Improvements in Anamorphotic Lens Systems.
Year: 1898

page_text: PROVISIONAL SPECIFICATION.
Improvements in Anamorphotic Lens Systems.
I, PAUL RUDOLPH, Doctor of Philosophy, of Jena, Grand-Duchy of
Saxe Weimar, German Empire, do hereby declare the nature of this invention to
be as follows:-
This invention has for its object to solve the following problem to obtain, by
means of a lens system, such an image of a plane object, that all the points of the
object be reproduced in that image by corresponding sharply defined points (the
pencils of rays from the points of the object being once more homocentrically
united in the plane of the image), the linear enlargements however, taken in two
diameters of the plane of the image, at right angles to each other, be unequal, so
that the relation which the object and image bear to each other must be described,
not as similarity but as affinity.
The practicability of obtaining such a reproduction, which whilst ensuring
homocentric junction of the rays in the image, results in the kind of dissimilarity
just referred to and in a corresponding regular distortion, an important question
for many purposes of the trade. But in view of the means which have hitherto
been employed in the production of optical images, the problem above stated must
appear insoluble; for all the arrangements for optical reproduction heretofore
adopted offer the following alternative. They either make a homocentric pencil
of rays proceeding from an object point to remain homocentric, each of the succes-
sive refracting or reflecting surfaces converting the homocentric pencil into another
pencil which is also homocentric. This is accomplished in respect not only of one
object point, but of all points of the right line representing the axis, and further-
more by virtue of geometrical necessity-each plane perpendicular to this straight
line is reproduced with equal enlargement in all directions, that is to say in a
strictly similar image. Or they bring about a distortion of the images by aban-
doning those conditions under which the refracted or reflected pencils of rays
remain homocentric, employing reflecting or refracting surfaces of double curvature
for the purpose of distorting the images; in which case, however, what corresponds
in the image to the points of the object, are not points but circles of diffusion, the
order of magnitude of which is the same as that of the areas of the pencils of rays
in action.
The first named case is exemplified in the ordinary optical systems, consisting of
centred spherical surfaces, in respect of the right line which forms the axis of the
system. The pencils of rays from all points of this axis after passing through the
system are homocentric once more, at least to that degree of approximation which
exists, in spite of the spherical aberration, after refraction by one spherical surface
only. By consequence there exists also junction of rays of the said degree of
approximation for the points of all elementary surfaces of the object space which
intersect the axis at right angles; but, at the same time, a necessary geometrical
consequence of such a union of rays is that the images of all those elementary sur-
faces show linear enlargements, which are strictly identical in all directions, so

page_text: that the image planes and the object planes are exactly similar. A distortion of
the image in relation to the object (.e. an enlargement varying in different direc-
tions) is therefore not feasible with a spherical system.
The second alternative case is realized by the optical devices which have hitherto
been employed in producing what is termed anamorphosis; such devices being
cylindrical or conical mirrors, and glasses with surfaces of cylindrical or irregular
shape, and their effects consisting in, enlargements which are unequal in the various
directions, or in irregular distortions of any kind. The disproportional enlarge-
ment effected by the said means depends upon such refracting or reflecting surfaces
as will convert homocentric pencils of rays, as they proceed from the points of the
object, into anacentric pencils, or pencils which in no part present a point as cross-
section (a focal point) but which always form two focal lines situated at right
angles to and at a greater or less distance from each other, while all their remaining
cross-sections are elliptical surfaces of diffusion varying in the ratio of their axes.
With all these devices a real or virtual image of the object only arises as far as
the pencils of rays in action are very narrow, so that their sectional areas (or
areas of diffusion) in the plane of the image also remain very narrow and lead to
no marked indistinctness of the image. An image built up with sharp points
cannot be obtained by the means in question.
Thus it is evident that with the devices hitherto available for producing images
the two requirements of the new problem: a homocentric union of the rays in the
points of the image and a disproportional enlargement of the image, exclude each
other. The following considerations, however, go to show that the two objects
named are not absolutely and unconditionally incompatible.
In the optical systems of the first named class (i.e. combinations of centred
spherical surfaces) the impossibility of unequally enlarging the images as to the
different directions, would seem to be an outcome of the symmetry of the system in
relation to its axis, since this symmetry alone suffices to prevent the effect from
varying in different planes intersecting each other in the axis. But in reality
even in systems of this description, the geometrical incompatibility of dispropor-
tional enlargement with a homocentric reunion of the rays of light is not a result
of the symmetrical character of the elements of the system. As a matter of fact
the incompatibility of these two attributes would be just as stringent a geometrical
necessity, were the points of the axis homocentrically reproduced not by centred
spherical surfaces but by any other refracting surfaces being entirely unsymmetrical
both in position and action. The mere fact that the points of the axis, in con-
tinuous succession, are reproduced sharply-i.c. by homocentric pencils of rays-
no matter how the homocentricity of the pencils were originated, is geometrically
conflicting with the requirement of a disproportional enlargement of an object
plane which is perpendicular to the axis. On the axis side this geometrical con-
tradiction is solely brought about by the supposition of a continuous reproduction
of the axis. Disproportional enlargement of the images and homocentric junction
of the rays of light in the points of the image would be two compatible attributes
of an optical system, if the second attribute were only required for isolated points of
the axis.
