PATENT ABSTRACT
Large aperture optical systems that are extremely well corrected over a large flat field and over a large spectral range are disclosed. Breathing and aberration variation during focusing are optionally controlled by moving at least two groups of lens elements independently. Aberration correction in general is aided by allowing the working distance to become short relative to the format diagonal. Field curvature is largely corrected by a steeply curved concave surface relatively close to the image plane. This allows the main collective elements to be made of low-index anomalous dispersion materials in order to correct secondary spectrum. In wide-angle example embodiments, distortion may be controlled with an aspheric surface near the front of the lens.

PATENT DESCRIPTION
CLAIM OF PRIORITY 
     This application is a continuation-in-part of and claims priority from U.S. patent application Ser. No. 12/229,031, filed Aug. 19, 2008 which issued as U.S. Pat. No. 7,733,581 on Jun. 8, 2010, and which claims priority from U.S. Provisional Application Ser. No. 60/966,036, filed on Aug. 24, 2007. U.S. Pat. No. 7,733,581 is incorporated by reference herein. 
    
    
     FIELD OF THE INVENTION 
     The present invention relates to the field of large aperture imaging optics, and more specifically relates to photographic, cinematographic and reconnaissance optical systems that are extremely well corrected over a wide range of f/numbers and over a large spectral range. 
     BACKGROUND ART 
     A long sought-after goal of photographic optical design is to provide a large aperture objective having an f/number of about f/1.5 or smaller that is well corrected for all aberrations over a large flat field and over a large spectral range. In addition, a goal for objectives used in cinematography is that they be corrected for “breathing.” Breathing is defined as a change in chief ray angle in object space as the lens is focused, and it causes objects to move radially in the image frame as they go in and out of focus. 
     The Petzval design form is capable of being well corrected at a large aperture over a small field of view, and has been used for many decades in applications such as cinema projection lenses, heads-up displays, and microscope objectives. Although the Petzval form is too limited for use as a general-purpose photographic or cinematographic lens, it provides important insights about large aperture lens design. In particular, a Petzval lens achieves excellent correction at large apertures by minimizing the power of the negative lens elements located far from the image plane. Petzval objectives often employ a negative field flattener located fairly close to the image plane. As a result, the positive lens elements can be made from low-index fluor crown glass or calcium fluoride crystal if desired in order to reduce or eliminate the secondary spectrum. 
     Examples of large aperture Petzval lenses include U.S. Pat. Nos. 2,649,021; 3,255,664; and 4,329,024. 
     The double-Gauss design form is very widely used for high-speed objectives that must cover a relatively wide field of view. However, double-Gauss designs rely on the use of positive elements made of high index crown glass in order to flatten the field, and as a result it is difficult or impossible to correct the secondary spectrum. Double-Gauss designs also have a strong tendency to suffer from oblique spherical aberration that severely limits off-axis performance at wide apertures. Vignetting is normally used to control the aberrated tangential rays, so high-speed double-Gauss designs have a characteristic strong illumination fall off coupled with a large amount of residual sagittal oblique spherical aberration. The oblique spherical aberration can be corrected to a large extent by relaxing the requirement for a large working distance and/or by introducing one or more aspheric surfaces into the design. Examples of high-speed double-Gauss designs include U.S. Pat. Nos. 2,012,822; 3,504,961; and 4,394,095. 
     Triplet derivatives, including the Sonnar and Ernostar design forms, have also been widely used in the past for high-speed objectives with small to moderate fields of view. These designs tend to have many of the strengths and shortcomings of the double-Gauss type designs, but are generally more suited to narrower fields of view. Examples include U.S. Pat. Nos. 1,975,678; 2,310,502; and 3,994,576. An interesting sub grouping of triplet derivatives designed mainly for television cameras is moderately well corrected for extremely large apertures of about f/0.7 and incorporate a meniscus doublet lens group with a very strong concave surface near the image plane. This meniscus group serves to flatten the field. These designs also have characteristics similar to Petzval lenses, particularly the minimization of negative power in large elements located far from the image plane. Examples of this type include U.S. Pat. Nos. 2,978,957; 3,300,267; 3,445,154; 3,454,326; and 3,586,420. 
     The reverse telephoto design form is useful for achieving both a large aperture and a wide field of view. However, the image quality for high-speed wide-angle examples is generally mediocre, and these lenses need to be stopped down substantially to achieve good results. An exception is found in microfilm objectives, such as U.S. Pat. Nos. 3,817,602 and 4,310,223; and microscope objectives such as U.S. Pat. No. 5,920,432. However in these cases a high image quality is achieved at the expense of image size. Examples of high-speed wide-angle reverse telephoto designs include U.S. Pat. Nos. 3,992,085; 4,025,169; 4,095,873; 4,136,931; and 5,315,441. 
     Despite the above efforts, virtually all high-speed photographic lenses designed to date are compromised by the need to balance large aberrations against each other. Most often these designs are notably soft in the outer sub-group of the image field, and must be stopped far down to achieve good performance. Accordingly, there is a need for high-speed optical systems that are both extremely well corrected and reasonably compact when scaled to an image diagonal of about 28 mm. In addition, for cinematographic applications, there is a need for such optical systems to be well corrected for breathing. 
     SUMMARY OF THE INVENTION 
     The present invention is directed to optical systems that are extremely well corrected over a wide range of f/numbers and a relatively large spectral range. More specifically, the present invention is directed to cinematographic and photographic objectives that are extremely well corrected over a range of f/numbers between f/0.6 and f/4, and are optionally well corrected for breathing. 
     The recent rise of electronic image capture in the fields of still photography and cinematography have opened up new challenges and opportunities for the optical systems used in these applications. In cinematography, for example, new high-quality digital cameras often dispense with an optical viewfinder in order to reduce noise and cost. As a result, the optics can take advantage of a shorter working distance to improve speed, optical correction, or both. However, the optical design may need to take into account any plane parallel filters in the optical path near the image plane. These include IR/UV blocking filters, anti-aliasing filters, and sensor cover glasses. The exit pupil of the optical design must also match the characteristics of the sensor as closely as possible. 
     In the present invention, an extraordinary degree of optical correction is achieved at a very large aperture by allowing the working distance to become relatively short, and also by employing a unique optical construction. The main part of this optical construction is a primary lens group (“primary group”), or PG, that in turn comprises a front negative-powered sub-group P 1 , followed by a positive-powered sub-group P 2 , followed by a negative-powered sub-group P 3 , and optionally followed by a final sub-group P 4  that can be either negatively or positively powered. In the discussion below, the focal length of primary group PG is called FG. Also in the discussion below, the focal lengths of sub-groups P 1 , P 2 , P 3 , and P 4  are called F 1 , F 2 , F 3 , and F 4 , respectively. 
     Sub-group P 1  has only a weak negative power, and its function is to help flatten the field as well as to reduce the chief ray angle as it enters the remaining sub-group of PG. Sub-group P 1  can also be usefully split into two parts that move separately during focusing to aid in aberration and breathing correction. Elements in sub-group P 1  that are near the aperture stop are also good locations for an aspheric surface to correct spherical aberration. 
     Sub-group P 2  has a strong positive power, and its focal length F 2  is somewhat less than FG. Sub-group P 2  is the main collective group of lenses within primary group PG. Since field curvature is corrected elsewhere within the lens it is not necessary to use high index glass for the positive elements in sub-group P 2 , so low index glass with anomalous dispersion can be used. This is analogous to the use of low-index anomalous-dispersion glass in the positive-powered elements of a Petzval type lens, where field curvature is typically corrected by the use of a field flattener located near the image plane. The ability to use low-index anomalous-dispersion glasses is critical for achieving the excellent color correction necessary for any fast lens that must be extremely well corrected. By judicious use of these anomalous-dispersion glasses it is possible to achieve good correction over a very broad waveband. Surfaces within sub-group P 2  that are near the aperture stop can provide a good location for an asphere to correct spherical aberration. 
     Sub-group P 3  is a moderately high-powered negative group whose primary function is to flatten the field and to correct astigmatism. The rearmost surface of sub-group P 3 , referred to below as SC, has a very short radius of curvature and is concave toward the image plane. Much of the field flattening correction takes place at surface SC. Rearmost surface SC is readily identified as being the first air-glass interface within primary group PG that is concave toward the image and that has a radius of curvature less than FG and in which the marginal ray height is substantially less than the maximum value of the marginal ray height in the system. The radius of curvature of rearmost surface SC is called R SC . The distance along the optical axis from surface SC to the image plane is called Z SC . 
     Rearmost surface SC is also located axially in such a way that the marginal ray height as it intersects this surface is small relative to its largest value. This condition may be expressed as 1.3&lt;y MAX /y SC &lt;4.0, where y MAX  is the maximum height of the marginal ray as it passes through the lens and y SC  is the height of the marginal ray as it intersects the surface SC. The large negative power of surface SC combined with the fact that the marginal ray height is relatively small at SC is what enables this particular surface to have such an important contribution to field curvature correction. 
     Sub-group P 4  is the remaining group of elements located on the image side of sub-group P 3 . This sub-group has relatively low power, which can be either positive or negative. The function of sub-group P 4  is primarily to help control the exit pupil location, to correct distortion, and to make final adjustments to astigmatism. Sub-group P 4  is optional and may be eliminated if requirements for off-axis image quality are not so stringent. 
     Example embodiments with an image diagonal of about 28 mm and a focal length of about 40 mm or larger may be solely comprised of primary lens group PG. Example embodiments with an image diagonal of about 28 mm and a focal length shorter than about 40 mm are best comprised of primary lens group PG plus a grouping of lens elements in front of PG. These elements in front of PG essentially act as a wide-angle afocal attachment to shorten the focal length and widen the field of view while allowing primary group PG to retain its basic structure. Of course, the added elements in front of PG need not function precisely as an afocal attachment, and can have a net negative or net positive power. Consequently, the object magnification for primary group PG, calculated as the slope of the marginal ray entering the primary group divided by the slope of the marginal ray exiting this group, can be negative, zero, or positive. The object magnification of primary group PG is called OBMG PG    
     Lenses for cinematography must be well corrected for a wide range of object distances, and in addition should be well corrected for breathing. Breathing is a phenomenon in which off-axis image points move radially within the image format as they go in and out of focus. A lens that is focused by simply moving the entire optical structure as a unit along the optical axis will suffer from breathing because the image format will subtend a larger angle for a distant object than it will for a close object. A lens corrected for breathing will have a chief ray angle that is constant for all focus settings. Breathing is defined by the expression:
 
 B =(100%)(chief ray angle−chief ray angle at infinity)/(chief ray angle at infinity),
 
where the chief ray angle is measured in object space.
 
     It is also desirable for cinematographic and general photographic objectives to have a variable soft focus feature so that a wide variety of special effects can be achieved. This is especially true of cinematographic objectives with a focal length of about 50 mm and longer, which are frequently used for close-up headshots and the like. The best method of introducing a soft focus is to vary the amount of spherical aberration in a controlled manner by an axial adjustment of one or more lens elements. Ideally a lens with a variable soft focus adjustment will also have a means for adjusting paraxial focus at the same time spherical aberration is changed. This will allow for the best balance of defocus and spherical aberration to achieve the desired soft focus effect. 
     In addition to cinematographic and general photographic applications, the present invention is well-suited to demanding reconnaissance, surveillance, and inspection applications. These applications generally require a narrower magnification range than cinematography or general photography. For example, lenses used for aerial reconnaissance are typically always used with a very distant object, and need not be optimized for close focusing. Surveillance lenses are also typically used with distant object distances, though perhaps a bit closer than the object distances encountered in aerial photography. Inspection lenses are typically used for close object distances, but only a narrow range of magnifications. 
     A lens designed according to the present invention for cinematographic or general photographic applications may be applied to applications requiring only a narrow range of magnifications simply because the broad magnification range of the design encompasses the magnification range required for the application at hand. However, it is generally be better to optimize an optical system to work best under the actual conditions in which it is used. So, for example, an aerial reconnaissance lens should be optimized to perform best at infinite or nearly infinite conjugates, and this performance need not be compromised to improve performance for closer object distances. 
     Surveillance and reconnaissance lenses are also often required to work well over an extended waveband that includes the near-infrared portion of the spectrum. This allows for improved imagery through haze and atmospheric scattering, and it also permits the use of specialized sensors for low-light applications. 
     Accordingly, an aspect of the invention is a large-aperture imaging optical system for forming an image of an object and having an f/number. The optical system includes a) a primary lens group PG having a focal length FG comprising in order from the object to the image:
         i) a front negative-powered lens sub-group P 1  having a focal length F 1 ;   ii) a positive-powered lens sub-group P 2  having a focal length F 2 ; and   iii) a negative-powered lens sub-group P 3  having a focal length F 3 .       

     The primary lens group PG contains a surface SC that is concave toward the image, the marginal ray height at surface SC being less than 0.8 times the maximum marginal ray height at any lens surface within PG. The f/number is in the range from f/0.6 to f/4. 
     Another aspect of the invention is a large-aperture imaging optical system having an f/number and that forms an image of an object. The optical system includes a) a primary lens group PG having a focal length FG comprising in order from the object to the image:
         i) a front negative-powered lens sub-group P 1  having a focal length F 1 ;   ii) a positive-powered lens sub-group P 2  having a focal length F 2 ;   iii) a negative-powered lens sub-group P 3  having a focal length F 3 ; and   iv) a lens sub-group P 4  having a focal length F 4  that can be either negatively or positively powered.       

