ANALYTIC FUNCTION OPTICAL COMPONENT

ELEMENT OPTIQUE POUR FONCTION ANALYTIQUE

English Abstract

ABSTRACT OF THE DISCLOSURE
Optical elements shaped in accordance with pre-
ferred analytic functions which permit the elements to be
relatively rotated about one or more pivots decentered
with respect to an optical axis to simulate the dioptric
action of a well corrected rotational lens element of
variable power that can be used to maintain focal setting
over a large range of object distances. The optical
elements can be used in pairs, singly with mirror images
of themselves, or they can be incorporated in more
elaborate systems to provide focusing action.

Claims

1. An analytic function optical component
comprising at least one transparent element that is
disposed along an optical axis, said element having on
one side a first surface that has a predetermined shape
which is at least a part of a surface of revolution and
on the other side thereof a second aspheric surface that
is nonrotationally symmetric and mathematically described
by a preselected polynomial equation having a nonzero
term of at least 4th order, said first and second
surfaces of said element being structured so that said
element can be displaced generally laterally relative to
said optical axis by rotation of said element about an
axis of revolution other than said optical axis such that
said second surface of said element operates to continu-
ously change certain optical properties of said component
as said element moves relative to said optical axis while
said first surface of said element remains optically
invariant, not effecting any changes in any optical pro-
perties of said component, as said element moves relative
to said optical axis.
2. The component of claim 1 wherein said first
surface is planar.
3. The component of claim 1 wherein said axis
of revolution is offset and parallel with respect to said
optical axis.
4. The component of claim 1 wherein said poly-
nomial equation describing said second surface is in
Cartesian coordinates of the form:
z = K1 (xy2+?x3)-K2x3y+K3xy3
where K1 is assignable constants and
K2 = -?K1 and K3 = - ?K1.
-47-

5. The component of claim 1 wherein said poly-
nomial equation describing said second surface is in
Cartesian coordinates of the form:
Z = K1(xy2+?x3)-K2x3y+K3xy3+K4x4+K5x2y2
where K1-K4 are assignable constants and K5 = 2K4 plus
residue conveniently set to zero.
6. An analytic function optical component
comprising at least one pair of transparent elements
that are disposed in tandem along an optical axis, said
elements each having on one side a first surface that
has a predetermined shape which is at least a part of
a surface of revolution and on the other side thereof a
second aspheric surface that is nonrotationally symmetric
and mathematically described by a preselected polynomial
equation having a nonzero term of at least 4th order,
said first and second surfaces of said elements being
structured so that said elements can be displaced
generally laterally relative to one another by rotation
of one or both of said elements about one or more axes of
revolution, other than said optical axis, such that said
second surfaces of said elements operate to continuously
change certain optical properties of said component as
said elements move relative to one another while said
first surfaces of said elements remain optically invari-
ant, not effecting any changes in any optical properties
of said component, as said elements move relative to one
another.
7. The component of claim 6 wherein said first
surface is planar.
8. The component of claim 6 wherein said axis
or axes of revolution is offset and parallel with respect
to said optical axis.
9. The component of claim 6 wherein said poly-
nomial equation describing said second surface is in
Cartesian coordinates of the form:
-48-

z = K1 (xy2+?x3)-K2x3y+K3xy3
where K1 is assignable constants and
K2 = - ?K1 and K3 = - ?K1
10. The component of claim 6 wherein said
polynomial equation describing said second surface is in
Cartesian coordinates of the form:
z = K1 (xy2+?x3)-K2x3y+K3xy3+K4x4+K5x2y2
where K1-K4 are assignable constants and K5 = 2K4 plus
residue conveniently set to zero.
11. An analytic function optical component
comprising a plurality of transparent elements that are
disposed in tandem along an optical axis, the outwardly
facing sides of the outer ones of said elements each
having a surface that has a predetermined shape which is
at least a part of a surface of revolution while the
inwardly facing sides of said outer elements and each of
the oppositely facing sides of the remaining ones of said
elements have an aspheric surface that is nonrotationally
symmetric and mathematically described by a preselected
polynomial equation having a nonzero term of at least 4th
order, said surfaces of said elements being structured so
that said elements can be displaced generally laterally
relative to one another by rotation of one or more of
said elements about one or more axes of revolution, other
than said optical axis, such that said nonrotationally
symmetric aspheric surfaces of said elements operate to
continuously change certain optical properties of said
component as said elements move relative to one another
while said outer surfaces of said outer elements remain
optically invariant, not effecting any changes in any
optical properties of said component, as said elements
move relative to one another.
12. The component of claim 11 wherein said
first surface is planar.
-49-

13. The component of claim 11 wherein said
axis or axes of revolution is offset and parallel with
respect to said optical axis.
14. The component of claim 11 wherein said
polynomial equation describing said second surface is in
Cartesian coordinates of the form:
z = K1(xy2+?x3)-K2x3y+K3xy3
where K1 is assignable constants and
K2 = - ?K1 and K3 = - ?K1.
15. The component of claim 11 wherein said
polynomial equation describing said second surface is in
Cartesian coordinates of the form:
z = K1 (xy2+?x3)-K2x3y+K3xy3+K4x4+K5x2y2
where K1-K4 are assignable constants and K5 = 2K4 plus
residue conveniently set to zero.
16. An optical system comprising an analytic
function optical component having at least one trans-
parent element that is disposed along an optical axis,
said element having on one side a first surface that has
a predetermined shape which is at least a part of a sur-
face of revolution and on the other side thereof a second
aspheric surface that is nonrotationally symmetric and
mathematically described by a preselected polynomial
equation having a nonzero term of at least 4th order,
said first and second surfaces of said element being
structured so that' said element can be displaced general-
ly laterally relative to said optical axis by rotation of
said element about an axis of revolution, other than said
optical axis, such that said second surface of said
element operates to continuously change certain optical
properties of said optical system as said element moves
relative to said optical axis while said first surface of
said element remains optically invariant, not effecting
any changes in any optical properties of said optical
-50-

system, as said element moves relative to said optical
axis.
17. The component of claim 16 wherein said
first surface is planar.
18. The component of claim 16 wherein said
axis of revolution is offset and parallel with respect to
said optical axis.
19. The component of claim 16 wherein said
polynomial equation describing said second surface is in
Cartesian coordinates of the form:
z = K1(xy2+?x3)-K2x3y+K3xy3
where K1 is assignable constants and
K2 = - ?K1 and K3 = -?K1.
20. The component of claim 16 wherein said
polynomial equation describing said second surface is in
Cartesian coordinates of the form:
z = K1(xy2+?x3)-K2x3y+K3xy3+K4x4+K5x2y2
where K1-K4 are assignable constants and K5 = 2K4 plus
residue conveniently set to zero.
21. An optical system comprising at least a
pair of analytic function optical components arranged in
tandem along an optical axis and each component having at
least one transparent element that is disposed along said
optical axis, said elements each having on one side a
first surface that has a predetermined shape which is at
least a part of a surface of revolution and on the other
side thereof a second, aspheric surface that is nonrota-
tionally symmetric and mathematically described by a
preselected polynomial equation having a nonzero term of
at least 4th order, said first and second surfaces of
said elements being structured so that said elements can
be displaced generally laterally relative to one another
by rotation of one or both of said elements about one or
more axes of revolution, other than said optical axis,
such that said second surfaces of said elements operate
-51-

to continuously change certain optical properties of said
optical system as said elements move relative to one
another while said first surfaces of said elements remain
optically invariant, not effecting any changes in any
optical properties of said optical system, as said
elements move relative to one another.
22. The component of claim 21 wherein said
first surface is planar.
23. The component of claim 21 wherein said
axis or axes of revolution is offset and parallel with
respect to said optical axis.
24. The component of claim 21 wherein said
polynomial equation describing said second surface is in
Cartesian coordinates of the form:
z = K1(xy2+?x3)-K2x3y+K3xy3
where K1 is assignable constants and
<IMG>.
25. The component of claim 21 wherein said
polynomial equation describing said second surface is in
Cartesian coordinates of the form:
z = K1(xy2+?x3)-K2x3y+K3xy3+K4x4+K5x2y2
where K1-K4 are assignable constants and K5 = 2K4 plus
residue conveniently set to zero.
26. The system of claim 21 wherein said
optical system includes at least one positive meniscus
lens element located along said optical axis and forward
of said pair of analytic function components.
27. The system of claim 21 wherein said
optical system includes one positive glass meniscus lens
element located along said optical axis and forward of
said pair of analytic function components.
28. The system of claim 21 wherein all
components thereof are comprised of a single material.
29. The system of claim 28 wherein said single
material is an optical plastic.
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30. An optical system comprising an analytic
function optical component having a plurality of
transparent elements that are disposed in tandem along an
optical axis, the outwardly facing sides of the outer
ones of said elements each having a surface that has a
predetermined shape which is at least a part of a surface
of revolution while the inwardly facing sides of said
outer elements and each of the oppositely facing sides of
the remaining ones of said elements have an aspheric
surface that is nonrotationally symmetric and mathemati-
cally described by a preselected polynomial equation
having a nonzero term of at least 4th order, said sur-
faces of aid elements being structured so that said
elements can be displaced generally laterally relative to
one another by rotation of one or more of said elements
about one or more axes of revolution, other than said
optical axis, such that said nonrotationally symmetric
aspheric surfaces of said elements operate to continuous-
ly change certain optical properties of said optical
system as said elements move relative to one another
while said outer surfaces of said outer elements remain
optically invariant, not effecting any changes in any
optical properties of said optical system as said
elements move relative to one another.
31. The component of claim 30 wherein said
first surface is planar.
32. The component of claim 30 wherein said
axis or axes of revolution is offset and parallel with
respect to said optical axis.
33. The component of claim 30 wherein said
polynomial equation describing said second surface is in
Cartesian coordinates of the form:
z = K1(xy2+?x3)-K2x3y+K3xy3
where K1 is assignable constants and
<IMG>
-53-

34. The component of claim 30 wherein said
polynomial equation describing said second surface is in
Cartesian coordinates of the form:
Z = K1(xy2+?x3)-K2x3y+K3xy3+K4x4+K5x2y2
where K1-K4 are assignable constants and K5 = 2K4 plus
residue conveniently set to zero.
35. An analytic function optical component
comprising a plurality of transparent elements that are
disposed in tandem along an optical axis, the outwardly
facing side of one of the outer ones of said elements
having a surface that has a predetermined shape which is
at least a part of a surface of revolution while the out-
wardly facing side of the remaining outer element and the
inwardly facing sides of said outer elements and each of
the oppositely facing sides of the remaining ones of said
elements have an aspheric surface that is nonrotationally
symmetric and mathematically described by a preselected
polynomial equation having a nonzero term of at least 4th
order, said surfaces of said elements being structured so
that said elements can be displaced generally laterally
relative to one another by rotation of one or more of
said elements about one or more axes of revolution, other
than said optical axis, such that said nonrotationally
symmetric aspheric surfaces of said elements operate to
continuously change certain optical properties of said
component as said elements move relative to one another
while said outer surface of said one outer element
remains optically invariant, not effecting any changes
in any optical properties of said component, as said
elements move relative to one another.
36. The component of claim 35 wherein said
first surface is planar.
-54-