The theorem herebefore stated and which is demonstrable by the most general
propositions of geometrical optics, may be briefly stated as follows: The in-
compatibility of a sharply defined reproduction and of a disproportional reproduction
in the case of the optical systems now in use results (quite apart from conditions
of symmetry) simply from the sharp reproduction in a consecutive order of the
points of the axis and it results, on the other hand, only from this uninterrupted
succession of the points supposed to be sharply reproduced. With a view to this
position the optical problem stated in the introductory part of this specification
may be embodied in the following question. Is it possible to devise a system of
refracting or reflecting surfaces, which, in relation to an axis attributed to it, has
the property of homocentrically transmitting pencils of rays from certain isolated
points of such axis, whilst rendering the pencils from all the remaining points of
.

page_text: the axis anacentrical or in other words converting them into pencils with two
focal lines which cross each other at right angles? A reply to the question thus
formulated and consequently a solution of the problem with which this invention
is intended to deal is furnished by the following demonstrations forming a supple-
ment to the well known propositions concerning the properties of optical pencils
of rays, which are based upon Malus' theories.
A ray of light A A' may pass, in accordance with the laws of refraction or
reflection, through a combination of surfaces fi
fk, no matter of what
shape or how inclined to the direction of the ray, though curved continuously, and
may be considered as the axis of reproduction. A homocentric pencil of rays
proceeding either really or virtually from any point of such axis in the object
space will then in general he transmitted as an anacentric pencil having no point
of junction in any part of the image space, but exhibiting, at two points of the
axis situated at a greater or less distance apart, focal lines extending at right
angles to such axis and crossing each other also at right angles. Within the limits
of this general mode of action the following four special cases have to be discerned
as geometrically possible and exhausting such possibility:
1. There may be no point sending forth a pencil of rays that is transmitted homo-
centrically, i.e. so that the two focal lines coincide in the same part of the axis A¹
and change to one common focal point. Then each plane in the object space at
right angles to the axis will be reproduced-in planes in the image space lying in
proximity to the two focal lines of the axial pencil-by circles of diffusion formed
by the pencils of rays transmitted and having varying enlargement and varying
distortion, but none of these object planes will be reproduced in sharply defined
points. In this case the system has, in regard to all parts of the object space, the
same effect as the well-known devices above mentioned for producing optical
anamorphoses.
2. Or there may be a point and only one point in the axis, the pencil of rays
proceeding from which is transmitted homocentrically. In that case, a plane at
right angles to the axis, will be reproduced, as to all points closely surrounding
the said axis, sharply in a plane image presenting a linear enlargement, which is
smaller or greater but always identical in all directions, so that the image is
exactly similar to the object. Thus in this case the system will, in regard to the
single object plane, have the same effect as a system of centred spherical surfaces;
whilst in regard to all other object planes its action will be like that of the ordinary
anamorphotic arrangements.
3. Or again, there may be two points on the axis, but not more than two,
proceeding from which a pencil of rays continues homocentric. In this case each
of both object planes (at the said points), but none other, will be sharply reproduced
in a corresponding image plane, but the enlargement will necessarily differ in the
directions of two diameters taken at right angles to each other; wherefore there.
will be no similarity between the image and object, instead of which a regular dis-
tortion will make them stand to each other in the relation of affinity. This distor-
tion, as regards the object plane at one of the aforesaid two points must necessarily
be exactly the reverse of that existing with reference to the object plane at the
second point, so that, if in the case of one plane the enlargements in two diameters
taken at right angles to each other be in the ratio of n: m, the corresponding
ratio for the other plane is m: n.
4. Or lastly there may he more than two points in the axis, whence the pencils
of rays remain homocentrical; in which case the same thing must happen in regard
to all points of the axis, and all object planes, which are perpendicular to the axis,
must be reproduced in continuous succession by sharp junction of rays, but also
with exact similarity. In this case the system, notwithstanding the lack of
symmetry in the construction of its elements and in their mode of action, will
produce in respect to the totality of the object space-exactly the same effect as
a system made up of centred spherical surfaces.
These four cases wholly exhaust all geometrically possible effects of a combina-

page_text: tion of reflecting or refracting surfaces of the most general kind, and we may now
reduce the problem, the solution of which is the object of the present invention,
to the following: To form a combination of refracting surfaces which individually,
all or part of them, will cause the pencils of rays which they transmit to become
anacentric whilst fulfilling by the joint effect the conditions of the third of the
possible contingencies above enumerated; or in other words, a combination posse3-
sing the peculiar property of ensuring a homocentric reunion of the rays proceeding
from two (but not more than two) points of the axis.
Of the infinite variety of possible arrangements capable of bringing about this
result, such only are practicable as fulfil the following conditions. 1. That the
axis of reproduction, if it should not continue rectilinear from the object space
up to the image space, shall remain within one and the same plane and 2, that
the planes of the principal curvatures of all refractive surfaces be situated only in
two planes both including the axis and perpendicular to each other. The first of
these conditions, moreover, involves the necessity for all refracting surfaces, that
the tangential planes at the points where these surfaces are traversed by the axis,
are situated at right angles, if not to the direction of the respective part of the
axis, yet to the plane containing the axis.