     The primary lens group PG contains a surface SC that is concave toward the image, the marginal ray height at surface SC being less than 0.85 times the maximum marginal ray height at any lens surface within PG. The f/number is in the range from f/0.6 to f/4. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1   a  is a layout drawing of an f/1.33 lens system showing magnification settings of 0.0× and −0.137× according to Example 1 of the present invention. 
         FIG. 1   b  are plots of MTF vs. Image Height at spatial frequencies of 10 cycles/mm, 20 cycles/mm and 40 cycles/mm for magnification settings of 0.0×, −0.032×, −0.064× and −0.137× over a waveband of 435 nm to 656 nm according to Example 1 of the present invention. 
         FIG. 1   c  are plots of Distortion and Astigmatism (S and T) vs. Image Height for magnification settings of 0.0×, −0.032×, −0.064× and −0.137× according to Example 1 of the present invention. 
         FIG. 1   d  is a plot of Relative Illumination vs. Image Height for apertures of f/1.33 and f/2.8 and a magnification of 0.0× according to Example 1 of the present invention. 
         FIG. 1   e  shows three plots of the amount of spherical aberration (mm) vs. Relative Pupil Height for three different soft-focus corrections (Type  1 , Type  2  and Type  3 ) that each involve axial motions of the third and fourth lens elements; 
         FIG. 2   a  is a layout drawing of an f/1.4 lens system showing magnification settings of 0.0× and −0.124× according to Example 2 of the present invention. 
         FIG. 2   b  are plots of MTF vs. Image Height at spatial frequencies of 10 cycles/mm, 20 cycles/mm and 40 cycles/mm for magnification settings of 0.0×, −0.032×, −0.064× and −0.124× over a waveband of 435 nm to 656 nm according to Example 2 of the present invention. 
         FIG. 2   c  are plots of Distortion and Astigmatism (S and T) vs. Image Height for magnification settings of 0.0×, −0.032×, −0.064× and −0.124× according to Example 2 of the present invention. 
         FIG. 2   d  is a plot of Relative Illumination vs. Image Height for apertures of f/1.4 and f/2.8 and a magnification of 0.0× according to Example 2 of the present invention. 
         FIG. 3   a  is a layout drawing of an f/1.4 lens system according to Example 3 of the present invention. 
         FIG. 3   b  are plots of MTF vs. Image Height at spatial frequencies of 10 cycles/mm, 20 cycles/mm and 40 cycles/mm for a magnification setting of 0.0× over a waveband of 435 nm to 656 nm according to Example 3 of the present invention. 
         FIG. 3   c  are plots of Distortion and Astigmatism (S and T) vs. Image Height for a magnification setting of 0.0× according to Example 3 of the present invention. 
         FIG. 3   d  is a plot of Relative Illumination vs. Image Height for apertures of f/1.4 and f/2.8 and a magnification of 0.0× according to Example 3 of the present invention. 
         FIG. 4   a  is a layout drawing of an f/1.33 lens system showing magnification settings of 0.0× and −0.094× according to Example 4 of the present invention. 
         FIG. 4   b  are plots of MTF vs. Image Height at spatial frequencies of 10 cycles/mm, 20 cycles/mm and 40 cycles/mm for magnification settings of 0.0×, −0.030×, −0.060× and −0.094× over a waveband of 435 nm to 656 nm according to Example 4 of the present invention. 
         FIG. 4   c  are plots of Distortion and Astigmatism (S and T) vs. Image Height for magnification settings of 0.0×, −0.030×, −0.060× and −0.094× according to Example 4 of the present invention. 
         FIG. 4   d  is a plot of Relative Illumination vs. Image Height for apertures of f/1.33 and f/2.8 and a magnification of 0.0× according to Example 4 of the present invention. 
         FIG. 5   a  is a layout drawing of an f/1.4 lens system showing magnification settings of 0.0× and −0.033× according to Example 5 of the present invention. 
         FIG. 5   b  are plots of MTF vs. Image Height at spatial frequencies of 10 cycles/mm, 20 cycles/mm and 40 cycles/mm for magnification settings of 0.0× and −0.033× over a waveband of 435 nm to 656 nm according to Example 5 of the present invention. 
         FIG. 5   c  are plots of Distortion and Astigmatism (S and T) vs. Image Height for magnification settings of 0.0× and −0.033× according to Example 5 of the present invention. 
         FIG. 5   d  is a plot of Relative Illumination vs. Image Height for apertures of f/1.4 and f/2.8 and a magnification of 0.0× according to Example 5 of the present invention. 
         FIG. 6   a  is a layout drawing of an f/1.4 lens system showing magnification settings of 0.0× and −0.05× according to Example 6 of the present invention. 
         FIG. 6   b  are plots of MTF vs. Image Height at spatial frequencies of 10 cycles/mm, 20 cycles/mm and 40 cycles/mm for magnification settings of 0.0× and −0.05× over a waveband of 435 nm to 656 nm according to Example 6 of the present invention. 
         FIG. 6   c  are plots of Distortion and Astigmatism (S and T) vs. Image Height for magnification settings of 0.0×, and −0.05× according to Example 6 of the present invention. 
         FIG. 6   d  is a plot of Relative Illumination vs. Image Height for apertures of f/1.4 and f/2.8 and a magnification of 0.0× according to Example 6 of the present invention. 
         FIG. 7   a  is a layout drawing of an f/1.4 lens system showing magnification settings of 0.0× and −0.065× according to Example 7 of the present invention. 
         FIG. 7   b  are plots of MTF vs. Image Height at spatial frequencies of 10 cycles/mm, 20 cycles/mm and 40 cycles/mm for magnification settings of 0.0× and −0.065× over a waveband of 435 nm to 656 nm according to Example 7 of the present invention. 
         FIG. 7   c  are plots of Distortion and Astigmatism (S and T) vs. Image Height for magnification settings of 0.0× and −0.065× according to Example 7 of the present invention. 
         FIG. 7   d  is a plot of Relative Illumination vs. Image Height for apertures of f/1.4 and f/2.8 and a magnification of 0.0× according to Example 7 of the present invention. 
         FIG. 8   a  is a layout drawing of an f/1.33 lens system showing magnification settings of 0.0× and −0.137× according to Example 8 of the present invention. 
         FIG. 8   b  are plots of MTF vs. Image Height at spatial frequencies of 10 cycles/mm, 20 cycles/mm and 40 cycles/mm for magnification settings of 0.0×, −0.032×, −0.064× and −0.137× over a waveband of 435 nm to 656 nm according to Example 8 of the present invention. 
         FIG. 8   c  is a plot of MTF vs. Image Height at a spatial frequency of 20 cycles/mm for temperatures of 0 C, 20 C and 40 C according to Example 8 of the present invention. 
         FIG. 8   d  are plots of Distortion and Astigmatism (S and T) vs. Image Height for magnification settings of 0.0×, −0.032×, −0.064× and −0.137× according to Example 8 of the present invention. 
         FIG. 8   e  is a plot of Relative Illumination vs. Image Height for apertures of f/1.33 and f/2.8 and a magnification of 0.0× according to Example 8 of the present invention. 
         FIG. 9   a  is a layout drawing of an f/1.4 lens system showing magnification settings of 0.0× and −0.064× according to Example 9 of the present invention. 
         FIG. 9   b  are plots of MTF vs. Image Height at spatial frequencies of 10 cycles/mm, 20 cycles/mm and 40 cycles/mm for magnification settings of 0.0× and −0.064× over a waveband of 435 nm to 656 nm according to Example 9 of the present invention. 
         FIG. 9   c  are plots of Distortion and Astigmatism (S and T) vs. Image Height for magnification settings of 0.0× and −0.064× according to Example 9 of the present invention. 
         FIG. 9   d  is a plot of Relative Illumination vs. Image Height for apertures of f/1.4 and f/2.8 and a magnification of 0.0× according to Example 9 of the present invention. 
         FIG. 10   a  is a layout drawing of an f/1.4 lens system showing magnification settings of 0.0× and −0.056× according to Example 10 of the present invention. 
         FIG. 10   b  are plots of MTF vs. Image Height at spatial frequencies of 10 cycles/mm, 20 cycles/mm and 40 cycles/mm for magnification settings of 0.0× and −0.056× over a waveband of 435 nm to 656 nm according to Example 10 of the present invention. 
         FIG. 10   c  are plots of Distortion and Astigmatism (S and T) vs. Image Height for magnification settings of 0.0× and −0.056× according to Example 10 of the present invention. 
         FIG. 10   d  is a plot of Relative Illumination vs. Image Height for apertures of f/1.4 and f/2.8 and a magnification of 0.0× according to Example 10 of the present invention. 
         FIG. 11   a  is a layout drawing of an f/1.4 lens system showing magnification settings of 0.0× and −0.037× according to Example 11 of the present invention. 
         FIG. 11   b  are plots of MTF vs. Image Height at spatial frequencies of 10 cycles/mm, 20 cycles/mm and 40 cycles/mm for magnification settings of 0.0× and −0.037× over a waveband of 435 nm to 656 nm according to Example 11 of the present invention. 
         FIG. 11   c  are plots of Distortion and Astigmatism (S and T) vs. Image Height for magnification settings of 0.0× and −0.037× according to Example 11 of the present invention. 
         FIG. 11   d  is a plot of Relative Illumination vs. Image Height for apertures of f/1.4 and f/2.8 and a magnification of 0.0× according to Example 11 of the present invention. 
         FIG. 12   a  is a layout drawing of an f/1.4 lens system according to Example 12 of the present invention. 
         FIG. 12   b  are plots of MTF vs. Image Height at spatial frequencies of 50 cycles/mm, 100 cycles/mm and 200 cycles/mm for an object at infinity over a waveband of 435 nm to 1000 nm according to Example 12 of the present invention. 
         FIG. 12   c  are plots of Distortion and Astigmatism (S and T) vs. Image Height for an object at infinity according to Example 12 of the present invention. 
         FIG. 12   d  is a plot of Relative Illumination vs. Image Height for an aperture of f/1.4 and an object at infinity according to Example 12 of the present invention. 
         FIG. 13   a  illustrates a cross-sectional layout of an example 150 mm focal length objective optimized for aerial reconnaissance. 
         FIG. 13   b  plots the MTF vs. Image Height at 50, 100 and 200 cycles/mm for an object at infinity for the objective of  FIG. 13   a.    
         FIG. 14   a  shows a cross-sectional layout of an example of a very fast 50 mm focal length objective optimized for low light applications. 
         FIG. 14   b  plots the MTF vs. Image Height at 10, 20 and 40 cycles/mm for an object at infinity for the objective of  FIG. 14   a.    
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     The present invention relates to the field of large aperture imaging optics. More specifically, the present invention relates to optical systems that are extremely well corrected throughout a large waveband over a large range of f/numbers. 
     In the Summary of the Invention section above, in the descriptions below and in the claims, the phrases “well-corrected” and “extremely well-corrected” in relation to the optical system of the present invention is understood in the art to mean that the collective effect of aberrations in the optical system are reduced to the point where the optical system is able to satisfactorily perform its particular imaging function. For example, in a photographic objective optical system according to the present invention, the Modulation Transfer Function (MTF) is an excellent and widely accepted means by which to judge the state of optical correction. 
     In particular, a photographic or cinematographic objective with a focal length greater than about 35 mm intended for a format diagonal of about 28 mm is considered to be well-corrected if the MTF values at 20 cycles/mm is approximately 80% or greater on-axis and is approximately 60% or greater at image heights less than or equal to 14 mm off-axis. 
     For wide-angle objectives the criteria for “well-corrected” are relaxed slightly in the outer part of the image field. Thus, a photographic or cinematographic objective with a focal length less than about 35 mm intended for a format diagonal of about 28 mm is considered to be well-corrected if the MTF values at 20 cycles/mm is 80% or greater on-axis and is approximately 50% or greater at image heights less than or equal to 14 mm off-axis. 
     An objective covering a format diagonal of about 28 mm can be considered to be “extremely well-corrected” if it fulfills the above conditions for off-axis field points and the MTF value at 40 cycles/mm is approximately 80% or greater on-axis. 
     EXAMPLE 1 
     Example 1, which is a 65 mm focal length objective for 35 mm format cinematography, is illustrated in  FIG. 1   a , which shows cross-sectional layouts at magnifications of 0 and −0.137×. 
     All of the element and group designations mentioned below are shown in  FIG. 1   a . The relative aperture is f/1.33, the image diagonal is 28 mm, and the diagonal field of view (FOV) is 24.3 degrees. 
     MTF vs. Image Height at 10, 20 and 40 cycles/mm for four different object distances is illustrated in  FIG. 1   b . These curves indicate that Example 1 is extremely well corrected at f/1.33, with MTF values at 40 cycles/mm greater than 80% near the optical axis in the middle part of the focusing range and very near 80% at the extreme ends of the focusing range. This extraordinary performance falls off very gradually to the corner of the field, and the S and T curves stay close together.  FIG. 1   c  shows distortion and astigmatism (Coddington curves) for the same four object distances used in  FIG. 1   b . Distortion is virtually zero at all magnifications, and astigmatism is also very well controlled.  FIG. 1   d  is a plot of relative illumination vs. image height at f/1.33 and f/2.8, and it indicates that the Example 1 design has extremely low illumination falloff for such a high-speed lens. 
     The primary group PG comprises the entire lens except for a plane parallel filter element  114 . As discussed above, primary group PG comprises a negative powered front sub-group P 1 , followed by a positive powered sub-group P 2 , followed by a negative powered sub-group P 3 , followed by a sub-group P 4  that can be either positively or negatively powered. 
     In Example 1, sub-group P 1  comprises two negative lens groups: a negative doublet  116  and a negative singlet  103 . The convex object-facing surface of element  103  is aspherical, and singlet  103  is made of S-BSL7 to ensure that manufacturing this aspherical surface will not be problematic. The doublet  116  uses high-index anomalous dispersion materials N-KZFS4 and S-NPH1 to advantage, and as a result aids in reducing secondary and tertiary chromatic aberrations. The surface of sub-group P 1  closest to the object is a concave surface. 
     Sub-group P 2  comprises two positive lens groups: a positive singlet  105  and a positive doublet  117 . The positive elements in sub-group P 2  are elements  105  and  107 . Both are made of low-index anomalous dispersion material S-FPL51. The single negative element  106  is made of a matching anomalous dispersion material N-KZFS4. Since most of the positive optical power for Example 1 resides in sub-group P 2 , the system as a whole is very well corrected for chromatic aberrations, and is in fact apochromatic over a waveband extending from the deep violet end of the visible spectrum to about 1000 nm in the near infrared. 
     Sub-group P 3  comprises a single negative powered doublet  118 . Anomalous dispersion materials S-FPL51y and N-KZFS4 are used for the individual lens elements  108  and  109 , respectively, which aids in the correction of secondary spectrum. The outer shape of doublet  118  is meniscus toward the image plane. The concave surface SC is strongly curved and therefore helps a great deal in correcting field curvature and astigmatism. 
     Sub-group P 4  is a fairly weak positive group comprising a positive doublet  119 , a negative singlet  112 , and a positive singlet  113 . Sub-group P 4  serves mainly to correct distortion and astigmatism, and to make fine adjustments to the exit pupil location. 
     Element  114  is a plane parallel plate that serves to model the effect of the anti-aliasing filter, the IR/UV filter, and the sensor cover plate that are commonly found in digital cameras. Element  114  is not intended as a precise model for any particular brand or model of camera, but rather is intended as a viable means for avoiding any filter-induced aberrations. In Example 1, element  114  has been made fairly thick, and in all likelihood would be too thick to accurately model a camera filter pack. However, in this case the thickness of element  114  could be reduced so that when a real filter pack is introduced the aberration balance is not disturbed. This is particularly important in digital photography and cinematography with extremely well corrected high-speed lenses because the filter pack thickness is likely to vary from camera to camera, and it will be a great advantage to be able to customize the lens for an individual camera simply by changing a filter in the rear. 
     Focusing from a distant to a close object is accomplished by moving sub-group P 1  and group  120  independently away from the image plane as illustrated by  FIG. 1   a . Element  113  is stationary with respect to the image plane. This complex focusing motion ensures that performance remains very high throughout the focusing range, and simultaneously ensures that breathing is almost zero. Breathing reaches a maximum value of only −0.09% at closest focus, which is almost undetectable. In Example 1, sub-group P 1  moves a relatively great distance during focusing because it is a relatively weak group. Although this results in a good aberration balance and loose tolerances it does also result in greater bulk as the lens is focused up close. 
     Soft focus correction is readily accomplished in Example 1 by a number of different methods, of which three are particularly useful. These three methods, which will be called Type 1, Type 2, and Type 3, respectively, all involve axial motions of the third and fourth lens elements  103  and  105 . The advantage of these three methods is that the two elements involved are located near the aperture stop, so the induced aberration is almost all spherical aberration, and it is added almost uniformly throughout the image field. 
     In Type 1, soft-focus element  103  is displaced axially. When the displacement is positive, meaning that element  103  moves toward the image plane, the spherical aberration correction is changed from its nominal well-corrected state to an over-corrected state. Over-corrected spherical aberration is effective in giving defocused foreground highlights a soft edge. When the displacement is negative, meaning that element  103  moves away from the image plane, the spherical aberration is changed from nominal to an under-corrected state. Under-corrected spherical aberration is effective in giving background highlights a soft edge. The top set of curves in  FIG. 1   e  shows the effect of moving element  103  by plus or minus 1 mm. An important feature of Type 1 soft-focus is that the change optical correction involves almost a pure change in spherical aberration with very little induced defocus. 
     In Type 2 soft-focus, element  105  is displaced axially. The induced spherical aberration is similar in magnitude but opposite in direction compared to Type 1. In other words, when the displacement of element  105  is positive the spherical aberration is changed from nominal to an under-corrected state rather than to an over-corrected state. Another important difference between Type 1 and Type 2 soft-focus is that a substantial amount of defocus is also induced with Type 2. The middle set of curves in  FIG. 1   e  shows the effect of moving element  105  by plus or minus 1 mm. 
     Type 3 soft-focus is a combination of Type 1 and Type 2 in which both element  103  and  105  are displaced axially. By taking advantage of the different defocus and spherical aberration inducing qualities of Type 1 and Type 2, Type 3 is able to achieve a wide range of spherical aberration states together with a specific amount of defocus. In general, a modest amount of defocus is beneficial when adding spherical aberration to a lens system to achieve a soft focus effect. The bottom set of curves in  FIG. 1   e  shows the effect of moving element  103  by minus 0.5 mm and element  105  by plus 0.5 mm. 
     EXAMPLE 2 
     Example 2, which is a 65 mm focal length objective for 35 mm format cinematography, is illustrated in  FIG. 2   a , which shows cross-sectional layouts at magnifications of 0 and −0.124×. All of the element and group designations mentioned below are shown in  FIG. 1   a . The relative aperture is f/1.4, the image diagonal is 28 mm, and the diagonal field of view (FOV) is 24.3 degrees. 
     MTF vs. Image Height at 10, 20 and 40 cycles/mm for four different object distances is illustrated in  FIG. 2   b . These curves indicate that Example 2 is extremely well corrected at f/1.4, with MTF values at 40 cycles/mm greater than 80% near the optical axis in the middle part of the focusing range and very near 80% at closest focus. This extraordinary performance falls off very gradually to the corner of the field, and the S and T curves stay close together.  FIG. 2   c  shows distortion and astigmatism (Coddington curves) for the same four object distances used in  FIG. 2   b . Distortion is less than 1% at all magnifications, and astigmatism is also very well controlled.  FIG. 2   d  is a plot of relative illumination vs. image height at f/1.4 and f/2.8, and it indicates that the Example 2 design has extremely low illumination falloff for such a high-speed lens. 
     The primary group PG comprises the entire lens except for a plane parallel filter element  216 . As discussed above primary group PG comprises a negative powered front sub-group P 1 , followed by a positive powered sub-group P 2 , followed by a negative powered sub-group P 3 , followed by a sub-group P 4  that can be either positively or negatively powered. In Example 2, sub-group P 1  comprises a single negative group: a negative doublet  218 . Doublet  218  uses high-index anomalous dispersion materials N-KZFS4 and S-NPH1 to advantage, and as a result aids in reducing secondary and tertiary chromatic aberrations. The surface of sub-group P 1  closest to the object is a concave surface. 
     Sub-group P 2  comprises three positive groups: a positive doublet  219 , a positive singlet  206  and a positive doublet  220 . The positive elements  204 ,  206  and  208  are made of anomalous dispersion materials S-FPL53, S-FPL53 and S-PHM52, respectively. The single negative element  207  is made of a matching anomalous dispersion material N-KZFS4. Since most of the positive optical power for Example 2 resides in sub-group P 2 , the system as a whole is very well corrected for chromatic aberrations, and is in fact apochromatic over a waveband extending from the deep violet end of the visible spectrum to about 1000 nm in the near infrared. The convex object side surface of doublet  219  is aspherical in order to control spherical aberration. 
     Sub-group P 3  comprises a single negative powered doublet  221 . Anomalous dispersion materials S-PHM52 and N-KZFS4 are used for the individual lens elements  209  and  211 , respectively, which aids in the correction of secondary spectrum. The outer shape of doublet  221  is meniscus toward the image plane. The concave surface SC is strongly curved and therefore helps a great deal in correcting field curvature and astigmatism. 
     Sub-group P 4  is a positive group comprising two positive doublets  222  and  223 . Sub-group P 4  serves mainly to correct distortion and astigmatism, and to make fine adjustments to the exit pupil location. 
     Element  216  is a plane parallel plate that serves to model the effect of the anti-aliasing filter, the IR/UV filter, and the sensor cover plate that are commonly found in digital cameras. As with the corresponding element  114  in Example 1, element  216  is not intended as a precise model for any particular brand or model of camera, but rather is intended as a viable means for avoiding any filter-induced aberrations. 
     Moving groups P 1 ,  224  and  223  independently away from the image plane as illustrated by  FIG. 2   a  accomplishes focusing from a distant to a close object. This complex focusing motion ensures that performance remains very high throughout the focusing range, and simultaneously ensures that breathing is almost zero. Breathing reaches a maximum value of only −0.01% in the middle portion of the focusing range. Compared with Example 1, sub-group P 1  moves a relatively small distance during focusing because it is a relatively weak group. This has advantages for packaging and handling. 
     EXAMPLE 3 
     Example 3, which is a 65 mm focal length objective for 35 mm format cinematography, is illustrated in  FIG. 3   a , which shows cross-sectional layout. All of the element and group designations mentioned below are shown in  FIG. 3   a . The relative aperture is f/1.4, the image diagonal is 28 mm, and the diagonal field of view (FOV) is 24.3 degrees. 
     MTF vs. Image Height at 10, 20 and 40 cycles/mm is illustrated in  FIG. 3   b . These curves indicate that Example 3 is extremely well corrected at f/1.4, with MTF values at 40 cycles/mm well above 80% over the majority of the image circle. This extraordinary performance falls off very gradually to the corner of the field, and the S and T curves stay extremely close together.  FIG. 3   c  shows distortion and astigmatism (Coddington curves). Both Distortion and astigmatism are virtually zero, and there is just a trace of field curvature.  FIG. 3   d  is a plot of relative illumination vs. image height at f/1.4 and f/2.8, and it indicates that the Example 3 design has extremely low illumination falloff. 
     The primary group PG comprises the entire lens. As discussed above, PG comprises a negative powered front sub-group P 1 , followed by a positive powered sub-group P 2 , followed by a negative powered sub-group P 3 , followed by a sub-group P 4  that can be either positively or negatively powered. 
     In Example 3, sub-group P 1  comprises a single negative element  301 . The nearly plano image-facing surface of element  301  is aspherical, and  301  is made of S-BSL7 to ensure that manufacturing this aspherical surface will not be problematic. The surface of sub-group P 1  closest to the object is a concave surface. 
     Sub-group P 2  comprises two positive groups: a positive singlet  303  and a positive triplet  312 . The positive elements in groups  303 ,  304 , and  306 , are made of anomalous dispersion S-FPL53, S-FPL53, and S-PHM52. The single negative element  305  is made of a matching anomalous dispersion material N-KZFS4. Since most of the positive optical power for Example 1 resides in sub-group P 2 , the system as a whole is very well corrected for chromatic aberrations, and is in fact apochromatic over a waveband extending from the deep violet end of the visible spectrum to about 1000 nm in the near infrared. 
     Sub-group P 3  comprises a positive powered singlet  307  and a negative powered singlet  308  to form an air spaced doublet. The combined power of elements  307  and  308  is negative. Ordinarily it would be advantageous to cement elements  307  and  308  together to loosen tolerances and improve transmission. However, the main purpose of Example 3 is to demonstrate that sub-group P 3  can comprise an air-spaced doublet instead of the cemented doublet used in other examples. Of course, it would also be possible to formulate sub-group P 3  as a cemented or air spaced triplet or quadruplet or even singlet without departing from the spirit of the present invention. Anomalous dispersion materials S-PHM52 and N-KZFS4 are used for the individual lens elements  307  and  308 , respectively, which aids in the correction of secondary spectrum. The outer shape of air-spaced doublet P 3  is meniscus toward the image plane. The concave surface SC is strongly curved and therefore helps a great deal in correcting field curvature and astigmatism. 
     Sub-group P 4  is a fairly weak positive group comprising a positive doublet  313 . Sub-group P 4  serves mainly to correct distortion and astigmatism, and to make fine adjustments to the exit pupil location. 
     EXAMPLE 4 
     Example 4 is similar in form to Example 2, but is scaled and optimized for a longer 125 mm focal length. Chromatic aberration correction is also improved, and is now superachromatic in the visible to near infrared range. Example 4 is illustrated in  FIG. 4   a , which shows cross-sectional layouts at magnifications of 0 and −0.094×. All of the element and group designations mentioned below are shown in  FIG. 4   a . The relative aperture is f/1.33, the image diagonal is 28 mm, and the diagonal field of view (FOV) is 12.8 degrees. 
     MTF vs. Image Height at 10, 20 and 40 cycles/mm for four different object distances is illustrated in  FIG. 4   b . These curves indicate that Example 4 is extremely well corrected at f/1.33, with MTF values at 40 cycles/mm greater than 80% near the optical axis except at the closest object distance. At the optimum object distance of about 3 to 4 meters the on-axis MTF at 40 cycles/mm exceeds 90% on-axis. This extraordinary performance falls off very gradually to the corner of the field, and the S and T curves stay reasonably close together.  FIG. 4   c  shows distortion and astigmatism (Coddington curves) for the same four object distances used in  FIG. 4   b . Distortion is virtually zero at all magnifications, and astigmatism is also very well controlled.  FIG. 4   d  is a plot of relative illumination vs. image height at f/1.33 and f/2.8, and it indicates that the Example 4 design has extremely low illumination falloff even at the widest aperture of f/1.33. 
     The primary group PG comprises the entire lens. As discussed above primary group PG comprises a negative powered front sub-group P 1 , followed by a positive powered sub-group P 2 , followed by a negative powered sub-group P 3 , followed by a sub-group P 4  that can be either positively or negatively powered. 
     In Example 4, sub-group P 1  comprises a single negative doublet  415 . This doublet uses high-index anomalous dispersion materials N-KZFS4 and S-NPH1 to advantage, and as a result aids in reducing secondary and tertiary chromatic aberrations. The surface of sub-group P 1  closest to the object is a concave surface. 
     Sub-group P 2  comprises three positive groups: a positive doublet  416 , a positive singlet  405  and a positive doublet  417 . The positive elements  404 ,  405  and  407  are made of S-FPL53, CaF2, and S-FPL53, respectively. The single negative element  406  is made of a matching anomalous dispersion material N-KZFS4. Since most of the positive optical power for Example 4 resides in sub-group P 2 , the system as a whole is very well corrected for chromatic aberrations, and is in fact superachromatic over a waveband extending from the near ultraviolet to the near infrared. 
     Sub-group P 3  comprises a single negative powered doublet  418 . Anomalous dispersion materials S-PHM52 and N-KZFS4 are used for the individual lens elements  409  and  410 , respectively, which aids in the correction of secondary spectrum. The outer shape of doublet  418  is meniscus toward the image plane. The concave surface SC is strongly curved and therefore helps a great deal in correcting field curvature and astigmatism. 
     Sub-group P 4  is a fairly weak positive group comprising positive doublets  419  and  420 . Sub-group P 4  serves mainly to correct distortion and astigmatism, and to make fine adjustments to the exit pupil location. 
     Moving groups P 1 ,  421 , and  420  independently away from the image plane as illustrated by  FIG. 4   a  accomplishes focusing from a distant to a close object. This complex focusing motion ensures that performance remains very high throughout the focusing range, and simultaneously ensures that breathing is almost zero. Breathing reaches a maximum value of only −0.78% at closest focus, which is very low for a longer focal length lens. 
     EXAMPLE 5 
     Example 5, which is a 14.5 mm focal length ultra wide-angle objective for 35 mm format cinematography, is illustrated in  FIG. 5   a , which shows cross-sectional layouts at magnifications of 0 and −0.033×. All of the element and group designations mentioned below are shown in  FIG. 5   a . The relative aperture is f/1.4, the image diagonal is 28 mm, and the diagonal field of view (FOV) is 89.12 degrees. Since this objective has a small amount of barrel distortion, it provides a somewhat larger field of view than its paraxial focal length of 14.5 mm would suggest. In this case the effective corrected focal length is equal to the image height of 14 mm divided by the tangent of the half angle of view (HFOV):
 