37. The component of claim 35 wherein said
polynomial equation describing said second surface is in
Cartesian coordinates of the form:
z = K1(xy2+?x3)-K2x3y+K3xy3
where K1 is assignable constants and
<IMG>
38. The component of claim 35 wherein said
polynomial equation describing said second surface is in
Cartesian coordinates of the form:
z = K1(xy2+?x3)-K2x3y+K3xy3+K4x4+K5x2y2
where K1-K4 are assignable constants and K5 = 2K4 plus
residue conveniently set to zero.
-55-

Description

i5S
6627
ANALYTIC FUNCTION OPTICAL COMPONENT
BACKGROUND OF THE INVENTION
This invention in general relates to optical
systems which are particularly suitable for use in
cameras and, in particular, to optical elements having
preferred shapes in the form of analytic functions which
permit their rotation about one or more decentered pivots
to maintain the focal setting of a photographic objective
over a large range of object distances.
From the earliest use of hand cameras, it has
been recognized that a change in object distance with
respect to the location of the camera objective lens
causes an inevitable but easily calculated change in
image distance which, if not compensated in some way,
leads directly to ~ degradation o~ image guality over the
chosen field of view. Everyone who has made use of
photographic objectives becomes aware of this fundamental
fact~ and indeed camera manufacturers have adopted
several convenient means for bringing the aerial image
and the sensitive photographic film or coated glass plate
into registration.
The most natural means, one employed from the
beginning, is simply the technique of moving the position
of the photographic objective along the optical axis for
the purpose of focusing. Generally, the film plane
remains stationary. However, there are cameras and
certainly telescopes with movable ~ilm or plate holders,
_ I _

particularly where the photographic equipment is large
and cumbersome. In either case, the distance from objec-
tive to film is changed in such a way that the image can
be focused onto a ground glass and thereafter onto a
substituted photographic emulsion or other form of light
sensitive surface.
In some cameras, particularly in the modern
era, it has proved to be convenient to restrict the
focusing movement to but a portion of the optical system,
generally but a single element or component. The movable
element or compone~t, however, is a mixed blessing inas-
much as the image quality may suffer as a result of the
displacement of the element or component from its optimum
position. Various aberrations that have been minimized
or balanced for good image quality under average condi-
tions reappear or become larger on displacement of an
element or component. Both lateral and longitudinal
chromatic aberrations may reappear, along with an
enhancement of spherical aberration, coma, and astigma-
tism. However, careful design has often resulted inpracticable systems with substantial range in object
distances and even in magnification, as for example, the
various forms of zoom systems now generally available.
Other forms of focusing have also been intro-
duced. It is possible to interchange lens elementsproviding a discrete change in dioptric po~ers to provide
a reduced focusing range for each, within which indivi-
dual range the image ~uality can remain reasonably
stable. The sequence of focusing ranges can then be made
to overlap in such a way that the convertible system can
be used over a large range of object distances. This
technical means becomes all the easier if the focusing
interchangeable elements are inherently of low dioptric
power, whether positive or negative. In this way, the
weak element in use at any given time interferes only

i55
slightly with the image quality and indeed may be used to
improve the quality if suitably located and shaped. With
moldable elements use can be made of an aspheric "touch-
up" to improve the image quality selectively within the
individual range. If the dioptric lens elements are
mounted onto a rotor or disc for easy interchange, the
rotor can be referred to as a set o~ Waterhouse
elements. Waterhouse discs have also been used from
long ago for aperture con~rol and for insertion of
readily interchangeable filters.
Still ar:other form of focusing involves the use
of liquid filled flexible cells which with changing pras-
sure can be made to perform weak dioptric tasks such as
focusing. Ordinarily, the changes in sagittae associated
with dioptric focusing of hand camera objectives are very
small, whether positive or negative, and for the usual
focal lengths can be measured in but a few dozens of
micrometers. It is necessary, however, that the deformed
flexible cell provide sufficiently smooth optical
surfaces for acceptable image quality after focusing has
been performed.
Still another form of focusing has been intro-
duced in U.S. Patent No. 3,305,294 issued to L. W.
Alvarez on February 21, 1967. In this device a pair of
deformed plates are moved transversely in equal but
opposite displacementsO The plates have the same shapes
but are opposed such that in the "null" position the
variations in thickness cancels and the two plates used
together have zero dioptric power. Polynomial expres-
sions are used to define the common aspheric shape andare strongly dependent on cubic terms in a power series
in two variables. The polynomial coefficients are care-
fully chosen to allow the plates to simulate by transmis-
sion and refractions the performance of a dioptric

lens. When the plates are moved transversely with
respect to one another, the ne~ effect is a simulation
of a bi-convex or bi-concave simple element thereby
providing for a continuous range of dioptric powers.
Even though the deformed plates of Alvarez produce
desirable variations in focal length within small space
requirements, the thin lens systems which they simulate
are not of themselves well-corrected for aberrations in
many applications.
In previous U. S. Patent, No.3,583,790, issued
to James G. Baker there has been provided transversely
slideable plates which are an improvement over the above
mentioned Alvarez plates in that focusing action can be
achieved while also correcting for aberrations. Here, it
has been shown that even one special lens element, one of
whose surfaces is plane and the other of a prefer~ed
polynomial shape, or more generally, of a shape defined
by a preferred analytic function, can be transversely
slid to effect focusing action while at the same time
minimizing certain aberrations. It has also been shown
that the refractive action of the sliding element, when
combined with the refractive action of a fixed , opposed
optical surface on a nearby fixed optical element, which
opposed surface is shaped in accordance with a preferred
analytic function, can be made to simulate the dioptric
action of a well-corrected rotational lens element of
variable power.
It has now heen ~ou~d that another type of
lateral motion can also be employed for effecting focal
action. Consequently~ it is a primar~ purpose of this
invention to set forth examples of how this novel kind of
lateral motion can be employed, with adequate optical
precision, to lead to a similar kind of simulation of a
rotational dioptric element of variable power.

It is a further pu~pose Oe this invention to
provide optical elements having preferred shapes which
make it possible to relatively rotate at least two of such
elements to maintain ocus according to the object dis-
tance.
It is yet another purpose of the present inven-
tion to provide optical elements having preferred shapes
in the form of analytic functions which permit the ele-
ments to be rotated about one or more pivots decentered
with respect to an optical axis to simulate the optical
action of variable dioptric and rotationally symmetric
aspheric elements.
It is still another purpose of the present
invention to provide two or more rotatable elements which
can have different analytic unction surfaces and still
simulate the optical action of combined vari~ble power
dioptric and aspheric rotational elements.
Other objects of the invention will in part be
obvious and will in part appear hereinafter. Accordingly,
the invention comprises the optical elements and systems
possessing the construction, combination and arrang0ment
of elements which are exemplified in the following
detailed description.
SUMMARY OF THE INVENTION:
. ~
This invention in general relates to optical
systems which are particularly suitable for use in
photographic cameras and, in particular, to optical
elements that are shaped in accordance with preferred
analytic functions or preferred polynomials which permit
3~ the elements to be relatively rotated about one or more
pivots decentered with respect to an optical axis to
simulate the dioptric and aspheric action of a well-
corrected rotational lens element o variable power which
--5--

;tj~j~
can be used to maintain ~ocal setting over a large range
of object distances or ~or more complex tasks. For this
purpose, the optical elements can be used singly with
mirror images of themselves, or in pairs, or they can be
incorporated in more elaborate systems to provide
focusing action.
The general shape of the analytic function
surfaces can be represented by preferred truncated poly-
nomial equations in two independent variables having the0 general form given by:
n ~ (n-j)
Zi = ~ ~ AijkXjyk
j=0 k=0
where i refers to surface numbers, Aijk represent co-
efficients, and x, y, and z refer to coordinates in a
preferred non-Cartesian coordinate system.
Several examples of the use of such elements
are described including systems which utilize a pair of
elements one of which is rotated about an axis of revolu-
tion offset and parallel with respect to a system optical
axis. In each of the examples, the novel element typi-
cally is a transparent element, preferably molded of a
suitable optical plastic. On one side of the element
there is provided a first surface that has a predeter-
mined shape which is at least a part of a surface of
revolution which f,or precision systems is preferably
plano and on the other side a second aspheric surface
that is asymmetric and can be described mathematically
by a preselected polynomial equation having at least one
nonzero term of at least fourth degree. The first and
second surfaces of the elements are structured so that
the element can be displaced generally laterally relative

t~
to the optical axis by rotation of the element about an
axis of revolution, other than the optical axis, such
that the second element surface operates to continuously
change certain optical performance characteristics as the
element moves relative to the optical axis while the
first element surface remains optically invariant, not
effecting any changes in any optical performance charac-
teristics, as the element moves relative to the optical
axls.
DESCRI PTI ON OF TH E DRAWI NGS:
The novel features that are considered
characteristic of the invention are set forth with
particularity in the appended claims. The invention
itself, hcwever, both as to its organization and method
of operation together with other objects and advantages
thereof will be best understood from the following
description of the illustrated embodiments when read in
connection with the accompanying drawings wherein like
numbers have been employed in the different figures to
denote the same parts and wherein:
Fig. 1 is a diagrammatic perspective view of
an optical element of the invention showing its general
features;
Fig. 2 is a diagrammatic perspective view
showing a pair of optical elements of the invention;
Fig. 3 is a perspective view illustrating
coordinate systems used in describing the novel surfaces
of the optical elements of the invention;
Fig. 4 is a diagrammatic cross-sectional view
illustrating on the right a pair of elements of the
invention and on the left a pair of rotationally
symmetric dioptric elements which are simulated by the
inventive elements;

S
Fig. 5 is a diagrammatic p]ot o certain system
related parameters; and
Figs. 6-11 are diagrammatic plan views of
optical systems embodying the present invention.
5OETAILED DESCRIPTION
As previously stated, this invention generally
relates to optical systems particularly suitable for use
- in cameras, and, more specifically, to optical elements
having preferred shapes, defined in form by preferred
polynomials or more generally by preferred analytic
functions. These novel elements are unlike the prior art
transversely slideable elements because their pre~erred
analytic function surfaces permit them to be rotated
about pivots decentered with respect to an optical axis
to maintain focal setting or to perform more elaborate
functions over a large range of object distances.
Figure 1 shows in diagrammatic fashion an
element 10 having the general characteristics represen-
tative of the invention. The element 10 is a thin,
transparent annular segment that can be mounted with any
suitable mechanical means for rotation about an axis, RA,
which is parallel to and offset with respect to an opti-
cal axis, OA. Along the optical axis, OA, there is shown
a nominally circular area 12 which may be taken to be an
aperture of maximum diameter in a designated plane in
which the element 10 more or less resides. Through the
transmission area 12 bundles of rays pass through the
element 10 when traveling from object to image space.
Around the periphery of element 10 is a section
14 of constant radial width that can be selectively moved
into optical alignment with transmission area 12 by
rotation of element 10 about the rotation axis, RA. One
surface (16) of the peripheral section 14, as shown here,
--8--

