L8 Mapping and realization by
optical instruments
Gauss' (1841) and Abbe's (1870/80) theory of mapping
The mapping of objects is a predominantly
mathematical task. Moreover, in essence, it has always been
treated as such, while viewing its physical aspects as a main
topic, that is, as a physical one to be treated mathematically.
Only Abbe 1840-1905 has changed this
relationship. He treats the task at first
as a purely mathematical one and then, after having solved it, asks whether and to what extent the
mathematical result can be realized
physically - that is,
by optical means.
Gauss - he finished off the initial approach
to the theory of mapping - started from a definite plan of
realization of the mapping. He assumed that there is given a
spherical surface and rays of light meeting it - that is, a
definite means of realization of the mapping, moreover, the fact that
light is refracted - and found further means to attain the
object, and an experimentally
determined law of refraction, obeyed by the rays. All these were items which take
into account experiments and from the start consider how the mapping is to be realized. Gauss
proceeded from a single refracting surface to the treatment of
the task of two faces - a lens, and from there
to systems of centred lenses. He always assumed that points
were lying sufficiently close to the axis of the system. In other
words: He starts from a certain approach by
which the mapping is to be achieved and from definite conditions under which it is to take place; he
then generalizes the conditions progressively and thus comes from
the mapping in a special case to a general theory of mapping.
This mental approach - an
inductive one - proceeds
from special cases to general cases.
However, a law
discovered in this manner can never be complete: You will never
know where lie the borders of the generalisation - also for the
law found by induction. Hence you can treat the theory of mapping
in this manner, but never reach results of real generality.
Results thus found rest always on special assumptions, that is,
they are only valid when these are fulfilled. The relationships
between the image point and the object point, which are thus
discovered, need not, for example, be correct if - in a special
case - it were to turn out that the assumed law of refraction did
not apply or that one is not dealing with spherical surfaces.
Abbe follows a new way. To start with, he
only talks of straight lines and the concept of mapping. He treats the mapping as a purely
geometrical task by postulating its one-one
relationship, in order to be able to apply the older concept
of collinearity. He only assumes that they arise through straight lines - he also calls them rays -, are realized point by point, so that to one point of the object there corresponds only one point of the image, and indeed
thus that to a group of rays through one point of the object
correspond rays all of which pass through the corresponding point
of the image. He assumes no more. - He demands from the mapping
that a given point of the object is represented
by a point of the image, that points,
which lie in the object along a straight
line, also do so in the image, indeed, in the same sequence next to
each other, and that points which lie in the object in a plane, do so in the image as well. Starting
from the demand that to a plane in the object corresponds a plane
in the image, Abbe examines: What mathematical
relations do
then exist between the points of the image and of the
object? He finds
four equations - mapping
equations - formulae
which comprise the essential properties of images, that is,
location and size ratios of the images. (They are valid, without
reference to any special approach to realization of the mapping
and reflect the essence
of mapping. They also
display what demands are to be met by a mapping instrument in the
ideal case; obviously, only those which do not contradict the
mapping equations, since otherwise they would not be compatible
with the nature of optical mapping. ) Hence Abbe proceeds from the general to the particular, that is,
he reasons deductively. Only after he has discovered the general mapping equations and
discussed their content, he attacks the question of realization
of a mapping. In this context (in his proof of the sine
condition), he has presented important results, which earlier
were not known. Gullstrand 1862-1930 1907 has opposed Abbe's assumption of collinearity in that the relationship,
mediated by the wave surface in general (for the context between
object and image space) is not
unique; as a consequence,
this assumption is generally admissible only for the special case
treated by Gauss (of the thread-like space about the
axis).
A space object represented on a
plane (Max von Rohr 1897)
If the mapping is
resolved in Abbe's sense as a mathematical task, that
is, there is assigned a point, a straight line and a plane in the
image to every point, to every straight line, to every plane in
an object , there corresponds eventually to a space object with
height, width and depth a space image with height, width and
depth - An image in
space! -
However, in reality, all images are plane images, irrespectively whether they are images made by man's
hand or by optical instruments like cameras or projectors.
Nevertheless, you acknowledge with every perspectively correct picture apart from height and
width also depth - in other words, in a plane image (it is almost always a plane)
- the space. Hence we ask: Is a mapping of a special object on a
plane possible?
We give the answer
in advance, it is no! A plane image, in which we recognize space, is only a mapping of
another plane image and this other one represents the space only in a projected image - it replaces it for our ability to
recognize to a certain
extent. The fact that
this is possible is related to the structure of our eye and the manner of our using it. The optical instrument does no more than enlarge or
reduce the projected image (the substitute representation of space).
The image in a plane comprises points and dispersion
circles. Only the points are strict images of certain
points of the spatial object; the dispersion circles represent only certain other of its points. The points are images of points in the plane,
conjugate to the image plane - according to von Rohr, who has introduced them during his extension of Abbe's basic theory into the theory of mapping, it is called
the focussing plane, the dispersion circles on the
image plane represent those points of the space object, which do not lie (beyond a certain mapping depth) in front or behind the focussing
plane. But - and this is decisive for the evaluation of an image
by the eye - as long as the dispersion circles do not exceed a certain magnitude, the
eye senses them as points and also as
long (that is, as long as that depth
is not exceeded) the figure consisting of points and
dispersion circles appears
only to consist of
points (appears therefore to be sharp,
although in actual fact it is not) and gives the eye as mapping also the points lying in front of and
at the back of the focussing plane. Only in this sense - and only in this sense - you may talk
of a mapping of a space object into a plane. However, the object actually represented in the image is,
as we will see, a plane projected figure, imagined to arise on
the focussing plane, which consists itself of points and
dispersion circles and represents the space object. Following von Rohr, it is called an object-sided
image.
How do the
dispersion circles arise and what relations exist between them
and the points as the representatives of which they are viewed?