 FC=IH /tan( HFOV )=14/tan(44.56)=14.2 mm
 
     MTF vs. Image Height at 10, 20 and 40 cycles/mm for four different object distances is illustrated in  FIG. 5   b . These curves indicate that Example 5 is well corrected at f/1.4, especially for such an extremely wide-angle objective. Performance falls off gradually to the corner of the field, and the S and T curves stay close together.  FIG. 5   c  shows distortion and astigmatism (Coddington curves) for the same two object distances used in  FIG. 5   b . Distortion is very low for an ultra-wide angle lens, and astigmatism is also very well controlled. The shape of the distortion curve indicates that under corrected (barrel) third order distortion is partially balanced by overcorrected fifth order distortion. This provides excellent straight-line rendition that is significantly better than a pure under corrected distortion of the same magnitude would provide.  FIG. 5   d  is a plot of relative illumination vs. image height at f/1.4 and f/2.8, and it indicates that the Example 5 design has low illumination falloff for a fast wide-angle lens. 
     The primary group PG comprises only the rear sub-group of Example 5. The front sub-group of the lens, comprising groups  528  and  529 , functions approximately as a wide-angle afocal attachment that outputs nearly collimated light into primary group PG. Group  528  has negative power and includes an asphere on the outermost object-side surface. Group  529  has positive power to roughly collimate the light output from group  528 . However, groups  528  and  529  together are not precisely afocal, and as a result the object magnification of primary group PG is −0.049×. As discussed above, primary PG comprises a negative powered front sub-group P 1 , followed by a positive powered sub-group P 2 , followed by a negative powered sub-group P 3 , followed by a sub-group P 4  that can be either positively or negatively powered. 
     In Example 5, sub-group P 1  comprises a single negative element  507 . The concave image-facing surface of element  507  is aspherical, and  507  is made of S-NSL3 to ensure that manufacturing this aspherical surface will not be problematic. The surface of sub-group P 1  closest to the object is a convex surface. 
     Sub-group P 2  comprises three positive groups: a positive doublet  523 , a positive triplet  524 , and a positive doublet  525 . The positive elements  510 ,  511 , and  513  are all made of low-index anomalous dispersion material S-FPL53, and positive element  515  is made of anomalous dispersion material S-PHM52. The negative elements  509 ,  512 , and  514  are all made of a matching anomalous dispersion material N-KZFS4. Since most of the positive optical power for Example 5 resides in sub-group P 2 , the system as a whole is very well corrected for chromatic aberrations, and is in fact apochromatic over a waveband extending from the deep violet end of the visible spectrum to the near infrared. 
     Sub-group P 3  comprises a single negative powered doublet  526 . Anomalous dispersion materials S-PHM52 and N-KZFS4 are used for the individual lens elements  516  and  517 , respectively, which aids in the correction of secondary spectrum. The outer shape of doublet  526  is meniscus toward the image plane. The concave surface SC is strongly curved and therefore helps a great deal in correcting field curvature and astigmatism. 
     Sub-group P 4  is a fairly weak positive group comprising a weak negative powered doublet  527  and a very weak positive powered meniscus singlet  520 . Sub-group P 4  serves mainly to correct distortion and astigmatism, and to make fine adjustments to the exit pupil location. The object-side surface of doublet  527  is aspherical in order to control higher order astigmatism. 
     Focusing from a distant to a close object is accomplished by moving groups  528  and  529  independently away from the image plane as illustrated by  FIG. 5   a . Primary group PG remains stationary with respect to the image plane, which substantially simplifies the mechanical design. This focusing motion ensures that performance remains very high throughout the focusing range, and simultaneously ensures that breathing is almost zero. Breathing reaches a maximum value of only −0.14% at closest focus, which is almost undetectable. 
     Because Example 5 has an extremely wide field of view any filter used in the conventional location on the object side of the front element would necessarily be very large. As an alternative, the filter can be placed inside the optical system closer to the aperture stop so that its size is reduced. Example 5 includes just such a filter: element  506 , which is located just in front of primary group PG. This location is particularly advantageous in this case because it is a nearly collimated air space. This means that the tolerances on the thickness of the filter can be very loose if desired. The filter can be placed on a turret along with a wide range of other filters. 
     EXAMPLE 6 
     Example 6, which is a 24 mm focal length wide-angle objective for 35 mm format cinematography, is illustrated in  FIG. 6   a , which shows cross-sectional layouts at magnifications of 0 and −0.05×. All of the element and group designations mentioned below are shown in  FIG. 6   a . The relative aperture is f/1.4, the image diagonal is 28 mm, and the diagonal field of view (FOV) is 60.5 degrees. 
     MTF vs. Image Height at 10, 20 and 40 cycles/mm for four different object distances is illustrated in  FIG. 1   b . These curves indicate that Example 6 is well corrected at f/1.4, with MTF values at 40 cycles/mm approaching 80% near the optical axis for longer object distances. This excellent performance falls off gradually to the corner of the field, and the S and T curves stay close together.  FIG. 6   c  shows distortion and astigmatism (Coddington curves) for the same two object distances used in  FIG. 6   b . Distortion is very low at all magnifications, and astigmatism is also very well controlled.  FIG. 6   d  is a plot of relative illumination vs. image height at f/1.4 and f/2.8, and it indicates that the Example 6 design has very low illumination falloff. 
     The primary group PG comprises only the rear portion of Example 6. The front portion of the lens, comprising elements  601 ,  602 ,  603 , and  604 , functions approximately as a wide-angle afocal attachment that outputs nearly collimated light into primary group PG. Elements  601  and  602  both have negative power, and element  601  includes as asphere on its object side surface to correct distortion. Elements  603  and  604  both have positive power, and combine to roughly collimate the light output from group elements  601  and  602 . However, the collimation is not perfect, and as a result the object magnification of PG is +0.088×. As discussed above, primary group PG comprises a negative powered front sub-group P 1 , followed by a positive powered sub-group P 2 , followed by a negative powered sub-group P 3 , followed by a sub-group P 4  that can be either positively or negatively powered. 
     In Example 6, sub-group P 1  comprises a single negative element  605  with a concave object-facing surface. 
     Sub-group P 2  comprises three positive groups: a positive doublet  622 , a positive triplet  623 , and a positive doublet  624 . The positive elements  608 ,  609 ,  611  and  613  are made of anomalous dispersion materials S-FPL53, S-FPL53, S-FPL51, and S-PHM52, respectively. The negative elements  610  and  612  are both made of a matching anomalous dispersion material N-KZFS4. Since most of the positive optical power for Example 6 resides in sub-group P 2 , the system as a whole is very well corrected for chromatic aberrations, and is in fact apochromatic over a waveband extending from the deep violet end of the visible spectrum to the near infrared. 
     Sub-group P 3  comprises a single negative powered doublet  625 . Anomalous dispersion materials S-PHM52 and N-KZFS4 are used for the individual lens elements  614  and  615 , respectively, which aids in the correction of secondary spectrum. The outer shape of doublet  625  is meniscus toward the image plane. The concave surface SC is strongly curved and therefore helps a great deal in correcting field curvature and astigmatism. 
     Sub-group P 4  is a fairly weak positive group comprising a positive triplet  626  and a negative doublet  627 . Sub-group P 4  serves mainly to correct distortion and astigmatism, and to make fine adjustments to the exit pupil location. 
     Focusing from a distant to a close object is accomplished by moving groups  628  and  629  independently toward and away from the image plane as illustrated by  FIG. 6   a . Element  601  is stationary with respect to the image plane, which means that the vertex length of the system as a whole remains constant during focusing. This focusing motion ensures that performance remains very high throughout the focusing range, and simultaneously ensures that breathing is well corrected. Breathing reaches a maximum value of 2.01% at closest focus. 
     EXAMPLE 7 
     Example 7, which is a 35 mm focal length objective for 35 mm format cinematography, is illustrated in  FIG. 7   a , which shows cross-sectional layouts at magnifications of 0 and −0.065×. All of the element and group designations mentioned below are shown in  FIG. 7   a . The relative aperture is f/1.4, the image diagonal is 28 mm, and the diagonal field of view (FOV) is 44.5 degrees. 
     MTF vs. Image Height at 10, 20 and 40 cycles/mm for four different object distances is illustrated in  FIG. 7   b . These curves indicate that Example 7 is extremely well corrected at f/1.4, with MTF values at 40 cycles/mm greater than 80% near the optical axis for all object distances. This extraordinary performance falls off very gradually to the corner of the field.  FIG. 7   c  shows distortion and astigmatism (Coddington curves) for the same two object distances used in  FIG. 1   b . Both distortion and astigmatism are well controlled.  FIG. 7   d  is a plot of relative illumination vs. image height at f/1.4 and f/2.8, and it indicates that the Example 7 design has low illumination falloff. 
     The primary group PG comprises only the rear sub-group of Example 7. The front sub-group of the lens, comprising elements  701  and  702 , functions roughly as a wide-angle afocal attachment for primary group PG. Elements  701  and  702  have negative and positive power, respectively, and element  701  includes as asphere on its convex object side surface to correct distortion. The object magnification of PG is −0.364× when the lens is focused on a distant object. As discussed above, primary group PG comprises a negative powered front sub-group P 1 , followed by a positive powered sub-group P 2 , followed by a negative powered sub-group P 3 , followed by a sub-group P 4  that can be either positively or negatively powered. 
     In Example 7, sub-group P 1  comprises a single negative element  703  with a concave object-facing surface. The convex image-facing surface of 703 is aspheric in order correct spherical aberration. Element  703  is made of S-BSL7 to enhance manufacturability. 
     Sub-group P 2  comprises five positive groups: two positive singlets  705  and  706 , followed by a positive doublet  717 , followed by two more positive singlets  709  and  710 . Elements  705 ,  706 ,  707 , and  710  are all made from low-index anomalous dispersion material S-FPL51, and element  709  is made from low-index anomalous dispersion material S-FPL53. The single negative element  708  is made of a matching anomalous dispersion material N-KZFS4. Since most of the positive optical power for Example 7 resides in sub-group P 2 , the system as a whole is very well corrected for chromatic aberrations, and is in fact apochromatic over a waveband extending from the deep violet end of the visible spectrum to the near infrared. 
     Sub-group P 3  comprises a single negative powered doublet  718 . Anomalous dispersion materials S-PHM52 and N-KZFS4 are used for the individual lens elements  711  and  712 , respectively, which aids in the correction of secondary spectrum. The outer shape of doublet  718  is meniscus toward the image plane. The concave surface SC is strongly curved and therefore helps a great deal in correcting field curvature and astigmatism. 
     Sub-group P 4  is a fairly weak positive group comprising a positive doublet  719 , a negative meniscus singlet  715 . P 4  serves mainly to correct distortion and astigmatism, and to make fine adjustments to the exit pupil location. 
     Focusing from a distant to a close object is accomplished by moving groups  701 ,  702  and PG independently away from the image plane as illustrated by  FIG. 7   a . This complex focusing motion ensures that performance remains very high throughout the focusing range, and simultaneously ensures that breathing is almost zero. Breathing reaches a maximum value of only −0.03% at closest focus, which is almost undetectable. 
     EXAMPLE 8 
     Example 8, which is a 65 mm focal length objective for 35 mm format cinematography, is illustrated in  FIG. 8   a , which shows cross-sectional layouts at magnifications of 0 and −0.137×. All of the element and group designations mentioned below are shown in  FIG. 8   a . The relative aperture is f/1.33, the image diagonal is 28 mm, and the diagonal field of view (FOV) is 24.3 degrees. 
     Example 8 is very similar to Example 1, with a key difference that it has been made athermal by the use of acrylic polymer for element  803 . One of the reasons for the very high optical performance of all of the examples in the present invention is that they are either apochromatic or superachromatic. This is brought about by extensive use of anomalous dispersion materials such as CaF2, S-FPL53, S-FPL51, and S-PHM52. Unfortunately, all of these materials have a high thermal expansion coefficient coupled with a large negative value for dn/dt. As a result, the focal plane will drift significantly with temperature. Although this focus drift can be dealt with by various active and passive mechanical means, it is most desirable to eliminate it by passive optical means. 
     Since most of the thermal drift mentioned above is caused by positive lens elements made of materials with a negative dn/dt, it follows that the thermal drift can be corrected by introducing one or more negative elements with an even larger negative value for dn/dt. Fortunately, such materials do exist in the form of optically transparent polymers such as acrylic. In the case of acrylic, its change of refractive index with temperature (dn/dt) is about an order of magnitude greater than that of the anomalous dispersion glasses listed above, so all that is necessary is to incorporate a relatively weak negative powered acrylic element. In Example 1 the third element has weak negative power and is made from S-BSL7, which has a roughly similar refractive index to acrylic. So, for Example 8, this third element was replaced by a similar element made from acrylic, and optimized for athermal performance over a broad temperature range.  FIG. 8   c  is a plot of MTF vs. Image Height at a spatial frequency of 20 cycles/mm for three different temperatures of 0 C, 20 C, and 40 C. No re-focusing is done for the different temperatures, and the mounting material is assumed to be aluminum.  FIG. 8   c  clearly shows that Example 8 is athermal in the sense that it maintains excellent optical performance at a large aperture over a wide temperature range without re-focusing. 
     MTF vs. Image Height at 10, 20 and 40 cycles/mm for four different object distances is illustrated in  FIG. 8   b . These curves indicate that Example 8 is extremely well corrected at f/1.33, with MTF values at 40 cycles/mm greater than 80% near the optical axis except at closest focus. This excellent performance falls off very gradually to the corner of the field, and the S and T curves stay close together.  FIG. 8   d  shows distortion and astigmatism (Coddington curves) for the same four object distances used in  FIG. 8   b . Distortion is virtually zero at all magnifications, and astigmatism is also very well controlled.  FIG. 8   e  is a plot of relative illumination vs. image height at f/1.33 and f/2.8, and it indicates that the Example 8 design has extremely low illumination falloff. 
     The primary group, PG comprises the entire lens except for a plane parallel filter element  814 . As discussed above, primary group PG comprises a negative powered front sub-group P 1 , followed by a positive powered sub-group P 2 , followed by a negative powered sub-group P 3 , followed by a sub-group P 4  that can be either positively or negatively powered. 
     In Example 8, sub-group P 1  comprises two negative groups: a negative doublet  816  and a negative singlet  803 . The convex object-facing surface of element  803  is aspherical, and  803  is made of acrylic polymer to achieve athermalization and to ensure straightforward manufacturing. Acrylic optical elements can be made either by direct diamond machining or by molding. The doublet  816  uses high-index anomalous dispersion materials N-KZFS4 and S-NPH1 to advantage, and as a result aids in reducing secondary and tertiary chromatic aberrations. The surface of P 1  closest to the object is a concave surface. 
     Sub-group P 2  comprises two positive groups: a positive singlet  805  and a positive doublet  817 . The positive elements in groups P 2 ,  805  and  807  are both made of low-index anomalous dispersion material S-FPL51. The single negative element  806  is made of a matching anomalous dispersion material N-KZFS4. Since most of the positive optical power for Example 8 resides in sub-group P 2 , the system as a whole is very well corrected for chromatic aberrations, and is in fact apochromatic over a waveband extending from the deep violet end of the visible spectrum to about 1000 nm in the near infrared. 
     Sub-group P 3  comprises a single negative powered doublet  818 . Anomalous dispersion materials S-FPL51y and N-KZFS4 are used for the individual lens elements  808  and  809 , respectively, which aids in the correction of secondary spectrum. The outer shape of doublet  818  is meniscus toward the image plane. The concave surface SC is strongly curved and therefore helps a great deal in correcting field curvature and astigmatism. 
     Sub-group P 4  is a fairly weak positive group comprising a positive doublet  819 , a negative singlet  812 , and a positive singlet  813 . Sub-group P 4  serves mainly to correct distortion and astigmatism, and to make fine adjustments to the exit pupil location. 
     Element  814  is a plane parallel plate that serves to model the effect of the anti-aliasing filter, the IR/UV filter, and the sensor cover plate that are commonly found in digital cameras. As in the analogous Element  114  in Example 1, Element  814  is not intended as a precise model for any particular brand or model of camera, but rather is intended as a viable means for avoiding any filter-induced aberrations. In Example 8, element  814  has been made fairly thick, and in all likelihood would be too thick to accurately model a camera filter pack. However, in this case the thickness of element  814  could be reduced so that when a real filter pack is introduced the aberration balance is not disturbed. This is particularly important in digital photography and cinematography with extremely well corrected high-speed lenses because the filter pack thickness is likely to vary from camera to camera, and it will be a great advantage to be able to customize the lens for an individual camera simply by changing a filter in the rear. 
     Focusing from a distant to a close object is accomplished by independently moving groups P 1  and  820  away from the image plane, as illustrated by  FIG. 8   a . Element  813  is stationary with respect to the image plane. This complex focusing motion ensures that performance remains very high throughout the focusing range, and simultaneously ensures that breathing is almost zero. Breathing reaches a maximum value of only 0.14% in the middle of the focusing range, which is almost undetectable. In Example 8, sub-group P 1  moves a relatively great distance during focusing because it is a relatively weak group. Although this results in a good aberration balance and loose tolerances it does also result in greater bulk as the lens is focused up close. 
     EXAMPLE 9 
     Example 9, which is a 35 mm focal length objective for 35 mm format cinematography, is illustrated in  FIG. 9   a , which shows cross-sectional layouts at magnifications of 0 and −0.037×. All of the element and group designations mentioned below are shown in  FIG. 9   a . The relative aperture is f/1.4, the image diagonal is 28 mm, and the diagonal field of view (FOV) is 43.6 degrees. 
     MTF vs. Image Height at 10, 20 and 40 cycles/mm for four different object distances is illustrated in  FIG. 9   b . These curves indicate that Example 9 is extremely well corrected at f/1.4, with MTF at 20 cycles/mm exceeding 90% over nearly the entire field of view and MTF at 40 cycles/mm exceeding 80% over a large central portion of the field of view. Performance falls off very gradually from the center to the corner of the field, and the S and T curves stay close together.  FIG. 9   c  shows distortion and astigmatism (Coddington curves) for the same two object distances used in  FIG. 9   b . Both distortion and astigmatism are very well corrected.  FIG. 9   d  is a plot of relative illumination vs. image height at f/1.4 and f/2.8, and it indicates that the Example 9 design has low illumination falloff. 
     The primary group PG comprises only the rear sub-group of Example 5. The front sub-group of the lens, comprising groups  901  and  916 , functions approximately as a wide-angle afocal attachment in front of primary group PG. Group  901  has negative power and includes an asphere on the outermost object-side surface. Group  916  has positive power to roughly collimate the light output from group  901 . However, groups  901  and  916  together are not precisely afocal, and as a result the object magnification of primary group PG is −0.288×. As discussed above, primary PG comprises a negative powered front sub-group P 1 , followed by a positive powered sub-group P 2 , followed by a negative powered sub-group P 3 , followed by a sub-group P 4  that can be either positively or negatively powered. 
     In Example 9, sub-group P 1  comprises a single negative element  904 . The convex image-facing surface of element  904  is aspherical, and  904  is made of S-BSL7 to ensure that manufacturing this aspherical surface will not be problematic. The surface of sub-group P 1  closest to the object is a concave surface. 
     Sub-group P 2  comprises five positive groups: two positive singlets  906  and  907 , a positive doublet  917 , and two positive singlets  910  and  911 . The positive elements  906 ,  907 ,  908 ,  910 , and  911  are all made of low-index anomalous dispersion material S-FPL53, and the negative element  909  is made of a matching anomalous dispersion material N-KZFS4. Since most of the positive optical power for Example 5 resides in sub-group P 2 , the system as a whole is very well corrected for chromatic aberrations, and is in fact apochromatic over a waveband extending from the deep violet end of the visible spectrum to the near infrared. 
     Sub-group P 3  comprises a negative powered singlet  912  made of anomalous dispersion material N-KZFS4, which aids in the correction of secondary spectrum. Element  912  is meniscus toward the image plane. The concave surface SC is strongly curved and therefore helps a great deal in correcting field curvature and astigmatism. 
     Sub-group P 4  is a fairly weak positive group comprising a positive powered singlet  913  and a negative powered singlet  914 . Sub-group P 4  serves mainly to correct distortion and astigmatism, and to make fine adjustments to the exit pupil location. 
     Focusing from a distant to a close object is accomplished by moving groups  901 ,  916 , and PG independently away from the image plane as illustrated by  FIG. 9   a . This complex focusing motion ensures that performance remains very high throughout the focusing range, and simultaneously ensures that breathing is almost zero. Breathing reaches a maximum value of only −0.23% at closest focus, which is almost undetectable. 
     EXAMPLE 10 
     Example 10, which is a 24 mm focal length wide-angle objective for 35 mm format cinematography, is illustrated in  FIG. 10   a , which shows cross-sectional layouts at magnifications of 0 and −0.056×. All of the element and group designations mentioned below are shown in  FIG. 10   a . The relative aperture is f/1.4, the image diagonal is 28 mm, and the diagonal field of view (FOV) is 60.5 degrees. 
     MTF vs. Image Height at 10, 20 and 40 cycles/mm for four different object distances is illustrated in  FIG. 10   b . These curves indicate that Example 10 is extremely well corrected at f/1.4, especially for a wide-angle objective. The MTF at 20 cycles exceeds 90% over nearly the entire field of view, and the MTF at 40 cycles/mm is approximately 80% over a large central region of the field of view. Performance falls off gradually to the corner of the field, and the S and T curves stay close together.  FIG. 10   c  shows distortion and astigmatism (Coddington curves) for the same two object distances used in  FIG. 10   b . Distortion is very low for a wide-angle lens, and astigmatism is also very well controlled.  FIG. 10   d  is a plot of relative illumination vs. image height at f/1.4 and f/2.8, and it indicates that the Example 10 design has low illumination falloff. 
     The primary group PG comprises only the rear sub-group of Example 10. The front sub-group of the lens, comprising groups  1001 ,  1002 , and  1003 , functions approximately as a wide-angle afocal attachment that outputs quasi-collimated light into primary group PG. Groups 1001 and 1002 have a combined negative power and include an asphere on the outermost object-side surface. Group  1003  has positive power to roughly collimate the light output from groups  1001  and  1002 . However, groups  1001  and  1002  together are not precisely afocal, and as a result the object magnification of primary group PG is −0.210×. As discussed above, primary PG comprises a negative powered front sub-group P 1 , followed by a positive powered sub-group P 2 , followed by a negative powered sub-group P 3 , followed by a sub-group P 4  that can be either positively or negatively powered. 
     In Example 10, sub-group P 1  comprises a single negative element  1004 . The convex image-facing surface of element  1004  is aspherical, and  1004  is made of S-BSL7 to ensure that manufacturing this aspherical surface will not be problematic. The surface of sub-group P 1  closest to the object is a concave surface. 
     Sub-group P 2  comprises five positive groups: two positive singlets  1006  and  1007 , a positive doublet  1017 , and two positive singlets  1010  and  1011 . The positive elements  1006 ,  1007 ,  1009 ,  1010 , and  1011  are all made of low-index anomalous dispersion material CaF2, also called Calcium Fluoride, and the negative element  1008  is made of a matching material S-BAL42. Since most of the positive optical power for Example 5 resides in sub-group P 2 , the system as a whole is very well corrected for chromatic aberrations, and is in fact apochromatic over a waveband extending from the deep violet end of the visible spectrum to the near infrared. 
     Sub-group P 3  comprises a single negative powered singlet  1012  made of anomalous dispersion material N-KZFS4, which aids in the correction of secondary spectrum. The concave surface SC facing the image plane is strongly curved and therefore helps a great deal in correcting field curvature and astigmatism. 
     Sub-group P 4  is a weak negative group comprising a positive powered singlet  1013  and a negative powered singlet  1014 . Sub-group P 4  serves mainly to correct distortion and astigmatism, and to make fine adjustments to the exit pupil location. 
     Focusing from a distant to a close object is accomplished by moving groups  1016  and PG independently toward and away from the image plane, respectively, as illustrated by  FIG. 10   a . The front group  1001  remains stationary with respect to the image plane. This focusing motion ensures that performance remains very high throughout the focusing range, and simultaneously ensures that breathing is well-corrected. Breathing reaches a maximum value of −2.42% at closest focus. 
     EXAMPLE 11 
     Example 11, which is a 14.4 mm focal length ultra wide-angle objective for 35 mm format cinematography, is illustrated in  FIG. 11   a , which shows cross-sectional layouts at magnifications of 0 and −0.037×. All of the element and group designations mentioned below are shown in  FIG. 11   a . The relative aperture is f/1.4, the image diagonal is 28 mm, and the diagonal field of view (FOV) is 91.3 degrees. Since this objective has a small amount of barrel distortion, it provides a somewhat larger field of view than its paraxial focal length of 14.4 mm would suggest. In this case the effective corrected focal length is equal to the image height of 14 mm divided by the tangent of the half angle of view (HFOV):
 
