both flints facing the eye is occasionally used. See Fig. 14.7 for an
example of a symmetrical eyepiece.
The Erfle eyepiece.
This eyepiece (Fig. 13.3c) is probably the most
widely used wide-field (±30°) eyepiece. The eye relief is long (0.8F), but
working distance is quite short. The Petzval sum is about 40 percent
less than the orthoscopic or symmetrical types because of the field-
flattening effect of the concave field lens surface, and distortion is
about the same as the orthoscopic (for the same angular field). The
type shown in Fig. 13.3c usually has undercorrected lateral color (for
use with erecting prisms) which can be reduced by use of an achro-
matic center lens as in Fig. 13.3d. Glasses used are usually dense bar-
ium crown and extra dense flint. An example of an Erfle eyepiece is
shown in Fig. 14.8.
Magnifiers and viewer lenses are basically the same as
eyepieces, with one notable exception: There is no fixed exit pupil. This
means that the eye is free to take almost any position in space and
therefore the aberrations of the magnifier must be insensitive to pupil
shift. For this reason, magnifiers tend to be symmetrical in configura-
tion. Two plano-convex lenses with convex surfaces facing or a sym-
metrical (Plössl) construction are common for better-grade magnifiers.
Where cost is important and a single element must be used, the fol-
lowing arrangements are good. If the eye is always close to the magni-
fier, use a plano-convex form with the plano surface toward the eye. If
the eye is always far from the magnifier, use a plano-convex form with
the convex surface toward the eye. If the eye position is variable, as in
a general-purpose magnifier, an equiconvex form is probably the best
compromise. Figure 14.5 is an example of a doublet magnifier.
Note that the eyepieces of instruments which use an electronic
image tube, such as the Sniperscope, fall into the category of magni-
fiers, since they are used to view a diffuse image on the phosphor sur-
face of the image tube. As such they must be designed so that they
perform well with the eye in a wide range of locations.
The optics of tabletop slide viewers, “head-up displays,” or HUDs,
and many simulators not only fall into this category but also share the
requirement that both eyes view the image through a single optical
system. Such systems are called biocular (as opposed to binocular sys-
tems, in which both eyes are used, but in which each eye views the
image through a separate optical train). In a biocular system one must
not only be concerned about the effects of eye motion but must also be
concerned about any disparity between the images as seen by the two
eyes. The convergence, divergence, and dipvergence (the vertical dif-
ference in direction) required of the eyes as they view the image must
be carefully considered in designing the system. Thus a biocular device
is designed for a pupil large enough to encompass both eyes plus any
head motion, although the image sharpness and resolution are deter-
mined by the aberrations of a pupil whose size is defined by that of the
Diopter adjustment (focusing) of eyepieces.
In binocular systems, one
eyepiece is usually focusable to compensate for any difference in focus
between the two eyes. The motion of the eyepiece is
where f is the eyepiece focal length, and D is the shift of the image
position in diopters (relative to the second focal point of the eyepiece—
where the eye is presumed to be located). The usual adjustment range
is ±4 diopters.
Erector systems come in all sizes and shapes. Occasionally
a single element may serve as an erector, or two simple elements in the
general form of a Huygenian eyepiece may be used, as in the terres-
trial eyepiece shown in Fig. 13.4a. This form of eyepiece is widely used
in surveying instruments, occasionally with an achromatic eyelens. A
popular erector for gun scopes is illustrated in Fig. 13.4b and consists
of a single element plus a low-power, overcorrecting doublet, often
meniscus in shape. Photographic objective systems are occasionally
used as erectors, symmetrical forms of the Cooke triplet, the Dogmar,
or the double-Gauss being the most popular. Probably the most wide-
ly used erector consists of two achromats, crown elements facing, with
a modest spacing between them.
As previously mentioned, erectors are usually designed in conjunc-
tion with either the eyepiece or objective of a telescopic system.
Considerable care should be taken in the first-order layout of any tele-
scope to be certain that the work load placed on the erector is not
impossibly large. The introduction of suitable field lenses is often nec-
essary to reduce the height of the principal ray at the erector, although
this does produce an undesirable increase in the Petzval curvature.
Note that many erectors have external pupils, often in the form of a
For most telescopic systems, the objective will be
an ordinary achromatic doublet, or one of the variations described in
The Design of Optical Systems: Particular
Sec. 12.5. A photographic-type objective may be used where a wide
field is desired, Cooke triplets and Tessars being the most commonly
used. A Petzval objective is useful when high relative apertures are
necessary; the construction of a Petzval objective (Sec. 13.3) is such
that its rear lens acts as a sort of field lens, and this characteristic is
occasionally useful. For high-power telescopes where it is desirable to
keep the system as short as possible, a telephoto type of construction
is valuable. The front component is an achromatic doublet and the rear
is a negative lens, either simple or achromatic. The focal length is usu-
ally 20 to 50 percent longer than the overall length of the objective.