~ i3t~
is planar and as such produces no change in its optical
ef~ect across the transrnission area 12 as element 10
rotates. Opposing the surface 16 is a~other surface, 18,
shown in exaggerated fashion and facing out of the
paper. The surface 18 is distorted in the shape of a
preferred analytic function that is selected according to
the teaching of the invention such that useful optical
changes are effected by the sur~ace 1~ as the element l0
rotates about the rotation axis, RA. It is the shape of
the surface 18, or similar surfaces, that is the crux of
the invention. As will be shown hereinafter, proper
selection of the shape of an analytic function surface
such as that of 18 allows an element such as 10 to be
used with other closely spaced similar elements to simu-
late rotationally symmetric dioptric or aspheric elementsof variable power. Moreover, such elements, when used in
combination, can be used with one or more fixed in place
and others rotating or all rotating about the same or
different axes whether in the same or in opposite direc-
tions to accomplish the tasks of simulation. Fig. 2, forexample, diagrammatically illustrates a pair of such
elements, designated at 20 and 22, adapted to oppositely
rotate about the displaced axis of rotation, RA. Such
elements can also be incorporated in more elaborate
systems as the examples to follow will illustrate.
To clear,ly understand the nature of the
analytic function surfaces of the invention and the
specific examples in which they may be incorporated, it
will first be necessary to describe various coordinate
systems which have been found convenient for defining
them along with a general description of important design
consideration~ in their use and, as well, a procedure by
which their shapes can be calculated and specified.

s
Figure 3 illustrates the various coordinate
systems found useful in analyzing, defining and speci-
fying the an~lytic function surfaces of the optical
elements of the invention. The first one, most conve-
S nient for facilitating manufacturing, is a cylindricalcoordinate system arranged in a transverse plane for the
treatment of the analytic surface mathematically such
that the origin of coordinates in the x-y plane lies at
the displaced assignable pivot point, the intersection of
the rotation axis, RA, with the x-y plane. The rotation
axis, RA, through the pivot point is taken to be parallel
to the optical axis, OA, and hence the geometry in the
transverse reference plane is of polar form and the ana-
lytic shape can be defined in principle in terms of
ordinary polar coordinates. That is, the coordinates of
a point, P, on the analytic surface in the transverse
plane are given by the coordinates r and phi. A
preferred shape in three-dimensional space in the polar
system is completely defined with the sagittal depth, z,
or departure from the transverse plane expressed as
functions of the polar coordinates in the transverse
plane.
Apart from the use of polar coordinates, use is
also made of at least two other coordinate systems. The
first of these is the x-y-~ system normally employed in
the course of optical design, whereby the optical axis,
OA, is taken to be the z-axis, the y-z plane is taken to
be the meridional plane and the x-axis, as the skew axis,
is perpendicular to either. As viewed by an observer who
is looking from the rear of an objective toward its
front, the ~alues of z are positive toward the observer,
the values of y are positive upwards in the meridional
plane, and the values of x are positive to the right.
Conversely, as seen from the front of an objective, one
notes that y-values are positive upwards, x-values are
--10--

i5.S
positive to the left and z-values are positive away from
the observer. For any given surface it is convenient to
locate the origin of coordinates at the vertex, namely,
at the point of intercept of the optical axis with the
surface.
In addition use can be made of an auxiliary
non-Cartesian frame of reference such that its origin o~
coordinates, as for the x-y-z system, lies on the optical
axis, OA, at the point of intersection of the optical
axis, OA, with the transverse re~erence (x-y) plane. In
some cases the respective origins in the two systems may
be related by a translational shift along the z-axis.
The new non-Cartesian frame of reference is
related approximately to the x-y-z system but of itself
involves arcs and radial extensions in the polar system.
Thus, distances along any radius from the pivot point
below in Fi~. 3, less the assigned radial distance from
the pivot point below to the origin of coordinates
(o~o,o), become the newly defined values of y. Positive
y-values are to the top of the origin of coordinates.
The lengths of arcs concentric about the pivot point, as
measured along the arcs from the y-axis, become the newly
defined values of x, and are positive to the left,
although still arcs. The quantity z is either the same
as z, or different only by a translational constant along
the z-axis.
It is, of course, feasible to map the above
defined Ibent' coordinates onto an auxiliary Cartesian
frame as a fourth system of coordinates which may perhaps
be convenient for mathematical studies. Mapping between
the polar system and the reference fourth Cartesian
system may involve certain mathematical poles, but as
long as the pivot point lies substantially outside of
the annular area of transmission (12) of the rotatory

surface, the mathematical pole present will not introduce
singularitie~, into the analytic nature of the functions
within the useful transmitting area. Indeed, for an
adjacent fixed element of a pair of optical elements of
the invention, the clear aperture normally will remain
circular as for any ordinary rotational surface. For the
transmitting area of a rotatory element, the circular
aperture spreads out into an annular area, or sectorized
annulus, providing therefore adequate transmission how-
ever the element may be rotated through its full range aspreviously explained.
In the case where a pair of adjacent elements
containing analytic shapes for the opposed surfaces are
to be rotated, each has a transmitting area in the form
o a sectorized annulus ~see Fig. 2), and likely it is
necessary for one to introduce a fixed circular defining
aperture nearby as, for example, the circle 19 in Fig.
2. In any event, focusing or other optical action takes
place when one of the two analytic shapes is rotated
about the displaced pivot point, or points where two
displaced axes of rotation are used, with respect to the
other, or both ~ith respect to the fixed circular aper-
ture centered on the optical axis, OA, nearby. That
is to say, in a more general case where both elements are
rotatory, one may have a pivot point below for one
element, as in Fig. 1, and a pivot point above for the
other element (not shown), or both pivot poi~ts together
but above or below, and similarly for other azimuths.
The differential geom~try o the individual
analytic shape can thus begin at the intersection of the
optical axis, OA, with the analytic unction surface,
which is the adopted origin of coordinates, and owing to
the complexities of any possibly closed analytic func-
tion, can instead be conveniantly described by a power

series in x and y. Furthermore, total exactness of
representation is not required and therefore the power
series may be truncated to polynomial forms. As long as
no pole or poles lie within the useful area of the
sectorized annulus, there will be no mathematical singu-
larity to deter, particularly because the inner bound is
comprised of a concentric arc about the offending singu-
larity at the pivot point. One can reasonably expect,
however, that the accuracy of representation of the
analytic shape of the surface by a truncated power series
or polynomial in x and y depends mathematically on the
relative proximity to the pole or singularity at the
origin in the polar system, even though the latter lies
entirely outside of the useful aperture. That is, one
will find that some of the coefficients in terms of the
parameters of the optical system will grow in magnitude
according to the reciprocal distance to the external pole
and powers thereof. In brief, one may say that an
analytic function defining the shape of the deformed
surface can be expressed in terms of a truncated power
series or polynomial form to any desired precision as
long as the function remains analytic wi~hin the useful
aperture and as long as sufficient terms are used.
Conversely, because of the existence of external poles,
the corresponding analytic function shapes expressed in
polar coordinates about such a pole become mathematical
forms not readily recognizable as power series but which
nevertheless adequately describe the shapes of the
deformed surfaces. These mathematical forms in the polar
frame of reference are also analytic and contain no
poles, and in general will represent what is meant by
referring to a working deformed surface as being of the
shape of a preferred analytic function.
-13-

~s~ 5
Because the relevant analytic expressions are
without poles and continuous within the respective trans-
mitting areas of the pairing of deormed surfaces, one
~ixed and the other variable in angle about the displaced
axis, RA, a mathematical procedure can be adopted for
representing the shape of each of the two surfaces, say
zi(x,y) and zi+l(x,y), in the form of a power series for
each but with undetermined coeficients. A procedure
then is established ~or the evaluation of all of these
coefficients to any desired order, where at first all
possible terms in power of x and y are present as will be
seen hereinafter.
However, before dealing with the mathematical
procedure, it is important at this point to set forth an
important goal of the invention which is the simulation
of the optical action of a closely adjacent pair of rota-
tional dioptric elements (see Fig. 4 on the left) with
possible aspheric powers amongst the four surfaces, by
the substitution of a closely adjacent pair of dioptric
elements o~ analytic function form as shown on the right
in Fig. 4. O.ne of the simulating pair is to be rotatory
about an eccentric pivot and have an outlying surace
which is plane/ the other which may be fixed has an
outlying surface which may be of a rotational dioptric
form with or without aspheric powers. Both these
elements have inner opposed surfaces that are to take on
the forms of anal~tic surfaces, and thereby perform the
same optical action as the pair to be simulated, as
nearly as possible. Furthermore, if for any givan set of
object distances, one finds a corresponding set of such
closely adjacent rotational dioptric elements, optimized
as may be desired for optical performance of an objective
over the aperture, field and spectrum, the rotational
dioptric elements varying not only in dioptric power but
-14-

t.)S
also in aspheric powers, distributed over their four
surfaces in a preferred way, one must then substitute a
rotatory motion of at least one of the inventive optical
elements about a displaced pivot point and about an axis
through the pivot point parallel to the optical axis, OA
in order to vary both the equivalent dioptric and aspher-
ic powers in simulation, as far as possible, of the
optical action of the first set. At the same time the
coefficients of the power terms must be evaluated for
both the rotatory and fixed surfaces that will assist
in optimizing the simulation. The target values of the
first dioptric set to be simulated must be determined by
the usual practices of optical design. The eYaluation of
the simulation, however, must proceed along mathematical
lines.
A first step, then, is to carry through a
detailed optical design of the desired system to be simu-
lated for an adopted mean object distance, either central
within the focusing range or decentered within the range
to favor distant scenic photographs, or more rarely, to
favor close-ups, according to the usage of a camera.
This detailed design must be planned in advance, however,
to incorporate at least one pair of adjacent surfaces
intended to be ultimately of polynomial or analytic form,
or at least inserted mathematically. If only one surface
of the simulating pair is to be rotatable about a dis-
placed axis, the s'imple element o which it is either the
~orward or rear surface must be at the start of plane
form for its other surface, or else the polynomial form
can be apportioned between the two sides. The rotatable
surface is to become of analytic or polynomial form. The
overall optical design may include, if found desirable,
dioptric and aspheric powers on either or both of the
inner-lying, opposed surfaces that are to become of
analytic or polynomial form, and the outlying surface