Under these restrictions all refracting surfaces, which are fit to be used in
forming a combination of the novel type may be reduced to three categories viz;
(a). Spherical surfaces centred in relation to the axis of reproduction as in
an ordinary lens system, so that, taken in pairs they form lenses of the usual kind.
(b). Cylindrical surfaces, the cylinder axes of which intersect the axis of repro-
duction perpendicularly and cross each other also at right angles.
(c). Planes which are at right angles to the plane of the axis of reproduction
and are inclined at any desired angle to this axis, so that, taken in pairs, they form
prisms, the principal section of which lies in the plane of the axis.
That all possible modifications of anacentric refraction may be ensured by
means of elements of these three kinds, is evidenced by the following considera-
tions. The refractive effect which is produced by a surface of double curvature
inclined to the axis at any angle, may in all cases be subdivided into the effect
of a plane limit surface, inclined to the axis at the said angle μ, and into the
effect of the infinitely thin lens, plane on one side and doubly curved on the other,
which might be split off by a cut parallel to the tangential plane. For this
inclined lens may, in regard to all refractive effects in the vicinity of its vertex,
be substituted another lens, also plane on one side and doubly curved on the other,
but of other principal curvatures and placed at right angles to the
axis. Such lens plane on one side and doubly curved on the other,
and placed at right angles to the axis, may furthermore, if the radii of its principal
curvatures are given, be replaced in an indefinite variety of ways by a centred
spherical surface or spherical lens and one or more centred cylindrical surfaces,
the cylinder axes of which are situated in the planes of the principal curvatures
of the lens to be replaced. Hence it will be understood that an optical system,
which acts as required for the solution of any specified form of the general task,
to which the invention relates, may be constructed without introducing into the
optical system other elements than centred spherical surfaces, centred cylindrical
surfaces or planes, that is to say: glass-bodies in the shape of spherical lenses,
cylindrical lenses or prisms. This fact is of practical importance considering
the technical difficulties of exactly working doubly curved surfaces, if they are
not, like the cylindrical surfaces, at the same time surfaces of revolution.
But apart from this technical advantage, the foregoing demonstration of the
sufficiency of the three simplest kinds of optical elements brings about another
advantage the computer is now enabled to deal with every concrete problem of
the kind in question just in the ordinary well known manner; for whenever a
plane object, located in a given point of the space, is to be sharply reproduced in a
plane at another predetermined point with a prescribed enlargement n in one
diameter and another given enlargement m in the diameter extending at right

page_text: angles to the first or, in other words, with a ratio of distortion n: m, all that is neces-
sary is to solve twice viz. with reference to the planes of both diameters-exactly
in the same manner the task familiar to every optician: to determine the radii of
curvature and the vertex points of the optical system of spherical surfaces which
will project at the second given point the image of an object situated at the first
given point, once with the enlargement n and then once again with the enlarge-
ment m. It will be understood that the radii of the two systems are in the present
case to be regarded as the radii of cylindrical surfaces, the axes of which are at
right angles to the given diameter of one or the other system, unless a surface of
one system and a surface of the other coincide in the same point of the axis and
be identical in magnitude and sign, so that they may be represented by one single
spherical surface. The possible insertion of prisms, side by side with lenses,
involves no material deviation from this rule, inasmuch as the displacement of the
image point and the change in the angle of convergence of a pencil of rays, which
are both effected by a plane surface inclined to the axis, may be determined in the
two principal planes in a like way as the corresponding effects of a spherical sur-
face, with the only difference that, instead of the degree of curvature, it is the
angle of inclination of the plane to the axis that has to be introduced into the
calculation.
$5
When by twice applying the method described, the problem has been solved in
regard to the prescribed points (object point and image point)-other circumstances"
connected with the object and conditions of employment of the construction being.
duly taken into account-the second pair of points where, as has been explained
above, a sharply defined reproduction takes place likewise, but with a reversed ratio
of enlargement m: n. may be searched for, because it is advisable to locate the
inlet and outlet "pupils" of the system at those points. They may be determined
without difficulty by making use of the fundamental formula, whereby the position
and size of the images, produced by ordinary lens systems, are determined after
having first found by calculation the focal lengths of the combination with refer-:
ence to the principal planes corresponding to the two diameters.
In calculating as described an anamorphotic system which presents sharp images,
it makes no difference whether reproduction by real or virtual junction of
rays in a predetermined image plane be required. Hence, systems of the
kind here contemplated may be constructed not only for photographic and
other purposes of projection, but also to act in the mode of a magnifying glass or.
of a telescope. All these widely different systems, however, will have one con-.
structional feature in common: the surfaces of double curvature of the system---
for instance cylindrical surfaces-cannot be confined to one point of the axis only;
on the contrary elements of that class are always to be provided at least at two
different points-unless there be among the elements a plane surface arranged
obliquely to the axis. For if-in the absence of such an oblique plane surface-
the anacentrically refracting surfaces of the system were all collected at one and
the same point of the axis, so that jointly they were approximately equivalent to
one single thin lens having doubly curved surfaces, the vertex point of such lens
would be one point of the axis in respect of which-no matter whether there be
spherical surfaces included in the system or not--there would take place homo-
centric junction of rays and identical enlargement in the various directions. This
vertex point therefore would be either the one single point of the second of the
four cases enumerated above, or one of more than two points as in the fourth case,
whilst the third case, upon which the solution of the problem stated is dependent,
would be excluded.