Either the Petzval or telephoto type of objective can be used as an
internal focusing objective (Fig. 13.5), where focusing is accomplished
by shifting the rear (inside) component, making a more easily sealed
instrument. Surveying instruments and theodolites conventionally
use the telephoto form with the focusing lens located about two-thirds
of the way from the front component to the focal plane so that the sta-
dia “constant” will remain constant as the instrument is focused.
Alignment telescopes use a positive focusing lens of high power placed
near the focal plane at infinity focus; thus, a modest shift of the focus-
ing lens toward the front component allows the system to be focused at
extremely short distances, or even on the objective itself. Note that any
system which works over a wide range of magnifications (as this type
Erector systems. (a) The four-element terrestri-
al erecting eyepiece. (b) Typical gunsight optical system. (c)
Symmetrical doublet erector.
of focusing lens does) should be designed so that the change of aberra-
tion contribution is small as the magnification is varied.
13.2 Microscope Objectives
Microscope objectives (Fig. 13.6) may be divided into three major class-
es: those designed to work with the object under a cover glass, those
designed to work with no cover glass, and immersion objectives, which
are designed to contact a liquid in which the object is immersed. All
types are designed by raytracing from the long conjugate to the short;
the effects of the cover glass (when used) must be taken into account
by including it in the raytrace analysis. Standard cover glass thickness
is 0.18 mm (0.16 to 0.19 mm, n = 1.523 ± 0.005, v = 56 ± 2).
Microscope objectives are designed to work at specific conjugates,
and their correction will suffer if they are used at other distances. For
cover glass objectives and immersion objectives, the standard distance
from object plane to image plane is 180 mm. For metallurgical types
(no cover glass), the standard distance is 240 mm. The chief effect of
changing the tube length or cover glass thickness from its nominal val-
ue is to overcorrect or undercorrect the spherical aberration; an objec-
tive which has been improperly adjusted at the factory may be
reclaimed by using a nonstandard tube length or cover glass if the
defect is not too serious.
Note that ordinary microscope objectives are designed to yield an
essentially perfect image, and aberrations (on axis at least) should be
reduced to well below the Rayleigh limit if at all possible. Micro-
objectives for projection or photography may be corrected with more
The Design of Optical Systems: Particular
Telescopic systems. (a) Typical surveying telescope with
negative focusing lens and terrestrial eyepiece. Note that the objective
is telephoto, in that its effective focal length is longer than the objec-
tive. (b) Alignment telescope. The strong positive focusing lens, when
shifted forward, allows the instrument to focus at extremely short dis-
emphasis on the outer portions of the field, depending on the exact
application for which they are intended.
These are usually ordinary achromatic doublets,
or occasionally three-element systems, as shown in Fig. 13.6a. The 32-
mm NA 0.10 or 0.12 is the most common and produces a magnification
of about 4×. A 48-mm NA 0.08 is also occasionally encountered. This
may be designed in exactly the same manner as the achromatic tele-
scope objective discussed in Secs. 12.4 and 12.5, except that the “object”
will be located at 150 mm (more or less) instead of at infinity.
As shown in Fig. 13.6b, these are usually
composed of two widely spaced achromatic doublets. The most common
objective is the 10×, 16 mm, which is available in several forms. The
ordinary achromatic 10× objective has an NA of 0.25 and is probably
the most widely used of all objectives. The divisible or separable
(Lister) version is designed so that it can be used as a 16-mm or, by
removing the front doublet, as a 32-mm objective. This is accomplished
at the sacrifice of astigmatism correction, since both components must
be independently free from spherical and coma and thus no correction
of astigmatism is possible. An apochromatic 16-mm objective is also
Microscope objectives. (a) Low-power achro-
matic doublet or triplet. (b) 10 × NA 0.25. (c) Amici
objective 20 × NA 0.5 to 40 × NA 0.8. (d) Immersion
objective. (e) Apochromatic 10 × NA 0.3. Shading indi-
cates fluorite (CaF
). (f) Apochromatic 50 × NA 0.95.
available with an NA of 0.3; fluorite (CaF
) is used in place of crown
glass to reduce the secondary spectrum.
The power layout for this type of objective is usually arranged so
that the product y is the same for each doublet; in this way the
“work” (bending of the marginal ray) is evenly divided. Conventionally
the second doublet is placed midway between the first doublet and the
image formed by the first doublet. (Note that the preceding refers to
raytracing sequence—in use the “second” doublet is near the object
to be magnified and the “first” doublet is nearer the actual image.)
This relatively large spacing allows the cemented surface of the second
doublet to overcorrect the astigmatism and flatten the field (assuming
the stop to be at the first doublet). This layout leads to a thin-lens
arrangement with the space about equal to the focal length of the
objective, the focal length of the first doublet approximately twice that
of the objective, and that of the second doublet about equal to that of
the objective. Note that this arrangement is similar to that of a high-
speed Petzval-type projection lens (see Fig. 13.24).