of the ixed element may have dioptric and aspheric
powers, as may be required. In special cases, the
erstwhile dioptric surface o~ the rotatable element may
be transferred or apportioned onto the outlying surface
of the fixed element which surace may already have
predetermined dioptric and aspheric powers.
Referring now to Figure 4, let us now consider
a mathematical reference line in x, y and z space that is
drawn parallel to the optical axis at some assigned value
of x and y. Ths reference line will not be identical to
any actual transmitted ray of light except at the optical
axis, but is only a mathematical convenience in that any
point on this line has the same values for x and y, vary-
ing only in z . An actual ray of light would instead be
refracted by the successive surfaces of the adjacent
simulating element pair, the inner opposed surfaces of
which are to become deformed in a preferred way. While
one might indeed use an actual ray of light as a refer-
ence, x and y would vary along the line segments between
successive surfaces. Any power series development in x,
y and z would have coefficients of considerable and
probably unnecessary complexity.
As a second step, therefore, it is expedient to
make use of the reference line parallel to the axis and
to calculate for the basic dioptric elements (on the
left in Fig. 4) the total optical thickness along this
reference line be~ween the point of intercept on the
first o the relevant four surfaces and the point of
intercept on the last of the four surfaces. The optical
thickness will in general consist of three line segments,
two of which are in the media and the central one in
air. The optical thickness along the reference line is
the simple summation of the geometrical line-segments
multiplied by the respective index of refraction. It is
-16-

5S
this optical thickness as a function of x and y, calcu-
lated analytically or the dioptric basic system in which
both elements are of rotational orm about the optical
axis, OA, that is to be reproduced by the summation o
the geometrical line-segments along the same reference
line, multiplied by the respective indices of reraction,
where the inner-lying paired dioptric surfaces o the
elements on the left in Fig. 4 are to be replaced by
polynomial or more generally by analytic shapes.
If the media are identical for both elements,
and if or convenience the outlying suraces are pro-
visionally plano, one may note that the line segment in
air, that is, the central segment, must have the same
geometrical length when the polynomial shapes are
employed as when the basic rotational dioptric elemants
are used. All that happens is that the location of
this air segment in z along the reference line shifts,
but in such a way that an increased thickness along the
reference line for one element i5 off-set by a decreased
thickness for the other. The to~al optical thickness
along the reference line is thus unchanged between the
value for the basic or mean dioptric system to be simu-
lated, and the substituted polynomial or analytic system
in null or unrotated position that in principle simulates
the basic dioptric system. It is this translational
movement in the z-direction, at any given value of x and
y of the central a'ir-segment between the original rota-
tional dioptric elements and the simulating polynomial
elements that comprises th0 crux of the entire proce-
dure. Indeed, it is this flexibility in the z-position-
ing of the air-segment that makes representation by the
polynomial system both possible and useful.
Once analytic function expressions are found
that accurately complete the simulation for a continuous

;Srj
array of radial distances from the optical axis, OA,
being single valued for the rotational system, but double
valued in x and y for the polynomial system, then it is
easy to see that relatively small aberrations that affect
the optical performance only slightly may be introduced
by the change-over to the polynomial system. By way of
illustration, it will be noted that the basic dioptric
system already has certain residual image errors and
distortion errors over the aperture, field and spectrum.
The simulating polynomial system likely introduces
further minute errors or aberrations caused by the neces-
sary and inherent shift of any given air-segment along
the z-axis fro~ its original and presumably optimum
position.
However, when the air-segment is small, as is
the case for closely adjacent surfaces, the new aberra-
tions are also small, being primarily a slight side-
stepping of the ray of light being refracted otherwise
this way and that through the optical system. There are
also somewhat smaller refractive errors caused by the
prismatic differential changes along any given ray from
slope errors at the points of intercept, as compared to
the slopes of the basic dioptric surfaces to be simu-
lated. These added-on aberrations in the polynomial or
simulating system are relatively small for objectives of
normal aperture, field and spectral coverage as for hand
camera applications, and become important only for larger
systems or for systems intended for the highest acuity.
Finally, there are additional small aberrations caused by
imperfections in the simulatio~ between the analytic rep-
resentation of the deformed surfaces and the rotational
dioptric surfaces of the basic system to be simulated,
caused principally by truncation of the power series
representations.
-18-

The second step initiated above now is the
algebraic task of representation o~ the dioptric to~al
optical thickness by a power series in x and y which in
the rotational space of the basic d;optric system expands
into a power series containing (x2+y2) to some power for
all variable terms. The power expression commonly used
takes the following form for a single rotational dioptric
surface:
z = _ C(X2+y2) + ~ (X2+Y2)2 + --
r
1 + ~ e2 ) C2 ( X2+y2 )
where c denotes the vertex curvature of the particular
dioptric surface and ~ is the first aconic coefficient.
The quantity e2 is the eccentricity squared for the
particular conic. For a spherical surface, both e2 and
are zero.
For purposes here the above expression can be
further expanded into a Taylor series, as follows:
z = 1 c (x2+y2) + [ ~ + 8 (1 - e2) C31 (X2+y2)2 + ...
If for the designation of the closely adjacent dioptric
surfaces of the basic system one uses the sub~cripts i
and (i +1), the geometrical length of the air-segment
along the reference line (refer to Fig.3) is simply the
difference of two such power series in the following
form: '
(Zi+l- Zi) = 2 (ci+l- ci)(x2+y2)t(Hi+l- Hi)(x2+y2)2+...
where for convenience the quantity H is a shorthand for
--19--

rjs
Hi = ~i + ~ e2)C3i,
and similarly for (i+l).
The polynomial surfaces on the other hand can
be expanded in x, y and z space as a generalized series,
as mentioned earlier, and at the start can be given the
foll~wing form:
n (n-j)
~ ~ '
Zi = ~ ~ Aijk x~ yk and similarly for (i+l).
j=o k=o
As in the case of the air-segment or the rotational
dioptric surfaces, the length of the geometrical air-
segment for the polynomial case can be written as the
simple difference of the z-values for the two closely
adjacent surfaces plus the axial separation.
For the particular case where the displaced
axis of rotation for the movable element is taken to be
parallel to the optical axis, OA, the x, y and z terms
and x, y and z terms are related to the polar representa-
tion in the following way:
y = r - a y = r cos ~ - a
x = r ~ x = r sin ~
Making use of these relationships, power series equiva-
lences are then found between the x, y and z, and the x,y and z spaces as follows:
x = x -- ~
6 ia + y)2
30 y = y _ _ x _ + x _ _ , .
2 ~a + y) 24 (a ~ y)3
-20-

; 5 -'~
These relationships can now be used to rede~ine the po~ler
series above for the rotational dioptric elements by
transformation into the polynomial space, as for the
general power series for the polynomial surfaces. Inas-
much as we are dealing not with the individual surfaceshapes but only with the differences, and indeed, in the
simplest case, only with the geometrical length of the
air-sesment, one can now equate the power series term
by term and thereby evaluate progressively all the
previously unknown coefficients.
At this point certain terms can be eliminated
or at least neglected because the~ are always zero, inas-
much as the equivalent terms in the ~ransformed series
for the rotational dioptric surfaces remain inherently
zero. There are also other terms of no immediate import
in that the geometrical length of the air-segment remains
the same, whatever value is assigned. These are terms
involving only the constant term ~i00 and terms involving
only powers in y which remain unaffected by rotational
changes described by y and therefore which subtract out
in the differencing. Such terms may as well be assigned
the value of zero and thereby eliminated. It may be,
however, that judicious use of assigned finite values for
such terms may allow for an easier fabrication of the
polynomial surfaces according to the methods used. More
importantly, such terms may be re-introduced in any later
computerized optimi'zation routines to effect possible
improvements in performance. That is to say, the
procedures made use of above are rendered tractable by
the use of the reference line which one has taken to be
parallel to the optical axis, OA. The exact optical
situation is more complex. An optimization routine when
properly performed will take into account any residual
improvements with respect to aperture, field and
-21-

;SS
spectrum, allowed by the use Oe non-z~ro or finite values
of these special terms in y or powers o~ y.
For the parallel case, it has been determined from
the above procedures that the following relationships
obtain:
Aioo = O (or other assignable value)
AilO =
Aiol = O (or other assignable value)
-
Ai20 = ~ 2 (Ci~l- Ci)
Aill
AiO2 = ~ 2 (ci+l- ci) + an assignable value
say ~ ~iO2
-
Ai30 ~i [ ( C i+l-c +l ) - ( C i- C i ) ] ( +_~
Ai21 = ~Ci+l~ Ci) 2(a+y)
Ail2 = ~ 2 [(c'i+l-ci~l)-(c i-ci)]
Aio3 = O (or other assignable value).
Ai40= (Ci+l-Ci~ 1 _ (Hi+l- Hi)
24(a+y32
Ai31= 1/6 [(c~i~l-ci+l)-(cli-ci)] _ _
(a+y)2
Ai22= -2 (~i+l- ~i)
Ail3=
Ai04= -(Hi+l- Hi) conveniently, + an assignable
value, say, ~ Aio4
-22-

;5S
Ai50= 1_[~Cli+l-ci~ (c i ci)l _ _
120 (a+y) 3 ~
--1 [(Hl i-~l- Hi+l)-(H i-Hi) 1--
( a+y)
Ai41= -(ci+l-ci) 1 _ + (Hi+l- Hi)
24(a+y)3 (a+y)
Ai32= - -- [(~ i+l-Hi+l)-(Hli-Hi)] _ 1
3 (a+y)
Ai23= (Hi+l- Hi ~ --
(a+y)
Ail4= -[(H i+l-Hi+l)-(H i-Hi)]
( a+y )
Aios= 0 (or other assigned value)
Ai+l,oo= O (or other assigned value = Ai,oo)
.
Ai+l~10
Ai+l~ol= 0 (or other assigned value = Ai~ol)
Ai+l, 20=
Ai+l~ll
Ai+l,02= 0 + an assigned value = ~ Aio2
-
Ai+1~30= - -- [(C'i*l-ci+l)-(C,i-Ci~]
6 (a+y)
Ai+1~21-
Ai~l~12= - -- ~(C i+l--Ci+l)-(C~i-Ci)]
2 (a+y)
Ai+l r 03= -~ an as5igned value = Aio3
_
Ai+l ~ 40=
Ai+l ~ 31= -- [ (c' i+l-Ci+l ) - (c' i-ci ) 1 1
6 (a+y)2
--23--