 FC=IH /tan( HFOV )=14/tan(45.63)=13.7 mm
 
     MTF vs. Image Height at 10, 20 and 40 cycles/mm for four different object distances is illustrated in  FIG. 11   b . These curves indicate that Example 11 is extremely well corrected at f/1.4, especially for such an extremely wide-angle objective. Performance falls off gradually to the corner of the field, and the S and T curves stay close together.  FIG. 11   c  shows distortion and astigmatism (Coddington curves) for the same two object distances used in  FIG. 11   b . Distortion is reasonably well corrected for an ultra-wide angle lens, and astigmatism is also very well controlled.  FIG. 11   d  is a plot of relative illumination vs. image height at f/1.4 and f/2.8, and it indicates that the Example 11 design has extremely low illumination falloff for a fast wide-angle lens. 
     The primary group PG comprises only the rear sub-group of Example 11. The front sub-group of the lens, comprising groups  1125 ,  1122  and  1126 , functions approximately as a wide-angle afocal attachment in front of the primary group PG. Group  1125  has negative power and includes an asphere on the outermost object-side surface. Groups 1122 and 1126 have a combined positive power to roughly collimate the light output from group  1125 . However, groups  1125 ,  1122  and  1126  together are not precisely afocal, and as a result the object magnification of primary group PG is −0.175×. As discussed above, primary PG comprises a negative powered front sub-group P 1 , followed by a positive powered sub-group P 2 , followed by a negative powered sub-group P 3 , followed by a sub-group P 4  that can be either positively or negatively powered. 
     In Example 11, sub-group P 1  comprises a single negative element  1110 . The convex image-facing surface of element  1110  is aspherical, and  1110  is made of S-BSL7 to ensure that manufacturing this aspherical surface will not be problematic. The surface of sub-group P 1  closest to the object is a concave surface. 
     Sub-group P 2  comprises five positive groups: two positive singlets  1112  and  1113 , a positive doublet  1124 , and two positive singlets  1116  and  1117 . The positive elements  1112 ,  1113 ,  1114 ,  1116 , and  1117  are all made of low-index anomalous dispersion material S-FPL53, and the negative element  1115  is made of a matching anomalous dispersion material N-KZFS4. Since most of the positive optical power for Example 5 resides in sub-group P 2 , the system as a whole is very well corrected for chromatic aberrations, and is in fact apochromatic over a waveband extending from the deep violet end of the visible spectrum to the near infrared. 
     Sub-group P 3  comprises a single negative powered singlet  1118  made of anomalous dispersion material N-KZFS4, which aids in the correction of secondary spectrum. Element  1118  is meniscus toward the image plane. The concave surface SC is strongly curved and therefore helps a great deal in correcting field curvature and astigmatism. 
     Sub-group P 4  is a fairly weak positive group comprising a positive powered singlet  1119  and a negative powered singlet  1120 . Sub-group P 4  serves mainly to correct distortion and astigmatism, and to make fine adjustments to the exit pupil location. 
     Focusing from a distant to a close object is accomplished by moving groups  1122  and  1126  independently toward the image plane as illustrated by  FIG. 11   a . Primary group PG and group  1125  remain stationary with respect to the image plane during focusing, which substantially simplifies the mechanical design. This focusing motion ensures that performance remains very high throughout the focusing range, and simultaneously ensures that breathing is almost zero. Breathing reaches a maximum value of only +0.16% at closest focus, which is almost undetectable. 
       1109  is a vignetting stop that restricts the lower rim rays in order to control both aberrations and illumination. 
     EXAMPLE 12 
     Example 12, which is a 200 mm focal length objective optimized for aerial reconnaissance, is illustrated in  FIG. 12   a , which shows a cross-sectional layout. All of the element and group designations mentioned below are shown in  FIG. 12   a . The relative aperture is f/1.4, the image diagonal is 28 mm, and the diagonal field of view (FOV) is 8.0 degrees. Example 12 is corrected over a waveband ranging from about 435 nm to 1000 nm, which is an especially useful waveband for reconnaissance and night-vision applications. 
     MTF vs. Image Height at 50, 100 and 200 cycles/mm for an object at infinity is illustrated in  FIG. 12   b . These curves indicate that Example 12 is extraordinarily well corrected at f/1.4, with MTF values at 100 cycles/mm greater than 80% over the entire image field. Example 12 does in fact meet the Rayleigh criterion for diffraction-limited performance at f/1.4 over most of the image field.  FIG. 1   c  shows distortion and astigmatism (Coddington curves) for an object at infinity. Both distortion and astigmatism are nearly zero.  FIG. 12   d  is a plot of relative illumination vs. image height at f/1.4, and it indicates that the Example 12 design has extremely low illumination falloff for such a high-speed lens. 
     The primary group PG comprises the entire lens. As discussed above, primary group PG comprises a negative powered front sub-group P 1 , followed by a positive powered sub-group P 2 , followed by a negative powered sub-group P 3 , followed by sub-group P 4  that can be either positively or negatively powered. 
     In Example 12, sub-group P 1  comprises a negative doublet  1215 . This doublet uses high-index anomalous dispersion materials N-KZFS4 and S-NPH1 to advantage, and as a result aids in reducing secondary chromatic aberrations. The surface of sub-group P 1  closest to the object is a concave surface. 
     Sub-group P 2  comprises six single-element lens groups: positive singlets  1203 ,  1204 ,  1205 ,  1208 , and  1209 ; and negative singlet  1206 . The convex object-facing surface of element  1203  is aspherical, and singlet  1203  is made of S-BSL7 to ensure that manufacturing this aspherical surface will not be problematic. The positive elements in sub-group P 2  are elements  1203 ,  1204 ,  1205 ,  1208 , and  1209 . All of these except  1203  are made of low-index anomalous dispersion materials. The single negative element  1206  is made of a matching anomalous dispersion material N-KZFS4. Since most of the positive optical power for Example 12 resides in sub-group P 2 , the system as a whole is very well corrected for chromatic aberrations, and is in fact superachromatic (i.e., with four color crossings) over a waveband extending from 435 nm to 1000 nm. 
     Sub-group P 3  comprises a single negative powered doublet  1216 . Anomalous dispersion materials S-PHM52 and N-KZFS4 are used for the individual lens elements  1210  and  1211 , respectively, which aids in the correction of secondary spectrum. The outer shape of doublet  1216  is meniscus toward the image plane. The concave surface SC is strongly curved and therefore helps a great deal in correcting field curvature and astigmatism. 
     Sub-group P 4  is a fairly weak positive group comprising a positive doublet  1217 . Sub-group P 4  serves mainly to correct distortion and astigmatism. 
     Since Example 12 is intended for aerial reconnaissance applications, it has been optimized for an object located at infinity. However, small focus adjustments can be made by moving the entire lens without significantly degrading the lens performance. 
     Optical Prescription Data 
     Tables 1a, 2a, 3a, 4a, 5a, 6a, 7a, 8a, 9a, 10a, 11a and 12a below provide optical prescription data for Examples 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12, respectively. The data provided includes surface number, radius of curvature, thickness, glass type, and the diameter of the clear aperture. OBJ refers to the object surface, IMA refers to the image surface, and STO refers to the aperture stop surface. Tables 1b, 2b, 4b, 5b, 6b, 7b, 8b, 9b, 10b, and 11b provide focusing data for Examples 1, 2, 4, 5, 6, 7, 8, 9, 10, and 11, respectively. In these Tables, OBMG refers to object magnification, and OBIM refers to the total distance from the object plane to the image plane. 
     Aspherical surfaces are expressed by the following equation:
 
 Z ( r )= r   2   /R ( R (1 +SQRT (1−(1 +k ) r   2   /R   2 )))+ C 4 r   4   +C 6 r   6   +C 8 r   8   +C 10 r   10   +C 12 r   12   +C 14 r   14   +C 16 r   16  
 
     Where Z is the displacement in the direction of the optical axis measured from the polar tangent plane, r is the radial coordinate, R is the base radius of curvature, k is the conic constant, and Ci is the i-th order aspherical deformation constant. Tables 1c, 2c, 3b, 4c, 5c, 6c, 7c, 8c, 9c, 10c, 11c and 12b provide aspheric surface data for examples 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12, respectively. 
     Values for the defining conditions for each of the eight examples are given in Table 13. A listing of refractive index at various wavelengths for all of the glass types used in the Examples is provided in Table 14. 
     