We consider first the eye as the
for us most important and most familiar optical instrument and
present it by a readily understood system. An object is the
centre of a hemi-sphere
of rays. However, only those rays
enter the eye, which dips into it, which pass through the pupil*. The other rays
are screened off by the iris. The pupil selects from the
hemi-sphere a sector, which enters the eye as a bundle (cone) of
rays (if several bundles of rays enter simultaneously from
different points, they have in the pupil a common cross-section).
In general, an arrangement for the limitation of
rays is called a blend. - Also, like the eye, every other optically
mapping instrument evaluates only bounded bundles of rays. Every
lens has only a limited size and is somehow enclosed. Whether it
is an eye or a magnifying glass or a telescope, you can speak
therefore of its blend and pupil.
* More accurately
: Through the opening, to be viewed as
the image of the pupil, which the cornea and aqueous
humor (of the eye) design of it in the aqueous humor.
The pupil is a general concept,
abstracted from the special case of the human eye. However, it
also has its general significance apart from optical instruments;
generally speaking, it specifies for all instruments, considered
to be at rest, the common cross-section of all those ray bundles
(out of the ray sphere) which actually contribute to an image.
The pupil, which in the object
space bounds the ray bundles coming from
the individual object points, effective for the mapping, is called the entry pupil (EP). It is the common cross-section
of all these ray cones. The pupil, which bounds the ray cones
proceeding to the individual image points is called
the exit pupil (AP). It is the common cross-section
of all these ray cones. The positions of the entry and exit
pupils relative to the object and image must be examined
specially for every optical instrument. We discuss particular cases
below.
In order to map on the retina
macula the point O (Fig. 702), all rays of the bundle must converge into a ray cone
with vertex on the retina. The retina macula is the spot
where the point O is mapped as the eye views
it. You can conceive the macula in an immediate
neighbourhood of the eye's axis as a plane perpendicular to it.
If the rays already intersect in front of or only behind
the retina, the rays intersect the retina in a circular spot
of light - a dispersion circle. If we imagine this section of the
retina to be a plane and the to this plane conjugate plane EE,
the points in EE, like O, have their images on
the retina (O'). However, points which like O1
and O2 do not lie on
this plane, do not have their image on the retina, but in front of or behind it (O'1 and O'2
); they are represented on the retina by dispersion circles O'1
and O'2. As the pupil becomes ever
smaller, also the dispersion circles becomes smaller. They reduce
more and more to the point, where the line through the centre of
the pupil and the vertex of the cone - called the principal ray of the bundle - meets the retina.
Due to the narrowing of the ray bundle, the image on the retina, which consists of points and
dispersion circles, thus approaches
as far as image sharpness is concerned
a real mapping - also of the points lying ahead of and behind
the plane EE. However, the blend achieves still more.
The pruning of a cone from the ray hemi-sphere by the blend and
the narrowing of the bundle by reduction of the blend's opening
also determines a direction. Hitherto, we were only able to talk quite generally and
indefinitely of the
relative positions of an
object and its image; now we have definite directions towards the blend and to and away from the image, which is
decisive for the manner in which the eye evaluates
an image on its retina and differentiates between directions
in space. The eye shifts - it projects the cause of the
stimulation, which it senses at O, along the line (principal ray) from O through the pupil's centre forwards
and outwards. Thereby occurs what is called perspective. We are now approaching the answer to the question: How is the mapping - or better still:
How is the mapping of a space object on a plane possible?
Plane-like image of space object
(on ME) and its
object-sided original (on EE)
We will now examine the direction
creating action of the blend and start from Abbe's perfect mapping (without ray limitation). Given anywhere a
space object (Fig. 703), we place through it a (vertical) plane - focussing plane (EE). Its intersection with the book's page is
the line EE, the line perpendicular to it its axis. The
point O lies in the focussing plane, the points O1
and O2 at the back and
in front of it. Somewhere in the image space, there is
a plane conjugate to it. We image that it has been found, that it
is parallel to EE like the frosted glass plate of a
camera, the camera is oriented at O, and focussed so
that O is mapped sharply in O'. We thus find
the plane the intersection of which with the book's page is ME,
which, following von Rohr, is called the frosted
glass plane (ME).
(The photographic chamber is here purely to assist the
reader's imagination; it is not an example!) The points O'1
and O'2 are conjugate to the
image points O1 and O2,
but the points O'1 and O'2
and all image points which
do not lie on ME are not perceptible on ME and
also not indicated by dispersion circles, because rays emanate from O1
and O2 in all directions,
that is, they are ray bundles of an unlimited opening. (Imagine
an object point to be the centre of a sphere and the sphere's
radii its rays). Hence the points lying outside EE
generate on ME infinitely large dispersion circles and for this reason
they are not even indicated on ME. We know from Fig. 681 and the following remarks that, if the dispersion
circles become small
enough, they can take the place of the
associated image points on ME, and that one can reduce their size arbitrarily by placing in front of ME a
narrow blend, which leads to a correspondingly small pupil. We do
this now and place it parallel to ME, centred on the
axis. We then obtain (like in Fig. 702 on the retina plane)
on ME a drawing of actual points and sufficiently small
dispersion circles (as images of the points lying in front of and
behind EE). Fig.
703 continues Fig. 702.
We have now projected on to ME a
figure consisting of points and sufficiently small dispersion
circles, which represent in a certain sense the points on EE
and to a certain depth in front and behind. However, it is not a mapping.Hence
we look now for the actual object,
mapped in the projected image on ME. For this purpose,
we interpret now ME as an object the EE,
conjugated to ME, as image and pursue, starting from the
figure in ME, in the opposite direction of the light
through the pupil the rays towards EE, in order to
discover the structure, conjugate to the figure in ME.