Ai+l~22=
A~ 13
Ai+l,04= 0 + an assiyned value = ~ Aio4
Ai+l,50= - [(c'i~l-ci+l)-(c i-ci)]
120 (a~y)3
_ 1 [(H'i+l- Hi+l)-(H i-Hi)
(a+y)
A~ 41= 2
Ai+1~32= -- -- [(H'i+l- Hi+l)-(H i-Hi)
3 (a+y) tt
Ai+l~23=
Ai+1~14= ~ [(~ i~l- Hi+l)-(H i-Hi)]
(a+y)~
Ai+ll05= + an assigned value = Aio5
From these equivalences and from the transfor-
mation expressions given above, the power series in x, y
and z space can be cast into polar representation in r
and phi. The externally located pole in its multiple
orders indicated by the transformation terms in
reciprocals of (a + y) and reciprocal powers of (a + y)
now are swallowed up in such a way that the coefficients
in the r and phi polar system no longer contain poles.
Indeed, the polar representation is no longer a fully
complete power series, but by equivalence, at least
through the order of the power series adopted, it does
represent the same surface shape as obtained earlier from
the analytic functions in x and y. There is no loss of
accuracy in the transformation to the polar system not
already inherent in the bent cartesian system. The polar
expression now becomes:
n (n-j)
Zi = ~ ~ Ai ik ri (p i (r-a)k =
j=0 k=0
and similarly for Zi+l = Zi+1
-24-

5~
One notes that in the developrnent of the
expressions for the A's, the variable quantity (a+yl
appears in the denominator in various powers for a number
of coefficients. The fact that such a term in y is
variable upsets the nature of the power series in the
normal representation. One can then develop any such
reciprocal term into a separate power series in y and
through series manipulations obtain augmented or adjusted
new coefficients such as, say, ~'s. The convergence can
be slow and hence one will be left with the situation
that the polar representation will be somewhat more
accurate than the readjusted power series truncated at
an adopted level. This discrepancy will reappear in
optimizations carried out in the x, y and z system
and then transformed back again into the x, y- and z
system or into the polar system~ An optimization routine
carried out in the polar system presumably in the long
run will yield the best results. Representation of the
optimum functional shape in the polar system will then
contain no poles and will be accurate in accordance with
the number of terms employad.
One can also transform the power series in x
and y for the polynomial surfaces into a similar power
series in x, y and z space in which the computer normally
operates. In this instance, it will have been necessary
beforehand to have expanded the reciprocals of (a+y) in
order for the results in the x, y, and z system to become
a proper power series with non-variable coefficients,
B's. For the case of the parallel displaced axis, one
can arrive at the following expression through accuracy
of the ~th order:
.,
-25-

~S~ t;5'~
Zi = Bi30x3 + Bil2xy2 ~ B13lx3y +Bil3xy3 + -----~--
where Bi30 = - - [(C'i+l-C~ (c i-Ci)
6 a
Bil2 = - -- [ (C i~l-ci+l ) - (c i-ci) ]
Bi31 = - -- [(C i~l-Ci+l) - (C~ i-Ci)]
6 a2
Bil3 = + - [(c i+l-ci~l) - (c'i-c
2 a2 ~
In the above expressions the factor 1/~ appears
at all points and ~ is the angular rotation of the
rotatable element about the displaced axis. For conve-
nience, further terms involving ~, ~3, etc. wherever
they appear, are neglected, but only on the assumption
that the angular value of ~ is small. If indeed, ~ is
small enough, the contribution of the higher order terms
will be small, or can be partially off-set by computer-
ized optizimation. In any event, if only the terms in
are retained, the appearance of ~ in the denominator
means that the designer can use a small value of ~ and
thereby one can determine a relatively strongly deformed
polynomial surface, or can use a larger value of ~ for a
weaker deformed surface, the latter endangering the
performance because of the larger contributions of the
higher order terms in powers of ~.
For purposes of clarity, it is emphasized that
~t and phi are in the same polar system. Phi as used
above is a variable in polar coordinates used to define
the actual stationary shape of the deformed surface in
polar coordinates. ~ on the other hand, is the angular
displacement of the pre-determined deformed shape for the
-26-
" .

movable or rotatable element. ~ is the particular value
of ~ that transforms the performance of the basic system
for an assigned mean object di-stance to compensation in
focus for movement of the object plane to some other
distance, possibly to infinity.
It is to be noted that in the above equivalen-
ces between coefficients, values of c and H are also
required for at least one outlyiny reciprocal object
distance other than the mean. These may be evaluated for
the rotational dioptric system at a second selected dis-
tance for the object plane, which conveniently may be for
infinity. In practice, however, the target values of c
and H can be evaluated for at least four object distan~
ces, one of which is infinity, and these values of H (or
~, if for convenience e2 = 1.0 in the evaluation) have
been plotted against cO Most usually, a parabolic curve
results, as shown in Fig. 5, whereas a linear relation-
ship is actually desired. A straight line is then to be
drawn in the best way possible through the parabola in
which process the performance at two selected distances
may be favored. The target values along the tilted
straight line, which may be the re-evaluated ~ (or ~)
for infinity if that is used, are then employed in the
evaluation of the equivalences. When all of this is
completed, step 3 is complete.
A further step, if desired, may be performed
by a computerized optimization routine, according to
the quality of perf,ormance desired. In this process,
an array of image points must be incorporated over the
full field, inasmuch as no two images are alike, and
the rotational relationships of the normal system become
suppressed. There are also variations over the pupil
that are non-rotational. The two patterns in aperture
and field will in ceneral lead to a much prolonged
computational routine, as compared to the requirements of
-27-

;55
a purely rotationaL systeln. ror convenience, ~here~orl?,
the optimization may he satisf~actorily performed in a
mean wavelength with usually adequate precision. In an
optical system requiring the utmost in performance, one
may wish to re-introduce all the terms previously Eound
to be zero, whether by convenience or by inherent nullity
in a rotational system, which the deEormed system no
longer Eollows. The re-introduced quantities may also
include the ~ Aiok. Indeed, all the possible terms in a
matrix in x and y may be employed, but many will be small
in magnitude. The computer burden may then be great.
~ final step may be to recast the fully
computer-optimized shapes in x and y back into the polar
system. In this concludin~ transEormation use should be
made of the exact relationships oE the transformations,
point by point, in order to avoid inaccuracies
re-introduced by inexact power series representation with
but a reasonable number of term~s. The final polar
representation may be of value mostly for fabrication
purposes, wherein the desired clear apertures of the
fixed and of the movable surfaces are clearly set Eorth.
The essential form required for each of the
analytic function surfaces of the invention to simulate
the variable portion of the rotational dioptric system
with one or more analytic Eunction elernents rotatable
about a common pivot is given in the x, y, z coordinate
system by: .
z = K1(xy2+-x3) - K2x3y+K3xy3
where K1 is an assignable constant
and K2 = ~ ~ Kl~ K3= - - Kl-
3a a
If the rotational system to be simulated includes
rotational aspheres, then the essential Eorm ~or each of
-28-

:~2~;5~
tile nnllytic ~unction~ de-ining tile !limu]i~lin-J ~ur~aceL;
become~:
z = Kl(xy2~-x3) - K2x3y~K3xy3-1~x4~sx2y~
where Kl to K4 are constants to he deterlnined.
and Ks = 2K4 plus residues conveniently set to zero.
From the foreyoing considerations, a number oE
optical systems have been designed, and several of these
will now be taken up as examples illustrating the inven-
tive concept. The first three examples are presented in
x,y,z coordinate system, while the last Eive are in the
x,y,z coordinate system. All are suitable for use as
variable Eocus photographic objectives and can be scaled
for use at a focal length of 125mm, have a speed of f/10
and an angular field coveraye of 50-degrees total based
on the format diagonal. Before presenting the examples
in detail, ho~ever, an explanation of the notation used
is appropriate.
The notation fd ordinarily refers to the
equivalent focal length of an optical system in the
wavelength d (58~.6 nm), as deEined paraxially for an
object plane at infinity. (For an optical system having
object and image space in the same medium, generally air,
the f and f' of gaussian optics become equal). Thus, Ed
becomes the same as the equivalent Eocal length Eor d
light, namely, the EFL.
I~ the mean object plane is at some assigned
finite distance, the practice of using the symbol Ed to
become a 'scale focal length' or more accurately, a scale
factor, has been adopted. If this designated object
distance is infinite, then Ed and fd become equal in the
first instance and mean the same thing, namely, a scale
focal length or F,FL applied to objects at infinity. The
quantity fd as a scale factor can also be adapted to
include distortion over a mean part o~ the Eield
(calibrated focal length Eor in~inity object).
29-

;55
~L .. , . . .- .
Even if the symbol fd applies or an assigned
mean object distance, the optical system will also have
the usual fd of gaussian optics, where for mathematical
purposes the object plane is shited temporarily to
infinity. For the type of focusing being done here, fd
is used for a fixed and assigned position of the image
plane, and fd for the concomitant but ever changing
system, while obtainable, is not actually being used in
the picture-taking process.
The meaning of all coefficients, whether for
rotational or non-rotational surfaces, has been explained
previously in the specification and will not be repeated
here. All values have been normalized with respect to
-
fd -
The first example is a pair of analytic
function elements combined with an objective as shown in
Fig. 6. The constructional data in tabular form is as
follcws:
~lement Separa~ions Clear
Number Sur~ace Radii Plastic Air Apertures Material
{ 1 0.1776 0.0286 0.200 Plexi
2 0.2707 ~ 0.0741 0.185
3 stop 0.0122 0.085 diaphragm
Il 4 -2.284 ~ 0.0131 0.111 Polycarbonate
plano ~ 0.0019 0.105
~S rII 6 2.18S ~* 0.0131 0.103 Polycarbonate
7 plano 0.0110 0.096
8 stop 0.8230 ~ 0.083 Iris 6 Shutter
Aspheric (rotational) ~ Pack focal distance ~de)
Polynomial ~non-rotational) Radius given is simulated for the mean
object distance.
fd=l-OOOO 51 (adopted mean objcct distance)- 20.96 fd~10.0
When one refocuses for infinite object distance by use of
a decentered transverse rotation of Element III by an
angular value of ~, one finds that
fd = 0.9983 tde held constant).
-30-
.,