       
         
               
             
               
               
               
               
               
             
               
               
               
               
               
             
           
               
                 TABLE 1a 
               
             
             
               
                   
               
               
                 Prescription Data for Example 1 
               
             
          
           
               
                 Surf 
                 Radius 
                 Thickness 
                 Glass 
                 Diameter 
               
               
                   
               
             
          
           
               
                 OBJ 
                 Infinity 
                 Infinity 
                   
                   
               
               
                  1 
                 −243.9697 
                 4 
                 N-KZFS4 
                 61.76092 
               
               
                  2 
                 206.7449 
                 5 
                 S-NPH1 
                 60.82355 
               
               
                  3 
                 422.6313 
                 1.949756 
                   
                 60.3132 
               
               
                  4 
                 81.99985 
                 10 
                 S-BSL7 
                 56.56122 
               
               
                  5 
                 71.78633 
                 28.64799 
                   
                 53.75262 
               
               
                 STO 
                 Infinity 
                 2.934045 
                   
                 54.7 
               
               
                  7 
                 113.8222 
                 13 
                 S-FPL51 
                 57.16058 
               
               
                  8 
                 −91.91932 
                 1.918958 
                   
                 57.46784 
               
               
                  9 
                 105.3766 
                 3 
                 N-KZFS4 
                 55.07957 
               
               
                 10 
                 41.14905 
                 14.88799 
                 S-FPL51 
                 52.07915 
               
               
                 11 
                 −93.91761 
                 0.25 
                   
                 51.56341 
               
               
                 12 
                 34.49819 
                 13 
                 S-FPL51Y 
                 43.1011 
               
               
                 13 
                 −194.7093 
                 2.5 
                 N-KZFS4 
                 38.62962 
               
               
                 14 
                 21.47767 
                 2.844639 
                   
                 30.07902 
               
               
                 15 
                 30.81431 
                 2.5 
                 N-KZFS4 
                 29.87583 
               
               
                 16 
                 16.63395 
                 7 
                 S-LAH53 
                 28.05017 
               
               
                 17 
                 28.83075 
                 2.756143 
                   
                 26.85199 
               
               
                 18 
                 47.79796 
                 2.5 
                 S-LAL14 
                 26.84398 
               
               
                 19 
                 31.35968 
                 4.043165 
                   
                 26.02517 
               
               
                 20 
                 79.29547 
                 5 
                 S-PHM52 
                 28 
               
               
                 21 
                 377.9263 
                 13.03858 
                   
                 28 
               
               
                 22 
                 Infinity 
                 5 
                 S-BSL7 
                 32 
               
               
                 23 
                 Infinity 
                 4.8 
                   
                 32 
               
               
                 IMA 
                 Infinity 
                   
                   
                 28.000 
               
               
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
               
               
               
             
           
               
                 TABLE 1b 
               
               
                   
               
               
                 Focusing and Breathing Data for Example 1 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 OBMG 
                 0.0 
                 −0.032 
                 −0.062 
                 −0.137 
               
               
                   
                 OBIM 
                 Infinity 
                 2163.4 
                 1174.8 
                 609.6 
               
               
                   
                 T0 
                 Infinity 
                 2000.0 
                 1000.0 
                 409.1 
               
               
                   
                 T3 
                 1.950 
                 12.745 
                 22.411 
                 44.023 
               
               
                   
                 T19 
                 4.043 
                 6.046 
                 7.816 
                 11.948 
               
               
                   
                 Breathing 
                 0.00% 
                 +0.03% 
                 +0.08% 
                 −0.09% 
               
               
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
             
               
               
               
             
           
               
                 TABLE 1c 
               
             
             
               
                   
               
               
                 Aspheric Surface Data for Example 1 
               
             
          
           
               
                   
                 Surf. # 
                 4 
               
               
                   
                   
               
             
          
           
               
                   
                 R 
                 82.000 
               
               
                   
                 k 
                 0.0 
               
               
                   
                 C4 
                 −1.727197e−6 
               
               
                   
                 C6 
                 −1.667576e−10 
               
               
                   
                 C8 
                 −7.766913e−13 
               
               
                   
                 C10 
                 5.730591e−16 
               
               
                   
                 C12 
                 4.667245e−19 
               
               
                   
                 C14 
                 −1.252642e−21 
               
               
                   
                 C16 
                 6.603991e−25 
               
               
                   
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
               
               
             
               
               
               
               
               
             
           
               
                 TABLE 2a 
               
             
             
               
                   
               
               
                 Prescription Data for Example 2 
               
             
          
           
               
                 Surf 
                 Radius 
                 Thickness 
                 Glass 
                 Diameter 
               
               
                   
               
             
          
           
               
                 OBJ 
                 Infinity 
                 Infinity 
                   
                   
               
               
                  1 
                 −55.04161 
                 3 
                 N-KZFS4 
                 50.67136 
               
               
                  2 
                 950.3995 
                 5 
                 S-NPH1 
                 52.03579 
               
               
                  3 
                 −528.7796 
                 3 
                   
                 52.52354 
               
               
                  4 
                 107.022 
                 5 
                 S-LAH53 
                 53.41207 
               
               
                  5 
                 153.2842 
                 8 
                 S-FPL53 
                 53.17786 
               
               
                  6 
                 −133.139 
                 10.68225 
                   
                 53.25474 
               
               
                 STO 
                 Infinity 
                 1.2 
                   
                 51.34386 
               
               
                  8 
                 90.4377 
                 12 
                 S-FPL53 
                 52.14142 
               
               
                  9 
                 −70.2512 
                 0.25 
                   
                 51.8957 
               
               
                 10 
                 325.283 
                 2.5 
                 N-KZFS4 
                 48.85419 
               
               
                 11 
                 40.1485 
                 10.5 
                 S-PHM52 
                 45.70621 
               
               
                 12 
                 −310.7279 
                 0.25 
                   
                 44.77577 
               
               
                 13 
                 32.86969 
                 11 
                 S-PHM52 
                 39.3514 
               
               
                 14 
                 −1017.895 
                 2.5 
                 N-KZFS4 
                 35.09814 
               
               
                 15 
                 17.80344 
                 3.320912 
                   
                 26.56843 
               
               
                 16 
                 31.75178 
                 2.5 
                 N-KZFS4 
                 26.46405 
               
               
                 17 
                 14.51134 
                 6 
                 S-LAH53 
                 23.59758 
               
               
                 18 
                 28.43016 
                 15.93711 
                   
                 22.35022 
               
               
                 19 
                 −41.12113 
                 7.458793 
                 S-PHM52 
                 23.69697 
               
               
                 20 
                 −22.20727 
                 3 
                 N-LASF9 
                 25.28074 
               
               
                 21 
                 −33.70055 
                 1 
                   
                 27.03097 
               
               
                 22 
                 Infinity 
                 5 
                 S-BSL7 
                 27.27129 
               
               
                 23 
                 Infinity 
                 4.8 
                   
                 27.47853 
               
               
                 IMA 
                 Infinity 
                   
                   
                 28.000 
               
               
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
               
               
               
             
           
               
                 TABLE 2b 
               
               
                   
               
               
                 Focusing and Breathing Data for Example 2 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 OBMG 
                 0.0 
                 −0.032 
                 −0.064 
                 −0.124 
               
               
                   
                 OBIM 
                 Infinity 
                 2127.7 
                 1131.3 
                 636.6 
               
               
                   
                 T0 
                 Infinity 
                 2000.0 
                 1000.0 
                 499.0 
               
               
                   
                 T3 
                 3.000 
                 4.813 
                 6.579 
                 9.929 
               
               
                   
                 T18 
                 15.937 
                 12.657 
                 9.946 
                 5.547 
               
               
                   
                 T23 
                 4.800 
                 10.101 
                 14.572 
                 22.004 
               
               
                   
                 Breathing 
                 0.00% 
                 −0.01% 
                 0.00% 
                 0.00% 
               
               
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
             
               
               
               
             
           
               
                 TABLE 2c 
               
             
             
               
                   
               
               
                 Aspheric Surface Data for Example 2 
               
             
          
           
               
                   
                 Surf. # 
                 4 
               
               
                   
                   
               
             
          
           
               
                   
                 R 
                 107.022 
               
               
                   
                 k 
                 0.0 
               
               
                   
                 C4 
                 −1.239477e−6 
               
               
                   
                 C6 
                 −1.464631e−10 
               
               
                   
                 C8 
                 −1.068124e−13 
               
               
                   
                 C10 
                 1.130896e−17 
               
               
                   
                 C12 
                 0.0 
               
               
                   
                 C14 
                 0.0 
               
               
                   
                 C16 
                 0.0 
               
               
                   
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
               
               
             
               
               
               
               
               
             
           
               
                 TABLE 3a 
               
             
             
               
                   
               
               
                 Prescription Data for Example 3 
               
             
          
           
               
                 Surf 
                 Radius 
                 Thickness 
                 Glass 
                 Diameter 
               
               
                   
               
             
          
           
               
                 OBJ 
                 Infinity 
                 Infinity 
                   
                   
               
               
                  1 
                 −84.5919 
                 5 
                 S-BSL7 
                 49.85981 
               
               
                  2 
                 −389.7759 
                 15.79056 
                   
                 49.35585 
               
               
                 STO 
                 Infinity 
                 1.5 
                   
                 50.8863 
               
               
                  4 
                 112.8608 
                 13 
                 S-FPL53 
                 52.70803 
               
               
                  5 
                 −55.78134 
                 15.72615 
                   
                 53.07695 
               
               
                  6 
                 88.85746 
                 11 
                 S-FPL53 
                 44.88271 
               
               
                  7 
                 −58.14963 
                 2.5 
                 N-KZFS4 
                 43.37819 
               
               
                  8 
                 705.2347 
                 7 
                 S-PHM52 
                 41.68126 
               
               
                  9 
                 −78.161 
                 0.5 
                   
                 40.71523 
               
               
                 10 
                 48.79291 
                 10 
                 S-PHM52 
                 35.37947 
               
               
                 11 
                 −110.2849 
                 1 
                   
                 30.88737 
               
               
                 12 
                 −81.14733 
                 2.5 
                 N-KZFS4 
                 30.03937 
               
               
                 13 
                 21.88237 
                 2.398791 
                   
                 26.28744 
               
               
                 14 
                 39.86111 
                 2.5 
                 N-KZFS4 
                 26.39182 
               
               
                 15 
                 17.11893 
                 10.51624 
                 S-LAH53 
                 26.1893 
               
               
                 16 
                 33.65242 
                 24.31896 
                   
                 24.40891 
               
               
                 IMA 
                 Infinity 
                   
                   
                 28.000 
               
               
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
             
               
               
               
             
           
               
                 TABLE 3b 
               
             
             
               
                   
               
               
                 Aspheric Surface Data for Example 3 
               
             
          
           
               
                   
                 Surface # 
                 2 
               
               
                   
                   
               
             
          
           
               
                   
                 R 
                 −389.776 
               
               
                   
                 k 
                 0.0 
               
               
                   
                 C4 
                 2.876369e−6 
               
               
                   
                 C6 
                 5.936459e−10 
               
               
                   
                 C8 
                 9.064682e−13 
               
               
                   
                 C10 
                 −1.958467e−16 
               
               
                   
                 C12 
                 0.0 
               
               
                   
                 C14 
                 0.0 
               
               
                   
                 C16 
                 0.0 
               
               
                   
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
               
               
             
               
               
               
               
               
             
           
               
                 TABLE 4a 
               
             
             
               
                   
               
               
                 Prescription Data for Example 4 
               
             
          
           
               
                 Surf 
                 Radius 
                 Thickness 
                 Glass 
                 Diameter 
               
               
                   
               
             
          
           
               
                 OBJ 
                 Infinity 
                 Infinity 
                   
                   
               
               
                  1 
                 −243.9697 
                 5 
                 N-KZFS4 
                 90.9493 
               
               
                  2 
                 206.7449 
                 8 
                 S-NPH1 
                 95.17419 
               
               
                  3 
                 422.6313 
                 2 
                   
                 97.01629 
               
               
                  4 
                 81.99985 
                 12 
                 S-PHM52 
                 100.6275 
               
               
                  5 
                 71.78633 
                 13 
                 S-FPL53 
                 101.196 
               
               
                  6 
                 Infinity 
                 24.8115 
                   
                 101.747 
               
               
                  7 
                 113.8222 
                 20.83024 
                 CaF2 
                 100.7282 
               
               
                  8 
                 −91.91932 
                 0.5 
                   
                 99.66903 
               
               
                  9 
                 105.3766 
                 5 
                 N-KZFS4 
                 91.88288 
               
               
                 10 
                 41.14905 
                 18.72651 
                 S-FPL53 
                 85.07208 
               
               
                 11 
                 −93.91761 
                 2 
                   
                 83.05963 
               
               
                 STO 
                 34.49819 
                 2 
                   
                 78.9513 
               
               
                 13 
                 −194.7093 
                 22 
                 S-PHM52 
                 69.69548 
               
               
                 14 
                 21.47767 
                 5 
                 N-KZFS4 
                 61.72373 
               
               
                 15 
                 30.81431 
                 6.641824 
                   
                 45.27939 
               
               
                 16 
                 16.63395 
                 5 
                 N-KZFS4 
                 45.20044 
               
               
                 17 
                 28.83075 
                 10.78727 
                 S-LAH53 
                 40.95277 
               
               
                 18 
                 47.79796 
                 21.30254 
                   
                 39.11442 
               
               
                 19 
                 31.35968 
                 20.5 
                 S-BSM16 
                 37.87622 
               
               
                 20 
                 79.29547 
                 6 
                 S-TIH1 
                 38.17191 
               
               
                 21 
                 377.9263 
                 9 
                   
                 38.54315 
               
               
                 IMA 
                 Infinity 
                   
                   
                 28.000 
               
               
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
               
               
               
             
           
               
                 TABLE 4b 
               
               
                   
               
               
                 Focusing and Breathing Data for Example 4 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 OBMG 
                 0.0 
                 −0.030 
                 −0.060 
                 −0.094 
               
               
                   
                 OBIM 
                 Infinity 
                 4231.0 
                 2239.5 
                 1500.0 
               
               
                   
                 T0 
                 Infinity 
                 4000.0 
                 2000.0 
                 1251.0 
               
               
                   
                 T3 
                 2.000 
                 9.462 
                 14.745 
                 20.782 
               
               
                   
                 T18 
                 21.303 
                 16.290 
                 12.846 
                 9.316 
               
               
                   
                 T21 
                 9.000 
                 17.467 
                 24.126 
                 31.076 
               
               
                   
                 Breathing 
                 0.00% 
                 +0.41% 
                 −0.19% 
                 −0.78% 
               
               
                   
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
             
               
               
               
             
               
               
               
               
             
           
               
                 TABLE 4c 
               
             
             
               
                   
               
               
                 Aspheric Surface Data for Example 4 
               
             
          
           
               
                   
                 Surface # 
                   
               
             
          
           
               
                   
                 4 
                 21 
               
               
                   
                   
               
             
          
           
               
                   
                 R 
                 185.301 
                 −70.903 
               
               
                   
                 k 
                 0.0 
                 0.0 
               
               
                   
                 C4 
                 −1.910272e−7 
                 4.626607e−7 
               
               
                   
                 C6 
                 −7.138951e−12 
                 −2.368491e−9 
               
               
                   
                 C8 
                 2.519699e−16 
                 5.512831e−12 
               
               
                   
                 C10 
                 −1.906009e−19 
                 −5.494228e−15 
               
               
                   
                 C12 
                 0.0 
                 0.0 
               
               
                   
                 C14 
                 0.0 
                 0.0 
               
               
                   
                 C16 
                 0.0 
                 0.0 
               
               
                   
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
               
               
               
             
               
               
               
               
               
               
             
           
               
                 TABLE 5a 
               
             
             
               
                   
               
               
                 Prescription Data for Example 5 
               
             
          
           
               
                   
                 Surf 
                 Radius 
                 Thickness 
                 Glass 
                 Diameter 
               
               
                   
                   
               
             
          
           
               
                   
                 OBJ 
                 Infinity 
                 Infinity 
                   
                   
               
               
                   
                  1 
                 375.5463 
                 5 
                 S-PHM52 
                 91.92377 
               
               
                   
                  2 
                 37.73174 
                 25.83448 
                   
                 65.79916 
               
               
                   
                  3 
                 −86.34626 
                 3.5 
                 S-FPL51 
                 65.73444 
               
               
                   
                  4 
                 57.95547 
                 2.008227 
                   
                 61.80289 
               
               
                   
                  5 
                 72.05063 
                 3.5 
                 S-NPH1 
                 61.81905 
               
               
                   
                  6 
                 42.61828 
                 12 
                 N-KZFS4 
                 59.68593 
               
               
                   
                  7 
                 101.2987 
                 11.86961 
                   
                 59.44564 
               
               
                   
                  8 
                 82.44051 
                 13 
                 S-LAH60 
                 63.52585 
               
               
                   
                  9 
                 −182.3396 
                 34.84959 
                   
                 62.7011 
               
               
                   
                 10 
                 Infinity 
                 5 
                 S-BSL7 
                 34.87959 
               
               
                   
                 11 
                 Infinity 
                 5 
                   
                 32.85024 
               
               
                   
                 12 
                 237.532 
                 5 
                 S-NSL3 
                 29.40529 
               
               
                   
                 13 
                 50.6145 
                 8.489515 
                   
                 26.37966 
               
               
                   
                 STO 
                 Infinity 
                 3.071435 
                   
                 26.78562 
               
               
                   
                 15 
                 102.3559 
                 3.5 
                 S-LAH63 
                 28.24707 
               
               
                   
                 16 
                 124.9423 
                 8 
                 S-FPL53 
                 28.47063 
               
               
                   
                 17 
                 −66.92142 
                 0.25 
                   
                 29.28444 
               
               
                   
                 18 
                 94.94834 
                 6 
                 S-FPL53 
                 29.33022 
               
               
                   