The start with, we realize: The pupil in the
image space (exit-pupil) is conjugate to such a figure in the
object space (entrance pupil). It is related to the one in front
of ME like an image to an object: The centres of both
pupils are conjugated as well as their rims. However, also the
rays from the centre of the object space pupil to O and O1
and O2 are conjugate to the rays
from the centre of the image space pupil to O' and O'1
and O'2. If these rays in Fig. 703 are sufficiently extended, you obtain on EE a
drawing consisting of points and dispersion circles, which is
conjugate to that on ME and completely the same apart
from its scale. Thus this
projected image on EE
is the actual (by
the image represented) object. Von Rohr calls it
the the object-sided
image and the conjugate
projected figure on ME the mapping-copy or mapping-image. The object-sided mapping represents with a view to
sharpness the space-embossment with an approximation to the, in Abbe's sense, perfect mapping which depends on the size of the dispersion
circles. The work of
optical instruments is the enlargement or reduction of the
object-sided image. It must
be noted with regard to the reproduction of the shape that the
dispersion circle, which we can conceive to be represented by its
centre, is expelled from the focussing plane by the (if
necessary, backwards extended) line linking the centre P of
the pupil to the object point. However, such a representation is
called a central projection or perspective. Thus, we can say: If you fix the
entrance-pupil of an optical instrument in front of an object,
that is, its distance and the direction of the axis of the
instrument, you determine thereby the perspective of the space
object.
We now know: The plane-like image of the space object is only a true
copy of another plane-like image comprising points and dispersion
circles (more accurately: points and circular spots) which represents the space embossment. How does it occur that the eye recognizes space
in the plane-like image? This is connected with the structure of the eye and the
manner in which it evaluates the image on the retina. An eye is
itself a mapping instrument; there forms on its retina a plane
image of the object which it views. With the aid of this image,
it sees the object. Note that is does not see
the image on its retina, it does not
know anything about it, it sees aided
by it. It also perceives with the aid of the plane-like image the space, in fact, as
follows: The image on the retina consists of points and dispersion
circles; as long as these circles do not exceed a certain size,
the eye perceives also them as points.
Perspective in unimpeded views
The stimulus sensed by the
retina at a point hit by a ray (or at the spot which it allows to
represent a point) is transposed by the eye to the principal ray
through the corresponding point of the retina (respectively, the
centre of the dispersion circle) and the centre of the pupil
forwards and outwards. Thus it projects
the retina image point by point like on to a plane and constructs point by point the object-sided mapping
- the copy of the retina image - and from it the object - the
space embossment - which is represented by the object-sided
mapping. Fig. 704 shows
how it executes this construction. A plane-like image of the cube
W, viewed by the eye a, arises on the retina.
The retina corresponds to the ground glass plane MM (Fig. 703) and the drawing l to the
projected figure on the focussing plane, which represents the
space embossment - the cube B. By the eye projecting
outwards the image on the retina - the mapping image of l
- it gains the perspective impression l of the
cube B. If the eye were to be located higher up or lower
down or further in front or in the back than in Fig. 704, the lines between the eye and the
corners of the cube or to the points in between the corners would
intersect the focussing plane at different angles, another
projected figure would
arise and also another retina image. We see: The
eye's position relative to objects is decisive for the
corresponding retina image and the associated projected image on
the focussing plane. Hence it is also decisive at what angle the
eye views the perspective drawing, corresponding
to the projected figure, in order to recognize
the space embossment, represented by the actual projected
image. Everyone knows how differently an object can appear,
depending on from where
it is viewed - more
accurately: How the pupil is located with respect to the object.
Only the delimitation
of the rays by
the blend and the pupil introduces preferred directions and enables representation of a space embossment on a plane. Only thereby arises
perspective perception by the eye and recognition of space on a
plane image.
In the case of the eye, two
basically different cases can occur. Depending on whether the eye
rests in its cavity or moves (stares or looks around), the direction which it perceives and
its manner of evaluation of different directions change. If is is
at rest (Fig. 705
below), it can only perceive directions which intersect at the centre of the pupil. If it looks around - turning in its
cavity about the eye's
pivotal point (about
13 mm behind the cornea's vortex) - it perceives within a certain
field of view all directions, which intersect in the eye's
pivotal point (Fig. 705
above). Hence for the eye at
rest, the centre of
the pupil (following Abbe) is the centre of perspective, for the looking around eye (following Christoph
Scheiner 1619 and Johannes Peter Müller 1801-1858) it is the eye's pivotal point.
In order to fix a point, you turn the eye ball so that
the image of the point arises at a certain point of the retina
(matula). And the eye sees
it in the direction
(actually: It sees it in the direction), which leads
outwards from the retina image through the centre of the pupil.
The eye at rest sees only the
fixed point sharp, everything else vaguely (in an indirect view): The moving eye
is directed in jerks at the different
points and sees then the fixed point sharply. The line through
the eye's point of rotation and the fixed point is called the view line. If it coincides with the axis of the
eye - an assumption which we will expect to be met -all lines of
view intersect in the eye's turning point, whence the eye's
turning point serves the surveying eye as centre of projection.
By bringing the eye to a
certain location relative to an object, a certain focussing plane is fixed. It is conjugate to the retina by the
position of the eye fixed in space; it takes here the place of
the frosted glass pane. However, this action also fixes the projection centre for the figure on
the focussing plane.