The practical solution of the problem, as above formulated, is feasible by means
of comparatively simple constructions, viz. either centred cylindrical lenses only
or spherical systems combined with cylindrical lenses, or with combinations of
prisms, the principal section of which contain the axis of the system.
two cylindrical lenses the
The simplest case
is
of
that
axes of
ה

page_text: which cross each other at right angles. This arrangement will be found service-
able, when the reproduction is performed by pencils of rays of narrow aperture
and the field of the image is only of moderate size. Such cylindrical lenses whether
incomposite or achromatized, are, in regard to their principal sections, attended
with defects equivalent to those of simple or achromatized spherical lenses.
anamorphotic system thus formed cannot, therefore, as regards reunion of rays,
be expected to do more than would an incomposite or achromatic spherical lens.
An
To improve the quality of the image and enhance the useful luminosity of such
a cylindrical combination, one or both of the cylindrical lenses may be replaced,
without any change in the principle of construction, by a set of cylindrical lenses,
either arranged singly or cemented together. And according to the special purpose
for which the combination is intended either positive sets only, or both positive
and negative sets, or else single lenses may be employed. Within the limits of the
just described cylindrical system any one of the known optical combinations may
be realized as, for example, any kind of objectives, eye-pieces, télescopes and micro-
scopes, which are intended to distort the image in a predetermined degree.
Where an image is to be produced by pencils of rays of larger aperture, it will be
advisable to construct the collecting means separate from the distorting means.
this end either systems of cylindrical lenses or combinations of prisms should be
placed in the path of the rays of a carefully corrected spherical object lens, giving
strictly similar image on a predetermined scale, which added combinations by
altering the convergence of the rays in the direction of one diameter will produce
the required distortion.
To
As regards the first of these different systems: cylindrical lenses in conjunction
with a set of spherical lenses, the cylindrical combination in its simplest form would
consist of one collecting cylindrical lens and one dispersing cylindrical lens, placed
at a definite distance apart. The focal lengths of the cylindrical lenses, in this
case, will be comparatively great, so that the effect that will be produced upon the
quality of the image projected by the spherical lens may be said to be insignificant.
It will of course, be understood that two such systems of cylindrical lenses, with
their respective axes crossing each other at right angles, may be used for the pur-
pose of varying the scale of reproduction of the spherical system at the same time
in two directions at right angles to each other, enlarging this scale in one direction
and diminishing it in the other so as to enhance the resulting degree of distortion.
In respect to the second of the two combinations above mentioned: spherical
lenses combined with a set of prisms, the principal section of which must contain
the axis of the spherical system--it will be observed that in general an achromatic
set of prisms, which in the plane of its principal section produces a divergence of
rays differing from that obtained in the plane at right angles thereto, would cause
the axis to turn in the plane of the principal section; where, however, Brewster's
straight-sighted set of prisms is employed, the position of the axis is not altered.
As such a set of prisms has the property of transmitting parallel pencils of rays
as parallel pencils, whilst making divergent pencils anacentric, the prisms must be
placed at a part of the system where the rays of light take a strictly parallel, or at
least very nearly a parallel, course. Exact parallelism of rays may be attained
by the employment of a symmetrical double objective in the inner air space of
which the rays may always be made to follow a parallel course for a predetermined
enlargement. The case of approximately parallel rays will occur where the
spherical system is constructed to produce a considerably enlarged image of the
object.
Datod this 19th day of Anvil 1908
JOHNSONS & WILLCOX,
47, Lincoln's Inn Fields, London, W.C.,
Agents.

page_text: Improvements in Anamorphotic Lens Systems.
I, PAUL RUDOLPH, Doctor of Philosophy, of Jena, Grand-Duchy of
Saxe-Weimar, do hereby declare the nature of this invention, and in what manner
the same is to be performed, to be particularly described and ascertained in and
by the following statement:-
This invention has for its object to solve the following problem: to obtain, by
means of a lens system, such an image of a plane object, that all the points of the
object be reproduced in that image by corresponding sharply defined points (the
pencils of rays from the points of the object being once more homocentrically
united in the plane of the image), the linear enlargements however, taken in two
diameters of the plane of the image, at right angles to each other, being unequal, so
that the relation which the object and image bear to each other must be described.
not as similarity but as affinity.
Fig. 1 is a diagrammatical representation of an optical system arranged according
to the invention.
Fig. 2 represents another system of this kind.
Fig. 3 shows a third system of the same kind.
Fig. 4 is an auxiliary representation, by means of which the problem solved by
the invention may be more easily understood.