;5S
The rotational coefficients are as follows:
Beta2 = 1.754 x 10 Beta4 a -1.528 X 101
Gamma2 a 7.031 x 10-3 Gamma~ = -2.862 x 10-2
Delta2 = -1.750 x 10-7 Delta4 = -4.491 x 10 9
5 Element II with its rear surace polynomial is ixed (no
rotation). Element III with its forward surface
polynomial rotates about an assigned displaced center
with the transversely displaced axis parallel to the
optical axis.
The rotational coefficients are given by:
As20 = -2.288 x 10-1 A620 =
A502 = -2-288 x 10-1 A602 =
As30 = 1.861 x 10 - 2/~ A630 = 1-861 X 10 - 2/~
A521 = 2.288 x 10-1 A621 =
15A512 = 5.584 x 10-2/~ A612 = 5.584 x 10-2/~
A~s40 = 1.906 x 10-2 A 640 =
As40 = -1.059 x 10 - 2 A640 =
A531 = -1.861 x 10-2/~ A631 = - 1.861 x 10-2/~
A522 = -2.118 x 10-2 A622 =
20A504 = -1.059 x 10-2 A604 =
A~ 550 = ~9.307 x 10-4/~ A~ 650 =~9-307 x 10-4/~r
A550 = 1.285 x 10 - 3/~ A650 = 1.285 X 10 - 3/~
A~541 = - 1.906 x 10-2 A 641~=
A541 = 2.118 x 10-2 A641 =
25A532 = 4.282 x 10-3/~ - A632 = 4.282 x 10-3/~
As23 = 2.118 x 10 - 2 A623 =
A514 = 6.423 ~x 10-3/~ A614 = 6.423 X 10-3/~
Where a prime (') is used to distinguish A-terms with two
parts, as in pages 22~ 23 and 24.
The second example is a pair of analytic
function elements combined with three elements as shown
in Fig. 7. Constructional data is as follows:

;S~
Eloment Separatlono Clear
Number Sureaco Radll Pla~tlc- Alr Apertures Materlal
r 1 0.1147 0.0242 0.196 Plexl
2 0.1502 ~ 0.0004 0.192
II 3 0.1339 0.0242 0.185 Plexi
4 0.1747 0.0792 0.171
~II 5 -0.2815 ~ 0.0054 0.071 Polycarbonate
6 plano ^~ 0.0004 0.066
IV 7 plano ~ 0.0054 0.066 Polycarbonate
8 plano 0.1228 0.070
V 9 1.0063 ~ 0.0205 0.250 Polycarbonate
4.511 0.5565 t 0.257
0 ' Aspheric ~rotational) I Back ~ocal distlnce ~dF)
Polynomlal (non-rotational); Radii given are as simulated Eor the mean
object distance.
~d = l.Oooo sl ~adopted mean object distance) = 13.55 ~d/10.1
When one refocuses for infinite object distance by use of
a decentered transverse rotation of Element IV by an
angular value of ~, one finds that
fd = 0-9987 (dF held constant)
Coefficients for rotational surfaces are as follows:
Beta2 v -1.412 x 101 Beta5 - -1.736 x 102 Betag - 9.233 x 10
Gamma2 = -9.123 x 10-3 Gammas = -1.892 x 10-2 Gammag - -9.670 x 10
Delta2 v -1.892 x 10-4 Delta5 ~ -4.442 x 10-5 Deltag - -1.695 x 10-
Element III with its rear surface polynomial is fixed
(no rotation). Element IV with its forward surface
polynomial rotates about an assigned displaced center
with the transversely displaced axis parallel to the
optical axis.
-32-

b;5~
Non-rotational coefEicients are given by:
A620 = A720 =
A602 - A702 =
A630 = 4.519 x 10-2/~ A730 = 4.519 x 10-
5A621 = A721 =
A612 = 1.356 x 10-~ 712 = 1.356 x 10-
A 640 = A 740 = 0
A640 A740 =
A631 = -4.519 x 10 2/~ A731 - -4.519 x 10-2/~
A622 = A722 =
A604 = A704 =
A'650 = -2.259 x 10-3/~ A'7so = -2.259 x 10-3/~
A6s~ = 3.234 x 10-1/~ A7so = -3.234 x 10-1/~
A 641 = A 741 =
A641 A741 =
A632 = -1.078 x 10/~ A732 = -1.078 x 10
A623 = A723 =
A614 = -1.617 x 10/~ A714 = -1.617 x 10
Where a prime (') is used to distinguish A-terms with two
parts, as on pages 22, 23 and 24.
The third example, and last in the x,y,z
system, is a pair of analytic function elements combined
with three elements as shown in Fig. 8. Constructional
data is:
Element Separations Clear
Number Sur~acq Radii Pla~tic Air Apertures Material
I 1 0.1183 0.0573 0.200 Plexi
2 0.2227 ~ 0.0767 0.175
II 3 -0.2'675 ~ 0.0097 0.077 Polycarbonate
4plano ~ 0.0004 0.073
30 III 5plano ~ 0.0097 0.073 Polycarbonate
6plano 0.1066 0.077
IV 7-2.405 0.0103 0.209 PlexL
80.9195 0.0206 0.220 Polycarbonate
9 -1.0332 ~ 0.5570 S 0.220
Aspheric (rotational) ~ Back ~ocal distance ~de~
~ Polynomial ~non-rotational); Radii ~iven are as simulated Eor the mean
35 obiect distance.
Fd ~ 1.0000 sl ~adoptad maan object distanca) ~ 13.61 ~d/10.0
-33-

When one re~ocuses for ininite object distance by use o
a decentered transverse rotation of Element III by an
angular value o~ 0, one finds khat
fd = 0-9996 (de held constant)
Rotational coef~icients are:
~ta2 ~ -2.095 x 101 ~eta3 ~ -1.642 x 102 P~etag - -1.216 x 10
Gamma2 5 -4.388 x 10~1 Gamma3 - 1.561 x 10-1 Gamli~ag ~ 7.548 x 101
Delta2 = -8.500 x 10~3 D~lta3 = -1.271 x 10-4 Deltag = 3.789 x 10
Epsilon2 z -8.876 x 10-5 Epsilon3 r OEpsilong ~S1.539 x 10-
Element II with its rear surface polynomia:l is fixed
(no rotation). Element III with its forward surface
polynomial rotates'about an assigned displaced center
with the transversely displaced axis parallel to the
optical axis.
15 Non-rotational coe~ficients are as follows:
~420 = A520 =
A402 = A502 =
A430 = 4-739 x 10-2/~A530 = 4.739 x 10-2/~
A421 = A521 =
20A412 = 1.422 x 10~1 ~ Asl2 = 1.422 x 10-1/~
A 440 = A 540 =
A44~ = 0.743 x 10 A540 =
A431 = -4.739 x 10-2/~ A531 = -4.739 x 10-2/~
A422 = 1.487 x 10 A522 =
A404 = 0-743 x 10 A504 =
A'4so = -2.370 x 10-3/~ A'sso = -2.370 x 10-3/~
A450 = -1.487 x 10-1/~ Asso = -1.487 x 10-1/~
A 441 = A'541
A441 = -1~487 x 10 A541 =
A432 = ~4-955 x 10-1/~ As32 = -4~955 x 10-1/~
A423 = -1.487 x 10 A523 =
A41~ = -7.433 x 10-1~ Asl4 = -7.433 x 10-1/~
Where a prime (') is used to distinguish A-terms with two
parts, as on pages 22, 23 and 24.
-3~-
.

;tj~
The next three examples, all in the x,y,z system
(with the pivot point lying in the ~x-axis), show a pair
of analytic unction elements combined with two other
elements as shown in Fig. 9. These three examples, the
4th, 5th and 6th in sequence, differ ~rom one another in
the value of their non-rotational coefficients, but
otherwise share the same base system as given by the
following constructional data:
RASE SYSTEM FOR 4th, 5th AND 6th EXAMPLES
Element Separations Clear
No. Surface Radii Plastic Air Apertures Material
1 10.1498 0.0426 0.201 Plexi
20.1,390 ~ 0.0856 0.172
2 3plano ~ 0.0120 0.094 Plexi
4base ~ 0.0122
3 5base ~ 0.0120 0.081 Plexi
6-0.0235 h 0.1073 0.084
4 70.0572 ~ O.nl22 0.193 Polystyrene
8plano 0.6813 t 0.200
Rotational Aspherics
Non-rotational (Polynomial) according to QP number
# Back Eocal distance
fd = 0.9843 sl ~adopted mean object distancet = 16.76 ~d/10.0
Rotational surface coeeEicients are:
beta2 ' 1.003 x 101 beta6 ~ -1.434 x 10 beta7 5 2.637 x 10
gamma2 = -6.781 x 10 gamma6 i -3.274 x 10-2 gamma7 = 1.084 x 10
delta2 - 1.485 x 10-3 delta6 - -1.099 x 10-4 delta7 ~ 1.192 x 10-1
epsilon2- 1.624 x 10-5 epsilon6 ~ -2.970 x 10-7 epsilon7- 1.233 x 10-3
Surfaces 4 and 5 are the principal polynomial
surfaces and have coefficient values to be designated.
The valu~s, of the non-rotational coeficients
for a fourth example to be used with the base system
description above and applied to surfaces 4 and 5 are as
follows:
-35-

~S;~tjtJ5
4th EXAMPLE NON-ROTATIONAL COEFFICIENTS
Rotatablc Surtace 4 FlKed Surt~c~ S
Coetl- Value Coottl- V~luq
ci~nt clent
1~420 " 0.17635 xlo~oo B520 ~ 0-33621 x 10~00
B411 ' 0.82193 X 10-ûl ~SII ' 0.83311 X 10-01
n402 n 0.17116 X 10-00 I)S02 ' 0-31073 X 10~00
1 0 B430 ' 0.96509 x 10~00 B530 ' 0.97725 x 10-00
84Zl ~ -0.14963 X 10~00 B521 " -0.17296 x 10-01
~412 ' 0-14764 X 10101 8S12 ' 0.15668 X In~OI
B403 ' -0.289a2 x 10~00 8503 ~ -0.25311 x 10~00
B440 = o-a4407 X 10-01 8540 - -0.263BI x 10~02
B431 - -0.14376 x 10~01 B531 ~ - 0.10465 X 10~01
B422 ' -0.38025 x 10~01 8S22 ~ -0.56745 x 10~02
n413 - -0.45028 X 10~01 8513 ~ -0.57384 x 10~01
3404 ~ 0.14459 x lOtOO B504 n -0.26319 x 10~02
a450 n -0. 17390 X 10~01 U550 ' -0.66473 X 10~01
B441 - 0.15724 X 10102 B541 ~ 0.16305 x 10+02
R432 ' -0.98287 x 10~02 8532 = -0.10847 X 10+03
R423 ~ - 0.15961 X 10 - 02 D523 ~ -0.16436 X lOfO2
8414 ~ 0.?4777 % 10~02 B514 = O.SOIlS X 10~02
~405 ~ 0.25328 X 10~02 D505 = 0.22133 x 10~02
n460 ' 0.96561 x 10-02 B560 ' 0.27799 X 10-01
8451 ' - 0.23303 x 10~00 B551 n - 0.23307 X 10~00
8442 ' -0.12419 X 10-01 B542 =0.42022 X 10-01
8433 ~ 0.13407 x 10~00 B533 ' 0.13409 X 10~00
B4z4 . 0.4-3734 X 10-02 8524 ~ 0.59314 X 10-01
8415 ~ 0.53554 x 10-01 8515 ~ 0.53576 X 10-01
B406 ' 0.99630 x 10-03 8S06 ' 0.19145 x 10-01
B470 ' 0.11685 X 10+00 B570 = 0.11238 X 10~00
8461 ~ O.12315 X 10-01 8S6l ~ O.12325 x ~0~01
B4s2 ~ 0.10226 x 10~02 ~552 = 0-10245 X 10-02
B443 ~ 0.15969 X 10100 B543 - 0.15946 X 10~00
B434 ~ -0.26285 X 10-01 1~534 ' -0.26229 X 10+01
8425 ~ -O.ZS506 X 10-00 8525 ~ -0.25555 X lO~ûO
8416 ' -0.27879 X 10~01 8516 ~ -0.28238 X 10~01
8407 ' -0.68930 X,10~00 a507 n -0.70525 x 10~00
' 8580 ~ 0.12113 X 10-02
B562 ~ -0.43173 x 10-04
B544 ~ 0.36196 X 10-03
8S26 ~ 0.289~0 X 10-03
esO8 n 0.10176 x 10-03
This example focuses from inf inity to approxi-
mately 25 inches with an offset distance of .570 inch for
ro~ating element II.
-36-