When you have understood all of
the preceding statements, you will also understand (Fig. 704) that you must look at every image, which has to be conceived as a projected one, from a definite point, in order to receive the same
impression of the object. The angles, at which the lines from the
eye to individual points of an object intersect the focussing
plane, differ when you raise or lower or shift you eye sidewards
relative to this point, that is, the projected images change for
every position of your eye. However, Fig. 704 tells you at the same time that, if you only displace
your eye in such a manner that the angles remain more or less the same, nothing can change in the appearance
of the object. This explains that the projected figure in single
eye view on the focussing plane can represent for you to a
certain degree the space embossment. It is a fact that you have
fixed the focussing plane
not only when your eye is merely
directed at an object,
but also when it is supplied with an optical instrument. In that
case, the centre of the entry pupil, corresponding to the
focussing plane, then lies somewhere in the optical instrument
and, indeed, it can be fixed by its construction (as in a
telescope and microscope) or only be determined by bringing your
eye's pupil into the instrument's image space (opera binoculars
or spectacles). (The object space pupil itself enters the eye and
is separated from the image space pupil by just a few
millimetres.)
Verant (von Rohr)
If the perspective of a distant
object is recorded photographically, its centre lies away from the image plane by the objective's focal
width, that is, in modern instruments only by 10 - 15 cm. Normal eyes cannot accommodate to such vicinity, whence images do not make the same impression on them
as the objects from the location of taking the photo.
However, if you look at the images through von Rohr's Verant lens
with the focal width of the objective of the camera, you see an image lying far away, which is completely similar to the objective side image and which yields
the direct view with the same clues for sensation of depth like
the object, when it is viewed from the location of taking the
image. The Verant lens (Fig. 706)is a magnifying glass of a special design with two
lenses; it was produced by the Zeiss works in
Jena following calculations by Max van Rohr at a
recommendation by Gullstrand 1862-1930 1903.
Peep-hole
perspective
In the formation of the
perspective in Fig. 704, the
eye is not restricted from looking about. However, perspective
occurs quite differently when the eye can only view a space (in
three dimensions) through a peep-hole. You then place your eye against the
opening and move your head to and fro, but, in order to have an
as large as possible field of view, you direct the eye so that
the line of viewing passes through the centre of the opening.
Then rotations of your eye in its cavity are coupled with the
rotations of your head, whence the resulting rotation of your eye
occurs about a point outside your eye's point of rotation. Your
view through a peep-hole is decided by the opening which fixes a perspective and your moving eye records
it. The centre of the opening becomes centre of the peep-hole
perspective, whence it lies outside
your eye.
Next
we discuss the
peep-hole perspective, which is most important for us! Your eye
looks through an auxiliary optical apperture just as through a peep-hole. For example, in the cases of an
astronomical telescope and a microscope, the peep-hole - exit
pupil - floats as a small bright circular disk close ahead of the
eyepiece frame in mid-air. You move the eye in front of the exit
pupil as in front of a peep-hole - von Rohr says key hole - to and fro and couple the rotations about its point of
rotation with
these displacements. Your eye cannot change
anything in the ray bundles reaching it through an astronomical
telescope or a microscope. It is different in the cases of
spectacles and opera glasses (Dutch telescopes) - the paths of
their rays are only set by the point of rotation of your
supported eye. For example, as you bring the point of rotation of
your eye into the image space of the Dutch telescope, you make
the lines of view into principal rays of the instrument and determine the path of its rays. Hence,
while Fig.
738 is correct for the
eye at rest, it is not so for the moving eye. It must be
augmented by Fig. 707. As
you look to the right (left) hand side, the head is displaced to
the right (left). Both movements (displacement of your head and
rotation of your eye) combine into a rotation about a point
between the exit pupil of the instrument and that of your eye. We
are also here dealing with an observation through a key hole
(using von Rohr's terminology).
In the case of a spectacle
lens, the field of view is bounded by the frame. The lines of
view of the moving eye become image-sided principal rays in the
spectacle lens, they intersect in the object space at a point P,
which is conjugate with the eye's point of rotation P'.
The position of the point of rotation of the eye with respect to
the lens determines the errors of oblique bundles which occur
during its usage. - The treatment of optical instruments in
connection with the moved eye is an extension of Abbe's theory. Its first suggestion for spectacles is due to
Fr. Ostwalt 1898, but then more extensively for the
Dutch telescope especially to Gullstrand 1902.
Physical mapping (Physical
image)
How are mappings created? In order to produce of a bright point P
(Fig. 708), the image of which you have found geometrically
at P', an actual image - that is again a bright point
- it is obviously insufficient that (straight) lines come from P and eventually
intersect at P'. This is not a physical
process. (Whether the image, arising physically, agrees completely with the geometrically constructed one or not is decided by the
ratio of the image forming wave length to the sizes of the
objects to be mapped.) According to the wave theory, the physical
process occurs as follows: The glowing point P emits waves of light - spherical waves WW - like A
in Fig.
307 emits
sound waves. As they spread, they encounter an arrangement (in Fig. 708 the lens L, in Fig. 307 a concave mirror), which cut from the
spherical wave a cone - a bundle of rays - and which transform
the cut out piece of spherical wave into a new spherical wave W'W'
so that it forms a new
excitation centre P'
(as in Fig.
307 the new excitation
centre B). This new centre P' is the image. (We must anticipate here and
refer to the refraction
of light). As a whole, the spherical wave emanating from P
should yield a sharp image point. However, the lens
cuts off a part of it and the sector of the spherical wave causes
near the geometrically constructed image point a distribution of light which, for example, generates
as image of the point P on the axis instead of the sharp
point P' a small bright circular spot, perpendicular to PP',
surrounded by bright and dark rings - a refraction diskette.
Instead of a point of light, you obtain a spot of light.
A circular lens opening yields a concentric diskette about P',
in which the brightness drops towards the rim to zero, with
several concentric rings, in which lies the brightness in the
same direction between zero and an outwards rapidly decreasing
maximum. Fig. 709
displays the brightness distribution at different locations of a
refraction diskette. The abscissae are the distances from the
centre of the diskette, the ordinates the brightness. Among the
bright rings, the first has still 1/60 the brightness at the
centre, the subsequent ones are much darker. Hence you can view
the image of a glowing point to be the central image alone. (Images of objects do not comprise therefore
points, but diskettes. This is in full agreement with the basic
theorem of physical optics: A finite amount
of light unified in a mathematical point
- whether a point of light or an image point - is
inconceivable.) We measure the size of the refraction diskette by
the size of the angle of the first dark ring; in the case of
small diskettes, which are here under consideration, there arises
on the retina of your eye for a 5 mm pupil a diskette which is
approximately equal to a cross-section of a plug (Fig. 725, Fig. 726).