The practicability, of obtaining such a reproduction, which, whilst ensuring
homocentric junction of the rays in the image, results in the kind of dissimilarity
above referred to and in a corresponding regular distortion, is an important question
for many purposes of the trade. But in view of the means which have hitherto
been employed in the production of optical images, the problem above stated must
appear insoluble; for all the arrangements for optical reproduction heretofore
adopted offer the following alternative. They either make a homocentric pencil
of rays proceeding from an object point to remain homocentric, each of the succes-
sive refracting or reflecting surfaces converting the homocentric pencil into another
pencil which is also homocentric. This is accomplished in respect not only of one
object point, but of all points of the right line representing the axis, and further-
more-by virtue of geometrical necessity-each plane perpendicular to this straight
line is reproduced with equal enlargement in all directions, that is to say in a
strictly similar image. Or they bring about a distortion of the images by aban-
doning those conditions under which the refracted or reflected pencils of rays
remain homocentric, employing reflecting or refracting surfaces of double curvature
for the purpose of distorting the images; in which case, however, what corresponds
in the image to the points of the object, are not points but circles of diffusion, the
order of magnitude of which is the same as that of the areas of the pencils of rays
in action.
The first named case is exemplified in the ordinary optical systems, consisting of
centred spherical surfaces, in respect of the right line which forms the axis of the
system. The pencils of rays from all points of this axis after passing through the
system are homocentric once more, at least to that degree of approximation which
exists, in spite of the spherical aberration, after refraction by one spherical surface
only By consequence there exists also junction of rays of the said degree of
approximation for the points of all elementary surfaces of the object space which
intersect the axis at right angles; but, at the same time, a necessary geometrical
consequence of such a union of rays is that the images of all those elementary sur-
faces show linear enlargements, which are strictly identical in all directions, so
that the image planes and the object planes are exactly similar. A distortion of

page_text: the image in relation to the object (i.e. an enlargement varying in different direc-
tions) is therefore not feasible with a spherical system.
The second alternative case is realized by the optical devices which have hitherto
been employed in producing what is termed anamorphosis; such devices being
cylindrical or conical mirrors, and glasses with surfaces of cylindrical or irregular
shape, and their effects consisting in enlargements which are unequal in the various
directions, or in irregular distortions of any kind. The disproportional enlarge-
ment effected by the said means depends upon such refracting or reflecting surfaces
as will convert homocentric pencils of rays, as they proceed from the points of the
object, into anacentric pencils, or pencils which in no part present a point as cross-
section (a focal point) but which always form two focal lines situated at right
angles to and at a greater or less distance from each other, while all their remaining
cross-sections are elliptical surfaces of diffusion varying in the ratio of their axes.
With all these devices a real or virtual image of the object only arises as far as
the pencils of rays in action are very narrow, so that their sectional areas (or
areas of diffusion) in the plane of the image also remain very narrow and lead to
no marked indistinctness of the image. An image built up with sharp points
cannot be obtained by the means in question.
Thus it is evident that with the devices hitherto available for producing images
the two requirements of the new problem; a homocentric union of the rays in the
points of the image and a disproportional enlargement of the image, exclude each
other. The following considerations, however, go to show that the two objects
named are not absolutely and unconditionally incompatible.
In the optical systems of the first named class (i.e. combinations of centred
spherical surfaces) the impossibility of unequally enlarging the images as to the
different directions, would seem to be an outcome of the symmetry of the system in
relation to its axis, since this symmetry alone suffices to prevent the effect from
varying in different planes intersecting each other in the axis. But in reality,
even in systems of this description, the geometrical incompatibility of dispropor-
tional enlargement with a homocentric reunion of the rays of light is not a result
of the symmetrical character of the elements of the system. As a matter of fact
the incompatibility of these two attributes would be just as stringent a geometrical
necessity, were the points of the axis homocentrically reproduced not by centred
spherical surfaces but by any other refracting surfaces being entirely unsymmetrical
both in position and action. The mere fact that the points of the axis, in con-
tinuous succession, are reproduced sharply-i.e. by homocentric pencils of rays--
no matter how the homocentricity of the pencils were originated, is geometrically
conflicting with the requirement of a disproportional enlargement of an object
plane which is perpendicular to the axis. On the other side this geometrical con-
tradiction is solely brought about by the supposition of a continuous reproduction
of the axis. Disproportional enlargement of the images and homocentric junction
of the rays of light in the points of the image would be two compatible attributes
of an optical system, if the second attribute were only required for isolated points of
the axis.
The theorem herebefore stated and which is demonstrable by the most general
propositions of geometrical optics, may be briefly stated as follows:-The in-
compatibility of a sharply defined reproduction and of a disproportional reproduction
in the case of the optical systems now in use results (quite apart from conditions
of symmetry) simply from the sharp reproduction in a consecutive order of the
points of the axis and it results, on the other hand, only from this uninterrupted
succession of the points supposed to be sharply reproduced. With a view to this
position the optical problem stated in the introductory part of this specification
may be embodied in the following question. Is it possible to devise a system of
refracting or reflecting surfaces, which, in relation to an axis attributed to it, has
the property of homocentrically transmitting pencils of rays from certain isolated
points of such axis, whilst rendering the pencils from all the remaining points of

page_text: and
the axis anacentrical or in other words converting them into pencils with two
focal lines which cross each other at right angles? A reply to this question
consequently a solution of the problem with which this invention
is intended to deal is furnished by the following demonstrations forming a supple-
ment to the well known propositions concerning the properties of optical pencils
of rays, which are based upon Malus' theories. .
