6 67.053 Z1 S7 48.154L4 1.400 1.834 37.3 S8 16.552 9.047 S9 -29.301L5 1.200 1.773 49.6 S10 37.735 0.200 S11 30.244L6 5.800 1.673 32.2 S12 -13.650L7 1.200 1.773 49.6 S13 -101.590 Z2Aperture 1.040 S14 651.451L8 2.500 1.834 37.3 S15 -75.022 Z3 S16 148.051L9 2.664 1.834 37.3 S17 -59.655 0.100 S18 17.272L10 3.600 1.517 64.2 S19 80.238 0.570 S20 -2295.605L11 6.060 1.785 26.1 S21 17.700 6.307 S22 47.296L12 2.000 1.805 25.5 S23 28.126 0.931 S24 29.714L13 6.000 1.517 64.2 S25 -40.167______________________________________ZOOM SPACING DATAEFL Z1 Z2 Z3 BFL FVD______________________________________ 35.90 mm 0.50 mm 10.79 mm 10.77 mm 49.46 mm 141.60 54.93 9.65 7.07 5.34 53.89 147.03 85.00 17.79 2.84 1.43 57.80 150.94101.42 20.76 0.60 0.70 58.53 151.67______________________________________

where L1-L13 are lens elements progressively from the object to the image end, S1-S25 are the lens surfaces from the object to image end, N.sub.D is the index of refraction of each lens element, and V.sub.D is the dispersion of each lens element as measured by its Abbe number.

9. A zoom lens as described in claim 1 scaled to an image frame of 24.times.36 mm described substantially as follows:

______________________________________ AXIAL DISTANCE SURFACE BETWEENLENS RADII(mm) SURFACES(mm) N.sub.D V.sub.D______________________________________ S1 87.132L1 2.500 1.805 25.5 S2 50.967L2 10.500 1.517 64.2 S3 -218.779 0.100 S4 38.314L3 4.000 1.618 55.2 S5 56.924 Z1 S6 117.702L4 1.400 1.834 37.3 S7 17.798 4.702 S8 -50.677L5 1.300 1.589 61.3 S9 49.929 0.100 S10 27.135L6 6.500 1.699 30.1 S11 -18.768L7 1.200 1.773 49.6 S12 65.374 Z2Aperture 1.700 S13 46.163L8 4.600 1.697 48.5 S14 -61.782 0.950 S15 -31.839L9 1.600 1.847 37.3 S16 -87.982 Z3 S17 2658.419L10 3.500 1.806 40.7 S18 -45.955 0.100 S19 55.327L11 5.400 1.487 70.4 S20 -30.009L12 1.300 1.805 25.5 S21 -166.729 10.527 S22 93.945L13 1.600 1.834 37.3 S23 37.521 1.956 S24 157.252L14 3.500 1.786 43.9 S25 -74.928______________________________________ZOOM SPACING DATAEFL Z1 Z2 Z3 BFLFVD______________________________________ 28.88 mm 1.75 mm 20.26 mm 16.96 mm 50.66 mm 174.73 mm 50.00 14.87 12.47 11.62 63.63177.70105.00 29.25 5.24 4.47 83.17197.24154.41 36.29 1.87 0.80 90.30204.37______________________________________

where L1-L14 are lens elements progressively from the object to the image end, S1-S25 are the lens surfaces from the object ot image end, N.sub.D is the index of refraction of each lens element, and V.sub.D is the dispersion of each lens element as measured by its Abbe number.

10. A zoom lens as described in claim 1 scaled to an image frame of 24.times.36 mm described substantially as follows:

______________________________________ AXIAL DISTANCE SURFACE BETWEENLENS RADII(mm) SURFACES(mm) N.sub.d V.sub.d______________________________________ S1 155.350L1 2.200 1.805 25.5 S2 62.701L2 9.300 1.517 64.2 S3 -159.115 0.100 S4 47.500L3 3.500 1.723 38.0 S5 73.500 Z1 S6 151.230L4 1.504 1.834 37.3 S7 20.622 5.415 S8 -38.920L5 1.400 1.598 61.3 S9 94.000 0.100 S10 37.400L6 7.700 1.699 30.1 S11 -19.236L7 1.100 1.773 49.6 S12 215.750 Z2Aperture 1.500 S13 109.780L8 3.300 1.743 49.2 S14 -144.100 0.100 S15 71.820L9 5.800 1.589 61.3 S16 -37.400L10 1.500 1.834 37.3 S17 -2090.000 Z3 S18 102.599L11 3.800 1.744 44.9 S19 -110.900 0.100 S20 57.000L12 5.500 1.487 70.4 S21 -49.500L13 1.500 1.805 25.5 S22 -350.000 10.306 S23 288.000L14 1.800 1.834 37.3 S24 35.570 1.663 S25 129.810L15 3.200 1.834 37.3 S26 -95.800______________________________________ZOOM SPACING DATAEFL Z1 Z2 Z3 BFLFVD______________________________________ 39.50 mm 8.97 mm 23.72 mm 24.87 mm 55.41 mm 187.60 mm 70.08 26.06 14.23 14.13 70.87199.93137.21 42.17 5.56 4.60 90.13217.10193.26 49.82 0.95 0.94 97.09223.94______________________________________