Coherence of waves of light
The possibility of generating a new concussion centre P' -
an image which (within a certain angular space) behaves like
a glowing point - rests on the fact that all the waves of light,
which meet in P', arrive there with the same phase. In order for them to be able do so and then
according to the wave theory support each other, they must emanate from P with the same phase (you say: They must be coherent). This physical
correlation, referred to as coherence (being of equal phase) expresses itself geometrically
as follows: All points, which are equally far away from
the centre of their generation, have at a given time the same
state of vibration. Indeed, the term equally far away relates
to the optical length of the distance. (Optical length is
understood to be the geometric
length, covered in the substance
concerned by the light, multiplied by its refraction index. The
optical length in Fig. 708 from P via a and a'
to P' is: (Pa x nair) + (aa'
x nglass) + (a'P x nair).)
Thus, all these points lie also during the refraction
of the spherical wave through the lens on a spherical
surface - the envelope of the spherical wave of the elementary
waves as in Fig. 301* above and also thus during the transformation of the
spherical wave WW into WW' and arrive with the
same phase at P'. The delimitation of the
spherical wave (about P) by the lens leaves in the image
space only the section W'W'; the action of this segment
on the point P' alone (according to Huygens'
principle) yields summation of the action of all elementary
waves. However, in the immediate neighbourhood of P',
these actions partially destroy each other, whence there
arises the distribution of brightness shown in Fig. 709. Thus the coherence persists in spite of refraction or
reflection; indeed, you can decompose a system of coherent waves
by reflection or refraction into two equal systems which also are coherent.
This discussion implies something which, while always
being fulfilled, should be especially referred to - the content
of the theorem named after Malus: If a bundle of rays is perpendicular to a plane and
hence also to all parallel planes, it retains this property also
after arbitrarily many refractions and reflections; indeed, the
path of the light between two such planes is the same for all
rays. Rays, emanating from a point, are perpendicular to all spherical
surfaces about this point, whence Malus' theorem
yields a property which they retain during all refractions and
reflections in spherical surfaces. The path of the light is the sum of
the products of the refraction ratio n of a substance
and the distance l covered in it. For the path of the
light - the optical length - holds the theorem of Fermat: If a ray
of light reaches B after an arbitrary number of
reflections and refractions from A, the sum of the
products nl is a limiting value, that is, it deviates
from the same sum for all infinitely neighbouring paths at most
by infinitely small second order terms. (The proofs of the
theorems of Fermat and Malus lie beyond the tasks of this book.)
If in Fig. 708 the waves emanating
from P are not coherent, they do not generate an image
point P, at which all their effects sum up, rather their action of light at P'
is distributed over a certain space. The waves
emanating from P are coherent whenever
P is a point-like independent source of light, say a
point-like spark. They are not coherent when the glowing point arises by
escaping through a point-like opening from a source of light, extended
in space and this opening acts as source of light. (The
individual points of the source of light, extended into space, are independent of each other and everyone of them
emits waves, which are not coherent with the waves from another
point of the source of light.) The image of such a glowing point
- of one which is not glowing on its own - arises in a quite
different manner as the image of a self-glowing point. For
example, this happens in a microscope: You observe, as a rule,
objects in light which falls on them. A source of light, extended
in space, radiates through the object. The irradiated object
generates a dispersion figure; the microscopes objective maps it in the rear
focussing plane of the objective and there arises in the plane,
conjugate to the objective plane (with respect to the object) as secondary
mapping, under certain conditions according to Huygens'
principle, an image of the object. (The opaque objects, which become
visible by means of diffusely reflected light, can be considered more or less to be glowing on their own.)
Mapping
errors
You use lenses to realize mappings. However,
neither objects nor ray bundles meet the necessary conditions for
mapping a point into a point by means of lenses; this demands
infinitely narrow bundles and a point on the optical axis. We can
map a plane only then into a plane when it is infinitely small.
However, infinitely narrow ray bundles are useless, because they
yield infinitely weak images -apart from the fact that diffraction makes them useless
- and mapping of infinitely small planes does not serve any
purpose. Hence those conditions limit mappimgs too much. However,
their boundaries can be extended.
Moreover: You can attain by linkage of several
optical systems (in microscopes, telescopes, etc. ) what a single
unit cannot achieve. Thus, instruments have been created, which
map very sharply objects of finite extent by bundles of finite
width. However, there arise in the process certain defects, which
Abbe
has called mapping
errors after comparing executable results with ideal ones. In essence, there are five types of errors:
1. In general, rays emitted by a point of an
object on the axis do not intersect on exit from the instrument
at the same point; the distance of the point of intersection of
an emerging ray from a point with the axis is called spherical deviation;
2. In general, the image of a plane element around a
point of the axis lies on two
curved surfaces, that is, two image
points correspond to everyone of the object; this phenomenon is
called astigmatism.
3. Even when you succeed in making image points
coincide, the surface of the image need not be a plane; this is
called image curvature.
4. If an image is plane, it can still be not similar;
this is called distortion.
5. The errors concerning the point on the axis can be
strengthened by neighbouring points, and indeed asymmetrically;
this called koma.