                 19 
                 −68.09017 
                 2.5 
                 N-KZFS4 
                 29.05241 
               
               
                   
                 20 
                 125.6468 
                 6 
                 S-FPL53 
                 28.91357 
               
               
                   
                 21 
                 −53.67274 
                 0.25 
                   
                 28.90076 
               
               
                   
                 22 
                 46.87171 
                 1.998782 
                 N-KZFS4 
                 30.21724 
               
               
                   
                 23 
                 22.65798 
                 10.04979 
                 S-PHM52 
                 29.8263 
               
               
                   
                 24 
                 −61.68191 
                 0.25 
                   
                 29.60787 
               
               
                   
                 25 
                 54.7172 
                 6.925449 
                 S-PHM52 
                 27.83196 
               
               
                   
                 26 
                 −31.5037 
                 1.998782 
                 N-KZFS4 
                 26.79483 
               
               
                   
                 27 
                 22.58101 
                 3.469186 
                   
                 22.81917 
               
               
                   
                 28 
                 111.9798 
                 2 
                 N-KZFS4 
                 22.79616 
               
               
                   
                 29 
                 22.78321 
                 4.624977 
                 S-PHM52 
                 23.29371 
               
               
                   
                 30 
                 84.68205 
                 2.394582 
                   
                 23.49102 
               
               
                   
                 31 
                 −47.88134 
                 2.649691 
                 S-BAL42 
                 23.52596 
               
               
                   
                 32 
                 −47.94859 
                 9 
                   
                 24.50782 
               
               
                   
                 IMA 
                 Infinity 
                   
                   
                 28.000 
               
               
                   
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
               
             
           
               
                 TABLE 5b 
               
               
                   
               
               
                 Focusing and Breathing Data for Example 5 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 OBMG 
                 0.0 
                 −0.033 
               
               
                   
                 OBIM 
                 Infinity 
                 606.1 
               
               
                   
                 T0 
                 Infinity 
                 400.0 
               
               
                   
                 T4 
                 2.008 
                 3.832 
               
               
                   
                 T9 
                 34.850 
                 29.133 
               
               
                   
                 Breathing 
                 0.00% 
                 0.14% 
               
               
                   
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
             
               
               
               
               
             
               
               
               
               
             
           
               
                 TABLE 5c 
               
             
             
               
                   
               
               
                 Aspheric Surface Data for Example 5 
               
             
          
           
               
                   
                 Surface # 
               
             
          
           
               
                   
                 1 
                 13 
                 28 
               
               
                   
                   
               
             
          
           
               
                 R 
                 375.546 
                 50.614 
                 111.980 
               
               
                 k 
                 0.0 
                 0.0 
                 0.0 
               
               
                 C4 
                 1.333707e−6 
                 1.204138e−5 
                 −3.462446e−5 
               
               
                 C6 
                 −2.060015e−10 
                 −2.759991e−9 
                 1.890542e−8 
               
               
                 C8 
                 4.106414e−14 
                 1.027888e−10 
                 −9.138490e−10 
               
               
                 C10 
                 1.924041e−18 
                 −2.113544e−13 
                 1.622290e−12 
               
               
                 C12 
                 0.0 
                 0.0 
                 0.0 
               
               
                 C14 
                 0.0 
                 0.0 
                 0.0 
               
               
                 C16 
                 0.0 
                 0.0 
                 0.0 
               
               
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
               
               
               
             
               
               
               
               
               
               
             
           
               
                 TABLE 6a 
               
             
             
               
                   
               
               
                 Prescription Data for Example 6 
               
             
          
           
               
                   
                 Surf 
                 Radius 
                 Thickness 
                 Glass 
                 Diameter 
               
               
                   
                   
               
             
          
           
               
                   
                 OBJ 
                 Infinity 
                 Infinity 
                   
                   
               
               
                   
                  1 
                 92.60746 
                 4 
                 S-BAL42 
                 75.04797 
               
               
                   
                  2 
                 45.74328 
                 12.78556 
                   
                 65.11839 
               
               
                   
                  3 
                 581.3462 
                 3.5 
                 S-FPL51 
                 65.0555 
               
               
                   
                  4 
                 47.48849 
                 54.99829 
                   
                 58.3797 
               
               
                   
                  5 
                 −89.49438 
                 5 
                 S-FSL5 
                 55.56219 
               
               
                   
                  6 
                 −60.55685 
                 0.5 
                   
                 55.98575 
               
               
                   
                  7 
                 61.06037 
                 7 
                 S-FPL51 
                 52.43241 
               
               
                   
                  8 
                 371.3011 
                 27.48415 
                   
                 51.47304 
               
               
                   
                  9 
                 −44.26416 
                 4 
                 S-BSL7 
                 36.28671 
               
               
                   
                 10 
                 −211.557 
                 2 
                   
                 35.68603 
               
               
                   
                 STO 
                 Infinity 
                 2 
                   
                 35.09028 
               
               
                   
                 12 
                 73.75098 
                 3.5 
                 S-LAH53 
                 36.12446 
               
               
                   
                 13 
                 86.92011 
                 8 
                 S-FPL53 
                 35.99807 
               
               
                   
                 14 
                 −94.0894 
                 0.2495871 
                   
                 36.29503 
               
               
                   
                 15 
                 58.71839 
                 6 
                 S-FPL53 
                 35.93812 
               
               
                   
                 16 
                 326.8157 
                 2.5 
                 N-KZFS4 
                 35.0456 
               
               
                   
                 17 
                 357.0363 
                 5.331577 
                 S-FPL51 
                 34.48392 
               
               
                   
                 18 
                 −87.41435 
                 0.345288 
                   
                 33.69834 
               
               
                   
                 19 
                 58.2085 
                 1.998782 
                 N-KZFS4 
                 31.28165 
               
               
                   
                 20 
                 24.90714 
                 6.282571 
                 S-PHM52 
                 28.73855 
               
               
                   
                 21 
                 179.6464 
                 0.4350781 
                   
                 27.50653 
               
               
                   
                 22 
                 30.76528 
                 4.092707 
                 S-PHM52 
                 25.80209 
               
               
                   
                 23 
                 100.0604 
                 1.998782 
                 N-KZFS4 
                 24.33576 
               
               
                   
                 24 
                 17.16774 
                 3.277854 
                   
                 21.73136 
               
               
                   
                 25 
                 83.63094 
                 2 
                 N-KZFS4 
                 21.7505 
               
               
                   
                 26 
                 18.66912 
                 10.91264 
                 S-PHM52 
                 22.01914 
               
               
                   
                 27 
                 −14.75671 
                 2 
                 N-KZFS4 
                 22.24207 
               
               
                   
                 28 
                 −250.6923 
                 4.307977 
                   
                 22.86026 
               
               
                   
                 29 
                 −27.73417 
                 2.502858 
                 S-PHM52 
                 23.04734 
               
               
                   
                 30 
                 −25.15099 
                 2 
                 N-KZFS4 
                 23.8395 
               
               
                   
                 31 
                 −41.74433 
                 9 
                   
                 25.0488 
               
               
                   
                 IMA 
                 Infinity 
                   
                   
                 28.000 
               
               
                   
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
               
             
           
               
                 TABLE 6b 
               
               
                   
               
               
                 Focusing and Breathing Data for Example 6 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 OBMG 
                 0.0 
                 −0.050 
               
               
                   
                 OBIM 
                 Infinity 
                 609.6 
               
               
                   
                 T0 
                 Infinity 
                 409.6 
               
               
                   
                 T2 
                 12.786 
                 19.625 
               
               
                   
                 T8 
                 27.484 
                 19.594 
               
               
                   
                 T31 
                 9.000 
                 10.050 
               
               
                   
                 Breathing 
                 0.00% 
                 2.01% 
               
               
                   
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
             
               
               
               
             
               
               
               
               
             
           
               
                 TABLE 6c 
               
             
             
               
                   
               
               
                 Aspheric Surface Data for Example 6 
               
             
          
           
               
                   
                 Surface # 
                   
               
             
          
           
               
                   
                 1 
                 12 
               
               
                   
                   
               
             
          
           
               
                   
                 R 
                 92.607 
                 73.751 
               
               
                   
                 k 
                 0.0 
                 0.0 
               
               
                   
                 C4 
                 3.457947e−7 
                 −2.202698e−6 
               
               
                   
                 C6 
                 1.407617e−10 
                 −6.299493e−10 
               
               
                   
                 C8 
                 −2.136379e−14 
                 2.204620e−12 
               
               
                   
                 C10 
                 1.804411e−17 
                 −2.816188e−15 
               
               
                   
                 C12 
                 0.0 
                 0.0 
               
               
                   
                 C14 
                 0.0 
                 0.0 
               
               
                   
                 C16 
                 0.0 
                 0.0 
               
               
                   
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
               
               
               
             
               
               
               
               
               
               
             
           
               
                 TABLE 7a 
               
             
             
               
                   
               
               
                 Prescription Data for Example 7 
               
             
          
           
               
                   
                 Surf 
                 Radius 
                 Thickness 
                 Glass 
                 Diameter 
               
               
                   
                   
               
             
          
           
               
                   
                 OBJ 
                 Infinity 
                 Infinity 
                   
                   
               
               
                   
                  1 
                 396.9664 
                 4 
                 S-BSL7 
                 48.4 
               
               
                   
                  2 
                 38.47405 
                 15.86035 
                   
                 43 
               
               
                   
                  3 
                 49.92867 
                 7 
                 S-TIM2 
                 40 
               
               
                   
                  4 
                 56.06739 
                 23.0195 
                   
                 38 
               
               
                   
                  5 
                 −42.09343 
                 4 
                 S-BSL7 
                 33.2 
               
               
                   
                  6 
                 −103.028 
                 2.372342 
                   
                 36.4 
               
               
                   
                 STO 
                 Infinity 
                 2 
                   
                 38.34528 
               
               
                   
                  8 
                 474.2342 
                 7 
                 S-FPL51 
                 40.71931 
               
               
                   
                  9 
                 −58.61468 
                 0.25 
                   
                 42.23332 
               
               
                   
                 10 
                 −198.4969 
                 7 
                 S-FPL51 
                 43.66711 
               
               
                   
                 11 
                 −49.81846 
                 0.25 
                   
                 44.79874 
               
               
                   
                 12 
                 126.6418 
                 11 
                 S-FPL51 
                 45.55004 
               
               
                   
                 13 
                 −52.1216 
                 2.5 
                 N-KZFS4 
                 45.35187 
               
               
                   
                 14 
                 −129.611 
                 0.25 
                   
                 45.4687 
               
               
                   
                 15 
                 721.1586 
                 8.5 
                 S-FPL53 
                 44.98233 
               
               
                   
                 16 
                 −61.33901 
                 0.25 
                   
                 44.38943 
               
               
                   
                 17 
                 44.12771 
                 9 
                 S-FPL51 
                 38.94397 
               
               
                   
                 18 
                 −486.2903 
                 0.25 
                   
                 35.92704 
               
               
                   
                 19 
                 130.5751 
                 9 
                 S-PHM52 
                 33.81175 
               
               
                   
                 20 
                 −35.0691 
                 2 
                 N-KZFS4 
                 33.81175 
               
               
                   
                 21 
                 18.9107 
                 4.417785 
                   
                 23.48 
               
               
                   
                 22 
                 −207.8419 
                 2 
                 N-KZFS4 
                 23.48 
               
               
                   
                 23 
                 30.18606 
                 7 
                 S-PHM52 
                 24 
               
               
                   
                 24 
                 −33.99464 
                 1.105499 
                   
                 24 
               
               
                   
                 25 
                 −24.05843 
                 2.5 
                 S-BSM4 
                 23 
               
               
                   
                 26 
                 −48.65227 
                 15.72706 
                   
                 24 
               
               
                   
                 IMA 
                 Infinity 
                   
                   
                 28.000 
               
               
                   
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
               
             
           
               
                 TABLE 7b 
               
               
                   
               
               
                 Focusing and Breathing Data for Example 7 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 OBMG 
                 0.0 
                 −0.065 
               
               
                   
                 OBIM 
                 Infinity 
                 655.1 
               
               
                   
                 T0 
                 Infinity 
                 500.0 
               
               
                   
                 T2 
                 15.860 
                 1.990 
               
               
                   
                 T4 
                 23.020 
                 41.598 
               
               
                   
                 T26 
                 15.727 
                 17.899 
               
               
                   
                 Breathing 
                 0.00% 
                 −0.03% 
               
               
                   
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
             
               
               
               
             
               
               
               
               
             
           
               
                 TABLE 7c 
               
             
             
               
                   
               
               
                 Aspheric Surface Data for Example 7 
               
             
          
           
               
                   
                 Surface # 
                   
               
             
          
           
               
                   
                 1 
                 6 
               
               
                   
                   
               
             
          
           
               
                   
                 R 
                 396.966 
                 −103.028 
               
               
                   
                 k 
                 0.0 
                 0.0 
               
               
                   
                 C4 
                 −3.585839e−7 
                 7.833468e−6 
               
               
                   
                 C6 
                 9.755781e−10 
                 2.834875e−9 
               
               
                   
                 C8 
                 −1.576946e−12 
                 −3.428422e−12 
               
               
                   
                 C10 
                 1.165119e−15 
                 9.842607e−16 
               
               
                   
                 C12 
                 0.0 
                 0.0 
               
               
                   
                 C14 
                 0.0 
                 0.0 
               
               
                   
                 C16 
                 0.0 
                 0.0 
               
               
                   
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
               
               
               
             
               
               
               
               
               
               
             
           
               
                 TABLE 8a 
               
             
             
               
                   
               
               
                 Prescription Data for Example 8 
               
             
          
           
               
                   
                 Surf 
                 Radius 
                 Thickness 
                 Glass 
                 Diameter 
               
               
                   
                   
               
             
          
           
               
                   
                 OBJ 
                 Infinity 
                 Infinity 
                   
                   
               
               
                   
                  1 
                 −283.9121 
                 4 
                 N-KZFS4 
                 61.6184 
               
               
                   
                  2 
                 204.1113 
                 5 
                 S-NPH1 
                 60.59697 
               
               
                   
                  3 
                 366.4659 
                 2 
                   
                 60.01507 
               
               
                   
                  4 
                 87.86912 
                 10 
                 ACRYLIC 
                 55.98527 
               
               
                   
                  5 
                 76.18259 
                 27.65287 
                   
                 53.48406 
               
               
                   
                 STO 
                 Infinity 
                 2.997092 
                   
                 54.48122 
               
               
                   
                  7 
                 140.7083 
                 13 
                 S-FPL51 
                 56.69821 
               
               
                   
                  8 
                 −87.01917 
                 2.082923 
                   
                 57.24739 
               
               
                   
                  9 
                 105.2124 
                 3 
                 N-KZFS4 
                 55.19328 
               
               
                   
                 10 
                 45.35061 
                 14.9561 
                 S-FPL51 
                 52.6903 
               
               
                   
                 11 
                 −88.21712 
                 0.25 
                   
                 52.05543 
               
               
                   
                 12 
                 35.79661 
                 13 
                 S-FPL51Y 
                 43.42701 
               
               
                   
                 13 
                 −150.139 
                 2.5 
                 N-KZFS4 
                 39.11433 
               
               
                   
                 14 
                 22.08163 
                 2.832599 
                   
                 30.50124 
               
               
                   
                 15 
                 28.92564 
                 2.5 
                 N-KZFS4 
                 30.16155 
               
               
                   
                 16 
                 16.95775 
                 7 
                 S-LAH53 
                 28.38056 
               
               
                   
                 17 
                 25.87959 
                 3.153191 
                   
                 26.86656 
               
               
                   
                 18 
                 47.66877 
                 2.5 
                 S-LAL14 
                 26.93462 
               
               
                   
                 19 
                 34.94616 
                 1.75774 
                   
                 26.2819 
               
               
                   
                 20 
                 96.94503 
                 5 
                 S-PHM52 
                 27.11023 
               
               
                   
                 21 
                 770.8764 
                 15.16361 
                   
                 27.11922 
               
               
                   
                 22 
                 Infinity 
                 5 
                 S-BSL7 
                 28.000 
               
               
                   
                 23 
                 Infinity 
                 4.8 
                   
                 27.62637 
               
               
                   
                 IMA 
                 Infinity 
                   
                   
                 28.000 
               
               
                   
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
               
               
               
             
           
               
                 TABLE 8b 
               
               
                   
               
               
                 Focusing and Breathing Data for Example 8 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 OBMG 
                 0.0 
                 −0.032 
                 −0.062 
                 −0.137 
               
               
                   
                 OBIM 
                 Infinity 
                 2163.0 
                 1174.4 
                 609.6 
               
               
                   
                 T0 
                 Infinity 
                 2000.0 
                 1000.0 
                 409.5 
               
               
                   
                 T3 
                 2.000 
                 12.880 
                 22.446 
                 44.067 
               
               
                   
                 T19 
                 1.758 
                 3.753 
                 5.539 
                 9.688 
               
               
                   
                 Breathing 
                 0.00% 
                 0.14% 
                 −0.09% 
                 0.02% 
               
               
                   
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
             
               
               
               
             
           
               
                 TABLE 8c 
               
             
             
               
                   
               
               
                 Aspheric Surface Data for Example 8 
               
             
          
           
               
                   
                 Surface # 
               
               
                   
                 5 
               
               
                   
                   
               
             
          
           
               
                   
                 R 
                 87.869 
               
               
                   
                 k 
                 0.0 
               
               
                   
                 C4 
                 −1.961709e−6 
               
               
                   
                 C6 
                 −1.791880e−10 
               
               
                   
                 C8 
                 −1.068037e−12 
               
               
                   
                 C10 
                 1.896598e−15 
               
               
                   
                 C12 
                 −2.843718e−18 
               
               
                   
                 C14 
                 2.457603e−21 
               
               
                   
                 C16 
                 −8.748159e−25 
               
               
                   
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
               
               
               
             
               
               
               
               
               
               
             
           
               
                 TABLE 9a 
               
             
             
               
                   
               
               
                 Prescription Data for Example 9 
               
             
          
           
               
                   
                 Surf 
                 Radius 
                 Thickness 
                 Glass 
                 Diameter 
               
               
                   
                   
               
             
          
           
               
                   
                 OBJ 
                 Infinity 
                 Infinity 
                   
                   
               
               
                   
                  1 
                 333.9197 
                 4 
                 S-FPL53 
                 58.22967 
               
               
                   
                  2 
                 38.30447 
                 20.67997 
                   
                 50.95781 
               
               
                   
                  3 
                 54.68269 
                 4 
                 S-TIH53 
                 50.85248 
               
               
                   
                  4 
                 50.56709 
                 6 
                 N-KZFS4 
                 48.6793 
               
               
                   
                  5 
                 63.96197 
                 29.92107 
                   
                 46.69942 
               
               
                   
                  6 
                 −26.59337 
                 4 
                 S-BSL7 
                 34.48456 
               
               
                   