where L1-L15 are lens elements progressively from the object to the image end, S1-S26 are the lens surfaces from the object to image end, N.sub.D is the index of refraction of each lens element, and V.sub.D is the dispersion of each lens element as measured by its Abbe number.

11. A zoom lens as described in claim 1 scaled to an image frame of 24.times.36 mm described substantially as follows:

______________________________________ AXIAL DISTANCE SURFACE BETWEENLENS RADII(mm) SURFACES(mm) N.sub.d V.sub.d______________________________________ S1 243.578L1 2.200 1.805 25.5 S2 58.863L2 8.500 1.517 64.2 S3 -121.610 0.100 S4 44.309L3 4.600 1.664 35.9 S5 99.626 Z1 S6 92.673L4 1.400 1.834 37.3 S7 17.808 5.358 S8 -36.818L5 1.400 1.517 64.2 S9 56.813 0.100 S10 27.885L6 6.800 1.699 30.1 S11 -17.481L7 1.100 1.773 49.6 S12 61.149 Z2Aperture 1.500 S13 46.648L8 4.100 1.603 60.7 S14 -56.897 0.900 S15 -31.060L9 1.500 1.834 37.3 S16 -56.434 Z3 S17 184.516L10 3.800 1.664 35.9 S18 -52.262 0.100 S19 52.773L11 5.500 1.487 70.4 S20 -32.229L12 1.500 1.805 25.5 S21 -157.768 9.691 S22 200.032L13 1.800 1.834 37.3 S23 37.027 1.692 S24 141.797L14 3.200 1.743 49.2 S25 -67.446______________________________________ZOOM SPACING DATAEFLZ1 Z2Z3 BFLFVD______________________________________ 28.88 mm 1.00 mm 22.71 mm 19.79 mm 55.01 mm 167.60 mm 66.5221.03 11.5710.90 79.41192.00135.0034.22 5.164.12 104.19216.77193.0040.61 1.950.94 115.22227.80______________________________________

where L1-L14 are lens elements progressively from the object to the image end, S1-S25 are the lens surfaces from the object to image end, N.sub.D is the index of refraction of each lens element, and V.sub.D is the dispersion of each lens element as measured by its Abbe number.

12. A lens according to claim 1 having at least one aspheric surface defined by the equation ##EQU2## where x is the surface sag at a semi-aperture distance Y from the axis A of the lens, C is the curvature of a lens surface of the optical axis A equal to the reciprocal of the radius of the optical axis, K is a conic constant.

13. A zoom lens as described in claim 12 scaled to an image frame of 24.times.36 mm described substantially as follows:

______________________________________ AXIAL DISTANCE SURFACE BETWEENLENS RADII(mm) SURFACES(mm) N.sub.d V.sub.d______________________________________ S1 -781.242L1 2.500 1.805 25.5 S2 156.226 0.105 S3 75.329L2 7.473 1.487 70.4 S4 -728.237 0.110 S5 58.221L3 8.716 1.491 57.2 S6 -1188.187 Z1 S7 64.040L4 1.600 1.834 37.3 S8 18.666 5.066 S9 -63.869L5 2.000 1.815 65.2 S10 18.672L6 3.537 1.805 25.5 S11 56.132 Z2Aperture 0.800 S12 40.929L7 2.500 1.564 60.8 S13 -356.168 1.575 S14 -22.864L8 1.500 1.517 52.2 S15 -47.470 Z3 S16 36.349L9 7.011 1.640 60.2 S17 -106.713 0.100 S18 24.204L10 6.572 1.678 55.5 S19 165.775L11 1.600 1.805 25.5 S20 20.436 3.220 S21 48.161L12 2.924 1.491 57.2 S22 -528.121______________________________________Surface S5 and S22 are asphericS5 S22______________________________________K = -1.000 D .2551 .times. 10.sup.-04 E .3653 .times. 10.sup.-07 F .2158 .times. 10.sup.-11 G .5188 .times. 10.sup.-12______________________________________ZOOM SPACING DATAEFL Z1 Z2 Z3 BFL FVD______________________________________ 28.50 mm 0.71 mm 14.95 mm 18.93 mm 40.12 mm 136.62 50.00 16.49 7.08 11.02 48.58 145.08102.00 33.46 0.83 0.30 60.60 157.10______________________________________

where L1-L12 are lens elements progressively from the object to the image end, S1-S22 are the lens surfaces from the object to image end, N.sub.D is the index of refraction of each lens element, and V.sub.D is the dispersion of each lens element as measured by its Abbe number.