Spherical aberration (sphere
deviation)
The spherical aberration in
lenses arises just as in concave mirrors. When the mono-chromatic
ray bundle in Fig. 710 meets
the lens, not all of the spherical shape of the lens refracted
rays lead to the same point on the axis, but those closer to the axis cc aim towards C,
those closer to the rim rr towards R. This
deviation of the points of intersection from each other is called
the deviation due to the spherical shape of the lens - spherical aberration (better also monochromatic). The rays incident between the rim and
the axis lead to points between R and C. Hence
there cannot arise a point-like image on the axis. While a plane QQ,
perpendicular to the axis, is met at C by the vertex of
a cone, C is surrounded by a circular disk, intersected
by the rays, the meeting points of which lie between R and
C and which from these points spread towards QQ.
Hence there arises around C a dispersion circle, filled with light. As consequence of
spherical aberration, there arises a burning area (as in the case
of spherical mirrors.
Image curvature. Abbe's sine condition (1853)
The removal of the spherical deviation on the axis characterizes the strict mapping of a point O
on the axis into another O'. Satisfaction of Abbe's sine
condition characterizes the strict mapping by means of bundles
with finite opening (Fig.
711) of an area element dq,
perpendicular at O, into a similar such element dq by
bundles, the openings of which are limited by a circular blend in
such a manner that the incident ray forms with the axis angles u
U, the exiting
rays angles u'
U'. This condition (of which there exists an elementary
derivation by Hoskin) is: sinu/sinu' = n'/n·b, that is:
The ratio of the sines of the conjugate angles must be constant,
and indeed equal to the product of the quotient of the refractive
indices and the lateral
enlargement b. The enlargement dq'/dq is important: In order
that the mapping system will map a plane area
element into a plane area element (you say: It is aplanatic), it must enlarge everywhere equally
strongly. The sine
theorem is the condition for the inditity of the enlargement by
the different sections of the mapping lens. The immense practical significance of the
fulfilment of this condition is indicated by the fact that all
useful microscope objects, which Abbe has tested, were empirically corrected for satisfaction of the sine
condition before the condition was at all identified.
Astigmatism of oblique bundles
Aberration only occurs for
bundles with finite opening, astigmatism also for infinitely
small bundles. A point, which is at a finite distance from the
axis like Qw in Fig. 712 emits an oblique bundle
to the lens; in general, it is not refracted towards a single point. This is readily understood by
taking into consideration that rays are normals to curved
surfaces (wave surfaces) and the normal at a surface point O is,
in general, not intersected by the normals of neighbouring points
at the same point, strictly speaking, we should now
ask: How do the normals to Qw's
wave surface, exiting from the lens, join up? You will expect an image point and look
for it to start with on Qw's
principal ray, which intersects the centre P of EP at
the angle w. However, the question: How do the neighbouring normals
join the principal normal at the surface point O? is answered differently by the theory of
surfaces. Imagine that the points A, B, B', A, A'
in Fig.
713 belong to a curved
surface element. Then the answer relating to this area element
is: In general, the neighbouring normal
passes the principal normal O obliquely, but there exist two pairs of exceptional normals (in A, A'
and in B, B'),
which really intersect the principal normal (O) at f1
and f2. Each of these pairs lie in a plane,
which also contains the principal normal (A'f1A
and B'f2B) - a principal cut - and these two principal cuts are
always perpendicular to each other. The answer to the present
optical question " How
do in Fig.
712 the normals of the
wave surface, belonging to Ow and
exiting from the lens, join?" is according to the theory of surfaces: In general, that is, during oblique ray incidence,
two image points Ow1 and
Ow2 belong to one object point.
By suitable shaping of the lenses, you can make them coincide at
the one point Ow1, that is, remove
the astigmatism of oblique bundles. Hence, in general, there does not exist a common
junction, whence there arises the term astigmatism (a =
not, stigma = point). If you place through a
refracted astigmatic ray bundle planes perpendicular to the
principal ray, there arise, depending on the location of the cut
and the delimitation of the bundle, on the refracting surface
figures of different shapes. Fig. 713 shows the forms which an initially
circularly bounded bundle assumes one after the other. You can
consider as image of a point the two focal lines C0C'0
and CC' - two short straight
lines, perpendicular to the axis of the bundle and mutually
perpendicular - they form a cross. However, they lie at a certain
distance from each other.
Delimitation of the opening of
ray bundles (Abbe)
While you can suppress spherical aberration and
astigmatism by means of several lenses, so that they hardly
influence the sharpness of the image even when the the bundle is
relatively wide and the object considerably extended., you must
nevertheless limit the opening of the bundle and the size of the
object to a certain extent. This happens through blends, opaque slides mostly with a circular opening, which
are fitted into the optical system so that the axis passes
perpendicularly through the centre of the opening. In Fig. 714, the blend BB delimits the opening of the
bundle emanating from P.
Blends serve different purposes. Already their
mere presence determines directions
in the image as well as in the object space. However, they
also determine by their position and size the openings of the
mapping bundles, the inclination of the principal rays to the
optical axis and the location where the refracting planes are met
by the mapping rays. And, in turn, they determine certain
properties of optical instruments like the content and the
visibility of the image, the correctness of the drawing of the
image, the enlargement and the intensity of the light of the
instrument. Only the ray path, determined by blends, informs
completely regarding the manner of effectiveness of an optical
instrument. The theory of ray delimitation is therefore a part of
the foundation of practical optics. Abbe has first developed it in
great generality and employed it technically. (The first
comprehensive presentation is due to Siegfrid Sczapski 1861-1907 1893).
You can compare a blend with a circular window
frame. We are concerned with the opening, not with the frame!