iS5
Values for the non-rotational sur~ace
coe~icients ~or a 5th system example are as follows:
5th EXAMPLE NON-ROTATIONAL COEFFICIENTS
Rotatable Sureace 4 Fixed Sureace 5
Coefei- Value Coe~ei- Value
c ient c i8nt
~420 ~ 0-31745 x 10~00 B520 = 0-49309 x 10+00
~411 = 0-19416 x 10-01 Bsll ~ 0.20374 x 10-01
B402 - 0.31725 x 10+00 B502 = 0-49261 x 10~00
B430 = 0.73235 x 10+00 B~30 = 0.76B28 x 10+00
B421 = -0.82315 x 10-01 Bs21 = -0.95244 x 10-01
B412 = 0.22562 x 10+01 B512 = 0.23585 x 10+01
B403 = ~0-3~735 x 10+01 Bso3 = -0.38551 x 10+00
B440 = -0.25889 x 10-01 Bs40 = -0.26488 x 10+02
B431 = 0.88754 x 10+00 Bs31 = 0.13355 x 10+01
~422 = -0-27066 x 10+01 Bs22 = -0.55631 x 10~02
B413 - - 0.46913 x 10+01 Bs~3 = -0.60622 x 10+01
B404 = - 0.31827 x 10+00 Bso4 = - 0.26780 x 10+02
B4So = 0.17338 x 10+02 B550 = 0.15100 x 10+02
B441 ~ 0.63013 x 10+01 Hs41 = 0.63070 x 10+01
B432 = -0.22539 x 10+03 Bs32 = -0.25300 x 10+03
B423 ~ - 0.19065 x 10~01 B523 = - 0.19098 x 10+01
B414 ~ 0.26757 x 10+03 Bsl4 = 0.2621S x 10+03
B4os ~ - 0.63023 x 10~01 B5os = -0.62411 x 10+01
B460 - 0.32409 x 10-02 B560 = 0.21386 x 10-01
B4sl - -0.18306 x 10+00 Bssl = -0.18308 x 10~00
B442 ~ -0.27969 x 10-02 Bs42 = 0.51649 x 10-01
B433 A 0.13301 x 10+00 ~533 - 0.13301 x 10+00
B424 ~ 0.92066 x 10-03 ~524 8 0-55366 x 10-01
B41s -- 0.4?979 x 10--01 Bsl5 = 0.47988 x 10-01
B406 ~ 0.22130 x 10 - b3 Bso6 ~ 0.18370 x 10-01
BS80 = 0.57177 x 10-04
BS62 = 0.22871 x 10-03
B544 = 0.34306 x 10-03
BS26 ~ 0.22871 x 10-03
B508 ~ 0-57177 x 10-04
~37 -

S 5 ~ ~
With these coefficients for surfaces 4 and 5,
the base system of Fig. 9 is focusable from infinity to
approximately 26 inches with a point distance of 0.6
inches and an angular excursion of the rotatable element
of approximately 50 along the arc between center points
of the outlying apertures.
The value of the non-rotational coefficients
for the 6th example, again referenced to Fig. 9 and the
ab ~ c~m, are as follows:
\
-3~-

t;55
6th EXAMPLE NON-ROTATIONAL COEFFICIENTS
Rotatable Surface 4 Fixed Sureace 5
Coe f fi- Value Coe ff i- Value
c lent cient
~3420= 0-46600 x ln+~0 B520 = 0-62957 x 10+00
B411 = 0.77017 x 10-01 Bsll = 0.78618 x 10 01
0 B402 = 0.46026 x 10+00 BsO2 5 0.62204 x 10+00
B430 = 0.11176 x 10+01 B530 = 0.11489 x 10+01
B421 = --0.82694 x 10-01 B521 = --0.91447 x 10-01
B412 = 0-24534 x 10+01 Bsl2 = 0.25386 x 10+01
B403 = 0-55940 x 10-01 Bso3 = 0.53522 x 10-01
B440 = 0.56821 x 10-01 Bs40 = 0.26405 x 10+02
B431 = 0.21750 x ;0+01 B531 = 0.25840 x 10+01
B422 = -0.49442 x 10+01 B522 = --0-57886 x 10+02
B413 = -0-69085 x 10+01 Bsl3 = -0.81771 x 10+01
B404 = -0-93669 x 10--01 B504 = --0.26554 x 10+02
B450 = 0.54783 x 10+01 B550 = 0.24635 x 10+01
B441 = 0.11101 x 10+02 B541 = 0.11114 x 10+02
B432 = -0.18539 x 10+03 Bs32 = -0.21302 x 10+03
B423 = -0.56624 x 10+01 B523 = -0.56598 x 10+01
B414 - 0.25269 x 10+03 B514 = 0.24719 x 10+03
B4os - -0.12350 x 10+02 B505 = --0-12256 x 10+02
B460 = 0-42151 x 10-02 Bs60 = 0.22366 x 10-01
13451= -0.18961 x 10+00 Bssl 5 -0.18964 x 10+00
B442 = -0.10823 x 10-01 Bs42 = 0.43619 x 10-01
B433 5 0.13247 x 10+00 Bs33 = 0.13249 x 10+00
B424 1 0.18329 x 10-02 Bs24 = 0.56275 x 10-01
B41s = 0.48S90 x 10--01 Bsl5 = 0.48709 x 10-01
B406 = 0-36013 x 10-03 3506 = 0-18510 x 10-01
BsgO =0.57177 x 10-04
Bs62 =0.22871 x 10-03
Bs44 =0.34306 x lû-03
526 =0.22871 x 10-03
B508 5 . 57177 x 10-04
This system also ~ocuses down to approximately
26 inches, but with a pivot distance of 1.0 inch.
- 3 9--

s ~.
An example of a three element all plexiglass
system is shown in Fig. 10 which has the following
constructional data:
7th EXAMPLE
S
ElementSeparations Clear
Number Surface Radii Medium Air Apertures Material
I l 0.1397 0.0485 0.200 Plexi
2 0.1503 ~ 0.0909 0.163
II 3 plano 0.0122 0.093 Plexi
4 base ~ 0.0150 0.086
0 III 5 base ~ 0.0122 0.085 Plexi
6 2.897 0.8006 ~ 0.092
*Rotational Aspheric #Back focal distance
**Non-rotational (polynomial) according to present
design.
~d = 0 9974
sl (adopted mean object distance) = 13.93 fd/10
Rotational coefficients are:
beta2 = 1.928 x 101
gamma2 = 1.018 x 102
delta2 = 1.191 x 105
epsilon2 = 1.258 x 105
Surfaces 4 and 5 are the polynomial surfaces,
although surface 3 can also be drawn upon as a polynomial
surface together with revised optimization over surfaces
3, 4 and 5 collectively.
It should be noted that surface 5 contains
implicitly dioptric power and corrective rotational
aspheric terms as ~ base system, all of which are there-
after contained within and replaced by the polynomial
coefficients given, as follows:
-40-

;5'~
Rotatable Sureace Flxe(~ Sur~ace
Coe ~ E i - Va lue Coe ~ E i - Va lu~
cient , cient
B410= ~0.17099 x 10-01 B510 = -0.11225 x 10-01
B401= 0-22753 x 10-02 B501 = 0.22097 x 10--02
B420= 0.16375 x 10+00 B520 = 0.93574 x 10+00
0 B411= 0.94617 x 10-01 B511 = 0.95766 x 10-01
B402= 0.15192 x 10+00 B502 5 0.92366 x 10+00
B430= 0.99257 x 10+00 B530 = 0.10052 x 10+01
B421= -0.21431 x 10+00 Bs21 = -0.24252 x 10+00
15 B412= 0.14404 x 10+01 Bsl2 = 0.14626 x 10+01
B403= -0.26486 x 10+00 Bso3 = -0.26536 x 10+00
B440= -0-95951 x 10-02 Bs40 = -0.16220 x 10+02
B431= --0.14827 x10+01 B531 = -0-11177 x 10+01
20 B422= 0.19776 x 10+01 Bs22 = -0.37242 x 10+02
B413= -0.68388 x 10+01 B513 = -0.58275 x 10~01
B404= 0.32939 x 10-01 BSo4 = -0.16024 x 10+02
B4sO= 0.30359 x 10+00 Bsso = -0.48918 x 10+01
25 B441= 0.16705 x 10+02 B541 = 0.17317 x 10+02
B432= -0.74565 x lOfO2 Bs32 = -0.85360 x 10+02
B423= -0.24712 x 10+02 B523 = -0.25217 x 10+02
B414= 0.63656 x 10+02 Bsl4 = 0.37524 x 10+02
30 B405= 0-16952 x 10~02 B5os = 0.13567 x 10~02
B460= 0-10214 x 10-01 B560 = 0.16309 x 10+04
B4sl= -0.24900 x 10+00 Bssl = -0.24904 x 10+00
B442= -0.13165 x 10-01 ~542 = 0-48277 x 10+04
B433= 0.14230 x 10+00 Bs33 = 0.14231 x 10+00
35 B424= 0.51671 x 10-02 Els24 = 0.48275 x 10+04
B415= 0.56869 x 10-01 Bsls = 0.56893 x 10-01
B406= 0.10566 x 10-02 B506 = 0.16295 x 10+04
B470= 0.12357 x 10+00 Bs70 = 0.11882 x 10+00
40 B461= 0.13049 x 10~01 Bs61 = 0.13060 x 10+01
8452= 0.10837 x 10+02 B5s2 = 0.10856 x 10+02
B443= 0.16926 x 10+00 B543 = 0.16902 x 10+00
B434= -0.27B65 x 10+01 Bs34 = -0.27805 x 10+01
B425= -0.27037 X 10+00 Bs2s = -0.27089 X 10+00
45 B416= --0.29553 x10+01 B516 = -0.29934 x 10+01
B407= -0.73066 x 10~00 Bso7 = -0-74757 x 10+00
B480" 0-12233 x 10-02 Bs~o = 0.45586 x 10+03
B471= -0.39020 x 10-03 Bs71 = -0.39020 x 10-03
50 B462= -0.28Bl9 x 10-03 B562 = 0.18234 x 10+04
B4s3a --0.77884 x10--04 B553 = --0.77884 x 10--04
B444= 0.20034 x 10--04 B544 = 0.27351 x 10+04
B43s~ 0.63118 x 10--04- Bs3s = 0.63118 x 10-04
B426~ 0.64329 x 10--04 aS26 n 0.18234 x 10+04
55 B417~ 0.62391 x 10--04 Bsl7 3 0.623q2 x 10--04
g408' 0.47259 x 10-04 H50g ~ 0.45585 x 10+03
As a last example, the 8th, there is shown in
Fig. 11 a four element focusable objective which utilizes
a glass replacement pair of elements, instead of the
front plexiglass element of the 7th example, to get
improved correction for chromatic aberration. ~hi 5
system has constructional data as ~ollows:
-41-