A ray of light A A', Fig. 4, may pass, in accordance with the laws of refraction or
reflection, through a combination of surfaces f₁ ·
fk, no matter of what
shape or how inclined to the direction of the ray, though curved continuously, and
may be considered as the axis of reproduction. A homocentric pencil of rays proceed-
ing either really or virtually from any point I of such axis in the object space will
then in general be transmitted as an anacentric pencil having no point of junction
in any part of the image space, but exhibiting, at two points P₁ and P of the.
axis A situated at a greater or less distance apart, focal lines extending at right
angles to such axis and crossing each other also at right angles. Within the limits.
of this general mode of action the following four special cases have to be considered
as geometrically possible and exhausting such possibility.
1. There may be no point P sending forth a pencil of rays that is transmitted
homocentrically, i.e., so that the two focal lines coincide in the same part of the
axis A and change to one common focal point P". Then each plane in the object
space at right angles to the axis will be reproduced-in planes in the image space
lying in proximity to the points P1 and P which belong to the two focal lines of
the axial pencil by circles of diffusion formed by the pencils of rays transmitted
and having varying enlargement and varying distortion, but none of these object
planes will be reproduced in sharply defined points P¹. In this case the system has,
in regard to all parts of the object space, the same effect as the well-known devices
above mentioned for producing optical anamorphosis.
2. Or there may be a point and only one point P in the axis A, the pencil of rays
proceeding from which is transmitted homocentrically. In that case, a plane at
this point P, at right angles to the axis, will be reproduced, as to all points closely
surrounding the said axis, sharply in a plane image at P1 presenting a linear
enlargement, which is smaller or greater but always identical in all directions, so
that the image is exactly similar to the object. Thus in this case the system will,
in regard to the single object plane, have the same effect as a system of centred
spherical surfaces; whilst in regard to all other object planes its action will be like.
that of the ordinary anamorphotic arrangements.
1
3. Or again, there may be two points, P, and P2, on the axis A, but not more
than two, proceeding from which a pencil of rays continues homocentric. In this
case each of both object planes (at the said points P, and P2), but none other, will
be sharply reproduced in a corresponding image plane at P₁₁ and P₂ respectively
on the axis A¹, but the enlargement will necessarily differ in the directions of
two diameters taken at right angles to each other; wherefore there will be no
similarity between the image and object, instead of which a regular distortion
will make them stand to each other in the relation of affinity. This distortion, as
regards the image plane at P, in comparison with the object plane at P₁, must
necessarily be exactly the reverse of that existing in the image plane at P₁₁ with
reference to the object plane at P₁, so that, if in the case of the plane at P₁', the
enlargements in two diameters taken at right angles to each other be in the ratio
of nm, the corresponding ratio for the plane at P is m: n.
4. Or lastly there may be more than two points in the axis A, whence the pencils
of rays remain homocentrical in which case the same thing must happen in
regard to all points of the axis, and all object planes, which are perpendicular to
the axis, must be reproduced in continuous succession by sharp junction of rays,
but also with exact similarity. In this case the system, notwithstanding the lack
of symmetry in the construction of its elements and in their mode of action, will
produce in respect to the totality of the object space-exactly the same effect as a
system made up of centred spherical surfaces.

page_text: These four cases wholly exhaust all geometrically possible effects of a combina-
tion of reflecting or refracting surfaces of the most general kind, and we may now
reduce the problem, the solution of which is the object of the present invention,
to the following: To form a combination of refracting surfaces which individually,
all or part of them, will cause the pencils of rays which they transmit to become
anacentric whilst fulfilling by the joint effect the conditions of the third of the
possible contingencies above enumerated; or in other words, a combination posses-
sing the peculiar property of ensuring a homocentric reunion of the rays proceeding
from two (but not more than two) points of the axis.
Of the infinite variety of possible arrangements capable of bringing about this
result, such only are practicable as fulfil the following conditions. 1. That the
axis of reproduction, if it should not continue rectilinear from the object space
up to the image space, shall remain within one and the same plane, and 2, that
the planes, of the principal curvatures of all refractive surfaces be situated only in
two planes both including the axis and perpendicular to each other. The first of
these conditions, moreover, involves the necessity for all refracting surfaces, that
the tangential planes at the points where these surfaces are traversed by the axis,
are situated at right angles, if not to the direction of the respective part of the
axis, yet to the plane containing the axis.
Under these restrictions all refracting surfaces, which are fit to be used in
forming a combination of the novel type may be reduced to three categories viz.:
(a) Spherical surfaces centred in relation to the axis of reproduction as in
an ordinary lens system, so that, taken in pairs they form lenses of the usual kind.