(The rays which encounter the frame, are retained.) You make it
into a component of the optical system and locate it, depending
on its purpose, in between the lenses or in front or at the back
of them. Wherever a blend is located in the object space, it
becomes itself an
object. whence the system must also
generate an image of it. This is very important, because also at
the blend and its image the object points are conjugated to the
image points. For example, the rays which have generated the image of
the opening's rim have passed through the border of the
opening; the rays, which map the opening of the
blend, through its opening. Other rays cannot reach the image of
the opening of the blend. In other words, the image of
a blend, generated by the system, itself acts as a blend
in the image space. Fig. 714 shows: With the image
C, which the lens L generates of the blend B, it generates itself yet one
blend. The blend's image acts like a real blend: Only such rays can
reach Q, which have passed through the opening of the
blend's image B, at the very best the rim's rays. The
image of the blend tells us about the opening of the emerging
ray bundle as the blend B tells us about the
opening of the incident bundle.
If we remove the blend BB in the
object space and place at the location of the blend's image
CC a real blend, conjugate with this image, the lens
generates in the returning direction of the light of this blend
an image at that location in the object space and in that size,
which previously the blend had occupied*. Then LL receives
a bundle which is opened as far as the lens. However, at the
best, again only the same rays reach Q as before. A
blend and its image have been represented in turn by an actual
blend, otherwise nothing has changed.
*In this case, in which
both have been assumed to be real, you would even be able to
catch them in the returning direction of the light.
Hence, whether you delimit the incident
or the exiting rays by a blend, the action is the same; by the
delimitation of the incident bundle also that of the refracted
one is given and conversely. For example, if you employ a
biconvex lens as a magnifying glass (Fig. 715), you hold it close
to your eye (II iris, pp pupil) and the object ab
close to the lens. The bundles, which fall, for example, from the
object point b on to the lens (dotted in Fig. 715), are opened as far as the lens admits. The entire
surface of the lens, facing the object, receives rays and through
the entire surface, facing your eye, rays emanate. However, only
those emerging rays are active
for you vision which encounter your
pupil pp. The figure only shows those which yet
encounter the rim pp of the pupil. You can identify
which of the rays from the lens become effective, that is,
contribute to the image, if you construct for ab
the image a'b'; the image is
virtual. upright and enlarged, because ab is at a
distance less than the focal length from the lens. Every object
point, for example b, sends an infinity of rays to the
lens, whence in the image points b', conjugate to them,
also an infinite number of them intersect. However, only those
become effective, which pass through the pupil pp, that
is, only the small cones of rays, the base of which is the
pupil's area. And what applies to b', also applies to
all other image points; thus the refracted ray bundles, which contribute to the image, are thereby characterized. They are cones
which have their vertices in individual image points and the
eye's pupil as a common cross-section, that is, as exit pupil.(In
the figure, the effective bundles are not specially marked.)
We consider next the roles of the image
of the iris and the eye's pupil. The eye's pupil collects
indirectly the rays as they leave the lens, however, directly
already those coming to it. For, indeed, it acts as if all of the
rays reaching the lens have been screened off, which would have
met behind the lens the eye beyond the pupil's rim. You survey the rays which,
for example, coming from a to the lens actually
contribute to the image by construction (Fig. 716) of the image I'I' to II. (The iris II
and the pupil is for the lens L an object; its
image is upright, virtual and enlarged, because it lies closer to
the lens than the focal point. An observer looking through the
lens into the eye would see the pupil enlarged.) The rim images p'p'
are conjugate to the rim points pp and, consequently, so are the rays escaping
from the lens, which, for example, belong to a' and pass
through the rim points pp, conjugate to those, which
come from the object point a conjugate to a' and
aim for the rim point images p'p'. Hence the pupil is
met after refraction by only
those rays, which prior to refraction aim
at the interior of the pupil image; in the extreme case, it is
its rim, for example, among the rays which come from a
to the lens only the small ray cone the base of which is I'I'.
All other rays meet the eye after refraction outside the pupil
and what holds for A applies to all object points. This pupil image
is the common cross-section of all incident ray bundles,
which contribute to the image, that is the entry pupil. In Fig. 715, they are the unhatched bundles from b'
and a'.
Field of vision and field of
vision blend
In order to survey the path of
the rays through a refracting system, you follow the rays which
pass through the centre of the pupil. Abbe calls them principal
rays (the
strong lines in Fig. 717) and
their path through the system the ray path.
The angle between the rays from the axis point a (Fig. 716) to two diametrically opposite points p'
p' on the rim of EP is called the aperture angle and the blend, which delimits
the opening of the mapping bundle, the aperture blend. - In order to obtain a useful image,
you must also delimit the size of the object. This is also
achieved by blends. However, for the time being, we will assume
that the object is so small that it does not demand special
blending. Only for an explanation of the concept of field of vision, we will return to Fig. 715 and show what determines the size of
the object to be mapped.
In order to become effective, the rays from an object point must aim
at EP; but they must also really meet the lens. The size of the lens is limited.
Object points, which lie with respect to the lens so that the
rays from them to EP bypass
the lens, can also not be mapped. Points which are placed so that
only a part of the rays aiming from them
at EP encounters the lens, are still mapped, but less brightly than those image points which
are reached by all the rays from the corresponding object
points to EP. These brightness conditions determine the size of the useful, that is,
sufficiently bright part of the image and thereby the field of vision. Its size depends therefore on its
definition with respect
to the brightness governing it.
The field of vision of a
magnifying glass, employed with an eye at rest, is typical for it
and so easily visible that it will be used for an explanation.
The AP of the magnifying glass (Fig. 717) is identical to the observer's pupil.
The rays to the pupil (Fig. 718 below) are those which come from the image ab through the magnifying glass (they
correspond to the rays of the unshaded
bundle in Fig. 715
above, which enter pp). The eye views through the lens'
frame S like through a circular hatch. We call the
radius of the frame p, that of the pupil p, the distance of the pupil from the
lens p, when the following applies:
1. The
straight lines, which (without intercepting each other between
the pupil and the rim of the magnifying glass) link the edge of the pupil and the edge of the lens and thus form the angle 2w
[tan w = (p - p)/d], delimit in the image the segment ab.