                  7 
                 −73.84637 
                 2.413273 
                   
                 37.60764 
               
               
                   
                 STO 
                 Infinity 
                 2 
                   
                 39.51109 
               
               
                   
                  9 
                 −1054.178 
                 8.946326 
                 S-FPL53 
                 41.47097 
               
               
                   
                 10 
                 −35.74606 
                 0.25 
                   
                 42.90604 
               
               
                   
                 11 
                 −614.98 
                 7.676709 
                 S-FPL53 
                 45.80089 
               
               
                   
                 12 
                 −47.17851 
                 0.25 
                   
                 46.41689 
               
               
                   
                 13 
                 87.01859 
                 13 
                 S-FPL53 
                 46.00835 
               
               
                   
                 14 
                 −47.23339 
                 2.5 
                 N-KZFS4 
                 45.21423 
               
               
                   
                 15 
                 −127.2815 
                 0.25 
                   
                 44.90176 
               
               
                   
                 16 
                 173.0087 
                 8.026098 
                 S-FPL53 
                 43.78873 
               
               
                   
                 17 
                 −78.90241 
                 0.25 
                   
                 42.71688 
               
               
                   
                 18 
                 34.08032 
                 9 
                 S-FPL53 
                 36.5209 
               
               
                   
                 19 
                 381.1721 
                 0.25 
                   
                 32.94332 
               
               
                   
                 20 
                 91.59234 
                 5.338591 
                 N-KZFS4 
                 31.6 
               
               
                   
                 21 
                 19.37844 
                 14.57205 
                   
                 25.8 
               
               
                   
                 22 
                 −56.03064 
                 6 
                 S-FPL53 
                 25.4 
               
               
                   
                 23 
                 −31.00777 
                 1.193065 
                   
                 25.52397 
               
               
                   
                 24 
                 −22.63342 
                 2.5 
                 N-KZFS4 
                 25.52397 
               
               
                   
                 25 
                 −34.92177 
                 9.0806 
                   
                 27 
               
               
                   
                 IMA 
                 Infinity 
                   
                   
                 28.000 
               
               
                   
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
               
             
           
               
                 TABLE 9b 
               
               
                   
               
               
                 Focusing and Breathing Data for Example 9 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 OBMG 
                 0.0 
                 −0.064 
               
               
                   
                 OBIM 
                 Infinity 
                 670.7 
               
               
                   
                 T0 
                 Infinity 
                 500.0 
               
               
                   
                 T2 
                 20.680 
                 4.107 
               
               
                   
                 T5 
                 29.921 
                 52.876 
               
               
                   
                 T25 
                 9.081 
                 11.345 
               
               
                   
                 Breathing 
                 0.00% 
                 −0.23% 
               
               
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
             
               
               
               
             
               
               
               
             
           
               
                 TABLE 9c 
               
             
             
               
                   
               
               
                 Aspheric Surface Data for Example 9 
               
             
          
           
               
                   
                 Surface # 
               
             
          
           
               
                   
                 1 
                 7 
               
               
                   
               
             
          
           
               
                 R 
                 333.920 
                 −73.846 
               
               
                 k 
                 0.0 
                 0.0 
               
               
                 C4 
                 3.394842e−7 
                 8.11606e−6 
               
               
                 C6 
                 3.05377e−10 
                 1.990386e−9 
               
               
                 C8 
                 −1.402218e−13 
                 −4.780505e−12 
               
               
                 C10 
                 6.990304e−17 
                 −3.430611e−16 
               
               
                 C12 
                 0.0 
                 0.0 
               
               
                 C14 
                 0.0 
                 0.0 
               
               
                 C16 
                 0.0 
                 0.0 
               
               
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
               
               
             
               
               
               
               
               
             
           
               
                 TABLE 10a 
               
             
             
               
                   
               
               
                 Prescription Data for Example 10 
               
             
          
           
               
                 Surf 
                 Radius 
                 Thickness 
                 Glass 
                 Diameter 
               
               
                   
               
             
          
           
               
                 OBJ 
                 Infinity 
                 Infinity 
                   
                   
               
               
                  1 
                 100.4234 
                 5 
                 S-FPL51 
                 73.48584 
               
               
                  2 
                 37.38913 
                 18.51934 
                   
                 60.32297 
               
               
                  3 
                 −147.4289 
                 4 
                 CaF2 
                 59.25265 
               
               
                  4 
                 55.04329 
                 25.08171 
                   
                 54.91074 
               
               
                  5 
                 216.8735 
                 8 
                 S-BAL42 
                 55.90533 
               
               
                  6 
                 −139.4082 
                 56.78947 
                   
                 55.69153 
               
               
                  7 
                 −40.97067 
                 4 
                 S-BSL7 
                 36.80999 
               
               
                  8 
                 −110.0568 
                 2.413273 
                   
                 39.47896 
               
               
                 STO 
                 Infinity 
                 2 
                   
                 41.62642 
               
               
                 10 
                 1349.394 
                 11 
                 CaF2 
                 45.6 
               
               
                 11 
                 −43.21708 
                 0.25 
                   
                 45.6 
               
               
                 12 
                 −473.4083 
                 10 
                 CaF2 
                 47.1 
               
               
                 13 
                 −53.85559 
                 0.25 
                   
                 47.32 
               
               
                 14 
                 119.5412 
                 2.5 
                 S-BAL42 
                 46 
               
               
                 15 
                 31.90913 
                 15 
                 CaF2 
                 44 
               
               
                 16 
                 −272.1249 
                 0.25 
                   
                 44 
               
               
                 17 
                 106.3444 
                 9 
                 CaF2 
                 44 
               
               
                 18 
                 −71.05028 
                 0.25 
                   
                 44 
               
               
                 19 
                 39.68814 
                 13 
                 CaF2 
                 38.4 
               
               
                 20 
                 −69.05873 
                 0.25 
                   
                 38.4 
               
               
                 21 
                 −185.5592 
                 2 
                 N-KZFS4 
                 34 
               
               
                 22 
                 25.56384 
                 13.18843 
                   
                 29 
               
               
                 23 
                 500.099 
                 6 
                 CaF2 
                 27.6 
               
               
                 24 
                 −37.13322 
                 1.193065 
                   
                 27.6 
               
               
                 25 
                 −26.3389 
                 2.5 
                 S-BAL42 
                 27.10797 
               
               
                 26 
                 −89.46392 
                 12.58599 
                   
                 27.8 
               
               
                 IMA 
                 Infinity 
                   
                   
                 28.000 
               
               
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
               
             
           
               
                 TABLE 10b 
               
               
                   
               
               
                 Focusing and Breathing Data for Example 10 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 OBMG 
                 0.0 
                 −0.056 
               
               
                   
                 OBIM 
                 Infinity 
                 605.0 
               
               
                   
                 T0 
                 Infinity 
                 380.0 
               
               
                   
                 T2 
                 18.519 
                 32.141 
               
               
                   
                 T6 
                 56.789 
                 41.689 
               
               
                   
                 T26 
                 12.586 
                 14.081 
               
               
                   
                 Breathing 
                 0.00% 
                 −2.42% 
               
               
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
             
               
               
               
               
             
               
               
               
               
             
           
               
                 TABLE 10c 
               
             
             
               
                   
               
               
                 Aspheric Surface Data for Example 10 
               
             
          
           
               
                   
                   
                 Surface # 
               
             
          
           
               
                   
                   
                 1 
                 8 
               
               
                   
                   
               
             
          
           
               
                   
                 R 
                 100.423 
                 −110.057 
               
               
                   
                 k 
                 0.0 
                 0.0 
               
               
                   
                 C4 
                 7.760058e−7 
                 7.048276e−6 
               
               
                   
                 C6 
                 1.290256e−10 
                 4.832011e−10 
               
               
                   
                 C8 
                 −5.36044e−14 
                 1.525344e−12 
               
               
                   
                 C10 
                 3.874459e−17 
                 −4.411433e−15 
               
               
                   
                 C12 
                 0.0 
                 0.0 
               
               
                   
                 C14 
                 0.0 
                 0.0 
               
               
                   
                 C16 
                 0.0 
                 0.0 
               
               
                   
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
               
               
             
               
               
               
               
               
             
           
               
                 TABLE 11a 
               
             
             
               
                   
               
               
                 Prescription Data for Example 11 
               
             
          
           
               
                 Surf 
                 Radius 
                 Thickness 
                 Glass 
                 Diameter 
               
               
                   
               
             
          
           
               
                 OBJ 
                 Infinity 
                 Infinity 
                   
                   
               
               
                  1 
                 127.4965 
                 5 
                 S-PHM52 
                 108.0221 
               
               
                  2 
                 43.87358 
                 31.87358 
                   
                 80.72893 
               
               
                  3 
                 −167.2721 
                 3.5 
                 S-FPL51 
                 80.45617 
               
               
                  4 
                 60.97883 
                 3 
                   
                 72.33558 
               
               
                  5 
                 77.11638 
                 3.5 
                 S-NPH1 
                 72.34854 
               
               
                  6 
                 43.80593 
                 12 
                 N-KZFS4 
                 68.04119 
               
               
                  7 
                 87.73247 
                 25.05135 
                   
                 67.54355 
               
               
                  8 
                 163.6522 
                 13 
                 S-LAH60 
                 71.87265 
               
               
                  9 
                 −154.9931 
                 1.804394 
                   
                 71.39477 
               
               
                 10 
                 177.1917 
                 4 
                 S-FPL53 
                 64.80238 
               
               
                 11 
                 41.56382 
                 25.38503 
                   
                 57.16877 
               
               
                 12 
                 70.88989 
                 4 
                 S-TIH53 
                 51.9796 
               
               
                 13 
                 85.19841 
                 6 
                 N-KZFS4 
                 50.51155 
               
               
                 14 
                 97.69889 
                 32.10155 
                   
                 47.98542 
               
               
                 15 
                 Infinity 
                 15 
                   
                 31 
               
               
                 16 
                 −28.37547 
                 4 
                 S-BSL7 
                 32.24222 
               
               
                 17 
                 −67.85317 
                 2.413273 
                   
                 35.2074 
               
               
                 STO 
                 Infinity 
                 2 
                   
                 37.23384 
               
               
                 19 
                 351.0629 
                 8.043594 
                 S-FPL53 
                 39.4388 
               
               
                 20 
                 −40.03792 
                 0.25 
                   
                 40.49308 
               
               
                 21 
                 −565.0694 
                 6.591541 
                 S-FPL53 
                 41 
               
               
                 22 
                 −47.81657 
                 0.25 
                   
                 41 
               
               
                 23 
                 90.31838 
                 9.455159 
                 S-FPL53 
                 40 
               
               
                 24 
                 −44.1493 
                 2.5 
                 N-KZFS4 
                 40 
               
               
                 25 
                 −173.1045 
                 0.25 
                   
                 40 
               
               
                 26 
                 143.7942 
                 6.476801 
                 S-FPL53 
                 39.8997 
               
               
                 27 
                 −75.13476 
                 0.25 
                   
                 39.39415 
               
               
                 28 
                 34.15295 
                 6.727187 
                 S-FPL53 
                 36 
               
               
                 29 
                 505.9532 
                 0.25 
                   
                 34.8 
               
               
                 30 
                 90.79729 
                 2.516471 
                 N-KZFS4 
                 33.4 
               
               
                 31 
                 20.62152 
                 6.357717 
                   
                 28.8 
               
               
                 32 
                 −222.4085 
                 6 
                 S-FPL53 
                 28.8 
               
               
                 33 
                 −30.19114 
                 0.648371 
                   
                 29 
               
               
                 34 
                 −26.87561 
                 2.5 
                 N-KZFS4 
                 29 
               
               
                 35 
                 −51.55495 
                 22.29939 
                   
                 30 
               
               
                 IMA 
                 Infinity 
                   
                   
                 28.000 
               
               
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
               
             
           
               
                 TABLE 11b 
               
               
                   
               
               
                 Focusing and Breathing Data for Example 11 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 OBMG 
                 0.0 
                 −0.037 
               
               
                   
                 OBIM 
                 Infinity 
                 609.6 
               
               
                   
                 T0 
                 Infinity 
                 334.6 
               
               
                   
                 T4 
                 3.000 
                 14.306 
               
               
                   
                 T7 
                 25.051 
                 21.027 
               
               
                   
                 T14 
                 32.102 
                 24.827 
               
               
                   
                 Breathing 
                 0.00% 
                 0.16% 
               
               
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
             
               
               
               
               
             
               
               
               
               
               
             
           
               
                 TABLE 11c 
               
             
             
               
                   
               
               
                 Aspheric Surface Data for Example 11 
               
             
          
           
               
                   
                 Surface # 
                   
               
             
          
           
               
                   
                 1 
                 10 
                 17 
               
               
                   
               
             
          
           
               
                   
                 R 
                 127.496 
                 177.192 
                 −67.853 
               
               
                   
                 k 
                 0.0 
                 0.0 
                 0.0 
               
               
                   
                 C4 
                 6.386489e−7 
                 6.302277e−7 
                 8.104566e−6 
               
               
                   
                 C6 
                 −5.204384e−11 
                 9.365987e−10 
                 2.137244e−9 
               
               
                   
                 C8 
                 7.139764e−15 
                 −3.328055e−13 
                 −2.210840e−12 
               
               
                   
                 C10 
                 1.397088e−18 
                 2.363550e−16 
                 −2.049639e−15 
               
               
                   
                 C12 
                 0.0 
                 0.0 
                 0.0 
               
               
                   
                 C14 
                 0.0 
                 0.0 
                 0.0 
               
               
                   
                 C16 
                 0.0 
                 0.0 
                 0.0 
               
               
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
               
               
             
               
               
               
               
               
             
           
               
                 TABLE 12a 
               
             
             
               
                   
               
               
                 Prescription Data for Example 12 
               
             
          
           
               
                 Surf 
                 Radius 
                 Thickness 
                 Glass 
                 Diameter 
               
               
                   
               
             
          
           
               
                 OBJ 
                 Infinity 
                 Infinity 
                   
                   
               
               
                  1 
                 −172.16 
                 7 
                 N-KZFS4 
                 135.6177 
               
               
                  2 
                 Infinity 
                 12 
                 S-NPH1 
                 141.6501 
               
               
                  3 
                 −766.15 
                 4.8 
                   
                 143.8727 
               
               
                  4 
                 451.74 
                 25 
                 S-BSL7 
                 147.7423 
               
               
                  5 
                 −218.85 
                 1.9 
                   
                 149.1813 
               
               
                  6 
                 300.00 
                 18 
                 CaF2 
                 147.2112 
               
               
                  7 
                 1676.30 
                 0.5 
                   
                 145.0360 
               
               
                  8 
                 196.12 
                 18 
                 CaF2 
                 141.8413 
               
               
                  9 
                 726.80 
                 5 
                   
                 138.5623 
               
               
                 10 
                 541.42 
                 8 
                 N-KZFS4 
                 135.5623 
               
               
                 11 
                 118.64 
                 44.7 
                   
                 126.5730 
               
               
                 STO 
                 Infinity 
                 4 
                   
                 128.6329 
               
               
                 13 
                 201.50 
                 25 
                 S-FPL53 
                 129.8978 
               
               
                 14 
                 −301.88 
                 0.5 
                   
                 128.8176 
               
               
                 15 
                 150.80 
                 25 
                 S-FPL53 
                 120.3746 
               
               
                 16 
                 −291.55 
                 11.6 
                   
                 116.7305 
               
               
                 17 
                 328.00 
                 25 
                 S-PHM52 
                 93.4670 
               
               
                 18 
                 −255.20 
                 20 
                 N-KZFS4 
                 79.2993 
               
               
                 19 
                 49.97 
                 20.2 
                   
                 57.0197 
               
               
                 20 
                 95.00 
                 11.6 
                 N-KZFS4 
                 54.7523 
               
               
                 21 
                 39.66 
                 13.6 
                 S-LAH53 
                 49.6368 
               
               
                 22 
                 93.31 
                 51.118 
                   
                 46.2380 
               
               
                 IMA 
                 Infinity 
                   
                   
                 28.0000 
               
               
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
             
               
               
               
             
           
               
                 TABLE 12b 
               
             
             
               
                   
               
               
                 Aspheric Surface Data for Example 12 
               
             
          
           
               
                   
                 Surf. # 
               
               
                   
                 4 
               
               
                   
               
             
          
           
               
                   
                 R 
                 451.74 
               
               
                   
                 k 
                 0.0 
               
               
                   
                 C4 
                 −7.071869e−08 
               
               
                   
                 C6 
                 5.824021e−13 
               
               
                   
                 C8 
                 −1.045519e−17 
               
               
                   
                 C10 
                 2.748538e−22 
               
               
                   
                 C12 
                 0.0 
               
               
                   
                 C14 
                 0.0 
               
               
                   
                 C16 
                 0.0 
               
               
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 13 
               
             
             
               
                   
               
               
                 Values for the Conditions for Each Example 
               
             
          
           
               
                   
                 F1/FG 
                 F2/FG 
                 F3/FG 
                 F4/FG 
                 y MAX /y SC   
                 Z SC /FG 
                 R SC /FG 
                 OBMG PG   
               
               
                   
                   
               
             
          
           
               
                 Ex. 1 
                 −3.576 
                 0.919 
                 −1.431 
                 2.256 
                 1.972 
                 0.761 
                 0.340 
                 0.000 
               
               
                 Ex. 2 
                 −1.646 
                 0.774 
                 −1.471 
                 1.815 
                 2.010 
                 0.754 
                 0.274 
                 0.000 
               
               
                 Ex. 3 
                 −3.202 
                 0.851 
                 −1.088 
                 2.241 
                 2.228 
                 0.611 
                 0.337 
                 0.000 
               
               
                 Ex. 4 
                 −2.109 
                 0.812 
                 −1.442 
                 1.587 
                 2.280 
                 0.634 
                 0.245 
                 0.000 
               
               
                 Ex. 5 
                 −3.611 
                 0.816 
                 −2.027 
                 −20.31 
                 1.990 
                 0.704 
                 0.659 
                 −0.049 
               
               
                 Ex. 6 
                 −2.121 
                 0.676 
                 −1.481 
                 5.308 
                 1.889 
                 0.727 
                 0.337 
                 0.088 
               
               
                 Ex. 7 
                 −3.568 
                 0.731 
                 −0.945 
                 8.747 
                 2.379 
                 0.893 
                 0.484 
                 −0.364 
               
               
                 Ex. 8 
                 −3.558 
                 0.914 
                 −1.408 
                 2.290 
                 1.938 
                 0.765 
                 0.340 
                 0.000 
               
               
                 Ex. 9 
                 −2.000 
                 0.671 
                 −0.990 
                 −8.742 
                 2.220 
                 0.868 
                 0.473 
                 −0.288 
               