(b) Cylindrical surfaces, the cylinder axes of which intersect the axis of repro-
duction perpendicularly and cross each other also at right angles.
(c) Planes which are at right angles to the plane of the axis of reproduction
and are inclined at any desired angle to this axis, so that, taken in pairs, they form
prisms, the principal section of which lies in the plane of the axis.
That all possible modifications of anacentric refraction may be, ensured by
means of elements of these three kinds, is evidenced by the following considera-
tions. The refractive effect which is produced by a surface of double curvature
inclined to the axis at any angle, may in all cases be subdivided into the effect
of a plane limit surface, inclined to the axis at the said angle p, and into the
effect of the infinitely thin lens, plane on one side and doubly curved on the other,
which might be split off by a cut parallel to the tangential plane. For this
inclined lens may, in regard to all refractive effects in the vicinity of its vertex,
be substituted another lens, also plane on one side and doubly curved on the other,
but of other principal curvatures and placed at right angles to the
axis. Such lens plane on one side and doubly curved on the other,
and placed at right angles to the axis, may furthermore, if the radii of its principal
curvatures are given, be replaced in an indefinite variety of ways by a centred
spherical surface or spherical lens and one or more centred cylindrical surfaces,
the cylinder axes of which are situated in the planes of the principal curvatures
of the lens to be replaced. Hence it will be understood that an optical system,
which acts as required for the solution of any specified form of the general task,
to which the invention relates, may be constructed without introducing into the
optical system other elements than centred spherical surfaces, centred cylindrical
surfaces or planes, that is to say: glass-bodies in the shape of spherical lenses,
cylindrical lenses or prisms. This fact is of practical importance considering
the technical difficulties of exactly working double curved surfaces, if they are
not, like the cylindrical surfaces, at the same time surfaces of revolution.
But apart from this technical advantage, the foregoing demonstration of the
sufficiency of the three simplest kinds of optical elements brings about another
advantage; the computer is now enabled to deal with every concrete problem of
the kind in question just in the ordinary well known manner; for whenever a
plane object, located in a given point of the space, is to be sharply reproduced in a
plane at another predetermined point with a prescribed enlargement n in one

page_text: diameter and another given enlargement m in the diameter extending at right
angles to the first or, in other words, with a ratio of distortion n; m, all that is neces-
sary is to solve twice viz. with reference to the planes of both diameters exactly
in the same manner the task familiar to every optician to determine the radii of
curvature and the vertex points of the optical system of spherical surfaces which
will project at the second given point the image of an object situated at the first
given point, once with the enlargement n and then once again with the enlarge-
ment m. It will be understood that the radii of the two systems are in the present
case to be regarded as the radii of cylindrical surfaces, the axes of which are at
right angles to the given diameter of one or the other system, unless a surface of
one system and a surface of the other coincide in the same point of the axis and
be identical in magnitude and sign, so that they may be represented by one single
spherical surface. The possible insertion of prisms, side by side with lenses,
involves no material deviation from this rule, inasmuch as the displacement of the
image point and the change in the angle of convergence of a pencil of rays,, which
are both effected by a plane surface inclined to the axis, may be determined in the
two principal planes in a like way as the corresponding effects of a spherical sur-
face, with the only difference that, instead of the degree of curvature, it is the
angle of inclination of the plane to the axis that has to be introduced into the
calculation.
When, by twice applying the method described, the problem has been solved
in regard to the prescribed points (object point P₁. and image point P₁₁)-other
circumstances connected with the object and conditions of employment of the con-
struction being duly taken into account-the second pair of points, P2 and P₁₂
in which as has been explained above, a sharply defined reproduction takes place
likewise, but with a reversed ratio of enlargement m: n, may be searched for,
because it is advisable to locate the inlet and outlet diaphragms of the system at
those points P2 and P₁₂. They may be determined without difficulty by making
use of the fundamental formula, whereby the position and size of the images,
produced by ordinary lens systems, are determined after having first found by
calculation of the focal lengths of the combination with reference to the principal
planes corresponding to the two diameters.
In calculating as described an anamorphotic system which presents sharp images,
it makes no difference whether reproduction by real or virtual junction of
rays in a predetermined image plane be required. Hence, systems of the
kind here contemplated may be constructed not only for photographic and
other purposes of projection, but also to act in the mode of a magnifying glass or
of a telescope. All these widely different systems, however, will have one con-
structional feature in common: the surfaces of double curvature of the system-
for instance cylindrical surfaces-cannot be confined to one point of the axis only;
on the contrary elements of that class are always to be provided at least at two
different points unless there be among the elements a plane surface arranged
obliquely to the axis. For if-in the absence of such an oblique plane surface-
the anacentrically refracting surfaces of the system were all collected at one and
the same point of the axis, so that jointly they were approximately equivalent to
one single thin lens having doubly curved surfaces, the vertex point of such lens
would be one point of the axis in respect of which--no matter whether there be
spherical surfaces included in the system or not-there would take place homo-
centric junction of rays and identical enlargement in the various directions. This
vertex point therefore would be either the one single point of the second of the
four cases enumerated above, or one of more than two points as in the fourth case,
whilst the third case, upon which the solution of the problem stated is dependent,
would be excluded.