(In order to survey the conditions in space,
you should image that Fig. 718 below has been rotated once about the line of view: The
line ab then yields a circular plane and the angle 2w
becomes the angle of the cone.) The pupil receives from each point of the interval ab,
respectively, of the circle with the diameter ab, a ray
cone, the base of which
is completely filled -
that is, the opening of which equals the entire pupil. The
central part ab of the object ab appears therefore to be equally bright and brightest. - The iris of the eye bounds
the opening of the bundles, which enter the pupil from each
points of ab: The Iris is the aperture blend.
2. The straight lines which come from the centre
of the pupil through the rim of the lens and form the angle 2W
(tan W = p/d), delimit in the
image on both sides of ab the segments aA and bB
(in the focussing plane in space a circular ring about the circle
ab): The pupil receives from each point of this region a
ray cone the opening of which in the most favourable case (the points a and b) equals that of the pupil, in the most unfavourable case (points A and B)
that of half the pupil - hence the brightness of the image
decreases steadily from a to A and from b to B;
A and B are only half as bright as a
and b.
3. The lines, which link the rims of the pupil
and lens and thereby form the angle w [tan w =
(p + p)/d], delimit in the image on
both sides of A and B the segments Aa and Bb (in the space angle a circular ring
about the previous one): The pupil receives from each point of
this region a ray cone, the opening of which in the most
favourable case (points A and B) equals half its opening; in the most unfavourable
case (points a
and b) it receives a single ray - the brightness decreases further from A to w and
from B to b to
complete darkness.
We
define as field of
vision angle the space angle between the principal rays, that is,
between the rays through the pupil's centre. Then A and B
in Fig. 718 are on the edge of the field of vision and the circle
on which they lie is the bound of all image points, which are at
least half as bright as the central section (ab). This
definition determines the field of vision by tan W = p/d,
that is, by the diameter of the lens' opening, whence the
lens' frame is the
field of view blend. However, it is so
only if, as in Fig.
718, the lens' opening is larger
than the eye's pupil. If it is smaller (Fig. 719), it becomes the opening blend and the iris becomes the
blend of the field of vision. Also in this case, you find by
linking by straight lines the pupil's edge, respectively its
centre, to the lens' edge the three angles w, W and
w and the three differently bright zones. However, as the
figure shows, no bundle coming from an image point can be opened
further than the lens'
frame, whence it is now also
the opening blend. The field of vision angle W is now
determined by tan W=p/d, that is, by the diameter of the pupil, that
is, the iris is now the
blend of the field of vision. Every
time, the larger is the
field of vision, the smaller
is d, that is, the closer
the magnifying lens is to the eye.
Effective blend of an optical
system
In the case of a single lens and a single blend, only one blend image
exists, whence the location and size of the entrance pupil and
exit pupil and the blend of the field of vision are unique. However, what happens when there are
several lenses and blends, that is, also several blend images? Given the optical system of Fig. 720, specified by S' and S"
(every lens imagined to be the representative of a part of the system S, that
is, of several lenses and blends) and blends in between the
lenses of the system S in any order as happens, for
example, in photographic double objectives. B1B2
is such a system. S' represents that part of the optical
system S, which precedes
the blend on the side
of the object, S" that part which follows it on the
side of the image.
Example: The optical apparatus
of the eye (Fig. 724). In
the complete system S, the iris J with the
pupil opening is the sole blend; on the object side, it is
preceded by the system of the aqueous humor A - cornea C
(as part of S'), on the image
side, it is followed by the system of the
crystalline lens L - vitreous humor Q (as part of S"). The
image, which its aqueous humor and cornea generate of it towards
the object side is the entrance pupil of the eye, the image of
the crystalline lens and vitreous humor of it towards the image
side (in the vitreous humor) the exit pupil of the eye. Strictly
speaking, we should say entry
pupil of the
eye, whereever we just speak of the pupil.
How do you find that blend, which makes the decision regarding the
opening of the mapping bundle - the effective blend? Answer:
Imagine every present blend mapped through its S'
towards the object side, that is, towards P1P2.
Then the opening of the mapping bundle is determined by that
blend, the image P1P2 of
which appears from the central object
point O at the smallest vision angle. The angle (2u)
is called the opening
angle of the system
and the base of the ray cone (with O at its vertex),
belonging to the space angle 2u, is called the inlet pupil. All rays in the object space
which aim at it can then pass through all other blend images on
the object side. The image P'1P'2,
generated by S' towards the image side,
then appears seen ahead of the central image point as well at a
smaller vision angle as every other one - it is the exit pupil of the system. All rays from the inlet
pupil into the exit pupil can pass through the blend images on
the image side.
In order to obtain a useful
image, the object must also be delimited. (Hitherto, in view of
our assumption , there was no need for special
blending.) In order to discover the blend which determined it in
the system, you proceed as has just been described. The blend
opening, the image of which - the inlet hatch - appears, seen from the centre of the inlet pupil P at the smallest angle, is
here decisive. This vision angle (= 2w) is called the
system's vision field
angle, the blend the vision
field blend; the latter delimits the external principal
rays. The image of the same blend opening, projected through its
successive part of the system towards the image space, then
appears from the centre of the exit pupil P likewise at
a smaller angle than all others. - Following von Rohr, the object-sided image of the field of view blend is
called the entry hatch (EL), the image-sided
one, the exit hatch (AL), because they
have for the optical instrument the same role as hatches for an
eye, looking from a room outside. If the entrance hatch coincides
with the object and the exit hatch is in the plane of the image,
the image is sharply delimited. However, if the object lies at a
certain distance from the entrance hatch, as in Fig. 718 above, where the rim of the magnifying glass becomes the entrance
hatch, then the image is brightest at its centre and uniformly
bright, but becomes gradually less bright towards the rim.
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