               
                 Ex. 10 
                 −2.723 
                 0.648 
                 −0.767 
                 −7.586 
                 2.102 
                 0.789 
                 0.546 
                 −0.210 
               
               
                 Ex. 11 
                 −2.243 
                 0.618 
                 −1.006 
                 12.399 
                 1.883 
                 0.876 
                 0.478 
                 −0.175 
               
               
                 Ex. 12 
                 −2.029 
                 0.677 
                 −0.515 
                 1.402 
                 2.712 
                 0.483 
                 0.250 
                 0.000 
               
               
                   
               
             
          
         
       
     
                                                                 TABLE 14                   Refractive Indices for Glasses Used in the Examples                    N F′     N e     N C′         Glass Type   Maker   λ = 480 nm   λ = 546 nm   λ = 644 nm                    N-KZFS4   Schott   1.623802   1.616638   1.609873       S-NPH1   Ohara   1.835745   1.816432   1.799572       S-BSL7   Ohara   1.522357   1.518250   1.514251       S-FPL51   Ohara   1.501575   1.498454   1.495433       S-FPL51Y   Ohara   1.501604   1.498465   1.495430       S-LAH53   Ohara   1.821104   1.810774   1.801169       S-LAL14   Ohara   1.706235   1.699788   1.693583       N-LASF9   Schott   1.870583   1.856501   1.843756       ACRYLIC   Ohara   1.498324   1.493801   1.489370       S-FPL53   Ohara   1.442214   1.439854   1.437559       CaF2   Schott   1.437268   1.434929   1.432673       S-PHM52   Ohara   1.625350   1.620327   1.615507       S-BSM16   Ohara   1.628151   1.622864   1.617775       S-TIH1   Ohara   1.736122   1.723096   1.711428       S-LAH63   Ohara   1.819895   1.809220   1.799323       S-LAH60   Ohara   1.851152   1.839322   1.828416       S-NSL3   Ohara   1.524856   1.520325   1.515981       S-BAL42   Ohara   1.590519   1.585467   1.580614       S-FSL5   Ohara   1.492672   1.489147   1.485688       S-TIM2   Ohara   1.633149   1.624087   1.615811       S-BSM4   Ohara   1.620577   1.615203   1.610050       S-TIH53   Ohara   1.874313   1.855040   1.838067                    
Applications and Examples with Larger and Smaller Apertures
 
     Although examples of the present invention are very well suited for optical objectives having a relative aperture of f/1.5 or faster, the invention is also suited for somewhat slower optical systems that must have an extraordinarily high level of optical correction and/or cover a very large image format. Such optical systems are typically required in applications such as aerial reconnaissance, where the image circle may range up to about 70 mm in diameter or even larger and the desired aperture may range from f/1.8 up to about f/4. 
     The present invention is also well suited for extremely fast optical objectives having a relative aperture of f/1.2 or faster. Here it is convenient to use numerical aperture, or NA, which is defined as NA=1/(2(f/number)). Thus, an f/1.2 lens would have NA=1/(2×1.2)=0.417. Such high speed systems are particularly useful for low light imaging, night vision systems, and other specialized photographic and cinematographic applications. In some cases it is desirable to have sharp imagery with a numerical aperture (NA) larger than 0.5 (f/1.0), and in extreme cases the numerical aperture may reach 0.7 to 0.85 (f/0.71 to f/0.59). 
     Thus, the optical systems of the present invention can generally have an f/number in the range from f/0.6 to f/4. However, certain designs may fall into sub-ranges of this larger range, such as f/1.8 to f/4 and f/1.8 down to f/0.7, or even f/1.5 down to f/0.7 or f/1.5 down to f/0.6. 
     Altering a given optical system design to have a reduced numerical aperture is fairly straightforward. This typically involves stopping down the lens, reducing the vignetting to some extent, and re-optimizing for the new aperture. Generally, the image quality is improved when this is done unless the starting point is already diffraction limited. The lens size will also generally decrease, especially if the lens center thicknesses are reduced as much as possible after stopping down to avoid unnecessarily thick edge thicknesses. So, for example, a person with ordinary skill in the art of lens design would be able to successfully transform an f/2 or f/2.5 design into an f/2.8 or f/4 design. 
     Altering a given design to have an increased numerical aperture is more difficult, but is generally possible with proper technique. Lens center thicknesses must often be increased to avoid having edge thicknesses that are too thin. Vignetting must sometimes be increased to avoid overly large lens elements and/or to deal with increased aberration. The alteration must often be done in small steps to avoid the situation that rays can&#39;t be traced through the system during the early phases of optimization. Thus, in transforming an f/0.71 design into an f/0.6 design, it may be necessary to perform intermediate optimizations at f/0.67 and f/0.63, for example. These techniques will be well understood by a person with ordinary skill in the art of lens design. 
     EXAMPLE 13 
       FIG. 13   a  illustrates a cross-sectional layout of an example 150 mm focal length objective optimized for aerial reconnaissance. All of the element and group designations mentioned below are shown in  FIG. 13   a . The relative aperture is f/2.5, the image diagonal is 70 mm, and the diagonal field of view (FOV) is 26.3 degrees. Example 13 is corrected over a waveband ranging from about 450 nm to 900 nm, which is an especially useful waveband for reconnaissance and night-vision applications. 
       FIG. 13   b  plots the MTF vs. Image Height at 50, 100 and 200 cycles/mm for an object at infinity for example 13. The MTF curves indicate that Example 13 is extraordinarily well corrected at f/2.5, with MTF values at 100 cycles/mm greater than 60% over the entire image field. Example 13 does in fact meet the Rayleigh criterion for diffraction-limited performance at f/2.5 over a large central portion of the image field. This lens is well suited to sensors having a pixel size of 2 microns or less, and in an example has a total image resolution of about 1 gigapixel (i.e., about 1000 megapixels) or more. 
     In Example 13, the primary group PG comprises the entire lens. As discussed above, primary group PG comprises a negative powered front sub-group P 1 , followed by a positive powered sub-group P 2 , followed by a negative powered sub-group P 3 , followed by sub-group P 4  that can be either positively or negatively powered. 
     Also in Example 13, sub-group P 1  comprises a negative doublet  1314  and singlet  1303 . Doublet  1314  uses high-index anomalous dispersion materials N-KZFS4 and S-NPH1 to advantage, and as a result aids in reducing secondary and tertiary chromatic aberrations. The surface of sub-group P 1  closest to the object is a concave surface. The object-facing surface of singlet  1303  is aspherical, and this element is made of S-BSL7 to ensure that manufacturing this aspherical surface will not be problematic. The function of the aspherical surface is primarily to correct spherical aberration. 
     Sub-group P 2  comprises two positive cemented doublets  1315  and  1316 , together with a positive singlet  1309 . The positive elements in sub-group P 2  ( 1305 ,  1308 , and  1309 ) are all made of the low-index anomalous dispersion glass S-FPL53. The two negative elements are made of a matching anomalous dispersion material N-KZFS4. Since most of the positive optical power for Example 13 resides in sub-group P 2 , the system as a whole is very well corrected for chromatic aberrations, and is in fact superachromatic (i.e., with four color crossings) over a waveband extending from 450 nm to 900 nm. 
     Sub-group P 3  comprises a negative powered singlet  1310 . The outer shape of 1310 is meniscus toward the image plane. The concave surface SC is strongly curved and therefore helps a great deal in correcting field curvature and astigmatism. 
     Sub-group P 4  is a negative group comprising a negative cemented doublet  1317 . Sub-group P 4  serves mainly to correct distortion and astigmatism. This group has a fairly high power compared to other examples disclosed in this invention having four sub-groups. This is due in part to the fact that the aperture requirements have been relaxed, so that the outer negative groups can have greater optical power, and hence a greater field-flattening contribution without harming the overall correction. 
     Since Example 13 is intended for aerial reconnaissance applications, it has been optimized for an object located at infinity. However, small focus adjustments can be made by moving the entire lens without significantly degrading the lens performance. 
     Most of the optical materials used in Example 13 are listed in Table 14 above. However, there are two materials used in this example that are not listed in Table 14. S-LAH59, manufactured by Ohara, has indices of refraction of 1.829266, 1.820166, and 1.811579 at wavelengths of 480 nm, 546 nm, and 644 nm, respectively. N-KZFS2, manufactured by Schott, has indices of refraction of 1.566119, 1.560823, and 1.555701 at wavelengths of 480 nm, 546 nm, and 644 nm, respectively. 
     The values for the defining conditions of Example 13 are as follows: F1/FG=−3.869; F2/FG=+0.600; F3/FG=−1.734; F4/FG=−0.974; y max /y sc =1.734; Z SC /FG=0.626; R SC /FG=0.398; and OBMG PG =0.000. 
     The optical prescription for Example 13 is given in Table 15a, below. Aspheric coefficients for surface #4 are given in Table 15b. 
     
       
         
               
             
               
               
               
               
               
             
               
               
               
               
               
             
           
               
                 TABLE 15a 
               
             
             
               
                   
               
               
                 Prescription Data for Example 13 
               
             
          
           
               
                 Surf 
                 Radius 
                 Thickness 
                 Glass 
                 Diameter 
               
               
                   
               
             
          
           
               
                 OBJ 
                 Infinity 
                 Infinity 
                   
                   
               
               
                  1 
                 −159.7966 
                 7 
                 S-NPH1 
                 69.64 
               
               
                  2 
                 −123.6288 
                 6 
                 N-KZFS4 
                 69.33 
               
               
                  3 
                 −3697.4493 
                 6.349026 
                   
                 67.98 
               
               
                  4 
                 154.0810 
                 12 
                 S-BSL7 
                 66.37 
               
               
                  5 
                 275.8622 
                 20.062832 
                   
                 64.84 
               
               
                 STO 
                 Infinity 
                 10 
                   
                 66.241 
               
               
                  7 
                 221.4204 
                 12 
                 S-FPL53 
                 68.51 
               
               
                  8 
                 −144.1959 
                 6 
                 N-KZFS4 
                 68.92 
               
               
                  9 
                 −187.7078 
                 0.298864 
                   
                 69.77 
               
               
                 10 
                 149.2193 
                 6 
                 N-KZFS4 
                 70.22 
               
               
                 11 
                 69.2649 
                 12 
                 S-FPL53 
                 69.82 
               
               
                 12 
                 616.6508 
                 11.473882 
                   
                 70.17 
               
               
                 13 
                 85.0340 
                 15 
                 S-FPL53 
                 73.78 
               
               
                 14 
                 −189.0097 
                 13.779707 
                   
                 73.30 
               
               
                 15 
                 88.0624 
                 8 
                 S-LAH59 
                 63.62 
               
               
                 16 
                 59.7944 
                 44.695137 
                   
                 58.13 
               
               
                 17 
                 −47.4458 
                 5 
                 N-KZFS2 
                 53.14 
               
               
                 18 
                 166.8544 
                 12 
                 S-LAH63 
                 58.75 
               
               
                 19 
                 −260.1443 
                 32.355036 
                   
                 60.70 
               
               
                 IMA 
                 Infinity 
                   
                   
                 70.00 
               
               
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
             
               
               
               
             
           
               
                 TABLE 15b 
               
             
             
               
                   
               
               
                 Aspheric Surface Data for Example 13 
               
             
          
           
               
                   
                 Surf. # 
               
               
                   
                 4 
               
               
                   
               
             
          
           
               
                   
                 R 
                 107.022 
               
               
                   
                 k 
                 0.0 
               
               
                   
                 C4 
                 −2.981230e−7 
               
               
                   
                 C6 
                 −1.733792e−11 
               
               
                   
                 C8 
                 −6.953303e−15 
               
               
                   
                 C10 
                 2.407175e−18 
               
               
                   
               
             
          
         
       
     
     EXAMPLE 14 
       FIG. 14   a  shows a cross-sectional layout of an example of a very fast 50 mm focal length objective optimized for low light applications. All of the element and group designations mentioned below are shown in  FIG. 14   a . The relative aperture is f/0.71, the image diagonal is 16 mm, and the diagonal field of view (FOV) is 18.2 degrees. Example 14 is superachromatically corrected over a large waveband ranging from about 435 nm to 1000 nm, which is an especially useful waveband for reconnaissance and night-vision applications. 
       FIG. 14   b  plots the MTF vs. Image Height at 10, 20 and 40 cycles/mm for an object at infinity. These curves indicate that Example 14 is extraordinarily well corrected at f/0.71, with particularly good correction near the optical axis. 
     The primary group PG comprises the entire lens. As discussed above, primary group PG comprises a negative powered front sub-group P 1 , followed by a positive powered sub-group P 2 , followed by a negative powered sub-group P 3 , followed by sub-group P 4  that can be either positively or negatively powered. 
     In Example 14, sub-group P 1  comprises a negative doublet  1414 . This doublet uses high-index anomalous dispersion materials N-KZFS4 and S-NPH1 to advantage, and as a result aids in reducing secondary and tertiary chromatic aberrations. The surface of sub-group P 1  closest to the object is a concave surface. 
     Sub-group P 2  comprises two positive cemented doublets  1415  and  1416 , together with positive singlets  1405 ,  1408 ,  1409  and  1410 . Most of the positive elements in sub-group P 2  are made of the low-index anomalous dispersion glass S-FPL53. The two negative elements  1404  and  1407  are made of anomalous dispersion materials S-NBH51 and N-KZFS4, respectively. Since most of the positive optical power for Example 13 resides in sub-group P 2 , the system as a whole is very well corrected for chromatic aberrations, and is in fact superachromatic (i.e., with four color crossings) over a waveband extending from 435 nm to 1000 nm. The doublet  1415  incorporates aspheric surfaces on both of its outer faces. This serves to correct both spherical aberration and spherochromatism. 
     Sub-group P 3  comprises a negative powered singlet  1411 . The outer shape of 1411 is meniscus toward the image plane. The concave surface SC is strongly curved and therefore helps a great deal in correcting field curvature and astigmatism.  1411  also includes an aspheric surface, which helps to correct coma and astigmatism. 
     Sub-group P 4  is a negative group comprising a very weak positive singlet  1412 . Sub-group P 4  serves mainly to correct distortion and astigmatism, and includes an aspheric surface to assist in this correction. 
     Most of the optical materials used in Example 14 are listed in Table 14 above. However, there are two materials used in Example 14 that are not listed in Table 14. S-NBH51, manufactured by Ohara, has indices of refraction of 1.765738, 1.754531 and 1.744240 at wavelengths of 480 nm, 546 nm, and 644 nm, respectively. S-TIH11, also manufactured by Ohara, has indices of refraction of 1.808412, 1.791919 and 1.777326 at wavelengths of 480 nm, 546 nm, and 644 nm, respectively. 
     The values for the defining conditions of Example 14 are as follows: F1/FG=−2.578; F2/FG=+1.125; F3/FG=−0.635; F4/FG=7.839; y max /y sc =4.985; Z SC /FG=0.303; R SC /FG=0.289; and OBMG PG =0.000. 
     The optical prescription for Example 14 is given in Table 16a. Aspheric coefficients for surfaces #4, 6, 18 and 20 are given in Table 16b. 
     
       
         
               
             
               
               
               
               
               
             
           
               
                 TABLE 16a 
               
             
             
               
                   
               
               
                 Prescription Data for Example 14 
               
             
          
           
               
                 Surf 
                 Radius 
                 Thickness 
                 Glass 
                 Diameter 
               
               
                   
               
               
                 OBJ 
                 Infinity 
                 Infinity 
                   
                   
               
               
                 1, STO 
                 −66.1622 
                 3 
                 N-KZFS4 
                 70.45 
               
               
                  2 
                 197.3219 
                 7 
                 S-NPH1 
                 78.27 
               
               
                  3 
                 −1609.8865 
                 5 
                   
                 79.19 
               
               
                  4 
                 756.7839 
                 18 
                 S-PHM52 
                 81.69 
               
               
                  5 
                 −60.8839 
                 5 
                 S-NBH51 
                 83.40 
               
               
                  6 
                 −92.7276 
                 19.231450 
                   
                 89.46 
               
               
                  7 
                 114.3918 
                 25 
                 S-FPL53 
                 98.78 
               
               
                  8 
                 −142.6890 
                 0.1 
                   
                 98.78 
               
               
                  9 
                 61.8502 
                 22 
                 S-FPL53 
                 83.65 
               
               
                 10 
                 −623.5548 
                 2.5 
                 N-KZFS4 
                 79.61 
               
               
                 11 
                 38.7974 
                 2.669583 
                   
                 63.75 
               
               
                 12 
                 41.3346 
                 20 
                 S-FPL53 
                 64.16 
               
               
                 13 
                 −194.2278 
                 0.1 
                   
                 63.11 
               
               
                 14 
                 39.5412 
                 9 
                 S-FPL53 
                 51.62 
               
               
                 15 
                 147.8627 
                 0.1 
                   
                 49.86 
               
               
                 16 
                 28.9170 
                 12 
                 S-FPL53 
                 39.96 
               
               
                 17 
                 212.3527 
                 0.1 
                   
                 32.03 
               
               
                 18 
                 58.2811 
                 2 
                 N-KZFS4 
                 27.68 
               
               
                 19 
                 14.4684 
                 1.983785 
                   
                 21.20 
               
               
                 20 
                 22.5127 
                 4.465207 
                 S-TIH11 
                 21.20 
               
               
                 21 
                 22.1455 
                 8.684122 
                   
                 18.24 
               
               
                 IMA 
                 Infinity 
                   
                   
                 16.00 
               
               
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
             
               
               
               
               
               
             
               
               
               
               
               
             
           
               
                 TABLE 16b 
               
             
             
               
                   
               
               
                 Aspheric Surface Data for Example 14 
               
             
          
           
               
                   
                 Surf. # 
               
             
          
           
               
                   
                 4 
                 6 
                 18 
                 20 
               
               
                   
               
             
          
           
               
                 R 
                 107.022 
                   
                   
                   
               
               
                 k 
                 0.0 
                 0.0 
                 0.0 
                 0.0 
               
               
                 C4 
                 −6.274019e−7 
                 −3.711145e−8 
                 −7.534748e−6 
                 −6.022078e−7 
               
               
                 C6 
                 −3.511426e−10 
                 −1.676246e−10 
                 5.880522e−8 
                 −7.811193e−8 
               
               
                 C8 
                 2.209673e−13 
                 9.128595e−14 
                 −2.905395e−10 
                 6.400339e−10 
               
               
                 C10 
                 −6.980152e−17 
                 −2.430516e−17 
                 7.553780e−13 
                 −6.496343e−12 
               
               
                 C12 
                 −8.751703e−22 
                 0.0 
                 0.0 
                 0.0 
               
               
                   
               
             
          
         
       
     
     It will be apparent to those skilled in the art that various modifications and variations can be made to the present invention without departing from the spirit and scope of the invention. Thus it is intended that the present invention cover the modifications and variations of this invention provided they come within the scope of the appended claims and their equivalents.