;t:,~
8th EXAMPLE
Element Sop~ratlon~ Cl~ar
Number Sur~ce R~dll Medlum Alr Aoe~turcs Mater~al
0.1450 0.0~92 0.200 SK-5 (Gla~s~
2 0.2694 0.004a 0.192
II 3 0.2670 0.0152 0.188 styrene
4 0.1503 ~ 0.0909 0.161
III 5 plano 0.0122 0.093 Plexi
6 base ^~ 0~0150 0.086
IV 7 base ~ 0.0122 0.085 Plexi
8 2.897 0.8006 1 0.092
10 *Rotational Aspheric #Back focal distance
**Non-rotational (polynomial) according to present
design.
fd = 1.0063
51 (adopted mean object distance) = 13.93 fd/10
Rotational surface coefficients are:
beta4 = 1.508 x lol
gamma4 = 1.363 x 101
delta4 - 1.030 x 105
epsilon4 = 1.046 x 105
Surfaces 6 and 7 are the polynomial surfaces,
although surface 5 can also be drawn upon as a polynomial
surface together with revised optimization over surfaces
5, 6 and 7 collectively.
It should be noted that surface 7 contains
implicitly dioptric power and corrective rotational
aspheric terms as a base system, all of which are there-
after contained within and replaced by the polynomial
coefficients given,,as follows:
-42~

5S
Rotatable Sur~ace Fixed Sur~ace
Coef~i- Value Co~ e ~ i - Value
cient cient
B610 = -0.17099 x 10-01 B710 = -0.17225 x 10-01
8601 = 0.22753 x 10-02 B701 = 0.22097 x 10-02
B620 = 0.16375 x 10+00 B720 = 0-93574 x 10+00
0 B611 = 0.94617 x 10-01 B711 = 0.95766 x 10-01
B602 = 0.15192 x 10+00 B702 = 0.92366 x 10+00
B630 = 0.99257 x 10+00 B730 = 0.10052 x 10+01
B621 = -0.21431 x 10+00 B721 = -0.24252 x 10+00
B612 = 0-14404 x 10+01 B712 = 0.14626 x 10+01
B603 = -0.26486 x 10+00 B703 = -0.26536 x 10+00
B640 = -0.95951 x 10-02 B740 = -0.16220 x 10+02
B631 = -0.14827 x 10+01 8731 = -0.11177 x 10+01
B622 = -0.19776 x 10+01 B722 = -0.37242 x 10+02
B613 = -0.68388 x 10+01 B713 = -0.58275 x 10+01
B604 = 0.32939 x 10-01 B704 = -0.16024 x 10+02
B650 = 0 30359 x 10+00 B750 = -0.48918 x 10+01
B641 = 0.16705 x 10+02 B741 = 0.17317 x 10+02
B632 = -0.74565 x 10+02 B732 = -0.85360 x 10+02
B623 = -0.24712 x 10+02 B723 = -0.25217 x 10+02
B614 = 0.63656 x 10~02 B714 = 0.37524 x 10+02
B605 = 0-16952 x 10+02 B705 = 0.13567 x 10+02
B660 = 0.10214 x 10-01 B760 = 0.16309 x 10+04
B651 = -0.24900 x 10+00 B751 = -0.24904 x 10+00
B642 = -0.13165 x 10-01 B742 = 0.48277 x 10+04
B633 = 0.14230 x 10+00 B733 = 0.14231 x 10~00
B624 = 0.51671 x 10-02 B724 ~ 48275 x 10+04
B615 = 0.56869 x 10-01 B715 = 0 56893 x 10-01
B606 = 0.10;66 x 10-02 B706 = 0.16295 x 10+04
B670 = 0.12357 x 10+00 B770 = 0.11882 x 10100
B661 = 0.13049 x 10+01 B761 = 0.13060 x 10~01
a652 = 0.10837 x 10102 B7s2 = 0.10856 x 10+02
B643 = 0.16926 x 10+00 B743 = 0 16902 x 10+00
B634 = -0.27865 x 10+01 B734 = -0 27805 x 10+01
B625 = -0.27037 x 10+00 B725 = -0.27089 x 10+00
B616 = -0.29553 x 10+01 B216 = -0.29934 x 10+01
B607 = -0.73066 x 10100 B707 = -0-74757 x 10+00
B680 = 0.12233 x 10-02 B780 = 0 45586 x 10+03
~671 = -0.39020 x 10-03 B771 = -0 39020 x 10-03
B662 = -0.28al9 x 10-03 B762 = 0.18234 x 10+04
B6s3 = -0.77884 x 10-04 B753 = -0.77884 x 10-04
B644 ~ 0.20034 x 10-04 B744 = 0.27351 x 10+04
B635 = 0.63118 x 10-04 B735 = 0.63118 x 10-04
B626 ~ 0.64329 x 10-04 B726 = 18234 x 10~04
B617 ~ 0.62391 x 10-04 B717 - 0 62392 x 10-04
~608 ~ 0.47259 x 10-04 B708 ~ 0.45585 x 10+03
Obviously, the foregoing examples may be
rescaled, and in this process it is required to redeter-
mine numerical values to go with the rescaling ~or the
new physical size. In the simplest application where all
numerical quantities are given in terms of physical
length, such as inches or millimeters, scaling to some
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5S
other value me~ns only applying some factor to all dimen-
sional quantities, such as radii, separations and clear
apertures.
There are other quantities which are non-
linear, however, such as the beta, gamma, delta, etc. ofthe aspheric coefficients. Inverse quantities such as
curvatures must be scaled inversely, etc. If in addi-
tion, one works from one unit length to another, such
that fd in one system transforms to some other value in
the other system, then the required scaling becomes more
elaborate. A check can always be made, however, by
transforming both systems through english or metric
measures as an intermediate device.
Let us suppose that we wish to transform a
system having fd = 1.250 to a numerical system having fd
= 1.000. We need only to divide all dimensional quanti-
ties by 1.250 or multiply by ~.800. Thus, fd =
1.250/1.250 becomes the desired fd = 1.000. If Rl =
2.500 for example, then in the scaled system Rl = 2.000,
20 etc. Or, Rl = 2.50 x 0.g00 = ~.000 directly.
Beta scales as the cube, gamma by the fifth
power, delta by the 7th, etc., but inversely. That is,
for the example above, beta scales by (0.800)3, gamma by
(0.800)5, delta by (o.800)7, etc.
It will be obvious to those skilled in the art
that changes may be made in the above-described embodi-
ments without departing from the scope of the invention.
But, in any optical~design making use of the invention at
least one pair of closely adjacent but opposed dioptric
surfaces of preferred analytic forms which lie respec-
tively on closely adjacent lens elements, must be
included. At least one of this pair of elements is then
rotated about a pivot point displaced by an assignable
distance from the optical axis and therefore transversely
about an axis parallel to the optical axis. An~ element
-44-

t>;,~;tj5;
so rotated must then employ either a plane surface for
its outlying surface such that the transverse rotation
about the displaced parallel axis causes no perceptible
change in the dioptric action of this transverse plane
surface, or in special cases one may apportion the work
of the analytic surface from its one original face,
instead, over both of the surfaces of the element.
Conversely, if one of the pair of elements is
to remain fixed in order that all focusing action is
effected by means of the transverse rotation of the other
about a displaced axis, then the outlying surface of the
fixed element may be designed as with any other outlying
dioptric surface or element of the optical system, and
may therefore have such rotational dioptric and aspheric
powers as may prove advantageous. In special cases one
may apportion the work of the fixed analytic surface
amongst the two surfaces of the element, in which
instance the action of the so apportioned analytic
surface becomes superimposed onto the base dioptric and
aspheric powers of the outlying surface of the fixed
element, whatever they may be. The converse also
applies.
In the most general case one may require that
both elements of a pair rotate individually about
assigned displaced pivot points and displaced parallel
axes, either equally but oppositely in sense of rotation
as a selected important variant, or with proportional
rotations of the same or opposite sense as may be
desired, or even non-linearly in dual rotations of the
same or opposite sense where such control may be
required. ~11 such transversely rotated elements must
therefore have either plane out-lying surfaces that
rotate in their own planes without optical effect, or
must have the work of the respective analytic surfaces
apportioned amongst the surfaces.
-45-

1~s~5S
Thus, it has been shown that with proper selec-
tion of parameters for the adjace~it analytic surfaces one
can effect the refractive action of a rotational dioptric
lens taken mathematically to be of variable power to an
S order of precision sufficient for hand photography with
cameras of moderate to small focal lengths and aperture-
ratios. It has also been shown that further corrections
normally effected by use of rotational aspheric terms on
one or more nearby fixed rotational surfaces can when
necessary be taken over into a further elaboration of the
analytic function defining the shape of the fixed surface
of the cooperative pair of surfaces. That is to say, if
one is dealing with the two analytic refractive surfaces
which already are aspheric in a broad sense, it may not
then be necessary to employ one or more other nearby
rotational aspherics. Instead, the action can be com-
bined with the fixed member of the pair of analytic
surfaces, which differ from one another for just that
purpose, the fixed analytic function containing therefore
superimposed rotat;onal dioptric and aspheric powers not
shared by the analytic shape of the movable or rotatory
opposed surface. Therefore, it is intended that all
matters contained in the above description or shown in
the accompanying drawings shall be interpreted as
illustrative and not in a limiting sense.
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