The practical solution of the problem, as above formulated, is feasible by means
of comparatively simple constructions, viz, either centred cylindrical lenses only
or spherical systems combined with cylindrical lenses, or with combinations of
prisms, the principal sections of which contain the axis of the system.

page_text: 7
The simplest case is that of two cylindrical lenses L, and L, (Fig. 1) the axes of
which cross each other at right angles. This arrangement will be found service-
able, when the reproduction is performed by pencils of rays of narrow aperture
and the field of the image is only of moderate size. Such cylindrical lenses whether
incomposite or achromatized, are, in regard to their principal sections, attended
with defects equivalent to those of simple or achromatized spherical lenses. An
anamorphotic system thus formed cannot, therefore, as regards reunion of rays,
be expected to do more than would an incomposite or achromatic spherical lens.
To improve the quality of the image and enhance the useful luminosity of such
a cylindrical combination, one or both of the cylindrical lenses may be replaced,
without any change in the principle of construction, by a set of cylindrical lenses,
either arranged singly or cemented together. And according to the special purpose
for which the combination is intended either positive sets only, or both positive
and negative sets, or else single lenses may be employed. Within the limits of the
just described cylindrical system any one of the known optical combinations may
be realized as, for example, any kind of objectives; eye-pieces, telescopes and micro-
scopes, which are intended to distort the image in a predetermined degree.
To
Where an angle is to be produced by pencils of rays of larger aperture, it will be
advisable to construct the collecting means separate from the distorting means.
this end either systems of cylindrical lenses or combinations of prisms should be
placed in the path of the rays of a carefully corrected spherical object lens, giving
strictly similar image on a predetermined scale, which added combinations by
altering the convergence of the rays in the direction of one diameter will produce
the required distortion.
As regards the first of these different systems: cylindrical lenses in conjunction
with a set of spherical lenses, the cylindrical combination in its simplest form would
consist of one collecting cylindrical lens and one dispersing cylindrical lens, placed
at a definite distance apart. The focal lengths of the cylindrical lenses, in this
case, will be comparatively great, so that the effect that will be produced upon the
quality of the image projected by the spherical lens may be said to be insignificant.
It will of course be understood that as in the example shown in Fig. 2-two such
systems of cylindrical lenses, L₁ L, and L, L4, with their respective axes crossing
each other at right angles, may be used for the purpose of varying the scale of repro-
duction of the spherical system S at the same time in two directions at right angles to
each other, L, L, enlarging this scale in one direction and L, L, diminishing it in the
other so as to enhance the resulting degree of distortion. It will be understood,
that L₁ may be cemented to La and L₂ to L or L, L, L3 and L₁ may all be
separate lenses; neither of which modifications would in the least affect the essen-
tial nature of the invention.
"In respect to the second of the two combinations above mentioned; spherical
lenses combined with a set of prisms, the principal section of which must contain
the axis of the spherical system-it will be observed that in general an achromatic
set of prismis, which in the plane of its principal section produces a divergence of
rays differing from that obtained in the plane at right angles,thereto, would cause
the axis to turn in the plane of the principal section; where, however, as in the
combination represented by Fig. 3. Brewster's straight-sighted set of prisms P' is
employed, the position of the axis is not altered. As such a set of prisms has the
property of transmitting parallel pencils of rays as parallel pencils, whilst making
divergent pencils anacentric, the prisms must be placed at a part of the system
where the rays of light take a strictly parallel, or at least very nearly a parallel,
course. Exact parallelism of rays may be attained by the employment of a
symmetrical double objective in the inner air space of which the rays may always
he made to follow a parallel course for a pre-determined enlargement. The case of
approximately parallel rays will occur where the spherical system as in Fig. 3 is
constructed to produce a considerably enlarged image of the object.

page_text: Below are given the constructional elements of one special form of each of the
novel anamorphotic arrangements exemplified in Fig. 1 to 3 of the drawings, the
letters of reference being the same as those used in the figures. Each system is
shewn in the two principal sections, but only diagrammatically, the distances and
dimensions of the constitutive parts being from want of space-not made pro-
portionate to the numeral values given in the examples below.
The letter S designates a spherical lens system; L indicates a cylindrical system;
a is the distance of the object, that of the image from the lens system, d the
distance between the two lenses and P a combination of prisms. A designates
the object, supposed to be a small disc; B and B are images of A in the two
principal sections of the anamorphotic system.
Distances:
a = 22 mm.
b = 220 mm.
Scale of reproduction:
B
=
20 × A.
B = 10 x A.
Having now particularly described and ascertained the nature of my said in-
vention and in what manner the same is to be performed, I declare that what I
claim is:-
An anamorphotic system of refracting surfaces realizing homocentric reunion
of the rays which proceed from each of two points of the axis and anacentricity of

page_text: the pencils of rays which proceed from all remaining points of the axis, essentially
as and for the purpose set forth.
Dated this 5th day of December 1898.
JOHNSONS & WILLCOX,
47, Lincoln's Inn Fields, London, W.C.,
Agents.