Ordinarily three sets of shapes for the two components can be found
for which spherical and coma are corrected. One form will be that of
the divisible objective, with the spherical and coma zero for each dou-
blet; this is usually the form with the poorest field curvature.
If the surface contribution equation for the spher-
ical aberration of a single surface is solved for zero spherical, three
solutions are found. One case occurs when the object and image are at
the surface and is of little interest. A second is of more value; when
object and image both lie at the center of curvature, there is no spher-
ical aberration introduced (and the axial rays are not deviated). The
third case, usually called the aplanatic case, allows the convergence of
a cone of rays to be increased (or decreased) by a factor equal to the
index without the introduction of spherical aberration. It occurs when
any of the following relationships are satisfied.
L′ = R
U′ = I
n′ + n
n′ + n
The Design of Optical Systems: Particular
Note that if any of the above are satisfied, all are satisfied, and that,
since no spherical is introduced, if L = l, then L′ = l′. It is also worth
noting that coma is zero for all three cases and that astigmatism is
zero for the first and third cases and overcorrecting between.
This principle is used in the “aplanatic front” of
an oil-immersion microscope. The object is immersed in an oil whose
index of refraction matches that of the first lens. R
(as shown in Fig.
13.7) is chosen to satisfy Eq. 13.1; this results in a hyperhemispheric
form for the first element. R
is chosen so that the image formed by R
is at its center of curvature; R
is chosen to satisfy Eq. 13.1. Note that
sin U is reduced by a factor of n at each element, and that the “apla-
natic front” reduces the numerical aperture of the cone of rays from a
large value (as high as NA = n sin U = 1.4) to a value which a more
conventional “back” system can handle.
The Amici objective (Fig. 13.6c) consists of a hyperhemispheric front
element combined with a Fig. 13.6b (Petzval) type of back combina-
tion. Since the Amici is usually a dry objective, the radius of the hyper-
hemisphere is frequently chosen somewhat flatter than that called for
by the aplanatic case to partially offset the spherical introduced by the
dry plano surface. The space between the hyperhemisphere and the
adjacent doublet is kept small to reduce the lateral color introduced by
the front element. The standard 4-mm 40× NA 0.65 to 0.85 objectives
are usually Amici objectives. The working distance (object to front sur-
face) is quite small in the Amici, to the order of a half millimeter. Since
there is a direct relationship between zonal spherical and working dis-
tance in this type of objective, the higher-NA versions tend to have
very short working distances.
The oil-immersion objective utilizes the full “aplanatic front” and
may be combined with a Fig. 13.6b type of back, as shown in Fig.
13.6d, or a more complex arrangement. Both the Amici and immersion
The aplanatic front.
The object is immersed in a fluid
whose index matches that of the
hyperhemispheric first element.
is an aplanatic surface. The
image formed by R
is at the
center of curvature of R
an aplanatic surface of the same
type as R
types are frequently designed with fluorite (CaF
) crowns to reduce or
eliminate secondary spectrum. Some of the new FK glasses can serve
the same purpose.
Note that although the aplanatic front is a classic textbook case,
departures from the exact aplanatic form are common. For example, it
is possible to find a meniscus lens of higher power than the aplanatic
case which will introduce overcorrected spherical. This not only
reduces the ray-bending work that the back elements must accom-
plish, but also reduces the correction load as regards spherical aberra-
tion (but not chromatic). Aplanatic-front objectives have a residual
lateral color resulting from the separation of the chromatically under-
corrected front and the overcorrecting back. Special compensating eye-
pieces with opposite amounts of lateral color are used to correct this
Flat-field microscope objectives.
The objectives shown in Fig. 13.6 are
all afflicted with a strongly inward-curving field. Such objectives can
yield extremely sharp images in the center of the field, but the deep
field curvature and/or astigmatism severely limit the resolution
toward the edge of even the relatively small field of the microscope.
Many flat-field types of objectives have their Petzval curvature
reduced by a thick-meniscus negative component placed in the long
conjugate. This may be an achromatized doublet as shown in Fig. 13.8,
or simply a thick singlet. The field-flattening effect is greater if the
negative-power element or surface is a large distance from the posi-
tive-power member. Often the balance of the objective is simply a stack
of positive components. The improvement in image quality at the edge
of the field is quite marked when compared to the standard type of
objective. Another desirable feature of this form of objective is a long
working distance from object to front lens. Note that this configuration
is the analog of the retrofocus or reversed telephoto camera lens. Many
flat-field objectives incorporate a construction similar to the thick-
meniscus doublets of the double-Gauss or Biotar form (see Fig. 13.14)
as a field-flattening device. Another technique is to convert the
The Design of Optical Systems: Particular
Achromatized negative doublet in a flat-field microscope objective.
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