14. A zoom lens as described in claim 12 scaled to an image frame of 24.times.36 mm described substantially as follows:

______________________________________ AXIAL DISTANCE SURFACE BETWEENLENS RADII(mm) SURFACES(mm) N.sub.d V.sub.d______________________________________ S1 195.772L1 2.500 1.805 25.5 S2 67.344 0.500 S3 66.244L2 7.000 1.613 54.6 S4 -411.314 0.100 S5 46.824L3 4.000 1.834 42.4 S6 66.472 Z1 S7 100.775L4 1.600 1.834 37.3 S8 21.211 5.851 S9 -153.496L5 2.000 1.481 69.6 S10 19.263L6 3.537 1.824 24.6 S11 45.706 Z2Aperture 0.800 S12 42.808L7 2.500 1.581 59.3 S13 -8183.436 1.575 S14 -24.474L8 1.500 1.776 40.0 S15 -37.379 Z3 S16 35.468L9 5.400 1.663 57.9 S17 -104.707 0.100 S18 26.104L10 4.800 1.692 54.4 S19 238.734L11 1.600 1.805 25.5 S20 23.704 5.602 S21 -402.429L12 2.924 1.491 57.2 S22 -402.429 0.200 S23 107.084L13 3.000 1.482 55.9 S24 -405.409______________________________________Aspheric Surface S22______________________________________ D .2455 .times. 10.sup.-04 E .5226 .times. 10.sup.-07 F -.6774 .times. 10.sup.-10 G .3605 .times. 10.sup.-12______________________________________ZOOM SPACING DATAEFL Z1 Z2 Z3 BFLFVD______________________________________ 28.51 mm 0.71 mm 20.48 mm 18.70 mm 38.51 mm 139.51 mm 50.00 18.89 9.95 11.04 46.17147.17 75.03 30.18 4.82 4.89 52.33153.33102.04 38.16 1.22 0.50 56.72157.72______________________________________

where L1-L12 are lens elements progressively from the object to the image end, S1-S24 are the lens surfaces from the object to image end, N.sub.D is the index of refraction of each lens element, and V.sub.D is the dispersion of each lens element as measured by its Abbe number.

15. A zoom lens as described in claim 12 scaled to an image frame of 24.times.36 mm described substantially as follows:

______________________________________ AXIAL DISTANCE SURFACE BETWEENLENS RADII(mm) SURFACES(mm) N.sub.d V.sub.d______________________________________ S1 122.608L1 2.500 1.805 25.5 S2 55.746 0.500 S3 54.539L2 10.600 1.517 64.2 S4 -249.144 0.100 S5 41.949L3 4.328 1.702 41.2 S6 75.883 Z1 S7 84.733L4 1.500 1.834 37.3 S8 17.252 5.757 S9 -52.546L5 1.300 1.773 49.6 S10 45.963 0.200 S11 29.582L6 6.738 1.673 32.2 S12 -17.554L7 1.200 1.773 49.6 S13 -532.786 Z2Aperture 1.000 S14 77.815L8 2.452 1.834 37.3 S15 -58.953 0.550 S16 -34.685L9 1.300 1.717 29.5 S17 -70.992 Z3 S18 33.486L10 4.446 1.720 50.3 S19 -174.911 0.100 S20 37.651L11 5.299 1.717 48.0 S21 -55.577L12 2.000 1.785 26.1 S22 23.790 5.147 S23 -32.331L13 2.300 1.491 57.2 S24 -110.442 0.200 S25 150.096L14 6.350 1.487 70.4 S26 -31.538______________________________________Aspheric Surface S24______________________________________ D .1995 .times. 10.sup.-04 E .5118 .times. 10.sup.-07 F -.1442 .times. 10.sup.-09 G .2947 .times. 10.sup.-12______________________________________ZOOM SPACING DATAEFL Z1 Z2 Z3 BLFFVD______________________________________ 28.60 mm 1.50 mm 15.87 mm 17.87 mm 41.80 mm 144.94 mm 50.00 17.14 8.62 9.48 50.91153.87 75.00 26.79 4.36 4.09 56.92159.89102.00 33.36 1.00 0.88 60.76163.76______________________________________

where L1-L14 are lens elements progressively from the object to the image end, S1-S26 are the lens surfaces from the object to the image end, N.sub.d is the index of refraction of each lens element, and V.sub.d is the dispersion of each lens element as measured by its Abbe number.